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Heterogeneous Media (heterogeneous + media)
Selected AbstractsRequired source distribution for interferometry of waves and diffusive fieldsGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2009Yuanzhong Fan SUMMARY The Green's function that describes wave propagation between two receivers can be reconstructed by cross-correlation provided that the receivers are surrounded by sources on a closed surface. This technique is referred to as ,interferometry' in exploration seismology. The same technique for Green's function extraction can be applied to the solution of the diffusion equation if there are sources throughout in the volume. In practice, we have only a finite number of active sources. The issues of the required source distribution is investigated, as is the feasibility of reconstructing the Green's function of the diffusion equation using a limited number of sources within a finite volume. We study these questions for homogeneous and heterogeneous media for wave propagation and homogeneous media for diffusion using numerical simulations. These simulations show that for the used model, the angular distribution of sources is critical in wave problems in homogeneous media. In heterogeneous media, the position and size of the heterogeneous area with respect to the sources determine the required source distribution. For diffusion, the sensitivity to the sources decays from the midpoint between the two receivers. The required width of the source distribution decreases with frequency, with the result that the required source distribution for early- and late-time reconstruction is different. The derived source distribution criterion for diffusion suggests that the cross-correlation-based interferometry is difficult to apply in field condition. [source] Three-dimensional models of elastostatic deformation in heterogeneous media, with applications to the Eastern California Shear ZoneGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2009Sylvain Barbot SUMMARY We present a semi-analytic iterative procedure for evaluating the 3-D deformation due to faults in an arbitrarily heterogeneous elastic half-space. Spatially variable elastic properties are modelled with equivalent body forces and equivalent surface traction in a ,homogenized' elastic medium. The displacement field is obtained in the Fourier domain using a semi-analytic Green function. We apply this model to investigate the response of 3-D compliant zones (CZ) around major crustal faults to coseismic stressing by nearby earthquakes. We constrain the two elastic moduli, as well as the geometry of the fault zones by comparing the model predictions to Synthetic Aperture Radar inferferometric (InSAR) data. Our results confirm that the CZ models for the Rodman, Calico and Pinto Mountain faults in the Eastern California Shear Zone (ECSZ) can explain the coseismic InSAR data from both the Landers and the Hector Mine earthquakes. For the Pinto Mountain fault zone, InSAR data suggest a 50 per cent reduction in effective shear modulus and no significant change in Poisson's ratio compared to the ambient crust. The large wavelength of coseismic line-of-sight displacements around the Pinto Mountain fault requires a fairly wide (,1.9 km) CZ extending to a depth of at least 9 km. Best fit for the Calico CZ, north of Galway Dry Lake, is obtained for a 4 km deep structure, with a 60 per cent reduction in shear modulus, with no change in Poisson's ratio. We find that the required effective rigidity of the Calico fault zone south of Galway Dry Lake is not as low as that of the northern segment, suggesting along-strike variations of effective elastic moduli within the same fault zone. The ECSZ InSAR data is best explained by CZ models with reduction in both shear and bulk moduli. These observations suggest pervasive and widespread damage around active crustal faults. [source] Parsimonious finite-volume frequency-domain method for 2-D P,SV -wave modellingGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2008R. Brossier SUMMARY A new numerical technique for solving 2-D elastodynamic equations based on a finite-volume frequency-domain approach is proposed. This method has been developed as a tool to perform 2-D elastic frequency-domain full-waveform inversion. In this context, the system of linear equations that results from the discretization of the elastodynamic equations is solved with a direct solver, allowing efficient multiple-source simulations at the partial expense of the memory requirement. The discretization of the finite-volume approach is through triangles. Only fluxes with the required quantities are shared between the cells, relaxing the meshing conditions, as compared to finite-element methods. The free surface is described along the edges of the triangles, which can have different slopes. By applying a parsimonious strategy, the stress components are eliminated from the discrete equations and only the velocities are left as unknowns in the triangles. Together with the local support of the P0 finite-volume stencil, the parsimonious approach allows the minimizing of core memory requirements for the simulation. Efficient perfectly matched layer absorbing conditions have been designed for damping the waves around the grid. The numerical dispersion of this FV formulation is similar to that of O(,x2) staggered-grid finite-difference (FD) formulations when considering structured triangular meshes. The validation has been performed with analytical solutions of several canonical problems and with numerical solutions computed with a well-established FD time-domain method in heterogeneous media. In the presence of a free surface, the finite-volume method requires 10 triangles per wavelength for a flat topography, and fifteen triangles per wavelength for more complex shapes, well below the criteria required by the staircase approximation of O(,x2) FD methods. Comparisons between the frequency-domain finite-volume and the O(,x2) rotated FD methods also show that the former is faster and less memory demanding for a given accuracy level, an attractive feature for frequency-domain seismic inversion. We have thus developed an efficient method for 2-D P,SV -wave modelling on structured triangular meshes as a tool for frequency-domain full-waveform inversion. Further work is required to improve the accuracy of the method on unstructured meshes. [source] Dynamic non-planar crack rupture by a finite volume methodGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2007M. Benjemaa SUMMARY Modelling dynamic rupture for complex geometrical fault structures is performed through a finite volume method. After transformations for building up the partial differential system following explicit conservative law, we design an unstructured bi-dimensional time-domain numerical formulation of the crack problem. As a result, arbitrary non-planar faults can be explicitly represented without extra computational cost. On these complex surfaces, boundary conditions are set on stress fluxes and not on stress values. Prescribed rupture velocity gives accurate solutions with respect to analytical ones depending on the mesh refinement, while solutions for spontaneous propagation are analysed through numerical means. An example of non-planar spontaneous fault growth in heterogeneous media demonstrates the good behaviour of the proposed algorithm as well as specific difficulties of such numerical modelling. [source] A practical grid-based method for tracking multiple refraction and reflection phases in three-dimensional heterogeneous mediaGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2006M. De Kool SUMMARY We present a practical grid-based method in 3-D spherical coordinates for computing multiple phases comprising any number of reflection and transmission branches in heterogeneous layered media. The new scheme is based on a multistage approach which treats each layer that the wave front enters as a separate computational domain. A finite-difference eikonal solver known as the fast-marching method (FMM) is reinitialized at each interface to track the evolving wave front as either a reflection back into the incident layer or a transmission through to the adjacent layer. Unlike the standard FMM, which only finds first arrivals, this multistage approach can track those later arriving phases explicitly caused by the presence of discontinuities. Notably, the method does not require an irregular mesh to be constructed in order to connect interface nodes to neighbouring velocity nodes which lie on a regular grid. To improve accuracy, local grid refinement is used in the neighbourhood of a source point where wave front curvature is high. The method also provides a way to trace reflections from an interface that are not the first arrival (e.g. the global PP phase). These are computed by initializing the multistage FMM from both the source and receiver, propagating the two wave fronts to the reflecting interface, and finding stationary points of the sum of the two traveltime fields on the reflecting interface. A series of examples are presented to test the efficiency, accuracy and robustness of the new scheme. As well as efficiently computing various global phases to an acceptable accuracy through the ak135 model, we also demonstrate the ability of the scheme to track complex crustal phases that may be encountered in coincident reflection, wide-angle reflection/refraction or local earthquake surveys. In one example, a variety of phases are computed in the presence of a realistic subduction zone, which includes several layer pinch-outs and a subducting slab. Our numerical tests show that the new scheme is a practical and robust alternative to conventional ray tracing for finding various phases in layered media at a variety of scales. [source] A one-way wave equation for modelling seismic waveform variations due to elastic heterogeneityGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2005D. A. Angus SUMMARY The application of a new one-way narrow-angle elastic wave equation to isotropic heterogeneous media is described. This narrow-angle finite-difference propagator should provide an efficient and accurate method of simulating primary body wave(s) passing through smoothly varying heterogeneous media. Although computationally slower than ray theory, the narrow-angle propagator can model frequency-dependent forward diffraction and scattering as well as the averaging effects due to smooth variations in medium parameters that vary on the sub-Fresnel zone level. Example waveforms are presented for the propagation of body waves in deterministic as well as stochastic heterogeneous 3-D Earth models. Extrapolation within deterministic media will highlight various familiar wave-diffraction and pulse-distortion effects associated with large-scale inhomogeneities, such as geometrical spreading, wavefront folding and creeping-wave diffraction by a compact object. Simulation within stochastic media will examine the effects of varying the correlation lengths of random heterogeneities on wave propagation. In particular, wave phenomena such as frequency-dependent forward scattering, the appearance of random caustics and the generation of seismic coda will be shown. [source] Wavelet Galerkin method in multi-scale homogenization of heterogeneous mediaINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2006Shafigh Mehraeen Abstract The hierarchical properties of scaling functions and wavelets can be utilized as effective means for multi-scale homogenization of heterogeneous materials under Galerkin framework. It is shown in this work, however, when the scaling functions are used as the shape functions in the multi-scale wavelet Galerkin approximation, the linear dependency in the scaling functions renders improper zero energy modes in the discrete differential operator (stiffness matrix) if integration by parts is invoked in the Galerkin weak form. An effort is made to obtain the analytical expression of the improper zero energy modes in the wavelet Galerkin differential operator, and the improper nullity of the discrete differential operator is then removed by an eigenvalue shifting approach. A unique property of multi-scale wavelet Galerkin approximation is that the discrete differential operator at any scale can be effectively obtained. This property is particularly useful in problems where the multi-scale solution cannot be obtained simply by a wavelet projection of the finest scale solution without utilizing the multi-scale discrete differential operator, for example, the multi-scale analysis of an eigenvalue problem with oscillating coefficients. Copyright © 2005 John Wiley & Sons, Ltd. [source] Non-local dispersive model for wave propagation in heterogeneous media: one-dimensional caseINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2002Jacob Fish Abstract Non-local dispersive model for wave propagation in heterogeneous media is derived from the higher-order mathematical homogenization theory with multiple spatial and temporal scales. In addition to the usual space,time co-ordinates, a fast spatial scale and a slow temporal scale are introduced to account for rapid spatial fluctuations of material properties as well as to capture the long-term behaviour of the homogenized solution. By combining various order homogenized equations of motion the slow time dependence is eliminated giving rise to the fourth-order differential equation, also known as a ,bad' Boussinesq problem. Regularization procedures are then introduced to construct the so-called ,good' Boussinesq problem, where the need for C1 continuity is eliminated. Numerical examples are presented to validate the present formulation. Copyright © 2002 John Wiley & Sons, Ltd. [source] Non-local dispersive model for wave propagation in heterogeneous media: multi-dimensional caseINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2002Jacob Fish Abstract Three non-dispersive models in multi-dimensions have been developed. The first model consists of a leading-order homogenized equation of motion subjected to the secularity constraints imposing uniform validity of asymptotic expansions. The second, non-local model, contains a fourth-order spatial derivative and thus requires C1 continuous finite element formulation. The third model, which is limited to the constant mass density and a macroscopically orthotropic heterogeneous medium, requires C0 continuity only and its finite element formulation is almost identical to the classical local approach with the exception of the mass matrix. The modified mass matrix consists of the classical mass matrix (lumped or consistent) perturbed with a stiffness matrix whose constitutive matrix depends on the unit cell solution. Numerical results are presented to validate the present formulations. Copyright © 2002 John Wiley & Sons, Ltd. [source] Optimal convergence properties of the FETI domain decomposition methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2007Y. Maday Abstract In this paper an original variant of the FETI domain decomposition method is introduced for heterogeneous media. This method uses new absorbing interface conditions in place of the Neumann interface conditions defined in the classical FETI method. The optimal convergence properties of the classical FETI method and of its variant are first demonstrated, both in the case of homogeneous and heterogeneous media. Secondly, novel and efficient absorbing interface conditions, which avoid rigid body motions, are investigated and analysed. Numerical experiments illustrate the dependence of the proposed method upon several parameters, and confirm the robustness and efficiency of this method when equipped with such absorbing interface conditions. Copyright © 2006 John Wiley & Sons, Ltd. [source] Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layerMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2008Clair Poignard Abstract We study the behaviour of steady-state voltage potentials in two kinds of bidimensional media composed of material of complex permittivity equal to 1 (respectively, ,) surrounded by a thin membrane of thickness h and of complex permittivity , (respectively, 1). We provide in both cases a rigorous derivation of the asymptotic expansion of steady-state voltage potentials at any order as h tends to zero, when Neumann boundary condition is imposed on the exterior boundary of the thin layer. Our complex parameter , is bounded but may be very small compared to 1, hence our results describe the asymptotics of steady-state voltage potentials in all heterogeneous and highly heterogeneous media with thin layer. The asymptotic terms of the potential in the membrane are given explicitly in local coordinates in terms of the boundary data and of the curvature of the domain, while these of the inner potential are the solutions to the so-called dielectric formulation with appropriate boundary conditions. The error estimates are given explicitly in terms of h and , with appropriate Sobolev norm of the boundary data. We show that the two situations described above lead to completely different asymptotic behaviours of the potentials. Copyright © 2007 John Wiley & Sons, Ltd. [source] Physical properties of rocks from the upper part of the Yaxcopoil-1 drill hole, Chicxulub craterMETEORITICS & PLANETARY SCIENCE, Issue 6 2004Y. Popov Thermal conductivity, thermal diffusivity, density, and porosity were measured on 120 dry and water-saturated rocks with a core sampling interval of 2,2.5 m. Nondestructive, non-contact optical scanning technology was used for thermal property measurements including thermal anisotropy and inhomogeneity. Supplementary petrophysical properties (acoustic velocities, formation resisitivity factor, internal surface, and hydraulic permeability) were determined on a selected subgroup of representative samples to derive correlations with the densely measured parameters, establishing estimated depth logs to provide calibration values for the interpretation of geophysical data. Significant short- and long-scale variations of porosity (1,37%) turned out to be the dominant factor influencing thermal, acoustic, and hydraulic properties of this post impact limestone formation. Correspondingly, large variations of thermal conductivity, thermal diffusivity, acoustic velocities, and hydraulic permeability were found. These variations of physical properties allow us to subdivide the formation into several zones. A combination of experimental data on thermal conductivity for dry and water-saturated rocks and a theoretical model of effective thermal conductivity for heterogeneous media have been used to calculate thermal conductivity of mineral skeleton and pore aspect ratio for every core under study. The results on thermal parameters are the necessary basis for the determination of heat flow density, demonstrating the necessity of dense sampling in the case of inhomogeneous rock formations. [source] A new multilevel algebraic preconditioner for the diffusion equation in heterogeneous mediaNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2010Yu Kuznetsov Abstract We develop and analyze a new multilevel preconditioner for algebraic systems arising from the finite volume discretization of 3D diffusion,reaction problems in highly heterogeneous media. The system matrices are assumed to be symmetric M -matrices. The preconditioner is based on a special coarsening algorithm and the inner Chebyshev iterative procedure. The condition number of the preconditioned matrix does not depend on the coefficients in the diffusion operator. Numerical experiments confirm theoretical results and reveal the competitiveness of the new preconditioner with respect to a well-known algebraic multigrid preconditioner. Copyright © 2010 John Wiley & Sons, Ltd. [source] 2D internal flux compatibility equation of the flux Green element method for transient nonlinear potential problemsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2010Akpofure E. Taigbenu Abstract This article presents the derivation and implementation of the normal directional flux compatibility equation (relationship) at internal nodes when the Green element formulation that consistently provides accurate estimates of the primary variable, and its normal directional derivative (normal flux) is applied in 2D heterogeneous media to steady and transient potential problems. Such a relationship is required to resolve the closure problem due to having fewer integral equations than the number of unknowns at internal nodes. The derivation of the relationship is based on Stokes' theorem, which transforms the contour integral of the normal directional fluxes into a surface integral that is identically zero. The numerical discretization of the compatibility equation is demonstrated with four numerical examples using the six-node quadratic triangular and the four and eight-node rectangular elements. The incorporation of triangular elements into the current formulation demonstrates that the internal compatibility equation can be successfully implemented on irregular grids. The direct calculation of the fluxes significantly enhances the accuracy of the formulation, so that high accuracy, exceeding that of the finite element method, is achieved with very coarse spatial discretization. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source] Optical characterization of concentrated dispersions: applications to laboratory analyses and on-line process monitoring and control,POLYMER INTERNATIONAL, Issue 9 2004H Buron Abstract Light scattering methods are often used to study the stability of suspensions or emulsions and to estimate the dispersed phase properties such as particle size and volume fraction. However, such optical methods often require a previous dilution of the dispersion because of a limited measurement range, and are then unable to give information about the real physical state of dense heterogeneous media. A new technology based on multiple light scattering analysis and called Turbiscan has been recently developed by a French company, Formulaction, to fill this gap and to characterize both diluted and concentrated dispersions. In the first part, we review the physical concepts of multiple light scattering by dispersions. In relation to the optical analyser Turbiscan, we present physical and statistical models for the radiative transfer in dense suspensions. In the second part, we investigate the influence of particle volume fraction and particle size (polystyrene latex bead suspensions) on the backscattered and transmitted light fluxes measured by Turbiscan. The experimental data are compared with results from the physical models. In the last section, we use the optical analyser Turbiscan Lab to detect and characterize various concentrated dispersions destabilization (coalescence, flocculation, creaming and sedimentation), and then the Turbiscan On Line to monitor and characterize an emulsification process under ultrasonic agitation. Copyright © 2004 Society of Chemical Industry [source] | |||||||||||||