Page 199 - Fundamentals of Reservoir Engineering

STABILIZED INFLOW EQUATIONS 137

1 ∂ ∂ p qµ
r =− (6.2)
2
rr ∂ r ∂ π rkh
e
and integrating this equation

∂ p q r 2
µ
=− + C 1 (6.3)
2
t ∂ 2r kh
π
e
where C 1 is a constant of integration. At the outer no-flow boundary ∂p/∂r is zero and
hence the constant can be evaluated as C 1 = qµ/2πkh which, when substituted in
equ. (6.3), gives

∂ p qµ 1 r
= − 2 (6.4)
r ∂ 2 kh r r e
π

Integrating once again

2
p r qµ r r
2
p p wf = 2kh lnr − 2 r e r w (6.5)

π
or

qµ r r
2
p − p wf = ln − 2 (6.6)
r
2kh r w 2r e
π
2
2
in which the term r/r is considered to be negligible. Equation (6.6) is a general
w
e
expression for the pressure as a function of the radius. In the particular case when
r = r e then
qµ r 1
p − p wf = ln e − + S (6.7)
e
2kh r w 2
π
This is the familiar well inflow equation under semi-steady state conditions and is
similar to that presented as equ. (4.27) for steady state flow. It can be transposed to
give the Pl relationship

q 2 kh
π
PI = = (6.8)
p − p wf r e 1
e
µ ln − + S
r w 2
in which the van Everdingen skin factor has been included as described in Chapter 4,
sec. 7. One unfortunate aspect concerning the application of this equation is that, while
both q and p wf can be measured directly, the outer boundary pressure cannot. It is
therefore more common to express the pressure drawdown in terms of p − p wf instead
of p e − p wf, since p, the average pressure within the drainage volume, can readily be
determined from a well test as will be shown in Chapter 7, sec. 7. To express the inflow