1674.pdf

A B S T R A C T

A M A

N

D A L

D

Y E

: Lat t ic e Bol tz

m

a n n

S

i

m

u

l

at ion o

f

N on-D _{a rc}

y F low in

S

p

h

e re

P

ack ings

(

Un

d

e r t

h

e

d

irect ion of

D

r

C

a s s T

M

i

ll

er

)

Lat t ic e-_B

o

l

tz

m

an n

m

o

d

e

l

ing was us ed to si

m

ula te sing

l

e

-p hase, non -D _{a r c}

y f

l

ow t hr oug

h fl

o

w

d

o

m

ains c onstructed ut i

l

izin_g a n a_{l g}o r_{i t}

_hm

t hat s_{up p}o rts t

h

e ge ne r_at ion o

f

a s_yste

m

o

f

s_p

h

e r_es

w i t h lo_g-_nor

m

a

ll

y

d

istri buted ra

d

i i a n

d

a

d

e

fi

ne

d

por o si t y

R

epre sentat ive e

l

e

m

e nta ry v o

l

u

m

e s

were

d

eter

m

in e

d

using t he pe r

m

ea

b

i

l

i t y an

d

in ert ia

l

c o e

ffi

cie nts o

f

a n on-

_d

_i

_m

ensio n al

m

o

m

e ntu

m

equ at io n

f

or a ra nge o

f

s_p

h

ere

-size

d

istri

b

ut io ns

P

oro u s

m

e

d

ia were gen erate

d f

or _po r o si t

i

e s

r ang ing fr o

m

0 3 to 0 6

fo r

the d i

ffe r

e nt s_{p h}e r e-_{siz e d istri but ions T he}

pa r a

m

ete r s o

b

taine

d

fr o

m

t

h

e n o n-_D

a rcy

f lo

w

m

o

d

e

l d

i

d

not

fi

t t

he

e xist ing pe r

m

e a

b

i l i t y an

d

in ert ial c or relat ions

pr o_po s e_{d b y}

C

a r

m

e n-

K

_{o ze n}

y,

R

u

m

p

f

-

G

_u

p te,Ergu n a nd ot he r s Bas e

d

on t he expe ri

m

enta

l d

ata

c o

ll

ected fr o

m

t

h

e

l

at t ic e-

B

o

l

tz

m

an n si

m

u

l

at io ns, e

m

p irica

l

c o r r e

la

t ions a re sug geste

d

t

h

at r e

l

ate

A C K N O

W

L E

_{D G}

M

E

N T S

F ir st I wa nt to t ha nk

m

y a

d

vis o r

D

r

M

i l ler an

d

Dr

G

r ay

fo r

ta

k

ing a chanc e on

m

e

I

d

eep l y a_{p p}r e ciate t

h

eir pat ie n c e and co ntri but io ns of t i

m

e, i de a s, and

f

u nd ing to

m

ake

m

y

M

a ster s e x_perie n c e

b

ot

h

_pro

d

uct ive an

d

st i

m

u

l

at in_g

I

wo u

ld

a

l

so

l

i

k

e to t

h

a n

k D

r

V

izu ete

f

or

ta

k

ing t he t i

m

e to be on

m

y t hesis c o

m m

i t te e

I a

m fo r e v e r

in

d

e

b

te

d

to

J

a

m

es

M

c

C l

ur e

f

o r a

l l

t

h

e

lo n

_g-su

ffe r

in_g

h

e

h

a

d

to _go t

h

r ou_g

h

to

h

el p

m

e u n

d

er stan

d

t

h

e pr ogr a

m m

in_g si

d

e o

f m

y rese arch He a ut hore

d m

uc

h

o

f

t

h

e co

d

ing t

h

at

went into

m

a

k

ing t

h

is wor

k

pos si

b

le an

d

to ok on t he u no

ffi

cia

l

r o

le

o

f

pro

b

le

m

f ixer along t he

wa_y

I

a

m

also grate

f

ul to

S

ar a

h Sh

el ton

f

o r a

l l

t

h

e e

m

ot io na

l

sup port

,

h

el p, a n

d

c o

m

r a

d

e rie she

ha s pr o vi

d

e

d

t hr oug ho ut t h is s o

m

et i

m

e s d i

ffi

c ul t pr o c e s s

I

jast

l

_y,

I

wo u

ld l

i

k

e to t

h

a n

k m

y

f

a

m i

l y

f

or a

ll

t

h

eir

l

ove a n

d

e n c o urage

m

e nt

F

or

m

y si

b U

n gs,

S

a

m

ant ha , Eric a,

an

d

Kevin,

m

y papa,

B

i

l l

, w

h

o

ha s

b

e en

m

y

b

i g ge st

f

an since t he

b

eg in ning, a

l

ways insp iring

m

e to

f

ol low

m

y

d

rea

m

s, an

d m

o st o

f

al l

m

y pa r ents. B i l l an

d

Pa

m

, w

h

o

h

ave

b

e en a c onstant s our c e o

f

su_{p p}ort - _e

m

_{ot io n}

al,

m

o ral, an

d

o

f

c our se f

i

n a ncia

l

- _t

h

r ou_g

h

out

m

y

stu

d

ent ca r e e r a n

d

t

h

is t

h

esis wo u

l d

certain

l

y n ot

h

a v e existe

d w

i t

h

o ut t

h

e

m

I

t is t ha n

k

s to t he

t wo o

f

t

h

e

m f

or ex posing

m

e to t

h

e wo rl

d

a n

d

i gni t ing

m

y ongoing se arch

f

or kn ow

le

d

ge -_{i t is to}

T

A

B L E O F C O

N

T E

N

T S

L

I S T

O F T A

B L

E S

v

L I S T

O F F I G U R E S

v

i

Intro

d

uct ion 1

Ba c

k

gro u n

d

1

M

et

h

o

d

s 4

G

ene r at ion of

M

e

d

ia 4

M

ul t i-R_ela_{x a}t i_{o n} T i

m

_e Lat t ice Bol tz

m

an n

S

che

m

e 5

R

e su

l

ts an

d

D is cus sion 6

D

evelop

m

ent o

f R

epr e se ntat ive

E

le

m

enta r_y

V

olu

m

e s

7

D

eriv e

d C

or r elat io ns

8

Pe r

m

ea

b

i

l

i t y

C

o r re

l

at ion

8

Ine rt ia

l C

o r r e

l

at ion 1 0

C

onc

l

usio n 1 1

L I

S T

O

F

T A B

L E

S

Ta

bl

e

1

G

ene r ated

M

e

d

ia

L I

S T O

F F I

G U R

E

S

F

_{i g}ure

1

C

o e

f fi

cie nt v a

l

ue s relat ive to t

h

e a s s o ciate

d

R E V v a

lu

e s o

b

tain e

d fo

r the pe r

m

e¬

a

b

i l i t y an

d

ine rt ial pa r a

m

ete r s

fo

r a r ange o

f

s_p

h

e r e _pa ck in_gs 7

2

C

o

m

_pa ris on bet we en t

h

e simu

l

ate

d da

ta a n

d

t he ex ist in_g e

m

p iric alrelat io ns 8

3

T

h

e pr opo s e

_d

_pe r

m

e a

b

il_{i t y} c o r r e

la

t ion _pr e

d

icts t he si

m

ulat ion

d

ata 9

4

C

o

m

pa riso n

b

et ween t

h

e simulate

d d

ata an

d

t he e x ist ing e

m

p iric a

l

r elat ions 1 0

I

n

t

r

o

d

u

c

t

i

o n

S

in_{g l}e _{p h}a s e

fl

ow in _por o u s

m

e

d

ia _{p l}a_ys a fu n

d

a

m

en tal r ole in

m

an_y n atural and industrial

pr o c e s s e s

W

h

i

l

e t

h

e _pri

m

a ry ap pro a c

h

es

f

or

d

es c ri

b

ing t

h

es e syste

m

s are ro o t e

d

in e

m

p iricis

m

,

r e c ent wo rk has fo cuse

d

on de rivin_{g t}r ad itiona

l m

odels fo r sing le p ha s e

fl

ow fr o

m fi

r s_{t p}rinc_{i p l}e s an

d

l ink in_g

m

a cr o s c o_{p i}c p

h

e n o

m

e n a w i t

h m

ic ro s c ale

fl

ow

b

e

h

a v

i

o r

₍

H

ass a niza

d

e

h

a n

d G

r a_y, 1 9 8 7, 1 9 8 8 ;

M

a an

d R

ut

h

, 1 9 9 4

)

T he cur re nt stat e of know le

d

ge categorizes sing le p ha s e flow into t wo _pri

m

a r_y re_{g i}

m

e s: a we ak ine rtia r e_{g i}

m

e c o r r e s_pond in_{g t}o s

m

a

l l fl

ow v elo ci t ie s in w h ich

v is c o u s

f

o r c e s c o

_{m p l}

ete

_{l y d}

o

m

in ate in ert ia

l f

o r c e s a n

d

a str o ng in e rt ia r e

g

i

m

e corr e s_po n

d

in_g

to

m

o

d

erate

fl

ow ve

l

o ci t ies in w

h

ic

h

in ert ia

l f

or ces c a n n ot

b

e n e_g

l

e cte

d

₍

M

a a n

d R

ut

h

, 1

9 9

4 ;

W

a n_g, T hauvin an

d M

o

h

a nt y, 1

9 99

; Four a r et a

l

, 2 0 0 4 ; Pan

fi l

o v an

d F

o ur a r, 2 0 0 6

)

T

he se t wo

r e_{g i}

m

e s a r e l inke

d

to t wo o

f

t

h

e

m

o stc o

m m

on_{l y} a_{p p l i}ed

m

odels,

D

a r c_y '

s

l

a

w

a nd t he Fo r ch

h

ei

m

e r

e_quat ion

S

tate o

f

t

h

e art n u

m

er

i

c a

l m

et

h

o

d

s pr ovi

d

e t

h

e op po rtu ni t y to stu

d

y

fl

ow in poro us

m

e

d

ia

d

ir e c_{t l y}

f

r o

m

t he

m

ic ros c ale T h is_pr o vi

d

e s t

h

e o_{p p}o rtu n_{i t y t}o a s s e s sdetai

l

s o

f

t

h

e

m

ic r o s c ale

fl

ow

stru cture in a

d d

i t io n to

m

acr o s c o_{p i}c

f

o r

m

s T

h

e

l

at t ic e Bol tz

m

ann

m

et

h

o

d h

a s_gaine

d

_po_pu

l

a ri t y

f

or si

m

ulat io n o

f

_po r o u s

m

e

d

ia a n

d

ot

h

e r c o

m

p lex syste

m

s

d

u e to c o

m

putat io n a

l

a

d

v a nta_ges

ov er

m

ore tr a

d

i t io n a

l fl

ui

d d

_yna

m

ic s a_{p p}r o a c

h

es, _pa rt ic ula r

l

_y in

h

a n

d l

in_g co

m

_p

l

e x

b

o u n

d

a r_y

c o n

d

i t io ns a n

d

para

ll

e

l

iz at io n

M

u

l

t i

R

e

l

a x at io n T i

m

e

(

M R T

)

lat t ic e

B

ol tz

m

an n sche

m

es hav e

b

ee n

d

e

m

onstr ate

d

to

h

ave

m

a n_y a

d

v anta_ge s o v er t

h

e

m

or e w i

d

e_{l y} u se

d

sin_g

l

e r e

l

axat ion t i

m

e

(

B G K

)

l

at t ic e Bo

l

tz

m

a n n

m

o

d

e

l

, inc

l

u

d

ing i

m

pr o ve

d

n u

m

erica

l

sta

bi li

t y a n

d

si

m

u

l

at io n o

f h

i g

h

R

eyno

ld

s n u

m b

e r

fl

ows

(

P

a n, Lu o and

M

i l le r, 2 0 0 6

)

T

h

e o ve r a

l l

go a

l

o

f

t

h

is wo r

k

is to a

d

v anc e t he i

m d

e r sta n

d

in_g o

f

sin_g

l

e

fl

u_{i d p h}a se

fl

ow

b

_y

pe r

f

or

m

in_g

m

icr o s c a

l

e

fl

ow si

m

ulat io ns usin_{g t}

h

e

l

at t ice Bol tz

m

a n n

m

et ho

d

Is otro_{p i}c _poro us

m

ed iu

m

s_yst e

m

s w i l l

b

e co nsi

d

ere

d f

or

R

e_yn o

ld

s n u

m b

e r s < 1 5 0 T

h

e spe ci

f i

c o

b

j

ect iv es o

f

t h is

wo r

k

are;

1 to invest_{i g}ate t

h

e

m

ost a_{p p}r o_priate

f

o r

m

o

f

a_{p p}ro xi

m

ate

m

o

m

e ntu

m

e_quat ions_;and

2

to

d

e v e

l

o_p re

l

at

i

o n s to est

im

ate

m

o

m

e ntu

m

e_qu at io n _para

m

ete r s

f

r o

m

po r o u s

m

e

d

iu

m

c

h

a r a cte rist ics

B

a

c

k

_g

r

o u n

d

D

a r cy's

l

aw

f

or o n e-

d

_i

m

_ensional

fl

_{ow in}

po r o us

m

e

d

iu

m

c a n

b

e e x

_p

r e s s e

d

in t

h

e

f

or

m

(

_G

ray

a n

d

M

i

l

ler, 2

0

0 6

)

:

w he r e p™ is t he

m

a c r o s c ale

fl

ui d pr e s sur e, p

™

is t he

m

a c r o s c ale flui d density, g

™

is the ex t ernal fo r c e pe r u ni t

m

a s s a ct ing on t

h

e

fl

ui d p ha s e in t he x d ir e ct ion, e is t he po r o si t y, j ;

™ is t he

b

a r_yc e ntric

m

a cro s c ale ve

l

oc_{i t y} o

f

t

h

e

fl

uid re

l

at ive to t

h

e s ol i

d

in t he x

di

rect io n , an

d R

is t he

m

o

m

entu

m

resista nc e c oe

f fi

cie nt T

h

e

fo

r

m

o

f R

is o

f

_princ

i

p

le

concer n

f

o r t

h

e stu

d

y o

f fl

ow in

po r o u s

m

ed ia and is

k

nown to

_d

epen

d

on _pr o_pe rt ies of t

he

_por o u s

m

e

d

iu

m

a nd

fl

uid a s wel l a s

tli e

R

e_yn ol ds n u

m

be r :

j

l" ' ^ '

w

h

e re /

!

"

'

is t

h

e

d

_yn a

m

ic v is cos_{i t y} a n

d d

is a c

h

ar acte rist ic _po re

le n

_{g t}

h

s ca

le

For

fl

ows in w

h

ic

h

ine rt ia

l f

or c e s

m

ay n ot be ne_g

l

e cte

d

, t he

m

o

m

entu

m

r e sista nc e co e

ffi

cient c an

b

e ap pr o ximate

d

b y

(

M

c

C lu

r e,

G

r ay an

d M

i l le r, 2 0 1 0

)

:

R

=

^

^

a

{

dn

₎

+

b

{m ) \

R

e

\

Y

(

3

₎

T

h

e c oe

ffi

cie nt a is related to t

h

e in trinsic _per

m

e a

b

il_{i t y} o

f

t

h

e _po r ous

m

e

d

iu

m

a n

d b

re

l

ates to the stren_{g t h} o

f

ine rt ial effe cts T

h

es e c o e

ffi

cie nts _pr o vide a

m

a c r o s c o_{p i}c

m

e a sur e o

f

t

h

e in

fl

uenc e

that _porou s

m

ediu

m

m

or_{p h}o

l

o_{g y} an

d

topo

l

og y e xert o n

_m

ic r o s cop ic

fl

ow

b

e

h

av ior, a n

d

are t he r e

f

or e pr e su

m

ed to de_pend o n an u nknow n s et of

d

imensionle s s

m

o rp holog ic al

m

e asur e s

9H

I

d

e nt i

fi

c at ion o

f

an ap propriate setof

m

easur e spe r

m

i tst

h

e co nstruct ion o

f

pre

d

ict ive co nst i tut ive

l

aws, an o

b

j

ect iv e w h ich ha s

b

e en _pur sue

d

e xte nsiv el y

b

y ot

h

e r aut ho r s

(

Te

k

, 1

95

7

;

G

e erts

m

a,

1 9

74

₎

I

n o r

de

r to si

m

_p

li

f

_y s u

b

s e_qu e nt a n a

l

ysis, a

d

ime n sio n

le s s

fo

r

m f

o r t

h

e

m

o

m

e ntu

m

equat

io n

ca n be c onstr u cted

by

substit ut in_g

E

_q 3 into

E

q 1 an

d m

u

l

t i p l y ing b y p

" ' (

f

/ {

p

^'

)

to o

b

tain:

F

o =

[

a

(

Dn

)

+

b

_(O

T ) \

R

e

\

\

R

e,

(

4

)

w

h

e r e t

h

e

d

i

m

ension

l

e s s

fo

r cin_g te r

m

is

d

e

fi

n ed a s:

F

^ =

7

^

{

-

^

+ P

'"

9

^

]

(

5

)

T

h

e

m

o st s_{i g}nificant a s_pe cts o

f

poro us

m

e

di

u

m m

orp

ho

lo

g y w i t

h

respect to pr e

di

ct io n o

f

t

h

e c oe

ffi

cients a an

d b

, a re t

h

e _porosi t y e an

d

t he s_pe ci

fi

c s urface a re a e *

Su

r

f

a c e-_to-_{v o}

l

_u

m

_e

rat io p

la

ys a ke_y ro

le

in

m

ic ros co_{p i}c

fl

ow _proce sses, a n

d

i ts ef fect ca n

b

e incorporate

d

into t he

d

imension

l

e s s

fl

ow e_quat ions _{b y} re

l

at in_{g i t}s va

l

ue to t

h

e len_{g t h} s c ale

fo

r t he s_yste

m

:

T he length defi n ed

b

_{y E q} 6 is the d ia

m

ete r of t he s_{p h}e r e w ith the e_quiv alen t s u rfa c e-_{t o}

v olu

m

e

r at io, t y p ic al l y known a s t he

Sa

ute r

d

ia

m

ete r

W

i t h t he ef fe ct s

d

u e to t he spe ci f ic sur

f

a c e a r ea a c c ou n ted fo r b y t

h

e

l

e n_{g t h} s c ale

d

ef inition,

e xist in_g c o r r e

l

at ions _{t y p}

i

c a

l l

_{y p}r ed ict t

h

e co ef

fi

cie nt va

lu

e s a a n

d b

as

f

u n ct

i

o ns o

f

t

h

e _po r o si t y

onl y T

h

e

m

o st we

l l

-

kn o

w _n o

f

t hese e xpressions is

E

r_gu n '

s equat io n, w

h

ic

h

spe ci f

i

e s

f

unct io nal

for

m

s

f

o r t

h

e

d

i

m

ensio nle s s _per

m

e ab i l i t y:

'

-

~

.

=

^

^

^

^

^

(

^

)

an

d

in ert ia

l

c o e

ffi

c

ie n

t:

b =

B

i

l

: :

i

l

₍

8

₎

W

h

e n t hese ex_pres sio ns are co

m

_pare

d

to t

h

os e as sociate

d

w i t

h

t he vo

lu

m

etric

fl

ow r ate, u s e o

f

t

h

e

R

eyn o

ld

s n u

m b

er a s

d

e

fi

ne

d b

y E k j 2 a c c ou nts

fo

r a

d

i

ff

e r ence o

f

1

_/

e in t he _pe r

m

ea

b

i l i t y

e xpr e s sion an

d

1

/

e^

in

t

h

e e xpres sio n

fo

r t

h

e ine rt ia

l

co e

ffi

cient Er_gun sug ge ste

d

t

h

at

A

=

150

a n

d

B = 1 7 5

₍

Er_gu n, 1

9

5 2

)

,

b

ut se v eral

d

i f

f

e r ent v alue s

f

o r t

he

c o e

f

fi cients ha v e be en _pr o_pos e

d

W

h

en

A

— 1

8

0,

E

q

7

is t

h

e w i

d

e

l

y us e

d C

a r

m

e n

-

K

o ze ny relat ion sh i p

(

B

e a r, 1

9

7

2

)

M

a cD ona

l d

et a

l d

eter

m

in e

d

a r a nge

f

o r the B c oe

ffi

cie nt, 1 8 <

B

<

4 0

,t

h

at ser v e

d

to

m

atc

h

experi

m

e nta

l

d

ata

f

ro

m

six

d

i

f fe

re nt _poro u s

m

e

d

iu

m m

o

d

e

l

s

(

M

ac

d

o n a

l d P

, 1

9 79

)

O

t

h

e r co r re

l

at io n

f

or

m

s

h

ave

b

ee n propos e

d

a

d d i

t ional

l

_{y t}o t

h

ese

m

o

d

i

fi

e

d

Er_gu n

f

o r

m

s

Fo r t

h

e c a se o

f D

a r cy

fl

ow ,

R

u

m

p

f

an

d G

up te pr e

d

ict t

h

e pe r

m

e a

bi l i

t y using t

h

e r e

l

at ions

h

i p

(

R

u

m

p

f

and

G

up te, 1

9

7 1

)

:

—

= -

i

- _e^ '^

₍

₉

₎

D

^ _{5 6}

^

^>

Pa n et al

f

ou n

d

t

h

at t

h

e

C

a r

m

en- Ko z en_y r e

l

at ions

h

i p u n

d

e r e st i

m

at e s t he pe r

m

eab i l i t y, a n

d

o

b

serv e

d d

ev iat

io n s

fr o

m

t

h

e

R

u

m

p

f

-

G

_u

p te re

l

at io n w

he n

syste

m

s o uts

i de

t

h

e ra nge o

f

e xper¬

i

m

e nta

l

sup po rt

fo r

t

h

is e x _pres sion we re co nsi

de

re

d

T

h

ey pr o_pose

d

a n al tern a tive cor re

la

t io n

fo r

m

in w

h

ich a n a

d d

i t io n a

l d

i

m

e n sio nle s s v a ria

bl

e, t

h

e r e

la

t iv e sta n

d

a r

d d

e viat io n a o ,wa s a

ls o

inclu

d

e

d

(

Pan,

H

i

l

_pert an

d M

i l le r, 2 0 0 1

)

:

-^

=

_/3

i e ^=

(

l +

_A

3 a

^

^

)

(

1 0

₎

M

o r e gene r a

l l

_y, t

h

e coe

f fi

cients in E q 4

m

ay

d

epe n

d

upon any in

d

epe ndent,

d

i

m

ensio n

l

e s s

m

easur e o

f

_po r ou s

m

e

d iu

m m

or_p

h

olo_{g y}

F

o r t

h

e c a s e o

f fl

ow in

h

eteroge n e o us, is otr op ic

m

e

d

ia

co

m

pose

d

of s_{p h}ere pa c

k

ings,we pr e su

m

e t

h

at t fi e

fl

ow

m

ay

b

e de scri

b

e

d

a

d

equatel y b y a s s u

m

in_g

d

epe n

d

e nc ie s o

f

t

h

e

fo r

m

:

b

_{(m )}

^ bj

{

t

)

b2

{a D

₎

,

(

1 2

)

w he r e :

a o -

^

-^

L

_ ,

(

1

3

)

in t

h

e c a se t hat t

h

e r a

d

i i a r e

l

o_g

-n or

m

a

l l

_y

d

istri

b

ute

d

w i t h

m

e an fi an

d

v a rianc e a

'

^

M

e

t

h

o

d

s

G

ene r a

t

i

o n o

f

M

e

d i

a

Su

r r o_gate _po ro us

m

e

d

ia we r e c onstructe

d

ut i l izin_g a c ol le ctiv e r e a r r an_ge

m

ent a

l

_go ri t h

m

t

ha

t gene rate s s_p

h

e re _pac

k

in_gs w i t h

l

og-_{no r}

m

_a

l l y

d

istri

b

ute

d

r ad i i an

d

a

d

e

fi

n e

d

po r o si t y T he a

l

_go¬

r

i

t

h m

is

b

a s e

d

o n o n e u s e

d b

y

W

i

ll ia

m

s a n

d P h

i

l

i pse to generate pa ck ings o

f

sp

h

e r e o cy

li

n

d

e r s

(W

i

l l

ia

m

s a nd

P h

i

l

i pse, 2 0 0 3

)

an

d f

o

l l

ows t

h

e s e

m

a

j

o r steps ;

1

A

s_yste

m

c ontaininga setn u

m b

er of sp he r e s o

f

t

h

e spe ci

fi

ed v a riance is c onstructe

d

w i t

h

in t

h

e a

ll

o c ate

d

do

m

ain

2

O

v e rla_ps bet we e n t

h

e s_{p h}er e s ar e e

l

i

m

inate

d

3

T he siz e o

f

t

h

e r a

d

i i is in cr e a se

d b

_y a c onsta nt

fa

cto r

S

teps

2

a n

d 3

a r_{e r epe}ate

d

u nt i

l

t

h

e s_yste

m

o

f

_pac

ke

d

s_p

h

er e s rea c

h

e s t

h

e

d

esir e

_d

_po ros_{i t y}

_B

_y ini t ia

l

izing an

d

r e s ca

l

ing t

h

e ra

d

i i in a n ap propriate way,sp

he

re pa c

k

ings

m

ay

be

ge n erate

d

w

i

t

h

lo

g-_{n o}r

m

a

ll

y

d

istr

ib

ute

d

ra

d

i i:

l

og

(

r

«

)

~ No r

m

a

l

(

M,( T

2

)

(

14

₎

T he va rianc e c r^ is _prov i

d

e

d

as a n in_pu_{t p}a r a

m

eter w

h

i

le

t

h

e

m

e a n _{n i}s

d

eter

m

ine

d b

_{y t h}e

fi

n a

l

pa ck ing po r o sity

To a c c ele rate c o nv_er_genc e

,the syste

m

o

f

sp he r e s is

d

iv i

d

e

d

into a s et of cel ls ea ch c o

m

po s ed o

f

equa

ll

_y-siz e

d

s u

b d

o

m

ains

E

a ch o

f

t

h

e c e

l l

s c ontain s a

l

ist o

f

t

h

e sp

h

e r e s w i t h ce ntr oi

d

s

l

o cate

d

w i t

h

in t

h

e

b

ou n

d

aries o

f

t

h

e c el l

F

or any g ive n s_p

h

e r e over

l

aps are o nl y con si

d

e r e

d f

r o

m

t

h

e ne_{i g}

hb

o rin_g cel

l

, t

h

ere

f

ore t he al go ri t h

m

runt i

m

e is acc e

l

e r ate

d b

y inc r easing t he n u

m b

er o

f

c e

l

ls be cau s e t

h

e

l

e ng t

h

o

f

t

h

e se ar c

h

_pat

h

w

h

e n c o

m

put ing over

la

ps is

d

e c r e ase

d

To pr e ve nt

ov e r si g

h

ts in t

h

e o v er

l

a_p co

m

putat io n in t

h

e e v e nt t

he

m

axi

m

u

m

sp

he

re ra

d

ius exce e

ds o n e

h

a

l f

t he c el l w i

d

t h, t

h

e

m

a x i

m

u

m

ra

d

ius is chec

ke

d

a

f

ter t

he

si

m

ulat io n is co

m

p

le

te a n

d

a

w

ar_nin_g

M

u

l

t

i

-

R

e

l

a

x

a

t

i

o n

T i

m

e

L

a

t t

i

c e

B

o

l

t

z

m

a n n

S

c

h

e

m

e

In t h iswo rk,we u tiliz e a t

h

r e e -

_d

_i

_m

ensiona

l

,nineteen ve

l

o ci t y v e ctor

(

D

3

Q

1 9

)

m

u

lt

i

-relaxat io n

t ime

(

M RT

)

f

o r

m

ulat ion o

f

the lattic e

B

oltz

m

an n

m

et hod to o

b

tain ste a

d

y state ve

l

o c_{i t y f i}el ds

f

o r a s equen c e of

R

e_yn o

l d

s nu

m b

e r s In t h is a_{p p}r o a ch, a s olut io n

f

o r t

h

e po r e

-s c a

l

e v elo ci t y

fi

e

l d

u

₍

x j

)

is o

b

taine

d

at e v enl y spa c e

l

at t ic e si te s x ; b y c o n si

d

e rin_{g t h}e e v olut ion o

f

a set o

f di

s c r ete

d

istribut ions

/

,)

(

x i

)

, w her e t

h

e in

d

ex g is a s s o ciate

d

w i t h a pa rticula r

d

is c r ete v e

l

oci t y:

{

0

,

0

,

0

}

^

_f

_or _g= 0

{

± l ,0,0

}

^

,

{

0,

± l ,

O

p

'

,

{

0

,0,± l

}

'

^

_f

o r 9 = 1,

2,

,

6

(

1 5

)

{

±

1,

± 1,

0

}

^

,

{

± 1,

0

,± 1

}

'

^

,

{

0,± 1,± 1

}

^

_f

_o

r _g =

7

,

8

, ,1

8

T he densit_y an

d m

o

m

en tu

m

ca n

b

e ex_pr e s s ed a s

l

ine a r c o

m

b inat ions o

f

t he

d

is c r ete

d

istribut ions :

18

8= 0

18

i

E v

<;

(

1 7

)

^

« = 0

W

i t

h

in t

h

e

M

R

T

f

r a

m

ewo r

k

, t

h

e

rela x at io n o

f

t

h

e

d

is c r ete

d

istribut

i

o n s to

w

ar

d

t

h

eir e_quilib¬

riu

m

va

l

u es is

m

o

d

e

l

e

d b

y co nsi

d

erin_g a s et o

f

e_qui

li

b

riu

m m

o

m

e nts

_/

_„, o

b

taine

d b

y a

l

in e ar

tra nsfo r

m

at ion of t he

d

istri but ions :

1 8

0

T

h

e values tr ans

f

o r

m

at ion

m

atrix

M

, , ,,, a r e c

h

o s en

b

ase

d

on a

G

r a

m

-

S

_c

h m

_i

dt

_ort hogo n a

li

z at io n

c onstr u cte

_d

usin_{g p}o_{l y}n o

m

ials of t he

d

is c r ete v elo ci t ies

_^

„

S

olut ion for t he

d

is c r ete

d

istri but ions

is provi

d

e

d b

_y a s su

m

in

_g

t

h

at e a c

h m

o

m

ent re

l

ax e s towa r

d

i ts e_quil ibriu

m

value

_f

^

at a r ate

s_pe ci

fi

e

d b

y a n a s s ociate

d

re

l

a x at io n _par a

m

eter

Aj

„

T h

e

f

or

m

o

f

t

h

e equili

b

riu

m m

o

m

e nts

/

f

^_''

,

tr ans

f

or

m

at ion

m

atrix

M

_q

^m, s J i

d

inver s e tr ans

f

or

m

at

i

o n

m

atrix A

/

*

f

ollo

w d

'

H

u

m

ier e s a n

d

G

inz

b

urg

₍

d

'Hu

m

ieres et a

l

,

20 0 2

)

T

h

is cor respo n

d

s to solut io n of t he e_qu at io n:

1 8

/

,

(

X, +

$

_^,t + 1

)

-

_/

^

(

x.,i

)

=

E K

m

^

r .

{

r j

-

_L

)

+

F

q,

(

1 9

)

w he r e

M

*

^^

a r e c o e

ffi

cients o

f

t he tr ans

f

o r

m

at ion

m

atrix inv e r s e an

d F

q is a c ontribut ion

d

ue to

an extern a

l f

or ce _g:

T he c onstant re

f

e r e n c e de n s_{i t y po} is s et to u ni t y a n

d

t he wei g

h

ts w_o = 1

_/

3, w ^ = 1

/

18 fo r

q = 1, ,6 an

d

Wq = 1

/

36 f

o r g = 7, . ,1 8 T

h

e s et o

f

r elaxat ion pa r a

m

ete r s a r e cho s en to

m

ini

m

iz e t he

_d

e_pendenc e o

f

_pe r

m

e a_{b i l i t y} on fluid visc o s_{i t y}

(

Pan, Luo an

d M

i l le r, 2 0 0 6

)

:

A

j =

A

j =

A

g =

A

io =

A

j i =

A

i 2 = -'^_{i s}= -

^

_{1 4}=

^

_{1 5}

= ~

i

(

2 1

)

A

₄ =

A

_g =

A

g =

A

i g =

A

i7 =

A

^g = a _{_}

\

' ' ^ ^

^

w he re t he pa r a

m

ete r r >

0 5

relate

d

to t he

k

ine

m

at ic vis c o s_{i t y} o

f

t he fluid:

. =

^

(

. -

i

)

(

2

3

)

T

o

d

e s c ribe no n-

D

_{ar c}

y

fl

ow , a seque nc e o

f

ste a

d

y

-_st_ate ve

l

oci t y

fi

e

ld

s are si

m

u

l

ate

d

w i t

hi

n

the _gene r ate

d m

ed ia using spe cified value s o

f

Fo to

d

rive t he

fl

ow in t

h

e x

d

ir e ct ion

O

nc e a

ste a

d

y state ve

l

oci t y

fi

e

ld h

as

b

e e n re a c

h

e

d f

o r a _{g i}v e n va

h

i e o

f F

o, t

h

e

m

acr o s cop ic

fl

ow v e

l

o ci t y

v " *

i

s c alculate

d

usin_g a

d

ens

i

t y-

w

_ei g h t_e

d

_{v o}l_u

m

e av er age o

f

t

h

e po r e-s ca

l

e veloc i t y

fi

el d u

(

x i

)

integrate

d

o v e r t he

fl

ow

d

o

m

ain

Q

,n >:

r

o

d

r

~

_^

^ ' "/ n_{- n}

.'.P -

^

•'

T he

m

a c r o.s c o_{p i}c v elo cit_{y i}s u s ed to c o

m

_pu te the

R

eynolds n u

m b

e r a s

d

e

fi

n e

d

b y E q 2

O

nc e

t

h

e s equen c e o

f

Fo v a

l

ue s a n

d

t

h

eir a s.so ciate

d R

eyn o

ld

s n u

m b

ers

h

a ve

b

e e n c o

m

pute

d

,

E

q

4

is

us_e

_d

to c alc ulate t

h

e pe r

m

e_a

_b

_{i l i t y} an

d

ine rt ia

l

c o e

f fi

cients a s s o ciated w i t

h

t

h

e ana

l

_yz ed

m

ed ia

R

e s

u

l

t

s

a n

d

D

i

s c

u

s s

i

o n

N

on-

D

_ar cian

fl

ow simu

l

at io ns we r e per

f

or

m

e

d b

y ru nning t

h

e

l

at t ic e

B

o

l

tz

m

a n n si

m

u

l

ator o n

the Tops ai l co

m

put ing syste

m

, a L in ux

b

a s e

d

clust er owned an

d

oper ated b y t he Univ e rsi t y o

f

No rt h

C

a r o

li

na T he c

l

uste r is c o

m

pose

d

o

f 52

0

c o

m

put ing no

d

es, ea c

h

equ

i

p pe

d w

i t

h 2

Inte

l

qu a

d

-c o re proce ssor s

(

M

o

d

e

l B

5

34

5

_/

C l

o v ertown

)

ru n ning at 2

3 G H

z a n

d

12

G B

o

f m

e

m

o ry

T

o

re

d

u ce si

m

u

l

at io n limes

,t

h

e

m

o

d

e

l

wa s r a n in pa ra

l l

e

l

o n

6

4 Tops ailcore s using

M

e ss age

P

as sage

Inter

f

a c e

(

M P I

)

Ta

bl

e 1:

G

ene r ate

d M

ed ia

Lo_gnormal D istri bu ted Va riance

, a ^ _P

oro sit y Ran_ge

T

'

0 38-0 60

0 1 036-_{0 5 4}

0 2 0 3 2.04 8

0 3 0 30-_{0 42}

v a r

i

a nce s, a

^

,

0

,

0

1,

0 2

, a n

d 0 3

o v er a ra nge o

f

poro si t ies, se e Ta

b

le 1

W

it

h

in ea c

h

po rosi t y

ran_ge, a s_{p h}e re _pa cl^in_{g w}a s _ge ne rate

d f

o r eve ry inc re

m

ent o

f 0.

0 1 To a ch ie v e sta

b

i l i t y,e a cho

f

t

h

e

pa clcin_gs ha

d m

e an coo r

d

inat ion n u

m b

e r s > 6 To ens u r e t

h

e e vo

l

ut ion o

f

a po ro us c ont in u u

m

,

eac

h

porous

m

e

d

ia s_yste

m

u se

d

w i t h in t he si

m

ula tio nsw a s a

m

ic ros c a

le

re_pr e s entat ive e

le

m

en t ary

v ohi

m

e

(

R E V

)

o

f

a

m

a c ros co_{p i}c s_yste

m

D

e ve

l

o

_p

m

e n

t

o

f

R

e

_p

r e s en

t

a

t i

ve

E l

e

m

e n

t

ar

_y

V

o

l

u

m

e s

T h

e

R

E

V h

a s to

b

e la r_ge e n o u_g

h

to c a_{p t}ure t he

m

a c ros cale _p

li

_ysics ofc onc e rn ,t

he r e

fo r e

t

h

e

non-

_D

a r c_y curve

m

ust

b

e in

d

epe n

d

e nt o

f b

ot

h

t

h

e

l

attice siz e o

f

t

h

e si

m

u

l

ate

d d

o

m

a

i

n a n

d

t

h

e

po re siz e o

f

t

h

e _gene ra t e

d m

e

d

ia

T

o

de v

e

l

o_p a por ous

m

e

d

iu

m

t

h

a t _{y i}e

ld

e

d

_gri

d

-

i

_n

d

_e pe n

d

e nt

r esu

l

ts, a syste

m

c ontaininga set n u

m

ber o

f

sp

h

e res wa s pac

k

ed an

d

t

h

e

l

at t ice siz e wa s in c rease

d

u nt i l t

h

e non-_Da

r cy cur v e c onver_ged to o ne

_d

ef

i

ni t ive solu_{t i}o n To e xte n

d

t h is s_yste

m

to a

d

o

_m

ain

in

de

_pen

d

ento

f

_po re str u cture,t he n u

m

be r o

f

s_{p h}e res wa s inc re

m

e ntal l y inc r e a s e

d

unt i lo nce a_gain

a

d

e

fi

ni t iv e solu tion

f

or t

h

e n o n-D _{ar c}

y cur ve wa s re ached T he r e

l

at ive intrinsic _pe r

m

ea

b

i

l

i t y a n

d

the in ert ial c o ef

fi

cients a r e s

_h

ow n a s

f

u nctio ns of s_{p h}e r e nu

m b

e r

fo r

e a ch v a rianc e of

l

o_gno r

m

al

d

istri

b

ute

d

r a

d

i i in

F

_{i g l}

o

a

^

~

0 5 0 0 I 15 0 0 2 0 0 0 2 5 0 0 3 0 O 0

N u m be rof S p he r e s

(

a

₎

Perme ab i l ity

3 5 0 0 4 0 O 0 4 5 0 0

MS-'

V

-..

2 0 0 0 2 5 00

Num be rof S p heres

(

b

₎

In er_{t i}al

4 0 00 4 50 0

F

i gure 1:

C

oe

ffi

cie n t v a

l

ue s r elat ive to t

he

as s o ciate

d R

E

V

value s o

b

tain e

d f

o r t he per

m

e a

b

i

l

_{i t y}

a n

d

in ert ial par a

m

eter s

fo r

a ra nge o

f

sp he r e pa c

k i

n_gs

T

a

bl

e 2 o ut l in es t

h

e n u

m b

er o

f

s_p

h

er es a n

d

c u

b

ic

l

at t ic e siz e s t

h

at were use

d

to gen erate a

R E V

po r o u s

m

ed iu

m

for e a ch v a ria nc e of t he

l

og-_{n o}r

m

al l y

d

istri buted r a

d

i i co nsi

d

ere

d

w i t

h

in t he

D

e r

i

ve

d C

or r e

l

a

t i

o ns

P

e r

m

e ab i

l i

t y

C

or relat

i

o n

E

x ist in _g c o r re

l

at ions

f

o r t

h

e

d

i

m

e nsion

l

e ss pe r

m

ea

b il

i t y were co

m

par e

d

to resu

l

ts o

b

tain e

d

b

a s e

d

o n s

im

u

l

at ions per

f

or

m

e

d

usin_g t

h

e

l

at t ic e

B

o

l

tz

m

a n n

m

et

h

o

d

P

o

i

nts o

b

tain e

d f

ro

m

si

m

u

l

at ion

d

i

d

n ot a

d

e_quate

l

y

fi

t a n_y o

f

t

h

e

f

u nct io n a

l f

or

m

s

d

esc ri

b

e

i

n t

h

e

b

ac

k

_gro u n

d

se ct

i

o n

f

or t he

f

u

l l

r an_geo

f

po r o s_{i t y} v a

l

u e s c onsi de re

d

F i g 2 shows how t he Ergun r elat ion de v iated s_{i g}ni fic an t_{l y f}r o

m

the si

m

ula tion resul ts for the

f

ul lrange o

f

_po r osi te s

,e spe cial

l

y at h i g

h

e r por o si t y v alu es

{

> 0 4 2

)

T

h

e

E

r_gu n r elat io n

d

eviat io n

wa s qua

l

it at iv el y

b

a s ed on a str ai g

h

t tu

b

e _ge o

m

etr_y o

f

t

h

e po r e spa ce an

d h

a s

b

e en

f

ou n

d

to

b

e

val i

d

o nl y in a ra nge o

f R

eyn ol

d

s nu

m b

ers, 0 <

R

e <

7

5

(

Ergu n, 1 9

5

2 ;

M

ac

d

ona

l

d

F

, 1 9 7 9

)

B

ase

d

o n t

h

e _pac

k m

_{g p}ara

m

eter s s e_{t b y t}

h

e

R

E

V

c a

l

culat ions

(

T

a

b

le

2

)

, t he ra nge o

f R

eyno

ld

s

n u

m b

e r s ach ie v e

d

w i t

h

in t

h

e si

m

ulat ion

d

i

ff

e r s

f

or e a c

h

va ria n c e T

h

e

R

eyno

ld

s n u

m b

er ra n_ge

is

0

<

R

e < 1 4

0 f

or t he

h

o

m

o_gen ous _pac

k

in_g

(

c r^ = 0

₎

, 0

< Re < 1 3 0

f

o r c r^ = 0 1,

0

<

R

e < 1 1 5

f

o r a^ = 0 2, an

d 0

<

R

e < 9 5

f

o r a

^ _{= 0 3}

_A

s t he

R

e_yn o

l d

s n u

m

ber r an

_g

e

f

or ea ch v ariance

incr eas es pa sse

d

t

h

e up

p

er

l

i

m

i t o

f

t

h

e Er_gu n relat ion, t

h

e d i

ff

er ence

b

et we e n t

h

e pe r

m

e ab i

l

i t y v alues

f

ro

m

t

h

e si

m

u

l

at

i

o n a n

d

t

h

ose _pre

d

icte

d b

y

E

r_gu n inc re as e s

Si mulated Data a nd Exist ing Co r r elat io n s :Pe rme ab il it y

E

b

£rgu n Ca r m a nKo z e ny R um ptOup t B

——_pa n e ta l . ^,0 A 0^=01 O o^ 'OZ

.

^

^

'

^

^

^

Po r o sit y

F

i gure 2:

C

o

m

pa ris on

b

et we en t

h

e si

m

u

l

ate

d d

ata a n

d

t

h

e e xist in_g e

m

p

i

ric a

l

r e

l

at io ns

P

a n et al

b

as e

d

t he

i

r _per

m

ea

b

i

l

i t y r e

l

at io n o n a set o

f fl

ow si

m

u

l

at io ns

w

i t

h

a porosi t y ra nge

o

f

0 3

3

< e < to 0 4 5

₍

P

an, H i l pe rt and

M

i l ler, 2

0

0 1

)

, cor r elat ing to t he range of po r o si t ie s t he

Ta

_b

le 2:

_R

epre sentat ive E

l

e

_m

e ntar_y

_V

olu

m

e s

f

or

_G

e nerate

d M

e

d

ia

Va ria nc e

,ij^ Numbe r o_{f S p h}e r e s

, jV^ Lat t ic e Dime nsio ns n ^ _M

e a n p ix els pe r grain d iamete r ,P

"

(i 15 0 0 3 *5' "

3 2 8 4

0 1 150 0 3 8 0'

3 0 0 2 3 00 0 460'

27 9 7 7

re

l

at ion c o r respon

d

s to t he si

m

ulate

d d

ata in _{F i g} 2

A

t the po r o si t ieso utsi de t he e xperi

m

e n tal

l

_y

su_{p p}o rted r an_ge, t he Pa n et a

l

re

l

at ion un

d

e r e sti

m

at es t he

m

e a sur ed pennea

b

i l i t ies

T he

C

a r

m

en-_{Ko z en}

y r elat io n

fi

t we

l

l to a r an_ge o

f

t

h

e

m

eas u re

d

per

m

e a

b

i l i t ie s, bu t

f

o r

l

ower

po r osi t

i

es

(

< 0 4

2

)

t

h

e re

l

at io n starts to ve er o

_{f f}

si

g

n i

fi

c an t

l

y

f

r o

m

t he

d

ata

(

F

i g 2

)

T h

e

l

owe r po r o si t y

d

eviat io n s ar e con sista nt w i t h t

h

e

fi

n

d

in_gs o

f

ot

h

e r s

(

P

an,

H

i l pert a n

d

M

i

l

le r ,

2

0

0 1

₎

a n

d h

ave

b

e en at tri

b

ute

d

to t

h

e

C

ar

m

en-

_K

_o

z enzy re

l

at ion o nl y

b

eing v a

l

i

d f

or

l

a

m

ina r

fl

ow

(

Prieur Du P

_l

e s sis an

d M

asl i yah, 1 9 9 1

)

,a n i

d

e a sup po rted b y the ine rtial pa r a

m

ete r r e sul ts

f

ro

m

t

h

e si

m

ulat ion

d

ata

F

i g 4 shows

h

ow t he inert ial cor rect ion to D a r c_y

'

s

l

aw, c o e

ffi

c ient

h

,

f

o r t he si

m

u

l

ate

d fl

ow in cr ea s e s a s porosi t y

d

e cr e a ses

T h

e

d

eviat ion

f

r o

m

Darcy

fl

ow at t

h

e

l

o

w

e r _porosi t y ra n

_g

e su

b

s e_qu e nt

l

y

d

e cr eases t

h

e v a

l

i

d

i t y o

f

t he

C

ar

m

en-

_K

_{o z e n}

y re

l

at io n.

T

h

e

R

u

m

_p

f

-

G

_u

p te re

l

at

i

o n

d

eviates t

h

e

m

ost

f

ro

m

t

h

e si

m

u

l

ate

d d

ata, overpre

d i

ct ing t

h

e

d

a ta

f

or po r o si t ie s >

3

8 an

d

u n

d

e r_pre

d

ict ing

f

or _po rosi t ie s < 3

8

T

h

e

d

ata

f

or

R

u

m

p

f

-

G

_u

p te

is

b

a s e

d

o n s_{p h}e r e pack ings w i t

h

r elat iv e stan

d

ar

d d

e viat

i

ons o

f

t he s_{p h}ere

-siz e

d

istri but ion,

C

T o, of 0 0 9

4

5, 0 3 2, and 0

3

2

7

o v e r a w i de r ange o

f

por o si t y

(

0

3

6 6 < e < 0 6 4

)

an

d

R eyn ol ds

n u

m

ber

[

0 <

R

e < 1 0 0

) (

M

ac

d

on al d F , 1 9 7

9

)

T h

e ct

^ = _{0 3}

_d

ata w i t

h

in 0 3

6

< e < 0 4 2 is t

h

e

o n

l

y si

m

u

l

at io n

d

ata t

h

at

fi

ts w i t

hi

n t

h

e e x

_p

eri

m

e nta

ll

y s u_{p p}orte

d

re

_g

i

m

e o

f

t

h

e

R

u

m

p

f

-

G

_u

p te r e

l

at io n

F i

g 2 s

h

ow s

h

ow t

h

e

d

ata w i t

h i

n t

h

e

R

u

m

p

f

-

G

_u

p te e xpe ri

m

e nts co r r e

l

ates to t

h

e

porosi t y ra nge w

h

ere

d

e viat io n s a re a

m

ini

m

u

m

, a n o

b

ser vat io n sup porte

d b

y

(

Pa n ,

H

i

l

pe rt a n

d

M

i

l l

er, 2

00

1

)

A

n exponent ia

l fi

t o

f

t

h

e

f

u

ll

r ange o

f

si

m

u

l

at ion

d

ata y ie

ld

e

d m

or e s at is

f

a cto r_y agree

m

e nt w i t

h

t

h

e si

m

u

l

ate

d d

ata T

h

e a ss ociate

d f

u n ct ion a

l f

o r

m

is :

oi

(

e

)

= ae

'

'''

,

(

2 5

)

w

_h

er e t

h

e be st-

_fi

_t

c o e

fi

i cie nt va

l

ue s ar e a — 1

9

4

₅

x 1

₀

^ a n

d

7 =

9 8

4

₆

_P

_{i g}

₃

co

_m

_pares t

h

e

pr opose

d

co r r e

l

at io n

m

o

d

e

l

to t

h

e si

m

u

l

ate

d d

ata

u o cc

Pr ed icted D ime ns io nle s s Pe rme a_{bi lity}

I

ne rt

i

a

l C

o r r e

l

at

i

o n

S

i

m

ulate

d

value s for t he ine rtial pa r a

m

ete r a r e sho

w

n in c o

m

_pa ris on w i t h v alue s _pr e

d

icted

b

y

E

q 8 in

F

i g 4

N

ei t

h

er o

f

t

h

e c oe

ffi

cients pr e

d

icte

d b

y Er_gu n an

d M

a c

d

o na

ld f

or t

h

e

E

r_gu n

f

u nct iona

l fo r

m

(

E q

8

)

_prov i

d

e a s at is

f

actor_y

fi

t to t

h

e si

m

u

la

t ion

d

ata, a n

d b

ot

h

re

l

at io n s

o ver_pr e

d

ict t

h

e

l

owe r ran_ge o

f

po rosi t ies

(

< 0 4

0

)

an

d

un

d

e r_pre

d

ict t he

h

i g her ra nge o

f

po r o si t ie s

(

>

0

4 0

₎

As d is c u s s ed w ith in the la st s e ct io n

f

or t he _pe r

m

e ab i l i t y c o ef

fi

cie nt, t he r ange of te sted

R

eyn o

ld

s n u

m b

er se e

m

s to

b

e a sour c e o

f

er r o r

b

et wee n t

h

e si

m

ulate

d

da ta an

d

t ho se pre

d

icte

d

b

_y t

h

e

E

rgu n r e

l

at ion T he

E

r_gun rela t

i

o n ha s a

l

s o

b

e e n

f

o u n

d

to

b

e va

l

i

d

onl y in poro si t ies

ra ng

i

ng 0

38

< e <

0 47

(

H

a_{p p}e

l

a n

d

Br en n er, 1

9

6

5

)

,cor r e

l

at ing to t

h

e r ange o

f da

ta w

h

er e t

he

Ergu n r elat ion sta rts to co n ve r_ge w i t

h

t

h

e si

m

u

l

ate

d d

ata be

fo r e

d

iv e r_{g i}n_g at a _po rosi t y o

f

0

3 5

Simul a ted Data and Exist ing Co r r elat io n s :In e rt ial Co ef f icie nt

^ 28

o

O 26

f

^

i

:

S

«

\

—

- Ergu n

Ma c d o n aldfl tal

■ o^ =a O

A 0^= 01

O 0 ^

=0 2

%

_%

'

'

_t

4

0 o

;> S.

■ ^-_{» . *}

Po r o sit y

F

_{i g}ur e 4:

C

o

m

pa ris o n bet we e n t he s

i m

ulate

d d

ata a n

d

t

h

e e xist in_g e

m

p irical re

l

at io ns

T h

e

M

a c

do

nal

d

et a

l

r e

l

at

i

o n is t

h

e

E

r_gu n r e

l

at ion

(

E

q 8

)

w i t

h

a

m

od i

fi

e

d

B co ef fi cient

base

d

on t

h

e c o

m

parat iv e a n a

l

ysis o

f

n u

m

e r ous e xpe ri

m

enta

l

r e sul ts, inclu

d

ing

R

u

m

p

f

-

_G

up te

T

h

e anal ysis a

ls

o to o

k

in a c c o i

m

t data

f

ro

m

a w i

d

e r ange o

f

po r o si t ies

₍

0

1 2 3 < e <

0

9 1

9

)

an

d

gr a n ular s

h

ape s an

d

siz e s w i t

h

t he _go alo

f d

eriving an

E

r_gu n r e

la

t io n t

h

at was ap p l ica

b

le

f

o r

pa c

k

in_gs o

f

no n-_s

p

h

e ric al gr ains

(

M

a c

d

onal

d

F , 1 9 7 9

)

M

uc

h

o

f

t he

lo

wer po r o si t y

d

ata

(

e < 4 0

)

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m

e

f

ro

m

ex _pe ri

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l

ar s

h

a_pe

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o

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d

an

d

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l m

ixtures, a n

d

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o

f

u n

d

is c

lo s e

d m

ate ria

ls

S

inc e t

h

e si

m

u

l

ate

d d

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l

y ta

ke s

in ac co u nt por ous

m

e

d

ia c o

m

_pose

d

o

f

s

m

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h

s_p

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h

n e ss co u

ld

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ain t

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arge

d

e viat

i

o n s

b

et we e n t

h

e in ert ia

l

para

m

ete r

pre

d

icte

d b

_{y t h}e

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a c

d

onal d et a

l

re

l

at ion an

d

t

h

e si

m

ulate

d d

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0

T

o provi

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e a

b

et ter

fi

t

f

o r t

h

e

f

u

l l

ra nge o

f

si

m

u

l

ate

d d

ata, n o n

-

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_i

n ea r

l

e ast squ ares wa s us ed

to fin

d

t he

b

e st

fi

t co e

ffi

cients to a _gene ra

l

ize

d

fo r

m

o

f

t

h

e Er_gi

m

r elat io n:

(

2 6

₎

w

h

e r e t

h

e c o e

f fi

cie nts ar e

B

= 4 2 1, 71 = 1 5

8

an

d

72 =

0

3 8

F

i g

5

co

m

pares t

h

e _propos e

d

co r re

l

at ion

m

o

d

elto the si

m

u

l

ate

d d

ata

■ 0^ =0 2

1 8 2 2 2 2 4 2 6 2 8 3 2 3 4 3 6 38

Pr ed icted In ert ial Pa r am e te r

F i gur e

₅

: T he pr opo s e

d

in ert ia

l

c o r r elat ion pre

d

icts t he si

m

ulat io n

d

ata

C

o n

c

l

u

s

i

o n

s

1 T

_h

e

f

ul

l

s et o

f

si

m

ulat ion

d

ata

f

ro

m

t

h

e lat t ice-_Bo

_l

_t

z

m

a n n

m

ode

l d

i

d

not

fi

t a n_y o

f

t

h

e ex

i

st ing pe r

m

e a

b

i

fi

t y cor re

l

at io ns,t

h

e Ergu n a n

d R

u

m

p

f

-

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_u

p te re

l

at ion sov e r _pre

d

ict

m

uc

h

o

f

t he _por o si t y r an_ge w h i le t

h

e Pan et al an

d C

a r

m

en-

K

_{o z en}

y r elations fi t

d

ata w i t

h

in certain po rosi t y ra nge s I t

w

a s o

b

s e r v e

d

t

h

at t

h

e

d

e viat io ns o

f

t

h

e si

m

u

l

at io n

d

ata

f

r o

m

t

h

e Pa n et a

l

a n

d R

u

m

p

f

-

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_u

p te r elat ions c o r r e s_pon

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to t he r ange of _po r o si t y v alue s

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ack ing e x_pe ri

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enta

l

su_{p p}ort w he r e a s t he

d i fT

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t

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e

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ar

m

e n-

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oze ny r e

l

ate to t

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f

t

h

e in ert ial para

m

eter I t was also note

d

t hat a s t

h

e

R

e_yn o

ld

s n u

m b

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d

pa st t he v al i

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d

i

ff

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b

et we e n t

h

e

m

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d

pe r

m

e a

b

i l i t y and t he

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e a_{b i fi t y g}r ew

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n e xpo nent ia

l

c o r relat ion

m

odel

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epen

d

e nt o n por o si t y wa s

d

e rive

d f

o r t

h

e

d

ata s et

2

T h

e in ert ia

l

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m

ete rs

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ro

m

t

h

e

fl

ow si

m

ulat io ns

fi

t t

h

e _genera

l

ize

d f

or

m

o

f

t

h

e

E

r_gu n

re

l

at ions but n ot a ny of t he e xist ing c o e

ffi

cie nts propose

d b

_{y E}r_gu n a n

d

M

ac

d

on al

d

A

gen era

l

iz e

d f

or

m

o

f E

r_gu n's ex_pre ss

i

o n wa s c o nstru cte

d

to _provi

d

e a

m

o r e sat

i

s

f

a cto ry

fi

t o

f

t

h

e si

m

ulate

d d

ata

R

e

f

e r e

n

c e s

Be a r,J 1 9 7 2 D yn a

m

ic s a

}

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luids in

P

o r ous

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e

di

a

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l

s evie r

d

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u

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, I

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b

ur

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,

M

Kr a

f

c zy

k

, P

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al le

m

an

d

a nd

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u

lt i p l

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-r elax at ion-t i

m

e

l

at t ice

B

o

lt

z

m

a nn

m

o

d

els in t

h

r ee

d

i

m

en sio n s " P h i

l

o s op

h

ic a

l

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o

_f

t

h

e

R

o_ya

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o ciet y o

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o n

d

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at

h

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m

at ica

l P h

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a n

d E

n_{g i}ne e rin_g

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cienc e s

3

6 0:4

37

- 4 5 1

Ergu n,

S

1

9

5 2 "

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l

uid flow t

h

ro u_g

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_pac

k

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colu

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ns "

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m

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l E

n_{g i}ne e rin_g

P

r ogr e s s 4 8:8

9

-

₉₄

Fo ur a r,

M

, G. Flad i l la,

R

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m

a n

d

and

C

M

oyne 2 0 0 4

"

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n t he non-_{l in}

e a r be

h

a vio r of

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m

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-p has e

fl

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d

t

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l

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ia "

A

dv anc e s in

W

ate r

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e s o u r c es 2

7

₍

6

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:

6 69

-

677

G

e erts

m

a,

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1 9

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Est i

m

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h

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fi i

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in ert ial r esistan c ein

fl

ui d

fl

o_{w t}

h

r ou_{g h p}o r ous

m

e

d

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En_{g p p} 2 1 1-_{2 1}

₆

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r ay,

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a nd

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06

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er

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l

_{l y}

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f

or

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ena in Po ro u s

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s :

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l

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v anc e s in

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9

(

1 1

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:1 7 4 5-1 7 6 5

H

a_{p p}e

l

,

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a n

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B

re nner 1

9 65

L

ow

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s

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y

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ng lewo o

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i

ff

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d

eh,

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a n

d

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G

G

r ay 1 9

87

"

H

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h

-

V

_{elo c}

i t y

F

low in Por o u s-

M

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d

_ia " _{Tr ans} po rt in

P

o r ous

M

e

di

a 2

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6

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:5

2

1-

_{53 1}

H

a s s a niza

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eh,

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a n

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88

"

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ep

ly t

o

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y

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ar a

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H

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h V

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in

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b

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ass a nizei

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h

a n

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ra_y "

T

r ans_port in

P

o r ou s

M

e

di

a

3

(

3

)

:

3

1

9

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_{32 1}

M

a,

H

an

d

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W

Rut h 1 9 9 4 "

A

Nu

m

e ric al-

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n al ysis o

f

t he Inte r

f

a cial Dr a_{g F}or c e

f

o r F luid

-F l

ow in

P

o r ous-

M

ed ia " Tr anspo rt in P o r o u s

M

e

d

ia 1 7

(

1

)

:

8 7

- 1

03

M

a c

d

o n al

d F

,

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l

-

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d S

,

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ow

K

Du

Ui

en F 1 9 7

9

"

F low t hro ug

h

Poro u s

M

e

d

ia - _{t he}

_E

rgu n

E

qu at ion

R

ev isi te

d

"

I

n

d E

n_g

C h

e

m

F

un

d

a

m

1

8

(

3

)

:199-2

08

M

c

C l

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J E

,

W

G

G

r ay a n

d C

T

M

i

ll

e r

2 0

1

0

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e_yo n

d A

nisotrop y:

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x a

m

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N

o n-

D

a r c_y

F l

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A

s_y

m m

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P

or ous

M

e

d

ia "

T Vans_port in

P

o r ous

M

e

di

a

84

₍

2

₎

:

53 5

-

5

4

8

Pan

,

C

,

L

-

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L

uo a n

d C

T

M

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200 6

"

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n eva

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on o

f l

at t ic e

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o

lt

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m

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f

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m fl

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ulat ion "

C

o

m

pute r s

&

F

l

uids

₃₅

₍

8-9

₎

:

₈₉₈

- 9 0

9

Pa n,

C

,

M

H ilpert a n

d C

T

M

ille r 2

00

1 "

P

or e-_{s cale rn o}

d

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satu rate

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m

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h

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6

4

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:

9

P

a n

fi l

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a n

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F

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20

06

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P h

ys

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sp

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ng o

f

n o n

l

in ea r e

f f

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i

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po ro u s

m

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ia "

A d

v anc e s in

W

ate r

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9

₍

1

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:

30

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Pr

i

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J

a nd

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F l

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h

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h I

sotr o_{p i}c

G

r an u

l

ar

P

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M

ed ia " Tr ans_port in Po r ou s

M

e

di

a

6

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7

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R

u

m

p

f

,

H

, an

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G

up te 19 71

"

E in

fl

i is s e

d

er Po ro si tat u n

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o r n_gro s senv ertei

l

u n_{g i}

m

W

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d

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P

ore n str o

m

un_g "

Ch

e

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ie

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T

ec

h

ni

k

4

3

:

3

6

7

-

_{3 75}

.

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,

M R

1

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5 7 "

D

evelo_p

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f

a gene r aUz ed Da r cy equat ion " Tr ans

A

I

M

E 2 1 0:

376

-

37 7

W

ang, X , F T

h

auvin an

d

K K

M

ohant y 1

999

"

No n-_Darc

y

fl

ow t hr ou_{g h} a nis otro_{p i}c _po r ous

m

e

d

ia "

C h

e

m

ic a

l E

n_{g i}ne e rin_g

S

cienc e

₅

4

₍

12

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:1

8

5 9-₁

_{86 9}

W

i

Ui

a

m

s,

S

a n

d A

P h i l i ps e 2 0 0

3

"

R

an

d

o

m

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ki

n_gs of s_{p h}e r e s a n

d

s_{p h}erocy l in

d

e r s si

m

ulated

b

y

m

e c

h

a nical co ntr a ct io n " P

h

ysic al

R

eview

E 6

7