# Dynamic Permeability of Fractal Porous Media

Abdellah Bouchendouka, Zine El Abiddine Fellah, Alexander S. Alexander S.

Balankin, Erick Ogam, Philippe Bindzi, Claude L. Depollier

► To cite this version:

Abdellah Bouchendouka, Zine El Abiddine Fellah, Alexander S. Alexander S. Balankin, Erick Ogam, Philippe Bindzi, et al. Dynamic Permeability of Fractal Porous Media. Journal of the Acoustical Society of America, In press. hal-05531443

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Submitted on 1 Mar 2026

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Dynamic Permeability of Fractal Porous Media

A. Bouchendouka,¹ Z.E.A. Fellah,¹ A.S. Balankin,² E. Ogam,¹ P. Bindzi,¹ and C.

Depollier³

¹Aix Marseille Univ, CNRS, Centrale Med, LMA, Marseille,
France

²Instituto Politécnico Nacional, SEPI-ESIME Zacatenco,
Unidad Profesional Adolfo López Mateos, Mexico City 07738,
Mexico

³Acoustics Laboratory of the University of Le Mans (LAUM), UMR 6613,
Institut d'Acoustique - Graduate School (IA-GS), CNRS, Le Mans University,
Le Mans, France

(Dated: 1 March 2026)
This work is devoted to theoretical and experimental studies of acoustic waves in air-saturated fractal porous media. The theoretical model of oscillatory flow through a porous medium is based on the concept of a fractal continuum in a non-integer dimensional space. This model allows us to obtain analytical solutions for the velocity field and frequency-dependent permeability. Accordingly, static and dynamic permeability are expressed as functions of fractal dimension numbers and characteristic length scales. To validate theoretical findings, we conducted two independent acoustic experiments with polyurethane open-cell foams having a fractal pore structure. High-frequency ultrasound measurements (in the range of tens to hundreds of kHz) were used to evaluate the fractal properties of the studied foams. The fractal parameters were determined by fitting the signals transmitted from the foam to theoretical output signals. These parameters were then used to predict the static permeability $k_0$. We found that the predicted values of $k_0$ are in good agreement with the experimental values obtained in low-frequency long acoustic waveguide (a few hundred Hz range) experiments. Thus, this work provides a robust and physically meaningful framework for describing acoustic wave propagation in fractal porous materials.
## I. INTRODUCTION

Natural materials, particularly porous media, inherently exhibit heterogeneity. A porous medium can be viewed as a region within Euclidean space filled with matter possessing continuous properties, where material properties, when averaged over relevant length and time scales, vary smoothly. Conventional homogenization techniques offer an effective means to model heterogeneous materials when the microstructure possesses translational symmetry and decoupled length scales (Allaire, 1992; Whitaker, 2013). However, porous media often display intricate structures characterized by statistical scale invariance across extensive ranges of length scales, significantly influencing transport phenomena (Adler and Thovert, 1993).

In this context, fractal geometry, common in natural porous substances such as sedimentary rocks, biological tissues, and engineered foams, is frequently employed to model scale-invariant materials. In natural media, self-similarity typically exists only between two distinct length scales, a lower and an upper cutoff, where such structures are referred to as pre-fractals. Investigating transport within these structures is challenging, as functions defined on fractals lack conventional differentiability (Balankin, 2015a; Balankin and Mena, 2023). Consequently, various methodologies have been proposed to handle fractals within a continuous framework. Among these, the non-integer dimensional space approach has received significant attention (Balankin and Mena, 2023; Tarasov, 2015b, 2025). This approach maps problems defined on fractals onto equivalent problems in a non-integer dimensional space, where the fractal is treated as a continuous matter region termed a fractal contin-
uum (Tarasov, 2005), later generalized to incorporate anisotropy, fractal connectivity, and
dynamics (Balankin and Mena, 2023; Ostoja-Starzewski, 2009).

A fractal continuum can be defined as a region of integer dimension (Balankin et al., 2016a) or fractional dimension (Balankin, 2020), equipped with a fractal norm, metric, measure, and suitable vector differential calculus. This allows mapping physical problems on fractals onto boundary value problems for fractal continua. Integer-dimensional fractal continua are applicable only to fractals where the random walker has an integer number of effective spatial degrees of freedom, which is generally not the case for porous media. The fractional number of effective spatial degrees of freedom can instead be accounted for within the non-integer dimensional space approach (Balankin, 2015b; 2020; Stillinger, 1977; Tarasov, 2015b). Therefore, in this work, porous media are modeled using a fractal continuum in a non-integer dimensional space.

A critical aspect of analyzing physical phenomena on fractals is vector calculus. Various definitions of operators such as gradient, divergence, curl, and Laplacian have been proposed in non-integer dimensional spaces (see Refs. (Balankin, 2020; Balankin et al., 2016a; Balankin and Mena, 2023; Li and Ostoja-Starzewski, 2009; Ostoja-Starzewski, 2009; Tarasov, 2014; 2015b)). These definitions differ in their physical interpretations, and distinct fractal media may require different modeling approaches (Balankin and Elizarraraz, 2013; Tarasov, 2014). For transport phenomena in fractal porous media, it is essential that the chosen framework properly incorporates the fractal characteristics of the medium; otherwise, the resulting predictions may be non-physical (Balankin and Elizarraraz, 2013). In this study, we employ the vector calculus model developed in (Balankin, 2015b; 2020; Balankin and Mena, 2023),
as the associated fractal dimensions have clear geometric interpretations and can be exper-
imentally measured in natural porous media. Other models (Tarasov, 2014; 2015b; 2025)
may be explored in future work for comparison.

This paper is devoted to the study of acoustic wave propagation within a fractal porous
medium. While transport phenomena in fractal porous media have been widely studied,
acoustic wave propagation remains largely unexplored, with only preliminary attempts re-
ported in Refs. (Berbiche et al., 2017; Bouchendouka et al., 2024b; Fellah et al., 2017, 2021a).
In these studies, operators defined in non-integer dimensional spaces are applied to homog-
enized macroscopic equations, which raises questions regarding physical consistency of the
obtained results.

In contrast, the present approach considers a statistical fractal structure spanning from
the microscopic scale to the representative elementary volume (REV) scale, assuming that
the wavelength is sufficiently large compared to the REV size and that scale separation
holds (Yu, 2008). The microscopic conservation equations governing wave propagation are
formulated within the fractal continuum in non-integer dimensional space framework, leading
to an effective medium characterized by a dynamic permeability that depends explicitly on
fractal dimensions. The significance of this work lies in its fundamentally different modeling
approach to acoustic wave propagation in complex pore geometries, offering new physical
insights and new modeling tools.

The remainder of this paper is organized as follows. Section II reviews classical models for
wave propagation in rigid porous media. Section III describes the structural characteristics
of fractal porous media and the relevant length scales. Section IV introduces the fractal
continuum and non-integer dimensional space framework. Section V derives the governing equations and analytical expression for the dynamic permeability. Section VI discusses experimental determination of the fractal dimensions and model validation. Section VII concludes the paper.

## II. WAVE PROPAGATION IN CLASSICAL POROUS MEDIA

Consider a fluid-saturated porous medium with bicontinuous solid and fluid networks subjected to harmonic excitation at frequency $\omega$. Assuming the solid phase remains motionless (thanks to its high density or elastic rigidity) and that wave propagation occurs solely through the fluid-filled voids characterized by density $\rho_f$, viscosity $\eta$, and bulk modulus $K_f$, the fluid motion is governed by the linearized Navier-Stokes, continuity, state, and energy equations. Initially formulated by Kirchhoff (Kirchhoff, 1868), this problem is inherently complex. To address this complexity, the porous medium is treated as a statistically homogeneous material with two distinct scales: a microscale $h$ and a macroscale $H$, and structural complexities at scale $h$ are averaged out (Allaire, 1992; Keller, 2017). Here, the wavelength $\lambda$ is considered to be around the same size of the macroscale $H$. The solution of this problem involves addressing the microscopic problem that characterizes visco-thermal effects using the classical two-scale asymptotic method (Keller, 2017). This approach treats the microscopic fields (velocity, temperature, pressure) as perturbations expanded in terms of $v = h/H$ (Sheng and Zhou, 1988; Zhou and Sheng, 1989). At first order, the fluid is assumed incompressible at the microscale. Solutions are typically obtained for extreme frequency regimes high and low (Allard and Atalla, 2009).
By averaging the microscopic fields (velocity and temperature) at these frequency extremes using spatial averaging techniques (Whitaker, 2013), effective properties are derived. Specifically, the effective density and bulk modulus are given by $\rho(\omega) = \rho_0\alpha(\omega)$ and $K(\omega) = K_f\beta(\omega)^{-1}$ (Allard and Atalla, 2009). The functions $\alpha(\omega)$ and $\beta(\omega)$, known as response functions, reflect the porous medium's structural influences on wave propagation, attenuation, and energy dissipation. These functions are defined by models such as the Johnson-Champoux-Allard-Lafarge (JCAL) (Champoux and Allard, 1991; Johnson et al., 1987; Lafarge et al., 1997). Specifically, the JCAL model provides:

$$ \alpha(\omega) = \alpha_{\infty} \left( 1 + \frac{2}{\Lambda} \varepsilon(\omega) \right), \qquad (1) $$

$$ \beta(\omega) = \gamma - (\gamma - 1) \left( 1 + \frac{2}{\Lambda'} \varepsilon'(\omega) \right)^{-1}, \qquad (2) $$

for high frequencies, and

$$ \alpha(\omega) = \phi \frac{\varepsilon(\omega)^2}{k_0}, \quad \beta(\omega) = \gamma - (\gamma - 1) \left( \phi \frac{\varepsilon'(\omega)^2}{k_0'} \right)^{-1}, \qquad (3) $$

for low frequencies. Here, $\gamma$ is the adiabatic index, $\varepsilon(\omega) = \sqrt{\eta/(j\omega\rho_f)}$ and $\varepsilon'(\omega) = \varepsilon(P_r\omega)$ represent the viscous and thermal complex skin depths, respectively, with $P_r$ being the Prandtl number. The model employs six parameters to describe the porous structure: the ideal fluid tortuosity $\alpha_{\infty}$, viscous and thermal characteristic lengths $\Lambda$ and $\Lambda'$, static viscous and thermal permeabilities $k_0$ and $k_0'$, and open porosity $\phi$. Additionally, the viscous and thermal dynamic permeabilities $k(\omega)$ and $k'(\omega)$ can be expressed in terms of $\alpha(\omega)$ and $\beta(\omega)$ as:
$$k(\omega) = \phi\varepsilon(\omega)^2\alpha(\omega)^{-1}, \quad k'(\omega) = \phi\varepsilon'(\omega)^2\left(\frac{\gamma - \beta(\omega)}{\gamma - 1}\right). \qquad (4)$$

In summary, the classical framework for acoustic wave propagation in rigid porous media, with large wavelengths, assumes statistical homogeneity and periodic smooth microstructures, thereby overlooking micro-scale structural complexities. In the latter case, fractal geometry offers a more sophisticated mathematical approach for describing transport phenomena in porous media, including wave propagation. The use of fractal geometry based models is becoming widespread (see Refs (Xiao et al., 2023; Zhang et al., 2023) and references therein), because of their accurate depiction of the intricate micro-structures of porous materials using a few set of fractal dimensions. The subsequent section discusses the motivation for employing fractal concepts and outlines the methodology for studying wave propagation in fractal porous media.

### III. FRACTAL POROUS MEDIA

Historically, porous media were modeled as statistically homogeneous structures possessing translational symmetry (see, for review, (Allaire, 1992; Keller, 2017) and references therein). Katz and Thompson (Katz and Thompson, 1985) were among the first to reveal that pore spaces in various sandstones display statistical scale invariance over three to four orders of magnitude, ranging from 10 Å to 100 µm. Since then, it has been found that many porous materials exhibit fractal features (see (Baghban et al., 2020; Wei and Xia, 2017) and references therein). In contrast to translation-invariant homogeneous structures, fractals
obey scale invariance, that is, their patterns repeat at different scales (Mandelbrot, 1983).

Nevertheless, natural fractals such as porous media exhibit statistical self-similarity between a lower cutoff $l_{min}$, and an upper cutoff $l_{max}$, that is

$$l_{min} \le r \le l_{max} \qquad (5)$$

where r is a fractal linear size. The fractal mass dimension D is widely employed in the literature to describe fractal porous media. The fractal mass dimension D quantifies the extent to which a structure fills space (see Appendix A for definition). For determining the porosity of a medium, the dimension D alone is sufficient. However, the fractal mass dimension does not capture information about connectivity, tortuosity, or the dynamic properties of fractal porous media. Importantly, two fractals with identical D values may display different tortuosity and dynamic behaviors (see refs (Balankin, 2015a; Balankin and Elizarraraz, 2012b; Balankin et al., 2022a; Yu, 2008; Yu and Cheng, 2002) and references therein). Therefore, modeling a specific porous medium requires knowledge of at least three independent fractional dimensions. Numerous additional dimensions exist to characterize fractal objects, each offering distinct insights. In addition to the mass fractal dimension, this study employs the connectivity (chemical) dimension $d_{\ell}$ and the spectral dimension $d_{s}$ (see Table II and Appendix A for detailed definitions). In this regard, to model the real porous medium structure as a fractal one, we need at least one dimension out of these dimensions to be different from the topological dimension $d_{T}$.
## A. Scale Separation, REV, and Modeling Assumptions

Typically, a porous medium is also characterized by two length scales: the microscopic scale $h$ and the macroscopic size $H$, where $h \ll H$. The microscopic scale is associated with the size of the building blocks of the porous medium. Therefore, the lower cutoff of fractal structure is $l_{min} \approx h$. Consequently, the relation between $H$ and $l_{max}$ defines two physically different scenarios. In the simplest case of

$$l_{min} = h \ll l_{max} = H, \qquad (6)$$

the fractal structure spans the entire sample (Balankin et al., 2022b), as is the case for exact fractals such as Menger sponge or Cantor dust. Conversely, if the fractal space is confined to the mesoscale, such that

$$l_{min} \approx h \ll l_{max} \ll H, \qquad (7)$$

then the porous medium appears homogeneous on larger scales (Adler and Thovert, 1993; Balankin, 2015a).

In this study, we focus on fractal porous media with pore space that obeys scale-dependent behavior within scales (7), which encompasses a wide range of real fractal porous materials. Flow streamlines in such materials exhibit scale invariance over the length range given by equation (5). Accordingly, to analyze wave propagation, we follow two steps:

1. Definition of a representative volume element (RVE). Let $\Omega$ denote a representative volume large enough to encompass the fractal structure, with $\Omega$ having linear size at least $l_{max}$. This volume must accurately represent the statistical properties of the medium and is assumed to be periodic on the macroscale $H \gg l_{max}$ (see Fig.1). The
volume Ω consists of two phases: the solid matrix Ωs and the pore space Ωf occupied
by fluid.

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2. Formulation of the physical problem. The fluid motion in Ω is governed by the laws of conservation of momentum and mass. In this work, thermal effects are not taken into account, since it is assumed that the properties of the fluid are independent of temperature and are not related to the momentum field. To account for the fractal geometry of the pore space, we use proven methods for studying wave propagation in fractal structures. These methods fall into two main categories. The first category includes classical approaches based on traditional hydrodynamic equations for pore networks, where the flow is described using traditional hydrodynamic equations applied to a fractal pore network (Adler and Thovert, 1993; Balankin et al., 2022b). Although these approaches provide numerical solutions using methods such as finite element analysis, they are often impractical for engineering applications, e.g., in hydrology and the petroleum industry (Li et al., 2022). Alternative approaches involve the use of modified (fractal) hydrodynamic equations (Balankin, 2025; Balankin and Elizarraraz, 2012a,b; 2013; Balankin and Mena, 2023; Balankin et al., 2017b; 2016b; Ostoja-Starzewski, 2009; Reis and Voller, 2019; Tarasov, 2014; Yang, 2022). In this study, the problems for saturated porous media with fractal pore spaces are mapped onto fractal continuum flows governed by non-conventional hydrodynamic equations. The next section discusses this approach in more detail.

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In summary, the fractal model developed in subsequent sections is valid provided that at least one of the following dimensions—the fractal mass dimension *D*, the connectivity

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<figure><img src="image_2.png" /><figcaption>FIG. 1: (Color online) Representation of the Representative Volume Element (RVE) encompassing the fractal structure of the porous medium.</figcaption></figure>

dimension $d_\ell$, or the spectral dimension $d_s$—differs from the topological dimension $d_T$. In addition, the model assumes a well-defined scaling regime between a lower cutoff $l_{\min}$ and an upper cutoff $l_{\max}$, with the wavelength considered being much larger than $l_{\max}$.

## IV. FRACTAL CONTINUUM IN NON-INTEGER DIMENSIONAL SPACE

Non-integer dimensional space (Calcagni, 2012; Stillinger, 1977), and fractal continuum (Balankin, 2015a; Tarasov, 2011) are mathematical frameworks widely used in physics and applied sciences. Like integer-dimensional spaces, a non-integer dimensional space can be defined by a set of axioms. In this way, the non-integer-dimensional space can be defined either as a subset of an integer-dimensional space or without any reference to embedding. Accordingly, the fractional dimension of space may or may not be related to the covering measure associated with the Euclidean metric. Moreover, the non-integer dimensional space may or may not be scale-invariant and may contain loops at all scales or have no loops (Balankin, 2020).
The connection between the concept of non-integer dimensional space and a certain class of non-commutative geometries was discussed in (Arzano et al., 2011; Balankin and Mena, 2023). Theoretical studies of physical phenomena in non-integer dimensional spaces were reported in the literature (see, for review, (Balankin and Mena, 2023; Calcagni, 2012; Fellah et al., 2017; Golmankhaneh, 2023; He, 1990; Naqvi and Fiaz, 2017; Palmer and Stavrinou, 2004; Rami, 2009; Zubair et al., 2012)).

In order to map the problems on fractals into a continuum framework, Tarasov (Tarasov, 2005) introduced the concept of a fractal continuum. In general, a fractal continuum can be defined as a region of integer or non-integer space filled with continuous matter and provided with a fractal metric, measure, and suitable calculus (Balankin, 2015a). Different models of fractal continua were developed and reported in the literature (Balankin, 2015b, 2020; Balankin et al., 2016a; Balankin and Elizarraraz, 2012a, 2013; Balankin and Mena, 2023; Li and Ostoja-Starzewski, 2009; Tarasov, 2014, 2015b). Further, these models were employed in studies of physical phenomena in different types of fractal materials (Balankin, 2015a, 2018, 2025; Balankin et al., 2017b; Herrera-Hernández et al., 2019; Tarasov, 2014).

In this work, to study acoustic wave propagation in a saturated medium with a fractal porous space, we use the model of fractal continuum flow developed by (Balankin and Mena, 2023). In this model, the spatial dimension of the fractal continuum flow $F^{\gamma}$, defined via the Gaussian integral

$$ \int_{F^{\gamma}} d\vec{l} \exp(-a\vec{l}^2) = \left(\frac{\pi}{a}\right)^{n_{\gamma}/2}, \qquad (8) $$
where a is a constant, and the space dimension $n_γ$ is equal to the number of effective spatial degrees of freedom of a point walker in the pore space, which is defined by Eq. (A8) in Appendix A.

The anisotropy of fractal porous space can be accounted for by the spatial anisotropy constraints $γ_i$, which are associated with the Laplacian in an anisotropic non-integer dimensional space (Palmer and Stavrinou, 2004):

$$ \Delta = \sum_{i=1}^{n} \left( \frac{\partial^2}{\partial x_i^2} + \frac{\gamma_i - 1}{x_i} \frac{\partial}{\partial x_i} \right), \qquad (9) $$

where n is the Euclidean dimension of the embedding space, while

$$ \sum_{i=1}^{n} \gamma_i = n_γ \leq n. \qquad (10) $$

The model of fractal continuum flow also involves the definitions of the metric induced by the fractal topology (Balankin, 2015a). Accordingly, the coordinate $l_i$ along axis i in the fractal continuum flow is related to the Cartesian coordinate $x_i$ in the embedding Euclidean space as

$$ l_i = c_i x_i^{\zeta}, \qquad (11) $$

where $\zeta$ is the metric exponent equal to

$$ \zeta = D/n_γ, \qquad (12) $$

while the geometric constants $c_i = \lambda_i l_{min}^{(1-\zeta)}$ account for the pore space geometry (Balankin et al., 2017b).
In (Balankin and Mena, 2023) it was argued that, due to non-commutative geometry, in a fractal continuum of non-integer dimension one can define two different del-operators

$$ \vec{\nabla}_s^\gamma = \sum_{i=1}^n \ell_i^{\gamma_i-1} \frac{\partial}{\partial \ell_i} \vec{e}_i, \qquad \vec{\nabla}_v^\gamma = \sum_{i=1}^n \ell_i^{1-\gamma_i} \frac{\partial}{\partial \ell_i} \vec{e}_i, \qquad (13) $$

acting on scalar and vector fields, respectively. Accordingly, basic differential operators can be expressed in terms of two del-operators (13) in a conventional way as follows

$$ \mathrm{Grad}_\gamma(\Phi) = \vec{\nabla}_s^\gamma \Phi, \qquad (14) $$

$$ \mathrm{Div}_\gamma(\vec{\Psi}) = \vec{\nabla}_v^\gamma \cdot \vec{\Psi}, \qquad (15) $$

$$ \Delta_\gamma \Phi = \vec{\nabla}_v^\gamma \cdot (\vec{\nabla}_s^\gamma \Phi), \qquad (16) $$

The definitions of key attributes of fractal porous media and the corresponding parameters of the fractal continuum flow model are summarized in Table II.

In the next section, these operators are used to describe wave propagation in air-saturated fractal porous media.

## V. WAVE PROPAGATION IN A FRACTAL POROUS MEDIUM

As explained in Section III, the study of wave propagation in fractal porous media requires defining the microscopic problem that describes viscous fluid motion under harmonic excitation while taking into account the fractal structure. This problem is governed by the
<table><caption>TABLE I: Dimension numbers and scaling exponents characterizing the pore space and the corresponding model of fractal continuum flow (The fractal structures mentioned in this table can be found in refs (Balankin and Mena, 2023; Balankin et al., 2022a,b)).</caption><thead><tr><th></th><th>D</th><th>d<sub>ℓ</sub></th><th>d<sub>s</sub></th><th>n<sub>γ</sub></th><th>ζ</th></tr></thead><tbody><tr><td>Non-fractal case</td><td>D = d<sub>T</sub> = n = 3</td><td>3</td><td>3</td><td>3</td><td>1</td></tr><tr><td>Inverse Menger sponge (IMS)</td><td>ln20/ln3 = 2.727</td><td>2.727</td><td>2.51</td><td>2.94</td><td>0.926</td></tr><tr><td>Percolation cluster in 3D (PC)</td><td>2.5236</td><td>1.834</td><td>1.327</td><td>2.341</td><td>1.0766</td></tr><tr><td>Inverse Sierpinski cube (ISC)</td><td>ln98/ln5 = 2.849</td><td>2.849</td><td>2.738</td><td>2.96</td><td>0.963</td></tr></tbody></table>

conservation laws—namely, conservation of momentum and mass—and is formulated within a representative fractal volume $\Omega$. This volume comprises a fractal pore space $\Omega_f$ and a non-fractal solid phase $\Omega_s$. In this case, the governing equations within $\Omega_f$ using the fractal continuum in a non-integer dimensional space discussed in Section IV, are written in the frequency domain as follows:

$$ \rho_f j \omega \vec{V} = -\text{Grad}_\gamma p + \eta \Delta_\gamma \vec{V} + \left(\eta' + \frac{\eta}{3}\right) \text{Grad}_\gamma \left(\text{Div}_\gamma \vec{V}\right) + \vec{f}, \quad (17) $$

$$ j\omega\rho = -\rho_f \text{Div}_\gamma \vec{V}, \quad (18) $$

$$ \vec{V} = \vec{0} \quad \text{on} \quad \partial\Omega_f = \partial\Omega_s, \quad (19) $$

where $\rho_f$ is the fluid density, $\vec{V}$ is the fluid velocity, $p$ is the pressure, $\eta$ is the dynamic viscosity, $\eta'$ is the bulk viscosity, $\vec{f}$ is a body force acting on the fluid, and $j = \sqrt{-1}$. The
<table><caption>TABLE II: Dimension numbers and scaling exponents characterizing the pore space and the corresponding model of fractal continuum flow.</caption><thead><tr><th>Dimension number or exponent</th><th>Symbol</th><th>Property</th><th>Definition</th></tr></thead><tbody><tr><td>Spatial dimension of embedding space</td><td>n</td><td>Number of mutually orthogonal coordinates</td><td>Euclidean dimension</td></tr><tr><td>Fractal dimension</td><td>D</td><td>Box-counting measure</td><td>Eq. (A1)</td></tr><tr><td>Connectivity dimension</td><td>d<sub>ℓ</sub></td><td>Connectivity measure</td><td>Eq. (A3)</td></tr><tr><td>Spectral dimension</td><td>d<sub>s</sub></td><td>Number of effective dynamical degrees of freedom</td><td>Eq. (A5) or Eq. (A6)</td></tr><tr><td>Dimension of non-integer space</td><td>n<sub>γ</sub></td><td>Number of effective spatial degrees of freedom</td><td>Eqs. (8) and (A8)</td></tr><tr><td>Spatial anisotropy constraints</td><td>γ<sub>i</sub></td><td>Constraints of effective spatial degrees of freedom along coordinate axes</td><td>Eqs. (9) and (10)</td></tr><tr><td>Metric exponent</td><td>ζ</td><td>Metric induced by the fractal topology</td><td>Eqs. (11) and (12)</td></tr></tbody></table>

<footnote n="263">263 operators Grad<sub>γ</sub>, Div<sub>γ</sub>, and Δ<sub>γ</sub> are defined by Eqs. (14), (15), (16), respectively.</footnote>

<footnote>264</footnote>
The response of the fractal porous medium to harmonic excitation is characterized by the fractal permeability tensor. This tensor can, in principle, be obtained by averaging Eqs. (17)-(18). The expected form of the fractal Darcy's law is:

$$ \langle \vec{V} \rangle = - \frac{k_{ij}^{(c)}(\omega)}{\eta} \mathrm{Grad}_{\gamma} \langle p \rangle, \qquad (20) $$

where $\langle \cdot \rangle$ denotes the averaging operator over the fractal pore space $\Omega_f$. The $k_{ij}^{(c)}$ is the characteristic permeability tensor of the fractal continuum, which depends on the frequency $\omega$, the fractional dimensions $D, d_s, d_\ell, n_\gamma$, and other relevant geometrical parameters.

Deriving a general expression for $k_{ij}^{(c)}$ from Eqs. (17)-(19) is a challenging task for future research (Balankin and Elizarraraz, 2012b). However, wave propagation in a fractal porous medium can still be studied by adopting a simplified approach. Specifically, we consider a unidirectional flow through a porous medium where the cells $\Omega$ are contain a fractal pore space (see Figs. 2 and 3). A similar concept has been explored in multiple studies (Balankin et al., 2017b; Bouchendouka et al., 2022; 2023; Tarasov, 2014), which investigated the flow of a "fractal" fluid.

## A. Unidirectional fractal continuum flow

The modeling approach for a one-dimensional oscillatory flow through a fractal pore space is illustrated in Figs. 2 and 3. We assume an oscillatory flow motion in direction $i = 3$ (along the length of the cell). For the fluid to travel through the fractal cell $\Omega$ from one side to the other, there must be a continuous, interconnected network of pores. This implies that
1 ≤ $d_s$ ≤ $d_ℓ$ ≤ $D$. Regarding the geometry of $\Omega$, we consider a cell of length $L$ with a rectangular cross-section of sides $2a$ and $2b$. The saturating fluid is a viscous Newtonian fluid. Additionally, we assume that the wavelength of sound in the fluid is much larger than the characteristic sizes of pores and grains in the medium, so scale separation still applies (Yu, 2008). This assumption ensures that the fluid may be considered incompressible on the scale of the pore sizes. The solid matrix is assumed to be rigid, this rigidity being due to either a very high density, very large elastic moduli, or both. The cell size $2a$ (or $2b$) can be interpreted as the REV size in our fractal homogenization model, or the largest pore diameter in the pore space, or the upper cutoff of the fractal structure $l_{max}$. On the other hand the length $l_{min}$ in Eq. (11) represents the smallest pore size or the lower cutoff. The typical values of these lengths for the porous samples considered in this study will be discussed in Section VI.

<figure><img src="image_3.png" /><figcaption>FIG. 2: (Color online) Representation of a fractal porous medium consisting of cells $\Omega$ that contain a fractal pore space.</figcaption></figure>

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<figure><img src="image_4.png" /><figcaption>FIG. 3: (Color online) Representation of a single fractal cell $\Omega$, its cross-sectional view, and its side view.</figcaption></figure>

By transforming from the Cartesian coordinates $(x_1, x_2, x_3)$ in $E^3$ to the fractional co-ordinates $(l_1, l_2, l_3)$, the flow of the fluid is considered continuous and homogeneous in $F^{n_\gamma}$.

Consequently, the dimensions of the homogeneous fractal cell are given by:

$$A_{\gamma} = c_{1}(2a)^{\zeta}, \quad B_{\gamma} = c_{2}(2b)^{\zeta}, \quad L_{\gamma} = c_{3}L^{\zeta}, \qquad (21)$$

where (21) are obtained using the effective coordinates represented by Eq. (11). The exponent $\zeta$ is defined in Table II, and $c_i$ are defined by Eq. (11). Moreover, as already highlighted in (Balankin et al., 2022a), the metric exponent $\zeta$ is the same in all directions, even if the medium is anisotropic. The anistropy of the fractal cell is accounted for by Eq. (10) and the coefficients $c_i = \lambda_i l_{\min}^{1-\zeta}$. Since the pore space is less connected in direction $i = 1$ compared to $i = 3$ (see Fig. 3), we assume that $\lambda_1 < 1$ while $\lambda_3 = 1$.
To obtain an analytical solution, we consider a unidirectional fractal continuum flow, then Eqs.(17)-(19) become:

$$ \left( \frac{\partial^2 v}{\partial l_1^2} + \frac{n_\gamma/3 - 1}{l_1} \frac{\partial v}{\partial l_1} \right) - \frac{v}{\varepsilon^2} = \frac{l_3^{3-1}}{\eta} \frac{\partial p}{\partial l_3}, \quad (22) $$

$$ l_3^{1-n_\gamma/3} \frac{\partial v}{\partial l_3} = 0, \quad (23) $$

$$ v = 0, \quad (l_1 = \pm \frac{A_\gamma}{2}), \quad (24) $$

where $v$ is the velocity in the $l_3$ direction, with $\varepsilon^2 = \eta/(j\omega\rho_f)$. Eq. (22) resembles the modified Bessel equation. Using the boundary condition (24), the solution is straightforward:

$$ v(l_1) = \left( -\varepsilon^2 \frac{l_3^{n_\gamma/3-1}}{\eta} \frac{\partial p}{\partial l_3} \right) \left[ 1 - \left( \frac{2l_1}{A_\gamma} \right)^{-\nu} \frac{I_\nu(l_1/\varepsilon)}{I_\nu(A_\gamma/[2\varepsilon])} \right], \quad (25) $$

where $I_\nu$ is the modified Bessel function of the first kind, and $\nu = (n_\gamma/3 - 2)/2$. The velocity profile represented by Eq.(25) reflect the fluid flow as it is confined in a fractal pore space. It is imperative to emphasize that a unidirectional fractal continuum flow in $F^{n_\gamma}$ is not the same as a unidirectional flow in $E^3$. The former preserves the complexity of the geometry as well as the dynamical processes through $D$, $d_\ell$, $d_s$, while the latter does not. In this regard, a unidirectional flow in $F^{n_\gamma}$ is still useful for modeling wave propagation in porous media.

To verify the mathematical validity of Eq.(25) we consider the non-fractal case ($D = d_\ell = d_s = n_\gamma = n = d_T = 3$), meaning no pore space in the cell, then Eq.(25) coincides exactly with the classical velocity profile of a unidirectional fluid flow in $E^3$ (Allard and Atalla, 2009):

$$ v = \left( -\frac{\varepsilon^2}{\eta} \frac{\partial p}{\partial x_3} \right) \left[ 1 - \frac{\cosh(x_1/\varepsilon)}{\cosh(a/\varepsilon)} \right], \quad (26) $$
## B. Dynamic permeability in fractal porous media

The characteristic dynamic permeability $k^{(c)}(\omega)$ in $F^{n_\gamma}$ is represented by Eq. (20) which can be obtained by averaging the velocity profile (25) over the fractal pore network. In order to obtain the averaged velocity, one needs to perform the integration in a non-integer dimensional space. For the integration we use the single variable measure in a non-integer dimensional space:

$$d\mu_i^\gamma = \frac{\pi^{\gamma_i/2}}{\Gamma(\gamma_i/2)} \mathfrak{I}_i^{\gamma_i-1} d\mathfrak{I}_i, \qquad (27)$$

Considering that the velocity profile in $F^{n_\gamma}$ is dependent on $\mathfrak{I}_1$ only, the average of Eq. (25) can be expressed as:

$$\langle v(\mathfrak{I}_1) \rangle = \frac{2 \frac{\pi^{n_\gamma/6}}{2\Gamma(n_\gamma/6)}}{2 \frac{\pi^{n_\gamma/6}}{\Gamma(n_\gamma/6)} \int_0^{A_\gamma/2} \mathfrak{I}_1^{n_\gamma/3-1} d\mathfrak{I}_1} \int_0^{A_\gamma/2} v \mathfrak{I}_1^{n_\gamma/3-1} d\mathfrak{I}_1. \qquad (28)$$

The result of the integration (28) is:

$$\langle v \rangle = \left( -\varepsilon^2 \frac{\mathfrak{I}_3^{n_\gamma/3-1}}{\eta} \frac{\partial p}{\partial \mathfrak{I}_3} \right) \left[ 1 - 2(\nu + 1) \frac{2\varepsilon I_{\nu+1}(A_\gamma/[2\varepsilon])}{A_\gamma I_\nu(A_\gamma/[2\varepsilon])} \right]. \qquad (29)$$

Following Refs (Balankin and Elizarraraz, 2012b; Balankin et al., 2016b), the apparent dynamic permeability of the real fractal porous medium is defined as:

$$k(\omega) = -\eta \frac{\phi \langle v \rangle}{\partial p / \partial x_3}. \qquad (30)$$

where $\phi = (l_{\min}/l_{\max})^{3-D}$ is the porosity of the fractal porous medium. For a unidirectional flow, the apparent dynamic permeability $k(\omega)$ and the characteristic dynamic permeability $k^{(c)}(\omega)$ in $F^{n_\gamma}$ are related as:

$$k(\omega) \propto k^{(c)}(\omega) L^{-\theta_m}, \qquad (31)$$
334 where $\theta_m$ is referred to as the momentum diffusion exponent (Balankin et al., 2016b).

335 Replacing Eq.(29) in Eq.(30), we can obtain an apparent permeability:

$$k(\omega) = \phi \epsilon^2 \frac{L^{-\theta_m}}{\zeta c_3^{-2\nu}} \left[ 1 - \frac{2(\nu + 1) I_{\nu+1}(\kappa^* \sqrt{j})}{\kappa^* \sqrt{j} I_{\nu}(\kappa^* \sqrt{j})} \right], \quad (32)$$

336 where the momentum diffusion exponent $\theta_m = \zeta(2 - n_\gamma/3) - 1$, and $\kappa^* = \frac{A_\gamma}{2} \sqrt{\frac{\rho\omega}{\eta}}$ is an effective Womersley number. Notice that in the classical case ($D = d_s = d_\ell = n_\gamma = n = d_T = 3$), Eq.(32) coincides exactly with the classical expression of the dynamic permeability (Allard and Atalla, 2009) in $E^3$:

$$k(\omega) = \phi \epsilon^2 \left[ 1 - \frac{\tanh(\kappa \sqrt{j})}{\kappa \sqrt{j}} \right], \quad (33)$$

<figure><img src="image_5.png" /><figcaption>FIG. 4: (Color online) Plot of the dynamic permeability defined by Eq.(32) with respect to frequency $\omega$ for different fractal structures represented in Table I, with $\lambda_1 = 0.5$ and $\lambda_3 = 1$ for all four fractal structures, and the ratio of the cutoffs $l_{\max} / l_{\min} = \phi^{1/(D-3)}$ varies for each structure and value of $\phi$.</figcaption></figure>

340 Fig. 4 plots the real and imaginary parts of the dynamic permeability $k(\omega)$ with respect to frequency $\omega$ for different fractal structures presented in Table I. From this figure, we see
a couple of clear trends stand out. Compared to the non-fractal case, all fractal geometries modify both the magnitude and frequency dependence of the response. Specifically, a reduction in the real part of $k(\omega)$ for all four fractal structures is observed, which is an expected result, as the fractal structure impedes the flow of the fluid more than a non-fractal structure does. Moreover, the peaks of $\mathfrak{S} k(\omega)$ occur at different values of the frequency. Thus, the frequency-dependent transition from viscosity-dominated (low frequencies) to inertia-dominated (high frequencies) flow regimes is heavily affected by the fractal structures.

Now we want to define the parameters from the Johnson-Koplik-Dashen (JKD) model (Johnson et al., 1987), namely the tortuosity $\alpha_\infty$, the viscous characteristic length $\Lambda$, and the static permeability $k_0$, taking into account the fractal pore space. The high frequency parameters $\alpha_\infty$ and $\Lambda$ are defined via the behavior of the dynamic permeability $k(\omega)$ as $\omega \to \infty$ and the low frequency permeability $k_0$ is defined as $\omega \to 0$. To obtain the expressions of these parameters, we can use the Hankel's expansions for the modified Bessel functions in the case of large arguments (high frequency), and for $\omega \to 0$, we take up to three terms in the expansion of the modified Bessel function in Eq. (32) (see Appendix B for more details). In this case, we can easily get the following definition of the different parameters:

$$ \alpha_{\infty} = \frac{\zeta L^{\theta_m}}{c_3^{2\nu}}, \quad \Lambda = \frac{A_{\gamma}}{2(\nu + 1)}, \quad k_0 = \frac{\phi c_3^{2\nu} L^{-\theta_m} A_{\gamma}^2}{16\zeta(\nu + 1)(\nu + 2)}. \qquad (34) $$

Notice that for the non fractal case ($D = n_\gamma = n = d_T = d_\ell = d_s = 3$ and $\nu = -1/2$), we get the classical definitions in $E^3$:
$$ \alpha_{\infty} = 1, \quad \Lambda = 2a, \quad k_0 = \phi \frac{a^2}{3}. \qquad (35) $$

## VI. EXPERIMENTAL VALIDATION AND PHYSICAL INTERPRETATION

In this section, we validate the developed model using two independent acoustic experiments. The first is a high-frequency experiment, in which ultrasonic signals transmitted through porous samples are measured within the 80–120 kHz bandwidth. The second is a low-frequency experiment operating in the 6–400 Hz range, which is appropriate for measuring the permeability $k_0$.

The rationale for employing these two experiments is the following: in the high-frequency experiment, we estimate the fractal model parameters—namely $D$, $d_\ell$, $d_s$, $l_{min}$, and $l_{max}$—by identifying the combination of values that yields a good agreement between the theoretical model and the measured transmitted signals using a least square method (see Appendix C for the procedure behind the identifiability of these parameters). Since we also have an analytical expression for $k_0$, this permeability can then be predicted by substituting the fractal parameters obtained from the high-frequency analysis into Eq. (34) for $k_0$. The predicted value of $k_0$ is finally compared against the experimental measurement obtained from the low-frequency experiment.

For the porous samples, we employ conventional plastic foams denoted as S1 and S2 with thicknesses 1.4 cm and 1.1 cm, respectively (see Fig. 6). It is known in the literature that open cell polyurethane foams exhibit a statistical fractal structure (Kimball and Frisch, 1991; Shi et al., 2006), due to their broad pore-size distribution and complex connectivity.
Furthermore, the model developed in this study applies to a broad range of natural porous media, including fibrous materials, open-cell plastic foams, and earth blocks, provided that the wavelength is significantly greater than the upper cutoff $l_{max}$. This assertion is supported by the fact that heterogeneous natural porous media are characterized by tortuous connectivity, such that $d_{min} \ge 1$ and $d_{\ell} \ne d_{T}$ (Balankin and Elizarraraz, 2012b) (see Appendix A for detailed definitions of $d_{min}$ and $d_{\ell}$), even when the pore space is geometrically Euclidean with $D = d_{T} = 3$. The current fractal continuum model is not justified only when the porous medium contains straight pore channels, and the pores are approximately equal in size.

According to the literature, for open-cell polyurethane foams, the upper cutoff $l_{max}$ corresponding to the largest pore size, typically ranges from 300 µm to a few millimeters. In contrast, the lower cutoff scale $l_{min}$, which represents the smallest pore size, generally lies between a few micrometers to a few hundreds of micrometers (Shi et al., 2006). In what follows, we present the details of the experimental measurements.

## A. High frequency measurements

In this subsection we would like to measure the various fractional dimensions employed in this study using ultrasonic transmitted waves by two plastic foams. Experiments were performed in air using a pair of broadband Ultran transducers, NCG100-D50 with a central frequency of 100 kHz with a frequency bandwidth 80 kHz-120 khz. Pulses of 400 V are provided by a 5058PR Panametrics pulser/receiver. We apply a band-pass filter of 30 kHz-300 to reduce low and high frequency noise. Electronic interference is eliminated by 6000 acquisition averages. (Setup illustrated in Fig. 5).
<figure><img src="image_6.png" /><figcaption>FIG. 5: (Color online) High frequency experiment</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIG. 6: (Color online) Microstructure of (left) foams $S_1$ and (right) foam $S_2$. The scale bar is 1000 µm, and images taken using Apenzel 300× coupled with an iPhone 16e.</figcaption></figure>

In the conducted experiment, the incident signal together with the transmitted signals measured for the two foams are presented in Fig. 7.

These foams can be characterized using the high-frequency limit of Eq. (32) (see Appendix B), expressed as

$$ \lim_{\omega \to \infty} k(\omega) = \phi \epsilon^2 \frac{L^{-\theta_m}}{\zeta c_3^{-2\nu}} \left[ 1 - \frac{4(\nu+1)}{A_\gamma} \epsilon \right]. \qquad (36) $$
387

This relation is used in the expression of the transmission coefficient (see Eq. (3) in
Ref. (Bouchendouka et al., 2024a)) to predict the transmission behavior of the two foams.

388

389

The comparison between experimental and theoretical results is displayed in Fig. 8, with the
fitting parameters summarized in Table III (see Appendix C for more details on parameter
estimation).

390

391

392

It is important to emphasize that, at sufficiently high frequencies, scattering from microstruc-
tural heterogeneities can bias the estimated parameters reported in Table III. To assess
whether scattering influences the HF experiment measurements, we estimate the frequency
$f_s$ at which scattering dominates, following the results from paper (Tournat et al., 2004).

393

394

395

396

The authors report that scattering dominates at $f_s = 80$ kHz for their foam sample. Taking
this into account and after some straightforward calculations (see Appendix C), we found
$f_s \approx 140$ kHz for the foam samples used in our study. The transducers used in our experi-
ment have a bandwidth of 80-120 kHz < $f_s$, so the influence of scattering on the estimated
parameters in Table III is expected to be negligible.

397

398

399

400

401

The values obtained satisfy the condition $1 \le d_s \le d_\ell \le D$, which defines path connected
fractal structures, more specifically permeable media (see ref (Balankin and Elizarraraz,
2012b)). This is in line with theoretical expectations, given that open-cell foams are path
connected. Furthermore, the estimated lower cutoff $l_{\min}$ and upper cutoff $l_{\max} \approx 2a$, cor-
responding respectively to the smallest and largest pore sizes, align well with the sizes
observed from the foams micro-structure (see Fig. 6). Moreover, these values are in good
agreement with typical values reported in the literature for polyurethane foams (see Sec-
tion 3 in Ref. (Shi et al., 2006)). The relatively high values of $D$, $d_\ell$, and $d_s$ (close to $d_T = 3$)

402

403

404

405

406

407

408
409 are expected because of the moderate attenuation observed in the transmitted signals (see 410 Fig. 7). High values of these parameters suggests moderate tortuosity and moderate energy 411 dissipation within the foam structure. Moreover, the high value of D = 2.98 suggests a 412 high porosity, which is characteristic of polyurethane foams. In general, for fractal porous 413 media, the porosity (Katz and Thompson, 1985) is given by $\phi = (l_{min}/l_{max})^{3-D}$. Using the fitted 414 parameters reported in Table III, we obtain $\phi = 0.9692$ for foam $S_1$ and $\phi = 0.9641$ for foam 415 $S_2$, both of which are typical values for polyurethane foams. Additionally, by substituting 416 the same parameters into the expressions of $\alpha_\infty$ and $\Lambda$ (defined in Eq. (34)), we obtain 417 $\alpha_\infty = 1.0396$ and $\Lambda = 267.86 \ \mu m$ for foam $S_1$, and $\alpha_\infty = 1.108$ and $\Lambda = 270 \ \mu m$ for foam $S_2$. 418 These results reflect the low tortuosity and moderate energy losses within both materials. 419

Overall, the fitted fractal model yields physically plausible values. This conclusion will 420 be further substantiated through low-frequency permeability measurements presented in the 421 following subsection.

<figure><img src="image_8.png" /><figcaption>FIG. 7: (Color online) Measured incident signal compared with the ultrasonic transmitted signals by foams S1 and S2.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>FIG. 8: (Color online) Comparison between experimental transmitted signal and theoretical signal simulated using Eq. (36) with $D, d_{\ell}, d_{s}, l_{\min}, l_{\max} = 2a$ and $\lambda_{1}$ treated as the fitting parameters (fitting procedure is explained in Appendix C).</figcaption></figure>

<table><caption>TABLE III: Summary of the results obtained from the high frequency measurements for the two foam specimens.</caption><thead><tr><th>Sample</th><th>D</th><th>d<sub>l</sub></th><th>d<sub>s</sub></th><th>l<sub>max</sub> (μm)</th><th>l<sub>min</sub> (μm)</th><th>λ<sub>1</sub></th><th>λ<sub>3</sub></th></tr></thead><tbody><tr><td>S<sub>1</sub></td><td>2.98</td><td>2.92</td><td>2.92</td><td>450</td><td>15</td><td>0.55</td><td>1</td></tr><tr><td>S<sub>2</sub></td><td>2.98</td><td>2.81</td><td>2.8</td><td>750</td><td>55</td><td>0.3</td><td>1</td></tr></tbody></table>

B. Low frequency measurements

The LF measurements used to extract the permeability are conducted using the setup shown in Fig. 9. A pulse with a setpoint width of 1.9 s and amplitude 150 mV is generated by a function generator (Agilent 33220 A, Agilent Technologies, Santa Clara, CA). This signal is amplified and sent to a loudspeaker mounted at the entrance of a long guide tube (length exceeding 100 meters) with an internal diameter of 2.5 cm, which contains the porous sample
under test. The loudspeaker converts the electrical signal into an acoustic wave of very short duration, that propagates through the tube and the porous material. The transmitted acoustic signal is detected by a microphone positioned just downstream of the sample. A very long coiled pipe is used as the termination, effectively preventing perturbations due to end reflections. The microphone is connected to a National Instruments (NI) PXI module (National Instruments, Austin, TX) for data acquisition, which interfaces with a PC. Final signal capture and processing are carried out using NI InstrumentStudio software.

<figure><img src="image_10.png" /><figcaption>FIG. 9: (Color online) Low frequency acoustic waveguide experiment</figcaption></figure>

The incident signal along the low frequency measured transmitted signals for the two foams are depicted in Fig. 10.

In this case, the permeability $k_0$ is directly measured using the experimental setup described above. In the low-frequency limit we have $k(\omega) = k_0$. The comparison between the experimental data and the reconstructed signal obtained in this case is presented in Fig. 11, and the corresponding values of $k_0$ yielding this agreement are summarized in Table IV. Let us denote this measured permeability as $k_0^{\text{measured}}$. Substituting the experimentally determined values of $D$, $d_\ell$, $d_s$, $l_{\text{min}}$, $l_{\text{max}}$ and $\lambda_1$ (see Table III) into the ex-
pression of $k_0$ derived from the fractal model (Eq. (34)), we obtain the predicted permeabilities listed in Table IV. The agreement between the measured and predicted permeabilities is very good, with deviations of approximately 3.8% for foam $S_1$ and 8.8% for foam $S_2$. The deviation $\Delta k_0$ between $k_0^{\text{measured}}$ and $k_0^{\text{predicted}}$ is calculated using the formula $\Delta k_0 = (|k_0^{\text{measured}} - k_0^{\text{predicted}}|)/k_0^{\text{measured}} \times 100$. This close correspondence between $k_0^{\text{measured}}$ and $k_0^{\text{predicted}}$ provides further validation of the fractal model proposed in this study.

Similar to the Johnson–Koplik–Dashen model, the fractal continuum model developed in this study relies on an effective medium description. However, the main difference lies in how they describe the porous medium geometry. In particular, the two approaches differ in how the macroscopic parameters $\alpha_\infty$, $\Lambda$, and $k_0$ are defined and interpreted, and consequently in how these parameters can be used in real-world applications. Classical homogenization theory describing wave propagation in porous media relies on the concept of a representative elementary volume (REV), whose size $L_{\text{REV}}$ must be sufficiently large for macroscopic descriptors $\alpha_\infty$, $\Lambda$, and $k_0$ to converge to constant values. Accordingly, these parameters are defined as

$$ \alpha_{\infty} = \frac{\langle \mathbf{v}_{\infty}^2 \rangle}{\langle \mathbf{v}_{\infty} \rangle^2}, \quad \Lambda = 2 \frac{\int_V \mathbf{v}_{\infty}^2 dV}{\int_{\partial V} \mathbf{v}_{\infty}^2 dS}, \quad \frac{k_0}{\phi} = \frac{\langle \mathbf{v}_0 \rangle \cdot \langle \mathbf{v}_0 \rangle}{\langle -\mathbf{v}_0 \cdot \nabla^2 \mathbf{v}_0 \rangle}, \quad (37) $$

where $\mathbf{v}_\infty$ and $\mathbf{v}_0$ are the high- and low-frequency velocity fields, respectively. These definitions implicitly assume that all geometric irregularities below the REV scale are averaged out. However, in highly heterogeneous media, this assumption is not always valid, as these parameters are known to exhibit scale dependence (Ghanbarian et al., 2021; Zhang et al., 2000). In contrast, the definitions of $\alpha_\infty$, $\Lambda$, and $k_0$ derived using the fractal continuum model (see Eq. (34)) explicitly account for the scale of observation through the geometric
cutoffs $l_{\min}$ and $l_{\max}$. Furthermore, the classical definitions depend on pore-scale velocity
fields $v_{\infty}$ and $v_0$, which are very hard (or even impossible) to measure directly and are
typically obtained via numerical simulations based on fully resolved 3D reconstructions of
the pore space. The fractal continuum model, by comparison, relies on well-defined pa-
rameters that can be estimated directly and robustly from microstructural analysis, making
it particularly attractive for industrial characterization, where full pore resolution is im-
practical or expensive. Finally, the fractal-based formulation provides explicit relationships
between macroscopic parameters $\alpha_{\infty}$, $\Lambda$, and $k_0$ and the size and geometry of the porous
medium. This allows one to predict how these parameters evolve as the sample size or fractal
dimensions ($D$, $d_{\ell}$, $d_s$) are varied, which is not accessible within the classical REV-based
framework.

Hence, the present model provides an alternative theoretical framework capable of predict-
ing acoustic behavior across frequencies and a practical tool for characterizing materials
based solely on geometrical information through fractal dimensions $D$, $d_\ell$, $d_s$, and other
geometrical parameters related to fractal structure. Thus, the fractal model developed in
this study is important for the design and optimization of porous materials with targeted
acoustic properties.

## VII. CONCLUSIONS

In this study, we analyzed wave propagation in fractal porous media by reformulating the
fundamental microscopic equations, specifically the conservation of momentum and mass,
based on the concept of fractal continuum in non-integer dimensional space. By considering
<figure><img src="image_11.png" /><figcaption>FIG. 10: (Color online) Measured incident signal compared with the low frequency transmitted signals by foams S1 and S2</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>FIG. 11: (Color online) Comparison between experimental transmitted signal and theoretical signal simulated using $k(\omega) = k_0$ with $k_0$ as the fitting parameter.</figcaption></figure>

unidirectional flow through a medium composed of fractal cells, we derived an analytical expression for the dynamic permeability $k(\omega)$. The resulting plots of $k(\omega)$ demonstrated that the fractal structure has a significant influence on the medium's permeability and on the visco-inertial transitions within it. These transitions are naturally captured by an effective Womersley number $\kappa^*$. Moreover, explicit formulas for the tortuosity $\alpha_\infty$, the viscous characteristic length $\Lambda$, and the static permeability $k_0$ were obtained, each depending on the box-counting dimension $D$, the chemical dimension $d_\ell$, and the spectral dimension
<table><caption>TABLE IV: Comparison between the measured (from low-frequency experiment) and predicted (Eq.(34)) static permeability $k_0$ for the two foam specimens.</caption><thead><tr><th>Sample</th><th>k<sub>0</sub><sup>measured</sup> (μm<sup>2</sup>)</th><th>k<sub>0</sub><sup>predicted</sup> (μm<sup>2</sup>)</th><th>(Eq.(34))</th><th>Δk<sub>0</sub> (%)</th></tr></thead><tbody><tr><td>S<sub>1</sub></td><td>5940</td><td>5752</td><td></td><td>3.8</td></tr><tr><td>S<sub>2</sub></td><td>6220</td><td>5715</td><td></td><td>8.8</td></tr></tbody></table>

$d_s$ and the characteristic lengths $l_{min}$ and $l_{max}$. This formulation enables direct estimation of these parameters by analyzing micro-structural images of the medium.

The model was validated through a two-step experimental approach. High-frequency ultrasonic measurements were employed to estimate the fractal parameters by fitting the transmitted signals from open-cell foam samples. This procedure yielded physically consistent values for the fractal dimensions $D$, $d_ℓ$, $d_s$, and the cutoff scales $l_{min}$ and $l_{max}$. Low-frequency experiments conducted with a guide-tube setup provided direct measurements of the static permeability $k_0$. The permeability predicted by the fractal model, using parameters extracted from the high-frequency analysis, showed excellent agreement with the low-frequency measurements, with deviations well within acceptable ranges. This cross-validation highlights the model's predictive capability and its ability to capture the key features of wave propagation in fractal porous media. Further validation using either direct measurements of fractal parameters or numerical simulations, can be explored in future work. Moreover, in this study, we used open-cell foams, but other porous materials, such as fabric or earth blocks, can be used to test model predictions. Finally, future work may
extend this framework to include thermal effects, allowing a more complete description of propagation in porous media.

In summary, we show that the fractal continuum model provides both predictive power and physical insight. This model can be used in various industrial applications, such as the design of acoustic materials with the desired properties.

## AUTHOR DECLARATIONS

### Conflicts of Interest

The authors have no conflicts of interest to disclose. They also affirm that their work adheres to ethical approval, indicating that their research does not involve experiments on animal subjects and/or human participants.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## APPENDIX A: DEFINITIONS OF FRACTAL DIMENSIONS

Scale invariance of a fractal object implies that the box-counting measure scales with the fractal linear size r as

$$N \propto \left( \frac{r}{r_0} \right)^D, \qquad (A1)$$
where N is the number of n-dimensional boxes of size $r_0$ needed to cover a fractal object with topological dimension $d_T$ and fractal (box-counting) dimension $D$, while $d_T < D \le n$. In a porous material, either the pore space or the solid matrix may obey the scaling relationship (A1). Besides, the pore-solid interface can also obey relation (A1) even if the matrix and pore space are not scale-invariant. Accordingly, there are three basic kinds of fractal porous media. Unlike mathematical fractals, porous materials obey the scaling relation (A1) only in a limited interval of the length scale

$$l_{\min} \le r_0 < r \le l_{\max}, \qquad (A2)$$

where $l_{\min}$ and $l_{\max}$ are the lower and the upper cutoffs of scale invariance, respectively (Feder, 2013; Patiño-Ortiz et al., 2024). Nevertheless, the scale invariance of a porous material can also be characterized using the box-counting dimension (Feder, 2013; Liu et al., 2024). The scale invariance strongly affects the transport processes in saturated porous media (Balankin et al., 2016a; 2022b; Hodaei et al., 2025; Kawabe et al., 2009; Ponge et al., 2014; Yu, 2008; Yu and Cheng, 2002)).

The box-counting dimension is important, but not sufficient to characterize the fractal features of fractal materials (Nakayama et al., 1994; Stanley, 1984). The key topological properties and metrological attributes of a fractal object must be quantified using a set of dimensional numbers that are typically distinct from the fractal and topological dimensions (Balankin, 2015a; Balankin et al., 2022a; Patiño-Ortiz et al., 2024). In particular, fractal connectivity can be quantified using the connectivity dimension, defined as

$$d_{\ell} = \lim_{\mathcal{N} \to \infty} \left[ \frac{\ln \mathcal{N}(\ell)}{\ln \ell} \right] \qquad (A3)$$
where $\mathcal{N}(\ell)$ is the number of points of the fractal space connected with an arbitrary chosen point inside a $d_l$-dimensional ball of diameter $\ell$ (Rammal et al., 1984; Suzuki, 1983). In the literature, the connectivity dimension (A3) is also called the intrinsic, chemical, or spreading dimension (Balankin, 2015a; Meakin et al., 1984; Schmutz, 1986; Vannimenus, 1984). The ratio between the fractal dimension D and the connectivity dimension $d_\ell$ is equal to the fractal dimension of geodesic paths $d_{min}$ (Balankin et al., 2022a). Accordingly, the minimum path between two points in the fractal scales with the Euclidean distance between these points as

$$ \ell \propto r^{d_{min}} \qquad (A4) $$

where $d_{min} = D/d_\ell$. In this way, the fractal ramification can be characterized by the topological Hausdorff dimension $D_{tH}$ (Balankin et al., 2017a) which was introduced by Balka et al. (Balka et al., 2015) via a natural combination of the definitions of topological and Hausdorff dimensions. However, unlike the topological dimension $d_T$, the topological Hausdorff dimension $D_{tH}$ is not invariant under homeomorphic transformations. The topologically invariant topological connectivity dimension $d_{tl}$ was defined in (Balankin et al., 2017a). Further it was shown that the topological connectivity dimension is related to $D_{tH}$, D, and $d_\ell$ as $d_{tl} = 1 + d_\ell(D_{tH} - 1)/D$ (Balankin, 2024). In this regard, it is also worth noting that in the case of totally disconnected fractals, such as Cantor dust, fractal points can be connected under certain physical assumptions (Balankin and Elizarraraz, 2012a; Golmankhaneh and Balankin, 2018). In this case, the fractal dimension of the minimal paths can be less than one (Balankin et al., 2022a), that is, $d_{min} = D/d_\ell < 1$, while $d_T = 0 < D < d_\ell = n$. Another important topological feature is the fractal loopiness (Balankin et al., 2022a; Havlin and
Ben-Avraham, 2002). Loops represent closed paths or cycles, and therefore they control the degrees of freedom of walkers in the fractal space (Balankin, 2015b). A point walker in Euclidean space has the same number of spatial and dynamic degrees of freedom, equal to the dimension of the space $n$. In contrast, the number of effective dynamic degrees of freedom of a point-like walker in fractal space can be less than or equal to the number of effective spatial degrees of freedom (Balankin, 2015b). Specifically, the number of effective dynamical degrees of freedom $d_s$ can be defined via the scaling asymptotic behavior of the probability that the random walker returns to its origin point after $t$ steps:

$$P \propto t^{-d_s/2} \qquad (A5)$$

see, for instance, ref (Burioni and Cassi, 2005). Therefore, $d_s$ determines the scaling properties of the eigenvalues of the Laplacian defined on the fractal and so it is equal to the Lagrangian dimension (Mosco, 1997). Consequently, the number of effective dynamical degrees of freedom can also be defined via the scaling of the density of low-frequency vibrational modes (Alexander and Orbach, 1982)

$$g(\omega) \propto \omega^{(d_s-1)}, \qquad (A6)$$

and so, the number of effective dynamical degrees of freedom $d_s$ is commonly called the spectral dimension (Rammal and Toulouse, 1983). Furthermore, many types of fractals obey the Alexander-Orbach relation $d_s = 2D/D_W$, where $D_W$ is the fractal dimension of random walk (Havlin and Ben-Avraham, 2002) defined via the scaling behavior of the mean squared displacement of random walker

$$\langle r^2 \rangle \propto t^{(2/D_W)} \qquad (A7)$$
The Einstein law relating the drift and diffusion of charge implies that $D_W = D + \rho$, where $\rho$ is the electrical resistance exponent (Burioni and Cassi, 2005). Recall that the electrical resistance exponent is defined via the scaling relations $\rho \propto L^{\rho}$, if $\rho > 0$, or $\rho = \rho_{\infty} - cL^{\rho}$, if $\rho < 0$, where $\rho$ is the electrical resistance of a sample with size $L$, while $\rho_{\infty}$ and $c$ are constants. On the other hand, the number of effective spatial degrees of freedom $n_{\gamma}$ can be defined as the number of independent directions determined by the fractal topology (Balankin et al., 2022a). In this regard, (Balankin, 2015b) has established that

$$n_{\gamma} + d_{s} = 2d_{\ell}, \qquad (A8)$$

and so the ratio of the numbers $n_{\gamma}$ and $d_{s}$ is equal to $n_{\gamma}/d_{s} = 2d_{\ell}/d_{s} - 1 = d_{W} - 1$, where $d_{W} = D_{W}/d_{\min} = d_{\ell} + \rho/d_{\min}$ is the fractal dimension of random walk with respect to the geodesic metric. The fractal dimension of random walk of loopless fractals is equal to $d_{W} = d_{\ell} + 1$, while $\rho = d_{\min}$ (Havlin and Ben-Avraham, 2002), whereas fractals with loops at all scales are characterized by $2 \le d_{W} < d_{\ell} + 1$, while $2 - D < \rho < d_{\min}$ (Balankin et al., 2022a).

## APPENDIX B: DERIVATION OF THE PARAMETERS $\alpha_{\infty}$, $\Lambda$, AND $K_0$

To determine the expressions of $\alpha_{\infty}$, $\Lambda$ and $k_0$ from expression (32) we follow the same procedure of Zwiker and Kosten (Zwikker and Kosten, 1949). First let us rewrite Eq.(32) as

$$k(\omega) = C \left[ 1 - \frac{2(\nu + 1) I_{\nu+1}(z)}{z I_{\nu}(z)} \right], \qquad (B1)$$
where $C = \phi\epsilon^2 L^{-\theta_m}/(\zeta c_3^{-2\nu})$, and $z = \kappa^*\sqrt{j} = A_\gamma/(2\epsilon)$. To determine the expression of $k(\omega) = k_0$ as $\omega \to 0$, we use the expansion of the modified Bessel function:

$$I_{\nu}(z) = \frac{(z/2)^{\nu}}{\Gamma(\nu + 1)} \left( 1 + \frac{z^2}{4(\nu + 1)} + \frac{z^4}{32(\nu + 1)(\nu + 2)} + \dots \right), \quad (B2)$$

$$I_{\nu+1}(z) = \frac{(z/2)^{\nu+1}}{\Gamma(\nu + 2)} \left( 1 + \frac{z^2}{4(\nu + 2)} + \frac{z^4}{32(\nu + 2)(\nu + 3)} + \dots \right). \quad (B3)$$

From this we get

$$\frac{I_{\nu+1}}{I_{\nu}} = \frac{z}{2(\nu + 1)} \left( 1 - \frac{z^2}{4(\nu + 1)(\nu + 2)} + O(z^4) \right). \quad (B4)$$

Plugging into Eq. (B1), and replacing $C$ and $z$ by their values yields

$$\lim_{\omega \to 0} k(\omega) = k_0 = \frac{\phi A_{\gamma}^2}{\zeta c_3^{-2\nu} 16(\nu + 1)(\nu + 2)} L^{-\theta_m}. \quad (B5)$$

In the case of high frequencies we use the large argument expansion for $I_\nu(z)$

$$I_{\nu}(z) \sim \frac{e^z}{\sqrt{2\pi z}} (1 + O(z^{-1})). \quad (B6)$$

From this expansion we take only the first term, thus resulting in $I_{\nu+1}(z)/I_\nu(z) = 1$ yielding:

$$\lim_{\omega \to \infty} k(\omega) = \phi \epsilon^2 \frac{L^{-\theta_m}}{\zeta c_3^{-2\nu}} \left[ 1 - \frac{4(\nu + 1)}{A_\gamma} \epsilon \right]. \quad (B7)$$

Comparing with the general expression of the dynamic permeability $k(\omega) = \phi\epsilon^2\alpha_\infty^{-1}(1 - 2/\Lambda\epsilon)$, we can identify $\alpha_\infty$ and $\Lambda$ accordingly:

$$\alpha_\infty = \frac{L^{\theta_m}}{\zeta c_3^{2\nu}}, \quad \Lambda = \frac{A_\gamma}{2(\nu + 1)}. \quad (B8)$$
APPENDIX C: ESTIMATION OF FRACTAL PARAMETERS FROM HIGH FREQUENCY EXPERIMENT

1. Extraction of fractal parameters via minimization

Although the parameter values were obtained by fitting the theoretical transmission to the HF measured transmitted signals, the parameters were not treated as unconstrained free variables; instead, they were estimated through a constrained inversion under independent priors and microstructural bounds.

First of all, the cost function is obtained using a simple relative root mean square error:

$$ \text{RMSE}(\boldsymbol{\theta}) = \sqrt{\frac{1}{N} \sum_{i=1}^{N} [S_{i}^{\text{exp}} - S_{i}^{\text{theo}}(\boldsymbol{\theta})]^{2}}, \qquad (\text{C1}) $$

$$ U(\boldsymbol{\theta}) = 100 \times \frac{\text{RMSE}(\boldsymbol{\theta})}{\max(S^{\text{exp}})}, \qquad (\text{C2}) $$

where $\boldsymbol{\theta} = (D, d_{\ell}, d_{s}, l_{min}, l_{max}, \lambda_{1}, \lambda_{3})$. Below we explain the procedure for parameter ex-traction

(i) **The upper and lower cutoffs:** The upper and lower cutoffs, $l_{min}$ and $l_{max}$, were estimated from microscopic images of the two foams. In fractal porous media, the lower cutoff is defined as the typical smallest pore size; below this threshold, the foam appears continuous, either as solid or void (Yu and Cheng, 2002). The upper cutoff corresponds to the largest typical pore size, beyond which the foam appears homogeneous (Yu and Cheng, 2002). Determining precise values for these cutoffs is challenging with the available microscopic images (see Fig.6), which were obtained
using an optical microscope. More accurate analysis would require higher-resolution images, such as those from scanning electron microscopy (SEM). Nevertheless, the current images provide a reasonable estimate for these parameters. Direct inspection indicates that, for foam S1, the lower cutoff $l_{min}$ ranges from a few microns up to 100 µm, while the upper cutoff $l_{max}$ is between 300 µm and 700 µm. A satisfactory fit was obtained using $l_{min}$ = 15 µm and $l_{max}$ = 450 µm for S1. For foam S2, the cells are noticeably larger than those in foam S1. Image analysis indicates that $l_{min}$ ranges from several tens of microns up to 200 µm, while $l_{max}$ varies between 600 and 900 µm. A satisfactory fit was achieved with $l_{min}$ = 55 µm and $l_{max}$ = 750 µm for this foam.

(ii) **Fractal mass dimension D**: The fractal mass dimension D measures how well the fractal pore space fills the porous medium in a given volume inside the medium. Therefore, it is directly linked to porosity via the famous relationship provided by Katz and Thompson (Katz and Thompson, 1985):

$$ \phi = \left( \frac{l_{min}}{l_{max}} \right)^{3-D} . \qquad (C3) $$

For open-cell polyurethane foams, the porosity is very close to 1 and generally falls in the range 0.9-0.99. Considering these values, the $l_{min}$ and $l_{max}$ for both foams, and using the relationship (C3), we obtain D values between 2.96 and 2.99. Therefore, in our fit, we set D to 2.98.

(iii) **Spectral and connectivity dimensions $d_s$ and $d_\ell$**: During the fitting of experimental signals, these two dimensions were treated as free parameters, but a constraint was imposed to determine their values. A permeable fractal medium, such as porous
media, is classified as a path-connected fractal, indicating the existence of a fully connected path that allows fluid to flow from one side to the other. For such fractals (Balankin, 2015b), the following relationship applies:

$$1 \le d_s \le d_\ell \le D. \qquad (C4)$$

The fitting procedure incorporated this condition, and any fit that failed to satisfy it was rejected as not physically meaningful for the samples used. Consequently, the possible values of $d_\ell$ and $d_s$ were restricted to those that meet this criterion. Furthermore, based on the definitions of $d_\ell$ and $d_s$, the obtained values can be physically interpreted. High values of these dimensions (approaching 3) are expected for open-cell polyurethane foams used in this study. Elevated $d_s$ and $d_\ell$ values indicate low energy loss and reduced tortuosity. This interpretation is supported by the experimental transmitted signals, which exhibit moderate attenuation for both foam samples.

(iv) **Geometrical constants $\lambda_1$ and $\lambda_3$**: The constants $\lambda_i$ in $l_i = \lambda_i l_{\min}^{1-\zeta} x_i^\zeta$ are geometrical constants of order unity. They account for the anisotropic structure of the fractal (see refs (Balankin and Mena, 2023; Balankin et al., 2017b)). In this study, $\lambda_3$ is set to one for the main flow direction. $\lambda_1 < 1$ shows there is less connection in the pore network along $i = 1$ than along $i = 3$ (see Fig. 3).

## 2. Scattering effects

At sufficiently high frequencies, when the acoustic wavelength approaches the characteristic microstructural length scale, microstructural scattering can influence coherent wave
propagation and potentially bias the extracted parameters. It is generally assumed that, for homogenization to be valid, the wavelength $\lambda$ must be much larger than the characteristic microstructural size $R$. While the low-frequency (LF) experiment satisfies this condition, it is necessary to determine whether it also holds in the high-frequency (HF) experiment. To address this, we estimate the frequency $f_s$ at which scattering effects become significant. Our calculations are based on the findings of the paper (Tournat et al., 2004). According to the experiments reported in that study, for the foam sample with permeability $k_0^{ref} = 18880 \mu m^2$, scattering effects become dominant at a frequency $f_s^{ref} = 80$ kHz (see figures 4-7 in (Tournat et al., 2004)). We will calculate the frequencies $f_s^{S1}$ and $f_s^{S2}$ at which scattering becomes significant for foams $S1$ and $S2$, using the results from (Tournat et al., 2004). To proceed, we assume that the geometry of the foams in our study and those in (Tournat et al., 2004) are sufficiently similar, such that universality applies and their properties can be related through appropriate scaling. Therefore, we expect the following scaling relationship to hold:

$$R^{S1} \sim a_1 R^{ref}, \quad R^{S2} \sim a_2 R^{ref}, \qquad (C5)$$

Here, $R^{S1}$, $R^{S2}$, and $R^{ref}$ denote the micro-structural sizes of foams $S1$, $S2$, and the foam used in (Tournat et al., 2004), respectively, while $a_1$ and $a_2$ represent scaling constants. Generally, permeability is proportional to $R^2$, the square of the characteristic micro-structural size; therefore, the following scaling relationship applies.

$$k_0^{S1} \sim a_1^2 k_0^{ref}, \quad k_0^{S2} \sim a_2^2 k_0^{ref}. \qquad (C6)$$
Scattering affects start dominating if $R/(2\pi\lambda) = 2\pi f_s R/c_0 \sim 1$, thus $f_s \sim 2\pi c_0/R$.

Therefore

$$f_s^{S1} \sim \frac{2\pi c_0}{R^{S1}} = \frac{2\pi c_0}{a_1 R^{ref}} = \frac{1}{a_1} f_s^{ref}, \qquad (C7)$$

$$f_s^{S2} \sim \frac{2\pi c_0}{R^{S2}} = \frac{2\pi c_0}{a_2 R^{ref}} = \frac{1}{a_2} f_s^{ref}. \qquad (C8)$$

Finally, estimating the scaling constants from Eq. (C6) as

$$a_1 = \sqrt{k_0^{S1}/k_0^{ref}} = 0.56, \quad a_2 = \sqrt{k_0^{S2}/k_0^{ref}} = 0.58, \qquad (C9)$$

and using $f_s^{ref} = 80$ kHz, yields $f_s^{S1} \approx 142$ kHz, $f_s^{S2} \approx 139$ kHz. Since the high-frequency experiments reported here are restricted to the bandwidth 80–120 kHz, i.e. below the frequencies $f_s^{S1}$ and $f_s^{S2}$, scattering effects are expected to remain weak and not significantly affect the extraction of fractal parameters reported in Table III.

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