Hostname: page-component-7dd5485656-jtdwj Total loading time: 0 Render date: 2025-10-29T19:20:41.763Z Has data issue: false hasContentIssue false
  • English
  • Français

Averaging of flows with capillary hysteresis in stochastic porous media

Published online by Cambridge University Press:  01 June 2007

BEN SCHWEIZER
BEN SCHWEIZER*
Affiliation:
Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland email: Ben.Schweizer@unibas.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

Fluids in unsaturated porous media are described by the relationship between pressure (p) and saturation (u). Darcy's law and conservation of mass provides an evolution equation for u, and the capillary pressure provides a relation between p and u of the form p∈ pc(u,∂tu). The multi-valued function pc leads to hysteresis effects. We construct weak and strong solutions to the hysteresis system and homogenize the system for oscillatory stochastic coefficients. The effective equations contain a new dependent variable that encodes the history of the wetting process and provide a better description of the physical system.

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 18 , Issue 3 , June 2007 , pp. 389 - 415
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Barbu, V. (1976) Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, The Netherlands.CrossRefGoogle Scholar
[2]Beliaev, A. (2001) Porous medium flows with capillary hysteresis and homogenization. In: International Conference on Homogenization and Applications to Material Sciences, September 15–19, 2001, University Timisoara, Romania, pp. 23–32.Google Scholar
[3]Beliaev, A. (2001) Unsaturated porous flows with play-type capillary hysteresis. Russian J. Math. Phys. 8 (1), 1–13.Google Scholar
[4]Beliaev, A. & Hassanizadeh, S. M. (2001) A theoretical model of hysteresis and dynamic effects in the capillary relation for two-phase flow in porous media. Transport Porous Media 43, 487–510.Google Scholar
[5]Beliaev, A. & Schotting, R. J. (2001) Analysis of a new model for unsaturated flow in porous media including hysteresis and dynamic effects. Comput. Geosci. 5 (4), 345–368.CrossRefGoogle Scholar
[6]Bourgeat, A. & Panfilov, M. (1998) Effective two-phase flow through highly heterogeneous porous media: Capillary nonequilibrium effects. Comput. Geosci. 2 (3), 191–215.Google Scholar
[7]Bourgeat, A. & Piatnitski, A. (2004) Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincaré Probab. Statist. 40 (2), 0246–0203.Google Scholar
[8]Collins, R. E. (1990) Flow of Fluids Through Porous Materials, Research & Engineering Consultants, CO, Englewood.Google Scholar
[9]Dal Maso, G. & Modica, L. (1986) Nonlinear stochastic homogenization. Ann. Mat. Pura Appl. 144, 347–389.CrossRefGoogle Scholar
[10]van Duijn, C. J. & Peletier, L. A. (1982) Nonstationary filtration in partially saturated porous media. Arch. Rat. Mech. Anal. 78 (2), 173–198.Google Scholar
[11]Jikov, V., Kozlov, S. & Oleinik, O. (1994) Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin.Google Scholar
[12]Kinderlehrer, D. & Stampacchia, G. (1980) An Introduction to Variational Inequalities and their Applications, Academic Press, New York.Google Scholar
[13]Kozlov, S. M. (1980) Averaging of random operators. Math. USSR Sbornik 37, 167–179.CrossRefGoogle Scholar
[14]Kröner, D. (1984) Parabolic regularization and behaviour of the free boundary for unsaturated flow in a porous medium. J. Reine Angew. Math. 348, 180–196.Google Scholar
[15]Kubo, M. (2004) A filtration model with hysteresis. J. Differential Equations 201 (1), 75–98.CrossRefGoogle Scholar
[16]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
[17]Schweizer, B. (2005) Laws for the capillary pressure in a deterministic model for fronts in porous media. SIAM J. Math. Anal. 36 (5), 1489–1521.Google Scholar
[18]Schweizer, B. (2005) A stochastic model for fronts in porous media. Ann. Mat. Pura Appl. 184 (3), 375–393.CrossRefGoogle Scholar
[19]Schweizer, B. (in press) Regularization of outflow problems in unsaturated porous media with dry regions. J. Differential Equations.Google Scholar
[20]Visintin, A. (1994) Differential Models of Hysteresis, Springer, Berlin.CrossRefGoogle Scholar