DE’E‘EEMMTEGN OF CAPELMRY CONEE‘C?‘§VE?? é}? ENSfiUEé’E‘EE WEGEES REE-EA FEW BeiGESTEERE CHARACTERISTICS Thesis {or fire Deg?“ of pit. D. EECSEGAS SEfi‘E {JEWEESEW Abdal Razziq Qazi 2927-3- LIBRARY TH 55:13 ' Michigan State Lhuyennqr This is to certify that the thesis entitled Determination of Capillary CwnductiVity of Unsaturated Porous Media From Moisture Characteristics presented by Abdul Raviq qui has been accepted towards fulfillment of the requirements for __P_'£L._D_.._ degree in M5 . fALKLWMW Major professorzl—s Datel7AV? 70 0-169 ABSTRACT DETERMINATION OF CAPILLARY CONDUCTIVITY OF UNSATURATED POROUS MEDIA FROM MOISTURE CHARACTERISTICS BY Abdul Raziq Qazi Conductivity function is very essential for pre- diction of moisture movement in unsaturated porous mater- ials. Some mathematical models have been prOposed in previous studies for analytical determination of unsat- urated capillary conductivity for porous materials from their moisture characteristics. Moisture characteristics used in these models were obtained by static method, whereas, in nature the varia- tion of moisture and capillary pressure in soil are gradual and dynamic. Therefore, in this study a dynamic method was develOped and used to determine moisture char— acteristics of three natural soils. Gamma ray attenua- tion technique has been studied and used to determine the varying moisture content of the soils. The mass Abdul Raziq Qazi attenuation coefficient of water has been found to vary with thickness of water. The curves of static and dynamic moisture char- acteristics have been found to differ from each other appreciably. Capillary conductivities calculated by various models from both dynamic and static moisture character- istic are compared with the experimentally determined capillary conductivities. It is observed that no model can accurately predict the conductivity for adsorption Brooks and Corey's model does not seem to work for fine textured soils and for other soils at higher moisture range without modification. A modification has been sug- gested. Kunze's equation seems to have the tendency of giving lower conductivity values at low moisture contents. For glass beads of Topp and Miller, it tends to predict higher values of conductivity at all moisture contents using dynamic data. It gives fairly accurate results for static data for the soils studied. Direct numerical integration of the original Burdine equation using dynamic capillary pressure and saturation data gives conductivity values that approximate the experimental results better than other mathematical models. Approved r f r ajo Approvedfi W .He of Department s DETERMINATION OF CAPILLARY CONDUCTIVITY OF UNSATURATED POROUS MEDIA FROM MOISTURE CHARACTERISTICS BY Abdul Raziq Qazi A THESIS Submitted to Michigan State University in partial fullfilment of requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1970 ACKNOWLEDGMENTS The author expresses deep appreciation to the department of Agricultural Engineering, especially to Professor E. H. Kidder, Committee chairman, members of the Committee, Dr. Merle Esmay, Professor L. V. Nothstine and particularly Dr. R. J. Kunze for his guidance. The author is also grateful to the department of Soil Science for the use of their facilities. The author is alSo thankful to his wife, Lorna, for all the efforts put forth by her during the period of this study. ii TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . LIST OF FIGURES O O O O O I O O O O O I O O I 0 INTRODUCTION 0 O O O O O O O O O O O O O O O 0 Need for Study 0 O O O O O O I O O O O O O O Sc0pe of Research . . . . . . . . . . . . . REVIEW OF LITERATURE . . . . . . . . . . . . . Use of Functions K(6), D(9) and C(6) in the Diffusion Equation and its Application . Methods of Determination of K(6), D(6) Functions . . . . . . . . . . . . . . . . Transient methods . . . . . . . . . . . . Computational techniques for K(6) measurements . . . . . . . . . . . . . Gamma Ray Attenuation Technique for Moisture Measurements . . . . . . . . . . Hydraulic Gradient and its Application to D'Arcy's Equation in Unsaturated Flow . . Measurement of Moisture Characteristics and conductiVity O O O O O I O O O O O O O 0 EXPERIMENTAL DESIGN AND PROCEDURE . . . . . . . Materials, Equipment and Methods . . . . . . Materials 0 O O O O O O O O O I O O O O 0 Equipment 0 O O O O O O O O O O O I O O 0 Calibration and response of tensiometer- transducer and recorder system . . Moisture content determination equipment iii Page ii U1 UH I-‘ < 11 11 20 32 38 41 44 44 44 49 53 54 Page Determination of mass absorption coeffi- cient . . . . . . . . . . . . . . . . . ~ 58 Methods . . . . . . . . . . . . . . . . . . . 60 Moisture characteristic determination . . 61 Conductivity measurements . . . . . . . . 64 PRESENTATION AND ANALYSIS OF DATA . . . . . . . . . 68 Theory . . . . . . . . . . . . . . . . . . . . . 68 Theory of gamma radiation, attenuation teChnique I O O O O O O I O O I O O O O O 68 Mathematical models used in the computation of conductivity function . . . . . . . . . 74 Presentation of Data and Analysis . . . . . . . 76 Dynamic method . . . . . . . . . . . . . . . 76 Dynamic method compared with static method . 78 Head loss between tensiometers and boundary I O O I O O I O O O O O O O O I O 78 K, e, and PC data. 0 o o o o o o o o o o o o o 98 Comparison of theoretical and experimental conductivity results ... . . . . . . . . . 118 DISCUSSION 0 O C O O O O I O I O I O O O I O O O I 133 Gamma Radiation Attenuation Technique . . . . . 133 Static and Dynamic Methods . . . . . . . . . . . 134 Experimental and Calculated Conductivity . . . . 135 CONCLUSIONS 0 I O O O O O O O O O O O I O O O O I O 14 2 REFERENCES 0 O O O O O O O O O . . . . . . . . . . 144 APPENDIX I o o o o o o o o o o o o o o o o 0 O 0 0 149 APPENDIX II o o o o o o o o o I O ' . . . . . . . . 153 APPENDIX III e o o o o o o o o o 0 ° ' . ' . . . . 157 iv LIST OF FIGURES Figure Page 1. Grain size distribution of porous materials used 0 O O O O O O O O O O I I O O O O O O O O 45 2. (a) Cross sectional view of the tensio- meter assembly (b) Cross sectional view of one end section and one tensiometer assembled in the pressure cell . . . . . . . . . . . . . 46 3. (a) Schematnzdiagram of the equipment (b) Side View of the pressure cell . . . . . . 50 4. Strain gage bridge of pressure transducer . . 52 5. Relationship between moisture contents and count rate . . . . . . . . . . . . . . . . 71 6. Verification of exponential law by plotting experimental values of moisture contents against calculated values . . . . . . . . . . 72 7. Variation of mass attenuation coefficient of water with increase in thickness of water 0 O O O O O O O I O I O O O O O O O O O 73 8. Variation of non-wetting, wetting and capillary pressures with time in medium sand for air pressure application rate of 1.65 mm./cm. H20. . . . . . . . . . . . . . 79 9. Variation of P Pw and PC . in medium sand for P = “35. 5w min. /cm. and AH = 25.0 cm. H20 CAB O O O O O O O O O O O O O O O O O 80 10. Variation of P and P in medium sand for R = 26. 7 nKin. Pycm. H2 5 and AH = 31.0 cmHgB 81 Figure Page 11. Variation of Pn' and P in medium sand for p = 37.1 fiin. Pycm. H2 8 and AH = 25. 0 cm. £8 0 O I O O O O O O O O I I I I O I O O O 82 12. Variation of P and P with time in fine sand for AH= 3?. OP ”R -$9. 6 min. /cm. H20 . . 83 13. Variation of PRW, Pw and PC in fine sand for RAP = 1. 795 m1 /cm. H20 . . . . . . . . . . . . 84 14. Variation of PPn, and P with time in fine sand for RA 8. 625 min. /8m. H20 . . . . . . . 85 15. Variation of P P and P with time for sandy loam withR w: 18.15 min. /cm. H20 and AH= 18 cm. H98 . . . . . . . . . . . . . . 86 16. Moisture characteristics by dynamic method and lag between P and Pn nw in medium sand for various RAP and AH values . . . . . . . . . . . 87 17. (a) Moisture characteristics by dynamic method and lag between PC and in for RAP=lo55 min./cm. (b) Comparison of moisture characteristics obtained by dynamic and static equilibrium method for medium sand . . . . . . . . . . . . . 88 18. Moisture characteristics by dynamic method and lag between Pc and in in fine sand for RAP = 19.6 min. /cm. H20 and AH = 16 cm. H20 . . . . . 89 19. Moisture characteristics by dynamic method and lag between PC and in in fine sand for R = 1.6 minO/cm. H20 0 O O O O I O O O O O O @P. O O 90 20. Moisture characteristic by dynamic method and lag between PC and in in fine sand for RAP = 37 sec./cm. H2 0 O O O O O O I O O O C I I O O O 91 21. Comparison of moisture characteristics by dynamic method, at various RAP values, with static equilibrium method for fine sand . . . . 92 22. Comparison of moisture characteristics by static and dynamic method for sandy loam . . . . . . . 93 23. Variation of hydraulic gradient between tensio- meters with flow velocity in medium sand . . . . 95 vi Figure 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. Variation of hydraulic gradient between tensiometers with flow velocity in fine sand Variation of hydraulic gradient between tensiometers with flow velocity in sandy loam O O O O O O O O O O O O O O O O O O 0 Experimental capillary conductivity versus moisture contents for medium sand . . . . Experimental capillary conductivity versus moisture contents for fine sand . . . . . Experimental capillary conductivity versus moisture content for sandy loam . . . . . Experimental capillary conductivity versus capillary pressure for medium sand . . . . Experimental capillary conductivity versus capillary pressure for fine sand . . . . Experimental capillary conductivity versus capillary pressure for sandy loam . . . . Variation of hydraulic conductivity of mono-dispersed beads with capillary pressure in rewet-redry loop . . . . . . - Variation of capillary conductivity with capillary pressure for aggregated beads for redry and rewet loop . . . . . . . . . Saturation versus capillary pressure for medium sand for desorption and adsorption Saturation versus capillary pressure for fine sand for desorption and adsorption Saturation versus capillary pressure for sandy loam desorption and adsorption process Saturation versus capillary pressure for mono-dispersed glasstxnxks rewet-redry loop Saturation versus capillary pressure for aggregated glass beads rewct-redry loop .- vii Page 96 100 101 102 103 104 105 106 108 109 110 111 112 Figure 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. Relationship between effective saturation and capillary pressure for medium sand . . . . . . Relationship between effective saturation and capillary pressure for fine sand . . . . . . Relationship between effective saturation and capillary pressure for sandy loam . . . . . Relationship between effective saturation and capillary pressure for mono-dispersed glass beads I I I I I I I I I I I I I I I I I I I Relationship between effective saturation and capillary pressure for aggregated glass beads. Area under the curve for relative conductivity computation for medium sand . . . . . . . . Area under the curve for relative conductivity computations for fine sand . . . . . . . . . . Area under the curve for sandy loam for relative conductivity computations . . . . . . Area under curve for mono-dispersed glass beads for relative conductivity computations Area under curve for aggregated glass beads for relative conductivity computations Conductivity comparison for medium sand desorption . . . . . . Conductivity comparison for medium sand adsorption . . . . . . . . . . . Conductivity comparison for fine sand desorption . . . . . . . . . . . . . . . . Conductivity Comparison for sandy loam desorption . . . . . . . . . . . Conductivity Comparison of mono-dispersed glass beads desorption . . . . . . . . . . . . Conductivity comparison of mono-dispersed glass beads adsorption . . . . . . viii Page 113 114 115 116 117 119 120 121 122 123 125 126 127 128 129 130 Figure Page 55. 56. Conductivity comparison for aggregated glass beads desorption . . . . . . . . . . . . 131 Conductivity comparison for aggregated glasstxxxks adsorption , , , , . . , . . . . . 132 ix INTRODUCTION Need for Study In order to understand, control and accurately pre— dict the water movement in soils due to drainage, evap- oration and infiltration after rainfall or irrigation one must understand the factors which control this move— ment. In agriculture this is important from the point of View of plant growth. In construction work it is important from the point of view of drainage of highways and seepage through walls, foundations and dams. To the petroleum industry it is important for removing oil from porous materials by displacement with either wetting or non-wetting fluids. This complex process of water movement in porous materials needs more and more attention in order to increase agricultural production for the growing popu- lation of the world. The increasing loss of agricultural land to buildings and roads makes it more crucial to make use of these unused lands. Deserts or marshlands both offer problems of water movement in soil--in one case the supplemental irrigation is needed and in the other case the drainage is important. Since D'Arcy discovered the relationship, over one hundred years ago, between flow velocity, hydraulic "gradient and hydraulic conductivity of porous material, many investigators have been attempting to develop mathematical models describing flows in saturated soils. They attempted first to describe the one dimensional flow and then the two dimensional flow and finally the unsaturated flow with limited boundary conditions. D'Arcy's equation is basically for steady state laminar incompressible flow. Flow of liquids in soils is 1am- inar and incompressible but not always steady. Richards in 1931 suggested a second order non-linear differential equation for general case by combining the D'Arcy law (V=KI) and the continuity equation V.V = de/dt. These kind of equations are converted into a system of linear equations in the finite difference form. Knowing soil moisture characteristics, the con— ductivity or diffusivity function and the initial and boundary conditions, the solution of these equations through a process of iteration with the help of a com— puter gives a relationship between water content, ten- sion, space and time. Thus the moisture conditions can be predicted at a given point at a certain time for certain problems. When the medium is unsaturated and conductivity K becomes a function of moisture content, 6, it is no longer a constant. The same is true for diffusivity 0(6) and water capacity C(G) which are related as K(e) = C(B)-D(6). Attempts have been made to determine K(6) from soil moisture characteristics but soil moisture characteristics exhibit hysteresis when the medium goes through the process of desorption and then adsorption. Thus the conductivity function computed from soil mois- ture characteristics will be affected by this phenomenon. Moisture characteristics which characterize the pore size distribution of the porous material depend upon the pro- cess used to empty or fill the pores. Usually pressure steps are applied successively to drain or fill the pores, achieving equilibrium state after each step at which point the tension or the capillary pressure cor- responds to the air pressure. Saturation of the media is accomplished by releasing the pressure in steps. In nature, however, the porous material is not saturated or desaturated by successive pressure steps but is in fact a gradual process which seems to differ from the step method. Scope of Research In this study, therefore, a gradual and dynamic method has been developed to vary the capillary pressure and moisture contents continuously, but gradually. The moisture contents are obtained by nondestructive method of gamma ray attenuation. The unsteady state of flow was maintained and capillary conductivity values corres- ponding to measured values of moisture contents and capillary pressure were determined. Results of the dy- namic data compared with the standard step method data are presented. Two mathematical models for determining the K(9) from moisture characteristics were investigated to see if the K(6) values predicted by these models fit the experimental K(6) values. Effects of the hysteresis on these models was analyzed. Effect of moisture content variations on the mass absorption coefficient of the water was studied. The exponential law for the attenuation of gamma radiation 137 was verified for a fixed geometry. A regression from Cs technique to correlate moisture contents with the count rate is developed and analyzed. The results from the study of effects of external hydraulic gradients and air pressure application rates on the wetting phase pressure and capillary pressure are presented. The effects of the head loss in the barriers as related to the suction gradient between two points in the soils are discussed. REVIEW OF LITERATURE Use of Functions K(e), D(9), and C(6) in the Diffusion Equation and its Application The heat diffusion equation and its application to unsaturated flow problems appear in early literature. The flow parameters—-capi11ary conductivity K(6), dif- fusivity D(6), and water capacity C(G) have been defined by the following relationships: Volume rate of flux per unit area is 39 '1‘”) is? “ 'me) “a? (1) where D(6) = K(6)/C(6), C(e) = ae/aH, H is hydraulic head (cm), L is length of soil column (cm) and 6 is volumetric moisture contents of the soil (cc/cc). By combining the continuity equation _ 36 Vov _' T with D'Arcy's equation V = K(0)I Richard (34) in 1939 obtained the following second order non-linear differential equation: v. [K(0)VH] = 13% (2) where (V.V) is the divergence of velocity vector V, t is time in seconds and I is the hydraulic gradient (cm/cm). The hydraulic gradient can be expressed as V¢ = V = — I H g (3) where ¢ is the total potential (cmZ/secz), and ¢ = w + gz, w is the capillary potential (cmZ/secz), gz is the gravitational potential (cmz/secz), g being the acceler- ation due to gravity and z the distance in the soil column along the vertical axis (cm). The intrinsic permeability k(6) (cm2) is defined as the property of the media for transmitting fluid irrespective of the nature of the fluid passing and is related to capillary hydraulic conductivity K(6) as K(6) = (pg/u) k(6) (4) where p is the density and u is the absolute viscosity of the fluid respectively. Substituting the relation- ships (3) and (4) in equation (2), results in the equa- tion for vertical flow as 30 _=3_ 3t 32 x E k(e) %% + %% k(6) - (5) ‘C If the change in the w is small and hence the change in moisture content is small, for horizontal flow the above expression reduces to 2 36 _ g 3 w e- [w 2:21 Equation (6) implies one dimensional horizontal flow in isotr0pic material with k(6) constant for a small change in moisture content. For vertical flow H = ¢/g, for horizontal flow H = W/g, D 1 3w _ 86 '3 me) §_3x - me) B—X (7) and 2 ae _ a e 'at ' ”‘9’ 5? ‘8) which is similar to the diffusion equation for one di- mensional heat flow. In order to predict moisture distribution after infiltration into the soil, Ashcroft gt_§1. (1) developed a technique for solving an implicit difference analogue of the diffusion equation. The diffusivity values were chosen to convert the nonlinear diffusion equation into a system of linear equations which were finally solved by Gaussian elimination technique. The equation used in this method was B — EL 22 '3? — 8x (D(Q) 3x) Q) which was converted to finite difference form as 99+1 —69 J 3 At = D(9?:i;§) ( §:{_ Og+l> /Ax -o(6?fij§) (03+1— 9?ti>/AX Ax where superscripts and subscripts denote time and space respectively. Further, substituting the initial and boundary cond1tions and approximating n+l/2 311/2 n=1/2 .* n 0 and D(8 jil/Z ej_+_l/2 by 311/2 by D(0* ) = D(6 ) j:1/2 where a is an arbitrarily chosen small constant and n . . . 8j+1/2 is an average of two succe551ve m01sture con— tents at x and x + Ax, the following equation results: * .n+l 1* n+1 -_ D 0. ). + E. ; . r ( 3:1/2) t'j-l (1+r D()j—1/2)+rD(tj+1/2)) 0] _ 2* ,n+l_ n where r I (At)/([x)2. The above system of linear equa- tIODS forms a coefficient matrix of the form bl cl W a2 b2 C2 a b m m 6* e ‘* where al — -rD( 1—1/2)’ bl — 1 + rD( 1-1/2) +rD(Bi+l/2) * and c1 = —rD(Bi+l/2). The above tri-diagonal system of equations was solved by Gau551an elimination technique for 8?. The solution so obtained was compared by them to the solution obtained by using Boltzmann transforma- tion. They reported that the two solutions gave similar results for infiltration into horizontal columns of semi- infinite uniform media as obtained experimentally. Hanks and Bowers(l6) used Similar techniques for flow upwards and downwards in vertical columns. They compared their results with other investigators and claim excellent agreement. However, the results were not verified experimentally. It was p01nted out by them that the soil mOisture characteristics and a rela- tionship between soil moisture content and diffu51vity must be known to use their method. Whisler and Klute (44) used the equation of the type 8h _ EL . ah 8K(h,z) C(h,Z) t _ 82 K(hrZ) 3‘2- + ———BZ 10 to develop a numerical solution to determine time and depth distribution of water content and pressure head during the infiltration process in vertical columns when both wetting and drying is taking place in some parts of the soil column. Thus hysteresis effects had to be considered. However, moisture characteristics used by them were arbitrary and C(h,z) values were ob- tained by differentiating the moisture characteristic curve; K(6) values were obtained by use of Millington and Quirk (29) formula and moisture characteristic curves. Whisler and Klute concluded that the position of wetting front was over estimated or under estimated if hysteresis was ignored depending upon whether the drainage or wett- ing curve was used. Staple (37) conducted a similar experiment and used a numerical technique to compute infiltration and redistribution of water in vertical soil columns. Hysteresis was incorporated by using conductivity and soil moisture tension data obtained from the desorption curve for the upper portion of the column which was desaturating; at the lower end of the column the adsorp- tion curve filled this need. Staple points out that use of diffusion equation involving D in the finite difference form predicted moisture profile for wetting which agreed better with experimental data than the equation using K. However, when accompanied by the hysteretic effects, the 11 error in the use of an equation involving K was small for drying part of the profile. Methods for Determination of K(6) and D(9) Functions It has been the goal of physicists, soil physicists and engineers to develop some efficient method for deter- mination of the flow parameters used in unsaturated flow studies. Experimental methods which have been developed to determine these parameters indirectly are less time consuming than direct methods. Some of the indirect methods involve determining the advance of a wetting front in a soil column and the moisture contents at various points along the axis of flow. Other methods in- volve determination of outflow data or moisture charac- teristics. Some methods involve both. These methods are briefly discussed. Transient methods 1/2 By using Boltzmann transformation 1 = xt_ Bruce and Klute (4) avoided the assumption of constant K(6) or 0(6) and equation (8) becomes 38 __ a as with the initial and boundary conditions as 12 where Si and as are initial and saturation moisture con- tents of the soil. Solution of Eris expression yields x Mex) = — 2%; ($99,: ei xde (9) which is evaluated from the plot of 9 versus x graphi- cally. Their experiments indicated that D(6) increases with moisture contents with a maximum value before sat- uration. Gardner (12) used a pressure plate outflow method to determine capillary conductivity K(6). He assumed that K(6) was constant for small changes in soil moisture tension and hence also for small changes in 6. It was also assumed that 6 is a linear function of soil moisture tension for a small increment in soil moisture tension -P i.e. 6=d+bP (10) Neglecting gravity, the one-dimensional flow equation (2) results in 66 _ l 6 3P 5? “ 35 ‘52 We) '3? which by equation (10) reduces to 6? pg 2 13 with boundary conditions, P(O,t) = 0, (BP/Bz)z=L = O, P (2L,t) = 0, P(z,0) = AP. Equation (11) was solved by method of separation of variables to yield A 00 £EE 2 Sin (nflz) exp (—a2Dt) 2L P(z,t) = II till-4 n l where a = nfl/ZL. Substituting 6 from equation (10) and defining _ ae _ as c _ b-W—Wthenb—E&K-ngD 9(z,t) = d + 4b$P 1 % Sin (nflz ) exp (—a2Dt) n=l 2L total moisture content is obtained by integrating the above expression which yields 00 L W(t) =jr A. e(z,t) dz=dV 8bA§V 2 $7 exp (-a2Dt) (12) 0 N n=1 n where A is cross sectional area and V is sample volume. The series converge rapidly with time. The initial water content of the soil becomes Wi = dV + bVAP, and final moisture content at t = w is W = dV. Total out- f flow for the process is Q = (wi «wf) = bVAP (13) 14 thus b can be calculated from experimental data. The equation for cumulative outflow at time t is _ _§_ .1. -2 Q(t) — QO 1 “2 n2 exp ( a Dt)] :3 ll M3 1 neglecting all but the first term of the series and tak- ing the logarithm yields _ _ 8Q _ 2 1n(QO Qt) — 1n (759) a Dt. Thus diffusivity D and the capillary conductivity K can be obtained by plotting the experimental data. Gardner (12) reported that the lower boundary condition P(6,t) was not always met since the resistance to flow between the sample and the porous plate was not negligible especially at higher moisture contents when large amounts of water were released. A logarithmic relationship be— tween conductivity and tension was suggested by Gardner as Log K = Log (a') - b' Log (P). (14) Kunze and Kirkhan1(23) gave a method of determin- ing D(6) and K(6) from experimental data without compli- cated computations. The technique takes into account the impedance of the porous plate and contact between the sample and the plate, and uses the initial outflow data for each pressure step thus reducing errors which may arise with time (due to air bubbles diffusing into 15 the plate). The technique makes use of the theoretical curves (Q/QO versus alth/LZ) plotted by using the equation 0_ = l — 2.. 2 “EN-0% Dt/Lz (15) 2 0? Q0 n=l q§(1 +a+an n where volumetric cumulative outflow at time t (Cm3) Q: QO = final cumulated outflow at equilibrium (Cm3) G“ = is obtained by solution of equation aah = Cota n D = diffusivity (sz/min) t = time (min) L = length of soil sample (Cm) a = ratio of plate impedance to soil slab impedance. Appropriate values of t when aiDt/L2 =1, give D = Lz/di t and K = (DAB/PZ-Pl), where change in moisture content corresponds to pressure change of (PZ-Pl). By these in— vestigations some doubts were cast on the application of diffusion theory, firstly, because piston action was anti- cipated and remedied by placing a capillary tube in the sample to convey high pressure from upper surface of sam- ple to the porous plate to remove moisture from the lower end first instead of the upper end and, secondly, because at low pressure the same values of K could not be obtained by using millipore filter and ceramic plates in spite of accounting for the impedance of the plates. l6 Nielsen gt_al. (30) experimentally investigated the applicability of diffusion equation by using oil or water as fluid entering the horizontal soil columns at different negative pressures. He assumed: (1) that D'Arcy's law is valid, i.e., the flux is proportional to water content gradient or pressure gradient for iso- thermal condition and (2) that there exists 1(6) such that 1(6) = xt'l/2 , where A is a single valued function of e, and x is the distance of the wetting front at time t from the starting end. The results of the above study question the validity of one of the above assumptions 1/2 was curvilinear because the relation between x and t for negative entry pressures larger than -2mb. Also, the predicted relationship between 9 and x was different from the measured values for higher negative pressures. They concluded that values of diffusivity calculated depend upon the boundary conditions at which the water enters the soil column and therefore those values cannot be used for the solution of diffusion equation for other boundary conditions. Gardner (13) and Doering (10) simplified the cal- culation of diffusivity by using one large pressure step instead of several small steps thus eliminating the need of the assumption of constant diffusivity by using in- stantaneous outflow from the pressure cell apparatus. The relationship used by them has the same theoretical foundation, equation (12), and is given as: 17 2 D = _—__.-4L gfl fl2(W-Wf) dt (16) where dW/dt is instantaneous outflow rate at time t, W is volume water content and Wf is the final equilibrium moisture content. Their results compared well with other methods. They found that the membrane impedance was not a significant factor except at saturation mois- ture content but boundary impedance had some effect when small pressure steps were used. For the multistep method the diffusivity values for negligible and non-negligible boundary impedance were not the same. Scatter of the points for diffusivity and conductivity versus moisture content was noticed in the case of multiple step method whereas in the one step method a smooth curve was ob- tained thus pointing out that the flow properties are affected by the applied pressure gradient. Many methods for determination of capillary dif- fusivity have been suggested as mentioned earlier but this parameter is still not completely understood. Kunze (21) believed that besides the loss of moisture content there are some other factors such as size and rate of the applied pressure which are time dependent and affect this parameter. He observed that in order to determine moisture status in a draining soil, moisture history and relationships of these flow parameters with soil moisture tension must be known. 18 Skaggs et al. (35) used the modified form of equation (14) of Gardner to relate conductivity K(h) with tension —h as K(h) = [(h/hl)a + b1'1 (17) with boundary and initial conditions 11:0 6:68 X=0 tio h = h. 6 = 6. x > O t = 0 1 1 h = h. 6 = 6. x = L t < t 1 1 - e where a, hl' and b are constants to be determined. 6S is saturated moisture contents, 6i and hi are initial moisture contents and tension respectively in the soil column, and te is time at which the wetting front arrives at the bottom of the column. The constant b is determined at the end of the experiment when steady flow is reached at which time h is assumed to be zero whereby equation (17) gives b = l/KO, and a and hl are determined by trial and error. Value of K(h) so deter- mined is used to solve the equation 6 3h K a—x [K(h) 8E] ' 3x . (18) | 0 EE ll C(h) in equation (18) is water capacity determined from the moisture characteristics for absorption. The results of equation (18) are compared with the experimental re— sults from flow into the vertical soil column. The 19 difference in the experimental results and that of the equation (18) is reduced by choosing another set of values for hl and a, until the difference is minimum. As a first approximation the value of a is taken as a = n, where n is the slope of a plot of log K(h) ver- sus log (-h). The basis for this choice of value of a is the equation K(h) = K $12)” for -h>Pb (19) given by Brooks and Corey (3) which will be discussed later in this chapter. KS is saturated conductivity, Pb is bubbling pressure and n is defined as pore size distribution index. According to equation (19) the slope of the plot of log K(h) versus log (-h) is a straight line with slope = n; equation (17) gives a straight line also with slope = a whose first value is thus taken as n. This method is described by the authors as approxi- mate since it uses equation (17) which is not universal in the sense that it does not hold true for all the soils. Secondly, two separate experiments are needed. The errors from both these experiments may accumulate as compared to some of the methods discribed later which only use the moisture characteristics. 20 Computational techniques for K(6) measurements Parallel model Navier Stokes equations are applied to the flow through the unsaturated porous media. The flow is as- sumed to be laminar and incompressible for Newtonian fluid. The Navier-Stokes equation for creeping fluid is VP = uVZZ. For flow in a capillary tube the equation gives E = 8% VP (20) and E = 5%:- VP (21) for thin film flow on a flat plate and _ -b2 u = 12? VP (22) for flow between parallel plates, where H is the one dimensional average velocity, r is the radius of the capillary tube, u is the coefficient of absolute vis- cosity, AP is the drop in piezometric pressure, d is the thickness of the film and b is the distance between two plates. When the above equations are applied to flow in porous materials they can be written as u = -——- VP (23) 21 where R is the hydraulic radius of the pore and is de- fined as cross sectional area divided by wetted peri- meter. If in equations (20), (21), and (22) the lengths r, d, and b are replaced by R, then values of kf become 2 for equation (20) and 3 for equations (21) and (22). Carman (7), from this observation, concluded that the value of the shape factor kf must lie between 2 and 3. However, the value of hydraulic radius R would differ considerably for the two cases, hence a mean value of R5 was suggested instead of R2. Another parameter called tortuosity T = (Le/L)2 was incorporated in the equation (23); Le is the distance of the tortuous path and L is the direct distance. Thus the equation (23) becomes The volume flux is uT L (24) where ¢ is the porosity and s is saturation defined by the relation S = 6/¢ where 6 is volumetric moisture con- tent. According to the theories advanced by Purcell (33) and later used by Burdine (6) the mean of R2 over the entire range of saturation can, when applicable, be ex- pressed in terms of the capillary rise equation 22 r = 2ygos c where Pc is the capillary pressure at saturation S, 9 is contact angle, Y is surface tension and r is pore radius. Then R the hydraulic radius is 2 Area = fir = r WEtted perimeter 2flr 2 R = Y cos 9 5 PC , s = If ds 0 then R25 = (1—%9§—9)2.]: ds c s 2 2 2_Yc056 Ids therefore R - ————§———- 0 P_7 (25) c Equation (24) resembled D'Arcy's Law V = Ki, therefore, the intrinsic permeability is given by 2 _ ¢SR (s) k " kaTs (26) T and R are.functions of saturation. Burdine (6) ob- tained an empirical expression relating tortuosity and saturation (W) - (175—) ‘27) T1 0 is tortuosity at S = l and Sr is the residual sat- uration, i.e., saturation point on the moisture charac- teristic curve after which no appreciable amounts of liquid will drain. (L/Le)2 = l/Ts varies from zero at 23 S = Sr to 1.0 at S = 1.0, because Le becomes infinitely long at low saturation, Sr' This was verified by Corey (9). Equation (27) gives 2 l-S T(S) = T(l.0) (FE?) substituting T(s) and R2 in equation (26) Burdine (6) obtained the expression for intrinsic permeability as 8-5 2 s k(s) = “F522" Y2 .0529 f 01—35 EfT(1.0) 0 PC (28) for unsaturated flow, and l _ 2 2 k ‘ +ka1.0)fl YC°S 8 f0 9‘37 (29) for saturated flow in isotropic media. The relationship __ s-s se ‘ H's: (30) is called effective saturation. All terms including tortuosity can be eliminated if relative conductivity kr is considered as the ratio kS/k = kr resulting in _ 2 3 ds 1 ds kr ‘ (53) f :3 :7 (31) 0 ? O C This equation can be integrated graphically or numberi- cally. However, Brook and Corey (3) obtained the follow- ing results 24 1 Se = (gh) for PC > Pb (32) c where A is the slope of the plot of log (Pc/pg) versus log (Se) and is termed the pore size distribution index of the medium, and Pb is defined as bubbling pressure of the medium at which the desaturation starts. 1Sub- stituting equation (32) expressed as PC = Pb(Se)x and the relation ds = (l-Sr)dse (obtained from equation (30)) in equation (28) and (29) we get s 2/1 k(s) = ¢y2 c0326 (Se)2 (1-sr) j‘e ase ka(1.0) ;;2‘ 0 se 2 2 _ 2+3) ¢Y C°s e (1 gr) (xég)(Se)‘Y“ (33) ka(1.0) Pb and gyz‘cosze (l-S ) A k = _2.I. (-—) ka(l.0) Pb 1+2 (34) 2+3) kr = (se) ‘T"' (35) or kr = $11)” (36) 25 where 0 = (2+3X/X) is the slope of the plot of log (kr) versus log (Po/pg). They observed good prediction of permeability by the above relationship except for uncon- solidated fine sand,GE No.13, and Touchet silt, GE NO. 3, and consolidated Berea Sandstone. They used the step method for determination of the soil moisture charac- teristics for desorption only. For nonconsolidated porous material they had to perform two separate experi- ments using different samples of the same materials, one to determine the moisture characteristic and the other for conductivity. They used hydrocarbons instead of water to avoid the swelling effects of water on porous media. 'Laliberte (25), (26) modified equation (38) by substituting _ S'Sr 2 T(1.0) - T(S) (113;) to obtain 2 2 1:2. k(s) = ¢Yk §?:)9 <1'§r> (1:2) (Se) 4 . (37) f Pb Where ¢ (l-Sr) may be denoted by ¢e and equation (37) may be written as k _ 12 COS2e 1+2 ¢e 1 ___ (s) ' kgfi(syv 3-7 (fi2) (Se) 1 (33) b 26 for saturated condition equation (38) becomes Y2 c0528 ¢e A k = ka(s)P137 (747% Laliberte assumed cos 8:1 and used values of Carman (7) for k = 2.5 and T = 2.0 to attain a dimensionless form f of equation (38) as 2 ¢e A _ :‘L'f (x17) 5 0 13% He used this equation to determine the effects of changes in ¢e or bulk density and noted that the decrease in bulk density or increase in ¢e was followed by an in- crease in permeability and decrease in bubbling pressure and the pore size distribution index. Series parallel model Child et al. (8), Marshall (28) and Millington (29) have given methods of calculating the capillary conduc- tivity from soil moisture characteristics. Child and Collis — George (8) used the equation for intrinsic permeability R X o f(o) 6rf(o) Gr (39) Which is based upon the probability of continuity of lPores. In this equation f(p) Sr is the cross sectional ‘area corresponding to the range of pore p to p + Sr and f(cJ 5r is the area corresponding to the range of pore 27 o to c + 6r. R is the largest pore size of interest that remains full of water, m is a matching factor ob- tained by matching calculated and experimental values of the permeability at a certain point. The computa- tion: are based on Table l of reference (8). Marshall (28) uses equation of the type ¢2n—2 2 _ 2 2 _____ _ 2 k - 8 (rl +3r2 + 5r3+ + (2n l)rn ) (40) where r1, r ----rn are radii of n equal classes of por— 2 osity of interest in decreasing order and ¢ is the por- osity of the porous material. If in equation (40) r = 2Y/pgh is substituted in terms of h, where Y is surface tension and r1 corresponds to hl’ hl

32 this effect was negligible. He also observed that the accuracy of the conductivity function depended upon the range of the moisture characteristic. A more complete moisture char- acteristic curve resulted in larger m values; m was smaller for fine textured soils as compared to coarse textured soils. The conductivities obtained from ad— sorption and desorption did not agree in all cases. Kunze (22) also suggested a more economical com- ;puter method of solving the equations of the type (43), iby'adding the terms of the summation series backwards from r: the smallest radius to r 2 the largest radius. 1 The calculation of Kn' the largest value of calculated 30 conductivity, by conventional computer programs requires (n-l) multiplications and additions; for K(n-l), (n-2) multiplications and additions are required, and so on. But if addition is started from the smallest radius going to the next higher radius, only four additions are needed for each K value. Thus the total number of additions and multiplications is reduced from n(n-l) to 4n. Brutsaert (5) gives a theoretical discussion of various models advanced for permeability calculations. Some of them have been discussed earlier in this chapter. Some of his views are quoted below: From the point of view of probability theory there is an interesting difference between the parallel and the series parallel model. In the later it is assumed by cutting and random rejoining, that the sizes of the pores in sequence are completely independent of one another. On the other hand, in the simple par- allel model it is assumed that the sizes of pores in sequence are completely dependent on one another; as a matter of fact each flow channel is assumed to have a uniform cross section over its whole length. Therefore the simple parallel model tends to over estimate the flow rate and the concept of tortuosity had to be introduced. In the parallel model the non- uniformity of the pores in the direction of the flow is taken care of by "tortuosity" while in series parallel model it is done by "cutting and rejoining." Intuitively it would appear that the sizes of the pore sequences are not completely independent either. Therefore, the cutting and rejoining may yield an underestimation-of the flow rate. Moreover, it is assumed that there is no bypassing of sequences of several pores, that the smaller pores in the sequence governs the flow rate, and that it remains uniformly narrow over its whole length. However, there are also several assumptions which tend to yield an overestimate of permeability and which may thus cancel the effect of above. The tubes 31 are assumed perfectly fitting, except in the model of Millington and Quirk, and straight without tortuosity.- They are assumed to have a regular or even circular cross section while they are in fact highly irregular. There are also many dead—end pores which do not con- duct water even though they are filled. In some cases the permeability is further over predicted because the_ porous medium has strong secondary structure. Often there is the possibility of non-Newtonian flow espec- ially in small pores when the medium has a high clay content. Another fact is that, while the size of the large pores in the sequence is assumed to govern the empty- ing suction, the size of the smallest is assumed to govern the flow rate. As mentioned, this may yield an underestimation of permeability at saturation. But when sequences consisting of large pores are emptied, the result is a large decrease in saturation and a disproportionately small decrease in permeabil- ity. The sequence of small pores with small pores have a permeability which is relatively higher, at least when one takes the average pore size of a sequence as reference. The net effect of all these assumptions seems to be that, as was shown in most experiments, the rela- tive permeability is overestimated at lower moisture contents by models of Childs and Collis-George and Marshall. Because Millington and Quirk assumed that the flow area in the individual pores decreases as the moisture content decreases, their method produces better results. He further indicated that for most of the models the relationship between the effective saturation Se and relative permeability kr can be simplified in the form c kr = (Se)a+ B where the parameters a, c, B vary for different models. For Brooks and Corey (3) model a = 3, c = 2 and B = A, which is the pore size distribution index. Thus the equation becomes 32 2;. B (44) For Child and Collis-George's model 2 2+ — kr = (Se) B (45) the exponent is one less than the exponent in Brooks and Corey's equation. For a uniform material B = A = w, I” thus Brooks and Corey's model reduces to f __ 3 L Kr — (S ) From this and from other models Brutsaert (5) concluded that the exponent depends on the pore size range and tortuosity. He also indicated that the two equations, (44) and (45), give different results because they represent different models. Gamma Ray Attenuation Technique for Moisture Measurements The attenuation of gamma rays has proved to be a very useful discovery in the soil moisture studies es- pecially for rapidly varying moisture contents. Its :major advantages are that it is nondestructive to the Sch.system, is very fast as compared to gravimetric Inethods, is very accurate and has high resolution. A beam of gamma ray energy is directed towards a spot in the soil column, part of which is attenuated 33 depending upon the density of the soil—water system. Some of the photons are scattered after collisions and the rest of the photons go through the soil. A detector picks up these photons and after amplification the re- sulting signal pulses are counted on a scaler. The source of gamma photons usually used is C5137, which has a peak value of .661 mev.l The source is con- tained in a lead shield centered against a collimating hole in the shield. The detector consists of a NaI crystal photoelectrically connected to a photomultiplier and a preamplifier. The detector is also collimated and 1C5137 has an half life of 30 years and emits both gamma and beta radiation. The specific gamma radiation constant in Roentgen/millicuries-hour at 1 cm. is 3.0. Exposure rate mR/hr at lm=.003. One Roentgen = 2.58 x 10"4 Coulomhs/K gm of dry air, and is unit of available radiation concentration at certain distance from the source. Rad. is unit of absorbed radiations, i.e. energy absorbed/ gm. of a material. One Rad. is equivalent of 100 ergs absorbed / gm., or 0.01 Jouls per K gm. Rem. (radiation equivalent man), represents the biological effectiveness of different kinds of radiation and includes the quality factor of the radiation. The quality factor for x,E3&y'rays is 1.0 and that of fast neutrons and pro- tons up to 10 mev. and a particles is 10.0. Larger qual- ity factor decreases the biological effectiveness. A millicurie represents the rate of atom disintegration- l mci = 107 x 3.7/sec. Allowable safe dosage for the human body should not exceed 100 m Rem/week or 5 Rem/ year. For gamma radiation for persons above 18 years 1 Rem is equivalent to l Rad. .03 m Rad/hour is well within the A.E.C. background radiation regulation. This amount of radiation is absorbed by a man facing a 100 mc source .in a 10 cm. lead shielding and standing at a distance of .10 cm. from the surface of the lead shielding. 34 shielded to prevent the counting of scattered photons. The scaler is set to measure a certain section of the energy spectrum usually very near the peak energy value (.611 to .711 mev.). The purpose is to discriminate against all lower energy photons outside of desired range which might have lost their energy slightly by collision with H atoms in the soil but were not absorbed completely. The photomultiplier is energized by a high voltage; this of course varies from detector to detector. The high voltage is supplied to the detector through the scaler. The equation I = IOe'D“X (46) is used to determine the moisture contents. I is the signal pulse count number measured in a certain time. I0 is the count number if an attenuating medium is ab- sent, 9 is the density of the medium, x the thickness of the medium and u is called the mass absorption co- efficient of the medium. Equation (46) can be adopted to the geometry. and the procedure followed in the experiment to be pre- sented later. Topp (38), (39) used a somewhat different technique VVitm.200 mc (millicure) source in hysteresis studies of 35 glass bead media. He used ionization chambers one of which acted as a detector chamber and the other a moni— tor chamber. Both these chambers converted the energy from the gamma rays into current. The difference of the two currents was amplified and recorded. The absorption by the sample was directly compared with a standard ab- sorber. The correction for any drift was applied to the recorder reading. This standard absorber was frequently used to check the drift and make corrections. Gardner (14) gives a description of the gamma radiation attenuation method. The relationship = ln(Nm/Nd) (47) -uws is used to determine moisture contents 6, where Nm are counts in a certain time coming out of the sample at any moisture contents, Nd are counts coming out of a dry sample in the same time, “w is mass attenuation co- efficient of the water for gamma rays and S is the thick- ness of the soil column. He also gives general speci- fication for setup and design of the geometry of the system. Considering a normal distribution for the gamma ray emission, randomness of the emission can be checked Jay the fact that the area under a normal curve covering <>ne standard deviation on either side of the mean value 36 is 67.8%. One standard deviation as estimated by Gard- ner (14) is approximately equal to square root of the counts. I m. (48) therefore the value of the counts measured in a certain time should be within i/Iglof the mean value 68% of the time or roughly 7 out of 10 readings on the scaler should be lat/f; where Ia is mean value. Gardner (14) also gives a rough estimate of 06, the standard devia- tion of moisture contents measured by this method as» _ 1 ° " WI: ‘4” which can be reduced by increasing either Im or count- ing time. Nutter (31) and Smith (36) used similar methods for moisture measurements. Smith mentions some investi- gators who achieved a precision of 0.006 gm. per cu. cm. for water contents from 0.05 to 0.40 gm. per cc. at dry density of 1.3 gm. per cc. Ligon (27) applied the same technique for field measurements of moisture contents variation with per— cipitation and drainage. He reports some drift in the System indicated by the check on standard Mg and plexi- EJlass absorbers. These checks were made before and after series of counts were taken. A mass absorption 37 coefficient of u = 0.0775 cmZ/gm was used for both soil and water; individual values of “w were not determined. The equation Isoil 1n _EH__- 0 = 103.81 - 32.21 I 31 (so) ln IBl mg for soil density measurement was used which required three readings, one for soil and two for standard absorbers. If the two standard absorber readings, taken before and after the soil reading at a particular depth, differed from each other by more than 5%, the scaler was recali- brated and the readings were taken again. The change in the moisture contents of the soil was calculated by the equation 7 _ I soil ln Isoil ln w = .4734 Ilmg = Img 1n Ipl 1n Ipl (51) ._ mg .4 l L mg -J 2 Klock (19) used the gamma ray attenuation method for moisture determinations in his conductivity studies. The theory and general technique was the same as used .by’other investigators except that he used 229 me. of 1Un241 for the reason that 60% of its radiation has 0.061 nuev. energy which required less shielding and less soil 'tflickness for proper attenuation. 38 Holland (17) investigated the possibilities of constructing calibration curves for moisture contents and radiation counts by use of regression technique. A beta guage was used for moisture measurements in leaves and a neutron guage was used in the soil moisture measurements. It was found that for fitting the regression equation y = a + bx by least squares method to the beta gauge data the de- pendent variate y (radiation counts) needed logrithmic transformation to fit the experimental data while neutron gauge data fitted better for a limited moisture range in soil with no transformations required. Hydraulic Gradient and Its Application to D'Arcy's Equation in Unsaturated Flow Vachaud (41) and Olsen gt_al. (32) studied the re- lationship between the flow velocities and suction grad- ients. Even though the techniques followed by both of these investigators are quite different from this study, their results are interesting to note. Vachaud used a transient flow technique. Initially dry soil was wetted in.a horizontal position, the moisture contents at various Imoints of the column were determined by the gamma ray attenuation technique. The distance and the time of aiérvance of the wetting front were noted. However, the 39 moisture contents versus suction relationship for wetting were determined in a separate experiment on the same soil by a suction plate method (step procedure). Moisture con- tent 6 versus time t relationship was plotted for each distance x, thus obtaining a family of curves. Equation of continuity for horizontal direction is 3 -36 2% = TE ‘52) and for a fixed moisture content an, den = 0, i.e. _ Ben Ben _ den—Wdt+—3—tdt—O. (53) Here q is the flow velocity; other variables are the same as used earlier. Combining equations (52) and (53) gives 12:39. 91‘. 3x 8x ' dt (54) where dx/dt = V is defined as the advancing velocity of a water content 6n and is given by the slope of the re- lationship x = f(0n,t). By integrating (54) one gets AV (6 - e (55) qk+l ' qk = k+l k) *wdiere qk=1 and qk correspond to the water contents ek+l and 0k. AV is the small change assumed in V. If 6i is 40 the initial water contents of the soil and 80 the final, then 90 is calculated by the total intake of water by the soil volume and 6i is determined before starting the flow. At t = o, qi = o; q0 corresponds to 90 and is determined at the end of the experiment by achieving a steady saturated flow. At a fixed point x, an increment AG was assumed .and corresponding At and Ax were calculated from the 0 versus t curves with x as parameter (figure 4, refer- ence (41).) Thus Aq was calculated from equation (55). The data of the family of curves (figure 4 of reference) of 9 versus t with x as parameter was replotted as 0 versus x with time as parameter (figure 5 of reference). Corresponding values of suctions obtained from a separate experiment were inserted along the 6 axis. Suction gradients were thus obtained from this family of curves for each tn at different values of 6. Lastly the suc- tion gradients were plotted against the flow velocity for each value of 6 (6 = 40% to 25%). At each 6 the re- lationship was indicated as a straight line proving that for K = constant, Val and hence D'Arcy's law is valid for unsaturated flow. Olson gt_gl. (32) carried similar studies for srteady state flow conditions for desaturation only. They also give a graphical solution for determining 41 flow velocity versus suction relationship. Their re- sults are very similar to the transient flow results discussed above. However, no moisture characteristics are reported in this report. Both the above mentioned studies, however, indi- cate that with decreasing moisture contents and flow velocity, the suction gradient increases but for fixed (6) or (-h) V is directly prOportional to hydraulic gradient. Measurements of Moisture Characteristics and Conductivity As mentioned earlier, most methods employed in the measurement of moisture characteristic use pressure steps. T0pp (39) in his hysteresis study of glass beads media designed two water pressure regulators to keep the water pressure atmospheric at the level of these regulators. Since no air pressure was used, the level of the above regulators was changed to change the capillary pressure and moisture contents of the soil samples. The whole procedure and equipment was quite elaborate and several trial runs were necessary to acquaint himself with some 0f the unpredictable occurrences. Capillary pressure and pressure gradients were directly recorded by two transducers and flow measurement was measured by mea- suring the pressure drOp in a capillary tube for both 42 inflow and outflow and then applying Poiseuille's law to the average dr0p. In fact, the average drOp was calibrated for flow velocity and recorded directly. Moisture contents were measured by gamma radia- tion method as mentioned earlier. Thus 0, V, I, Pc and K were computed. His results and the results of other investigators have shown hysteresis in the 0 versus Pc and PC versus K relationship but only a small hystere- sis was encountered in 6 versus K relationship. The independent domain theory when applied did not predict the desirable results. Topp gE_§l. (39) investigated the difference in shapes of moisture characteristics of a sand for desat- uration by several methods of obtaining the desaturation curve. The results reported show that the curve obtained by applying static pressure steps and achieving equilib— rium after every stOp, crosses the curves obtained by applying small pressure steps (of the order of a cm) while not achieving any equilibrium (semi dynamic method). These curves differed from each other considerably. The lepe of static equilibrium method curve became steeper than the curve of the other method at tension of -30 cm. It flattened at about -50 cm. of tension and appeared to cross the other curve again at tension of -57 cm. This indicated that the desaturation curves of Semddynamic method seem to retain more moisture initially 43 than the curves of the static equilibrium method. Finally, however, the dynamic curve becomes steeper and tends to join the static equilibrium curve. TOpp et al. also re- ported the results of the rate of the pressure application on the shape of the curves. Three rates were compared, two of the curves were similar but the third was slightly different. Another curve was obtained by achieving steady state of flow in the soil after each large pressure step. This curve lies near the static equilibrium curve. Apparently large steps in both cases gave similar shape of curve. EXPERIMENTAL DESIGN AND PROCEDURE Materials! Equipment and Methods Materials Three types of soils were used. Thegrain size distribution of these soils is given in Figure (1). Soil one is medium sand (Sphinx series), soil two is a fine sand (Ottawa series) and soil three is a sandy loam (Hillsdale series). The moisture characteristics of these materials as obtained by the static equilibrium method (pressure steps) and dynamic method (gradual pressure application and release) are given in Figures (17): (21) and (22). The aluminum pressure cell (H), Figure (2b) used in the experimental work had an inside diameter of 12.8 cm. and a height of 11.65 cm. with an air inlet (E) and two tensiometers (C) and (D) set 4 cm. apart. The cell had two end sections (G); each section consists of a NRWP millpore 1 micron filter (K)l ‘ lMillpore filter NRWP 142 is made of nylon, has ii pore size of 1 micron, porosity of 63%, suggested bub- ling pressure of 12 psi., thickness of 150 + 10 microns. 1“llters were obtained from Millpore Corporation, Bedford, Massachusetts . 44 ‘ercent Finer D 45 100 --""‘ L- L 1 L ‘ -o—o—o—o— Medium Sand “*‘°—‘*—<*'Fine Sand ’ ~0—0—0—O- Sandy Loam 10.0 b J l.() _ L I l L l l L l l i l 0 .1 . 2 . 3 .4 . 5 C 6 .7 . 8 . 9 l. 0 Diameter m.m. Figure 1.-- Grain size distribution of porous materials used. 46 Figure 2. (a) Cross sectional view of the tensiometer assembly. (b) Cross sectional view of one end section A,B C,D BR and one tensiometer assembled in the pressure cell. Outlets for end sections Tensiometers‘l outlets TOp and bottom ring of tensiometer Air pressure inlet Top ring of end section End sections Bottom plate of end section Rubber gaskets Aluminum cylinder Bolts Millipore filter Porous plate Space to receive water Screws O-ring seals Porcelain ring Center of soil solumn T1,T2 Tens iometers 47 manEwmm¢ HHOU wusmmmum Any 1HU| u umumaoflmcwe Amy #13-.. 48 supported on a very porous polyethylene disc (pore size 120 micron) 0.5 cm. thick.1 This combination of filter and porous disc rests against a collar in a corrosion re- sistant brass bottom plate (61). In the back of the porous plate is a small space (M) with two outlets (A,A). The plate and filter combination are secured between the tOp brass ring (F) and the bottom brass plate (G) by six metal screws (N). A rubber gasket (GS) serves as an air seal. The two end sections (G) of the cell can be easily assembled together by four bolts and wing nuts (J). The saturated conductivity of the filter and porous disc combination varies between 0.01 to 0.04 cm./min. The air entry value of this combination was 11 cm. of Hg. It was observed that the saturated conductivity of porous disc- filter combination decreased with time as fine soil parti- cles settled on the surface of the filter. However, this combination proved much better than a plain porcelain plate of the same air entry value, but a much lower con- ductivity, or a porvic filter and porcelain combination that were used in earlier trial runs. Details of tensiometers (T1 and T2) are shown in Figures 2a and b. The brass holder, which holds the lPolyethylene porous discs, Cat. No. F-1255, 120 mm. diameter, 5/10" thickness, pore size of 120 microns, were ground to smooth surface with 0.5 cm. thickness. These were obtained from Bel Art Products, Pequannock, New Jersey. 49 porcelain ring (Q),} has an upper (BRl) and a lower part (BR2)' O-ring (P) is wedged between the porcelain ring and BRl' BR2 pushes against two O-rings, one in contact with the porcelain ring and the other in contact with the lower part. The brass holder with porcelain and O-rings in place is held together with six screws. Water moving through the porcelain ring may be withdrawn from the tensiometers through copper tubing (C and D). The tubing (C and D) goes out of the aluminum cylinder wall and is fastened to the wall with a nut and a rubber gasket. Equipment The experimental equipment is shown in the sche- matic diagram Figure (3). The tensiometer outlet tubes are connected to a transducer (T) through a 3-way valve (U). The transducer used is a strain gage type Dynisco Model PT 14-01 with a maximum range of 15 psi. The liquid pressure and electrical connections of the trans- ducer are shown in Figures (3) and (4). Four l.5v. long life telephone batteries (Y) in series provide a six volt excitation voltage to the bridge of the strain gage. lThe porcelain rings were cut from porcelain plates five inches in diameter having air entry value (factory suggested) of 3.5-4.6 psi. Actual tested value was 8-9 cm. of Hg, pore diameter of 9.2-12 microns, porosity of 39.4% and flow rate for water of 3-5 cc./sec/in under 20psi. head. These plates were obtained from Coors Por- celain Co., 600 9th St., Golden, Colo. 50 Figure 3. (a) Schematic diagram of the equipment. K x 2: <3 c: 0% m H N (b) Side view of the pressure cell. Inflow Mariotte burrete Outflow Mariotte burrete Tensiometer connections to 3-way valve Air pressure inlet Constant temperature and humidity chamber Jacks Pressure transducer 3-way valve Detector Recorder Sealer/analyzer Batteries Lead shielding Source holder Source Pressure regulator Air inlet Multispeed transmission Driving motor Manometer Collimation plug 51 Hamu whammwum ADV D: ucmsflummxm may no smnwmsv oflumsmsom Amy w 52 .Hmospmcmup whommmum mo mmoflun mmmm cflmuum "I I l l l l | I I l l I Hmpuoomu OB J————--_--———— .v musmflm 53 The strain gage is attached to the dry side of the trans- ducer diaphram, i.e., the side which is open to atmos- pheric pressure. Full range sensitivity of the trans- ducer is .918 uv/v. Recorder (W) used was Sargent recorder model S-72150. It is a versatile recorder as it has many ranges (1.25 millivolts or microamperes to 2500 volts or milliamperes.) Accuracy is .1% or 20 uv. whichever is smaller. Chart speed can be varied from 1/3 inch to 12 inches/min. Pen speed is 1.8 seconds full scale. D.C. power is supplied by duracell mercury batteries supported by two dry cells placed inside the recorder. The useful features of this recorder are: (1) easy standardization at any time, (2) quick zero setting at any point of the chart by a displacement knob and (3) provision for rolling the chart in a backward direction for quick scanning of the data. Calibration and response of tensiometer-transducer and recorder system.--This system had to be calibrated quite often by directly connecting a column of water to the transducer and raising or lowering it and observing the change in signals on the recorder. It was seen that although the results were exactly alike when positive or negative pressure was applied, yet after some time because of the weakening of batteries, the signal voltage decreased; therefore, the variation had to be checked occasionally and the system recalibrated. The variation was 7 mv/10 cm. 54 of water pressure for new batteries to 6.2 mv/lO cm. of water pressure when batteries were replaced after approxi- mately six months. The time constant and response of the system was determined by applying a sudden air pressure step to the pressure cell with saturated soil inside while inflow and outflow were closed. A pressure of 29.3 cm. of water was applied and was recorded as 27.5 cm. of water on the chart in 1.2 minutes giving a time constant of 0.43 minutes.1 Experimentally tC corresponding to AP = 63.2% of P was .4 minute. The volume of water required for a displacement of the transducer diaphram was measured as lcc/250 cm. of water pressure. Moisture content determination equipment.--One 137 hundred mc. of CS was used as gamma ray source. It was shielded in a cylindrical lead bucket 8 inches in diameter and 8 inches high. It was designed according to the equation A exp(-ut) a D(R,t) = 2.134 13w. t) 3‘3— Bo MR; 1This follows the equation AP/P = (1 - eEE) where A? is the recorded change in pressure in time t, th is time constant and P is the applied pressure step. If t = t0 is put in the ab ve equation the following equation re- sults: Ap = P(l - 2) = .632 P. Similarly for t equal to 2' 3: ahd 4 time constants the recorded pressure will be 86'5%: 95% and 98.2% of the applied pressure respectively. HOWever' the reSponse becomes much slower at lower moisture contents . 55 given by Gardner (14).1 In this equation D(R,t) is dose rate in millirad per hour at a distance R in cm. from the source of strength A in mc. and energy E0 in mev; t is the thickness of shielding in cm. and u the attenuation coefficient in cm-l. B(u,t) is buildup factor; “a is energy absorption mass-attenuation coefficient for the source energy and the material which is to absorb the dosage; and p is density of the shielding material. Source (No. 2 in Figure 3) is held by a bolt (No. 1 in Figure 3) screwed in a sleeve in the shielding. Thus it can be raised or lowered for minor adjustments of the source against the collimation plug (No. 8) which is 4 1/2 inches long with a 6 mm. circular hole drilled through it. The plug was first molded then machined. This size of collimation gave better results than other collimation sizes. The collimated gamma rays were di- rected at a spot (R) on the pressure cell in between the tensiometers. The detector (V) was shielded against scattered radiation by a 4 1/2-inch thickness of lead in all directions. A 3/16-inch hole was drilled in the lead for collimation in front of the detector and the two 1For 0.661 mev. gamma radiation from C3137, ua/E as measured in tissues is 0.317 cmZ/gm., u is 1.134 cm’ , B(u,t) for lead thickness of 3 to 20 cm. is 1.2 + 0.13t. When A is 100 and R = 10, D(R,t) = 0.1 mrad at the sur- face of the lead and 0.03 mrad at 10 cm. from the surface of the lead. Four inches radius and 8 inches height for the bucket was thus a safe value. 56 collimation holes were aligned so that a maximum count was obtained. This was achieved by rotating (Z), lower- ing or raising jack (8) and bolt (No. l). The detector used was a modified Nuclear Chicago model D8 100 with a 1 1/2 inch NaI crystal connected to a photomultiplier and preamplifier. The detector was con- nected to a Nuclear Chicago model 8725 scalar-analyser. The scaler was calibrated to represent one mev. for full scale of 10 volts baseline. The base was set at 0.611 mev. A narrow differential counting mode was used in which case the full scale setting of 10 on the window scans only 10% of the base. Since the energy scale was calibrated to represent 1 mev. for full scale baseline of 10 volts, the window width corresponded to 0.1 mev. Thus, a window setting of 10 scanned the energy Spectrum from .611 to .711 mev. and rejected all other energies. The high voltage required to excite the scintillation NaI crys- tal was of the order of 910 volts. This voltage had to be readjusted occasionally in order to maintain a peak value of counts. The readjustment was necessitated by the drift in the baseline and/or amplifier or window. Need of this readjustment was determined by obtaining counts at several voltages higher and lower than the volt- age in use previously. If the counts thus obtained were higher at a different voltage, this voltage was used in subsequent moisture measurements. 57 In order to account for the randomness of the gamma ray emission, ten one-minute counts were taken for slow runs of conductivity measurements and 3 to 4 one- minute counts were taken for fast runs of moisture charac- teristic determinations. Maximum c/m (counts per minute) for the fixed geometry were 351,000. The fixed geometry in this case was 4 1/2-inch thickness of lead shielding around the source with a 6-mm. collimation hole followed by a 6-inch air space and then another 4 1/2-inch lead shield for the detector with a 3/16-inch collimation hole. When the cell replaced the air space the c/m were of the order of 60,000/minute. However, the last digit is drop- ped in calculations as it varied considerably for con- stant moisture content. This does not affect the accur— acy of results appreciably. The values of 0 are calculated from the equation (64), Chapter IV: I l m m s uws IS 'where am is moisture content corresponding to Im and as is saturated moisture content corresponding to 15' “w and S are mass attenuation coefficient of water and thickness of soil, respectively. This equation involves the ratio (Im/Is) which will give correct results up to 4 decimal places even if the last digit in c/m is drOpped. Plotting the graphs for more than 3 decimal places for 6 is not 58 very practical anyway. Theoretical standard deviation for counts of the order of 59,000 to 60,000 at saturation is of the order of i 243 (Oe= /Is). Actual variation was well within this range. Theoretical standard deviation in moisture content at saturation is 1 o = , e uws7f; or o = 1 = + 00444 or i 0.44% which is quite reasonable. Determination of mass absorption coefficient uw.-- In literature different values of uw are reported; it is therefore necessary to determine the values of “w for the fixed geometry to be used. Four cubical cells of dimen- sions 2.85 x 4.80 x 5 cm. each were constructed from a 1/8 inch thick plexiglass plate. The cells were placed in between the two collimators at fixed positions. Seven one—minute counts were taken with the empty cells in the normal position. Then the first cell was filled with water and seven more one-minute counts were taken. Simi- larly, counts per minute for the remaining cells were taken after filling each. Using equation 1 _0 s: m 'o H 59 the mass attenuation coefficient of water was determined. In this equation 3 is the dimension 2.85 cm. through which the gamma rays pass. IO and II are the average of seven one-minute counts taken before and after filling a cell. The effect of the order in which the cells were filled did not seem to effect the “w value. But higher values of “w resulted when the number of cells filled was increased, (uw varied from 0.072 to .075), thus indicating that “w increases with thickness of water. The value of “w to be used in equation (56) was therefore obtained by using the aluminum cell (H, Figure 3), and taking counts before and after filling the cell with water. Here 5 is 12.8 cm.; “w obtained was 0.078 cm.2/gm. The counter of the sealer/analyser is insensitive during the short interval in which the pulse builds up and this time is called dead time. A correction may be applied to obtain the true counts by equation Imot I = _ (57) m (t Imodt) where dt is the dead time per pulse, Imo are the observed counts in time t and Im is the corrected value. The analy- ser used had a 1.5 u sec. pulse pair resultation time and total dead time for minimum moisture contents (maximum c/m) 60 never exceeded Imodt = 0.111 sec. Therefore, correc- tions were not necessary. Methods The pressure cell was packed with dry soil in layers of one inch. It was then tapped to settle the soil. When the cell became full it was wetted by imbibi- tion with deaerated water containing .1% phenol for several hours after which some additional soil was needed to fill the cell. The soil level in the cell was kept slightly above the rim of the cell before assembling the three parts of the cell together to preclude the possi- bility of further settlement or consolidation of the soil later. After clamping all outlets, counts were taken at different levels of the cell to check packing irregulari- ties. No appreciable difference was observed in count rates. Both end sections were flushed and then positive water pressure was applied to the lower end of the pres- sure cell. Outflows from the end sections were shut off while maintaining positive water pressure at the lower end section. When water started flowing out of the ten- Siometers, the cell was laid on its side with tensiometer outlets pointing upwards, then the tensiometers were con- nected to a suction device to remove air. The tensio- meters were clamped and the cell was connected as shown in Figure (3). 61 All other connections were made as shown in Fig- ure (3). The equipment enclosed in the rectangle (I) was set inside a constant humidity and temperature chamber1 and kept at a dry bulb temperature of 72°F. and wet bulb temperature'of 66°F. resulting in a relative humidity of 77 per cent. Initially saturated conductivities were deter- mined in a stainless steel permeameter. A sample from this permeameter was used to determine the saturated moisture content by oven drying. Saturated moisture content of the same soil used in the pressure cell were also determined at the end of the series of experiments by taking samples from the middle of the cylinder and oven drying them. This moisture content was lower than the saturated moisture content obtained from the permea- meter sample. However, the saturated moisture content as obtained from the actual pressure cell sample was used in equation (56) to determine 6m, since IS corresponding to as was available. Saturated conductivities obtained in the pressure cell were also lower than those obtained by permeameter. Moisture characteristics determination.--At first a water column was connected to one of the two outlets (C) * . lAminco Air, Cat. No. 4-5478D and 4-5479D. Sup- Eilied by American Instrument Co., Inc., 8030 Georgia Ave., 31 lversprings , Md . 62 of the transducer. The other end of this water column was placed in level with the center of the cell corres- ponding to spot (R) in Figure (3). At this point the recorder pen was centered on the recorder chart estab- lishing a zero gage pressure or a reference line on the chart. The tensiometer (C) was reconnected, and a mar- riot burrette (Al) was brought to the level of (R) and inflow was allowed. All other outlets were clamped. Valve (U) was so positioned that both tensiometers were connected to the transducer giving average pressure be- tween two tensiometers. The air pressure inlet E was left open to let any air in the cell escape.- These con- ditions were maintained until the pen reached the center of the chart, a position originally established to indi- cate zero gage pressure or zero suction. After achieving equilibrium state, inflow from (Al) was clamped and the dripping point of (Bl) was raised to the level (R) and outlet (B1) was unclamped. Then the air pressure hose was connected at (E). Now the air pressure could be ap- plied to the soil either in steps (static method) or gradually (dynamic method) to change the moisture con- tent. For static equilibrium, steps of approximately 10 cm. of water pressure were applied with an air pressure regulator and were noted on the manometer (7). When equilibrium was reached, as indicated by the fact that the recorder pen came back to the center of the chart and 63 outflow stopped, another step was applied. For the first few steps the equilibrium could be achieved in a few hours but for higher pressure, equilibrium state was reached after several days depending upon the kind of soil and range of pressures. The same process was fol— lowed for rewetting the soil. Several gamma counts were taken after each equilibrium state to calculate 0m. Cap— illary pressure values were assumed at the corresponding values at equilibrium. For the dynamic experiments the regulator stem was rotated at a controlled rate by the device shown by numbers (3), (5) and (6) Figure (3). Number (6) is a D.C. motorl controlled by a speed control.2 The motor has two shafts. The main shaft is geared to the second shaft reducing its speed from 10 to one. This speed can be further controlled by the minarik speed control which can reduce the speed of the second shaft from a maximum of 173 RPM to zero RPM continuously. Through this control the direction of rotation of the motor can also be re- versed to reduce air pressure in the cell. The motor is 1The motor is type NSH-12 R, 115 v. D.C., 0.36 amps., 1726 RPM max. 1/50 H.P. reducer motor output shaft 173 RPM speed reduction 10:1, torque 49 in lbs. Supplied by Bodine Electric Co., Chicago. 2Speed control and converter from A.C. 60 C. 115 v, to D.C. 115 v. Model SH-12. Supplied by Minarik Elec- tric Co., Los Angeles, California 90013. 64 further geared to a multispeed transmissionl (No. 5, Fig- ure 3) with a choice of 12 speeds, the fastest equaling the speed of the motor shaft. It can be reduced to 1/5000 by choosing the slowest gear speed. In between each step of gear speed any other speed can be chosen with the help of the Minarik speed control. Theoretically it is possible to obtain almost any RPM desired. Each revolution of the regulator shaft corresponds to 51.4 cm. of water pressure. Once a speed is chosen, it is kept constant for that entire saturation and desaturation process. Several such speeds were selected by trial and error for the three $0113. The equation, PC = Pw - in, was used to A , given by a manometer and compute capillary pressure Pc from the air pressure P or non wetting phase pressure, in P absolute pore water or wetting phase pressure recorded WI by the tensiometer—transducer-recorder system. PA and in are used interchangeably. Values of 8 were computed from the average of several gamma—ray counts, half of them taken before and half of them taken after Pw and PA were recorded. Conductivity measurements.--Experiments were also run for three soils to determine the experimental values lMultispeed transmission has 12 gear speeds and a nuetral. Neutral is used by applying pressure steps by manually rotating the regulator stem to get the desired pressure step indicated by the manometer. Supplied by Harvard Apparatus Co., Dover, Mass. 65 of K(Pc) along with 0 and PC for each soil. The values for the conductivity functions were determined as follows: initial saturated conductivity was determined for zero Pc and (g by keeping both inflow and outflow levels above the soil level R and establishing a saturated steady flow in the soil. Steady state flow was achieved when inflow and outflow were equal. Conductivity could be calculated by noting the head loss between tensiometers. A certain RPM value was chosen to affect an air pressure change. As the pressure started increasing, the inflow decreased and the outflow increased depending upon the pressure rate and the external levels of inflow and outflow points or the external hydraulic gradient. If this external gradient was too low,outflow could occur from both inflow and outflow outlets. If the gradient was too large, K(Pc) could not be measured accurately because of the dependency of K on Pc‘ Thus the external hydraulic gradient was chosen so that it took into account the above considerations and gave inflow and outflow values relatively close to each other (within lcc/min. in this study). The average flux was used in D'Arcy's equation to determine K(PC). The choice of the pressure rates was governed by: (1) maximum time needed for each set of readings, (2) maximum desirable -PC and (3) differ- ence between inflow and outflow values. If the pressure rate was too fast, Pw became larger and eventually would 66 go out of range of the recorder. The maximum value of PA itself was limited by the air entry value of the filter disc combination and the air entry value of the tensio- meters. If either of these air entry values was exceeded, air bubbles were forced into the water systems resulting in erroneous measurements. Rapid pressure changes caused larger differences in inflow and outflow quantities. Therefore, pressure rates for conductivity experiments were kept much lower than in other experiments in which only the moisture characteristic determination was the objective. P,P w inflow, outflow and gamma counts were re- A’ corded for an interval At. The internal head loss in soil APC was determined by positioning valve (U) to first connect one and then the other tensiometer. These values were averaged over the period At to correspond to a time t1 + At/2, where tl was time of previous averaged read- ings. K(e) was calculated from K(e) = V/I, I - APC/CAl where APC is the head loss in terms of chart reading and C is calibration constant. Al is the distance between the tensiometers and V is the average of inflow and out- flow velocities. Pw was given by the average of the two tensiometer readings divided by calibration constant. Short programs were written to solve the equations A A O |-' K(e) = %E« (cm/min) :3” 1 Im 0 = 0 --——— ln —— cc/cc m s uws Is and PC = %§ - 2(MR-50) cm of water on an electronic computer. Here Q is average inflow and outflow in time t (min.). A is crossectional area of flow. CR is chart reading CR/C = Pw’ MR is manometer reading. Initially both legs of a manometer were set at 50. PRESENTATION AND ANALYSIS OF DATA Theory Theory of gamma radiation attenuation technique. When count rate per minute was plotted against ex- perimentally determined values of moisture content of a soil (contained in similar containers and saturated to different degrees of moisture content) on a semi-log paper, a straight line resulted as shown in Figure (5). This indicates that a logarithmic relationship of the form 0 = -K 1n I (58) exists between 0, the moisture content, and I, the count rate. Since the value of I varies with geometry, source strength, type of soil and collimation, a reference mois- ture content as of known value corresponding to Is for a fixed geometry is necessary. Saturated conditions can be used for this purpose when 0 = -K 1n Is. (59) S By subtracting equation (59) from (58) the equation 68 69 8 = BS - K(ln I -1nIS) (60) or S results, where K is the slope of the plot. Examination of the exponential law I = ID e"me (61) for attenuation of gamma rays shows that K is the recipro- cal of the product of S, the thickness of the soil, and u the mass attenuation coefficient, as follows: w, For wet soil contained in an aluminum cylinder I = I exp-s @ (62) O + Guw)-ZS soil usoil AuApA where p, u, s are density, mass attenuation coefficient and thickness of the respective material. I is the count rate recorded with attenuation and I0 is the count rate recorded without any attenuation. When the soil is saturated, equation (61) can be written as I = I exp-s (p (63) S O U + 0 uw)-ZS soil soil 5 AuApA Dividing equation (62) by(63)and taking logarithm gives I 1n IE. 1n exp s(9uw + esuw) (64) = -suw (6 - 65) 70 or _1_ U) (.0 I: H Therefore, if §%; = K, the exponential law can be used for moisture content determination. It was further veri- fied by plotting the experimentally determined values of 6 against the calculated values as shown in Figure (6). Disregarding possible experimental errors, the plot is practically a straight line. The mass attenuation coefficient for water can be determined for the same fixed geometry by taking count rates before and after filling the experimental cylinder with water. For the empty cylinder the exponential law can be written as I = I exp-ZtDA A- (55) For the cylinder full of water the same law is expressed as -2tDAU I2 = I exp A - Spwuw - (66) Dividing equation (65) by (66) and taking logarithm re- sults in: 71 +4 3X10 P ZXl0+4F .5x10+4- c3 "-1 E ‘\ 2 +4 8 10 ' o I H l e l 10 20 30 0 - % Volumetric moisture contents Figure 5.--Relationship between moisture contents and count rate. Pi 72 30 — 20- 10 umetric calculated moisture contents 6 -Vol c io 20 3o 46 ee-% Volumetric experimental moisture contents Figure 6.--Verification of exponential law by plotting experimental values of moisture contents against calculated values. “guns; “0 UCQNfiUJ-umwmwfidannv :nJ_UlN~J~.~nVUU1 nannyp: .l 3~a 73 .076 .075 .074 .073 .072 - mass attenuation coefficient of water 2.8 5.6 8.4 11.2 Ll Thickness of water interposed (cm.) Figure 7.-—Variation of mass attenuation coefficient of water with increase in thickness of water. 74 The fact that a change in the sample thickness results in the change of the mass attenuation coefficient is demon- strated by Figure (7). The mass attenuation coefficient was determined by increasing the thickness of water by successively filling four plastic cells placed between the source and detector and computing “w after each addi- tional cell was filled. Mathematical models used in computation of conductivity function Three mathematical equations were used for con- ductivity function determination. Burdine's equation 2 s l Kr = (Se) Jr -J¥; j, -§%-for PC > Pb (67) 0 PC 0 PC was solved by numerical integration (Appendix 1) for both 1 adsorption and desorption. In this equationjr ds/Pc2 re- 0 presents the total area under curves in Figures (44), (45), s (46), (47), (48) and] ds/PC2 is the fractional area under 0 the curve corresponding to different values of Pc and S. The purpose of summing the areas directly is to avoid the use of relationshi “l§'= ~l-(s )2/A involvin A and P because A and Pb cannot be correctly estimated for adsorp- tion curves. In a different approach instead of using the experi- mental moisture characteristic data, the functions 75 .1. _ __1_ S‘Sr 2H for p > p (68) P 2 - P 2 l-Sr c I c b and d 12 = 1 exp _ c (1:: ) for Pc < PI (69) PC PS2 1 generated the relationship between the saturation and the capillary pressure. Brooks and Corey's equation (68) generates a curve shown dotted in Figure (44) and there— fore, does not represent the experimental data completely. Function (69) is, therefore, a necessity because at s = s and P = P the curve changes its curvature from I I concave downwards to concave upwards. At s = SI 1 _ 1 _ 1 e-c 2 ’ __2 ’ 2 Pc PI PS PS2 ... C = ‘11’1 (——2-) . PI The value of d = 0.4 was determined by trial and error. Theoretically PS is capillary pressure at saturation when S = l and 76 For adsorption the value of PS is 0 for the three soils investigated in this study. Therefore, in order to make the program (Appendix II) function properly, a finite value of PS greater than zero has to be used. The rela- tive conductivity is computed by substituting the func— in tion (68) for PC > P and function (69) for Pc < P I I equation (67). Capillary conductivity is calculated as K = CSTD. x K r by numerical integration, where CSTD is the experimental value of conductivity corresponding to 05. Equation (43) of Kunze K _ -2 -2 -2 2 s 30 n y ¢ + pgu (h1 3h2 —2 +--(2n-l)hn ) so (70) was also used and the results are compared in Figures (49) to (56). Presentation of Data and Analysis Dynamic Method Figures (8) to (16) represent the dynamic processes of desorption and adsorption. The air pressure (PA) or non- wetting phase pressure and the wetting or absolute pore water pressure (PW) are plotted as functions of time; the difference of these two pressures, i.e., the capillary I 77 pressure is also shown. Pore water pressure can be either positive or negative and is affected both by the externally applied hydraulic head and also by the time rate of air pressure application RAP' As RAP increases from 1.65 min./cm. in Figure (8) to RAP = 25.5 min./cm. in Figure (9) for the same soil , maximum value of Pw decreases from 34 cm. H20 to 11.0 cm. H20. Therefore, to achieve the same maximum value of Pc' the maximum value of PNw is larger for faster runs than slower runs. This can be ob- served by comparing Figures (9) and (11) where AH is the same but RAP differs. Figure (9) and (10) show the ef- fect of increasing AH. As a result of increase in AH, Pw is higher in Figure (10) as compared to Figure (9). The same general behavior was observed in the other two soils. In fine sand (Figures (12), (13) and (14)) Pw decreased as RAP increased from 0.625 min./cm. to 19.6 min./cm. It was also observed that Pw does not increase as much in fine sand as in medium sand and increases the least in sandy loam (Figure(15)). Figures (16), (17a), (18), (19) and (20) show the lag between the air pressure and the capillary pressure. In fast runs it was noted that in became zero before PC became zero (Figure (8)) and that it took a long time afterwards for the soil to satu- rate suggesting that it took a long time to replace the entrapped air from the pores. The lag between in and Po 78 for sandy loam is negligible and is not shown in Figure (22). Dynamic method compared with static method In Figure (17b), (21) and (22) the dynamic data are compared with the static data. In all cases it is noted that the two moisture characteristics do not follow the same path. These results agree with the results of Topp gE_al. (40). It is significant that when different RAP are compared, as in Figures (17b) and (21), the de- sorption process follows approximately the same path. In Figure (17b) the maximum Pc achieved was the same for all RAP values; hence, it is not unusual that the adsorption process also follows the same path. Since maximum Pc values in Figure (21) were different, adsorption curves were also different. Head loss between tensiometers and the boundary Figures (23), (24), and (25) show the variation of head loss between the two tensiometers installed in the soil with flow velocity in the three soils studied. Even though the curves for adsorption and desorption are not the same, they do follow a similar pattern. The flow velocity in the soil can be expressed as 79 .0mm .EO\.EE mm.a Mo mpwu GOMDMOAHQQM unammmnm Ham How pawn Esfltmfi CH mEHu spas mmusmmmum mumaaflmmo was mcflu903 .msfluum3lsos mo GOHDMHHM>II.m musmflm .oH m. o A.sHEV wasp oom omw( com com OOH q . fi _ _ . 4 3 m . madmmmnm mnmaaflmmo u om 3 unannoum mmmnm mcfluums u m unannoum mmmnm msfluumzso: u 3cm .Eo\.cws mo.a u mfim 3G om om om (02H °mo) ma pus ov on 00 ooaa 80 .so\.cws m.mm mfl com A.:HEV mEHu com com .ONm 00¢ .80 o.mN oom Ed cow was m How pawn ESwUmE cw mfiflp nvw3 om was 3m .BCm mo SOHDMAHm>aI. 3C - a 1 O N m .Eo\.cwE m.mm A.Eov mam>ma onmuso was 3odusfl swmzuun mocouommflo wusmwh O V‘ O ('1 O N O H C) O H (.3 N C) 0 fl' m 0 :n (OZH 'mD) “a pus ”“a '°a m4 0.0mm .wo owmm n ma can o~m .EO\.GHE m.mm u m How comm anome as m was m . m mo soaumanm>ll.ca mnzmflm l.anev was“ ocov comm cccm comm ccom coma coca com d d - - , u 5 81 m.ao\.ans s.m~ .so o.Hm oNI om (cm 82 m .EU\.¢HE H.5m a .ONm .ao o.m~ noma ccmzom 3a a mo conumsnm>--.aa musmsm H mm How psmm Edmoms SH mafia cuss m cam m . A.assv was» 0004 comm ooom comm ooom coma oooa com d q - d1 4 J_( A .so\.css H.5m u .80 c.mm .om .ONm .so\.cna m.ma mm 3 .3: u m .o.mH u 34 How UQMm mean as was» cuss om can m m mo soflumflum>ll.ma musmflm oml c¢l cmu cm: cal 83 A.cssv.mssu comm . onWm - a comsir///,ooma . com cos _ 4 — _ J... i & N o m .Eo\.cwE m.ma CEO OImH cm ow cm om (OZH °mo) Md ’Mud ’od 84 Ad .0 N m .EO\.saE mmh.a u m Mom pawn scam as om was 3m .zsm mo coflumwum>ll.ma musmwm A.cssv wasp omm can own son cos oma om ov a. . — q 3 J - owl om . .04: o l on. o . m .om o 3cm 0 6 [CV .so\.cfls mas.s u mam . .oo (02H 'm0) Md pus Mom Ucmm mcflm GH mEHu nuHB cm can m omH ova oNH m .Eo\.:HE mNm.o n mll.va mnsmflm .ON A.GHEV mEHu ooa cm cm ow cN 85 .Eo\.sHE mNm.o - u u d — '4‘. ow -ccH (OZH 'mO) “a pus M“a '°a 86 Comm 080 ma n me new ems .so\.css cmm .50 ms n ma ma.ma n mém £ua3 Emoa mpsmm so“ 08H“ nufl3 om was 3m .3sm mo coHuMHHm>II.mH musmflm A.csac mans ooov comm coom comm ooom coma OOOH com . o T q A . _ W q . NH] . o O O o o. o noml Dd c U5 . .cvl M o D. u ['qu (JDf' AU A C} c a p i 3 d m... o M o - IO? ml 33 . m m o a 0.10m _CH . m can .Eo\.css ma.ma u mam . . NH 87 mQSHM> SQ .muflm @4m MDOHHm> HOW Uflwm ESHGGE Ca 3cm can om comauoo and cam oocume UHEmswo an moflumflumuomumno musumwozll.ma muomflm 3C 0 Aon .Eov muowmmnm “Hm m can whammmum Nunaaflmmo m can QNI cm: owI cm: can ONI omu ceI 0H1 ONI omI owl omI 1 - . 4 q q a _ . 1 q 4 . — NH1 1 (b O Nrc T NV \ a \ m \l . \\ \09C 3 m NJCI O m . VAT Q \C I x I \ \. m V. ‘\ MW I AV \ \ 3 \ o o \ . mmma. r \. 1 . o m - \ \ E \ 31 \ 3 k \. r \ t m: x n . \ e .m. om: .. j o I c 1 mm . , u T in. 1 .m em. I m w m o m % . N .Eo\.cflE o m t _ was .5 T? I. Hi cmm .2 8 \.GH8 m.mN n mdm N EO\.CHE m.mN n m 0 m cm N . . . r N . I O m So am H mo 60 0 mm 0 mo 0 m SO ma I m< om r 88 0 - % moisture content (cm.3/cm.3) / / ./ (b) (a) / Z—¢>—o—-o—StatJ.C ‘ #0 ///°/ WEAP=%75i RAP = 1.65 min./cm. fl £"5.AP= ° . 1 ¢.// -o—o—o-RAP=25-7 8 1 IL I 1 1 J l -40 -30 -20 -10 0 —50 -40 -30 ~20 -10 Pc capillary pressure and in air pressure (cm. H20) Figure 17. (a) Moisture characteristics by dynamic method and lag between PC and in for RAP = 1.65 min./cm. (b) Comparison of moisture characteristics ob- tained by dynamic and static equilibrium method for medium sand. 89 aH = 16 cm. H20 0 O R = 19.6 min./cm. 0 AP H20 ° 1 0,14 / / ‘ /, . / - // ./ ’ .// / / . ,4 / in , / . 1 P . / c o // ./ / ,/ _ // / l_ L l 1 L .L _l ‘50 -50 —40 -30 -20 -10 >26 24 22 20 18 PC capillary pressure and in air pressure (cm. H20) Figure 18.--Moisture characteristics by dynamic method and lag between Pc and in in fine sand for RAP = 19.6 min./cm. H20 and AH = 16 cm. 2 0 - % moisture content (cc/cc) H O- 90 o .on .Eo\.:HE o.a u mmm now comm ocflm CH 3cm was m smmzumo oma cam conume UHEmczo mo moflpmflumuomumzo mnopmflozII.mH musmflm CHI ONI oml owl oml oml owl v . . . LJI .Iro.|.olh \. ONI . 0m \ \ w \\\\ o \»s / . . m CNN. \0 \ .w \x .\ t \\ o m. \.. \ m N \ x V t w .\ x e \. _\ u \ ..\ mom-.\ \\ m 1.. \ % \ .mmf 0. e . o . . 0mm .EU\.cfls m.a u mum om. 91 8 - % moisture content (cc/cc) .on .Eo\.omm mm H mam How pawn was“ as 3cm was om smokumn mud cam oozumfi OHEmsmp an Usumflumuomsmno mHsumHozII.om Gunman 3 0 AONm .Eov muommmnm Ham cm pom ousmmonm wumaaflmmo m OH! ONI oml ovl oml owl OBI owl om! OOHI q q d _ 1 u 5 q q - lllollloll-ll\ .4m\o \ \ \ \ O\ \ \\ \. \. ONm .EO\.Omm mm H mflm. 92 .osmm scam How ponumfi Edmunflawswm oaumum nuflz .mmsam> 8 - % moisture content (cm.3/cm.3) mdm msoHHm> um .conumfi anmsmo an moflumaumuomumco unsumHOE mo sOmHHmmEooII.HN mHsmHm NH HN MN mN 5N mN oHI _ ACNE .Eov musmmmum mundaflmmo m omI q oMI U o¢| owl owl OOHI ONHI J 3 . a \\\\\||o .\\\\ \\w.\ \ O m . .EU\.G«E Hm.um¢m.|4IdIATl€I 0 ONE .EO\.G._..E ofnflmdm IOIOIII on .EO\.s..nE m.m.numdmulololclol GOSUOZ mwflm Idl¢llTl¢l 93 8-% volumetric moisture content (co/cc) vH.I ma ma cN NN vN mN mN .EMOH Npcmm MOM oonumE Deannao pom oeumum an moaumflumuomumno musumflofi mo COmHHmmEOUII.NN musmflm Aon .EOV mndmwmum mumaaflamo om oaI cNI OMI ch omI owl chI omI omI coal oHHI cNHI omal oval omHI .omHI l _ q _ . . _ _ q _ _ _ _ I a . N O 3 .EO o.ma u m .EO\.cHe mH.mH u mam ponumfi OHEMGNU Mom 0mm conuma UHEmcmo.IIIVIIIOIII e059: 033m IIJIIaII. 94 v = K(8)I or Ahs and AH = Ahs + 2Ahb + 2Ahf + 2Ahp + Aht (71) where AH is the total external applied head, and Ahs, Ahb, Ahf, Ahp, Aht are losses in soil, boundary between soil and supporting filter-plate combination, filter, plate and tubing. Head loss in the tubing can be calcu- lated from 2 _ EV— 72 Aht — f D 2g ( ) where f is the friction factor for laminar flow. D is diameter of tubing in ft., v is kinematic viscosity and L is the length of the tubing. hf in the tubing was calculated as 0.000032 ft. or .000974 cm. of water which is negligible; therefore, the main causes of head loss are the soil, supports and boundary. The head loss de- termined experimentally in 4 cm. of soil ranged between 0.2 to 3.2 cm. of water, or 0.6 to 9.6 cm. of water in the full length of soil, the head loss being greater in finer soils. The total applied head AH ranged from 16.0 to 31.0 cm. of water for different soils. From the Figures (23), (24) and (25) it is ob- served that the head loss in the soil between the 95 0.04 002 b O .01 Vfi P 0.003 ~ ° A V - Flow Velocity (cm./min.) 0.002 ' D —~0-—0- Desorption -——A—-A——-Adsorption 0.001 ' 7L 1 1 1 l I g L L 1 0 .05 .1 .15 .2 .25 Hydraulic gradient I = %%-(cm./cm.) Figure 23.--Variation of hydraulic gradient between tensiometers with flow velocity in medium sand. 96 .01 r 0 o O "' O ’7 o a .,.I I. E \. é o 3 1 >. .p -:-l o O o H 8’ 3 .001 '- o o H m I - o > G o ’ o r -—O——O-—Desorption -—4F-4>—-Adsorption .0002 ' 1 l .2 .3 .4 A11 A2 (cm./cm.) Hydraulic gradient I = Figure 24.--Variation of hydraulic gradient between tensiometers with flow velocity in fine sand. 97 b.— 02 L 01 - ’7 L I: -a E \. a L .IJ ea 0 o H b m > 3 Fl ‘7' > 001? L —O— O— Desorption L _A_A— Adsorption )- 1 1 J I J 5 O 3 O 4 O 5 O 6 O 7 O 8 Gradient I = %% (cm./cm.) Figure 25.--Variation of hydraulic gradient between tensio- meters with flow velocity in sandy loam. 98 tensiometers, and thus in the whole length of the soil, first increases as the velocity decreases but then de- creases with a continuing decrease in velocity. The pattern is similar in adsorption and desorption, if the direction of the pathway is ignored. Equation (72) suggests that initially when ve- locity is high, the head loss in the filter-plate combin- ation is very high as compared to head loss in the soil. This is attributed to the very small pore diameters in the filter-plate. Head loss being proportional to the square of the velocity, it decreases considerably in the filter- plate combination as the velocity decreases. The total head loss AH being constant throughout the run, Ahs in- creases as large pores are emptied. After the moisture content is decreased, the boundary between the soil and the filter-plate combination may also contribute to head loss. With further decreases in moisture content, it ap- pears that the head loss in the boundary becomes very large. Perhaps this is the result of separation of soil and filter or a loss of water film continuity. K, 8 and Pc data Figures (26) to (28) present the experimental data for K(8)-8 relationships and Figures (29) to (33) present data for K(Pc)-Pc relationships. 99 K(8) capillary conductivity tm./min.) .04 .02 , . -— "-1 r P . o :5 I. 'U c o I- o L >« 36 H .003). H -a m m o ,4 I (D ".002¥ M —-o—o— Desorption .001 —4L—A—-Adsorption l i L 15 20 25 30 8 - % moisture content (cc./cc.) Figure 28.--Experimenta1 capillary conductivity versus moisture content for sandy loam. 102 .7 ’ A ' A . c' - k E \, .4I- ’ E 3 o 3‘ A ’S' -a .2 _ . .|J U 0 9 "U I: O A o if m .l : S . ' ‘ -H m . m o no . L 21.04 - M ‘ . .02 ' O - A —o—o—Desorption -4L4L—Adsorption . A l g l -30 -20 -10 PC ~ capillary pressure (cm. H20) Figure 29.—-Experimenta1 capillary conductivity versus capillary pressure for medium sand. 103 0.04 P ___o ’0 Lb ' o P O 0.02 " c3 '2 \ A (a) O -0.01- . a . u . H > -r-( 4., )- o :3 o p 8 A U .004 ~ f: o m p H A H -:-l 8* A 0.002 * —<>4&—- Desorption 0? A —4hqs—- Adsorption E . .001 1 1 e I ! ~50 ~40 ~30 ~20 ~10 PC ~ capillary pressure (cm. H20) Figure 30.~-Experimenta1 capillary conductivity versus capillary pressure for fine sand. 104 0.05 ’ 0.04 t O _ ’t 0.03 . c .a E \. é 0 02, o 3 a .1) H > W4 ‘8 5 0.01 U c o 0 >. u m H T: A 5340.004 . 0 o ”b 9:. m 0.002 ’ 0.001 ’ I J l L l l 1 ~120 ~100 ~80 ~60 ~40 ~20 PC ~ capillary pressure (cm. H20) Figure 31.-~Experimenta1 capillary conductivity versus capillary pressure for sandy loam. K ~ conductivity (cm./min.) O H .001‘ n J L 1 I l 4 -70 ~60 ~50 ~40 ~30 -20 ~10 0 PC ~ capillary pressure (cm. H20) Figure 32o--VAriation of hydraulic conductivity of mono~ dispersed glass beads with capillary pressure in rewet and redry loop. 106 6x10" J4x10- 32x10— 8x10— 4x10- 2x10- 10‘ ’6x10- l L l J L _1_ _1 ~80 ~70 ~60 ~50 ~40 ~30 ~20 ~10 PC ~ capillary pressure (cm. H20) Figure 33.-~Variation of capillary conductivity with capillary pressure for aggregated glass beads for redry and rewet loop. (cm./min.) K ~ capillary conductivity 107 Figures (32) and (33) represent Topp's (38) experiments on mono-dispersed and aggregated glass beads. Topp's data was obtained by a dynamic method using tension supplied by a water column instead of air pressure. One point of difference between the data obtained from three natural soils in this study and Topp's glass bead data is that his adsorption curves for glass beads approach a horizontal asymptote as the capillary pressure becomes zero. But ad- sorption curves of the soils used in this investigation approach zero pressure with a sharp slope. Topp's data is included in this investigation to broaden the scope of testing the mathematical models for conductivity. From these figures it is observed that hysteresis exists when curves are plotted for K versus Pc but little hysteresis was exhibited in plots of 9 versus K. In the case of fine sand the adsorption and desorption curves even cross each other. Figures (34) to (38) are plots of moisture char- acteristics using saturation S = 0/¢ instead of 8. These plots are needed to make the first estimate of Sr' the residual saturation by extrapolation as shown. Sr is to be used later for plotting Se versus Pc on log-log scales (Figures (39) to (43). Here Se’ called the effective saturation, is equal to (§:§:). From these plots the values of Pb and )I are estimated and used in equation (68) for computation of conductivity function. 108 60 50' EL ‘ 0 - 0 312 m ¢ — s 7 ° . Residual saturation Sr = .11 E o I; 40’ “*>——°‘—‘ Dynamic data 2 -—qG—a*—+- Static data 3’: m u m >. 30" H m H H -r-I Q m o I 20" o m I 10“ 0.1 . 5 ~ saturation (8/0) Figure 34.~~Saturation versus capillary pressure for medium sand for desorption and adsorption. 109 100 r 4 ¢ = 8s = 0.29 E; S = 0.32 mNBOI— r 5' 8 m -—0—-0- Dynamic data 3 -A-4L-'Static data 0160" (I) m I.) m >. L4 ,‘3 F4 40 ’ -H 04 m o l o “‘20- I ~ I l I J .3 .4 .5 .6 s - saturation Figure 35.~~Saturation versus capillary pressure data for fine sand for desorption and adsorption. ~P ~ capillary pressure C 160 140 120 100 60 40 20 80H- 1 110 __o__o——.Dynamic data -—A-—A-— Static data .1 .2 .3 .4 .5 .6 .7 .8 .‘9 1 5 ~ saturation Figure 36.-~Saturation versus capillary pressure for sandy loam desorption and adsorption. 111 60’ 50’ 40 - capillary pressure _PC 20' 5 ~ saturation Figure 37.-~Saturation versus capillary pressure for mono-dispersed glass beadsrewet-redry loop. 100’ 190’ o N a: .80' E 3 0,70- H :3 3’. (- 0’60 14 o. a! 50 F (U f... 33 I. o, 40 (U U ' 30 r U D; I 20 . 10 L 0 3 ~ saturation Figure 38.-~Saturation versus capillary pressure for aggregated glass beads rewet-redry loop. 113 uorndzosea ]; uoradzospv .vcmm Esflvme Mom musmmmum uumHHHmmo 6cm coflumuduMm w>fluommmm cmm3umn mflcmcoflumammnu.mm musmfim a Oflumum Intlldallldll UHEmamo .1: 0'0] .. HH. 4 r b P 0 b Q p ‘ om om ow om om oa w v m o.H musmmmnm annaaflmmo I um I , e - S nexnnes aArqoegga no; 114 mHSmmmHm mumHHHmmu cam .Ufimm GGflM HON bvh. men. mmmaoo. Hm.H Nm.N rm. Nvmoo. Hmmooo. w H mm mtnv 1mm oaumum owfimcha . uorquospv uorquosaa mm . um . . IN IM- oflmum F « ounamcaalollol Iv . 10. o is. o o . . .. w! W cm ofi‘ m muammmhm aumaafimmo I coHumusuMm m>Huowmmm cmm3umn maanOHumemII.ov mudmflm uorneanes 8AT338;J3 115 — PC - capillary pressure (cm. H20) $0 29 30 40 5060 70 80 100 -‘ I 1‘1 . ' o ' A . A A O c .3 " A ' ”06' 'A ‘ m H v o s - ~ 3.4 S = 0.34 m .33 A‘ . é —0—o—Dynamic 3.2- +A—Static I . m . m .l 8 Dynamic Static "S Pb 89 122 g* l 8 sz .000127 .000067 3 A 2.43 8.5 c .3 IE) 5.8 49 u l 8 rd . ¢ A .317 l 63 Figure 4l.--Relationship between effective saturation and capillary pressure for sandy loam. - Effective saturation S 116 .8!- O O Sr=0003 '6 P Desorption Pb = 39 .4 - l P 2 = .000657 b .3 - A = 5.04 2 _ Adsorption Pb = 20.5 L Pb2 = 00248 01" A = 2.52 -| L l l I l 10 20 30 4O 6O 80 - PC - capillary pressure (cm. H20) Figure 42.--Relationship between effective saturation and capillary pressure for mono-dispersed glass beads. S - effective saturation e 117 - PC - capillary pressure (cm. H20) 20 30 49 59 7? $00 200 o .8 ' '6 '- S_. = 0008 Desorption .4 - Pb = 33.0 '3? 1 i = .000917 P b '2 ” A = 3.17 Adsorption = 19.4 01 I. -l—2 = .00266 .08 . b 006i’ A = 2.12 .04 * Figure 43.--Relationship between effective saturation and capillary pressure for aggregated glass beads. 118 Comparison of theoretical and experimental conductivity results The theoretical results for conductivity function obtained by three different methods are presented separ- ately for adsorption and desorption to avoid confusion because for each experimental desorption or adsorption curve six calculated curves are obtained, three for static and three for dynamic data. The plots of only K versus 8 are compared. These plots are shown in Figures (49) to (56). Figures (44) to (46) are plots of Eli versus 8 for the three natural soils, for both adsorgtion and de- sorption. Figure (47) and (48) are plots for Topp's glass beads experiments. In the adsorption process (arrows up- wards) it may be noted that the area under the adsorption curves is greater than the area under desorption curves. In Figures (49) to (56) CSTD. is the measured value of capillary conductivity near saturation. (FMT)D and(FMT)S are the matching factors for dynamic and static processes respectively. The matching factor is the ratio of the measured conductivity to calculated conductivity near saturation. (FMB) is the measure of the deviation of the matched calculated conductivity from the measured CODdUCtiVity at minimum moisture content, that is: Measured conductivit (ME) = ( v) , (calculated conductivity).(FMT) Therefore (FMBXX‘FMT) gives matching factor at 8min. .0084 11% s 2 ds (S ) ‘—‘Z .007‘ e j; PC K = ___—_________ r l d s I f .2 0 c .006- _1_2 = _1_2 we)” Pc Pb .005 - . D o N H o 9' .004 \7 . Medium sand .003 -“ , —o-—0- Dynamic -H—o— Static ,I~‘ ' 1 i X i i x .001 - E \ i \K a \ J5 g \ “I5i ' * 0 0 c 7 l. .8 .6 .4 .2 0.0 S - saturation Figure 44.--Area under the curve for relative conductivity computation for medium sand. 120 .007: T) I, I. A .006: Fine sand -—o——o——Dynamic .00 -—A——A——Static .00 ‘ I" i .4 o m .1 .00, I i .00:l’ o ‘.\O 0.0 . \‘ L 1.0 .8 .6 .4 S - saturation Figure 45.--Area under the curve for relative conductivity computation for fine sand. 121 .007 ' A .006 ' .005 ’ Sandy loam —o—o-— Dynamic '_A_A_ . .004 . Static O 003 .002 .001 00. .4 S - saturation Figure 46.--Area under the curve for sandy loam for relative conductivity computations. 122 .007 » mono-dispersed 0 glass beads 0006 " .005 - .004 — N r4 0 m .003 v .002 F .001 - o 0-0 s : s 2e 1.0 .8 .6 .4 .2 S - saturation Figure 47.--Area under curve for mono-dispersed glass beads for relative conductivity computations. 123 .007 . .006 ‘ -005 ‘ Aggregated glass beads 0002 '- 1.0 .8 .6 .4 .2 S - saturation Figure 48.——Area under curve for aggregated glass beads for relative conductivity computations. 124 Experimental data ‘{*<*<>' Model la Burdine Equation 2 3 ds 1 ds K = CSTD x (s ) .10—2 47 PC C Dynamic -4n4s—- Static -¥—-X— 1b For P > PI Brooks and Corey equation C Z/A s S-Sr K - CSTD s 2) JC -Sr ~ d8 ‘ x ( e 1 *2/1 Jf s—sr ds 0 l-Sr For PC < PI suggested modification 8 > S I S 1-8 0.4 f exp {-C) Iz—S— dS K = CSTD x (s 2) ° I as e l l-S 0.4 f exp ('C) -l—_-S— d8 0 I Dynamic -—A—Ar- Static -H— 2 Millington and Quirk equation as modified by Kunze. K 2 -2 K = 33 307 n ¢ (h. 2+3h KSC pgu 1 -2 -l -2 2 ---(2n ) hn ) Dynamic I-CF{}- Static —1|—1l— Legend for Figures (49) to (55) for the comparison of experimental conductivity with theoretical conductivity. *I 125 .4- .2r .5 «4 E \\ E 01 p 8 L >‘.08 4.! '9 'H .06 p 4.1 o s '2 04 I- 8 ’ Medium sand I . Desorption M O CSTD = 0.8 (FMT)D = .143 .02 - °‘ _ ‘ / (FMB)D — 4.31 000 J I (FMT)S = .143 /' (FMB)S = .877 .fi 1 1 l 1 10 20 30 40 8 - % volumetric moisture content Figure 49.--Conductivity comparison for medium desorption. sand (cm./min.) K - conductivity 126 C §.. Figure 50.--Conductivity adsorption. .8 h o i 0 °/ 0 / .4 r ., O o I 2 p o o .1 — 0 ° 0 0 Medium Sand 0 o Adsorption ° 0 CSTD = 0.8 ' (FMT) == .058 .04 - g” .2 D (FMB)D = 12.22 c (FMT)S = .033 ° (FMB)S = 15.18 0 a o .02 — P0 .01 l :9 l 1 l 15 20 25 30 6 - % volumetric moisture content comparison for medium sand 127 .O4L fine sand desorption »~ L é '02 CSTD = 0.06 '2 (FMT)D = .170 \. é (FMB)D = 1.23 .8 (FMT)S = .133 31'01” (FMB)S = 2.82 ’ "-4 > -a 43 U :3 c c o 0 ..004- M .OOZF O .001 I I I l I 4 ‘L I l __l 19 20 21 22 23 24 25 26 27 28 8 - % volumetric moisture content Figure 51.--Conductivity comparison for fine sand desorption. 128 .08 . .06 P fl 0‘8 .04 r G i sandy loam desorption . CSTD = 0.06 .02 '- ,~ (FMT)D = .574 8 (FMB) = 1.198 0 .H D E (FMT)S = .210 E .01 — (FMB)S = 2. m JJ H > -H 4J o 5 U 8 o .004 — I M . o .002 - ' o ’08 O o .001 .0007 E . a 20 25 30 8 - % volumetric moisture content Figure 52.--Conductivity comparison for sandy loam desorption. 129 .4r mono-dispersed .2 glass beads desorption CSTD = 0.9 l L- FMT = .611 FMB .217 .06 - A 004 .- c5 "-1 E. \. E .02 _ 3 >3 3'? S .01. H 4.) o :3 U L- g .008 o ' .004'” m .002 . .001 - l .0006 L O o .0004 - A a 21 l l ! 4 5 10 15 20 6 — % volumetric moisture contents Figure 53.-—Conductivity comparison of mono-dispersed glass beads desorption. Eh. (cm./min.) - conductivity r. .01 ' .04 h .01 .006 ' .004 ' .002 b .001 “ Figure 130 mono—dispersed glass beads adsorption CSTD = 0.9 FMT = .302 FMB = .633 l 1 I 1 I 5 10 1'5 20 25 0 - % volumetric moisture contents 54.--Conductivity comparison of mono-dispersed glass beads adsorption. 131 1'0“ aggregated beads /// desorption - ’6 CSTD = 2.0 .4F FMT = .572 FMB = .400 02" ,fi .1_ E \ . E3 .06» >1 .04~ 4J -r-{ > '3 o .02p 5 c c o 0 .01- I x .006* 0 ' .004» ” o ”I .002. o .001- ‘ o .0006’ ' .0004- P g L I I 0 20 30 40 50 6 - % volumetric moisture contents Figure 55.—-Conductivity comparison for aggregated glass beads desorption. .06. .02- .01 K - conductivity (cm./min.) .006' .004" .002’ .001_ .0006. .0004, Figure 132 aggregated beads adsorption o CSTD = 2.0 ‘ FMT = .259 FMB = .933 l ;; I I 1 10 20 30 40 50 8 - % volumetric moisture contents 56.--Conductivity comparison for aggregated glass beads adsorption. DISCUSSION Gamma Radiation Attenuation Technique Although this technique is very useful for the determination of varying moisture contents, there are some factors which must be considered for its success- ful application. The detector must be shielded with lead as much as the source itself. Whereas the shielding of the source is necessary for personal safety, the shielding of the detector is necessary in order to prevent the counting of scattered radiation. When a strong CS137 source was used (lOOmci) and the shielding was not adequate around the detector, it was noticed that even though the moisture content of the soil changed, the change in the count rate was not appreciable; therefore, accurate calculation of the moisture content was not possible. Additional shield- ing of the detector solved this problem. This factor must be considered in using this technique in the field. Another factor is prOper collimation of the gamma beam both at the source and at the detector. 133 134 Static and Dynamic Methods As in the studies of Topp et_§l, (40) it has been observed in these investigations also that moisture char- acteristics obtained by dynamic and static methods (Fig- ures (17b), (21),(22)) differ from one another. It seems that more water drains in the beginning of the desorption process by the static method than by the dynamic method. The possible reason for this could be that when a certain -P pressure step P 1 is applied, all pores of radii r2 to 2 r1 corresponding P to P tend to drain simultaneously, 2 1 whereas when pressure is changed gradually, only the largest pore starts draining and in doing so creates dis- continuities in the porous material making it difficult for the next smaller pore to drain. Therefore, the amount of water drained is smaller in the dynamic process initially. After a few pressure steps the same kind of discontinu- ities caused a slow down of drainage by the static method and therefore, the slope of the moisture characteristic curve tends to decrease. Different rates of pressure application did not seem to affect the draining process and give approximately the same desorption curve for a particular soil. It was also observed that the adsorption curves did not become horizontal near saturation as reported by Topp (38). It approached saturation very steeply. 135 The plots in Figures (9), (12) and (15) for medium sand, fine sand and sandy loam show that a higher value of maximum Pc is achieved for sandy loam than for fine sand and similarly maximum PC is higher in the case of fine sand than in medium sand. The reason for this is that Pw did not build up as fast in sandy loam as in medium sand. Pw built up fastest in medium sand. This indicates that Pw builds up faster in coarse material than in fine material. In the case of medium sand (Fig- ure (9)), it was observed that after in = 42 cm. H20, the increase in Pw was approximately the same as the in- crease in in. The reason for this appears to be that in fine textured soils the increase in Pw is alleviated by drainage of the excess water. Apparently in the coarser textured soils the film contact is lost, and hence there is no pathway out of the tensiometer in order to relieve this pressure. Higher values of the external hydraulic gradient AH, are necessary in coarse material than in fine material to be able to measure Ah, the internal head loss in soil. Experimental and Calculated Conductivity The experimental values of conductivity plotted against moisture content (Figures (26)—(28)) show some 136 scatter of data for adsorption and desorption processes. This scatter has been observed by Topp (38) but was not attributed to hysteresis. This observation is further strengthened in this study. In the case of medium sand the adsorption curve is below the desorption curve, in fine sand the two curves cross each other and in sandy loam the adsorption curve is above the desorption curve. The scatter seems to be due to experimental errors. Dotted lines in these figures show the average of the adsorption and desorption conductivity values. P16ts in Figures (26), (27) and (28) are reproducible with the same degree of accuracy. Three runs for medium and fine sands produced similar results. For sandy loam (Figure (28)), only one run was made. The experimental conductivity data when repro- duced for comparison with the calculated values in Figures (49) to (56) are shown by open circles for both desorption and adsorption. Figures (29) to (33) show the plots of conductivity versus the capillary pressure. In these plots hysteresis is well marked. This has been shown by several investigators previously. Conductivity K(6) may be assumed as a unique function of moisture content but not of capillary pressure, because for each value of Pc there are two values of 0, one for desorption and another 137 for adsorption; therefore, for each value of Pc there are two values of conductivity K. Figures (49) to (56) give a comparison of experi- mental and calculated conductivities of five materials. Comparisons for the adsorption process of fine sand and sandy loam are not included because the calculated values compared very poorly with the experimental values. CSTD, the saturated conductivity, and matching factors for dy- namic and static data are given on these plots. (FMT)D is the matching factor for the dynamic process and is the ratio of the experimental to calculated conductivities near saturation ; (FMB) is a similar ratio at 5min after multiplying the calculated conductivity values with (FMT). For the adsorption process none of the equations gave accurate results. By using Burdine or Brooks and Corey models, the calculated values are much smaller than the experimental values. The total area f ds/PC2 shown in Figures (44) to (48) is much greater for adso p- tion than desorption; therefore, the ratio fog ds/PC 2 .[1 ds/Pc2 is smaller for adsorption than.desorption and the resulting calculated conductivity K is smaller. The Brooks and Corey equation Se = (;E)A or —£— = -l— (Se)2/A c PCZ P02 assumes a unique relationship between Se and Pc but actually the same equation can not represent both the adsorption and desorption processes. If the above rela- tionship is to be used for the adsorption process, a 138 different value of 1 will have to be used. The use of Brooks and Corey's technique for the determination of 1 for the adsorption process is not possible because the plot of log (PC) versus log (Se) is not a straight line but is a curve (Figures (39) to (43)). Even if a straight line is fitted through the adsorption data, it gives a different value of Pb' Theoretically, there should be only one value of bubbling pressure for a particular soil. Therefore, this approach does not hold true for the ad— sorption process. Millington and Quirk's equation predicted higher values of conductivities for adsorption than the equa- tions discussed above; however, the predicted values still differed very much from the experimental values. It appears, therefore, that the theory behind all of these approaches is not applicable to the adsorption process because the mechanics of the adsorption process differ from that of the desorption process. Adsorption of liquids by porous media is affected by air entrapped in the pores but this is not a significant factor in de- sorption. Use of Millington and Quirk's equation with matching factors (as shown in Figures (49) to (56)) re- sulted in different matching factors for the upper and lower parts of the conductivity curves which indicate that the curves obtained from calculated values are not parallel 139 to the experimental curves. Generally, the curves matched better at low moisture content with the exception of the desorption curves for Topp's glass beads. The calculated values of conductivity are larger than the experimental values at saturation and when multiplied by the matching factor the calculated values became smaller than the mea- sured values at low moisture contents. For the desorption process, the difference between the experimental values and the calculated values of con- ductivity, even though not negligible, were not as much as for adsorption. _1_ sz used to estimate 3&7 and this was substituted in Burdine's c . l _ 2/1 Brook's and Corey's equation 5;: — (Se) was equation (67) to compute conductivity. The approximation _l_.= .1? (se)2/A is shown by a dotted line in Figure (44). Pc2 Pb It does not exactly represent the experimental data. This approximation may give good results for those soils which have plots of Fifi-versus S concave downwards. For soils having these plots either partially or completely concave upwards as shown in Figures (44) to (48) this approxima- tion did not yield good results. For such soils approxi- mation by equation (69) yielded better results (See Figure (44) Open squares) for dynamic process. For the above reasons conductivity values calcu- lated by use of the original Burdine equation (67) did not match the values calculated by substituting equation 140 (68) and (69) in equation (67) (See Figure (49), Open triangles and crossed circles). However, when the use of equation (67) is reduced, i.e., in plots of SEE-versus S, when the curves became concave upwards (See dark circles for static desorption (Figure (44), triangles and circles for desorption in Figures (45) to (48)), results of Burdine equation and that of equation (69) substituted in equation (67) gave similar results. These results are shown in Figure (49) dark triangles and crosses for static desorption and open triangle and crossed circles in Fig- ures (52), (53) and (55) for dynamic desorption. It, therefore, seems more desireable either to use equation (67) directly or when the curves of Figures (44) to (49) are concave upwards, equation (69) may be substituted in equation (67) and then integrated for the conductivity calculation. From the above analysis it may be concluded that the direct numerical integration of Burdine's equation, which is now possible with the aid of a computer, gave better results for dynamic desorption process. Equation (70) gave better conductivity values from static desorp- tion data than from the dynamic desorption data (see closed squares compared with open squares in Figures (49) to (52)). In the case of glass beads it overestimated the conductivity values at all moisture contents. This probably was due to assumptions on which the development 141 of Millington and Quirk's equation is based, i.e., cutting the physical flow model at planes perpendicular to the direction of flow and then random rejoining, thus, over- estimating the flow area and hence the conductivity. This is contrary to what had been anticipated since Millington and Quirk assumed that the flow area decreased with a decrease in moisture content. In spite of these limitations, equation (70) in- volves no such parameters as Sr' 1, Pb’ PS etc. and, therefore, is easy to use. Sr is important in the use of Burdine's equation and in equation (68). Values of Sr cannot be correctly estimated unless complete moisture characteristics are available. Lambda (1) can be estimated only if the relationship log PC versus log (Se) is a straight line. From Figures (34) to (38) it may be observed that the moisture characteristic curves are extended be- yond the experimental values to properly estimate the values of Sr' CONCLUSIONS 1. Dynamic moisture characteristics can be sat- isfactorily obtained by gamma radiation attenuation tech- nique with proper shielding and collimation. Mass attenuation coefficient of water varies with the thickness of water interposed between the source and detector. Logarithmic relationship between the count rate and moisture content of the porous material is valid for gamma radiation attenuation technique. 2. The moisture characteristics obtained by the dynamic method differ from those obtained by the static method. The use of different pressure application rates in the dynamic method gave very similar results. Moisture characteristics were obtained for medium sand by raising air pressure as fast as 1 cm. of water in 1.65 min. For finer soils such as fine sand and sandy loam this rate can be higher. In coarse soils such as medium sand the drainage of water from the soil is considerably impeded after a capillary pressure of 30 cm. H20; whereas in sandy loam higher capillary pressure values can be reached before the drainage slows down. 142 143 3. The existing theories for the determination of capillary conductivity from moisture characteristics cannot predict the capillary conductivity values for the adsorption process. For the desorption process Burdine's equation gives better results for the dynamic process. Brooks and Corey's approximation for determination of Pc from Se does not match experimental data; therefore, its use does not produce the same results as direct integra- tion of Burdine's equation. However, when this approxima- tion is replaceable by equation (69) (when Eii-versus S curves are concave upwards for desorption), the calculated values of conductivity by Burdine's equation and by using equation (69) along with equation (67) give similar re- sults. Equation (70) is more convenient to use, but the results obtained by employing dynamic data do not match experimental conductivity values. Using static data in equation (70) gives results which fit better with the ex- perimental conductivity data. Generally, the conductivity values predicted for low moisture content by this equation require a smaller matching factor, showing that the calcu— lated conductivities are higher than the experimental conductivity values. 10. REFERENCES Ashcroft, G. L. 1962 Gaussian elimination tech- nique for solving the diffusion equation for moisture movement in unsaturated soil. Ph.D. Thesis, Oregon State University. Microfilm copy available from University Microfilms Inc., Ann Arbor, Mich. 79 p. Bauer, L. D. 1956 Soil Physics. 3d ed. New York. John Wiley & sons, Inc. 489 p. Brooks, R. H., and A. T. Corey. 1964 Hydraulic properties of porous media. Hydrology paper no. 3. Colorado State University. 27 p. Bruce, R. R. and A. Klute. 1956 Measurement of soil moisture diffusivity. Soil Sci. Soc. Amer. Proc. 20:458-462. Brutsaert, Wilfried. 1967 Some methods of cal- culating unsaturated permeability. Am. Soc. Ag. Eng. Trans. 10, 3:400-404. Burdine, N. T. 1953 Relative permeability cal- culations from pore size distribution data. Am. Inst. Mining Metullurgical Engrs. Petro Trans. 198:71-78. Carman, D. C. 1937 Fluid flow through grannular beds. Inst. Chem. Engrs. (London) 15:150-156. Child, E. C., and N. Collis-George. 1950 Perme- ability of porous materials. Proc. Roy. Soc. (london) A201:392-405. Corey, A. T. 1954 Interrelation between gas and oil relative permeabilities. Producers Monthly, Vol. XIX, NO. 1. Doering, E. G. 1964 Soil water diffusivity by one step method. Soil Science 90:322-326. 144 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 145 Ferguson, H., and W. H. Gardner. 1962 Water con- tent measurement in soil columns by Gamma-ray adsorpsion. Soil Sci. Soc. Amer. Proc. 26: 11-14. Gardner, W. R. 1956 Calculation of capillary con- ductivity from pressure plate outflow data. Soil Sci. Soc. Amer. Proc. 20:317—321. Gardner, W. R. 1962 Note on the separation and solution of diffusion type equations. Soil Sci. Amer. Proc. 29:485-487. Gardner, W. R. 1965 Water contents by gamma-ray and neutron attenuation. Methods of Soil Analy- sis, C. A. Black, ed. p. 114-125. Hanks, R. J. 1968 A numerical method for estimat- ing infiltration, redistribution, drainage and evaporation of water from soil. Paper no. 68- 214 presented at 1968 annual meeting of Am. Soc. Ag. Eng. 17 p. Hanks, R. J., and S. A. Bowers. 1962 Numerical solution of moisture flow equation for infil- tration into layered soil. Soil Sci. Soc. Amer. Proc. 26:530-534. Holland, D. A. 1969 Construction of calibration curve for determining water contents from radi— ation counts. Jour. Soil Sc. 20:132-140. Jackson, R. D., R. J. Reginato, and C. M. H. von Bavel. 1963 Examination of pressure-plate out- flow method for measuring capillary conductivity. Soil Sci. 96:249-256. Klock, G. O. 1968 Pore size distributions as measured by mercury intrusion method and their use in predicting permeability. Ph.D. Thesis, Oregon State University. Microfilm copy avail- able from University Microfilms Inc., Ann Arbor, Mich. 91 p. Klute, A. 1965 Laboratory measurements of hydraulic conductivity of unsaturated soils. Methods of Soil Analysis, C. A. Black, ed. p. 253-261. Kunze, R. J. 1965 Unsaturated flow in drainage problem. Drainage for efficient crop production conference, Chicago. Published by ASAE. p. 40- 45. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 146 Kunze, R. J. 1969 A rapid method for summing series of terms in calculating hydraulic con- ductivity. Soil Sc. no. 2, 108:149-150. Kunze, R. J., and D. Kirkham. 1962 Simplified accounting for membrane impedance in capillary conductivity determinations. Soil Sci. Soc. Amer. Proc. 26:421-426. Kunze, R. J., G. Uehara and K. Graham. 1968 Factors important in the calculation of hydraulic conductivity. Soil Sc. of Amer. Proc. no. 6, 32:761-765. Laliberte, G. E., R. H. Brooks and A. T. Corey. 1968 Permeability calculated from desaturation data. Am. Soc. Civil Eng. Proc., Journal of the Irrigation and Drainage Division, IRI, March 1968, 57-71. Laliberte, G. E. 1966 Properties of unsaturated porous media. Ph.D. Thesis, Colorado State University, Fort Collins, Colorado. Microfilm copy available from University Microfilms Inc., Ann Arbor, Mich. 121 p. Ligon, J. T. 1968 Evaluation of gamma transmis- sion method for determining soil water balance and evapotransporation. Paper no. 68-220, pre- sented at 1968 annual meeting of Am. Soc. Agr. Eng., Utah State University, Logan, Utah. 28 p. Marshall, T. J. 1968 A relation between perme— ability and size distribution of pores. Soil Science 9:1-8. Millington, R. J. and J. P. Quirk. 1960 Perme- ability of porous solids. Trans. Faraday Soc. 57:1200-1207. Nielsen, D. R., J. W. Biggar, and J. M. Davidson. 1962 Experimental consideration of diffusion analysis in unsaturated flow problems. Soil Sci. Soc. Amer. Proc. 26:107-111. Nutter, L. W. 1967 Field measurements of soil water contents in sito by gamma attenuation. Presented to Forestry Section, Michigan Academy of Science, Arts and Letters. 6 p. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 147 Olsen, T. C. and D. Swartzendruber. 1967 Velocity- gradient relationships for steady—state unsat- urated flow of water in nonswelling artificial soils. Paper presented at Division S—l, Soil Sc. Soc. of Amer., Nov. 9, 1967, Washington D. C. 28p. Purcell, W. R. 1949 Capillary pressures—-their measurements using mercury and calculation of permeability therefrom. Am. Inst. Mining Metullurgical Engrs. Petr. Trans. 186:39-48. Richard, L. A. 1931 Capillary Conduction of liquids through porous medium. Physics 1:318- 333. Skaggs, R. W., E. J. Monk and L. F. Hugins. 1969 An approximate method for defining the hydraulic conductivity-pressure potential relationship for soils. Paper no. 69-742 presented at 1969 winter meeting of Am. Soc. Ag. Eng., Chicago, 16 p. Smith, E. M., T. H. Taylor, and S. W. Smith. 1967 Soil moisture measurement using gamma trans- mission techniques. Am. Soc. of Ag. Eng. Trans. 10, no. 2:205-208. Staple, N. J. 1966 Infiltration and redistribu- tion of water in verticle columns of loam soil. Soil Sci. Soc. Amer. Proc. 30:553-558. Topp. G. C. 1964 Hysteretic moisture character- istics and hydraulic conductivities for glass bead media. Ph.D. Thesis, University of Wis- consin. Microfilm copy available from Univer- sity Microfilms, Inc., Ann Arbor, Mich. 86 p. Topp, G. C. 1969 Soil water hysteresis measured in a sandy loam and compared with the hysteretic domain model. Soil Sc. Soc. Amer. Proc. no. 5, Sept.-Oct., 1969, 30:645-651. Topp. G. C., A. Klute and D. B. Peters. 1967 Comparison of water'content pressure-head data obtained by equilibrium steady-state and un- steady-state methods. Soil Sci. Soc. Amer. Proc. 31:312—314. Vachaud, Georges. 1967 Determination of hydraulic conductivity of unsaturated soils from analysis of transient flow data. Water Resource Research, 1967. 3:697—705. 42. 43. 44. 45. 148 Van, J. C. Improved outflow barriers for perme- ability measurements. Can. J. Soil Sc. 49: 261-262, Je 1969. Watson, K. K. 1967 Response behavior oftensio- meter pressure transducer system under condi- tions of changing pore air pressure. Soil Sc. 104:439-443. ' Whisler, F. D., and A. Klute. 1964 The numerical analysis of infiltration, considering hystere- sis, into a vertical column at equilibrium under gravity. Soil Sci. Soc. Amer. Proc. 29:489- 494. Wyllies, M. R. J., and M. B. Spangler. 1952 Application of electrical resistivity measure- ments to problems of fluid flow in porous media. Bulletin, Am. Assoc. Petr. Geol., 36: 359-403. APPENDIX I Computer Program and Data for Numerical Integration of Burdine Equation 2 3 ds 1 ds K = CSTD- (s ) . e ’4; PC2 /fo PC2 149 l" 13228200 2215.0 +..~In.a Ixnca St.,.nIn.» I10.» .\1.~In.e I.~.uze»..u.~._0c . 032.8200 0 or 00 .._.uz~».bc.knch .0» Z.~nn doc ~J.$n~ m 00 uozeezcu 00 czecrm+2~.uzekuae+ecozek «4.20“ 00 CO m.o*0220hm+.2.ku.zcozek 2+2uz a 22I24.\..2.kI.2.».uozhokm m ~+uz.okm\.achus. . ~4\.2.»noz~o»m K ~J\omu~2 .\.c<0.xom.*o#. #42000 00 m»m.qpm.mem.mkm._bm.km.co F2200 ca .o.cum~.»I mm< u»_>L»unozoo m>~kaumsuao .awIL.\AamImcumm.>e~moaou\maoemaozum 003mmman zo.knaomuouu >e~>~kuoozoo omeaaohqm «cemo . >F~mcaonnala .Oznk w>—kumamma aou waDmmmau ZOHhaaomwo omeO “mudomqu. £404 >OZO A<3o .434Fb0. ozOhx~2~0mcozqm ED_owZ ~Noaxuzeamvoz02OZOAWJOZ P 1 s-s 2/1 C I f r d 5 Sr I-Sr and 0.4 CSTD x (S )2 .[S exp(-c) l-S ds e S I-SI r K= ' l l-S 0.4 ‘[ exp(-c) (I:§_) ds '5 I r for Pc < PI CLlC = C CLZC = 2/1 PB2=._L P 2 b DTHETA = 03/10,000 AKSjjsexperimental value of saturated conductivity corresponding to Saturation ST 153 m0C00|qum , "00.0m0 00*..Scoaauxncoooa «r 2000.0040.>.u.2u+a”.0000".200000 0 002.200 0 00 b .fim 0k 00 Am&.U~JU.~m.WVZDu+Auvfloaflu.".0000 Q cobohnumlmvhh melmum oomonun d 00 oCNA~UOOOQ WOCCC0+FMNV ~COC."¢C a COHNE mDZ~PZCU w ~.m.na~0.tnmla2eumcu~ ~0.I._IZCOMuazcnm 002.002 2 cc ch.Ikmna~.0m NJU\.Ku0040 .m**~a\m**mu.UOJ pm.my<.Sm.am.\. amI.J.0m..u.J.mmm X*.Jcoaauqn.4.h.2~camczuu._cazmk 2.00. «1 00 mm or 00 .ma.ofiso._m.ancamezouuxncazwe ma.mu.m~.~mlxncamcue 2.2u0 «a 00 m0*.ncooaauxncooaa wozakzou .ma.u~40.”m.mczou+.nsooaau21.oooa a" o» 00 ANma.UNJU.>.mvzu+.ncooaun“5.0060 0.00.0n22mImcue . moswum 0".~u~ «a 00 .omnnvooan m.mmn 0" 0 h V . 0w (— bu Cu K— Nm @— EH «a CH —~ cm 155 0N0. om. 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