Skip to main content
SpringerLink
Log in

A multidomain discretization of the Richards equation in layered soil

  • ORIGINAL PAPER
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

We consider the Richards equation on a domain that is decomposed into nonoverlapping layers, i.e., the decomposition has no cross points. We assume that the saturation and permeability functions are space-independent on each subdomain. Kirchhoff transformation of each subdomain problem separately then leads to a set of semilinear equations, which can each be solved efficiently using monotone multigrid. The transformed subdomain problems are coupled by nonlinear continuity and flux conditions. This nonlinear coupled problem can be solved using substructuring methods like the Dirichlet–Neumann or Robin iteration. We give several numerical examples showing the discretization error, the solver robustness under variations of the soil parameters, and a hydrological example with four soil layers and surface water.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alt, H.W., DiBenedetto, E.: Nonsteady flow of water and oil through inhomogeneous porous media. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12, 335–392 (1985)

    Google Scholar 

  2. Alt, H.W., Luckhaus, S.: Quasilinear elliptic–parabolic differential equations. Math. Z. 183, 311–341 (1983)

    Article  Google Scholar 

  3. Alt, H.W., Luckhaus, S., Visintin, A.: On nonstationary flow through porous media. Ann. Math. Pura Appl. 136, 303–316 (1984)

    Article  Google Scholar 

  4. Arbogast, T., Wheeler, M., Zhang, N.-Y.: A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media. SIAM J. Numer. Anal. 33, 1669–1687 (1996)

    Article  Google Scholar 

  5. Bastian, P., Ippisch, O., Rezanezhad, F., Vogel, H.J., Roth, K.: Numerical simulation and experimental studies of unsaturated water flow in heterogeneous systems. In: Jäger, W., Rannacher, R., Warnatz, J. (eds.) Reactive flows, diffusion and transport, pp 579–598. Springer (2005)

  6. Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.: A generic interface for adaptive and parallel scientific computing. Part II: implementation and tests in DUNE. Computing 82(2-3), 121–138 (2008)

    Article  Google Scholar 

  7. Bastian, P., Buse, G., Sander, O.: Infrastructure for the coupling of Dune grids. In: Kreiss, G., Lötstedt, P., Målqvist, A., Neytcheva, M. (eds.) Proceedings of ENUMATH 2009, pp 107–114. Springer (2010)

  8. Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications (1988)

  9. Bergamaschi, L., Putti, M.: Mixed finite elements and Newton-type linearizations for the solution of Richards’ equation. Int. J. Numer. Meth. Eng. 45(8), 1025–1046 (1999)

    Article  Google Scholar 

  10. Berninger, H.: Domain decomposition methods for elliptic problems with jumping nonlinearities and application to the Richards equation. PhD thesis, Freie Universität Berlin, 2007

  11. Berninger, H.: Non-overlapping domain decomposition for the Richards equation via superposition operators. In: Bercovier, M., Gander, M.J., Kornhuber, R., Widlund, O. (eds.) Domain decomposition methods in science and engineering XVIII, volume 70 of LNCSE, pp 169–176. Springer (2009)

  12. Berninger, H., Sander, O.: Substructuring of a Signorini-type problem and Robin’s method for the Richards equation in heterogeneous soil. Comput. Vis. Sci. 13(5), 187–205 (2010)

    Article  Google Scholar 

  13. Berninger, H., Kornhuber, R., Sander, O.: On nonlinear Dirichlet–Neumann algorithms for jumping nonlinearities. In: Widlund, O.B., Keyes, D.E. (eds.) Domain decomposition methods in science and engineering XVI,volume 55 of LNCSE, pp 483–490. Springer (2007)

  14. Berninger, H., Kornhuber, R., Sander, O.: Convergence behaviour of Dirichlet–Neumann and Robin methods for a nonlinear transmission problem. In: Huang, Y., Kornhuber, R., Widlund, O., Xu, J. (eds.) Domain decomposition methods in science and engineering XIX,volume 78 of LNCSE, pp 87–98. Springer (2010)

  15. Berninger, H., Kornhuber, R., Sander, O.: Fast and robust numerical solution of the Richards equation in soil, homogeneous. SIAM J. Numer. Anal. 49(6), 2576–2597 (2011)

    Article  Google Scholar 

  16. Berninger, H., Kornhuber, R., Sander, O., Holst, M., Widlund, O.: Heterogeneous substructuring methods for coupled surface and subsurface flow. In: Bank, R., Xu, J. (eds.) Domain decomposition methods in science and engineering XX, volume 91 of LNCSE, pp 427–434. Springer (2013)

  17. Berninger, H., Loisel, S., Sander, O.: The 2-Lagrange multiplier method applied to nonlinear transmission problems for the Richards equation in heterogeneous soil with cross points. SIAM J. Sci. Comp. 36(5), A2166–A2198 (2014a)

    Article  Google Scholar 

  18. Berninger, H., Ohlberger, M., Sander, O., Smetana, K.: Unsaturated subsurface flow with surface water and nonlinear in- and outflow conditions. Math. Models Methods Appl. 24(5), 901–936 (2014b)

    Article  Google Scholar 

  19. Brezzi, F., Gilardi, G.: Functional spaces, chapter 2 (part 1). In: Kardestuncer, H., Norrie, D.H. (eds.) Finite element handbook, pp 1.29–1.75. Springer (1987)

  20. Brooks, R.J., Corey, A.T.: Hydraulic properties of porous media. Technical Report Hydrology Paper No. 3, Colorado State University, Civil Engineering Department, Fort Collins (1964)

  21. Burdine, N.T.: Relative permeability calculations from pore-size distribution data. Petr. Trans. Am. Inst. Mining Metall. Eng. 198, 71–77 (1953)

    Google Scholar 

  22. Cancès, C., Pop, I.S., Vohralík, M.: An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow. Math. Comp. 83, 153–188 (2014)

    Article  Google Scholar 

  23. Chavent, G., Jaffré, J.: Dynamics of fluids in porous media. Elsevier Science (1986)

  24. Darcy, H.: Les Fontaines Publiques de la Ville de Dijon. Appendice. - Note D.Dalmont, Paris (1856)

  25. Ekeland, I., Temam, R.: Convex analysis and variational problems. North–Holland (1976)

  26. Eymard, R., Gutnic, M., Hilhorst, D.: The finite volume method for Richards equation. Comput. Geosci. 3 (3–4), 259–294 (1999)

    Article  Google Scholar 

  27. Farthing, M.W., Kees, C.E., Coffey, T.S., Kelley, C.T., Miller, C.T.: Efficient steady-state solution techniques for variably saturated groundwater flow. Adv. Water Resour. 26(8), 833–849 (2003)

    Article  Google Scholar 

  28. Forsyth, P.A., Kropinski, M.C.: Monotonicity considerations for saturated-unsaturated subsurface flow. SIAM J. Sci. Comput. 18(5), 1328–1354 (1997)

    Article  Google Scholar 

  29. Fuhrmann, J.: Zur Verwendung von Mehrgitterverfahren bei der numerischen Behandlung elliptischer partieller Differentialgleichungen mit variablen Koeffizienten. PhD thesis, TU Chemnitz–Zwickau (1994)

  30. Fuhrmann, J., Langmach, H.: Stability and existence of solutions of time-implicit finite volume schemes for viscous nonlinear conservation laws. Appl. Numer. Math. 37(1–2), 201–230 (2001)

    Article  Google Scholar 

  31. Gander, M.J.: Optimized Schwarz methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006)

    Article  Google Scholar 

  32. Helmig, R., Weiss, A., Wohlmuth, B.I.: Variational inequalities for modeling flow in heterogeneous porous pressure, media with entry. Comput. Geosci. 13(3), 373–389 (2009)

    Article  Google Scholar 

  33. Hornung, U.: Numerische Simulation von gesättigt-ungesättigten Wasserflüssen in porösen Medien. (German). In: Albrecht, J., Collatz, L., Hämmerlin, G. (eds.) Numerische Behandlung von Differentialgleichungen mit besonderer Berücksichtigung freier Randwertaufgaben, volume 39 of Int. Ser. Numer. Math., pp 214–232 (1978). Birkhäuser

  34. Kees, C.E., Farthing, M.W., Howington, S.E., Jenkins, E.W., Kelley, C.T.: Nonlinear multilevel iterative methods for multiscale models of air/water flow in porous media. In: Binning, P.J., Engesgaard, P.K., Dahle, H.K., Pinder, G.F., Gray, W.G. (eds.) Proceedings of computational methods in water resources XVI, p 8, Copenhagen (2006)

  35. Kees, C.E., Farthing, M.W., Dawson, C.N.: Locally conservative, stabilized finite element methods for flow, variably saturated. Comput. Methods Appl. Mech. Engrg. 197(51–52), 4610–4625 (2008)

    Article  Google Scholar 

  36. Klausen, R.A., Radu, F.A., Eigestad, G.T.: Convergence of MPFA on triangulations and for Richards’ equation. Int. J. Numer. Meth. Fl. 58(12), 1327–1351 (2008)

    Article  Google Scholar 

  37. Kröner, D.: Numerical Schemes for Conservation Laws. Wiley–Teubner (1997)

  38. Leoni, G., Morini, M.: Necessary and sufficient conditions for the chain rule in \(W_{loc}^{1,1}(\mathbb {R}^{N};\mathbb {R}^{d})\) and \(BV_{loc}(\mathbb {R}^{N};\mathbb {R}^{d})\). J. Eur. Math. Soc. (JEMS) 9(2), 219–252 (2007)

    Article  Google Scholar 

  39. Li, H., Farthing, M.W., Dawson, C.N., Miller, C.T.: Local discontinuous Galerkin approximations to Richards equation. Adv. Water Resour. 30(3), 555–575 (2007)

    Article  Google Scholar 

  40. Lui, S.H.: A Lions non-overlapping domain decomposition method for domains with an arbitrary interface. IMA J. Numer. Anal. 29, 332–349 (2009)

    Article  Google Scholar 

  41. Manzini, G., Ferraris, S.: Mass-conservative finite volume methods on 2-d unstructured grids for the Richards equation. Adv. Water Resour. 27(12), 1199–1215 (2004)

    Article  Google Scholar 

  42. Marcus, M., Mizel, V.J.: Complete characterization of functions which act, via superposition, on Sobolev spaces. Trans. Amer. Math. Soc. 251, 187–218 (1979)

    Article  Google Scholar 

  43. Pop, I.S., Schweizer, B.: Regularization schemes for degenerate Richards equations and outflow conditions. Math. Mod. Meth. Appl. S. 21, 1685–1712 (2011)

    Article  Google Scholar 

  44. Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (1999)

  45. Radu, F., Pop, I.S., Knabner, P.: Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards’ equation. SIAM J. Numer. Anal. 42(4), 1452–1478 (2004)

    Article  Google Scholar 

  46. Radu, F.A., Pop, I.S., Knabner, P.: ALGORITMY 2002 (Proceedings 16th Conference on Scientific Computing), 58–66 (2002)

  47. Rawls, W.J., Ahuja, L.R., Brakensiek, D.L., Shirmohammadi, A.: Infiltration and soil water movement, chapter 5. In: Maidment, D.R. (ed.) Handbook of Hydrology (1993). McGraw–Hill

  48. Richards, L.A.: Capillary conduction of liquids through porous mediums. Physics 1, 318–333 (1931)

    Article  Google Scholar 

  49. Sander, O.: Geodesic finite elements in spaces of zero curvature. In: Cangiani, A., Davidchack, R.L., Georgoulis, E.H., Gorban, A., Levesley, J., Tretyakov, M.V. (eds.) : Proceedings of ENUMATH 2011, pp 449–457. Springer (2013)

  50. Schneid, E., Knabner, P., Radu, F.: A priori error estimates for a mixed finite element discretization of the Richards’ equation. Numer. Math. 98(2), 353–370 (2004)

    Article  Google Scholar 

  51. Schweizer, B.: Regularization of outflow problems in unsaturated porous media with dry regions. J. Differ. Equations 237(2), 278–306 (2007)

    Article  Google Scholar 

  52. Sochala, P., Ern, A., Piperno, S.: Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows. Comput. Methods Appl. Mech. Engrg. 198, 2122–2136 (2009)

    Article  Google Scholar 

  53. San Soucie, C.A.: Mixed finite element methods for variably saturated subsurface flow. PhD thesis, Rice University (1996)

  54. Woodward, C.S., Dawson, C.N.: Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media. SIAM J. Numer. Anal. 37(3), 701–724 (2000)

    Article  Google Scholar 

  55. Younes, A., Fahs, M., Belfort, B.: Monotonicity of the cell-centred triangular MPFA method for saturated and unsaturated flow in heterogeneous porous media. J. Hydrol. 504, 132–141 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, CP 64, 1211, Genève 4, Switzerland

    Heiko Berninger

  2. Freie Universität Berlin, Institut für Mathematik, Arnimallee 6, 14195, Berlin, Germany

    Ralf Kornhuber

  3. Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52062, Aachen, Germany

    Oliver Sander

Authors
  1. Heiko Berninger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ralf Kornhuber
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Oliver Sander
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Heiko Berninger.

Additional information

This work was supported by the BMBF–Programm “Mathematik für Innovationen in Industrie und Dienstleistungen”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Berninger, H., Kornhuber, R. & Sander, O. A multidomain discretization of the Richards equation in layered soil. Comput Geosci 19, 213–232 (2015). https://doi.org/10.1007/s10596-014-9461-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-014-9461-8

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation