Joe Koebbe

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.

Equivalent Permeability and Simulation of Two-Phase Flow in Heterogeneous Porous Media

The problem of calculating equivalent grid block permeability tensors for heterogeneous porous media is addressed. The homogenization method used involves solving Darcys equation subject to linear boundary conditions with flux conservation in subregions of the reservoir and can be readily applied to unstructured grids. The resulting equivalent permeability tensor is stable as defined relative to G-convergence. It is proposed to use both conforming and mixed finite elements to solve the local problems and compute approximations from above and below of the equivalent permeability, respectively. Comparisons with results obtained using periodic, pressure and no-flux boundary conditions and the renormalization method are presented. A series of numerical examples demonstrates the effectiveness of the methodology for two-phase flow in heterogeneous reservoirs.

Numerical Simulation and Homogenization of Diphasic Flow in Heterogeneous Reservoir

By mean of the so called homogenization theory, see for instance [16], we derive mathematically rigorous “effective” reservoir equations from exact local equations of incompressible two-phase flow (miscible or immiscible) in a heterogenous reservoir. The main result is that “effective” equations are exactly of the same type as the original ones. In general cases the effective permeability tensor is given only as a mathematical limit. In some special cases where there is some additional knowledge on the heterogeneities repartition as for instance a spatial periodic repartition, we may really compute the effective parameters and then numerically compare both behaviour in a heterogeneous or in a homogenized reservoir. In [1] and [12], we have presented some simulation of stratified medium; in [2] and [12] we have presented several simulations on spatially periodic heterogeneities. Herein we are presenting only one of such a simulation to illustrate our results.