*Oil & Gas Science and Technology - Rev. IFP*Vol. 49 (1994), No.4, pp. 345-358

## Expression en termes d'énergie pour la perméabilité absolue effective. Application au calcul numérique d'écoulements diphasiques en milieu poreux

### Expression in Energy Terms for Absolute Effective Permeability. Application to the Numerical Computing of Two-Phase Flows in Porous Media

Université de Douala

Le cadre de ce travail est le calcul des paramètres pétrophysiques effectifs d'un milieu poreux hétérogène pour le simulateur de réservoirs pétroliers. Après le choix d'un modèle d'écoulement dans un milieu poreux hétérogène comportant une microstructure périodique nous rappelons brièvement les grandes étapes de la méthode des échelles multiples pour l'homogénéisation de ce modèle. Cela nous conduit à la formule classique d'homogénéisation de la perméabilité absolue. Par la suite nous présentons une démarche originale permettant de passer de cette formule classique à une formule plus simple (d'un point de vue numérique) s'exprimant en termes d'énergie dissipée par les forces de viscosité locales et caractérisant le milieu hétérogène périodique considéré. Nous démontrons ensuite, sous certaines hypothèses, l'égalité entre les énergies dissipées par les forces de viscosité associées respectivement à l'écoulement local et à l'écoulement macroscopique. Nous terminons par la présentation de quelques résultats numériques concernant des modèles d'écoulement diphasique incompressible.

Abstract

This project falls within the general framework of computing the effective petrophysical parameters characterizing a heterogeneous medium when it is considered from a macroscopic viewpoint as opposed to a description on the local scale. The concept of scale inevitably appears as soon as the concept of heterogeneity is broached. Depending on the applications planned, it is easy to define different observation scales of natural porous media. The local scale is a small in which the porous medium may be considered to be continuous, and in which the hydrodynamic equations are written for the fluid phase. At this scale, the elements making up the medium are sufficiently small porous volumes (compared to the dimensions of medium) to be considered as points (in the mathematical sense), but large enough to encompass pores with different and varying diameters. The macroscopic scale corresponds to a scale in which the local petrophysical parameters are averagedfor volumes liable to contain several geologic structures, such as sand, limestone and clay. The average parameters considered are constants that can be used to make an overall (or macroscopic) description of flow in the domain occupied by the porous medium. These are the average parameters that are called the effective petrophysical (or homogenized) parameters. They are used to simulate petroleum reservoirs. After having chosen a flow model in a heterogeneous porous medium containing a periodic microstructure, we briefly review the major phases in the multiple-scale method for homogenizing this model. This leads us to a conventional formula giving the coefficients for absolute homogenized permeability (Eq. 26). Then we describe an original procedure for going from the conventional formula to a simpler formula (from the numerical standpoint) expressed in terms of energy dissipated by local viscosity forces and characterizing the periodic medium being considered. In this part of the project, an essential phase is the formulating of so-called local equations in a form that better brings out their physical meaning by a judicious change in the unknown function. The integral transformation that results, in the equation for homogenized coefficients, opens up the way to obtaining the above-mentioned simple formula (Eq. 34). We then show, given various assumptions, the equality between the energies dissipated by viscosity forces associated respectively with local and macroscopic flows (Theorems 3 and 4). Theorem 3 is actually a specific case of Theorem 4, which in turn is used to interpret all the homogenized coefficients given by Eq. 34. This project ends with an application to the numerical analysis of an incompressible water/oil two-phase flow that is horizontal and twodimensional, in a heterogeneous porous medium with a periodic structure (Figs. 1a and 1b). We have assumed the effects due to capillary forces to be negligible, and we have considered relative permeabilities depending continuously on water saturation, with these relative permeabilities being linear piece by piece (Fig. 1c). It is well known that single-phase flow in a porous medium is described correctly by Darcy's law, in which absolute permeability (spatially variable) plays the role of diffusion coefficients. However, the situation is much more complex for multiphase flows in which, in the absence of an unquestionably established theory, experimental observations lead us to assume that the flow of each fluid phase is described locally by an equation of Darcy's lawtype in which absolute permeability is replaced by the product of absolute permeability and the relative permeability of the phase being considered. In [1] and [3] and more explicitly in [6], we show that for incompressible two-phase flow, if we assume capillary forces to be negligible, then the homogenizing of the Absolute Permeability x Relative Permeabilityproduct is equal to Absolute Homogenized Permeability x Relative Homogenized Permeability . Hence the numerical computing of water/oil flow requires determining the homogenized absolute and relative permeabilities. In our numerical simulations of this flow, the very conventional Upstreampetroleum method has been used for the homogenized relative permeabilities, while Eq. (34) (result proven) was used to determine the homogenized absolute permeability. We compare our results with those obtained by a so-called algebraic equation (concerning the homogenized absolute permeability) used in some big industrial codes. In our simulations, the reference solutions are obtained with a fine mesh pattern (of the domain) in which local values of petrophysical parameters are used. The results of the simulations are gathered in Figs. 2, 3 and 4. These results show that Eq. 34 can be used to obtain, for the situations investigated, very satisfactory homogenized solutions (i. e. closer to the reference solutions) when compared to Eq. 49.

*© IFP, 1994*