With the development of shale-gas exploration and exploitation, it is necessary to study the 3D spatial distribution of shale-gas fractures for research on shale rock physics. Because of the limitation of instruments, accurate shale slice is discontinuous in depth, and the minimum interval between adjacent slices is inconsistent with horizontal resolution of digital cores. These are the main factors which hamper accuracy improvement of fracture representation and physical modeling for digital cores. In order to study the 3D spatial distribution of fractures, we doubled the vertical slices increasing the vertical resolution to make it consistent with the horizontal resolution. The curvelet transform and projection onto convex sets (POCS) method are used to achieve the reconstruction of 3D digital cores. The curvelet transform is a sparse transform which has been widely used in seismic data denoising and interpolation and image denoising. The POCS method is an efficient method for seismic data interpolation and can be used in the reconstruction of 3D digital cores. This method is applied on each vertical slice and the 3D digital cores can be obtained after all the vertical slices are processed. Besides, the proposed method is superior to the spgll method. With the proposed method, we achieve the 3D digital cores from the cores which were sampled one per two slices in depth for the 3D volume of sandstone obtained by X ray scanner. The reconstruction result is consistent with the original one and superior to the spgll method, which proves the validity and superiority of the proposed method. Then the proposed method is applied to the shale cores, of which the 2D horizontal slices are obtained using focused ion beam scanning electron microscopy (FIB-SEM). Because of instrumental limitations, the vertical resolution is almost halved compared with the horizontal resolution. With the proposed method, we can double the horizontal slice in depth, which can help improve the vertical resolution to make it consistent with the horizontal one, weakening the discontinuity of shale slices in depth caused by the instrument limitation, resulting in a clearer fracture distribution. 3D digital cores are reconstructed from the 2D shale slices based on the projection onto convex sets (POCS) method in the curvelet domain. The sand reconstruction test of 3D digital cores demonstrates that the proposed method is more suitable for rock slice reconstruction with high efficiency and accuracy compared with the popular spgll method. The shale reconstruction test of 3D digital cores from nano-scale shale slices obtained by FIB-SEM indicates that vertical spatial distribution of fractures is more clear and the minimum interval between adjacent vertical slices is basically consistent with the horizontal resolution after reconstruction, which can lay the foundation for the subsequent data processing and related simulation analysis of shale digital cores. 3D digital cores reconstruction numerical tests on 2D sand and shale slices demonstrate the validity of the proposed method. The methods based on direct 3D datasets and efficient sparse transform will be developed in future work.
