Characterizing pore-scale structure-flow correlations in sedimentary rocks using magnetic resonance imaging

Quantitative, three-dimensional (3D) spatially resolved magnetic resonance flow imaging (flow MRI) methods are presented to characterize structure-flow correlations in a 4-mm-diameter plug of Ketton limestone rock using undersampled k- and q-space data acquisition methods combined with compressed sensing (CS) data reconstruction techniques. The acquired MRI data are coregistered with an X-ray microcomputed tomography (μCT) image of the same rock sample, allowing direct correlation of the structural features of the rock with local fluid transport characteristics. First, 3D velocity maps acquired at 35 μm isotropic spatial resolution showed that the flow was highly heterogeneous, with ∼10% of the pores carrying more than 50% of the flow. Structure-flow correlations were found between the local flow velocities through pores and the size and topology (coordination number) associated with these pores. These data show consistent trends with analogous data acquired for flow through a packing of 4-mm-diameter spheres, which may be due to the microstructure of Ketton rock being a consolidation of approximately spherical grains. Using two-dimensional and 3D visualization of coregistered μCT images and velocity maps, complex pore-scale flow patterns were identified. Second, 3D spatially resolved propagators were acquired at 94 μm isotropic spatial resolution. Flow dispersion within the rock was examined by analyzing each of the 331 776 local propagators as a function of observation time. Again, the heterogeneity of flow within the rock was shown. Quantification of the mean and standard deviation of each of the local propagators showed enhanced mixing occurring within the pore space at longer observation times. These spatially resolved measurements also enable investigation of the length scale of a representative elementary volume. It is shown that for a 4-mm-diameter plug this length scale is not reached.

Figure 1

Schematic of the pulse sequence used to acquire high-resolution velocity maps. It combines a pulsed gradient spin echo (PGSE) sequence with a rapid acquisition with relaxation enhancement (RARE) imaging sequence used for velocity and spatial encoding, respectively. The velocity encoding is achieved by a pair of unipolar gradients of strength $g$ and duration δ, separated by an observation time, Δ. The spatial encoding is achieved using the frequency-encoding gradient, $G_z$, and the phase-encoding gradients, $G_x$ and $G_y$. A RARE factor ($N_{\mathrm{RF}}$) is the number of 180 ° refocusing pulses in an echo train. $t_{\mathrm{e}}$ is the echo spacing.

Figure 2

Schematic of the APGSTE-RARE pulse sequence used to acquire 3D spatially resolved propagators. It consists of a 13-interval APGSTE sequence and a RARE imaging sequence. The displacement encoding was achieved by gradients $g$, which can be applied in any of the three orthogonal directions ($z$, $x$, or $y$). In this work, the displacement-encoding gradient was applied parallel to the superficial flow direction ($z$). The spatial encoding was achieved by the frequency-encoding gradient, $G_z$, and phase-encoding gradients, $G_{x,y}$.

Figure 3

(a) Pore-scale 3D magnitude flow map generated from the $x$-, $y$-, and $z$-velocity components showing the main flow channels through the Ketton rock plug. Velocity distributions of (b) $z$- and (c) $y$-velocity components; the $x$-velocity component has a similar distribution of velocities to the $y$-velocity component.

Figure 4

Structure-flow correlations in Ketton rock. (a) The fractional summed $z$-velocities (in the superficial flow direction) within pores with a summed $z$-velocity equal to or greater than (the parametric variable) $i$, $v_z^{\mathrm{sum}}\left(i\right)/v_z^{\mathrm{sum},\mathrm{tot}}$, plotted as a function of the fraction of the number of pores, $N_{\mathrm{p}}\left(i\right)/N_{\mathrm{p}}^{\mathrm{tot}}$ (―), and the volume of pores, $V_{\mathrm{p}}\left(i\right)/V_{\mathrm{p}}^{\mathrm{tot}}$ ($┄$), carrying this flow. (b) The average mean $z$-velocities, ${\overline{v}}_z^{\mathrm{m}}$, within pores plotted as a function of the average radius of these pores for different coordination numbers; the size of the red dots represents coordination numbers from 2 to 8 in ascending order. (c) Correlation between the average mean pore velocities, ${\overline{v}}_z^{\mathrm{m}}$, and coordination numbers for all pores. (d) Correlation between the average mean pore velocities, ${\overline{v}}_z^{\mathrm{m}}$, and coordination numbers for pores with local Reynolds numbers greater than the mean local Reynolds number of 0.064. The solid blue lines represent the regression fits to the data.

Figure 5

Structure-flow correlations in a pack of spherical beads. (a) The fractional summed $z$-velocities (in the superficial flow direction) within pores with a summed $z$-velocity equal to or greater than (the parametric variable) $i$, $v_z^{\mathrm{sum}}\left(i\right)/v_z^{\mathrm{sum},\mathrm{tot}}$, plotted as a function of the fraction of the number of pores, $N_{\mathrm{p}}\left(i\right)/N_{\mathrm{p}}^{\mathrm{tot}}$ (―), and the volume of pores, $V_{\mathrm{p}}\left(i\right)/V_{\mathrm{p}}^{\mathrm{tot}}$ ($┄$), carrying this flow. (b) The average mean $z$-velocities, ${\overline{v}}_z^{\mathrm{m}}$, within pores plotted as a function of the average radius of these pores for different coordination numbers; the size of the red dots represents coordination numbers from 2 to 10 in ascending order. (c) Correlation between the average mean pore velocities, ${\overline{v}}_z^{\mathrm{m}}$, and coordination numbers for all pores. (d) Correlation between the average mean pore velocities, ${\overline{v}}_z^{\mathrm{m}}$, and coordination numbers for pores with local Reynolds numbers greater than the mean local Reynolds number of 0.82. The solid blue lines represent the regression fits to the data.

Figure 6

A 3D section of $1\mathrm{m}{\mathrm{m}}^3$ extracted from the coregistered 3D MRI velocity vector map and μCT image of Ketton rock. The displayed 3D velocity vectors were obtained by combining z-, x-, and y-velocity components; arrows represent the direction of flow, and the size and color of these arrows represent the magnitude of the flow. The superficial flow direction was along the $z$-axis.

Figure 7

(a), (b) 2D $xy$-slice images extracted from a 3D coregistered $z$-velocity map and μCT image of Ketton rock. Four regions have been highlighted by the red boxes which identify different flow patterns in the the rock: (1) adjacent positive and negative flow in a large pore; (2) stagnant water in a moldic pore; (3) high-velocity flow channel in a large pore; and (4) concentrated flow in the narrow parts of pores. Regions (c) 1 and (d) 3 are also visualized as coregistered 3D velocity ($x$, $y$, $z$) vector maps.

Figure 8

(a) Global propagator, obtained by summation of all 331 776 individual, per-voxel propagators (―) and the expected Gaussian propagator ($┄$) for water in the absence of flow. The gray, vertical line represents a cutoff between the “flowing” and “stagnant” water. Based on this cutoff, the (b) stagnant and flowing water images were generated and coregistered with the high-resolution μCT image of Ketton rock. The displayed colors in (b) represent the average of colormaps of MRI and μCT images.

Figure 9

(a) Global propagators, obtained by summation of all 331 776 local propagators of the $\mathrm{\Delta }=150\mathrm{ms}$ (―) and 900 ms ($—$) data sets. (b–f) Five examples of local, per-voxel propagators, plotted on the same scale as the propagators in (a).

Figure 10

2D histograms showing the mean and standard deviation of all local propagators in the rock, for (a) $\mathrm{\Delta }=150\mathrm{ms}$ and (b) $\mathrm{\Delta }=900\mathrm{ms}$. Both mean and standard deviation are rescaled using the values calculated for the global propagator at the given Δ. The color scale corresponds to the ${\log }_{10}$ of the total integral I of all the propagators within each bin. The black dots show the exact position of 0.1% of the total number of local propagators, randomly picked from the data set. The red dot indicates the position $\mu /{\mu }_{\mathrm{global}}=1$, $\sigma /{\sigma }_{\mathrm{global}}=1$ for reference.

Figure 11

Rescaled mean $\mu /{\mu }_{\mathrm{global}}$ and standard deviation $\sigma /{\sigma }_{\mathrm{global}}$ of per-voxel propagators at different coarse-grained spatial resolutions. The highest resolution of 94 μm ($•$) equals that of the acquired data. The acquired data have been coarse-grained to 562 μm ($◼$), 1.125 mm ($▼$) and 2.25 mm ($◆$) isotropic spatial resolutions. The result for 2.25 mm spatial resolution represents a single voxel and is therefore found at position (1,1).