Page 189 - Fundamentals of Reservoir Engineering

CHAPTER 5

THE BASIC DIFFERENTIAL EQUATION FOR RADIAL FLOW IN A POROUS
MEDIUM

5.1 INTRODUCTION

In this chapter the basic equation for the radial flow of a fluid in a homogeneous porous
medium is derived as

1 ∂ kρ ∂ p = φρ ∂ p (5.1)
c
r
r ∂ r µ r ∂ t ∂

This equation is non-linear since the coefficients on both sides are themselves
functions of the dependent variable, the pressure. In order to obtain analytical
solutions, it is first necessary to linearize the equation by expressing it in a form in
which the coefficients have a negligible dependence upon the pressure and can be
considered as constants. An approximate form of linearization applicable to liquid flow
is presented at the end of the chapter in which equ. (5.1) is reduced to the form of the
radial diffusivity equation. Solutions of this equation and their applications for the flow
of oil are presented in detail in Chapters 6 and 7. For the flow of a real gas, however, a
more complex linearization by integral transformation is required which will be
presented separately in Chapter 8.

5.2 DERIVATION OF THE BASIC RADIAL DIFFERENTIAL EQUATION

The basic differential equation will be derived in radial form thus simulating the flow of
fluids in the vicinity of a well. Analytical solutions of the equation can then be obtained
under various boundary and initial conditions for use in the description of well testing
and well inflow, which have considerable practical application in reservoir engineering.
This is considered of greater importance than deriving the basic equation in cartesian
coordinates since analytical solutions of the latter are seldom used in practice by field
engineers. In numerical reservoir simulation, however, cartesian geometry is more
commonly used but even in this case the flow into or out of a well is controlled by
equations expressed in radial form such as those presented in the next four chapters.
The radial cell geometry is shown in fig. 5.1 and initially the following simplifying
assumptions will be made.

a) The reservoir is considered homogeneous in all rock properties and isotropic with
respect to permeability.

b) The producing well is completed across the entire formation thickness thus
ensuring fully radial flow.

c) The formation is completely saturated with a single fluid.