Exact and approximate reflection/transmission responses from a layer containing vertical fractures

Abstract

SUMMARY Analyses of vertical fractures are of great interest in characterizing the fluid flow and minimum in situ stress direction in reservoirs. Long-wavelength equivalent orthorhombic (ORT) media typically characterize the anisotropy induced by a set of vertical parallel fractures or two sets of vertical and mutually orthogonal fractures embedded into a transversely isotropic medium with a vertical symmetry axis (VTI). Reflection and transmission (R/T) responses quantify wave amplitude variations in 1-D media and help to reveal the model property enclosing the heterogeneity. Conventionally, the R/T responses are analysed for an interface bounded by two half-spaces. However, for a plane wave travelling through a subsurface layer, the wave scattering effects at the top and bottom of the layer interact with each other. For a continuous infinite ORT space cut in two halves along the horizontal symmetry plane, we focus on the plane wave R/T responses from an ORT layer that is placed between the two halves, where the azimuths of the vertical symmetry plane in the layer and in the upper and lower half-spaces are identical. The R/T coefficient modelling method can be found in many publications for the ORT layer with an arbitrary finite thickness. We decompose the exact R/T coefficients into series expansions that correspond to different orders of intrabed multiples in the ORT layer. Under the weak-contrast assumption for the ORT half-spaces and the ORT layer, we use the anisotropic background medium to obtain the first-order R/T coefficient approximations and second-order reflectivity approximations. There is no constraint for the middle layer thickness in the obtained first-order reflectivity approximations. In the proposed first-order transmissivity approximations and second-order reflectivity approximations, the layer thickness is assumed to be thin to obtain appropriate approximations for a few wave modes. The isotropic background medium is also considered for weakly anisotropic models to obtain simpler approximations that facilitate parametric analyses. For the ORT layer with its thickness much smaller than the propagating wave's wavelength, the influences of the layer thickness on R/T coefficients can be inspected conveniently from the derived approximations. Particularly, the R/T coefficients are analysed for the model which would be a homogeneous VTI medium, if the vertical parallel fractures were absent from the middle layer. Numerical tests demonstrate that the proposed R/T coefficient approximations perform well for the thin ORT layer. The approximation accuracy decreases when the thin layer thickness increases.