Transient Electromagnetic Logging (TEL)

• TEL is a well-logging technique that employs pulsed currents and receiver coil measurements to characterize subsurface formations via induced electromagnetic fields.
• It uses frequency-domain modeling and cylindrical or horizontally layered media frameworks to accurately simulate responses in complex borehole environments.
• Optimized sine-transform filtering and normalized Bessel functions reduce required frequency samples, enabling efficient inversion and real-time formation analysis.

Searching arXiv for papers on transient electromagnetic logging and related multilayer cylindrical EM modeling. Transient electromagnetic logging (TEL) is a well-logging method in which a down-hole transmitter coil is driven with a pulsed current and the induced voltage is measured in a receiver coil, both coaxial with the borehole. It is of great significance for detecting deep formation structures outside the wellbore, and its forward modeling commonly proceeds through frequency-domain calculations followed by time-domain conversion (Li, 25 Aug 2025). In current arXiv literature, TEL is treated both in horizontally layered media and in multilayer cylindrical conductive configurations that also arise in eddy current tube testing, with the borehole, casing, cement, and formation represented as concentric regions (Theodoulidis et al., 2024).

1. Geophysical scope and model classes

TEL is used to probe conductive structure beyond the immediate wellbore environment. In the multilayer cylindrical setting, the physical configuration is a transmitter-receiver coil system inside a multi-layer cylindrical conductive configuration; in a real borehole, the multilayer model represents fluid (innermost), casing, cement sheath, invaded zone, and undisturbed formation (Theodoulidis et al., 2024). In the formulation for cylindrically layered formations, a four-zone tool model is also explicitly considered: mandrel, borehole fluid, steel casing, and formation (Li, 25 Aug 2025).

A recurring modeling distinction is between horizontally stratified media and cylindrically stratified media. For electromagnetic field computations in horizontally stratified media, mature Hankel transform methods already enable precise solutions. For cylindrically stratified media, by contrast, oscillatory integrals involving cosine functions must be evaluated (Li, 25 Aug 2025). This division is central to TEL numerics because the inversion and forward-modeling machinery depends strongly on whether the medium is planar or concentric cylindrical.

The measurement target is the transient response of the receiver coil. In the cylindrical multilayer literature, the quantity measured is the open-circuit induced voltage in the receiver, written in the Laplace domain as

$$V_R(s) = -\,s\,N_R^2\,W_T\,W_R\,\int_0^\infty A_\theta(r_R)\,dr,$$

with the source specified by the transmitter current $I_T(t)$ and its Laplace transform $I_T(s)$ (Theodoulidis et al., 2024). In the sine-transform-based TEL formulation, the time-domain output is denoted $V(t_m)=Y[m]$ after frequency-domain receptivity has been converted to time samples (Li, 25 Aug 2025).

2. Governing electromagnetic formulation

A standard TEL formulation uses the magnetic vector potential $\mathbf A$ with only an azimuthal component $A_\theta(r,z,t)$. Maxwell’s equations then reduce to the scalar diffusion–Helmholtz equation in cylindrical coordinates,

$$\frac{\partial^2 A_\theta}{\partial r^2} +\frac1r\frac{\partial A_\theta}{\partial r} -\frac{A_\theta}{r^2} +\frac{\partial^2 A_\theta}{\partial z^2} -\mu\,\sigma\,\frac{\partial A_\theta}{\partial t} =-\,\mu\,J_{s,\theta}(r,z,t),$$

with Faraday’s law and the Ampère–Maxwell law written using $\mathbf B=\mu\nabla\times\mathbf A$ and $\mathbf H=\nu\mathbf B$, $\nu=1/\mu$, while neglecting displacement current in conductors (Theodoulidis et al., 2024).

After Laplace transformation in time and separable expansion in $I_T(t)$0, the field in each cylindrical layer satisfies

$I_T(t)$1

For a model with $I_T(t)$2 concentric regions $I_T(t)$3, each layer is characterized by electrical conductivity $I_T(t)$4, relative permeability $I_T(t)$5, and interfaces at $I_T(t)$6, with $I_T(t)$7 (Theodoulidis et al., 2024). The general radial solution uses modified Bessel functions,

$I_T(t)$8

Interface conditions enforce continuity of $I_T(t)$9 and of $I_T(s)$0. These conditions can be cast into a $I_T(s)$1 transfer matrix, and in the air region one obtains the reflection coefficient

$I_T(s)$2

where $I_T(s)$3 (Theodoulidis et al., 2024). This transfer-matrix structure is the basis for stable semi-analytical TEL forward models in concentric cylindrical geometries.

3. Frequency-to-time conversion by sine transform

In one recent TEL formulation, the time-domain field is represented as a Fourier sine transform,

$I_T(s)$4

using the fact that for transient EM logging the response is odd in time (Li, 25 Aug 2025). After the change of variables $I_T(s)$5 and $I_T(s)$6, the continuous transform reduces to a convolution on an exponential grid,

$I_T(s)$7

Truncation to a finite digital filter of length $I_T(s)$8 with $I_T(s)$9 gives the 125-point sine transform filter,

$V(t_m)=Y[m]$0

or, in physical time,

$V(t_m)=Y[m]$1

The coefficients $V(t_m)=Y[m]$2, $V(t_m)=Y[m]$3, are obtained by solving the discrete convolution system for a set of test functions and then optimizing for minimal aliasing and truncation error (Li, 25 Aug 2025).

The reported purpose of the filter design is twofold: the high-frequency tail of $V(t_m)=Y[m]$4 is sufficiently suppressed, minimizing aliasing when sampling in $V(t_m)=Y[m]$5, and the finite-length truncation error in the convolution sum remains below $V(t_m)=Y[m]$6 relative in most practical TEL scenarios (Li, 25 Aug 2025). Compared to a direct inverse FFT or naïve numerical inverse Fourier integration, the exponential sampling in $V(t_m)=Y[m]$7 and $V(t_m)=Y[m]$8 plus the optimized filter reduce the number of frequency samples needed from thousands to 125; the filter’s side-lobes are controlled to below $V(t_m)=Y[m]$9; numerical tests report sub-0.1% relative error in time-decay curves; and the sine-transform approach remains stable even for long-time signals $\mathbf A$0, where Laplace inversion such as Stehfest typically degrades (Li, 25 Aug 2025).

4. Cylindrical forward solvers and convergence acceleration

For cylindrically layered formations, one central task is evaluation of an oscillatory cosine integral. The axial magnetic field is written as

$\mathbf A$1

so that the core numerical problem is

$\mathbf A$2

on $\mathbf A$3 (Li, 25 Aug 2025). The semi-infinite interval is partitioned into finite segments $\mathbf A$4, and on each segment an $\mathbf A$5-point Gauss-Legendre rule is applied. Partial sums

$\mathbf A$6

are then accelerated toward the true integral value using Levin–Sidi extrapolation, including Sidi’s $\mathbf A$7-algorithm in recursive form. In practice, $\mathbf A$8 Gauss points per subinterval and up to $\mathbf A$9 extrapolation orders suffice to drive the tail error below $A_\theta(r,z,t)$0 (Li, 25 Aug 2025).

A separate but closely related multilayer cylindrical strategy replaces the continuous integral over $A_\theta(r,z,t)$1 by a discrete sum over eigenvalues $A_\theta(r,z,t)$2 and weights $A_\theta(r,z,t)$3, after truncating the infinite axial domain to $A_\theta(r,z,t)$4 with Dirichlet boundary conditions (Theodoulidis et al., 2024). In that framework,

$A_\theta(r,z,t)$5

For large $A_\theta(r,z,t)$6, the modified Bessel functions $A_\theta(r,z,t)$7 and $A_\theta(r,z,t)$8 overflow or underflow, so normalized ratios such as

$A_\theta(r,z,t)$9

are introduced so that exponential factors cancel (Theodoulidis et al., 2024).

Taken together, these formulations show that TEL forward modeling in cylindrical media is not tied to a single transform strategy. One line of work emphasizes direct handling of oscillatory cosine integrals with Gauss-Legendre quadrature and Levin–Sidi extrapolation, while another emphasizes domain truncation, normalized transfer matrices, and Laplace-domain modal summation (Li, 25 Aug 2025, Theodoulidis et al., 2024).

5. Time-domain reconstruction, algorithmic workflow, and validation

The sine-transform TEL workflow is organized as follows: for each exponentially sampled frequency $$\frac{\partial^2 A_\theta}{\partial r^2} +\frac1r\frac{\partial A_\theta}{\partial r} -\frac{A_\theta}{r^2} +\frac{\partial^2 A_\theta}{\partial z^2} -\mu\,\sigma\,\frac{\partial A_\theta}{\partial t} =-\,\mu\,J_{s,\theta}(r,z,t),$$0, the spectral field $$\frac{\partial^2 A_\theta}{\partial r^2} +\frac1r\frac{\partial A_\theta}{\partial r} -\frac{A_\theta}{r^2} +\frac{\partial^2 A_\theta}{\partial z^2} -\mu\,\sigma\,\frac{\partial A_\theta}{\partial t} =-\,\mu\,J_{s,\theta}(r,z,t),$$1 is computed by matching boundary conditions in each cylinder; the oscillatory integral is evaluated over subintervals with Gauss-Legendre quadrature and accelerated with Levin–Sidi; the frequency-domain receptivity is formed as $$\frac{\partial^2 A_\theta}{\partial r^2} +\frac1r\frac{\partial A_\theta}{\partial r} -\frac{A_\theta}{r^2} +\frac{\partial^2 A_\theta}{\partial z^2} -\mu\,\sigma\,\frac{\partial A_\theta}{\partial t} =-\,\mu\,J_{s,\theta}(r,z,t),$$2; and a 125-point sine-filter convolution yields the time-domain samples $$\frac{\partial^2 A_\theta}{\partial r^2} +\frac1r\frac{\partial A_\theta}{\partial r} -\frac{A_\theta}{r^2} +\frac{\partial^2 A_\theta}{\partial z^2} -\mu\,\sigma\,\frac{\partial A_\theta}{\partial t} =-\,\mu\,J_{s,\theta}(r,z,t),$$3, with output $$\frac{\partial^2 A_\theta}{\partial r^2} +\frac1r\frac{\partial A_\theta}{\partial r} -\frac{A_\theta}{r^2} +\frac{\partial^2 A_\theta}{\partial z^2} -\mu\,\sigma\,\frac{\partial A_\theta}{\partial t} =-\,\mu\,J_{s,\theta}(r,z,t),$$4 (Li, 25 Aug 2025).

The reported computational complexity for the spatial integration step is $$\frac{\partial^2 A_\theta}{\partial r^2} +\frac1r\frac{\partial A_\theta}{\partial r} -\frac{A_\theta}{r^2} +\frac{\partial^2 A_\theta}{\partial z^2} -\mu\,\sigma\,\frac{\partial A_\theta}{\partial t} =-\,\mu\,J_{s,\theta}(r,z,t),$$5, with a typical count of $$\frac{\partial^2 A_\theta}{\partial r^2} +\frac1r\frac{\partial A_\theta}{\partial r} -\frac{A_\theta}{r^2} +\frac{\partial^2 A_\theta}{\partial z^2} -\mu\,\sigma\,\frac{\partial A_\theta}{\partial t} =-\,\mu\,J_{s,\theta}(r,z,t),$$6 frequencies $$\frac{\partial^2 A_\theta}{\partial r^2} +\frac1r\frac{\partial A_\theta}{\partial r} -\frac{A_\theta}{r^2} +\frac{\partial^2 A_\theta}{\partial z^2} -\mu\,\sigma\,\frac{\partial A_\theta}{\partial t} =-\,\mu\,J_{s,\theta}(r,z,t),$$7 subintervals $$\frac{\partial^2 A_\theta}{\partial r^2} +\frac1r\frac{\partial A_\theta}{\partial r} -\frac{A_\theta}{r^2} +\frac{\partial^2 A_\theta}{\partial z^2} -\mu\,\sigma\,\frac{\partial A_\theta}{\partial t} =-\,\mu\,J_{s,\theta}(r,z,t),$$8 points $$\frac{\partial^2 A_\theta}{\partial r^2} +\frac1r\frac{\partial A_\theta}{\partial r} -\frac{A_\theta}{r^2} +\frac{\partial^2 A_\theta}{\partial z^2} -\mu\,\sigma\,\frac{\partial A_\theta}{\partial t} =-\,\mu\,J_{s,\theta}(r,z,t),$$9 direct kernel evaluations. Levin–Sidi adds $\mathbf B=\mu\nabla\times\mathbf A$0, and the filtering step is $\mathbf B=\mu\nabla\times\mathbf A$1. Memory stores $\mathbf B=\mu\nabla\times\mathbf A$2 for all $\mathbf B=\mu\nabla\times\mathbf A$3 at about $\mathbf B=\mu\nabla\times\mathbf A$4 double values plus the filter. Frequency loops are embarrassingly parallel, subinterval quadrature sums per $\mathbf B=\mu\nabla\times\mathbf A$5 can be threaded, and final convolution for each time sample is also parallel (Li, 25 Aug 2025).

A complementary multilayer cylindrical implementation computes the inverse Laplace transform of $\mathbf B=\mu\nabla\times\mathbf A$6 by three strategies: numerical inversion via FFT plus $\mathbf B=\mu\nabla\times\mathbf A$7-algorithm, Stehfest series for short times $\mathbf B=\mu\nabla\times\mathbf A$8, and the pole-residue theorem for long times $\mathbf B=\mu\nabla\times\mathbf A$9 (Theodoulidis et al., 2024). The empirical crossover time is reported as

$\mathbf H=\nu\mathbf B$0

The trade-off is explicit: Stehfest converges rapidly for $\mathbf H=\nu\mathbf B$1 but fails at large $\mathbf H=\nu\mathbf B$2, while pole-residue captures the slow decay with only a few dominant poles but omits early-time high-frequency content; the hybrid scheme yields full accuracy across all $\mathbf H=\nu\mathbf B$3 (Theodoulidis et al., 2024).

Reported validation results are summarized below.

Scenario	Reported result	Conditions
Homogeneous formation	All relative errors $\mathbf H=\nu\mathbf B$4	$\mathbf H=\nu\mathbf B$5–$\mathbf H=\nu\mathbf B$6, source–receiver $\mathbf H=\nu\mathbf B$7, $\mathbf H=\nu\mathbf B$8–$\mathbf H=\nu\mathbf B$9
Open-hole cylindrical case	Peak Error $\nu=1/\mu$0; runtime $\nu=1/\mu$1	Four-zone tool model
Cased-hole cylindrical case	Peak Error $\nu=1/\mu$2; runtime $\nu=1/\mu$3	Four-zone tool model

For the homogeneous formation tests, the maximum relative errors over time are reported as $\nu=1/\mu$4, $\nu=1/\mu$5, $\nu=1/\mu$6, and $\nu=1/\mu$7 for resistivities $\nu=1/\mu$8, $\nu=1/\mu$9, $I_T(t)$00, and $I_T(t)$01, respectively, over a time window $I_T(t)$02 to $I_T(t)$03 with 50 log-spaced points (Li, 25 Aug 2025). For the cylindrical tests, convergence plots show that with at least 16 Gauss nodes and at least 10 extrapolation orders, the oscillatory integral error falls below $I_T(t)$04; runtimes were measured on a single $I_T(t)$05 core, and frequency-parallel runs scale nearly linearly (Li, 25 Aug 2025).

6. Interpretation, applications, and limitations

The reported practical motivation for accurate TEL forward modeling is inversion. Sub-0.1% accuracy across a wide resistivity range is stated to ensure that inversion of TEL logs will not be biased by forward-modeling error, and the reduction in frequency-sample count from roughly 1000+ to about 125 cuts CPU time by an order of magnitude (Li, 25 Aug 2025). In the multilayer cylindrical literature, matching measured $I_T(t)$06 to the semi-analytical response $I_T(t)$07 is described as a route to inversion for formation resistivity, casing thickness, cement integrity, and invasion depth (Theodoulidis et al., 2024).

The application domain includes both open-hole and cased-hole wells. The sine-transform-based method is reported to remain robust in cased-hole scenarios with extreme contrasts, where standard Hankel-transform methods fail, and the integrated methodology is described as supporting reliable interpretation of field TEL logs for more accurate detection of reservoir boundaries, fluid encroachment, and casing integrity (Li, 25 Aug 2025). The multilayer cylindrical framework further states that the speed and numerical stability of the algorithms allow rapid forward-model evaluation and Jacobian computation, making real-time inversion or look-up-table calibration feasible in a downhole tool (Theodoulidis et al., 2024).

Several limitations are made explicit. The exponential sampling rate $I_T(t)$08 is optimized for the time window $I_T(t)$09–$I_T(t)$10; different windows may require re-derivation of the filter (Li, 25 Aug 2025). For highly dispersive dielectric formations, more frequency samples or a specialized filter may be needed (Li, 25 Aug 2025). In practice, one should monitor the Levin–Sidi residual, and if

$I_T(t)$11

exceeds $I_T(t)$12, the subinterval count or extrapolation order should be increased (Li, 25 Aug 2025). A plausible implication is that TEL inversion quality is tightly coupled to transform design, convergence control, and the physical appropriateness of the cylindrical or horizontally layered model selected for the borehole environment.