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

TL;DR

This paper introduces a rigorous geometric approach to analyze nonlinear Forchheimer flows in porous media, establishing a transformation linking geometric surfaces and pressure distributions, with applications in reservoir engineering.

Contribution

It develops a novel geometric transformation connecting constant mean curvature surfaces to pressure graphs in nonlinear flow models, advancing analytical tools for reservoir analysis.

Findings

Proved existence of a geometric transformation relating CMC surfaces and pressure graphs.

Established a relationship between CMC graph equations and g-Forchheimer equations.

Provided analytical methods for evaluating reservoir parameters.

Abstract

Over the past few years, we developed a mathematically rigorous method to study the dynamical processes associated to nonlinear Forchheimer flows for slightly compressible fluids. We have proved the existence of a geometric transformation which relates constant mean curvature surfaces and time-invariant pressure distribution graphs constrained by the Darcy-Forchheimer law. We therein established a direct relationship between the CMC graph equation and a certain family of equations which we call $g$-Forchheimer equations. The corresponding results, on fast flows and their geometric interpretation, can be used as analytical tools in evaluating important technological parameters in reservoir engineering.