  Simulation of fracturing processes in porous rocks can be divided into two
main branches: (i) modeling the rock as a continuum which is enhanced with
special features to account for fractures, or (ii) modeling the rock by a
discrete (or discontinuous) approach that describes the material directly as a
collection of separate blocks or particles, e.g., as in the discrete element
method (DEM). In the modified discrete element (MDEM) method, the effective
forces between virtual particles are modified in all regions, without failing
elements, so that they reproduce the discretization of a first order finite
element method (FEM) for linear elasticity. This provides an expression of the
virtual forces in terms of general Hook's macro-parameters. Previously, MDEM
has been formulated through an analogy with linear elements for FEM. We show
the connection between MDEM and the virtual element method (VEM), which is a
generalization of FEM to polyhedral grids. Unlike standard FEM, which computes
strain-states in a reference space, MDEM and VEM compute stress-states directly
in real space. This connection leads us to a new derivation of the MDEM method.
Moreover, it gives the basis for coupling (M)DEM to domain with linear
elasticity described by polyhedral grids, which makes it easier to apply
realistic boundary conditions in hydraulic-fracturing simulations. This
approach also makes it possible to combine fine-scale (M)DEM behavior near the
fracturing region with linear elasticity on complex reservoir grids in the
far-field region without regridding. To demonstrate the simulation of hydraulic
fracturing, the coupled (M)DEM-VEM method is implemented using the Matlab
Reservoir Simulation Toolbox (MRST) and linked to an industry-standard
reservoir simulator.
