Environmental Engineering Reference
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ˆ
F
ʱ
∂
∂
S
F
,ʱ
ˁ
F
,ʱ
c
F
,
k
ʱ
+∇·
ˁ
F
,ʱ
c
F
,
k
ʱ
v
F
,ʱ
+
d
F
,
k
ʱ
t
ʱ
=
I
F
,
k
ʱ
+
F
mF
,
k
ʱ
+
F
fF
,
k
ʱ
,
(132)
ʱ
ˆ
f
ʱ
∂
∂
S
f
,ʱ
ˁ
f
,ʱ
c
f
,
k
ʱ
+∇·
ˁ
f
,ʱ
c
f
,
k
ʱ
v
f
,ʱ
+
d
f
,
k
ʱ
t
ʱ
=
I
f
,
k
ʱ
+
F
mf
,
k
ʱ
−
F
fF
,
k
ʱ
,
(133)
ʱ
where
ˆ
f
are the porosities of the matrix blocks, the large fractures, and
the small fractures, respectively. Under the assumption of quasi steady-state flow,
the mass transfer flux terms can be defined according to Eq. (
126
)as
ˆ
m
,
ˆ
F
, and
F
mF
=−
ʛ
Fm
k
m
ˁ
μ
(
p
F
−
p
m
) ,
(134)
p
f
p
m
,
F
mf
=−
ʛ
fm
k
m
ˁ
μ
−
(135)
p
F
−
p
f
,
F
fF
=−
ʛ
fF
k
f
ˁ
μ
(136)
for large fracture-matrix, small fracture-matrix, and large fracture/small fracture
interactions. Following Warren and Root (
1963
),
, while for
the
f
-
F
interaction, the shape factor for one-dimensional small fractures is defined
by (Wu et al.
2004
)
ʛ
Fm
=
ʛ
fm
=
ʛ
A
fF
d
fF
,
ʛ
fF
=
(137)
where
A
fF
is the total large-fracture and small-fracture connection area per unit vol-
ume of rock and
d
fF
is the characteristic distance between large and small fractures.
Again the phase velocity in each medium
v
M
,ʱ
f
, is given by Darcy's
law (
10
). An energy conservation equation for each medium is also required if the
effects of heat transfer are important. However, these equations can be combined into
a single equation for the temperature if thermal equilibrium is assumed for the entire
system. This gives a total of 3
,forM
=
m
,
F
,
1 differential equations. The system is closed
by adding to the flow equtions 3 saturation constraints, 3
P
concentration constraints,
3
P
(
n
+
P
)
+
equilibrium conditions as
given by relations (
123
), (
124
), (
125
), and (
101
), respectively. As a final remark, for
deformation-dependent flows, the body force in Eq. (
42
) will have the form
−
3 capillary pressure relationships, and 3
n
(
P
−
1
)
=
ˆ
m
p
m
+
ˆ
F
p
F
+
ˆ
f
p
f
I
F
,
(138)
where
p
m
,
p
F
, and
p
f
are the pressures exerted by the fluid on the pores of the
three media. The extension of the present triple-continuum model to a unified
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