Here we note that our closure equation will be homogeneous in ˜c
h
if the gradient of the regional average concentration is zero. For this reason we have identified the two terms
involving this gradient as the sources of the ˜c
h
-field. An analogous form can be derived for the q-region transport
equation, and the two will be connected by the interfacial boundary conditions.
On the basis of eqns 4, 5 and 8, we see that the boundary conditions take the form
B:C:1 c
h h
¼ c
q q
, at A
hq
45 B:C:2
n
hq
·D
p h
·= c
h h
¼ n
hq
·D
p q
·= c
q q
, at A
hq
46 and when we use the decompositions given by eqn 11, we
shall obtain the boundary conditions in terms of the desired spatial deviation concentrations, ˜c
h
and ˜c
q
. This leads us to the closure problem as follows.
2.2 Closure problem
Periodicity : ˜c
h
r þ ,
i
¼ ˜c
h
r
, ˜c
q
r þ ,
i
¼ ˜c
q
r
, i ¼ 1
, 2
, 3
ð 47eÞ
Average : ˜c
h h
¼ ,
˜c
q q
¼ ð
47f Þ Here it should be clear that all the sources, or the non-
homogeneous terms in this boundary value problem, can be expressed in terms of the two concentration gradients
and the concentration difference, i.e.
Sources : =
c
h h
h
, =
c
q q
q
, c
q q
q
¹ c
h h
h
ÿ
47a
47b
47c
47d 68
A. Ahmadi et al.
At this point we have replaced the original problem by a set of large-scale averaged equations and a local-scale closure
problem involving the large-scale variables and the spatial deviations. Our objective now is to obtain an approximate
solution of this problem. Following ideas developed in the treatment of heat transfer in porous media
27,29–32
, or in dealing with the flow of a slightly compressible fluid in a
heterogeneous porous
medium
33,34
, this
suggests representations for the spatial deviation concentrations of
the form ˜c
h
¼ b
hh
·= c
h h
h
þ b
hq
·= c
q q
q
þ r
h
c
q q
q
¹ c
h h
h
ÿ ð
48aÞ ˜c
q
¼ b
qh
·= c
h h
h
þ b
qq
·= c
q q
q
þ r
q
c
q q
q
¹ c
h h
h
ÿ ð
48bÞ
in which we refer to b
hh
, b
qh
, r
q
, etc., as the closure variables. In terms of these closure variables, there are
three closure problems that result from eqns 47, and the first of these is given by
Problem I
=·
v
b h
b
hh
þ ˜ v
bh
¼ = · D
p h
·=b
hh
ÿ þ =· ˜
D
p h
¹ J
¹ 1
h
c
hh
49a B:C:1
b
hh
¼ b
qh
at A
hq
49b B:C:2
n
hq
·D
p h
·=b
hh
þ n
hq
·D
p h
¼ n
hq
·D
p q
·=b
qh
at A
hq
49c
=· v
b q
b
qh
ÿ ¼ =· D
p q
·=b
qh
ÿ ¹ J
¹ 1
q
c
qh
49d Periodicity :
b
hh
r þ ,
i
¼ b
hh
r
,
b
qh
r þ ,
i
¼ b
qh
r
, i ¼ 1
, 2
, 3
ð 49eÞ
Average :
b
hh h
¼ ,
b
qh q
¼ 49f
Here we have used the vectors c
hh
and c
qh
to represent the inter-region flux terms according to
c
hh
¼ ¹ 1
V
`
Z
A
hq
n
hq
· v
b h
b
hh
¹ D
p h
·=b
hh
¹ ˜ D
p h
dA 50a
c
qh
¼ ¹ 1
V
`
Z
A
qh
n
qh
· v
b q
b
qh
¹ D
p q
·=b
qh
ÿ dA
50b and these are related by
c
hh
¼ ¹ c
qh
50c The second closure problem is related to the source,
= c
q q
q
, and it is given by
Problem II
=·
v
b h
b
hq
¼ = · D
p h
·=b
hq
ÿ ¹ J
¹ 1
h
c
hq
51a B:C:1
b
hq
¼ b
qq
, at A
hq
51b B:C:2
n
hq
·D
p h
·=b
hq
¼ n
hq
·D
p q
·=b
qq
þ n
hq
·D
p q
at A
hq
51c
=· v
b q
b
qq
ÿ þ ˜v
bq
¼ = · D
p q
·=b
qq
ÿ þ =· ˜
D
p q
¹ J
¹ 1
q
c
qq
51d
Periodicity : b
hq
r þ ,
i
¼ b
hq
r
,
b
qq
r þ ,
i
¼ b
qq
r
, i ¼ 1
, 2
, 3
ð 51eÞ
Average :
b
hq h
¼ ,
b
qq q
¼ 51f
In this case the two constant vectors are defined by
c
hq
¼ ¹ 1
V
`
Z
A
hq
n
hq
· v
b h
b
hq
¹ D
p h
·=b
hq
dA 52a
c
qq
¼ ¹ 1
V
`
Z
A
qh
n
qh
· v
b q
b
qq
¹ D
p q
·=b
qq
¹ ˜ D
p q
dA 52b
and they are related by
c
hq
¼ ¹ c
qq
52c The third closure problem originates with the exchange
source, c
q q
q
¹ c
h h
h
, and it takes the form
Problem III
=·
v
b h
r
h
¼ = · D
p h
·=r
h
ÿ ¹ J
¹ 1
h
a
p
53a B:C:1
r
h
¼ r
q
þ 1
, at A
hq
53b B:C:2
n
hq
·D
p h
·=r
h
¼ n
hq
·D
p q
·=r
q
at A
hq
53c
=· v
b q
r
q
ÿ ¼ =· D
p q
·=r
q
ÿ þ J
¹ 1
q
a
p
53d Periodicity :
r
h
r þ ,
i
¼ r
h
r
, r
q
r þ ,
i
¼ r
q
r
, i ¼ 1
, 2
, 3
ð 53eÞ
Average : r
h h
¼ ,
r
q q
¼ 53f
Here the mass transfer coefficient, a, is defined by a
p
¼ ¹ 1
V
`
Z
A
hq
n
hq
· v
b h
r
h
¹ D
p h
·=r
h
dA
¼ þ 1
V
`
Z
A
qh
n
qh
· v
b q
r
q
¹ D
p q
·=r
q
ÿ dA
54 Heterogeneous porous media V
69
Rather than work directly with the closure variables, r
h
and r
q
, it is convenient to define new variables according to s
h
¼ r
h
, s
q
¼ r
q
þ 1
55 in order to represent the closure problem for the exchange
coefficient in terms of a continuous closure variable. Under these circumstances we express the third closure problem
as follows.
Problem III9
=· v
b h
s
h
¼ = · D
p h
·=s
h
ÿ ¹ J
¹ 1
h
a
p
56a B:C:1
s
h
¼ s
q
, at A
hq
56b B:C:2
n
hq
·D
p h
·=s
h
¼ n
hq
·D
p q
·=s
q
at A
hq
56c
=· v
b q
s
q
ÿ ¼ =· D
p q
·=s
q
ÿ þ J
¹ 1
q
a
p
56d Periodicity :
s
h
r þ ,
i
¼ s
h
r
, s
q
r þ ,
i
¼ s
q
r
, i ¼ 1
, 2
, 3
ð 56eÞ
Average : s
h h
¼ ,
s
q q
¼ 1
56f In this case the mass transfer coefficient takes the form
a
p
¼ ¹ 1
V
`
Z
A
hq
n
hq
· v
b h
s
h
¹ D
p h
·=s
h
dA 57
These closure problems are similar to those that have been solved previously by Quintard and Whitaker
30,35,36
, Fabrie et al.
37
and Quintard et al.
32
, and they can be used to determine the coefficients that appear in both the two-
equation model and the one-equation model that was developed in Part IV. The major difference between this
development and previously studied two-equation models is associated with the spatial variations of the dispersion
tensors due to their dependence on velocity fluctuations. As a consequence, new diffusive source terms appear in
the closure problem in the form of the divergence of the deviation of the dispersion tensors. The derivation of
the closure problem for the one-equation, equilibrium model is presented in Appendix A.
In order to develop the closed forms of eqn 23a, we substitute the representation for ˜c
h
given by eqn 48a and make use of the change of variable indicated by eqn 55 to
obtain e
h
1 þ K
h
ÿ J
h
] c
h h
h
]t þ =
· J
h
v
b h
n o
h
c
h h
h
h i
¹ = · d
h
c
h h
h
¹ c
q q
q
ÿ ¹ u
hh
·= c
h h
h
¹ u
hq
·= c
q q
q
¼ = · D
pp hh
·= c
h h
h
ÿ þ =
· D
pp hq
·= c
q q
q
ÿ ¹ a
p
c
h h
h
¹ c
q q
q
ÿ ð
58Þ Here the various coefficients are defined by
d
h
¼ J
h
˜ v
bh
s
h
¹ D
p h
·=s
h h
59a
u
hh
¼ ¹ 1
V
`
Z
A
hq
n
hq
· v
b h
b
hh
¹ D
p h
·=b
hh
¹ ˜ D
p h
dA 59b
u
hq
¼ ¹ 1
V
`
Z
A
hq
n
hq
· v
b h
b
hq
¹ D
p h
·=b
hq
dA 59c
D
pp hh
¼ J
h
D
p h
· I þ =b
hh
ÿ ¹ ˜v
bh
b
hh h
59d
D
pp hq
¼ J
h
D
p h
·=b
hq
¹ ˜ v
bh
b
hq h
59e
a
p
¼ ¹ 1
V
`
Z
A
hq
n
hq
· v
b h
s
h
¹ D
p h
·=s
h
dA 59f
In order to obtain the closed form of the q-region transport equation, we follow the above development from eqn 23b
to arrive at
e
q
1 þ K
q
ÿ J
q
] c
q q
q
]t þ =
· J
q
v
b q
q
c
q q
q
¹ = · d
q
c
h q
q
¹ c
h h
h
ÿ ¹ u
qh
·= c
h h
h
¹ u
qq
·= c
q q
q
¼ = · D
pp qh
·= c
h h
h
ÿ þ =
· D
pp qq
·= c
q q
q
ÿ ¹ a
p
c
q q
q
¹ c
h h
h
ÿ ð
60Þ The coefficients in this case are analogous to those given
by eqns 59, and for completeness we list them as
d
q
¼ J
q
˜ v
bq
s
q
¹ D
p q
·=s
q q
61a
u
qq
¹ 1
V
`
Z
A
qh
n
qh
· v
b q
b
qq
¹ D
p q
·=b
qq
¹ ˜ D
p q
dA 61b
u
qh
¼ ¹ 1
V
`
Z
A
qh
n
qh
· v
b q
b
qh
¹ D
p q
·=b
qh
ÿ dA
61c 70
A. Ahmadi et al.
D
pp qq
¼ J
q
D
p q
· I þ =b
qq
ÿ ¹ ˜v
bq
b
qq q
61d
D
pp qh
¼ J
q
D
p q
·=b
qh
¹ ˜ v
bq
b
qh q
61e
a
p
¼ ¹ 1
V
`
Z
A
qh
n
qh
· v
b q
s
q
¹ D
p q
·=s
q
ÿ dA
61f In the next section we shall present results for the coeffi-
cients given by eqns 59 and 61. The large-scale equations, eqns 58 and 60, represent a
generalized version of two-equation models for describing dispersion and adsorption in such systems, and it is interest-
ing to discuss the theoretical status of the linear mass exchange term in these equations. On the basis of the
assumptions we have made, the concentration deviations given by eqns 48, coupled with the Darcy-scale problem
given by eqns 47, represent a simplified closure scheme for the large-scale averaged equations associated with the h-
and q-regions. A general solution would involve a more complicated expression for the exchange between the two
effective media, and the retention of the transient form of the closure problem
38
. In the next section we test the present theory versus numerical experiments obtained for the case
of stratified systems.
3 NUMERICAL EXPERIMENTS FOR STRATIFIED SYSTEMS
In this section, we present a complete analysis of the strati- fied system illustrated in Fig. 3 in the absence of adsorption
effects. This system has a behaviour typical of the two- region models that have been studied previously
24,39–41
, while being simple enough to allow for precise analysis.
We first obtain Darcy-scale solutions that will serve as numerical experiments for a comparison with theoretical
predictions.
3.1 Local problem
