A pore-scale numerical model for flow through porous media

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS

Int. J. Numer. Anal. Meth. Geomech., 23, 881}904 (1999)

A PORE-SCALE NUMERICAL MODEL FOR FLOWTHROUGH POROUS MEDIA

YI ZHUs, PATRICK J. FOX*,t AND JOSEPH P. MORRISA

School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA

SUMMARY

A pore-scale numerical model based on Smoothed Particle Hydrodynamics (SPH) is described for modelling#uid #ow phenomena in porous media. Originally developed for astrophysics applications, SPH is extendedto model incompressible #ows of low Reynolds number as encountered in groundwater #ow systems. In thispaper, an overview of SPH is provided and the required modi"cations for modelling #ow through porousmedia are described, including treatment of viscosity, equation of state, and no-slip boundary conditions.The performance of the model is demonstrated for two-dimensional #ow through idealized porous mediacomposed of spatially periodic square and hexagonal arrays of cylinders. The results are in close agreementwith solutions obtained using the "nite element method and published solutions in the literature. Copyright( 1999 John Wiley & Sons, Ltd.

Key words: smoothed particle hydrodynamics; numerical modelling; #ow through porous media; hydraulicconductivity; Darcy's law

1. INTRODUCTION

Although the geometry of voids and solids is inherently discrete at the microscale, #uid #owthrough porous media has been traditionally characterized using a continuum approach basedon Darcy's law

v"!ki, (1)

where v is discharge velocity, k is hydraulic conductivity, i is hydraulic gradient, and boldface typesigni"es a vector or tensor quantity. Equation (1) is written in terms of macroscopic quantitiesand does not describe #ow within an individual pore. Strictly speaking, Darcy's law represents thestatistical equivalent of the Navier}Stokes equations applied to viscous laminar #ow throughporous media. This equivalency permits the development of solutions to problems within the

*Correspondence to: P. J. Fox, School of Civil Engineering, Purdue University, West Lafayette, Indiana 47907, USA. Tel.:(765)494-0697; fax: (765) 496-1364; e-mail: [email protected] Research AssistanttAssociate ProfessorAPost-Doctoral Fellow

Contract/grant sponsor: Air Force O$ce of Scienti"c ResearchContract/grant number: F49620-96-1-0020

CCC 0363}9061/99/090881}24$17.50 Received 13 October 1997Copyright ( 1999 John Wiley & Sons, Ltd. Revised 8 July 1998

framework of potential #ow theory. In essence, viscous e!ects are taken into account by Darcy'slaw and the #ow can thereafter be treated as frictionless.

Darcy's law has been successfully used for many applications, owing to its reduced order incomparison to the Navier}Stokes equations. However, to improve available continuum #owmodels, a better understanding of the fundamental physics which govern #ow and transportprocesses is required. One means by which this understanding can be achieved is through thedevelopment of pore-scale #ow models that more accurately represent the geometry of #uid andsolid phases within porous media.

Considerable theoretical, numerical, and experimental research has been devoted to thedevelopment of pore-scale models for #ow through porous media. Early theoretical works1}3solved the Stokes equations for #ow through dilute periodic arrays of spheres and cylinders.Building on this work, other solutions have been obtained for similar problems with simple mediageometry.4}9 Pore-scale #ow has been studied extensively using numerical techniques, includingthe "nite di!erence method,10 "nite element method,11}15 and boundary integral method.16,17Lattice-gas automata18}20 and lattice-Boltzmann21,22 methods have also been used to simulateporous media #ow, as have hydraulic network models based on a simple representation of poresystem geometry.23,24 A few models have been developed to handle solid grain shapes thatdeviate from simple cylinders or spheres.15}17,25 Some work has also been completed on thereconstruction of actual porous media for #ow models using digital image processing23,26 andcomputer microtomography.27}29

Smoothed Particle Hydrodynamics (SPH) can also be used to investigate pore-scale #owphenomenon in porous media. First developed for astrophysical applications,30,31 SPH has onlyrecently been applied to low Reynolds number incompressible #ows.32,33 SPH is a fully Lagran-gian computational #uid dynamics technique in which the numerical solution is achieved withouta grid. Using this approach, #uid velocity and pressure distributions, discharge velocity, and #uidparticle pathlines can be computed, as well as other information that would be di$cult orimpossible to observe experimentally. SPH has a number of advantages over competing numer-ical techniques. Mobile solid boundaries are di$cult to incorporate into more conventionalmethods, which require either continual remeshing of the domain or complicated algorithmicmodi"cations. The meshless nature of SPH simpli"es the simulation of mobile34 or evendeformable boundaries. The Lagrangian nature of SPH simpli"es the inclusion of extraphysical e!ects at a #uid}#uid boundary. For example, it is possible to simulate immiscible#uids with SPH,35 which is of crucial importance to modelling the mobility of non-aqueousphase liquids within a solid matrix. Most methods su!er from an increase in complexitywhen extended to three-dimensional problems. The SPH algorithm remains essentiallyunchanged when considering one, two, or three dimensions. In addition, most formulations ofSPH guarantee local conservation of mass, momentum, and energy. This is not typically thecase with competing methods, such as the "nite element method. While SPH is versatile, errorscan sometimes be larger than those obtained using grid-based methods tailored for speci"cproblems. Moreover, SPH can be computationally expensive for certain applications, although atcomparable resolutions, the computational expense of SPH is comparable with conventionalmethods.

The objective of this paper is to present a pore-scale model for #ow through porous mediadeveloped using SPH. An overview of SPH is "rst presented, followed by the necessary modi"ca-tions for treatment of quasi-incompressible viscous #ow. The model is described and its perfor-mance is illustrated for two-dimensional creeping #ow through spatially periodic square and

882 YI ZHU, P. J. FOX AND J. P. MORRIS

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 23, 881}904 (1999)

hexagonal arrays of cylinders. The results are compared with Darcy's law, solutions obtainedusing the "nite element method, and other solutions available in the published literature. Finally,conclusions are drawn and the scope of future work using the model is outlined.

2. OVERVIEW OF SPH

The standard approach to SPH is reviewed by Benz36 and Monaghan.37 In SPH, a compressible#uid is represented by a "eld of disordered particles that move with the local #uid velocity (Figure1). Each particle is mathematically treated as an interpolation point at which #uid properties arecomputed as weighted sums of values from neighbouring particles. To illustrate, consider a "eldquantity A(r) expressed by

A(r)"P A(r@)d (r!r@) dr@ (2)

where r and r@ are position vectors and d is the Dirac delta function. If we replace d with aninterpolation kernel,= (r, h), we obtain an integral interpolant, A

*(r), of the function,

A*(r)"P A(r@)=(r!r@, h) dr@ (3)

The kernel typically takes the form

= (r, h)"1

hpf A

D r Dh B (4)

Figure 1. Sphere of in#uence for particle a

A PORE-SCALE NUMERICAL MODEL FOR FLOW THROUGH POROUS MEDIA 883


where p is the number of dimensions for the problem and h is the smoothing length. It has thefollowing properties:

P=(r!r@, h) dr@"1, (5)

and

limh?0=(r!r@, h)"d (r!r@) (6)

For numerical work, A*(r) is approximated by a summation interpolant, A

4(r), over the particle

"eld:

A4(r)"+

b

mb

ob

Ab=( Dr!r

bD , h) (7)

where m is mass, o is density, and "eld quantities at particle b are denoted by subscript b. Thequantity m

b/o

bis the inverse of the number density at particle b and can be considered as the

volume of #uid associated with particle b.Using the above concepts, SPH equations governing #uid motion can be obtained. For

example, the #uid density at particle a, oa, may be evaluated by direct particle mass summation as

oa"+

b

mb=

ab, (8)

where

=ab"=(r

ab, h) (9)

and

rab"r

a!r

b(10)

Other expressions for derived "eld quantities at the particles are obtained by summationinvolving the kernel and/or its derivatives. As derivatives can be obtained by ordinary di!erenti-ation, there is no need for a grid. For example, the gradient and divergence of A(r) can be obtainedby

+A4(r)"+

b

mb

ob

Ab+=( Dr!r

bD , h) (11)

and

+ 'A4(r)"+

b

mb

ob

Ab'+=( Dr!r

bD , h) (12)

respectively.

3. SPH FOR INCOMPRESSIBLE FLOW

Groundwater #ow is generally regarded as incompressible since the bulk #uid velocity is muchsmaller than the corresponding speed of sound. The solution to an incompressible #ow problem is



achieved by solving the following mass and momentum conservation equations throughout the#ow domain:

+ ' u"0 (13)

du

dt"!

1

o+p#v+2u#g (14)

where d/dt"L/Lt#u '+ is the Lagrangian or material derivative, u is velocity, t is time, p ispressure, v is kinematic viscosity, and g is gravitational acceleration. SPH is well suited to modelcompressible #ows because, in SPH, the #uid is driven by local density #uctuations at theparticles. Monaghan34 extended SPH to incompressible #ow problems involving free surface#ows for high Reynolds numbers and free-slip boundary conditions. The extension of SPH to lowReynolds number incompressible #ows calls for the following additional modi"cations, some ofwhich are discussed by Morris et al.32

3.1. Equation of state

SPH cannot model a truly incompressible #uid. Rather, an incompressible #ow must beapproximated by solving for the #ow of a nearly incompressible, or quasi-incompressible, #uid.As a result, an equation of state is required in the form of

p"f (o). (15)

In the case of water, the actual state equation could be used but this would result in a prohibitive-ly small time step for numerical stability (by the CFL condition38). Instead, simulations areperformed using the following arti"cial state equation:

p"c2o (16)

where c is a numerical speed of sound. Although c"1470 m/s for water (standard pressure andtemperature), a much smaller value ((0)1 m/s) was chosen for the SPH simulations describedherein to increase the numerical time step. This value is low enough to be practical and yet highenough to limit #uid density #uctuations to about 1 per cent for creeping #ow. A similarapproach has been used in grid-based techniques to model steady incompressible #ows.39}41

Considering force balance in equation (14), Morris et al.32 found that c2 should be comparablewith the largest of

u20/j, vu

0/¸

0j, F¸

0/j , (17)

where u0

and ¸0

are the velocity and length scales, respectively, for the problem, F is themagnitude of the driving body force (see Section 3.4), and j is de"ned by

j"*o/o0

(18)

where *o is the maximum density di!erence in the #ow and o0

is the initial #uid density. A valueof j"0)01 was chosen for this study because SPH kernel interpolation is only accurate to withinapproximately 1 per cent. Equation (17) provides a "rst estimate of c2 for a given problem. To



obtain a better estimate, an initial simulation is run at low resolution to "nd the distance *¸ inthe direction of the driving body force between the locations of minimum and maximum #uiddensity. The third term of equation (17) then becomes,

o0F*¸/*o (19)

3.2. Conservation of mass

To simulate free surface incompressible #ows with SPH, Monaghan34 evolved particle densit-ies according to the following SPH equation for mass conservation:

doa

dt"+

b

mbuab'+

a=

ab(20)

where +adenotes the gradient with respect to the coordinates of particle a and

uab"u

a!u

b(21)

Equation (20) is derived from the mass conservation equation for a compressible #uid:

do/dt"!o+ ' u (22)

Using equation (20), density can be evolved concurrently with particle velocities and other "eldquantities, thus signi"cantly reducing the computational e!ort. Although equation (20) does notconserve mass exactly (equation (8) does, provided the total number and mass of particles areconstant), direct particle summation can be used intermittently during a simulation to preventsigni"cant &&drift'' in particle masses.

3.3. Conservation of momentum

3.3.1. Pressure gradient acceleration. Momentum conservation can be satis"ed by an in"nitenumber of symmetric forms of the pressure gradient acceleration term given as37

!A1

o+pB

a

"!+b

mb A

pa

oeao2~eb

#

pb

o2~ea

oebB +

a=

ab(23)

where e may take any value. The following form of the pressure gradient acceleration term (e"1)provides certain advantages for problems involving contact discontinuities and was used for thesimulations described herein:

!A1

o+pB

a

"!+b

mb A

pa#p

boaobB +

a=

ab(24)

3.3.2. Viscosity. Most implementations of SPH employ an arti"cial viscosity that was "rstintroduced to permit the modelling of strong shocks.36,37 Although this formulation has beenused to model real viscosity, it can produce inaccurate velocity pro"les for low Reynolds number#ows. Other expressions have been proposed to model real viscous forces; however, their



implementation requires nested summations over the particles and hence twice the computa-tional e!ort42,43 or second derivatives of the kernel which introduce sizable errors at lowresolution.44

Morris et al.32 modelled viscous di!usion using the following expression:

(l+2u)a"A

ko

+2uBa

"+b

mb(k

a#k

b)r

ab'+

a=

aboaob(r2ab#0)01h2)

uab

(25)

where k is dynamic viscosity. Equation (25) is based on a similar SPH expression used byMonaghan45 to model heat conduction. The expression conserves linear momentum exactly,while angular momentum is only approximately conserved. If the kernel takes the form ofequation (4), equation (25) simpli"es to

(l+2u)a"+

b

mb(k

a#k

b)u

aboaob

A1

Drab

DL=

abLr

abB (26)

Substituting equations (24) and (26) into equation (14), the SPH form of the momentum equationis

dua

dt"!+

b

mb A

pa#p

boaobB +

a=

ab#+

b

mb(k

a#k

b)u

aboaob

A1

D rab

DL=

abLr

abB#g (27)

3.4. Dynamic pressure

For low Reynolds number #ows, local variations in pressure gradient which force #uid motioncan be very small in comparison to the hydrostatic pressure gradient. This is of special signi"-cance to SPH since pressure is obtained using an explicit function of density and is only accurateto about 1 per cent. Consequently, for many problems, it is simpler to model the dynamicpressure, p

d, de"ned as,

p$"p

5!p

)(28)

where p5and p

)are the total and hydrostatic pressures, respectively. Since pressure appears in the

Navier}Stokes equations only as a gradient, the e!ect of p)

is that of a body force:

!

1

o+p

t"!

1

o+p

d!

1

o+p

)(29)

Substituting equation (29) into equation (14) gives

du

dt"!

1

o+p

$#l+2u#g!

1

o+p

)"!

1

o+p

$#l+2u#F (30)

where F is the net body force driving the #ow de"ned by

F"g!1

o+p

)(31)

For simplicity, p is used in the following sections to denote p$. Also, it is the dynamic pressure that

is modelled by equation (16) and used in equation (27).



3.5. Boundary conditions

No-slip #uid-solid boundary conditions are needed to correctly model #ow through porousmedia. Previously proposed techniques for the simulation of both free-slip34,46}49 and no-slipboundaries44 may result in signi"cant errors for low Reynolds number incompressible #ows.Morris et al.32 used real SPH particles to create no-slip boundaries. In this approach, solidboundaries are created by initially placing SPH particles on a regular lattice and designatingthose particles which fall within solid objects as boundary particles. These particles contribute tothe usual SPH expressions for density and momentum, and their own densities are also evolved.In addition, boundary particles are assigned arti"cial velocities such that antisymmetry in thevelocity "eld is created across the boundary surface. Although the approach of Morris et al.32 isan improvement for low Reynolds number #ows, the suggested placement of boundary particlepositions leads to signi"cant errors for low porosity media.

The method used for this study is similar to that proposed by Morris et al.,32 except thatboundary particles are placed in layers parallel to the surface. Figure 2 illustrates the concept fora curved boundary. For each #uid particle a, the normal distance, d

a, to the boundary is

calculated. This normal de"nes a tangent plane (a line in two dimensions) from which the normaldistance, d

B, to each boundary particle B is calculated. The velocity of particle a is extrapolated

across the tangent plane, assuming zero velocity on the plane itself, giving each boundary particlethe velocity:

uB"!

dB

da

ua

(32)

If the boundary is in motion, uashould be replaced by the #uid velocity relative to the boundary.

The arti"cial velocity uB

is used to calculate viscous forces, whereas the actual boundary velocityis used to evolve boundary particle positions and densities.

If a #uid particle closely approaches a boundary surface during the course of a simulation,potentially large arti"cial velocities for boundary particles may result. To prevent this problem,

Figure 2. No-slip boundary for curved surface



dais bounded according to

da"max Ada

,J3

4*xB (33)

where *x is the initial nearest neighbour distance between #uid particles (see Figure 4).

3.6. Interpolation kernel

The interpolation kernel is used to calculate weighted sums of #uid properties at a point. Theuse of di!erent kernels for SPH is analogous to the use of di!erent "nite di!erence operators forthe "nite di!erence method. Although most SPH applications employ a cubic spline kernel, thefollowing quintic spline kernel,50 here normalized for two dimensions, results in less noise in thepressure and velocity "elds,

=(r, h)"7

478 nh2

igjgk

(3!s)5!6(2!s)5#15(1!s)5 if 0)s(1

(3!s)5!6(2!s)5 if 1)s(2

(3!s)5 if 2)s(3

0 if s*3

(34)

where

s"Dr D/h (35)

By using a quintic spline, particles interact within a distance of 3h (Figure 1). This produces #uidproperty estimates that are smoother than those obtained using a cubic spline. The numericalsolution also has better stability properties as a result.51 The quintic spline is, however, morecomputationally expensive than the cubic spline by approximately a factor of two.

3.7. Time integration

Explicit time integration is performed using the predictor}corrector method52 with the timestep, *t, limited by stability constraints. The CFL condition38 requires

*t)0)25h

c(36)

Additional constraints arise from the magnitude of particle accelerations f,37

*t)0)25 mina SA

h

faB (37)

and viscous di!usion

*t)0)125 h2/l (38)

Equation (38) is based on the usual condition for an explicit "nite di!erence method simulatingdi!usion. For simulations having high resolution (small h) or large viscosity, equation (38) istypically the dominant time constraint.



4. POROUS MEDIA FLOW SIMULATIONS

SPH was used to simulate two-dimensional creeping #ow of water (o0"103 kg/m3,

l"10~6 m2/s) through spatially periodic porous media. A spatially periodic medium has a unitcell that is repeated in one, two, or three dimensions. These structures were historically the "rstporous media to be studied and also provide certain advantages for the numerical treatment ofboundaries to the computational domain. The primary objective of the simulations was to verifythe results of the model through a comparison with other available solutions.

4.1. Spatially periodic porous media

Flow through two spatially periodic porous media, shown in Figure 3, was simulated for thisstudy. The media are composed of uniform solid cylinders of radius R that are arranged in squareand hexagonal lattices. The unit cell for the square lattice has side length ¸ and the unit cell for

the hexagonal lattice has side lengths ¸1

and ¸2"J3¸

1. The porosity, n, of each medium is

n"<7/<

#%--, (39)

where <7and <

#%--are the void volume and total volume of the unit cell, respectively.

For the square lattice, simulations were performed for ¸"1)2 mm and n"0)3, 0)4, 0)5 and 0)6(corresponding to R"0)566, 0)524, 0)479, and 0)428 mm, respectively). Simulations for thehexagonal lattice were performed for ¸

-"1)2 mm and n"0)3, 0)4, 0)5 and 0)6 (corresponding to

R"0)527, 0)488, 0)446 and 0)399 mm, respectively). The cylinder diameters used in this study fallwithin the particle size range for medium sand.

4.2. Initialization

For each #ow simulation, #uid particles are initially placed throughout the computationaldomain in a hexagonal arrangement. Each particle is assigned in initial density o

0"103 kg/m3,

Figure 3. Geometry for spatially periodic porous media: (a) square lattice, and (b) hexagonal lattice



Figure 4. Initial hexagonal arrangement of #uid particles

an initial velocity of zero, and a mass ma"o

0<a, where <

ais the volume associated with particle

a (Figure 4). Particles that fall within solid cylinders are deleted. The chosen number of #uidparticles is governed by the desired resolution and the computational expense. Larger numbers ofparticles produce more accurate results but also increase computation time. For porous mediasimulations, it is recommended that each pore throat be spanned by at least 15 #uid particles. Asa result, the computation time for problems having small porosity (i.e. narrow pore throats)dramatically increases due to the larger number of particles and the reduced numerical time step(by equations (36)}(38)).

Boundary particles are placed on solid cylinders using a pseudo-hexagonal arrangement(Figure 2). The "rst layer of boundary particles are positioned on the perimeter with an equalspacing of approximately *x (depending on R). A second layer of an equal number of particles is

then placed at a distance of J3*x/2 from the boundary. A third layer of particles is positionedsimilarly and the process is continued until boundary particles "ll an annular zone havinga thickness of at least 6h. Particles farther than 6h from the boundary surface are not neededbecause they do not contribute to the calculations. Once the boundary particles are positioned,each is assigned an initial density of o

0"103 kg/m3 and a mass consistent with its contributing

volume.The positioning of boundary particles disrupts the initial hexagonal arrangement of #uid

particles in the problem domain. To account for changes in particle density near the boundaries,the initial density #uid (#uid particles and boundary particles) is recalculated using directsummation (equation (8)) Once particle densities are corrected, a body force F is applied toinitiate #uid motion in the x-direction. F is related to the hydraulic gradient by

i"F/g (40)

where g is the gravitational constant (9)81 m/s2). Using this approach, pressure gradient-driven#ow through lattices can be simulated. Periodic boundary conditions are applied to all #uidquantities and the #ow is driven by the e!ective body force.

4.3. Execution

Once the body force is applied to the system, particle densities are evolved according toequation (20) and particle accelerations are computed using equation (27). The smoothing length



h is chosen as 2*x for the quintic kernel. To ensure stability of the integration scheme, thetime step is limited according to the conditions set forth in Section 3.7. Particle velocitiesand positions are updated using the predictor}corrector method and the particles are movedaccording to

dr/dt"u (41)

No-slip boundary conditions are simulated by assigning arti"cial velocities to the boundaryparticles using equation (32).

During a simulation, linked list data structures53 are used to identify neighbouring particleswithin a distance of 3h. The computational domain is divided into square cells having side lengthsof 3h and a list of particles belonging to each cell is created. A particle located within a given cellthen considers interactions only with particles in neighbouring cells.

Although computationally less expensive, equation (20) does not conserve mass exactly andmay introduce errors over the course of a simulation. To correct for this, particle densities areupdated every 50 time steps by direct summation (equation (8)). Using this procedure, computa-tional speed is maintained and #uid mass is conserved as well.

Periodic unit cell boundaries are created through the use of solid cylinder and #uid particleimages (Figure 5), and by wrapping #uid particles around the #ow domain when theyleave a boundary (Figure 6). Image particles and image cylinders are created within a distanceof 3h from the unit cell boundary to provide the necessary neighbours for particles withinthe cell.

Figure 5. Image #uid particles and image cylinders for square lattice



Figure 6. Wrapping #uid particles around a unit cell

4.4. Flow calculations

During the course of each simulation, volumetric #ow rate in the x-direction, qx, is calculated

by summing particle volumes passing out of the right side of the unit cell over a given time period.The discharge velocity, v

x, is calculated by dividing q

xby the gross cross-sectional area of the unit

cell perpendicular to F (¸ for the square lattice, ¸2

for the hexagonal lattice). The hydraulicconductivity, k

x, and intrinsic permeability, kM

x, are calculated at steady state as

kx"!v

x/ix

(42)

and

kMx"k

xl/g (43)

where ix

is the hydraulic gradient in the x-direction. The maximum #uid velocity in thex-direction, u

x,.!9, and the cylinder drag force in the x-direction, F

Dx, are also computed for each

simulation. FDx

is the x-component of the sum of boundary particle forces on the cylinder.A dimensionless drag force, FM

Dx, is de"ned by

FMDx

"FDx

/kvx

(44)

For each simulation, Reynolds number, Re, and Mach number, Ma, are calculated at steady stateas

Re"2Rvx/l (45)

and

Ma"vx/c (46)



Table I. Summary of results for square lattice

Simulation 1 2 3 4 5 6 7 8Porosity, n 0)6 0)5 0)5 0)5 0)5 0)5 0)4 0)3Cylinder radius, R (mm) 0)428 0)479 0)479 0)479 0)479 0)479 0)524 0)566Number of SPH particles 4344 6394 6394 6394 6394 6394 16006 43156Body force, F (m/s2) 0)049 0)049 0)0392 0)0245 0)0098 0)0049 0)049 0)049Hydraulic gradient, i 0)005 0)005 0)004 0)0025 0)001 0)0005 0)005 0)005Speed of sound, c (m/s) 0)045 0)045 0)04 0)032 0)02 0)014 0)045 0)045Discharge velocity, v

x(m/s) 3)11]10~4 1)27]10~4 1)02]10~4 6)40]10~5 2)53]10~5 1)32]10~5 3)91]10~5 4)96]10~6

Hydraulic conductivity, kx

(m/s) 0)0621 0)0255 0)0256 0)0256 0)0253 0)0264 0)00782 0)000992Intrinsic permeability, kM

x(m2) 6)34]10~9 2)60]10~9 2)61]10~9 2)61]10~9 2)58]10~9 2)69]10~9 7)98]10~10 1)01]10~10

Dim. cylinder drag force, FMDx

214 547 541 543 548 535 1790 14165Reynolds number, Re 0)2662 0)1217 0)0977 0)0613 0)0242 0)0126 0)041 0)0056Mach number, Ma 0)0069 0)00282 0)00255 0)002 0)00127 0)00094 0)00087 0)00011

Table II. Summary of results for hexagonal lattice

Simulation 1 2 3 4 5 6 7 8Porosity, n 0)6 0)5 0)5 0)5 0)5 0)5 0)4 0)3Cylinder radius, R (mm) 0)399 0)446 0)446 0)446 0)446 0)446 0)488 0)527Number of SPH particles 7700 11440 11440 11440 11440 11440 19824 24656Body force, F (m/s2) 0)049 0)049 0)0392 0)0245 0)0098 0)0049 0)049 0)049Hydraulic gradient, i 0)005 0)005 0)004 0)0025 0)001 0)0005 0)005 0)005Speed of sound, c (m/s) 0)045 0)045 0)04 0)032 0)02 0)014 0)045 0)045Discharge velocity, v

x(m/s) 3)15]10~4 1)53]10~4 1)23]10~4 7)65]10~5 3)08]10~5 1)58]10~5 6)56]10~5 2)18]10~5

Hydraulic conductivity, kx

(m/s) 0)063 0)0307 0)0308 0)0306 0)0308 0)0315 0)0131 0)00435Intrinsic permeability, kM

x(m2) 6)43]10~9 3)13]10~9 3)14]10~9 3)12]10~9 3)14]10~9 3)21]10~9 1)34]10~9 4)44]10~10

Dim. cylinder drag force, FMDx

185 393 387 391 390 379 904 2795Reynolds number, Re 0)2514 0)1365 0)1097 0)0682 0)0275 0)0141 0)064 0)023Mach number, Ma 0)007 0)00341 0)00308 0)00239 0)00154 0)00113 0)00146 0)00048

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(1999)

5. RESULTS

Numerical simulations were performed on a Sparc 20 workstation and an IBM RISC Sys-tem/6000 Computer Cluster. Tables I and II present a summary of the results obtained forthe square and hexagonal lattices at steady state. The small values of Re correspond to creeping#ow conditions and the small values of Ma indicate that each simulated #ow was quasi-incompressible.

5.1. Model behavior

For any SPH simulation, #uid particles will eventually become disordered as they movethrough the #ow domain. Due to the low Reynolds numbers, however, steady-state conditionswere achieved in a considerably shorter time for the numerical simulations in this study. The longterm performance of the model was investigated for each cylinder lattice for n"0)5 andF"0)049 m/s2. The initial and "nal particle positions are shown in Figures 7 and 8 for the squareand hexagonal lattices, respectively. The model was run for 39 000 time steps for the square latticeand 27 506 steps for the hexagonal lattice. In each case, the #uid particles were fully disordered atthe end of the simulation. Figure 9 shows u

x,.!9and FM

Dxas a function of time for each simulation.

The values quickly reached steady state and thereafter exhibited small #uctuations. The corres-ponding plot of average discharge velocity, computed for 0)1 s time intervals, is shown inFigure 10. Steady-state discharge velocity was also achieved quickly, however the #uctuations aremore pronounced because of the discrete process by which #uid mass leaves the unit cell. Figures9 and 10 illustrate that the model is numerically stable well after steady-state conditions arereached.

Each problem was also solved using a Finite Element Method (FEM) program for steadyincompressible viscous #ow. In terms of computation time, SPH reached steady state faster than

Figure 7. Particle positions for square lattice (n"0)5): (a) initial positions, and (b) "nal positions. Fluid and boundaryparticles are shown in black and grey, respectively



Figure 8. Particle positions for hexagonal lattice (n"0)5): (a) initial positions, and (b) "nal positions. Fluid and boundaryparticles are shown in black and grey, respectively

Figure 9. Maximum particle velocity and dimensionless cylinder drag force vs. time for square and hexagonal lattices



Figure 10. Discharge velocity vs. time for square and hexagonal lattices

FEM for a similar number of particles and nodes. However, a much longer time was required forthe SPH #uid particles to become fully disordered. Velocity and dynamic pressure distributionsfrom the two methods are compared by plotting values within one nearest neighbour distance(*x) of the four paths (A, B, C, D) shown in Figure 3. Figure 11 shows pro"les of #uid velocity inthe x-direction, u

x, and dynamic pressure for paths A and B, respectively, through the square

lattice unit cell for n"0)5 and F"0)049 m/s2. Corresponding plots for paths C and D throughthe hexagonal lattice unit cell are shown in Figure 12. For both lattices, the values are in closeagreement with typical errors of about 5 per cent. The plots show that the SPH results are lesssmooth than those obtained using the FEM. This &noise' is due to particles moving past eachother, which is then ampli"ed by the relatively sti! equation of state. In general, values obtainedusing SPH show more variability as the compressibility of the #uid decreases. The plots alsoindicate that computed pressures for the hexagonal lattice are not fully realized near thestagnation points of the cylinders. This is similar to the "ndings of Morris et al.32 which showedthat, for a similar number of particles and nodes, SPH does not fully capture pressure extrema onsolid boundaries. The FEM better captures the extrema since grid-stretching increases resolutionin the vicinity of a cylinder. The SPH and FEM solutions are, however, in close agreement for thebulk of the #ow.

5.2. Hydraulic conductivity

Four additional simulations were performed for each cylinder lattice for n"0)5 andF"0)0392, 0)0245, 0)0098 and 0)0049 m/s2. Figure 13(a) shows that discharge velocity varieslinearly with hydraulic gradient for each lattice, which is in agreement with Darcy's law. Thecomputed values of k

xare 0)0255 m/s for the square lattice and 0)0307 m/s for the hexagonal

lattice. At n"0)5, the hexagonal lattice has a higher hydraulic conductivity than the squarelattice because of the larger #ow channels in the hexagonal lattice unit cell. Values of k

xare

plotted as a function of n for both lattices in Figure 13(b). Noting the linearity of plots in



Figure 11. Comparison of SPH and FEM results for square lattice: (a) velocity pro"le for path A, and (b) dynamicpressure pro"le for path B

Figure 13(a), values of kx

for porosities other than 0)5 were calculated using one data point(F"0)049 m/s2). As n increases, k

xincreases for both lattices and the di!erence in k

xfor the

lattices decreases.Values of kM

x/R2 and FM

Dxare plotted as a function of porosity in Figure 14. One additional

solution has been included in Figure 14(a) for the square lattice at n"0)558. Solutions obtainedusing the FEM as well as published results from previous studies,4,8,13}15 which were obtainedfor various values of R, are also shown for comparison. Values are in good agreement (themaximum di!erence is 5 per cent), indicating that SPH is capable of producing results that arecomparable with those obtained using other methods. Figure 14(a) also indicates that kM

x/R2 is

a unique function of n for each periodic lattice. Figure 14(b) shows that cylinder drag forcedecreases sharply with increasing porosity.



Figure 12. Comparison of SPH and FEM results for hexagonal lattice: (a) velocity pro"le for path C, and (b) dynamicpressure pro"le for path D

Figure 13. (a) Discharge velocity vs. hydraulic gradient, and (b) hydraulic conductivity vs. porosity for square andhexagonal lattices



Figure 13. (Continued)

Figure 14. (a) kMx/R2, and (b) dimensionless cylinder drag force vs. porosity for square and hexagonal lattices



6. CONCLUSIONS AND FUTURE WORK

Necessary extensions have been implemented and tested which allow Smoothed Particle Hydro-dynamics (SPH) to model incompressible #ow through porous media. Test results con"rm thatthe proposed modi"cations to the equation of state, viscosity formulation, boundary conditions,and interpolation kernel result in a method which is stable and accurate. Simulations of #owthrough two periodic porous media show discharge velocity proportional to hydraulic gradient,as required by Darcy's law. In addition, the solutions are in close agreement with values obtainedusing the "nite element method and published solutions in the literature.

Future work will include a study of hydraulic conductivity and hydrodynamic dispersionproperties of various idealized porous media. Di!erent media can be created by varying the sizes,shapes, and con"gurations of the solid particles within the unit cell. Surface tension has beenincorporated into the model,35 which will permit the investigation of multiphase #ow throughporous media. Extension of the method to three dimensions is straightforward in theory, but mayrequire a parallel version of the code.

ACKNOWLEDGMENT/DISCLAIMER

This work was sponsored by the Air Force O$ce of Scienti"c Research, USAF, under grantnumber F49620-96-1-0020. The views and conclusions contained herein are those of the authorsand should not be interpreted as necessarily representing the o$cial policies or endorsements,either expressed or implied, of the Air Force O$ce of Scienti"c Research or the U.S. Government.

APPENDIX I. NOTATION

A "eld quantityA

*integral interpolant of A

A4

summation interpolant of Ac speed of sound, L/Td normal distance to boundary, Lf general function, particle acceleration, L/T2

F body force, L/T2

FD

cylinder drag force, ML/T2

FMD

dimensionless cylinder drag forceg gravitational constant, L/T2

h smoothing length, Li hydraulic gradientk hydraulic conductivity, L/TkM intrinsic permeability, L2

¸ side length of square unit cell, L¸0

typical length scale, L¸1, ¸

2side lengths of hexagonal unit cell, L

m particle mass, MMa Mach numbern porosityp pressure, dynamic pressure, M/LT2

p$

dynamic pressure, M/L T2



p)

hydrostatic pressure, M/L T2

pt

total pressure M/L T2

q volumetric #ow rate, L3/TR cylinder radius, LRe Reynolds numbers kernel indext time, Tu #uid velocity, L/Tu0

typical velocity scale, L/Tv discharge velocity, L/T< volume associated with a particle, L3

<#%--

volume of unit cell, L3

<7

void volume of unit cell, L3

= kernel function, 1/Lpx, y coordinates, L

Greek letters

d Dirac delta function*¸ distance between maximum and minimum #uid densities in direction of driving body

force, L*t time step, T*x initial nearest neighbour distance of #uid particles, L*o maximum allowable density di!erence, M/L3

e, j variablesk dynamic viscosity, M/L Tl kinematic viscosity, L2/To density, M/L3

o0

initial density, M/L3

p number of dimensions

Operators

+ gradient operator, 1/L

Subscripts

a quantitity associated with #uid particle ab quantity associated with #uid particle bB quantity associated with boundary particle Bx quantity in x-direction

¹ensors and vectors

F body force, L/T2

g gravitational acceleration, L/T2



i hydraulic gradientk hydraulic conductivity, L/Tr, r@ position, Lu #uid velocity, L/Tv discharge velocity, L/T

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A pore-scale numerical model for flow through porous media