Rules of upscaling for rock physics transforms: Composites of randomly and independently drawn elements

The tight trends (transforms) between two rock physical properties obtained in the physical or digital laboratory on cm- or mm-sized samples may hold for a composite constructed of these samples. One requirement is that all the elements of the composite obey the same trend. Another requirement is that the composite is spatially uncorrelated and, hence, approximately isotropic. The final requirement is that the underlying elemental transforms are approximately linear. The methods we use to address this exportability of trends for the dynamic elastic moduli (or the elastic-wave impedance) versus porosity, permeability versus porosity, and electrical resistivity versus porosity are, respectively, theoretical elastic bounding, numerical reservoir-scale fluid flow, and electrical current simulations. A practical implication is that if an elastic property of a large volume is determined from remote sensing, be it seismic or cross-well data, its average porosity can be estimated using the transforms established at a much smaller scale. Then, this average porosity can be translated into the hydraulic and electrical properties once again, using the elemental transforms established in the physical or digital laboratory.