Effective Relative Permeabilities Based on Momentum Equations with Brinkmann Terms and Viscous Coupling
Authors: Qiao Yangyang , Andersen Pål Østebø , Ahmadpour Sadegh 
Published: June 2023

Abstract
The relative permeability expresses the mobility reduction factor when a fluid flows through a porous medium in 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 Brinkmann 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 (core flooding) and counter-current spontaneous imbibition. In the former a constant water injection rate is applied, and capillary forces 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 Brinkmann terms, unique saturation dependent relative permeabilities are obtained for the two flow modes that depend on the flow mode. Now, with Brinkmann 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 utilize the JBN method for forced imbibition to calculate relative permeabilities from pressure drop and average saturation. Both flow setups are parameterized with literature data and sensitivity analysis is performed.
During core flooding, Brinkmann 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 Brinkmann 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. Brinkmann effects vanish after sufficient distance traveled resulting in the unique saturation functions as a limit. Unsteady state relative permeabilities (based on transient data from single phase injection) differ from steady state relative permeabilities (based on steady state data from co-injection of two fluids) because the Brinkmann terms are zero at steady state. During spontaneous imbibition, higher effect from the Brinkmann 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 of the imbibition profile. Local relative permeabilities approached the unique saturation functions without Brinkmann terms deeper in the system because phase velocities (involved in the Brinkmann terms) decrease with distance. In both systems, scaling and simulations demonstrate that the relative change in relative permeabilities due to Brinkmann terms increases with Brinkmann coefficient, permeability and inverse squared distance from the inlet.