Lattice Boltzmann framework for accurate NMR simulation in porous media

Nuclear magnetic resonance (NMR) responses of fluids saturating porous media arise from complex relaxation-diffusion dynamics of polarized spins. These constitute a sensitive probe of the microstructure and are described by the Bloch-Torrey equations. An NMR simulation framework based on an augmented lattice Boltzmann method aimed at the fine-scale resolution of nuclear polarization density is presented. The approach encapsulates the time evolution of the full magnetization vector and naturally incorporates the mechanisms of diffusional transport. Spin dephasing mechanisms are fully resolved at tomogram voxel scale to account for magnetic field inhomogeneity. The approach is validated against analytical solutions of spin-echo decays for simple pore geometries. An application to a nano-computed-tomography image of chalk with inhomogeneous internal fields yields $T_2$ spectral measures in good agreement with experiment and illustrates the spatial pore-scale dynamics of net magnetization. Findings establish the feasibility of the framework for pure diffusion and present an approach vector to modeling the evolution of magnetization under flow conditions.

Figure 1

D3Q27 basis $\left\{{\mathit{e}}_{0:26}\right\}$ illustrated in 3D perspective projection with respective weights $\left\{w_{0:26}\right\}$: twenty-six neighbors in addition to the rest particle.

Figure 2

Diagram of finite-difference phase increment operation on a local magnetization vector $q_i$ in ${\mathbb{R}}^3$.

Figure 3

Illustration of NMR-LBM surface relaxation bounce-back principle with a D2Q5 lattice example (boundary voxels are depicted in gray; fluid voxels are in white).

Figure 4

Analytical solutions alongside simulated spin-echo intensity decays under restricted diffusion conditions for [(a) and (d)] planar, [(b) and (e)] circular, and [(c) and (f)] spherical geometries (1-, 2-, and 3-spheres, respectively).

Figure 5

Comparison of NMR-LBM and RW decay data alongside fast diffusion limit (FDL) analytical models for spherical pores $5\mu$m in diameter for various Cartesian discretizations: (a) $d=20\epsilon , \epsilon =250$ nm; (b) $d=40\epsilon , \epsilon =125$ nm; (c) $d=60\epsilon , \epsilon =83.3$ nm; and (d) $d=80\epsilon , \epsilon =62.5$ nm.

Figure 6

Three-dimensional image of the chalk subsample digital segmentation used for simulations: cubic voxels of side length $\epsilon =64$ nm, “0” labels for void space, and “1” labels for chalk solid matrix.

Figure 7

Spectral profiles of (a) experimental acquisitions and (b) NMR-LBM simulations and (c) random-walk simulations on chalk at 2 MHz; $t_E=100\mu$s and $t_E=1000\mu$s.

Figure 8

Spectral profiles of (a) experimental acquisitions and (b) NMR-LBM simulations on chalk at 43 MHz; $t_E=100\mu$s and $t_E=1000\mu$s.

Figure 9

Clockwise from top right: Two-dimensional $xy$ plane slices through the ${380}^3$ transverse magnetization density map taken during along a CPMG sequence at 4, 159, and 318 echoes at $t_E=100\mu$s; 2D $xy$ slice through the chalk tomogram at the same $z$ coordinate. Chalk solid matrix is illustrated in white and void is in black.

Figure 10

Cross section through a subsection of a micro-CT tomogram of the chalk sample at a resolution of $\epsilon =688$ nm with a field of view of ${1200}^2$ voxels, illustrating both the predominant matrix porosity, and larger-scale “vugs” as well as larger-scale solid grains.