Abstract

Many rock surfaces are (weakly) charged, due to the surface chemistry of the rock. Counterions, which neutralize this charge, form a charge cloud in the fluid immediately adjacent to the rock surface. When the fluid flows, these ions are convected, and electric fields are set up to ensure that electrical neutrality is maintained; the generated potential is referred to as streaming potential. We study numerically the streaming potential generated by flow along a small cylindrical capillary, meant to represent a porous medium at pore scale. In the case of single-phase flow, it is well known that the potential difference ΔΦ between the two ends of the capillary is proportional to the pressure drop Δp in the fluid. This relationship holds for streaming potentials in complex porous media such as rocks. The goal of this work is to investigate the effect of a second phase on the relation between ΔΦ and Δp by considering the presence of a drop in the capillary. The streaming potential is modified by the change of resistivity of the pore, as well as by the change in the convective electric current along the capillary wall.

If gravity effects are neglected, the drop motion (hydrodynamics) is determined by three independent parameters: the size a of the undeformed drop relative to that of the capillary, the viscosity ratio λ between the drop phase and the wetting phase, and the capillary number Ca, which measures the relative importance of viscous and capillary forces. Assuming that the drop phase is a perfect insulator, and that the drop surface is uncharged, the electrokinetic problem, which yields the streaming potential, is solely determined by the hydrodynamics. The results indicate that the difference ΔΦ-Δp (in dimensionless variables) is always positive. Since this quantity is independent of the pore length (unlike ΔΦ and Δp), it is an interesting quantity to measure experimentally to characterize the impact of the second phase on the streaming potential. Novel asymptotic analysis provides theoretical predictions for the streaming potential in the case of a vanishingly small spherical droplet, as well as for large drops with vanishingly small surface tension (large capillary number).