Review of Theories and a New Classification of Tortuosity Types

Abstract

Many different definitions of tortuosity can be found in literature. In addition, also many different methodologies are nowadays available to measure or to calculate tortuosity. This leads to confusion and misunderstanding in scientific discussions of the topic. In this chapter, a thorough review of all relevant tortuosity types is presented. Thereby, the underlying concepts, definitions and associated theories are discussed in detail and for each tortuosity type separately. In total, more than 20 different tortuosity types are distinguished in this chapter. In order to avoid misinterpretation of scientific data and misunderstandings in scientific discussions, we introduce a new classification scheme for tortuosity, as well as a systematic nomenclature, which helps to address the inherent differences in a clear and efficient way. Basically, all relevant tortuosity types can be grouped into three main categories, which are (a) the indirect physics-based tortuosities, (b) the direct geometric tortuosities and (c) the mixed tortuosities. Significant differences among these tortuosity types are detected, when applying the different methods and concepts to the same material or microstructure. The present review of the involved tortuosity concepts shall serve as a basis for a better understanding of the inherent differences. The proposed classification and nomenclature shall contribute to more precise and unequivocal descriptions of tortuosity.

FormalPara Remark

Although tortuosity is related to porous media transport, it must be emphasized that the purpose of this chapter is not to give a review on transport equations and associated simulation of transport in porous media. For such topics we refer to dedicated books (e.g., Bird et al. [1]).

2.1 Introduction

Over the last 100 years, many different approaches were developed, how tortuosity can be defined and measured. A unifying concept for tortuosity is still lacking and therefore it is not easy to understand the difference between these numerous existing tortuosity types. Furthermore, there exists no suitable nomenclature that helps to address specific tortuosity types in a clear and simple way. In this chapter, the classical concepts and theories of all relevant tortuosity types are reviewed, and a new nomenclature is introduced.

2.1.1 Basic Concept of Tortuosity

For a given porous medium, tortuosity (τ) is basically defined as the ratio of effective path length (L_eff) over direct path length (L₀) through the considered porous medium ([2,3,4,5]), i.e.

$$ \tau = \frac{{L_{eff} }}{{L_{0} }}. $$

(2.1)

Figure 2.1 schematically illustrates L_eff and L₀. Thereby, the direct path length (L₀) is easily captured, since it is the sample dimension in transport direction. For theoretical treatment of tortuosity, one then simply has to find a suitable definition of the effective path length (L_eff), and for practical application one simply has to find a suitable method to measure effective path length.

Fig. 2.1

Illustration of three different concepts of hydraulic flow velocity

2.1.2 Basic Challenges

Unfortunately, definition and measurement of the effective path length (L_eff) are not as simple as it appears at a first glance, which explains the emergence of numerous different tortuosity concepts over time. As will be shown in Chap. 3, the different types of tortuosities often reveal significantly different values when applied to the same microstructure. These differences and the underlying proliferation of concepts are usually not properly accounted for in the description of tortuosity, e.g., in appropriate papers and conference presentations. This frequently deficient description of tortuosity is partly caused by the fact that there exists no suitable nomenclature for the different types of tortuosities. In addition, very often, researchers in this field are not aware of the inherent differences between the various tortuosity types. This lack of awareness often becomes the source of confusion in scientific debate, and it can also be the source of misinterpretation of acquired data. The basic challenges in this field are thus to foster the awareness in the scientific community for the systematic differences between tortuosity types, and to introduce a useful classification scheme and nomenclature that can then serve as a basis for more precise descriptions and for clearer scientific discussions of the topic.

2.1.3 Criteria for Classification

Two main criteria will be used to classify the different tortuosity types:

2.1.3.1 Method of Determination

Initially, there were no suitable methods available for direct measurement of effective path lengths (L_eff) from the microstructure. Thus, for a long period, tortuosity was calculated indirectly, using information such as effective transport property and porosity. Over the last two decades, new methods for 3D analysis became available, including micro- and nano-tomography, 3D image processing and numerical simulation. Nowadays, these methods enable us to measure tortuosity and effective path lengths (L_eff), respectively, directly from the 3D microstructure. Researchers can now choose from a multitude of direct and indirect methods. Understanding the underlying systematic differences between those tortuosity types is crucial, for example, to make a sensible choice of methods and concepts when planning an investigation of porous media.

2.1.3.2 Concept of Definition

The definition of tortuosity can be approached from different viewpoints. For example, the effective path length (L_eff) can be considered as a purely geometric characteristic of the microstructure, which is independent from the involved transport process. The geometric tortuosity is thus an intrinsic material property, and it is typically determined with quantitative 3D image analysis.

An alternative viewpoint for the definition of tortuosity and description of effective path length focuses on the tracking of a microscopic particle on its way through the porous material. Thereby, it makes a difference whether the transport process is, for example, viscous flow or diffusion. This viewpoint leads to so-called physics-based definitions of tortuosity, which are considering both, the impact of material structure, and the impact of the involved transport process.

Nowadays, a multitude of physics-based tortuosity types (hydraulic, electric, diffusional, thermal) as well as many different geometric tortuosity types (medial-axis, skeleton, geodesic, percolation path etc.) are available. Our aim is to provide a profound understanding of the inherent differences between these tortuosity types.

2.1.4 Content and Structure of This Chapter

In this chapter, we describe the underlying concepts, definitions, and theories for all relevant tortuosity types. Thereby, the evolution of emerging definitions and concepts is presented in a chronological (historical) order. The concept of tortuosity was initially introduced in context with the Carman-Kozeny equations for flow in porous media. All other tortuosity types then evolved and diverged from there. Different branches of physics-based tortuosities (i.e., hydraulic, electric, and diffusional tortuosity types) unfolded in parallel over a long period. These physics-based branches are described in separate subsections. The geometric tortuosity types, which appeared more recently, are then presented in the following subsections. The classification of tortuosity is made even more complicated because there exist also mixed tortuosity types. They are mixed in the sense that they fulfil both criteria, for classification as physics-based, and also for classification as geometric tortuosity.

To establish the basis for more precise descriptions of tortuosity, we thus present a new classification scheme that uses the two mentioned criteria (i.e., method of determination and concept of definition) for a meaningful distinction of all existing tortuosity types. This results in no more than three main categories: (a) direct geometric, (b) indirect physics-based and (c) mixed tortuosity types. This classification scheme also serves as a basis for a systematic and specific nomenclature, which aims to provide all relevant information that is necessary for a scientifically correct treatment of the different tortuosity types.

2.2 Hydraulic Tortuosity

2.2.1 Classical Carman-Kozeny Theory

2.2.1.1 Capillary Tubes Model by Kozeny

For porous media, the volumetric flow (Q) induced by a pressure gradient (ΔP/L₀) can be described by Darcy's law from 1856 [6]

$$ Q = v_{s} A = - \frac{\kappa A}{\mu } \frac{\Delta P}{{L_{0} }}, $$

(2.2)

with superficial flow velocity (v_s), cross-section area (A), dynamic viscosity (μ) and permeability (κ). All resistive effects of the microstructure are implicitly and indistinguishably contained within the permeability (κ). ‘In early times’, when 3D methods for microstructure investigation were not yet available, flow and its relationship to the underlying microstructure were modeled based on a simplified geometrical model consisting of a bundle of parallel tubes (i.e., equivalent channel model, see Kozeny [5]).

Capillary flow in a straight tube can be described with the Hagen-Poiseuille equation

$$ v_{c} = - \frac{{r^{2} }}{8 \mu } \frac{\Delta P}{{L_{0} }}, $$

(2.3)

with capillary velocity (v_c, also called interstitial velocity) and tube radius (r). The comparison of Eqs. 2.2 and 2.3 reveals that permeability (κ) in tube models scales with r²/8.

For models where capillary tubes are not straight, the effective length of the capillary flow path (L_eff) is larger than the direct length (L₀), which leads to a reduction of the effective pressure gradient. To correct this effect, Kozeny introduced the notion of hydraulic tortuosity, which he defined as the ratio of the effective hydraulic path length over the direct length (i.e., τ_hydr = L_{eff_hydr}/L₀). This leads to

$$ v_{c} = - \frac{{r^{2} }}{8 \mu } \frac{\Delta P}{{L_{{eff_{hydr} }} }} = - \frac{{r^{2} }}{{8 \mu \tau_{hydr} }} \frac{\Delta P}{{L_{0} }}. $$

(2.4)

To adapt Poiseuille's description of a single tube for equivalent tubes (as analogy for porous media), it is necessary to also consider the impact of pore volume fraction on superficial velocity and associated volume flow. According to Dupuit's relation, superficial velocity (v_s in Eq. 2.2 for porous media flow) is equal to the capillary velocity (v_c in Eq. 2.4 for tube flow) multiplied by porosity (i.e., v_s = v_c ε). In analogy to Darcy's law, the equation for volume flow in the capillary tubes model thus becomes

$$ Q = v_{s} A = - \frac{{r^{2} \varepsilon A}}{{8 \mu {\rm T}_{hydr} }} \frac{\Delta P}{{L_{0} }}. $$

(2.5)

The distinct notation of tortuosity factor (T, instead of τ) in Eq. 2.5 originates from a later extension of Dupuit's relation by Carman [2], which is discussed below in context with Eq. 2.14.

Permeability strongly depends on the effective hydraulic radius (r_hydr), which represents a tube equivalent radius that is characteristic for the overall viscous drag. Kozeny introduced the hydraulic radius as the ratio of area open to flow (in a 2D cross-section perpendicular to flow) over the perimeter of this area exposed to flow. For a given volume of porous media, the hydraulic radius can also be defined as the ratio of the pipes volume open to flow over the corresponding surface area of these pipes. For a single straight tube, the hydraulic radius is thus half of the tube radius (r_{hydr_tube} = πr²L/2πrL = r/2).

In a more generalized description for porous media, the volume-to-surface ratio is rewritten as the ratio of porosity over specific surface area per volume (r_{hydr_K} = ε/S_V = r/2, with subscript K for Kozeny). For non-circular tubes, Kozeny additionally introduced a shape correction factor (c_K). This leads to the well-known semi-empirical Kozeny equation [5]

$$ Q = v_{s} A = - \frac{{r_{{hydr_{K} }}^{2} }}{{c_{K} }}\frac{ \varepsilon }{{{\rm T}_{hydr} }}\frac{ A}{{\mu }}\frac{\Delta P}{{L_{0} }} = - \frac{{\varepsilon^{3} A}}{{c_{K} S_{V}^{2} {\rm T}_{hydr} \mu }} \frac{\Delta P}{{L_{0} }}. $$

(2.6)

For the specific case of circular tubes, the Kozeny factor c_K is equal to 2. For non-circular tube cross-sections, shape correction factors in the range from 1.5 to 2.6 were specified based on experimental data.

Combining Eq. 2.6 with Eq. 2.2, we obtain an expression for permeability in terms of porosity, Kozeny factor, specific surface area (per volume) and hydraulic tortuosity factor. This expression, also called Kozeny equation in the literature, reads as

$$ \kappa = \frac{{\varepsilon^{3} }}{{c_{K} S_{V}^{2} {\rm T}_{hydr} }}. $$

(2.7)

2.2.1.2 Packed Spheres Model by Carman

In 1937, Carman [2] presented some important modifications of Kozeny's equations, in order to describe permeability in granular materials (instead of a bundle of parallel tubes). For this purpose, Carman considered a simplified geometrical model of packed spheres. Specific surface area per total volume (S_V) is replaced by surface area per solid volume (a_V), which then requires the solid volume fraction (1 − ε) as a correction term. For mono-sized spheres, the surface area per solid volume (a_V) can be written as a function of the particle diameter (a_V = 6/D_p). For non-spherical particles, the hydraulic radius needs to be corrected with a shape factor (c_C, with subscript C for Carman). With these corrections for shape (c_C) and solid volume fraction (1 − ε), one obtains

$$ r_{hydr\_C} = \frac{{c_{C } \varepsilon }}{{a_{V} \left( {1 - \varepsilon } \right)}} = \frac{{c_{C } D_{p } \varepsilon }}{{6 \left( {1 - \varepsilon } \right)}}. $$

(2.8)

Permeability of a packed spheres model (as analogy for granular materials) is thus described with the Carman-Kozeny equation

$$ \kappa = \frac{{r_{hydr\_C}^{2} \varepsilon^{2} }}{{2 {\rm T}_{hydr} }} = \frac{{c_{C}^{2} \varepsilon^{3} }}{{2 a_{V}^{2} \left( {1 - \varepsilon } \right)^{2} {\rm T}_{hydr} }} = \frac{{c_{C}^{2} D_{p}^{2} \varepsilon^{3} }}{{72 \left( {1 - \varepsilon } \right)^{2} {\rm T}_{hydr} }}. $$

(2.9)

Thereby the Kozeny factor (c_K) for tube shapes becomes obsolete and can be replaced with a constant value of 2. The Carman factor (c_C) for correction of non-spherical particle shapes was determined experimentally for grain-sorted powders, whereby values in the range from 0.28 (for mica) to 1 (for spherical particles) were obtained.

2.2.1.3 Different Concepts of Flow Velocity

Carman [2] pointed out that the comparison of Eq. 2.2 (Darcy, porous media, superficial velocity) with Eq. 2.3 (Hagen-Poiseuille, tube flow, capillary velocity) requires a careful consideration of the involved velocities. In principle, velocity can be defined as the ratio of path length over characteristic residence time (t) during which a particle is travelling from inlet to outlet. Three different velocities must be distinguished in context with porous media flow, which was later also discussed by Epstein in 1989 [3]. From the relationship between the three flow velocities, a new definition of hydraulic tortuosity as well as an extension of Dupuit’s relation can be deduced, as illustrated in Fig. 2.1:

• (a) Capillary velocity (v_c)

The interstitial microscopic capillary velocity (v_c) used in Eq. 2.3 for tubes is also called intrinsic velocity for porous media. The notion of capillary velocity is based on a microscopic consideration of particles travelling through porous media (or through a single tube). Their capillary velocity can be described as the ratio of the effective tortuous path length, denoted by L_{eff_hydr}, over the residence time (t), i.e.

$$ v_{c} = \frac{{L_{eff\_hydr} }}{ t}. $$

(2.10)

Thereby, L_{eff _hydr} is interpreted as a mean length, which is characteristic for a large number of particle pathways. Subsequently, homogenized 'mean' properties of locally defined quantities are denoted with angle brackets, i.e., <x> denotes the homogenized mean of x. For complex porous media, it was not possible for a rather long time to measure the mean length of hydraulic transport pathways or streamlines (<L_{eff_hydr}>). However, a simpler approach to determine the mean capillary velocity <v_c> without measuring L_{eff_hydr} was later presented by Duda et al. [7] and Matyka and Koza [8]. These authors derived mean velocity <v_c> based on velocity vector fields computed with transport simulations at pore scale. These authors then used <v_c> as a basis for calculating the volume averaged tortuosity in an elegant way (see the discussion in context with Eq. 2.18).

• (b) Axial velocity (v_x)

The interstitial axial velocity (v_x) is based on a macroscopic observation of flow in porous media, where only the direct path length (L₀), i.e., the sample length between inlet and outlet planes is known. The residence time (t) is the same as for the microscopic observation related to capillary velocity. Then, v_x is given by the ratio of direct path length over residence time, i.e.

$$ v_{x} = \frac{{L_{0} }}{ t}. $$

(2.11)

Capillary velocity (v_c) and interstitial axial velocity (v_x) are equivalent only for the case of a single, straight tube, where L_{eff_hydr} = L₀. In all other cases capillary velocity is higher than axial velocity (i.e., v_c, ≥ v_x).

• (c) Superficial velocity (v_s)

Finally, in a porous media, the macroscopic superficial velocity (v_s) in Darcy's law (Eq. 2.2) can be deduced from the ratio of volume flow over cross-section area (i.e., v_s = Q/A). According to Dupuit's relation, the macroscopic superficial velocity in porous media (v_s) can also be obtained from the interstitial axial velocity (v_x, e.g., in a single tube or in porous media consisting of tubes) with correction of the volume effect using porosity (ε). In this way one obtains

$$ v_{s} = \frac{Q }{{ A}} = v_{x} \varepsilon = \frac{{L_{0} }}{ t} \varepsilon . $$

(2.12)

The careful distinction of three different flow velocities leads to two main conclusions:

First, from the ratio of Eq. 2.10 over Eq. 2.11, Carman obtained two equivalent definitions of hydraulic tortuosity,—first as ratio of mean path lengths and second as ratio of mean velocities, i.e.

$$ \tau_{hydr} = \frac{{\left\langle {L_{eff\_hydr} } \right\rangle }}{{ L_{0} }} = \frac{{\left\langle {v_{c} } \right\rangle }}{{\left\langle { v_{x} } \right\rangle }}. $$

(2.13)

For a long time, conventional definitions of tortuosity focused on the ratio of path lengths. However, with the rise of numerical simulations the consideration of mean velocity components has gained importance as an equivalent definition for tortuosity (see Eq. 2.18).

Second, by combining Eqs. 2.10–2.13, Carman also obtained

$$ v_{s} = v_{c} \varepsilon \frac{1}{{\tau_{hydr} }}, $$

(2.14)

as an extension of Dupuit's relation. Substituted in Eq. 2.5, tortuosity was thus introduced for a second time in context of flow equations (i.e., first, for the correction of pressure gradient by Kozeny and second, for the correction of velocity by Carman). Consequently, the meaning of the tortuosity factor (T) in Eqs. 2.5–2.9 must be redefined as hydraulic tortuosity by a power of 2 (see [2, 3, 9]), i.e.,

$$ {\rm T}_{hydr} = \left( {\frac{{L_{eff\_hydr} }}{{L_{0} }}} \right)^{2} = \tau_{hydr}^{2} . $$

(2.15)

2.2.2 From Classical Carman-Kozeny Theory to Modern Characterization of Microstructure Effects

2.2.2.1 Limitations of the Carman-Kozeny Approach

Th Carman-Kozeny equations describe two main transport limitations arising from the microstructure. First, viscous drag induced by wall friction is captured with the hydraulic radius (r_hydr). Second, non-viscous effects are attributed to reduced pore volume fraction and/or to increased length of transport pathways. These non-viscous effects are described with dimensionless microstructure descriptors for porosity (ε) and hydraulic tortuosity (τ_hydr). Variations of tortuous path lengths affect both, superficial velocity, and effective pressure gradient, and thus, tortuosity appears with a power of 2 in the Carman-Kozeny equations.

The Carman-Kozeny equations were introduced at a time when tomography and 3D image analysis were not yet available and therefore hydraulic radius and tortuosity could not be measured directly from the microstructure. As a loophole to this problem, Carman considered a simplified geometrical model consisting of mono-sized spheres as an analogy for the complex pore structure in granular media. For this simplified model the hydraulic radius can be described with easily accessible geometric descriptors (ε, S_V, a_V, D_p), as summarized in previous sections. However, the determination of hydraulic tortuosity remained a major problem. Based on geometric analysis of streamlines in a packed bed of spheres, Carman proposed to use a constant value of $\sqrt 2$ for τ_hydr.

Experimental validations confirmed that the Carman-Kozeny equations are capable to predict permeability and flow reasonably well for simple granular media consisting of mono-sized spheres. For non-spherical particles, Carman introduced a shape correction factor (c_C), which must be fitted for different particle shapes and size distributions separately. It turns out that the uncertainties of permeability predictions with the Carman-Kozeny equations increase with geometric complexity of the granular material (e.g., for non-spherical particles, for wide particle size distributions and for anisotropic particle packing and grain orientation). In the meanwhile, numerous studies have shown that the semi-empirical Carman-Kozeny approach is highly uncertain for materials with complex microstructures (see e.g., [10,11,12,13]). Despite these uncertainties, the Carman-Kozeny equations are still widely used for the study of granular materials such as battery electrodes (materials science) and sandstones (geoscience), where they give reasonable predictions of permeability and flow. As will be discussed in Chap. 5, new equations using new morphological descriptors from 3D analysis have been presented in literature, which provide reliable predictions of flow and permeability also for porous media (granular and non-granular) with more complex microstructures.

2.2.2.2 Controversy About (Un)realistic Values for Hydraulic Tortuosity

Much effort was expended to visualize the streamlines of flowing liquids in porous media and to estimate the associated streamline tortuosity. Already in 1956, Carman [14] was able to visualize streamlines by injecting dye into dense packed glass spheres. With this experiment he demonstrated that on average the streamlines diverge from the direction of macroscopic flow by an angle of about 45°. Based on these observations Carman concluded that the hydraulic streamline tortuosity (τ_{hydr_streamline}) in porous granular media must be approximately $\sqrt 2$. Thus, in early theoretical work, hydraulic tortuosity was often replaced by a constant value of $\sqrt 2$.

Contrariwise, in experimental work, tortuosity is usually calculated indirectly from relative properties at macroscopic scale. A relative property is defined as ratio of the effective property (e.g., effective electrical conductivity (σ_eff) of a porous medium saturated with electrolyte) over the intrinsic property of the transporting medium (e.g., intrinsic conductivity of the pure electrolyte (σ₀)), which results in σ_rel = σ_eff/σ₀. A simple relationship between microstructure and the macroscopic relative property is then often assumed, according to which, for example, the relative conductivity (σ_rel) depends only on porosity (ε) and electrical tortuosity (i.e., σ_rel = ε/τ_ele²). Hence, when relative conductivity is known from experiment or simulation, the indirect electrical tortuosity can then be calculated easily (τ_{indir_ele} = $\sqrt {(\varepsilon /\sigma_{rel} )}$). By assuming the same simple relationship for relative diffusivity, the indirect diffusional tortuosity can be determined in the same way, i.e., by τ_{indir_diff} = $\sqrt {(\varepsilon /D_{rel} )}$.

For flow and permeability, the micro–macro relationship is more complex since it involves additional microstructure descriptors for the viscous effects (i.e., r_hydr). For example, the Carman-Kozeny formulations could be used for calculation of indirect hydraulic tortuosity (i.e., by combining and reformulating Eqs. 2.6, 2.7, 2.9, 2.15). The resulting expression for indirect hydraulic tortuosity then reads as follows

$$ \tau_{indir\_hydr} = \sqrt {\frac{{r_{hydr}^{2} \varepsilon }}{\kappa }} = \sqrt {\frac{{\varepsilon^{3} }}{{C_{K} S_{V}^{2} \kappa }}} . $$

(2.16)

However, this equation is rarely used because the involved descriptors are more difficult to determine. For simplicity, the indirect 'hydraulic' tortuosity is thus often calculated with the same simple approach as described above for relative conductivity or relative diffusivity. It is important to note, that the computed values that are reported in literature for indirect tortuosities are usually much higher than $\sqrt 2$, and sometimes even up to 20 [15,16,17,18].

In Chap. 3, we present an extensive collection of empirical data from literature, which is the basis for a systematic comparison of different tortuosity types. This collection of literature data illustrates a clear mismatch between the relatively high values ($\gg$ 2) for indirect tortuosities (calculated from known effective properties) versus relatively low values in the range of $\sqrt 2$ for streamline-tortuosities (from simulated 3D flow patterns). The latter fits well with the predictions from Carman [14]. A possible explanation for this mismatch is given below in Sect. 2.2.2.4.

2.2.2.3 New Methods for Characterization of Hydraulic Tortuosity

(a) Hydraulic streamline tortuosity (τ_{mixed_hydr_streamline})

Over the last two decades considerable progress was achieved in tomography, 3D image processing and pore scale modeling. This allows for a computation of the 3D geometry of streamlines based on simulated flow fields and the associated effective path lengths can be described statistically (see e.g., [19,20,21,22,23]), as schematically illustrated in Fig. 2.2. However, to extract a physically relevant mean value for the effective path length (L_eff), the question arises how the single streamlines must be counted in the statistical analysis? Bear [24] and Clennell [9] argued that hydraulic streamline tortuosity should be calculated by weighting the streamlines with the corresponding fluid fluxes, i.e.,

$$ L_{eff\_weighted} = \frac{{\mathop \sum \nolimits_{i} L_{i} w_{i} }}{{\mathop \sum \nolimits_{i} w_{i} }} $$

(2.17)

Fig. 2.2

Illustration of streamline tortuosity in a simplified structure (i.e., channel with constant width). For statistical analysis and for determination of a mean effective path length (<Leff>), the individual streamlines must be weighted, which is a challenging task. Different streamline patterns are shown for viscous flow (bottom left), for bulk diffusion (bottom right) and for electrical conduction (top left). Colors of the streamlines represent the underlying velocity field. Bottom middle shows the center-streamlines for flow (blue) and diffusion (red) and compares them with the medial axis (black line). From this comparison the following order of tortuosities can be estimated: τ_{dir_medial_axis} > τ_{mixed_hydraulic_SL} > τ_{mixed_diff_SL (=} τ_{mixed_ele_SL)}

where w_i represents a weighting factor for the flux represented by streamline i. In the meanwhile, several weighting approaches were presented in literature (see e.g., [19,20,21,22,23, 25]). For a detailed discussion we refer to Duda [7], who concluded that these different weighting approaches lead to inconsistent results. In particular, circular Eddy-currents may impose a significant source of error. Finally, statistical analysis of streamline geometry is computationally expensive, which is a further drawback of this type of tortuosity.

Remark I

For more complex microstructures, this order may be different.

Remark II

Simulations of electrical conduction with Ohms law and bulk diffusion with Fick’s law are mathematically identical. Hence, the electrical and diffusional streamline tortuosities are identical.

(b) Hydraulic volume averaged tortuosity (τ_{mixed_hydr_Vav})

A much easier method to compute hydraulic tortuosity was then presented by Matyka and Koza [8] and Duda et al. [7], based on earlier work from Koponen et al. [20]. Instead of focusing on the challenging analysis and weighting of streamlines, their method is based simply on the integration of local vector components from the 3D velocity field:

$$ \tau_{mixed\_hydr\_Vav} = \frac{{\left\langle {v_{c} } \right\rangle }}{{\left\langle { v_{x} } \right\rangle }} = \frac{{\mathop \smallint \nolimits_{V}^{{}} v_{c} \left( r \right) d^{3} r}}{{\mathop \smallint \nolimits_{V}^{{}} v_{x} \left( r \right) d^{3} r}} $$

(2.18)

$$ \approx \frac{{\mathop \sum \nolimits_{k = 1}^{n} \frac{1}{n} \sqrt {v_{x} \left( k \right)^{2} + v_{y} \left( k \right)^{2} + v_{z} \left( k \right)^{2} } }}{{\mathop \sum \nolimits_{k = 1}^{n} \frac{1}{n} v_{x} \left( k \right)}} = \frac{{\mathop \sum \nolimits_{k = 1}^{n} \sqrt {v_{x} \left( k \right)^{2} + v_{y} \left( k \right)^{2} + v_{z} \left( k \right)^{2} } }}{{\mathop \sum \nolimits_{k = 1}^{n} v_{x} \left( k \right)}}, $$

where n is the number of discrete control volumes with equal volume (e.g., voxels from tomography and from the simulated flow field, respectively).

It must be emphasized, that this definition of hydraulic tortuosity is compatible with an alternative definition from Carman (see Eq. 2.13 and Fig. 2.1), who described tortuosity also as the ratio of capillary velocity (v_c) over interstitial axial velocity (v_x). According to Matyka and Koza [8], <v_c> is the 'average magnitude of the intrinsic velocity over the entire pore volume' (i.e., mean capillary velocity) and <v_x> represents the 'volumetric average of the velocity component parallel to the macroscopic flow direction' (i.e., the mean interstitial axial velocity). The mean values are obtained by integration of local properties (i.e., vectors components) at each point r in a discretized (mesh- or voxel-based) velocity field, which is obtained from numerical simulation of flow. The vector components in a flow field are schematically visualized in Fig. 2.3.

Fig. 2.3

Visualization of principle approach for computation of volume averaged tortuosity (τ_{mixed_hydr_Vav} or τ_{mixed_diff_Vav}). Transport simulations of flow (left) and diffusion (right) are performed for a zic-zac channel of constant width. The color code in the plots of streamlines and flux represent the local variation of transport velocities. Note the difference in the flow velocity pattern between flow and diffusion. The volume averaged tortuosity is based on the integration of the velocity vector components in local flow direction (v_c, i.e., capillary velocity) and in direct flow direction (v_x, i.e., axial velocity)

Compared to the streamline approach, the volume-averaged approach has several important advantages:

• Neither streamline extraction nor weighting of streamlines are necessary.

• Problems with eddy currents are solved in an elegant way.

• Implementation is relatively easy, and computation is relatively cheap.

• This method not only holds for fluid flow, but also for other transport processes such as diffusion and electrical or thermal conduction.

In literature, this type of tortuosity is called area (2D) or volume (3D) averaged tortuosity. For 2D-cases it was introduced by Koponen, 1996 [20]. For 3D-cases it was first applied in 2011 by Matyka and Koza [8], Duda et al. [7] and Ghassemi and Pak [26]. Since then, it is increasingly used for characterization of all kinds of porous media (see e.g., [27,28,29,30,31,32,33,34]).

Throughout the present article, the volume averaged as well as the streamline tortuosities are denoted as ‘mixed' tortuosities (i.e., τ_{mixed_hydr_Vav}, τ_{mixed_hydr_streamline}). The term 'mixed' emphasizes the fact that this category incorporates 'mixed' information. First, it includes geometric information from 3D analysis of simulated flow fields. Second, it also includes physics-based information from simulation of a specific transport process (i.e., flow, diffusion, or conduction). Thereby, the mixed information is neither determined directly from microstructure nor indirectly from effective or relative properties. (Note: A new tortuosity-classification with direct, indirect, and mixed tortuosities is introduced in Sect. 2.5, see Fig. 2.8).

2.2.2.4 New Microstructure Descriptors for Bottleneck Effect and Constrictivity

As mentioned in Sect. 2.2.2.2, the values measured for mixed tortuosities (i.e., volume averaged tortuosity (_{mixed_hydr_Vav}) and streamline tortuosity (τ_{mixed_hydr_streamline})) are roughly compatible with Carman's estimation of hydraulic tortuosity (ca. $\sqrt 2$). In contrast, the relatively high values for indirect tortuosities (τ_{indir_ hydr} or τ_{indir_ele}) reported in literature indicate that the effective path lengths are overestimated with this approach. By computing tortuosity indirectly from effective transport properties, other limiting effects in addition to path lengths are also included in the calculation of the indirect tortuosity, which explains the obvious overestimation of tortuosity and path lengths. In particular, the limitations arising from narrow bottlenecks are not addressed separately with the indirect approach. The omission of the bottleneck effect is also a major shortcoming of the Carman-Kozeny theory.

(a) Constrictivity in idealized microstructures by Petersen

It was shown by Petersen [35] in 1958 that the retarding impact of varying cross-sections for flow in a straight tube can be described with a so-called constrictivity factor (β_Petersen), which he defined as the ratio of cross-section areas at open (A_max) and at constricted locations (A_min), i.e.,

$$ \beta_{Petersen} = \frac{{A_{{{\text{max}}}} }}{{ A_{{{\text{min}}}} }} = \frac{{r_{{{\text{max}}}} ^{2} }}{{r_{{{\text{min}}}}^{2} }}, $$

(2.19)

where $r_{{{\text{min}}}}$ and $r_{{{\text{max}}}}$ denote the radii of the disk-shaped cross-sections A_min and A_max, respectively. For the simple case of a constricted pipe, the bottleneck effect and the associated microstructure descriptors (β, r_min, r_max) are visualized in Fig. 2.4.

Fig. 2.4

Illustration of a bottleneck in a circular tube. Petersen introduced constrictivity as a correction parameter in transport equations, which accounts for the resistive effects of bottlenecks in pipe flow

(b) Constrictivity in complex microstructures by Holzer

In recent years, it became more and more accepted that the bottleneck effect (i.e., constrictivity) is an important retarding effect for transport in porous media, which needs to be considered separately from and/or in addition to the path lengths effect (i.e., ‘true’ tortuosity) (see e.g., [36,37,38,39,40,41,42,43]). However, until recently there were no methods available to quantify constrictivity (β) from complex microstructures. A suitable method was then introduced by Holzer et al. [38] in 2013, which was later formalized in the framework of stochastic geometry [44]. Thereby, the average sizes of bulges and bottlenecks are obtained from two different size distribution curves, see Münch and Holzer [45]:

• (a) continuous pore size distribution (c-PSD), for which there is a one-to-one relationship with the granulometry function [46], is used to characterize the size distribution of bulges, and

• (b) a geometrically defined mercury intrusion pore size distribution (MIP-PSD, also called porosimetry curve) is used to characterize the size distribution of bottlenecks.

With the MIP-PSD method, the 3D distance map is modeled in such way that the pore/phase sizes are evaluated in transport direction from inlet to outlet plane. Thereby, the pore/phase sizes can only become smaller in transport direction and therefore the smallest bottleneck ‘upstream’ represents a hard constraint for the maximum size in the ‘downstream’ domains. This geometric treatment is comparable with the pressure loss that occurs upon viscous flow in porous media, which is also inverse proportional to the involved bottleneck sizes (r_min², respectively). Further details on c-PSD and MIP-PSD can be found in Münch and Holzer [45], and for further reading about constrictivity see [37, 41, 44].

Figure 2.5 (left) illustrates the concept of the two size distribution methods. The mean radii corresponding to the volumetric 50% quantiles (i.e., r₅₀) of these two pore size distributions are considered as mean effective sizes for bulges (r_{50_cPSD} = r_max) and bottlenecks (r_{50_MIP_PSD} = r_min), respectively. Constrictivity (β) is then defined as the ratio of the squared effective bottleneck radius (r_min) over the squared effective bulge radius (r_max), which is the inverse of Petersen's definition of constrictivity, i.e.,

$$ \beta = \frac{{A_{{{\text{min}}}} }}{{ A_{{{\text{max}}}} }} = \frac{{r_{{{\text{min}}}} ^{2} }}{{r_{{{\text{max}}}}^{2} }} = \frac{1}{{\beta_{Petersen} }}. $$

(2.20)

Fig. 2.5

Constrictivity (β) in materials with complex microstructures can be derived based on two different pore size distribution (PSD) methods, which is illustrated for an SOFC electrode consisting of porous LSC (modified after Holzer et al. [38]). To describe the resistive effects from narrow bottlenecks towards electrical transport in the solid phase (i.e., LSC), the same geometric descriptors (i.e., PSD, β) and associated image processing tools can be used as for the mass transport resistance in the pore phase. Two 3D images on the left side represent so-called distance maps with color coded radii of the LSC-phase, from which the c-PSD and MIP-PSD curves on the right side are derived. The continuous phase/pore size distribution curve (c-PSD) captures the size of bulges in the contiguous LSC-phase network (red curve). In analogy, the PSD curve from mercury intrusion porosimetry (MIP-PSD) captures the sizes of bottlenecks in the same contiguous LSC-phase network (blue curve)

It is important to note that r_min and r_max do not describe minimum and maximum radii of a pore structure, but they represent mean values of two different size distribution curves capturing the sizes of either narrow bottlenecks (MIP-PSD, r_min) or wide bulges (c-PSD, r_max). For further reading about constrictivity we refer to [37, 41, 44].

The fact that constrictivity, still today, is often not included in the traditional transport equations explains the relatively high values that are typically obtained when calculating indirect tortuosity from effective transport properties, e.g., with Eq. 2.16 [38, 41]. In Sects. 2.2.2.2 and 2.2.2.3, a striking discrepancy between mixed tortuosities (with characteristic values in the range of $\sqrt 2$) and indirect tortuosity (with values typically > 2) was described. This discrepancy can mainly be attributed to the exclusion of constrictivity (bottleneck effect) from the calculation of indirect tortuosity, which is well documented by Wiedenmann et al. [42]. Indirect tortuosities that are derived from relative or effective properties (e.g., permeability, conductivity) are thus often also interpreted as fudge factors (or structure factors), because they represent an overall resistive effect of the microstructure, which is not or only partly related to the lengths of transport pathways (see e.g., Clennell [9]).

In summary, based on improved methods for 3D image analysis over the last two decades, new descriptors for microstructure characteristics have become available. These innovations include new types of tortuosities (direct-geometric and mixed types) and new approaches for measuring characteristic length, hydraulic radius, bottleneck size and constrictivity. Based on these new descriptors also new expressions for the quantitative relationship between microstructure and effective properties (permeability, conductivity, diffusivity) could be formulated. In contrast to the Carman-Kozeny equations, these new expressions have a high prediction power also for materials with complex microstructures. The mathematical description of micro–macro relationships for transport in porous media and how these equations evolved over time is reviewed in Chap. 5.

2.3 Electrical Tortuosity

2.3.1 Indirect Electrical Tortuosity

The concept of electrical tortuosity (τ_ele) was developed since ca. 1940 (see Archie [47]) in parallel to the hydraulic tortuosity concept. The electrical tortuosity describes resistive effects of the microstructure, which limit the effective electrical conductivity (σ_eff) and, equivalently, increase the effective electrical resistance (R_eff) in porous media.

It must be emphasized that electrical conduction in porous media can take place, either, through the pore phase, which is saturated with a liquid electrolyte (whereas the solid phase is insulating). Or, alternatively, the electrical conduction can take place through the solid phase, which is illustrated in Fig. 2.6 for an SOFC-electrode with conductive LSC-phase. In the following section, we consider the case of a porous material saturated with a liquid electrolyte. Ohm's law can be used to describe the electrical flux (J_ele) in the saturated porous media, i.e.,

$$ J_{ele} = \frac{1 }{{R_{eff} }} \frac{\Delta U}{{L_{0} }} = \sigma_{eff} \frac{\Delta U}{{L_{0} }}, $$

(2.21)

with the potential gradient (ΔU/L₀) as driving force. The resistive formation factor (F_R) was then defined as the ratio of effective electrical resistance of the porous media (R_eff) over the intrinsic resistance of the electrolyte (R₀):

$$ F_{R} = \frac{{R_{eff} }}{{ R_{0} }} = \frac{{\sigma_{0} }}{{ \sigma_{eff} }} = \frac{1}{{ \sigma_{rel} }} = \frac{1}{ M}. $$

(2.22)

The inverse of the formation factor is the relative conductivity (σ_rel), which is also called microstructure (M)-factor. The difference between effective and intrinsic properties is due to the mentioned resistive effects of the underlying microstructure. According to Archie's law [47], which reads as

$$ F_{R} = \frac{1}{{\varepsilon^{m} }}, $$

(2.23)

the formation factor can also be described as a power law of porosity (ε) with a so-called empirical cementation exponent (m).

Archie's law is widely used in geo- and soil-science. However, it has limited validity because it relates effective transport properties and associated formation factor to a single microstructure characteristic (i.e., porosity) and it ignores all other morphological effects. In an alternative approach by Wyllie and Rose, 1950 [48] the formation factor was described with an additional microstructure characteristic, namely the so-called structural factor (X_ele), defined by

$$ F_{R} = \frac{{X_{ele} }}{\varepsilon } = \frac{{\tau_{ele}^{2} }}{\varepsilon }. $$

(2.24)

In analogy with Carman's formulation for flow, the structural factor (X_ele) was also considered as an equivalent of the (electrical) tortuosity factor (T_ele = τ_ele²). This relationship is nowadays more commonly formulated as

$$ \sigma_{eff} = \frac{{\sigma_{0} \varepsilon }}{{\tau_{ele}^{2} }}. $$

(2.24b)

Hence, the electrical tortuosity (τ_{indir_ele}) can be obtained indirectly by plugging experimental results for the formation factor (or relative conductivity) and porosity into Eq. 2.24, which leads to

$$ \tau_{indir\_ele} = \sqrt {F_{R} \varepsilon } = \sqrt {\frac{\varepsilon }{{\sigma_{rel} }}} . $$

(2.25)

This kind of indirect tortuosity (sometimes also called formation tortuosity) is nowadays very prominent because it can be obtained easily from numerical simulations of electric conduction, e.g., with commercial software like GeoDict from Math2Market [49] or with open-source software such as TauFactor from Imperial College London [50]. It must be emphasized that the determination of this indirect electrical tortuosity does not consider any geometric information. It is therefore by no means a measure for the true length of transport pathways. Whenever the tortuosity is afterwards used to determine effective conductivities and/or the resistive effects of microstructure on effective conductivity, respectively, this is a good way to define tortuosity, though.

Katsube et al. [15] performed an extensive investigation on shales that were saturated with electrolyte. The measured values of F_R were in the range from 140 to > 17,000 and porosities were in the range from < 0.01 to 0.1. The corresponding indirect electrical tortuosities took values in the range from 3.4 to 12. Katsube et al. [15] interpreted these values for τ_{indir_ele} as unrealistically high based on geometric considerations. This pessimistic interpretation is in accordance with Carman's 45° argument for the streamlines and associated estimation of $\sqrt 2$ for streamline tortuosity. Katsube et al. concluded that the empirical results for the indirect electrical tortuosities are unrealistically high because other important microstructure effects (in addition to ε, τ) are not yet included in Eqs. 2.24 and 2.25. Owen [51] and Dullien [52] argued that the influence from narrow bottlenecks needs to be considered as an additional resistive effect. The formation factor was thus redefined by adding constrictivity (β) from Eq. 2.20 (see also [36]), which results in

$$ F_{R} = \frac{{\tau_{ele}^{2} }}{\varepsilon \beta }. $$

(2.26)

It must be emphasized that the method for direct measurement of constrictivity from porous media, as presented above in Sect. 2.2.2.4, was only introduced in 2013 (Holzer et al. [38]). Because of a lack of suitable methods, constrictivity was thus not considered—until recently—as a separate microstructure characteristic in the calculation of indirect electrical tortuosity. Consequently, unrealistically high values (> 3) are often reported in literature for the indirect electrical tortuosity. These high tortuosity values must be interpreted as mixed information that includes resistive effects not only from tortuous path lengths but also from narrow bottlenecks.

Nevertheless, nowadays it is more and more accepted that the bottleneck effect and constrictivity should be considered separately from the path length effect. Hence, indirect tortuosity is now sometimes also calculated based on a separate treatment of constrictivity (see e.g. He et al. [53]), i.e.,

$$ \tau_{indir\_ele\_II} = \sqrt {\frac{\varepsilon \beta }{{\sigma_{rel} }}} . $$

(2.27)

2.3.2 Mixed Electrical Tortuosities

Today, the recent progress in numerical simulation and 3D image processing opens new possibilities for the computation of other (mixed) types of electrical tortuosity. In analogy to the hydraulic tortuosity discussed above in Sect. 2.2.2.3, also the electrical tortuosity can be extracted from simulated 3D fields of electrical flux. This approach provides either streamline (τ_{mixed_ele_Streamline}) or volume averaged tortuosities (τ_{mixed_ele_Vav}) (see Matyka and Koza [8] and Duda et al. [7]). For the volume averaged tortuosity, the equation can be rewritten as follows:

$$ \tau_{{mixed\_{\varvec{ele}}\_Vav}} = \frac{{\left\langle {v_{c} } \right\rangle }}{{\left\langle { v_{x} } \right\rangle }} = \frac{{\mathop \smallint \nolimits_{V}^{{}} v_{c} \left( r \right) d^{3} r}}{{\mathop \smallint \nolimits_{V}^{{}} v_{x} \left( r \right) d^{3} r}} $$

(2.18)

Furthermore, it should be noted, that electrical conduction and associated electrical tortuosity are not limited to porous media saturated with electrolyte. As mentioned earlier, the same principles can be used to describe electrical conduction in solid phases of a porous medium or a composite and thereby analyzing the electrical tortuosity of the conducting phase (e.g., electrical conduction in cermet electrodes of solid oxide fuel cells).

2.4 Diffusional Tortuosity

2.4.1 Knudsen Number

This section mainly deals with tortuosity in context with molecular diffusion (also called bulk diffusion), which is the dominant process in systems where chemical interactions between particles (ions, molecules) are negligible. Typically, this is the case for electrolytes with a high level of dilution and/or with an ideal, inert tracer. Also, gas transport at low pressure in macro- and mesoporous media is often dominated by molecular diffusion. Contrariwise, in systems where advection or surface effects (adsorption or dispersion) are important, molecular diffusion may not be the dominant transport process anymore.

The Knudsen number (K_n) is used to distinguish between different diffusion regimes in porous media. K_n is defined by the ratio of mean free path length (λ) over the characteristic length (L_c), i.e.,

$$ K_{n} = \frac{\lambda }{{L_{c} }}. $$

(2.28)

The mean free path length (λ) depends on pressure, temperature and on the effective cross-sectional area of the gas species. Typically, it is in the range of 30–200 nm. For example, for air at room temperature and ambient pressure λ is 68 nm [54].

The characteristic length (L_c) is an ill-defined property, but it is usually considered as being equivalent to the characteristic pore radius. Hence, for gas transport in nanoporous media with r₅₀ < 10 nm, the Knudsen number is much larger than 1. In this case, we say that we are in the Knudsen diffusion regime, which is controlled by molecule-wall collisions [55].

For gas transport in macro-porous media, K_n is < 1, meaning that we are in the regime of bulk molecular diffusion, which is controlled by molecule–molecule collisions. For all liquid electrolytes, λ is very small (nm or smaller) so that diffusion in porous media is usually controlled by bulk diffusion.

For Knudsen numbers close to 1, both transport phenomena must be considered. Bosanquet's approximation [55] or the Dusty Gas Model [56] are then often used to model transport in this mixed regime.

2.4.2 Bulk Diffusion

2.4.2.1 Indirect Diffusional Tortuosity

For molecular or bulk diffusion (K_n < 1), Fick's first law can be used to describe diffusional flux (J_D), which is driven by a concentration gradient (Δc/L₀). The flux in porous media directly scales with diffusivity (D_eff), which is an effective property of the system under consideration (see e.g., Satterfield and Sherwood [57]). More precisely,

$$ J_{D} = - D_{eff} \frac{\Delta c}{{L_{0} }}. $$

(2.29)

The effective diffusivity (D_eff) of a porous medium depends on the intrinsic diffusivity (D₀) of pure electrolyte or pure gas, respectively, and on the resistive effects from the microstructure. The transport limitations from obstacles in the microstructure are quantitatively expressed by the relative diffusivity (D_rel, also called microstructure factor or M-factor (see also Eq. 2.22)), which is a dimensionless characteristic, i.e.,

$$ D_{eff} = D_{0} M = D_{0} D_{rel} . $$

(2.30)

Initially, all resistive effects from the underlying microstructure were attributed to the diffusional tortuosity factor (D_rel = 1/T_diff, with T_diff = τ_diff²), see [57]. Still today some authors prefer this definition of tortuosity, which can then be considered as a global transport resistance (e.g., Elwinger et al., [58]). However, it was recognized very early that diffusion depends on different microstructure effects, which are associated with path length variations as well as with pore volume variations. Hence, D_rel was then defined in an analogous way as the relative electrical conductivity, including porosity in addition to tortuosity (in analogy to σ_rel in Eqs. 2.22 and 2.25, see [3, 48, 59]), i.e.,

$$ D_{rel} = \frac{\varepsilon }{{\tau_{diff}^{2} }}. $$

(2.31)

This leads to the frequently used indirect diffusional tortuosity defined by

$$ \tau_{indir\_diff} = \sqrt {\frac{\varepsilon }{{D_{rel} }}} . $$

(2.32)

Note that the mathematical treatment for numerical simulation of bulk diffusion (Fick's law) and electrical conduction (Ohm's law) is identical. In both approaches the Laplace equation is solved. For completion, it is mentioned here that this analogy also applies to Fourier's law of heat conduction (Q_thermal = − λ_eff (ΔT/L₀); with Q_thermal = thermal flux, λ_eff = effective heat conduction). It follows that the impact of pore structure on the effective properties of all three processes (bulk diffusion, electric and thermal conduction) must be identical. In fact, it was reported from several experimental studies that the same values are obtained for indirect electrical and diffusional tortuosities when the same porous media was analyzed [60,61,62].

Furthermore, in a similar way as discussed previously for the indirect electrical tortuosity, also unrealistically high values for indirect diffusional tortuosity were often reported in empirical studies of diffusion. These high values must be attributed again to the fact that the bottleneck effect is not treated as a separate resistive effect. Following van Brakel and Hertjes [36] and in analogy to the electrical tortuosity, this can be improved by introducing constrictivity (β) to the equation for relative diffusion, i.e.,

$$ D_{rel} = \frac{\varepsilon \beta }{{\tau_{diff}^{2} }}. $$

(2.33)

Nevertheless, still today the indirect diffusional tortuosity is often calculated based on Eq. 2.32, without considering constrictivity separately. We conclude that diffusional tortuosity for systems with Kn < 1 (bulk diffusion described with Fick's law) is in principle identical with electrical tortuosity (described by Ohmic conduction), and therefore the limiting effects of pore structures are the same (as discussed by Clennell [9]).

The indirect diffusional tortuosity in Eq. 2.32 can be derived in many ways depending on the method by which the relative diffusivity D_rel is determined. For example, D_rel can be obtained from diffusion experiments (which can be denoted as τ_{indir_diff_exp}). Very often, D_rel is obtained from simulation of bulk diffusion in a 3D model representing the pore microstructure (which can be specified as τ_{indir_diff_bulk}). Alternatively, D_rel can be determined with random walk simulation. The random walk simulation is briefly described below for Knudsen diffusion (Sect. 2.3.3, Eq. 2.34), but of course it can be applied in a very similar way also for the computation of D_rel in the bulk diffusion regime. By substituting D_rel from random walk simulation into Eq. 2.32 we obtain a third type of indirect diffusional tortuosity (i.e., τ_{indir_diff_Rwalk}).

2.4.2.2 Mixed Diffusional Tortuosities

(a) Streamline and volume averaged tortuosities (τ_{mixed_diff_Streamline}, τ_{mixed_diff_Vav})

In analogy to electrical conduction, more sophisticated tortuosity types can be determined nowadays based on numerical simulation of diffusional flux and 3D image processing. The resulting streamline- and volume averaged tortuosities (τ_{mixed_diff_Streamline}, τ_{mixed_diff_Vav}) represent more rigorous measures of the diffusive path lengths, since they do not mix with a hidden bottleneck effect, as it is usually the case for indirect tortuosity (τ_{indir_diff}). For the volume averaged diffusional tortuosities (see Matyka and Koza [8] and Duda et al. [7]), the mean capillary velocity <v_c> and the mean axial <v_x> velocity can be computed from simulated vector fields. The corresponding equation can be rewritten in analogy to the volume averaged electrical or hydraulic tortuosity:

$$ \tau_{{mixed\_{\varvec{diff}}\_Vav}} = \frac{{\left\langle {v_{c} } \right\rangle }}{{\left\langle { v_{x} } \right\rangle }} = \frac{{\mathop \smallint \nolimits_{V}^{{}} v_{c} \left( r \right) d^{3} r}}{{\mathop \smallint \nolimits_{V}^{{}} v_{x} \left( r \right) d^{3} r}}. $$

(2.18)

(b) Random walk tortuosity (τ_{mixed_diff_Rwalk})

In context with diffusional mass transport, random walk methods can be used to simulate Brownian motion of particles. From the random walk simulation, a statistical measure for the displacement of moving particles (i.e., the mean square displacement, MSD) can be extracted. The MSD is proportional to the product of time and intrinsic diffusivity (MSD = f (D₀ t)). In porous media, the particle diffusion is hindered by the obstacles of the pore wall. This limiting effect is quantitatively captured by the random walk tortuosity (τ_{mixed_diff_Rwalk}), which is defined as the ratio of MSD in free space over MSD in the porous medium.

In principle, the movements in each direction sum together and therefore, the MSD can be decomposed into the Axial Square Displacements (ASD). The axial tortuosities in x-, y-, and z-directions (τ_{x/y/z_mixed_diff_Rwalk}) can then be calculated from the corresponding ASDs. Pytrax is a simple and efficient random walk implementation for calculating the directional tortuosity from 2D and 3D images (see Tranter et al. [63,64,65,66,67,68]).

2.4.3 Knudsen Diffusion

In nanoporous materials with K_n $\gg$ 1 (i.e., in the Knudsen regime), gas diffusion is controlled by collisions with the pore walls. Numerically, this process can be simulated with the random walk method (see e.g., Babovsky [69]). For each particle the corresponding diffusivity can be calculated from displacement length and travel time. For simulations that are based on a large number of particles and sufficiently long travelling time these calculations result in a homogenized effective Knudsen diffusivity (D_{eff_Kn}). The relative Knudsen diffusivity (D_{rel_Kn}) is then again defined as the ratio of effective over intrinsic Knudsen diffusivity, i.e.,

$$ {\text{D}}_{{{\text{rel}}\_{\text{Kn}}}} { } = \frac{{{\text{D}}_{{{\text{eff}}\_{\text{Kn}}}} }}{{{\text{D}}_{{0\_{\text{Kn}}}} }}{ } = \frac{{\upvarepsilon }}{{{\uptau }_{{\text{indir}\_{\text{Kn}}}}^{2} { }}}. $$

(2.34)

Similar as the relative diffusivity in the bulk diffusion regime, also the relative Knudsen diffusivity is a dimensionless property, which describes the resistive impact of pore microstructure against transport. The corresponding indirect Knudsen tortuosity (τ_{indir_Kn}) is obtained by

$$ {\uptau }_{{{\text{indir}}\_{\text{Kn}}}} { } = { }\sqrt {\frac{{\upvarepsilon }}{{{\text{D}}_{{{\text{rel}}\_{\text{Kn}}}} }}} . $$

(2.35)

Note that such computed values for relative Knudsen diffusivity (D_{rel_Kn}) and for the indirect Knudsen tortuosity (τ_{indir_Kn}) strongly depend on the intrinsic Knudsen diffusivity (D_{0_Kn}, sometimes also called apparent diffusivity (D_a)). D_{0_Kn} itself can be computed using characteristic length (L_c) and thermal velocity (v_th = k_b T/m), i.e.,

$$ D_{0\_Kn} = \frac{1}{3} L_{c} v_{th} . $$

(2.36)

The characteristic length (L_c) is a rather ill-defined property, which is somehow related to pore size distribution and to the average pore size, respectively. The uncertainty associated with L_c propagates into the relative Knudsen diffusivity and into the associated Knudsen tortuosity (τ_{indir_Kn}), which is critically discussed by Zalc et al. [70].

Unfortunately, to the best of our knowledge, there is currently no method available for a more direct analysis of Knudsen tortuosity based on effective path lengths (L_eff/L₀), as it is the case for bulk molecular diffusion, for electric conduction and also for viscous flow (see the discussion of mixed-streamline and -volume averaged tortuosities). Knudsen tortuosity is thus generally determined indirectly and therefore it is difficult to understand Knudsen tortuosity as a resistance that is related to distinct morphological features of the pore structure and to the corresponding length of transport pathways.

In literature, the interpretation of Knudsen tortuosity in context with gas diffusion in nanoporous media is highly controversial. For example, Ferguson et al. [71] used different methods (Random walk, FVM) for modeling transport at continuum scale and in the Knudsen regime, which enabled to compute both tortuosities (i.e., for bulk diffusion and for Knudsen diffusion). Also, Gao et al. [72] compared different modeling approaches (Knudsen, Dusty Gas and Oscillator models) that were used to characterize diffusivity and tortuosity in nanoporous media. Gao et al. emphasizes that the resulting tortuosity is highly dependent on the definition of the characteristic pore size (i.e., characteristic length) and on other experimental parameters (chemical species, temperature, pressure). Therefore, Gao et al. [72] suggested using the coordination number (i.e., average number of vertices that are connected to the nodes) instead of the Knudsen tortuosity to describe the impact of nanoscale microstructure on diffusive transport. This approach leads to more stable results and the physical and geometric interpretations of the coordination number are clearer than the indirect Knudsen tortuosity.

2.4.4 Limitations to the Concept of Diffusional Tortuosity

In nanoporous materials, transport mechanisms very often consist of a superposition of several processes such as bulk molecular diffusion, Knudsen diffusion, viscous flow, adsorption, and surface diffusion (see the examples in [53, 72]). For such cases with a mixed transport mechanism, it seems no longer possible to maintain the initial tortuosity concept as proposed by Kozeny and Carman, which is based either on the ratio of path lengths (L_eff/L₀) or on the ratio of velocity components (<v_c>/<v_x>), see Eqs. 2.13 and 2.18. It is obvious that the indirect tortuosity, which is then usually applied also for mixed transport mechanisms, has limited value in explaining distinct (geometric) microstructure effects. Also, in this case the indirect tortuosity must be rather understood as a fudge factor that describes the bulk resistive effects from microstructure and should not be interpreted as a measure for effective path lengths.

In very fine-grained, nanoporous media the molecular radii of gas species and the thickness of the surface adsorption layer can be in a similar range as the pore radii. In this case, variations of molecular radii and thickness of adsorption layers can become equally important for effective transport as the pore size and pore structure. Tortuosity in such systems is nowadays often determined with dedicated methods of numerical modeling (e.g., molecular dynamics [53]) or experimental characterization (e.g., NMR [58]), but the link with the initial geometrical tortuosity concept (i.e., with path lengths) is often not clear, since other physical effects (e.g., adsorption) may become dominant for diffusion.

2.5 Direct Geometric Tortuosity

The tortuosity types discussed in previous sections cannot be considered as strictly geometric characteristics of the microstructure. The indirect (hydraulic, diffusional, electrical, thermal) tortuosities are derived from effective properties. They are thus rather interpreted as fudge factors or as parameters that describe the bulk resistive effects of the microstructure. The mixed (streamline, volume-averaged, random walk) tortuosities are derived from the vector fields resulting from transport simulations. The mixed tortuosities for the same 3D microstructure thus vary with the simulated transport mechanism (i.e., diffusion, flow, conduction etc.). Therefore, also the mixed tortuosity types do not represent a direct geometric description of the microstructure itself. In contrast, geometric tortuosity includes a whole group of tortuosity types, which entirely depend on the pore morphology, and which are therefore determined directly from 3D images representing the pore microstructure.

The steadily increasing importance of geometric tortuosity types is triggered by the recent progress in the fields of tomography and 3D image analysis, which is summarized in Chap. 4. Many different approaches to measure geometric path lengths and associated geometric tortuosities can be found in literature. The following section describes the most prominent examples.

It must be emphasized that geometric tortuosities do not consider any information regarding the transport mechanisms. This is in contrast, for example, to streamline and volume averaged tortuosities, which are based on simulations of conduction, diffusion, or flow. This is also in contrast to indirect tortuosities, which are extracted from effective properties that are related to specific transport mechanisms.

2.5.1 Skeleton and Medial Axis Tortuosity

The medial axis tortuosity (τ_{dir_medial_axis}) can be considered as a prototype for the family of geometric tortuosities. Therefore, in literature it is often just called 'geometric tortuosity' without further specification (see e.g., Stenzel et al. [41]). In our nomenclature, we allocate it to the group of ‘direct’ tortuosities since it is derived directly from the 3D microstructure by image analysis, in contrast to the indirect tortuosity types. The computation of medial axis tortuosity is based on several rather complex image-processing steps, which are illustrated in Fig. 2.6: The raw data from tomography is first segmented into its constituent phases. The example in Fig. 2.6a shows a fuel cell anode with the phases nickel, Gd-doped ceria, and pores [73]. For each phase a medial axis skeleton (MAS, see Chap. 5 of Soille [74] for details) is then produced (Fig. 2.6b, c). The shortest pathways through the MAS network, which connect couples of inlet and outlet points, are found e.g., with the help of the Dijkstra algorithm. For propagation algorithms on graphs to determine shortest path lengths, the reader is referred to Jeulin et al. [75]. The voxel-based skeleton is then transformed into a 3D graph, i.e., into a network representation consisting of vertices (nodes) and edges (branches) between them. A small portion of a 3D graph is shown in Fig. 2.6d. Note that further information can be attached to edges and vertices in form of additional characteristics of the local microstructure (e.g., local pore size and bottleneck size of each segment, coordinates, and coordination nr of each node etc.). The analysis of the 3D graph is computationally cheap. It reveals a distribution of medial axis tortuosities for each space direction (Fig. 2.6e). Based on graph analysis, tortuosity information of each pathway can be combined with local pore characteristics, such as the paths orientation. The example from Keller et al. [76, 77] in Fig. 2.6f shows a stereographic projection of pore path orientations in an anisotropic clay rock. The path orientation information is combined with the medial axis tortuosities of each path. Tortuosity values are indicated with a color code. In this example, the pathways parallel to the bedding plane (yz-direction) have lower medial axis tortuosities (2.7, dark blue) compared to the pathways perpendicular to the bedding plane (x-direction: up to 13, yellow), which of course has a strong impact on the anisotropy of the macroscopic permeability.

Fig. 2.6

Illustration of image processing steps, which are the basis for computation of medial axis tortuosity (τ_{dir_medial_axis}): a FIB-SEM tomography and phase segmentation of a cermet anode for solid oxide fuel cells (SOFC) with pores (black), nickel (gray) and Gd-doped ceria (CGO, white) from Holzer et al. [73], b zoom-in of a, c medial axis skeletons (MAS) of each phase, d schematic illustration of a graph consisting of branches/edges and nodes/vertices (e.g. used for analyzing path lengths), e graph analysis, i.e. statistical analysis of paths lengths and associated medial axis tortuosities (mean τ) for the nickel phase in the main transport-direction (z) and f stereoplot showing anisotropic distribution of tortuosities in a clay rock (d + f from Keller et al. [76, 77])

Reproducibility of medial axis tortuosity among different research groups may be a challenge, since there exist many different methods for skeleton extraction from 3D voxel data (see Soille [74] for a detailed description of different skeletonization algorithms). Throughout the present paper, we use the following nomenclature: If the skeleton is not a medial axis representation, then we speak of τ_{dir_skeleton} instead of τ_{dir_medial_axis}. The procedure for generation of a medial axis skeleton (MAS) can be found e.g., in Lindquist et al. [78]. MAS generation is typically based on an iterative erosion process called topological thinning. The resulting skeleton consists of lines or curves with a thickness of one voxel. These curves are always located in the center of the pore bodies. As previously mentioned, the shortest pathways between couples of inlet and outlet points can then be algorithmically computed, e.g., by means of the Dijkstra algorithm [79]. It turns out that for complex pore structures a robust skeletonization procedure is challenging.

Efforts have been made to develop algorithms for an accelerated computation of shortest pathways in skeleton networks. Besides the propagation algorithm in Jeulin et al. [75] mentioned above, TESAR (Tree-structure Extraction algorithm delivering Skeletons that are Accurate and Robust) was introduced by Sato et al. [80]. Thereby the distance field used in the Dijkstra algorithm is modified, such that the shortest pathways in a pore network can be found in a fast and reliable way. This algorithm thus also reveals a medial axis skeleton (MAS). It was later implemented in commercial software, such as Avizo Fire (thermofischer.com), which was used by many authors as a basis for graph analysis to compute medial axis tortuosity (see e.g., [41, 73, 76, 77, 81, 82]; examples are shown in Fig. 2.6). Keller et al. [76] also presented a detailed description of the transformation steps from a voxel-based skeleton, via a 3D graph to a vector-based list of segments and nodes (assigned with local pore characteristics, such as effective lengths, orientation, pore size and bottleneck size).

Open-source software solutions for skeletonization and extraction of medial axes are available e.g., as Matlab code (Tort3D described by Al-Raoush and Madhoun [83]) or in ImageJ/Fiji (see https://imagej.net/Skeletonize3D and imagej.net/AnalyzeSkeleton).

Furthermore, Thiedemann et al. [84] also presented methodological details that can be used for an extension of medial axis tortuosity towards weighting of pathways for their fluxes. This approach is based on a detailed analysis of the 3D graph including various local structural characteristics (see also Jungnickel [85] for a general reference to graph theory). Thereby the bottleneck size, which limits the flux of a specific pathway, is used for weighting the segments and local path lengths, which in turn affects the overall statistics of effective path lengths and the associated medial axis tortuosity. The weighting with bottleneck size can be understood intuitively as an analogue of the previously discussed weighting of streamlines by flux. A similar approach was recently also described by Nemati et al. [25].

In summary, the computation of medial axis tortuosity is based on various complex image-processing steps (e.g., skeleton extraction, 3D graph analysis). Therefore, its implementation is complex, and its computation may be time consuming. A major disadvantage is the fact that skeleton extraction can be done in many ways, which may lead to different results for the same type of geometric tortuosity. Moreover, it becomes difficult to interpret the extracted skeletons as transport networks in case of high porosities. Nevertheless, based on fundamental graph theory, this approach allows combining tortuosity with other local characteristics (bottleneck size, pore path orientation, connectivity), which are important for understanding the influence of microstructure on effective transport properties.

2.5.2 Path Tracking Method (PTM) Tortuosity

The path tracking method (PTM) was introduced by Sobieski et al. [86,87,88]. It allows for a fast computation of a geometric tortuosity (i.e., τ_{dir_PTM}) that is very similar to the skeleton approach discussed in the previous section. However, the PTM algorithm is only applicable for microstructures consisting of packed spheres. It identifies tetragonal structures formed by four neighboring spheres. These tetragons represent an approximation of the interstitial pore space. The algorithm then finds the shortest pathways through the material by connecting gravity centers in the base triangles of neighboring tetragons. The resulting pathways from PTM are similar to the shortest pathways in a network skeleton. Despite this similarity, it must be emphasized that the pathways used for PTM differ in general from the medial axis representation.

In [87], a comparison of the PTM tortuosity (τ_{dir_PTM}) with the hydraulic volume averaged tortuosity (τ_{mixed_hydr_Vav}) is presented, which shows almost identical results. However, whereas the geometric PTM approach is fast and easy, the hydraulic volume averaged tortuosity (based on e.g., LBM simulations) is computationally expensive and time consuming. Hence, PTM is a fast method for measuring geometric tortuosity in 3D models of packed spheres or particles.

2.5.3 Geodesic Tortuosity

The concept of geodesic tortuosity (τ_{dir_geod}) relies on the geodesic metric in image data introduced by Lantuéjoul [89] and is used, e.g., in [41, 90, 91] to compute the corresponding geodesic tortuosity. The geodesic tortuosity is based on a statistical analysis of shortest path lengths (L_eff) from inlet- to outlet-planes divided by the sample length (L₀). For this purpose, the shortest path lengths are defined in terms of geodesic distances within the set of those voxels that represent the transporting phase (see e.g., Stenzel et al. [41]). Figure 2.7 illustrates the difference between geodesic (red) and medial axis (green) tortuosity. For the medial axis tortuosity there is only one starting point per pore body in the inlet plane, whereas for the geodesic tortuosity all voxels of the transporting phase in the inlet plane are taken as starting points. With increasing distance, the numerous pathways starting from different seed points concentrate on a few geodesic tracks. In both cases (i.e., geodesic and medial axis tortuosities) the shortest pathways can be calculated using the Dijkstra algorithm [79, 92]. However, whereas in the case of medial axis tortuosity the shortest pathways are restricted to the centerlines, for the case of geodesic tortuosity all voxels of the transporting phase are interpreted as vertices of a graph, which are connected to their neighboring voxels (26-neighborhood). Thus, in average, the geodesic pathways are shorter than the medial axis pathways. Note that—due to discretization errors on the voxel grid—single geodesic pathways might be longer than the corresponding paths along the skeleton. In [41] the following relationship

$$ \tau_{dir\_geodesic} = 0.76{ }\tau_{dir\_median\_axis} $$

(2.37)

Fig. 2.7

Schematic illustration of geometric pore pathways used to measure path lengths for a medial axis tortuosity, b geodesic tortuosity and c percolation path tortuosity. The percolation method finds the pathways with the least constriction (i.e., pathway, along which the largest possible sphere can migrate from inlet- to outlet-planes. In contrast, the medial axis and geodesic methods find the shortest pathways for each couple of inlet- and outlet-points, independent from the corresponding bottleneck dimensions. Note: Pathways from medial axis and percolation methods are indicated as dotted lines with green and blue colors, respectively, as guides to the eye

has been empirically derived by linear regression, using 43 virtual microstructures generated by a specific type of a 3D stochastic microstructure model. The coefficient of determination R² quantifying the goodness of fit for linear regression was equal to 0.81. Note that R² is between 0 and 1, where 1 indicates a perfect fit.

A mathematical formalization of geodesic tortuosity in the framework of random sets, a key object in stochastic geometry and mathematical morphology for microstructure characterization [46, 93], was recently provided by Neumann et al. [44, 94], while a slightly modified version of geodesic tortuosity was presented by Barman et al. [11].

2.5.4 Fast Marching Method (FMM) Tortuosity

The fast-marching method (FMM) tortuosity is very similar to the geodesic tortuosity, in the sense that it also considers geodesic distances within the voxel space of a given phase. The FMM algorithm is based on the simulation of a propagating front from inlet- to outlet-plane, which is described e.g., by Vicente et al. [95]. In particular, FMM solves the following Eikonal equation (see Sethian et al. [96,97,98,99])

$$ \left\| {\nabla T\left( x \right) } \right\|F\left( x \right) = 1, (F\left( x \right) > 0) $$

(2.38)

on the voxel grid, where T(x) is the arrival time at location x and F(x) is the speed of the front. For each voxel the minimum arrival time is computed and by considering the speed of the front, this results in a distance map representing the shortest path lengths (L_FMM). For each pixel in the outlet plane, the corresponding FMM tortuosity can be calculated by dividing L_FMM through the sample length (L₀).

Jørgensen et al., 2011 [100, 101] describes the FMM method in context with microstructure analysis of SOFC electrodes. Thereby, FMM is also used to extract additional (local) information of the transporting phase network, such as the distribution of interface distances, distribution of characteristic path diameters and identification of dead-end pores. A further application of FMM was presented by Taiwo et al. [102] for battery electrodes. A recipe for the implementation of FMM can be found in Appendix E (supplementary info) of Hamann et al. [103].

In summary, the FMM tortuosity is very similar to the geodesic tortuosity, in the sense that it also finds the shortest (geodesic) pathways within the voxel space representing the transporting phase. In addition, it is computationally cheap and relatively fast.

2.5.5 Percolation Path Tortuosity

Percolation path tortuosity is based on an algorithm that finds the pathway(s) with the least constricting bottleneck(s) (i.e., with the largest minimum bottleneck size). Hence, this algorithm allows the largest possible sphere(s) to travel from inlet- to outlet-plane and, at the same time, it finds the shortest path through the network for this sphere. Tortuosity is then defined as ratio of percolation path length over direct length (τ_{dir_percolation} = L_percolation/L₀). This method is, for example, implemented in the GeoDict Software (www.math2market.com). Thereby it is possible not only to calculate the percolation path for a single largest sphere but also for a defined number (n) of largest spheres. Hence, it enables us to find the n least constricting pathways and it calculates the corresponding mean tortuosity.

As shown in Fig. 2.7c, the blue sphere (with radius corresponding to the least constricting bottleneck) cannot pass through the narrow bottlenecks of the direct pathway (left) and hence it must take a deviation (right pathway). Percolation tortuosity is thus often larger than medial axis tortuosity, which takes more direct pathways through narrow bottlenecks.

The percolation pathways capture the maximum possible opening, which can be intuitively associated with a pathway of high flux. In contrast, the pathways for medial axis and geodesic tortuosities capture their shortest pathways regardless of the bottleneck radius, and therefore medial axis and geodesic tortuosities do not represent characteristics that can be related to pathways of high flux. Considering percolation path tortuosity for varying radii reveals interesting insights on porous microstructures going beyond the information gained by geodesic tortuosity. This is demonstrated using an example of paper-based materials in [46, 104].

2.5.6 Pore Centroid Tortuosity

For the pore centroid method (see e.g., [81, 101]), the 3D image volume is processed as a stack of 2D images. In each 2D section the position of the center of mass is determined for the transporting phase (e.g., pores). These centers are then tracked in transporting direction perpendicular to the 2D images, which results in one single tortuous centroid path. The centroid tortuosity is then calculated as the ratio of the effective centroid path length (L_eff) over the sample thickness (L₀). The pore centroid method is a quick and simple method, which is, e.g., implemented in the Avizo Software (www.thermofischer.com). For increasing volume fractions, the mass center approaches the image center and thus, pore centroid tortuosity goes to one. One can think of simple examples for microstructures with low values of centroid tortuosity (close to one), where the actual transportation paths are highly tortuous. Hence, the relevance of the centroid tortuosity in context with microstructure—property relationships is highly uncertain.

Finally, further approaches for the extraction of geometric tortuosity from 3D images can be found in literature. They are usually based on distance propagation and/or shortest path algorithms (see examples in [105,106,107]).

2.6 Tortuosity Types: Classification Scheme and Nomenclature

2.6.1 Classification Scheme

The above-presented review reveals a multitude of different tortuosity types. However, in literature, in conference presentations and in associated scientific discussions dealing with tortuosity, the type of tortuosity under consideration is very often not properly defined. This lack of information often becomes the source of confusion and misunderstanding. For clarification, we propose to use a rough classification scheme with only three main categories of tortuosities. For a more precise specification, we introduce a systematic nomenclature that builds on the simple classification scheme. The nomenclature aims to provide all relevant details that are necessary for proper interpretation of the specific tortuosity under consideration. Both, the classification scheme and the detailed nomenclature approach are illustrated in Fig. 2.8.

Fig. 2.8

Tortuosity classification scheme (top) and tortuosity nomenclature (bottom). Three main classes of tortuosity can be identified based on the method of determination: direct τ, mixed τ and indirect τ. For precise tortuosity nomenclature, additional information on the underlying concept of definition (geometric, physics-based) and on the effective property characterization method (for indirect τ) is added in the second and third terms. The classification scheme and nomenclature should help avoiding confusion in the discussion and interpretation of tortuosity

The classification of tortuosity (Fig. 2.8, top) is based on two main criteria:

(a) The method of determination:

The method of determination is in most cases either based on a direct geometric analysis of the microstructure using tomography and 3D image analysis (called direct τ). Alternatively, tortuosity can be deduced indirectly from effective transport properties (called indirect τ).

(b) The concept of definition:

The concept of definition is in most cases either based on the assumption that tortuosity and associated path lengths (L_eff) are geometric properties of the microstructure (called geometric τ) or, alternatively, the definition emphasizes that tortuosity is a function of the transport process under investigation, such as viscous flow, diffusion, or electric conduction (called physics-based τ).

It turns out that the method of determination and the concept of definition are generally linked with each other in a specific way, which leads to a reduction to three main categories of tortuosity. The first category consists of direct, geometric tortuosity types. The second category consists of indirect, physics-based tortuosity types. And the third category consists of mixed types, including streamline and volume averaged tortuosities (i.e., τ_{mixed_phys_streamline}, τ_{mixed_phys_Vav}). The definition of mixed types emphasizes both, the dependency on the transport process (i.e., physics-based) as well as the geometric aspect of path lengths and associated tortuosity. Characterization of mixed types is challenging, because the required information cannot be obtained directly from 3D image analysis of the microstructure. First, it requires some numerical 3D simulation to compute a transport-specific volume field of velocities or fluxes. In a second step, 3D image analysis is then used to extract the lengths of streamlines or the volume-averaged ratio of vector components from these volume fields (see Sect. 2.2.2.3).

2.6.2 Nomenclature

Based on our classification scheme we introduce a new tortuosity nomenclature (Fig. 2.8, bottom), which consists of 2 or 3 terms. With the first term we describe the method of determination, which is one of three main categories (i.e., direct, mixed, or indirect determination of tortuosity).

The nomenclature then contains additional information on the underlying concept of definition. With the second (and third) term we specify details related to the underlying concept of definition. This can be either, a specification of the geometric analysis (for direction tortuosities), or a specification of the involved transport process (for indirect tortuosities). For mixed tortuosities, it is necessary to specify both, the geometric analysis, and the involved transport process. In the following section, the nomenclature is rigorously applied to all relevant tortuosity types, which are treated in this book.

2.6.2.1 Category I: Direct Geometric Tortuosities (τ_{dir_geom})

The name includes a first term (dir), which emphasizes direct 3D analysis of the microstructure, e.g., from tomography data. The second term (i.e., ‘geom’ alias medial axis, PTM, percolation, geodesic, FMM or pore centroid) specifies the geometric concept or method. The following examples of direct geometric tortuosities were discussed previously in this section:

• τ_{dir_medial_axis} and τ_{dir_skeleton}

• τ_{dir_PTM} (path tracking method)

• τ_{dir_percolation}

• τ_{dir_geodesic}

• τ_{dir_FMM} (fast marching method)

• τ_{dir_pore_centroid.}

Note: In a recent review by Fu et al. (2020) [66] different names are used for geometric tortuosities. τ_{dir_geodesic} is termed 'direct shortest path searching method' (DSPSM) and τ_{dir_skeleton} is termed 'skeleton shortest path searching method' (SSPSM).

2.6.2.2 Category II: Mixed Tortuosities (τ_{mixed_phys_streamline}, τ_{mixed_phys_Vav}, τ_{mixed_diff_ Rwalk})

The first term (mixed) defines the category. Mixed tortuosities are neither calculated directly from a morphological analysis of the pore structure, nor are they determined indirectly from effective properties. Typically, mixed tortuosities are obtained by a multi-step process, which starts with a specific simulation of transport, and which is then complemented with an additional postprocessing step (i.e., geometric analysis) of the simulation output. This simulation output can be, for example, a velocity field from simulation of flow, conduction, or diffusion. The output can also consist of many random walkers, which are obtained by simulation of diffusion.

The second term (physics-based) thus contains information about the specific transport process under consideration. This can be either viscous flow (hydr), bulk diffusion (diff), electric or thermal conduction (ele, therm).

The third term then describes the geometric method, which is either based on the analysis of streamlines or volume-averaged vector components (i.e., ratio of velocity vector components) or lengths of random walkers:

• τ_{mixed_hydr_streamline}

• τ_{mixed_diff_streamline}

• τ_{mixed_ele_streamline}

• τ_{mixed_therm_streamline}

• τ_{mixed_hydr_Vav}

• τ_{mixed_diff_Vav}

• τ_{mixed_ele_Vav}

• τ_{mixed_therm_Vav}

• τ_{mixed_diff_Rwalk}.

Note: In literature, tortuosities belonging to categories II (mixed) and III (indirect) are often not distinguished and all of them are called 'flux-based' or 'physical' (see e.g., Tjaden et al. [65] and Fu et al. [66]). Usually the flux-based hydraulic tortuosities are calculated from streamlines or velocity fields (i.e., they belong to the category II: mixed tortuosities), whereas the flux-based electrical and diffusive tortuosities are typically calculated from the corresponding effective properties (i.e., they belong to category III: indirect tortuosity).

2.6.2.3 Category III: Indirect Physics-Based Tortuosities (τ_{indir_phys_ sim,} τ_{indir_phys_exp})

The first term (indirect) defines the category. The second term (physics-based) contains information about the specific transport process under consideration. The physical nature of transport is either electrical conduction (ele), thermal conduction (therm), bulk diffusion (diff), Knudsen diffusion (Kn) or hydraulic flow (hydr):

• τ_{indir_ele_sim or_exp}       τ_{indir_ele} = $\sqrt {(\varepsilon /\sigma_{rel} )}$ or = $\sqrt {(1/\sigma_{rel} )}$ or = $\sqrt {(\varepsilon \beta /\sigma_{rel} )}$

• τ_{indir_therm}       τ_{indir_therm} = $\sqrt {(\varepsilon /K_{rel} )}$ or = $\sqrt {(1/K_{rel} )}$ or = $\sqrt {(\varepsilon \beta /K_{rel} )}$

• τ_{indir_diff}       τ_{indir_diff} = $\sqrt {(\varepsilon /D_{rel} )}$ or = $\sqrt {(1/D_{rel} )}$ or = $\sqrt {(\varepsilon \beta /D_{rel} )}$

• τ_{indir_Kn}       τ_{indir_Kn} = $\sqrt {(1/D_{Kn\_rel} )}$ (see Eq. 2.35)

• τ_{indir_hydr}       τ_{indir_hydr} = $\sqrt {(rh^{2} \,\varepsilon /\kappa )}$ (see Eq. 2.16).

The value of the effective property used as input for indirect tortuosity may be different if it is determined by simulation or with an experimental approach (see e.g., the discussion of apparent tortuosity-discrepancy for Li ion batteries by Usselgio-Viretta et al. [108]). Therefore, we recommend adding a third term that describes the nature of effective property input ('sim' for simulation or 'exp' for experimental).

It is well documented that the indirect tortuosity is dependent on the method by which the underlying effective property is determined. Examples for experimental characterization approaches are diffusion cells, tracer diffusion experiments, measurements of electrical resistance and formation factor, electrochemical experiments (EIS), flow-cell experiments for gases or liquids. It is clear that the measured transport properties may strongly depend on experimental parameters, which then also affects the indirect tortuosity. In a similar way, methodological details of transport simulation will influence the resulting effective properties and associated indirect tortuosity. These aspects cannot be captured by nomenclature and should therefore be described separately.

Furthermore, values obtained for indirect tortuosity are also heavily dependent on the underlying mathematical expression, which describes the relationship between the effective property (input) and tortuosity (output). In most cases, this relation is of the same type as Eqs. 2.25 and 2.32 (e.g., τ_{indir_ele} = $\sqrt {(\varepsilon /\sigma_{rel} )}$). However, different relationships can be found in literature, either without consideration of porosity (τ_{indir_ele_inclPore} = $\sqrt {(1/\sigma_{rel} )}$), where inclPore means that the pore volume effect is included in this indirect tortuosity calculation) or with separate consideration of constrictivity in addition to porosity (see Eq. 2.27: τ_{indir_ele_exBN} = $\sqrt {(\varepsilon \,\beta /\sigma_{rel} )}$, where exBN means that the bottleneck effect is excluded from this indirect tortuosity calculation). Hence, for electrical conduction, details of the indirect tortuosity calculation can be expressed as follows:

• τ_{indir_ele} = $\sqrt {(\varepsilon /\sigma_{rel} )}$       (standard case: Eq. 2.25)

• τ_{indir_ele_inclPore} = $\sqrt {(1/\sigma_{rel} )}$

• τ_{indir_ele_exBN} = $\sqrt {(\varepsilon \,\beta /\sigma_{rel} )}$.

The cases for thermal conduction and diffusion can be treated analogously. In any case, it is recommended that if the indirect tortuosity for diffusion and conduction is not derived with the standard expression (Eqs. 2.25 and 2.32) this should be emphasized explicitly, and the corresponding mathematical expression should be declared. Finally, it must be emphasized that the indirect tortuosity for flow is rarely used, because it requires knowledge of the hydraulic radius (see Eq. 2.16).

Hence, the indirect tortuosity can be determined in many ways, which makes comparisons difficult. To avoid confusion, 'indirect tortuosity' always calls for some detailed specifications (in addition to nomenclature) regarding a) the underlying method used to determine the effective property by experiment or simulation, and b) the underlying equation, which relates indirect tortuosity with effective property, and which is used to calculate the indirect tortuosity. Without these specifications, 'indirect tortuosity' is an ill-defined characteristic.

2.7 Summary

Tortuosity is generally defined as ratio of the mean effective path length over the direct path length (τ =  <L_eff>/ <L₀>). However, determination of the mean effective path length in complex disordered microstructures is not an easy task. Therefore, many different concepts, definitions and methods of characterization can be found in literature. This diversity often leads to confusion and misinterpretation. More than 20 different tortuosity types are presented and discussed in the present review.

In order to enable a precise description of a specific tortuosity, a new and systematic tortuosity nomenclature is presented in this chapter. Figure 2.8 can be used as a guide that helps to find the correct name of a specific tortuosity type. This nomenclature is based on a classification scheme that uses two distinctive criteria:

• (a) The method of determination distinguishes tortuosities that are either calculated indirectly from effective transport properties or directly from 3D images representing the microstructure under investigation.

• (b) The concept of definition makes a distinction between physics-based tortuosities (i.e., hydraulic, electric, diffusional, thermal τ) or geometric tortuosities. The latter represent intrinsic properties of the microstructures, and they are thus independent from the involved transport process.

It turns out that all relevant tortuosity types can be grouped into three main categories:

• I: Indirect physics-based tortuosities (τ_{indir_phys})

• II: Direct geometric tortuosities (τ_{dir_geom})

• II: Mixed tortuosities (τ_{mixed_phys}).

I: The indirect physics-based tortuosities describe bulk resistive effects from the microstructure towards a specific transport process (i.e., viscous flow, electric and thermal conduction, bulk and Knudsen diffusion). The indirect physics-based tortuosities typically overestimate the true path length because they are calculated from the corresponding effective properties, which also include other limiting effects from the microstructure (e.g., bottlenecks).

When using the Carman-Kozeny equation for prediction of hydraulic flow in porous media, very often the indirect hydraulic tortuosity is fixed at a value of $\sqrt 2$ (based on geometric arguments from Kozeny). This approach may work well for granular materials consisting of packed spheres, but it typically fails to predict the transport properties of more complex microstructures. For the latter cases, a more sophisticated consideration of tortuosity is required.

II: The direct geometric tortuosities include a group of metrics that are capable to provide realistic estimations of the mean path length (i.e., τ_{dir_geom}, with ‘geom’ = medial axis, skeleton, geodesic, fast marching method FMM, path tracking method PTM, percolation path or pore centroid). It must be emphasized that the direct geometric tortuosities consider the path length as an intrinsic property of the microstructure, which is independent from the transport process.

There is also another important aspect to be considered in context with the direct geometric tortuosities. In order to capture the entire resistive effect from the microstructure, one has to consider also other morphological effects such as the bottleneck effect (constrictivity) and viscous drag at the pore wall (hydraulic radius), in addition to the path length effect (geometric tortuosity).

III: The mixed tortuosities represent the most advanced descriptions of the path length effect. Typically, the mixed tortuosities are based on a numerical simulation of the involved transport process by using as an input the discretized 3D microstructure (e.g., from tomography or from stochastic geometry modelling). Geometric analysis can then be performed on the simulated flow fields. This approach reveals mixed tortuosities that are calculated either from volume averaged velocity vectors, or from mean path lengths of streamlines and/or random walkers.

The volume averaged tortuosity (τ_{mixed_phys_Vav}) is perceived as a particularly promising type of tortuosity. It is based on the integration of local velocity vectors over the simulated flow field. Thereby, two specific vector components are considered: one parallel to the local, microscopic flow direction (v_c) and one parallel to the direct, macroscopic flow direction (v_x). In this way, an alternative definition of tortuosity is obtained, which is the ratio of the mean capillary velocity over the mean axial velocity (τ_{mixed_phys_Vav} =  <v_c>/<v₀>). This approach is computationally cheap and geometrically simple. It provides a reliable description of the mean effective path length also for complex microstructures, and it captures the specific impact of the involved transport process.