Stress tensor determination by modified hydraulic tests on pre-existing fractures:Method and stress constraints

The hydraulic testing of pre-existing fractures(HTPF)is one of the most promising in situ stress measurement methods,particularly for three-dimensional stress tensor determination.However,the stress tensor determination based on the HTPF method requires at least six tests or a minimum of 14-15 tests(under different conditions)for reliable results.In this study,we modified the HTPF method by considering the shear stress on each pre-existing fracture,which increased the number of equations for the stress tensor determination and decreased the number of tests required.Different shear stresses were attributed to different fractures by random sampling;therefore,the stress tensors were obtained by searching for the optimal solution using the least squares criterion based on the Monte Carlo method.Thereafter,we constrained the stress tensor based on the tensile strength criterion,compressive strength criterion,and vertical stress constraints.The inverted stress tensors were presented and analyzed based on the tensorial nature of the stress using the Euclidean mean stress tensor.Two stress-measurement campaigns in Weifang(Shandong Province,China)and Mercantour road tunnel(France)were implemented to highlight the validity and efficiency of the modified HTPF(M-HTPF)method.The results showed that the M-HTPF method can be applied for stress tensor inversion using only three to four tests on pre-existing fractures,neglecting the stress gradient.The inversion results were confined to relatively small distribution dispersions and were significantly reliable and stable due to the shear stresses on the fractures and the stress constraints employed.The M-HTPF method is highly feasible and efficient for complete stress tensor determination in a single borehole.

Vertical variations of wave-induced radiation stress tensor

The distributions of the wave-induced radiation stress tensor over depth are studied by us- ing the linear wave theory, which are divided into three regions, i. e., above the mean water level, be- low the wave trough level, and between these two levels. The computational expressions of the wave-in- duced radiation stress tensor at the arbitrary wave angle are established by means of the Eulerian coordi- nate transformation, and the asymptotic forms for deep and shallow water are also presented. The verti- cal variations of a 30°incident wave-induced radiation stress tensor in deep water, intermediate water and shallow water are calculated respectively. The following conclusions are obtained from computations. The wave-induced radiation stress tensor below the wave trough level is induced by the water wave parti- cle velocities only, whereas both the water wave particle velocities and the wave pressure contribute to the tensor above the wave trough level. The vertical variations of the wave-induced radiation stress ten- sor are influenced substantially by the velocity component in the direction of wave propagation. The dis- tributions of the wave-induced radiation stress tensor over depth are nonuiniform and the proportion of the tensor below the wave trough level becomes considerable in the shallow water. From the water surface to the seabed, the reversed variations occur for the predominant tensor components.

Gedanken Experiment for Fluctuation of Mass of a Graviton, Based on the Trace of GR Stress Energy Tensor-Pre Planckian Conditions that Lead to Gaining of Graviton Mass, and Planckian Conditions That Lead to Graviton Mass Shrinking to 10-62 Grams

We will be looking at the energy of a graviton, based upon the Stress energy tensor, and from there ascertaining how fluctuations in early universe conditions impact the mass of a graviton. Physically the mass of the graviton would be shrinking right after Planck time and presumably it would be going to its equilibrium value of about 10-62 grams, for its present day value. It, graviton mass, would increase up to the Plank time of about 10-44 seconds. Note that the result that graviton mass shrinks to 10-62 grams for its present day value works only for relic gravitons.

STUDY ON THE GENERALIZED PRANDTL-REUSS CONSTITUTIVEEQUATION AND THE COROTATIONALRATES OF STRESS TENSOR

In this paper, the generalized Prandtl-Reuss (P-R) constitutive equations of elastic-plastic material in the presence of finite deformations through a new approach are studied. It analyzes the generalized P-R equation based on the material corotational rate and clarifies the puzzling problem of the simple shear stress oscillation mentioned in some literature. The paper proposes a modified relative rotational rate with which to constitute the objective rates of stress in the generalized P-R equation and concludes that the decomposition of total deformation rate into elastic and plastic parts is not necessary in developing the generalized P-R equations. Finally, the stresses of simple shear deformation are worked out.

Moment tensor and stress inversion solutions of acoustic emissions during compression and tensile fracturing in crystalline rocks

We investigate the accuracy and robustness of moment tensor(MT)and stress inversion solutions derived from acoustic emissions(AEs)during the laboratory fracturing of prismatic Barre granite specimens.Pre-cut flaws in the specimens introduce a complex stress field,resulting in a spatial and temporal variation of focal mechanisms.Specifically,we consider two experimental setups:(1)where the rock is loaded in compression to generate primarily shear-type fractures and(2)where the material is loaded in indirect tension to generate predominantly tensile-type fractures.In each test,we first decompose AE moment tensors into double-couple(DC)and non-DC terms and then derive unambiguous normal and slip vectors using k-means clustering and an unstructured damped stress inversion algorithm.We explore temporal and spatial distributions of DC and non-DC events at different loading levels.The majority of the DC and the tensile non-DC events cluster around the pre-cut flaws,where macro-cracks later develop.Results of stress inversion are verified against the stress field from finite element(FE)modeling.A good agreement is found between the experimentally derived and numerically simulated stress orientations.To the best of the authors’knowledge,this work presents the first case where stress inversion methodologies are validated by numerical simulations at laboratory scale and under highly heterogeneous stress distributions.

Explicit Algebraic Stress Model for Three-Dimensional Turbulent Buoyant Flows Derived Using Tensor Representation

An explicit algebraic stress model (EASM) has been formulated for two-dimensional turbulent buoyant flows using a five-term tensor representation in a prior study. The derivation was based on partitioning the buoyant flux tensor into a two-dimensional and a three-dimensional component. The five-term basis was formed with the two-dimensional component of the buoyant flux tensor. As such, the derived EASM is limited to two-dimensional flows only. In this paper, a more general approach using a seven-term representation without partitioning the buoyant flux tensor is used to derive an EASM valid for two- and three-dimensional turbulent buoyant flows. Consequently, the basis tensors are formed with the fully three-dimensional buoyant flux tensor. The derived EASM has the two-dimensional flow as a special case. The matrices and the representation coefficients are further simplified using a four-term representation. When this four-term representation model is applied to calculate two-dimensional homogeneous buoyant flows, the results are essentially identical with those obtained previously using the two-dimensional component of the buoyant flux tensor. Therefore, the present approach leads to a more general EASM formulation that is equally valid for two- and three-dimensional turbulent buoyant flows.

THE STRESS-ENERGY TENSOR AND POHOZAEV'S IDENTITY FOR SYSTEMS

Utilizing stress-energy tensors which allow for a divergence-free formulation, we establish Pohozaev＇s identity for certain classes of quasilinear systems with variational structure.

Three-dimensional stress variation characteristics in deep hard rock of CJPL-Ⅱ project based on in-situ monitoring

In deep hard rock excavation, stress plays a pivotal role in inducing stress-controlled failure. While the impact of excavation-induced stress disturbance on rock failure and tunnel stability has undergone comprehensive examination through laboratory tests and numerical simulations, its validation through insitu stress tests remains unexplored. This study analyzes the three-dimensional stress changes in the surrounding rock at various depths, monitored during the excavation of B2 Lab in China Jinping Underground Laboratory Phase Ⅱ(CJPL-Ⅱ). The investigation delves into the three-dimensional stress variation characteristics in deep hard rock, encompassing stress components and principal stress. The results indicate changes in both the magnitude and direction of the principal stress during tunnel excavation. To quantitatively describe the degree of stress disturbance, a series of stress evaluation indexes are established based on the distances between stress tensors, including the stress disturbance index(SDI), the principal stress magnitude disturbance index(SDIm), and the principal stress direction disturbance index(SDId). The SDI indicates the greatest stress disturbance in the surrounding rock is 4.5 m from the tunnel wall in B2 Lab. SDIm shows that the principal stress magnitude disturbance peaks at2.5 m from the tunnel wall. SDId reveals that the largest change in principal stress direction does not necessarily occur near the tunnel wall but at a specific depth from it. The established relationship between SDI and the depth of the excavation damaged zone(EDZ) can serve as a criterion for determining the depth of the EDZ in deep hard rock engineering. Additionally, it provides a reference for future construction and support considerations.

Spatio-temporal variation of the stress field in the Wenchuan aftershock region

Focal mechanism solutions and centroid depths of 312 M≥4 aftershocks from the 2008 Wenchuan earthquake sequence have been derived by CAP （Cut and Paste） method from broadband waveform data with relatively high signal-to-noise ratio （SNR）. Following this, we have analyzed the distribution of focal depths and the stress tensors, as well as the types of focal mechanisms. The major results are： （1） different cross-sections show that the depth ranges of the aftershocks at the southern and northern ends of the aftershock area along the Longmenshan fault zone are wider than those on the central segment, where rare M≥4 aftershocks occurred at depths shallower than 10 kin. The main faults trend to the NW on the southern and central segments, and for the northern segment, no dominant trend direction has been determined; （2） stress tensor distribution demonstrates that the majority of the aftershock areas on the cross-section along the major axis are mainly under compressive stress perpendicular to the profile; however, for the areas near Lixian, Beichuan, Qingchuan and the shallow parts of its northern segment, large principal stress components are parallel to the major axis profile direction. On the cross-sections perpendicular to the major axis, the three areas above can be divided into two parts： one with dominantly compressional stress near the major faults of the Longmenshan fault zone on the SE side, and the other with NE-direction push along the fault zone on the NW side; （3） the stress tensor distribution in map view is very similar to those on the vertical cross-sections. In map view, the orientation of the principal compressional stress axis $1 on the central segment of the aftershock area presents an SE-trending arc shape; （4） the stress tensor slices at different depths show that the orientation of S1 axis mainly changes on the central segment and at the northern end, indicating that the two segments have different seismogenic structures at different depths; （5） with the exception of the northern end of the aftershock region, the orientation of the $1 axis changes little during the early and late stages, illustrating the seismogenic structures are relatively stable; （6） preliminary analyses for the seismogenic structures at the northern end indicated that deeper strike-slip quakes occurred on the ENE-striking branch at first, and then the NNE-striking branch faults at the northern end were activated and generated a series of relatively shallow strike-slip earthquakes due to subsequent stress-triggering; （7） the aftershock triggering mechanism that occurred near Lixian is different between the shallow and deep depths, and between the early and late stages, indicating that the main faults and the branch faults responsible for aftershocks are at different depths. Consequently, the relaxation effect of the main shock particularIy impacts the branch faults.

Moment tensor inversion of the November 6, 1988 MS=7.6, Lancang-Gengma, China,earthquake using long-period body-waves data

Moment tensor inversion was carried out to myert the source mechanism and source time function of the Ms=7.6November 6. 1988, Lancang-Gengma. Yunnan Province, Chin4 earthquake. Waveforms of long-period bodywaves recorded by China Digital Seismograph Network (CDSN) were used in the inversion. The inverted resultshows one nodal plane of right-lateral strike-slip faulting and another of left-lateral strike-slip faulting and a simplesource time function of a duration of about 15 s and scalar seismic moment of 6.4x 102oN-N-m From the geologicaldata and tectonic settings and also from field observations and epicentral distribution of aftershocks, the nodalplane striking in the azimuth of 313° is preferred as the fault plane. The pressure axis lies almost horizontally innorth-south direction.

THEORETICAL MODEL OF EFFECTIVE STRESS COEFFICIENT FOR ROCK/SOIL-LIKE POROUS MATERIALS

Physical mechanisms and influencing factors on the effective stress coefficient for rock/soil-like porous materials are investigated, based on which equivalent connectivity index is proposed. The equivalent connectivity index, relying on the meso-scale structure of porous material and the property of liquid, denotes the connectivity of pores in Representative Element Area （REA）. If the conductivity of the porous material is anisotropic, the equivalent connectivity index is a second order tensor. Based on the basic theories of continuous mechanics and tensor analysis, relationship between area porosity and volumetric porosity of porous materials is deduced. Then a generalized expression, describing the relation between effective stress coefficient tensor and equivalent connectivity tensor of pores, is proposed, and the expression can be applied to isotropic media and also to anisotropic materials. Furthermore, evolution of porosity and equivalent connectivity index of the pore are studied in the strain space, and the method to determine the corresponding functions in expressions above is proposed using genetic algorithm and genetic programming. Two applications show that the results obtained by the method in this paper perfectly agree with the test data. This paper provides an important theoretical support to the coupled hydro-mechanical research.

Stress and fabric in granular material

It has been well recognized that, due to anisotropic packing structure of granular material, the true stress in a specimen is different from the applied stress. However, very few research efforts have been focused on quantifying the relationship between the true stress and applied stress. In this paper, we derive an explicit relationship among applied stress tensor, material-fabric tensor, and force-fabric tensor; and we propose a relationship between the true stress tensor and the applied stress tensor. The validity of this derived relationship is examined by using the discrete element simulation results for granular material under biaxial and triaxial loading con- ditions.

Regional stress field around the Taigu fault zone in Shanxi Province,China

A comprehensive study on regional stress field around the Taigu fault zone in Shanxi Province, China, was performed in this study. To get a better understanding of the present-day stress status in this area, 31 focal mecha- nisms of ML 〉3 earthquakes since 1965 were compiled, and the best stress tensor was then inverted based on the database. Additionally, magnetic fabrics along the Taigu fault zone were investigated to get an indication of the regional stress field in the past. Our results show that the present-day stress field around the Taigu fault zone is characterized by astable NW-SE extension with a strike- slip component, consistent with the geological surveys and recent GPS data. Results from magnetic fabrics indicate that the orientations of principal stress axes from magnetic fabrics of sedimentary rocks in Neogene coincide to the orientations of principal stress axes from focal mecha- nisms. The south segment of the Taigu fault displays more complicated magnetic fabrics and more activity of mod- erate earthquakes. It is connected with the Mianshan west fault and intersects with NW-SE striking Fenyang fault and the north fault of the Lingshi uplift at the south edge of Taiyuan basin. This may be the area needing more atten- tion in terms of seismic risk along the Taigu fault.

Moment tensor inversion for focal mechanism of the Beibuwan earthquakes

Two earthquakes of Ms=6.0 and Ms=6. 1 consecutively occurred on December 31, 1994 and January 10, 1995 in Beibuwan region, China. By using the generalized reflection-transmission coefficient matrix and the discrete slowness integration method in the calculation of Green's functions, we obtained the focal mechanisms of these earthquakes using long-period waveforms of regional body waves recorded by the China Digital Seismograph Network (CDSN) by means of moment tensor inversion method in frequency domain. The results inverted indicate that the focal mechanisms of these two earthquakes were similar to each other. Their principal compressional stresses are in NW-SE direction and principal tensional stresses are in NE-SW direction. It turns out that the occurrence of the two earthquakes was controlled by the same tectonic environment related to the collision of the Philippine Plate and the Eurasian Plates. On the other hand, the results imply that the stress field in the seismogenic region has a significant change after the Ms=6.0 earthquake. It may be proposed that the occurrence of the Ms=6. 1 earthquake could be related to the stress field adjustment caused by the Ms=6.0 earthquake.

Summation and decomposition of principal stresses in the crust

In the compilation of World Stress Map, 9％ of data comes from overcoring and hydraulic fracturing measurement, 23％ from borehole breaking off, 63％ from earthquake focal mechanism, and 5％ from young geological investigation (Zoback, et al, 1989).……

Moment tensor inversion of focal mechanism for the aftershock sequence of 1982 Lulong M_S =6.1 earthquake

Using the moment tensor inversion method, we calculate the focal mechanisms of the aftershock sequence of the Ms=6.1 Lulong earthquake occurred on October 19, 1982 in Hebei Province. We found that the pressure axis in Lulong basin is nearly in the east-west direction with an azimuth of N74°E. However, in the north of the basin the stress axis changes to N43°E; and in some places near the center of the basin it changes to the northwest that is almost perpendicular to the P axis obtained by us from those events around the basin. This feature illuminates that in Lulong earthquake sequence, the stress direction is different in different parts of crustal structure, which shows that the tectonic movement in Lulong region is complex. This is because that Lulong is located in the eastern part of Chinese mainland and is subject to the compression of Japanese Sea Basin driven by the Pacific Plate. On the other hand, nipped by the Yanshan and North China blocks, Lulong is obviously restricted by the block boundaries.

Determinations of directions of the mean stress field in Sichuan-Yunnan region from a number of focal mechanism solutions

Based on the spatial orientation and slip direction of the fault plane solutions, we present the expression of corresponding mechanical axis tensor in geographic coordinate system, and then put forward a method for calculating average mechanical axis tensor and its eigenvalues, which involves solving the corresponding eigenequation. The method for deducing mean stress field from T, B, and P axes parameters of a number of focal mechanism solutions has been verified by inverting data of mean stress fields in Fuyun region and in Tangshan region with fitting method of slip direction, and both results are consistent. To study regional average stress field, we need to choose a population of focal mechanism solutions of earthquakes in the massifs where there are significant tectonic structures. According to the focal mechanism solutions of 256 moderate-strong earthquakes occurred in 13 seismic zones of Sichuan-Yunnan region, the quantitative analysis results of stress tensor in each seismic zone have been given. The algorithm of such method is simple and convenient, which makes the method for analyzing tectonic stress field with large amount of focal mechanism solution data become quantified.

Extended intrinsic mean spin tensor for turbulence modelling in non-inertial frame of reference

We investigate the role of extended intrinsic mean spin tensor introduced in this work for turbulence modelling in a non-inertial frame of reference. It is described by the Euclidean group of transformations and, in particular, its significance and importance in the approach of the algebraic Reynolds stress modelling, such as in a nonlinear K-ε model. To this end and for illustration of the effect of extended intrinsic spin tensor on turbulence modelling, we examine several recently developed nonlinear K-ε models and compare their performance in predicting the homogeneous turbulent shear flow in a rotating frame of reference with LES data. Our results and analysis indicate that, only if the deficiencies of these models and the like be well understood and properly corrected, may in the near future, more sophisticated nonlinear K-ε models be developed to better predict complex turbulent flows in a non-inertial frame of reference.

Moment tensor inversion of near-source broadband data

Moment tensor inversion was carried out for small and moderate earthquakes with near-source broadband data recorded by a temporal small-aperture network consisting of three component accelerographs. Accelerograms were integrated twice to produce displacement seismograms and filtered by a Butterworth band-pass filter. Green's functions were calculated for a homogeneous semi-infinite elastic medium in the inversion. Direct P,S and converted SP phases were identified for the inversion with reference to the synthetic seismograms. Through the moment tensor inversion it is demonstrated that with very simple structure model and selected phases, one is able to retrieve the source mechanism and the seismic moment for small events, andthe source mechanism for moderate events. It is also demonstrated that the technique described in this studyis convenient for the determination of source mechanismal and stress state in dealing with numbers of small andmoderate earthquakes without detailed knowledge about the structure. As an application of the present technique, moment tensors of 15 aftershocks of the Ms 6. 1 earthquake occurred on April 18, 1985 in Luquan,Yunnan Province, China were retrieved. The inversion results show that these events can be reasonably modeled by a predominant double couple. It can be found from the distribution of principal stress axes and the average moment tensor that the pressure axis in this area lies horizontally in the NNW direction, which is consistent with the results obtained from previous studies. It implies that the occurrence of the Luquan earthquake sequence is tectonically related to the relative collision motion between the Indian and Eurasian Plates.