1. Numerical Simulation of Dispersion of Density Dependent Transport in Heterogeneous Stochastic Media MSc.Nooshin Bahar Supervisor: Prof. Manfred Koch

2. Figure from Hornberger et al. (1998) Generalization of Darcy’s column  h/L = hydraulic gradient q = Q/A Q is proportional to  h/L q = - K grad h

3. Darcy’s law grad h qequipotential line grad hq Isotropic Kx = Ky = Kz = K Anisotropic Kx, Ky, Kz K f =k.ρg/μ

4. time Diffusion and Dispersion

5. Illustration of transport

6. Sea Water intrusion Transition Zone: Relative Densities of sea water Tides Pumping wells The rate of ground water recharge Hydraulic characteristics of the aquifer

7. 2D, Saturated porous media Flow Equation : Transport Equation :,

8. 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 We UNDERSTAND: at microscopic level We MEASURE, PREDICT..at macroscopic level. quasi one-dimensional laminar flow with a constant water flow

9. 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

10. 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

11. Popular models for salt water intrusion SUTRA (Voss, 1984),Saturated- Unsaturated TRAnsport SEAWAT HST3D FEFLOW MODFLOW

12. 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.

13. Focus on last works Koch (1993, 1994) Koch and Voss (1998 Koch and Zhang (1998 Koch 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 λ c are proportional to the correlation length λ x, 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?

14. Transversal and Longtidinual macrodispersivity Welty et al., 2003: Gelhar and Axness, 1983:

15. Lateral Dispersion H =  lnk ² · x A T ~  lnk ², x , 1/u Repetitions in high Concentrations and high Heterogeneity??

16. Model design Mean, Variance, Correlations Q Q 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

17. 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-023.90E-09-3.26518 5G 0.0021 0.0283982.90E-09-3.56144 5 0.00175 0.0137091.40E-09-4.28968 5F 0.00135 0.012731.30E-09-4.36379 7 0.00098 0.0038193.90E-10-5.56776 8 0.00062 0.0018611.90E-10-6.28689 6 0.00049 0.0009791.00E-10-6.92874 9S 0.00038 0.0005095.20E-11-7.58267 9H 0.00028 0.0004014.10E-11-7.82034 GEBA 0.00013 0.0001271.30E-11-8.96896

19. Statistical Properties of packing Sand packVariance of lnkMean (Υ g =Lnk)λxλx λyλy 12.240.0040.250.075 22.240.0010.250.075 33.150.0010.250.075 43.150.0010.250.025 53.150.0010.25 Different realization Interpretaion of Vriogeram

20. Kg =0.004, σ2=2.24, λx=0.25, λy=0.075 Kg =0.001, σ2=3.15, λx=0.25, λy=0.075

21. Stable system C s = 250 ppm C f =0 V=4 m/s ɸ =0.44 λ x =0.25 λ y =0.075 Y=lnk= -12.50 σ2= 2.24 Stable system C s = 250 ppm C f =0 V=4 m/s ɸ =0.44

22. Y=-13.50, Var=5, λx= 0.25, λy=0.075 Y=-13.50, Var=5, λx= 0.025, λy=0.025 Different Correlations

23. 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)

24. Thank you Vielen Dank سپاسگزارم