Modeling the Drag Forces of Porous Media Acoustics

The drag forces controlling the amount of relative flow induced in a fluid-saturated porous material by a mechanical wave are modeled here from first principles. Specifically, analytical expressions are derived for the drag force in material models that possess variable-width pores; Le., pores that have widths that vary with distance along their axis. The dynamic (complex, frequency-dependent) permeability determined for such a variable-width pore model is compared to estimates made using the models of Johnson, Koplik, and Dashen (JKD) and of Biot. Both the JKD model and the Biot model underestimate the imaginary part of the dynamic permeability at low frequencies with the amount of discrepancy increasing with the severity of the convergent/divergent flow; Le., increasing with the magnitude of the maximum pore-wall slope relative to the channel axis. It is shown how to modify the JKD model to obtain proper lowfrequency behavior; however, even with this modification, discrepancies still exist near the transition frequency that separates viscous-foree-dominated flow from inertial-forcedominated flow. The amount of discrepancy is again a function of the severity of the convergent/divergent flow (maximum pore-wall slope). INTRODUCTION This paper is concerned with modeling the fluid flow induced in a fluid-saturated porous material by a mechanical wave. The material type to be considered is characterized as having continuously distributed fluid and solid phases; Le., it is assumed that no isolated pockets of one phase are completely surrounded by the other phase. One example of this material type wouid be a packing of solid grains. As a compressional wave propagates through such a material, it both generates a pressure gradient in the fluid phase and accelerates the solid framework of the material. These two forces drive an accelerated flow of fluid relative to the solid frame. For a shear wave, the relative flow is driven only 92 Pride by the acceleration of the frame. As the fluid flows, traction forces are set up on the fluid/solid interface that act to resist the flow. These traction forces are called "drag forces." The maguitude of the relative flow is determined by the balance between the driving forces of the wave, the drag forces, and the inertial forces. The drag forces are usually modeled by assuming the pore space to consist of a collection of constant-width flow channels; Le., flow channels whose widths do not vary with distance along their axis (e.g., Biot, 1956b; Bedford, Costley and Stem, 1984; Yamamoto and Turgot; 1988). In this work, allowance is made for variable-width flow channels. The mechanical waves considered here have wavelengths much larger than the grains and pores comprising the material. Therefore, the response of interest is that which has been averaged over volumes much larger than a characteristic pore size but smaller than the wavelengths. Upon carrying out such a volume average on the force balance equations obeyed by the fluid and solid phases of a porous material, Pride, Gangi, and Morgan (1992) have obtained the following coupled equations of motion &ii. _ OW PB = '1 . T B PI 8t at PI OW &ii. ¢ at -d = -'lPI -PI 8t' where d is the drag force defined by (1)