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INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech ., 2003; 27 :565583 (DOI: 10.1002/nag.286)

Numerical solutions for ow in porous media

J.G. Wang 1, n ,y , C.F. Leung 2 and Y.K. Chow 2

1 Tropical Marine Science Institute, National University of Singapore, Singapore 119260, Singapore2 Department of Civil Engineering, National University of Singapore, Singapore 119260, Singapore

SUMMARY

A numerical approach is proposed to model the ow in porous media using homogenization theory. Theproposed concept involves the analyses of micro-true ow at pore-level and macro-seepage ow at macro-level. Macro-seepage and microscopic characteristic ow equations are rst derived from the Navier Stokes equation at low Reynolds number through a two-scale homogenization method. Thishomogenization method adopts an asymptotic expansion of velocity and pressure through the micro-

structures of porous media. A slightly compressible condition is introduced to express the characteristicow through only characteristic velocity. This characteristic ow is then numerically solved using a penaltyFEM scheme. Reduced integration technique is introduced for the volumetric term to avoid mesh locking.Finally, the numerical model is examined using two sets of permeability test data on clay and one set of permeability test data on sand. The numerical predictions agree well with the experimental data if constraint water lm is considered for clay and two-dimensional cross-connection effect is included forsand. Copyright # 2003 John Wiley & Sons, Ltd.

KEY WORDS : homogenization method; characteristic ow; penalty method; permeability; constraint waterlm; two-dimensional effect; anisotropy

1. INTRODUCTION

Problems involving seepage in porous media are important in many elds [17]. These includeseepage and consolidation problems in land reclamation [1] and underground waste disposal ingeotechnical engineering [2], lter problems in chemical engineering [3, 10], and bio-uid ow inbioengineering of human body [4]. Macro-analyses are generally carried out to tackle theaverage pore water pressure and velocity in porous medium [5]. However, macro-analysis hastwo major disadvantages. Firstly, it can only obtain the macro-pseudo velocity and pore waterpressure within the entire porous media. The true velocity and pore water pressure at pore levelis difficult or impossible to determine. In geotechnical engineering, the true velocity at pore levelis an important factor in the design of lter that protects the sand from boiling and piping [6, 7].

Contract/grant sponsors: National University of Singapore; contract/grant number RP3940674 Contract/grantsponsors: National Science and Technology Board of Singapore

Received 1 December 2001Revised 10 January 2003Copyright # 2003 John Wiley & Sons, Ltd.

y E-mail: [email protected]

n Correspondence to: J.G. Wang, Tropical Marine Science Institute, National University of Singapore, Singapore119260, Singapore

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Secondly, macro-analysis typically employs the coefficient of permeability in Darcys law toexpress the micro-structure effect of porous medium on micro-hydraulic ow resistance [8, 9].

Many studies have been carried out to investigate the effect of micro-structures on ow inporous media [2, 3, 611]. For example, Kim et al. [11] studied Darcys law within arepresentative volume element (RVE) through volume averaging theorem. A closed-form of Darcys law was developed from the interaction of pore water and solid particles. Their methodstill has two issues remaining to be solved. The rst is the relationship between the complicatedmicro-structures of porous media and the coefficient of permeability. In particular, theboundary condition for the RVE is not clear if the computation at micro-scale level is carriedout. The second is that both macro-scale problem and micro-scale problem are not clearlydened [10]. In the 1970s, a two-scale homogenization theory was developed based onasymptotic expansions [8]. The homogenization theory can consider the behaviourssimultaneously at both macro- and micro-scale levels. The rst important result usinghomogenization theory was obtained by Sanchez-Palencia in 1974 on ow through porousmedia [8]. Since then, most researches focused on a formal expansion of NavierStokesequation, for example in Reference [12]. Numerical approaches are also proposed [13, 14]. Lee

et al. [14] proposed a mixed mode of pressure and velocity for a solute transfer and dispersivityproblem. Numerical methods for homogenization theory are successful in composite materialsbecause their micro-structures are designed and determined. However, the situation ingeotechnical engineering is a little different. This is because the micro-structure of soil massesis usually unknown and only index such as void ratio or weight fraction [15] can be measured.Some schemes such as Sierpinski carpets [13] or well-random media [16] were proposed topartially solve this problem in computation. However, these models are not capable of handlingvariable soil conditions. In this paper, the concept of equivalent particle size, which can bedetermined experimentally, is proposed to join tackle the computation for a variety of soilconditions.

The mixed mode of velocity and pressure for incompressible NavierStokes equation has itslimitation. For example, the zero divergence condition of velocity eld will produce a discrete

algebraic system equation with zero diagonal terms if Galerkins formulation is used. Thislimitation of mixed mode can be circumvented by the penalty method as only velocities areincluded in the weak form. The number of unknowns is largely reduced [17], too. One drawbackfor the penalty method is the ill-conditioning of the system matrix if penalty parameter is notappropriately selected. When penalty parameter is sufficiently large, the integration forvolumetric term should be carefully treated. Otherwise, mesh locking may occur for somemeshes. The reduced integration approach [18] can avoid mesh locking. In the authorsknowledge, the penalty method has not been applied to the characteristic function of uid owin porous media.

This paper proposes a numerical approach to model the ow of porous media from micro-analysis. The coefficient of permeability of clay and sand is numerically computed. Thisnumerical approach is based on the two-scale asymptotic analysis of uid channels in porous

media. It is capable of handling micro-true ow analysis at pore-level and macro-seepageanalysis simultaneously. This is useful to understand the behaviours of seepage at both macro-and micro-scale levels. As a preliminary, pore wall is assumed to be rigid in this paper. Thecharacteristic equation at micro-scale level is numerically solved through a penalty method andreduced integration scheme. In order to carry out the computation in micro-scale level, anequivalent particle size was introduced into the unit cell. This largely simplies the complicated

Copyright # 2003 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech . 2003; 27 :565583

J. G. WANG, C. F. LEUNG AND Y. K. CHOW566

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micro-structures of porous media and makes the computation applicable in practice. As theequivalent particle size is determined from a pair of macro-scale experimental data, thegeometry of the pores is partially expressed by the equivalent particle. Using this numericalapproach, the micro-structural effects such as constraint water lm in clay and multi-dimensional effect in sand are studied. The numerical predictions are compared withexperimental data.

This paper is organized as follows. Firstly, the micro-uid ow is described through Navier Stokes equation at low Reynolds number along micro-pore channels. A two-scale homogeniza-tion theory based on asymptotic expansion is introduced to derive the macro-seepage equationand microscopic characteristic equation. The similarity between the Stokes ow and thecharacteristic ow is discussed. This similarity is helpful to construct the solution structures of the characteristic ow from the known solution of Stokes ow. Secondly, a penalty niteelement approach is developed for the solution of the characteristic equation for incompressibleow. Reduced integration is adopted for the volumetric term in order to avoid mesh locking.Thirdly, the proposed numerical model is veried by the comparison of the permeability of twosets of experimental data on clay and one set of experimental data on sand. The effects of

constraint water lm for clay and two-dimensional cross-connection for sand are investigatedand nally, concluded.

2. MICRO-CHANNEL FLUID FLOW IN POROUS MEDIA

Micro-channel uid ow in porous media is usually assumed to follow the NavierStokes owfor low Reynolds number. However, boundary conditions for such a problem are complicated.For the micro-channel uid ow, the governing equation and boundary conditions can begenerally expressed as follows:Equation of motion

@ P @ xi m

@2V i@ xi@ x j r X i 0 1

Continuity equation (incompressible case )

@V i@ xi

0 2

Boundary conditions

V i 0 on G

V i %

V V i on GF

or

P %

PP on GF 3

where V i is the true velocity in the pore channel of porous media, P the pore water pressure, mthe coefficient of viscosity, r the density of water, r X i the unit body force, G the uidsolidinterface, and xi is the ith component of co-ordinates.

%

V V i and %

PP are the prescribed velocity andpressure on uid external boundary, GF ; respectively. Index i is a free index and index j is a


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dummy index that denotes the summation. For two-dimensional ow i; j 1; 2 and for three-dimensional ow i; j 1; 2; 3:

An exact solution is normally required to evaluate the above equations along all porechannels in the porous media. This requires huge computation time because of complexity of micro-channels. An approximation of the true uid ow should include the main properties of uid ow in porous media at both macro- and micro-levels. Homogenization theory based onasymptotic expansions is an appropriate tool to describe the two-scale properties simulta-neously. In the homogenization approach, a local problem at micro-scale describes the micro-characteristic ow, and a macro-seepage problem at macro-scale analyses the average porepressure and velocity. The micro- and macro-ows are linked through the homogenizedcoefficients of permeability.

3. ASYMPTOTIC EXPANSION

3.1. Periodic expansion with scale parameter

Figure 1 expresses a typical porous medium domain of ow. The micro-channels arecomplicated and simplied one-dimensional (1D) or two-dimensional (2D) unit cells can beregarded as the fundamental cells. A scale parameter e is employed to link the unit cell andwhole ow domain:

y xe

4

where x are global co-ordinates and y local co-ordinates. True variables such as velocity andpressure are hence functions of e; denoted as V ei x and P ex: They can be expressed withfollowing asymptotic expansions:

V ei x e2V 0i x ; y e3V 1i x ; y

P ex P 0x ; y e P 1x ; y 5

The V ai x ; y and P ai x ; y a 0; 1; 2; . . . are functions of global co-ordinates x and local co-ordinates y. The Y-periodicity is expressed as

P ex ; y P ex ; y Y

V ai x ; y V ai x ; y Y

6

Body force of uid is expressed using the same expansion:

X ei x X 0i x e

1 X 1i x ; y e2 X 2i x ; y 7

Mathematical chain rules for scale parameter e are as@

@ xi)

@

@ xi

@

@y k

@y k

@ xi

@

@ xi

1

e

@

@y i8

for the rst-rank derivatives and

@2

@ xi@ xi)

@2

@ xi@ xi

2e

@2

@ xi@y i

1e2

@2

@y i@y i9

for the second-rank derivatives.



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The motion equation in Equation (1) is discussed. A e-series polynomial is obtained fromEquation (5) to (9). This polynomial can be expressed in the following form:

X1

a 1

. ea 0 10

Equation (10) should be held for any e; which implies that the coefficients denoted by . arezeros for all order e: The coefficients at different orders e express different physical meanings.For example, the coefficient for the e 1-term is

@ P 0

@y i 0 in Y F 11

where Y F is the uid domain in a unit cell. Equations (11) implies that P 0x ; y P 0x: Inanother word, the zero-order pore pressure is independent of local co-ordinates y. This result isof special importance because P 0xis the leading term of pore water pressure. This suggests that

the homogenization method be applicable for this case.A recursive relation is developed through the coefficient of e0-term. The governing equationfor the higher-order pore water pressure P 1 is obtained as

@ P 0

@ xi@ P 1

@y i m

@2V 0i@y k @y k

r X 0i 0 in Y F 12

The same expansion is applied to the continuity equation given by Equation (2):

e @V 0i@y i

e2 @V 0i

@ xi

@V 1i@y i 0 in Y F 13

This expansion is held for any scale parameter e: For the e1

-term, incompressible condition isagain obtained at micro-level as follows:

@V 0i@y i

0 in Y F 14

The recursive formula is obtained from the e2-term:

@V 0i@ xi

@V 1i@y i

0 in Y F 15

The boundary condition at the uidsolid interface, G; can be also expanded as follows:

e2V 0i e3V 1i 0 on G 16

The above equation is decomposed into

V 0i 0; V 1i 0; . . .on G 17


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The @ P 0=@ xk r X 0k k 1; 2; 3is a function in terms of x co-ordinates only. The two-scale co-ordinates are separated as follows:

V 0i @ P 0

@ xk r X 0k

vk i

P 1 @ P 0

@ xk r X 0k p k

k 1; 2; 3

18

where vk i and p k are the characteristic functions of velocity and pressure, respectively.

Normalization of the equations denoted by Equations (12) and (14) yields a characteristicequation for unit cell as well as no-slip and periodic conditions. This characteristic equation istermed as a local problem. The average equation over whole unit cell domain is called a macro-seepage problem.

3.2. Local problem

Equation of motion

@p k

@y i m

@2vk i@y j@y j

dik 0 k 1; 2; 3 19

Continuity equation (incompressible case )

@vk i@y i

0 k 1; 2; 3 20

Boundary conditions

vk i 0 on G

p k x ; y p k x ; y Y

vk i x ; y vk i x ; y Y ) periodicity condition on GF21

It is helpful to compare above governing equations and boundary conditions of vk i with theNavierStokes equations for velocity V i :

(1) NavierStokes ow expressed by Equation (1) has true body force, while characteristicow expressed by Equation (19) has characteristic body force f dik g: This characteristic bodyforce is not a physical force. It exists only for micro-characteristic ow when the macro-ow isalong that direction. Therefore, the characteristic body force is also termed as pseudo-bodyforce.

(2) Incompressible condition is true for both characteristic ow and NavierStokes ow. The

uid at micro-scale is still incompressible.(3) Both ows have no-slip boundary condition along the uidsolid interface G: This is the

physical description for the no-slip condition at both macro- and micro-scale levels.(4) However, boundary conditions are different on the uid boundary GF : The NavierStokes

ow gives velocity %

V V i; pressure %

PP or shear stress (a mixed boundary), while the characteristicow requires only periodicity on the articial uid boundary GF :



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Therefore, both ows should have similar structure of solutions. This similarity of solutionstructures provides a clue to pursue an analytical or numerical solution of characteristic owfrom the solution of NavierStokes ow.

3.3. Macro-problemSeepage equation

At macro-scale, the main concern is to obtain the average velocity or pore water pressure in aunit cell. The well-known Darcys law can be obtained through a volume averaging of truevelocity in a unit cell,

*

V V 0i : Particularly, the averaging of Equation (18) is given as*

V V 0i 1jY jZ Y F V 0i dV K ij r X 0 j @ P

0

@ x j 22where the permeability tensor, K ij ; is the volume average of characteristic velocity given as

K ij 1jY jZ Y F v ji dV 23

where jY j is the volume of the unit cell which includes the fraction of solid matrix. The

permeability of porous media is a macro-tensor. It is completely computed from the micro-owvelocity, v ji ; in terms of volume averaging or porosity of the porous media. Equation (23) istherefore a linkage between macro- and micro-analyses. The macro-seepage problem is obtainedthrough an average procedure of the continuity equation of Equation (15):

@@ xi

K ij@ P 0

@ x jr X 0 j 0 24

The permeability tensor can be determined analytically through characteristic ow withinmicro-uid channels. The key issue for the characteristic ow is how to compute thecharacteristic function v ji : A numerical procedure based on penalty nite element method isproposed.

4. WEAK FORMS OF LOCAL PROBLEM AND ITS DISCRETIZATION

4.1. Weak forms of local problem

A bulk modulus l is introduced to replaces the original incompressible condition with a slightlycompressible condition as follows:

@vk i@y i

p k

l 0 25

If l is big enough, the approximation is sufficiently accurate. For example, whenl ! 1 ; p k =l ! 0: Using Equation (25), the equation of motion Equation (19) can be

expressed as

l@2vk j

@y i@y j m

@2vk i@y j@y j

dik 0 26

Equation (26) signicantly reduces the complexity of the computation procedure because onlycharacteristic velocity is included. Its weak form is obtained by applying a weight function dvi


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(dvi 0 on G) that only varies with the fast variable y and is the Y-periodicity, dviy dviy Y:

Z Y F

l@2vk j

@y i@y j m

@2vk i@y j@y j !dvi dV

Z Y F

dvidik dV 27

By applying the GreenGauss theorem to the second term at the left-hand side, one gets

Z Y F m @2vk i

@y j@y jdvi dV I @Y F m @v

k i

@y jn jdvi d s Z Y F m@v

k i

@y j@dvi

@y jdV 28

The boundary term is zero because

I @Y F m@vk i

@y jn j dvi d s Z G m @v

k i

@y jn j dvi d s Z GF m @v

k i

@y jn j dvi d s 29

Since the rst term vanishes due to no-slip condition ( dvi 0 on G), and the second termvanishes due to periodicity on uid boundary GF : Similarly, the rst term of Equation (27)becomes as

Z Y F l @2vk j

@y i@y jdvi dV I @Y F l @v

k j

@y jnidvi d s Z Y F l @v

k j

@y j@dvi

@y idV Z Y F l @v

k j

@y j@dvi

@y idV 30

Thus, the weak form of Equation (26) is nally expressed as

Z Y F l @vk j

@y j@dvi

@y idV Z Y F m @v

k i

@y j@dvi

@y jdV Z Y F dvk dV 0 31

This is a weak form of the local problem with penalty method. When the bulk modulus l (aspenalty parameter) is determined, the characteristic velocity vk i can be uniquely solved.

4.2. Discretization of characteristic velocity

Fluid domain Y F is divided into n sub-domains, whose sub-domain Y s is an element. Eachelement has m nodes. The characteristic velocity in an element is approximated in terms of thenodal characteristic velocity V k ij and shape function F j :

vk i ffiXm

j1

V k ij F j 32

The weight function is approximated using the same shape function:

dvi ffiXm

j1

dV ij F j 33

where dV ij is the weight function at the jth node.Equation (31) can be discretized into

Kv Kvp Vk F k k 1; 2; 3 34

where the stiffness matrix associated with shear part is

K vrs Z Y s m @F r@y j @F s@y j dV 35Copyright # 2003 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech . 2003; 27 :565583


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and the stiffness matrix associated with volume part is

K vprs Z Y s l @F r@y i @F s@y j dV 36The characteristic loading induced by characteristic body force is

F k si dik Z Y s F s dV 37Special care should be taken for the numerical integrations of Kv and Kvp : Otherwise, meshlocking will take place for non-triangle elements. In this paper, the reduced integrationprocedure [18] is used for stiffness Kvp ; while normal integration is applied to stiffness Kv:

5. NUMERICAL PROCEDURES AND EXAMPLES

The micro-structures of porous media are complex. This paper simplies these micro-structuresinto 1D ow and 2D ow models as shown in Figure 1. With these models, the effect of bulkmodulus is studied. This numerical method is then applied to study the permeability of clay andarticial sand. For clay, the effect of constraint water lm around particles is studied in detail.

Figure 1. Seepage through porous media.


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For sand, multi-dimensional effect is studied by comparing 1D results with 2D results. Finally,anisotropic ow is numerically analysed.

5.1. Numerical procedure and computational parameter

Isoparametric elements with 8-nodes are used in computations. Shear stiffness, Kv; isnumerically treated by 3 3 point Gaussian integration, while volumetric stiffness, Kvp ; isnumerically treated by 2 2 point Gaussian integration. No-slip condition is imposed on uid solid boundaries. In addition, periodic condition is applied on articial uiduid boundariesGF : Material parameter includes water viscosity m=r g 1:2 10 2 cm s at 15 8C : Figure 2 showsthe effect of penalty parameter (bulk modulus) l on the coefficient of permeability for one-dimensional ow. Two-dimensional isotropic ow yields similar results. When l is small, theerror of numerical results is big. However, numerical results approach to a stable value whenl5 10: Figure 3 shows 1D and 2D ow patterns when l 10: These results indicate that theproposed numerical procedure is reasonable as long as l is larger than 10. Hence in subsequentcomputations, the penalty parameter is taken as 10.

5.2. Permeability of clay and constraint water lm effect

5.2.1. Equivalent particle size . In the rst example, the coefficient of the permeability of clay isstudied using the experimental data from Casteleiro [19] who had carried out seven types of experiments to determine the coefficient of permeability of dredged clay. These experiments

Figure 2. Effect of penalty parameter on permeability.



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include conventional consolidation, single load consolidation, slurry consolidation, directpermeability, gravity drainage, vacuum drainage, and eld inltration tests. Figure 4 plots thecoefficients of permeability against void ratio, e; of the dredged clay.

A micro-structural model termed unit cell is necessary to compute the micro-ow andthe coefficient of the permeability for a given void ratio using the proposed numericalapproach. The 1D unit cell shown in Figure 1 has two important geometrical parameters:equivalent particle size d and void ratio e: The coefficient of the permeability is generallyexpressed as

k f d ; e; m 38

As the coefficient of viscosity is a constant at a given temperature, the equivalent particle size

can be back calculated through a single experimental data of k ; e: For this model, a typicalexperimental data is taken as k ; e 1:565 10 8 m=s; 3:5115 : The equivalent particle size isdetermined to be 7 :66 10 5 cm based on this set of data. Using the equivalent particle size, thecoefficient of the permeability can be computed for various void ratios. The numerical resultsare represented by thick solid line in Figure 4. Similarly, 2D isotropic unit cell model as shownin Figure 1 is also used to predict the coefficient of the permeability. It is found that the 2D

Figure 3. Flow patterns for 1D and 2D ows.


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results are slightly lower than the 1D ones. Thus multi-dimensional effect is insignicant forthis clay.

Numerical results agree reasonably well with experimental data for the clay with void ratiolarger than 3. However, numerical predictions are higher than experimental data for the claywith void ratio smaller than 3. We take another set of k ; e 3:76 10 10 m=s; 1:5 toreanalyse the model. This new pair of experimental data roughly represents the mean of conventional consolidation tests (majority of these test data have void ratio less than 3).

The dash line as shown in Figure 4 is the numerical predictions. Although the modelpredicts reasonably well for void ratio smaller than 3, it clearly under-predicts thepermeability with larger void ratio. Equivalent particle can include some range of void ratios;however, it cannot cover whole void ratios. These discrepancies may be due to the effect of constraint water lm.

5.2.2. Effect of constraint water lm . Constraint water lm refers to the adhesive water lmaround a particle. This water lm is of special importance to the materials with small particlessuch as clay, because it forms a typical two-layer micro-structure [20]. As established earlier,one-dimensional ow applies for clay. The effect of constraint water lm is simulated by adistribution of coefficient of viscosity along normal direction as shown in Figure 5. Thisdistribution is contrast to a constant viscosity all ow channels. The viscosity coefficient

increases considerably within some distance from its partial surface. The water lm narrowsow channels, thus reducing the permeability of clay. The reduction ratio depends on pore sizes.The smaller the pore size is, the larger the reduction ratio of ow channels. For clay with thesame particle size, smaller void ratio yields greater reduction. The measured void ratio of clay isnot effective to free water passages. An effective void ratio is dened as that void ratio for freewater passages. If the thickness of constraint water lm is Dd ; the effective void ratio e is then

0 2 4 86 101E-012

1E-011

1E-010

1E-009

1E-008

1E-007

1E-006

Void ratio e

C o e

f f i c i e n t o f p e r m e a

b i l i t y k ( m / s )

Present work (water film 0.2 m)

Present work (No water film)using (k,e)=(1.565x10 -8m/s, 3.51)

by Casteleiro (1975)

Present work (No water film)using (k,e)=(3.76x10 -10m/s, 1.5)

Conventional consolidationSingle load consolidation

Slurry consolidationDirect permeabilityGravity drainage

Vacuum drainage

Field infiltration

Figure 4. Comparison of numerical predictions with Casteleiros experimental data.



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determined by

e d 01 Dd d 0 Dd

39

and the equivalent particle size is

d d 0 Dd 40

where superscript 0 refers to the state without water lm Dd 0: A simple closed-formsolution is obtained for the coefficient of the permeability in one-dimensional ow:

k r g3m

d 01 Dd 3

d 0 d 0141

where d 0

d 01 is the characteristic size of a unit cell. When d

01 Dd 4 0; there is no ow passageand k 0: This implies that a critical void ratio exists. When the actual void ratio is below this

critical void ratio, the coefficient of the permeability is zero. At this time, constraint water lmcompletely blocks ow channels.

The thickness of constraint water lms varies with constituents of clay. Polubarinova-Kochina [20] found that the thickness should not exceed 0 :5 mm : A thickness of 0 :2 mm waterlm is assumed in this example. The numerical predictions which include constraint water lmsare denoted by the dash dot line in Figure 4. The agreement is good between numerical andexperimental k values for the entire range of void ratios. This shows that the effect of constraintwater lm plays an important role for clay with relatively small void ratio. Because theequivalent particle size of clay is small, the water lm can block the ow passage heavily.

The permeability of remolded clays has been experimentally studied by Carrier and Beckman

[21] who conducted stress-controlled and constant rate-of-deformation consolidometer tests. Atotal of 52 samples are represented, of which 22 are phosphatic, 13 are dredged materials, and 17are remolded natural clays. Figure 6 compares the CarrierBeckmans experimental data withthe numerical results with/without water lm. The equivalent particle size is 5 :7 mm from testpoint k ; e 1:77 10 8 m=s; 5:0: The water lm is taken to be 0 :2 mm thick. In general, thenumerical results agree reasonably well with experimental data. The effect of water lm is only

Figure 5. Double water lm model.


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noticeable when the void ratio of clay is small. Numerical results using 2D isotropic unit cellmodel are close to 1D results. This conrms that 1D model is sufficient to predict thepermeability of clay.

5.3. Permeability of sand and two-dimensional effect

Articial uniform sand named as WignerSeitz grains was tested by Carman [22]. The

equivalent particle size is to be 0 :1135 cm if the data point k ; e 4:694 105

m=s; 1:4437 isused. The experimental data are plotted in Figure 7. A small void ratio is taken so as to avoidpossible multi-dimensional effect. The numerical results agree generally well with experimentaldata except for void ratio larger than 3. As sand has much bigger pore size than clay, the waterlm effect is not important. For sand, a major factor for this discrepancy may be from the multi-dimensional effect. This effect refers to the ow resistance at inter-connective pores. Aconceptual 2D model is proposed to investigate the multi-dimensional effect.

The real ow channel in porous media is not always 1D. The zigzag channel ows interactwith each other, thus increasing ow resistance. A simple 2D unit cell is conceived in Figure 1.Channels cross each other and their widths are d 1 and d 2: Characteristic particle size is assumedto be d and d 1 is equal to d 2 for an isotropic ow. The void ratio e is

e 1 d 1d

2

1 42

The coefficient of permeability can be determined if particle size and void ratio are known. Thisrelation can be used to determine the equivalent particle size d if k ; e is known. When d isknown, the coefficient of the permeability can be determined for any void ratio. Figure 8compares the distribution of characteristic velocity at void ratios e 0:331 and 1 :25;

0 5 10 15 20 25 301E-012

1E-011

1E-010

1E-009

1E-008

1E-007

1E-006

1E-005

Experimental data by Carrier-Beckman(1984)No water film

0.2 m water film

C o e


b i l i t y k ( m

/ s )

Void ratio e

Figure 6. Comparison of numerical predictions with CarrierBeckmans experimental data.



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respectively. When the pore size is small, crossing effect is small and ignorable as illustrated inFigure 8(a). However, the effect cannot be ignored when pore sizes are big. For a bigger pore asshown in Figure 8(b), the cross-effect is signicant and hence one-dimensional ow solution isnot adequate to predict the coefficient of the permeability accurately. The numerical results

Void ratio e

C o e


b i l i t y k ( m / s )

0 5 10 15 201E-007

1E-006

1E-005

1E-004

1E-003

1E-002

Test data (Wigner-Seitz Grain)

1D numerical solution

2D isotropic solution

Figure 7. Comparison of numerical predictions with WignerSeitz grain experimental data.

Figure 8. Cross-connection effect for different void ratios. (a) Void ratio 0:331 and (b) void ratio 1:25:


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with/without cross-effects are all plotted in Figure 7. It is evident that the 2D model is able topredict the coefficient of permeability for all void ratios. In summary, constraint water lmaffects the pore ow for small particles in small void ratio zone, while multi-dimensional effectaffects the pore ow for big particles in large void ratio zone.

5.4. Effect of particle shape and anisotropic ow

Anisotropy of permeability is an important topic in practice. An anisotropic numerical model isshown in Figure 9. The model varies ow channel d 2 while keeping ow channel d 1 constant.Such a simulation is valid for clay as the shape of clay components is relatively at [23]. Aparameter study is carried out on a unit cell with 13 mm 13 mm in size. This unit cell representsa 2D isotropic ow with clay particles of 11 mmd 1by 11 mmd 2and ow channel of 2 mm wide.By varying d 2 and keeping d 1; the numerical permeability along the d 1 and d 2 channels areshown in Figure 9. Two ndings are made from the numerical results. Firstly, the coefficient of the permeability along d 1 channel is almost constant due to same pore size in this direction. Thisimplies that the ow in this direction is hardly affected by the ow in other directions. Secondly,the coefficient of the permeability along d 2 channel increases with pore size d 2: When pore sizesin both directions are the same, the permeability becomes identical. This is the case for isotropic

ow. A limit case for d 2 ! 0 should be specially noted. For this limit case, the pore ow is onlyalong d 1 channel. From the view of micro-analysis, an anisotropic ow is decomposed into twoone-dimensional ows, as void ratio and characteristic particle size are all along that direction.In other words, one-dimensional numerical model is sufficient to describe a two-dimensionalpermeability if a directional porosity and a directional particle size are introduced into thepresent numerical model.

Shorter particle size ( m)

C o e


b i l i t y k ( m

/ s )

d1 direction

Conceptual microstructure of soils (Anisotropic 2D flow)

d 2 d i r e c t i o n

10.0 10.2 10.4 10.6 10.8 11.0

2.0E-009

4.0E-009

6.0E-009

8.0E-009

1.0E-008

1.2E-008

1.4E-008

d2

d1d 1 = 2 m

Size of unit cell= 13 m 13 m

L o n g e r s

i z e

Shorter size

Figure 9. Anisotropic permeability due to anisotropic uid channels.



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6. CONCLUSIONS

A numerical model for the ow in porous media is proposed and the coefficient of permeabilityis then computed. The characteristic equation at micro-level and seepage equation at macro-level are developed from a two-scale homogenization process of the NavierStokes ow inporous media. A penalty nite element approach is put forth to solve the characteristic equationat micro-level. The coefficients of permeability computed from the proposed approach arecompared with experimental data for clays and sands. The effect of constraint water lm isstudied for clay while two-dimensional ow effect is studied for sands. Finally, anisotropy of permeability is discussed through the current numerical model. From these analyses, followingconclusions can be drawn.

Macro-seepage equation and microscopic characteristic equation at low Reynolds number areobtained from the NavierStokes equation through a two-scale homogenization method. TheStokes ow and micro-level characteristic ow have some similarity in solution structures. Thissimilarity indicates that characteristic ow is just a special case of NavierStokes ow. Theirdifferences include body force and articial boundary condition.

Penalty method is an efficient numerical approach in solving the characteristic equation if thepenalty parameter adopted is sufficiently large. As only characteristic velocity is included in theweak form, penalty method minimizes unknowns in FEM. Reduced integration is applied totreat the volumetric integration in order to avoid mesh-locking for non-triangular element.

Constraint water lm of clay has a vital effect on the coefficient of the permeability when thevoid ratio is small. However, the multi-dimensional effect can be ignored for clay because of small pore sizes. Constraint water lm blocks ow channels and thus reduces the permeability of clay. Examples reveal that water lm effect is signicant when void ratio is less than 2. If thecharacteristic particle size of clay is taken as the particle size plus the thickness of constraintwater lm, one-dimensional model of unit cell is sufficiently accurate to predict the coefficient of permeability of clay.

Multi-dimensional effect becomes important for sand when void ratios are large as larger

cross-channels produce greater resistance. On the other hand, constraint water lm effect isestablished to be insignicant for sand.

NOMENCLATURE

d equivalent particle sizee void ratio F k si characteristic loading K ij permeability tensor of porous media K vrs element shear stiffness matrix K vprs element volume stiffness matrix

k coefficient of permeabilitym nodal number in an elementn element number P ; P ex pore water pressure

%

PP pore water pressure at uid boundary P 0x leading term of pore water pressure


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PI plastic indexPL plastic limitp k characteristic pressureV i ; V ei x true velocity

%

V V i velocity at uid boundary*

V V 0i macro-pseudo velocity of porous media owV k ij Nodal characteristic velocityvk i characteristic velocityx global co-ordinates xi ith component of coordinatesr X i unit body forceY domain of a unit celly local co-ordinatesY F uid domain in a unit cellY s sub-domain or element domaine scale parameter

m coefficient of viscosityr density of waterG uidsolid interfaceGF uid boundary or articial uid boundaryl bulk modulus or penalty parameterdij characteristic body force or pseudo-body forcedvi weight functiondV ij weight function at the j th node.F j shape functionDd thickness of the constraint water lm

ACKNOWLEDGEMENTS

The funding for the research study is provided by research grant RP3940674 funded by the NationalUniversity of Singapore and the National Science and Technology Board of Singapore.

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