The Equations of Motion of Fluids in Porous Media: I. Propagation Velocity of Pressure Pulses
Authors: Foster W.R. , McMillen J.M. , Odeh A.S. 
Published: December 2007

Abstract
The complete equations of average linear momentum balance for a single-phase fluid in an incompressible, homogeneous, porous medium are derived. The derivation begins with Euler's equation of motion /or a continuum and uses an integral transform recently developed by Slattery. For steady flow of compressible, Newtonian fluid, the usual equations of motion result. For transient flow, the space-time description of the pressure is determined in the lowest approximation by the telegrapher's equation. From the analysis a new phenomenological coefficient results which connects the viscous traction to the derivative of the linear momentum density. The magnitude of this coefficient determines the velocity of sound through the pore structure in this approximation to the pressure field.

Introduction
The modification of Darcy's law of momentum balance for steady, single-phase flow through porous media has been discussed for many years, The first modification was suggested by Forcheimer who added terms of higher order in the velocity. These can be expected to appear because the underlying microscopic equations of momentum balance are themselves nonlinear in the point velocity field. The Reynolds tensor pvv, which represents the convective flux of momentum density, appears in the momentum balance equation. Only in rectilinear flow (parallel stream lines) does the divergence of this tensor vanish. Since the steady flow stream lines in most porous media are not parallel, nonlinear dependence of the pressure gradient on the velocity should naturally appear. This nonlinearity has nothing to do with turbulence in the ordinary sense of random fluctuations in the pressure and velocity fields. It arises simply because the stream lines converge and diverge, even for steady flows. Klinkenberg demonstrated that the permeability coefficient in Darcy's law depends on the absolute pressure or, alternatively, on the density field. however, because he neglected inertial terms of the Forcheimer type, his correction coefficient could not be represented by a constant but tended toward a constant as the velocity decreased. Forcheimer's and Klinkenberg's modifications can be combined in a rigorous way to account for both inertia and slip during steady flow. This will be shown in a future paper. The transient change of pressure in porous media has been described by the diffusion equation. This form results from eliminating the velocity and density fields from a combination of the equations of motion in the form of Darcy's law, the continuity equation and an equation of state. Fatt suggested that the cause of deviations from the prediction of the diffusion equation for pressure transients lies not in the choice of Darcy's law as the equation of motion but on the existence of dead-end pores which might invalidate the averaged equation of continuity. On the other hand, Oroveanu and Pascal noted that the time derivative of the momentum density must be included in the equations of motion since this measures the local rate of change of momentum density. Their differential equation for pressure is the telegrapher's equation (neglecting gravity). However, their form of this equation predicts that the speed of pressure propagation through the pore structure is the same as that through the bulk fluid. M. K. Hubbert attempted a derivation of Darcy's law by volume averaging the Navier-Stokes equations. Since these equations represent momentum balance at a point within an open set of points containing the fluid itself, Hubbert's volume averaging cannot lead to terms involving transfer of momentum between the fluid and the walls of the pores.