Effective Relative Permeabilities and Capillary Pressure for One-Dimensional Heterogeneous Media

The paper presents an analytical construction of effective two-phase parameters for one-dimensional heterogeneous porous media, and studies their properties. We base the computation of effective parameters on analytical solutions for steady-state saturation distributions. Special care has to be taken with respect to saturation and pressure discontinuities at the interface between different rocks. The ensuing effective relative permeabilities and effective capillary pressure will be functions of rate, flow direction, fluid viscosities, and spatial scale of the heterogeneities.The applicability of the effective parameters in dynamic displacement situations is studied by comparing fine-gridded simulations in heterogeneous media with simulations in their homogeneous (effective) counterparts. Performance is quite satisfactory, even with strong fronts present. Also, we report computations studying the applicability of capillary limit parameters outside the strict limit.     

A Variational Model of Disjoining Pressure: Liquid Film on a Nonplanar Surface

Variational methods have been successfully used in modelling thin liquid films in numerous theoretical studies of wettability. In this article, the variational model of the disjoining pressure is extended to the general case of a two-dimensional solid surface. The Helmholtz free energy functional depends both on the disjoining pressure isotherm and on the shape of the solid surface. The augmented Young–Laplace equation (AYLE) is a nonlinear second-order partial differential equation. A number of solutions describing wetting films on spherical grains have been obtained. In the case of cylindrical films, the phase portrait technique describes the entire variety of mathematically feasible solutions. It turns out that a periodic solution, which would describe wave-like wetting films, does not satisfy Jacobi’s condition of the classical calculus of variations. Therefore, such a solution is nonphysical. The roughness of the solid surface significantly affects liquid film stability. AYLE solutions suggest that film rupture is more likely at a location where the pore-wall surface is most exposed into the pore space, and the curvature is positive.     

Homogenization of two-phase flow: high contrast of phase permeabilityHomogénéisation d'écoulement diphasique : grand contraste de perméabilité d'une phase

The steady-state two-phase flow non-linear equation is considered in the case when one of phases has low effective permeability in some periodic set, while on the complementary set it is high; the second phase has no contrast of permeabilities in different zones. A homogenization procedure gives the homogenized model with macroscopic effective permeability of the second phase depending on the gradient and on the second order derivatives of the macroscopic pressure of the first phase. This effect cannot be obtained by classical (one small parameter) homogenization. To cite this article: G.P. Panasenko, G. Virnovsky, C. R. Mecanique 331 (2003).     

A Steady-State Upscaling Approach for Immiscible Two-Phase Flow

The paper presents a model for computing rate-dependent effective capillary pressure and relative permeabilities for two-phase flow, in 2 and 3 space-dimensions. The model is based on solving the equations for immiscible two-phase flow at steady-state, accounting for viscous and capillary forces, at a given external pressure drop. The computational performance of the steady-state model and its accuracy is evaluated through comparison with a commercial simulator ECLIPSE. The properties of the rate-dependent effective relative permeabilities are studied by way of computations using the developed steady-state model. Examples presented show the dependence of the effective relative permeabilities and capillary pressures, which incorporate the effects of fine scale wettability heterogeneity, on the external pressure drop, and thereby on the dimensionless macro-scale capillary number. The effective relative permeabilities converge towards the viscous limit functions as the capillary number tends to infinity. Special cases, when the effective relative permeabilities are rate-invariant, are also studied. The applicability of the steady-state upscaling algorithm in dynamic displacement situations is validated by comparing fine-gridded simulations in heterogeneous reservoirs against their homogenized counterparts. It is concluded that the steady-state upscaling method is able to accurately predict the dynamic behavior of a heterogeneous reservoir, including small scale heterogeneities in both the absolute permeability and the wettability.     

Coarse Scale Capillary Pressure in Heterogeneous Reservoirs

The problem of capillary pressure upscaling and generation of initial water saturation data on the simulation scale consistent with the underlying geological model is addressed analytically. The approach is based on the concept of random spatial variations of permeability and porosity. We have revised the previously published expression for the coarse scale capillary pressure (Desbarats, Water Resour Res 31, 281–288) and rigorously derived a new one avoiding the unnecessary assumptions. Both expressions are evaluated by their comparison to the directly averaged realizations of the corresponding random fields. Generally, the new expression is superior to the one previously published. The important features of the analytical coarse scale capillary pressure expression are the dependence of the endpoint water saturation on the drainage pressure, or equivalently, on the elevation over the free water level and an additional multiplier taking into account the variation of rock properties within the coarse scale grid-block. George A. Virnovsky was the main editor of this article and we, the co-writers, want to honour his memory. He died suddenly on 12 March 2008 before this article was published. George was an internationally recognized scientist especially in the areas of multi-phase upscaling, interpretation of special core analysis and pore scale modelling.