Page 227 - Fundamentals of Reservoir Engineering

OILWELL TESTING 165

Further, by defining a modified version of the dimensionless time as

r 2 kt
t = t w = (7.26)
D
DA
A φµ cA
equ. (7.25) can be expressed in its more common form as

4A
π
p(t ) = 1 2 ln + 2 t DA (7.27)
D
D
γ Cr w 2
A
The necessity for, and usefulness of, this dimensionless time t DA will be illustrated later
in the chapter.
No attempt can yet be made to define a p D function appropriate to describe the
3
pressure drop at the wellbore during the late transient period. Ramey and Cobb have
shown, however, that for a well situated at the centre of a regular shaped drainage
area, for instance, a circle, square or hexagon, the late transient period is of extremely
short duration and under these circumstances it is possible to equate equs. (7.23) and
(7.27) to determine the approximate time at which the change from transient to semi-
steady state conditions will occur, i.e.

4t 4A
1 ln D ≈ 1 ln + 2 t π
2 2 2 DA
γ γ Cr w
A
which may be expressed as either

r 2 4t r / A
2
π
Ct w ≈ e D w (7.28)
AD
A
or
4t
π
Ct ≈ e DA (7.29)
ADA
Solving equ. (7.28) for t D will give an approximate solution for the dimensionless
2
transition time which is dependent both on the ratio r /A and C A. Solving equ. (7.29) for
w
t DA, however, will give a dimensionless transition time which is only dependent on the
shape factor. The solution of equ. (7.29), for C A ≈ 31, is

kt
t DA = ≈ 0.1 (7.30)
φµ cA

so that for a well draining from the centre of one of the regular drainage area shapes
mentioned, a fairly abrupt change from transient to semi-steady state flow occurs for a
value of t DA ≈ 0.1, irrespective of the size of the area being drained. This in part
explains the usefulness of expressing dimensionless times in terms of t DA rather than
t D. The real time when the transition occurs can be determined by solving equ. (7.30)
explicitly for t.