Comparison of Finite Difference Schemes for Water Flow in Unsaturated Soils

Flow movement in unsaturated soil can be expressed by a partial differential equation, named Richards equation. The objective of this study is the finding of an appropriate implicit numerical solution for head based Richards equation. Some of the well known finite difference schemes (fully implicit, Crank Nicolson and Runge-Kutta) have been utilized in this study. In addition, the effects of different approximations of moisture capacity function, convergence criteria and time stepping methods were evaluated. Two different infiltration problems were solved to investigate the performance of different schemes. These problems include of vertical water flow in a wet and very dry soils. The numerical solutions of two problems were compared using four evaluation criteria and the results of comparisons showed that fully implicit scheme is better than the other schemes. In addition, utilizing of standard chord slope method for approximation of moisture capacity function, automatic time stepping method and difference between two successive iterations as convergence criterion in the fully implicit scheme can lead to better and more reliable results for simulation of fluid movement in different unsaturated soils.

Authors:

• H. Taheri Shahraiyni
• B. Ataie Ashtiani

Keywords:

• Finite Difference methods
• Richards equation
• fullyimplicit
• Crank-Nicolson
• Runge-Kutta.

References:

[1] R. Haverkamp, M. Vauclin, J. Touma, P. J. Wierenga, and G. Vachaud,
"A comparison of numerical simulation models for one-dimensional
infiltration," Soil Soc. Am. J., vol. 41, pp. 285-293, 1977.
[2] M. A. Celia, B. T. Bouloutas, and R. L. Zarba, "A general mass
conservative solution for the unsaturated flow equation," Water Resour.
Res., vol. 26, pp. 1483-1496, 1990.
[3] K. Rathfelder, and L. M. Abriola, "Mass conservative numerical
solutions of the head based Richards equation," Water Resour. Res., vol.
30, pp. 2579-2586, 1994.
[4] P. S. Huyakorn, and T. D. Wadsworth, "FLAMINCO: a three
dimensional finite element code for analyzing water flow and solute
transport in saturated-unsaturated porous media," Geo. Trans. Inc.,
Herndon,VA, 1985.
[5] K. Huang, B. P. Mohanty, and M. T. Van Genuchten, "A new
convergence criterion for the modified Picard iteration method to solve
the variably saturated flow equation," Journal of Hydrology, vol. 78, pp.
69-91, 1996.
[6] S. O. Magnuson, R. G. Baca, and A. J. Sondrup, "Independent
verification and benchmark testing of the PORFLO-3 computer code,"
version 1.0, Idaho Natl. Lab., Idaho, Rep. EGG-BG-9175, 1990.
[7] A. I. El-Kadi, and G. Ling, "The Curant and Peclet number criteria for
the numerical solution of Richards equation," Water Resour. Res., vol.
29, pp. 3485-3494, 1994.
[8] M. Th. Van Genuchten, "A closed form equation for predicting the
hydraulic conductivity of saturated soils," Soil Soc. Am. J., vol. 44, pp.
892-898, 1980.