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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 .

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Numerical Simulation of Dispersion of Density Dependent Transport in Heterogeneous Stochastic Media

MSc.Nooshin Bahar

Supervisor: Prof. Manfred Koch


Generalization of Darcy’s column

h/L = hydraulic


q = - Kgrad h

Q is proportional

to h/L

q = Q/A

Figure from Hornberger et al. (1998)


Darcy’s law

q = - Kgrad h



equipotential line

grad h


grad h


Kx = Ky = Kz = K


Kx, Ky, Kz

sea water intrusion
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


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)


quasi one-dimensional laminar flow with a constant water flow

The ratio between the longitudinal and the transverse dispersion coefficients varies with the dispersion regime

  • We UNDERSTAND: at microscopic level
  • We MEASURE, macroscopic level.

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
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
Popular models for salt water intrusion
  • SUTRA (Voss, 1984),Saturated- Unsaturated TRAnsport
  • HST3D
  • 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
Focus on last works

Koch (1993, 1994)

Koch and Voss (1998

Koch and Zhang (1998

Koch and Betina (2001: 2006)


  • 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 macrodispersivity
Transversal and Longtidinual macrodispersivity

Welty et al., 2003:

Gelhar and Axness, 1983:

lateral dispersion
Lateral Dispersion

H = slnk² · lx

AT~slnk², lx, 1/u

Repetitions in high Concentrations and high Heterogeneity??

model design
Model design

Mean, Variance, Correlations


Initial Concentration

Specified Pressure( Boundry Conditions)

p ( z) = rh (c = 0 ) * g * z

Mesh Structure(392*98)

Time steps

Each element: 2.5 *1.25 cm


statistical properties of packing
Statistical Properties of packing

Different realization

Interpretaion of Vriogeram


Kg =0.004, σ2=2.24, λx=0.25, λy=0.075

Kg =0.001, σ2=3.15, λx=0.25, λy=0.075


Stable system

Cs= 250 ppm


V=4 m/s




Y=lnk= -12.50

σ2= 2.24

Stable system

Cs= 250 ppm


V=4 m/s



Different Correlations

Y=-13.50, Var=5, λx= 0.25, λy=0.075

Y=-13.50, Var=5, λx= 0.025, λy=0.025

  • 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