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Two-phase flow equations with a dynamic capillary pressure

Published online by Cambridge University Press:  21 September 2012

JAN KOCH ,
ANDREAS RÄTZ  and
BEN SCHWEIZER
JAN KOCH
Affiliation:
Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, D-44227 Dortmund, Germany emails: jan.koch@tu-dortmund.de, andreas.raetz@tu-dortmund.de, ben.schweizer@tu-dortmund.de
ANDREAS RÄTZ
Affiliation:
Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, D-44227 Dortmund, Germany emails: jan.koch@tu-dortmund.de, andreas.raetz@tu-dortmund.de, ben.schweizer@tu-dortmund.de
BEN SCHWEIZER
Affiliation:
Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, D-44227 Dortmund, Germany emails: jan.koch@tu-dortmund.de, andreas.raetz@tu-dortmund.de, ben.schweizer@tu-dortmund.de
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Abstract

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.

Keywords

Two-Phase flowCapillary hysteresisFinite-element scheme
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 1 , February 2013 , pp. 49 - 75
Copyright
Copyright © Cambridge University Press 2012

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References

[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, 335–392.Google Scholar
[2]Alt, H. W. & Luckhaus, S. (1983) Quasilinear elliptic-parabolic differential equations. Math. Z. 183 3, 311–341.Google Scholar
[3]Alt, H. W., Luckhaus, S. & Visintin, A. (1984) On nonstationary flow through porous media. Ann. Mat. Pura Appl. 136 4, 303–316.CrossRefGoogle Scholar
[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, 1669–1687.Google Scholar
[5]Bagagiolo, F. & Visintin, A. (2000) Hysteresis in filtration through porous media. Z. Anal. Anwendungen 19 4, 977–997.Google Scholar
[6]Bagagiolo, F. & Visintin, A. (2004) Porous media filtration with hysteresis. Adv. Math. Sci. Appl. 14 2, 379–403.Google Scholar
[7]Bastian, P. & Helmig, R. (1999) Efficient fully coupled solution techniques for two-phase flow in porous media. Adv. Water Resour. 23 3, 199–216.Google Scholar
[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, 487–510.Google Scholar
[9]Buzzi, F., Lenzinger, M. & Schweizer, B. (2009) Interface conditions for degenerate two-phase flow equations in one space dimension. Analysis (Munich) 29 3, 299–316.CrossRefGoogle Scholar
[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.Google Scholar
[11]Cancès, C. & Pierre, M. (2011) An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field. HAL: hal-00518219.Google Scholar
[12]Carrillo, J. (1994) On the uniqueness of the solution of the evolution dam problem. Nonlinear Anal. 22 5, 573–607.CrossRefGoogle Scholar
[13]Chen, Z. (2001) Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution. J. Differ. Equ. 171 2, 203–232.Google Scholar
[14]Chen, Z. (2002) Degenerate two-phase incompressible flow. II. Regularity, stability and stabilization. J. Differ. Equ. 186 2, 345–376.Google Scholar
[15]Davis, T. A. (2004) Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30 2, 196–199.Google Scholar
[16]Kröner, D. & Luckhaus, S. (1984) Flow of oil and water in a porous medium. J. Differ. Equ. 55 2, 276–288.Google Scholar
[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, 33–64.Google Scholar
[18]Lenzinger, M. & Schweizer, B. (2010) Two-phase flow equations with outflow boundary conditions in the hydrophobic-hydrophilic case. Nonlinear Anal. 73 4, 840–853.Google Scholar
[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, 1561–1577.Google Scholar
[20]Otto, F. (1997) L 1-contraction and uniqueness for unstationary saturated-unsaturated porous media flow. Adv. Math. Sci. Appl. 7 2, 537–553.Google Scholar
[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, 431–450.CrossRefGoogle Scholar
[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), 1452–1478 (electronic).Google Scholar
[23]Schweizer, B. (2007) Averaging of flows with capillary hysteresis in stochastic porous media. European J. Appl. Math. 18 3, 389–415.Google Scholar
[24]Schweizer, B. (2007) Regularization of outflow problems in unsaturated porous media with dry regions. J. Differ. Equ. 237 2, 278–306.Google Scholar
[25]Schweizer, B. (2012a) The Richards equation with hysteresis and degenerate capillary pressure. J. Differ. Equ. 252 10, 5594–5612.Google Scholar
[26]Schweizer, B. (2012b) Instability of gravity wetting fronts for Richards equations with hysteresis. Interfaces Free Boundaries 14, 37–64.CrossRefGoogle Scholar
[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, 2523–2528.Google Scholar
[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, 215–244.Google Scholar
[29]Vey, S. & Voigt, A. (2007) AMDiS – adaptive multidimensional simulations. Comput. Visual. Sci. 10, 57–67.CrossRefGoogle Scholar
[30]Visintin, A. (1994) Differential Models of Hysteresis. Applied Mathematical Sciences, Vol. 111, Springer-Verlag, Berlin, Germany.Google Scholar