Estimation of fracture parameters from reflection seismic data - Part I: HTI model due to a single fracture set

The simplest effective model of a formation containing a single fracture system is transversely isotropic with a horizontal symmetry axis (HTI). Reflection seismic signatures in HTI media, such as NMO velocity and amplitude variation with offset (AVO) gradient, can be conveniently described by the Thomsen-type anisotropic parameters epsilon ((V)), delta ((V)). and gamma ((V)). Here, we use the linear slip theory of Schoenberg and co-workers and the models developed by Hudson and Thomsen for penny-shaped cracks to relate the anisotropic parameters to the physical properties of the fracture network and to devise fracture characterization procedures based on surface seismic measurements. Concise expressions for epsilon ((V)), delta ((V)),and gamma ((V)), linearized in the crack density, show a substantial difference between the values of the anisotropic parameters for isolated fluid-filled and dry (gas-filled) penny-shaped cracks. While the dry-crack model is close to elliptical with epsilon ((V)) approximate to delta ((V)), for thin fluid-filled cracks epsilon ((V)) approximate to 0 and the absolute value of delta ((V)) for typical V-S/V-P ratios in the background is close to the crack density. The parameters epsilon ((V)) and delta ((V)) for models with partial saturation or hydraulically connected cracks and pores always lie between the values for dry and isolated fluid-filled cracks. We also demonstrate that all possible pairs of epsilon ((V)) and delta ((V)) occupy a relatively narrow triangular area in the area in the [epsilon ((V)), G delta ((V))]-plane, which can be used to identify the fracture-induced HTI model from seismic data. The parameter delta ((V)), along with the fracture orientation, can be obtained from the P-wave NMO ellipse for a horizontal reflector. Given delta ((V)), the NMO velocity of a dipping event or nonhyperbolic moveout can be inverted for epsilon ((V)). The remaining anisotropic coefficient, gamma ((V)), can be determined from the constraint on the parameters of vertically fractured HTI media if an estimate of the V-S/V-P ratio is available. Alternatively, it is possible to find gamma ((V)) by combining the NMO ellipse for horizontal events with the azimuthal variation of the P-wave AVO gradient. Also, we present a concise approximation for the AVO gradient of converted (PS) modes and show that all three relevant anisotropic coefficients of HTI media can be determined by the joint inversion of the AVO gradients or NMO velocities of P- and PS-waves. For purposes of evaluating the properties of the fractures, it is convenient to recalculate the anisotropic coefficients into the normal (Delta (N)) and tangential (Delta (T)) weaknesses of the fracture system. If the HTI model results from penny-shaped cracks, Delta (T) gives a direct estimate of the crack density and the ratio Delta (N)/Delta (T) is a sensitive indicator of fluid saturation. However, while there is a substantial difference between the values of Delta (N)/Delta (T) for isolated fluid-filled cracks and dry cracks, interpretation of intermediate values of Delta (N)/Delta (T) for porous rocks requires accounting for the hydraulic interaction between cracks and pores.