Seismic dispersion and attenuation in saturated porous rocks with aligned fractures of finite thickness: Theory and numerical simulations — Part 1: P-wave perpendicular to the fracture plane

Abstract

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.

Effects of Coupling Between Wave‐Induced Fluid Flow and Elastic Scattering on P‐Wave Dispersion and Attenuation in Rocks With Aligned Fractures

“…Therefore, we can conclude that our theoretical model gives reasonable values of P ‐wave velocities in the low‐ and high‐frequency limits, as well as at intermediate frequencies. Furthermore, the predictions of our theoretical model for the FB‐WIFF caused P ‐wave dispersion and attenuation are consistent with those given by Guo, Rubino, et al (). These validate the correctness of our theoretical model.…”

“…Figure compares their results with our theoretical model. The results of Guo, Rubino, et al (); red dashed lines) are generally in good agreement with those given by our theoretical model (red solid lines). The primary discrepancies are due to the slightly different P ‐wave velocities in the low‐ and high‐frequency limits of FB‐WIFF.…”

“…This results in the slightly different fracture shapes considered in our model and that in the linear slip theory. The shape difference can result in the discrepancies for the calculated velocities (Guo, Rubino, et al, ). The other reason is the fracture interactions.…”

“…Instead of specifying the shapes of the background pores, the second type of models describe the background properties using the macroscopic parameters (such as porosity and permeability). This allows us to employ the Biot‐Gassmann theory to study the FB‐WIFF mechanism, for which quite a few models have been developed (e.g., Ba et al, , , ; Brajanovski et al, ; Fu et al, ; Galvin & Gurevich, ; Guo et al, , ; Gurevich et al, ; Kong et al, ). The effects of FB‐WIFF on the P ‐wave dispersion and attenuation are studied in detail using these models.…”

Seismic Dispersion and Attenuation in Saturated Porous Rock With Aligned Slit Cracks

“…Equations and are identical to equations and , respectively. This confirms that the asymptotic behavior for the 2‐D and 3‐D cases at high frequencies is the same, which was also observed by Guo, Rubino, Barbosa, et al (), Guo, Rubino, Barbosa, et al (), and Guo, Rubino, Glubokovskikh, et al () by comparing the 2‐D numerical simulations with the 3‐D theoretical predictions. This is because at high frequencies the condition │ k 2 a │ ≫ 1 and the fluid diffusion length is much smaller than the crack length for both slit and penny‐shaped cracks.…”