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

Abstract

Summary 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 can be properly modeled in reservoir simulators. In this work, non-Darcy flow of an incompressible fluid is modeled using a physically representative (Bryant et al. 1993) 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). A new friction factor correlation is developed for laminar flow in converging/diverging ducts using the numerical solution to the Navier-Stokes equations. The new equation is used to describe flow in pore throats of the network model. The permeability and non-Darcy coefficient, β, are determined for these isotropic and anisotropic media in which Forchheimer's equation is applicable. The numerical model is compared to existing experimental data and appears to be at least as successful as correlations for predicting the non-Darcy coefficient in isotropic media. Furthermore, limitations to Forchheimer's equation at both low and high velocities are determined and discussed.