• Received : 2015.03.15
  • Accepted : 2015.05.26
  • Published : 2015.06.25


Modeling water flow in variably saturated, porous media is important in many branches of science and engineering. Highly nonlinear relationships between water content and hydraulic conductivity and soil-water pressure result in very steep wetting fronts causing numerical problems. These include poor efficiency when modeling water infiltration into very dry porous media, and numerical oscillation near a steep wetting front. A one-dimensional finite element formulation is developed for the numerical simulation of variably saturated flow systems. First order backward Euler implicit and second order Crank-Nicolson time discretization schemes are adopted as a solution strategy in this formulation based on Picard and Newton iterative techniques. Five examples are used to investigate the numerical performance of two approaches and the different factors are highlighted that can affect their convergence and efficiency. The first test case deals with sharp moisture front that infiltrates into the soil column. It shows the capability of providing a mass-conservative behavior. Saturated conditions are not developed in the second test case. Involving of dry initial condition and steep wetting front are the main numerical complexity of the third test example. Fourth test case is a rapid infiltration of water from the surface, followed by a period of redistribution of the water due to the dynamic boundary condition. The last one-dimensional test case involves flow into a layered soil with variable initial conditions. The numerical results indicate that the Crank-Nicolson scheme is inefficient compared to fully implicit backward Euler scheme for the layered soil problem but offers same accuracy for the other homogeneous soil cases.


  1. R. Jiwari: A hybrid numerical scheme for the numerical solution of the Burgers' equation, Computer Physics Communications, 188 (2015), 59-67.
  2. A. Verma, R. Jiwari and S. Kumar: A numerical scheme based on differential quadrature method for numerical simulation of nonlinear Klein-Gordon equation, International Journal of Numerical Methods for Heat & Fluid Flow, 24(2.6) (2014), 1390-1404.
  3. R. Jiwari, S. Pandit and R. C. Mittal: A Differential Quadrature Algorithm to Solve the Two Dimensional Linear Hyperbolic Telegraph Equation with Diriclet and Neumann Boundary Conditions, Applied Mathematics and Computation, 218 (2012), 7279-7294.
  4. S. Pandit, M. Kumar and S. Tiwari: Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients, Computer Physics Communications, 187(2015), 83-90.
  5. D. Sharma, R. Jiwari and S. Kumar: A comparative study of Modal matrix and finite elements methods for two point boundary value problems, Int. J. of Appl. Math. and Mech., 8(3.4) (2012), 29-45.
  6. D. Sharma, R. Jiwari and S. Kumar:Numerical solutions of two point boundary value problems using Galerkin-Finite element method, Int. J. of Nonlinear Sciences, 13(2.1)(2012), 204-210.
  7. S. Shukla, M. Tamsir and V. K. Srivastava: Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method, Computer Physics Communications, 183 (2012), 600-616.
  8. B. Brunone, M. Ferrante, N. Romano and A. Santini: Numerical simulations of one-dimensional infiltration into layered soils with the Richards' equation using different estimates of the interlayer conductivity, Vadose Zone Journal, 2(2003), 193-200.
  9. M. Celia, E. Bouloutas and R. Zarba: A general mass-conservative numerical solution for the unsaturated flow equation, Water Resour. Res., 26(1990), 1483-1496.
  10. V. Lima-Vivancos and V. Voller: Two numerical methods for modeling variably saturated flow in layered media, Vadose Zone Journal 3(2004), 1031-1037.
  11. J. Simunek, M. Sejna, H. Saito, M. Sakai and M. T. van Genuchten: 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.
  12. J. van Dam and R. Feddes: Numerical simulation of infiltration, evaporation and shallow groundwater levels with the Richards' equation, Journal of Hydrology, 233(2000), 72-85.
  13. B. Belfort and F. Lehmann: Comparison of equivalent conductivities for numerical simulation of onedimensional unsaturated flow, Vadose Zone Journal, 4(2005), 1191-1200.
  14. H. Li, M. Farthing and C. Miller: Adaptive local discontinuous Galerkin approximation to Richards' equation, Adv. Water Resour., 30(2007), 1883-1901.
  15. R. L. Cooley: Some new procedures for numerical solution of variably saturated flow problems, Water Resour. Res., 19(1983), 1271-1285.
  16. E. O. Frind and M. J. Verge: Three-dimensional modeling of groundwater flow systems, Water Resour. Res., 14(2.4)(1978), 844-856.
  17. R. G. Hills, I. Porro, D. B. Hudson and P. J. Wierenga: Modeling of one dimensional infiltration into very dry soils: 1. Model development and evaluation, Water Resour. Res., 25(1989), 1259-1269.
  18. P. S. Huyakorn, S. D. Thomas and B. M. Thompson: Techniques for making finite elements competitive in modeling flow in variably saturated media, Water Resour. Res., 20(1984), 1099-1115.
  19. P. S. Huyakorn, E. P. Springer, V. Guvanasen and T. D. Wadsworth: A three dimensional finite element model for simulating water flow in variably saturated porous media, Water Resour. Res., 22(1986), 1790-1808.
  20. S. P. Neuman: Saturated-unsaturated seepage by finite elements, J. Hydraul. Div. ASCE, 99(HY12)(1973), 2233-2250.
  21. P. Ross: Efficient numerical methods for infiltration using Richards' equation, Water Resour. Res., 26(1990), 279-290.
  22. F. Lehmann and P. H. Ackerer: Comparison of iterative methods for improved solutions of the fluid flow equation in partially saturated porous media, Transport in Porous Media, 31(1998), 275-292.
  23. C. Paniconi and M. Putti: A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems, Water Resour. Res., 30(1994), 3357-3374.
  24. L. Bergamaschi and M. Putti: Mixed finite elements and Newton-type linearizations for the solution for the unsaturated flow equation, Int. J. Numer. Meth. Eng., 45(1999), 1025-1046.<1025::AID-NME615>3.0.CO;2-G
  25. C. Fassino and G. Manzini: Fast-secant algorithms for the non-linear Richards Equation. Communications in Numerical Methods in Engineering, 14(1998), 921-930.<921::AID-CNM198>3.0.CO;2-0
  26. D. Kavetski, P. Binning and S. W. Sloan: Noniterative time stepping schemes with adaptive truncation error control for the solution of Richards' equation, Water Resour. Res., 24(2002), 595-605.
  27. A. A. Aldama and C. Paniconi: An analysis of the convergence of Picard iterations for implicit approximations of Richards' equation, in Proceedings of the IX International Conference on Computational Methods inWater Resources, edited by T. F. Russell, R. E. Ewing, C. A. Brebbia, W. G. Gray, and G. F. Pinder, pp. 521-528, Computational Mechanics Publications, Billedca, Mass., 1992.
  28. W. Brutsaer: A functional iteration technique for solving the Richards' equation applied to two-dimensional infiltration problems, Water Resour. Res., 7(2.5)(1971), 1583-1596.
  29. C. R. Faust: Transport of immiscible fluids within and below the unsaturated zone: A numerical model, Water Resour. Res., 21(2.3)(1985), 587-596.
  30. C. Paniconi, A. A. Aldama and E. F. Wood: Numerical evaluation of iterative and noniterative methods for the solution of the nonlinear Richards equation, Water Resour. Res., 27(1991), 1147-1163.
  31. R. H. Brooks and A. T. Corey: Hydraulic properties of porous media, Hydrology Paper No.3, Civil Engineering, Colorado State University, Fort Collins, CO, 1964.
  32. M. T. van Genuchten: A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils, Soil Sci. Soc. Am. J., 44(1980), 892-898.
  33. C. T. Miller, G. A. Williams, C. T. Kelley, and M. D. Tocci: Robust solution of Richards' equation for non uniform porous media, Water Resour. Res., 34(1998), 2599-2610.
  34. D. Kavetski, P. Binning and S.W. Sloan: Adaptive backward Euler time stepping with truncation error control for numerical modelling of unsaturated fluid flow, Int. J. Numer. Meth. Eng., 53(2001a), 1301-1322.
  35. C. M. F. D'Haese, M. Putti, C. Paniconi and N. E. C. Verhoest: Assessment of adaptive and heuristic time stepping for variably saturated flow, Int. J. Numer. Meth. Fluids, 53(2007), 1173-1193.
  36. V. Casulli and P. Zanolli: A Nested Newton-type algorithm for finite volume methods solving Richards' equation in mixed form, SIAM J. Sci. Comput., 32(2010), 2255-2273.
  37. M. S. Islam and M. K. Hasan (2014): An application of nested Newton-type algorithm for finite difference method solving Richards' equation, IOSR Journal of Mathematics, 10(2014), 20-32.
  38. K. Rathfelder and L. M. Abriola: Mass conservative numerical solutions of the head-based Richards' equation, Water Resour. Res., 30(9)(1994), 2579-2586.
  39. M. D. Tocci, C. T. Kelley, and C. T. Miller: Accurate and economical solution of the pressure-head form of Richards' equation by the method of lines, Adv. Water Resour., 20(1)(1997), 1-14.
  40. C. T. Miller, C. Abhishek and M. Farthing: A spatially and temporally adaptive solution of Richards' equation, Adv. Water Resour., 29(2006), 525-545, 2006.
  41. M. S. Islam and M. K. Hasan: 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, 2(2013), 328-346.
  42. C. E. Kees and C. T. Miller: Higher order time integration methods for two-phase flow, Adv. Water Resour., 25(2.1)(2002), 159-77.
  43. M. S. Islam and M. K. Hasan: An investigation of temporal adaptive solution of Richards' equation for sharp front problems, IOSR Journal of Mathematics, 10(2.1)(2014), 106-117.
  44. D. McBride, M. Cross, N. Croft, C. Bennett and J. Gebhardt: Computational modeling of variably saturated flow in porous media with complex three-dimensional geometries, Int. J. Numer. Meth. Fluids, 50(2006), 1085-1117.
  45. F. Marinelli and D. S. Durnford: Semi analytical solution to Richards' equation for layered porous media, J. Irrigation Drainage Eng., 124(1998), 290-299.