Permeability-porosity relations from single image of natural rock: Subsampling approach

Natural rock is heterogeneous at all scales. This challenges the digital rock physics concept since the images of natural rock are mm-sized, while the results should be applicable at much larger spatial scales. To address this challenge we subsample a digital volume and compute the permeability of its subvolumes, as well as that of the entire volume, using the lattice-Boltzmann method (LBM). We reassemble the resulting values in 3D and use Darcy's flow simulator to re-compute the permeability of the entire host volume. In spite of significant heterogeneity of the subvolume properties, the Darcy-upscaled entire-volume permeability falls very close to that LBM-computed. Next, by using random reconstructions of the subvolume permeability matrix, we obtain physically meaningful permeability versus porosity transforms that can arguably be used at a scale much greater than that of a digital CT-scan image. This simple principle “to see a trend in a grain of sand” is the novel contribution of our work. Arguably, it can be implemented using images of drill cuttings collected within an interval in a well. We discuss examples of this principle for three basic rock types: high-porosity unconsolidated sand, medium-porosity sandstone, and medium-porosity carbonate.