Irregular elastic half-plane gravity problem

Regional stress field as determined from electromagnetic radiation in a tunnel

Traction modifies the contact area and the vertical and horizontal stress distributions beneath the tyre

A detailed understanding of the present-day stress state is key to understanding naturally occurring seismicity and safely and successfully conducting subsurface operations. For example, in the case of geothermal energy applications, the role of faults and fractures in both productivity and induced seismicity critically depends on the present-day stress state. In addition, knowledge of the subsurface stress state is also of significant importance to mitigate drilling risks. Here, in particular, the least principal stress controls the maximum allowable wellbore pressure before the drilled formation is unintentionally fractured. In its simplest form, the state of stress can be described by the magnitude of vertical stress and two horizontal stresses and their azimuthal orientations. Ideally, the state of stress includes the counteracting effect of pore fluid pressure (short: pore pressure) and is described as the effective stress tensor (effective stress is the difference between stress and pore pressure).In this study, we investigate the magnitude of the least principal stress (minimum horizontal stress) in the North Alpine Foreland Basin in SE Germany using stress measurements such as Formation Integrity (FIT) and Leak-Off Tests (LOTs).Whilst pore pressure magnitudes have been extensively studied and published in numerous publications in the North Alpine Foreland Basin in SE Germany, knowledge of the prevailing least principal stress is still quite limited, particularly in overpressured formations. So far, only subsets of the available FIT/LOT data, mainly concentrated around Munich, have been investigated. Recently, additional FIT/LOT data became available covering greater depths (up to 4,2 km) and overpressured formations. We investigate this new dataset in combination with data from previous studies to establish a minimum horizontal stress gradient model, which considers both pore pressure and rock type. To do so, we consider the ratio between the measured minimum horizontal and vertical effective stress using a previously established pore pressure magnitude model. The resulting effective stress ratio model is tested against the least principal stress measurements of deep geothermal wells in the study area's hydrostatically and overpressured regions, showing that considering both lithological and pore pressure variations is necessary to predict the least principal stress magnitudes. The established model can be used to improve the efficiency and safety of future drilling campaigns in the study area and can also serve as an input for mechanical subsurface modelling, e.g. for a better understanding of deformation or natural and induced seismicity. 

Stress analysis of a finite plate with a rectangular hole subjected to uniaxial tension using modified stress functions

An analytical method for solving gravity-induced stresses in slope

Diffraction of P1, SV, or Rayleigh waves by an arbitrary-shaped canyon in an unsaturated half-plane

Here, analytical studies have been carried out to determine the optimal values of effective parameters on the stress concentration factor around a cutout using genetic algorithm. Optimum designs of single lamina as well as symmetric laminates with 4, 8 and 12 layers of graphite/epoxy and glass/epoxy plates containing a circular cutout with various sizes are presented. The work focuses on extending the analytical solution given by Greszczuk to determine the stress distribution in multilayered composite plates subjected to arbitrary in-plane loadings. This is achieved by introducing an arbitrary oriented uniaxial, biaxial and shear loading conditions into Greszczuk solution. In order to mimic as much as possible the real structural behavior, the finite-width correction factor given by Tan is used. Effective parameters on stress distribution around the circular cutout in composite plates considered as design variables include: load angle, fiber orientation, cutout size and stacking sequence of the laminate. The objective function in this study is the minimization of maximum stress concentration around the cutout which is calculated by the present analytical solution. The first ply failure load predicted using Tsai–Wu criterion is maximized for both single lamina and symmetric laminates. Also, the weight of the plates is minimized by increasing the hole size to width ratio. The results obtained by the present analytical solution compare favorably with those obtained using complex variable method. For laminated plate subjected to shear loading, the stress concentration factor decreased drastically by 48.79% compared to a single lamina. The failure load is also increased in most of the loading cases. The results also showed that the genetic algorithm code converges rapidly in most of the cases. The accuracy, quickness, low computational cost and the simplicity of the present solution encourage the designers to use it in practical applications.

The effect of geo-stress fields on macroscopic hydro-geological conditions or microcosmic permeability of water-bearing media should follow some laws or principles. Cases study and tests show that: (1) At macro-geologic large scale, deformed and crashed rocks which were induced by geo-stress fields changing provided space for groundwater storage and flow. Groundwater adjusts water-bearing space and dilatants fractures by flowing and press transferring. Coupling of liquid and solid can be implemented for rocks and groundwater. Although tectonic fields witness several times of change and build-up in geological time, stress fields forming regional tectonic framework are coherent with seepage fields, orientation of the maximum horizontal stress demonstrates main seepage directions. (2) At macro-geologic middle scale, zones of stresses changing sharply, quite low stresses, stress or shear concentration can be used to show locations and types of main fractures, zones of geo-stresses changing equably can be acted as normal base media zones of tri-porosity media. (3) At micro-geologic small scale, tri-porosity media include fractured rocks, porous rocks and capillary rocks. Investigations indicate that porosity or permeability is functions of effective stresses, and porosity or permeability changing rules of porous rocks with variation of effective stresses can be described as the index model, the model of power exponent functions is suitable for those of fractured rocks, the model of the second power parabola for capillary rocks. The porosity and permeability loss in fractured rocks, which are greater than that in porous rocks, are shown by calculation of effective compressive coefficient and closing pressure in cracks. The calculations can also explain the mechanism why porosity changes are always larger than permeability changes. It is proved by the thick wall cylinder theory that the second power parabola relation between porosity or permeability loss and effective stresses for capillary rocks is correct.

The complex variable method is used for an analytical solution for an arbitrary cavity in a homogeneous elastic half-plane. Proposing a method for solving the coefficients of the function by conformally mapping the region that contains an arbitrary cavity onto a circular ring utilizing a mapping function, which can be used for arbitrary shape cavity. Considering the stress-free condition of the upper surface, the relationship between two analytic functions is derived; consequently, the problem of solving two analytic functions is attributed to solving one analytic function, and the solve process for the analytic function in general case is given in detail, finally, achieving the solution of an arbitrary shape cavity in an elastic half-plane. The effects of cavity shape and depth were analyzed to the required terms of analytic function under a given accuracy. Examples for a vertical-wall semicircle cavity and a horseshoe cavity are given through the present method, and verification example is given by ANSYS software.

The frictionless contact problem for an elastic layer resting on an elastic half plane is considered. The problem is solved by using the theory of elasticity and integral transformation technique. The compressive loadsPandQ(per unit thickness in direction) are applied to the layer through three rigid flat punches. The elastic layer is also subjected to uniform vertical body force due to effect of gravity. The contact along the interface between elastic layer and half plane is continuous, if the value of the load factor,λ, is less than a critical value, . In this case, initial separation loads, and initial separation points, are determined. Also the required distance between the punches to avoid any separation between the punches and the elastic layer is studied and the limit distance between punches that ends interaction of punches is investigated for various dimensionless quantities. However, if tensile tractions are not allowed on the interface, for the layer separates from the interface along a certain finite region. Numerical results for distance determining the separation area, vertical displacement in the separation zone, contact stress distribution along the interface between elastic layer and half plane are given for this discontinuous contact case.

Geomechanical modeling using finite element method for prediction of in-situ stress in Krishna–Godavari basin, India

To understand the stress regime in the shallow portion of the Nankai Trough seismogenic zone, we carefully investigated the resistivity image logs at IODP Site C0009 that penetrated to 1600 meters below the sea floor (mbsf) in the central Kumano forearc basin. During the riser drilling, several downhole measurements were run including image logs, caliper and comprehensive geophysical logs sets. Borehole resistivity images obtained by a wireline‐logging tool were reprocessed in order to eliminate image artifacts generated by sticky tool movements, etc. The constraints on the possible magnitude and orientation of horizontal principal stresses are provided from the borehole images, rock strength, and logging data. The absence of borehole breakouts above 1285 mbsf represents a small difference (<10 MPa) between horizontal principal stresses and reflects the normal faulting and strike–slip faulting stress regimes. A stable, continuous breakout was observed in the resistivity image logs suggest that the stress state is in a strike–slip faulting and tending to reverse faulting regime below 1285 mbsf. The differential horizontal stress (up to 15 MPa) is significantly larger than in the shallow portion of the borehole (SHAMX ≫ Sv > Shmin).

Based on the comprehensive analysis of wellbore characteristics in a deep shale gas field, the in-situ stress state of the shale reservoir was assessed in this study for the Longmaxi formation in the Dingshan area, Southwestern China. The data obtained from hydraulic fracturing, drilling-induced fractures, and in-situ core testing were used to determine the magnitude and direction of the maximum principal horizontal stress in five wells. Besides, hydraulic fracturing and cross-multipole array acoustic logging (XMAC) were employed to determine the vertical variation of the in-situ stress. Based on the logging interpretation and mechanical test results, the spatial distribution of rock mechanical parameters in the Dingshan area was assessed by the amplitude variation versus offset (AVO) seismic inversion. A 3D heterogeneous mechanical inversion model was realized via the ANSYS and CATIA3D finite element software packages, providing the area in-situ stress field simulation. The depth, fault strike, and position change effects on the main stress, horizontal stress difference, and horizontal stress difference coefficient were numerically simulated. The results show that the maximum principal stress azimuth was mainly concentrated in the NE20°-NE80° sector. Moreover, the development zone of natural fractures was related to the area with the highest principal stress differences. The maximum principal stress variation in the study area was mainly in the compressive range from −135 to −45 MPa, gradually increasing from east to west and south to north. The stress type mainly depended on the depth, fault strike, and rock mechanical parameters, while the stress difference and stress difference coefficient near the fault structure were relatively small. This study’s findings are considered instrumental in improving the borehole stability, determining the casing setting point, and optimizing the well location in deep shale reservoirs with similar geological conditions.

Uncertainty in the maximum principal stress estimated from hydraulic fracturing measurements due to the presence of the induced fracture

The impact of shallow tunnel construction on the surrounding environment is often considered as a symmetric half-plane problem with circular holes. In this research, the analytical solutions of the ground stresses and displacements of a shallow circular tunnel in an elastic half-plane under arbitrary distributed loads on ground surface were derived, based on the complex variable method. Then, an application was implemented to analyze the potential plastic zone induced by shallow tunneling adjacent to the ground surface structures. The verification of the results obtained from the proposed analytical prediction model was carried out using the numerical simulations. Additionally, the influences of different boundary condition (different magnitudes and ranges of arbitrary distributed loads and different symmetric boundary conditions of the tunnel perimeter) on the distribution characteristics of the potential plastic zones were analyzed. In general, the results showed that the larger the pile loads and the closer the relative position between the tunnel and distributed loads, the more distinct the coalesced trends of the potential plastic zones around the tunnel and the potential plastic zones around the distributed loads.