Phillip Andrew Cilli, Mark Chapman

Joint electrical-elastic rock physics modelling can be instrumental in lowering uncertainty in subsurface reservoir characterisation. Typical electrical-elastic cross-property models, however, are empirical or require an intermediate step of porosity estimation to link a rock's electrical and elastic moduli, which can be error-prone away from well controls. Another outstanding issue in electrical-elastic modelling is the challenge of predicting a rock's shear modulus and ultimately Vp/Vs ratio from electrical measurements. By reformulating an existing electrical differential effective medium (DEM) theory to embed ellipsoidal pores into a background of matrix material, rather than the typical method of embedding ellipsoidal grains into a background of fluid, we express the model in terms of the geometrical function R, which is present in other electrical ellipsoidal inclusion models. This reformulation is consistent with two other effective conductivity models and shares its geometrical function R with the electrical self-consistent approximation (SCA) model, providing a new mathematical link between electrical DEM and SCA models. Combining this reformulated electrical DEM model and a pre-existing elastic DEM model, we obtain expressions for a rock's effective elastic moduli with respect to effective conductivity. This method is analogous to the more common electrical-elastic modelling stratagem where an electrical model is substituted into an elastic model or vice versa through their shared independent variable, porosity, which is rendered a dummy variable in the process. Modelling the elastic moduli of clay-bearing sandstones using public domain laboratory measurements, there seems to be a weak sensitivity of the model's single parameter, equivalent pore aspect ratio, to clay volume fraction. Furthermore, the uncertainty in shear modulus modelling from conductivity measurements seems weakly sensitive to clay content. By employing the Gardner empirical velocity-density relation for sandstones, we forward model Vp and Vs from electrical measurements in the absence of porosity and density measurements, with accuracy comparable to the Han (1986) empirical Vp/Vs model for mixed sandstones. Our proposed cross-property DEM method generalises mathematically to relate any two of a composite's elastic moduli, electrical conductivity, electrical permittivity, thermal conductivity, magnetic permeability, and diffusion constant due to the equivalence of these properties in inclusion modelling by the universality of the Laplace equation, which underpins the models' constructions. This generalisation of the cross-property DEM model to numerous physical properties leads to a testable hypothesis: the cross-property DEM parameter, aspect ratio, is (or is not) universal when linking a given rock's various physical properties.