Propagation of Biot slow waves in heterogeneous pipe networks: Effect of the pipe radius distribution on the effective wave velocity and attenuation

摘要

This paper extends a previous study of the harmonic (or AC) flow of a compressible fluid through a single, elastic, thick-wall pipe. The model previously developed is used to investigate propagation of pore-scale Biot slow waves through heterogeneous one-, two- and three-dimensional networks of pipes. A novel method is applied to the results of the network simulations to numerically determine the dispersion equation of the upscaled Biot slow waves and investigate its dependence on pore-scale heterogeneity. As a function of frequency, the phase velocity of the macroscale Biot slow waves displays an S-shaped curve, increasing from zero at low frequencies (i.e., nonpropagative regime) to C_x at high frequencies (i.e., propagative regime with С_x lower than the sound velocity in the fluid). The transition between these two regimes is marked by the inflection point at the frequency x_B (whereis inversely proportional to the length scale A characteristic of fluid flow and permeability). The high-frequency phase velocity С depends on the dimensionality of the network considered and decreases with increasing heterogeneity. The wave attenuation (as measured by the inverse quality factor) also presents an S-shaped curve, decreasing from 2 (i.e., critical damping) to zero, with the same inflection point at This behavior is approximately independent on the pore radius distribution, provided that x_B (or the corresponding fluid flow length scale A) is held constant. A mechanism based on wave scattering and interferences of forward and backward traveling (pore-scale) Riot slow waves is proposed to explain the observations.