An Image-Based Explicit Matrix-Free FEM Implementation with Lumped Mass for NMR Simulations

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 in...

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Bibliographic Details
Published in:Transport in porous media 2023-03, Vol.147 (1), p.35-57
Main Authors: Bez, Luiz F., Leiderman, Ricardo, Souza, André, Azeredo, Rodrigo B. de V., Pereira, André M. B.
Format: Article
Language:English
Subjects:
Civil Engineering
Classical and Continuum Physics
Computed tomography
Computer simulation
Earth and Environmental Science
Earth Sciences
Finite element method
Geotechnical Engineering & Applied Earth Sciences
Hydrogeology
Hydrology/Water Resources
Industrial Chemistry/Chemical Engineering
Lumping
Magnetic fields
Mass matrix
Mathematical models
Medical imaging
NMR
Nuclear magnetic resonance
Pore size
Pore size distribution
Simulation
Time integration
Time marching
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Summary: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 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. Performance of three explicit time-integration schemes is studied numerically with different coefficients, grid sizes, and time-step sizes. The lumping of the mass matrix allows the implementation of a fully explicit scheme while adding negligible amounts of numerical error.
ISSN:0169-3913
1573-1634
DOI:10.1007/s11242-022-01894-1