Numerical modeling for transient electromagnetic responses on a 2.5-dimension model

The transient electromagnetic method (TEM) is widely used in mineral and oil exploration as well as water exploration, engineering and environment geophysics exploration. Numerical modeling for transient electromagnetic responses of a 3-D source over 2-D geoelectric model is so-called 2.5-D problem, in which the conductivity σ, dielectric permittivity ɛ and magnetic permeability μ of the 3-D geoelectric model are invariant along the strike direction. The 3-D problem can be converting into 2-D problem in Fourier domain by applying the Fourier transform to the electromagnetic field with respect to the strike direction. Thus the computing time is reduced compare with the real 3-D problem. And it is more accurate than pure 2-D problem since the source are three-dimensional. This paper deals with the forward numerical modeling for central-loop electromagnetic method on a 2.5-D problem using finite element method. Basic procedures of the 2.5-D forward modeling algorithmare: firstly, carry out the Laplace transform and the Fourier transform to the partial differential equations of E and H vectors in the 3-D spatial field, converting the problem into 2-D partial differential equations for scalar E and H; secondly, turn 2-D boundary-value problems into 2-D problem of calculus of variations, and use finite element technique to seek for numerical solution; and thirdly, take the inverse Fourier transform and the inverse Laplace transform to obtain transient responses of the electric field. In order to check up the algorithm’s validity, we apply it to compute the three-layer models (H-type section and K-type section) and four-layer model (HK-type section), and compare with corresponding analytical solution on the layered earth. Relative errors are less than 3%. In addition, we implement computation for several typical 2-D plate models. Results show the algorithm in this paper is valid. Main characteristics of the algorithm established in this paper are as following: (1) Rearrange the EM field equations from six to two to yield coupled equations with parameter variable wavenumber and simplify the components from six to two along-strike fields in the Fourier transform domain, in which only two components of electric and magnetic primary field are needed computing when the secondary field algorithm is directly used. (2) A 2-D finite element method is adopted in which two diagonal lines are increased in each rectangular net to form the triangular networks. The unknown variable at the central node of the rectangular net has been eliminated by the Gauss elimination. So the complex 2-D geoelectric section can be modeling more exactly but the computation quantity is much less than that in rectangular network. So not only the computation accuracy is high, but also the computation time is not increasing much. (3) The new G-S inverse Laplace transform method based on the delay theorem of Laplace transform is used for the rapid calculation of the transient electromagnetic response with dense samples.