A dynamic reporting grid method for investigating slightly compressible fluid flow in low-permeability media

Abstract

Darcy's Law is a foundational equation for modeling fluid flow through porous rock and other media. It is a constitutive law that relates fluid velocity to the pressure gradient, scaled by the ratio of permeability (the rock's ability to transmit fluids) to viscosity (the fluid's resistance to flow). As a stand-alone equation, Darcy's Law is sufficient to model steady-state flow of incompressible fluids, but it can be extended to account for compressible fluids and time-dependent flow. This extension to a transient flow model is fundamental to both analytical and numerical approaches in fields such as reservoir engineering. When rock permeability is low, as in unconventional reservoirs, laboratory studies indicate that Darcy's Law does not accurately predict the observed flow velocity [10, 27, 38], and nonlinear behavior is observed. Modified versions of Darcy's Law have been proposed for low-permeability samples, including formulations that introduce a threshold pressure gradient [14, 15, 38]. This dissertation develops transient flow equations for low-permeability media by replacing Darcy's Law with a modified version that incorporates a threshold pressure gradient. The equations are solved numerically using a dynamic reporting grid method, which numerically replicates threshold pressure gradient behavior. This method provides an adaptable framework for modeling threshold-driven flow and enables investigation of scaling effects and nonlinear terms.