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Numerical Simulation of Dispersion of Density Dependent Transport in Heterogeneous Stochastic M edia. MSc.Nooshin Bahar Supervisor: Prof. Manfred Koch . Generalization of Darcy’s column. h/L = hydraulic gradient. q = - K grad h. Q is proportional to h/L . - PowerPoint PPT Presentation
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Numerical Simulation of Dispersion of Density Dependent Transport in Heterogeneous Stochastic Media
MSc.Nooshin Bahar
Supervisor: Prof. Manfred Koch
Figure from Hornberger et al. (1998)
Generalization of Darcy’s column
h/L = hydraulic gradient
q = Q/A
Q is proportionalto h/L
q = - K grad h
q = - K grad h
Darcy’s law
grad h
q equipotential line
grad hq
IsotropicKx = Ky = Kz = K
AnisotropicKx, Ky, Kz
Kf=k.ρg/μ
time
Diffusion and Dispersion
Illustration of transport
Sea Water intrusion
Transition Zone:
• Relative Densities of sea water• Tides• Pumping wells• The rate of ground water recharge• Hydraulic characteristics of the aquifer
2D, Saturated porous media
Flow Equation:
Transport Equation:
,
Heterogeneity is known to produce
(Dagan, 1989; Dentz et al., 2000; Dentz and Carrera, 2003; Cirpka and Attinger, 2003)
dispersion
The ratio between the longitudinal and the transverse dispersion coefficients varies with the dispersion regime
0 0.5 1
z (m
)
0.1686 0.8414
F(z)zs
zs+s
zs+2s
zs+3s
zs-s
zs-2s
zs-3s
f(z)
tD2zerf12
1c
z,xc)z(FT0• We UNDERSTAND: at microscopic level
• We MEASURE, PREDICT..at macroscopic level.
quasi one-dimensional laminar flow with a constant water flow
Where are groundwater models required?
• For integrated interpretation of data• For improved understanding of the functioning of aquifers• For the determination of aquifer parameters• For prediction• For design of measures• For risk analysis• For planning of sustainable aquifer management
Modeling process
Conceptual Model (Model Geometry, Boundaries,…)
Mathematical Model Numerical Model Code Verification Model Validation Model Calibration Model Application Analysis of uncertainty and stochastic modeling Summery, conclusion and reporting
Popular models for salt water intrusion
• SUTRA (Voss, 1984),Saturated- Unsaturated TRAnsport• SEAWAT• HST3D• FEFLOW• MODFLOW
Sutra The model is two-dimensional and can be applied either aerially or cross-section to
make a profile model.
The equations are solved by a combination of finite element and integrated finite difference methods.
The coordinate system may be either Cartesian or radial which makes it possible to simulate phenomena such as saline up-coning beneath a pumped well.
It permits sources, sinks and boundary conditions of fluid and salinity to vary both spatially and with time
It allows modelling the variation of dispersivity when the flow direction is not along the principal axis of aquifer transmissivity.
Focus on last works
Koch (1993, 1994)Koch and Voss (1998Koch and Zhang (1998Koch and Betina (2001: 2006)
Questions
• Decrease of AT with increasing of flow velocity• Increase of AT with variance and correlation length Decreasing and increasing of investigated Correlation length ?• Increase of AT with concentration, variance, correlation length Their effects on AL?• Decrease of AT with increasing velocity injection Consider of law AT in high variance: Law correlations, law consentration and high velocity (4 m/s)?• The wave lengths λc are proportional to the correlation length λx, but independent of the concentration
differences and the flow velocities, and dispersivities? Keep advection and how creates to prevent upward or downward flow? Morphology of fingure instabilities and keep?
Transversal and Longtidinual macrodispersivityWelty et al., 2003:
Gelhar and Axness, 1983:
Lateral Dispersion
u = 4 m/d
0.0E+00
4.0E-04
8.0E-04
1.2E-03
1.6E-03
2.0E-03
0 0.04 0.08 0.12
H (m)
A T (m
)
c = 250 ppmc = 5000 ppmc = 35000 ppmc = 100000 ppm
H = slnk² · lx
AT ~ slnk², lx , 1/u
Repetitions in high Concentrations and high Heterogeneity??
Model design
Mean, Variance, Correlations
Q
QInitial ConcentrationSpecified Pressure( Boundry Conditions)p ( z) = rh (c = 0 ) * g * zMesh Structure(392*98)Time stepsEach element: 2.5 *1.25 cm
Available sands and their properties (10 kind of sand)
Sand name (Dorfner)
Dm(m) Kf (m2/s) K(m2) Ln(kf)
3 0.0027 3.82E-02 3.90E-09 -3.26518
5G 0.0021 0.028398 2.90E-09 -3.56144
5 0.00175 0.013709 1.40E-09 -4.28968
5F 0.00135 0.01273 1.30E-09 -4.36379
7 0.00098 0.003819 3.90E-10 -5.56776
8 0.00062 0.001861 1.90E-10 -6.28689
6 0.00049 0.000979 1.00E-10 -6.92874
9S 0.00038 0.000509 5.20E-11 -7.58267
9H 0.00028 0.000401 4.10E-11 -7.82034
GEBA 0.00013 0.000127 1.30E-11 -8.96896
0.11100
10
20
30
40
50
60
70
80
90
100 SAND 8
GEBA
SAND6
SAND 7
SAND 5 G
SAND 5
SAND 5 F
SAND 9 S
SAND 9 H
Grain Diameter
Perc
ent fi
ner (
%)
0 0.0005 0.001 0.0015 0.002 0.0025 0.0030
5E-10
0.000000001
0.0000000015
0.000000002
0.0000000025
0.000000003
0.0000000035
0.000000004
0.0000000045
f(x) = 0.000496870002330544 x^1.99423041861703R² = 0.983433443011476
dm(m)
Perm
eabi
lity(
m2)
0 1 2 3 4 5 6 7 8 9 100.00E+00
5.00E-03
1.00E-02
1.50E-02
2.00E-02
2.50E-02
X(m)
Perm
eabi
lity
(m2/
s)
Statistical Properties of packing
Sand pack Variance of lnk Mean (Υg=Lnk) λx λy
1 2.24 0.004 0.25 0.075
2 2.24 0.001 0.25 0.075
3 3.15 0.001 0.25 0.075
4 3.15 0.001 0.25 0.025
5 3.15 0.001 0.25 0.25
-9.1 -8.1 -7.1 -6.1 -5.1 -4.1 -3.102468
101214161820
lnkf
Different realizationInterpretaion of Vriogeram
Kg =0.004, σ2=2.24, λx=0.25, λy=0.075
-3.25
-3.5 -4.25
-4.5 -5.5 -6.25
-7 -7.5 -8 -90
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.5
1
1.5
2
2.5
3
Variogeram Sill:2.30, kg:0.004, Sample variance:2.24
x-directiony-direction
lag distance
Sem
i var
ioge
ram
Kg =0.001, σ2=3.15, λx=0.25, λy=0.075
-3.25 -3.5 -4.25 -4.5 -5.5 -6.25 -7 -7.5 -8 -90
0.05
0.1
0.15
0.2
0.25
freq
.
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
3
3.5
4
x-directionLogarithmic (x-di-rection)y-direction
lag distance
Sem
ivar
ioge
ram
Stable systemCs= 250 ppmCf=0V=4 m/sɸ=0.44
-0.2
-1.665334536937...
0.2
0.4
0.6
0.8
1
1.2
1.4
Y=lnk =-13.5, Var=2.24
2m4m6m8m
salt fraction c/c0
Dept
h (m
)
-0.200.20
0.601.00
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Y=-12.50,Var=2.24
2m4m6m8m
salt fraction c/c0de
pth(
m)
-0.2 0.2 0.6 10
0.2
0.4
0.6
0.8
1
1.2
1.4
Y=-13.5,Var=3.15
2m4m6m8m
salt fraction
Dept
h (m
)
λx=0.25λy=0.075Y=lnk= -12.50σ2= 2.24Stable systemCs= 250 ppmCf=0V=4 m/sɸ=0.44
Y=-13.50, Var=5, λx= 0.25, λy=0.075
Y=-13.50, Var=5, λx= 0.025, λy=0.025
Different Correlations
0.000.200.400.600.801.001.200
0.2
0.4
0.6
0.8
1
1.2
2m4m6m8m
salt fraction c/c0
dept
h(m
)
0.00 0.20 0.40 0.60 0.80 1.00 1.200
0.2
0.4
0.6
0.8
1
1.2
2m4m6m8m
salt fraction c/c0
dept
h(m
)
References
• Stochastic Subsurface Hydrology(Gelhar,1993)• Seawater intrusion in coastal aquifers (Bear et al., 1999)• Saltwater upconing in formation aquifers (Voss and Koch, 2001)• Variable -density groundwater flow and solute transport in heterogeneous (Simmon, 2001)• Laboratory Experiments and Monte Carlo Simulations to Validate a Stochastic Theory of
Tracer- and Density-Dependent Macrodispersion (Betina and Koch, 2003)• Monte Carlo Simulations to Calibrate and Validate Stochastic Tank Experiments of
Macrodispersion of Density-Dependent Transport in Stochastically Heterogeneous Media (Koch and Betina, 2005)
• Pore-scale modeling of transverse dispersion in porous media (Branko Bijeljic and Martin J. Blunt,2007)
• Investigated effects of density gradients on transverse dispersivity in heterogeneous media (Nick, 2008)
• Heterogeneity in hydraulic conductivity and its role on the macro scale transport of a solute plume: From measurements to a practical application of stochastic flow and transport theory (Sudicky, 2010)
Thank you Vielen Dank سپاسگزارم