Mathematical Frame-work for the Productivity Index of Wells with Fast Forchheimer (non-Darcy) Flow in Porous Media

E. Aulisa [1], A. Ibragimov[1], P. Valko [2], and J. Walton[3]
[1] Texas Tech University, Department of Mathematics and Statistics
[2] Texas A & M University, Department of Petroleum Engineering
[3] Texas A & M University, Department of Mathematics

In the current paper, similar features for fast or turbulent, non-linear flows subjected to Forchheimer equations are analyzed.

Under some hydrodynamic and thermodynamic constrains, it has been shown that there exists the so called pseudo-steady state regime for Forchheimer flows in reservoir media. This invariant is called diffusive capacitance, which is the mathematical model for the well Productivity Index.

Obtained results allow computing the well Productivity Index by resolving a single steady state boundary value problem for a second order quasi-linear elliptic equation.

Analytical and numerical studies, obtained using COMSOL Multiphysics, highlight some new relations for the productivity index of the well in linear and nonlinear cases.