Uncertainty quantification on the macroscopic properties of

Uncertainty quantification on the macroscopic properties of heterogeneous porous media Peng Wanga , Huali Chena , Xuhui Mengb , Xin Jianga , Dongbin Xiuc , Xiaofan Yangd,b,∗ a School

of Mathematics and System Sciences, Beihang University, China Computational Science Research Center, Beijing, China c Department of Mathematics, Ohio State University, Columbus, USA d State Key Laboratory of Earth Surface Processes and Resource Ecology, Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China b Beijing

Abstract Pore-scale simulation is an essential tool to understand complex physical process in many environmental problems. However, structural heterogeneity and data scarcity render the porous medium and in turn, its macroscopic properties, uncertain. Meanwhile, direct numerical simulation of the medium at the fine scale often incurs high computational cost, which further limits one’s effort to quantify the parametric uncertainty over those macroscopic properties. To address such challenge, we propose a novel framework to compute the probabilistic density function (PDF) of the macroscopic property, based on the generalized polynomial chaos (gPC) expansion method and the Minkowski functionals. To illustrate the effectiveness of our approach, we conduct numerical experiments for one macroscopic property, the permeability, and compare its PDF with that obtained from Monte Carlo simulations. Both two- and three-dimensional cases show that our framework requires much fewer realizations while maintaining the desired accuracy. Keywords: pore-scale modelling, uncertainty quantification, Minkowski functionals, generalized polynomial chaos expansion

∗ Corresponding

author: [email protected]

Preprint submitted to Journal of LATEX Templates

August 28, 2018

1. Introduction Flow and transport in porous media are fundamental in many applications, from groundwater remediation, CO2 sequestration, to polymer electrolyte fuel cells [1, 2]. With recent development and proliferation of computing power 5

and resources, pore-scale modeling has become an essential tool in the study of porous media [3] and a series of numerical approaches have since been proposed [4, 5]. For a comprehensive review of pore-scale imaging, modeling and simulation, please refer to the recent book of Blunt [2]. There are three major challenges in porous media research: the heterogene-

10

ity of the pore-scale morphology (structure), the multiscale nature (both in time and space) and the associated uncertainties in experiment and computation [6]. Small-scale heterogeneity not only affects properties at the macroscopic scale [7], but also incurs high computational cost on pore-scale direct numerical simulations (DNS). Meanwhile, insufficient site characterization further exacerbates

15

this problem. In spite of continuous progress in the state-of-the-art imaging techniques, from X-ray Computed Tomography (XCT) [8], micro-Computed Tomography (micro-CT) [9] to Magnetic Resonance Imaging (MRI) [10], measurement and interpretation limitations on imaging resolution, image processing algorithm and etc., are ubiquitous and cannot be entirely eliminated. In conclu-

20

sion, heterogeneity and data scarcity render the microscopic structure of porous media and thus its macroscopic properties uncertain. To address similar issues due to parametric uncertainty in other research fields, the scientific community has since developed uncertainty quantification (UQ) tools. Among those methods, the Multi-Level Monte Carlo (MLMC)

25

method [11] and the variance reduction techniques [12], have been variously applied to stochastic homogenization (e.g.

the Darcy or the heat conduc-

tion equation) with random heterogeneous diffusion coefficients. Lately, Icardi and co-authors [13] introduced MLMC to pore-scale simulations and computed the mean and variance of permeability of random porous media. As a non30

intrusive method, the MLMC offers easy implementation and considerably re-

2

duces the overall computational cost by preconditioning fine-scale results with their coarse-scale counterparts, which are generally cheaper to obtain. In addition, as a type of Monte Carlo (MC) technique, the convergence rate of MLMC is independent of the number of random parameters. In other words, the num35

ber of realizations that MLMC takes to converge for porous media described by a hundred random parameters would be the same as those described by ten random parameters, albeit each realization of the former would take much longer to conduct. This is a great advantage over other UQ methods inhibited by the “curse of dimensionality”.

40

In practice, Monte Carlo-based techniques are often used to obtain the ensemble mean and variance of a system state, which can be used to estimate the system’s average response and to measure an associated prediction error, respectively. However, those two statistical moments are insufficient for risk assessment [14] that typically focuses on the probability of rare events, e.g., the

45

distribution tail. On the other hand, the actual distribution of a macroscopic pore property may not be fully described by its mean and variance alone. Although recent works have partially addressed the issue [15, 16], the demand for full statistical information of the macroscopic property, such as its probabilistic density function (PDF) or cumulative density function (CDF), renders most

50

Monte Carlo-based techniques impractical, due to their slow convergence rates for higher statistical moments. To alleviate such problem, we propose an alternative UQ framework to compute the PDF of a macroscopic property A of the porous medium. Our approach is based on the generalized polynomial chaos (gPC) expansion method

55

and constructs a surrogate model of A in the probability space of Minkowski functionals, which characterize the random geometric structure of pore samples. Through the example of one macroscopic property, the permeability (calculated from pore-scale simulation), we demonstrate that such combination of gPC and Minkowski functionals could overcome the “curse of dimensionality” and also

60

reduce the overall cost in obtaining the permeability PDF. The remainder of the paper is organized as below: we first set up the overall 3

problem (Section 2) and then formulate our uncertainty quantification framework in Section 3, which includes the basic concepts of Minkowski functionals (Section 3.1), the gPC method (Section 3.2) and the existing numerical code 65

for pore-scale flow simulation (Section 3.3). Using two- and three-dimensional examples, Section 4 presents the PDF results and compare them with those obtained from MLMC. Key findings of the new framework are concluded in Section 5. 2. Problem formulation

70

The macroscopic properties (A) of porous media are the manifestation of pore-scale microscopic structures, which are often defined by the spatial distribution of pore size, shape and number, Z(x). To obtain such properties, one often needs to solve a set of coupled governing equations at different scales, A = G [Z(x)]. For example, the permeability K for incompressible flow through

75

porous media is calculated by first solving the Navier-Stokes (N-S) equations for velocity and pressure at the microscopic scale (pore scale): ∇ · u = 0,

(1a)

  1 ∂t u + ∇ · (uu) = − ∇p + ν∇ · ∇u + (∇u)T + F, ρ0

(1b)

and is then derived from Darcy’s law at the mesoscopic scale: K=−

ρ0 νuD ∇pD

(2)

Here u and p are the pore-scale velocity and pressure field; density and kinematic viscosity of the fluid are denoted by ρ0 and ν, respectively; F represents the 80

external force field and the superscript

T

is the transposition operator; uD and

pD are the volume-average of the velocity and the pressure at the pore space, respectively. 3. Uncertainty quantification framework Pore-scale parametric uncertainty due to heterogeneity and data scarcity 85

would propagate to its macroscopic property A, via the governing equations 4

G(·). To quantify such uncertainty, one can treat these parameters as random fields, e.g., they not only vary in physical domain, but also in the probability space. Unless specified otherwise, subsequent work is conducted on a complete probability space (Ω, F , µ), in which Ω represents the collection of events, F is 90

a σ-algebra on sets of Ω, and µ denotes a probability measure on F . All random variables are assumed to have finite second moments. For a given set of porous media samples, we propose the following general framework to compute the probabilistic density function (PDF) of its macroscopic property A:

95

1. Determine the Minkowski functionals m of all pore samples and obtain their statistical distributions Fm (m′ ). 2. Construct the generalized polynomial chaos expansion of the macroscopic property A in the probability space of the Minkowski functionals Fm (m′ ). 3. Based on the gPC expansion order, select a subset of pore samples and

100

calculate their macroscopic property A using appropriate pore-scale simulation approach. 4. Determine the gPC expansion coefficients ai from the sample pairs (m, A). 5. Generate a number of A from the gPC model and obtain its PDF from the histogram.

105

To demonstrate the general procedures above, in subsequent work, we compute the PDF of one macroscopic property, the permeability K; and employ an existing pore-scale flow simulation code to solve the Navier-Stokes equation (1) and Darcy’s law (2). The entire procedure could be executed in both serial and parallel, which has been well-performed on Tianhe-II supercomputers. Without

110

loss of generality, the procedures could also be applied to compute the PDF of other macroscopic properties, such as the reaction rate if transport process is involved. 3.1. Minkowski functionals A Minkowski functional represents a notion of distance on a linear space and

115

is introduced to describe the geometric structure of a given porous media sample 5

in its binary image [17, 18]. To reconstruct a three-dimensional porous structure with identical permeability as that of the original sample [19], four Minkowski functionals m = {m1 , m2 , m3 , m4 } are required, namely, the total volume of the pores m1 , the surface area of the boundary between porous and solid phase m2 , 120

the mean curvature m3 of the interface and the Euler characteristic m4 :  Z Z  Z 1 1 1 1 1 ds, m3 (X) = m2 (X) = ds, m4 = + ds, (3a) 2 δX r1 r2 4π δX r1 r2 δX where δX denotes the pore surface, ds is the surface element, r1 and r2 represent the minimum and maximum radius of curvature for the surface element ds, respectively. One can physically interpret those four Minkowski functionals (3) as below:

125

m1 , the total volume of the pores, can be alternatively defined as porosity; the surface area m2 denotes a surface density relevant for the interaction of solutes at the interfaces; the mean principal curvature m3 is a description of the pore shape, which regulates the energy density of the wetting fluid in porous media or the pressure difference between wetting and non-wetting phase; and lastly,

130

the Euler characteristic m4 is a measure for connectivity of the pores. We note here that for a two-dimensional sample, only three Minkowski functionals, the porosity, surface area (now boundary length) and Euler characteristics are needed. As reported in previous studies [19, 20], one can use the Minkowski function-

135

als (3) to reconstruct a porous structure with identical macroscopic property as that of the original sample, although slight geometric differences at the microscopic scale exsit between the reconstructed and original sample: pore size in the reconstructed medium would be larger than the original one, as illustrated in Fig. 1 [20].

140

In practice, porous media samples may come in various image sizes and thus the Minkowski functionals are often normalised by the volume of the image Vω [21]: m′1 =

m1 , Vω

m′2 =

m2 , Vω

m′3 = 6

m3 , Vω

m′4 =

m4 , Vω

(4)

(a)

(b)

(c)

(d)

Figure 1: (Top) Three-dimensional porous media of 256 × 256 × 256 pixels: (a) an original sample of the HASYLAB synchorton sand sample; (b) a reconstructed image from the same Minkowski functionals values using 3D-spherical Boolean model. (Bottom) Two-dimensional porous media of 800 × 800 pixels: (c) the cross-section image of the original sand sample above; (d) a reconstructed image from the same Minkowski functionals values using circular Boolean model [20].

In subsequent analysis, we drop the primes to facilitate presentation. Detailed derivations of the Minkowski functionals in two-dimensional and three145

dimensional cases are shown in Appendix Appendix A and one can also refer to previous studies [22, 20]. 3.2. gPC method The generalised polynomial chaos (gPC) expansion method [23] is an extension of the seminal work on Hermite polynomial chaos [24] and has become one of the most widely used uncertainty quantification methods. Based on multi-dimensional approximation theory, it constructs a N -th order surrogate model KN (m) to approximate the relation between system random inputs (the

7

Minkowski functionals m) and system output (the permeability K): K(m) ≈ KN (m) =

N X

ai Φi (m) =

|i|=0

|i|=0

Here i = (i1 , . . . , iNm ) ∈

N X

m NN 0

a(i1 ,...,iNm ) φi1 (m1 ) · · · φiNm (mNm ). (5)

stands for a multi-index, Nm is the dimension of

the random parameters and |i|= i1 + · · · + iNm . The symbol φiNm (mNm ) repre150

sents the iNm -th degree Wiener-Askey polynomials of random variable mNm . To facilitate subsequent presentation, we employ the popular graded lexicographic order and use single index to express the gPC expansion [1]. To be specific, the graded lexicographic order of the four Minkowski functionals is shown in Table 1. The multi-variate N -th degree gPC basis Φi (m) = φi1 (m1 ) · · · φiNm (mNm ) are orthogonal polynomials: Z E[Φi (m)Φj (m)] = Φi (m′ )Φj (m′ )dFm (m′ ) = δi,j , Ω

155

0 ≤ |i|, |j| ≤ N

(6)

where Fm (m′ ) = Prob(m1 ≤ m′1 , . . . , mNm ≤ m′Nm ) is the joint probability

distribution of the Minkowski functionals and m′ = {m′1 , . . . , m′Nm } are their possible values.

We also note that δi,j represents the multi-variate Kronecker delta function  N m  1 if i = j , Y d d δi,j = δid ,jd = (7)  0 if id 6= jd . d=1

Using the orthogonality relationship (6), one can determine the polynomial

basis {Φi (m)} of the random parameters m. For example, Gaussian distribu160

tion corresponds to Hermite polynomials while uniform is related to Legendre polynomials. Detailed account of hypergeometric polynomials and the Askey scheme can be found in [26, 27, 23]. We further note that the accuracy of the gPC method can be measured in the popular L2 -norm using classical approximation theory: Z kK(m) − KN (m)k2L2 := |K(m′ ) − KN (m′ )|2 dFm (m′ ) → 0, Ω

Ω

8

N → ∞, (8)

|i|

Multi-index i

Single index k

0

(0 0 0 0)

1

1

(1 0 0 0)

2

(0 1 0 0)

3

(0 0 1 0)

4

(0 0 0 1)

5

(2 0 0 0)

6

(1 1 0 0)

7

(1 0 1 0)

8

(1 0 0 1)

9

(0 2 0 0)

10

(0 1 1 0)

11

(0 1 0 1)

12

(0 0 2 0)

13

(0 0 1 1)

14

(0 0 0 2)

15

(3 0 0 0)

16

(2 1 0 0)

17

(2 0 1 0)

18

···

···

2

3

Table 1: The graded lexicographic ordering of the four random Minkowski functionals, m = {m1 , m2 , m3 , m4 } [25].

9

whose convergence rate is in tandem with the smoothness of system output, e.g., K(m). In other words, sufficient accuracy can be achieved with a low165

degree gPC expansion KN for a relatively smooth K(m) with considerably low computational cost. To implement the gPC method and solve for the gPC coefficients ai , one may employ the stochastic collocation method [28], which allows the independent use of existing programming code/solver of the underlying deterministic system, e.g.

170

“non-intrusive” [29] and thus is easy to implement. In essence, the stochastic collocation method aims to compute the gPC coefficients ai with Nq quadrature nodes: E[Φi (m)K(m)] ai = = E[Φ2i (m)]

PNq

nq =1

Φi (m|nq ) K(m|nq ) fm (m′ |nq ) w|nq E[Φ2i (m)]

,

(9)

where m|nq refers to the Minkowski functionals evaluated at the nq -th quadrature node, ω|nq and fm (m′ |nq ) are the corresponding weight and joint proba175

bilistic density function (PDF) of m, respectively. The quadrature nodes and weights are predetermined once the quadrature rule and level are set. Lastly, the permeability values at the quadrature nodes, K(m|nq ), can be obtained via pore-scale flow simulations of the pore space realization of m|nq . 3.3. Pore-scale flow simulation

180

To calculate the permeability, we employ the multi-relaxation-time lattice Boltzmann equation model (MRT-LBE) [30] to obtain the pore-scale flow field. It is noted that other numerical schemes could also be used in the current framework to solve the Navier-Stokes equation (1). The MRT-LBE is selected here just for its easy implementation and availability.

185

The evolution equation of the MRT-LBE model can be expressed as below: fi (x + ci δt , t + δt ) − fi (x, t) =

− M−1 ΛM


 ij

 fj (x, t) − fjeq (x, t) , (10)

where i, j are the discrete velocity index, fi (x, t) is the particle distribution function at position x and time t at velocity ci , δt represents the time step,

10

M and Λ denotes the transform matrix and the diagonal relaxation matrix, respectively. The equilibrium distribution function fieq is defined as    ci · u (ci · u)2 u2 Af δt S : (ci ci − c2s I) . + − + fieq = ωi ρ + ρ0 c2s 2c4s 2c2s 2c2s 190

(11)

Here ρ is a variable related to pressure p = ρc2s and Af is a tuneable parameter related to the fluid viscosity ν. The shear rate [30] is represented by S, cs is the lattice sound speed and I is the identity matrix. For the discrete velocity ci , we employ the D2Q9 (two-dimensional-ninevelocity) and the D3Q19 (three-dimensional-nineteen-velocity) models for twoand three-dimensional cases, respectively, which are in the form as follows:    c(0, 0), i = 0,      
π π ci = (12a) c cos (i − 1) 2 , sin (i − 1) 2 , i = 1, . . . , 4,      
  2c cos (i − 5) π + π , sin (i − 5) π + π , i = 5, . . . , 8, 2 4 2 4    c(0, 0, 0),   ci = c(±1, 0, 0), c(0, ±1, 0), c(0, 0, ±1)     c(±1, ±1, 0), c(±1, 0, ±1), c(0, ±1, ±1)

i = 0,

i = 1, . . . , 6,

(12b)

i = 7, . . . , 18,

where c = δx /δt is the lattice speed, δx and δt are lattice spacing and time step, respectively. Finally, the variables related to the fluid pressure and velocity are obtained as: ρ=

b−1 X

fi (x, t), ρ0 u =

ci fi (x, t),

(13)

i=0

i=0

195

b−1 X

Details of the pore-scale MRT-LBE model can be found at the works of [30] and can be shared per request.

4. Results and Discussion In this section, we demonstrate the effectiveness of our probabilistic framework by calculating the permeability PDF for a set of two- and three-dimensional 200

porous samples, respectively. The pore structure of samples are numerically generated using the quartet structure generation set (QSGS) method [31, 32] for 11

its easy implementation and that its morphological features closely resembling the forming progress of many real porous media. The code includes four input parameters: the core distribution probability, growth probability in each given 205

direction, porosity, e.g., the first Minkowski functional (m1 ) and phase number. As outlined in Appendix Appendix B, the QSGS iterative procedure is running until reaching the desired porosity, which is set as m1 = 0.65, representative of silt, in our analysis. Without loss of generality, the proposed uncertainty quantification framework can also incorporate other structure generation codes

210

and does not have a prior assumption on the sample porosity m1 and thus can be applied to a wide range of porosity, including realistic sediment samples. A total of Nr = 4000 realizations of the Minkowski functionals are generated for two- and three-dimensional cases, respectively. In turn, we employ quadruple core CPU to obtain their corresponding permeability values from MRT-LBE.

215

In our numerical examples, we compute the gPC coefficients ai from Eqn. (9) using permeability values K whose pore space realizations fall close to the exact values of the quadrature nodes. To be specific, we introduce a relative error defined as a normalized Euclidean distance between the nr -th realization m|nr

220

and the nq -th quadrature node m|nq : m1 |nr − m1 |nq mNm |nr − mNm |nq ,..., ǫ|nr ,nq = m1 |nq mNm |nq

,

(14)

2

We then identify the “closest” samples, i.e. ǫ|nr ,nq < 3%, and substitute their permeability values K(m|nr ) for K(m|nq ) in Eqn. (9). To construct the gPC expansion KN (m), we employ the stochastic collocation method and Gauss-Patterson quadrature in Eqn. (9). The GaussPatterson quadrature is a type of Smolyak sparse grid whose relaxation on

225

boundary-value samples considerably reduces the overall number of samples required for convergence in high dimensional parameter space Nm [33, 34]. In Appendix Appendix C, we provide more details about the Gauss-Patteson quadrature rule and the free numerical codes [35] are employed to compute the integral (9) in subsequent examples. Without loss of generality, readers could

230

also replace the Gauss-Patterson quadrature with other quadrature rules. 12

4.1. Two-dimensional structure We start by generating 4000 binary images of 400×400 pixels (grids) with the QSGS code (Fig. 2). For a two-dimensional structure with specified porosity (m1 ), one needs to compute the other two Minkowski functionals (boundary length and Euler characteristics) for each sample.

Figure 2:

A two-dimensional 400 × 400-pixel sample of porous media generated from the

QSGS code [31] with fixed porosity m1 = 0.65. 235

As shown in Fig. 3, we found that both m2 and m4 follow the Beta distributions: fm (m′ ; α, β) =

m′α−1 (1 − m′ )β−1 , B(α, β)

(15)

where fm (m′ ) is the marginal probability density function (PDF) of m, m′ is its possible values in the outcome space and B(·) denotes the Beta func240

tion; α and β are the distribution’s shape parameters and their values can be determined with maximum likelihood method. Our findings are verified with Kolmogorov−Smirnov (KS) test of 10% significance level, whose results, along with detailed values of the parameters distributions are listed in Table 2. We note here that the Beta distribution (15) corresponds to Jacobian polynomials.

245

13

(a)

′

m2 B(10.29, 10.25)

100

′

fm2 (m2 )

80

60

40

20

0

0.07

0.075

0.08

0.085

0.09

0.095

0.1

0.105

′

m2

(b)

′

m4 B(10.35, 10.26)

1,250

′

fm4 (m4 )

1,000

750

500

250

0

4.5

5

5.5 ′

m4

6

6.5 ×10-3

Figure 3: Probabilistic density functions (PDFs) of (a) boundary length fm2 (m′2 ; 10.29, 10.25); and (b) Euler characteristics fm4 (m′4 ; 10.35, 10.26), computed from 4000 pore samples of 400× 400 cells with porosity m1 = 0.65.

Minkowski Functionals

Distribution Type

Interval

KS-test

m2

B(10.29, 10.25)

[0.0040, 0.0069]

0.279

m4

B(10.35, 10.26)

[0.0661, 0.1061]

0.406

Table 2: Statistics of the Minkowski functionals (m2 , m4 ) from 4000 pore samples of 400 × 400 cells.

14

For the case of two-dimensional porous media, we employ Gauss-Patterson quadrature of level l = 3 and use Nq = 17 quadrature nodes to construct a two-variate (Nm = 2) fifth order gPC expansion (N = 5): KN (m) =

5 X

a(i2 ,i4 ) φi2 (m2 ) φi4 (m4 ).

(16)

i2 +i4 =0

Figure 4 presents the permeability PDFs computed from the fifth order gPC model and 4000 MCS realizations. It is clear that the gPC expansion provides a good match to the MCS result. The mean and variances of the permeability are found to be 0.4309 and 0.022774 for the gPC method, and 0.4312 and 0.024088 250

for MCS, respectively. However, only 17 samples are required for the gPC model to reach the same accuracy as 4000 realizations of MCS. 3.5 3

MCS GPC

′

fK (K )

2.5 2 1.5 1 0.5 0 0

0.2

0.4

0.6

K

0.8

1

′

Figure 4: The permeability PDF fK (K ′ ), computed from 4000 MCS realizations and those from the fifth order gPC expansion with Gauss-Patterson quadrature of level 3, respectively.

4.2. Three-dimensional structure The performance of our UQ framework is further tested for three-dimensional porous medium, which is more practical in applications. Again, 4000 samples of 255

size 200 × 200 × 200 voxels (cells) are generated using the QSGS code (Fig. 5).

15

Figure 5: A three-dimensional 200 × 200 × 200-pixel sample of porous media generated from the QSGS code [31] with porosity m1 = 0.65.

Figure 6 shows that statistics of the three Minkowski functionals (surface area, mean curvature and Euler characteristics) are found to resemble the Beta distribution (15), e.g. m2 ∼ Beta(m′2 ; α2 , β2 ), m3 ∼ Beta(m′3 ; α3 , β3 ), m4 ∼

Beta(m′4 ; α4 , β4 ), verified with Kolmogorov Smirnov (KS) test of 10% signif-

260

icance level. Detailed values of the distribution and their KS test results are shown in Table 3. Minkowski Functionals

Distribution Type

Interval

KS-test

m2

B(6.17, 6.18)

[0.3055, 0.3165]

0.2534

m3

B(10.40, 10.41)

[−0.0255, −0.0188]

0.3918

m4

B(13.92, 13.92)

[−0.0018, −0.0007]

0.1347

Table 3: Statistics of the Minkowski functionals (m2 , m3 , m4 ) from 4000 pore samples of 200 × 200 × 200 cells.

For the case of three-dimensional porous media, we employ Gauss-Patterson quadrature of level l = 4 and use Nq = 111 quadrature nodes to construct a three-variate Nm = 3 fourth order gPC expansion (N = 4): KN (m) =

4 X

a(i2 ,i3 ,i4 ) φi2 (m2 ) φi3 (m3 ) φi4 (m4 ).

i2 +i3 +i4 =0

16

(17)

(a)300

′

m2 B(6.17, 6.18)

200

′

fm2 (m2 )

250

150 100 50 0

0.306

0.308

0.31

0.312

0.314

0.316

′

m2

(b)

′

m3 B(10.40, 10.41)

600

′

fm3 (m3 )

500 400 300 200 100 0 -0.025

-0.024

-0.023

-0.022

-0.021

-0.02

′

m3

(c) 6,000

′

m4 B(13.92, 13.92)

4,000

′

fm4 (m4 )

5,000

3,000 2,000 1,000 0 -1.6

-1.5

-1.4

-1.3

-1.2 ′

m4

-1.1

-1 ×10-3

Figure 6: Probabilistic density functions (PDFs) of (a) surface area fm2 (m′2 ; 6.17, 6.18); (b) mean curvature fm3 (m′3 ; 10.40.10.41) and (c) Euler characteristics fm4 (m′4 ; 13.92, 13.92),

17× 200 cells with porosity m1 = 0.65. computed from 4000 pore samples of 200 × 200

The permeability PDFs fK (K ′ ), computed from our framework and 4000 MCS realizations, respectively, are illustrated in Fig. 7. One can see that there is a good agreement between the two PDF results. We further note that the 265

first two statistical moments of K are 5.7112 × 10−4 and 3.4334 × 10−9 from the gPC expansion, while they are 5.822 × 10−4 and 3.9310 × 10−9 from MCS.

Again, for the same level of accuracy, the gPC method reaches with much less samples, only 111 quadrature points. 8000

MCS GPC

′

fK (K )

6000

4000

2000

0 3.5

4

4.5

5

5.5

6

K

′

6.5

7

7.5

8 ×10-4

Figure 7: The permeability PDF fK (K ′ ), computed from 4000 MCS realizations and that from the fourth order gPC expansion with Gauss-Patterson quadrature of level 4, respectively.

We note here that for a desired accuracy, a rise in the number of random 270

Minkwoski functionals (from two to three) would lead to an increment in the number of realizations (from 17 to 111). This is expected since the gPC approach relies on Karhunen-Loeve expansion and would require higher orders and thus more terms to be truncated should the number of random variables increases.

5. Conclusion 275

We developed a novel approach of the gPC expansion-based uncertainty quantification method for the analysis of macroscopic property of porous materials. By employing the Minkowski functionals to describe the random structure 18

of the pore, our framework is capable of constructing a polynomial chaos expansion of the property in the probability spaces of those functionals. The accuracy 280

and robustness of solutions of our method were investigated via the PDF of one such macroscopic property, the permeability. Our analysis leads to the following conclusions: • The proposed approach provides an alternative uncertainty quantification framework to obtain the statistical information of a random porous media

285

property. • At the same accuracy level, the new method requires much less samples (an order of magnitude) than Monte Carlo simulations and thus significantly reduces the overall computational cost. • The proposed approach would become less advantageous if the number of

290

random variables describing the porous structure increases. • By computing the macroscopic property’s PDF, our framework goes beyond conventional uncertainty quantification tools that focus on obtaining the mean and variances, and enables one to perform probabilistic risk assessment often defined by rare events, e.g. the probability distribution

295

tails. • Consisting of Minkowski functionals and generalized polynomial chaos method, the general framework could also be further extended to obtain the full statistical information of the random porous media’s other macroscopic properties, such as the reaction rate if considering the transport

300

processes.

Acknowledgement P. Wang, H. Chen and X. Jiang were partially funded by National Key Research and Development Program of China (Grant No. 2017YFB0701702), the National Natural Science Foundation of China (Grant No. 11571028) and 19

305

the Recruitment Program of Global Experts; X. Meng and X. Yang were funded by the Special program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) and the Recruitment Program of Global Experts.

Appendix A. Minkowski functionals for random porous samples 310

In a binary image of porous media sample, we first identify each pixel (twodimensional system) or voxel (three-dimensional system) x as pore (0) or solid (1). The pore space is collectively denoted as S w . The Minkowski functionals can be computed using the Ohser-M¨ ucklin estimator [22]. Appendix A.1. Minkowski functionals for two-dimensional sample

315

For a 2D binary image, integral geometry provides three Minkowski functionals, porosity, surface area and the Euler characteristic [17]. As shown in Fig. A.8 (a), a lattice cell of 2 × 2 = 4 pixels can be identified for each location

within a 2D image. Hence there are 24 = 16 possible configurations for each lattice and the configuration in (x0 = 0, x1 = 1, x2 = 1, x3 = 0) Fig. A.8 (a) is 320

denoted with binary coding q(0110) = q(6). To compute porosity, a lattice cell is counted if it contains one pore 0 out of the four vertices, while the other three are accounted for neighbouring cells. m1 = where Nc =

P15

i=0 Iq (i)

7 1 X Iq (2i + 1), Nc i=0

(A.1)

is the total number of lattice cells and Iq (i) is the

indicator function. Iq (i) =

  0,

 1,

configuration i is solid phase

(A.2)

configuration i is pore

The surface area can be computed as: m2 =

7 15 π X 1 X Iq (i) I{xg1(ν) ∈S w } I{xg2(ν) ∈S / w}, 8Nc ν=0 rν i=0

20

(A.3)

Figure A.8: (a) A 2D lattice cell with spacing h and configuration q(0110) = q(6); (b) the prominent directions in a 2D quadratic-lattice.

where ν denotes the eight prominent directions dictated by a 2D-quadratic lattice, as shown in Fig. A.8 (b). xg1(ν) and xg2(ν) is the initial and end vertex of direction ν, respectively, while rν = |xg2(ν) − xg1(ν) | stands for their Euclidean distance. I{f (·)} represents the indicator function which equals one if 325

the function f (·) holds true. Finally, we take h as the lattice spacing and calculate the Euler characteristics as " 3  # 3 X X 1 m4 = Iq (2i ) − Iq (15 − 2i ) + 2 Iq (6) + Iq (9) . 4Nc h2 i=0 i=0

(A.4)

Appendix A.2. Minkowski functionals for three-dimensional sample 330

For a 3D binary image, our basic lattice cell becomes a 2 × 2 × 2 = 8 cube

with 28 = 256 possible configurations. Again, we use binary coding for each configurations, such as Fig. A.9 (a) in which q(11100110) = q(103).

21

The first three Minkowski functionals can be computed as below: m1

=

127 1 X Iq (2i + 1), Nc i=0

25 255 4 X cν X Iq (i) I{xg1(ν) ∈S w } I{xg2(ν) ∈S / w}, Nc ν=0 rν i=0

m2

=

m3

= 2π

25 X

ν=0

cν P ν .

(A.5) (A.6) (A.7)

Here ν now represents the twenty-six prominent directions (three cell edges, six 335

face diagonals and four spatial diagonals in both directions) for a 3D lattice cell in Fig. A.9 (b); cν is the weight of corresponding voronoi cell associated with each direction.

Figure A.9: (a) A 3D lattice cell with spacing h and configuration q(11100110) = q(103); (b) half the rose of relevant directions in a 3D lattice cell.

We further note that Pν denotes the contribution from Euler characteristics: If ν is the cell edge direction in Fig. A.10 (a); " 255 X 1 Iq (i) I{xp1(ν) ∈S w } I{xp2(ν) ∈S Pν = / w } I{xp3(ν) ∈S / w } I{xp4(ν) ∈S / w} 2 Nc h i=0 −

255 X

#

Iq (i) I{xp1(ν) ∈S w } I{xp2(ν) ∈S w } I{xp3(ν) ∈S w } I{xp4(ν) (A.8a) ∈S / w}

i=0

22

340

If ν is the spatial diagonal direction in Fig. A.10 (b); " 255 X 1 Pν = √ Iq (i) I{xp1(ν) ∈S w } I{xp2(ν) ∈S / w } I{xp3(ν) ∈S / w } I{xp4(ν) ∈S / w} 2 2Nc h i=0 −

255 X

Iq (i) I{xp1(ν) ∈S w } I{xp2(ν) ∈S w } I{xp3(ν) ∈S w } I{xp4(ν)(A.8b) ∈S / w}

i=0

If ν is the face diagonal direction in Fig. A.10 (c); " 255 X 1 Pν = √ Iq (i) I{xp1(ν) ∈S w } I{xp2(ν) ∈S / w } I{xp3(ν) ∈S / w} 3Nc h2 i=0 −

#

255 X

#

Iq (i) I{xp1(−ν) ∈S w } I{xp2(−ν) ∈S w } I{xp3(−ν) ∈S . / w }(A.8c)

i=0

where xp1(ν) , xp2(ν) , xp3(ν) , xp4(ν) denotes the vertices of the plane (or triangles) to which direction ν belongs.

Figure A.10: An example of the prominent planes to estimate contributor Pν in a 3D lattice cell: (a) a cuboidal face; (b) a spatial diagonal rectangle; (c) two face diagonal triangles.

Finally, the fourth Minkowski function, the Euler characteristics, is com-

23

345

puted as m4

=

255  1 X Iq (i) · I{x0 ∈S w } − I{x0 ∈S w } I{x1 ∈S w } − I{x0 ∈S w } I{x2 ∈S w } 3 Nc h i=0

− I{x0 ∈S w } I{x4 ∈S w } + I{x0 ∈S w } I{x1 ∈S w } I{x2 ∈S w } I{x3 ∈S w }

+ I{x0 ∈S w } I{x2 ∈S w } I{x4 ∈S w } I{x6 ∈S w } + I{x0 ∈S w } I{x1 ∈S w } I{x4 ∈S w } I{x5 ∈S w }  − I{x0 ∈S w } I{x1 ∈S w } I{x2 ∈S w } I{x3 ∈S w } I{x4 ∈S w } I{x5 ∈S w } I{x6 ∈S w } . (A.9) Appendix B. Quartet structure generation set The quartet structure generation set (QSGS) method is an iterative procedure to generate porous media samples: 1. Locate the cores of the growing phase in the grid system randomly with a 350

core distribution probability, Pd , which is smaller than the volume fraction of the solid phase. Each cell in the grid will be assigned a random number by a uniform distribution function from 0 to 1. Each cell whose random number is smaller than Pd will be specified as a core. 2. Enlarge every element of the existing solid phase to its neighbouring cells in

355

all directions based on each given directional growth probability, Pi , where i represents the direction. Again for each growing element, new random numbers will be assigned to its neighbouring cells. The neighbouring cell in direction i will becomes part of the solid phase if its random number is no greater than Pi .

360

3. Repeat step (2) until the volume fraction of the solid phase reaches its given value. Interested readers can refer to earlier studies [31, 32] for more details.

Appendix C. Gauss-Patterson quadrature rule The Gauss-Patterson quadrature is a nested sparse grid which requires only a subset of samples of the full tensor grids. Based on Smolyak’s formula, one

24

can express the integral of a multi-variate function fm (m′1 , . . . , m′Nm ):     X N − 1 m Nm  Q1k ⊗ · · · ⊗ Q1k f |nq , (−1)l+Nm −|k|−1  Ql f = Nm 1 |k| − l l≤|k|≤l+Nm −1 (C.1) m where QN represents the Gauss-Patterson quadrature operator at level l ∈ N0 l 365

for Nm number of random parameters. k = (k1 , . . . , kNm ) is a multi-index, |k| = k1 + · · · + kNm and Q1kNm is the Gauss-Patterson quadrature operator at level kNm for the Nm -th random variable. The tensor product operator is denoted by the symbol ⊗. We note here that the precision of a quadrature operator is determined by its quadrature level.

370

In summary, the Gauss-Patterson quadrature at high-dimensional paramm eters space QN is determined by its counterparts at one-dimensional space l

Q1kNm . Their expression and precision at each quadrature level have been well studied and documented. In this paper, we employ the numerical codes [35] to compute the integral (9).

375

References [1] J. Bear, Dynamics of Fluids in Porous Media (Elsevier, New York, 1972). [2] M. J. Blunt, Multiphase Flow in Permeable Media: A Pore-Scale Perspective (Cambridge University Press, 2017). [3] M. J. Blunt, B. Bijeljic, H. Dong, et al., Adv. Water Resour. 51, 197-216

380

(2013). [4] S. Molins, Rev. Mineral. Geochem. 80, 461-481 (2015). [5] X. Yang, Y. Mehmani, W. A. Perkins, et al., Adv. Water Resour. 95, 176-189 (2016). [6] T. Scheibe, E. Murphy, X. Chen, et al., Groundwater 53, 38-56 (2015).

385

[7] Y. Al-Khulaifi, Q. Lin, M. J. Blunt, et al., Environ Sci. Technol. 51, 41084116 (2017). 25

[8] D. Wildenschild, A. P. Adrian and P. Sheppard, Adv. Water Resour. 51, 217-246 (2013), [9] J. P. Pereira Nunes, M. J. Blunt, and B. Bijeljicm, J. Geophys. Res. Solid 390

Earth 121, (2016). [10] X. Yang, T. D. Scheibe, M. C. Richmond, et al., Adv. Water Resour. 54, 228-241 (2013). [11] Y. Efendiev, C. Kronsbein, and F. Legoll, Multiscale Model. Simul. 13, 11071135 (2015).

395

[12] X. Blanc, C. Bris, and F. Legoll, Phil. Trans. R. Soc. A 374, (2016). [13] M. Icardi, G. Boccardob, and R. Tempone, Adv. Water Resour. 1, 123-456 (2016). [14] D. M. Tartakovsky, Geophys. Res. Lett. 34, L05404 (2007). [15] C. Bierig and A. Chernov, J. Comput. Phys. 314, 661-681 (2016).

400

[16] M. Giles, T. Nagapetyan and K. Ritter, SIAM/ASA J. Uncertainty Quantification, 3, 267-295 (2015). [17] K. R. Mecke, Statistical Physics and Spatial Statistics (Springer, 2000). [18] K. R. Mecke and D. Stoyan, Morphology and Condensed Matter (Springer 2002).

405

[19] P. Lehmann, M. Berchtold, B. Ahrenholz, et al., Adv. Water Resour. 31, 1188-1204 (2008). [20] M. Berchtold, Ph.D thesis, Swiss Federal institute of technology Zurich, 2008. [21] H. J. Vogel, U. Weller and S. Schluter, Computers and Geosciences, 36,

410

1236-1245 (2010).

26

[22] J. Osher and F. M¨ ucklich, Statistical analysis of microstructures in materials science (J. Wiley & Sons, 2000). [23] D. Xiu and G. Karniadakis, SIAM J. Sci. Comput. 24, 619-644 (2002). [24] R. Ghanem and P. Spanos, Stochastic Finite Elements: a Spectral Ap415

proach (Springer-Verlag, 1991). [25] D. Xiu, Numerical methods for stochastic computations (Princeton University Press, 2010). [26] W. Schoutens, Stochastic Processes and orthogonal polynomials (SpringerVerlag 2000).

420

[27] R. Koekoek and R. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue (Tech. Report 98-05, 1998). [28] L. Mathelin and M. Hussaini, A stochastic collocation algorithm for uncertainty analysis (Tech. Report NASA/CR-2003-212153, 2003). [29] L. Giraldi, A. Litvinenko, D. Liu, et al., SIAM J. Sci. Comput. 36, A2720-

425

A2744 (2014). [30] X. Meng and Z. Guo, Phys. Rev. E 92, 043305 (2015). [31] M. Wang, J. K. Wang, N. Pan, et al., Phys. Rev. E 75, 036702 (2007). [32] M. Wang and N. Pan, J. Comput. Phys. 228, 5978-5988 (2009). [33] T. Gerstner and M. Griebel, Numer. Alg. 18, 209-232 (1998).

430

[34] E. Novak and K. Ritter, Constructive Approx. 15, 499-522 (1999). [35] J. Burkardt, http://people.sc.fsu.edu/jburkardt/m_src/patterson_rule/patterson_rule.html.

27