Tortuosity (τ) is widely recognized as a key concept for transport in porous media, which describes the impact of pore structure on the effective transport properties. Tortuosity is a dimensionless parameter that depends on the windedness of transport pathways. The increase of path lengths due to tortuous pore morphology can contribute significantly to the transport resistance. This concept was introduced almost 100 years ago by Kozeny in 1927 [1]. Since then, tortuosity has been widely applied in various research disciplines, such as chemical engineering, geoscience, materials science, and life science. Uncountable studies dealing with tortuosity have been performed, using different methodologies (physical theory, laboratory experiments, numerical simulations, 3D imaging and image processing) and combinations thereof. It is beyond the scope of this book to present a detailed review of all these studies. For this purpose, we refer to excellent review articles on tortuosity in specific (e.g., by Clennell [2], Ghanbarian et al. [3], Shen and Chen [4], Tjaden et al. [5]) and on transport in porous media in general (e.g., [6,7,8,9,10,11,12,13,14,15]).

In order to explain the focus of this paper, it must be emphasized that until now there exists no unifying theory for the tortuosity concept. Therefore, the discussion of tortuosity bears considerable potential for confusion. Many different definitions of tortuosity have been presented, depending either on the characterization method (direct geometric 3D analysis by tomography and image processing vs. indirect calculation from effective properties) and/or on the underlying transport mechanism (flow, diffusion and conduction). Confusion is amplified by the fact that many different tortuosity-terms are in use (see Table 1.1). Unfortunately, in many cases there exists no clear definition for these terms and moreover, a globally accepted classification scheme as well as a systematic nomenclature for the different tortuosity types are missing.

Table 1.1 List of tortuosity (Ï„) terms from literature
Full size table

As will be discussed in this book, the different tortuosity terms have distinct meanings and can therefore not be used interchangeably. However, the meaning of a specific tortuosity term is often strongly related to the methodology by which tortuosity is determined. Therefore, the topic of tortuosity must be discussed in context with the corresponding methodologies (i.e., 3D imaging and image analysis, transport simulation or laboratory experiments) and their continuing development.

Initially, the basic theories on tortuosity (e.g., Carman-Kozeny equations) were developed at a time when direct measurement of tortuosity by means of tomography and 3D image analysis was not possible. Therefore, tortuosity was determined indirectly—usually from effective transport properties that were measured experimentally. This led to a certain gap between theoretical descriptions, which are based on considerations of path lengths in simplified geometric models (e.g., in bundles of tubes or in packed spheres), and empirical investigations, which derive tortuosity values indirectly from bulk effective properties. Hence, different definitions for tortuosity evolved over time, depending on the basic approach (theory vs. experiment vs. modeling), depending also on the field of research and on an associated ‘school of thinking’ (e.g., petro-physics vs. electrochemistry), and depending also largely on the availability of certain characterization techniques (e.g., computational methods for pore scale modeling or techniques for 3D analysis by tomography and image processing).

Over the last two decades significant progress was achieved in high-resolution tomography as well as in stochastic modeling and numerical simulation of 3D image data representing the morphology of microstructures. These methodological improvements open new possibilities for studying microstructure-property relationships, in general, as well as for measuring tortuosity directly from the microstructure by means of 3D analysis and transport simulation. Due to the availability of new methods, it is now possible to compare different tortuosity concepts and establish correlations between the different tortuosity types. These new possibilities are the basis for the present review, which is structured as follows:

In Chap. 2, the classical theories and concepts of tortuosity (starting with the Carman-Kozeny equations), as well as the underlying definitions for the most important tortuosity types are presented in a chronological (historical) order. At the end of Chap. 2, a new classification-scheme is introduced together with a systematic tortuosity-nomenclature. Three main categories are distinguished: direct geometric tortuosities, indirect physics-based tortuosities and mixed (i.e., geometric and physics-based) tortuosities. This classification may help to avoid confusion in future debates.

In Chap. 3, empirical data from literature is collected and compared. The collection includes more than 2000 data-points (i.e., tortuosity-porosity-couples) from 70 studies in various fields such as geology, battery and fuel cell research. Thereby, experimental approaches are considered as well as investigations that are based on computational modeling and simulation. The collection of literature data represents the basis for an empirical description, which shows how tortuosity varies for different types of materials and microstructures. Furthermore, in many of these studies different types of tortuosities are measured for the same materials. These datasets enable to define a relative order among the different tortuosity types. More precisely, it turns out that for a given material, the values of certain types of tortuosities tend to be systematically lower than the values of other tortuosity types. This comparison of different tortuosity types results in a surprisingly clear and consistent pattern.

In Chap. 4, modern methods for microstructure characterization and associated calculation approaches for tortuosity are reviewed. Chapter 4 is structured according to the workflow, which is typical for this kind of microstructure characterization. First, an overview of modern tomography methods is presented with a special emphasis on recent innovations and on current trends. Subsequently, calculation approaches by image analysis and by transport simulation are discussed for all three tortuosity categories: direct geometric, indirect physics-based and mixed tortuosities. In addition, an extensive list with available software packages for image processing, which include codes for the computation of specific tortuosity types, is presented. Finally, modern methods of stochastic geometry used for virtual materials testing are discussed in context with their applications in Digital Materials Design (DMD) and Digital Rock Physics (DRP), which are all strongly associated with the investigation of tortuosity.

In Chap. 5, it is discussed how the recent progress in tomography, 3D image analysis, microstructure modeling and virtual materials testing can be used for a thorough understanding of microstructure-property relationships. Based on modern 3D characterization techniques, the effects from tortuous pathways can now be distinguished from other microstructure effects, such as the limitations arising from narrow bottlenecks and from the friction at pore walls. New morphological descriptors were introduced for the bottleneck effect (i.e., constrictivity), for the wall friction effect (i.e., hydraulic radius) and also for the path length effect (i.e., tortuosity). Consequently, new formulas describing the complex relationships between microstructure and effective transport properties have been established recently. The evolution of morphological descriptors and associated formulas describing the micro–macro relationships are reviewed in Chap. 5. For porous media with random microstructures, these new formulas have a higher prediction power compared to traditional equations from the literature, such as e.g., the Carman-Kozeny equation for viscous flow.

Finally, a summary and conclusions are presented in Chap. 6.