Page 245 - Fundamentals of Reservoir Engineering

OILWELL TESTING 183

part of the plots represents both the pure transient flow period, equ. (7.43), and the late
transient period and it is not worthwhile trying to distinguish between the two.

Equation (7.44) is interesting since it reveals how the Dietz shape factors were
9
originally determined. Dietz, whose paper on pressure analysis was published some
years after that of MBH, evaluated the relationship expressed in equ. (7.44) for the
specific value of t DA = 1, thus

4kh *
π
t
p D(MBH) ( DA = ) 1 = ( p − p ) = ln C A (7.45)
qµ t = 1
DA
Values of In C A (and hence C A ) could be determined as the ordinate of the MBH charts
for each separate plot corresponding to the value of t DA = 1, and these are shown in
fig. 6.4. In some cases of extreme well asymmetry, late transient flow conditions still

prevail at t DA = 1 (e.g. fig 7.13) and in these cases the linear trend of p D(MBH) versus t DA
must be extrapolated back to the specific value t DA = 1 to determine the correct shape
factor. The usefulness of the Dietz shape factors in the formulation of equations
describing semi-steady state flow, for which they were derived, has already been amply
illustrated in this text.
The importance of equ. (7.42) for generating dimensionless pressure functions for a
variety of boundary conditions and for any value of the flowing time cannot be
overemphasised. It is rather surprising that this equation has been lying dormant in the
literature since 1954, the date of the original MBH paper, with its full potential being
largely unrealised. It appears in disguised form in many papers and even in the classic
6
Matthews, Russell, SPE monograph (equ. 10.18, p. 109), yet it was not presented in
the simple form of equ. (7.42) of this text until it was highlighted in a brief J.P.T. Forum
8
article in 1973 by Cobb and Dowdle . The latter use a slightly modified form of the
equation in which the right hand side of equ. (7.42) is expressed strictly as a function of
t DA, thus

4A
t
t
p D ( DA ) = 2 t DA + 1 2 ln t DA + 1 2 ln − 1 2 p D(MBH) ( DA ) (7.46)
π
r γ w 2
In application to general oilwell test analysis, any rate-time-pressure sequence can be
analysed using the following general equations

2kh n
π
t
p −
j
( i p wf n ) = ∆ q p D ( D n − t D j 1 ) + q S (7.31)
n
µ j1 −
=
in which p D ( D n − t D j 1 ) = p D () can be evaluated using either equ. (7.42) or (7.46), for
t
t′
D
−
dimensionless time arguments t′ or t′ , respectively and as will be shown in
D
DA
Chapter 8, with slight modification, the same combination of equations can also be
applied to gas well testing. Theoretically, at least, the use of equ. (7.42) to quantify the
p D function in equ. (7.31) removes the problem of trying to decide under which flowing
condition p D should be evaluated because it is valid for all flowing times. Even if t DA
exceeds the maximum value on the abscissa of the MBH chart, the plots are all linear
at this point, and therefore p D(MBH) can readily be calculated by linear extrapolation. For