Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 2, No. 2, 2022 

 

6 

Construction of a model for calculating average 
tortuosity of porous media based on optimal fractal 
structure 

Lijie Cai 1, Xiaoyin Wang2, * 

1 School of Mathematical Sciences, Tiangong University, Tianjin 300387, China 
2 School of Textile Science and Engineering, Tiangong University, Tianjin 300387, China 

* Corresponding author: Xiaoyin Wang 

 

Abstract: The distribution of pores and skeletons in porous media materials are irregular, and its distribution of curved pore 

channels is uneven and complex. The bending degree of pores affects the internal heat transfer process. The tortuosity is not only 

a parameter to describe the bending degree of pore channels, but also is one of the important structural basic parameters of porous 

media materials. According to the porosity of the porous media materials, Sierpinski carpet model is established to characterize 

its structures. The calculation model of the average tortuosity of the porous media materials is derived by using the average value 

of the route of the heat flow through Sierpinski carpet. In order to find the best fractal unit structure and the number of iterations 

in the average tortuosity calculation model, an optimization discriminant model is established. The results of the average 

tortuosity calculation model of this model and the other three scholars are compared, and their average relative errors are 

calculated to be all less than 1 %, indicating that the average tortuosity calculation model established in this study is effective 

and reliable. The average tortuosity prediction model is related to the fractal unit structure and stage of the carpet, and there is 

no empirical constant in this model. It can be used to calculate the average tortuosity of other porous media materials, providing 

a new idea for calculating the average tortuosity. 

Keywords: Average tortuosity; Sierpinski carpet; Porous media materials; Heat flow paths. 

 

1. Introduction 

Porous media materials with solid phase as the solid 

skeleton, with gas or liquid or gas-liquid two-phase for the 

pore structure, and its pore structure is interconnected and 

complex. When studying the permeability, electrical 

conductivity and thermal conductivity of porous media 

materials, it is necessary to consider the macroscopic 

transport properties of porous media in their microstructure 

[1-3].  

For example, in the study of thermal conductivity of porous 

media materials, in addition to porosity affects the heat 

transfer inside the porous media materials, the bending degree 

of the pores also affects the heat transfer process inside the 

porous media materials. Because of the irregular distribution 

of pores and skeletons in porous media materials, the 

distribution of curved pore channels is uneven and complex. 

Therefore, the tortuosity is not only a parameter to describe 

the bending degree of pore channels, but also is one of the 

important structural basic parameters of porous media 

materials [4]. Moreover, tortuosity is also widely used in 

energy conversion and storage systems such as oil 

exploitation, carbon sequestration, fuel cells, and lithium-ion 

batteries. It is also used as an effective geometric parameter 

to describe ion transport in porous material electrodes [5-6] 

and an important parameter index for designing fixed bed 

reactors [4]. 

At present, the methods of calculating tortuosity are 

experimental measurement, numerical simulation and 

theoretical analysis. Numerical simulation methods are 

random walk numerical simulation [7-8,15], corrugated pore 

structure model (CPSM) [9-11], maximum sphere method 

[12], lattice gas simulation method (LG) [13], FLUENT 

numerical simulation [4], etc. These methods don’t require 

too much theoretical basis and they can be calculated by 

software. However, these methods usually have one or more 

empirical constants and some errors, and the calculation 

process is complex. In order to make the calculation simple, 

some scholars use the theoretical analysis model to calculate. 

Such as Lanfrey et al. [16] derives the theoretical model of 

the tortuosity of the random filling of the same spherical 

particles, which can be used to calculate the tortuosity of 

porous media materials with porosity between 0.36 and 0.45. 

However, when calculating materials with low porosity or 

high porosity, the experimental error is large. Some scholars 

have also studied the microstructure of porous media 

materials based on fractal theory. For example, Yuan Pei et al. 

[14] bases on the accurate self-similar Sierpinski carpet model, 

which can be used to calculate the tortuosity of porous media 

materials with porosity between 0.75-1. Similarly, Du et al. 

[4] obtains the relationship between tortuosity and porosity by 

changing the geometric shape of the regular polygon solid 

matrix in the Sierpinski carpet. The results are not much 

different from the FLUENT numerical simulation results. 

However, when the number of regular polygon edges 

increases, the tortuosity does not change much, and only the 

tortuosity of porous media materials with porosity below 0.9 

can be calculated. However, the above scholars are based on 

the Sierpinski carpet with a fractal unit of 3 3  to establish 

the tortuosity calculation model, which is not friendly to the 

calculation of the tortuosity of porous media materials with 

high porosity. Porous media materials with different porosity 

can be characterized by different fractal unit structures. So 

finding the best fractal unit structure is what this paper needs 

to consider.  

Basing on the previous theoretical research and considering 



 

7 

the porosity of porous media is different. Since Sierpinski 

carpet is used to characterize its internal microstructure, its 

fractal unit structure should not be static. Therefore, in order 

to calculate the average tortuosity of aerogels or porous media 

materials suitable for various porosities, this paper is based on 

the average tortuosity calculation model of Sierpinski carpet 

with self-similar 3 3  of LI Jianhua et al. [17]. The average 

tortuosity calculation model of Sierpinski carpet with fractal 

unit structure of k k  is derived. The optimal fractal unit 

structure and stage of porous media materials are found to 

determine the average tortuosity calculation model. This 

study provides a new idea for calculating the average 

tortuosity of porous media materials with high porosity. 

2. Establish an average tortuosity 
model 

2.1. Sierpinski carpet fractal process of 

different fractal units 

Since the internal pores of porous media materials are 

disordered and complex, Sierpinski carpet can be used to 

simulate it. Changing the fractal unit structure of Sierpinski 

carpet can better characterize its internal structure. At present, 

the most basic Sierpinski carpet is composed of 3 3  fractal 

unit structure. However, since the Sierpinski carpet with 

fractal unit structure of 3 3   is not suitable for 

characterizing porous media materials with high porosity, its 

maximum porosity is only 0.8889. In order to find the average 

tortuosity calculation model suitable for porous media 

materials with high porosity, this paper further extends the 

Sierpinski carpet based on 3 3  fractal unit structure. And 

we are used to the Sierpinski carpet composed of k k  

fractal unit structure to characterize porous media materials. 

Moreover, because of the diversity of Sierpinski carpets 

composed of even fractal units, this paper only considers 

Sierpinski carpets composed of odd fractal units. When 

1k   is a positive integer and odd, the Sierpinski carpet with 

the fractal unit structure k k  starts as a square with a side 

length of 1. In the first iteration, a square with a side length of 

1/ k  is removed from the center of the square with a side 

length of 1. In the second iteration, the remaining 2 1k −  

sub-square shapes with a side length of 1/ k  are removed 

from their central 
2(1/ )k   sub-squares as in the first 

iteration. Therefore, the schematic diagram of the Sierpinski 

carpet with the fractal unit structure k k  is as follows. The 

black area represents the solid phase (porous material 

skeleton) area, and the blank area represents the gas phase 

(pore) area. 

 

Fig. 1 Sierpinski carpet with fractal unit of k k . 

Figure 1 (a) is the basic fractal unit fractal, Figure 1 (b) is 

the fractal unit after the first iteration, and Figure 1 (c) is the 

fractal unit after the second iteration. 

2.2. Calculation of average tortuosity based on 

heat flow paths 

The tortuosity i  is defined as: 

ei
i

i

L

L
 =

 

(1) 

where eiL
 is the true length of the curved pore channel; 

iL
 is the linear length at both ends of the curved pore channel.  

The average tortuosity   is defined as: 

1

N

i

i

N



 ==


 

(2) 

where, N   is the number of connected curved pore 

channels. 

The average tortuosity is closely related to the fractal 

dimension or porosity of the material [11-13]. In this paper, 

the Sierpinski carpet with a fractal unit of k k   is 

constructed by the porosity of the porous medium material, 

and then the Sierpinski carpet with a fractal unit of k k  is 

used to construct the average tortuosity model. The Schematic 

of heat flow paths is as follows:  

 

Fig. 2 Schematic of heat flow paths of Sierpinski carpet with fractal 
unit of k k . 

Fig.2 (a) is a zeroth-stage Sierpinski carpet with a fractal 

unit of k k , whose heat conduction direction is set from 

left to right, and the actual length of the curved pore channel 

is equaled to the length of the straight line at both ends of the 

linked curved pore channel, then its average tortuosity is 

(0) 1 =  . Fig.2 (b) is a 1st-stage Sierpinski carpet with a 

fractal unit of k k  , whose heat flow line has twists and 

turns in the middle sub-square, and the law of heat flow line 

needs to be further analyzed at this position. Fig.2 (c) is a 2nd-

stage Sierpinski carpet with a fractal unit of k k , whose 

streamline has twists and turns in multiple sub-squares, and 

its formation process needs to be deduced according to the 

1st-stage Sierpinski carpet. 

 
Fig. 3 Schematic of flow paths through the 1st-stage Sierpinski 

carpet with fractal unit of k k .  



 

8 

Figure 3 is the 1st-stage Sierpinski carpet 's Schematic of 

heat flow paths. By simulating its streamline, we are found 

that the heat flow diagram can be divided into k  parts from 

top to bottom. That is a row of squares to be a part. Except for 

part ( 1) 2k +  , the remaining pore channels are 

approximately straight. And the average tortuosity of this part 

1k −  is 

𝜏1 = 𝜏2 = ⋯ = 𝜏𝑘+1
2

−1
= 𝜏𝑘+1

2
+1
= ⋯ = 𝜏𝑘

=
∑ 1𝑁
𝑖=1

𝑁
= 1 

(3) 

The tortuosity of part ( 1) 2k +   is more complex. 

Among them, the flow path of A B→  is the longest, and 

the average tortuosity is 

1

2
1

12

2

1
1 2

12

1

k
e

A B

k

k

L
kk

L k


+

→

+

+

+ 
+

= = =

 

(4) 

The streamline of C D→  is approximately 1. Therefore, 

the average tortuosity of part ( 1) 2k +   takes the longest 

and shortest averages as 

1

2

1
1

2 1

2 2
k

k

kk

k
 +

+
+

+
= =

 

(5) 

Therefore, the average tortuosity of the 1st-stage Sierpinski 

carpet with a fractal unit of k k is 

2

2

2 1
1

2 12(1)
2

k
k

kk

k k


+
+ −

+
= =

 

(6) 

Since the average tortuosity of zeroth-stage Sierpinski 

carpet with a fractal unit of k k  is (0) 1 = ，the average 

tortuosity of the 1st-stage Sierpinski carpet with a fractal unit 

of k k  can be rewritten as  

2

2

2 1
(1) (0)

2

k

k
 

+
=

 

(7) 

where (0)   represent the average tortuosity of the 

zeroth-stage Sierpinski carpet with a fractal unit of k k , 

and (1)   represent the average tortuosity of the 1st-stage 

Sierpinski carpet with a fractal unit of k k . 

 
Fig. 4 Schematic of flow paths through the 2nd-stage Sierpinski 

carpet with fractal unit of k k . 

Figure 4 is the 2nd-stage Sierpinski carpet 's Schematic of 

heat flow paths. By simulating its streamline, the 2k   sub-

square is divided into 2k  parts. It is found that except for the 

[ 1 2]k th+( )  sub-square, the average tortuosity of the other 

2 1k −  sub-square is 1/ k  of the 1st-stage Sierpinski carpet. 

That is 

𝜏1 = ⋯ = 𝜏𝑘2+1
2

−1
= 𝜏𝑘2+1

2
+1
= ⋯ = 𝜏𝑘2

=
1

𝑘
𝜏（1） 

(8) 

the average tortuosity of the [ 1 2]k th+( )  sub-square is 

1

2

1 1 1
2

32 1 1
2 2

k
k k k

k
  +

 + +

= （）= （）

 

(9) 

The 2nd-stage Sierpinski carpet with a fractal unit of 

k k   is then divided into k   parts from top to bottom. 

Excepting for part ( 1) 2k + . The average tortuosity of the 

other 1k −  parts is 

𝜏1 = 𝜏2 = ⋯ = 𝜏
𝑘+1
2

−1 = 𝜏
𝑘+1
2

+1 = ⋯ = 𝜏𝑘

= 𝑘 ×
1

𝑘
𝜏（1） = 𝜏（1） 

(10) 

Since the average tortuosity of the [( 1) 2]k th+  part of 

the 2nd-stage Sierpinski carpet is similar to that of the 1st-

stage Sierpinski carpet. The average tortuosity of this part is 

𝜏
𝑘+1
2 =

3

2𝑘
𝜏（1）+ (𝑘 − 1) ×

1

𝑘
𝜏（1）

=
2𝑘 + 1

2𝑘
𝜏(1) 

(11) 

Therefore, the average tortuosity of the 2nd-stage 

Sierpinski carpet with a fractal unit of k k is 

2

2

2 1
(1) ( 1) (1)

2 12(2) (1)
2

k
k

kk

k k

 
 

+
+ −

+
= =

 

(12) 

Where (1)   represent the average tortuosity of the 1st-

stage Sierpinski carpet with a fractal unit of k k  , and 

(2)   represent the average tortuosity of the 2nd-stage 

Sierpinski carpet with a fractal unit of k k . 

By Eq. (7) and Eq. (12), and so on, the average tortuosity 

of the 1st-stage Sierpinski carpet is 
2

2

2 1
( 1) ( )

2

k
n n

k
 

+
+ =

 

(13) 

where 0,1,2,n =  . represent the stages of the 

Sierpinski carpet with a fractal unit of k k . 

Moreover, since (0) 1 = , the average tortuosity model of 

Sierpinski carpet with a fractal unit of k k   can also be 

written as 

𝜏(𝑛) =
2𝑘2 + 1

2𝑘2
𝜏(𝑛 − 1) = (

2𝑘2 + 1

2𝑘2
)2𝜏(𝑛 − 2) = ⋯

= (
2𝑘2 + 1

2𝑘2
)𝑛𝜏(0) = (

2𝑘2 + 1

2𝑘2
)𝑛 

(14) 

where 1k    is a positive integer and is odd, n   is a 

positive integer and ( )n   is the average tortuosity of the 

nth- stage Sierpinski carpet with a fractal unit of k k . 



 

9 

2.3. Optimization of the best fractal unit 

structure 

The porosity of porous media materials can be used to 

construct Sierpinski carpet. And in turn, Sierpinski carpet can 

also be used to characterize the internal microstructure of 

porous media materials. When the porosity of Sierpinski 

carpet is closer to the porosity of porous media materials, the 

characterization effect is better. Therefore, we are found the 

best fractal unit structure by the difference between the 

porosity after Sierpinski carpet iteration and the real porosity. 

When the difference between them is smaller, the internal 

structure of the porous media material is not only better, the 

average tortuosity calculated by the model is but also more 

accurate. The true porosity of porous media materials is set to 

0 . In the 1st-stage Sierpinski carpet with a fractal unit of 

k k , since the solid phase portion accounts for 21/ k , its 

porosity is: 
2

1 2 2

1 1
1

k

k k


−
= − =

 

(15) 

After n iterations, the porosity of Sierpinski carpet is: 
2

2

1
( )n

n

k

k


−
=

 

(16) 

The difference between the true porosity of the porous 

media material and the porosity of the Sierpinski carpet 

iteration is set to be t, which is defined as:  
2

0 0 2

1
( )n

n

k
t

k
  

−
= − = −

 

(17) 

where 1k    is a positive integer and odd, n   is a 

positive integer, 0  is the true porosity of the porous media 

material and n  is the porosity of the nth- stage Sierpinski 

carpet with a fractal unit of k k . 

Here, the dimension (k) and the number of iterations (n) of 

the optimal fractal unit structure are obtained by establishing 

an optimization model: 
2

0 2

1
min ( )nk

t
k


−

= −
 

(18) 

2

0 2

 . .

1

(

  1      

    

)

 

1 n

s k is a positive integer and is odd

is a positi

t

ve i

k

ntegn

k

er









−







,

，

。
 

(19) 

From the optimization model, k and n can be obtained and 

the average tortuosity can be calculated. 

2.4. Verification of the average tortuosity 

calculation model 

The validity of the model is tested according to the three 

more accurate models in the current average tortuosity study. 

Koponen [13] is used the Lattice Gas Simulation (LG) 

method to calculate the average tortuosity of low Reynolds 

number flows in a two-dimensional matrix formed by 

randomly placed fully overlapped rectangles. The formula is 

0.8(1 ) 1K = − +
 

(20) 

where 0.8 is the fitting constant, 


  is the porosity of 

porous media, and K   is the average tortuosity of porous 

media. 

Comparing it with the results of the average tortuosity 

calculation model established in this paper, the relative error 

is 

| |
100%

K

Kerror
 



−
= 

 

(21) 

Yuan Pei et al. [14] are derived the average tortuosity 

calculation model by n iterations of the self-similar Sierpinski 

carpet model: 

3 1

2 2
YP = −

 

(22) 

where 0.8 is the fitting constant, 


  is the porosity of 

porous media, and YP
  is the average tortuosity of porous 

media. 

Comparing it with the results of the average tortuosity 

calculation model established in this paper, the relative error 

is 

| |
100%

YP

YPerror
 



−
= 

 

(23) 

Du Peng et al. [4] are summarized the relationship between 

tortuosity and porosity in Sierpinski carpets at low Reynolds 

numbers as: 

1 (1 2)(1 )DP = − − −
 

(24) 

where 0.8 is the fitting constant,    is the porosity of 

porous media, and DP  is the average tortuosity of porous 

media. 

Comparing it with the results of the average tortuosity 

calculation model established in this paper, the relative error 

is 

| |
100%

DP

DPerror
 



−
= 

 

(25) 

Therefore, the validity of the average tortuosity calculation 

model established in this paper can be tested by observing the 

size of 
K

error  ,
YP

error   and 
DP

error  . In this paper, 

when the three relative errors are less than 0.1 %, the average 

tortuosity calculation model of porous media materials is 

effective. 

3. Application and test of the average 
tortuosity model 

3.1. Application of the average tortuosity 

model 

In this paper, five aramid nanofiber aerogels (ANFAs) 

samples were selected. The preparation method was used 

KOH / DMSO to deprotonate aramid fibers to obtain aramid 

nanofiber dispersions, and then was used water to reduce their 

structures. Finally, aramid nanofiber aerogels were obtained. 

The fractal dimensions at different magnifications are close, 

and ANFAs is statistical self-similarity [18]. According to the 

porosity of 5 samples, the number of iterations of Sierpinski 

carpet with 10 different fractal unit structures can be 

calculated respectively. The following table shows: 

This paper expects to use small dimension fractal unit 

structure to characterize porous media materials in the case of 

small error. Taking sample 1 as an example, the difference 

between its true porosity and the porosity of Sierpinski carpet 



 

10 

with fractal unit structure of 21 21  is only 0.002039. And 

the relative error is 0.21 %, which can well characterize the 

internal microstructure of ANFAs. Therefore, this paper only 

lists the Sierpinski carpet data of 10 fractal unit structures, 

without considering the higher-dimensional fractal unit 

structure. 

Table 1. The number of iterations of 5 samples under Sierpinski 
carpets with different fractal structures 

 

By Eq. (18) and Eq. (19), the best fractal unit structures of 

the five samples are 17 17 , 17 17 , 17 17 , 19 19 , 

9 9  , and the number of iterations are 3, 3, 3, 4, 1, 

respectively. According to Eq. (14), the average tortuosity of 

the five samples under the Sierpinski carpet of the 

corresponding fractal unit structure can be calculated. The 

results are shown in Table 2: 

Table 2. The average tortuosity calculation results of 5 samples 

 

According to Table 2, when the porosity is larger, the 

average tortuosity is gradually reduced in general case. 

However, if the porosity gap is below 0.03 %, the average 

tortuosity remains unchanged. 

3.2. Model validation 

The average tortuosity calculated based on Table 2 is 

compared with the average tortuosity calculation model 

established by Koponen, Yuan Pei, Du Peng and others. From 

the Eq. (21), Eq. (23) and (25), it can be obtained: 

Table 3. Comparison of calculated results between this model and 
other tortuosity prediction models 

 

According to the comparison of the results of the tortuosity 

prediction model in this paper with the other three prediction 

models, it is found that their average relative errors are 

0.2674 %, 0.0362 % and 0.1234 % respectively. Therefore, 

the average tortuosity calculation model in this paper is 

effective and reliable. Moreover, it can be used to calculate 

the tortuosity of high porosity materials such as aerogels. The 

model in this paper is only related to the fractal unit structure 

of Sierpinski carpet and the number of iterations. It not only 

has no empirical constant, but also can be used as a new 

method to calculate the average tortuosity. 

4. Conclusions 

According to the porosity of the porous media materials, 

Sierpinski carpet model was established to characterize its 

structures. The calculation model of the average tortuosity of 

the porous media materials was derived by using the average 

value of the route of the heat flow through Sierpinski carpet. 

In order to find the best fractal unit structure and the number 

of iterations in the average tortuosity calculation model, an 

optimization discriminant model is established. 

According to the data of 5 samples, the results of the 

average tortuosity calculation model of this model and the 

other three scholars are compared, and their average relative 

errors are calculated to be 0.2674 %, 0.0362 % and 0.1234 % 

respectively, indicating that the average tortuosity calculation 

model established in this study is effective and reliable. 

The average tortuosity prediction model established in this 

paper has no empirical constant, which reduces the subjective 

error of model calculation. It can be used to calculate the 

average tortuosity of other porous media materials, providing 

a new idea for calculating the average tortuosity. 

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