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Impact of heterogeneity upon the accuracy of the Eikonal solution using the Fast Marching Method

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Abstract

The Fast Marching Method (FMM) is an efficient tool for the solution of the Eikonal equation to characterize frontal propagation in heterogeneous and anisotropic media. Previous studies have applied the FMM in solving the Eikonal equation to obtain the “diffusive time of flight” (DTOF), which is used to characterize the pressure front propagation in subsurface porous media. In the DTOF (τ) based one-dimensional flow simulation, the accuracy of the pressure solution relies significantly upon the drainage volume characterization and the DTOF calculation, both of which are influenced by the reservoir heterogeneity. We study first order discretization schemes of the Eikonal equation in porous media with different levels of heterogeneity and determine the impact of the corresponding DTOF solutions upon the one-dimensional flow simulation. The local solution of the Eikonal equation is formulated based on a piecewise uniform approximation of the DTOF gradient that satisfies a causality requirement within each simplex that provides the available nodal DTOF values. A homogenization approach is developed for use within the local solution for corner point grid cells, in which effective properties are obtained across adjacent grid cells. Comparison of the computational cost and error from different discretizations of the Eikonal equation and the corresponding pressure solution with increasing heterogeneity, demonstrates the need for increased angular resolution compared to the most usual cell centered 5-point discretization. The study also demonstrates the need for more accurate solutions in the near well region, suggesting a multi-stencil approach with higher angular and linear resolution near the well, and a less accurate and less costly calculation in the bulk of the computational grid.

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Acknowledgements

We would like to thank members of the Texas A&M University MCERI (Model Calibration and Efficient Reservoir Imaging) Joint Industry Project and the Energi Simulation foundation for their financial support. We also acknowledge the support of Schlumberger for the use of their reservoir simulation software ECLIPSE.

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  1. Chengdu University of Technology, Chengdu, Sichuan, China

    Chen Li

  2. Texas A&M University, College Station, Texas, USA

    Chen Li & Michael J. King

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Correspondence to Chen Li.

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The authors acknowledge funding by the MCERI JIP at Texas A&M University and support by the Energi Simulation foundation, but otherwise affirm that there are no competing financial or non-financial interests associated with this work.

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Li, C., King, M.J. Impact of heterogeneity upon the accuracy of the Eikonal solution using the Fast Marching Method. Comput Geosci 27, 465–484 (2023). https://doi.org/10.1007/s10596-023-10204-7

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