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 "ß" have been comprehensively reviewed; nine correlations found suitable for use in this study for technical reasons. A new dimensionless number (Be) relating ß, 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. Intoduction Traditionally, when mentioned, non-Darcy behavior means gas flow in reservoirs and the attention goes to the Forchheimer's equation. On the other hand, despite that Darcy equation has been around for about one and a half century and essentially based on empirical experimental approach with no mathematical foundation, until today, it is considered the corner stone of fluid flow in porous medium. The uncertainties usually associated with the predictions of commercial reservoir simulators mostly attributed to a number of factors such as poor quality and lack of in-put data, unrealistic history matching, assumptions encountered, etc. It is only lately that the basic governing equations of flow behavior become questioned. In fact, transmissibility and so saturations distribution within a reservoir at any time and coordinates are direct functions of pressure predicted by the diffusivity equation employed. As it controls the evaluation of other parameters, if this pressure is in doubt then the ultimate simulator predictions are consequently misleading. Non-Darcy flow has been encountered in oil and gas reservoirs produced by vertical and horizontal wells alike. The severity of the non-Darcy effect is indeed affected by the velocity of the flow and both physical properties of the flowing fluid and the characteristics of the medium. The productivity of a horizontal well has been affected by the non-Darcy behavior around the well bore. To include this high flow rate effect on the productivity model of a horizontal well the flow in the near wellbore has been assumed normal to the well trajectory and a radial flow region is supposed to take place around the wellbore. The impact of high flow rate region is predicted to occur near the wellbore and as it gets away from the wellbore in the flow field the effect vanishes. It has been found that non-Darcy effect, if occurred, in a horizontal well under radial or pseudo radial flow cases become significant in a range of 10 to 20 feet around the wellbore, beyond that the effect become insignificant and practically fades out1.