Sedimentary Porous Materials as a Realization of a Stochastic Process

Abstract

Statistical variation in the properties of porous materials, including sedimentary rocks, has been recognized for over half a century. In early studies, determinations were made of measures of central tendency and variation in the sizes of pores and grains. Some work was directed to measuring probability density functions of pore- or grain-size distributions. The effort needed for accurate determination of such functions precluded their widespread use. Despite the early recognition of the statistical nature of porous materials, fluid-flow models of porous media continue to be based on simple deterministic models of spherical particle or capillary tube assemblages. Such models are inadequate for the study of fine details in pore-grain structure or fluid flow at the particulate level. These models are equally unsuited for studies of particle transport and sedimentation or for the characterization of sediment and sedimentary rocks. The models provide no basis for accurate lithologic description or analysis.

Recently, experimental and theoretical tools have been developed for the description of porous materials as a realization of a stochastic or random process. Methods of Fourier optics and optical data processing, both digital and analog, combined with the theory of random processes, constitute the basis for a new approach to analyzing and simulating mathematically porous materials.

Representation of porous material as a realization of a random process implies that each sample in a population will be different from others in a point-to-point sense, yet be similar in average properties. If the process that generates a given sample (a particular realization) is the same for three coordinate directions, the material is considered isotropic and homogeneous. The concept of statistical homogeneity thus is related to the equivalence of the random process for all coordinate directions. Similarly, it is possible to describe precisely trends in statistical parameters through long or short intervals by unequal variation in these random processes for the three coordinate directions. As a demonstration of the sufficiency of the parameters extracted from a porous material, it is possible to simulate mathematically a structure with all the attributes of a natural or synthetic porous material.

Fourier transforms of images of the pore-grain structure are useful for classifying pore structure numerically and for determining the pore- or grain-size distribution in a sample quickly, cheaply, and with precision. These same transforms permit, through optical filtering, the analysis and synthesis of “idealized” porous materials. Such analysis or synthesis can be done with optical image analysis equipment in either digital or analog form. The necessary equipment for optical data processing as a method of pore-structure or grain-size analysis is relatively inexpensive, although radically different from traditional measurement devices.