Ck-Reconstruction of Surfaces from Partial Data

In this paper, we study the problem of constructing a smooth approximant of a surface defined by the equation z = f(x ₁, x ₂), the data being a finite set of patches on this surface. This problem occurs, for example, after geophysical processing such as migration of time-maps or depth-maps. The usual algorithms to solve this problem are picking points on the patches to get Lagrange's data or trying to get local junctions on patches. But the first method does not use the continuous aspect of the data and the second one does not perform well to get a global regular approximant (C ¹ or more). As an approximant of f, a discrete smoothing spline belonging to a suitable piecewise polynomial space is proposed. The originality of the method consists in the fidelity criterion used to fit the data, which takes into account their particular aspect (surface's patches): the idea is to define a function that minimizes the volume located between the data patches and the function, and which is globally C ^k. We first demonstrate the new method on a theoretical aspect and numerical results on real data are given.     

A three-dimensional Hoek–Brown failure criterion based on an elliptical Lode dependence

A new three-dimensional (3D) Hoek–Brown (HB) failure criterion based on an elliptical Lode dependence is proposed to describe failure of rocks and concrete under multiaxial stress states. This criterion not only inherits all benefits of the classical HB criterion that is developed for the triaxial compression (TXC) of rocks but also accounts for the effect of the intermediate principal stress. It is capable of representing the strength difference between the triaxial extension (TXE) and TXC with the introduction of an additional coefficient k (0.5 ≤ k ≤ 1.0), which can be derived from TXE tests or taken as 0.53 for rocks in cases where the TXE test data is unavailable. Other two material constants (m_i and σ_ci) involved in this criterion can be obtained from TXC tests. Additionally, the failure surface of this criterion is smooth and convex on the deviatoric stress plane when 0.5 < k ≤ 1.0. The new criterion achieves very good fit to the test data of TXC/TXE, biaxial compression, and polyaxial compression (PXC) on a wide variety of rock materials and concrete, reported in the literature. Comparison of the new criterion with an existing 3D HB criterion based on the same Lode dependence has demonstrated that the new criterion performs better than the latter for test data of rock and concrete under multiaxial stress states except for PXC test data of one rock type. Finally, the influence of values of k on the accuracy of the new criterion is discussed.     

The limitations of three-dimensional kinematic vorticity analysis

The kinematic vorticity number (W_k) can be calculated for three-dimensional as well as two-dimensional geologic deformations. For steady-state deformations, W_k can be correlated to and analyzed in terms of finite strains. The analysis shows that assumptions commonly made for two-dimensional deformations are not applicable to three-dimensional deformations. A single W_k describes an infinite number of three-dimensional deformations. Further, even knowledge of flow apophyses orientation, instantaneous stretching axes orientation, and/or W_k are not sufficient to describe deformation. Three-dimensional deformations also require knowledge of the deformation ‘type’ or boundary conditions of deformation (e.g. transpression). Hence, in addition to being difficult to estimate, the value of knowing W_k for three-dimensional deformations is greatly reduced compared with plane strain. The most useful methods of determining W_k from naturally deformed rocks are presented.     

Ck-Reconstruction of Surfaces from Partial Data

In this paper, we study the problem of constructing a smooth approximant of a surface defined by the equation z = f(x ₁, x ₂), the data being a finite set of patches on this surface. This problem occurs, for example, after geophysical processing such as migration of time-maps or depth-maps. The usual algorithms to solve this problem are picking points on the patches to get Lagrange's data or trying to get local junctions on patches. But the first method does not use the continuous aspect of the data and the second one does not perform well to get a global regular approximant (C ¹ or more). As an approximant of f, a discrete smoothing spline belonging to a suitable piecewise polynomial space is proposed. The originality of the method consists in the fidelity criterion used to fit the data, which takes into account their particular aspect (surface's patches): the idea is to define a function that minimizes the volume located between the data patches and the function, and which is globally C ^k. We first demonstrate the new method on a theoretical aspect and numerical results on real data are given.     

Algorithm of Calculation of Lyapounov Coefficients for Analysis of Chemical Autooscillations, as Applied to Calcite Crystallization Model

The numerical algorithm of calculation of Lyapounov coefficients (L _k) of any order is developed. The apparatus of analytical calculations is not used in this algorithm. The proposed algorithm is of use for usual computer languages and allows us to find the numerical value of L _k for any k and to make complete qualitative analyses of dynamic models on the plane.     

A unified criterion for initiation of sediment motion and inception of sheet flow under water waves

A unified criterion is developed for initiation of non‐cohesive sediment motion and inception of sheet flow under water waves over a horizontal bed of sediment based on presently available experimental data. The unified threshold criterion is of the single form, U_o = 2πC[1 + 5(T_R/T)²]^?1/4, where U_o is the onset velocity of sediment motion or sheet flow, T is wave period, and C and T_R are the coefficients. It is found that for a given sediment, U_o initially increases sharply with wave period, then gradually approaches the maximum onset velocity U_o = 2πC and becomes independent of T when T is larger. The unified criterion can also be extended to define sediment initial motion and sheet flow under irregular waves provided the significant wave orbital velocity and period of irregular waves are introduced in this unified criterion.     

Simulation of discrete cracks driven by nearly incompressible fluid via 2D combined finite-discrete element method

In this work, we propose a novel hydraulic solver in order to simulate key mechanisms that control fluid-driven cracks in the framework of the combined finite-discrete element method (FDEM). The main innovative aspect of the present work is the independence of the fluid's critical time step size with the fracture opening. This advantage is extremely important because it means that very fine meshes can be used around areas of interest, such as boreholes, without penalizing the computational cost as fractures propagate (ie, open) and the fluid flows through them. This is a great advantage over other recently introduced approaches that exhibit a dependency of the time step in the form of Δt_crit ∝ (l/a)² where l is the element size and a is the fracture aperture. This paper presents a series of benchmark cases for the proposed solver. The rationale adopted by the authors was to benchmark and validate the implementation of the hydraulic solver in an incremental fashion, starting from the simplest cases and building in complexity. The results shown in this work clearly demonstrate that the proposed approach is able to reproduce analytical results for fluid flow through a single crack. The results presented in this paper also demonstrate that the new approach is robust enough to deal with complex fracture patterns and complex geometries; the obtained fluid-driven fracture patterns in the vicinity of a borehole certainly stand to the scrutiny of human visual perception.     

Sheath fold development in monoclinic shear zones

We use numerical simulations to investigate the evolution of sheath folds around slip surfaces in simple‐shear‐dominated monoclinic shear zones. A variety of sheath fold shapes develops under general shear, including tubular folds with low aspect ratio eye patterns and tongue‐like structures showing bivergent flanking structures in sections normal to the sheath elongation, which may potentially lead to confusing shear sense interpretations. Not all investigated monoclinic flow end‐members lead to the development of sheath folds sensu stricto (folds with apical angle <90°). The aspect ratio of the eye patterns, R_yz, correlates with the ratio between the principal strain in the Y‐direction and the smaller of the principal strains in the X–Z plane, and thus it could be used in strain analysis.     

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.     

Peculiarities of α-element abundances in Galactic open clusters

A catalog compiling the parameters of 346 open clusters, including their metallicities, positions, ages, and velocities has been composed. The elements of the Galactic orbits for 272 of the clusters have been calculated. Spectroscopic determinations of the relative abundances, [el/Fe], for 14 elements synthesized in various nuclear processes averaged over data from 109 publications are presented for 90 clusters. The compiled data indicate that the relative abundances of primary α elements (oxygen and magnesium) exhibit different dependences on metallicity, age, Galactocentric distance, and the elements of the Galactic orbits in clusters with high, elongated orbits satisfying the criterion (Z_max² + 4e²)^1/2 > 0.40 and in field stars of the Galactic thin disk (Z_max is the maximum distance of the orbit from the Galactic plane in kiloparsec and e is the eccentricity of the Galactic orbit). Since no systematic effects distorting the relative abundances of the studied elements in these clusters have been found, these difference suggest real differences between clusters with high, elongated orbits and field stars. In particular, this supports the earlier conclusion, based on an analysis of the elements of the Galactic orbits, that some clusters formed as a result of interactions between high-velocity,metal-poor clouds and the interstellar mediumof theGalactic thin disk. On average, clusters with high, elongated orbits and metallicities [Fe/H] < -0.1 display lower relative abundances of the primary a elements than do field stars. The low [O, Mg/Fe] ratios of these clusters can be understood if the high-velocity clouds that gave rise to them were formed of interstellar material from regions where the star-formation rate and/or the masses of Type II supernovae were lower than near the Galactic plane. It is also shown that, on average, the relative abundances of the primary a elements are higher in relatively metal-rich clusters with high, elongated orbits than in field stars. This can be understood if clusters with [Fe/H] > -0.1 formed as a result of interactions between metal-rich clouds with intermediate velocities and the interstellar medium of the Galactic disk; such clouds could form from returning gas in a so-called “Galactic fountain.”     

Preference ratios for mercury and other chemical elements in the Madeira River, Brazil

Samples of water, suspended solids, and bottom sediments from the Madeira River, Rondônia state, Brazil, were physically and chemically analyzed to investigate the actual Hg mobilization in the aquatic environment and compare it with that of other heavy metals and elements in the area. Two dimensionless Hg preference ratios were defined, expressing (1) the ratio of Hg and other elements in the liquid phase divided by the ratio of Hg and other elements in bottom sediments (P_l.phase) and (2) the ratio of Hg and other elements in the particulate matter divided by the ratio of Hg and other elements in bottom sediments (P_s.solids). These preference ratios are useful for comparing Hg transport in three different phases (liquid, particulate matter, and bottom sediments). They also were applicable to any analyzed element in the area studied, because they generated an almost constant value when the maximum calculated was divided by the minimum (P_l.phase=2931; P_s.solids=84) and because of their sensitivity to the dominance of sorption processes by Fe oxides and hydroxides. Mercury could be transported preferentially to other analyzed elements in the particulate phase only if its concentration reached values at least 10⁴-fold higher than those expected or quantified in the area.     

Crystal Structure of Natural Non-metamict Ti- and Fe~(2+)-rich Chevkinite-(Ce)

1 Introduction Chevkinite groups can be assigned to the chevkinite-(Ce) subgroup and perrierite-(Ce) subgroup in accord with the angle β : β ≈ 100o for the chevkinite subgroup and β ≈ 113o for the perrierite subgroup. Chevkinite-(Ce), polykovite-(Ce) and Maoniupingite (new mineral No. 017 of 2003) belong to the former subgroup, while renjeite and matsubaraite belong to the latter group. As strontio-chevkinite is a Sr-analogue of perrierite, usually the natural chevkinite-(Ce) group min…     

A note on fault reactivation

Reactivation of existing faults whose normal lies in the σ₁^′σ₃^′ plane of a stress field with effective principal compressive stresses σ₁^′ >σ₂^′ >σ₃^′ is considered for the simplest frictional failure criterion, τ = μσ_n^′ = μ(σ_n − P), where τ and σ_n are respectively the shear and normal stresses to the existing fault, P is the fluid pressure and μ is the static friction. For a plane oriented at θ to σ₁^′, the stress ratio for reactivation is (σ₁^′/σ₃^′) = (1 + μ cot θ)/(1 − μ tan θ). This ratio has a minimum positive value at the optimum angle for reactivation given by (1/μ) but reaches infinity when θ = 2θ^*, beyond which σ₃^′ < 0 is a necessary condition for reactivation. An important consequence is that for typical rock friction coefficients, it is unlikely that normal faults will be reactivated as high-angle reverse faults or thrusts as low-angle normal faults, unless the effective least principal stress is tensile.     

Probabilistic Stability Evaluation of Oppstadhornet Rock Slope,Norway

Probabilistic analyses provide rational means to treat the uncertainties associated with underlying parameters in a systematic manner. The stability of a 734-m-high jointed rock slope in the west of Norway, the Oppstadhornet rock slope, is investigated by using a probabilistic method. The first-order reliability method (FORM) is used for probabilistic modeling of the plane failure problem in the rock slope. The Barton–Bandis (BB) shear strength criterion is used for the limit state equation. The statistical distributions of the BB criterion parameters, for which comprehensive data were collected and statistically analyzed, are determined by using distribution fitting algorithms. The sensitivity of the FORM model for the BB criterion is also investigated. It is found that the model is most sensitive to the mean value of the residual friction angle (ϕ _r) and least sensitive to the mean value of the slope angle (β _f). It is also found that the standard deviation of joint compressive strength (JCS) causes the greatest difference in the reliability index, which has the least sensitivity to the change in the mean and standard deviation of joint roughness coefficient (JRC).     

Studying the effect of non‐spherical micro‐particles on Hoek–Brown strength parameter mi using numerical true triaxial compressive tests

The strength parameter m_i in the Hoek–Brown strength criterion is empirical and was developed by trial and error. To better understand the fundamental relationship between m_i and the physical characteristics of intact rock, this paper presents a systematic study of m_i by representing intact rock as a densely packed cemented particle material and simulating its mechanical behavior using particle flow modeling. Specifically, the three‐dimensional particle flow code (PFC3D) was used to conduct numerical true triaxial compression tests on intact rock and to investigate the effect of non‐spherical micro‐particle parameters on m_i. To generate numerical intact rock specimens containing non‐spherical micro‐particles, a new genesis process was proposed, and a specific loop algorithm was used based on the efficiency of the process and the acceptability of generated specimens. Four main parameters—number, aspect ratio, size, and shape—of non‐spherical micro‐particles were studied, and the results indicated that they all have great effect on m_i. The strength parameter m_i increases when the number, aspect ratio, or size is larger or the shape becomes more irregular, mainly as a result of the higher level of interlocking between particles. This confirms the observations from engineering experience and laboratory experiments. To simulate the right strength parameter m_i, it is important to use appropriate non‐spherical micro‐particles by controlling these four parameters. This is further demonstrated by the simulation of two widely studied rocks, Lac du Bonnet granite and Carrara marble. Copyright © 2014 John Wiley & Sons, Ltd.     

Unifying Batch-Dissolution Kinetics for Salts: Probing the Back Reaction for Gypsum and Calcite by Means of the Common-Ion Effect

Recent success in fitting the shrinking object model for dissolution kinetics to biogenic silica, silica gel, simple salts, sucrose and gypsum prompted this study of the effects of common ions upon gypsum dissolution kinetics. Middle-ground dissolutions were mainly studied, in which shrinkage of the surface area, S, is significant, and the system approaches, but does not reach, saturation, c _sat. Dissolution was monitored by conductimetry. At a constant ionic strength of 0.060 M, the net rate for gypsum dissolution is given by \textNet \textRate = k_\textb ·S ·(c_\textsat - c ) {\text{Net}}\,{\text{Rate}} = k_{\text{b}} \cdot S \cdot (c_{\text{sat}} - c ) , where k _b is a rate constant, and c can be expressed alternatively in terms of either [Ca²⁺], [SO₄ ²⁻] and [ε_±], that part of the electrolyte concentration contributed by gypsum dissolution, or as the equivalent total concentrations of these species, for example, [SO₄ ²⁻]_T. The presence of either calcium or sulphate as a common ion slows dissolution, and the effect of this upon c _sat, k _b and k _f, the forward rate constant, is discussed. Contrary to previous experience, it is emphasised that each fitting of the shrinking object model demands its own value of the Solubility of gypsum, c _sat, which can be derived from the Solubility Product. This experience with gypsum is aligned with previous work on calcite, to develop a unified approach to the batch dissolution of salts. It highlights serious deficiencies in the way earlier common-ion experiments were conceived and enacted, and in particular with the rate equation of Sj?berg (Geochim Cosmochim Acta 40:441–447, 1976) for calcite above a pH of 7. Common-ion experiments are shown to be crucially important for probing the back reaction to dissolutions and might be applied to the far bigger problem of silicate-mineral dissolution, where ‘non-linear kinetics’ are often observed.     

High-pressure electronic absorption spectroscopy of natural and synthetic Cr<Superscript>3+</Superscript>-bearing clinopyroxenes

Comparison of polarized optical absorption spectra of natural Ca-rich diopsides and synthetic NaCrSi2O6 and LiCrSi2O6 clinopyroxenes, evidences as vivid similarities, as noticeable differences. The similarities reflect the fact that in all cases Cr3+ enters the small octahedral M1-site of the clinopyroxene structure. The differences are due to some iron content in the natural samples causing broad intense near infrared bands of electronic spin-allowed dd transitions of Fe2+(M1, M2) and intervalence Fe2+/Fe3+ charge-transfer transition, and by different symmetry and different local crystal fields strength of Cr3+ in the crystal structures. The positions of the spin-allowed bands of Cr3+, especially of the low energy one caused by the electronic 4 A 2g → 2 T 1g transition, are found to be in accordance with mean M1–O distances. The local relaxation parameter ε calculated for limCr 3+ → 0 from the spectra and interatomic á Cr - O ñ \left\langle {Cr - O} \right\rangle and á Mg - O ñ \left\langle {Mg - O} \right\rangle distances yields a very high value, 0.96, indicating that in the clinopyroxene structure the local lattice relaxation around the “guest” ion, Cr3+, deviates greatly from the “diffraction” value, ε = 0, than in any other known Cr3+-bearing systems studied so far. Under pressure the spin-allowed bands of Cr3+ shift to higher energies and decrease in intensity quite in accordance with the crystal field theoretical expectations, while the spin-forbidden absorption lines remain practically unshifted, but also undergo a strong weakening. There is no evident dependence of the Racah parameter B of Cr3+ reflecting the covalence of the oxygen-chromium bond under pressure: within the uncertainty of determination it may be regarded as practically constant. The values of CrO6 octahedral modulus, k\textpoly\textloc k_{\text{poly}}^{\text{loc}} , derived from high-pressure spectra of natural chromium diopside and synthetic NaCrSi2O6 kosmochlor are very close, ~203 and ~196 GPa, respectively, being, however, nearly twice higher than that of MgO6 octahedron in diopside, 105(4) GPa, obtained by Thompson and Downs (2008). Such a strong stiffening of the structural octahedron, i.e. twice higher value of k\textCr3 + \textloc k_{{{\text{Cr}}^{3 + } }}^{\text{loc}} comparing with that of k\textMg2 + \textloc k_{{{\text{Mg}}^{2 + } }}^{\text{loc}} , may be caused by simultaneous substitution of Ca2+ by larger Na+ in the neighboring M2 sites at so-called jadeite-coupled substitution Mg2+ + Ca2+ → Cr3+ + Na+. It is also remarkable that the values of CrO6 octahedral modulus of NaCrSi2O6 kosmochlor obtained here are nearly twice larger than that of 90(16) GPa, evaluated by high-pressure X-ray structural refinement by Origlieri et al. (2003). Taking into account that the overall compressibility of the clinopyroxene structure should mainly be due to the compressibility of M1- and M2-sites, our k\textCr3 + \textloc k_{{{\text{Cr}}^{3 + } }}^{\text{loc}} -value, ~196 GPa, looks much more consistent with the bulk modulus value, 134(1) GPa.     

Using the Maximum Effective Moment Criterion to Interpret Quartz <c>‐Fabric Patterns

The Maximum Effective Moment (MEM) criterion predicts that the initial orientation of ductile shear zones and shear bands is ~55° relative to the maximum principal stress axis (σ1) and that the kinematic vorticity number (Wk) is ~0.94. These preferred orientations should be reflected in the pattern of quartz -fabrics in shear zones and shear bands. Common quartz -fabrics in plane strain can be divided into low-temperature (L) and high-temperature (H) fabrics, with each group showing three patterns. A steady flow with a constant value of Wk≈0.94 gives rise to L-1 and H-1 patterns, which are commonly characterized by a single axis girdle normal to the shear zone and a single -point maximum parallel to the shear zone.Once the conjugate set develops, L-1 and H-1 have opening angles of ~70° and ~110°, respectively. L-2 and H-2 are asymmetric patterns associated with variable deformation partitioning and vorticity values of 0< Wk<0.94. In contrast, L-3 and H-3 are symmetric patterns associated with 100% deformation partitioning and Wk=0. The opening angle in quartz -fabrics is implicitly linked to the temperature during deformation. The opening angle is ~70° at low temperature and ~110° at high temperature. However, a linear correction between the opening angle and the temperature cannot be established. During deformation partitioning, synthetic shear bands form earlier than antithetic bands and are more easily developed. This may result in opening angles of <70° for low-temperature fabrics and of >110° for high-temperature fabrics. The following criteria can be used to recognize reworked shear zones that have experienced multiple orogenic phases and changes in the stress state: 1) the initial Wk is larger or smaller than ~0.94; 2) the change in Wk is abrupt, rather than progressive; 3) inconsistent shear senses are inferred for the different phases of deformation; and 4) a negative value of Wk is found in reworked shear zones.     

Collapse loads for rectangular foundations by three-dimensional upper bound limit analysis using radial point interpolation method

A three-dimensional kinematic limit analysis approach based on the radial point interpolation method (RPIM) has been used to compute collapse loads for rectangular foundations. The analysis is based on the Mohr-Coulomb yield criterion and the associated flow rule. It is understood that the internal plastic power dissipation function and flow rule constraints can be expressed entirely in terms of plastic strain rates without involving stresses. The optimization problem has been solved on basis of the semidefinite programming (SDP) by using highly efficient primal-dual interior point solver MOSEK in MATLAB. The results have been presented in terms of the variation of the shape factors with changes in the aspect ratio (L/B) of the footing for different values of soil internal friction angle (ϕ). Computations have revealed that the shape factors, s_c and s_q, due to effects of cohesion and surcharge increase continuously with (1) decrease in L/B and (2) increase in ϕ. On the other hand, the shape factor s_γ, due to the effect of soil unit weight, increases very marginally with an increase in L/B up to (1) ϕ = 25° for a rough footing and (2) ϕ = 35° for a smooth footing. Thereafter, for greater values of ϕ, the variation of s_γ with L/B has been found to be quite similar to that of the factors s_c and s_q. The variations of (1) nodal velocity patterns, (2) plastic power dissipation, and (3) maximum plastic shear strain rates have also been examined to interpret the associated failure mechanism.     

Dynamic Model of Fracture Normal Behaviour and Application to Prediction of Stress Wave Attenuation Across Fractures

Summary.  The purpose of this paper is to establish a dynamic constitutive model of fracture normal behaviour, based on laboratory tests of artificial fractures cast by cement mortar. A series of tests are systematically carried out under quasi-static (10⁻¹ MPa/s) up to highly dynamic (10³ MPa/s) monotonic loading conditions. The normal stress-fracture closure response is measured at different loading rates. Based on the measured curves, a nonlinear (hyperbolic) dynamic model of fracture normal behaviour, termed as dynamic BB model, is proposed. The dynamic model is modified from the existing BB model of static normal behaviour of fractures by taking into account the loading-rate effect. Two important dynamic parameters of fractures, FSC _d (dynamic fracture stiffness constant, which describes the incremental ratio of dynamic initial stiffness) and FCC _d (dynamic fracture closure constant, which describes the decremental ratio of dynamic maximum allowable closure), are identified. They indicate the quantitative degree of loading-rate effect on fracture normal behaviour subjected to dynamic loads. For practical application, the new model is incorporated into the Universal Distinct Element Code (UDEC) and subsequently, UDEC modelling of normally incident P-wave transmission across single fractures with the dynamic BB model is conducted. Wave transmission coefficient is obtained for various combinations of fracture dynamic parameters, as well as different wave amplitudes and frequencies. The numerical results show that wave transmission coefficient for a fracture with the dynamic BB model is greater than that for a fracture with the static BB model. In addition, a fracture with higher values of FSC _d and FCC _d leads to higher transmission (lower attenuation). Author’s address: J. Zhao, Ecole Polytechnique Federale de Lausanne (EPFL), Rock Mechanics Laboratory, 1015 Lausanne, Switzerland