Transmission of Elastic P-waves across Single Fractures with a Nonlinear Normal Deformational Behavior

Summary This paper presents a theoretical study on normally incident elastic P-wave transmission across single dry fractures with a nonlinear normal deformational behavior. The effects of nonlinear fracture normal behavior on P-wave transmission are examined without the mixture of fracture shear behavior. The linear displacement discontinuity model for wave propagation across fractures is extended to a nonlinear model – the hyperbolic elastic model (BB model). Numeric solutions of magnitudes of transmission (|T _non|) and reflection (|R _non|) coefficients, for normally incident P-wave transmission across the nonlinear deformable fractures, are obtained and related to the closure behavior of fractures. Parametric studies are conducted to acquire an insight into the effects of the nonlinear fracture normal deformation on P-wave transmission, in terms of initial normal stiffness and the ratio of current maximum closure to maximum allowable closure of the fractures, as well as the incident wave amplitude and frequency. Comparisons between the linear and nonlinear models are presented. It is shown that, |T _lin| and |R _lin| for the linear model are special solutions of |T _non| and |R _non| for the nonlinear model, when the incident wave amplitude is so low that the current maximum closure of fracture incurred during the wave transmission is much smaller, relative to the maximum allowable closure. In addition, the nonlinear fracture behavior gives rise to a phenomenon of higher harmonics during the wave transmission across the fracture. The higher harmonics contribute to the increase of |T _non| from |T _lin|.     

Quantification of Aperture and Relations Between Aperture,Normal Stress and Fluid Flow for Natural Single Rock Fractures

Accurate quantification of rock fracture aperture is important in investigating hydro-mechanical properties of rock fractures. Liquefied wood’s metal was used successfully to determine the spatial distribution of aperture with normal stress for natural single rock fractures. A modified 3D box counting method is developed and applied to quantify the spatial variation of rock fracture aperture with normal stress. New functional relations are developed for the following list: (a) Aperture fractal dimension versus effective normal stress; (b) Aperture fractal dimension versus mean aperture; (c) Fluid flow rate per unit hydraulic gradient per unit width versus mean aperture; (d) Fluid flow rate per unit hydraulic gradient per unit width versus aperture fractal dimension. The aperture fractal dimension was found to be a better parameter than mean aperture to correlate to fluid flow rate of natural single rock fractures. A highly refined variogram technique is used to investigate possible existence of aperture anisotropy. It was observed that the scale dependent fractal parameter, K _v, plays a more prominent role than the fractal dimension, D _a1d, on determining the anisotropy pattern of aperture data. A combined factor that represents both D _a1d and K _v, D _a1d × K _v, is suggested to capture the aperture anisotropy.     

A coupled compressible flow and geomechanics model for dynamic fracture aperture during carbon sequestration in coal

This paper presents the development of a discrete fracture model of fully coupled compressible fluid flow, adsorption and geomechanics to investigate the dynamic behaviour of fractures in coal. The model is applied in the study of geological carbon dioxide sequestration and differs from the dual porosity model developed in our previous work, with fractures now represented explicitly using lower-dimensional interface elements. The model consists of the fracture-matrix fluid transport model, the matrix deformation model and the stress-strain model for fracture deformation. A sequential implicit numerical method based on Galerkin finite element is employed to numerically solve the coupled governing equations, and verification is completed using published solutions as benchmarks. To explore the dynamic behaviour of fractures for understanding the process of carbon sequestration in coal, the model is used to investigate the effects of gas injection pressure and composition, adsorption and matrix permeability on the dynamic behaviour of fractures. The numerical results indicate that injecting nonadsorbing gas causes a monotonic increase in fracture aperture; however, the evolution of fracture aperture due to gas adsorption is complex due to the swelling-induced transition from local swelling to macro swelling. The change of fracture aperture is mainly controlled by the normal stress acting on the fracture surface. The fracture aperture initially increases for smaller matrix permeability and then declines after reaching a maximum value. When the local swelling becomes global, fracture aperture starts to rebound. However, when the matrix permeability is larger, the fracture aperture decreases before recovering to a higher value and remaining constant. Gas mixtures containing more carbon dioxide lead to larger closure of fracture aperture compared with those containing more nitrogen.     

Hydromechanical properties and alteration of natural fracture surfaces in the Soultz granite (Bas-Rhin, France)

Natural fractures are characterized by rough surfaces and complex fluid flows. A large distribution of apertures (residual voids) within their walls and the presence of contact points (in situ normal loads) produce heterogeneous flows (channeling). The resulting permeabilities, porosities or fluid–rock exchange surfaces cannot be realistically modeled by parallel and smooth plate models. Four natural fractures are sampled at different depths and degrees of alteration in the Soultz sandstone and granite (EPS1 drillhole, Soultz-sous-Forêts, Bas-Rhin, France). The fracture surfaces are measured with mechanical profilometry and maps of asperity heights (XYZ). Resulting local apertures (XYe) are then calculated. A statistical study of the surface profiles (XZ) show that the fractures are more or less rough and tortuous according to the types of alteration. Altered samples are characterized by smoother surfaces of fractures. Such differences imply that (i) the average fracture aperture is not representative for the whole fracture and that (ii) the different local apertures should be integrated in hydraulic and mechanical models. A hydraulic model (finite difference calculations) of fluid flow, taking into account the elastic closure (Hertz contact theory) of fractures with depth, is used. Maps of contact points and relative local loads within the fracture planes are compared to flow maps. They show different channeling of fluid flows. Strongly altered fractures are characterized by homogeneous fluxes despite the presence of numerous contact zones during the closure of fracture. By contrast, fresh fractures develop, increasing fluid flow channels with depth.Fracture closure (increasing normal stress) does not systematically increase the channeling of fluid flow. There is evidence for a general smoothing out of the irregularities of the fracture walls due to precipitation of secondary minerals, indicating that the cubic law can be commonly valid, also at great crustal depth but this validity depends on the degree of fracture alteration. Mineralogical and geochemical observations, thus, should be taken into account to perform more accurate permeability calculations and models of fluid circulation in fracture networks.     

P‐wave transmission across fractures with nonlinear deformational behaviour

Stress wave attenuation across fractured rock masses is a great concern of underground structure safety. When the wave amplitude is large, fractures experience nonlinear deformation during the wave propagation. This paper presents a study on normal transmission of P‐wave across parallel fractures with nonlinear deformational behaviour (static Barton–Bandis model). The results show that the magnitude of transmission coefficient is a function of incident wave amplitude, nondimensional fracture spacing and number of fractures. Two important indices of nondimensional fracture spacing are identified, and they divide the area of nondimensional fracture spacing into three parts (individual fracture area, transition area and small spacing area). In the different areas, the magnitude of transmission coefficient has different trends with nondimensional fracture spacing and number of fractures. In addition, the study reveals that under some circumstances, the magnitude of transmission coefficient increases with increasing number of fractures, and is larger than 1. Copyright © 2006 John Wiley & Sons, Ltd.     

Experimental and modelled mechanical behaviour of a rock fracture under normal stress

Summary  The mechanical and hydromechanical behaviour of isolated rock joints is of prime importance for a correct understanding of the behaviour of jointed rock masses. This paper focuses on the mechanical behaviour of a fracture under normal stress (fracture closure), using approaches based on both experimentation and modelled analysis. Experimental closure tests were carried out by positioning four displacement transducers around a fracture, leading to results which tended to vary as a function of transducer location. Such variations can be explained by the non-constant void space distribution between both walls of the fracture. The present study focuses on the importance of transducer location in such a test, and on the significant role played, in terms of mechanical response, by the morphology of the fracture surfaces. An analytical mechanical model is then developed, which takes into account the deformation of surface asperities and of the bulk material surrounding the fracture; it also includes the effects of mechanical interaction between contact points. The model is validated by simulating the behaviour which is very similar to experimental observations. Various parametric studies (scale effect, spatial distribution of contact points) are then carried out. The study of scale effects reveals a decrease in the normal stiffness with increasing fracture size. Finally, analysis of the role of various mechanical parameters has shown that the most influential of these is Young’s modulus corresponding to the bulk material surrounding the joint. Many applications, such as geothermal fluid recovery from fractures, could benefit from these results. Correspondence: Antoine Marache, Université Bordeaux 1, GHYMAC, Av. des Facultés, 33405 Talence Cedex, France     

Evolution of fluid pressure and fracture propagation during contact metamorphism

Abstract Rock fracture enhances permeability and provides pathways through which fluids migrate. During contact metamorphism, fluids contained in isolated pores and fractures expand in response to temperature increases caused by the dissipation of heat from magmas. Heat transport calculations and thermomechanical properties of water-rich fluids demonstrate (1) that thermal energy is a viable mechanism to produce and maintain pore fluid pressure (P_f) in a contact metamorphic aureole; (2) that the magnitude of P_f generated is sufficient to propagate fractures during the prograde thermal history (cause hydrofracture) and enhance permeability; and (3) that P_f-driven fracture propagation is episodic with time-scales ranging from years to thousands of years. Because P_f dissipation is orders of magnitude faster than P, _f buildup, P_f oscillations and cyclical behaviour are generated as thermal heating continues. The P_f cycle amplitude depends on the initial fracture length, geometry and the rock's resistance to failure whereas the frequency of fracture depends on the rate of heating. Consequently, oscillation frequency also varies spatially with distance from the heat source. Time series of fluid pressures caused by this process suggest that cyclical fracture events are restricted to an early time period of the prograde thermal event near the intrusive contact. In the far field, however, individual fracture events have a lower frequency but continue to occur over a longer time interval. Numerous fracture cycles are possible within a single thermal event. This provides a provisional explanation for multiple generations of veins observed in outcrop. P _f cycling and oscillations may explain several petrological features. If pore fluids are trapped at various positions along a pressure cycle, the large amplitude of P_f variations for small fractures may account for different pressures recorded by fluid inclusions analysed from a single sample. P_f oscillations, during a single thermal episode, also drive chemical reactions which can produce complex mineral textures and assemblages for discontinuous reactions and/or zoning patterns for continuous reactions. These can mimic polymetamorphic or disequilibrium features. Temporal aspects of fracture propagation and permeability enhancement also constrain the likely timing of fluid flow and fluid-mineral interactions. These data suggest that fluid flow and fluid-mineral reactions are likely to be restricted to an early period in the prograde thermal history, characterized by high P_f coincident with relatively high temperatures, fracture propagation and consequent increases in permeability. This early prograde hydration event is followed by diffusional peak metamorphic reactions. This relationship is evident in the complex mineralogical textures common in some metamorphosed rocks.     

Correlations Developed for Estimation of Hydraulic Parameters of Rough Fractures Through the Simulation of JRC Flow Channels

The hydro-mechanical response of fractured rock masses is complex, due partly to the presence of fractures at different scales. Surface morphology has a significant influence on fluid flow behaviour of a fracture. Different empirical correlations and statistical models have been proposed to estimate the equivalent hydraulic aperture and determine the pressure drop along a fracture. However, the existing models suffer from not being adequately generalised to be applicable to a wide range of real fracture surfaces. To incorporate the effect of profile roughness in the hydro-mechanical behaviour of fractured rock masses, the joint roughness coefficient (JRC) is the most widely used empirical approach. However, the average JRC of two fracture walls in fluid flow analysis, as is a common practice, appears to be inappropriate. It will be shown how different combinations of pairs of JRCs could lead to a similar JRC value. Also, changing the position of the top and bottom walls of a fracture can significantly change the hydraulic response of the fracture while the average JRC is identical in both cases. In this paper, correlations are developed which are based on the simulation of JRCs using estimated fluid flow parameters of 2D fractures can be estimated. In order to widen the application range of the correlations, JRC flow channels were generated: these are 2D channels with their top and bottom walls being made from two of the JRC profiles. To estimate the JRC of linear profiles, a correlation developed between JRC and a newly developed Riemannian roughness parameter, D _R1, is proposed. Considering ten JRC profiles, a total of 100 JRC flow channels were generated. In order to only investigate the effect of surface roughness on fluid flow, the minimum closure between the top and bottom walls of JRC flow channels were considered to be constant. Three cases with minimum closures of 0.01, 0.05 and 0.10 cm were considered in this study. All JRC flow channels were subjected to fluid analysis using FLUENT software. Based on these results, correlations were developed between the geometrical and hydraulic properties of flow channels. Analysis of several real fractures demonstrated the applicability of these correlations.     

Modeling spatial fracture intensity as a control on flow in fractured rock

Spatial fracture intensity (P ₃₂, fracture area by volume) is an important characteristic of a jointed rock mass. Although it can hardly ever be measured, P ₃₂ can be modeled based on available geological information such as spatial data of the fracture network. Flow in a mass composed of low-permeability hard rock is controlled by joints and fractures. In this article, models were developed from a geological data set of fractured andesite in LanYu Island (Taiwan) where a site is investigated for possible disposal of low-level and intermediate-level radionuclide waste. Three different types of conceptual models of spatial fracture intensity distribution were generated, an Enhanced Baecher’s model (EBM), a Levy–Lee Fractal model (LLFM) and a Nearest Neighborhood model (NNM). Modeling was conducted on a 10 × 10 × 10 m synthetic fractured block. Simulated flow was forced by a 1% hydraulic gradient between two vertical x–z faces of the cube (from North to South) with other boundaries set to no-flow conditions. Resulting flow vectors are very sensitive to spatial fracture intensity (P ₃₂). Flow velocity increases with higher fracture intensity (P ₃₂). R-squared values of regression analysis for the variables velocity (V/V _max) and fracture intensity (P ₃₂) are 0.293, 0.353, and 0.408 in linear fit and 0.028, 0.08, and 0.084 in power fit. Higher R ² values are positively linked with structural features but the relation between velocity and fracture intensity is non-linear. Possible flow channels are identified by stream-traces in the Levy–LeeFractal model.     

基于离散颗粒模型的岩体结构面动-静态刚度系数对比的数值模拟研究

岩体工程计算分析中结构面刚度系数是至关重要的力学参数,计算分析的精度和可靠程度与这个参数密不可分,结构面刚度系数取值仍然是一个难点.岩体中应力波传播至结构面处将会发生反射和透射现象,利用应力波透射系数可反演结构面动态刚度系数.本文从细观力学角度运用颗粒离散元方法,开发分段线性接触模型及应力波吸收边界模型,开展宏观岩体中...     

A Connectivity Index for Discrete Fracture Networks

Connectivity is an important measure for assessing flow transport in rock, especially through fractures. In this paper, rock fracture systems are modelled by a discrete fracture model simulated by a marked point process. A connectivity index is then introduced to quantify the connectivity between any two points in space. Monte Carlo simulation is used to evaluate the connectivity index for stationary cases and relationships between the connectivity index and the parameters of the discrete fracture model are analysed. The average number of intersections per fracture, X_f, and the fracture intensity, P12 (P32), are calculated and the relationships between these parameters and the connectivity index are investigated, concluding that X_f is the more suitable parameter for the classification of rock mass flow properties. The relationships between the connectivity index and the percolation state of the fractured medium are also discussed. An edge correction is briefly discussed and a practical example is used to demonstrate the method of computing the connectivity index.     

Assessment of using borehole-closure data to determine the constitutive behavior of salt

Borehole closure data have often been used to determine the constitutive behaviour of rock salt. This is an inverse problem and analysis of the data is confused by the fact that borehole closure is a structural response that does not uniquely determine constitutive behaviour since both the stress and deformation depend on the constitutive behaviour. This application brief assesses the conclusion of a recent study that the transient power law, ? = K τ^mtⁿ, which includes no term to account for steady-state deformation, is a better constitutive model for salt than are models that account for steady-state straining, and that an empirical model developed from the borehole closure data could be extrapolated to give conservative predictions of long-term closure of single or widely spaced openings.     

Simulations of hydro-fracking in rock mass at meso-scale using fully coupled DEM/CFD approach

The paper deals with two-dimensional (2D) numerical modelling of hydro-fracking (hydraulic fracturing) in rocks at the meso-scale. A numerical model was developed to characterize the properties of fluid-driven fractures in rocks by combining the discrete element method (DEM) with computational fluid dynamics (CFD). The mechanical behaviour of the rock matrix was simulated with DEM and the behaviour of the fracturing fluid flow in newly developed and pre-existing fractures with CFD. The changes in the void geometry in the rock matrix were taken into account. The initial 2D hydro-fracking simulation tests were carried out for a rock segment under biaxial compression with one injection slot in order to validate the numerical model. The qualitative effect of several parameters on the propagation of a hydraulic fracture was studied: initial porosity of the rock matrix, dynamic viscosity of the fracking fluid, rock strength and pre-existing fracture. The characteristic features of a fractured rock mass due to a high-pressure injection of fluid were realistically modelled by the proposed coupled approach.

     

Role of heterogeneities in jointing (fracturing) of geological media: Numerical analysis of fracture mechanisms

Fracture process is investigated using finite-difference simulations with a new constitutive model. It is shown that both geometry and fracture mechanism itself depend on the preexisting heterogeneities that are stress concentrators. In the brittle regime (low pressure, P), Mode-I fractures propagate normal to the least stress σ₃ from the imposed weak zones. At high P, shear deformation bands are formed oblique to σ₃. At intermediate values of P, the fracture process involves both shear banding and tensile cracking and results in the initiation and propagation of pure dilation bands. The propagating band tip undulates, reacting on the failure mechanism changes, but its global orientation is normal to σ₃. The σ₃-normal fractures are joints. There are thus two types of joints resulting from Mode-I cracking and dilation banding, respectively. The obtained numerical results are in good agreement with and explain the results from previous similar experimental study.     

Occurrence of springs in massifs of crystalline rocks,northern Portugal

An inventory of artesian springs emerging from fractures (fracture springs) was conducted in the Pinh?o River Basin and Morais Massif, northern Portugal, comprising an area of approximately 650 km². Over 1,500 springs were identified and associated with geological domains and fracture sets. Using cross-tabulation analysis, spring distributions by fracture sets were compared among geological environments, and the deviations related to differences in rock structure and, presumably, to differences in deformational histories. The relation between spring frequencies and rock structures was further investigated by spectral determination, the model introduced in this study. Input data are the spring frequencies and fracture lengths in each geological domain, in addition to the angles between fracture strikes and present-day stress-field orientation (θ). The model's output includes the so-called intrinsic densities, a parameter indexing spring occurrence to factors such as fracture type and associated deformational regime and age. The highest densities (12.2 springs/km of lineament) were associated with young shear fractures produced by brittle deformation, and the lowest (0.1) with old tensional and ductile fractures. Spectral determination also relates each orientation class to a dominant structural parameter: where spring occurrence is controlled by θ, the class is parallel to the present-day stress-field orientation; where the control is attributed to the length of fractures, the spring occurrence follows the strike of large-scale normal faults crossing the region. Electronic Publication     

Natural fracture profiles,fractal dimension and joint roughness coefficients

Summary Thirteen natural rock profiles (Barton and Choubey, 1977) are analyzed for their fractal properties. Most of the profiles were found to approximate fractal curves but some also showed features of specific wavelengths and amplitudes superimposed on fractal characteristics. The profiles showed fractal dimensions from 1.1 to 1.5 covering a range of selfsimilar and self-affine curves. The analysis results suggest a negative correlation between fractal dimension,D, and amplitude,A. Joint roughness coefficients (JRC) show a positive correlation with amplitude,A, and a negative correlation with fractal dimension,D. A numerical model of fracture closure is used to investigate the effects of different profile characteristics (D, A and sample size) on the nature of dilation and contact area, using the natural profiles and synthetic fractional Brownian motion profiles. Smooth profiles (low JRC, highD, lowA) display many small contact regions whereas rough fractures (high JRC, lowD, highA) display few large contact areas. The agreement with published experimental data supports the suggested correlations between JRC and the fractal parameters,A andD. It is suggested that observed scale effects in JRC and joint dilation can be explained by small differential strain discontinuities across fractures, which originate at the time of fracture formation.     

Fracture systems in normal fault zones crosscutting sedimentary rocks,Northwest German Basin

Field studies of fracture systems associated with 58 normal fault zones crosscutting sedimentary rocks were performed in the Northwest German Basin. Fracture orientations, densities, apertures and lengths, as well as fault zone structural indices, were analysed separately for fault damage zones and host rocks. The results show a pronounced difference between carbonate and clastic rocks: mainly in carbonate rocks we found presence of clear damage zones, characterized by higher fracture densities than in the host rocks. While the maximum aperture is similar for both units, the percentage of fractures with large apertures is much higher in the damage zones than in the host rocks.Based on laboratory measurements of Young's moduli and field measurements of fracture densities, we calculate effective stiffnesses E_e, that is the Young's moduli of the in situ rock masses, within the normal fault zones. Compared with carbonate rocks, E_e computed for clastic-rock damage zones decreases significantly less due to lower fracture densities. We conclude that normal fault zones in carbonate rocks have more profound effects on enhancing permeability in fluid reservoirs than those in clastic rocks. The results are of great importance for modelling the hydromechanical behaviour of normal fault zones in subsurface fluid reservoirs.     

Numerical simulation and evaluation of the fracturing and tight gas production with a new dimensionless fracture conductivity (FCD) model

Hydraulic fracturing is an essential technology for the development of unconventional resources such as tight gas. The evaluation of the fracture performance and productivity is important for the design of fracturing operations. However, the traditional dimensionless fracture conductivity is too simple to be applied in real fracturing operations. In this work, we proposed a new model of dimensionless fracture conductivity (F_CD), which considers the irregular fracture geometry, proppant position and concentration. It was based on the numerical study of the multistage hydraulic fracturing and production in a tight gas horizontal well of the North German Basin. A self-developed full 3D hydraulic fracturing model, FLAC3D^plus, was combined with a sensitive/reliability analysis and robust design optimization tool optiSLang and reservoir simulator TMVOCMP to achieve an automatic history matching as well as simulation of the gas production. With this tool chain, the four fracturing stages were history matched. The simulation results show that all four fractures have different geometry and proppant distribution, which is mainly due to different stress states and injection schedule. The position and concentration of the proppant play important roles for the later production, which is not considered in the traditional dimensionless fracture conductivity F_CD. In comparison, the newly proposed formulation of F_CD could predict the productivity more accurately and is better for the posttreatment evaluation.

     

Extensional aspects of earthquake induced ruptures determined by an analysis of fracture bifurcation

Numerical and experimental investigation of symmetric fracture bifurcation (Kalthoff, 1972), has shown that for forks with small branch angles α<α_c, where α_c is approximately 14°, the propagation of the branches tends to enlarge the angle. For forks with larger branch angles, α>α_c, the propagation of the branches tends to diminish the angle. Forks with the critical angle α_c will propagate in their original direction. Kalthoff theorized that the branch angle changes as a function of K_I/K_II, where K_I and K_II are the stress intensity factors for tensile and shear (sliding) modes, respectively, and K_I is considerably larger than K_II. In this study I test the hypothesis that this fracture mechanic theory applies to the analysis of fault bifurcation in the crust, particularly in cases of rapid fracture.Fractures produced during the 1968 earthquake at the Coyote Creek fault in California are intensively branched and an example of rapid rupture. The angular behaviour of the branching ruptures in eight forks follows Kalthoffs theory unusually well. This implies that fracture at the surface was dominated by the tensile mode. Additional observations that support this implication are: series of prominent ruptures which show openings (of 20–30 mm per rupture), the symmetrical and bilateral forking, the high-intensity and angular shapes of individual branches, the opening of grabens associated with several bifurcations, lack of bifurcation in the southern break of the Coyote Creek fault, and the patterns of en echelon fractures which reflect mixed mode surfacial rupture.Hence, contrary to previous interpretations, according to field evidence and fracture mechanic theory, the fault bifurcation and opening along the Coyote Creek fault are not compatible with local tension caused by the primary shear. Fracture probably occurred by different mechanical modes at depth and at the surface. While faulting may have originated by shear at depth, rupture at the surface was dominated by far-field tension associated with NE-SW extension in South California. The present model predicts the directions of fracture propagation along the fault.     

Single phase water flow through rock fractures

Flow analysis plays a major role in various geotechnical applications, and the understanding of flow mechanisms is essential for the development of a hydro-mechanical flow model suitable for underground excavations in rock. Discrete flow analysis through discontinuities is reviewed including empirical and analytical flow models. The influence of external loading on joint deformation and single-phase flow show that the surface roughness and aperture size are the prime factors influencing flow rate. Nevertheless, the idealization of natural fractures as smooth parallel plate joints is still followed in many numerical models, because of the simplicity of the cubic law when applied to fracture networks. A numerical study of water flow through a network of joints employing Universal Distinct Element Code (UDEC) is used to quantify the effects of joint orientation and external stress acting on idealized joints.It is found that, for the same joint spacing, the flow rate into an excavation depends on the boundary block size (A_b) relative to the excavation size (A_e). The inflow becomes excessive if A_b/A_e is less than 4, but becomes very small if A_b/A_e exceeds 8.