A goal-oriented adaptive finite element method for 3D resistivity modeling using dual-error weighting approach

Abstract

A goal-oriented adaptive finite element (FE) method for solving 3D direct current (DC) resistivity modeling problem is presented. The model domain is subdivided into unstructured tetrahedral elements that allow for efficient local mesh refinement and flexible description of complex models. The elements that affect the solution at each receiver location are adaptively refined according to a goal-oriented posteriori error estimator using dual-error weighting approach. The FE method with adapting mesh can easily handle such structures at almost any level of complexity. The method is demonstrated on two synthetic resistivity models with analytical solutions and available results from integral equation method, so the errors can be quantified. The applicability of the numerical method is illustrated on a resistivity model with a topographic ridge. Numerical examples show that this method is flexible and accurate for geometrically complex situations.

Keywords

adaptive finite element / dual-error weighting approach / unstructured mesh / 3D resistivity

References

[1] Babuška I., Rheinboldt W. C. A-Posteriori Error Estimates for the Finite Element Method. International Journal for Numerical Methods in Engineering, 1978, 12(10): 1597-1615.

[2] Bank R. E., Xu J. C. Asymptotically Exact a Posteriori Error Estimators, Part II: General Unstructured Grids. SIAM Journal on Numerical Analysis, 2003, 41: 2313-2332.

[3] Barrett R., Berry M., Chan T. F., . Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2006 Philadelphia: Society for Industrial and Applied Mathematics

[4] Blome M., Maurer H. R., Schmidt K. Advances in Three-Dimensional Geoelectric Forward Solver Techniques. Geophysical Journal International, 2009, 176(3): 740-752.

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

[6] Huang J. G., Ruan B. Y., Bao G. S. Finite Element Method for IP Modeling on 3-D Geoelectric Section. Earth Science—Journal of China University of Geosciences, 2003, 28(3): 323-326.

[7] Hvoždara M., Kaikkonen P. The Boundary Integral Calculations of the Forward Problem for D.C. Sounding and MMR Methods for a 3-D Body near a Vertical Contact. Studia Geophysica et Geodætica, 1994, 38(4): 375-398.

[8] Key K., Weiss C. Adaptive Finite-Element Modeling Using Unstructured Grids: The 2D Magnetotelluric Example. Geophysics, 2006, 71(6): G291-G299.

[9] Geophysics, 2007, 72 2

[10] Li Y. G., Pek J. Adaptive Finite Element Modelling of Two-Dimensional Magnetotelluric Fields in General Anisotropic Media. Geophysical Journal International, 2008, 175(3): 942-954.

[11] Li Y. G., Spitzer K. Finite Element Resistivity Modelling for Three-Dimensional Structures with Arbitrary Anisotropy. Physics of the Earth and Planetary Interiors, 2005, 150(1–3): 15-27.

[12] Li Y. G., Spitzer K. Three-Dimensional DC Resistivity Forward Modelling Using Finite Elements in Comparison with Finite-Difference Solutions. Geophysical Journal International, 2002, 151(3): 924-934.

[13] Ovall J. S. Duality-Based Adaptive Refinement for Elliptic: [Dissertation], 2004 San Diego: University of California

[14] Ovall J. S. Asymptotically Exact Functional Error Estimators Based on Superconvergent Gradient Recovery. Numerical Mathematics, 2005, 102: 543-558.

[15] Penz S., Chauris H., Donno D., . Resistivity Modelling with Topography. Geophysical Journal International, 2013, 194(3): 1486-1497.

[16] Pridmore D., Hohmann G. W., Ward S. H., . An Investigation of Finite-Element Modeling for Electrical and Electromagnetic Modelling Data in Three Dimensions. Geophysics, 1981, 46: 1009-1024.

[17] Qiang J. K., Luo Y. Z. The Resistivity FEM Numerical Modeling on 3D Undulating Topography. Chinese J. Geophys., 2007, 50(5): 1606-1613.

[18] Ren Z. Y., Tang J. T. 3D Direct Current Resistivity Modeling with Unstructured Mesh by Adaptive Finite-Element Method. Geophysics, 2010, 75(1): H7-H17.

[19] Ruan B. Y., Xiong B., Xu S. Z. Finite Element Method for Modeling Resistivity Sounding on 3D Geoelectric Section. Earth Science—Journal of China University of Geosciences, 2001, 26(1): 73-77.

[20] Rücker C., Günther T., Spitzer K. Three-Dimensional Modelling and Inversion of DCResistivity Data Incorporating Topography-I. Modelling. Geophysical Journal International, 2006, 166(2): 495-505.

[21] Sasaki Y. 3-D Resistivity Inversion Using the Finite-lement Method. Geophysics, 1994, 59(12): 1839-1848.

[22] Si H. TETGEN: A 3D Delaunay Tetrahedral Mesh Generator, 2003

[23] Tang J. T., Wang F. Y., Xiao X., . 2.5-D DCResistivity Modeling Considering Flexibility and Accuracy. Journal of Earth Science, 2011, 22(1): 124-130.

[24] Wang W., Wu X. P., Spitzer K. Three-Dimensional DC Anisotropic Resistivity Modelling Using Finite-Elements on Unstructured Grids. Geophysical Journal International, 2013, 193(2): 734-746.

[25] Weiss C. J. A Matrix-Free Approach to Solving the Fully 3D Electromagnetic Induction Problem. 71st Annual International Meeting, SEG, Expanded Abstracts, 2001

[26] Wu X. P., Wang T. T. A 3-D Finite Element Resistivity Forward Modeling Using Conjugate Gradient Algorithm. Chinese J. Geophys., 2003, 46(3): 428-432.

[27] Xu S. Z. The Finite Element Method in Geophysics, 1994 Beijing: Science Press

[28] Xu S. Z., Liu B., Ruan B. Y. The Finite Element Method for Solving Anomalous Potential for Resistivity Surveys. Chinese J. Geophys., 1994, 37(S2): 511-515.

[29] Zhou B., Greenhalgh S. A. Finite Element Three-Dimensional Direct Current Resistivity Modelling: Accuracy and Efficiency Considerations. Geophysical Journal International, 2001, 145: 679-688.

[30] Zienkiewicz O. C., Taylor R. L. The Finite Element Method (5th Ed.), 2000