Statistical reconstruction of three-dimensional porous media from two-dimensional images

Abstract

A method of modeling the three-dimensional microstructure of random isotropic two-phase materials is proposed. The information required to implement the technique can be obtained from two-dimensional images of the microstructure. The reconstructed models share two-point correlation and chord-distribution functions with the original composite. The method is designed to produce models for computationally and theoretically predicting the effective macroscopic properties of random materials ͑such as electrical and thermal conductivity, permeability and elastic moduli͒. To test the method we reconstruct the morphology and predict the conductivity of the well known overlapping sphere model. The results are in very good agreement with data for the original model. ͓S1063-651X͑97͒02309-X͔ PACS number͑s͒: 47.55. Mh, 44.30.ϩv, 81.05.Rm, 61.43.Bn Predicting the macroscopic properties of composite or porous materials with random microstructures is an important problem in a range of fields ͓1,2͔. There now exist largescale computational methods for calculating the properties of composites given a digital representation of their microstructure ͑e.g., permeability ͓3,4͔, conductivity ͓3-5͔, and elastic moduli ͓6͔͒. A critical problem is actually obtaining an accurate three-dimensional description of this microstructure ͓3,7,8͔. For particular materials it may be possible to simulate microstructure formation from first principles. Generally this relies on a detailed knowledge of the physics and chemistry of the system, the accurate modeling of each material requiring a significant amount of research. Where such information is unavailable an alternative is to directly ͓9-15͔ or statistically ͓3,4,8,16-21͔ reconstruct the microstructure from experimental images.Several techniques of direct reconstruction have been implemented. A composite can be repeatedly sectioned and imaged, and the results combined to reproduce a threedimensional digital image of the microstructure ͓9-11͔. For porous materials, time-consuming sectioning can be avoided by using laser microscopy ͓12͔ which can image pores to depths of around 150 m. Recent microtomography studies have also directly imaged the three-dimensional microstructure of porous sandstones ͓13,14͔ and magnetic gels ͓15͔. The complexity and restrictions of these methods provide the impetus to study alternative reconstruction methods.Based on the work of Joshi ͓16͔, Quiblier ͓17͔ introduced a method of generating a three-dimensional statistical reconstruction of a random composite. The method is based on matching statistical properties of a three-dimensional model to those of a real microstructure. A key advantage of this approach is that the required information can be obtained from a two-dimensional image of the sample. Recently the method was applied to the reconstruction of sandstone ͓4,8,18,19͔ and a material composed of overlapping spheres ͓3͔. Computations of the permeability and conductivity ͓3,4,18͔ of the reconstructed images underestimate experimental data by around a factor of ...

Characterising the Morphology of Disordered Materials

“…A number of statistical models have been proposed for reconstructing porous media from statistical information [1,9,20,41,42,44,45]. These methods, based on different underlying model microstructures, are generated in such a manner that they match the observed two-point statistical properties of the rock (see p. 93 in [11]).…”

“…One system gave an excellent match to the global Minkowski functionals of the sandstone samples. Here the three different Boolean ensembles (IOS C , ROS (2) , and OSC) and a fourth stochastic model for Fontainebleau sandstone, based on a Gaussian reconstruction given by 5 intersecting 1-level-cut Gaussian models [44], are compared for the accuracy of reconstruction by evaluating the Minkowski functions. The Gaussian kernel used for the intersection reconstruction is given by the Fourier transform pair g(x) and ρ(k) = −F (k)/(4πk) with…”

“…The three length scale parameters are obtained by a best fit procedure to minimize the non-linear least squares error [44],…”

“…To generate matching Gaussian random field (GRF) models we employ the field-field correlation function [44,57] …”

Multi-scale fracture of random heterogeneous materials

“…Many natural and man-made heterogeneous materials have a random internal structure comprised of two or more phases. There are two challenging problems in the study of a heterogeneous material: (1) modelling the microstructure of the material (Quintanilla and Torquato 1997;Roberts 1997;Grigoriu 2003;Rahman 2008); and (2) predicting the effective properties, such as transport, thermal, mechanical properties (Ferrante and Graham-Brady 2005;Rahman and Chakraborty 2007), including fracture performance (Chakraborty and Rahman 2008), from this microstructural information. Although research in solving these problems has exploded in recent years, fundamental progress in this field has been slow, if not hampered, due to, first, a lack of rigorous mathematical models capable of characterising complex microstructures.…”