Numerical simulation of seismic activity by the grid-characteristic method

Seismic activity in homogeneous and layered enclosing rock masses is studied. A numerical mechanical-mathematical model of a hypocenter is proposed that describes the whole range of elastic perturbations propagating from the hypocenter. Synthetic beachball plots computed for various fault plane orientations are compared with the analytical solution in the case of homogeneous rock. A detailed analysis of wave patterns and synthetic seismograms is performed to compare seismic activities in homogeneous and layered enclosing rock masses. The influence exerted by individual components of a seismic perturbation on the stability of quarry walls is analyzed. The grid-characteristic method is used on three-dimensional parallelepipedal and curvilinear structured grids with boundary conditions set on the boundaries of the integration domain and with well-defined contact conditions specified in explicit form.     

Numerical investigation of seismic attenuation in fluid-saturated fractured media: An extended view of the Skempton coefficient

The interpretation of data from seismic exploration gains from consideration of hydro-mechanical processes in particular when the waves propagate through fluid-filled fractured porous rocks. We specifically investigate attenuation processes in porous rock containing complex fracture networks with the main objective to identify effective hydro-mechanical material properties by computational homogenization. To this end, we generate stochastic fracture ensembles and approximate fractures as ellipses with low aspect ratio (1/100). In other words, the fractures constitute thin and long hydraulic conduits with high permeability. Full periodic boundary conditions are employed on the small scale. We observe a strongly heterogeneous pressure propagation in the fracture network and the surrounding rock. The properties of the macroscopic substitute model are derived by means of a volume averaging technique. We identify a dependence of effective properties on fluid saturation and amount of fractures. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Body Wave Propagation and Site Effects

The interaction between seismic radiation and near-surface geology plays a role of primary importance in the risk hazard mitigation especially due to the so called site effects. Such kind of phenomena depends on the geological conditions that may cause spatial variability of ground motion, such as producing an amplification of the seismic signal at defined frequencies. For this reason the analysis of site effects plays a key role in the seismic risk mitigation and the numerical simulation represents a powerful tool in this field. This work contemplates the simulation of seismic signals and analysis of their power spectra, varying the intensity of the impulse and its location in the domain, as well as the number of layers and/or their physical properties in order to guess a possible relation between the maximum of peak amplification and the geometrical and physical characteristics of the medium where seismic waves propagate. The seismic wave propagation is described by means of an Initial Boundary Value Problem on a three-dimensional domain with an arbitrary topography, whose numerical integration, obtained by implementing the Finite Element Method, provides the analyzed seismic signals. We present several cases and analyze the corresponding power spectra in order to recognize the maximum frequency involved in the soil shaking amplification.     

Simulation of elastic wave propagation in geological media: Intercomparison of three numerical methods

For wave propagation in heterogeneous media, we compare numerical results produced by grid-characteristic methods on structured rectangular and unstructured triangular meshes and by a discontinuous Galerkin method on unstructured triangular meshes as applied to the linear system of elasticity equations in the context of direct seismic exploration with an anticlinal trap model. It is shown that the resulting synthetic seismograms are in reasonable quantitative agreement. The grid-characteristic method on structured meshes requires more nodes for approximating curved boundaries, but it has a higher computation speed, which makes it preferable for the given class of problems.     

Numerical electroseismic modeling: A finite element approach

Electroseismics is a procedure that uses the conversion of electromagnetic to seismic waves in a fluid-saturated porous rock due to the electrokinetic phenomenon. This work presents a collection of continuous and discrete time finite element procedures for electroseismic modeling in poroelastic fluid-saturated media. The model involves the simultaneous solution of Biot’s equations of motion and Maxwell’s equations in a bounded domain, coupled via an electrokinetic coefficient, with appropriate initial conditions and employing absorbing boundary conditions at the artificial boundaries. The 3D case is formulated and analyzed in detail including results on the existence and uniqueness of the solution of the initial boundary value problem. Apriori error estimates for a continuous-time finite element procedure based on parallelepiped elements are derived, with Maxwell’s equations discretized in space using the lowest order mixed finite element spaces of Nédélec, while for Biot’s equations a nonconforming element for each component of the solid displacement vector and the vector part of the Raviart-Thomas-Nédélec of zero order for the fluid displacement vector are employed. A fully implicit discrete-time finite element method is also defined and its stability is demonstrated. The results are also extended to the case of tetrahedral elements. The 2D cases of compressional and vertically polarized shear waves coupled with the transverse magnetic polarization (PSVTM-mode) and horizontally polarized shear waves coupled with the transverse electric polarization (SHTE-mode) are also formulated and the corresponding finite element spaces are defined. The 1D SHTE initial boundary value problem is also formulated and approximately solved using a discrete-time finite element procedure, which was implemented to obtain the numerical examples presented.     

Modeling 1-D elastic P-waves in a fractured rock with hyperbolic jump conditions

The propagation of elastic waves in a fractured rock is investigated, both theoretically and numerically. Outside the fractures, the propagation of compressional waves is described in the simple framework of 1-D linear elastodynamics. The focus here is on the interactions between the waves and fractures: for this purpose, the mechanical behavior of the fractures is modeled using nonlinear jump conditions deduced from the Bandis–Barton model classically used in geomechanics. Well-posedness of the initial-boundary value problem thus obtained is proved. Numerical modeling is performed by coupling a time-domain finite-difference scheme with an interface method accounting for the jump conditions. The numerical experiments show the effects of contact nonlinearities. The harmonics generated may provide a nondestructive means of evaluating the mechanical properties of fractures.     

The propagation of the energy of elastic waves in anisotropic media

The particular features of the propagation of the seismic energy of elastic waves in anisotropic media with four constants of elasticity, depending on the directions of motion of the waves and the ratios of the constants of elasticity for all practical media of the anisotropy class considered, are investigated. A direct connection is established between the formation of acute-angled edges on the fronts of quasi-transverse waves from point sources and the distinctive features of the propagation of the energy of the waves under certain conditions for the constants of elasticity.     

Dynamic Wave Propagation in Porous Media Semi-Infinite Domains

The problem of dynamic wave propagation in semi-infinite domains is of great importance, especially, in subjects of applied mechanics and geomechanics, such as the issues of earthquake wave propagation in an infinite half-space and soil-structure interaction under seismic loading. In such problems, the elastic waves are supposed to propagate to infinity, which requires a special treatment of the boundaries in initial boundary-value problems (IBVP). Saturated porous materials, e. g. soil, basically represent volumetrically coupled solid-fluid aggregates. Based on the continuum-mechanical principles and the established macroscopic Theory of Porous Media (TPM) [1, 2], the governing balance equations yield a coupled system of partial differential equations (PDE). Restricting the discussion to the isothermal and geometrically linear case, this system comprises the solid and fluid momentum balances and the overall volume balance, and can be conveniently treated numerically following an implicit monolithic approach [3]. Therefore, the equations are firstly discretised in space using the mixed Finite Element Method (FEM) together with quasi-static Infinite Elements (IE) at the boundaries that represent the extension of the domain to infinity [4], and secondly in time using an appropriate implicit time-integration scheme. Additionally, a stable implementation of the Viscous Damping Boundary (VDB) method [5] for the simulation of transient waves at infinity is presented, which implicitly treats the damping boundary terms in a weakly imposed sense. The proposed algorithm is implemented into the FE tool PANDAS and tested on a two-dimensional IBVP. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of seismic exploration problems in the Arctic region by applying the grid-characteristic method

The goal of this paper is the numerical solution of direct problems concerning hydrocarbon seismic exploration on the Arctic shelf. The task is addressed by solving a complete system of linear elasticity equations and a system of acoustic field equations. Both systems are solved by applying the grid-characteristic method, which takes into account all wave processes in a detailed and physically correct manner and produces a solution near the boundaries and interfaces of the integration domain, including the interface between the acoustic and linear elastic media involved. The seismograms and wave patterns obtained by numerically solving these systems are compared. The effect of ice structures on the resulting wave patterns is examined.     

Mathematical modeling of seismic and acousto-gravitational waves in a heterogeneous earth-atmosphere model

A numerical-analytical solution to problems of seismic and acoustic-gravitational wave propagation is applied to a heterogeneous Earth-Atmosphere model. The seismic wave propagation in an elastic half-space is described by a system of first order dynamic equations of the elasticity theory. The propagation of acoustic-gravitational waves in the atmosphere is described by the linearized Navier-Stokes equations. The algorithm proposed is based on the integral Laguerre transform with respect to time, the finite integral Bessel transform along the radial coordinate with a finite difference solution of the reduced problem along the vertical coordinate. The algorithm is numerically tested for the heterogeneous Earth-Atmosphere model for different source locations.     

Models of unidirectional propagation in heterogeneous excitable media

Two differential equation models of excitable media (threshold and recovery kinetics) with solutions that exhibit unidirectional propagation are presented. It is shown that unidirectional propagation in heterogeneous excitable media with non-oscillatory kinetics can be initiated from homogeneous initial data. Simulations on a reaction-diffusion model with FitzHugh-Nagumo kinetics and spatially heterogeneous parameters yields a rotating wave on a one-dimensional circular spatial domain. An ordinary differential equation model with four semi-coupled excitable cells and heterogeneous parameters is analyzed to determine a critical parameter region over which unidirectional propagation may occur.     

Finite element approximation of coupled seismic and electromagnetic waves in fluid‐saturated poroviscoelastic media

This work presents a collection of global and iterative finite element procedures for the numerical approximation of coupled seismic and electromagnetic waves in 2D bounded fluid‐saturated porous media, with absorbing boundary conditions at the artificial boundaries. The equations being analyzed are the coupled Biot's equations of motion and Maxwell equations in the diffusive range. Both seismoelectric and electroseismic coupling are simultaneously included and analyzed in the model. The case of compressional and vertically polarized shear waves coupled with the transverse magnetic polarization (PSVTM‐mode) is analyzed in detail, including the derivation of a priori error estimates on the global finite element procedure and results on the convergence of a domain decomposition iterative algorithm. Later, the corresponding results for the case of horizontally polarized shear waves coupled with the transverse electric polarization (SHTE‐mode) are stated. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

On the velocity of the Rayleigh wave propagation along curvilinear surfaces

To investigate the propagation of Rayleigh waves on curvilinear boundaries, wave propagation along cylindrical and spherical surfaces is considered. For elastic media with indicated boundaries, exact solutions of equations of elasticity theory are constructed and the asymptotics of Hankel and Legendre functions are used. On the basis of the results obtained, a conjecture is made concerning the dependence of the velocity of the Rayleigh wave on a small curvature of the route and on a small curvature in the perpendicular direction. Bibliography: 7 titles.     

Tile patterns in excitable media subject to non-solenoidal flow fields

The propagation of spiral waves in excitable media subject to a non-solenoidal advective field which satisfies the no-penetration condition on the boundaries of the domain is studied numerically, and it is shown that, depending on the amplitude and spatial frequencies of the velocity field, the spiral wave may be distorted highly, break up into a number of smaller spiral waves, or exhibit polygonal shapes or tile patterns. These patterns reflect the symmetry/asymmetry of the velocity field and are characterized by thick regions of high concentration at stagnation points where the velocity gradient is largest, and thin ones which are parallel to the velocity vector. It is also shown that the advective field distorts the spiral wave by decreasing its thickness where the velocity is largest due to the stretching of the wave, and by increasing it at the stagnation points where the curvature of the wave is largest.     

Extended finite element modeling of deformable porous media with arbitrary interfaces

In this paper, an enriched finite element method is presented for numerical simulation of saturated porous media. The arbitrary discontinuities, such as material interfaces, are encountered via the extended finite element method (X-FEM) by enhancing the standard FEM displacements. The X-FEM technique is applied to the governing equations of porous media for the spatial discretization, followed by a generalized Newmark scheme used for the time domain discretization. In X-FEM, the material interfaces are represented independently of element boundaries and the process is accomplished by partitioning the domain with some triangular sub-elements whose Gauss points are used for integration of the domain of elements. Finally, several numerical examples are analyzed, including the dynamic analysis of the failure of lower San Fernando dam, to demonstrate the efficiency of the X-FEM technique in saturated porous soils.     

2‐D solitary waves in thermal media with nonsymmetric boundary conditions

Optical solitary waves and their stability in focusing thermal optical media, such as lead glasses, are studied numerically and theoretically in (2 + 1) dimensions. The optical medium is a square cell and mixed boundary conditions of Newton cooling and fixed temperature on different sides of the cell are used. Nonlinear thermal optical media have a refractive index which depends on temperature, so that heating from the optical beam and heat flow across the boundaries can change the refractive index of the medium. Solitary wave solutions are found numerically using the Newton conjugate‐gradient method, while their stability is studied using a linearized stability analysis and also via numerical simulations. It is found that the position of the solitary wave is dependent on the boundary conditions, with the center of the beam moving toward the warmer boundaries, as the parameters are varied. The stability of the solitary waves depends on the symmetry of the boundary conditions and the amplitude of the solitary waves.     

Propagation of Seismic Waves in Block Fluid Media

The propagation of seismic waves in block two- and three-dimensional fluid media is investigated. For these media, effective models, which are anisotropic fluids, are established. Formulas for the velocities of wave propagation in these fluid media are derived and analyzed. Special investigation is conducted in the cases where blocks with different fluids alternate along the coordinate axes or where blocks filled with a fluid are surrounded by blocks with another fluid. In both cases, the dependence of the wave velocities in the entire medium on the differences of the densities and the wave velocities in fluid blocks is studied. Bibliography: 9 titles. Dedicated to P. V. Krauklis on the occasion of his seventieth birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 124–146.     

Numerical-analytical simulation of wave fields for complex subsurface geometries and structures

In this paper, we propose an analytical method of modeling seismic wave fields over a wide range of geophysical media: elastic, inelastic, anisotropic, anisotropic-inelastic, porous, random-inhomogeneous, etc., at very large distances. Since no finite-difference approximations are used, no grid dispersion occurs in computing wave fields for arbitrary media models and observation points. An analytical solution representation in the spectral domain makes it possible to analyze the wave field by parts, specifically, to obtain primary waves. A program of computing the wave fields has been developed, and a simulation of water waves and seismic “ringing” of the Moon has been carried out. The phenomenon of a monotonic shift of the resonance to the lower frequency area with increasing distance of recording is explained. This phenomenon was detected in some experiments with a seismic vibrator.     

Propagation of Seismic Waves in Block Elastic-Fluid Media. III

The propagation of seismic waves in block two- and three-dimensional media is investigated. These media are composed of identical cells in which there are several fluid blocks and one elastic block. For these media, effective models, which are anisotropic fluids, are established. Formulas for the velocities of propagation in these fluids are derived and investigated. A special investigation is carried out in the cases where the elastic block occupies almost the entire cell or where the relative volume of the elastic block is very small. Bibliography: 9 titles. Dedicated to P. V. Krauklis on the occasion of his seventieth birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 147–160.