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The unstable dynamic propagation of multistage hydrofracturing fractures leads to uneven development of the fracture network and research on the mechanism controlling this phenomenon indicates that the stress shadow effects around the fractures are the main mechanism causing this behaviour. Further studies and simulations of the stress shadow effects are necessary to understand the controlling mechanism and evaluate the fracturing effect.

In the process of stress-dependent unstable dynamic propagation of fractures, there are both continuous stress fields and discontinuous fractures; therefore, in order to study the stress-dependent unstable dynamic propagation of multistage fracture networks, a series of continuum-discontinuum numerical methods and models are reviewed, including the well-developed extended finite element method, displacement discontinuity method, boundary element method and finite element-discrete element method.

The superposition of the surrounding stress field during fracture propagation causes different degrees of stress shadow effects between fractures and the main controlling factors of stress shadow effects are fracture initiation sequence, perforation cluster spacing and well spacing. The perforation cluster spacing varies with the initiation sequence, resulting in different stress shadow effects between fractures; for example, the smaller the perforation cluster spacing and well spacing are, the stronger the stress shadow effects are and the more seriously the fracture propagation inhibition arises. Moreover, as the spacing of perforation clusters and well spacing increases, the stress shadow effects decrease and the fracture propagation follows an almost straight pattern. In addition, the computed results of the dynamic distribution of stress-dependent unstable dynamic propagation of fractures under different stress fields are summarised.

A state-of-art review of stress shadow effects and continuum-discontinuum methods for stress-dependent unstable dynamic propagation of multiple hydraulic fractures are well summarized and analysed. This paper can provide a reference for those engaged in the research of unstable dynamic propagation of multiple hydraulic structures and have a comprehensive grasp of the research in this field.

The purpose of this study is to investigate the unstable propagation of parallel hydraulic fractures induced by interferences of adjacent perforation clusters and thermal diffusion. Fracture propagation in the process of multistage fracturing of a rock mass is deflected owing to various factors. Hydrofracturing of rock masses in deep tight reservoirs involves thermal diffusion, fluid flow and deformation of rock between the rock matrix and fluid in pores and fractures.

To study the unstable propagation behaviours of three-dimensional (3D) parallel hydraulic fractures induced by the interferences of adjacent perforation clusters and thermal diffusion, a 3D engineering-scale numerical model is established under different fracturing scenarios (sequential, simultaneous and alternate fracturing) and different perforation cluster spacings while considering the thermal-hydro-mechanical coupling effect. Stress disturbance region caused by fracture propagation in a deep tight rock mass is superimposed and overlaid with multiple fractures, resulting in a stress shadow effect and fracture deflection.

The results show that the size of the stress shadow areas and the interaction between fractures increase with decreasing multiple perforation cluster spacing in horizontal wells. Alternate fracturing can produce more fracture areas and improve the fracturing effect compared with those of sequential and simultaneous fracturing. The larger the temperature gradient between the fracturing fluid and rock matrix, the stronger the thermal diffusion effect, and the effect of thermal diffusion on the fracture propagation is significant.

This study focuses on the behaviours of the unstable dynamic propagation of 3D parallel hydraulic fractures induced by the interferences of adjacent perforation clusters and thermal diffusion. Further, the temperature field affects the fracture deflection requires could be investigated from the mechanisms; this paper is to study the unstable propagation of fractures in single horizontal well, which can provide a basis for fracture propagation and stress field disturbance in multiple horizontal wells.

Moderately thick circular cylindrical shells are widely used as supporting structures or storage cavities in structural engineering, rock engineering, and aerospace engineering. In practical engineering, shells often work with micro-cracks or defects. The existence of micro-crack damage may result in the disturbance of dynamic behaviours and even induce accidental dynamic disasters. The free vibration frequency and mode are important parameters for the dynamic performance and damage identification analysis. In particular, stiffness weakening of the local damage region leads to significant changes in the vibration mode, which makes it difficult for the mesh generated in the conventional finite element method to capture a high-precision solution of the local oscillation.

In response to the above problems, this study developed an adaptive finite element method and a crack damage characterisation method for moderately thick circular cylindrical shells. By introducing the inverse power iteration method, error estimation, and mesh subdivision refinement technique for the analysis of finite element eigenvalue problems, an adaptive computation scheme was constructed for the free vibration problem of moderately thick circular cylindrical shells with circumferential crack damage.

Based on typical numerical examples, the established adaptive finite element solution for the free vibration of moderately thick circular cylindrical shells demonstrated its suitability for solving the high-precision free vibration frequency and mode of cylindrical shell structures. The any order frequency and mode shape of cracked cylindrical shells under the conditions of different ring wave numbers, crack locations, crack depths, and multiple cracks were successfully solved. The influences of the location, depth, and number of cracks on the disturbance of dynamic behaviours were analysed.

This study can be used as a reference for the adaptive finite element solution of free vibration of moderately thick circular cylindrical shells with cracks and lays the foundation for further development of a high-performance computation method suitable for the dynamic disturbance and damage identification analysis of general cracked structures.

With the development of fracturing technology, the research of multi-well hydrofracturing becomes the key issue. Frac-hits in multi-well hydrofracturing has an important effect on fracture propagation and final production of fractured well; in the process of hydrofracturing, there are many implement parameters that can affect frac-hits, and previous studies in this area have not systematically targeted the influence of a single parameter on multi-well hydrofracturing. Therefore, it is of great significance to study the occurrence rule and influence of frac-hits for optimizing the design of fracturing wells.

Based on the proposed numerical models, the effects of different fracturing implement parameters (perforation cluster spacing, well spacing and injection rate) on frac-hits are compared in numerical cases. Through the analysis of fracture network, stress field and microseismic, the effects of different fracturing implement parameters on frac-hits and connections are compared.

The simulation results show that the effect of perforation cluster spacing and well spacing on frac-hits is greater than that of injection rate. Smaller well spacing makes it easier for fractures between adjacent wells to interact with each other, which increases the risk of frac-hits and reduces the risk of fracture connections. Smaller perforation cluster spacing results in larger individual fracture lengths and greater deflection angles, which makes the possibility of frac-hits and connections greater. The lower the injection rate, the lower the probability of frac-hits.

In this study, the influence of different fracturing implement parameters on frac-hits and connections in multi-well hydrofracturing is studied, and the mechanism of frac-hits and connections is analyzed through fracture network, stress field and microseismic analysis. Different simulation results are compared to optimize fracturing well parameter design and provide reference for engineering application.

Multi-well hydrofracturing is a key technology in engineering, and the evaluation, control and optimization of the fracturing network determine the recovery rate of unconventional oil and gas production. In engineering terms, altering well spacing and perforation initiation sequences changes fracture propagation behavior. Fracture propagation can result in fracture-to-fracture and well-to-well interactions. This may be attributed to the interference between fractures caused by squeezing of the reservoir strata. Meanwhile, the stratal movement caused by the propagation of the fractures may lead to either the secondary fracturing of wells with primary fractures or perforation to begin fracturing. Besides, the stratal compression and squeeze of multi-well hydrofracturing will cause earthquakes; the fracture size is different owing to the different fracturing scenarios, and the occurrence of induced microseismic events is still unknown; microseismic events also affect fracture orientation and deflection. If the mechanism of the above mechanical behavior cannot be clarified, optimizing the fracture network and reduce the induced microseismic disaster becomes difficult.

In this study, combined finite element-discrete element models were used to simulate the multi-well hydrofracturing. Numerical cases compared the fracture network, dynamic stratal movement and microseismic events at 50, 75 and 100 m well spacings, respectively, and varying initiation sequence of multiple horizontal wells.

From the results, fracture propagation in multi-well hydrofracturing may simulate the propagation and deflection of adjacent fractures and induce fracture-to-fracture and well-to-well interactions. As the well spacing increases, the effect of fracturing-induced stratal movement and squeezing deformation decrease. In alternate fracturing, starting from a well located in the middle can effectively reduce the influence of stratal movement on fracturing, and the fracturing scenario with cross-perforation can minimize the influence of stratal movement. The stratal movement between multiple wells is positively correlated to microseismic events, which behaviors can be effectively weakened by reducing the strata movement.

The fracture network, thermal-hydro-mechanical coupling, fracturing-induced stratal movement and microseismic events were analyzed. This study analyzed the intersection and propagation behavior of fractures in multi-well hydrofracturing, which can be used to evaluate and study the mechanism of hydrofracturing fracture network propagation in multiple horizontal wells and conduct fracture optimization research to form an optimized hydrofracturing scheme by reasonably arranging the spacing between wells and initiation sequences of perforation clusters.

This study presents a novel hp-version adaptive finite element method (FEM) to investigate the high-precision eigensolutions of the free vibration of moderately thick circular cylindrical shells, involving the issues of variable geometrical factors, such as the thickness, circumferential wave number, radius and length.

An hp-version adaptive finite element (FE) algorithm is proposed for determining the eigensolutions of the free vibration of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation. This algorithm first develops the established h-version mesh refinement method for detecting the non-uniform distributed optimised meshes, where the error estimation and element subdivision approaches based on the superconvergent patch recovery displacement method are introduced to obtain high-precision solutions. The errors in the vibration mode solutions in the global space domain are homogenised and approximately the same. Subsequently, on the refined meshes, the algorithm uses higher-order shape functions for the interpolation of trial displacement functions to reduce the errors quickly, until the solution meets a pre-specified error tolerance condition. In this algorithm, the non-uniform mesh generation and higher-order interpolation of shape functions are suitable for addressing the problem of complex frequencies and modes caused by variable structural geometries.

Numerical results are presented for moderately thick circular cylindrical shells with different geometrical factors (circumferential wave number, thickness-to-radius ratio, thickness-to-length ratio) to demonstrate the effectiveness, accuracy and reliability of the proposed method. The hp-version refinement uses fewer optimised meshes than h-version mesh refinement, and only one-step interpolation of the higher-order shape function yields the eigensolutions satisfying the accuracy requirement.

The proposed combination of methodologies provides a complete hp-version adaptive FEM for analysing the free vibration of moderately thick circular cylindrical shells. This algorithm can be extended to general eigenproblems and geometric forms of structures to solve for the frequency and mode quickly and efficiently.

This study aimed to overcome the challenging issues involved in providing high-precision eigensolutions. The accurate prediction of the buckling load bearing capacity under different crack damage locations, sizes and numbers, and analysing the influence mechanism of crack damage on buckling instability have become the needs of theoretical research and engineering practice. Accordingly, a finite element method was developed and applied to solve the elastic buckling load and buckling mode of curved beams with crack damage. However, the accuracy of the solution depends on the quality of mesh, and the solution inevitably introduces errors due to mesh. Therefore, the adaptive mesh refinement method can effectively optimise the mesh distribution and obtain high-precision solutions.

For the elastic buckling of circular curved beams with cracks, the section damage defect analogy scheme of a circular arc curved beam crack was established to simulate the crack size (depth), position and number. The h-version finite element mesh adaptive analysis method of the variable section Euler–Bernoulli beam was introduced to solve the elastic buckling problem of circular arc curved beams with crack damage. The optimised mesh and high-precision buckling load and buckling mode solutions satisfying the preset error tolerance were obtained.

The results of testing typical examples show that (1) the established section damage defect analogy scheme of circular arc curved beam crack can effectively realise the simulation of crack size (depth), position and number. The solution strictly satisfies the preset error tolerance; (2) the non-uniform mesh refinement in the algorithm can be adapted to solve the arbitrary order frequencies and modes of cracked cylindrical shells under the conditions of different ring wave numbers, crack positions and crack depths; and (3) the change in the buckling mode caused by crack damage is applicable to the study of elastic buckling under various curved beam angles and crack damage distribution conditions.

This study can provide a novel strategy for the adaptive mesh refinement for finite element analysis of elastic buckling of circular arc curved beams with crack damage. The adaptive mesh refinement method established in this study is fundamentally different from the conventional finite element method which employs the user experience to densify the meshes near the crack. It can automatically and flexibly generate a set of optimised local meshes by iteratively dividing the fine mesh near the crack, which can ensure the high accuracy of the buckling loads and modes. The micro-crack in curved beams is also characterised by weakening the cross-sectional stiffness to realise the characterisation of locations, depths and distributions of multiple crack damage, which can effectively analyse the disturbance behaviour of different forms of micro-cracks on the dynamic behaviour of beams.

This study aims to analyse the deep resource mining that causes high in situ stress, and the disturbance of tunnelling and mining which may induce large stress concentration, plastic deformation and rock strata compression deformation. The depth of deep resources, excavation rate and multilayered heterogeneity are critical factors of excavation disturbance in deep rock. However, at present, there are few engineering practices used in deep resource mining, and it is difficult to analyse the high in situ stress and dynamic three-dimensional (3D) excavation process in laboratory experiments. As a result, an understanding of the behaviours and mechanisms of the dynamic evolution of the stress field and plastic zone in deep tunnelling and mining surrounding rock is still lacking.

This study introduced a 3D engineering-scale finite element model and analysed the scheme involved the elastoplastic constitutive and element deletion techniques, while considering the influence of the deep rock mass of the roadway excavation, coal seam mining-induced stress, plastic zone in the process of mining disturbance of the in situ stress state, excavation rate and layered rock mass properties at the depths of 500 m, 1,500 m and 2,500 m of several typical coal seams, and the tunnelling and excavation rates of 0.5 m/step, 1 m/step and 2 m/step. An engineering-scale numerical model of the layered rock and soil body in an actual mining area were also established.

The simulation results of the surrounding rock stress field, dynamic evolution and maximum value change of the plastic zone, large deformation and settlement of the layered rock mass are obtained. The numerical results indicate that the process of mining can be accelerated with the increase in the tunnelling and excavation rate, but the vertical concentrated stress induced by the surrounding rock intensifies with the increase in the excavation rate, which becomes a crucial factor affecting the instability of the surrounding rock. The deep rock mass is in the high in situ stress state, and the stress and plastic strain maxima of the surrounding rock induced by the tunnelling and mining processes increase sharply with the excavation depth. In ultra-deep conditions (depth of 2,500 m), the maximum vertical stress is quickly reached by the conventional tunnelling and mining process. Compared with the deep homogeneous rock mass model, the multilayered heterogeneous rock mass produces higher mining-induced stress and plastic strain in each layer during the entire process of tunnelling and mining, and each layer presents a squeeze and dislocation deformation.

The results of this study can provide a valuable reference for the dynamic evolution of stress and plastic deformation in roadway tunnelling and coal seam mining to investigate the mechanisms of in situ stress at typical depths, excavation rates, stress concentrations, plastic deformations and compression behaviours of multilayered heterogeneity.

Simultaneous hydrofracturing of multiple perforation clusters in vertical wells has been applied in the stimulation of hydrocarbon resources reservoirs. This technology is significantly impeded due to the challenges in its application to the multilayered reservoirs that comprise multiple interlayers. One of the challenges is the accurate understanding and characterization of propagation and deflection of the multiple hydraulic fractures between reservoirs and embedded interlayers.

Numerical models of the tight multilayered reservoirs containing multiple interlayers were established to study hydrofracturing of multiple perforation clusters and its influencing factors on unstable propagation and deflection of hydraulic fractures. Brittle and plastic multilayered reservoirs fully considering the influences of different in situ stress ratio and physical attributes for reservoir and interlayer strata on propagations of hydraulic fractures were investigated. The combined finite element–discrete element method and mesh refinement strategy were adopted to guarantee the accuracy of stress solutions and reliability of fracture path in computation.

Results show that the shear stress fields between adjacent multiple hydraulic fractures are superposed to cause fractures deflection. Stress shadows induce the shielding effects of hydraulic fractures and inhibit fractures growth to emerge unstable propagation behaviors, and a main single fracture and several minor fractures develop. As the in situ stress ratio increases, hydraulic fractures more easily deflect toward the direction of maximum in situ stress, and stress shadow and mutual interaction effects between them are intensified. Compared to brittle reservoir, plastic-enhanced reservoir may limit fracture growth and cannot form long fracture length; nevertheless, plastic properties of reservoir are prone to induce more microseismic events with larger magnitude.

The obtained fracturing behaviors and mechanisms based on engineering-scale multilayered reservoir may provide effective schemes for controlling and estimating the unstable propagation of multiple hydraulic fractures.

Supercritical CO₂ (SC–CO₂) fracturing is a potential technology that creates a complex fracturing fracture network to improve reservoir permeability. SC–CO₂-driven intersections of the fracturing fracture network are influenced by some key factors, including the disturbances generated form natural fractures, adjacent multi-wells and adjacent fractures, which increase the challenges in evaluation, control and optimization of the SC–CO₂ fracturing fracture networks. If the evaluation of the fracture network is not accurate and effective, the risk of oil and gas development will increase due to the microseismicity induced by multi-well SC–CO₂ fracturing, which makes it challenging to control the on-site engineering practices.

The numerical models considering the thermal-hydro-mechanical coupling effect in multi-well SC–CO₂ fracturing were established, and the typical cases considering naturally fracture and multi-wells were proposed to investigate the intersections and connections of fracturing fracture network, shear stress shadows and induced microseismic events. The quantitative results from the typical cases, such as fracture length, volume, fluid rate, pore pressure and the maximum and accumulated magnitudes of induced microseismic events, were derived.

In naturally fractured reservoirs, SC–CO₂ fracturing fractures will deflect and propagate along the natural fractures, eventually intersect and connect with fractures from other wells. The quantitative results indicate that SC–CO₂ fracturing in naturally fractured reservoirs produces larger fractures than the slick water as fracturing fluid, due to the ability of SC–CO₂ to connect macroscopic and microscopic fractures. Compared with slick water fracturing, SC–CO₂ fracturing can increase the length of fractures, but it will not increase microseismic events; therefore, SC–CO₂ fracturing can improve fracturing efficiency and increase productivity, but it may not simultaneously lead to additional microseismic events.

The results of this study on the multi-well SC–CO₂ fracturing may provide references for the fracturing design of deep oil and gas resource extraction, and provide some beneficial supports for the induced microseismic event disasters, promoting the next step of engineering application of multi-well SC–CO₂ fracturing.