scholarly journals Electromagnetic induction in a fully 3‐D anisotropic earth

Geophysics ◽  
2002 ◽  
Vol 67 (4) ◽  
pp. 1104-1114 ◽  
Author(s):  
Chester J. Weiss ◽  
Gregory A. Newman
Keyword(s):  
Finite Difference ◽  
Electromagnetic Induction ◽  
Water Saturation ◽  
Research Problem ◽  
Analytic Solutions ◽  
Electrical Anisotropy ◽  
Staggered Grid ◽  
Ongoing Research ◽  
Em Induction ◽  
D Model

The bulk electrical anisotropy of sedimentary formations is a macroscopic phenomenon which can result from the presence of porosity variations, laminated shaly sands, and water saturation. Accounting for its effect on induction log responses is an ongoing research problem for the well‐logging community since these types of sedimentary structures have long been correlated with productive hydrocarbon reservoirs such as the Jurassic Norphlet Sandstone and Permian Rotliegendes Sandstone. Presented here is a staggered‐grid finite‐difference method for simulating electromagnetic (EM) induction in a fully 3‐D anisotropic medium. The electrical conductivity of the formation is represented as a full 3 × 3 tensor whose elements can vary arbitrarily with position throughout the formation. To demonstrate the validity of this approach, finite‐difference results are compared against analytic and quasi‐analytic solutions for tractable 1‐D and 3‐D model geometries. As a final example, we simulate 2C–40 induction tool responses in a crossbedded aeolian sandstone to illustrate the magnitude of the challenge faced by interpreters when electrical anisotropy is neglected.

Optimized staggered-grid finite-difference operators using window functions

2018 ◽  
Vol 15 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Ying-Jun Ren ◽  
Jian-Ping Huang ◽  
Peng Yong ◽  
Meng-Li Liu ◽  
Chao Cui ◽  
...  
Keyword(s):  
Finite Difference ◽  
Staggered Grid ◽  
Difference Operators ◽  
Window Functions

Fourth‐order finite‐difference P-SV seismograms

Geophysics ◽  
1988 ◽  
Vol 53 (11) ◽  
pp. 1425-1436 ◽  
Author(s):  
Alan R. Levander
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Fourth Order ◽  
Equations Of Motion ◽  
Dispersion Analysis ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Order System ◽  
Surface Boundary ◽  
Lamb’S Problem

I describe the properties of a fourth‐order accurate space, second‐order accurate time, two‐dimensional P-SV finite‐difference scheme based on the Madariaga‐Virieux staggered‐grid formulation. The numerical scheme is developed from the first‐order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga‐Virieux staggered‐grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic‐elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free‐surface or within a layer and to satisfy free‐surface boundary conditions. Benchmark comparisons of finite‐difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite‐difference and reflectivity solutions for elastic‐elastic and acoustic‐elastic layered models.

scholarly journals Improved convergence and stability properties in a three-dimensional higher-order ice sheet model

2011 ◽  
Vol 4 (3) ◽  
pp. 1569-1610
Author(s):  
J. J. Fürst ◽  
O. Rybak ◽  
H. Goelzer ◽  
B. De Smedt ◽  
P. de Groen ◽  
...  
Keyword(s):  
Velocity Field ◽  
Finite Difference ◽  
Three Dimensional ◽  
Ice Sheet ◽  
Higher Order ◽  
Velocity Fields ◽  
Force Balance ◽  
Staggered Grid ◽  
Comparison Results ◽  
Resultant Velocity

Abstract. We present a novel finite difference implementation of a three-dimensional higher-order ice sheet model that performs well both in terms of convergence rate and numerical stability. In order to achieve these benefits the discretisation of the governing force balance equation makes extensive use of information on staggered grid points. Using the same iterative solver, an existing discretisation that operates exclusively on the regular grid serves as a reference. Participation in the ISMIP-HOM benchmark indicates that both discretisations are capable of reproducing the higher-order model inter-comparison results. This allows a direct comparison not only of the resultant velocity fields but also of the solver's convergence behaviour which holds main differences. First and foremost, the new finite difference scheme facilitates convergence by a factor of up to 7 and 2.6 in average. In addition to this decrease in computational costs, the precision for the resultant velocity field can be chosen higher in the novel finite difference implementation. For high precisions, the old discretisation experiences difficulties to converge due to large variation in the velocity fields of consecutive Picard iterations. Finally, changing discretisation prevents build-up of local field irregularites that occasionally cause divergence of the solution for the reference discretisation. The improved behaviour makes the new discretisation more reliable for extensive application to real ice geometries. Higher precision and robust numerics are crucial in time dependent applications since numerical oscillations in the velocity field of subsequent time steps are attenuated and divergence of the solution is prevented. Transient applications also benefit from the increased computational efficiency.

scholarly journals A parallel finite‐difference approach for 3D transient electromagnetic modeling with galvanic sources

Geophysics ◽  
2004 ◽  
Vol 69 (5) ◽  
pp. 1192-1202 ◽  
Author(s):  
Michael Commer ◽  
Gregory Newman
Keyword(s):  
Magnetic Field ◽  
Finite Difference ◽  
Electric Field ◽  
Electric Fields ◽  
Time Derivative ◽  
Stable Solution ◽  
Parallel Computer ◽  
Staggered Grid ◽  
Electromagnetic Modeling ◽  
Transient Electromagnetic

A parallel finite‐difference algorithm for the solution of diffusive, three‐dimensional (3D) transient electromagnetic field simulations is presented. The purpose of the scheme is the simulation of both electric fields and the time derivative of magnetic fields generated by galvanic sources (grounded wires) over arbitrarily complicated distributions of conductivity and magnetic permeability. Using a staggered grid and a modified DuFort‐Frankel method, the scheme steps Maxwell's equations in time. Electric field initialization is done by a conjugate‐gradient solution of a 3D Poisson problem, as is common in 3D resistivity modeling. Instead of calculating the initial magnetic field directly, its time derivative and curl are employed in order to advance the electric field in time. A divergence‐free condition is enforced for both the magnetic‐field time derivative and the total conduction‐current density, providing accurate results at late times. In order to simulate large realistic earth models, the algorithm has been designed to run on parallel computer platforms. The upward continuation boundary condition for a stable solution in the infinitely resistive air layer involves a two‐dimensional parallel fast Fourier transform. Example simulations are compared with analytical, integral‐equation and spectral Lanczos decomposition solutions and demonstrate the accuracy of the scheme.

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