Parameter Estimation in a Model for Tracer Transport in One-Dimensional Fractals - Selected Topics of Computational and Experimental Fluid Mechanics - page 235

U ad x 1 − α D ad x − α − ʸ f 1 ( x D ) =

− ʸ

(17)

D D

D ad x 1 − α − ʸ f 2 ( x D ) = . (18)

D

Equation ( 16 ) is discretized by using a time-averaged finite-difference method
(Crank-Nicolson) for the time and backward and symmetric finite differences for
the first and second order spatial partial derivatives, respectively. This is

C n + 1
j

C j

1

C j

−

f 1 j

C j − 1 + C n + 1

j

C n + 1
j

= − − = (19)

−

ʴ t D 2 ʴ x D

2 C j − 1 −

1

f 2 j

2 C j C j + 1 + C n + 1

j

2 C n + 1
j

C n + 1
j

+

1 −

+ ,

− +

(ʴ

x D )

2

which after grouping terms and doing some algebra is
ʳ j − ʲ j C n + 1

1 + 1

ʲ j C n + 1

− ʲ j C n + 1 − ʳ j + 2

1 =

(20)

j − j j +

ʲ j − ʳ j C j − 1 + 1

ʲ j C j

+ ʲ j C j + 1 + ʳ j − 2

ʴ t D f 1 j
2

ʴ t D f 2 j
2

where

2 .
The last equation allows matrix representation so that the solution to the direct
problem (Eq. 16 ) can be expressed as

ʳ j = and ʲ j =

ʴ x D

(ʴ x D )

C n + 1 A − 1 bC n

= , (21)

where A and b are tridiagonal spatial-dependent matrices.

3 Parameter Estimation Procedure

Themethodology for determining the fitting parameters has been derived by (a) defin-
ing adequate dimensionless variables and fitting parameters, (b) specifying proper
validity ranges for the parameter values, (c) providing an objective function and a
suitable optimization method, and (d) performing a robustness analysis of the whole
procedure. These issues will be described below. The use of synthetically generated
tracer breakthrough data is essential in the robustness analysis, since it provides
control on the procedure and allows the quantification of the fitting goodness.