Results

Fig. 6 shows the relative diffusivity in the wetting and non-wetting phases as a function of wetting fluid saturation for the nonoverlapping sphere model (System a). Clearly there is a strong dependence of relative diffusivity on saturation. For instance, there is a significant decrease in diffusivity (in either phase) at or around a wetting phase fraction of about 50%. Here D_ri is about 0.25. (For the case of System c, D_ri  0.15 when S=0.5 see Fig. 7)

Figure 6: Relative diffusivity curves for both the wetting (triangles) and non-wetting (squares) phase in System a. The saturation, S, corresponds to the wetting phase. The dashed lines correspond to asymptotic approximations in text. The solid lines are fits to equations 16 and 17.

Figure 7: Relative diffusivity curves for both the wetting (triangles) and non-wetting (squares) phase in System c. The saturation, S, corresponds to the wetting phase.

It was found that values of D_ri drop off much more quickly than the phase fraction of the fluid. Note, for a tube geometry, D_r is proportional to S. For the case of the non-wetting fluid, D_ri goes to zero because the non-wetting fluid becomes disconnected as isolated blobs of NW fluid form in the pore space when the fraction of non-wetting fluid decreases. In the low saturation of wetting fluid regime, the wetting fluid fills the regions near neighboring spheres where there is more surface area per unit volume of pore space (hence reducing energy). As a result, the wetting fluid has difficulty forming a connected path except for a possible thin film. Regardless, any contribution to the conductivity due to a presence of a very thin film would be so small as to be negligible. Indeed, in real porous rocks, the wetting fluid initially resides in small isolated imperfections of the pore-solid surface.

Four fluid saturation regimes are clearly identifiable which correspond to endpoints of the relative diffusivity curves. First, let us consider conduction in the wetting fluid. In the regime of high wetting saturation we can imagine that the non-wetting fluid begins to form little spherical droplets as the non-wetting phase fraction increases. Small perturbations to electric fields by nonconducting spherical objects is well understood and can be calculated using a cluster expansion approach [18]. To second order in volume fraction, c, of nonconducting solid the conductivity is given by / _o = 1 - 3/2c +  0.558 c² . In Fig. 6 we plot this equation in the regime of high wetting saturation. The agreement in the high saturation regime is very good but as c increases, higher powers of c become important. Also, the morphology of the non-wetting fluid becomes less well approximated by spherical inclusions.

In the regime of low wetting saturation, the fluid begins to probe the surface tortuousity as it fills in regions containing the smallest pores. Hence it is not expected that the conductivity will increase rapidly with saturation in this regime. Consider the case of two neighboring spheres in contact with a controlled amount of wetting fluid. As the amount of wetting fluid is increased it will accumulate more so in the region near the point of contact of the spheres in order to reduce the total surface energy. For such a system there is no conducting path until the interstitial region is filled. For a three dimensional bead pack of uniform sized spheres constructed in such fashion, the critical saturation S_c at which a connected path forms is 1/3. In the spirit of the Pade approximation [15] we fit the data to an empirical polynomial function,

D_ri=a(S-S_c )+ b(S-S_c ) ² +c (S-S_c )³ , (16)

where a, b and c must depend on the slopes at (and) the endpoint values of the D_r curve. Unfortunately, there is no theoretical prediction for the slope of D_r in the low wetting saturation regime. Also one would have to accurately determine S_c to complete the fit which is beyond the scope of our calculations (also the possibility of a thin conducting layer is ignored here). Nevertheless, it was found that the above simple polynomial function fit our data quite well. Fig. 6 includes a fit of the above equation to the data. While the function can be adjusted to make a good fit over the given data set, it is likely that a careful fitting very near the percolation threshold may be weak due to finite size and resolution effects.

Now consider the case of conductivity in the non-wetting phase at low wetting phase saturation. As expected, the conductivity decreases as the fraction of non-wetting fluid decreases. Since the wetting fluid prefers to fill the interstitial regions and coat the solid pore surface, we may think of increasing the degree of wetting fluid saturation as effectively reducing the porosity somewhat akin to a grain consolidation effect [11] (i.e. fixed spheres whose radius gradually increases). The conductivity of such a system is well characterized by  =  ⁿ, where n is in the range of -1.5 to -2. Since we are describing our conductivity in terms of saturation, we may write, $\sigma = \phi_{o}^n S^n$, where $\phi_o$ _o is the porosity. Therefore, D_ri =  / _o = S ⁿ ( Archie's 2nd law) [8]. In Fig. 6, we include a plot of the previous equation for the non-wetting phase at low wetting saturation, with very good agreement for n $n \approx -1.7$  -1.7.

As mentioned previously, as the degree of saturation of the non-wetting phase decreases, the non-wetting fluid will eventually form a set of disconnected blobs such that it no longer percolates (see Fig. 4). This scenario reduces to a type of percolation problem describing clusters of fluid which are limited to residing in the pore space. Here, near the percolation threshold, the conductivity should scale as (S-S_c )^t, where S_c is the saturation of the non-wetting phase needed for the non-wetting blobs to percolate. In general, S_c depends on the class of pore structure studied and should increase as the porosity is reduced and the tortuousity of the pore space increases. An accurate determination of S_c for different porous media can be difficult in that finite size effects need to be considered [2]. Further, if the non-wetting fluid had been directly injected into the porous medium instead of filling the pores by the phase separation process a totally different fluid morphology would have resulted with the non-wetting phase generally remaining connected. It is likely the local degree of saturation would not be uniform throughout the medium leading to other finite-size effects. In this case, the ingress of non-wetting fluid is more akin to an invasion percolation or non-wetting fluid invasion model [7] [17]. Regardless, in System a it was found that the non-wetting phase became disconnected at $S \simeq 0.27$ to 0.3 (of the non-wetting phase). It is well known that the conductivity of overlapping spheres has the following scaling behavior:    (  - _c ) ^t, where $\phi_c$ _c [10] is the critical porosity at which the pore space becomes disconnected (no longer percolates) and t is the critical exponent. For the overlapping sphere model, $t \simeq 2.4$, while artifacts from digitizing can lower t to 2 [10]. Our model is not the same as the overlapping sphere model, because the non-wetting phase resides in a subspace of the porous region which will deform the shape of the non-wetting blobs. Also, restricting the non-wetting fluid to a subspace should affect the percolation threshold. On the other hand, the critical exponent, if in the same universality classes (i.e. values of critical exponents typically fall into groups called universality classes [17]), would be unchanged for our model. It is found that D_ri, at saturations near the percolation threshold of the non-wetting blobs can be described by a power law. Reasonably good fits were obtained with our data using t $t \approx 2.2$ 2.2. While we cannot accurately determine t and S_c with this simulation we believe that our results are consistent with such scaling assumptions. It remains to be seen whether our model is in the same universality class as the overlapping sphere model.

To fit data over the entire range of saturation where the non-wetting fluid percolates, the following empirical equation worked reasonably well

$$D_{ri}=(1-S)^{n}(\frac{S_c-S}{S_c})^m$$ (17)

where S is the wetting phase fraction. Clearly D_ri  1 as  S  0, and D_ri   0 as  S  S_c . In general, the exponents can be chosen to match the slopes of D_ri at S=1 and S=S_c. A reasonable first choice for the exponents m and n is the percolation exponent and the Archie's law exponent respectively. Fig. 6 shows a fit to the data. Here good fits was obtained for n=-1.7 and m=2.2.

In Fig. 8 we show D_ri for the case where the contact angle $\theta =90^o$  = 90ºdegrees in System b. Note that in this case neither phase percolates below 50 percent saturation. Included in Fig. 8 is data for the case of completely wetting/non-wetting. At high saturations there is a small but noticeable difference in the initial slopes between the two cases (more so in the wetting regime). As the saturation level decreases, the case where $\theta =90^o$  = 90º decreases much more rapidly as the fluids form disconnected regions. The bi-continuity of the two components is more easily maintained in the perfect wetting/nonwetting case because each fluid then is limited to the pore surface and center of the pore respectively. Is is found that the empirical polynomial function (Eq. 16) fit the data well. It is interesting that in Archie's original paper [8] it was noted that the scaling behavior of the conductivity was not strongly dependent on the fluids that filled the pore space, even if the fluids (gas and oil) filled the pore space in a different manner. This result is not too surprising in light of the fact that the present study shows that the scaling behavior of conductivity at high saturations of either wetting or non-wetting fluid is not very different. In other cases it is clear that Archie's 2nd law will break down, especially when the non-wetting phase nears its percolation threshold and when neither fluid preferentially wets a surface.

Figure 8: Relative diffusivity associated with System b for the case of perfect wetting/non-wetting fluids (triangles) and when neither fluid preferentially wets ( $\theta =90^o$ = 90º ) the surface (squares).