Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

BIVARIATE NUMERICAL MODELING OF THE FLOW THROUGH POROUS SOIL

  • S. JELTI (Mohamed first University) ;
  • A. CHARHABIL (Sorbone Paris Nord University) ;
  • A. SERGHINI (Department of Informatic, Mohamed first University) ;
  • A. ELHAJAJI (ENCGJ, Chouaib Doukkali University) ;
  • J. EL GHORDAF (University of Sultan Moulay Slimane)
  • Received : 2022.02.27
  • Accepted : 2022.07.20
  • Published : 2023.03.30
Download PDF
⟨ Previous Next ⟩

Abstract

The Richards' equation attracts the attention of several scientific researchers due to its importance in the hydrogeology field especially porous soil. This work presents a numerical method to solve the two dimensional Richards' equation. The pressure form and the mixed form of Richards' equation are solved numerically using a bivariate diamond finite volumes scheme. Euler explicit scheme is used for the time discretization. Different test cases are done to validate the accuracy and the efficiency of our numerical model and to compare the possible numerical strategies. We started with a first simple test case of Richards' pressure form where the hydraulic capacity and the hydraulic conductivity are taken constant and then a second test case where the hydrodynamics parameters are linear variables. Finally, a third test case where the soil parameters are taken according the Van Gunchten empirical model is presented.

Keywords

Acknowledgement

The authors are grateful to the reviewers for their helpful suggestions which have deeply improved the quality of this paper.

References

  1. H. Alt, S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z. 183 (1983), 311-341. https://doi.org/10.1007/BF01176474
  2. V. Baron, Y. Coudiere, P. Sochala, Adaptive multistep time discretization and linearization based on a posteriori error estimates for the Richards' equation, Appl. Numer. Math. 112 (2017), 104-125. https://doi.org/10.1016/j.apnum.2016.10.005
  3. M. Berardia, M. Vurroa, The numerical solution of Richards' equation by means of method of lines and ensemble Kalman filter, Math. Comput. Simul. 125 (2016), 38-47. https://doi.org/10.1016/j.matcom.2015.08.019
  4. H. Berninger, R. Kornhuber, O. Sander, Fast and robust numerical solution of the Richards' equation in homogeneous soil, SIAM J. Numer. Anal. 49 (2014), 2576-2597. https://doi.org/10.1137/100782887
  5. K. Brenner, C. Cances, Improving Newton's method performance by parametrization: the case of the Richards' equation, SIAM J. Numer. Anal. 55 (2017), 1760-1785. https://doi.org/10.1137/16M1083414
  6. K. Brenner, D. Hilhorst, H. Vu-Do, The generalized finite volume SUSHI scheme for the discretization of Richards' equation, Vietnam J. Math. 44 (2017), 557-586. https://doi.org/10.1007/s10013-015-0170-y
  7. R.H. Brooks and A.T. Corey, Hydraulic Properties of Porous Media, Hydrology Papers 3, Colorado State University, Fort Collins, 1964, pp 27.
  8. M. Celia, E. Bouloutas, General mass-conservative numerical solutions for the unsaturated flow equation, Water Resour. Res. 26 (1990), 1483-1496. https://doi.org/10.1029/WR026i007p01483
  9. C. Chavez-Negrete, D. Santana-Quinteros, and F. Dominguez-Mota, A Solution of Richards' Equation by Generalized Finite Differences for Stationary Flow in a Dam, Mathematics 9 (2021), 1604.
  10. B. Cumming, T. Moroney, I. Turner, A mass conservative control volume finite element method for solving Richards' equation in heterogeneous porous media, Numer. Math. 51 (2011), 845-864. https://doi.org/10.1007/s10543-011-0335-3
  11. B. Deng, J. Wang, Saturated-unsaturated groundwater modeling using 3D Richards' equation with a coordinate transform of nonorthogonal grids, Appl. Math. Model. 50 (2017), 39-52. https://doi.org/10.1016/j.apm.2017.05.021
  12. R. Eymard, M. Gutnic, D. Hilhorst, The finite volume method for Richards' equation, Comput. Geosci. 3 (1999), 259-294. https://doi.org/10.1023/A:1011547513583
  13. L. Fengnan, Y. Fukumoto and X. Zhao, A Linearized Finite Difference Scheme for the Richards Equation Under Variable-Flux Boundary Conditions, J. Sci. Comput. 83 (2020).
  14. P.A. Forsyth, Y.S. WuK, Pruess Robust numerical methods for saturated-unsaturated flow with dry initial conditions in heterogeneous media, Adv. Water. Res. 18 (1995), 25-38. https://doi.org/10.1016/0309-1708(95)00020-J
  15. L. Guarracino, F. Quintana, A third-order accurate time scheme for variably saturated groundwater flow modelling, Communications in Numerical Methods in Engineering 20 (2004), 379-389. https://doi.org/10.1002/cnm.680
  16. J. Kacur, On a solution of degenerate elliptic-parabolic systems in Orlicz-Sobolev spaces I, Math. Z. 203 (1990), 153-171. https://doi.org/10.1007/BF02570728
  17. G. Marinoschi, Functional Approach to Nonlinear Models of Water Flow in Soils, 1st edition, Mathematical Modelling: Theory and Applications, Springer, 2010.
  18. M. Kuraz, P. Mayer, T. Dagmar, An adaptive time discretization of the classical and the dual porosity model of Richards' equation, J. Comput. Appl. Math. 233 (2010), 3167-3177. https://doi.org/10.1016/j.cam.2009.11.056
  19. G. Manzini, S. Ferraris, Mass-conservatie finite volume methods on 2-D unstructured grids for the Richards' equation, Adv. Water Resour. 27 (2004), 1199-1215. https://doi.org/10.1016/j.advwatres.2004.08.008
  20. I.S. Pop, F. Radu, P. Knabner, Mixed finite elements for the Richards' equation: linearization procedure, J. Comput. Appl. Math. 168 (2004), 365-373 https://doi.org/10.1016/j.cam.2003.04.008
  21. M.A. Pour, M.M. Shoshtari and A. Adib, Numerical Solution of Richards Equation by Using of Finite Volume Method, World Applied Sciences Journal 14 (2011), 1838-1842.
  22. L.A. Richards, Capillary conduction of liquids through porous mediums, Physics 1 (1931), 318-333. https://doi.org/10.1063/1.1745010
  23. I. Rees, I. Masters, A.G. Malan, R.W. Lewis, An edge-based finite volume scheme for saturated-unsaturated groundwater flow, Comput. Methods Appl. Mech. Eng. 193 (2004), 4741-4759. https://doi.org/10.1016/j.cma.2004.04.003
  24. M.S. Slam, Sensitivity analysis of unsaturated infiltration flow using head based finite element solution of Richards' equation, Matematika 3 (2017), 131-148.
  25. D. Svyatskiy, K. Lipnikov, Second-order accurate finite volume schemes with the discrete maximum principle for solving Richards' equation on unstructured meshes, Adv. Water Resour. 104 (2017), 114-126.
  26. A. Szymkiewicz, Modelling Water Flow in Unsaturated Porous Media: Accounting for Nonlinear Permeability and Material Heterogeneity, Springer Science & Business Media, Berlin, Germany, 2013.
  27. F. Otto, L1-contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differ. Equ. 131 (1996), 20-38. https://doi.org/10.1006/jdeq.1996.0155
  28. C. Van Duyn, L. Peletier, Nonstationary filtration in partially saturated porous media, Arch. Ration. Mech. Anal. 78 (1982), 173-198. https://doi.org/10.1007/BF00250838
  29. M. 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
  30. F. Yasuhide, L. Fengnan, Z. Xiaopeng, A Finite Difference Scheme for the Richards Equation Under Variable Flux Boundary, 1st International Symposium on Construction Resources for Environmentally Sustainable Technologies, CREST 2020 Fukuoka, Japan, Lecture Notes in Civil Engineering, 2021, 231-245.