Understanding the properties of cracked rocks is of great importance in scenarios involving

Wave propagation is controlled by the effective rock properties. Wave velocity and attenuation can be estimated from seismic data in scenarios such as seismic exploration, seismology, borehole measurements and tomography. Rock physics could then be used to estimate different rock properties, such as mineral composition, elastic moduli, the presence of a fluid, and pore space connectivity (and hence permeability) from seismic measurements. Thus, investigation of how cracks and fluids affect seismic properties has many practical applications. In activities including nuclear waste disposal,

Cracks and grain-scale discontinuities are the key rock parameters which control effective elastic and hydraulic properties of such rocks. Many studies show that seismic waves exhibit significant dispersion and attenuation in cracked porous rocks due to pore-scale fluid flow

Fluid flow due to a passing wave may happen at different scales: at the wavelength scale, at the mesoscopic scale and at the pore scale

At the pore scale, a passing wave induces fluid pressure gradients which occur between interconnected cracks, as well as between cracks and stiffer pores. Such pressure gradients force fluid to move between different cracks and pores until the pore pressure equilibrates throughout the connected pore space. This phenomenon, known as squirt flow

Dispersion and attenuation caused by squirt flow can be simulated numerically by solving the coupled fluid–solid deformation at the pore scale using Lamé–Navier and Navier–Stokes equations with appropriate boundary conditions and then calculating effective frequency-dependent viscoelastic properties. During the last decades, many studies used numerical methods to solve mechanical problems

We numerically simulate squirt flow in three dimensions and calculate frequency-dependent effective stiffness moduli using the finite-element method to solve the quasistatic Lamé–Navier equations coupled to the linearized quasistatic Navier–Stokes equations

This paper is organized as follows. First, we briefly describe the numerical methodology. Then, we describe the numerical model and show the numerical results – frequency-dependent effective stiffness moduli. After, by solving the Christoffel equation, we evaluate the angle-, azimuth- and frequency dependent velocities of the model. Lastly, we quantify the anisotropy strength of the models analyzing the conventional Thomsen-type anisotropy parameters and also by adopting another scalar parameter.

We consider that at the pore scale, a rock is composed of a solid material (grains) and a fluid-saturated pore space (cracks). The numerical methodology is described by

The fluid phase is described by the quasistatic linearized compressible Navier–Stokes momentum equation

The COMSOL Multiphysics partial differential equation module is used for implementing Eqs. (

Two 3D numerical models are constructed, which consist of a pore space embedded into an elastic solid grain material (Fig.

A fine, regular mesh is used inside the crack to accurately account for dissipation, while in the grain material the mesh is coarser (Fig.

Sketch illustrating two flat cylinders representing two cracks. The blue region represents the pore space saturated with a fluid and the transparent gray area corresponds to the solid grain material. In the first model, the two cracks are disconnected as illustrated by the upper right sketch. In the second model, the two cracks are connected as illustrated by the lower right sketch.

One crack embedded into an isotropic background induces a transverse isotropy (five independent components of the stiffness tensor, e.g.,

The symmetry of the saturated numerical model with connected cracks is tetragonal (Fig.

Material properties of the numerical model.

Let us first consider the geometry shown in Fig.

For the model with disconnected cracks, the effective stiffness moduli are (in Voigt notation)

The effective stiffness moduli of the two models are different. Zero values are written if the value is below 0.0002 GPa (i.e., up to numerical precision). The

Sketch illustrating the element's size distribution for the model with connected cracks. The element's size in the crack is

Here and later on we deal only with a liquid-saturated pore space. The liquid has properties of glycerol (Table

In the model with disconnected cracks, the fluid pressure in the cracks is the same in all three regimes, which corresponds to the unrelaxed state in the model with connected cracks. The unrelaxed state can be interpreted as the elastic limit because there is no fluid flow between the cracks, and the effective properties of the two models (connected and disconnected cracks) are the same, as will be shown in the next subsection.

Figure

Note that the width of the inverse quality factor peak (at half amplitude) for the components

Snapshots of the fluid pressure

In the model with disconnected cracks, all components of the stiffness tensor

Figure

Figures

Numerical results for the connected (C) and disconnected (D) crack models: real part of the

Due to the symmetry of the model, the behaviors of the P-, SV-, SH-wave phase velocities in the

First, we quantify the Thomsen-type anisotropic parameters

Thomsen-type anisotropic parameters (

P-wave phase velocity versus phase angle in the

SV-wave phase velocity versus phase angle in the

SH-wave phase velocity versus phase angle in the

Figure

In the

In the

Thomsen-type anisotropic parameters in the

The universal elastic anisotropy index

Figure

The anisotropy measure in bulk

The universal elastic anisotropy index measure

Thomsen-type anisotropic parameters provide a very detailed description of the velocity anisotropy in different planes. Most importantly, only a limited number of the stiffness tensor coefficients are needed to calculate

The analysis of two sets of anisotropic measures shows that (i) the overall anisotropy of the model with connected cracks (Fig.

In this study, we numerically solve a coupled fluid–solid deformation problem at the pore scale. If we consider the mesoscopic scale scenario and use Biot's (1941) equations, the fluid flow effects on the effective moduli are equivalent to that of the coupled elastic Stokes equations (as in the present study), as it was shown by

In summary, fluid flow effects on seismic anisotropy are nonlinear with a possible increase or decrease in the elastic anisotropy at different frequencies. These two extreme cases, the maximum negative and the maximum positive

Numerical simulations are useful but analytical models are especially attractive since they help us to better understand the physics and do not require sophisticated numerical simulations. The limitations of the analytical solutions are restricted to simple pore space geometry, and some assumptions related to physics are needed to derive the closed form analytical formulas. Such a comparison of the numerical results against an analytical solution has been performed by

We have numerically calculated the frequency-dependent elastic moduli of a fluid-saturated porous medium caused by squirt flow. The considered 3D numerical models consist of two perpendicular connected or disconnected cracks embedded in a solid grain material. Cracks are represented by very flat cylinders filled with a fluid. Grains are described as a linear isotropic elastic material, while the fluid phase is described by the quasistatic linearized compressible Navier–Stokes momentum equation.

We showed that seismic velocities are azimuth, angle and frequency dependent due to squirt flow between connected cracks. The resulting elastic frequency-dependent anisotropy was analyzed by using the Thomsen-type anisotropic parameters and the universal elastic anisotropy index. The latter is a scalar parameter which can be used to analyze the overall anisotropy of the model and its divergence from the closest isotropy. We showed that the seismic anisotropy may locally decrease as well as increase due to squirt flow in one specific plane. However, the overall anisotropy of the model mainly increases due to squirt flow between the cracks towards low frequencies. In the model with disconnected cracks, no fluid flow occurs, and thus the effective properties of the model correspond to the elastic limit. The elastic limit is equivalent to the high-frequency limit for the model with connected cracks. Seismic velocities are only azimuth and angle dependent as for a fully elastic material, and they are independent of frequency. In summary, squirt flow does affect effective mechanical properties of cracked rocks and thus seismic velocity anisotropy. Given that seismic anisotropy variations with frequency are very sensitive to the pore space geometry and material properties, it is difficult to make a general prediction. According to our study, the effective frequency-dependent response of a cracked medium is different in different planes. The local response (in a certain plane) is controlled by crack orientation, which is the key parameter. The magnitude of the frequency-dependent response (i.e., the dispersion and attenuation) is controlled by crack compliances, crack porosity and their fluid content. (Dry or liquid-saturation conditions will cause completely different behavior.) Most importantly, crack porosity is a very important parameter in fluid-saturated rocks (contrary to dry rocks) since it defines the volume of fluid which may flow due to wave propagation, causing wave attenuation and dispersion.

Let us consider a cuboid, volume

The mixed test for the

The mixed test for the

In the other four planes, the normal component of the displacement is set to zero, other components are free. Then, using the following equation

Thomsen-type anisotropic parameters

Assuming that one deals with an anisotropic frequency-dependent effective fourth-rank stiffness tensor

The double contraction of the scalar product (quadruple contraction) of Eqs. (

If the effective stiffness tensor is isotropic, then

In geophysics, the separation of the elastic anisotropy measure in bulk and shear modes is necessary because rocks might exhibit different frequency dependencies due to bulk and shear deformations. Therefore, in analogy to the universal elastic anisotropy index measure

These two parameters,

Similarly, the Voigt and Reuss shear moduli are (in Voigt notation)

Equations (

This is a theoretical study. Data are available from the authors upon reasonable request.

YA performed the numerical simulations and wrote the article. The idea of this project was first inspired by the paper by

The authors declare that they have no conflict of interest.

This research is funded by the Swiss National Science Foundation, project number 172691. Yury Alkhimenkov thanks Germán Rubino (CONICET, Centro Atómico Bariloche, Argentina) for fruitful discussions on the frequency-dependent anisotropy due to fluid flow, Nicolás D. Barbosa (University of Geneva, Switzerland) for fruitful discussions regarding the polarity change of the P-wave velocity with frequency and Irina Bayuk (Russian Academy of Sciences, Russia) for enlightening discussions regarding the fourth-rank tensor averaging and the elastic symmetry classes.

This research has been supported by the Swiss National Science Foundation (grant no. 172691).

This paper was edited by Susanne Buiter and reviewed by Yves Gueguen and Vishal Das.