Abstract
Simulation of fracturing processes in porous rocks can be divided in two main
branches: (i) modeling the rock as a continuum enhanced with special features to account for fractures, or
(ii) modeling the rock by a discrete (or
discontinuous) modeling technique that describes the material directly as an assembly 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 linear FEM for
linear elasticity. This provides an expression of the virtual forces in terms of general Hook's macroparameters.
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 traditional FEM to polyhedral grids. Unlike standard FEM, which computes strain-states
in 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 domains
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
simulation of hydraulic fracturing, the coupled (M)DEM-VEM method is implemented in the Matlab
Reservoir Simulation Toolbox (MRST) and linked to an industry-standard reservoir simulator. Similar
approaches have been presented previously using standard FEM, but due to the similarities in the
approaches of VEM and MDEM, our work is a more uniform approach and extends previous work to
general polyhedral grids for the non-fracturing domain.
branches: (i) modeling the rock as a continuum enhanced with special features to account for fractures, or
(ii) modeling the rock by a discrete (or
discontinuous) modeling technique that describes the material directly as an assembly 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 linear FEM for
linear elasticity. This provides an expression of the virtual forces in terms of general Hook's macroparameters.
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 traditional FEM to polyhedral grids. Unlike standard FEM, which computes strain-states
in 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 domains
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
simulation of hydraulic fracturing, the coupled (M)DEM-VEM method is implemented in the Matlab
Reservoir Simulation Toolbox (MRST) and linked to an industry-standard reservoir simulator. Similar
approaches have been presented previously using standard FEM, but due to the similarities in the
approaches of VEM and MDEM, our work is a more uniform approach and extends previous work to
general polyhedral grids for the non-fracturing domain.
