Uncertainty quantification on the macroscopic properties of heterogeneous porous media

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 efforts to quantify the parametric uncertainty over those macroscopic properties. To address this challenge, we propose a framework to compute the probabilistic density function (PDF) of the macroscopic property based on the generalized polynomial chaos expansion method and the Minkowski functionals. To illustrate the effectiveness of our approach, we conduct numerical experiments for one macroscopic property, namely the permeability, and we 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.

Figure 1

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

Figure 2

A two-dimensional $400\times 400$ pixel sample of porous media generated from the QSGS code [31] with fixed porosity $m_1=0.65$.

Figure 3

PDFs of (a) boundary length $f_{{\mathrm{m}}_2}(m_2^{\prime };10.29,10.25)$ and (b) Euler characteristics $f_{{\mathrm{m}}_4}(m_4^{\prime };10.35,10.26)$, computed from 4000 pore samples of $400\times 400$ cells with porosity $m_1=0.65$.

Figure 4

The permeability PDF $f_{\mathrm{K}}\left(K^{\prime }\right)$, computed from 4000 MCS realizations and those from the fifth-order gPC expansion with Gauss-Patterson quadrature of level 3, respectively.

Figure 5

A three-dimensional $200\times 200\times 200$ pixel sample of porous media generated from the QSGS code [31] with porosity $m_1=0.65$.

Figure 6

PDFs of (a) surface area $f_{m_2}(m_2^{\prime };6.17,6.18)$, (b) mean curvature $f_{m_3}(m_3^{\prime };10.40.10.41)$, and (c) Euler characteristics $f_{m_4}(m_4^{\prime };13.92,13.92)$, computed from 4000 pore samples of $200\times 200\times 200$ cells with porosity $m_1=0.65$.

Figure 7

The permeability PDF $f_{\mathrm{K}}\left(K^{\prime }\right)$, computed from 4000 MCS realizations and that from the fourth-order gPC expansion with Gauss-Patterson quadrature of level 4, respectively.

Figure 8

(a) A 2D lattice cell with spacing $h$ and configuration $q\left(0110\right)=q\left(6\right)$; (b) the prominent directions in a 2D quadratic lattice.

Figure 9

(a) A 3D lattice cell with spacing $h$ and configuration $q\left(11100110\right)=q\left(103\right)$; (b) half the rose of relevant directions in a 3D lattice cell.

Figure 10

An example of the prominent planes to estimate contributor ${\mathcal{P}}_{\nu }$ in a 3D lattice cell: (a) a cuboidal face; (b) a spatial diagonal rectangle; (c) two face diagonal triangles.