Drain-spacing formula for transient-state flow with ellipse as an initial condition JACEK UZIAK1 and SIETAN CHIENG2 xDepartment of Machine Theory and Automatics, Academy of Agriculture, Lublin, Poland; and 2Bio-Resource Engineering Department, University ofBritish Columbia, Vancouver, BCf Canada V6T1W5. Received1 June 1988, accepted 12 December1988. Uziak, J. and Chieng, S. 1989. Drain-spacing formula for transient state flow with ellipse as an initial condition. Can. Agric. Eng. 31: 101-105. A new solution of the differential equation describing the drainage problem was presented. An initial condition in the form of an ellipse, approximated by a negativeexponentialfunction, was used. From this solution, a new formula was proposed for drain-spacing cal culations. It was found that the proposed formula is more general for drain-spacing calculations. Two well-known drain-spacing formulae, Glover's equation (Van Schilfgaarde 1974), and Glover-Dumm's (Dumm 1960, 1964) equation, are two special cases of the proposed Y= 0 (for x = 0 and X = L) (2) where L = drain spacing. The initial condition is much more complicated and three types of initial water table shapes, as shown in Fig. 2 (Dass and MorelSeytoux 1974), are usually assumed: (a) A constant water table height, Y0, exists everywhere between the two adjacent drains except above the drains where the water table abruptly drops to zero (to drain level). formula. (for 0 < X < L where t = 0) (3) INTRODUCTION Within the pastthreedecades, a considerable amountof research on drainage problems has been done. So far, drainageproblems have been divided into steady-stateand transient-state flow con ditions. A steadystate exists when the boundaries and flow rates of a system do not changewithtime. Otherwise, a transient state exists. As steady state seldom exists under actual field condi tions, solutions using the transient-state condition should be adopted. However, owingto their simplicity, steady-state equa tions basedon elliptical initialconditions have been widely used in subsurface drainage design. Currently, there is no transientstate drain-spacing equation basedon elliptical initial conditions. Drain-spacings calculated from steady and nonsteady stateequa tions cannot be meaningfully compared. This study addresses thisneedby using the ellipse-initial conditionin transient-state flow and developing a new drain-spacing equation. (b) The water table shape is a fourth degree parabola with the following form: Y= -^ (L3X - 3L2 X2 + 4LX3 -2X4) (for t = 0) (4) L4 (c) The water table is a parabola with the inflexionpoint given by the expression: v 16*o X1 (L-X)2 (for t = 0) (5) The initial water table shape with a fourth degree parabola expressed by Eq. 4 is most widely used. Using this initial con ditionand the boundarycondition in the form of Eq. 2, the solu tion of Eq. 1 in the following form is obtained (Luthin 1978): Y = 192r0 v nV-8 n2ir2at exp n=l,3,5, TRANSIENT FLOW rnrX Transient or nonsteady-state conditionsoccur when the ground (6) sin water table fluctuates with time; therefore, the hydraulic head is changing constantly. The linearized differential equation for transient flow, as derived on the basis of Dupuit-Forchheimer assumptions, can be expressed as (Luthin 1978): Atmid-spacing, X=L/2, the water table height canbe expressed as: 192 £ Ym = dy (1) (-D az2-8/tt2 « = 1,3,5,... dx2 dt ir2n2at where: a = K.D/f K = soil-saturated hydraulic conductivity, D d Y0 / = = = = average depth of flow region = d + Y0/2, depth to impermeable layer below drain, watertable heightat mid-spacing as shown in Fig. 1, drainable porosity or specific yield, t = time, and Y = water table height above the datum. The solution of Eq. 1 depends on the initial and boundary con ditions. The boundary conditions are relatively simple and are generally set up as follows: CANADIAN AGRICULTURAL ENGINEERING (7) exp ELLIPSE AS AN INITIAL CONDITION In the steady-state condition it is assumed thatthe recharge and drainage rates are equal, and the hydraulic head does not vary with time. This situation is described by Hooghoudt (Luthin 1978) as an ellipse equation with a semi-major axis of d2 + 8L2 AK and a semi-minor axis of fd2 + RL AK 101 nonsteady-state condition when the rechargeand drainage rates JL GROUND are not balanced. Hooghoudt's equation and other steady-state SURFACE equations have been used widely in subsurface drainage design. It therefore seems logical to assume the ellipse as an initial con WATER TABLE y when t » 0 y when t > 0 DRAIN IMPERMEABLE dition for transient-state conditions. This enables the compar ison of drain-spacings calculated from two different approaches (i.e., steady and nonsteady) based on the same initial condition (i.e., ellipse). The equation of this ellipse-condition can be expressed as: (X-L/2)2 + L2(Y0+d)2 4[(Y0+d)2-d2] LAYER (X+d)2 (Y0 + d)2 = (8) 1 Figure 1. Definition sketch for drainage problem. Unfortunately, because of the mathematical complications, it is extremely difficult to obtain the analytical solution of Eq. 1 in the normal manner using an ellipse as the initial condition. For this reason the authors approximate the ellipse by the wellestablished negative-exponential function as: Y k e>o T I I I -^X DRAIN I - (for 0 < X < LI2) Y = A{\ - exp (-B(L-X)) I DRAIN Y = A{\ - exp (-BX)) 1 (for LI2 < X < L) (9a) (9b) From Eqs. 8, 9a and 9b the following expressions are obtained: L • (a) Y0 (Y0 + d)2 1 — exp I DRAIN ^b 1 B = (10) 2(Y0 + d)2 (11) d2-L Figure 3 compares the shapes of ellipses (Eq. 8) andnegativeexponential functions (Eqs. 10 and 11). The agreement between the curves depends on the value of d/Y0. Comparisons were done for d/Y0 = 0 to d/Y0 = oo. The best agreement is found for d/Y0 = 2. Although the ratio of d/Y0 could vary from 0 to infinity in theory, a range of 0-5 is more frequently encoun tered in practice. As d/Y0 = 2 is about half-way between 0 and 5, it is felt that the proposed negative-functions can be used to adequately approximate the ellipse. Substituting equations 10and 11 into Eqs. 9a and9b, yields: T Y0 Y = Y0 + d 1 — exp d DRAIN 1 —exp ( — 2 (Y0 + d)2X Ld2 (12a) (for 0 < X < LI2) Figure 2. Three initial water table conditions studied by Dass and Morel-Seytous. Y0 + d 1 — exp 2 (Y0 + d)A (L-Xj where R is the drainage coefficient or recharge rate. These are shown in Fig. 1. The steady-state condition becomes a 102 1 —exp ( — Ld2 (for LI2 < X < L) (12b) UZIAK AND CHIENG SOLUTION OF TRANSIENT STATE EQUATION A solution of the transient state equation (Eq. 1) which satis fies the boundary conditions (Eq. 2) and proposed ellipse ini i+2o tial condition (Eqs. 12a and 12b), is obtained as follows: 4F0 y = exp - 1+^0\2 d exp air2n2t - (14) =1,3,5, l-exp where Ym is the water table height above drains at mid-spacing, -1 + at time t. Figure 4 shows the relationship between the dimensionless parameters Y/Y0 and KDt/fL2 for different ratios of d/Y0. As rnr - (-1) nic a comparison, the same relationship for the U.S. Bureau of Reclamation's equation (Eq. 6) is also included. It can be seen from Fig. 4 that when d/Y0 increases the time required for „2_2 4 1 + ^0\4 4 + nV 2 1 + Y0 exp - i+ water table height to drop from Y0 to Y decreases. When the drains are placed on the impermeable layer (d=0), the value of d/Y0 is zero and Eq. 13 becomes: 2oV d av2n2t ft Equations 12a and 12b which have been used as initial con exp - (13) sin ditions for Eq. 13 are an approximation of an ellipse given by Eq. 8. It can be seen in Fig. 3 that the agreement between the original ellipse's function and the proposed approximation appears to be very good. Therefore, Eq. 13 can be treated as a solution of Eq. 1 with an ellipse as the initial water table shape. Since the main interest is in the height of the water table at the midpoint between the drains, we can obtain the following expression for y, at x = L/2: n-\ 00 41o Ym = E ■♦?)' 1—exp (-D =1,3,5,, nir - (-1) nir 4 1 + + nV Y = onr2n2t 47n E n = l,3,5,. Texp(n nirX (15) and the water table height at mid-spacing can be obtained as: n-\ Ym = 4Y0 °° 1 D (-1) 2 7exp(- air2n2t (16) n=l,3,5,... Equations 15 and 16 are the solutions of Eq. 1 with an initial condition in the form of Eq. 3 discussed above (Carslaw and Jaeger 1959). In fact, when d=0 is substituted into Eq. 12, we can obtain Y= Y0 for all values of X, except for X=0 and X—L. This result is exactly the same condition as described by Eq. 3. It is interesting to find that the difference between the pro posed solution (Eq. 13) and the USBR's equation is less than 2%. 0.8- 0.6- 0.4- 0.2- Figure 3. Shapes of ellipse and of negative-exponential functions for different d/Y0 values. CANADIAN AGRICULTURAL ENGINEERING 103 NEF WHEN d/yQ - 0 GLOVER-DUMM (d/j^= 0.9166) NEF WHEN d/yQ - 0 0.8- NEF WHEN d/yQ • 5 NEF WHEN d/y0 • °° 0.6- i 0.4- 0.2- 0.0001 KDVflf Figure 4. Relationship between Y/Y0 and KDt/fL for different d/Y0 values. NEW DRAIN-SPACING FORMULA It has been found that the sum of the second and remaining terms in the series, in Eq. 14, is very small and can be neglected. This gives the approximate solution of Eq. 14, by taking only the first term of the series, which yields: 1974). It was interesting to find that when the ratio of d/Y0 = 0.9100, is used in Eq. 18, the following equation is obtained: 1 4Fn ym = It should be mentioned that Eq. 19 is exactly the same as the well-known Glover's drain-spacing equation (Van Schilfgaarde l-exp[-(l + y0/J)2] l* 2 4(l+ Yo/d)4 +r2 (XT t In (1.15 x-2 (Y+Yo/d)2 exp (~(\-YQld)2 exp (-air2t/L2) (17) From Eq. 17, a new formula for drain-spacing calculation is developed as follows: air t •2 _ (18) 1%. CONCLUSION xym(4co2 + 7r2)[l-exp(-o3)] where a = (1 + Y0/d)2. In the case of d = 0 (i.e., co approaches infinity) Eq. 18 becomes: L2 = In (4Y0/irYJ In (1.27Y0/Ym) (19) When d approaches infinity, the value of co approached unity and Eq. 18 becomes: r2 8r0[2 + xexp (-1)] In 7rrm(4 + 7r2)(l-exp(-l)) air t In (0.91 IVI'm) 104 A new solution of the differential equation (Eq. 13) describing the transient-state drainage problem was presented. The ellipse initial condition was used and was approximated by a negative exponential function in the form of Eq. 12. Good agreement between the negative exponential function and the true ellipse was found. Owing to this good agreement, the solution obtained with this negative exponential function as an initial condition could be considered as the solution with a true ellipse as the initial condition. cnr2t _ -) Equation 21 is almost the same as the Glover-Dumm formula reported by Dumm (1960). The only difference between Eq. 21 and Glover-Dumm's equation is the "constant" term in the equation (i.e., 1.15 in Eq. 21 and 1.16 in Glover-Dumm). The difference between these two equations is found to be less than 8K0w[2o3 + ir exp ( —«)] In (21) L2 = (20) It was found that when the ellipse was used as the initial con dition, its shape was influenced by the ratio of the semi-major and semi-minor axes. For this reason, the shape of the initial water table will depend not only on water table height at midspacing but also on the distance to the impermeable layer below drains. These features are also represented by the negative exponential function. UZIAK AND CHIENG The new drain-spacing formula (Eq. 18) proposed in this paper can beconsidered as ageneral formula for drain-spacing calculations. It was found that two well-known drain-spacing formulae, Glover's equation (Van Schilfgaarde 1974) and Glover-Dumm's (Dumm 1960, 1964) equation, are covered by this new formula. They can be considered as the solution of two special cases of the proposed formula as given in Eqs. 19 REFERENCES CARSLAW, H. S. and J. C. JAEGER. 1959. Conduction of heatin solids. 2nd ed. Oxford University Press, London, U.K. 510 pp. DASS, P. and H. J. MOREL-SEYTOUX, 1974. Sub surface drainage solutions by Galerkin's Method. J. Irr. Drain. and 21, respectively. Div., ASCE 100 (IR1):1-15. Currently, a nonsteady-state drain-spacing equation based on elliptical initial conditions is not available. Drain spacings calculated from steady and nonsteady-state equations cannot be meaningfully compared. This need was successfully addressed in this study by using the ellipse-initial condition in nonsteady-state flow and developing a new drain-spacing DUMM, L. D. 1960. Validity and use of the transient flow equation. CANADIAN AGRICULTURAL ENGINEERING concept in subsurface drainage. Paper No. 60-717 presented at ASAE meeting, Memphis, TN. 4-7 Dec. 1960. DUMM, L. D. 1964. Transient flow concept in subsurface drainage: its validity and use. Trans. ASAE (Am. Soc. Agric. Engrs.) 7(2):142-145, 151. LUTHIN, N. L. 1978. Drainage engineering. Robert E. Krieger Publishing Co., Huntington, NY 281 pp. VAN SCHILFGAARDE, J. 1974. Non-steady flow to drains. In Drainage for agriculture. Agronomy No. 17. Am. Soc. Agron. 677 South Segoe Road, Madison, WI. pp. 245-265. 105

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