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

Abstract

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 $R_{{i}}$ (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 $\lambda _\mathrm{wo}$ and $\lambda _\mathrm{ow}$ 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 $I>0$ 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 $\hat{{k}}_{\mathrm{rw}} /\hat{{\mathrm{k}}}_{\mathrm{ro}}$ 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 $R_{{i}}$ and the fluid–fluid interaction term R.

Appendix

A1 Positive and negative counter-current relative permeabilities

Equation 45 shows that negative values for counter-current relative permeabilities may occur for certain combination of input parameter values. For both expressions, we note that the denominator is always positive and only the negative sign corresponds to adding negative terms in the nominator. The only way to have negative $k_{\mathrm{rw}}$ is to have counter-current flow (“-”) and low water saturations ($s_{\mathrm{w}}- s_{\mathrm{o}} < 0$). If the fluid–fluid interaction is weak, $k_{\mathrm{rw}}$ can stay positive for all saturations. In particular, it follows from Eq. 45 that a positive $k_{\mathrm{rw}}$ for given saturations is equivalent to have

$$\begin{aligned} I<\frac{I_\mathrm{o} }{\mu _\mathrm{w} s_\mathrm{o} S_\mathrm{o}^{-\beta } \left( {s_\mathrm{o} -s_\mathrm{w} } \right) }, \end{aligned}$$

(A1)

which means that if

$$\begin{aligned} I<\frac{I_\mathrm{o} }{\mu _\mathrm{w} (1-s_{\mathrm{wr}} )(S_\mathrm{o}^*)^{-\beta }\left( {1-s_{\mathrm{or}}^ -s_{\mathrm{wr}} } \right) } \end{aligned}$$

(A2)

(where $S_{\mathrm{o}}^*=\frac{0.5-s_{\mathrm{or}}}{1-s_{\mathrm{or}} -s_{\mathrm{wr}}}$ is the scaled saturation evaluated when the saturations are equal), $k_{\mathrm{rw}}$ will be positive for all saturations. By similar derivation, we obtain that:

$$\begin{aligned} I<\frac{I_\mathrm{w} }{\mu _\mathrm{o} (1-s_{\mathrm{or}} )(S_\mathrm{w}^*)^{-\alpha }\left( {1-s_{\mathrm{or}} -s_{\mathrm{wr}} } \right) }, \quad S_\mathrm{w}^*=\frac{0.5-s_{\mathrm{wr}}}{1-s_{\mathrm{or}} -s_{\mathrm{wr}}}, \end{aligned}$$

(A3)

produces strictly positive counter-current $k_{\mathrm{ro}}$.