Multiphase electrokinetic coupling: Insights into the impact of fluid and charge distribution at the pore scale from a bundle of capillary tubes model

A bundle of capillary tubes model is used to investigate multiphase electrokinetic coupling during the flow of water and an immiscible second phase such as air or oil. The charge on the surface of each capillary is assumed constant, and the impact of charge distribution at the pore scale is investigated by calculating the relative streaming potential coupling coefficient assuming that the diffuse part of the electrical double layer is (1) much less than the capillary radius ("thin") and (2) comparable to the capillary radius ("thick"). The relative coupling coefficient generally decreases with decreasing water saturation, falling to zero at the irreducible water saturation. In the limit of a thin double layer, the relative coupling coefficient at partial saturation is independent of capillary size distribution and depends upon the wettability of the capillaries only if surface electrical conductivity is significant and the irreducible water saturation is small. In the limit of a thick double layer, the relative coupling coefficient depends upon the capillary size distribution and wettability. If water is the only phase that contains an excess of charge, the relative coupling coefficient can be described in terms of the water relative permeability, relative electrical conductivity, and relative excess countercharge density transported by the water. This latter quantity increases with decreasing water saturation in water-wet models and decreases with decreasing water saturation in oil-wet models. It does not scale inversely with water saturation as has been assumed previously and depends upon the pore scale distribution of fluid and charge.