Functional regression-based fluid permeability prediction in monodisperse sphere packings from isotropic two-point correlation functions

http://www.diva-portal.org

Postprint

This is the accepted version of a paper published in Computational materials science. This

paper has been peer-reviewed but does not include the final publisher proof-corrections or

journal pagination.

Citation for the original published paper (version of record):

Röding, M., Svensson, P., Loren, N. (2017)

Functional regression-based fluid permeability prediction in monodisperse sphere

packings from isotropic two-point correlation functions

Computational materials science, 134: 126-131

https://doi.org/10.1016/j.commatsci.2017.03.042

Access to the published version may require subscription.

N.B. When citing this work, cite the original published paper.

Permanent link to this version:

Functional regression-based fluid permeability

prediction in monodisperse sphere packings from

isotropic two-point correlation functions

Magnus R¨odinga,b,c,∗_{, Peter Svensson}a_{, Niklas Lor´en}a,b,d

a

RISE Bioscience and Materials, G¨oteborg, Sweden b

VINN Excellence Center SuMo Biomaterials, Chalmers University of Technology, G¨oteborg, Sweden

c

School of Energy and Resources, UCL Australia, University College London, Adelaide, Australia

d

Department of Applied Physics, Chalmers University of Technology, G¨oteborg, Sweden

Abstract

We study fluid permeability in random sphere packings consisting of imperme-able monodisperse hard spheres. Several different pseudo-potential models are used to obtain varying degrees of microstructural heterogeneity. Systematically varying solid volume fraction and degree of heterogeneity, virtual screening of more than 10,000 material structures is performed, simulating fluid flow using a lattice Boltzmann framework and computing the permeability. We develop a well-performing functional regression model for permeability prediction based on using isotropic two-point correlation functions as microstructural descrip-tors. The performance is good over a large range of solid volume fractions and degrees of heterogeneity, and to our knowledge this is the first attempt at us-ing two-point correlation functions as functional predictors in a nonparametric statistics/machine learning context for permeability prediction.

Keywords: Granular materials, sphere packings, permeability, correlation

functions, functional regression

1. Introduction

In adsorption, filtration, separation, chromatography, and catalysis applica-tions, understanding the impact of microstructural morphology of random, het-erogeneous, porous materials on effective transport properties is key [1, 2, 3, 4, 5]. Indeed, establishing quantitative structure-property relationships and

determin-5

ing the importance of different three-dimensional morphological characteristics is a prerequisite for targeted optimization of a microstructure, and hence fine-tuning of a material for a specific purpose [6]. In e.g. gel chromatography, spherical gel particles are packed into a column through which solutes in liquid suspension flows. The separation of the solutes is determined largely by the flow

10

around the gel particles, which is a function of microstructure.

There is generally a lack of good analytical expressions for quantitative structure-transport relationships [7], although there are empirical expressions like the Kozeny-Carman equation for estimating permeability [8, 9] which can sometimes be rather useful. Exact prediction of transport properties,

effec-15

tive diffusivity and permeability, in random, heterogeneous, porous materials requires in principle complete knowledge of the microstructure in three dimen-sions i.e. the geometry of the solid-liquid interface. Complete microstructural information in three dimensions is not accessible for real materials; hence, there is a great interest in finding useful proxies i.e. pieces of limited microstructural

20

information that capture the essential features.

A well-investigated set of microstructural descriptors are the n-point cor-relation (probability) functions for n = 1, 2, ..., introduced by Brown [10] for estimating effective transport properties in random materials. The set of all these correlation functions provide complete microstructural information, but

25

are in practice increasingly unavailable for increasing n [11]. Two-point and three-point correlation functions and microstructural parameters derived from them have been used to establish lower and upper bounds for effective transport coefficients. This includes both effective diffusivity (including other physical pro-cesses mathematically analogous to effective diffusivity, like electrical

ity, thermal conductivity, and magnetic permeability) [12, 13, 14, 15, 16, 17, 18] and fluid permeability [19, 20, 21, 22, 23, 24]. In two fairly recent papers [25, 26], a microstructural parameter extracted from the three-point correlation function and introduced by Torquato [16] is used in prediction of effective diffusivity in monodisperse and polydisperse hard sphere systems.

35

In this work, we are concerned with permeability in random sphere packings consisting of monodisperse, impermeable, solid, hard spherical particles. This is a highly interesting case e.g. for the design of materials for chromatography, separation and catalysis. Several different pseudo-potential models are used to obtain varying degrees of microstructural heterogeneity i.e. varying the

distri-40

bution of pore space sizes. Systematically varying solid volume fraction and degree of heterogeneity, virtual screening of more than 10,000 material struc-tures is performed, simulating fluid flow using a lattice Boltzmann framework and computing the permeability. It is evident that the permeability will de-pend not only on the solid volume fraction (and on the specific surface which

45

is perfectly correlated with the solid volume fraction in this setting) but also on the microstructure. We develop a functional regression model for perme-ability prediction based on using isotropic two-point correlation functions as microstructural descriptors. The performance is good over a large range of solid volume fractions and degrees of heterogeneity, and to our knowledge this is the

50

first attempt at using two-point correlation functions as functional predictors in a nonparametric statistics/machine learning context for permeability prediction.

2. Results and discussion

2.1. Microstructure generation

Statistically homogenous and isotropic sphere packing microstructures are generated using a hard sphere monte carlo algorithm. A target solid volume

fraction φ is chosen uniformly distributed in the range 0.10 ≤ φ ≤ φRCP, where

φRCP ≈ 0.64 is the random close packing limit [27]. Monodisperse spheres

sim-ulation domain [0, L]3 _{with periodic boundary conditions, possibly with some}

overlaps. The initial simulation domain length L is chosen such that the initial solid volume fraction (given that no particles overlap) is φstart = min (φ, 0.55),

where φ is the target solid volume fraction. The reason for this choice is that it is increasingly more difficult to remove overlaps for increasing solid volume fractions, which will be further discussed below. The relation between solid volume fraction and simulation domain length is

L = 4πN

3φ 1/3

R. (1)

After the initialization, the system is relaxed i.e. all overlaps are removed by sequentially translating the spheres by normal distributed displacements with

zero mean and standard deviation σt. The amount of overlap is characterized

by a system energy function, E = N X i=1 N X j=i+1 max 0, (Ri+ Rj)2− kxi− xjk2
, (2)

where Ri = Rj = 1 are the radii of spheres i and j and xi and xj are their

55

positions. The energy between two non-overlapping particles is zero and the energy between two overlapping particles is a quadratic function of the amount of overlap. Only translations which lead to a decrease in E are accepted. The

standard deviation σt is initially 0.05 but is then adaptively adjusted in each

sweep, striving for an acceptance probability target value of 0.25. After relaxing

60

the system (i.e. after the system energy has reached E = 0, which is possible because the solid volume fraction is set to a value below the random close packing limit, hence all overlaps can be resolved), it is equilibrated by performing 100 sweeps at E = 0. If φ ≤ 0.55, we are now done with generating a basic structure and proceed to obtain controlled heterogeneities as described below.

65

However, if φ > 0.55, the simulation domain is gradually compressed in constant increments of φ, ∆φ = 5 · 10−5_{, until the target solid volume fraction is reached,}

enforcing that E = 0 at each step before compressing further. The system is then equilibrated again by performing 100 sweeps at E = 0. All these steps

are performed using the same algorithm for random translations as described

70

above.

To obtain controlled heterogeneities, realizations of periodic random pseudo-potentials f are generated and the spheres perform random displacements (with the constraint that E = 0) using the same algorithm as above until the total potential of the system,

P =

N

X

i=1

f (xi) , (3)

is minimized (it is possible that the optimization of P converges to a local but not global minimum, i.e. a metastable state, but this fact does not compromise our purpose of generating these structures because the wide range of heterogeneities is nevertheless obtained). Three different types of pseudo-potentials are used: For type (I), we use a uniformly random number M (between 1 and 128) of uniformly distributed attractive quadratic point potentials centered in the points ym, f (x) = − M X m=1 kx − ymk−2. (4)

For types (II) and (III), realizations of Gaussian random fields are simulated on

a 1283 _{grid using fast Fourier transform methods [28]. For type (II), we use a}

covariance function family taken from Euclidean quantum field theory [28, 29], with the density of the measure given by the Fourier transform of the covariance function being

γ (p) =1 + p2k1 + p2k2 + p2k3

l−n

(5) for k = 1, n = 1.765, and l = 1.5. Briefly, the square root of this density is multiplied by independent white noise on a grid and inverse Fourier transformed to yield a random periodic function, the spatial scale of which is varied to generated different degrees of heterogeneity. For type (III), we use a Mat´ern covariance function family [30] with

γ (p) = 8π 3 2Γ ν +3 2
2 ν_νν Γ (ν) l2ν  2ν l2 + 4π 2 _p2 1+ p 2 2+ p 2 3
−ν−3 2 (6) using ν = 5 and l = 0.2, and Γ is the gamma function. As for type (II), the spa-tial scale is varied to generate different degrees of heterogeneity. A considerable

range of degrees of heterogeneity is hereby systematically explored. We refrain from getting into more technical detail about the Gaussian random fields, but

75

proceed to showing examples of pseudo-potentials and the corresponding gen-erated sphere packing microstructures in Fig. 1, showing one structure from each class, all with solid volume fraction φ = 0.15. In total, in excess of 10,000 sphere packing microstructures are generated in this fashion. Data sets with both uniformly distributed, random solid volume fractions φ as well as fixed

80

solid volume fractions φ = 0.10, 0.15, ..., 0.55 are generated, the latter to study the impact of microstructure isolated from solid volume fraction and specific surface. The algorithms are implemented in Julia [31, 32]. The computational time required to generate the structures is 0.6 h on average.

2.2. Flow simulations

85

The generated microstructures are converted to binary voxel structures of

size 1923 _{voxels, that in turn are converted to geometric surface data}

(trian-gulated surfaces which is necessary for input into the flow simulation software; however, the flow simulations are performed on a voxel grid). The fluid flow through the structures is then computed using a lattice Boltzmann method im-plementation [33, 34, 35], a numerical framework for solving partial differential equations based on the simulation of local particle populations. Briefly, the Navier-Stokes equations for pressure-driven flow are solved for the steady state. No-slip boundary conditions (of bounce-back type) on the solid/liquid interface and periodic boundary conditions orthogonal to the primary flow direction are used. The flow is driven by constant pressure difference boundary conditions across the structure [36], and a linear gradient is used as initial condition. We

use the two relaxation time collision model with the free parameter λeo = ₁₆3,

which guarantees that the computed permeability is independent of the

relax-ation time (and thus the viscosity) [37]. The relaxrelax-ation time τ = −1

λe is kept

at 1.25. After convergence to steady state flow, the permeability κ is obtained from Darcy’s law,

¯

u = −κ∆p

where ¯u is the average velocity, ∆p the applied pressure difference, µ the dy-namic viscosity, and d the length of the microstructure in the flow direction. For sufficiently small Reynolds numbers (Re < 0.01), the permeability is indepen-dent of the fluid and the pressure difference and a property of the microstructure. Equivalently, the same condition assures that the velocity is indeed proportional

90

to the pressure difference. Computed permeabilities are rescaled to units of R2

(squared sphere radii). The computational grid is 1923_{points. Fig. 2 illustrated}

the result of a flow simulation in one of the structures. For the entire data set, the obtained permeabilities cover about 3.7 orders of magnitude.

2.3. Two-point correlation functions

95

We characterize the microstructures by two-point correlation functions. Let I(x) be the indicator function for the void phase i.e.

I(x) =      1 if x in void 0 otherwise . (8)

Then, the two-point correlation function for a statistically homogeneous mate-rial is

S2(r) = hI(x)I(x + r)i, (9)

an average over all x. For a statistically homogeneous and isotropic material, the (isotropic) two-point correlation function is only dependent on distance, not direction, and is

S2(r) = hI(x)I(x + rˆr)i, (10)

an average over all x and all unit directional vectors ˆr. It holds that S2(0) = 1−

φ and S2(∞) = (1 − φ)

2

; here S2(r) only contains volume fraction information.

For 0 < r < ∞ however, S2(r) contains microstructural information as well.

The two-point correlation function is estimated using a monte carlo approach [38, 39] in the range r ∈ [0, 20]. A point x is selected randomly in the cubic

100

simulation domain, and along a random direction ˆr, I(x + rˆr) is evaluated for

’draws’ provides a precise estimate of S2(r). This computation is implemented

in Julia [31, 32]. In Fig. 3, some examples of isotropic two-point correlation functions are shown.

105

2.4. Functional regression

There are a plethora of ways in which the isotropic two-point correlation functions could be used as input data for permeability prediction, including ba-sically any machine learning and/or regression approach such as support vector machines, artificial neural networks, generalized linear models, and so on. How-ever, given the functional structure of the S2(r) data, functional regression is a

natural choice because it exploits the continuity as a function of r. We consider

permeability prediction as a regression problem with functional predictors S2(r)

and scalar responses log10κ, and formulate the functional regression model

log10κ = β0+ Z ∞ 0 β1(r) S2(r) dr + Z ∞ 0 β2(r) S22(r) dr, (11)

which is the purely quadratic (without interactions) special case of the general functional quadratic regression model from [40]. We employ the classical squared

second-derivative penalty [41] on both β1(r) and β2(r), effectively attempting

to solve the minimization problem

110 min β0,β1,β2  log10κ − β0− Z ∞ 0 β1(r) S2(r) dr − ... Z ∞ 0 β2(r) S22(r) dr 2 + ... λ " Z ∞ 0  ∂2_β 1(r) ∂r2 2 dr + Z ∞ 0  ∂2_β 2(r) ∂r2 2 dr # . (12)

In practice, we solve the linear least squares problem      log10κ 0m×1 0m×1      =      1n×1 S2∆r S22∆r 0m×1 Λ 0m×m 0m×1 0m×m Λ           β0 β₁ β₂      . (13)

Here, log10κis the n × 1 vector of (log-)permeabilities, S2is the n × m matrix

the discretization spacing of S2, β1 and β2are the m × 1 discretized regression

coefficient functions, and Λ is the second-derivative m× m penalty matrix equal to Λ= λ1/2∆r−3/2      1 −2 1 · · · 0 . .. 0 · · · 1 −2 1      . (14)

The penalty parameter λ is selected so as to minimize the mean squared vali-dation error in a repeated random sub-sampling (monte carlo) cross-valivali-dation procedure. The full data set is randomly divided into a training set (70 %) and a validation set (30 %), the linear least squares problem is solved for a range of λ values, and the λ value minimizing the mean squared error averaged over

115

the monte carlo replicates (NMC = 103), λopt = 4.31 × 10−8, is selected. For

λ = λopt, we perform a final fit using the full data set and compute the 95 %

confidence interval for β0 and 95 % point-wise confidence bands for β1(r) and

β2(r) using bootstrapping (NBS = 103) [42]. The procedure is implemented

in Matlab (Mathworks, Natick, MA, US). The constant term is estimated to

120

ˆ

β0= −4.01 (95 % confidence interval [−3.91, −4.11]). The results for β1(r) and

β2(r) are shown in Fig. 4 together with the mean squared error function used

for penalty parameter selection. The results indicate clearly that the two-point

correlation functions S2(r) are most informative for small r. This comes as no

surprise considering their general appearance shown in Fig. 3 and the fact that

125

the fluid ’experiences’ the microstructure most pronounced at a scale of up to a few sphere radii.

The relative prediction error

η = κ − ˆκ

κ (15)

as well as its absolute value kηk are used to quantify error. The latter is

0.028±0.033 (mean ± standard deviation). The ratio ˆκ/κ is 1.001±0.047 (mean

± standard deviation). In Fig. 5, the estimated vs true (simulated)

permeabil-130

ities are shown for the full data set where 0.10 ≤ φ ≤ φRCP, sorted according

vol-ume fractions φ = 0.10, 0.15, ..., 0.55 are shown, and further elaborated upon in Tab. 1, showing mean and standard deviation of both relative error η and

absolute value of relative error kηk. We see that the permeability prediction

φ hηi std (η) hkηki std (kηk) 0.10 -0.013 0.088 0.060 0.066 0.15 0.019 0.050 0.042 0.033 0.20 0.001 0.049 0.037 0.032 0.25 -0.003 0.047 0.034 0.032 0.30 -0.001 0.036 0.026 0.021 0.35 0.004 0.038 0.025 0.029 0.40 0.009 0.037 0.022 0.031 0.45 0.003 0.032 0.020 0.025 0.50 -0.004 0.026 0.019 0.018 0.55 -0.001 0.021 0.017 0.013

Table 1: Summary of relative prediction error for fixed solid volume fractions φ = 0.10, 0.15, ..., 0.55, showing mean and standard deviation of both relative error η and absolute value of relative error kηk.

135

is impressively consistent over several orders magnitude; however, for decreas-ing solid volume fractions and increasdecreas-ing permeabilities, there is an increase in prediction variability and hence in absolute value of relative error. This is ex-pected, because even though all structures are statistically isotropic and hence can in principle be described by the isotropic two-point correlation functions,

140

particular realizations may be somewhat anisotropic. This is most pronounced for low solid volume fraction structures that correspond to large permeabilities. Notably, the range of permeabilities for a fixed solid volume fraction goes up to 1.16 orders of magnitude (for φ = 0.10), and still reasonable prediction accuracy is obtained. From the results for the fixed solid volume fractions, we note that

145

the method can rather accurately predict changes in permeability induced by microstructural changes for fixed solid volume fraction and specific surface.

3. Conclusion

In this work, we develop a functional regression model for permeability pre-diction based on using isotropic two-point correlation functions as

microstruc-150

tural descriptors. We study the performance of the prediction model using data sets from virtual screening of more than 10,000 random sphere packings consist-ing of impermeable monodisperse hard spheres, in which fluid flow is simulated using a lattice Boltzmann framework. The prediction model perform well over almost 4 orders of magnitude difference in permeability, induced by varying

155

solid volume fraction as well as the degree of microstructural heterogeneity. In particular, the model can accurately predict changes in permeability induced by microstructural changes even for fixed solid volume fraction and specific surface. In conclusion, to our knowledge this is the first attempt at using two-point cor-relation functions as functional predictors in a nonparametric statistics/machine

160

learning context for permeability prediction, which provides a powerful method for extracting transport-relevant microstructural descriptors. Classical, short, analytical formulae will remain relevant for understanding the problem of mass transport; nevertheless, complex machine learning-type approaches like ours using different types of information provides complementary perspectives and

165

insights.

Acknowledgements

The financial support of the VINN Excellence Centre SuMo Biomaterials as well as the Swedish Foundation for Strategic Research project ”Material structures seen through microscopy and statistics” is acknowledged. Tobias

170

Geb¨ack is acknowledged for aiding with the theory and methods behind the flow simulations. The simulations were in part performed on resources at Chalmers Centre for Computational Science and Engineering (C3SE) provided by the Swedish National Infrastructure for Computing (SNIC).

References

175

[1] S. Torquato, Random heterogeneous materials: microstructure and macro-scopic properties, Springer, 2013.

[2] J. Giddings, Dynamics of chromatography: principles and theory, CRC Press, 2002.

[3] M. Sahimi, Heterogeneous Materials I: Linear transport and optical

prop-180

erties, Vol. 22, Springer, 2003.

[4] J. K¨arger, D. Ruthven, D. Theodorou, Diffusion in nanoporous materials,

Wiley, 2012.

[5] J. Rouquerol, F. Rouquerol, P. Llewellyn, G. Maurin, K. Sing, Adsorption by powders and porous solids: principles, methodology and applications,

185

Academic press, 2013.

[6] T. M¨ullner, K. Unger, U. Tallarek, Characterization of microscopic

disor-der in reconstructed porous materials and assessment of mass transport-relevant structural descriptors, New Journal of Chemistry 40 (2016) 3993– 4015.

190

[7] L. Masaro, X. Zhu, Physical models of diffusion for polymer solutions, gels and solids, Progress in polymer science 24 (1999) 731–775.

[8] J. Kozeny, ¨Uber kapillare leitung des wassers im boden:(aufstieg,

ver-sickerung und anwendung auf die bew¨asserung), Sitz. Ber. Akad. Wiss,

Wien, Math. Nat. (Abt. Iia) 136a (1927) 271–306.

195

[9] P. Carman, Fluid flow through granular beds, Transactions - Institution of Chemical Engineeres 15 (1937) 150–166.

[10] W. Brown, Solid mixture permittivities, The Journal of Chemical Physics 23 (1955) 1514–1517.

[11] S. Torquato, Random heterogeneous media: microstructure and improved

200

bounds on effective properties, Applied mechanics reviews 44 (1991) 37–76. [12] Z. Hashin, S. Shtrikman, A variational approach to the theory of the ef-fective magnetic permeability of multiphase materials, Journal of applied physics 33 (1962) 3125–3131.

[13] A. Sen, S. Torquato, Effective conductivity of anisotropic two-phase

com-205

posite media, Physical Review B 39 (1989) 4504–4515.

[14] M. Beran, Use of the vibrational approach to determine bounds for the effective permittivity in random media, Il Nuovo Cimento 38 (1965) 771– 782.

[15] G. Milton, Bounds on the electromagnetic, elastic, and other properties of

210

two-component composites, Physical review letters 46 (1981) 542.

[16] S. Torquato, Effective electrical conductivity of two-phase disordered com-posite media, Journal of applied physics 58 (1985) 3790–3797.

[17] Y. Jiao, S. Torquato, Quantitative characterization of the microstructure and transport properties of biopolymer networks, Physical biology 9 (2012)

215

036009.

[18] J. Thovert, I. Kim, S. Torquato, A. Acrivos, Bounds on the effective proper-ties of polydispersed suspensions of spheres: An evaluation of two relevant morphological parameters, Journal of applied physics 67 (1990) 6088–6098. [19] S. Prager, Viscous flow through porous media, Physics of Fluids 4 (1961)

220

1477–1482.

[20] H. Weissberg, S. Prager, Viscous flow through porous media. ii. approxi-mate three-point correlation function, Physics of Fluids 5 (1962) 1390–1392. [21] H. Weissberg, S. Prager, Viscous flow through porous media. iii. upper bounds on the permeability for a simple random geometry, Physics of Fluids

225

[22] J. Berryman, G. Milton, Normalization constraint for variational bounds on fluid permeability, The journal of chemical physics 83 (1985) 754–760. [23] J. Berryman, Bounds on fluid permeability for viscous flow through porous

media, The journal of chemical physics 82 (1985) 1459–1467.

230

[24] J. Rubinstein, S. Torquato, Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds, Journal of fluid mechanics 206 (1989) 25–46.

[25] H. Liasneuski, D. Hlushkou, S. Khirevich, A. H¨oltzel, U. Tallarek,

S. Torquato, Impact of microstructure on the effective diffusivity in random

235

packings of hard spheres, Journal of Applied Physics 116 (2014) 034904. [26] D. Hlushkou, H. Liasneuski, U. Tallarek, S. Torquato, Effective diffusion

coefficients in random packings of polydisperse hard spheres from two-point and three-point correlation functions, Journal of Applied Physics 118 (2015) 124901.

240

[27] A. Donev, I. Cisse, D. Sachs, E. Variano, F. Stillinger, R. Connelly, S. Torquato, P. Chaikin, Improving the density of jammed disordered pack-ings using ellipsoids, Science 303 (2004) 990–993.

[28] A. Lang, J. Potthoff, Fast simulation of gaussian random fields, Monte Carlo Methods and Applications 17 (2011) 195–214.

245

[29] J. Glimm, A. Jaffe, Quantum physics. A functional integral point of view, Springer, 1981.

[30] B. Mat´ern, Spatial variation, in: Lecture Notes in Statistics, Springer, 1986. [31] J. Bezanson, S. Karpinski, V. Shah, A. Edelman, Julia: A fast dynamic

language for technical computing, CoRR (2012) abs/1209.5145.

250

[32] J. Bezanson, A. Edelman, S. Karpinski, V. Shah, Julia: A fresh approach to numerical computing, CoRR (2014) abs/1411.1607.

[33] T. Geb¨ack, A. Heintz, A lattice boltzmann method for the advection-diffusion equation with neumann boundary conditions, Communications in Computational Physics 15 (2014) 487–505.

255

[34] S. Wass´en, R. Bordes, T. Geb¨ack, D. Bernin, E. Schuster, N. Lor´en, A.-M. Hermansson, Probe diffusion in phase-separated bicontinuous biopolymer gels, Soft matter 10 (2014) 8276–8287.

[35] T. Geb¨ack, M. Marucci, C. Boissier, J. Arnehed, A. Heintz, Investigation of the effect of the tortuous pore structure on water diffusion through a

260

polymer film using lattice boltzmann simulations, The Journal of Physical Chemistry B 119 (2015) 5220–5227.

[36] Q. Zou, X. He, On pressure and velocity boundary conditions for the lattice Boltzmann BGK model, Physics of fluids 9 (1997) 1591–1598.

[37] I. Ginzburg, F. Verhaeghe, D. dHumieres, Study of simple hydrodynamic

265

solutions with the two-relaxation-times lattice Boltzmann scheme, Com-munications in Computational Physics 3 (2008) 519–581.

[38] P. Smith, S. Torquato, Computer simulation results for the two-point prob-ability function of composite media, Journal of computational physics 76 (1988) 176–191.

270

[39] P. Smith, S. Torquato, Computer simulation results for bounds on the effective conductivity of composite media, Journal of applied physics 65 (1989) 893–900.

[40] F. Yao, H.-G. M¨uller, Functional quadratic regression, Biometrika 97 (2010)

49–64.

275

[41] I. Good, R. Gaskins, Nonparametric roughness penalties for probability densities, Biometrika 58 (1971) 255–277.

[42] B. Efron, R. Tibshirani, An introduction to the bootstrap, CRC press, 1994.

(a)

(b)

(c)

Figure 1: (SINGLE COLUMN FIGURE) Representative two-dimensional slices of pseudo-potentials and the corresponding generated microstructures with 4096 impermeable spherical granules, showing examples of (a) an attractive quadratic point potential, (b) a Gaussian random field of type II, and (c) a Gaussian random field of Mat´ern type (type III), all with solid volume fraction φ = 0.15.

Figure 2: (SINGLE COLUMN FIGURE) An example of simulated steady state flow through a type II microstructure with solid volume fraction φ = 0.20.

0 5 10 15 20

r

S

(r)

Figure 3: (SINGLE COLUMN FIGURE) Examples of isotropic two-point correlation func-tions, evaluated using a monte carlo approach for m = 250 equidistant values of r for some different microstructures. For r = 0 and r = ∞, the two-point correlation functions only con-tain volume fraction information; for 0 < r < ∞, they concon-tain microstructural information as well.

10-10 10-5 100 10-4

10-3 10-2

Mean squared error

-20 0 20 1

estimate

r

estimate

Figure 4: (SINGLE COLUMN FIGURE) Results of model fit, showing (a) mean squared error vs penalty parameter λ, indicating the optimal λopt= 4.31 × 10−8, (b) estimated β1(r) with 95 % point-wise confidence bands, and (c) estimated β2(r) with 95 % point-wise confidence bands.

10-3 10-2 10-1 100 101 Estimated permeability 10-3 10-2 10-1 100 101 10-6 10-4 10-2 100 Relative error 10-3 10-2 10-1 100 101 Permeability 10-3 10-2 10-1 100 101 (a) (b) (c) (d) (e) (f)

Figure 5: (DOUBLE COLUMN FIGURE) Prediction results for the full data set where solid volume fractions are uniformly distributed in the range 0.10 ≤ φ ≤ φRCP, showing (a-c) the predicted permeability as a function of the true permeability for microstructure types (I)-(III), and (d-f) relative prediction errors (positive errors are red, negative errors are yellow) as a function of the true permeability for microstructure types (I)-(III).

10-3 10-2 10-1 100 101

Permeability

Estimated permeability

Figure 6: (SINGLE COLUMN FIGURE) Prediction results for the fixed solid volume fractions φ= 0.10, 0.15, ..., 0.55, showing (a-c) the predicted permeability as a function of the true permeability for φ = 0.10 (red circles), φ = 0.15 (yellow circles), φ = 0.20 (green circles), φ= 0.25 (blue circles), φ = 0.30 (purple circles), φ = 0.35 (red squares), φ = 0.40 (yellow squares), φ = 0.45 (green squares), φ = 0.50 (blue squares), and φ = 0.55 (purple squares).