Exact seismic velocities for TI media and extended Thomsen formulas for stronger anisotropies

INTRODUCTION

Thomsen's weak anisotropy formulation (Thomsen, 1986) was originally designed for media having vertical transversely isotropic (VTI) symmetry, but clearly applies equally well to any other TI media (for example HTI) with only very minor technical changes related to how the orientation of the axis of symmetry is labelled in Cartesian coordinates. This formulation is also independent of the natural mechanism producing the anisotropy, whether it be due to layering, or horizontal fractures, or randomly oriented vertical fractures, or some other source. So the method has wide applicability for use in exploration problems. However, when the approximate results of the Thomsen's original formulation are compared to known exact results for the same VTI media, it is easy to see that there are some deficiencies. In particular, for VTI media, the vertically polarized (SV) shear wave will always have a peak (or possibly a trough, for some fairly rare types of anisotropic media) somewhere in the range $0 \le \theta \le \pi/2 = 90^o$. Thomsen's weak anisotropy formulation always puts this extreme point (either minimum or maximum) exactly at $\theta = \pi/4 = 45^o$. However, as I show here, the $\theta = 45^o$ angular location never actually occurs for any interesting degree of VTI anisotropy; instead $\theta \to 45^o$ (by which I mean the extreme point approaches but never reaches $45^o$) for extremely weak anisotropy -- e.g., very low horizontal crack density is one example of this. In an effort to determine whether it might be possible to improve on Thomsen's approximation, I have found that a relatively small modification of Thomsen's formulas places the extreme $v_{sv}$ point at nearly the right angular location, and also typically (though not universally) improves the overall fit of both $v_{sv}(\theta)$ and $v_p(\theta)$ to the exact VTI curves. The ultimate cost of this improvement is negligible since the data required to estimate the location of the extreme point are exactly the same as the data used to determine Thomsen's other parameters for weak anisotropy. The method can also be used with only minor technical modifications for media having horizontal transversely isotropic (HTI) symmetry, such as reservoirs having aligned vertical fractures. The paper focuses on the general theory and uses other recent work relating fracture influence parameters (Sayers and Kachanov, 1991; Berryman and Grechka, 2006) to provide some useful examples of the applicability of the new method. Other choices of the various possible applications of the new method will appear in later publications.

The main result of the paper -- from which all the subsequent results follow -- is a new, more compact, and more intuitive way of writing the quantity $\zeta(\theta)$ [appearing here in equation 12]. This quantity has its extreme value at almost the same location as that of the quasi-SV-wave phase velocity, and this angular location is very easy to determine.

The following section reviews the standard results for wave speeds in a VTI medium, and also presents the Thomsen weak anisotropy results. The next section presents the analysis leading to the extended (i.e., improving on Thomsen) anisotropy formulation, which allows the wave speed formulas to reflect more accurately the correct behavior near the extremes (greatest excursions from the values at normal incidence and near horizontal incidence). Then, the next section shows how to determine the value of $\theta_m$ (the incidence angle that determines where the extreme $SV$-wave behavior occurs) from the same data already used in Thomsen's formulas. Furthermore, normal moveout corrections are recomputed for the new formulation, and it is found that the results are identical to those for Thomsen formulation; thus, no new corrections are needed near normal incidence. Finally, to illustrate the results, models of VTI and HTI reservoirs having vertical fractures are computed using the new wave speed formulation and compared to prior results. Appendix A computes the quasi-SV-wave speed at $\theta = \theta_m$ exactly, and also at two levels of approximation in order to have values to check against the corresponding results in the main text. Appendix B discusses how to get HTI results simply and directly from VTI results, both for the exact wave speeds and for the new approximate wave speed formulas. The final section of the main text presents an overview and suggests some possible applications of the results.

Exact seismic velocities for TI media and extended Thomsen formulas for stronger anisotropies