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Analysis of the Influence of a Natural Fracture Network on Hydraulic Fracture Propagation in Carbonate Formations 总被引：2，自引：1，他引：1
A new experimental model has been designed to simulate the influence of a natural fracture network on the propagation geometry of hydraulic fractures in naturally fractured formations using a tri-axial fracturing system. In this model, a parallel and symmetrical pre-fracture network was created by placing cement plates in a cubic mold and filling the mold with additional cement to create the final testing block. The surface of the plates will thus be weakly cemented and form pre-fractures. The dimension and direction of the pre-fractures can be controlled using the plates. The experiments showed that the horizontal differential stress $\Updelta \sigma$ and the angle $\Updelta \theta$ between the maximum horizontal principal in situ stress and the pre-fracture are the dominating factors for the initiation and propagation of hydraulic fractures. For $\Updelta \theta = 90^\circ$ and $\Updelta \sigma \ge 2{\text{ MPa}}$ or $\Updelta \theta = 60^\circ$ and $\Updelta \sigma \ge 4{\text{ MPa}}$ , the direction of the initiation and propagation of the hydraulic fractures are consistent with or deviate from the normal direction of the pre-fracture. When the hydraulic fractures approach the pre-fractures, the direction of the hydraulic fracture propagation will be consistent with the normal direction of the pre-fracture. Otherwise, the hydraulic fracture will deflect and perpendicularly cross the parallel and symmetric pre-fracture network. For $\Updelta \theta = 90^\circ$ and $\Updelta \sigma < 2{\text{ MPa}},\,\Updelta \theta = 60^\circ$ , and $\Updelta \sigma < 4{\text{ MPa}}$ or $\Updelta \theta = 45^\circ$ and $\Updelta \sigma = 4 - 8{\text{ MPa}}$ , before the hydraulic fracture and the pre-fractures intersect, the direction of the hydraulic fracture propagation remains unchanged, and the pre-fractures open or dilate when the hydraulic fracture propagates to the intersection point, forming a complicated hydraulic fracture network with the propagation region of the overall hydraulic fracture network taking the shape of an ellipse. In this condition, the complexity level of the hydraulic fracture is controlled by the net pressure, the compressive normal stress acting on the pre-fractures, the shearing strength and the cohesion strength of the planes of weakness. The conclusions of this research are inconsistent with the formulation of the approach angle that has been widely accepted by previous studies. The principle of hydraulic fracture propagation is that it follows the least resistance, the most preferential propagation, and the shortest propagation path. 相似文献
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古构造应力场是构造动力学研究中的一个重要内容,且断层滑动数据古应力反演已经成为古构造应力场恢复研究中比较常用的重要方法之一。近年来,断层滑动数据古应力反演方法研究和应用取得了一系列重要进展,但有关反演结果的解释仍存歧义,反演结果的影响因素及其误差范围等并未得到深入研究与定量分析。本文总结指出,影响断层滑动数据古应力反演结果的主要因素包括变形体制、剪切破裂面类型、断层面的形态以及地质体内薄弱面的存在等。在此基础上,分别对新生断层和先存薄弱面滑动数据的古应力反演综合误差进行了定量分析。研究指出,在可以大致厘定变形体制或误差在允许范围内的前提下,将断层滑动数据反演结果解释为应力状态是合理可行的。各种因素导致的反演误差定量分析表明,同一期构造应力场形成的破裂面滑动数据的古应力方位反演误差最大不超过35°。换言之,在没有证据表明存在不同期次的应力作用情况下,主应力方位变化小于35° 的应力状态,可以划归同一期应力场。 相似文献
4.
A two-dimensional brittle creep model for rock provides insight into the initiation of shear fracture along weak discontinuities in rock. The model accounts for material heterogeneity and introduces the concept of a mesoscopic renormalization to capture the cooperative interaction between cracks in the transition from distributed to localized damage. A series of shear creep tests on rock with discontinuities were performed to simulate the initiation and propagation of crack along a pre-existing weakness under sustained shear stress and normal stress. The investigation showed that shear stress level and the normal stress level might have significant effect on the long-term behavior of rock with weak discontinuities. Moreover, a case study of rock slope instability was also investigated, where the numerically simulated instability failure of rock slope with discontinuities showed that both tensile and shear damage at the weakest elements are the trigger for the failure surface initiation in the rock slope. Once damage occurs, redistributed stress concentrations would then intensify fracture propagation and coalescence within these damage zones, leading to the progressive development of a failure surface. Moreover, failure surface extending is not only dominated by the properties and the position of discontinuities but also influenced remarkably by the complex interaction between existing discontinuities and fracture propagation. The results are of general interest because they can be applied to the investigation of time-dependent instability in rock masses, to the mitigation of associated rock hazards in rock engineering, and even to a better understanding of the physical phenomena governing the stability of rock slope. 相似文献
5.
Stability of a levee and its foundation soil during flood is governed by seepage forces. In this study, the seepage forces in the flow domain have been determined using conformal mapping technique. Stability of foundation soil has been studied for a levee with and without cut-off wall on permeable soil of infinite depth. In situ effective normal and shear stresses due to self-weight, additional induced stresses due to levee structure and ponded water on upstream sloping face have been determined by assuming soil to be elastic. Besides, components of seepage stress acting at a point on two orthogonal planes, i.e., on vertical and horizontal planes, are considered and the Mohr’s circle is drawn. Mohr–Coulomb failure envelope is drawn for the foundation soil with known effective cohesion ( $c^{\prime}_{1}$ c 1 ′ ) and angle of internal friction ( $\varphi^{\prime}_{1}$ φ 1 ′ ), and then the stability of the foundation soil has been assessed. Probable zone of general shear failure considering seepage flow has been identified. Zones vulnerable to foundation failure for different width of levee and depth of cut-off wall have been computed. The hydraulic unstable zone in downstream side of the levee resembles a log spiral. The provision of cut-off wall restricts the progress of failure zone only in the downstream side; otherwise, the zone of failure will enter the foundation of the levee. The stability of the foundation soil of an existing ring levee along the rivers Ganga, Sone and Punpun, near Patna (India), has been evaluated. 相似文献
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Liao Jianxing Gou Yang Feng Wentao Mehmood Faisal Xie Yachen Hou Zhengmeng 《Acta Geotechnica》2020,15(2):279-295
Although hydraulic fracturing has been massively studied and applied as a key technique to enhance the gas production from tight formations, some problems and uncertainties exist to accurately predict and analyze the fracture behavior in complex reservoirs, especially in the naturally fractured reservoirs like shale reservoirs. This paper presents a full 3D numerical model (FLAC3D) to study hydraulic fracturing behavior under the impact of preexisting orthogonal natural fractures. In this numerical model, the hydraulic fracture propagation direction is assumed perpendicular to the minimum principal stress and activated only by tensile failure, whereas the preexisting natural fractures can be activated by tensile or shear failure or a combination of them, and only tensile failure can open the natural fracture as well. The newly developed model was used to study the impact of preexisting orthogonal natural fractures on hydraulic fracturing behavior, based on a multistage hydraulic fracturing operation in a naturally fractured reservoir from the Barnett Shale formation, northwest of Texas in USA. In this multistage operation, two more representative stages, i.e., stage 1 with a relatively large horizontal stress anisotropy of 3.3 MPa and stage 4 with a comparatively small one of 1.3 MPa, were selected to conduct the simulation. Based on the numerical results, one can observe that the interaction between hydraulic and natural fracture is driven mainly by induced stress around fracture tip. Besides, the horizontal stress anisotropy plays a key role in opening the natural fracture. Thus, no significant opened fracture is activated on natural fracture in stage 1, while in stage 4 an opened fracture invades to about 90 m into the first natural fracture. Conversely, the hydraulic fracture length in stage 1 is much longer than in stage 4, as some fluid volume is stored in the opened natural fracture in stage 4. In this work, the shear failure on natural fractures is treated as the main factor for inducing the seismic events. And the simulated seismic events, i.e., shear failure on natural fractures, are very comparable with the measured seismic events.
相似文献7.
Property and behaviour of sand–pile interface are crucial to shaft resistance of piles. Dilation or contraction of the interface soil induces change in normal stress, which in turn influences the shear stress mobilised at the interface. Although previous studies have demonstrated this mechanism by laboratory tests and numerical simulations, the interface responses are not analysed systematically in terms of soil state (i.e. density and stress level). The objective of this study is to understand and quantify any increase in normal stress of different pile–soil interfaces when they are subjected to loading and stress relief. Distinct element modelling was carried out. Input parameters and modelling procedure were verified by experimental data from laboratory element tests. Parametric simulations of shearbox tests were conducted under the constant normal stiffness, constant normal load and constant volume boundary conditions. Key parameters including initial normal stress ( $ \sigma_{{{\text{n}}0}}^{\prime } $ ), initial void ratio (e 0), normal stiffness constraining the interface and loading–unloading stress history were investigated. It is shown that mobilised stress ratio ( $ \tau /\sigma_{\text{n}}^{\prime } $ ) and normal stress increment ( $ \Updelta \sigma_{\text{n}}^{\prime } $ ) on a given interface are governed by $ \sigma_{{{\text{n}}0}}^{\prime } $ and e 0. An increase in $ \sigma_{{{\text{n}}0}}^{\prime } $ from 100 to 400 kPa leads to a 30 % reduction in $ \Updelta \sigma_{\text{n}}^{\prime } $ . An increase in e 0 from 0.18 to 0.30 reduces $ \Updelta \sigma_{\text{n}}^{\prime } $ by more than 90 %, and therefore, shaft resistance is much lower for piles in loose sands. A unique relationship between $ \Updelta \sigma_{\text{n}}^{\prime } $ and normal stiffness is established for different soil states. It can be applied to assess the shaft resistance of piles in soils with different densities and subjected to loading and stress relief. Fairly good agreement is obtained between the calculated shaft resistance based on the proposed relationship and the measured results in centrifuge model tests. 相似文献
8.
Lutz Kübler 《Physics and Chemistry of Minerals》1985,12(6):353-362
Single crystals of hexagonal and monoclinic pyrrhotite, Fe1?xS, have been experimentally deformed by uniaxial compression at 300 MPa confining pressure, and at a strain rate of 1 × 10?5 s?1 in the temperature range from 200° C to 400° C. Very high anisotropy characterizes the mechanical behaviour of the crystal structure. During compression parallel to thec-axis, when no slip system may be activated, the maximum strength is observed. One or two degrees of non-parallelism between [c] and σ1 results in slip on the basal plane, illustrating the very low resistance of the lattice against shear in this plane. At σ1 Λ(0001)=45°, i.e. when maximum resolved shear stress is attained on the basal plane, the strength reaches a minimum. Thecritical resolved shear stress (CRSS) increases from less than 4.7 MPa at 400° C to 52 MPa at 200° C. A new slip system, \((10\overline 1 0)\parallel \left\langle {1\overline 2 10} \right\rangle \) prism slip, is described. It is activated only at high angles (>70°) between σ1 and [c]. The CRSS of the prism slip ranges from 7 MPa (400° C) to 115 MPa (200° C). Twinning on \((10\overline 1 2)[(10\overline 1 2):(1\overline 2 10)]\) , earlier reported by several authors, has been produced only at the highest temperature either as secondary feature during pressure release (compression ‖[c]) or in heterogeneously strained areas (compression ⊥[c]). As twinning and prism slip attain their maximum values of the Schmidt factor under nearly equal stress conditions it is postulated that the former of the two deformation modes has the higher shear resistance. 相似文献
9.
Baotang Shen Hyung-Mok Kim Eui-Seob Park Taek-Kon Kim Manfred W. Wuttke Mikael Rinne Tobias Backers Ove Stephansson 《Rock Mechanics and Rock Engineering》2013,46(1):135-151
This paper describes a boundary element code development on coupled thermal–mechanical processes of rock fracture propagation. The code development was based on the fracture mechanics code FRACOD that has previously been developed by Shen and Stephansson (Int J Eng Fracture Mech 47:177–189, 1993) and FRACOM (A fracture propagation code—FRACOD, User’s manual. FRACOM Ltd. 2002) and simulates complex fracture propagation in rocks governed by both tensile and shear mechanisms. For the coupled thermal-fracturing analysis, an indirect boundary element method, namely the fictitious heat source method, was implemented in FRACOD to simulate the temperature change and thermal stresses in rocks. This indirect method is particularly suitable for the thermal-fracturing coupling in FRACOD where the displacement discontinuity method is used for mechanical simulation. The coupled code was also extended to simulate multiple region problems in which rock mass, concrete linings and insulation layers with different thermal and mechanical properties were present. Both verification and application cases were presented where a point heat source in a 2D infinite medium and a pilot LNG underground cavern were solved and studied using the coupled code. Good agreement was observed between the simulation results, analytical solutions and in situ measurements which validates an applicability of the developed coupled code. 相似文献
10.
土体中的水力劈裂破坏机理存在两种观点：张拉破坏机理和剪切破坏机理。前人已经对其进行了大量地研究,并分别基于两种破坏机理提出了起劈压力的理论计算式,但是对其适用性没有进一步探讨。分别分析三向应力状态下土体水力劈裂的张拉破坏和剪切破坏判定准则,并基于Mohr-Coulomb屈服准则分析两种破坏机理的适用条件。分析结果表明,土体的水力劈裂是张拉破坏还是剪切破坏与小主应力σ3和不排水抗剪强度cu的大小密切相关,当小主应力σ3较大且不排水抗剪强度cu较小时,土体的劈裂多受剪切破坏机理控制;反之,土体的劈裂为张拉破坏。最后通过已有文献中的试验结果验证了所提出的两种破坏机理适用条件的合理性。 相似文献
11.
Deformation-induced stacking defects in dolomite have been characterised following examination at the cation sublattice level using high-resolution electron microscopy at 500kV. Slip on c (≡{0001}) is observed to produce stacking faults, often de-localised laterally, which are terminated by partial dislocations with Burgers vectors of the form 1/3 [1 \(\overline 1 \) 00]: a model for the faulted dolomite lattice has been constructed which agrees with the image appearance. Slip on f (≡{10 \(\overline 1 \) 2}) produces long planar faults which are established as not being stacking faults, in the normal sense, since there appear to be no offsets of the cation sublattice across the faults, nor any general indication of any terminating partial dislocations: it is proposed that the contrast arises from rotational disorder in CO3 groups which has resulted from the prior passage of partial dislocations during deformation. 相似文献
12.
A Bonded Particle Model Simulation of Shear Strength and Asperity Degradation for Rough Rock Fractures 总被引：3，自引：3，他引：0
Mohammad Sadegh Asadi Vamegh Rasouli Giovanni Barla 《Rock Mechanics and Rock Engineering》2012,45(5):649-675
Different failure modes during fracture shearing have been introduced including normal dilation or sliding, asperity cut-off and degradation. Attempts have been made to study these mechanisms using analytical, experimental and numerical methods. However, the majority of the existing models simplify the problem, which leads to unrealistic results. With this in mind, the aim of this paper is to simulate the mechanical behaviour of synthetic and rock fracture profiles during direct shear tests by using the two-dimensional particle flow computer code PFC2D. Correlations between the simulated peak shear strength and the fracture roughness parameter D R1 recently proposed by Rasouli and Harrison (2010) are developed. Shear test simulations are carried out with PFC2D and the effects of the geometrical features as well as the model micro-properties on the fracture shear behaviour are studied. The shear strength and asperity degradation processes of synthetic profiles including triangular, sinusoidal and randomly generated profiles are analysed. Different failure modes including asperity sliding, cut-off, and asperity degradation are explicitly observed and compared with the available models. The D R1 parameter is applied to the analysis of synthetic and rock fracture profiles. Accordingly, correlations are developed between D R1 and the peak shear strength obtained from simulations and by using analytical solutions. The results are shown to be in good agreement with the basic understanding of rock fracture shear behaviour and asperity contact degradation. 相似文献
13.
Radomir Petrović 《Contributions to Mineralogy and Petrology》1973,41(2):151-170
Coherency stress and coherency strain energy generated by Na+?K+ ion exchange in alkali feldspars are calculated using an isotropic model, and deformation of single crystals of alkali feldspars exposed to molten alkali chlorides at \(P_{H_2 O} \) < 1 bar is described. Coherency stress in alkali feldspars can reach 10–20 kb. When it is large, partial relaxation by fracture and/or plastic deformation takes place under anhydrous conditions, but temporary build-up of stress is unavoidable even under hydrothermal conditions. Because of coherency strain energy, a thin layer of an end-member alkali feldspar produced by cation exchange on a grain of the other end-member alkali feldspar would be unstable with respect to dissolution. Therefore, under hydrothermal conditions one end-member alkali feldspar replaces the other by dissolution and precipitation. The mechanism of the reaction $$Na_x K_{1 - x} AlSi_3 O_{8_{(feld.)} } + yK^ + \rightleftharpoons Na_{x - y} K_{1 + y - x} AlSi_3 O_{8_{(feld.)} } + yNa^ + $$ is primarily controlled by \(P_{H_2 O} \) and by ΔK/(Na + K), the difference between the equilibrium value and the initial value of the atomic K/(Na + K) ratio of the feldspar. When ¦ΔK/(Na + K)¦ is small, the reaction proceeds by cation exchange. When ¦ΔK/(Na + K)¦ is large, cation exchange still occurs if \(P_{H_2 O} \) is very low, but under hydrothermal conditions replacement by dissolution and precipitation occurs. 相似文献
14.
Correlation of large displacement drained shear strength of landslide soils measured by direct shear and ring shear devices 总被引：2，自引：0，他引：2
Shriwantha Buddhi Vithana Shinya Nakamura Seiichi Gibo Anshun Yoshinaga Sho Kimura 《Landslides》2012,9(3):305-314
Based on a laboratory study of drained direct shear tests of remoulded, pre-cut and polished specimens and drained ring shear tests of uncut and remoulded specimens of slip surface materials of five landslide soils, significant correlations of the mobilized shear strength parameters, cohesion (c, c r) and internal friction angle (Φ, Φr), are proposed. The investigated soils consisted of the slip surface material belonging to tuffaceous clay, mudstone, loess and siltstone. Most of the previous studies on residual shear strength measured by reversal direct shear and ring shear devices have obtained significantly different results using the two devices, even when pre-cut and polished specimens were used in the direct shear device. In this study, the residual shear strength values of remoulded specimens measured by a ring shear device are shown to significantly correlate with the drained large displacement shear strength values of remoulded specimens, which were measured using pre-cut and polished specimens in a direct shear device. The correlation between the cohesion measured in the two shear devices is expressed by the linear relationship, $ {c_{\text{r}}} = 0.{7394}c - {6}.{6857} $ , while the correlation between the friction angle measured in the two devices is expressed by the linear relationship, $ {\Phi_{\text{r}}} = {1}.0{852}\Phi - {6}.0{247} $ . The proposed linear correlations for effective cohesion (c′) and effective friction angle (Φ′) have yielded significant coefficients of determination within an effective normal stress range of 30–150?kN/m2. 相似文献
15.
工程开挖面附近卸荷扰动区的岩体,受结构面和拉应力共同影响作用,其变形和破坏具有拉剪复合特征。为研究节理岩体的拉剪力学特性,基于颗粒离散元法针对共面断续节理岩体开展了系列数值模拟研究。通过假设粒间接触的力学参数服从Weibull分布表征岩体的非均质性,探讨了非均质性、均质度、法向拉应力和节理连通率对节理岩体拉剪强度和破坏模式的影响。研究表明:拉剪应力条件下非均质性节理岩体主要沿阶梯型破裂面破坏,剪应力-水平位移曲线可以分为线性变形阶段、非线性变形阶段、峰值及峰后阶段;随均质度提高,节理岩体的剪切强度逐渐增加且提升幅度逐渐减弱,趋于均质岩体,岩体中微裂纹由弥散型分布向破裂面集中;节理岩体峰值剪切强度和法向拉应力的大小呈非线性负相关关系;岩体剪切强度随节理连通率增加而显著降低。 相似文献
16.
The paper presents an innovative numerical approach to simulate progressive caving of strata above a longwall coal mining panel. A proposed Trigon logic is incorporated within UDEC to successfully capture the progressive caving of strata which is characterized by fracture generation and subsequent propagation. A new damage index, D, is proposed that can quantify regions of both compressive shear and tensile failure within the modelled longwall. Many features of progressive caving are reproduced in the model and found to fit reasonably well with field observations taken from a case study in the Ruhr coalfield. The modelling study reveals that compressive shear failure, rather than tensile failure, is the dominant failure mechanism in the caved strata above the mined-out area. The immediate roof beds act like beams and can collapse in beam bending when vertical stress is dominant or in beam shear fracture when horizontal stress is dominant. The proposed numerical approach can be used to guide the design of longwall panel layout and rock support mechanisms. 相似文献
17.
Kais Aridhi Sabri Aridhi Mohamed Abdoullah Ould Bagga Saâdi Abdeljaouad Fouad Zargouni Eric Mercier 《Arabian Journal of Geosciences》2013,6(5):1591-1599
The imbrication’s area in northern Tunisia is the most external segment of Alpine range, where several associated folds types with thrust ramps are recognized within imbricate units beneath Numidian front slope. Their presence help to understand thrusting mechanisms installation through studied area. In fact, this zone was considered as a result of Paleogene gravitary slop (Kujawski (Ann Miner Géol Tunis (24):281, 1969); Caire (Ann Min Géol Tunis 26:87–110, 1973); Rouvier 1977), which is proved to be affected by major deep decollement, given rise to various structures, some are propagation folds, specific of foreland front, limited to this area, and those in more external position: Tunisian Atlas (Creusot et al. (C R Acad Sci Paris 314(Sér II):961–965, 1992); Ouali and Mercier (PII: S0191-8141(97):00048-5, 1997); Ouali 1984; Ahmadi et al. (J Struct Geol 28:721–728, 2006)). Various categories of fold ramps could be identified: frontal folds ramp NE–SW and others as lateral or oblique ramp with NW–SE trend (Aridhi et al. (C R Geosci 343:360–369, 2011)). The relation between various structures has been used as recognition tools of thrusting sequences and to propose a new deformation chronology. Delimited outcropping of these structures between two both parallel faults strikes with regional displacement, leads to interpret these faults as cogenetic tear faults of propagation thrusts; this fault separates two domains with different deformation styles from each other side. 相似文献
18.
Widely extended, cation stacking faults in experimentally deformed Mg2GeO4 spinel have been studied using transmission electron microscopy (TEM). The faults lie on {110} planes. The displacement vector is of the form \(\frac{1}{4}\left\langle {1\bar 10} \right\rangle \) and is normal to the fault plane. The partial dislocations which bound the stacking fault have colinear Burgers vectors of the form \(\frac{1}{4}\left\langle {1\bar 10} \right\rangle \) which are normal to the fault plane. 相似文献
19.
Present-day stress orientations in the Northern Perth Basin have been inferred from borehole breakouts and drilling-induced tensile fractures observed on image logs from eight wells. Stress indicators from these wells give an east – west maximum horizontal stress orientation, consistent with stress-field modelling of the Indo-Australian Plate. Previous interpretations using dipmeter logs indicated anomalous north-directed maximum horizontal stress orientations. However, higher-quality image logs indicate a consistent maximum horizontal stress orientation, perpendicular to dominant north – south and northwest – southeast fault trends in the basin. Vertical stress was calculated from density logs at 21.5 MPa at 1 km depth. Minimum horizontal stress values, estimated from leak-off tests, range from 7.4 MPa at 0.4 km to 21.0 MPa at 0.8 km depth: the greatest values are in excess of the vertical stress. The maximum horizontal stress magnitude was constrained using the relationship between the minimum and maximum horizontal stresses; it ranges from 8.7 MPa at 0.4 km to 21.3 MPa at 1 km depth. These stress magnitudes and evidence of neotectonic reverse faulting indicate a transitional reverse fault to strike-slip fault-stress regime. Two natural fracture sets were interpreted from image logs: (i) a north- to northwest-striking set; and (ii) an east-striking set. The first set is parallel to adjacent north- to northwest-striking faults in the Northern Perth Basin. Several east-striking faults are evident in seismic data, and wells adjacent to east-striking faults exhibit the second east-striking set. Hence, natural fractures are subparallel to seismically resolved faults. Fractures optimally oriented to be critically stressed in the present-day stress regime were probably the cause of fluid losses during drilling. Pre-existing north- to northwest -striking faults that dip moderately have potential for reactivation within the present-day stress regime. Faults that strike north to northwest and have subvertical dips will not reactivate. The east-striking faults and fractures are not critically stressed for reactivation in the Northern Perth Basin. 相似文献
20.
José M. Carcione Juan E. Santos Stefano Picotti 《Rock Mechanics and Rock Engineering》2012,45(5):929-942
The triaxial nature of the tectonic stress in the earth’s crust favors the appearance of vertical fractures. The resulting rheology is usually effective anisotropy with orthorhombic and monoclinic symmetries. In addition, the presence of fluids leads to azimuthally varying attenuation of seismic waves. A dense set of fractures embedded in a background medium enhances anisotropy and rock compliance. Fractures are modeled as boundary discontinuities in the displacement u and particle velocity v as $[{\varvec{ \kappa}}\cdot {\bf u} + {\varvec{\eta}} \cdot {\bf v} ],$ where the brackets denote discontinuities across the fracture surface, ${\varvec{\kappa}}$ is a fracture stiffness, and ${\varvec{\eta}}$ is a viscosity related to the energy loss. We consider a transversely isotropic background medium (e.g., thin horizontal plane layers), with sets of long vertical fractures. Schoenberg and Muir’s theory combines the background medium and sets of vertical fractures to provide the 13 complex stiffnesses of the long-wavelength equivalent monoclinic and viscoelastic medium. Long-wavelength equivalent means that the dominant wavelength of the signal is much longer than the fracture spacing. The symmetry plane is the horizontal plane. The equations for orthorhombic and transversely isotropic media follow as particular cases. We compute the complex velocities of the medium as a function of frequency and propagation direction, which provide the phase velocities, energy velocities (wavefronts), and quality factors. The effective medium ranges from monoclinic symmetry to hexagonal (transversely isotropic) symmetry from the low- to the high-frequency limits in the case of a particle–velocity discontinuity (lossy case) and the attenuation shows typical Zener relaxation peaks as a function of frequency. The attenuation of the coupled waves may show important differences when computed versus the ray or phase angles, with triplication appearing in the Q factor of the qS wave. We have performed a full-wave simulation to compute the field corresponding to the coupled qP–qS waves in the symmetry plane of an effective monoclinic medium. The simulations agree with the predictions of the plane-wave analysis. 相似文献