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![]() | Modeling 3-D anisotropic fractal media | ![]() |
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In the geophysical world we often deal with heterogeneous media whose inhomogeneities are caused by the presence of two different types of material with different mechanical properties. A typical example is the case of a stratified formation of shale embedded in sandstone. In fluid flow and reservoir engineering problems, the rock samples are generally composed of a matrix and pore space. Continuously random fields are therefore inadequate to describe randomness in similar settings. I seek to describe a random field in which the medium can be represented as a two-state model. This new field is called a binary field and the process of deriving the binary field from the continuous field is called ``binarization'' (Holliger et al., 1993). The problem is to relate the statistics of the binary field to those of the continuous field. Holliger et al. (1993) gave a brief description of their mapped two-dimensional binary field which I apply in a straightforward generalization to the three-dimensional problem.
To illustrate the effects of ``binarizing'' a continuous field, let's consider two examples of random fields with Gaussian and exponential autocorrelation functions, respectively. In the first example I simulate a randomly-stratified medium. The second example is a realization of a random medium with statistically isotropic homogeneous inclusions. I like to analyze the change in the medium properties by comparing the autocorrelation function of the distribution before and after ``binarization''. For better observation, I limit the analysis to the study of the correlation function along one axis, i.e, in the x-direction.
Figure shows the averaged 1-D correlation
function along the x-axis
for the randomly layered medium. The solid curve displays the autocorrelation
of the continuous field; the dashed one represents the autocorrelation
of the ``binary'' field. The two functions are noticeably different from one
another; the slope near the origin is greatly increased after ``binarization''
indicating
a rougher distribution compared to the continuous case. Figure
shows
the same observations for the isotropic random field with Gaussian
autocorrelation; again the roughness
of the field has increased as indicated by the steepening in the slope of the
autocorrelation.
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layered
Figure 6. Synthetic continuous random field with apparent layering and Gaussian autocorrelation; ![]() ![]() ![]() |
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lay-bin
Figure 7. Synthetic binary field derived from the continuous realization of a layered random field with Gaussian autocorrelation. |
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isotropic
Figure 8. Synthetic continuous random field with isotropic Gaussian autocorrelation function; ![]() ![]() ![]() |
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iso-bin
Figure 9. Synthetic binary field derived from the continuous realization of a random field with Gaussian autocorrelation. |
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lay-auto
Figure 10. Autocorrelation functions of the continuous (solid lines) and binary (dashed lines) fields for the layered random medium with exponential correlation. |
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iso-auto
Figure 11. Autocorrelation functions of the continuous (solid lines) and binary (dashed lines) fields for the isotropic random medium with Gaussian correlation |
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![]() | Modeling 3-D anisotropic fractal media | ![]() |
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