Computation of Inter-Nodal Permeabilities for Richards Equation

  • Adam SzymkiewiczEmail author
Part of the GeoPlanet: Earth and Planetary Sciences book series (GEPS)


An important part of all finite difference and many finite volume discretization schemes developed for multiphase flow equations is the approximation of the average permeability value between two neighbouring nodes. Various averaging techniques are presented in this chapter, with particular focus on the case of one-dimensional unsaturated flow in a homogeneous medium, for which accurate inter-nodal permeability estimations based on steady flow analysis are available. It is shown that the relation between capillary and gravity forces at the scale of a single grid cell has key importance for the choice of the averaging scheme. An averaging method developed by the author for one-dimensional flow is presented in detail, and its extensions to heterogeneous materials and multidimensional problems are discussed. Implications for two-phase flow modelling are also considered.


Relative Permeability Coarse Grid Pressure Head Average Scheme Unsaturated Flow 
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  1. 1.
    Aziz K, Settari A (1979) Petroleum reservoir simulation. Applied Science, EssexGoogle Scholar
  2. 2.
    Baker D (1995) Darcian weighted interblock conductivity means for vertical unsaturated flow. Ground Water 33(3):385–390. doi: 10.1111/j.1745-6584.1995.tb00294.x CrossRefGoogle Scholar
  3. 3.
    Baker D (2000) A Darcian integral approximation to interblock hydraulic conductivity means in vertical infiltration. Comput Geosci 26(5):581–590CrossRefGoogle Scholar
  4. 4.
    Baker D (2006) General validity of conductivity means in unsaturated flow. J Hydrol Eng 11(6):526–538CrossRefGoogle Scholar
  5. 5.
    Baker D, Arnold M, Scott H (1999) Some analytic and approximate Darcian means. Ground Water 37(4):532–538. doi: 10.1111/j.1745-6584.1999.tb01139.x CrossRefGoogle Scholar
  6. 6.
    Belfort B, Lehmann F (2005) Comparison of equivalent conductivities for numerical simulation of one-dimensional unsaturated flow. Vadose Zone J 4(4):1191–1200CrossRefGoogle Scholar
  7. 7.
    Berg P (1999) Long-term simulation of water movement in soils using mass-conserving procedures. Adv Water Resour 22(5):419–430CrossRefGoogle Scholar
  8. 8.
    Brunone B, Ferrante M, Romano N, Santini A (2003) Numerical simulations of one-dimensional infiltration into layered soils with the Richards’ equation using different enstimates of the interlayer conductivity. Vadose Zone J 2(2):193–200Google Scholar
  9. 9.
    Burzyński K, Szymkiewicz A (2011) Unstructured finite-volume meshes for two-dimensional flow in variably saturated porous media. TASK Q 15(3):1001–10,014Google Scholar
  10. 10.
    Celia M, Binning P (1992) A mass conservative numerical solution for two-phase flow in porous media with application to unsaturated flow. Water Resour Res 28(10):2819–2828CrossRefGoogle Scholar
  11. 11.
    Celia M, Bouloutas E, Zarba R (1990) A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour Res 26(7):1483–1496CrossRefGoogle Scholar
  12. 12.
    Desbarats A (1995) An interblock conductivity scheme for finite difference models of steady unsaturated flow in heterogeneous media. Water Resour Res 31(11):2883–2889CrossRefGoogle Scholar
  13. 13.
    Fagerlund F, Niemi A, Odén M (2006) Comparison of relative permeabilityfluid saturationcapillary pressure relations in the modelling of non-aqueous phase liquid infiltration in variably saturated, layered media. Adv Water Resour 29(11):1705–1730. doi: 10.1016/j.advwatres.2005.12.007 Google Scholar
  14. 14.
    Fletcher C (1991) Computational techniques for fluid dynamics 1. Fundamental and general techniques. Springer, BerlinCrossRefGoogle Scholar
  15. 15.
    Forsyth P, Kropinski M (1997) Monotonicity considerations for saturated–unsaturated subsurface flow. SIAM J Sci Comput 18(5):1328–1354CrossRefGoogle Scholar
  16. 16.
    Forsyth P, Wu Y, Pruess K (1995) Robust numerical methods for saturated–unsaturated flow in heterogeneous media. Adv Water Resour 18(1):25–38CrossRefGoogle Scholar
  17. 17.
    Fuhrmann J, Langmach H (2001) Stability and existence of solutions of time-implicit finite volume schemes for viscous nonlinear conservation laws. Appl Numer Math 37(1–2):201–230Google Scholar
  18. 18.
    Fučik R, Mikyška J, Beneš M, Illangasekare T (2007) An improved semi-analytical solution for verification of numerical models of two-phase flow in porous media. Vadose Zone J 6(1):93–104CrossRefGoogle Scholar
  19. 19.
    Gastó J, Grifoll J, Cohen Y (2002) Estimation of internodal permeabilities for numerical simulations of unsaturated flows. Water Resour Res 38(12):1326CrossRefGoogle Scholar
  20. 20.
    Guarnaccia J, Pinder G, Fishman M (1997) NAPL: simulator documentation. Environmental Protection Agency, USAGoogle Scholar
  21. 21.
    Haverkamp R, Vauclin M (1979) A note on estimating finite difference interblock hydraulic conductivity values for transient unsaturated flow. Water Resour Res 15(1):181–187CrossRefGoogle Scholar
  22. 22.
    Helmig R, Huber R (1998) Comparison of Galerkin-type discretization techniques for two-phase flow in heterogenous porous media. Adv Water Resour 21(8):697–711CrossRefGoogle Scholar
  23. 23.
    Kavetski D, Binning P, Sloan S (2001) Adaptive time stepping and error control in a mass conservative numerical solution of the mixed form of Richards equation. Adv Water Resour 24(6):595–605CrossRefGoogle Scholar
  24. 24.
    Kees C, Miller C (2002) Higher order time integration methods for two-phase flow. Adv Water Resour 25(2):159–177CrossRefGoogle Scholar
  25. 25.
    Kirkland M, Hills R, Wierenga P (1992) Algorithms for solving Richards equation for variably saturated soil. Water Resour Res 28(8):2049–2058CrossRefGoogle Scholar
  26. 26.
    Kueper B, Frind E (1991) Two-phase flow in heterogeneous porous media 1. Model development. Water Resour Res 27(6):1049–1057CrossRefGoogle Scholar
  27. 27.
    Lassabatere L, Angulo-Jaramillo R, Cuenca R, Braud I, Haverkamp R (2006) Beerkan estimation of soil transfer parameters through infiltration experiments—BEST. Soil Sci Soc Am J 70(2):521–532CrossRefGoogle Scholar
  28. 28.
    Lima-Vivancos V, Voller V (2004) Two numerical methods for modeling variably saturated flow in layered media. Vadose Zone J 3(3):1031–1037Google Scholar
  29. 29.
    Manzini G, Ferraris S (2004) Mass-conservative finite volume methods on 2-d unstructured grids for the Richards’ equation. Adv Water Resour 27(12):1199–1215CrossRefGoogle Scholar
  30. 30.
    Miller C, Williams G, Kelley C, Tocci M (1998) Robust solution of Richards equation for nonuniform porous media. Water Resour Res 34(10):2599–2610CrossRefGoogle Scholar
  31. 31.
    Pei Y, Wang J, Tian Z, Yu J (2006) Analysis of interfacial error in saturated–unsaturated flow models. Adv Water Resour 29(4):515–524CrossRefGoogle Scholar
  32. 32.
    Romano N, Brunone B, Santini A (1998) Numerical analysis of one-dimensional unsaturated flow in layered soils. Adv Water Resour 21(4):315–324CrossRefGoogle Scholar
  33. 33.
    Ross P (2003) Modeling soil water and solute transport—fast simplified numerical solutions. Agron J 95(6):1352–1361CrossRefGoogle Scholar
  34. 34.
    Schnabel R, Richie E (1984) Calculation of internodal conductances for unsaturated flow simulations. Soil Sci Soc Am J 48(5):1006–1010CrossRefGoogle Scholar
  35. 35.
    Šimnek J, Šejna M, Saito H, Sakai M, van Genuchten M (2008) The HYDRUS-1D software package 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, RiversideGoogle Scholar
  36. 36.
    Srivastava R, Guzman-Guzman A (1995) Analysis of hydraulic conductivity averaging schemes for one-dimensional, steady-state unsaturated flow. Ground Water 33(6):946–952. doi: 10.1111/j.1745-6584 CrossRefGoogle Scholar
  37. 37.
    Sunada D, McWhorter D (1990) Exact integral solutions for two phase flow. Water Resour Res 26(3):399–413CrossRefGoogle Scholar
  38. 38.
    Szymkiewicz A (2007) Numerical simulation of one-dimensional two-phase flow in porous media. Arch Hydro-eng Environ Mech 54(2):117–136Google Scholar
  39. 39.
    Szymkiewicz A (2009) Approximation of internodal conductivities in numerical simulation of 1D infiltration, drainage and capillary rise in unsaturated soils. Water Resour Res 45:W10403CrossRefGoogle Scholar
  40. 40.
    Szymkiewicz A, Helmig R (2011) Comparison of conductivity averaging methods for one-dimensional unsaturated flow in layered soils. Adv Water Resour 34(8):1012–1025CrossRefGoogle Scholar
  41. 41.
    Touma J, Vauclin M (1986) Experimental and numerical analysis of two-phase infiltration in a partially saturated soil. Transp Porous Media 1(1):27–55CrossRefGoogle Scholar
  42. 42.
    Tracy F (2006) Clean two and three-dimensional analytical solutions of Richards’ equation for testing numerical solvers. Water Resour Res 42:W08503CrossRefGoogle Scholar
  43. 43.
    van Dam J, Feddes R (2000) Numerical simulation of infiltration, evaporation and shallow groundwater levels with the Richards equation. J Hydrol 233(1):72–85CrossRefGoogle Scholar
  44. 44.
    Warrick A (1991) Numerical approximation of Darcian flow through unsaturated soil. Water Resour Res 27(6):1215–1222CrossRefGoogle Scholar
  45. 45.
    Warrick A, Yeh TC (1990) One-dimensional, steady vertical flow in a layered soil profile. Adv Water Resour 13(4):207–210CrossRefGoogle Scholar
  46. 46.
    Zaidel J, Russo D (1992) Estimation of finite difference interblock conductivities for simulation of infiltration into initially dry soils. Water Resour Res 28(9):2285–2295CrossRefGoogle Scholar
  47. 47.
    Zhang X, Ewen J (2000) Efficient method for simulating gravity-dominated water flow in unsaturated soils. Water Resour Res 36(9):2777–2780CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Faculty of Civil and Environmental EngineeringGdansk University of TechnologyGdanskPoland

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