research article lattice boltzmann simulation of...

Research ArticleLattice Boltzmann Simulation of Permeability and Tortuosityfor Flow through Dense Porous Media

Ping Wang

Department of Engineering Mechanics State Key Laboratory for Structural Analysis of Industrial EquipmentDalian University of Technology Dalian 116024 China

Correspondence should be addressed to Ping Wang wangpdluteducn

Received 17 December 2013 Revised 26 May 2014 Accepted 26 May 2014 Published 20 July 2014

Academic Editor Shaoyong Lai

Copyright copy 2014 Ping Wang This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Discrete element method (DEM) is used to produce dense and fixed porous media with rigid mono spheres Lattice Boltzmannmethod (LBM) is adopted to simulate the fluid flow in interval of dense spheres To simulating the same physical problem thepermeability is obtained with different lattice number We verify that the permeability is irrelevant to the body force and the medialength along flow direction The relationships between permeability tortuosity and porosity and sphere radius are researchedand the results are compared with those reported by other authors The obtained results indicate that LBM is suited to fluid flowsimulation of porous media due to its inherent theoretical advantages The radius of sphere should have ten lattices at least and themedia length along flow direction should be more than twenty radii The force has no effect on the coefficient of permeability withthe limitation of slow fluid flow For mono spheres porous media sample the relationship of permeability and porosity agrees wellwith the K-C equation and the tortuosity decreases linearly with increasing porosity

1 Introduction

Fluid flow through porous media has been widespreadconcern It plays an important role in the seepage drainageand oil engineering Engineers often need to predict the fluidflow rate of water or oil in porous media A simple empiricalmethod called Darcyrsquos law is often used [1] the formula reads

119896119894119895

= minus120583

119901119895

⟨V119894⟩ (1)

where V119894is outflow speed in the direction 119894 ⟨⟩ represents

volume average 120583 is the viscosity coefficient of the fluid 119901119895is

the pressure gradient in direction 119895 and 119896119894119895is the permeability

matrix Permeability matrix is assumed to be diagonal andpositive definite The elements of permeability matrix canbe measured by experimental methods or estimated bycomputational methods

Many factors affect permeability such as porous mediastructure fluid physical properties and degree of fluidsaturation This paper studies the impact of porous mediastructure only The traditional predictive research is usually

concerned with the relationship between permeability andporosity In addition to that specific surface of the particleshape of the particle and tortuosity also have a significantimpact on permeability But effects of these factors are hard oreven impossible to measure accurately these problems needpore-scale simulation of fluid flow in porous media

Because of the complexity of fluid flow in porous mediaexcept for some very simple question it is difficult toget the analytical solution With the development of theporous media geometry visualization computing power andcomputational methods it is possible to simulate directlyfluid flow in pore-scale porous media

TheLBMmethodhas been applied to this field since it wasintroduced in 1989 Ghassemi and Pak [2] studied the effectsof permeability and tortuosity on flow through particulatemedia by LBM and DEM the relationships between perme-ability and tortuosity with other parameters such as particlesdiameter grain specific surface and porosity were identifiedThe results show that the relationship between permeabilityand porosity of the porous media agreed the K-C formulawell but there is a bit deviation in the high porosity region

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 694350 7 pageshttpdxdoiorg1011552014694350

2 Mathematical Problems in Engineering

tortuosity almost linearly decreases with the porosity and theparticle diameter does not affect tortuosity any longer after itbecomes large enough

Sun et al [3] introduced a multiscale methodMicrostructural attributes such as occludedconnectedporosity and geometrical tortuosity are extracted usingnew computational techniques to form digital images ofporous media and a LBMFEM scheme is used to obtainhomogenized effective permeability at specimen scaleThe results show that the permeability depends not onlyon the porosity but also on the connectivity and thetortuosity

Experimental measurements and LBM simulation onsamples of packed particle beds were researched by Videlaet al [4] Due to the resolution limit when the particlediameter is greater than 500microns the experimental resultsand numerical results are consistent The impact of therelaxation time on permeability was also studied Vidal etal [5] verified that the LBM numerical permeability of theporous media of spherical particles agrees with experimentaldata very well When the size distribution of sphericalparticles disperses to a certain extent the results start todeviate fromK-C formula and a modified formula was givenaccording to the results Jeong [6] studied the micro flowsthrough granular porousmedia at various Knudsen numbersthe correlation between the permeability the porosity andthe Knudsen number is derived a new fitting formula isgiven and the effect of rarefaction on the permeability is alsodiscussed van der Hoef et al [7] simulated fluid flow pastmono- and bidisperse random arrays of spheres with LBMthe calculated permeability is consistent with experimentalresults and a new drag force formula is given The effectof the arrangement of the particles and particle shape onpermeability anisotropy was researched by Stewart et al[8]

In this work the permeability and tortuosity of fluidflow in three-dimensional dense porous media have beensimulated and analyzed The effects of porosity and the sizeof particle have been studied The DEM is described inSection 2 Numerical model of LBM is present in Section 3Calculation results are given in Section 4 and the Conclusionin Section 5

2 Discrete Element Method andPorous Media Generation

21 Discrete Element Method (DEM) DEM is an explicitnumerical method for modeling motion of distinct particlesSince this method was introduced by Cundall and Strack [9]it has become a powerful tool and has been used to researcha wide range of problems in science and engineering

In DEM the granular porous media are regarded as anassembly of distinct rigid particles When overlap occursbetween particles (or particle andwall) the contact force sep-arates them from each other Using the force-displacementlaw the force system of each particle can be summed Thedisplacement and velocity of each particle can be calculatedby Newtonrsquos second law of motion

The force-displacement law gives the relationshipbetween the contact force and the overlap between particles(or particle and wall) by the following equations

Δ119865119899= 119870119899Δ119899 Δ119865

120591= 119870120591Δ120591 (2)

where Δ119865 119870 and Δ are increments of force stiffness andoverlap betweenparticles respectivelyThe superscripts119899 and120591 represent normal and tangential direction of overlap plane

The contact forces consist of normal force and tangentialforce The tangential force is related to the normal forceby the Coulomb frictional law If 119865

120591

119901119894le 120583119904119865119899

119901119894 relative

translation motion will not occur Here 120583119904is the sliding

friction coefficient between the contact particles and119865120591

119901119894119865119899119901119894

are the normal and tangential forces on particle 119901 in position119894

Newtonrsquos second law of motion describes the relationshipbetween the acting forces and movement of particle Forsphere 119901 the law reads

119898119901119901119894

= sum(119865120591

119901119894+ 119865119899

119901119894) minus 119865119889

119901119894

119868119901

120579119901119894

= sum[(F120591119901times R119901)119894] minus 119872

119889

119901119894

(3)

where 119898119901 119868119901are the mass and inertia of sphere particle 119901

119901119894

and 120579119901119894

are linear and angular accelerations in direction119894 119865119889119901119894 119872119889119901119894

are the global damp force and moment R119901is the

vector from the center of sphere to the contact point

22 Porous Media Generation Using DEM The first step ofmodeling is producing dense assembly of granular particlesWe retrieve it with opening LAMMPS code The details areshown in Figures 1 and 2 A domain whose size is 100 times

100 times 1000 by lattice unit is created first Periodic boundarycondition is used in the two horizontal directions In verticaldirection rigid and impassable planes are set at the top andbottom boundaries

The rigid particles are created in a small region nearthe top boundary first in order to reduce the probabilityof overlap the ratio of particles volume to domain is set toten percent The position of each particle will be determinedwhen it does not overlap with any other particles Becausethe percentage of particles is low this process is easy toimplement

After the loose assembly is created in the specific regionit starts to fall down under gravity (Figure 1) When theparticles leave the initial region totally new particles will becreatedThis process is repeated until enough particles appearin computational domain The particles interact with otherparticles and the dense assembly of particles forms undergravity at last

To get loose porous media with a desired porosity theradii of particles are shortened with same scale but theamount and positions of particles are unchanged

Mathematical Problems in Engineering 3

Gravity

Periodic directions

Figure 1 Generation of dense porous media

3 Lattice Boltzmann Method andSimulation of Fluid Flow

31 LBM The simplest LBM model is known as the BGK-Boltzmann model It was first introduced by Bhatnagar et al[10] The evolutional equation reads

119891120572(r + e

120572120575119905 119905 + 120575119905) minus 119891

120572 (r 119905) = minus1

120591[119891120572 (r 119905) minus 119891

119890119902

120572(r 119905)]

(4)

where 119891120572 r 119905 and e

120572are probability distribution function of

particle in direction 120572 space position time and microscopicvelocity respectively 119891

120572moves from r to r + e

120572in a time

step The right term indicates the colloids function thatis relaxation function Probability distribution function ofparticle approaches 119891

119890119902

120572(r 119905) (the local equilibrium distribu-

tion function) 119891119890119902120572

(r 119905) is the function of local density andvelocity The relaxation factor 120591 indicates relaxation speedrate and is related to fluid viscosity coefficient by 120591 = 3] +

05 gt 05 Every evolution procedure consists of two stages apropagation stage and a collision stage Under limitation oflow fluid flow the steady solution of (4) is the solution ofNavier-Stokes equations

The final fluid flow state is evidently influenced byboundary conditions The interface of the particles and fluidis assumed as a no-slip boundary The simplest methodwhich is called half-way bounce-back method is adoptedhere The pressure drop is implemented by placing a bodyforce on fluid Periodic boundary condition is used in alldirections The D3Q19 model is used to calculate the fluidflow among the interval of porous media (Figure 3) Thismodel includes a stationary part and eighteen moving partsThey are e

120572 = 119864

119879i j k119879 here e

120572 = e

0 e1 e2 e

18

and 119864 = 0 1 minus1 0 0 0 0 1 minus1 1 minus1 1 minus1 1 minus1 0 0 0 0

0 0 0 1 minus1 0 0 1 minus1 minus1 1 0 0 0 0 1 minus1 1 minus1

0 0 0 0 0 1 minus1 0 0 0 0 1 minus1 minus1 1 1 minus1 minus1 1

Local equilibrium distribution function is given by

119891119890119902

120572= 120588119908120572(1 +

e120572sdot u

1198882119904

+(e120572sdot u)2

21198884119904

minus1199062

21198882119904

) (5)

where 120588 119888119904 u and 119908

120572are density sound velocity velocity

vector and constant coefficient respectively The soundvelocity equals radic33 and the constant coefficients are

119908119894=

1

3 119894 = 0

1

18 119894 = 1 minus 6

1

36 119894 = 7 minus 18

(6)

New density 120588(r 119905) and velocity k(r 119905) of fluid can beextracted fromprobability distribution function of particle byequations

sum

120572

119891120572= 120588 sum

120572

119891120572e120572= 120588u (7)

32 Simulation of Fluid Flow Using LBM As input conditionassembly of dense particles created by DEM is transmittedinto LBM code Lattice is signed by TRUE if it does not locateinside any particle FAULSE if it does All particles are fixedduring simulation procedure and they interact with fluid bybounce-back boundary condition [11]

Lattice should be fine enough to catch the details of theparticles We place ten lattices along the radius of spherein our simulation The computer load is quite big and anelementary lattice structure is 240 times 100 times 100

In flow direction additional layers are added to the endsof domain The path of fluid flow is shown in Figure 4 Peri-odic boundary condition is used in the other two directions

A uniform body force drives fluid to flow the GZSmodelis adopted in our simulation [11] All the simulations startfrom a zero velocity filed convergence to the steady solutionis examined by the following criterion

error =sum (

10038161003816100381610038161003816k119905+1 minus k119905

10038161003816100381610038161003816)

sum |k119905|lt 10minus6

(8)

The summation covers all the fluid lattice sites and super-script denotes the time steps

4 Results and Discussions

The permeability of the porous media can be computed byformula (1) from steady flow of fluidThe tortuosity is definedby119879 = sum |k(r)| sum |V

119894(r)| where k(r) and V

119894(r) are the velocity

vector and the velocity component in flow direction 119894 atposition 997888

119903 respectively

Permeability and Lattice Number of the Radii The space isdispersed into regular lattice point in the LBM The porousmedia particles are described by the lattice point In order to


10R

10R

20R

45R

Figure 2 Computational domain and the whole sample

X

Y

Z

12

4

3 7

8

9

10

0

11

1312

14

15

16

17

18

5

6

Figure 3 Particle distribution function on a D3Q19 lattice

Additional layers Additional layersPacked particles

Force

Figure 4 Sketch of fluid flow path

accurately capture the shape of the particles enough latticesmust be placed along the radius of the particle The more thelattices are the more accurate the particles are depicted butit also brings the problem of excessive calculationThereforewe need to find a balance point between accuracy andcomputation burden

In this paper the lattice unit is applied always To sim-ulating the same physical problem the key criteria numbersof simulations and real physical problem should be matchedThese key criteria numbers are reference length 119871

119903 reference

density 120588119903 and reference velocity 119906

119903 They are read

119871119903=

1198711015840

119871 120588

119903=

1205881015840

120588 119906

119903=

1198881015840

119904

119888119904

119871119903119906119903=]1015840

] (9)

here 119871 120588 119905 and ] are length density time and viscosity inlattice unit respectively and119871

10158401205881015840 1199051015840 ]1015840 and 1198881015840

119904are correspond

parameters in physical unit respectivelyTherefore the viscosity and body force must be adjusted

simultaneously The permeability is obtained with differentlattice number Figure 5 shows the relationship between

permeability and the number of lattices among the radiusthe permeability enlarges with increase of lattice numberFor controlling the computational time (consideringmachinememory and convergence time) ten lattices are placedalong the radial direction (red data points) According toVidela et alrsquos [4] work the difference between experimentalmeasurements and numerical simulations is less than 30when the ratio of diameter to resolution is bigger than 25 Ifthe ratio is less than 20 (about 175 and 125) the differenceis very big Our numerical results are consistent with Videlarsquosconclusion

Permeability and Media Length Along the flow direction thesample of porousmedia should guarantee enough lengthThepermeability of the media should be independent of medialength in the flow direction Fluid flow in porous mediafor different lengths of sample is simulated (see Figure 2)Figure 6 shows the variations of permeability versus themedia length in the flow direction From the figure when theratio of channel length to radius is more than 20 variation ofpermeability is very smallThismedia length is used in all oursimulations (red data points)

Permeability and Body Force Similarly the permeability ofporous media should not depend on the body force (pressuregradient) Figure 7 depicts the permeability results of thesame sample under different body force The permeabilityremains unchanged when the body force is small the per-meability is significantly increasing when the body forcebecomes larger It should be noted that maximum speed is00124 when the body force is 31623119890 minus 4 In the followingcalculation the body force is taken as 1119890 minus 5 (the red datapoint in Figure 7)

Permeability and Porosity About relationship between poros-ity (void ratio) and permeability researchers proposed somedifferent equations Carman introduced and extended the K-C equation [12] the equation reads

119896 = 981 times 104 1

11987820119862119862minus119870

1198903

1 + 119890= 1198621

1198903

1 + 119890 (10)

1198621is a constant parameter which is related to the specific

surface area tortuosity and fluid physics and 119890 = 120576(1 minus 120576) isvoid ratio Ghassemirsquos simulation gave 119896 = 119862

2(11989042

(1+119890)) [2]In order to study the influence of porosity on permeability aseries of porousmedia samples with uniform sphere particlesis simulated by LBM All the samples are made of sphereparticles with the same diameter but they are different in


4 6 8 10 12 14 16024

026

028

03

032

034

Lattices in a radius

Perm

eabi

lity

Figure 5 Variations of permeability versus lattices along radius

5 10 15 20 25 30 35 40 45 50027

028

029

03

031

032

033

LR

Perm

eabi

lity

Figure 6 Variations of permeability versus length

porosity In Figure 8 the relationship between regularizationof the permeability 119896

1015840= 119896119862 and void radio is shown The

curve fits well with the K-C equation Using least squaresmethod we have 119896 = 119862

3(1198903(1 + 119890)) = 18424(119890

3(1 +

119890)) When the void ratio is less than 1 the three resultsare similar for large porosity our numerical results arealso consistent with K-C equation but are different fromGhassemirsquos results Ghassemirsquos simulation is based on two-dimensional porous media and his model cannot capture allthe physical characteristics of real porous media the authorelaborated this fact in his paper

Permeability and Particle Size Figure 9 depicts the relation-ship between permeability and particle size the porosityremains constant (120576 = 05) in all samples According to theK-C equation the permeability is inversely proportional tothe square of the specific surface area For the homogeneoussphere porous media the permeability should be propor-tional to the square of the diameter From our results the

02821

02822

02823

02824

02825

02826

02827

Force

Perm

eabi

lity

10minus6

10minus5

10minus4

10minus3

Vel = 392e minus 4

Vel = 00124

Figure 7 Variations of permeability versus body force

05 1 15 20

05

1

15

2

25

3

Void ratio

Nor

mal

ized

per

mea

bilit

y

Kozenc-CarmanGhassemiNumerical simulation

Figure 8 Variations of permeability versus porosity

relationship between the permeability and the radius does notseem to comply with this rule In fact it is because the rangeof particle radius is narrow and the permeability is basicallyproportional to the radius

Tortuosity and Porosity For the relationship between tortu-osity and porosity different researchers have given differentresults also Koponen and Ghassemirsquos results show that thetortuosity decreases linearly when porosity increases [13]Yet Boudreau gave the nonlinear relationship [14] Diasbelieves that exponential relationship is better [15] Forporous media of uniform particles Figure 10 depicts therelationship between tortuosity and porosity The tortuosityessentially decreases linearly with increasing porosity Butour result is much lower than the absolute value of Koponenand Ghassemirsquos results The reason is that the dead end will


9 95 10 105 11 115085

09

095

1

105

11

115

12

125

13

135

Radius

Perm

eabi

lity

Figure 9 Variations of permeability versus particle radius

035 04 045 05 055 06 065108

11

112

114

116

118

12

Porosity

T

Numerical simulationminus042 lowast porosity + 135

Figure 10 Variations of tortuosity versus porosity

not form in the three-dimensional porous media samplesFor example when the porosity is equal to 05 Koponenand of Ghassemirsquos results are 14 and about 133 respectivelyour result is 114 For the slope of the curve the resultsof our calculations (about minus042) and Ghassemi (minus045)are relatively close and far from Koponenrsquos result (minus08)The porous media in Koponenrsquos paper are composed bysmall rectanglesThe channel among porous media has moretwists and turns and the permeability is more sensitive tothe porosity Therefore when the porosity increases thetortuosity decreases faster

5 Conclusion

DEM and LBM are used to simulate the fluid flow in thedense porous media the ability of LBM to resolve this

problem is confirmedThe relationship between permeabilitytortuosity and porosity and particle size is researched In ourcalculation all the sample particles are uniform size rigidand fixed high porosity samples are created through rescalingof dense samples We obtain the following conclusions

LBM is suitable for the simulation of flow in porousmedia because of its inherent advantages The results showthat the radius of the sphere should not be less than tenlattices the ratio of channel length to radius should be largerthan 20 Under the limitation of low speed body force haslittle influence on the calculation results If relaxation timeis equal to one the convergence can be achieved quicklyor the convergence is extremely slow For porous mediawith uniform spheres the relationship of permeability andporosity basically meets the K-C formula and the tortuositydecreases linearly with increasing porosity

Simulation of samples with a wide range of particle sizewill be run in further work and the permeability calculationof nonuniform and nonspherical particle samples will bestudied also

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The project was supported by the National Natural ScienceFoundation (11102034) and the Important National Scienceamp Technology Specific Project (2011ZX02403-004)

References

[1] HDarcyDetermination des Lois drsquo Ecoulement de lrsquoeau aTraversle Sable Les Fontaines Publiques de la Ville de Dijon VictorDalmont Paris France 1856

[2] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[3] W C Sun J E Andrade and J W Rudnicki ldquoMultiscalemethod for characterization of porous microstructures andtheir impact on macroscopic effective permeabilityrdquo Interna-tional Journal for Numerical Methods in Engineering vol 88 no12 pp 1260ndash1279 2011

[4] A R Videla C L Lin and J D Miller ldquoSimulation of satu-rated fluid flow in packed particle bedsmdashthe lattice-Boltzmaanmethod for the calculation of permeability from XMT imagesrdquoJournal of the Chinese Institute of Chemical Engineers vol 39 no2 pp 117ndash128 2008

[5] D Vidal C Ridgway G Pianet J Schoelkopf R Roy andF Bertrand ldquoEffect of particle size distribution and packingcompression on fluid permeability as predicted by lattice-Boltzmann simulationsrdquo Computers and Chemical Engineeringvol 33 no 1 pp 256ndash266 2009

[6] N Jeong ldquoAdvanced study about the permeability for micro-porous structures using the lattice Boltzmann methodrdquo Trans-port in Porous Media vol 83 no 2 pp 271ndash288 2010


[7] M A van der Hoef R Beetstra and J A M KuipersldquoLattice-Boltzmann simulations of low-Reynolds-number flowpast mono- and bidisperse arrays of spheres results for thepermeability and drag forcerdquo Journal of Fluid Mechanics vol528 pp 233ndash254 2005

[8] M L Stewart A L Ward and D R Rector ldquoA study of poregeometry effects on anisotropy in hydraulic permeability usingthe lattice-Boltzmann methodrdquo Advances in Water Resourcesvol 29 no 9 pp 1328ndash1340 2006

[9] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[10] P L Bhatnagar E P Gross andMKrook ldquoAmodel for collisionprocesses in gases I Small amplitude processes in charged andneutral one-component systemsrdquo Physical Review vol 94 no3 pp 511ndash525 1954

[11] Z L Guo and C G Zheng Theory and Applications of LatticeBoltzmann Method Science Press Beijing China 2009

[12] P C Carman ldquoFluid flow through granular bedsrdquo Transactionsof the Institution of Chemical Engineers vol 15 pp 150ndash166 1937

[13] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porousmediardquoPhysical Review E StatisticalPhysics Plasmas Fluids and Related Interdisciplinary Topicsvol 56 no 3 pp 3319ndash3325 1997

[14] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[15] R Dias J A Teixeira M Mota and A Yelshin ldquoTortuosityvariation in a low density binary particulate bedrdquo Separationand Purification Technology vol 51 no 2 pp 180ndash184 2006

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tortuosity almost linearly decreases with the porosity and theparticle diameter does not affect tortuosity any longer after itbecomes large enough

Sun et al [3] introduced a multiscale methodMicrostructural attributes such as occludedconnectedporosity and geometrical tortuosity are extracted usingnew computational techniques to form digital images ofporous media and a LBMFEM scheme is used to obtainhomogenized effective permeability at specimen scaleThe results show that the permeability depends not onlyon the porosity but also on the connectivity and thetortuosity

Experimental measurements and LBM simulation onsamples of packed particle beds were researched by Videlaet al [4] Due to the resolution limit when the particlediameter is greater than 500microns the experimental resultsand numerical results are consistent The impact of therelaxation time on permeability was also studied Vidal etal [5] verified that the LBM numerical permeability of theporous media of spherical particles agrees with experimentaldata very well When the size distribution of sphericalparticles disperses to a certain extent the results start todeviate fromK-C formula and a modified formula was givenaccording to the results Jeong [6] studied the micro flowsthrough granular porousmedia at various Knudsen numbersthe correlation between the permeability the porosity andthe Knudsen number is derived a new fitting formula isgiven and the effect of rarefaction on the permeability is alsodiscussed van der Hoef et al [7] simulated fluid flow pastmono- and bidisperse random arrays of spheres with LBMthe calculated permeability is consistent with experimentalresults and a new drag force formula is given The effectof the arrangement of the particles and particle shape onpermeability anisotropy was researched by Stewart et al[8]

In this work the permeability and tortuosity of fluidflow in three-dimensional dense porous media have beensimulated and analyzed The effects of porosity and the sizeof particle have been studied The DEM is described inSection 2 Numerical model of LBM is present in Section 3Calculation results are given in Section 4 and the Conclusionin Section 5

2 Discrete Element Method andPorous Media Generation

21 Discrete Element Method (DEM) DEM is an explicitnumerical method for modeling motion of distinct particlesSince this method was introduced by Cundall and Strack [9]it has become a powerful tool and has been used to researcha wide range of problems in science and engineering

In DEM the granular porous media are regarded as anassembly of distinct rigid particles When overlap occursbetween particles (or particle andwall) the contact force sep-arates them from each other Using the force-displacementlaw the force system of each particle can be summed Thedisplacement and velocity of each particle can be calculatedby Newtonrsquos second law of motion

The force-displacement law gives the relationshipbetween the contact force and the overlap between particles(or particle and wall) by the following equations

Δ119865119899= 119870119899Δ119899 Δ119865

120591= 119870120591Δ120591 (2)

where Δ119865 119870 and Δ are increments of force stiffness andoverlap betweenparticles respectivelyThe superscripts119899 and120591 represent normal and tangential direction of overlap plane

The contact forces consist of normal force and tangentialforce The tangential force is related to the normal forceby the Coulomb frictional law If 119865

120591

119901119894le 120583119904119865119899

119901119894 relative

translation motion will not occur Here 120583119904is the sliding

friction coefficient between the contact particles and119865120591

119901119894119865119899119901119894

are the normal and tangential forces on particle 119901 in position119894

Newtonrsquos second law of motion describes the relationshipbetween the acting forces and movement of particle Forsphere 119901 the law reads

119898119901119901119894

= sum(119865120591

119901119894+ 119865119899

119901119894) minus 119865119889

119901119894

119868119901

120579119901119894

= sum[(F120591119901times R119901)119894] minus 119872

119889

119901119894

(3)

where 119898119901 119868119901are the mass and inertia of sphere particle 119901

119901119894

and 120579119901119894

are linear and angular accelerations in direction119894 119865119889119901119894 119872119889119901119894

are the global damp force and moment R119901is the

vector from the center of sphere to the contact point

22 Porous Media Generation Using DEM The first step ofmodeling is producing dense assembly of granular particlesWe retrieve it with opening LAMMPS code The details areshown in Figures 1 and 2 A domain whose size is 100 times

100 times 1000 by lattice unit is created first Periodic boundarycondition is used in the two horizontal directions In verticaldirection rigid and impassable planes are set at the top andbottom boundaries

The rigid particles are created in a small region nearthe top boundary first in order to reduce the probabilityof overlap the ratio of particles volume to domain is set toten percent The position of each particle will be determinedwhen it does not overlap with any other particles Becausethe percentage of particles is low this process is easy toimplement

After the loose assembly is created in the specific regionit starts to fall down under gravity (Figure 1) When theparticles leave the initial region totally new particles will becreatedThis process is repeated until enough particles appearin computational domain The particles interact with otherparticles and the dense assembly of particles forms undergravity at last

To get loose porous media with a desired porosity theradii of particles are shortened with same scale but theamount and positions of particles are unchanged


Gravity

Periodic directions




119891120572(r + e

120572120575119905 119905 + 120575119905) minus 119891

120572 (r 119905) = minus1

120591[119891120572 (r 119905) minus 119891

119890119902

120572(r 119905)]

(4)

where 119891120572 r 119905 and e




120572in a time


119890119902


tion function) 119891119890119902120572




120572 = 119864

119879i j k119879 here e

120572 = e

0 e1 e2 e

18





119891119890119902

120572= 120588119908120572(1 +

e120572sdot u

1198882119904

+(e120572sdot u)2

21198884119904

minus1199062

21198882119904

) (5)

where 120588 119888119904 u and 119908



119908119894=

1

3 119894 = 0

1

18 119894 = 1 minus 6

1

36 119894 = 7 minus 18

(6)


sum

120572

119891120572= 120588 sum

120572

119891120572e120572= 120588u (7)





error =sum (

10038161003816100381610038161003816k119905+1 minus k119905

10038161003816100381610038161003816)


(8)







119903 respectively



10R

10R

20R

45R


X

Y

Z

12

4

3 7

8

9

10

0

11

1312

14

15

16

17

18

5

6



Force




119903 reference



119871119903=

1198711015840

119871 120588

119903=

1205881015840

120588 119906

119903=

1198881015840

119904

119888119904

119871119903119906119903=]1015840

] (9)


10158401205881015840 1199051015840 ]1015840 and 1198881015840








119896 = 981 times 104 1

11987820119862119862minus119870

1198903

1 + 119890= 1198621

1198903

1 + 119890 (10)



2(11989042



4 6 8 10 12 14 16024

026

028

03

032

034


Perm

eabi

lity


5 10 15 20 25 30 35 40 45 50027

028

029

03

031

032

033

LR

Perm

eabi

lity





3(1198903(1 + 119890)) = 18424(119890

3(1 +



02821

02822

02823

02824

02825

02826

02827

Force

Perm

eabi

lity

10minus6

10minus5

10minus4

10minus3

Vel = 392e minus 4

Vel = 00124


05 1 15 20

05

1

15

2

25

3

Void ratio

Nor

mal

ized

per

mea

bilit

y






9 95 10 105 11 115085

09

095

1

105

11

115

12

125

13

135

Radius

Perm

eabi

lity


035 04 045 05 055 06 065108

11

112

114

116

118

12

Porosity

T




5 Conclusion







Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Gravity

Periodic directions




119891120572(r + e

120572120575119905 119905 + 120575119905) minus 119891

120572 (r 119905) = minus1

120591[119891120572 (r 119905) minus 119891

119890119902

120572(r 119905)]

(4)

where 119891120572 r 119905 and e




120572in a time


119890119902


tion function) 119891119890119902120572




120572 = 119864

119879i j k119879 here e

120572 = e

0 e1 e2 e

18





119891119890119902

120572= 120588119908120572(1 +

e120572sdot u

1198882119904

+(e120572sdot u)2

21198884119904

minus1199062

21198882119904

) (5)

where 120588 119888119904 u and 119908



119908119894=

1

3 119894 = 0

1

18 119894 = 1 minus 6

1

36 119894 = 7 minus 18

(6)


sum

120572

119891120572= 120588 sum

120572

119891120572e120572= 120588u (7)





error =sum (

10038161003816100381610038161003816k119905+1 minus k119905

10038161003816100381610038161003816)


(8)







119903 respectively



10R

10R

20R

45R


X

Y

Z

12

4

3 7

8

9

10

0

11

1312

14

15

16

17

18

5

6



Force




119903 reference



119871119903=

1198711015840

119871 120588

119903=

1205881015840

120588 119906

119903=

1198881015840

119904

119888119904

119871119903119906119903=]1015840

] (9)


10158401205881015840 1199051015840 ]1015840 and 1198881015840








119896 = 981 times 104 1

11987820119862119862minus119870

1198903

1 + 119890= 1198621

1198903

1 + 119890 (10)



2(11989042



4 6 8 10 12 14 16024

026

028

03

032

034


Perm

eabi

lity


5 10 15 20 25 30 35 40 45 50027

028

029

03

031

032

033

LR

Perm

eabi

lity





3(1198903(1 + 119890)) = 18424(119890

3(1 +



02821

02822

02823

02824

02825

02826

02827

Force

Perm

eabi

lity

10minus6

10minus5

10minus4

10minus3

Vel = 392e minus 4

Vel = 00124


05 1 15 20

05

1

15

2

25

3

Void ratio

Nor

mal

ized

per

mea

bilit

y






9 95 10 105 11 115085

09

095

1

105

11

115

12

125

13

135

Radius

Perm

eabi

lity


035 04 045 05 055 06 065108

11

112

114

116

118

12

Porosity

T




5 Conclusion







Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10R

10R

20R

45R


X

Y

Z

12

4

3 7

8

9

10

0

11

1312

14

15

16

17

18

5

6



Force




119903 reference



119871119903=

1198711015840

119871 120588

119903=

1205881015840

120588 119906

119903=

1198881015840

119904

119888119904

119871119903119906119903=]1015840

] (9)


10158401205881015840 1199051015840 ]1015840 and 1198881015840








119896 = 981 times 104 1

11987820119862119862minus119870

1198903

1 + 119890= 1198621

1198903

1 + 119890 (10)



2(11989042



4 6 8 10 12 14 16024

026

028

03

032

034


Perm

eabi

lity


5 10 15 20 25 30 35 40 45 50027

028

029

03

031

032

033

LR

Perm

eabi

lity





3(1198903(1 + 119890)) = 18424(119890

3(1 +



02821

02822

02823

02824

02825

02826

02827

Force

Perm

eabi

lity

10minus6

10minus5

10minus4

10minus3

Vel = 392e minus 4

Vel = 00124


05 1 15 20

05

1

15

2

25

3

Void ratio

Nor

mal

ized

per

mea

bilit

y






9 95 10 105 11 115085

09

095

1

105

11

115

12

125

13

135

Radius

Perm

eabi

lity


035 04 045 05 055 06 065108

11

112

114

116

118

12

Porosity

T




5 Conclusion







Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4 6 8 10 12 14 16024

026

028

03

032

034


Perm

eabi

lity


5 10 15 20 25 30 35 40 45 50027

028

029

03

031

032

033

LR

Perm

eabi

lity





3(1198903(1 + 119890)) = 18424(119890

3(1 +



02821

02822

02823

02824

02825

02826

02827

Force

Perm

eabi

lity

10minus6

10minus5

10minus4

10minus3

Vel = 392e minus 4

Vel = 00124


05 1 15 20

05

1

15

2

25

3

Void ratio

Nor

mal

ized

per

mea

bilit

y






9 95 10 105 11 115085

09

095

1

105

11

115

12

125

13

135

Radius

Perm

eabi

lity


035 04 045 05 055 06 065108

11

112

114

116

118

12

Porosity

T




5 Conclusion







Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










9 95 10 105 11 115085

09

095

1

105

11

115

12

125

13

135

Radius

Perm

eabi

lity


035 04 045 05 055 06 065108

11

112

114

116

118

12

Porosity

T




5 Conclusion







Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article lattice boltzmann simulation of...

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