MULTISCALE RELATIONSHIPS BETWEEN FRACTURE LENGTH...

... APERTURE, DENSITY AND PERMEABILITYby:Dr. Shlomo P. NeumanDepartment of Hydrology and Water ResourcesUniversity of Arizona, Tucson, Arizona 85721ABSTRACT:Fractured rocks exhibit a hierarchical structure which renders their attributes scale-dependent. In particular available data indicate a tendency for fracture length scales to be distributed according to a power law, average fracture aperture to be given by a power of the fracture length scale, and fracture density as well as log permeability to behave as random fractals. To date, no consistent theoretical relationship has been developed between fracture type (as categorized, for example, by length scale and/or aperture) and corresponding fractal attributes (such as density and log permeability). We explore multiscale relationships between these fracture categories and attributes on the basis of a theory recently proposed by Neuman (2003), which allows linking them in a formal way. Analyzing the available data in light of this theory allows us to demonstrate that, for fractures having length scale L, (a) the variance of any fractal attribute grows as a positive power of L, (b) the same variance decreases as a negative power of the smallest length scale sampled, (c) for nominal parameters that are most representative of values inferred from field data, the variance of fracture densities increases quadratically with L, rendering their standard deviation linearly proportional to L, (d) for such nominal parameters log permeability variance increases as , (e) for a given L the variance of log permeability is proportional to that of fracture density, the constant of proportionality being a (positive, zero or negative) power of L, and (f) the standard deviation of log permeability is proportional to a positive power of the average aperture, where . The underlying theory contains explicit expressions for the mean, variance, variogram and integral (spatial correlation) scale of a statistically anisotropic fractal attribute truncated by upper and lower length scale cutoffs and/or internal lacunae. The attribute may have a Gaussian distribution, in which case it forms fractional Brownian motion (fBm), or a zero-mean symmetric Levy stable distribution, in which case it forms fractional Levy motion (fBm), the latter distribution exhibiting heavier tails than does the former. Our expression for the mean attribute of a truncated hierarchy of fracture length scales, in terms of the mean attributes associated with individual scales, may yield meaningful representations of the overall density or porosity of a truncated fracture hierarchy. However, it generally does not yield equivalent or effective values of permeability. Instead, the latter are defined on the basis of equivalent or effective forms of Darcy's law the derivation of which typically requires simulating fluid flow through the hierarchy. We mention briefly the effective permeability of a box embedded in a hierarchical medium, and associated measures of uncertainty, develop by Di Federico et al. (1999) on the basis of the scaling theory on which this talk is partly based. Based on Neuman, S.P., Multiscale relationships between fracture length, aperture, density and permeability, Geophys. Res. Lett., L22402, doi:10.1029/2008GL035622, 2008.

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