Traction Boundary Integral Equations Research Articles

Numerical study of simultaneous growth of multiple hydraulic fractures from a horizontal wellbore combining dual boundary element method and finite volume method

Horizontal well and multi-stage hydraulic fracturing have been extensively used to extract unconventional oil/gas. The fracture pattern created from the multiple fractures determines the surface area of the reservoir in contact with the wellbore, thus influencing the overall quality of the multi-stage fracturing. Therefore, a good understanding of the HF propagation behavior is of importance for designing the perforation layout. In this paper, a new fully coupled hydraulic fracturing model is developed by combining the dual boundary element method (DBEM) that describes the rock matrix deformation and the finite volume method (FVM) that simulates the fluid flow in the fracture. The DBEM applies displacement boundary integral equation to one crack surface and traction boundary integral equation to the other. The model presented is first validated against the existing semi-analytical and numerical solutions for single and two fractures in the infinite domain. Then the impact of spacing, in-situ stress difference and injection parameters on the competitive growth behavior between multiple HFs is investigated. A parameter formulated in terms of the ratio of fluid volume allocated to different fractures is proposed to be the indicator of simultaneous or preferential propagation. The results show that less isotropic in-situ stress and a larger fluid viscosity or injection rate tend to facilitate simultaneous growth of multiple fractures, in which the influence of the in-situ stress difference is enhanced while that of the injection parameters is weakened when increasing the fracture spacing.

Numerical study of hydraulic fracturing near a wellbore using dual boundary element method

The pattern and path of hydraulic fracture (HF) from a wellbore circumference have a great impact on subsequent formation of complex fracture networks highly desired for permeability enhancement of unconventional reservoirs. In this paper, a new fully coupled hydraulic fracturing model is developed in the framework of the dual boundary element method (DBEM) that describes the deformation of rock matrix and the finite volume method that simulates the fluid flow in the fracture and the distribution of injection rate among fractures. The DBEM applies displacement boundary integral equation to one crack surface and traction boundary integral equation to the other one. The model presented is first validated against the existing semi-analytical and numerical solutions for single fracture in an infinite domain, and shows excellent agreements with a relative error less than 1% for injection pressure and half fracture length. Then the problem of multiple HFs including natural fractures (NFs) cutting the wellbore is investigated. It is shown that a larger product of fluid viscosity and injection rate or a small in-situ stress difference can smooth the fracture path and enhance the fracture opening. In the presence of multiple fractures, the HF and NF competes in growth, and the results show that the initial permeability and inclination angle of NF can alter the propagation preference from HF to NF. In particular, a large fluid viscosity and a more isotropic stress condition will propagate both NFs and HFs simultaneously.

An improved singular curved boundary integral evaluation method and its application in dual BEM analysis of two- and three-dimensional crack problems

In this paper, an improvement is made to an efficient direct method for numerical evaluation of high order singular curved boundary integrals. Then this improved singular integral evaluation method is employed to solve the strongly and hypersingular integrals involved in the dual boundary element method (Dual BEM), which combine the use of displacement and traction boundary integral equations to solve crack problems in a single domain formulation. The singular integral evaluation method is carried out based on a parameter plane expansion and radial integral approach, this paper proposed a new strategy for treating the singular radial integral, which plays a vital role in this method. In isoparametric coordinate system, the singular curved boundary integral is mapped into a singular square plane integral in intrinsic coordinates, then the radial integration method (RIM) is employed to transform the singular square plane integral into a regular line integral over the contour of intrinsic square plane and a singular radial integral over the path from source point to the contour of intrinsic square plane. A singularity isolation technique is utilized to divide the singular radial integral into two parts, the regular radial integral can be evaluated normally using Gauss quadrature and the singular radial that can be evaluated analytically by expanding the non-singular part of the integrand function into a power series. Compared with conventional local interpolation approach to deal with the singular radial integral, the newly proposed method has a more rigorous mathematical derivation, and can achieve more stable and precise results. Based on the successful implementation of direct evaluation of singular boundary integrals, Dual BEM is successfully applied to solve two- and three-dimensional elastic crack problems including straight and curved crack paths with continuous or discontinuous elements. Two different approaches, geometrical extrapolation method and J-integral method are used in the evaluation of stress intensity factors. Several numerical examples are given to validate effectiveness of the presented method.

Boundary integral equations in the frequency domain for interface linear cracks under impact loading

The linear crack between two dissimilar elastic isotropic half-spaces under normal pulse loading is considered. The system of boundary integral equations for displacements and tractions in the frequency domain is derived from the dynamic Somigliana identity and adapted to solve the problem in the time domain. The numerical convergence of the method with respect to the number of the Fourier coefficients is proved. The effects of material properties of the bimaterial on the distribution of stress intensity factors (opening and transverse shear modes) are presented and analysed.

BIEM analysis of a graded nano‐cracked elastic half‐plane under time‐harmonic waves

AbstractThis work addresses the scattering of time‐harmonic waves by a finite, blunt nano‐crack in a graded isotropic bulk material with a free surface. The mechanical model is based on classical elastodynamic theory and describes the elastic, isotropic, graded half‐plane with quadratically varying material parameters along the depth coordinate. This model is extended to nano‐mechanics by using non‐classical boundary conditions and localized constitutive equations at the interface between crack and matrix material following the Gurtin‐Murdoch surface elasticity theory. The computational technique employed is based on the non‐hypersingular traction boundary integral equation method (BIEM) using a closed form Green's function for a half‐plane consisting of a functionally graded material (FGM). The dependences of the diffracted and scattered waves and of the local stress concentration fields on key problem parameters such as size and depth of the embedded crack, surface elasticity effects, type and characteristics of the incident wave, and the dynamic interaction phenomenon between the crack and the free‐surface boundary are all examined through parametric studies.

Dynamic Fracture Analysis of Plates Loaded in Tension and Bending Using the Dual Boundary Element Method

The purpose of this paper is to solve dynamic fracture problems of plates under both tension and bending using the boundary element method (BEM). The dynamic problems were solved in the Laplace-transform domain, which avoided the calculation of the domain integrals resulting from the inertial terms. The dual boundary element method, in which both displacement and traction boundary integral equations are utilized, was applied to the modelling of cracks. The dynamic fracture analysis of a plate under combined tension and bending loads was conducted using the BEM formulations for the generalized plane stress theory and Mindlin plate bending theory. Dynamic stress intensity factors were estimated based on the crack opening displacements.

Boundary element method analysis for mode III linear fracture mechanics in anisotropic and nonhomogeneous media

AbstractWe present here a Boundary Element Method implementation for anti‐plane fracture mechanics in nonhomogeneous and anisotropic media. The results are obtained using two different formulations based on displacement and traction Boundary Integral Equations (BIEs). The crack tips are discretized with special quarter‐point boundary elements. The main crack configurations are studied to determine the stress intensity factors (SIFs). Results for homogeneous and isotropic/anisotropic cases are compared with available solutions in the literature, showing good results. For the case of exponential nonhomogeneity, numerical results are obtained for moderately stiffer media and for softer media.

An efficient iteration approach for nonlinear boundary value problems in 2D piezoelectric semiconductors

• An efficient iteration approach is proposed to analyze boundary value problems in 2D PSCs. • By analyzing a plane problem, the proposed approach is verified with finite element method . • By analyzing an elliptical hole problem, the stress concentration phenomenon of the edge of elliptical hole is studied. • When an elliptical hole existing in PSCs is narrow enough, the mechanical-electric properties in PSCs remain unchanged. • The coupling effect among the mechanical field, electric field and electric current field is proved. Based on initial nonlinear constitutive equations, we establish the extended displacement and traction boundary integral equations for a piezoelectric medium with a volume electric charge, along with electron and electric current density boundary integral equations for a conductor with a volume electric current. Then, an iterative approach is proposed for investigation of boundary value problems in two-dimensional piezoelectric semiconductors (PSCs). Compared with extended displacements obtained by finite element analysis, this approach is validated via a rectangular PSC under extended external loads. Furthermore, as a numerical example, extended displacements across an elliptical hole in a rectangular PSC are investigated. It is shown that there is a stress concentration near the elliptical hole, which is closely dependent on its shape.

Dual boundary element method for analyzing three-dimensional cracks in layered and graded halfspaces

This paper presents a dual boundary element analysis of three-dimensional cracks in layered and graded halfspaces. The fundamental solution of a multilayered solid is used to develop the dual boundary element method so that only the external boundary surface and the crack surface need to be discretized while the material interfaces do not need to be discretized. Infinite boundary elements and crack-tip discontinuous elements are introduced to consider the far-fields of a layered halfspace and capture the crack-tip behavior, respectively. Special attentions are given to strongly singular and hypersingular integrals in the discretized displacement and traction boundary integral equations. For square-shaped, penny-shaped and elliptical cracks located in a homogeneous halfspace, the stress intensity factors obtained with the present formulation are in very good agreement with existing numerical results and closed-form solutions. The square-shaped cracks in horizontally layered halfspaces and the penny-shaped and elliptical cracks in graded halfspaces are analyzed. Results show that the material heterogeneity in layered and graded halfspaces can have a profound effect on the stress intensity factors.

Analysis for T-stress of cracks in 3D anisotropic elastic media by weakly singular integral equation method

This paper proposes an efficient numerical procedure for computing T-stresses of cracks in three-dimensional, linearly elastic, infinite media. The technique is established in a broad framework allowing a medium made of generally anisotropic materials and cracks of arbitrary shape and under general loading conditions to be treated. A pair of weakly singular, weak-form, displacement and traction boundary integral equations is utilized to formulate the key equations governing the unknown crack-face fields. Besides the basic benefits such as the reduction of spatial dimensions of the solution space and the efficient treatment of unbounded domains and remote boundary data, use of such integral equations in the formulation offers additional positive features including the computational simplicity resulting from the weakly singular nature of all involved integrals and the involvement of a complete set of crack-face displacement fields. A weakly singular symmetric Galerkin boundary element method together with the special near-front approximation is utilized to solve for the unknown relative crack-face displacement whereas the sum of the crack-face displacement is obtained from the displacement boundary integral equation for cracks via the Galerkin technique. The latter step is one of the essential aspects of the present study that provides the direct means for determining the T-stress data in terms of the sum of the crack-face displacement in the neighborhood of the crack-front. An extensive numerical study is conducted for various scenarios and a selected set of results is reported to demonstrate both the accuracy and capability of the proposed technique.

Dynamic fracture of a nano-cracked finite exponentially inhomogeneous piezoelectric solid

Aim of the study is to develop, verify and apply in simulations an efficient non-hypersingular traction boundary integral equation method for solution of anti-plane dynamic problem of a finite exponentially inhomogeneous piezoelectric solid with a nano-crack. The modeling approach is in the frame of continuum mechanics, wave propagation theory, the Gurtin and Murdoch surface elasticity theory and linear fracture mechanics. The simulations reveal the dependence of the stress concentration factors on the electromechanical coupling, on the type and characteristics of the dynamic load, on the position-dependent material properties, on the surface elasticity, on the size effect and on the wave–nanocrack–material gradient interaction in a bounded solid.

Dynamic stress intensity factors of interacting branched cracks

In the present work the boundary element method (BEM) is applied to analysis of statically and dynamically loaded infinite plates with multiple stationary branched cracks. The material of the plates is linear-elastic, homogenous and isotropic. In the applied BEM approach the displacement and traction boundary integral equations are used simultaneously for nodes on crack surfaces. Contrary to the finite element method (FEM) in the BEM numerical solutions are obtained by discretization of external boundaries and crack surfaces. The dynamic problem is solved by using the Laplace transform method and the solution in the time domain is computed by the Durbin numerical inversion method. Numerical examples of multiple branched cracks in infinite plates subjected to static and dynamic loadings are presented. An influence of orientation, distances between cracks and the number of cracks on static and dynamic stress intensity factors (SIF) is studied.

Dynamic dual boundary element analyses for cracked Mindlin plates

In this paper, a new dual boundary element formulation is presented for dynamic crack problems in Mindlin plates. The displacement and traction boundary integral equations are derived in the Laplace frequency domain to allow for a boundary-only formulation. The cracked plate is modelled with the dual boundary element method and dynamic plate bending stress intensity factors are evaluated. Four benchmark examples are presented including mode I and mixed mode deformation. Such stress intensity factors obtained are shown to be in excellent agreement with finite element results as well as published results.

Dynamic Response of a Cracked Viscoelastic Anisotropic Plane Using Boundary Elements and Fractional Derivatives

Abstract The aim of this study is to develop an efficient numerical technique using the non-hypersingular, traction boundary integral equation method (BIEM) for solving wave propagation problems in an anisotropic, viscoelastic plane with cracks. The methodology can be extended from the macro-scale with certain modifications to the nano-scale. Furthermore, the proposed approach can be applied to any type of anisotropic material insofar as the BIEM formulation is based on the fundamental solution of the governing wave equation derived for the case of general anisotropy. The following examples are solved: (i) a straight crack in a viscoelastic orthotropic plane, and (ii) a blunt nano-crack inside a material of the same type. The mathematical modelling effort starts from linear fracture mechanics, and adds the fractional derivative concept for viscoelastic wave propagation, plus the surface elasticity model of M. E. Gurtin and A. I. Murdoch, which leads to nonclassical boundary conditions at the nano-scale. Conditions of plane strain are assumed to hold. Following verification of the numerical scheme through comparison studies, further numerical simulations serve to investigate the dependence of the stress intensity factor (SIF) and of the stress concentration factor (SCF) that develop in a cracked inhomogeneous plane on (i) the degree of anisotropy, (ii) the presence of viscoelasticity, (iii) the size effect with the associated surface elasticity phenomena, and (iv) finally the type of the dynamic disturbance propagating through the bulk material.

A simplified evaluation of the mechanical energy release rate of kinked cracks in piezoelectric materials using the boundary element method

Based on the concept of the hoop field intensity factors of an initial crack prior to any kink, an apparent hoop mechanical (strain) energy release rate (MERR) is defined to approximate the MERR of a piezoelectric crack with an infinitesimal kink at any arbitrary angle. The validity and the efficiency of the simplified approximation are examined by numerical examples using the boundary element method (BEM). The generalized crack-opening-displacements or displacement jumps are computed by the traction boundary integral equations (BIEs). By using the displacement extrapolation method, the crack-tip field intensity factors of any arbitrarily kinked crack in linear piezoelectric materials are obtained and the BEM results are validated by comparing them with the available reference analytical results. Then, the differences between the conventional field intensity factors and MERR of an infinitesimally kinked crack and the hoop field intensity factors and hoop MERR of the main crack prior to any kink are numerically analyzed. Finally, the crack propagation in an infinite linear piezoelectric material is numerically simulated. The paths of the crack growth are predicted by adopting four different fracture criteria, namely, the maximum hoop stress intensity factor (SIF) and MERR fracture criteria for the main crack-tip before the next propagation, and the maximum KI and MERR fracture criteria for the kinked tip of the main crack with an infinitesimal branch at an arbitrary kinking angle evaluated by using a trial crack extension technique. The comparisons among these results show that the present simplified approximation can efficiently provide a sufficient accuracy for numerical simulation of crack growth in linear piezoelectric materials.

A synoptic approach to the seismic sensing of heterogeneous fractures: From geometric reconstruction to interfacial characterization

A non-iterative waveform sensing approach is proposed toward (i) geometric reconstruction of penetrable fractures, and (ii) quantitative identification of their heterogeneous contact condition by elastic waves in the frequency domain. To this end, the fracture support Γ (which may be non-planar and unconnected) is first recovered without prior knowledge of the interfacial condition by way of the recently established approaches to non-iterative waveform tomography of heterogeneous fractures, e.g. the methods of generalized linear sampling and topological sensitivity. Given suitable approximation Γ̆ of the fracture geometry, the jump in the displacement field across Γ̆ i.e. the fracture opening displacement (FOD) profile is computed from remote sensory data via a regularized inversion of the boundary integral representation mapping the FOD to remote observations of the scattered field. Thus obtained FOD is then used as input for solving the traction boundary integral equation on Γ̆ for the unknown (linearized) contact parameters. In this study, linear and possibly dissipative interactions between the two faces of a fracture are parametrized in terms of a symmetric, complex-valued matrix K(ξ) collecting the normal, shear, and mixed-mode coefficients of specific stiffness. To facilitate the high-fidelity inversion for K(ξ), a 3-step regularization algorithm is devised to minimize the errors stemming from the inexact geometric reconstruction and FOD recovery. The performance of the inverse solution is illustrated by a set of numerical experiments where a cylindrical fracture, endowed with two example patterns of specific stiffness coefficients, is illuminated by plane waves and reconstructed in terms of its geometry and heterogeneous (dissipative) contact condition.

An interaction integral and a modified crack closure integral for evaluating piezoelectric crack-tip fracture parameters in BEM

To evaluate the crack-tip field intensity factors of a piezoelectric crack with any inclined angle, the current widely-used interaction integral method (I-integral) is here extended to the boundary element applications under some coordinate transformations. As well, a new modified crack closure integral method (MCCI) is proposed by considering the discontinuous quarter-point singular elements for the crack-face discretization arising from the dual boundary element method (BEM). This dual BEM involves the strongly singular displacement boundary integral equations (BIEs) for the external boundary and the hypersingular traction BIEs for the crack faces. The crack-tip fracture parameters evaluated by the I-integral and MCCI are verified by the existing analytical solutions and meanwhile, compared with those results achieved by the classical displacement extrapolation method and the J-integral. Three examples are presented to show the high accuracy of the interaction integral method and the improvement of MCCI for the piezoelectric crack problems.

Grid-based volume integration for elasticity: Traction boundary integral equation

A volume integral algorithm for the non-homogeneous 3D elasticity traction boundary integral equation is presented. The body force volume integral is exactly split into a relatively simple boundary integral, together with a remainder volume integral that can be evaluated using a regular grid of cuboid cells covering the problem domain. Of particular importance for (inelastic) fracture analysis is that the volume integral over the regular grid is computed without explicit knowledge of the domain boundary, including the fracture surface. A Galerkin approximation is employed, and the numerical implementation is validated by solving body force elasticity problems with known solutions.

Modeling of multiple crack propagation in 2-D elastic solids by the fast multipole boundary element method

In this paper, a fast multipole boundary element method (BEM) is presented for modeling crack propagation in two-dimensional (2-D) linear elastic solids. A dual boundary integral equation (BIE) formulation using a linear combination of the displacement and traction BIEs is applied to model cracks in this BEM. Constant boundary elements are used to discretize the BIEs and the fast multipole method (FMM) is applied to accelerate the solution of the BEM system of equations. Numerical examples of multiple crack propagation in 2-D elastic domains and under cyclic loading, including perforated plates with multiple holes and cracks, are presented to show the effectiveness and efficiency of the developed fast multipole BEM.

A New and Efficient Boundary Element-Free Method for 2-D Crack Problems

An efficient boundary element-free method is established for 2-D crack problems by combining a pair of boundary integral equations and the moving-least square approximation. The displacement boundary integral equation is collated on the on-crack boundary, and a new traction boundary integral equation is applied on the crack surface without the separate consideration of the upper and lower sides. In virtue of integration by parts, only singularity in order 1/r is involved in the integral kernels of new traction boundary integral equation, which brings convenience to the numerical implementation. Meanwhile, the integration by parts produces the new variables, the displacement density, and displacement dislocation density, and they are the coexisting unknowns along with the displacement and displacement dislocation. With the high-order continuity of the moving-least square approximation, these new variables are directly approximated with the nodal displacement or displacement dislocation, and the final system of equations contains the unknowns of nodal displacements and displacement dislocations only. The boundary element-free computational scheme is established, and several examples show the efficiency and flexibility of the proposed method.