Displacement Discontinuity Research Articles

Complex fracture network simulation and optimization in naturally fractured shale reservoir based on modified neural network algorithm

Creating the optimum Complex Fracture Network (CFN) during hydraulic fracturing by propagating Hydraulic Fractures (HF) stimulating and connecting the existing Natural Fractures (NF) is of great importance to fractured shale reservoirs' development. However, it yet haven't obtained enough research. Facing this urgent demand, a comprehensive CFN configuration, and optimization numerical investigation, considering geomechanics and economic effects, is proposed. An HF propagation model is constructed coupling the Displacement Discontinuity Method (DDM) and the fluid flow function, together with a power-law probability NF pattern, to simulate the CFN propagation considering the rock and fluid mechanics. Then the CFN productivity in shale reservoirs is simulated by the Embedded Discrete Fracture Model (EDFM) with shale gas flow model, in which the complexly distributed fractures and the adsorption and slippage effect of shale gas are described. Based on the above, a framework for CFN optimization is developed, in which the net present value (NPV) is adopted as the objective function to consider the economic viability. A state-of-the-art Neural Network Algorithm (NNA) is employed and further improved into a Modified Neural Network Algorithm (M-NNA) in this study, which presents 89% optimization searching accuracy tested by the complex multi-extreme benchmark function, far better than the 28% accuracy of the original PSO algorithm. The M-NNA is then adopted as the optimization algorithm in the CFN optimization framework. Finally, as an example, a case of CFN propagation and production is simulated under the coupling effects of geomechanical and fluid parameters, results show that the CFN provides nearly 90% of the total gas production; furthermore, the CFN for aligning and alternating fracturing placements are simulated and optimized considering the geological and economical influence, the results demonstrate that the alternating fracturing placement constructs better CFN configuration and leads to higher economic revenue.

Petroleum rock mechanics: An area worthy of focus in geo-energy research

Petroleum rock mechanics: An area worthy of focus in geo-energy research

Study on CO2 foam fracturing model and fracture propagation simulation

CO2 foam fracturing fluid has the advantages of water saving and environmental protection, which has been widely used in unconventional oil and gas reservoir. However, there are still many technical difficulties in fracture propagation model and numerical calculation method of CO2 foam fracturing. In this paper, a CO2 foam fracturing fracture propagation model with temperature-pressure-phase coupling is established. Physical parameters of CO2 are calculated by Span-Wagner method, and the finite difference and displacement discontinuity methods are used to solve the model. Moreover, we compare the results of this model with the field measured data, KGD model and EFRAC-3D model to verify the model. The computation results show that in the process of fracturing, improving the CO2 foam quality can significantly enhance the fracturing effect. When the quality increased from 0.5 to 0.8, the fracture width raised by more than 2 times. In addition, the fracture propagation is significantly affected by injection temperature. With the increase of injection temperature, fracture width decreases continuously, and if the CO2 foam is supercritical phase state, it is not conducive to increase the fracture width.

Effective elasticity of a medium with many parallel fractures

Summary We consider an alternative way of obtaining the effective elastic properties of a cracked medium. Similarly, to the popular linear-slip model, we assume flat, parallel fractures, and long wavelengths. However, we do not treat fractures as weakness planes of displacement discontinuity. In contrast to the classical models, we represent fractures by a thin layer embedded in the background medium. In other words, we follow the Schoenberg-Douma matrix formalism for Backus averaging, but we relax the assumptions of infinite weakness and marginal thickness of a layer so that it does not correspond to the linear-slip plane. To represent the properties of a fracture, we need a fourth order elasticity tensor and a thickness parameter. The effective tensor becomes more complicated, but it may describe a higher concentration of parallel cracks more accurately. Apart from the derivations of the effective elasticity tensors, we perform numerical experiments in which we compare the performance of our approach with a linear-slip model in the context of highly fractured media. Our model becomes pertinent if filled-in or empty cracks occupy more than one per cent of the effective medium.

An extended layerwise spectral finite element framework for delamination growth simulation in laminated composite strips

A computational framework is presented for the delamination propagation analysis in laminated composite strips that combines an extended high-order layerwise model and a spectral finite element with high-order spatial approximation in the plane of the structure. The extended layerwise laminate mechanics approximate the through-thickness fields using Hermite cubic splines, while the discontinuities in displacement induced by a delamination crack are treated as generalized damage degrees of freedom (DOFs). The layerwise mechanics are integrated into a novel spectral strip finite element, which involves integration points collocated with the nodes, thus providing accurate nodal predictions of the interlaminar stresses on the interface and ahead of the delamination crack tip. Strain energy release rate is predicted by adapting the VCCT method to rely exclusively on the damage DOFs. Finally, an iterative solution method is developed that takes advantage of the damaged DOFs to quickly predict the crack propagation without remeshing. The proposed numerical scheme is applied to mode I, II and mixed-mode delamination fracture problems and is validated vs. reference literature solutions, experimental results and standard 2D plane-strain FE models. The numerical results ultimately illustrate the capacity of the present method to provide accurate predictions with moderately coarse meshes and high computational speed.

Stress intensity factors and T-stresses by boundary integral equations: 3D statics

This paper presents the boundary integral equation method (BIEM) for the stress intensity factors and elasticity T-stresses evaluation in 3D problems. Flat rectangular, elliptic, penny-shaped cracks and rectangular crack on a cylindrical surface have been investigated. The hyper-singular integrals are treated with the Taylor’s series expansion of the kernel, and the Chebyshev polynomials of the second kind are used to solve the integral equations numerically. The stress intensity factors (SIFs) on the crack front are obtained by the coefficients of the Chebyshev polynomials. In order to verify the solutions by BIEM, the finite element method (FEM) with ABAQUS is conducted. The efficiency and convergence of the BIEM are observed in three examples. Comparisons are made with the analytical solutions in the stress intensity factor handbook and numerical solutions using the displacement discontinuity method (DDM).

Contribution of crystal orientation and grain boundary compliance to low shear velocity observed near base of polar ice sheets

SUMMARYSeismic shear wave velocity (S-velocity) shows a decrease towards the base of ice sheets in Antarctica and Greenland that is not accompanied by a corresponding decrease in compressional velocity (P-velocity). This decrease has been interpreted as arising from liquid water below the melting point (pre-melt water) at grain boundaries, but the lack of a corresponding decrease in P-velocity has not been explained. Representing grain boundaries as displacement discontinuities allows the change in P- and S-velocities to be written as functions of the normal and shear compliance of the grain boundaries. This allows the normal-to-shear compliance ratio of the grain boundaries to be constrained, and seismic anisotropy resulting from a partial orientation of grain boundaries to be estimated. This approach demonstrates that the observed reduction in S-velocity with no significant decrease in P-velocity near the base of ice sheets in Antarctica and Greenland can be explained by pre-melt water at small aperture grain boundaries. Such water may enable sliding along the grain boundaries and so may enhance creep of ice near the base of ice sheets. If stress state is anisotropic the aperture of water-containing grain boundaries may vary with azimuth, with the most open grain boundaries oriented with strikes perpendicular to least compressive stress. Microcracks and fractures may be treated also as displacement discontinuities and, together with oriented grain boundaries, may contribute to shear wave splitting as observed in West Antarctica in a fast-moving ice stream.

DDFS3D: A set of open-source codes leveraging hybrid 3D displacement discontinuity method and fictitious stress method to simulate fractures

Underground engineering is becoming increasingly sophisticated in recent years, exemplified by shale gas recovery, nuclear waste disposal and enhanced geothermal systems. Displacement Discontinuity Method (DDM) and Fictitious Stress Method (FSM) are two branches of boundary element method which can efficiently and accurately simulate fractures and openings embedded in large spatial domains. Nevertheless, in literature, studies engaging DDM/FSM to perform the aforementioned hot-topic research are rare, and one of the key reasons lies in the inconvenient access to DDM-FSM programs. In this study, we provide DDFS3D, a set of open-source codes incorporating the commonly used constant triangular and quadrilateral elements to implement 3D DDM, FSM and the hybrid of the two. The constant triangular element is reproduced from early studies, while the constant quadrilateral element is an independent work presented in this study. Gaussian quadrature and Hadamard finite-part integral are utilized to evaluate the regular and strongly singular integrals, respectively. To the authors’ best knowledge, DDFS3D is the first and the only open-source codes for 3D DDM-FSM implementation. We hope it could promote the application of DDM-FSM and advance the role of DDFS in coping with the ever-increasing new challenges in underground engineering.

Analysis of a penny-shaped crack with semi-permeable boundary conditions across crack face in a 3D thermal piezoelectric semiconductor

In this paper, we study a penny-shaped crack model with electrically and thermally semi-permeable boundary conditions in a three-dimensional transversely isotropic piezoelectric semiconductor. An extended displacement discontinuity boundary element method together with an iterative process is proposed to analyze the penny-shaped crack model. The extended displacement discontinuities across crack face, electric displacement and heat flux along an inner crack cavity, as well as extended stress intensity factors near crack front are obtained via the proposed method. The effects on extended intensity factors near crack front are discussed, including boundary conditions across crack face, applied loads and initial electron concentration. It is shown that boundary conditions across crack face significantly affect extended stress intensity factors near crack front. This implies that a larger initial electron concentration can lead to electrical failure.

Three-dimensional crack propagation and inclusion-crack interaction based on IGABEM

The isogeometric boundary element method (IGABEM) is developed to simulate the crack propagation and the inclusion-crack interaction in 3D infinite isotropic medium. The influence of complex shape inclusions on the stress intensity factors (SIFs) along the crack front is studied from the aspects of shape, stiffness, size and position. The non-uniform rational B-spline (NURBS) basis functions can be used to accurately describe the geometric shapes of inclusions and cracks, and the displacement, traction, and discontinuous displacement fields also can be approximated by the same NURBS basis functions. During crack propagation, the normal and tangential vectors of the crack boundary can be uniquely solved. Three examples verify the accuracy and effectiveness of the proposed method. The results show that the SIFs can be calculated accurately even using the single point formula, and the crack propagation process is stable and the path is smooth.

An exact solution for a partially clamped rectangle with a crack

The article deals with a boundary value problem for a rectangle whose horizontal sides are rigidly clamped, and the ends are free. In the centre of the rectangle, a vertical cut is made on which a discontinuity of the longitudinal displacements is given. An exact solution to the problem is constructed in the form of series in Papkovich–Fadle eigenfunctions. First, the corresponding boundary value problem for an infinite clamped strip is solved, then the solution for a rectangle is superimposed on this solution, with the help of which the boundary conditions at its ends are satisfied. Examples are given in which discontinuities of three types are considered which differ in the smoothness of the discontinuity contour near its ends.

Fracture Hits and Hydraulic-Fracture Geometry Characterization Using Low-Frequency Distributed Acoustic Sensing Strain Data

The propagation process and geometry of hydraulic fractures depend on complex interactions among the induced fractures and the pre-existing rock fabric, the heterogeneous rock properties, and the stress state. Accurate characterization of the resulting complex hydraulic-fracture geometry remains challenging. Fiber-optic-based distributed acoustic sensing (DAS) measurements have been used for monitoring hydraulic fracturing in adjacent treatment wells. DAS requires an optical fiber attached to the wellbore to transmit the laser energy into the reservoir. Each section of the fiber scatters a small portion of the laser energy back to a surface sensing unit, which uses interferometry techniques to determine strain changes along with the fiber. DAS data in offset wells fall in the low-frequency bands, which has been proven to be a powerful attribute for the characterization of the geometry of hydraulic fractures. Numerous recently published field examples demonstrate the potential of low-frequency DAS (LF-DAS) data for the detailed characterization of the hydraulic fracture geometry. Understanding the fracture-induced rock deformation associated with LF-DAS signals would be beneficial for the better interpretation of real-time data. However, interpretation of LF-DAS measurement is challenging due to the complexity of the subsurface conditions, in addition to potential unanticipated completion issues such as perforation failure, stage isolation failure, etc. All current research efforts focus on the qualitative interpretation of field data.In this study, we quantified the hydraulic fracture propagation process and described the fracture geometry by developing a geomechanical forward model and a Green’s function-based inversion model for the LF-DAS data interpretation, substantially enhancing the value of the LF-DAS data in the process. The work has a significant transformative potential, involving a tool package with developed forward and inversion models that can provide crucial insights for the optimization of hydraulic-fracturing treatments and reservoir development. Methodology The tool package can be used directly in the field to interpret LF-DAS data and monitor hydraulic fracture propagation. Raw data from the field measurement can be automatically processed. The geomechanics forward model we developed can quantify and analyze the strain-rate response from the LF-DAS measurements based on the 3D displacement discontinuity method. Fracture hits are detected by calculating three 1D features along the channel (location) axis, i.e., the maximum strain rate, the summation of strain rates, and the summation of strain-rate amplitudes. Channels with fracture hits usually exhibit significant peak values of these three features. We proposed general guide-lines for fracture-hit detection based on the quantitative analysis of strain/strain-rate responses during the multistage fracturing treatment. The details of the forward model can be found in Liu et al. (SPE 202482, 204457, AMRA-2020-1426). Additionally, we developed a novel Green’s function-based inversion model to qualify fracture width and height based on the determined fracture hits. The strain field that is estimated from the integration of the strain rates measured by the LF-DAS data along the offset monitoring well is related to the fracture widths through a geomechanics Green’s function. The resulting linear system of equations is solved by the least-square method. Details can be found in Liu et al. (SPE 204158, 205379, 204225).

Embedded PZT sensors for monitoring formation and crack opening in concrete structures

An application for sensing the stress-induced localization of damage in the form of a discrete crack in a concrete substrate is developed using embedded PZT patches. Stress-wave measurements are obtained through the stages of damage localization leading to crack formation, propagation, and opening of the discrete, localized crack in the concrete. A self-compensating attenuation factor is proposed to quantify the changes in stress waves produced by the material discontinuity introduced by cracking in concrete. The attenuation factor measurements provide a sensitive indication of displacement discontinuity introduced by a crack in the stress wave path. Displacement discontinuity in concrete associated with a crack opening smaller than 10 μm in the loaded state is detected in the attenuation factor measurements even after the removal of stress. The embedded PZT sensors can continuously monitor concrete structures for indications of crack formation before visual indication.

The compressive failure analysis of rock-like materials by experimental and numerical methods

Investigating the crack propagation mechanism is of paramount importance in analyzing the failure process of most materials. This process may be exposed during each kind of loading on the materials. In this work, the cracking mechanism in rock-like materials is studied using the numerical methods and compared with the experimental test results. However, the mechanism of crack growth in brittle materials such as rocks is influenced by different parameters. This research work focuses on the effect of the initial crack angles on the crack growth paths of these materials. Some cubic samples containing pre-existing cracks are tested in compression by considering different flaw orientations. The specimens are made of cement, water, and sand. Moreover, the mentioned process is numerically simulated using three different methods: the finite difference method for discontinuous bodies or discrete element method, the displacement discontinuity method, and the versatile finite element method. The micro-parameters for simulation are gained by the trial-and-error procedure for the discrete element method. Eventually, the crack growth paths observed in the experiments are compared with the numerically simulated models. The results obtained show that these central cracks propagate in two ways, which are dependent on their initial angle. By increasing the initial crack angle to greater than 30° (α > 30°), the wing crack path moves further away from the initial crack, and by decreasing α to smaller than 30° (α < 30°), only the shear cracks are initiated. Therefore, the validity and accuracy of the results are manifested by comparing all the corresponding results obtained by different methods. Based on these results, it can generally be concluded that the strength of the cubic (rock material) specimens increases with increase in the crack angles with respect to the applied loading direction.

Computational modeling of cracks different forms in three-dimensional space

All celestial bodies have defects in the form of cracks. Some of the bodies are large. Colliding with such objects can be dangerous. Therefore, the problem of grinding such bodies remains very urgent. In order to avoid large financial costs, you need to know where to strike in order to achieve the greatest effect. This paper studies the behavior of internal defects under various loads. A three-dimensional elastic medium weakened by a system of plane parallel cracks was examined in this study. A method of boundary elements called the method of discontinuous displacements, was chosen as the numerical method. The advantage of this method is that only the crack boundary is divided into elements, which allows the use of a small (compared to other numerical methods) amount of computer memory. The code was implemented in C ++. The interaction of two circular cracks was investigated depending on the size of the cracks and their relative locations. Calculating the singular integral is required when solving problems involving one crack. In case of several cracks, such integrals are not analytically incalculable. In this study, we present a method that does not require the calculation of singular integrals.

3D displacement discontinuity analysis of in-situ stress perturbation near a weak faul

A numerical investigation utilizing the 3D displacement discontinuity method is performed to examine the stress perturbations and induced displacements near a weak fault with arbitrary orientations and dip, assuming zero shear stress and normal displacement. The in-situ stress field near the fault is taken as known and varied with depth. The modelling is constructed based on indirect boundary integral equations. In this work, the fault plane is first modelled as a rectangular plane with negligible thickness between the adjacent surfaces. The fault plane is then divided into numerous rectangular boundary elements with imposed shear singularities on the surface, which is normal to the fault plane to simulate a traction-free scenario. The numerical results of the total induced stresses and displacements are then compared to the existing solutions of a penny-shaped crack for validation purpose. With validated results, the paper moves on to the discussion of various factors that have impacts on the induced stress and displacements, including: aspect ratio which is defined by strike over dip; orientation of the strike on the horizontal ground surface; as well as dip. The boundary integration method with modification is also used to model an elliptical distribution of singularities with inner, corner, and edge elements to accommodate more complex shape of a discontinuity; small differences are observed. Cited as: Chai, Y., Yin, S. 3D displacement discontinuity analysis of in-situ stress perturbation near a weak fault. Advances in Geo-Energy Research, 2021, 5(3): 286-296, doi: 10.46690/ager.2021.03.05

Transport and Localization of Elastic Waves in Two‐Dimensional Fractured Media: Consequences on Scattering Attenuation

AbstractWe present numerical simulations of elastic wave transport in two‐dimensional fractured media. Natural fracture systems, following a power‐law length scaling, are modeled by the discrete fracture network approach for geometrically representing fracture distributions and the displacement discontinuity method for mechanically computing fracture‐wave interactions. The model is validated against analytical solutions for wave reflection, transmission, and scattering by single fractures, and then applied to solve the wavefield evolution in synthetic fracture networks. We find that the dimensionless angular frequency ῶ = ωZ/κ plays a crucial role in governing wave transport, where ω, Z, and κ are the angular frequency, seismic impedance, and fracture stiffness, respectively. When ῶ is smaller than the critical frequency ῶc (≈5), waves are in the extended mode, either propagating (for small ῶ) or diffusing by multiple scattering (for intermediate ῶ); as ῶ exceeds ῶc, waves become trapped, entering either the Anderson localization regime (kl* ≈ 1) in well‐connected fracture systems or the weak localization regime (kl* &gt; 1) in poorly‐connected fracture systems, where k is the incident wavenumber and l* is the mean free path length. Consequently, the inverse quality factor Q−1 scales with ῶ obeying a two‐branch power‐law dependence, showing significant frequency dependence when ῶ &lt; ῶc and almost frequency independence when ῶ &gt; ῶc. Furthermore, when ῶ &lt; ῶc, the wavefield exhibits a weak dependence on fracture network geometry, whereas when ῶ &gt; ῶc, the fracture network connectivity has an important impact on the wavefield such that strong attenuation occurs in well‐connected fracture systems.

An evaluation of MITC and ANS elements in the nonlinear analysis of shell structures

The recently emerged idea of incorporating higher-order strain or displacement discontinuities into standard finite element interpolations has triggered the development of robust techniques, which allow efficient modeling of large deflections and rotations. A number of studies on efficient elements with embedded high-order discontinuities, including mixed interpolation tensorial components (MITC) and assumed natural strain (ANS) elements, were published during the past decade. It was verified that local enrichments of the displacement and/or strain interpolation can enhance the responses of displacements and stresses by efficient models. The multitude of methods proposed in the literature calls for a comparative study that would present the diverse approaches in a unified framework, point out their common features and differences, and find their limits of applicability. In this regard, by proposing two robust triangular elements based on the higher-order strain field, mixed and strain-based approaches are compared. The strengths and weaknesses of individual formulations are elucidated by analyzing the convergence behavior. After determining the optimal condition, the nonlinear theoretical analysis of functionally graded (FG) shells has been performed by employing the weighted arc-length method. The equilibrium path for load-bearing factor versus post-buckling capacity, the error rate path and the residual force norm are also presented in different material and boundary conditions.

A 3D thermo-hydro-mechanical coupling model for enhanced geothermal systems

Enhanced geothermal systems (EGS) is currently the most efficient method to mine heat from hot dry rock (HDR) reservoirs. The coupled thermo-hydro-mechanical (THM) effect therein is of critical importance to predict the long-term performance of EGS. A novel 3D THM coupling model integrating displacement discontinuity method (DDM), finite volume method (FVM) and finite element method (FEM) is developed in this study to aid such prediction. The present model is first validated against analytical solutions, and then applied to a penny shape EGS to simulate a 30-year prolonged injection. The simulation results reveal that EGS maintains high productivity during the early injection stage but once thermal drawdown takes place, the EGS productivity degrades rapidly. The fracture aperture model employed in the present study is capable of simulating hydraulic fractures in which the fluid pressure is large enough to overcome the normal stress exerted to the fracture faces and thus jack the fracture open. In contrast, most early studies adopted a compressive aperture model, where a substantial part of the normal stress is borne by fracture gouge/roughness. The strength of DDM + FVM + FEM scheme lies in the efficiency and versatility of simulating fractures. Our study indicates that the DDM + FVM + FEM scheme has high potential to serve as a competent alternative to performing the THM coupling simulation in EGS.

Variable subset DIC algorithm for measuring discontinuous displacement based on pixel-level ZNCC value distribution map

Digital image correlation (DIC) is one of the most used techniques for measuring deformation. It is widely used in mechanical performance testing, structural health monitoring and many other fields. The standard subset based DIC technology assumes that the deformation is continuous. However, discontinuous deformation such as cracks and dislocations are quite common in practical applications. The measurement accuracy of DIC in these areas will be significantly reduced, or even measurement failure. This is a limitation for the application of DIC. To overcome this problem, this paper proposes a variable subset DIC algorithm. Taking the zero-mean normalized cross-correlation (ZNCC) function as the correlation criterion, the concept of pixel-level ZNCC value distribution map is proposed. The proposed algorithm can automatically select the most suitable subset for matching under the guidance of the pixel-level ZNCC value distribution map. This method has the advantages of good flexibility, strong applicability, and high precision.