Finite element three-dimensional direct current resistivity modelling: accuracy and efficiency considerations

SUMMARY The finite element method is a powerful tool for 3-D DC resistivity modelling and inversion. The solution accuracy and computational efficiency are critical factors in using the method in 3-D resistivity imaging. This paper investigates the solution accuracy and the computational efficiency of two common element-type schemes: trilinear interpolation within a regular 8-node solid parallelepiped, and linear interpolations within six tetrahedral bricks within the same 8-node solid block. Four iterative solvers based on the pre-conditioned conjugate gradient method (SCG, TRIDCG, SORCG and ICCG), and one elimination solver called the banded Choleski factorization are employed for the solutions. The comparisons of the element schemes and solvers were made by means of numerical experiments using three synthetic models. The results show that the tetrahedron element scheme is far superior to the parallelepiped element scheme, both in accuracy and computational efficiency. The tetrahedron element scheme may save 43 per cent storage for an iterative solver, and achieve an accuracy of the maximum relative error of <1 per cent with an appropriate element size. The two iterative solvers, SORCG and ICCG, are suitable options for 3-D resistivity computations on a PC, and both perform comparably in terms of convergence speed in the two element schemes. ICCG achieves the best convergence rate, but nearly doubles the total storage size of the computation. Simple programming codes for the two iterative solvers are presented. We also show that a fine grid, which doubles the density of a coarse grid, will require at least 2 7 =128 times as much computing time when using the banded Choleski factorization. Such an increase, especially for 3-D resistivity inversion, should be compared with SORCG and ICCG solvers in order to find the computationally most efficient method when dealing with a large number of electrodes.

[1]  F. S. Grant,et al.  IP AND RESISTIVITY TYPE CURVES FOR THREE‐DIMENSIONAL BODIES , 1969 .

[2]  J. Coggon.  Electromagnetic and electrical modeling by the finite element method , 1971 .

[3]  O. Zienkiewicz.  The Finite Element Method In Engineering Science , 1971 .

[4]  Irshad R. Mufti,et al.  FINITE‐DIFFERENCE RESISTIVITY MODELING FOR ARBITRARILY SHAPED TWO‐DIMENSIONAL STRUCTURES , 1976 .

[5]  D. Kershaw.  The incomplete Cholesky—conjugate gradient method for the iterative solution of systems of linear equations , 1978 .

[6]  A. Dey,et al.  Resistivity modeling for arbitrarily shaped three-dimensional structures , 1979 .

[7]  Gerald W. Hohmann,et al.  Topographic effects in resistivity and induced-polarization surveys , 1980 .

[8]  G. W. Hohmann,et al.  An investigation of finite-element modeling for electrical and electromagnetic data in three dimensions , 1981 .

[9]  M. Okabe,et al.  BOUNDARY ELEMENT METHOD FOR THE ARBITRARY INHOMOGENEITIES PROBLEM IN ELECTRICAL PROSPECTING , 1981 .

[10]  Gene H. Golub,et al.  Matrix computations , 1983 .

[11]  H. T. Holcombe,et al.  Three-dimensional terrain corrections in resistivity surveys , 1984 .

[12]  E. Mundry.  GEOELECTRICAL MODEL CALCULATIONS FOR TWO‐DIMENSIONAL RESISTIVITY DISTRIBUTIONS* , 1984 .

[13]  M. A. Ajiz,et al.  A robust incomplete Choleski‐conjugate gradient algorithm , 1984 .

[14]  B. James.  Efficient microcomputer-based finite-difference resistivity modeling via Polozhii decomposition , 1985 .

[15]  Owe Axelsson,et al.  A survey of preconditioned iterative methods for linear systems of algebraic equations , 1985 .

[16]  U. Das,et al.  RESISTIVITY AND INDUCED POLARIZATION RESPONSES OF ARBITRARILY SHAPED 3‐D BODIES IN A TWO‐LAYERED EARTH* , 1987 .

[17]  Hans Rudolf Schwarz,et al.  Finite Element Methods , 1988 .

[18]  Shi‐zhe Xu,et al.  An integral formulation for three-dimensional terrain modeling for resistivity surveys , 1988 .

[19]  Peter N. Shive,et al.  Singularity removal: A refinement of resistivity modeling techniques , 1989 .

[20]  Hiromasa Shima,et al.  Two-dimensional automatic resistivity inversion technique using alpha centers , 1990 .

[21]  M. Papadrakakis,et al.  Improving the efficiency of incomplete Choleski preconditionings , 1991 .

[22]  Stephen K. Park,et al.  Inversion of pole-pole data for 3-D resistivity structure beneath arrays of electrodes , 1991 .

[23]  Alex Marcuello,et al.  2-D resistivity modeling; an approach to arrays parallel to the strike direction , 1991 .

[24]  Hiromasa Shima,et al.  2-D and 3-D resistivity image reconstruction using crosshole data , 1992 .

[25]  Douglas W. Oldenburg,et al.  Approximate inverse mappings in DC resistivity problems , 1992 .

[26]  3-D Inversion In Subsurface Electrical Surveying—I. Theory , 1994 .

[27]  Robert G. Ellis,et al.  The pole-pole 3-D Dc-resistivity inverse problem: a conjugategradient approach , 1994 .

[28]  Yutaka Sasaki,et al.  3-D resistivity inversion using the finite-element method , 1994 .

[29]  Jie Zhang,et al.  3-D resistivity forward modeling and inversion using conjugate gradients , 1995 .

[30]  R. Barker,et al.  Least-squares deconvolution of apparent resistivity pseudosections , 1995 .

[31]  K. Spitzer.  A 3-D FINITE-DIFFERENCE ALGORITHM FOR DC RESISTIVITY MODELLING USING CONJUGATE GRADIENT METHODS , 1995 .

[32]  R. Barker,et al.  Rapid least-squared inversion of apparent resisitivity pseudosections by a quasi-Newton method , 1996 .

[33]  Matthew J. Yedlin,et al.  Some refinements on the finite-difference method for 3-D dc resistivity modeling , 1996 .

[34]  Zhou Bing,et al.  Explicit expressions and numerical calculations for the Fréchet and second derivatives in 2.5D Helmholtz equation inversion , 1999 .