The impact of grain-scale elastic and viscoelastic changes on seismic wave propagation [electronic resource]
Keywords: Geophysics.

Naturally occurring rocks are typically composed of various constituents with varied elastic properties, such as gases (CO2, methane, vapor, etc.), low-viscosity liquids (water, oil, etc.), high-viscosity liquids (heavy-oil, magma, kerogen, etc.), and solids (quartz, feldspar, calcite, etc.). The primary objective of this thesis is to identify the fundamental physical laws which govern the sensitivity of seismic velocities and effective rock stiffness to grain-scale changes in rock constituents. Analytical solutions of macroscopic physical laws are developed, probed, benchmarked, and analyzed with numerical simulations of already established grain-scale physics at complex pore boundaries using the finite element method (FEM). Also, we suggest approximations to the exact solutions, since sometimes direct measurements of the required parameters may not be available. For fluid and solid substitution, which is one of the most fundamental problems in rock physics, we find that the exact solution requires parameters that depend on pore geometry; thus, substitution is non-unique if only pore-fill volume fraction is known. We also prove that the classical Gassmann's bulk modulus equation is exact for solid substitution if compression-induced mean stresses (pressure) in initial and final pore solids are homogeneous, and either the shear modulus of the substituted solid does not change or no shear stress is induced in pores. Using the new exact substitution equations, we interpret that predicting solid-filled rock stiffness from a dry rock stiffness measurement requires more information (i.e., assumptions about the pore shape) as compared to predicting the same from a fluid-saturated rock stiffness. We also derive substitution relations for the P-wave modulus, assuming S-wave velocity or shear modulus is not known; this is a common practical problem. For the general case of solid substitution, the exact P-wave modulus substitution equation depends on usually unknown parameters. However, for fluid substitution, fewer parameters are required, and the dependence of exact substitution on these unknown parameters reduces with increase in Poisson's ratio of the mineral in rock frame. Thus we find that P-wave modulus fluid substitution, in the absence of shear velocity, can be performed with relatively higher confidence for rocks with calcite/dolomite frame (such as carbonates) as compared to those with quartz frame (such as sandstones). Since information on pore geometry is seldom available, we present four embedded-bound constructions for fluid and solid substitution that are based on realizable materials. In the limiting case of pore fluids, for bulk modulus, two of these constructions reduce to the bounds of Gibiansky and Torquato, which illustrates that those bounds are optimum. The first two constructions correspond to a homogeneous pore stiffness and predict the smallest change in modulus. The third construction prediction corresponds to a pore space with heterogeneous stiffness, and predicts much larger change in modulus. We also extend our exact substitution relations to substitute one or more phases in multimineralic isotropic rocks. These new solutions are also equivalent to relaxing the assumption of unchanging rock microstructure upon substitution -- a core assumption in the current models. Both the pore-filling phase and rock microstructure can change due to diagenesis, dissolution, precipitation, partial freezing or melting, etc., and these situations can be modeled using the new formulation. Approximate bounds for the change in effective rock stiffness upon change in pore geometry are also developed which are in good agreement with laboratory and numerical examples; these bounds depend only on initial effective stiffness, properties of constituents, and volume fractions of constituents. For high viscosity fluids (such as heavy-oil, magma, kerogen, etc.) Biot theory has consistently failed to reproduce laboratory measured dispersion. Over the years, grain-scale dispersion mechanisms such as squirt (local-flow) and shear-relaxation have been more successful in explaining the measured dispersion. We present a new method to quantify the combined high-frequency effects of squirt and shear-dispersion (solid-squirt) on the elastic properties of rocks saturated with viscous fluids. Viscous fluid at high-frequencies is idealized as an elastic solid of finite shear modulus, hydraulically locked in stiff and soft pores at high-frequencies. This method entails performing solid substitution in stiff pores of a dry rock frame that is unrelaxed due to solid-filled soft pores. The unrelaxed frame stiffness solutions require information on the pressure dependency of the rock stiffness and porosity. This method does not have any adjustable parameters, and all required inputs can be directly measured. With various laboratory and numerical examples, we note that accounting for combined effects of squirt and shear-dispersion is necessary to explain laboratory measured velocities of rocks saturated with fluids of high viscosity. Predictions of the new method are in good agreement with laboratory data. Finally, we present a simple approach to model effective creep and relaxation functions of organic-rich shales. We find that model curves corresponding to mixing mineral inclusions in kerogen background better fit both dynamic and static laboratory measurements when compared to those corresponding to mixing kerogen inclusions in mineral background. We find that the creep time exponents are anisotropic and depend on boundary conditions of rock deformation. Often it is not possible to directly measure all time exponents; thus we present a simple set of empirical relations which can yield crude estimates of unmeasured time exponents starting with those measured directly.