Page 9 - Transmissibility Corrections and Grid Control for Shale Gas Numerical Simulation

VA - DF: Transmissibility Corrections and Grid Control for Shale Gas Numerical Simulation p 8/23
contributions are distributed over each surface. This leads to a global linear system involving
interaction coefficients that has to be solved to insure the boundary condition at the producing
surfaces. If the problem at hand can be reduced to 2D (fully perforating vertical fracture for
example), an analytical solution to the corresponding system is used, leading to a very fast
transmissibility calculations scheme.
Let us start by considering two adjacent cells of a k-orthogonal grid, in a homogenous
reservoir (Figure 8).
Figure 8 – Two adjacent cells of a k-orthogonal grid
The flux between the two cells is classically written as a function of the pressure drop through
the use of a constant transmissibility:
 
j i ij ij PPT Q  

1
where i P and j P are the average pressures in each cell.
The usual linear assumption on the pressure field leads to the following standard expression
for transmissibility:
ij
ij
ij L
Fk
T


Above, ij F is the surface of the face between the two cells, and ij L is the distance between the
two nodes.
If the pressure field cannot be considered linear in the two cells, another expression has to be
found. Let us express the average pressure in each cell (of volume i v ) :
  
Ci
i i Pdv v P /1
Alternatively, Darcy’s law gives:
   

  
Fij Fij
ij SdP
k
Sdu Q
  

The transmissibility can hence be expressed as:
 

  


Cj
j
Ci
i
Fij
ij Pdv v Pdv v
SdP
k T
/1 /1
