Environmental Engineering Reference
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steady-state matrix-fracture transfer. Its dimension is the inverse of a squared length.
For instance, assuming a cubic block of size
a
, Warren and Root (
1963
) derived the
following expression for the shape factor
4
N
(
N
+
2
)
ʛ
=
,
(127)
a
2
where
N
is the number of flow dimensions.
In most modern simulators the expression for
is based on a simplified
representation of the dual-medium fractured system, where fracture and matrix grids
are identical and superposed. Each cell is a discretized element of volume including
both fracture and matrix blocks and contains a number of identical matrix blocks,
which are assumed to be parallelepipeds of volume
abc
. Therefore, an expression
for the matrix-fracture transfer flux per unit volume of matrix rock is
F
mf
,
k
ʱ
6
1
abc
F
mf
,
k
ʱ
=
f
i
,
k
ʱ
,
(128)
i
=
1
where the
f
i
,
k
ʱ
through each of
the six faces of the block. This term is usually split up into two contributions due
to convection-diffusion and diffusion-dispersion effects (Sarda et al.
1997
). Since
the matrix-fracture flow rate is governed by the matrix permeability, the absolute
permeability used for the calculation of these flow rates is given by
k
m
,
i
, the matrix
permeability in the face direction
i
, which allows for permeability anisotropy through
different values of
k
m
,
i
in each direction. An expression for the fluxes
f
i
,
k
ʱ
is given
by Sabathier et al. (
1998
), where for the anisotropic case the matrix-fracture coupling
transmissibility between a cell of the matrix grid and the corresponding one of the
fracture grid is defined as
represent the mass flux of component
k
in phase
ʱ
4
k
m
,
x
a
2
k
m
,
y
b
2
k
m
,
z
c
2
ʛ
k
m
,
i
=
+
+
.
(129)
If deformation of the matrix-fracture system is important, the body force term in
Eq. (
42
) for the solid displacements is of the form
=
ˆ
m
p
m
+
ˆ
f
p
f
I
F
,
(130)
where
p
m
and
p
f
are, as in Eq. (
126
), the mean pressure exerted by the fluid on the
pore matrix and pore fractures, respectively. A description of deformation-dependent
flow models of various porosities and permeabilities relevant to the characterization
of naturally fractured reservoirs is given by Bai et al. (
1993
).
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