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