MODIFICATION TO DARCY-FORCHHEIMER MODEL DUE TO PRESSURE-DEPENDENT VISCOSITY: CONSEQUENCES AND NUMERICAL SOLUTIONS
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2017 by Begell House, Inc. The standard Darcy model is based on a plethora of assumptions. One of the key assumptions is that the drag coefficient is constant. However, there is irrefutable experimental evidence that viscosities of organic liquids and carbon dioxide, for example, depend on the pressure. Experiments have also shown that the drag varies nonlinearly with respect to the velocity and its gradient at high flow rates. The flow characteristics and pressure variation under varying drag are both quantitatively and qualitatively different from that of a constant drag. Motivated by experimental evidence, we consider an application of the Darcy model where the drag coefficient depends on both the pressure and velocity. We focus on major modifications to the Darcy model based on the Barus formula and Forchheimer approximation in this paper. The proposed modifications to the Darcy model result in nonlinear partial differential equations that are not amenable to analytical solutions. To this end, we present a mixed finite element formulation based on the variational multiscale (VMS) formulation for the resulting governing equations. We also illustrate how to recover local mass conservation through a postprocessing technique based on convex optimization.With the proposed modifications to the Darcy model and the associated finite element framework, we 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 reported in the literature.
