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.

The transient EM step response of a dipping plate in a conductive half‐space

Geophysics ◽  
1992 ◽  
Vol 57 (9) ◽  
pp. 1116-1126 ◽  
Author(s):  
James E. Hanneson
Keyword(s):  
Half Space ◽  
Free Space ◽  
Early Time ◽  
Step Response ◽  
Current Loop ◽  
Late Time ◽  
Electromagnetic System ◽  
University Of Toronto ◽  
Transient Electromagnetic ◽  
Induction Process

An algorithm for computing the transient electromagnetic (TEM) response of a dipping plate in a conductive half‐space has been developed. For a stationary [Formula: see text] current loop source, calculated profiles simulate the response of the University of Toronto electromagnetic system (UTEM) over a plate in a 1000 Ω ⋅ m half‐space. The objective is to add to knowledge of the galvanic process (causing poloidal plate currents) and the local induction process (causing toroidal currents) by studying host and plate currents with respect to surface profiles. Both processes can occur during TEM surveys. Plates are all [Formula: see text] thick with various depths, dips, and conductances. Calculated host and plate currents provide quantitative examples of several effects. For sufficiently conductive plates, the late time currents are toroidal as for a free‐space host. At earlier times, or at all times for poorly conducting plates, the plate currents are poloidal, and the transitions to toroidal currents, if they occur, are gradual. At very late times, poloidal currents again dominate any toroidal currents but this effect is rarely observed. Stripped, point‐normalized profiles, which reflect secondary fields caused by the anomalous plate currents, illustrate effects such as early time blanking (caused by noninstantaneous diffusion of fields into the target), mid‐time anomaly enhancement (caused by galvanic currents), and late time plate‐in‐free‐space asymptotic behavior.

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.

The effect of a conductive half‐space on the transient electromagnetic response of a three‐dimensional body

Geophysics ◽  
1985 ◽  
Vol 50 (7) ◽  
pp. 1144-1162 ◽  
Author(s):  
William A. SanFilipo ◽  
Perry A. Eaton ◽  
Gerald W. Hohmann
Keyword(s):  
Half Space ◽  
Free Space ◽  
Eddy Currents ◽  
Three Dimensional ◽  
Vertical Component ◽  
The Body ◽  
Electromagnetic Response ◽  
Loop Current ◽  
Transient Electromagnetic ◽  
Space Response

The transient electromagnetic (TEM) response of a three‐dimensional (3-D) prism in a conductive half‐space is not always approximated well by three‐dimensional free‐space or two‐dimensional (2-D) conductive host models. The 3-D conductive host model is characterized by a complex interaction between inductive and current channeling effects. We numerically computed 3-D TEM responses using a time‐domain integral‐equation solution. Models consist of a vertical or horizontal prismatic conductor in conductive half‐space, energized by a rapid linear turn‐off of current in a rectangular loop. Current channeling, characterized by currents that flow through the body, is produced by charges which accumulate on the surface of the 3-D body and results in response profiles that can be much different in amplitude and shape than the corresponding response for the same body in free space, even after subtracting the half‐space response. Responses characterized by inductive (vortex) currents circulating within the body are similar to the response of the body in free space after subtracting the half‐space contribution. The difference between responses dominated by either channeled or vortex currents is subtle for vertical bodies but dramatic for horizontal bodies. Changing the conductivity of the host effects the relative importance of current channeling, the velocity and rate of decay of the primary (half‐space) electric field, and the build‐up of eddy currents in the body. As host conductivity increases, current channeling enhances the amplitude of the response of a vertical body and broadens the anomaly along the profile. For a horizontal body the shape of the anomaly is distorted from the free‐space anomaly by current channeling and is highly sensitive to the resistivity of the host. In the latter case, a 2-D response is similar to the 3-D response only if current channeling effects dominate over inductive effects. For models that are not greatly elongated, TEM responses are more sensitive to the conductivity of the body than galvanic (dc) responses, which saturate at a moderate resistivity contrast. Multicomponent data are preferable to vertical component data because in some cases the presence and location of the target are more easily resolved in the horizontal response and because the horizontal half‐space response decays more quickly than does the corresponding vertical response.

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.

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.

Approximate calculations of the transient electromagnetic response from buried conductors in a conductive half‐space

Geophysics ◽  
1984 ◽  
Vol 49 (7) ◽  
pp. 918-924 ◽  
Author(s):  
J. D. McNeill ◽  
R. N. Edwards ◽  
G. M. Levy
Keyword(s):  
Half Space ◽  
Free Space ◽  
Target Location ◽  
Electromagnetic Coupling ◽  
Practical Interest ◽  
Electromagnetic Response ◽  
Galvanic Current ◽  
Transient Electromagnetic ◽  
Space Modeling ◽  
Space Response

The transient electromagnetic (TEM) response from a conductive plate buried in a conductive half‐space and energized by a large‐loop transmitter is investigated in a heuristic manner. The vortex and galvanic components are each calculated directly in the time domain using an approximate procedure which ignores the electromagnetic coupling present in the complete solution. In modeling the vortex and galvanic current flows, the plate is replaced with a single‐turn wire loop of appropriate parameters and a distribution of current dipoles, respectively. The results of calculations of the transient magnetic field at the surface of the earth are presented for a few selected cases of practical interest. The relative importance of the vortex and galvanic components varies with the half‐space resistivity. The vortex component dominates if the half‐space is resistive, in which case free‐space algorithms suffice for numerical modeling. Furthermore the measured responses give much useful information about the target, and large depths of exploration should be achieved. As the half‐space resistivity decreases, a significant half‐space response is observed, caused by currents induced in the half‐space itself. This response can be very large. Spatial variations in it caused by relatively small changes in resistivity, i.e., geologic noise, obscure the response from deep targets making them difficult to detect. The effect of the half‐space is also to delay, distort, and reduce the vortex component in comparison with the free‐space response. The behavior of the galvanic component is determined by the haft‐space current flow. The presence of this component explains the large enhancement of overall target response seen at early times over relatively resistive ground and the departure from an exponential decay seen over more conductive ground, again with respect to responses predicted by free‐space modeling. In more conductive ground the galvanic component completely dominates the vortex component, resulting in the loss of useful diagnostic information. Although target location and depth can still be determined, target shape and orientation are poorly defined. Because of galvanic current saturation good conductors are difficult to distinguish from poor ones.

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