The Poroacoustics node introduces several fluid models for modeling the propagation of acoustic waves in porous materials. After some general remarks about fluid models and the rigid and limp regime approximations, these models are discussed in this section:

The poroacoustics fluid models are equivalent fluid models that mimic the behaviors of a full Poroelastic Material model, which is defined by Biot’s theory. A poroacoustics fluid model is based on describing the frequency-dependent effective fluid density ρ(ω) and the effective fluid bulk modulus K(ω) of the combined equivalent fluid-solid system (saturating fluid and porous matrix). The description of these models includes the losses associated with the propagation of acoustic waves in porous materials. An equivalent fluid model is computationally less demanding than the full poroelastic model. However, it is only physically correct for certain choices of material parameters. Most poroacoustic models are only valid in the rigid or limp porous matrix approximations.

In the rigid porous matrix limit, the matrix is assumed to be so stiff that it does not move (sometimes referred to as a motionless skeleton model). In this case it is assumed that in Biot’s theory u = 0, which yields a wave equation with complex density and bulk modulus. In a rigid porous material the Biot-Willis coefficient is equal to the porosity αB = εP. All the poroacoustic fluid models are based on defining the rigid effective density ρrig (see Ref. 9).

The limp porous matrix limit is the opposite of the rigid assumption. It is used to model materials where the stiffness of solid phases (the porous matrix) is so weak that it cannot support free, structure-borne wave propagation (neither longitudinal nor transverse). That is, the stiffness (in vacuo bulk stiffness) is very small compared to air such that the solid phase motion becomes acoustically significant. If it is light enough, the solid phase still moves because it is “dragged along” by the fluid motion; a limp porous material model is also an equivalent fluid model because it only features a single longitudinal wave type. Typically, the limp assumption can be applied to very light weight fibrous materials (less than 10 kg/m3) if these are not stiffened by the injection of binder material. In the limp case, it is assumed that the stress tensor vanishes and that in Biot’s theory, the Biot-Willis coefficient is αB = 1. The limp density is related to the rigid density by a mixture model (see Ref. 10)

where ρf is the fluid density, ρd is the drained porous matrix density, ρav is the average effective density, and ρlimp is the resulting effective limp density.

We show below that all implemented models of poroacoustics apply for all fluids (gases and liquids) except the purely empirical Delany-Bazley-Miki model which has been obtained from fitting to measurement data.

The applicability of the poroacoustics models to general fluids is not described in the literature, but follows directly from repeating the published derivations without assuming the saturating fluid to be an ideal gas. The only difference is that one needs to use the following general expression for the ratio of specific heats γ (express γ in terms of other material parameters) which is valid for any fluid (for gases γ is typically a known parameter)

where α0 is the isobaric thermal expansion coefficient, T0 is the background quiescent temperature, ρ0 is the background quiescent density, Cp is the specific heat at constant pressure, β0 is the isothermal compressibility (inverse isothermal bulk modulus KT), and c is the speed of sound. This result comes from thermodynamics.

To show the general applicability of the poroacoustics models, we here rederive the equivalent bulk modulus for the simplest case of the Zwikker-Kosten model for a general fluid. All other implemented models (again except the Delany-Bazley-Miki model) are extensions of this result (see Ref. 9), and the equivalent bulk moduli can therefore in general be written as

for any of these models, where the frequency response Ψ(ω) is specific to each model. Consequently, the following derivation is sufficient to show the general result.

A relationship between the pressure p and the density ρ is needed to define the effective bulk modulus, and the equation of state provides such a relationship. Instead of relying on the ideal gas equation, we here use the following general equation of state

which merely stipulates that the density ρ is a function of both pressure p and temperature T. The temperature-dependence is nonnegligible since we are dealing with small pores, and acoustics in porous materials belong to thermoviscous acoustics (see Thermoviscous Acoustics Interfaces). Assuming the acoustic fields are small harmonic perturbations (denoted by a subscript 1) about a quiescent reference state (denoted by subscript 0)

and Taylor-expanding the general equation of state Equation 2-46 to first order, we find

where βT is the isothermal compressibility.

In order to calculate the effective bulk modulus from Equation 2-45 using the equation of state Equation 2-48, a relationship between T1 and p1 is derived from the linearized energy equation (see Theory Background for the Thermoviscous Acoustics Branch)

where Dth is the thermal diffusivity, k is the thermal conductivity, and Q is a volumetric heat source. To proceed with the solution of this equation, it is generally assumed that (i) the pore can be modeled as a hollow cylinder, (ii) the contributions from radial variations dominate over axial variations so it suffices to solve the radial problem while assuming no axial variations Ref. 9, and (iii) the pore radius is much smaller than the wavelength, whereby the pressure does not change noticeably across the radius. By these assumptions the pressure p1 can be treated as a constant in Equation 2-49 and the equation has only radial variations:

The cross-sectional average solution under these assumptions, and with the boundary condition of zero acoustic temperature T1 on the pore walls at r = R, is

where Wo is the Womersley number (see the note below), and Pr is the Prandtl number, measuring the relative width of the viscous boundary layer thickness to the thermal boundary layer thickness.

With the solution from Equation 2-50, the pressure can now be expressed as a function of the density using the first-order equation of state Equation 2-48:

From this we obtain the following expression for the equivalent bulk modulus (see Equation 2-45)

where the ratio of specific heats γ is defined in Equation 2-43 for a general fluid and K0 is the isentropic bulk modulus of the fluid. This formula Equation 2-53 is mathematically equivalent to the normal Zwikker-Kosten formula for the equivalent bulk modulus (see Ref. 15), but it has been derived for a general fluid and not just an ideal gas.

We emphasize that from this result for a general fluid follows the general formula Equation 2-44, in this particular case with

with the Womersley number Wo containing the frequency dependence.

where ρf is the fluid density, f is the frequency, and Rf is the flow resistivity. Several predefined sets of the coefficients Ci exist. They are the classic Delany-Bazley model, the Miki model (see Ref. 9, section 2.5 and Ref. 22), the Qunli, several variants of the Mechel model for different configurations, the Komatsu model, and a so-called Modified Champoux and Allard model (see Ref. 23). These are all empirical models based on fitting the two complex functions to measured data for the complex wave number kc and complex specific acoustic impedance Zc. All the models are applicable for materials with a porosity εp close to 1. The applicability of the different model parameters is listed in Table 2-10 and also Ref. 23 for further details.

Zwikker-Kosten is one of the earliest equivalent fluid models for porous materials (Ref. 15). It is a rigid frame model defined by the complex (rigid) density

where Hr is the hydraulic radius of the pores (for straight cylindrical pores Hr = a = radius) and Wo is the Womersley number (see Equation 2-51 and the note below). Wo is related to the ratio between viscous penetration depth δv and the hydraulic radius. δv gives the scale of the viscous boundary layer thickness (see Theory Background for the Thermoviscous Acoustics Branch for details). The bulk modulus is given by

where pA denotes the ambient pressure, ρf the fluid density, γ the ratio of specific heat, Pr the Prandtl number, µ the dynamic viscosity, Cp the heat capacity at constant pressure, and k the coefficient of thermal conduction. J0 and J1 are Bessel functions of the first kind. The factor γ pA is the isentropic bulk modulus (K0 = γ pA). The free parameters of the pores are the porosity εP and the hydraulic radius Hr.

The Attenborough model is also based on the cylindrical-like pore assumption. It is a so-called four parameter semi-empirical model. The model is an extension of the Zwikker-Kosten model and adds two more input parameters. It accounts for the tortuosity (high-frequency limit) τ∞, which is related to the orientation of the pores relative to the propagation direction. The hydraulic diameter of the pores is replaced by an expression that includes the flow resistivity Rf, and a fitting parameter b, (this parameter is related to the anisotropy of the pores). See Ref. 9 and Ref. 16. The equivalent density and bulk modulus are defined as

where pA denotes the ambient pressure, ρf the fluid density, γ the ratio of specific heat, Pr the Prandtl number, µ the dynamic viscosity, Cp the heat capacity at constant pressure, and k the coefficient of thermal conduction. J0 and J1 are Bessel functions of the first kind. The variable s' (anisotropy factor) is derived from other material parameters and is related to the Womersley number (see Equation 2-51):

Here, ω denotes the angular frequency. The four parameters needed (when the fluid is air at room temperature) are the porosity εP, the tortuosity τ∞, flow resistivity Rf, and the fitting parameter b (dimensionless, close to 1). The fitting parameter b is tabulated for certain well-defined pore cross-sections in Table 2-11.

The viscous characteristic length of the model Lv can also be defined by

with δv the viscous penetration depth. This length is related to the pores’ circular cross section radius R (for a cylinder) and the thickness of the viscous boundary layer. The parameter b = 1/s, where s is the viscous characteristic length parameter. See the following models:

Note that the tortuosity is related to the angle θ between the cylindrical pores and the direction of propagation of the wave, by

The Wilson model is a generalization of the analytical models for porous materials with constant cross section and parallel pores. This model is intended to match the middle frequency behavior of a porous material (see Ref. 9, Ref. 17, and Ref. 18). It is not a good model for ω tending to 0 or infinity. The equivalent density and bulk modulus are given by

where τvor denotes the vorticity-mode relaxation time, τent the entropy-mode relaxation time, ρ∞ the infinity-frequency limit for the density, K∞ the infinity-frequency limit for the bulk modulus, and γ is the ratio of specific heats. These are the four free parameters. With appropriate choices for the relaxation parameters, the Wilson model can be fitted to mimic all the models described here. For example, setting τvor = 2.54/Rf and τent = 3.75/Rf, the equations mimic the Delany-Bazley model (see Cox and D’Antonio Sec. 5.4.4, Ref. 18).

Here τ∞ denotes the (high-frequency limit) tortuosity (it is called q2 in the Wilson’s paper Ref. 17), εp the porosity, ρf the fluid density, l a characteristic pore dimension, and Pr is the Prandtl number.

The Johnson-Champoux-Allard (JCA) porous matrix model is defined by the following equivalent rigid densities ρrig(ω) and equivalent bulk modulus K(ω):

Here τ∞ is the tortuosity factor (high-frequency limit), ρf is the fluid density, εp is the porosity, Rf is the flow resistivity, μ is the dynamic viscosity, pA is the quiescent pressure, γ is the ratio of specific heats, Lv is the viscous characteristic length, Lth is the thermal characteristic length, and Pr is the Prandtl number. The viscous characteristic length is related to the viscous characteristic length parameter s by

Here s is a pore geometry dependent factor between 0.3 and 3.0 (for example 1 for circular pores, 0.78 for slits)

The Johnson-Champoux-Allard-Lafarge (JCAL) model introduces corrections to the bulk modulus thermal behavior at low frequencies that is not captured by the JCA model (see Ref. 13). The equivalent density is the same as in the JCA model. The correction is to the bulk modulus and is given by

with the introduction of the new parameter, k’0, which is the static thermal permeability (SI unit: m2). For measurements and details of this parameter see Ref. 11 and Ref. 12 for examples.

The Johnson-Champoux-Allard-Pride-Lafarge (JCAPL) model further extends the JCAL models by introducing a static viscous τ0 and thermal τ'0 tortuosity, which both introduce low frequency corrections to the JCAL and JCA models. See Ref. 14 and Ref. 9.

where the new parameter is the static viscous tortuosity τ0 (dimensionless). The viscous permeability is defined as k0 = μ/Rf (SI unit: m2).

The complex bulk modulus K is given by:

where the new parameter is the static thermal tortuosity τ'0 (dimensionless).

The Williams EDFM model is a so-called equivalent density fluid model. The model is derived under the assumption that the bulk and shear moduli of the frame of the porous material are negligible, see Ref. 24 for further details. This is why the model pertains to sediments. The effective bulk modulus Keff and an effective density ρeff defined my the model are given by

where the subscript “gr” pertains to the grains and the subscript “f” to the saturating fluid. The porosity is denoted εp, the observed mixture density ρmix, the tortuosity , the dynamic viscosity μ, the angular frequency ω, and the permeability of the sediments κ. The function F is a function of the Womersley number Wo defined as

where Jn(x) is the Bessel function of the first kind of order n and Hr is the hydraulic radius. Based on these the effective complex speed of sound and density are given as

where θf, Kf, and ρf, are the fluid’s volume fraction, adiabatic bulk modulus, and density, respectively; and θi, Ki, and ρi, are the inclusion’s volume fractions, adiabatic bulk moduli, and densities, respectively. And again the complex speed of sound and density are defined as