| Hydraulic fracturing or fracking is one of the key problems in various engineering communities including exploitation of shale oil and gas,and geothermal,sequestration of CO2 and nuclear waste,and design and safety control of high concrete dams.Understanding the mechanisms of the fracking is highly essential for optimization of structural design,improvement of exploitation efficiency,and protection of natural environment.Rock or concrete under triaxial compression is commonly featured as brittle fracture,while its fluid-driven fracture is quasi-brittle,with non-negligible fracture process zones（FPZ）around crack tips.Classical numerical methods of fracking are typically based on the assumption of brittle fracture,which are consequently inapplicable for quasi-brittle fracking.Meanwhile,they are usually cumbersome to simulate three-dimensional,multi-crack,and heterogeneous fracking problems,due to the need of remeshing or crack tracking.Therefore,it is highly desirable to develop efficient numerical methods for fracking to gain deep insights into the mechanisms of multi-crack and multi-scale fluid-driven fracture in quasi-brittle materials,with an aim to provide reliable tools of numerical analyses for the practice of engineering.Motivated by the above aim,this study conducted theoretical and algorithmic research for modelling of fracking in quasi-brittle materials such as rock and concrete,based on the unified phase-field theory（also known as the phase-field regularized cohesive zone model）and fluid mechanics of porous media.A hydro-mechanical fully coupled cohesive damage phase-field model and a numerical algorithm of multiphysics coupling finite element method were proposed and extended to mesoscale,three-dimensional,dynamic,and non-isothermal fracking problems.The main content of this research is concluded as:（1）Quasi-static cohesive phase-field model and numerical algorithm for fracking in quasi-brittle materials:A hydro-damage-mechanical fully coupled cohesive phase-field method is proposed for quasi-static fracking modelling.In this method,a unified fluid continuity equation with a modified Darcy-Poiseuille law,based on the Biot’s poroelastic theory,is used for simultaneous modelling of fluid flow in both fractures and porous media.Pressure-dependent displacement and phase-field governing equations are derived based on the variational and thermodynamic principles to simulate fluid-driven solid deformation and fracture.The weak forms of the governing equations of the displacement-pressure-damage fields are derived using the Galerkin method,which are then coupled in a staggered way and solved by an alternative minimization（AM）Newton-Raphson iterative algorithm.The method is first used to model one-dimensional consolidation of porous media,fluid flow in cracks,fluid-driven single-/multi-crack propagation,with simulated results being insensitive to the mesh size and phase-field length scale,and agreeing well with analytical and published numerical results.Experiments of hydraulic fracturing of concrete are then simulated.It is found that fluid flowing in the FPZ has significant effects on the pattern of crack propagation and the peak fluid pressure.Brittle and quasi-brittle fracking are modelled by controlling the permeability of FPZ,and the results considering the effect of FPZ are in good agreement with experimental data.Horizontal wellbore fracking problems with parallel hydraulic cracks and random natural fractures are then simulated,with the effects of spacing,number,and angle of perforations investigated in detail.It is found that hydraulic cracks are deviated due to mutual influences which are weakened as the increase of spacing of perforations.（2）Numerical modelling of mesoscale fracture and/or fracking in heterogeneous quasi-brittle materials:Fracking is a multi-scale fracture problem,where the crack path is highly affected by material’s heterogeneity at mesoscale.Therefore,the developed method in（1）is combined with the mesoscale models of Weibull fields and random aggregates to understand the mechanisms of mesoscale fracking in heterogeneous materials.1800samples of Weibull random fields are generated for Monte Carlo simulations to investigate the effects of the variance and characteristic length of random fields.It is found that the simulated fluid pressure curves and crack paths are highly different due to the effect of randomness.The mean of peak pressures is decreased with the increase of the variance of random fields,while a larger characteristic length leads to a higher mean.Additionally,the simulation of fracking in random aggregate models shows hydraulic cracks tend to propagate along matrix-aggregate interfaces,and the predicted final crack path agrees reasonably with the result modelled by the cohesive zone model.Parametric studies on fluid rates and viscosities show the increase from 1.0 m Pa s to 0.1 Pa s of the fluid viscosity leads to the 41%rise of the peak pressure,but the fluid rate has less influence due to the use of quasi-static models.Afterwards,a 3D unified phase-field of fracture modelling is developed based on the2D computer codes,and used to model mesoscale fracture of concrete for validation.A 3D modelling method of fracking is then developed by combining the 3D framework with the2D theory in（1）.The method is used to simulate 3D fracking in horizontal well models,natural fracture models,random field models,and random aggregate mesoscale models.It is found that hydraulic cracks propagate in non-planar ways,due to the effects of confining stresses,mutual influences of cracks.Before the crack propagation,there are complex evolutions of FPZ around the crack tip of the quasi-brittle mesoscale models.（3）Dynamic cohesive phase field model and numerical algorithm for fracking in quasi-brittle materials:High-rate dynamic or pulsing fracking can generate more complex fracture networks than low-rate cases,with improved efficiency of fracking and exploitation.However,quasi-static models are inapplicable for the modelling of dynamic fracking.Therefore,a phase-field method of dynamic fracking is developed by combining the quasi-static model of（1）with the implicit Newmark algorithm.The method is first used to simulate dynamic fracture in homogeneous and heterogeneous materials under impact loadings,with simulated crack paths and loading-displacement curves agreeing well with the results modelled by the explicit unified phase-field model and experimental data.Dynamic consolidation of porous media and dynamic fracking of concrete are then simulated,where the predicted curves of pressure and displacement agree well with analytical solutions and numerical data simulated by the peridynamics model.And the peak pressure and crack path are close to the low-rate experimental result,as the gradual decrease of fluid injection rates.Additionally,high-rate and cyclic dynamic fracking problems are modelled,and parametric studies are conducted.It is found that faster fluid injection and smaller fracture energy leads to complex propagation patterns of hydraulic cracks with bifurcation.The elastic modulus and tensile strength have no evident influence on crack propagation.The decrease of the cyclic period from 1.0 ms to 0.01 ms results in the 27%reduction of the peak pressure.Lastly,3D fracking problems of horizontal wells with multiple boreholes are successfully simulated,based on the 3D computational framework of（2）.（4）Non-isothermal cohesive phase field model and numerical algorithm for fracking in quasi-brittle materials:Understanding the mechanisms of thermally-induced crack propagation,and temperature evolutions of fractured rocks and fluids is essential for geothermal exploitation.Therefore,a thermal-hydro-mechanical-damage cohesive phase-field method and numerical codes are developed by introducing the thermal energy of fluids and solids into the thermodynamic framework of（1）for non-isothermal fracking.In this method,a new sigmoid interpolation function is proposed to characterise the transition of fluid and temperature fields between cracks and porous media for permeability,thermal conductivity,and specific heat capacity.Unified governing equations of the fluid and temperature fields can thus be successfully built for modelling of fluid flow,thermal conduction and advection in fractured porous media.The method is first used to simulate quenching of high-temperature aluminium plates and thermal conduction and advection in cracking porous media,with simulated crack propagation and temperature evolution agreeing well with experimental and analytical data.High-temperature fracking experiments of hot-dry rocks are then simulated using mesoscale models of random fields.It is found that the initial thermal stress or damage around the rock borehole after heating is the major contributor to the decrease of peak pressures,which makes the rock be fractured easier.Single-/multi-crack fracking and fluid circulation problems in 3D models of geothermal systems with considerable complex natural fracture networks,are lastly modelled as an application for geothermal exploitation.It is found that natural fractures with weak tensile strength and fracture energy,have deviation effects on hydraulic cracks,causing complex crack paths.The production temperature of the production well connected with the injection well decreases faster than the unconnected one,and larger pressure gradients between the production well and the injection well make the production temperature drop more quickly.In conclusion,the developed numerical methods of this research are capable of modelling 2D/3D,single-/multi-crack,quasi-static/dynamic,isothermal/non-isothermal problems of hydraulic fracturing,with little sensitivity on the mesh size and phase-field length scale,with no need of remeshing,crack tracking,and criteria of crack initiation and propagation.It thus has the potential to be used for exploitation of shale oil/gas,geothermal energy,and safety control and design of hydraulic structures. |
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