# Three-dimensional DC Resistivity Modeling using Galerkin Finite Element Method Composed by Tetrahedral Elements

## Authors

• Wahyu Srigutomo Physics of Earth and Complex System, Physics Department, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha No. 10, Bandung 40132
• Hairil Anwar Physics of Earth and Complex System, Physics Department, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha No. 10, Bandung 40132
• Eleonora Agustine Geophysics Department, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Jalan Raya Bandung-Sumedang Km. 21, Jatinangor, Sumedang
• Harry Mahardika Physics of Earth and Complex System, Physics Department, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha No. 10, Bandung 40132

## Keywords:

apparent resistivity, conjugate gradient method, forward modeling, Galerkin finite element, Wenner configuration.

## Abstract

Successful interpretation of DC resistivity data depends on the availability of a proper forward modeling scheme. In this study, a three-dimensional DC resistivity forward modeling scheme was developed using the finite element method. The finite element equations were obtained using a weakened form of the weighted-residual method called the Galerkin method. Discretization of the modeling domain was carried out by dividing it into smaller three-dimensional blocks and subdividing each block into five tetrahedral elements. A linear interpolation function was employed and elemental linear equations were set up, followed by formation of global matrix systems of equation and incorporation of proper boundary conditions. The conjugate gradient method was applied to solve the global system of equations, which in this study was proven to be more efficient than a direct solver, contributing to a 67% time reduction. Using a Wenner array configuration, comparison with theoretical calculation of the electric potential for a homogeneous model yielded a relative error of 3.66%. To confirm the applicability of this forward modeling scheme, apparent resistivity profiles for several basic three-dimensional subsurface resistivity models were compared with the analytical profiles, yielding an acceptable level of fitting.

## References

Dieter, K., Paterson, N.R. & Grant, F.S., IP and Resistivity Type Curves

for Three-Dimensional Bodies, Geophysics, 34, pp. 615-632, 1969.

Okabe, M., Boundary Element Method for the Arbitrary Inhomogeneities Problem in Electrical Prospecting, Geophys. Prospect., 29, pp. 39-59, 1981.

Xu, S.Z., The Boundary Element Method in Geophysics, Geophysical Monograph Series Number 9, SEG, 2001.

Furman, A., Warrick, A.W. & Ferre, T.P.A., Electrical Potential Distributions in Response to Applied Current in a Heterogeneous Subsurface, Solution for Circular Inclusions, Vadose Zone Journal, 1(2), pp. 273-280, 2002.

Mufti, I.R., Finite-Difference Resistivity Modeling for Arbitrarily Shaped Two-Dimensional Structures, Geophysics, 41, pp. 62-78, 1976.

Dey, A. & Morrison, H.F., Resistivity Modeling for Arbitrarily Shaped Three-Dimensional Structures, Geophysics, 44(4), pp. 753-780, 1979.

Zhang, J., Mackie, R.L. & Madden, T.R., 3-D Resistivity Forward Modeling and Inversion Using Conjugate Gradients, Geophysics, 60(5), pp. 1313-1325, 1995.

Coggon, J.H., Electromagnetic and Electrical Modeling by the Finite Element Method, Geophysics, 36(1), pp. 132-155, 1971.

Pridmore, D.F., Hohmann, G.W., Ward, S.H. & Sill, W.R., An Investigation of Finite-Element Modelling for Electrical and Electromagnetical Data in Three Dimensions, Geophysics, 46, pp. 1009-1024, 1980.

Sasaki, Y., 3-D Resistivity Inversion Using the Finite-Element Method,

Geophysics, 59(12), pp. 1839-1848, 1994.

Li, Y. & Spitzer, K., Three-Dimensional DC Resistivity Forward Modelling Using Finite Elements in Comparison with Finite-difference Solutions, Geophys. J. Int., 151, 924-934, 2002.

Rucker, C., Gunther, T. & Spitzer, K., Three-Dimensional Modelling and Inversion of Dc Resistivity Data Incorporating Topography I: Modelling, Geophys. J. Int., 166, pp. 495-505, 2006.

Jin, J.M., The Finite Element Method in Electromagnetics, 3rd ed., John Wiley & Sons, Inc., 2014.

Bhattacharya, B.B. & Shalivahan, Geoelectric Methods Theory and Applications, McGraw Hill Education (India) Private Limited, 2016.

Telford, W.M., Geldart, L.P. & Sheriff, R.E., Applied Geophysics,

Cambridge University Press, pp. 522-577, 1990.

Polycarpou, A.C., An Introduction to the Finite Element

Method in Electromagnetism, Morgan & Claypool, 2006.

Reddy, J.N, An Introduction to the Finite Element Method, McGraw- Hill, 1993.

Blome, M., Maurer, H.R. & Schmidt, K., Advances in Three-Dimensional Geoelectric Forward Solver Techniques, Geophysical Journal International, 176(3), pp. 74-752, 2009.

Gunther, T., Rucker, C. & Spitzer, K., Three-Dimensional Modelling and Inversion of DC Resistivity Data Incorporating Topography - II: Inversion, Geophysical Journal International, 166(2), pp. 506-517, 2006.

Song, T., Liu, Y. & Wang, Y., Finite Element Method for Modeling 3D Resistivity Sounding on Anisotropic Geoelectric Media, Mathematical Problems in Engineering, Article ID 8027616, pp. 1-12, 2017.

Spitzer, K. & Wurmstich, B., Speed and Accuracy in 3D Resistivity Modeling, in Three-Dimensional Electromagnetics, eds. Oristaglio, M. & Spies, B., no. 7 in Geophysical Developments, Society of Exploration Geophysicists, 1999.

Holmes, M.H., Introduction to Scientific Computing and Data Analysis (Texts in Computational Science and Engineering), Springer International Publishing Switzerland, 2016.

Lowry, T., Allen, M.B. & Shive, P.N., Singularity Removal: A Refinement of Resistivity Modeling Techniques, Geophysics, 54(6), 766-774, 1989.

Zhao, S. & Yedlin, M.J., Some Refinements on the Finite-Difference Method for 3-D DC Resistivity Modeling, Geophysics, 61(5), 1301-1307, 1996.

Pan, K. & Tang, J., Advance 2.5-D and 3-D DC Resistivity Modelling Using an Extrapolation Cascadic Multigrid Method, Geophys. J. Int., 197, pp. 1459-1470, 2014.

Reynolds, J.M., An Introduction to Applied and Environment

Geophysics, John Willey & Sons, pp. 418-488, 1997.

2019-08-31

## How to Cite

Srigutomo, W., Anwar, H., Agustine, E., & Mahardika, H. (2019). Three-dimensional DC Resistivity Modeling using Galerkin Finite Element Method Composed by Tetrahedral Elements. Journal of Engineering and Technological Sciences, 51(4), 516-536. https://doi.org/10.5614/j.eng.technol.sci.2019.51.4.5

Articles