Linear Flow of Incompressible Fluids, Steady State

Figure 8.4 represents linear flow through a body of constant cross section, where both ends are entirely open to flow and where no flow crosses the sides, top, or bottom. If the fluid is incompressible, or essentially so for all engineering purposes, then the velocity is the same at all points, as is the total flow rate across any cross section; thus, in horizontal flow,

Figure 8.4 Representation of linear flow through a body of constant cross section.

Separating variables and integrating over the length of the porous body,

For example, under a pressure differential of 100 psi for a permeability of 250 md, a fluid viscosity of 2.5 cp, a formation volume factor of 1.127 bbl/STB, a length of 450 ft, and a cross-sectional area of 45 sq ft, the flow rate is

In this integration, B, q, μ, and k were removed from the integral sign, assuming they were invariant with pressure. Actually, for flow above the bubble point, the volume, and hence the rate of flow, varies with the pressure, as expressed by Eq. (8.2). The formation volume factor and viscosity also vary with pressure. Fatt and Davis have shown a variation in permeability with net overburden pressure for several sandstones.⁴ The net overburden pressure is the gross less the internal fluid pressure; therefore, a variation of permeability with pressure is indicated, particularly in the shallower reservoirs. Because these effects are negligible for a few hundred-psi pressure difference, values at the average pressure may be used for most purposes.