Higher Order Displacement Discontinuity Method Research Articles

An Efficient Model for Hydraulic Fracture Height Growth Considering the Effect of Bedding Layers in Unconventional Shale Formations

Summary Understanding and quantifying the effect of the weak bedding layers on the hydraulic fracture height growth in shale formations has always been challenging during the hydraulic fracturing design and operations. This work introduces a comprehensive and nonintrusive hydraulic fracture height growth model that accounts for both formation rock properties and the weak interfaces and can be incorporated into pseudo-3D hydraulic fracture simulators. The formation rock properties are handled by the equilibrium fracture height model. The shear slippage of the weak bedding layers is simulated by an efficient 2D higher-order displacement discontinuity method (HDDM), and the effectiveness on the hydraulic fracture height is quantified by correcting the stress intensity factors (SIFs) at the hydraulic fracture tips. The model is applied in the Permian Basin Wolfcamp formation to quantitatively investigate the effect of bedding layers on the hydraulic fracture height growth. Numerical studies show that the shear slippage of the beddings can significantly slow down the hydraulic fracture height growth and that the shear fracture toughness of the bedding layers and the spacing between the laminations have a considerable impact on the effectiveness of the shear slippage on the fracture height growth. Larger shear fracture toughness and smaller spacing of the bedding layers add more resistance to the fracture height propagation and require larger pressure to breakthrough. Additional results show that the effect of bedding layers is more significant in formations with low-stress contrast but less obvious in formations with high-stress contrast. The impact of the landing depth on the hydraulic fracture height growth is also investigated to provide crucial insights into the optimization of the landing depth in laminated formations.

A New Higher Order Displacement Discontinuity Method Based on the Joint Element for Analysis of Close-Spacing Planar Fractures

Summary The 2D displacement discontinuity method (DDM) has been widely used to characterize the hydraulic fracture geometry and the induced in-situ stresses in the oil and gas industry owing to its simplicity and accuracy. As smaller fracture spacing is used by multistage fracturing, the constant DDM (CDDM) loses its accuracy in predicting the fracture behaviors, especially for the inner fractures in a stage where they are subjected to the strong stress shadowing effect. In this paper, the 2D higher order DDM (HDDM) based on the joint elements was developed to overcome this limitation. The higher order displacement discontinuity intensively increases the accuracy of CDDM but maintains the same amount of computation time by using patched-element pattern. The joint elements are introduced to simultaneously determine the opening, shearing, and closing of each fracture element based on the stress boundary condition, which can avoid the “negative width” of the inner fractures given by CDDM which are mechanically closed under the strong stress shadowing effect. The developed 2D joint element HDDM (JE-HDDM) gives the same results with the CDDM when the fracture spacing is relatively large, but shows its outperformance in both efficiency and accuracy over the CDDM in predicting the displacement discontinuities and induced in-situ stresses in close fracture-spacing case.

Direct tensile strength measurement of granite by the universal tensile testing machine

The direct tensile strength of a typical hard rock like granite is measured by a novel apparatus known as compression-to-tensile load transfer (CTLT) device. The rock specimen is prepared in form of a slab containing a central hole and placed in the universal testing machine where the direct tensile stress can be applied to this specimen by implementing a special type of load transferring device which converts the applied compressive load to that of the tensile during the test. In the present work, some typical hard rock specimens of granite are specially prepared and tested in the laboratory to measure their direct tensile strengths. Then, a new load converting device implemented in the universal tensile testing machine is used to cause the rock specimen to be subjected to a direct tensile loading during the test. The compressive load was applied to the transferring device at the rate of 0.02 MPa/s. Numerical modeling of the tested specimens were accomplished using the discrete element method (DEM) and the higher order displacement discontinuity method (HODDM). The tensile failure of granite rock mainly occurs along the horizontal axis. The experimental results were in a good accordance with DEM results and HODDM outputs.

Applicability of the classical fracture mechanics criteria to predict the crack propagation path in rock under compression

A higher order displacement discontinuity method (HDDM) with quadratic elements is used in this study to evaluate the validity of classical fracture criteria in order to predict crack propagation path under mixed mode loading I–II in the case of a crack in an infinite body. Crack propagation path is predicted here by defining three stress states, uniaxial tensile stress (UTS), uniaxial compressive stress (UCS) and biaxial compressive stress (BCS). Comparing with experimental results, it was found that by increasing the crack inclination angle, the length of crack propagation path increases and the slope of this path decreases. For BCS state, G-criterion almost loses the validity and can be replaced with σ-criterion. S-criterion is strongly dependent on Poisson’s ratio and it is recommended to study fracture under brittle and ductile behaviour. Due to the complexity of crack tip element equations, the effect of this element on the results has been studied in detail and it was found that by considering the mid-point of the last element as a reference point to evaluate displacement discontinuities (Ds and Dn), there is no need to use special crack tip element. This procedure facilitates the modelling and shortens the solution.

A Comprehensive Numerical Study of Hydraulic Fracturing Process and Its Affecting Parameters

Production of hydrocarbons from shale formations has been made possible mainly due to hydraulic fracture (HF) technology. It increases the permeability of reservoir rocks around a well by connecting fractures and improving conductivity. HF behavior especially in presence of natural fractures have recently given much attention in studies. This paper investigates HF propagation and its associated parameters in various conditions. A higher order displacement discontinuity method is used to achieve higher accuracy in the results. First, behavior of crack opening displacement (COD) of an HF i.e. HF width in various conditions is studied. COD is a key parameter in determination of an HF operation success. It is proportional to production rate of oil and gas wells and provides a path for proppant entrance into the fractures. An equation considering many important parameters, based on numerous numerical modellings of various mechanical and geometrical effects on COD is proposed with coefficient of determination and standard error of 94.35% and 4.37 × 10−4 respectively. The next part of the paper studies the HF propagation in a naturally fractured reservoir. These natural fractures alter the stress fields and hence affect the propagation of a hydraulic fracture. In fact, it is shown that in certain orientations of hydraulic fractures and natural fractures, the effect of natural fractures disappear or completely changes propagation path. Using a combination of several interaction criteria, a new modeling of HFs and NFs interaction is presented. The modellings showed that spacing and angle of intersection can significantly affect HFs propagation. The results of COD and HF propagation in presence of natural fractures may be considered in HG design and primary orientation of perforated fractures.

On the accuracy of higher order displacement discontinuity method (HODDM) in the solution of linear elastic fracture mechanics problems

The higher order displacement discontinuity method (HODDM) utilizing special crack tip elements has been used in the solution of linear elastic fracture mechanics (LEFM) problems. The paper has selected several example problems from the fracture mechanics literature (with available analytical solutions) including center slant crack in an infinite and finite body, single and double edge cracks, cracks emanating from a circular hole. The numerical values of Mode I and Mode II SIFs for these problems using HODDM are in excellent agreement with analytical results (reaching up to 0.001% deviation from their analytical results). The HODDM is also compared with the XFEM and a modified XFEM results. The results show that the HODDM needs a considerably lower computational effort (with less than 400 nodes) than the XFEM and the modified XFEM (which needs more than 10000 nodes) to reach a much higher accuracy. The proposed HODDM offers higher accuracy and lower computation effort for a wide range of problems in LEFM.

Numerical investigation of effect of crack geometrical parameters on hydraulic fracturing process of hydrocarbon reservoirs

Hydraulic fracturing (HF), as a stimulation technique in petroleum engineering, has made possible the oil production from reservoirs with very low permeability. The combination of horizontal drilling and multiple HF with various perforation angles has been widely used to stimulate oil reservoirs for economical productions. Despite the wide use of HF, there are still ambiguous aspects that require more investigation. Therefore, optimizing the geometry of the initial fractures using numerical methods is of high importance in a successful HF operation. Different geometrical parameters of the initial HF cracks including patterns, spacings, crack lengths, and perforation phase angles were modeled using the higher order displacement discontinuity method (HODDM) in horizontal and vertical oil wells. Several well-known issues in HF such as crack interference and crack arrest were observed in certain patterns of the HF cracks. Also the best possible arrangements of the HF cracks were determined for a better production. The results obtained were verified by the in-situ measurements existing in the literature. In addition, the best perforation phase angle in vertical wells was investigated and determined.

Simulating the propagation of hydraulic fractures from a circular wellbore using the Displacement Discontinuity Method

Hydraulic fracturing as applied to well stimulation is analyzed. Propagation of one or more hydraulic fractures from a wellbore that is oriented with either the maximum or intermediate principal in situ stress is studied. A Higher Order Displacement Discontinuity Method (HODDM) with special crack tip elements is used for achieving higher accuracy in simulation of the process. The effect of fluid pressure inside pre-existing fractures is considered. A uniformly distributed internal pressure, which implies injection of a fluid with a very low viscosity is assumed. On the onset of initiation, the crack may reorient itself as it extends from the wellbore, until it becomes aligned with the preferred direction for crack propagation relative to the far-field stresses. Propagation of multiple cracks has been considered to study their interaction. In such cases, some cracks are unable to extend at all, or they arrest after limited propagation, but others can propagate in length relative to earlier developed fractures and even may intersect each other. Also two common perforation phases are studied and best orientations for each one is determined.

Fracture analyses of different pre-holed concrete specimens under compression

Crack propagation processes in specially prepared concrete discs and rectangular specimens containing a single cylindrical hole or multiple holes of varying diameters have been studied both experimentally and numerically. In this research, the cracks coalescence paths in Brazilian disc and rectangular specimens made from rock-like material containing multi-holes are investigated. These concrete specimens are specially prepared from an appropriate mixture of Portland Pozzolana Cement (PPC), fine sands, and water. The pre-holed Brazilian discs and rectangular specimens are experimentally tested under compression. The breakage load in the ring type disc specimens containing an axial hole with varying diameters is measured and the distribution of the induced lateral stress is obtained. The mechanism of cracks propagation in the wall of the ring type specimens is also studied. In the case of multi-hole Brazilian disc and rectangular specimens, the cracks propagation and cracks coalescence are also investigated. These experiments are numerically modeled by a modified higher order displacement discontinuity method. It has been shown that the corresponding experimental and numerical results are in good agreement with each other. The results presented in this research validate the accuracy and applicability of these crack analyses procedures.

Propagation mechanism of neighboring cracks in rock-like cylindrical specimens under uniaxial compression

The mechanism of cracks propagation and coalescence of neighboring cracks existing in precracked rock like cylindrical specimens has been studied experimentally and numerically by considering multiple cracks in the middle part of each specimen. The pre-cracked cylindrical specimens of rock like materials are experimentally tested under uniaxial compression in a rock mechanics laboratory. Then these experiments are numerically simulated by a modified higher order displacement discontinuity method (HODDM) using cubic displacement discontinuity elements and three special crack tip elements for crack tip behavior to increase the accuracy of the Mode I and Mode II stress intensity factors (SIFs) obtained based on linear elastic fracture mechanics (LEFM) theory. The crack propagation and coalescence paths of the internal inclined crack are estimated by implementing a suitable iteration algorithm of incremental crack length extension in a direction predicted by using the maximum tangential stress criterion. The numerical and experimental results are compared showing the validity, applicability and accuracy of the present work. Finally, a numerical simulation has been accomplished to study the effect of interaction among neighboring cracks on the SIFs.

Simulation of crack coalescence mechanism underneath single and double disc cutters by higher order displacement discontinuity method

The present research is focused on the numerical crack coalescence analysis of the micro-cracks and cracks produced during the cutting action of TBM disc cutters. The linear elastic fracture mechanics (LEFM) concepts and the maximum tangential stress criterion are used to investigate the micro crack propagation and its direction underneath the excavating discs. A higher order displacement discontinuity method with quadratic displacement discontinuity elements is used to estimate the stress intensity factors near the crack tips. Rock cutting mechanisms under single and double type discs are simulated by the proposed numerical method. The main purposes of the present modeling are to simulate the chip formation process of indented rocks by single and double discs. The effects of specific disc parameters (except speed) on the thrust force F r, the rolling force F r, and the specific energy E S are investigated. It has been shown that the specific energy (energy required to cut through a unit volume of rock) of the double disc is less than that of the single disc. Crack propagation in rocks under disc cutters is numerically modeled and the optimum ratio of disc spacing S to penetration depth P d (i.e. S/P d ratio) of about 10 is obtained, which is in good agreement with the theoretical and experimental results cited in the literature.

A boundary element analysis of crack-propagation mechanism of micro-cracks in rock-like specimens under a uniform normal tension

In this work, the mechanism for fracture of brittle substances such as rocks under a uniform normal tension is considered. The oriented straight micro-cracks are mostly created in all the polycrystalline materials resulting from the stress concentrations. The present work focuses on the interactions of the pre-existing micro-cracks, which can grow and propagate within a rock-like specimen. The micro-crack initiation and propagation in rock-like specimens is investigated using the Fortran Code TDDCRACK 2D , which is a 2D displacement discontinuity method (DDM) for crack analysis, a boundary element computer code based on the linear elastic fracture mechanics (LEFM) theory. In the present work, a higher order DDM is used to implement special crack tip elements for estimation of the stress intensity factors (SIFs) and crack initiation angles for the wing-crack problems initiated at different angles from the original micro-crack tips in an infinite specimen under a uniform tension.

A coupled experimental and numerical simulation of rock slope joints behavior

In this research, a coupled numerical-experimental analysis of crack propagation, crack coalescence, and breaking process of jointed rock slopes is performed by studying the mechanical behavior of pre-cracked brittle substances considering the specially prepared rock-like specimens (rock-like specimens are specially prepared by a proper mixing of Portland Pozzolana Cement (PPC), fine sand, and water in a rock mechanics laboratory) and natural rock slopes, simultaneously. The numerical analyses are accomplished using a numerical code based on the higher-order displacement discontinuity method (HODDM). A cubic displacement discontinuity variation along each boundary element is assumed to evaluate the mode I and mode II stress intensity factors (SIFs). Based on the LEFM theory, the maximum tangential stress criterion is implemented in the proposed code for predicting the crack initiation and its direction of propagation (crack coalescence path). Experimental tests are accomplished to evaluate the crack propagation process, crack coalescence, and final breakage path of rock-like specimens containing three parallel cracks (simulating the natural joint set in a rock slope). There is a good agreement between the numerical and experimental results obtained from the tested specimens demonstrating the accuracy and effectiveness of the proposed approach.

Erratum to: Cracks coalescence mechanism and cracks propagation paths in rock-like specimens containing pre-existing random cracks under compression

The mechanism of cracks propagation and cracks coalescence due to compressive loading of the brittle substances containing pre-existing cracks (flaws) was modeled experimentally using specially made rock-like specimens from Portland Pozzolana Cement (PPC). The breakage process of the specimens was studied by inserting single and double flaws with different inclination angles at the center and applying uniaxial compressive stress at both ends of the specimen. The first crack was oriented at 50° from the horizontal direction and kept constant throughout the analysis while the orientation of the second crack was changed. It is experimentally observed that the wing cracks are produced at the first stage of loading and start their propagation toward the direction of uniaxial compressive loading. The secondary cracks may also be produced in form of quasi-coplanar and/or oblique cracks in a stable manner. The secondary cracks may eventually continue their propagation in the direction of maximum principle stress. These experimental works were also simulated numerically by a modified higher order displacement discontinuity method and the cracks propagation and cracks coalescence were studied based on Mode I and Mode II stress intensity factors (SIFs). It is concluded that the wing cracks initiation stresses for the specimens change from 11.3 to 14.1 MPa in the case of numerical simulations and from 7.3 to 13.8 MPa in the case of experimental works. It is observed that cracks coalescence stresses change from 21.8 to 25.3 MPa and from 19.5 to 21.8 MPa in the numerical and experimental analyses, respectively. Comparing some of the numerical and experimental results with those recently cited in the literature validates the results obtained by the proposed study. Finally, a numerical simulation was accomplished to study the effect of confining pressure on the crack propagation process, showing that the SIFs increase and the crack initiation angles change in this case.

Experimental and numerical analysis of Brazilian discs with multiple parallel cracks

The pre-cracked Brazilian disc specimens of rock-like materials (Portland Pozzolana cement (PPC), fine sands, and water) are especially prepared in a rock mechanics laboratory to study the breaking process of brittle solids. The Brazilian discs may contain one, two, three, four, and five (parallel) center slant cracks (45° to the horizontal) under compressive line loading. The breaking load of the pre-cracked disc specimens is measured showing that as the number of cracks increases, the final breakage load of the specimen decreases. The experiments are carried out under compression (just like the Brazilian tests used for measuring the indirect tensile strength of intact rocks). It has been experimentally observed that the wing cracks are produced at the first stage of loading and start their propagation toward the direction of compressive line loading in the pre-cracked Brazilian discs. The same specimens are numerically simulated by a higher order displacement discontinuity method (HDDM). The effect of bridge area and orientation of cracks on the cracks coalescence and breakage path of the pre-cracked Brazilian discs specimens are simultaneously studied.

Numerical modeling of hydraulic fracture propagation and reorientation

Hydraulic fracture propagation, the prevailing method of reservoir stimulation, is controlled by the in situ reservoir stress state. This controls hydraulic fracture orientation and, for the most part, the created hydraulic fracture geometry. In this study, the boundary element method based on the displacement discontinuity formulation is presented to solve general problems of hydraulic fracturing propagation under biaxial loading. The crack tip element and a higher order displacement discontinuity method are used to study the hydraulic fracture propagation and reorientation. The maximum tangential stress criterion (or σ-criterion) is used to find the fracture path. The effect of fluid pressure and stress difference on the hydraulic fracture propagation is studied too. The developed numerical program (2DFPM) for modelling the reorientation process is verified by comparing its result with those of analytical and experimental tests, where a good agreement was observed. The numerical analysis showed that hydraulic fracture reorientation is effected remarkably by stress difference, fluid pressure and hydraulic fracture inclination.

On the HDD analysis of micro crack initiation, propagation, and coalescence in brittle materials

Micro cracks in brittle materials seem to be the main cause of their breakage under various loading conditions. In the present paper, the numerical and experimental analyses of initiation, propagation, and coalescence of micro cracks have been investigated. A higher order displacement discontinuity method with special crack tip elements is used to estimate the mixed-mode (mode I and mode II) stress intensity factors near the tips of the preexisting micro cracks. The propagation direction of micro cracks is estimated by using the maximum tensile stress criterion which is based on the linear elastic fracture mechanics concept. Some experiments are also carried out to observe the crack coalescence and crack propagation path of micro cracks. These experiments also verify the validity of the numerical results. For example, the crack initiation angles obtained numerically are compared with the corresponding experimental results showing the validity of these numerical results. Some of the numerical and experimental results are also compared with the corresponding results given in the literature.

A coupled numerical–experimental study of the breakage process of brittle substances

Pre-existing cracks in brittle substances seem to be the main cause of their breakage under various loading conditions. In the present paper, a coupled numerical–experimental analysis of crack propagation, cracks coalescence, and breakage process of brittle substances such as rocks and rock-like samples have been studied. The numerical analyses are accomplished using a numerical code based on the Higher order Displacement Discontinuity Method for Crack (HDDMCR2D) analysis. A quadratic displacement discontinuity variation along each boundary element is assumed to evaluate the Mode I and Mode II stress intensity factors. Based on the linear elastic fracture mechanics theory, the maximum tangential stress criterion (i.e., a mixed mode fracture criterion) is implemented in the HDDMCR2D code for predicting the crack initiation and its direction of propagation (cracks propagation path). Some numerical and analytical problems in finite and infinite planes are solved numerically by the proposed numerical method, and the results are compared in different tables illustrating the accuracy and validity of the numerical results. Experimental tests are also being done to evaluate the final breakage path and cracks initiation and cracks coalescence stresses in rock-like specimens containing two random cracks. The numerical and experimental results obtained from the tested specimens show a good agreement between the corresponding values and demonstrate the accuracy and effectiveness of the approach.

Numerical Crack Analysis of Blunt Rock Indenters by an Indirect Boundary Element Method

Linear elastic fracture mechanics principles are widely applied for the analysis of crack problems in rock fracture mechanics. Rock indentation is an important and complicated problem among rock engineering issues. In this paper, in addition to the fracture criterion of maximum tangential stress adjacent to crack tip, the higher order displacement discontinuity method (which is a version of the indirect boundary element method) has been used for modeling the crack propagation mechanism under blunt indenters. In order to achieve more accurate results, higher order boundary elements i.e. quadratic elements, has been used to calculate displacement discontinuities and also to reduce the singularities of stress and displacement fields near the crack tip, the special crack tip elements has been used to calculate the stress intensity factors (SIF) at the crack tips. In this modeling, the effect of crack angle on stress intensity factors has been investigated. The numerical results of stress intensity factors obtained from some example problems were compared to the theoretical and experimental results cited in the literature which always show a percentage error less than one percent. The simulated results may pave the way for increasing the efficiency of mining and drilling by improving the design of tools and indentation equipments.

Advanced Theoretical and Numerical Models of Failure Mechanism and Fracture Propagation Around the Boreholes in Hydro‐Fracturing Process of Hot Dry Rock Reservoirs

This paper describes progress on the development of theoretical models required for studying failure mechanism, crack initiation and growth around the boreholes driven by hydrofracturing processes in Hot Dry Rock (HDR) reservoirs of geothermal energy. Due to the importance of the stress intensity factor concept (K) in Fracture Mechanics, some advanced modeling techniques for accurate and fast determination of K for relevant problems are proposed. Alternative tools to deal with stress intensity factor determination are developed and assessed from the points of view of accuracy and computational cost. We concentrate on residual strength, crack initiation and crack growth as a means to model and understand experimentally observed behaviors. Several modeling methods such as compounding and weight function techniques, and boundary and finite element modeling for stress intensity factor calculation are discussed. Further to reviews of those techniques, work performed included (i) developing alternative solutions to deal with boundary‐to‐boundary interaction when using the compounding technique, (ii) relating the precision of K calculations with the level of precision of the crack opening displacement of a reference solution, in order to assess the precision of weight function technique, (iii) modeling relevant geometries using the finite element method (FEM), (iv) working on the implementation of direct stress intensity factor K determination in the Higher Order Displacement Discontinuity Method (HODDM), and (v) developing tools to deal with residual stress fields around the boundary of the hydraulically pressurized boreholes.