Page 215 - Fundamentals of Reservoir Engineering

OILWELL TESTING 153

x 1 = e − S
e -s s s
s = x

s s s
(a) (b) (c)

∞ − S
ei (x) = e ds
x s

x
(d)

Fig. 7.2 The exponential integral function ei(x)

where the number 0.5772 is Euler's constant, the exponential of which is denoted by

γ = e 0.5772 = 1.781

and therefore equ. (7.8) can be expressed as

ei(x) ≈− ln (γ ) x for x < 0.01 (7.9)

The separate plots of ei(x) and −In(γx), in Fig. 7.3, demonstrate the range of validity of
equ. (7.9). The significance of this approximation; is that reservoir engineers are
frequently concerned with the analysis of pressures measured in the wellbore, at r = r w.
2
Since in this case x φµ= cr / 4kt , it is usually found that for measurements in the
w
wellbore, x will be less than 0.01 even for small values of t. Equation (7.6) can then be
approximated as

qµ 4kt
p = p = p − ln
w rt wf i 4kh γφ µ cr w 2
π
Or, if the van Everdingen mechanical skin factor is included as a time independent
perturbation (ref. Chapter 4, sec. 7), then

qµ 4kt
p wf = p − ln + 2S (7.10)
i
4kh γ φ µ cr w 2
π
As expected for this transient solution there is no dependence at all upon the area
drained or well position with respect to the boundary since for the short time when
equ. (7.10) is applicable the reservoir appears to be infinite in extent.