Sandbox:Fluid flow in porous media

Introduction

In this initial chapter on fluid flow in porous media, we begin with a discussion of the differential equations that are used most often to model unsteady-state flow. Simple statements of these equations are provided in the text; the more tedious mathematical details are given in Appendix A for the instructor or student who wishes to develop greater understanding. The equations are followed by a discussion of some of the most useful solutions to these equations, with emphasis on the exponential-integral solution describing radial, unsteady-state flow. An appended discussion (Appendix B) of dimensionless variables may be useful to some readers at this point.

The chapter concludes with a discussion of the radius-of-investigation concept and of the principle of superposition. Superposition, illustrated in multiwell infinite reservoirs, is used to simulate simple reservoir boundaries and to simulate variable rate production histories. An approximate alternative to superposition, Horner's "pseudoproduction time," completes this discussion.

The ideal reservoir model

To develop analysis and design techniques for well testing, we first must make several simplifying assumptions about the well and reservoir that we are modeling. We naturally make no more simplifying assumptions than are absolutely necessary to obtain simple, useful solutions to equations describing our situation – but we obviously can make no fewer assumptions. These assumptions are introduced as needed, to combine (1) the law of conservation of mass, (2) Darcy's law, and (3) equations of state to achieve our objectives. This work is only outlined in this chapter; detail is provided in Appendix A and the References.

Consider radial flow toward a well in a circular reservoir. If we combine the law of conservation of mass and Darcy's law for the isothermal flow of fluids of small and constant compressibility (a highly satisfactory model for single-phase flow of reservoir oil), we obtain a partial differential equation that simplifies to

if we assume that compressibility, c, is small and independent of pressure; permeability, k,is constant and isotropic; viscosity, μ, is independent of pressure; porosity, ϕ, is constant; and that certain terms in the basic differential equation (involving pressure gradients squared) are negligible. This equation is called the diffusivity equation;the term 0.000264k/ϕμc is called the hydraulic diffusivity and frequently is given the symbol η.

Eq. 1.1 is written in terms of field units. Pressure, p, is in pounds per square inch (psi); distance, r, is in feet; porosity, ϕ, is a fraction; viscosity, μ, is in centipoise; compressibility, c, is in volume per volume per psi [c = (l/ρ) (dρ/dp)]; permeability, k,is in millidarcies; time, t, is in hours; and hydraulic diffusivity, η, has units of square feet per hour.

A similar equation can be developed for the radial flow of a nonideal gas:

where z is the gas-law deviation factor.

For simultaneous flow of oil, gas, and water,

where c_t is the total system compressibility,

and the total mobility λ_t is the sum of the mobilities of the individual phases:

In Eq. 1.4, S_o refers to oil-phase saturation, c_o to oil-phase compressibility, S_w and c_w to water phase, S_g and c_g to gas phase; and c_f is the formation compressibility. In Eq. 1.5, k_o is the effective permeability to oil in the presence of the other phases, and μ_o is the oil viscosity; k_g and μ_g refer to the gas phase; and k_w and μ_w refer to the water phase. Because the formation is considered compressible (i.e., pore volume decreases as pressure decreases), porosity is not a constant in Eq. 1.3 as it was assumed to be in Eqs. 1.1 and 1.2.