Is It Possible to Characterize the Geometry of a Real Porous Medium by a Direct Measurement on a Finite Section? 1: The Phase-Retrieval Problem

Abstract

Macroscopic transport properties of natural porous media, such as the permeability tensor \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{K} }\), are the sum of uncountable microscopic events. Understanding how these microscopic events come together to yield a descriptive property, such as \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{K} }\), is facilitated if a set of porous descriptors Pii=1,N can be measured that synthesize all the critical microscopic properties of a real porous medium and that can be related to the computed \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{K} }\). The main difficulty lies in choosing the correct Pii=1,N. A description of the microgeometry of a porous medium by a set of Pii=1,N will be declared adequate if synthetic numerical porous media generated from the Pii=1,N possess the same \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{K} }\) as the real medium. This paper is another step for creating such synthetic porous media. The candidate geometrical descriptor P1 considered herein is the autocorrelation function (ACF) measured directly on a sample of finite size v included in a porous medium of size V; V ≫ v. Capitalizing on the phase retrieval problem (retrieving an object from knowledge of its Fourier modulus) encountered in general imaging, it is shown that there exists a one-to-one relation between a digital thin section and its ACF. This is demonstrated using an iterative procedure, the Error Reduction/Hybrid Input Output algorithm, that allows one to recover uniquely, to within a pixel, a finite image from its ACF. A theoretical implication of this is that a direct measurement on a finite image cannot characterize the geometry of a porous medium. Yet, in stochastic modeling, quasi-infinite numerical porous media are commonly generated from acquired ACF. Such quasi-infinite stochastic porous media must include a structural noise as a practical consequence of the unicity of the relation between an image and its ACF. To correctly interpret the relation between \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{K} }\) and the microgeometry, it becomes necessary to verify that this nonimposed structural noise does not control the output \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{K} }\) of the numerical simulations.