A Predictive Pore-Scale Model for Non-Darcy Flow in Anisotropic Media

Abstract

Abstract Non-Darcy flow is often observed near wells and in hydraulic fractures where relatively high velocities occur. In these regions an empirical model, Forchheimer's equation, is used in place of Darcy's law. It includes a quadratic correction to the linear model and has been adequately fit to many experimental data sets, while found to be insufficient in others. Furthermore, a number of numerical and theoretical approaches have shown limitations of the Forchheimer model in the laminar flow regime. It is important to understand the applicability of Forchheimer's equation and to be able to obtain good predictions of macroscopic properties so that nonlinear flow may be properly modeled in reservoir simulators. In this work non-Darcy flow is modeled using a physically-representative1 pore-scale network model. Quantitative and predictive results are obtained using both computer-generated porous media and real sandstones digitized through x-ray computed microtomography (XMT). The permeability and non-Darcy coefficient, b, are determined for these isotropic and anisotropic media where Forchheimer's equation is applicable. The numerical model is compared to existing experimental data and good agreement is found. Furthermore, limitations to Forchheimer's equation at both low and high velocities are determined and discussed. Introduction Darcy's Law is a constitutive model that adequately describes the slow flow of Newtonian fluids in porous media. Darcy's law is strictly valid for stokes flow (Re = 0), but is usually applicable in engineering applications for Re < 1. It is generally acceptable to use Darcy's Law for modeling flow in reservoirs and aquifers, because the low matrix permeability results in low velocities. However, higher velocities are often observed in fractures and near wellbores; a more complicated constitutive model is necessary to describe flow in these cases.