5.1 Introduction

In this chapter, empirical relationships between morphological characteristics (porosity, tortuosity, constrictivity, hydraulic radius) and macroscopic transport properties (effective conductivity, effective diffusivity, and permeability) are described. Based on the rapid progress of analytical techniques (i.e., 3D imaging, image processing, stochastic simulation, numerical modeling, cloud computing) the prediction power of such equations has improved considerably over the last decade. The newest formulations are now capable to predict micro–macro relationships for many different types of materials and microstructures (e.g., granular, fibrous, cellular, and platy microstructures) in a reliable way. Consequently, these empirical relationships do have some general meaning although they are not derived from a rigorous theoretical basis.

It must be emphasized that microstructure effects limiting the transport in porous media can be investigated in different ways and with different methodologies. In this context, transport simulations based on 3D microstructure models are particularly well suited to predict effective transport properties of porous media. However, the transport simulations themselves cannot replace the valuable information from micro–macro relationships. For example, in materials engineering these empirical relationships provide a unique basis for controlled microstructure optimization and associated materials design, which cannot be replaced by transport simulations. In contrast, for settings with complex transport mechanisms, the concept of tortuosity typically comes to its limits and sophisticated transport simulations are better suited to study the properties of porous media. This is for example the case when diffusion and flow are coupled with physico-chemical reactions at the pore walls (e.g., with adsorption and/or chemical reactions, see discussion in Chap. 4.6). Hence, empirical micro–macro relationships and transport simulations represent different and complementary approaches for studying the properties of porous media, which do not replace each other.

As an introduction to this chapter, recall that the review of tortuosity, performed in this book, basically reveals two main schools of thinking, where tortuosity is either determined indirectly or, alternatively, it is directly based on morphological descriptors of the underlying 3D microstructure.

Indirect physics-based tortuosity

In a traditional approach, tortuosity is interpreted as the predominant microstructure effect. Following this approach, tortuosity can be determined indirectly, assuming a relatively simple relationship with effective properties (see e.g., Eq. 2.31: Deff/D0 = Drel = ε/τindir_diff2). However, the values obtained for indirect tortuosity are consistently higher than what would be expected from geometric considerations of e.g., streamlines. Hence, indirect tortuosity clearly overestimates the lengths of (shortest) transport pathways, which indicates that it includes other microstructure effects such as the limiting influence from narrow bottlenecks. Hence, the indirect tortuosity can be interpreted as bulk microstructure resistance,—normalized by the pore volume. Thus, this information is often used as valuable input for macro-homogeneous models, which intend to describe the bulk microstructure resistance.

Direct geometric tortuosity

An alternative ‘geometric' school of thinking is focusing on geometric and mixed tortuosities, which provide reliable estimations of the lengths of transport paths. In order to establish quantitative relationships between microstructure and effective transport properties, it is then necessary to capture all relevant microstructure characteristics—not only the path length effect that is described with direct geometric or mixed tortuosities. This approach thus requires an additional effort, e.g., for determining constrictivity and eventually also hydraulic radius. The quantitative micro–macro relationships obtained by this approach provide a deeper understanding of the relevant morphological effects, which represents the basis for a purposeful microstructure optimization and materials design.

It must be emphasized that the geometric school of thinking profits a lot from the recent progress in 3D image analysis, stochastic geometry and virtual materials testing. These new options for 3D analysis are applied with the aim to establish quantitative micro–macro relationships for porous media. In the following sections selected results of such investigations are summarized first for conductivity and diffusivity, and subsequently also for flow and permeability.

5.2 Quantitative Micro–Macro Relationships for the Prediction of Conductivity and Diffusivity

In this section, micro–macro relationships describing the limiting effects of microstructure on conductivity and diffusivity are reviewed. Thereby, we typically consider transports in porous media like gas diffusion or liquid conduction. However, it must be emphasized that transport in solid phases of a composite material (e.g., ionic or electric conduction in a cermet electrode) is suspended, in principle, to the same microstructure limitations as the conductive or diffusive transport in porous media and can therefore be described with the same morphological characteristics and mathematical equations (see discussion in Chap. 4.5.2).

The progress made in the investigation of micro–macro relationships for conductivity and diffusivity is schematically illustrated in Fig. 5.1. In the following description we follow step by step this illustration from bottom to top.

Fig. 5.1
A chart exhibits an overview of the predictions of conductive and diffusivity studies. 1950, methods only for simple geometries. 2000, progress in 3 D analysis gives access to microstructure characteristics in complex media. 2010, stochastic geometry. 2020, quantitative micro-macro relationship.

Illustration of progress in microstructure characterization and virtual materials testing. It shows how the evolving 3D methods help to improve quantitative micro–macro relationships, which nowadays enable reliable predictions of effective diffusivity and conductivity even for materials with complex microstructures

Full size image

Already a long time ago it was recognized that transport in porous media is limited not only by the lengths of tortuous pathways but also by narrow bottlenecks (see e.g., Owen [1]). In 1974, van Brakel and Hertjes [2] thus postulated a micro–macro relationship for conductivity and diffusivity, which includes constrictivity (β) as well as tortuosity (Eq. 2.33: Drel = εβ/τ2). Unfortunately, at that time, constrictivity as well as direct geometric or mixed tortuosities could not be determined for complex microstructures. Nevertheless, for the simple case of straight tubes (τdir_geometric = 1) with varying cross-sections, it was shown by Petersen in 1958 [3] that the retarding impact of bottlenecks can be described by the ratio of the constricted cross-sectional area (Amin) over the ‘bulged’ cross-sectional area (Amax). This simple pipe-flow model, which is illustrated at the bottom of Fig. 5.1, led to the definition of constrictivity according to Eq. 2.18 (β = rmin2/rmax2). However, in the last century the practical relevance of all theories dealing with resistive effects from bottlenecks (constrictivity) and/or path lengths (geometric tortuosity) was strongly limited, since there were no suitable 3D methods available for a quantitative morphological characterization.

With the introduction of FIB-tomography in 2004 [4], 3D imaging of porous media at sub-μm scales became possible. As a next step, suitable tools for quantitative 3D image analysis were required. Two methods to quantify the size distributions of pore bulges and bottlenecks in complex disordered microstructures were introduced by Münch and Holzer [5]. Thereby the continuous pore size distribution (cPSD) was used to characterize the size distribution of pore bulges. Note that the cPSD uses the concept of granulometry functions, which were introduced in [6]. Going beyond granulometry functions, a method for a geometry-based 3D simulation of mercury intrusion porosimetry (MIP) was introduced in [5]. The MIP-PSD (sometimes also called ‘porosimetry’) reveals the size distribution of bottlenecks. Typical examples of cPSD and MIP-PSD curves are shown in Fig. 5.1.

It was then recognized by Holzer et al. [7] that the 50% quantiles (i.e., r50) of these two pore size distribution curves can be considered as mean effective sizes for bulges (r50_cPSD = rmax) and for bottlenecks (r50_MIP_PSD = rmin), which can be substituted in Eq. 2.18 (β = rmin2/rmax2). In this way, a quantitative method based on 3D analysis was found for the characterization of constrictivity, which also works for materials with complex microstructures. A formal definition of constrictivity in the framework of stochastic geometry was recently provided in [8].

Using experimental data for determining effective properties, as well as constrictivity and geometric tortuosity from 3D analysis, it was soon found that van Brakels equation (Eq. 2.33: Drel = εβ/τ2) is not very precise in predicting the effective diffusivity. This finding led to the question, which type of equation must be used to describe the relationship between microstructure characteristics and effective diffusivity (Deff) or effective conductivity (σeff), respectively. In a series of studies [9,10,11,12], the following equations were considered as possible candidates:

$${\upsigma }_{rel} ;{ }D_{rel} = M = d\varepsilon^{a} \beta^{b} /\tau^{c}$$
(5.1)
$${\upsigma }_{rel} ;{ }D_{rel} = M = \varepsilon^{a} \beta^{b} /\tau^{c}$$
(5.2)
$${\upsigma }_{rel} ;{ }D_{rel} = M = d\varepsilon^{a}$$
(5.3)
$${\upsigma }_{rel} ;{ }D_{rel} = M = d\varepsilon^{a} /\tau^{c}$$
(5.4)
$${\upsigma }_{rel} ;{ }D_{rel} = M = d\varepsilon^{a} \beta^{b}$$
(5.5)
$${\upsigma }_{rel} ;{ }D_{rel} = M = d\varepsilon^{a} \beta^{b} /\tau^{2}$$
(5.6)
$${\upsigma }_{rel} ;{ }D_{rel} = M = \varepsilon^{a} \beta^{b} /\tau^{2}$$
(5.7)
$${\upsigma }_{rel} ;{ }D_{rel} = M = \varepsilon^{a1 - a2\beta } /\tau^{c}$$
(5.8)
$$\sigma_{rel} ;D_{rel} = M = \varepsilon^{a}$$
(Archie's law 2.23)
$${\upsigma }_{rel} ;{ }D_{rel} = M = \varepsilon /\tau^{2}$$
(2.31)
$${\upsigma }_{rel} ;{ }D_{rel} = M = \varepsilon \beta /\tau^{2}$$
(2.33)

According to the ‘geometric’ school of thinking, in all these equations τ is thought as a direct geometric tortuosity (τdir_geom, either geodesic, medial axis or skeleton tortuosity). The prediction power of these equations was investigated thoroughly through a statistical approach of error minimization in [9,10,11,12]. For this purpose, models from stochastic geometry were used to generate a large number of 3D microstructures with varying characteristics and effective properties. 3D image analysis was used to compute the microstructure characteristics (ε, β, τdir_geom, rmin, rmax). Numerical transport simulation was exploited to determine effective diffusivity and/or conductivity (Drel, σrel) for each virtual 3D microstructure generated by stochastic models. The unknown exponents (a, b, c, d) in the above-mentioned equations were then determined by means of error minimization. As a quality criterion for the predictive capabilities of the above-mentioned equations, the mean absolute percentage error (MAPE) was used in [9], where

$$MAPE \left( {M_{sim} , M_{predict} } \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \frac{{\left| {M_{sim,i} - M_{predict,i} } \right|}}{{M_{sim,i} }} \cdot 100\%$$
(5.9)

Thereby, M (microstructure-factor) stands for the relative properties (with respect to Drel or σrel), which were determined in two ways: a) either by numerical simulation (e.g., Msim = Deff_sim/D0), or b) by substituting the values obtained from 3D image analysis for the microstructure characteristics (ε, β, τdir_geom) into the equation under consideration (i.e., Mpred, for example, Mpred = dεaβb/τc in Eq. 5.1). The exponents a, b, c, d were then fitted for each equation (see list of equations above) in order to minimize the corresponding MAPE (i.e., the absolute value of the difference Msim − Mpred). For the most relevant equations, the values of the fitted pre-factor and exponents as well as the prediction errors (MAPE) are summarized in Table 5.1.

Table 5.1 Summary of the most important micro–macro relationships (for conductivity and diffusivity) obtained by virtual materials testing, including fitted pre-factor (d) and exponents (a, b, c) for porosity (ε), constrictivity (β) and tortuosity (τdir_geometric) and the corresponding mean absolute percentage error (MAPE) as quality criterion
Full size table

In a first publication of this series of investigations (Gaiselmann et al. [9]), it was shown that for the traditional equations (i.e., Eqs. 2.31, 2.32, 2.33 without any fitting of exponents) the prediction power becomes significantly better when constrictivity is considered as a relevant microstructure effect, in addition to geometric tortuosity and porosity (cf. Eq. 2.31, where MAPE = 625% versus Eq. 2.33, where MAPE = 37%). Note that the prediction power can be further improved when using equations with fitted pre-factor and exponents (Eqs. 5.1–5.7). From all the equations under consideration, Eq. 5.1 (and Eq. 5.2) revealed the best results with a MAPE of 25% (and 28%, respectively). Then Eq. 5.1 reads as

$$M_{{pred}} = ~2.35~\varepsilon ^{{1.57}} \beta ^{{0.71}} /\tau _{{dir\_skeleton^{2.3}}}.$$
(5.1a)

Again, equations including constrictivity in addition to geometric tortuosity reveal the best predictions. These findings underline the importance of the bottleneck effect. It must be emphasized that the skeleton tortuosity (τdir_skeleton) was used throughout [9]. However, other types of geometric tortuosity should be tested as well.

The study presented by Stenzel et al. [10] is based on the same methodologies as in [9]. However, the results of [10] extend the investigations of [9] in the sense that the prediction power of these equations was compared for different types of geometric tortuosity. It turned out that the prediction power with geodesic tortuosity (τdir_gedodesic) is better than with skeleton tortuosity. For example, the prediction error (MAPE) for Eq. 5.2 improved from 28% with τdir_skeleton to 19% with τdir_geodesic. The prediction formula then takes the form

$$M_{{pred}} = \varepsilon ^{{1.15}} \beta ^{{0.37}} /\tau _{{dir\_geodesic^{4.39}}}.$$
(5.2b)

Moreover, modified definitions of constrictivity were investigated in [10]. They are based on considerations that the relevant sizes of bulges (rmax) and bottlenecks (rmin) do not necessarily correspond to the 50% quantiles (r50) of the cPSD and MIP-PSD curves, respectively. Other quantiles (r0, r10, r25, r50, r75, r90) from cPSD and MIP-PSD curves were thus considered as possible candidates for characteristic sizes of bottlenecks (rmin) and bulges (rmax). By varying the combination of these candidates, Stenzel et al. [10] obtained 35 different rmin/rmax-ratios as possible definitions for constrictivity (e.g., β = r25_MIP-PSD2/r75_cPSD2). The impact of these different definitions on the prediction power was then tested for five equations (Eqs. 2.31, 2.33, 5.1, 5.2, 5.7) using the MAPE as a quality criterion. In short, it turned out that the initial definition of constrictivity (i.e., β = r50_MIP-PSD2/r50_cPSD2) from Holzer et al. [7] gives the most reliable and most precise predictions for all equations. It is thus proposed to maintain the initial definition of constrictivity.

Up to this stage, the statistical analyses included only a moderate number of 105 [9] and 43 [10] different 3D microstructure models, respectively. In an extensive simulation study by Stenzel et al. [11] the number of virtual 3D models was increased to 8119. With this big data approach, it was confirmed that effective diffusivity and conductivity are well predicted by Eq. 5.2b. The corresponding MAPE decreased further to 13.6% due to the better statistical data basis. However, it also became clear that Mpred and associated effective properties are underestimated by Eq. 5.2b for highly porous materials (i.e., for microstructures with M > 0.7). Better predictions, in particular for M > 0.7, could be achieved with methods from machine learning, namely random forests, and neural networks, which reveal MAPEs of 8.5% and 8.9%, respectively. However, these tools from machine learning generally do not provide a clear physical interpretation of the microstructure influence on effective transport properties. Therefore, random forests and neural networks could not be used as a basis for microstructure optimization.

In a recent paper by Neumann et al. [12], it was recognized that the problem of Eq. 5.2b for microstructures with M > 0.7 originates from an overestimation of the bottleneck-effect at high porosities (particularly in the limit when ε tends to 1). A modified equation (Eq. 5.8) was thus proposed, where constrictivity appears in the exponent of porosity. In this way, constrictivity acts as a correction factor for the effective pore volume, but the effect of β becomes negligible when porosity is close to 1. The 8119 virtual 3D microstructures from [11] were then used as a basis for testing the prediction power of Eq. 5.8b in [12]. Overall, this resulted in a MAPE of 18.3%, which is—when taking all structures into account—not better than the MAPE of Eq. 5.2b obtained in [11]. The prediction formula derived in [12] reads as follows:

$$M_{{pred}} = \varepsilon ^{{1.67 - 0.48\beta }} /\tau _{{dir\_geodesic^{5.18}}}.$$
(5.8b)

However, Eq. 5.8b has the advantage that - in contrast to Eq. 5.2b—it is consistent with theoretical results in the dilute limit, i.e., in the case when the obstacles of the transport process vanish. For materials exhibiting high porosities with M > 0.7, the predictions obtained by means of Eq. 5.8b are much better than those obtained by Eq. 5.2b. Nevertheless, for low porosity materials with M < 0.05, the predictions by Eq. 5.8b are even worse. Therefore, when considering only structures with M > 0.05, the MAPE for Eq. 5.8b improves significantly to 10.3%. Hence, the higher the M-value, the better the prediction power of Eq. 5.8b.

From the numerous equations that were evaluated statistically, the following three favorite equations remain (marked with bold emphasis in Table 5.1):

  • Eq. 5.1a from [9] gives the best predictions when skeleton tortuosity is used.

  • Eq. 5.2b from [10] gives the best results with geodesic tortuosity, but only for microstructures with M < 0.7.

  • Eq. 5.8b from [12] also gives good results with geodesic tortuosity. In particular, this equation should be used for highly porous materials with M > 0.7 (but not for low porosity materials with M < 0.05).

The prediction power of these equations was also validated experimentally for different porous materials by means of tomography (FIB-SEM and μCT). The validation was done by comparing the predicted properties (Mpred) from 3D image analysis either with results from experimental characterization (Mexp) or from simulations (Msim) or both. In this way, it has been shown in [9, 10, 12] that the equations considered in these papers give good results for SOFC cermet electrodes, including gas diffusivity in the pores as well as electrical conductivities of the solid phases.

Furthermore, Eq. 5.2b was experimentally validated for very different types of microstructures, such as sintered ceramic membranes [13], fibrous GDL in PEM fuel cells [14] and even for open cellular materials (unpublished data). These results confirm that the established micro–macro relationships are rather general in the sense that they are capable to predict effective properties for a wide spectrum of microstructures, the morphologies of which differ significantly from those used for deriving the microstructure-property relationships. This generality may be surprising, considering the fact that the predictions are based on three volume-averaged parameters (ε, β, τdir_geometric) only. However, these findings also indicate that these three parameters indeed capture all microstructure effects, which are relevant for conduction and diffusion, to a large extent. Exceptions are discussed in [15].

Due to the progress in microstructure characterization as well as in mathematical and numerical 3D modeling, many researchers are now considering the distinct morphological limitations that can be described with geometric tortuosity, constrictivity and phase volume fractions. Hence, the geometric school of thinking is permanently expanding. For example, quantitative relationships between ε–β–τdir_geometric and effective transport properties are the basis for recent investigations of Li-ion batteries [16], polymer films [17], tight gas reservoirs [18], sandstones [19, 20], packed beds in oil combustion [21], biomaterials/bone tissue [22] and general packings of slightly overlapping spheres [15], see also [23] where large-scale statistical learning has been performed for the prediction of effective diffusivity in porous materials using 90,000 artificially generated microstructures.

5.3 Quantitative Micro–Macro Relationships for the Prediction of Permeability

The evolution of quantitative micro–macro relationships for the prediction of permeability is schematically illustrated in Fig. 5.2. Basically, this evolution can be subdivided into an early period, when methods for 3D-analysis were not yet available, and a recent period after the year 2000, when micro- and nano-tomography, 3D-image processing as well as stochastic 3D modeling became available.

Fig. 5.2
A chart exhibits the timeline of quantitative micro–macro relations. 1850, single tube. 1927, parallel tube model. 1937, mono-sized spheres. 1986, rocks. 2016, porous ceramics. 2020, stochastic model.

Evolution of quantitative micro–macro relationships for prediction of permeability in porous media

Full size image

The basic principles describing the limiting effects, which arise from the underlying microstructure, were introduced by Kozeny in 1927 [24] and Carman in 1937 [25] (see also the discussion of hydraulic tortuosity in Sect. 2.2). The theories of Carman and Kozeny [24, 25] are based on simplified geometrical models, which serve as analogues for realistic pore structures. The bundle-of-tubes model [24, 26] and the sphere-packing model [25] are of particular importance. However, the morphological descriptors determined for such simplified models cannot easily be transferred to more complex microstructures. A large amount of literature that has been published since the work of Kozeny and Carman thus intends to improve the prediction power of Carman-Kozeny-type equations and to make them applicable for more realistic materials models with complex microstructures. Early examples are given by Panda and Lake (1994) [27] for poly-dispersed granular media, and by Costa (2006) [28] for fractal pore geometries.

Over the last two decades, the progress in 3D imaging and image processing opened new possibilities to quantify the relevant microstructure characteristics (geometric and mixed tortuosity, bottleneck radius, constrictivity). The availability of new morphological descriptors also led to new expressions for the micro–macro relationships that were presented in literature. Basically, all equations (the classical and new ones) can be reduced to a representation of permeability (к) by the simple product of characteristic length and M-factor, i.e.,

$$\kappa { } = { }L_{char} M .$$
(5.10)

Thereby, the term of characteristic length (Lchar) accounts for wall friction effects, which are captured by the squared hydraulic radius (rhc2). Note that the characteristic length term has a major impact on permeability. For example, when the hydraulic radius (rhc) of a porous material changes from 10 to 0.1 μm due to pore clogging, the corresponding permeabilities (к) decrease by 4 orders of magnitudes, e.g., from 5 10–11 to 5 10–15 m2 (assuming a constant M-factor of 0.5). Hence, the precise determination of the characteristic length and hydraulic radius (Lchar, rhc) is of major importance for a reliable prediction of permeability.

The second term in Eq. 5.10 is the microstructure M-factor for flow. It accounts for all the other transport limitations, except for wall friction. Thereby, the effects of a) varying pore-volume fractions, b) transport path lengths and c) bottlenecks can be described by dimensionless characteristics (ε, β, τ). The resulting M-factor then takes values between 0 and 1.

In the following sections, important expressions that were proposed for prediction of permeability are briefly reviewed in chronological order. The evolution of these expressions is also visualized in Fig. 5.2 from bottom to top. For better comparison, all expressions are reformulated such that the two terms for characteristic length and M-factor are clearly distinguished.

5.3.1 Bundle of Tubes Model

Kozeny’s description of flow [24] from 1927 is based on the consideration of a bundle of tubes (see Chap. 2.2). Flow in a single, straight tube can be described by the Hagen-Poiseuille law (Eq. 2.3). From comparison with Darcy’s law (Eq. 2.2) for porous media, it follows that ‘permeability’ of a single pipe depends only on the radius, i.e.,

$$\kappa_{pipe-flow} = L_{char} = r^{2} / 8.$$
(5.11)

Kozeny [24] then introduced a general definition for the hydraulic radius by

$$r_{hc} = \frac{Vol\,\, open\,\, to\,\, flow}{{wetted\,\, surface}} = \frac{{\pi r^{2} L}}{\pi 2r L} = \frac{r}{2},$$
(5.12)

which is the ratio of the tube volume open to flow over the surface area that is wetted by the fluid. The combination of Eqs. 5.11 and 5.12 leads to the following 'pipe-flow permeability' given by

$$\kappa_{pipe-flow} = L_{char} = r_{hc}^{2} / 2.$$
(5.13)

For porous media consisting of bundles of tubes, the hydraulic radius and associated volume-to-surface ratio from Eq. 5.12 can also be written in terms of porosity over specific surface area per volume (SV, with units m2/m3), i.e.

$$r_{hc} = \frac{\varepsilon }{{S_{V} }} .$$
(5.14)

The characteristic length term for a bundle of tubes thus becomes

$$L_{char} = \frac{1}{2} \left( {\frac{\varepsilon }{{S_{V} }}} \right)^{2} = \frac{1}{{c_{K} }} \left( {\frac{\varepsilon }{{S_{V} }}} \right)^{2} .$$
(5.15)

Thereby, the Kozeny factor (cK) for circular tubes is equal to 2, in accordance with Eq. 5.13. For non-circular tubes the cK-values vary in the range from 1.6 (for triangular tubes) to 3 (for rectangular tubes with a high aspect ratio).

Furthermore, Kozeny pointed out in [24] that the superficial velocity (vs) in Darcy’s macroscopic description of porous media flow (and thus also for a bundle of tubes) is different from the capillary velocity (vc) in Poiseuille's microscopic description of flow in a single pipe. According to Dupuit's relation the two velocities are linked by porosity, i.e.,

$$v_{s} = v_{c} \varepsilon .$$
(5.16)

Kozeny [24] thus introduced porosity as part of the M-factor for flow. In addition, from the comparison of the models for sinusoidal and straight tubes model, it follows that the lengths of the pathways increase from L0 to Leff. Consequently, the pressure gradient (in Hagen-Poiseuille's and Darcy's law) needs to be corrected accordingly from ∆P/L0 to ∆P/Leff. In this context, Kozeny [24] introduced the tortuosity concept with the definition of tortuosity (τ = Leff/L0) in order to describe the path length effect on the pressure gradient. This reads as

$$\frac{{{\Delta }P}}{{L_{eff} }} = \frac{1}{{{\uptau }_{hydraulic} }}\frac{{{\Delta }P}}{{L_{0} }}.$$
(5.17)

Note that Kozeny’s formulation of the M-factor thus includes corrections for porosity and path lengths (tortuosity), which leads to

$$M_{pred} = \frac{\varepsilon }{{{\uptau }_{hydraulic} }}.$$
(5.18)

Combining the M-factor (Eq. 5.18) and the characteristic length term (Eq. 5.15) gives the full Kozeny equation, which describes permeability for a bundle of tubes-model by

$$\kappa_{Kozeny} = \frac{1}{{c_{K} }}\left( {\frac{\varepsilon }{{S_{V} }}} \right)^{2} \frac{\varepsilon }{{\tau_{hydraulic} }},$$
(5.19)

with Kozeny’s shape factor cK = 2 for circular tubes. It must be noted, that at the time it was not yet possible to distinguish between direct geometric, indirect physics-based or mixed tortuosity. But from the qualitative descriptions, it becomes clear that Kozeny was using the concept of effective path length (such as streamlines), which is best described by mixed tortuosity.

5.3.2 Sphere Packing Model

In Carman’s work [25] from 1937, the Kozeny equation (Eq. 5.19) was modified such that it describes flow in a packed bed of spheres (see Sect. 2.2.1.2). For a simplified model with mono-sized spheres, it is straightforward to determine the specific surface area of the spheres (SP, with P for particle) per solid volume (VP) of the spheres (aV = SP/VP). In order to obtain the specific surface area per total volume of the porous material (SV = SP / Vtot), a correction for the solid volume fraction (1 − ε) is required, i.e.,

$$S_{V} = a_{V } \left( {1 - \varepsilon } \right),$$
(5.20)

In geometrical models for non-spherical particles, an additional shape correction factor (cC, the so-called Carman factor) was introduced by Carman [25]. At the same time, Kozeny’s correction factor (Eq. 5.15) for tube shape becomes redundant, and thus cK is replaced with the constant 2. This leads to Carman’s characteristic length term

$$L_{char} = \frac{1}{2} \left( {\frac{{c_{C} \varepsilon }}{{a_{V} \left( {1 - \varepsilon } \right) }}} \right)^{2}$$
(5.21)

for granular media consisting of mono-sized objects. According to Carman [25], the shape factor (cC) takes values in the range from 1 (for spheres) down to 0.28 (for platy minerals, mica). The surface-to-volume ratio (av) for packing of mono-sized objects (spheres, particles) is often written in terms of the particle diameter (av = 6/Dp), which then leads to

$$L_{char} = \frac{{c_{C}^{2} D_{p}^{2} \varepsilon^{2} }}{{72\left( {1 - \varepsilon } \right)^{2} }}.$$
(5.22)

Kozeny [24] correctly recognized that the pressure gradient in granular media must be corrected for the effect of path length and, therefore, tortuosity was introduced for this correction (see Eq. 5.17). However, it was Carman [25] who realized that the increase of path lengths also has an effect on the computed (superficial) flow velocity, and he therefore extended Dupuit’s relationship (cf. Eq. 5.16) for the influence of tortuosity, which leads to

$$v_{s} = v_{c} \varepsilon \frac{1}{{\tau_{hydraulic} }}.$$
(5.15b)

In this way, tortuosity was introduced a second time in the M-factor by Carman [25], which leads to

$$M_{pred} = \frac{\varepsilon }{{{\uptau }_{hydraulic}^{2} }}.$$
(5.23)

Thereby, τ2 is also called tortuosity factor (T).

Combining the M-factor (Eq. 5.23) with the characteristic length term (Eq. 5.21) gives the full Carman-Kozeny equation

$$\kappa_{C - K} = \frac{1}{2} \left( {\frac{{c_{C} \varepsilon }}{{a_{V} \left( {1 - \varepsilon } \right) }}} \right)^{2} \frac{\varepsilon }{{\tau_{hydraulic}^{2} }} = \frac{{ c_{C}^{2} D_{p}^{2} \varepsilon^{2} }}{{72\left( {1 - \varepsilon } \right)^{2} }} \frac{\varepsilon }{{\tau_{hydraulic}^{2} }},$$
(5.24)

which describes permeability for a packed bed of mono-sized particles. In the appendix of [25], a geometrical model for packed spheres is presented, which enables one to estimate the length of flow streamlines. Based on these geometric considerations, Carman argued in [25] that the streamline tortuosity in most granular media must be close to √2. Following this argumentation, the value for tortuosity is thus often fixed at τ = √2. The M-factor for flow then simplifies to ε/2. In the simplest form for mono-sized spheres (cC = 1) the Carman-Kozeny equation reduces to

$$\kappa_{C - K} = \frac{1}{2}\left( {\frac{\varepsilon }{{S_{V} }}} \right)^{2} \frac{\varepsilon }{2} = \frac{{ D_{p}^{2} \varepsilon^{2} }}{{72\left( {1 - \varepsilon } \right)^{2} }} \frac{\varepsilon }{2}.$$
(5.25)

Note that the Carman-Kozeny equation was developed at a time when methods for 3D imaging and image analysis were not yet available and therefore morphological descriptors were used, which are relatively easy to access (ε, SV, Dp, τ = √2). Still nowadays, this equation is widely used by the research community. However, it must be emphasized that the applicability of the Kozeny (Eq. 5.19) and the Carman-Kozeny (Eq. 5.25) equations are limited to simple microstructures such as the bundle-of-tubes model and the (mono-sized) packed-spheres model. Already for relatively small deviations from these idealized geometries (e.g., non-circular tubes or non-spherical grains) specific correction factors (cK, cC) must be fitted, which introduce considerable uncertainties. For more complicated materials with microstructure architectures that are significantly different from sphere or particle packing, the prediction power of the Carman-Kozeny equation decreases drastically.

Despite these drawbacks, the Carman-Kozeny equation (Eq. 5.25) is often applied also for the study of more complex microstructures such as dispersed granular materials and even foams and fibrous materials. However, it was shown by many authors that the predictions obtained by the Carman-Kozeny equation are highly uncertain for such complex microstructures [29,30,31,32]. Big efforts were undertaken to modify the Carman-Kozeny equation in order to improve the prediction power also for more realistic (complex) microstructures, e.g., in [27, 28]. In principle, most of these modifications still use the same relatively simple morphological descriptors (ε, SV, Dp, τ = √2).

In this context, it is worth to critically consider, which microstructure effects are reliably captured with the Carman-Kozeny equation, and which are not. From the above description, it can easily be recognized that the M-factor in the Carman-Kozeny equation is rather simple. Important microstructure effects resulting from the variation of path-lengths (constant tortuosity, τ = √2) and bottlenecks (no constrictivity included) are not captured accurately. However, the strength of the Carman-Kozeny equation is clearly the description of characteristic length and hydraulic radius, respectively, which enables to capture the wall friction effects in (mono-sized) granular media quite well.

5.3.3 Determination of Characteristic Length and M-factor by Laboratory Experiments

Katz and Thompson (1986) [33] presented an experimental solution for measuring characteristic length and M-factor (Mexp), which enables to predict permeability of complex porous media. In order to measure the characteristic length, it was proposed to use mercury intrusion porosimetry (MIP). The pore size distribution curve (MIP-PSD) typically shows a steep rise, the corresponding radius of which can be interpreted as ‘break-through radius’ (rMIP). When the pressure is raised to the break-through range, a large portion of the pore space is filled almost instantaneously with liquid mercury. The domain, which is filled with mercury, thus represents a contiguous pore network. Katz and Thompson [33] defined the inflection point of the MIP-PSD curve (convex-concave transition) as break-through radius (rMIP). They argued that rMIP is a characteristic quantity of the pore network, which has a significant influence on flow and permeability, and which can thus be interpreted as an equivalent of the hydraulic radius (rhc). Based on experimental evidence, a constant of 1/226 was determined in [33] as part of this definition of the characteristic length, i.e.,

$$L_{char} = \frac{{r_{hc}^{2} }}{2} = \frac{{r_{MIP}^{2} }}{226}.$$
(5.26)

Moreover, it was argued in [33] that there are additional effects from pore morphology and connectivity, which may have the same limiting influence on flow as they have on electrical conductivity (i.e., non-viscous/non-frictional effects). Consequently, it has been proposed in [33] that the M-factor for flow could be determined based on experimental measurements of effective electrical conductivity (i.e., using porous media saturated with an electrolyte, whereby σ0 denotes the intrinsic conductivity of the electrolyte). Doing so, an experimental M-factor was obtained, which is defined by

$$M_{exp} = \frac{{\sigma_{eff} }}{{\sigma_{0} }} = \sigma_{rel} .$$
(5.27)

Katz and Thompson [33] thus proposed to predict permeability from Lchar and Mexp, using the relationship

$${\upkappa }_{Katz - Thompson} = L_{char} \frac{{\sigma_{eff} }}{{\sigma_{0} }} = \frac{{\left( {{\text{r}}_{MIP} } \right)^{2} }}{226} \frac{{\sigma_{eff} }}{{\sigma_{0} }}.$$
(5.28)

Note that both characteristics, Lchar and Mexp, are easily accessible with standard experimental methods.

5.3.4 Determination of Characteristic Length and M-factor by 3D Image Analysis

With the advent of 3D imaging at sub-μm resolution (e.g., by FIB-SEM tomography in 2004 [4]), it became possible to quantify specific morphological characteristics in complex microstructures. Hence, direct geometric and mixed tortuosities are nowadays accessible from 3D image analysis and can be used to describe the effects resulting from variations of path lengths. Similarly, constrictivity (β) is accessible and can be used to describe the bottleneck effect. Based on these characteristics (ε, β, τ), new expressions for the relationship between microstructure characteristics and conductivity/diffusivity could be established. As described in Sect. 5.2, Mpred_conductivity was determined by Stenzel et al. [10, 11] by using modern methods of stochastic geometry, virtual materials testing and statistical error minimization (compare Eq. 5.2b: Mpred = εa βb/τgeodesicc, with a = 1.15, b = 0.37, c = 4.39).

In analogy to the paper of Katz and Thompson [33] reviewed in Sect. 5.3.3, Holzer, et al. [13] argued that the M-factor of conductivity from Stenzel et al. [10] (Eq. 5.2b) can be used as a first approximation for the M-factor of flow. In addition, for the effective lengths term, two different definitions for hydraulic radius (rhc_I, rhc_II) were proposed in [13]. Note that the two different definitions for hydraulic radius led to two different equations for the prediction of permeability (кpred_I, кpred_II).

The first approach presented in [13] uses the classical definition of the hydraulic radius (i.e., the ratio of porosity over specific surface area per volume). However, in contrast to the initial Carman-Kozeny approach, specific surface area (SV) is not determined from the characteristic sphere diameter (Dp), but it is determined directly from the complex microstructures using 3D image analysis, i.e.,

$$r_{hc\_I} = x_{I} \frac{\varepsilon }{{S_{V} }},$$
(5.29)

where XI is a fitting parameter (Note: XI is treated as a constant that is independent from pore morphology). Based on detailed investigations of sintered porous ceramics in [13], the effective properties (к, σeff) were determined by experiments (кexp), pore scale simulation (кsim) as well as 3D imaging/image analysis (кpred_I, SV, β, ε, τdir_geodesic). By error minimization (кpred_I − кsim and кpred_I − кexp) a value of XI = √2 was estimated. This leads to

$$L_{char\_I} = \frac{{{\varvec{r}}_{hc}^{2} }}{8} = \frac{1}{4} \left( {\frac{\varepsilon }{{S_{V} }}} \right)^{2}$$
(5.30)

as a description of the characteristic length. Permeability (кpred_I) can thus be obtained from the combination of Lchar_I (Eq. 5.30) with the M-factor for conductivity from Stenzel et al. [10] (Eq. 5.2b) by

$$\kappa_{pred\_I} = \frac{1}{4} \left( {\frac{\varepsilon }{{S_{V} }}} \right)^{2} \frac{{\varepsilon^{1.15} \beta^{0.37} }}{{\tau_{dir\_geodesic}^{4.39} }}.$$
(5.31)

For the second approach considered in [13], it was argued that the hydraulic radius can also be defined as convex combination of the mean size of bottlenecks (rmin) and the mean size of pore bulges (rmax), i.e.,

$$r_{hc\_II} = x_{II} r_{min} + \left( {1 - x_{II} } \right) r_{min} .$$
(5.32)

(We refer to Münch and Holzer [5] for the determination of rmin and rmax, respectively). Using results from 3D image analysis (кpred_II, rmin, rmax, β, τdir_geod), numerical simulation (кsim) and experimental characterization (кexp, for validation) as well as applying error minimization, a value of 0.5 was obtained for xII, which leads to

$$L_{char\_II} = \frac{{r_{hc}^{2} }}{8} = \frac{{\left( {0.5{ }r_{min} + 0.5{ }r_{max} } \right)^{2} }}{8}.$$
(5.33)

Permeability (кpred_II) is thus predicted by a combination of Lchar_II (Eq. 5.33) with the M-factor for conductivity (see Eq. 5.2b). More precisely,

$$\kappa_{pred\_II} = \frac{{\left( {0.5r_{min} + 0.5r_{max} } \right)^{2} }}{8} \frac{{\varepsilon^{1.15} \beta^{0.37} }}{{\tau_{dir\_geodesic}^{4.39} }}.$$
(5.34)

Both approaches (кpred_I, кpred_II) were tested with fibrous materials of a gas diffusion layer (GDL) in PEM fuel cells [14]. In-situ time-lapse tomography (μ-CT) was used to capture the changing 3D water-distribution upon ongoing imbibition. A good agreement was obtained between the predicted permeabilities (кpred_I, кpred_II) based on 3D characterization with simulated permeabilities (кsim) based on a numerical 3D flow model. Thereby, the predictions obtained by кpred_II (Eq. 5.34) resulted in smaller differences to кsim, compared to the predictions by кpred_I (Eq. 5.31).

Note that the prediction formula for permeability кpred_II (Eq. 5.34) presented by Holzer et al. [13] is similar to the prediction proposed by Katz and Thompson [33], in the sense that both approaches use MIP-PSDs (rmin and rMIP, respectively) for determining the hydraulic radius (cf. Eqs. 5.26 and 5.32). Furthermore, in both approaches, the M-factor is determined from effective/relative conductivity (cf. Eqs. 5.2b and 5.27). The difference is that physical experiments are used by Katz and Thompson [33], while the approach by Holzer et a l. [13] is based on 3D image analysis.

5.3.5 Determination of Characteristic Length and M-factor by Virtual Materials Testing

Using the same expressions as in Holzer et al. [13] for rhc_I (= xI ε/SV), rhc_II (= xII rmin + (1 − xII) rmax) and Mpred (=εaβb/τc), the corresponding constants and exponents (xI, xII, a, b, c) were determined recently by means of stochastic geometry and virtual materials testing (see Neumann et al., 2020 [12]). Thereby, the 8119 different 3D microstructures from Stenzel et al. [11] served as a basis for big data analysis. It must be emphasized that in this approach the results obtained with respect of the fitting of constants used in rhc (xI or xII) and of the exponents used in Mpred (a, b, c) are not independent of each other, since the fitting is performed with one simultaneous error minimization procedure. The resulting M-factor is thus specifically fitted for flow and permeability, respectively (i.e., Mpred_K). This approach is thus more specific than the permeability predictions of Holzer et al. [13] and Katz and Thompson [33] (see previous sections), where the M-factors are derived from electrical conductivity (i.e., Mpred_conductivity).

Using virtual materials testing (i.e., property prediction by 3D analysis and numerical simulation) and error minimization, Neumann et al. [12] obtained a constant of xI = 2.08 for the classical definition of the hydraulic radius (rhc_I = xI ε/SV). The resulting description of the characteristic length (Lchar) is thus very similar to the one from Kozeny [24] for circular pipes with cK = 2, i.e.,

$$L_{char\_I} = \frac{{r_{hc}^{2} }}{8} = = \frac{1}{8} \left( {2.08\frac{\varepsilon }{{S_{V} }}} \right)^{2} = 0.54 \left( {\frac{\varepsilon }{{S_{V} }}} \right)^{2} = \frac{1}{{c_{K} }} \left( {\frac{\varepsilon }{{S_{V} }}} \right)^{2} .$$
(5.35)

Furthermore, for the prediction of permeability considered by Neumann et al. [12], the exponents of Mpred_KI are significantly different from those in Mpred_cond for conductivity (see Eq. 5.2b, proposed in Stenzel et al. [11]). The fitting revealed a higher exponent for porosity and lower exponent for tortuosity, which leads to

$$M_{pred\_\kappa I} = \frac{{\varepsilon^{3.56} \beta^{0.78} }}{{\tau_{dir\_geodesic}^{1.67} }}.$$
(5.36)

The full equation for the prediction of permeability (кpred_I) is then given by

$$\kappa_{pred\_I} = 0.54\left( {\frac{\varepsilon }{{S_{V} }}} \right)^{2} \frac{{\varepsilon^{3.56} \beta^{0.78} }}{{\tau_{dir\_geodesic}^{1.67} }}.$$
(5.37)

For the second case, where the hydraulic radius is determined based on pore size analysis (i.e., rhc_II = xII rmin + (1 − xII) rmax), the virtual materials testing revealed a relatively high value of 0.94 for XII (compared to 0.5 that was estimated in [13]). The corresponding characteristic length is then given by

$$L_{char\_II} = \frac{{r_{hc}^{2} }}{8} = \frac{{\left( {0.94r_{min} + 0.06r_{max} } \right)^{2} }}{8}.$$
(5.38)

This result indicates that the hydraulic radius is almost identical with the mean radius of bottlenecks (rmin from MIP-PSD), which is very similar to the definition of the hydraulic radius proposed by Katz and Thomson [33], where rhc ≈ rMIP.

The M-factor (Mpred_KII) for permeability that is obtained from the fitting procedure for кpred_II is quite different to the M-factors for кpred_I (see Eq. 5.36) and for conductivity (see Eq. 5.2b), i.e.,

$$M_{pred\_\kappa II} = \frac{{\varepsilon^{2.14} \beta^{ - 0.05} }}{{\tau_{dir\_geodesic}^{2.44} }} \approx \frac{{\varepsilon^{2.14} }}{{\tau_{dir\_geodesic}^{2.44} }} .$$
(5.39)

For кpred_KII the exponent for constrictivity (βb) is slightly below 0, which is counterintuitive from a physical point of view. In [12], it was shown that the statistical error (MAPE) for кpred_KII is almost identical when comparing Eq. 5.39 with β−0.05 and Eq. 5.39 without constrictivity (i.e., the case β0). Hence, constrictivity drops out from Eq. 5.39. This can be explained by the fact that in кpred_KII, the bottleneck effect is already contained (as rmin) in the characteristic length term (Lchar_II). The equation for кpred_KII thus becomes

$$\kappa_{pred\_II} = \frac{{\left( {0.94r_{min} + 0.06r_{max} } \right)^{2} }}{8} \frac{{\varepsilon^{2.14} }}{{\tau_{dir\_geodesic}^{2.44} }}.$$
(5.40)

The statistical analysis performed by Neumann et al. [12] shows that the prediction powers of кpred_I and кpred_II are almost identical. The MAPE is 34.5% for both permeability approaches.

5.4 Summary

The virtual materials testing approach presented by Neumann et al. [12] is based on a statistical analysis of more than 8000 different 3D scenarios from stochastic microstructure modeling, which cover a wide range of microstructures and effective properties. Due to this large data basis, the proposed equations have a rather general character, since they are capable to predict permeability for various kinds of materials even with very complex microstructures. For example, μCT-data from cellular, foam like-structures was used in [12] to demonstrate the high prediction power of Eqs. 5.37 and 5.40 for materials, which have not been used to fit the parameters in these prediction formulas.

For comparison, the Kozeny and Carman-Kozeny equations were derived from parallel-tube and packed-spheres models with idealized geometries. Consequently, the prediction powers of these traditional equations are strongly limited and not really applicable to more complex microstructures. An important difference of recently proposed expressions compared to the traditional Carman-Kozeny approach is the introduction of constrictivity in the M-factor. More precisely, it is one of the main shortcomings of the Carman-Kozeny approach, that the limiting effect resulting from narrow bottlenecks is not properly addressed.

The equations for the prediction of permeability (and conductivity) have also improved due to a better description of path length effects. Carman [25] proposed to use a constant value of √2 for τhydraulic. In the approach proposed by Neumann et al. [12], geodesic tortuosity is used. For the 8000 3D microstructures, geodesic tortuosity varies between 1.05 and 2.4 (see Figs. 3.10b and 3.10c). The variation of τdir_geodesic is particularly large (1.2–2.4) for structures with low porosity (ε < 0.25). Hence, neither can tortuosity be considered as a constant (√2), as proposed by Carman [25], nor is tortuosity a simple function of porosity, as proposed in widely used tortuosity-porosity relationships (e.g., the Bruggeman relationship, see also discussion in Chap. 3).

For complex microstructures the permeability can only be predicted in a reliable way with suitable descriptions of geometric or mixed tortuosity and other relevant characteristics (effective porosity, constrictivity, hydraulic radius) gained from 3D analysis, see also recent study by Prifling et al. [23], where large-scale statistical learning has been performed for the prediction of permeability in porous materials using 90,000 artificially generated microstructures.

Modern methodologies of 3D analysis open new possibilities for the precise characterization of all transport relevant microstructure characteristics (i.e., ε, β, τ, rh, rmin, rmax, SV). Based on these characteristics, the effective transport properties (conductivity, diffusivity, permeability) can be predicted with a high prediction power.

The most important micro–macro relationships for prediction of effective conductivity and diffusivity are Eq. 5.2b (precise for microstructures with M < 0.7) and Eq. 5.8b (precise for high porosity materials with M > 0.7). For prediction of permeability, Eqs. 5.37 and 5.40 have the highest prediction power.

These four equations are all based on the direct geodesic tortuosity, which is independent from the transport process. It is possible, that the prediction power can be further improved when a mixed physics-based tortuosity is used (i.e., τmixed_phys_Vav), which combines the transport specific information with the precise geometric analysis of the corresponding path lengths.

It must be emphasized, that the benefit of these quantitative expressions is not only that they can be used to estimate the effective transport properties of porous media. This task can also be fulfilled with dedicated experiments or with numerical simulations. But the micro–macro relationships very much help to understand, which microstructure feature and which microstructure effect (i.e., pore volume fraction, constrictive bottlenecks, tortuous pathways, viscous drag at pore walls) represents the dominant transport limitation. In this way, these expressions provide important information to materials engineers, which is necessary for a purposeful optimization of the microstructure.