Aligned vertical fractures, HTI reservoir symmetry, and Thomsen seismic anisotropy parameters for polar media

Sayers and Kachanov (1991) defined crack-influence parameters that are shown to be directly related to Thomsen (1986) weak-anisotropy seismic parameters for fractured reservoirs when the crack/fracture density is small enough. These results are then applied to the problem of seismic wave propagation in polar (i.e., non-isotropic) reservoirs having HTI seismic wave symmetry due to the presence of aligned vertical fractures and resulting in azimuthal seismic wave symmetry at the earth’s surface. The approach presented suggests one method of inverting for fracture density from wave-speed data. It is also observed that the angular location θex of the extreme value (peak or trough) of the quasi-SV-wave speed for VTI occurs at an angle determined approximately by the formula tan2 θex ' tan θm = [(c33− c44)/(c11− c44)], where θm is an angle determined directly (as shown) from the cij elastic stiffnesses, whenever these are known from either quasi-static or seismic wave measurements. Alternatively, θex is given in terms of the Thomsen seismic anisotropy parameters by tan θex ' ( [v2 p(0) − v2 s(0)]/[(1 + 2 )v2 p(0)− v2 s(0)] 1/4 , where = (c11 − c33)/2c33, v2 p(0) = c33/ρ, and v2 s(0) = c44/ρ, with ρ being the background inertial mass density. The axis of symmetry is always treated here as the x3-axis for either VTI symmetry (due, for example, to horizontal cracks), or HTI symmetry (due to aligned vertical cracks). Then the meaning of the stiffnesses is derived from the fracture analysis in the same way for VTI and HTI media, but for HTI the wave speeds relative to the earth’s surface are shifted by 90o in the plane perpendicular to the aligned vertical fractures. Skempton’s (1954) coefficient is used as a general means of quantifying the effects of fluids inside the fractures. Explicit formulas for Thomsen’s parameters are also obtained for either drained or undrained fractures resulting in either VTI or HTI symmetry of the reservoir.