SOLUTE DISPERSION IN SOIL A Thesis Presented to The Faculty of Graduate Studies of The University of Guelph by KOFI BUDU LARYEA In partial fulfilment of requirements for the degree of Doctor of Philosophy © Kofi Budu Laryea, 1979 University of Ghana http://ugspace.ug.edu.gh $ £ J S ' < L 3 2 - /VLst£»eL|v G W E ? 2 ? University of Ghana http://ugspace.ug.edu.gh LARYFA UNIVERSITY OF GUELPH surname Koft Budu 080415400 FACULTY OF GRADUATE STUDIES given names I .D . numbei Land Resource Science Ph.D. department degree CERTIFICATE OF APPROVAL (DOCTORAL THESIS) We, the undersigned, hereby certify that the thesis presented by the above- named candidate in partial fulfilment of the requirements for the degree of Doctor of Philosophy is worthy of acceptance and may now be formally submitted to the Dean of Graduate Studies. Title: ^Chairman, Doctoral Examination Committee <£i-\ - U l U 'External Examiner Department Member c iv ■/" rvispry Committee m Supervisory Committee Date: S T * * ? Received by: _________________________ Dean of Graduate Studies Date: University of Ghana http://ugspace.ug.edu.gh TARYEA UNIVERSITY OF GUELPH • sumane Kot’i Budu 980A15400 FACULTY OF GRADUATE STUDIES given names I.D. numbei Land Resource Science Ph.D. department degree REPORT OF DOCTORAL EXAMINATION COMMITTEE The undersigned, having been duly appointed as provided in the Regulations to be the Doctoral Examination Committee for the above-named candidate, have examined the candidate upon the thesis presented to us. Title: ______________ We hereby report both the defence and the thesis to be: . Chairman/ Doctoral Examination Committee , ... 1 ” External-" Examiner J iii _>-'1 .Departgner^ t Member iv Supervisory £cmnuttee v 0 > c /0 , S L Supervisory Committee Date of Doctoral Examination: DIS SAT UNS □ □ 1 3 " □ □ m □ □ i z f □ □ □ □ REPORT OF DEPARTMENT CHAIRMAN On the basis of the foregoing, I report the decision of the Doctoral Examination Committee to be ______ <- dateDepartment Chairman/ Please note thabvche candidate is deemed to have passed if not more than one of the five Examiners votes negatively. An abstention is regarded as a negative vote. A unanimous vots is required for a distinguished ratinq. The rating, applyinc to both 'the thesis and thj c'afencs, is also noted on the Certificate of Approval. Received by: ____________ Dean of Graduate Studies date University of Ghana http://ugspace.ug.edu.gh ABSTRACT SOLUTE DISPERSION IN SOIL K. B. Laryea, Ph.D Supervisor: University of Guelph, 1979 Dr. D. E. Elrick An experimental study of solute transport during one-dimensional sorption of a solution of KC1 into a uniformly packed horizontal soil column of Akuse clay, (a tropical clay loam from Ghana which is dominantly montmorillonitic) and Brookston clay of the Huron Catena (dominantly illitic) is reported in this thesis. A hydrodynamic equation based on the existing theories of irreversible thermodynamics is developed to incorporate the effect of anion exclusion in the usual hydrodynamic equation for one-dimensional flow of solute in soil. This equation is used to calculate the. longitudinal dispersion coefficient D^ in horizontal infiltration experiments where anion exclusion effects were observed. Also, the analysis of hydrodynamic dispersion during one-dimensional horizontal flow developed by Smiles eit al. (1978) is extended to include gravitational effects present during one-dimensional vertical infiltration by using the power series form of solution develop­ ed by Elrick _et a^ L. (1979). Simulation methods using computer programs written in system/360 Continuous System Modeling Program (CSMP) are used to solve the equations used in this study. The longitudinal dispersion coefficient is in all the analyses presented in this study, assumed to be independent of the Darcy flux and a function of the water content only. In all experiments, both the soil solution content, the chloride concentration and the potassium concentration preserved similarity in University of Ghana http://ugspace.ug.edu.gh terms of distance divided by square root of time. This confirmed the assumption that the longitudinal dispersion coefficient is effectively independent of the Darcy flux for the conditions of the experiments reported in this study. These results were found to be consistent with the studies of Saffman (1959) , Pfannkuch (1963) and Scotter and Raats (1970). The observed incomplete piston-like displacement of the ante­ cedent water content suggests the presence of stagnant or 'immobile' water fractions in the soil columns considered. The derived longitudinal dispersion coefficient for Cl agreed fairly well with the derived longitudinal dispersion coefficient for K . This is an indication that under the experimental conditions imposed, the equilibrium adsorption isotherm used to describe the adsorption term in the hydrodynamic equation for solutes which interact with soil particles, is adequate. Simulation of water content, chloride concentration and concen- + - 1/2 tration of K as a function of X(=xt ) using derived data D(X), Dg (X) for Cl and Dg (X) for K+ showed very good agreement with experimental data. The simulated c(X) for Cl also showed excellent agreement with calculation of c(X) using a program for the analytical solution. Water content profile and chloride concentration profile obtained for vertical infiltration experiments indicated good agreement with theoretical water content and Cl concentration profile simulated with CSMP. The theoretical chloride concentration profiles showed a progress­ ively dispersed 'front' with long infiltration time periods whereas the theoretical water content profiles for various time periods showed a sharp and abrupt wetting front. University of Ghana http://ugspace.ug.edu.gh iACKNOWLEDGMENTS The author is profoundly grateful to Dr. D. E. Elrick for his invaluable suggestions, his guidance and for sharing his ideas and knowledge throughout the course of this study. Regular discussions with Drs. P. H. Groenevelt, B. Kay and C. T. Corke who served as membeir of his research committee, have been very rewarding and he is very grateful for that as well as for their useful suggestions during the preparation of this manuscript. He also appreciates very much the help of Drs. S. S. Wang of the Institute of Computer Science, University of Guelph, and D. E. Smiles of C.S.I.R.O. division of Environmental Mechanics, Australia. In addition, appreciation is extended to the Vice-Chancellor, Dean of Faculty of Agriculture and the Head of Department of Soil Science, University of Ghana, for granting him leave of absence to enable him to pursue this study. To his wife Charity, and children Cynthia, Gwendolene and Kwame Budu, the author expresses his gratitude for the many sacrifices they made during his long absence from the house. To Mrs. Mei-Fei Elrick, Mr. and Mrs. Norbert Baumgartner, the Faculty, staff and graduate students of the Department of Land Resource Science, the author is very thankful for their friendliness and help in diverse ways which made his stay in Guelph a happy one. To Dr. J. C. M. Shute who was the director of the defunct Ghana- Guelph Project, the author expresses his appreciation for his understanding and encouragement throughout the author's stay in Canada. The author would also like to thank Ms. Sue Patterson who shared in the arduous but crucial task of typing the manuscript. University of Ghana http://ugspace.ug.edu.gh Finally, the writer would like to acknowledge most gratefully, financial assistance offered by the Canadian International Development Agency through both the Ghana-Guelph Project and the Association of Universities and Colleges of Canada (AUCC). Furthermore, financial assistance in the form of assistantship granted by the Department of Land Resource Science is very much appreciated. University of Ghana http://ugspace.ug.edu.gh iii TABLE OF CONTENTS ACKNOWLEDGEMENTS 1 TABLE OF CONTENTS 111 LIST OF FIGURES v±i:i LIST OF TABLES xiid LIST OF SYMBOLS ^ CHAPTER 1 - INTRODUCTION 1 CHAPTER 2 - REVIEW OF LITERATURE 6 2.1 Dispersion in capillary tubes 2.2 The classical statistical models 12 2.3 Solute transport in inert porous media 16 2.4 Transport with exchange or adsorption in porous medium 23 CHAPTER 3 - THEORY 28 3.1 Convective-dispersive transport equations for horizontal infiltration of solution 28 3.2 Solution of equations 3.8a and 3.8b for Dg 33 3.3 Inclusion of salt sieving in the convective- dispersive transport equation for solutes 39 3.4 Dispersion during vertical infiltration 58 CHAPTER 4 - MATERIALS AND METHODS 63 4.1 Soils 63 4.2 Saturation of exchange complex of soils with calcium 65 Page University of Ghana http://ugspace.ug.edu.gh Page 65 66 67 68 69 69 71 84 86 87 89 98 109 Time of equilibration experiments Determination of adsorption isotherms High energy moisture characteristics Determination of hydraulic conductivity Chemical analyses Horizontal infiltration experiments 4.8.1 Determination of concentration of K+ in solution during the infiltration experiment 4.8.2 Experimental verification of the reliability of the method of estimating concentration of K+ in solution during the infiltration experiment 4.8.3 Determination of concentration of chloride in solution for the infiltration experiments Computer programming method used to analyse data 4.9.1 Description of the computer program for calculating the soil moisture diffusivity D(0) and simulation of 0(A) using the calcu­ lated D(6) values 4.9.2 Description of the computer program for calculating dispersion coefficient D (A) s and simulation of c(A) using the calculated D (A) values s 4.9.3 Description of the computer program for calculating dispersion coefficient D (A) for K+ and simulation of c(A) using the calculated D (A) values s iv University of Ghana http://ugspace.ug.edu.gh 4.9.4 Description of the computer program for calculating the dispersion coefficient Dg for chloride in the case where anion exclusion occurs 4.9.5 Description of the computer program for calculating ^ , xs > and 0)g for water and salt flow 4.9.6 Description of CSMP program for simulating water content profiles for various time periods for vertical infiltration of water and salt 4.9.7 Description of CSMP program for simulating concentration of chloride profiles for various time periods for vertical infiltration of water and salt CHAPTER 5 - RESULTS AND DISCUSSION 5.1 Relative adsorption rates and adsorption isotherms 5.2 Horizontal infiltration with KC1 solution 5.3 Effect of salt exclusion in the infiltration experiments 5.4 Derived data from the horizontal infiltration experiments 5.5 Simulated 0(A) and c(A) results for horizontal infiltration experiments 5.6 Simulated water content and salt content profiles for vertical infiltration Page 120 130 151 156 162 170 185 195 214 244 University of Ghana http://ugspace.ug.edu.gh CHAPTER 6 - REFERENCES APPENDIX A APPENDIX B APPENDIX Cl APPENDIX C APPENDIX D APPENDIX E APPENDIX F APPENDIX G APPENDIX H APPENDIX I CONCLUSIONS - Development of the ordinary differential equations for y > 4> and oj for water flow as given by w W W Philip (1957) - Development of the ordinary differential equations for x s > 4>s and 0) for salt flow as given by Elrick et a^ L. (1979) Computer program for calculating K(9) - Algorithm for the simulation of 0(A) from the calculated D(A) values - Algorithm for the simulation of c(A) from the calculated D (A) values s CSMP listing for calculating soil water diffusivity D and simulation of 0(A) for Akuse clay CSMP listing for calculating dispersion coefficient for chloride and simulation of c(A) using (i) analytical and (ii) computer solution for Akuse clay - CSMP listing for calculating dispersion coefficient D^ for K+ and simulation of c(A) from computed Dg (A) values for Akuse clay - CSMP listing for calculating x> and aj for water and salt flow for Akuse clay - CSMP listing for simulating water content profiles for various time periods for vertical infiltration vi Page 271 275 284 293 302 308 315 318 321 325 329 University of Ghana http://ugspace.ug.edu.gh vii APPENDIX J of water and salt for Akuse clay 335 CSMP listing for simulation of concentration of chloride profiles for various time periods for vertical infiltration of water and salt for Akuse clay 338 Page University of Ghana http://ugspace.ug.edu.gh viii 2.0 2.1 3.1 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 FIGURE LIST OF FIGURES DESCRIPTION Capillary tube conceptual model Types of breakthrough curves for miscible displacement Schematic diagram indicating the exclusion zone, thick­ ness of film of water and equilibrium concentration Set up for the horizontal infiltration experiment Some of the perspex sections aligned with a rod First section in a column D showing the 0-ring seal, inlet section E showing the two ports for air and solution and a smaller perspex section F Schematic diagram of part of the sectioned column of soil CSMP listing for calculating soil water diffusivity D as a function of water content 0 and simulation of water content as a function of A from the calculated D(0) values for Brookston clay CSMP listing for calculating dispersion coefficient Ds for Cl- and simulation of c(A) from the computed DS (A) values using (i) the analytical solution and (ii) the computer solution in the text (Brookston clay) CSMP listing for calculating dispersion coefficient Dg for KT1" and simulation of concentration of in solution as a function of A for Brookston clay CSMP listing for calculating dispersion coefficient Ds for Cl- in the case where anion exclusion occurs (Akuse clay) CSMP listing for calculating x> and to f°r water and salt flow (Brookston clay) CSMP listing for simulating water content profiles for various time periods for vertical infiltration of water and salt (Brookston clay) CSMP listing for simulating concentration of Cl profiles for various time periods for vertical infil­ tration of water and salt (Brookston clay) 13 20 55 72 74 76 79 90 99 110 123 136 152 Page 158 University of Ghana http://ugspace.ug.edu.gh ix 5.1 5.2 5.3 5. 4a 5.4b 5.5a 5.5b 5.5c 5. 6a 5.6b 5. 6c 5.7a 5.7b 5. 8a 5.8b 5.9a 5.9b FIGURE Quantity of K+ adsorbed at different times + 2+ K adsorption isotherm on soil fractions of Ca saturated Akuse clay “t* 2 + K adsorption isotherm on soil fractions of Ca - saturated Brookston clay Bulk density distribution in soil columns used for horizontal infiltration (Akuse clay) Bulk density distribution in soil columns used for horizontal infiltration (Brookston clay) - 1/2 Soil water content versus A(=xt ) for Akuse clay Concentration of Cl in solution versus A for Akuse clay Concentration of K+ in solution versus A for Akuse clay Water content 0 versus A for Brookston clay Concentration of Cl in solution versus A for Brookston clay Concentration of K+ in solution versus A for Brookston clay Reduced mean pore water velocity versus A for Akuse clay Reduced mean pore water velocity versus A for Brookston clay Bulk density distribution in soil columns used for horizontal infiltration in Akuse clay (dry) where salt sieving was observed Bulk density distribution in soil columns used for horizontal infiltration for Caledon fine sandy loam Water content versus A for Akuse clay (dry) where salt sieving was observed Concentration of Cl in solution versus A for Akuse clay (dry) where salt sieving was observed DESCRIPTION 163 166 168 172 172 177 177 177 179 179 179 182 182 186 186 190 Page 190 University of Ghana http://ugspace.ug.edu.gh X5.9c 5. 9d 5.10a 5.10b 5.11a 5.11b 5.12a 5.12b 5.13a 5.13b 5.14a 5.14b 5.15 5.16a 5.16b 5.17a 5.17b 5.18a 5.18b FIGURE 9c versus X for Akuse clay (dry) where salt sieving was observed Concentration of K+ in solution versus X for Akuse clay in the case where salt sieving was observed DESCRIPTION Page 190 190 Water content 6 versus X for Caledon fine sandy loam 192 Concentration of Cl in solution versus X for Caledon fine sandy loam 192 Dispersion coefficient Ds for both Cl and K+ versus water content for Akuse clay 196 Dispersion coefficient Ds for both Cl and K+ versus water content for Brookston clay 198 Soil water diffusivity versus reduced water content for Akuse clay 204 Soil water diffusivity versus reduced water content for Brookston clay 205 Soil water characteristic curves for Akuse clay 206 Soil water characteristic curves for Brookston clay 206 Hydraulic conductivity versus reduced water content for Akuse clay 209 Hydraulic conductivity versus reduced water content for Brookston clay 209 Dispersion coefficient for Cl versus 0 for Akuse clay (dry) where salt sieving was observed 212 Simulated and experimental 0 versus X for Akuse clay 220 Simulated and experimental 0 versus X for Brookston clay 220 Simulated Cl , experimental Cl and Cl concentration from analytical solution versus X for Akuse clay 227 Simulated Cl , experimental Cl and Cl concentration from analytical solution versus X for Brookston clay 227 Simulated K+ and experimental K+ concentration versus X for Akuse clay 235 Simulated K+ and experimental K+ concentration versus X for Brookston clay 235 University of Ghana http://ugspace.ug.edu.gh 5.19a 5.19b 5.19c 5.19d 5.20a 5.20b 5.20c 5.20d 5.21a 5.21b 5.22a 5.22b 5.23a 5.23b 5. 24a FIGURE 5.24b xi Reduced water content and reduced Cl concentration versus X for Akuse clay Reduced water content and reduced Cl concentration versus X for Akuse clay Reduced water content and reduced Cl concentration versus ip for Akuse clay Reduced water content and reduced Cl concentration versus oj for Akuse clay Reduced water content and reduced Cl concentration versus X for Brookston clay Reduced water content and reduced Cl concentration versus x f°r Brookston clay Reduced .water content and reduced Cl concentration versus \p for Brookston clay Reduced water content and reduced Cl concentration versus w for Brookston clay Theoretical and experimental water content profiles at t=7200s for Akuse clay Theoretical and experimental Cl concentration profiles at t=7200s for Akuse clay Theoretical and experimental water content profiles at t=19260s for Akuse clay Theoretical and experimental Cl concentration profiles at t=19260s for Brookston clay Theoretical water content profiles for various time periods for Akuse clay Theoretical Cl content profiles for various time periods for Akuse clay Theoretical water content profiles for various time periods for Brookston clay Theoretical Cl content profiles for various time periods for Brookston clay DESCRIPTION 242 242 242 242 247 247 247 247 263 263 265 265 267 267 269 Page 269 University of Ghana http://ugspace.ug.edu.gh xii FIGURE DESCRIPTION Cl Computer program for calculating K(0) C2 Alternate computer program for calculating K(0) 1C Schematic representation of 0(A) function with the other two 0 curves generated by RVELO and RVELAO 2C A schematic plot of THETA2 and THETAA at A=0 versus their corresponding RVELO and RVELAO ID Schematic representation of c(A) function depicting the other two c curves generated by VELCO and VELCAO 303 305 309 311 Page 316 University of Ghana http://ugspace.ug.edu.gh xiii TABLE 3.1 4.1 4.2 5.1 5.2 5.3 5.4 5.5a 5.5b 5. 6a 5.6b 5.7a 5.7b 5.8a LIST OF TABLES Ordinary differential equations for water and solute transport Some chemical and physical characteristics of the soils used Measured and computed concentration of K+ in solution using the method proposed in the text (Brookston clay) Summary of experimental conditions imposed during horizontal and vertical infiltration experiments Chloride salt balance for horizontal infiltration with Akuse clay (dry initially) Computer output showing positive Ds values due to inclusion of salt sieving term in hydrodynamic equation Computer output showing both positive and negative Ds values due to omission of salt sieving term from the hydrodynamic equation Computer output for soil water diffusivity D, experi­ mentally determined water content and simulated water content (Akuse clay) Computer output for soil water diffusivity D, experi­ mentally determined water content and simulated water content (Brookston clay) Computer output for experimental c(A) for Cl , c(A) from analytical solution and simulated c(A) for Akuse clay Computer output for experimental c(A) for Cl , c(A) from analytical solution and simulated c(A) for Brookston clay Computer output for simulation of concentration of K+ in solution using calculated D (A) values for Akuse clay Computer output for simulation of concentration of K+ in solution using the calculated D (A) values for Brookston clay Computer output for y j ^ > (*) > X > ^ and oj for Akuse clay w w s s s Page 61 64 86 171 194 215 217 223 225 230 232 238 240 249 University of Ghana http://ugspace.ug.edu.gh xiv 5.8b 5.9a 5.9b 5.10a 5.10b 5.11a 5.11b TABLE Page Computer output for Y , ij; , w , x > 'P and w ^or Brookston clay w w s s s 252. Computer output showing how well the boundary conditions listed in Table 4.3 were met for Akuse clay 253 Computer output showing how well the boundary conditions listed in Table 4.3 were met for Brookston clay 255 1/2 3/2 2 Contribution of At , Xw*-’ ^ w11 anc^ t0 t'le vertical distance moved by water for Akuse clay 258 1/2 3/2 2 Contribution of At , Xst’ ^ s*1 an<^ ^st to the vertical distance moved by chloride for Akuse clay 259 1/2 3/2 2 Contribution of At , X w t > an(i ^wt to the vertical distance moved by water for Brookston clay 260 1/2 3/2 2 Contribution of At , Xst> fet wst to the vertical distance moved by chloride for Brookston clay 261 University of Ghana http://ugspace.ug.edu.gh LIST OF SYMBOLS XV a radius of a cylinder in the Taylor model, L b half the thickness of liquid films, L b thickness of the exclusion layer, L ex 3 c concentration in solution, keq./L 3 c ^ - concentration of chloride in solution, keq./L 3 c^_ concentration of potassium in solution, keq./L 3 c^ local equilibrium concentration, keq./L 3 c ^ equilibrium concentration at inlet end of soil column, keq./L 3 c q concentration at A=0, keq./L 3 c mean measured concentration, keq./L m reduced concentration in solution, dimensionless 2 D soil water diffusivity, L ft 2 hydrodynamic dispersion coefficient, L /t 2 D molecular diffusion coefficient, L /t m 2 D dispersion coefficient, L /t 2 g gravitational acceleration, L/t h matric potential, L 2 flux of component i, M/L t j° diffusion flux, L/t v , j macroscopic volume flux, L/t keq kilo-equivalents kg kilograms K hydraulic conductivity, L/t 1 litre L length University of Ghana http://ugspace.ug.edu.gh phenomenological coefficient relating the flux of component due to a force on component j meter milliequivalent milliequivalent molar 2 pressure within the liquid phase, M/Lt Peclet number 2 flux of salt, keq/L t distance measured radially from the centre of the capillary tube, L universal gas constant, ergs/mol. C time, t amount of solute adsorbed per mass of soil, keq./M absolute temperature, T velocity of the fluid at a radial distance from the axis of the tube, L/t maximum velocity at the axis of the tube, L/t pore water velocity, L/t 1/2 reduced pore water velocity, L/t Darcy flux, L/t local velocity of component i, L/t 3 partial specific volume of the constituent k, L /H space variable in the fixed coordinate system, L depth, L correction term, dimensionless University of Ghana http://ugspace.ug.edu.gh xvii 0 volumetric water content, dimensionless 0 reduced volumetric water content, dimensionless X horizontal distance from source divided by square root of 1/2 time, L/t 2 2 Pj. total thermodynamic potential of component k, L /t c . , ~ T 2 . 2 U concentration ui-pendent part of u> L /t V kinematic viscosity, L^/t 2 it osmotic pressure, M/Lt 3 p bulk density, M/L 3 density of component k, M/L 3 O entropy production, M/Lt T cr reflection coefficient, dimensionless volume fraction of component k, dimensionless local volume fraction of component k, dimensionless distance divided by time for component k, L/t 3/2ik distance divided by time to power 1.5 for component k, L/t 2 0)^ distance divided by time squared for component k, L/t University of Ghana http://ugspace.ug.edu.gh CHAPTER 1 INTRODUCTION The understanding and description of the simultaneous transport of solutes and water through porous media is of vital importance to science and to mankind in general. Almost all the models employed to describe the movement of solutes in porous media make use of the solute dispersion phenomenon. Consequently, understanding solute dispersion is central to the prediction of the fate of solutes during movement through porous media. In modern agriculture, frequent applications of large quanti­ ties of water soluble materials like fertilizers and pesticides are made on the surface of the soil. Part of these materials remains in the root zone and part is carried underground by the flowing water. In order to estimate the magnitude of the hazard posed by some of these chemicals, it is important to investigate the processes that control their movement from the soil surface through the rootzone to the ground water table. Drainage, evaporation, infiltration and plant uptake studies all involve movement of chemicals through the soil. Also, the practice of leaching salts from surface layers of soil, either to lower depths or out of the soil profile, involves movement of chemicals. In addition, there are a number of other disciplines which are involved with the transport of chemicals through porous media and hence are concerned with the phenomenon of dispersion in porous media. For 1 University of Ghana http://ugspace.ug.edu.gh 2example, petroleum engineers interested in secondary recovery processes and sanitary engineers need to know more about chemical and microbio­ logical transport in soils, particularly when dealing with problems concerning waste disposal, dumps, land fills and septic tanks. Also, hydrologists need to know more about solute dispersion since they usually use tracers to study groundwater movement and aquifer parameters. Fried and Combarnous (1971) in their comprehensive review have described salt water intrusion into coastal aquifer and seepage from industrial refuse heaps and decanting tanks in which the basic information of dispersion is necessary in order to predict the movement of the various chemicals. Furthermore, as pointed out by Fried and Combarnous (1971), "... the historical development of dispersion theories offers a good background of the various modeling techniques of porous media." As a consequence, a study of dispersion is important not only because of its contribution to solving problems of practical value but also because of its importance in the design of modelling experiments. Before proceeding further, it is necessary to qualitatively define dispersion. Dispersion refers to all the physical phenomena which govern the evolution of a transition zone when a fluid originally in a porous medium is displaced by another, both fluids being miscible. Many physical phenomena contribute to dispersion, namely, diffusion, velocity distribution within one pore geometry, velocity distribution between different pore geometries in a medium, dead end pores, flow of fluid in most pores taking place at an angle relative to mean direction of water flow and salt sieving effects (Kemper, 1960; Krupp et al., 1972). Traditionally, investigations in movement of chemicals by dis­ persion through soils have been carried out mainly by analysing the University of Ghana http://ugspace.ug.edu.gh 3concentration of the effluent, usually by fraction collection in labora­ tory experiments at the bottom of a column of soil in a miscible dis­ placement apparatus (Day, 1956; Day and Forsythe, 1957; Nielsen and Biggar, 1961; Elrick and French, 1966). The behaviour of the chemical within the column was then interpreted by examining the concentration of * _ the ion specie in the effluent by using a breakthrough curve . In a few cases, tracer concentrations have been determined, in situ, as for example the conductivity measurements of Miller and King (1966), the "gamma emitting radioisotope" method of Shalhevet and Yaron (1967), the radiation technique used by Corey et al. (1970) and also by Kirda et al. (1974). These in situ measurements, however, have inherent problems due principally to the relative thickness of the measuring probe in comparison with the dimensions of the soil column. Literature to date reveals that in many of the-studies concerning solute movement, the tracer was assumed to be inert with respect to its environment, so that interactions with the soil matrix were often ignored. In addition, most of these studies have been limited to situations in which the soil or porous material is either completely saturated with solution or has a constant unsaturated water content, with the result that the convective motion was always assumed to be steady state. Very little work has been done which considered the behaviour of a reactive solute in a plane within the soil column. Biggar and Nielsen (1963), Kay and Elrick (1967) considered the case of adsorption during •k A breakthrough curve is the relationship between the ratio of concen­ tration c/c0 and the number of pore volumes V/V0 , where c is the concentration of the solute in the effluent, c0 is the concentration of the solute in the displacing fluid, V is the volume of the effluent collected and VD is the volume of the porous medium occupied by the fluid. University of Ghana http://ugspace.ug.edu.gh 4movement but again these authors interpreted the behaviour of the adsorbed species by considering the concentration of the effluent. Quite recently, Smiles et al, (1978) and Smiles and Philip (1978), using horizontal infiltration measurements, have shown that both the salt concentration in solution and volumetric moisture content were - 1/2 unique functions of xt , where x is the horizontal distance from the absorbing surface and t is the absorption time. Flrick _et al. (1979) used the pertubation procedure of Philip (1957) to solve the partial differential equation for salt flow during vertical infiltration of a solution of salt into a soil column. The purpose of the present study is to investigate the dispersion phenomenon in two clay soils with a view to greater understanding of predicting the fate of solutes during flow in porous media. The research programme was designed to: (a) extend the method of Smiles _et al. (1978) to study the movement of a reacting solute (potassium in this case) during horizontal infiltra­ tion in soil; (b) modify the convective-dispersion equation to include salt sieving and to use this modified equation to calculate the dispersion coefficient in a flow system where salt sieving occurs; (c) simulate the water and salt distribution in the horizontal soil column using the calculated dispersion coefficients from the horizontal infiltration experiments and a computer program written in system 360 Continuous System Modeling Program (CSMP); (d) simulate the water and salt distribution during vertical infiltration for a specified time by using the data from the horizontal University of Ghana http://ugspace.ug.edu.gh 5infiltration experiments and the mathematical analysis of Philip (1957) and Elrick et al. (1979). University of Ghana http://ugspace.ug.edu.gh CHAPTER 2 REVIEW OF LITERATURE Soil systems, like all porous media, are comprised of voids and solids which are not necessarily randomly distributed in size and space. Consequently, flow of fluids in the structure of soils extremely difficult to investigate using mathematical analyses. In order to enable mathematical treatment, the true system is usually re­ placed by a fictitious space which is "small enough to meet practical purposes and complex enough to allow accurate description "(Fried and Combarnous, 1971). This simpler space termed a conceptual model, is generally a set of characteristics derived from experimental or theore­ tical considerations which are mathematically related. A conceptual model can be physical, mathematical or statistical in nature. Fried and Combarnous (1971) define physical models as those in which relation­ ships between characteristics are derived from physical experiments while mathematical models derive their relationship between character­ istics from a set of equations; for example, partial differential equations. Many researchers, in seeking to understand and to determine dis­ persion, have developed various conceptual models of transport of soluble matter in soils. While some of these models include the effects of both hydrodynamic dispersion and molecular diffusion, others consider only hydrodynamic dispersion assuming the effects due to molecular diffusion to be minimal. What is more, most of these models have restrictive 6 University of Ghana http://ugspace.ug.edu.gh 7assumptions thus reducing their applicability to many porous media. Historically, two main developments in the study of dispersion may be delineated; namely, (i) development of conceptual models to facilitate understanding of the dispersion process, and (ii) laboistory and recently,field experiments aimed at utilizing one or the other of the conceptual models. The review of literature which forms the substance of this chapter will cover some of the impor­ tant landmarks in the development of both the conceptual models and laboratory experimentation. 2.1 Dispersion in Capillary Tubes Due to a similarity between a pore in a porous medium and the cylindrical space in a capillary tube, the first theoretical works on dispersion considered diffusion studies in capillary tubes. Among the early workers, Taylor (1953, 1954) formulated an expression for the description of the dispersion of salt based on observations of the dispersion of soluble matter in solvent flowing slowly through a capillary tube. In a cylinder of radius a, depicted in figure 2.0, the distri­ bution of velocities U(r) in a cross section is parabolic and obeys the relationship r2 2 U(r) = 2Uq (1 - /a ) (2.1a) where U(r) is the velocity at a distance r from the axis of the tube University of Ghana http://ugspace.ug.edu.gh and is the maximum velocity on the axis of the cylinder. Suppose a capillary tube is filled with one fluid and a second fluid is injected at one end of the tube. Assuming the two fluids are of the same viscosity, flow is laminar and diffusion effects are of no significance, then the concentration of the displacing fluid in the effluent stream is determined by integrating the flow equation for laminar conditions. Injected fluid will not appear in the effluent stream until one-half of the tube volume is injected. For continued injection Perkins and Johnston (1963) give the instantaneous average concentration of the effluent as: V 2 V Vf = i - ( ~ ) ; (when Vd > ^ ) (2.1b) d where V. is the volume fraction of injected fluid in the effluent; V f t is the total volume of the tube and V, is the volume of the fluid d injected. In actual case, molecular diffusion and hydrodynamic dispersion will cause mixing along the interface. As a consequence, the zone of mixing will be narrower than calculated from the convection effects alone because of transverse (or lateral) diffusion. Theoretical equations derived by Taylor (1953) showed that if one fluid was displaced by another fluid under conditions where diffusion could nearly damp-out radial concentration variations, then a symmetrical longitudinal mixed zone would be established. The mixed zone would travel with the mean speed of the injected fluid and would be dispersed as if there were a constant dispersion coefficient k, which University of Ghana http://ugspace.ug.edu.gh 9Is related to molecular diffusion coefficient D by the relation: d 2TT2 a U E ( 2 . 2 ) 48D d where U equals 1/2 U . m o In deriving equati^ 2.2, Taylor assumed that: (i) the changes 'in concentration due to transport by convection along the tube occur in such a short time that the effect of longitudi­ nal molecular diffusion may be neglected, and (ii) radial differences in concentration are quickly smoothed out by molecular diffusion compared with the time necessary for appreci­ able effects due to convective transport to appear. It has to be mentioned that k in equation 2.2 is a function of velocity squared. Aris (1956) generalised Taylor's approach to irregularly shaped capillary tubes in which the molecular diffusion coefficient varies with concentration, and local velocity distributions may not be parabolic. Forming moments of various order of concentration distribution, Aris developed his equations which he solved by the method of induction. He found that the moments behaved asymptotically like the moments of a normal law of probability and thus inferred that the mean concentration is dispersed according to a Gaussian distribution about a point moving at the mean velocity of flow. The variance of this distribution was used to obtain the effective diffusion coefficient K as: University of Ghana http://ugspace.ug.edu.gh 10 K = D + ct[(aV ) / D ,] (2-3) a m a In equation 2.3, a is a dimensionless number depending on the cross section of the capillary tube (for a circular cross section a = 1/48), is the mean velocity of flow, and a is the radius of the cross section. The works of Taylor (1953, 1954) and Aris (1956) define dis­ persion as more than just a molecular diffusion problem. Other geometrical models based on the capillary tube theory have been used in the past to derive flow equations. De Josselin de Jong (1958) and Saffman (1959, 1960) have both introduced random networks of capillary tubes which in some cases yield interesting qualitative results. Saffman's model which is a general case of De Josselin de Jong's, consists of a network of randomly oriented and distributed straight pores, the dimensions of which are comparable with size of pores in real porous medium and in each of which the flow is uniform. In this case, the path of each fluid particle may be considered as a random walk so that the length, direction and duration of each step are random variables. The values of the dispersion coefficient are deduced from calculation of probability distribution function of the displacement of a particle after a given time. Saffman's model was the first to account for the existence of three main domains of dispersion, namely, pure molecular diffusion, molecular diffusion and hydrodynamic dispersion of equal importance, and hydrodynamic dispersion and molecular diffusion interfering with the latter being the smaller of the two. In addition to these, Saffman's work indicates the difference University of Ghana http://ugspace.ug.edu.gh 11 in importance between lateral and longitudinal dispersion. He predicts in his work that for Peclet numbers (fluid velocity x characteristic pore size/molecular diffusion coefficient of solute in bulk solution) less than 1 , the dispersivities are essentially equal to the coefficient of molecular diffusion of the solute through the medium. In other words, the effects of convection on mixing at the interface are essentially negligible. As pointed out by Peck (1971), convection may still be dominant in solute transport at Peclet numbers less than 1 because for -9 2 -1 a solute with molecular diffusivity of 10 m s in a medium with characteristic pore diameter of 10 m (a very fine sand or silt) , the Peclet number is less than 1 and mixing at the interface is essentially -4 -1 by molecular diffusion for fluid velocities less than about 10 m s . 4 In about 10 s, convective transport would move solutes a distance of about 1 m and in the same period molecular diffusion may transport the -3 solute a distance of about 10 m. Bear _et_ a l . (1968) and Bachmat (1969) have improved the random capillary models of De Josselin de Jong and Saffman by averaging micro­ scopic quantities (microscopic scale is the scale of the pore) in a repre­ sentative element of volume REV at a point to build a continuum at the macroscopic level (the scale of dispersion phenomenon and Darcy's Law). Their model fits experimental results closer than Saffman's model and provides a general equation for nonhomogeneous media and nonhomogeneous fluids. All the capillary models, however, experience the same diffi­ culties as does the Kozeny-Carman's development of a formula for calculating the hydraulic conductivity of a soil in that precise theory for simple University of Ghana http://ugspace.ug.edu.gh 12 geometry Is forced to fit the complex internal architecture of a porous medium. 2.2 The Classical Statistical Models Einstein's (1937) theory of random walk has been extended by Giddings (1959) who indicated that the spread of a solute as a result of continuous change in flow velocity of a volume element from an initially horizontal zone, obeyed the laws of statistics and could be characterised by a parameter which he called an "eddy diffusion coefficient" (now referred to as hydrodynamic dispersion coefficient). Following Giddings' work and under the assumption that (i) the porous medium under consideration is homogeneous and isotropic, (ii) events occurring in one elemental time step are independent of events occurring in any other time step, and (iii) the movement of a solute obeys the laws of laminar flow, Scheidegger (1974) presents two types of models, namely, a random walk model and a random media model. In the former, randomness is directly ascribed to the fluid particles while in the latter flow occurs along random channels. The random walk model as proposed by Scheidegger is based on an analogy with Brownian motion. A particle of fluid is first considered to be very small such that it cannot be separated into separate channels during passage in a porous medium. Then for a fixed time T, one asks for the probability P(x,T) of finding a particle at the spot x at time T. The total time is split into small time steps t during which the progress University of Ghana http://ugspace.ug.edu.gh F i g u r e 2. 13 : C a p i l l a r y tube c o ncep tu a l model . University of Ghana http://ugspace.ug.edu.gh 14 Figure 2.0 University of Ghana http://ugspace.ug.edu.gh 15 of the particle is considered a random variable. The sum of these elemental progresses yields the position of the particle at time t. Scheidegger (1974) used the central limit theorem which established conditions under which sums of independent random variables are asymp­ totically normally distributed to show that function P(x,T) is Gaussian. Hen^~- it is a solution of a diffusion-type equation given for a co-ordinate system (x',T') moving with mean pore velocity of flow as The change of variables x = x'+UT' and t = T' was used to transform equation 2.4 to co-ordinates (x,t), thus obtaining displacing fluid in a miscible displacement is equal to the probability of finding a particle of displacing fluid at that point. K in equations 2.4 and 2.5 is a scalar called the dispersion coefficient. From experi­ mental observations that when the concentration gradient is perpendicular to the mean flow (lateral dispersion) the dispersion coefficient is different from the dispersion coefficient observed when the concentration gradient is parallel to the mean flow (longitudinal dispersion), Scheidegger (1974) then generalised equation 2.5 to show that K is a (2.4) 3P 3t 8c 3t (2.5) According to the law of large numbers, the concentration in a University of Ghana http://ugspace.ug.edu.gh 16 tensor and not a scalar. 2.3 Solute Transport in Inert Porous Media Porous media such as soils are not well represented as collec­ tions of capillary tubes since the flow paths are not canalized but are extensively cross linked. Thus in each pore in a soil several flc ’ paths nearly coalesce and then diverge. Most of the dispersion i n soils, therefore, comes from the meandering of the streamtubes through the complex structure of the medium and not from only the existence in each pore of a velocity profile as in a tube. Laboratory experiments aimed at studying dispersion in soils have been carried out in the past. A common technique of most of the experimental work is to compare theore­ tical curves obtained from one of the existing models or a new model with experimental breakthrough curves. If the two curves match, it is presumed that the assumptions made for the derivation of the model also hold for the experimental situation. Most experiments are conducted with certain restrictive conditions, the usual conditions being that: (i) the dispersion coefficient which is a tensor is reduced to its principal coefficients; i.e., the lateral dispersion coefficient and the longitudinal dispersion coefficient with boundary conditions chosen such that one of these coefficients can be neglected. k (ii) the flow is made uniform and fingering prevented, and k Fingering is a distortion of the boundary between displacing and dis­ placed fluids frequently caused by differences in density and viscosity of the fluids. University of Ghana http://ugspace.ug.edu.gh 17 (iii) generally the initial concentration profiles are step functions. An early experimental work which demonstrated the essence of the process of dispersion without mathematical treatment was reported by Griffiths (1911) who observed that a tracer injected into a stream of water spreads out in a symmetrical manner about a plane in the cross- section which mo,.=s with the mean speed of flow. As pointed out by Taylor (1953), (reviewed earlier in section 2.1) Griffith's observation is rather startling for two reasons. First is that since water moves at twice the mean speed near the centre of the pipe and the patch of colour at the mean speed, the clear water in the middle must approach the colour patch, absorb colour as it passes into it and then lose colour as it passes out,finally leaving the patch as perfectly clear water. The second remarkable feature is that the colour patch spreads out symmetrically from a point which moves with the mean speed of the fluid in spite of the fact that the distribution of velocity over the section, which gives rise to this dispersion, is highly unsymmetrical. Burd and Martin (1923) observed that when distilled water is used to displace a solution from a packed column of saturated soil, the displaced solution had an almost constant composition until a substantial fraction of this solution had been displaced, then the concentration decreased gradually instead of decreasing abruptly to zero as one might expect from the sharp interface that was initially established between the original soil solution and the displacing water phase. This mani­ festation of dispersion was referred to as hydrodynamic dispersion by Day (1956) who demonstrated that layers of salt water displaced through a water saturated column of sand by fresh water, were distributed in University of Ghana http://ugspace.ug.edu.gh 18 accordance with the statistical theory developed by Scheidegger (1954). One of the most well known introductions of miscible displace­ ment technique to the field of Soil Science was made by Nielsen and Biggar in the series of papers, Nielsen and Biggar (1961, 1962, 1963) and Biggar and Nielsen (1962, 1963). Nielsen and Biggar (1962) indicated that any experimentally measured breakthrough curve in a porous medium may have the characteristic of one or a combination of the five break­ through curves shown in Figure 2.1. If there is no interaction between the solvent, solute and the porous medium, the following relationship should hold for Figure 2.1a, 2.1b, and 2.1c, irrespective of the shape of the breakthrough curve. Q V c (1 - ) dt = 1 (2 .6 ) c . i o 3 -1 where Q is the mean flow rate in m s t is time in s V is the volume of the porous material occupied by the o fluid in c is the concentration of the solute found in the effluent e • , - 3 in keq. m c^ is the concentration of the solute in the displacing -3 fluid in keq. m The area under the breakthrough curve up to one pore volume is, under these conditions, equal to the area above the curve beyond one pore University of Ghana http://ugspace.ug.edu.gh 19 volume. Mathematically, this requirement is expressed as Q V o (1 - — ) dt c . c i (2.7) When there is interaction between the solute and soil solids such as adsorption process, the breakthrough curve would be translated to the right of one pore volume as in Figure 2.Id. Processes within the soil which result in an increase of solute concentration or incomplete mixing throughout the soil solution would, as shown in Figure 2.1e, give a breakthrough curve translated to the left of one pore volume. Consequently, processes such as dissolution of slightly soluble salts present in porous media, incomplete mixing due to dead end pores would all translate the breakthrough curve to the left of one pore volume. Using Aiken clay and Oakley sand, Nielsen and Biggar (1961, 1962, 1963) concluded that physical differences among soils contribute to the shape and position of breakthrough curves. depicted for the effects of anion exclusion and for processes within the soil which result in increase of solute concentration due to dissolution of salts present in the soil (Fig. 2.1e) by Nielsen and Biggar (1962) is inaccurate. When salt sieving occurs, one expects, as pointed out by Nielsen and Biggar (1962) a breakthrough curve translated to the left of one pore volume. The relative concentration c /c. after one pore e i r volume, however, would be smaller than 1.0 (see Figure 2.If). In the It should be pointed out that the type of breakthrough curve University of Ghana http://ugspace.ug.edu.gh 20 F i g u r e 2 .1 : Types o f b re ak th r o u gh c u r v e s f o r m i s c i b l e d i s p l a c em en t (a to e were adapted from N i e l s e n and Bi g g a r , 1962). University of Ghana http://ugspace.ug.edu.gh 1.0 (a) 0.5 1 Pore volume Piston flow Pore volume Chemical reaction,precipitation or exchange Figure 2.1 Pore volume Pore volume I-1 Longitudinal dispersion Extremely wide range in velocity distribution Pore volume Exclusion of solute or velocity distribution with velocities near zero University of Ghana http://ugspace.ug.edu.gh Re la tiv e co nc . ce /c j 22 (f) Pore vokme Expected curve for salt exclusion Figure 2.1 Pore volume Expected curve for processes which increase solute concentration University of Ghana http://ugspace.ug.edu.gh 23 case where processes within the soil result in increase of solute concentration, one expects relative concentration greater than 1.0 after one pore volume (Figure 2.1g). If there is a constant source which causes the increase of the solute (i.e. greater quantities of soluble salts present), then d c^ in Figure 2.1g will flatten above 1.0. On the other hand, if the source is limiter1, c/c will, after increasing e for some period beyond 1 .0 , creep back to 1 .0 . Other workers, for example Elrick and French (1966), Elrick et al. (1966), Corey (1966), Corey ej: _al. (1967), have also extensively employed the miscible displacement technique to study movement of mostly non-interacting solutes such as chloride or nitrate in porous media. 2.4 Transport with Exchange or Adsorption in Porous Medium The dispersion phenomenon may be complicated by the occurrence of clay in the porous medium because surface reactions such as adsorp­ tion may occur. Adsorption is a process in which matter is extracted from one phase and concentrated at the surface of a second and as such it is termed a surface phenomenon (Weber, 1972). For most systems three principal types of adsorption may be identified: physical adsorption resulting from van der Waals attraction; chemical adsorption and exchange adsorption resulting from electrostatic attractions to charged surface sites on the solid. The factors affecting adsorption include the specific surface area of the solid, the concentration and the nature of competing ions, pH and the nature of the adsorbent. Generally, adsorption and/or University of Ghana http://ugspace.ug.edu.gh 24 exchange reactions are of prime importance in most agricultural soils where cations usually interact with the negatively charged clay surfaces or replace other cations already on the clay surface through cation exchange reaction. The diffusion-like description of dispersion in porous media has in the past been modified to include the effects of simple reversible exchange or adsorption of solute on the medium so that the equation of continuity for the exchange of solute between a mobile phase and a soil column is usually taken as 9c 3c 5 c 1 3S . + D - D — jr - ~ (.2.a; 3t m 3x s j 2 y 3t where is the mean pore velocity Y is the void fraction and S is the amount of solute adsorbed per unit volume of the packed soil. At a point in a porous medium system, adsorption results in the removal of solutes from solution and their resulting concentration at the surface of the solid, to such time as the concentration of the solute remaining in solution is in dynamic equilibrium with that at the surface. 1 3S The term (— -5— ) in equation 2.8 represents the exchange of solute betweeny o t the phases and is evaluated either by using a finite rate of adsorption equation or by introducing the concept of instantaneous equilibrium when values of S at each position, x, would be given by the adsorption isotherm S = f(c). Lapidus and Amundson (1952) have considered both cases of University of Ghana http://ugspace.ug.edu.gh 25 equilibrium adsorption and kinetic reaction for a semi-infinite column (0 5 X < ») where a continuous solute flux at the surface was treated as a boundary condition (c = constant at X = 0). The rate equations used were S = k c + k^ (2.9) in which k^ and k^ are constants, and H - V - k4 (2'10) where k^ and k^ are the rate constants. Equation 2.9 indicates the condition of local equilibrium between the fluid and solid phase at all points in the porous medium and also linearity of the adsorption isotherm. Equation 2.10, however, is a non-equilibrium first order kinetic relationship and implies that the rate of adsorption is finite. Following Lapidus and Amundson (1952), Houghton (1963) has treated mathematically the dispersion phenomenon assuming an equilibrium between the adsorbed and solution phases and a non-linear isotherm of the polynomial form S = f(c) = KQ + Kxc + K2c2 (2.11) As indicated by Houghton, this form of isotherm has the advantage that it is concave to the S-axis for positive , concave to c-axis for negative and linear for equals zero. has been introduced so University of Ghana http://ugspace.ug.edu.gh 26 that equation 2.11 might be used to approximate highly curved isotherms 3 4 without the use of higher order terms such as K^c , K^c Adopting the rate equations and mathematical solution of Lapidus and Amundson (1952), Banks and Ali (1964) presented experimental information in which the rate constants k„ and k, were calculated, 3 4 but failed to check for agreement between the theoretical solution and the experimental data on grounds that the theoretical solution is complex. Hashimoto _et al. (1964) have also obtained a solution similar to the mathematical solution of Lapidus and Amundson (1952). This solution, which includes the longitudinal mixing process as well as equilibrium and linear adsorption during one dimensional flow, was later used by Kay and Elrick (1967) to show that at low flow rates the movement of lindane, a pesticide, in their porous material was reasonably well described by their chromatographic model. However, at high pore water velocities, Kay and Elrick (1967) and also Davidson and Chang (1972) have shown that there is considerable deviation between their mathematical models and experimental data. These two studies suggested that for high pore water velocities the use of a kinetic rate equation to describe the adsorption process may be necessary. Nonetheless, examining various adsorption models usually employed for the description of the adsorption process, Van Genuchten et al. (1974) found kinetic adsorption-desorption models to be inadequate when predicting the mobility of pesticides for high average pore water velocities. They indicate, however, that these kinetic models agreed with the equilibrium adsorption model when used to describe pesticide effluent concentration distribution at low pore water velocities. University of Ghana http://ugspace.ug.edu.gh 27 In the past, other types of isotherms, for example, the Freundlich model and Langmuir model and also first order kinetic reactions, have been used with differing degrees of success (Lindstrom and Boersma, 1970; Swanson and Dutt, 1973; Selim and Mansell, 1976; Selim e_t aA. , 1977) . In many practical situations, other chromatographic models b?ve been applied (Bower et a l ., 1957; Gardner and Brooks, 1957; Biggar and Nielsen, 1967). In some of these applications these models have been quite successful but other investigators have attributed discrepancies between theory and experimental data to changes of the cation exchange capacity with pH (Thomas and Coleman, 1959) and the neglect of dispersion due to convection and diffusion (Biggar and Nielsen, 1963). It appears that the theory of the transport of a solute subject to adsorption or exchange has not been adequately tested in transient flow systems. The conclusion may, however, be drawn from examination of the partial differential equations describing the transport of solutes which interact with the soil matrix, that adsorbed and ex­ changed materials are transported less rapidly than those which are not. University of Ghana http://ugspace.ug.edu.gh CHAPTER 3 THEORY 3.1 Convective-dispersive equation for horizontal infiltration of water and salt The basic assumption usually made when considering the problem of simultaneous transfer of solute and water is that the transport of the soluble matter ,‘.s governed principally by molecular diffusion (thermal motion within the soil solution) and convection (viscous movement of the solution). Macroscopically, diffusion of solutes in a uniform body of water may be described by Fick's first law which states that the rate of transfer of a diffusing substance through a unit area of section is proportional to the concentration gradient measured normal to the surface through which diffusion is taking place. In differential form this law is Js ■ -Dd -2 -1 where J is the flux of solute keq. m s s D. is the Diffusion Coefficient of the solute in d 2 -1 water m s c is the concentration of the solute in solution keq. m ^ x is the horizontal coordinate m. Equation 3.1 is applicable under conditions of no liquid flow. In soils the diffusion coefficient D^ is less than the equivalent 28 University of Ghana http://ugspace.ug.edu.gh 29 coefficient in a free water system. Also, for practical purposes, the diffusion coefficient in a clay-water system may be considered indepen­ dent of the salt concentration and dependent only on the water content (Kemper and van Schaik, 1966). Generally, it is assumed that the macroscopic transport by convection must take into account the average flux of water as well as the mechanical or hydrodynamic dispersion. Experimental and theoretical works of Perkins and Johnston (1963) and Passioura (1971) have shown that the magnitude of the hydrodynamic dispersion coefficient in a given porous medium depends on the average flow velocity and that under steady and saturated conditions D, may often be taken to be proportional h to the first power of the average flow velocity, (Bear, 1961; Perkins and Johnston, 1963). This is stated in the form where k in centimeters is an experimental constant depending on the characteristics of the porous material (especially the pore size D, (V ) = klV h s !s (3.2) distribution) and is in centimeters per second. When the Darcy flux, v, is used, equation 3.2 is of the form k v 6 where 6 is the volumetric moisture content. The joint effect of diffusion and convection is combined to formulate the mathematical expression for the flux in one dimensional University of Ghana http://ugspace.ug.edu.gh 30 flow of solute as: J s v c - 6 [Dh (v) + Dd (0)] ~ (3.3a) J s (3.3b) where Dg is the combined diffusion-hydrodynamic dispersion coefficient (referred to in the rest of this thesis as the dispersion coefficient). cussions and experiments. In equation 3.3b a tacit assumption has been made that a single constant can be used to describe both the diffusive and dispersive effects. Examination of the extreme cases, when either effect may be neglected, clarifies this. Thus, when v equals zero, D, (v) = 0, and so D equals D, (Bear, 1961; Perkins and Johnston, 1963). n s d At high Darcy fluxes, the effect of molecular diffusion is normally neglected and D then equals D, . The structure of D for intermediate s h s situations, for example where D, is approximately equal to D, , is not d n very clear. However, for D. » D, , additivity has been claimed on d n theoretical grounds (Beran, 1957). This assumption is often extended to a wide range of D , to D, ratio: D./D, < 50 (Perkins and Johnston, 1963). d n n d In this study, apart from using a single term to describe the phenomenon of hydrodynamic dispersion and molecular diffusion and their possible interaction, an assumption is also made that D^ is 0-dependent but effectively independent of the Darcy flux (Smiles et al>, 1978; Smiles and Philip, 1978; Elrick e_t aJ-. , 1979). For non-steady state or transient conditions the equation The nature and values of Dg have been the subject of many dis- University of Ghana http://ugspace.ug.edu.gh 31 describing time and space relationships of the flux of solute is derived through the use of equation 3.3b and the continuity equation 9J (3.4) 3t 3x 3M s 3 where M (keq/m ) is the total quantity of solute in the bulk volume of soil. For a non-interacting solute M may be expressed as M = 0c (3.5) For solutes that interact with the soil particle surfaces, however, M may be expressed as M = 0c + pS (3.6) _3 where p (kg m ) is the bulk density of the soil and S is the amount of solute associated with the solid phase in units of quantity of solute per mass of soil (i.e. keq/ per kilogram of soil). Consequently, the continuity equations for non-interacting and interacting solutes become,respectively ^T 3(0c) s , “T r ~ ~ - » r (3-7a) r\ T 3(0c + pS) _ s 3t “ 9x (3'7b) which when substituted into equation 3.3b with the assumption of the University of Ghana http://ugspace.ug.edu.gh 32 dependence of D only on 9 yields for one dimensional horizontal flow: s [0D (6 ) |^] ~ (3,8b)at dt dx S ox dx The one-dimensional flow of water may be described by the Darcy equation of the form v = -D(0) (3.9) 2 -1 where D is the soil moisture diffusivity ( m s ) and is equal to K _3iJ> , K is the hydraulic conductivity (ms ■*") and ip is the matric 30 potential (m). In equation 3.9, the assumption has been made that there is a unique relationship between 0 and ip in the moisture characteristic curve which is not true because of hysteresis. However, if we consider only the sorption or only the desorption curve, then Darcy's equation in this form may be used. Substitution of the continuity equation for water flow H ■ - I into equation 3.9 yields 90 _ 3 rn/'Q'i 30-, , It ~ "3x [D(e) 9x (3-11*) University of Ghana http://ugspace.ug.edu.gh 33 which can be transformed into the following ordinary differential equation: (3.11b) where X = xt The solution of equation 3.11b for D(8) subject to following initial and boundary conditions 3.2 Solution of Equations 3.8a and 3.8b for Dg It is observed that any analytical or numerical solution of equations 3.8a and 3.8b and subsequent prediction of the concentration distribution with time from these equations will require information on the magnitude and the dependence of the dispersion coefficient on 6 , v and c. One of our objectives, as indicated earlier, is to solve equations 0 (x,t) = 0 for x > 0 , t = 0 , A -*-00 n } (3.12) 0(x,t) = 0 for x = 0 , t > 0 , A = 0 s has been given by Bruce and Klute (1956) as 0 D(0) 1 dA 2 d0 Ad0 (3.13) 0n University of Ghana http://ugspace.ug.edu.gh 34 3.8a and 3.8b for D , design experiments to measure indirectly, and use these D values to simulate the concentration distribution, s Equations 3.8a and 3.8b may be expanded as +9it +pi v t f - cI Substitution of the continuity equation 3.10 into equations 3.14a and 3.14b and use of a further assumption that in equation 3.14b the bulk density p is constant during horizontal infiltration (that is, there is no swelling), gives | f = y - [9D ( 9 ) I s ] - V I s- C3t dx S dx dx | £ + p | l 3 [6D (0) |£] - v | ^ C 9t dt dx S dx dx The initial and boundary conditions of interest for equations 3.15a and 3.15b are: 0 (x,t) = 9 , c(x,t) = c for x _ 0 , t = 0 (I n n 0 (x, t) = 0 , c(x,t) = c for x = 0 , t > 0 ('■ s o Condition 3.16a indicates that initially the soil column has for all values of x > 0 , a constant moisture content 0 and a constant concentra- n . 14a) .14b) .15a) .15b) • 16a) ,16b) University of Ghana http://ugspace.ug.edu.gh 35 tion of salt, c . Condition 3.16b implies that a solution would be n applied at the input end x = 0 at time t = 0 and for all times t > 0 and that the solution would be applied in such a way as to maintain the soil at the input end of the column at a constant moisture content 0g and a constant salt concentration c . o cmiles et^ ad. (1978) have shown that both the water and salt concentration profile preserve similarity during horizontal infiltration in terms of distance divided by the square root of time. Accordingly, - 1/2 the substitution X = xt was shown to remove both x and t from equation 3.15a as follows: 3c 3X _ _3_ ... 3c_ _3X_, _ v jk; _3A^ 3X 3t 3x L s'1' } 3X 3x 3X 3x lx> - « 1 / 2 1 Substitution of equation 3.9 into equation 3.17 for v gives 0X t 1 3c _ t-1 3 3c ... 30 3X 3c 3X 2 "3X “ 3X [6Ds (0) 3X] + D(9) 3X 3^ 3X 3^ 0X t 1 3c t-1 3 rQTS 3c, , 36 3c T IX " Jx [eDs (9) + D(0) 3X 3X -k [9V 8> I f ) + < f - D(e> §> t - 0 which may be written as Tx [msm if1 + ^ f ■ ° <3.18b) University of Ghana http://ugspace.ug.edu.gh 36 where g(8) = 0X + 2D(9) -jr = CLA Xd0 (3.18c) n The extreme right hand term in equation 3.18c can be obtained by substituting equation 3.13 into 3.18c for D(0). Transformation of equation 3.15b in a similar way yields the ordinary differential equation of the form £ [0Ds < 8 ) § ] t f f - 0 (3.19) -1/2 In view of the transformation X = xt , conditions 3.16a and 3.16b may be written as 3 , c = c for X ->■ °° (3.20a) n n 3 , c = c for X = 0 (3.20b) s o This implies also that for 9 = 0 and c = c , 4r = 0 and = 0. n n dX dX Equation 3.18b is linear in c and may be solved in the explicit form subject to 3.20a and 3.20b to obtain (Smiles et al., 1978) ° °o M(X) n o with University of Ghana http://ugspace.ug.edu.gh However, we may solve equation 3.18b for Dg (0) as follows: If y = 0Ds(0) , tl.'n dy s(9) dc- ^ w h i c h upon integration yields g (0) dc Since for X -+ °°, dc/dX = 0 we get D (0) = s JL 1 dX 2 6 dc g (0) dc (3.22) Dg(0) may be calculated either from equation 3.22 using a numerical technique or, as was done in this study, from equation 3.18b directly using C.S.M.P. In the case of reactive solutes (equation 3.19), we require a relationship between the quantity of solute adsorbed per gram of soil S and concentration of solute in solution c. A number of models of adsorption and exchange have been proposed, some representing equilibrium between c and S and other representing time-dependent adsorption processes University of Ghana http://ugspace.ug.edu.gh 38 (Boast, 1973). In this study, the model used assumes an equilibrium by programming the computer to fit c and S values provided, with a non­ linear function. This aspect of the thesis is considered in detail in Chapter 4. However, another approach was used and a non-linear poly­ nomial similar to that of Houghton (1963) but in logarithmic scale was fitted to the adsorption isotherm and upon substitution into equation 3.19, the resulting equation solved for Dg as follows: between c and S. D (0) was calculated from equation 3.19 using C.S.M.P- s The non-linear polynomial in logarithmic scale was of the form log S (3.23) where k^, k^ and k^ are constants. From equation 3.23 2 S = 10^"kl l0g C + k2log c " k3] 2 .n [-k.In c + k.lnc - k_] 10 4 5 3 (3.24) In equation 3.24, k^ = k^/5.304 and k,_ = k2/2.303 dS dc [In 1 0 ( )] [10 c - 2k.Inc + k is dt 2.303 c (k - 2k Inc) (10 o 4 (3.25) Substituting equation 3.25 into equation 3.15b and transformation with - 1/2 X = xt yields University of Ghana http://ugspace.ug.edu.gh 39 .-k.ln c+k_lnc-k ] , [9D„(6) # ] = - | ( 0X + 2 ‘3°3P ■ (k5-2k4lnc)(1 0 [ )}^XdX s dX n/’O'i “ D(6) dX dX (3.26) dc • • •Substitution of y = 0D -rr- into equation 3.26 and integrating gives s dA X X-*00 {ex + 2 •303pA(k -2k Inc)( [~k4ln c+k5 lnc k3]) c 3 4 lu Xd0 } dc dc which because at X -»■ », -rr = 0 we obtain aA Ds(0) = 1 1 dX 2 e dc {ex + l^i°3pX (k 2k inc)( [ k4ln c+k5lnc k3 ]) c 5 4 10 Xd6 } dc (3.27) 3.3 Inclusion of Salt Sieving in the Convective-Dispersive Transport Equation for Solutes The existence of an osmotic gradient as a driving force on the li­ quid in soils and clays has been recognized for some time (Kemper and Rollins, 1966; Kemper and Letey, 1968; Groenevelt and Bolt, 1969). This phenomenon is usually attributed to the exclusion of anions from negatively charged surfaces of soil pores. The clay or soil then behaves as a University of Ghana http://ugspace.ug.edu.gh 40 "leaky" semi-permeable membrane. The effect of anion exclusion on flow of anions in soils has also been studied (Thomas and Swoboda, 1970; •k Krupp, Biggar and Nielsen, 1972). The contribution of osmosis and anion exclusion to the simultaneous transport of salt and water in un­ saturated soils under transient conditions has recently been explored by Bresler (1973, 1978). Bresler (1978) developed two separate equations to describe the rate of change of water content with time and the rate of change of salt concentration with time. It is interesting to note that the reflection coefficient a (termed the "osmotic efficiency coefficient" in Bresler's paper) appears in the equation describing water movement whereas his equation describing the movement of salt omits the effects of salt-sieving, which is the counterpart of osmosis. In this section of the study, basic equations in irreversible thermodynamics are used to develop the governing partial differential equation describing transient one-dimensional simultaneous transfer of salt and water. The objective in this study is to examine the movement of Cl during horizontal infiltration of KC1 solution into a soil whose 2+ colloidal complex is saturated with Ca . In this development, sub­ scripts 1 and 2 will be used to refer to the salts KC1 and CaCl0 respectively. The rate of entropy production (a), or the dissipation function, To,(the rate of entropy production times the temperature), is equal to the sum of the products of the fluxes and forces in the system. The appropriate equation representing the dissipation function for the movement of water due to salt concentration gradient in soil. University of Ghana http://ugspace.ug.edu.gh 41 system under consideration is: (3.28a) where T is the temperature (K) -1 -3 -1O is the rate of entropy production (k^ m s K ) -2 -1 is the flux of the component k (kg m s ) is the total thermodynamic potential of component k (i.e. 2 - 2 chemical potential plus external force fields ( m s ) Equation 3.28a is written for the system we are considering as: where subscripts w, si and s2 refer to water, salt 1 (KC1) and salt 2 (CaC^) respectively. At constant temperature Vii for water and solutes can be written (3.28b) as: Vy = V VP + Vyc w w (3.29a) (3.29b) (3.29c) where is the specific volume of constituent k (volume of k/mass of i 3 ik, m kg ) University of Ghana http://ugspace.ug.edu.gh 42 c y k is the concentration dependent part of the total thermo- - 2 2 dynamic potential y (m s- ) -1 -2 P is the pressure within the liquid phase (kg m s ). In equations 3.29a, 3.29b and 3.29c the gravity component of y has been neglected and so any derivations using the equations are restricted to horizontal flow. Substitution of equations 3.29a, 3.29b and 3.29c into equation 3.28b yields: T« - - V V ? + 5V - JS1 < V ?P + - J,2<5 S27P + VI,CS2> - -JVVP - V S , - JS15«CS1 - Js2VuCs2 (3'30»> V —1 where j is the total volume flux (ms ) defined as jV = j V + j _V . + j -V . (3.30b) J Jw w Jsl si s2 s2 In neutral salt solutions the concentration dependent parts of the thermodynamic potential of the different components are connected via the Gibbs-Duhem equation: Pw VyCw + Psl VyCsl + Ps2 V^Cs2 = ° <3-31> where is the bulk density of constitutent k (mass of k/volume _3 of mixture; kg m ). Equation 3.31 gives: University of Ghana http://ugspace.ug.edu.gh 43 P P c S1 c S2 c Vy = i Vyc Vy w p s, p s„w 1 w I which upon substitution into equation 3.30a gives Ps P To = -jvVP+j t - ± Vy^ + 7 ^ V y h - j VyJ - j Vy^ w S 1 w 2 S 1 S 1 2 2 = "jV VP - (jSl" V s ^ V\ " (Js2 ‘ VwPs2} VyS2 (3' In equation 3.32a, v is the linear velocity of water (ms ). For the w rest of the development, v^ _ is the linear velocity of constituent k. The flux j^ _ of constituent k is given by the relationship: Jk = PkVk Equation 3 . 32a therefore becomes: Ta = -jVVP - p (v -v )VyC - p (v -v )VyC (3. J 1 S1 w 1 2 S 2 W S 2 The osmotic potential gradient may be defined as: -pwVyC - J PkVMk = w k=sl,s2 The total osmotic pressure gradient (Vtt) may now be split into two parts (Vtt = Vtt^ + Vt^ ) , each part being the effect of the presence of one of the two salts. Then, according to Van't Hoff's law: 32a) 32b) University of Ghana http://ugspace.ug.edu.gh 44 Vtt1 = n ^ W c and Vir = n 2RTVc2 Thus, TO = -jvVP - (b (v ,-v )n RTVc, - d> (v „-v )n„RTVc„ (3.33) Tw si w 1 1 Tw s2 w 2 2 In equation 3.33, R is the gas constant in ergs per mole per °K. T is the temperature (K). -3 c^ is the concentration of constituent k (moles m ). If c^ is 3 + 2+ measured in equivalents/m of K and Ca , then n^ (the number of particles the salt will dissociate into) is 2 for salt 1 and 3 for salt 2. ffl, is the volume fraction of constituent k (= p, V, ). Tk rk k Therefore, To = -jVVP - j ^ n RTVc - j ^ R T V ^ (3.34a) To = -jVVP - j1DVTrl - j 2DVtt2 (3.34b) where and ^ 2 are che diffusion flux of salt 1 and salt 2 respectively and are defined as jlD = U vsl - V (3.35a) University of Ghana http://ugspace.ug.edu.gh 45 j?D = <> (v , - v ) (3.35b) 2 rw s2 w The dissipation function has been transformed from the expression as stated in equation 3.28 to the expression in equation 3.34a because the forces in the latter are more directly measurable. Since the flux of constituent k is given by the relationship: ’ equation 3.30b becomes: jV = p v V + p 1v . V + p v „ V which w w w si si si s2 s2 s2 upon further substitution of the volume fraction of constituent k given by the relationship Pk = \ Vk yields: j = (b v + d ) v +(j)0v rw w Tsl si rs2 s2 Therefore, A = jV ~ ((^ slv sl + 1’s2Vs2 ) (3-36) Substitute equation 3.36 into equations 3.35a and 3.35b. J 1D = t V s l - [j V - ^ s l Vs l + i 2V s 2 ) ] University of Ghana http://ugspace.ug.edu.gh 46 (3.37a) (3.37b) The terms „v „ and (J) .v in equations 3.37a and 3.37b appear S t - S Z S-L SJ- to be necessary in order to account for the possibility that the two salts have a common anion (or cation), in particular when the diffusion flux is monitored according to the movement of that common ion. It should be pointed out that both „v „ are very small in S J- S JL S Z S Z comparison to the other terms in the equations. from equation 3.34b according to the following general guideline: every flux appearing in the dissipation equation is set to be a linear homo­ geneous function of every force appearing in that same equation. Thus: The transport equations for water and salt are now composed . v (3.38a)] (3.38b) . D (3.38c) where L is called the phenomenological coefficient and represents the constant of proportionality between the flux and the force. is University of Ghana http://ugspace.ug.edu.gh directly related to the hydraulic conductivity, is the coefficient relating the flow of water conjugate to an osmotic pressure gradient (capillary osmosis) and L is the salt sieving coefficient, that is the coefficient representing the movement of the solute relative to the solvent as a pressure gradient is applied in the absence of concentra­ tion gradients. In general, L is a coupling coefficient relating flux of entity i due to the force conjugate to entity j. At this stage, only the movement of salt 1 will be considered. V Substitution of j from equation 3.38a into equation 3.37a gives: jlD = » W + Osl^sl + *S2VS2 - (- V * - W ^ l ' W V Therefore, % + *sl)vsl + ^s2Vs2 = * 1 - H VP - S D l ^ l - LVD2V772 (3'39) Substitute equation 3.38b into equation 3.39 for jj^s <♦» + + ’ - V l TO - LB1W 1 - W 2 - \ ,p - LVD1VTT1 LVD2V7T2 Therefore, University of Ghana http://ugspace.ug.edu.gh 48 As a first approximation, ”LDD2^Tr2 whicl1 is a secon^ order interaction term and may be very small in magnitude in comparison with the other terms, is omitted. Assuming that the semi-permeable properties of the medium are the same for salt 1 and salt 2 , then L^di = an<3 + ^ 7T2 = '^7T’ t l^e tota-*- osmotic potential gradient. It must be pointed out that when two solutes are "different" (in dissociation or in size of particles), the semi- permeable properties of the soil with respect to the two solutes are different; for example, for a mixture of carbowax and KC1 the semi- permeable properties of the medium will be very different for the two solutes. (3.41) Equation 3.41 with Lyp-^ set equal to Ly|-p becomes: (3.42) Multiply equation 3.42 through by c^: (3.43) University of Ghana http://ugspace.ug.edu.gh 49 Equation 3.3b for unidimensional flow of salt and water may be written as: q = -0D Vc - c K VH (3.44) s s where VH is the hydraulic potential gradient (mm )^ and K is the hydraulic conductivity (ms "^) . Now comparing equations 3.43 and 3.44 it is observed that: (i) Ci[(*w + The continuity equation for water flow is written as: University of Ghana http://ugspace.ug.edu.gh 53 — - 8 mrfn —at " ^ [D(e) 9x c 3 0 Therefore, the term -r— on the left hand side of equation 3.53 cancels a t c 9 rD(0) 99, ^ u , L J on the right hand side: 9_9c _ ^ 9c _9_ reD 9c, _ _9_ ,0cK(9)_ 9tt, 9t x 9t 9x S 9x 9x gp 9x w D(9) 98 9c _ 9 facD(9) 96, -,1, 9x 9x 9x L 9x Let X = xt so that 9X/9x = t , 9X/9t = - t ^ . Substitution of X into equation 3.53 gives: X9 9X dc _ ex dc _ _d_ ,GD dc_, _ _d_ ,qcK(9) drr, 2 dX 2 dX dX 1 S dXJ dX ^ gp dX w , D(9) d9 dc d acD(0) d0 ... + d x d x ' d x ( d x } (3-54) It is necessary at this stage to derive an approximate relation­ ship between 0 and concentration of Cl in solution,c,or water content 6 in order to estimate a . Substitution of the diffusion flux jj = — - jV , into equation 3.45b yields e jsl/c - jV a = - C J ] j Vtt=0 where c^ is the local equilibrium concentration. jVc Vtt=0 e University of Ghana http://ugspace.ug.edu.gh 54 JslBut c = (— ) m - V r ,i Vtt=0 where c is the mean local concentration, m Therefore, c a = 1 - — (3.55) c e Alternatively, relation 3.55 is derived by considering that at the inlet end of our soil column, the equilibrium concentration is known (c . = 1 .0 ) ei because the liquid there is directly in contact with an "equilibrium pool" (i.e. the solution in the inlet compartment). c is also measured for m the first section of the soil column and so O at A=0 can be estimated. •k At some position A , even though the equilibrium concentration is not ■k known, the moisture content 0 is known. Now consider figure 3.1 where A represents the surface of a soil pore, b is the thickness of the exclusion layer where c=0 and b is the thickness of the film of water on the surface. Then b 0 ex ex a = — = — (3.56) Also, c (b - b ) e______ ex Cm b Therefore, University of Ghana http://ugspace.ug.edu.gh 55 F i g u r e 3- S chemat i c d i a g r am i n d i c a t i n g the e x c l u s i o n zone, t h i c k n e s s o f f i l m o f w a te r and e q u i l i b r i u m c o n c e n t r a t i o n . University of Ghana http://ugspace.ug.edu.gh 56 Figure 3.1 University of Ghana http://ugspace.ug.edu.gh 57 c b — = 1 e x ce b b c and a = ~ = 1 - -E (3.57) b c e which is similar to equation 3.55. Since c . is known for the inlet ex section, and a at A=0 is calculated using equation 3.57, 0 (assumed constant) can be estimated by substituting the a calculated for A=0 into * equation 3.56. Thus at position A 0 (A) = H P (3.58) dc Designating 0D -rr- = y, equation 3.54 becomes s dA dy _ ex dc _ _0A d c d_ ,CTcK(0) dir _ ... dj3 _dc dA 2 dA 2 dA dA 1 gp dA ; 1 j dA dA w d_ ,0cD(0) d^ . dA C dA ; Multiply through by dA and integrate, noting that the initial and boundary conditions formulated in equations 3.20a and 3.20b imply that at 9 = 0 , n J A J c = c , and — r = 0, —t- = 0 and hence y = 0 . Thus,n dA dA n y fC •c ’C 0ex 2 Xdc - ~ 0 Adc + acK(0) ^ - gPT7 dA w D(8) |f dc + c c c acD(0) d0 dA It follows that, University of Ghana http://ugspace.ug.edu.gh 58 c 7 = 1 (9exX " 0X - 2D(0) |f)dc + acK(9) dTr gPT7 dX W acD(9) d6 dX n Therefore c c (3.59) n Equation 3.59 was modified and solved using a computer program written in System 360 CSMP language (see figure 4.8 and section 4.9.4). 3.4 Dispersion During Vertical Infiltration Assuming unidimensional flow during vertical infiltration, the flow velocity, v, is given by: where z is the vertical space coordinate taken positive downward. Sub­ stitution of equation 3.10 (with z instead of x) into 3.61 gives the equation describing the distribution of water content with time for vertical flow: v = K(0) - D(9) oz (3.61) (3.62) The following initial and boundary conditions apply: University of Ghana http://ugspace.ug.edu.gh 59 z > 0 , t = 0 0 , t > 0 (3.63a) (3.63b) Substitution of equation 3.61 into 3.15a (with z instead of x) yields the equation describing the rate of change of concentration of nonreactive solute with time for vertical flow: It ‘ If) - « 0> - 1 1 If |f = j; [9D (0) |§] + D(0) |f |f - K(0) |f (3.64) dt a Z S dZ dZ oZ dZ The initial condition for the solute is: c = c , z > 0 , t = 0 (3.65a) n As discussed by Smiles et aJ. (1978) two forms of boundary conditions are possibly appropriate, namely the constant concentration boundary condition: c = c q , z = 0, t > 0 (3.65b) or the flux boundary condition: - 6Dg ( 0 ) |f = v ( c q - c) , z = 0, t > 0 (3.65c) University of Ghana http://ugspace.ug.edu.gh 60 As shown by Philip (1957) and later used by Elrick e_t al. (1979) , equations 3.62 and 3.64 rewritten with z as the dependent variable are of the form: lz 3K(0) _ 3 r D(0) . ,,v ~ 3t "30 - 30 [^730 ] (3>66) and Both equations 3.66 and 3.67, subject to conditions 3.63a, 3.63b, 3.65a, 3.65b, have solutions in power series of the form: z(0) = A(0)t1/2 + Xw (9)t + ipw (0)t3/2 + tow (0)t2 + ... (3.68) z(c) = A(c)t^ 2 + x (c)t + ib (c)t3/2 + ca (c)t2 + ... (3.69) s ' s s Details of the mathematical treatment for obtaining the coefficients X , x> w etc. as given by Philip (1957) and Elrick e_t al. (1979) are provided in Appendices A and B. The resultant ordinary differential equations for X , ip and a) are summarized in Table 3.1. In equations 3.72 through to 3.77 it is observed that x> $ and 03 are the dependent variables and that 0 or c is the independent variable. Consequently it is appropriate to designate x> and to for water flow with a subscript, w, (thus we have x^ 7 ip^ and co^ ) and to use subscript 's' for salt flow, i.e. , x > 4> and co . It is pertinent to mention thats s s Xw (0) and Xs(c) are not the same. Also, ^ ipg and j4 a) - Physically this must be so since the salt front always lags behind the water front. University of Ghana http://ugspace.ug.edu.gh TABLE 3.1. Ordinary Differential Equations for Water and Solute Transport Variable Water Solute * -1 ■ 55 ID §i (3-71) * x » - 1 P (e> 3 T 1 + T p <3- 72) [!■«») 3 T + K f (3 T > 2 <3' 80) V 9> ■ eV 9> S (3 F )2 <3' 81 ) dip dx dtp dx R(0) = Q(G) [2 ~j7~ “ 'jf ] 0.82) Rs (0) = Qs (0) [2 i f " d f ] (3‘83) From Elrick et al. (1979). University of Ghana http://ugspace.ug.edu.gh 62 The theory presented in this chapter was tested experimentally using two clay soils which have relatively high cation exchange capa­ cities. Chapter 4 which follows, describes in detail the experiments conducted. It is, however, pertinent at this stage to introduce briefly the experiments conducted to test the theory presented for (a) transport of solutes which interact with the soil matrix, (b) flow systems vhere anion exclusion occurs and (c) for vertical movement of water and chloride in soils. Preliminary experiments were conducted to: (i) ascertain if the assumption of instantaneous equilibrium between the solid and solution phases is valid; (ii) determine the adsorption isotherm for these soils; (iii) verify the method used to estimate the concentration of K+ in solution during the horizontal infiltration experiments, and (iv) determine the soil moisture characteristic curves for sorption and desorption. Horizontal infiltration experiments were conducted to obtain 0(A) and c(A) data which were used to calculate D (0). The converse s problem of predicting c(A) from the calculated D (0) data was also done using C.S.M.P. Also, the theory developed for the calculation of D (0) in a flow system where anion exclusion occurs, was tested. Lastly, using the theory developed for vertical infiltration and input data 0(A), c(A) (from the horizontal infiltration experiments) and K(0) data, both water and chloride content profiles for vertical infiltration of water and solutes, were predicted. University of Ghana http://ugspace.ug.edu.gh CHAPTER 4 MATERIALS AND METHODS 4.1 Soils The soils used in this study are: (i) the Akuse clay loam which is a tropical black clay normally found under coastal savannah vegetation in Ghana. This soil is usually developed over basic gneiss (hornblende gneiss) and occurs on very gentle upland topography. It is dark grey to black, is sticky and plastic when wet but becomes hard and compact with cracks when dry. The dominant clay mineral is smectite (Oteng, 1976). Samples from 0-15 cm depth were used; (ii) the Brookston clay loam, an Orthic humic gleysol, is a poorly drained member of the Huron catena. This soil is usually developed over clay rich glacial till. The dominant clay mineral in this soil is illite. Samples of 0-15 cm depth, collected from Essex county in Southern Ontario, were used in this study; (iii) Caledon fine sandy loam, a grey brown podzolic from East half of concession l,Lot number 30,of Erin Township in Southern Ontario was used to check the concentration measurements from the chloride analysis. This soil is dark greyish brown and usually developed on fine sands overlying coarse gravel. Samples from 0-15 cm depth were used. Pertinent physical and chemical characteristics of these soils are given in Table 4.1. 63 University of Ghana http://ugspace.ug.edu.gh TABLE 4.1. Some chemical and physical characteristics of the soils used. CEC Exchangeable cations PH % Particle size analysis Saturated C a ^ Mg"1”1" K+ Na+ in Or- Hydraulic me/ me/ me/ me/ me/ 0 .01M g a m e % % / Conductivity 100 g 100 g 100 g 100 g 100 g CaCl2 1:2 Car- bon Sand Silt Clay -1 ms Akuse Brookston 29.24 28.09 0.069 0.23 0.67 6.35 1.19 47.98 18.31 33.71 1.54 x 10-5 26.16 19.0 5.33 0.51 0.22 6.10 3.19 17.0 48.0 35.0 4.31 x 10 - 6 O' ■P- University of Ghana http://ugspace.ug.edu.gh 65 4.2 Saturation of exchange complex of soils with calcium One kilogram of sieved soil was placed in a Buchner funnel. Two litres of 0.5M calcium nitrate adjusted to pH 7.0 were leached through the soil very slowly, stopping the flow periodically by fitting a rubber stopper to the outlet of the funnel, in order to ensure effective exchange of cations. Samples of the effluent were collected after the last portion of the calcium nitrate solution had been allowed to saturate the soil overnight. The effluent collected was analysed for potassium to ascertain if most of the local exchangeable K+ were replaced by calcium. The soil was then washed with two litres of deionised water after which it was air-dried, ground and sieved to pass through a 0.25 mm sieve with square holes (wire woven). 4.3 Time of equilibration experiments The purpose of this experiment was to determine the time required for equilibrium conditions to be established between the quantity of K+ adsorbed by soil and K+ in solution in order to allow sufficient time for equilibrium to be achieved in the 'batch-shaking' procedure used k for the adsorption isotherm determination. In addition, it was neces­ sary to ascertain if the assumption of instantaneous equilibrium used to develop equation 3.8b is sufficiently justified. Five grams of soil (in triplicate) were placed in 15 ml plastic k An adsorption isotherm is the relation between the amount of solute per unit weight adsorbed by the porous material in equilibrium with concen­ tration of the same species of solute in solution. University of Ghana http://ugspace.ug.edu.gh 6 6 centrifuge tubes. Five millilitres solution of KC1 adjusted to pH 7.0 and containing 1.0 m.e. K+ was added to each tube and equilibrated on a reciprocating shaker for the following time periods: 5, 10, 15, 30, 60, 120 and 180 minutes. At the end of each shaking period, the set of tubes for that treatment was centrifuged for 5 minutes and the supernatant solution decanted and stored for analysis of K+ using the atomic absorption spectrophotometer. 4.4 Determination of Adsorption Isotherms Quantitative determination of potassium adsorption by soils from solution is essential to an understanding of the contribution of adsbrp- tion to the transport of potassium in soils. The conventional method, "batch equilibration method", which involves measurement of concentra­ tion changes in a solution of known volume after shaking with an adsorbent of known weight was used. Five grams of the sieved Ca-saturated soil were weighed into a 15 ml plastic centrifuge tube. Five millilitres of 0.001M, 0.002M, 0.003H, 0.004M, 0.005M, 0.006M, 0.008M, 0.01M, 0.02M, 0.03M, 0.04M, 0.06H, 0.08M, 0.1M, 0.2M, 0.4M, 0.6M, 0.8M and 1.0M KC1 solution adjusted to pH 7.0 were added to each tube and equilibrated for three hours on a reciprocating shaker. The tubes were then centrifuged for 5 minutes and the super­ natant analysed for both K+ using the atomic absorption spectro­ photometer and Cl using a method similar to the Technicon Auto­ analyzer Industrial method no. 99-70W. The concentrations of K+ and Cl~ in the original stock solutions were analysed and by difference between this measured concentration of the stock solution and the concentration University of Ghana http://ugspace.ug.edu.gh 67 of the solution after equilibration, the amount of K+ or Cl adsorbed per gram of soil was calculated and plotted against the equilibrium concentration. k 4.5 High energy moisture characteristics A Buchner funnel with a sintered ^late (porosity 4) connected by flexible tubing to a hanging column of water and a horizontal 10 ml burette was set up with the surface of the sintered plate at a matric potential of -120.0 cm ^ 0 . The initial volume of water in the hori­ zontal burette was noted. A known mass of soil (which was subsequently used in the infiltration experiments), of known moisture content was poured into the funnel and tapped gently to a predetermined volume so that the bulk density at packing was also known. The horizontal burette ensured that sorption occurred at a constant matric potential of -12 0 . 0 cm. When there was no further uptake of water by the soil (i.e. when the meniscus of water in the horizontal burette was constant) the volume of water absorbed was read off. The volumetric moisture content of the soil at this matric potential was obtained by adding the volume of water adsorbed to the original water in the soil before packing into the funnel and expressing it as a ratio to the total volume into which the soil was packed. The complete sorption cycle from a moisture potential of -120.0 cm H^0 to moisture potential of zero was thus obtained by determining the equilibrium moisture content for successive 20 cm reductions in the height of the hanging column (i.e. 20 cm increases in matric potential). Once at k High energy refers to matric potentials from slightly greater than zero to approximately - 1 2 0 cm H^O. University of Ghana http://ugspace.ug.edu.gh 6 8 matric potential of zero (measured relative to the middle of the soil sample) the draining cycle was determined on the same sample by lowering the hanging column in 20 cm steps. A check on cumulative quantities of water taken up and subsequently released by the soil was obtained by sampling at matric potential of -120 Ho0. 4.6 Determination of Hydraulic Conductivity The saturated hydraulic conductivity of the Ca-saturated soils used in the horizontal infiltration experiments was determined with a constant head permeameter. The unsaturated hydraulic conductivity, K(0), was obtained from the sorption moisture characteristic curve and the moisture dif­ fusivity data obtained from horizontal infiltration experiments through the relationship K(0) = D (9) / || (4.1) where D is the soil water diffusivity and 3h/39 is the slope of the moisture characteristic curve. The computation of the K(9) values was done with a computer pro­ gram written in system 360 CSMP (see Appendix Cl for description). University of Ghana http://ugspace.ug.edu.gh 69 4.7 Chemical Analyses Cation exchange capacity and exchangeable cations were determined by standard methods (Chapman, 1965). Organic matter was determined by the method of Walkley-Black (1934) and pH in 0.01M Ca C ^ by the method of Peech (1965). 4.8 Horizontal Infiltration Experiments The infiltration column (Figure 4.1) used in the present study is similar to that used by Elrick e_t al. (1979). It consisted of a cylinder (A), 0.45 m long and an internal diameter 0.0285 m which was assembled from lucite sections made up of two lengths (refer to fig. 4.2 and D and F of fig. 4.3). The inlet region up to about the middle of the column was made of sections 0.0199 m long (fig. 4.3D) and from the middleto the end made of sections 0.0098 m long (fig. 4.3F). The first section of the column (fig. 4.3E) has two entry ports (one for entry of solution and the other to aid in the escape of air). Also a glass bead base (approximately 0.002 m thick) was pro­ vided in this section to facilitate the establishment of the boundary condition at x=0 . The second section in the column (fig. 4.3D) has a groove in which a 0.035 m (internal diameter) lurene 0-ring is embedded to ensure a watertight seal. In order to ensure proper alignment of the sections, University of Ghana http://ugspace.ug.edu.gh 70 an acrylic rod in addition to two small rods (fig. 4.2) were used. These rods were removed once alignment of the sections was obtained and the frame made to hold these sections has securely been tightened (fig. 4.1). Uniformity of packing of the column of soil was achieved by pouring the soil into the column evenly and tapping at the same time as the soil was being poured. With the Akuse soil, two initial moisture contents were invest­ igated, namely air-dry water content of 0.05 and a moistened soil of water content 0 = 0.10. Only the moistened soil of water content 0.12 was investigated in the case of the Brookston soil whilst the Caledon sand used to check the concentration analysis of chloride had an initial moisture content of 0.001. All the experiments were carried out in a laboratory in which the temperature varied between 21 and 23 C. The most critical phase of the experiment was the start. Initially, the burette (B) was raised to the same level as the top of the horizontal column of soil. The burette was then opened and the stop watch started simultaneously. After about 30 s, the initial positive inflow head was lowered to approximately -1 mbar and at the same time the port for the escape of air was closed. The infiltrating solution in all experiments was 1.0M KC1. The cumulative volume of solution as well as the distance to the wetting front was recorded as a function of the square root of time as an initial University of Ghana http://ugspace.ug.edu.gh 71 check on the preservation of similarity with regard to water flow. Data were obtained from experiments terminated at different elapsed times. At the end of each infiltration experiment, the burette was closed and the final time recorded quickly. The bolts securing the frame were then loosened and with a small wooden plank placed on top of the column, it was knocked down thus slicing the column into sections. The moist soil from each section was quickly weighed in tared moisture dishes and then divided into two subsamples, half of which was used for gravimetric moisture content determination and then for the chloride determination. This initial weight of the wet soil in each section was used to calculate the oven dry equivalent of the soil in each section for the bulk density calculation, once the moisture content on weight basis was obtained on half of the sample from each section. The other half of the wet soil from each section was weighed into a tared 15 ml centrifuge tube and used for the determination of concentration of potassium in solution during the infiltration experi­ ment. 4.8.1 Determination of concentration of K+ in solution during the infiltration experiment Since the water content distribution of the horizontal column of soil was, for most of the sections, drier than saturation water content, it was not possible to directly determine the concentration of K+ in solution University of Ghana http://ugspace.ug.edu.gh 72 FIGURE 4.1 Set up for the Horizontal Infiltration Experiments (see text for description) Legend: A Horizontal column of soil B 10 ml burette University of Ghana http://ugspace.ug.edu.gh University of Ghana http://ugspace.ug.edu.gh 74 FIGURE 4.2 Some of the Sections Used for the Soil Column An acrylic rod is used to ensure good alignment of the sections. University of Ghana http://ugspace.ug.edu.gh 75 University of Ghana http://ugspace.ug.edu.gh 76 FIGURE 4.3 D: Lucite section with an O-ring to near the inlet end where soil is E: Inlet section showing the inlet air vent. F: A thin section. avoid leakage near saturation, tube and the University of Ghana http://ugspace.ug.edu.gh 77 University of Ghana http://ugspace.ug.edu.gh 78 by extracting the soil solution through centrifugal filtration as was done, for example, by Soon and Miller (1977). A method was therefore developed to estimate the concentration of K+ in solution during the horizontal infiltration experiment. The weighed half of the moist soil from each section of the soil column was e-lilibrated with 5.0 ml of 0.5M KC1 solution on a recipro­ cating shaker for 3 hours, after which the contents of each tube were centrifuged. The supernatant solution was then stored for determination of K+ using Atomic Absorption Spectrophotometer. The concentration of K+ in solution was then estimated from this equilibrium solution. The theoretical consideration for the estimation of the concentra­ tion of K+ in solution is as follows: consider a horizontal column which has been sectioned at the end of an infiltration experiment with KC1 as the infiltrating solution (fig. 4.4). The analysis which follows holds whether a sub-sample of the section was used or all the moist soil in the section was used. For simplicity, we will consider the case where the whole wet soil in each section was used. There is in each section, therefore, Ma_^ which represents K+ adsorbed on colloid during infiltration and the native K+ (meq) and Ms_^ , representing quantity of K+ in solution (meq) during infiltration. Let us equilibrate each section with a known volume (say 5 ml) of a KC1 solution whose concentration is higher than that of the soil solution after infiltration (an estimate of the concentration of KC1 to use in this second equilibration may be obtained from the adsorption isotherm). We now have in the system Ma^ which represents the quantity of K adsorbed during equilibration with the new solution of KC1 (meq) University of Ghana http://ugspace.ug.edu.gh 79 F i g u r e k . k : S chema t i c d i a g ram o f p a r t o f a s e c t i o n e d column o f s o i l . University of Ghana http://ugspace.ug.edu.gh SECTION KC1 SOLUTION AND SOIL SO ILDS University of Ghana http://ugspace.ug.edu.gh 81 and Ms^ representing the quantity of K+ in solution after the second equilibration (meq). Let Mg represent the amount of K+ in the original volume of solution used for the second equilibration. The total quantity (meq) of K+ adsorbed in both the infiltration experiment, the second equilibration and the native K+ on the soil before the infiltration experi­ ment are therefore represti.ted by the sum of Ma^ and Ma^ which may be given the svmbol Ma . In this notation, 'a' stands for adsorbed, 1i ' xe for infiltration, 's' for solution and 'e' for equilibration with another concentration. The mass balance for this system then is given by Ma. + Ms. + H = M a . + Ms (A. 2a) x x e le e which for convenience may be written as M a . + M s . = Ma. + M s - M (4.2b)x x xe e e Now, in order to calculate Ms^ and hence the concentration of K+ in solution during the infiltration experiment, Cs_^ , we represent: (i) the meq. K+ adsorbed per gram of soil during the infiltration experi­ ment and the meq. K originally on the soil before infiltration, by Sa^ Sa = (me/g) (4.3) i Mass of soil solids (ii) the meq. K in equilibrium solution per litre of solution associated with Sa_^ , by Cs_^ University of Ghana http://ugspace.ug.edu.gh 82 Ms . Cs . = — ------7-1 7 7— (me/1) (4.4) 1 Volume of solution ( in sample say ) x 10 3 y cm (iii) the meq. K' in the adsorbed phase per gram soil after both infiltration and subsequent equilibration with another KC1 solution of different concentration, by Sa. le Sa. = Ma. /Mass of soil solids (me/g) (4.5) le le (iv) the meq. K in solution per litre after the equilibration with another KC1 solution, by Cs . le _3 Cs . = Ms. /total volume of solution x 10 (me/1) (4.6) xe xe /• c, \(i.e. 5+y) Cs, is measured using the Atomic Absorption Spectrophotometer, and there- xe fore Ms^^may be obtained from equation 4.6. In addition to equations 4.2a to 4.6, we require the equation for the adsorption isotherm which in this study was given by a quadratic equation in the logarithmic form (equation 3.24) in order to solve the six equations with six unknowns. Using equation 3.24 and the calculated Cs. from equation 4.6, xe we obtain Sa and hence Ma. by substitution into equation 4.5. Since M xe xe J H e is known, the righthand side ofthe mass balance equation 4 .2b is known and may be represented by E. Thus Ma^ = E - Ms^ (4.7) For the infiltration experiment, equation 3.24 may be written as University of Ghana http://ugspace.ug.edu.gh 83 [-k log^Cs. + k^logCs^ - k^] Sa = 10 1 which upon i further substitution from equations 4.3 and 4.4 yields . M s . M s . Ma. [-knlog"(-— -------j------ ~) + k log(— ------1------ ~) - k ] l , „ 1 Volume of , n-3 2 ° Volume of in-3 3— 7 TZT = 1 0 , X 10 X 10Mass of soil solution solution solids (4.8) Substituting for Ma^ on the lefthand side of equation 4.8 from equation 4.7 we obtain „ m - M s . M s . Si [-k, log (tt-;------h------ “) + k^logCTT-;----- 1----------~------------ - 1 Volume of , -3 2 Volume of n - 3 3 Mass of 10 , ' x 10 , . x 10, . , solution solution soil solids (4.9) Equation 4.9 is an implicit equation and was solved for Ms^ by an iterative procedure facilitated by a programmable calculator. Once a value for Ms^ was obtained, it was substituted into equation 4.4 to get Cs^, the con­ centration of K+ in solution during the infiltration experiment. The assumptions inherent in this method of estimating the concen­ tration of K+ in solution during horizontal infiltration are that, (i) equilibrium or pseudoequilibrium conditions exist between the quantity of solute adsorbed and the concentration in the solution phase such that the equilibrium adsorption isotherm is applicable; (ii) the adsorption isotherm is unique when only the adsorption process is considered so that when the concentration in solution is increased through the second equilibration with fresh solution of KC1 of known concentration, the new equilibrium that is established follows the original adsorption isotherm curve; and University of Ghana http://ugspace.ug.edu.gh 84 (iii) the adsorption isotherm curve is not influenced by moisture con­ tent so that the equation for the adsorption isotherm may be used to estimate the concentration of K in solution for the low water contents of the soil column. 4.8.2 Experimental verification of the reliability of the method of estimating concentration of K+ in solution during horizontal infiltration In order to verify the reliability of the method proposed in Section 4.8.1 for estimating the concentration of K+ in solution during horizontal infiltration experiments, 5.0 g of Ca-saturated Brookston clay whose air-dry moisture content had previously been determined, were first equilibrated with 5 ml of 1.0M KC1 solution in a 25 ml centrifuge tube for two hours. Eight replicate samples were used. After equilibration, three of the tubes were centrifuged and the supernatant solution analysed for K+ using Atomic Absorption Spectrophotometer. The con­ centration of K+ obtained from these three replicates served as the measured equilibrium concentration of K+ in the initial equilibration which represents the equilibrium concentration in the horizontal infil­ tration experiment. At this stage the volume of liquid is the sum of the initial air-dry water content and the 5.0 ml of 1.0M KC1 solution added. Five milliliters of 0.5M KC1 were then added to the contents of each of the remaining five centrifuge tubes and equilibrated again for two more hours on a reciprocating shaker, after which the tubes were centrifuged and the supernatant solution analysed for K . The concentration of K~*" in this second equilibration was used to estimate the concentration of the first University of Ghana http://ugspace.ug.edu.gh 85 equilibrium solution and compared with the measured concentration for this first equilibration. SAMPLE CALCULATION The calculation was done for the verification experiment described in Section 4.8.2 and is similar to that employed i. estimating the con­ centration of K+ in solution in the infiltration experiments. Mass of soil solids = 4.565 g 3 Volume of water in air dry soil = 0.435 cm Volume of 1.0M KC1 added = 5.0 cm3 3 Therefore volume of liquid in system at first equilibration = 5.435 cm Sa. 1 Ma. M s . T cTcT meq/g, Cs = ^ meq/I 5.435 x 10 J Ma . Ms. SaiP = raq/g. Cs-,-n= To = 645.78 meq/l 16 10.435 x 10 Ms. = 6.7387 meq le Ma. = 0.8229 meq le Me (measured) = 2.4616 meq Therefore, from equation 4.7, Ma. + Ms. = (6.7387 + 0.8229) - 2.4616 i i = 5.10 meq. From equation 4.9 we obtain the equation University of Ghana http://ugspace.ug.edu.gh 8 6 M s . M s . 5.10 = Ms. + 4.565 (. -0.10291og (----- 1— rr) + 0.74441og( To) 1 10 5.435x10 5.435x10 - 2.0231 ) Msi = 4.2519 Csi = 782.32 Sa. = 0.1858 l The measured equilibrium concentration of K and the concentration of K calculated using the technique and theoretical procedure outlined above, are given in Table 4.2. It is observed that the agreement between measured and calculated concentration of K in solution is good. TABLE 4.2 Measured and computed concentration of K in solution using the method proposed in the text (Brookston clay) Replicate Number Calculated Concentration (keq/m3) Measured Equilibrium Concentrations (keq/m3) % Error = (Calc-Measured)xlOO Measured 1 _ 0.7801 2 - 0.7801 3 - 0.7801 4 0.7823 +0.3 5 0.7823 +0.3 6 0.7823 +0.3 7 0.7823 +0.3 8 0.7823 +0.3 4.8.3 Determination of concentration of Cl in solution during the infiltration experiment Each of the oven dried subsamples from the determination of gravimetric water content of each section was shaken with 150 ml distilled University of Ghana http://ugspace.ug.edu.gh 87 water for 30 minutes on a reciprocating shaker. The extract was centri­ fuged and chloride in the clear supernatant solution determined using a method similar to the Technicon Auto Analyzer Industrial method no. 99-70W. 4.9 Computer programming method used to analyse data The computation of the moisture diffusivity D, the dispersion coefficient Dg and subsequent simulation of (i) moisture content 0 as a function of A(=x//t) and (ii) concentration of Cl or K+ in solution c as a function of A using the computed values of D and D^ were done with a computer program written in System 360 CSMP (I.B.M. 1972) whose state­ ments also obey FORTRAN conventions. Each program consists of three segments: INITIAL, DYNAMIC and TERMINAL. All operations specified in the INITIAL part of the program are carried out prior to the computation or simulation and are not repeated every time step. The operations in the DYNAMIC section, however, are per­ formed repeatedly and updated for each elapsed time-interval during the period of simulation. Finally, the TERMINAL part consists of calculations or decisions which are to be made only after the dynamic procedure has been completed. It must be noted that segmenting a program is not manda­ tory in CSMP and that if a program is not segmented, the computer auto­ matically assumes the entire program to be DYNAMIC. A concise description of CSMP is given in the User Manual (I.B.M. 1972). Detailed explanations of the principles and procedures of CSMP have been provided by Brennan and Silberberg (1968) and Speckhart and Green (1976) . CSMP models for simulation of some soil physical processes have been published by Wierenga and de Wit (1970) and also Hillel (1977). Therefore, only the principal University of Ghana http://ugspace.ug.edu.gh 8 8 features of CSMP language which were found to be useful in this study will in general terms be discussed. Perhaps the most powerful and important features of CSMP which make it particularly amenable to this study are the statements which cause (i) integration to be performed with time, and (ii) differentiation to be performed with redpect to time. The general form of statements for these mathematical operations are: Y = INTGRL(IC,X) (A.10) X = DERIV(IC,Y) (4.11) Equation 4.10 calculates the output variable Y by integrating the differ­ ential function X, with the initial condition that Y at the start of computation is equal to 1C. Using a METHOD statement, a choice may be made among seven numerical integration techniques, two of which use the variable time step of integration whilst the others use a fixed time step of integration. Equation 4.11, on the other hand, calculates the derivative output X by differentiating the integrated function Y with the initial condition that X at the start of computation is equal to IC. Another feature of CSMP which was frequently used in this study is the calculation of definite integrals. Definite integrals were per­ formed by instructing the computer to use the final time step value of the variable integration. This specification is put in the TERMINAL section. For an illustration of this feature, consider the calculation of equation 4.12, where t is the final time of integration. University of Ghana http://ugspace.ug.edu.gh 89 r t ft Ydt = Ydt - t n ^ Ydt (4.12) The definite integral on the right hand side of equation 4.12, designated KON, may be evaluated by specifying to the computer to set KON=B where B = INTGRL(IC,Y) 4.9.1 Description of the computer program for calculating the soil moisture diffusivity D(0) and simulation of 0(A) using the calculated D(0) values The actual program for Brookston clay written in System 360 CSMP is presented in fig. 4.5 (see also Appendix E for a similar program for Akuse soil). The governing equations are equations 3.11b and its solu­ tion (equation 3.13). Integration in CSMP is performed with respect to time. Renaming time equal to lambda, however, enables integration to be performed with respect to lambda. The integral is rewritten so that integration is carried out 11 with respect to A instead of 0 as follows: Ad0 in equation 3.13 n University of Ghana http://ugspace.ug.edu.gh 90 FIGURE 4.5. CSMP listing for calculating soil water diffusivity D as a function of water content 0 and simulation of - 1/2 water content as a function of A(=xt ) from the calculated D(9) values for Brookston clay. University of Ghana http://ugspace.ug.edu.gh 91 * * * * CONT INUOUS SYSTEM MODELING PR k A M * * * * ______________________ . . . * * * VERS IO N ____________________________ ____ ________ T I T L E C A LCU LAT IO N CF S O I L MC ISTURE C I F F U S I V I T Y AND S IM U LA T IO N OF >______________TTTI F * A T ^ P CflNJ T FN .l AS A F UN CT.I11N.. _QF i AMrtuA . LortU iDK.ST C K CL A Y )__________________ * U N IT S * KG =K I LOG HA MS ______________ * Mj=M£.T_ERS — -------------------------------------------- * S=SECONDS * GLOSSARY OF SYMBOLS ______________ ♦ DTHL A.n=DER I VAT I.V.E. QF T > F THFTA.~LA .MBDA.-FlJNCT I CiN AT I AMRO A F ^ l - A i -S _______ * ZERO * T HE T AO= W AT ER CONTENT AT LAMBDA EUUALS ZERO _____________ .* J_AMDAN=U-AMBDA._.AT. JUE.1NJ.TX. . X -JH K T 1M J ________________ . — . _____ ____ * THET A N = IN I T I AL WATER CCNTENT * T H L A = I NTERPOL ATED CURVE FCR TH ETA—LAMdDA FUNCT ION ______________4______R V F L =TH F PRODUCT GF S G I I v. A l ER D I FF U S I y I T Y AN u T>-F ^ r R I ^ A T i y g __________ * OF THE THETA -LAMBDA FUNCT1GN * RVELO=RVEL AT LAMBDA ECLALS ZERC ________ ____ _______S > s £ £ l±. .JfATJER H L F F U ^ X V - I J J l __ ... - * R D IV = D E R IV A T IV E OF RVEL W ITH RESPECT TO LAMBDA * RVELAO=RVELO M U L T I P L IE D BY 1 . 0 0 31 ______________£_____ BVE1-A=HALF PA T R GF RV F 1 THR f lUGh T Hr F ATT. Ik 1 . u-*Q i 11^. c y.gj, ^ ______________ * R D IV A =H A L F P A IR OF R D IV THRCUGh T h £ FACTL tk 1 • j 1 * T HE T A 2 = S IM U L A T ED WATER CONTENT ______________ Sl____ M EXAA= t iA l_ E _ i5 A XR _DB-T-HEXA2.- XHR£ -UG ta - IH£ -J ^AC -TC lR -U T.flOJ.___________________ * DR VEL-=CORRECT ICN FACTOR * DTHETA= D I F F E R E N C t BETWEEN THETA2 AND THETAA ______________£______^1 n P F s R A T i n n p t h f n i f f f r f n t . f r f t * f f n t h - t a a A ^ n t h = t a p r n the -_________ * D IF FERENCE BETWEEN RV EL AC AND RVELO * T BT HLA=T ABLE CF WATER CCNTENT AND CORRESPOND !n G LAMBDA VALUES__________ RE iJAJAE^.XM£sJ_AMaDA_______________________________________________ I N I T I A L PARAMETER DTHLAO = - 5 • C , T H E T A O = * 5 3 « L AMUAN=1 . 6 2 E - 0 3 * . . . ________________________ T H r T AN= l -» !-»■ R.VE-l. Cte— I »i»L.i ■= * L C-S J . »-!==;.■>■!■=-______________ R V E L A O = R V E L C * I 1 * 0 + D E L ) FUNCT ION TB THL A = ( 0 . 0 . . 5 3 > , < 1 . 0 E - 0 4 » • 5 2 5 ) * ( 3 . ) E - 0 4 * . 5 2 ) , . . . - . t a ^ E ^ A ^ . 5 4-^54-i ---- ( 1 . 1E—03 » . 4 9 ) . ( 1 . 2 E - 0 3 * . 4 8 ) , ( 1 . 3 E - 0 3 * . 4 7 3 ) , < 1 . 4 E - 0 3 . • <*C3 ) , . . . ( 1 . 5E—03 * . 4 4 5 ) . ( 1 . 5 5 E - G 3 * . 4 3 5 ) , (1 . o E - j 3 * . * 2 ) , ( 1 . 6 5 E - , 5 , . 4 " ) , . . . ________________________ * 1« 7 E - 0 3 . i 3 7 ) . t U 7 5 F - C 3 . » 3 ? 5 ) . I ) . 7 a = ~ ^ . . ^ a ) . f 1 . f l = - Q 3 . . ,£ 3 « ) -------- -------- ! ( 1 . 6 1 E—0 3 * . 1 65 )» ( 1 . 8 2 S - C 3 , . 1 2 ) » (1 . d c - J j , . 1 2 ) . ( 2 . P E - ' ' 3 , . 1 2 ) | DYNAM IC D T H L A = D E R IV (D T H L A O ,T F L A ) T H E T A 1 - I N T G R H T H E T A O .CTH LA ) ■________________________ C ^ I M G R H Q . G t T M l A )________________________________________________________________________ K ON 1=THETAN♦LAMD A N E = ( T H L A * L A M 6 C A ) - K O N l F F - F - C __________________________________________________________________ ______________________ G =FF+K 0N 2 D=—. 5 * G / D T H L « ________________________ NU5QRT______________________ ___________________________________________________________________ I F ( J • LE • 2 ) GC TO 10 j SORT ________________________ RVEL= I NTGRL I C.VF1 f l . B n i V i _________________ _ _________________________________ ____ R VELA = IN T G R L (R V E L A O .R C IV A ) R D I V=—LAMBDA *RVE L * . 5 / C \ _______________________ RD I V A = -L AM BDA * W Fi a * - S / r. ____________________________________________________ THE TA 2 = I NTG R L ( TH E T AQ # R V E L /D ) T HET A A = I NT G R L (THETAO * R V E L A /D ) NOSORT 10 CONT INUE University of Ghana http://ugspace.ug.edu.gh TERM INAL . _____________________ T T M T p r I KIT T M = 1 Qg.l = ---------------- P R IN T G # T H E T A A *0 * THET A 1 * T H L a • THET A 2 I F ( J * * j E « 3 ) GC Tu I S . j .ss j+_l - - - KON2=C GO TO 20 >----------------------------- L5_____l=_ULL-------------------------------------------------------------------------------------------------------------------------- I F { A B S ( T H E T A 2 -T H E T A N ) . L T . . DO 001 ) STOP D T HET A=THET AA—THETA 2 ______ - S LOPE=DTHET-A /DSL - / -RV£LC ------ - ...................... D R V E L = ( T H E T A 2 - T H E T A N ) /S LO PE R VELO=RV ELU—CRVEL _______________ av~i AUcRvPi r.*/ 1. ft+nFi >_____________________________ WRITE ( 6 , ICO ) THETAA ,D « T HL A » THET A1 * T HET A 2 100 F ORMAT( / / / * THET A A • D * T h LA * T H E T A 1 * THET A2 = ' * 5 - 1 5 * 6 ) WR ITE ( f a , 10 1 ) RVELO * K-VELAO — 101 F uRM A T ( / / * CALL RERUN W ITH R V E LC . R V E L A O - 2 E 20 * 7 ) 2 0 i F ( J . G T . l O ) ST CP __________________________________3 F ( I. «.C»I • ___S-T-lJ-E_________________________________________________________________________ ____ CA L L RERUN END ________ - STOP - OUTPUT V A R IA B L E SEQUENCE R V F L A O T H-l. A_____ D - lU L - a ___ T .HF .T.AJ___ C--------------------K f M ----------- E____________EJ£-----------------G--------------------D----------------- Z Z 0 0 0 4 R O IV RVEL RD I VA RVE_A Z 2 CO 0 6 THETA 2 Z Z C J 1 0 THETAA Z Z O O i l Z Z 0 0 1 2 J K 0N 2 I DT HET A S LCPE DRVEL RVEl O RVELAO OUTPUTS IN PU TS PARAMS I NTEGS + MEM B LKS FORTRAN DAT A CCS 3 3 ( 5 0 0 ) 71 ( 1 4 0 0 ) 1 2 ( 4 0 0 ) 64- 0 = 6 ( 3 0 0 ) 3 9 ( b J J ) 13 ENDJOB University of Ghana http://ugspace.ug.edu.gh 93 Using integration by parts u = X du = dX dv = d0 Therefore, ■0 0 Xd0 = uv - 0 0 n n •0 9 Xd0 = 0X - 0 0n vdu 0dX 0X - 0 X n n 0dX (A.13) Since integration by the computer necessarily always commences from time=0 (in this case X=0), I 0dX was then rewritten such that integration JX would start from zero as follows: rX =c» n fx rX =°° n 0dX = 0dX + 0dX (4.14) 0 0 X University of Ghana http://ugspace.ug.edu.gh 94 f0 -A =oo fAn Ad0 = 0A - 0 A + 0dA - 0dA n n 0 0 0 (4.15) The governing equation 3.13 for the calculation of D therefore becomes D = - ^ [9A - 0 A + 2 d0 n n A =» n 0dA 0dA (4.16) The Dvalues obtained from equation 4.16 were then used to simulate 0(A) by multiplying the second term of equation 3.11b by ^ ^ and rearranging it as follows: A . m m — dA [D(0) dA A_ D(0 ) _d0 2 D(0) dA (4.17) J A Letting y = D(0) = RVEL (4.18) equation 4.17 becomes RDIV = - ---- — — dA 20(0) (4.19) The INITIAL section begins with specifiable constants DTHLAO, the derivative of the 0(A) function at A=0; THETAO,the moisture content at A=0; LAMDAN, the final A where the initial moisture content is THETAN; A 0dA but because it is not known at 0 K0N2 which is the definite integral this stage, is set equal to 1.0 in the PARAMETER section. The actual value of K0N2 is used after the first set of calculations through CALL RERUN and the specification K0N2=C in the TERMINAL section. The University of Ghana http://ugspace.ug.edu.gh 95 counters J and I are initialised in the INITIAL section. Also the relation between RVELAO and RVELO is defined. It has to be mentioned that the multiplying factor may be made smaller, for example 1 .0 0 0 0 1 , in which case the value of .0001 used for the slope must be changed to .00001 in the algorithm at the TERMINAL section. (Steps in the algorithm are given in Appendix C.) Function TBTHLA is a table of volumetric water content 0 versus lambda (A) read from a curve obtained from the horizontal infil­ tration experiment. In this table, the first values of each bracket are X values (independent variables) and the second values are the corresponding moisture contents (dependent variables), arranged such that the independent variables are in a descending order of magnitude. In the DYNAMIC section, the following calculations are made and updated at each time step during the simulation: (1) Volumetric moisture content is read from the table of 0(A) provided in FUNCTION TBTHLA through a Lagrange quadratic interpolation between points from: THLA = NLFGEN(TBTHLA, LAMBDA) (4.20) where NLFGEN designates the arbitrary non linear function generator of CSMP for tabular pairs of x, y coordinates. (2) The derivative, d0/dA which is the rate of change of the volumetric water content with lambda: DTHLA = DERIV(DTHLAO, THLA) (4.21) (3) The derivatives of the 0(A) function computed in equation 4.21 are integrated again, as a check on the differentiation procedure: THETAI = INTGRL(THETAO,DTHLA) (4.22) rX (4) The integral 0dA is evaluated: 0 University of Ghana http://ugspace.ug.edu.gh 96 C = INTGRL(0.0, THLA) (4.23) (5) KON1 which is equal to the product of 9n and X^ is calculated and then (0X - 9 X ) evaluated: n n K0N1 = THETAN * LAMDAN (4.24a) E = (THLA * LAMBDA) - K0N1 (4.24b) X 0d\) is then calculated: 0 FF = E - C (4.25) (6) (9X - 0 X - n n (7) The terms in brackets on the righthand side of equation 4.16 are calculated: G = FF + K0N2 (4.26) where K0N2, initially set equal to 1.0, is the definite integral Xn=°° 0dX . 0 (8 ) The soil moisture diffusivity is then calculated by dividing the product of -.5 and G by the derivative d0/dX(DTHLA): D = -,5*G/DTHLA (4.27) So far, the program only calculates D(9) or D(X). There is then specification NOSORT which is necessary any time an IF statement is invoked or if the computer is required to perform operations in the sequence listed. The IF statement listed in fig. 4.5 requires that computation should bypass the statements immediately following the IF statement for the first two iterations. It is only when the counter J - 3 that simula­ tion of 0(X) is started. This stipulation is necessary because the value of the definite integral K0N2 is known only when J - 1 and so the correct values of D for simulation are obtained when J £ 1. (9) Calculation of y (RVEL) by integrating, dy/dX (RDIV) (equation 4.19) is executed: University of Ghana http://ugspace.ug.edu.gh 97 RVEL = INTGRL(RVELO,RDIV) (4.28) RVELO is first estimated and specified in the PARAMETER section. Suc­ cessive values of RVELO are calculated using the algorithm in the TERMINAL section, until either (a) the absolute value of the difference between the simulated moisture content at FINTIM and the initial moisture content is let.>< than .00001, (b) J is greater than 10 or (c) counter I is greater than 6 . (10) Equation 4.19 is used to calculate dy/dA (RDIV) RDIV = -LAMBDA * RVEL * .5/D (4.29) (11) Simulated 0, designated THETA2, are obtained from the integration of equation 4.18: THETA2 = INTGRL(THETAO, RVEL/D) (4.30) As explained in Appendix C, equations 4.28, 4.29 and 4.30 are paired by multiplying the initial condition RVELO of equation 4.28, by a factor which in this example was 1.0001. Consequently we have RVELA = INTGRL(RVELAO, RDIVA) (4.31) RDIVA = -LAMBDA*RVELA*.5/D (4.32) THETAA = INTGRL(THETAO, RVELA/D) (4.33) The TERMINAL section includes statements of the total simulation time to be run (FINTIM) , the time intervals for printing the output (PRDEL) and the list of output variables to be printed. This section also specifies that after the first iteration where J=0 and KON2=1.0, the value of K0N2 in the second and subsequent iterations should be set equal to the last value obtained in the calculation of equation 4.23. In addition, this section contains the algorithm for the computation of RVELO, RVELAO and other iterations. University of Ghana http://ugspace.ug.edu.gh 98 When there is no METHOD statement in the TERMINAL as with this program, the method of integration used is the Runge-Kutta fourth order variable time step procedure. Finally, the TERMINAL section has two WRITE statements. The first write statement specifies that values of 5 output variables in E-format with a total of fifteen digits and a 6 decimal place should be printed for each iteration. The second write statement specifies that RVELO and RVELAO values used for each iteration should be printed. 4.9.2 Description of the computer program for calculating dispersion coefficient D (A) and simulation of c(A) using the calculated D (A) values s lating c(A) for Brookston clay, also written in system 360 CSMP, is given in fig. 4.6 and also Appendix F which is a similar program for Akuse clay. Many of the features in this program are similar to those of the preceding program. Enough of the features, however, are new to warrant a complete (albeit partially repetitive) explanation. The governing equations are equations 3.18b and 3.18c. Using equation 4.15, g(6) in equations 3.18b and 3.18c is written as The computer program for calculating D for Cl and then simu- •A=°° A g(0) = 6A - [0A - 0 A + 0dA - 0dA] n n J 0 J 0 A =c° An 0 A 0dA + 0dA (4.34)n n 0 0 For very large numbers or small numbers the format involving the letter E is used in CSMP, for example 1.7647E—14 is equivalent to 1.7647x10”-^. University of Ghana http://ugspace.ug.edu.gh 99 FIGURE 4.6 CSMP listing for calculating dispersion coefficient for Cl and simulation of c(A) from the computed Dg (A) values using (i) the analytical solution, and (ii) the computer solution described in the text. University of Ghana http://ugspace.ug.edu.gh 100 / f t f t f t f tCONT INU 3US SYSTEM MODEL I No PROGRAM * * * * * * * V F R S I f lN .*.3. . . * * * . .............................. . ......... ..... . _________ T I T L E D IS P E R S IO N C O E F F IC IE N T AND S IM U L A T IO N J f CUN C . ( BROOKSTON C LAY ) / * K E U = K IL G - E Q U I V AL ENT S * M=METERS * GLOSSARY OF SYMBOLS ♦ DTHLAO=DER I V A T IV E OF T H ETA -LAMBDA FU NC T IO N AT LANeDA * * D T H L A =D E « IV A T IV E OF THETA - L AM BDA FUNCT ION * THE T A Q=-VOLUME TR I C WATER CONTENT AT LAMBDA EQUALS ZERO * ft THET AN= I N I T I AL WATER CONTENT ft THL A = IN TE RPOLATEO CURVE FOR T H ETA—L AM BOA FUNC T ION VFL C = THr P fiiTR l , .C T J jF (n AT FR CPINT F NT . QTSPPPsT i lN . m c F F I T T S KIT ft AND THE D E R IV A T IV E OF T l-E CONTR AT ION—LAMBDA F oN C T IO N * VELCO=VELC AT LAMoDA E C lA L S ZEPC VEL CAQsVELC n MLjL T I P I I F R RY l . n o n n i ft SEE 0 = CONCENT RAT I ON OF O L C R I D E IN SO LUT ION AT oAMEDA EQUALSft ZEROft «;ff n= t n i T t ai r r w r F N T P a t t h n o f r w r c r n ? i n mm c r n * DSEEL A =DER I VA T I VE OF C ONCENTRAT I ON- LAMBDA FUNCT IO N , - i ITH RESPECTft TO LAMBDA * EQUALS ZERO * SEEL A = IN TER PO LAT ED CURVE FOR CCNCENTRAT ION -LAMBDA FUNCT ION _ * * T B TH LA =TA B LE OF WATER CONTENT AND CORRESPONDING LAMBDA VALUESft T BS EL A= TABLE OF CONCENTRAT ION CF CHLOR IDE I N SO LU T IO N AND ft D = S O IL WATER D I F F J S I V I T Y * D IV = D E R IV A T 1 VE OF VELC M T H RESPECT TO L a MBDA « ft SEE l=CONCENTRAT IO N OF CHLOR IDE CALCULATED US IN G THE A N A LY T IC A Lft SO LU T I ON__ ftft DVELC=CORRECT ICN FACTORft SLOPE = R A T I 0 OF THE D IF FERENCE BETWEEN SEE A AND SEE2 TO THE * ft D S E E =D IF F E RENCE BETWEEN SEE2 ANC SEEA AT F l N T I V RENAME T IM E = L AMBCA I-N IX -LAL .. ............................ ................ - ..... .......... .............. - .............................- ____ - ... ................ .......... - - .................. I NCUN O S E L A L = - 1 , 0 ,S EEC= . 97 , SEEN = . t 0 = . V 5 L C O = - . 2 6 7 8 2 3 2 5 - 4 PARAMETER DTHL A3 = - 5 . 0 , T H E TAO = . 5 3 . LA r t JAN = 1 . B 2 E - 0 3 , J = 0 , I = 0 ________________CDl^ S.TAl^T T H r T a l c . 1 ? ■ K | - N ? = 1 .-1 .K,1 N.1 = I . I . K I 1 N 4 S I . L . H FI = . ' ! £ ~ 1____________ VELCAU = V E LC O * ( 1. 0 +DE L ) FUNCT I ON TOTHL A= ( 0 . 0 , . 5 3 ) , ( 1 . 0 E - 0 4 . . 3 2 .5 i , I 3 . I E - 0 4 , . 5 2 ) . . . . --------------------- 1-5. CE>_C 4 , . 5 1SJ .1 Z . ' J E - C t . - . 5 . U . i S . - £ - J 4 J - . -U 1 . L E -C 3 - . * -4 95 J ..... ( 1 . 1 E—03 . . 4 9 1 . ( 1 . 2 5 - 3 3 . . 4 8 1 , ( 1 . 3 5 - 0 o , . 4 7 j ) , ( 1 . 4 £ - 0 3 , . 4c-3 ) . . . . ( 1 . 5E—0 3 , . 4 4 5 ) . ( 1 . 5 5 E—0 3 » . 4 3 5 ) , ( 1 . o E - C 3 . . 4 2 ) , ( 1 . 6 5 5 - ' J . . - ) . ----------------------< 1 . 7 F - 0 3 . ■ --t 7 I . I I ■ 7 5 - - r .1 . ■ ~-i?5 1 . ( 1 . 7 fl = -.-).H . . ^ ) . ( 1 . HF- n ^ . a I -------- ( 1 . 6 1 E -0 3 , . 1 € 5 ) , ( 1 . 3 2 E —0 3 , ■ 1 2 ) . (1 • 9 E— } 3 » . 1 2 J . ( 2 . 0 E —0 3 . . 1 2 ) FUUCT1UN TdSEL A= ( 0 . c , . 57 ) , ( 1 . OE-C-4 , . 9 6 9 9 ) . ( 2 . OE—0 4 . . 9 6 9 3 ) . . . . ls a c»97 l s ..as-a 4 . . . . s o ____ _______ ( 6 . 0 E - 0 4 , . 9 6 9 4 ) , ( 7 . 0 E- 0 4 , . 9 6 9 3 ) , ( d . C . c - , 4 , , v o « ) . . . . ( 9 . 0 E - 0 4 . . 9 6 ) , l l . 0 E - 0 3 , . 9 5 ) . ( l . l f c - j j , . < y 3 ) . l l . 2 E - 0 J . . 8 < 9 5 ) , . . « ----------------------1 1 ■ a r i F - f l J , . , . p. 7 ) . [ 1 . 5 ) . ( ! . — 1 ■ . 7 r , ■_______________________ ( 1 . 4E -C 3 . . to 4 1. (1 . 4 2 E -C J . . 5 4 ) , ( 1 . 4 9 ) , C 1 . 4 4 E - 03 . . 4 0 ) . . . . ( 1 . 4 5 E —0 3 . . J 5 ) » ( 1 . 4 t o E—0 3 , . 3 0 ) . ( 1 . 4 7 5 —J J , . 2 5 ) . ( 1 . 4 3 5 —C 3 . . ? C ) * . » . ( 1 . 4 9 E - 0 3 , . 1 e> ,1 1 . 5r F-f 3 , . 1 5 ) . ( 1 . 5 2 5 - 0 3 , . 1 2 ) , ( 1 . 54E - ‘‘ 3 ■ . 9 ) . . . . ( 1 . 5 5 E - 0 3 , . 0 e s ) . ( 1 . 5 6 E - C 3 . . 0 7 5 ) , ( 1 . 5 7 5 - 0 3 . . 0 6 5 ) . . . . University of Ghana http://ugspace.ug.edu.gh 101 ( 1 . 5 8 c - 0 3 I • Ota) *( 1 • 5 y 5 E — 0 2 , * 0 5 ) * J 4 ) « • • • _______( - m . i i n? --n«.v.i .n . 7r-.^ » > n i s y ( i • 7 5 E - 0 3 * • C 1 ) * ( 1 « 8 2 E—C 3 * « 0 0 5 ) » ( 1 . * = - 0 3 , . 0 0 1 ) , ( 2 . 0 c - 3 , j . 0 ) DYNAMI C — -T-HL A=N1_F-GEN C T.£iTHLA.*.L AMSOA ) _ ............. ............ ..................... S EELA=NLFGEN (TBS EL A , LAMBDA ) C l HETA-=( THL A— T HE T A N ) / { T hE TAO—TH6T AN ) \ i- i _____________________________________________________________ _________ O TH LA =DER IV < DT HL A 0 , T 1-L A ) DSEEL A= D E R I V ( D S E L A O , S E E L A ) _____________________ _X=4-NT-GRL T-Hl. A ) _______________ _______________ ,____________________ _____________—................ ...................... K C N l = THETAN*LAMD AN E = ( TH LA *LAMBDA > -K O N l «=<=--c-r G=FF + K0N2 0 = —• 5 * G /D T HLA ___________ _____ ___LATHH = iT H LA *L AM tJ .D A ) + (-2-.G * O * 0 T H L A ) __ GS= OS EEL A * L AT HH H S= I NTGRL ( 0 . 0 .G S ) M C, 1 =K , IN ” —H HS2 —♦ 5 * H S 1 T H D S =HS2 /DS EELA ~ . —D-S=JTHD-S/ T-HL A -.......................... ----- NOSCRT I F ( I * L T • 3 ) GC TC 11 I F f T H H s * F ( 1 . .i* ) ;^f i TQ . .1* SORT A H = L A TH H /T H DS _________________ ___BH = J N T-GR-H-O« 0-* AM -)_______________. _____ - - ................ . C H = E X P ( - * 5 * B > - ) MH = I N T G R L ( v • C *C H /T H D S ) P MH - MH/K HN4 .............................. ........ . SEE1 = R M H * { SEEN -SEEQ >+ SEEC V E LC = IN TG R L ( VELCO *RD IV ) -----------------------------------------V E LC A = lN T oR L - (V E L CAO* R C IV A ) - ------- R D I V ——( L AT HH *VEL C ) / ( 2 * 0 * T H D S ) R D I V A = - ( L A T H h * V E L C A ) / ( 2 * 0 * T H D S ) r f f ? = r N iT^a i i « : F F n . V " i r / T H n « ; i S E E A = IN T G R L ( SEEO » V E L C A /T H D S ) NOSORT 4 0 - -CONTINUE TERM INAL METHOD S IM P PK1NT THI A . D . T H D S . D S . A H .S E F i A .SE~1 » ^ E - P T IM ER F I NT IM = 1 « 3 2 E —0 3 * DELT = 5 . 0 E - 0 t o , P R J i L = j • c a E—05 1=1 + 1 --------------------------- K 0N 2=C KCN3=HS I F ( I «LT • 4 ) GO TO 2 0 K HN4sMH I F ( A 8 S ( S E E 2 - S E E N ) . L T . • OCOC.'U ) S T j P DSEE =SEEA -S E E 2 ------------------ _ __s l l p e = d s e e / o e l / v e l c c ............ D V E LC = (S E E 2—bE EN ) /S LO P E VELCO=VELCO -CVELC v f i r .A ' teU F i r r . * i i . n + n F i i W R I T E t f c * 1 0 0 ) K O N 2 .K O N 2 »K0N4 1 00 F ORMAT( / / / * KOK2 • KON3 , K C N 4 = • , 3 E 1 5 * 6 ) -------------— - WRI TE ( b , 1 01 ) D SEE .S L C F E .D V E L C 101 F O R M A T ! / / / * DSEE * S LOPE *DVE LC = • , 3 E 1 5 . u ) W R I TE ( 6 * 1 0 3 ) VE LCO ,VELCAO V-------------------------- L 0 -3------ F t J R MA T ( //•__ CAL I p c a . i N Ui T T H V F l f T . W - i .‘ A . i g t . ^ r r - 7 1 20 I F ( 1 #G T . 1 0 ) ST CP C A LL RERUN END S T O P University of Ghana http://ugspace.ug.edu.gh 102 Designating y = 0D 4^ S Cl A (4.35) equation 3.18b becomes dy = _ g(8) dc dA 2 dA g (e) t dA (4.36) dc y =0 because equations 3.20a and 3.20b imply that -tt- at A is zero, n dA n Also, because integration commences from time = 0, equation 4.36 was rewritten such that integration would start at A=0 as follows: y = f[ g(6) g dA A g(0) f dA ] (4.36) Once y was obtained, 0D and hence D were obtained by substitution into s s equation 4.35 to give D = ^ s 0 dc (4.37) Simulation of c(A) from the calculated D values is achieved by s multiplying the second term on the lefthand side of equation 3.18b by 0D /0D and rearranging terms as follows: s s _d_ rqn dc , g(0) CDs dc dA L s dA J 2 0D dA ° s (4.38) Substitution of equation 4.35 yields University of Ghana http://ugspace.ug.edu.gh 103 _Z_ (4.39) dX 2 0D Equations 4.39 and 4.35 are used in the algorithm to simulate c(X). In addition, c(X) data are also calculated using the analytical solution equation 3.21. The first specification in ^ e program is to rename time=lambda so that integration and differentiation would be performed with respect to lambda. The initial section then begins with constants specified in CSMP data statements INCON, PARAMETER and CONSTANT. These three data statements are completely equivalent and are used to assign numerical values. In figure 4.6, the following parameters or constants are assigned numerical values: (1) DSELAO: derivative of c(X) function at X=0. This is estimated from the first two data points of the c(X) curve which was used to provide data points for FUNCTION TBSELA. (2) SEEO: concentration of chloride in solution at X=0. (3) SEEN: initial concentration of chloride in the dry soil before commencement of horizontal infiltration (i.e. concentration of chloride in solution at X=X (LAMDAN). n dc (4) VELCO: first guess of the value of 0D -rr- at X=0. s d A (5) DTHLAO: the derivative of 0(X) at X=0. (6) THETAO: the volumetric water content at X=0. (7) THETAN: initial volumetric moisture content of soil before hori­ zontal infiltration was started (i.e. moisture content at X=X ). (8) K0N2: the definite integral -i nA =°° 0dX in equation 4.34. 0 University of Ghana http://ugspace.ug.edu.gh 104 (9) KON3: the definite integral ■A =° n g(9)4f dA in equation 4.36. (10) KON4: the definite integral M(°°) = in equation 3.21. X =° [qT exP ('5 0 s I n eD 10 s dX)] dX As was done in the case of the simulation of 0(A) using D(0) values, K0N2 and K0N3 are initially assigned value of 1.0 but the actual values of K0N2 and K0N3 are used after the first iteration as specified in the TERMINAL section. (11) The I and J counters are initialised. Two tables are provided for 0(X) and c(X) which respectively are labelled FUNCTION TBTHLA and FUNCTION TBSELA. In the DYNAMIC section of the program, volumetric water content 9, and concentration of Cl in solution are respectively determined by qua­ dratic interpolation of FUNCTION TBTHLA and FUNCTION TBSELA through the statements: THLA = NLFGEN(TBTHLA, LAMBDA) (4.40) SEELA = NLFGEN(TBSELA, LAMBDA) (4.41) The derivatives d0/dA and dc/dA are also performed respectively, using the statements: DTHLA = DERIV(DTHLAO, THLA) DSEELA = DERIV(DSELAO, SEELA) (4.42) (4.43) University of Ghana http://ugspace.ug.edu.gh 105 Equation 4.34 is then computed with the following statements: C = INTGRL(0.0, THLA) (4.44) K0N1 = THETAN*LAMDAN (4.45) B = K0N1+C (4.46) LATHH = B-K0N2 (4.47) Equations 4.44, 4.45, 4.46 and 4.47 compute respectively the terms A rA 0dA, 0 A , (0 A + 0dA) and g(0). K0N2 of equation 4.47 is the 0 n n n n J ^ definite integral An=oo 0dA 0 Equation 4.36 is then solved by integrating the product of g(0) and dc/dA as follows: GS = DSEELA*LATHH (4.48) HS = INTGRL(0.0, GS ) (4.49) HS1 = K0N3-HS (4.50) HS2 = .5*HS1 (4.51) dc The product g(0) in equation 4.48 is integrated using the statement in equation 4.49. K0N3 in equation 4.50 represents the definite integral A =°° dc g "JT dA and is assigned the value 1.0 for the first iteration. Like o dX K0N2, the correct magnitude of K0N3 is used in subsequent iterations. University of Ghana http://ugspace.ug.edu.gh 106 Having solved equation 4.36, 0Dg and hence Dg are computed by the state­ ments : THDS = HS2/DSEELA (4.52) DS = THDS/THLA (4.53) Since the magnitude of K0N2 and K0N3 were for the first iteration assumed to be 1.0, more than one iteration is required to calculate the correct value of D^. The IF statements in the program ensure that: (1) at least three iterations are carried out before simulation is started, so that calculation of c(A) data is done with the correct D s values, and (2) avoid dividing by zero when calculating DIV, in case Dg is zero or M(°°) = K0N4 is zero. Concentration of Cl as a function of X is first calculated using the analytical solution (equation 3.21). In this section of the program, the term g(9)/0Dg is calculated using the statement: AH = LATHH/THDS (4.54a) Equation 4.54a is then integrated and each integral value multiplied by -0.5 and exponentiated through the statements: BH = INTGRL(0.0,AH) CH = EXP(-.5*BH) (4.54b) (4.54c) University of Ghana http://ugspace.ug.edu.gh 107 CH in equation 4.54c is divided by 0Dg and integrated by the statement: The FINTIM value of MH is the value of the definite integral M(°°) in equation 3.21. MH is divided by M(°°) which is designated KG/'* to obtain the righthand side of equation 3.21. Because the value of K0N4 is not known initially, it is assigned the value of 1.0. The specification, K0N4=MH in the TERMINAL section allows the FINTIM value calculated in equation 4.54d to be assigned to K0N4. Concentration of Cl in solution as a function of X is then calculated using the statement: that of figure 4.5. y and dy/dX in equations 4.38 and 4.39 are designated VELC and DIV respectively. These are paired up through VELCO, the initial value of VELC so that MH = INTGRL(0.0,CH/THDS) (4.54d) RMH = MH/K0N4 (4.54e) SEEI = RMH*(SEEN-SEE0)+SEE0 (4.54f) which is equivalent to c (4.54g) The simulation part of the program (figure 4.6) is similar to University of Ghana http://ugspace.ug.edu.gh 108 DIV = -(LATHH*VELC)/(2.0*THDS) (4.54h) DIVA = -(LATHH*VELCA)/(2.0*THDS) (4.55) VELC = INTGRL(VELCO,DIV) (4.56) VELCA = INTGRL(VELCAO,DIVA) (4.57) Equation 4.5^. is the CSMP version of equation 4.39. Integration of equation 4.39 which is done with the statement in equation 4.56 gives y (VELC). VELCA, whose initial value VELCA0= VELCO x 1.00001 is the half pair of VELC and it is used to generate the half pair of SEEO, namely SEEA and also DIVA (the half pair of DIV) through the algorithm in the TERMINAL section. Rearrangement of equation 4.38b and solving for c gives rX c = c + y/0D dX o s A=0 which is written in CSMP language as: SEE2 = INTGRL(SEEO,VELC/THDS) (4.58a) SEEA = INTGRL(SEEO,VELCA/THDS) (4.58b) The TERMINAL section of figure 4.6 stipulates that Simpson's method of integration should be used to calculate all integrations. The total _3 simulation time (FINTIM=1.82x10 ), the time interval for printing the output PRDEL=3.64x10 ^ and the output variables to be printed are also specified. Statements instructing the computer to set the FINTIM value of computations in equations 4.44, 4.49 and 4.54d as K0N2, K0N3 and K0N4 University of Ghana http://ugspace.ug.edu.gh 109 (i.e. the definite integrals included in this section. •An=°° 0dA, ■An- g(0) atld M(°°)) are also dA An algorithm (see details in Appendix D) similar to that for the simula­ tion of 0(A) in figure A.5, which enables new values of VELCO and VELCAO to be calculated and used in subsequent iterations, is put under this TERMINAL section. Finally there are three WRITE statements, the format of two of which specifies that the FINTIM values of K0N2, K0N3, K0N4, DSEE, SLOPE and DVEL be printed. The format of the third WRITE statement specifies that the new values of VELCO and VELCAO to be used in the next iteration should be printed. By examining these printed values of VELCO, a better estimate of VELCO than what was originally specified can be made if the first estimate does not converge quickly enough. 4.9.3 Description of the computer program for calculating dispersion coefficient D (A) for K+ and simulation of c(A) using the s calculated D (A) values s Like the other models used in this study, the calculation of D^(A) for K' and the subsequent simulation of c(A) using the calculated Dg(^) values was programed in System 360 CSMP (Figure 4.7 and Appendix G). The governing equations are equations 3.19, 3.20a, 3.20b and 4.34. Letting y = 0Dg (4.59) equation 3.19 yields University of Ghana http://ugspace.ug.edu.gh 110 FIGURE 4.7 CSMP listing for calculating dispersion coefficient for K+ and simulation of concentration of K+ in - 1/2 solution as a function of A(=xt ) for Brookston clay. University of Ghana http://ugspace.ug.edu.gh I l l * * * *C 0NT INU 3US SYSTEM MODELING PROGRAM**** * * * VERSICN l . j * * * T ITLE DS FOR POTASSIUM AND S IMULATICN CF CONCENTRATI ON (BROOKSTCN CLAY) * UNITS * KEQ=KILO-EQUIVALENTS * KG—K l LOG RAMS ... ...................... * M=METERS * S=SECONDS * GLOSSARY OF SYMBOLS ___________________ * L A M D AN = LAM BD A AT I N F I N I T Y ( = F I N T IM ) * T H L As= I N T E R PO LA TE D CURVE FOR T H E T A -u AMBDA F U N C T IO N * V E L K = PRODU CT CF THE WATER CONTENT * 0 1 S PER 5 1 ON C O E F F I C I E N T AND * THE D E R I V A T I V E OF THE C C N T R A T IC N - L A M B D A F U N C T IO N * V E L K C = V E L K AT LAMBDA E C L A L S ZERC *_____ V E L K A = H A L F PA I R OF VE L K T t -ROUGh T HE F A CTOR 1 . C 0 C 1 __________ * V E L K A C = V E L K C M U L T I P L I E D EY l o O O O l * S E E 0 = CONCE N TRAT I ON OF F C T A S S IU N I N S O L U T IO N AT l AMBDA EQ UA LS * Z E R O .......................................................... ...... .............. . ........................ * S E E N = IN I T I A L C C N C E N TR A T IO N O F P O T A S S IU M IN S O L U T IO N IN THE * M O IS T S O IL _ *_____ DSE L A 0=D E R IV A T IV E OF CC N CEN TR A T I O lM -L AMBD A F U N C T IC N AT L AM 3D A * ’ E Q U A LS ZERO * S E E L A = IN T E R P O L A T E D CURVE FOR CC NCENT R A T I UN—LAM ED A F U N C T IO N * T BT H L A C T A B LE OF WA T E_R__ CC N JE N T AND _CGRRESP UND I N G LAMBDA V A LU E S * T 8 S E L A = T A B L E OF CO NCENT RAT IO N CF~ P O T A S S IU M I N S O L U T IO N AND * CO RR ESPO ND IN G LAM BDA V A LU E S * A D S I S C = A D S O R P T IO N I S O T I - c R M _______________________________________________________ _______ * S S = I N TERPO L AT E C CURVE FCR K I L O -E Q U I V A LE .JT UF P O T A S S IU M * ADSORBED PER K ILG G R AM S C I L * IN T E R P O L A T IO N IS W IT H R ESPEC T * TO LAM BDA THROUGH T H E ACS C RPT I C N I SO T HE R M ................................... * D S S O = D E R IV A T IV E OF SS AS A FU N C T IO N OF LAM BDA A T LAM BDA EQ UALS * ZERO * D SE E L A = D E R IV A T IV E O F CONCEN T R A T IO N -L A M B D A R E L A T IO N ______________ * D S S = D E R IV A T IV E OF SS W IT H R ESPEC T TO LAM BDA * D = S O I L WATER D I F F U S I V I T Y * D T H L A = D E R IV A T IV E O F WATER CONTENT W IT H R E S P EC T TC LAM BDA * C = IN T E G R A L V A LU E OF T h E T A FROM LAM BDA E JU A i_S z. ERC TO LAM BDA * K 0 N 2 = D E F IN IT E IN T E G R A L VALUE OF T H E T A F r tJ M LAM BD A EQUALS ZERO TO _ * LAM DAN ( = L AMBD A AT F I N T I M ) __________________________________ _ * D S = D I S P E R S I ON COErF I C I E K t “ * D IV = D E R IV A T IV E OF V E LK W ITH R ESPEC T TO LAM BDA * D IV A = H A L F P A IR OF D IV T hR C UG H THE FAC TOR 1 . 0 C 0 1 * SEE 1 = S IM U L A T E C C O N C EN TR A T IO N OF P C T A S S IU M I in S O LU T IO N * S E E A = H A L F P A IR CF S E E 1 THROUGH THE FAC TO R l . C O O l * D S E E = D I F F E R E NCE EETWEEN SEEA AtsC SEE1 AT F I N T I M ____________________________ * S S LO P E =R A T IO O F DSEE TO THE D IF F E R E N C E SETWc-EN V E L KAO AND V E LKO * D P V E L = C O R R E C T IC N F A C TOR RENAME T IM E = L A M B C A I N I T I A L IN C O N V E L K 0 = - . 2 9 2 6 4 3 4 E - 0 4 , F L A 6 = 0 • S E E N = 0 . 0 . T H E T A N = .1 2 , 6 0 = . 1 9 3 E - ? __________PARAMETER L AMDAN = 1 . 8 2E-C3.DSELAC=—I »D»DSS3=-1 . C.RHO-1 . 0 6 e £ 0 3 C O NSTAN T K 0 N 2 = 1. 0 ,K C N 3 = I . O , D E L = .O O 0 1 » I = 0 . J = C , S E E C = . 9 6 5 . . . - D T H LA O =—5 . C V E L K A 0 = V ELJ2:>) . ( 3 . 0 E - 0 4 • . 5 2 ) T . . . ( 5 . 0 E - 0 4 , . 5 1 5 ) , ( 7 . 0 E - C 4 ♦ . 5 1 ) • ( 9 . 0 E - 0 4 , . 5 0 ) • ( I . C E - 0 3 . . 4 9 5 ) * . . . ____________ ( 1 . I E - 0 3 » . 4 9 ) . C 1 . 2 E - 0 3 . . 4 8 ) . ( 1 . 3 E - Q 3 , . < * 7 3 ) . t 1 . 4 E - 0 3 . . 4 6 3 ) . . . . ____ ( l . S E - 0 3 . . 4 4 5 ) , ( 1 . 5 5 E - C E , . 4 3 E ) , ( 1 . 6 2 - 0 3 , . 4 2 ) . ( 1 . 6 5 E - C 3 . , 4 ? ) , . . . ( 1 * 7 E— 03 * « 3 7 ) » ( 1 • 75E— C3• * 3 2 5 ) , ( 1 • 7 8 E - 0 3 . • 2 8 ) . ( 1 * 8 E - 0 3 , * 2 3 S ) . . . . ( 1 . 8 1 E - 0 3 , . 1 8 5 ) , ( 1 . 8 2 E - 0 3 , . 1 2 ) , ( 1 . s * E - 3 3 . . 1 2 ) . ( 2 . 0 E - G 3 . . 1 2 ) F U N C T IO N T B S E L A = ( 0 . 0 » . 9 6 5 ) . < l . O E —0 4 » « 9 6 4 V ) • ( 2 . CE—0 4 . * 9 6 4 8 ) » « • e University of Ghana http://ugspace.ug.edu.gh 112 ( 3 . 0 E - 0 4 . . 9 6 * 7 ) , ( 4 . 0 E - 0 4 . . 9 6 > . ( 5 . 0 E - 0 + , . 9 5 b ) . ( t . O E - 0 4 , . 9 4 ) , . . i { 7 . OE -04 , . 9 1 ) . <8 . C E - 0 4 , . 8 5 ) , < 8 . 5 E - 0 4 , . 8 0 ) , I 9 . 0 E - 4 . . 7 0 ) . . . . < 9 . 3 E - 0 4 t * 6 0 ) • <9• 4 E— 0 4 , • 5 5 ) • ( 9 . 5 E - 0 4 , o 4 7 ) , ( 9 * 6E - 0 4 , e4 0 ) »»•» { 9 . 7 E - 0 4 , . 3 0 . ( 9 . 8E -0 4 . . 2 4 ) , ( 9 . 9E -0 4 , . I S ) ,< 1 . 0 E - 0 3 . . 1 4 ) , . . . ( 1 . 0 1 E - 0 3 . . 1 2 5 ) , ( 1 • 02E—03 , . 1 CS) , ( 1 . 0 3 E -0 3 , , 09 ) ( 1 . 04E -0 3 , . 0 7 5 ) , ( 1 . 0 5 E - 0 3 , . 0 0 5 ) , ( 1 . d o £ - 0 3 . . 0 5 5 ) , . . . C1 . 0 7 E - 0 3 . . 0 4 5 ) . < 1 . 0 P E -C 3 , . 0 4 ) , ( l . 0 9 E - 0 3 . . 0 3 5 ) . . . . < 1 * 1 E—0 3 • * 0 3 ) * ( 1 • A 1 E - C 3 . . 0 2 5 ) , ( 1 . 1 2 E - 0 3 , * 0 2 3 ) . . . . I 1 . 1 3E -0 3 . . C2) ,( 1 . 1 7 E - 0 3 , . 01 ) , ( 1 . 2 E - 0 3 , . 0 0 6 ) , . . . ( 1 . 2 1 E - 0 3 . . 0 0 5 9 ) , ( 1 . 2 1 9 E -0 3 , . 0 0 5 8 2 ) • ( 1 • 2 1 9 4E -0 2 • . 0 0 5 8 l b ) , . . . ( 1 . 2 2 E - 0 3 , . 0058).., ( 1 • 2 3 E -C 3 , .C C 5 7 ) ___ ( 1 * 2 4 E - 0 3 . • 0 0 5 6 ) . ( 1 . 2 E E - 0 3 , # C 0 5 5 ) , ( l * 2 6 E - 0 3 • * 0 0 5 4 ) • ( 1 . 27E—0 3 * « 0 0 5 3 ) . ( 1 . 2 e E - 0 3 , . 0 0 5 2 ) . ( 1 . 2 9 E - 0 3 , . 0 C 5 1 ) . . . . ( 1 . 3C E—03 * . 0 0 5 ) , . ( 1 . 4 E - 0 3 , . 0 0 4 ) , { 1 . 5 E - C 3 , . 0 0 3 ) . ( l . o E - 0 3 , . 0 0 2 ) , ( 1 . 7 E - 0 3 , . 0 0 1 ) , . . . ( 1 . 8 2 E - 0 3 . 0 . 0 ) . ( 2 . 0 E - C 3 , 0 . 0 ) FUNCTION AD S ISO=( 1 « 0 E - 0 4 , 1 • OE-O6 ) , ( 2 * 0 E - 0 4 , 2 • 5 E - 0 6 ) . . . c ( 5 . 0 E - 0 4 . 5 . 6 E - 0 5 ) , ( 1 . CE—0 3 • 9 • 5E—0 6 ) , ( 2 . 0 5 - 0 3 . 1 . S S E - 0 5 ) . . . . ( 5 . 0 E - 0 3 . 2 . 8E -05 ) , ( . 0 1 . 4 . 2 E - C 5 ) , ( . C21, 6 . 0 5 - 0 5 ) , ( . 0 5 . 8 . 8 E - C 5 ) , 99< ( . 1 , . 1 1 5 E - 0 3 ) , C.2 , . 1 4 3 E -0 3 ) , ( . 3 , . 1 5 5 E - 0 3 ) , ( . 5 , . 1 7 5 E - 0 3 ) , . . . ( . 7 , . 1 8 5 E - 0 3 ) , ( . 9 , . 1 5 E -0 3 ) , ( l . C , . 1 9 5 E - C 3 ) DYNAMIC T HLA= NLF GEN( TBTHL A,LAVECA) S EELA=NL FGEN ( T ES ELA.LAMEDA) SS=NLFGEN( ACSISO ,SEEL/> ) _________DTHLA=DERI V ( DT HL A G, T h L A )______________________________________ DSEELA= DERI V ( DSE L AD » £ EELA) DSS=DERI V ( DSSO »S S ) A A= I NT GRLJ S EEO .O S EE L A ).............. ................... . C = I NT GRL ( 0 • 0 * T HL A ) BB=INTGRL( 8 0 ,DSS) _________KON1—THETAN*LAMDAN________________________________________________ E = ( T H L A * L AM E D A ) - K C N 1 F F = E—C G=FF + K0N2 ..................................._............................ , D = - . 5 *G /D TH LA El=LAMBDA*RhC*DSS_____L ATHAD= ( . 5 * T hLA«LAMBC/>tD*DTHLA) *O S £ £ L * t ( , p « E l )________ G1=1NTGRL( 0 ,C .LATHAD) G2=K0N3-G1 T H D S= G2/ DSE ELA............... . DS=THDS/THLA NOSORT ___________ I F ( FLAG . £ Q .Q . AND . I . L T . 2 ) GO TC 10______________________________ IFCTHDS. EQ. 0 . 0 )' GO TC 10 SORT VELK= INTGRL(VELKO .D IV ) VELKA=INTGRL( VELK AO >C IVA ) D I V = ( - .5 *TH LA *LAMBDA * VELK/THCS ) - ( D *VELK*DTHi_ A /T hDS ) - . 5 * E 1 _________ D 1 V A= . 5 * ThLA*L A MBDA * VELKA/THDS)- ( D*VELKA*DTHL* /THDS ) - . 5 * E l S EE I s I NT GRL(SEED.VELK/ThDS) SEEA=I NT GRL(SEED > VELK^/THD S ) NOSORT 10 CONTINUE TERMINAL _________ PRINT D .E .THLA .LATHAC .THDS. 0 £ . SEELA. SEE 1________________________ TIMER F IN T1M -1 . 8 2 E -0 3 ,PR DE L = 3 . 6 4 E ~ 0 5 ,O E LM IN = 1 .8 2 E -2 0 IF < P L AG .EQ .1) GC TO IE 1=1 + 1 K O N 2 = C ............................ .............................. ........................ K0N3=G1 _________ I F ( I . GE . 3 > FLAG= 1______ __________________________________________________ Wk I T E ( 6 . 1 0 0 ) K C N 2 . K 0 N 3 1 0 0 F O R M A T ! / / / * KON2 . KON£ = • . 2 E 1 5 . 6 ) IF tFLAG .EG i . 0 ) GO TO 2 0 15 J = J + 1 University of Ghana http://ugspace.ug.edu.gh 113 I F ( A B S ( S E E l - S E E N ) . L T . 1 . 0 E - 1 0 ) S TOP _________________________D SE E = S E EA—SEE 1________________________________________________________ S S L O P E = D S E E / D E L / V E L K C D R V E L = ( S E E l - S E E N ) / S S L C P E V E L K O = V E L K O—DRVEL V E L K A O = V E L K C * ( 1 . 0 + D E L ) WRI TE ( 6 « 101 ) D S E .SSLCPE .DRVEL __________________1 0 1 FO RMAT C / / / ' D S E E . S S L C P E , D R V E L = » . 3 E 1 5 . 6 ) ________________ W R I T E ( 6 . 1 0 2 ) V E L K G . V E L K A C 1 0 2 F ORM AT ( / / ' C A LL RERUN W I T H V E L K C . V E L K A O = « , 2 E 2 0 • 7 > 20 I F C I . S T j . 1 0 ) STOP ... ........................ . ....... I F I J . G T . 7 ) STOP C A L L RERUN ______________ END____________________________________ ______________________ _________________________ STOP OUTPUT V A R I A B L E SEQUENCE VELKAO SEELA D SE EL A AA THLA C SS DS S 66 El DTHL A K0N1 E FF G C LATHAD G1 G2 THDS DS______ ZZ0007 D I V VE L K DI VA VELKA ZZOOM SEEl ZZC'O 12 SEE A ZZ00 14 ZZ 0 015 I K0N2 K0N5 FLAG J DSEE s s l g p e ' CRVEL VELKC VELKAO OUTPUTS INPUTS PARAMS INTEGS + MEM BLKS FORTRAN DATA CCS 4 - 6 ( 5 0 0 1 0 7 ( 1 4 0 0 ) 2 2 ( 4 0 0 ) 3 + 0= e ( 3 0 0 ) 5 5 ( o O O 26 ENDJOB i University of Ghana http://ugspace.ug.edu.gh 114 (4.60) which upon integration and noting that equations 3.20a and 3.20b imply that dc/dX at X is zero and therefore y =0, we obtain n n Let B (X) = g(0) + Xp Therefore, y = 2 1 X X B(X)dX - B(X)dX ] 0 0 Substitution of equation 4.59 into 4.61 gives (4.61) 6D - i f [ s 2 dc X B(X)dX B(X) dX ] 0 (4.62) and D = 6D /9 (4.63) s s As with the case for the simulation of concentration of Cl in solution as a function of X, the simulation of c(X) for K' was done by multiplying the second term of equation 3.19 by 0Dg/6Ds and substituting equation 4.59 to obtain University of Ghana http://ugspace.ug.edu.gh 115 i z = _ g(Q) y Ip (4 64) dA 2 0D 2 dA Equations 4.59 and 4.64 are used in the algorithm to simulate c(A) for K . Time was renamed lambda by the specification RENAME TIME=LAMBDA, so that integration and differentiation which are normally carried out with respect to time by the CSMP package would be performed in this case, with respect to lambda. The INITIAL section of this program also dc specifies constants such as VELKO (=0D — r- at A=0); SEEN (= c the initial s dA n concentration of K+ in solution in the soil before commencement of hori­ zontal infiltration with KC1 solution); SEEO(= c , the concentration of K+ o in solution at A=0); THETAN (=0 , the initial soil moisture content on n volumetric basis); BO which is the keq. K+ adsorbed per kg. soil at A=0 and is estimated from the adsorption isotherm and concentration of K+ in solution at A=0; and LAMDAN(=A , the value of A in the dry soil just n adjacent to the wetting front. Other constants which are specified are: (1) DSELAO: derivative of c(A) at A=0 which is estimated from the first two data points of the c(A) curve used for FUNCTION TBSELA; (2) DSSO: dS/dA at A=0; (3) RHO: the bulk density of the soil in the column; rA (4) K0N2: the definite integral n0dA 0 *n (5) K0N3: the definite integral | B(A)dA in equation 4.61. J0 For the first iteration when 1=0, K0N2 and K0N3 are not known and so are assigned arbitrary value of 1.0. Subsequent iterations, however, use the actual values of K0N2 and K0N3 because of the statement in the TERMINAL section setting K0N2 and K0N3 equal to the FINTIM value of their respective University of Ghana http://ugspace.ug.edu.gh 116 variable time step integrals. The counters I and J are initialised and the relationship between VELKAO and VELKO through the multiplicative factor 1.0001 is also defined followed by tables TBTHLA, TBSELA and ADSISO which are experimental data points for 0(A),c(A) and SS(c) respectively. In the DYNAMIC section, quadratic interpolation of TBTHLA, TBSELA and ADSISO provides values of volumetric moisture content, concentration of K+ in solution, and amount of K+ adsorbed for each lambda step, respectively, through the statements: THLA = NLFGEN(TBTHLA, LAMBDA) (4.65) SEELA = NLFGEN(TBSELA, LAMBDA) (4.66) SS = NLFGEN(ADSISO, SEELA) (4.67) Equation 4.67 provides SS(A) through the adsorption isotherm and c(A) interpolation. The derivatives dc/dA and dS/dA are calculated respectively through the statements: DSEELA = DERIV(DSELAO, SEELA) (4.68) DSS = DERIV(DSSO, SS) (4.69) Equations 4.68 and 4.69 are integrated to serve as a check on the accuracy of the calculation of dc/dA and dS/dA, through the statements: AA = INTGRL(SEEO, DSEELA) (4.70) University of Ghana http://ugspace.ug.edu.gh BB = INTGRL(BO, DSS) (4.71) The soil water diffusivity, D, is then calculated with statements similar to those in figure 4.5. C = INTGRL (0.0, THLA) (4.72a) K0N1 = THETAN*LAMDAN (4.72b) E = (THLA*LAMBDA)-K0N1 (4.73a) FF = E-C (4.73b) G = FF+K0N2 (4.74a) D = -.5*G/DTHLA (4.74b) The second term XpdS/dX of equation 4.60 and then equation 4.61 are calculated with the statements: El = LAMBDA*RHO*DSS (4.75) LATHAD= (.5*THLA*LAMBDA+D*DTHLA)*DSEELA+(.5*E1) (4.76) G1 = INTGRL(0.0, LATHAD) (4.77) G2 = K0N3-G1 (4.78) In equation 4.76, (0X + 2Dd0/dX) which is equal to g(X) in equation 4.60 has been substituted so that in mathematical form we have L4TBAD - f fx + f & University of Ghana http://ugspace.ug.edu.gh 118 0D and D are calculated with the statements: s s (4.79a) (4.79b) THDS = G2/DSEELA DS = THDS/THLA Simulation of c(A) follows when FLAG=1 after three iterations to obtain D, 9D and D . Statements following the NOSORT specification s s (figure 4.7) ensure that three iterations are performed and also avoids division by zero if 9D is zero. In order to simulate the c(A) for K+ , s (0A + 2Dd0/dA) is substituted for g and dy/dA is designated DIV in equation 4.64. Thus equation 4.64 becomes DIV = - d # J L _ Ap ds (4 80) 20D dA 0D 2 dA s s Integration of DIV with respect to A then gives y which is designated VELK. In CSMP, equation 4.80 and its subsequent integration are calculated using the statements: DIV = (~.5*THLA*LAMBDA*VELK/THDS) - (D*VELK*DTHLA/THDS) - .5*E1 (4.81) VELK = INTGRL(VELKO, DIV) (4.82) The concentration of K+ as a function of A is obtained by integrating equation 4.59, that is: re r A dc c0 y/0D dA s A=0 University of Ghana http://ugspace.ug.edu.gh 119 co + y/0D dX s (4 .83) X=0 Equation 4.83 is calculated with the statement: SEEl = INTGRL(SEEO, VELK/THDS) (4.84) As was done with the preceding simulations in this study, equations 4.81, 4.82 and 4.84 were paired through the factor 1.0001 of VELKO as follows: VELKA = INTGRL(VELKAO, DIVA) (4.85) DIVA = (-.5*THLA*LAMBDA*VELKA/THDS)-(D*VELKA*DTHLA/THDS) -.5*E1 (4.86) SEEA = INTGRL(SEEO,VELKA/THDS) (4.87) In the TERMINAL section of figure 4.7, the total simulation time -3 (FINTIM=1.82x10 ), the time interval for printing the output, the minimum integration step DELMIN=1.82E-20, and the output variables to be printed, are specified. An IF statement specifies to the computer to start using the algorithm for simulation after three iterations when FLAG is set equal to 1. The algorithm for simulating c (X) for K+ is similar to that employed in the simulation of c(X) for Cl (see Appendix D for details). Simulation stops if the difference between the simulated concentration at FINTIM and c is less than l.OxlO-10. If this n convergence is not obtained after either ten iterations on the I counter or seven iterations on the J counter, simulation is then terminated. Examination of the WRITE output for VELKO and VELKAO then enables a better University of Ghana http://ugspace.ug.edu.gh 120 guess of VELKO which will ensure rapid convergence. 4.9.4 Description of the computer program for calculating the dispersion coefficient D for chloride in the case where anion exclusion s occurs The governing equations are 3.54, 3.55 and 3.58. It must be pointed out that dn/dX in equation 3.54 is the total osmotic pressure gradient. Its conversion to dc/dA using Van't Hoffs Law may over­ estimate the effect of osmosis in the experiments considered in this study. This over-estimation was actually observed in our experiments because preliminary calculations indicated that the osmosis term on the righthand side of equation 3.54 dominated all other terms. Because the total osmotic pressure gradient is not known in our experiments, equation 3.54 was approximated by multiplying the second term in brackets on the righthand side by a factor 3 defined as: g _ C Cexpected c The assumption is made in the formulation of 3 that in the wet zone where the moisture content is near saturation, compensation between ions occurs so that osmotic pressure in this region may be very small or negligible. In the horizontal infiltration experiment in this study, it was observed that the concentration distribution of K+ was for all practical _3 —1/2 purposes zero at A = 1.2x10 ms and so the chloride present between -3 -3 -1/2 A = 1.2x10 and 1.82x10 ms (the wetting front) is presumably linked University of Ghana http://ugspace.ug.edu.gh 121 with Ca (see figure 5.9). Therefore, drr/dA in equation 3.54 was con­ verted to c by using Van't Hoff's equation so that we have dir/dA = n RT dc/dA 3 with n = 3/2 and c is the chloride concentration in keq. Cl /m . Equation 3.54 therefore becomes: 2+ 0A dc , 1 ex dc d 3RTOcK(0) dc T dl ~ dA " dA L(0Ds 2gp B) dAw (4.88a) Designating (0Dg - gj = y in equation 4.88a we have w X ft dy 0A dc , “ ex dc d0 dc , d , d0 s dA = - T dA + — dA " D(6) dA dA + dA (GcD(0) dA > which when integrated with the initial and boundary conditions formulated in equations 3.20a and 3.20b gives: A r0A dc _ A0 L 2 dA 2 dA ex dc /Q. d0 dc d . d0 , , ,, +D(0) d A d X ' d A (CTcD(9) d A )] dX (HI - HI2 + H2 - H3) dA (4.88b) where 0A d£ 2 dA University of Ghana http://ugspace.ug.edu.gh 122 TT1 0 _ ex dc H12 ~ 2 dA H2 = D(0) dO dc dX dX H3 . A <0CB (EQUAT IO N 4 . 8 9 ) * M S E EO =M EAS URE D C H L O R I D E C O N C E N T R A T I O N AT L AM BD A EQUA LS Z ERO * T H E TAO =SO IL WATER CONTENT AT LAMBDA EQUALS ZERO * S IGMA=THE R E F LE C T IO N C C E F F I C I ENT * TH ETA =TAB LE OF S O I L WATER CONTENT VERSUS LAMBDA I * ____ _SEE=T ABL E _OF__C.HLCRXDE. -C.CiSLCENXRATI ON I N SO LU T I OJ^_AND_________ _ ___________ * C OR RE S P ON D I NG L AM BD A V A L U E S * M POT = T A BL E OF M A T F I C P O T E N T I A L AND C O RR E S P ON D I NG WATER CON T EN T * VALUES * SEE XP ^TABLE OF EXPECTEC CHLOR IDE CONCENTRAT ION VERSUS LAMBCA * MRC=WATER RETENT ION CURVE * LAMBDA * T HETEX=WATER CCNTENT OF THE EXC LUS ION ZONE * MOLES=CONCENTRAT ION OF CHLOR IDE I K SO LUT ION * SEELA=CONCENTRAT ION OF CHLOR IDE I N M I L L I - EQU I V AL ENT PEP. * C U B IC CENT IMETERS OF SC LUT ION _R ENAME__TJ M f = L j AMDA ___________________________________________________________________________________________________ _ _ _ I N I T I A L INCON SE EO =1 . 0 E - 0 3 . R = f i . 2 1 4 3 2 E 0 7 . T = 2 9 6 . 0 . MRC0 = 4 . 6 3 7 4 E C 2 PARAMETER H 0 3 = - 2 * 6 9 2 3 E -C 6 * SE E N = E - 0 3 . T H L A O —— 1 • 0 • M O LO = - 1 • 0 E - 0 4 C CNSTANT K O N 2 = l . 0 , K 0 N 4 = 1 . 0 . KCN5 = 1 . 0 E - 0 3 »TH ETAN= . 0 5 * • . . L A MD AN = 1 . 8 2 E - 0 1 , M S E E 0 = . S I E - 0 3 . T H E T A O = . 4 9 5 , R HOW= 1 . 0 , A C C = 9 8 0 . 6 ECIN C-T10.M—TM£TA= ( -0 0 « . 4 9 5 ) . ( 5 « Q F - 0 3 . . A Q ) f ( l . ft i=- n ? . - r , - .______ ( 2 . 0 E - 0 2 . . 4 8 5 ) t ( 3 . 0 E —0 2 . . 4 8 3 ) . ( 4 . 0 E - 0 2 . . 4 8 ) • ( 6 . 0 E - 0 2 , . 4 7 5 ) . . . . ( 8 . O E - 0 2 » » 4 7 ) • (1 « 0 E— 0 1 , * 4 6 5 ) . (• 1 1 . . 4 6 ) . ( • 1 2 » « 4 5 E ) . » « c ( . 1 3 * . 4 4 5 ) . ( . 1 3 6 . . 4 4 ) . ( . 1 4 . . 4 3 5 ) . ( . 1 5 * . 4 2 ) # . . . ( . 1 5 5 , . 4 0 5 ) , ( . 1 6 . . 3 9 ) , ( . 1 6 5 . . 3 6 7 5 ) . . . . ( . 1 7 . . 3 3 7 5 ) . ( . 1 7 5 . . 2 9 ) . ( . 1 8 . . 1 9 ) . . . . ( . J L B 2 ^ ^ j0-5J. ,_ I_U_9^ .J05J_________________________________________________________________ FUNCT ION S E E = ( 0 . 0 , • 9 3 ) . ( . 0 1 » . 9 2 9 ) . ( . 0 2 . . 9 2 8 ) . ( . 0 3 , . 9 2 6 ) . . . . ( . 0 4 . . 9 2 4 ) , ( . 0 5 , . 9 2 2 ) , ( . 0 6 , . 9 2 ) . ( . 0 7 . . 9 1 ) . ( . 0 8 , . 8 9 5 ) , . . . ( * 0 8 5 • • 8 9 ) , ( . 0 9 , • 8 7 5 > . ( . 0 9 5 * « 8 6 5 ) # ( * l , « 8 5 ) t C * 1 0 £ * . 8 4 ) , . . . ( . 1 1 . . 8 2 ) . ( . 1 2 » . 8 1 ) , ( . 1 2 , . 8 2 ) * ( . 1 4 . . 8 3 ) * , . . University of Ghana http://ugspace.ug.edu.gh 125 C . 1 5 , . 6 4 5 ) , ( . 1 5 5 . . 8 6 5 ) * ( . 1 P , . 9 0 ) * ( . 1 6 5 , . 9 5 ) » . . . __________ ( • 1 7 , 1 , 0 1 ) , ( . 1 7 5 , 1 . 1 4 ) , ( . 1 7 6 , 1 . 2 ) , ( . 1 8 2 * . 0 1 .)_________________________ FUNCT ION SEEXP= ( O . C , . 9 3 ) » ( . 0 1 * . 9 2 9 ) » ( . 0 2 , . 9 2 8 ) * ( . C 3 , . 9 2 6 ) , . * . ( • 0 4 * • 9 2 4 ) , ( # 0 5 * « 9 2 2 ) * ( ® 0 6 * * 9 2 ) , ( • 0 7 * * 9 1 ) * ( • 0 8 * • 8 9 5 ) , e » * ( . 0 8 5 , . 8 9 ) , ( . 0 9 , . 8 7 5 ) • ( . 0 9 5 * , 8 6 5 ) * ( . 1 • . 8 5 ) , ( . 1 C5 * . 8 4 ) . . . . ( . 1 1 , « 8 2 ) * ( . 1 2 * . 8 ) * ( . 1 3 * . 7 9 ) * ( . 1 4 * * 7 8 ) • ( . 1 5 * . 7 7 ) * ( « 1 6 * « 7 6 ) * c c e ( . 1 7 , . 7 5 5 ) , ( . 1 7 6 * . 7 5 ) »( . 1 8 2 * . 0 1 ) _____________ FUNCT I ON MPQT= (* 2 8 * 1 2 0 . C ) * ( . 2 8 5 * 1 1 5 . 0 ) * ( * 2 9 3 * 1 1 0 . 0 ) , . . . __________ ( * 2 9 7 5 * 1 0 5 * 0 ) * (e 3 0 5 , 1 CO e 0 ) , ( • 3 1 , 9 5 * 0 ) * { • 3 1 7 5 * 9 C * 0 ) * o e e ( . 3 2 5 * 8 5 . 0 ) , ( . 3 3 5 * 8 0 . 0 ) * ( . 3 4 2 * 7 5 . 0 ) * ( . 3 5 3 * 7 0 . 0 ) . . . . ( . 3 6 3 * 6 5 . 0 ) * ( . 3 7 5 * 6 0 » C ) * ( . 3 8 7 5 * 5 5 . 0 ) * ( . 4 0 3 * 5 0 . C ) * o • • ( . 4 1 7 5 , 4 5 . 0 ) * ( . 4 3 5 , 4 0 . 0 ) * ( . 4 5 , 3 5 . 0 ) , ( , 4 o 5 * 3 0 . 0 ) * . . . ( . 4 8 3 , 2 5 . 0 ) * ( . 4 9 5 * 2 0 . 0 ) * ( . 5 1 , 1 5 . 0 ) * ( . 5 1 7 5 * 1 0 .C ) * . . . _____________ ( • 5 2 5 * 5 * 0 ) t j . 5 3 . 0 >0 )_______________________________________________ ___ DYNAMIC THL A ^AFGEN ( ThETA , LAMDA ) S E E LA =A FG EN (S EE , L A M D A ) MRC-=NLFGE N( MPOT, TH LA ) S E E X P T = A FG E N (S E E X P *L A V D A ) ____________A = D ER I V ( THL AC * THL A )_________________________________________________________________ DC =D ER IV (M RC0 ,M RC ) D MR C=—D C /A C= I NTGRL ( 0 • 0 . T H L A ) KON1=THETAN *LAMDAN E = ( T H L A * L A M D A )—KON1 ______ E F =E ^C________________________________________ G =FF+K 0N2 D = - . 5 * G / A HCOND=D/DMRC NOSORT I F ( I . L E . 3 ) GC TO 5 _____________ SORT_______________________________________________________________________________________ MCLES=K 0N 5 * SEELA C 1 =D E R IV (M O LO ,M O L E S ) S IG MAO=1 .O -M S E EO /S E E C TH ETEX =TH ETAC * SI G MAC S IGM A = THE TEX /TH L A _____________B E T A = (S E E L A -S E E X P T ) y S E E L A ________________________________________________________ AH 3=S IGMA *MG LES * D *A H 1 = « 5 * T H L A * L A M D A *C 1 H 1 2 = . 5 * T H E T E X * L A M D A * C 1 H 2 = D *A *C 1 H3 = DER I V ( HO 3 * Ah3 ) ______ H 4 = H 1 - H 1 _ 2 + H ^ H 3 __________ H5= INTGRL ( 0 • 0 * H4 ) Y = K C N 4 -H 5 H 7 = Y /C 1 H8= ( 1 • 5 * R * T *M 0 L E S * S IG M A * H C O N C * B E T A ) / ( ACC *RHOh ) THDS=H7+H8 _____________ D S= TJd D-SZ_THL_A__________________________________________________________________ _______ NOSORT 5 CONT INUE TERM INAL METHOD S IM P PR IN T T H L A ,B E T A , S E E X P T ,H 2 , Y * H 7 * H8 *DS — -__________I_LMER F I NT I M = l *8 2 E —0 1 . P RDEI.. = 3 . 6 4 E - 0 3 . DEL T s 1 . 1 5 7 5 F -Q 4 _______________ 1 = 1+1 K0N2=C I F ( I . L E . 4 ) GC TO 20 J = J + l K 0N 4=H 5_______ WR IT E ( 6 . IO C ) K DM P . K f l K ^ i ______________________ 1 0 0 F ORMAT( / / / * K O N 2 * K O N 4 = « , 2 E 1 5 . 6 ) 2 0 I F O . G T . 6 ) STOP I F ( J , G T ♦ 4 ) STOP C A L L RERUN END STOP_____________________________________ _______________________ University of Ghana http://ugspace.ug.edu.gh 126 CONSTANT: 1. SEEO: concentration of Cl in moles per cm in the infiltrating solution 2. R : gas constant in g/moles.K 3. T : temperature K 4. MRCO: dH/dA at A=0 and H is the uiatric potential from the moisture retention curve provided as a table in FUNCTION MPOT 5. AH3: the product of a , c, D and the derivative d0/dA of the 0(A) curve given as a table in FUNCTION THETA 3 6. SEEN: initial concentration of chloride in the air dry soil 8. THLAO: d0/dA at A=0 9. MOLO: 3 dc/dA at A=0 (c in moles/cm ) 10. THETAN: initial moisture content A the definite integral 0dA11. K0N2: J0 12. ACC: acceleration due to gravity 13. RHOW: density of water fAn 14. K0N4: the definite integral PdA in equation 4.89 J o 3 3 15. K0N5: factor to convert concentration in meq/cm to moles/cm 16. LAMDAN: A at FINTIM 17. MSEEO: measured concentration at A=0. Tables THETA, SEE, and MPOT which are experimental data points for 9(A), c(A), and H(0) respectively, are provided. Table fore expected University of Ghana http://ugspace.ug.edu.gh 127 as a function of X, obtained by extrapolation of the linear portion of the experimental c(X) curve is given in FUNCTION SEEXP. In the DYNAMIC section linear interpolation of Tables THETA, SEE, SEEXP and quadratic interpolation of Table MPOT provide values of volu­ metric water content, concentration in solution and expected c for each lambda step respectively, through the statements: THLA = AFGEN(THETA, LAMBDA) (4.91) SEELA = AFGEN(SEE, LAMBDA) (4.92) SEEXPT = AFGEN(SEEXP, LAMBDA) (4.93) MRC = NLFGEN(MPOT, THLA) (4.94) The derivatives d0/dX, dH/dX and dh/d9 are computed respectively through the statements: A = DERIV(THLAO, THLA) (4.95) DC = DERIV(MRCO, MRC) (4.96) DMRC = -DC/A (4.97) Then moisture diffusivity D is calculated (see description in sections 4.9.1, 4.9.2 and 4.9.3). Hydraulic conductivity is then computed by dividing D by dH/d0: HCOND = D/DMRC (4.98) An IF statement following equation 4.98 allows at least three University of Ghana http://ugspace.ug.edu.gh 128 iterations to be carried out so that the correct values of the moisture diffusivity are obtained since the correct value of K0N2 is only known after the first iteration when K0N2 is set equal to the last value of C 3 in figure 4.8. Concentrations measured in meq/cm are then converted 3 into moles/cm and the derivatives of the resulting c(A) curve computed with the following statements: MOLES = K0N5*SEELA (4.99) Cl = DERIV(MOLO, MOLES) (4.100) a at A=0 is computed from equation 3.55, then 6^ is computed using the calculated a at A=0, after which equation 3.58 is used to calculate cr(A). The following statements are used to calculate a at A=0, 0 , and cr(A), ex respectively: SIGMAO = 1.0-MSEE0/SEE0 (4.101) THETEX = THETA0*SIGMA0 (4.102) SIGMA = THETEX/THLA (4.103) The correction factor (3 defined by equation 3.60a and the last term in brackets on the righthand side of equation 3.60b (viz: CTc D(6)d9/dA) are computed through the statements: BETA = (SEELA-SEEXPT)/SEELA (4.104) AH3 = SIGMA*MOLES*D*A (4.105) HI, H12, H2 and H3 in equation 4.88b are computed using the following University of Ghana http://ugspace.ug.edu.gh 129 statements: HI = . 5*THLA*LAMBDA*C1 (4.106) HI2 = .5*THETEX*LAMBDA*C1 (4.107) H2 = D*A*C1 (4.108) H3 = DERIV(H03,AH3) (4.109a) H4 = H1-H12+H2-H3 (4.109b) H4 is then integrated with respect to X and y (in equation 4.88) computed with the statements: H5 = INTGRL(0.0, H4) (4.110) Y = KON4-H5 (4.111) fXn K0N4 is the definite integral PdX in equation 4.89. For the first J0 iteration its value is not known and so it is given a value of 1.0 speci­ fied in the PARAMETER statement. For subsequent iterations, the correct value of K0N4 is used through the statement specified in the TERMINAL section that K0N4 should be set equal to H5. The first and second terms in the brackets on the righthand side of equation 4.90a are computed through the statements: H7 = Y/Cl (4.112) H8 = (1.5*R*T*M0LES*SIGMA*HC0ND*BETA)/(ACC*RH0W) (4.113) Lastly, 0D^ and D^ are calculated with the statements: University of Ghana http://ugspace.ug.edu.gh 130 THDS = H7+H8 (4.114) DS = THDS/THLA (4.115) The TERMINAL section of figure 4.8 specifies that Simpson's integration method should be used for all integration procedures. The total simulation time (FINTIM=1.82E-01), the time interval for prJ-'ting the output and the integration steps to be used are also specified in the TERMINAL section. There is also the specification for K0N2 and K0N4 to be set equal to last values of C and H5, respectively. A WRITE statement indicating that values of K0N2 and K0N4 used for each iteration be printed, is also specified in this section. 4.9.5 Description of the computer program for calculating y , ip r u ,v7 W W y , iii and to for water and salt flow As s s The governing equations are equations 3.72 through to 3.83 which were all recasted with X as the independent variable, thus obtaining their corresponding equations shown in Table 4.2a. In recasting with A as the independent variable, the original definition of P and P^ were used. R and R are rewritten as: s (4.116) (4.117) The advantage with recasting the equations with X as the indepen­ dent variable is that experimentally determined values of c(X) and 0(X) University of Ghana http://ugspace.ug.edu.gh TABLE 4.2a. Recasted Form of the Ordinary Differential Equations 3.72 to 3.83 with A as the Independent Variable Water Solute d0 *w dA dA lU dA dA J dA (4.118) ''0X- ° dA dA K) dA dAt0Ds dA dA 1 dc dXs, (4.119) d6 2 ^w dA ^ [D ^ d0 d^w dA dA dA - Q ] •a d| , , , dljj (4.120) ( X " ° f - d f + ^ " ^ 6Ds t - d f - Qsl(4-12l) d6 _ _d_ . d6 d<1>w _ , w dA d A dA dA ] (4.122) <26“s " ° f T5T + < X - f x ^ s f X ^ - \K4.123) 131 University of Ghana http://ugspace.ug.edu.gh 132 TABLE A.3. Initial and Boundary Conditions for Equations 4.118 to 4.123 Equation Initial and Boundary Conditions 4.118 a =o , e=eo , v o A=A , 0=0 , Dd0 dXw n n dA dA 4.124a 4.124b 4.119 A=0, 0=0 , c=c . x =0 o o As A=A , 0=0 , c=c , 0D dc ^ s n n n ®dA dA 4.125a 4.125b 4.120 A=0, 0=0 , ip =0 o w A=A , 0=0 , Dd0 d^w - Q=0, Q=0 n n dA dA 4.126a 4.126b 4.121 A=0, 0=0 , c=c » 41 =0 o o s difj A=A , 0=0 , c=c , 0D dc s - Q =0, Q =0 n n n * d \ dA s S 4.127a 4.127b 4.122 A=0, 0=0 , to =0 o w doo A=A , 0=0 , Dd0 w - R=0, R=0 n n dA dA 4.128a 4.128b 4.123 A=0, 0=0 , c=c , O) = 0 o o s dioA=A , 0=0 , c=c , 0D dc s - R =0, R =0 n i^ a ^t s 4.129a 4.129b University of Ghana http://ugspace.ug.edu.gh 133 can be used to calculate D , D , Y > ' l J > a)> X > ^ and and hence com- S W S S S pute the moisture content and concentration profiles for different time periods. Alternatively, if D(9) and 0Dg values are available, 0(A) and c(A) can be simulated using the program outlined in figure 4.6 and then calculate x> ip and to for both water and salt flow. The relation between 0 and x is not unique. Consequently, recasting the equations with 0 as the independent variable presents problems with computer simulation. The initial and boundary conditions used to solve equations 4.118 to 4.123 are given in Table 4.3. The terms in brackets on the righthand side of equations 4.118 to 4.123 are designated y , y^, y^> etc. so that we have: d0 dXw (4.130) (4.131) (4.132) (4.133) (4.134) R s (4.135) which upon substitution into equations 4.118 to 4.123 yield: University of Ghana http://ugspace.ug.edu.gh 134 dA d9 \j dA dK dA (4.136) dA = Ox _ - D TT d0 dXw dA dA - K) dc dA (4.137) ^ 3 dA 3 , d0 2 w dA (4.138) dy 4 dA ,3 d0 d^w (2 ^ s “ ° dA 7 T + Q) dc dA (4.139) dy 5 dA 2“ 7Tw dA (4.140) ^ 6 dA ,»Q d0 dU)w dc " (29aJs “ D dA + R) dA (4.141) Integration of equations 4.136 to 4.141 with respect to A yields y-,i Yn > y, > Yc an<3 y r Which in the computer program written in system 1 Z J 4 j b 360 CSMP (figure 4.9) are designated RVEL, RVELS, QVEL, QVELS, SVEL and SVELS, respectively. dy./dA, dy„/dA, dy./dA, dy./dA, dy /dA and dy,/dA 1 z j 4 d o are also respectively designated RDIV, RDIVS, QDIV, QDIVS, SDIV and SDIVS. Integrating equations 4.130 to 4.135 and using the initial conditions formulated in equations 4.124a, 4.125a, 4.126a, 4.127a, 4.128a and 4.129a, we obtain: *w D d0 dA y2 dA _ 0D dc 0 s dA (4.142) (4.143) University of Ghana http://ugspace.ug.edu.gh 135 [(y3 + Q)/d if ] dA <(^ + v /9ds ! i 1 dA [(y + R)/D || ] dX + R.)/eDs IX 1 dl (4.144) (4.145) (4.146) (4.147) J0 As with the preceding programs, the first specification in the computer program used to calculate x^, Xg > and 0Jg (see figure 4.9 and also Appendix H) is to rename time=lambda so that integra­ tion and differentiation would be performed with respect to lambda. The INITIAL section then begins with constants specified in CSMP data statements INCON, PARAMETER and CONSTANT, which are all used to assign numerical values. In figure 4.9 the following are assigned numerical values: (1) RVELN, RVELNS, QVELN, QVELNS, SVELN, SVELNS: equal to y , v2 > y3, y^, y,. and y^, respectively, of equations 4.130 to 4.135, evaluated at A=Xn (FINTIM). These specify the boundary conditions formulated in equations 4.124b, 4.125b, 4..126b, 4.127b, 4.128b and 4.129b. (2) SEELAO: the derivative of c(A) at A=0 (3) SEEO: the concentration of chloride in solution at A=0 (4) SEEN: the initial concentration of chloride in the moist soil, c n (5) DCONDO: the derivative of hydraulic conductivity as a function of University of Ghana http://ugspace.ug.edu.gh 136 FIGURE 4.9 CSMP listing for calculating x > and oj for water and salt flow (Brookston clay). University of Ghana http://ugspace.ug.edu.gh 137 * * **CONTINUOUS SYSTEM MODELING PRUGRAM**** * * * VERSICN l . J * * * T ITLE C H I«PS I « AND CMEGA FOR WATER AND SALT FLOW (6R0CKST0N CLAY) _* UNITS ------------------------------------------------------------------------------------------------------------------------------------------ * KEQ=KILO-EOUI VALENTS * KG=KILOGRAMS * M=MSTERS * S=SECONDS * GLOSSARY OF SYMBOLS * R VELN= PRODL1CT CF SOU___W ATER. D I FFU.SI V1 TY «... PER.I VA-LI.V-E.-f)E faAXE_E__________ * CONTENT WITH RESPECT TC LAMBDA AND THE DERIVAT IVE CF CHI OF * WATER WITH RESPECT TO LAMBDA AT LAMDAN (=LAMeDA AT F INT IM ) * UVE LN=PRODUCT OF SO IL WATER D IF FU S IV IT Y . DERIVATIVE OF WATER * CONTENT WITH RESPECT TC LAMBDA AND THE DERIVATIVE GF PS I FCP <- WATER WITH RESPECT TO LAMBDA MINUS Gc EVALUATED AT LAMDAN * ( =LAMBDA AT F IN T IM )________________________________________________________________ * U=PRODUCT OF SOIL WATER D IFFUS IV ITY , DERIVATIVE CF WATER * CUNTENT WITH RESPECT TC LAMBDA AND THE DERIVATIVE OF CHI OF * WATER SQUARED * R=Q MULTIPL IED BY THE DIFFERENCE BETWEEN TWICE ThE DERIVATIVE <= OF PSI FOR WATER WITH RESPECT TC CHI FOR WATER AND THE _£____ DFP IVAT IVF n F f H I FOR WflTFR WITH RESPECT TQ LAtfSCA______________________ * US=THE PRODUCT OF WATER CONTENT , THE DERIVATIVE CF CHLORIDE * CONCENTRATION WITH RESPECT- TO LAMBDA .AND THE DERIVATIVE * OF CHI FOR SALT WITH RESPECT TC LAMBDA SUUARED * PS=QS MULT IPL IED BY THE DIFFERENCE BETWEEN TWICE THE DERIVATIVE * OF PSI FOR SALT WITH RESPECT TC CHI FOR SALT AND THE _£____ D E-R I VATI VF Qr.-C.HI _.F flR—*~ALT WITH RESPECT TP LAMBDA_________________________ ^ SVE L N=THE PRODUCT OF SC IL WATER D IF FUS IV ITY • T HE DERIVATIVE * OF WATER CONTENT WITH RESPECT TO LAMBDA AND I'HE DERIVATIVE CF * OMEGA FOR WATER WITH RESPECT TC LAMBDA MINUS R EVALUATED AT * LAMDAN ( =LAMSDA AT F INT IM ) * R VEL NS=THE PRGCUCT OF WATER CONTENT .THE DISPERSICN COEFFIC IENT _£____ T.HE-..D.ER...IVA.,TJ.V,E- OF. THE ..Ci-JLLRIDJE XaKCENTRAT ILN .« IT h RESPECT TC LAMBlD.A . * AND THE DERIVATIVE OF C h i FOR SALT WITH RLSPECT TC LAMBDA * EVALUATED AT LAMDAN ( =LAMEDA AT F IN T IM ) * QVELNS=THE PRCDUCT OF W AT ER CONTENT .THE DISPERSICN COEFFIC IENT. * THE DERIVATIVE OF CHLORIDE CONCENTRATION WITH RESPECT TO * LAMBDA AND THE DERIVATIVE OF PSI FUR SALT WITH RESPECT TO _S____ L. AM SPA ...-M.1N.U.5. QS-£V.A.LU.A.T.ED ..AI ...LAMQ.AfM.. ..(.~L.AM.BD.A- A.T. - F .I-NT-LMJ_________________ SVELNS=THE PRLDUCT CF WATER CONTENT , THE DISPERSICN COEFFIC IENT, THE DERIVATIVE OF THE CHLORIDE CONCENTRATION WITH RESPECT TC LAMBDA AND THE DERIVATIVE OF CMEGA OF SALT WITH RESPECT TO LAMBDA EVALUATED AT LA^CAN { =LAMBDA AT F IN T IM ) SEELAO=DERIVATIVE OF THE CHLORIDE CONCENTkATlON WITH RWSPECT TC .LAMBDA. EVALUATED .AT LAMP AN (.- .LAMBDA AT F I NT 1M.)_______________________ SCEL= CHLORIDE CONCENTRATION AT LAMBDA LUUALS tERC SEE N = IN IT IA L CHLCPIDE CCNCENTRATI ON OF THE MOIST SOIL DCONDO=DERIVATIVE OF THE HYDRAULIC CONDUCTIVITY WITH RESPECT TO LAMBDA EVALUATED AT LAMDAN ( = LAMDA AT F IN T IM ) RVEL=THE PRODUCT OF SO IL WATER DI FFUS I \J I T Y » T HE DERIVATIVE OP ■-W.ATS.R CONTENT h l T H RESPECT. TQ. LAMBDA ..AND. THE. P_EELL.YAX.LVE . QF________ CHI OF WATER WITH RESPECT TO LAMBDA R VE L 0=RVEL EVALUATED AT LAMBDA EQUALS ZERO QVEL=THE PRODUCT OF SO IL WATER D IFFUS IV ITY ,THE DERIVATIVE CF WATER CONTENT WITH RESPECT TO LAMBDA AND THE DERIVATIVE OF PSI OF WATER WITH RESPECT TO LAMBDA .vlINUS Q -Q..V£L Q.=QV£1— ’EY ALUATSP . AT L AMSP.&. EQUALS ____________________________ SVEL=THE PRODUCT OF SO IL WATER D IFFUS IV ITY .THE DERIVATIVE CF WATER CONTENT WITH RESPECT TO LAMBDA ANU THE DERIVATIVE OF OMEGA FOR WATER WITH RESPECT TC LAMBDA MINUS R SVELO=SVEL EVALUATED AT LAMBDA EQUALS ZERO University of Ghana http://ugspace.ug.edu.gh 138 R V E LS = T H E PRODUCT UF WATER C O N T E N T , THE D IS P E R S IO N C O E F F IC IE N T THF D F P T V A T T V F OF T HF C H L P R ID F CH IMCE NTR AT I PIN a T T I" P F S P C f T T f l __________ LAM BD A AND TH E D E R IV A T IV E OF C h i FOR S A LT W IT H R ESPEC T TO L AMBOA V E L 0 = R VELS E V A LU A T E D AT L AMBDA EU U A LS ZERO Q V E LS = TH E PRODUCT OF WATER C O N TEN T , THE D IS P E R S IO N C O E F F IC IE N T , TH E D E R IV A T IV E OF TH E C K L C R ID E C O N C EN TR A T IO N W IT H R ESPEC T TC LAMBDA AND THE PER i.VAT IVE. QF PS1 FOR SALT WITH RESPECT TO_____________ LAM BD A M IN U S OS Q V E LO S =Q V E LS E V A LU A TE D AT LAM BDA EQ UALS ZERO S V E LS = T H E PRODUCT O F T H E WATER C O N TEN T , THE D IS P E R S IO N C O E F F IC IE N T THE D E R IV A T IV E OF T H E C F L C R ID E C O N C EN TR A T IO N W IT H R ESPEC T TC L AM BD A AND TH E D E R IV A T IV E OF ONEGA FOR S A L T W IT H R ESPEC T TC L AM BD A M IN U S RS_____________________________________________________ SVE L CS= S V E LS E V A LU A TE D *T LAM BDA EQ UALS ZERO T H LA O = T H E D E R IV A T IV E OF TH E WATER CONTENT W IT H R ESPECT TO LAM BD A E V A LU A T E D A T L A V EDA EQ U A LS ZERO THE TAO =W ATER CCNTENT A T LAM BDA EQUALS ZERO DPS 1 0 = T HE D E R IV A T IV E OF P S I FOR WATER W IT H R E S P E C T TO LAM BDA E V A LU A T E D AT L AM5D A E Q U A LS ZERC________________________________________________ D C H I0 = T H E D E R IV A T IV E CF C H I FOR WATER W IT H R E S P E C T TO LAM BDA E V A LU A T E D AT LAMEDA EQ U A LS ZERC OME GAO =THE D E R IV A T IV E CF CMEGA CF WATER W IT H R E S FE C T TO LAM BD A E V A LU A T E D A T LAM ED A EQ U A LS ZERO D C H IS O = T H E D E R IV A T IV E OF C H I FC R S A L T W IT H R E S P E C T TC uAM BDA E VAI-. DAT ED AT LAMBDA E Q I.A L S ZERC________________________________________________ DPS IS O = T H E D E R IV A T IV E OF P S I FC R S A L T W ITH R E S P E C T TC L AM 8D A E V A LU A T E D AT LAMBDA EQ U A LS ZERC K 0 N 2 = T H E IN T E G R A L V A LU E OF TH E WATER CONTENT FROM LAM BDA EQ UA LS ZERO TO LAM BD AN ( = L AMBDA AT F IN T IM ) G =TH E SUM OF THETA T IM E S LAM BD A AND TW IC E THE S O IL WATER -D..1E.C U S L .V .I.I Y_ .M .U L I I.?J ,J .£ .D _jB.Y TH E -D .E R .iy .A T . lV j. P E _M A J £ R .-C^ t t T .£ ia X -M I l t d ___ P E S P E C T TO LAM EDA K 0 N 3 = T H E IN T E G R A L V A LU E OF THE PRODUCT OF o ANC THE D E R IV A T IV E OF THE C H L O R ID E C D N C EN TR A T IO N W IT H R E S P EC T TC LAM BDA FRCM LAM BD A E Q U A LS ZERO TO LAMCAN ( = LAM BD A AT F IN T IM ) T H E T A N = IN IT IA L SC I L WATER CONTENT T H E T A = T A B L E O F WATER CCN TEN T V ERSUS LAM BDA C O N D = T A B LE OF H Y D R A U L IC C O N D U C T IV IT Y VERSUS W ATER CONTENT T HL A = L IN E A R IN T E R P O L A T IO N OF T H E T A -L A M B D A E X P E R IM E N T A L DATA P R O V ID E D I N F U N C T IO N TH E T A S E E L A = L IN E A R IN T E R P O L A T IO N OF E X P E R IM E N T A L C O N C EN TR A T IO N HC0ND = N G N -L IN E A R IN T E R F C LA T IO N OF H Y D R A J L IC C O N D L C T IV IT Y V ER SU S LAM BD A THRDUGH T l-E D E R IV E D H Y D R A U L IC C O N D U C T IV IT Y VERS TH E T A P R O V ID E D IN F U N C T IC N CONC A = D E R IV A T IV E OF WATEP C CNTENT W ITH R ESPEC T TO LAMBDA C = IN T E G R A L V A LU E O F T H E T A FROM LAM BDA EQ U A LS ZER C TO LAM DAN ( = L A M B D A A T F I N T I M ) ___________________________________________________________________ D = S O IL WATEP D I F F U S IV IT Y C T H E T A = D IM E N S IC N L E S S WATER CONTENT C S E E = D IM E N S IO N LE S S C H L C R ID E C O NC EN TR A T1JN C 1 = T HE D E R IV A T IV E OF T H E C H LO R ID E CONCENT RAT I 0 N W IT H R ESPECT TO LAMBDA _D .£=D 1 SP_ERS_I_QN CO E= F IC I F NT_________________________________________________________ THDS=THE PRODUCT OF THE WATER CCNTENT AND THE C1SPERSION * COEFF IC IE NT * RDIV= THE DERI VAT IV E OF Y1 WITH RESPECT TO LAMBCA IN EQUATION 4 . 136 * QDIV= THE DER I VAT IVE OF Y3 WITH RESPECT TU LAMBDA IN EQUATION 4 . 133 SD1 V= THE DERI VAT IVE OF Y5 WITH RESPECT TO LAMBDA IN EQUATICN «. . 1 40 * RDIV S=THE _DER I VATT VE QF Y2 W.1JH RESPECT T D 1 AMBDA TN EQUATICN . 1 3 7 Q D I V S = THE D E R IV A T IV E OF Y 4 W IT H R E S P EC T TO LAM BD A IN E Q U A T IC N 4 „ 1 3 P S D IV S = T H E D E R IV A T IV E OF Y 6 W IT h R ESPEC T T U LAM BD A IN E Q U A T IC N 4 . 1 4 1 C H I = C H I FOR WATER P S I s P S I FOR WATER University of Ghana http://ugspace.ug.edu.gh 139 $ OME G A=OM:iG A F Cfi WATER -4____ CHfg|=r.HT FHR SALT--------------------------------------------------------------------------------------------------------------- * PS IS=PS I FOR SALT * OMEGAS=GMEGA FOR SALT RENAME TIME=L AMD A IN I T IA L INCON DEL=. 0 0 0 0 1 » RV ELN= C .O iQ V EL N= 0 ® 0 » SVEi_N= 0 * 0 * o o • ___________ R V E L N S = 0 .0 .Q V E L N S = 0 . 0 . S VEL NS = 0 . 3 --------- ------------------------------------------------------------------------------ SEELAO=—1 .0 , SEEO=. 9 7 • SEEN=. 0 0 0 1 8 • DCONDO=-6• 9 6 1 S E -0 3 * . RVELO=—• 3 2 6 814 E - 0 5 * QVELG=o 5 2 0 0 0 2 2E - 0 8 . SVELO=. 1 2 1 4 9 0 3E - 1 0 * . . . V E LO = - . 5 948 184E-0 7 .QVELCS= - .2 2 7 0 6 9 5 E - 1 0 . SVELCS= . 7 7 4C 96 5E -13 PARAMETE R TH LA0= -5o0 . THETA0= • 5 3 »LAMDAN=1«8 2 E - 0 3 . o • • D C H I0 = , 9 8 6 2 1E—02 , DPS IC = 8 . 3 0 9 9 E - 0 5 , OMEGAG=. 6 3 8 1 9 E - 0 6 , . . . _________ PCMISQ = £_* 21 6e£-C 2..*-DE.5I SP=5*. 7.e- lE-Qt,_______________________________________ CONSTANT THETAN= o 1 2 , KCN2=1 e 0 . K 0 N 3 = i . 0 . I = 0 , J = 0 FUNCTI ON THETA=C r . 0 , .S3 ) *C 1 . CE -04 . . 5 2 5 ) , C3. C E - 0 4 , . 5 2 ) , . . . ( 5 . 0 E - 0 4 , • 5 1 5 )»C 7 o 0 E— C 4 » • 5 1) . ( 9 . 0 E - 0 4 , . 5 0 > * ( l c 0 c - 0 3 . « 4 9 5 ) , oeo ( 1 . I E - 0 3 , . 4 9 ) . C l . 2 E - 0 2 . . 4 8 ) . C 1 . 3 E - 0 3 . . 4 7 3 ) * ( 1 . 4 E - 0 3 , . 4 6 3 ) . . . . C 1 . 5E -C 3 , . 4 4 5 ) ,( 1 • 55E —0 3 , . 4 3 5 ) . < 1 . 6E -03 . . 4 2 ) * ( I . 6 5 E - 0 3 , . 4 0 ) , . . . ____________ ( l « 7 E - 0 i f 3 7 ) . (1 « 7 5 E - 0 3 ^ 3 2 5 ) .C l « 7 3 £ - 0 J » « 2 f i> . ( 1 . P F — Q 5 . . ? ? F ) ----- ( 1 . 8 1 E - 0 3 . . 185 ) . ( 1 . 8 2 E - r 3 , . 1 2 ) , C l . 9 E - 0 3 . . 1 2 ) ,C 2 . 0 E -C 2 , . 12 ) FUNCTION SEE=( On 0 *e 97 ) . ( 1 • 0 E - 0 4 . . 9 6 9 9 ) , C 2* 0 E - 0 4 , o 9698 ) . . » t ( 3 . Q E -0 4 . . 9 6 5 7 ) . ( 4 . 0 E - 0 4 . . 9 6 96 ) . CS• 0 S - 0 4 » . 9 o 9 5 ) . . . . ( 6 . 0E -C 4 , . 9 6 9 4 ) , ( 7 . O E -0 4 , . 9 6 5 3 ) . < 8 . O E -0 4 * . 9 6 52 ) . . . . ( 9e OE—04 » * 9 6 > . C1 . OE-0 3 , . 9 5 ) * C l . I E - 0 3 , . 9 3 ) . C 1 . 2 E - 0 3 , . 8 9 5 ) . . . . _________ I 1 . 25<=-0.3 ■.■»■£? ).,.{ 1 . ( 1 - 3 aF -03 . . 7t>5 >. . . . ____________________ ( 1* 4E -03 > « 6 4 ) i (1 o 42E— C3■ •5& )«C 1«43£—03» «49 ) . ( 1 • 4 4 E -0 3 •« 40 ) *e « « ( 1 . 4 5 E - 0 3 , . 3 5 ) , ( 1 . 4 6 E - 0 2 . . 3 0 ) , ( 1 . 4 7 E - 3 3 , . 2 5 ) . C 1 . 4 8 E - 0 3 . . 2 0 ) . . . . C1 . 4 9 E -C 3 . . 1 8 ) . C l . S 0 E - 0 3 . . 1 5 ) . C 1 . 5 2 E - 0 3 » . 1 2 ) » ( 1 . 5 4 E - 0 3 * . 0 5 } . . . . { 1 * 5 5 E - 0 3 . . 0 8 5 ) , C1 . 5 6 S -C 3 , . 0 7 5 ) . C1 . 5 7 E - 0 3 , . 0 6 5 ) , . . . C 1 . 5 8E -0 3 . . C6 > , ( 1. 5 9 5 E - C 3 , . OS ) . C1 . 6 1 5 E -C 3 . . 0 4 ) , . . . ( 1 . 7 5 E - 0 3 , . 0 1 ) . ( 1 . 8 2 E - 0 3 . . 0 0 5 ) . C l . 9 E - 0 3 , .0 01 ) . C 2 . 0 E - 3 .C .C ) FUNCTION COND=(. 1 2 * 1 - 5 6 E - 0 9 ) * C. 1 6 1 * 2 . 0E -0 9 ) . { . 2 C 2 . 2 . 7 2 E -C 5 ) * « .o C. 2 4 3 , 4 . C E -0 9 ) . ( . 2 8 4 . 6 . 7 2E -0 9 ) . C. 3 2 5 , 1 . 6E -0 8 ) , C . 3 6 6 , 6 . 4 E - 0 8 ) ( . 3 8 2 4 . 1 . 0 E - 0 7 ) , ( . 4 0 7 , 1 . 9 2 E - 07) . ( . 4 2 3 4 , 2 * 8 4 E -0 7 ) . • „ o ( . 4 4 8 , 4 . 8 E— 07 ) , ( . 4 7 2 6 , 8 . 8 E - 0 7 ) . C. 4 d 9 . 1 . ^ 4 E - 0 6 ) . . . . _________ ( . 5 L ^ f i . ? . f l R F - r , 6 ) . I . S 3 .4 . >_____________________________________________ RVELA0=R VEL C+( 1 .0+DEL ) QVELA0=QVELG*( 1 . 0+DEL ) SVELAO=SVELD*(1. 0+DEL) VELAO=VELG* C1 .0+ DEL ) QLAOS=QVELOS*( 1 . C+DEL ) _________ VELSAQsSVELCSgn .O + DEI >______________________________________________________ D YNAMIC T HL A=AFG EN( ThETA , LAMDA) HCGND = NL FGE N ( CGND,T HLA) DHCOND=DERIV( DCONDO. HCOND) A=DERIV (THLAO.THLA) _________ C = INTGRL C3 .0 .TH LA )____________________________________________________________ KON1=LAMDAN*THETAN E = C THL A#L AM CA)—K G N1 F F = E—C G—FF+KON2 D = - . 5 * G / A --------------------C T H E T A = (T H L A -T h £ T A N ) / t THETA .Q -J.HE .T A N )_____________________________________________ CSEE=( SEELA-SEEN ) / ( SEEG—SEEN ) SEELA = AFGENC SEE,LAMDA) C1 = DERIVC SEELAG, SEELA) LATHH=(THLA#LAMDA)+ (2«0*D *A ) GS=C1*LATHH -------------------- H-5= IN T G R L ( 0 .O .C S 1____________________________________________________________________________ HS1=K0N3—HS THDS= .5 *HS1/C1 DS=THDS/THL A NOSORT University of Ghana http://ugspace.ug.edu.gh 140 1F ( Jo LEe3 ) GO TO 10 _________ TF I THDS. FQ. Q.O ) GO TL -1.0------------------------------- SORT RD IV = (C H I# A )—DHC OND RD IV A = (CH IA *A )—D HCOND RVEL=INTGRL (RVE^O .RD IV ) RVELA=INTGRL(R VE LAO » RC I V A ) >____________________ R 1 sRVF-L/ C D»AJ------------------------- -------------------------------------------------------- 0 8 1 =RVEL A / ( D*A ) CH I= INTGRL( 0 . 0 . B 1) CH IA= INTGRL( 0 - 0 . BB1) DDCH I=DER IV( DCHI0 »CH I ) Q=D*A*DDCHI*DDCHI _____________________ GDI V=1 . 5 * PS.1*A_______________________________________________ QL> I VA=1 .5 *PS1A *A QVEL= INTGRL{ GVEL O .QD 1 V ) QVELA=INTGRL(QVELAO, QC1VA) CC1= - » a ) * C l______ QDIVAS=( ( 1 « S *THLA*PS I AS) - ( D *A *DPS I ) + U ) * C l OVELS=INTGRL(QVELOS.CCIVS) Ci VELAS=I NTGRL < Q_ AOS * GDI V AS-) CC=(QVELS+QS)/* C 1 SDIVAS—( ( 2 . C*THLA+MEGAAS > - ( 0 * A*DOMEGA) * R ) * C 1 _____________________ SV£LS=I NTGRL ( S VE LOS . S C I Vfi )_______________________________ SVELAS=I NT GRL( VE L SAO. SD IVAS) DD=( SVEL S + R S ) / (THDS*C1) DDA=(SVELAS+RS) / ( THDS*C1) OMEGAS=INTGRL( 0 . 0 .DD ) MEGAAS=INTGRL( 0 . 0 ,DDA ) V____________________ NDSflRT________________________________ __________________ _ 10 CONTINUE TERMINAL METHOD SIMP TIMER F I NT1M=1 * 3 2 E - 0 3 .PRDEL=3 • 64E- 0 5 . DELT= I • 0E -05 University of Ghana http://ugspace.ug.edu.gh 141 PRINT CT HET A »CHI ,P S I • CME GA * CSEE » CHi S • PS IS * OMEGAS _______I Ft J . GF.4J G£—ID L5--------------------------------------------------------------------- J = J + i KON2=C K0N3=HS GO TO 2C 15 1 = 1+1 ______ I F ( ABS ( RVEL N—RVEi_} ,«.E.G. STOP .______________________ I F C ABS ( Q VEL N—Q VE L ) .EC «O .C ) STOP I F ( ABS( SVEL N—S VE L )«EG«OcC) STCP IF (ABS (RVELNS -RVELS ) . E Q . 0 . 3 ) STCP I F ( ABS(QVEL NS—QV E L S ) .EO .O .O ) STCP I F ( ABS(SVELNS-SVELS) . E Q . 0 . 0 ) STCP ______ D VEL=R VEL A— P. VEI_______________________________ _______________ DVEL1=QVELA—QVEL OVEL2=SVELA-SVEL DVELS=RVELA S—RVELS 0 VELS1=QVELAS-QVELS DVELS2=SVELAS-SVELS ______ SLOPE=DV EL /DEL /R V FLD_______________________________________ SLOPE1=DVEL1/DEL/QVELC SLOPE2=DVEL2/DEL/SVELC P.SLOPE=DVEL S/DEL /VELC q s lo p e= d v e l s i / d e l / qve lcs SSLOPE=DVEL S2/DE L/SVELCS ______ ORVFt =r RVFI K-R\/=( ) / <; i rPF__________________________________ DQVEL=(QVELN-QVZL>/SLCPEl DSVEL=(SVELN-SVEL ) /SLCPE2 DRVELS=(RVELNS-RVELS J/RSLOPE DQVELS=(QVELNS-QVELS>/QSLOPE D SVEL S= < SVELNS-SVELS)/SSLOPE ________a v - 1 n = c v n r n - n p v - i______________________________________________________ QVELO—QVELO +DOVEL SVELO=SVELG+DSVEL V EL 0= VEL 0 +D R VELS Q VE LOS=GVEL C S+DQV ELS SVELOS=SVELCS + DS VELS ______ RVFi A11 =R V EL Qy X J1 . f> +D FI 1_____________________________________ QVELAC=QVELC*( l .O+DEL ) SVELAO=SVELG*( I .O+DEL ) VE LA0=VEL0 * ( 1 • O+DEL) QLACS=QVELOS*( 1 . O+DEL) VEL SAO = SVEL CS* < 1 .O+DEL) ______ W.R.l-T.£..L6..t.-I.P.l.I_fi-YLL.C..i..R_V-.£LAC_________________________________ 101 F ORMAT( / / ' CALL RERUN WITH RVELC• RVELA0=■ . 2E2C . 7> W R IT E ( £ * 1 0 2 ) QVELO.QVELAO ' 102 FO RM AT t / / ' CALL RERUN IvilTH QVELC , QVEL A0= • . 2 £20 . 7 ) WRITE( 6 * 1 0 3 ) SVELO.SVELAO 103 F GRMAT( / / • CALL RERUN WITH SVELC, SVELAG = • , 2 E20 . 7) ______ W R IT ER . VE_Q.Vg l.aD _____________________ 104 F ORMAT( / / ' CALL RERUN WITH VELO, V E L A O = 2 E 2 0 • 7 > W R IT E ( 6 * 1 0 5 ) QVELCS.CLACS 105 FORMAT( / / * CALL RERUN WITH QVELCS. QLAOS=• . 2 E 2 0 . 7 ) W R IT E (6 » 1 0 6 ) SVELOS. VELSAO 106 F ORMAT( / / • CALL RERUN WITH SVELCS. VELSA0 - • , 2E2C . 7) 2._Q___ I F 1 J i i f . T n 1 5 ) STCP_____________________________________________ I F ( 1 * G T . 13 ) STCP CALL PERUN END STOP University of Ghana http://ugspace.ug.edu.gh 142 lambda evaluated at X=0 (6) RVELO, VELO, QVELO, QVELOS, SVELO, SVELOS: are y , y ^ y , y^, y5 and y , respectively, evaluated at X=0. These values are initially es- 6 timated and the algorithm in the TERMINAL section calculates actual values which ensure that the boundary conditions stipulated in equations 4.124b, 4.125b, 4.126b, 4.127b, 4.128b 'nd 4.129b in the TERMINAL section are satisfied. (7) THLAO: the derivative of 9(A) estimated at A=0 (8) THETAO: moisture content at X=0 (9) LAMDAN: FINTIM lambda, i.e. X or X n r (10) DCHIO, DPSIO, 0MEGA0: are respectively, d^/dX, dtf) /dX, daWdX evaluated at X=0 (11) DCHISO, DPSISO: are, respectively, dxg/dX, dip /dX at X=0 (12) THETAN: initial water content of the moist soil (13) K0N2: the definite integral (14) K0N3: the definite integral XF 9dX 0 g(0)|f dX 0 dX The use of K0N2 and K0N3 has been explained in section 4.9.2 which deals with simulation of c(X) from calculated D(X) and D (^) values. RVELO, VELO, QVELO, QVELOS, SVELO, and SVELOS are each multiplied by 1.00001 thus obtaining RVELAO, VELAO, QVELAO, QLAOS, SVELAO and VELSAO which are the initial values of the half pair of y^, y^, y^> y^> y^ and y^. Tables of concentration versus lambda (FUNCTION SEE), moisture content versus lambda (FUNCTION THETA) and hydraulic conductivity versus moisture content (FUNCTION COND) are provided. In these tables values of the independent University of Ghana http://ugspace.ug.edu.gh 143 variable, for example lambda in the case of FUNCTIONS SEE and THETA, are listed first in the brackets with the corresponding dependent vari­ able. The DYNAMIC section consists basically of seven subprograms which are executed through the sorting capability of CSMP algorithm. The first subroutine calculates the moisture diffusivity D and then the dispersion coefficient D^ and is similar to the program already described in section 4.9.2. An IF statement after the statement for calculating the D^ (figure 4.9) ensures that at least three iterations are carried out to obtain the correct D and D values to be used in the other six sub- s routines. The second subroutine calculates Xw * dy^/dX in equation 4.136 is designated RDIV and equation 4.108 written in CSMP statement form as: RDIV = (CHI*A) - DHCOND (4.148) Here, DHCOND as already specified in the first subroutine is the deriva­ tive of the hydraulic conductivity with respect to X done through the statements: HCOND = NLFGEN(COND, THLA) (4.149a) DHCOND = DERIV(DCONDO, HCOND) (4.149b) Integration of RDIV gives y^ which is designated RVEL: RVEL = INTGRL (RVELO, RDIV) (4.150) University of Ghana http://ugspace.ug.edu.gh 144 In equation 4.150, RVELO which is the value of y^ at A=0 is not known and so a guess is first made. The integrand in equation 4.142 designated B1 is then calculated through the statement: B1 = RVEL/(D*A) (4.151) B1 is then integrated to obtain xw through the statement: CHI = INTGRL(0.0, Bl) (4.152) RDIV, RVEL, Bl and CHI are paired using the multiplying factor 1.00001 to obtain RVELAO as specified in the INITIAL segment of the program. The following statements are the half pairs of RDIV, RVEL, Bl and CHI: RDIVA = (CHIA*A) - DHCOND (4.153) RVELA = INTGRL(RVELAO, RDIVA) (4.154) BB1 = RVELA/(D*A) (4.155) CHIA = INTGRL(0.0, BB1) (4.156) At this stage the derivative of x^, (i.e. dXw /dA) is calculated using the statement: DDCHI = DERIV(DCHIO, CHI) (4.15/) This is followed by calculation of 0 in equation 4.120 and defined in 3.80: University of Ghana http://ugspace.ug.edu.gh 145 Q = D*A*DDCHI*DDCHI (4.158) The third subroutine calculates ib . dy„/dA in equation 4.138 w 3 is designated QDIV and equation 4.138 written in a CSMP statement as: QDIV = 1.5*PSI*A (4.159) Integration of QDIV yields y^ which is designated OVEL: QVEL = INTGRL(QVELO, QDIV) (4.160) The integrand in equation 4.144 designated CC1 is calculated by the statement: CC1 = (QVEL+Q)/(A*D) (4.161) CC1 is then integrated to obtain ip through the statement: PSI = INTGRL(0.0, CC1) (4.162) As was done with the second subroutine, ODIV, QVEL, CC1 and PSI in this third subroutine are also paired through the multiplier 1.00001 used to obtain OVELAO from QVELO. The half pairs are: ODIVA = 1.5*PS1A*A QVELA = INTGRL(QVELAO, QDIVA) (4.163) (4.164) University of Ghana http://ugspace.ug.edu.gh 146 CC2 = (QVELA+Q)/(A*D) (4.165) PS1A = INTGRL(0.0, CC2) (4.166) The derivative of ij; with respect to X designated DPS1 and then R in equation 4.122 (defined in equation 3.82) are respectively calculated as follows: DPS1 = DERIV(DPS10, PSI) (4.167) CC3 = (2.0*DPS1/DDCHI) - DDCHI (4.168) R = Q*CC3 (4.169) The fourth subroutine calculates ui . dy^/dX in equation 4.140 is designated SDIV and equation 4.140 written as: SDIV = 2.0*OMEGA*A (4.170) Equation 4.170 is integrated to obtain y^ designated SVEL: SVEL = INTGRL(SVELO, SDIV) (4.171) The integrand in equation 4.146 is calculated with the statement: DD1 = (SVEL+R)/(A*D) (4.172) Integration of DD1 gives to : w OMEGA = INTGRL(0.0, DD1) (4.173) University of Ghana http://ugspace.ug.edu.gh 147 SDIV, SVEL, DD1 and OMEGA are paired as was done with the preceding sub­ routines as follows: SDIVA = 2.0*OMEGAA*A (4.174) SVELA = INTGRL(SVELAO, SDIVA) (4.175) DD2 = (SVELA+R) / (A*jj) (4.176) OMEGAA = INTGRL(0.0, DD2) (4.177) do) w —tt- is calculated through the statement: d A DOMEGA = DERIV(OMEGAO, OMEGA) (4.-178) The fifth, sixth and seventh subroutines calculate xg , and ojg , respectively. dy^/dA in equation 4.137 is calculated with the statement: RDIVS = ((THLA*CHIS)-(D*A*DDCH1)-HCOND)*C1 (4.179) Integrating RDIVS gives y^: RVELS = INTGRL(VELO, RDIVS) (4.180) The integrand in equation 4.143 designated BB is calculated and then integrated to obtain Xs : University of Ghana http://ugspace.ug.edu.gh 148 BB = RVELS/(THDS*C1) (4.181) CHIS = INTGRL(0.0, BB) (4.182) The half pairs of RDIVS, RVELS, BB and CHIS are obtained through VELAO which is 1.00001 multiplied by VELO: RDIVAS = ((THLA*CHIAS)-(D*A*DDCHI)-HCOND)*C1 (4.183) RVELAS = INTGRL(VELAO, RDIVAS) (4.184) BBA = RVELAS/(THDS*C1) (4.185) CHIAS = INTGRL(0.0, BBA) (4.186) d>^/dX and Qg defined in equation 3.81 are computed using the statements: DCHIS = DERIV(DCHISO, CHIS) (4.187) QS = THDS*C1*DCHIS*DCHIS (4.188) Equation 4.189 calculates dy^/dX which when integrated (equation 4.190) gives y^ which is designated QVELS: QDIVS = ((1.5*THLA*PSIS)-(D*A*DPS1)+Q)*C1 (4.189) QVELS = INTGRL(QVELOS, QDIVS) (4.190) The integrand in equation 4.145 is computed and integrated to obtain ip through the statements: CC = (QVELS+QS)/(THDS*C1) (4.191) University of Ghana http://ugspace.ug.edu.gh 149 PSIS = INTGRL(0.0, CC) (4.192) The half pairs of QDIVS, QVELS, CC and PSIS given by QLAOS are: QDIVAS = ((1.5*THLA*PSIAS)-(D*A*DPS1)+Q)*C1 (4.193) QVELAS = INTGRL(QLAOS, QDIVAS) (4.194) CCA = (QVELAS+QS)/(THDS*C1) (4.195) PSIAS = INTGRL(0.0, CCA) (4.196) The derivative of with respect to X, di|> /dX and R^ defined in equation 3.83 and recasted with X as the independent variable in equation 4.117 are computed through the statements: DPSIS = DERIV(DPSISO,PSIS) (4.197) RS = QS*((2.0*DPSIS/DCHIS)-DCHIS) (4.198) Equation 4.141 with dy^/dX designated SDIVS, is computed for each X step and then integrated through the statements: SDIVS = ((2.0*THLA*OMEGAS)-(D*A*DOMEGA)+R)*C1 ‘ (4.199) SVELS = INTGRL(SVELOS,SDIVS) (4.200) The integrand in equation 4.147 is computed and then integrated to obtain a) as follows: s DD = (SVELS+RS)/(THDS*C1) (4.201) University of Ghana http://ugspace.ug.edu.gh 150 OMEGAS = INTGRL(0.0, DD) (4.202) Using VELSAO (=SVELOS*l.00001) the half pairs of SDIVS, SVELS, DD and OMEGAS are obtained with the statements: SDIVAS = ((2.0*THLA*MEGASS)-(D*A*DOMEGA)+R)*C1 (4.203) SVELAS = INTGRL(VELSAO, SDIVAS) (4.204) DDA = (SVELAS+RS)/(THDS*C1) (4.205) MEGAAS = INTGRL(0.0, DDA) (4.206) In the TERMINAL section of figure 4.9 the stipulation is made that Simpson's integration method should be used to perform all integrations. -3 The total simulation time (FINTIM=1.82x10 ), the time interval for printing the output, and the integration step DELT are also specified. An IF statement then enables computation of the first three iterations to be performed with K0N2 set equal to the last value of C and K0N3 equal to the value of HS at FINTIM. For iterations greater than or equal to 4, the algorithm for computation of new values of RVELO, QVELO, SVELO, VELO, QVELOS and SVELOS for the next iteration is used. This algorithm is similar to that used for the simulation of 0(A) and c(A), details of which are outlined in Appendices C and D. The boundary conditions formulated in equations 4.124b, 4.125b, 4.126b, 4.127b, 4.128b and 4.129b are specified in an IF statement, requiring computation to stop if the _3 simulated y ^ y2 > y y , y a n d y& at FINTIM (A=l.82x10 ) is zero. The rest of the algorithm first calculates the difference between the pair RVELA and RVEL (the other subroutines are similar, for example University of Ghana http://ugspace.ug.edu.gh 151 DEVLI = QVELA-QVEL). This difference, designated DVEL is divided by 0.OOOOlxRVELO to obtain the slope. The difference between 0.0 and the computed y^ at FINTIM divided by the slope gives the correction factor DRVEL which is added to the value of RVELO from the preceding iteration to get a new RVELO for subsequent iteration. Similarly, DVEL1, DVEL2, DVELS1 and DVELS2 are the differences between QVELA and Q.'^L, SVELA and SVEL, RVELAS and RVELS, QVELAS and QVELS, SVELAS and SVELS, respectively. DVEL1, DVEL2, DVELS, DVELS1 and DVELS2 are divided by DEL multiplied by QVELO, SVELO, VELO, QVELOS and SVELOS, respectively, (DEL=.00001) to obtain SL0PE1, SL0PE2, RSLOPE, QSLOPE and SSLOPE which are used to cal­ culate the correction factors DQVEL, DSVEL, DRVELS, DQVELS, DSVELS (see TERMINAL section of figure 4.9). New values of QVELO, SVELO, VELO, QVELOS and SVELOS are thus computed and used for subsequent simulation by adding the correction terms to the preceding values of QVELO, SVELO, Six WRITE statements specify to the computer to print values of RVELO, RVELAO, QVELO, QVELAO, SVELO, SVELAO, VELO, VELAO, QVELOS, QLAOS, SVELOS and VELSAO used for each iteration following the previous simula­ tion. The last statement enables simulation to be halted if after 13 iterations simulated y , y , y , y , y or y at FINTIM fails to converge 1 2 3 4 5 6 to zero. 4.9.6 Description of CSMP program for simulating water content profiles for various time periods for vertical infiltration of water and salt The governing equation is equation 3.68. The actual program written in system 360 CSMP is given in figure 4.10 (see also Appendix I for program University of Ghana http://ugspace.ug.edu.gh 152 FIGURE 4.10 CSMP listing for simulating water content profiles for various time periods for vertical infiltration of water and salt (Brookston clay). University of Ghana http://ugspace.ug.edu.gh 153 AAAiLCOlJXXhlUaUS SXSX£ M MOHFl I NtV PRnGRAM.tii.Jii*--------------------------------------------------- ---------------- | i * * * VERS ION 1 . 3 * * * T I T L E SO IL WATER CCNTEMT PROF ILE W ITH T IME (dROOKSTCN CLAY) ! * U N ITS \______________ 4____ KG=K I LQGRAMS— ------------------------------------------------------------------------------------------------------------------------------------------------- * M=METERS * S=SECONDS * GLOSSARY OF . SYMBOLS ................................................ ......... .................. ............... .... —. - - * C THETA=D1 MENS IONLr SS WATER CONTENT * THETAO=SO IL WATER CONTENT AT LAMBDA EQUALS ZERO . CHI EQUALS ZERO, i__________s.____ RS I ._E.QUALS_ZE.RQ*.. AND. OM£6A EQUALS . ZERO___________________________ _______ * T H E T A N = IN I T I A L SO IL WATER CONTENT I * T = T I ME v THETA = SQ IL -WATER CONTENT * L AMDA=TABLE OF LAMBDA AND CORRESPONDING WATE=c CONTENT VALUES * CH ITH=TABL E OF CHI FOR WATER AND CORRESPONDING WATER CONTENT VALUES t ______ * _____RS.I TH=T ABL E O F .P S I FOR WATER . AND .CORRESPONDING.—WATER, jCONTENT VALUES I RENAME T IME^CTHETA INCCN T H E T A C = . 5 3 , T H E TAN = .12 | ..................... PARAMETER. T .=13£a . . 0 * 7 2 , 0 0 . . 0 * . 1 9 2 6 0 . .0., 216Q.0.* OJ FUNCT ION L AMD A = ( 0 . 0 , 1 . 8 2 E - 0 3 ) . ( . I 5 8 5 , 1 . 6 1 E - 0 3 ) . . . . C. 2 8 0 5 , 1 . 8 E - C 3 ) * ( . 3 9 0 2 . 1 . 7 8 E - 0 3 ) * ( . 5 . 1 . 7 5 E -G 3 ) . . . . _______________ i . . .6 .098 ,1 • _ 7 f - r 03 . l t . (a 662S*J .^65Err03_ )- *_L*Z31 { . 7 6 8 3 , 1 . 5 5 E—03 ) . ( . 7 9 2 7 , 1 . 5 E - 0 3 ) • ( . 8 3 o 6 , 1 . 4 E - 0 3 > * . . . ( . 8 6 0 9 8 , 1 . 3 E - 0 3 ) , ( . 8 7 8 1 . 1 . 2 E - 0 3 ) . ( . 9 0 2 4 , 1 . 1 E - 0 3 ) . . . . { • 9 1 4 6 • 1 * .0E —03.). t. t. . .92£>8 » 9 ..jQJL—X).4J. 9512.*Z^0.E.—04 ) » . • • > ( . 9 6 3 4 . 5 . 0 c - 0 4 ) , { . 9 7 5 6 . 3 . C E - 0 4 ) , ( . 9 8 7 8 * 1 . O E - 0 4 ) . . . • ( 1 . 0 . 0 . 0 ) _________________________________ FJJNC X IO N _ C tilJLH=_LfL* .0 .,.3 ,5U9£^Qx>l » { . 2 5 Q 3 . .A l_6ZE~a 6± , «.. ( . 3 7 0 4 9 , 3 . 4 8 6 E - 0 6 ) . ( . 4 4 3 6 6 . 3 . 5 5 4 6 E - 0 6 ) » ( . 5 0 t > 1 5 » 3 * 6 2 3 4 E - 0 6 ) *« o » ( . 5 4 6 1 . 3 . 6 9 1 4E—0 6 ) . ( . 5 8 6 0 5 * 3 . 7 5 8 4 E - 0 6 > . ( . b 4 7 2 2 . 3 . 8 8 8 7 E - 0 6 ) . . . . . ... .( . 6 . 9 4 6 3 , 4 . 0 1 2 2 E - 0 6 ) . X . 7 3 0 1 . 5 , 4 . . 1 2& 2E -Q 6 J *1*7571.7., 4 * . 2 3 6 5 E - 0 6 ) » . . . ( . 7 7 8 6 3 , 4 . 3 3 7 5 E - 0 6 ) , ( . 7 9 6 0 2 , 4 o4 3 2 4 E - 0 6 ) * ( . 8 1 2 , 4 . 5 2 1 1 E - 0 6 ) , . . . { . 8 3 5 9 7 , 4 . 6 3 6 7 E - 0 6 ) * ( . 8 5 4 . 4 . 7 6 2 E - C o ) . ( . 8 6 3 5 2 , 4 . 8 5 8 9 E - C 6 ) , . . . ;_______________________ (_. 8 7 7 8 4 ,.4_» 91.1 E - 0 6J , ( . . 8 8 2 2 , 4 . , 5 23_7Ejt.06.)^XoJ3-6_6_bi,3_._933_2E- C 6 ) , . . c { . 8 9 1 0 7 , 4 . 9 3 9 2 E - 0 6 ) , ( . 8 9 5 5 1 . 4 . 9 4 1 8 E - 0 o ) • ( . 8 9 9 9 £ , 4 . 9 4 0 7E - 0 6 ) . . . . ( . 9 0 5 6 3 , 4 • 9 2 7 9 E - 0 6 ) , ( . 9 1 4 5 1 . 4 . 8 7 5 5 E - 0 6 ) , ( . 9 2 3 3 9 , 4 . 7 8 4 9 E - 0 6 ) . . . . 1 ( . 9 3 0 0 5 . 4 . 6 8 98 E -0 . 61 , ( * .9411 5 , 4 . 4 7 5 7 .E -06 J * t . 9 .500 2 . 4 . 2 5 0 6 E - 0 6 ) . . . . \ ( . 9 6 0 6 1 . 3 . 6 1 0 6 E - 0 6 ) * ( . 9 7 0 6 * 2 . 8 0 2 1 £ - 0 t > ) * ( . 9 8 0 5 9 , 1 . 7 8 2 4 E - 0 6 ) . . . . ( . 9 9 1 1 2 * 6 e 7 4 7 2 E - 0 7 ) * C. 9 9 7 7 8 , 1 . 7 9 0 4 E - 0 7 ) , { 1 . 0 , 0 . 0 ) !_______________________ F U N.CXLON—P.SUE Hj=L.O . 0 . 8 * 3 1 3 2 E - 0 9 ) . ( . 2 5 8 5 4 , 7 . 6 2 9 2 E - 0 9 ) , _________ _______ ( . 3 7 0 4 9 , 7 ■ 6 0 3 3 E - 0 9 ) . ( . 4 4 3 6 6 , 7 . 5 7 8 6 E - 0 9 ) * ( . 5 0 6 1 5 * 7 . 5 5 3 4 E - 0 9 ) * . . . ( . 5 4 6 1 , 7 . 5 3 3 4E—0 9 ) * ( . € 4 7 2 2 , 7 . 5 2 6 E - 0 9 ) , ( . 6 9 4 6 3 , 7 . 5 8 0 8 E - 0 9 ) . . . . | ( . 7 3 0 1 5 * 7 . 6 8 6 6 E - 0 9 ) , ( . 7 7 8 6 3 * 8 i 0 3 9 2 E - 0 9 ) . 8 1 2 , 8 . 5 3 7 6 E -C 9 ) . . . . ' ( . 8 1 9 9 9 * 8 . 6 8 7 3 E - 0 9 ) * ( . 8 3 5 9 7 . 9 . 0 2 1 4 E - 0 9 ) * ( . 8 4 95 6 * 9 . 584- IE —OS ) . . . * j ( . 8 6 2 3 1 , 1 . 0 2 0 2 £ - 0 8 ) . ( . 8 7 4 7 4 . 1 . 1 1 0 2 E - 0 8 ) , ( . 8 8 6 6 3 . 1 . 1 8 2 6 E - 0 8 ) , . . . ________________________t . 8 9 5 5 1 . 1 . 2 3 2 8 E - 0 8 ) ( . 9 0 3 4 1 . 1 . 2 83 S E - 0 d ) * ( . 91 00 7 * 1 . 3 5 8 7 E - 0 8 ) * . . .___________ ____ ( . 9 2 1 1 7 , 1 • 4 7 7 7 E - 0 8 ) . ( . 9 3 0 0 5 , 1 . 5 6 1 B E - 0 8 ) , { . 9 4 1 1 5 * 1 . 6 4 3 4 E - 0 8 ) , . . . ( . 9 5 0 0 2 , 1 . 6 8 0 8 E - 0 8 ) , { . 9 5 2 8 4 , 1 . 6 8 S 4 E - 0 8 ) , { . 9 5 3 9 5 . 1 . 6 9 0 5 E - C 8 ) . . . . ! ..................... ( * 9 5 5 0 6 , 1 . 6 9 . 1 4 E-0.8.) • ( . 9 5 6 1 7 , 1 . 6 9 1 I E - 0 8 > , ( • 9 5 7 2 6 * 1 . 6 8 9 4 E - 0 8 ) * . . . ( . 9 6 0 6 1 , 1 . 6 7 6 2 E - 0 8 ) * C. 9 6 5 0 5 . 1 . 6 3 2 4 E - 0 8 ) . . . . ( . 9 7 0 6 , 1 . 52 4 5 E - 0 8 ) , ( . 9 7 5 0 4 , 1 . 3 8 7 3 E - 0 8 ) , ( . 9 8 0 5 9 * 1 . 1 3 94 c - 0 8 ) * « . « _______________________L * -9 .8502 , 8 . 6 9 03E - 0 9 ) , ( . 9 911 2 . 5 .1 8 2 7 E - 0 9 ) * ( . 9 9 7 7 8 * 1 . 49_47E -_Q_9j_,_._._.________________ ( 1 . 0 , 0 . 0 FUNCT ION MEGATH=( 0 . 0 , 1 . 5 6 9 3 E - 1 1 ) . ( . 2 5 8 5 4 , 1 . 4 0 6 3 E - 1 1 ) , . o . ...................... - S . 3 , 7 0 . 4 9 3.Q8.7.£=_1-1.).,.1 • 443.6.6 * 1 *.3.679.c—.11 J 50.61.5., .I .. ..2.485E— 1 1 ) . , . . . . ( . 5 8 6 0 5 . 1 . 3 0 3 2E -1 1) , ( . 6 2 0 5 9 , 1 . 2 7 6 7 E - 1 1 ) , ( . 6 7 3 8 5 , 1 . 2 1 8 E - 11 ) . . . » ( . 7 1 2 3 9 , 1 . 1 5 1 5 E - 1 1) . ( . 7 5 7 1 7 , 1 . 0 4 2 9 E - 1 1 ) * ( . 8 0 4 0 1 . 8 . 6 1 5 7 E - 1 2 ) . . . . ( . 8 8 6 6 3 . 7 . 2 1 6 5 E - 1 2 ) . ( . 9 0 3 4 1 , 9 . 3 9 * 4 E - l 2 ) , ( . 9 1 0 0 7 , 1 . 1 8 6 8 E - 11 ) . . . . ( . 9 2 1 1 7 . 1 . 7 2 4 3 E - 1 1 ) , ( . 9 3 C 0 5 , 2 . 2 6 4 3 E - 1 1 ) , { . 9 4 1 1 5 . 3 . 0 5 3 3 E - 1 1 ) , . . . ( . 9 5 0 0 2 , 3 . 7 3 4 2 E - l 1 ) , ( . 9 6 0 6 1 . 5 . 2 1 1 9 E - 1 1 ) . < . 9 a 5 0 5 , 5 . 7 6 5 2 E - 1 1 ) . . . . ( . 9 7 0 6 . 6 . 2 4 1 2 E - 1 1 ) * ( . 9 7 1 7 1 . 6 . 2 9 5 6 E - 1 1 ) . ( . 9 7 2 8 2 , 6 . 3 3 3 2 E - 11 ) . . . . University of Ghana http://ugspace.ug.edu.gh 154 ( • 9 7393 » 6 * 3 524E - 1 I ) . ( . 9 7 5 0 4 , 6 . 3 5 1 4 E - 1 1) , ( . 9 7 7 2 6 . 6 . 2 8 2 E - 11 ) , o , < . Q f t O S Q . f i . Q 7 Q 3 E - 1 1 ) „_L *_9 .B 5 lO 2 -.-S-. 1 0 .a 9 E - . i l ) ., ( , 9 9 I.I.2 ..3...4 9 8 6 E - „ L U _ , . ( . 9 9 7 7 8 , 1 . 122E-1 1 ) . ( 1 . 0 . 0 . 0 ) A1=NLFGEN(LAMDA, CTHETA) A2=NLFG=N(CHITH. CTHETA.) . . . A3 = NLFGEN(PSITH, CTHETA) A4=NLFGENCMEGATH.CTHETA > ------------------------------------------ B l = A I *SQRT_ f JT_) _ _ --------------------------------------------------------------------- B2=A2*T B 3 = A 3 * ( T * * l . 5 ) -B4=A * *T *T —- ....... ........... ....... ..... .................. 2 = B1+82+83+04 THE TA = CT HETA#{ THETAO-THETAN) +THETAN ______________________________T T M FR F T N T T M =1 m 0 J P R D F l = ? . O F - Q ? . f i l l T O F l = f t . Q F - Q ? ___________________________ P R T P L T Z LABEL MOISTURE CONTENT PROFILE WITH TIME . J?.R1NT....THETA.„A1 ,A2 *01 . E 2 . B 3 . B 4 . 2 . . . . . END ! STOP O U T P U T V A R I A B L E S E Q U E N C E A 4 B4 A3 B3 A2 B2 A1 B1 Z THETA OUTPUTS INPUTS PARAMS INTEGS + MEM BLKS FORTRAN DATA CDS 1 4 ( 5 0 0 ) 4 3 ( 1 4 0 0 ) 1 0 ( 4 0 0 ) 0+ 0= 0 ( 3 0 0 ) 1 1 ( 6 0 0 ) 54 University of Ghana http://ugspace.ug.edu.gh 155 used for Akuse clay). This program is slightly different from previous programs in this study in that it is not segmented into INITIAL, DYNAMIC and TERMINAL sections. This segmentation is not used in this program because the multiple run capability of the PARAMETER card which is used in this program to calculate water content profiles for the time periods specified does not require segmentation of the program. Time is renamed CTHETA ( 0) which is equal to (6-8 )/(6 -0 ) n o n where 0 is the moisture content for a particular X, or to ; 0q is the moisture content at X=0, Y =0, it) =0 or to =0 and 0 is the initialw w n soil water content. The advantage with using the dimensionless water content in this program is that its minimum value is zero and maximum value is 1.0 and so simulation which must necessarily commence at zero proceeds without difficulties. On the other hand, if 0 is used, the minimum value 0 is not necessarily zero, thus requiring extrapolation n so that computing starts at zero time. In the INCON data card, two constants are specified, namely 0^ (THETAO) and 0 (THETAN). The PARAMETER data card specifies four time n periods T equal to 360s, 7200s, 19260s and 21600s for each of which the water content profile with depth is computed. The use of PARAMETER T = (360.0, 7200.0, 19260.0, 21600.0) will therefore cause the program to run first with T=360.0, second with T=7200.0, then with T=19260.0 and finally with T=21600.0. Four tables are provided. The first, FUNCTION LAMBDA is a table of X ( 0 ) with the first number in each bracket being the water content value and the second number, the corresponding X value. FUNCTION CHITH is a table of ¥ ( Q ). Again the first number in each parenthesis is University of Ghana http://ugspace.ug.edu.gh 156 the independent variable, water content and the second number is the dependent variable y . The other two Tables are FUNCTION PSITH which is ip ( 0 ) and FUNCTION MEGATH, oj ( 0 ), A non linear quadratic w w interpolation is used between data points provided by the four tables: Al = NLFGEN ''LAMBDA, CTHETA) (4.207) A2 = NLFGEN(CHITH,CTHETA) (4.208) A3 = NLFGEN(PSITH,CTHETA) (4.209) A4 = NLFGEN(MEGATH,CTHETA) (4.210) Each of the terms on the righthand side of equation 3.68, that is 1/2 3/2 2 Xt , X _ t, tJj t and to t is calculated as follows: w w Bl = A1*SQRT(T) (4.211) B2 = A2*T (4.212) B3 = A3*(T**1.5) (4.213) B4 = A4*t *T (4.214) The depth z is then the sum of Bl , B2, B3 and B4. A PRTPLT statement gives an output of a plot of 0 as a function of z. In addition THETA, Al, A2, Bl, B2 , B3, B4 and z are printed. 4.9.7 Description of CSMP program for simulating concentration of Cl profiles for various time periods for vertical infiltration of water and salt The governing equation for simulation of concentration of Cl University of Ghana http://ugspace.ug.edu.gh 157 profiles is equation 3.69. The program used for the simulation is given in figure 4.11 (see also Appendix J which provides the data for Akuse clay). This program, like the one presented in figure 4.10 for water content profile with time, is not segmented into INITIAL, DYNAMIC and TERMINAL sections. Time is renamed CSEE, C.[=( c- c )/(c -c )] where c is the concen- z n o n tration of Cl , corresponding to a particular L V , i or to ; c is the s s s o concentration of Cl at \ = 0 , Y =0, ib =0 and u) =0; and c is the initial s s s n concentration of Cl in the soil. In the data card INCON, two constants, namely c^ (SEEO) and c^ (SEEN) are specified. Four time periods viz: 360s, 7200s, 19260s, and 21600s are specified in the PARAMETER card. Like the program in figure 4.10, the use of PARAMETER T = (360.0, 7200.0, 19260.0, 21600.0) enable the program to run first with T = 360.0, second with T = 7200.0, then with T = 19260.0 and finally with T = 21600.0. The following four tables for A(C„), x (C7), ip (C7) and to (C9) A. S z S Q Z. respectively are provided: FUNCTION CLAMDA, FUNCTION CHIS, FUNCTION PSIS and FUNCTION OMEGAS. In these tables, the independent variable C^ is listed first in each of the parentheses. The dependent variable is the second value in the bracket. A Lagrange quadratic interpolation is used between the data points given in the four tables: Al = NLFGEN(CLAMDA, CSEE) A2 = NLFGEN(CHIS, CSEE) A3 = NLFGEN(PSIS, CSEE) University of Ghana http://ugspace.ug.edu.gh 158 FIGURE 4.11 CSMP listing for simulating concentration of Cl profiles for various time periods for vertical infiltration of water and salt (Brookston clay). University of Ghana http://ugspace.ug.edu.gh 159 * * * *C 0NT INU 3US SYSTEM MODELING PROGRAM**** * « * VE RSICN 1 . 3 * * * T IT LE CHLORIDE CONCENTRATION PROFILES WITH TIME (BROOKSTON CLAY) -*____ UNI I S _____________ ___________________________________________________________________ * KG=KILOGRAMS * M=METERS * KEQ=KILO—E OUIVALENTS * S=S ECONDS * GLOSSARY OF SYMBOLS *____ C SF E =D I ME N SIO NLE SS CONCENTRA7 ICN____________________________________________ * S EE 0=CONCE NTR ATI ON OF CHLORIDE AT LAMBDA EQUALS ZERO , CHI FCR * SALT EQUALS ZERO, PSI FCR SALT EQUALS ZERO. AND CMEGA FOR SALT * EQUALS ZERO * SEE N = IN IT IA L CCNCENTRATIGN OF ChLCRIDE IN THE MC 1ST SOIL * CLAMOA=TABLE OF LAMBDA AND CORRESPONDING CONCENTRATION VALUES * C..HI. ■£= .T ABLE . QF CHI F OR .S I L T A ND CORRE S P O N D ING. CQN CFNT.R.AT I ON V A1 UE&. * PS IS=TABLE OF PSI FOR SALT AND CORRESPONDING CCNCENTRAT ION VALU"S * OMEGAS=T ABLE CF OvlEGA FCR SALT AND CORRESPONDING CONCENTR AT ICN * VALUES * T=TIME * SEE =C CNCENTRA T I ON OF CHLORIDE IN SOLUTION RF NIA Mg T IMFs rSFF_________________________________ INCON SE E 0 = • 9 7 .S EEN=• COO18 PARAMETER T = ( 3 62 . 0 , 7 2 0 0 . 0 , 1 9 2 6 0 . 0 , 2 1 o 0 0 . 0 ) FUNCTION C LAMDA=( C . C . l . 8 2E -0 3 ) • ( 6 . 3 1 0 a E - 0 3 » 1 . 8C18E -C3 ) . . . . ( 8* 9914E—0 3 . 1 .7 6 5 4 E - 0 3 ) , < 1 o2 2 S I E - 0 2 . 1 • 7 2 9 E - 0 3 ) , . * . ( 1 . 6 5 5 3 E - 0 2 ,1 .6 9 2 6 E - 0 2 ) • ( 2 . 3 9 9 4 E - 0 2 , 1 . 6 5 6 2 E -C 3 ) , . . . _( 3 . 776F - 02 . 1 .6 19 BF-Q2 ) . ( f i . Q345E -Q2 . ) I __________________ ( e . 9 0 0 6 E - 0 2 , 1 , 54 7 E - 0 2 ) . ( . 1 0 9 9 4 , 1. 5 2 8 8 E - 0 3 ) . . . . ( . 1 7 7 9 9 , 1 . 4 S 2 4E -P 3 ) . ( . 3 2 9 7 7 . 1 . 4 5 & E - 0 3 ) . < . 5 5 8 68 ,1 . 41 9 6E -0 3 ) , . . . ( . 7 0 3 0 4 . 1 * 3 632E- 0 3) , .( . *7 .9258, 1 .34 to8E -03> . ( • 83 762 . 1 • 3 10 4 E - 02 ) , . . . ( . 8 7 4 6 2 . 1 . 2 74E-0 3 ) , ( . 9 1 2 6 6 , 1 .21 9 4E -0 3 ) , ( . 9 2 8 8 , 1 . 183E -C3 ) . . . . ( . 9 4 1 9 4 , 1. 1466E— 03 ) . ( . 9 6041 • 1 . 0 9 2E -0 3 ) . ( . 9 7 1 6 6 . 1 . 0 3 7 4 E - 0 3 ) . . ce ___________ ( >981 15.. 9...f i .2f iF-0 4 ) . i . .9fl.fi66 . 9 . . I F -Q4 ). 9.92.9 2-t 6 . . .S S 4 F - 0 4 ) . . . ._________ ( . 9 9 7 3 7 , 8 . 1 9 E - 0 4 ) , { . 9 9921 . 7 . 6 4 4 E-C 4 ) , ( , 9 9 9 4 , 5 . 824E-04 > . . . . ( . 9 9961 , 3 . 8 22E -0 4 ) , ( . 9 9 9 81 , 1 . 8 2 E - 0 4 ) , ( . 9 9 9 9 1 . 9 . I E - 0 5 ) . . . . ( 1 . 0. 0. 0 ) FUNCTION C H IS = ( 0 . 0 . 2 . C 8 5 9 E - 0 6 ) , ( 7 . 6 5 0 9 E - 0 3 . 2 * 3 9 E - 0 6 ) . . . . ( 8 . 9 9 1 4 E - 0 3 .2 .4 1 7 9 E -0 6 ) . ( 1 . 2 2 9 1 E - 0 2 . 2 . 4 o 5 9 E -Oo ) . . . . ____________( 1 . 6 5 5 3 E - 0 2 . 2 .5? 5 aE -C 6 ) . (2 . 3 9 c4 g - c ? . 2 . a 3 8 5 E -Of. ) . . . .________________ ( 3 . 7 7 6 E - 0 2 , 2 . 5 6 5 E - 0 6 ) . ( 5 . 9 3 4 5E—0 2 , 2 * 5 8 5 E—06 ) . . . . ( 8 . 9006E —0 2 . 2 . 6 0 0 3E -0 6 ) , ( . 1 38C9 , 2 . 6 1 0 8 E - 0 6 ) . . . . ( . 1 7 7 9 9 . 2 . 6 1 4 I E - 0 6 ) . ( * . 2 3 5 9 4 , 2 *6 16E -Oo ) . 1 . 3 2 9 7 7 . 2 . 6 1 5 3 E - 0 6 ) . o * * ( . 5 5 8 6 8 , 2 . 6 0 71E - 0 6 ) » ( . 7 0 3 0 4 , 2 . 5 9 0 2 E - 0 o ) • ( . 7 9 2 5 8 , 2 . 567 3 E - 0 6 ) . . . . ( . 8 3 7 6 2 . 2 . 5 3 9 5E -0 6 ) , ( . 6 7 4 6 2 . 2 . 5 C 7 2E -0 6 ) . ( . 9 0 3 2 6 . 2 . 4 7 04E -0 6 > ♦ . . . _________ ( * S5.SQ 7 ,2.3.1 2flE-Q..t>l . ( . 9 67 91 , 2 , 23 -09E -06 ) , a ».«_ ( . 9 7 9 1 7 . 2 . 1 4 1E-C 6 ) , ( . 9 8 4 9 , 2 . C 4 4 E - 0 o ) , ( . 9 9 0 4 7 , 1 . 9 4 1 2 E -0 6 ) . . . . ( . 9 9 3 9 2 . 1 o 8 69E—0 6 ) . ( . 9 9 7 3 7 ,1 . 7 9 3 7E -0 6 ) . ( . 9 9 9 2 1 . 1 . 6 7 5 5 E - 0 6 ) . e « . ( . 9 9 9 3 1 , 1 . 4 7 7 7E -0O ) . ( . 9 9 9 4 , 1 . 2 7 9 5 E - 0 6 ) . ( . 9 9 9 51 » 1 . 0 4 1 1 E - 0 6 ) . . . . ( . 9 9 9 6 1 , 8 . 4 2 0 4 E - 0 7 ) , ( . 9 9 9 7 ,6 . 4 2 4 6E -0 7 ) , ( • 9 9 9 8 , 4 . 0 2 2 7 E - 0 7 ) , . . . ( . 9 9 9 9 1 , 2 . 0 1 4 6 E - 0 7 ) , ( 1 . 0 . 0 . 0 ) ---------------- FUNCTION PS I„,S = (3 . C . 3., eeZ5E-rQ.S.J ».£7. qS.Q9S.-Q3., ^ 1 .S £1E -C9J^ - . - . - . _______ ( 1 . 0 4 1 4 E - 0 2 . 5 . 2 S 0 1 E - 0 9 ) , ( 1 • 4 1 6 8 E - 0 2 , 5 * 3 547E- 09 ) , . . . ( 2 . 3 9 9 4 E -0 2 , 5 . 45 0 8 E -0 9 ) , ( 3 . 7 76E -0 2 . 5 . 50 38E-C- 9 ) . . . . < 7 , 1787E -02 » 5 .56 8 9E—09) , ( . 109 9 4 , 5 , 6 0 59E -0 9 ) ( . 1 7 7 9 9 , 5 . 6 4 0 6 E - 0 9 ) . ( . 3 2 9 7 7 , 5 .6 7 5 E - 0 9 ) , ( • o 5 o 6 8 • 5 . 7 0 3 9 E - 0 9 ) , . . . ( . 7 0 3 0 4 , 5 . 7 224E -C9 ) , ( . 7 4 9 9 5 , 5 . 7 2 8 6 E - 0 9 ) , ( . 7 9 2 5 8 . 5 . 7 3 2 2 E - 0 9 ) , . . . — — ------- U JU S 11 5,»7.3 41E-? 9 ) , .I ■ 6376^ 1 S.B7 3 4 1 £ - 0 9 ) , . t o 857.73 , £ . 7315£~0_? 1_,... ( . 8 9 1 5 1 . 5 . 7 1 9E-0 9 ) * ( . 9 1 2 6 6 , 5 . 6 9 5 9 L - 0 9 ) , ( . 9 2 8 8 , 5 . 6 6 2 7E -0 9 ) . . . . ( . 9 4 1 9 4 . 5 . 6 19 I E - 09 ) , ( . 9 5 5 0 7 , 5 . 5 6 1 7 E - 0 9 ) , ( . 9 6 791 , 5 . 447 2E -0 9 ) . . o» ( . 9 7 9 1 7 , 5 . 2 9 9 1 E— 0 9 ) , ( .9830 3 , 5 . 1 8 1 4E -0 9 ) , ( . 9 8 8 o 6 , 4 . 9 8 1 6 E - 0 9 ) , . . . ( . 9 9 2 1 9 , 4 . 82 99E -0 9 ) , ( . 9 9 5 6 5 . 4 . 6 6 1 6E -0 9 ) , ( . 9 9 91 . 4 . 4 7 3 2 E - 0 9 ) . . . . University of Ghana http://ugspace.ug.edu.gh 160 ( * 9 9 9 3 , 3 * 7 7 6 E - 0 9 ) , ( • 9 9 9 4 . 3 • 2 7 5 6 E - 0 9 ) , < . 9 9 9 5 , 2 . 6 7 1 9 E - 0 9 ) , . . . _______________ I .QQQn.? . l f t f i f i F -P q ) . t . 9SS7. .1 . .6S62E -09 1 . t . QSS f l . l .P ar 1F-PQ) ------------ < . 9 9 9 9 1 , 5 . 2 2 3 1 E - 10 ) , ( o9 9 9 9 4 » 3 • 1 3 9 4 E - 1 0) • ( I * 0 * 0 * 0 ) FUNCTION OMEGAS=( 0 . 0 * 6 . 8 6 8 2 E—1 2 ) , ( 6 . 3 1 0 5 E - 0 3 , 9 . S 2 8 4E -1 2 ) , . . . ( 8 . 9 9 1 4 E -C 3 . 9 . 3 4 0 4 E - 12) , ( 1 . 2 2 9 1 E - 0 2 . 9 . 3 8 7 4 E - 12 ) . . . . ( 1 * 6553E—02 , 9 o44 1 2E—1 2 ) , ( 2 . 9 6 8 6 E - 0 2 , 9 . S 2 1 3 E - 12 ) . . . . (4-. 7968E -C2 .9 . 56 7 6 E -1 2 ) , ( 7 . 1 7 87E -0 2 • 9 . 5 9 8 8 E - 12 > . . . . >_____________________< , 10994 , 9« ,6 l75E - l, 2J - ._ ( - „ -17?99 ,9« ,62a6E -12 ) . ( . 3 2 Q 7 7 . 0 . 6 S & 7 E - 1 2 ) . » .» ( . 5 5 8 6 8 , 9 . 6 9 7 5 E - 1 2 ) , ( . 7 0 3 0 4 • 9 . 7 4 2 3 E - l 2 ) . ( . 7 9 2 5 6 * 9 . 7 9 2 3 E - 1 2 ) . . . . ( . 8 3 7 6 2 , 9 . 8 4 7 5 E - 12) • ( . 8 7 4 6 2 , 9 . 9 0 4 9 E - 1 2 ) • ( . 9 0 3 2 8 * 9 . 9 6 0 5 E - I 2 ) » . . . ( * 9 2 20 5 , 1 * 0 0 1 1 E - 1 1 ) , ( . 9 3 5 3 7 , 1 . 0 055E-11 ) , ( . 94-85 1 . 1 . 0 0 8 5 E - 1 1 ) , . . . ( . 9 5 5 0 7 , 1 . 0 0 9 I E - 1 1) , < . 9 6 4 1 6 , 1 *0 0 8 3E - l 1 ) , ( . 9 7 1 6 6 , 1 . 0 0 4 9 E - 1 1 ) , . . . ( * 9 7 9 1 7 , 9 * 9 7 4 9 E - 12 ) , ( . 9 8 4 9 . 9 * 7 9 7 7 E - 1 2 ) • ( • 9 9 0 4 7 , 9 * 5 3 3 5 E - 1 2 ) , * • * _____________________ (_._993.9.2,«_9 .29-4.8E--1.2-)... ( . 9 9 7 3 7 . 8 .9 94 ZE -1 2 ) . ( . 9 9 9 I_9.e . 17 3 E - 1 2 ) . . . . ( . 9 9 9 2 5 , 8 . 0 444E- 12) , ( . 9 9 9 3 1 * 7 . 4 6 9E -1 2 ) » ( . 9 9 9 4 * 6 . 5 0 2 6E -1 2 ) . . . . ( * 9 9 9 5 1 , 5 . 3 2 9 1 E - l 2 ) , ( . 9 9 9 6 1 , 4 . 3 3 7 7 5 - 1 2 ) , ( . 9 9 97 * 3 . 3 3 2 3 E - 1 2 ) , . . . ( . 9 9 9 8 1 , 2 . 1C 48E -1 2 ) , ( . 9 9 9 9 1 , 1 . 0 6 2 1 E -1 2 ) , ( . 9 9 9 9 4 . 6 . 3 9 6 5 E - 1 3 ) , . . . (1*0.0.0) A1 = NLFGEN( CLAMDA , CSEE ) __________________________A 2 = NLFGEN(Ch lS tCSEF )_________________________________________________________ A3=NLFGEN(PSIS ,CSEE) A 4= NLFGE N( D MEGAS, CSEE) B1=A1*SQRT(T ) B2=A2*T B 3 = A 3 * { T * * l . 5 ) _____________________.3A=A4-*.T.*-T________________________ _______________________________________________ 2 = 51+82+ B3+B4 SEE=CSEE*(SEEO-SEEN)+SEEN TIMER F I NT I M=1 .!> , PRDEL=2 .0 E - 0 2 , OUT DEl_=4 .OE-O 2 PRTPLT Z LABEL CONCENTRATION PROFILE WITH TIME &BJ_NI-^SE£.«.A-l-aA2j.B-l . 3 2 iB3 «B4 . Z_____________________________________________ END ST CP OUTPUT VARIABLE SEQUENCE A4 B4 A3 B3 A2 B2 A l B l Z SEE CUTPUTS INPUTS PARAMS INTEGS + MEM BLKS FORTRAN DATA CCS 1 4 ( 5 0 0 43(14C*C) 1 0 ( 4 0 0 ) C+ 0= C (3 0 0 ) 1 1 ( 6 0 0 ) 64 University of Ghana http://ugspace.ug.edu.gh 161 A4 = NLFGEN(OMEGAS, CSEE) 1/2 3/2 2 The terms At , Xsc > llJs t an^ “g1- -'-n equation 3.69 are calculated respectively through the statements: Bl = A1*SQRT(T) B2 = A2*T B3 = A3*(T**1,5) B4 = A4*T*T The sum of Bl, B2, B3 and B4 gives the depth z. A PRTPLT statement gives a plot of C^ as a function of z and this plotting is labelled con­ centration profile with time. Print output for CSEE,A1, A2, Bl, B2, B3, B4 and z are also obtained. University of Ghana http://ugspace.ug.edu.gh CHAPTER 5 RESULTS AND DISCUSSION 5.1 Relative Adsorption Rates and Adsorption Isotherms Two models are usually used to describe the adsorption and ex­ change phenomena in the convective-dispersive hydrodynamic equation. One of the models represents an equilibrium between the concentration of a chemical in solution and that on the adsorbed phase (Bower et_ al., 1957; Lapidus and Amundson, 1952; Kay and Elrick, 1967; Lindstrom and Boersma, 1970; Lai and Jurinak, 1971). The other models the rate of approach to equilibrium (Bower et al., 1957; Lapidus and Amundson, 1952 • Lindstrom et al., 1971). In the horizontal infiltration experiments considered in this study it was necessary to ascertain which of these two models best describes the adsorption or exchange phenomenon. The relationship between the keq. K+ adsorbed per kg. soil plotted against various equilibration times (figure 5.1) indicates that equilibrium was established between the adsorbed and solution phases of potassium in less than 600 s. One would expect progressively increasing quantities of K+ adsorbed with increasing equilibration time periods if a slow kinetic type of reaction pertains in the adsorption process of K+ in these soils. Figure 5.1, therefore, appears to indicate an establishment of an almost instantan­ eous equilibrium and thus justifies the use of an equilibrium type of model to describe the adsorption term in the hydrodynamic equation for the movement of reactive solutes in the soils used in this study. The adsorption isotherms for Akuse and Brookston clays plotted as kiloequivalent K adsorbed per kilogram soil against equilibrium 162 University of Ghana http://ugspace.ug.edu.gh 163 FIGURE 5.1. Quantity of K+ adsorbed at different time periods. University of Ghana http://ugspace.ug.edu.gh 10^ S ( ke q. K* ad so rb ed pe r kg so il) 164 3.0 t •- 2.0 1.0 • A KU SECLAY A BROOKSTON CLAY 3600 7200 1 0800 TIME (s) Figure 5.1 University of Ghana http://ugspace.ug.edu.gh 165 concentration (in kiloequivalent K per cubic meters of solution) are given in figures 5.2 and 5.3. It is observed that over the concentration —A 3 range of 1 .0x10 ' to 1.0 keq/m , the adsorption isotherms of the two soils do not obey the Freundlich's equation of the form Sa(jg = kc11, where k and n are constants. In our horizontal infiltration experiments 3 where concentration in the soil column can vary from 1.0 ic~ri/m at the inlet to zero at the wetting front, it was necessary to fit equations to the adsorption isotherms. The isotherms were fitted with polynomial equation of the logarithmic form to obtain S = (-0.12051og2c+0.12921ogc-3.6124) for Akuse clay 2 and S = ^Q(-0,10291og“c+0.12681ogc-3.7163) for Brookston clay. These equations were used to calculate the concentration of K+ in solu­ tion for the horizontal infiltration as outlined in Chapter 4. It has to be mentioned that the coefficients in these two equations for the adsorption isotherm will be different if the units are changed, thus if the isotherms are plotted in milliequivalent of K+ adsorbed per gram soil versus milliequivalents K+ in solution per litre, the equations for Akuse clay and Brookston clay become, respectively, S = 10(-0.12051og2c+0.85201ogc-2.084) and S = io<'_0 -1 0 2 9 1oS2c+0 -7444logc-2.0231) In the computation of the dispersion coefficient D for k"'" using University of Ghana http://ugspace.ug.edu.gh 166 FIGURE 5.2. + 2+K adsorption isotherm on soil fractions of Ca - saturated Akuse clay. University of Ghana http://ugspace.ug.edu.gh S (m e K * ad so rb ed pe r g so i I ) EQUILIBRIUM SOLUTION c (keq./m3) EQUILIBRIUM SOLUTION (me lONI Figure 5.2 S( ke q. K* ad so rb ed pe r kg so il) University of Ghana http://ugspace.ug.edu.gh 168 FIGURE 5.3. + 2+ K adsorption isotherm on soil fraction of Ca saturated Brookston clay. University of Ghana http://ugspace.ug.edu.gh S( m e K* ad so rb ed pe r g so il) EQUILIBRIUM SOLUTION c (keq./m3) Figure 5.3 S (k eq . K * ad so rb ed pe r kg so il ) University of Ghana http://ugspace.ug.edu.gh 170 a computer program written in system 360 CSMP data points from figures 5.2 and 5.3 provided FUNCTION ADSISO which was interpolated, and deriv­ ative performed to obtain dS/dX used in the computation. Lastly, it must be mentioned that, in a preliminary experiment, it was observed that no Cl was adsorbed when the same concentration “4 3 range of 1.0x10 to 1.0 keq/m KC1 was equilibrateu with known weights of the two soils used in this study. The low concentration of chloride -3 -1/2 measured for lambda values between zero and 1.1x10 mS cannot therefore be ascribed to chloride adsorption by amorphous aluminum and iron oxides in these soils. 5.2 Horizontal Infiltration with KC1 solution Figures 5.4a and 5.4b present the bulk density profiles in the soil columns used for the horizontal infiltration experiments for Akuse clay and Brookston clay, respectively. The low mean bulk densities of 3 3 3 3 1.03x10 kg/m and 1.07x10 kg/m for Akuse clay and Brookston clay, respectively, are the bulk densities which could be reproduced fairly well in replicate infiltration runs. Any attempt to pack these moist soils (initial moisture content of 0.10 and 0.12 for Akuse clay and Brookston clay, respectively) to higher bulk densities resulted in un­ even bulk density distribution in the columns which could not be repro­ duced for subsequent infiltration runs. Figures 5.5 and 5.6 present the 8 , and c ^ profiles of the horizontal experiments listed in Table 5.1. The smoothed curves of University of Ghana http://ugspace.ug.edu.gh TABLE 5.1. Experimental conditions imposed during horizontal and vertical infiltration experiments Soil Type of Infiltration Initial volu­ metric water Content 0n Initial Cl Concentration keq/m3 Initial K+ Concentration keq/m3 Mean Bulk Density in Soil Column 10~3p kg/m^ Measured Sorptivity 103 SORP. / 2 m/s Time t s Akuse 0.099 0.00023 0.0 1.021 2,615 Clay Horizontal 0.105 0.00024 0.0 1.030 }l.749 3,745 0.107 0.00022 0.0 1.024 5,558 0.110 0.00025 0.0 1.046 9,330 Vertical 0.101 0.0003 0.0 1.041 - 7,200 Brook­ ston 0.124 0.005 0.0 1.06 3,600 Clay Horizontal 0.118 0.005 0.0 1.07 }0.6518 7,200 0.122 0.005 0.0 1.07 14,400 0.120 0.005 0.0 1.06 21,600 Vertical 0.124 0.0002 0.0 1.08 19,260 Akuse Clay Horizontal 0.053 0.001 0.0 1.266 3,600 (dry 0.054 0.002 0.0 1.271 }0.7182 5,400 soil 0.054 0.002 0.0 1.271 10,800 initially) 0.050 0.002 0.0 1.266 14,400 University of Ghana http://ugspace.ug.edu.gh 172 FIGURE 5.4. (a) Bulk density distribution in soil columns used for the horizontal infiltration (Akuse clay). (b) Bulk density distribution in soil columns used for the horizontal infiltration (Brookston clay). University of Ghana http://ugspace.ug.edu.gh BULK DENSITY 10‘3 P (kg/m3) BULK DENSITY 1Q‘3 P (kfl/m3) University of Ghana http://ugspace.ug.edu.gh 174 best fit were drawn by eye. The first observation from figures 5.5 and 5.6 is that as was found by Smiles et al. (1978), Smiles and Philip (1978) and also by Elrick et_ al. (1979) (under review), the water content, chloride content and potassium content profiles all preserve -1/2similarity in terms of A=xt , reasonably well. Variability in packing the soil columrs (figure 5.4a and 5.4b), problems of air entrapment and losses of small amounts of solution during division of soil sample from each section for Cl and K+ determinations, likely account for the variability in the data. Both the cumulative volume of solution as 1/2 well as the distance to the wetting front as functions of t for each experimental run gave excellent straight lines which passed through the origin. However, due to the experimental difficulties listed above, the slopes of these lines differed very slightly from one column to another. Generally, it is accepted that the water flow equation is valid, so we use the uniqueness of 0(A) as a guide to provide a basis for examination of the c^_(A) and c^(A) relationships in figures 5.5 and 5.6 and con­ clude that the c^_(A) and c^ .+ (A) curves are unique and therefore the assumption that Dg be considered only as a function of 0 is justified. Justification of this assumption is also confirmed by the work of Smiles et_ al • (1978), Smiles and Philip (1978) and Elrick et_ al_. (1979). We next observe that the "salt front" for Cl lags behind the infiltration front (cf. figures 5.5a and b and also figures 5.6a and b). This effect, due to "piston flow", is observed because if there is hydro- dynamic dispersion and the initial water content of the soil >0 , the original water and the encroaching solution mix across the solute pene­ tration depth resulting in the development of a transition zone in which University of Ghana http://ugspace.ug.edu.gh 175 the concentration varies from c to c„. The salt front or transition n 0 zone usually lags behind the infiltration front because at least some of the antecedent soil water is pushed ahead of the salt front. This result is consistent with the previous studies of Warrick et_ al. (1971); Kirda et al. (1973, 1974); Ghuman et al. (1975); Smiles et al. (1978) and Smiles and Philip (1978). Shalhevet and Reiniger (1964) and Terkeltoub and Babcock (1971) also used this principle of "piston flow" to formulate leaching strategies. The notional plane X = X ' , about which chloride dispersion occurs may be defined by the material balance equation (c - c) dX o 0 (c - c ) dX (5.1) n Similarly, the plane X = X* defining the plane of separation, if the water initially in the soil column were perfectly displaced by the en­ croaching solution, is defined by the material balance equation r0(X*) X* 6 (X*) = Xd0 (5.2) These two planes are identified in the ccl_(X) and 6 (X) plots (figures 5.5a, b and 5.6a, b). One expects these two planes to coincide exactly (X'=A*) if all the initial water content is swept entirely ahead of the salt front, that is if there is perfect piston flow without any mixing University of Ghana http://ugspace.ug.edu.gh 176 or transition zone. For the soils under consideration in this study, these two planes for X' and X* do not correspond exactly as was reported by Smiles and Philip (1978) for the sand/kaolin mixture. The possible explanation to this is partly due to mixing which is reflected in a dispersed salt front compared with the sharp, abrupt water front (cf. figures 5.5a and 5.5b and also 5.6a and 5.6b) and also partly due to anion exclusion, and "immobile" water fraction in dead end pores in these aggregated clays whose aggregates are still visible even after grinding and sieving to pass through 0.25 mm sieves (Fatt et al., 1960; Coats and Smith, 1964; Philip, 1968; van Genuchten and Wierenga, 1976; Gaudet et al., 1977). Also, defining mean pore water velocity u in terms of the Darcy flux v and volumetric water content 0 as: substitution of v = -D dO/dx, transformation with the Boltzmann trans- “1/2 d0 1 form X = xt and further substitution of -D -jr- = -r Xd0 yields dA z J g u v /0 (5.3) n ■0 u 20 Ad0 (5.4) 0 n University of Ghana http://ugspace.ug.edu.gh 177 FIGURE 5.5. (a) Experimental moisture content data points 9(A), (b) experimental solution concentration data points for Cl and (c) experimental solution concentration data points for K+ Cg+CA). •k 1 The vertical broken lines identify A and A planes calculated as described in the text (Akuse clay initially moist). University of Ghana http://ugspace.ug.edu.gh CONCENTRATION OF K 'IN SOLUTION (keq ./m 3) CONCENTRATION OF C l" IN SOLUTION (keq ./m 3 ) ,suj )X c0 l WATER CONTENT 0 (m3/™3 ) o *-• 5o co « b> o> 178 University of Ghana http://ugspace.ug.edu.gh 179 FIGURE 5.6. (a) Experimental moisture content data points 6 (A), (b) experimental solution concentration data points for Cl c cl_(A), and (c) solution concentration data points for K+ c^ _|_(A). * ' The vertical broken lines identify A and A planes calculated as described in the text (Brookston clay). University of Ghana http://ugspace.ug.edu.gh Figure 5.6 CONCENTRATION O f K ' lN SOLUTION Ik e q ./m 3! CONCENTRATION OF C r IN SOLUTION |k eq ./m 3 | MOISTURE CONTENT 0 (m3/m 3 > University of Ghana http://ugspace.ug.edu.gh 181 If we further define a reduced mean pore water velocity u* as 1/2u* = ut , equation 5.4 becomes u* (X) = 20 Ad0 (5.5) Computation of the reduced mean pore water velocity from figures 5.5a and 5.6a reveals that the reduced mean pore water velocity tends to increase very slightly to a maximum at A* (figures 5.7a and 5.7b). The maximum reduced mean pore water velocity appears to coincide with the region where the dispersion coefficient is minimum. This latter finding is unexpected and presumably relates to our initial assumption of non-dependence of Dg on the pore water velocity. The concentration of Cl profile for Akuse clay (figure 5.5b) extrapolated to A=0 , indicates that this concentration is about 8% lower 3 than the inlet concentration of 1.0 keq/m . This is attributed to chloride exclusion which is discussed in the next section. The concen­ tration profile of K (figure 5.5c) for Akuse, extrapolated to A=0, on the other hand, shows an increase of about 9%. It was expected, as observed in figures 5.6b and 5.6c, that both K+ and Cl concentrations at A=0 will be equal, in order to preserve electroneutrality. This difference, however, cannot be explained. As a result of adsorption, the potassium content profiles for both Akuse clay and Brookston clay (figures 5.5c and 5.6c) lag greatly University of Ghana http://ugspace.ug.edu.gh 182 FIGURE 5.7. (a) Reduced mean pore water velocity versus X for Akuse clay. (b) Reduced mean pore water velocity versus X for Brookston clay. University of Ghana http://ugspace.ug.edu.gh REDUCED MEAN PORE WATER VELOCITY 103u* (ms*K ) > * C C/3 e 8 i University of Ghana http://ugspace.ug.edu.gh Figure 5.7b REDUCED MEAN PORE WATER VELOCITY 10V (ms'K ) CD JO O O V ) H O 2 f8I University of Ghana http://ugspace.ug.edu.gh 185 behind both the 'chloride front' and the infiltration front. At A=3.0xl0 - 1/2 + ms , there is practically no K in solution for Akuse clay even though -3 the infiltration front is at A=4.56x10 . Similarly, for Brookston clay, -3 -1/2 +at A-l.17x10 ms there was no measurable amount of K in solution -3 even though the infiltration front is at A=l.82x10 . Comparison of figvres 5.5b and 5.5c appear to indicate a sharper 'concentration front' for K+ than for Cl in the case of Akuse clay whilst both concentration front for K+ and Cl for Brookston are not distinguishably different. 5.3 Effect of salt exclusion in the infiltration experiments The set of horizontal infiltration data which show the effects of salt sieving were obtained with infiltration of KC1 solution into 3 3 air dried Akuse clay at volumetric water content of 0.05 m /m and would be referred to as Akuse (dry) to distinguish it from the results in the previous section for Akuse clay initially packed moist into the columns. To facilitate comparison, especially, of the concentration of Cl profile, data for air dried Caledon sandy loam (66.0% sand, 30.5% silt and 3.5% clay) are also presented. The bulk density profile (figure 5.8a) for the soil columns initially packed with air dried Akuse clay, shows a fairly uniform packing with slightly higher bulk densities 3 3 at the inlet end than the mean bulk density of 1.26x10 kg/m . Also, because packing was done with air dried soil, reproducible packing was obtained with higher bulk densities in these experiments than those reported for the infiltration runs conducted with moist Akuse clay. The Caledon sandy loam was packed to a mean bulk density of 1.57x10 3 kg/m (figure 5.8b). The inlet end, however, showed higher bulk density University of Ghana http://ugspace.ug.edu.gh 186 FIGURE 5.8. Bulk density profile for soil columns used in the horizontal infiltration experiment for: (a) Air-dried Akuse clay 0 = 0.05 (b) Air-dried Caledon fine sandy loam 0 = 0.004 University of Ghana http://ugspace.ug.edu.gh 2.0 6 AKUSE a 14400s O 10800s ■ 5400s □ 3600s 1.5 Q Z> 6 ° ^J1 'it b o te^ Litr-i 1.0 0.4 Figure 5.8a University of Ghana http://ugspace.ug.edu.gh TB" a *■ J _______I_______I_____ _l_______I_______I_______I0.8 1.2 1.6 2.0 103A (m s '1*) 187 University of Ghana http://ugspace.ug.edu.gh BU LK DE NS IT Y 10 '3 ? (k g/ m 3) 2.0 r o 1.5 - 1.0 0 CALEDON D 7200s ■ 5400s 1.0 2.0 103X(ms'%) 3.0 4.0 Figure 5.8b University of Ghana http://ugspace.ug.edu.gh 189 3 3 of up to 1.7x10 kg/m . The plot of 0, ccl-> ®cci- ’ and CK+ aS a funct:i-on °f ^ for Akuse (dry) are presented in figures 5.9a, 5.9b, 5.9c and 5.9d, respectively. It is observed from figure 5.9a that 0 at A=0 of 0.495 is lower than that for the same soil packed moist (figure 5.5a). This difference is the result of air entrapment which is lessened when the soil is initially moist. Also, the difference in A values between figures 5.5a and 5.9a for Akuse (packed initially moist) and that packed initially dry is firstly the result of the difference in bulk density 3 3 3profiles (mean for Akuse moist is 1.03x10 kg/m compared with 1.26x10 3 kg/m for Akuse dry) and secondly due to the fact that the hydraulic conductivity for moist soil is higher than that for dry soil. Figures 5.9b and 5.9c present some interesting features which we attribute to salt sieving effects. The concentration of Cl at A=0 3 is significantly lower than the input concentration of 1.0 keq/m . At -3 -1/2 A=l.lxl0 ms the concentration distribution starts to increase to 3 a peak greater than 1.0 keq/m just behind the infiltration front. This trend was not expected for in a soil with a low initial water content so that there is very little or no mixing during unsteady unsaturated flow phenomenon induced by sudden availability of KC1 solution at the surface of the soil column, one expects no transition zone but a constant concentration equal to the concentration of the solution at the source as was obtained for Caledon sandy loam (figure 5.10b). The low concen­ tration of Cl at A=0 in figure 5.9b could not be due to adsorption of Cl because adsorption isotherm conducted showed no Cl adsorption for all the concentration range used. It appears then that some of the Cl University of Ghana http://ugspace.ug.edu.gh 190 FIGURE 5.9. (a) Volumetric water content 0 versus A for Akuse (dry). (b) Concentration of Cl in solution versus X for Akuse (dry). (c) Concentration of Cl per bulk volume of soil versus X for Akuse (dry). (d) Concentration of K+ in solution as a function of X for Akuse (dry) University of Ghana http://ugspace.ug.edu.gh Fiqure 5.9 CONCENTRATION OF C l' IN SOLUTION DURING SORPTION Ikeq.. n.3) CONCENTRATION OF K*IN SOLUTION (keq./m3) WATER CONTENT 6 (m3 m3) 9 c c |Jkeq./m3) T6I University of Ghana http://ugspace.ug.edu.gh 192 FIGURE 5.10. (a) Volumetric water content as a function of X for Caledon sandy loam. (b) Concentration of Cl in solution versus X for Caledon sandy loam. University of Ghana http://ugspace.ug.edu.gh CONCENTRATION OF C l" IN SOLUTION (keq ./m 3) University of Ghana http://ugspace.ug.edu.gh WATER CONTENT 0 lm3/(n3 > University of Ghana http://ugspace.ug.edu.gh 194 was excluded at the inlet end. The chloride ions which entered the column of soil, however, were confined to the central more rapidly moving region of each flow path and thereby hastened its passage. At the wetting f r o n t where water contents are low, the water together with the Cl moved between wedge shaped volumes passing through thin films which sieved the chloride ions thus resulting in accumulation of Cl just behind the infiltration front. Chloride salt balance for the set of experiments reported here was calculated by integrating a plot of 0c versus the horizontal distance x for each infiltration run using CSMP. The integrated area multiplied by the surface area of the soil column gave the total quantity of Cl in the soil column after infiltration was terminated. This was compared with the total amount of Cl in the cumulative, quantity of solution that entered the soil column. Table 5.2 which summarizes this comparison indicates that on the average about 7.03% Cl was excluded from the soil column. TABLE 5.2. Chloride Salt Balance for Horizontal Infiltration With Akuse Clay Packed Dry Infiltration Run # A keq. Cl in soil column after in­ filtration xl0-6 B keq. Cl added through cumula­ tive water in­ filtrated xl0-6 % Cl excluded, i.e. (B-A)IOO B 1 26.04 27.78 6.26 2 29.26 32.0 8.57 3 45.92 48.82 5.94 4 52.16 56.3 7.35 University of Ghana http://ugspace.ug.edu.gh 195 Comparison between figures 5.9b, 5.9c and 5.9d indicate that at -3 -1/2 +A=1.4xl0 ms , there is practically no K in solution whereas the Cl 'front' is very close to the infiltration front of -3 -1/21.82x10 ms . It may be inferred, therefore, that the chloride -3 -3 -1/2 present between A~1.0xl0 and A=l.82x10 ms (wetting front) is 2+ presumably linked with Ca . Conversion of dn/dA in equation 3.54 to dc/dA was done using Van't Hoff's law, thus obtaining dir/dA = n RT dc/dA, where n for uni-divalent salts is 3. However, because concentration c is 3 measured in equivalents of Cl per m , n = 3/2. Also it must be pointed out that dir/dA is the total osmotic pressure gradient. Consequently conversion to dc/dA estimates only the concentration gradient due to Cl alone and may over-estimate the osmotic effect. In our experiments the factor $ defined earlier was introduced, assuming that in the wet zone where water content is near saturation, compensation between ions occurs so that osmotic pressure gradient in this region may be neglected. Again, the concentration of K+ at A=0 exceeds that of the source by about 5%. 5.4 Derived Data From the Horizontal Infiltration Experiments The derived data for the dispersion coefficient D as a function s of 0, calculated for both Cl and K using CSMP and equations 3.11b and 3.19 are presented in figures 5.11a and 5.11b. It must be reiterated that the most accurate data in these figures lie between 0=0. 4 9 5 and University of Ghana http://ugspace.ug.edu.gh 196 FIGURE 5.11a. Dispersion coefficient for Cl and K+ plotted as a function of volumetric water content (Akuse clay - moist). University of Ghana http://ugspace.ug.edu.gh D s (m ‘ 197 10** AKUSE 10-5 10-« 10*7 * 1 10-8 10 ',-9 _l_ _L 0.1 0.2 0.3 0.4 MOISTURE CONTENT (m3/m3) 0.5 0.6 Figure 5.11a University of Ghana http://ugspace.ug.edu.gh 198 FIGURE 5.11b. Dispersion coefficient for Cl and K+ plotted as a function of volumetric water content (Brookston clay). University of Ghana http://ugspace.ug.edu.gh Ds (m 199 10',-3 BROOKSTON 10*4 10-5 k D s for Cl “ 10-6 p . A • I 1 1 10“' A * ▲ 10-8 JL JL JL 0.1 0.2 0.3 0.4 MOISTURE CONTENT (m3/m3 ) 0.5 0.6 Figure 5.11b University of Ghana http://ugspace.ug.edu.gh 200 0=0.40 for Akuse clay and 0=0.49 to 0=0.37 for Brookston clay. Values at the extremes must be less reliable because at these extremes dc/dX tends to zero so that dX/dc becomes infinitely large and also fX dcg - r r dX tends to zero. Bearing in mind these limitations, however, •v d A n it is observed from figures 5.11a and 5.11b that the dispersion coeffi­ cient ^ata for Cl agree fairly well with those for K+ . If adsorption is properly and adequately described, it is to be expected that Dg for Cl should be identical with D for K+ . This stems from the argu- s ment that if we infiltrate a solution of KC1 into an inert porous medium, for example sand or glass beads, the calculated for Cl in such a system will be the same as that for K+ . Considering that the molecular diffusivity of KC1 in water is -9 2 -1 about 1.917x10 m s , it is concluded that the dispersion coefficient in the soil columns is large and that dispersion in these soil columns is mainly the result of hydrodynamic effects rather than diffusion. It is therefore necessary to re-examine the initial assumption of non­ dependence of Dg on the pore water velocity or Darcy flux made in this study. In keeping with Saffman's (1959) analysis, Pfannkuch (1963) found that Dg was independent of Peclet number P^ provided that P^ < 1 P = u 1/D (5.6) e m In equation 5.6, u is the mean pore water velocity, 1 is a characteristic pore dimension and D^ is the molecular diffusivity of the salt in solu­ tion. The Peclet number appropriate to the sorption experiments is University of Ghana http://ugspace.ug.edu.gh 201 formed in a similar way as was done by Smiles and Philip (1978) as follows: u is first identified with the maximum reduced mean pore •k velocity u of equation 5.5 so that equation 5.6 becomes P = u*t 1/2 1/D (5.7) e m Furthermore, as was done by Saffman (1959), Pfannkuch (1963), and Smiles and Philip (1978), we take 1 as the 'mean diameter of the grains', and estimate 1 from the saturated hydraulic conductivity K through the use o of the Kozeny-Carman relation (Carman, 1939), which yields 1 = 6/5 (l-eo) [vKo/(g0o3) ]1/2 (5.8) Equation 5.8 is that of Smiles and Philip (1978) and V is the kinematic viscosity and g is the acceleration due to gravity. Thus 1=2.38x10 m for Akuse clay and 1=1.04x10 m for Brookston clay is inferred from ”5 -1 — 5 the value of K of 2.0x10 ms and 4.0x10 ms for Akuse clay and o Brookston clay, respectively (see the hydraulic conductivity versus moisture content curve, figures 5.14a and 5.14b). Taking D for KC1 m -9 2 - 1 * -3 -1/2 * -4 as 1.917x10 m s , u =1.72x10 ms for Akuse clay and u =6.42x10 - 1/2 ms for Brookston clay (see figures 5.7a and 5.7b), we obtain upon substitution into equation 5.7: - 1/2 - 21.37t for Akuse clay and -1/2 - 3.5 t for Brookston clay, so that for Akuse clay University of Ghana http://ugspace.ug.edu.gh 202 21.37t < 1 i.e. t > 7.6 minutes and for Brookston clay 3.5t ~ 1 i.e. t > 12.2 seconds For these soils, therefore, and under the experimental conditions imposed, is independent of vs except during the initial period of about 10 minutes. The computed soil moisture diffusivity, D, plotted as a function of 0 = (0-0 )/(0 -6 ) for Akuse clay (moist) and Brookston clay are n o n given in figures 5.12a and 5.12b respectively. The high energy moisture characteristic curves for Akuse clay and Brookston clay, used to calcu­ late the hydraulic condutivity K, data (figures 5.14a and 5.14b) are presented in figures 5.13a and 5.13b. The slight differences between water contents at matric potential of zero and X=0 (figure 5.5a and 5.6a) is due to air entrapment invariably encountered in horizontal infiltra­ tion experiments. These D( 0 ), D (0) and K( 0 ) data were used to simu- s late moisture content and salt content profiles for both horizontal infiltration and vertical infiltration. The dispersion coefficient data for Akuse clay where salt sieving effects are observed (figure 5.15 and Table 5.3 ) was computed from equation 3.54 and the CSMP listing in Chapter 4. In Table 5.3, -1 / ? THLA is the water content 0, BETA is g defined in Chapter 4, SEEXPT is X (Hl-Hl2+H2-H3)dX, H7 is J A j the expected c , H2 is D tt- , Y is thedA dA x Y dX/dc, H8 is the osmosis term 3RTcaK(0)g dispersion 2gpw coefficient. As indicated earlier the introduction of g assumes that in the wet region osmosis is very small and that osmotic effects University of Ghana http://ugspace.ug.edu.gh 203 FIGURE 5.12. (a) Soil moisture diffusivity as a function of 0 = (0-9 )/(0 -0 ), Akuse clay (moist), n o n (b) Soil moisture diffusivity as a function of 0 = (0-0 )/(0 -0 ), Brookston clay, n o n University of Ghana http://ugspace.ug.edu.gh D (m 204 Figure 5.12a University of Ghana http://ugspace.ug.edu.gh D (m io*3r 205 BROOKSTON 10"4 10 -5 10-1 10-7 / / • / 10 '-8 w (60 -en) 1.0 Figure 5.12b University of Ghana http://ugspace.ug.edu.gh 206 FIGURE 5.13. (a) High energy moisture characteristic curves for wetting (w) and draining (d) cycles, Akuse clay. (b) High energy moisture characteristic curves for wetting (w) and desorption (d) cyles, Brookston clay. University of Ghana http://ugspace.ug.edu.gh 0.10 0^ 20 0.30 0.40 0.50 0.60 0.70 M OISTURE CONTENT (m 3/m 3) MATRIC POTENTIAL (m) A K U SE University of Ghana http://ugspace.ug.edu.gh MA TR IC PO TE NT IA L (in ) 208 MOISTURE CONTENT (it^/m3 ) Figure 5.13b University of Ghana http://ugspace.ug.edu.gh 209 FIGURE 5.14. (a) Hydraulic conductivity K as a function of 0 = (0-0 )/(6 -0 ) for Akuse clay, n o n (b) Hydraulic conductivity K as a function of 0 = (0-0 )/(0 -0 ) for Brookston clay. University of Ghana http://ugspace.ug.edu.gh Figure 5.14a H YDRAULIC CONDUCTIV ITY (m s*1 > -si m 01 J i--------1------------------ 1-------------------1------------------ 1 I I \ I V \ \ © l a>o ot i CD ■2-— A K U SE University of Ghana http://ugspace.ug.edu.gh Figure 5.14b H YDRAULIC CONDUCT IV ITY (ms ’1 ) © 1 o® (D1 CD3
0. Introduction of the salt sieving and osmosis terms result in positive values for all A values. However, for greater reliability of data presented in figure 5.15, it is necessary to devise a method to determine the total salt concentration gradient during the infiltration run. Note that the Dg values in Table 5.4 are slightly larger than those in Table 5.3. This is so because X 9 ex dc H12 (= — -— -jy ) is zero when salt sieving effect is not included in the calculation of D . Data in figure 5.15 for D (with salt sieving s s effect accounted for) plotted as a function of 9 shows a minimum at 9=0.45. Smiles et al. (1978) observed a similar minimum in D^ plotted against 9 for their experimental soil. This minimum in D^ may be attributed to an artifact introduced by the correction term $, especi­ ally when we consider that the extrapolated curve (figure 5.9b) for expected c is arbitrary. 5.5 Simulated 9(A) and c(A) Results for Horizontal Infiltration Experiments The simulated 9(X) relationship calculated from D(X) results which were in turn derived from smoothed experimental 9 (X) data is presented in figures 5.16a and 5.16b for Akuse clay and Brookston clay, respectively. Tables 5.5a and 5.5b are the computer output of the data University of Ghana http://ugspace.ug.edu.gh 215 TABLE 5.3. Computer output showing positive Dg values due to inclusion of salt sieving term in equation 3.54. Explanation of the other output headings is in the text. University of Ghana http://ugspace.ug.edu.gh 8 0 8 5 3 5 E — 0 1 CD •>! N S to ■P> ~ o oj n o O O' W 03m m ro ro t i l l o o o o a- O' O' o i w -J 111 O 01 N o ft CD 01 05 ro O O' W tt1 rn ro ro ro ro ro I I I i I I O O O o o o ■f> ft OJ oi •-* ® O' \0 0i o o> ro m m m O' o-i*05 * O lro m m r ■ v- o ■ ro 03 IOI ft o i m m 01 'X) 05 O' M Om m m vf t ' Oa3 a>a3 ^ j > i r f O ' 0' t n o i o i o . c > ( » c » J G j r o ! \ ) r o * - » - H * > i ( >j o • • • • • • • • • • • • • • • • • • • • • • • • • • • r ^ * * > i ( t ) 0 0 ' N « i / i ' - ( H > o s u o ( M o ' O i i i ' - a n » o i u a . o > 0' o U ' i o j > o i M m a i M o ^ i j ( j 1o i > ^ H * ' ( i i w u ! > o o ) o z ^ O 0' N ® ( > O f f r D ( U > o ( J ' M ( J l f t o 9i | \ ) ( l ) f i O ( M \ ) O O O o o o o o o o o o o o o o o o o o o o o o o o o o of n r n r f i r n f n r r m m r n m m m r n m r n m m m m —. ^ ™ I I I I I I I I I I I I I I 0 0 0 0 0 0 O O O O O * ... m r n r n m r o m m m m i I 1 I 1 I 1 .................................. o < I * I 1 I 1 l l 1 i 1 1 1 1 1 1 1 ( 1 i 1 O O O O O O O O O O O O O O O O O O O O O O O O O O r o r v r o r o r o r o t o r o r o r o r o r o r o r o r o r o r o r o r o r o r o r o r o r o o i o j O i r o r \ j o i o i o j 0j f t . ( ! - f t f t f t * - * > . * > f > f t f t f t f t o ro ro O M W S O CD O' to 0 O O'm rn rn rn1 1 1 I O O O O 01 -4 05 ro ro ^ a «o ro O' O CDrn rn rn 1 I I 0 0 0 o >- ro o *- f* ■P- OJ i - ft O' ftm ro rn ro o j o j . O' ro ->i » O' o *-* ► O O' o cro in m 1 ft oi 01 01 -d * Ci o< ro cd -*j O ft CD O'ro rn ro ro f t f t f t f t f t f t f t f t f t j > f t f t f t f t . * vO a I I 1 I I 1 I 1 1 I I I 0 0 0 0 0 0 0 0 0 0 0 0 010' 0‘ 0' 0*C>*^*J ' M O ^ U I I O - O i n C 6 C D x l J 01< i O I W ' - 0 «0 0 > - ' - O v o i o r o f ' O J 4> i n 5i ' j a i i o o - ^ u j > ( f l ^ > i C D > o c D * j C D o o i romromrornrorornrnmmrnrommmrororomroromrom 1 1 1 I I I 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 I 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 w a s 0 oj ro 01 a -J m ro ro a> >o -o x O - 01 r CD 0. O > O' O' o • > » ................................. ■ 1 ■ • 1 1 1 1 i 1 i O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O ) O' OJ> -* M <0 -C- — m rn 1 to o O' o> cn03 ft ff> Ul OJ Oj tO •— ■ ' J O J O O O O O O O O O O O O O O O O O O O O O O O O O O O O > . . • . . . • • • . . • 0 . 0 . • * . • . • • • • . . . • • . . . . ■ 0' - ' i 0 0 >- r0 0^ £ ' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 m O Oi in O O' -g > 0 0 0 I I I I I o ' > j O t O - ' J - t - O J f t O l O l s fl M o (1 O ft ftin m hi m hi ro in m l t 1 i l I I I 0 0 0 0 0 0 0 0 rorororororooioi 0 0 0• • . < 0 0 0 1 ill irt tA tft 10 to ro to 0/ m 05 05 o o o m 01 * f \ c O ' « > t o ^ O ‘ Cr> u t o o ^ t . N N ^ a ' - N C ' U N O ' O O j ' C c a a ' O a IT fi ^ ^ -si £ M C' H rt A M A h fo ft ^ in ^ sio > r c 00 ' < ; s ^ r 0 ^ j 0' 0 o o w o * - r 0 a f \ / ' g C j ' 0 - ^> i f l j N i c C t f H o o i o i r . u u o N C - w ^ M t - s ^ o a N rororororomrororomrnnirnronirnroromrn'Tirorofn I 1 1 1 1 i 1 I I I I 1 1 1 I I 1 1 1 1 1 I 1 I O O O O O O O O O O O O O O O O O O O O O O O G o i o i o ^ o ’ o i o ’ o i o ' o ' o ' o ' o ' c c o ' o ' o ' a o - o ' o ' a a p O to * l! O) OJ o ir » .f) m -0 (I * M -*i CD vTl O ft to ro *-13 « O OJro ro ro ro >- ft ft01 D ►“ >J JJ I J Min ro ro m 1 1 1 1 0 0 0 0 1 l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 o j o j o j o i o j o j < i j o j r o i \ j r o f o t o r o i \ > ( o r o r o r o r ' j i \3 <0 m n> in ^ u iv i m if) m n m f t in £ t i j u i ' l \ ) * - 4 0 0 l O l M O a i d ' 0 > * l j l O ' 0' I O O C D 0 0 1 ' i l C I ) O W N O ' 0) l O O O m * - ' C o a i * - r o i o o i » - - J O i * - i - i \ ) - j > - { i i c ' i C D ^ r o r n r o r o m m r o r o m r o n m r o r n m m m r o r o m r o m l I 1 l 1 l l l l 1 1 l l 1 1 1 1 1 l 1 1 O O O ^ O O O O O O O O O O O O O O O O O0' ^ 0' 0' 0' 0' C ' C ' 0' 0' f7' 0' 0' 0' 0' ' ? ' C > 0' C ' 0' 0' I i I t I r o M M w i o r o w w w i o M N m m m » * » ~ o o o o w w v w t ' ' U N r O O l U l l / I O O i O l ' O ^ t i i o i a ) ^ N o i r o r o c D r \ > o - M O i o r o O J o I I I I I I I I I 0 0 0 0 0 0 0 0 00'rt'C'C'0''T‘0‘0'<">' ** M W 10 i — ro v o r u O J C D ' v i u i ^ o j ' - — •(il 0"O ►* IJl * 0) 'fl B (I U IJ S t I o i r o r o » - o a : n 3 f t ^ j > - c > a 3 - « i f t r SM**0)'OUOl'i)N'4N0)0'< in ro ro ro ro ro ro m rn m ro ro ro rn r 1 1 1 1 1 1 i 1 1 1 I 1 1 1 O O O O O O O O O O O O O O C ft h oj (*j oj ( j ( j oj ro to 10 ro m ro r 1 cd 01 r ' n. 01 < 1 m ro r - IW “ O O S O' ffi W I I “ ! *n (d Oi >c ^ I O' OJ OJ ->d o 1 rn ro rn ro ro 1 I I l 1 ro — - O O O O O O O O O O O O O O O O ' co ft rv> o o o 0 0000000000000 0 -a O ' O l C B f c - f t C ' O J r o - J ' T ' O ' I., 1 m m ro in ro in m ro ro m ro 1 1 I 1 1 I I I I 1 I O O O O O O O O O O o N M N ro M M 01 01 u 01 1,1 O O O O O O O O O O O O O O 1 O O O O O O O O O o O O O O GC 1 oj ro . 1 w m o m ft *— r>> oi r3 U» M OJ ro t/l >0 w - j ► - rn o*. (/I •T) ft I .1 < 3) O' ->l - ! ft 03 Ul •{) '■a o' >n ft •. m m in m r t i l l O O O o < ro ro ro ro r oj oj 10 u 1 rn (j . • *- 01 t ■ to - r r j « ' > M H i - M i - ( l| | u ( l| 0i l MO ' f t l A l O l U I 0' O l l o r 0 M M N N W N I V ) l \ ) M t f t f t J > f t - I I O O - 1 ro ro r 1 O' -g <0 to 1 ro ro o ro I M ^ 9 O 1 M -J - J il)1 rn ro m 'n vO 123 O OJ cd >n m 03 U1 CD O' >- ro cd o' vo rn in in m l l l l 0 0 0 0 o j o j o j Oj - tu - - » 1 oj ro o ( 1 ro N - . o ( j n> r 1 rn ro m 1 I I I I I I I I I I 1 n m in in m ft 111 » f t Id 01 l\> . * - Q 5 a ; o i a : o i i f l ( < j i \ 3 0 n i n 0 ' < ' O ' . p - j •- f t CD+ s ' OJ i i j a J o i O ' — o o ' o i ' o r o rT''0 U10' tV3O r 0 '0 r0 f t - ' T' I V'0 ^ ' C3 m ro rn n i ro in '-n m m rn m ro m m ■ I 1 t t l I l I 1 t l l 1 1 1OO O O O O O O O O O O O O O o» o o o o o o o o o o o o o o o o i r o r o r o i o i o r o i o i v r o r o r o r o r o i o r o r o 913 University of Ghana http://ugspace.ug.edu.gh 217 TABLE 5.4 Computer output showing both positive and negative Dg values because salt sieving term was omitted by setting o = 0 . University of Ghana http://ugspace.ug.edu.gh e O H S J S F -o i . M - M M ,0 !(1 ro ft » • o i« -J o .c- cr •— ui ca ro o O' to cd p- o O' to 03 c-m ci m in ii m m rn m m I I I I i I I i i I O O O o o o o *"> o 6j 01 I (.1 M C ^ . •7- <£> U C 'n O' to < rn rn m i iu ro t\> "J W o o s s — *• O' w r« m m m m I I i i i i O O 03 ui ~ to •s. vc c. I i I I I « > o ■"> O ">00000000 f o o' r0 O o < m m m i1 I l o o o 0 0> • ' — cr. < > O' to ( » o o < i rn ni r I O' ft’ I in U’ cn £ p- i‘i oj ro ro to — ; oj O' ■ ro co > o o i m in o c> ro >o cti — o> c- o o £ • - £ a ro o. -Ow. -*> o (J| W 0- u O' to -J O 0 '" 'O O O O O O O O Om in in rn m m m m m rn m1 i i i i i i i i i i O O O O O O O O O O O N M N N M M N W I \ j U( ) J . . . r to C o j> •J u o i to i oi u oi o ■ ■ p -P- i> #• 4> <2- . i P ft- -O ^ i v c i \ ) ( . n s a o - ‘ N W u u * ' e ‘ f ‘ 0 i ( . n ( ; 0 i 0 ' ( ? / > c 0 ' 0 ' C ' 0 ' 0 . - > i - s i e n h ^ o' tt o n ^ i;. o >i a w io u >- 0 ' M C C ' C N * U - 0 ’ O ^ > - i - 0 ' W 0 ) s U ' * N o J > U I \ ) f - O i f l f f i ' B i O ' t O ' O t l J ^ O ' t O O ' 0 ' ) l i l O ^ O ) 0 ' f f l O M I » l i l f > U l O ' ' J ( l l i l ) A’n m m m m m m m in rn ni m m m m in m m m in m m in m m rn m «J ^ N CH 0) '1 "J O' Ul P 61 to i 3 a CD CO <33 I I r i i i iii i > 0 0 0 - 3 2 0 0 0 0 0 0 0 0 0 I I I I I I I O O v O O O O * i \ 3 0 i f r u ) 0 ' - j c i > ,o ,i ) ' - f o w o j ^ u i o - s ^ y i f r G i t o H o o o ^ - o i T - f r i *»* iy ui ^ w O' s o o cd 'i « u u oi 7 m m rn m m in m m in ni in rn rn .n in m i ) I I I I I i i I I I I i i i O O o O O O O O O O ' j O w O O O p P oj a N CD N C- .n rn < i I O O < •cox *- Cr: r Uj O > . to i l O W i O l G l N - S U ' ^ o O O O i * -u C, -«J to »0 (P M O O' O' 01 O' N »0 O >-■ to O P- o o O O O I too' i i s ‘- > i a * n ' u o o . ' i o u u i ( i ' » ' j >0 >0 ui N N - O Ol W 01 - o IN) * w ■& UI (J1 ( 3 f ( > » i C ' i > 0 ( I O M T i H ^ ( D t o O O o * t mi m >n ni in in ni ui hi in 'U in in in in m iit m hi I I I I I I I I I I I I I I I I I I Io o n o o o o o o o o o o o O O O OO M ^ H - M i - K M f O M M I U M N W N M O l O l o o o o o o o o o o o o o o u o ■> o o o a * > O O O O O O O O O O O O O O O O ^ O O O O O f o ' i ' j ' j o i ( i ) O' CD O Olto — ^0 f>-cd o to jo to /n m m ni in I t o -fr ■ >i ^ 3 O' O • i m m i O' (J CD tom in 03 >0 CD to O O' O' o . I I I I I I I I I I I I ■ » 0 0 0 0 0 0 0 0 0 0 0 0 0 O' >0 ->l 61 -vl ■to CD O 61 I■£ CD ■C- to irn m rn m i J 03 i*0 i -o i o O to to CO m m m o e- o0 61 to J> in rn1 I o o — totototototo i\j to ro to U l O ^ 0 ) O - > - l \ ) 0 l 6 f r ^ ^ 0 ' o a ^ ' - o i o c o u i w ^ ' j t M i o0 p a> to in ® >- p c- j ro m a oin m m m m m m in rn m m rn in in 1 i I i i I i i I i i I I i o o o o o o o o o o o o o o if) • • to to ~j co0 — , Oj CDrn rn i1 i o o ■ to to ro to oj m I i I I I i I I I l I I I l I i06iin0'6jto6itofc-* -P- •£■ 61 61 6J 6j 6> 0 01 61 1 1 S ►- Ol 6J a Cl 01 01 61 S -J N ff C •vISNO'C O' 01 U U Cj ( ^ to 03 to ►* r Oi 4> CD O a a. O' a m ni r - m i f m m m m m m ......................................................I I 0 o CJI 01 1 o. to to -c >r c uo/C ^M o a 1 m i t i o o I I I I I o O O O O U! UO'. IT-^CC'CO^O. U^OC ^ *>*•■15 C ( i u ^ a - ' i u i t o o o ( f n O . ' ^ O M - O ' f l ' O O l S a it m m m m m m rn m it m rn rn I i I I I I I I I I I I I o o o o o o o o o o o o o i n u i u i w u i ^ ^ u i ^ - u i u i ^ ' " ■P 6 U 01 Ui Ul61 to O * OJ >1vO 01 4- - * +• I I I I I I I i to O *- ( • n C Oj ■ i r; it. m ni m i I i I I I I i I i I I i ^to * o- o. to | C 03 a C Cl0. 1 f N S M O 03 N ’ fn m m m I i i i i i i o w oi oi io toO' 61 O' o- O >0 ft to CD -> 03>0 O' O' 6l m m a> co >oto vO O' to o -p m m m to to to r o to oi ( cd — cr. * — in to ( m m m r I i i i i I I i i to to w ■ vfl O' o ) l , ' > - v^ ( S , | ^ t o ^ t o u - * “ oO 'OOJMu i \ ) to to to to to i'J to to to to ro to to 6 J - f r - o f l ' 0 6 J N v0'0>ioto»-ui ' 0UiUiCD m m m m m m r n m r n i n m m i n m I I I i i I I I I i i I I i O O O O O O O O O O T O O OO'C'. 0'C'0'0'0',1'0'0'0'0‘ ^ O' * ui < O' CO • m m r I Ul 6J — ■ Ul •{> CO i C' 61 O i m m m O' in i d to o o n ro - ui— m m m "n i i o — ►“ *— ( O O' 03 S - -J 4> 6J to OJ O' a ■; i t in m i I I l O O O : i i\> to m j2- .I I I I 1 I I O' to CD O > ->1 ' OJ O' < to C: - 1C- -J -C1 vD »0 O I 10 >1 >0 f> N O i i n m m m • n to w roto to -0 OJ Ul to O T1- O in m m i i i ro w to oj ui cd — o -j p CD -e- Ui O to OJ n in ni in I l I i to O O '0 to 03 ts O' Ul I - ->J 03 O' *- Ul to UJ '0 ui to OJ►-* CD OJ to -J* -vi OJin hi in m m in in l I i i i i i o o o o o o o OJ Ul OJ >- (d I h- Oi O' Ul O J* 'J o .n in i to m ( j to to ►- *- > ‘ to to to w w to U « O O 'O S C‘ 01(J^^0 lu6J- f r0J0J(J l t ) 'DC' < ' boi>i to i nui toC0O'0Jtoa)O' -P6i>j0 ' ^cD0JUiv 0 ' 6 l ® “ ® 0 ' |- O U l ^ “ l ^ t oO ' 'OCOO-JU lC ' - too*i )-(s->iCD^6JCB' - '0 6 JC'-e- — u i ' J 'Oo t oO ' i m hi ,n m in ui mi in m in m in 'n m n -n :n ,n ;n m in f I I I I i I I i I i i I I i I i i I i i i O O O O O O O O O O O O O O O O O O O O O l 6J6J6JU0J0 iw t o t o t o t oMMi \ ) t o t o t owto t o t o r O o o o o o o mCi o o o o o o o o o o o o o o o o o o o o o a o o o o o o o o o o o o o o o o o o o o o o o o o o 33 0 0 0 0 “> 0 0 0 0 3 0 T o o J O O O O O O O O O O O 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 3 0 0 0 0 > C9 H 1 ( 1 ( 1 1 1 1 1 1 1 1 O — N0 ' ( l ' 0 ' i t ) i 0 ' 0 ' 0 ' 0 J » 'O i f l N I \ ) f r l M 6 1 0 JM> i S ^ ^ 0 ' 0 ' 0 >WN I ' }MN t oN | <0 to a cd I O' 03 0 ►- \n ^ 01 to P to O P <0 P 'J3 Ul O u; O' P oj o,n m m m in1 I i i i O' l / i ^O' inuiui -ocD lilff'00''IUIM '0'J o ' “ 0'C0'0“*~*'»l0' 0 ' ( j i s ^ t o « 0 ' 0 i > m m n rn m m m rn in t i I I I I I i I p oi ro • - ito o co >0 i 6J O O' t o I P O' -J t o • i n i n i n m i i i i i O O o > < 61 oi Oj ui < i io a ■ ® S P O >0 I P -J 61 I m m m I i I — to ui ui ro w 0 (J o to to 03rn m m1 I I o •> o OJ OJ 01 •&ojwO'Q03n»iO' f in)CD-' j 'Oc O vO C ' t o a 3 ^ O ' ' O O ' C D 0 J « O - M - O' *- -0 l i O S - M C O O ' t o t o C S I toO'»- toO' -vj raCD*-ou6J- f r -m m m m m m m rn in m m m rn r 1 Ul Ui i to rn I I I I I I I a p o -« i/> • — (J 03 - j I CB * to Ji O' to t> a i m m m ‘n 812 University of Ghana http://ugspace.ug.edu.gh 219 plotted in figures 5.16a and 5.16b, included to show the degree of agreement between experimental and simulated 0(A) data. In Tables 5.5a and 5.5b, THLA is the experimental water content and THETA2 is the simulated water content. d0/dA which was calculated with the DERIV algorithm of CSMP was integrated to obtain THETA1 which is compared with both THLA and THETA2 to ascertain the accuracy of the calculation of d0/dA. Interpolation of the smoothed experimental 0(A) data was done by Lagrange quadratic interpolation algorithm of CSMP. These interpolated 0(A) data (THLA in Tables 5.5a and 5.5b) are also plotted in figures 5.16a and 5.16b. It is observed that excellent agreement is obtained between simulated 0(A) and experimental 0(A) data. Given D(0) or D(A) therefore, 0(A) can be predicted or simulated with a very good degree of accuracy using the computer program described in Chapter 4. Simulated c(A) relationship computed from Ds (^ -) data which were calculated from smoothed experimental c(A) data for chloride is given in figures 5.17a and 5.17b for Akuse clay and Brookston clay, respectively. c(A) relationship computed using CSMP program for the analytical solu­ tion (equation 3.21) is also plotted in figures 5.17a and 5.17b which also show simulated c(A) data obtained with constant values of 0D s chosen from upper, middle and lower Dg range of figures 5.11a and 5.11b. Tables 5.6a and 5.6b also present computer output of the data plotted in figures 5.17a and 5.17b. Tables 5.6a and 5.6b have been included to show clearly the minor differences (which do not appear in the graphical representations, figures 5.17a and 5.17b) between c(A) data calculated analytically, c(A) interpolated from experimental data and c(A) com­ puted non-analytically using the algorithm described in Appendix D. In University of Ghana http://ugspace.ug.edu.gh 220 FIGURE 5.16. (a) Simulated water content as a function of X (Akuse clay). Interpolated experimental data points are provided for comparison. (b) Simulated water content as a function of X (Brookston clay). University of Ghana http://ugspace.ug.edu.gh U iu/ju) 0 0.6 AKUSE 0 . 5 - ■ • ’ * . 0.4 0.3 0.2 • INTERPOLATED DATA POINTS FROM SMOOTHED EXPERIMENTAL DATA POINTS SIMULATED DATA POINTS 0.1 1.0 2.0 3.0 4.0 5.0 103 X Figure 5.16a 221 University of Ghana http://ugspace.ug.edu.gh 9 (nr /m ) BROOKSTON 0.5 0.4 0.3 0.2 0.1 ■ • ■ • INTERPOLATED DATA POINTS FROM SMOOTHED EXPERIMENTAL DATA POINTS ■ SIMULATED DATA POINTS J_______I_______I_______I_______I-------1-------1-------1-------1 0.4 a.8 U 1.2 1.6 2.0 103X(ms ) Figure 5.16b 222 University of Ghana http://ugspace.ug.edu.gh 223 TABLE 5.5a. Computer output for soil moisture diffusivity D, experimentally determined water content THLA and simulated water content THETA2. THETA1 is the integration of d0/dX (which was done with the DERIV algorithm of CSMP) to serve as a check on the accuracy of DERIV algorithm of CSMP (Akuse clay). University of Ghana http://ugspace.ug.edu.gh University of Ghana http://ugspace.ug.edu.gh 225 TABLE 5.5b. Computer output for soil moisture diffusivity D, experimental water content THLA and simulated water content THETA2. THETA1 is the integration of dG/dA. to serve as a check on the accuracy of the DERIV algorithm of CSMP (Brookston clay). 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K) f r O' n m m n in rni i I l i l 'j o c> b o o fr J: t P £ • i • .3 <0 fo <0 «0 5 '• fO fr Ul a o fr Or ** vo .0 ro y£> ro P O' • n in in m inI I I I Ij ' L> O CD fr 0 o m m 1 i ^ O' O' b u i u i O Ul p 03 ^ • - u i as ro a 3 O' ro b f r o3 O O P O O n m m fn m rni i i h i i 3 O O b O O fr fr p P fr 0 O' f r fr l\) Ul Ul Si \0 01 n m m Ii IO O f r fr fr fr f r !fr f r -fr fr Ul O' H 0> D 03 f r b O' *- j O' s tn o *-n m m rn m rnI l l I I I 3 o o b o o fr f r 4> Ifr f r 4S. n u ui ■» o o »l Ul >J >1 f r \) bl Si i l III m I I I 3 0 0 t f r OJ D Ul iO M s o o a> 0 f r Uln m m l I o oy O' O' ■1 L • • L 0 o ^ Kj ? to *- m 33 CD vO b M n i in in 1 1 I !l 3 0 0 0 Ul 0 »- fn m rn 1 io c> C r ' «■' O p o c s f r u i In u oi ft lO O b O O sj vo »- Ul o, S T' cr o O' cn o.): *“» t— CD 03 OJ n m m fn m rnI I i l I i •> o o o o o 0 'O U o o o ■J vo *- UJ U’ >1>i cn -j b -o fr ui ui ro ui >jin in -*n in in I I !l I l O' O' ro ro to , <0 xft iD r> ro fo Ui ►- Ul iC n m m m l I » • • b a • fr fr fr -fr Ul UlJ> 'O \C U!> o o ji o fr b ►- fr o -fr u i Ui a >o n m m rn rn m i i 1 4 1 J3 o o b o o t P p r> p v. Ul U> UJ O vC in fu I I o o m m i Ji 01 Ul b ' UI UI • * « • d b *— fo ro w n ro L vr 03 *- ro t i fr u> in m fn in in l I ;l f l o o o o '> ^ i - V- * - in (n jn i P (3 H |lj Ul (.1 N H h O (0 j) » >0 C> >- w n in in rn m in I l I l l o C3 Ul O s 0 Ul M bt3 f r O - f r 03 fO UI *0 0 cd f r b O' ro3 O O P O On m m fn m m 1 I l i l I I3 C O O O O fr fr fr Ifr fr fr O' a « • Ul Ul > O' O' n ui ui 3 0 ^ si a> o to uj cn O' to p vO O' n m rn rn m m i I }l i I 3 0 0 0 0 0fr fr fr fr fr fr n ui ui i» cn ui > • • • • * » M H M M fr tn O' Ki cn *o J1 s i O' O' f r f rfr 01 & O) Uj n in m Hi h i in l I f ;l I I 3 O O P O O H M H M H h I ; r r r r r o ro ro |,j ro ro 0 vo vc o o 03 >i s i a> o -g oj 3 -J s i U O Ol n m m fn m m 1 i 1 ,1 i I 3 0 0 0 0 ° ,n cn oi in O’ ui y, u u Ui O' v i—• I-* * - »- c. o ki a- *•* S C ti, fr fr J O' - J (L vo fr ii (ii m fn m n: i I i l i3 O b «' O Ul 01 01 Ul « • • • F Ul O' si CD iC si O' O' f r f r UlJ> >J OJ vO *■» •- n rn rn fn in nr ' I I !l I I 3 'i O O vj n tn in in Oi in Ol O' M t t >C c -si O' Ol f r f r .vi f r 01 O' co to n in i'i f i in in I i i i i 3 • ■> P ■* >.> j* cn cn • • fo ro o Oj fr - ro ui 3 oj cn u in in l I s O O > lo (O OJ SJ CD n m m m m m i i jl i ID O p O O n cn cn in oi O' ji fo ■ n m i I i n rn rn i i i < - '< 3 0 o, jn to to X' 0- n rn ‘ f ^ o n in o N) 3 01 UJ fr P i fn I >* 2E Ol 01 ro ro uj fr ro O' ni ro rn m i i c - o cn tn i • ^ ■ ro ro i •- fo I si CD i o ro i n m i I 3 O ( n oi cn • • H o ro U) X > co o Pi ji - o H 03 O >in in > I l ! 3 0 0 ) Jl 01 Ol - ro — :03 CD ; fr Ul Ii m m I I I '3 0 0: cr n u- u . 1 • • bo ro Oj i t j o fn *i ^ v, H a o *> n m rn f* I I c o ! U M OJ CD O *- O I i in cn i • o u > ro Oj tn' cd o ni J *-* O *H1 a o > i in rn fo I I ;> o o • tT i.m nF M il H 1Q J_S X U R £ n r = f im » iT » a m .) M u m i t t . u i f EJC S 1 N T a liH J t CO NTEN T AS A FU N C TIO N OF LA M B D A . (BR U O K STO N C LA V ) University of Ghana http://ugspace.ug.edu.gh 227 FIGURE 5.17. (a) Simulated chloride content as a function of X . Interpolated experimental data points are provided for comparison (Akuse clay). (b) Simulated chloride content as a function of X (Brookston clay). University of Ghana http://ugspace.ug.edu.gh CO NC . OF Cl IN SO LU TI ON (k eq ./ m 3) t.0 * » y ° s - ° - * -°* -°«-OA - ° * -°* • • o - * t o o . . 04Ck^ ^ a ° □ ■ " o o • EXPERIMENTAL a a O ANALYT ICAL A A D A SIMULATED DATA WITH VARIABLE S D s a b A SIMULATED DATA WITH 0 D S -C O N ST .’ 7.2997 x 10*5 & D i.e. D s = 1.4092 x 1 0 ‘4 A ° ■ D SIMULATED DATA WITH 6 D s - CONST .- 7.8787 x 1 0 ‘7 * a ° i.e. D s - 1.5851 x 1 0 -® *, o □ SIMULATED DATA WITH 8 D S = C O N ST ." 2.4862 x 10*8 * * d a i.e. D s - 5.8288 x 1 0 '8 ■ * eg _L J__________________ l__________________"l ■ h , a °. 1.0 2.0 3.0 4.0 10 A (m s ) AKUSE J Figure 5.17a 228 University of Ghana http://ugspace.ug.edu.gh CO NC . OF Cl IN SO LU TI ON (k eq ./ m 3) 1.0 aoa AO# a O n ■ °*0»,□ ■ OAcuo BROOKSTON□ A ° ■ O q • EXPERIMENTAL ■ ° » o ANALYT ICAL□ a □ ^ A □ o ^ a SIMULATED DATA WITH 0 D s - CONST = 2.9459 * 1 0 '5 i.e. 5.5871 * 1 0 '5 A D ” □ SIMULATED DATA WITH 6 D s = CONST = 1.2408 x 10‘7 i.e. 2.5302 x 1 0 '7 SIMULATED DATA WITH 0 D s = CONST.’ 6.723B x 10 ‘9 A □ ■ o A a ° ■ 4 1-6031 x 10-8 A □ 0.4 0.8 1.2 1.6 2.0 103A (m s '14) Figure 5.17b 229 University of Ghana http://ugspace.ug.edu.gh 230 TABLE 5.6a. Computer output showing experimental c(A), c(A) from analytical solution (equation 3.21) and c(A) simulated non-analytically using the algorithm in Appendix D (Akuse clay). University of Ghana http://ugspace.ug.edu.gh 4* * * 45. -f> O CD 0 CDm m 1 i o o L«J oj N U UJ ^ ^ • ■ • • • O O U N o - O O' VO -p- - O o t j i a c o u 0 4) Ol N ‘N c 1 rn i m m m i l I I I i o o o o o o 01 ro •- 1 s ro 'C ■ -'J O' Cl • 0> 4> ro ■ m m m i i i i o o o 1U OJ OJ ( to «-* O'jm m I o o U Oj U OJ o ^ ro ro o 43 — O' OJ — ro ro o s43 Q} O O \0in m ip in rn l I l I I o o o o o N 0* O' O' O' o ai m m ro m • • • • • • o a ro oj ai oj01 >0 O' f M 0 N o U ro Oi ud -sm ro m ro ro 1 i i i Io o o o a vc ro ro co ro O 0J 0i Oj O' CP © a c c. ~ p H ^ OJ H Cl & n m n I i i o o o ro ro a CDNO -O IS -O J fO— — ^ oioi-o'cro'^ rovcoo O'OmO'icn^cjWO'D 4> ro o ro o N o cd O' m m m m m m m m m m i l i I t i I 1 i I o o o o p o o o o o U U O IU U O J O J U U O J ^ 4* ^ -P £• £•i • • • • 0J £• d O' O' -«4 0J O’ N O >J W \j ro O' 0 M -P (T ^ I I I I I Io o o o o o O' O' O' o' ui cn e> ro ^ -4 ro - vc ro oi o £ •'J >0 *- O &> 4> f> Oj -4m ro m m m mi i i i i i 0 *ro. •* -s fo |\}U Vi N < rn m rn i1 i I O O p < ro ro b ■ a - n> oj oi *• • • • H* s O' & *»• o - a o QC ■- \Q rn m m i i I o o o ■s >i >» 4> £. ^ a ro ro >o vO O' o -p -g ** P ro o oj ro O' 'j «- O' ro O' p ro rn m rn ro m mI I I I I l o o o o o o ►- ru ro ~ ro ro o*. m m ro ro ro £ O l\J Cl -P W■>j ro oj cn ro oi o *- ->i ro fo rn in in nT hi mi I l l I l l 0 o o o o o 01 cn oi ui oi cn ru w * x» u (ji* • • • • <- S P U O' OJ vO t\> ro ro oS « >J -f> O1 sM « P* O 10 m m m m m mi i i i i i O O O 3 0 0 ->4 -4 N S 01 0! O © -J ►- >- £ Oi O Cw ro n 1 -Nj >4 O' ro r\> r\> to ro ro• • • « • • >o ro s o> oi -c* ro oi ci O'■>J O' OJ fo ro © ro o> p m rn m m m rn I I i I I i 0 o o o o o01 OJ OJ 01 CJ 01 P P P P 01 Cl• • • • • • i£ £ \C tQ o o oi oi «si a *- oi h* >1 o ro o> ^ »- a p •- m m ro m rn ml i i I I I o o o o o o iv ro c . o i ro ro 0 o o O' s roUl M O 0"0 01- O fO Cl -4 — J' op 43 Clm m iii m in rn 1 l I i I I o o o o o o ■p cn o i o i cn cn oj >- >4 0s Cn m i I i I I ro ro ro r\> ro ~ Oi ro i - n ro ro »- o a ro o cc rn rn m I i i o o o o j o i o j ro cd ro ro oi• • • • . oi r\) o »- a ■p i\j O' o ro oi -p ro o 430 ro -<4 ci O' i m m rn rn ro. i 1 i i I iO O O O Q I ■*J N 'J S O' i ►- *- - *- O' O' • • • • • O' O' O' O' »- ►-a cn ro ro oi ro oi ro ro oi o q*4 I© ►- s I- ^ m m m rn m mI i I i I lo o o o o o O' O' O' O' O' O' ( i I i i t oi ci oi ci ci cn o o o o o oP 01 O N 03 O P — £■ 01 -4 ro ro O' *-* 'O o m m m m m m t i l l ) o o o o o o ro •- O' O' o» n• • • • • _ 01 «- d 'O f\) l> o o o « c ■> O' ro ro O' u O' ro ro ro cn *«ju«l ui ni .in ni in I I ( I I I 0 O O O O O ^ ^ o» ;o» cn oi i IJ ro n, o Oi O' oj ro ro oi ro o> O' ro O' ci 01 oi o “ CD h 01 -p- 01 P O'm m m m m m i i i i l _ O O O O o •*> O' ->i d cn oi Oi OJ O IO 01 Ci sO O *Q ro N <6 CD 'O O U rn rn m rn i I l l o o o c O' P P P N O m m t I 0 o >0 vc o — N O' 01 O' p ro m m m 1 I r 0 o o 01 01 01 r o s e 01 O; ro Oi p cn c O' ■o ro *- o *o ro 0 ro o' » io o rn m m m m rn1 I i I I i a o o o o o Oj Oj Oj OJ Oj 01 01 01 CT 01 0) 01• • • • • • 0 * - * - » - t- M O' ro ro p o rn m ro m m m1 I i i i i o o o o o o • • • • • • ■ o o o o o oS> £> £> Oi ro o oj •- ro 0- vQ * ro * oi om m mi in mi mi i l l i l i o o c o o o Cl P p P ci **j o o O'• « • • • • ro ^ ro * « cd p ro \o oj c “ o oj o ^ - O' oi ro oi oi m m rn m ro ro I I i I i 0 o o o o o 01 oi 01 d 01 01 ro ro — oi ^ s 10 O' «o OJ 01 o 0 o. O' P «C 01 ^ o< *- o m ro m rn m m I I I I l I 0 o o o o op p p p 1 I I I I I Qs Ol ro Ol — sO 01 ro p — fo p OJ OJ OJ s> *- - i- ro 'O *- «- — *- — - - N. ►- ro *- ro IO ro to ro ro tv ro ro • • • • • • • • • • • • • • • « • > 01 ■o OJ Ol Ol ro ro O' tn ro ** -4 >4 (n o nv ro ro 4> Cl -4 p o O' ro ro Ol 01 ro j> O' Cl f t -4 ■si o >— 43 o *■ O' Ol O' u ro >0 p O Ol Ol p Cl p "4 ro i— O' to ro s ■Si Oj OJ Cl -4 cn ro o 43 U o p Cl 01 43 O' ro Oi IO Oj ro ro o 01 ro ro *- p »—Oj p 4> ro 01 O' o p vO o *-* (V O cn 01 ro vO 01 ro fo ►— ro O' cn ffi ro -»> o O' -0 IS­O' u <0 01 cn N Ol OJ ro 0! vO O' o 01 o Oi >0 p 'O >0 <3 -4 O' ro 0) P O' ro O ro ro ro ■{> ro p 01 ro ro O' S Oj ■p C' 4> ro N oi o ro 0* ■'* rn m rn rn m 1 m rn rn m m ro ro m m m m ro ro m m m ro ro rn ro ro m ro ro ro ro rn m ro m m ro ro rn ro ro ro m ro m m ro ro m => o o o o o o o o o o o o o o o o o p o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o op p p p Oj OJ 01 ro ro ro ro ro ro ro ro ro ro ro ro fO ro o ro ro o *- *■* ►-* N“ *“ ** *— *“ *— ** ** *“ *■* ■“ ** c o O' ' j Oj p cn O' O' -*j "4 ■si ro CD ro ro ro ro ro ro vO in vO «o \£) vO o & r> \C 43 43 43 & 43 • • • • • • • • • • • • • • • • • • • • • * • • a • • • • • » ■ • « cn b o ro "4 o O' -4 O' ro ■s •-* P s o no 01 01 O' >4 ro o o O o o ►- i— ** *- — ►—►* r- ►—►- *- >— pm H* ►- ro rno o p ro ro Ol o N >— ro ro ro ro OJ ►- o ro ro -4 O' O' OJ p O' *» -4 ~4 ro ro ro ro ro ro ro CD ro >0 <0 43 O — ro 01 p 01 O' -«J ro >0 o o — ro 01 p cn O' ■*4 ro 43 o ro o Oj O' IV p -4 01 'C o 01 O' o O' ro o p O' £■ 01 s *4 O >c ro ro ■*4 O' 01 p OJ ro o <0 ro -4 O' Cl ■o OJ 01 ro o >■n m m ro rn m T] ro ro m rn m m ro rn ro ro ro ro1 ro m m ro m1 m ro m rn rn1 ro mI m m ro ro rn ro ro ro ro ro rn m ro ro ro m ro ro ro m o o o o o o o o o o o o o o o o o o o o o o o o o o o O o o o o o o o o o o o o o o o o o o o o o o o Ol p OJ ro ro •“ *■* *■“ *“ *■* *“ ** *“ ** N* *“ ** ►- *■* H- *- "■* •"* ** *"* *“ ’- N- *”* ■“ *“ *- ** ** •“ *“ O' O' o> -4 Oj p 01 O' O' N ro ro ro ro ro ro ro ro \0 >0 \0 vO •0 <43 >0 vO >0 \0 »0 'O 'O 43 'O vC >0 43 ■0 >0 43 43 43 • • • • • • • * • • • • • • • • • • • • • • • • • • • • • • • • • 0)ro o ro ■N - i O' "4 O' ro >1 p s o ro OJ 01 O' ro •c o o o o ►— i— H* ►- ►—«— >—►- >-* *- ►- t—«-* »-* •— ►- •— pm *- ro m O' -4 p a 01 01 o N ►—ro ro hi "j a 01 sO a ro N O' O' •-* OJ p O' •— -4 >1 ro a ro ro CD ro a ro ro a ro 43 vO o >0 >0 43 <0 >0 43 o mp o O' ro p- ro -p O s cn ■— CD cn cn ro ■p* O' OJ a i— OJ o o CJ s to ro ro 0^ o ►- ro Ol 01 01 O' ■>i ro O »— ro OJ ■P- 01 O' >1 ro O o •—p o p o. u ro N a Ol ro ro O' 01 O' O' O' O' ro o -j s O' nO 01 O' p OJ OJ N ro o O 'O ro ro ro "•4 O' O' d o 4> 01 01 ro ro o 1? m ro rn m rn m m (T| ro m rn ro ro ro ro rn T ro m rn ro rn m ro rn rn ro m m ro rn ro ro ro ro ro m m ro rn m m ro ro ro ro ro ro m ro o o i o o o o o o O o o o o o o o o O o o o o lo o o o o o o o o o o o o o o o lo o o o o o o o o o o o o 01 p OJ ro to M. m *“ M *- ■“* *“ ** *— »- N* H- ’■* * ' *“ **• *■* *"* *"■ *■“ H* Mi: ** ** *“ h- *" *■* — M Cl O' O' 1 «-* Oj p 01 O' O' s N ro ro ro ro a ro ro ro \0 vO <0 <0 <0 *0 43 VO <0 *0 'O <0 <0 >0 >0 'O >0 vO vO 'O 4) *0 o to -j 0> O' fo >i p ■^1 o ro OJ ui O' ro »o O o o o * * *4 •M — — *- N- ** pm — to ro p ro OJ ro p 01 o •4 fo ro ro -4 a u ro a O' o> OJ p O' ■*4 -4 ro ro a ro ro ro a ro ro ro ro »o <0 O >0 O 43 *3 \D O mp Ol ro O' 01 O >1 cn ►- -P- 01 ro p O' 01 IO ro o O 01 s ro ro a >0 o »- ro 01 <>■ -& 01 O' N ro <0 o 1— rv Ol 4> cn O' CD >43 O ro o O' 1— o a Oi w* p t— ro 01 o p 01 p p O' ro cn O' cn ro ro 01 OJ ro ro »— p—o o >0 CO ro -4 O' O' 01 01 P Ol 01 ro pm >—O J1 m m m m m rn m m rn m mi m m m m1 ml ro ro ro m m m m ro m rn m ro rn m m mt m ro m1 m mI ro rol ro1 mI ro m ro m rn m rn m m o o o o o o a o o o o o o o o o o O o o o o O o o o o o o o o o o o o o o o o o o o o o o o o o o o O c i p 01 ro ro *- ** *»• ** |— *- •• *—*■* *-* ** *- *■ *■* *“■ •“* ** ** ►- ** *** *• •“ ** ** *“ *“ T £Z - m « ro IO - o o H to n tr «c o m o O' cn p u o ro ro O' -p- ro o oro m ro -ro m m l i I l i i o o o o o o Oj oi Oj Oi P p P Oi oi O' -4 ro oi *---g cn ro o O' « o v- oi ro ro rn m m ro ro i i i - i i i o o o o o o o o o o £• \D vO P O) ** *- 01 O P ojP N j - P Oj roMl Ml m Ml Ml III 1 1 1 1 ( 1 0 o o o o op p J> p p rn fi (ji O' in ft p ro ro ro O' O' oj oi o ro O' - O' « IO o ci a -j »o *- ro rn m m m m ro1 I I I i l 0 a o o o o 01 Ol 01 cn Cl 01 ro ro u ro ~ oj oi ci ro ^ ro >- N Ci - o 'J o rn rn m I I i 3 0 0 > P P I I I I I 1 ro oi c- J m m rn i i I I o o O I 4S- * 4> ■ s O' o» p oi ro • • • • • * ro oj ■*> oi o- 'O ro n O' 4> u O' *> ro o ro O' 0 o o o o o ro m rn rn ro rn1 I I l i l O o o o o o& & o P P 01 01 01 01 01 01 *- ro ro ro ro cd vo o •- ro oi ro o cn p oj ro ro p oj ■<> oi O' rn rn rn rn ro rn l I I I I i o o o O o o ■c 'O ro a ro ro CD P 01 N O' *4p - o ro >4 ro 0 ro to u ii mi mi rn rn 1 I l i l 0 o o o o o01 oi O’ 01 01 01 0 in oi f t in ni • • • • • « B S 01 P M S•- ro o ~ s ro-f> O' O' o — n — ci o fo m rn rn m ro rn 1 I i l i I o o o o o o cn 01 01 01 01 d t- *- Ol Oj oj O' ro ro rn ro . 1 I o o c p p p M - fo 0 . 0 -G0 ^ oi N O >0ro m rr 1 i io c o p p p I I I I I I >0 o• • i a *- o . ro ro a o 0 o m m1 i o o 01 d 01• • • H ro ro ro x* cr o r o o > vC Orn m m I I l o o o ro ro ro . • . c a a ■'i ro ro 01 N P ro u aMl III Ml l I i 0 o o cn oi oi i f t O' **J• • • -i 01 >1 «- Xvo a- ro o*j a h w ro a a m m ro l i i 0 o o01 d 01 . • a c. i N P ' ro m I I O o p * D IS P E R S IO N - A N A LY TIC A L AND N O N -A N A LY T IC A L S C LU T IO N S (A K U S E ) S IM P University of Ghana http://ugspace.ug.edu.gh 232 TABLE 5.6b. Computer output showing experimental c(A), c(A) from analytical solution (equation 3.21) and computer solution for c(A) using the algorithm in Appendix D (Brookston clay). University of Ghana http://ugspace.ug.edu.gh D IS P E R S IO N - A N A LY TI C A L AN D N O N -A N A LY T IC A L S C LU T IO N S (U R O U KS TO N ) SI M P IN T E G R A T IO N •M m* _ o o o O o o o o o o o o o o 01 LU UJ LU ; UJ UJ UJ UJ UJ UJ UJ UJ UJ LU UJ UJ o © OJI CO in N ro O' in •M s ro O' in OJ o O' O' CO co CD N N o vO <0 in in <* LU o O' 0< O' O' O' 0> 01 O' O' O' O' o O' O' UJ N £ © © vo vO © vO 'O 43 vO vo >0 >o in • • • O' 0> O' O' O' O' O' O' O' O' O' O' O' O' O' -< _ o c* o o e o c o c* o o c o o 01 UJ 1 UJ ujIuj 1 UJ UJ 1 UJ UJ UJ UJ UJ LU UJ UJ 1 LU o © OJ CO in n rO O' in N <■0 O' m o O' O' CO co CO N N vO vO vO in in 0 © © © vO vO vO vO \D >0 o © vO rO O' © o O' O' ro co CD N N N o 0 o o © vO o o <3 vO vO o O0 n O' © in n O' OJ in N I o < • • • • 04 i CN! 1 Ol 1 OJ 1 OJi OJ 1 OJ 1 OJ 1 OJ 1 OJ i OJ 1 T 1 1 i m in m m w in in in in in in m in s> LO o o o o o o o o o o o o o o O © j j UJ UJ UJ © UJ UJ UJ UJ UJ u UJ UJ JJ © O' ro fr o © fr ** —< © o w* ro <}• vO <*■ S) © o in 'O o O' o fr © co ro 10 OJ OJ co m <* N N s. •f)O n N tn © ■0 'Si O' ro ro rO 0» <0 o • • m fr fr fr fr ro <* ro LO m in in in in in in in in m m in inO o C-' o © o o o o o o o o c UJ UJ UJ UJ UJ UJ UJ UJ UJ UJ UJ UJ UJ UJ LU CO CO CO O' O' m CO OJ ~D O' N m OJ ro (Sin N CO O' o vC OJ ro O' N OJ N O' a n N r-< N © in co ■tf CC in ■0 'C CO x © 10 U) TO ro Ol ro o ro OJ —< OJ OJ c ro H • • • CVJ 04 IVJ OJ CM OJ O! 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The moisture content (THLA), moisture diffusivity (D), 0D (THDS), D (DS) and g/0D (AH) n S S S and g = 0A - d.0 Ad9 = 9A + 2D -jy ) are also presented. wn As observed in figures 5.17a, 5.17b, Tables 5.6a and 5.6b, agreement between the analytical solution, c(A) data simulated non- analytically with variable Dg (^) and experimental c(A) data is very good. Large deviations occur, however, when c(A) data is simulated using con- _ s 2 —1 stant values of 0D . For constant low value of 0D =2.486x10 m s s s _8 2 —1 (i.e. D =5.829x10 m s ) the agreement with experimental c(A) data -3 -1/2 for Akuse clay is good up to A=3.0xl0 ms , after which point sub­ stantial deviations occur. For Brookston clay the low 0D value ofJ s -9 2 - 1 -8 2 -1 6.72x10 m s (i.e. D =1.603x10 m s ) shows good agreement with s -3 -1/2 smoothed experimental data up to A=1.0xl0 ms , after which devia­ tions occur. It appears, therefore, that even though simulation using constant low value of 0D^ predicts c(A) data similar to experimental c(A) from A=0 until A close to the wetting front, in general constant value of 0D^ as was suggested by Smiles and Philip (1978) does not predict the c(A) relationship very well for the soils considered in this study. Figures 5.18a and 5.18b present the simulated c(A) data for K+ obtained from Dg (A) relationship derived from smoothed experimental 0(A) and c(A) curves (figures 5.5a, 5.5c, 5.6a and 5.6c). Computer output for the data presented graphically in figures 5.18a and 5.18b are given in Tables 5.7a and 5.7b. Also printed in Tables 5.7a and 5.7b are the soil water diffusivity D, water content (THLA), (f- ^ 4r) designated I dA 2 dA University of Ghana http://ugspace.ug.edu.gh 235 FIGURE 5.18. (a) Simulated K+ content as a function of X. Interpolated experimental data points are provided for comparison (Akuse clay). (b) Simulated K content as a function of X (Brookston clay). University of Ghana http://ugspace.ug.edu.gh CO NC . OF K+ IN SO LU TI ON (k eq ./ m 3) 1.5 i a u d i d i d i 1.0 0.5 AKUSE a ■ □ ■ INTERPOLATED DATA □ S IMULATED DATA P * 1.0 2.0 3.0 4.0 5.0 103 X (m s 'H ) Figure 5.18a 236 University of Ghana http://ugspace.ug.edu.gh CO NC . OF K* IN SO LU TI ON (k eq ,/ m 3) to o a a d cm a a □ B D BROOKSTON 1.0 0.5 cq 1 a “Q ■q a □ ■ INTERPOLATED DATA □ S IMULATED DATA B "□ J------- 1-------1_______I_______I__ - I n I_______I I_______| 0.4 0.8 1.2 1.6 2.0 103 A(ms ) Figure 5.18b 237 University of Ghana http://ugspace.ug.edu.gh 238 TABLE 5.7a. Computer output for simulation of concentration of K+ in solution using calculated values for Akuse clay. University of Ghana http://ugspace.ug.edu.gh 6£Z PO TASSIUM AND SIM U LATIO N OF CCNCEN TRATIO N (AKUSE C LAY) RKS IN TE G R A TIO N University of Ghana http://ugspace.ug.edu.gh 240 TABLE 5.7b. Computer output for simulation of concentration of K+ in solution using calculated Dg values for Brookston clay. University of Ghana http://ugspace.ug.edu.gh i i - i^^ oz i ’o r.o dc cc e*» «'«0 cc -a to oe c r * o- = ~ i3 A* jn ‘ n dc ns s4 na no 7 7 7 7* 7 7 “ 7 7 7 T 7 7 7 7 7 7 7 7 7 O .if •o O' CD GD S4 sJ O' O' n\ tn o'. 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O' CJ Ol 01 CJ 'O to O' CD Ol ro ty o o 00 to O' o ft CD to O' o ft 01 ft o o to to o o o Cj 05 i to ro 01 CJ o Cl •£ ft -J Oi o o o o o o o O o o ft •o CD o 05 o io o OJ >1 o i to o >0 o S ft >0 -0 *n Ti rn rn ,n m1 T m |T1 •TI rn in ai TI m rn rn in 1 rn m rn m m mirn rr m 01 m mi m 7 in in m 01 m in in o o r> o o o o 3 O o o o l ■> o O o o o o o o o D O 'O o o o o o o o n o o o o o o •o ft * ft to OJ' 01 Ol to 01 ro CJ. < J w OJ u rj i w to to to " ■“ * ** fc' “ *" *■* *■* OJ J\ CD _ >_ _ rc so OJ OJ 00 ft ■p- ft tn Ol O' Oi w o» O' _ toOl O' ■0 CD a> CD | (X) <0 •O >0 >o >0 o vO o 9 » • ■x< O' CD to tn •n OJ -»i o ffi tn o ro O' o 01 o cn O IV) O -J O' •— 0' o (rj o -J *• •0 r.i OD to to CD m ; ft to ; X> o 01 1 S CJ tn O' (0 vO , ft O' CD OJ CD ft >o 01 ■0- to 01 0- N ~4 >1 a- O' Ol •o tn O' to ft ft ft ' CD ft to ■vJ O'' - s >o :to CD CD s O' ft ►— 33 ©• cr O' vf> o Ol ■c to ro ft Ol O >0 ro '0 to Cl ft 01 ro i tn Cl s •CD 01 ns to s ro7 n ■n T m -n rn T7 m rn m 17 n H m rn m ■ m m m m m m ;m m 111 rn m imm m m nn m m ;in m m o -J oft o*• oOJ Z o . Oto oOJ oOj oOJ ■ 03 01 o(0 OOj o -03 o (*j o w o to o ro a to o -0 1 -0 1 o o i o -0 1 -0 1 o o o o o o o o — t o f t " C - C D * - i - t - * - « * - » - t oa r *-<005,0- i f l O ^ U l P l T L i l O M 0 - t - in in rn m in mi ’n n in in rn m l I I I I l I l I l l I o o o o o o o o o o a o C- ‘ M l\) f ) M M l\) tO W CJ I) O o w l \ ) U P ( J '® o r J O ui W >- M O' (jl - O >• ( B a i u t - g w c f f l oI'fnrnmmmmmntn I I 1 l I I I I Ilooooooaoa ioc *-^ -s:totototototoitooiOJ : to it O' s a «io •* is > J OJ OJ to Cj < (D O S <0 .D <0 o >0 N O CD CD tfl lil m (j 'ifl m /n & n -j rn M ro I I I I I I I I I I I > 00000000-000 ■ «, a <» and iIj „ respectively. 2 w s (d) Reduced water content 0 and reduced concentration C„ as functions of a) and oj , respectively. 2 w s (Akuse clay) University of Ghana http://ugspace.ug.edu.gh F igure 5.19 243 University of Ghana http://ugspace.ug.edu.gh 244 LATHAD, 0D (THDS), D , interpolated experimental concentration of K (SEELA) and simulated concentration of K+ (SEEl). The interpolated experimental data obtained with the NLFGEN (non linear function genera­ tor) algorithm of CSMP which interpolates between data points using Lagrange quadratic interpolation is plotted in figures 5.18a and 5.18b for comparison. It is observed from Tables 5.7a, 5.7b and figure” 5.18a and 5.18b that simulated and experimental values of c(A) for K+ agree very well. Finally, we conclude that given D^(A) or 0^(0) and D(A) or D(9), the computer programs described in Chapter 4 simulated 0(A), c(A) for both Cl and K+ which agree very well with experimentally determined data. Also excellent agreement is obtained between simulated c(A) for Cl and c(A) calculated using a computer program for the analytical solution. 5.6 Simulated Water Content Profile and Salt Content Profiles for Vertical Infiltration The simulated water content profile with depth and chloride content profile with depth for various time periods was based on the first four terms of the expansion z(0) = A (0)t1/2 + xw (6)t + ipw (0)t3/2 + tow (0)t2 ; and z(c) = A(c)t1/2 + x (c)t + ijj ( c ) ^ 2 + co (c)t2 . s s s Figures 5.19a, 5.19b, 5.19c and 5.19d present the 0 (A), C2 (A), University of Ghana http://ugspace.ug.edu.gh 245 G(v ) , C.(y ), © (Uj ), 0(ip ) and C.(oj ) data for Akuse clay. In these w Z S W W Z S plots, 0 is the reduced water content 0, defined as 0 = (0-0 )/(0 -0 ) n o n and the reduced concentration CL is defined similarly as C„ = (c-c )/(c -c ).I 2 n o n Similar graphical representations for Brookston clay are presented in figures 5.20a, 5.20b, 5.20c and 5.20d. The 0(A) and plots are the smoothed experimental data interpolated with the CSMP algorithm NLFGEN which provides a Lagrange quadratic interpolation between experi­ mental data points given in a FUNCTION card. As mentioned earlier, the wetting front is abrupt and sharp whilst the 'salt front' which lags behind the wetting front is dispersed. The xw » Xs > 4*w j i’s > ww and “g data (Tables 5.8a and 5.8b) were obtained with the computer program described in Chapter 4 and the experi­ mental data inputs 0(A), c(A) and hydraulic conductivity as a function of water content K(0) (figures 5.5a, 5.5b, 5.6a, 5.6b, 5.14a and 5.14b). Figures 5.19b and 5.20b indicate that for 0 or C ^ , the Xw values are greater than Xs with the former attaining a maximum near saturation (for example, maximum for Akuse clay is at 0 = 0.9) whilst Xg attains a maximum at low chloride concentrations corresponding with the drier region in the soil column (e.g. for Akuse clay maximum Xs is at C2=0.02). Similar trends are observed with 0 (lb ) and C~(ili ); 0(u) ) and C. (to ) w 2 Ts w 2 w relationships. In all cases, the maximum for \p and to for water occurs w w near higher water contents whilst the maximum for or tog for salt occurs at lower salt contents corresponding to low water contents. Table 5.9a and 5.9b show how well the boundary conditions listed in Tables 4.3 were met for Akuse clay and Brookston clay, respectively. Definitions for RVEL, QVEL, SVEL, RVELS, QVELS and SVELS in Table 5.9a University of Ghana http://ugspace.ug.edu.gh 246 and 5.9b are as already given in the text (see equations 4.130 to 4.135). In Tables 5.9a and 5.9b the soil water diffusivity D and dispersion coefficient DS are also printed. Note that the FINTIM values for RVEL, QVEL, SVEL, RVELS, QVELS and SVELS, even though not zero, are very small indeed. For example, RVEL values in Table 5.9a are generally of -5 -7 -3 -1/2 the order of magnitude 10 to 10 but that at A = 4.56 x 10 ms is 10”13. It is pertinent at this stage to compare the magnitudes of A, X, ip and to for water and salt (see Tables 5.8a and 5.8b). It is observed -3 that for Akuse clay A is of the order of magnitude 10 , the order of *“5 —8 magnitude of x is 10 whilst and to are of the magnitude 10 ' and 10 respectively. Similarly, the order of magnitude of X , Xj 4> and _ 3 —XI to for Brookston clay are 10 , 1 0 , 10 and 10 , respectively. This is an indication that, as expected, the dominant terms in the series expansion are A and As to whether the expansion may be trun­ cated to the first two terms to obtain a good approximation depends on 1/2 3/2 2 the magnitudes of the terms At , \t, tfjt and tot calculated with the computer program in figures 4.10 and 4.11. Representative data 1/2 3/2 2 showing the percent contribution of At , xc> tyt anc* <*>t to the depth of vertical infiltration and the vertical distance moved by chloride are presented in Tables 5.10a, 5.10b, 5.11a and 5.11b. The percent contribution of each of the terms in these tables was obtained 1/2 3/2 2 by dividing At , xt> t|Jt , or tot value for a particular value of 0 = (9-0 )/0 -9 ) or C, = (c-c )/(c -c ) by the depth z corresponding n o n 2 n o n to that 0 or C^ and multiplying by 100. Data in Tables 5.10a, 5.10b, 5.11a and 5.11b indicate that at the onset of and during the initial University of Ghana http://ugspace.ug.edu.gh 247 FIGURE 5.20. (a) Reduced water content 0 = (0-9 )/(0 -0 ) and n o n reduced concentration C„ = (C-C )/(C -C ) plottedl n o n as functions of A. (b) Reduced water content 0 and reduced concentration C2 plotted as functions of and Xs > respectively. (c) Reduced water content 0 and reduced concentration C„ as functions of lb and ip , respectively. 2 w s (d) Reduced water content 0 and reduced concentration C plotted against u> and to , respectively. z w s University of Ghana http://ugspace.ug.edu.gh l 0 3 X |m s ',/s) 106 X (m s ‘ ') F igure 5.20 University of Ghana http://ugspace.ug.edu.gh or C BROOKSTON w Qo - en 1-----1-----1_____i_____i 4 .0 6 0 8.0 10^uj(ms'^) 248 University of Ghana http://ugspace.ug.edu.gh 249 TABLE 5. a. Computer output for Y , y , ip , ip , co and co v s w s w £ for Akuse clay. 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Contribution of At , x t, if) t and u) t to the vertical distance moved by water (Brookston clay) Time 0 (e-0n) Depth z At1/2 % Contri­ bution to z V % Contri­ bution to z * t3/2 rw % Contri­ bution to z 2 % Contri­ bution to z<6o - V m 0) t w 360 0.05 0.10 0.50 0.80 0.98 3.579xl0~2 3.572x10 3.456x10 y 2.982x10 1 5.077x10 3.447xl0~2 3.441x10 2 3.32 xlO 2.816x10 4.324x10 96.32 96.34 96.08 94.43 85.16 1.259xl0~3 1.252x10 1.302x10 1.604x10 6.655x10 3.52 3.51 3.77 5.38 13.11 5.588x10” ^ 5.498x10 5.161x10 5.693x10 7.99 xlO 0.16 0.15 0.15 0.19 1.57 1.993x10“*? 1.952x10 1.750x10 1.140x10 7.841x10 0.006 0.006 0.005 0.004 0.150 7200 0.05 0.10 0.50 0.80 0.98 0.1852 0.1846 0.1799 0.1636 4.293x10 0.1542 0.1539 0.1485 0.1259 ? 1.934x10 83.27 83.35 82.57 77.00 45.04 2.519x10~2 2.505x10 2 2.604x10 2 3.207x10 1.331x10 13.60 13.57 14.48 19.61 31.00 4.998x10~3 4.917x10 * 4.616x10 5.092x10 -1 7.147x10 2.70 2.66 2.57 3.11 16.65 7.972x10“^ 7.808x10 7 7.001x10 4.561x10 3.137x10 0.43 0.42 0.39 0.28 7.31 19260 0.05 0.10 0.50 0.80 0.98 0.3471 0.3458 0.3377 0.3173 0.1209 0.2521 0.2517 0.2429 0.2060 3.162x10 72.64 72.79 71.91 64.92 26.15 6.738xl0~2 6.700x10 6.965x10 „ 8.579x10 3.560x10 19.41 19.38 20.62 27.04 29.44 2.187xl0_2 2.151x10 2 2.020x10 „ 2.228x10 2 3.127x10 6.30 6.22 5.98 7.02 25.85 5.704x10“3 5.587x10 ^ 5.010x10 ^ 3.264x10 2.244x10 1.64 1.62 1.48 1.03 18.56 21600 0.05 0.10 0.50 0.80 0.98 0.3757 0.3743 0.3656 0.3449 0.1388 0.2670 0.2666 0.2572 0.2181 „ 3.349x10 71.07 71.22 70.35 63.24 24.13 7.556xl0_2 7.514x10 2 7.811x10 2 9.621x10 2 3.993x10 20.11 20.08 21.37 27.90 28.77 2.597xl0_2 2.555x10 2 2.399x10 2 2.646x10 2 3.714x10 6.91 6.83 6.56 7.67 26.76 7.175xl0_3 7.028x10 't 6.301x10 4.105x10 2 2.823x10“ 1.91 1.88 1.72 1.19 20.34 260 University of Ghana http://ugspace.ug.edu.gh 1/2 3/2 2 TABLE 5.11b. Contribution of At , x t, ip t and a) t to the vertical distance moved by chloride (Brookston clay) S Time Depth % % % % c-c z At1/2 Contri­ Contri­- „ t 3/2 s Contri­‘ m t2 Contri­ s c. -c o n m bution to z '^ St bution to z bution to z CO s bution to z 360 0. 05 0.10 0.50 0.80 0.98 3.125xl0_2 3.013x10 2.805x10 p 2.642x10 ^ 1.959x10 3.028x10"^ 2.916x10 2 2.707x10 2 2.546x10 1.878x10 96.90 96.76 96.51 96.35 95.90 9.287x10~^ 9.375x10 9.396x10 7 9.229x10 7 7.665x10 2.97 3.11 3.35 3.49 3.91 3.783x10";? 3.825x10 3.893x10 3.916x10 3.604x10 0.12 0.13 0.14 0.15 0.18 1.240x10_£ 1.246x10 ? 1.255x10 1.270x10 1.290x10 0.004 0.004 0.005 0.005 0.007 7200 0.05 0.10 0.50 0.80 0.98 0.1579 0.1531 0.1439 0.1363 0.1031 0.1354 0.1304 0.1211 0.1138 8.399x10 85.78 85.19 84.17 83.52 81.50 1.857xl0~2 1.875x10 2 1.879x10 2 1.846x10 2 1.533x10 11.76 12.25 13.06 13.54 14.87 3.383x10"^ 3.421x10 3.482x10 3.502x10 * 3.224x10 J 2.14 2.24 2.42 2.57 3.13 4.962x10“^ 4.985x10 7 5.021x10 5.080x10 7 5.161x10 4 0.31 0.33 0.35 0.37 0.50 19260 0.05 0.10 0.50 0.80 0.98 0.2895 0.2820 0.2671 0.2545 0.1962 0.2215 0.2133 0.1980 0.1862 0.1374 76.50 75.64 74.13 73.15 70.03 4.968xl0_2 5.016x10 2 5.027x10 4.937x10 4.101x10 17.16 17.79 18.82 19.40 20.90 1.480xl0~2 1.497x10 2 1.523x10 2 1.532x10 2 1.410x10 5.11 5.31 5.70 6.02 7.19 3.551x10“ ^ 3.567x10 3.593x10 ^ 3.635x10 ^ 3.693x10 1.23 1.27 1.35 1.43 1.88 21600 0.05 0.10 0.50 0. 80 0.98 0.3123 0.3044 0.2887 0.2753 0.2129 0.2346 0.2258 0.2097 0.1972 0.1455 75.10 74.20 72.64 71.62 68.34 5.572xl0~2 5.625x10 5.638x10 2 5.537x10 2 4.599x10 17.84 18.48 19.53 20.11 21.61 1.758x1O-2 1.778x10 2 1.809x10 2 1.820x10 2 1.675x10 5.63 5.84 6.27 6.61 7.87 4.466x10“^ 4.487x10 4.519x10 4.572x10 4.645x10 1.43 1.47 1.57 1.66 2.18 University of Ghana http://ugspace.ug.edu.gh 262 In a preliminary test of the analysis in section 3.4 and the resultant ordinary differential equations presented in Table 3.1, com­ puted water content profiles and computed chloride content profiles are compared with experimentally determined water content and chloride content profiles for vertical infiltration for t=7200s for Akuse clay (figures 5.21a and 5.21b) and for t=19260s for Brookston clay (figures 5.22a and 5.22b). The slight difference in theoretical and experimentally determined curves may be attributed to the slight difference in initial water content of the soils used for the horizontal infiltration experi­ ment and those for the vertical infiltration experiments; and also to slight differences in bulk densities between soil columns used for horizontal infiltration experiments and those for vertical infiltration experiments. In spite of these problems, it is observed that there is a fairly good agreement between theoretical z(0) and z(c) profiles and the experimentally determined z(0) and z(c) data. The theoretical water content profiles and chloride content pro­ files for various time periods for Akuse clay and Brookston clay are presented in figures 5.23a, 5.23b, 5.24a and 5.24b, respectively. It is observed that the water content profiles (figures 5.23a and 5.24a) show a sharp, abrupt wetting front whereas the chloride content profiles (figures 5.23b and 5.24b) exhibit a dispersed 'salt front'. However, the effect of gravity tends to make 'salt front' in the vertical infiltration less dispersed than the horizontal infiltration (cf. figures 5.17a, 5.17b with figures 5.23b and 5.24b). Examination of the theoretical chloride content profiles (figures 5.23b and 5.24b) reveals that the chloride content profile has a sharp front at small time periods t=360s. The front then disperses at time periods greater than the initial period. University of Ghana http://ugspace.ug.edu.gh 263 FIGURE 5.21. (a) Theoretical and experimental water content profiles at t=7200s (Akuse clay). (b) Theoretical and experimental chloride content profiles at t=7200s (Akuse clay). The theoretical curves were obtained by employing the method of Philip (1957) and the method of Elrick et al. (1979). University of Ghana http://ugspace.ug.edu.gh EI -3 (5)=-g.~gn- w e0-en AKUSE I = 7200s □ THEORETICAL ■ EXPERIMENTAL 1 J) "bu J5- □ □ □ □ □ □ □ a 0 0 0 1 3 0 □ □ □ Figure 5.21 DE PT H (rn ) University of Ghana http://ugspace.ug.edu.gh t - 7200s o THEORETICAL ■ EXPERIMENTAL AKUSE University of Ghana http://ugspace.ug.edu.gh 265 FIGURE 5.22. (a) Theoretical and experimental water content profiles at t=19260s (Brookston clay). (b) Theoretical and experimental chloride content profiles at t=19260s (Brookston clay). University of Ghana http://ugspace.ug.edu.gh (h)= e--gn- ^ e0-eno .2 .4 .6 .8 1.0 0 o • o .101~ o BROOKSTON 1 = 19260s O • PREDICTED Q O EXPERIMENTAL q % E. x .20 ?0 .30 h o• o • s •o • • O Figure 5.22 O • &° tP 0^ 3 o 0 o University of Ghana http://ugspace.ug.edu.gh BROOKSTON I = 19260s • PREDICTED O EXPERIMENTAL University of Ghana http://ugspace.ug.edu.gh 267 FIGURE 5.23. (a) Theoretical moisture content profiles for various time periods (Akuse clay). (b) Theoretical chloride content profiles for various time periods (Akuse clay). University of Ghana http://ugspace.ug.edu.gh DE PT H (m ) 3D □ Q 0 □ O u CO “1--------- T----------1----------1--------3* A KU SE • 360s o 7200s * 14400s O 21600s >0000000000°° O o i i i r <) O O O AKU SE • 360s o 7200s * 14400s O 21600s Figure 5,23 University of Ghana http://ugspace.ug.edu.gh 269 FIGURE 5.24. (a) Theoretical water content profiles for various time periods (Brookston clay). (b) Theoretical chloride content profiles for various time periods (Brookston clay). University of Ghana http://ugspace.ug.edu.gh w eu - en .4 .6 ( 5 ) = J L lS q_ 1.0 T T .10 □ □ □ D D D .20 BROOKSTON • 360s □ 7200s i 19260s O 21600s .30 O ko ) 0 0 o o o o o o o o O o o .40 Figure 5.24 University of Ghana http://ugspace.ug.edu.gh BROOKSTON • 360s □ 7200s A 19260s O 21600s University of Ghana http://ugspace.ug.edu.gh CHAPTER 6 CONCLUSIONS In this study, simultaneous transport of water and solutes during one dimensional horizontal and vertical infiltration has been examined from the point of view of experimentation, mathematical analysis presented by Smiles et al. (1978), Smiles and Philip (1978) and Elrick et al. (1979), and simulation of the water and chloride content profiles using CSMP. Porous media, especially soils, are extremely complex and for a given solute species of interest in a given system, many physical and/or chemical processes can occur simultaneously, greatly complicating the analysis. Such processes include adsorption or ion exchange, vari­ ous chemical and physical reactions, precipitation, anion exclusion and dispersion. Of these, dispersion, anion exclusion, adsorption or ion exchange processes have been dealt with in this study. The one-dimensional horizontal infiltration experiments with KC1 solution conducted in this study have shown that the water, the chloride and the potassium concentration profiles preserved similarity in terms of the distance divided by square root of time. This observa­ tion, reported earlier by Scotter and Raats (1970) for studies with NaCl, has recently been confirmed by the studies of Smiles et al. (1978) and Smiles and Philip (1978). The experimental results and the foregoing analysis in this study also lead to the conclusion that the transport of KC1 during unsaturated flow in the soils studied, may be described by a velocity-independent dispersion coefficient. This conclusion is 271 University of Ghana http://ugspace.ug.edu.gh 272 consistent with the studies of Saffman (1959), Pfannkuch (1963), Smiles et al_. (1978), and Smiles and Philip (1978) and represents a very good simplification of the formulation, analysis and prediction of solute transport in such systems. It is observed in this study that the chloride concentration 'front' lags behind the infiltration front. This partial piston-like displacement of the initial water content of the soil column by the encroaching KC1 solution was also observed by Warrick et al. (1971), Kirda e_t al. (1973, 1974), Smiles et al. (1978) and Smiles and Philip (1978). The conclusion by Smiles and Philip (1978) that the whole initial water content is swept in its entirety by the encroaching KC1 solution, however, was not observed in this study. The observed effect of salt exclusion in Akuse soil whose clay fraction is almost entirely made up of smectite (Acquaye, 1973) is quite significant. Similar observations regarding anion exclusion have been made by Thomas and Swoboda (1970) and Warrick et al. (1971). The theoretical analysis presented in this study is adequate to obtain positive Dg values provided the concentration of all salts in solution has been measured. Further experimental studies into this aspect of solute movement is necessary. The soils studied show that equilibrium adsorption isotherms for these two soils describe the adsorption phenomenon fairly well as observed from the agreement of D values obtained for Cl and for K . s This is consistent with the observation by van Genutchen et al. (1974) that at low pore water velocities the equilibrium model for adsorption isotherm describes the adsorption phenomenon adequately. University of Ghana http://ugspace.ug.edu.gh 273 The computer programs presented in this study, especially the algorithm, are very useful and predict very well, the soil water content, concentration of Cl in solution and concentration of K+ in solution as functions of distance divided by the square root of time. In the case of Cl , the c(A) relationship computed with both the analytical solution and the non-analytical solution provided by the algorithm, agree very well with experimentally determined c(A) data. Provided with D (0) s and D(0) or D (A) and D(A), the water content and concentration of Cl s as a function of A(=x//t) can be simulated to a good degree of accuracy using the computer programs described in this study. The analysis developed by Smiles et_ a]^ . (1978) to describe hydrodynamic dispersion during one-dimensional infiltration of a solution into soil which was extended by Elrick et_ al. (1979) to describe one­ dimensional vertical infiltration, has been used in this study to show 1/2 that the power series solution in t for concentration, similar to that developed by Philip (1957) for water, is a powerful mathematical tool for describing both water and salt content profiles during vertical infiltration of water and salt into a soil with constant initial water and salt content. Theoretical data for both water and chloride content profiles agree fairly well with experimental data. This approach developed by Elrick _et_ aJL. (1979) presents a new way of describing simultaneous movement of water and solutes in soils under conditions of changing water contents and fits more closely the natural conditions encountered in the field. Theoretical water content profiles show a transition zone of uniform water content and an abrupt, sharp wetting front of low water content. This abrupt, sharp wetting front is also University of Ghana http://ugspace.ug.edu.gh 274 observed visually during the experiments and has been reported earlier by other workers like Bodman and Colman (1944), Youngs (1957) and Childs (1964). The theoretical chloride content profiles, however, show a dispersed 'salt front'. The experimentally determined X and the calculated X j ^ aELC^ w _3 _ g —1 0 were of the order of magnitude 10 , 1 0 , 10 and 10 , respectively, 3 _^ — § ” * 1 1 for Akuse clay and 10 , 1 0 , 10 and 10 , respectively, for Brookston clay. This is an indication that the dominant terms in the series expansion are X and x- Derived data for the percent contribution 1/2 3/2 2 of each of the terms Xt > Xc> 4*t and cot to the distance moved vertically downward by both water and chloride leads to the conclusion that at the onset and during the early stages of infiltration the 1/2 dominant terms are Xt and Xt which together contribute between 95 to 99% of the depth at which water or salt moved. At longer duration of 1/2 infiltration, the Xt and xt terms, even though still dominant at the wetting zone, diminish slightly and contribute on the average about 80 to 92% to the depth at which water or salt moved at the wetting zone. University of Ghana http://ugspace.ug.edu.gh LITERATURE CITED !■ Acquaye, D. K. 1973. Factors determining the potassium supplying power of soils in Ghana. Proceedings of the 10th Colloquium of the International Potash Institute. Abidjan/Ivory Coast. 51-58. 2. Aris, R. 1956. On the dispersion of a solute in a fluid flowing through a tube. Proc. Roy. Soc. Lond. A235:67-77. 3. Bachmat, Y. 1969. Hydrodynamic dispersion in a saturated homo­ geneous porous medium at low Peclet numbers and nonhomogeneous solution. Water Resour. Res. 5:139-143. 4. Banks, R. B. and I, Ali. 1964. Dispersion and adsorption in porous media flow. J. Hydraulics Div. A.S.C.E. 90 (HY5) 13-31. 5. Bear, J. 1961. Some experiments in dispersion. J. Geophys. Res. 66:2455-2467. 6. Bear, J., D. Zaslavsky and S. Irmay. 1968. "Pysical Principles of Water Percolation and Seepage." Unesco, Paris. 7. Beran, M. J. 1957. Dispersion of soluble matter in flow through granular media. J. Chem. Phys. 27: 270-274. 8. Biggar, J. W. and D. R. Nielsen. 1962. Miscible displacement. II. Behaviour of tracers. Soil Sci. Soc. Am. Proc. 26:125-128. 9. Biggar, J. W. and D. R. Nielsen. 1963. Miscible displacement. V. Exchange processes. Soil Sci. Soc. Am. Proc. 27:623-627. 10. Biggar, J. W. and D. R. Nielsen. 1964. Chloride-36 diffusion during stable and unstable flow through glass beads. Soil Sci. Soc. Am. Proc. 28:591-595. 11. Biggar, J. W. and D. R. Nielsen. 1967. In irrigation of Agricultural Lands. Eds. R. M. Hagan e_t _al. Am. Soc. Agron. Madison, Wisconsin. 254-274. 12. Boast, C. W. 1973. Modeling the movement of chemicals in soils by water. Soil Sci. 115:224-230. 13. Bodman, G. B. and E. A. Colman. 1944. Moisture and energy condi­ tions during downward entry of water into soils. Soil Sci. Soc. Am. Proc. 8:116-122. 14. Bower, C. A., W. R. Gardner and J. 0. Goertzen. 1957. Dynamics of cation exchange in soil columns. Soil Sci. Soc. Am. Proc. 21:20-24. 275 University of Ghana http://ugspace.ug.edu.gh 276 15. 16. 17. 18. 19. 20 . 21 . 22. 23. 24. 25. 26. 27. 28. 29. Brennan, R. D. and M. Y. Silberberg. 1968. The system 360 Continuous System Modeling Program. Simulation 11:301-308. Bresler, E. 1973. Anion exclusion and coupling effects in non­ steady transport through unsaturated soils. I. Theory. Soil Sci. Soc. Am. Proc. 37:663-669. Bresler, E. 1978. Theoretical modeling of mixed-electrolyte solution flows for unsaturated soils. Soil Sci. 125:196-203. Bruce, R. R. and A. Klute. 1956. The measure of soil moisture diffusivity. Soil Sci. Soc. Am. Proc. 20:458-462. Burd, J. S. and J. C. Martin. 1923. Water displacement of soil and the soil solution. J. Agric. Sci. 13:265-295. Carman, P. C. 1939. Permeability of saturated sands, soils and clays. J. Agric. Sci. 29:262-273. Chapman, H. D. 1965. Cation exchange capacity. In C. A. Black (ed.), Methods of Soil analysis. Part 2. Agronomy 9:891-901. Childs, E. C. 1964. The ultimate moisture profile during infil­ tration in a uniform soil. 97:173-178. Coats, K. H. and B. D. Smith. 1964. Dead end pore volume and dispersion in porous media. Soc. Pet. Eng. J. 14:91-99. Corey, J. C. 1966. Miscible displacement of nitrates and chloride through soil columns. Unpub. Ph.D thesis, Iowa State University, Ames. Corey, J. C., D. R. Nielsen and D. Kirkham. 1967. Miscible dis­ placement of nitrate through soil columns. Soil Sci. Soc. Am. Proc. 31:497-501. Corey, J. C., R. H. Hawkins, R. F. Overman and R. E. Green. 1970. Miscible displacement measurements within laboratory columns using the gamma-photoneutron method. Soil Sci. Soc. Am. Proc. 34:854-858. Davidson, J. M. and R. K. Chang. 1972. Transport of picloram in relation to soil physical conditions and pore water velocity. Soil Sci. Soc. Am. Proc. 36:257-261. Day, P. R. 1956. Dispersion of a moving salt water boundary advancing through saturated sand. Trans. Am. Geophys. Union. 37:595-601. Day, P. R. and W. M. Forsythe. 1957. Hydrodynamic dispersion of solutes in the soil moisture stream. Soil Sci. Soc. Am. Proc. 21:477-480. University of Ghana http://ugspace.ug.edu.gh 277 30. De Josselin de Jong, C. 1958. Longitudinal and transverse diffusion in granular deposits. Trans. Am. Geophys. Union 39(l):67-74. 31. Einstein, H. A. 1937. Thesis, E.T.H. Zurich. Cited by F. Helfferich. 1962. Ion exchange, p. 486, McGraw-Hill Book Co., N.Y. 32. Elrick, D. E., K. T. Erh and H. K. Krupp. 1966. Applications of miscible displacement techniques to soils. Water Resour. Res. 2:717-727. 33. Elrick, D. E. and L. K. French. 1966. Miscible displacement patterns on disturbed and undisturbed soil cores. Soil Sci. Soc. Am. Proc. 30:153-156. 34. Elrick, D. E., K. B. Laryea and P. H. Groenevelt. 1979. Hydro- dynamic dispersion during infiltration of water by soil. Soil Sci. Soc. Am. J. (with reviewers). 35. Fatt, I., R. G. Goodknight and W. A. Klikoff. 1960. Non-steady state fluid flow and diffusion in porous media containing dead end pore volume. J. Phys. Chem. 64:1162-1168. 36. Fried, J. J. and M. A. Combarnous. 1971. Dispersion in porous media. Adv. in Hydroscience 7:169-282. 37. Gardner, W. R. and R. H. Brooks. 1957. A descriptive theory of leaching. Soil Sci. 83:295-304. 38. Gaudet, J. P., H. Jegat, G. Vachaud and P. J. Werenga. 1977. Solute transfer, with exchange between mobile and stagnant water, through unsaturated sand. Soil Sci. Soc. Am. J. 41: 665-671. 39. Ghuman, B. S., S. M. Verma and S. S. Prihar. 1975. Effect of application rate, initial soil wetness and redistribution time on salt displacement by water. Soil Sci. Soc. Am. Proc. 39:7-10. 40. Giddings, J. C. 1959. 'Eddy' diffusion in chromatography. Nature 184:357-358. 41. Griffiths, A. 1911. On the movement of a coloured index along a capillary tube, and its application to the measurement of the circulation of water in a closed circuit. Proc. Phys. Soc. Lond. 23:190-197. 42. Groenevelt, P. H. and G. H. Bolt. 1969. Non-equilibrium thermo­ dynamics of soil-water system. J. Hydrol. 7:358-388. University of Ghana http://ugspace.ug.edu.gh 278 43. Hashimoto, I., K. B. Deshpande and H. C. Thomas. 1964. Peclet numbers and retardation factors for ion exchange columns. Ind. Eng. Chem. Fundamentals 3:213-218. 44. Hillel, D. 1977. Computer simulation of soil water dynamics. A compendium of recent work. IDRC-082e. Ottawa. 45. Houghton, G. 1963. Band shapes in non-linear chromatography with axial dispersion. J. Phys. Chem. 67:84-88. 46. IBM Corporation. 1972. System/360 Continuous System Modeling Program. User's Manual, 5th edition, GH20-0367-4. Data Processing Division, IBM, White Plains, N.Y. 10604. 47. Kay, B. D. and D. E. Elrick. 1967. Adsorption and movement of lindane in soils. Soil Sci. 104:314-322. 48. Kemper, W. D. 1960. Water and ion movement in thin films as influenced by the electrostatic charge and diffuse layer of cations associated with clay mineral surfaces. Soil Sci. Soc. Am. Proc. 24:10-16. 49. Kemper, W. D. and J. B. Rollins. 1966. Osmotic efficiency coefficient across compacted clays. Soil Sci. Soc. Am. Proc. 30:529-534. 50. Kemper, W. D. and J. C. van Schaik. 1966. Diffusion of salts in clay-water systems. Soil Sci. Soc. Am. Proc. 30:534-540. 51. Kemper, W. D. and J. Letey. 1968. Solute and solvent flows as influenced and coupled by surface reactions. Int. Congr. Soil Sci. Trans. 9th (Adelaide, Aust.) 1:223-241. 52. Kirda, C., D. R. Nielsen and J. W. Biggar. 1973. Simultaneous transport of chloride and water during infiltration. Soil Sci. Soc. Am. Proc. 37:339-345. 53. Kirda, C., D. R. Nielsen and J. W. Biggar. 1974. The combined effects of infiltration and redistribution on leaching. Soil Sci. 117:323-330. 54. Krupp, H. K., J. W. Biggar and D. R. Nielsen. 1972. Relative flow rates of salt and water in soil. Soil Sci. Soc. Am. Proc. 36:412-417. 55. Lai, Sung-Ho and J. J. Jurinak. 1971. Numerical approximation of cation exchange in miscible displacement through soil columns. Soil Sci. Soc. Am. Proc. 35:894-899. 56. Lapidus, L. and N. R. Amundson. 1952. Mathematics of adsorption in beds. VI. The effect of longitudinal diffusion in ion exchange and chromatographic columns. J. Phys. Chem. 56: 984-988. University of Ghana http://ugspace.ug.edu.gh 279 57. Lindstrom, F. T. and L. Boersma. 1970. Theory of chemical transport with simultaneous sorption in a water saturated porous medium. Soil Sci. 110:1-9. 58. Lindstrom, F. T., L. Boersma and D. Stockard. 1971. A theory on the mass transport of previously distributed chemicals in a water saturated sorbing porous medium: isothermal cases. Soil Sci. 112:291-300. 59. Miller, S. F. and C. J. King. 1966. Axial dispersion in liquid flow through packed beds. Am. Inst. Chem. Eng. J. 12:767-773. 60. Nielsen, D. R. and J. W. Biggar. 1961. Miscible displacement in soils: I. Experimental information. Soil Sci. Soc. Am. Proc. 25:1-5. 61. Nielsen, D. R. and J. W. Biggar. 1962. Miscible displacement in soils. III. Theoretical considerations. Soil Sci. Soc. Am. Proc. 26:216-221. 62. Nielsen, D. R. and J. W. Biggar. 1963. Miscible displacement in soils. IV. Mixing in glass beads. Soil Sci. Soc. Am. Proc. 27:10-13. 63. Oteng, J. ¥. 1976. Fixation and some adsorption-desorption characteristics of ammonium and potassium. Unpub. Ph.D. Thesis, Univ. of Guelph. 64. Passioura, J. B. 1971. Hydrodynamic dispersion in aggregated media. I. Theory. Soil Sci. 111:339-344. 65. Peck, A. J. 1971. Transport of salts in unsaturated and saturated soils. In Salinity and Water Use (eds.) T. Talsma and J. R. Philip, pp. 109-123. MacMillan Press Ltd., 66. Peech, M. 1965. Hydrogen ion activity. In Methods of soil analysis (eds.) C. A. Black et^ al. Vol. 2. Am. Soc. of Agron. pp.914-926. Inc. Madison, Wis. 67. Perkins, T. K. and 0. C. Johnston. 1963. A review of diffusion and dispersion in porous media. Soc. Petrol. Engrs. J. 70-84. 68. Pfannkuch, H. 0. 1963. Contribution a l'etude des deplacements des fluides miscibles dans un milieu poreux. Rev. Inst. Fr. Petrol. 18:215-270. 69. Philip, J. R. 1957. Numerical solution of equations of the diffusion type with diffusivity concentration-dependent. II. Aust. J. of Phy. 10:29-42. University of Ghana http://ugspace.ug.edu.gh 280 70. Philip, J. R. 1968. Diffusion, dead-end pores and linearised absorption in aggregated media. Aust. J. Soil Res. 6:21-30. 71. Russo, D. and E. Bresler. 1977b. Effect of mixed Na/Ca solutions on the hydraulic properties of an unsaturated soil. Soil Sci. Soc. Am. J. 41:713-717. 72. Saffman, P. G. 1959. A theory of dispersion in a porous medium. J. Fluid Mech. 6:321-349. 73. Saffman, P. G. 1960. Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries. J. Fluid Mech. 7:194-208. 74. Scheidegger, A. E. 1954. Statistical hydrodynamics in porous media. J. Appl. Phys. 25:997-1001. 75. Scheidegger, A. E. 1974. The Physics of Flow Through Porous Media. Univ. of Toronto Press, Toronto, Canada. 76. Scotter, D. R. and P. A. C. Raats. 1970. Movement of salt and water near crystalline salt in relatively dry soil. Soil Sci. 109:170-178. 77. Selim, H. M. and R. S. Mansell. 1976. Analytical solution of the equation for transport of reactive solutes through soils. Water Resources Res. 12:528-532. 78. Selim, H. M., J. M. Davidson and P. S. C. Rao. 1977. Transport of reactive solutes through multi-layered soils. Soil Sci. Soc. Am. J. 41:3-10. 79. Shalhevet, J. and P. Reiniger. 1964. The development of salinity profiles following irrigation of field crops with saline water. Israel J. Agric. Res. 14:187-196. 80. Shalhevet, J. and B. Yaron. 1967. Ion distribution, moisture content and density of soil columns measured with gamma radiation. Soil Sci. Soc. Am. Proc. 31:153-156. 81. Smiles, D. E., J. R. Philip, J. H. Knight and D. E. Elrick. 1978. Hydrodynamic dispersion during absorption of water by soil. Soil Sci. Soc. Am. J. 42:229-234. 82. Smiles, D. E. and J. R. Philip. 1978. Solute transport during absorption of water by soil: Laboratory studies and their practical implications. Soil Sci. Soc. Am. J. 42:537-544. 83. Soon, Y. K. and M. H. Miller. 1977. A centrifugal filtration method for isolating rhizocylinder solution. Soil Sci. Soc. Am. J. 41:143-144. University of Ghana http://ugspace.ug.edu.gh 281 84. Speckhart, F. H. and W. L. Green. 1976. A guide to using CSMP - The Continuous System Modeling Program. A program for simulating physical systems. Prentice-Hall Inc., Englewood Cliffs, New Jersey. 85. Staverman, A. J. 1951. The theory of measurement of osmotic pressure. Rec. Trav. Chim. Pays Bas 70:344-352. 86. Swanson, R. A. and G. R. Dutt. 1973. Chemical and physical pro­ cesses that affect atrazine movement and distribution in soil systems. Soil Sci. Soc. Am. Proc. 37:872-876. 87. Taylor, G. 1953. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. Roy. Soc. Ser. A219, 186-203. 88. Taylor, G. 1954. Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion. Proc. Roy. Soc. Ser. A225, 473-477. 89. Terkeltoub, R. W. and K. L. Babcock. 1971. A simple method for predicting salt movement through soil. Soil Sci. 111:182-187. 90. Thomas, G. W. and N. T. Coleman. 1959. A chromatographic approach to the leaching of fertilizer salts in soils. Soil Sci. Soc. Am. Proc. 23:113-116. 91. Thomas, G. W. and A. R. Swoboda. 1970. Anion exclusion effects on chloride movement in soils. Soil Sci. 110:163-166. 92. van Genuchten, M. Th., J. M. Davidson and P. J. Wierenga. 1974. An evaluation of kinetic and equilibrium equations for the prediction of pesticide movement through porous media. Soil Sci. Soc. Am. Proc. 38:29-35. 93. van Genuchten, M. Th. and P. J. Wierenga. 1976. Mass transfer studies in sorbing porous media. I. Analytical solutions. Soil Sci. Soc. Am. J. 40:473-480. 94. Walkley, A. and I. A. Black. 1934. An examination of the Degtjaveff method for determining soil organic matter and a proposed modification of the chromic acid titration method. Soil Sci. 34:29-38. 95. Warrick, A. W., J. W. Biggar and D. R. Nielsen. 1971. Simultaneous solute and water transfer for an unsaturated soil. Water Resour. Res. 7:1216-1225. 96. Weber, W. J. 1972. Physicochemical Processes for Water Quality Control. Wiley-Interscience (John Wiley & Sons Inc.), N.Y. University of Ghana http://ugspace.ug.edu.gh 2 8 2 97. Wierenga, P. J. and C. T. de Wit. 1970. Simulation of heat trans­ fer in soils. Soil Sci. Soc. Am. Proc. 34:845-847. 98. Youngs, E. G. 1957. Moisture profiles during vertical infiltration. Soil Sci. 84:283-290. University of Ghana http://ugspace.ug.edu.gh APPENDICES 283 University of Ghana http://ugspace.ug.edu.gh 284 APPENDIX A Development of the ordinary differential equations for x> w as given by Philip (1957) Following Philip's (1957) notation, the horizontal space coordi­ nate in equation 3.11a will be assigned x' and the vertical coordinate of equation 3.53 given x instead of z. The identity: is introduced to make x' and x (equations 3.11a and 3.53 with x' and x for space coordinates respectively) the dependent variables giving: (A2) (A3) Subtraction of A3 from A2 gives which may further be algebraically rearranged to obtain _3_ (x-x') 3t A rn f 3x'/3e-3x/36 i 1 _|K 39 LU (3x/30) (3x736) J ~ 30 _3_ (x-x') 39____________"i 7 _3K Ox/30) (3x730) 11 30 University of Ghana http://ugspace.ug.edu.gh 285 Substitution of y = x-x' yields JZ = iL r n 3y/36 n _ M at 30 L u Ox/30) (3x730 J 30 Therefore, = i _ r _ D 9 y i e 3 6 1 _ M (A4) 3t 30 L D 30 3x 3x' 30 V 30 30 3y 30 30 f,r'1 N0te: ^x ~ "Ix1 = "30 "3x "3x' (A5) Approximation 1: Let and so y = y' Equation A4 becomes: lz' _ 3_ r V , _ 9K ( 6) 3t ' 30 [ D(3x '} 30 ] 30 C } 1/9 1/2 Let x ’=Xt ; therefore 3x'/3X = t ii' _ 3_ D 30 23y' 3K at 30 Lt 3X 30 J 30 K n Ordinary differential equation for x : Let X - y't 1 Therefore, |2- = X and |Z’ = t (A8) University of Ghana http://ugspace.ug.edu.gh 286 Substitute A 8 into A7 J- r 5. flix2 IzllX t + 11. , 36 1 t 3A 3x 39 39 ' r D ,39 2 3x 3K 30 1 t 3X 30 c J 30 = fe 1 + § where P(0) = D(6) (4 f ) 2 (A1 0 ) dA Ordinary differential equation for iJj: Starting from equations A4 and A 6 , subtract A 6 from A4 to get: 3 , 3 rT> r 3y 30 30 ,30 .2 3y'-i , 3t (y y ) " ~ 30 [D ( 3Q 3x 3x. (gx .) 3e > ] (All) Let z = y-y'. 3z _ 3_ rn _36. r 1 0. 3y1 -i-i 3t 30 L 3x' 1 3x " 3x' 36 n 3z _ 3 rn 36. ,3y 3y \ n 3t 30 [ 3x' 3x “ 3x (M 2 ) University of Ghana http://ugspace.ug.edu.gh 3 v "v ^ (gx " g i) on the righthand side of equation A12 is expanded alge­ braically as follows: 1 x _ ix' = ix _ i x 1 , i x ’ _ ix' 3x 3x' 3x 3x 3x 3x' i* + ix' _ ix' since i*. = 3(y-y’) = ix _ ix' 3x 3x 3x1 3x 3x 3x 3x 3z dj_' 3 x ' _ 3 y ' 3x 3 x ' 3x 3 x ' 3z 3y_' .3x' 3x 3x' 3x 1) ,3x' ,3x' 3x n 3(x '-x ) 3y nce “ 1) = = - 3x ix _ ix' 3x 3x' 3z _ ix 3y' 3x 3x 3x' Substitute A13 into A12 to obtain 3z _ i_ , _30_ j. _3z _ 3x ix' t i 3t 36 LU 3x' 1 3x 3x 3x' }i a • o T 3z 3z' ix 3y' iApproximation 2: Let — = — , , ^ , z = z (A13) (A14) Equation A14 then becomes: University of Ghana http://ugspace.ug.edu.gh 288 — = 1£ _ 1 d6 (A16) 9x' 3X 9x' t1 /2 dX 1®. = d6 = i d6 (A17) 3y' dx 3y' t dx = iZ M _ t dx 1 d0 1/2 dx (A18) 3x' 30 3x' t d0 1/2 dX C dX Substitute A16, A17, A18 into A15 to obtain: 3z' _ _3_ , D d0 r z^_'_30_ _ , ,dX\2 i, 3t = 30 ltl/2 dX 1 30 3x' M X ; J _L- i i \ 30 L 1/2 dX 1 1/2 dX 30 dX 3_ D_ d0 r L 1 /O *■ 1_ r° Is.' _ t-1/2-r, d0 ,dx.2 30 Lt dX 30 dX dX Therefore, 3z' _ 3 rP 3z* 1/2 (A19v 3^ " 30 [7 30 _ C Q] ( 19) Where Q(0) = D(0) || ( g -)2 z ' u r , 3/2 3z' 3. .1/2 Let i|j = ^ j ~2 ; therefore, z = ipt and 7^7 = -j^ t 3 z S z ' d i b _ 3/2 d^ ^ Ls0’ 30 3i|j d0 d0 which upon substitution into A19 yields: University of Ghana http://ugspace.ug.edu.gh 289 vpt1/2 3_ 30 [f t3/2 || - t1/2 Q] Therefore, 3 i d ~2 d0 [P (6) d\jj ae Q(0) (A20) Ordinary differential equation for a) : Start with equations A1A and A15. A15 is subtracted from A14 to obtain 3_ (z-z') 3t 3_ 36 [D i i t 3x' 1 3z' 3x 3x ,) - (■ ■izix.' _ (izVw 3x 3x' 3x ) ] Let w = z-z' 3w 3t 3_ 36 [D 36 3x' { 0 _ N _ (iz lz'' _ f i z V ■ l3x 3x' ;3x 3x )> (A21) (|J - ) in equation A21 may be rearranged as: 3z _ _3z' _ _3z _3z_' 3z' _ 3z' 3x 3x' 3x 3x 3x 3x' 3w _3z ' 3x' _ 3z' 3x 3x' 3x 3x' 3w 3x 3z' ,3x' 3x' '•9x 1) 3x' n _ 3x' 3x 3(x’-x) 3vSince ^ - 1 - ^ ^ ^ --- = - gZ University of Ghana http://ugspace.ug.edu.gh 290 — - — ' = liL _ jL5.' lZ (A22) 9x 3x' 3x 3x' 3x Substitute A22 into A21 to get: Iii = rD M 3z' _ ,3y 3y_' _ i (A23) 3t 36 lU 3x 3x 3x' 3x (3x 3x' { d x ' ] } t J K J I j g - (|j',)2 in equation A23 is rearranged as: iz iz' _ ( i z \ 2 = iz ' flz „ i z \ 3x 3x! 3x' 3x' 3x 3x' Because y = x-x' and z = y-y', , , , 3y 3z 3y'y = z+y a n d ^ = ^ + ^ Therefore, h . 1 z' _ clz' 2 = iz ' clz _ iz' -v = lz 'flz + lz' _ lz '■) 3x 3x' 3x' 3x' 3x 3x' 3x' 3x 3x 3x' And so, I z i z ’ _ (& .'■ 3x 3x' 3x'' 3y' ,3z _ _3y _3y_'. 3x' 3x 3x 3x' (A24) Since lZ ' _ i Z ' = 3 * ' 1 Z ’ _ | Z ' bince 3x 3x 3x 3x' 3x' i Z ’ (3x’ _ n 3x' 3x 1} University of Ghana http://ugspace.ug.edu.gh 291 = ix' a(x’-x) 3x' 3x = _ ix' ix 3x' 3x Substitute A24 into A23 to obtain — = i_ rn — ( (— - — ix~i _ ix' /!z _ ix ix' \ 1 3t 39 L 3x' 1 3x 3x' 3x 3x'^3x 3x 3x,; ; J Approximation 3: 3w _ 3w' 3y _ 3y' 3z _ 3z' , 3w 3w' Let 3x “ "Ix’ 5 3x ~ 3x' ’ 3x ~ "Ix1 End It " 3t Therefore: iw' _ i_ rn 36. r /3w' _ _3z' 3x _ ix /is. _ (iX ) 2 } i (A25) 3t “ 30 1 3x’ 1 3xr 3x' 3x' 3x’ ^3x' ^3x,; J 1 ; Now, x' = X t ^ 2, y' = xc > z ' = Therefore, 3z' 3z_' 30. = 3/2 di(; __ 1_ d6 _ t dtj; 3x' 30 3x' d0 1/2 dX dX ^ ; 3w' = 3w' 30 _ _ 1__ 3w' _30 3x' 30 3x' 1/2 30 3X (A27) and University of Ghana http://ugspace.ug.edu.gh 292 !*: = -4 - ix = 1/2 dx (A28) 3x' 4.1/2 dX dX Substitute A26, A27, A28 into A25 Jw' _ 3_ __ D d O f 1 3w' 36 t d^ 1/2 dx 3t 30 Ltl/2 dX 1 1/2 30 3X dX t dX tl/2 § (t S - ) > i — ' = fD d6 2 3w' _ d0 di dx + D d6 dx.3 , . 3t 30 Lt M X ; 30 Z t U dX dX dX + tU dX M X J ( A ‘ Letting R(0) - D(S) § g [2 f - (f* ) 2 ], equation A30 becomes: 3w' 3 rP 3w' 3t 30 [t 30 ] ( ^ T *. w ' ' OLet a) = — r- -r— = 2 lot t 9t .. 3w' _ 3w' 3a) 2 doo M s o ’ 36 = 3^ 30 = C d6 Equation A31 becomes: 2o)t 36 [t 11 d0 tR ^ Therefore, 2w = -^ [P(0) ^ - R(0) ] (A32) University of Ghana http://ugspace.ug.edu.gh 293 APPENDIX B Development of the ordinary differential equations f°r X> tyj w for salt movement as given by Elrick et_ a l . (1979) The procedure developed by Elrick et_ al. (1979) for vertical flow of salt parallels very closely the development by Philip (1957) for water flow given in Appendix A. The identity (refer to equation Al) used previously is employed to transform equations 3.55 and 3.15a so that x' and x become the dependent variable as follows: 3x _3c _ 3_ r 9Ds 1 3c ... 30_ 3c. Tr 3c 3t 3x 3c 3x/3c 3x 3x 3x 3x + 3^ - K - - h [3^ (B1) and , _3x' _3c _ 3_ , 9Ds , 3c _36 dc_ 3t 3x' 3c 3x'/3c 3x' 3x' 3x' 3x' P(9) _ _ 3_ f 9Ds 1 3t 3x’/30 3c 3x'/3c U5Z; Subtraction of B2 from B1 and substitution of y = x-x' yields It must be mentioned that as was done in Appendix A, University of Ghana http://ugspace.ug.edu.gh Approximation 1: Let S = ! £ and y=y ’ The second term on the lefthand side of equation B3 becomes 30 3y = 30 jfc' = 30 3^' 30 = 30 2 3^' 3x' 3x 3x' 3x' 3x' 30 3x' 3x' 36 The term on the righthand side is also rearranged as 3 rfiri 3c _|x, = 3_ rpm ,3c 3y' 3c . = 3_ ffi ,3c ,2 3y’ 3c s 3x' 3x 3c s 3x' 3c 3x' 3c s 3x' 3c so that substitution of B5 and B6 into B3 yields 3y' n ,30 ^ 2 3x’ _ 3 ,3c ,2 3y'. 37 Hx 36 “ K 37 [6D s (3^') 37 1 Now, let x'=At1/2, P(0) = D(0)(~)2 Therefore, 30 d0 3X _ d0 1 3x' dX 3x’ dX 1/2 Equation B7 therefore becomes: University of Ghana http://ugspace.ug.edu.gh Ordinary differential equation for x : y 1 9vT 3y1 dy 9y' Let X = f - • Therefore, ^ = x , ^ = t ^ and - g = Therefore, equation B8 becomes 9X _ M . t - K(6) = - [P t ^ ] A t d0 t dc s dc Ordinary differential equation for ip: We start with equation B3 and B7. 30 2 9v? Noting that D (-g^ ,) in equation B7 may be rewritten as n 36 30 n 30 3y' , . D t o ' t o ' 30 " ° 3^ ' 3 ^ ’ and a l s ° ’ 0D ,3c ^ 3^’ 3c 3c a*’ _ 3c 3 ^ s 3x 3c °Us 3x 3x' 3c 0DS 3x’ 3x' equation B7 is subtracted from B3 to obtain University of Ghana http://ugspace.ug.edu.gh 296 e3(y-y') _ D i i (h . _ ix! 3t 3X ' 3x 3x ,) = f [SD 3c s ox ox 8x ,) ] Let z y -y Therefore, ail. _ n l i r ii _ ix' \ = i_ rpm 9t 9x' 3x 9x' 3c s 9x' 9c ,3y _ 9y' 9x 9x ,) 1 9y 9y' 9z 9y 9y' . c . . 1.. "3x ~ "9x' = 9l ~ 9x 9x' (refer t0 equatl0n A13> Therefore, n 9 z n 3 0 r 9 z ®3F " ° 3^' 9x 9y 9y1 3x 3x' T" t0D3c s 3c 3x' 3z _ _9x 9y' 9x 9x 9x ,) ] Approximation 2: 9z Let -r—3x 3z' 9x' 9y _ 9y' 9x 9x' and Equation B12 becomes: l i r9z.’ r lxV i _ 9_ ' “ 3x J 3c“ ° 9^' [ 9x' 3c Js 3x' ,3z' "3x' Now x' = Atl/2, y' = Xt, P(0) = D (0) (4t")2 (BIO) (BID (B12) (B13) The second term on the lefthand side of equation B13 upon substitution of A16, A17, A18 becomes: University of Ghana http://ugspace.ug.edu.gh Similarly, the term on the righthand side of equation B13 is rearranged as follows: 3 ffm — (— ' s9x' 9x' 9x 9x9c 3_ r 9Ds ( d z ' 9c _ A 2 9c L 1/2 dX 9c 9x' M X ; ; J 9 r s dc r 1 dc 9z' _ ,dy. 2 ■, 9c Ltl/2 dX tl/2 dX 9c W S 6D 3c [— — (|y )2 0D t1/2 4r-(4f)2dX 9c (7t) 1dX dX = 1_ 3z' _ 11 /2 9c 1 t 9c ys where Qg (0) = 0Dg (e) ^ ( g ) 2 ] (B15) Substitution of B14 and B15 into equation B13 yields University of Ghana http://ugspace.ug.edu.gh 298 Also 1*' - i i 1 M - j-3/2 di ’ 86 dip 36 d0 3z 1 3z ' d<|J 3/2 diJj and -r— = — — 3L = t ~r^ 3c 3ip dc dc Equation B16 therefore becomes: P 3/2 d^ _ 1/2 _ 3_ _ s 3/2 di _ 1/2 Lt d6 3c lt dc V Therefore, I e t - p ( 6 ) f + Q (0) - f j [p s <0) f - Qs ce> ] Ordinary differential equation for ca: This is started by subtracting equation B13 from B12, to obtain: „3(z-z') 36 , ,3_z 3z' _ ,dj_ dj_' _ ,3y ’1.2 3t ° 3x’ 3x 3x' O x 3x' 3x' ; j r0D l£ { ') _ M IZ.' _ (IX')2) } 3c 1 s Sx'1 l3x 3x' O x 3x' O x ' ; ; Let w = z-z' _ n — r i iz' O x ’ n 2 \ i D 3 x ' 3 x 3 x ' 3 x 3 7 ' " ( 3 ^ ' } } 1 = i_ feD l£ f (-Is. _ 3z' x O x ly' ,3y\ 2 , 3c lyDs 3x' 1 3x Sx'5 (3l 3x' “ (3i'> } } ] (B17) (B18) University of Ghana http://ugspace.ug.edu.gh 299 In equation B18, 3z 3z' _ 3w 3z' 3y , , . .~9n. *3x ~ 3x' 3^ “ 31' ai (refer t0 e^ atlon A22> Also, from equation A24, 3y 3y' /ix'x2 3y' r ix' 9y i t d x ' ~ ( 5 x ' } = 3 ? [ 3^ • 3x' ] And so, substitution of equation A22 and A24 into B18 yields: a3w ^36 r,3w 3z' 3y, 3y' Sz 3^' 3^ N •, ^ “ °3l’ [ (37 " 3 ? h P ~ 3x’ (3^ ‘ W 3x ] i_ r0D i£ { _ 9z' lX) _ lX' (is. _ Ix' iX ) } 3c 1 s Sx'1 3x 3x' 3x 3x' ^3x 3x' 3x } ! Approximation 3: 3w _ 3w' 3y _ _3^ ' _3z_ _ 3z1 Let 3x “ 3x' ’ 3x _ 3x' 5 3x 3x’ Equation B19 becomes: q3w' 39 f ,3w’ 3z' 3y'. 3x' ( _3z_' r3y'x2 , at" U 3x' 3x' 3x' 3x' _ 3x' 3x’ ^3x'; 1 J J L r6D l £ { (iSL' _ is.' ' IX ' ) _ I x ' { I * ' _ ( i i ' ) 2 }} i 3c L s 3x' 3x' 3x' 3x' 3x’ l3xf l3xl} J (B19) (B20) Now, x' = Xt1/2, y' = xt, z' = ^t3//? University of Ghana http://ugspace.ug.edu.gh Substitute A26, A27, A28 into B20 to obtain: ©22' _ d6 r 1 3wf 30 d£ 1/2 dx _ 1/2 dx _ (^u2 , 3t tl/2 dX Ltl/2 36 3X dX dX dX 1 dX W /J = ffin 1 1c f 1 3w'dc _ dij; l/2dx _ 1/2 dx, d£ _ ^dX-,2-, 3c L s tl/2 dX fcl/2 3c dX dX dX dX^ dX M X ; Therefore, ^w'_ {D d0 2 3w' _ d0 di dx + d0 dx 3 _*.7.c>.0.5.1 a 1 . . 4 * . 7 3 1 ( . 4 5 * . 6 8 2 9 ) * ( . 5 * . 6 4 6 3 ) » ( . 5 5 . . 6 0 9 8 ) * ( . t>0 £>, . 5 8 5 4 ) * ( . 6 5 5 . . 5 6 1 ) , C . 7 5 . . 5 2 4 4 ) , ( . 85 , . 4 8 7 g ) . ( . 9 . . 4 7 S o ) . ( . 9 3 , . 4 6 J 4 ) , ( 1 . : , . 4 a l 2 > , . ( 1 . 0 7 5 . 1 4 3 9 ) . ( l o 1 5 , . 4 2 6 6 ) . ( l » 2 , « 4 1 4 b ) _____________________________________ DYNAMIC C THET A=NL FGEN( MR C » S W S ) _______________ .T.0.1 F.fL=NLFG= N (DIEJE *.CT..t- ET AJ ____ ____ __________________ DMRC=DdRIV( MRCO, CTHcT/») K=—( THET A 3 -THE TAN ) *TC IFF*DMRC ______________________THLA=CTHE T A * (T H IT AO -ThETAN ) +THETAN_____________________ TERMINAL PRINT DMPC, CTHETA .T H LA .T D IF F .K _____ ___________ .TJMER. F.J.N.TI M=1 J 2.^P.RD.eL=2J,.4c*.C2__________________ _______ END STOP OUTPUT VARIABLE SEQUENCE CTHET A DMRC TD IFF K THLA OUTPJTS INPUTS PARAMS INTEGS + MEM BLKS FORTRAN DATA CCS 9 ( 5 3 ' ) 3 3 (1 4 0 0 ) 7 (4 0 0 ) 0+ 0= C (3 0 0 ) t> (6 0 J ) 17 University of Ghana http://ugspace.ug.edu.gh 305 FIGURE C2 Alternate computer program using X as the variable time, for calculating hydraulic conductivity as a function of water content. University of Ghana http://ugspace.ug.edu.gh 306 * * * * CONTINUE US SYSTEM MODELING PROGRAM**** -- ***_ .VERSJCN_-L. J * * * ------------------------------ ------ ------- IT LE CALCULATION OF WDRAULIC CONDUCTIVITY (BROOKSTON CLAY) TB_TJHLA=TABLE CF WATER CCNTENT AND CORRESPONDING LAMBDA VALUES------ S WS=TABLE OF SC IL WATER SUCTION AND CORRESPONDING WATER CONTENT VALUES XHLA=JNTERi?QLATED ..WAT.ER.-.CQ.NJ.ELN.X--A.ND ._L_AMBD.A ..VALUES -------------- ------ MRC=INTERPOLATED SO IL WATER SUCTION AND LAMBDA VALUES THROUGH THE FUNCTION SKS A= fiE rUYA IXVE OF WATER CCNTENT WITH JiESPECT TO LAMBDA_____________ __ DC=DERIVATIVE CF SO IL WATER SUCTION WITH RESPECT TO LAMBDA DMRC=DERIVATIVE 0= SO IL WATER SUCTION WITH RESPECT TO to AT ER ... -CDN.TENX. ______ _ ______________________ ___________ _____ C=INTEGRAL VALUE OF THETA FROM LAMBDA EQUALS ZERC TO LAMBDA D =S 0 IL WATER D2FFJ S IV I TV H C O N D A U L - I . c CONDUCT IV ITY_______________________________________________ C TH ET A=DIMENSIONLE S S WATER CONTENT K0N2=THE D E F IN ITE INTEGRAL VALUE GF THETA FROM LAMBDA EQUALS ._.ZER0_..T0 J-AMDAJS. ( =L AMBDA. AT...F.J NT IM ) .. . _ _ ....— . - RENAME TIME=LAMBDA IN IT IA L I-NC.ON_ J&EXAD=. S3 . THET AN= KO N 2= l.C ,LAM DAN = I.Q 2E -03 * 1 = 0 FUNCTION TBTHL A=( 0 . 0 • , 5 3 ) • ( I . 0E -Q 4 • * 5 2 5 ) • ( 3 . 0 E - 0 4 . . 5 2 ) • • . • .... £S.*.0E-04ju.SX5.1JiI.7 .*0E -TJ}4 ,*51J .^19A0 £ -0 4 :a.^B0J.AC.l*J3E~D3J...495J_»...x__ ( 1 . I E - 0 3 , . 4 9 ) , (1 . 2 E - 0 3 . . 4 0 ) , < 1 . 3 E -0 3 . . 4 7 3 ) • ( I . 4 E -0 3 . . 4 6 3 ) • . . . ( 1 . 5 E -0 3 , . 4 4 5 ) , ( 1 •5 5E—0 3 • . 4 3 5 ) • ( 1 . 6 E -0 3 • . 4 2 ) . { 1 . 6 5 E -0 3 , . 4 0 ) , . . . _________LX *JZE -0 i,_ ..37J »X J ji7 5E ^QA jla3 .2 ^J .i ( X.7.8E.- 0 ..3..«mZ.a±.*±.1.3&TLQ3 ( 1 . 8 1 E -0 3 , . 1 e 5 )# ( 1 .8 2E—0 3 • . 1 2 ) • ( 1 . 9 E -0 3 . . 1 2 ) . ( 2 .CE -C 3 . . 1 2 ) FUNCTION SWS=(.2 8 , 1 . 2 ) . ( . 3 . 1 . 1 ) . ( . 3 0 5 * 1 . 0 ) . ( . 3 1 5 . . 9 ) , . . . I , 3 2 5 . . 81 . ( . . 3 4 . . 7 .X . • C * 2 5 5 « ^ 6 X , ( . 3 7 . 5 ^ . 5 X ^ X . ^ . . 4 X ) . . X * 4 3 5 . . 3 . ) a . . . . . . ( . 4 5 . . 2 5 5 ) . ( . 4 6 7 5 * . 2 ) . ( . 4 8 . . 15) » ( . 4 9 5 , . 1) . ( .51 » . 0 5 ) . . « . ( . 5 3 , 0 . 0 ) D Y N A M I C THLA=NLFGEN(TBTHLA.LAMBDA) MRC=NLFGEN(SWS.THLA) DC = DER1V(MRCO,M3 C) DMRC=—DC/A C= I NT GRL(O .C .THLA ) KON1=THE T AN *LA W3 A N E=(THLA*LAMBDA) - KON1 .FF—E—C ............ ........................ ......................................... ................................... ......... G=FF+K0N2 D = - .5 *G /A C THE TA= ( THL A—T HE T AN ) ✓ (THET AO-THET AN ) TERMINAL ...............................................PRINT MRC.DC.DMRC .CTHETA .THLA.HCOND .D ................ .................................. . TIMER F I NT IM = 1 .8 2 E -0 3 ,PRDEL=3. 6 4 E -0 5 1=1+1 K0N2=C ................. ..END... I F ( I .G T .6 ) STOP CALL RERUN STOP OUTPUT VARIABLE -SEQUENCE. C HCOND A MRC CTHETA I DC KON2 D MRC K ON 1 E ZZ0003 University of Ghana http://ugspace.ug.edu.gh 307 content 9 versus lambda (X = xt ) and soil matric suction versus water content, respectively. Time in this program is renamed lambda and soil water diffusivity D calculated (see description in Section 4.9.1). Interpolated values for soil matric suction SWS and lambda is provided through the statement -1/2 MRC = NLFGEN(SWS,THLA) The derivatives, d0/dX and dh/dX are calculated respectively using the statements A = DERIV(THLAO,THLA) DC = DERIV(MRCO,MRC) din d A The derivative dh/d6 (= • -tk-) is calculated using the statement: dA do DMRC = -DC/A The hydraulic conductivity K is then calculated through the statement: HCOND = D/DMRC Values of K(9) for low moisture contents were obtained by extrapolation of the K(9) curve. University of Ghana http://ugspace.ug.edu.gh 308 APPENDIX C Algorithm for the simulation of 0(A) from calculated D(A) values Assuming the objective is to simulate 0(A) relationship in fig. 1c (curve c), RVEL0(=D d0/dA at 1=0) is guessed. RVELAO is thus ob­ tained through the multiplicative factor 1.0001. Using equations 4.28 through to 4.33, RVELO and RVELAO generate THETA0 and THETA*. respect­ ively. The difference between these two generated 0(A) functions, designated DTHETA, is then calculated, followed by the calculation of the slope which is done using the usual mathematical equation for a slope as follows: The slope m, of the relationship between THETA2, THETAA and their respective RVELO and RVELAO (fig. 2c) is given by the equation y2 ~ Y1 = THETAA - THETA2 , m x, - xx RVELAO - RVELO ^ 1 But RVELAO-RVELO = RVELO x 1.0001 - RVELO = RVELO(1.0001 - 1.0) = 0.0001 RVELO Therefore, SLOPE = DTHETA/(.0001 x RVELO) (C2) The correction factor, DRVEL, required for the calculation of a 'new' RVELO, is the difference between the FINTIM value of the simulated moisture content and the initial moisture content 0 divided by the n ’ slope: University of Ghana http://ugspace.ug.edu.gh 309 Figure lc: Schematic representation of S(X) function with two 6 curves generated by RVELO and RVELAO. University of Ghana http://ugspace.ug.edu.gh 310 \ University of Ghana http://ugspace.ug.edu.gh 311 F i g u r e 2c: S chemat ic p l o t o f THETA2 and THETAA a t X = 0 v e r s u s t h e i r c o r r e s p o n d i n g RVELO and REVELAO. University of Ghana http://ugspace.ug.edu.gh 312 THETA2 THETAA THETAN RVELO RVELAO Figure 2c. University of Ghana http://ugspace.ug.edu.gh 313 DRVEL = (THETA2-THETAN)/SLOPE This relationship for DRVEL is obtained from the consideration of fig- 20 as follows: * By the theorem of similar triangles, ABC and Ax x^ are congruent. Therefore, RVELAO - RVELO X3 " X1 THETAA - THETA2 THETA2 - THETAN From equation Cl RVELAO - RVELO = 1 THETAA - THETA2 SLOPE Therefore, DRVEL = x3 - x1 = (THETA2 - THETAN)/SLOPE The 'new' RVELO to be used for the next iteration is the old RVELO minus DRVEL. In CSMP, this is written as: RVELO = RVELO - DRVEL * ,Theorem states that if corresponding angles of two triangles are equal, then their corresponding sides are proportional. University of Ghana http://ugspace.ug.edu.gh 314 The specification in the algorithm (fig. 4.5) that iteration should stop when simulated THETA2 has converged sufficiently so that the absolute value of the difference between FINTIM value of THETA2 and 6 n (THETAN) is less than 0.00001, ensures that the simulation stops at that accuracy. Failure to converge, the specification on the counters, that is, IF(J.GT.10)STOP and IF(I.GT.6)STOP will terminate the simulation. University of Ghana http://ugspace.ug.edu.gh 315 APPENDIX D Algorithm for the simulation of c(A) for the calculated D (A) valuess The development of the algorithm for the simulation of c(A) is similar to that for 9(A) given in Appendix A. The main difference is dc that VELCO (equivalent to RVELO in Appendix C) is equal to 0Dg -jr at A=0. Here, the objective is to simulate c (A) relationship in fig. ID (curve f). VELCO is guessed. This guess yields VELCAO because of the multiplying factor 1.00001. Equations 4.54 through to 4.58b are then used to simulate SEE2 and its half pair SEEA, respectively. DSEE which is the difference between these two generated c (A) functions is calculated. Then the slope which from equations similar to Cl and C2 is equal to DSEE/(.00001 x VELCO) is computed. The correction factor, DVELC, is then obtained from the difference between the simulated c at FINTIM and the initial concentration c^ (specified in the PARAMETER section) divided by the slope. The 'new' VELCO to be used for the next iteration is ob­ tained by subtracting the correction factor from the 'old' VELCO. When simulated concentration values converge sufficiently the simulation is terminated due to the specification for simulation to stop when the absolute value of the difference between SEE2 and SEEN is less than .00005. Otherwise the stop specifications on the counters I and J will terminate simulation. University of Ghana http://ugspace.ug.edu.gh 316 F i g u r e Id : S chema t i c r e p r e s e n t a t i o n o f c ( X ) f u n c t i o n d e p i c t i n g the two c cu r v e s g ene ra t ed by VELCO and VELCAO. University of Ghana http://ugspace.ug.edu.gh 317 Figure Id . University of Ghana http://ugspace.ug.edu.gh 318 APPENDIX E CSMP listing for calculating soil water diffusivity D and simulation of 0(A) for Akuse clay University of Ghana http://ugspace.ug.edu.gh 319 * * * * C ON T I N UD U S S Y S T E M MODEL I N o P R J G R A M * * * * T I T L E C A L C U L A T I O N OF S O I L M C IS T U R E D I F F j S I V I T Y AND S IM U L A T I O N OF * U N I TS * K G = K I L O G R A MS* * S = S ECON DS * G LO S S AR Y OF SYMBOLS * D TH I AQ=DFR I V A T I V F . D F T H F T H F T A - 1 AMBDA F U M C T i n N AT 1 AMHD4 P I3 I .A I ^ * Z E RO * T H E T AO =W A T E R CONTENT A T LAM BD A E Q U A L S ZERO * LAMDA fM = l_AMBDA AT I N F I N I T Y ( s s F I K T T M J * T H E T A N = I N I T I A L WATER C C M E NT * T H L A = l N T E R P O L A T EC CURVE FOR T H E T A —L AM BD A F U N C T IO N R V F L = T H £ _PRGDI.CT O F S O I L W A T rR n i F F U S I V I T V ' AMD T h F D FP T V A T T V r OF T H E T H E T A - L A M B D A F U N C T IO N * R V E L C = R V E L AT LAM BD A E C L A L S Z E R C n s s n i l WATER n i F F U S f V I T Y * R D I V = D E R I V A T I V E OF R V E L W I T H RE S PEC T TO L AM B D A * R V E L A O = R V E L 0 N I L T I P L I E C BY 1 . C C 0 1 * R V F l A=HAL_F P A I R . D r _RVEi T H R n i l f i H T Hc F A f T H W l _ n n n i I N RUC l A C * R D I V A = H A L F P A I R OF R D I V THROUGH TH E F A C TO R 1 • J 3 0 1 * T H E T A 2 = S IM U L A T E D WATER CONTENT * * DRVEL -=CORRECT I C N F A C TO R * D TH E T A = D I F F E R E N C E BETWEEN T H E T A 2 AND TH E TA A S L O P E s R A T l f l . TiF THF n T F F F B F N lf F RFTkJFFN T H F T A A AN.D T H F T A P T r , T H - * D I F F E R E N C E BETWEEN R V E L A O AND RVELO * T B T H L A =T A B L E OF WATER CC N TEN T AND C O R R E S P O N D IN G L AMBDA V A L U E S I N I T I A L P AR AM E TE R D T HL AO = — 1 . 0 E 0 1 »T H E T AO = • 5 2e>, L AMD AN = 4 • 5 6 E - 0 3 , . . • R V E L A 0 = R V E L C * ( 1 * 0 + D E L ) F U N C T IO N T B T H L A= { 0 * 0 . . 5 2 6 ) . ( 1 . f . E - 0 4 , . 5 2 5 ) , ( 2 . C 6 - 0 4 , . 5 2 4 ) . . . . ( 2 . 4 E - 3 , • 5 0 4 ) • ( 2 . 5 E - 3 « « 5 0 3 ) * ( 2 . 6 c - 3 . . 5 . > ) , ( 3 . J E - 2 . * 4 9 5 ) * • . . ( 3 . 2 E - 3 * . 4 3 7 5 ) . ( 3 . 4 - E - 3 . . 4 8 0 ) . ( 3 . 5 E - 3 , . 4 7 5 ) . ( 3 . 6 E - o , . 4 7 0 ) . . . . ( 4 . l - E - 3 * . 4 2 0 ) . ( 4 . 2 E — 2 ♦ . 4 0 ) . { 4 . 3 E - 3 . . 3 7 0 ) . ( 4 . 4 < ' E - 3 . . 3 3 ) * . . . ( 4 . 4 6 E - 3 * . 2 9 5 ) > ( 4 . 5 0 E - 3 * . 2 5 0 ) . ( 4 . 5 6 E - 3 » * 1 0 ) . ( 5 . C E - J . o l O ) T H L A = N L F G E N ( T B T H L A , L A M B D A ) D T H L A = D E R I V (D T H L A O . T h - L A ) C = I N T G R L ( 0 . O t T H L A ) KON1 = T H E T A N * L A M D A N F F = E - C G—F F + K O N 2 D = - . 5 # G / D T H L A NOSORT I F { J « L E • 2 ) GC TO 10. s n n T R VEL= I NT GRL (R V E L O .R D lV ) R V c L A = I NTGRL (R VE LAO , R C IV A ) p D T V=-_LJIMBD A.*R_V? L * R / r _____ R O IV A = -L A M B D A *R V E LA * . 5 /C T HET A 2 = I NT G RL( THETAO , R V E L /D ) T H £ T A A = IN T G R L (T H E T a O ,R V E L A /D ) NOSORT 10 CONT INUE University of Ghana http://ugspace.ug.edu.gh 320 _______________ T E R M 1 M AL_______________________________________ !_____________________________________ ’----------------------------------------------------------- T I M E R F I N T 1 M = 4 . 5 6 E - 0 2 • P R D E L = 9 • 1 2 E - j 5 P R I N T G , T H E T A A * D » T H E T f i 1 . T H L A . T H E T A 2 _ . - I F ( J • G E • 3 ) G C T O 1 5 J = J + 1 K 0 N 2 = C >------------------------------------------------------ GX i— J O — 2 0 - . — - _________ — 1 5 1 = 1 + 1 I F ( A B S < T H E T A 2 - T H E T A N ) . L T . . O O C O l ) S T O P - O T .H E T - A = T H E T A A - T H E T - A 2 ................................. S L O P E = D T H E T A / D E L / R V E L C D R V E L = ( T H E T A 2 - T H E T A N ) / S L O P E _________________________________ -R V F 1 G = R V F I n - H R V F I ____________________________________________________________________________ R V E L A O = R V E L C * ( 1 . O + D E L ) W R I T E ( 6 * 1 0 0 ) T H E T A A * D * T H L A « T I - E T A l . T H E T A 2 1 0 0 F D R M A T t / / / * T - H E T - A A * D * T H L A • T H E T A 1 • T H E T A 2 = J ~ » L 1 5 * . o ) W R I T E ( 6 , 1 0 1 ) R V E L U , R V E L A O 1 0 1 F O R M A T { / / * C A L L R E R U N W I T H R V E L C , R V E L A U = 1 , ^ E 2 0 • 7 ) ___________________ 2 -0_____________I F t J . f i T . 1 0 J___________ S J -Q 5 _______________________________________________________ I F ( I . G T . b ) S T O P C A L L R E R U N ___________________E N D _____ _ - ................ - - - - - - - - - - S T O P O U T P U T V A R I A R . L F — S E .Q U F -N jC E_____________________________________________________________________________________________ R V E L A O T H L A D T H L A T H E T A 1 C K C M E F F o 0 Z Z 0 0 0 4 R O I V R V E L R D I V A R V E L A Z Z 0 0 0 8 T H E T A 2 x _ Z 0 0 i 0 T H E T A A Z Z C 0 1 1 Z Z 0 0 X 2 _ J . K 0 N 2 - - 1 - - ----------- D T h £ . X - A S L £ .F £ ~ - . .~ D R \ jE i- - - R V E i - U R V £ L A U O U T P U T S I N P U T S P A R A M S I N T E G S + M E M B L K S F O R T R A N D A T A C D S 7J ( 1 4 3 3 )_____1 2 ( fiL±____ 0 = ___£lX 3_0_Q _J ZS lU xX H ________ IA__________ £NDjOB---------- .----- ------ University of Ghana http://ugspace.ug.edu.gh 321 APPENDIX F CSMP listing for calculating dispersion coefficient Dg for Cl and simulation of c(X) using (i) analytical solution and (ii) computer solution (Akuse clay) University of Ghana http://ugspace.ug.edu.gh 322 ****CONTINUOUS SYSTEM MODEL IN J HRJGRAM **** *** VE-RS.ICIwl.J-*** - -....... — - T IT LE DISPERSION COEFFIC IENT AND S INULAT IO im OF CONC» (AKUSE CLAY)_* I IM y Tg _____ _ * KEQ-K ILO -EQU I VALENTS * M=METERS * S=SECGNDS ■ . - ....... ......................... .................. * GLOSSARY OF SYMBOLS * D THLAO=DERIVATIVE OF T hET A—LAMEDA FUNCTION AT LAV EDA —c-------- rn t iA i s -7.FBn ___________ __ __________ ____ _ ___________________________ _________ * DTHLA=DERIVATIVE OF THETA -LAMEDA FUNCTION * THFTAO=VOLUMETRIC WATER CCNTENT AT LAMBDA EujU^LS ZERO * L AMDAN=LAMBDA AT IN F IN IT Y ( = F1NT IM ) — * THET A N= J NI T IA L WATER CCNTENT * T HL A=INTERPOL ATEC CURVE FOP THETA-LAMdD* FUNCTION - * -------- V.E1 C— THE. PRniil.r.T OF WATFP r.QNT £ KT . aJ^jsg^LSJ-Obt COE PP I £ TrMT_________ * AND THE DERIVATIVE OF ThE CONT R AT I UN-LAMdl>« FUNCTICN * VEL CO = VELC AT LAM3 DA EQUALS ZERO * VELCA£=VELCO MULT IPL IED £Y l.QOOO-1 * SEE 0=C0NCE NTRAT I ON OF CHLORIDE IN oO LoT luN AT LAVEOA EQUALS * ZERO - * -------- SFF.N- UiXT-LAl r TMfF MTB A T I r N HF r w D im - T u M.11^7 c r Ti ____________ * DSEELA=DERIVATIVE OF CCNCENTRATI ON-LaMBJA FUNCTICN * IT h R E S c E C T * TO LAMBDA * --DSELAO=OER1VATIV£ OF. CCIvCENTRATION-LAMaDA J=UNCTI-CN AT LAMBDA * EQUALS ZERO * SEELA = I NTERPOL AT ED CURVE F CR CCNCENT RAT 1 tJlv-LAMcD FUNCTION -Jt---------1 ATHH—T.HE ...ii—F.LNCT I iJJnj Q ^F IKFn I K - CliA T T. i:J _________________________ * TBTHLA=TABLE CF WATER CCNTENT AND CuRRc-SPuND 1N C LAMBDA Va LLES * TBSELA=TABLE OF CONCENTRATION OF CHLORIDE IN oCLLT ICN AND * . CORRESPONDING LAMBDA VALUES - * 0 = S 0 IL WATER D IFFU S IV ITY * 0 IV = DERIVAT I V E OF VELC UITH RESPECT TC L AMaDA _S__________________ A T -n CCNCFNIT R AT ITN Q F r n T im SLJ-jUXXOia._______________ * SEE 1 = C0NCENTRAT1 ON OF ChLCRIDE CALCULATED USIwG ThE ANALYTICAL. * SOLUTION ▼ ~ SEEA=HALF PA IR UF -SEE2 THROUGH THE FACTUii I .Q C O O l * DVELC=CURRECTION FACTOR * SLOPE=RATIO OF THE DIFFERENCE BETWEEN 5E -A AND SEE2 TC THE _*-------- D.I F F.r-RENCE_..a?_T. WEEN. VZ 1._ CAC AND VEL.CQ______________________________________ * DSEt=DIFFERENCE BETWEEN SEE2 AND SCEA AT F IN T1V RENAME TIME=L AMD DA IN IT IA L • I NC ON THETA N— i 13 « LAM CAN = ,V .56E — 0 3 , KDN*d= 1 • 'J»K,l.N3=: 1«C* KON<* = 1 . C PARAMETER 0 SELAO= - 1 . C , SFEU=.5 2 » DT HLAJ = —1 . . E . 1 . T hFT A 0= •6 2 C -----------------CONSTANT S~ EN-«n fin d ft = = _______________________________________ VEL C0=—•7 1 7 9S15E — 0 4 , D E L = .3 0 0 0 1 VEL CAO=VELCC*( 1 .O+DEL ) —------------- E.UNCT1 ON T 3T H LA = I0 .0 » « 52o ) . ( l . O E - J ^ , . 1 2 .0 E -34 . . 52<+ ) ( b .C E - 4 , . 5 2 ) • ( 1. 2 E -3 * . S I S ) , ( 1 . 6 5 - 3 , . u l ) , ( 2 . a = -3 » .5 - 5 ) , . . ( 2 o 4 E - 3 , * 5 0 4 ) , ( 2 . 5 E - 3 , . S 0 3 ) , ( 2 . c £ - 3 , . 5 0 ) • I 3 , 0E- 3 * • * 9 5 ) * • ----------------- 1 3 . 2r-r3,.-..4d7.5J U .3 .«4£ -3« .4 .601 d O i . L* t. 2....r.E-3, I I 3 • 7 E—3 , . 4 6 4 ) , (3 . B E -3 , .4 5 5 ) . I 3 . SE -3 , . + 4- 7z>) , { 4 . C E -3 . .4 3 5 ) ( A , I E - 3 . « 42 0 ) * . 2E—3 * . 4 C ) * { 4 .3 £ -3 . .3 7 0 ) , (4 ♦ HO E -3 , . 3 3 ) . . X 4 . 46E -3 .* ,~2&.5) . ( 4 . 5 0 E - 3 , ..250 ) . ( 4 .-5 oE -3 , . 1 - ) . 15 . C E -o . . 1 l. J FUNCTION TBSEL A= ( 0 . 0 . . 9 2 ) . ( 1 • DE-04 , . 91 9 9 ) , I c • 0 E -0 * * • 9 19d"I ( 3 . 0 E -0 4 , .9 1 9 7 ) , ( 4 . 0 E - 0 4 , . 9 1 9 o ) , ( O .0 E -0 4 . .9 1 5 5 ) - I f : . . Q f - 0 4 . . 9 1 9 40^-X 7 . OF — 4 . . 9 1 G . l i - I h . .W 1 £2. > ( 9 .0 E -0 4 > .9 1 9 1 ) . ( 1 .0 E-~ 0 3 . . 9 1 9 ) , C l . l c— J 3 . • y 1 tt ^ ) * C 1 .2E -0 3 * .S > 16Q ). ( 1 . 3E-C 3 . . 91 67 ) , ( 1 . ^ E -0 3 , . 9 1 do ) ( 1 • 5E -0 3 , . 9 1 6 5 ) , ( 1 .6 E —03» * 91 64 ) , ( 1 . 7 E -O o . • 9 1 t»3 ) ( 1 . 8E--J3 , . 9 1 8 2 ) , ( 1 . 9 E - 0 3 , . 9 i e i ) . ( 2 . J c l-0 3 . . 9 1 8 ) . University of Ghana http://ugspace.ug.edu.gh 323 ( 2 . 1 E - 0 3 * , 9 1 7 9 ) , ( 2 . 2 E - 0 3 , . 9 1 ) . . —------------------------------------------l ^ t A F r O a . . ^ . g, ) . ( ? , h F - ^ , . e i ; n . ( j ) . j ? » fc> f i - •? '» . . f l a ) . ------- ( 2 • VE -0 3 , . 8 7 ) , ( 3 . C E—C 3 , . 8 6 ) , ( 3 . I E - 0 3 , . « 4 ) , { j . i f - r j , . b 2 ) , . . . ( j * 3E -0 3 * . 8 0 ) , ( 3 ® A S -0 3 , « 7 7 ) • ( 3 ® 6 = - J d , # 7 j o ) . ( 3 * 6 5 -0 3 * * o 9 o >»■*.« ----------------------------------- X-3u*.7E~-:i>3,-.o 45)-.(.3 -*Jd£^.a3..-,5&5-)-*43^L~.-t^3 .*-^v-) 3 - 6 - — ( 4 . I E - 0 3 , . 2 0 ) . ( A . 1 2 E -C 3 . .1 7 ) , (4 • X 4 c - JJ , . 14 ) . C 4 • 1 69 .-3 . • 12 ) , * * < ( 4 . 1 B E - 0 3 , . 0 9 ) , ( 4 . 2 E - C 3 , . 0 7 5 ) , { 4 * 2 2 c : - J o , , ) c J , ( 4 . 2 * E - 3 * • ) » . . . ( 4 • 3toE— 03 *» 0 0 8 )» ( 4 * 3 8 E —C 3 ,*0C £ ) »(**• <+£—03» , 0 0 4 ) , • • • ( 4 . 4 2 E -0 3 » . C02)» ( 4 . 4 4 E— C3 ,•0 "> 0 S ) , ( ^ . 4 u £ - u 3 * . 0 C C 7 ) , . . . ( 4 . 5 2 E -0 3 , . 000 1> . < 4 . 5 4 E -0 3 , . OOOOd) , 14 « 5oE—03 » .0 0 0 ")6 )» . . . ( 4 * 5 7 E~ 0 3 1 .0 0 0 0 5 ) . ( 4 .S 8 E - r .3 , .00CG4) DYMAMTr THLA=NLFGEN( TBTHLA. LA *BDA ) S EELA=NL F GEN(TBS ELA.LAMBDA) ------------------------------ jCJHE.T-A= ( THLA -THLTA fU /^TJ-XTAD -TK£T^NJ CSEE=( SEELA-SEEN ) /(S EED -S EEN ) DTHLA=DERIV C H =EXP (-*S *EH ) RMH-MH/K0N4 SEE l = RMH * ( S E EN—S E EO ) ■+ S E fc 0 V ELCA= I NTGRL (VE i.CAO .RC IVA) RDI V=“ (LATHH*VELC ) / ( 2 .0 #THDS ) SEE2=INTGRL I SEED, VELC/THDS ) SEEiA = I NT GRL (SEEO , VELC/>/THDS) 10 CONTINUE TERMINAL PRI NT THLA, D.THDS *DS , AH , SEELA .SEE 1 , i»££2 T I MER F I NT I M=4 . 5 6 E - 03 , CELT = S • OE-0 6 ,PRDEL=9* 1 2E -C5 K0N2=C K0N3=HS KUN4=MH IF ( ABS(SEE2—SEEN) *LT . .0 00001 ) STuP ------------------------------------DJi.= F-SFF a- ^ p f ? SL0PE=DSEE/DEL/VELCC DV£LC=(SEE2-SE£N ) /SLCFE V £LCO=VELCO-DVELC VELCAU=VELCC*{ 1 . 0 +DEL ) University of Ghana http://ugspace.ug.edu.gh 324 ---------- 1-0-0 F C1R.MAT U J-/ • WRITEC 6. 101 XDA2_L.KJlN^ .Krii.iL= -1 2Z0005 AH 3r4 CH zzt:- ic Mh RD IV VELC RDI V A VELC-c ZZ0014 SEE2 ZZ0016 SEEA RMH SEEl ZZCC1? I KUK2 KC-N3 zzoors K0N4 OSLE SLOPE DVELC v e lc c VELCAC - - OUTPUTS INPUTS PARAMS I NT E GS 4 MEM BLKS F 0 •=:Trt AN DAT* CCS s i t s a o i n o r u a o ) 14 ( 4 .1 1 )--__ * ip, -i-> ^ ^ / n 1 »\ -ENDJOB- University of Ghana http://ugspace.ug.edu.gh 325 APPENDIX G CSMP listing for calculating dispersion coefficient Dg for K+ and simulation of c(A) from the computed DS(A) values for Akuse clay University of Ghana http://ugspace.ug.edu.gh 326 ****CONTINUDUS SYSTEM MODELING PROGRAM**** * * * VERSION _E DS FOR POTASSIUM AND SIMULATICN CF CONCENTRATION (AKUSE .CLAY) UNI TS KEQ—KILO—E QUIV^LENT S K G=K I LOG RAMS M=M ETERS S = S ECONDS GLOSSARY _ O F S_YM3CL_S _____________________ L AM DAN=LAM BDA .CT IN F IN IT Y ( = F IN T IM J THLA=JNTERPOLATEC CURVE FOR T H ETA-i_ AMoDA FUNCTION V EL K= PRODUCT CF THE WATER CONTENT .D IoPERS IUN COEFFIC IENT AND THE DER IVATIVE OF ThE CCNTRATICN —LAMriD A FUNCTION VELKO=VELK AT LAMBDA ECLALS ZEFC V ELKA=HALF _ PA 1F OF VELK T HROUGH THE FACTOR 1 *0 CO 1 V ELKA C=VELKO M LLT IPL IEC EY 1 .0 0 0 1 S EEO= CONCE NTR AT I ON OF FCT^SSIUM IN SOLUTION AT i_ AMBOA EQUALS ZEPO SEE N = IN IT IA L CCNCENTRAT ICN OF FGTASSIUM IN SOLCTICN IN THE MOIST SOIL DSFl A r=DER IVAT IVE O F CCNCFNTRATION-LAMODh FUNCTI CN AT LAMcD i EQUALS ZERO SEEL A= I NT ER POL AT ED CURVE F CR CCNCENTRAT ION—i_AMEDA FUNCTICN T3THLA=TABLE CF fcATER CCNTENT AND CORRESPOND1NG LAMBDA VALUES TBSEL A=T ABLE CF CONCENTRATION CF POTASSIUM IN SOLUTION AND CORRESPONDING LAMBDA VALUES ...A0SJ.S0=AD^.0RPl.XQ l5i-JLaim iERii________________ S S=INTERPOLAT E C CURVE FCR K ILO -EQU IVALENT OF POTASSIUM ADSGRBED PER KILOGRAM S C IL . INTERPOLATION IS WITH RESPECT TO LAMBDA THRCUGh ThE ACSCRPTICN ISOTHERM DSSC=DERIVATIVE CF SS AS AFUNCT ION OF LAMBDA AT LAMBDA EQUALS ZERO DSEELA=D£F.l VA.T.I VE._a£__C-Cii.C£J^LB.A T J ON —L ABUP A._KEl AT ICN DSS=DEP.I VATI VE OF SS InITH RESPECT TO LAMdDA D= SC IL WATER D IF FU S IV ITY DTHLA=DERIVATIVE OF WATER CONTENT * I T t t RESPECT TC LAMBDA C = I NT EGRAL VALUE OF THETA FROM LAMBDA EJUALS ZERC TC LAMBOA K 0N2=DE F IN IT E INTEGRAL VALUE OF THETA FROM LAMED-* EQUALS ZERC T LAMDAN .t=LAM3DA AT ELKXJKJ_____________________ DS=DISPERSION COEFFIC IENT D IV = DERI VATI VE OF VEl.K k IT h RESPECT TO LAMoOA D IVA=HALF PA IR CF D IV THRCUGH THE FACTOR 1 .0 0 0 1 SEE1=SIMULATEC CONCENTRATION OF PCTASSIUM IN SOLUTION SEE A=HALF PA IR OF SEE1 THROUGH THE FACTOR 1.CCC1 D ^FF^C I FFFF.ENCE BETWEEN SEEA ANC..SEEI . A1__FI NT.1M _ . _ SSLOPE=RATIO CF DSEE TO THE DIFFERENCE oETWcEN VELKAO AN'C V ELK J DRVEL=CORRECTICN FACTOR RENAME TIME=LAM ECA T I AL INCON VELK0=—.53 3958S E— C4 • FLAG= C * SEEN = 0 • 0 . TH ET A N = .1 0 > .» *. R n= . 2 4F - Ci?, . 1 AMDA . 5 6 F — 0 5 . C EL= .0 0 01 . S£ EO=i ----_ _ PARAMETEP D S E LA 0 = -2 .E 1E -0 1 » D S £0=—2 . 3E—0 5 * RHO = 1 . C3E03 CONSTANT K0N2= lo O .K O N 3 = 1 .0 .I= C » J= 0 .D T H LA O = - l .0EC1 VELKAO=VELKC*(1. O + D E L ) FUNCTION TBTHL A= ( 0 . 0 » . 5 2 6 ) . ( 1 .0 E -0 4 » .5 2 5 ) t ( 2 • 0 E - 04 , • 524 > , ( o . 0 E -4 , . S£ ) . ( 1 . 2 E—3 , . 5 1 5 ) . ( 1 . 8 E - 3 . . o l ) • ( 2 . < iE -3 * .5 0 5 ) . . ( 3 . 2 E - 3 , , 4 8 7 5 ) . ( 3 . 4 E - 3 , o * 8 0 ) , ( 3 .5 E -3 » ^ 4 7 5 ) . ( 3 . C 5 - 3 , .4 7 0 ) ( 3 . 7 E - 3 . .4 G 4 ) » (2 . 6 E - 3 . . 4 5 5 ) . ( 3 . 9E -3 . . •+<+ 75 ) , ( 4 . C E -3 , . 4 3 5 ) ( 4 . I E - 3 , .4 2 0 ) . < 4 .2 E -2 . . 4 0 ) . ( 4 . 3 E - 3 . .3 7 0 ) . < A • 4 0 F - 3 . .3 3 ) » ( 4 . 4 6 E -3 , .2 9 5 ) , ( 4 .5 0 E - 2 , .2 5 0 ) , { 4 . 5 6 E - 3 , .1 0 ) . ( 5 .0 E - 2 » .1 C ) University of Ghana http://ugspace.ug.edu.gh 327 FUNCTION Tb SEL A=( C.O i 1 *0 9 ) » ( ** • OE— y<* * 1 . ) » ( 8 . GE— 0 4 * 1 . 0 5 5 8 ) * . . • ( 1 . 2 E - 3 , 1 e0B97 >, ( 1 .4 E -2 , j . o„fcL?ei , (A ,6 6 5 - 3 * 1 . 0 8 ) . . . . ( 1 . 8 0 E -3 , 1 . C£) , . . . ( 1 .9 E - 3 , 1 . 0 3 ) , (2e OE-3 . * 9 9 ) * ( 2 . 1 E - 3 . . 9 J ) > ( 2 . 1 4 E -2 , * 9 0 ) , . * * ( 2 . 2 E -3 , .8 1 ) , < 2 . 2 4 E - 3 , . 7 0 ) , ( 2 . 2 6 E - 5 , ,o O > • 1 2 . 2 8 E -3 , . 5 0 ) . . . . ( 2 . 3 0 E - 3 , .4 0 ) . ( 2 . 3 4 E - 2 , . 3 0 ) . ( 2 . 4 2 E -3 * . 2 0 ) . ( 2 . 5 C E - 3 . . 1 3 ) . . . . ( 2 .5 4 E -3 , . 1 0 ) , (2 . 6 0 E -2 ♦ .0 7 ) , C 2 .7 0 E -3 , . 0 S ) , ( 2 . 8 C E - 3 , . 0 3 ) , . . . ( 2 . 9E -3 , . 0 2 5 ) , (2 . 0 E -2 , . C £3J ,J.2 . 2 E -3 * • 021 ) . ( 3 . 4 E -2 . . C I S ) . . . . ( 3 • 6E—3 , e 015 ) , (3 • 6 E— 3 , • 0 1 ) , ( 4 * 5 c £ - J , l . O E - 4 ) , ( 5 . 0 E - 3 , 0 . 0 ) FUNCTION AD S ISO =C 1 .C E -C 4 .1 . 0 E -C 6 ) . ( 2 . 7 H -0 * , 2 . 5 E -0 6 ) * . . . ( 3 . 8E -0 4 » 3 . SE— 06 ) , ( 6 • 6 E -04 » 6 • CE-Ofc) • { 1• 4 E -0 3 . 1« C E -0 5 ) . . . . ( 2 .9 E —0 3 ,1 . 9E -05 ) , (4 . 3E -C3 • 2 . £E-Ot>) . ( 7 . OE-C 3 , 3 . EE -05 ) . . . . ( 1 . OE-C2 , 4 . 4E -C 5 ) , { 1 . SE -C 2, 6 . C E -0 5 ) , < 3 . OE-02 .7 .E E -05 ) . . . . ( © 0 6 8 . • 1 1 E - 03 )_»(_ o jg E.r 0_3J_» !_• 4 5 0^.1 , I * 6 » •22E— 0 3 ) . . . . ( 1 . 0 . . 24E -C -2) . (1 . 0 9 , . 24 7E—C 3 ) DYNAMIC T HL A=NLF GEN ( TBTHL A»L>*> SEEA=INTGRL (SEEO » VELK A/~\ HDS ) NOSORT 10 CONTINUE TERMINAL PRINT pJ.E ,T hLA JL-AmACJJTJ-iOS. DS i SEELA^-SEE 1 T IM E * F lN T IM = 4 . 5 6 E -0 2 ,FPDEL=9 . 12E -D 5 , DELM1N=4 . E6E -20 IF (F L A G .LQ . 1 ) GO TO IE 1 = 1+1 KON2=C K0N3=G1 1 F ( 1 «G£*.3 J ..FLAG=_1 - - W R ITE (6 , IOC.) KCN2 * K 0 N 2 100 F CRMAT( / / / • KON2 , KON3 —' , 2E 1 5 • 6 ) 1F ( F LAG .EQ .0 ) GG TO 20 15 J=J+1 I F ( ABS (SEE l—SEEN) .L T . 1 . 0 E -1 0 ) STOP _ OSEE=SE£.A=-_S£.El__________________________________________ -- - SSLCPE=DSEE/DEL/VELKC DRVEL=(SEE l-SEEN ) /SSLCFE V ELKO=VELKO—DRVE L VE LKAC=VELKC *(1 ,0+DEL) University of Ghana http://ugspace.ug.edu.gh 328 w R IT E ( 6 * 1 0 1 ) DSEE.SSLCPE.DRVEL 101 F OR MAT ( / / / • OSEE_. SSLCFE *pRVEL=„’.» 3 E 1 5 .6 )_ W R IT E ( 6 ♦ 1 0 2 ) VELKC.VELKAC 102 FORMAT( / / 1 CALL RERUN WITH VELKG. VELKAU=• • 2 E2 0 .7 ) £ 0 1 F ( I #GTe 10 ) ST0=> 1 F ( J • GT. 7 ) STOP CALL RERUN END __ ___ STOP GUTPUT VELK.AC VARIABLE SEQUENCE SEELA DSEELA AA THLA C SS DSS SB E l DTHLA KON1 E FF G D LATMAD G1 T HD S DS ZZ0007 D IV VEL K _ DIVA VELKA ZZ0011 _S_EE1 __7.ZCC 13 SEE A ZZCC 1 4 ZZCC15 1 K0N2 K0N3 FLAG J DSEE SSL uFE CRVEL VELKC VE LKAC OUTPUTS INPUTS PARAMS INTEGS + MEM BLKS FORTRAN DATA CDS 4 6 (5 0 0 ) 1 0 7 (1 4 0 0 ) 2 2 ( 4 0 0 ) 8+ 0= 8 (3 0 0 ) 5 5 (6 0 0 ) 30 ENDJOB University of Ghana http://ugspace.ug.edu.gh 329 APPENDIX H CSMP listing for calculating x> 'P an IE a iS lY U X * „D B B X V jM iM E ._ C £ _ J ifA .T E R ___ - * CONTENT WITH RESPECT TC LAMBDA AND THE DER IVAT IVE OF CHI OF * WATER WITH RESPECT TO LAMBDA AT LAMDAN { =LAMBDA AT F IN T IM ) * QVELN=PRODUCT.CF SO IL WATER D IFFUSIV-ITY .. DER IVATIVE OF WATER * CONTENT WITH RESPECT TO LAMBDA AND THE DER IVAT IVE OF PS I FOR * WATER WITH RESPECT TO LAMEDA MINUS Q EVALUATED AT LAMDAN _(=LAMBDA.. AT FJ.NTJM). ______ . * Q=P RO DUCT OF SO IL WATER D IF FU S IV IT Y .D ER IVAT IVE OF WATER * CONTENT WITH RESPECT TC LAMBDA AND THE DER IVAT IVE OF CH I OF *___ WATER .SQUARED..,............... . ....................................... .................. ..... „ ... * R=Q MULT IPL IED BY THE DIFFERENCE BETWEEN TWICE ThE DER IVAT IVE * OF PS I FOR WATER WITH RESPECT TC CHI FOR WATER AND THE DERLVAT.I VE_0_E__CH.I Jr_DR__WATEB. W_I TH RESPECT TO LAMBDA...................... ............. .. . * QS= THE PRODUCT OF WATER CONTENT * THE DER IVAT IVE CF CHLORIDE * CONCENTRATION WITH RESPECT TO LAMBDA »AND THE DERIVATIVE OF ...CHI. FOR ...SALT ..WIXH..i?£EPEC.T .TC..LAMBDA ..SQUARED * RS=QS MULTIPL IED BY THE DIFFERENCE BETWEEN TWICE THE DERIVATIVE * OF PS I FOR SALT WITH RESPECT TC CHI FOR SALT AND THE *____ D.EBLVATJLVE—QE—CHI FOR SALT W ITH RESPECT TO LAMBDA_____________________ _ .._ * SVELN=THE PRODUCT OF SO IL WATER D IF FU S IV IT Y .THE DERIVATIVE * OF WATER CONTENT WITH RESPECT TO LAMBDA AND THE DERIVATIVE CF * . ... OMEGA F.OR WATER ..WITH RESPECT... T.C.. LAMBDA.. MI.NLUS .R EVALUATED AT * LAMDAN ( =L AMBDA AT F IN T IM ) * RVELNS=THE PRODUCT CF WATER CONTENT .THE O ISPERSICN COEFFIC IENT * THE PEP IV ATJ.YE_OF. THE CHLORIDE CONCENTRATION W lTh RESPECT . TC LAMBDA * AND THE DERIVATIVE OF C h i FOR SALT WITH RESPECT TC LAMBDA * EVALUATED AT LAMDAN ( = LAMBDA AT F IN T IM ) * QVELNS=THE PRODUCT OF WATER CONTENT .THE D ISPERSION COEFFIC IENT, * THE DER IVAT IVE OF CHLORIDE CONCENTRATION WITH RESPECT TO * LAMBDA AND THE DER IVATIVE OF PS I FOR SALT WITH RESPECT TO _*____ L.AMBDA_M_INJJS _ _Q 5 _EVALUAX.E.Q_AT. LAMDAN ( =LAMBDA AT F IN T IM ) ____________ * SVELNS=THE PRODUCT OF WATER CONTENT .THE DISPERSICN COEFFIC IENT, * THE DER IVAT IVE OF THE ChLORIDE CONCENTRATION WITH RESPECT TC * LAMBDA AND THE DE3 I VAT I VE ..OF .OMEGA OF .SALT W ITh RESPECT TO * LAMBDA EVALUATED AT LAMDAN ( = L AMBDA AT F IN T IM ) * SEELAO=DERIVATIVE OF ThE CHLORIDE CONCENTRATION WITH RWSPECT TC * LAMBDA EVALUATED AT L A MDAN ( - L AMBDA AT F IN T IM )____________________________ * SEE0=CHLORIDE CONCENTRATION AT LAMBDA EQUALS ZERO * SEE N = IN IT IA L CHLORIDE CCNCENTRAT I ON OF THE MOIST SOIL * DCONDQ=DERIVATIVE OF THE HYDRAULIC CONDUCTIVITY WITH RESPECT TO * LAMBDA EVALUATED AT LAMCAN ( =LAMDA AT F IN T IM ) * RVEL=THE PRODUCT OF SO IL WATER D IF F U S IV IT Y , THE DER IVATIVE CF * WATER CONTENT WITH RESPECT TO LAMBDA AND THE DERIVATIVE OF____________ * CHI OF WATER WITH RESPECT TO LAMBDA * RVELO=RVEL EVALUATED AT LAMBDA EQUALS ZERO * . Q VEL=THE PR0D.UCT,.QF_.,S.0 I L WATER. DI.FF.U_SJ V.I TY_.__»THE 0 EP I.V AT I V E .. CF * WATER CONTENT WITH RESPECT TO LAMBDA AND THE DERIVATIVE OF * PS I OF WATER WITH RESPECT TO LAMBDA MINUS' Q _*--------QVELQ^QV£L_Ey_ALUATED AT L AMBDA EQUALS ZERO_________________________________ * SVEL= THE PRODUCT OF SO IL WATER D IF FU S IV IT Y .THE DERIVATIVE OF * WATER CONTENT WITH RESPECT TO LAMBDA AND THE DERIVATIVE OF * OMEGA FOR WATER WITH RESPECT TC LAMBDA MINUS R * SVELO=SVEL EVALUATED AT LAMBDA EQUALS ZERO University of Ghana http://ugspace.ug.edu.gh 331 RVELS=THE PRODUCT OF WATER CONTENT, THE DISPERSION COEFFIC IENT—THE DERIVAXLttE-O.E-T.HE. ChLORIDE .CQNCENTRAT rnw u IT h RESPECT._XC------ LAMBDA AND THE DERIVATIVE OF CHI FOR SALT WITH RESPECT TO LAMBDA VEL0= RVELS EVALUATED AT .LAMBDA .EQUALS „ZERQ ____ QVELS=THE PRODUCT OF WATER CONTENT .THE D ISPERSICN COEFFIC IENT, THE DER IVAT IVE OF THE ChLORIDE CONCENTRATION WITH RESPECT TC —L A M B D A A N D—THE D E R I V A T I V E OF P S I F O R SA I T W I T H R E S P F C T T O __________________ LAMBDA MINUS QS QVELOS=QVELS EVALUATED AT LAMBDA EQUALS ZERO SVELS=THE PRO.D.U.CT...OF...T.HE WA.T.ER„.COKTENT^__T.id£_D.ISe£RSICN .COEEE.ICI.ENT.. THE DER IVAT IVE OF THE ChLCRIDE CONCENTRATION W lTb RESPECT TO LAMBDA AND THE DER IVAT IVE OF OMEGA FOR SALT WITH RESPECT TC LAMBDA M INUS RS____________________________________________________________________ SVELOS=SVELS EVALUATED AT LAMBDA EQUALS ZERO THLAO=THE DER IVATIVE OF THE WATER CONTENT WITH RESPECT TO . .LAMBDA ..Ei/ALUA.TED ..AT .LAMEDA EQUALS ZERO......................................._ ......................... T HETA 0=WATER CONTENT AT LAMBDA EQUALS ZERO DPSIO=THE DERIVATIVE OF PS I FOR WATER WITH RESPECT TO LAMBDA F V A I U A T F O A T L A M B D A F O i l f l l ^ 7 F R n _________________________________________________________________ DCHI0=THE DER IVATIVE OF CHI FOR WATER WITH RESPECT TO LAMBDA EVALUATED AT LAMBDA EQUALS ZERC ...OMEGA 0= THE .DERIV.AT1 .V£ OF .OMEGA. OF WATER WITH RESPECT TO LAMBDA EVALUATED AT LAMBDA EQUALS ZERO DCHISO=THE DERIVATIVE OF CHI FOR SALT WITH RESPECT TO LAMBDA „_EV ALUAT_ED._AT LAMED A £ Q.U.ALS Z.EB C. _________________________ DPSISO= THE DERIVATIVE CF PSI FCR SALT WITH RESPECT TO LAMBDA EVALUATED AT LAMEDA EQUALS ZERC K0N2=THE I NTE.GRAL...VALJJ£_DE....T.HE...WATER CONTENT .F.ROM LAMBDA EQUALS ZERO TO LAMBDAN (=LAMBDA AT F IN T IM ) G=THE SUM OF THETA TIMES LAMBDA AND TWICE THE SO IL WATER D IF F USTVITY MhLT P L l E RESPECT TO LAMBDA KQN3= THE INTEGRAL VALUE CF THE PRODUCT OF G AND ThE DER IVATIVE OF. THE .CHLORIDE .ON CENT RATION .WITH RESPE£-T.-_TjQ LAMBDA FROM LAMBDA EQUALS ZERO TO LAMDAN ( =LA MBDA AT F IN T IM ) THE TA N = IN IT IA L SO IL WATER CONTENT _JS££=TABLE_ OF CHLORIDE C0N.C£NT_RAI1.ON. .V.ERS.U..S.J-AMBDA__________________ T HET A=T ABL E OF WATER CCNTENT VERSUS LAMBDA CQND=TABLE OF HYDRAULIC CONDUCTIVITY VERSUS WATER CONTENT T HL A = L I NEAR I.NTERPOLAT I.CN OF THET A -L AMBDA EXPERIMENTAL DATA PROVIDED IN FUNCTION THETA S EE LA =L I NEAR INTERPOLATION OF EXPERIMENTAL CONCENTRATION _J£EB S.U S_L AMBDA_..DATA _. PRO.V_IDED__.LN __FU NCXLON_S££______________________ HCO ND = NON—L IN EAR I NTERPCLATI ON OF HYDRAULIC CONDUCTIVITY VERSUS LAMBDA THROUGH ThE DERIVED HYDRAULIC CONDUCTIVITY VERSUS THETA PROVIDED IN .FUNC T ION ..COND A=DER IVATIVE OF WATER CCNTENT ‘ w ITH RESPECT TO LAMEDA C=INTEGRAL VALUE OF THETA FROM LAMBDA EQUALS ZERC TO l AMDAN _L=L.AM 8_QA . A T.JFIKLIM .}.______________________________________________________________ D=SO IL WATER D IF FU S IV IT Y CTHET A=DIMENS IONLE SS WATER CONTENT CSEE = DIMENSIONLESS ..CHLORIDE CONCENTRATION C1 = THE DERIVATIVE OF THE CHLORIDE CONCENTRATION WITH RESPECT TO LAMBDA _D_S =_D IJ5.P.ERS 1.0 N C.O.EF. FJ CI£_NT_______________________________________________________ THD S= THE PRODUCT OF THE WATER CCNTENT AND THE DISPERSION COEFFIC IENT RD.i.V=THE D.ERI.VAT.iyE....QF Y 1 .JU T K RESPECT. .TU...LAMP.DA—IN..EQU AT.ION. .4.. 136 . QDIV=THE DERIVATIVE OF Y3 WITH RESPECT TO LAMBDA IN EQUATION 4 .1 3 S SOIV=THE DERIVATIVE OF Y5 WITH RESPECT TO LAMBDA IN EQUATICN 4 .1 4 0 —RDJJISs The PER IVATI VE OF Y2 WITH RESPECT TO LAMBDA IN EQU AT I CN__4.«_13_7 QDIVS=THE DERIVATIVE OF Y4 WITH RESPECT TO LAMEDA IN EQUATION 4«139 SDIVS—THE DERIVATIVE OF Y6 WITH RESPECT TO LAMEDA IN EQUATION 4 .1 4 1 CH I=CH I FOR WATER PS I=PS I FDR WATER University of Ghana http://ugspace.ug.edu.gh 332 □ ME GA=OMEGA FOR WATER ^0±LLS=.Ct±I—ELQ.B S.AL.T._____ * P S I S = P S I FOR S A LT * OME GAS=OMEGA FOR SALT RENAME T I ME=.L AMD A IN IT IA L I N C O N D E L = « 0 0 0 0 1 . R V E L N = 0 * 0 . Q V E L N = 0 * 0 . S V E L N = 0 * 0 . * * * R V E L ^0= j^ j A Z 1 _ 5 E -_ 0 A j _Q V E LC = . 3 6 6 5 B E - 0 7 . S V E LD = .b 0 4 4 2 E—1 0 . DCH 10 = 1 . 384 E -0 2 , DPS I 0=1 . 51 0 3 E -0 4 , DCHI S 0= . 4 58 1 5E -02 PARAMETER THLAC=—1 . 0 E01. J = 0 . THETAO=. 5 2 0 . I = 0 ,K C N 3= 1 .0 . ,.SEEQ=. 92 »SEEN=*0 0 006-».SEELAB=-J.«.0-____ ________ _________ ____ CONSTANT THETAN= * 1 0 . K0N2=1•0 . LAMDAN = 4 *56E— 03 * oo * DCONDO=—5 . 1 096E - 03 * R VEL NS=0. 0 .QVELNS=0. 0 . SVELNS=0 .C . . -D-P-SXS0 = JL._QB_6.£-05.ji OMEGA0=1_. 3 8 6 3 E -0 6 . . . . ______________ VE L0=—.3 2 9 8 2E—0 6 ,QVELCS=.1 5 2 4 9 5 5 -0 9 .SVELQS= . 334 872E -10 FUNCTION THETA=( 0 . 0 . . 5 2 6 ) . ( 1 . 0 E - 4 , .5 2 5 ) . { 2 . C5 - 4 . .5 2 4 ) . . . ( 6 c 0 . E - 4 . , . * 52.) . ( l e 2 E - 3 • • 5 1 5 ) • ( l * 8 E - 3 , * . 5 1 ) . ( 2 . 2 E - 3 . * 5 .05. ) . . . . . ( 2 . 4 c— 3 » . 5 0 4 ) . ( 2 . 5 E— 3 » •50 3 ) . < 2 . 6 E -3 • . 5 0 ) . C3 . O E -3 • . 4 9 5 ) , . ( 3 . 2 E - 3 . . 4 8 7 5 ) . ( 3 . 4 E - 3 . .4 8 0 ) • ( 3 » 5 E -3 » • 4 7 5 ) , ( 3o 6 E -3 .p 4 7 0> . ..(.3 . 7 E -3 ». . 4 64 i . , C5 . 8.E-3.,. .4 5 5 ) j { 3 . 9 E -3 . . 44 7 5 ) . ( 4 . 0 E - 3 . «43.5). ( 4 . 1 E—3 . . 4 2 0 ) , ( 4 . 2E—3 * .4 0 ) ,< 4 . 3 E - 3 . . 3 7 0 ) . ( 4 . 4 0 E - 3 . . 3 3 ) , . ( 4 . 4 6 E - 3 . * 2 9 5 ) . ( 4 . 5 0 E - 3 . * 2 5 0 ) , { 4 . 5 6 E -3 , . 1 0 ) . ( 5 . O E - 3 . . 10 ) .FUNCTION SE.E= 10...C.»...92.) . Ll_. 0E -C 4 ., . 9.1 991^1Z»G E -0 4 . .9.198.) . ( 3 . 0 E - 0 4 . . 9 1 9 7 ) . ( 4 , 0 E - 0 4 . • 9 1 9 6 ) » ( 5 . 0 E - 0 4 . • 9 1 9 5 ) , « « » < 6 o OE—0 4 •* 9 1 9 4 ) •{ 7 . 0 E-04 . . 9 1 9 3 ) . ( 8 . 0 E - 0 4 . . 9 1 9 2 ) . . . . I S ... 0.E-C 4 . . .3 1 9 ,1 ) , LX ,.0 .E -0.3.,. 9 1 SJL,X1._1E- 0 3j_._21 8 . S J ___ ( 1 . 2 E - 0 3 . c . 9 1 8 8 ) . C 1 * 3 E—0 3 , . 9 1 8 7 ) . ( 1 . 4 E - 0 3 , . 9 1 8 6 ) . . . . ( 1 . 5 E - 0 3 . .9 1 S 5 ) . ( 1 . 6 E -0 3 . .9 1 8 4 ) . ( 1 . 7 E - 0 3 . . 9 1 8 3 ) . . . . . 1 1 . 8 E r 03 » » 9.1 82.).*.L.l.o.9.E-£a.».* 9.1 a U *.L2 • 0 E t03 ., o 9.1 3 ) *.» o.«................ ( 2 . I E - 0 3 , . 9 1 7 9 ) , ( 2 .2 E - 0 3 . .9 1 ) , . . . ( 2 .4 E -C 3 , . 9 0 5 ) ,C 2 . 6 E - 0 3 . . 9 0 ) . ( 2 . 7 E -0 3 , . 8 9 ) , ( 2 . 8 E -0 3 . . 8 8 ) . ( 2 . 9 E - 0 3 . . 8 7 ) . _(_3_. .0_£=0_3_»_,_8.6.1. ( 3 . IE—0 3 . . 8 4 ) . ( 3 .2 c—03 .,..._a.2_)_, . ( 3 . 3 E - 0 3 . . 8 0 ) . ( 3 . 4 E -0 3 . . 7 7 ) . ( 3 . 5 E -0 3 , . 7 3 5 ) . ( 3 . 6E -0 3 . . 6 9 5 ) , . . ( 3 * 7E—0 3 . * 6 4 5 ) . { 3o 8E— 0 3 , * 5 8 5 ) , ( 3 * 9E—0 3 , • 5 0 ) . ( 4 * 0 E - 3 . * 3 8 ) , o Bo ( 4 . IE —0 3 . . .2 0 ) , (4 . 12.E—03 • ..1 7 )_• 14...-1AE—03.j C.4 ..16= -3 ., . l 2 .)j . . ( 4 .1 8 E - 0 3 . . O S ). ( 4 .2 E—C 3 . .0 7 5 ) . ( 4 . 2 2 E - 0 3 . . 0 6 ) . ( 4 . 2 4 E - 3 . . 0 4 ) , * ( 4 .2 6 E -0 3 . . 0 3 ) , ( 4 . 2 8 5 - 0 3 , . 0 2 ) , ( 4 . 3 S - 0 3 . . 0 1 5 ) , ( 4 .3 « E -0 3 , . 0 1 ) , JL 4., 3 6E -0 3 . . . 0.0 8J » L4 . 3..8E-C.3,-. 0.0 _____________ (4 * 42E—0 3 . • 0 0 2 ) . ( 4 0 4 4 E - 03 . * 0 009 ) , C4 • 4 o E -0 3 , • 0 0 0 7 ) , • • c ( 4 .4 6 8 8 E -0 3 . .0 0 0 6 ) . ( 4 . 4 8 E -0 3 . . 0 0 0 5 ) . ( 4 . 5E -0 3 . .C 0 0 3 ) . . . . ( 4 .5 2 E - 0 3 , ..0001 ) , . ( 4 . 5 4 E -0 3 . . 000 08 ) , ( 4.. 5 6 E -0 3 , * 0C006) »«o.a ( 4 .5 7 E -0 3 . .0 0 0 05 ) . ( 4 .5 8 E -0 3 , .00 004 ) FUNCTION COND=(* 1 . 3 . 6 E -0 8 ) . ( . 142 o . 3 . 9 E -0 8 ) , ( . 1 8 5 2 , 4 . 2 E - 0 8 ) , . . . ( • 2 2 7 .8 4 •_ 8.5 —0 8 _ C e_27 04 r 6 •O E - 0 0 ) .J. • 2 9 6 .7 * 8 5 - 0 8 ) . <>313*1 iO 5 - 0 7 ) , ( . 3 3 . 1 . 4 5 - 0 7 ) . ( . 3 4 71 . 2 .0 E -0 7 ) , ( . 3 5 5 6 , 2 . 5E -C 7 ) . . . . ( .3 8 9 7 .6 . 0 E - 0 7 ) , ( . 3 9 8 2 , 7 . 6 5 - 0 7 ) , ( * 4 1 5 2 , 1 .2 E -0 6 )•<> •• ( . 4 2 3 8 . 1 • 5£ —0 6 ) • ( . 4 4 0 8 . 2 .3 E -0 6 ) , ( . 4 5 7 8 , 3 . 5E -0© ) . . . . ( . 4 7 4 9 . 5 . 4 E -0 6 ) . ( .4 8 3 4 . 6 . 7E -C 6 ) . ( . 4 9 1 9 , 8 . 4E - 06 ) . . . . ( * 5 1 7 5 , 1 * 6E —05 )» ( * 5 2 6 , 2 * 0 E -0 5 ) RVELA0=RVELO*( 1. 0+DEL ) ____________________________________________________ QVELAD=QVELO*(1 . 0+DEL) S VELAO=SVELC*( 1 . 0 +D E L ) . VEL AO=VELO* ( 1.. 0+DEL ) QLAOS»QVELOS*( 1* 0 +DEL ) VEL SAO=S VEL OS*(1 .0 +D E L ) DYNAM I C___________________________________ T HL A=AFG EN( T hETA. LAMDA) HCOND = NLFGE N(COND.THLA) DHCOND=DERI V_(D.CDNDO, HCOND.)... A=DERI V ( THLAO ,THLA) C = I NT GRL( 0 • C .TH LA )--------- !!_Q-N I =LAMJ2AN.* T H E ! ______ E=(THLA*LAMDA)—KON1 FF—E—C G=FF+K0N2 D——. 5*G /A University of Ghana http://ugspace.ug.edu.gh 333 CTHETA=(THLA—THET AN )/(THETAO—THET AN) JC£.EE=( -£EEJLA=£EEbUV..(.S-EEO=S££m_________ SEELA=AFGEN(SEE. LAMD A ) C1=DER IV ( SE ELAC* S EELA) LATHH=(THLA*LAMDA) + ( 2 .0 *D *A ) GS=C1*LATHH HS=INTGRL(0 . 0» GS ) H S1 =KQNu3r±l£________________________________ THDS=«5*HS1/C1 DS=THDS/THL A ... NO SORT.................. . .. ................... .......... I F ( J . L E . 3 ) GO TO 10 I F ( THDS. EQ . C • f ’ ) GO TO 10 SORT RDI V=(CH I*A )-DHCOND RD IVA=( C H IA ^A )—DHCOND ..................... .Ry.Ei.sJLNT.GRL.1RVELQ .R D IV ) RVELA=INTGRL(RVE L AO , RCIVA) B 1 ~R V E L /(D *A ) B B l= R V E LA /(D *A ) CHI = IN TG RL (0 < 0 ,B 1 > CH IA= INTGRL( 0 . 0 . BB 1 ) ................... DDCHI=DERI.V (DCH IO .CH I ) Q=D*A*DDCHI*DDCHI Q D IV = l*5 *P S I*A 0 VE L= I NT G RL ( QVELO .QD IV ) Q VELA=INTGRL( QVELAO * GCIVA) ...........C C 1=1 Q V EL+Q..}/.(. A * D.).... ............ .................................................. .................. C C 2= (Q V ELA +G )/(A *D ) P S I= I NTGRL( 0 .0 .CC l 1) PS IA = IN TG RL( 0 . 0 . CC2) . . . . DPS I=D E R IV (D P S IO ,P S I ) C C 3= (2 . 0*DP SI/DDC H I ) —CDCHI .. R~Q*.CC3.......................................................................................... S D IV = 2 .0 *OM EGA*A SD IV A=2 • 0 * 0 MEGAA *A SVEL= I NT GRL(SVELO .SD IV ) SVE LA = 1NTGRL(S VE L AO » SDI VA) DD1=(SVEL+R ) / ( A *D ) DD2 = ( S V E LA +R )/(A *D ) ............ ......... .. 0 MEGA=INTGRL ( 0 • DD1 ) OMEGAA = I NTGRL( 0*> 0 ,DD2 ) DQMEGA=DERIV( OMEGAO. CVEGA) R D IV S = ((TH LA *C H IS ) - ( D *A*DDCHI )-HCOND )*C l RD IVAS=( ( THL A * CH I AS ) - (C *A # D D C H >-HCOND)*C l RVELS=INT GRL( VELO »RDI VS) R VE L A S= I NTG RL ( VE L AO , R C IV AS ) ! BB=RVELS /(THDS*C1) CH IS= INTGRL( 0 .0 , BB) CH IAS= INT GR L ( 0 .C , BBA) DCH IS=DER IV (DCH ISD .CH IS ) ........... QS=THDS*C1*DCHIS*DCHIS i QD IVS=( ( 1 .5 *T H L A *P S IS ) - ( D*A*DPS I ) + Q )*C1 QVELS=INT GRL( QVE LOS, QDI VS) Q VELAS=I NT G RL( QL A OS» Q C IV AS ) ....CC=(QVELS+QSJ./..(.THDS*C1)......... ............................................... CCA=(QVELAS+QS)/( THDS*C1) PS IS = IN TG RL( 0 .0 , CC) PS IAS= INTGRL( 0 .P • CCA) DPSIS=DERIV (DPSI.SO, PS IS ) RS=QS*(( 2 .0 *D P S I S /D C H IS )—DCH IS ) SDI VS -( ( 2 .0 *TH L4 *OM EG AS )-( D* A*DOME GA ) +R ) *C 1 S D IV AS= (( 2 . 0*THLA*MEGAAS) - ( D*A*COMEGA) +R) *C 1 University of Ghana http://ugspace.ug.edu.gh 334 SVELS=INT GRL(SVE LOS » SOI VS) ----------------S_VZL^S^LNXGRLCVZ1._S.A O .SD IVAS l__________________________________ DD=(SVELS+RS )/(THDS *C1) DDA=(SVELAS+RS) / ( THDS*C1) DMEGAS=INTGRL(0, 0 .DD> MEGAAS=I NTGRL(0 . 0 ,DDA ) NOSORT 15 C-Q-N.T-I NUE ______________________________________________________ _ TERMINAL METHOD SIMP -TIMER F I NTI M=4.56E-QJ3 • PRDEL = S_«.1-2E.—H5.»D£LI=.2.-.aSE-0.6........ PRINT CTHETA,CHI » PS I » CMEGA *CSEE »CHIS»PS IS * OM EGAS IF C J .G E .4 ) GO TD 15 ______ JL=Jjfci_ ________________ ___________________ KON2—C K0N3=HS GO TO. 2.0 ....... 15 1=1+1 IF (A B S (R V E LN -R V E L ).E Q .O .O ) STOP J F ( ABS (QVELN-QVEL )».E_C#0*0 ) _ STOP .____________________________ IF (A B S (S V E LN -S V E L ).E C .C .C ) STCP IF (A BS (R VE LN S -R V E LS )•EQ «0 *0 ) STCP -__ IF ( ABS.( QVEL NS-.Qy.ELS ).♦ EQ.».0.«.0 ). .STCP........................................ ............ I F ( ABS(SVEL NS—SVELS) .EQ .O .O ) STCP DVEL=RVE LA— RVEL ______ 0 V.E.L.1= Q y E L A.—Q V_EI______________________________________ ____________ DVEL2=SVELA—SVEL D VELS =RV ELA S-R VE L S DVELS l =Q VEL AS-r.QVELS..................... ............................. DVE LS2=SVEL AS—SV ELS SLOPE=DVEL/DEL/R VELO ______ SLOPE! H-D_VELA/DEL_/_QVEL.C__________________________________________ SLOPE2=DVEL2/DEL/ SVE LC RSL OPE=DVELS/DEL/VEL C _..QSLOPE= D.VEL S 1/.DE L /Q V.E LC S. .................................... _. SSL0PE=DVELS2/DEL/SVELCS DRVEL=(RVELN-RVELJ/SLCPE ______ DJay.EL=l QYEL.N=fl_VE L J / SL.C.PE I ______________________________________ DSVEL=(SVELN—SVEL)/SLCPE2 DRVELS—( RVELNS—RVEL S ) /RSLOPE DQVELS=(QVELNS—QVELS) /QSLOPE .................. DSVEL S=( SVELNS-SVELS)/SSLOPE R VELO=RVELO+DRVEL PV-ELO=QyELO+DO.VEL_________________________________ ______________ SVELO=SVELO +DSVEL VELG=VELO+D RVELS QVEL0S=QVELOS+DQVELS .......... SVELOS=SVELCS+DSVELS RVELAO=RVELC*(1 .0 +DEL ) ______ QVELAO=QVELC*< 1. O+DEL )__________________________________________ SVELAO=SVELC*(le O+DEL) VELAO=VELO* { 1.04-DEL) OLAOS=QVELOS*< l t O+DEL ) __ . VELSAO=SVELCS*(1 .O + DEL) W R IT E (6 ,1 0 1 ) RVELO.RVELAO JLOJL FOR MAT ( / / * CALL RERU N . WITH R VEL C . RVELAO= ' »2E20 . 7 )______ WRI TE (6 » 102 ) QVELO.QVELAO 102 FORMAT( / / * CALL RERUN WITH QVELC>QVELAO=* . 2E 2 0 • 7 J WRI TE( 6 . 1 03 ) S.VELO.SVELAO 103 FORMAT{ / / • CALL RERUN 'WITH SVELC. SVELAO=• *2 E 2 0 .7 ) WRITE(6 » 1 0 4 ) VEL 0 »VELAC _1Q 4. F ORMAT ( / / ’ CALL. RERUN W ITH_.Y ELO.VELAO= 1 . 2E2C . 7 ) _________ WRITE( 6 * 1 0 5 ) QVELOS.QLAOS 105 F ORMAT( / / 1 CALL RERUN WITH QVELOS• QLAOS=* . 2 E 20 .7 ) W R IT E (6 .1 0 6 ) SVELOS. VELSAO 106 FORMAT( / / • CALL RERUN WITH SVELCS . VELSAO=-» . 2E20 . 7) 20 I F ( J . GT.1 5 ) STOO ------------ I F { I . GT. .Jl3J._S.TQP_________________________________________ — ---------- CALL RERUN END STOP University of Ghana http://ugspace.ug.edu.gh 335 APPENDIX I CSMP listing for simulating water content profiles for various time periods for vertical infiltration of water and salt (Akuse clay) University of Ghana http://ugspace.ug.edu.gh 336 ^ * * * * C 0 N T i Nu3 u s SYSTEM M0OEl“ I Nt» PR0GRAM* : * * * * * * VERSION 1 .3 * * * T IT LE SO IL WATER CONTENT PROFILE WITH TIME (AKUSE CLAY)>______£_OMULS_________ _____________________________ * KG= K 1LOGRAMS * M=METERS * S = SE CGNDS * GLOSSARY OF SYMBOLS * C THETA=DI MENS IONLE S S WATER CONTENT ____________ * ____ TH£T.AC = S01I---- WATER CONTENT AT LAMBDA EQUALS ZERO. CH 1 Evi UAL £ ZgRQ.____ * PS I EQUALS ZE RO» AND OMEGA EQUALS ZERO * THE7 AN=1N IT IA L SCI L WATER CONTENT * T = T I ME * T HE T A= SO IL ^A TER CONTENT * L AMDA=T ABL E OF LAMBDA AND CORRESPONDING WATER CONTENT VALUES _________________ * ______C .H I TH = TABL E Q F CHI FOR W.ATER AND C f i R.R F S PflND.XN . .W-A T r .^ CQNTFNT V A L ' i r . S . * PS IT H=T ABL E OF PSI FOR WATER AND CORRESPONDING WATEP CCNTENT VALUES RENAME TIME = CT HET A INCCN THETAC = .6 2 6 *THETAN=-*10 PARAMETER T = (3 6 0 . 0 , 7 2 0 0 .0 .1 4 4 0 0 .0 .2 1 6 0 0 .0 ) FUNCTION LAMCA = £ . 1 , 4 . 5 6E -0 3 ) , ( . 2 5 , 4 . 5 E - 0 3 ) . . . . _______________________ i . ? S S . .4 > a A c - m . I . 3 3 . 4 . 4 F -n - ^ \ . ( = -------- -------------------------------------------- < « 4 ,4 .2 E - 0 3 ) , ( *42 ,4 * I E - 0 3 ) , ( e 43 5 , 4 «s0E -03) , • c . ( . 4 4 7 5 , 3 . 9 E -0 3 ) , ( .4 5 5 , 3 . 8 E - 0 3 ) . ( . 4 6 4 , 3 . 7 E - 0 3 ) . . . . ( . 4 7 ,3 • 6E—03 J , ( . 4-7£3-,-3*SE” 03 ) . ( . 4a . 3 . 4 E -03 ) . . . . ( , 4 8 7 5 , 3 . 2 5 - 0 3 ) , ( . 4 9 5 , 3 . 0 E -0 3 > , < .5 . 2 .6 E - 0 3 ) . . . . ( . 5C3 » 2 • 5 E -C 3 ) , < . 5C4 , 2 . 4E -C3 ) , ( .5 0 5 ,2 .2 E -C 3 ) . . . . _____________________ ( . ^ P a . ? . n c - n 4 ) . / l . f lc -O A 1-------------------------------------- FUNCTION CHITH=( 0 .0 , 1 . 67E—0 5 ) . ( .1 , l . Oy0c>E-0t> ) . . . . ( . 2 14 , 1 . 725E— 05 ) , ( . 3 5 7 2 , 1 . 8 6 2 2E -G 5 ) , ( . 3 6 7 7 6 , 1 .5 2 8 2 E -0 5 ) , » • o ( .4 1 0 0 8 , 1 .9 90.2E- 05-)-,- ( .. 4.262.4 , 2 .0 4t> 9 E -0 5 ) ( . 4 4 4 8 , 2 . 1 2 1 E -0 5 ) , . . . ( .4 5 2 7 2 , 2 . 1 628E-C 5 ) , ( . 4 604 7 ,2 .1 9 9 1 E -C o ) ( • 6-6986 , 2 • 2 408E - 0 5 ) , ( * 4 7 4 4 4 , 2 o 2 6 1 2E -0 5 ) ( • 4 7 6 7 2 ,2 o 2 6 9 2 E -0 5 ) , • . . I . 4 7 9 . 2 . ?7S 7F -Q 51 . ( . hROQfi. ? . P flO fiF -pS ) .1 .4 H ? h 7 .? .? f l4 !E - r 5 V . ------ ( .4 8 4 3 8 , 2 . 2 8 6 3 5 - 0 5 ) , ( . 4 8 6 0 9 , 2 .2 87E —05) , ( .4 8 7 6 ,2 * 2 8 6 I E - 0 5 ) ,«»e ( .6 5 1 2 2 , 2 .2 7 9 4 E - 0 5 ) , ( . 5 0 0 0 2 , 2 . 1 5 4 9E -0 5 ) ( . 5 0 38 3 , 2 . 08C2E —0 5 ) , . . . ( . 5 0 4 8 3 . 1 . 9 9 4 4 E -C 5 ) . ( . 5 0 6 8 5 , 1 .9 0& S E -0 5 ) • ( .5 C 57 , 1 .7 6 4 6 E—0 5 ) . . . . ( • 5 1 0 9 4 , 1 • 673E—0 5 ) , ( • 5 1 2 8 4 . i •5 0 79E—0 5 ) , ( •5 1 3 9 9 .1 * 4 0 12E—0 5 ) . . . . ( . 5 1 5 1 2 , 1 .2 8 8 5 E -0 5 ) , ( .5 1 6 2 6 ,1 .1 6 9 7 E—05 ) , ( . 6 1 7 4 , 1 . 0 448E—0 5 ) , . . . ( .^ 1 8F.4. 9 .1 3C 9F -0 6 ) . ( . 5 2 0 0 7 .7 .2 566 “ — J 6 ) . ( , 5 2 23 = . 4 -6 9 1 4 = - 0 t l ^ « --------------------------------- ( . 5 2 4 1 8 , 2 .4 3 9 1 E -0 6 ) , ( . 5 2 5 5 4 ,6 .2 7 7 4 E -3 7 ) , ( . 5 2 6 , C .O ) FUNCTION PS ITH = (C -.C , 5 .7 E -0 8 ) . ( . 1 , 5 . 6 9 1 4 E -CS ) , . . . ( «214 ,5 * 6064E -0S ) ,.( • 3 E 7-2 .-5 *2 574 E— 0 8 ) , I • 3s? 77 6 , 5 . 1 1 9 72 -0 8 ) . . . . ( . 41008, 5 . o s s s e - o s ) , ( .4 2 6 2 4 , 5 .C 2 5 E -0 8 ) , ( . 4 4 4 9 ,5 .1 3 1 4E—C 8 ) . . . . ( . 4 5 2 7 2 ,5 .2 SE— 09 ) , ( • 4 6 0 4 7 ,5c475E—0U ) . ( • 4 6 9 8 6 ,5 *8 5 9 6 5 -0 8 ) , o •» *. { . ^7444 . 6 .1 5 ^ r,F - ■*>«). i .4 7 6 7 2 . 6 . 3 1 42F—0 B ) . 1 . 4 7 9 . < - .4 7 71E -flR l . . . . ( . 4 8 r9 t> ,6 .6 4 4 5 E - r 8) , { . 4 8 2 6 7 ,6 .8 1 4 & E -0 8 ) , ( . 4 8 43 6 , 6 . 9 8 6 7 E -0 e ) , . . . ( c 4 8 6 0 9 ,7 e1602E -0 8 ) , ( •4 8 7 8 ,7 * 3 3 3 9 E - 0 8 ) , ( .4 9 1 2 2 , 7 . 6777E—0 8 ) . . . . ( . 5 0 0 0 2 , 9 . 2 9 6 9 E -0 6 ) , ( . 5 0 3 8 3 , S .7 C 5 4 - -0 B ) , ( .5 C 482 , 1 .0C 2 3 E -C 7 ) , . . . ( . 5 0 6 8 5 , 1 . 0 2 6 5E -0 7 ) • ( o 5 0 9 7 ,1 * 0 3 6 2 E -0 7 ) , ( • 5 1 0 94 , 1 o 03E—07 ) , * o c ( . 5 1 2 8 4 , 1 . 0 0 2 7 E -0 7 ) , ( . 5 1 3 9 8 , 5 .7 5 1 6 E -D 8 ) , ( . 5 1 5 1 2 ,9 . 3 8 2 9 5 - ^ 8 ) , . . . ( . 5 1 6 2 6 . f l . 9 13 8 F - f R) . ( . 51 74 . ft . 3 3 6FiF — fiS 1 . ( . 5 1 8 5 4 . 7 . 6 3 3 8 F - n S ) . . . . ... .... . . ( o5 2 0 0 7 ,6 e4 632E -0 8 ) , ( .5 2 2 3 5 , 4 .5 2 2 8 E—03 ) . ( . 5 2 4 1 8 , 2 . 5 0 1 E - 0 8 ) . . . . ( . 5 2 5 5 4 , 6 . 7 2 5 3 E -0 9 ) , ( . 5 2 6 . 0 . 0 FUNCTION MEGAT H= ( 0 * 0 , 7 *36E—11 ) , ( a 1 , 7 *3 6 i>9E— 11 ) , • o • ( . 2 1 4 , 7 .7 9 1 7E -11 ) , ( . 3 5 7 2 ,6 .5 8 1 6 E—1 1 ) , ( . 3877 6 , 5 . 7-503 E - l 1 ) , . . . j ( . 4 1 0 0 8 , 4 .6 2 5 9 E - 1 1 ) , ( . 4 2 6 2 4 , 3 .2 2 E - 1 1 ) , ( . 4 4 4 8 . 7 .5704E—1 2 ) , . . . V J . 4 5 2 7 2 . - 8 . . ( . 4 6047 . - 2 .4 2 5 F -1 1 ) . ( .4 6 9 P f. . - 4 .3 2 4 4 = -1 1 ) , . ... { . 4 7 4 4 4 , - 5 . 2 2 1 4E — 11 ) , ( . 4 7 6 7 2 . - 5 . 6 4 2 3 E - 1 1 ) , . . . ( e 4 7 9 , —5® 75 SE—11 ) , ( • 4 6 0 9 6 ,—5 * 8 6 5 9 E -1 1 ) , ( • 4 8 2 67 5« 8784E—11 ) , • • = ( . 4 8 4 3 8 , - 5 . 791 7r - 11 ) , ( . 4 8 6 0 9 , - 5 .5 9 4 E -1 1 ) , ( . 4 8 7 6 , - 5 . 2 7 8 8 E -1 1 ) . . . . ( . 4 9 1 2 2 , - 4 .2 7 2 5 E - 11 ) , ( . 5 0 0 0 2 . 7 . 1 5 0 2 E - 1 1 ) . ( . 5 0 3 8 2 . 1 . 2 7 1 7 E - 1 0 ) University of Ghana http://ugspace.ug.edu.gh 337 ( .5 0 4 83 . 1 . e 577 E- 1 0 ) . ( . 5 0 6 6 5 . 2 .4 522E -1D ) , ( . 5 0 9 7 , 3 . 1 7 9 7 E - I 0 ) , 0 94.-Z...5.849E-..1.0 > . . 4^_l_Sfc7E-l C > . I . 5 1 2 2 .S .4 . 4 85Q.F- 1H4 ( c 5 l5 1 2 » 4 e 7 239 E- 10) . < • £ 1 62 6 . 4 •8 8 69E -1 0 ) , ( * 5 1 7 4 ,4 « 9 5 8 3 E -1 0 ) . ( • 5 1 8 5 4 . 4 .9 1 1 9 E -1 0 ) . ( . 5 2 0 0 7 .4 .6 0 3E -1 0 ) » ( . 5 2 2 3 5 . 3 . 6 3 5 6 E -1C ) . ( • 5 2 4 1 8 .2 .2 0 1 E —1 0 ) . ( . £ 2 5 5 4 . 6 . 3 2 1 6E -1 1 ) . ( .5 2 6 .0 ® C ) THETA=CTHETA*( THETAC-THETAN) +THETAN A 1=NLFGE N( L AMD A ,THETA ) A 2 = N L F G E N ( C H I T H . T H E T A )__________________________________________________________ A 3= NL FGE N (PS ITH . THETA) A4=NLFGEN(MEGATH.THETA) B1=A1*SQRT(T ) B2=A2*T B 3= A3 * ( T 1 *5 ) B 4 = A 4 * T * T ___________________________________________________________ Z=81+B2+B3+B4 TIMER F I NTIM = 1 .D . PRDEL=2. 0 E -0 2 . CUTDEL=4. O h -02 PRTPLT 2 LABEL MOISTURE CONTENT PROFILE WITH TIME PRINT THETA ,A1 ,A2 ,B1 ,B 2 .B 3 ,B 4 ,2 cMn GUTPUT VARIABLE SEQUENCE THETA A4 B4 A3 B3 A2 B2 A l B1 Z ..CUTPUTS .XNg-U-X-S___ PARA.MS___INTS.GS t .-MEM B.LKS FORTRAN QA.T..A.. CO-S- 1 4 (5 0 0 ) 4 3 (1 4 0 0 ) 1 0 (4 0C ) C+ C= C (3 0 0 ) U (bO O ) 55 END JOB University of Ghana http://ugspace.ug.edu.gh 338 APPENDIX J CSMP listing for simulating concentration of Cl profiles for various time periods for vertical infiltration of water and salt (Akuse clay) University of Ghana http://ugspace.ug.edu.gh 339 #***C0NT INUDUS SYSTEM MODELING PRJGRAM**** * * * VERSICN l . j T IT LE CHLORIDE CONCENTRATION PROFILES WITH TIME (AKUSE CLAY) S________ *_____ U.NI TS___________________________ __________________________________________________ * KG=KILOGRAMS * M=METERS * KEQ=KIL0-EQU IVALENTS * S=SECONDS * GLOSSARY OF SYMBOLS. --------------------- * --------CSEE=DIMENSIONLESS CONCENTRATICN ________________________________________ * S EE 0=C0NCE NTRAT ICN OF O LCR ID E AT LAMBDA EUUALS ZERO . CHI FCR * SALT EOUALS ZERO* PS I FCR SALT EQUALS ZERO, ANC CMEGA FOR SALT * EQUALS ZERO * SEE N = IN IT IA L CCNCENTRATICN OF ChL CR1DE IN THE MOIST SOIL * CLAMDA=TABLE OF LAMBDA AND CORRESPONDING CONCENTRATION VALUES --------------------- * --------CHI S = TABLE QF CHI FOR SJLT AND CO ER = SPQ.>101 NG CCNCgNTRAT I CN V f lL iI rs * PSIS=TABLE OF PS I FOR SALT AND CORRESPONDING CONCENTRATION VALUES * 0 ME GAS=T ABLE CF OVIEGA FCR SALT AND CORRESPONDING CONCENTRA T I CN * VALUES * T =T IM E * S EE —C CNCENTRA T ION OF CI-LCRIDE IN SOLUTION _RF NAME T IM E = CS£E_________________________________________ _ ________________________________________ INCCN SEEO= o92 ,S EEN = . C0006 PARAMETER T= (363 . 0 « 7 2 00 . 0 , 1 44 DO .0 , 2 1 o 0 0 . 0 ) FUNCTION CL AMOA=(-ClOO0 4 .4 . S 8E -03 ) • ( .0 0 0 0 5 .4 * 5 7E -0 3 ) . . .< , ( . 0 0 0 0 6 , 4 .5 6E -03 ) . ( . 0 0 0 0 8 , 4 .5 4 E - 0 3 ) , I . 0 0 0 1 , 4 . 5 2 E -0 3 ) . . . . ( . 0 0 0 3 , 4 . 5 E - 0 3 ) , ( .O O 0 S .4 .4 8 E -O 3 ) , ( . 0 0 0 o , 4 . 4 o 88E -0 3 ) , . . . _____________________ i ^ Q O a 7 , 4 . 4 f i F - Q 3 ) . ( . Q 0 a Q . 4 . A A F - Q 3 | . I , Q f l ? . 4 . A A t=—f . 3 \ ------------------------------ ( . 0 0 4 , 4 . 4 E - 03 ) . ( .0 0 6 . 4 . 3 8 c - 02 ) , ( . C0 8 , 4 . 3 6 E -0 3 ) . . . . ( . 0 1 . 4 . 3 4 E - 0 3 ) »( . 0 1 5 . 4 . 2E -03 > , ( . 0 2 , 4 . 2 8 E -0 3 > , C . 0 3 ,4o26E—0 3 ) *e e a J. a0 4* 4 *-24E— 03 ) . ( 0 6 - , - 4 .2 2 5 -0 3 ) , ( . 0 7 5 , 4 . 2 E - 0 3 ) . ( .0 9 , 4 . 1 8 E -3 ) , . . . ( . 1 2 , 4 . 1 6 E -C 2 ) . ( . 1 4 , 4 . 1 4 E -0 3 ) , ( . 1 7 . 4 . 1 2 E -0 3 > , ( . 2 , 4 . 1 E - 0 3 ) , . . . ( . 3 8 , 4 . 0 E - 0 3 ) , U 5 0 , 3 . SE —03 ) , ( o 5 8 5 . 3 . 8 E -0 3 ) , ( . 6 4 5 . 3 .7 E - 0 3 ) , . . . _____________________ ( . 6 9 5 .3 . 6E -Q 2 ) . ( . 735 . 2 . 5E -C3 ) . ( . 77 . 3 . + F.-Q3 1 ____ ( . 8 2 , 3 . 2E—0 3 ) , ( # 8 4 » 3s I E - 03 ) . ( » 8 6 . 3 . 0 5 - 0 3 ) . ( • 8 7 , 2 . 9 E -0 3 ) . o .e ( . 8 8 .2 .8 E - 0 3 ) . ( . 8 9 , 2 .7 E -0 3 ) , ( . 9 . 2 . 6 E - 5 3 ) . ( . 9 0 5 , 2 . 4 E -0 3 ) . . . . ( . 9 1 . 2 . 2 E— 0 3 ) . ( . 9 1 7 9 . 2 . I E - 0 3 ) . . . . ( . 9 1 8 , 2 . OE—03 ) , ( . 9 1 8 1 . 1 . 9 E - 0 3 ) , ( .9 1 8 2 ,1 . 8 E -0 3 ) . . . . ( . 9 1 8 3 , 1 . 7E —0 3 ) . ( . 9 1 8 4 , 1 .6 E -C 3 ) , ( . 9 1 8 5 , 1 .5 E -0 3 > . . . . _____________________ ( .9 1 8 6 . 1 . 4 E - 0 3 ) . ( . 9 1 8 7 . 1 .3 E -C 3 ) . ( . 9 IB S . 1 . 2 E - 03 ) . ■c .________________ ( .9 1 8 9 ,1 . 1 E—0 3 ) , ( .9 1 5 ,1 . 0 E - 0 3 ) , ( . 9 1 9 1 , * . OE-O4) . . . . ( . 9 1 9 2 , 8 .C E—0 4 ) , ( . 9 1 9 2 , 7 .OF—04) , ( . 9 1 9 4 , fc .O E -04 ) , . . . ( . 9 1 9 5 , 5 . OE—04 ) , ( . 9 1 9 6 , 4 .0 E -C 4 ) , ( . 9 1 9 7 , 3 . 0 E - 04 ) . . . . ( . 9 1 9 8 , 2 .OE —0 4 ) , ( . 9 1 9 9 , 1 . 0 E -0 4 ) , ( . 9 2 ,0 .C ) FUNCTION C H 1S = (6 .O E -0 5 , 1 .2 5 8 4 E -0 5 ) , ( 1 . 5 6 0 3 E -0 4 , 1 .2 9 2 4E -0 5 ) _____________________ ( . 0 1 1 ,1 .3 3 7 3 E— 05 1 . ( .0 3 9 6 .1 .3 5 01 £ - 0 5.) . ( . 1 30 4 . 1 . 2S79E.-05J-. --------- ( .2 7 48 8 , 1 . 3 6 19E - 0 5 ) . ( . 4 7 4 0 8 . 1 .3 6 2 E -0 5 ) , ( . 5 5 9 1 6 ,1 .3 5 9 1 E - 0 5 ) . . c o ( .6 2 1 4 8 , 1 .3 5 4 2 E -0 5 ) , ( .6 9 3 8 . 1 .3 4 3 3 fc -0 5 ) , ( .7 3 0 5 ? , 1 .3 3 3 7 E -0 5 ) . . . . ( . 7 4 7 0 4 , 1 .3 2 8 I E - 0 5 ) , ( . 7 6 3 , 1 .2 2 2 1 E -0 5 ) , ( . 7 7 7 0 8 , 1 .3 1 5 5 E -C 5 ) . . . . ( . 7 9 1 3 6 . 1 .3 0 8 3 E -0 5 ) » ( .8 0 3 3 6 . 1 .3 0 0 6 E -0 5 ) , ( . 8 1 2 * e , 1 . 2 9 2 4 E -0 £ ) , « . * ( .8 2 1 6 .1 . 2 8 3 8 E -0 5 ) . ( .8 3 9 8 4 .1 .2 6 5 E -0 5 ) . ( . 9 0 0 0 2 , 1 .1 2 2 9E -C 5 ) . . . . _________________________________( . 9 0 4 5 8 . 1 .0 577E - 0 5 ) . ( . 9 0 9 1 4 ,S .8 898E -06 ) . ( . 9 1 Za£ iS b - ia & 5E = fl t ) a -cl. ( . 9 1 8 1 8 , 8 . 1 404E -0 6 ) . ( . 9 1 8 3 1 . 7 .5 4 2 2 E -0 o ) , ( . 9 1 8 5 4 . 6 . 5 4 1 7 E -06 ) . . . . ( .9 1 8 6 8 , 5 . 9 392 E- 0 6 ) • ( .9 1881 , 5 .3 3 4 8 E -0 6 ) , ( . 9 1 89 5 ,4 . 7 2 8 4 E - '' 6 > t . 9 1 9 0 9 ,4 .1 1 9 9 E -0 6 ) , ( . 9 1 9 2 2 , 3 .5 0 9 2 E -0 6 ) , ( . 9 1 9 4 1 ,2 . 6 9 1 2 E - 0 6 ) , . . . ( . 9 1 9 0 4 ,1 .6 6 2 3 E -0 6 ) . { .9 1 9 8 2 .8 .3 3 7 1 = -0 7 ) , ( . 9 1 9 9 £ , 2 . 0 8 9 2E -C 7 ) ( . 9 2 . 0 . 0 ) V----------------------------------EJJiHCTI ON PS 15 = (S .Q E -Q5» 4 , 15 £ -Q 8 ) i ( j ,a o t f3 t--Q 4-i _S_.-.£.Z£2E -0 9 ) » . . . -------- ( . 0 1 1 , 4 . 188 4 E -C S ) , ( . 0 3 9 fc ,4 .1 556E -0 8 ) , ( . 1 3 0 4 , 4 . 2 1 6 7 E -0 8 ) . . . . ( . 2 7 4 8 8 , 4 .2 3 9 8 E -0 8 ) , ( . 4 7 4 0 8 , 4 .2 7 5 6 E -0 8 ) , ( . 5 5 9 1 £ , 4 . 2 9 9 3 E -0 8 ) , . . . ( . 6 2 1 4 8 .4 .3 2 2 4 E -0 8 ) » ( .6 9 3 8 .4 . 3 5 4 5 E -0 8 ) , ( . 7 3 0 5 2 , 4 . 3 7 3 3 E -0 8 ) , . . . ( . 7 4 7 0 4 . 4 . 3 8 1 5 E -0 8 ) , ( . 7 6 3 . 4 . 388 8 E -0 8 ) , ( . 7 7 7 o 8 , 4 o 3 9 4 9E -0 8 ) , ooc University of Ghana http://ugspace.ug.edu.gh 340 { .7 9 1 3 6 , 4 .3 9S7E -0 8 ) , { .8 0 3 3 6 , 4 .4 0 3 X c~08 ) . ( . 8 1 24 8 , 4 . 405 I E - 06 ) , . . . , -------------------------------------------- ( . 8 2 1 6» 4 . 4 0 5 6 E —2 fl ) . ( . £3S_84_. 4 . . 4 Q 11£•— 0 8 ) . ( . 0 0 0 0 2 . 4 . 1 a 4 E - Q 8 ) ------ ------ ( c 9 0 4 5 8 , 4o 0 1 58 E— 0 8 ) , ( « 9091 4 ,3 * 8 194E -08 ) , C * 9 1 79 £ , 3 . 56E -0 8 ) , . . . ( . 9 1 8 1 8 , 3 . 18 7 9E -0 8 } , ( .91831 , 2 . 9 6 2 7E -0 8 ) , ( . 9 1 8 5 4 , 2 . 5 8 3 6E -C 8 ) , . . . ( . 9 1 8 6 8 ,2 .3 5 3 8 E -0 8 ) , ( . 9 1 8 8 1 . 2 o1218 E -0 3 ) , ( * 9 1 8 9 5 . lo 8 8 7 6 E -0 6 ) , ooe ( .9 1 9 0 9 ,1 .6 5 0 9 E -0 8 ) , ( . 9 1 9 2 2 , 1 .4 1 1 7 E -0 8 ) , ( . 9 1 9 4 1 , 1 .0 8 8 5 E -0 8 ) , . . . ( . 9 1 9 6 4 , 6 . 7 7 1 9E -0 9 ) . ( . 9 1 9 8 2 , 3 . 4 1 6 6 E -0 9 ) , ( . 9 1 9 9 = , 8 . 6 009 c - 1 0 ) » . . . >_______________________ ( , 9 2 . 0 * 0 1_________ _____________________________________________________________________ FUNCTION OMEGAS=(6 .0 6 - 0 5 , 3 .1 8 7 1 £ - 0 8 } • ( 1 .5 6 0 3 £ -C 4 , 3 .8 0 C 4 E -1C ) , . . ( . 0 1 1 , 5 . 9 9 2 7 E -1 1 ) , ( • 0 3 9 6 .5 * 4 S E -1 1 ) , ( * 1 3 0 4 . 5 * 3 1 4 6 E -1 1 ) , * * e ( . 2 7 4 8 8 , 5 . 2 5 2 7 E -1 1) , ( .4740 8 , 5 .2 D8&E-11 ) , ( . 5t»9 11 , 5 . 2 0 0 3E - 1 1 ) , . . . ( .6 2 1 4 8 , 5 .2 0 6 5 E -1 1) , ( . 6 9 3 8 , 5 . 2 5 C 9E -11) , ( .7 3 0 52 , 5 . 3 079E -1 1 ) , . . . ( , 7 4 7 0 4 ,5 * 3 4 5 2 E -1 1 ) , ( * 7 6 3 , 5 .3 8 8 6 E -1 1 ) , ( . 7 7 7 6 8 , S .43 8 5 E -1 1 ) , . . . --------------------------------------------- L_.JZ-9 . 1 3 6 , 5 . 4 9 4 2 E - 1 1 ) . ( . 8 C 5 3 6 . 5 . 5 5 5 8 E - 1 1 ) . i . 8 1 2 4 8 . 5 . 6 2 1 9 ? - ! 1 t --------- ( . 8 2 1 6 ,5 .6 9 22E -1 1 ) , ( * 83984 , 5 »84 4 8 E -11 ) , ( * 90 0 02 ,£ *6 7 5 3 E -1 1 ) , * e - ( .9 0 4 5 8 , 6 .8 032E - 1 1 ) , ( .9091 4 , 6 .8 2 6 c - 11 ) , ( .9 1 795 ,6 . 5 6 0 9E - 1 1 ) , . . . ( . 9 1 8 1 8 , 5 .9 8 4 I E - 1 1 ) , ( . 9 1 8 3 1 ,5 .6 2 7 E - 1 1 ) , ( , 9 1 8 5 4 , 5 . 0 0 9 9 E -1 1 ) , . . . ( * 9 1 8 6 8 ,4 * 6 24 1E—11 ) , (*9 1 8 8 1 ,4 .2 2 4 4 E -1 1 ) , ( . 9 1 8 5 £ ,3 .8 0 9 1 E - 11 ) , . . . ( . 9 1 9 0 9 , 3 . 3 7 7 9 E -1 1) , ( . 9 1 9 2 2 , 2 . 9 2 9 6 E -1 1) , ( . 9 1 9 4 1 , 2 . 3 r 2 5 E - l1 ) , . . . ____________________ ( » 9-1.9.6 4 .. 1 n 4b Q2E—1 1 ) . ( «_S 1 5 B2 . 7 . 55 99E - 1 2 ) . ( . 91 9