Continuation Methods for the Simulation of Rock Fracture with Cohesive Elements

"Rock fracture simulation most often presents non-linear behaviour due to loss of material integrity and strain concentration in the fracture zone. These strains generate local instabilities that propagate to the global system of equilibrium equations, generating convergence problems of the numerical solution. Often, in such cases, analyses in the post-critical regime are unfeasible. To overcome the numerical difficulties associated with problems of stiffness loss in the post-critical regime control methods, also called continuation methods, are employed. Among them are the arc-length method, the energy control method and the indirect displacement control method. These methods are employed in conjunction with a Newton- Raphson scheme for the solution of non-linear systems of equations. However, depending on the system instability, convergence in not guaranteed even with very small increments. In the present work these control techniques are investigated in combination with cohesive elements to simulate rock fracture. The constitutive model employed with the cohesive elements represents stiffness degradation through a damage law, which leads to serious convergence difficulties as reported in the literature. Here the continuation methods mentioned are applied to a Mode I rock fracture problem and to hydraulic fracture simulations. The effectiveness of the different continuation methods is compared.INTRODUCTIONThe process of rock fracture induced by the fluid pressure is called hydraulic fracturing. The propagation of a hydraulic fracture is a complex process, which is basically defined by the mechanical deformation of the rock, the fluid flow within the fracture and the fracture propagation itself (Mokryakov, 2011). In the works of Bendezu et al. (2013), Chen (2012) and Carrier et al. (2012), vertical hydraulic fracture propagation is modeled by cohesive elements with traction-separation law. The softening behavior of rock in post-critical regime is demonstrated by experimental testing and is characterized mainly by high strains and low stresses (Crowder and Bawden, 2004). This nonlinear behaviour can arise due to material nonlinearity, such as micro-cracking and damage, leading to softening behaviour (Chandrakant, S.D., 2012). The cohesive fracture model considers in its formulation the softening behaviour that most rocks present in post-critical regime with propagation criteria."