DOI: https://doi.org/10.26089/NumMet.v21r326

Numerical estimation of electrical resistivity in digital rocks using GPUs

Authors

  • T.S. Khachkova
  • V.V. Lisitsa
  • G.V. Reshetova
  • V.A. Tcheverda

Keywords:

digital rock physics
finite-difference method
iterative methods
electrical resistivity
numerical upscaling

Abstract

We present a numerical algorithm for computing the electric field in digital rock samples and estimating their electrical resistivity (conductivity). The main peculiarity of the algorithm is its applicability tostrongly heterogeneous models including partially saturated and multi-mineral rock samples. The algorithm is based on the iterative Krylov-type solver preconditioned by the inverse Laplace operator for homogeneous media. The preconditioner is computed using the spectral method in directions orthogonal to the direction of the main electric current, whereas the series of 1D problems are solved by the Thomas algorithm. We implement the algorithm using GPUs, which allows us to use a single GPU to solve the problems for samples whose size is up to 4003 voxels.

New computational technologies

Downloads

Published

2020-09-27

Issue

Vol. 21 (2020): Issue 3.

Section

Methods and algorithms of computational mathematics and their applications

Author Biographies

T.S. Khachkova

Trofimuk Institute of Petroleum Geology and Geophysics of SB RAS
• Researcher

V.V. Lisitsa

Trofimuk Institute of Petroleum Geology and Geophysics of SB RAS
• Head of Laboratory

G.V. Reshetova

The Institute of Computational Mathematics and Mathematical Geophysics of SB RAS (ICM&MG SB RAS)
• Leading Researcher

V.A. Tcheverda

Trofimuk Institute of Petroleum Geology and Geophysics of SB RAS
• Head of Laboratory

References

  1. Ya. V. Bazaikin, D. R. Kolyukhin, V. V. Lisitsa, et al., “Effect of CT-Image Scale on Macro-Scale Properties Estimation,” Tekhnol. Seismorazved., No. 2, 38-47 (2016).
  2. R. V. Vasilyev, K. M. Gerke, M. V. Karsanina, and D. V. Korost, “Solution of the Stokes Equation in Three-Dimensional Geometry by the Finite-Difference Method,” Mat. Model. 27 (6), 67-80 (2015) [Math. Models Comput. Simul. 8 (1), 63-72 (2016)].
  3. K. V. Voronin and S. A. Solovyev, “Solution of the Helmholtz Problem Using the Preconditioned Low-Rank Approximation Technique,” Vychisl. Metody Programm. 16, 268-280 (2015).
  4. K. A. Gadylshina, T. S. Khachkova, and V. V. Lisitsa, “Numerical Modeling of Chemical Interaction between a Fluid and Rocks,” Vychisl. Metody Programm. 20, 457-470 (2019).
  5. A. V. Likhachov, “Allocation of Three Brightness Levels on a Noisy Image,” Vychisl. Metody Programm. 21, 180-186 (2020).
  6. D. A. Neklyudov, I. Yu. Silvestrov, and V. A. Tcheverda, “A 3D Helmholtz Iterative Solver with a Semi-Analytical Preconditioner for Acoustic Wavefield Modeling in Seismic Exploration Problems,” Vychisl. Metody Programm. 15, 514-529 (2014).
  7. G. V. Reshetova and T. S. Khachkova, “A Numerical Method to Estimate the Effective Elastic Moduli of Rocks from Two- and Three-Dimensional Digital Images of Rock Core Samples,” Vychisl. Metody Programm. 18, 416-433 (2017).
  8. A. A. Samarskii, The Theory of Finite Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).
  9. S. A. Solovyev, “Application of the Low-Rank Approximation Technique in the Gauss Elimination Method for Sparse Linear Systems,” Vychisl. Metody Programm. 15, 441-460 (2014).
  10. T. S. Khachkova, Ya. V. Bazaikin, and V. V. Lisitsa, “Use of the Computational Topology to Analyze the Pore Space Changes during Chemical Dissolution,” Vychisl. Metody Programm. 21, 41-55 (2020).
  11. Y. Al-Khulaifi, Q. Lin, M. J. Blunt, and B. Bijeljic, “Pore-Scale Dissolution by CO_2 Saturated Brine in a Multimineral Carbonate at Reservoir Conditions: Impact of Physical and Chemical Heterogeneity,” Water Resour. Res. 55 (4), 3171-3193 (2019).
  12. Y. Al-Khulaifi, Q. Lin, M. J. Blunt, and B. Bijeljic, “Pore-Scale Dissolution by CO_2 Saturated Brine in a Multi-Mineral Carbonate at Reservoir Conditions: Impact of Physical and Chemical Heterogeneity,” (2019)
    doi 10.5285/52b08e7f-9fba-40a1-b0b5-dda9a3c83be2 . Cited August 15, 2020
  13. F. O. Alpak, B. Riviere, and F. Frank, “A Phase-Field Method for the Direct Simulation of Two-Phase Flows in Pore-Scale Media Using a Non-Equilibrium Wetting Boundary Condition,” Computat. Geosci. 20, 881-908 (2016).
  14. H. Andr854, N. Combaret, J. Dvorkin, et al., “Digital Rock Physics Benchmarks - Part I: Imaging and Segmentation,” Comput. Geosci. 50, 25-32 (2013).
  15. H. Andr854, N. Combaret, J. Dvorkin, et al., “Digital Rock Physics Benchmarks - Part II: Computing Effective Properties // Comput. Geosci. 50, 33-43 (2013).
  16. Y. Bazaikin, B. Gurevich, S. Iglauer, et al., “Effect of CT Image Size and Resolution on the Accuracy of Rock Property Estimates,” J. Geophys. Res.: Solid Earth 122 (5), 3635-3647 (2017).
  17. M. Belonosov, V. Kostin, D. Neklyudov, and V. Tcheverda, “3D Numerical Simulation of Elastic Waves with a Frequency-Domain Iterative Solver,” Geophysics 83 (6), T333-T344 (2018).
  18. N. Dugan, L. Genovese, and S. Goedecker, “A Customized 3D GPU Poisson Solver for Free Boundary Conditions,” Comput. Phys. Commun. 184 (8), 1815-1820 (2013).
  19. H. Edelsbrunner and J. Harer, Computational Topology: An Introduction (AMS Press, Providence, 2010).
  20. K. M. Gerke, M. V. Karsanina, and R. Katsman, “Calculation of Tensorial Flow Properties on Pore Level: Exploring the Influence of Boundary Conditions on the Permeability of Three-Dimensional Stochastic Reconstructions,” Phys. Rev. E 100 (2019).
    doi 10.1103/PhysRevE.100.053312
  21. E. Haber, U. M. Ascher, D. A. Aruliah, and D. W. Oldenburg, “Fast Simulation of 3D Electromagnetic Problems Using Potentials,” J. Comput. Phys. 163 (1), 150-171 (2000).
  22. E. Haber and U. M. Ascher, “Fast Finite Volume Simulation of 3D Electromagnetic Problems with Highly Discontinuous Coefficients,” SIAM J. Sci. Comput. 22 (6), 1943-1961 (2001).
  23. P. Iassonov, T. Gebrenegus, and M. Tuller, “Segmentation of X-Ray Computed Tomography Images of Porous Materials: A crucial Step for Characterization and Quantitative Analysis of Pore Structures,” Water Resour. Res. 45 (2009).
    doi 10.1029/2009WR008087
  24. H. Johansen and Ph. Colella, “A Cartesian Grid Embedded Boundary Method for Poisson’s Equation on Irregular Domains,” J. Comput. Phys. 147 (1), 60-85 (1998).
  25. J. L. Jodra, I. Gurrutxaga, J. Muguerza, and A. Yera, “Solving Poisson’s Equation Using FFT in a GPU Cluster,” J. Parallel Distr. Comput. 102, 28-36 (2017).
  26. A. Kameda, J. Dvorkin, Y. Keehm, et al., “Permeability-Porosity Transforms from Small Sandstone Fragments,” Geophysics 71 (2006).
    doi 10.1190/1.2159054
  27. M. V. Karsanina and K. M. Gerke, “Hierarchical Optimization: Fast and Robust Multiscale Stochastic Reconstructions with Rescaled Correlation Functions,” Phys. Rev. Lett. 121 (2018).
    doi 10.1103/PhysRevLett.121.265501
  28. V. Kostin, S. Solovyev, A. Bakulin, and M. Dmitriev, “Direct Frequency-Domain 3D Acoustic Solver with Intermediate Data Compression Benchmarked Against Time-Domain Modeling for Full-Waveform Inversion Applications,” Geophysics 84 (2019).
    doi 10.1190/geo2018-0465.1
  29. C. Madonna, B. Quintal, M. Frehner, et al., “Synchrotron-Based X-Ray Tomographic Microscopy for Rock Physics Investigations,” Geophysics 78 (2013).
    doi 10.1190/geo2012-0113.1
  30. A. Pleshkevich, D. Vishnevskiy, and V. Lisitsa, “Sixth-Order Accurate Pseudo-Spectral Method for Solving One-Way Wave Equation,” Appl. Math. Comput. 359, 34-51 (2019).
  31. E. H. Saenger, M. Lebedev, and D. Uribe, “Analysis of High-Resolution X-Ray Computed Tomography Images of Bentheim Sandstone under Elevated Confining Pressures,” Geophys. Prospect. 64 (4), 848-859 (2016).
  32. V. Shulakova, M. Pervukhina, T. M. Müller, et al., “Computational Elastic Up-scaling of Sandstone on the Basis of X-Ray Micro-Tomographic Images,” Geophys. Prospect. 61 (2), 287-301 (2013).
  33. K. Stüben, “A Review of Algebraic Multigrid,” J. Comput. Appl. Math. 128 (1-2), 281-309 (2001).
  34. U. Wollner, A. Kerimov, and G. Mavko, “Scale and Boundary Effects on the Effective Elastic Properties of 2-D and 3-D Non-REV Heterogeneous Porous Media,” J. Geophys. Res.: Solid Earth 123 (7), 5451-5465 (2018).
  35. X. Zhan, L. M. Schwartz, M. N. Toksöz, et al., “Pore-Scale Modeling of Electrical and Fluid Transport in Berea Sandstone,” Geophysics 75 (2010).
    doi 10.1190/1.3463704

License

Copyright (c) 2020 Numerical methods and programming

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.