Background:
The mathematical model describes a tracer experiment that was conducted at the lake
Gaardsjon in Sweden, to investigate acidification of groundwater pollution. To conduct the experiment, a catchment of 1.000
m^{2} was covered by a roof. A tracer impulse consisting of Lithium-Bromide was applied with
steady state flow conditions. Tensiometer measurements of the tracer concentration were documented in a distance of 40
m from the center of the covered area.
The Mathematical Model:
The diffusion equations proposed by Van Genuchten and Wierenga are chosen to analyze the diffusion process and to get a simulation model.
A two-domain approach was selected in form of two equations, to model the mobile and the immobile part of the system.
The first one is needed for the so-called immobile part, that is the mass transfer orthogonal to the flow
direction, and the second one describes the diffusion of the flow through the soil by convection and
dispersion:
C_{im}(x,t)_{t} = p_{im}(C_{m}(x,t) - C_{im}(x,t))
C_{m}(x,t)_{t} = D_{m} C_{m}(x,t)_{xx} - V_{m} C_{m}(x,t)_{x} - p_{m}(C_{m}(x,t) - C_{im}(x,t))
The subindices 't' and 'x' or 'xx', respectively, denote the partial differentiation with respect to time variable t and space variable x. We suppose homogeneous initial values C^{im}(x,0) = C_{m}(x,t) = 0 and boundary conditions of the form
V_{m} (C_{m}(0,t) - C_{0}) - D_{m} C_{m}(0,t)_{x} = 0_{ } , if t < t_{0} ,
- V_{m} C_{m}(L,t) + D_{m} C_{m}(L,t)_{x} = 0
at the right boundary x=L . C_{m}(x,t) and C_{im}(x,t) are the tracer concentrations, D_{m} is the dispersion coefficient, and V_{m} the volume. D_{m},_{ }p_{m}, and p_{im} are the unknown parameters to be estimated. Fitting criterion for which measurements are available, is
V_{m} C_{m}(L/2,t) - D_{m }C_{m}(L/2,t)
Literature:
1. Andersson F., Olsson B. eds. (1985): Lake Gadsjon. An Acid Forest Lake
and its Catchment, Ecological Bulletins, Vol. 37, Stockholm
2. Schittkowski (2002):
Numerical Data Fitting in Dynamical Systems - A Practical Introduction with
Applications and Software,
Kluwer
Academic Publishers
Implementation:
The complete solution of a data fitting problem is described
in six
steps:
Results:
Then you would like to take a look at reports and graphs:
- parameter values
- experimental data versus fitting criterion
- surface plot of state variable
Model equations (or use your own favorite editor):
Measurement data (or use import function for text file or Excel):
Parameters, tolerances, and start of a data fitting run:
Numerical results (computed by the least squares code DFNLP):
Report about parameter values, residuals, performance, etc. (or export to Word):
Experimental data versus fitting criterion (also available for Gnuplot):
Surface plot of state variable (also available for Gnuplot):