Oscillating Flow of a Compressible Fluid through Deformable Pipes and Pipe Networks: Wave Propagation Phenomena

Abstract

Similarly to blood pulse propagation in the artery system, oscillating flow can propagate as a wave in fluid-saturated pipes, networks of pipes or, by extension, in porous media, if the fluid is compressible and/or the pipes are elastically deformable. First, propagation of flow waves generated in a semi-infinite pipe by harmonic pressure oscillations at the pipe entrance is analyzed. The dispersion equation is derived, allowing determination of the phase velocity and quality factor as functions of frequency. Wave reflections at the end of a finite-length pipe and ensuing interferences between forward and backward traveling waves are then examined. Because of fluid storage in the pipe, the amplitude of the AC volumetric fluxes entering and exiting the pipe at its upstream and downstream ends are not equal. Thus, two different, upstream and downstream, frequencydependent, AC hydraulic conductivities are introduced. Superposed on the classic viscous-inertial flow transition (controlled by the value of the pipe radius), these complex-valued parameters show another transition between an interference-free regime at low frequencies and a strong interference regime above a critical frequency that roughly scales as the pipe length. Because of attenuation, the flow wave interferences tend to gradually weaken with increasing frequencies. Finally, the single pipe model is used to investigate fluid flow waves through pipe networks with results very similar to those described above. The flow waves analyzed here are akin to the Biot's slow P waves and their propagation properties could affect seismic soundings in some geological settings.

Frequency-Dependent Streaming Potential of Porous Media—Part 1: Experimental Approaches and Apparatus Design

“…which is consistent with Bernabé [14][15][16] and Charlaix et al [17]. The limit of (2) as ω → 0 is simply r 2 /8, which is consistent with the steady-state permeability of a tube with radius r given by Poiseuille's law κ DC = πr 4 /8A tube for a single tube, where A tube = πr 2 .…”

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

“…In order to help distinguishing scales, the microscale and macroscale phase velocities will be hereafter indicated by lower and uppercase letters, respectively. As expected, the single pipe model does allow propagation of fluid flow waves that can be identified with pore‐scale Biot slow waves [ Bernabé , 2009]. If the long‐wave approximation is assumed, the following dispersion equation is obtained: where J 0 and J 1 denote Bessel functions of the first kind, κ 2 = i ω ρ F / η , c ( ω ) is the phase velocity of the pore‐scale Biot slow wave in the pipe, r p the pipe radius and V S the shear wave velocity in the solid walls.…”

“…Motivated by the blood circulation studies, I developed a model of AC flow of a compressible fluid through a single cylindrical conduit inside an elastic solid of very large lateral extent (see Bernabé [2009]; in contrast, arteries are modeled as thin‐wall pipes [ Zamir , 2000]), the idea being that this single pipe model can be used to investigate AC flow through heterogeneous pipe networks and, by extension, porous rock. The work reported here can be viewed as an attempt to upscale the phase velocity of the Biot slow wave from the microscale (i.e., single pipe) to the macroscale (i.e., network of pipes).…”

“…If the long‐wave approximation is assumed, the following dispersion equation is obtained: where J 0 and J 1 denote Bessel functions of the first kind, κ 2 = i ω ρ F / η , c ( ω ) is the phase velocity of the pore‐scale Biot slow wave in the pipe, r p the pipe radius and V S the shear wave velocity in the solid walls. Obviously, equation (3) can be recast in the form of a fourth degree algebraic equation in c ( ω ) [ Bernabé , 2009]. However, the form above makes it easier to verify numerically that, in the limit of a rigid solid (i.e., V S ≫ c F ), equation (3) is nearly identical to equations (1) and (2) with Λ ≈ r p and α = 1 as expected for a straight conduit oriented in the flow direction (and using the fact that, as a consequence of Poiseuille formula, the ratio k /ϕ for a single pipe contained in a finite solid volume is equal to r p 2 /8).…”

Effects of Fluid Rheology and Pore Connectivity on Rock Permeability Based on a Network Model

“…The compressibility of the fluid is α = (1/ρ f )∂ρ f /∂p. Equations (1) and (4) can be simplified under the assumption of long wavelengths (Bernabé, 2009a). Assuming that the wave propagates in the z direction, u, v, and p have the form of u(r,…”