Environmental Engineering Reference
In-Depth Information
Q
M
,ʱ
ʱ
where
denotes the volumetric injection/production rate of phase
in medium
=
,
M
m
per unit volume of the reservoir. This rate is positive in production and
negative in injection.
The phase velocity in both media
v
M
,ʱ
f
is expressed using Darcy's law (
10
) with
the mobility tensor of phase
as defined by relation (
11
). However, we note that
the absolute permeability of the medium
k
is the equivalent permeability of the flow
continuum under consideration, that is, the permeability of the medium that is scaled
up to the simulated flow scale. For instance, in the dual-continuummodel of (Warren
and Root
1963
), the permeability of the fracture large-scale continuum is obtained
by homogenizing the transport equations on the fracture network, assuming imper-
vious boundaries. On the other hand, if heat transfer is important, we need to add
two additional equations for the specific internal energy
U
M
,
k
ʱ
ʱ
in each medium.
These equations have the same form of Eq. (
89
), except that now the term
Q
M
,
k
ʱ
is
a source/sink or fracture-matrix interaction term for energy. If we assume thermal
equilibrium between the fluids and the media, these two equations can be combined
into a single equation for the common temperature
T
. To close the system, addi-
tional equations must be added to the flow equations. These are the two saturation
constraints
S
M
,ʱ
=
1
,
(123)
ʱ
the 2
P
composition constraints
n
c
M
,
k
ʱ
=
1
,
(124)
k
=
1
the 2
P
−
2 capillary pressure relationships
p
M
,
c
ʱʲ
=
p
M
,
c
ʱ
−
p
M
,
c
ʲ
,
(125)
and the 2
n
(
P
−
1
)
equilibrium conditions of the form given by relation (
101
)for
M
=
f
.
Thematrix-fracture transport flux
m
,
in Eqs. (
120
) and (
121
) can be expressed
either by the direct application of Darcy's law between matrix and fractures with the
explicit input of matrix block dimensions or the input of a shape factor that takes
into account the size and shape of the matrix blocks. For instance, considering a
single-phase, quasi-steady-state flow, Warren and Root (
1963
) expressed
F
mf
,
k
ʱ
F
mf
as
p
f
p
m
,
k
m
ˁ
μ
F
mf
=−
ʛ
−
(126)
where
k
m
is the matrix permeability (assumed to be isotropic),
p
f
and
p
m
the
fracture and matrix pressures,
ʛ
the shape factor. This factor is a constant matrix-fracture exchange factor that
depends only on the geometry and characteristic size of the matrix blocks assuming a
ˁ
the fluid density,
μ
the fluid viscosity, and
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