Multiple Parallel Fractures Research Articles

Interpretation of low-frequency distributed acoustic sensing data based on geomechanical models

Distributed acoustic sensing (DAS) is a technology that enables continuous, real-time measurements along the entire length of a fiber-optic cable. The low-frequency band of DAS can be used to analyze hydraulic fracture geometry and growth. In this study, the low-frequency strain waterfall plots with their corresponding pumping curves were analyzed to obtain information on fracture azimuth, propagation speed, number of fractures created in each stage, and restimulation of preexisting fractures. We also use a simple geomechanical model to predict fracture growth rates while accounting for changes in treatment parameters. As expected, the hydraulic fractures principally propagate perpendicular to the treated well, that is, parallel to the direction of maximum horizontal stress. During many stages, multiple frac hits are visible, indicating that multiple parallel fractures are created and/or reopened. Secondary fractures deviate toward the heel of the well, likely due to the cumulative stress shadow caused by previous and current stages. The presence of heart-shaped tips reveals that some stress and/or material barrier is overcome by the hydraulic fracture. The lobes of the heart are best explained by the shear stresses at 45° angles from the fracture tip instead of the tensile stresses directly ahead of the tip. Antennas ahead of the fracture hits indicate the reopening of preexisting fractures. Tails in the waterfall plots provide information on the continued opening, closing, and interaction of the hydraulic fractures within the fracture domain and stage domain corridors. The analysis of the low-frequency DAS plots thus provides in-depth insights into the rock deformation and rock-fluid interaction processes occurring close to the observation well.

Experimental Research on Dynamic Response of Layered Medium under Impact Load

Stress waves propagating through multiple parallel fractures are attenuated due to the multiple wave reflections and transmissions at the fractures. The dynamic response of a layered medium under an impact load is systematically studied by a model test in this paper. The effects of the impact energy, number of joints, thickness of the medium and joint layer, material properties of the cementation layer, and other factors of the dynamic stress propagation and attenuation of layered media are analyzed. Studies have shown that Sadovsky’s empirical formula fits the attenuation law of vertical peak velocity well, and the obtained attenuation coefficient k and index b can be used for reflecting the dynamic response characteristics of layered media. The attenuation coefficient k is positively correlated with the impact height and negatively correlated with the plate thickness in a single-layer plate impact test. The thickness of the medium layer is the main factor affecting the vibration response of the layered medium in the impact test of the multi-layer slab. The medium thickness, the number of medium layers, and the number of joints have little influence. The different cementation materials and the change of cementation thickness will have a great influence on the dynamic response of layered media, when the cementation layer is filled with joint surfaces for the impact test of multi-layer plates.

Spatial Deflection of Parallel Hydraulic Fractures and Induced Shear Stress Disturbance Under Different Perforation Cluster Spacing Considering Thermal Effects

Reliable estimation of fracture network length and morphology in hydrofracturing is crucial for controlling and optimizing fracturing effects. Hydraulic fracture propagation will be affected by a variety of factors to produce deflection, resulting in different fracture network morphology. To study the spatial deflection behaviours of multiple parallel hydraulic fractures, three-dimensional engineering-scale numerical model for multistage fracturing is established to study the induced shear stress disturbance and unstable spatial propagation behavior of hydraulic fractures under different perforation cluster spacing. In the model, the thermal diffusion, fluid flow and deformation of rock between the rock matrix and fluid in pores and fractures are considered to describe the thermal-hydro-mechanical coupling. In this study, the results show that the thermal effect between fracturing fluid and rock matrix is an important factor affecting fracture propagation, and thermal effects can increase induced shear stress area and promote fracture propagation. The induced shear stress disturbance caused by fracture propagation is superimposed in multiple fractures, resulting in stress shadow effect and spatial deflection of parallel fractures. The stress shadow areas and the spatial deflection of parallel hydraulic fractures will increase with the decrease of multiple perforation cluster spacing.

Experimental study on the multiple fracture simultaneous propagation during extremely limited-entry fracturing

Horizontal well with multi-stage fracturing is one of the most effective stimulation methods for unconventional reservoirs, e.g. tight oil/gas or shale. To maximize reservoir stimulation volume (SRV), tighter fracture spacing and fewer perforations are distributed in one stage during extreme limited-entry fracturing (XLEF) in recent years. However, the fracture geometries and injection pressure curve are not clear when multiple fractures with close spacing were created simultaneously in the perforated wellbore during XLEF. This study investigated the multiple fracture simultaneous propagation in the XLEF perforated wellbore based on the true tri-axial fracturing experiments. Critical factors of horizontal stress difference (HSD), the number of perforation clusters, helical/in-plane perforated method, number of perforations per cluster and fracturing fluid flowrate were investigated in detail. The results showed that, firstly, compared to one fracture produced by the helical perforated method, XLEF with the in-plane perforated method has a higher breakdown pressure and could simultaneously create multiple transverse fractures. Secondly, longitudinal fractures and a small number of curved transverse fractures occurred simultaneously under lower HSD conditions, while multiple parallel transverse fractures could be created under high HSD conditions. Thirdly, increasing the number of perforations per cluster will reduce perforation cluster effectiveness, and increasing the number of clusters will lead to the merging of multiple fractures. Finally, three relationships between pressure response and fracture geometries during XLEF, e.g. single transverse fracture, multiple transverse fractures, co-existence of longitudinal and transverse fractures, have been revealed. This study provides a meaningful perspective for the multiple fracture propagation in the perforated wellbore, which could help the field fracturing design during XLEF.

Experimental study on plugging performance and diverted fracture geometry during different temporary plugging and diverting fracturing in Jimusar shale

Temporary plugging and diverting fracturing (TPDF) is one promising technology to form complex fracture networks in unconventional oil and gas resources, such as Jimusar shale in China. The key to the success of this technology depends on two processes: creating a tight plug within previously-created fractures using temporary plugging agents (TPA), and then forming diverted fractures after plugging. Many scholars have studied the plugging law of TPA under different channels aiming at the first process. However, the second process of fracture re-direct to form fracture networks after effective plugging has not been investigated clearly, especially in shale formations. Based on Jimusar shale, this paper investigated the plugging performance of fibers and particles and diverting fracturing behavior under three different TPDF application scenarios: near-wellbore in-plane TPDF, multistage TPDF in the horizontal wellbore, and far-field TPDF within fractures. A series of experiments were carried out under different stress difference conditions and TPA types using true tri-axial fracturing system. Experimental results show that: firstly, in near-wellbore TPDF experiments, the plugging pressure peaks would be increased with horizontal stress difference (HSD) while the deflection angle between the diverted fracture and the initial fracture would decrease with HSD. Once the HSD is over 7.5 MPa, the diverted fractures would be nearly parallel to the initial fractures. Secondly, in far-field TPDF within 2 mm propped fractures, 1 wt% pure fibers cannot form the effective plug in any fractures regardless of the HSD, while the combination of 1 wt% fibers and 1 wt% particles could effectively plug all fractures. The diverted fractures could be only created as branch fractures at the middle of the initial fractures, but the diverted fractures would be merged into the initial fractures when HSD is over 7.5 MPa. Thirdly, in multistage TPDF in the horizontal wellbore, the initial fracture almost initiates from only one slot and becomes one transverse fracture, but multiple parallel transverse fractures after plugging could happen but not always occur. The diverted fracture may also form a longitudinal fracture under low HSD because the slots have been plugged by TPAs. In addition, influenced by bedding and natural fractures, more bedding fractures are activated after plugging, which forms a more complex fracture network. Our experimental results could deepen the understanding of TPDF in Jimusar shale reservoir and also provide fundamental for field design in similar shale reservoirs.

Numerical simulation of fracture failure characteristics of rock mass with multiple parallel fractures under seepage stress coupling

Numerical simulation of fracture failure characteristics of rock mass with multiple parallel fractures under seepage stress coupling

Modeling discrete fractures in continuum analysis and insights for fracture propagation and mechanical behavior of fractured rock

Abstract This paper presents a procedure for modeling discrete fracture initiation, deformation, and propagation in continuum analysis of rock, and provides numerical simulations to evaluate performance of the procedure against analytical solutions and laboratory data. The procedure is based on aggregating the mechanical behavior of evolving discrete fractures and unfractured rock and provides specific information about fracturing; such as when and where fractures may occur and the type (tensile or shear), geometrical attributes, and aperture of each fracture. By storing fracture geometry as continuum properties that are used internally to generate geometrical parameters needed for analysis, the procedure models the geometry and mechanical behavior of discrete fractures explicitly in continuum analysis without including the fractures in the model grid. Numerical simulations are described for typical laboratory loading conditions such as triaxial compression and extension, and Brazilian testing. The results are compared with laboratory data, analytical solutions, and established concepts to make a case that the procedure performs satisfactorily and can be relied upon for understanding and predicting rock mechanical behavior. Furthermore, the paper discusses insights based on the simulation results for understanding rock fracturing; mechanical behavior of fractured rock; and large-scale behavior such as localization of rock deformation, stress control of fracture mechanisms in failure zones, and conditions for occurrence of multiple parallel fractures.

Velocity Amplification of Seismic Waves Through Parallel Fractures Near a Free Surface in Fractured Rock: A Theoretical Study

To determine the dynamic demand for support design under rockburst conditions, one of the most important issues is the prediction of ground motion parameters at the site of interest. Field monitoring has shown that the peak ground motion at the surface of an excavation in fractured rock is preferentially amplified compared to the motion in solid rock at a similar distance from the source. However, the traditional scaling laws used in rock support design do not account for the effect of free surface (excavation) and fracturing of rock. Recent studies have shown that high ground motion might be generated when a seismic wave crosses through fractures near a free surface in fractured rocks which is very complex and is not well understood. In this paper, particle velocity amplification was theoretically studied by investigating the dynamic interaction between seismic wave and multiple fractures near a free surface using the method of characteristics and the displacement discontinuity model. A harmonic load was applied on a model with a fractured zone near a free surface to investigate this phenomenon. After the harmonic wave propagated normally through multiple parallel fractures, the velocity amplification factor (VAF) was calculated as a function of the ratio of the magnitude of the peak particle velocity at the free surface of the model to the peak input velocity. The VAF can be as high as 3.77 and varies depending on the state of the fractured rock and the characteristics of the seismic wave. Parameter studies were conducted to investigate the effects of seismic load and multiple fractures on wave propagation, especially in terms of the wave frequency, the fracture spacing, the number of fractures and the stiffness of fractures. The results have proved that the interaction of the seismic wave and multiple fractures near the free surface strongly influences the ground motion. Quantitative relationships between the various influential factors and the corresponding VAF were developed. It is anticipated that such relationships can provide criteria to improve the current design procedures and help mining engineers to improve their rock support practice for rockburst-prone areas.

Hydraulic fracturing initiation and near-wellbore nonplanar propagation from horizontal perforated boreholes in tight formation

Tight formations, which are rich in reserves in China, have a high economic development value. Perforated fracturing would need to be used widely to exploit these formations because of reservoir characteristics such as low permeability, low porosity and high density, resulting in high fracture pressure, limited fracture propagation, and complex fracture geometry. To address these issues, the mechanism of fracture propagation under different construction conditions must be understood and described. Therefore, the effect of different perforation parameters (shot density, shot length, shot diameter, shot phase, and shot interval spacing) and different horizontal stress contrasts on fracture geometry, initiation, and propagation from perforated horizontal wells in tight formations is studied based on numerous triaxial hydraulic fracturing simulation experiments. Three types of fracture geometry are observed in these experiments: A single flat fracture can be created by an initiated perforation or more initiated perforations with good connectivity, spiral-shaped fractures are generated by a large perforation diameter or high perforation density, and multiple-parallel fractures are induced by a high perforation phase. Moreover, each fracture type has certain pressure behaviors corresponding to a fracture curve shape. The perforations are likely to initiate when the angle between the perforation and maximum horizontal stress is small, and the initiation position is most likely at the base of the perforation. It is difficult to predict the initiation order of perforations; however, the first initiated perforation can be ensured by the profile of the relief pressure. In addition, a higher horizontal stress contrast value results in more initiated perforations, lower fracture pressure, and shorter breakage time. Ultimately, a large perforation diameter, high perforation density, and perforation phase of 60° should be chosen for fracturing at low fracture pressure and simple fracture geometry in tight formations.

Heat recovery from multiple-fracture enhanced geothermal systems: The effect of thermoelastic fracture interactions

Abstract This study investigates the effect of thermoelastic interactions between multiple parallel fractures on energy production from a multiple-fracture enhanced geothermal system. A coupled thermo-hydro-mechanical finite element model has been developed that accounts for non-isothermal fluid flow within the fractures, conductive heat transfer in the rock matrix, and the mechanical deformation of the matrix. The model results show that the matrix deformation significantly increases the interactions between the two adjacent fractures. Matrix contraction due to the cooling of the matrix increases the fracture aperture in the adjacent fracture, and facilitates the creation of favourable flow pathways between the injection and production wells. These flow paths reduce the energy production from the system. The effects of fracture spacing, reservoir temperature gradient and mechanical properties of the rock matrix on the production temperature and the net production energy are investigated. It is shown that the spacing calculated based on the assumption of rigid matrix (constant uniform aperture) are too small, and in order to account for the thermoelastic interactions, the spacing between fractures should be further increased to maximise the net energy production from the system. Otherwise, the multiple-fracture system fails to improve the energy recovery from the geothermal reservoir, as initially intended.

Validating the ability of the discontinuous deformation analysis method to model normal P‐wave propagation across rock fractures

SummaryThe effects of fractures on wave propagation problems are increasingly abstracting the attention of scholars and engineers in rock engineering field. This study aims to fully validate the ability of discontinuous deformation analysis (DDA) to model normal P‐wave propagation across rock fractures. The effects of a single fracture and multiple parallel fractures are all tested. The results indicate that DDA can accurately reflect the fracture effects, including the fractures stiffness, the fracture spacing and the fracture number, and the effects of incident wave frequency on one‐dimensional P‐wave propagation problems. Thus, DDA is able to deal well with normal incident P‐wave propagation problems. Copyright © 2017 John Wiley & Sons, Ltd.

Perturbation analysis for predicting the temperatures of water flowing through multiple parallel fractures in a rock mass

In this paper, we revisit this problem of heat extraction from an enhanced geothermal system (EGS) with multiple parallel fractures, which provide more surface area and conductive paths between injection and production wells, by employing a semi-analytical method. A dimensionless formulation is obtained first to find the controlling parameters. By using the Laplace transformation, the original partial differential equations (PDEs) are converted into a series of ordinary differential equations (ODEs) that can be solved analytically. A dimensionless parameter, the Peclet number Pe, which reflects the rate of the heat removed by the flowing fluid in the fracture from the surrounding formation, is then used as the small parameter for the regular perturbation analysis. The approximate solutions at higher order are obtained for the rock and fluid temperatures. In addition, numerical results based on a fourth-order Runge–Kutta (R–K) method are presented and match well with these by perturbation analysis. Our solutions also match the exact solutions, which exist for specific conditions. A few leading terms in the asymptotic expansion are sufficient to provide accurate results. The results demonstrate that optimal values of the fracture number and spacing can be found to maximize the productivity and lifetime of an EGS system.

Shale Reservoirs Multi-Fracture Fracturing Technique and Studies on Reservoirs Stresses

Theoretical studies have shown that the generation of the hydraulic fractures reduces or even reverses the stress anisotropy near the fractures and results in increasing the complexity of fractures. A finite element model was established in which the pore pressure elements were used to simulate the behavior of porous media and the pore pressure cohesive elements were adopted to catch the characters of hydraulic fractures. A special fracturing manner was adopted to create complicated fracture networks by reducing or even reversing the stress anisotropy between fractures. The geometries of hydraulic fractures, strains, stresses, pore pressure distributions and fluid pressures within the fractures are obtained. The results of the model are fit well with the corresponding theoretical data. The simulation results show that the stress anisotropy is reduced by the generation of the hydraulic fractures, multiple parallel transverse fractures of horizontal well even reverse the stress anisotropy in some place of the reservoir. The simulation results validate the feasibility of the theoretical studies and the expected complex network fractures could be created by adopting special fracturing manner.

Simulation of Stress Changes of Shale Reservoirs during Hydraulic Fracturing to Create Fracture Networks

Theoretical studies have shown that the generation of hydraulic fractures reduces or even reverses the stress anisotropy between the fractures and results in increasing the complexity of fractures. A finite element model was established in which the pore pressure element was used to simulate the behavior of porous media and the pore pressure cohesive element was adopted to catch the characters of hydraulic fracture. A special fracturing manner was adopted to create complicated fracture networks by reducing or even reversing the stress anisotropy between fractures. The geometries of hydraulic fractures, strains, stresses, pore pressure distributions and fluid pressures within the fractures are obtained. The results of the model are fit well with the corresponding theoretical data. The simulation results show that the stress anisotropy is reduced resulting from the generation of the hydraulic fracture, multiple parallel transverse fractures of horizontal well further reduce or even reverse the stress anisotropy in some place of the reservoir. The simulation results validate the feasibility of the theoretical studies and the expected complex network fractures could be created by adopting the special fracturing manner.

Effect of Cemented Natural Fractures on Hydraulic-Fracture Propagation

This article, written by Senior Technology Editor Dennis Denney, contains highlights of paper SPE 160197, ’Examining the Effect of Cemented Natural Fractures on Hydraulic-Fracture Propagation in Hydrostone-Block Experiments,’ by B. Bahorich, J.E. Olson, SPE, and J. Holder, The University of Texas at Austin, prepared for the 2012 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 8-10 October. The paper has not been peer reviewed. Microseismic data and coring studies suggest that hydraulic fractures interact heavily with natural fractures, creating complex fracture networks in naturally fractured reservoirs such as the Barnett shale, Eagle Ford shale, and Marcellus shale. Because direct observations of subsurface hydraulic-fracture geometries are incomplete or nonexistent, properly scaled experimental research and computer modeling based on realistic assumptions are used to help understand fracture-intersection geometries. Observations of cores from the Barnett and other shale plays suggest that most natural fractures are cemented. The most significant finding of this research was that fracture-intersection geometries are complex. Results showed that bypass, separation of weakly bonded interfaces, diversion, and mixed-mode propagation are likely in hydraulic-fracture intersections with cemented natural fractures. Introduction The resulting geometry from hydraulic-fracturing treatments in rocks having natural-fracture weaknesses, typically pumped from multiple injection zones spaced along lengthy horizontal wells, is not clear. The best direct evidence of fracture geometry is from mine-back studies in which realistic fracturing fluids were injected into realistic wellbores. But those studies often were at very shallow depths or in igneous or metamorphic rock. These types of studies report hydraulic-fracture diversion along natural planes of weakness, such as natural fractures and bedding planes. Multiple fractures can propagate simultaneously and parallel to one another, but, eventually, under many circumstances, a single fracture plane will dominate. Coring across hydraulic-fracture trends has shown that such multiple parallel fractures exist in tight gas sandstones. Laboratory experiments also investigate the physics of hydraulic-fracture complexity, with the advantage of better boundary-condition control and near-complete observation of fracture geometry. The drawback of laboratory experiments is potential issues with proper scaling to ensure that results are applicable to field operations. The objective of this work was to examine the nature of hydraulic-fracturing interaction with naturally cemented or sealed pre-existing fractures. This investigation was motivated by observations in formations similar to the Barnett shale in the Fort Worth basin in Texas, where most natural fractures are observed to be filled with calcite or quartz. Inclusions (glass slides, Berea sandstone slides, and gypsum-plaster slides) were embedded in gypsum-plaster blocks to represent cemented natural fractures. Linear and crosslinked gel was injected into the blocks under a true triaxial-stress state in which the hydraulic-fracture path was designed to intersect the embedded inclusions (natural fractures). Results showed a variety of interactions between hydraulic and natural fractures that could improve understanding of the subsurface process and provide rea-son to develop more-accurate hydraulic-fracturing models.

The effect of a blast on the mandible and teeth: transverse fractures and their management

Transverse mandibular fractures caused by blast waves have given us a better understanding of the physical and pathophysiological effects on this anatomical region. The external contour of the mandible, being rounded in parts (en face) and flat in others (lateral) will experience different effects of the blast. Impact of a spherical blast wave will inflict transverse lines of fractures on the mandibular body, and may be associated with transverse shearing of teeth at the cementoenamel junction. Fractures of the mandible caused by blast are different from those in the same region caused by any type of civilian trauma; these are the vertical to the longitudinal axis unlike comminuted fractures caused by shrapnel or bullet injuries. Fractures of the mandibular body by blasts are single or multiple parallel fractures in the lower border, between the apices of the roots and the mandibular lower border. Fractures are broken off by acceleration or deceleration of the blast wave.

A further study of P-wave attenuation across parallel fractures with linear deformational behaviour

When a wave propagates through fractured rock masses, it is greatly attenuated (and slowed) due to the presence of fractures. In the situation where dynamic responses are induced by a far field explosion or vibration, the magnitude of the stress wave is generally too small to mobilize nonlinear deformation of the fractures, so linear deformational behaviour is adopted in studying wave attenuation. Cai and Zhao [Effects of multiple parallel fractures on apparent wave attenuation in rock masses. Int J Rock Mech Min Sci 2000;37(4): 661–82.] used the method of characteristics to study P-wave attenuation across linear deformational fractures by considering interfracture wave reflections. However, the frequency characteristic of the incident wave (a one-cycle sinusoidal wave) was overlooked in their previous study. The present study treats the incident wave as the sum of a series of harmonic waves in the frequency domain. Results and wave phenomena in the previous study are improved and better understood. When the incident wavelength is substantially greater than fracture spacing, an equivalent medium method is developed to explain the wave phenomena in such a situation. In addition, the uncertainty of back analyzing fractured structures and the effects of wave frequency are examined. These results are in accordance with conclusions in the field of geophysics.

Elastic wave propagation in and permeability for rocks with multiple parallel fractures

Elastic wave propagation in and permeability for rocks with multiple parallel fractures

Galileo views of the geology of Callisto

Callisto was imaged by the Galileo spacecraft on 10 orbits, half of which involved close flybys, producing images with resolutions as high as 15 m/pixel. The Valhalla, Asgard, and Adlinda multi-ring structures were imaged, along with two features which might also be large impacts, Utgard and Heimdall (both provisionally named). The high albedo interior zones of these structures are relatively rugged and include small knobs. The interior ring zone of Valhalla includes outward facing scarps, as discovered on Voyager images, while the inner zones of the other, smaller multi-ring structures include grabens and inward facing scarps. The outer ring zones consist of grabens which are sinuous and include multiple parallel faults and fractures. High albedo areas associated with the fractures had been suggested previously as resurfacing zones; they are now seen to consist of numerous, closely spaced fractures. Crater size frequency distributions for the units associated with multi-ring structures suggest that all were formed essentially contemporaneously with their respective impact events. Galileo images show that surface degradation occurs on a variety of scales, including large landslides from the rims of craters and apparent sub-decameter surface modifications, leading to the removal of small carters in some areas. Similar styles of degradation are not observed on Ganymede, suggesting fundamental differences in the properties of the surface materials. Catenae (crater chains) were also observed; some appear to be secondaries in which their primary source crater can be identified, while others are probably the result of the breakup of primary objects, as previously proposed. Although most of Callisto’s surface consists of ancient cratered plains, some small ‘smooth’ areas had been suggested to result from cryovolcanic resurfacing. Representative bright and dark areas were viewed in high resolution; bright areas were found to be rugged at the 10s of meters scale, show no indications of flow features, and are not considered to be cryovolcanic. Some smooth dark areas appear to mantle and embay the surrounding cratered plains. Although this is consistent with possible cryovolcanic processes, it is not definitive evidence of endogenic processes. The extension of imaging data, both in coverage and resolution, enable about 90% of the surface to be mapped photogeologically and gives new insight into the nature of surface features and the timing of their development. Cratered plains form the oldest recognizable unit on Callisto, but subtle differences in color and crater frequencies suggest regional differences in its development. The formation of Adlinda, Asgard, Valhalla, Heimdall, and Lofn crater appear to follow in that order. The morphology of associated features is consistent with a differentiated interior for Callisto and the presence of a thin icy lithosphere at the time of the impacts.

Effects of multiple parallel fractures on apparent attenuation of stress waves in rock masses

Abstract Stress waves propagating through multiple parallel fractures are attenuated (and slowed) due to multiple wave reflections and transmissions at the fractures. This paper presents a theoretical study and the UDEC modeling on the effects of multiple parallel planar fractures on the apparent attenuation of normally incident one-dimensional elastic waves. The case of normal incidence of waves is studied, because we want to remove the influence of the incident angle and to focus purely on the effects of stiffness, spacing and number of parallel fractures. In the theoretical study, an approach is developed to explicitly take into account the attenuative effect of each fracture with the displacement discontinuity model, and to implicitly consider complex interfracture multiple wave reflections with the method of characteristics. The intrinsic attenuation mechanisms are neglected so that the attenuative effects of the multiple reflections can be concentrated on. This approach does not lose the discreteness of wave attenuation at individual fractures, and avoids the difficulty in explicitly determining the complex process of superposition of multiple reflected and transmitted wave fields. With this approach, a set of recurrence equations with respect to particle velocities before and after the fractures are established. These equations are numerically solved with sufficient accuracy. In analysis, there is no restriction to large fracture spacing or to small fracture spacing compared with wavelength. The magnitude of transmission coefficient (∣TN∣) for waves transmitting normally across a set of multiple parallel fractures is calculated as a function of the ratio (ξ) of fracture spacing to wavelength, for different normalized stiffness (k/zω) and for a different number of fractures. Parametric studies are conducted to examine the effects of multiple parallel fractures on wave attenuation, especially in terms of the spacing and the number of fractures. It is shown that the dependence of ∣TN∣ on the fracture spacing and the fracture number is governed by ξ. In addition, the effects of multiple wave reflections on ∣TN∣ are quantitatively discussed for different values of ξ. In the numerical modeling, the same problem is studied with a discontinuum-based numerical method, termed as the Discrete Element Method (DEM). The Universal Distinct Element Code (UDEC) is used to model one-dimensional wave propagation in rock masses containing no fractures, a single planar fracture and multiple parallel planar fractures. The modeling results of the transmission coefficient are compared with the theoretical solutions. An agreement between them has been achieved. It is verified that the UDEC is capable of modeling one-dimensional wave propagation across multiple parallel fractures.