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Dynamic Data Analysis – v5.12.01 - © KAPPA 1988-2017
Chapte
r 2 – T heory- p30/743
To the finite radius solution we can add wellbore storage and skin in Laplace space:
Adding wellbore storage & skin:
)(
1
)(
1
)(
u puS uC
u puS
u
u p
FRD
D
FRD
wfD
The problem is once again transferred to real space using the Stehfest algorithm. The behavior
is studied in the next section. As for IARF we will use an interesting late time approximation:
Late time approximation:
S
r
r
t
t
p
eD
eD
D
D wfD
4
3
ln
2
2
This will be the origin of the pseudo-steady state behavior. Converting the dimensionless
radius back dimensionless variables to physical variables we get the PSS equation:
PSS equation:
S
rC
A
t
A
r
t
p
wA
D
w
D wfD
4045 .0
ln
2
1
2
2
2
C
A
is the Dietz shape factor, characteristic of the shape of the reservoir and the position of the
well in the reservoir. In the case of a well at the center of a closed circle we get C
A
=31.62.
2.E.3
Pseudo-Steady State flow
At late time the pressure response of a well flowing at the center of a homogeneous reservoir
of circular shape will be approximated by:
Circular PSS equation:
4
3
ln
2. 141
03723 .0
2
w
e
e t
i
r
r
S
kh
qB
t
hrc
qB
p tp
We see that the slope of this linear approximation is inversely proportional to the square of the
circle radius. More generally the slope will be inversely proportional to the reservoir area.
In the case where the reservoir thickness is not the constant h the slope will be inversely
proportional to the volume of the reservoir (or the drainage area). In other words quantifying
the PSS slope will provide an estimate of the reservoir volume and therefore the reserves.
The figures below show the drawdown response of a vertical well with wellbore storage and
skin in a homogeneous reservoir and under three boundary configurations: an infinite reservoir
(green), a circular reservoir of 6,000 ft of radius (blue) and an even smaller reservoir with a
radius of 3,000 ft (red). Responses are shown on a linear scale (left) and a loglog scale (right)
together with the Bourdet derivative defined in the PTA chapter.
During production, when the boundary is detected, the behavior deviates from Infinite Acting
Radial Flow to reach Pseudo-Steady State according to the equation above. PSS is
characterized by linearity between the pressure change and the elapsed time on the linear
scale.
On the loglog plot PSS is characterized by a unit slope of the Bourdet derivative. Though it is
slower, the pressure change also tends towards merging with the pressure derivative on the
same unit slope.