Magnetization exchange in a single pore due to diffusion in internal fields: Simulation and experiment

Nuclear magnetic resonance measurements of spin relaxation are used in studies of liquid-saturated porous media where multidimensional relaxation correlations probe interpore diffusion and other transport processes. However, the magnetic susceptibility contrast between the solid and liquid results in pore-scale field inhomogeneities that influence all measurements of transverse $T_2$ relaxation time. In a previous publication we conjectured that experimentally observed exchange phenomena can correspond to intrapore diffusion between localized volumes of coherent magnetization generated by these internal gradients, rather than interpore diffusion. Here, we use a finite-element method to explore the decay of magnetization in a single pore (interstice between spheres) for the Carr-Purcell-Meiboom-Gill and two-dimensional $T_2\text{−}{T}_2$ exchange experiments. These simulations permit direct visualization of the time-dependent distribution of magnetization in the pore. The chosen grain size and susceptibility contrast are matched to values for Bentheimer sandstone, allowing for a comparison of simulated and experimental results. Despite the simplicity of the model pore geometry, the salient features of the magnetization decay observed experimentally are reproduced in simulation. We demonstrate that intrapore magnetization transfer can explain observations of diffusive exchange in porous materials characterized by a monomodal pore size distribution and large internal gradients.

Figure 1

Schematic of diffusion regimes. The dark gray regions indicate the conjectured range of the asymptotic diffusion behaviors [18, 20]: short time (ST), motional averaging (MAV), or localization (LOC). The light gray region indicates the conjectured preasymptotic regime. The diffusion behaviors explored in the present (and prior [19]) study are indicated by the $\times$ symbols. The consequence of increasing either $B_0$ or $t_{\mathrm{e}}$ is indicated by the respective arrows.

Figure 2

Pore geometry created in the comsol multiphysics software. Six identical spheres are arranged to define an interstice (a). A seventh sphere, highlighted red in (a), defines the closed pore in (b). This pore is encapsulated by a cylinder of rock in (c). The rock is defined as solid with a specified magnetic susceptibility. The pore is defined as liquid with specified magnetic susceptibility, diffusion coefficient, and bulk relaxation time. The “pore” surface is defined as a partially absorbing boundary with relaxivity (absorption strength) ${\rho }_2$. See online version for references to color.

Figure 3

Example of the magnetic field calculated in a model pore. The static field was $B_0=2\mathrm{T}$ ($z$ axis) with a susceptibility contrast of $\mathrm{\Delta }\chi =5.28\times {10}^{-6}$ between the liquid-filled pore and the solid rock. A two-dimensional (2D) plane ($z=0$) is defined for illustration (a); the deviation in magnetic field in this plane is expanded in (b) for clarity. The corresponding internal gradient map is shown in (c), where the image resolution is determined by the finite-element mesh.

Figure 4

Distributions of internal gradient generated by simulation in a single pore (solid lines) and estimated experimentally in Bentheimer sandstone (dashed lines) [19]. The abscissa have been normalized by the static magnetic field strengths of (a) $B_0=42\mathrm{mT}$, (b) $282\mathrm{mT}$, (c) $2\mathrm{T}$, (d) $4.7\mathrm{T}$, and (e) $9.4\mathrm{T}$. The vertical gray line in each panel indicates the theoretical value of $g_{\max }$ from Eq. (7).

Figure 5

Distributions of effective transverse relaxation time $T_{2,\mathrm{eff}}$ (a)–(e) measured experimentally on water-saturated Bentheimer sandstone [19] and (f)–(j) calculated by finite-element simulation in an isolated pore. The static magnetic field strength increases as (a), (f) $B_0=42\mathrm{mT}$, (b), (g) $282\mathrm{mT}$, (c), (h) $2\mathrm{T}$, (d), (i) $4.7\mathrm{T}$, and (e), (j) $9.4\mathrm{T}$. The area under each distribution has been normalized to unity. The legend in (f) indicates $t_{\mathrm{e}}$ and applies to all plots.

Figure 6

Data collapse plots after [40] (a)–(e) measured experimentally on water-saturated Bentheimer sandstone [19] and (f)–(o) calculated by finite-element simulation in an isolated pore. The data in (k)–(o) are the data in (f)–(j) replotted with an adjusted echo time scaling. The static magnetic field strength increased as (a), (f), (k) $B_0=42\mathrm{mT}$, (b), (g), (l) $282\mathrm{mT}$, (c), (h), (m) $2\mathrm{T}$, (d), (i), (n) $4.7\mathrm{T}$, and (e), (j), (o) $9.4\mathrm{T}$. For the experimental data, the CPMG decay at $42\mathrm{mT}$ and $t_{\mathrm{e}}=2\mathrm{ms}$ was taken as a proxy for “gradient-free” relaxation and used for normalization. In simulation a true gradient-free magnetization decay was calculated ($\mathrm{\Delta }\chi =0$). The vertical gray lines indicate the expected time of departure from unity according to Eq. (11). Color has been used to differentiate data acquired at different $t_{\mathrm{e}}$. [CPMG decays measured at certain echo times on the permanent magnet systems suffered artifactual enhanced decay due to coupling of power-line fields with the magnet poles. These data have been omitted from (a) and (b).] See online version for references to color.

Figure 7

Simulated magnetization exchange in an isolated pore. $T_2\text{−}{T}_2$ exchange plots were calculated with storage times of (b) $t_{\mathrm{s}}=5\mathrm{ms}$, (c) $50\mathrm{ms}$, and (d) $500\mathrm{ms}$ for $B_0=2\mathrm{T}$ and $t_{\mathrm{e}}=4\mathrm{ms}$. The contour intervals are the same in each plot. The dashed diagonal lines indicate equality of the $T_{2,\mathrm{eff}}$ axes. The projected marginal $T_{2,\mathrm{eff}}$ distribution is shown in (a) for reference. The spatial distribution of coherent magnetization in a 2D plane [defined in Fig. 3] is shown after storage times of (e) $t_{\mathrm{s}}=0$, (f) $t_{\mathrm{s}}=5\mathrm{ms}$, (g) $50\mathrm{ms}$, and (h) $500\mathrm{ms}$. In each case, transverse relaxation occurred for $4t_{\mathrm{e}}=16\mathrm{ms}$ before the magnetization was stored. The color bar applies to (e)–(h). See online version for references to color.

Figure 8

Increase in $T_2\text{−}{T}_2$ off-diagonal peak amplitude (integral volume) $M_{\mathrm{xp}}$, normalized by the total magnetization $M_{\mathrm{tp}}$, as a function of storage time. The solid line is a fit to the data using a solution to Eq. (5) with $M_{\mathrm{A}}=M_{\mathrm{B}}$, $R_{1,\mathrm{A}}=R_{1,\mathrm{B}}=0$, $R_{2,\mathrm{eff},\mathrm{A}}=33{\mathrm{s}}^{-1}$, and a range of $R_{2,\mathrm{eff},\mathrm{B}}=1.1$–$10{\mathrm{s}}^{-1}$.

Figure 9

Simulated magnetization in an isolated pore at $B_0=9.4\mathrm{T}$. The $T_{2,\mathrm{eff}}$ distribution in (a) was generated with $t_{\mathrm{e}}=2\mathrm{ms}$, repeated from Fig. 5. The spatial distribution of coherent magnetization in a 2D plane [defined in Fig. 3] at time $4t_{\mathrm{e}}=8\mathrm{ms}$ is shown in (b). The spatially resolved magnetization is nonzero everywhere. The color bar applies to (b). See online version for references to color.