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

Abstract

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 at high frequencies (i.e., propagative regime with C lower than the sound velocity in the fluid). The transition between these two regimes is marked by the inflection point at the frequency ωB (where ωB is inversely proportional to the length scale Λ characteristic of fluid flow and permeability). The high‐frequency phase velocity C 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 ωB. This behavior is approximately independent on the pore radius distribution, provided that ωB (or the corresponding fluid flow length scale Λ) is held constant. A mechanism based on wave scattering and interferences of forward and backward traveling (pore‐scale) Biot slow waves is proposed to explain the observations.