Frequency dependent P-wave speed in a porous medium with aligned fractures

Abstract

In an elastic medium, fractures can be modeled as thin layers, the elastic stiffnesses of which approach zero as the volume fraction of the fractures hf→0. This yields linear slip theory [M. Schoenberg, J. Acoust. Soc. Am. 68, 1516–1521 (1980)], shown to be a robust way to account for the acoustic effect of fracturing. From Norris’ [J. Acoust. Soc. Am. 94, 359–370 (1993)] dispersion relation for alternating porous layers, fractures must be modeled similarly in porous media. However, another fracture parameter of great interest is permeability. If fracture permeability is taken to be O(hf−1) or taken to be independent of hf, one arrives at a dispersion relation for the fast P-wave dependent on porous background properties and a real excess compliance which takes the fractures into account. However, if fracture permeability is assumed to be small, and is taken to be O(hf), the P-wave dispersion dependence on background parameters remains the same, but the term accounting for the fractures is frequency dependent. Only in the zero frequency limit does this result agree with that of the other two cases.