Comparison of Fluid Pressure Wave between Biot Theory and Storativity Equation

Abstract

Compressibilities of pore fluid and rock skeleton affect pressure profile and flow velocity of fluid in aquifers. Storativity equation is often used to characterize such effects. The equation suffers from a disadvantage that at infinite large frequency, the predicted velocity of fluid pressure wave is infinitely large, which is unrealistic because any physical processes need certain amounts of time. In this paper, Biot theory is employed to investigate the problem. It is shown that the key equations of Biot theory can be simplified to storativity equation, based on low-frequency assumption. Using Berea sandstone as an example, we compare phase velocity and the quality factor between Biot theory and storativity equation. The results reveal that Biot theory is more accurate in yielding a bounded wave velocity. At frequency lower than 100 kHz, Biot theory yields a wave velocity 8 percent higher than storativity equation does. Apparent permeability measured by fluid pressure wave (such as Oscillatory Hydraulic Tomography) may be 14 percent higher than real permeability measured by steady flow experiments. If skeleton is rigid, Biot theory at very high frequencies or with very high permeabilities will yield the same velocity as sound wave in pure water. The findings help us for better understanding of the physical processes of pore fluid and the limitations of storativity equation.

Rayleigh wave and well head response to calculate porosity in the Edwards aquifer of south-central Texas, USA

“…The elastic model in the Appendix has two assumptions: (1) the rocks are undrained during the passage of a transient Rayleigh wave such that the Gassmann (1951) equation is applicable; (2) the well head adequately represents fluid pressure in the matrix surrounding the well. For assumption (1), dimensional analysis using the Biot theory (Biot 1956a,b) has revealed that the response of saturated rocks to seismic waves is dependent on dimensionless frequency (Li 2020a), where and are water density, permeability, water viscosity and angular frequency, respectively. Low frequency ( ) will cause to tend to zero.…”

“…Strictly speaking, aquifer storativity (specific storage) is defined as , where and are porosity, compressibility coefficients of water, skeleton and solid material, respectively (Li et al 2020a). As a reminder, this expression is slightly more accurate than the one in Domenico & Schwartz (1997) in considering the compressibility of solid material.…”

“…The definition of storativity in previous studies on aquifer response to Rayleigh wave (Shih 2009;Shih et al 2013;Folnagy et al 2013;Sun et al 2018) required estimation of the aquifer thickness. Actually, as aforementioned, the standard storativity (Li et al 2020a) can be well got via porosity.…”

“…The major assumption of the above elastic model is that the aquifer matrix is undrained (no water is allowed to transfer between pores). Nevertheless, dimensional analysis (Li et al 2020a) has shown that a low frequency ( ) is almost equivalent to undrained rock (Darcy permeability ) for the fast fluid pressure wave. Therefore, the elastic model is accurate at the first order of approximation.…”

A new squirt coefficient to simulate P‐wave attenuation

“…Biot (1956aBiot ( , 1956b theory was more accurate than the storativity equation for water-pressure diffusion in hydrogeology; the storativity equation is the governing equation of water pressure in aquifers with the consideration of water compressibility and skeleton compressibility (Domenico & Schwartz, 1997;Li et al, 2020a). Under the assumptions of low frequency and zero confining pressure, Li et al (2020a) showed that Biot theory can be simplified to the storativity equation. Biot (1956a) used Darcy permeability (k D ) in hydrogeology and static fluid viscosity (μ) in fluid mechanics to quantify viscous stress between solid and fluid.…”