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,
, between the two fluids is introduced
which effectively perturbs the equilibrium velocity [12]
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:
Gii'a= 2G; |
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Gii'a= G; |
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Gii'a= 0; | i=i' |
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 180o, 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
Here s is taken as one or zero depending on whether the region is solid or pore respectively and Wia is adjusted so that the fluid is either wetting or non-wetting (positive or negative).
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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].
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In Fig. 5 we show the case where neither
fluid preferentially wets ( = 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.
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