Relative Permeability

We next present a sample calculation of the relative permeability for the 22% porosity Fontainebleau sandstone. In this case, the pore space is filled with two fluids. One fluid preferentially wets the solid surface and the second fluid is non-wetting. The degree of saturation, $\Theta_w$_w is V_w / V_p, where the V_w is the volume of the wetting phase in the pore space and V_p is the volume of the pore space. Although there is debate as to the correct formulation of the macroscopic two phase flow equations [14], we use the following empirical relation to describe the response of a multiphase fluid system to an external driving force:

$$\vec{v}_1=-\frac{K_{12}}{\mu_2}\nabla P_2 - \frac{K_{11}}{\mu_1} \nabla P_1$$ (10)

$$\vec{v}_2=-\frac{K_{21}}{\mu_1}\nabla P_1 - \frac{K_{22}}{\mu_2} \nabla P_2$$ (11)

Here the K_ij are the components of a permeability tensor and the applied pressure gradient on each fluid component $\nabla P_i$ is from a simple body force, $\nabla P= \rho g$, where g is an acceleration constant. The average velocity of each fluid component is given by $\vec{v}_1$ and $\vec{v}_2$. The forcing can be applied to each phase separately allowing determination of the off-diagonal terms in the permeability tensor. The viscosity µ _i is the same for both fluids. Relative permeability data is usually presented in terms of constant capillary number, $C_a=\frac{\mu v}{\gamma}$, where $\gamma$ is the interfacial surface tension. For our body force driven fluids, we can define an effective capillary number, $C^*_a$, by replacing v with the Darcy velocity so that $C^*_a= \frac{\mu <v>}{\gamma}=\frac{k \rho g}{\gamma}$. Figure 5 shows the relative permeability of the $\phi=22~\%$ = 22% rock for the cases of $C^*_a= 7.5\times 10^{-4}$ = 7.5 x 10^-4 and 7.5 x 10^-5.

Figure 5: Relative permeabilities of 22% porosity Fontainebleau sandstone versus wetting fluid saturation, $\Theta _W$_W. The solid and dashed lines correspond to $C^*_a= 7.5\times 10^{-4}$ = 7.5 x 10^-4 and $C^*_a =7.5\times 10^{-5}$ = 7.5 x 10^-5 respectively. The lower curves correspond to the off-diagonal elements of the permeability tensor with the * denoting the case where the nonwetting fluid is driven.