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Averaging of stochastic evolution equations of transport in porous media

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Abstract

A study is made of the problem of averaging the simplest one-dimensional evolution equations of stochastic transport in a porous medium. A number of exact functional equations corresponding to distributions of the random parameters of a special form is obtained. In some cases, the functional equations can be localized and reduced to differential equations of fairly high order. The first part of the paper (Secs. 1–6) considers the process of transport of a neutral admixture in porous media. The functional approach and technique for decoupling the correlations explained by Klyatskin [4] is used. The second part of the paper studies the process of transport in porous media of two immiscible incompressible fluids in the framework of the Buckley—Leverett model. A linear equation is obtained for the joint probability density of the solution of the stochastic quasilinear transport equation and its derivative. An infinite chain of equations for the moments of the solution is obtained. A scheme of approximate closure is proposed, and the solution of the approximate equations for the mean concentration is compared with the exactly averaged concentration.

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Literature cited

  1. M. I. Shvidler, Statistical Hydrodynamics of Porous Media [in Russian], Nedra, Moscow (1985).

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  2. M. I. Shvidler, “Dispersion of flow in a porous medium,” Izv. Akad Nauk SSSR Mekh. Zhidk. Gaza, No. 4, 65 (1976).

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  3. M. I. Shvidler, “Averaging of the equations of flow transport in porous media with random inhomogeneities,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 1, 68 (1985).

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  4. V. I. Klyatskin, Stochastic Equations and Waves in Randomly Inhomogeneous Media [in Russian], Nauka, Moscow (1980).

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Authors and Affiliations

  1. Moscow

    P. V. Indel'man & M. I. Shvidler

Authors
  1. P. V. Indel'man
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  2. M. I. Shvidler
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Additional information

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–136, September–October, 1985.

We are grateful to A. I. Shnirel'man for pointing out the possibility of obtaining an averaged equation in the case of a velocity distribution in accordance with a Cauchy law.

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Indel'man, P.V., Shvidler, M.I. Averaging of stochastic evolution equations of transport in porous media. Fluid Dyn 20, 775–784 (1985). https://doi.org/10.1007/BF01050092

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  • DOI: https://doi.org/10.1007/BF01050092

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