Simplified Seismic Modelling of Fractured Rock: How Effective is the Localised Effective Medium Compared to Explicit Representation of Individual Fractures

Abstract

Seismic waves can be an effective probe to retrieve fracture properties particularly when measurements are coupled with forward and inverse modelling. These seismic models then need an appropriate representation of the fracturing. The fractures can be modelled either explicitly, considering zero thickness frictional slip surfaces, or by considering an effective medium which incorporates the effect of the fractures into the properties of the medium, creating anisotropy in the wave velocities. In this work, we use a third approach which is a hybrid of the previous two. The area surrounding the predefined fracture is treated as an effective medium and the rest of the medium is made homogeneous and isotropic, creating a Localised Effective Medium (LEM). LEM can be as accurate as the explicit but more efficient in run-time. We have shown that the LEM model can closely match an explicit model in reproducing waveforms recorded in a laboratory experiment, for wave propagating parallel and perpendicular to the fractures. The LEM model performs close to the explicit model when the wavelength is much larger than the element size and larger than the fracture spacing. By the definition of the LEM model, we expect that as the LEM layer becomes coarser the model will start approaching the effective medium result. However, what are the limitations of the LEM and is there a balance between the stiffness, the frequency and the thickness, where the LEM performs close to an explicit model or approaches the effective medium model? To define the limits of the LEM we experiment varying fracture stiffness and source frequency. We then compare for each frequency and stiffness the explicit and effective medium with five models of LEM with different thickness. Finally, we conclude that the thick LEM layers with lower resolution perform the same as the thinner and finer resolution LEM layers for lower frequencies and higher fracture stiffness.

Special Issue on Rock Fracturing and Fault Activation: Experiments and Models

“…They found that the amount of permeability increase with damage and the injection pressure are two major factors governing the stability of the modelled faults and fault branching. Parastatidis et al (2021) demonstrate that localized effective medium (LEM) models are an accurate and efficient method to retrieve fracture properties from inverse modelling of seismic waves in comparison to less effective explicitly modelled zero-thickness frictional slip surfaces or less accurate effective medium models. They note that the thickness of the localized effective medium layer representing the fracture becomes important when the frequency is high, and the normal stiffness is low.…”

The effects of joint spacing on transmission characteristics of stress waves through layered composite rock masses

“…[34][35][36] Compared with rock masses with a single joint, multiple transmissions and reflections in layered rock masses make wave propagation more complex. [37][38][39][40][41][42] For layered homogeneous rock masses with the same medium in front of and behind joints, the effect of multiple transmissions and reflections between joints on wave transmission characteristics is related to joint spacing. 10,11 When a wave propagates through a layered homogeneous rock masses with large joint spacing, the multiple transmissions and reflections between joints were ignored to obtain an analytical expression of the transmission coefficient.…”

“…Compared with rock masses with a single joint, multiple transmissions and reflections in layered rock masses make wave propagation more complex 37–42 10,11 .…”

Jointed Rock Masses as Metamaterials - Implications for Railway Tunnel Vibrations

“…Parastatidis [28] investigated the effects of using equivalent materials on the transmission of stress waves through joints, using a finite difference (FD) model. Three different material models were Fig.…”

“…( 5) and ( 6) in further detail by breaking modelling assumptions, such as symmetrical boundaries parallel to wave propagation, infinite joints, and plane waves. In addition to this, equivalent material models, as previously used by Parastatidis [28], are investigated to determine whether they exhibit resonance characteristics. The outcomes are appraised in the context of rail vibrations.…”