Page 214 - Fundamentals of Reservoir Engineering

OILWELL TESTING 152

qµ Ce − s
4kh = 2
π

which yields, as r (and therefore s) tends to zero

qµ
C 2 = 4kh
π

Equation (7.5) can now be integrated between the limits t = 0 (s → ∞ ) and the current
value of t, for which s = x; and p i (initial pressure) and the current pressure p.

i.e.

p qµ x e − s
dp = 4kh s ds
i p π ∞

which gives

∞ − s
qµ e
p r,t = p − ds (7.6)
i
4kh
π
φµ cr 2 s
x =
4k t
Equation (7.6) is the line source solution of the diffusivity equation giving the pressure
p r,t as a function of position and time.
The integral

∞ e − s ∞ e − s
s ds = s ds (7.7)
x φµ cr 2
x =
4kt
is a standard integral, called the exponential integral, and is denoted by ei(x).

Qualitatively, the nature of this integral can be understood by considering the
component parts, fig. 7.2.

The integral of curve (c) between x and ∞ will have the shape shown in fig. 7.2 (d).
Thus ei (x) is large for small values of x, since the ei-function is the area under the
graph from the particular value of x out to infinity (i.e. the shaded area in curve (c) of
fig. 7.2) and, conversely, small for large values of x. The ei-function is normally plotted
on a log-log scale and such a version is included as fig. 7.3. From this curve it can be
seen that if x < 0.01, ei (x) can be approximated as

ei(x) ≈− ln x − 0.5772 (7.8)