Environmental Engineering Reference
In-Depth Information
3.1 Fitting Parameters
As can be seen in Eq. (
12
), there are four fitting parameters:
U
ad
,
D
ad
,
.
However, as will be seen below, one of them,
U
ad
, depends on the other parameters,
and thus only three parameters are actually present. The fitting parameters are:
α
, and
ʸ
•
Fractal length dimension
,
. It is the effective fractal dimension of length along
the flow direction. As shown in Eqs. (
10
) and (
11
), it affects both the tracer velocity
and its dispersion. The Euclidean limit is
α
α
=
1.
•
Connectivity index
,
. It is a dynamical parameter related to the tracer dispersion
in the fractal structure. In the Euclidean limit,
ʸ
ʸ
=
0, the dispersion coefficient
remains constant, whereas for
ʸ>
0 it decreases with the travelled distance.
•
Dimensionless dispersion coefficient
,
D
ad
. It is a dimensionless parameter asso-
ciated to the dispersion phenomenon in the fractal media at the inter-well scale.
•
Dimensionless velocity
,
U
ad
. It is a dimensionless effective tracer velocity in the
fractal porous medium. This velocity can be expressed in terms of the other para-
meters by considering the time the tracer pulse requires to sweep (fill) the whole
fractal volume between
x
=
x
w
and
x
=
L
. Further, if we choose the reference
time (
ʔ
t
) as this total sweep time, it follows that
1
α
.
U
ad
=
(22)
This relationship reduces the amount of fitting parameters to three:
α
,
ʸ
, and
D
ad
.
3.2 The Inverse Problem
The objective function (OF) is written in terms of the synthetic data
{
C
D
i
,
t
D
i
}
and
the model prediction
{
C
(
t
D
i
;
α,
D
ad
,ʸ)
}
in the least square sense as
N
C
D
i
−
D
ad
,ʸ)
2
OF
(α,
D
ad
,ʸ)
=
C
(
t
D
i
;
α,
,
(23)
i
=
1
<α
≤
where the parameters are constrained according to themodel formulation: 0
1,
0
5. The last equation describes a hyper-surface in
the parameter space. The inverse problem consists in finding the optimal values of
the parameters that minimizes the OF (Eq. (
23
)) on the hyper-surface. In doing so, a
constrained optimization algorithm is needed.
<
D
ad
≤
0
.
1 and 0
≤
ʸ
≤
0
.
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