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Density driven flow in porous media: How accurate are our models?

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## Density driven flow in porous media: How accurate are our models?

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**Density driven flow in porous media: How accurate are our**models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss Federal Institute of Technology, Zurich, Switzerland**Contents**• Examples of density driven flow in aquifers • Equations • Formulation • Special features of density driven flows • Benchmarks • Analytical and exact solutions • Experimental benchmark: Grid convergence • Experimental benchmark: Fingering problem • Upscaling issues • Conclusions**Density driven flows in groundwater resources management**• Sea water intrusion • Salt water upconing under freshwater lenses (both on islands and inland) • Salt water fingering under playa lakes and saltpans • Flow around salt domes (nuclear waste repositories) • Brine injection • Leachate from waste deposits • Even the ordinary tracer experiment...**Saltwater Intrusion**Salt water Fresh water**Formation of toe**Fresh water Salt water**Saltwater Upconing**Fresh water Salt water**Example: Salt Water Upconing on Wei Zhou Island**Thesis Li Guomin**Freshwater Lens**Upconing**200 km**Salt fingers on islands in the Okavango Delta**Schematic cross section of an island**Transpiration Trona saltcrust Evaporation Increasing salinity of GW Increasing salinity of GW gravity vs. upward flow**Instability on the Islands**instable stable kf= 10-5 m/s, uET=10-8 m/s Critical Wooding Number:**t=900 d cmax=11 mg/l**t=6000 d cmax=54 mg/l t=8500 d cmax=75 mg/l t=16800 d cmax=235 mg/l t=32500 d cmax=350 mg/l t=66000 d cmax=350 mg/l t=2900 d cmax=30 mg/l t=12400 d cmax=110 mg/l t=25000 d cmax=350 mg/l t=46500 d cmax=350 mg/l Simulation of fingering**Flow in the vicinity of a salt dome**Recharge Discharge Salt water - fresh water interface Top of salt dome With density difference No density difference**Basic Equationsexpressed in mass fraction c and pressure p**• Mass balance total mass • Mass balance salt • Darcy law • Dispersion tensor • Constitutive relationships • Boundary conditions (many combinations) e.g. Possible simplification: Boussinesq approximation**Features of density driven flow**• Non-linearity • Consistency problem of boundary conditions • Rotational flow with closed streamlines • Plus all difficulties known from advective- dispersive transport**Flow in porous media and rotation**Darcy-flow in heterogeneous porous media is rotational Example: kf But we still have: In density flow, rotation is non-trivial: closed streamlines For constant k/m Rotational when r not parallel to**Numerical solution and testing of codes**• Analytical solutions • Exact solutions • Inter-code comparison • Experimental benchmarks • Grid convergence All computations are made with d3f, a density flow model using unstructured grids, finite volume discretization, multigrid solver, error estimator, automatic local refinement/coarsening, parallel computing**Idea of „exact“ solution**(steady state) Pressure equation Salt mass fraction equation Assume any differentiable functions p(x,z), c(x,z) Assume any domain Assign function values as first kind boundary conditions on boundary of that domain**Plug functions into flow equations**Pressure Salt mass fraction Right-hand sides are not zero: They are interpreted as source-sink terms So analytical expressions are exact solution for problem with - these source-sink terms and - first kind boundary conditions with given function values Only good if source-sink terms are small and do not dominate the problem**Analytical expressions for „exact“ solution**(steady state) Pressure Salt mass fraction Values in example tuned to make sources/sinks small: t=20, s=12, h=.14, b=1, r0=1, g=1, Dx=0.1, Dz=0.02, xs=1, zs=-0.1 In PDE: n=1, g=1, k=10, m=1, Dm=1, r/c=0.1, r(c)=r0+ r/c c=1+0.1c**Analytical**Pressure Values between 0 and 1.13 p units Salt mass fraction Values between 0 and 1 c units Plugged into equations for c and p**Source-sink distributions**Red: max. input Turquoise: 0 input Total mass Salt mass Red: max. input Light blue: 0 input Blue: output**Computed (with 4 grid levels)**Pressure Values between 0 and 1.13 p units Salt mass fraction Values between 0 and 1 c units**Error**Pressure Red : computed value too large by 0.004 % Blue: computed value too small by 0.005 % Salt mass fraction Red : computed value too large by 0.007 % Blue: computed value too small by 0.006 %**Experimental benchmark**• 3D transient experiment in box with simple boundary and initial conditions • Measurement of concentration distribution in 3D with Nuclear Magnetic Resonance Imaging • Measurement of breakthrough curves Drawback: Test of both model equations and mathematics Way out: Construction of a grid convergent solution inspired by the physical experiments**Experimental setup**• Cube filled with glass beads of diameter 0.7 mm • Size of model 20*20*20 cm3 • Injection of dense fluid on bottom center hole • Application of base flow via top corner holes • In unstable case: Injection from below and rotation • All parameters measured except transverse dispersivity, diffusion coefficient**Experimental setup continued**20 cm 20 cm unstable situation stable situation Salt water**NMR images of diagonal section: stable situation at low**concentration contrast Injection Equilibration Flushing End**NMR images of diagonal section: stabel situation at high**concentration contrast Injection Equilibration „Entraining“ End**Two modes**No density contrast Large density contrast**Experimental breakthrough curves**Low contrast High contrast**Comparison of computed and measured breakthrough curves**Low contrast High contrast Choice of parameters within intervals given through measurements of those**Comparison of concentrations along diagonal section**Low contrast case, end of experiment**Comparison of concentrations along diagonal section**High contrast case, end of experiment**unit 10-2**Grid convergence: Low contrast case Level # grid points 0 8 1 27 2 125 3 729 4 4,913 5 35,937 6 274,625 7 2,146,689 Dx at level 7: 1.56 mm**Grid convergence: High contrast case**unit 10-2 Level # grid points 0 8 1 27 2 125 3 729 4 4,913 5 35,937 6 274,625 7 2,146,689 8 16,974,593 Dx at level 8: 0.78 mm**Error of grid-convergent solutions**Low contrast High contrast**NMRI of vertical and horizontal section: fingering**experiment Horizontal section Vertical section**Causes for poor perfomance**• Numerical dispersion smoothes out fingers and eliminates driving force • Initial perturbance not known well enough • Start of fingers on microlevel, not represented by continuum equations**Influence of heterogeneity on density flow**• Homogeneous Henry problem • Heterogeneous Henry problem**Definition of Henry problem: Homogeneous aquifer**Hydraulic conductivity 1E-2 m/s, effective diffusion coefficient 1.886E-05 m2/s Boundary conditions: Left fresh water flux given at 6.6E-05 m/s Right hydrostatic salt water, salt mass fraction 0.0357 kg/kg