ANALYSIS OF SEEPAGE FLOW IN IRRTGATTON. FURROWS Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY YOU-TSAI HUNG 1969 f‘.':1Ci-“1igar;. H.136 I.;21I‘Vf?lfyll)l vex 'I This is to certify that the thesis entitled ANALYSIS OF SEEPAGE FLOW IN IRRIGATION FURROWS presented by YOU-TSAI HUNG has been accepted towards fulfillment of the requirements for Ph.D. degree in Agricultural Engineering ,. ajor professor ll! nah/M3 a}. W? 0-169 BINDING IV . nuts & snns' ' any; anuunvnuc. I T {m mmrogr.nmmr!!" L'— } p-nh ABSTRACT ANALYSIS OF SEEPAGE FLOW IN IRRIGATION FURROWS BY You-Tsai Hung The main purpose of this study was to apply the concepts of the Darcy's and Philip's approaches to flow problems in irrigation furrows. The effective use of irrigation water depends on a knowledge of the behavior of water flow and the distribution of water content in the soil. Proper water distribution in the root zone enables crops to exhibit maximum growth and production. In previous studies, there has been little information on the distribution of water content in the cross-section of the ridge where the root systems of crops develop. However, the behavior of water and its movement in the direction perpendicular to the furrow plays an important role in studies dealing with irrigation water utilization and times of irrigation. You-Tsai Hung To obtain a better understanding of the behavior of soil water in the furrow cross section a test box was made as a model. Two different constant depths of water (2-inch and 4-inch) were maintained to determine: 1. The maximum movement of the wetting front in the unsaturated region of the ridge, 2. The distribution of water content in the ridge cross section at a given time under unsteady state condition, 3. The validity and applicability of the flow equa— tions to predict the movement of the wetting front and the distribution of water content in the unsaturated region. Three theoretical equations have been derived to meet the requirements of the purpose of this study. The equations are; 1. Equation of the saturated-unsaturated boundary fi‘ra' — v-' r " “ — - ——-- ____-————l¥ You-Tsai Hung 2. Equation describing horizontal inflitration _ _ 2 3 KoA 1/2 1/2 X1 - (l R )[(2+3Ai)(eoe:)] t 3. Equation describing vertical infiltration 3K X=X+ 0 3/2 3/2 2 2 1 gored [ClRJ (KR ) + CZRJ_2/3(XR ) + R ]t 2/3 Experimental results show a good agreement with the computed results for vertical infiltration and agree quite well at the earlier stage during horizontal infil— tration. Other findings from this study are; l. The water movement in the unsaturated region is independent of the water depth over the bottom of the furrow, but dependent on the position where water surface hits the side slope of the furrow and on the position of the saturated- unsaturated boundary, 2. The contribution of the gravity term is dependent on the parameters KC, 90, ed, and Qi and indepen— dent of As and K. You-Tsai Hung The capillary conductivities computed by using K = D C in which D was determined by the equa- tion derived by Bruce and Klute (1956) have shown satisfactory results, The maximum vertical water movement at a given time occurred at the point where the wetting front intersects the saturated-unsaturated boundary. ,4 by first - Z we... fiajor E ofessor Approved by w“ M Department Chairman ANALYSIS OF SEEPAGE FLOW IN IRRIGATION FURROWS BY You-Tsai Hung A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1969 .. '1 ' { c/Ik/ - ' ‘7' .I f} 1}, a- /./ ‘ (1‘. v ACKNOWLEDGEMENT ciation to his major professor, Dr. George E. Merva of the Agricultural Engineering Department under whose guidance, supervision, and unfailing interest this work was done. The author is pleased to acknowledge the other graduate committee members: Professor Ernest H. Kidder, Dr. Orlando B. Andersland, Dr. Raymond J. Kunze, and Dr. Norman L. Hills for their suggestions and guidance during the phases of the graduate program. Special thanks are due also to Mr. Gerald H. The most heartfelt thanks go to the writer's wife, yueh-chin, and to his children, whose faith that some day the requirements for the degree might be completed was of constant encouragement. ii TABLE ACKNOWLEDGEMENTS. . . . . LIST OF SYMBOLS . . . . . LIST OF FIGURES . . . . . INTRODUCTION. . . . . . . REVIEW OF LITERATURE. . . OF CONTENTS EXPERIMENTAL SETUP AND PROCEDURE. . . THEORETICAL DEVELOPMENTS. RESULTS AND DISCUSSION. . CONCLUSIONS . . . . . . . O O 0 O O O RECOMMENDATIONS FOR FUTURE STUDY. . . REFERENCES. . . . . . . . iii Page ii iv ix 21 27 52 67 7O 71 LIST OF SYMBOLS constant cross sectional area characteristic length constant constant constant half bottom width of the furrow constant integration constants constant water capacity diffusivity constants cumulative infiltration functions of o and B 2+3Ai Gauge pressure head depth of water in the furrow iv List of Symbols.--Cont. total head or total potential gravitational head capillary rise intake rate final intake rate function of B in the T plane modified Bessel's function capillary hydraulic conductivity saturated hydraulic conductivity constant hydraulic condictivity at 9i coefficient of permeability real number parameter real number total pore space of soil gauge pressure applied pressure total volume of water quantity of seepage 3/2 List of Symbols.-—Cont. < ucoumli.a .mam .xoa ummu may mo 3mfl> xommll.m .mfim 23 A sheet of filter paper covered the bottom of the container. Two-inch thick layers of soil were placed in the container and 60 blows of a wooden rod were applied to each layer to obtain a uniform soil density. The final average dry density of the soil was 1.63 g/bc. A half furrow cross-section with a 1-inch wide bottom and 3 to 5 lepe was formed for each test. A constant water level tank was constructed to control the water level in the furrow. Two plastic tubes connected the tank to the furrow to act as a siphon and to supply water continuously. At the beginning of each test a volume of water was poured in the furrow to re- duce errors due to an insufficient supply of water from the siphon. One—inch square grids were marked on the front face of the plexiglass container (Figure 1). The wetting front relating to each elapsed time was obtained from a series of pictures taken at selected intervals through- out the test. Forty—eight one-inch diameter holes spaced one-inch apart were drilled on the back side of the con- tainer (Figure 2). The holes were sealed by 48 circular plexiglass plugs mounted on a plexiglass sheet and 24 attached to the container to contain the test sample. Forty-eight plexiglass tubes one-inch in diameter and one—inch long were mounted on another sheet in order to obtain soil samples after each test. Each test ran about 6 to 7 hours or until the movement of the wetting front was fairly slow. As soon as a test was completed the plexiglass sheet having the round pieces of plexiglass was removed and the plexiglass tubes plugged into the holes. Forty—eight soil samples were thus obtained for the determination of the water content. The water contents on an oven dry basis were determined by weighing on an electrical balance with an accuracy of 0.00005 g. These were then converted into volumetric basis. The relationship between water content and soil water suction was obtained by using a tube consisted of a series of rings with a diameter and length of 1-9/16 and 1 inches reSpectively. The rings were held together by masking tape. The tube was placed vertically with one end immersed in a constant water level, and the other end exposed to the air. When the capillary rise approached equilibrium, the rings were separated and the average 25 water content in each ring was determined. This relation- ship is plotted in Figure 5. To obtain diffusivity D, the same tube used for determining the water content vs. soil water suction was used. The tube was placed horizontally and a constant head device was installed to maintain water level constant at the center line of the tube. The positions of the wetting front and the corresponding elapsed times were recorded and the average water content in each ring was determined. The diffusivity was then obtained from a formula derived by Bruce and Klute (1956) 5x ."1 e (---—'69 .) f xl dee (29) . xl 9i 1 D(9) = - a; where; t = time (total elapsed time), x = the position of the wetting front from the beginning of the tube, 9. = initial water content. The capillary conductivity was calculated from the formula K=DC (30) 26 where; K = capillary conductivity, D = diffusivity, .. 39. . Cw - 5H, water capaCity, H = soil water suction. A Constant-Head Method was used to determine the saturated hydraulic conductivity (KO) of the soil samples which was calculated from Ko = (Q/Aat) (SL/Ht) (31) where; total volume of water flowing through the test soil sample, Q t = time, SL = length of the sample, A = cross sectional area of the sample, H = total head. THEORETICAL DEVELOPMENT Consider the trapezoidal cross-section (Figure 3) of an ideal irrigation furrow in which the soil is homo- geneous and isotrOpic. To facilitate the theoretical development relating to the flow of soil water in the furrow ridge assume; 1. vapor and solid phases of soil water are ne— glected, 2. temperature of both the soil water and the soil mass are constant, 3. the soil water is incompressible, 4. the flow of soil water is laminar, and 5. the wetting process only is being considered (water content is a single-valued function of soil suction). Water flowing into the cross-section of the furrow ridge produces two phenomena; i.e., saturated and 27 Unsaturated region 28 L~ 23 5 H l —..;.-¥-.=~ _.| ’— i l :1 1.....1 saturated-insaturated boundary Saturated region 9.. J T i Fig. 3.--Trapezoidal cross-section 29 unsaturated flows. The flow in the saturated region is described by Darcy's law which is a simplified form of the Navier-Stokes equation under certain assumptions (see literature review), while the flow in the unsaturated region is described by the diffusion equation which is obtained by combining the equation of continuity and Darcy's equation. Both the saturated and the unsaturated regions of Figure 3 increase with time. However, the e: ‘I E .0 saturated—unsaturated boundary (line of seepage) for the saturated region becomes constant when the saturated flow reaches a steady state condition, while the unsaturated flow remains an unsteady state. The movement and the distribution of soil water in the unsaturated region can be obtained by finding; 1. the position of the boundaries (bf and gk), 2. the solutions of the diffusion equation exterior to bf and gk. Due to the symmetry of the boundaries with the X axis, only the boundary bf need be considered in the 2 following derivation. 30 SATURATED FLOW REGION To determine the position of the boundary bf which is the line of seepage for a steady state flow, one can use Zhukovsky's function (l949r1950) ~a-.-r_.tp o where; ) C)= Zhukovsky's function, t Z = x1 + 1X2, w = o + iy, B, A = real valued constants. One can separate equation (32) into real and imaginary parts, viz. cos 1' (33) X 4. I3: = Aqu6 sin g (34) Where o and y are velocity potential and stream functions respectively. Note that one can substitute —w for y and Symmetrical about x2 axis. for x since equation (33) and equation (34) are 31 Define the velocity potential as P = — X c) Ko(r+ 2) (35) where; Ko = hydraulic conductivity, and the subscript indicates the conductivity at saturation, P = gauge pressure, r = specific weight of water. Since the pressure on the boundary bf is atmosPheric, i.e., P equals zero, equation (35) becomes The free surface (bf) of the saturated region must satisfy the following two equations L- --x2 + K — O 0 or -L 37 x2 - K ( ) o and __EL 38) where q is quantity of seepage. 32 From equation (37), when x2 is zero, %—-is also 0 zero. However, in equation (34) for this case A and ¢/fi e are not zero so cos ( g') = o. (39) Using equation (38) and equation (39) one can obtain _ 9—. _ cos ( 2’3) 0 then g—=- 7in: +o.+1.12....) 25 (2n + 1)2. (_ _ or q = -(2n + l)BW. (40) Let n equal zero, then equation (40) becomes q = -Br or _ _Q (41) B _ 7 After substituting equation (41) into equation (33). also making use of equation (36) and equation (38), one has 33 x_.£_._ -KX7r/q 1 2K0 — As 0 2 , (42) From Figure 3, when x equals B, x equals zero 1 2 then - _ _Q_ A “ B 2K 0 and _ .9..- _ _§_. ’K 7x /q xl 2K0 — (B 2K )e o 2 . (43) Equation (43) enables one to obtain the position of the saturated boundary bf under a steady state condi- tion. The value of q can be obtained by using Vedernikovfs (1960) equation under the assumption that the quantity of capillary water in the unsaturated region is negligible. Vedernikov's equation is = ' 44 q KO(ZB + A H1) ( ) where; A' = 2 f2(0.61,- f1(0.6)/bos or ' (45) tan O1r J2 1r/2 - f2(0.6) H1 = depth of water in the furrow, o o =‘-, w 6 a a value along the real axis of the complex plane (T plane) under the transformation 2.2. plane into T plane, where z and w dw from . .13. F.) 34 are defined in equation (32), H II functions of o and 5, J2 = function of B in the T plane. Vedernikov provided a graphical solution for q given in Figure 4. 4.4 4.0 3.6 3.2 2.8 2.4 2.0 1.6 m - cot d o 5 10 15 20 213/111 . Fig. 4.--ZB/H1 vs. A' (After Vedernikov) 1b? Q “I 4. 35 UNSATURATED FLOW REGION The flow of water in an unsaturated soil obeys both Darcy's law and the law of conservation of matter. For flow in the rectangular Cartesian coordinate system, the law of conservation of matter yields the equation of continuity which expresses the fact that the sum of the rates of flow into and out of an element of the soil added to the rate of change of storage of the soil ele— ment equals zero. The equation of continuity can be written in tensor form = 46 akUk + 809 O ( ) where; the rate at which water crosses a section of unit area perpendicular to the flow direction, WF: 9 - water content expressed as the volume of water per unit volume of soil, t = time, a do — _§t’ One can generalize Darcy's law in tensor form (47) =-K. .49 Uk Jkaj 36 where; 7: ll jk capillary conductivity, a function of e, the potential, x. = the distance in the flow direction. n ”A. a .I For an isotropic and homogeneous media, j equals k and ij equals K therefore equation (47) becomes, Uj = ~K8j¢ (48) After substituting equation (48) into equation (46), one obtains 8 e = a.(Ka..¢) (49) 0 J 3 Equation (49) is the well-known non-linear diffur. sion equation. The potential ¢ can be defined as ¢ = p/r + h ‘ (50) where; p = gauge pressure (negative), r = Specific weight of water, h = gravitational potential. If one differentiates equation (50) with respect to xj, one obtains 37 1 63¢ = E ojp + ojh. (51) The pressure in the unsaturated soil is negative. Denot- ing p/r by H, enables one to write a.¢ = o.H + a.h J J J (52) Let the gravitational components in the Cartesian coordinate system be gj = g ojh. (53) The coordinate system is oriented so that gj = 2 is in the direction of gravity, thus 52h becomes unity and alh becomes zero since gj = l is in the direction perpendic- ular to the direction of gravity. For horizontal flow 51h = 3&1 = 0, thus, equation (52) becomes §2_. _ §§_ 8x1 _ 5x1 and under the assumption that i equals 1, equation (49) becomes §§__a_ 5i (54) at _ 6x1 (K5X1)' 38 For vertical flow 52h = 5;; = -1, so that equation (52) is §g_,= 5H _1 x2 SK and equation (49) becomes 88 8 8H 8K —- .. (K—-) - - (55) at 5x2 5x2 8x2 By using Childs and Collie-George's (1950) concept on of diffusivity D = K55, one can transform equation (54) into 59 _d_ at ' ax (D §Q_) (56) 1 ax1 for horizontal infiltration. For vertical infiltration the same concept enables one to write 69 a 59 fig ——=-——(D -—)- (57) 5t 5x2 5x2 8X2 where x2 is positive downward. 39 A. Horizontal Infiltration Philip (1955) deve10ped an iterative method for which convergence is rapid and simpler than the Crank- Henry's (1949) method used by Klute (1952) for the solu- tion of equation (56) subjected to the conditions 9 = 9 X1 > O t = 0 1 (58) e = 9 x = 0 t Z O o 1 with: 9i = initial water content, 9 = water content at saturation. 0 Philip used the Boltzmann transformation a) = X t—l/Z (59) to reduce equation (56) to the ordinary differential equa- tion de “’1 .9. £32.) (60, ____.= ( 2&31 dbl dbl which can be integrated to yield 40 subject to the conditions 9:9,a) =0. (62) The author uses the functional form for the solu- tion of ml similar to that suggested by Brutsaert (1968a) (D=l-R (63) R — — d (64) l - Sd e - ed according to Irmay's (1954) concept where; S = , 9/90 9 = water content, 90 = water content at saturation, sd = part of the voids filled with dead water, i.e., Gd/GO. 9d = dead water content or water content at which K tends to zero. Brutsaert concluded that m equals (n - l), in Therefore, which n is assumed to be 3 (see equation 66). m equals 2 and equation (63) becomes 41 Irmay (1954) proposed the following expression for the relationship among the unsaturated (K), and saturated hydraulic (KO) conductivities and water content s - sd 3 K = Ko (1 - s ) d or K = K R3. (66) o Brutsaert (1966) presented one of the most satis- factory equations describing the relationship between water content and soil water suction, i.e. A - S 67 R ' [As + (-H)T] ( ) where; A . n = parameters, H - soil water suction (negative value). From Brutsaert's inspection of wetting characteristics for a number of different soils, he found that n is close to unity. Thus, equation (67) reduces to _ (68) R = 1.5/(AS H). 42 Substituting the differentiation of equations (64) and (65) in equation (61) results in 2 4 D = (R - R /3 + RAi) (69) where; 3 A. = -R. + R./3' and i 1 1 Ri = (9i — 9d)/'eO - 9d). The difusivity D, defined by Childs and Collis- George can be better expressed by differentiating equa- tion (64) and equation (68) along with equation (66) to Obtain O S /( 0 d) ( ) Substituting equation (70) into equation (56), one obtains 5R 5 [ 5R 5t* 8x1 5x1 subject to the conditions R = R. x > O t* = O 1 1 (72) R = l X = O t* _>_ O with; (73) 43 Brutsaert (1968a) suggested that a function of the type of equation (69) may provide better description of (K/KOAS)(%E) than does R. Therefore, equation (71) can be represented equally well by 4 an a 2 __ an at* = aXII‘R ‘ 3 + RAi)oxl] (74’ subject to the same conditions as in equation (72). Equa- tion (74) can therefore be transformed into 4 5R 5 2 a a; 5?'= S§“'[(R ' 3 T RAi) 8x 1 (75) 1 l with; T = 13/12 + 3Ai)]t* (76) SC > K A r = (2 +33A )(e °_Se )t. (77) i o d From equation (59), equation (63), and equation (77). it follows that K A l 1 (78) x = (1 - R2)[(2 +331 )(9 0-39 )1 /2t /2 1 i o d and K A 2 3 o s 1/2. ml = (1 - R )[(2 + 3Ai)(90 _ ed)l .‘l‘l. .. L 44 This is the solution of equation (56), which enables one to calculate an xl correSponding to an elapsed time t by assuming R = R. = d. (79) The water content distribution can be calculated by knowing the values of xl/tl/z. B. Vertical Infiltration For equation (57) Philip (1957a) obtained the series solution mi 1:“2 (80) X II “M8 i 1 The series converges very rapidly so that retaining only the first two terms is accurate enough for practical pur— poses, i.e. 1/2 (81) x2 a cult + (th in which m1 is the same as defined previously in the dis- cussion of horizontal infiltration. It is assumed that 45 x = x + x’ (82) 2 1 2 where x5 is called the residual. x1 is given from the solution of horizontal infiltration. Philip (1957a) rewrote equation (56) and equation (57) with 9 as the independent variable i ‘1 ox i __l._ 9. §§_. a ' at ' 88(D 8x1) (83) 5x 2 _ a as _ 8K 7 at ‘ 89(D 8x2) 59' (84) After subtracting equation (84) from equation (83) one obtains OX’ OX’ J=§‘(D-§9—J)+§S (85) at 59 ox ox 58 1 2 and by using the approximation if; = if; (86) 5x2 8x1 equation (85) is converted to 5"; a as. 2 93:21 + as (87) a?" 55 [D (5.1) a. a. 46 where xg’is written for x5, since the use of approximation (86) implies that equation (87) can not give the exact value for x5. The transformation a -1 m2 - x2 t (88) can reduce equation (87) to the ordinary differential equation db ‘3 __Z. 95 (1)2 = 35 [10(8)de ] + d9 (89) where; Me) = 1:(2 (90) 051(9) = x1 t"l/2 (91) One obtains, by integrating equation (89) subject to the condition _ = (93) 9 — 90 wz 0 where; K. = the value of K at 9 = 91' 1 Three functions used in the problem of horizontal infiltration are also used here. They are: 47 R = (s - sd)/(1 — sd) = (e - admeo - ed). K = K"R3, and o 21 II its/(As - H). Substituting these functions into equation (57) . . 45.74 4T- gives (2 + 3A1) (2 + 3A1) aR3 8R 8 8R — = [ R -----1 - (94) OT 5X2 3 6x2 3A8 5x2 subject to the conditions R = R x2 > O T = O 1 (95) R = 1 x2 = 0 1‘2 0 One can assume a new form I! = (96) x2 ur and compare equation (88) and equation (96) to obtain .E LL = (”21' (97) where; 3 KoA T = ( _ )t. (2 + 3A1) 90 ed (89) as Under this assumption one can express equation 48 -1. d_ as. e 2 u " T* dR(R dR) + Xg-R (98) subject to the conditions -1.éu R dR = 0 for R = Ri' (99) u = o for R = 1. (100) where; T* = lZ/G G =2+3Ai. One can now rearrange equation (98) in the form of a regular differential equation * Rzn”'- Ru’ — T*uR3 = - §_§R5. (101) S The complementary solution of equation (101) can be ob— tained by setting the right hand side of the equation equal to zero, i.e. R2u”_ RH,’ _ T*R3LL = 0 (102) Equation (102) is reducible to a modified Besselfls equation with a solution of the form a. ql 91 n = R 1[C1J§(XR ) + CZYV(AR )1 (103) 49 where; cl = 1, >. = (2/3) W v = 2/3, q1 = 3/2. Since v is not an integer, equation (103) can be written (.c = R[c1JV(xR3/2) + c2J_v(>.R3/2)]. (104) The particular solution of equation (101) can be found by the method of undetermined coefficients. The solution is assumed to have the form 00 u = z BmRm. (105) Substituting up, u;, and ug'into equation (101) one can obtain = (106) B2 G/As and B = B .= B = ,.. B = 0. Therefore, the particular 0 l 3 m solution is = (G/A )R2. (107) u s _ 3295-3 50 The complete solution of equation (101) can be written as M = “c + up or _ 3/2 3/2 . )1 _ RlclJv(>.R ) + c2J_v().R ) + (G/AS)R]. (108) To determine C1, one can use equation (99) and consider R equal to R1' The value of Cl thus obtained is 4/3R +3. . C [N J_2/3(x) 3595 (5 ) .1 . (109) 112/3 2/3J (— ) [2. 2173J _2/3(>.) + 3. 3595(2 -) J2/3(>.)R. ] The coefficient C2 can be obtained by using equa— tion (100) [1 + C1J213(>\)] 11 C = 7 J (i) ' ( O) 2 -2/3 From equation (97), one can obtain 3K 0 3/2 3/2 2 = ———————- XR ) + C2 RJ (1 R )+R 1 111 “’2 so - ed [ClR J2/3‘ -2/3 ( ) Therefore, the equation describing vertical in- filtration can be written 51 + wzt (81) where; )( KoAs )11/2 9 - e o d w = defined by equation (111). 3 2 (l-R)[( 2 + 3Ai 8 |—l ll One can assume R equal to Ri to obtain the wetting front correSponding to each elapsed time t, then solve for R or 9 by using the total elapsed time t and assuming values of x2. RESULTS AND DISCUSSION The constants in the solutions of the differential equations were obtained for this experiment. They are: The average value of Ko equal to 0.0406 inch per minute was determined from the Constant-Head Method. An approximate value for Ko equal to 0.0425 inch per minute was also obtained by ex- trapolation from the curve of 9 vs. K in Figure 8. K0 was assumed to equal 0.0406 inch per minute for computational purposes. Average value of 90 was assumed to be 0.334. Average value of 9i was assumed to be 0.003. ed was determined as follows. First, from Figure 8 as K tends to 0.0000133 inch per minute, the value of 9d was equal to 0.032. Since the value of K was so small, it was assumed that no flow occurred. Second, from capillary rise tests, the water 52 53 content within top half inch was about 0.032. Third, from Wyckoff and Botset's test results it was determined that ed equaled 0.1 9 , i.e., 0 9d = 0.033. A trial value of AS equal to 7 was obtained from a straight line making an angle of -45° with the (AS - H) axis where R ranges from 0.6 to 1.0 in the lOg-log plot of Figure 7. If one makes use of Figure 4 and Ko' q's for both 2-inch and 4-inch water depths can be found equal to 0.532 and 0.9634 in3 /min respectively. Final computational equations for this study are: To determine the position of the saturated- unsaturated boundary: (a) For a 2—inch depth of water -2.222 e'0°24OX2 + 6.552 (112) X II (b) For a 4-inch depth of water -4.195 e'0'132x2 + 11.865 (113) :4 II Suction H (in) 54 17T— 15... 10... O 1 l 1 l 1 l l 0 .05 .10 .15 .20 .25 .30 7734 Water Content (in3/in3) Fig. 5.-—Soil water content vs. suction. 55 mm. 0 Om> H xuu.o .msm ON. _ ma. OH (u?) Ix ON 56 .1 .09 .08 .07 .06 60:) 45678910 A - H (in) s Fig. 7.--R vs. (As - H) 57 x .m> o)n.m .osm Acflécd IOHX K m ON _ _ OH OH. O N om. ow. (Eur/Eur) e iillil 58 2. For horizontal infiltration; (a) To describe the position of the wetting front _ 1/2 x1F - 1.088 t (114) (b) To describe the water content distribution x — 1.098(1 — R2) tl/2 (115) 19 3. For vertical infiltration; (a) To describe the position of the wetting front X2F = 1.088 tl/2 + 0.0553 t (116) (b) To describe the water content distribution at time equal to 360 minutes X - 41.1279 - 26.1265R2 + 35.2915R3 20— 5 6 8 -52.4783R + 7.6683R - 5.7063R + 0.6362R9 — 0.3011Rll. (117) The Modified Bessel functions were expanded and only the first two terms were taken to calculate the co- efficient of 0.0553 in equation (116). Equation (117) was obtained from equation (81) by taking the eight terms using the power of R up to eleven. 59 The saturated—unsaturated boundaries for both the 2-inch and 4-inch water depths were calculated with the help of the 3600 CDC computer. The observed lines of equiwater contents at saturation selected from one of each treatment are shown in Figures 9 and 10. Both figures show that the computed and observed curves are in good agreement. In Figure 11, x1 was the maximum distance measured from the line x1 = B to the point where the wetting front at time t, while x2 was measured from the line x2 = 0 to the point where the wetting front intersects the saturated- unsaturated boundary. Values of x1 and x2 were determined through two treatments and no significant differences were noted in the values of X1 observed at the same elapsed time for both treatments and also for the observed values of x2. For horizontal infiltration during the earlier stages the observed values agree with the computed values but they deviate after one hour. The deviation may be due to air entrapment laterally on the side of the test box. For vertical infiltration, the observed values agree quite closely with the computed values. 60 Observed I I 1.——-Theoretical Fig. 9.--Saturated-unsaturated boundary for 2-inch depth of water. 61 -—— Observed - -—’— l Theoretica1-—fi Fig. 10.—-Saturated-unsaturated boundary for 4—inch depth of water. Maximum X1 or x2 (in) 62 301. 25" Theoretical vertical // infiltrat' / 20'—' ion IL—-Observed I 7 / / / / 15.. // Theoretical hori- , zontal infiltration // /I / ’,/ 10__. AL—Observed ///// / I / d f 5_ ' Note. x1 measure rom x1 - B x2 measured from x2 ='0 0 . l . _l 0 5 10 15 20 t1/2 (minl/Z) tl/2 Fig. 11.-—Maximumx1 and x2 vs. 63 mN H .GHE omm u u um m .m> x ESEAxMZII.NH .mwm Acwv ax Edfiflxmz ON ma OH m m H _ _ _ _ DCOEUMOHU £0Cfll¢ usefiuounu Sucflum an EOHM ”NH—DMMUE Hun «QUOZ OH. ON. Om. Ov. (Eur/Eur) e 64 N .248 oom u u 06 o .m> xu).ms .oam Acuv ax mN ON ma OH m O . _ _ _ _ o |.oH. uses—amok”. nucwle .- uaoauouuu socwlm .x l ow. ’1/[’ II; on 0 i7 Ov. (Eur/Eur) e 65 The observed water distributions along the line of maximum horizontal movement at time equal to 360 minutes for both treatments are drawn ianigure 12 for comparison with the computed values. Figure 12 shows a very good agreement for x1 less than or equal to 10 in- ches, but less satisfactory agreement when x1 exceeds 10 inches. The error can be explained by the argument previously given in the discussion of the movement of horizontal wetting front. In Figure 13, the computed curve was obtained by combining equation (116) and equation (117) with the same R at t equal to 360 minutes for both treatments. The values of x2 were calculated by assuming x1 and R values. The observed values of water contents distribution for the treatments agree very well with the computed values. Deviations between the observed and computed values could also result from errors in the parameters As' 90' ed' The dead water content (ad) is due to some water 9.. K, K , and q. 1 o being retained in the sharp angles between particles. It might be replaced by a filler of solid or incompres- sible material without altering the flow. The relative 66 saturation (R) is a function of water content which offers potential energy to the movement of soil water. The po- tential energy is produced by the difference between the present water content (9) and some minimum water content which stops water movement. This minimum water content was defined by Irmay as the dead water content (ed). The relative saturation (R) as stated in this study was Irmay's concept which defines R = (9 - Gd)/(eo — 9d). Defined in this manner (R) is physically more meaningful as compared to Brutsaert's concept in which he ignored the stagnant water content in the infiltration equations. The coefficient of ml obtained from his equations was greater than that from this study. Therefore the values computed from Brutsaert's equations would be expected to deviate from the observed values to a greater extent. study. CONCLUSIONS The following conclusions may be drawn from this The maximum movement of the wetting front at a given time can be determined from the saturated- unsaturated boundary by solving equations (78) and (81) under the assumption that R equals Ri’ The maximum vertical movement occurred at the time where the wetting front intersects the predicted saturated-unsaturated boundary. When the initial water content (Si) is less than dead water content (9d) there is practically no soil water flow. The soil water starts moving if the water content in the soil exceeds the dead water content. From these results, it can be concluded that the relative saturation (R) has the form indicated in this study. An advantage 67 68 of this assumption is that there is no restric— tion on the initial water content. As long as the saturated—unsaturated boundary is obtained from the water depth in the furrow, the distribution of water content in the unsaturated region depends simply on the position of the saturated-unsaturated boundary. The equations (43), (78), and (81) derived from this study can be used to estimate the wetting front at a given time. Irmay's formula on the relationship among the water content, capillarity and saturated hy- draulic conductivities has been shown to be valid. The contribution of the gravity term wzt was de- pendent on the parameters KO, 0d, 90, and 91 but independent of AS and K. The experimentally determined capillary conduc- tivities using K = D Cw as derived by Bruce and Klute have given satisfactory results. IO. 69 Equation (78) with R as a variable can be used to find the water content distribution from the saturated region out to the distance of maximum horizontal water movement and the positions of x2 can be found by substituting the corresponding values of the water content in equation (81). Equation (78) with R = Ri can provide information on the maximum distance the wetting front can penetrate to reach the root zone of a plant so that the duration of irrigation can be determined if the depths of water in the irrigation furrow are chosen. goal: RECOMMENDATIONS FOR FUTURE STUDIES Future studies in this area should have as their The influence of various shapes of furrow cross sections on the soil water penetration, Improvements in the techniques of measuring the parameters AS, 8. 90. 9., 0 cially the development of an accurate method to determine the dead water content (ed). 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Gostekhizdat, Moscow. kl .«uiifiuitb.zg.f«muz L... 1th .r