Alain Bourgeat

Two-phase flow

In practical applications, the problem of modeling two-phase flow behavior is of greatest importance. In petroleum engineering the so-called enhanced oil recovery (EOR) process is based on displacing a fluid (oil) by another one (gas or water, for example). In soil science, water and contaminant or water and air are involved in many underground flows in the unsaturated zone. Domestic gas, before delivery, is kept in natural reservoirs and moved by water and a cushion gas.

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

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.

Asymptotic homogenization of laminated piezocomposite materials

Abstract The objective of this paper is to apply the technique of asymptotic homogenization to determine the effective elastic, piezoelectric and dielectric moduli of a laminated piezocomposite medium with a periodic structure. Each periodic cell of the medium can possess any finite number of piezoelectric layers. The general formulae obtained are a generalization of those that appear in chapter 5 of Pobedria (Pobedria, B. E. (1984) Mechanics of Composite Materials . Moscow State University Press, Moscow (in Russian)) and involve both cases of Newnhams connectivity theory (Newnham, R. E., Skinner, D. P. and Cross, L. E. (1978) Connectivity and piezoelectric-pyroelectric composites. Materials Research Bulletin 13 , 525–536) for layered piezoelectric media. We calculate explicitly overall effective characteristics for three examples of such layered media. For the particular case of a binary layered medium, connected in parallel, with transversely isotropic constituents such formulae transform exactly to the formulae for effective constants obtained by Benveniste et al. (1992) in which a different method of homogenization was used. Finally, we apply these results to a piezocomposite material and obtain new piezoelectric with better global properties for hydrophone applications.

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

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.

Numerical simulation and homogenization of 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.

Effective model of two-phase flow in a porous medium made of different rock types

We consider the behavior of immiscible and incompressible two-phase flows in a porous medium made of several rock types. Each rock type, is defined by one porosity, one absolute rock permeability tensor, two relative permeability curves and one capillary pressure curve. Using two-scale convergence, we get an homogenized model which governs the global behavior of the flow. In this model, macroscopic equations are generally coupled with local equations; but under some assumptions, these two problems can be decoupled.

Homogenization of a polymer flow through a porous medium

We derive the constitutive laws that connect the velocity and the pressure gradient in stationary incompressible purely viscous non-Newtonian ows through porous media. Starting from the stationary non-Newtonian Navier-Stokes system, with viscosity obeying Carreau and power laws, and using the homogenization theory, we obtain a variety of laws according to the size of the Reynolds number. It appears that there is some critical Reynolds number such that after reaching it the ow regime in porous media changes from a linear, Darcys type law, to a nonlinear and nonlocal law. We prove uniqueness for the homogenized problems. The proof of convergence is given under the usual assumption of periodic geometry. The main tools for getting the laws in nonlinear ow regimes are the two-scale convergence, L r-estimates for the pressure and monotonicity methods. The convergence for the linear ow regime is obtained using the energy method and precise estimates for the viscosity as function of strain tensor.

Singular Double Porosity Model

We consider the linear parabolic equation describing the transport of a contaminant in a porous media crossed by a net of infinitely thin fractures. The permeability is very high in the fractures but very low in the porous blocks. We derive the homogenized model corresponding to a net of infinitely thin fractures, by means of the singular measures technique. We assume that these singular measures are supported by hyperplanes of codimension one. We prove in a second step that this homogenized model could be obtained indistinctly either by letting the fracture thickness, in the standard double porosity model, tend to zero, or by homogenizing a model with infinitely thin fractures.

A HOMOGENIZED MODEL OF AN UNDERGROUND WASTE REPOSITORY INCLUDING A DISTURBED ZONE

The mathematical model describing the leaking of an underground waste repository should include the multiscale geometry and the large variation of the geological coefficients. Numerical simulations for performance assessments using such a local and detailed model are unrealistic, and there is a need to replace this local model (mesoscopic model) by a global one (macroscopic model). After introducing a small parameter

NONLINEAR EFFECTS FOR FLOW IN PERIODICALLY CONSTRICTED CHANNEL CAUSED BY HIGH INJECTION RATE

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