The flow of a gas at any radius r of Fig. 8.8, where the pressure is p, may be expressed in terms of the flow in standard cubic feet per day by Substituting in the radial form of Darcy’s law, Separating variables and integrating, or Finally, The product μz has been assumed to be constant for the derivation of Eq. (8.23).… Continue reading Radial Flow of Compressible Fluids, Steady State

Equation (8.3) is again used to express the volume dependence on pressure for slightly compressible fluids. If this equation is substituted into the radial form of Darcy’s law, the following is obtained: Separating the variables, assuming a constant compressibility over the entire pressure drop, and integrating over the length of the porous medium,

Although the pore spaces within rocks seldom resemble straight, smooth-walled capillary tubes of constant diameter, it is often convenient and instructive to treat these pore spaces as if they were composed of bundles of parallel capillary tubes of various diameters. Consider a capillary tube of length L and inside radius ro, which is flowing an incompressible fluid of μ viscosity… Continue reading Flow through Capillaries and Fractures

Consider two or more beds of equal cross section but of unequal lengths and permeabilities (Fig. 8.7, depicting flow in series) in which the same linear flow rate q exists, assuming an incompressible fluid. Obviously the pressure drops are additive, and (p1 – p4) = (p1 – p2) + (p2 – p3) + (p3 – p4) Figure 8.7 Series flow in linear beds. Substituting… Continue reading Permeability Averaging in Linear Systems

The rate of flow of gas expressed in standard cubic feet per day is the same at all cross sections in a steady-state, linear system. However, because the gas expands as the pressure drops, the velocity is greater at the downstream end than at the upstream end, and consequently, the pressure gradient increases toward the downstream… Continue reading Linear Flow of Compressible Fluids, Steady State

The equation for flow of slightly compressible fluids is modified from what was just derived in the previous section, since the volume of slightly compressible fluids increases as pressure decreases. Earlier in this chapter, Eq. (8.3) was derived, which describes the relationship between pressure and volume for a slightly compressible fluid. The product of the flow… Continue reading Linear Flow of Slightly Compressible 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… Continue reading Linear Flow of Incompressible Fluids, Steady State

Now that Darcy’s law has been reviewed and the classification of flow systems has been discussed, the actual models that relate flow rate to reservoir pressure can be developed. The next several sections contain a discussion of the steady-state models. Both linear and radial flow geometries are discussed since there are many applications for these… Continue reading Steady-State Flow

Reservoir flow systems are usually classed according to (1) the compressibility of fluid, (2) the geometry of the reservoir or portion thereof, and (3) the relative rate at which the flow approaches a steady-state condition following a disturbance. For most engineering purposes, the reservoir fluid may be classed as (1) incompressible, (2) slightly compressible, or… Continue reading The Classification of Reservoir Flow Systems

In 1856, as a result of experimental studies on the flow of water through unconsolidated sand filter beds, Henry Darcy formulated the law that bears his name. This law has been extended to describe, with some limitations, the movement of other fluids, including two or more immiscible fluids, in consolidated rocks and other porous media.… Continue reading Darcy’s Law and Permeability