Aufgabe 3 Zusatz (PDF 114Kb)

Technische Universität Carolo Wilhelmina
Institut für Geoökologie
Abt. für Hydrologie und Landschaftsökologie
Prof. Dr. M. Schöniger
WS 2005/06
Trainings-Aufgabe 3
Zusatzaufgaben, Hinweise
In the context of unsaturated flow the basic formulation involves as unknown variables both,
the fluid pressure head Ψ and the saturation s. For these is only one balance equation
available, the basic RICHARDS equation:
∂s(ψ )
+ ∇ ⋅ [K r (s )K (∇ψ + ∇z )] − Q = 0
∂t
To convert one variable to the other a constitutive relationship in form of the capillary
pressure head-saturation function is additionally needed.
Basic Equation (Program Feflow)
The mass conservation equation of a fluid in a variably saturated media is given by
∂ψ
∂s (ψ )
+ε
+∇⋅q = Q
(1) S o⋅s (ψ ) ⋅
∂t
∂t
The fluid motion is described by the DARCY equation written in the form
(2) q = − K r (s )K (∇h + χe ) = − K r (s ) ⋅ K [∇ψ + (1 + χ )e]
with:
h
ψ
s(ψ)
q
z
t
S0
ε
γ
ϒ
Kr(s)
K
χ
e
Q
= ψ + z, hydraulic (piezometric) head;
pressure head, (ψ>0) saturated medium, ψ≤0 unsaturated medium);
saturation, (0<s≤1, s=1 if medium is saturated);
DARCY flux vector;
elevation above a reference level;
time;
= εγ + (1-ε)ϒ, specific storage due to fluid and medium compressibility;
porosity,
fluid compressibility;
coefficient of skeleton compressibility;
relative hydraulic conductivity (0<Kr≤1, Kr = 1 if saturated at s = 1);
tensor of hydraulic conductivity for the saturated medium (anisotropy);
buoyancy coefficient including fluid density effects;
gravitational unit vector;
specific mass supply.
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Technische Universität Carolo Wilhelmina
Institut für Geoökologie
Abt. für Hydrologie und Landschaftsökologie
Prof. Dr. M. Schöniger
WS 2005/06
Constitutive relationships are additionally required (1) for the saturation s as a function of the
pressure (capillary) head ψ, as well as its inverse, the pressure head ψ as a function of the
saturation s, and (2) for the relative hydraulic conductivity Kr as a function of either the
pressure head ψ or the saturation s. The empirical relationships are used for the present study
(BEAR & BACHMAT 1991 and VOGEL ET AL. 1996) see exercise number three.
Some information about the parameter fitting tool FE-LM2
(Levenberg-Maequardt Local Minimizer)
What is FE-LM2 and when is it useful?
DIERSCH 2002
Data-plot views associated with the same data set, shown with point-selection rubber box
The FEFLOW modeling system incorporates a number of constitutive relations such as
saturation vs. pressure and hydraulic conductivity vs. saturation for unsaturated flow or
absorption isotherms and reaction kinetics for reactive transport. Various parametric
expressions are provided in FEFLOW to describe these constitutive relations. It remains for
the user to select the best suitable expression and to identify the respective parameter set. The
parameter fitting tool FE-LM2 was designed to assist in this task, typically known as "curvefitting."
Built around a Levenberg-Marquardt Local Minimizer, FE-LM2 provides instantaneous
graphical feedback of a best fit when switching between parametric models and during usercontrolled gradual changes in the weighting of selected data points. Possible local minima in
the optimization function can be immediately recognized and subsequently circumvented by
modifying the parameter ranges and optimization starting values through the user interface.
As a parameter is gradually changed or explicitly set via the interface controls, the displayed
shape of the corresponding model curve continuously tracks the user input.
Coupled models that share parameters can be fitted simultaneously to their respective data
sets with graphical feedback provided for each set. The catalog of parametric models provided
in FE-LM2 includes all those implemented in FEFLOW.
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Technische Universität Carolo Wilhelmina
Institut für Geoökologie
Abt. für Hydrologie und Landschaftsökologie
Prof. Dr. M. Schöniger
WS 2005/06
Parameter estimation by residual minimization
The Levenberg-Marquardt algorithm minimizes a function q defined as the sum of weighted,
squared residuals according to
q=
N
i
( )
wi ri 2
∧
y − y(x i , p )
ri = i
si
where xi, yi, σi, ri, and wi are the measured independent dependent variables, measurement
standard deviation, residual, and weighting factor, respectively, at point ‘i’. The total number
∧
of data points is N. The estimated dependent value is obtained from a model function y (x,p)
utilizing the parameter vector p.
Dividing the difference between measured and estimated dependent variable by the
measurement standard deviation σi results in normalized, unitless residuals and makes qvalues better suitable to compare. Measurement standard-deviation values are either supplied
by the user or will be estimated from the respective data set. Weighting factors are included to
permit data-point or data-set weighting that is unrelated to the certainty of the measured
values, for example, to compensate variations in measurement density within a set. It can also
be used for compensation of different number of data points among various sets.( more
information, see DIERSCH 2002).
Where do you find the Feflow tool FE LM2?
Press on Info boxes, you get the Quick Access Menu and then press on Parameter fitting.
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Technische Universität Carolo Wilhelmina
Institut für Geoökologie
Abt. für Hydrologie und Landschaftsökologie
Prof. Dr. M. Schöniger
WS 2005/06
Instructions in designing superelements as the geometric representation of the study
area: Feflow´s Mesh Editor (see: FEFLOW Mesh Design.htm)
?
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