26 / 743
Dynamic Data Analysis – v5.12.01 - © KAPPA 1988-2017
Chapte
r 2 – T heory- p26/743
As the line source, the infinite reservoir conditions arrives at B(u)=0. The only difference is
that we now calculate the well equation at r
D
=1 and not r
D
=0. This gets us to a different value
for A(u):
From the inner boundary condition:
u Kuu
uA
1
1
)(
If we define the function:
x xK
xK
xK
1
0
1
0
The finite radius solution at the well is
u K
u
up
D
1
0
1
)(
In the wellbore storage and skin solution we eliminate q
D
and p
wfD
in the top equations, we
take the Laplace transform, etc, etc. In fact, we can make the generic demonstration that, if
we know the Finite Radius solution of a given problem in Laplace space, wellbore storage and
skin can be added by the following transform:
We obtain a modified Bessel equation and solve for inner and outer boundaries; a term K
1
appears corresponding to the finite radius well condition, and we finally get the following
results:
Adding wellbore storage & skin:
) ,1(
1
) ,1(
1
)(
u puS uC
u puS
u
u p
FRD
D
FRD
wfD
In the homogeneous infinite case:
u KS uC
u KS
u
u p
D
wfD
1
0
1
0
1
1
)(
The problem is then solved in real space by taking the inverse Laplace transform using
Stehfest numerical algorithm. Though the solution is more complex, the difference between
this solution and the Exponential Integral solution will stabilize when wellbore storage effects
vanish, the residual difference remaining is the skin factor. This links to the IARF by:
After wellbore storage:
S
t
E t
p
D
i
D wfD
4
1
) (
Dimensionless IARF: for
100
D
t
S
t
t p
D
D wfD
2
80907 .0 ln
2
1