Page 175 - Well Logging and Formation Evaluation

Reservoir Engineering Issues 165

r e = drainage radius, in m
r w = wellbore radius
k = permeability, in m 2
m= viscosity, in Pa.

For a situation in which the boundary is sealed and the pressure is drop-
ping linearly with time, a semi-steady state exists that has the solution:

PP w = ( *m ( * p*2 k h)) ( * ln r r w ) - 3 4 ). (11.4.6)
*
-
e (
Q
Note that P and P w both vary with time, but their difference remains
constant. Since the wellbore radius is always very small compared with
the drainage radius, it is possible mathematically to study the theoretical
case in which the wellbore radius becomes infinitessimally small, whose
solution is:

*
P i - P f = Q*m ( * p*4 k h)* E x) (11.4.7)
(
i
where P f is the pressure of the flowing phase, E i(x) denotes the exponen-
tial integral, and

2
x =k * r 4 t *

k = f m**Ck .

The exponential integral, E i(x), is given by:

g
!
*
(
Ex) =- - ln ( x) - ( S - x) n n n
i
2
where g is Euler’s constant (1.781). If t > 25*f*m*C*r /k, the summa-
tion term becomes less than 0.01 and can be ignored; then we can give
the approximate version of equation 11.4.7 as:
2
P i - P f = -( *m ( * *p4 k * h )*ln ( * *g k r 4 * t). (11.4.8)
Q
2
Solving for t/r :
2
(
tr = ( 14)* * *exp 4 ( * * k h* D P Q*m ) (11.4.9)
gk
p
*
where DP = P i - P f.