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Over the past few years, we developed a mathematically rigorous method tostudy the dynamical processes associated to nonlinear Forchheimer flows forslightly compressible fluids. We have proved the existence of a geometrictransformation which relates constant mean curvature surfaces andtime-invariant pressure distribution graphs constrained by theDarcy-Forchheimer law. We therein established a direct relationship between theCMC graph equation and a certain family of equations which we call$g$-Forchheimer equations. The corresponding results, on fast flows and theirgeometric interpretation, can be used as analytical tools in evaluatingimportant technological parameters in reservoir engineering.

Geometric Methods in the Analysis on Non-linear Flows in Porous Media (Preliminary Version)