Page 10 - Transmissibility Corrections and Grid Control for Shale Gas Numerical Simulation
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VA - DF: Transmissibility Corrections and Grid Control for Shale Gas Numerical Simulation
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And the well index is given by:
Fiw
Ci
i
Fiw
SdP S Pw Pdv
v
SdP
k WI
/1
If the fluid compressibility is further neglected, the pressure field becomes harmonic and can
be expressed in terms of Greens function representation. For a source point located at
s
x
, we
define a free space function by
s
s
s
x x x
xxG
/) ( ) ;(
0
. Assuming a distribution of source
points, the free space function can be integrated to obtain the resulting potential field:
S
s
s
xdxxG x
) ;(
)(
0
This field being directly proportional to the uncompressible pressure field, the formulae for
transmissibility and well index can be used directly with the potential in place of the pressure.
3.
Solution Procedure
The evaluation of the potential field is done differently depending on the complexity of the
problem. If the problem at hand is two-dimensional (fully perforating fractures for example),
an analytical solution can be found by conformal mapping techniques. When the problem
becomes three-dimensional, a numerical evaluation procedure must be used to evaluate the
potential function. In this case the surface is discretized in terms of elementary surfaces and
the linearity of the problem at hand allows for direct superposition of individual contribution to
obtain the resulting potential field.
Another difficulty arises from the required boundary condition at the producing surfaces: the
superposition of singularity distribution with the same strength leads to a solution equivalent
to the uniform flux solution, i.e. the pressure is not constant along the producing area, on each
elementary surface. In order to obtain a constant boundary pressure condition (infinite
conductivity fracture for example), a numerical procedure must be involved to compute
distribution strength on each elementary surface.
For two-dimensional problems, conformal mapping techniques give closed form solutions for
given geometry and boundary conditions. In the present case, we can consider the potential
between co-focal ellipses with a degenerate inner ellipse given by:
f
X
b a
yx
ln ,
, where
f
X
is the half segment length and
1
2
2
2
2
b
y
a
x
2 2
22
2
2
2
2
2
4
2
1
f
f
f
Xy
X y x
X y x
b
, and
2
2
b X a
f
This solution can be readily used for fully perforating fractures. Note however that in multi-
fracture cases the superposition of an array of such solutions would lead to a solution that
would not fulfil the constant pressure condition at the fracture faces. In this case, the potential
field is built from superposition of local potential fields (no interferences).
