Page 200 - Fundamentals of Reservoir Engineering

STABILIZED INFLOW EQUATIONS 138

equation in these terms requires the determination of the volume averaged pressure
within the radial cell as

e r
pdV

p = w r (6.9)
e r
dV
w r

and since dV = 2πrhφdr, equ. (6.9) can be expressed as

e r
p2 rh dr
π
φ
w r
p = 2 2
π (r − r )hφ
w
e
or
2 e r
p = 2 prdr
2
(r − r ) w r
e
w
2
2
2
2
2
2
−
and since r − r = r (1 r /r ) r ,then
≈
e
w
e
e
w
e
2 e r
p = 2 prdr (6.10)
r e w r
The pressure in the integrand of equ. (6.10) is obtained from equ. (6.6) which is a
general expression for p as a function of r. Substituting the latter in equ. (6.10) gives
2
2 qµ e r r r
pp = . r ln − 2 dr (6.11)

−
wf
r e 2 2kh w r r w 2r e
π
The first term in the integrand is evaluated using the method of integration by parts, i.e.
e r r r 2 r e r e r 1 r 2
rln dr = ln − dr
w r r w 2 w r w r w r r 2
r 2 r e r r 2 e r
= ln −
2 w r w r 4 w r
r 2 r r 2
≈ e ln e − e
2 r w 4
while the integration of the latter term in equ. (6.11) gives

e r r 3 r r e r 2
4
2 dr = 2 ≈ 8
e
w r 2r e 8r e w r