When a seismic wave travels through a fluid-saturated porous reservoir containing aligned fractures, it induces oscillatory fluid flow between the fractures and the embedding background medium. Although there are numerous theoretical models for quantifying the associated seismic attenuation and velocity dispersion, they rely on certain assumptions, such as infinitesimal fracture thickness and dilute fracture concentration, which rarely hold in real reservoirs. The objective of this work is to overcome some of these limitations and, therefore, improve the applicability of the available theoretical models. To do so, we extend existing models to the finite fracture thickness case for P-waves propagating perpendicular to the fracture plane using the so-called branching function approach. We consider three types of fractures, namely, periodically and randomly spaced planar fractures, as well as penny-shaped cracks. The extended unified model is then tested by comparing with corresponding numerical simulations based on Biot’s theory of poroelasticity. We consider two cases of 2D rock samples with aligned elliptical fractures, one with low fracture density and the other with high fracture density. The results indicate that the influence of the finite fracture thickness on seismic dispersion and attenuation is small at low frequencies when the fluid pressure has enough time to equilibrate between the fractures and background medium. However, this effect is significant at high frequencies when there is not sufficient time for the fluid pressure equilibration. In addition, the theoretical predictions of the penny-shaped crack model are found to match the numerical simulation results very well, even under relatively high fracture density. Analyses of stress distributions suggest that the small discrepancies found between theoretical predictions and numerical simulations are probably due to fracture interactions. In a companion paper, we will extend the analysis for considering the full stiffness matrix and anisotropic properties of such rocks.

The detection and characterisation of domains of intersecting fractures are important goals in several disciplines of current interest, including exploration and production of unconventional reservoirs, nuclear waste storage, CO2 sequestration, and groundwater hydrology, among others. The objective of this study is to propose a theoretical framework for quantifying the effects of fracture intersections on the frequency‐dependent elastic properties of fluid‐saturated porous and fractured rocks. Three characteristic frequency regimes for fluid pressure communication are identified. In the low‐frequency limit, fractures are in full pressure communication with the embedding porous matrix and with other fractures. Conversely, in the high‐frequency limit, fractures are hydraulically isolated from the matrix and from other fractures. At intermediate frequencies, fractures are hydraulically isolated from the matrix porosity but can be in hydraulic communication with each other, depending on whether fracture sets are intersecting. For each frequency regime, the effective stiffness coefficients are derived using the linear‐slip theory and anisotropic Gassmann equations. Explicit mathematical expressions for the two characteristic frequencies that separate the three frequency regimes are also determined. Theoretical predictions are then applied to two synthetic 2D samples, each containing two orthogonal fracture sets: one with and another without intersections. The resulting stiffness coefficients, Thomsen‐style anisotropy parameters, and the transition frequencies show good agreement with corresponding numerical simulations. The theoretical results are applicable not only to 2D but also to 3D fracture systems and are amenable to being employed in inversion schemes designed to characterise fracture systems.

Numerous theoretical models have been proposed for computing seismic wave dispersion and attenuation in rocks with aligned fractures due to wave-induced fluid flow between the fractures and the embedding background. However, all these models rely on certain assumptions, for example, infinitesimal fracture thickness or dilute fracture concentration, which rarely hold in real reservoirs and, thus, limit their applicability. To alleviate this issue, theoretical models for periodically or randomly spaced planar fractures and penny-shaped cracks were recently extended by the authors to the case of finite fracture thickness for P-waves propagating perpendicular to the fracture plane. Theoretical predictions under low and relatively high fracture density were then assessed by comparing with corresponding numerical simulations. However, the case of arbitrary incidence angles as well as the behaviors of S-waves remained unexplored. In this work, we therefore extended the prediction results to the full stiffness matrix through two theoretical approaches. The first approach uses an interpolation between the low- and high-frequency limits using a relaxation function obtained from the normal-incidence solution. The second approach is based on the linear slip theory with a frequency-dependent fracture compliance. Both derivations rely on the fact that all the stiffness coefficients are controlled by the same relaxation function. With the full stiffness matrix, anisotropic seismic properties can then be studied. P- and S-wave velocities and attenuations at different frequencies and incidence angles and also corresponding anisotropy parameters are calculated for one synthetic 2D rock sample. The results indicate that the predictions provided by the two theoretical approaches are in good agreement with each other and also indicate a good agreement with the corresponding numerical simulations. The extended theoretical models presented in this work are easy to apply and computationally much cheaper than numerical simulations and, hence, can be used in the seismic characterization of fractured reservoirs.

Fractures often play an important role in controlling the fluid flow in hydrocarbon reservoirs. When the seismic wave propagates through media containing fracture corridors, significant scattering dispersion and attenuation can occur. In this work, we study the P-wave dispersion and attenuation due to the scattering caused by 2-D fluid-saturated aligned fractures with finite thickness, which are embedded in an isotropic elastic background medium. Using the Foldy approximation and the representation theorem, the P-wave dispersion and attenuation are related to the displacement discontinuities across the fractures. These fracture displacement discontinuities are obtained from the boundary conditions and the P-wave dispersion and attenuation can thus be calculated. A numerical example shows that the fracture thickness has significant influence on the dispersion and attenuation, especially in the low-frequency regime when the fracture size is smaller than the seismic wavelength. The effects of the fluid bulk modulus are also significant, which are opposite to those of the fracture thickness. However, the effect of the fluid viscosity is found to be negligible for the studied configurations. To validate the proposed model, the theoretical predictions are compared with ultrasonic measurements on fractured samples. The comparison shows overall good agreement between theory and experiment. This work reveals the important influence of fracture thickness and saturating fluid properties on the P-wave scattering dispersion and attenuation. Hence, it shows a potential to extract these parameters from seismic data.