Modeling 3-D anisotropic fractal media

Introduction

Our understanding of the physical phenomena occurring in the earth always involves the study of the medium itself. Unfortunately, the earth offers an unusually complicated medium in which heterogeneities are observed at every scale. Sometimes the problem is too difficult to deal with deterministically but it turns out to be quite simply treated by statistical methods. Solutions to the one-dimensional problem have targeted the study of reflectivity series obtained from well logs. Seismic impedance can be modeled as a special type of Markov chain, one which is constrained to have a purely exponential correlation function (Godfrey et al., 1980). The two-dimensional problem has gained a lot of attention in the recent years from studies of seismic scattering in heterogeneous media, e.g., (Wu and Aki, 1985; Holliger et al., 1993; Goff and Jordan, 1988; Frankel and Clayton, 1986; Holliger and Levander, 1992). Three dimensional simulations can be used in fluid flow experiments (Popovici and Muir, 1989).

This paper presents a method for simulating three-dimensional anisotropic random fields using second order-statistics. The method was introduced by Goff and Jordan (1988) to model a two-dimensional seafloor morphology. I have considered the cases of random media characterized by Gaussian, exponential and von Karman correlation functions. I use the von Karman functions as a generalizations to the exponential correlation functions in modeling random sequences. This type of correlation function was first introduced by von Karman (1948) for characterizing the random velocity field of a turbulent medium. It has since been frequently used in the statistical literature, studies of turbulence problems, e.g.(Tatarski, 1961), and studies of random media such as wave scattering, e.g.(Chernov, 1960). The von Karman functions were identified specifically as belonging to the class of continuous correlation functions (Matern, 1970). Holliger et al. (1993) used von Karman covariance functions to model binary fields and defined ``binarization'' as a mapping of all values in a continuous field to just two values of the new field. I have employed their technique to model two state models (i.e, rock/pore or sandstone/shale) from continuous realizations and test the increase in medium roughness through the ``binarization'' process.

Modeling 3-D anisotropic fractal media