Numerical Simulation of Non-Darcy Flow Utilizing the New Forchheimer's Diffusivity Equation

Abstract

Abstract Recently, it became evident that non-Darcy flow occurs not only in gas reservoirs, fractured reservoirs and multi-permeability systems within oil reservoirs experienced non-linearity due to non-Darcy flow behavior. Most reservoir simulators currently used encountered false predictions due to their dependency on the traditionally used diffusivity equation. This paper introduces alternative diffusivity equation to replace the one derived from Darcy's law. The new equation was derived from the commonly known as Forchheimer's equation which is basically Darcy's equation plus an inertia term to account for high velocity fluid flow in porous medium. Mathematical derivation of the diffusivity equation based on Forchheimer equation has been presented in a previous paper by the authors. The newly derived diffusivity equation has been numerically simulated. Correlations used to estimate the non-Darcy coefficient "β" have been comprehensively reviewed; nine correlations found suitable for use in this study for technical reasons. A new dimensionless number (Be) relating β, velocity, density and viscosity has been introduced to differentiate between Darcy and non-Darcy flow in porous medium for any rock type and any flowing fluid. Evidences show that this new dimensionless number cannot be considered a declaration of turbulence flow in porous medium rather the energy loss is contributed to the nature of both flowing fluid and the porous medium. The point of deviation from the Darcian behavior to the non-Darcian behavior has been found at Be = 0, for practical use it has been determined that Be = 0.0526 at 5% deviation from Darcy's linear trend. A range of permeability from 1 md to 1000 md with porosity changing accordingly has been verified with the new model, velocity as low as 0.0001 cm/sec and as high as 700 cc/sec has been tested as well. Both Darcy and non-Darcy behaviors have been identified for the domain of testing, and the numerical model has proven of good agreement in all cases.