Three-phase capillary pressure and relative permeability relationships in mixed-wet systems

A simple process-based model of three-phase displacement cycles for both sp reading and non-spreading oils in a mixed-wet capillary bundle model is pre sented. All possible pore filling sequences are determined analytically and it is found that the number of pore occupancies that are permitted on phys ical grounds is actually quite restricted. For typical non-spreading gas/oi l/water systems, only two important cases need to be considered to see all types of allowed qualitative behaviour for non-spreading oils. These two ca ses correspond to whether water or gas is the 'intermediate-wetting' phase in oil-wet pores as determined by the corresponding contact angles, that is , cos theta (o)(gw) > 0 or cos theta (o)(gw) < 0, respectively. Analysis of the derived pore occupancies leads to the establishment of a number of rel ationships showing the phase dependencies of three-phase capillary pressure s and relative permeabilities in mixed-wet systems. It is shown that differ ent relationships hold in different regions of the ternary diagram and the morphology of these regions is discussed in terms of various rock/fluid pro perties. Up to three distinct phase-dependency regions may appear for a non -spreading oil and this reduces to two for a spreading oil. In each region, we find that only one phase may be specified as being the "intermediate-we tting' phase and it is only the relative permeability of this phase and the capillary pressure between the two remaining phases that depend upon more than one saturation. Given the simplicity of the model, a remarkable variet y of behaviour is predicted. Moreover, the emergent three-phase saturation- dependency regions developed in this paper should prove useful in: (a) guid ing improved empirical approaches of how two-phase data should be combined to obtain the corresponding three-phase capillary pressures and relative pe rmeabilities; and (b) determining particular displacement sequences that re quire additional investigation using a more complete process-based 3D pore- scale network model.