Considering poroelastic media containing periodically distributed parallel fractures, we numerically quantify the effects that fractures with variable aperture distributions have on seismic wave attenuation and velocity dispersion due to fluid pressure diffusion (FPD). To achieve this, realistic models of fractures are generated with a stratified percolation algorithm which provides statistical control over geometrical fracture properties such as density and distribution of contact areas. The results are sensitive to both geometrical properties, showing that an increase in the density of contact areas as well as a decrease in their correlation length reduce the effective seismic attenuation and the corresponding velocity dispersion. Moreover, we demonstrate that if equivalent physical properties accounting for the effects of contact areas are employed, simple planar fractures can be used to emulate the seismic response of fractures with realistic aperture distributions. The excellent agreement between their seismic responses was verified for all wave incidence angles and wave modes.

Fractures in rocks occur in a wide range of scales (from microscale to continental), and their identification and characterisation are important tasks for several areas such as oil and gas exploration and extraction, production of geothermal energy, nuclear waste disposal and civil engineering works, among others

Several authors have studied fracture-related FPD effects on seismic attenuation and velocity dispersion

Fractures can be conceptualised as two uneven surfaces in contact, which produce variable separation between their boundaries or walls

In this work, we follow the workflow proposed by

To study attenuation and dispersion of seismic waves in a fluid-saturated rock with parallel and periodically distributed fractures, we model fractures as poroelastic media embedded in a homogenous poroelastic background. The aperture of the modelled fractures can be spatially variable. In the present work, we refer as open regions of the fracture to the zones where the fracture walls are not in contact (non-zero aperture) and are filled with a highly permeable and porous material. Contact areas (zero aperture), on the other hand, are represented by a porous material having the same properties as the background medium. We define the density of the contact areas as the ratio between the area of the fracture walls in contact and the area of the entire fracture. In this work, we use similar material properties to those employed by

Assuming that the prevailing wavelengths are much larger than the fracture aperture and spacing, we can obtain the effective seismic properties of the fractured medium by performing oscillatory relaxation tests on a representative elementary volume (REV) of the medium. We solve the quasi-static poroelastic equations given by

Material properties.

Biot's (1941) equations in the space–frequency domain and in absence of external forces are

We solve the system of Eqs. (1) and (7) employing Eq. (3) and using the finite element method. Considering auxiliary functions to represent the variables between nodes, the “weak” formulation is obtained by combining the differential equations (“strong” form) with natural (undrained) boundary conditions in an integral form, which allows for reducing the order of the spatial derivatives

In order to numerically analyse the general impact that fractures with variable aperture produce on seismic attenuation and velocity dispersion, we first consider fractures with simple geometries and distributions of contact areas. Then, we extend the investigation to fractures having realistic aperture distributions and perform a sensitivity analysis of the effective seismic response of the fractured medium in terms of density and correlation length of contact areas.

We first consider simple fracture models for illustrating general effects of contact areas distributions on the P-wave modulus normal to the fractures and the associated seismic attenuation. The numerical model is a cube of 4 cm sides having a horizontal fracture crossing its centre, that is, normal to the vertical (

The numerical results for the real part of the effective P-wave modulus and seismic attenuation normal to the fracture are presented in Fig.

For better understanding the impact of contact area distributions on the seismic responses shown in Fig.

Fracture apertures with regular

Real part of P-wave modulus (

Real part of the vertical (

In order to analyse the seismic response of realistic fractures, we perform numerical simulations considering fractures with variable aperture distributions generated following the stratified percolation approach of

Figure

Fracture aperture distributions generated using a stratified percolation workflow

Real part of P-wave modulus (

A common assumption in analytical models is that fracture compliance depends on the fracture volume and the distribution of the fracture microstructure, such as mean crack radius

Fracture with binary aperture distributions derived from fracture models A to D in Fig.

In the analysis presented above, each fracture is one realisation of a pseudo-random generation process with given contact area density and correlation length. In order to analyse the variability of the results, we generated several fracture models with the same characteristics as model B (Fig.

In studies on the effective seismic response of fractured media, a fracture is frequently represented as a thin compliant layer of constant thickness (e.g.

For estimating the equivalent properties of the fractures, we follow the linear slip theory

To support the consideration of our models as realistic, the fracture compliances presented in Table

Fracture normal and shear excess compliances and equivalent properties of a fracture represented as a poroelastic thin layer.

After computing the equivalent dry bulk and shear moduli and porosity for each fracture (Eqs. 12–16), as well as the mean aperture, we employ the analytical solution of

Real part of P-wave modulus (

To further verify and generalise the validity of the equivalent fracture model of constant thickness, we extended the methodology presented by

Real part of the components of the effective stiffness matrix as functions of frequency. Circles correspond to numerical simulation for fracture models A, B and D with variable aperture (VA) distributions shown in Fig.

Figure

Obtaining information on the hydraulic and mechanical behaviours of fractures by means of their seismic responses is an ultimate goal in fracture characterisation.

We also showed that the density and correlation length of contact areas control the normal and shear fracture compliances (Fig.

The aim of the present contribution was to analyse the effects of variable aperture distributions of 3-D fracture models on FPD between fractures and background. To do so, we numerically quantified the effective frequency-dependent stiffness matrix coefficients and seismic attenuation for realistic fracture models representing REVs of media containing periodically distributed parallel fractures. Our fracture models were characterised by aperture distributions generated using a stratified percolation algorithm and accounting for different densities and correlation lengths of contact areas. We showed that for a given density of contact areas, fractures with correlated distributions of contact areas (i.e. highest correlation length) exhibit higher P-wave modulus dispersion and seismic attenuation than those with a low correlation. Furthermore, lower P-wave modulus dispersion and seismic attenuation were observed when increasing contact area density for a given correlation length. This compensatory effect allows fractures with highly different aperture distributions to produce similar seismic responses. Moreover, although the effects of distribution of contact areas on the P-wave modulus are maximal at the low-frequency limit, these distributions also play an important role at the high-frequency limit. We also observed that, if the distribution of contact areas is fixed, fracture mean aperture (which controls fracture volume) dominates the seismic response due to FPD effects, while the variable aperture in the open regions of the fracture has a negligible influence.

Finally, we demonstrated that a simple fracture geometry such as thin layers with constant aperture and appropriate equivalent physical properties produces the same effective anisotropic seismic response of fractures with a much more intricate geometry. The equivalent elastic properties can be obtained from the excess fracture compliances, determined according to the linear slip theory, as long as the effects of contact areas are accounted for. Our results validate the use of simple models of fractures having constant thickness for numerically simulating the effects of fractures with realistic geometries which, in turn, can significantly reduce computational cost and overcome meshing limitations.

All numerical results are reproducible by solving the equations and boundary conditions described in this work and using the fracture models provided in the Supplement. Numerical results can also be shared by contacting the first author.

To generate the fracture models shown in Fig.

The supplement related to this article is available online at:

This work is part of the PhD project of SL, conducted under the supervision of BQ. The paper was prepared by SL with contributions from all co-authors.

The authors declare that they have no conflict of interest.

Simón Lissa and Beatriz Quintal thank Holger Steeb and Eva Caspari for insightful discussions. J. Germán Rubino acknowledges financial support from Agencia Nacional de Promoción Científica y Tecnológica (PICT 2017-2976) and a visit to the University of Lausanne financed by the Fondation Herbette.

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

This paper was edited by Tarje Nissen-Meyer and reviewed by Junxin Guo and one anonymous referee.