Microgeometry induced non-uniqueness in the elastic and electrical properties of rocks
Keywords: Geophysics.

Most naturally occurring rocks are complex, heterogeneous composites and the exact micro-geometry of a rock sample is seldom known in its entirety. This lack of information about the unknown, exact micro-geometry of a rock translates into uncertainty or non-uniqueness in the effective rock properties that are micro-geometry dependent, such as elastic moduli, electrical conductivity, dielectric permittivity and fluid permeability among others. In this dissertation we explore important areas of rock-physics where micro-geometry induced non-uniqueness is not commonly or adequately addressed, estimating the practical impact of the unaccounted non-uniqueness and suggesting schemes to remedy that. We study the elastic properties of randomly oriented single-phase poly-crystals, noting the often ignored fact that most isotropic mineral elastic moduli reported in published literature usually and arbitrarily correspond to the mid-point of existing poly-crystal property bounds, developed to account for micro-geometry induced non-uniqueness. The magnitude of the usually unaccounted non-uniqueness or uncertainty in effective elastic moduli, given by the width of the bounds, increases with increasing elastic anisotropy of the constituent mineral crystals. Using a self-consistent scheme, we develop a model to study the impact of crystal shape and relative lattice orientation on moving the effective poly-crystal elastic moduli within the bounds. For aggregates of crystals with transverse isotropy, we use Monte-Carlo simulations and a regression tree based sensitivity analysis to demonstrate the relationship between fractional bound widths of effective elastic moduli and single crystal elastic parameters. Additionally, we show that aggregates of layered laminates represent an example of poly-crystal micro-geometry that may violate the embedded bounds for elastic solid substitution. We also study elastic properties of preferentially oriented single-phase poly-crystals. For a given orientation distribution function (ODF) of the constituent crystals, effective elastic properties are most commonly computed as ODF weighted averages of the single crystal elastic stiffness tensor (Voigt scheme, corresponding to uniform strain) or the single crystal elastic compliance tensor (Reuss scheme, corresponding to uniform stress). In previously published literature authors mostly use one scheme or another without systematically evaluating the impact of this choice on the estimated poly-crystal properties. Focusing on composites with rotational symmetry, we demonstrate that elastic moduli and anisotropy parameters estimated using Voigt and Reuss schemes can be considerably different from each other, the difference representing micro-geometric non-uniqueness and generally increasing with increasing elastic anisotropy of the constituent crystals. Using Monte-Carlo simulations and a regression tree based sensitivity analysis we demonstrate the impact of single-crystal elastic properties on the difference between the Voigt and Reuss estimates of the effective Thomsen's anisotropy parameters in a poly-crystal of rotational symmetry. We study cross-property relations between the electrical conductivity and the bulk modulus of rocks, demonstrating some inadequacies in popular one-to-one conductivity-modulus relationships and the importance of factoring in porosity when using or interpreting cross-property relations. We propose the use of the narrowest rigorous cross-property bounds due to Gibiansky and Torquato as an alternative, wherein the range of effective bulk modulus predicted for a given value of rock conductivity (and vice versa) captures the inherent micro-geometric non-uniqueness of a composite. Based on digital and laboratory data we obtain empirical constraints on Gibiansky and Torquato\textsc{\char13}s rigorous cross-bounds to make them narrower for some common reservoir rocks, such as brine filled sandstones and carbonates. Finally, we demonstrate the use of the constrained cross-bounds to estimate the range of the Archie cementation factor of a formation when presented with large-scale elastic (e.g., seismic) and electrical (e.g., CSEM) surveys. Finally, we study the problem of electrical fluid substitution - predicting the change in the effective electrical conductivity of a composite when one constituent conducting phase is substituted with another while the micro-geometry remains fixed. We demonstrate that the substitution problem is inherently non-unique due to variations in micro-geometry. We extend the concept of the embedded bounds, developed initially for elastic solid substitution, to obtain rigorous equations for substitution bounds on the electrical conductivity of two-phase, three-dimensional isotropic composites. We prove that when the conductivity contrast between composite phases is high, estimates from the popular Archie's law correspond approximately to the upper bound on the change of conductivity due to substitution. Inclusion modeling suggests that vuggy or poorly-connected pore space could account for conductivity changes smaller than predicted by Archie's law. Comparison of the conductivity substitution bounds with brine-saturated sandstone data of varying clay content reveals that the position of measured data within the conductivity substitution bounds can be indicative of the effective clay content in shaly sand samples.