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

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$${\varvec{\nabla }}P$$Pand the fluid velocityv. The results indicate that when the Reynolds number Re is low enough that the Darcy’s law holds, the magnitude$$\omega _z$$ωzof the vorticity is nearly zero. As Re increases, however, so also does$$\omega _z$$ωz, 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 velocityv, given by,$$-{\varvec{\nabla }}P=(\mu /K_e)\textbf{v}+\beta _n\rho |\textbf{v}|^2\textbf{v}$$-P=(μ/Ke)v+βnρ|v|2v, provides accurate representation of the numerical data, where$$\mu$$μand$$\rho$$ρare the fluid’s viscosity and density,$$K_e$$Keis the effective Darcy permeability in the linear regime, and$$\beta _n$$βnis a generalized nonlinear resistance. Theoretical justification for the relation is presented, and its predictions are also compared with those of the Forchheimer’s equation.

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Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Transport in Porous Media
Medium: X Size: p. 795-812
["p. 795-812"]
Sponsoring Org:
National Science Foundation
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