'Iliumlull:minimalJillian # L 3 1293 00080 4231 LIBRARY Michigan State University This is to certify that the dissertation entitled EXPERIMENTAL AND THEORETICAL ANALYSES 0F CABILLARY RISE THROUGH POROUS MEDIA presented by George C; Zalidis' has been accepted towards fulfillment of the requirements for PhD Crap & Soil Seiences degree in U erro'fessor 7 Date ///:3~/// f87 MSU is an Affirmaliw Action/Equal Opportunity Institution 012771 ——.——————-__._ —_ _ _-___ MSU LlBRARlES RETURNING MATERIALS: RTéce in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. dififbhflv - JllL'gSQléS! w EXPERIMENTAL AND THEORETICAL ANALYSES 0F CAPILLARY RISE THROUGH POROUS MEDIA By George C. Zalidis A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Crop and Soil Sciences 1987 ABSTRACT EXPERIMENTAL AND THEORETICAL ANALYSIS OF CAPILLARY RISE THROUGH POROUS MEDIA By George C. Zalidis The objectives of this study were 1) to examine the process of capillary rise through soils, 2) to test numerical solutions of Richards' equation in describing upward water flow in soils and 3) to use the experimental and theoretical analyses in a case study. For this purpose capillary rise experiments were performed under near constant temperature conditions using Metea sandy loam soil (Arenic Hapludalts) packed in columns of different bulk density. Adsorption soil moisture characteristic curves were obtained from these experiments. The data were analysed and information for understanding the physics of the phenomenon was gained. All curves exhibited a theta-straight region, followed by a parabolic curve flattened at higher auctions. The parabolic curve could be separated into three different zones. The different zones suggested transitions in the water content-potential relationships. These transitional zones can be associated with relative percentages of filled pores and thickness of water films. It was found necessary to use a power function in each of the three zones to successfully fit the adsorption data. Conductivity and diffusivity functions were determined from adsorption data. FINDIT was tested against a finite-difference scheme and experimental data. When moisture characteristics, cumulative infiltrations and fluxes were used as criteria, the agreement between the two models and experimental data was very good. FINDIT performed excellently using internally generated derivatives and predicted extremely accurate experimental profiles. It agreed well with a simple algebraic equation for calculating cumulative infiltration. To test the influence of a wetting agent in capillary rise, both treated and untreated soils were used. When experimental and simulated wetting curves of these soils were analysed, the treated soil exhibited higher cumulative infiltration and rise of wetting front but lower water capacity. A change in the kinetics of capillary rise was observed when wetting films approach a critical thickness which resulted in a significant slow down of the wetting process. At equilibrium, water content gradients serve as the upward driving force but are countered by gravity. The wetting curves obtained from capillary rise can be used to determine capacity and intensity factors in addition to proportionality coefficients and flux factors of soil water . To my parents ii ACKNOWLEDGEMENT The completion of this thesis would not have been possible without the educational support provided by the Department of Crop and Soil Sciences of Michigan State University and the financial support of the late George Bouyoucos whose contribution to my opportunity will always be remembered. I would also like to thank the Aristotelian University of Thessaloniki, Greece, and especially my colleagues from the Department of Soil Science and Land Reclamation for their trust and support. I would like to express my thanks to all my committee members; to my major adviser Dr. Raymond J. Kunze for his guidance, support, and encouragement throughout the completion of my thesis; to Drs. B. Ellis, J. Ritchie, R. Wallace and P. Rieke for their help and support. I would also like to thank Dr. D. L. Nofziger for sending me a copy of his model without which the modeling task would have been more difficult. I owe special thanks to my parents and my brother for their support and encouragement which began from my youth, to all my friends and their wives and especially to Alex Kopatsis and his wife whose support, encouragement and inspiration went beyond the limits of our friendship. I also owe many thanks to my fiancee Connie for her unlimited love and understanding. iii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES . LIST OF SYMBOLS . INTRODUCTION . CHAPTER I: CAPILLARY RISE IN METEA SOIL. 1. INTRODUCTION . 2. MATERIALS AND METHODS . 2.1 MATERIALS 2.2 PREPARATION OF THE FLOW SYSTEM . 2.2.1 SAMPLE PREPARATION . . 2.2.2 PACKING OF THE SOIL COLUMNS 2.2.3 THE FLOW SYSTEM . . . 2.3 EXPERIMENTAL METHODS 3. RESULTS AND DISCUSSION 4. SUMMARY . LIST OF REFERENCES. CHAPTER II: SIMULATING UPWARD WATER MOVEMENT INTO METEA SOIL . 1. INTRODUCTION . 1.1 THEORETICAL DEVELOPMENT - FINITE DIFFERENCE MODEL 1.2 THEORETICAL DEVELOPMENT - FINDIT . 1.3 MATERIALS AND METHODS. 2. RESULTS AND DISCUSSION iv Page vi viii 13 13 13 13 15 19 21 23 42 43 46 A6 54 58 6O 61 3. SUMMARY . . . . . . . . . . . . . . . . . . . . . . 74 LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . 76 CHAPTER III: SOIL WATER CONTENT AND SOIL WATER POTENTIAL AS DRIVING FORCES . . . . . . . . . . 79 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . 79 2. MATERIALS AND METHODS . . . . . . . . . . . . . . . 84 w RESULTS AND DISCUSSION . . . . . . . . . . . . . . 86 4. SUMMARY . . . . . . . . . . . . . . . . . . . . . . 99 LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . 101 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . 103 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . 118 TABLE # 10 11 12 13 14 LIST OF TABLES PAGE Characterization of soil columns used for capillary rise experiments. . . . . . . . . . . 18 Values for air entry pressure h and slopes b1 calculated with Equation [7] for three dierrent zones. . . . . . . 28 Cumulative infiltration from FINDIT and piecemeal integration, ip n’ for Metea sandy loam soil. . . . . . .73 Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 1. . . . . . . . . . .103 Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 2. . . . . . . . . . .104 Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 3. . . . . . . . . . .105 Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 4. . . . . . . . . . .106 Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment S. . . . . . . . . . .107 Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 6. . . . . . . . . . .108 Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 7. . . . . . . . . . 109 Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 10. . . . . . . . . . 110 Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 11.. . . . . . . . . .111 Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 8. . . . . . . . . . .112 Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 9. . . . . . . . . . .113 vi 15 16 17 18 Adsorption soil water content-elevation Metea sandy loam soil-Experiment 12.. Adsorption soil water content-elevation Metea sandy loam soil-Experiment 13. Adsorption soil water content-elevation Metea sandy loam soil-Experiment l4. Adsorption soil water content-elevation Metea sandy loam soil-Experiment 15.. vii data for data for data for data for .114 115 116 .117 Figure # 1 Cumulative particle size distribution curve for Metea sandy loam soil . 2 Schematic diagram of the soil column packer . 3 Experimental apparatus used for capillary rise experiments . 4 Experimental water adsorption characteristic curve of soil A . 5 Experimental water adsorption characteristic curve of soil B . 6 Fitting capillary rise data on a log-log scale . 7 Comparison between experimental and fitted wetting curves for soil A . 8 Comparison between experimental and fitted wetting curves for soil B . 9 Experimental retention curves obtained through capillary rise and desorption experiments for Metea soil 10 Diffusivity and conductivity functions for two replications of soil A . 11 Diffusivity and conductivity functions for two replications of soil B . 12 Comparison of the conductivity function calculated from capillary rise using Equation [8] and Haverkamp's Equation [12]. . . . . . 13 Simulated curves generated with FINDIT and experimental wetting curves for two replications of soil A 14 Simulated curves generated with FINDIT and experimental wetting curves for two replications of soil B 15 Experimental wetting curves for Metea soil for different bulk densities . . . . . . . . . . . . . . 16 Moisture profiles calculated with FINDIT for Metea sandy LIST OF FIGURES loam soil at indicated time values viii Page 14 16 20 24 2S . 26 30 .31 .33 34 35 . 37 . 38 . 39 40 62 17 18 19 20 21 22 23 24 25 26 27 28 29 3O 31 32 The specific, average and derivative fluxes calculated at the soil-plate interface with FINDIT . Moisture profiles calculated with the implicit scheme for Metea sandy loam soil at indicated time values . Comparison of moisture profiles obtained with the implicit scheme and FINDIT at indicated time values Comparison of capillary rise experimental data with moisture profiles obtained with the two models for Metea sandy loam soil. Comparison of specific fluxes at soil-plate interface calculated with FINDIT and the implicit scheme . Cumulative infiltration 1 versus time t obtained with FINDIT, the implicit scheme and experimental capillary rise data . . . . . . . . . . . . . . . . . . . . . Cumulative infiltration ratios obtained with FINDIT and the implicit scheme for capillary rise . Diffusivity and conductivity functions for treated and un- treated Metea sandy loam soil . Simulated wetting curves for treated and untreated soils using the implicit scheme and FINDIT Simulated water profiles for treated and untreated soils at 30 and 600 min. Experimental wetting curves for treated and untreated soils . The water content gradients for treated and untreated Metea sandy loam soil . The change of water capacity with water content for treated and untreated Metea sandy loam soil . Experimental wetting front-time curves for treated and untreated soils . The time dependence of cumulative infiltration for treated and untreated soils . Comparison of diffusivity functions generated using calculated water capacity and water flow experiments in horizontal and vertical soil columns . ix . 64 65 67 68 69 71 72 87 88 89 90 .92 94 .95 97 . 98 i+1/2, j+1/2 1.3 LIST OF SYMBOLS Meaning arbitrary constant soil-moisture diffusivity (cmz/min) D as function of 6 acceleration due to gravity (cm/secz) cumulative infiltration (cm) matric component of cumulative infiltration gravitational component of cumulative infiltration indices referring to average quantities summation indices unsaturated hydraulic conductivity (cm/min) saturated hydraulic conductivity (cm/min) calculated conductivity for a specified moisture content (cm/min) measured saturated conductivity (cm/min) calculated saturated conductivity (cm/min) hydraulic conductivity at moisture content 61 (cm/min) K as a function of 6 hydraulic conductivity at initial moisture content flux density (ms/m2 min) sorptivity (cm/minl/z) X dh/dz ah/ax, ah/ay, ah/az 66/82 69/3: BK/az infiltration time (min) total infiltration time (min) square root of time infiltration rate (cm/min) matric component of infiltration rate gravitational component of infiltration rate wetting distance in vertical direction (cm) wetting distance in horizontal direction (cm) vertical component of flow attributed to gravity volumstric moisture content at saturation (m /m ) volumetric moisture content (ms/m3) gravimetric moisture content (cm3/g) initial moisture content (m3/m3) 9 as a function of x and t specific water capacity hydraulic gradient in one-dimensional space hydraulic gradients in x-, y- and 2- directions volumetric moisture content gradient volumetric moisture content rate of change with time hydraulic conductivity gradient air entry potential total potential (cm or kpa) matric potential xi A(6) X! W, ‘0’ "'9 fm sfl 8V gravitational potential matric potential gradient in three- dimensional space pneumatic pressure head vector differential operator small change in flux distance or time Boltzmann transformation (cm/minl/z) A as a function of 8 parameters which are single-valued functions of 9 density of water (g/cc) surface tension (dynes/cm - g/sec2) integral sign (operator) specific flux average flux summation sign xii INTRODUCTION The upward movement of water into and through soil is of great importance and concern to the agricultural and petroleum industries. Knowledge of the rise of water above a water table and the changes of soil water content in sub-irrigation of plants, evaporation, and rise of the water table are necessary for good water and land management. Mathematical equations are helpful in the prediction of water content changes and rise of the wetting front in the soil profile. The complex nature of the soil-water system makes it difficult to specify directly the forces acting on soil water. The description of soil water movement depends not only on the forces residing in the soil but also on the amount of water present. Both diffusion and convection may occur in liquid and gaseous phases while the solid matrix determines the path length of diffusing molecules and the cross-sectional area available for these processes. Understanding the mechanism by which water moves upward through soil is extremely important for promoting good soil-plant relationships. Capillary rise of water in soil has been observed for a long period of time; however, only few experimental and theoretical investigations have been published. Like infiltration and drainage, 'capillary rise of water in soil is a process which is driven towards equilibrium. Equilibrium conditions are ideal for measuring physical parameters of the soil 1 2 like proportionality coefficients, fluxes, water capacity and soil water potential. Under such conditions these parameters are more representative in characterizing the hydraulic properties of a given soil than measurements taken prior to equilibrium conditions. The mathematical equation for describing liquid water movement combines the continuity and Darcy equations and was derived by Richards (1931). Because of the strong non-linearity of this partial differential equation, a general analytical solution has not been found. Numerical solutions, however, may be used to obtain solutions of the partial differential equation. An iterative numerical method based on flux terms (FINDIT) and a finite-difference implicit scheme were used to obtain the solutions in this study. The objectives of this study were: 1). To examine the mechanism of capillary rise, an adsorption process, and compare it with the desorption process of water retention in soils 2). To determine the conductivity and diffusivity functions which are necessary inputs to both FINDIT and finite-difference equations to model capillary rise. 3). To examine how well the two numerical methods based on Richards' equation describe upward water movement. 4). To use capillary rise as an experimental method to test the efficacy of wetting agent treated soils to improve water retention and water movement through the soil matrix. C H A P T E R I CAPILLARY RISE IN METEA SOIL 1. INTRODUCTION Capillary rise is defined as the wetting of a soil from a lower elevation; hence wetting of soil is opposed by the force of gravity. Wollny (1885) was the first to publish his experiments on capillary rise. The well known equation for the height of rise, h, in capillary tubes is: h - 21 cos(¢)/ gpr ................................ [1] where 1 is the surface tension, p the density of the liquid, g the acceleration due to gravity, r the radius of the tube and m the angle of wetting. This formula explains the rise of water in soils, the smaller the pore diameter the higher the rise of water. Pores in coarse soils are larger than in fine ones, thus water rise will be lower than in fine textured soils with smaller pores. The wetting of soils through capillary rise is frequently characterized by the pF- curve, i.e, the relation between the negative log of tension in water films and the water content. Peerlkamp and Boekel (1960) gave some examples of soils where the moisture content of the pF-curve at 3 4 saturation remained the same with increasing negative pressure (tension). McLaughlin (1924) observed that the zone of maximum moisture content was appreciably above the water table, frequently referred to as theta-straight zone. Thereafter, the moisture content decreased gradually with increasing tension. Although the capillary tube theory is applicable to most soil systems, the energetic theory of soil moisture as introduced by Buckingham (1907) and improved by Gardner (1922) is more appropriate. Buckingham (1907) attributed the rise of water in soil to differences in the surface curvature of soil water masses lying between adjacent soil grains. The resulting increasing tension in the water masses is supposed to lift water from the reservoir at the bottom of the soil column, move it through the connecting films, and use part of it at least, in increasing the radius of curvature of the water masses immediately above the last wetted grain. Water ceases to rise when the upward and downward forces in the wetted region are in equilibrium. According to Keen (1924) the following three rather distinct moisture horizons exist in a capillary system: "the first of these should be marked by complete saturation, the next by a gradual decrease of water content and the last by a region of incomplete and decreasing moisture content approaching the initial condition of the soil.” Wadsworth (1931), plotted log-rise of the wetting front against log-time and suggested that capillary rise is a compound process dominated in the first phase by an action which may result in complete filling of pore spaces, and in the later phase by an action which 5 results in incomplete filling of pore spaces and a continuous, uniform decrease in moisture content with height. When coarse materials are used, the transition between these two processes can be noted by inspection of the log-log plot of rise versus time curve. With fine materials, inspection of the moisture distribution is required. subsequent further movement of water into the dry zone is at a very slow rate some of which may be attributed to vapor movement. Swartzendruber (1955) used Poiseuille's analysis, assuming an idealized soil model consisting of parallel, non-interconnected capillary tubes. He found that it is possible within limits to predict from the intersection of behavior of capillary rise curves the behavior of the capillary adsorption coefficient (sorptivity) as it is affected by changes in pore radii, surface tension and wetting angle. Because of favorable qualitative comparisons between the capillary tube theory and experimental data, he suggested that the capillary tube theory can be refined to determine the wetting angle which is now a difficult but determinable soil physical property. Temperature plays an important role in all water flow phenomena in soils. Surface tension decreases with increasing temperature. Therefore, an unsaturated soil at a given potential holds less water at higher than at a lower temperature. This results in an increased rate of water uptake and lower pressure potentials throughout the column when temperature decreases. A rise in temperature causes the water potential to increase whereas a drOp in temperature has the reverse effect. Nielsen, et. a1. (1962) found deviations from a linear square root of time versus distance 6 relationship when wetting soil in a horizontal mode. Contrarily, the results of Stockinger, et. al. (1965) taken in a constant temperature room showed linearity and they concluded that fluctuations in temperature are an additional driving force for water movement,causing the deviations from linearity in Nielsen's work. The macroscopic definition of hydraulic conductivity stems from Darcy's law: q - -KVh .......................................... [2] where q is the specific discharge vector and Vh is the gradient of the hydraulic head. For unsaturated soils (0(08), Equation [1] is of a limited applicability in the field (Braester et. al., 1971). The dependence of K upon the fluid and matrix properties and the sequence of wetting and drying processes is more complex than in the saturated state. Darcy's law, however, is still generally used as a working tool and applied to a wide range of unsaturated flow problems. Because water potential is difficult to measure, measurements of water potential distributions are not always made, and instead, water content measurements are made and are frequently recorded in the literature. However, to use Darcy's law with such data the moisture characteristic must be generated from models or other characteristics of the soil. The quality of water used when measuring water content profiles in soil columns is a major concern. Childs and Collie-George (1950) reported that it was impossible to maintain the initial rate of flow in saturated soils when tap water was used because 7 the dissolved air coming out of solution in the soil pores reduced the moisture content and permeability of the soil. The trouble did not arise with unsaturated soils because the continuous air filled pore space provided a leak path for the escape of the momentarily trapped air soon after its emergence from solution. In unsaturated soils the use of boiled water is not necessary. On the other hand, for unsaturated experiments which last for long periods of time, like capillary rise experiments, the use of deaired water can create differences in fluxes with time as the deaired water will change to become aerated with time. Thus the use of regular tap, not deaired water, is recommended. Precautions should be taken to prevent water loss or evaporation through acrylic plastic and other impermeable construction materials. This is particularly important for small values of soil-water flux (Willis, et. al. 1965, Nielsen, et. al. 1972). Soil water content distributions may be measured nondestructively using gamma-attenuation techniques (Ferguson and Gardner, 1963; Rawlins and Gardner, 1963; Davidson, et. al., 1963) or neutron attenuation (Gardner and Calissendoff, 1967). Gravimetric sampling has been successfully used but many samples are required. To obtain a soil column with sufficient uniformity offers the greatest difficulty (Cassel, et. al. 1968). The soil packer developed by Jackson, et. a1. (1962) allows several columns to be packed to the same average bulk density and to nearly the same bulk density distribution. It is very useful to describe water flow using water content as a driving force. Combining Darcy's law with the continuity 8 equation and applying the chain rule for the one dimension horizontal flow yields: 86/8t - 8/82 [K(9) ah/ae 89/82] ................... [3] Childs and Collis—George (1950) defined diffusivity, D, as: 0(9) - K(6) dh/de ................................. [4] Using [4], [3] takes the form: 86/8t - 8/82 [D(9) 89/82] ........................ [5] and [2]: q - -D(9)V8 ....................................... [6] Both unsaturated conductivity and diffusivity functions decrease as water content decreases with the former showing a sharper reduction. This results from the fact that as water content is reduced, the larger pores empty first. According to the Hagen- Poiseuille law, small pores conduct water much less readily than large pores. In addition, the path of the flow becomes more tortuous as the soil desaturates. These factors have been successfully incorporated in models which give very good descriptions of unsaturated hydraulic conductivity and diffusivity as functions of water content (Brooks and Corey, 1966; Campbell, 1974) or as 9 a function of soil water potential (Gardner and Mayhugh, 1958). The relationship between matric potential and water content is described by the retention curve after equilibrium is established and the wetting front ceases its upward movement. For a matric potential h, where h < he’ the air entry potential, the retention curve is described by the function (Campbell, 1975): -b h - he(9/Ss) ................................... [7] Here 6 is the water content, as is the saturated water content and b is the slope of the log(h) vs. log(e/Gs) plot. Hence, the height above the water table versus the corresponding moisture content data are used to find he and b values. By plotting moisture retention data on a log-log scale and fitting a straight line to the data, the values of he and b are found. Most of the models characterizing the unsaturated hydraulic conductivity function use semi-empirical formulas. Campbell (1974) used the formula: x - Ks(e/es)“' ..................................... [a] and he suggested that m - 2b+3. This is identical to that of Brooks and Corey (1966). Equation [8] combined with [7] gives: n K - Ks(he/h) ..................................... [9] where n - 2+3/b. 10 Values of diffusivity can be obtained through water characteristic data and known conductivity values: Di+l/2 - K1+1/2 (hi-1 - h1)/(91-1 - 91) ........... [10] Equation [10] is the difference form of equation [4]. Simple forms have been derived to express diffusivity as a function of water content. Gardner and Mayhugh (1958) used an expression for the diffusivity D of the form: D - Dn exp[b (G-Gn)] .............................. [11] where Dn is the diffusivity at the lower boundary with a moisture content 8n and b is an empirical constant. Haverkamp, et. al. (1977) used analytical forms for unsaturated conductivity as a function of matric potential h given as: B K-AKs/(A+h) ................................. [12] and water content: a - a(9 - e )/(a + hb) + e ..[13] s r r .................... where the parameters A, B, a, and b are obtained through a non- linear parameter estimation. Experimental methods for measuring the functions are also available. The disadvantages of the 11 experimental methods are: a) The functions are soil specific and their measurements are difficult and time consuming and b) different methods yield different results. Thus it is unclear which method best represents the hydraulic characteristics of the field (Ragab, et. al., 1981). Bruce and Klute (1956) first solved for D with Equation [5], introduced by Childs and Collis-George (1948). Using Boltzmann's transformation (Boltzmann, 1894), they let 8 be given by: 6 - f(A) .......................................... [14] where A is a function of distance x and time t defined by: A - xc'1/2 ......................................... [15] with initial and boundary conditions for the moisture flow in the horizontal tube: (1) 6 - 9i for x--> w ........................... [15a] (2) 9 - OS for x-O .............................. [15b] the solution of [5] with respect to D is given by: 9x 0(9) - 1/(de/cu)9 (-1/2) I Ada ................... [16] e 1 Methods for determining D(8) based on the nonsteady-state l2 and the Boltzmann transform are described by Bruce and Klute (1956). The water content gradient, determined by gravimetric sampling at a fixed time in a horizontal infiltration flow system, was used to calculate the diffusivity function. Whisler, et. al. (1968) described a method in which the water content was measured as a function of time at a fixed position. This procedure required a nondestructive method of determining the water content in the soil column, such as gamma attenuation. The sorptivity method (Dirksen, 1975) entails the determination of the soil water diffusivity function D(9) by means of a series of one-dimensional adsorption experiments. All the above described methods determine diffusivity for wetting soils and are applicable to capillary rise and other adsorption experiments. Gardner (1956) devised a method for computing the diffusivity D(6) and conductivity K(9) functions using water outflow from soil placed in a pressure plate apparatus applicable only to desorption experiments. 13 2. MATERIALS AND METHODS 2.1 MATERIALS The Metea sandy loam soil (Arenic Hapludalts; sandy over loamy, mixed, mesicl) was used in this study. The A-horizon of the Metea soil is a dark, sandy loam, approximately 10 cm thick. It has a very rapid permeability in the upper profile and moderate in the lower profile. The water holding capacity is described as moderate. / Disturbed samples of soils were taken between 0 and 10 cm depth from Michigan State University Soils Research Farm in East Lansing, located in the north central portion of Ingham County between 42 and 43 degree latitude and 84 and 85 degree longitude. The particle density of the soil is 2.624 Mg m-3. To minimize the problems during the packing procedure, a soil was selected with low clay content. The cumulative particle size distribution is shown in Figure l. 2.2 PREPARATION OF THE FLOW SYSTEM 2.2.1 SAMPLE PREPARATION 1Soil Survey of Ingham County, Michigan, United States Department of Agriculture and Soil Conservation Service in cooperation with Michigan Agricultural Experimental Station. 1977. 14 :o» Ewe. 3:3» ages. .2 25:0 cozantfifi on.» Eczema m wN 2:3 33:35 22:3. 90.. NN v. _ l. a .9“. row now you row row on: efiewemed enuennwng 15 Soil samples were evenly spread over laboratory benches to obtain air-dryness and later screened through a 2-mm sieve followed by a screening through a l-mm sieve. The screening was done to help obtain the same average bulk density between soil columns and a uniform bulk density distribution within each column. 2.2.2 PACKING OF THE SOIL COLUMNS Fifty or more two-cm cylindrical glass sections were fastened to each other by tape to form a long cylinder with an additional 10 l-cm cylindrical sections fastened to one end. The sections were numbered and arranged in sequential order forming a cylinder one meter or more in length. The column was packed in a vertical position and closed from the bottom with a rubber stopper. To overcome the difficulty of uniform packing, several techniques were tested, all relating to the addition of soil to the column. A special packing device, designed by Dr. A. T. Corey of Colorado State University and modified by H. M. Kar-Kuri (1983), was first tested without satisfactory results. A major problem was the continuous clogging of the screen and the unsteady flux of soil into the column. This prompted a modification of the device. A sketch of the device is shown in Figure 2. The upper part of the receptacle consisted of a storage vessel with a moveable door to control the flux of the soil. The 45 degree angle helped to keep the flux constant and to prevent clogging of the screen. The 2-mm diameter screen in the lower part of the device helped in the random distribution of soil particles by scattering particles after striking the screen. 16 .353. £83.00 son 9: *0 50.50% uszmzum .N .9”. 33:33 323:. 6:73 .o «@5330 coocoefi .— om? EDD.N|¢ Tu coat m£>oE+l [7:5 17 Before packing began, the screen was positioned approximately 20 cm above the rubber stopper at the end of the empty cylinder. This position was maintained constant by moving the cylinder with an elevating jack throughout the filling process. The purpose of this extensive effort in filling the tube was to attain uniform bulk density within and between different columns. Through the height of fall and the striking of the screen by soil particles, the packing device maintained a continuous and uniform flow of soil from the source into the column and thereby produced a homogeneous distribution of particles across the entire soil column surface area. Once the column was filled, the desired bulk density was obtained by dropping the column on the stoppered end from a height of 10 cm, 50 to 150 times, depending on the desired bulk density. The greater number of drops produced a higher bulk density. As the soil consolidated, additional soil was added. Because the variability of bulk density was larger at the ends of the column, a 20-cm section was removed from each end after the consolidation procedure was complete. The observation of identical wetting curves is the strongest evidence of similar bulk density distibutions and bulk density averages - similar soils generate similar wetting curves. Experiments based on similar wetting curves were conducted to test the hypothesis that soil columns packed in the same manner will exhibit similar bulk density distributions and bulk density averages. If the hypothesis is accepted, the two columns exhibiting the same water characteristic curves are considered a pair of replicates. In these experiments the desired average bulk density was 18 achieved by controlling the energy applied to the system through the number of drops. In a prelimenary experiment when using 48 rings and striving for an average bulk density of 1.36 Mg m-3, the standard deviation was $0.02 within the rings of column. An F-test at the 0.01 level showed no significant statistical difference. Out of fifteen capillary rise experiments which were run in the laboratory the following three pair of replicates shown in Table 1 below were arbitrarily selected for this study and are identified as experiments 4, 5, 6, 7, 12 and 13 in the Appendix. TABLE 1. Characterization of soil columns used for capillary rise experiments. Average No. of No. of Bulk denfity Standard Sail Realism sirens rinse _t1s_m missus l 100 48 1.331 $0.039 Soil A* 2 100 48 1.332 $0.045 l 150 48 1.432 $0.020 Soil 8* 2 150 48 1.406 $0.024 1 so 48 1. 354*“ to . oss Soil C** *** 2 50 48 1.370 $0.057 * Soils in the following text are going to be referred to by their names as defined above ** Soil which had gone through one cycle of wetting and drying in the cylinder *** The bulk density was calculated after the wetting and drying cycle Soil columns labeled C had gone through one previous cycle of wetting and drying in cylinders before the capillary rise experiment to test the hypothesis that reorientention of soil particles may 19 change the bulk density and the flow dynamics. Discontinuities in the soil matrix can be minimized following the first wetting and drying cycle because particles are more firmly attached to neighboring particles. 2.2.3 THE FLOW SYSTEM The apparatus used for the capillary rise experiments is shown in Figure 3. The water supply system contained a constant-head burette of 250-ml capacity and divisions of 1 ml facilitating measurement of water uptake by the soil. The soil column consisted of 2-cm and l-cm sections of clear glass tubing 3.60 cm in diameter taped together to form a 1-meter long cylinder. Alignment of the column was accomplished with a trapezoidal-shaped notch cut into a block of wood. A meter stick was taped to the column to measure wetting front positions. A ceramic plate with relatively large pores (air entry value less than 5 cm of suction) and low resistance to the water flux was attached to the bottom part of the column and positioned to contact a free water surface exactly at the interface between the plate and the soil. Hence, the ceramic plate was aligned at x-0 and zero pressure was applied. The upper part of the packed column was plugged with a small amount of glass wool and a rubber stopper with a hole in it, allowing air to escape as the water front advanced through the column. When the water front stopped moving, equilibrium was assumed. Then, the negative matric potential at any point in the wetted column was assumed to be equal to the distance of that water above the water table. 20 3235393 om: 33.2.30 3. now... 3:33an. 323E_3axm .m .9". 21 Horizontal infiltration experiments were also run to determine the diffusivity function, applicable to capillary rise. During these experiments the potential was maintained at ~0.14 kpa by a fritted glass head plate (Nielsen and Philips, 1958) attached to one end of the column and placed at x-O. To start the experiment, the ceramic plate was filled with water and the desired potential (-0.14 kpa) was applied to the plate before it was placed in contact with the end of the soil column. The small negative potential was necessary to keep water from dripping from the bottom of the plate. 2.3 EXPERIMENTAL METHODS Two soil columns were prepared for each analysis. Fifteen columns were prepared as outlined previously and each pair of replicates was run simultaneously for comparison purposes. Thus, both were under the same temperature regime, ensuring that the two experimentally generated profiles were as alike as experimentally possible. For this study the wetting agents used were tap water and distilled water with CaSOa added to make a 0.01 N solution. The water was not deaired because this served no useful purpose. Water temperature was recorded at all times. The average temperature was 2260 with a standard deviation of $2C°. In both horizontal and capillary rise experiments, water entering‘ the columns was measured volumetrically in the constant head burette for a given time and distance to the wetted front was measured at the same time with the meter stick. When flow had 22 proceeded for the desired time or distance (in capillary rise until equilibrium was established) the water supply was cut off and the column quickly segmented into its 2- and 1-cm sections. The water content of each section was determined gravimetrically (98) and converted to volumetric values (6v) by using the bulk density of the corresponding section. Measurement of wetting front distances versus the square root of time and soil moisture distribution profiles were made for both experiments. Soil water diffusivity values (defined in equation [16]) versus water content were obtained by the method of Bruce and Klute (1956). Soil water potential values needed for generating capillary conductivity functions were obtained from the capillary rise experiments. Experimental desorption characteristic curves were obtained using pressure plate and pressure cooker equipment following the standard procedure. An ATT 6300 personal computer with an 8087 math co—processor and a 20-megabyte hard disk was used for data analysis. All the programming was done using Microsoft FORTRAN 77, version 3.2, 1984, and the FORTRAN IMSL math library. 23 3. RESULTS AND DISCUSSION Six of the fifteen experiments, as indicated in Table 1, were selected for discussion. Equation [7] was used to fit the adsorption characteristic data for each experiment. Figures 4 and 5 show the experimental wetting curves for two replications of two soils with different bulk densities. One can see that: 1) the upper part is parabolic and flattens out at the end; 2) the wetting front stopped rising at a height of about 60 cm even though experiments continued for more than 7000 minutes; and 3) the bulk densities of the two replications are in good agreement as could be inferred from the similarity of the two curves and agrees well with data shown in Table 1. This supports the hypothesis that columns packed in the same manner will exhibit similar wetting curves. By fitting the parabolic part of the curves or segments of it on a log-log scale to a straight line, the values for the air entry pressure he and the slope b were found. Using these values, smoothed analytical wetting curves were generated and compared with experimental data. Figure 6 shows this fitting procedure for soil B. As the literature suggests (Campbell, 1985), if Figure 6 had been generated from desorption data, it would have been one slope with the metric potential decreasing monotonically by the same factor (slope) as the water content decreased. In capillary rise three different slopes could be extracted from each curve, which means matric potential decreased monotonically but by different 24 09w .< :03 Lo 3330 33.33033 3398.3 33; 3.38390 .3 .9... om_.o 2.6 - w3\3 8.0 on .o _ t _ o—d _ N 838.3»: _ 8:823. TE 2. 2.33.28 .33 < =om 3230 3.53.08 .03.... _ - O O I l O P cop (3-01wdx) |°!lual°cl- 25 cod .m =8 .0 0330 0.333033 3.33300 33; .BcoEtmaxu .m .9... m0\0 00.0 2.0 2.0 8.0 2.0 N 3:00:03. 0 .. _ 3:00:00”. 0 . e . . T 0 r d . o . m. H . nu... 0: $410.28 0:33 H w n o m zom ... m. r 8 lop \/ .. a. 00m w . \“QO I nu- . o... . {we 000 00 . 000 r . .. o . . H 32:00 03.0.9: .022. H r h h . p . — . — n — - t cop 26 3.00m @0700. 0 :0 300 mm: b05000 0:5... .m .0: «.03 02.. To Md who :0 _ _ x nmemnnd + hhmnnmé I r x «N53,... + m—mmnm... n. > x mmmafin + muons; u r TE 02 $73.28 0:33 m =8 (Z_Ol*°d>i) lonuaiod 60"- 27 scales (slopes) as the water content decreased. From looking at Figures 4 and 5 one can say that during capillary rise two obvious changes in the slope of pF curves are taking place: 1) in the theta-straight portion there is a change in potential but no change in water content; 2) in the parabolic portion there is both a change in potential and water content. By plotting results from Figure 4 on a larger scale in Figure 6, three transitional stages of wetting were detected in the parabolic portion and Equation [7] was applied to each of the three different regions. These transitional stages are associated with the number and size of soil pores filled and with the thickness of water films on soil particles. When the films become critically small, the wetting process stops because contact is not made with adjacent soil particles. Table 2 shows the values for air entry pressure, he, and slopes b1, obtained using Equation [7] in three different zones. The minimum, maximum, mean and standard deviation values of he and b1 for the first nine experiments were calculated to delineate bounds due to sample variation and experimental error. The other six experiments were omitted because additional treatments were applied to the soil columns. If two or more profiles were to be compared, the agreement of the parameters for a given moisture range was considered acceptable when the values were within two standard deviations. When the parameters were outside of this bounding range, a significant difference in profile shape was detected. Because soil column densities were varied purposely, the standard deviations were larger than they would be if densities were uniform. Figures 7 and 8 show experimental data when compared with l) 28 TABLE 2. Values for air entry pressure h and slopes b calculated with Equation [7] for three different zones. Average Log10 Experiment Bulk density he b1 b2 b3 # Mg m.3 cm 1 1.3860 1.2669 1.7147 0.6380 0.4205 2 1.3400 1.0446 2.0293 0.8521 0.4677 3 1.3450 1.0638 1.8875 0.9501 0.2035 4* 1.4320 1.0120 3.1965 1.3051 0.3595 5 1.4060 1.0842 2.1471 1.0905 0.1101 6 1.3320 1.0444 2.0874 0.4520 0.1669 7 1.3310 1.0293 2.2571 0.9496 0.3459 8** 1.1600 0.9725 1.4760 1.1361 0.3569 9 1.1500 1.0101 1.6967 1.2438 0.2437 10 1.2000 0.4855 2.4446 1.2446 0.6352 11 1.2300 0.4422 2.4836 1.3244 0.5892 12*** 1.3700 0.7828 1.8802 0.6395 0.3021 13 1.3540 0.7738 1.8888 0.5132 0.4034 14 1.3300 0.6276 2.2208 1.1260 0.5916 15 1.3400 0.7441 2.1128 1.5021 0.4008 Minimum**** 1.1500 0.9725 1.4760 0.4520 0.1101 Maximum 1.4300 1.2669 3.1965 1.3051 0.4677 Mean 1.3166 1.0586 2.0547 0.9574 0.2971 Std. Dev. $0.0849 $0.0797 $0.4668 $0.2627 $0.1143 * Experiment numbers 4 and 5 are labeled soil B, 6 and 7 are labeled soil A and 12 and 13 are labeled soil C ** Experiments 8 and 9 were soil columns described in Chapter 3 *** Experiments 12, 13, 14 and 15 experienced one cycle of wetting and drying in the cylinder **** The following data refer to experiments 1-9 only 29 fitted data using Equation [7] for the entire curve and 2) fitted data using Equation [7] by segmenting the curve into three different slopes as suggested in Figure 6. The curves with the square symbols in Figures 7 and 8 are typical for fitting desorption data; hence Equation [7] when used only as a single slope curve is unable to fit experimental capillary rise data. These figures illustrate that with three slopes the agreement between fitted and experimental data is excellent. It is obvious in the initial part of the adsorption wetting curves that all soil particles are saturated in the theta-straight region. Experimental data of Figures 4 and 5 show clearly the existence of this region. In the parabolic part where the curves flattened the wetting front stopped rising because of two possible circumstances 1) the net driving force (the difference between the metric and gravitational gradients) approached zero and/or 2) a discontinuity occurred in the soil matrix. Such discontinuities may be formed continuously during the wetting process just ahead of the wetting front. When films are thick, the wetting process may not be restricted. However, as films become thinner, contact cannot be made with adjacent dry particles and the wetting process may be hindered or stopped completely. Philip (1957a), using Moore's (1939) data, chose a parabolic shaped water profile without any theta-straight part. Green and Ampt (1911) working with downward adsorption data suggested an orthogonal water profile. An orthogonal profile would be one where the theta- straight line is extended for some distance and then intercepted by a perpendicular line to form a rectangle. Figures 4 and 5 suggest that a 30 .< :00 3.. 00230 0:50; 00:: 0:0 .0.:0E.._0ax0 :0050: 30:00:30 S .0... ATEnEV 32:00 333 o.me:.0> 0Y0 . ond 0N6 o...o 00.0 r _ r 0:20 0:30; _0~:0Etoux0 00... b05000 I o 000.» 0:0 0:7,: 02:0 0:50; 00...... film 0000.0 00:... 9.0: 0250 0:50; 00:... I In imp rem -Nn 0:058:00: 03 .0 0oo3>< . [we 7:. 0: Sunbeam... 0.30 3.51. .n-.m0\0.o;u..0og .822 < :8 -00 INN (3-0 l*°d>i) Ionuaiocl- 31 .m. :00 :0. 00230 9.53; 00:: 0:0 3.38.3wa 30.300 30.3930 .m .9... ATEMEV 33:00 33; 0.:.0E:.o> 0.4.0 onto cud 3.0 00.0 _ r L l 0230 0:30; .3:0E...00x0 0m... b05000 I 000.» 0:0 05»: 02:0 0:50: 003... mlm 0000.0 00:: 0503 02:0 0:50; 00...... I 0:038:00: 05 .0 0mo._0>< 7:. 0: «0.10.88 530 3.5.... 0-103.210»? .002. m .60 o in new rvm 1N»... 10m 1mm row (3-0 L*°d)i) Ionuawd— 32 theta-straight region, followed by a parabolic curve flattened at the end, is the most realistic approach to fit capillary rise data. Both standard wetting and drying procedures for calculating moisture characteristic curves may be done using pressure plate equipment. The adsorption wetting curves from capillary rise experiments do not appear to be compatible with the standard procedure of constructing such curves. Figure 9 shows the incompatibility between retention curves for Metea soil when obtained through desorption experiments and through capillary rise experiments. This incompatibility may be explained by a discontinuity taking place in the soil column during capillary rise. Conductivity and diffusivity functions may be generated from the experimental wetting curves. For the conductivity function in this study Equation [8] was used where the parameters b1 were taken from Table 2. Campbell (1974) suggested first that a pore interaction term can be included to make m - 2b+3 instead of 2b+2 as used by Brooks and Corey (1966). Also in this study m - 2b+4 was tested to account for more tortuosity. Equation [8] worked best for fitting capillary rise data with m - 2b+3. The diffusivity function was obtained using Equation [10] where the slope dh/de was obtained from the fitted wetting curves using three different slopes. Figures 10 and 11 show the diffusivity and conductivity functions for two replications of two different bulk density soils. Diffusivity functions, although not similar and smooth, are still sufficiently accurate to reproduce the experimental profiles in Figures 7 and 8. When smooth diffusivity functions with the one slope fitting procedure were used as an input to the model, there was a significant discrepancy 33 :00 0030.2 :0: 0808.303 3380000 0:0 00.: 02.300 330:5 00:.030 00830 33:03: 6808.305 .0 .9... “78:8. 3:03:00 3302, 0.308:.0> 00.0 00.0 P 0N0 0 — .0 00.0 _ L w 7:. 02 2.:"0050 0.30 0:038:00: 0.3 :0 0oo:0>< < _.0m 00.: 02:30 80:: 0330 0:33: no 0230 33:80: :0303000 3808.390 00 l O s 0009 (z—Olwdx) logiueiod_ O O O F o illllj l 1 L .< ..o .0 000303.00: 05 :00 20:83 0.2.3080 08 0.28:5 .0. .0... ATEnEV 0:00:00 :30; 0.:00E:.o> 1111111 J 0....0 00.0 00.0 9.0 00.0 _ _ . 050.0 1:: 0: 8.. ”0.2% 0.30 $900 < :00 m .0020 .0000; r 5:00:00: acoooalbsgavcoo III 80803: 0201038080 I H 00002.00: Boomalbigta aim . _ 808.32 090.00.00.05 9.0 m _ . _ u 0000.0. (I-ulwzwa) CI .m ..0 30 0:038:00: 05 :0: 20:82 0.2.2.008 0:0 03.03:: .: 0.: 072:005 3:03:00 :030; 0.:3080.o> 00.0 00.0 00.0 2.0 00.0 . _ _ _ 0.00.0 500$ E 0: a... 0.2% v.30 . .... u . 30.0.0 . 0 :00 x . . \10500: n m . n . m.0020 w m w. u . rw 008 0L . ".0000; 008.0.u 5308.00: acooomlbsgavcoo I 508.39. 02.703.03.080 I 5303.00: 300001333035 min 508.32 020-0233 To . II'I' . 0000.0— (.-u1w,wa) a 36 between experimental and simulated profiles. Diffusivity data from capillary rise experiments did not exhibit an exponential increase with water content. Thus, Gardner and Mayhugh's (1958) semi-empirical model, Equation [11], could not be used successfully to fit capillary rise data, and it is not recommended for use in the capillary rise mode. Haverkamp et. a1. (1977) introduced analytical equations for conductivity and water content as described in Equations [12] and [13]. Through a non-linear parameter determination using Marquardt's least square technique, the parameters a and b for Equations [12] and [13] were determined and used as input in the finite-difference model of one-dimensional water movement in soils developed by D. L. Nofziger (1985). Typical results of this fitting procedure are shown in Figure 12. The goodness of fit differed from experiment to experiment, depending on the spread of the data. Because different equations (8 and 12) were used for calculating K in the FINDIT and finite-difference procedures, it was difficult to get the K functions to agree. This discrepancy between the input functions used by FINDIT and the finite difference-scheme is particularly noticeable in the low moisture range. Figures 13 and 14 illustrate the experimental and simulated (FINDIT) wetting curves for two replications of two different bulk density soils. Experimental and simulated wetting characteristic curves compared very well because both curves are located within the bounding range. Hence, FINDIT appears to be very capable in modeling capillary rise in soils, but not without difficulty. Figure 15 shows the experimental profiles at equilibrium for 37 0 LD .H F :030000 0.0E0M..__0>0 0:0 mm :03000m. 0:.00 00.: 0... 00 E03 0030.30.00 530:3 33300080 0:: :0 80:00:60 .3 .0... .00 7.0%.. 60:30.... 00 0¢ on ON 0— 0 _ . — p — p h p _ :— 0000.0 w 0.08.0 h H n 0:5 02 00.90.28 0.30 m w .0. ..0m w0_00.0 : . H H m m . v voowoo w "000:0 m mm; :030000 0.0800305... 0:.0: 003... III w H0. :030000 052. 00.: 02:08 E8: 0030.30.00 I u n . . _ + _ p : h . . n 0000.3 (-ugm ma) )4 38 .< :8 ho 20.60.38 05 .2 823 9.30; 6.58.330 96 :02... 5.; 038050 amino 3.2365 . m4 .9... A7815 23:00 .30; 2.2082; ovd omd 8.6 3.0 cod . 3.33 .BcoEtomxw I o 2.3:. 3.0.386 ale 15 :3 rpm row Inn IN¢ €033.32 03 we omo..o>< 9. < _.om "-5 o2 nn._ua_2% x3e 18 :.E ooo~flp .0 3:35 she _ _ p _ 05 (3-0 L*°d)|) Ionuawd- 43.9 .m :00 .6 20.60.69. 03 02 00200 90.30; 6.008.830 0:0 :02: £3, 038200 8:8 032.06% . i .9... A7815 28:00 08.0; 0.3080_0> 9.0 omd ow..o 0:038:08 03 *0 omEo>< m =8 72 as. 9.70.23 0.50 52 0000...; 8 8.220 2.6 cod 2.33 3.005.390 II- o 2:80 3.0.286 one 1 L1 _ [pm d m. a -8 w m... Inn \uN ma IN+ H O -0... n0 ( 10m :8 40 ’ 60.0.0000 0:30 6000:.0 000. =00 0062 .5". 00.200 00.06; 6608.003”. . m— .0... ATEnEV 60600 0063 0.:0E:_0> 00.0 0m.0 0N..0 0 «.0 00.0 «00.60.69. 05 6 000003. 0.E 000nup 6 02.600 7:. 9.. 3.7.3.0 =8 .1. 7E 0: «0.7.510 60 I vs 0: mpg—nan. .< :8 010 Inc on (z_ouod>1) Ionuawd- 41 different bulk density soils. Soils A and B, although having similar wetting curves, were found to be different when bound analyses was used as criteria. Soil B was always outside the bounds at the four points of interest. Table 2 supports this hypothesis by indicating that soil B had the highest bulk density. Soil C which had gone through one cycle of wetting and drying held more water in the theta- straight region but held significant less water than soils A or B at all other potential values. When compared with the bound analyses,it was found to be a soil of a total different character. Assuming that the available energy (potential) is associated with the number and size of soil pores filled with water, reorientation of soil particles during the first wetting and drying cycle apparently increased the effective number of large pores and decreased the effective number of small pores. 42 4. SUMMARY Capillary rise experiments were performed under nearly constant temperature in columns packed at different bulk densities with Metea sandy loam soil. Through these relatively simple experiments the wetting moisture characteristic curves were obtained. All curves exhibited a theta-straight region, followed by a parabolic curve flattened at higher suctions. The parabolic curve when plotted on log- log scale could be separated into three different zones in all 15 samples tested. The different zones suggest transitions in the water content potential relationships. These transition stages can be associated with relative percentages of filled pores and thickness of water films. At equilibrium, the matric and gravitational forces in the wetted soils were equal but of opposite sign. Contrary to desorption data, a power function had to be used in each of the three zones described above to successfully fit the experimental adsorption data. Conductivity and diffusivity functions were determined from experimental adsorption data using Campbell's, Haverkamp's and Gardner's formulas. Simulated (FINDIT) and experimental capillary profiles were compared and showed good agreement. LIST OF REFERENCES LIST OF REFERENCES Boltzmann, L. 1894. Zur integrationder diffusionsgleichung bei variablen diffusion koeffizienten. Annalen der physik und chemie, 53:959-964. Braester, C., G. Dagan, S. Neuman, and D. Zaslavsky. 1971. A Survey of Equations and Solutions of Unsaturated Flow in Porous Media. Technion project no. AlO-SWC-77, Haifa. Israel Institute of Technology. Brooks, R. H., and A. T. Corey. 1966. Hydraulic Properties of Porous Media. Hydrology paper no. 3, Colorado State University, Ft. Collins. Bruce, R. R., and A. Klute. 1956. The Measurement of Soil Moisture Diffusivity. Soil Sci. Soc. Am. Proc., 20:458-462. Buckingham, E. 1907. Studies of the Movement of Soil Moisture. U. S. D. A. Bur. Soils Bull. 38. U.S. Government Printing Office, Washington, D.C. Campbell, G. S. 1974. A Simple Method for Determining Unsaturated Conductivity from Moisture Retention Data. Soil Sci., 117:311- 314. Campbell, G. S. 1985. oi h si s wi h asic a s or Modelg for Soil Plant SystemI Chapters 5, 6, 7, and 8, ppz40-97. Elsevier, New York. Cassel, D. K., A. W. Warrick, D. R. Nielsen, and J. W. Biggar. 1968. Soil Water Diffusivity Values based upon Time Dependent Soil-Water Content Distributions. Soil Sci. Soc. Amer. Proc., 32:774-777. Childs, E. C., and N. Collis-George. 1948. Soil Geometry and Soil Water Equilibria. Discuss. Faraday Soc., 3:78-85. Childs, E. C., and N. Collis-George. 1950. The Permeability of Porous Materials. Proc. Roy. Soc. London, Sere. A., 201:392-405. Davidson, J. M., D. R. Nielsen, and J. W. Biggar. 1963. The Measurement and Description of Water Flow through Columbian Silt Loam. Hilgardia, 34:601-617. Dirksen, C. 1975. Determination of Soil Water Diffusivity by Sorptivity Measurements. Soil Sci. Soc. Am. Proc., 39:22-27. ‘13 44 Ferguson, A. H., and W. H. Gardner. 1963. Diffusion Theory Applied to Water Flow Data Obtained Using Gamma Ray Absorption. Soil Sci. Soc. Amer. Proc., 27:243-246. Gardner, W., and J. A. Widtsoe. 1922. The Movement of Soil Moisture. Soil Sci., 11:215-232. Gardner, R. W. 1956. Calculation of Capillary Conductivity from Pressure Plate Outflow Data. Soil Sci. Soc. Am. Pr. 20:317-320. Gardner, W. R., and M. S. Mayhugh. 1958. Solutions and Test of the Diffusivity Flow Equation for the Movement of Water in Soil. Soil Sci. Soc. Am. Proc., 22:197-201. Gardner, W. H. and C. Calissendorff. 1967. Gamma Ray and Neutron Attenuation in Measurement of Soil Bulk Density and Water Content. Reprinted from "Isotope and Radiation Techniques in Soil Physics and Radiation Studies." International Atomic Energy Agency, Vienna. Green, W. H., and Ampt, G. A. 1911. Studies on Soil Physics: 1. Flow of Air and Water through Soils. J. Agr. Sci., 4:1-24. Haverkamp, R., M. Vauclin, J. Touma, P. J. Wierenga, and G. Vachaud. 1977. A Comparison of Numerical Simulation Models for One- Dimensional Infiltration. Soil Sci. Soc. Am. Proc., 41:285-294. Jackson, R. D., R. J Reginato, and W. E. Reeves. 1962. A Mechanical Device for Packing Soil Columns. U. S. D. A. - A. R. 8., 41-52. Kar-Kuri, H. M. 1983. Testing of the One-Dimensional Infiltration Equation on Some Michigan Soils. M. S. Thesis, (Crop and Soil Science), Michigan State University, East Lansing, MI. Keen, B. A. 1924. On the Moisture Relations of Ideal Soil. J. Agr. Sci., 14:170-177. McLaughlin, W. W. 1924. The Capillary Distribution of Moisture in Soil Columns of Small Cross Section. U. S. D. A., Dept. Bul., 1221:1-22. Moore, R. E. 1939. Water Conduction from Shallow Water Tables. Higardia, 6:383-426. Nielsen, D. R., and R. E. Phillips. 1958. Small Fritted Glass Bead Plates for Determination of Moisture Retention. Soil Sci. Soc. Am. Proc., 22:574-575. Nielsen, D. R., J. W. Biggar, and J. M. Davidson. 1962. Experimental Consideration of Diffusion Analysis in Unsaturated Flow Problems. Soil Sci. Soc. Am. Proc., 26:107-111. 45 Nielsen, D. R., R. D. Jackson, J. W. Cary, and D. D. Evans. 1972. 5911 Water, Chapter 4, ppz64-91, American Society of Agronomy and Soil Science Society of America, Madison, Wisconsin. Nofziger, D. L. 1985. Interactive Simulation of One-Dimensional Water Movement in Soils: User's Guide. Institute of Food and Agricultural Sciences, University of Florida. Peerlkamp, P. R., and P. Boekel. 1960. Moisture Retention by Soils. Versi. Mede. Comm. Hydr. 0nderz. T. N. 0., 5. Philip, J. R. 1957a. Numerical Solution of Equation of the Diffusion Type with Diffusivity Concentration-Dependent. II. Aust. J. Physics, 10:29-42. Ragab, R., J. Feyen , and D. Hillel. 1981. Comparative Study of Numerical and Laboratory Methods for Determining the Hydraulic Condition Function of a Sand. Soil Sci., 131:375-388. Rawlins, S. L., and W. H. Gardner. 1963. A Test of the Validity of the Diffusion Equation for Unsaturated Flow of Soil Water. Soil Sci. Soc. Amer. Proc., 27:507-511. Stockinger, K. R., E. R. Perrier, and W. D. Fleming. 1965. Experimental Relations of Water Movement in Unsaturated Soils. Soil Sci., 100:124-129. Swartzendruber, D. 1955. Anomalies in Capillary Intake as Explained by Capillary Rise Experiments. Soil Sci. Soc. Proc., 20:453-458. Wadsworth, H. A. 1931. Further Observation upon the Nature of Capillary Rise through Soils. Soil Sci., 32:417-434. Whisler, F. D., A. Klute, and D. B. Peters. 1968. Soil Water Diffusivity from Horizontal Infiltration. Soil Sci. Soc. Am. Proc., 32:6-11. Willis, W. 0., D. R. Nielsen, and J. W. Biggar. 1965. Water Movement Through Acrylic Plastic. Soil Sci. Amer. Proc., 29:636-637. Wollny,E. 1885. Untersuchungen uber die kapillare Leitung des Wassers. forsch. Geb. Agr. Phys., 8:206-220. C H A P T E R II SIMULATING UPWARD WATER MOVEMENT INTO METEA SOIL 1. INTRODUCTION Buckingham's (1907) "capillary potential and capillary conductivity" represented two basic steps in the development of modern ideas of soil water movement. The concept of soil water movement as a diffusion phenomenon was explicitly proposed by Childs (1936a) who studied the hypothesis of constant diffusivity. Since the diffusivity varies greatly with moisture content, it is not surprising that results do not accord well with observation. In 1950, Childs and Collis-George developed concentration-dependent diffusivity. This work implied the existence of a general partial differential equation describing liquid phase water movement in a porous medium which would prove to be the parabolic second order diffusion type. In this case the pressure potential is a single-valued function of the moisture content . Klute (1952), following Richards' (1931) approach, applied the continuity requirement to the vector form of Darcy's law for a homogeneous medium and derived the equation which is the basis 46 47 for the mathematical study of moisture flow in unsaturated porous media. This equation, as derived by Klute (1952), expressed in volumetric terms with the liquid density assumed constant can be written as: ae/at - V(KVh) ........................................ [17] where V is the vector differential operator. When the total potential is separated into gravitational and matric components, as is assumed throughout the following analysis, Equation [17] may be written: ae/at - V(KVh) + aK/az ................................ [18] When diffusivity is introduced into Equation [18], the following equation is derived: ae/ac - V(Dve) + aK/az ................................ [19] Kirkham and Feng (1949) found that the diffusion equation is not an acceptable mathematical model for the movement of water in unsaturated soil when diffusivity is constant. Gardner and Mayhugh (1958) reported good agreement between experimental data obtained from solutions of Equation [19] when diffusivity was considered a function of moisture content. The authors' success depended largely upon the Boltzmann transformation which converts Equation [18] into an ordinary 48 differential equation and gives transformed boundary conditions which are as simple as the non-transformed boundary conditions. However, the Boltzmann transformation is limited to problems of horizontal flow with a uniform initial moisture content and a semi-infinite uniform medium. Problems still remain with finite media and arbitrary initial moisture concentrations that cannot be handled using the Boltzmann transformation. These problems have not been treated very extensively in the literature mainly because of the difficulties encountered in obtaining analytical solutions for non-linear differential equations. Numerical approaches seem to provide the best method of attack although some analytical and analogue techniques have been used. As Philip (1967) stated one can use three techniques for solving non-linear equations like Equation [19]: 1) A computer solution using numerical methods, 2) Quasi-analytical and analytical solutions using methods of mathematical analysis to establish the basic form of the solution, and 3) Analytically based estimations of integral properties of the solution. Quasi-analytical solutions of the non-linear Fokker-Plank equation in one dimensional form, subject to restrictive and simple initial and boundary conditions, were developed by Philip (1957a) and more recently by Parlange (1971). The predictions appear generally to be in agreement with short time laboratory observations for situations where the air phase is free to escape in advance of the wetting front. Long time predictions are not yet possible without changing equations and making modifications. Analytical solutions for non-linear equations have some 49 assumptions which do not always hold under real conditions. Also field description of transient isothermal flow of water into nonswelling unsaturated soil is highly complicated since the initial and boundary conditions are usually not constant while the soil characteristics may vary with time and space. In view of this, most efforts have in recent years been concentrated on seeking numerical solutions. Freeze (1969) presented a list of papers with finite difference solutions for one-dimensional infiltration equations with their initial and boundary conditions. Each of these solutions employs different forms of the non-linear Fokker- Plank equation and different ways of discretization. Using the specific water capacity C(h), Equation [18] can be transformed into: C(h) 6h/at - 6/62[K(h)(8h/az - cos(a))] ............... [20] where a is the angle between the direction of flow and the vertical downward direction. In the saturated zone Equation [20] becomes Laplace's equation, provided the soil is isotropic and homogeneous. The value of h varies from positive values in the saturated zone to negative values in the unsaturated zone. Equation [20] can be solved using explicit or implicit methods (Haverkamp, et. al., 1977). In the explicit method a series of linearized independent equations is solved directly. In the implicit method, on the other hand, a system of linearized equations has to be solved. For a given grid point at a given time, the values of the coefficients C(h) and K(h) may be expressed either he 50 from their values at the preceding time step (explicit linearization) or from a prediction at time (t + 1/2At) using an implicit linearization method described by Douglas and Jones (1963). Equations [17] to [20] are valid only if one assumes the h(0) relationship to be unique. Due to the strong non-linearity, there exists no general analytical solution. A specific solution for Equation [19] was first obtained by Philip (1957b). The solution is valid only for the case of infiltration in a homogeneous semi-infinite column satisfying the boundary conditions: t_<_0 220 e - e n tZO z-O 9 - 60.. ................................... [203] In a later paper (Philip, 1958) Equation [19] was solved for the conditions: t<0 220 h-hn :20 z-O h-ho ........................................ [20b] where ho could take positive values corresponding to an infiltration experiment with submersion. Philip's method led to a solution of a power series in tl/z, (Philip, 1955, 1957a, 1957b), having the form: 2 1/2 +.... fm(9)tm/ ........ [21] 4/2 2 - At + xt + wt3/2 + 0t 51 where A, x, ¢. w,...,fm are single-valued functions of 0, subject to conditions [19a]. Equation [21] provides a theoretical formula for obtaining values of the vertical distance z versus 0, useful for predicting wetting distances in vertical columns. The solution of Equation [21] is accurate for short time intervals; however, for long time intervals it fails to converge and loses accuracy. To avoid this, Philip (1957c) used a matching procedure to empirically link the short time solution with that of the long time. The t-range of convergence is dependent upon the soil characteristic and the initial and boundary conditions. Philip (1957d) compared theoretical moisture distribution curves with the experimental curves of Bodman and Coleman (1944) who divided the soil moisture profile into four zones: saturated, transition, transmission and a wetting zone. Philip's analysis predicted all these zones with the exception of the transition zone. Apparently in this zone diffusivity is not a unique function of moisture content due to air entrapment. By introducing the relation: (az/at)e - -(ae/ac)z (az/ae)t ......................... [22] Equation [19] may be transformed into: az/at + 6/80 [DEG/32] - aK/ae ......................... [23] which is valid only if the K(0) relationship is unique. Parlange uh 52 (1971) has proposed a quasi-analytical solution for Equation [23] with the following initial and boundary conditions: t<0 220 9-9“ tZO z-O q-k-Dae/az-qo ............................... [24] One of Parlange's (1985) recent solutions for capillary rise has the form: 2K12t/52 - -(1-a)'1[2K11/52 + 1n(1-a'1 + a-lexp(-2aKli/82))] ................ [25] where K1 is KO-Kn, the difference between the conductivities at 0-00 and 9-9“, respectively, and sorptivity S, which can be estimated when soil properties are known as a function of 6 as given by Parlange (1975) is: 6o 82 - f (9 +9 -29 ) D de [26] o 1+1/2 n j+1/2 ...................... 9 n Philip's (1957a) earlier definition of sorptivity was given by: 00 s - f Ade ............................................ [26a] 6 n Both Equations [26] and [26a] give very nearly the same results for 8. Cumulative infiltration i is the volume of water that moves into the surface of the soil profile over a specified time and can be S3 expressed as: 00 1 - f zde + Knt ...................................... [27] 9 n By reducing [21] to only two terms, Philip attempted to describe an all-encompassing, simple infiltration equation well suited to the needs of applied hydrology. The two term algebraic equation is given by Philip (l957e): 1/2 1 - St + At .......................................... [28] where A - A'+ K n 60 A' - f 209 ............................................. [28a] 9 The parameter A is not easily calculated. Models as presented by Philip (1969) and Kunze (1982) show that A' ranges from 0.378 to 0.67 K . o The infiltration rate can be obtained by differentiating Equation [28] with respect to time and setting Vo-ai/at to give: 1/2 vo - 1/28t' + A ..................................... [29] q .51 CO: 7 re. “V av! 55a 54 In both Equations [28] and [29] S is treated as constant while A is treated as variable corresponding to the gravitational component of flux. Sorptivity, is constant in horizontal flow but should not be expected to hold in vertical flow. In the vertical flow one should expect it to be the other way around because the gravitational part of the flux is the result of the constant gravitational field which does not change in our frame of reference. Philip (l957f, 1958) studied the influence of the initial moisture content and the water depth, h, on the infiltration rate, cumulative infiltration and the shape of the moisture profile. The infiltration rate decreased while the advance of the wetting front increased at higher initial moisture content. He also found an increase in the depth of the saturated zone with larger h. In one of his papers (1958) he introduced the aspect of a tension saturated zone. He defined the zone as one where the volumetric moisture content is equal to that at h-ho, but in which h assumes a non-zero negative value. In this study this zone is going to be called the theta-straight region, but it has also been referred to as "capillary fringe" in research articles and textbooks. 1.1 THEORETICAL DEVELOPMENT - FINITE-DIFFERENCE MODEL The finite-difference model was developed by D. L. Nofziger (1985) for IBM compatible personal computers. The model is based on Equation [20] and can be used for finite or semi-infinite 55 soil systems. In the finite case, the initial conditions are: h(z,t) - hinitial for t - 0 and 0 < z < L ......... [30] where L is the length of the soil system. In the semi-infinite case, the initial conditions are: h(z,t) - h for t - 0 and 0 < z ............ [31] initial At time t-O and any later time, one of the following boundary conditions can be imposed at the upper boundary (z-O): BC#1. Constant Potential ho: h(0,t) - hO BC#2. Constant Flux Density qo: -[K(h)(8h/6z - cos(a))] - qo for z-O BC#3. Mixed Type: -[K(h)(ah/az - cos(a))] - qo for z - 0; t_<_tO h(0,t) - ho for z - 0; t>to ...... [32] where ho and q0 represent the specified potential and flux at the soil surface, respectively, and to is the time at which the soil at z-O reaches a potential ho. The rainfall boundary condition #3 has qo equal to the rainfall rate and hO equal to zero. The flux is positive for water entering the soil system at z-0 and negative for water leaving the soil at z-0. For the finite soil system at time 56 t-O, one of the following boundary conditions can be imposed at the lower boundary (z-L): BC#4. Constant Potential h : h(L,t) - h L L BC#5. Constant Flux Density qL: -[K(h)(ah/az - cos(a))] - qL for z-L BC#6. Mixed Type: -[K(h)(8h/az - cos(a))] - qL for z - L; tSt h(0,t) - hL for z - L; t>t where hL and qL represent the specified potential and flux at z-L, respectively, and tL is the time at which the soil at z-L reaches a potential hL' The defined soil hydraulic properties equal the 9(h) and K(h) functions. The discrete experimental data obtained from capillary rise experiments were fitted using the analytical functions from Haverkamp et. al. (1977) through a non-linear parameter determination as described in the first chapter. An implicit finite-difference scheme with explicit linearization was used. In this scheme the partial differential equation takes the form: C(1.J>‘h<1.3+1>'h<1.3)’/At ' l/Az [K(1+1/2,j)((h(1+1,j+1)’h(1,3+1)]/°2)'°°s (3)) 'K(1-1/2,j)((h(1,j+1)°h(i-1,j+1)l/Az)'°°s (a))] 57 where h(1,3) ' h(iAz,jAt)’ C(1.J) ' C(h<1.J>" - [K01 + K01 )))]/2. )>)1/2 .............. [34] K(in/2.3) (1.3) (1-1-v.1) (i+1,j - [K(h + K(h K(id/2.1) (1.1 and At and A2 are the mesh sizes in time and depth, respectively. The finite-difference equation above is used for all interior mesh points of the soil system. Special forms of this equation are used to represent the boundary conditions selected. This results in a system of simultaneous equations which must be solved for each time step At. One equation applies to each mesh point in depth. Since each equation involves only three unknowns, the system of equations defines a tridiagonal matrix which is relatively easily solved for up to 200 equations on the personal computer. All floating point calculations are performed in double precision (16 decimal digits). The software contains an algorithm to determine the initial mesh sizes in time and depth. Frequently the mesh size in time and depth can be increased as the time increases. The model contains an algorithm to adjust the mesh size based on the mass balance and the depth of wetting. In the following discussion the term 'implicit scheme' will be used for the finite difference implicit model. 1.2 THEORETICAL DEVELOPMENT - FINDIT 58 Equation [19] in its one-dimensional form can describe capillary rise with initial and boundary conditions: 0(z,t) - On for z>0, t-O 6(z,t) - 60 for z-O, t>0 0(z,t) - 6n for z-->w, t>0 .......................... [35] At the moment the water surface is brought into contact with the soil, 0n changes to 60 at the lower soil boundary, while the rest of the semi-infinite height soil column remains at an. The calculations of the distance zJ at all 91 values are given by the following two recursion equations as they were derived by Kunze (1984): 1+1 J 03+1/2 +Dj+1/2 de / [a/ac(f (z/t)t de)+Kj+1/2-Kn] ............ [36] an and Gan/2 6341/2 f (z/t) d6 - f (z/t) dG-zj+1/t d0 .................... [37] e e n n For capillary rise the flux at the free water surface Vo can be partitioned into matric and gravitational components (Kunze and Nielsen, 1982, 1983) to give: vo - vmo + v5o ........................................ [38] 59 or can be partitioned at any moisture content: vJ - v‘“J + v8 ........................................ [39] J where a negative sign proceeding the second term on the right side means that gravitational force opposes the matric force. Because flux is the time derivative of cumulative infiltration, equation [39] can take the form (see Kunze and Nielsen, 1982, 1983): 63+1/2 vj+1/2 - a/ac [f (z/t)t d0] ......................... [40] 0 n where fluxes are now evaluated at the average water content of a 6 interval indicated by 1/2 in the subscript and integration limit, consistent with Dj+l/2 and Kj+l/2’ the average diffusivity and conductivity, respectively. As Kunze and Nielsen (1982) showed, Equation [40], which is part of the recursion formula [39], is broken down and evaluated in terms of two derivative products, namely: 65+1/2 03+1/2 93+1/2 a/at [f (z/t)t d0] - f z/t d0 + ca/at (f z/t d9) ...... [41] e e e n n n The actual or specific flux in Equation [41] is calculated from the average flux plus the change of the average flux. This calculation is the central feature of FINDIT procedure. For capillary rise the 60 specific flux is always less than the average flux. Equation [41] must converge for all 0j+1/2 values in the moisture profile before a solution is acceptable. Present criteria require that: 6.1+1/2 -4 Aj+1/2 < (10 ) ta/ac (f z/t d0) ..................... [42] 9 n where Aj+1/2 is the change in successive iterative calculations of the far right-hand-side term of Equation [41]. 1.3 MATERIALS AND METHODS A series of capillary rise experiments were performed in the laboratory and the experimental data were fitted to obtain the 0(h), K(0) and D(0) functions used as input to Equations [34] and [36]. Details of the measuring techniques were reported in the first chapter. 61 2. RESULTS AND DISCUSSION Figure 16 shows the moisture profiles for capillary rise calculated with FINDIT over wide range of time values. It is apparent from the curves that : 1) FINDIT can cope with a large range of time values; 2) the wetting process slows down appreciably with time as was shown in the first chapter; and 3) after long periods of time equilibrium was established. FINDIT approximated the experimental moisture characteristic curve very well as was shown in Chapter 1. Kunze et. al. (1985) also showed that FINDIT approximated the wetting characteristic curve calculated from the analytical steady state solution of Richards' equation ie. the solution of Equation [19] without the left hand side term, as was discussed by Philip (1966) and Parlange and Aylor(l972). These two cases are the best evidence that the recursion formulas and the iterative process of FINDIT are not only rigorous, but also highly sensitive to the laws governing upward water flow in soils. The accuracy in determining the wetting heights is mainly dependent on the accuracy of numerical calculations of the fluxes as given by Equation [41]. This equation essentially breaks down the actual or specific flux (left hand side of the equation) into average flux and the change in average flux or the derivative term. The flux terms have a physical significance. The average flux corresponds to the average velocity of soil water whereas the actual flux corresponds to the velocity at a specific time. The cumulative infiltration at time t may be calculated as a function of average flux, vav’ with the 62 .00:6> 000.0 0060.00. 6 =00 Eco. 6000 0062 000 0020 5.; 0228.8 00.020 0.30.22 .00 .00 A7015 60600 006; 0.06.0:6> 00.0 on .0 0N0 0 — .0 00.0 b — — — o s... 2... To. / l../.. . . ION I. s... 8.... . Ion . -9. 0 0...: 08.!“ r low “Q 88... . . 82 . 00 10h P _ h _ (we) aouoisgq 63 following algebraic equation: 1 - vav t ......................................... [43] The derivative term corresponds to change of the matric field since the gravitational field is constant. The second term on the right hand side of Equation [41] was calculated numerically using the three point derivative, ie. at T+AT, T and T-AT. The derivatives at TiAT were determined from the derivative at T multiplied by the respective average flux terms at TiAT and divided by the average flux at T. Figure 17 shows the change in the different flux components over time for a given moisture content at the soil surface where the moisture content is saturation. Hence the fluxes for any moisture content could be represented as was done for saturation in Figure 17. The derivative curve is negative, continuous and smooth and has a similar shape as the average flux term which is to be expected based on the method of calculations. As stated earlier, the specific flux is always less than the average flux but both decrease with time. In any case their difference is always a negative quantity represented by the flux derivative. Figure 18 illustrates the moisture profiles calculated with the implicit finite-difference scheme. The implicit scheme also showed the ability to cope with a wide range of time values. The model was stable for AT-lx10°4 min and AZ-.5 cm. These time and space mesh sizes were chosen by a special algorithm supplied by the model. Limitations of the finite-difference approach include errors in calculating the mass balance. Here the quantity of water entering or 64 .002: 0:3 000.006. 06.0 :00 0c: 6 0066060 00x3: 0262000 000 000006 00.0000 00.. .5 .0... A060 06.0 oorom . 00.0w . 00_N¢ 00.0w 00¢ — 0 r! ’- 11.. . . n . Paco-o It!!! 1716!!! III}! [112/ um. [I/ _ x I o . 1P/ -980 o n_u a... mm A .1, Wm... . .0... //4 K .m m... - If r -880 ...x It O m.o . r a w 0 I , 0 0. Ar 3.63000 I W. 1x2: 0.0.000m and 000.6 /\ .3: 000005 0.0 p — . b . _ b p p 65 000.? 06: 006065 6 :00 600. 6000 0062 02 060600 06:00: 05 £3 020.028 8.080 83202 .0. .9... ATEnEV 60600 006; 060.060.; 0....0 00.0 00.0 2.0 00.0 b L n — o ' . 1, 0 as 0.....0 . :00 52 02a. . 0 10+ '00 £5 000E . r.00 . 5.: 0000.0 . p _ . 5.: 002.3 00. (we) aoumsgo 66 leaving the soil surface may not be the same if determined from surface fluxes and from changes in water content especially for short time values. However, no oscillation of the results was observed in the calculations. The profiles generated by the two models were compared for short and long time values in Figure 19. The agreement was acceptable over most of the moisture range. In the low moisture range the agreement was not as good as expected. Discrete values of the diffusivity and conductivity functions were supplied at mid-moisture intervals for FINDIT. Haverkamp's analytical formulas, Equations [12] and [13], were used to supply the functions for the implicit scheme. The non- linear parameter determination routine used in conjunction with Haverkamp's equation was not able to successfully fit the conductivity function in the low moisture range. Hence, the discrepancy in the conductivity function resulted in the higher wetting front predictions when simulation was done with the implicit scheme. FINDIT approximated the experimental profile very well as was shown in Figures 13 and 14. At equilibrium, the implicit scheme again simulated the experimental profile well in the saturation to 25% moisture range but not very well in the drier region as the bound analysis suggests in Figure 20. The largest discrepancy was in the lowest water content region as expected. The specific fluxes from FINDIT and the implicit scheme were compared in Figure 21 . The agreement was very good for the whole time range; however, the flux simulated with the implicit scheme was lower than the flux simulated with FINDIT for time values more than 350 min. For shorter times the flux from the implicit scheme was higher than 67 .002? 06: 0062?: 6 :02: 0:0 2:28 0.2.95 20 £5 005060 00:60 003066 6 000600600 .3 .9... An-EnEV 60600 606; 0_:060_0> EH0 00.0 00.0 2.0 00.0 P — — o r T -0~ 55 007.0 002.... . m. 080 on - we... . 55 00.3 .6: E. w / - O 8 0+ \01 _E 000.". .002: . M. -00 080 on - SE 0000...“. £0: E. — 68 ...00 600. x0000 0062 08 0.0006 05 05 5.; 000.060 00:60 0030.06 5.; 060 6606.006 00... 0005000 6 60.60600 .0m .9... A7660 60600 606; 05060.; 00.0 onrd 0N..0 0 P..0 00.0 0200 00.30: 660600000 olo 08200 0.3.0.0. film . 002—“. I c_.E 0005-5 0 V isle V U 8 (um) 9000;ng V I 69 I I rTIITTV‘IY'VIVVVIVVTrW'TWrrTYIVVYIUVII 060600 0.0.06. 05 0:0 :02... 5.; 006.0060 000t06. 06.0 ..00 6 093: 0.0.0000 6 60.60600 .5 .0... 350 0&0 000. 09 062.00 0.0.05. 0-0 .002...— film 0. . 0000.0 . 1 ”-0050 . 0003.0 H0080 090.0 .0080 0000.0 008.0 0000.0 .2080 . 000 —.0 (uguJ/wo) xnl; owgoads Etc 56’. 70 from FINDIT. This is in agreement with the cumulative infiltration generated over time for the two models, illustrated in Figure 22. Although cumulative infiltration obtained with FINDIT was always lower, its final value is nearly that of the implicit scheme. Cumulative infiltration could be considered as the area under the wetting profiles as shown in Figure 19. The area under the profile obtained with the implicit scheme was larger than the area obtained with FINDIT resulting in higher cumulative infiltration. The experimental infiltration in Figure 22 was lower than both simulated infiltration profiles; part of this difference may be ascribed to errors in experimental techniques, however FINDIT was able to approximate the experimental infiltration profile better than the implicit scheme. To further explore the agreement between cumulative infiltration simulated with the two models, the cumulative infiltration ratios were plotted as a function of time in Figure 23. Acceptable agreement was observed over both short and long time values but a maximum difference of 12.56% was observed in the mid-time range. Both models have limitations in describing water movement in soils. The fact that the models are designed for one-dimensional flow in homogeneous soils means that they will not precisely describe flow in heterogeneous soils where flow is likely to be three dimensional. Computational speed may be a limitation mainly with the implicit scheme. This problem was greatly reduced by the use of the 8087 math co-processor. The vapor phase was not incorporated in either models. How much it contributed in the low moisture range has not been assessed. Piecemeal integration is a routine to check FINDIT for accuracy 7'1 .060 00.0 02.000 660600008 000 062.00 6.06. 0.: .002... £.3 000.060 0 060 0:00? . 6.66.6. 026.0600 .00 0.... 6.60 06: 0000 0000 0000 0000 000 P o . b . p . p b p . 0.0 .002: 4 060600 0.0..06. o .060 66060090 I I .. 0N . _. I 0.0 10.0 10.0 0.0.. (W) 1 72 +0+wé .00... 02.300 5. 0528 6.32. 2.. 25 Ez: 5.; 02.38 8:8 8.08:6. 2.6355 .8 .9... 6.6. 065 .000_ .00. .0. _ . 00A. 1 1005. \t \ 00” .. WJo \ .0. \ It!» t 00— d \ ox V. \ l: \ ((10 \\ . /or \ P 1 4r]... \ 1x -0... /9!/K P _ 0N; 73 TABLE 3. Cumulative infiltration from FINDIT and piecemeal integration, 1p n’ for Metea sandy loam soil. Time (min) FINDIT(cm) ip,n (cm) Difference 13 2.4413 .3811 .0602 100 5.2632 .1917 .0715 240 5.9290 .8809 .0481 400 6.5526 .4871 .0655 600 6.8721 .8682 .0039 800 7.1558 .1299 .0259 1000 7.3314 .3286 .0028 1500 7.6907 .6820 .0087 2000 7.9076 .9007 .0069 2500 8.0710 .0660 .0050 3000 8.1879 .1838 .0041 3500 8.2856 .2821 .0035 4000 8.3596 .3567 .0029 4500 8.4252 .4226 .0026 5000 8.4766 .4745 .0021 5500 8.5228 .5209 .0019 6000 8.5599 .5580 .0019 6500 8.5951 .5934 .0017 7000 8.6233 .6218 .0015 74 of the derivative calculations. It presents another way of determining the cumulative infiltration beside the standard cumulative infiltration calculated by FINDIT; however data generated by FINDIT are necessary. If the two do not agree, the derivative calculations are in error. The piecemeal integration routine calculates the cumulative infiltration, in,p’ at the present time, Tn, as a function of the specific flux, sflp, summed over the interval AT and added to the accumulated flux obtained at Tn-l: m 1 - v T _ +[<1/m> 2 sfl 1 AT ------------- [441 n,p avn_1 n l p-1 P The sflp values for the AT time interval were calculated using a cubic spline technique. The results obtained are shown in Table 2 for m-20. Data illustrate that the agreement between cumulative infiltration obtained with FINDIT and piecemeal integration, 1n,p’ is excellent suggesting that the calculation of the derivative is accurate. 3. SUMMARY FINDIT supplies the user with two physically based components of the specific flux, the average flux and the flux derivative. It is from the latter two fluxes that the specific flux at any moisture content can be calculated and the wetting profiles generated. Average flux can be used to calculate cumulative infiltration at any time with FINDIT. In the previous chapter it was shown that profiles obtained with 75 FINDIT were in excellent agreement with experimental profiles. Here FINDIT is compared with an implicit finite-difference scheme. The profiles from the two models showed fair agreement with FINDIT approximating the experimental profile better at higher auctions. Specific soil water fluxes agreed very well over time. When cumulative infiltration was used as a criterion, the agreement between the two models did not exceed a maximum difference of 12.56%. Hence, the overall agreement of the two models was found to be acceptable for most capillary rise work. Cumulative infiltration from FINDIT and piecemeal integration routines compared excellently. Lack of agreement would have suggested that the derivative function was not calculated accurately. LIST 01“ REFERENCES LIST OF REFERENCES Bodman, G. B., and E. A. Colman. 1944. Moisture and Energy Conditions during Downward Entry of Water into Soils. Soil Sci. Amer. Proc., 8:116-122. Buckingham, E. 1907. Studies of the Movement of Soil Moisture. U. S. D. A. Bur. Soils Bull. 38. U.S. Government Printing Office, Washington, D.C. Childs, E. C. 1936a. The Transport of Water Through Heavy Clay Soils: I. J. Agr. Sci., 26:114-127. Childs, E. C. 1936b. The Transport of Water Through Heavy Clay Soils: II. J. Agr. Sci., 26:527-545. Childs, E. C., and N. Collis-George. 1950. The Permeability of Porous Materials. Proc. Roy. Soc. London, Sere. A., 201:392-405. Douglas, J. J., and B. F. Jones. 1963. On Predictor-Corrector Method for Nonlinear Parabolic Equation. J. Siam., 11:195-204. Freeze, A. 1969. One-Dimensional Vertical, Unsteady, Unsaturated Flow above a Recharging or Discharging Ground-Water Flow System. Water Resour. Res., 5:153-171. Gardner, W. R., and M. S. Mayhugh. 1958. Solutions and Test of the Diffusivity Flow Equation for the Movement of Water in Soil. Soil Sci. Soc. Am. Proc., 22:197-201. Haverkamp, R., M. Vauclin, J. Touma, P. J. Wierenga, and G. Vachaud. 1977. A Comparison of Numerical Simulation Models for One- Dimensional Infiltration. Soil Sci. Soc. Am. Proc., 41:285-294. Kirkham, D., and Feng, G. L. 1949. Some Tests of the Diffusion Theory and Laws of Capillary Flow in Soils. Soil Sci., 67:29- 40. Klute, A. 1952. Some Theoretical Aspects of the Flow'of'Hater in Unsaturated Soils. Soil Sci. Soc. Amer. Proc., 16:144-148. Kunze, R. J., and D. R. Nielsen. 1982. Finite Difference Solutions of the Infiltration Equation. Soil Sci., 134:81-88. Kunze, R. J., and D. R. Nielsen. 1983. Comparison of Soil Water Infiltration Profiles Obtained Experimentally and by Solution of Richards' Equation. Soil Sci., 135:342-349. 76 77 Kunze, R. J., and H. M. Kar-Kuri. 1984. Gravitational Flow in Infiltration. Proc. of the Nat. Conf. on Advance in Infilt., 12-13 Dec., Am. Soc. of Agric. Eng., St. Joseph, Mich., pp.l4-23. Kunze, R. J., J. Y. Parlange, and C. W. Rose. 1985. A Comparison of Numerical and Analytical Techniques for Describing Capillary Rise. Soil Sci., 139:491-496. Nofziger, D. L. 1985. Interactive Simulation of One-Dimensional Water Movement in Soils: User's Guide. Institute of Food and Agricultural Sciences, University of Florida. Parlange, J. Y. 1971a. Theory of Water Movement in Soils: 1. One Dimensional Absorption. Soil Sci., 111:134-137. Parlange, J. Y., and D. Aylor. 1972. Theory of Water Movement in Soils: 9. The Dynamics of Capillary Rise. Soil Sci., 114179- 81. Parlange, J. Y. 1975. Convergence and Validity of Time Expansion Solution: A Comparison to Exact and Appoximate Solution. Soil Sci. Soc. Am. Proc., 39:1-6. Parlange, J. Y., W. L. Hogarth, J. F. Boulier, J. Touma, R. Havercamp, and G. Vachaud. 1985. Flux and Water Content Relation at the Soil Surface. Soil Sci. Soc. Am. J., 49:285-288. Philip, J. R. 1955. Numerical Solutions of Equations of the Diffusivity Type with Diffusivity Concentration-Dependent. Trans. Faraday Soc., 51:885-892. Philip, J. R. 1957a. Numerical Solution of Equation of the Diffusion Type with Diffusivity Concentration-Dependent. II. Aust. J. Physics, 10:29-42. Philip, J. R. 1957b. The Theory of Infiltration: 1. The Infiltration Equation and its Solution. Soil Sci., 83:345-357. Philip, J. R. 1957c. The Theory of Infiltration: 2. The Profile of Infinity. Soil Sci., 83:435-448. Philip, J. R. 1957d. The Theory of Infiltration: 3. The Moisture Profile and Relation to Experiment. Soil Sci., 84:163-187. Philip, J. R. l957e. The Theory of Infiltration: 4. Sorptivity and Algebraic Infiltration Equations. Soil Sci., 84:257-264. Philip, J. R. 1957f. The Theory of Infiltrationz5. The Influence of the Initial Water Content. Soil Sci., 84-329-339. Philip, J. R. 1958,‘ The Theory of Infiltration: 6. Effect of Water Depth over Soil. Soil Sci., 85:278-286. 78 Philip, J. R. 1966. The Dynamics of Capillary Rise. UNESCO Symposium on Wat. in the Unsat. Zone, II:559-578. Philip, J. R. 1967. Sorption and Infiltration in Heterogeneous Media Australian J. Soil Res., 5:1-10. Philip, J. R. 1969. Theory of Infiltration. Adv. in Hydrosci., 5:215-305. Richards, L. A. 1931. Capillary Conduction of Liquids Through Porous Mediums. Physics, 1:318:333. C H A P T E R III SOIL WATER CONTENT AND SOIL WATER POTENTIAL AS DRIVING FORCES 1. INTRODUCTION The potential concept of soil water is of primary importance in determining the state and movement of water in soil. The soil water potential is defined as the amount of work that a unit quantity of water in an equilibrium soil-water system is capable of doing when it moves to a pool of water in the reference state at the same temperature. Buckingham (1907) first introduced the concept of soil water potential without mentioning any relationship between his equation and Darcy's law. W. Gardner and Widtsoe (1922), were the first to include a gravity component in defining the potential for the soil solution. L. A. Richards (1928) defined a total potential, h, in reference to the soil solution as the sum of two components: h-h +2 ............................................ [45] m in which hm is the capillary potential or matric potential often referred to as tension head and z is the gravitational potential. Richards was the first to recognize that Darcy's equation applies to flow in unsaturated soil and he observed that K depends on 79 80 moisture content. Edlefsen and Anderson (1943) also noted that the constituent water in a soil solution may be transported in response to concentration gradients as well as in response to gradients of a solution potential. There are shortcomings in using the total potential concept as Corey and Klute (1985) point out in their review. The most important problem is that the osmotic potential, which refers to the water component only, has been added to pressure and gravitational potentials of the soil solution. This is why the total potential cannot be a valid potential for liquid flow and why the net transport of water may be in any direction with respect to a gradient of the quantity defined. Another problem with the potential concept is that the effects of both gravity and pressure cannot be uniquely defined for a solution of varying density. Isothermal transport of water in the solution phase of a porous medium can be considered at both the microscopic and macroscopic levels. Convection and diffusion may be formulated in terms of a linear rate law, stating that flux is proportional to the moving force. When convection, also called mass flow, is taking place the entire mass of water is moving from a zone of higher pressure to a zone of lower pressure in the context of total soil solution pressure. In the case of diffusion on the other hand, the moving force is the gradient of concentration or water content, and it causes the molecules of the unevenly distributed water content to migrate from a zone of higher to lower concentration even while the liquid phase as a whole may remain stationary. According to Corey and Klute (1985) there are three 81 requirements for thermodynamic equilibrium in any system: 1) thermal equilibrium; this requires that the temperature of the system remains uniform, 2) mechanical equilibrium; this requires that there be no net force tending to produce convection acting on any part of the system, and 3) chemical equilibrium; this requires that there be no net diffusion transport or chemical reaction of any components in the system. Thus, the constant soil solution potential is a necessary, but not a sufficient condition for thermodynamic equilibrium of water in a soil-water system. The relation between soil water content and the soil solution potential is important in characterizing the hydraulic properties of soil as well as solving the flow equation for soil water. This relationship is identified as the water retention function or soil moisture characteristic. The function relates a capacity factor, the water content, 9, to an intensity factor, h, the energy state of the soil water. The water retention function is primarily dependent upon the texture or particle-size distribution of the soil and the structure or arrangement of these particles. Also, the organic matter content and the composition of the solution phase play an important role in determining the retention function (Salter & Williams, 1965; Richards & Weaver, 1944; Croney & Coleman, 1954). The traditional method of determining the water retention function involves establishing a series of equilibria between water in the soil sample and a body of water at a known potential. Each data point (ev’hm) constitutes one point on a retention function. In laboratory measurement the soil-water system is in 82 hydraulic contact with the body of water via a water-wetted porous plate or membrane. At each equilibrium, the volumetric water content, 6v, of the soil is determined and paired with a value of the matric pressure head, hm, determined from the pressure in the body of water and the gas pressure phase of the soil. The conventional way to determine the retention function experimentally is by obtaining adsorption or desorption scanning curves. A drainage curve is mapped by establishing a series of drainage equilibria starting from a zero pressure head. A wetting curve is obtained by equilibrating samples, bringing them to some degree of saturation from a lower water content. The retention function is hysteretic, i.e. the water content at any given pressure head for a wetting soil is always less than that for a draining soil (Poulovasilis, 1962; Pavlakis & Barden, 1972). Soils of relatively rigid structure have retention curves where the water content is nearly constant (theta-straight region) as the matric pressure head is decreased from zero. In this region the slope dG/dhm is zero. In all cases a decrease of water content requires the entry of air (Klute, 1986). In determining the retention function through a pressure cell apparatus, the total pressure head is given by: h—h +h ........................................... [46] —a m where hm is the matric pressure head and ha is the gas phase pressure or the pneumatic pressure head. When the soil sample is at equilibrium, the pneumatic pressure head is positive, and the 83 matric pressure head has an equal negative value. When the gas pressure is released, it is assumed that the matrix pressure is unchanged. Klute (1986) showed that this assumption is not totally valid for two reasons. First, the isolated bodies of entrapped air in the solution phase of the sample will expand when the reduction of the gas phase pressure is transferred to the liquid phase. This expansion of entrapped air will affect the curvature of the interface between the solution phase and the continuous gas phase affecting the matrix pressure head. Second, disturbance of the soil matrix is possible when the sample is removed from the chamber causing the matrix pressure head to increase. During the upward isothermal infiltration process, commonly called capillary rise, there is a point where the gravitational and matrix components of the potential are equal but of opposite sign. At that time all points in the soil solution are at equilibrium. Here the height above the water table is the matric potential hm. The matric potential hm and the corresponding volumetric water content 8v are the data pair at one point of the retention function. Assuming osmotic potential is negligible, the retention function can be obtained via a simple capillary rise experiment. Three forces influence the movement of soil solution into a porous medium: a) gravity, which is constant and may be either a positive or negative force relative to a reference elevation, b) adhesion and, c) cohesion. The last two, collectively known as surface tension, are the two constituents of the matrix water potential. Bouyoucos (1925) reported that if columns of dry clay are wetted with either water or kerosene, the organic liquid at the end of 84 three days will have risen four or five times as high as the water in spite of its lower surface tension. He explained this phenomenon as the inability of the colloidal material to absorb the kerosene and consequently reduce the effective pore space. Pekishek, et. a1. (1962) and Sunderman and Banbury (1978), have shown that wetting agents have either no effect or an adverse effect on infiltration into wettable, hydrophilic, soils. A wetting agent (surfactant) applied to a hydrophilic .soil where the wetting angle is already zero and no further decrease is possible, is going to reduce the effective pore space and thus have'an adverse effect. On the other hand, a decrease of the wetting angle and a reduced surface tension causes higher infiltration rates in non-wetting soils. The rules of wetting a soil that apply to infiltration also apply to capillary rise. Hence, when the soil is water repellent (hydrophobic), the addition of a wetting agent thus will increase the amount of water adsorbed and the height that water can rise in the soil matrix (Bond, 1968). 2. MATERIALS AND METHODS The soil selected for this study was Metea Ap whose physical properties were described in Chapter 1. The hydrophobic character of the soil was presumed but not assessed. The average bulk densities were 1.15 and 1.16 Mg m.3 with standard deviations of 10.023 and 30.022 for the treated and the untreated columns, respectively, and are listed as columns 8 and 9 in Table 2. The profiles at equilibrium were determined experimentally and simulated using the FINDIT model as described in the first chapter. The diffusivity and conductivity 85 functions were used as input to the model and were obtained as outlined in Chapter 1. Capillary rise experiments were conducted in the same way as described in the first chapter. In addition, soil which was labeled 'treated' received 2048 ppm of the blended non-ionic soil wetting agent, Aqua-Gro, manufactured by Aquatrols Corporation of America and dried before it was packed in the column. Both columns were wetted only with tap water in the capillary rise experiments. 86 3. RESULTS AND DISCUSSION The diffusivity and conductivity functions for the wetting agent treated and untreated soils are shown in Figure 24. The functions are not significantly different for the two soils. Figure 25 shows the simulated wetting curves using the finite-difference scheme and FINDIT at equilibrium. The wetting curves obtained with the two models showed good agreement over the wetter half of the moisture range and they are in agreement with the observations made in Chapter 2, Figure 20. The simulated wetting characteristic curves were higher for the wetting agent treated soil with both models. These results are supported in Figure 26 where simulated water profiles are shown for the treated and untreated soils at 30 and 600 minutes, respectively. The capillary rise was higher for the wetting agent treated soil at all moisture contents except close to saturation. Hence the trend of the wetting characteristics of the two columns observed at equilibrium (7000 minutes) in Figure 25 could already be observed at short times. This suggests that the treated soil had a greater adhesion capability for water during the entire wetting process. Figure 27 shows treated and untreated experimental capillary rise profiles at 7000 minutes. At this point the soil water system was at equilibrium for all practical purposes and no further movement of the water front would be observed. At each point the gravitational potential was equal but of opposite sign to the matric potential producing the wetting characteristic curve. The 000p. . .60 600. .0000 0062 060060 000 00 00.; 03 000.600. 0.2600000 000 0.20005 Eu .0... ATEnEV 000.000 006; 0.0.060.o> 0¢r.0 0n..0 0N..0 9.0 00.0 illLl l l L 1 111111 L 1 =8 208... .6 02.3050 .7- ..0... 0300000 .0 0.5000000 I =8 0380 .0 00.2.0.0 0.0 =0» 0300.000 .6 3.30008 010 r UI'UU' r I l V 1 IUUIITU l IIUTUT ' ' I'VTUU U o 30.0 005.0 002.0 0000.0 P (,-uguu two) 0 .88 .002... 000 060000 0.0..06. 0.... 00.00 0:00 0060.500 000 0060: 00. 00000 00.30: 006.06.m .mm .9... A7606V 60.000 006; 0.0660_0> 00.0 00.0 owd 3.0 00.0 - b — — 8.9 662.00 0.0.66.1...» 0060.... I 662.00 00:06:60 00626: I 00201.8 028000 T. 00201.8 .038... .T- .000 55 82"., so 02080 . (0cm) Ionuawd omow 89 0.6 000 000 on 6 0:00 0060060 000 0060.; 00. 00.006 0063 006.06% .mm .0. .0 ATEAEV 60600 0063 0.0660.o> 00.0 00.0 8.0 2.0 00.0 — _ — . 8.0 .. r 1.- - -000 .11.! I]... I I II. 0.5 on H T 109.0 I¢N.o SE oomnu and =00 0062.0: sis . ..00 0060.: min 10+ 0 ...00 00626: all ...00 0060.... mum . — _ _ — (Lu) 0ng bomdoo 90 .m_.0m 03003:: 0:0 00.09.. .5. 00230 00.30; 6.008.305 SN .9... ATEnEV «02:00 00.03 0.0.0E:_0> 0.”.0 omd omd o 9.6 00.0 3:05 _0~coE..oaxol..0» 03095.5 min 2:80 .0uch.0oaxol..0n 0302.. all 5.: 000?. 00 8520 cod INS”. 4&6 land into and (Dam) Ionuaaocl ouww 91 treated soil was wetted to higher elevations than the untreated one over the entire moisture range with the exception of saturation. The difference in the wetting curves may be explained by stating that the treated soil held more water at any given potential. It is known from the literature (Sunderman, undated) that when a wetting agent is used on a hydrophobic soil the soil wets to a higher elevation. The bound analyses shown in Figure 27 suggests that the treated wetting curve is significantly higher than the untreated one although the bound analyses is based on samples of much higher bulk density (1.32 Mg m-3). The bulk densities of the treated and untreated soil in Figure 27 were 1.15 and 1.16 Mg m-3, respectively. The low bulk densities greatly increase the probability of discontinuity formation and reinforces the fact that treatment effect did occur. The derivatives (dB/d2) of the experimental profiles from Figure 27 are calculated and plotted in Figure 28. The resulting gradients are larger for the treated than for the untreated soil indicating that the treated soil had a stronger driving force (dB/d2) to move water into soil as described by Equation [6]. This also enabled water to be adsorbed to higher elevations. Figure 28 demonstrates that the driving force for the treated soil was greater over the entire moisture range. The water in the treated soil experienced a stronger driving force (dB/d2) throughout the soil profile in spite of the higher moisture content for a given potential. When thermal gradients do not exist, the water content gradient can be interpreted to be the upward driving force although countered by gravity in the case of capillary rise. The contribution of other soil solution constituents is negligible in the total pressure of soil 92 ._.00 0.02 0000 00.02 00.08.00 000 00.00... 00. 0.00.090 .00.000 00.0; 0.: .mm .0... 9-00000 .0308 00.0; 0....0E3_0> r 00.0 00.0 0A0 owd 00.0 _.00 00.08.03 to. 0000. 00.2.0 I :0» 00.00... to. 0000. 003.00 I... 000.ml r000.¢l -0000: :000.Nl Ioooél [000.0 000.. (kw) ZP/ep 93 water. Water capacity is defined as the amount of water adsorbed or released by soil per unit change of energy. The lower the water capacity (C) for a given soil the less water is adsorbed by the soil per given amount of energy and the greater the diffusivity (Equation [h]), thus exhibiting a greater flux (Equation [6]). The treated soil exhibited the lower water capacity as it is seen in Figure 29, an indication that it is able to conduct the soil water faster through its profile. The water capacity in the theta-straight region of both profiles is zero which is not shown in Figure 27. Experimental data in Figure 30 shows the wetting front rises higher and at a faster rate with the wetting agent treated soil. This is in agreement with data from experimental and simulated profiles for all time periods, again an indication that the wetting agent has influenced the contact angle between the water and the soil matrix. Wadsworth (1931) suggested that from the plotting of log capillary rise-log time curves, two processes may be observed. The first, where soil eventually becomes saturated, is suggested by the straight line part on the left hand side of the curve in Figure 30. The straight-line part suggests that the wetting front moved as a function of the square root of time. The second process dominates at large time with a much slower rate of wetting front movement than observed initially. Both the treated and the untreated soils have curves with the same shape. A change in the rate of adsorption took place at the inflection point of the curves in Figure 30 which is close to 15 cm and 80 min resulting in a change of slope at this point. A change in the kinetics of the adsorption reaction can be ._.00 0000. x0000 00.02 00.02.03 000 00.00... to. .00.000 00.0; 5.: 3.00000 0.03 .0 000000 00.. .00 .0... A7810. .0308 00.0; 0.0.0E:_o> 0.3.0 0m .0 0N..0 0 .0 00.0 b 00.00:. .0020 0 00.08.00 0 ...000. . 00: 0000". . 000.0 P up/ep 00.00— .m_.00 00.08.00 000 00.00... to. 00200 000.....00: 000.03 6.008.090 .00 .0... AEEV 000... 00.0. 0m: 0. =00 00.00.. 0 =00 00.08.00 0 _ _ _ l O 00— (wo) 300.1; Bumem ;o 3813 96 inferred from this change of slope. It is reasonable to assume that there exists a critical energy level—pore size relationship, dependent upon the wetting agent used and the particle size distribution of a given soil. Energy reductions beyond this threshold level caused the capillary rise process to slow down significantly. This agrees well with the changes in the slope of pF curves which were observed in the first chapter where equilibrium data suggested three different zones. Figure 30 supports the existence of different macroscopic flow characteristics in two zones. In the first zone the flow is associated with the linear part of the curve in Figure 30. In the second zone, flow is significantly slowed down as shown by the parabolic curve of the upper arm. In the final stage the wetting process stopped completely after 7000 minutes. From a practical point of view, whenever prediction of upward water flow has to be made, knowledge of the inflection point or the height of the theta-straight region can be very useful. Figure 31 supports the higher rise of the wetting front for the treated soil illustrated in Figure 30 by showing that water uptake was larger for the wetting agent treated soil. Differences in cumulative infiltration became evident as early as 5 minutes after the capillary rise was initiated. Thereafter the cumulative infiltration differences between the treated and untreated soil continued to increase until equilibrium was achieved. Diffusivity functions were generated by several different methods as shown in Figure 32 to demonstrate the capability of the different techniques for calculating this parameter. Diffusivity functions generated with Equation [10] compared well with diffusivity functions 97 , .0000 00.00.00: 000 00.000. 00. 00..o0._.00. 02.03050 .0 0000000000 000.. 000. .3” .0... 2. 2:0 00. 00. 00 00 00 on 0 . 0 . _ . . . . . . . O Illllollflllfl11 >1 pl 0 I! 01 9 IIOII 0 0 0 n - -00 . =8 0380...: I . =00 00.000 0. ole . . (ww) uogmmyug eAgiolnwng 98 00.0200 :00 .00_.00> 000 .0 005000 0_ 0.0080098 30: 00.0; 000 000000 00.03 00.0328 @000: 00.0.0000 002.003 0.020320 .0 000009000 .Nm .9..— ATEnEV .0308 00.0; 0_0.0E:_0> 00.0 00.0 and cud 2.0 8.0 _ _ _ _ _ OPOO.O n. woos... 1 1009.0 f x r r m m . u 1008 . 000.20 052 000 00:51:00 00.02.00 I . . 3.00000 00.03 00:20. 02000000000 00.02.? a H 000. b05000 00:000. 00.22.001.00 00.09.... c H . £00000 00.03 00:20. 00.0000003000 00.02.00 0 . out 00.0000 00:000. 00.000000I000 00.02.02 0 m _ _ P _ — .. ooocd . (Pungwa) g 99 obtained by 1) fitting experimental capillary rise data as described in Chapter 1 and 2) Bruce and Klute's horizontal transient method. Hence, any of these methods appear to be acceptable in generating the D(6) function. It is known from classical physics that physical parameters may be characterized in terms of: 1) capacity, 2) intensity, 3) proportionality coefficient and 4) flux. In heat, these factors are heat capacity, temperature, thermoconductivity and heat flux, respectively. In case of soil water the capacity factor is the water capacity C(e), the change in water content per unit change of potential, the intensity factor is the water potential or the energy state of the soil, proportionality constants are the diffusivity and conductivity functions and flux is the water velocity. From the wetting characteristic curves obtained through capillary rise experiments, the specific water capacity, the intensity factor, h, the proportionality constants and the flow velocity were successfully obtained. The importance of these parameters can be recognized in Equations [2] and [6] in addition to characterizing the physical properties of a given soil. 4. SUMMARY The hypothesis that the use of wetting agent can influence the flow of water into the soil matrix was tested as a case study of capillary rise. The treated soil had a higher cumulative infiltration and held more water at a given potential as shown from 100 the experimental and simulated adsorption characteristic curves. It also conducted water faster because of the presence of stronger adsorption forces resulting from decreased surface tension. A change in the kinetics of capillary rise was observed which resulted in a significant slow down of the process. Water gradients serve as the upward driving force but are countered by gravity. When equilibrium is attained, these are equal but of opposite sign. The wetting curves obtained from capillary rise can be used to determine capacity and intensity factors in addition to proportionality coefficients and flux factors of soil water. LIST OF REFERENCES 101 LIST OF REFERENCES Bond, R. D. 1968. Water Repellent Sands. 9th Int. Congr. Soil Sci. Trans., 1:339-347. Bouyoucos, G. J. 1925. The Role of Colloids in Soil Moisture. Colloid Symposium Monog., 2:126-134. Bruce, R. R., and A. Klute. 1956. The Measurement of Soil Moisture Diffusivity. Soil Sci. Soc. Am. Proc., 20:458-462. Buckingham, E. 1907. Studies of the Movement of Soil Moisture. U. S. D. A. Bur. Soils Bull. 38. U.S. Government Printing Office, Washington, D.C. Corey, A. T., and A. Klute. 1985. Application of the Potential Concept to Soil Water Equilibrium and Transport. Soil Sci. Soc. Am. J., 49:3-11. Croney, D., and J. D. Coleman. 1954. 5011 Structure in Relation to Soil Suction (PF). J. Soil Sci., 5:75-84. Edlefsen, N. E., and Anderson, A. B. C. 1943. Thermodynamics of Soil Moisture. Hilgardia, 15:31-298. Carder, W., and J. A. Widtsoe. 1921. The Movement of Soil Moisture. Soil Sci., 11:215—232. Klute, A. 1986. Water Retention: Laboratory Methods. In A. Klute (ed). etho o 0 a1 , Part 1.2:nd ed. Agronomy, 9:635-660. Pavlakis, George, and L. Barden. 1972. Hysteresis in the Moisture Characteristics of Clay Soil. J. Soil Sci., 23:350-361. Pekishek, R. E., J. Osborn, and J. Letey. 1962. The Effects of Wetting Agents on Infiltration. Soil Sci. Soc. Amer. Proc., 26:595-598. Poulovasilis, A. 1962. Hysteresis of Pore Water: An Application of the Concept of Indepedent Domains. Soil Sci., 93:405-412. Richards, L. A. 1928. The Usefulness of Capillary Potential to Soil Moisture and Plant Investigations. J. Agr. Res., 37:719- 742. Richards, L. A., and L. R. Weaver. 1944. Moisture Retention by some Irrigated Soils Related to Soil Moisture Tention. J. Agric. 102 Rec., 69:215-235. Salter, P. J., and J. B. Williams. 1965. The Influence of Texture on the Moisture Characteristics of Soils. Part 1: A Critical Comparison of Techniques for Determining the Available Water Capacity and Moisture Characteristic Curve of a Soil. J. Soil Sci., 16:1-15. Sunderman, H. D., and E. E. Banbury. 1978. Performance of Soil Wetting Agents in Northwestern Kansas. Agron. Abst., p. 184. Sunderman, H. Undated. Soil Wetting Agents: Their Use in Crop Production. North Central Extention Publication 190. Wadsworth, H. A. 1931. Further Observation upon the Nature of Capillary Rise through Soils. Soil Sci., 32:417-434. APPENDIX 103 TABLE 4. Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 1.* Elevation (h) Wate o 9 3 -3 cm m m 16.000 0.420 18.000 0.410 20.000 0.392 22.000 0.370 24.000 0.360 26.000 0.350 28.000 0.337 30.000 0.331 32.000 0.292 34.000 0.289 36.000 0.281 38.000 0.279 40.000 0.277 42.000 0.275 44.000 0.255 46.000 0.251 48.000 0.250 50.000 0.240 52.000 0.227 54.000 0.223 56.000 0.220 58.000 0.214 59.000 0.212 60.000 0.210 61.000 0.205 62.000 0.200 63.000 0.195 64.000 0.190 65.000 0.185 66.000 0.180 67.000 0.178 68.000 0.173 69.000 0.165 70.000 0.160 71.000 0.158 72.000 0.150 73.000 0.130 0.095** * Average bulk density-1.386 Mg m-3; No of drops-100 ** Initial moisture content 104 TABLE 5. Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 2.* E v o W t o t 6 -3 cm m m 10.000 0.410 12.000 0.370 14.000 0.360 16.000 0.340 18.000 0.330 20.000 0.320 22.000 0.310 24.000 0.280 26.000 0.270 28.000 0.260 30.000 0.250 32.000 0.240 34.000 0.230 36.000 0.210 38.000 0.200 40.000 0.190 42.000 0.170 44.000 0.160 46.000 0.140 47.000 0.120 0.095** * Average bulk density-1.340 Mg m-3; No of drops-100 ** Initial moisture content content 105 TABLE 6. Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 3.* §1s222120_1hi Ha;2r_sgntsnt_fl§l 3 -3 cm mm 8.000 0.420 10.000 0.410 12.000 0.400 14.000 0.380 16.000 0.350 18.000 0.340 20.000 0.330 22.000 0.310 24.000 0.300 26.000 0.290 28.000 0.270 30.000 0.260 32.000 0.250 34.000 0.230 36.000 0.220 38.000 0.210 40.000 0.200 42.000 0.190 44.000 0.180 46.000 0.160 47.000 0.150 48.000 0.120 0.095** * Average bulk density-1.345 Mg m-3; No of drops-100 ** Initial moisture content 106 TABLE 7. Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 4.* Elevation (h) Wate tent 6 3 -3 cm m m 8.090 0.420 10.210 0.410 12.200 0.400 14.180 0.380 16.190 0.370 18.170 0.360 20.160 0.340 22.160 0.330 24.140 0.320 26.130 0.310 28.110 0.290 30.080 0.270 32.080 0.260 34.090 0.250 36.090 0.240 38.080 0.230 40.100 0.220 42.100 0.210 44.110 0.200 46.130 0.190 48.180 0.110 0.095** * Average bulk density-1.432 Mg m-3; No of drops-150 ** Initial moisture content 107 TABLE 8. Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 5.* v o h Wate t 6 3 -3 cm m m 10.450 0.420 12.560 0.410 14.660 0.380 16.760 0.360 18.870 0.340 20.990 0.330 23.110 0.320 25.260 0.300 27.340 0.290 29.420 0.280 31.550 0.270 31.610 0.260 35.670 0.240 37.800 0.230 39.880 0.220 41.920 0.210 43.990 0.200 46.090 0.190 48.220 0.180 50.360 0.160 51.370 0.110 0.095** * Average bulk density-1.406 Mg m-3; No of drops-150 ** Initial moisture content 108 TABLE 9. Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 6.* .Elgxatien_lhl Es£2r_22ntsn£_l§i -3 cm mm 10.250 0.440 12.320 0.410 14.420 0.390 16.470 0.370 18.570 0 350 20.670 0.330 22.700 0 320 24.800 ‘0.300 26.920 0.290 29.050 0.280 31.190 0.270 33.270 0.260 35.360 0.250 37.480 0.240 39.540 0.238 41.610 0.230 43.730 0 220 45.810 0.210 47.860 0.200 49.920 0.190 52.030 0 180 54.150 0.160 56.290 0 150 57.310 0.120 0.095** * Average bulk density-1.332 Mg m-3; No of drops-100 ** Initial moisture content 109 TABLE 10. Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 7.* Elevation (h) Wate o e (cm) (Inn3 111-3) 10.150 0.440 12.200 0.400 14.160 0.390 16.290 0.370 18.270 0.350 20.260 0.330 22.260 0.320 24.240 0.310 26.230 0.300 28.230 0.290 30.220 0.280 32.200 0.270 34.190 0.260 36.150 0.250 38.150 0.240 40.160 0.230 42.160 0.220 44.150 0.210 46.170 0.200 48.180 0.190 50.180 0.180 52.200 0.170 54.250 0.160 55.340 0.150 56.470 0.140 57.500 0.130 58.000 0.110 0.095** * Average bulk density-1.331 Mg m'3; No of drops-100 ** Initial moisture content 110 * TABLE 11 . Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 10.** Eloxogion (h) Wage; oontent (9) 3 -3 cm m m 6.010 0.434 8.010 0.292 10.030 0.248 11.990 0.247 13.970 0.246 15.970 0.232 18.000 0.205 19.930 0.202 21.950 0.194 23.960 0.188 25.840 0.172 27.830 0.161 29.790 0.148 31.710 0.126 32.000 0.120 0.095*** * Average bulk density-1.200 Mg m-3; No of drops-150 ** Soil particle diameter greater than 1 mm *** Initial moisture content 111 * TABLE 12 . Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 11.** ELEVEELOD (b) Water oonoent (9) 3 -3 cm m m 6.070 0.442 8.030 0.279 9.960 ' 0.267 11.950 0.243 13.920 0.244 15.900 0.234 17.880 0.220 19.890 0.209 21.890 0.184 23.900 0.182 25.920 0.183 27.930 0.174 29.970 0.156 31.990 0.129 32.500 0.120 0.095*** * Average bulk density-1.230 Mg m-3; No of drops-150 ** Soil particle diameter greater than 1 mm *** Initial moisture content 112 TABLE 13. Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 8.* 81mm W61 e e 9 3 -3 cm m Ill 8.490 0.450 10.550 0.400 12.660 0.350 14.700 0.330 16.800 0.310 18.910 0.280 20.930 0.270 23.030 0.250 25.160 0.240 27.280 0.230 29.420 0.210 31.510 0.200 33.590 0.190 35.710 0.180 37.780 0.170 39.840 0.160 41.960 0.150 44.050 0.140 46.090 0.110 0.095** 3 * Average bulk density-1.160 Mg m- ; No of drops-50 ** Initial moisture content 113 * TABLE 14 . Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 9.** Elevation (h) Wate c ent 9 3 -3 cm m m 8.030 0.450 10.080 0.420 12.040 0.390 14.160 0.370 16.150 0.340 18.130 0.320 20.140 0.310 22.120 0.290 24.110 0.280 26.110 0.270 28.090 0.260 30.080 0.250 32.060 0.240 34.030 0.230 36.030 0.220 38.040 0.210 40.040 0.200 42.020 0.190 44.050 0.180 46.050 0.170 48.060 0.160 50.080 0.150 52.120 0.140 53.220 0.110 0.095*** *' Average bulk density-1.150 Mg m-3; No of drops-50 ** $011 which was treated with wetting agent' *** Initial moisture content 114 * TABLE 15 . Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 12.** Elovation (h) Wa er ent 6 -3 cm m m 5.973 0.461 7.938 0.382 9.862 0.360 11.857 0.321 13.821 0.303 15.806 0.269 17.790 0.262 19.794 0.253 21.798 0.238 23.803 0.228 25.827 0.217 27.831 0.207 29.875 0.199 31.899 0.187 33.883 0.184 35.848 0.178 37.832 0.173 39.816 0.165 41.801 0.156 43.825 0.143 45.829 0.133 47.833 0.123 49.857 0.102 0.095*** * Average bulk density-1.340 Mg m-3; No of drops-50 ** Soil which had gone through one cycle of wetting and drying in the cylinder *** Initial moisture content 115 * TABLE 16 . Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 13.** ev on Wate co ent 6 3 -3 cm m m 6.013 0.467 8.017 0.396 10.031 0.349 11.996 0.321 13.980 0.300 15.974 0.280 18.008 0.249 19.933 0.251 21.957 0.236 23.961 0.223 25.846 0.222 27.831 0.202 29.795 0.198 31.710 0.195 33.685 0.187 35.639 0.182 37.624 0.177 39.688 0.168 41.712 0.164 43.736 0.154 45.829 0.143 47.833 0.135 0.095*** * Average bulk density-1.354 Mg m-3; No of drops-50 ** Soil which had gone through one cycle of wetting and drying in the cylinder *** Initial moisture content 116 * TABLE 17 . Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 14.** Elsxetign_ihl Estsr_sgntsnt_£§l 3 -3 cm mm 3.949 0.491 6.052 0.399 8.037 0 380 10.081 0.327 12.045 0.317 14.168 0.288 16.153 0.268 18.137 0.254 20.141 0 247 22.126 0.231 24.110 0.226 26.095 0.224 28.079 0.205 30.083 0.201 32.068 0.196 34.032 0 190 36.036 0.180 38.040 0.172 40.045 0.166 42.029 0 148 44.053 0.146 46.057 0.126 47 000 0.120 0.095*** * Average bulk density-1.370 Mg m-3; No of drops-50 ** Soil which had gone through one cycle of wetting and drying in the cylinder *** Initial moisture content 117 * TABLE 18 . Adsorption soil water content-elevation data for Metea sandy loam soil-Experiment 15.** 1ev h Wate t 9 3 -3 cm m m 4.247 0.446 6.390 0.403 8.493 0.375 10.557 0.320 12.660 0.314 14.704 0.283 16.808 0.264 18.911 0.246 20.935 0.245 23.039 0.226 25.162 0.217 27.285 0.208 29.428 0.202 31.512 0.187 33.595 0.183 35.719 0.173 37.782 0.171 39.846 0.162 41.969 0.149 43.954 0.121 0.095*** * Average bulk density-1.354 Mg m'3; No of drops-50 ** Soil which had gone through one cycle of wetting and drying in the cylinder *** Initial moisture content LIST OF REFERENCES 118 BIBLIOGRAPHY General References Ahuja, L .R. 1983. Modeling Infiltration into Crustreol Soils by the Green and Ampt Approach. Soil Sci. Am. J., 47:412-418. Ashcroft, C., D. D. Marsh, D. D. Evans, and L. Boersma. 1962. Numerical Method for Solving the Diffusion Equation: Horizontal Flow in Semi-Infiltration Media. Soil Sci. Soc. Am. Proc., 26:522-525. Bally, R. J. 1966. Determination of the Coefficients of Water Migration through Soils. UNESCO Symposium on Wat. in the Unsat. Zone, 1:245-256. Bodman, G. B., and E. A. Colman. 1944. Moisture and Energy Conditions during Downward Entry of Water into Soils. Soil Sci. Amer. Proc., 8:116-122. Boltzmann,L. 1894. Zur integrationder diffusionsgleichung bei variablen diffusion koeffizienten. Annalen der physik und chemie, 53:959-964. Bond, R. D. 1968. Water Repellent Sands. 9th Int. Congr. Soil Sci. Trans., 1:339-347. Bontarenko, N., and S. Nerpin. 1966. Influence of Viscous Plastic Properties of Water on its Equilibrium and Transfer in Unsaturated Zone. UNESCO Symposium on Wat. in the Unsat. Zone, 2:440-451. Bouyoucos, G. J. 1925. The Role of Colloids in Soil Moisture. Colloid Symposium Monog., 2:126-134. Braester, C., G. Dagan, S. Neuman, and D. Zaslavsky. 1971. A Survey of Equations and Solutions of Unsaturated Flow in Porous Media. Technion project no. A10-SWC-77, Haifa. Israel Institute of Technology. Brooks, R. M., and A. T. Corey. 1966. Hydraulic Properties of Porous Media. Hydrology paper no. 3, Colorado State University, Ft. Collins. Bruce, R. R., and A. Klute. 1956. The Measurement of Soil Moisture Diffusivity. Soil Sci. Soc. Am. Proc., 20:458-462. Brutsaert, W. 1974. More on an Approximate Solution for Non- 119 Linear Diffusion. Water Resource Res., 10:1251-1257. Brutsaert, W. 1976. The Concise Formulation of Diffusive Sorption of Water in a Dry Soil. Water Resource Res., 12:1118-1124. Brutsaert, W. 1979. Universal Constraints for Scaling the Exponential Soil Water Diffusivity. Water Resource Res., 10:1251-1252. Brutsaert, W., and R. N. Weisman. 1970. Comparison of Solutions of a Nonlinear Diffusion Equation. Water Resource Res., 6:642- 644. Buckingham, E. 1907. Studies of the Movement of Soil Moisture. U. S. D. A. Bur. Soils Bull. 38. U.S. Government Printing Office, Washington, D.C. Burejev, L. N., and Z. M. Burejeva. 1966. Some Numerical Method for Solving Problems of Non-Steady Seepage in Non- Homogeneous Anisotropic Soils. UNESCO Symposium on Wat. in the Unsat. Zone, 11:500-525. Campbell, G. S. 1974. A Simple Method for Determining Unsaturated Conductivity from Moisture Retention Data. Soil Sci., 117:311- 314. Campbell. G. S. 1985. W W fioii Piano Sygoen. Chapters 5, 6, 7, and 8, pp:40-97. Elsevier, New York. Carey, J. W. 1967. Experimental Measurements of Soil Moisture Hysteresis and Entrapped Air. Soil Sc., 104:174-180. Cassel, D. R., A. W. Warrick, D. R. Nielsen, and J. W. Biggar. 1968. Soil Water Diffusivity Values based upon Time Dependent Soil-Water Content Distributions. Soil Sci. Soc. Amer. Proc., 32:774-777. Childs, E. C. 1936a. The Transport of Water Through Heavy Clay Soils: I. J. Agr. Sci., 26:114-127. Childs, E. C. 1936b. The Transport of Water Through Heavy Clay Soils: II. J. Agr. Sci., 26:527-545. Childs, E. C., and N. Collis-George. 1948. Soil Geometry and Soil Water Equilibria. Discuss. Faraday Soc., 3:78-85. Childs, E. C., and N. Collis-George. 1950. The Permeability of Porous Materials. Proc. Roy. Soc. London, Sere. A., 201:392-405. Clothier, B. E., et. al. 1983. Diffusivity and One Dimensional Absorption Experiments. Soil Sci. Soc. Am. J., 47:641-644. Corey, A. T., R. 0. Slatyer, and W. D. Kemper. 1967. Comparative 120 Terminologies for Water in Soil-Plant Atmosphere System. In R. M. Hagan, et. al., (ed)., Irrigation in Agricultural Soils. Am. Agronomy, 11(22):427-445. Corey, A. T., and A. Klute. 1985. Application of the Potential Concept to Soil Water Equilibrium and Transport. Soil Sci. Soc. Am. J., 49:3-11. Croney, D., and J. D. Coleman. 1954. Soil Structure in Relation to Soil Suction (PF). J. Soil Sci., 5:75-84. Cushman, J. and D. Kirkham. 1978. A Two-Dimensional Linearized view of One-Dimensional Unsaturated Flow. Water Resour. Res., 14(2):319-323. Davidson, J. M., D. R. Nielsen, and J. W. Biggar. 1963. The Measurement and Description of Water Flow through Columbian Silt Loam. Hilgardia, 34:601-617. Davidson, J. M., et. a1. 1966. Soil Water Diffusivity and Water Content Distribution During Outflow Experiment. UNESCO Symposium on Wat. in the Unsat. Zone, I:214-223. Dirksen, C. 1975. Determination of Soil Water Diffusivity by Sorptivity Measurements. Soil Sci. Soc. Am. Proc., 39:22-27. Douglas, J. J., and B. P. Jones. 1963. On Predictor-Corrector Method for Nonlinear Parabolic Equation. J. Siam., 11:195-204. Edlefsen, N. E., and Anderson, A. B. C. 1943. Thermodynamics of Soil Moisture. Hilgardia, 15:31-298. Elrick, D. E., and M. J. Robin. 1981. Estimating the Sorptivity of the Soils. Soil Sci., 132:127-133. Ferguson, A. H., and W. H. Gardner. 1963. Diffusion Theory Applied to Water Flow Data Obtained Using Gamma Ray Absorption. Soil Sci. Soc. Amer. Proc., 27:243-246. Freeze, A. 1969. One-Dimensional Vertical, Unsteady, Unsaturated Flow above a Recharging or Discharging Ground-Water Flow System. Water Resour. Res., 5:153-171. Freshley, M. D., A. E. Reisenaner, and G. W.Gee. 1985. Sensitivity Analysis Applied to Unsaturated Flow Modeling of a Retorted Oil Shale Pile. Soil Sci. Am. J., 49:28-34. Gardner, W., and J. A. Widtsoe. 1922. The Movement of Soil Moisture. Soil Sci., 11:215-232. Gardner, R. W. 1956. Calculation of Capillary Conductivity from Pressure Plate Outflow Data. Soil Sci. Soc. Am. Pr. 20:317-320. Gardner, W. R., and M. S. Mayhugh. 1958. Solutions and Test of the 121 Diffusivity Flow Equation for the Movement of Water in Soil. Soil Sci. Soc. Am. Proc., 22:197-201. Gardner, W. H. and C. Calissendorff. 1967. Gamma Ray and Neutron Attenuation in Measurement of Soil Bulk Density and Water Content. Reprinted from "Isotope and Radiation Techniques in Soil Physics and Radiation Studies." International Atomic Energy Agency, Vienna. Green, W. H., and Ampt, G. A. 1911. Studies on Soil Physics: 1. Flow of Air and Water through Soils. J. Agr. Sci., 4:1-24. Green, R. E. and J. C. Corey. 1971. Calculation of Hydraulic Conductivity: A Further Evaluation of Some Predictive Methods. Soil Amer. Proc., 3523-8. Grismer, M. E. 1986. Pore Size Distribution and Infiltrations. Soil Sc., 141:249-260. Hanks, R. J., and S. A. Bowers. 1962. Numerical Solutions of the Moisture Flow Infiltration into Layered Soils. Soil Sci. Soc. Am. Proc., 26:530-534. Hanks, R. J., and G. L. Ashcroft. 1980. e o 5. Chapter 2, pp:20-59. Springer-Verlag, Berlin. Haverkamp, R., M. Vauclin, J. Touma, P. J. Wierenga, and G. Vachaud. 1977. A Comparison of Numerical Simulation Models for One- Dimensional Infiltration. Soil Sci. Soc. Am. Proc., 41:285-294. Hillel. D. 1980. WWW Chapter 7. 1391123- 162. Academic Press, New York. Ijjas, I. 1966. Effect of Compactness and Initial Moisture Content of the Soil on the Process of Capillary Rise. UNESCO Symposium on Wat. in the Unsat. Zone, 11:547- 559. Irmay, S. 1966. Solutions of the Non-Linear Diffusivity Equation with a Gravity Term in Hydrology. UNESCO Symposium on Wat. in the Unsat. Zone, 2:478-499. Jackson, R. D. 1963. Temperature and Soil Water Diffusivity Relations. Soil Sci. Soc. Am. Pr., 27:363-366. Jackson, R. D. 1964. Water Vapor Diffusivity in Relatively Dry Soil: 1. Theoretical Consideration and Sorption Experiments. Soil Sci. Soc. Am. Proc., 28:172-176. Jackson, R. D. 1964. Water Vapor Diffusivity in Relatively Dry Soil: II. Desorption Experiments. Soil Sci. Soc. Am. Proc., 28:464-466. Jackson, R. D. 1964. Water Vapor Diffusivity in Relatively Dry Soil: III. Steady States Experiments. Soil Sci. Soc. Am. Proc., 122 28:467-470. Jackson, R. D. 1964. Water Vapor Diffusivity in Relatively Dry Soil: IV. Temperature and Pressure Effects on Sorption Diffusivity Coefficients. Soil Sci. Soc. Am. Proc., 29:144- 148. Jackson, R. D., R. J Reginato, and W. E. Reeves. 1962. A Mechanical Device for Packing Soil Columns. U. S. D. A. - A. R. 8., 41-52. Kar-Kuri, H. M. 1983. Testing of the One-Dimensional Infiltration Equation on Some Michigan Soils. M. S. Thesis, (Crop and Soil Science), Michigan State University, East Lansing, MI. Keen, B. A. 1924. On the Moisture Relations of Ideal Soil. J. Agr. Sci., 14:170-177. Kirkham, D., and Feng, G. L. 1949. Some Tests of the Diffusion Theory and Laws of Capillary Flow in Soils. Soil Sci., 67:29- 40. Kirkham, D., and W. L. Powers. 1972. va ed Chapters 6 and 7, pp:235-378. Wiley Interscience, New York. Klute, A. 1952. Some Theoretical Aspects of the Flow of Pater in Unsaturated Soils. Soil Sci. Soc. Amer. Proc., 16:144-148. Klute, A. 1986. Water Retention: Laboratory Methods. In A. Klute (ed). MW. Part 1.2:nd ed. Agronomy. 9:635-660. Klute, A. and C. Dirksen. 1986. Hydraulic Conductivity and Diffusivity: Laboratory Methods. A. Klute (ed). M2£h2Q§_2f;§211 Annlyoig, Part 1.2:nd ed. Agronomy, 28:687-732. Kobayashi, H. 1966. Theoretical Analysis and Numerical Solution of Unsaturated Flow in Soil. UNESCO Symposium on Wat. in Unsat. Zone, II: 429-438. Kuiper, L. R. 1979. Comment on "A Two-Dimensional Linearized View of One-Dimensional Unsaturated-Saturated Flow" by John Cushman and Don Kirkham. Water Resour. Res., 15(3):719-720. Kunze, R. J., and D. Kirkham. 1962. Simplified Accounting for Membrane Impedance in Capillary Conductivity Determinations. Soil Sci. Soc. Am. Pr., 26:421-426. Kunze, R. J., and D. R. Nielsen. 1982. Finite Difference Solutions of the Infiltration Equation. Soil Sci., 134:81-88. Kunze, R. J., and D. R. Nielsen. 1983. Comparison of Soil Water Infiltration Profiles Obtained Experimentally and by Solution of Richards' Equation. Soil Sci., 135:342-349. 123 Kunze, R. J., and H. M. Kar-Kuri. 1984. Gravitational Flow in Infiltration. Proc. of the Nat. Conf. on Advance in Infilt., 12-13 Dec., Am. Soc. of Agric. Eng., St. Joseph, Mich., pp.14-23. Kunze, R. J., J. Y. Parlange, and C. W. Rose. 1985. A Comparison of Numerical and Analytical Techniques for Describing Capillary Rise. Soil Sci., 139:491-496. Lenhard, R. J., and R. H. Brooks. 1985. Comparison of Liquid Retention Curves with Polar and Nonpolar Liquids. Soil Sci. Am. J., 49:816-821. McLaughlin, W. W. 1924. The Capillary Distribution of Moisture in Soil Columns of Small Cross Section. U. S. D. A., Dept. Bul., 1221:1-22. Moore, R. E. 1939. Water Conduction from Shallow Water Tables. Higardia, 6:383-426. Mualem, Y. 1986. Hydlaulic Conductivity of Unsaturated Soils. Predictions and Formulas. A. Klute (ed). Methods of Soil Analysis, Part 1.2:nd ed. Agronomy, 31:799-823. Nakano, M. Y., Amemiya, and K. Fujii. 1986. Saturated and Unsaturated Hydraulic Conductivity of Swelling Clays. Soil Science, 141:1-6. Nielsen, D. R., and R. E. Phillips. 1958. Small Fritted Glass Bead Plates for Determination of Moisture Retention. Soil Sci. Soc. Am. Proc., 22:574-575. Nielsen, D. R., J. W. Biggar, and J. M. Davidson. 1962. Experimental Consideration of Diffusion Analysis in Unsaturated Flow Problems. Soil Sci. Soc. Am. Proc., 26:107-111. Nielsen, D. R., and J. W. Biggar. 1982. 1molioo§1ono_of_yooo§o Zono go Woto: Booouzce nanogeneno, Sc. Basis of Water Res. Mgt., National Research Council, National Academic Press, Washington D.C. Nielsen, D. R., R. D. Jackson, J. W. Cary, and D. D. Evans. 1972. fioil_flo;o;i Chapter 4, pp:64-91, American Society of Agronomy and Soil Science Society of America, Madison, Wisconsin. Nimko, J. R., and E. E. Miller. 1986. The Temperature Dependence of Isothermal Moisture vs. Potential Characteristics of Soils. Soil Sci. Am. J., 50:1105-1113. Nofziger, D. L. 1985. Interactive Simulation of One-Dimensional Water Movement in Soils: User's Guide. Institute of Food and Agricultural Sciences, University of Florida. Onstand, C. A., T. C. Olson, and L. R. Stone. 1973. An 124 Infiltration Model Tested with Monolith Moisture Measurements. Soil Sci., 116:13-17. Packer, J. C., J. B. Kool, and M. Th. Van Genuchten. 1985. Determining Soil Hydraulic Properties from One-Step Outflow Experiments by Parameter Estimation: 1. Theory and Numerical Studies. Soil Sci. Soc. Am. J., 49:1348-1354. Packer, J. C., J. B. Kool, and M. Th. Van Genuchten. 1985. Determining Soil Hydraulic Properties from One-Step OutFlow Experiments by Parameter Estimation: II. Experimental Studies. Soil Sci. Soc. Am. J., 49:1354-1359. Parlange, J. Y. 1971a. Theory of Water Movement in Soils: 1. One Dimensional Absorption. Soil Sci., 111:134-137. Parlange, J. Y. 1971b. Theory of Water Movement in Soils: 2. One- Dimensional Infiltration. Soil Sci., 111:170-174. Parlange, J. Y., and D. Aylor. 1972. Theory of Water Movement in Soils: 9. The Dynamics of Capillary Rise. Soil Sci., 114:79- 81. Parlange, J. Y. 1973. Theory of Water Movement in Soils: 10. Cavaties with Constant Flux. Soil Sci., 116:1-7. Parlange, J. Y. 1975. A Note on the Green and Ampt Equation. Soil Sci., 119:466-467. Parlange, J. Y. 1975. Convergence and Validity of Time Expansion Solution: A Comparison to Exact and Appoximate Solution. Soil Sci. Soc. Am. Proc., 39:1-6. Parlange, J. Y. 1975a. Theory of Water Movement in Soils: 11. Conclusion and Discussion of Some Recent Developments. Soil Sci., 119:158-61. Parlange, J. Y. 1975b. On Solving the Flow Equation in Unsaturated Soils by Optimization. Soil Sci. Soc. Am. Proc., 39:415-418. Parlange, J. Y. 1980. Water Transport in Soils. Ann. Rev. Fluid Mech., 12:77-102. Parlange, J. Y. 1982. The Three-Parameter Infiltration Equation. Soil Sci., 133:337-341. Parlange, J. Y., W. L. Hogarth, J. F. Boulier, J. Touma, R. Haverkamp, and G. Vachaud. 1985. Flux and Water Content Relation at the Soil Surface. Soil Sci. Soc. Am. J., 49:285-288. Pavlakis, George, and L. Barden. 1972. Hysteresis in the Moisture Characteristics of Clay Soil. J. Soil Sci., 23:350-361. 125 Peck, A. J. 1966. Diffusivity Determination by a New Outflow Method. UNESCO Symposium on Wat. in the Unsat. Zone, 1:191-202. Peerlkamp, P. R., and P. Boekel. 1960. Moisture Retention by Soils. Versi. Mede. Comm. Hydr. Onderz. T. N. 0., 5. Pekishek, R. E., J. Osborn, and J. Letey. 1962. The Effects of Wetting Agents on Infiltration. Soil Sci. Soc. Amer. Proc., 26:595-598. Philip, J. R. 1955. Numerical Solutions of Equations of the Diffusivity Type with Diffusivity Concentration-Dependent. Trans. Faraday Soc., 51:885-892. Philip, J. R. 1957a. Numerical Solution of Equation of the Diffusion Type with Diffusivity Concentration-Dependent. II. Aust. J. Physics, 10:29-42. Philip, J. R. 1957b. The Theory of Infiltration: 1. The Infiltration Equation and its Solution. Soil Sci., 83:345-357. Philip, J. R. 1957c. The Theory of Infiltration: 2. The Profile of Infinity. Soil Sci., 83:435-448. Philip, J. R. 1957d. The Theory of Infiltration: 3. The Moisture Profile and Relation to Experiment. Soil Sci., 84:163-187. Philip, J. R. 1957a. The Theory of Infiltration: 4. Sorptivity and Algebraic Infiltration Equations. Soil Sci., 84:257-264. Philip, J. R. 1957f. The Theory of Infiltrationz5. The Influence of the Initial Water Content. Soil Sci., 84-329-339. Philip, J. R. 1958a. The Theory of Infiltration: 6. Effect of Water Depth over Soil. Soil Sci., 85:278-286. Philip, J. R. 1958b. The Theory of Infiltration: 7. Soil Sci., 85:333-337. Philip, J. R. 1964. Similarity Hypothesis for Capillary Hysteresis in Porous Media. J. of Geophysical Res., 69:1553-1562. Philip, J. R. 1966. A Linearization Technique for the Study of Infiltration. UNESCO Symposium on Wat. in the Unsat. Zone, 11:471-478. Philip, J. R. 1966. The Dynamics of Capillary Rise. UNESCO Symposium on Wat. in the Unsat. Zone, 11:559-578. Philip, J. R. 1967. Sorption and Infiltration in Heterogeneous Media Australian J. Soil Res., 5:1-10. Philip, J. R. 1969. Theory of Infiltration. Adv. in Hydrosci., J]... _-,_ . AL. 34‘ 126 5:215-305. Poulovasilis, A. 1962. Hysteresis of Pore Water: An Application of the Concept of Indepedent Domains. Soil Sci., 93:405-412. Raats, P. A. C. 1966. Development of Equations Describing Transport of Mass, Momentum, and Energy in Soils. UNESCO Symposium on Wat. in the Unsat. Zone, 2:535-546. Ragab, R., et. a1. 1981. Comparative Study of Numerical and Laboratory Methods for Determining the Hydraulic Condition Function of a Sand. Soil Sci., 131:375-388. Rawlins, S. L., and W. H. Gardner. 1963. A Test of the Validity of the Diffusion Equation for Unsaturated Flow of Soil Water. Soil Sci. Soc. Amer. Proc., 27:507-511. Richards, L. A. 1928. The Usefulness of Capillary Potential to Soil Moisture and Plant Investigations. J. Agr. Res., 37:719-742. Richards, L. A. 1931. Capillary Conduction of Liquids Through Porous Mediums. Physics, 1:318:333. Richards, L. A. 1965. Physical Conditions of Water in Soil. Agronomy, 9:128-139. Richards, L. A., and L. R. Weaver. 1944. Moisture Retention by some Irrigated Soils Related to Soil Moisture Tention. J. Agric. Rec., 69:215-235. Rode, A. A. 1966. Hydrological Properties and Moisture Regime in Unsaturated Zone. UNESCO Symposium on Wat. in Unsat. Zone, 1:33-46. Rose, D. A. 1966. Water Transport in Soils by Evaporation and Infiltration. UNESCO Symposium on Wat. in the Unsat. Zone, 1:170-181. Rubin, J. 1966. Numerical Analysis of Pointed Rainfall. UNESCO Symposium on Wat. in the Unsat. Zone, II:440-451. Salter, P. J., and J. B. Williams. 1965. The Influence of Texture on the Moisture Characteristics of Soils. Part 1: A Critical Comparison of Techniques for Determining the Available Water Capacity and Moisture Characteristic Curve of a Soil. J. Soil Sci., 16:1-15. Sander, C., and J. Y. Parlange. 1984. Water and Air Movement in Soils: An Application of Brutsaert's and Optimization Tech. Soil Sci., 138:198-202. Sposito, C., and W. A. Jury. 1985. Inspection Analysis in the Theory of Water Flow through Unsaturated Soil. Soil Sci. Am. J., 49:791-798. 127 Stockinger, K. R., E. R. Perrier, and W. D. Fleming. 1965. Experimental Relations of Water Movement in Unsaturated Soils. Soil Sci., 100:124-129. Sunderman, H. D., and E. E. Banbury. 1978. Performance of Soil Wetting Agents in Northwestern Kansas. Agron. Abst., p. 184. Sunderman, H. Undated. Soil Wetting Agents: Their Use in Crop Production. North Central Extention Publication 190. Swartzendruber, D. 1955. Anomalies in Capillary Intake as Explained by Capillary Rise Experiments. Soil Sci. Soc. Proc., 20:453-458. Taylor, S. A., and G. Ashcroft. 1972. Physical anohology, W. H. Freeman, San Francisco, CA. Wadsworth, H. A. 1931. Further Observation upon the Nature of Capillary Rise through Soils. Soil Sci., 32:417-434. Warrick, D. 0., Lomen, and S. R. Yates. 1985. A Generalized Solution to Infiltration. Soil Sci. Soc. Am. J., 49:34-38. Wesseling, J. and K. E. Wit. 1966. An Infiltration Method for the Determination of the Capillary Condition of Undisturbed Soil Cores. UNESCO Symposium of Wat. in the Unsat. Zone, 1:223-234. Whisler, F. D., A. Klute, and D. B. Peters. 1968. Soil Water Diffusivity from Horizontal Infiltration. Soil Sci. Soc. Am. Proc., 32:6-11. Willis, W. 0., D. R. Nielsen, and J. W. Biggar. 1965. Water Movement Through Acrylic Plastic. Soil Sci. Amer. Proc., 29:636-637. Wind, G. P. 1961. Capillary Rise and Some Applications of the Theory of Moisture for Land and Water Management Research. Wagenengen, the Netherlands, Technical Bulletin #22. Wind, G. P. 1966. Capillary Conductivity Data Estimated by a Simple Method. UNESCO Symposium on Wat. in the Unsat. Zone, 1:181-191. Wollny,E. 1885. Untersuchungen uber die kapillare Leitung des Wassers. forsch. Geb. Agr. Phys., 8:206-220. Youngs, E. G. 1957. Moisture Profiles During Vertical Infiltration. Soil Sci., 84:283-290.