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The Darcy model is based on a plethora of assumptions. One of the mostimportant assumptions is that the Darcy model assumes the drag coefficient tobe constant. However, there is irrefutable experimental evidence thatviscosities of organic liquids and carbon-dioxide depend on the pressure.Experiments have also shown that the drag varies nonlinearly with respect tothe velocity at high flow rates. In important technological applications likeenhanced oil recovery and geological carbon-dioxide sequestration, oneencounters both high pressures and high flow rates. It should be emphasizedthat flow characteristics and pressure variation under varying drag are bothquantitatively and qualitatively different from that of constant drag.Motivated by experimental evidence, we consider the drag coefficient to dependon both the pressure and velocity. We consider two major modifications to theDarcy model based on the Barus formula and Forchheimer approximation. Theproposed modifications to the Darcy model result in nonlinear partialdifferential equations, which are not amenable to analytical solutions. To thisend, we present mixed finite element formulations based on least-squaresformalism and variational multiscale formalism for the resulting governingequations. The proposed modifications to the Darcy model and its associatedfinite element formulations are used to solve realistic problems with relevanceto enhanced oil recovery. We also study the competition between the nonlineardependence of drag on the velocity and the dependence of viscosity on thepressure. To the best of the authors' knowledge such a systematic study has notbeen performed.

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