Kinematical measurement of hydraulic tortuosity of fluid flow in porous media

Kinematical measurement of hydraulic tortuosity

of °uid °ow in porous media

Y. Jin

School of Resource and Environment

Henan Polytechnic University

Jiaozuo, Henan 454003, P. R. China

Key Laboratory of Biogenic Traces & Sedimentary Minerals

of Henan Province, 2001 Century Avenue

Jiaozuo, Henan 454003, P. R. [email protected]

J. B. Dong*, X. Li† and Y. Wu‡

School of Resource and Environment

Henan Polytechnic University

Jiaozuo, Henan 454003, P. R. China*[email protected]

†[email protected]‡[email protected]

Received 28 March 2014

Accepted 11 June 2014

Published 4 July 2014

It is hard to experimentally or analytically derive the hydraulic tortuosity (�) of porous

media °ow because of their complex microstructures. In this work, we propose a kinematical

measurement method for � by introducing the concept of local tortuosity, which is de¯ned as

the ratio of °uid particle velocity to its component along the macro °ow. And then, thecalculation model of � is analytically deduced in terms of that � is the mean value of the local

tortuosity. To avoid the impact from the singularity of local tortuosity, the velocity is nor-

malized, and � is then approximated by the ratio of the mean normalized velocity to the

average value of its component along the macro-°ow direction. The new estimation method isveri¯ed by °ow through di®erent types of porous media via the lattice Boltzmann method,

and the relationships between permeabilities and tortuosities obtained by di®erent methods

are examined. The numerical results show that tortuosity by the novel approach is in goodagreement with the existing theory, and the kinematic de¯nition of hydraulic tortuosity is

also proven.

Keywords: Porous media °ow; hydraulic tortuosity; Lattice Boltzmann method; velocity ¯eld;

Kozeny-Carmon equation.

PACS Nos.: 47.11.Qr, 47.56.þr, 47.10.�g.

International Journal of Modern Physics C

Vol. 26, No. 2 (2015) 1550017 (21 pages)#.c World Scienti¯c Publishing CompanyDOI: 10.1142/S0129183115500175

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http://dx.doi.org/10.1142/S0129183115500175

1. Introduction

Fluids transport through porous media is of great economic importance in various

areas. Permeability, one of the basic transport properties, synthetically describes

how a °ow is retarded by the microstructure of porous media,1,2 and its estimation is

associated with some more fundamental and well-de¯ned parameters solely deter-

mined by the geometry of porous media.

Because of the complexity and disorder of microstructures in natural media,

permeability prediction remains a long-standing problem with great practical value.

The previous experimental results and theoretical research3–5 showed that porous

materials have their permeabilities determined by their structural parameters,

among which, the Kozeny–Carman (KC) equation is considered the most famous

one.6 Based on an equivalent assumption between porous medium and a bundle of

capillaries of equal length and constant cross-section, KC equation relates perme-

ability to four structural parameters as follows:

�as ¼’3

��2S2; ð1Þ

where �as is the analytical permeability, ’ is the porosity, S is the speci¯c surface

area, � is the shape factor and � is the hydraulic tortuosity de¯ned as

� ¼ Le

L� 1: ð2Þ

In Eq. (2), Le is the mean length of real °ow paths and L is the straight-line

distance through porous media in the direction of the macroscopic °ow. According to

Eq. (1), if ’3=�S2 is ¯xed, then �2 is inversely proportional to the permeability,

which has been found to agree well with the experimental results for random packs of

monodisperse granules.3–5,7

Although the KC equation has been validated widely, its practical application is

seriously hindered by the di±culties in measuring the fundamental parameters.

Until recently, only the porosity of a porous medium could be experimentally

measured easily. Its speci¯c surface area and Kozeny constant, however, could only

be measured with a nontrivial experimental setup. Actually, the Kozeny constant

strongly depends on the porosity as well as on the microstructures of porous media,

and has been proven not to be a constant.8 For an arbitrary porous medium,

measurements of � and � seem to be an impossible mission, especially for the

hydraulic tortuosity7 due to the complex microstructure and its ambiguous de¯-

nition. To quantify � , di®erent de¯nitions have been proposed experimentally or

analytically, and that is why a lot of scholars doubted whether � is a fundamental

attribute of pore space, but just a parameter to reduce the bias between experi-

mental data and theoretical estimation.4 Of course, such doubts were always re-

moved by the evidences from di®erent applications of the hydraulic tortuosity and

its overloaded version,5,9–12 and the confusions are all resulted from di±culties in

the measurement of � .

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Due to the di±culties in measuring hydraulic tortuosity experimentally or ana-

lytically, a lot of researchers tried to solve the problem via numerical methods. With

the development of computational techniques and advancement of theories, the

°uid velocity ¯eld can now be numerically determined even in a quite complex

geometry,2,5,7,13–17 which realizes the numerical characterization of the tortuous na-

ture of °uid °ow in principle.7 Among the computational approaches, the newly

developed lattice Boltzmann methods (LBM) is broadly accepted as the most e±cient

solution for its advantages and capability in dealing with complex °uids °ows.18–20

In this paper, we focus on themethodology of the hydraulic tortuositymeasurement

for arbitrary porous media. A kinematical calculation model is theoretically deduced

according to the de¯nition of hydraulic tortuosity as well as the latest research

results.7,14,15 Then, based on the stable °ow velocity ¯eld simulated by LBM with

incompressible °ow assumption, the new tortuosity calculation model is veri¯ed by

di®erent benchmark porousmedia °ow. Some conclusions are drawn in the last section.

2. A New Method to Measure Hydraulic Tortuosity Quantitatively

2.1. Basic concepts and existing methods

The notion of tortuosity was ¯rst proposed by Carman6 as a fundamental structural

parameter in his semi-empirical formula to predict the permeability (Eq. (1)).

Owning to its ambiguous concept, the hydraulic tortuosity has several de¯nitions, all

of which are either qualitative or conceptual, such as Eq. (2).5

In spite of no consensuses in the de¯nition of � , the e®orts devoted to its esti-

mation or measurement have never been reduced because of the increasing appli-

cations. Generally, all the attempts could be cataloged into two types: the

experiment-based methods and the analytical methods. For the former, all hydraulic

tortuosity estimation models are derived from physical or numerical experimental

results, so they are convincing but lack universality because they always contain

some empirical parameters with unclear physical meaning.5,13,21–25 As to the ana-

lytical methods, even though the tortuosity de¯nitions possess strong physical

meaning, they are all obtained from the porous media with particular setting9,21,26–29

or geometrical assumptions,32–34 thus their universality will be reduced and usually

fail to work for an arbitrary porous space.

Till now, there is no universal de¯nition of hydraulic tortuosity for arbitrary pore

space, and that should be ascribed to the validity of calculation methods and the

highly arbitrary geometry of microstructures of porous media. So, if the measure-

ment of tortuosity is possible, we will remove our focus from the de¯nition of � to the

prediction of permeable capacity of porous media.

Since Kozeny, much work has been done to facilitate the numerical and analytical

measurement of hydraulic tortuosity.2,5,7,13–17 Of the methods thus developed, some

are based on geometrical approaches,32,33 while others make use of kinematical char-

acteristics of °uids. Actually, hydraulic tortuosity is the collective behavior of °uids

particles which will keep adjusting their movement instead of fully obeying the local


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structure in pore space. Thus, the deviation between the real °uids °ow paths and

those characterized by the geometrical structure is inevitable. So, in this study, we

adopt a kinematical average to estimate the hydraulic tortuosity of porous media °ow.

According to the kinematical characteristics of °uid, when we discretize a whole

pore spaceD into small °uid particles with the same size, all particles will possess the

same mass because of the incompressible assumption. In a statistical manner, the

macroscopic property � can then be determined by7

� ¼ h�xih�X

x ið3Þ

in which h� � � i denotes a spatial average over the whole pore space, �x is the local

velocity of lattice in position x, and �Xx is the X component of �x along the direction

of the macroscopic °ow. The full dissertation version of Eq. (3) is given by

� ¼P

x2Dp�xP

x2Dp�Xx

: ð4Þ

Based on Eq. (3), Duda reported their method for determining the hydraulic

tortuosity for arbitrary porous media °ow via LBM.7 Actually Eq. (3) or Eq. (4) will

result in biased estimation of hydraulic tortuosity, and detailed discussion is in the

following subsection.

2.2. A new method to measure hydraulic tortuosity

As a basic property characterizing the tortuous °ow, � is a dimensionless constant for a

certain porous medium, regardless of the °uid properties such as mass density and

dynamic viscosity. The quantity of hydraulic tortuosity, therefore should be deter-

mined by the geometrical layout of pore space and governed only by the basic Navier–

Stokes equation. From the statistical point of view, the value of � is the mean deviation

between the real paths of °uid particles and the straight-line °ows along the macro-

scopic °ow. In an arbitrary porous medium, the °ow directions of °uid particles will

keep changing because of the microstructure even when they have a ¯xed expectation.

By nominating local tortuosity the elongation of °uid particle °ow (located at x)

and denoting it by � ðxÞ, we get

� ðxÞ ¼�x�Xx

: ð5Þ

As shown in Eq. (5), local tortuosity is dimensionless and independent. Its

quantity value depends only on the direction of particle velocity but not on the

velocity magnitude. Actually, � ðxÞ results from the constraint of local geometry and

the governing of the Navier–Stokes equation. Based on the concept of local tortu-

osity, we will derive a new hydraulic tortuosity below in detail.

For a °ow tube Fi in a certain porous medium (where i denotes the ith tube, and

all tubes make up the void space (Dp) of the porous medium), there exist a lot of � ðrÞ

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ðr 2 FiÞ varying with the local geometry. The average tortuosity in domain Fi is then

given by

�Fi¼Xr2Fi

�r�Xr

,Xr2Fi

1 ¼ �r�Xr

� �: ð6Þ

From the mathematical viewpoint, the tortuosity in Fi calculated by Eq. (4) is not

equal to that by Eq. (6) because of the di®erences in velocity magnitude caused by

the °uid properties and velocity direction by the local solid–liquid boundary, except

when Fi is a straight tube where � ðrÞ keeps invariant. Taking into account the whole

pore space, only when all °ow paths are straight and parallel or the angles between Fi

and the direction of the macroscopic °ow satisfy the supplementary relation, can

Eqs. (4) and (6) be equal. Apparently, such a condition can be hardly guaranteed for

an arbitrary porous medium.

Additionally, the hydraulic tortuosity of °ows in porous media is relative to the

distribution of pore size17,33 and the microstructures. That is, an Fi with a wide

channel but a small hydraulic tortuosity will bring about a small hydraulic tortu-

osity, thus °uids may easily get passing, and vice versa. So, the hydraulic tortuosity

should read

� ¼XFi

!i�� Fi¼XFi

!i

Xr2Fi

�r�Xr

,Xr2Fi

1; ð7Þ

whereP

F!i ¼ 1, and F is the set of all °ow tubes in the porous media fFig.For the whole pore space, the macro/statistical property � should read as Eq. (8).

� ¼Xr2Dp

� ðrÞ

,Xr2Dp

1 ¼ h�x=�Xx i: ð8Þ

Meanwhile, the relationship described by Eq. (9) is obviously satis¯ed for certain

porous medium.

Xr2Dp

� ðrÞ ¼XFi2F

�Fi�Xr2Fi

1

!: ð9Þ

Substituting Eq. (9) into Eq. (8), we obtain the following relationship:

� ¼Xr2Dp

� ðrÞ

,Xr2Dp

1 ¼XFi2F

�Fi�Xr2Fi

1

!,Xr2Dp

1

¼XF

�Fi�P

r2Fi1P

r2Dp1¼XF

! 0i�Fi

; ð10Þ

whereP

F!0i ¼ 1 is satis¯ed apparently. The pore size e®ect on � is then included

implicitly in the tortuosity estimation model in the form of weighted coe±cients,

which are ignored by Eq. (3).


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By the bottom-to-up strategy, Eq. (10) calculates the hydraulic tortuosity in a

cause-e®ect way. Compared to the idea which simply takes °ow tortuosity as the

ratio of h�xi=h�Xx i without eliminating the e®ects from °uids properties and pore size

distribution,6,13 Eq. (10) fully embodies the physical background in the estimation

process. So, a more universal calculation model is then given by

� ¼RDp

�x�XxdðDpÞR

DpdðDpÞ

¼ �x�Xx

� �: ð11aÞ

In practice, one singular value from local tortuosities will seriously a®ect the result

of Eq. (11a). That is, if the angles between �x and �Xx are close to �=2, then

� ðxÞ ! 1. To overcome such abnormalities and maintain the bottom-to-up deriva-

tion manner, in this study, the magnitudes of all �x are normalized to 1, and the

corresponding �Xx is then scaled to ��X

x proportionally. Thus, the e®ects from °uid

properties are eliminated and the tortuous geometry of real °ow streams is preserved.

Due to the statistical characteristics, � can be approximated by

� � 1

h��Xx i

; ð11bÞ

where �x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�X

x Þ2 þ ð�Yx Þ2

p, �X

x ð�Yx Þ is the component along (perpendicular to) the

direction of the macroscopic °ow. In Eq. (11a) �Xx takes an absolute value to ensure

the positiveness of � ðxÞ, while in Eq. (11b) �Xx takes the scalar value. That is, if �X

x has

a direction along the macroscopic °ow, its value will be positive; otherwise, it will be

negative, and then scaled to ��Xx for normalization.

3. Lattice Boltzmann Algorithm and Test Porous Models

3.1. Lattice Boltzmann methods

The LB algorithm is a full discretization version of the Boltzmann equation. Be-

cause the algorithm describes the macroscopic °uid °ow via the collective behavior

of many ¯ctitious molecules but not via full molecular dynamics, it is referred to as

a mesoscopic description of microscopic physics. Space D is discretized in terms of a

regular lattice with spacing �x, time t in terms of a time step �t, and velocity space

in terms of a small set of velocities ci that are chosen such that ci�t is a vector

connecting two nearby lattice sites.30 For example, the popular D2Q9 model35

employs nine velocities, corresponding to zero and the four nearest and four next-

nearest neighbors on a simple-cubic lattice. The central quantities on which the

algorithm operates are the populations fiðx; tÞ, representing the mass density

corresponding to velocity ci, such that the local mass density �ðx; tÞ at site x at

time t is given by

�ðx; tÞ ¼Xi

fiðx; tÞ: ð12Þ

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Similarly, the momentum density Mðx; tÞ is obtained as the ¯rst velocity mo-

ment,

Mðx; tÞ ¼Xi

fiðx; tÞci ð13Þ

and the average °ow velocity of the lattice is given by

uðx; tÞ ¼ Mðx; tÞ�ðx; tÞ : ð14Þ

In the novel hydraulic tortuosity calculation model Eq. (11), �x is the absolute

value of uðx; tÞ at size x, and �Xx is the component of �x along the direction of the

macro °ow.

The algorithm is then described by the lattice Boltzmann equation

fiðxþ ci�t; tþ �tÞ ¼ f �i ðx; tÞ ¼ fiðx; tÞ þ �ðffðx; tÞgÞ: ð15Þ

The collision operator �i modi¯es the populations at the site (ffig denotes the setof all populations at the site), such that mass and momentum are conserved. The

conservation equations therefore readXi

�i ¼Xi

�ici ¼ 0: ð16Þ

This results in a set of post-collisional populations f �i , which are then propagated

to the neighboring sites. The algorithm thus satis¯es some important requirements

for °uid °ow simulations, such as the mass and momentum conservation and locality

of °uid °ows, but lacks Galilean invariance due to the ¯nite number of velocities. Full

rotational symmetry is also lost, which makes the recovery of isotropic momentum

transport unable. To overcome these shortcomings, Chen and Qian18,35 proposed a

simpli¯ed collision function based on the single relaxation time model, and named it

as Lattice Bhatnagar–Gross–Krook Boltzmann model (LBGK). The BGK collision

operator reads as

�iðffðx; tÞgÞ ¼�t

�LBM½f eq

i ðx; tÞ � fiðx; tÞ�; ð17Þ

where �LBM is a dimensionless relaxation time, and f eqi ðx; tÞ is a quasiequilibrium

distribution function.

In the test cases, the two-dimensional LBGK model is used. This model uses a

D2Q9 lattice model with 19 discrete velocities. The velocity directions of the D2Q9

Lattice Boltzmann BGK model are de¯ned as

ci ¼ c�

ð0; 0Þ i ¼ 0;

ðcos �i; sin �iÞ �i ¼ ði� 1Þ �2

i ¼ 1 � 4;

ffiffiffi2

p ðcos �i; sin �iÞ �i ¼ ði� 5Þ �2þ �

4i ¼ 5 � 8

8>>>><>>>>:

ð18Þ


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the equilibrium distribution function f eqi is taken as

f eqi ðx; tÞ ¼ !i�ðx; tÞ 1þ 3

u � cic2

þ 9ðu � ciÞ22c4

� 3u2

2c2

� �ð19Þ

and the kinetic viscosity of the °uid () is given by

¼ ð�LBM � 0:5Þ �2x

3�t; ð20Þ

where c ¼ �x=�t, a characteristic lattice velocity de¯ned by the grid spacing �x.

Denoting cs as the sound speed, we can get the relation c2s ¼ffiffiffiffiffiffiffiffiRT

p ¼ c2=3 based on

the constrains of conservation and isotropy, and then rewrite Eq. (19) as

f eqi ðx; tÞ ¼ !i�ðx; tÞ 1þ u � ci

c2sþ ðu � ciÞ2

2c4s� u2

2c2s

� �:

Equation (20) imposes a constraint on the choice of �LBM, which must be greater

than 0.5 to guarantee the physical correctness of the viscosity. It is also known

that, owing to the nonlinearity of the NS equation, a �LBM value close to 0.5 will

produce unstable numerical behavior.36 In our study, we choose �LBM ¼ 1:0, and

thus take the kinetic viscosity of 0.16667. The weights !i > 0 are normalized

such thatP

i!i ¼ 1. This notation is presented here to emphasize that, for the

symmetry reason, the weights depend only on the absolute values of speed ci, but

not on its direction. Furthermore, the weights can be adjusted in such a way that

f eqi satis¯es X

i

f eqi ðx; tÞ ¼ �ðx; tÞ; ð21Þ

Xi

f eqi ðx; tÞui ¼ M: ð22Þ

For the D2Q9 model, this implies !i ¼ 1=36 for the next-nearest neighbors, !i ¼1=9 for the nearest neighbors and !i ¼ 4=9 for the rest population.

Boundary conditions signi¯cantly in°uence on the stability, e±ciency and accu-

racy of numerical computation. In a natural porous medium, the physical solid–°uid

interface is extremely complex, resulting from the disordered and extremely com-

plicated microstructures. Thus, at such a solid–°uid interface exists almost no net

°uid motion.20 The physical boundary condition at solid–°uid interfaces in an ar-

bitrary porous medium can, therefore, physically approximate the no-slip boundary

condition.

In the LBM, the physical no-slip boundary condition is usually realized as the

bounce-back rule. For simplicity and without loss of generality, the complete bounce-

back scheme is used to simulate the no-slip boundary condition at solid–°uid

interfaces, which requires that when a particle distribution streams to a solid

boundary node, it scatters completely back to the node where it comes with the

opposite velocity vector.31

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3.2. Test porous media and numerical simulation settings

To validate Eq. (11b), the relationships between permeability and tortuosities by

di®erent methods are analytically compared. Four cases corresponding to di®erent

models for 2D porous media are examined from simple to complex, with sample

width L ¼ 1mm and height H ¼ 1mm. Case I is a capillary model with an inclined

straight channel (Fig. 1(a)). Cases II and III are the capillary models with curved

channels respectively with the boundary being serpentine (U-shaped) and sin-type

(a) (b)

(c) (d)

Fig. 1. Test porous media to validate Eq. (11). (a) The inclined channel model with rotate angle a and

width h; (b) the U-shaped channel with step depth b and width h; (c) the curved channel model with the

sin-type boundary; (d) random deposited circles model.


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(Figs. 1(b) and 1(c)). Case IV is the circles random distribution model without

overlap (Fig. 1(d)).

Fluid °ow through the porous medium de¯ned above is simulated via the LBGK

model. Darcy's law is used in combination with the given pressure gradient and °uid

viscosity to determine permeability �LBM in dimensionless lattice units.

� �P

LLBM

¼

�LBM

huxi; ð23Þ

where LLBM is the spatial lattice resolution along the direction of the macroscopic

°ow in the lattice discretization system, ��P=LLBM the pressure gradient in the LB

system, the °uid viscosity, ux the component of lattice average velocity along the

direction of the macro °ow (Eq. (14)), and h� � � i denotes the spatial average over thepore space.

The conversion between dimensionless �LBM and �ns in real physical units is

straightforward by multiplying with the square of the e®ective length of a voxel side,

shown as

�ns ¼ �LBM

L

LLBM

� �2

: ð24Þ

In Eq. (24), L is the length of porous samples in real physical units, here is set to

be 1mm. To keep Darcy's law valid, the pore scale Re must be small enough, i.e. the

pressure gradient should be small enough if the kinematic viscosity and pore size

are ¯xed during the LBM simulations. Here, we employ the maximum relative

errors function (Eq. (25)) to determine the range of �P that validates the Darcy's

law.

MRER ¼ maxj�as � �nsj

�as

; ð25Þ

where as, ns respectively indicate the analytical and numerical solutions, �as and �ns

are de¯ned in Eq. (1) and Eq. (24), respectively.

The numerical results show that �as and �ns make no obvious di®erence when

�P < 0:2. Thus, here we choose �P ¼ 0:00005Pa; meanwhile, to void the errors

from the discretization resolution, the lattice size takes 800� 800 lattice units (l.u.)

for all test porous media with physical size 1� 1mm2 (H ¼ 1mm, L ¼ 1mm).

For convenience, geometric tortuosity is denoted as �g, �Duda as the tortuosity

calculated by Eq. (4) and �Novel as the result from Eq. (11b). The analytical per-

meability estimated by Eq. (1) with tortuosity z is denoted as �anðzÞ.

4. Numerical Results and Discussion

In this section, we will study how Eq. (11) behaves for °ows through various

geometries with increasing complexity, and discuss all the test models. For all the

testing cases in LBM simulations, dynamic viscosity , lattice size of the testing

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models, dimensionless relaxation time �LBM, and pressure gradient all remained

unchanged as afore-mentioned. �Duda and �Novel are both calculated based on the

velocity ¯elds when the LBM simulations reach a steady state, �g geometrically.

4.1. The inclined channel and validation of LBM simulation

The simplest model of a porous medium approximates a bunch of straight pipes, with

each pipe inclined at an angle a to the macroscopic °ow direction. In this case, all

streamlines are of the same length and �g ¼ 1= cos a geometrically. A series of two-

dimensional con¯gurations of an inclined channel are constructed with di®erent

channel width h and angel a. The relation among the tortuosities from di®erent

de¯nitions is depicted in Fig. 2.

As shown in Fig. 2, the hydraulic tortuosities (�Duda and �Novel) all agree well with

the values expected for geometric considerations (�g) even for a narrow channel.

Small errors observed in our simulations should stem from the discretization errors

and staircased approximation e®ect of the channel boundaries in the LBM method

(a) (b)

(c) (d)

Fig. 2. Hydraulic tortuosity � calculated by di®erent methods in a channel with aperture d (lattice units)

rotated by an angle a. The solid line shows the geometric tortuosity 1/cos a.


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(see Fig. 3), which cause slight underestimation of the hydraulic tortuosity. This test

indicates that Eq. (3) is equal to Eq. (11) for straight °ow channels.

With Eq. (1), we get �asð�Þ=� ¼ ’3=ð�2S2Þ. To validate the LBM, we investigate

the relation between �ns and �asð�Þ=� by respectively taking the value of �g, �Duda

and �Novel (Fig. 4).

Figure 4 indicates that the slope of the best ¯tted linear model to the relation

between �ns and �as approximates to 3, which is the expectation of shape factor and

consistent with the reports in Refs. 37 and 38. The shape factors derived by �Duda

and �Novel are slightly higher than that by �g, because the hydraulic tortuosity (�Duda

and �Novel) is a bit underestimated due to the staircased e®ect (Fig. 3). Thus, the

validity of the LBM is veri¯ed.

4.2. The U-shaped channel

The U-shaped channel models are constructed, as shown in Fig. 1(b), with the LBM

simulation settings the same as in Sec. 3.1. In this model, �g ¼ ðLþ 2bÞ=L, S ¼2ðLþ 2bÞ=ðL�HÞ and ’ ¼ d�g=H. Comparison of the tortuosities between �g, �Duda

and �Novel for various values of b and h is shown in Fig. 5.

A linear dependency of geometric and hydraulic tortuosities on b is expected as

shown in Fig. 5, and clearly geometric tortuosity (�g) is generally larger than the

hydraulic ones (�Novel and �Duda), caused by the corner e®ects. Because the °uids °ow

tends to seek a shorter path owing to the kinematic characteristics,1 such derivation

becomes apparent as the channel width increases. This phenomenon indicates that,

in a narrow channel, the tortuosity e®ect is mainly a®ected by the geometrical

constraints; while in a wide channel, the kinematical e®ect will become prominent.

To give a convincing explain to such a phenomenon convincingly, the KC equation is

employed to examine the relation between permeability and tortuosity indirectly.

Fig. 3. (Color online) The staircased e®ect in LBM simulations at the channel boundaries near the inlet

and outlet boundaries.

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(a) (b)

(c)

Fig. 4. Relationships between �ns and �as=� (short by �as) for straight-line porous media.

(a) (b)

Fig. 5. Hydraulic tortuosity � calculated by di®erent methods in a U-shaped channels as a function of

relative step depth b=H for di®erent apertures d (lattice units). The solid line shows the geometric tor-

tuosity ðLþ 2bÞ=L for the same porous media.


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By substituting Eqs. (23), (24) into Eq. (1), we can get

� S2huxi’3�P

L2

LLBM

¼ 1

��2¼ �nsS

2

’3ð26Þ

which implies that �nsS2’�3 is proportional to ��2 because of constant shape factor �

for a certain porous medium. The relations between �nsS2’�3 and ��2 calculated by

di®erent methods is plotted in Fig. 6.

It is noticeable that relation between � �2Novel and �nsS

2’�3 is better than that

between the other two. This result, however, is not enough to determine which

method is better in measuring the hydraulic tortuosity for porous media °ow. We

then calculate the maximum relative errors (MRER, see Eqs. (24), (25)) and average

relative errors, de¯ned byP j�as � �nsj=

P�as, for the three di®erent relationships.

The results are listed in Table 1.

In terms of the maximum relative errors, average relative errors and correlation

coe±cients, the hydraulic tortuosity calculated by the new method is clearly better

than that by the geometric method and by Duda et al.7

4.3. The sin-type curved channel

Here a more complex channel model, the sin-type curved model as shown in Fig. 1(c),

is established to examine the newly proposed hydraulic tortuosity calculation

method. In this model, amplitude A is changed from 60 to 160 lattice units at a step

of 30 l.u., and the channel boundaries are set by y ¼ H=2 d=2þ A sin 2�!x. The

tortuosity is changed by altering frequency !, amplitude A or channel width d.

Geometric length Le of the °ow path takes the cumulative arc-length of the dis-

cretized sin-type curve with sample length L=LLBM, and �g is then obtainable from

Eq. (2). Using the simulation parameters mentioned above, the °ows through the

sin-type curved porous medium are simulated by the LBGK method and the

(c) (d)

Fig. 5. (Continued )

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hydraulic tortuosities are then calculated. Figure 7 shows the relationships between

the tortuosity (�g, �Duda and �Novel) and shape factors (A, !) of the sin-type curved

boundary, which again kinematically veri¯es the de¯nition opinion of hydraulic

tortuosity.

Notably, �Novel is smaller than �g and �Duda in general. To validate the new

method, the correlations between � �2� and �nsS2’�3 are examined again (Fig. 8).

(a) (b)

(c)

Fig. 6. Relation between �nsS2’�3 and � �2� in U-shaped channels. �� is calculated by di®erent methods

and denoted by T . d represents the channel width (lattice unit).

Table 1. The correlative parameters between � �2� and �nsS2’�3 for

porous media represented by the U-shaped channel, where �� is the hy-draulic tortuosity by di®erent methods.

Correlation between Correlation % Maximum % Average

� �2� and �nsS

2’�3 coe±cient (R2) relative error relative error

�g 0.9799 11.3121 6.1931

�Duda 0.9903 3.8542 2.8413�Novel 0.9958 1.3245 1.0267


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According to the correlation coe±cients, �nsS2’�3 has a higher relativity with

� �2Novel than with �g and �Duda, and the data °uctuation in Fig. 8(b) is slighter than in

Figs. 8(a) and 8(c). Meanwhile, the linear model best ¯tted to the relation between

� �2Novel and �nsS

2’�3 has a shape factor of 2.997, which, as expected, approximates

to 3, and is consistent with the result of smooth channels.37,38

4.4. Porous media randomly deposited by monosized circles

To validate the generality of Eq. (11b), we calculate the hydraulic tortuosity of °uid

°ows in the porous media randomly ¯lled with monosized circles without overlap. By

altering the number of ¯lled circles, the porosity will be changed. No-slip boundary

conditions are imposed on the top and bottom walls and periodic boundary condi-

tions are assumed on the inlet and outlet sides. Some LBM simulation results are

illustrated in Fig. 9.

(a) (b)

(c) (d)

Fig. 7. Hydraulic tortuosity � calculated by di®erent methods in a sin-type channel as function of

frequency (!) and amplitude A for di®erent apertures d (lattice units).

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For the porous media ¯lled by monosized circles, porosity ’ and speci¯c area S

can be respectively calculated by Eqs. (27) and (28).

’ ¼ HL� nAs

HLð27Þ

S ¼ nPs

HLð28Þ

where As and Ps are respectively the area and perimeter of single solid particles, and

n is the circle number.

Figure 10 illustrates the relationship between ’3=� 2�S2 (short by �as) and �ns

directly calculated from Eq. (24) based on the LBM simulated °ow ¯eld.

According to the best linear ¯tted results, the value of shape factor (�) can be

expected to be 2.5 by the novel hydraulic tortuosity calculation method, consistent

with that proposed in Ref. 39 for beds of spherical particles. When using the method

(a) (b)

(c)

Fig. 8. Relations between � �2� and �nsS2’�3 in sin-type channels, with �� calculated by di®erent methods

and denoted by T .


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(a) (b)

Fig. 10. Relation between �ns directly calculated based on the velocity ¯eld simulated by the LBM and

that by ’3=� 2�S2 (denoted by �as for short), �� represents the hydraulic tortuosity by di®erent methods.

Circles indicate the experimental data, and solid lines the best ¯tted linear models.

Fig. 9. Normalized velocity ¯eld of °ow through porous media randomly ¯lled by monosized circles in

radius of 16 l.u.s at di®erent porosities when the LBM simulation reaches a stable state. The porosity of

(a), (b), (c) and (d) is 0.7549, 0.6564, 0.5579 and 0.4731, respectively.

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by Duda et al.7 the shape factor will be higher than 2.5, indicating the overestimation

of hydraulic tortuosity but underestimation of porous media permeability.

Meanwhile, previous work shows that there is some dependency of the tortuosity

on the porosity, and di®erent de¯nitions between � and ’ have thus been proposed.

Boudreau40 reported a relationship (Eq. (29)) based on an empirical observation in

unlithi¯ed ¯ne-grained sediments.

�2 ¼ q� p lnð’2Þ: ð29ÞBy numerical simulations of the Lattice Gas Automata (LGA) method, Koponen

et al.13 found that � is a function of ’ in the range of ’ 2 ½0:5; 1�, which follows

� ¼ pð1� ’Þ þ q; ð30Þwhere p is a justifying parameter, and in their ¯ndings, p ffi 0:8.

Comiti and Renaud22 approximated the tortuosity-porosity relation by Eq. (31),

and by numerical studying, Matyka15 con¯rmed such a relationship and concluded

that in Eq. (31), p takes a value of 0.8 with the microscopic model of a porous

medium arranged as a collection of freely overlapping squares.

� ¼ q� p ln’: ð31ÞIn Eqs. (29)–(31), q takes the value 1.0, quantifying the tortuosity of straight-line

°ow along the direction of macro °ow. And p a parameter to ¯t experimental result

to the prede¯ned tortuosity–porosity relationship. Based on the above knowledge,

we ¯t the hydraulic tortuosities (�Novel and �Duda) to the tortuosity–porosity relation

mentioned above. All the results indicate that no matter what tortuosity–porosity

relationship is taken, the correlative degree between �Novel and ’ is better than that

between �Duda and ’, as shown in Table 2.

Meanwhile, we ¯nd the tortuosity–porosity relationship, with � calculated in terms

of °uid kinematics, is not in good agreement with the de¯nitions by Boudreau40 or by

Comiti and Renaud,22 because of the ¯tted values of q and p. The hydraulic tortu-

osity–porosity relationship, however, agrees well with the model proposed by Kopo-

nen13 (see Fig. 11). It is also noticeable that the correlation between �Novel and ’ is

better than that between �Duda and ’, with p ffi 0:8, well consistent with that by

Koponen13 and in good agreement with the result by Feranie.41

Table 2. The best parameters of the relationship between porosity and tortuosity ¯tted toEqs. (29)–(31).

Tortuosity–porosity relationship �� p q Correlation coe±cient (R2)

Equation (29) �Novel 0.6579 1.1842 0.9901

�Duda 0.7173 0.9463 0.9777

Equation (30) �Novel 0.8002 1.0454 0.9904�Duda 0.9119 0.9340 0.9742

Equation (31) �Novel 0.4845 1.1133 0.9898

�Duda 0.5541 1.0104 0.9810


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Taking into account the performance of both tortuosity–permeability and tor-

tuosity–porosity relationships, the validity of Eq. (11b) is veri¯ed.

Even through, we must point out that the validation of Eq. (11b) should be

constrained in laminar °ow ¯tting Darcy's law, because the examining model of KC

equation used in this study. At the same time, we focus our attention on the hy-

draulic tortuosity measuring method as well as the calculation model. The de¯nition

of � for a certain porous medium with particular geometrical settings is beyond the

scope of this study.

5. Conclusion

Following the bottom-to-up strategy, a novel method is kinematically derived, and

then the LBM simulations are carried out to investigate the tortuous e®ect from

porous media with di®erent setting-ups. The relation between tortuosity and per-

meability is examined, and the validity of the newly proposed method is veri¯ed by

comparative analyses. This study shows that the tortuosity–porosity relationship is

in good agreement with � � pð1� ’Þ þ 1, with p being close to 0.8. The analysis

results reveal that the KC theory prefers the tortuosity with the kinematical average

of real °ow paths, not the geometrical approach.

Acknowledgment

This work is supported by National Natural Science Foundation of China (Grant

No. 41102093, 41072153), CBM Union Foundation of Shanxi Province of China

(Grant No. 2012012002) and Doctoral Scienti¯c Foundation of Henan Polytechnic

University (Grant No. 648706).

Fig. 11. Relation between tortuosity and porosity in Eq. (30).

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Kinematical measurement of hydraulic tortuosity of fluid flow in porous media

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