stochastic geometry and transport in porous media - · pdf fileintroduction characteristics...

IntroductionCharacteristics

Simulating pore spacesResults

Discussion

Stochastic geometry and transport in porousmedia

Hans R. Kunsch

Seminar fur Statistik, ETH Zurich

February 15, 2007, Reisensburg

Hans Kunsch, ETHZ Stochastic geometry and porous media



Discussion

Coauthors

Thanks to the coauthors of this paper:P. Lehmann, A. Kaestner, H. Fluhler, Institute of terrestrialecosystems, ETH ZurichM. Berchtold, Seminar fur Statistik, ETH ZurichB. Ahrenholz, J. Tolke, M. Krafczyk, Computer applications incivil engineering, TU Braunschweig.

The work is part of the research project “Fimotum = Firstprinciples-based models for transport in unsaturated media”,joint by ETH Zurich, TU Braunschweig and University ofStuttgart.A special issue of Advances in Water Resources is planned forresults from Fimotum.




Discussion

Overview1 Introduction2 Characteristics

Minkowski functionalsOther characteristics

3 Simulating pore spacesSimulated annealingBoolean modelsThresholded Gaussian models

4 ResultsComparing other geometric characteristicsPermeabilityWater retention curves

5 Discussion




Discussion

Background

The goal of soil physics is to describe and predict the transportof water, air and solutes through the pores between the soilsurface and the aquifer.This is a largely unsolved problem: The pore space is notknown for applications and too complex for solving thehydrodynamic equations (Navier-Stokes).To some extent, continuum theory (Richards equation) isapplicable. It requires a parametrization of the relationsbetween saturation and pressure and between hydraulicconductivity and pressure. These are purely empirical.




Discussion

Aim of the Fimotum project

The Fimotum project aims to improve the understanding of theinfluence of the pore structure at the microscale on transportproperties.

It uses recent progress in imaging technology: We have3d-binary images of two sand samples (coarse and fine). Animage consists of 8003 voxels of size (11µ)3.

It also uses progress in the numerical solution of theNavier-Stokes equation with complicated boundary conditions(Lattice Boltzmann method).




Discussion

Coarse sand image




Discussion

Permeability and water retention curve

For this measured pore space, we can compute the saturatedpermeability by solving the Navier-Stokes equation.Permeability is viscosity times flow rate divided by appliedpressure gradient. Saturated means that the pore space iscompletely filled with the fluid.With a pore network model, we can also compute the waterretention curve, i.e. saturation as a function of applied pressure(starting with the saturated condition).




Discussion

Our main questions

For a better understanding, we would like to find simplegeometric characteristics of the microstructure of porous mediathat determine the transport properties of the medium.

When we have candidates for such characteristics, we want togenerate artificial pore spaces with prescribed values of thesecharacteristics and compute the permeability and the waterretention curves for them.As prescribed values, take those of the real sand probes andvary them in a systemtic manner.




Discussion


Minkowski functionals

It is clear that transport properties of a porous medium dependon the volume, the surface (because of friction at the solidwalls) and some measures of connectivity.

The Euler characteristic is a measure of 3d connectivity.Because of the Crofton formula, the mean curvature is ameasure of average 2d connectivity.

Hence we use the 4 Minkowski functionals as our candidatesfor the geometric characteristics. The Hadwiger theorem canbe seen as an indication that they might be sufficient.




Discussion


Minkowski functionals for openings

One can argue that when draining a porous medium byincreasing the suction, pores of different size are emptied atdifferent suctions. This means the Minkowski functionals not forall pores, but for those with a given diameter are relevant.

Hence consider the Minkowski functionals of the morphologicalopenings with balls of radius r as a function of r . We call thesethe Minkowski functions.




Discussion


Chord length

Another characteristic that might be relevant is the chord lengthdistribution because this also describes how frequent pores ofdifferent width are.

In the following, we simulate pore spaces with prescribedvalues for the 4 Minkowski functionals. We then use theMinkowski functions and the chord length distribution as anindication how different the simulated and the real pore spacesare.




Discussion

Simulated annealingBoolean modelsThresholded Gaussian models

Simulated annealing I

Assume we have m functionals F1, . . . , Fm and m real numbersf1, . . . , fm and an observation window W . We want to generateone or many pore spaces P ⊂ W such that

Fj(P)

vol(W )= fj (j = 1, . . . , m).

This can be done by minimizing the cost function

J(P) =∑

αj

∣∣∣∣ Fj(P)

vol(W )− fj

∣∣∣∣ .

Simulated annealing is a stochastic minimization algorithmwhich avoids being trapped in local minima.




Discussion


imulated annealing II

Simulated annealing proceeds iterativelyChoose a simple modification of P ′ = T (P) randomly (e.g.flipping the values of two voxels).If J(P ′) ≤ J(P), always accept the modification. Otherwise,accept with probability

exp(−β(J(P ′)− J(P)).

Increase β and iterate.Practical issues are the choice of the basic modifications, of theweights αj and of the speed with which we increase β with thenumber of iterations.




Discussion


Simulated annealing III

In theory, with an appropriate choice of β →∞, we end upsampling from the uniform distribution on the set {P; J(P) = 0}.

In the literature, successes with simulated annealing arereported. Our own experience is less positive. If we match onlythe Minkowski functionals, we often obtain strange“non-stationary” P. Using additional functionals, we haveproblems with convergence even though we worked only in 2d .




Discussion


Boolean models I

For Boolean models, we can express the four (specific)Minkowski functionals in terms of the intensity of the germs andthe 3 expected Minkowski functionals of the typical convexgrain. (Davy 1978, Mecke 2000). Moreover, this relation isinvertible.

Hence we need to construct grain distributions with prescribedvalues of the expected Minkowski functionals.




Discussion


Boolean models II

We use the Boolean model for the solid phase and chooseellipsoids for the grains. For a fast algorithm there are at leasttwo possibilities:Choose ellipsoids where two half axes are the same, and takethe two different half axes to be independent (Arns et al., 2003).Choose ellipsoids of the form Z · E0 where E0 is fixed and Z israndom (our solution).Open questions: Can we obtain all possible specific Minkowskifunctionals with ellipsoid grains ? How do we obtain the “leasteccentric” grains?




Discussion


Thresholded Gaussian models

Consider a Gaussian stationary and isotropic random field(Z (x)) with correlation function ρ and the thresholded binaryfield

Ξ(x) = 1[a,b](Z (x)).

Results of Adler show that the four specific Minkowskifunctionals depend only on the thresholds a, b and the secondderivative of ρ at the origin. Hence we can obtain in this wayonly a 3d subset of the 4d space of all specific Minkowskifunctionals. By chance (?) the values of the real sand imagesare very close to this 3d subset.




Discussion

Comparing other geometric characteristicsPermeabilityWater retention curves

Real sand




Discussion


Boolean model, 4 functionals matched




Discussion


Boolean model, increased surface




Discussion


Gaussian model, 4 functionals matched




Discussion


Chord length distributions

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 500 1000 1500

chord length [micron]

po

re v

olu

me f

racti

on

[-]

measured

optimized

surface 75%

surface 110%

surface 120%

surface 150%

surface 200%

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 500 1000 1500

chord length [micron]

po

re v

olu

me f

racti

on

[-]

measured

optimized

volume 87.5%

volume112.5%

curvature 50%

curvature 200%

connectivity 50%

connectivity 150%




Discussion


Porosity as a function of pore size

0

0.1

0.2

0.3

0.4

0.5

0 100 200 300 400

pore diameter [micron]

vo

lum

e [

-]

measured

optimized

surface 75%

surface 150%

surface 120%

surface 150%

surface 200%

0

0.1

0.2

0.3

0.4

0.5

0 100 200 300 400


vo

lum

e [

-]

measured

optimized

volume 87.5%

volume 112.5%

curvature 50%

curvature 200%

connectivity 50%

connectivity 150%




Discussion


Surface as a function of pore size

0

5

10

15

20

25

30

35

0 100 200 300 400pore diameter [micron]

su

rface [

mm

-1]

measured

optimized

volume 87.5%

volume 112.5%

curvature 50%

curvature 200%

connectivity 50%

connectivity 150%

0

5

10

15

20

25

30

35

0 100 200 300 400


are

a [

mm

-1]

measured

optimized

surface 75%

surface 110%

surface 120%

surface 150%

surface 200%




Discussion


Euler char. as a function of pore size

-100

300

700

1100

1500

0 100 200 300 400


Eu

ler

ch

ara

cte

ris

tic

[m

m-3

]

measured

optimized

volume 87.5%

volume 112.5%

curvature 50%

curvature 200%

connectivity 50%

connectivity 150%

-100

300

700

1100

1500

0 100 200 300 400


Eu

ler

ch

ara

cte

ris

tic

[m

m-3

]

measured

optimized

surface 75%

surface 110%

surface 120%

surface 150%

surface 200%




Discussion


Permeability for real and simulated pore spaces

Values for solid phase, relative to real sand probe

origin volume surface curvature Euler permeability10−12 m2

model 0.92 1.0 1.0 1.0 194.6model 1.0 1.0 1.0 1.5 162.0model 1.0 1.0 1.0 0.5 161.3model 1.0 1.0 1.0 1.0 149.0

real 1.0 1.0 1.0 1.0 149.0model 1.0 1.0 2.0 1.0 148.5model 1.0 0.75 1.0 1.0 144.9model 1.0 1.0 0.5 1.0 143.8model 1.0 1.2 1.0 1.0 105.8model 1.08 1.0 1.0 1.0 79.5




Discussion


Water retention curves I

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50

|matric potential head| [cm]

wate

r satu

rati

on

[-]

measured

optimized

volume 87.5%

volume 112.5%

curvature 50%

curvature 200%

connectivity 50%

connectivity 150%

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50

|matric potantial head| [cm]

wate

r satu

rati

on

[-]

measured

optimized

surface 75%

surface 110%

surface 120%

surface 150%

surface 200%




Discussion


Water retention curves II

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50|matric potential head| [cm]

wat

er s

atur

atio

n [-]

Scan

Bool

Gauss




Discussion

Discussion

Boolean models are not suitable for these images, even whenwe are satisfied with a rough approximation. ThresholdedGaussian models are better, but less flexible.Minkowski functionals alone are not sufficient for transportproperties.Can we match Minkowski functions ? Are they sufficient fortransport properties ?How would packing of hard or soft grains with random shapeand size work ? Could we relate the properties of the graindistribution to transport properties ?


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