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

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.