A new computational model is proposed for propagation of low-amplitude seismic waves through a pre-stressed anisotropic medium. The model provides clues to kinematics and dynamics of these waves forming under the effect of initial stress. The governing differential equations are formulated in terms of velocity, stress tensor and small rotation of an element of the medium. The equations are solved numerically by means of the rotated staggered-grid method, and the approach is tested in numerical examples.
The new computational model for the seismic wave propagation in the initially prestressed media is proposed, the governing equations of which are written in terms of velocities, stress tensor and small rotations of element of the medium. The properties of wavefields in the prestressed medium are studied and some examples showing anisotropy of prestressed state are discussed. The staggered grid numerical method is developed for solving the governing equations of the model and numerical example is presented.
A basis for the development of numerical framework to modelling seismic wave propagation through prestressed zones is proposed. The governing equations for elastic waves in prestressed media written in stress-velocity formulation supplemented by the small rotation tensor equation are presented. The simplified version of equations is considered, assuming that the initial strain of prestressed state is small. The stress-induced anisotropy of unidirectionally compressed or stretched medium is studied. The rotated staggered grid technique is modified for solving formulated equations and the problem of interaction of the prestressed layer in a homogeneous half plane with elastic waves is solved as a two-dimensional numerical test problem. It is shown that the influence of this layer on the wave field can be significant. The existence of prestressed zones must be taken into account for forward modelling and inversion of seismic waves, especially in seismic travel-time tomography.
The paper presents an original approach to numerical simulation of sonic logging for tilted transversely isotropic media with attenuation. We suggest a new way to introduce attenuation to the stiffness tensor components for anisotropic media, so that the quality factors of the elastic waves are of a given quantity. The numerical approach itself is based on the Lebedev finite difference scheme on staggered grid applied to cylindrical coordinate system which is more efficient than well-known rotated staggered grid scheme. The numerical experiments, presented in the paper, were done to illustrate an impact of anisotropy and attenuation on a sonic logging, for example appearance of azimuthal component of the displacement velocity even in case of volumetric source acting at the axis of a fluid-filled borehole.
The paper presents an original approach to numerical simulation of sonic logging for tilted transversely isotropic media with attenuation. We suggest a new way to introduce attenuation to the stiffness tensor components for anisotropic media, so that the quality factors of the elastic waves are of a given quantity. The numerical approach itself is based on the Lebedev finite difference scheme on staggered grid applied to cylindrical coordinate system which is more efficient than well-known rotated staggered grid scheme. The numerical experiments, presented in the paper, were done to illustrate an impact of anisotropy and attenuation on a sonic logging, for example appearance of azimuthal component of the displacement velocity even in case of volumetric source acting at the axis of a fluid-filled borehole.
This paper presents Lebedev scheme (LS) for simulation of waves' propagation in arbitrary anisotropic elastic media. This is second order explicit finite-difference scheme on staggered grids and is natural generalization of well known Virieux one used for
On the base of the application of a finite-difference approximation of an initial boundary value problem for elastic wave equations (velocity/stress formulation), numerical method and its algorithmic implementation have been developed in order to perform a computer simulation of sonic logging. The very general statement is dealt with – surrounding medium allowed to be 3D vertical transversely isotropic heterogeneous with attenuation and source can be located at any point inside or (VTI) outside the well. To provide the most precise description of the sharpest interface of the problem – the interface of the well, we formulate the problem in cylindrical coordinates with axis directed along the well. In order to truncate area of computations two approaches are used: classical version of Perfectly Matched Layer (PML) and extension on the base of optimal grid. Implementation of parallel computations is done via Domain Decomposition Data exchange between Processor Units is performed with the help of Message Passing Interface library. Results of numerical experiments for VTI media are presented and discussed.
Linear model for elastic waves propagation through prestressed media based on the general theory of finite deformations was proposed,. The governing equations in terms of velocities, stress and small rotations are formulated in the form of the first order partial differential equathions system. Presented a kinematic characteristic of the model and numerical simulathion algorithm.