A Study On Adaptive Finite Element Solution Of Phase-Field Models For Hydraulic Fracturing

The phase-field model based on the variational approach is a new method for the simulation of complex fractures.Unlike the traditional discrete fracture models,the phase-field approach uses a continuous variable to describe the discontinuous problem.A phase-field variable d is introduced to indicate whether the material is damaged.One of the major advantages of this model is that the initiation and propagation of cracks are completely determined by a coupled system of partial differential equations based on the energy functional and no additional calculation is needed to determine crack propagation.Another advantage is that the generation and propagation of fracture networks do not require explicitly tracking fracture interfaces.This enables the phase-field model to handle complex cracks,crack propagation,and creation of new cracks more easily.In the phase-field modeling,the decomposition of the strain tensor into tensile and compressive components is essential for the success of modeling of brittle fracture but results in a non-smooth elastic energy and stronger nonlinearity in the governing equation.This makes the governing equation that is more difficult to solve.In particular,Newton’s iteration often fails to converge during the computations.Three regularization methods are proposed to smooth out the decomposition of the strain tensor.Numerical examples of fracture propagation under quasi-static load demonstrate that all of the methods can effectively improve the convergence of Newton’s iteration for relatively small values of the regularization parameter but without comprising the accuracy of the numerical solution.On the other hand,induced crack model for the initially existing cracks is commonly used to define the initial crack conditions due to its ability to deal with complex initial cracks.However,energy split may lead to a small remaining stiffness in the totally damaged zone,which violates the crack boundary condition.The critically damaged zone determined by the critical parameter is proposed to overcome that unrealistic crack boundary condition.Meanwhile,an improved volumetric-deviatoric split method is proposed that is based on the volumetric-deviatoric split method while taking the critical damage zone into account.Numerical results show that Our method is more straightforward and provides comparable results with existing literature but with less computational costs.A moving mesh finite element method is studied for the numerical solution of a phase-field model for brittle fractures.A metric tensor based on the Hessian of the phase-field variable d is considered and provide the information of the size,shape,and orientation of mesh elements needed in the mesh adaption procedure.The moving mesh partial differential equation(MMPDE)approach is employed to dynamically track crack propagation.Numerical results show that the moving mesh finite element method is able to adaptively concentrate the mesh elements around propagating cracks and handle multiple and complex crack systems.Compared to a uniform mesh,the CPU time for adaptive mesh is reduced by nine-tenth,and the number of required mesh element is also greatly reduced.Finally,the effects of the fluid pressure in the fracture based on the phase-field model are considered.A phase-field model of hydraulic fracturing coupled with the phase field,displacement field and fluid pressure field is established.The adaptive finite element solution of the phase-field model is achieved in the complex fracture system for hydraulic fracturing,such as fracture reorientation,interaction and interferenceIn this study,some innovative understandings and breakthroughs have been made in the optimization of the algorithm of the phase field model.The MMPDE moving mesh method has been applied to solve the governing equations,which makes it easier to implement complex crack simulation under cross-scale engineering conditions.The study will contribute to the theoretical basis and technical support for the development of hydraulic fracturing technology for unconventional reservoirs.