3D Anisotropic Media Research Articles

A New 3D Marine Controlled-Source Electromagnetic Modeling Algorithm Based on Two New Types of Quadratic Edge Elements

This study proposes a new algorithm for a higher-order vector finite element method based on two new types of second-order edge elements to solve the electromagnetic field diffusion problem in a 3D anisotropic medium. To avoid source singularity in the quasistatic variant of Maxwell’s function, a secondary field formulation was adopted. The modeling domain was discretized using two types of quadratic edge hexahedral elements, which were obtained using the edge unification method to reduce variables on each side of two conventional quadratic edge elements. Compared with the traditional quadratic element, the number of unknowns that needed to be solved was significantly reduced. The sparse linear equation of the finite element system was solved using an open-source direct solver called MUMPS. The numerical results demonstrated that the proposed algorithm has the same level of accuracy as the conventional vector finite element method and has a significant advantage over it in terms of computational cost.

FDwave3D: a MATLAB solver for the 3D anisotropic wave equation using the finite-difference method

Seismic modeling plays an important role in geophysics and seismology for estimating the response of seismic sources in a given medium. In this work, we present a MATLAB-based package, FDwave3D, for synthetic wavefield and seismogram modeling in 3D anisotropic media. The seismic simulation is carried out using the finite-difference method over the staggered grid, and it is applicable to both active and passive surveys. The code package allows the incorporation of arbitrary source mechanisms and offers spatial derivative operators of accuracy up to tenth-order along with different types of boundary conditions. First, the methodological aspects of finite-difference method are briefly introduced. Then, the code has been tested and verified against the analytical solutions obtained for the homogeneous model. Further, the numerical examples of layered and overthrust models are presented to demonstrate its reliability.

A discontinuous Galerkin fast-sweeping eikonal solver for fast and accurate traveltime computation in 3D tilted anisotropic media

We tackle the challenging problem of efficient and accurate seismic traveltime computation in 3D anisotropic media by applying the fast-sweeping method to a discontinuous Galerkin (DG)-based eikonal solver. Using this method leads to a stable and highly accurate scheme, which is faster than finite-difference schemes for a given precision, and with a low computational cost compared to the standard Runge-Kutta DG formulation. The integral formulation of the DG method also makes it easy to handle seismic anisotropy and complex topographies. Several numerical tests on complex models, such as the 3D SEG advanced modeling model, are given as illustration, highlighting the efficiency and the accuracy of this new approach. In the near future, these results will be used together with accurate solvers for seismic amplitude and take-off angle computation to revisit asymptotic inversion (traveltime/slope tomography) and imaging approaches (quantitative migration involving amplitudes and angles).

3D CSAMT modelling in anisotropic media using edge-based finite-element method

ABSTRACTControlled-source audio-frequency magnetotellurics (CSAMT) is an important geophysical technique. Numerous studies have shown that the electrical anisotropy in the Earth cannot be ignored because it would probably lead to misinterpretations of electromagnetic data. It is necessary and meaningful to study CSAMT responses in three-dimensional (3D) electrical anisotropic media; therefore, we have developed an edge-based finite element method for 3D CSAMT forward modelling in generalised anisotropic media. The total electric field in this approach is decomposed into a primary electric field and a secondary electric field, and the Galerkin weighted residuals method is adopted to obtain the variational equation. The accuracy of this algorithm was initially validated by comparing solutions with those obtained in previous work on a 3D arbitrary anisotropic model. We then studied the responses of an oblique source as well as equatorial and axial configurations and the tensor source for anomalies with different Euler’s angles. Several meaningful conclusions can be derived from this work and a synthetic model is developed; the results also confirm the validity of previous conclusions. This study shows that a tensor source is necessary for a CSAMT survey in 3D anisotropic media, and the anisotropic parameters of anomalies have complex and significant influences on CSAMT responses.

Forward modeling and inversion of tensor CSAMT in 3D anisotropic media

Tensor controlled-source audio-frequency magnetotellurics (CSAMT) can yield information about electric and magnetic fields owing to its multi-transmitter configuration compared with the common scalar CSAMT. The most current theories, numerical simulations, and inversion of tensor CSAMT are based on far-field measurements and the assumption that underground media have isotropic resistivity. We adopt a three-dimensional (3D) staggered-grid finite difference numerical simulation method to analyze the resistivity in axial anisotropic and isotropic media. We further adopt the limited-memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) method to perform 3D tensor CSAMT axial anisotropic inversion. The inversion results suggest that when the underground structure is anisotropic, the isotropic inversion will introduce errors to the interpretation.

A novel inverse method for identification of 3D thermal conductivity coefficients of anisotropic media by the boundary element analysis

This paper presents a new numerical inverse method for identifying the six components of the thermal conductivity tensor of a 3D anisotropic medium with an arbitrary shape. The unknowns are inversely calculated using heat conduction problems with extra information at some boundary sampling points. For better stability, the inverse method uses data supplied from more than one steady-state heat conduction problem. Since all sampling points are taken to be on boundary surfaces for the sake of easy access, the boundary element method (BEM) for direct calculation is employed for the sensitivity analyses. The off-diagonal components of the thermal conductivity tensor must satisfy three nonlinear inequality constraints, which make the inverse analysis even more challenging. To overcome this difficulty, the inverse problem is formulated in terms of the principal thermal conductivities along with the rotation angles of principal axes, by which the three constraints disappear. For the inverse analysis, the damped Gauss–Newton method is adopted for the optimization process. In the end, numerical examples are presented, showing that the proposed method can yield reliable solutions even in cases with relatively large measurement error and with initial guesses far from the exact solution.

A Novel Analysis Procedure for Estimating Thickness-Independent Horizontal and Vertical Permeabilities From Pressure Data at an Observation Probe Acquired by Packer-Probe Wireline Formation Testers

Summary This paper presents a new spherical-flow cubic-analysis procedure for estimating horizontal and vertical permeabilities solely from pressure-transient data acquired at an observation probe of a dual-packer-probe wireline formation tester (WFT). The procedure applies for all inclination angles of the wellbore in a single-layer, 3D anisotropic porous medium. It does not require one to know the formation thickness or to obtain radial flow. It is shown that the procedure provides unique estimates of horizontal and vertical permeabilities from observation-probe pressure data obtained along both vertical and horizontal wellbores. However, for slanted-well cases, the analysis procedure yields two possible solutions for the horizontal and vertical permeabilities. Therefore, one must use a priori information on permeabilities, from sources such as core data or pretests, to help identify the appropriate solution for the slanted-well cases. For the more general case of a 3D anisotropic medium, except for the vertical wellbores, an estimate of the anisotropy ratio and the azimuth angle of the well are also required as inputs for determining the values of horizontal and vertical permeability with our analysis procedure. We illustrate the applicability of the proposed analysis procedure by considering two synthetic and two field packer-probe data sets from vertical and slanted wellbores.

SGBEM–FEM coupling for analysis of cracks in 3D anisotropic media

AbstractA weakly singular symmetric Galerkin boundary element method (SGBEM) is coupled with the standard finite element method (FEM) in order to establish an accurate and efficient numerical technique for analysis of fractures in three‐dimensional, anisotropic, linearly elastic media. In the strategy, the weakly singular SGBEM developed by Rungamornrat and Mear (textitInt. J. Solids Struct. 2008; 45:1283–1301; Comput. Methods Appl. Mech. Engrg 2008; 197:4319–4332) is utilized to model a small‐scale region containing the crack while the (possibly large and complex) compliment region is treated by the FEM. The coupled technique exploits the positive features of both methods; the SGBEM proves to be a convenient and highly accurate method for obtaining mixed‐mode stress intensity factors along the crack front, whereas the FEM is very efficient for modeling large‐scale problems in the absence of cracks. An important aspect of the formulation and implementation of the technique is that continuity of displacement and traction across the interface between the SGBEM and FEM regions is enforced in a weak sense. This allows the two regions (one modeled by the SGBEM and the other by the FEM) to be discretized independently without the need for the resulting meshes to conform on the interface separating the regions, and this flexibility in the discretization process leads to a significant reduction in the modeling effort. To demonstrate the utility and accuracy of the technique, several boundary value problems involving both embedded and surface breaking cracks are treated, and it is shown that the coupled technique yields highly accurate stress intensity factors that exhibit only a slight dependence upon mesh refinement. Copyright © 2010 John Wiley & Sons, Ltd.

A Split-Step Algorithm for Effectively Suppressing the Numerical Dispersion for 3D Seismic Propagation Modeling

In this article, we propose a 3D split-step algorithm (SSA) for seismic- wave simulation. We first transform the wave equations in 3D anisotropic media into a system of first-order partial differential equations with respect to time t. Then we use the multidimensional high-order interpolation method to approximate the high-order spatial derivatives, so that we obtain a system of semidiscrete ordinary differential equations (ODEs). Finally, the third-order implicit Adams method and truncated dif- ferentiator series method are applied to solve the semidiscrete ODEs. We provide the theoretical study on the properties of the 3D SSA, such as stability criteria, theoretical error, numerical error, numerical dispersion, and computational efficiency. We also compare some seismic modeling results of this method against those of some high-order finite-difference schemes. Theoretical analysis and numerical tests show that the 3D SSA is third-order accurate in time and fourth-order accurate in space. However, its computational costs and memory requirements are much less than those of the fourth-order Lax-Wendroff correction method and the fourth-order staggered- grid method. Using a multilayer elastic model with large velocity contrasts and free surface, we compare the result of the 3D SSA with that of the discrete-wavenumber method. We also present the synthetic seismograms in the 3D three-layer isotropic medium, the wave-field snapshots in the 3D two-layer medium, and the 3D trans- versely isotropic medium with a vertical symmetry axis. All these promising numer- ical results illustrate that the 3D SSA can suppress effectively the numerical dispersion caused by discretizing the wave equations when too few sampling points per mini- mum wavelength are used or when models have large velocity contrasts between adjacent layers, further resulting in both increasing the computational efficiency and saving the storage space when big grids are used.

3D plane-wave migration in tilted coordinates: A field data example

We have extended isotropic plane-wave migration in tilted coordinates to 3D anisotropic media and applied it on a Gulf of Mexico data set. Recorded surface data are transformed to plane-wave data by slant-stack processing in inline and crossline directions. The source plane wave and its corresponding slant-stacked data are extrapolated into the subsurface within a tilted coordinate system whose direction depends on the propagation direction of the plane wave. Images are generated by crosscorrelating these two wavefields. The shot sampling is sparse in the crossline direction, and the source generated by slant stacking is not really a plane-wave source but a phase-encoded source. We have discovered that phase-encoded source migration in tilted coordinates can image steep reflectors, using 2D synthetic data set examples. The field data example shows that 3D plane-wave migration in tilted coordinates can image steeply dipping salt flanks and faults, even though the one-way wave-equation operator is used for wavefield extrapolation.

Fast solution of the elasticity problem for a planar crack of arbitrary shape in 3D-anisotropic medium

A planar crack of arbitrary shape in a 3D-anisotropic elastic medium subjected to an arbitrary external stress field is considered. An efficient numerical method of the solution of the problem is proposed. The problem is reduced to an integral equation for the crack opening vector on the crack surface. For discretization of this equation, Gaussian (radial) approximation functions centered at a system of nodes that covers the crack surface are used. For such functions, the elements of the matrix of the discretized problem are calculated in a quasi analytical form that involves standard non-singular integrals. If the node grid is regular, the matrix of the discretized system has Teoplitz’s structure, and the Fast Fourier Transform algorithm may be used for the calculation of matrix-vector products with such a matrix. It accelerate substantially the process of the iterative solution of the discretized system. Examples of the solutions for a circular crack in a transversally isotropic elastic medium are presented.

A weakly-singular SGBEM for analysis of cracks in 3D anisotropic media

A weakly-singular, symmetric Galerkin boundary element method (SGBEM) is developed for analysis of fractures in three-dimensional, anisotropic linearly elastic media. The method constitutes a generalization of that by Li et al. [S. Li, M.E. Mear, L. Xiao, Symmetric weak-form integral equation method for three-dimensional fracture analysis, Comput. Methods Appl. Mech. Engrg. 151 (1998) 435–459] for isotropic media, and is based upon a pair of weak-form displacement and traction integral equations recently established by Rungamornrat and Mear [J. Rungamornrat, M.E. Mear, Weakly-singular, weak-form integral equations for cracks in three-dimensional anisotropic media, Int. J. Solids Struct. 45 (2008) 1283–1301]. The formulation involves only weakly-singular kernels and, as a consequence, standard C 0 elements can be employed in the numerical discretization. The kernels possess a relatively simple form (for general anisotropy) which involves an equatorial line integral like that associated with the displacement fundamental solution. Despite their simple form, it is still necessary to evaluate these kernels efficiently in the numerical implementation; to do so, certain symmetry properties associated with the specific material type under consideration are exploited, and a simple yet accurate interpolation strategy is introduced. Another important aspect of the numerical treatment is the use of a special crack-tip element which allows highly accurate mixed-mode stress intensity factor data to be extracted as a function of position along the crack front. This crack-tip element was originally introduced by Li et al. [S. Li, M.E. Mear, L. Xiao, Symmetric weak-form integral equation method for three-dimensional fracture analysis, Comput. Methods Appl. Mech. Engrg. 151 (1998) 435–459] in their treatment of isotropic media, and here it is extended to allow evaluation of the stress intensity factors for generally anisotropic media. The element involves “extra” degrees of freedom associated with the nodes along the crack front (with these quantities being directly related to the stress intensity factors), and it is capable of representing the asymptotic behavior in the vicinity of the crack front to a sufficiently high order that relatively large crack-tip elements can be utilized. Various examples are treated for cracks in an unbounded domain and for both embedded and surface-breaking cracks in a finite domain, and it is demonstrated that very accurate results can be obtained even with relatively coarse meshes.

Optimally Accurate Nearly Analytic Discrete Scheme for Wave-Field Simulation in 3D Anisotropic Media

We present a new numerical method for elastic wave modeling in 3D isotropic and anisotropic media, which is called the 3D optimal nearly analytic dis- crete method (ONADM) in this article. This work is an extension of the 2D ONADM (Yang et al., 2004) that models acoustic and elastic waves propagating in 2D media. The formulation is derived by using a multivariable truncated Taylor series expansion and high-order interpolation approximations. Our 3D ONADM enables wave propa- gation to be simulated in three dimensions through isotropic and anisotropic models. Promising numerical results show that the error of the ONADM for the 3D case is less than those of the conventional finite-difference (FD) method and the so-called Lax-Wendroff correction (LWC) schemes, measured quantitatively by the root- mean-square deviation from analytical solution. The seismic wave fields in the 3D isotropic and anisotropic media are simulated and compared with those obtained by using the fourth-order LWC method and exact solutions for the acoustic wave case. We show that, compared with the conventional FD method and the LWC schemes, the ONADM for the 3D case can reduce significantly the storage space and compu- tational costs. Numerical experiments illustrate that the ONADM provides a useful tool for the 3D large-scale isotropic and anisotropic problems and it can suppress effectively numerical dispersions caused by discretizing the 3D wave equations when too-coarse grids are used, which is the same as the 2D ONADM. Numerical modeling also implies that simultaneously using both the wave displacement and its gradients to approximate the high-order derivatives is important for both decreasing the nu- merical dispersion caused by the discretization of wave equations and compensating the important wave-field information included in wave-displacement gradients.

Green's function in plane anisotropic bimaterials with imperfect interface

The fundamental solutions or Green's functions for 2D or 3D anisotropic media with imperfect interface remain a challenging problem. In this paper, a general method is presented for the rigorous solution for the 2D Green's function in an anisotropic elastic bimaterial subject to a line force or a line dislocation. Most significant is the fact that the bonding along the bimaterial interface is considered to be homogeneous imperfect. Specifically, the tractions are continuous but the displacements are discontinuous and proportional, in terms of interface stiffness parameters, to their respective traction components. Using complex variable techniques, the basic boundary-value problem for two analytic vector functions is reduced to a coupled linear first-order differential equation for a single analytic vector function defined in the lower half space. The coupled linear differential equation for the single analytic vector function can be subsequently decoupled into three independent linear first-order differential equations for three newly defined analytic functions. Closed-form solutions for the 2D Green's function are derived in terms of the exponential integral. Unlike previous works which involve some sort of inverse transform method to obtain the physical quantities from the transform domain, the key feature of the present method is that the physical quantities can be readily calculated without the need to perform any inverse transform operations.

An approach to computing the dispersion of wavespeed for the most general 3D anisotropic media

In this paper,we develop a method to calculate the dispersion of seismic wave speed (phase velocity and group velocity) for a general anisotropic medium, which is defined by twenty-one elastic moduli. The solution includes,as special cases, the isotropic and transversely isotropic problems. We apply the plane-wave analysis to the general 3D anisotropic medium and obtain explicit expressions for three eigenvalues (phase velocities) and their corresponding group velocities, which are the propagation speeds of the wavefronts and the energy fluxes (ray-paths) of one qP wave and two qS waves. Basing on the solutions,we show that the phase and group-velocity vectors generally have different directions and they depend on twenty-one elastic moduli and the direction cosines of the incident wave. As examples of using the eigenvalue solutions, we numerically calculate the phase velocities and the group velocities for an isotopic medium, a VTI-medium and a qTI-medium. Two real models (clay shale and phenolic) were used for moduli selection. These results clearly show that the wave speeds vary with the azimuthal angle and the vertical angle of the incident wave, as well as the elastic moduli. This means that the solutions may be applied to investigation of kinematic features of real samples of rocks and the sensitivity of the wavespeed to each elastic modulus. We also show the application of the eigenvalue solutions to the 2D/3D ray tracing in the most general anisotropic media.