Stability analysis of the Biot/squirt models for wave propagation in saturated porous media

TL;DR

This paper analyzes the stability of the Biot/squirt models for wave propagation in saturated porous media, revealing conditions under which the models are unstable due to exponentially growing solutions, and confirming stability in specific cases.

Contribution

It demonstrates the conditions leading to instability in BISQ models and clarifies stability criteria, challenging the widespread use of the models without such considerations.

Findings

Models can have exponentially exploding solutions when the squirt-flow coefficient is negative or complex.

The squirt-flow coefficient can have non-zero imaginary parts for some experimental parameters.

The 3D isotropic BISQ model is stable when the squirt-flow coefficient is positive.

Abstract

This work is concerned with the Biot/squirt (BISQ) models for wave propagation in saturated porous media. We show that the models allow exponentially exploding solutions, as time goes to infinity, when the characteristic squirt-flow coefficient is negative or has a non-zero imaginary part. We also show that the squirt-flow coefficient does have non-zero imaginary parts for some experimental parameters. Because the models are linear, the existence of such exploding solutions indicates instability of the BISQ models. This result calls on a reconsideration of the widely used BISQ theory. Furthermore, we demonstrate that the 3D isotropic BISQ model is stable when the squirt-flow coefficient is positive. In particular, the original Biot model is unconditionally stable where the squirt-flow coefficient is 1.