Pore-Size Measurement from Eigenvalues of Magnetic Resonance Relaxation

Nonground eigenvalues are widely disregarded in magnetic resonance relaxation measurements of porous media due to difficulties involved in their measurement, detection, and the derivation of physically meaningful parameters from them. Such nonground eigenvalues may be experimentally observed in relaxation measurements, such as the relaxation correlation of $T_1-T_2$, and yield information on the pore size and surface relaxivity of porous media without calibration through other independent measurements. Nonground eigenvalue analysis of $T_1-T_2$ measurements on Berea sandstone undertaken at three static magnetic fields produces pore sizes consistent with those obtained through x-ray microtomography and SEM measurements. Similar agreement is found for a Bentheimer sandstone with a more complex pore geometry. A phase-encoding imaging variant of this method measures the imbibition confinement-size profile in Berea sandstone. It is suggested that the existence of nonground eigenmodes may be much more prevalent in simple magnetic resonance relaxation measurements than previously considered. Therefore, it is possible to measure pore size by matching numerical Brownstein-Tarr solutions with those of experiments in a wide variety of samples and magnetic resonance methods.

Figure 1

Surface relaxation at the pore surface enhances magnetic resonance relaxation rate in a sandstone rock sample saturated with an aqueous phase. Excited ¹H nuclei $◯1$ diffuse to the pore surface $◯2$, where they relax by paramagnetic ions, free electrons, other nuclear spins, or homonuclear dipole-dipole coupling $◯3$, and subsequently return to the bulk of the pore $◯4$. This process is significantly affected by mass diffusivity and surface relaxivity.

Figure 2

Magnetization $M_z(\mathbf{r},t)$ for (a) fast-diffusion regime, $N_{\mathrm{BT},1}=1.6\times {10}^{-3}$, ${\rho }_1=0.2\mu \mathrm{m/s}$, and (b) intermediate-diffusion regime, $N_{\mathrm{BT},1}=1.6$, ${\rho }_1=200\mu \mathrm{m/s}$. Magnetization profile reduces homogeneously for (a) fast-diffusion regime. Magnetization profile of (b) intermediate-diffusion regime, however, is the summation of several eigenfunctions with their corresponding relaxation times. In both cases, $l=20\mu \mathrm{m}$, $T_{1b}=3\mathrm{s}$, and $D=2.5\times {10}^{-9}{\mathrm{m}}^2\mathrm{/s}$. $50$ eigenvalues and 1000 profile points are used in numerical calculations. Magnetization profiles are for (a) $t=0$, $0.1$, 1, $3$, and $20$ s and (b) $t=0$, $0.0003$, $0.001$, $0.0032$, $0.01$, $0.032$, $0.06$, $0.1$, $0.16$, and $1$ s. First three eigenfunctions and eigenvalues of (b) are shown in Fig. 3. Video of Fig. 2 is available as Supplemental Material [35].

Figure 3

First three eigenfunctions of the intermediate-diffusion regime system of Fig. 2. For the ground eigenstate $n=0$, ${\xi }_{1,0}=0.7910$, $T_{1,0}=0.0626\mathrm{s}$, $A_{1,0}=1.1016$, $I_{1,0}=0.9903$; for the first nonground eigenstate $n=1$, ${\xi }_{1,1}=3.3744$, $T_{1,1}=0.0035\mathrm{s}$, $A_{1,1}=-0.1282$, $I_{1,1}=0.0088$; and for the second nonground eigenstate $n=2$, ${\xi }_{1,2}=6.4074$, $T_{1,2}=0.0010\mathrm{s}$, $A_{1,2}=0.0379$, $I_{1,2}=0.0007$. Gray dashed and dotted lines show relaxation of the eigenfunction at $t=2$ and $5\mathrm{ms}$, respectively. It is only at early times that it is possible to observe the contribution of nonground eigenvalues, as they relax much faster than the ground eigenvalue.

Figure 4

INVSESPI method is a one-dimensional phase-encoding imaging variant of $T_1-T_2$ to measure confinement size in porous media.

Figure 5

Visualization of pore spaces in Berea (left) and Bentheimer (right) sandstones by x-ray microtomography. Boxes are $1000\times 1000\mu {\mathrm{m}}^2$ with a height of $860\mu \mathrm{m}$. It is readily apparent that Bentheimer has a larger pore size than Berea. X-ray microtomography data are available from Ref. [54] and are described in Ref. [55].

Figure 6

$\mathrm{Lo}{\mathrm{g}}_{10}[I(T_{1,p},T_{2,q})]$ corresponding to the first three eigenvalues, $P00$, $P11$, and $P22$_, of a planar geometry with $l=10\mu \mathrm{m}$, ${\rho }_1=200\mu \mathrm{m/s}$, ${\rho }_2=300\mu \mathrm{m/s}$, $D=2.5\times {10}^{-9}{\mathrm{m}}^2\mathrm{/s}$, and $T_{1b}=T_{2b}=3\mathrm{s}$ with a Gaussian noise of ${\sigma }_{\mathrm{noise}}=0.002$ at three regularization parameters of $\alpha =0.001$, $1$, and $1000$. Intensity is logarithmically scaled in the range of $[-4,-2]$ and exact coordinates of contributing time constants are shown as red circles. No nonground eigenvalue is observed at $\alpha =1000$. At $\alpha =0.001$, the first nonground eigenvalue is observed very close to the theoretically exact coordinates, $(T_{1,1},T_{2,1})=(3.5\mathrm{ms},3.3\mathrm{ms})$. Second eigenvalue may not be observed, even at small regularization parameters, since its corresponding intensity is smaller than ${\sigma }_{\mathrm{noise}}.$ White boxes and text show the intensity contribution and logarithmic mean of the coordinates of observed peaks.

Figure 7

$\mathrm{Lo}{\mathrm{g}}_{10}[I(T_{1,p},T_{2,q})]\mathrm{from}{T}_1-T_2$-correlation map of a Berea sandstone sample at 8.5 MHz with three regularization parameters $\alpha =0.01$, $1$, and $100$. Only the ground eigenvalue, $P00$, is observed at $\alpha =100$. Ground eigenvalue peak pearls, splits into smaller peaks, at $\alpha =0.01$. It is possible to observe nonground eigenvalues $P11$ and $P22$ at sufficiently small regularization parameter of $\alpha =0.01$ (see the Supplemental Material [35]). Relative intensities of $P00$, $P11$, and $P22$ peaks are $0.831$, $0.0728$, and $0.0473$, respectively.

Figure 8

$T_1-T_2$-correlation map of Berea sandstone at (a) 2.2, (b) 8.5, and (c) 98.8 MHz; all values are normalized before inversion and inverted with a regularization parameter of $\alpha =10$. Slope of the ground eigenvalue changes from one at 2.2 MHz to two at 98.8 MHz. This change in slope is due to fast transverse relaxation at 98.8 MHz due to high surface relaxivity or internal gradients. Similar pore sizes are calculated from three measurements. $T_1-T_2$-correlation map of Berea sandstone at 2.2 and 98.8 MHz may be found in Ref. [30] and the Supplemental Material [35], respectively.

Figure 9

$T_1-T_2$-correlation map of a Bentheimer sandstone sample at $2.2\mathrm{MHz}$ with three regularization parameters, $\alpha =0.01$, $1$, and $100$. $Lpp$ and $Spp$, respectively, refer to eigenvalues of relaxation for large and small pores. Large pores are associated with intergrain porosity and small pores are associated with water in feldspar fragments and clay regions. Volume of water in the intergrain space is approximately $12$ times that of the volume of water associated with feldspar fragments and clays. Nonground eigenvalues are marked with white circles of L11 and S11. See the Supplemental Material [35] for the $T_1-T_2$-correlation map as a function of $\alpha$.

Figure 10

Volumetric probability density of pore diameters obtained from SEM (dashed line) and XMT (solid line) for Berea (left) and Bentheimer (right) sandstones. Pore diameters from magnetic resonance measurements are shown as gray rectangles (NMR), with widths showing the perceived uncertainty and height showing the relative volume of each pore size for Bentheimer sandstone.

Figure 11

Confinement size (solid line) and signal amplitude (dashed line) in a Berea core plug undergoing the imbibition process (left) and the $T_1-T_2$-relaxation-correlation map of the sample at a position of $54\mathrm{mm}$. Water is injected from the bottom of the core plug, at small position values. Decreasing apparent pore size is observed due to partial saturation of pores closer to the top side of the core plug. At data discontinuity, the method is unable to detect the nonground eigenvalue and measure the confinement size. Markers show measurement quantities, while the lines are a guide for the eye.

Figure 12

$T_2$ distribution of Berea sandstone and its estimated ground and nonground eigenvalues at $2.2\mathrm{MHz}$. $T_2$ distribution (solid line) is measured using the CPMG method with an interecho spacing of $300\mu \mathrm{s}$. Varying l and ${\rho }_2$ and using a planar geometry for solving eigenvalues of the relaxation-diffusion equation lead to estimated (dashed line) contributions to the $T_2$ distribution by the dominant ground eigenvalues and a smaller nonground eigenvalue peak. Shortest relaxation-time peak is ascribed to surface water in the roughness of the pores and clay agglomerates.