Page 191 - Fundamentals of Reservoir Engineering

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW 129

1 ∂ V
c =−
V ∂ p

and since

m
ρ =
V

then the compressibility can be alternatively expressed as

m
∂
ρ ρ 1 ρ∂
c =− = (5.4)
m ∂ ρ ρ ∂ p

and differentiating with respect to time gives

∂ p ∂ ρ
cρ = (5.5)
t ∂ t ∂

Finally, substituting equ. (5.5) in equ. (5.3) reduces the latter to

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

This is the basic, partial differential equation for the radial flow of any single phase fluid
in a porous medium. The equation is referred to as non-linear because of the implicit
pressure dependence of the density, compressibility and viscosity appearing in the
coefficients kρ /µ and φ cρ. Because of this, it is not possible to find simple analytical
solutions of the equation without first linearizing it so that the coefficients somehow
lose their pressure dependence. A simple form of linearization applicable to the flow of
liquids of small and constant compressibility (undersaturated oil) will be considered in
sec. 5.4, while a more rigorous method, using the Kirchhoff integral transformation, will
be presented in Chapter 8 for the more complex case of linearization for the flow of a
real gas.

5.3 CONDITIONS OF SOLUTION

In principle, an infinite number of solutions of equ. (5.1 ) can be obtained depending on
the initial and boundary conditions imposed. The most common and useful of these is
called the constant terminal rate solution for which the initial condition is that at some
fixed time, at which the reservoir is at equilibrium pressure p i, the well is produced at a
constant rate q at the wellbore, r = r w. This type of solution will be examined in detail in
Chapters 7 and 8 but it is appropriate, at this stage, to describe the three most
common, although not exclusive, conditions for which the constant terminal rate
solution is sought. These conditions are called transient, semi-steady state and steady
state and are each applicable at different times after the start of production and for
different, assumed boundary conditions.

a) Transient condition