  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
formalism and variational multiscale 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.
