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GENERATING TWO-STATE MODELS

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.

layered
layered
Figure 6.
Synthetic continuous random field with apparent layering and Gaussian autocorrelation; $a_x=5$, $a_y=80$, $a_z=80$.
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lay-bin
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
isotropic
Figure 8.
Synthetic continuous random field with isotropic Gaussian autocorrelation function; $a_x=15$, $a_y=15$, $a_z=15$.
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iso-bin
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
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
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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next up previous [pdf]

Next: CONCLUSIONS Up: Chemingui: Fractal media Previous: Modeling seismic impedances

2013-03-03