Expansion of a high-pressure gas sphere in a porous medium II: Fluid compressible

Abstract

The work presented in an earlier paper [D. Epstein, J. Geophyos. Res. 72, 3701–3710 (1967)] on the expansion of a high-pressure gas bubble in a fluid-saturated porous medium, with the fluid treated as incompressible, is extended by treating the fluid as slightly compressible. Two limiting cases are considered: If the particulate matter is carried along with the fluid, the porous medium acts as a dense liquid, with parameters determined by simple functions of the density and compressibility of the fluid and solid particles, and the usual equations of motion apply. However, if the relative velocity of the fluid with respect to the solid is essentially equal to the fluid velocity, the motion is assumed to be governed by a compressible, generalized Darcy equation. From this equation a pair of integrodifferential equations for the bubble radius as a function of time are derived. The form of the radius-time curve depends on a nondimensional resistance factor m? = (1/2) β?/c, a combination of quantities characteristic of the fluid, the porous medium, and the gas bubble. The transition regime between ideal and Darcy flow is determined as a function of this parameter. For very small values, the bubble pulsates about the equilibrium raius ?. With increasing m?, the ratio of radiated acoustic energy to energy dissipated decreases, until the Darcy limit is reached, where bubble pulsation no longer occurs, and the expansion radius approaches ? monotonically. Subject Classification: [43]30.60; [43]25.60.