A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid-Fluid and Fluid-Fluid Interactions

A novel model is presented for estimating steady-state co- and counter-current relative permeabilities analytically derived from macroscopic momentum equations originating from mixture theory accounting for fluid-fluid (momentum transfer) and solid-fluid interactions (friction). The full model is developed in two stages: first as a general model based on a two-fluid Stokes formulation and second with further specification of solid-fluid and fluid-fluid interaction terms referred to as (i = water, oil) and R, respectively, for developing analytical expressions for generalized relative permeability functions. The analytical expressions give a direct link between experimental observable quantities (end point and shape of the relative permeability curves) versus water saturation and model input variables (fluid viscosities, solid-fluid/fluid-fluid interactions strength and water and oil saturation exponents). The general two-phase model is obeying Onsager's reciprocal law stating that the cross-mobility terms and are equal (requires the fluid-fluid interaction term R to be symmetrical with respect to momentum transfer). The fully developed model is further tested by comparing its predictions with experimental data for co- and counter-current relative permeabilities. Experimental data indicate that counter-current relative permeabilities are significantly lower than corresponding co-current curves which is captured well by the proposed model. Fluid-fluid interaction will impact the shape of the relative permeabilities. In particular, the model shows that an inflection point can occur on the relative permeability curve when the fluid-fluid interaction coefficient which is not captured by standard Corey formulation. Further, the model predicts that fluid-fluid interaction can affect the relative permeability end points. The model is also accounting for the observed experimental behavior that the water-to-oil relative permeability ratio is decreasing for increasing oil-to-water viscosity ratio. Hence, the fully developed model looks like a promising tool for analyzing, understanding and interpretation of relative permeability data in terms of the physical processes involved through the solid-fluid interaction terms and the fluid-fluid interaction term R.