Relative permeability

We next present a sample calculation of the relative permeability for the 22 % porosity Fontainebleau sandstone. Although there is debate as to the correct formulation of the macroscopic two phase flow equations [10], we use the following empirical relation to describe the response of a multiphase fluid system to an external driving force:

$$\vec{v}_1=-\frac{K_{12}}{\mu_2}\nabla P_2 - \frac{K_{11}}{\mu_1} \nabla P_1$$ (9)

$$\vec{v}_2=-\frac{K_{21}}{\mu_1}\nabla P_1 - \frac{K_{22}}{\mu_2} \nabla P_2$$ (10)

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 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 $C^*_a= \frac{\mu <v>}{\gamma}=\frac{k \rho g}{\gamma}$. Below is a plot, Fig. 5, of the relative permeability of the = 22 % $C^*_a= 7.5\times 10^{-4}$ 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 $C^*_a= 7.5\times 10^{-4}$ and $C^*_a =7.5\times 10^{-5}$ 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.

Clearly, as the forcing decreases the relative permeability decreases. Also note that at lower wetting saturation the relation K₁₂=K₂₁ holds fairly well. This is the well known Onsager relation. However, as the wetting phase increases, this relation appears to break down. In this regime, the nonwetting phase is beginning to form disconnected blobs that do not respond in a linear fashion to the applied force due to pinning effects as the nonwetting blobs are pushed through the smaller pores.

The LB code can be easily extended to model three or more fluid components. As a simple test case we considered a three component system with each component having a pore volume fraction of 1/3. In addition, two phases were made non-wetting while the third was wetting. A forcing was applied to one of the nonwetting phases. We found, over the range $7.5\times10^{-4}<C^*_a<3.75\times10^{-3}$, there was an approximately 30 % to 35 % decrease in flow of the driven nonwetting fluid as compared to the case where 1/3 fluid was non wetting and 2/3 was wetting. Presumably, this decrease can be understood as the non-driven nonwetting phase interfering with the flow of the driven nonwetting phase as the fluids move through narrow channels.