3-D EM MODELLING USING THE COMPACT FINITE ELEMENT METHOD

Abstract

Numerical methods employed for 3-D electromagnetic (EM) forward modelling are : Finite Element Method (FEM), Integral Equation Method (IEM) and a Hybrid Method (HM) - Compact Finite Element Method (CFEM) - that amalgamates the positive features of the first two methods. FEM has the advantage of being able to deal with an arbitrary resistivity distribution. However, as the whole earth must be modelled^necessitating discretization of a large domain, the resulting stiffness matrix becomes rather large. IEM, on the other hand, can handle only confined targets buried in a layered earth. Here only the target need be discretized which makes the resulting coefficient matrix much smaller. In this case, however, we have the problem of singularities and of distributing the artificial charges arising at cell boundaries from the assumption of constant EM field in each cell. CFEM like IEM, can handle only confined targets and requires discretization of only the target alongwith an element thick veneer of the host medium. FEM is then used to solve for secondary fields and IEM to generate necessary boundary values. The two operations can be performed iteratively, in which case the coefficient matrix is banded, or a direct scheme can be developed where the coefficient matrix is not banded. In CFEM the coefficient matrix is not large and there are niether singularities nor artificial charges generated at the element boundaries because the EM field varies over the element. Although FEM is more versatile, this advantage becomes inconsequential as it can not be exploited in the present day computing environment. This thesis deals with the development of three CFEM algorithms- HYBRIDC, SANGAM and SAMAYA. The first two enable one to compute solutions in the frequency domain and the third one in time domain. A brief outline of the thesis is as follows. (vii) In the introductory chapter, the importance of numerical forward model ling in data inversion is discussed. The wide range of applications of EM methods is next presented. The scarcity of 3-D EM modelling results, parti cularly in time domain, is highlighted in the state-of-the-art review of the EM numerical modelling. In chapter-II the EM theory is discussed in brief, the ranges of physical parameters : conductivity, dielectric permittivity and magnetic permeability encountered in the earth are discussed with a view to simplifying the general EM equations. Also presented in brief is an account of different EM response functions that are computed from observations for interpretation purposes. In chapter-Ill, a classification of different numerical methods is pre sented. Weighted Residual Methods, in general, and FEM and IEM, in particular are discussed next. Finally, CFEM is discussed and its two variations, iterative and direct schemes, are formulated. In chapter-IV, is presented an account of the development of the three CFEM algorithms HYBRIDC, SANGAM and SAMAYA that originated from an iterative HM algorithm HYB3D developed by Lee et.al. (1981, Geophysics, H, 796-805). The adapted version of HYB3D was named as HYBRIDB. The search routines, developed to eliminate the repetition in the computation of Green's functions that occurred in HYBRIDB, and the resulting modifications that led to the development of HYBRIDC are discussed next. HYBRIDC needs an order of magnitude less time than HYBRIDB. The development of SANGAM which implements direct scheme of CFEM and not the iterative one imple mented by HYBRIDB and HYBRIDC, is discussed next. For the number of nodes that can be handled presently, SANGAM needs an order of magnitude less time than HYBRIDC. The results of a comparative study of the perfor mances of HYBRIDB, HYBRIDC and SANGAM are then presented. Finally, (viii) the development of SAMAYA is discussed. In chapter-V the results of various studies made on SANGAM and SAMAYA are presented. The studies performed on SANGAM pertain to (i) mesh convergence, (ii) convergence of responses of a 3-D body, as its strike-length is progressively increased, to that of the corresponding 2-D body, (iii) no resistivity contrast case, (iv) reciprocity test and (v) comparison of SANGAM results with those obtained by using other algorithms. The studies conducted on SAMAYA pertain to i) identification of the range of frequencies for which frequency domain response need be computed, ii) identification of the minimum number of frequencies per decade necessary for stable transformation from frequency to time domain, iii) comparison of the transform routine results for a 1-D model with those obtained from a direct time domain algorithm for 1-D study, iv) comparison of the response of elongated 3-D body with those of a 3-D thin sheet like body, v) comparison of SAMAYA results with corres ponding results obtained by using an IEM algorithm of San Filipo et.al. (1985, Geophysics, 50, 798-809, 1144-1162), vi) the overburden effect and vii) the bore hole study. Finally, in chapter-VI it is concluded 'that these algorithms offer new reliable software packages for forward 3-D EM modelling. Their possible use in catalogue preparation and development of definitive features of a 3-D anomaly is highlighted. Also discussed here is the possible future scope of work in the directions such as development of new matrix solvers exploiting more efficiently the sparsity and inherent structure of the matrix, use of a smaller set of digital filters in convolution operation and recourse to pipe lining and parallel computing wherever possible.