A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

https://doc.rero.ch/record/9768/files/Renard_Ph.-_Calculating_equivalent_permeability_20080903.pdf

Calculating equivalent permeability: a review

A general form of the double porosity model of single phase flow in a naturally fractured reservoir is derived from homogenization theory. The microscopic model consists of the usual equations describing Darcy flow in a reservoir, except that the porosity and permeability coefficients are highly discontinuous. Over the matrix domain, the coefficients are scaled by a parameter $\epsilon $ representing the size of the matrix blocks. This scaling preserves the physics of the flow in the matrix as $\epsilon $ tends to zero. An effective macroscopic limit model is obtained that includes the usual Darcy equations in the matrix blocks and a similar equation for the fracture system that contains a term representing a source of fluid from the matrix. The convergence is shown by extracting weak limits in appropriate Hilbert spaces. A dilation operator is utilized to see the otherwise vanishing physics in the matrix blocks as $\epsilon $ tends to zero.

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Derivation of the double porosity model of single phase flow via homogenization theory

The periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C.R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99–104] (with the basic proofs in [Proceedings of the Narvi...

/pdf/the-periodic-unfolding-method-in-homogenization-zlpt4ljiw9.pdf

The Periodic Unfolding Method in Homogenization

In this paper the theory of two-scale convergence (introduced by G. Nguetseng \ref[SIAM J. Math. Anal. 20 (1989), no. 3, 608--623; and developed later by G. Allaire \ref[SIAM J. Math. Anal. 23 (199 ...

Two-scale convergence

This note deals with localized approximations of homogenized coefficients of second order divergence form elliptic operators with random statistically homogeneous coefficients, by means of “periodization” and other “cut-off” procedures. For instance in the case of periodic approximation, we consider a cubic sample [0 ,ρ ] d of the random medium, extend it periodically in R d and use the effective coefficients of the obtained periodic operators as an approximation of the effective coefficients of the original random operator. It is shown that this approximation converges a.s., as ρ →∞ , and gives back the effective coefficients of the original random operator. Moreover, under additional mixing conditions on the coefficients, the rate of convergence can be estimated by some negative power of ρ which only depends on the dimension, the ellipticity constant and the rate of decay of the mixing coefficients. Similar results are established for approximations in terms of appropriate Dirichlet and Neumann problems localized in a cubic sample [0 ,ρ ] d .

/pdf/approximations-of-effective-coefficients-in-stochastic-3l4t9z9rxe.pdf

Approximations of effective coefficients in stochastic homogenization

It is well known that the modelling of physical processes in strongly inhomogeneous media leads to the study of differential equations with rapidly varying coefficients. More precisely, if the scale of inhomogeneity of the medium is of order ε, then the coefficients of the differential operators are of the form a(s~x), where the function a depends on the microstructure. For small ε, a direct numerical solution of such problems is for all practical purposes impossible, and this necessitates the construction of averaged models. The main goal of homogenization theory is the analysis of these averaged models s ε -» 0, with particular emphasis on the establishment of convergence to a limit in an appropriate sense and the derivation of averaged equations for the limit. One expects such results if there is a certain regularity in the microstructure, present for example when the coefficients are periodic, almost periodic, or are homogeneous random fields.

Stochastic two-scale convergence in the mean and applications

In this paper, we justify by periodic homogenization the double-porosity model for immiscible incompressible, two-phase flow. The volume fraction of the fissured part and the nonfissured part are kept positive constants and of the same order. The scaling is such that, in the final homogenized equations, the less permeable part of the matrix contributes as a nonlinear memory term. To prove the convergence of the total velocity and of the “reduced” pressure, we use the two-scale convergence since it seems to be appropriate for the problem, even though it would be possible to work with periodic modulation. However, in the final step, the degenerate ellipticity prevents the use of the two-scale convergence method and leads us to use periodic modulation.

Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow

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

We consider the two-phase flow through a dual-porosity medium, characterized by a period of heterogeneity ω, a ratio of global permeabilities ∈K, and a ratio of the order of capillary forces ∈c. The limit when ω tends to zero at different values of ∈K and ∈c gives four classes of global behavior, differing by the type of elementary flows at the one-cell level. We propose a diagram of their predominance. A macro-scale model is constructed by formal homogenization techniques for one of these classes; it shows a nonlinear kinetic relationship for the averaged capillary pressure functions, and leads to a decomposition for the effective phase permeability tensors. A capillary relaxation time is explicitly determined.

Alain Bourgeat

Papers

Approximations of effective coefficients in stochastic homogenization

Stochastic two-scale convergence in the mean and applications

Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow

Two-phase flow in heterogeneous porous media

Effective two-phase flow through highly heterogeneous porous media: Capillary nonequilibrium effects