Abstract
NMR techniques are key in the study of porous reservoir rock, both experimentally and numerically. The \(T_2\) relaxation process, the most common application of NMR, measures the loss of coherence of transversal magnetization and strongly depends on the fluid/matrix interaction—thus providing useful insights into the pore size distribution of a rock sample. The pore space is often studied at the μm scale through the use of micro-CT images, which are formed by stacks of images with hundreds of thousands of pixels each, posing significant challenges to numerical simulations. In this paper, we present an image-based, fully explicit, and matrix-free finite element implementation for the simulation of the \(T_2\) relaxation process that is capable of handling such large problems. The utilized mathematical model considers relaxation due to bulk effects and surface relaxivity, not taking into account the effects of magnetic field gradients. We propose the usage of stable time marching schemes that use hyperbolization as means of acquiring stability with large time-steps. We compare the numerical performance of different time-marching schemes, showing that the application of the Leap-Frog method in a hyperbolized form of the equation can give the best trade-off between memory use and numerical convergence. Additionally, we show that the use of a lumped mass matrix allows for a fully explicit and simpler implementation while adding negligible amounts of numerical error.
Article Highlights
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A novel image-based and matrix-free methodology combined with explicit stable methods permits the solution of large problems with low time and low memory consumption.
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Performance of three explicit time-integration schemes is studied numerically with different coefficients, grid sizes, and time-step sizes.
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The lumping of the mass matrix allows the implementation of a fully explicit scheme while adding negligible amounts of numerical error.
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References
Beltrachini, L., Taylor, Z.A., Frangi, A.F.: A parametric finite element solution of the generalised Bloch–Torrey equation for arbitrary domains. J. Magn. Reson. 259, 126–134 (2015). https://doi.org/10.1016/j.jmr.2015.08.008
Beltrachini, L., Taylor, Z.A., Frangi, A.F.: An Efficient Finite Element Solution of the Generalised Bloch-Torrey Equation for Arbitrary Domains. In: Fuster, A., Ghosh, A., Kaden, E., Rathi, Y., Reisert, M. (eds) Computational Diffusion MRI. Mathematics and Visualization. Springer, Cham (2016)
Benavides, F., Leiderman, R., Souza, A., Carneiro, G., Bagueira, R.: Estimating the surface relaxivity as a function of pore size from NMR T2 distributions and micro-tomographic images. Comput. Geosci. 106, 200–208 (2017). https://doi.org/10.1016/j.cageo.2017.06.016
Bergman, D.J., Dunn, K.-J., Schwartz, L.M., Mitra, P.P.: Self-diffusion in a periodic porous medium: a comparison of different approaches. Phys. Rev. E 51, 3393–3400 (1995). https://doi.org/10.1103/PhysRevE.51.3393
Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65–98 (2017). https://doi.org/10.1137/141000671
Brownstein, K.R., Tarr, C.E.: Importance of classical diffusion in NMR studies of water in biological cells. Phys. Rev. A 19, 2446–2453 (1979). https://doi.org/10.1103/PhysRevA.19.2446
Burden, R.L., Faires, J.D.: Numerical Analysis. The Prindle, Weber and Schmidt Series in Mathematics. Nineth edition, PWS-Kent Publishing Company, Boston (2010)
Coates, G.R., Xiao, L., Prammer, M.G.: NMR logging: principles and applications. 234 pages. Haliburton Energy Services, Houston, (1999)
Du Fort, E.C., Frankel, S.P.: Stability conditions in the numerical treatment of parabolic differential equations. Mat. Tables Other Aids Comput. 7(43), 135–152 (1953). https://doi.org/10.1090/S0025-5718-1953-0059077-7
Dunn, K.J., Bergman, D.J., Latorraca, G.A.: Nuclear Magnetic Ressonance: Petrophysical and Logging Applications. Pergamon, Handbook of geophysical exploration (2002)
Fordham, E.J., Kenyon, W.E., Ramakrishnan, T.S., Schwartz, L.M., Wilkinson, D.J.: Forward models for nuclear magnetic resonance in carbonate rocks. Log Anal 40, 260–270 (1999)
Galloway, D., Ivers, D.: Slow-burning instabilities of Dufort–Frankel finite differencing. ANZIAM J (2021). https://doi.org/10.21914/anziamj.v63.15640
Hughes, T.J.R., Levit, I., Winget, J.: An element-by-element solution algorithm for problems of structural and solid mechanics. Comput. Methods Appl. Mech. Eng. 36(2), 241–254 (1983). https://doi.org/10.1016/0045-7825(83)90115-9
Lucas-Oliveira, E., Araujo-Ferreira, A., Trevizan, W.A., Fortulan, C.A., Bonagamba, T.J.: Computational approach to integrate 3D X-ray microtomography and NMR data. J. Magn. Reson. 292, 16–24 (2018). https://doi.org/10.1016/j.jmr.2018.05.001
Mohnke, O., Klitzsch, N.: Microscale simulations of NMR relaxation in porous media considering internal field gradients. Vad. Zone J. 9(4), 846 (2010). https://doi.org/10.2136/vzj2009.0161
Myshetskaya, E.E., Tishkin, V.F.: Estimates of the hyperbolization effect on the heat equation. Comput. Math. Math. Phys. 55, 1270–1275 (2015). https://doi.org/10.1134/S0965542515080138
Nguyen, D., Li, J.R., Grebenkov, D., Le Bihan, D.: A finite elements method to solve the Bloch–Torrey equation applied to diffusion magnetic resonance imaging. J. Comput. Phys. 263, 283–302 (2014). https://doi.org/10.1016/j.jcp.2014.01.009
Nguyen, V.D., Jansson, J., Hoffman, J., Li, J.R.: A partition of unity finite element method for computational diffusion MRI. J. Comput. Phys. 375, 271–290 (2018). https://doi.org/10.1016/j.jcp.2018.08.039
Ryu, S. (2009). Effects of spatially varying surface relaxivity and pore shape on NMR logging. In: SPWLA 49th Annual Logging Symposium, pp. 25-28, , Edinburgh, Scotland, paper BB
Schwartz, L.M., Johnson, D.L., Mitchell, J., Chandrasekera, T.C., Fordham, E.J.: Modeling two-dimensional magnetic resonance measurements in coupled pore systems. Phys. Rev. E 88, 032813 (2013). https://doi.org/10.1103/PhysRevE.88.032813
Senturia, S.D., Robinson, J.D.: Nuclear spin-lattice relaxation of liquids confined in porous solids. Soc. Petrol. Eng. J. 10(03), 237–244 (1970). https://doi.org/10.2118/2870-PA
Wilkinson, D.J., Johnson, D.L., Schwartz, L.M.: Nuclear magnetic relaxation in porous media: the role of the mean lifetime \(\tau ({\rho}, D)\). Phys. Rev. B (1991). https://doi.org/10.1103/PhysRevB.44.4960
Acknowledgements
This research was carried out in association with the ongoing R &D project registered as ANP n\(^o\) 21289–4, “Desenvolvimento de modelos matemáticos, estatísticos e computacionais para o aperfeiçoamento da caracterização petrofísica de reservatórios por Ressonância Magnética Nuclear (RMN)” (UFF/Shell Brasil/ANP), sponsored by Shell Brasil under the ANP R &D levy as “Compromisso de Investimentos com Pesquisa e Desenvolvimento”. The authors also recognize the support from CAPES, CNPq and FAPERJ.
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LFB contributed to software, methodology, validation, investigation, data curation, writing—original draft, visualization. André MBP contributed to conceptualization, writing—review & editing, supervision. RL contributed to writing—review & editing, project administration, supervision. AS contributed to methodology, writing—review & editing. RB de VA contributed to writing—review & editing, project administration.
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Bez, L.F., Leiderman, R., Souza, A. et al. An Image-Based Explicit Matrix-Free FEM Implementation with Lumped Mass for NMR Simulations. Transp Porous Med 147, 35–57 (2023). https://doi.org/10.1007/s11242-022-01894-1
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DOI: https://doi.org/10.1007/s11242-022-01894-1