1 Introduction

The internal structure of porous materials, including the geometry and the connectivity of pores, has a direct influence on the flow of fluid in the medium, and it should be taken into account when determining the permeability of such a material. One of the parameters that are often used is tortuosity, which was first introduced to porous media studies by Carman (1937).

Despite the extensive use of the notion of tortuosity in describing flow in porous media, its definition or interpretation is not unique. The geometrical tortuosity \((T_\mathrm{Geo})\) is the average path length per unit streamwise displacement of all fluid particle in a representative element volume (REV), disregarding the different velocities of particles moving along these path lines. It can be alternatively interpreted as the ratio between the shortest path of interconnected points in pore fluid space to the straight distance between these points (Bear 1972). The kinematic tortuosity \((T_\mathrm{Kin})\) is the average distance travelled per streamwise displacement in a time interval \([t,t+\delta t]\) by all fluid particles in the REV at time t (Clennell 1997; Duda et al. 2011). The dynamic tortuosity \((T_\mathrm{Dyn})\) is the average distance travelled per streamwise displacement in a time interval \([t,t+\delta t]\) by all fluid particles in the REV within the time interval \([t,t+\delta t]\) (Johnson et al. 1987; Bear 1972; Scheidegger 1972). On the other hand, the hydraulic tortuosity factor \(\tau _\mathrm{KC}\) in the Kozeny–Carman equation (Carman 1937; Kozeny 1927) is the ratio of the effective hydraulic path length \((L_\mathrm{e})\) to the straight line distance (L) in the flow direction. Even though \(\tau _\mathrm{KC}\) may be very different from the geometrical tortuosity (Clennell 1997), \(\tau _\mathrm{KC} =L_\mathrm{e} /L=1/T_\mathrm{Geo}\) is generally accepted for one-dimensional streamwise flow. Nevertheless, the geometrical tortuosity is important as it facilitates the determination of average flow velocity streamwise (as discussed by Carman 1937). In this meaning, the geometrical tortuosity can be related to both the kinematic and the dynamic tortuosities. It should be noted that tortuosity is also defined for other phenomenon in porous media, such as electrical tortuosity and diffusive tortuosity (see, e.g., Dullien 1992; Ghanbarian et al. 2013).

According to the capillaric model (Scheidegger 1972), for a porous medium having a series of parallel straight capillaries of arbitrary cross-sectional shapes, based on Hagen–Poiseuille equation for laminar Newtonian flow, the permeability of the porous medium can be expressed as

$$\begin{aligned} k=\frac{\phi D_\alpha ^2}{16\psi }=\frac{\phi D_\alpha ^2 }{16C_0 \tau ^{2}} \end{aligned}$$
(1)

where \(\phi \) is the porosity of the material, \(D_\alpha \) is the “average” hydraulic diameter, \(C_0\) is a shape factor (\(C_0 =2\) for a circular pipe flow and 3 for flow between two parallel plates), and \(C_0 =2\)–3 can be used for granular soils in general (Dullien 1992). The tortuosity factor \(\tau \) is the same as the Kozeny–Carman hydraulic tortuosity factor \(\tau _\mathrm{KC}\). The Kozeny–Carman constant \(\psi =C_0 \tau ^{2}\) is a function of pore structure, including pore shape and connectivity. The Kozeny theory yields a same expression for permeability (Kozeny 1927; Scheidegger 1972), with the hydraulic diameter being considered as that for a virtual stream tube. More specifically, \(D_\alpha \) is considered as the average pore size on a plane perpendicular to stream lines and is determined as the ratio between the total area of pore cross sections to their circumferences.

By using the volume averaging technique, Bear and Bachmat (1986, 1990) derive the tortuosity tensor and the permeability tensor for porous media. More specifically, the tortuosity tensor is the static moment of oriented areal elements of fluid on the boundary of the REV per unit volume of the fluid phase within the REV. For some specific cases, Diedericks and Du Plessis (1995) modifies Bear and Bachmat’s definition of tortuosity tensor by incorporating the differences between the macroscopic flow and the local streamwise directions. Unfortunately, the theoretically derived permeability tensor by Bear and Bachmat (1986, 1990) may not be applicable any more when using the definition by Diedericks and Du Plessis (1995).

Even though numerous studies have been performed on flow in porous media, the term tortuosity is still a source of misunderstanding and sometimes causes confusions. The objective of this paper is to identify the connection between the tortuosity defined by Bear and Bachmat and the geometrical tortuosity, the kinematic tortuosity and the hydraulic tortuosity factor in the Kozeny–Carman equation. To achieve this goal, the microscopic analysis of transport phenomena in porous media carried out by Bear and Bachmat (1986, 1990) is revisited first. A modification to the tensorial quantity \(\alpha _{ij}\), which is a second-rank tensor characterizing the geometry configuration of the particle–fluid interfaces in terms of its unit normal vector, and hence the permeability tensor is made to better account for the characteristics of the fluid–solid interfaces. The results show intrinsic connection between different definitions of tortuosity and how the interaction between the fluid and solid phases has significant influence on the permeability tensor. The anisotropy of tortuosity is characterized using the directional distribution of pore voids. The lower and upper bounds of the tortuosity of granular soils are obtained from the analyses. The results are found generally consistent with published data in the literature.

2 Review of Bear and Bachmat’s Homogenization Approaches

2.1 Two Versions of Bear and Bachmat’s Momentum Balance Equation

In the original work of Bear and Bachmat (1986, 1990), the momentum balance equation for flow in a saturated granular material is described using an approach of local volume averaging that takes into account the mass balance on both macro- and micro-levels. However, Bear and Bachmat (1986, 1990) provide two versions of analyses, which yield different results for the influence of geometry and internal structure of a porous material.

Fig. 1
figure 1

a REV of a porous medium and b a unit sphere

Full size image

For a unit sphere enclosing a representative element volume (REV) of the material consisting of \(\alpha \)-(fluid) and \(\beta \)-(solid) phases as shown in Fig. 1a, let \(V_0\) be the volume of the REV, \(S_{0}\) the surface area of the sphere (with the radius of \(R = 1\)) encompassing the REV, \(S_{\alpha \alpha }\) the area around the REV that intersects the fluid, \(S_{{\beta }{\beta }}\) the area around the REV that crosses the particles, and \(S_{\alpha {\beta }}\) the total area of the particles/fluid interfaces in the REV. The total areas enclosing the unit sphere and the fluid are then \(S_0 = S_{\alpha \alpha } + S_{\beta \beta }\) and \(S_{0\alpha } = S_{\alpha \alpha } + S_{\alpha \beta }\), respectively. Bear and Bachmat (1986, 1990) define the tortuosity tensor \(T_{\alpha ij}^*\) as

$$\begin{aligned} T_{\alpha ij}^*=\frac{1}{V_\alpha }\int _{S_{\alpha \alpha }} {\mathring{x}_{i} \nu _{\alpha j} \mathrm{d}S} \end{aligned}$$
(2)

where \(\mathring{\mathbf{x}} =\mathbf{r}-\mathbf{r}_0\) in which \(\mathbf{r}\) and \(\mathbf{r}_0\) are the position vectors of a point on \(S_{\alpha \alpha }\) and the center of the REV, respectively; \(\nu _{\alpha j}\) is the unit vector defining the outer normal of \(S_{\alpha \alpha }\), as illustrated in Fig. 1a. \(\mathbf{T}_\alpha ^{*}\) expresses the total static moment of the oriented areal element comprising the \(S_{\alpha \alpha }\)-surface, with respect to planes passing through the centroid of the REV, per unit volume of the \(\alpha \)-phase within \(V_0\). Mathematically, the tortuosity tensor \(T_{\alpha ij}^{*}\) is the outcome of the modified rule (Bear and Bachmat 1986; Bear and Sorek 1990) for averaging the spatial derivative of fluid pressure \(p^{\alpha }\), based on the notion of microscopic zero Laplacian \(\nabla ^{2}p^{\alpha }=0\) in \(V_\alpha \) (i.e., p does not have an extremum distribution). Physically, Bear and Bachmat (1990) interpret \(T_{\alpha ij}^{*}\) as a quantity that “transforms the local body force into the macroscopic one.”

Based on the volume averaging, the macroscopic momentum balance equation of the fluid in a porous medium is expressed as

$$\begin{aligned} \phi \bar{{\rho }}^{\alpha }\frac{D_\alpha \overline{V_j^m }^\alpha }{Dt}=\frac{\partial \phi \overline{\tau _{\alpha ij} }^\alpha }{\partial x_i}+\frac{1}{V_0}\int _{S_{\alpha \beta }}\tau _{\alpha ij} \nu _{\alpha i} \mathrm{d}S-\phi \left( \frac{\partial \bar{{p}}^{\alpha }}{\partial x_i}+\bar{{\rho }}^{\alpha }g\frac{\partial z}{\partial x_i} \right) T_{\alpha ij}^{*} \end{aligned}$$
(3)

where \(\bar{{\rho }}^{\alpha }\) is the average mass density of the \(\alpha \)-(fluid) phase, \(\bar{{p}}^{\alpha }\) stands for the pressure, and \(\tau _{\alpha ij}\) is the viscous stress tensor that is the deviator of the stress tensor. The surface integral in the second term of the r.h.s. of Eq. (3) expresses “the viscous resistance, or viscous drag force exerted by the solid phase on the flow fluid at their contact surface within the REV, per unit volume of porous medium” (Bear and Bachmat 1990).

It should be noted that Eq. (3) was obtained by Bear and Bachmat (1990) assuming for the Navier–Stokes microscopic momentum balance that drag and inertial forces at the solid–fluid interface are much smaller than surface and body (gravity) forces. By extending the assumption made by Bear and Bachmat (1990), Levy et al. (1999) and Sorek et al. (2005) extended Eq. (3) when accounting for other Navier–Stokes microscopic terms through the solid–fluid interface. More specifically, an inertial term, in which the Forchheimer tensor is used to address the exchange of inertia at the microscopic solid–fluid interface, is added to Eq. (3).

Bear and Bachmat (1986, 1990) use two methods to estimate the surface integral on the r.h.s of Eq. (3). In Bear and Bachmat (1986), \(\tau _{\alpha ij}\) is determined from the constitutive relation of the fluid as

$$\begin{aligned} \tau _{\alpha ij} =\mu _\alpha \left( {\frac{\partial V_i^m}{\partial x_j}+\frac{\partial V_j^m}{\partial x_i}} \right) \end{aligned}$$
(4)

in which \(\mu _{\alpha }\) is the dynamic viscosity of the fluid and \(V_i^m\) is the fluid velocity vector at any point in the close proximity of the \(S_{\alpha \beta }\)-surfaces. The traction on the \(S_{\alpha \beta }\)-surface is then

$$\begin{aligned} \tau _{\alpha ij} \nu _{\alpha j} =\mu _\alpha \left( {\frac{\partial V_i^m}{\partial x_j}+\frac{\partial V_j^m}{\partial x_i}} \right) \nu _{\alpha j} =\mu _\alpha \left( {\delta _{ij} +\nu _{\alpha i} \nu _{\alpha j}} \right) \frac{\partial V_j^m}{\partial s_v} \end{aligned}$$
(5)

with \(\partial V_i^m /\partial s_v\) being the velocity gradient in the direction perpendicular to the \(S_{\alpha \beta }\)-surfaces. It can be shown that the normal component of the traction on the \(S_{\alpha \beta }\)-surfaces is

$$\begin{aligned} \tau _\alpha ^N =\tau _{\alpha ij} \nu _{\alpha i} \nu _{\alpha j}= & {} \mu _\alpha \frac{\partial V_j^m}{\partial s_v}\left( {\delta _{ij} +\nu _{\alpha i} \nu _{\alpha j}} \right) \nu _{\alpha i}\nonumber \\= & {} 2\mu _\alpha \frac{\partial V_j^m}{\partial s_v}\nu _{\alpha j} \end{aligned}$$
(6)

Equation (6) implies a nonzero normal velocity gradient in the direction \(\varvec{\upnu }_\alpha \) in the immediate vicinity of the \(S_{\alpha \beta }\)-surfaces, which may be questionable for incompressible flow.

The second term of the r.h.s of Eq. (5) is the surface traction component in direction \(\varvec{\upnu }_\alpha \). As a result, the surface integral of the r.h.s of Eq. (3) is computed as

$$\begin{aligned} \frac{1}{{V_{0}}}\int _{{S_{{\alpha \beta }}}} \,\tau _{{\alpha ij}} \nu _{{\alpha i}} \mathrm{d}S= & {} \frac{{\mu _{\alpha }}}{{V_{0} }}\int _{{S_{{\alpha \beta }}}} \,\frac{{\partial V_{i}^{m} }}{{\partial s_{v}}}\left( {\nu _{{\alpha i}} \nu _{{\alpha j}} + \delta }_{{ij}} \right) \nu _{{\alpha i}} \mathrm{d}S \nonumber \\= & {} \mu _{\alpha } \left( {\delta _{{ij}} + {\overline{{\nu _{{\alpha i}} \nu _{{\alpha j}} }}}^{\,{\alpha \beta }}} \right) \frac{1}{{V_{0}}}\int _{{S_{{\alpha \beta }}}} \,\frac{{\partial V_{i}^{m}}}{{\partial s_{v}}}\mathrm{d}S \end{aligned}$$
(7)

Bear and Bachmat (1986) further suggest

$$\begin{aligned} \frac{1}{{V_{0}}}\int _{{S_{{\alpha \beta }}}} \,\frac{{\partial V_{i}^{m}}}{{\partial s_{v}}}\mathrm{d}S \simeq \frac{{\overline{{V_{{si}} |_{{S_{\,{\alpha \beta }}}}}}}^{\,{\alpha \beta }} - {\overline{{V_{i}^{m}}}}^{\,\alpha }}{{\Delta _\mathrm{c} }}\frac{{S_{{\alpha \beta }}}}{{V_{0}}} \simeq \frac{{{\overline{{V_{i}^{m}}}}^{\,\beta } - {\overline{{V_{i}^{m} }}}^{\,\alpha }}}{{\Delta _\mathrm{c}}}\frac{{S_{{\alpha \beta }} }}{{V_{0}}} = C_{\alpha } \phi _{\alpha } \frac{{{\overline{{V_{i}^{m}}}}^{\,\beta } - {\overline{{V_{i}^{\,m} }}}^{\alpha }}}{{\Delta _{\alpha }^{2}}} \end{aligned}$$
(8)

where \({\overline{V_i^m}}^\alpha \) is the average velocity of the \(\alpha \)-phase on the \(S_{\alpha \beta }\)-surface (i.e., the local flow velocity parallel to the \(S_{\alpha \beta }\)-surface) and \({\overline{V_i^m}}^\alpha -{\overline{V_i^m}}^\beta \) is the velocity of the \(\alpha \)-phase relative to the \(\beta \)-phase. The characteristic length \(\Delta _\mathrm{c}\), which is a distance from the solid wall to the interior of the fluid phase in the opposite direction of \(\varvec{\upnu }_\alpha \), is assumed to be \(\Delta _\mathrm{c} =\Delta _\alpha /C_\alpha \) with \(C_\alpha \) being a macroscopic dimensionless shape factor and \(\Delta _\alpha \) the hydraulic radius defined as the ratio of fluid volume to the fluid–solid interface area (i.e., \(\Delta _\alpha =V_\alpha /S_{\alpha \beta }\)). The relation between the hydraulic diameter \(D_\alpha \) and the hydraulic radius \(\Delta _\alpha \) is \(D_\alpha =4\Delta _\alpha \).

In the second method, Bear and Bachmat (1990) assume that the total viscous resistance consists of two parts: (a) the resistance from the internal friction of the fluid and (b) the resistance from the friction at fluid–solid interface, which causes drag on the fluid–solid interface. Different from the expression in Eq. (5), the drag (i.e., the traction) on the fluid–solid interface (at any point in the close proximity of the \(S_{\alpha \beta }\)-surfaces) is determined based on the gradient of tangential velocity as follows:

$$\begin{aligned} \tau _{\alpha ij} \nu _{\alpha j} =\mu _\alpha \frac{\partial V_i^{mT} }{\partial s_v}=\mu _\alpha \frac{\partial V_j^m}{\partial s_v }\left( {\delta _{ij} -\nu _{\alpha i} \nu _{\alpha j}} \right) \end{aligned}$$
(9)

Different from Eq. (7), the average velocity gradient \(\partial V_i^m /\partial s_v\) in the direction perpendicular to the \(S_{\alpha \beta }\)-surfaces is evaluated via the following approach that is different from Eq. (8):

$$\begin{aligned} \frac{1}{{V_{0}}}\int _{{S_{{\alpha \beta }}}} \,\frac{{\partial V_{i}^{m}}}{{\partial s_{v}}}\mathrm{d}S \simeq \underbrace{{\frac{1}{{V_{0}}}\int _{{S_{{\alpha \beta }}}} \,\frac{{V_{{si}}^{*} |_{{S_{{\alpha \beta }}}} - V_{i}^{m} |_{\Delta } }}{\Delta }\mathrm{d}S}}_{{{\text {tangential velocity}}}} \simeq \frac{{\Sigma ^{{\alpha \beta }}}}{{\Delta _\mathrm{c}}}\underbrace{{\left( {{\overline{{V_{{sj}}^{{}} |_{{S_{{\alpha \beta }} }}}}}^{{\alpha \beta }} - {\overline{{V_{j}^{m}}} }^{\alpha }} \right) }}_{\text {Fluid velocity (not tangential)}}\alpha _{{ij}}^{-} \end{aligned}$$
(10)

in which \(\alpha _{ij}^- =\delta _{ij} -{\overline{\nu _i \nu _j}}^{\alpha \beta }\).

In this method, Bear and Bachmat (1990) on the one hand clearly state that \(S_{\alpha \beta }\)-surface is a material surface with respect to the fluid’s mass, and hence, \((V_j^m -V_{sj}|_{S_{\alpha \beta }})v_i =0\) in its immediate neighborhood. Herein a “material surface with respect to an E-quantity” is surface such that no E-quantity crosses it. On the other hand, the local velocity component in the tangential plane of the \(S_{\alpha \beta }\)-surface is determined as the projection of the average fluid velocity on the plane, i.e., \(V_{si}^{*} |_{S_{\alpha \beta }} -V_i^m |_\Delta =({V_j^m -V_{sj} |_{S_{\alpha \beta }}} )({\delta _{ij} -\nu _{\alpha i} \nu _{\alpha j}} )\). Such a treatment implies that the local fluid velocity \(V_i^m\) is not necessarily parallel to the \(S_{\alpha \beta }\)-surface and results in the \(\alpha _{ij}^-\) term in Eq. (10). This introduces some inconsistency on the local level.

The above two approaches by Bear and Bachmat (1986, 1990) yield two versions of the macroscopic momentum balance equation

$$\begin{aligned} {\overline{{\mu }}}_\alpha ^\alpha \alpha _{ij}^\pm C_\alpha \phi _0 \frac{{\overline{{V}}}_{\alpha j}^\alpha -{\overline{{V}}}_{\beta j}^\beta }{\Delta _\alpha ^2}=-\phi _0 T_{\alpha ij}^{*} \left( {\frac{\partial {\bar{{p}}}^\alpha }{\partial x_j}+ {\bar{{\rho }}}^\alpha g\frac{\partial z}{\partial x_j}} \right) \end{aligned}$$
(11)

in which \(\phi _{0}\) is the porosity of the material defined as \(\phi _0 =\phi _\alpha =V_\alpha /V_0\) for saturated porous media and \(C_{\alpha }\) represents a shape factor related to the geometry of pore spaces. The quantities \(\alpha _{ij}^+\) and \(\alpha _{ij}^-\), which are second-rank tensors characterizing the geometry configuration of \(S_{\alpha \beta }\) in terms of its unit normal vector, are defined as

$$\begin{aligned} \alpha _{{ij}}^{\pm } = \delta _{{ij}} \pm \beta _{{ij}} ,\,\,\,\,\,\,\beta _{{ij}} = \frac{1}{{S_{{\alpha \beta }} }}\int _{{S_{{\alpha \beta }}}} \,\nu _{{\alpha i}} \nu _{{\alpha j}} \mathrm{d}S \end{aligned}$$
(12)

where \(\delta _{ij}\) is the Kronecker delta function, \(\nu _{\alpha i}\) the unit normal vector of \(S_{\alpha \beta }\) (as shown in Fig. 1a). The tensor \(\beta _{ij}\) describes the orientation of solid particles. For an assembly of spherical particles, \(\beta _{ij}\) becomes \(\beta _{ij} =\delta _{ij} /3\). Bear and Bachmat (1990) alternatively interpreted \(\alpha _{ij}^\pm \) as a quantity that introduces “the effect of the configuration of the solid–fluid surface in the term that transforms part of the force resisting the flow at a point to an averaged resistance force at the fluid–solid interface.” Finally, the permeability tensor can be readily obtained from Eq. (11) as

$$\begin{aligned} k_{ij} =\frac{\phi _\alpha \Delta _\alpha ^2}{C_\alpha }G_{\alpha ij}^{*} ;\quad G_{\alpha ij}^{*} =\left( \alpha _{il}^\pm \right) ^{-1}T_{\alpha lj}^{*} \end{aligned}$$
(13)

For an isotropic material, \(\alpha _{ij}^\pm \) and \(T_{\alpha ij}^{*}\) in Eq. (13) can be expressed as \({\alpha _{ij}^\pm =a^{\pm }\delta }_{ij}\) and \(T_{\alpha ij}^{*} =T_0^{*} \delta _{ij}\), respectively. It follows that

$$\begin{aligned} k_{ij} =k\delta _{ij} , \quad k=\phi _\alpha \Delta _\alpha ^2 \left( C_\alpha a^{\pm }\right) ^{-1}T_0^{*} \end{aligned}$$
(14)

A comparison of Eq. (14) with the expression of k in Eq. (1) yields an equivalent hydraulic tortuosity factor with the same meaning as that in the Kozeny–Carman equation (Guo 2012):

$$\begin{aligned} \tau _{\mathrm{Bear86}} =\sqrt{a^{+}/T_0^{*}} , \quad \tau _{\mathrm{Bear90}} =\sqrt{a^{-}/T_0^{*}} \end{aligned}$$
(15)

where the subscripts “Bear86” and “Bear90” stand for hydraulic tortuosity factor corresponding to the methods in Bear and Bachmat (1986, 1990), respectively. It has been shown that under some conditions the resulting hydraulic tortuosity factor \(\tau \) in the meaning of \(\tau _{\mathrm{Bear90}}\) may be smaller than unity, which seems questionable (Ahmadi et al. 2011).

2.2 Inconsistency in Tortuosity Computation: Two Simple Examples

As the first example, let us examine a simple case when a flow line crosses the boundary of a porous medium obliquely, as shown in Fig. 2a, b. For a macroscopic flow in the horizontal direction, the different definitions of tortuosity give the same value of \(T_\mathrm{Geo} =T_\mathrm{Kin} =T_\mathrm{Dyn} =\cos \theta =L/L_\mathrm{e}\) with \(L_\mathrm{e}\) being the effective flow length. The correct Kozeny–Carman hydraulic tortuosity factor \(\tau _\mathrm{KC} =L_\mathrm{e} /L=1/\cos \theta \) is also obtained from the analysis of laminar Newtonian flow between two parallel plates. From the definition for tortuosity by Bear and Bachmat in Eq. (2), the tortuosity component of flow for the cases in Fig. 2a, b is \(T_0^*=1\). It is found that, except the tortuosity derived by Bear and Bachmat (1986, 1990), the three other definitions give the same value of tortuosity.

For the macroscopically horizontal flow shown in Fig. 2a, b, the \(\alpha _{ij}^\pm \) tensors defined in Eq. (12) are simplified as scalars \(a^{\pm }\). Referring to Fig. 2a, \(\nu _{\alpha 1} =-\sin \theta \), \(S_{\alpha \beta } =L/\cos \theta \). According to Eq. (12), the component of \(\beta _{ij}\) corresponding to the macroscopically horizontal flow is determined as

$$\begin{aligned} \beta _{11} =\frac{\cos \theta }{L}\int _0^L {\left( {\sin ^{2}\theta \frac{\mathrm{d}x}{\cos \theta }} \right) } =\sin ^{2}\theta \end{aligned}$$

which yields \(a^{\pm }=1\pm \beta _{11} =1\pm \sin ^{2}\theta \). The same result is obtained for flow shown in Fig. 2b. The Kozeny–Carman hydraulic tortuosity factor is then obtained from Eq. (13) by calculating \(k_{11}\), with \(\tau _{\mathrm{Bear90}} =\sqrt{a^{-}/T_0^{*}}=\cos \theta \) based on the method in Bear and Bachmat (1990) and \(\tau _{\mathrm{Bear86}} =\sqrt{a^{+}/T_0^{*} }=\sqrt{1+\sin ^{2}\theta }\) based on Bear and Bachmat (1986), respectively. The result \(\tau _{\mathrm{Bear90}} =\cos \theta \) seems questionable since it implies that the effective length \(L_\mathrm{e}\) is shorter than the length L of the porous medium. Even though \(\tau _{\mathrm{Bear86}}\) is larger than unity, it is different from the expected value of \(\tau _\mathrm{KC} =L_\mathrm{e} /L=1/\cos \theta \).

Fig. 2
figure 2

A porous medium with (a, b) a flow line cross the boundary obliquely, and (c) a M-shaped flow channel

Full size image

The second example to examine is a porous medium with a M-shaped flow channel as shown in Fig. 2c. Since the average interstitial flow velocity is the same at all cross sections of the flow channel, the different definitions of \(T_\mathrm{Geo} , T_\mathrm{Kin}\) and \(T_\mathrm{Dyn}\) give the same value of \(T_0^*=L/L_\mathrm{e}\), which corresponds to \(\tau _\mathrm{KC} =L_\mathrm{e} /L\) with \(L_\mathrm{e} =L+2\left( {L_1 +L_2} \right) \). According to the definitions in Eqs. (2) and (12), it can be shown \(\beta _{11} =2(L_1 +L_2)/L_\mathrm{e}\), with the following geometry-related quantities:

$$\begin{aligned} T_0^*=\frac{L}{L_\mathrm{e}}=\frac{L}{L+2\left( {L_1 +L_2} \right) },\quad a^{\pm }=1\pm \frac{2(L_1 +L_2)}{L_\mathrm{e}} \end{aligned}$$
(16)

The Kozeny–Carman hydraulic tortuosity factors based on the two approaches of Bear and Bachmat are

$$\begin{aligned} \tau _{\mathrm{Bear90}} =\sqrt{\frac{a^{-}}{T_0^{*}}}=1, \quad \tau _{\mathrm{Bear86}} =\sqrt{\frac{a^{+}}{T_0^{*}}}=\sqrt{\frac{L_\mathrm{e} +2(L_1 +L_2)}{L}} \end{aligned}$$
(17)

respectively. Similar to the previous case, both \(\tau _{\mathrm{Bear86}}\) and \(\tau _{\mathrm{Bear90}}\) underestimate the hydraulic tortuosity factor \(\tau _\mathrm{KC} =L_\mathrm{e} /L\).

The above simple examples reveal some inconsistencies between the tortuosity \(T_0^*\) (or the equivalent hydraulic tortuosity factor \(\tau \)) in the derivations of Bear and Bachmat (1986, 1990) and other definitions or the Kozeny–Carman hydraulic tortuosity factor. In both cases examined, \(\tau _{\mathrm{Bear86}}\) and \(\tau _{\mathrm{Bear90}}\) underestimate the hydraulic tortuosity factor defined as \(\tau _\mathrm{KC} =L_\mathrm{e} /L\).

3 A Modification to Bear–Bachmat’s Method

In this section, a different approach is proposed to estimate the surface integral on the r.h.s of Eq. (3). Referring to Fig. 3, let \(\tau _{\alpha ij}\) be the viscous stress tensor in the vicinity of the fluid–solid interface \(S_{\alpha \beta }\), \(t_{\alpha j}\) stands for the traction vector on \(S_{\alpha \beta }\), \(t_{\alpha j}^T\) and \(t_{\alpha j}^N\) represent the normal and tangential components of \(t_{\alpha j}\) relative to the local tangent plane of \(S_{\alpha \beta }\), and \(v_{\alpha i}\) denotes unit vector representing the normal of \(S_{\alpha \beta }\). When the local flow direction is different from the macroscopic flow direction, the interaction between the solid and fluid phases results in nonzero normal traction components \(t_{\alpha j}^N\) on \(S_{\alpha \beta }\). The traction vector \(t_{\alpha j}\) and its components in the normal and tangential directions of \(S_{\alpha \beta }\) are expressed as:

$$\begin{aligned} t_{\alpha i} =\tau _{\alpha ij} \nu _{\alpha j} ,\quad \tau _{\alpha i}^T =\tilde{\alpha }_{ij} t_{\alpha j} ,\quad \tau _{\alpha i}^N =(\delta _{ij} -\tilde{\alpha }_{ij})t_{\alpha j} \end{aligned}$$
(18)

in which \(\tilde{\alpha }_{ij} =\delta _{ij} -v_{\alpha i} v_{\alpha j}\). It follows that

$$\begin{aligned} \tau _{\alpha i} =(\tilde{\alpha }_{ij})^{-1}\tau _{\alpha j}^T \end{aligned}$$
(19)

Since the \(S_{\alpha \beta }\)-surface is a material surface with respect to the fluid’s mass, the fluid flow in its immediate neighborhood must satisfy \((V_j^m -V_{sj} |_{S_{\alpha \beta } })v_i =0\), or \(V_i^m =V_i^{mT}\). The tangential traction \(t_{\alpha j}^T\) on the \(S_{\alpha \beta }\)-surface is next related to the tangential velocity gradient in the close vicinity of the \(S_{\alpha \beta }\)-surface by

$$\begin{aligned} t_{\alpha i}^T =\mu _\alpha \frac{\partial V_i^m}{\partial s_\nu } \end{aligned}$$
(20)

The surface integral in the second term of the r.h.s. of Eq. (3) can be calculated as

$$\begin{aligned} \frac{1}{{V_{0}}}\int _{{S_{{\alpha \beta }}}} \,\left( {\mu _{\alpha } \frac{{\partial V_{i}^{m}}}{{\partial s_{\nu }}}} \right) \mathrm{d}S= & {} \frac{1}{{V_{0}}}\int _{{S_{{\alpha \beta }}}} \,t_{{\alpha i}}^{T} \mathrm{d}S = \frac{1}{{V_{0}}}\int _{{S_{{\alpha \beta }}}} \,{\tilde{\alpha }}_{{ij\,}} t_{{\alpha j}} \mathrm{d}S \\= & {} \frac{1}{{V_{0}}}\int _{{S_{{\alpha \beta }}}} \, {\tilde{\alpha }}_{{ij}} \tau _{{\alpha jk}} \nu _{{\alpha k}} \mathrm{d}S \simeq \frac{{\alpha _{{ij}}^{-}}}{{V_{0}}}\int _{{S_{{\alpha \beta }} }} \,\tau _{{\alpha jk}} \nu _{{\alpha k}} \mathrm{d}S \end{aligned}$$

Referring to Eq. (8) for the average gradient of velocity in the close proximity of the \(S_{\alpha \beta }\)-surfaces, the second term of the r.h.s. of Eq. (3) is given as

$$\begin{aligned} \frac{1}{{V_{0}}}\int _{{S_{{\alpha \beta }}}} \,\tau _{{\alpha jk}} \nu _{{\alpha k}} \mathrm{d}S \simeq \left( {\alpha _{{ij}}^{-}} \right) ^{{- 1}} \frac{1}{{V_{0}}}\int _{{S_{{\alpha \beta }}}} \,\left( {\mu _{\alpha } \frac{{\partial V_{i}^{m}}}{{\partial s_{\nu }}}} \right) \mathrm{d}S \simeq \left( {\alpha _{{ij}}^{-}} \right) ^{{- 1}} \mu _{\alpha } \Sigma ^{{\alpha \beta }} \frac{{{\overline{{V_{i}^{m}}}}^{\beta } - {\overline{{V_{i}^{m} }}}^{\alpha }}}{{\Delta _\mathrm{c}}} \end{aligned}$$

As a result, the macroscopic momentum balance equation in Eq. (11) becomes

$$\begin{aligned} {\bar{{\mu }}}_\alpha ^\alpha \left( \alpha _{ij}^-\right) ^{-1}C_\alpha \phi _0 \frac{ {\bar{{V}}}_{\alpha j}^\alpha - {\bar{{V}}}_{\beta j}^\beta }{\Delta ^{2}}=-\phi _0 T_{\alpha ij}^{*} \left( {\frac{\partial {\bar{{p}}}^\alpha }{\partial x_j}+ {\bar{{\rho }}}^\alpha g\frac{\partial z}{\partial x_j}} \right) \end{aligned}$$
(21)

The corresponding expression for the permeability tensor is the same as that in Eq. (13) such that

$$\begin{aligned} k_{ij} =\frac{\phi _\alpha \Delta _\alpha ^2}{C_\alpha }G_{\alpha ij}^{*} ;\quad G_{\alpha ij}^{*} =\alpha _{il}^- T_{\alpha lj}^{*} \end{aligned}$$
(22)

The difference between Eq. (21) and the original work of Bear and Bachmat (1990) is summarized as follows. Referring to Eq. (10), the normal component of traction in the immediate neighborhood of the \(S_{\alpha \beta }\)-surfaces is determined as

$$\begin{aligned} \tau _\alpha ^N =\tau _{\alpha ij} \nu _{\alpha i} \nu _{\alpha j}= & {} \mu _\alpha \frac{\partial V_i^{mT}}{\partial s_v}\nu _{\alpha i} =\mu _\alpha \frac{\partial V_j^m}{\partial s_v}\left( {\delta _{ij} -\nu _{\alpha i} \nu _{\alpha j}} \right) \nu _{\alpha i} \\= & {} \mu _\alpha \frac{\partial V_j^m}{\partial s_v}\left( {\delta _{ij} \nu _{\alpha i} -\nu _{\alpha i} \nu _{\alpha j} \nu _{\alpha i}} \right) =0 \end{aligned}$$

which means that the tractions on the fluid–solid interface are only in the tangential plane of the \(S_{\alpha \beta }\)-surface. However, the assumption that the local velocity component is the projection of the average fluid velocity on this plane implies nonzero flow component perpendicular to the \(S_{\alpha \beta }\)-surface. As such, one may argue that the results from Bear and Bachmat (1990) are statically admissible solutions. The resulted tortuosity and permeability tensors can be considered as the lower bound of tortuosity and upper bound for the permeability, respectively.

Fig. 3
figure 3

Surface traction on the fluid–solid interface \(\hbox {S}_{\upalpha \upbeta }\)

Full size image

In the modified method, the tangential component, \(t_{\alpha j}^T\), of the local traction at any point on the \(S_{\alpha \beta }\)-surface is determined from the viscous shear stress tensor \(\tau _{\alpha ij}\). The local tangential velocity gradient is determined based on \(t_{\alpha j}^T\), with the constraint \((V_j^m -V_{sj} |_{S_{\alpha \beta }})v_i =0\) being satisfied in the immediate neighborhood of the \(S_{\alpha \beta }\)-surface. Moreover, Eq. (20) implies zero normal velocity gradient in the direction \(\varvec{\upnu }_\alpha \) in the immediate vicinity of the \(S_{\alpha \beta }\)-surfaces, since

$$\begin{aligned} t_{\alpha i}^T \nu _i =\mu _\alpha \nu _i \frac{\partial V_i^m }{\partial s_\nu }=\mu _\alpha \frac{\partial V_i^{mN}}{\partial s_\nu }=0 \end{aligned}$$

As a result, the derived tortuosity and permeability tensors using the modified method can be considered as kinematically admissible solutions, which correspond to the upper bound of tortuosity and the lower bound for the permeability

In summary, the proposed modification to Bear–Bachmat’s momentum balance equation is based on the concept of streamwise channel flow assumption, which implies that the average intrinsic velocity is the same along a streamwise channel. Owing to the variation of flow direction along the stream line, the tractions on the solid–fluid interface may not be parallel to the \(S_{\alpha \beta }\)-surface. The viscous drag force on the \(S_{\alpha \beta }\)-surface is considered as the tangential traction component, which depends on the viscosity of the fluid and the velocity gradient in the normal direction of \(S_{\alpha \beta }\), as shown in Eq. (20).

As a special case, for isotropic porous materials or one-dimensional macroscopic flow, the equivalent Kozeny–Carman hydraulic tortuosity factor based on the above modification, or Eq. (22), is:

$$\begin{aligned} \tau _\mathrm{cor} =1/\sqrt{a^- T_0^{*}} \end{aligned}$$
(23)

in which \(a^-\) and \(T_0^{*}\) have been defined previously. Equation (23) gives the correct Kozeny–Carman hydraulic tortuosity factors for the special example cases in Fig. 2. Since both \(a^-\) and \(T_0^{*}\) have already been determined from the analysis in Sect. 2.2, the following results are readily obtained:

  • Example 1: \(a^{-}=\cos ^{2}\theta \), \(T_0^*=1\), \(\tau _\mathrm{cor} =1/\cos \theta \)

  • Example 2: \(T_0^*=a^{-}=\frac{L}{L_\mathrm{e}}=\frac{L}{L+2\left( {L_1 +L_2} \right) }\); \(\tau _\mathrm{cor} =\frac{L_\mathrm{e}}{L}\)

It can be further shown that Eq. (23) yields the correct hydraulic tortuosity factor for streamwise flow in parallel channels.

4 Upper and Lower Tortuosity Bounds of Isotropic Porous Media

This section discusses the lower and upper bounds for the tortuosity of homogeneous porous media with randomly distributed pores or solid particles. The tortuosity factor in the following discussions is generally referred to as the Kozeny–Carman hydraulic tortuosity unless otherwise specified.

For an isotropic porous medium with random distribution of solid particles, Guo (2012) and Ahmadi et al. (2011) derived the relation between \(T_{\alpha ij}^{*}\) and \(\alpha _{ij}^-\) as

$$\begin{aligned} \alpha _{ij}^- =\frac{1}{3}\left( 2\delta _{ij} +T_{\alpha ij}^{*}\right) \end{aligned}$$
(24)

For an isotropic medium, the \(\alpha _{ij}^-\) and \(T_{\alpha ij}^{*}\) tensors are simplified as \(\alpha _{ij}^- =a^{-}\delta _{ij} , T_{\alpha ij}^{*} =T_{\alpha 0}^{*} \delta _{ij}\) with \(T_{\alpha 0}^{*} =\phi _0\), \(a^{-}=({2+T_{\alpha 0}^{*}} )/3\). According to the analyses in the previous sections, \(\tau _{\mathrm{Bear90}} =\sqrt{a^{-}/T_0^{*}}\) in Eq. (15) and \(\tau _\mathrm{cor}\) in Eq. (23) are the lower and upper tortuosity bounds, \(\tau ^\mathrm{LB}\) and \(\tau ^\mathrm{UB}\), respectively. It follows that

$$\begin{aligned} \tau ^\mathrm{LB} =\sqrt{\frac{a^{-}}{T_{\alpha 0}^{*} }}=\sqrt{\frac{2+\phi _0}{3\phi _0}} , \quad \tau ^\mathrm{UB} =\frac{1}{\sqrt{a^{-}T_0^{*}} }=\sqrt{\frac{3}{\left( {2+\phi _0} \right) \phi _0}} \end{aligned}$$
(25)

Figure 4a compares \(\tau ^\mathrm{LB}\) and \(\tau ^\mathrm{UB}\) with measured data in the literature (Wyllie and Gregory 1955; Currie 1960; Koponen et al. 1996, 1997; Salem and Chilingarian 2000; Mota et al. 2001; Dias et al. 2006; Matyka et al. 2008; Lanfrey et al. 2010; Duda et al. 2011). One observes that the experimental tortuosity factors are mostly between the theoretical lower and upper bounds given in Eq. (25). It is interesting to note that the expressions for \(\tau ^\mathrm{LB}\) and \(\tau ^\mathrm{UB}\) in Eq. (25) are almost identical to \(\tau =(\phi _0)^{-p}\) with \(p = 0.4\) and 0.6, respectively. In other words, \(\tau ^\mathrm{LB}=(\phi _0)^{-0.4}\) and \(\tau ^\mathrm{UB}=(\phi _0)^{-0.6}\) can be used approximately as the lower and upper bounds of tortuosity factors for granular materials with all connected, uniformly distributed pore voids.

Fig. 4
figure 4

Estimated hydraulic tortuosity factor for an isotropic porous medium

Full size image

In the data presented in Fig. 4a, it is worthwhile to take a close look at the test results of Salem and Chilingarian (2000) on various materials. Their experimental data show that the hydraulic tortuosity factor of all tested materials with porosity \(\phi >30\,\%\), including glass spheres having 3–8 mm diameter, is bounded by \(\tau ^\mathrm{LB}\) and \(\tau ^\mathrm{UB}\). However, glass spheres and fine sand with different fractions of grain sizes (ranging from 0.02 to 8 mm in diameter with the particle diameter ratio up to 400) and low porosity (10.2–29.4 %) tend to have lower tortuosity than \(\tau ^\mathrm{LB}\). This is likely owing to local variations in pore size relative to the size of REV that are not included in Eqs. (2) and (3). Conceptually, replacing one of the larger grains shown in Fig. 1a with an assembly of smaller grains of the same solid volume \(V_\beta \) will increase the local pore size and hence yield different results. At higher porosities, differences will be muted because the REV size is larger to begin with. In addition, the presence of stagnant water by large grains may also attribute to the lower than \(\tau ^\mathrm{LB}\) tortuosity in dense packing. Evidences of stagnant water for flow in porous medium can be found in Masad et al. (2000), Duda et al. (2011) and Yazdchi et al. (2011).

We next examine the published experimental results of Dias et al. (2006), who examined the influence of particle diameter ratio on the hydraulic tortuosity of binary particulate materials. The tortuosity factor in Dias et al. (2006) is \(\tau _\mathrm{KC}\) back-calculated from experimental permeability test results using the Kozeny–Carman equation by assuming the shape factor \(C_{0} = 2\) for the packing of spheres. The materials used were binary particulates of different particle diameter ratios D / d and varying volume fractions of large particles. The relation \(\tau _\mathrm{KC} =\phi _0^{-p}\) matches the experimental data well, but the exponent p varies with the particle diameter ratios D / d. For mono-sized particulate bed and small particle diameter ratios, the value of p is always 0.5. With the increase in particle diameter ratio D / d (up to 53.8) and the volume fraction of large size particles (up to 0.65), the value of p gradually approaches 0.4, as shown in Fig. 4b. Obviously, the experimental results of Dias et al. (2006) generally support \(\tau ^\mathrm{LB}=(\phi _0)^{-0.4}\) and \(\tau ^\mathrm{UB}=(\phi _0)^{-0.6}\). It should be mentioned, however, the data in Fig. 4b show that larger D / d ratio could result in \(p < 0.4\) at certain volume fraction combinations. This is consistent with the test results in Fig. 4a by Salem and Chilingarian (2000) when the maximum particle diameter ratio is up to 400.

When following the approach in Bear and Bachmat (1986), \(\alpha _{ij}^+\) can be related to \(T_{\alpha ij}^{*}\) via \(\alpha _{ij}^+ =(4\delta _{ij} -T_{\alpha ij}^{*})/3\). For isotropic porous media, one has \(a^{+}=({4-T_{\alpha 0}^{*}} )/3\), \(T_{\alpha 0}^{*} =\phi _0\). The resulting tortuosity factor is

$$\begin{aligned} \tau _{\mathrm{Bear86}} = \sqrt{\frac{a^{+}}{T_{\alpha 0}^{*} }}=\sqrt{\frac{4-\phi _0}{3\phi _0}} \end{aligned}$$
(26)

As illustrated in Fig. 4, the tortuosity factor \(\tau _{\mathrm{Bear86}}\) based on the approach in Bear and Bachmat (1986) is slightly smaller but close to the theoretical upper bound \(\tau ^\mathrm{UB}\). However, neither the static nor the kinematic boundary conditions on the \(S_{\alpha \beta }\)-surfaces are satisfied for this solution.

5 Concluding Remarks

This paper presents theoretical upper and lower bounds for the tortuosity of isotropic porous materials with random distribution of voids and solid particles. The key to the success in this paper is Bear and Bachmat’s homogenization approach for the momentum balance equation (Bear and Bachmat 1990, 1986) and the method to characterize the pore structure of porous media in Guo (2012). For isotropic porous media, the theoretical analysis yields explicit expressions for the upper and lower bounds of tortuosity as functions of porosity, which are in agreement with experimental data in the literature. The proposed method can be readily extended to the analysis for anisotropic porous media, by taking into account the directional distribution of pores characterized by the linear porosity using the method in Guo (2012).