Conductive Half-space Research Articles

Singularity of lithosphere mass density over the mid-ocean ridges and implication on floor depth and heat flow

The relation of heat flow and floor depth across the mid-ocean ridges versus lithosphere age can be described by linear functions of square root of age according to plate thermal conductive Half Space Models (HSM). However, one of the long-standing problems of these classical models is the discrepancies between predicted and observed heat flow and floor depth for very young and very old lithosphere. There have been several recent attempts to overcome this problem: one model incorporates temperature- and pressure-dependent parameters and the second model includes an additional low-conductivity crustal layer or magma rich mantle layer (MRM). Alternatively, in the current paper, the ordinary density of lithosphere in the plate conductive models is substituted with a reduction of lithosphere density towards axis that features the irregularity and nonlinearity of plates across the mid-ocean ridges. A new model is formulated incorporating the new form of density for predicting both peak heat flow and floor depth. Simple solutions of power-law forms derived from the model can significantly improve the predicting results of heat flow and floor depth over the mid-ocean ridges. Several datasets in the literature were reutilized for model validation and comparison. These datasets include both earlier datasets used for original model calibration and the more recently compiled high-quality datasets with both sedimentary and crustal loading corrections. The results indicate that both the heat flow and the slope (first order-derivative) of sea floor approach infinity (undifferentiability or singularities) around the mid-ocean ridges. These singularities are partially due to the boundary condition as it has been already known in the literature and partially to the reduction of density of lithosphere as discovered for the first time in the current research.

Smoke rings re-visited: an analytic solution for the transient electric field produced by a vertical magnetic dipole on or in a homogeneous conductive half-space

For a vertical magnetic dipole (VMD) on or under the surface of a conductive half-space, explicit analytic formulae, not Hankel transform expressions, have been derived for the transient electric field induced below the ground, and the corresponding Hertz potential. These formulae permit rapid calculation of surface and underground dB/dt and B-field components, as well as of the induced current distribution. The solution allows the current produced by the surface reflection to be isolated, revealing how the interaction with the ground surface is responsible for the downward diffusion of the smoke ring. The mathematical form of the solution has also permitted confirmation of a prediction that the current distribution from a buried dipole in the limit of late time is a scaled version of the current distribution produced by a VMD on the ground surface.

Stability Analysis of Discontinuous Galerkin Time-Domain Method for Conductive Media

In this study, the effect of conductivity on the stability of the discontinuous Galerkin time-domain (DGTD) method, which uses Runge-Kutta (RK) or leap-frog (LF) (time-averaging, time-forward, and time-backward) schemes to integrate the Maxwell equations in time, is studied. As a test case, transient reflection from one-dimensional conductive half-space, for which the analytical solution exists, is considered. Numerical results demonstrate that the LF-DGTD schemes, which use time-averaging and time-forward, are significantly more stable and faster than the RK-DGTD scheme for problems involving conductive materials.

Analytic formulas for complete and incomplete first-order TEM moments below ground in a conductive half-space

In the resistive limit, discrete conductors give rise to magnetic dipole fields. Magnetostatic modeling of time integrals, or moments, of transient electromagnetic (TEM) data therefore offers a means for fast approximate 3D modeling and inversion of TEM data sets. In our approximate inversion scheme, the net TEM moment response is represented as the combination of discrete conductor and uniform host responses. The inversion algorithm first estimates a homogeneous host conductivity, and then it subtracts the host response and fits the residual moment data by adjusting the conductivities of cells comprising a 3D rectangular mesh. To expedite calculation of the host response, we have derived analytic formulas for first-order TEM moments produced on and under the ground by a horizontal electric dipole on the surface of a homogeneous conductive half-space. We present analytic expressions for idealized all-time “complete” moments, or resistive limits, as well as for realizable finite-time “incomplete” moments. The moments produced by an arbitrary horizontal polygonal loop are determined by combining contributions from appropriately oriented electric dipoles. Downhole TEM moments computed with the new expressions reveal substantial differences between incomplete and complete moments when early time data are excluded and between step and impulse response incomplete moments. The role of the formulas in the first stage of our moment-based 3D inversion scheme is illustrated via analysis of downhole TEM data recorded at Santander, Peru. The host conductivity of best fit for early time B-field moments is 2.40 mS/m, consistent with apparent conductivities derived from ground TEM data recorded in the same area.

Effect of Fractional Parameters on Plane Waves of Generalized Electromagneto–thermoelastic Diffusion in a Rotating Half-space

This paper studies the effect of score parameters on the dynamic response of a thermally conductive half-space at a constant angular velocity and with variable material properties. The problem is formulated in the context of fractional order generalized thermoelastic diffusion theory. The exact expressions for the dimensionless results are derived and graphed.

Plane electromagnetic wave propagation of pulsed periodic field in conductive half-space

Nowadays, application of pulsed periodic fields in various technological devices is of interest, in particular for the influencing on a conductive media. In this paper solution of the problem of the propagation of a plane electromagnetic wave of pulsed periodic field created by a planar inductor, in the conductive half-space is presented. The solution is obtained in the form of Fourier series as a function of a complex argument. The differential and integral characteristics of the “inductor-conductive half-space” system for different values of the inductor time constant are determined. A comparative analysis of the obtained results with the electromagnetic characteristics of plane sinusoidal electromagnetic wave propagation process in conductive half-space is made.

Characteristics of Eddy Current Attenuation and Thickness Measurement of Metallic Plate

In eddy current testing, the law of attenuation of eddy current (EC) is of great concern. In conductive half space under the excitation of uniform magnetic field, the EC density decreases exponentially in the depth direction. However, in conductor with finite thickness tested by coil, the distribution of EC in the depth direction is more complicated. This paper studies the characteristics of EC attenuation in metallic plate of finite thickness. Simulation results show that there is an EC reflection at the bottom of plate, which changes the law of EC attenuation. A new concept, namely the equivalent attenuation coefficient, is proposed to quantify the speed of EC attenuation. The characteristics of EC attenuation are utilized to explain the nonmonotonic relation between coil voltage and plate thickness. Procedure of selecting frequency is discussed. Thereafter, measurement of plate thickness is carried out and accurate result is obtained.

Generalized Magnetothermoelastic Interaction for a Rotating Half Space

A generalized magnetothermoelasticity, in the context of Lord-Shulman theory, is employed to investigate the interaction of a homogeneous and isotropic perfect conducting half space with rotation. The Laplace transform for time variable is used to formulate a vector-matrix differential equation which is then solved by eigenvalue method. The continuous solution of displacement component while the discontinuous solutions of stress components, temperature distribution, induced magnetic and electric field have been analyzed in an approximate manner using assymptotic expansion for small time. The graphical representations also prove this continuity and discontinuity of the solutions.

Wave transforms of transient electromagnetic field in conductive earth

The study aims at finding a stable method for transformation of time-domain electromagnetic diffusion field to an electromagnetic wavefield. Two ways of transformation are considered: singular-value decomposition (SVD) and Tikhonov’s regularizations. Transformation is applied to TEM responses of a conductive half-space, a conductive S film, and to a series of horizontally layered models. The wave transforms are used to plot travel-time curves and to estimate the velocity of EM field propagation.

Natural boundary conditions and nearsurface non-homogeneity in nonferromagnetic electro conductive half-space and layer

Within the framework of the local nonhomogeneous electroconductive solid model the regularities of near surface non-homogeneity in half-space and layer are studied. Two characteristic sizes are inherent to this non-homogeneity. It is shown that in a free of force load body the values of surface stress and surface, charge are uniquely defined by physical parameters of the material and the body. The electric double layer is the result of taking into account the structural non-homogeneity of material and the forces of Coulomb interaction.

An Analytical Model of an I-Cored Coil With a Circular Air Gap

Probes used in eddy current testing contain ferrite cores of various shapes. In this paper, an axially symmetric mathematical model of an I-cored coil with an air gap was shown. The coil was placed above a two-layered conductive half-space and the solution domain was truncated radially. Using the truncated region eigenfunction expansion method, expressions describing the magnetic vector potential and the impedance of such a coil were derived. The calculations were carried out in MATLAB, both for an air-cored coil and the I-cored one containing an air gap. The results were verified using the COMSOL Multiphysics package in which the finite-element method is applied and the values show a very good agreement. Moreover, impedance components for the I-cored coil with and without an air gap were compared. The difference between the two types was significant, particularly for low-frequency values.

A High-Order Time Domain Discontinuous Galerkin Method with Orthogonal Tetrahedral Basis for Electromagnetic Simulations in 3-D Heterogeneous Conductive Media

AbstractWe present a high-order discontinuous Galerkin (DG) method for the time domain Maxwell's equations in three-dimensional heterogeneous media. New hierarchical orthonormal basis functions on unstructured tetrahedral meshes are used for spatial discretization while Runge-Kutta methods for time discretization. A uniaxial perfectly matched layer (UPML) is employed to terminate the computational domain. Exponential convergence with respect to the order of the basis functions is observed and large parallel speedup is obtained for a plane-wave scattering model. The rapid decay of the out-going wave in the UPML is shown in a dipole radiation simulation. Moreover, the low frequency electromagnetic fields excited by a horizontal electric dipole (HED) and a vertical magnetic dipole (VMD) over a layered conductive half-space and a high frequency ground penetrating radar (GPR) detection for an underground structure are investigated, showing the high accuracy and broadband simulation capability of the proposed method.

Transient electromagnetic fields of a buried horizontal magnetic dipole

Electromagnetic surveys using a vertical transmitter loop are common in land, marine, and airborne geophysical exploration. Most of these horizontal magnetic dipole (HMD) systems operate in the frequency domain, measuring the time derivative of the induced magnetic fields, and therefore a majority of studies have focused on this subset of field measurements. We examine the time-domain electromagnetic response of a HMD including the electric fields and corresponding smoke rings produced in a conductive half-space. Cases of a dipole at the surface and buried within the earth are considered. Results indicate that when the current in the transmitter is rapidly switched off, a single smoke ring is produced within the plane of the vertical transmitter loop, which is then distorted by the air-earth interface. In this situation, the circular smoke ring, which would normally diffuse symmetrically away from the source in a whole space, is approximately transformed into an ellipse, with a vertical major axis at an early time and a horizontal major axis at a late time. As measured from the location of the transmitter, the depth of investigation and lateral footprint of such a system increases with burial depth. It is also observed that the electric field measured in the direction of the magnetic dipole only contains a secondary response related to the charge accumulation on any horizontal conductivity boundaries because the primary field is always absent. This field component can be expressed analytically in terms of a static and time-varying field, the latter term adding spatial complexity to the total horizontal electric field at the earth surface at early times. Applications of this theoretical study include the design of time-domain induction-logging tools, crossborehole electromagnetic surveys, underground mine expansion work, mine rescue procedures, and novel marine electromagnetic experiments.

Calculation of the surface effect in the ferromagnetic conductor with the harmonic electromagnetic field

The authors of the paper have obtained formulas for analytical calculation of the constants with the harmonic electromagnetic field, which characterize the surface layer (a skin layer) of the ferromagnetic conductors in case of heating and nonlinear magnetic properties, which can be used for practical calculation of the electromagnetic screens, rotors of the electrical machines and inductive heating installations. A nonlinear dependence of the magnetic induction on the magnetic tension of the ferromagnetic conductor is replaced by one or two linear sections. It is considered that the skin layer of the conductor has constant quantities of the specific conductivity and averaged temperature. Linear electrodynamics equations are solved for the conductive half-space. Parameters of the ferromagnetic conductor's surface layer are calculated: magnetic permeability, the thickness of the skin layer and its averaged temperature, exposure time of the electromagnetic field on the conductor with the established maximum temperature on the conductor's surface, pressure of the field on the conductor and its resistance, inductivity of the internal magnetic field in the conductor, the thermal energy capacity. The methods credibility is confirmed with the concurrence of the resistance and inductiviry of the ferromagnetic conductor with analogous quantities from other methods.

E-Cored Coil With a Circular Air Gap Inside the Core Column Used in Eddy Current Testing

The problem of an axially symmetric E-cored coil with a circular air gap inside the core column located above a two-layered conductive half-space is presented. The truncated region eigenfunction expansion method is used to derive expressions describing the magnetic vector potential of the filamentary coil. The final expressions for the impedance of the rectangular cross-sectional coil are obtained, and calculations for various frequency values are carried out. The results are compared with those from the COMSOL package, which shows a very good agreement.

On smoke rings produced by a loop buried in a conductive half-space

The transient electromagnetic (TEM) response of a wire loop at the surface of the earth has been studied in detail for several decades. When current in the loop is rapidly switched off, an identical system of current, commonly referred to as a “smoke ring,” is induced in the conductive subsurface and diffuses downward and outward into the earth. What has not been studied as thoroughly is the response of a loop buried within a conductive medium, as might be the case in the marine environment or when TEM prospecting systems are placed in mine tunnels or horizontal boreholes. We have examined the TEM response of a horizontal wire loop buried within a conductive half-space. Expressions were derived for the azimuthal electric field and corresponding vertical and radial magnetic fields. Our results showed that when the loop is far from the earth-air interface, a single smoke ring system diffuses radially outward from the transmitter, while the electric field and corresponding current density decays away in the vertical direction. As the loop approaches the interface, the smoke ring system diffuses radially at early times, but gradually, the complex image of the loop in the air produces a system of secondary azimuthal electric fields, which, when combined with the primary field, adds a vertical component to the field diffusion. At late times, the field behavior reduces to the well-known surface case and the maximum current system diffuses downward at a constant angle of 26° with respect to the plane of the loop. We concluded that it was the effect of the interface that produces the downward migration of a smoke ring system, whereas the outward migration is mainly generated by the primary field.

Electrical resistivity structure under the western Cosmonauts Sea at the continental margin of East Antarctica inferred via a marine magnetotelluric experiment

The western Cosmonauts Sea, off the coast of East Antarctica, was a site of rifting of the Gondwana supercontinent and subsequent early seafloor spreading. To improve our understanding of the breakup of Gondwana, we conducted a marine magnetotelluric experiment to determine the electrical resistivity structure within the uppermost several hundred kilometers beneath the western Cosmonauts Sea. Magnetotelluric response functions at two sites, obtained after considering possible influences of non-plane magnetic field sources, suggest that these responses include distortions by topographic variations and conductive anomalies around the observation sites. Three-dimensional forward modeling confirmed that these distortions due to topographic variations and a thin (∼2-km thick) conductive layer immediately under the sites (mostly sediments) are severe. Furthermore, three-dimensional forward modeling to investigate the resistivity structure at deeper depths revealed an upper resistive layer (≥300 Ω-m), with a thickness of <100 km, and an underlying conductive half-space (∼10 Ω-m). The upper resistive layer and the underlying conductive structure most likely represent dry and water/melt-rich oceanic upper mantle, respectively. The upper resistive layer may be thinner than anticipated under the old seafloor of the study area (likely >90 Ma), and may suggest a conductive anomaly in the upper mantle produced by mantle convection and/or upwelling.

Contribution of the source velocity to the scattering of electromagnetic fields caused by airborne magnetic dipoles

The velocity of controlled airborne sources of electromagnetic geophysical surveys plays an additional role in the scattering of the fields by the earth. Therefore, it is necessary to investigate its contribution in the space and time variation of secondary electromagnetic fields. The model of a vertical magnetic dipole moving at a constant speed along a horizontal line in the air and above a homogeneous conductive half-space constitutes a first approach to stress the kinematic aspect and determine the difference between the fields due to an airborne and a static source. The magnetic moment of the source is equal to 104 A m2, its height is 120 m, and the horizontal and vertical separations between it and the receiver are, respectively, equal to 100 and 50 m: these values of the model are typical of towed-bird airborne TDEM surveys. We employed four values for the common velocities of source and receiver (0, 60, 80, and 100 m s−1), four values of the conductivity of the half-space (0.5, 0.1, 0.05, and 0.01 S m−1), and two causal source currents (box with periods of 80 and 10 ms and periodic with frequency values of 12.5 and 100 Hz). The results demonstrate that the relative velocity between source and medium yields a measurable variation compared to the static condition. Therefore, it must be taken into consideration by compensating the discrepancy in measured data employing the respective theoretical result. The results also show that it is necessary to adjust the concepts of time and frequency domain for electromagnetic measurements with traveling sources.

3D finite-element forward modeling of electromagnetic data using vector and scalar potentials and unstructured grids

We present a finite-element solution to the 3D electromagnetic forward-modeling problem in the frequency domain. The method is based on decomposing the electric field into vector and scalar potentials in the Helmholtz equation and in the equation of conservation of charge. Edge element and nodal element basis functions were used, respectively, for the vector and scalar potentials. This decomposition was performed with the intention of satisfying the continuity of the tangential component of the electric field and the normal component of the current density across the interelement boundaries, therefore finding an efficient solution to the problem. The computational domain was subdivided into unstructured tetrahedral elements. The system of equations was discretized using the Galerkin variant of the weighted residuals method, with the approximated vector and scalar potentials as the unknowns of a sparse linear system. A generalized minimum residual solver with an incomplete LU preconditioner was used to iteratively solve the system. The solution method was validated using five examples. In the first and second examples, the fields generated by small dipoles on the surface of a homogeneous half-space were compared against their corresponding analytic solutions. The third example provided a comparison with the results from an integral equation method for a long grounded wire source on a model with a conductive block buried in a less conductive half-space. The fourth example concerned verifying the method for a large conductivity contrast where a magnetic dipole transmitter-receiver pair moves over a graphite cube immersed in brine. Solutions from the numerical approach were in good agreement with the data from physical scale modeling of this scenario. The last example verified the solution for a resistive disk model buried in marine conductive sediments. For all examples, convergence of the solution that used potentials were significantly quicker than that using the electric field.

The field of the vertical electric dipole immersed in the heterogeneous half-space

The field of the vertical electric dipole (VED) immersed in the heterogeneous conductive halfspace (sea) is analyzed in time domain. In the near field of the source, the amplitudes of the electric and magnetic components of the field are proportional to power 3/2 and power 5/2 of the conductivity of the medium, respectively. After termination of the transmitter pulse, all the VED components decay with time as ∼1/t5/2. The possibility of applying the VED field for estimating the electrical properties of the offshore geological sections is demonstrated.