Abstract
We analyze a stochastic model for the motion of fronts in two-phase fluids and derive upscaled equations for the capillary pressure. This extends results of [11], where the same law for the capillary pressure was derived under an assumption on typical explosion patterns. With the work at hand we remove that assumption and show that in the stochastic case the upscaled equations hold almost surely.
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Auriault, J.-L., Sanchez-Palencia, E.: Remarques sur la loi de Darcy pour les écoulements biphasiques en milieu poreux. J. Mc. Thor. Appl. (Numéro spécial “Modélisation asymptotique d’écoulements de fluides”), 141–156 (1986)
Bourgeat, A.: Two-phase flow. In: Homogenization and porous media, ed. by U. Hornung, pp. 95–127. Berlin, Heidelberg, New York: Springer 1997
Cortesani, G.: Asymptotic behaviour of a sequence of Neumann problems. Commun. Partial Differ. Equations 22, 1691–1729 (1997)
Dal Maso, G., Modica, L.: Nonlinear stochastic homogenization. Ann. Mat. Pura Appl. 144, 347–389 (1986)
Jikov, V., Kozlov, S., Oleinik, O.: Homogenization of differential operators and integral functionals. Berlin, Heidelberg, New York: Springer 1994
Kozlov, S.M.: Averaging of random operators. Mat. USSR Sb. 37, 167–179 (1980)
Leverett, M.C.: Steady flow of gas-oil-water mixtures through unconsolidated sands. Trans. AIME 132, 149 (1938)
Mikelic, A., Paoli, L.: On the derivation of the Buckley-Leverett model from the two fluid Navier-Stokes equations in a thin domain. Comput. Geosci. 1, 59–83 (1997)
Sanchez-Palencia, E.: Non homogeneous media and vibration theory. Lect. Notes Phys., vol. 17. Berlin, Heidelberg, New York: Springer 1980
Schweizer, B.: Laws for the capillary pressure via the homogenization of fronts in porous media. Habilitationsschrift an der Ruprecht-Karls-Universität Heidelberg 2002
Schweizer, B.: Laws for the capillary pressure in a deterministic model for fronts in porous media. To appear in SIAM J. Math. Anal.
Witting, H., Müller-Funk, U.: Mathematische Statistik II. Stuttgart: Teubner 1995
Wright, S.: On the steady flow of an incompressible fluid through a randomly perforated porous medium. J. Differ. Equations 146, 261–286 (1998)
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Schweizer, B. A stochastic model for fronts in porous media. Annali di Matematica 184, 375–393 (2005). https://doi.org/10.1007/s10231-004-0122-8
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DOI: https://doi.org/10.1007/s10231-004-0122-8