Page 193 - Fundamentals of Reservoir Engineering

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW 131

The constant referred to in equ. (5.7) can be obtained from a simple material balance
using the compressibility definition, thus

dp dV
cV =− =− q (5.8)
dt dt

dp q
or =− (5.9)
dt cV

which for the drainage of a radial cell can be expressed as

dp q (5.10)
dt =− cr hφ
2
π
e
This is a condition which will be applied in Chapter 6, for oil flow, and in Chapter 8, for
gas flow, to derive the well inflow equations under semi-steady state conditions, even
though in the latter case the gas compressibility is not constant.

One important feature of this stabilized type of solution, when applied to a depletion
1
type reservoir, has been pointed out by Matthews, Brons and Hazebroek and is
illustrated in fig. 5.3. This is the fact that, once the reservoir is producing under the
semi-steady state condition, each well will drain from within its own no-flow boundary
quite independently of the other wells.

For this condition dp/dt must be approximately constant throughout the entire reservoir
otherwise flow would occur across the boundaries causing a re-adjustment in their
positions until stability was eventually achieved. In this case a simple technique can be
applied to determine the volume averaged reservoir pressure

pV i
i
p res = i (5.11)
V i
i
in which

th
V = the pore volume of the i drainage volume
i
th
and p = the average pressure within the i drainage volume
i
Equation (5.9) implies that since dp/dt is constant for the reservoir then, if the variation
in the compressibility is small