Representative-volume sizing in finite cylindrical computed tomography by low-wavenumber spectral convergence

TL;DR

This study tackles the challenge of representative-volume sizing in finite cylindrical μCT scans of Thalassinoides-bearing rocks, where slow axial drift and nonstationarity bias traditional convergence assessments. It introduces a two-step workflow: (i) detrend the segmented burrowsity field to obtain a near-stationary fluctuation field, and (ii) apply a covariance/spectrum-based convergence test, the \\widehat{C}-test, on nested cylinders to determine the minimum representative diameter and axial height. The method yields D_REV ≈ 93 mm and H_REV ≈ 83 mm for the analyzed core, providing a principled domain for reproducible correlation-scale reporting and connectivity-sensitive property estimation in digital-rock workflows. By tying REV sizing to the low-wavenumber stability of the isotropic spectrum, the approach offers a transparent, auditable pathway to quantify long-wavelength connectivity from CT data, with code and data openly available. The framework is poised to improve the robustness of downstream proxies and upscaling analyses in heterogeneous, connectivity-dominated rock fabrics.

Abstract

Choosing a representative element volume (REV) from finite cylindrical $μ$CT scans becomes ambiguous when a key field variable exhibits a slow axial trend, because estimated statistics can change systematically with subvolume size and position rather than converging under simple averaging. A practical workflow is presented to size an REV under such nonstationary conditions by first suppressing axial drift/trend to obtain a residual field suitable for second-order analysis, and then selecting the smallest analysis diameter for which low-wavenumber content stabilizes within a prescribed tolerance. The approach is demonstrated on \textit{Thalassinoides}-bearing rocks, whose branching, connected burrow networks impose heterogeneity on length scales comparable to typical laboratory core diameters, making imaging-based microstructural statistics and downstream digital-rock proxies highly sensitive to the chosen subvolume. From segmented data, a scalar ``burrowsity'' field--capturing burrow-related pore spaces and infills--is defined, and axial detrending (with optional normalization) is applied to mitigate acquisition drift and nonstationary trends. Representativeness is then posed as a diameter-convergence problem on nested inscribed cylinders: the two-point covariance and its isotropic spectral counterpart $\widehat{C}$ are estimated, and the smallest diameter at which the low-wavenumber plateau becomes stable is selected. Applied to a segmented \textit{Thalassinoides} core, the method identifies a minimum analysis cylinder of approximately $D_{\mathrm{REV}}\approx 93~\mathrm{mm}$ and $H_{\mathrm{REV}}\approx 83~\mathrm{mm}$, enabling reproducible correlation-scale reporting and connectivity-sensitive property estimation.

Figures (4)

• Figure 1: Core sample of Thalassinoides rock used in the analysis. The analyzed CT dataset corresponds to a segmented cylindrical core with dimensions $488\times 488\times 160$ voxels (transverse size $N_x=N_y=488$ pixels and axial length $N_z=160$ slices). The physical specimen dimensions were $D_{\mathrm{s}}=180~\mathrm{mm}$ in diameter and $H_{\mathrm{s}}=310~\mathrm{mm}$ in height, implying voxel spacings $\Delta x=\Delta y=D_{\mathrm{s}}/488\simeq 0.369~\mathrm{mm/pixel}$ and $\Delta z=H_{\mathrm{s}}/160\simeq 1.94~\mathrm{mm/slice}$.
• Figure 2: Axial detrending of the slice-wise phase-fraction signal. Top: slice-wise phase fraction $\phi(z_k)$ (black) and the corresponding moving-average trend $\bar{\phi}(z_k)$ (red) computed with window width $w$ (in slices). Bottom: residual $r(z_k)=\phi(z_k)-\bar{\phi}(z_k)$ (blue), used to quantify how much low-frequency drift remains after detrending and to guide the choice of $w$ via residual diagnostics (e.g., excess kurtosis).
• Figure 3: Excess kurtosis $K_{\mathrm{ex}}(w)$ of the detrended axial residual as a function of the moving-average window width $w$. Because a Gaussian residual has $K_{\mathrm{ex}}=0$, zero-crossings of $K_{\mathrm{ex}}(w)$ are used as a practical criterion for selecting a detrending scale. Among the candidate windows where $K_{\mathrm{ex}}(w)\approx 0$, the largest window is chosen to suppress slow axial drift most strongly while maintaining a residual that is approximately Gaussian.
• Figure 4: Slice-wise statistics for a Thalassinoides core. (a) Boolean slice $B_{\mathrm{slice}}$ (burrow/phase pixels in black, background in white) with a concentric sampling circumference of radius $r$ overlaid in red. (b) Enclosed area fraction $\langle \phi\rangle(r)=\langle B_{\mathrm{slice}}\rangle_{|\mathbf{x}|\le r}$ as a function of radius. (c) Two-point covariance $C(r)$ computed on the slice within an inscribed circular region. (d) Radial spectrum $\widehat{C}(k_r)$ obtained from $C(r)$ using the 2D isotropic Hankel transform.