density driven flow in porous media: how accurate are our models? wolfgang kinzelbach institute for...
TRANSCRIPT
![Page 1: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/1.jpg)
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
![Page 2: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/2.jpg)
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
![Page 3: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/3.jpg)
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...
![Page 4: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/4.jpg)
Saltwater Intrusion
Salt waterFresh water
![Page 5: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/5.jpg)
Formation of toe
Fresh water Salt water
![Page 6: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/6.jpg)
Saltwater Upconing
Salt waterFresh water
40
1
![Page 7: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/7.jpg)
Example: Salt Water Upconing on Wei Zhou Island
Thesis Li Guomin
![Page 8: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/8.jpg)
Upconing
Freshwater Lens
![Page 9: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/9.jpg)
Alternative Extraction Strategies
![Page 10: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/10.jpg)
Salinity backflow from Chotts in Tunisia
Oasis
![Page 11: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/11.jpg)
200 km
Salt fingers on islands in the Okavango Delta
![Page 12: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/12.jpg)
Islands and salt crusts in Delta
![Page 13: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/13.jpg)
Schematic cross section of an island
Transpiration
Evaporation
Increasing salinity of GW
Trona saltcrust
Increasing salinity of GW
gravity vs. upward flow
![Page 14: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/14.jpg)
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
2
4
6
8
10
12
14
16
18
20
k in units of u/D m-1
Criti
cal W
oodi
ng N
umbe
r
instable
stable
![Page 15: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/15.jpg)
Simulation of fingering
20 40 60 80 100
20
40
60
80
100
t=900 d cmax=11 mg/l
20 40 60 80 100
20
40
60
80
100
t=2900 d cmax=30 mg/l
20 40 60 80 100
20
40
60
80
10020 40 60 80 100
20
40
60
80
100
t=6000 d cmax=54 mg/l
20 40 60 80 100
20
40
60
80
100
t=8500 d cmax=75 mg/l
20 40 60 80 100
20
40
60
80
100
t=12400 d cmax=110 mg/l
20 40 60 80 100
20
40
60
80
100
t=16800 d cmax=235 mg/l
20 40 60 80 100
20
40
60
80
100
t=25000 d cmax=350 mg/l
20 40 60 80 100
20
40
60
80
100
t=32500 d cmax=350 mg/l
20 40 60 80 100
20
40
60
80
100
t=46500 d cmax=350 mg/l
20 40 60 80 100
20
40
60
80
100
t=66000 d cmax=350 mg/l
![Page 16: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/16.jpg)
Flow in the vicinity of a salt dome
Recharge Discharge
Top of salt dome
Salt water -fresh waterinterface
No density difference With density difference
![Page 17: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/17.jpg)
• 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
![Page 18: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/18.jpg)
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
![Page 19: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/19.jpg)
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
![Page 20: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/20.jpg)
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
![Page 21: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/21.jpg)
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
![Page 22: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/22.jpg)
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
![Page 23: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/23.jpg)
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
![Page 24: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/24.jpg)
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
![Page 25: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/25.jpg)
Source-sink distributions
Total mass
Salt mass
Red: max. inputTurquoise: 0 input
Red: max. inputLight blue: 0 inputBlue: output
![Page 26: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/26.jpg)
Computed (with 4 grid levels)
Salt mass fraction
Pressure
Values between 0 and 1.13 p units
Values between 0 and 1 c units
![Page 27: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/27.jpg)
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 %
![Page 28: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/28.jpg)
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
![Page 29: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/29.jpg)
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
![Page 30: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/30.jpg)
Experimental setup continued
20 cm 20 cm
Salt water
stable situation unstable situation
![Page 31: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/31.jpg)
Stable situation: Low concentration contrast
![Page 32: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/32.jpg)
NMR images of diagonal section: stable situation at low concentration contrast
Injection
EndFlushing
Equilibration
![Page 33: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/33.jpg)
NMR images of diagonal section: stabel situation at high concentration contrast
Injection Equilibration
„Entraining“ End
![Page 34: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/34.jpg)
Two modes
No density contrast Large density contrast
![Page 35: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/35.jpg)
Experimental breakthrough curves
Low contrast
High contrast
![Page 36: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/36.jpg)
Comparison of computed and measured breakthrough curves
Low contrast
High contrast
Choice of parameters within intervals given through measurements of those
![Page 37: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/37.jpg)
Comparison of concentrations along diagonal sectionLow contrast case, end of experiment
![Page 38: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/38.jpg)
Comparison of concentrations along diagonal sectionHigh contrast case, end of experiment
![Page 39: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/39.jpg)
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
![Page 40: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/40.jpg)
Grid convergence: Low contrast case
![Page 41: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/41.jpg)
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
![Page 42: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/42.jpg)
Grid convergence: High contrast case
![Page 43: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/43.jpg)
Error of grid-convergent solutions
Low contrast
High contrast
![Page 44: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/44.jpg)
Unstable case: NMRI data
![Page 45: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/45.jpg)
NMRI of vertical and horizontal section: fingering experiment
Vertical section Horizontal section
![Page 46: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/46.jpg)
Fingering: Experiment vs. Computation
![Page 47: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/47.jpg)
Finger velocity: comparison of experiment and computation
![Page 48: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/48.jpg)
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
![Page 49: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/49.jpg)
Influence of heterogeneity on density flow
• Homogeneous Henry problem
• Heterogeneous Henry problem
![Page 50: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/50.jpg)
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
![Page 51: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/51.jpg)
Solution of Henry problem: Homogeneous aquifer
Relative concentration contours between 0 (left) and 1 (right) in steps of 0.1
![Page 52: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/52.jpg)
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
![Page 53: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/53.jpg)
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?
![Page 54: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/54.jpg)
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
![Page 55: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/55.jpg)
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
![Page 56: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/56.jpg)
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
![Page 57: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/57.jpg)
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
![Page 58: Density driven flow in porous media: How accurate are our models? Wolfgang Kinzelbach Institute for Hydromechanics and Water Resources Engineering Swiss](https://reader035.vdocument.in/reader035/viewer/2022070306/5516308f550346a2308b5fd2/html5/thumbnails/58.jpg)
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