Unified theory of global flow and squirt flow in cracked porous media

Abstract

Approximations for frequency-dependent and complex-valued effective stiffness tensors of cracked porous media (saturated with a single fluid) are developed on the basis of an inclusion-based model (the T-matrix approach to rock physics) and a unified treatment of the global-flow and squirt-flow mechanisms. Essentially, this study corrects an inconsistency or error related to fluid-mass conservation in an existing expression for the t-matrix (wave-induced deformation) of a communicating cavity, a cavity that is isolated with respect to stress propagation (through the solid matrix) but that can exchange fluid mass with other cavities because of global and/or local pressure gradients associated with passage of a long viscoelastic wave. An earlier demonstration of Gassmann consistency remains valid because the new theory of global flow and squirt flow (which also takes into account solid mechanical effects of stress interaction by us-ing products of communicating t-matrices associated with two-point correlation functions of ellipsoidal symmetry) only differs from an earlier version by a correction term that goes to zero in the low-frequency limit. If the unified model is applied to the special case of a model involving a single set of spheroidal cavities (having the same aspect ratio and orientation), the results become identical with those obtained using a special theory of global flow that predicts that at zero frequency the cavities will behave as though they are isolated with respect to wave-induced fluid flow (in accordance with Gassmann’s formulas) and that at high frequencies, they will behave as though they are dry. Our theory predicts that there will be a continuous transition from a global-flow-dominated system (characterized by a negative velocity dispersion) to a squirt-flow-dominated system (characterized by a positive velocity dispersion) if one begins with a single set of cavities and then introduces a distribution of shapes and/or orientations that gradually becomes wider (more realistic).