A mathematically rigorous method of homogenization is presented and used to analyze the equivalent behavior of transient flow of two incompressible fluids through heterogeneous media Asymptotic qpansions and H-convergence lead to the definition of a global or effective model of an equivalent homogeneous reservoir Numerical computations to obtain the homogenized coefficients of the entire reservoir have been carried out via a finite element method Numerical experiments involving the simulation of incompressible two-phase flow have been performed for each heterogeneous medium and for the homogenized medium as well as for other averaging methods The results of the simulations are compared in terms of the transient saturation contours, production curves, and pressure distributions Results obtained from the simulations with the homogenization method presented show good agreement with the heterogeneous simulations

Two-phase flow in heterogeneous porous media

A mathematically rigorous method of homogenization is presented and used to analyze the equivalent behavior of transient flow of two incompressible fluids through heterogeneous media Asymptotic expansions and H-convergence lead to the definition of a global or effective model of an equivalent homogeneous reservoir Numerical computations to obtain the homogenized coefficients of the entire reservoir have been carried out via a finite element method Numerical experiments involving the simulation of incompressible two-phase flow have been performed for each heterogeneous medium and for the homogenized medium as well as for other averaging methods The results of the simulations are compared in terms of the transient saturation contours, production curves, and pressure distributions Results obtained from the simulations with the homogenization method presented show good agreement with the heterogeneous simulations

Numerical Simulation and Homogenization of Two-Phase Flow in Heterogeneous Porous Media

The aim of this paper is to analyze a model of a degener- ate nonlinear system arising from immiscible, compressible, two-phase, three-dimensional flows occurring in porous media. A degenerate weighted formulation is introduced to take into account the compress- ibility and the degeneracy. Two existence results of such a degenerate weak solution are introduced. The first result concerns the existence of solutions under a reasonable assumption on the capillary pressure. This condition allows the degeneracy only where one of the two phases is exclusively present. The second result establishes, for suitable initial data, the existence of solutions when the degeneracy occurs where one or the other phase is exclusively present. Nevertheless, for suitable ini- tial data, a classical weak solution is obtained when the degeneracy is not too strong and occurs only where the injected phase is exclusively present.

/pdf/on-a-degenerate-parabolic-system-for-compressible-immiscible-2oygfo74ab.pdf

On a degenerate parabolic system for compressible, immiscible, two-phase flows in porous media

We develop a multi-length-scale (multifractal) theory for the effect of rock heterogeneity on the growth of the mixing layer of the flow of a passive tracer through porous media. The multifractal exponent of the size of the mixing layer is determined analytically from the statistical properties of a random velocity (permeability) field. The anomalous diffusion of the mixing layer can occur both on finite and on asymptotic length scales.

A multi-length-scale theory of the anomalous mixing-length growth for tracer flow in heterogeneous porous media

http://web.math.pmf.unizg.hr/~jurak/paper-JDE-version21juin.pdf

An existence result for a coupled system modeling a fully equivalent global pressure formulation for immiscible compressible two-phase flow in porous media

We discuss a three-dimensional displacement model of one miscible compressible fluid by another in a porous medium. The motion is modeled by a nonlinear system of parabolic type coupling the pressure and the concentration. We give an existence result of weak solutions for a model with diffusion and dispersion, using the Schauder fixed point theorem. We also study a model in the absence of diffusion and dispersion. The system becomes of parabolic-hyperbolic type, the existence of global weak solutions is then obtained through a compensated compactness argument.

Mathematical analysis for compressible miscible displacement models in porous media

Kinetic formulation for a transport equation with memory

Global weak solutions for a parabolic system modeling a one-dimensional miscible flow in porous media

We introduce a kinetic formulation for a transport equation arising from an homogenization problem. We give existence, uniqueness and finite speed propagation results

Etude d'une equation de transport a memoire

We study the asymptotic behavior, with respect to high Peclet numbers, of the solutions of the nonlinear elliptic-parabolic system governing the displacement of one incompressible fluid by another, completely miscible with the first, in a porous medium. Using compensated compactness techniques, we obtain the existence of a global weak solution to the nonlinear degenerate elliptic-parabolic system modelling the flow when the molecular diffusion effects are neglected.

https://www.ems-ph.org/journals/show_pdf.php?issn=0232-2064&vol=23&iss=2&rank=8

Asymptotic Behavior of the Solutions of an Elliptic-Parabolic System Arising in Flow in Porous Media

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