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 waterFresh water

Formation of toe

Fresh water Salt water

Saltwater Upconing

Salt waterFresh water

40

1

Example: Salt Water Upconing on Wei Zhou Island

Thesis Li Guomin

Upconing

Freshwater Lens

Alternative Extraction Strategies

                        

 

 

Salinity backflow from Chotts in Tunisia

Oasis

200 km

Salt fingers on islands in the Okavango Delta

Islands and salt crusts in Delta

Schematic cross section of an island

Transpiration

Evaporation

Increasing salinity of GW

Trona saltcrust

Increasing salinity of GW

gravity vs. upward flow

Instability on the Islands

ET

f

u

k Number Wooding

Critical Wooding Number:TEM Evaluation

Island-TransectTDS center

(mg/l)TDS layer 2

(mg/l)norm. density

contrast (-)Wooding

No.(-)Thata-1 22425.00 1352.94 0.0147 14.74

Thata-3 9425.00 878.59 0.0060 5.98

Mosupatsela-1 8645.00 1495.31 0.0050 5.00

Mosupatsela-3 5200.00 1803.28 0.0024 2.37

Tshwene-1 19110.00 1135.50 0.0126 12.57

Tshwene-2 19110.00 1135.50 0.0126 12.57

Lebolobolo-1 13000.00 1014.03 0.0084 8.38

Lebolobolo-2 26000.00 1887.59 0.0169 16.86

Lebolobolo-3 7800.00 886.25 0.0048 4.84

Monyopi-1 2665.00 1118.03 0.0011 1.08

Monyopi-2 2665.00 747.22 0.0013 1.34

Atoll-3 15600.00 2172.15 0.0094 9.39

Kwena-1 9750.00 1480.68 0.0058 5.78

Montsana-1 1625.00 367.65 0.0009 0.88

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

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

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20

k in units of u/D m-1

Criti

cal W

oodi

ng N

umbe

r

instable

stable

Simulation of fingering

20 40 60 80 100

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100

t=900 d cmax=11 mg/l

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t=2900 d cmax=30 mg/l

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10020 40 60 80 100

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t=6000 d cmax=54 mg/l

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t=8500 d cmax=75 mg/l

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t=12400 d cmax=110 mg/l

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t=16800 d cmax=235 mg/l

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t=25000 d cmax=350 mg/l

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t=32500 d cmax=350 mg/l

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t=46500 d cmax=350 mg/l

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t=66000 d cmax=350 mg/l

Flow in the vicinity of a salt dome

Recharge Discharge

Top of salt dome

Salt water -fresh waterinterface

No density difference With density difference

• Mass balance total mass

• Mass balance salt

• Darcy law

• Dispersion tensor

• Constitutive relationships

• Boundary conditions (many combinations)

Basic Equationsexpressed in mass fraction c and pressure p

volq s)c()v(t

)n(

volqqm scccDDnvccn

)())((t

)(

)gp(K

v

lk

lkijklij vvav

D,

1

),(),( pcpc

sf c

c

c

c

11

11

maxmax

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 rotationalExample:

)gp(k

v

gk

gpk

v

)())()((

But we still have:

In density flow, rotation is non-trivial: closed streamlines

Rotational when not parallel to g

0)k/v( f

For constant k/

kfv hkv f

0 v

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 usingunstructured 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

0),(),,((1 zxczxpL

0)),(),,((2 zxpzxcL

Assume any differentiable functions p(x,z), c(x,z)Assume any domainAssign function values as first kind boundary conditionson boundary of that domain

Plug functions into flow equations

Pressure

Salt mass fraction

),()),(),,((1 zxfzxczxpL

),()),(),,((2 zxgzxpzxcL Right-hand sides are not zero: They are interpreted as source-sink termsSo 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

)z)xx()x3(()zzg)(z1()z,x(p ssx0z0

1

222

2

x )zx()z3x(hz2

5.4e1)z,x(c

Values in example tuned to make sources/sinks small: =20, =12, h=.14, =1, 0=1, =1, x=0.1, z=0.02, xs=1, zs=-0.1In PDE: n=1, g=1, k=10, =1, Dm=1, /c=0.1, (c)=0+ /c c=1+0.1c

Analytical

Salt mass fraction

Pressure

Values between 0 and 1.13 p units

Values between 0 and 1 c units

Plugged into equations for c and p

Source-sink distributions

Total mass

Salt mass

Red: max. inputTurquoise: 0 input

Red: max. inputLight blue: 0 inputBlue: output

Computed (with 4 grid levels)

Salt mass fraction

Pressure

Values between 0 and 1.13 p units

Values between 0 and 1 c units

Error

Pressure

Salt mass fraction

Red : computed value too large by 0.004 % Blue: computed value too small by 0.005 %

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

Salt water

stable situation unstable situation

Stable situation: Low concentration contrast

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

Injection

EndFlushing

Equilibration

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 sectionLow contrast case, end of experiment

Comparison of concentrations along diagonal sectionHigh contrast case, end of experiment

Level # grid points 0 8 1 27 2 125 3 729 4 4,913 5 35,937 6 274,625 7 2,146,689

Grid convergence: Low contrast case

unit 10-2

2

,1

,11

2

,1

1,1 , jijijiji cccc

x at level 7: 1.56 mm

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 8 16,974,593

Grid convergence: High contrast case

unit 10-2

jijijiji cccc ,2

,12

2

,2

1,2 ,

x at level 8: 0.78 mm

Grid convergence: High contrast case

Error of grid-convergent solutions

Low contrast

High contrast

Unstable case: NMRI data

NMRI of vertical and horizontal section: fingering experiment

Vertical section Horizontal section

Fingering: Experiment vs. Computation

Finger velocity: comparison of experiment and computation

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/sBoundary 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

Heterogeneous Henry problem: Permeability distribution

Lognormal distribution, exponential autocorrelationArithmetic mean 1.68E-9 m2, Geometric mean 1.02E-9 m2 Variance of log(k) = 1Corr. lengths: horizontal 0.05 m, vertical 0.05 mRed: 3.5E-08 m2, Blue: 2.6E-11 m2

Heterogeneous Henry Problem:Concentration distribution

Relative concentration contours between 0 (left) and 1 (right) in steps of 0.1Eff. diffusion coefficient as in homogeneous case

Question: Are there equivalent effective parameters to mimickmain effect of heterogeneity in a homogeneous model?

Heterogeneous Henry ProblemComparison (zero local dispersion)

Heterogeneous caseOnly diffusion

Homogeneous case:Permeability equal arithmetic mean of heterogeneous caseOnly diffusion

Homogeneous case:Permeability equal geometric mean of heterogeneous caseOnly diffusion

Heterogeneous Henry ProblemComparison (zero local dispersion)

Heterogeneous caseonly diffusion

Homogeneous case:Permeability equal geometric mean of heterogeneous caseonly diffusion, zero dispersion

Homogeneous case:Permeability equal geometric mean of heterogeneous case, diffusion plus macrodispersion after Gelhar&Axness

Heterogeneous Henry ProblemComparison (with local dispersion)

Heterogeneous caseDiffusion + local dispersion

Homogeneous case:Permeability geometric mean Diffusion + local dispersion

Homogeneous case:Permeability geometric meanDiffusion + 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 flow towards a fixed concentration leads to „boundary layer“. Dispersion at upper right boundary and at stagnation point is „upstream diffusion“: c/c0 = 1 – exp (-x/L)

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