Adequate numerical prediction and forecasting of reservoir performance rely on the knowledge of relative permeabilities. In gravity driven processes the flow is complex, consisting of co-current and counter-current flows. This work describes the characteristics of gravity driven flow and provides generalized permeabilities (or mobilities) for such flow. The results can be used in improved numerical simulation of gravity drainage processes.
The standard multiphase relative permeability approach used in existing reservoir simulators is sufficient for describing steady state processes for either co- or countercurrent flow. However, in the case of complex three dimensional flow that combines both co- and counter-current components, such as that in gravity drainage processes, transport coefficients have more complex form and should be addressed through matrix formulation. Furthermore, in the case of non-steady state counter-current flow in gravity drainage, the effective mobility becomes a more complex function of steady state relative permeabilities and the standard approach no longer represents the physics of the process accurately.
During the last decade, new matrix formulation of phase mobilities has been extensively discussed in the literature. The theory is mature and its implementation is long overdue. The new momentum equations are a natural extension of the standard formulation with proper representation of cross influences between flowing phases. The theory provides improved flow description as compared to the standard relative permeability concept in modeling complicated non-steady state processes such as gravity drainage. This is due to:
correct description of multiphase flows,
easy incorporation into existing multiphase simulators,
being a natural extension of existing standard relative permeability theory.
A description of multiphase flow in porous media through matrix representation was initiated by Rose (1969). A significant number of papers on the subject for example, Kalaydjian (1987, 1990), Spanos, et al (1988), Rose (1990a, 1993), Goode et al (1993), Bentsen et al (1993, 1994) along with many others, have appeared since then. All these works emphasized effects of viscous lubrication of two phases simultaneously flowing in porous media, so called Yuster effect. The significance of the cross terms in flow equations and even their existence was, until recently, only a theoretical statement. The experimental work by Dullien and Dong (1996) illustrated that there is no doubt that pressure gradient in one phase can drag the other phase, even if the pressure gradient in the second phase is zero. Recent theoretical work has shown that viscous friction between phases is not the only reason that gives rise to non-zero cross-coefficients in the transport equations. Capillary coupling between phases can lead to similar effects (Babchin and Yuan, 1997). This paper discusses application of matrix formulation of transport equations to the description of gravity drainage processes.
We start our consideration with a one dimensional gravity drainage situation as shown in Figure 1. Due to fluids incompressibility and the restricted volume of the cell, the volumetric fluxes of oil and water are equal in magnitude and opposite in direction. Each of these fluxes can be represented through Darcy approach:
