Summary
The main purpose of the present work is to predict the permeability of a porous medium from its three-dimensional (3D) porous structure network. In this work, 3D porous structure is reconstructed by the truncated Gaussian method using Fourier transform and starting from a 2D binary image obtained from a thin section of a porous sample. The skeleton of the 3D porous structure provides a way of visualizing the graph of the pore network. It is determined using a thinning algorithm, which is conceived to preserve topology. It gives both visual and quantitative information about the connectivity of the pore space, the coordination number for every node and local hydraulic radius. Once the network of the pore structure is obtained, the macroscopic transport properties, such as the permeability, can be predicted. The method is applied to a 500 mD Berea sandstone and the predicted permeability is in good agreement with the experimental value and empirical correlations.
Introduction
The prediction of equilibrium and transport properties of porous media is a long-standing problem of great theoretical and practical interest, particularly in petroleum reservoir engineering.1 Past theoretical attempts to derive macroscopic transport coefficients from the microstructure of porous media entailed a simplified representation of the pore space, often as a bundle of capillary tubes.1–3 These models have been widely applied because of their convenience and familiarity to the engineers. But they do have some limitations. For example, they are not well suited for describing the effect of the pore space interconnectivity and long range correlation in the system. Network models have been advanced to describe phenomena at the microscopic level and have been extended in the last few years to describe various phenomena at the macroscopic level. These models are mostly based on a network representation of the porous media in which larger pores (pore bodies) are connected by narrower pores (pore throats). Network models represent the most important and widely used class of geometric models for porous media.2 A network is a graph consisting of a set of nodes or sites connected by a set of links or bonds. The nodes can be chosen deterministically or randomly as in the realization of a Poisson or other stochastic point process. Similarly the links connecting different nodes may be chosen according to some deterministic or random procedure. Finally, the nodes are dressed with convex sets such as spheres representing pore bodies, and the bonds are dressed with tubes providing a connecting path between the pore bodies. The original idea of representing a porous structure by a network is rather old, but it was only in the early 1980s that systematic and rigorous procedures were developed to map, in principle, any disordered rock onto an equivalent random network of bonds and sites. Once this mapping is complete one can study a given phenomenon in porous media in great detail.3 Dullien1 reviewed the details of various pore-scale processes, including detailed descriptions of many aspects of network models. The most important features of pore network geometry and topology that affect fluid distribution and flow in reservoir rocks are the pore throat and pore body size distributions, the pore body-to-pore throat size aspect ratio and the pore body coordination number.4 These data have been tentatively assumed in the previous works. The extension of these techniques to real porous media has been complicated by the difficulty in describing the complex three-dimensional (3D) pore structure of real porous rocks. Information about the pore structure of reservoir rocks is often obtained from mercury intrusion and sorption isotherm. Mercury intrusion and sorption isotherm data provide statistical information about the pore throat size distribution, or, more correctly, the distribution of the volumes that may be invaded within specified pore throat sizes. Advanced techniques such as microcomputed tomography5 and serial sectioning6,7 do provide a detailed description of the 3D pore structures of rocks.
Recently, image analysis methods used over pictures of highly polished surfaces of porous materials (e.g., Refs. 8-10), taken with an electron scanning microscope have been used to describe the porous structure. Image analysis techniques such as opening (2D and 3D)11,13 and median line graphs (2D)13 were developed. Information on porous structure is obtained from the analysis of 2D binary images. For isotropic media, a 3D microstructure may be reconstructed from any statistically homogeneous 2D section. The general objective of a reconstructed porous structure is to mimic more closely the geometry of real media. This method has been previously applied to the prediction of important petrophysical and reservoir engineering properties, such as permeability8 and formation factor14 with reasonable success. Thovert et al.15 used the reconstructed porous structure and developed thinning algorithms to obtain the graph of the 3D pore structure. Some topological characteristics such as the number of loops were derived. Bakke and O/ren16 generated 3D pore networks based on numerical modeling of the main sandstone forming geological processes. Absolute and relative permeability were computed for a Bentheimer sandstone. However, although their algorithms worked well on their models, the problem of connectivity preservation for a 3D thinning algorithm appears to be only correctly taken into account by Ma,17 who proposed sufficient conditions for providing a 3D thinning algorithm to preserve connectivity.
Statistics of highly heterogeneous flow fields confined to three-dimensional random porous media