Research On Complex Resistivity Forward And Inversion With Finite Element Method And Its Application

Complex resistivity method is based on the characteristics of SIP parameters ofrocks and ores, and the multiple parameters provide more information for evaluatinganomalous bodies. However, the inverse problem of SIP has not been resolved well,and has seriously restricted the application and development of SIP. In this paper, the2.5D and3D SIP numerical simulations using finite element method were studied, anda large-scale2.5D SIP inversion algorithm for application was proposed and finished.The basic boundary value problem of SIP is derived by using Maxwell’s equationand Cole-Cole model. From the standard variational principle, the generalizedvariational principle is derived which is the basic theory of3D vector finite elementanalysis in this paper. At last, the Galerkin finite element method is introduced, whichis the basic theory of2.5D node finite element analysis in this paper.Based on the node finite element method, the2.5D complex resistivity forward isstudied. From the basic boundary value problem of SIP, the frequency domain electricand magnetic fields coupled partial differential equations were derived, and they werediscretized by Galerkin method. Considering the effect of topography, the solvedregion is divided into small units with isoparametric element，and the pseudo-deltafunction is used to improve the simulation precision of electromagnetic fields near thesource region. Considering the computational efficiency, the non-zero elements ingeneral coefficient matrix is stored directly by using1D compressing method and theIC-CBCG method is used in solving forward equations. The rule of selecting effectivewave numbers is proposed for carrying out inverse Fourier Transform. The apparentresistivity and apparent phase of2D complex resistivity model were calculated, and itscharacteristics were also analyzed.Based on the vector finite element method, the3D complex resistivity forwardmodeling is studied. In order to improve the precision of numerical simulation in thenear source region, the total field is decomposed into the background and secondaryfields. The field analysis formula of layered media under the excitation of long wiresource is derived, thus the high-precision calculation of the background field is achieved. For the secondary field, the rectangular elements are used to split the solvedregion, and vector interpolation function is used to complete the unit analysis, whichavoids the “pseudo-solution” in node finite element method. The same scheme as2.5Dforward is used in coefficient matrix storage. The3D forward method is verified andsome typical model’s responses are analyzed by SIP3D numerical simulation.A reliable and practical2.5D combined inversion algorithm is proposed in thispaper, which used amplitudes and phases of electric fields of multi-arrangements. Themodel’s smoothness and parameter variation range as two kinds of constraints areintroduced into2.5D SIP inversion, thus the multiplicity of inversion is efficientlyimproved. The analytical expression of the sensitivity matrix is derived with the partialderivatives of the electric field, and calculated by the reciprocity theorem. Thealgorithm can perform a large-scale inversion and take advantage of geologicalinformation. Finally, the inversion procedure was used in Anhui SIP measured data,and the result shows good consistence with known drilling data and CSAMT inversionresult.