Interaction Potential

In order to model the phase separation of fluids in porous media an interaction between the fluids is needed to drive them apart. Here a force, $\frac{dp^i}{dt}({\bf x})$, between the two fluids is introduced which effectively perturbs the equilibrium velocity [12]

$$n^i({\bf x}){\bf v}^{'}({\bf x})=n^i {\bf v}({\bf x})+ \tau_i \frac{dp^i}{dt}({\bf x})$$ (11)

where v' is the new velocity used in the equations 5 and 6.

We use a simple interaction that depends on the density of each fluid, as follows:

$$\frac{dp^i}{dt}({\bf x})=-n^i({\bf x})\sum_{i'}^ {S}\sum_{a}G_{ii'}^{a} n^{i'}({\bf x}+{\bf e}_a) {\bf e}_a$$ (12)

					Gii'a= 2G;			$\vert{\bf e}^a\vert = 1$
					Gii'a= G;			$\vert{\bf e}^a\vert = \sqrt{2}$
					Gii'a= 0;			i=i'

where G is a constant which controls the strength of interaction. The forcing term is then related to the density gradient of the fluid. It has been shown that the above forcing term will drive the phase separation and naturally produce an interfacial surface tension effect consistent with the Laplace law boundary condition [2] where at the boundary between two fluids there is a pressure drop proportional to the local curvature of the interface.

At the point where the fluid-fluid interface meets a solid, a contact angle, is defined by the planes tangent to the fluid-fluid interface and the fluid-solid interface (Fig. 2). For  = 90º neither fluid preferentially wets the surface. When  = 0º or 180^o, the fluids are wetting and nonwetting respectively. To model fluids with wetting or non-wetting properties, with respect to the solid phase, a fluid-solid interaction is included in equation 12

$$-n^i({\bf x})\sum_{a}W_{i}^{a} s({\bf x}+{\bf e}_a) {\bf e}_a$$ (13)

Here s is taken as one or zero depending on whether the region is solid or pore respectively and W_i^a is adjusted so that the fluid is either wetting or non-wetting (positive or negative).

Figure 2: A static contact angle, , defined where the fluid-fluid interface meets a solid surface, is obtained as the result of the balance of interfacial surface tension forces .

In our simulations the pore space is initially saturated with a homogeneous mixture of two fluids with a given mass ratio. The fluids then separate until reaching an equilibrium state. Fig. 3 shows the final position of each phase in the overlapping sphere model for a wetting/non-wetting mixture. Here the degree of saturation of each phase is equal. Note that the wetting fluid covers the solid surface and tends to fill the smaller pores. The non-wetting fluid lies mostly in central parts of the pores. For the above saturation, both the wetting and non-wetting phase form percolating networks through the pore space. As the wetting phase saturation is decreased, the wetting fluid will typically form a thin layer on the solid surface probing the surface tortuousity. Due to numerical resolution limits, we cannot accurately calculate the diffusivity in this low saturation regime. In contrast, as the non-wetting phase fraction saturation decreases, the non-wetting fluid begins to form disconnected regions of isolated clusters or "blobs" of non-wetting fluid (see Fig. 4). In this saturation regime, diffusive transport in the nonwetting phase should be consistent with percolation ideas [7].

Figure 3: Image of phase-separated binary mixture of fluids (50/50 mixture) in porous media (system a). The red region represents the wetting fluid and the blue region is the non-wetting fluid.

Figure 4: Image of a phase-separated binary mixture of fluids (20/80 mixture by volume) in porous media (System a). The red region represents the wetting fluid (80 percent) and the blue region is the non-wetting fluid (20 percent). In this low non-wetting saturation the non-wetting fluid forms disconnected blobs.

In Fig. 5 we show the case where neither fluid preferentially wets ( $\theta =90^o$ = 90º) the solid. Note the dramatic difference in morphology of the two fluids from that shown in Fig. 3. Here the two fluids appear to isolate themselves into local regions. In this case, it may be more difficult for the two fluids to form a bicontinuous phase through the pore space since neither fluid lies solely along the solid surface or in the middle of the pore. After the system has relaxed to an equilibrium position, each fluid is labeled and given a conductance of either 1 or zero, while the solid phase is assigned zero conductance. The conductivity is then determined in each separate fluid.

Figure 5: Image of phase separated binary mixture (50/50 mixture by volume) in a porous medium (system a). Here neither fluid preferentially wets the pore surface.