Co-Current Spontaneous Imbibition in Porous Media with the Dynamics of Viscous Coupling and Capillary Back Pressure

Abstract

Abstract This paper presents a numerical study of water displacing oil by combined co-current / counter-current spontaneous imbibition (SI) of water displacing oil from a water-wet matrix block exposed to water at one side and oil at the other. Counter-current flows can induce a stronger viscous coupling than during co-current flows leading to deceleration of the phases. Even as water displaces oil co-currently the saturation gradient in the block induces counter-current capillary diffusion. The extent of counter-current flow may dominate the domain of the matrix block near the water-exposed surfaces, while co-current imbibition may dominate the domain near the oil-exposed surfaces implying that one unique effective relative permeability curve for each phase does not adequately represent the system. As relative permeabilities are routinely measured co-currently it is an open question whether the imbibition rates in the reservoir (depending on a variety of flow regimes and parameters) will in fact be correctly predicted. We present a generalized two phase flow model based on momentum equations from mixture theory that can account dynamically for viscous coupling between the phases and the porous media due to fluid-rock interaction (friction) and fluid-fluid interaction (drag). These momentum equations effectively replace and generalize Darcy's law. The model is parameterized using experimental data from the literature. We consider a water-wet matrix block in 1D that is exposed to oil on one side and water on the other side. This setup favors co-current SI. We also account for the fact that oil produced counter-currently into water must overcome the socalled capillary back pressure, which represents a resistance for oil to be produced as droplets. This parameter can thus influence the extent of counter-current production and hence, viscous coupling. This complex mixture of flow regimes implies that it is not straightforward to model the system by a single set of relative permeabilities, but rather relies on a generalized momentum equation model that couples the two phases. In particular, directly applying co-currently measured relative permeability curves gives significantly different predictions than the generalized model. It is seen that at high water-to-oil mobility ratios, viscous coupling can lower the imbibition rate and shift the production from less counter-current to more co-current as compared to conventional modelling. Although the viscous coupling effects are triggered by counter-current flow, reducing or eliminating counter-current production via the capillary back pressure does not eliminate the effects of viscous coupling that take place inside the core, which effectively lower the mobility of the system. It was further seen that viscous coupling can increase the remaining oil saturation in standard co-current imbibition setups.