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A semi-analytical model for predicting the mechanical responses around a non-circular wellbore by using complex variable method

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GEOMECHANICS AND GEOPHYSICS FOR GEO-ENERGY AND GEO-RESOURCES 2025年第1期11卷 1-23页

作者： Zhai, Wenbao Zhang, Haoze Yang, Zixuan Chen, Zhaowei Wu, Bisheng Zhou, Jiaxing Cheng, Shaoyi Tan, Peng Natl Engn Res Ctr Oil & Gas Drilling & Complet Tec Beijing Peoples R China CNPC Engn Technol R&D Co Ltd Beijing Peoples R China Tsinghua Univ State Key Lab Hydrosci & Engn Beijing Peoples R China Tsinghua Univ Key Lab Hydrosphere Sci Minist Water Resources Beijing Peoples R China Tsinghua Univ Dept Hydraul Engn Beijing Peoples R China

A good knowledge of in situ stresses, including their orientations and magnitudes at depth, is of critical importance in extracting unconventional oil and gas resources. In this paper, an efficient semi-analytical model is developed for accurately predicting the maximum horizontal principal stress magnitude based on the wellbore breakout shapes, which, unlike the vertical and minimum horizontal stresses, cannot be measured directly. In this paper, a semi-analytical model is developed based on the complex variable method to calculate the stresses and displacements around a non-circular wellbore and the Mohr-Coulomb criterion is used to evaluate the failure zones around the wellbore. The non-circular wellbore shape is directly used in the model for stress calculation, and the model requires only the outline coordinates of the hole, the elastic modulus of the material, and the magnitude and direction of the far-field stresses. The stresses obtained from the proposed semi-analytical model based on the complex variable method are compared against those obtained from the commercial finite element package ABAQUS, and the model is in excellent agreement with the finite element model for stress fields, showing its accuracy and efficiency. In addition, the model runs in a few seconds on a laptop computer, thus providing an effective technique for calculating stresses and displacements around a non-circular wellbore. It can be found that the stress distribution around the non-circular wellbore predicted from this model is different from those around a circular wellbore, especially when the wellbore has a large elongation. The proposed method will be useful to study the effect of the wellbore shape on the stress distribution around the wellbore and predict the evolution of wellbore breakout.

关键词： Mechanical responses Non-circular wellbore Semi-analytical solution complex variable method

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Ground displacements and stresses around a non-uniformly deforming circular tunnel in elastic media using complex variable method

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ACTA GEOTECHNICA 2023年第4期18卷 2169-2193页

作者： Jain, Hemant Kumar, Jyant Indian Inst Sci Civil Engn Dept Bengaluru 560012 India

A closed-form analytical solution using complex variables has been determined to predict ground displacements and stresses around a non-uniformly deforming single circular tunnel in a linearly elastic weightless and ponderable (weighty) medium. For the weightless medium, the solution has been derived with the satisfaction of (1) stress-free conditions on the boundary of the elastic half-space and (2) the displacements' boundary condition on the tunnel periphery. While for the ponderable medium, the solution has been obtained with the satisfaction of (1) geostatic stress conditions, (2) stress-free ground surface, and (3) non-uniformly deforming tunnel periphery but with an inclusion of the resultant buoyant force. The ground volume loss (GVL) is prescribed as an input in the form of a Gaussian distribution of the non-uniform radial displacements around the tunnel using the Fourier series. The displacements and stresses have been both evaluated. An elastic plane strain finite elements (FE) analysis has also been performed. The ground displacements from the proposed analytical formulation and the FE analysis were compared with each other and a close comparison between the two solutions has been clearly noted. The comparisons of the obtained solutions have also been provided with those observed in field from the literature. The proposed analytical formulation will form an accurate benchmark solution for determining the stresses and displacements in a linear elastic medium for a non-uniform deforming circular tunnel with a specified magnitude of GVL.

关键词： Analytical solution complex variable method Finite elements analysis Ground settlements Tunnels

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Effect of stacking sequence on thermal stresses in laminated plates with a quasi-square cutout using the complex variable method

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STRUCTURAL ENGINEERING AND MECHANICS 2021年第2期77卷 245-259页

作者： Chaleshtari, Mohammad H. Bayati Khoramishad, Hadi Iran Univ Sci & Technol Adhesively Bonded & Sandwich Struct Res Lab Sch Mech Engn Tehran 16846 Iran

In this research, the influence of the laminate stacking sequence on thermal stress distribution in symmetric composite plates with a quasi-square cutout subjected to uniform heat flux is examined analytically using the complex variable technique. The analytical solution is obtained based on the thermo-elastic theory and the Lekhnitskii's method. Furthermore, by employing a suitable mapping function, the solution of symmetric laminates containing a circular cutout is extended to the quasi-square cutout. The effect of important parameters including the stacking sequence of laminates, the angular position, the bluntness, the aspect ratio of cutout, the flux angle and the composite material are examined on the thermal stress distribution. It is found out that the circular shape for cutout may not necessarily be the optimum geometry for all stacking sequences. The finite element analysis results are used to validate the analytical solution.

关键词： stacking sequence thermal stress quasi-square cutout symmetric laminated composite complex variable method

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An analytical model for predicting the mechanical behavior of a deep lined circular pressure tunnel by using complex variable method

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GEOMECHANICS AND GEOPHYSICS FOR GEO-ENERGY AND GEO-RESOURCES 2021年第4期7卷 1-28页

作者： Guo, Zhijie Wu, Bisheng Jia, Shanpo Tsinghua Univ Dept Hydraul Engn State Key Lab Hydrosci & Engn Beijing 100084 Peoples R China Northeast Petr Univ Inst Unconvent Oil & Gas Daqing 163318 Peoples R China

A good understanding of the mechanical behavior around a tunnel is of importance for tunnel design. In this paper, an analytical model is developed to study the mechanical behavior, i.e. stress, displacement and stability, of a deep lined tunnel by combining the complex variable method with the Biot theory. The stress function is expressed in terms of two complex potential functions and the coefficients of the complex potential functions are determined by using the boundary conditions at the inner surface of the liner, rock/liner interface and far-field. The results from the proposed method are validated against those obtained from commercial finite element software, showing excellent agreement. Some scenarios are carried out to study the mechanical behavior around the tunnel, i.e. the stresses and displacements at the rock/liner interface and in the rock. In the end, the Mohr-Coulomb criterion is used to evaluate the stability of the lined tunnel. The results show that the effect of pore pressure on the failure behavior around the tunnel is significant as the smaller effective compressive stresses resulting from the pore pressure tend to cause smaller failure zone. This analytical model provides an efficient way to calculate the mechanical behavior and predicts the stability around the tunnel and can be extended easily to the case where the tunnel is non-circular.

关键词： Lined circular tunnel complex variable method Pore pressure Mohr-Coulomb criterion

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Influence of constant total hydraulic head on pore pressure and water inflow of grouted tunnel calculated by complex variable method

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TUNNELLING AND UNDERGROUND SPACE TECHNOLOGY 2023年第1期136卷

作者： Meng, Wei He, Chuan Zhou, Zihan Yan, Qixiang Yang, Wenbo Guo, Deping Chen, Ziquan Chongqing Jiaotong Univ Sch Civil Engn Chongqing 400074 Peoples R China Southwest Jiaotong Univ Key Lab Transportat Tunnel Engn Minist Educ Chengdu 610031 Si Chuan Peoples R China Xuzhen Railway Co Ltd Zhaotong 657900 Yunnan Peoples R China

Understanding the distribution of pore pressure and predicting water inflow are critical in tunnel engineering. In this study, the numerical solutions obtained by ABAQUS are compared with the theoretical solutions derived using the complex variable method (CVM). This comparison clarifies the influence of assuming a constant total hydraulic head along the interface between the grouted zone and soil on the theoretical solutions of pore pressure and water inflow calculated by the CVM. Results show that assuming a constant total hydraulic head can cause the negative and positive deviations with approximately equal areas in the total hydraulic head regardless of the permeability of the grouted zone and soil. Consequently, the theoretical solution invariably approaches the numerical solution for water inflow even when the deviation of the total hydraulic head is considerable. Relative permeability, kr/kg (where kg and kr represent the permeability coefficient of the grouted zone and soil, respectively), has a significant influence on the deviation of pore pressure. When the relative permeability is large, the pore pressure deviation is acceptable. As the relative permeability increases or the water depth (hw) decreases, the total hydraulic head and pore pressure deviations decrease gradually.

关键词： Constant total hydraulic head Pore pressure Water inflow complex variable method Grouted tunnel

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Modified complex variable method for displacement induced by surcharge loads and shallow tunnel excavation

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JOURNAL OF ENGINEERING MATHEMATICS 2020年第1期123卷 1-18页

作者： Lin, Luo-bin Chen, Fu-quan Li, Da-yong Fuzhou Univ Sch Civil Engn 2 Xueyuan Rd Fuzhou 350116 Peoples R China

In geo-engineering, mechanical problems of surcharge loads acting on ground surface and shallow tunnel excavation are often encountered. When the complex variable method is applied, such problems turn to non-zero resultant issues and generally result in infinite displacement singularity in geomaterial at infinity, which violates the fact that displacement components at infinity should be zero in field observations. The displacement singularity at infinity caused by non-zero resultants corresponds non-zero coefficients of logarithmic items in complex potentials. To eliminate the singularity at infinity, the symmetrical method is adopted so that the far-field displacement obtained from the modified displacement solution conducted by the complex variable method would be logically identical to the reality. Two fundamental cases of single surcharge load acting on ground surface and single tunnel excavation are presented to verify the modified displacement solution, and good agreements are observed. The main advantage that the proposed solution eliminates displacement singularity at infinity and makes far-field displacement logically accord with field observation is presented by comparing to the original solution. The drawback that the proposed solution is dependent on modified depth is discussed as well.

关键词： complex variable method Displacement Symmetrical modification Surcharge loads Shallow tunnel excavation

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complex variable method for equivalence of the elliptical inhomogeneity to Eshelby's elliptical inclusion under remote loading

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ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK 2019年第6期70卷 1-16页

作者： Chen, Y. Z. Jiangsu Univ Div Engn Mech Zhenjiang 212013 Jiangsu Peoples R China

Two boundary value problems are formulated in this paper. Among them, one is for elliptical inhomogeneity embedded in an infinite matrix and other is for Eshelby's elliptical eigenstrain inclusion in an infinite matrix. In both problems, the same remote loading is applied. Both problems are solved by using the complex variable method. In the first problem, the solution for stresses in the inhomogeneity depends on the elastic constants on two phases and the remote loading. In addition, the solution for stresses in the Eshelby's elliptical eigenstrain inclusion depends on the elastic constants for the matrix or the inclusion and the remote loading. It is known that the stress components in the inclusion are uniform for two problems. Letting the stress components in the inclusion for two problems be the same, the equivalent eigenstrains are evaluated, which has a linear relation with respect to the remote loading.

关键词： Inhomogeneity Eshelby's inclusion complex variable method Closed-form solution Numerical analysis

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Thermal stress analysis of orthotropic plate containing a rectangular hole using complex variable method

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EUROPEAN JOURNAL OF MECHANICS A-SOLIDS 2019年 73卷 212-223页

作者： Jafari, Mohammad Jafari, Mohammad Shahrood Univ Technol Mech Engn Dept Shahrood Iran Shahrood Univ Technol Fac Mech & Mechatron Engn Shahrood Iran

One of the most important issues in the design of engineering structures is the prevention of structural failure due to the stress caused by geometric discontinuities. Hole geometry is one of the effective parameters in the distribution of stress and displacement around it. In this study, using the two-dimensional thermoelastic theory and based on the Likhnitskii' complex variable technique, the stress analysis of orthotropic infinite plate with a circular hole under a uniform heat flux is developed to the plate containing a rectangular hole. To achieve this goal, a conformal mapping function is used to map an infinite plate containing a rectangular hole into the outside of a unit circle. Stress and displacement distributions around the rectangular hole in an orthotropic infinite plate are investigated in thermal steady state condition. The plate is under uniform heat flux at infinity and Neumann boundary conditions and thermal-insulated condition at the edge of the hole are considered. The rotation angle of the hole, fiber angle, the flux angle, bluntness and the aspect ratio of hole size are important parameters investigated in the present study. The obtained results show that these parameters have a significant effect on the stress and displacement distributions around the rectangular hole.

关键词： Infinite orthotropic plate Rectangular hole complex variable method Thermal stress

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APPLICATION OF THE HYBRID complex variable method TO THE ANALYSIS OF A CRACK AT A PIEZOELECTRIC-METAL INTERFACE

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JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES 2018年第4期13卷 587-605页

作者： Govorukha, Volodymyr Kamlah, Marc Oles Honchar Dnipro Natl Univ Dept Computat Math Dnipro Ukraine Karlsruhe Inst Technol Inst Appl Mat Eggenstein Leopoldshafen Germany

A plane strain problem for an electrically conducting interface crack between linear transversely isotropic piezoelectric and isotropic elastic conductor materials under remote mechanical loading is considered. The attention is focused on a hybrid complex variable method which combines the Stroh formalism for piezoelectric materials with the Muskhelishvili formalism for conducting isotropic elastic materials. This method is illustrated in detail for the open crack model and the contact zone crack model. Using special presentations of mechanical quantities via sectionally analytic functions, a combined Dirichlet- Riemann and Hilbert boundary value problem is formulated and solved analytically. Stress intensity factors as well as the crack tip energy release rate are found in a clear analytical form. Furthermore, transcendental equations for the determination of the realistic contact zone length and the location of the first interpenetration point have been obtained. A significant influence of the external mechanical loading on the crack opening and the stresses as well as the contact zone and interpenetration region lengths is observed. The dependencies of the mentioned values on the intensities of the mechanical loading are presented in tables and associated diagrams.

关键词： Piezoelectric-metal joint complex variable method electrically conducting interface crack contact zone

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Revisit of an elliptic hole problem by using complex variable method

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JOURNAL OF THE CHINESE INSTITUTE OF ENGINEERS 2016年第1期39卷 121-130页

作者： Chao, Ching-Kong Chen, Fu-Mo Natl Taiwan Univ Sci & Technol Dept Mech Engn Taipei Taiwan Nan Kai Univ Technol Dept Mech Engn Nantou Taiwan

In this paper, the problem of an elliptic hole embedded in an infinite plate interacting with an arbitrary point load is revisited by using the complex variable method. Based on analytical continuation theorem, the continuity conditions across the interface are automatically satisfied in a straightforward manner. It is shown that the solution for an infinite domain with an elliptic hole can be obtained from the solution of the corresponding homogeneous problem merely by a simple algebraic expression. This relation is universal in the sense of being independent of the loading considered. The solution of the corresponding homogeneous problem is considered as the principal part of the complex potentials while the complementary part of the complex potentials can be obtained by using analytical continuation theorem. Different expressions of the complementary part of the complex potentials are presented in this paper which are all proved to be the same result.

关键词： elliptic hole complex variable method analytical continuation

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