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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

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density driven flow in porous media how accurate are our models

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
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
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...
slide4

Saltwater Intrusion

Salt water

Fresh water

formation of toe
Formation of toe

Fresh water

Salt water

saltwater upconing
Saltwater Upconing

Fresh water

Salt water

slide11

200 km

Salt fingers on islands in the Okavango Delta

schematic cross section of an island
Schematic cross section of an island

Transpiration

Trona saltcrust

Evaporation

Increasing salinity of GW

Increasing salinity of GW

gravity vs.

upward flow

slide14

Instability on the Islands

instable

stable

kf= 10-5 m/s, uET=10-8 m/s

Critical Wooding Number:

slide15

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
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 equations expressed in mass fraction c and pressure p
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

slide18

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
slide19

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
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

slide21

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

slide22

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

slide23

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

slide24

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

slide25

Source-sink distributions

Red: max. input

Turquoise: 0 input

Total mass

Salt mass

Red: max. input

Light blue: 0 input

Blue: output

slide26

Computed (with 4 grid levels)

Pressure

Values between 0 and 1.13 p units

Salt mass fraction

Values between 0 and 1 c units

slide27

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
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
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
Experimental setup continued

20 cm

20 cm

unstable situation

stable situation

Salt water

nmr images of diagonal section stable situation at low concentration contrast
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
NMR images of diagonal section: stabel situation at high concentration contrast

Injection

Equilibration

„Entraining“

End

two modes
Two modes

No density contrast

Large density contrast

slide35

Experimental breakthrough curves

Low contrast

High contrast

comparison of computed and measured breakthrough curves
Comparison of computed and measured breakthrough curves

Low contrast

High contrast

Choice of parameters within intervals given through measurements of those

slide37

Comparison of concentrations along diagonal section

Low contrast case, end of experiment

slide38

Comparison of concentrations along diagonal section

High contrast case, end of experiment

slide39

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

slide41

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
Error of grid-convergent solutions

Low contrast

High contrast

nmri of vertical and horizontal section fingering experiment
NMRI of vertical and horizontal section: fingering experiment

Horizontal section

Vertical section

causes for poor perfomance
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
Influence of heterogeneity on density flow
  • Homogeneous Henry problem
  • Heterogeneous Henry problem
slide50

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

slide51

Solution of Henry problem: Homogeneous aquifer

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

slide52

Heterogeneous Henry problem:

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

slide53

Heterogeneous Henry Problem:

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?

slide54

Heterogeneous Henry Problem

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

slide55

Heterogeneous Henry Problem

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

slide56

Heterogeneous Henry Problem

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
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
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