High-resolution X-ray-computed-tomography (CT) images are increasingly used to numerically derive petrophysical properties of interest at the pore scale—in particular, effective permeability. Current micro-X-ray-CT facilities typically offer a resolution of a few microns per voxel, resulting in a field of view of approximately 5 mm3 for a 2,0482 charge-coupled device. At this scale, the resolution is normally sufficient to resolve pore-space connectivity and calculate transport properties directly. For samples exhibiting heterogeneity above the field of view of such a single high-resolution tomogram with resolved pore space, a second low-resolution tomogram can provide a larger-scale porosity map. This low-resolution X-ray-CT image provides the correlation structure of porosity at an intermediate scale, for which high-resolution permeability calculations can be carried out, forming the basis for upscaling methods dealing with correlated heterogeneity.
In this study, we characterize spatial heterogeneity by use of overlapping registered X-ray-CT images derived at different resolutions spanning orders of magnitude in length scales. A 38-mm-diameter carbonate core is studied in detail and imaged at low resolution—and at high resolution by taking four 5-mm-diameter subsets, one of which is imaged by use of full-length helical scanning. Fine-scale permeability transforms are derived by use of direct porosity/permeability relationships, random sampling of the porosity/permeability scatter plot as a function of porosity, and structural correlations combined with stochastic simulation. A range of these methods is applied at the coarse scale. We compare various upscaling methods, including renormalization theory, with direct solutions by use of a Laplace solver and report error bounds. Finally, we compare with experimental measurements of permeability at both the small-plug and the full-plug scale.
We find that both numerically and experimentally for the carbonate sample considered, which displays nonconnecting vugs and intrafossil pores, permeability increases with scale. Although numerical and experimental results agree at the larger scale, the digital core-analysis results underestimate experimentally measured permeability at the smaller scale. Upscaling techniques that use basic averaging techniques fail to provide truthful vertical permeability at the fine scale because of large permeability contrasts. At this scale, the most accurate upscaling technique uses Darcy's law. At the coarse scale, an accurate permeability estimate with error bounds is feasible if spatial correlations are considered. All upscaling techniques work satisfactorily at this scale. A key part of the study is the establishment of porosity transforms between high-resolution and low-resolution images to arrive at a calibrated porosity map to constrain permeability estimates for the whole core.