A Method for Determining the Exponent Value in a Forchheimer-Type Flow Equation

Abstract

This article presents a direct method for determining the exponent in a Forchheimer-type flow equation. Forchheimer proposed the following expression as the "law of flow" in porous media: P n = au + bu, .................................(1) L where p/ L = pressure gradient u = volumetric velocity (flow rate per unit area) a, b = constants for a particular porous media n = velocity exponent It is well established in oil and ground-water flow that velocities are generally sufficiently low such that Darcy's law is usually valid as the law of flow. That is, P = cu,........................................(2) L where c = a constant. In those cases where Darcy's law is not valid, Muskat states that the empirical flow data can be expressed in all cases by an equation of the form of Eq. 1, with n 2 when flow velocity is sufficiently high to be in the turbulent flow region. Eq. 1 is also applicable when flow is laminar, but kinetic energy losses are large relative to viscous forces. Although it is not the purpose of this article to discuss the range of validity of either Eqs. 1 or 2, it should be noted that the application of Eq. 1 is appropriate more often than is commonly recognized. Many high-rate Middle Eastern wells, particularly those in fractured reservoirs, are described by Eq. 1, as first reported by Baker. Using Eq. 2 in place of Eq. 1 can result in errors on the order of 50 percent or more in predicting well rates. It is well documented in the literature that a rate equation based on Eq. 1 also applies to many gas wells. Nevertheless, the use of the empirical Rawlins-Schellhardt backpressure flow equation remains common. Cornelson has recently presented a discussion of the potential error in predicting gas rates and reserves when the Rawlins-Schellhardt equation is used in place of Eq. 1 for the flow equation. In addition to reservoir flow considerations, well screens, perforations, and other near-wellbore effects generally cause the pressure loss to be proportional to n th power of the flow rate. Investigators in the field of ground-water hydraulics note that n may deviate significantly from 2 and should be determined separately for each well. In such cases, n is usually determined by trial-and-error curve fitting of well-test data. A direct solution for n, as well as the associated flow coefficients, is possible using the following procedure. Run the well-flow test such that at least three pressure drawdown values are obtained for three rates at equal flow times. P. 883