  Nano-CT (computerized tomography) has emerged as a non-destructive
high-resolution cross-sectional imaging technique to effectively study the
sub-$\mu$m pore structure of shale, which is of fundamental importance to the
evaluation and development of shale oil and gas. Nano-CT poses unique
challenges to the inverse problem of reconstructing the 3D structure due to the
lower signal-to-noise ratio (than Micro-CT) at the nano-scale, increased
sensitivity to the misaligned geometry caused by the movement of object
manipulator, limited sample size, and a larger volume of data at higher
resolution. In this paper, we propose a scalable double regularization (SDR)
method to utilize the entire dataset for simultaneous 3D structural
reconstruction across slices through total variation regularization within
slices and $L_1$ regularization between adjacent slices. SDR allows information
borrowing both within and between slices, contrasting with the traditional
methods that usually build on slice by slice reconstruction. We develop a
scalable and memory-efficient algorithm by exploiting the systematic sparsity
and consistent geometry induced by such Nano-CT data. We illustrate the
proposed method using synthetic data and two Nano-CT imaging datasets of
Jiulaodong (JLD) shale and Longmaxi (LMX) shale acquired in the Sichuan Basin.
These numerical experiments show that the proposed method substantially
outperforms selected alternatives both visually and quantitatively.
