Numerical manifold method for thermo-mechanical coupling simulation of fractured rock mass

As a calculation method based on the Galerkin variation,the numerical manifold method(NMM)adopts a double covering system,which can easily deal with discontinuous deformation problems and has a high calculation accuracy.Aiming at the thermo-mechanical(TM)coupling problem of fractured rock masses,this study uses the NMM to simulate the processes of crack initiation and propagation in a rock mass under the influence of temperature field,deduces related system equations,and proposes a penalty function method to deal with boundary conditions.Numerical examples are employed to confirm the effectiveness and high accuracy of this method.By the thermal stress analysis of a thick-walled cylinder(TWC),the simulation of cracking in the TWC under heating and cooling conditions,and the simulation of thermal cracking of the SwedishÄspöPillar Stability Experiment(APSE)rock column,the thermal stress,and TM coupling are obtained.The numerical simulation results are in good agreement with the test data and other numerical results,thus verifying the effectiveness of the NMM in dealing with thermal stress and crack propagation problems of fractured rock masses.

A phase-field model for simulating the propagation behavior of mixed-mode cracks during the hydraulic fracturing process in fractured reservoirs

A novel phase-field model for the propagation of mixed-mode hydraulic fractures,characterized by the formation of mixed-mode fractures due to the interactions between fluids and solids,is proposed.In this model,the driving force for the phase field consists of both tensile and shear components,with the fluid contribution primarily manifesting in the tension driving force.The displacement and pressure are solved simultaneously by an implicit method.The numerical solution's iterative format is established by the finite element discretization and Newton-Raphson(NR)iterative methods.The correctness of the model is verified through the uniaxial compression physical experiments on fluid-pressurized rocks,and the limitations of the hydraulic fracture expansion phase-field model,which only considers mode I fractures,are revealed.In addition,the influence of matrix mode II fracture toughness value,natural fracture mode II toughness value,and fracturing fluid injection rate on the hydraulic fracture propagation in porous media with natural fractures is studied.

Fourth-order phase-field modeling for brittle fracture in piezoelectric materials

Failure analyses of piezoelectric structures and devices are of engineering and scientific significance.In this paper,a fourth-order phase-field fracture model for piezoelectric solids is developed based on the Hamilton principle.Three typical electric boundary conditions are involved in the present model to characterize the fracture behaviors in various physical situations.A staggered algorithm is used to simulate the crack propagation.The polynomial splines over hierarchical T-meshes(PHT-splines)are adopted as the basis function,which owns the C1continuity.Systematic numerical simulations are performed to study the influence of the electric boundary conditions and the applied electric field on the fracture behaviors of piezoelectric materials.The electric boundary conditions may influence crack paths and fracture loads significantly.The present research may be helpful for the reliability evaluation of the piezoelectric structure in the future applications.

Extended finite element-based cohesive zone method for modeling simultaneous hydraulic fracture height growth in layered reservoirs

In this study,a fully coupled hydromechanical model within the extended finite element method(XFEM)-based cohesive zone method(CZM)is employed to investigate the simultaneous height growth behavior of multi-cluster hydraulic fractures in layered porous reservoirs with modulus contrast.The coupled hydromechanical model is first verified against an analytical solution and a laboratory experiment.Then,the fracture geometry(e.g.height,aperture,and area)and fluid pressure evolutions of multiple hydraulic fractures placed in a porous reservoir interbedded with alternating stiff and soft layers are investigated using the model.The stress and pore pressure distributions within the layered reservoir during fluid injection are also presented.The simulation results reveal that stress umbrellas are easily to form among multiple hydraulic fractures’tips when propagating in soft layers,which impedes the simultaneous height growth.It is also observed that the impediment effect of soft layer is much more significant in the fractures suppressed by the preferential growth of adjoining fractures.After that,the combined effect of in situ stress ratio and fracturing spacing on the multi-fracture height growth is presented,and the results elucidate the influence of in situ stress ratio on the height growth behavior depending on the fracture spacing.Finally,it is found that the inclusion of soft layers changes the aperture distribution of outmost and interior hydraulic fractures.The results obtained from this study may provide some insights on the understanding of hydraulic fracture height containment observed in filed.

Improved Staggered Algorithm for Phase-Field Brittle Fracture with the Local Arc-Length Method

The local arc-length method is employed to control the incremental loading procedure for phase-field brittle fracture modeling.An improved staggered algorithm with energy and damage iterative tolerance convergence criteria is developed based on the residuals of displacement and phase-field.The improved staggered solution scheme is implemented in the commercial software ABAQUS with user-defined element subroutines.The layered system of finite elements is utilized to solve the coupled elastic displacement and phase-field fracture problem.A one-element benchmark test compared with the analytical solution was conducted to validate the feasibility and accuracy of the developed method.Our study shows that the result calculated with the developed method does not depend on the selected size of loading increments.The results of several numerical experiments show that the improved staggered algorithm is efficient for solving the more complex brittle fracture problems.

Effect of fracture fluid flowback on shale microfractures using CT scanning

The field data of shale fracturing demonstrate that the flowback performance of fracturing fluid is different from that of conventional reservoirs,where the flowback rate of shale fracturing fluid is lower than that of conventional reservoirs.At the early stage of flowback,there is no single-phase flow of the liquid phase in shale,but rather a gas-water two-phase flow,such that the single-phase flow model for tight oil and gas reservoirs is not applicable.In this study,pores and microfractures are extracted based on the experimental results of computed tomography(CT)scanning,and a spatial model of microfractures is established.Then,the influence of rough microfracture surfaces on the flow is corrected using the modified cubic law,which was modified by introducing the average deviation of the microfracture height as a roughness factor to consider the influence of microfracture surface roughness.The flow in the fracture network is simulated using the modified cubic law and the lattice Boltzmann method(LBM).The results obtained demonstrate that most of the fracturing fluid is retained in the shale microfractures,which explains the low fracturing fluid flowback rate in shale hydraulic fracturing.

Fracture propagation and evolution law of indirect fracturing in the roof of broken soft coal seams

Indirect fracturing in the roof of broken soft coal seams has been demonstrated to be a feasible technology.In this work,the No.5 coal seam in the Hancheng block was taken as the research object.Based on the findings of true triaxial hydraulic fracturing experiments and field pilot under this technology and the cohesive element method,a 3D numerical model of indirect fracturing in the roof of broken soft coal seams was established,the fracture morphology propagation and evolution law under different conditions was investigated,and analysis of main controlling factors of fracture parameters was conducted with the combination weight method,which was based on grey incidence,analytic hierarchy process and entropy weight method.The results show that“士”-shaped fractures,T-shaped fractures,cross fractures,H-shaped fractures,and“干”-shaped fractures dominated by horizontal fractures were formed.Different parameter combinations can form different fracture morphologies.When the coal seam permeability is lower and the minimum horizontal principal stress difference between layers and fracturing fluid injection rate are both larger,it tends to form“士”-shaped fractures.When the coal seam permeability and minimum horizontal principal stress between layers and perforation position are moderate,cross fractures are easily generated.Different fracture parameters have different main controlling factors.Engineering factors of perforation location,fracturing fluid injection rate and viscosity are the dominant factors of hydraulic fracture shape parameters.This study can provide a reference for the design of indirect fracturing in the roof of broken soft coal seams.

Effect mechanism of seepage force on the hydraulic fracture propagation

The flow of fluid through the porous matrix of a reservoir rock applies a seepage force to the solid rock matrix.Although the seepage force exerted by fluid flow through the porous matrix of a reservoir rock has a notable influence on rock deformation and failure,its effect on hydraulic fracture(HF)propagation remains ambiguous.Therefore,in this study,we improved a traditional fluid–solid coupling method by incorporating the role of seepage force during the fracturing fluid seepage,using the discrete element method.First,we validated the simulation results of the improved method by comparing them with an analytical solution of the seepage force and published experimental results.Next,we conducted numerical simulations in both homogeneous and heterogeneous sandstone formations to investigate the influence of seepage force on HF propagation.Our results indicate that fluid viscosity has a greater impact on the magnitude and extent of seepage force compared to injection rate,and that lower viscosity and injection rate correspond to shorter hydraulic fracture lengths.Furthermore,seepage force influences the direction of HF propagation,causing HFs to deflect towards the side of the reservoir with weaker cementation and higher permeability.

Borehole stability in naturally fractured rocks with drilling mud intrusion and associated fracture strength weakening:A coupled DFN-DEM approach

Borehole instability in naturally fractured rocks poses significant challenges to drilling.Drilling mud invades the surrounding formations through natural fractures under the difference between the wellbore pressure(P w)and pore pressure(P p)during drilling,which may cause wellbore instability.However,the weakening of fracture strength due to mud intrusion is not considered in most existing borehole stability analyses,which may yield significant errors and misleading predictions.In addition,only limited factors were analyzed,and the fracture distribution was oversimplified.In this paper,the impacts of mud intrusion and associated fracture strength weakening on borehole stability in fractured rocks under both isotropic and anisotropic stress states are investigated using a coupled DEM(distinct element method)and DFN(discrete fracture network)method.It provides estimates of the effect of fracture strength weakening,wellbore pressure,in situ stresses,and sealing efficiency on borehole stability.The results show that mud intrusion and weakening of fracture strength can damage the borehole.This is demonstrated by the large displacement around the borehole,shear displacement on natural fractures,and the generation of fracture at shear limit.Mud intrusion reduces the shear strength of the fracture surface and leads to shear failure,which explains that the increase in mud weight may worsen borehole stability during overbalanced drilling in fractured formations.A higher in situ stress anisotropy exerts a significant influence on the mechanism of shear failure distribution around the wellbore.Moreover,the effect of sealing natural fractures on maintaining borehole stability is verified in this study,and the increase in sealing efficiency reduces the radial invasion distance of drilling mud.This study provides a directly quantitative prediction method of borehole instability in naturally fractured formations,which can consider the discrete fracture network,mud intrusion,and associated weakening of fracture strength.The information provided by the numerical approach(e.g.displacement around the borehole,shear displacement on fracture,and fracture at shear limit)is helpful for managing wellbore stability and designing wellbore-strengthening operations.

Numerical modeling of fracture propagation of supercritical CO_(2)compound fracturing

The exploitation of shale gas is promising due to depletion of the conventional energy and intensification of the greenhouse effect.In this paper,we proposed a heat-fluid-solid coupling damage model of supercritical CO_(2)(SC-CO_(2))compound fracturing which is expected to be an efficient and environmentally friendly way to develop shale gas.The coupling model is solved by the finite element method,and the results are in good agreement with the analytical solutions and fracturing experiments.Based on this model,the fracture propagation characteristics at the two stages of compound fracturing are studied and the influence of pressurization rate,in situ stress,bedding angle,and other factors are considered.The results show that at the SC-CO_(2)fracturing stage,a lower pressurization rate is conducive to formation of the branches around main fractures,while a higher pressurization rate inhibits formation of the branches around main fractures and promotes formation of the main fractures.Both bedding and in situ stress play a dominant role in the fracture propagation.When the in situ stress ratio(δ_(x)/δ_(y))is 1,the presence of bedding can reduce the initiation pressure and failure pressure.Nevertheless,it will cause the fracture to propagate along the bedding direction,reducing the fracture complexity.In rocks without bedding,hydraulic fracturing has the lengthening and widening effects for SC-CO_(2)induced fracture.In shale,fractures induced at the hydraulic fracturing stage are more likely to be dominated by in situ stresses and have a shorter reorientation radius.Therefore,fracture branches propagating along the maximum principal stress direction may be generated around the main fractures induced by SC-CO_(2)at the hydraulic fracturing stage.When the branches converge with the main fractures,fracture zones are easily formed,and thus the fracture complexity and damage area can be significantly increased.The results are instructive for the design and application of SC-CO_(2)compound fracturing.

Modeling the Interaction between Vacancies and Grain Boundaries during Ductile Fracture

The experimental results in previous studies have indicated that during the ductile fracture of pure metals,vacancies aggregate and form voids at grain boundaries.However,the physical mechanism underlying this phenomenon remains not fully understood.This study derives the equilibrium distribution of vacancies analytically by following thermodynamics and the micromechanics of crystal defects.This derivation suggests that vacancies cluster in regions under hydrostatic compression to minimize the elastic strain energy.Subsequently,a finite element model is developed for examining more general scenarios of interaction between vacancies and grain boundaries.This model is first verified and validated through comparison with some available analytical solutions,demonstrating consistency between finite element simulation results and analytical solutions within a specified numerical accuracy.A systematic numerical study is then conducted to investigate the mechanism that might govern the micromechanical interaction between grain boundaries and the profuse vacancies typically generated during plastic deformation.The simulation results indicate that the reduction in total elastic strain energy can indeed drive vacancies toward grain boundaries,potentially facilitating void nucleation in ductile fracture.

Influence of High-Density Bedding Plane Characteristics on Hydraulic Fracture Propagation in Shale Oil Reservoir

The existence of high-density bedding planes is a typical characteristic of shale oil reservoirs.Understanding the behavior of hydraulic fracturing in high-density laminated rocks is significant for promoting shale oil production.In this study,a hydraulic fracturing model considering tensile failure and frictional slip of the bedding planes is established within the framework of the unified pipe-interface element method(UP-IEM).The model developed for simulating the interaction between the hydraulic fracture and the bedding plane is validated by comparison with experimental results.The hydraulic fracturing patterns in sealed and unsealed bedding planes are compared.Additionally,the effects of differential stress,bedding plane permeability,spacing,and the friction coefficient of the bedding plane are investigated.The results showed that a single main fracture crossing the bedding planes is more likely to form in sealed bedding planes under high differential stress.The decrease in bedding plane permeability and the increase in the friction coefficient also promote the fracture propagating perpendicular to the bedding planes.Shale with high-density bedding planes has a poorer fracturing effect than that with low-density bedding planes,as the hydraulic fracture is prone to initiate and propagate along the bedding planes.Moreover,higher injection pressure is needed to maintain fracture propagation along the bedding.An increase in bedding density will lead to a smaller fracturing area.Fracturing fluid seepage into the bedding planes slows shale fracturing.It is recommended that increasing the injection flow rate,selecting alternative fracturing fluids,and employing multi-well/multi-cluster fracturing may be efficient methods to improve energy production in shale oil reservoirs.

Study on the Impact of Massive Refracturing on the Fracture Network in Tight Oil Reservoir Horizontal Wells

Class III tight oil reservoirs have low porosity and permeability,which are often responsible for low production rates and limited recovery.Extensive repeated fracturing is a well-known technique to fix some of these issues.With such methods,existing fractures are refractured,and/or new fractures are created to facilitate communication with natural fractures.This study explored how different refracturing methods affect horizontal well fracture networks,with a special focus on morphology and related fluid flow changes.In particular,the study relied on the unconventional fracture model(UFM).The evolution of fracture morphology and flow field after the initial fracturing were analyzed accordingly.The simulation results indicated that increased formation energy and reduced reservoir stress differences can promote fracture expansion.It was shown that the length of the fracture network,the width of the fracture network,and the complexity of the fracture can be improved,the oil drainage area can be increased,the distance of oil and gas seepage can be reduced,and the production of a single well can be significantly increased.

Numerical investigation of geostress influence on the grouting reinforcement effectiveness of tunnel surrounding rock mass in fault fracture zones

Grouting is a widely used approach to reinforce broken surrounding rock mass during the construction of underground tunnels in fault fracture zones,and its reinforcement effectiveness is highly affected by geostress.In this study,a numerical manifold method(NMM)based simulator has been developed to examine the impact of geostress conditions on grouting reinforcement during tunnel excavation.To develop this simulator,a detection technique for identifying slurry migration channels and an improved fluid-solid coupling(FeS)framework,which considers the influence of fracture properties and geostress states,is developed and incorporated into a zero-thickness cohesive element(ZE)based NMM(Co-NMM)for simulating tunnel excavation.Additionally,to simulate coagulation of injected slurry,a bonding repair algorithm is further proposed based on the ZE model.To verify the accuracy of the proposed simulator,a series of simulations about slurry migration in single fractures and fracture networks are numerically reproduced,and the results align well with analytical and laboratory test results.Furthermore,these numerical results show that neglecting the influence of geostress condition can lead to a serious over-estimation of slurry migration range and reinforcement effectiveness.After validations,a series of simulations about tunnel grouting reinforcement and tunnel excavation in fault fracture zones with varying fracture densities under different geostress conditions are conducted.Based on these simula-tions,the influence of geostress conditions and the optimization of grouting schemes are discussed.

A method for predicting the water-flowing fractured zone height based on an improved key stratum theory

In the process of using the original key stratum theory to predict the height of a water-flowing fractured zone(WFZ),the influence of rock strata outside the calculation range on the rock strata within the calculation range as well as the fact that the shape of the overburden deformation area will change with the excavation length are ignored.In this paper,an improved key stratum theory(IKS theory)was proposed by fixing these two shortcomings.Then,a WFZ height prediction method based on IKS theory was established and applied.First,the range of overburden involved in the analysis was determined according to the tensile stress distribution range above the goaf.Second,the key stratum in the overburden involved in the analysis was identified through IKS theory.Finally,the tendency of the WFZ to develop upward was determined by judging whether or not the identified key stratum will break.The proposed method was applied and verified in a mining case study,and the reasons for the differences in the development patterns between the WFZs in coalfields in Northwest and East China were also fully explained by this method.

Multi-domain equivalent method for prediction of elastic modulus of complex fractured rock mass

Reliable estimation of deformation and failure behaviors of fractured rock mass is important for practical engineering design.This study proposes a multi-domain equivalent method for fracture network to estimate the deformation properties of complex fractured rock mass.It comprehends both the advantages of the discrete fracture network model and the equivalent continuum model to capture the features of discontinuities explicitly while reducing computational intensity.The complex fracture network is stochastically split into a number of subfracture networks according to the domain,length or angle.An analytical solution is derived to infer theoretically the relationship between the elastic moduli of the original complex fractured rock mass and the split subfractured rock masses by introducing a correction term based on the deformation superposition principle.Numerical simulations are conducted to determine the elastic moduli of split subfractured rock masses using universal distinct element code(UDEC),while the elastic modulus of the original model is estimated based on the currently proposed analytical relationship.The results show that the estimation accuracy with the current domainbased splitting model is far superior compared to those with the other two splitting models.Thus,the estimation method of elastic modulus of complex fractured rock mass based on domain splitting mode of fracture network is identified as the multi-domain equivalent method proposed in this paper.The reliability of this method is evaluated,and its high computational efficiency is demonstrated through exemplification with regard to different geometric configurations for stochastically artificial discrete fracture network.The proposed multi-domain equivalent method constructs the theoretical framework except for the regression analysis hypothesis compared to the density-reduced model equivalent method.

A new approach for flow simulation in complex hydraulic fracture morphology and its application:Fracture connection element method

Efficient flow simulation and optimization methods of hydraulic fracture morphology in unconventional reservoirs are effective ways to enhance oil/gas recovery.Based on the connection element method(CEM)and distribution of stimulated reservoir volume,the complex hydraulic fracture morphology was accurately described using heterogeneous node connection system.Then a new fracture connection element method(FCEM)for fluid flow in stimulated unconventional reservoirs with complex hydraulic fracture morphology was proposed.In the proposed FCEM,the arrangement of dense nodes in the stimulated area and sparse nodes in the unstimulated area ensures the calculation accuracy and efficiency.The key parameter,transmissibility,was also modified according to the strong heterogeneity of stimulated reservoirs.The finite difference and semi-analytical tracking were used to accurately solve the pressure and saturation distribution between nodes.The FCEM is validated by comparing with traditional numerical simulation method,and the results show that the bottom hole pressure simulated by the FCEM is consistent with the results from traditional numerical simulation method,and the matching rate is larger than 95%.The proposed FCEM was also used in the optimization of fracturing parameters by coupling the hydraulic fracture propagation method and intelligent optimization algorithm.The integrated intelligent optimization approach for multi-parameters,such as perforation number,perforation location,and displacement in hydraulic fracturing is proposed.The proposed approach was applied in a shale gas reservoir,and the result shows that the optimized perforation location and morphology distribution are related to the distribution of porosity/permeability.When the perforation location and displacement are optimized with the same fracture number,NPV increases by 70.58%,which greatly improves the economic benefits of unconventional reservoirs.This work provides a new way for flow simulation and optimization of hydraulic fracture morphology of multi-fractured horizontal wells in unconventional reservoirs.

A modified single edge V-notched beam method for evaluating surface fracture toughness of thermal barrier coatings

The surface fracture toughness is an important mechanical parameter for studying the failure behavior of air plasma sprayed(APS)thermal barrier coatings(TBCs).As APS TBCs are typical multilayer porous ceramic materials,the direct applications of the traditional single edge notched beam(SENB)method that ignores those typical structural characters may cause errors.To measure the surface fracture toughness more accurately,the effects of multilayer and porous characters on the fracture toughness of APS TBCs should be considered.In this paper,a modified single edge V-notched beam(MSEVNB)method with typical structural characters is developed.According to the finite element analysis(FEA),the geometry factor of the multilayer structure is recalculated.Owing to the narrower V-notches,a more accurate critical fracture stress is obtained.Based on the Griffith energy balance,the reduction of the crack surface caused by micro-defects is corrected.The MSEVNB method can measure the surface fracture toughness more accurately than the SENB method.

Blount Method in the Management of Supracondylar Fractures of the Humerus in Children

Supracondylar fractures of the humerus are the most common traumatic elbow injury in children. Several techniques are described in their support. Blount’s method is one of the widely used orthopedic means. The objective of our study was to evaluate the results of supracondylar fractures of the humerus treated by this method in our department. This was a descriptive, longitudinal study with the prospective collection over a period of 12 months from May 2018 to April 2019. It focused on children aged 0 to 15 years treated in the Orthopedic Traumatology Department of the CHU of Kati for a supracondylar fracture of the humerus in extension. A minimum follow-up period of 6 months was necessary for inclusion. 12 patients were collected. The sociodemographic, anatomopathologic, therapeutic and evolutionary aspects were studied. The results were evaluated according to Flynn’s criteria. The male sex was predominant with a ratio of 2. The average age was 5.5 years. The average admission time was 12.5 hours. Domestic accidents were the main etiology (66.66%) and Type II fracture was the most frequent with 58.33%. The duration of immobilization was 42 days for 11 patients. Complications were 1 case of secondary displacement and 1 case of 12&#730 flexion deficit. The average Baumann angle was 72&#730. The result was satisfactory for all our patients. Supracondylar fractures of the humerus are common in children, especially boys. The method of Blount when it is well carried out by respecting the indications gives good results.

Virtual fracture test of asphalt mixture based on discrete element method

In order to study fracture behaviors of asphalt mixtures, virtual tests of the two-dimensional（2D） microstructure based on the discrete element method（ DEM） are designed. The virtual structure of the 2D digital specimen of asphalt mixture is generated based on a particle generation program, in which the gradation and the irregular shapes of aggregates are considered. With the 2D digital specimens, a DEM-based mixture model is established and center-point beam fracture simulation tests are conducted by the DEM. Meanwhile, a series of calibration tests are carried out in laboratory to evaluate the DEM model and validate the methods of virtual fracture tests. The test results indicate that the fracture intensity of asphalt mixtures predicted by the DEM matches very well with the intensity obtained in laboratory. It is concluded that the microstructural virtual tests can be used as a supplemental tool to evaluate fracture properties of asphalt mixtures.