Page 226 - Fundamentals of Reservoir Engineering

OILWELL TESTING 164

or using the same value of p D (t D ) = 35.71 in conjunction with equ. (7.19) (Darcy units)
to determine p wf as 193.5 atm. This example also illustrates that although equations
may be developed using dimensionless pressure functions, conversion can easily be
made at any stage to obtain the real pressure.

c) The majority of technical papers on the subject of pressure analysis, at least those
written since the late sixties, generally have all equations expressed in dimensionless
form. It is therefore hoped that by introducing and using dimensionless variables in this
text the engineer will be assisted in reading and understanding the current literature.
To illustrate the application of dimensionless variables, the constant terminal rate
solution of the radial diffusivity equation derived in sec. 7.3 for transient and
semisteady state conditions, will be expressed in terms of dimensionless pressure
functions.

The transient solution is

qµ 4 kt
p wf = p − In + 2S (7.10)
i
4kh γ φ µ cr w 2
π
which may be re-arranged as

2kh 4t
π
(p − p ) = 1 2 ln D + S
i
wf
qµ γ
and therefore, from the defining equation for p D (t D ), equ. (7.19), it is evident that
4t
t
p D () = 1 2 ln D (7.23)
D
γ
which is also frequently expressed as

t
p D () = 1 2 (ln t + 0.809) (7.24)
D
D
In either case p D (t D) is strictly a function of the dimensionless time t D. For semi-steady
state conditions equ. (7.13) can be expressed as

2 π kh 4A kt r 2
(p − p ) = 1 2 ln 2 + 2π 2 w + S
wf
i
qµ γ Cr w φ µ cr w A
A
or
2 π kh 4A r 2
(p − p ) = 1 2 ln 2 + 2 t π D w + S
i
wf
qµ γ Cr w A
A
and therefore, applying equ. (7.19)

4A r 2
p(t ) = 1 2 ln 2 + 2 t π D w (7.25)
D
D
γ Cr w A
A