Discontinuity development patterns and the challenges for 3D discrete fracture network modeling on complicated exposed rock surfaces

Natural slopes usually display complicated exposed rock surfaces that are characterized by complex and substantial terrain undulation and ubiquitous undesirable phenomena such as vegetation cover and rockfalls.This study presents a systematic outcrop research of fracture pattern variations in a complicated rock slope,and the qualitative and quantitative study of the complex phenomena impact on threedimensional(3D)discrete fracture network(DFN)modeling.As the studies of the outcrop fracture pattern have been so far focused on local variations,thus,we put forward a statistical analysis of global variations.The entire outcrop is partitioned into several subzones,and the subzone-scale variability of fracture geometric properties is analyzed(including the orientation,the density,and the trace length).The results reveal significant variations in fracture characteristics(such as the concentrative degree,the average orientation,the density,and the trace length)among different subzones.Moreover,the density of fracture sets,which is approximately parallel to the slope surface,exhibits a notably higher value compared to other fracture sets across all subzones.To improve the accuracy of the DFN modeling,the effects of three common phenomena resulting from vegetation and rockfalls are qualitatively analyzed and the corresponding quantitative data processing solutions are proposed.Subsequently,the 3D fracture geometric parameters are determined for different areas of the high-steep rock slope in terms of the subzone dimensions.The results show significant variations in the same set of 3D fracture parameters across different regions with density differing by up to tenfold and mean trace length exhibiting differences of 3e4 times.The study results present precise geological structural information,improve modeling accuracy,and provide practical solutions for addressing complex outcrop issues.

Automatic identification of rock discontinuity and stability analysis of tunnel rock blocks using terrestrial laser scanning

Local geometric information and discontinuity features are key aspects of the analysis of the evolution and failure mechanisms of unstable rock blocks in rock tunnels.This study demonstrates the integration of terrestrial laser scanning(TLS)with distinct element method for rock mass characterization and stability analysis in tunnels.TLS records detailed geometric information of the surrounding rock mass by scanning and collecting the positions of millions of rock surface points without contact.By conducting a fuzzy K-means method,a discontinuity automatic identification algorithm was developed,and a method for obtaining the geometric parameters of discontinuities was proposed.This method permits the user to visually identify each discontinuity and acquire its spatial distribution features(e.g.occurrences,spac-ings,trace lengths)in great detail.Compared with hand mapping in conventional geotechnical surveys,the geometric information of discontinuities obtained by this approach is more accurate and the iden-tification is more efficient.Then,a discrete fracture network with the same statistical characteristics as the actual discontinuities was generated with the distinct element method,and a representative nu-merical model of the jointed surrounding rock mass was established.By means of numerical simulation,potential unstable rock blocks were assessed,and failure mechanisms were analyzed.This method was applied to detection and assessment of unstable rock blocks in the spillway and sand flushing tunnel of the Hongshiyan hydropower project after a collapse.The results show that the noncontact detection of blocks was more labor-saving with lower safety risks compared with manual surveys,and the stability assessment was more reliable since the numerical model built by this method was more consistent with the distribution characteristics of actual joints.This study can provide a reference for geological survey and unstable rock block hazard mitigation in tunnels subjected to complex geology and active rockfalls.

A non-contact measurement method for rock mass discontinuity orientations by smartphone

Smartphones are usually packed with a large number of features.An increasing number of researchers are paying attention to the technological capabilities of smartphones,which is a new topic and research interest.This paper proposes a method using smartphones and digital photogrammetry to measure the discontinuity orientation of a rock mass.Smartphone photos satisfying a certain overlap rate provide an efficient method for generating point cloud models of rock outcrops based on image matching.Using the target and the generated point cloud model allows for determining actual geographic coordinates and the measurement of discontinuity orientations.The method proposed has been applied to two different study areas.The discontinuity orientations measured by the proposed method are compared with those measured by the manual method in two cases.The results show a good agreement,verifying the reliability and accuracy of the proposed method.The main contribution of this paper is to use knowledge of coordinate rotation to determine the actual geographic location of the model through a square target.The equipment used in this study is simple,and photogrammetric field surveys are easy to carry out.

Topography of the 660-km discontinuity within the Izu-Bonin subduction zone and evidence of slab penetration near the Bonin Super Deep Earthquake(~680 km)

The Izu-Bonin subduction zone in the Northwest Pacific is an ideal location for understanding mantle dynamics such as cold lithosphere subduction. The slab produces a lateral thermal anomaly, inducing local topographic changes at the boundary of a post-spinel phase transformation, considered to be the origin of the ‘660-km discontinuity.’ In this study, the short-period(1–2 Hz) S-to-P conversion phase S660P was used to obtain the fine-scale structure of the discontinuity. More than 100 earthquakes that occurred from the 1980s to the 2020s and were recorded by high-quality seismic arrays in the United States and Europe were analyzed. A discontinuity in the ambient mantle with an average depth of ~670 km was found beneath the 300–400-km event zone in the northern Bonin region near 33°N. Meanwhile, the ‘660-km discontinuity’ has been pushed upward, away from the slab, possibly because of a hot upwelling mantle plume. In the central part of the subduction zone, the 660-km discontinuity is depressed to an average depth of(690 ± 5) km within the slab at approximately 150 km below the coldest slab core, indicating a(300 ± 100) ℃ cold anomaly estimated using a post-spinel transformation Clapeyron slope of(-2.0 ± 1.0) MPa/K. In southern Bonin near 28°N, the discontinuity was found to be further depressed at an average depth of(695 ± 5) km below the deepest event and with a focal depth of ~550 km. The discontinuity is located where the slab bends abruptly to become sub-horizontal toward the west-southwest. Near the zone of the isolated Bonin Super Deep Earthquake, which occurred at ~680 km on May 30,2015, the discontinuity is depressed to ~700 km, suggesting a near-vertical penetrating slab and an S-to-P conversion in the coldest slab core, where a large low-temperature anomaly should exist.

Regression discontinuity design and its applications to Science of Science:A survey

Purpose:With the availability of large-scale scholarly datasets,scientists from various domains hope to understand the underlying mechanisms behind science,forming a vibrant area of inquiry in the emerging“science of science”field.As the results from the science of science often has strong policy implications,understanding the causal relationships between variables becomes prominent.However,the most credible quasi-experimental method among all causal inference methods,and a highly valuable tool in the empirical toolkit,Regression Discontinuity Design(RDD)has not been fully exploited in the field of science of science.In this paper,we provide a systematic survey of the RDD method,and its practical applications in the science of science.Design/methodology/approach:First,we introduce the basic assumptions,mathematical notations,and two types of RDD,i.e.,sharp and fuzzy RDD.Second,we use the Web of Science and the Microsoft Academic Graph datasets to study the evolution and citation patterns of RDD papers.Moreover,we provide a systematic survey of the applications of RDD methodologies in various scientific domains,as well as in the science of science.Finally,we demonstrate a case study to estimate the effect of Head Start Funding Proposals on child mortality.Findings:RDD was almost neglected for 30 years after it was first introduced in 1960.Afterward,scientists used mathematical and economic tools to develop the RDD methodology.After 2010,RDD methods showed strong applications in various domains,including medicine,psychology,political science and environmental science.However,we also notice that the RDD method has not been well developed in science of science research.Research Limitations:This work uses a keyword search to obtain RDD papers,which may neglect some related work.Additionally,our work does not aim to develop rigorous mathematical and technical details of RDD but rather focuses on its intuitions and applications.Practical implications:This work proposes how to use the RDD method in science of science research.Originality/value:This work systematically introduces the RDD,and calls for the awareness of using such a method in the field of science of science.

Machine learning-based classification of rock discontinuity trace:SMOTE oversampling integrated with GBT ensemble learning

This paper presents a hybrid ensemble classifier combined synthetic minority oversampling technique(SMOTE),random search(RS)hyper-parameters optimization algorithm and gradient boosting tree(GBT)to achieve efficient and accurate rock trace identification.A thirteen-dimensional database consisting of basic,vector,and discontinuity features is established from image samples.All data points are classified as either‘‘trace”or‘‘non-trace”to divide the ultimate results into candidate trace samples.It is found that the SMOTE technology can effectively improve classification performance by recommending an optimized imbalance ratio of 1:5 to 1:4.Then,sixteen classifiers generated from four basic machine learning(ML)models are applied for performance comparison.The results reveal that the proposed RS-SMOTE-GBT classifier outperforms the other fifteen hybrid ML algorithms for both trace and nontrace classifications.Finally,discussions on feature importance,generalization ability and classification error are conducted for the proposed classifier.The experimental results indicate that more critical features affecting the trace classification are primarily from the discontinuity features.Besides,cleaning up the sedimentary pumice and reducing the area of fractured rock contribute to improving the overall classification performance.The proposed method provides a new alternative approach for the identification of 3D rock trace.

A modified method of discontinuity trace mapping using three-dimensional point clouds of rock mass surfaces

This paper presents an automated method for discontinuity trace mapping using three-dimensional point clouds of rock mass surfaces.Specifically,the method consists of five steps:(1)detection of trace feature points by normal tensor voting theory,(2)co ntraction of trace feature points,(3)connection of trace feature points,(4)linearization of trace segments,and(5)connection of trace segments.A sensitivity analysis was then conducted to identify the optimal parameters of the proposed method.Three field cases,a natural rock mass outcrop and two excavated rock tunnel surfaces,were analyzed using the proposed method to evaluate its validity and efficiency.The results show that the proposed method is more efficient and accurate than the traditional trace mapping method,and the efficiency enhancement is more robust as the number of feature points increases.

An extended displacement discontinuity method for analysis of stress wave propagation in viscoelastic rock mass

An extended displacement discontinuity method (EDDM) is proposed to analyze the stress wave propagation in jointed viscoelastic rock mass (VRM).The discontinuities in a rock mass are divided into two groups.The primary group with an average geometrical size larger than or in the same order of magnitude of wavelength of a concerned stress wave is defined as 'macro-joints',while the secondary group with a high density and relatively small geometrical size compared to the wavelength is known as 'micro-defects'.The rock mass with micro-defects is modeled as an equivalent viscoelastic medium while the macro-joints in the rock mass are modeled explicitly as physical discontinuities.Viscoelastic properties of a micro-defected sedimentary rock are obtained by longitudinally impacting a cored long sedimentary rod with a pendulum.Wave propagation coefficient and dynamic viscoelastic modulus are measured.The EDDM is then successfully employed to analyze the wave propagation across macro-joint in VRM.The effect of the rock viscosity on the stress wave propagation is evaluated by comparing the results of VRM from the presented EDDM with those of an elastic rock mass (ERM) from the conventional displacement discontinuity method (CDDM).The CDDM is a special case of the EDDM under the condition that the rock viscosity is ignored.Comparison of the reflected and transmitted waves shows that the essential rock viscosity has a significant effect on stress wave attenuation.When a short propagation distance of a stress wave is considered,the results obtained from the CDDM approximate to the EDDM solutions,however,when the propagation distance is sufficiently long relative to the wavelength,the effect of rock viscosity on the stress wave propagation cannot be ignored.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TOWARD THE SUPERPOSITION OF CONTACT DISCONTINUITY AND SHOCK WAVE FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave is established under some smallness conditions. To do this, we first construct a new viscous contact wave such that the momentum equation is satisfied exactly and then determine the shift of the viscous shock wave. By using them together with an inequality concerning the heat kernel in the half space, we obtain the desired a priori estimates. The proof is based on the elementary energy method by the anti-derivative argument.

A state-of-the-art review of automated extraction of rock mass discontinuity characteristics using three-dimensional surface models

In the last two decades,significant research has been conducted in the field of automated extraction of rock mass discontinuity characteristics from three-dimensional(3D)models.This provides several methodologies for acquiring discontinuity measurements from 3D models,such as point clouds generated using laser scanning or photogrammetry.However,even with numerous automated and semiautomated methods presented in the literature,there is not one single method that can automatically characterize discontinuities accurately in a minimum of time.In this paper,we critically review all the existing methods proposed in the literature for the extraction of discontinuity characteristics such as joint sets and orientations,persistence,joint spacing,roughness and block size using point clouds,digital elevation maps,or meshes.As a result of this review,we identify the strengths and drawbacks of each method used for extracting those characteristics.We found that the approaches based on voxels and region growing are superior in extracting joint planes from 3D point clouds.Normal tensor voting with trace growth algorithm is a robust method for measuring joint trace length from 3D meshes.Spacing is estimated by calculating the perpendicular distance between joint planes.Several independent roughness indices are presented to quantify roughness from 3D surface models,but there is a need to incorporate these indices into automated methodologies.There is a lack of efficient algorithms for direct computation of block size from 3D rock mass surface models.

Bearing capacity of foundations on rock slopes intersected by non-persistent discontinuity

Rocks encountered in foundations of heavy structures are invariably intersected by discontinuities(joints).In the past,several studies have been performed by researchers to incorporate the effect of fully persistent joints in the assessment of the load-carrying capacity of rocks.However,in the field,the joints are non-persistent,and an assumption of full persistency will underestimate the capacity.Recently,Shaunik&Singh have studied the influence of non-persistency,number of joint segments and discontinuity orientation on the strength behaviour of rock specimens(Shaunik and Singh,2019).Bell’s approach can be used to obtain the bearing capacity of shallow foundations placed in jointed rocks.In the present study,results of the experimental work(Shaunik and Singh,2019)conducted by Shaunik&Singh have been used to suggest expressions by extending Bell’s approach for computing bearing capacity of the foundation placed near the crown of a rock slope.Easy to use design charts are also presented for field application.Finally,a real-life problem from Indian Garhwal Himalayas is considered,and the approach suggested in this study is utilised to obtain the bearing capacity of a bridge foundation as a function of uniaxial compressive strength(UCS)of intact rock,joint friction,spacing and orientation of joint,nonpersistency and number of joint segments.

CONVERGENCE RATE OF SOLUTIONS TO STRONG CONTACT DISCONTINUITY FOR THE ONE-DIMENSIONAL COMPRESSIBLE RADIATION HYDRODYNAMICS MODEL

This paper is concerned with a singular limit for the one-dimensional compress- ible radiation hydrodynamics model. The singular limit we consider corresponds to the physical problem of letting the Bouguer number infinite while keeping the Boltzmann number constant. In the case when the corresponding Euler system admits a contact discontinuity wave, Wang and Xie （2011） [12] recently verified this singular limit and proved that the solution of the compressible radiation hydrodynamics model converges to the strong contact 1 discontinuity wave in the L∞-norm away from the discontinuity line at a rate of ε1/4, as the reciprocal of the Bouguer number tends to zero. In this paper, Wang and Xie＇s convergence rate is improved to ε7/8 by introducing a new a priori assumption and some refined energy estimates. Moreover, it is shown that the radiation flux q tends to zero in the L∞-norm away from the discontinuity line, at a convergence rate as the reciprocal of the Bouguer number tends to zero.

ZERO DISSIPATION LIMIT TO CONTACT DISCONTINUITY FOR THE COMPRESSIBLE NAVIER-STOKES SYSTEM OF GENERAL GAS

The zero dissipation limit to the contact discontinuities for one-dimensional com- pressible Navier-Stokes equations was recently proved for ideal polytropic gas （see Huang et al. [15, 22] and Ma [31]）, but there is few result for general gases including ideal polytropic gas. We prove that if the solution to the corresponding Euler system of general gas satisfying （1.4） is piecewise constant with a contact discontinuity, then there exist smooth solutions to Navier-Stokes equations which converge to the inviscid solutions at a rate of k1/4 as the heat-conductivity coefficient k tends to zero. The key is to construct a viscous contact wave of general gas suitable to our proof （see Section 2）. Notice that we have no need to restrict the strength of the contact discontinuity to be small.

Effects of Area Discontinuity at Nozzle Inlet on the Characteristics of Self-resonating Cavitating Waterjet

The current research on self-resonating cavitating waterjet（SRCW） mainly focuses on the generation mechanism and structure optimization.Researches relating to the influences of disturbances at nozzle inlet on the characteristics of the jet are rarely available.In order to further improve the performance of SRCW,effects of area discontinuity（enlargement and contraction） are experimentally investigated using three organ-pipe nozzles.Axial pressure oscillation peak and amplitude as well as aggressive erosion intensity of the jet are used to evaluate the effects.The results reveal that area enlargement and contraction affect the peak differently,depending on the inlet pressure,nozzle geometry,and standoff distance;while area contraction always improves the amplitude regardless of these factors.At inlet pressures of 10 MPa and 20 MPa,area discontinuity improves the peak at almost all the testing standoff distances,while this only happens at smaller standoff distances with the inlet pressure increased to 30 MPa.The capability of area discontinuity for improving the amplitude is enhancing with increasing inlet pressure.Moreover,the cavitation erosion ability of the jet can be largely enhanced around the optimum standoff distance,depending on the type of area discontinuity and nozzle geometry.A preliminary analysis of the influence of area discontinuity on the disturbance waves in the flow is also performed.The proposed research provides a new method for effectively enhancing the performance of SRCW.

EXISTENCE OF MINIMISERS FOR A CLASS OF FREE DISCONTINUITY PROBLEMS IN THE HEISENBERG GROUP H^n

The purpose of this paper is to prove existence of minimisers of the functional where Ω is an open set of the Heisenberg group Hn, K runs over all closed sets of Hn, u varies in C_H^1(Ω\ K), α,β> 0,q≥1, g ∈ Lq(Ω) ∩ L∞(Ω) and f : R2n→R is a convex function satisfying some structure conditions (H1)(H2)(H3) (see below).

P-wave tomography of crust and upper mantle under Southern California: Influence of topography of Moho discontinuity

We use 146 422 P-wave arrival times from 6 347 local earthquakes recorded by the Southern California SeismicNetwork to determine a detailed three-dimensional P-wave velocity structure at 0～35 km depth. We have takeninto account the Moho depth variations, which were obtained by seismological methods. Checkerboard tests sug-gest that our inversion results are reliable. Our models provide new information on regional geological structuresof Southern California. At shallow depths P-wave velocity structure correlates with surface geological features andexpresses well variations of surface topography of the mountains and basins. The velocity structure at each layer ischaracterized by block structures bounded by large faults. Ventura Basin, Los Angeles Basin, Mojave Desert, Pen-insular Ranges, San Joaquin Valley, Sierra Nevada, and Salton Trough show respectively all-round block. SanAndreas Fault becomes an obvious boundary of the region. To its southwest, the velocity is higher, and there arestrong heterogeneity and deeper seismicity; but to its northeast, the velocity is lower and shows less variation thanto the southwest, the seismicity is shallower. To investigate the effect of the Moho geometry we conducted inver-sions for two cases: one for flat Moho geometry, another for a Moho with lateral depth variations. We found thatthe topography of the Moho greatly affects the velocity structure of the middle and lower crust. When the Mohotopography is considered, a more reasonable tomographic result can be obtained and the resulting 3-D velocitymodel fits the data better.

Regionalized difference of the 660 km discontinuity beneath Izu-Bonin

Digital waveform data recorded by the vertical component short period stations at the American networks of SCSN, NCSN and PNSN and three components broadband stations at the Germany and Swiss networks and arrays of GRFN, GRSN and SDSNet for the events between 1981 and 2000 under Izu-Bonin are used as data sets. The N-th root slant stack method was used to pick up the SdP phase converted at the velocity interface beneath source and the regionalized difference of the 660 km discontinuity beneath Izu-Bonin is studied. It is found that while the dip angles of the subducting slab and the maximal depths of sources increase gradually from 35N to 26N, the 660 km discontinuity appears regionalized differences. The discontinuity exists at 660 km while there is no effect from subducting slab, but it is depressed to the depth of 720 km while there are obvious effects. The dispersion of converted points is still an unsolved problem which maybe result from the complex structure of the discontinuity, converted phases which were misjudged, or the assumption of one dimensional spherical earth model.

Failure strength and fracture characteristics of rock with discontinuity under indirect tension

Large-scale discontinuities can significantly affect the mechanical properties of rock masses.However,the tensile behavior of rock discontinuities is often less investigated.To study the statistical characteristics of failure strength and fracture characteristics of rock discontinuities,Brazilian disc tests were conducted on limestone specimens with a single natural discontinuity at different load-discontinuity angles(β).In this study,β=0°andβ=90°correspond to the discontinuity parallel and perpendicular to loading direction,respectively.The results show that Brazilian failure strength(BFS)can reasonably represent the tensile strength of rock with discontinuities,by comparing the BFS and tensile stress in the disc center at peak force.The two-parameter Weibull distribution can capture the statistical BFS characteristics of rock discontinuities parallel to loading direction(β=0°)and at different loaddiscontinuity angles(β≠0°).All specimens with discontinuity at different load-discontinuity angles show more plastic deformational behaviour than intact rock specimen.With increasingβ,the mean BFS of limestone with discontinuity increases before reaching a plateau atβ=45°.The single plane of weakness theory best explains the BFS of fractured limestone withβ.Only a specific segment of preexisting rock discontinuity could affect the fracture process.Whenβ=0°,interfacial cracks and alternative cracks formed.Whenβ≠0°,mixed failure mode with shear and tensile failure occurred,particularly whenβ=30°andβ=60°.The findings can contribute to better understanding the failure and fracture characteristics of rock with discontinuities,particularly the interaction of pre-existing discontinuities with stress-induced fracturing.

On the accuracy of higher order displacement discontinuity method(HODDM) in the solution of linear elastic fracture mechanics problems

The higher order displacement discontinuity method(HODDM) utilizing special crack tip elements has been used in the solution of linear elastic fracture mechanics(LEFM) problems. The paper has selected several example problems from the fracture mechanics literature(with available analytical solutions) including center slant crack in an infinite and finite body, single and double edge cracks, cracks emanating from a circular hole. The numerical values of Mode Ⅰ and Mode Ⅱ SIFs for these problems using HODDM are in excellent agreement with analytical results(reaching up to 0.001% deviation from their analytical results). The HODDM is also compared with the XFEM and a modified XFEM results. The results show that the HODDM needs a considerably lower computational effort(with less than 400 nodes) than the XFEM and the modified XFEM(which needs more than 10000 nodes) to reach a much higher accuracy. The proposed HODDM offers higher accuracy and lower computation effort for a wide range of problems in LEFM.

Spatial variations of the 660-km discontinuity in the western Pacific subduction zones observed from CEArray triplication data

We examined the spatial variation of velocity structures around the 660-kin discontinuity at the western Pacific subduction zones by waveform modeling of triplication data. Data from two deep earthquakes beneath Izu-Bonin and Northeast China are used. Both events were well recorded by a dense broadband seismic network in China （CEArray）. The two events are located at approximately the same distance to the CEArray, yet significant differences are observed in their records： （1） the direct arrivals traveling above the 660-km discontinuity （AB branch） are seen in a different distance extent： -29° for the NE China event, -23° for Izu-Bonin event; （2） the direct （AB） and the refracted waves at the 660-km （CD branch） cross over at 19.5° and 17° for the NE China and the Izu-Bonin event, respectively. The best fitting model for the NE China event has a broad 660-km discontinuity and a constant high velocity layer upon it; while the Izu-Bonin model differs from the standard IASP91 model only with a high velocity layer above the 660-km discontinuity. Variations in velocity models can be roughly explained by subduction geometry.