Page 218 - Fundamentals of Reservoir Engineering

OILWELL TESTING 156

The constant terminal rate solution of the radial diffusivity equation during the late
transient flow period is too complicated to include at this stage. A simplified method of
obtaining this solution will be described in sec. 7.6.

Once semi-steady state conditions prevail the solution can be determined by adding
the simple material balance equation for the bounded drainage volume

p −
cAhφ ( i ) p = qt (7.12)

to the semi-steady state inflow equation

qµ 4A
pp wf = 2kh ½ ln γ C r 2 + S (6.22)
−
π
Aw
to give

qµ 4A kt
p wf = p − 2kh ½ ln γ Cr 2 + 2π φ µ cA + S (7.13)
i
π
Aw

In this equation p is the current average pressure within the drainage boundary and C A
is the Dietz shape factor introduced in Chapter 6, sec.5. The magnitude of C A depends
on the shape of the area being drained and also upon the position of the well with
respect to the boundary.

Theoretically, for the constant terminal rate solution, the rate q in equs. (7.12) and
(6.22) is the same. In practice, it is sometimes difficult to maintain the production rate of
a well constant over a long period of time and therefore, the current rate in equ. (6.22)
may differ from the average rate which is implicitly used in material balance, equ.
(7.12). In this case the rate in equ. (7.12) is set equal to the current, or final flow rate,
and the flowing time is expressed as an effective flowing time, where

Cumulative Production
t = Effective flowing time = (7.14)
Final flow rate

Use of the effective flowing time is therefore simply a method for equalising the rates
and preserving the material balance and is frequently used in pressure analysis, as will
be described later.

Even though no equation for describing the pressure decline during the late transient
flow period has yet been developed, equs. (7.10) and (7.13), which are appropriate for
transient and semi-steady state flow, can be usefully employed by themselves in well
test analysis.

Well testing involves producing a well at a constant rate or series of rates, some of
which may be zero (well closed in), while simultaneously taking a continuous recording
of the changing pressure in the wellbore using some form of pressure recording device.
The retrieved record of wellbore pressure as a function of time can be analysed in
conjunction with the known rate sequence to determine some or all of the following
reservoir parameters: