- Volume 20 Issue 3
DOI QR Code
A MASS LUMPING AND DISTRIBUTING FINITE ELEMENT ALGORITHM FOR MODELING FLOW IN VARIABLY SATURATED POROUS MEDIA
- ISLAM, M.S. (DEPARTMENT OF MATHEMATICS, SHAHJALAL UNIVERSITY OF SCIENCE& TECHNOLOGY)
- Received : 2016.05.31
- Accepted : 2016.09.09
- Published : 2016.09.25
The Richards equation for water movement in unsaturated soil is highly nonlinear partial differential equations which are not solvable analytically unless unrealistic and oversimplifying assumptions are made regarding the attributes, dynamics, and properties of the physical systems. Therefore, conventionally, numerical solutions are the only feasible procedures to model flow in partially saturated porous media. The standard Finite element numerical technique is usually coupled with an Euler time discretizations scheme. Except for the fully explicit forward method, any other Euler time-marching algorithm generates nonlinear algebraic equations which should be solved using iterative procedures such as Newton and Picard iterations. In this study, lumped mass and distributed mass in the frame of Picard and Newton iterative techniques were evaluated to determine the most efficient method to solve the Richards equation with finite element model. The accuracy and computational efficiency of the scheme and of the Picard and Newton models are assessed for three test problems simulating one-dimensional flow processes in unsaturated porous media. Results demonstrated that, the conventional mass distributed finite element method suffers from numerical oscillations at the wetting front, especially for very dry initial conditions. Even though small mesh sizes are applied for all the test problems, it is shown that the traditional mass-distributed scheme can still generate an incorrect response due to the highly nonlinear properties of water flow in unsaturated soil and cause numerical oscillation. On the other hand, non oscillatory solutions are obtained and non-physics solutions for these problems are evaded by using the mass-lumped finite element method.
- H. F.Wang and M. P. Anderson: Introduction to Groundwater Modeling: Finite Difference and Finite Element Methods, Freeman, San Francisco, 1982.
- C. Zheng and G. D. Bennett: Applied Contaminant Transport Modeling: Theory and Practice, Van Nostrand Reinhold, New York, 1995.
- R. L. Cooley: Some new procedures for numerical solution of variably saturated flow problems,Water Resour. Res., 19 (1983), 1271-1285. https://doi.org/10.1029/WR019i005p01271
- S. P. Neuman: Saturated-unsaturated seepage by finite elements, J. Hydraul. Div. ASCE, 99 (1973), 2233-2250.
- P. C. D. Milly: A mass-conservative procedures for time-stepping in models of unsaturated flow, Adv. Water Resources, 8 (1985),32-36. https://doi.org/10.1016/0309-1708(85)90078-8
- M. B. Allen and C. L. Murphy: A finite element collocation method for variably saturated flow in two space dimension, Water Resour. Res, 23 (1986), 1537-1542.
- M. A. Celia, E. T. Bouloutas, and R. L. Zarba: A General mass-conservative numerical solution for the unsaturated flow equation, Water Resour. Res., 26 (1990), 1483-1496. https://doi.org/10.1029/WR026i007p01483
- R. Haverkamp, M. Vauclin, J. Touma, P. Weirenga and G. Vachaud: Comparison of numerical simulation models for one-dimensional infiltration. Soil Sci. Soc. Am. J., 41 (1977), 285-294. https://doi.org/10.2136/sssaj1977.03615995004100020024x
- H. N. Hayhoe: Study of the relative efficiency of finite difference and Galerkin techniques for modeling soilwater transfer, Water Resour. Res.,14(1) (1978), 97-102. https://doi.org/10.1029/WR014i001p00097
- 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. https://doi.org/10.1029/WR020i008p01099
- 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. https://doi.org/10.1029/WR022i013p01790
- M. A. Celia, L. R. Ahuja and G. F. Pinder: Orthogonal collocation and alternating-direction procedures for unsaturated flow problems, Adv. Water Resour., 10 (1987), 178-187. https://doi.org/10.1016/0309-1708(87)90027-3
- E. O. Frind and M. J. Verge: Three-dimensional modeling of groundwater flow systems, Water Resour. Res., 14 (1978), 844-856. https://doi.org/10.1029/WR014i005p00844
- 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. https://doi.org/10.1029/WR025i006p01259
- P. Ross: Efficient numerical methods for infiltration using Richards' equation, Water Resour. Res., 26 (1990), 279-290. https://doi.org/10.1029/WR026i002p00279
- 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 in Water Resources, edited by T. F. Russell, R. E. Ewing, C. A. Brebbia, W. G. Gray, and G. F. Pinder, 521-528, Computational Mechanics Publications, Billerica, Mass, 1992.
- 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. https://doi.org/10.1002/(SICI)1097-0207(19990720)45:8<1025::AID-NME615>3.0.CO;2-G
- 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. https://doi.org/10.1023/A:1006555107450
- P. A. Forsyth, Y. S. Wu, and K. Pruess:Robust numerical methods for saturated-unsaturated flow with dry initial conditions in heterogeneous media, Adv. Water Resour., 18 (1995), 25-38. https://doi.org/10.1016/0309-1708(95)00020-J
- M.R. Kirkland, R.G. Hills, P.J. Wierenga: Algorithms for solving Richards equation for variably saturated soils, Water Resour. Res., 28 (1992) 2049-2058. https://doi.org/10.1029/92WR00802
- C.W. Li: A simplified Newton method with linear finite elements for transient unsaturated flow, Water Resour. Res., 29 (1993) 965-971. https://doi.org/10.1029/92WR02891
- K. Rathfelder and L.M. Abriola: Mass conservative numerical solutions of the head-based Richards equation, Water Resour. Res., 30 (1994), 2579-2586. https://doi.org/10.1029/94WR01302
- 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. https://doi.org/10.1016/S0309-1708(96)00008-5
- 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. https://doi.org/10.2136/sssaj1980.03615995004400050002x
- K. S. Zadeh, S.B. Shah: Mathematical modeling and parameter estimation of axonal cargo transport, J. Comput. Neurosci., 28 (2010), 495-507. https://doi.org/10.1007/s10827-010-0232-9
- K. S. Zadeh: Parameter estimation in flow through partially saturated porous materials, J. Comput. Phys., 227(24) (2008), 10243-10262. https://doi.org/10.1016/j.jcp.2008.09.007
- K. S. Zadeh: A mass-conservative switching algorithm for modeling flow in variably saturated porous media, J. Comput. Phys., 230 (2011), 664-679. https://doi.org/10.1016/j.jcp.2010.10.011
- 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. https://doi.org/10.1029/91WR00334
- H. Matthies and G. Strang: The solution of nonlinear finite element equations, Int. J. Numer. Methods Eng., 14 (1979), 1613-1626. https://doi.org/10.1002/nme.1620141104
- C. T. Miller, C. Abhishek and M. Farthing: A spatially and temporally adaptive solution of Richards' equation, Adv. Water Resources., 29 (2006), 525-545. https://doi.org/10.1016/j.advwatres.2005.06.008
- 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.
- M. S. Islam: Selection of internodal conductivity averaging scheme for unsaturated flow in homogeneous media, International J. of Engineering, 28 (2015), 490-498.
- M. S. Islam: Consequence of Backward Euler and Crank-Nicolson techniques in the finite element model for the numerical solution of variably saturated flow problems, J. KSIAM 19(2) (2015), 197-215.
- 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. https://doi.org/10.1029/98WR01673
- C. E. Kees and C. T.Miller: Higher order time integration methods for two-phase flow, Adv.Water Resources, 25(2) (2002), 159-177. https://doi.org/10.1016/S0309-1708(01)00054-9
- 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) (2014), 106-117. https://doi.org/10.9790/5728-1022106117