Open Access
Issue
ESAIM: M2AN
Volume 55, Number 5, September-October 2021
Page(s) 2101 - 2139
DOI https://doi.org/10.1051/m2an/2021047
Published online 13 October 2021
  1. A. Ait Hammou Oulhaj, C. Cancès and C. Chainais-Hillairet, Numerical analysis of a nonlinearly stable and positive control volume finite element scheme for Richards equation with anisotropy. ESAIM: M2AN 52 (2018) 1532–1567. [Google Scholar]
  2. B. Andreianov, C. Cancès and A. Moussa, A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs. J. Funct. Anal. 273 (2017) 3633–3670. [Google Scholar]
  3. T. Arbogast, M. Obeyesekere and M.F. Wheeler, Numerical methods for the simulation of flow in root-soil systems. SIAM J. Numer. Anal. 30 (1993) 1677–1702. [Google Scholar]
  4. T. Arbogast, M.F. Wheeler and N.-Y. Zhang, A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media. SIAM J. Numer. Anal. 33 (1996) 1669–1687. [Google Scholar]
  5. S. Bassetto, C. Cancès, G. Enchéry and Q.H. Tran, Robust Newton solver based on variable switch for a finite volume discretization of Richards equation, In: Finite Volumes for Complex Applications IX – Methods, Theoretical Aspects, Examples edited by Examples R. Klöfkorn, E. Keilegavlen, F.A. Radu and J. Fuhrmann Vol. 323 of Springer Proceedings in Mathematics & Statistics (2020) 385–394. [Google Scholar]
  6. S. Bassetto, C. Cancès, G. Enchéry and Q.H. Tran, On several numerical strategies to solve Richards’ equation in heterogeneous media with Finite Volumes. Working paper or preprint (2021) https://hal.archives-ouvertes.fr/hal-03259026. [Google Scholar]
  7. K. Brenner and C. Cancès, Improving Newton’s method performance by parametrization: the case of the Richards equation. SIAM J. Numer. Anal. 55 (2017) 1760–1785. [Google Scholar]
  8. K. Brenner, C. Cancès and D. Hilhorst, Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure. Comput. Geosci. 17 (2013) 573–597. [Google Scholar]
  9. K. Brenner, M. Groza, L. Jeannin, R. Masson and J. Pellerin, Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media. Comput. Geosci. 21 (2017) 1075–1094. [Google Scholar]
  10. K. Brenner, R. Masson, E.H. Quenjel and J. Droniou, Total velocity-based finite volume discretization of two-phase Darcy flow in highly heterogeneous media with discontinuous capillary pressure. IMA Journal of Numerical Analysis (2021) https://doi.org/10.1093/imanum/drab018. [Google Scholar]
  11. K. Brenner, R. Masson and E.H. Quenjel, Vertex approximate gradient discretization preserving positivity for two-phase Darcy flows in heterogeneous porous media. J. Comput. Phys. 409 (2020) 109357. [Google Scholar]
  12. R.H. Brooks and A.T. Corey, Hydraulic properties of porous media. Hydrol. Paper 7 (1964) 26–28. [Google Scholar]
  13. C. Cancès, Nonlinear parabolic equations with spatial discontinuities. Nonlinear Diff. Equ. Appl. 15 (2008) 427–456. [Google Scholar]
  14. C. Cancès, Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities. ESAIM: M2AN 43 (2009) 973–1001. [Google Scholar]
  15. C. Cancès and C. Guichard, Convergence of a nonlinear entropy diminishing control volume finite element scheme for solving anisotropic degenerate parabolic equations. Math. Comput. 85 (2016) 549–580. [Google Scholar]
  16. C. Cancès, F. Nabet and M. Vohralk, Convergence and a posteriori error analysis for energy-stable finite element approximations of degenerate parabolic equations. Math. Comput. 90 (2021) 517–563. [Google Scholar]
  17. 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. [Google Scholar]
  18. C. Chainais-Hillairet, J.-G. Liu and Y.-J. Peng, Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis. ESAIM: M2AN 37 (2003) 319–338. [EDP Sciences] [Google Scholar]
  19. G. Chavent and J. Jaffré, Mathematical models and finite elements for reservoir simulation: single phase, multiphase and multicomponent flows through porous media. In: Vol. 17 of Studies in Mathematics and its Applications. North-Holland, Amsterdam (1986). [Google Scholar]
  20. Z. Chen and R.E. Ewing, Fully discrete finite element analysis of multiphase flow in groundwater hydrology. SIAM J. Numer. Anal. 34 (1997) 2228–2253. [Google Scholar]
  21. Z. Chen and R.E. Ewing, Degenerate two-phase incompressible flow. III. Sharp error estimates. Numer. Math. 90 (2001) 215–240. [Google Scholar]
  22. K. Deimling, Nonlinear Functional Analysis. Springer-Verlag, Berlin (1985). [Google Scholar]
  23. H.-J.G. Diersch and P. Perrochet, On the primary variable switching technique for simulating unsaturated–saturated flows. Adv. Water Resour. 23 (1999) 271–301. [Google Scholar]
  24. J. Droniou and R. Eymard, The asymmetric gradient discretisation method, edited by C. Cancès and P. Omnes. In: Finite Volumes for Complex Applications VIII – Methods and Theoretical Aspects. Vol. 199 of Springer Proc. Math. Stat. Springer, Cham (2017) 311–319. [Google Scholar]
  25. G. Enchéry, R. Eymard and A. Michel, Numerical approximation of a two-phase flow in a porous medium with discontinuous capillary forces. SIAM J. Numer. Anal. 43 (2006) 2402–2422. [Google Scholar]
  26. B.G. Ersland, M.S. Espedal and R. Nybø, Numerical methods for flow in a porous medium with internal boundaries. Comput. Geosci. 2 (1998) 217–240. [Google Scholar]
  27. R. Eymard and T. Gallouët, #-convergence and numerical schemes for elliptic problems. SIAM J. Numer. Anal. 41 (2003) 539–562. [Google Scholar]
  28. R. Eymard, M. Gutnic and D. Hilhorst, The finite volume method for Richards equation. Comput. Geosci. 3 (1999) 259–294. [Google Scholar]
  29. R. Eymard, T. Gallouët, R. Herbin, Finite volume methods, In: Techniques of Scientific Computing (Part 3) edited by P.G. Ciarlet and J.-L. Lions. Vol. VII of Handbook of Numerical Analysis. North-Holland, Elsevier, Amsterdam (2000) 713–1018. [Google Scholar]
  30. R. Eymard, T. Gallouët, R. Herbin, M. Gutnic and D. Hilhorst, Approximation by the finite volume method of an elliptic-parabolic equation arising in environmental studies. M3AS: Math. Models Meth. Appl. Sci. 11 (2001) 1505–1528. [Google Scholar]
  31. R. Eymard, R. Herbin and A. Michel, Mathematical study of a petroleum-engineering scheme. ESAIM: M2AN 37 (2003) 937–972. [Google Scholar]
  32. R. Eymard, T. Gallouët, C. Guichard, R. Herbin and R. Masson, TP or not TP, that is the question. Comput. Geosci. 18 (2014) 285–296. [Google Scholar]
  33. R. Eymard, C. Guichard, R. Herbin and R. Masson, Gradient schemes for two-phase flow in heterogeneous porous media and Richards equation. Z. Angew. Math. Mech. 94 (2014) 560–585. [Google Scholar]
  34. 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. [Google Scholar]
  35. K. Gärtner and L. Kamenski, Why do we need Voronoi cells and Delaunay meshes?In: Numerical Geometry, Grid Generation and Scientific Computing edited by V.A. Garanzha, L. Kamenski and H. Si. Lecture Notes in Computational Science and Engineering. Springer International Publishing, Cham (2019) 45–60. [Google Scholar]
  36. V. Girault, B. Riviere and L. Cappanera, A finite element method for degenerate two-phase flow in porous media. Part II: Convergence. Journal of Numerical Mathematics (2021). https://doi.org/10.1515/jnma-2020-0005. [Google Scholar]
  37. H. Hoteit and A. Firoozabadi, Numerical modeling of two-phase flow in heterogeneous permeable media with different capillarity pressures. Adv. Water Resour. 31 (2008) 56–73. [Google Scholar]
  38. M.R. Kirkland, R.G. Hills and P.J. Wierenga, Algorithms for solving Richards equation for variably saturated soils. Water Resour. Res. 28 (1992) 2049–2058. [Google Scholar]
  39. J. Leray and J. Schauder, Topologie et équations fonctionnelles. Ann. Sci. École Norm. Sup. 51 (1934) 45–78. [Google Scholar]
  40. F. List and F.A. Radu, A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20 (2016) 341–353. [Google Scholar]
  41. F. Marinelli and D.S. Dunford, Semianalytical solution to Richards equation for layered porous media. J. Irrig. Drain. Eng. 124 (1998) 290–299. [Google Scholar]
  42. D. McBride, M. Cross, N. Croft, C. Bennett and J. Gebhardt, Computational modelling of variably saturated flow in porous media with complex three-dimensional geometries. Int. J. Numer. Meth. Fluids 50 (2006) 1085–1117. [Google Scholar]
  43. I.S. Pop, F.A. Radu and P. Knabner, Mixed finite elements for the Richards’ equation: linearization procedure. J. Comput. Appl. Math. 168 (2004) 365–373. [Google Scholar]
  44. F.A. Radu and W. Wang, Convergence analysis for a mixed finite element scheme for flow in strictly unsaturated porous media. Nonlin. Anal.: Real World Appl. 15 (2014) 266–275. [Google Scholar]
  45. F.A. Radu, I.S. Pop and P. Knabner, Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards’ equation. SIAM J. Numer. Anal. 42 (2004) 1452–1478. [Google Scholar]
  46. L.A. Richards, Capillary conduction of liquids through porous mediums. Physics 1 (1931) 318–333. [Google Scholar]
  47. M.T. van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Amer. J. 44 (1980) 892–898. [Google Scholar]

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