Solid viscosity of fluid-saturated porous rock with squirt flows at seismic frequencies

Abstract

We have developed a macroscopic model for a two-phase medium (solid porous rock frame plus saturating pore fluid) with squirt flows based on Lagrangian continuum mechanics. The model focuses on improved physics of rock deformation, including explicit differential equations in time domain, causality, linearity, frequency-independent parameters with clear physical meanings, and an absence of mathematical internal or memory variables. The approach shows that all existing squirt-flow models can be viewed as microscopic models of viscosity for solid rock. As in existing models, the pore space is differentiated into compliant and relatively stiff pores. At lower frequencies, the effects of fluid flows within compliant pores are described by bulk and shear solid viscosities of the effective porous frame. Squirt-flow effects are “Biot consistent,” which means that there exists a viscous coupling between the rock frame and the fluid in stiff pores. Biot’s poroelastic effects associated with stiff porosity and global flows are also fully included in the model. Comparisons with several squirt-flow models show good agreement in predicting wave attenuation to approximately 1 kHz frequencies. The squirt-flow viscosity for sandstone is estimated in the range of [Formula: see text], which is close to field observations. Because of its origins in rigorous mechanics, the model can be used to describe any wavelike and transient deformations of heterogeneous porous media or finite bodies encountered in many field and laboratory experiments. The model also leads to new numerical algorithms for wavefield modeling, which are illustrated by 1D finite-difference waveform modeling.