Two-Scale Concurrent Topology Optimization with Multiple Micro Materials Based on Principal Stress Direction

Inversion of borehole breakout orientation data : J Geophys Res V96, NB12, Nov 1991, P20093–20107

We have performed a laboratory study of cryogenic fracturing for improving oil/gas recovery from low-permeability shale and tight reservoirs. Our objective is to develop well stimulation techniques using cryogenic fluids, e.g. liquid nitrogen (LN) to increase permeability in a large reservoir volume surrounding wells. The new technology has the potential to reduce formation damage created by current stimulation methods as well as minimize or eliminate water usage and groundwater contamination. The concept of cryogenic fracturing is that sharp thermal gradient (thermal shock) created at the rock surface by applying cryogenic fluid can cause strong local tensile stress and initiate fractures. We developed a laboratory system for cryogenic fracturing under true triaxial loading, with a liquid nitrogen delivery/control and measurement system. The loading system simulates confining stresses by independently loading each axis up to about 5000 psi on 8"×8"×8" cubes. Both temperature in boreholes and block surfaces and fluid pressure in boreholes were continuously monitored. Acoustic and pressure-decay measurements are obtained before and at various stages of stimulations. Cubic blocks (8"×8"×8") of Niobrara shale, concrete, and sandstones have been tested, and stress levels and anisotropies are varied. Three schemes are considered: gas fracturing without cryo-stimulation, gas fracturing after low-pressure cryogen flow-through, gas fracturing after high-pressure flow-through. Pressure decay results show that liquid nitrogen stimulation clearly increases permeability, and repeated stimulations further increase the permeability. Acoustic velocities and amplitudes decreased significantly following cryo-stimulation indicating fracture creation. In the gas fracturing without the stimulation, breakdown (complete fracturing) occurs suddenly without any initial leaking, and major fracture planes form along the plane containing principal stress and intermediate stress directions as expected theoretically. However, in the gas fracturing after cryogenic stimulations, breakdown occurred gradually and with massive leaking due to thermal fractures created during stimulation. In addition, the major fracture direction does not necessarily follow the plane containing principal stress direction, esp. at low confining stress levels. In tests, we have observed that cryogenic stimulation seems to disrupt the internal stress field. The increase of borehole temperature after stimulation affects the permeability of the specimen. While a stimulated specimen is still cold, it keeps high permeability because fractures remain open and local thermal tension is maintained near the borehole. When the rock becomes warm again, fractures close and permeability decreases. In these tests, we have not used proppants. Overall, fractures are clearly generated by low and high-pressure thermal shocks. The added pressure of the high-pressure thermal shocks helps to further propagate cryogenic fractures generated by thermal shock. Breakdown pressure is significantly lowered by LN stimulation with breakdown pressure reductions up to about 40% observed.

An analysis of tensile principal stress below an HQ core stub was carried out with a finite element method for eleven lengths of core in 25 stress conditions under which core discing is likely to occur. The direction of the semi-axial tensile principal stress, which has an inclination angle relative to the core axis of less than 45°, was analyzed in detail for the central part of the core. Based on the results, a method for estimating three-dimensional directions of in-situ stress from the height at the periphery of the end-surface of a disc was proposed. Main results obtained in this study are summarized as follows:1) The direction of the semi-axial tensile principal stress is fairly uniform in a wide area of the central part of the core and does not depend on the length of core appreciably. Accordingly, the direction of the semi-axial tensile principal stress is considered to govern the geometry of the end-surface of a disc.2) The azimuth of the semi-axial tensile principal stress in the central part of the core approximately coincides with that of the minimum principal stress, while the inclination relative to the core axis (φm) is smaller than that of the minimum principal stress (φ3). A method for estimating φ3 from φm was proposed, which gave an error of less than 25 %.3) When the difference between the normal stresses perpendicular to the core axis (σx-σy) is great, a saddle-shaped disc is formed and the shape becomes more distinct with the increase in the stress difference. However the saddle shape becomes less distinct as the length of core decreases.4) By assuming that the end-surface of a disc is inclined in the direction of the semi-axial tensile principal stress in the central part of the core, a method for estimating three-dimensional directions of in-situ stress from the height at the periphery of the end-surface of a disc was proposed.

Sumatra, Indonesia, is situated in a highly seismically active region. The Sumatra Fault Zone, which spans 1,900 kilometers, runs along the island’s spine and develops within or around active volcanic arcs. The area of interest is located at the boundary between the Indo-Australian and Eurasian tectonic plates, resulting in high pressure and increasing the risk of earthquakes. The Sumatra Fault Segment Renun is characterized by a strike-slip type fault with a west-northwest to southeast orientation in the northern part of the island. This study aims to identify variations in the principal stress direction with depth in the subduction zone, the Sumatra Fault Segment Renun, and transition zone between two structures mentioned. The study used earthquake data from 1977 to 2023, which contains 72 events with focal mechanisms dominated by strike-slip. The earthquakes’ depths were categorized into three groups: 10-40 km, 41-60 km, and 61-100 km. Iterative joint inversion method was applied to determine the rotation of stress that occurred by utilizing data on focal mechanisms to estimate stress within the Earth’s crust. The results included a principal stress diagram with P/T (Pressure/Tension) axis, confidence level of principal stress, Mohr circle, and shape ratio. The principal stress direction (σ1) and shape ratio for the three depth categories are: north-south with a ratio of 0.4-0.55, southwest with a ratio of 0.7-0.9, and west-northwest to southeast with a ratio of 0-0.38. This suggests that the orientation of principal stress in Sumatra Fault Segment Renun is predominantly north-south at shallow depths of 10-40 km, which is positively correlated with the Sumatra Fault and indicates that this stress drives the movement of the Sumatra Fault Segment Renun with a west-northwest to southeast orientation.

The inversion of earthquake focal mechanisms is one of the few tools available for determining principal stress directions at seismogenic depths. Various methods have been proposed for performing such inversions. For three of the most commonly used methods, including one that has been proposed by Jacques Angelier, we discuss the physical assumptions and the error determination and then we propose an extension for one of the methods. All four methods are then applied for evaluating the stress field in the Upper Rhine graben. They are applied to seismic data recorded with a temporary monitoring network that was deployed 12 hours after the magnitude Mw = 4.4 Sierentz earthquake, which occurred on July 15, 1980. While differences in principal stress directions can be as much as 28° depending on the method used for the principal stress direction determination (orientation of the minimum principal stress has been found to range from N051°E with a 27° plunge to N090° E with a 20° plunge), the 90% confidence level associated with each method varies from 11° to 27°. Moreover, these various methods yield fairly diverse values for the R factor that characterizes relative differences between principal stress magnitudes (from R = 0.7 with a 0.2 90% confidence level to R = 0.3 with a 0.2 90% confidence level). Furthermore all three methods leave some focal mechanisms unexplained. These are then declared to be the result of heterogeneity and are not considered for the inversion. It is concluded that earthquake focal mechanisms inversions lack resolution for stress field evaluation at depth if no proper attention is given to the event independence hypothesis. When proper attention is given to this hypothesis, a resolution of the order of 15° may be achieved. The minimum principal stress orientation derived with these various focal mechanisms inversions differs by 4 to 36° from the orientation determined from borehole breakouts observed in Basel, in a 5 km deep well (N054°E ± 14°), located some 20 km from Sierentz. The solution that fits best borehole breakout observations is that which satisfies the minimum number (three) of prerequisite physical assumptions.

Undrained cyclic behavior of soft marine clay involved combined principal stress rotation

We use the locations and source mechanisms of oceanic intraplate earthquakes to test the hypothesis that the strength of oceanic fracture zones is less than that of normal oceanic lithosphere. If fracture zones are weaker than the surrounding lithosphere, they should be sites of concentrated intraplate seismicity, and the lithospheric stress field should be perturbed in their vicinity. The 77 earthquakes selected for the study have well‐determined focal mechanisms and epicenters in regions where fracture zones are well mapped. We have searched for dependence of faulting style, fault orientation, or principal stress direction on the distance from the nearest fracture zone. If fracture zones were generally weaker than the surrounding lithosphere, one of the principal horizontal stresses would be oriented nearly perpendicular to the fracture zone; we find no evidence that principal stresses near fracture zones are oriented preferentially in this manner. There is a slight tendency for earthquakes to occur near fracture zones, and patterns of fault orientation and sense of slip support the view that differential cooling and horizontal contraction on fracture zones may contribute seismogenic stress. These patterns are not demonstrable at high significance, however. If oceanic transform fault zones are weak relative to the surrounding lithosphere, these results suggest that the strength of the fracture zones increases within a few million years after cessation of transform motion to levels comparable with that of normal oceanic lithosphere.

Previous investigations have suggested that the direction of least principal stress in the western United States has shifted from northerly to easterly between 50 to 30 Ma. An effort is made to more tightly constrain the time of this shift using igneous dike‐host rock systems in which the isotopic age of both host and dike is known and differs by less than 10 Ma. In such systems, the possibility that magma intruded along old fractures unrelated to the state of stress at the time of intrusion is minimized and the axis of least principal stress is considered to be normal to the strike of the subvertical dike during the time interval between the ages of the dike and host rock. These systems indicate that a northerly least principal stress orientation persisted more recently than previously believed, until probably after 30 Ma, possibly about 26 Ma, and conceivably as recently as 18 Ma in some areas. The data are insufficient and too loosely bracketed in time to decide how fast the change in state of stress occurred in any one place and if by systematic rotation or abrupt shift. However, the change could be diachronous—earlier in the southeast and later in the northwest. Major easterly trending magmatic and tectonic elements of Tertiary age in the Great Basin of Utah and Nevada are compatible with the dike data. However, numerous metamorphic core complexes and detachment fault terranes seem to indicate more or less easterly extension throughout the time period represented by the dikes. This inconsistency is not clearly resolvable, but may reflect differences between the regional plate stress field and gravitational stresses in a crust overthickened by Mesozoic compression.

In total, 55 undrained cyclic torsional shear tests were conducted at a constant mean confining stress, σ0m, constant intermediate principal stress ratios, b = (σ2 − σ3)/(σ1 − σ3), and principal stress directions, α. The cyclic resistance of loose, isotropically consolidated specimens exhibited a dip in strength, with the lowest values for α being 45°, an angle at which α was oriented such that the potential shear band directions dovetailed with the plane having the weakest strength (i.e. horizontal plane). For a given α, the cyclic resistance decreased to the lowest values for b = 0·5, at which the value of the principal stresses was different in the three major directions. For dense specimens, the direction of the major principal stress had no significant influence on the cyclic resistance. For loose, anisotropically consolidated specimens, higher cyclic resistances were observed at the point of approaching the direction of the major principal stress during the consolidation and loading stages.

Concurrent multi-scale optimization of hybrid composite plates and shells for vibration

The Suki–Kinari hydropower station is situated in a landform with high mountains and deep valleys in northwest Pakistan. Affected by some geological conditions such as topography and faults, the in situ stress distribution of the project area is highly complex. To determine the in situ stress associated with this project, hydraulic fracturing in situ stress tests are conducted in four representative boreholes. Generally, the relation of the three principal stresses is σ H > σ z > σ h . However, the measured stress directions associated with the headrace tunnel and the powerhouse area are clearly different. The direction of the maximum horizontal principal stress of the headrace tunnel is NNE–NEE, whereas that of the underground powerhouse area is NNW–NWW. To identify the mechanism responsible for this difference, the present study simulates the stress field in the project area based on the three-dimensional finite element regression analysis method. Two theories are proposed on the basis of the numerical simulation results. On the one hand, the direction of the maximum horizontal principal stress in the rock mass above the valley elevation is controlled mainly by topography. On the other hand, the fault will disturb the stress field, causing noticeable changes in the stress direction and value of the fault-affected zone. By comparing the characteristics of the stress direction distribution at three elevations, the direction of maximum horizontal principal stress of the shallow strata is determined to be affected mainly by the topography and faults, and topography may be the most important factor. Moreover, with an increase in strata depth, the influence of the topography and faults on the stress direction decreases gradually. After reaching a particular strata depth, the direction of maximum horizontal principal stress tends to be the same as that of the regional tectonic stress, i.e., NNE.

Based on the unified strength theory, the elastic-plastic stress analysis is carried out under the premise that the first principal stress in two cases of radial stress and circumferential stress, and the formula of the radius of the surrounding rock and the plastic zone is derived. Tunnel surrounding rock has three states which including the elastic state, the elastoplastic state that the maximum principal stress is the radial stress and the elastoplastic state that the maximum principal stress is the circumferential stress. When the elastic-plastic analysis was carried out on the tunnel, the state of the surrounding rock should be judged and the correct formula is chosen to calculate. Analysis results show that the intermediate principal stress is benefit to the rock to give full play to the strength of the potential, so as to improve the stability of surrounding rock of the tunnel and the smaller intermediate principal stress coefficient, the higher sensitivity of the surrounding rock stability to the intermediate stress. When the inside pressure is less than the second critical stress and increasing the hole pressure is beneficial to improving the stability of surrounding rock, while the hole pressure is greater than the second critical stress, the stability of surrounding rock will reduce with the increase of the hole pressure.

Directional-dependent properties of the soil, like shear strength, stiffness and hydraulic conductivity, are known as anisotropy in soils. Shape and size of the soil particles and void distribution as microstructure characteristics and external factors such as stress history, environmental and geological conditions, and present stress condition can be the causes of the anisotropy in soils. In this paper, the behaviour of soil has been studied in stress-strain plain under monotonic anisotropic loading to investigate the effect of induced anisotropy on brittleness index of soil sample. The brittleness index of the soil is defined as the difference between the ultimate and peak shear strength divided by the peak shear strength of the soil. The two major parameters describing induced anisotropy or anisotropic loading are intermediate principal stress (b) and principal stress direction (α) which are representative of the difference between intermediate, maximum and minimum principal stresses and the rotation angle of the principal stresses’ axis, respectively. This paper only takes the effect of intermediate principal stress with the values of 0.25, 0.5, 0.75. In addition, the soil is in the unsaturated state with the saturation degree of 80% using the constant water (C.W.) method.

The stress state is the basis for evaluating soil deformation and strength. To study the 3-D stress state of the soil, a 3-D stress device based on fiber Bragg grating (FBG) and a microelectromechanical system (MEMS) was designed to measure the 3-D stress state, principal stress, and direction of principal stress. First, the FBG 1-D earth pressure cell is developed and calibrated. Based on the calculation relationship between the normal stress and the 3-D stress state, and considering the device rotation, the 3-D stress device was developed and verified in the large-scale triaxial test. The results show that the range of the 1-D earth pressure cell is 1 MPa, and the linearity is within 2.5% FS. The device can effectively measure soil’s 3-D stress, principal stress, and direction. The measured results are consistent with the reference and theoretical results. The device provides a new method for effectively measuring 3-D stress in soil.

Concurrent multi-material and multi-scale design optimization of fiber-reinforced composite material and structures for minimum structural compliance