Diffusion of electromagnetic fields into a two‐dimensional earth: A finite‐difference approach

Geophysics ◽  
1984 ◽  
Vol 49 (7) ◽  
pp. 870-894 ◽  
Author(s):  
M. L. Oristaglio ◽  
G. W. Hohmann
Keyword(s):  
Finite Difference ◽  
Half Space ◽  
Small Time ◽  
Two Dimensional ◽  
Time Step ◽  
Small Peak ◽  
Transient Electromagnetic ◽  
The Earth ◽  
Air Interface ◽  
Equation Boundary

We describe a numerical method for time‐stepping Maxwell’s equations in the two‐dimensional (2-D) TE‐mode, which in a conductive earth reduces to the diffusion equation. The method is based on the classical DuFort‐Frankel finite‐difference scheme, which is both explicit and stable for any size of the time step. With this method, small time steps can be used at early times to track the rapid variations of the field, and large steps can be used at late times, when the field becomes smooth and its rates of diffusion and decay slow down. The boundary condition at the earth‐air interface is handled explicitly by calculating the field in the air from its values at the earth’s surface with an upward continuation based on Laplace’s equation. Boundary conditions in the earth are imposed by using a large, graded grid and setting the values at the sides and bottom to those for a haft‐space. We use the 2-D model to simulate transient electromagnetic (TE) surveys over a thin vertical conductor embedded in a half‐space and in a half‐space with overburden. At early times (microseconds), the patterns of diffusion in the earth are controlled mainly by geometric features of the models and show a great deal of complexity. But at late times, the current concentrates at the center of the thin conductor and, with a large contrast (1000:1) between conductor and half‐space, produces the characteristic crossover and peaked anomalies in the surface profiles of the vertical and horizontal emfs. With a smaller contrast (100:1), however, the crossover in the vertical emf is obscured by the halfspace response, although the horizontal emf still shows a small peak directly above the target.

ADI-FDTD Method for Two-Dimensional Transient Electromagnetic Problems

2016 ◽  
Vol 19 (1) ◽  
pp. 94-123 ◽  
Author(s):  
Wanshan Li ◽  
Yile Zhang ◽  
Yau Shu Wong ◽  
Dong Liang
Keyword(s):  
Fdtd Method ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Electromagnetic Problems ◽  
Transient Electromagnetic ◽  
Te Mode ◽  
The Earth ◽  
Air Interface ◽  
Alternating Direction ◽  
Difference Time

AbstractAn efficient and accurate numerical scheme is proposed for solving the transverse electric (TE) mode electromagnetic (EM) propagation problem in two-dimensional earth. The scheme is based on the alternating direction finite-difference time-domain (ADI-FDTD) method. Unlike the conventional upward continuation approach for the earth-air interface, an integral formulation for the interface boundary is developed and it is effectively incorporated to the ADI solver. Stability and convergence analysis together with an error estimate are presented. Numerical simulations are carried out to validate the proposed method, and the advantage of the present method over the popular Du-Fort-Frankel scheme is clearly demonstrated. Examples of the electromagnetic field propagation in the ground with anomaly further verify the effectiveness of the proposed scheme.

A finite‐element solution for the transient electromagnetic response of an arbitrary two‐dimensional resistivity distribution

Geophysics ◽  
1986 ◽  
Vol 51 (7) ◽  
pp. 1450-1461 ◽  
Author(s):  
Y. Goldman ◽  
C. Hubans ◽  
S. Nicoletis ◽  
S. Spitz
Keyword(s):  
Finite Element ◽  
Sparse Matrix ◽  
Equation System ◽  
Electromagnetic Response ◽  
Implicit Time ◽  
Two Dimensional ◽  
Time Step ◽  
Transient Electromagnetic ◽  
Exact Boundary ◽  
The Time Domain

We present a numerical method for solving Maxwell’s equations in the case of an arbitrary two‐dimensional resistivity distribution excited by an infinite current line. The electric field is computed directly in the time domain. The computations are carried out in the lower half‐space only because exact boundary conditions are used on the free surface. The algorithm follows the finite‐element approach, which leads (after space discretization) to an equation system with a sparse matrix. Time stepping is done with an implicit time scheme. At each time step, the solution of the equation system is provided by the fast system ICCG(0). The resulting algorithm produces good results even when large resistivity contrasts are involved. We present a test of the algorithm’s performance in the case of a homogeneous earth. With a reasonable grid, the relative error with respect to the analytical solution does not exceed 1 percent, even 2 s after the source is turned off.

Some Practical Considerations in the Numerical Solution of Two-Dimensional Reservoir Problems

1968 ◽  
Vol 8 (02) ◽  
pp. 185-194 ◽  
Author(s):  
J.E. Briggs ◽  
T.N. Dixon
Keyword(s):  
Finite Difference ◽  
High Speed ◽  
Three Dimensional ◽  
Simultaneous Equations ◽  
Two Dimensional ◽  
Time Step ◽  
Finite Difference Approximations ◽  
Large Sets ◽  
Adi Methods ◽  
Three Dimensional Models

Abstract A study was made of numerical techniques for solving the large sets of simultaneous equations that arise in the mathematical modeling of oil reservoir behavior. It was found that noniterative techniques, such as the Alternating Direction Implicit (ADI) method, as well as some other finite difference approximations, produce oscillatory or unsmooth results for large time steps. Estimates of time step sizes sufficient to avoid such behavior are given. A comparison was made of the Point Successive Over-Relaxation (PSOR), Two-Line Cyclic Chebyshev Semi-Iterative SOR (2LCC), and iterative ADI methods, with respect to speed of solution of a test problem. It was found that, when applicable, iterative ADI is fastest for problems involving many points, while 2LCC is preferable for smaller problems. Introduction With the advent of high speed, large memory, digital computers, there has been an increasing emphasis on the development of improved methods for simulating and predicting reservoir performance. Two-dimensional, three-phase reservoir models with various combinations of PVT effects, as well as gravity and capillary forces, are common throughout the industry. Such models are also available through consulting firms, to anyone desiring to use them. Three-dimensional models will probably be practical in only a few years. We conducted a study of some of the numerical methods used for solving the large sets of simultaneous equations that arise in such models. A typical set of equations for a reservoir model is shown below: ..............(1) .............(2) .............(3) ..............(4) where a = 5.615 cu ft/bbl. In addition to Eqs. 1 through 4, one also would have to specify the conditions at the boundaries of the reservoir or aquifer being studied. Equations such as these are normally approximated by finite difference techniques and solved numerically because of their complexity. In deciding how to solve such equations, a number of decisions must be made. It is not our intention to cover all facets of the problem, but rather to concentrate on one of the important aspects, such as solving Eq. 1. SPEJ P. 185ˆ

The influence of a conductive host on two‐dimensional borehole transient electromagnetic responses

Geophysics ◽  
1984 ◽  
Vol 49 (7) ◽  
pp. 861-869 ◽  
Author(s):  
Perry A. Eaton ◽  
Gerald W. Hohmann
Keyword(s):  
Electric Field ◽  
Half Space ◽  
Time Window ◽  
Time Derivative ◽  
Finite Width ◽  
Late Time ◽  
Time Behavior ◽  
Two Dimensional ◽  
Peak Response ◽  
Transient Electromagnetic

We have computed transient borehole electromagnetic (EM) responses of two‐dimensional (2-D) models using a direct and explicit finite‐difference algorithm. The program computes the secondary electric field which is defined as the difference between the total field and the primary (half‐space) field. The time derivative of the vertical magnetic field in a borehole is computed by numerical differentiation of the total electric field. These models consist of a thin horizontal conductor with a finite width, embedded in a conductive half‐space. Dual line sources energized by a step‐function current lie on the surface of the half‐space and simulate the long sides of a large rectangular loop. Numerical results substantiate several important features of the transient impulse response of such models. The peak response of the target is attenuated as the resistivity of the host decreases. A sign reversal in the secondary electric field occurs later in time as the resistivity of the host decreases. The peak response and the onset of late‐time behavior are delayed in time as well. Secondary responses for models with different host resistivities (10–1000 Ω-m) are approximately the same at late time. If the target is less conductive, the effects of the host, i.e., the attenuation and time delay, are less. It is readily apparent that there exists a time window within which the target’s response is at a maximum relative to the half‐space response. At late time the shape of the borehole anomaly due to a thin conductive 2-D target appears to be independent of the conductivity of the host. The late‐time secondary decay of the target is neither exponential nor power law, and a time constant computed from the slope of a log‐linear decay curve at late time may be much larger than the actual value for the same target in free space.

scholarly journals 2D TESTING OF A FINITE DIFFERENCE SCHEME FOR MODELLING OF A VISCOUS DIFFUSION PROCESS IN COMPRESSIBLE GAS

2020 ◽  
Vol 22 (5) ◽  
pp. 128-131
Author(s):  
V.V. Nikonov ◽  
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Initial Conditions ◽  
Stokes Problem ◽  
Velocity Fields ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Time Step ◽  
Acceptable Accuracy ◽  
Compressible Gas

Viscous subproblem of direct numerical simulation of compressible gas is solved. This subproblem is tested on the two-dimensional problem of impulse start of a flat plate (Stokes’ problem). Three calculations were made with the different initial conditions and velocity fields were obtained. The numerical results are compared with the solution of Stokes’ problem. Analyzing the results, we can conclude that in order to achieve acceptable accuracy, it suffices to choose a time step according to the rule that the author formulated in his earlier works.

A Three-Dimensional Version of the Free Surface Capturing Discretization for Staggered Grid Finite Difference Schemes: Implementation into StagYY

2021 ◽  
Author(s):  
Paul Tackley
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Three Dimensional ◽  
Small Time ◽  
Staggered Grid ◽  
Finite Difference Schemes ◽  
Finite Volume Scheme ◽  
Two Dimensional ◽  
Simulation Code ◽  
Free Surface Capturing

<p>In order to treat a free surface in models of lithosphere and mantle dynamics that use a fixed Eulerian grid it is typical to use "sticky air", a layer of low-viscosity, low-density material above the solid surface (e.g. Crameri et al., 2012). This can, however, cause numerical problems, including poor solver convergence due to the huge viscosity jump and small time-steps due to high velocities in the air. Additionally, it is not completely realistic because the assumed viscosity of the air layer is typically similar to that of rock in the asthenosphere so the surface is not stress free.  </p><p>In order to overcome these problems, Duretz et al. (2016) introduced and tested a method for treating the free surface that instead detects and applies special conditions at the free surface. This avoids the huge viscosity jump and having to solve for velocities in the air. They applied it to a two-dimensional staggered grid finite difference / finite volume scheme, a discretization that is in common use for modelling mantle and lithosphere dynamics. Here I document the application of this approach to a three-dimensional spherical staggered grid solver in the mantle simulation code StagYY. Some adjustments had to be made to the two-dimensional scheme documented in Duretz et al. (2016) in order to avoid problems due to undefined velocities for certain boundary topographies. The approach was applied not only to the Stokes solver but also to the temperature solver, including the implementation of a mixed radiative/conductive boundary condition applicable to surface magma oceans/lakes.</p><p><strong>References</strong></p><p>Crameri, F., H. Schmeling, G. J. Golabek, T. Duretz, R. Orendt, S. J. H. Buiter, D. A. May, B. J. P. Kaus, T. V. Gerya, and P. J. Tackley (2012), A comparison of numerical surface topography calculations in geodynamic modelling: an evaluation of the ‘sticky air’ method, Geophysical Journal International,189(1), 38-54, doi:10.1111/j.1365-246X.2012.05388.x.</p><p>Duretz, T., D. A. May, and P. Yamato (2016), A free surface capturing discretization for the staggered grid finite difference scheme, Geophysical Journal International, 204(3), 1518-1530, doi:10.1093/gji/ggv526.</p>

Two‐dimensional transient electromagnetic responses

Geophysics ◽  
1985 ◽  
Vol 50 (12) ◽  
pp. 2849-2861 ◽  
Author(s):  
Jopie I. Adhidjaja ◽  
Gerald W. Hohmann ◽  
Michael L. Oristaglio
Keyword(s):  
Magnetic Field ◽  
Half Space ◽  
Time Domain ◽  
The Body ◽  
Two Dimensional ◽  
Total Field ◽  
Secondary Field ◽  
Field Solution ◽  
Transient Electromagnetic ◽  
Simple Boundary

The time‐domain electromagnetic (TEM) modeling method of Oristaglio and Hohmann is reformulated here in terms of the secondary field. This finite‐difference method gives a direct, explicit time‐domain solution for a two‐dimensional body in a conductive earth by advancing the field in time with DuFort‐Frankel time‐differencing. As a result, solving for the secondary field, defined as the difference between the total field and field of a half‐space, is not only more efficient but is also simpler and eliminates several problems inherent in the solution for the total field. For example, because the secondary field varies slowly both in space and time, it can be modeled on a coarse grid with large time steps. In addition, for a simple body the field is local; therefore, because the field can be assumed to satisfy a simple boundary condition in the earth computation is greatly simplified. Our tests show that for the same accuracy, the secondary‐field solution is roughly five times faster than the total‐field solution. We compute and analyze the magnetic field impulse response for a suite of models, most of which consist of a thin body embedded in a conductive half‐space—with or without overburden. The results indicate the conductive half‐space will both delay and attenuate the response of the body and even obscure it if the conductivity contrast is small. The results also suggest that the conductive host can alter the decay rate of the response of the body from its free‐space counterpart. Our results for multiple bodies illustrate the importance of early‐time measurements to obtain resolution, particularly for measurements of the horizontal magnetic field. The vertical magnetic field, however, can be used to infer the dip direction of a dipping body by studying the migration of the crossover. The results for models which include overburden show that the effect of a conductive overburden, in addition to the half‐space effect, is to delay the response of the body, because the primary current initially tends to concentrate and slowly diffuse through the overburden, and does not reach the body until later time. This effect also complicates the early‐times profiles, becoming more severe as the conductivity of the overburden is increased.

Asymptotic expansions for transient electromagnetic fields

Geophysics ◽  
1982 ◽  
Vol 47 (1) ◽  
pp. 38-46 ◽  
Author(s):  
T. Lee
Keyword(s):  
Half Space ◽  
Electromagnetic Fields ◽  
Asymptotic Expansions ◽  
Free Space ◽  
Layered Structures ◽  
Two Dimensional ◽  
Host Medium ◽  
Transient Electromagnetic ◽  
Axisymmetric Structure ◽  
Transient Voltage

Asymptotic expansions may be derived for transient electromagnetic (EM) fields. The expansions are valid when [Formula: see text] is less than about 0.1. Here l, σ, [Formula: see text], and t are the respective lengths, conductivities, permeabilities of free space and time. Cases for which asymptotic expansions are presented include (1) layered grounds, (2) axisymmetric structure, and (3) two‐dimensional (2-D) structures. In all cases the transient voltage eventually approaches that of the host medium alone, the ratio of anomalous response to the half‐space response being proportional to [Formula: see text]. Here v is equal to 0.5 for layered structures and 1.0 for 2-D or 3-D structures.

Theoretical seismograms for the two welded quarter-planes

1972 ◽  
Vol 62 (2) ◽  
pp. 619-630
Author(s):  
Dan Loewenthal ◽  
Z. Alterman
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Half Space ◽  
Line Source ◽  
Two Dimensional ◽  
Impulsive Force ◽  
Space Model ◽  
Finite Difference Technique

abstract The finite difference technique developed by Alterman and Rotenberg (1969) and Alterman and Loewenthal (1970) is applied to the problem of obtaining the motion of a two-dimensional half-space model, consisting of two homogeneous and isotropic welded quarter-spaces of different materials. Two source conditions are considered: an internal impulsive line source acting inside one of the quarter-planes, and an impulsive force acting perpendicular to the free surface.

scholarly journals A finite‐difference, time‐domain solution for three‐dimensional electromagnetic modeling

Geophysics ◽  
1993 ◽  
Vol 58 (6) ◽  
pp. 797-809 ◽  
Author(s):  
Tsili Wang ◽  
Gerald W. Hohmann
Keyword(s):  
Magnetic Field ◽  
Finite Difference ◽  
Execution Time ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Three Dimensional ◽  
Staggered Grid ◽  
Transient Electromagnetic ◽  
The Earth ◽  
The Magnetic Field

We have developed a finite‐difference solution for three‐dimensional (3-D) transient electromagnetic problems. The solution steps Maxwell’s equations in time using a staggered‐grid technique. The time‐stepping uses a modified version of the Du Fort‐Frankel method which is explicit and always stable. Both conductivity and magnetic permeability can be functions of space, and the model geometry can be arbitrarily complicated. The solution provides both electric and magnetic field responses throughout the earth. Because it solves the coupled, first‐order Maxwell’s equations, the solution avoids approximating spatial derivatives of physical properties, and thus overcomes many related numerical difficulties. Moreover, since the divergence‐free condition for the magnetic field is incorporated explicitly, the solution provides accurate results for the magnetic field at late times. An inhomogeneous Dirichlet boundary condition is imposed at the surface of the earth, while a homogeneous Dirichlet condition is employed along the subsurface boundaries. Numerical dispersion is alleviated by using an adaptive algorithm that uses a fourth‐order difference method at early times and a second‐order method at other times. Numerical checks against analytical, integral‐equation, and spectral differential‐difference solutions show that the solution provides accurate results. Execution time for a typical model is about 3.5 hours on an IBM 3090/600S computer for computing the field to 10 ms. That model contains [Formula: see text] grid points representing about three million unknowns and possesses one vertical plane of symmetry, with the smallest grid spacing at 10 m and the highest resistivity at 100 Ω ⋅ m. The execution time indicates that the solution is computer intensive, but it is valuable in providing much‐needed insight about TEM responses in complicated 3-D situations.

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