Numerical investigation of geostress influence on the grouting reinforcement effectiveness of tunnel surrounding rock mass in fault fracture zones

Grouting is a widely used approach to reinforce broken surrounding rock mass during the construction of underground tunnels in fault fracture zones,and its reinforcement effectiveness is highly affected by geostress.In this study,a numerical manifold method(NMM)based simulator has been developed to examine the impact of geostress conditions on grouting reinforcement during tunnel excavation.To develop this simulator,a detection technique for identifying slurry migration channels and an improved fluid-solid coupling(FeS)framework,which considers the influence of fracture properties and geostress states,is developed and incorporated into a zero-thickness cohesive element(ZE)based NMM(Co-NMM)for simulating tunnel excavation.Additionally,to simulate coagulation of injected slurry,a bonding repair algorithm is further proposed based on the ZE model.To verify the accuracy of the proposed simulator,a series of simulations about slurry migration in single fractures and fracture networks are numerically reproduced,and the results align well with analytical and laboratory test results.Furthermore,these numerical results show that neglecting the influence of geostress condition can lead to a serious over-estimation of slurry migration range and reinforcement effectiveness.After validations,a series of simulations about tunnel grouting reinforcement and tunnel excavation in fault fracture zones with varying fracture densities under different geostress conditions are conducted.Based on these simula-tions,the influence of geostress conditions and the optimization of grouting schemes are discussed.

Extended finite element-based cohesive zone method for modeling simultaneous hydraulic fracture height growth in layered reservoirs

In this study,a fully coupled hydromechanical model within the extended finite element method(XFEM)-based cohesive zone method(CZM)is employed to investigate the simultaneous height growth behavior of multi-cluster hydraulic fractures in layered porous reservoirs with modulus contrast.The coupled hydromechanical model is first verified against an analytical solution and a laboratory experiment.Then,the fracture geometry(e.g.height,aperture,and area)and fluid pressure evolutions of multiple hydraulic fractures placed in a porous reservoir interbedded with alternating stiff and soft layers are investigated using the model.The stress and pore pressure distributions within the layered reservoir during fluid injection are also presented.The simulation results reveal that stress umbrellas are easily to form among multiple hydraulic fractures’tips when propagating in soft layers,which impedes the simultaneous height growth.It is also observed that the impediment effect of soft layer is much more significant in the fractures suppressed by the preferential growth of adjoining fractures.After that,the combined effect of in situ stress ratio and fracturing spacing on the multi-fracture height growth is presented,and the results elucidate the influence of in situ stress ratio on the height growth behavior depending on the fracture spacing.Finally,it is found that the inclusion of soft layers changes the aperture distribution of outmost and interior hydraulic fractures.The results obtained from this study may provide some insights on the understanding of hydraulic fracture height containment observed in filed.

Tensile Fractures and in situ Stress Measurement Data Constraints on Cretaceous-Present Tectonic Stress Field Evolution of the Tanlu Fault Zone in Shandong Province,North China Craton

Tectonic stress fields are the key drivers of tectonic events and the evolution of regional structures.The tectonic stress field evolution of the Tanlu fault zone in Shandong Province,located in the east of the North China Craton(NCC),may have preserved records of the NCC’s tectonic history.Borehole television survey and hydraulic fracturing were conducted to analyze the paleo and present tectonic stress fields.Three groups of tensile fractures were identified via borehole television,their azimuths being NNW-SSE,NW-SE and NE-SW,representing multiple stages of tectonic events.Hydraulic fracturing data indicates that the study region is experiencing NEE-SWW-oriented compression and nearly-N-Soriented extension,in accordance with strike-slip and compression.Since the Cretaceous,the orientation of the extensional stress has evolved counterclockwise and sequentially from nearly-NW-SE-oriented to NE-SW-oriented and even nearly N-S-oriented,the stress state having transitioned from strike-slip-extension to strike-slip-compression,in association with the rotating and oblique subduction of the Pacific Plate beneath the NCC,with the participation of the Indian Plate.

Formation mechanism of fault accommodation zones under combined stress in graben basin:Implications from geomechanical modeling

A fault accommodation zone is a type of structure that is defined as regulating displacement and strain between faults structure.Increasing numbers of fault accommodation zones are being identified in graben basins,indicating the potential exploration target and petroleum accumulation areas.This study aims to analyze the formation mechanism and development of fault accommodation zones under combined stress by a numerical simulation method considering geomechanical modeling.Using three-dimensional(3-D)seismic interpretation and fractal dimension method,exampled with the Dongxin fault zone,the fault activity and fault combination pattern were conducted to quantitatively characterize the activity difference in fault accommodation zones.Combined with mechanical experiment test,a geomehcanical model was established for fault accommodation zones in a graben basin.Integrating the paleostress numerical simulations and structural physical simulation experiment,the developmental characteristics and genetic mechanism of fault accommodation zones were summarized.Influenced by multi movements and combined stresses,three significant tectonic evolution stages of the Dongxing Fault Zone(DXFZ)were distinguished:During the E_(s)^(3)sedimentary period,the large difference in the stress,strain,and rupture distribution in various faults were significant,and this stage was the key generation period for the prototype of the DXFZ,including the FAZ between large-scale faults.During the E_(s)^(2)sedimentary period,the EW-trending symmetric with opposite dipping normal faults and the NE-SW trending faults with large scale were furtherly developed.The junction area of two secondary normal faults were prone to be ruptured,performing significant period for inheriting and developing characteristics of fault accommodation zones.During the Es1 sedimentary period,the high-order faults in the DXFZ exhibited the obvious fault depressions and strike-slip activity,and the fault accommodation zones were furtherly inherited and developed.This stage was the molded and formative period of the FAZ,the low-order faults,and the depression in the DXFZ.

Fracture network characterisation of the Balmuccia peridotite using drone-based photogrammetry,implications for active-seismic site survey for scientific drilling

The presence of discontinuities(e.g.faults,fractures,veins,layering)in crystalline rocks can be challenging for seismic interpretations because the wide range of their size,orientation,and intensity,which controls the mechanical properties of the rock and elastic wave propagation,resulting in equally varying seismic responses at different scales.The geometrical characterisation of adjacent outcrop discontinuity networks allows a better understanding of the nature of the subsurface rocks and aids seismic interpretation.In this study,we characterise the discontinuity network of the Balmuccia peridotite(BP)in the IvreaeVerbano Zone(IVZ),northwestern Italy.This geological body is the focus of the Drilling the Ivrea eVerbano zonE(DIVE),an international continental scientific drilling project,and two active seismic surveys,SEismic imaging of the Ivrea ZonE(SEIZE)and high-resolution SEIZE(Hi-SEIZE),which aim to resolve the subsurface structure of the DIVE drilling target through high-resolution seismic imaging.For fracture characterisation,we developed two drone-based digital outcrop models(DOMs)at two different resolutions(10^(-3)-10 m and 10^(-1)-10^(3)m),which allowed us to quantitatively characterise the orientation,size,and intensity of the main rock discontinuities.These properties affect the seismic velocity and consequently the interpretation of the seismic data.We found that(i)the outcropping BP discontinuity network is represented by three more sets of fractures with respect to those reported in the literature;(ii)the discontinuity sizes follow a power-law distribution,indicating similarity across scales,and(iii)discontinuity intensity is not uniformly distributed along the outcrop.Our results help to explain the seismic behaviour of the BP detected by the SEIZE survey,suggesting that the low P-wave velocities observed can be related to the discontinuity network,and provide the basic topological parameters(orientation,density,distribution,and aperture)of the fracture network unique to the BP.These,in turn,can be used for interpretation of the Hi-SEIZE seismic survey and forward modelling of the seismic response.

Advances in seismological methods for characterizing fault zone structure

Large earthquakes frequently occur along complex fault systems.Understanding seismic rupture and long-term fault evolution requires constraining the geometric and material properties of fault zone structures.We provide a comprehensive overview of recent advancements in seismological methods used to study fault zone structures,including seismic tomography,fault zone seismic wave analysis,and seismicity analysis.Observational conditions limit our current ability to fully characterize fault zones,for example,insufficient imaging resolution to discern small-scale anomalies,incomplete capture of crucial fault zone seismic waves,and limited precision in event location accuracy.Dense seismic arrays can overcome these limitations and enable more detailed investigations of fault zone structures.Moreover,we present new insights into the structure of the Anninghe-Xiaojiang fault zone in the southeastern margin of the Qinghai-Xizang Plateau based on data collected from a dense seismic array.We found that utilizing a dense seismic array can identify small-scale features within fault zones,aiding in the interpretation of fault zone geometry and material properties.

Semi-analytical solution for mechanical analysis of tunnels crossing strike-slip fault zone considering nonuniform fault displacement and uncertain fault plane position

The tunnel subjected to strike-slip fault dislocation exhibits severe and catastrophic damage.The existing analysis models frequently assume uniform fault displacement and fixed fault plane position.In contrast,post-earthquake observations indicate that the displacement near the fault zone is typically nonuniform,and the fault plane position is uncertain.In this study,we first established a series of improved governing equations to analyze the mechanical response of tunnels under strike-slip fault dislocation.The proposed methodology incorporated key factors such as nonuniform fault displacement and uncertain fault plane position into the governing equations,thereby significantly enhancing the applicability range and accuracy of the model.In contrast to previous analytical models,the maximum computational error has decreased from 57.1%to 1.1%.Subsequently,we conducted a rigorous validation of the proposed methodology by undertaking a comparative analysis with a 3D finite element numerical model,and the results from both approaches exhibited a high degree of qualitative and quantitative agreement with a maximum error of 9.9%.Finally,the proposed methodology was utilized to perform a parametric analysis to explore the effects of various parameters,such as fault displacement,fault zone width,fault zone strength,the ratio of maximum fault displacement of the hanging wall to the footwall,and fault plane position,on the response of tunnels subjected to strike-slip fault dislocation.The findings indicate a progressive increase in the peak internal forces of the tunnel with the rise in fault displacement and fault zone strength.Conversely,an augmentation in fault zone width is found to contribute to a decrease in the peak internal forces.For example,for a fault zone width of 10 m,the peak values of bending moment,shear force,and axial force are approximately 46.9%,102.4%,and 28.7% higher,respectively,compared to those observed for a fault zone width of 50 m.Furthermore,the position of the peak internal forces is influenced by variations in the ratio of maximum fault displacement of the hanging wall to footwall and the fault plane location,while the peak values of shear force and axial force always align with the fault plane.The maximum peak internal forces are observed when the footwall exclusively bears the entirety of the fault displacement,corresponding to a ratio of 0:1.The peak values of bending moment,shear force,and axial force for the ratio of 0:1 amount to approximately 123.8%,148.6%,and 111.1% of those for the ratio of 0.5:0.5,respectively.

Unconfined compressive strength and failure behaviour of completely weathered granite from a fault zone

Understanding the strength characteristics and deformation behaviour of the tunnel surrounding rock in a fault zone is significant for tunnel stability evaluation.In this study,a series of unconfined compression tests were conducted to investigate the mechanical characteristics and failure behaviour of completely weathered granite(CWG)from a fault zone,considering with height-diameter(h/d)ratio,dry densities(ρd)and moisture contents(ω).Based on the experimental results,a regression mathematical model of unconfined compressive strength(UCS)for CWG was developed using the Multiple Nonlinear Regression method(MNLR).The research results indicated that the UCS of the specimen with a h/d ratio of 0.6 decreased with the increase ofω.When the h/d ratio increased to 1.0,the UCS increasedωwith up to 10.5%and then decreased.Increasingρd is conducive to the improvement of the UCS at anyω.The deformation and rupture process as well as final failure modes of the specimen are controlled by h/d ratio,ρd andω,and the h/d ratio is the dominant factor affecting the final failure mode,followed byωandρd.The specimens with different h/d ratio exhibited completely different fracture mode,i.e.,typical splitting failure(h/d=0.6)and shear failure(h/d=1.0).By comparing the experimental results,this regression model for predicting UCS is accurate and reliable,and the h/d ratio is the dominant factor affecting the UCS of CWG,followed byρd and thenω.These findings provide important references for maintenance of the tunnel crossing other fault fractured zones,especially at low confining pressure or unconfined condition.

Stress tensors and quantification of fracture patterns to analyze connectivity and potential fluid flow in a mesa landform of the Northern Andes

This work applies stress tensors inversions and quantification of fracture patterns along the Mesa de Los Santos, in the Eastern Cordillera of Colombia, to better understand the potential fluid flow. It thus contributes to the conceptual hydrogeological model. The area was subdivided into three blocks, separated by the NW Potreros and the Los Santos faults, having minor inner faults of different orientations. This separation facilitates the analysis of the fractures measured in the field, which in general show high dip angles and a conjugate geometry in the northern block, tension fractures(Mode I) in the central block, and a random distribution in the southwestern block. WinTensor treatment of slickensides yielded a maximum horizontal stress(SHmax) of 111o, which coincides with the WNW-ESE tensor observed from the conjugate and tension joints. We then used Frac Pa Q to generate interpolation maps of fracture intensity and density. The maps show the highest values in the central block and the lowest in the northern block,where the precipitation is higher, causing intensive rock weathering and homogenization of the fracture planes. Although the highest values of connectivity by line are found to the south of the mesa, we suggest the possibility of greater flow from the recharge zone(NE) along bedding planes and open NW-SE fractures.

A Gel-Based Solidification Technology for Large Fracture Plugging

Fault fractures usually have large openings and considerable extension. Accordingly, cross-linked gel materials aregenerally considered more suitable plugging agents than water-based gels because the latter often undergo contaminationvia formation water, which prevents them from being effective over long times. Hence, in this study, aset of oil-based composite gels based on waste grease and epoxy resin has been developed. These materials havebeen observed to possess high compressive strength and resistance to the aforementioned contamination, therebyleading to notable increase in plugging success rate. The compressive strength, thickening time, and resistance toformation water pollution of these gels have been evaluated indoors. The results show that the compressivestrength of the gel can reach 11 MPa;additionally, the related gelation time can be controlled to be more than3 h, thereby providing a safe construction time;Invasion of formation water has a small effect on the gel strengthand does not shorten the thickening time. All considered performance indicators of the oil-based gel confirm itssuitability as a plugging agent for fault fractures.

Implications for fault reactivation and seismicity induced by hydraulic fracturing

Evaluating the physical mechanisms that link hydraulic fracturing(HF) operations to induced earthquakes and the anticipated form of the resulting events is significant in informing subsurface fluid injection operations. Current understanding supports the overriding role of the effective stress magnitude in triggering earthquakes, while the impact of change rate of effective stress has not been systematically addressed. In this work, a modified critical stiffness was brought up to investigate the likelihood, impact,and mitigation of induced seismicity during and after hydraulic fracturing by developing a poroelastic model based on rate-and-state fraction law and linear stability analysis. In the new criterion, the change rate of effective stress was considered a key variable to explore the evolution of this criterion and hence the likelihood of instability slip of fault. A coupled fluid flow-deformation model was used to represent the entire hydraulic fracturing process in COMSOL Multiphysics. The possibility of triggering an earthquake throughout the entire hydraulic fracturing process, from fracturing to cessation, was investigated considering different fault locations, orientations, and positions along the fault. The competition between the effects of the magnitude and change rate of effective stress was notable at each fracturing stage. The effective stress magnitude is a significant controlling factor during fracturing events, with the change rate dominating when fracturing is suddenly started or stopped. Instability dominates when the magnitude of the effective stress increases(constant injection at each fracturing stage) and the change rate of effective stress decreases(the injection process is suddenly stopped). Fracturing with a high injection rate, a fault adjacent to the hydraulic fracturing location and the position of the junction between the reservoir and fault are important to reduce the Coulomb failure stress(CFS) and enhance the critical stiffness as the significant disturbance of stresses at these positions in the coupled process. Therefore,notable attention should be given to the injection rate during fracturing, fault position, and position along faults as important considerations to help reduce the potential for induced seismicity. Our model was verified and confirmed using the case of the Longmaxi Formation in the Sichuan Basin, China, in which the reported microseismic data were correlated with high critical stiffness values. This work supplies new thoughts of the seismic risk associated with HF engineering.

Determining the surface fault-rupture hazard zone for the Pazarcık segment of the East Anatolian fault zone through comprehensive analysis of surface rupture from the February 6,2023,Earthquake(Mw 7.7)

Following surface rupture observations in populated areas affected by the KahramanmaraşEarthquake(Mw 7.7)on February 6th,2023,along the Pazarcık segment of the East Anatolian Fault Zone(EAFZ),this study presents novel insights into physical criteria for delineating surface fault-rupture hazard zones(SRHZs)along ruptured strike-slip faults.To achieve this objective,three trench studies across the surface rupture were conducted on the Pazarcık segment of the EAFZ to collect field data,and earthquake recurrence intervals were interpreted using Bayesian statistics from previously conducted paleoseismological trenchings.The results of the proposed model indicate that the Pazarcık segment produced five significant surface-rupturing earthquakes in the last∼11 kyr:E1:11.13±1.74 kyr,E2:7.62±1.20 kyr,E3:5.34±1.05 kyr,E4:1.82±0.93 kyr,and E5:0.35±0.11 kyr.In addition,the recurrence intervals of destructive earthquakes on the subject in question range from 0.6 kyr to 4.8 kyr.Considering that the last significant earthquake occurred in 1513,the longest time since the most recent surface fault rupturing earthquake on this particular segment was 511 years.These results indicate that,in terms of the theoretical recurrence interval of earthquakes that can create surface ruptures on the Pazarcık segment,the period in which the February 6,2023,earthquake occurred was within the end of the expected return period.As a result,the potential for a devastating earthquake in the near future is not foreseen on the same fault.Finally,the SRHZ proposed for the Pazarcık section of Gölbaşıvillage was calculated as a 61-meter-wide offset on the fault lineament to reduce the negativities that may occur in the ruptured area in the future.It is recommended to take into account this width in the settlement of this area and nearby areas.

Characteristics of transfer zones under the influence of preexisting faults and regional stress transformation:Wenchang A subsag,Zhujiang River Mouth Basin,northern South China Sea

A transfer zone in rift basins preserves important information on regional tectonic evolution and plays significant roles in hydrocarbon accumulation.Based on the systematic analysis of 3D seismic data and hydrocarbon accumulation conditions,the geometry,kinematics,and reservoir control of a large synthetic overlapping transfer zone in the south of the Wenchang A subsag in the Zhujiang(Pearl)River Mouth basin were investigated.Results indicate that the development and evolution of the transfer zone was controlled by the interaction between pre-existing faults and regional stress transformation.The intense rifting of the main faults of the transfer zone controlled the development of source rocks and faultcontrolled slope break paleogeomorphology.The strike-slip overprint since the Oligocene is conducive to the formation of a large-scale fault-anticline trap,and the secondary faults in the transfer zone contribute to the hydrocarbon transportation.The conjugate intersection area of the NE-and NW-trending faults offers more opportunity for hydrocarbon migration and accumulation.

A method for predicting the water-flowing fractured zone height based on an improved key stratum theory

In the process of using the original key stratum theory to predict the height of a water-flowing fractured zone(WFZ),the influence of rock strata outside the calculation range on the rock strata within the calculation range as well as the fact that the shape of the overburden deformation area will change with the excavation length are ignored.In this paper,an improved key stratum theory(IKS theory)was proposed by fixing these two shortcomings.Then,a WFZ height prediction method based on IKS theory was established and applied.First,the range of overburden involved in the analysis was determined according to the tensile stress distribution range above the goaf.Second,the key stratum in the overburden involved in the analysis was identified through IKS theory.Finally,the tendency of the WFZ to develop upward was determined by judging whether or not the identified key stratum will break.The proposed method was applied and verified in a mining case study,and the reasons for the differences in the development patterns between the WFZs in coalfields in Northwest and East China were also fully explained by this method.

Response of underground pipeline through fault fracture zone to random ground motion

It is assumed that a pipeline is laid through a vertical fault fracture zone, and is excited by seismic ground motion modelled as stationary stochastic process. For horizontal incidence of waves, the cross-PSD （Power Spectral Density） function is developed using wave propagation theory, while for vertical incidence of waves the cross-PSD function is composed by auto-PSD model, coherence model and site response model. As the seismic input, the eross-PSD function is used to calculate the the axial and lateral seismic responses of underground pipeline through the fracture zone. The results show that the incident directions of seismic waves, width and soil property of the fracture zone have great influence on underground pipeline. It is suggested that the flexible joints with appropriate stiffness should be added into the pipeline near the interfaces between the fracture zone and the surrounded media.

Imaging Rock Density Distribution beneath Liwa Fracture Zone in the Southern Part of the Great Sumatran Fault System, Indonesia

We have imaged rock density distribution beneath Liwa fracture zone in the southern part of the the Sumatran Fault Zone by modelling and inverting Bouguer gravity data in two-and three-dimensional environments, respectively. The purpose of this study is aimed to figure out the subsurface distribution of rock densities associated with subsurface basement structure representing the evidence of trans-tensional tectonic product in the SF. In the gravity modeling, to eliminate distortions to the measured gravity values before modelling and inverting the data, Bouguer anomalies obtained in field measurements are reduced to the horizontal plane of z = +800 m as a representation of the average elevation in Liwa. For the inversion, we used algorithm implementing depth-and minimum volume weighting parameters in order to obtain a smooth model with better vertical resolution. The two-dimensional models show clearly surface topography of the basement rocks and the presence of normal faults. The reduced Bouguer anomaly of +800 m elevation shows the presence of structural lineaments extending primarily in a northwest-southeast direction, parallel to Sumatran Fault Zone and older graben faults showing a negative flower structure. From the three-dimensional inversion, the model illustrates an increase of density contrast, lower values being found near the surface and higher values in the deeper parts. The lower density contrast of 0.15 to 0.3 g/cm3 found in the rock groups at depths of 2 km and less is characteristic of relatively homogeneous and poorly compacted rocks. Rocks with moderate to high density contrast (>1.0 g/cm3) are recognized at depths of over 2 km. This model suggests a change of basement morphology as a function of depth, and delineates structural lineaments extending in northwest-southeast direction. This study supports the previous thought that Liwa area is underlain by graben structures, formed by trans-tensional tectonic events. Higher-density Tertiary volcanic breccia and lower-density Quaternary volcanic products of the Ranau Formation form the basement rocks and the overlying younger sediments, respectively.

Fracture suppression at steel/concrete connection zones by ECC

In order to avoid brittle fracture failure, a ductile engineered cementitious composite （ECC） was attempted in steel/concrete connection zones to replace normal concrete. The influence of the ECC material ductility on connection failure modes and structural performance was investigated via the pushout test of stud/ECC connection, the pullout test of two-dimensional anchor bolt/ECC connection and the finite element modeling （FEM）. The experimental results suggest that the micromechanically designed ECC with a tensile ductility 300 times that of normal concrete switches the brittle fracture failure mode to a ductile one in steel connection zones. This modification in material behavior leads to higher load carrying capacity and structural ductility, which is also confirmed in FEM investigation. The enhancement in structural response through material ductility engineering is expected to be applicable to a wide range of engineering structures where steel and concrete come into contact.

In-situ observations of damage-fracture evolution in surrounding rock upon unloading in 2400-m-deep tunnels

The damage-fracture evolution of deep rock mass has obvious particularity,which is revealed in 2400-mdeep tunnels by field tests.The evolution of the excavation damaged zone depth is consistent with that of the fractured zone depth.The ratio of the excavation damaged zone depth to the excavation fractured zone depth is greater than 2.0 in a rock mass with both high strength and good integrity,but less than1.5 in a rock mass with lower strength or poor integrity.Zonal disintegration in a rock mass with high strength and fair integrity is more likely to occur when it contains more than two groups of primary fractures in damaged zones.Fractures develop outward in zonal disintegration but are totally different from the single-zone fracture,in which the fractures develop inward,and it is the starting position of the fractured zone when the excavation surface of the middle pilot is 7–9 m close to the pre-set borehole and it stops after the excavation surface of the baseplate is 11–14 m away.The most intense evolution occurs around 2–4 m from the pre-set borehole in the sidewall expansion stage.The research results provide a reference for the monitoring scheme and support design of CJPL-Ⅲin its future construction.

A fracture model for assessing tensile mode crack growth resistance of rocks

Evaluating the fracture resistance of rocks is essential for predicting and preventing catastrophic failure of cracked structures in rock engineering.This investigation developed a brittle fracture model to predict tensile mode(mode I)failure loads of cracked rocks.The basic principle of the model is to estimate the reference crack corresponding to the fracture process zone(FPZ)based on the maximum normal strain(MNSN)ahead of the crack tip,and then use the effective crack to calculate the fracture toughness.We emphasize that the non-singular stress/strain terms should be considered in the description of the MNSN.In this way,the FPZ,non-singular terms and the biaxial stress state at the crack tip are simul-taneously considered.The principle of the model is explicit and easy to apply.To verify the proposed model,laboratory experiments were performed on a rock material using six groups of specimens.The model predicted the specimen geometry dependence of the measured fracture toughness well.More-over,the potential of the model in analyzing the size effect of apparent fracture toughness was discussed and validated through experimental data reported in the literature.The model was demonstrated su-perior to some commonly used fracture models and is an excellent tool for the safety assessment of cracked rock structures.