Recent Developments on the Numerical Upscaling and Homogenization of the Quasi-Static Maxwell's Equations

Luz Angelica Caudillo-Mata, Eldad Haber Geophysics Department, The University of British Columbia. 4013-2207 Main Mall, Vancouver, B.C., Canada. Z. C. V6R 1Z4 Key words: Finite Volume, Quasi-static Maxwell's Equations, Optimization, Upscaling, Homogenization, Exploration Geophysics. Abstract: Mineral exploration has exploited the application of mathematical modelling and inversion methods to electromagnetic data by creating a thoughtful workflow that assists in the identification of potential geological targets, the understanding of the larger scale stratigraphy and structure in which a deposit might be located, or delineating finer scale detail in an existing deposit. [1] In recent years, electromagnetic modelling and inversion techniques based on finite volume and finite elements have been studied extensively due to their usefulness in theory as well as in practice [2]. Although the theoretical foundation for these methods is straight-forward, it can face major difficulties when used to simulate realistic situations. One of the fundamental issues is modelling the vast heterogeneity of geological targets in terms of scale, magnitude and anisotropy. Robust and accurate simulations require very fine meshes, especially when the earth is highly heterogeneous. Such meshes are difficult-to-work-with and may lead to very expensive-to-compute simulations when considering large earth-multiscale scenarios. For instance, geological characterizations typically contain on the order of 1e7 to 1e8 cells. These models, which are referred as fine models, represent geological variations on very fine scales vertically, though their areal resolution is still relatively coarse [3]. Numerical upscaling is a mathematical procedure that strive to develop coarse scale models to accurately approximate fine scale ones. Therefore, it is a useful resource to alleviate the computational cost. Upscaling of Maxwell's equations presents big challenges such as choosing the appropriate upscaling method and the assessment of risk and uncertainty in geological characterizations. In this work, we present insights on how to build an efficient framework that allows for homogenization and upscaling of the Maxwell's Equations' coefficients trying to address the challenges mentioned above. One example on geological characterization is provided. [1] D. Oldenburg and D. Pratt, 'Geophysical inversion for mineral exploration: A decade of progress in theory and practice,' Proceedings of exploration, 2007. [2] S. H. Ward and G. W. Hohmann, "Electromagnetic theory for geophysical applications", Electromagnetic Methods in Applied Geophysics, Issue 1. pp. 131-311, 1998. Soc. Expl. Geophys. [3] L. Durlofsky, 'Upscaling of geocellular models for reservoir flow simulation: a review of recent progress,' 7th International Forum on Reservoir Simulation Bühl/Baden-Baden, Germany, pp. 1-58, 2003.