Category: Single-Phase Fluid Flow In Reservoirs

By applying appropriate boundary and initial conditions, particular solutions to the differential equation derived in the preceding section can be discussed. The solutions obtained pertain to the transient and pseudosteady-state flow periods for both slightly compressible and compressible fluids. Since the incompressible fluid does not exist, solutions involving this type of fluid are not discussed. Only… Continue reading Transient Flow

The radial diffusivity equation, which is the general differential equation used to model time-dependent flow systems, is now developed. Consider the volume element shown in Fig. 8.12. The element has a thickness Δr and is located r distance from the center of the well. Mass is allowed to flow into and out of the volume element during a period… Continue reading Development of the Radial Diffusivity Equation

Many producing formations are composed of strata or stringers that may vary widely in permeability and thickness, as illustrated in Fig. 8.10. If these strata are producing fluid to a common wellbore under the same drawdown and from the same drainage radius, then Figure 8.10 Radial flow in parallel beds. Then, canceling, This equation is the same… Continue reading Permeability Averages for Radial Flow

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