A variational approach to analyze a natural fault with hydraulic fracture based on the strain energy density criterion

A simplified analytical model of the interaction between a hydraulic fracture and an existing natural fault is developed. The mechanical activation of the natural fault as a result of contact with a pressurized fracture is described for plane strain conditions and quasi-static fracture propagation approximation. Using a variational approach, the normal and shear stresses, as well as the boundaries of the open and sliding zones along the fault, are predicted for three stages of the fracturing process (fracture approaching, coalescence, and fluid penetration). An accumulated concentration of shear stress at the tip of the fault’s sliding zone is shown to create sufficient tensile stress to initiate a new tensile crack on the opposite side of the fault, provided either the differential in situ stress is low or the friction coefficient is sufficiently large. The results of direct numerical simulation of the fracture interaction fit the model predictions made from the strain energy density fracture criterion.     

Evaluation of nonlocal approaches for modelling fracture near nonconvex boundaries

Integral-type nonlocal damage models describe the fracture process zones by regular strain profiles insensitive to the size of finite elements, which is achieved by incorporating weighted spatial averages of certain state variables into the stress–strain equations. However, there is no consensus yet how the influence of boundaries should be taken into account by the averaging procedures. In the present study, nonlocal damage models with different averaging procedures are applied to the modelling of fracture in specimens with various boundary types. Firstly, the nonlocal models are calibrated by fitting load–displacement curves and dissipated energy profiles for direct tension to the results of mesoscale analyses performed using a discrete model. These analyses are set up so that the results are independent of boundaries. Then, the models are applied to two-dimensional simulations of three-point bending tests with a sharp notch, a V-type notch, and a smooth boundary without a notch. The performance of the nonlocal approaches in modelling of fracture near nonconvex boundaries is evaluated by comparison of load–displacement curves and dissipated energy profiles along the beam ligament with the results of meso-scale simulations. As an alternative approach, elastoplasticity combined with nonlocal and over-nonlocal damage is also included in the comparative study.     

A phase-field modeling approach of hydraulic fracture in saturated porous media

Continuum porous media theories, extended by a diffusive phase-field modeling (PFM) approach, introduce a convenient and efficient tool to the simulation of hydraulic fracture in fluid-saturated heterogeneous materials. In this, hydraulic- or tension-induced fracture occurs in the solid phase. This leads to permanent local changes in the permeability, the volume fractions of the constituents as well as the interstitial-fluid flow. In this work, the mechanical behaviors of the multi-field, multi-phase problem of saturated porous media, such as the pore-fluid flow and the solid-skeleton deformation, are described using the macroscopic Theory of Porous Media (TPM). To account for crack nucleation and propagation in the sense of brittle fracture, the energy-minimization-based PFM procedure is applied, which approximates the sharp edges of the crack by a diffusive transition zone using an auxiliary phase-field variable. Furthermore, the PFM can be implemented in usual continuum finite element packages, allowing for a robust solution of initial-boundary-value problems (IBVP). For the purpose of validation and comparison, simulations of a two-dimensional IBVP of hydraulic fracture are introduced at the end of this research paper.     

Phase-field modeling of hydraulic fracture

In this work a theoretical framework implementing the phase-field approach to fracture is used to couple the physics of flow through porous media and cracks with the mechanics of fracture. The main modeling challenge addressed in this work, which is a challenge for all diffuse crack representations, is on how to allow for the flow of fluid and the action of fluid pressure on the aggregate within the diffuse damage zone of the cracks. The theory is constructed by presenting the general physical balance laws and conducting a consistent thermodynamic analysis to constrain the constitutive relationships. Constitutive equations that reproduce the desired responses at the various limits of the phase-field parameter are proposed in order to capture Darcy-type flow in the intact porous medium and Stokes-type flow within open cracks. A finite element formulation for the solution of the governing model equations is presented and discussed. Finally, the theoretical and numerical model is shown to compare favorably to several important analytical solutions. More complex and interesting calculations are also presented to illustrate some of the advantageous features of the approach.     

Self-similar solution for deep-penetrating hydraulic fracture propagation

The propagation of a vertical hydraulic fracture of a constant height driven by a viscous fluid injected into a crack under constant pressure, is considered. The fracture is assumed to be rectangular, symmetric with respect to the well, and highly elongated in the horizontal direction (the Perkins and Kern model). The fracturing fluid viscosity is assumed to be different from the stratum saturating fluid viscosity, and the stratum fluid displacement by a fracturing fluid in a porous medium is assumed to be piston-like. The compressibility of the fracturing fluid is neglected. The stratum fluid motion is governed by the equation of transient seepage flow through a porous medium.A self-similar solution to the problem is constructed under the assumption of the quasi-steady character of the fracturing fluid flow in a crack and in a stratum and of a locally one-dimensional character of fluid-loss through the crack surfaces. Crack propagation under a constant injection pressure is characterized by a variation of the crack sizel in timet according to the lawl(t)=l _o (1+At)^1/4, where the constantA is the eigenvalue of the problem. In this case, the crack volume isVl, the seepage volume of fracturing fluidV _f l ³, and the flow rate of a fluid injected into a crack isQ ₀l ^–1.     

Simulation of proppant transport at intersection of hydraulic fracture and natural fracture of wellbores using CFD-DEM

Proppants transport is an advanced technique to improve the hydraulic fracture phenomenon, in order to promote the versatility of gas/oil reservoirs. A numerical simulation of proppants transport at both hydraulic fracture (HF) and natural fracture (NF) intersection is performed to provide a better understanding of key factors which cause, or contribute to proppants transport in HF–NF intersection. Computational fluid dynamics (CFD) in association with discrete element method (DEM) is used to model the complex interactions between proppant particles, host fluid medium and fractured walls. The effect of non-spherical geometry of particles is considered in this model, using the multi-sphere method. All interaction forces between fluid flow and particles are considered in the computational model. Moreover, the interactions of particle–particle and particle–wall are taken into account via Hertz–Mindlin model. The results of the CFD-DEM simulations are compared to the experimental data. It is found that the CFD-DEM simulation is capable of predicting proppant transport and deposition quality at intersections which are in agreement with experimental data. The results indicate that the HF–NF intersection type, fluid velocity and NF aperture affect the quality of blockage occurrence, presenting a new index, called the blockage coefficient which indicates the severity of the blockage.     

水力劈裂问题的态型近场动力学建模

将态型近场动力学理论引入水力劈裂问题的模拟。构建了能反映岩土类材料准脆性断裂特征的态型近场动力学本构模型,并在物质点间相互作用力模型中加入等效水压力项,以实现在新生裂纹面上跟踪施加水压力。同时,考虑裂纹面间的接触,引入物质点间的短程排斥力作用,并设计了相应的接触算法。通过自编程序将模型和算法应用于含初始裂纹、不含初始裂纹以及含坝基软弱结构面的混凝土重力坝在高水头作用下的水力劈裂过程模拟,并与扩展有限元等模拟结果对比,验证了本文模型和算法的可行性和准确性。     

Gas-condensate flow in the vicinity of a hydraulic fracture

The problem of gas-condensate flow in the vicinity of a production well with a hydraulic fracture is considered. In the matrix, the flow is assumed to be three-dimensional, and at the fracture, it is assumed to be two-dimensional. It is shown that, for steady-state flow, the problem is split into a physicochemical problem (of phase transitions) and a filtration problem (of determining the pressure field). Numerical solutions are constructed for a rectangular fracture with finite and infinite conductivities. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 128–136, May–June, 2008.     

An asymptotic solution for fluid production from an elliptical hydraulic fracture at early-times

The early-time transient flow during the start-up of fluid production from a porous medium by a well intersected by a vertical elliptical hydraulic fracture is studied using an asymptotic analysis. The analysis is focused on the situation of practical interest where the fracture conductivity is high so that production from the fracture dominates. The first three terms in a short-time asymptotic expansion for the production rate during constant-pressure production, and for the well-pressure during constant-rate production, are obtained. It is shown that the fracture tip starts to influence the production rate only when the dimensionless time is increased to the square of the reciprocal of the dimensionless fracture conductivity. The asymptotic results also show that geometric factors of an elliptical fracture introduce non-negligible corrections to the so-called bilinear flow in the early times, which were previously erroneously associated with the effect of the fracture tip.     

Flow field modelling near a well with a conductive fracture

In the technology of oil recovery the oil production rate can be increased by generation of a vertical sand-filld conductive fracture on the wall of the well. Oil diffuses through the conductive fracture to the well. In this paper the seepage flow and isothermal deformation fields in both the formation and fracture and the oil production rate at the well are studied by modelling the formation as an infinite poroelastic medium saturated with a one-phase compressible fluid. The fracture is treated as a one-dimensional poroelastic medium. Darcy flows are considered in both the formation and fracture. The plane strain condition is imposed. Our solution is obtained numerically by a finite element method based on a variational principle. The accuracy of the analysis is studied by comparison of the numerical solutions of some problems with their analytical solutions. Since we are dealing with the transient flow problem of an infinite region, an extrapolation technique is employed to find the finite element solution. The production rate of a well with the conductive fracture is compared with that of a well without the conductive fracture.     

CDPM2: A damage-plasticity approach to modelling the failure of concrete

A constitutive model based on the combination of damage mechanics and plasticity is developed to analyse the failure of concrete structures. The aim is to obtain a model, which describes the important characteristics of the failure process of concrete subjected to multiaxial loading. This is achieved by combining an effective stress based plasticity model with a damage model based on plastic and elastic strain measures. The model response in tension, uni-, bi- and triaxial compression is compared to experimental results. The model describes well the increase in strength and displacement capacity for increasing confinement levels. Furthermore, the model is applied to the structural analyses of tensile and compressive failure.     

Three-dimensional ductile fracture analysis with a hybrid multiresolution approach and microtomography

A modified-JIC test on CT (compact tension) specimens of an alloy (Ti-Modified 4330 steel) was carried out. The microstructure (primary and secondary inclusions) in the fracture process zone and fracture surface are reconstructed with a microtomography technique. The zig-zag fracture profile resulting from nucleation of microvoid sheets at the secondary population of inclusions is observed. Embedding the experimentally reconstructed microstructure into the fracture process zone, the ductile fracture process occurring at different length scales within the microstructure is modeled by a hybrid multiresolution approach. In combination with the large scale simulation, detailed studies and statistical analysis show that shearing of microvoids (the secondary population of voids) determines the mixed mode zig-zag fracture profile. The deformation in the macro and micro zones along with the interaction between them affects the fracture process. The observed zig-zag fracture profile in the experiment is also reasonably captured. Simulations can provide a more detailed understanding of the mechanics of the fracture process than experiments which is beneficial in microstructure design to improve performance of alloys.     

On the problem of fluid leakoff during hydraulic fracturing

While a hydraulic fracture is propagating, fluid flow and associated pressure drops must be accounted for both along the fracture path and perpendicularly, into the formation that is fractured, because of fluid leakoff. The accounting for the leakoff shows that it is the main factor that determines the crack length. The solved problem is useful for the technology of hydraulic fracturing and a good example of mass transport in a porous medium. To find an effective approach for the solution, the thin crack is represented here as the boundary condition for pore pressure spreading in the formation. Earlier such model was used for heat conduction into a rock massif from a seam under injection of hot water. Of course, the equations have other physical sense and mathematically they are somewhat different. The new plane solution is developed for a linearized form that permits the application of the integral transform. The linearization itself is analogous to the linearization of the natural gas equation using the real gas pseudo-pressure function and where the flux rates are held constant and approximations are introduced only into the time derivatives. The resulting analytical solution includes some integrals that can be calculated numerically. This provides rigorous tracking of the created fracture volume, leakoff volume and increasing fracture width. The solutions are an advance over existing discreet formulations and allow ready calculations of the resulting fracture dimensions during the injection of the fracturing fluid.     

A 2-D lattice model for simulating the failure of paper

A new two-dimensional network model is proposed as a micromechanics model to simulate paper’s failure process due to sequentially breakages of fibers and/or bonds. Paper is approximated as a network composed of fibers any two of which link to each other by their intersecting point, namely so-called bond. Fibers distribute along three particular directions, leading to network’s macro-level isotropy. In the framework of finite element method, nodes correspond to fiber-to-fiber bonds, while elements are fiber segments between every two neighboring nodes and described by Timoshenko beam theory. Element breaks when its equivalent internal tensile stress reaches the tensile strength of fiber. Strength of nodes, i.e. fiber-to-fiber bonds is assumed to be dependant on shearing interaction between fibers, considering the dominant interaction is shearing in a plane problem. Numerical examples show the model’s capacity of reflecting basic failure characteristic in paper. Influences of fiber length and the ratio of fiber strength to bond strength are analyzed in detail.     

Aeroelastic study of flexible flapping wings by a coupled lattice Boltzmann-finite element approach with immersed boundary method

In this paper, the behavior of two-dimensional symmetric flapping wings moving in a viscous fluid is investigated. Harmonic motion is applied to idealize flying organisms with flexible wings and extensive testing is carried out to investigate the resultant flight behavior related to the ability to take-off or accelerate the flapping wing system away from a starting location. Special attention is paid to analyze the effect of the main mechanical parameters, as well as the effect of lateral wind on flight performances. Moreover, aiming to investigate the possible benefits of flying in flocks, a couple of synchronously flapping wings is considered in addition to the single arrangement. The numerical simulations are performed by solving the fluid–structure interaction problem through a strongly coupled partitioned approach. Fluid dynamics are modeled at the mesoscopic scale by the lattice Boltzmann method. The resulting macroscopic quantities are derived, as usual, based on the statistical molecular-level interpretation.Wings are modeled by geometrically nonlinear, elastic beam finite elements and structure dynamics is solved by the time discontinuous Galerkin method. Fluid–structure interface conditions are handled using the immersed boundary method. The resultant numerical approach combines simplicity and high computational efficiency. A Monte Carlo simulation strategy is employed to characterize the flight behavior subjected to lateral wind. Various scenarios are discussed.     

Self-similar solutions of the problem of formation of a hydraulic fracture in a porous medium

The problem of hydraulic fracture formation in a porous medium is investigated in the approximation of small fracture opening and inertialess incompressible Newtonian fluid fracture flow when the seepage through the fracture walls into the surrounding reservoir is asymptotically small or large. It is shown that the system of equations describing the propagation of the fracture has self-similar solutions of power-law or exponential form only. A family of self-similar solutions is constructed in order to determine the evolution of the fracture width and length, the fluid velocity in the fracture, and the length of fluid penetration into the porous medium when either the fluid flow rate or the pressure as a power-law or exponential function of time is specified at the fracture entrance. In the case of finite fluid penetration into the soil the system of equations has only a power-law self-similar solution, for example, when the fluid flow rate is specified at the fracture entrance as a quadratic function of time. The solutions of the self-similar equations are found numerically for one of the seepage regimes.     

Effective-medium treatment of flow through anisotropic fracture system — Improved permeability estimates using a new lattice mapping

The permeability tensor of a fractured reservoir, which will typically be anisotropic because of the presence of stress, is an important parameter to be taken into account when formulating a production strategy for the reservoir. Extensive computational effort is involved in calculating the permeabilities of model fracture systems by solving the fluid flow equations through finite realisations of the systems, and this renders a search for alternative techniques worthwhile. An attractive approach is to perform a rough mapping of the fracture system onto a lattice so that effective medium theory can be applied. For isotropic systems that are well-connected, this technique works well, but it gives increasingly poor results as the degree of anisotropy increases. In this contribution, a refinement of the lattice mapping is presented that incorporates an important aspect of the randomness present in the original system. This greatly increases the applicability of the technique.     

Nonlinear dynamics and stability of microstructures in a lattice model for ferroelastic materials

With the view of understanding how precise macroscopic properties of a material emerge from the underlying physics of homogeneous microstructures, a lattice model which can describe complex non-linear patterns made of elastic domains and interfaces is proposed. On the basis of a two-dimensional lattice model involving non-linear and competing interactions the dynamics of microstructure formation is examined. The emphasis is placed especially on an instability mechanism of a strain band producing localized domains. The influence of applied forces and dissipative effects on the dynamics of two perpendicular strain bands is studied. The results are interpreted as a microtwinning in crystalline alloys. The physical conjectures are checked by means of numerical simulations performed directly on the microscopic system.

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Sommario Si propone un modello reticolare che può descrivere complessi arrangiamenti fatti di domini elastici ed interfacce. Sulla base di un modello bidimensionale in cui sono presenti interazioni contrastanti e nonlineari si esamina la dinamica della formazione di microstrutture. L'accento è posto sui meccani'smi di instabilità che determinano bande di deformazione localizzata. Si studia l'influenza delle forze applicate e degli effetti dissipativi sulla dinamica di due bande perpendicolari e si interpretano i risultati come un microtwinning in leghe cristalline. Si verificano le congetture fisiche per mezzo di simulazioni numeriche del modello microscopico.

     

An energy-based crack growth criterion for modelling elastic–plastic ductile fracture

A crack growth criterion is derived based on the Griffith energy concept and the cohesive zone model for modelling fracture in elastic–plastic ductile materials. The criterion is implemented in the finite element context by a virtual crack extension technique. An automatic modelling of the ductile fracture process is realised by combining a local remeshing procedure and the criterion. The validity of the derived criterion is examined by modelling a compact tension specimen.