Page 213 - Fundamentals of Reservoir Engineering

OILWELL TESTING 151

Equation (5.20) can be expressed with respect to this new variable as

1 d r dp ∂ s ∂ s φµ c dp ∂ s
r ds ds r ∂ r ∂ = k ds t ∂

and using equs. (7.2) and (7.3), this becomes

1 φµ cr d φµ cr 2 dp =− φ µ cr 2 dp
r 2k t ds 2k t ds 2k t ds

which can be simplified as

d s dp s dp
ds ds =− ds

or
dp d dp dp
+ s = − s
ds ds ds ds

This is an ordinary differential equation which can be solved by letting

dp p′
ds =

Then

dp′
p′ + s = − sp′
ds (7.4)
+
dp′ (s1 )
=− ds
p′ s
Integrating equ. (7.4) gives

ln p′ =− ln s − s + C 1

or

e − s
p′ = C 2 s (7.5)

where C 1 and C 2 are constants of integration and C 2 can be evaluated using the line
source boundary condition

∂ p qµ dp ∂ s dp
lim r = = r = 2s
π
r → 0 r ∂ 2 kh ds r ∂ ds
therefore,