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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...

Example: Salt Water Upconing on Wei Zhou Island

Thesis Li Guomin

Upconing

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

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

(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

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

Red: max. input

Turquoise: 0 input

Total mass

Salt mass

Red: max. input

Light blue: 0 input

Blue: output

Pressure

Values between 0 and 1.13 p units

Salt mass fraction

Values between 0 and 1 c units

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

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

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

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

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

Solution of Henry problem: Homogeneous aquifer

Relative concentration contours between 0 (left) and 1 (right) in steps of 0.1

Permeability distribution

Lognormal distribution, exponential autocorrelation

Arithmetic mean 1.68E-9 m2, Geometric mean 1.02E-9 m2

Variance of log(k) = 1

Corr. lengths: horizontal 0.05 m, vertical 0.05 m

Red: 3.5E-08 m2, Blue: 2.6E-11 m2

Concentration distribution

Relative concentration contours between 0 (left) and 1 (right) in steps of 0.1

Eff. diffusion coefficient as in homogeneous case

Question: Are there equivalent effective parameters to mimick

main effect of heterogeneity in a homogeneous model?

Comparison (zero local dispersion)

Heterogeneous case

Only diffusion

Homogeneous case:

Permeability equal arithmetic mean of heterogeneous case

Only diffusion

Homogeneous case:

Permeability equal geometric mean of heterogeneous case

Only diffusion

Comparison (zero local dispersion)

Heterogeneous case

only diffusion

Homogeneous case:

Permeability equal geometric mean of heterogeneous case

only diffusion,zero dispersion

Homogeneous case:

Permeability equal geometric mean of heterogeneous case, diffusion plus

macrodispersion after Gelhar&Axness

Comparison (with local dispersion)

Heterogeneous case

Diffusion + local dispersion

Homogeneous case:

Permeability geometric mean

Diffusion + local dispersion

Homogeneous case:

Permeability geometric mean

Diffusion + macrodispersion after Gelhar&Axness

Effective dispersion parameters

- Stable situation with flow against direction of density gradient
- effective longitudinal dispersivity given by
- Gelhar & Welty, A11(with density gradient) < A11(without density gradient)
- Unstable situation with flow in direction of density gradient
- at sufficiently large Wooding/Raleigh number, dispersivities grow to
- infinity due to fingers forming
- Horizontal flowtowards a fixed concentration
- leads to „boundary layer“. Dispersion at upper right boundary and at
- stagnation point is „upstream diffusion“: c/c0 = 1 – exp (-x/aL)

Upstream diffusion

Conclusions

- Density flow of increasing importance in groundwater field
- Tests for reliability of codes are available
- Density flow especially with high contrast numerically demanding: Grid convergence may require millions of nodes
- Numerical simulation of fingering instabilities

still inadequate

- Heterogeneities can be handled by effective

media approach in situation without fingering

- New numerical methods are in the pipeline

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