Simulation of multi-scale fracture propagation
Micro-enriched continuum damage mechanics
We formulated the ``Differential Stress Induced Damage'' (DSID) model according to the framework of thermodynamic irreversible processes. Flow rules are expressed with the energy release rate conjugate to damage, as opposed to stress in most rock damage models. The damage criterion depends on principal stress differences. Non-elastic deformation due to damage is computed by an associate flow rule to capture the development of crack-induced strains in the main directions of damage.
Pictures: Xu and Arson, 2014; Jin and Arson, 2015
Then we formulated the Discrete Wing Crack Damage (DWCD) model to couple micro-mechanics and Continuum Damage Mechanics (CDM) principles. At the scale of the Representative Elementary Volume (REV), damage is obtained by integrating crack densities over the unit sphere, which represents all possible crack plane orientations. The unit sphere is discretized into 42 integration points. The damage yield criterion is expressed at the microscopic scale. The projection of shear stresses into a set of tensile forces allows predicting the occurrence of wing cracks at the tips of pre-existing defects. A hardening law is introduced to account for subcritical crack propagation, and non-associated flow rules are adopted for damage and irreversible strains induced by residual crack displacements after unloading. The DWCD model depends on only 6 constitutive parameters which all have a sound physical meaning and can be determined by direct measurements in the laboratory. The DWCD model can be used to predict a non linear-stress/strain relationship, damage-induced anisotropy, unilateral effects, the occurrence of irreversible strains, the apparent increase of strength and ductility in compression when the confinement increases and the increasing hysteresis on unloading-reloading paths as damage increases. Last but not least, the DWCD model provides realistic values of yield stress and strength in tension and compression. This is a significant advancement in the theoretical modeling of brittle solids. Future work will be devoted to the prediction of crack coalescence and to the modeling of the material response with interacting micro-cracks.
Numerical models coupling fracture and damage zone propagation
Pictures: Jin, Xu, Busetti and Arson, 2016 |
We proposed a numerical method that couples a Cohesive Zone Model (CZM) and a Finite Element - based Continuum Damage Mechanics (CDM) model. The CZM represents a mode II macro-fracture, and CDM Finite Elements (FE) represent the damage zone of the CZM. The coupled CZM/CDM model can capture the flow of energy that takes place between the bulk material that forms the matrix and the macroscopic fracture surfaces. We used the DSID model for the CDM Finite Elements and we calibrated it against triaxial compression tests performed on Bakken shale, so as to reproduce the stress/strain curve before the failure peak. Based on a comparison with Kachanov's micro-mechanical model, we confirm that the critical microcrack density value equal to 0.3 reflects the point at which crack interaction cannot be neglected. The CZM is assigned a pure mode II bilinear cohesive law. The cohesive shear strength of the CZM is calibrated by calculating the shear stress necessary to reach a CDM damage of 0.3 during a direct shear test. We find that the shear cohesive strength of the CZM depends linearly on the confining pressure. Triaxial compression tests are simulated, in which the shale sample is modeled as a FE CDM continuum that contains a predefined thin cohesive zone representing the idealized shear fracture plane. The shear energy release rate of the CZM is fitted in order to match to the post-peak stress/strain curves obtained during experimental tests performed on Bakken shale. We find that the energy release rate depends linearly on the shear cohesive strength. |
We then used the calibrated shale rheology to simulate the propagation of a meter-scale mode II fracture. Under low confining pressure, the macroscopic crack (CZM) and its damaged zone (CDM) propagate simultaneously (i.e. during the same loading increments). Under high confining pressure, the fracture propagates in slip-friction, i. e. the debonding of the cohesive zone alternates with the propagation of continuum damage. The computational method is applicable to a range of geological injection problems including hydraulic fracturing and fluid storage, and should be further enhanced by the addition of mode I and mixed mode (I+II+III) propagation.
Evolution of damage around a fracture propagating in mode II under low confining stress (left) and high confining stress (right). Pictures: Jin, Xu, Busetti and Arson, 2016
Simulation of the formation of fracture patterns around caverns and boreholes
The DSID model was implemented in a Finite Element program to simulate the anisotropy of stiffness and deformation induced by the propagation of tensile and compressional micro-cracks around pressurized boreholes and in the Excavation Damaged Zone. Simulations of borehole pressure changes in a pre-fractured rock mass were performed with pre-damaged Finite Elements and with joint elements. Current modeling efforts aim at combining these two numerical techniques in order to predict crack coalescence and subsequent borehole spalling.
Borehole pressurization. Pictures: Xu, Busetti and Arson, 2014
Excavation Damaged Zone: vertical stress (left) and horizontal damage (right). Pictures: Xu and Arson, 2015
Borehole drainage with viscous fluid flow in a prefractured rock mass. Fractures modeled with joint elements. Pictures: Jin, Pouya and Arson, 2015