Effective Relative Permeabilities Based on Momentum Equations with Brinkman Terms and Viscous Coupling

Abstract

Summary The relative permeability expresses the mobility reduction factor when a fluid flows through a porous medium in the presence of another fluid and appears in Darcy’s law for multiphase flow. In this work, we replace Darcy’s law with more general momentum equations accounting for fluid-rock interaction (flow resistance), fluid-fluid interaction (drag), and Brinkman terms responding to gradients in fluid interstitial velocities. By coupling the momentum equations with phase transport equations, we study two important flow processes—forced imbibition (coreflooding) and countercurrent spontaneous imbibition. In the former, a constant water injection rate is applied and capillary forces are neglected, while in the latter, capillary forces drive the process and the total flux is zero. Our aim is to understand what relative permeabilities result from these systems and flow configurations. From previous work, when using momentum equations without Brinkman terms, unique saturation-dependent relative permeabilities are obtained for the two flow modes that depend on the flow mode. Now, with Brinkman terms included, the relative permeabilities depend on local spatial derivatives of interstitial velocity and pressure. Local relative permeabilities are calculated for both phases utilizing the ratio of phase Darcy velocity and phase pressure gradient. In addition, we use the Johnson-Bossler-Naumann (JBN) method for forced imbibition (with data simulated under the assumption of negligible capillary end effects) to calculate interpreted relative permeabilities from pressure drop and average saturation. Both flow setups are parameterized with literature data, and sensitivity analysis is performed. During coreflooding, Brinkman terms give a flatter saturation profile and higher front saturation. The saturation profile shape changes with time. Local water relative permeabilities are reduced, while they are slightly raised for oil. The saturation range where relative permeabilities can be evaluated locally is raised and made narrower with increased Brinkman terms. JBN relative permeabilities deviate from the local values: The trends in curves and saturation range are the same but more pronounced as they incorporate average measurements, including the strong impact at the inlet. Brinkman effects vanish after sufficient distance traveled, resulting in the unique saturation functions as a limit. Unsteady state (USS) relative permeabilities (based on transient data from single-phase injection) differ from steady-state (SS) relative permeabilities (based on SS data from coinjection of two fluids) because the Brinkman terms are zero at SS. During spontaneous imbibition, the higher effect from the Brinkman terms caused oil relative permeabilities to decrease at low water saturations and slightly increase at high saturations, while water relative permeability was only slightly reduced. The net effect was a delay in the imbibition profile. Local relative permeabilities approached the unique saturation functions without Brinkman terms deeper in the system because phase velocities (involved in the Brinkman terms) decreased with distance. In both systems, scaling and simulations demonstrate that the relative change in relative permeabilities due to Brinkman terms increases with the Brinkman coefficient, permeability, and inverse squared distance from the inlet.