IJE TRANSACTIONS A: Basics Vol. 28, No. 4 (April 2015) 490-498   

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M. Sayful Islam
( Received: November 08, 2014 – Accepted: March 13, 2015 )

Abstract    The nonlinear solvers in numerical solution of water flow in variably saturated soils are prone to convergence difficulties. Many aspects can give rise to such difficulties, like very dry initial conditions, a steep pressure gradient and great variation of hydraulic conductivity occur across the wetting front during the infiltration of water. So, the averaging method applied to compute hydraulic conductivity between two adjacent nodes of the computational grid is one of the most important issues influencing the accuracy of the numerical solution of one-dimensional unsaturated flow equation i.e., Richards’ equation. A number of averaging schemes such as arithmetic, geometric, harmonic and arithmetic mean saturation have been proposed in the literature for homogeneous soil. The resulting numerical schemes are evaluated in terms of accuracy and computational time. It can be seen that the averaging scheme in the framework of arithmetic approach favorably to other methods for a range of test cases.


Keywords    Richards’ equation, Variably saturated flow, Internodal conductivity, Infiltration. Finite difference.


چکیده    روشهای حل غیر خطی برای حل عددی جریان آب در خاکهای اشباع شده احتمالا با مشکلات متعددی همراه است. عوامل مختلفی می توانند باعث افزایش این مشکلات شوند که از جمله آنها شرایط اولیه بسیار خشک است که موجب گرادیان فشار با شیب زیاد و تغییرات شدید هدایت هیدرولیکی در قسمت مرطوب در طی نفوذ آب می شود. بنابراین، روش متوسط گیری که برای محاسبه هدایت هیدرولیکی بین دو نقطه مجاور در آرایه محاسباتی استفاده می شود یکی از مسائل مهمی است که روی دقت حل عددی معادلات جریان غیر اشباع تک-بعدی مانند معادله ریچارد تاثیر می گذارد. الگوهای متوسط گیری مختلفی از جمله آریتمیک، ژئومتریک، هارمونیک و اشباع متوسط آریتمیک در متون برای خاکهای همگن ارائه شده است. الگوهای عددی به دست آمده از لحاظ دقت و زمان محاسبه مورد بررسی قرار می گیرند. ملاحظه می شود که الگوی متوسط گیری در چارچوب روش اریتمیک برای تعدادی از موارد بررسی شده به طور مطلوبی به روشهای دیگر نزدیک است.



1.        Baker, D. Darcian weighted interblock conductivity means Brooks, R. H. and Corey, A. T., ‘’Properties of porous media affecting fluid flow’’, Journal of Irrigation and Drainage Engineering Vol. 92(IR2), (1966), 61-88.

2.        van Genuchten, M. T., ‘’A closed-form equation for predicting the hydraulic conductivity of unsaturated soils’’,  Soil Science Society of America Journal, Vol. 44, (1991), 892-898.

3.        Hills, R. G., Hudson, I. Porro, D. and Wierenga, P., ‘’Modeling of one dimensional infiltration into very dry soils: 1. Model development and evaluation’’, Water Resources Research, Vol. 25, (1989), 1259–1269.

4.        Celia, M.A.,  Bouloutas, E.T. and Zarba, R.L., ‘’A general mass conservative numerical solution for unsaturated flow equation’’, Water Resources Research, Vol. 26, (1990), 1483–1496.

5.     Huang, K., Mohanty, B.P., Leij, F.J., and  van Genuchten, M.T., ‘’Solution of the nonlinear transport equation using modified Picard iteration’’, Advances in Water Resources, Vol. 21, (1998), 237–249.

6.     Brandt, A., Bresler, E., ‘’Diner, N., Ben-Asher, L., Heller, J. and Goldman, D.: Infiltration from a trickle source: 1. Mathematical models’’Soil Science Society of America Journal, Vol. 35, (1971), 675-689.

7.     Day, P. R.  and Luthin, J. N., ‘’A numerical solution of the differential equation of flow for a vertical drainage problem’’, Soil Science Society of America Journal, Vol. 20, (1956), 443-446.

8.     Freeze, R. A., ‘’The mechanism of natural groundwater recharges and discharges 1. One-dimensional, vertical, unsteady, unsaturated flow above a recharging and discharging groundwater flow system’’, Water Resources Research, Vol. 5, (1969), 153-171.

9.     Samani, H. M. V. and Kolahdoozan, M., ‘’Mathematical model of unsteady groundwater flow through aquifers and calibration via a nonlinear optimal technique’’, International Journal of Enginnering, Vol. 11(1), (1998), 1-13.

10.   Haverkamp, R., Vauclin, M., Touma, J. and Wierenga, P. J., Vachaud, G., ‘’A Comparison of numerical simulation models for one-dimensional infiltration’’, Soil Science Society of America Journal, Vol. 41, (1977), 285-295.

11. Cooley, R.L., ‘’A finite difference method for unsteady flow in variably saturated porous media: application to a single pumping well’’, Water Resources Research, Vol. 7, (1971), 1607-1625.

12.   Kirkland, M. R., Hills, R. G. and Wierenga, P. J., ‘’Algorithms for solving Richards’ equation for variably saturated soils’’, Water Resources Research, Vol. 28, (1992), 2049-2058.

13.   Rubin, J., ‘’Theoretical analysis of two-dimensional, transient flow of water in unsaturated and saturated soils’’, Soil Science Society of America Journal, Vol. 32, (1984), 607-615.

14.   Freeze, R.A., ‘’Three dimensional transient, saturated-unsaturated flow in a groundwater basin’’, Water Resources Research, Vol. 5, (1971a), 153-171.

15.   Freeze, R.A., ‘’Influence of the unsaturated flow domain on seepage through earth dams’’, Water Resources Research, Vol. 7, (1971b), 347-366.

16.   Brunone, B., Ferrante, M., Romano, N. and Santini, A., ‘’Numerical simulations of one-dimensional infiltration into layered soils with the Richards’ equation using different estimates of the interlayer conductivity’’, Vadose Zone Journal, Vol. 2, (2003), 193–200.

17.   Lima-Vivancos, V. and Voller, V., ‘’Two numerical methods for modeling variably saturated flow in layered media’’, Vadose Zone Journal, Vol. 3, (2004), 1031–1037.

18.   Simunek, J., Sejna, M., Saito, H., Sakai, M. and van Genuchten, M., ‘’The HYDRUS-1D software pack-age for simulating the one-dimensional movement of water, heat and multiple solutes in variably-saturated media’’, Version 4.0. Department of Environmental Sciences, University of California Riverside, Riverside, California, (2008).

19.   van Dam, J. and Feddes, R., ‘’Numerical simulation of infiltration, evaporation and shallow groundwater levels with the Richards equation’’, Journal of Hydrology, Vol. 233, (2000), 72–85.

20.   Belfort, B. and Lehmann, F., ‘’Comparison of equivalent conductivities for numerical simulation of one-dimensional unsaturated flow’’, Vadose Zone Journal, Vol. 4, (2005), 1191–1200.

21.   Li, H., Farthing, M. and Miller, C., ‘’Adaptive local discontinuous Galerkin approximation to Richards’ equation’’, Advances in Water Resources, Vol. 30, (2007), 1883–1901.

22.   Baker, D., ‘’Arnold, M. and Scott, H.: Some analytic and approximate Darcian means’’, Ground Water, Vol. 37, (1999), 532–538.

23.   Gast´o, J., Grifoll, J. and Cohen, Y., ‘’ Estimation of internodal permeabilities for numerical simulations of unsaturated flows’’, Water Resources Research, Vol. 38, (2002), 1326.

24.   Haverkamp, R. and Vauclin, M., ‘’A note on estimating finite difference interblock hydraulic conductivity values for transient unsaturated flow’’, Water Resources Research, Vol. 15, (1979), 181–187.

25.   Miller, C., Williams, G., Kelley, C. and Tocci, M., ‘’Robust solution of Richards’ equation for nonuniform porous media’’, Water Resources Research, Vol. 34, (1998), 2599–2610.

26.   Schnabel, R. and Richie, E., ‘’Calculation of internodal conductances for unsaturated flow simulations’’, Soil Science Society of America Journal, Vol. 48, (1984), 1006–1010.

27.   Srivastava, R. and Guzman-Guzman, A., ‘’Analysis of hydraulic conductivity averaging schemes for one-dimensional, steady-state unsaturated flow’’, Ground Water, Vol. 33, (1995), 946–952.

28. Szymkiewicz, A., ‘’Approximation of internodal conductivities in numerical simulation of 1D infiltration, drainage and capillary rise in unsaturated soils’’, Water Resources Research, Vol. 45, (2009), W10403.

29.   Zaidel, J.  and Russo, D., ‘’Estimation of finite difference interblock conductivities for simulation of infiltration into initially dry soils’’, Water Resources Research, Vol. 28, (1992), 2285–2295.

30.   Mansell, R., Ma, L., Ahuja, L. and Bloom, S., ‘’Adaptive grid refinement in numerical models for water flow and chemical transport in soil’’, Vadose Zone Journal, Vol. 1, (2002), 222–238.

31.   Miller, C., Abhishek, C. and Farthing, M., ‘’A spatially and temporally adaptive solution of Richards’ equation’’, Advances in Water Resources, Vol. 29, (2006), 525–545.

32.   Ross, P., ‘’Efficient numerical methods for infiltration using Richards’ equation’’, Water Resources Research, Vol. 26, (1990), 279-290.

33.   Williams, G., Miller, C. and Kelley, C., ‘’Transformation approaches for simulating flow in variably saturated porous media’’, Water Resources Research, Vol. 36, (2000), 923–934.

34.   Warrick, A. W., ‘’Numerical approximations of Darcian flow through unsaturated soil’’, Water Resources Research, Vol. 27(6), (1991), 1215-1222.

35.   Baker, D., ‘’Darcian weighted interblock conductivity means for vertical unsaturated flow’’, Ground Water, Vol. 33, (1995), 385–390.

36.   Baker, D., ‘’A Darcian integral approximation to interblock hydraulic conductivity means in vertical infiltration’’, Computers and Geosciences, Vol. 26, (2000), 581–590.

37.   Toufig, M. M., ‘’Seepage with nonlinear permeability by least FEM’’, International Journal of Engineering, Vol. 15, (2002), 125-134.

38.   Islam, M. S. and Hasan, M. K., ‘’Accurate and economical solution of Richards' equation by the method of lines and comparison of the computational performance of ODE solvers’’, International Journal of Mathematics and Computer Research, Vol. 2(2), (2013), 328-346.

39.   Mualem, Y., ‘’A new model for predicting the hydraulic conductivity of unsaturated porous media’’, Water Resources Research, Vol. 12, (1976), 513-522.

40.   Tocci, M. D., Kelley, C. T. and Miller, C. T., ‘’Accurate and economical solution of the pressure-head form of Richards’ equation by the method of lines’’, Advances in Water Resources, Vol. 20(1), (1997), 1-14.

41.   D’Haese, C. M. F., Putti, M. ,Paniconi, C. and Verhoest, N. E. C., ‘’Assessment of adaptive and heuristic time stepping for variably saturated flow’’, International Journal for Numerical Methods in Fluids, Vol. 53, (2007), 1173–1193.

42.   Kavetski, D., ‘’Binning, P. and Sloan, S. W. Adaptive backward Euler time stepping with truncation error control for numerical modelling of unsaturated fluid flow’’, International Journal for Numerical Methods in Engineering, Vol. 53, (2001a), 1301-1322.

43.   Kavetski, D., Binning, P. and Sloan, S. W., ‘’Adaptive time stepping and error control in a mass conservative numerical solution of the mixed form of Richards’ equation’’, Advances in Water Resources, Vol. 24, (2001b), 595-605.  

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