*2.5. FEM-DEM*

FEM-DEM is a hybrid approach that combines the advantages of DEM with finite element modelling (FEM) to reduce the computational load. The DEM component is applied within model micro-scale sections, and the stresses resultant from the DEM are then imposed within the FEM, which models the macro-scale domain [38]. This method is most applicable if the domain can be discretised into high- and low-sensitivity subdomains. It should also be mentioned that FEM-DEM codes are available that combine these two methods for different purposes [27,28].

#### *2.6. Equivalent Continuum Methods (ECM)*

Equivalent continuum methods assume that within a representative element volume (REV) the medium can be treated as homogeneous. Instead of defining fractures individually and modelling the strength degradation, this means that the properties of the domain are generalised and modelled as a continuum. The region of interest can then be discretised based on the regions of high and low fracture density, allowing for the modelling of the fractured and virgin zones of the reservoir [18,39]. The use of an ECM representation makes it simpler to apply well established numerical approaches such as the finite element method (FEM) to solve governing equations that describe the physical processes and behaviour of the medium.

Most of previous applications of ECM are focused on soils and near-surface materials [29–32], rather than for modelling hydraulic fracturing in rocks at a greater depth. This may be due to the general availability of a wide range of empirical methods to define the properties of fractured materials at shallow depths [40,41].

A novel ECM method introduced in this paper attempts to address the necessity of optimised hydraulic fracturing modelling. The introduced method in this paper uses the dependency of the permeability on void ratio to define the hydraulic fracture propagation. This method allows for an easier convergence due to uniform mesh dimensions and the exploitation of an already present coupled fluid flow and continuum relations.
