Electrical formation factor versus porosity coarse-scale transforms from microscopic digital images: Example-based study

The question posed in this work is whether and how elemental data obtained on small and often sub-representative fragments of natural rock can be exploited to generate a physically valid transform between two physical properties useable at a coarser spatial scale, such as the reservoir scale. The pair of such properties targeted here is the porosity and electrical formation factor. We use the process-based upscaling to prove, by example, that a coarse-scale transform between these two variables can be obtained from a dataset computed on microscopic digital volumes of natural rock, including unconsolidated sand, medium-porosity sandstones, and low- and high-porosity carbonates. This upscaling method is based on constructing a coarse-scale (effective) rock volume from the elemental volumes with a subsequent simulation of a relevant physical process in this effective object, specifically the electrical current. A numerical simulation of Laplace equation governing electrical current is used to compute the effective resistivity. The effective porosity is simply the arithmetically averaged porosity of the elements. By using hundreds of random realizations of the effective object, we generate tight formation factor versus porosity transforms for each of the natural rock samples under examination. Because the proof offered here is by-example, it is not general. However, the method offered is general and, as such, can be used to address the same question as posed for various situations.