Modeling the Spectral Features of Acoustic Noise Produced by Gas Flow in Rock Samples Based on the Theory of Ensemble of Interconnected Pore Resonators

Abstract

A new model describing the pore space as an ensemble of interconnected Helmholtz resonators is proposed. This model makes it possible to improve the description of spectral peculiarities of the experimentally recorded acoustic noise during gas flow through a porous medium. The results of the resonance frequency calculation are presented by the example of the pore space model of the Indiana Limestone. Microvortexes in the pores are considered as the main mechanism of acoustic noise generation by gas flow. The presented numerical simulations on COMSOL Multiphysics show that the generation of microvortexes begins when the Reynolds number in the pores reaches 1 to 10.

APPENDIX

Let us derive the formula for hydraulic resistance to gas flow for the system of three capillaries shown in Fig. A1: one capillary of radius R and length H is connected with two capillaries (throats) of a smaller radius r and length h. Let the pressure on the left end of the system (x = 0) and on the right end (x = 2h + H) be p₁ and p₄, respectively.

We assume that the flow in capillaries is laminar and described by the Poiseuille formula:

$${{q}^{v}}\left( x \right) = - \frac{{\pi {{\zeta }^{4}}\left( x \right)}}{{8\mu }}\frac{{\partial p}}{{\partial x}},$$

((A.1))

where ζ is the capillary radius and µ is the fluid viscosity (gas).

In the case of a steady flow, due to mass conservation,

$${{q}^{m}} = \rho \left( x \right){{q}^{v}}\left( x \right) = {\text{const}}{\text{,}}\,\,\,\,\forall x,$$

((A.2))

where superscript m denotes the mass flow rate and superscript v denotes the volumetric flow rate; ρ is the fluid density.

We also assume that gas is ideal and the flow process is isothermal. Then, the equation of state has the form:

$$\rho \left( p \right) = {{\rho }_{{{\text{ref}}}}}\frac{{{{p}_{{{\text{ref}}}}}}}{p},$$

((A.3))

where ρ_ref is the gas density at a certain reference pressure p_ref (e.g., atmospheric pressure p_ref = p_atm).

The substitution of (A.1) and (A.3) into (A.2) and subsequent integration between the limits x = 0 and x = x* yields

$${{q}^{m}} = \frac{{\pi {{\rho }_{{{\text{atm}}}}}}}{{8\mu }}{{\left[ {\int\limits_0^{x^*} {\frac{{dx}}{{{{\zeta }^{4}}\left( x \right)}}} } \right]}^{{ - 1}}}\frac{{{{p}^{2}}\left( {x{\text{*}}} \right) - p_{1}^{2}}}{{2{{p}_{{{\text{atm}}}}}}},$$

((A.4))

where p₁ = p(x = 0).

Conservation of the mass flow rate in cross sections 2 and 3 gives the system of two equations for two unknown pressures p₂ and p₃:

$$\begin{gathered} \left( {p_{2}^{2} - p_{1}^{2}} \right) = \alpha \left( {p_{3}^{2} - p_{2}^{2}} \right), \\ \left( {p_{4}^{2} - p_{3}^{2}} \right) = \alpha \left( {p_{3}^{2} - p_{2}^{2}} \right), \\ \end{gathered} $$

((A.5))

where $\alpha = {{\left( {\frac{R}{r}} \right)}^{4}}\frac{h}{H}.$

The final relationship between the mass flow rate and pressure on the left and right ends of the capillary system has the following form:

$${{q}^{m}} = \frac{{\pi {{\rho }_{{{\text{atm}}}}}}}{{8\mu }}\frac{{{{r}^{4}}}}{h}\frac{\alpha }{{1 + 2\alpha }}\frac{{p_{4}^{2} - p_{1}^{2}}}{{2{{p}_{{{\text{atm}}}}}}}.$$

((A.6))

In the case when 2α $\gg$ 1, formula (A.6) simplifies to

$${{q}^{m}} = \frac{{\pi {{\rho }_{{{\text{atm}}}}}}}{{8\mu }}\frac{{{{r}^{4}}}}{{2h}}\frac{{p_{4}^{2} - p_{1}^{2}}}{{2{{p}_{{{\text{atm}}}}}}}.$$

((A.6*))

As follows from (A.6), the relative contribution of the capillaries to the total hydraulic resistance of the system is inversely proportional to ratio of their radius to the fourth power and directly proportional to their length ratio. Thus, given similar lengths of capillaries, the hydraulic resistance of the system is almost completely determined by capillaries with a small radius.