Page 11 - 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
p 10/23
For complex three-dimensional problems, such as slanted wells or partially penetrating
fractures, the two-dimensional solution cannot be used, and closed form three-dimensional
solutions are limited to very simple geometries.
The source configuration, composed of either well segments or individual fractures, is
subdivided into subsections onto which a constant source distribution is considered (Figure 9):
Figure 9 – Fracture description
We have:
i
i
i
i
i
rec
u i
HLFM
M
) , , , , (
) (
Where
rec
u
is the panel unitary source solution as given in Appendix A. Assuming vertical
fractures, the two first images are accounted for by super-imposing the corresponding
solutions:
i
i
i
i
D
i
rec
u
i
i
i
U
i
rec
u
i
i
i
i
rec
u i
HLFM
HLFM
HLFM
M
) , , ,
, (
) , , ,
, (
) , , , , (
) (
With
U
i
F
and
D
i
F
corresponding to the images of the section center with respect of the upper
and lower bound (horizon) respectively.
The determination of the
i
coefficients is done differently depending on the boundary
conditions insured at the producing surfaces. If a uniform flux condition is desired, these
coefficients are all set to unity resulting in a varying potential along the producing surface. In
this case the segmentation is reduced to a single element and the computation is
straightforward. In the more complex case where a constant potential at the surface is
required, corresponding to the infinite conductivity condition, the equality of potential between
a set of points chosen at the center of each subsection is expressed. This results in a linear
system that has to be solved in order to obtain these coefficients.
A similar technique is employed in the case of wells with multiple segments and/or slanted
geometry where the unitary panel solution
rec
u
is replaced by unitary segment solution
Seg
u
as
given in Appendix A.
