structural geology, propagation mechanics and - Stanford School of ...

k
2d
m b
j = , (7)
lb
2d
m
k
b
+
+ l
k
m
where lb is the CB thickness. In addition, dlm in Equation (6) is replaced by m b , the
center to center distance (see Figure 7.4b). The presence of k
d + l / 2
b -1 in the denominator of
Equation (7) results in kj < km. It is in this way that the permeability-barrier effect of the
CBs enters the matrix-CB transmissibility and thus the flow equations. For flow between
two CB control volumes, Equation (6) is again applied, with quantities as defined in
Figure 7.4c. We note that permeability anisotropy in the CBs (i.e., different permeability
along and across the bands) could be readily modeled within the current formulation,
though this is not necessary here because CB permeability is assumed to be isotropic.
Writing the discrete mass balance ( q + q + q = −Qˆ
) for each cell, and expressing the qj
1
2
via Equation (5), gives the finite volume representation for Equation (1). Solution of the
resulting linear system provides the cell-center pressure unknowns.
In the case of two-phase flow, finite volume discretization of Equations (2) and (3)
entails mass conservation statements for each component (phase). The net flux out of the
control volume in this case is balanced by both the source and accumulation terms. The
flux of a particular phase (e.g., water) across edge j is now given by:
w
w w
q = λ T ( p − p )
(8)
j
w
j
l
m
w
w
where λw is the upstream-weighted phase mobility (λw = krw/µw) and p and p are the
water pressures in blocks l and m. The transmissibility Tj
is the same for both phases and
is again given by Equation (6). Expressing the discrete mass balances for each component
using Equation (8) and introducing time discretizations for the accumulation terms
provides the finite volume representation. Pressure and saturation for each cell at each
time step can then be computed.
3
The two-point flux approximation described above is applied for all connections. This
is strictly valid only when the grid is “orthogonal,” which in this context means that dlm
(the line connecting the two cell centers) is orthogonal to the shared interface.
Nonorthogonality (or full-tensor effects, not an issue here since permeability is taken as
179
l
m