Open AccessThe effect of fluid pressure on wave speeds in a cracked solid
Author(s) -
Hudson J. A.
Publication year - 2000
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1046/j.1365-246x.2000.01239.x
Subject(s) - isotropic solid , isotropy , mechanics , anisotropy , materials science , stress (linguistics) , porosity , fluid pressure , amplitude , boundary (topology) , geometry , geology , composite material , physics , mathematics , mathematical analysis , optics , philosophy , linguistics
In a porous or cracked elastic solid, the effective stress (defined in terms of the loads applied to the solid part of the outer boundary) and effective strain (defined in terms of the displacements at the solid part of the outer boundary) occurring in small‐amplitude deformations are connected by a linear relation along with the pressure within the fluid occupying the pores and cracks. We derive here a formula of this kind for a static system in which enough time is allowed for pressure to be equalized throughout the fluid (on the assumption that all pockets of fluid are connected in some way). The formula depends on the overall stiffnesses relating stress to strain for the same material with the fluid removed (dry or empty cracks and pores). For undrained conditions where no fluid is allowed to enter or leave the body, the pressure is directly related to the effective stress and strain, and the Gassmann relations are obtained relating the stiffnesses for an isotropic material in dry and undrained conditions. For an anisotropic material, the Brown–Korringa relations are recovered. Externally imposed stresses and fluid pressure distort the material structure and influence the wave speeds of elastic waves. The main way in which this occurs is in changing the aspect ratios of flat cracks, the most compliant part of the microstructural geometry. This effect on the wave speeds is studied here both in terms of crack closure, with corresponding changes in crack number density, and in variations in crack aspect ratios. The principal way in which the latter influences the wave speeds is through the fluid incompressibility factor in the formula for the properties of materials with connected cracks. An increase in aspect ratio of the cracks is equivalent to a reduction in the bulk modulus of the fluid. This effect is apparent in the limits of both high frequencies, when the material behaves as if the cracks were isolated, and low frequencies, when undrained conditions apply.