Towards realistic flow modelling. Creation and evaluation of two-dimensional simulated porous media: An image analysis approach

Abstract

This work is part of an attempt to quantify the relationship between the permeability tensor (K) and the micro-structure of natural porous media. A brief account is first provided of popular theories used to relate the micro-structure toK. Reasons for the lack of predictive power and restricted generality of current models are discussed. An alternative is an empirically based implicit model whereinK is expressed as a consequence of a few “pore-types” arising from the dynamics of depositional processes. The analytical form of that implicit model arises from evidence of universal association between pore-type and throat size in sandstones and carbonates. An explicit model, relying on the local change of scale technique is then addressed. That explicit model allows, from knowledge of the three-dimensional micro-geometry to calculateK explicitly without having recourse to any constitutive assumptions. The predictive and general character of the explicit model is underlined. The relevance of the change of scale technique is recalled to be contingent on the availability of rock-like three-dimensional synthetic media. A random stationary ergodic process is developed, that allows us to generate three-dimensional synthetic media from a two-dimensional autocorrelation functionr(λ _x ,λ _y ) and associated probability density function∈ _β measured on a single binary image. The focus of this work is to ensure the rock-like character of those synthetic media. This is done first through a direct approach:n two-dimensional synthetic media, derived from single set (∈ _β ,r(λ _x ,λ _y )) yieldn permeability tensorsK ⁱ _i-1,n (calculated by the local change of scale) of the same order. This is a necessary condition to ensure thatr(λ _x ,λ _y ) and∈ _β carry all structural information relevant toK. The limits of this direct approach, in terms of required Central Process Unit time and Memory is underlined, raising the need for an alternative. This is done by comparing the pore-type content of a sandstone sample andn synthetic media derived fromr(λ _x ,λ _y ) and∈ _β measured on that sandstone-sample. Achievement of a good match ensures that the synthetic media comprise the fundamental structural level of all natural sandstones, that is a domainal structure of well-packed clusters of grains bounded by loose-packed pores.