Article

ORIGINAL PAPER

Study of the numerical simulation of tight sandstone gas molecular diffusion based on digital core technology

Hong-Lin Zhu1,2 • Shou-Feng Wang3 • Guo-Jun Yin3 • Qiao Chen1,2 • Feng-Lin Xu2 • Wei Peng4 • Yan-Hu Tan1 • Kuo Zhang1

Received: 9 June 2017 / Published online: 20 January 2018 � The Author(s) 2018. This article is an open access publication

Abstract Diffusion is an important mass transfer mode of tight sandstone gas. Since nano-pores are extensively developed in the

interior of tight sandstone, a considerable body of research indicates that the type of diffusion is mainly molecular diffusion

based on Fick’s law. However, accurate modeling and understanding the physics of gas transport phenomena in nano-

porous media is still a challenge for researchers and traditional investigation (analytical and experimental methods) have

many limitations in studying the generic behavior. In this paper, we used Nano-CT to observe the pore structures of

samples of the tight sandstone of western of Sichuan. Combined with advanced image processing technology, three-

dimensional distributions of the nanometer-sized pores were reconstructed and a tight sandstone digital core model was

built, as well the pore structure parameters were analyzed quantitatively. Based on the digital core model, the diffusion

process of methane molecules from a higher concentration area to a lower concentration area was simulated by a finite

volume method. Finally, the reservoir’s concentration evolution was visualized and the intrinsic molecular diffusivity

tensor which reflects the diffusion capabilities of this rock was calculated. Through comparisons, we found that our

calculated result was in good agreement with other empirical results. This study provides a new research method for tight

sandstone digital rock physics. It is a foundation for future tight sandstone gas percolation theory and numerical simulation

research.

Keywords Tight sandstone gas � Nano-CT � Digital core � Molecular diffusion � Numerical simulation

1 Introduction

Tight sandstone gas (TSG) generated in tight reservoirs is

one of three major types of unconventional energy. TSG

has been found widely distributed worldwide and has great

potential for exploration. It has a tight matrix and in gen-

eral has a nanoscale pore system in which natural gas exists

and migrates with a mechanism different from conven-

tional gas reservoirs. Studies (Schloemer and Krooss 2004)

have demonstrated that natural gas diffuses under the

action of a concentration field and percolates under the

action of a pressure field among tight matrix pores and

throats, and that natural gas is released from micro- and

nano-pores through three major procedures: desorption,

diffusion, and percolation. Therefore, the study of desorp-

tion and diffusion mechanisms of natural gas in nanoscale

tight sandstone pores is significantly important to the

evaluation and development of natural gas. However, there

have been very few reports about quantitative studies of

Edited by Jie Hao

& Qiao Chen [email protected]

& Yan-Hu Tan [email protected]

1 Chongqing Institute of Green and Intelligent Technology,

Chinese Academy of Sciences, Chongqing 400714, China

2 State Key Laboratory of Oil and Gas Reservoir Geology and

Exploitation, Southwest Petroleum University,

Chengdu 610500, Sichuan, China

3 Oil and Gas Engineering Research Institute, Jilin Oilfield

Company, PetroChina, Jilin 138099, China

4 Shu’nan Gas Mine, Southwest Oil and Gas Field Branch

Company, PetroChina Co., Ltd, Luzhou 646300, Sichuan,

China

123

Petroleum Science (2018) 15:68–76 https://doi.org/10.1007/s12182-017-0210-1(0123456789().,-volV)(0123456789().,-volV)

diffusion and mass transfer in gas reservoirs from a

microscale/nanoscale perspective although the molecular

diffusion effect is a key process of the migration of tight

sandstone gas.

The first diffusion equation was presented by Adolf Fick

as an empirical equation where a diffusion coefficient was

used to represent the speed of mass diffusion in diffusion

media. Today, a large number of studies on the determi-

nation of the diffusion coefficient of natural gas in cores

have been conducted by domestic and international schol-

ars (Bird et al. 2014; Wang et al. 2014; Liu et al. 2012; Jian

et al. 2012); diffusion equations have been optimized and

are capable of serving at high temperature and high pres-

sure; and diffusion coefficient values ranging from 10 -12

to

10 -5

m 2 /s have been obtained (Bera et al. 2011; Bing et al.

2013; Kelly et al. 2015; Nozawa et al. 2012). Ertekin et al.

(1986) described Knudsen flow under the action of a

concentration field using Fick’s Laws of Diffusion in

combination with the description of gas flow in tight pores

and deduced an empirical equation for the calculation of

gas diffusion coefficients through the replacement of Gra-

ham’s Law of Diffusion using the Jones and Owens

experimental data. Liu (2013) and Wu et al. (2012)

employed a molecular simulation method to study the

diffusion characteristics of CH4 and CO2. Curtis et al.

(2012) built a microscale two-dimensional finite element

model based on the diffusion studies of other porous

materials to study the diffusion mechanism of natural gas

in micro/nano-pores. Wu et al. (2015) deduced a mathe-

matical percolation model using the infinitesimal method

which is different from Darcy percolation and applies to

the micro-study of tight gas in matrix pores. Javadpour and

Moghanloo (2013) built a methane molecule migration

model and further discussed the contribution of molecular

diffusion to gas production.

Preceding studies have brought some results and pro-

gress, but these macrotheories and methods are insufficient

for the description and interpretation of a micro-mecha-

nism. There are still some unprecedented technical chal-

lenges to reproduce the migration of tight sandstone gas in

micro/nano-pores for studies on percolation mechanisms.

The nano-pore structure of tight sandstone reservoirs has

very critical influence on gas accumulation and mass

transfer, so the modeling of pores is directly related to the

accuracy of numerical simulation results. However, in

existing studies, models are just simplified ideal models

which do not represent the real pore structure of tight gas

reservoirs. In this study, a refined digital model of the pore

structure of tight sandstone reservoirs was built using

advanced digital core technology (Guo et al. 2016; Chen

et al. 2015; Liu et al. 2014, 2017; Ni et al. 2017; Yin et al.

2016), and a numerical simulation of molecular diffusion

was conducted on the basis of the model. This has provided

new ideas for the study of the micro-mechanisms of

molecular diffusion of tight gas.

2 TSG digital core modeling

2.1 Nano-CT experiment

Computed tomography (CT) is a technology used to non-

destructively detect the internal structure of an object. It is

today’s most practical and accurate method of building 3D

digital cores. CT identifies the pores and skeleton based on

the fact that components of different densities in the core

absorb different amount of X-rays. In this study, a ZEISS

Xradia 800 Ultra X-ray microscope (Fig. 1a) was used for

3D imaging. The experimental sample is u25 9 50 mm cylindrical sample. Samples were dried before scanning.

For the u25 9 50 mm sample, to improve the sample resolution as far as possible, the scanning area of the

experimental sample is u70 9 70 lm. The sampling res- olution of this Nano-CT is down to 50 nm, which enables

the microscope to take 1022 two-dimensional section

images (Fig. 1b) with 1016 9 1024 pixels for one sample.

Each voxel has a spatial resolution of 65 nm. By super-

imposing these images in sequence, 3D grayscale images

can be obtained (Fig. 1c), representing the microstructure

of the tight sandstone samples.

2.2 Image processing

Various types of system noise exist in these grayscale

images of cores, decreasing image quality and negatively

affecting the subsequent quantitative analysis, so the first

step of image processing is to increase the signal–noise

ratio (SNR) using a filtering algorithm. For a 3D image, the

commonly used filtering algorithms include low-pass linear

filtering, Gaussian smoothing, and median filtering. A

comparison of the filtering effects among the three algo-

rithms has been made, and a median filter was employed in

this study. In a grayscale image filtered and processed with

the median filter, the transition between the pores and

skeleton is natural; the boundary is smooth; and important

characteristics are also retained in the image (as shown in

Fig. 2). For better identification and quantization of the

pores and skeleton, binary classification is required for the

grayscale image using the image segmentation method.

The segmentation threshold is crucial to binary classifica-

tion, so it must be selected carefully. In this paper, the

porosity of the core sample has already been determined

through Nano-CT scan, so the best segmentation threshold

obtained on the basis of the porosity can be used for image

segmentation. Based on the measured porosity, the

Petroleum Science (2018) 15:68–76 69

123

segmentation threshold k* is obtained through the follow-

ing equation:

f k�ð Þ ¼ min f kð Þ ¼ u � Pk

i¼IMIN p ið Þ PIMAX

i¼IMIN p ið Þ

( )

ð1Þ

where u is the porosity of the core, k is the gray threshold, the maximum and minimum gray values of the image are

IMAX and IMIN, respectively, the number of voxels with a

gray value of i is p(i), the voxels with a gray value lower

than the gray threshold represent pores, the remaining

represents the skeleton. Based on the final value searched,

k* = 56, which is used as the segmentation threshold,

binary images are obtained after segmentation. As shown

in Fig. 3, the blue color represents the pores and the black-

colored background represents the skeleton. If necessary,

the mathematical morphological algorithm can be used for

further refined processing by removing the independent

voxels using open surface operations and filling the small

holes using closed surface operations to connect the

neighboring voxels.

2.3 3D surface reconstruction and pore structure quantitative analysis

In this paper, the CT images measured 1016 9 1024 pixels. To

reflect macropore structures and macroscopic properties, and

taking into account the amount of reconstruction data generated

and the associated computational burden, in this paper, a com-

promise involved cutting the CT images. Selecting a represen-

tative elementary volume (REV, the smallest core unit that can

characterize the macroscopic physical properties of a core

effectively) is crucial to the follow-up study in this paper.

Fig. 1 The process of Nano-CT scanning technology. a Xradia 800 Ultra, b one of the CT slices, c 3D grayscale images

Fig. 2 The slice after median filtering Fig. 3 The result of binarization (pores are blue)

70 Petroleum Science (2018) 15:68–76

123

Repeated experiments show that when the size of a digital core

model is 500 9 500 9 500 voxels, its porosity is almost

unaffected by size (Fig. 4), indicating that the REV size can

characterize the macroscopic physical properties of TSG.

The Marching Cube algorithm (Lorensen and Cline

1987) is used to obtain a triangle mesh set from the 3D data

cube of the image processing results and an illumination

model is used to render the triangle meshes. Then, a 3D

surface image of the core is formed. In this way, a 3D

digital model of tight sandstone cores is built (Fig. 5).

In the digital model built in the above step, most of the

pores are closely contacted. It is very hard to identify the

boundary of each pore, which negatively affects the subse-

quent quantitative statistics of pore size distribution.

Therefore, the boundary of each pore must be identified and

labeled. In this study, the fast watershed algorithm is used for

boundary detection. Using the image as the geo-scientific

topography, the gray value of each pixel on the image as the

sea level elevation of the pixel, each local minimum and its

affected areas as a catchment basin, and the boundary of the

catchment basin as the watershed, this algorithm has enabled

the identification of each pore and generated a spatially

labeled graph (as shown in Fig. 6) labeling each pore inde-

pendently; in this way a single pore boundary can be iden-

tified easily, which enables convenient collection of pore

data for quantitative analysis. As long as the volume of each

pore is determined, the porosity of the digital core can be

obtained through calculation. In the meantime, assuming

that the volume of a sphere is approximately equal to the

volume of pores at the corresponding location, the equivalent

pore size of each pore can be determined through Eq. (2) and

a pore size distribution histogram can be obtained on the

basis of the final statistics (Fig. 7).

Deq ¼ 3 ffiffiffiffiffiffiffiffiffiffiffiffi 6Vpore

p

r

ð2Þ

It is shown in Fig. 7 that the pore radius of this TSG

sample is mainly distributed in the range of 0.43–1

microns. At the current resolution, few 80-nm-sized pores

are captured. The widespread distribution of sub-micron

and nanoscale pores is the root cause of the difficulty of

developing such gas reservoirs. Table 1 shows that the

porosity obtained through calculation is slightly lower than

the measured porosity. This error is mainly caused by

image smoothing, because removing the small holes affects

the calculation of porosity to some extent.

3 Numerical simulation of molecular diffusion

3.1 Theoretical foundation

3.1.1 Fick’s first law: definition of molecular diffusion

Molecular diffusion is a process whereby dissolved mass is

passively transported from a higher chemical energy state

to a lower chemical energy state through random molecular

0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0 0 100 200

Cube length, voxel

P or

os ity

300 400 500 600 700

Expand the cube length progressively The calculation porosity change with cube length

(a) (b)

Fig. 4 The schematic diagram of REV analysis. a Expand the cube length progressively, b the calculation porosity change with cube length

Fig. 5 The digital core model (3D reconstruction of pores)

Petroleum Science (2018) 15:68–76 71

123

motion. Steady state diffusion of a chemical species in free

solution can be described empirically using Fick’s first law:

j~¼ �D � r~c ð3Þ

where j~ is the solute mass flux (in mol m-2 s-1); D is the

diffusion coefficient of the solute in the solvent (in

m 2

s -1

); c is the concentration of the solute in the solvent

(in mol m -3

).

3.1.2 Fick’s second law

The partial differential equation describing transient dif-

fusion in a homogeneous (only one solid phase), saturated

(the void space of the material is filled by the solvent),

porous medium can be developed from the Fick’s first law

and conservation of mass. This equation is called Fick’s

second law and is written as follows:

oc

ot ¼ �D � r~c ð4Þ

To simulate a molecular diffusivity laboratory mea-

surement, a classical experiment is suggested, which is

based on the double reservoir test. Two reservoirs having

the same volume VR are positioned on each side of the

sample in a chosen direction. The other directions are

closed with impervious planes, so that no diffusion occurs.

The initial concentrations of the reservoirs are different:

Cin (t) and Cout (t). The sample is initially filled with the

solution at Cin (t0), t0 being the instant when the experiment

starts. At time t = t0, the reservoirs are connected to the

sample and the diffusion process starts. The influence of

gravity is neglected, only passive diffusion is considered,

not advection.

Considering these boundary conditions, Fick’s second

law governs the diffusion and defines the concentration

field in the sample. The concentration of the reservoirs also

evolves since they have a finite volume VR. By default, VR is supposed to be 100 times higher than the void space

volume in the sample. Let us note b = Vvoidspace

VR , the ratio of

the void space volume and the reservoir volume. The fol-

lowing equations govern the concentration in the

reservoirs:

VR oCinðtÞ

ot ¼ D

Z

Sin

r~c � n~dS ð5Þ

VR oCoutðtÞ

ot ¼ �D

Z

Sout

r~c � n~dS ð6Þ

where Sin and Sout are the faces of the sample where the

reservoirs are connected.

Once the diffusion process starts, the concentration in

the sample quickly evolves and the exchanges with the

reservoirs are asymmetrical. This transient state is then

replaced by an established state, when the exchanges with

the reservoirs are equal.

This established state is characterized by the fact that oCinðtÞ

ot = -

oCoutðtÞ ot

. Once this state is attained, the concen-

tration in the reservoirs will continue to vary until they

reach an equilibrium concentration c!. The difference of

these concentrations, Cout(t) - Cin(t), follows an expo-

nential law:

CoutðtÞ � CinðtÞ ¼ p � expð�k2tÞ ð7Þ

where p and k2 are constant coefficients to be determined. An analytic solution to this problem is suggested:

Fig. 6 The quantification and characterization of pore structure in digital coal model label image of pores

0%

20%

40%

60%

80%

100%

0

50

100

150

200

250

0. 08

0. 43

0. 78

1. 13

1. 48

1. 83

2. 17

2. 52

2. 87

3. 22

3. 57

3. 92

4. 27

4. 62

F re

qu en

cy

Pore radius, μm

Frequency

Cumulative %

Fig. 7 The pore size distribution histogram

Table 1 The comparison of porosity results

The calculated porosity, % The measured porosity, %

9.37 9.64

72 Petroleum Science (2018) 15:68–76

123

cðX; tÞ ¼ A cos kffiffiffiffiffiffiffi

Dapp p � 1 � �

sin kffiffiffiffiffiffiffi Dapp

p � �

2

6 6 4

3

7 7 5 cos

k ffiffiffiffiffiffiffiffiffi Dapp

p X

! 2

6 6 4

þ sin k ffiffiffiffiffiffiffiffiffi Dapp

p X

!#

expð�k2tÞ þ c1

ð8Þ

where c(X, t) is the local concentration at position X and

time t, A is a constant coefficient.

Knowing this solution must verify the previous

hypothesis of flux equality ( oCinðtÞ

ot = -

oCoutðtÞ ot

), the follow-

ing equation is derived:

�k2 cos kffiffiffiffiffiffiffi

Dapp p � 1 � �

sin kffiffiffiffiffiffiffi Dapp

p � � ¼ Dappb

k ffiffiffiffiffiffiffiffiffi Dapp

p ð9Þ

which links the k2 coefficient to the apparent diffusivity Dapp of the sample.

To sum up:

1. A first transient state during which the diffusion

process starts must be achieved before an established

state appears.

2. Once the established state has begun, the difference of

the reservoir concentrations follows an exponential

law. Therefore, the slope of the linear curve followed

by ln(Cout (t) - Cin (t)) can be estimated easily.

3. This slope is k2, the exponential coefficient, which is related to the apparent diffusivity Dapp.

3.1.3 Volume averaged form of Fick’s law

The effective molecular diffusivity tensor gives global

information about the diffusion capabilities of the material.

A change of scale to get equations valid for the entire

volume is necessary. The method of volume averaging is a

technique that accomplishes a change of scale. Its main

goal is to spatially smooth equations by averaging them

over a volume.

This theory develops a closure problem that transforms

the Fick equations to a vectorial problem; closure variable

b !

is used to state the concentration perturbation in a new

problem:

r2 b !

¼ 0 ð10Þ

When the problem is solved, it is possible to compute

the dimensionless diffusivity tensor defined as:

/ D !!

Dsolution ¼ / I!

! þ

1

Vf

Z

Sfs

nfs �! b

! dS

� �

ð11Þ

where / is the porosity; Dsolution is the bulk solution dif- fusivity; Vf is the volume of fluid; Sfs is the area of the

fluid–solid interface; nfs �! is the normal to the fluid–solid

interface directed from the fluid to the solid phase.

3.1.4 Boundary conditions

In the molecular diffusivity experiment simulation based

on the solution of Fick’s equations, the rate of reaction of

the solid is assumed to be zero: there is no reaction

occurring at the fluid–solid interface. Then the boundary

condition at fluid–solid interface is:

� nfs�! � rc �!

¼ 0 ð12Þ

where nfs �! is the normal to the fluid–solid interface

directed from the fluid to the solid phase.

Besides this fluid–solid interface condition, a one-voxel-

wide plane of solid is added on the faces of the image that

are not perpendicular to the main diffusion direction. This

allows isolation of the sample from the outside.

Boundary conditions at inlet and outlet require knowl-

edge of the concentrations in the reservoirs. These con-

centrations evolve over time:

Vr oCin

ot ¼ Z

Sin

nSin ��! � r

! cdS ð13Þ

Vr oCout

ot ¼ �

Z

Sout

nSout ��! � r

! cdS ð14Þ

when Sin and Sout are, respectively, the input and output

face of the sample; nSin ��!

and nSout ��!

are, respectively, the

normal to the input and output face.

In the molecular diffusivity tensor calculation by vol-

ume averaging, the vectorial problem that is solved in this

case is closed by imposing periodic boundary conditions to

b !

and the geometry. The fluid–solid interface condition

has the following similar form:

� nfs�! � r !

b !

¼ nfs�! ð15Þ

Then, we used a finite volume method to solve the

equation systems. The finite volume method (FVM) is a

method for representing and evaluating partial differential

equations in the form of algebraic equations. Similar to the

finite difference method or finite element method, values

are calculated at discrete places on a meshed geometry.

‘‘Finite volume’’ refers to the small volume surrounding

each node point on a mesh. In the finite volume method,

volume integrals in a partial differential equation that

contain a divergence term are converted to surface inte-

grals, using the divergence theorem. These terms are then

evaluated as fluxes at the surfaces of each finite volume.

Because the flux entering a given volume is identical to

Petroleum Science (2018) 15:68–76 73

123

that leaving the adjacent volume, these methods are con-

servative. Another advantage of the finite volume method

is that it is easily formulated to allow for unstructured

meshes. The method is used in many computational fluid

dynamics packages. In this paper, the discretization

scheme assumes that the voxel is isotropic (cubic). Once

discretized, the closure equation system can be written as

Ax = b, A being a sparse, symmetric matrix. The equation

system is solved using a fully implicit method (matrix

inversion). The PETSc (Portable, Extensible Toolkit for

Scientific Computation) library is used for the direct res-

olution of the linear system. An iterative resolution with a

conjugate gradient and ILU (Incomplete lower and upper

triangular factorization method) preconditioner is

Fig. 8 Visualization of the concentration field in an experiment simulation with molecular diffusion

Table 2 The computed results of molecular diffusivity

X direction, 10 -3

m 2

s -1

Y direction, 10 -3

m 2

s -1

Z direction, 10 -3

m 2

s -1

0.042 0.056 0.051

0

0.05

0.10

0.15

0.20

0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

M ol

ec ul

ar d

iff us

iv ity

Porosity

Maxwell

Weissberg

Numerical simulation

Analytic solution

Fig. 9 Comparison of empirical laws and numerical simulations for the determination of molecular diffusivity with respect to the porosity

74 Petroleum Science (2018) 15:68–76

123

performed. The convergence criterion used is the relative

decrease of the residual l2-norm.

3.2 The result of molecular diffusivity simulation

The default parameters simulate an experiment along the Z axis

with a concentration in the input reservoir at initial time of

1711 mol m -3

(Cin (t) = 1711 mol m -3

) and the concentra-

tion in the output reservoir at initial time being null (Cout (t) = 0). The default solution bulk diffusivity is 1 m

2 s -1

.

Then, the direction of the molecular diffusion can be adjusted to

X, Y, or Z direction. If several directions are selected, the

computations will be done successively. The concentration

values used as boundary conditions of the experiment can also

be modified. Modifying these values will not change the

molecular diffusivity, which is intrinsic to the porous medium.

It will only modify the output concentration field.

The resulting visualization in X, Y, and Z direction,

respectively, should look like Fig. 8. We can observe the

decrease of the concentration from the input reservoir (in

yellow), to the output reservoir (in light blue).

With these parameters, we compute the full intrinsic

diffusivity tensor. A full tensor computation requires three

computations, each equivalent in time and memory con-

sumption to one experiment simulation. Experiment sim-

ulation in each direction gives the following results (shown

in Table 2).

3.3 Comparison results

We base our validation on several studies reporting the

results of molecular diffusivity experimental measurements

and empirical laws aimed at computing the molecular

diffusivity of a material. Glemser (2008) reported the fol-

lowing analytical estimations of molecular diffusivity with

respect to the porosity /:

Maxwell 1881ð Þ : / D

Dsolution ¼

2/ 3 � /

ð16Þ

Weissberg 1963ð Þ : / D

Dsolution ¼

/ 1 � lnð/Þ=2

ð17Þ

We compare our values to the values computed with

empirical laws [Eqs. 16, 17)] and the analytic solution. All

these results are displayed in Fig. 9. It can be seen from

Fig. 9 that our computed result is in good agreement with

the other results as well as the analytic solution.

4 Conclusions

1. Nano-CT scanning can be used to photograph the true

pore structure of tight sandstone. The image

segmentation method based on experiment-measured

porosity is established to binarize the digital image of

tight sandstone.

2. The pore size distribution is obtained based on the

digital core model of tight sandstone. It has been found

that the radii of pores of this sample are mainly

distributed in the range of 0.43–1 microns, and few

80-nm-sized pores are captured at the current

resolution.

3. By the use of finite volume method, the molecular

diffusion process of gas in real pore space can be

simulated visually based on the digital core model of

tight sandstone, and the diffusion coefficient can also

be calculated, which is in good agreement with the

others’ empirical laws results.

With the development of computer technology, digital

rock physics will become an important technical means to

participate in the exploration and development of tight oil

and gas. Our study provides a new research method for

digital rock physics of tight sandstone. It is a foundation

work for the future tight gas percolation theory and

numerical simulation research.

Acknowledgements This study is supported by Open Fund (PLN1506) of State Key Laboratory of Oil and Gas Reservoir

Geology and Exploitation, Chinese National Natural Science Foun-

dation (41502287), Chongqing Basic and Frontier Research Projects

(CSTC2015JCYJBX0120), Chongqing City Social Undertakings and

Livelihood Protection Science and Technology Innovation Special

Project (CSTC2017SHMSA120001), Chongqing Land Bureau Sci-

ence and Technology Planning Project (CQGT-KJ-2017026,CQGT-

KJ-2015044,CQGT-KJ-2015018, CQGT-KJ-2014040).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://crea

tivecommons.org/licenses/by/4.0/), which permits unrestricted use,

distribution, and reproduction in any medium, provided you give

appropriate credit to the original author(s) and the source, provide a

link to the Creative Commons license, and indicate if changes were

made.

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