Modification to Darcy model for high pressure and high velocity applications and associated mixed finite element formulations

The Darcy model is based on a plethora of assumptions. One of the most important assumptions is that the Darcy model assumes the drag coefficient to be constant. However, there is irrefutable experimental evidence that viscosities of organic liquids and carbon-dioxide depend on the pressure. Experiments have also shown that the drag varies nonlinearly with respect to the velocity at high flow rates. In important technological applications like enhanced oil recovery and geological carbon-dioxide sequestration, one encounters both high pressures and high flow rates. It should be emphasized that flow characteristics and pressure variation under varying drag are both quantitatively and qualitatively different from that of constant drag. Motivated by experimental evidence, we consider the drag coefficient to depend on both the pressure and velocity. We consider two major modifications to the Darcy model based on the Barus formula and Forchheimer approximation. The proposed modifications to the Darcy model result in nonlinear partial differential equations, which are not amenable to analytical solutions. To this end, we present mixed finite element formulations based on least-squares (LS) formalism and variational multi-scale (VMS) formalism for the resulting governing equations. The proposed modifications to the Darcy model and its associated finite element formulations are used to solve realistic problems with relevance to enhanced oil recovery. We also study the competition between the nonlinear dependence of drag on the velocity and the dependence of viscosity on the pressure. To the best of the authors’ knowledge such a systematic study has not been performed.