Nuclear-magnetic-resonance relaxation rates for fluid confined to closed, channel, or planar pores

Fast-field-cycling nuclear-magnetic-resonance (FFC NMR) experimentation measures the spin-lattice relaxation rate $T_1^{-1}=R_1$ as a function of NMR frequency $f$. It is a proven technique for probing the nanoscale dynamics of ${}^1\mathrm{H}$ spins over multiple timescales. In many porous systems, fluid is confined to quasi-zero-dimensional (closed), quasi-one-dimensional (channel), or quasi-two-dimensional (planar) pores. Expressions are presented for $R_1\left(f\right)$ providing simulated dispersion curves for closed, channel, and planar pores where relaxation is associated with fluid movement relative to fixed relaxation centers in the solid. It is shown that fluid confined to nanosized (1–5 nm) spaces can be identified by submillisecond relaxation times for any geometry. The shape and magnitude of $R_1\left(f\right)$ is shown to be sensitive to pore geometry at low frequency only if relaxation is dominated by the motion of pore bulk fluid. Relaxation in most porous material is dominated by slow-moving surface fluid. Here, the pore geometry can only be distinguished if the relaxation center density is known a priori and then only at very low frequency. Systems containing mixtures of closed, channel, and planar pores of similar characteristic dimension $h$ would present as three peaks at low frequency with closed pores providing the largest $R_1$ and planar pores the smallest. Pore size and shape variability in real systems is shown to diminish the ability to distinguish the three peaks. We show that the ratio $T_1/T_2$, where $T_2$ is the spin-spin relaxation time, is a complex function of $h$, the surface diffusion time constant ${\tau }_{\ell }$, and NMR frequency for $f>1$ MHz. It is shown that measurements of $T_1/T_2$ at 20 MHz in cement paste and hydrocarbon rock capture information on both ${\tau }_{\ell }$ and $h$.

Figure 1

A schematic diagram is presented of a cuboidal closed pore in which the fluid is contained in a region $h_1\times h_2\times h_3$. The upper and lower panels show the projections of the spin-pair vectors in the $(x,z)$ and $(y,z)$ planes, respectively, at $t=0$ (${\mathbf{r}}_{\mathbf{0}}$) and at time $t$ ($\mathbf{r}$). The dashed lines (red) represent the effective layer of paramagnetic spin density of size $3h_1\times 3h_2$ placed a distance $d$ from the fluid volume. The cross (red) indicates a paramagnetic impurity and acts as the origin of the spin-pair coordinate system $(x,y,z)$ for the particular spin pair illustrated.

Figure 2

A schematic diagram of a cuboidal closed pore of dimensions $h_1\times h_2\times h_3$ is presented showing the bulk fluid environment. The diagram shows the $x-z$ plane. The darker dashed line (red) represents the effective layer of paramagnetic spin density and has dimensions $3h_1\times 3h_2$ as justified in the text. Equivalent paramagnetic layers are shown as lighter dashed lines (pink).

Figure 3

A schematic diagram of a cuboidal closed pore of dimensions $h_1\times h_2\times h_3$ is presented showing the surface fluid environments. The diagram shows the $x-z$ plane. The three assumed non-mixing surface layer ensembles are labelled A, B and C. The dashed line (red) represents the effective layer of paramagnetic spin density and has dimensions $3(h_1+2\delta )\times 3(h_2+2\delta )$.

Figure 4

A schematic diagram of a channel pore of infinite length in $x$ and with a rectangular cross section of dimensions $h_2\times h_3$ is presented. The upper and lower figures show the projections on the spin-pair vectors in the $(x,z)$ and $(y,z)$ planes, respectively, at $t=0$ (${\mathbf{r}}_{\mathbf{0}}$) and at time $t$ ($\mathbf{r}$). The cross (red) indicates a paramagnetic impurity and dashed lines (red) represent the effective layer of paramagnetic spin density placed a distance $d$ from the pore.

Figure 5

Schematic diagram illustrating random orientation of identically sized (a) closed cubic pores, (b) channel pores with square cross section, and (c) planar pores.

Figure 6

(a) The dipolar correlation function $G\left(t\right)$ and (b) the spin-lattice relaxation rate $R_{1b}$ are presented for a system of fluid-filled cubic closed pores. In (a), the vertical lines indicate the maximum magnitude of the gradient.

Figure 7

(a) The dipolar correlation function $G\left(t\right)$ is presented for a system of fluid-filled channel pores with a square cross section. Guide slopes labeled $t^{-3/2}$ and $t^{-1/2}$ are provided as $G(t\to \infty )$ for 3D and 1D motion, respectively. Results for the line channel are also presented. The corresponding spin-lattice relaxation rate $R_{1b}$ is presented in (b). The ${\omega }^{-1/2}$ form at low frequencies is shown for comparison. Parameters are indicated in Table 1.

Figure 8

(a) The dipolar correlation function $G\left(t\right)$ is presented for a system of fluid-filled planar pores of thickness $h$. Other parameters are indicated in Table 1. A guide slope labeled $t^{-2}$ is provided representing the long-time limit of $G\left(t\right)$. In (b), the corresponding spin-lattice relaxation rate $R_{1b}$ is presented.

Figure 9

(a) The dipolar correlation function $G\left(t\right)$ is presented for bulk fluid contained in a closed (orange dotted), channel (red dashed) and planar (green solid) pores of characteristic dimension $h=3$ nm. In (b), the corresponding spin-lattice relaxation rate $R_{1b}$ is presented.

Figure 10

(a) A schematic diagram of the system used to calculate $R_1$ comprising a combination of surface and bulk fluid. This schematic example is for a closed or channel pore of square cross section. Presented in (b) is a representation of a model pore with a mean dimension of 3 nm and a standard deviation of 1 nm.

Figure 11

The spin-lattice relaxation rate $R_1$ dispersion is presented for closed cubic pores (green curves with the largest $R_1$), channel pores with square cross-section (red curves) and planar pores (blue curves with the smallest $R_1$). In each case, the short dash (upper) and long dash (lower) curves are for nano-sized dimensions of 2 nm and 4 nm respectively. The solid lines (almost coincident) are for a nano-sized dimension of 3 nm but where the dispersion curves for the channel and closed pores assume different paramagnetic impurity densities as described in the text.

Figure 12

The figure provides a simulated spin-lattice relaxation rate $R_1$ response from a system containing equal volumes of fluid in closed, channel and planar pores at a frequency $f=100$ kHz. The Gaussian curves are estimated for $h=3\pm 1$ nm pores for closed cubic pores (green and at right), channel pores of square cross-section (red and central) and planar pores (blue and at left) with the combined response (black).

Figure 13

The ratio $T_1/T_2$ is presented as a function of frequency for a selection of values of ${\tau }_{\ell }$ and $h$ for a planar pore. The vertical bars are at a spot frequency of 20 MHz and represent the range of $T_1/T_2$ ratio observed in cement pastes (lower bar) and hydrocarbon rock (upper bar).