The Transition from Darcy to Nonlinear Flow in Heterogeneous Porous Media: I—Single-Phase Flow

Abstract

Using extensive numerical simulation of the Navier–Stokes equations, we study the transition from the Darcy’s law for slow flow of fluids through a disordered porous medium to the nonlinear flow regime in which the effect of inertia cannot be neglected. The porous medium is represented by two-dimensional slices of a three-dimensional image of a sandstone. We study the problem over wide ranges of porosity and the Reynolds number, as well as two types of boundary conditions, and compute essential features of fluid flow, namely, the strength of the vorticity, the effective permeability of the pore space, the frictional drag, and the relationship between the macroscopic pressure gradient $${\varvec{\nabla }}P$$ ∇ P and the fluid velocity v. The results indicate that when the Reynolds number Re is low enough that the Darcy’s law holds, the magnitude $$\omega _z$$ ω z of the vorticity is nearly zero. As Re increases, however, so also does $$\omega _z$$ ω z , and its rise from nearly zero begins at the same Re at which the Darcy’s law breaks down. We also show that a nonlinear relation between the macroscopic pressure gradient and the fluid velocity v, given by, $$-{\varvec{\nabla }}P=(\mu /K_e)\textbf{v}+\beta _n\rho |\textbf{v}|^2\textbf{v}$$ - ∇ P = ( μ / K e ) v + β n ρ | v | 2 v , provides accurate representation of the numerical data, where $$\mu$$ μ and $$\rho$$ ρ are the fluid’s viscosity and density, $$K_e$$ K e is the effective Darcy permeability in the linear regime, and $$\beta _n$$ β n is a generalized nonlinear resistance. Theoretical justification for the relation is presented, and its predictions are also compared with those of the Forchheimer’s equation.

Assessing Rheology Effects and Pore Space Complexity in Polymer Flow Through Porous Media: A Pore‐Scale Simulation Study

“…A common approach in the literature is to use Reynolds numbers to identify the creeping (laminar) flow and mark the beginning of the inertial flow regime (Koch & Hill, 2001;Sederman et al, 1998). For simple grain-based models of porous materials, the onset of inertial effects for the flow of Newtonian fluids is usually expressed in terms of Reynolds numbers based on the mean grain diameter as a characteristic length (Arbabi & Sahimi, 2024). For more complex systems, such as natural rock materials, a common approach is to define the Reynolds number as follows (Beavers & Sparrow, 1969;Muljadi et al, 2016;Wood et al, 2020)…”