Non-uniqueness in capillary pressure-saturation-relative permeability relationships for two-phase flow in porous media: Interplay between intensity and distribution of random micro-heterogeneities

The two-phase flow behaviour in porous media is determined on the basis of capillary pressure-saturation-relative permeability relationships (P^c-S-K_r). These relationships are highly non-linear and obtained by laboratory experiments on porous samples, typically around 10-12 cm in length. It is normally assumed that these samples are homogeneous; however it is well-known that this is in fact not the case and that even at this scale micro-scale heterogeneities exist. Two-phase flow experiments on soils with different properties (e.g., particle and pore size distribution, permeabilities, etc) result in different P^c-S-K_r relationships implying that they cause non-uniqueness in these curves. Recent work has shown that the presence of the micro-heterogeneities has a significant effect on the measured P^c-S-K_r relationships and they cause non-uniqueness in these relationships. In the previous work in this area, the micro-heterogeneity effects on the P^c-S-K_r relationships have been analysed in a number of contexts, e.g., uniformly distributed heterogeneities (simplified cases), various binary sand combinations, hydraulic parameters (e.g., entry pressure, permeability), boundary conditions, etc. There is also some evidence that the intensity and distribution of the micro-heterogeneities affect the P^c-S-K_r relationships. In the present work we use numerical simulations to investigate further the nature of these effects, in particular how the interplay between the intensity and random distribution of micro-heterogeneities affect the P^c-S-K_r relationships. Seven randomly heterogeneous patterns have been defined. These domains represent coarse sand media with fine sand blocks embedded in them. The domain size () has been chosen so that it represents a typical laboratory scale device. The results of the simulations show that it is particularly important to take into account both the intensity and distribution of heterogeneity when determining the effective P^c-S-K_r relationships of a sample. Further, there is a complex interplay between the intensity and distribution of micro-scale heterogeneities which determines the P^c-S-K_r curves. This observation is in contrast to the results of domains with uniformly distributed heterogeneities. We have found that in general if the intensity of heterogeneity is high; the irreducible wetting phase saturation (S_iw) of the sample is also high. The direction of flow and the orientation of the samples also have significant effects. For example, the injection of an immiscible phase from the top (vertically downward) of water saturated porous domain leads to a lower S_iw than injecting on horizontal plane. On the other hand, injection from the bottom (vertically upwards) leads to a higher S_iw. As expected, the distribution of heterogeneity has a significant effect on the saturation distribution and the shape of the P^c-S-K_r curves. However, we show that if the heterogeneities are distributed in such a way that they are closer to the boundary of injection, the irreducible wetting phase saturation is higher.