Skip to main content Accessibility help

Two-phase flow equations with a dynamic capillary pressure


We investigate the motion of two immiscible fluids in a porous medium described by a two-phase flow system. In the capillary pressure relation, we include static and dynamic hysteresis. The model is well established in the context of the Richards equation, which is obtained by assuming a constant pressure for one of the two phases. We derive an existence result for this hysteresis two-phase model for non-degenerate permeability and capillary pressure curves. A discretization scheme is introduced and numerical results for fingering experiments are obtained. The main analytical tool is a compactness result for two variables that are coupled by a hysteresis relation.

Hide All
[1]Alt, H. W. & DiBenedetto, E. (1985) Nonsteady flow of water and oil through inhomogeneous porous media. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4th series) 12 3, 335392.
[2]Alt, H. W. & Luckhaus, S. (1983) Quasilinear elliptic-parabolic differential equations. Math. Z. 183 3, 311341.
[3]Alt, H. W., Luckhaus, S. & Visintin, A. (1984) On nonstationary flow through porous media. Ann. Mat. Pura Appl. 136 4, 303316.
[4]Arbogast, T., Wheeler, M. F. & Zhang, N.-Y. (1996) A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media. SIAM J. Numer. Anal. 33 4, 16691687.
[5]Bagagiolo, F. & Visintin, A. (2000) Hysteresis in filtration through porous media. Z. Anal. Anwendungen 19 4, 977997.
[6]Bagagiolo, F. & Visintin, A. (2004) Porous media filtration with hysteresis. Adv. Math. Sci. Appl. 14 2, 379403.
[7]Bastian, P. & Helmig, R. (1999) Efficient fully coupled solution techniques for two-phase flow in porous media. Adv. Water Resour. 23 3, 199216.
[8]Beliaev, A. Y. & Hassanizadeh, S. M. (2001) A theoretical model of hysteresis and dynamic effects in the capillary relation for two-phase flow in porous media. Transp. Porous Media 43 3, 487510.
[9]Buzzi, F., Lenzinger, M. & Schweizer, B. (2009) Interface conditions for degenerate two-phase flow equations in one space dimension. Analysis (Munich) 29 3, 299316.
[10]Cancès, C., Choquet, C., Fan, Y. & Pop, I. (2010) Existence of Weak Solutions to a Degenerate Pseudo-Parabolic Equation Modeling Two-Phase Flow in Porous Media. Technical Report, Eindhoven University of Technology, CASA.
[11]Cancès, C. & Pierre, M. (2011) An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field. HAL: hal-00518219.
[12]Carrillo, J. (1994) On the uniqueness of the solution of the evolution dam problem. Nonlinear Anal. 22 5, 573607.
[13]Chen, Z. (2001) Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution. J. Differ. Equ. 171 2, 203232.
[14]Chen, Z. (2002) Degenerate two-phase incompressible flow. II. Regularity, stability and stabilization. J. Differ. Equ. 186 2, 345376.
[15]Davis, T. A. (2004) Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30 2, 196199.
[16]Kröner, D. & Luckhaus, S. (1984) Flow of oil and water in a porous medium. J. Differ. Equ. 55 2, 276288.
[17]Lamacz, A., Rätz, A. & Schweizer, B. (2011) A well-posed hysteresis model for flows in porous media and applications to fingering effects. Adv. Math. Sci. Appl. 21 1, 3364.
[18]Lenzinger, M. & Schweizer, B. (2010) Two-phase flow equations with outflow boundary conditions in the hydrophobic-hydrophilic case. Nonlinear Anal. 73 4, 840853.
[19]Mikelić, A. (2010) A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure. J. Differ. Equ. 248 6, 15611577.
[20]Otto, F. (1997) L 1-contraction and uniqueness for unstationary saturated-unsaturated porous media flow. Adv. Math. Sci. Appl. 7 2, 537553.
[21]Radu, F. & Pop, I. S. (2011) Mixed finite element discretization and Newton iteration for a reactive contaminant transport model with non-equilibrium sorption: Convergence analysis and error estimates. Comput. Geosci. 15 3, 431450.
[22]Radu, F., Pop, I. S. & Knabner, P. (2004) Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards' equation. SIAM J. Numer. Anal. 42 (4), 14521478 (electronic).
[23]Schweizer, B. (2007) Averaging of flows with capillary hysteresis in stochastic porous media. European J. Appl. Math. 18 3, 389415.
[24]Schweizer, B. (2007) Regularization of outflow problems in unsaturated porous media with dry regions. J. Differ. Equ. 237 2, 278306.
[25]Schweizer, B. (2012a) The Richards equation with hysteresis and degenerate capillary pressure. J. Differ. Equ. 252 10, 55945612.
[26]Schweizer, B. (2012b) Instability of gravity wetting fronts for Richards equations with hysteresis. Interfaces Free Boundaries 14, 3764.
[27]Selker, J. S., Parlange, J.-Y. & Steenhuis, T. S. (1992) Fingered flow in two dimensions. Part 2. Predicting finger moisture profile. Wat. Resour. Res. 28 9, 25232528.
[28]van Duijn, C. J., Pieters, G. J. M. & Raats, P. A. C. (2004) Steady flows in unsaturated soils are stable. Transp. Porous Media 57 2, 215244.
[29]Vey, S. & Voigt, A. (2007) AMDiS – adaptive multidimensional simulations. Comput. Visual. Sci. 10, 5767.
[30]Visintin, A. (1994) Differential Models of Hysteresis. Applied Mathematical Sciences, Vol. 111, Springer-Verlag, Berlin, Germany.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed