Statistical scaling of pore-scale Lagrangian velocities in natural porous media

We investigate the scaling behavior of sample statistics of pore-scale Lagrangian velocities in two different rock samples, Bentheimer sandstone and Estaillades limestone. The samples are imaged using x-ray computer tomography with micron-scale resolution. The scaling analysis relies on the study of the way $q\text{th}$-order sample structure functions (statistical moments of order $q$ of absolute increments) of Lagrangian velocities depend on separation distances, or lags, traveled along the mean flow direction. In the sandstone block, sample structure functions of all orders exhibit a power-law scaling within a clearly identifiable intermediate range of lags. Sample structure functions associated with the limestone block display two diverse power-law regimes, which we infer to be related to two overlapping spatially correlated structures. In both rocks and for all orders $q$, we observe linear relationships between logarithmic structure functions of successive orders at all lags (a phenomenon that is typically known as extended power scaling, or extended self-similarity). The scaling behavior of Lagrangian velocities is compared with the one exhibited by porosity and specific surface area, which constitute two key pore-scale geometric observables. The statistical scaling of the local velocity field reflects the behavior of these geometric observables, with the occurrence of power-law-scaling regimes within the same range of lags for sample structure functions of Lagrangian velocity, porosity, and specific surface area.

Figure 1

Images of the void space (white) along cross-sectional planes orthogonal to the mean flow direction ($x$ axis) for (a) Bentheimer sandstone and (b) Estaillades limestone.

Figure 2

Computed normalized flow fields for (a) Bentheimer sandstone and (b) Estaillades limestone. Results are depicted as the ratios of the magnitude of velocity $\left|\mathit{U}\right(\mathit{x}\left)\right|$ at the voxel centers divided by the average pore velocity magnitude $U_{av}$ and represented on a logarithmic colorized scale (average flow is from left to right).

Figure 3

Sample structure functions of absolute increments of $\varphi \left(x\right)$, $\mathcal{S}\left(x\right)$, and $U_x\left(\mathit{x}\right)$ computed at lags $s_x$ along the $x$ axis for the Bentheimer sandstone and (a) $q$ = 1, (b) $q$ = 2, and (c) $q$ = 3.

Figure 4

(a) Sample structure functions $S_N^q\left(s_x\right)$ of absolute increments of $U_x$ versus $s_x$ and (b) $S_N^{q+1}$ versus $S_N^q$ for the Bentheimer sandstone and $q$ = 1, 2, 3, 4. Vertical dashed lines demarcate breaks in power-law-scaling regimes. Logarithmic scale regression lines indicating extended power-law scaling and the corresponding relations between $S_N^{q+1}$ versus $S_N^q$ are given in (b).

Figure 5

Scaling exponent ξ($q$) evaluated as a function of $q$ by the MM and ESS for the Bentheimer sandstone sample. The slope of the dashed straight line passing through ξ(1) and the origin (equation listed) provides an estimate of the Hurst exponent.

Figure 6

Sample structure functions of absolute increments of $\varphi \left(x\right)$, $\mathcal{S}\left(x\right)$, and $U_x\left(\mathit{x}\right)$ computed at lags $s_x$ along the $x$ axis for the Estaillades limestone and (a) $q$ = 1, (b) $q$ = 2, and (c) $q$ = 3.

Figure 7

Sample structure functions of order $q$ = 1, 2, 3 of absolute increments of (a) $\varphi \left(x\right)$ and (b) $\mathcal{S}\left(x\right)$ for two resolution levels of the XCT reconstruction, i.e., $l^{\prime }$ = 7.0 μm (labeled Low Res and plotted with symbols) and $l$ = 3.3 μm (labeled High Res and plotted as dashed lines).

Figure 8

(a) Sample structure functions $S_N^q\left(s_x\right)$ of absolute increments of $U_x$ versus $s_x$ and (b) $S_N^{q+1}$ versus $S_N^q$ for the high-resolution Estaillades limestone and $q$ = 1, 2, 3, 4. Vertical dashed lines demarcate the two identified power-law-scaling regimes. Logarithmic scale regression lines indicating extended power-law scaling and corresponding relations between $S_N^{q+1}$ versus $S_N^q$ are given in (b).

Figure 9

Scaling exponent ξ($q$) evaluated as a function of $q$ by the MM and ESS for the high-resolution Estaillades limestone sample for scaling regions (a) I and (b) II. The slope of the dashed straight lines passing through ξ(1) and the origin (equations listed) provides estimates of the Hurst exponents associated with the two scaling regimes.