Mathematics (Jan 2024)
Second-Order Approximate Reflection Coefficient of Thin Interbeds with Vertical Fractures
Abstract
The horizontal fractures in the strata will close in the compaction effect of overlying strata, while the vertical cracks are widely developed, which can be equivalent to HTI (transverse isotropy with a horizontal axis of symmetry) medium. When an S-wave propagates into HTI media, the shear wave will divide into two types of waves: a fast S-wave and slow S-wave. When the strata of HTI are thin and overlapping, called the thin interbeds model, the wave field exhibits complex primary reflections, converted waves, and multiples. We introduce a new second-order approximation of the total reflection coefficient, with the incidence angle lower than the critical angle in thin-interbed HTI media using a recursive algorithm. We verify the effectiveness of the second-order approximation by analyzing the energy of multiples. Comparing the second-order approximate solution that degenerates the HTI medium into isotropic and Kennett’s exact solution, we find that our solution has an accuracy of over 99.9% in any azimuth, with the incidence angle lower than the critical angle under P-wave incidence. However, our solution of the SP wave field is suitable for incidence azimuth angles between 0–75° and 120–180°, with the lowest accuracy occurring at an incidence angle of 25° and a relative error of 6.4%. The approximate solution in the SS wave field has the same applicable range as the SP wave, with the maximum error of 6.3% occurring at the incident angle of 1°. This new second-order approximate formula for the total reflection coefficient of thin interbeds composed of HTI helps us to understand the reflection characteristics of complex thin interbeds. It also lays a theoretical foundation for the development of AVO (Amplitude Versus Offset) analysis and inversion techniques for lithological and stratigraphic oil and gas reservoirs.
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