c

Stanford Geothermal Program I n t e r d i s c i p l i n a r y Research

i n Engineering and Ear th Sciences Stanford Univers i ty Stanford , C a l i f o r n i a

THE EFFECTS OF TEMPERATURE AND PRESSURE O N

ABSOLUTE PERMEABILITY OF SANDSTONES

c

by Muhammadu Aruna

A p r i l 1 9 7 6

This r e s ea r ch was c a r r i e d out under Research Grant GI- 34925

by t h e National Science Foundation

ACKNOWLEDGEMENT -

T h e a u t h o r would l i k e t o express h i s s i n c e r e apprec i - I

a t i o n t o h i s a d v i s e r , D r . Henry J. Ramey, Jr., for h i s i n- I

s t r u c t i o n and guidance throughout t h e r e s e a r c h work.

made it p o s s i b l e f o r t h e a u t h o r t o come t o S tanfo rd U n i v e r s i l y

and complete h i s g radua te s t u d i e s s u c c e s s f u l l y .

He I

I I

I I

The a s s i s t a n c e and p r a c t i c a l sugges t ions from t h e

f a c u l t y of t h e Department of Petroleum Engineering, p a r t i c u -

l a r l y from Dr. W . E . Brigham and D r . S u l l i v a n S . Marsden, 1 are g r a t e f u l l y acknowledged. The coopera t ion I enjoyed from

I

I

I

I

t h e department secretaries, e s p e c i a l l y from A l i c e Mansouria4, I w i l l n o t be f o r g o t t e n . I

Many thanks are due t o m y f r i e n d s , p a r t i c u l a r l y Rorio

A r i h a r a , H. K . Chen, and Paul Atkinson, f o r t h e i r h e l p f u l

comments and Timely h e l p through the days of exper imenta l w&k.

~

I

C r e d i t f o r m d i f i c a t i o n of t h e a p p a r a t u s b u i l t by Wei4

braadt and Cass6 i s due Jon Grim, t h e Petroleum Engineering1

Dezs-tnent mach in i s t .

F i n a l l y , Dr. F. Ca.ss6 reviewed t h i s manuscript and ma& I

E?-- : ? s l p f u l sugg3st ions ' . ~

?>lis w s ~ k was fund.ed under Nat ional Science Foundatiovl

pm= _- ' 7 - . 3L . . - r - G I- 3 4 9 2 5 . I

i

This r e p o r t was prepared o r i g i n a l l y as a

d i s s e r t a t i o n sub,mitted to t h e Department of

Petroleum Engineering and the Committee on the

Graduate D iv i s i on of S tanford Un ive r s i t y i n

p a r t i a l f u l f i l l m e n t o f the requirements f o r t he

degree of Doctor of Philosophy.

ii

DEDICLTED TO MY PARENTS AND

ALL THE GOOD PEOPLE I N MY LIFE

iii

I

J

J

J

J

J

ABSTUCT

c

The s t a n d a r d procedurle f o r de t e rmin ing t h e p e r m e a b i l i t y

of porous media acco rd ing t:o API Code Mo. 2 7 i s based on t h e I

fundamental assumption that : , as long as v i s c o u s f l o w p r e v a i l s , I '

t h e a b s o l u t e p e r m e a b i l i t y of a porous medium i s a p r o p e r t y of

t h e medium, and i s independlent of t h e f l u i d used i n i t s deter-11

mina t ion , s l i p e f f ec t be ing t aken i n t o accoun t i n t h e case of

gas f l o w . Abso lu t e p e r m e a b i l i t y h a s , t h e r e f o r e , been t r a d i t i o d -

a l l y measured a t room c o n d i t i o n s , w i th t h e assumpt ion t h a t it I

changes o n l y w i t h overburden p r e s s u r e and n o t w i t h t e m p e r a t u r e +

R e s u l t s o b t a i n e d a t room tempera ture may t h e n be used t o p r e -

d i c t performance a t r e s e r v o i r c o n d i t i o n s a f te r c o r r e c t i o n f o r

r e d u c t i o n by stress e f fec t s .

I

1 ~

1 1

I I

Although t h i s assumption i s t r u e for m o s t f l u i d s , re-

s u l t s of r e c e n t a b s o l u t e pe rmeab i l i t y measurements of water

f low through porous media a t h igh t empera tu re s and h i g h

overburden p r e s s u r e s d i f f e r from room c o n d i t i o n v a l u e s .

I

N o t

only was t h e a b s o l u t e p e r m e a b i l i t y t o water a t h i g h c o n f i n i n g

p r e s s u r e lower t h a n t h a t to1 o t h e r f l u i d s used a t room tempera-

t u r e , b u t also, t h e r e was a s i g n i f i c a n t p e r m e a b i l i t y r e d u c t i o n ~l

a t e l e v a t e d t e m p e r a t u r e s .

An e x i s t i n g permeaneter w a s modified t o e n a b l e f l o w of I I d i f f e r e n t f l u i d s t h rough s e p a r a t e flow l i n e s .

water, a wh i t e m i n e r a l o i l , n i t r o p e n and 2- octanol were the

D i s t i l l e d

iv

f l u i d s used, and t es t s were c a r r i e d o u t on n a t u r a l conso l i-

d a t e d sands tones and unconso l ida ted s i l i c a sand.

With t h e excep t ion of w a t e r , t h e a b s o l u t e p e r m e a b i l i t i e s

of t h e cores t o o t h e r f l u i d s showed l i t t l e o r no t empera tu re

dependence.

t e m p e r a t u r e i n c r e a s e w a s a t t r i b u t e d t o i n t e r a c t i o n between

The r e d u c t i o n i n p e r m e a b i l i t y t o w a t e r wi th 1 1

water and s i l i ca .

systems o r i n thermal r ecove ry p r o c e s s e s which c a u s e l a r g e

changes i n format ion t empera tu re s , t h e effect of t e m p e r a t u r e s l

on absolute p e r m e a b i l i t y should be cons ide red i n e n g i n e e r i n g

c a l c u l a t i o n s . I

I n t h e case of water f l o w t h rough geothermd

I

V

1

TASLE OF CONTENTS - Page

c

c

L

ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . i v

ABSTRACT, . . . . . . . . . . . . . . . . . . . . . . . V

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . v i i

L I S T OF TABLES. . . . . . . . . . . . . . . . . . . . X

L I S T OF FIGURES . . . . . . . . . . . . . . . . . . . x i

CFAPTERS

1. INTRODUCTION. . . . . . . . . . . . . . . . 1

2. FLOW I N POROUS YEDIA. . . . . . . . . . . . 4

2 . 1 Darcy 's Law. . . . . . . . . . . . . . 4

2 . 2 Darcy's Law for Gas Flow . . . . . . . 5 '

2.3 S l i p Phenomena i n G a s F low . . . . . . 6 1 ;

2 . 4 Visco- Inert ial Flow. . . . . . . . . . 7

3.

4.

LITERATURE. . . . . . . . . . . . . . . . . 9

EXPERIMENTAL EQUIPMENT. . . . . . . . . . . 15 1

4 . 1 General D e s c r i p t i o n . . . . . . 1 5

4 . 2 C o r e Holder. . . . . . . . . . . . . . 1 8 ~

4 . 3 A i r B a t h and Temperature Control . . . 1 8 ' 4 . 4 Liquid and Gas Sources . . . . . . . . 20

4 . 5 Liq,uid Pumps . . . . . . . . . . . . . 2 1

4 . 6 Hydraulic Hand Puny. . . . . . . . . . 2 1

4 . 7 Heat Exchangers. . . . . . . . . . . . 21

vi

Table of Conten ts , con t inued

4.8 Backpressure Valve . . . . . 4.9 Flow Rate Measuring Devices.

4 .9- 1 Liquid F l o w

4.9- 2 Gas F l o w

4- 10 Pressure Recording Devices . PROCEDURE . . . . . . . . . . . . 5.

6 .

5.1

5.2

5 . 3

Core P r e p a r a t i o n . . . . . . 5 . 1 - 1 Conso l ida t ed Mass i l lon Sand-

s t o n e s

5.1- 2 Unconsol idated O t t a w a Sand

E s t a b l i s h i n g Run Cond i t i ons . . Measurements and C a l c u l a t i o n s .

5 . 3- 1 Liquid F l o w

5.3- 2 Gas Flow

ANALYSIS OF RESULTS AND DISCUSSION.

6 . 1

6 .2

6.3

6.4

6.5

Water Flow . . . . . . . . . . . 6 . 1- 1 Conso l ida t ed Sandstones

6.1- 2 Unconsol ida ted Sand

G a s Flow . . . . . . . . . . . 6 . 2- 1 Conso l ida t ed Sandstones

6.2- 2 Unconsol ida ted Sand

O i l Flow . . . . . . . . . 2-Octanol Flow . . . . . . Discuss ion . . . . . . . .

7 . C ONC LU S I O N S AND R.EC OIWENDAT I O N S

8 . REFERENCES. . . . . . . . . . .

.

Page

2 2

2 2

2 2

23

23

25

25

25

2 6

2 7

28

2 8

37

4 1

4 1

4 1

4 3

48

50

5 6

56

58

62

6 5

67

J

3

3

J

v i i

c

Table of C o n t e n t s , c o n t i n u e d Page

9. APPENDICES. . . . . . . 70

9 . 1 L i s t of T a b u l a t e d Data . - . . . . . 71

9 . 2 C o r e Data. . . . . . . . . . . 9 0

9 .3 D e r i v a t i o n s of E q u a t i o n s . . . . . 9 4

9 . 4 L i s t of Manufacturers . . . . . . . . 9 9

1 0 . NOMENCLATURE. ,, . . . . . . . . . 102

v i i i

Table

'LI'ST OF TABLES - I I

I I

Page

1

2

3

10

11

1 2

13

14

15

16

D e n s i t y and V i s c o s i t y of Water vs .

V i s c o s i t y of Ni t rogen vs. Temperature . . . 73

1 Temperature . . . . . . . . . . . . . . . . . 72

! I V i s c o s i t y of Chevron . O i l No. 3 vs. Temperature . . . . . . . . . . . . . . . . . 7 4

. . . . D e n s i t y of 2-Octanol vs. Temperature. 75 I

V i s c o s i t y of 2-Octanol v s . Temperature. . . . 7 6 ,

F l o w D a t a f o r Mass i l lon Sandstone Core N o . Water Flow. 77

F low D a t a for Mass i l lon Sandstone Core N o . Water Flow. 78

Flow D a t a for Mass i l lon Sandstone Core No. Water Flow. 80

Flow Data for Mass i l lon Sandstone Core N o .

2, . . . . . . . . . . . . . . . . . 1 1 3 , . . . . . . . . . . . . . . . . . 1 4, . . . . . . . . . . . . . . . . . i I 5,

N i t r o g e n Flow . . . . . . . . . . . . . . . . 8 1 I

F l ow D a t a for Mass i l lon Sandstone Core N o . 6 , N i t rogen Flow . . . . . . . . . . . . . . . . 8 2

Flow Data for P?assi l lon Sandstone Core N o . 1 0 , O i l Flow. . . . . . . . . . . . . . . . . 8 4

Flow Data for Mass i l lon Sandstone Core N o . I 11, O i l Flow. . . . . . . . . . . . . . . . . 85 I

Flow Data f o r Unconsol idated O t t a w a S i l i c a I Sand Core N o . 1 4 , Water Flow. . . . . . . . . 86

I

Flow Data f o r Unconsol idated O t t a w a S i l i c a Sand Core No. 1 5 , Water Flow. . . . . . . . . 87

F low Data for Unconsolidated O t t a w a S i l i c a Sand Core N o . 1 6 , Nitrogen Flow and Water F l o w . . . . . . . . . . . . . . . . . . . . . 88

Flow Data for Unconsolidated O t t a w a S i l i ca Sand Core No. 1 7 , 2-Octanol Flow. 89

I ! . . . . . . ' I

ix

c

LIST OF FIGUP,ZS -

F i g u r e Page -

1 Photograph of Apparatus . . . . . . . . . . . 16

2 Details of Assembly i n t h e A i r Bath . . . . . 16

3 Schematic Diagra.m of Apparatus . . . . . . . . 1 7

4 Schematic Diagram of t h e Core Holder . . . . . 19

5 Dens i ty of Water1 vs . Temperature of 2 0 0 p s i g 31

6 Water V i s c o s i t y vs . Temperature a t 2 0 0 p s i g . 3 2

7 Dens i ty vs . Temperature for Chevron White Oil No. 3 a t 14.7 p s i . . . . . . . . . . . 33

8 V i s c o s i t y v s . Temperature for Chevron White O i l N o . 3 . . . . . . . . . . . . . . . . . 34

9 Dens i ty of 2-Octanol vs . Temperature a t 1 4 . 7 p s i . . . . . . . . . . . . . . . . . . . . . 35

1 0 V i s c o s i t y of 2-Octanol vs . Temperature. . . . 36 ~

11 V i s c o s i t y of Ni t rogen vs . Temperature . . . 39

1 2 Water P e r m e a b i l i t y vs. Temperature , Mass i l lon Sandstone Core No. 2 . . . . . . . . . . . . 42

13 Water Pe rmeab i l i t y v s . Temperature , Mass i l lon Sandstone Core No. 3 . . . . . . . . . . . . 44

1 4 Water Pe rmeab i l i t y v s . Temperature a t a Cons tan t Conf in ing P r e s s u r e of 3000 p s i g , Mass i l lon Sandstone Core No. 3 . . . . . . . . 45

1 5 Water Pe rmeab i l i t y vs. Temperature , Mass i l lon Sandstone Core No. 4 . . . . . . . . . . . . . 4 6

1 6 Water Pe rmeab i l i t y v s . Temperature , Unconsoli- d a t e d O t t a w a S i l i c a Sand Core N o . 1 4 . . . . . 47

1 7 Water Perrr-eahil i ty vs . Temperature , O t t a w a S i l i c a Sand Core No. 1 5 . . . . . . . . . . 4 9

X

F i g u r e

1 8

19

20

2 1

2 2

23

2 4

2 5

2 6

,

J

L i s t df €Figures, c o n t i n u e d

Apparent P e r m e a b i l i t y vs . X e c i p r o c a l Mean P r e s s u r e a t S e v e r a l Tempera tures for M a s s i l l o n Core No. 5. . . . . . . . . . . Modif ied V i s c o- I n e r t i a l Graph, M a s s i l l o n C o r e N o . 5 w i t h Ni t rogen Flow . . . . . . . .

Page

51

52

Apparen t P e r m e a b i l i t y vs. I i e c i p r o c a l Mean P r e s s u r e a t S e v e r a l Temperatures for M a s s i l l o n S a n d s t o n e Core N o . 6 , Conf in ing P r e s s u r e = 1 0 0 0 p s i g . . . . . . . . . . . . . . . . . . 53

Apparen t P e r m e a b i l i t y v s . R e c i p r o c a l Mean P r e s s u r e a t S e v e r a l Temperatures for Mas- s i l l o n Sands tone Core 110. 6 , C o n f i n i n g Pres- s u r e . 3000 p s i g . . . . . . . . . . . . . . . . 5 4

Apparen t P e r m e a b i l i t y vs. R e c i p r o c a l Mean P r e s s u r e a t S e v e r a l Temperatures for M a s s i l l o n S a n d s t o n e Core No. 6 , Conf in ing P r e s s u r e = 4 0 0 0 p s i g . . . . . . . . . . . . . . . . . . 5 5

Apparen t P e r m e a b i l i t y vs . R e c i p r o c a l Mean P r e s s u r e a t S e v e r a l Temperatures for O t t a w a S i l i c a Sand Core No. 1 6 , Conf in ing P r e s s u r e = 2000 p s i g . . . . . . . . . . . . . . . . . . 57

O i l P e r a e a b i l i t y 'vs . TeEpera ture for M a s s i l l o n Core So. 10 . . . . . . . . . . . . . . . . . 59

O i l P s r n e a b i l i t y vs. Temperature for M a s s i l l o n CcFe KO. 11 . . . . . . . . . . . . . . . . . 60

2 - O c ~ z n o l P e r m e a b i l i t y vs. Tempera ture f o r Uncc r - so i ida t ed O t t a w a S i l i c a Sand Core N o . 1 7 6 1

J

J

J

J

J

1,. L?VIiODUCTION

A l o n g he ld conc lus ion t h a t t h e a b s o l u t e p e r m e a b i l i t y

of a porous medium i s a c o n s t a n t de t e rmined only by t h e

s t r u c t u r e of t h e medium i n q u e s t i o n i s t he s u b j e c t of t h i s

s t u d y .

m e a b i l i t y of porous media have shown t h a t t h e a b s o l u t e , ~

p e r m e a b i l i t y t o water f o r c e r t a i n s a n d s t o n e c o r e s v a r i e s w i t +

t h e l e v e l of con f in ing p r e s s u r e as w e l l as w i t h tempera ture .

For w a t e r f low, pe rmeab i l i t y r e d u c t i o n s of up t o 6 0 % were

observed when t empera tu re was i n c r e a s e d from 7OoF t o 30OoF.

This important d i s c o v e r y m y have s i g n i f i c a n t r a m i f i -

Experiments des igned t o measure t h e a b s o l u t e per-

I

I

I

1 c a t i o n s i n many o i l r ecove ry by thermal p roces ses .

j e c t i o n of h o t water and steam i n t o o i l r e s e r v o i r s , under-

ground combustion, i n j e c t i o n of f l u i d s i n t o w e l l s , t h e pro-

d u c t i o n of geothermal energy , and t h e d i s p o s a l o f atomic

The i n- 1

I

waste p r o d u c t s i n porous fo rma t ions a l l c a u s e changes

f o r m a t i o n t empera tu re s .

I n r e s e r v o i r e n g i n e e r i n g , a b s o l u t e pe rmeab i l i t y

b a s i c parameter which h a s o f t e n been measured a t room

i n I

i s a

con- ~1

d i t i o n s , w i t h t h e i m p l i c i t assumption t h a t on ly c o n f i n i n g

p r e s s u r e affected t h e r e s u l t .

known r e a c t i o n s between water and c l a y s . 1

f o r e , a s i n g l e va lue of a b s o l u t e p e r m e a b i l i t y throughout a

r a n g e of r e s e r v o i r t empera tu re s has been used in r e s e r v o i r

e n g i n e e r i n g c a l c u l a t i o n s .

t (Of c o u r s e w e i gno re t h e w e 1 4

H i t h e r t o , t h e r e-

Weinbrandtl ' found foir water f l o w , t h a t t h e a b s o l u t e

pe rmeab i l i t y of c o n f i n e d , f i r e d sands tone cores w a s s t r o n g l y

temperature dependent . Casse v e r i f i e d t h e Weinbrandt r e s u l t .

I n r a i s i n g t h e t empera tu re l e v e l of a f i r e d c o n s o l i d a t e d

sandstone core under a c o n f i n i n g p r e s s u r e of 2 0 0 0 p s i from

room tempera ture t o 3OO0F, h e observed a r e d u c t i o n i n a b s o l u t e

pe rmeab i l i t y of as much as 65%. With mine ra l o i l flow, or

i n e r t gas f low, he found t h a t t empera tu re had no a p p r e c i a b l e

e f f e c t on a b s o l u t e permeabilt i t y .

t h a t i n h i s w a t e r f low exper iments th rough s y n t h e t i c cement-

conso l ida t ed sand cores, he d i d n o t observe t empera tu re

effects on a b s o l u t e p e r m e a b i l i t y .

con f in ing p r e s s u r e s .

2

Recent ly , Ar iha ra2 r e p o r t e d

However, h e a p p l i e d l o w

C a s s g l sugges t ed t h a t c lay- water i n t e r a c t i o n caused t he

pe rmeab i l i t y r e d u c t i o n he observed. However, Greenberg, e t al.

i n 1 9 6 8 r e p o r t e d d a t a on t he p e r m e a b i l i t i e s t o water o f core

samples a r t i f i c i a l l y c o n s o l i d a t e d w i t h pheno l i c r e s i n o r by

s i n t e r i n g . The g e n e r a l t r e n d showed s l i g h t t o moderate de-

creases i n p e r m e a b i l i t y w i t h i n c r e a s i n g tempera ture .

f i n i n g p r e s s u r e w a s a p p l i e d to the core.

observed changes t o m i c r o - s t r u c t u r a l re- arrangements i n t h e

m a t r i x geometry of t h e sampl-es which had a rough and i r r e g u l a r

s u r f a c e . They observed no changes i n p e r m e a b i l i t i e s f o r

N o con-

They a t t r i b u t e d the

I

samples wi th r e l a t i v e l y smooth s u r f a c e s .

The purpose of t h i s work w a s (1) t o v e r i f y t h e r e s u l t s

of p rev ious s t u d i e s , ( 2 ) t o ex tend t h e p rev ious work t o o t h e r

i systems, and ( 3 ) t o i n v e s t i g a t e t h e r ea son fo r t empera tu re

- 2 -

d

J

J

J

J

4

J

J

J

J

d

- 3 -

\-

I.

c.

c

L

-- I ,

effects on a b s o l u t e pe rmeab i l i ty .

c l u e as t o what a c t u a l l y caused t h e p e r m e a b i l i t y t o d e c r e a s e

w i t h i n c r e a s i n g t e m p e r a t u r e , it was dec ided t o u s e c l a y- f r e e

rocks and o t h e r p o l a r l . iqu ids , such as 2- octanol .

I n o r d e r t o p r o v i d e a

2. FLOW IN POROUS WDIA

7 I n 1856 , as a r e s u l t of expe r imen ta l s t u d i e s on t h e f l o

of water through unconso l ida t ed sand f i l t e r beds , Henry Darcy k

formulated a flow l a w which now b e a r s h i s name. T h i s l a w hag

been extended t o d e s c r i b e ,

of o t h e r f l u i d s , i n c l u d i n g

conso l ida t ed r o c k s and 0 t h

2 . 1 Darcy's' L a w

w i th some l i m i t a t i o n s , t h e movemeqt

t w o or more immiscible f l u i d s , i n

r porous media.

Darcy's law, f o r t h e h o r i z o n t a l , v i s cous f l o w o f a f l u i d

i n a l i n e a r medium, s ta tes t h a t t h e flow v e l o c i t y of a homo-

geneous f l u i d i s p r o p o r t i o n a l t o t h e p r e s s u r e g r a d i e n t , and

i n v e r s e l y p r o p o r t i o n a l t o t h e f l u i d v i s c o s i t y , or:

(2-

where v i s t h e v e l o c i t y i : n cm/sec, a_ i s t h e f l o w ra te i n cc/

sec, p i s t h e f l u i d v i s c o s i t y i n c p , dp /ds i s t h e p r e s s u r e

g r a d i e n t i n a t m / c m , t a k e n i n t h e f i r e c t i o n of f l o w , and k

i s t h e rock p e r m e a b i l i t y , d a r c i e s .

m e a b i l i t y i s one i n which a f l u i d of one c e n t i p o i s e v i s c o s i t j

w i l l move a t a v e l o c i t y of one c e n t i m e t e r p e r second under a

p r e s s u r e g r a d i e n t of one a tmosphere p e r cen t ime te r .

A rock of one darcy pe r-

Darcy's l a w a p p l i e s o n l y i n t h e r e g i o n of l aminar f l o w .

For " t u r b u l e n t " or non-laminar f lows which occu r a t h i g h e r

v e l o c i t i e s , t h e p r e s s u r e g r a d i e n t i n c r e a s e s a t a g r e a t e r r a t e

t h a n does t h e flow r a t e .

- 4 -

A c o r r e l a t i o n produced by Fancher , L e w i s , and Barnes 5

may be used t o e s t i v a t e t h e r e p i o n of v i s c o u s flow f o r a

porous medium.

media when a modi f ied Reynolds' number

I n g e n e r a l , 9 a r c y ' s l a w is v a l i d f o r porous

vd P R e = - IJ

i s less t h a n one.

t h e f l u i d v i s c o s i t y , and d i s t h e d i ame te r of the ave rage

g r a i n s ize .

v i s v e l o c i t y , p is f l u i d d e n s i t y , 1.1 i s

A r e c e n t s t u d y by Geertsma' showed t h a t t h e ave rage

g r a i n s i z e s h o u l d be r ep l aced by a r a t i o of p e r m e a b i l i t y t o ' '

p o r o s i t y t o p r o v i d e a b e t t e r c o r r e l a t i o n .

The r equ i r emen t t h a t t h e p e r m e a b i l i t y be determined

for c o n d i t i o n s of v i scous flow i s b e s t s a t i s f i e d 7 by o b t a i n i h g

d a t a a t s e v e r a l flow rat:es and g raph ine f l o w ra te v e r s u s prtels-

s u r e drop fo r l i q u i d and f o r gas by p l o t t i n g the p roduc t of ~

* where p1 , mean f l o w r a t e and pore p r e s s u r e v e r s u s p1 2 - p2 i s t h e upst ream p r e s s u r e and p2 i s the downstream p r e s s u r e .

For c o n d i t i o n s of v i scous flow, t h e d a t a shou ld p l o t a s t r a i g h t

l i n e , p a s s i n g th rough t h e o r i g i n . Turbulence i s i n d i c a t e d I

by c u r v a t u r e of t h e p l o t t e d p o i n t s .

2 . 2 Darcy 's Law f o r G a s Flow

I n t h e case of f low of gases , t h e f l o w ra te i s no t

c o n s t a n t , b u t i n c r e a s e s w i t h t h e p r e s s u r e d rop , a c c o r d i n g t o I

Boyle 's l a w . T h e i n t e g r a t e d form of Darcy 's l a w which des- I

c r i b e s h o r i z o n t a l l inear . pas flow, under s t e a d y s t a t e , i s o -

thermal c o n d i t i o n s i s : i

- 5-

( 2 - 3 )

2 . 3 S l i p Phenomena i n G a s Flow

Kundt and Warburg* f i i ~ s t showed t h a t a l a y e r of gas

nex t t o a s o l i d s u r f a c e m!y have a f i n i t e v e l o c i t y w i t h

r e s p e c t t o t h e w a l l .

g ive a g r e a t e r ra te t h a n would be computed from P o i s e u i l l e ' s

l a w .

w a l l . T h i s " s l i p "

phenomena i s dependent upon t h e mean free p a t h of t h e gas

molecules ; t h e r e f o r e , gas pe rmeab i l i t y should b e a f u n c t i o n

of t h e factors c o n t r o l l i n g t h e mean f ree p a t h . These factors

are p r e s s u r e , t e m p e r a t u r e , and t h e n a t u r e of t h e gas i t s e l f .

When t h e mean f r e e p a t h s ar*e small, e.g. , a t h i g h p r e s s u r e s ,

t h e p e r m e a b i l i t y t o gas should be expected t o approach t ha t

I n case o f c a p i l l a r y f l o w , t h i s would

I n ef fec t , t h e g a s stream l l s l i p s " w i t h r e s p e c t t o t h e

Th i s i s n o t t h e case fo r l i q u i d f l o w .

t o l i q u i d s . T h i s i d e a w a s supported by t h e Klinkenburg 9

s tudy .

Based upon a t h e o r e t i c a l a n a l y s i s of t h e s l i p phenomenon

i n c i r c u l a r c a p i l l a r i e s , arid assuming a n ana logous b e h a v i o r

i n a porous medium, Klinkenberg' developed the r e l a t i o n s h i p

between t h e p e r m e a b i l i t y of a porous medium t o g a s and t o a

n o n r e a c t i v e l i q u i d as follows:

(2-411

I I.

2

J

J

J

J

4

J

J

J

..J

.J

where ka i s t h e apparent: o r observed p e r m e a b i l i t y t o g a s , k

i s t h e a b s o l u t e pe rmeab i l i t y t o gas a t h i g h p r e s s u r e s (p re-

sumed e q u a l t o t h e a b s o l u t e p e r m e a b i l i t y t o a s i n g l e l i q u i d

phase) , X i s the mean fr5ee p a t h of t h e g a s m l e c u l e s , r is -

t h e r a d i u s of a capi l lax-y (assumed t o be c o n s t a n t ) , and c

i s a p r o p o r t i o n a l i t y f a c t o r . The mean f r ee p a t h can be

expressed as:

where d i s a c o l l i s i o n diameter, n i s t h e c o n c e n t r a t i o n of

molecu les p e r u n i t volume, N i s Avogadro's N u m b e r , p, i s the

mean p r e s s u r e , T i s a b s o l u t e t empera tu re , and R i s t h e un i - ~

v e r s a 1 gas c o n s t a n t . F ~ o m Ea_. 2- 5, Eq. 2-4 becomes:

b k = k(l+- 4c'T ) = k(l+-) 2 m r N d pm

a Pm

c.

where b is r e f e r r e d t o as t h e Kl inkenberg factor , and is

u s u a l l y t a k e n as a c o n s t a n t f o r a g iven gas and a given

c porous medium. From Eq . 2- 6, t h e Kl inkenberg factor appear\$

d i r e c t l y p r o p o r t i o n a l t o t empera ture .

2 . 4 V i s c o- I n e r t i a l Flow - Darcy ' s l a w and t h e preced inp e q u a t i o n s are v a l i d when

c o n d i t i o n s of v i scous f low p r e v a i l . A t h i g h flow rates, f L b w

has been shown t o be desc r ibed by a q u a d r a t i c equa t ion . Su

an e q u a t i o n was proposed by Forchheimer'' and modif ied by

C o r n e l l and K a t z l l as fo l lows :

-7-

Dranchuk and Kolada12 have shown how t o d e l i n e a t e t h e visco-

i n e r t i a l f low r e g i o n from flow measurements, and how t o

determine t h e a b s o l u t e p e r m e a b i l i t y of a r o c k c o r e (see

Appendix 9 . 3 ) .

The v i s c o - i n e r t i a l f l o w d e s c r i b e d by Eq . 2-7 is some-

The d e p a r t u r e from t imes r e f e r r e d t o as "tur lbulent" flow.

laminar f low d e s c r i b e d by Eq. 2-7 i s n o t caused by physical

f a c t o r s which (cause t u r b u l e n t flow i n p i p e . T h i s phenomenon I

i s u s u a l l y r e f e r r e d t o as "non-Darcy," " i n e r t i a l , " or "visco-r

i n e r t i a l " f low. We s h a l l use t h e l a t t e r term. I

I

J

J

J

J

J

J

Y

-8-

I

.J

..

3. LITE-WTURE

I n 1 9 3 7 , Muskat4 s t a t e d t h a t t h e a b s o l u t e p e r m e a b i l i t y

of a porous medium "is t h u s a c o n s t a n t determined o n l y by

t h e s t r u c t u r e of t h e medium i n q u e s t i o n and i s e n t i r e l y i n d e t

pendent of t h e n a t u r e o f t h e f l u i d . " Also , t h e s t a n d a r d p r a t

cedure f o r de te rmin ing t h e p e r m e a b i l i t y of porous medium

according t o API Code No. 2 7 1 5 ( f i r s t e d i t i o n , October 1935)

w a s based on t h e fundamental assumption t h a t as long as t h e

ra te of flow w a s p r o p o r t i o n a l t o t h e p r e s s u r e g r a d i e n t , t h e I

p e r m e a b i l i t y c o n s t a n t of a porous medium w a s a p r o p e r t y of

t h e medium and w a s independent o f t h e f l u i d used i n i t s detdLmi-

n a t i o n .

~

... A major s t u d y by Klinkenberg appeared i n 1 9 4 1 . Klinkefi-

9 b e r g performed l i q u i d p e r m e a b i l i t y measurements on J e n a

f i l t e r s i n o r d e r t o a v o i d c l a y s w e l l i n g o r e r o s i o n . The

were not conf ined n o r w e r e v a r i a t i o n s i n t e m p e r a t u r e studiedl.

From h i s a n a l y s i s o f gas, f low through t h e same c o r e s , he t h e p e -

f o r e suppor ted t h e n o t i o n t h a t t h e p e r m e a b i l i t y c o n s t a n t of

a porous medium w a s a p r o p e r t y of t h e medium, and w a s indepen-

d e n t of t h e f l u i d used. I n a d d i t i o n , he w a s t h e f i r s t pe r sdn ~

t o e x p l a i n s l i p phenomena.

I n 1 9 4 3 , Grunberg and Nissan13 s t u d i e d t h e e f f e c t of

f low of aqueous s o l u t i o n s th rough l imes tone and sands tone

c o r e s ove r a r a n g e of t e m p e r a t u r e s . Four s o l u t i o n s w e r e t e s t e d :

-9-

(1) d i s t i l l e d water, (2) a 2 % n-amyl a l c o h o l s o l u t i o n , and

( 3 ) two sodium c h l o r i d e s o l u t i o n s ( 0 . 9 6 0 and 0.614N).

cho ice of t h e s e s o l u t i o n s was made so as t o obse rve t h e i n-

f l u e n c e o f s u r f a c e forces on t h e f l o w of l i q u i d s t h rough

porous media.

The

The core tempera tures were v a r i e d from 6OC I

t o 3OoC. 1

Pe rmeab i 1. it y d ecr e a:; e d w i t h i n c re as i ng temper a t u r e f o r 1 1

a l l of t h e four, aqueous s o l u t i o n s .

p e r m e a b i l i t y t e m p e r a t u r e curves wi th approximate ly t h e same

s l o p e . The slope of t h e graphs o f p e r m e a b i l i t y v s . tempera-

t u r e w a s 0 .8 mcV0C. Thus:

A l l f o u r gave l i n e a r

1

k = if - 0.8(t,OC) ( 3-1

I

As ~

where a i s a characteristic c o n s t a n t o f t he l i q u i d used.

t h e f l u i d v i s c o s i t y w a s cons idered p r o p e r l y i n c a l c u l a t i n g I

p e r m e a b i l i t y , t:hey concluded t h a t v i s c o s i t y w a s n o t t h e on ly 1 p r o p e r t y i n f l u e n c i n g f low.

2 I

I n a d d i t i o n , a log- log graph of

k/k , vs.(-) lJ gave a s t r a i g h t l i n e . kl i s a base p e r m e a b i l i t y ; 1 Pa

1.1 i s f l u i d v i s c o s i t y , p i s f l u i d d e n s i t y , and Q i s s u r f a c e

t e n s i o n .

1

A c o n c l u s i o n reached w a s t h a t the e f f e c t i v e c r o s s- s e c t i t

under v i s c o u s flow was d i f f e r e n t f o r d i f f e r e n t l i q u i d s due t i d i f f e r e n c e s i n s u r f a c e energy, and t h u s d i f f e r e n c e s i n t h e

t h i c k n e s s of adsorbed l a y e r s .

I n 1 9 4 6 , Calhoun and Yus te r " p r e s e n t e d r e s u l t s of flow

through a r t i f i c i a l porous bod ie s , some made of p y r e x g l a s s

and o t h e r s of f'used q u a r t z . I No c o n f i n i n g p r e s s u r e was a p p l i e

-10-

c.

L

They d i s a g r e e d w i t h Grunberg's and Ni s san ' s r e s u l t s and con-

firmed Kl inkenbe rg ' s r e s u l t s .

Calhoun and Yuster found that water gave a s l i g h t l y

lower va lue for pe rmeab i l i t y t h a n benzene. A s u s p i c i o n of

a n e l e c t r o - k i n e t i c e f fec t w a s r u l e d o u t because a d d i t i o n of

a t race of H C 1 or C a C 1 2 t o t h e water caused no appa ren t i n-

crease i n p e r m e a b i l i t y .

b i l i t y as low as t h a t of water, and, w i t h this hydrocarbon

l i q u i d a n e l e c t r o - k i n e t i c e f fec t should have been a b s e n t .

Calhoun and Y u s t e r were unable t o g i v e an e x p l a n a t i o n of

t h i s anomaly. They were t h e f i r s t t o obse rve a dependency od Kl inkenberg ' s fac tor , b on t empera tu re . 1

Furthermore, naphtha gave a permea-

I n t r y i n g t o correlate Klinkenberg b v a l u e s a t d i f f e r e n t

t e m p e r a t u r e s , Calhoun and Yus te r assumed t h a t e q u a l mean fred

c.

c. I

p a t h s would o c c u r a t equ iva l en t molecu la r c o n c e n t r a t i o n s a t ,

d i f f e r e n t t e m p e r a t u r e s , A gas a t p r e s s u r e p1 and T1 should ~

.

.

c

have t h e same molecu la r c o n c e n t r a t i o n as t h e same g a s a t a P:1 p2 I p r e s s u r e pz and T2, i f ~1 were e q u a l t o T. Therefore, e q u i j

2 v a l e n t p e r m e a b i l i t y v a l u e s should e x i s t f o r the same g a s , whcin

two d i f f e r e n t t empera tu re s were used, a t a mean p r e s s u r e p,

and t e m p e r a t u r e T1, and a t mean p r e s s u r e p (-1 a t a temperad

t u r e of Tz. T h i s c o r r e L a t i o n a t d i f f e r e n t t e m p e r a t u r e s w a s ' n o t i nc luded i n e i t h e r Kl inkenberg ' s or Grunberg and Nissan'B

T1 T2

p r e s e n t a t i o n . 18

I n t h e l a t e 1960s, Greenberg, e t a1.,3 and Weinbrandt

a g a i n observed a p e r m e a b i l i t y dependency on t empera tu re levea .

Greenberg, e t a l . , d i d not r e s t r a i n t h e i r cores and observed i I

-11-

16 o n l y small t empera ture e f fec ts . Weinbrandt , i n h i s e x p e r i-

ments on t h e e f fec t of t empera tu re on r e l a t i v e and a b s o l u t e

p e r m e a b i l i t y of sands tones , s u b j e c t e d h i s cores t o 2000 p s i

c o n f i n i n g p r e s s u r e . H e a l so f i r ed t h e cores a t 94OoF p r i o r

t o use t o o x i d i z e o r g a n i c matter i n t h e cores and t o d e a c t i - ' I

v a t e t h e c l a y s . H i s exper iments f o r Boise sands tone core

samples a t room t empera tu re and a t 175OF showed t h a t w i t h a n

i n c r e a s e i n t empera ture : (1) t h e i r r e d u c i b l e water saturatiofa

i n c r e a s e d ; ( 2 ) t h e r e s i d u a l o i l s a t u r a t i o n decreased ; ( 3 )

t h e r e l a t i v e p e r m e a b i l i t y t o w a t e r a t f lood- out i n c r e a s e d ; (

t h e r e l a t i v e perrneabi1it:y t o o i l i n c r e a s e d ; ( 5 ) t h e r e l a t i v e

p e r m e a b i l i t y iqatio, kw/ko, d e c r e a s e d ; and ( 6 ) t h e a b s o l u t e

p e r m e a b i l i t y to water dec reased . Casse v e r i f i e d the Wein-

b r a n d t f i n d i n g s and s u b s t a n t i a l l y extended t h e work.

.1 I

The ef fec t of mechanical stresses on t h e p e r m e a b i l i t y

of r o c k s has heen s t u d i e d by s e v e r a l i n v e s t i g a t o r s . I n 196711 I I

Wilhelmi and Somerton17 i n v e s t i g a t e d t h e e f f e c t of overburdefl 1

p r e s s u r e on rock p e r m e a b i l i t y .

by F a t t and Davis 1 8 i n 1!352, and r e p o r t e d t h a t p e r m e a b i l i t y ' They confirmed ea r l i e r work

a t 1 5 , 0 0 0 p s i c o n f i n i n g p r e s s u r e cou ld be 2 5 t o 6 0 % smaller

t h a n t h e p e r m e a b i l i t y a t a z e r o c o n f i n i n g p r e s s u r e , dependin PP I t

on t h e t y p e of rocks s t u d i e d . G e n e r a l l y speaking, t h e h i g h e l

t h e p e r m e a b i l i t y , t h e h i g h e r t h e pe rcen tage o f r e d u c t i o n .

About 6 0 % of t h e t o t a l r e d u c t i o n occu r red du r ing t h e f i r s t ~

3000 p s i c o n f i n i n g p r e s s u r e .

In 1963!, G r a y , --- e t a1 ,

I

I a l so measured t h e e f f e c t of 19

overburden p r e s s u r e on p e r m e a b i l i t y . P e r n e a b i l i t y r e d u c t i o n

was shown t o be a f u n c t i o n of t h e r a t i o of r a d i a l t o a x i a l

- 12-

J

J

J

J

J

J

J

J

J

J

J

stress, wi th t h e maximum r e d u c t i o n o c c u r r i n g under a uniform

stress, i . e . , when t h e a x i a l stress is e q u a l t o t he r a d i a l

stress.

c o n d i t i o n s of uniform s t ~ e s s .

The p r e s e n t work r e p o r t e d h e r e w a s accompanied undeth

Zoback and Bye r l ee (1975aI2' showed t h a t due t o t h e

presence of compres s ib l e matr ix material i n a r e l a t i v e l y

incompress ib le g r a n u l a r framework, t h e p e r m e a b i l i t y o f sand-

s t o n e i s n o t s imply a f u n c t i o n of e f f e c t i v e stress, b u t i s

h i g h l y s e n s i t i v e t o changes i n po re p r e s s u r e .

~

The ef fec t of t h e r m a l stresses on r o c k p r o p e r t i e s and

t h e effect of overburden p r e s s u r e on p o r o s i t y have a l s o been

of p a r t i c u l a r i n t e r e s t .

Somerton, e t al . ,* 'L h e a t e d a number of s a n d s t o n e s t o

about 150OOF under b o t h a tmospher ic and s i m u l a t e d r e s e r v o i r

p r e s s u r e s . P e r m e a b i l i t y w a s measured a t room tempera tu re

u s ing a s t a n d a r d a i r perimeameter, b e f o r e and a f te r h e a t i n g I

t h e samples. No p e r m e a b i l i t y changes were r e p o r t e d i n t he

range of 75-350°F. A t t empera tu re s w e l l above 5OO0F, t h e y

showed that permanent s t i r u c t u r a l damage and decompos i t ion ofi 1

rock minerals o c c u r r e d due t o thermal stresses.

Wyble22, working om t h e effect o f a p p l i e d p r e s s u r e on

sands tone ' s p r o p e r t i e s , observed asympto t ic d e c r e a s e s i n con-

d u c t i v i t y , p o r o s i t y , and p e r m e a b i l i t y o v e r a 0 p s i t o 3,500

p s i range . A t 2 0 0 0 p s i c o n f i n i n g p r e s s u r e , about a 1 0 % reduc-

t i o n i n p o r o s i t y w a s obslerved, and beyond t h i s , t he p o r o s i t y

va lue remained e s s e n t i a l l y c o n s t a n t .

-13-

I n summry , many s t u d i e s i n d i c a t e t h e l i k e l i h o o d of

a b s o l u t e p e r m e a b i l i t y of r o c k s be ing a f u n c t i o n of tenpera-

t u r e beyond the ef fec ts caused by a change i n the v i s c o s i t y

of the f l u i d .

i n d i c a t e d an effect o f major importance for r e s t r a i n e d c o r e s 4

I

The r e c e n t works of Weinbrandt’‘ and Cass6 1 1

N o obvious e x p l a n a t i o n e x i s t e d .

s t u d y w a s a v e r i f i c a t i o n of and s t u d y of poss ib le e x p l a n a t i o o s

of these effects . 1

The main purpose of t h i s I

I

-14-

i

I.

c.

4. EXPERIMENTAL EQUIPMENT

The e x p e r i m e n t a l eciuipment used w a s s i m i l a r t o t h a t .. -

desc r ibed p r e v i o u s l y by Cass6’ and Weinbrandt 1 6 . f i c a t i o n s were made i n t h e appara tus , however.

of t h e a p p a r a t u s f o l l o w s .

4 . 1 Genera l D e s c r i p t i o h

Some modi- 1

A d e s c r i p t i o n

I

Fig . 1 i s a photogriaph of t h e l a b o r a t o r y , and F i g .

2 p r e s e n t s t h e d e t a i l s of t h e a i r ba th assembly. A schemat ic

diagram o f t h e a p p a r a t u s i s shown i n Fig. 3 . T h e flow l i n e s ,

h e a t exchangers , and f i t t i n g s were c o n s t r u c t e d o f 316 s t a i n l e

s tee l material.

Three p a r a l l e l flow l i n e s were c o n s t r u c t e d for- gas ,

6S

o i l , water, or 2- oc tano l i n j e c t i o n . For each exper iment , the1

d e s i r e d f l o w l i n e w a s connected t o t h e c o r e ; a new c o r e be ing

used f o r each exper iment . Liquid flow w a s s u p p l i e d by a pul-

s a t i n g pump, and gas f l o w w a s supp l i ed from h i g h p r e s s u r e

c y l i n d e r s and d e l i v e r e d t‘hrough p r e c i s i o n p r e s s u r e r e g u l a t o r s .

Liquid f low r a t e th:rough t h e c o r e w a s measured by

a weighing b a l a n c e .

gas flow r a t e w a s measured e i t h e r by t h e use of a bubble f i l m 1

flowmeter o r by a Wet T e s t Meter.

a t t h e i n l e t f a c e of t h e c o r e , and t empera tu re w a s r eco rded

con t inuous ly d u r i n g t h e r u n s . The core h o l d e r assembly was

p laced i n an air b a t h t h a t maintained r u n t e m p e r a t u r e . Mea-

surements of u p s t r e a m p r e s s u r e and p r e s s u r e drop across t h e

Depe:nding upon t h e l e v e l of f low ra te ,

A thermocouple w a s p laced

-15-

f i !I. 1. PhotoF,raph o f A p p a r a t u s I

c.

W I

- 17-

I I I

c o r e were accomplished by the use of p r e s s u r e t r a n s d u c e r s

connected t o p r e s s u r e i n d i c a t o r s and r e c o r d e r s .

The d e s i r e d c o n f i n i n g p r e s s u r e l e v e l was a t t a i n e d

by us ing a h y d r a u l i c hand pump. Chevron No. 15 heavy O i l

was used as a c o n f i n i n g f l u i d and a "Viton A" r u b b e r s l e e v e ,

l / S " t h i c k , s e p a r a t e d t h i s f l u i d from t h e core. I

A d e s c r i p t i o n of the major components of t h e a p p a r a t u s

follows.

4 .2 Core H o l d e r - - -_

Fig . 4 s 'hows the d e t a i l s of t h e c o r e h o l d e r which

i s a Hassler r u b b e r sleeve t y p e . The r o c k specimen t o be

s t u d i e d i s h e l d i n a "Viton A" rubbe r s l e e v e , 1/8" t h i ck ,

between an ups t ream p l u g , which i s immobile, and a downstrean

p lug which moves h o r i z o n t a l l y and a d j u s t s t o t h e core l e n g t h .

The upstream p2ug h a s two p r e s s u r e t a p s A and D , one tap B

f o r i n l e t f l o w : , and a thermocouple w e l l C .

E i t h e r l i q u i d o r gas p r e s s u r e can be a p p l i e d t o

t h e Viton s l e e v e , b u t thrloughout t h e s e exper iments , Chevron

White O i l No. 1.5 w a s used . Both a x i a l and r a d i a l c o n f i n i n g

p r e s s u r e s are a.pplied s imu l t aneous ly t o the core because the

downstream plug, i s mobile and s u b j e c t t o t h e c o n f i n i n g p r e s-

s u r e . A p e r f o r a t e d aluminum t u b e i s f i t t e d around t h e core

r u b b e r s l e e v e t o p r e v e n t l a t e ra l deformat ion of c o r e d u r i n g

load ing .

1 . 9 4 6 i n . OD by 3 .0 i n . long.

4 . 3 A i r Bath and Temperature Con t ro l

The s l e e v e w a s i~ t h i c k c y l i n d e r 1 . 2 2 5 i n . I D by

I

An a i r b a t h w i t h a working space of 2 4 c u b i c f ee t

housed t h e core h o l d e r assemblv. Four kilowatts of power may -18-

J

J

.d

N O- w o o .- 0 0. a + I

Ex

, - a .

1 I

Icy 9 00 00 + 1 * a

be a p p l i e d t o t h e h e a t i n g elements by an A P I model 4010

power pack and a n API model. 2 2 8 t empera ture c o n t r o l l e r . The

thermocouple t h a t c o n t r o l s t h e ou tpu t of the heater c o n t r o l l e r

can be p laced a t any l o c a t i o n i n s i d e t h e oven, bu t t h e most

e f f i c i e n t h e a t i n g c y c l e w a s found t o r e s u l t when t h e thermo-

couple s e n s o r w a s f a s t ened t i g h t l y t o t h e core h o l d e r . A f a n

provided adequa te a i r c i r c u l a t i o n and the a i r b a t h w a s equipped'

~

I I

with a l i g h t and a window. ~

Temperature was monitored by a 24- point thermocouple I

r e c o r d e r wi th i ron- cons tan t an thermocouples . Two thermocouple$'

were a c t u a l l y used. One thermocouple measured t h e t empera ture I

of t h e c o r e , and t h e o t h e r measured t h e a i r b a t h temperature .

The a i r b a t h t empera ture reached t h e d e s i r e d t e s t tempera ture

i n about 1- 1 / 2 hou r s , but a.t least a n o t h e r four hour s were

r e q u i r e d f o r t h e rock specimen t o r each t h e t e s t temperature .

4 .4 L iqu id and Gas' Sources - A d e a e r a t e d l i q u i d was s t o r e d i n a 4000 cc c a p a c i t y

vacuum f l a s k . A Lapp "Microflo" P u l s a f e e d e r pump w a s used

t o pump t h e l i q u i d through t h e core . The pump h a s a d i a l

i n d i c a t o r , c a l i b r a t e d i n 1 0 0 0 increments which enab led f i n e

ad jus tmen t s i n f low r a t e t o be made.

w e r e used: w a t e r , o i l , and 2- octanol .

Three t y p e s o f l i q u i d s

Gas flow through t h e samples w a s s u p p l i e d from high

p r e s s u r e c y l i n d e r s , and reg,ula ted by a two- s tage a d j u s t-

able r e g u l a t o r equipped wi th a r e l i e f va lve . The i n i t i a l

c y l i n d e r p r e s s u r e w a s 2 5 0 0 p s i , and d e l i v e r y p r e s s u r e s ranged

from 5 p s i t o a maximum o f 4 0 0 p s i .

- 20 -

c

4 . 5 Liquid Pumps

The pumps w e r e a Lapp "Microflo" P u l s a f e e d e r t ype w i t h

a d i a l i n d i c a t o r c a l i b r a t e d i n 1 0 0 0 increments .

created l a r g e p r e s s u r e p u l s a t i o n s when d e l i v e r i n g a t h igh

Both pumps I

I

pres su re s . Accumulators, i n s e r t e d a long t h e f low l i n e s , I

e l im ina t ed t h e s e p u l s a t i o n s . A s t eady f low cou ld b e e s t ab-

l i s h e d and c o n s t a n t p r e s s u r e s maintained a t bo th ends of the

co re .

4 .6 Hydrau l ic Yand P u r 2

An Enerpac hvdraul-ic hand pump wi th a r ange of 0 t o

1 0 , 0 0 0 p s i provided adjustment o f t h e c o n f i n i n g p r e s s u r e .

The pump w a s connected to t h e c o r e h o l d e r and Chevron White

Mineral O i l No, 1 5 was used t o o b t a i n t h e d e s i r e d c o n f i n i n g

p re s su re .

4.7 Heat Exchangers

To ma in t a in t h e t empera ture of t h e f lowing f l u i d con-

s t a n t d u r i n g a r u n , large r e s e r v o i r s were i n s t a l l e d on each1

f low l i n e i n s i d e t h e oven be fo re t h e c o r e h o l d e r . Cold flu1

e n t e r e d a t t h e bottom of t h e r e s e r v o i r . Because of t h e larj

I

s i z e of t h e r e s e r v o i r arid t h e s m a l l f low rates that were

used w i th l i q u i d f low, ho t f l u i d l e f t from the t o p . The

tempera ture o f t h e l i q u i d s l e a v i n g t h e r e s e r v o i r and e n t e r i d g

I

I t h e c o r e w a s measured and found t o remain c o n s t a n t a t t h e 1

d e s i r e d test t empera ture du r ing t h e e n t i r e r u n . I I

The gas r e s e r v o i r i n s i d e t h e oven w a s f i l l e d wi th s t a i n - I

less s t e e l wool. Th i s improved h e a t exchange and enabled

t h e gas t o r e a c h t h e t e s t t empera ture b e f o r e e n t e r i n g t h e cbre .

-21-

The e f f l u e n t f r o m the c o r e and t h e a i r b a t h was cooled

p r i o r t o f l o w r a te de te rmina t ion . This w a s accomplished by

l e t t i n g t h e e f f l u e n t flow through a c o i l immersed i n a con-

s t a n t room tempera ture water b a t h .

4 .8 Backpressure Valve

~

Backpressure was r e g u l a t e d by means of a f i n e meter ing

n e e d l e va lve . This v a l v e , i n a d d i t i o n t o be ing used t o c h a n g e / I I

t h e f l o w r a t e , se rved t o main ta in a s u f f i c i e n t l y h igh p r e s s u r e I

i n t h e system t o keep t h e o p e r a t i n g l i q u i d i n the l i q u i d I

s ta te .

mean pore p r e s s u r e could be mainta ined. Mainta in ing a con-

s t a n t mean pore p r e s s u r e l e v e l w a s impor tan t because exper i-

By a d j u s t i n g t h e va lve and t h e pump r a t e , a c o n s t a n t i

, mental work by Zoback*' showed t h a t a change i n mean pore

p r e s s u r e could affect a b s o l u t e p e r m e a b i l i t y t o an even g r e a t e r

e x t e n t t h a n d i d a change i n c o n f i n i n g p r e s s u r e . A c o n s t a n t

mean pore p r e s s u r e of 2 0 0 ps i w a s ma in ta ined fo r a l l l i q u i d

r u n s .

,

l i

4.9 Flow Rate Measuring Devices

4 .9- 1 Liquid F low: A t s t e a d y s t a t e , t h e mass f l o w ra te

w a s c o n s t a n t throughout t h e system. The mass f l o w ra te w a s

measured by weighing small volumes of t h e e f f l u e n t l i q u i d by

a n a n a l y t i c a l ba lance over a known p e r i o d of trime. Repeated

measurement pe rmi t t ed checking d e t e r m i n a t i o n of s t e a d y s t a t e .

T h e vo lumet r i c flow ra te w i t h i n t h e core was determined as

t h e ratio of t h e mass flow ]?ate t o t h e d e n s i t y of t h e l i q u i d

a t r u n t empera tu re and mean p r e s s u r e . The flow ra te could

- 2 2-

L

be changed by a d j u s t i n g t h e pump ra te and /o r by a d j u s t i n g

t h e need le valve.

4.9-2 Gas Flow: Gas p r e s s u r e w a s r e g u l a t e d upstream

by a two-stage p r e s s u r e r e g u l a t o r and g a s flow w a s r e g u l a t e

downstream by means of a need le v a l v e . T h e cho ice of t he

a p p r o p r i a t e flowmeter w a s made a c c o r d i n g t o t h e l e v e l of f l

i . e . , laminar flow (low r a t e ) , or v i s c o - i n e r t i a l f low ( h i g h

r a t e ) . A bubble f i l m f lowmeter w a s used for a l l l aminar f l

measurements and a Wet T e s t Meter was used f o r v isco- iner t :

f l o w measurements. A s t o p watch, g radua ted i n d i v i s i o n s of

0 .2 second, w a s used as a t i m i n g d e v i c e .

I

I The bubble f i l m f lowmeter w a s t h e m o s t a c c u r a t e of th$/

two d e v i c e s , and was used t o check t he Wet T e s t Meter a t lod

flow rates . The bubble f i l m flowmeter w a s made as follows. I

The v e r t i c a l p a r t of a Y-shaped t u b e i s plunged j u s t up t o

t h e t h r o a t i n a soap so l .u t ion (see F ig . 1). The g a s f l o w

t o be metered e n t e r s through one of t h e branches and bubble$'

i n t o t h e other branch arid t h e n up i n t o t h e v e r t i c a l b u r e t t e

The flow ra te i s ob ta ined by measuring the t i m e it t a k e s f o

a bubble t o t r a v e r s e a known volume i n t he b u r e t t e . T h i s

method e s s e n t i a l l y provj-ded f l o w r a t e measurements a t room

c o n d i t i o n s because t h e p r e s s u r e r e q u i r e d t o d i s p l a c e t h e

bubbles was always less t h a n 0 . 0 1 p s i .

4 . 1 0 P r e s s u r e Recording Devices

1

I

Upstream p r e s s u r e and p r e s s u r e d r o p a c r o s s t h e c o r e I

w e r e measured w i t h a Pace Model KP15 d i f f e r e n t i a l p r e s s u r e

- 2 3 -

t r a n s d u c e r and a Pace '-%del C D 2 5 t r a n s d u c e r i n d i c a t o r . The

a p p r o p r i a t e p l a t e (1 p s i , 5 p s i , 25 p s i , 1 0 0 p s i , o r 500 p s i )

w a s used f o r t h e working range of pressure o r p r e s s u r e d rop .

An e l e c t r o n i c two-pen mult i- range r e c o r d e r , "Chesse l l ,I' was

connec ted t o t h e i n d i c a t o r and p rov ided a permanent r ecord

of t h e p r e s s u r e .

c a l i b r a t e t h e p r e s s u r e r e c o r d i n g d e v i c e s .

A Barne t t Dead Weight Tes te r w a s used t o I l i

J

J

J

J

.J

'i

-

- 2 4 -

L

5. PROCEDURE

Core samples t o be s tud i ed were h e l d , complete ly sat&-

a t e d wi th t h e d e s i r e d l i q u i d , i n a Hassler, r u b b e r s l e e v e

type c o r e h o l d e r , i n an a i r ba th .

ba th and t h a t o f t h e core were t hen brought t o t h e d e s i r e d

l e v e l . Flow w a s e s t a b l i s h e d , and a f te r r e a c h i n g a s teady-

s t a t e c o n d i t i o n , t h e necessa ry p r e s s u r e measurements and f l b w

rates were measured.

e f f e c t o f t empera tu re on f l u i d v i s c o s i t y and d e n s i t y , abso-

l u t e p e r m e a b i l i t y o f t h e core w a s computed a t run t empera t

The l i q u i d s used were water, Chevron White O i l N o . 3 , and 2

The t empera tu re of t h e a i r

I

With t h e necessa ry c o r r e c t i o n s f o r thlh I

o c t ano l . Ni t rogen was t h e gas used. I

5 . 1 Core P r e p a r a t i o n

I

The t w o t y p e s of ‘rocks used i n t h i s s t u d y were conso l I

da ted Mass i l l on Sandstones , and unconso l ida ted O t t a w a S i l i c h

Sand.

i .e . , exper iments N o . 2 and No. 1 4 , t h e cores were f i r e d f o r

a t least 1 2 hou r s i n a 5OO0C fu rnace so as t o o x i d i z e any

o rgan ic matter presef i t .

With t h e excep t ion of t h e f i r s t run w i t h each t y p e ,

5 .1- 1 Consc l ida t ed Massi l lon Sandstones : The c o r e s

were c u t wi th a one-inch diameter diamond d r i l l , trimmed on

a l a t h e , e x t r a c t e d i n a Dean S t a r k t ype a p p a r a t u s , and ign iyed

i n a 5OO0C fu rnace .

i n Ref. 1.

F u l l d e t a i l s o f t h e p r o c e s s are given

- 2 5-

2

A f t e r i g n i t i o n , thle c o r e s were s a t u r a t e d under vacuum

w i t h t h e d e s i r e d f lowing l i q u i d . The d i f f e r e n c e i n weight

a t 1 0 0 % l i q u i d s a t u r a t i o n and a t o t a l l y d r y c o n d i t i o n w a s J

used t o c a l c u l a t e t h e c o r e p o r o s i t y .

For n i t r o g e n f low, t h e a i r - s a t u r a t e d c o r e w a s mounted I I

i n t h e core h o l d e r and f l u s h e d w i t h s e v e r a l pore volumes of

n i t r o g e n . I

i

I

Details of t h e p h y s i c a l and m i n e r a l o g i c a l p r o p e r t i e s 1 ~

i I

Unconsolidated O t t a w a Sand: The sand was s i e v e d 1 j

periment No. 1 4 , t h e sand t h a t passed through s i e v e N o .

of t h e s e c o r e s appear i n Appendix 9 . 2 .

5.1-2

and s e p a r a t e d i n t o uniforbmly graded s i z e s . For one run, ex-

35 J

b u t r e t a i n e d i n s i e v e No. 4 5 w a s used. Fo r all t h e o t h e r

r u n s , the sand s i z e t h a t passed through s i e v e No. 8 0 b u t was ,

d

r e t a i n e d i n s i e v e No. 1 0 0 w a s used. This g i v e s average sand I

I !

' 1

g r a i n s i z e s of 0.385 mm and 0.156 rrun r e s p e c t i v e l y .

sand w a s i g n i t e d a t 50OoC.

The

4

The c o r e load ing and assembly programwas as follows:

T h e upstream p lug was i n s e r t e d i n t o t h e rubber s l e e v e and

t i g h t l y t i e d t o it w i t h 20 gage s t a i n l e s s s tee l w i r e . With 1 1 the assembly h e l d v e r t i c a l l y u p r i g h t , a No. 400 s t a i n l e s s I

4

s t e e l mesh, one inch i n d i a m e t e r , w a s s e a t e d on t h e upstream

I I

Y p l u g , and t h e sand w a s poured i n t o t h e s l e e v e slowly. When

t h e sand was about 2 inches h i g h i n t h e s l e e v e , ano the r

s imilar mesh w a s p laced on t o p of sand pack and t h e down-

stream p lug was in t roduced and a l s o f a s t e n e d i n t h e same

manner. The assembly w a s t h e n p laced i n the core h o l d e r and

I ~

3

-26-

u

aga in we l l compacted and t i g5 t ened wi th t h e h e l p of a v i ce .

The f i n a l o v e r a l l measurtenent o f t h e co re h o l d e r w a s used t o

calculate t h e a c t u a l l e n g t h of t h e co re i n p l a c e .

of t h e c o r e w a s t h e same as t h a t of t h e end p l u g s (1- inch

d i ame te r ) .

The diameter

With t h e core h o l d e r placed i n t h e a i r b a t h and con-

nected t o t h e flow l i n e , t h e c o r e and t h e f low l i n e were

complete ly evacuated and, while under vacuun?, f low w a s s t a rdkd

so as t o comple te ly s a t u r a t e t h e core and f i l l the l i n e s w i t l

t h e l i q u i d . 1 I

The d e t a i l s of t h e p h y s i c a l and m i n e r a l o g i c a l p r o p e r t i e s

of t h e s e c o r e s a l s o appear i n Appendix 9 .2 .

5 .2 E s t a b l i s h i n g Run Condi t ions

With t h e core i n t h e c o r e h o l d e r , the e n t i r e asserrbly

w a s p laced i n a c r a d l e i n t h e a i r ba th and a l l necessa ry

connec t ions w e r e made t o t h e flow l i n e s , p r e s s u r e t a p s , and

c o r e t empera tu re probe.

Conf in ing p r e s s u r e w a s t hen a p p l i e d s lowly by pumping

Chevron White O i l N o . 1 5 around t h e c o r e s l e e v e , care being

t a k e n t o d i s p l a c e a l l t h e a i r i n t h i s l i n e by opening t h e

c o r e end of t h e l i n e dur ing t h e i n i t i a l s t a g e of pumping.

A d e c r e a s i n g c o n f i n i n g p r e s s u r e wi th t i m e u s u a l l y i n d i c a t e d

leakage o f t h e c o n f i n i n g f l u i d i n t o t h e c o r e and flow sys tem,

To avoid l e a k a g e , a 2 0 gage s t a i n l e s s s tee l w i r e w a s always

t i e d around t h e rubbe r s l e e v e between t h e end p l u g s and t h e

1

rock b e f o r e assembly.

A i r t h a t might have been t rapped i n t h e c o r e and t h e

flow system w a s removed by connect ing a vacuum pump t o -27-

t h e ou t f l ow end of t h e s y s t e m , and app ly ing vacuum f o r a t

l e a s t f i v e hou r s . Before t h e vacuum pump w a s d i sconnec ted ,

f low w a s s t a r ted so t h a t t h e whole sys tem w a s complete ly

f i l l e d w i t h t h e l i q u i d t o be used.

The assembled system w a s t h e n hea t ed t o t h e desired ~

r u n t empera tu re a f t e r takfing room c o n d i t i o n measurements. I

The a i r b a t h t empera ture I?eached t h e desired t e s t tempera ture

i n about 1-1/2 hour s , but a t l eas t a n o t h e r f o u r hours were

i

r e q u i r e d for t h e rock specimen t o r each t h e t e s t temperature .

During t empera tu re changes

t he rma l expansion or contr3act i on depending upon whether h e a t i d g

t h e c o n f i n i n g f l u i d underwent

or c o o l i n g . Repeated manual ad jus tments were made t o keep l

t h e overburden p r e s s u r e w i t h i n t h e d e s i r e d l e v e l .

When e q u i l i b r i u m temperature was r eached , f l u i d flow

w a s s t a r t e d and cont inued f o r about a n hour b e f o r e measure-

ments w e r e t a k e n . I

Repeated measurement:s of t empera tu re , p r e s s u r e s and I

I

flow rate w e r e made a t r e g u l a r t i m e i n t e r v a l s . Steady s ta te

w a s assumed when no change w a s observed i n t h e r ead ings and

t h e s t e a d y v a l u e s were recorded. A l l f l o w ra te measurements ~

were t a k e n a t room conditi .ons.

5 .3 Measurements and C a l ~ c u l a t i o n s ! I

5.3- 1 Liqu id Flow: Three k inds of l i q u i d s were used

i n t h i s work: d i s t i l l e d w a t e r , m i n e r a l o i l , and a l coho l . i

I Tap w a t e r w a s d i s t i l l e d i n a Barnstead s t a i n l e s s s t ee l

s t i l l . The o i l used w a s Chevron White Minera l O i l No. 3 ,

and w a s commercial ly a v a i l a b l e . The p h y s i c a l p r o p e r t i e s of 1 I

J

J

J

4

- 2 8-

t h e o i l v a r i e d s l i g h t l y from one c o n t a i n e r t o ano the r .

a l c o h o l used w a s 2- octanol ( P r a c t i c a l ) . Before u s e , each

l i q u i d w a s d e a e r a t e d under a vacuum of about 0.3 p s i a and

f i It e r e d .

The

To keep the f l u i d i n t h e l i q u i d state a t h i g h tempera+

t u r e s , t h e e x i t f lowing p r e s s u r e was k e p t c o n s t a n t at 2 0 0

p s i f o r a l l r u n s .

dampen pump p r e s s u r e p u l s a t i o n s and also he lped t o keep t h e

f lowing p r e s s u r e from changing d r a s t i c a l l y when t h e flow wag

s t a r t e d or stopped.

An accumula tor , charged t o 200 p s i , h e l p e p

Darcy's l a w f o r v i s c o u s f l o w i n a h o r i z o n t a l and l i n e a r I I i porous medium is : I

where q i s i n cc/sec, k is i n d a r c i e s , 1.1 i n cp , A is i n c m 2 l i and i n a tdcm.

Eq. 5- 1 can be expressed as:

Lvw k = - 14700 a- PAP

where k i s i n m d , Ap i s i n p s i , w i n gm/sec, p i n gm/cc, and! I

t h e o t h e r u n i t s and symbols are as d e f i n e d p r e v i o u s l y .

equa t ion was used i n all c a l c u l a t i o n s .

T h i a

I n a d d i t i o n t o u s i n g Qefs. 5 and 6 t o estimate t h e

r e g i o n of v i scous flow w l h e r e p o s s i b l e , c o n d i t i o n s of viscousl ~

f l o w were a l so determined by o b t a i n i n g da ta a t s e v e r a l f l o w ,

r a t e s a t each t e m p e r a t u r e l e v e l and graphing f l o w ra te , q,

- 2 9-

versus p r e s s u r e d rop , p . F o r c o n d i t i o n s of v i s c o u s flow,

t h e d a t a should f a l l on a s t r a i g h t l i n e , p a s s i n g through t he

o r i g i n .

ward c u r v a t u r e of t h e p l o t t e d p o i n t s f r o m the s t r a i g h t l i n e .

The onse t of non--Darcy f l o w was i n d i c a t e d by a down-

It i s necessa ry t o know the v i s c o s i t y and d e n s i t y of

each l i q u i d a t each working t empera tu re l e v e l . Water d e n s i t y '

and v i s c o s i t y v e r s u s t e m p e r a t u r e were found i n t h e Steam i

T a b l e s z 3 and are p r e s e n t e d i n F i g s . 5 and 6 , and i n Table 1, 1

Appendix 9 .1 .

A c a p i l l a r y t u b e v i scomete r w a s c o n s t r u c t e d t o measure

o i l and a l c o h o l (2-octanol.) v i s c o s i t i e s v e r s u s t empera tu re

a t t h e working p r e s s u r e l e v e l of 200 p s i . The procedure f o r

t h e s e measurements i s pres ,ented i n d e t a i l i n Appendix 9.3.

1 1

The r e s u l t s of t h e s e measurements checked v e r y c l o s e l y w i t h I !

t h e d a t a supp l i ed by t h e manufac tu re r s and t h o s e found i n

Iief. 39. The d e n s i t y of a l c o h o l a t v a r i o u s t empera tu res w a s

measured wi th a Bingham t y p e pycnometer.

o i l was measured a t 6OoF and t he tables i n R e f . 2 4 were used

t o estimate t h e d e n s i t i e s a t h i g h e r t empera tu res . R e s u l t s

The d e n s i t y of t h e ' 1 ~

,

are shown i n F i g s . 7 th rough 1 0 , and Tab les 3 through 5 i n I Appendix 9 .1 .

-30-

c.

0 0

0 !n tn 00 0 00 - -

0 0 -

0

0 In M

0 0 M

0 Ln N

0 0 cv

0 L n 4

0 0 d

0 Ln

-31-

0

0

110 01 9 0,8 017 016

5

4

013

01 2

011 I I I I

?do I

60 100 150 200 250 TEMPERATURE, O F

'=i~. F. !!ater l l i s c o s i t y v s . T e m D e r a t u r e a t 2 9 0 n s i g

- 3 2 -

O1 85

OI84

0183

O1 82

O1 81

0.80

o1 79

OI78

\

0

0

\ \

0

100 150 200 lEMPERATURE, O F

250 300

F i g . 7. 9ensityvs.Temperature for Chevron White O i l ‘*e. 3 - 3 3 -

w a 5- a n I-

w c3 L

w v)

u v)

0 ' 0

W

>

a a I-

a cn 2 0 Q:

I V

- z 0

0 0 0 o m L n r f M N 4

o m x ) I \ c o m a- M 4

S3XOlS IlN33 All S O X I r\ 3 I l V W 3 N I ]

6.k

F ~ F . 8 . Viscosity vs. Temperature for Chevron White O i l No. 3 -34-

c 0 \

0 \

\ 0 \ 0

'\ 0

50 100 150 200 250 300 TEMPERATURE, O F

Fip. 9 . q e n s i t y of 2-nctanol vs . TemDerature a t 1 4 . 7 psi.

-35-

10 9 8 7 6

5

4

3

2

0

0

5

0

\

4 I I I 4 I I

300 1 i I

50 100 150 200 250 TEMPERATURE, O F

Fi.p. 10. l ’ i scos i tv of 2 - k t a n o l vs. Temerature at 1 4 . 7 n s i

-36-

5.3-2 Gas Flow: Gas w a s supp l i ed from h igh p r e s s u r e

c y l i n d e r s , and flow r a t e cou ld be r e g u l a t e d upstream of t h e

c o r e by a two- stage a d j u s t a b l e r e g u l a t o r equipped wi th a re-

l i e f va lve , and downstretsm. hy means of a needle valve. Most

experiments involved larrhar flow i n o r d e r t o o b t a i n a b s o l u t e

pe rmeab i l i t y and t h e Kliiikenberg s l i p f a c t o r . One visco-

i n e r t i a l run w a s made. I

I n a d d i t i o n t o measuring p re s su re d rop , Ap, a s e p a r a t e I

I

1

t r a n s d u c e r w a s used t o measure the upstream p r e s s u r e , pl .

From t h e s e va lues , t h e mean p res su re pm =

I

AP p1 - 7 was corn-

pu ted , and t h e d i f f e r e n c e (p12-p22) w a s c a l c u l a t e d as 2pmAp.

Low flow ra tes w e r e measured w i t h a bubble f i l m t ype

flowmeter and a Wet T e s t Yeter w a s used f o r h igh flow rate

measurements. Atmospheric p r e s s u r e and room temperature

were a l s o recorded t o pe:rmit c o r r e c t i o n s t o f lowing condi-

t ions . I

The i n t e g r a t e d form o f Darcy's l a w which desc r ibed

s t eady h o r i z o n t a l l i n e a r gas flow under i s o t h e r m a l c o n d i t i o n $

is: I

where :

= gas f low r a t e a t room c o n d i t i o n s , cc/sec 9,

k = a b s o l u t e p e r m e a b i l i t y , d a r c i e s

A = c r o s s s e c t i o n a l area of porous medium, c m

L = l eng th of porous medium, c m

2

= room t e m p e r a t u r e , OK Ta -37-

T = f l o w i n g t empera tu re , OK

= room p r e s s u r e , a t m . abs . Pa Ap = p r e s s u r e drop across porous Fedium, atrr..

pm = mean p r e s s u r e w i t h i n porous medium, a t r . abs.

v = g a s v i s c o s i t y a t T and pm, c p

z = mean gas c o m p r e s s i b i l i t y fac tor

I -

Ni t rogen w a s t r e a t e d as an i d e a l g a s and t a k e n as 1

because of t h e l o w working p r e s s u r e range .

s i d e of Eq. 5-3 w a s d iv ided by 1 4 , 7 0 0 because , i n t h e l a b o r a

t o r y , p r e s s u r e s were measured i n p s i and p e r m e a b i l i t i e s calc - l a t e d i n m i l l i d a r c i e s . Thus:

The r i g h t hand

7 " -L1J. Qa P aT AAPPmTa

k, md =: 1 4 , 7 0 0

Gas v i s c o s i t y ve r sus t empera tu re i s g iven by Suther lanQ's 25 formula .

For n i t r o g e n , N 2 :

-4 1 . 5 1.3.85(10 I T - - - VN2 102+T

where T i s i n 0 K and V is i n cp. Nitrogen v i s c o s i t y v e r s u s , I

t empera tu re a t atm0spheri.c p r e s s u r e i s p r e s e n t e d on F i g . 11 1

and i n Table 2, Appendix 9 . 1 . No s i g n i f i c a n t i n c r e a s e i n

n i t r o g e n v i s c o s i t y occurs from a tmospher ic p r e s s u r e t o the I

o p e r a t i n g l e v e l s , 2 0 0 ps i . , used i n t h i s s tudy . I

I I E q s . 5-4 and 5-6 were used i n gas f l o w c a l c u l a t i o n s

for v i s c o u s f l o w . See Appendices, s e c t i o n 9.3-2, f o r a n a l y s i s

- 3 8 -

Lo CN 0 - 0

cv CN 0 - 0

0 N 0

0 -

d 3 ‘AlIS03SII\

00 t-l 0 - 0

CD 4 0 - 0

0 0 M

0 L n cu

0 0 (v

0 I n 4

0 L n

- 39-

of v i s c o - i n e r t i a l flow. A l i n e a r graph of (p12-p22), ( p s i a ) 2 , where p1 i s upstream p r e s s u r e ( p s i a ) , p2 i s downstream pres-

s u r e ( p s i a ) , ve r sus g a s flow r a t e , g, (cc/sec) , w a s used t o

determine c o n d i t i o n s for v i s c o u s flow.

-40-

6 . ANALYSIS O'F P,ESULTS AND DISCUSSION

The fo l lowing p r e s e n t s t h e r e s u l t s found f o r t h e

ef fect of tempera ture l e v e l and c o n f i n i n g p r e s s u r e on s i n g l e *

phase f l o w i n sands tones . Conf in ing p r e s s u r e had t h e e f fec t

of r e d u c i n g t h e a b s o l u t e p e r m e a b i l i t y o f sands tones f o r a l l

t y p e s of f l u i d s used. S l i p factor f o r gas flow w a s found t o

be t empera tu re dependent as p r e d i c t e d by theory . A change

( d e c r e a s e ) i n a b s o l u t e p e r n e a b i l i t y w i t h tempera ture increasles

w a s observed only i n t h e case o f water flow.

6 . 1 Water Flow

Both conso l ida ted and unconso l ida ted sands tones were

used i n t h i s s tudy.

6 . 1- 1 Consolidated Sandstones : The f i r s t experiment

was c a r r i e d o u t w i t h a core t h a t was h e a t e d t o 3OO0C for 3

h r s and allowed t o cool o v e r n i g h t b e f o r e use. F i g . 1 2 pre-

s e n t s t he r e s u l t . The rock w a s f i r s t h e l d a t 1 0 0 0 p s i con- I

f i n i n g p r e s s u r e , and t h e a b s o l u t e p e r m e a b i l i t y was recorded

w i t h t empera tu re i n c r e a s i n g t o 30OoF. The rock was t h e n ,

al lowed t o cool t o room t e m p e r a t u r e . Confining p r e s s u r e

w a s i n c r e a s e d t o 2 0 0 0 p s i and t h e p r o c e s s o f h e a t i n g and

measurement r epea ted .

p o r t e d h e r e i n i s ra.ted a t 5 0 0 0 p s i a t 35OoF s a f e l y , but a

maxinum of 4 0 0 0 p s i and 3OO0F w a s used.

I

The core h o l d e r used i n t h e work re-

-41-

0

i

I 0

\ 0

.I I I 1

0 0 Ln

0 0 a-

0 0 M

0 0 cv

- 4 2 -

0 0 1

0 Ln M

! 0 ; 0 M

O Ln 4

0 0 4

0 m

~ ~~ ~ ~~

~

It can be seen from t h e graph t h a t a t each l e v e l o f

c

..

conf in ing p r e s s u r e , t h e a b s o l u t e p e r m e a b i l i t y t o water de-

creased a t an approximate rate of -1.0 nd/OF.

The nex t series of experiments w e r e performed on cores

f i r e d a t 5OO0C i n o r d e r to i n v e s t i g a t e h y s t e r e s i s and r e p r o-

d u c i b i l i t y of r e s u l t s . The r e s u l t s are shown on F i g s . 1 3 ,

1 4 , and 15 . A l l showed a decrease of a b s o l u t e p e r m e a b i l i t y

t o water w i t h t e m p e r a t u r e i n c r e a s e , b u t t h e s l o p e dec reased

wi th i n c r e a s i n g t empera tu re .

t o - 1.6 & / O F w a s observed.

An average s l o p e of - 1 .3 rnd/OF

The r e s u l t s a l s o show tha t t he

tempera ture effect w a s e s s e n t i a l l y r e v e r s i b l e (see Fig . 1 3 ) .

The f irst c o o l i n g c y c l e for t h e Mass i l lon Sandstone Nol

3 (Fig . 14) a t 3000 p s i g c o n f i n i n g p r e s s u r e i n d i c a t e d h y s t e r d -

s is . On t h e f i rs t c o o l i n g c y c l e , t h e pern-!eability i n c r e a s e d ~

w i t h r e s p e c t t o t h e f i r s t : h e a t i n g c y c l e .

a g a i n , r e s u l t s fo l lowed t:he f i r s t c o o l i n g r u n , i n d i c a t i n g telr-

p e r a t u r e r e v e r s i b i l i t y arid good r e p r o d u c i b i l i t y . No r e a s o n

for t h e i n c r e a s e i n p e r m a b i l i t y a f te r t h e first h e a t i n g r u n

w a s found.

On h e a t i n g and cool/ing

I

6.1-2 Unconsol ida ted Sand: The f i rs t experiment i n t h q s

series w a s r u n w i t h unconso l ida ted O t t a w a S i l i c a Sand of ,

average g r a i n s i z e of 0 .385 mm, This core w a s n o t s u b j e c t e d ~

t o h e a t t r e a t m e n t and was loaded as o u t l i n e d i n t h e procedur 3 , s e c t i o n 5 . Data were not: ob ta ined f o r the c o o l i n g c y c l e s .

The r e s u l t s a re shown i n F ig . 1 6 . Confining p r e s s u r e s of 500i,

1 0 0 0 , and 1500 p s i g were used. Absolute p e r m e a b i l i t y t o watw 1

- 43-

I ,

- 4 4 -

0 In M

0 0 M

0 m cv

0 0 cv

0 m 4

0 0 4

0 0 L n

0 0 M

0 0 cv

0 LA

0 0

I

J

J

3

3

4

U

-

c3 v) Q

0 0 0 M II

U

- v) cn w (1: Q

c3 z z L z 0 V

H

I

I I 1 I

0 0 4-

0 0 M

0 0 cv

0 0 -I

-45-

c(

v) CL

0 0 cv II

z

E 4

Gi

I 1 I I I

0 0 u3

0 0 Ln

0 0 a-

0 0 M

0 0 w

0 In M

0 0 M

0 In cv

0 0 cv

0 In 4

0 0 4

0 Ln 0 0

h ic,

’. M

4 r l 44

‘A11 1 I IfW3Wt13d - 4 6 -

v) n m 00

O M 0 - c v o II II

E W t r r e n - Z o n

a z v )

0 0 0

%

cv N

0 c3 0 00' 11

0 0 0

% a- ll

0 0 0

0 1

%

- 4 7 -

0 0

0 In M

0 0 M

0 L n cv

0 0 cv

0 In ll

0 0 11

0 Ln

0 0 0

decreased w i t h a n i n c r e a s e i n c o n f i n i n g p r e s s u r e .

of change was - 0.5 rnd/OF a t a11 l e v e l s of c o n f i n i n g p r e s s u r e c

As t h e a b s o l u t e p e r m e a b i l i t y of t h i s core w a s very h igh ,

2 2 , 0 0 0 md a t 500 p s i g co:nf in ing p r e s s u r e , t h e average sand

g r a i n s i z e was reduced for t h e nex t exper iment .

f o r methods of c o n t r o l o f p e r m e a b i l i t y and p o r o s i t y of syn-

t h e t i c sands .

6 .2 Gas F l o w

"he rate

( 1 1

See Ref. 26 1

1

1 Gas f low experiments were conducted a t t h r e e d i f f e r e n t 1

Abso- t e m p e r a t u r e l e v e l s and a range of c o n f i n i n g p r e s s u r e s .

l u t e p e r m e a b i l i t i e s were measured w i t h n i t r o g e n flow under

c o n d i t i o n s of v iscous flow and graphed as a f u n c t i o n of the

r e c i p r o c a l mean p r e s s u r e on a c o n v e n t i o n a l Klinkenberg graph.

One run w a s conducted i n t h e v i s c o - i n e r t i a l flow reg ion for

a c o n s o l i d a t e d sands tone .

gas flow d a t a are given i n Table 9 and i n Appendix ( s e c t i o n

9 . 3- 2 ) .

t e m p e r a t u r e s but for t h e same c o n f i n i n g p r e s s u r e , e x t r a p o l a t e d

t o t h e same i n f i n i t e p r e s s u r e v a l u e .

g raphs are, however, p robably n o t t r u e Klinkenberg s l i p coef-

I 1 1 Details of t h e a n a l y s i s of t h i s 1

I n a l l c a s e s , t h e a b s o l u t e p e r m e a b i l i t i e s a t d i f f e r e a t

I ! The s l o p e s o f t h e s e

f i c i e n t s . i

combina t ion of both s l i p and mean p o r e p r e s s u r e- c o n f i n i n g p r e s *

s u r e stress e f f e c t s on p e r m e a b i l i t y . Although t h e c o n f i n i n g

p r e s s u r e was h e l d c o n s t a n t a t 1000. psi o r more, t h e pore p res-

s u r e changed from n e a r l y a tmospher i c t o s e v e r a l hundred p s i

for each ser ies of r u n s .

w a s n o t a major o b j e c t i v e of t h i s s t u d y .

t i o n from t h e s e graphs is t h a t t e m p e r a t u r e has no apparen t effect

on a b s o l u t e pe rmeab i l i ty to gas.

The high s l o p e s are a r e s u l t of t h e e f f e c t s of a

I I

' I 3 e t e r m i n a t i o n of s l i p c o e f f i c i e n t s

The va luab le informadl

-49-

0 L n M

0 0 M

0 L n cv

0 0 N

0 Ln 4

8 0 4

0 In 0 r3 0

CD 3

0

- 4 9 -

Figs . 18 and 19, r e s p e c t i v e l y . On Fig. 1 8 , t h e Klinkenberg

s l i p f a c t o r s a t 6 8 , 1 5 0 , and 25OoF are 2 . 2 6 4 , 3.587, and 4.351

p s i , r e s p e c t i v e l y .

s l i p f a c t o r depends upon temperature. T h e apparent permea- I

b i l i t i e s a t d i f f e r e n t temperatures and p r e s s u r e s ex t rapo la ted

This shows t h a t t h e apparent Klinkenberg

t o t h e same va lue of 9 6 0 md and agree reasonably w i th t h e v i s c

i n e r t i a l a n a l y s i s value of 932 md shown on Fig. 1 9 . The turbud

l e n c e factor 8 , which may be obtained from Fig . 1 9 , was 1 . 7 x 7 10 ft-l, and i s i n agree,ment with a c o r r e l a t i o n proposed i n

R e f . 2 7 .

i n t h e s e r u n s .

The conf in ing p ressure was he ld c o n s t a n t at 1 0 0 0 p s i 1

Another se r ies o f experiments w a s conducted on Massillon i

! I

1 1

Sandstone Core N o . 6 a t room, 15OoF, and 25OoF tempera ture

l e v e l s and a t conf in ing p ressures of LOOO, 3000, and 4000 psig . !

Resu l t s are shown on Figs, , 20- 22, and are similar t o those of 1 1

Fig. 18. Absolute permeabi l i ty decreased w i t h an i n c r e a s e i n , conf in ing p r e s s u r e ; and at: t h e same l e v e l of conf in ing pressurq

t h e apparen t p e r m e a b i l i t i e s ex t rapo la ted t o t h e same i n f i n i t e

p r e s s u r e va lue apparent s l i p f a c t o r s a t room and 15OoF tempera-i

I

t u r e l e v e l s w e r e e s s e n t i a l l y t h e same for a l l l e v e l s o f con-

ni.trop,cn flow a n d wstcr flow for t h c consol i d c i t e d

c o r e s w e r e d i f f e r e n t . Pe rmeab i l i t i e s t o water a t

perature ranged between 550 rrd and 450 md f o r a l l -50-

6 . 2 - 1 Consolidatetl Sandstones: The r e s u l t s of t h e

first run i n t h e viscous and v i s c o- i n e r t i a l f low reg ions on

I Massi l lon Consolidated Sandstone Core Yo. 5 are shown on

f i n i n g p r e s s u r e , but higher a t 2 5OoF tempera ture . I

The v a l u e s of a b s o l u t e p e r m e a b i l i t i e s ob t a ined w i t h I

sandstone

room t e m -

of t h e

I

~

u, 0 0 Ln N

II I-

L L L L 0 0 0 0 0 L n c D I+

II II

t- I-

\

U

v) a 0 0 0 4

II w p: 3 v) v) w p:

(3 z z L z 0 u

n

U

U

0 0

0 0 cn

(7W ‘AlIlI€fW3W~3d lN3tlWddb -51-

1.4

1.2

n 1.0 cv I

I- LL v

'0 0.8 4

0.6

0,4

0.2

0

,

- 5 7 -

% I 3 L

0 LI 3

0 4 Ln cv ca

n - 0

-

I

E Ln

0 - a

3 W E 3

Ef

0 - u

@ a , P ' S I 0 *o

a 0 0 E a

M 0

- w u w cr:

0 0 m

Q 0 co

0 0 h

-53-

t-I

u- 0 Lo

0

0

0 0 0 -I

0 0 cn

0 0 00

- 5 4 -

co 0

-

d CI]

d

J

J

.J

J

J

J

4

J

J

-

LL

0 Ln 1

0

w LT 3 t- LT w a

I-

a

5

L L

C 3

CV

0

Ln

L L L L L L 0 0 0

0 0 0 r \ m m d c v 0 . 0

0 0 m

aW 'AlIlI8V3Wkl3d lN3klVddv

-55-

0 0 I\

J

conf in ing p r e s s u r e l e v e l s , whi le a b s o l u t e p e r m e a b i l i t i e s t o

n i t r o g e n were cons ide rab l ly h i g h e r and ranged from 9 6 0 md t o

800 md fo r t h e same c o n d i t i o n s .

omena caus ing t h e permea .b i l i ty r e d u c t i o n w i t h water f l o w a t

d This s u g g e s t s tha t t h e phen-

room t empera tu re may also be r e s p o n s i b l e for the p e r m e a b i l i t y

r e d u c t i o n a t h i g h t e m p e r a t u r e s . The Mass i l l on Sandstone has 4

a low c l a y c o n t e n t , and had been f i r e d at 50OoC. i

6.2- 2 Unconsol ida ted Sand: Fig . 23 shows r e s u l t s ob-

t a i n e d wi th f i r e d s i l i c a sand.

w a s a p p l i e d , and t h e t empera tu re l e v e l s used were 68OF, 150af,

and 25OoF. The r e s u l t s were s imilar t o t h o s e for t h e consol,!-

da t ed sands tones . 4

A 2000 p s i g c o n f i n i n g p r e s s u r e J i

I

P e r m e a b i l i t y t o water w a s a l s o measured f o r t h i s core.'

While t h e e x t r a p o l a t e d a b s o l u t e p e r r e a b i l i t y t o n i t r o g e n w a s /

4,260 md, t h e p e r m e a b i l i t y t o water was o n l y 2 , 1 2 7 md a t roq

tempera ture .

6 . 3 O i l F low

J

1" ~

.J

~

Because of t h e d i f f e r e n c e between t h e r e s u l t s o b t a i n e d

a t e l e v a t e d t e m p e r a t u r e s w i th water f l o w and gas f l o w , s i m i l g r

exper iments w e r e r u n w i t h ano the r f l u i d . Chevron White O i l ~

J

No. 3 w a s chosen for i t s h i g h e r v i s c o s i t y and i t s non- polar

c h a r a c t e r i s t i c s

v i s c o s i t y p o l a r f l u i d and may i n t e r a c t w i t h t h e r o c k s u r f a c e l

Experiments (see 3.ef. 1) had a l r eady been c a r r i e d o u t w i t h

j

as opposed t o water, which i s an i n t e r m e d i d t e

d

Chevron White O i l No. 1 5 , which i s very v i scous . A s i n water

f l o w or gas f low expe r imen t s , t h e combined ef fec t of temperait I

I 4

t u r e and overburden p r e s s u r e was i n v e s t i g a t e d . ~

-56-

0 0 J

In 3

11 t- I- a

0 0 0

L n 3

0 0 co a- 3

0 0 Lo -3

0 0 0 N

a- 3

- 5 7 -

Figs . 2 4 and 25 show t h e r e s u l t s wi th o i l f low for

c o r e s No. 1 0 and 11. O i l flow w a s cofiducted on ly on con-

s o l i d a t e d sands tones .

a f f e c t e d by c o n f i n i n g p r e s s u r e , as expec ted , b u t t h e r e w a s

on ly a s l i g h t d e c r e a s e i n a b s o l u t e p e r m e a b i l i t y t o o i l w i t h

tempera ture i n c r e a s e . H y s t e r e s i s e f f ec t s i n c o o l i n g c y c l e s

were more pronounced a t high c o n f i n i n g p r e s s u r e s of 3000 p s i

and 4 0 0 0 p s i .

Absolu te p e r m e a b i l i t y t o o i l w a s

Pe rmeab i l i t y l e v e l s a t room tempera tu re a g r e e c l o s e l y 1 I

with t h o s e o b t a i n e d w i t h n i t r o g e n flow.

rock- o i l i n t e r a c t i o n i s b e l i e v e d t o be r e s p o n s i b l e for the

d i f f e r e n c e between water and o i l p e r m e a b i l i t i e s . T h i s sug- 1 g e s t s a d e f i n i t e i n t e r a c t i o n between water and sands tone , I

The absence of

i I

conso l ida t ed o r unconso l ida t ed . I

6 . 4 2-Octanol Flow I

I n view of t h e results w i t h water, it appeared

u s e f u l t o run s i m i l a r expe r imen t s wi th a n o t h e r polar l i q u i d .

2- octanol a l c o h o l w a s chosen.

Experiments were conducted w i t h commercial g rade 2-

o c t a n o l , and t h e r e s u l t s are p re sen ted i n F ig . 2 6 . The un-

I conso l ida t ed O t t a w a S i l i c a Sandcore used was f i r ed a t 50OoC.

The average sand g r a i n s i z e w a s 0 . 1 5 6 mm. The c o n f i n i n g

p r e s s u r e a f f e c t e d t h e a b s o l u t e p e r m e a b i l i t y as expec t ed , b u t ~

t empera ture appeared t o c a u s e a. small i n c r e a s e i n a b s o l u t e

pe rmeab i l i t y . I n a d d i t i o n , t h e r e w a s a permanent r e d u c t i o n

i n a b s o l u t e p e r n e a b i l i t y o n each c o o l i n g c y c l e .

a re d rama t i c when compared t o t h e water d a t a on F igs . 1 6 and,117.

I

i

I

These r e s u l d s

-58-

J

J

J

J

J

i )

.J

4

-4

J

.l

O

G O n d

0

0

0 0 m

0 0 00

0 0

0 0 to

0 0 M

0 In N

0 L n 4

0 0 4

0 L n

0 0 L n

c(

cn a 0 0 cv

z W r a

a

i i

I I I I I

Q 0 0 4

0 0 0 0 00 R

aW ' A l I l I 8 V 3 W W d

- 6 0 -

0 0 L o

0 0 M

0 m cv

L L 0

rl rl

0 z

0 In 4

0 0 4

0 in

0 0 4

J

J

J

J

J

J

J

J

J

J

J

0

0

a l

v) v)

o w “CY w a a

(3 O Z 0- O Z 4“

0 0 N

II - v) v) W E Q,

W CY 0 a, z w = a

e

\ 0 0 e

a 0

- 0

II w N & v)

z U

a c 3 c1 z cn a

I I I I l

0 0 0 0 4

-l

0 0 0

cn -l

0 0 0

00 T

0 3 0

r\ -l

0 0 0

L o %

0 In 4

0 0 d

0 L n

0 0 0 In

%

-61-

I ,

I -

The s l i g h t i n c r e a s e i n p e r m e a b i l i t y t o o c t a n o l w i t h tempera-

t u r e i n c r e a s e f o r bo th h e a t i n g and c o o l i n g r u n s emphasizes

t h e impor tance of t h e dec rease i n k with t e m p e r a t u r e i n c r e a s e 3

for water. Recent d i s c u s s i o n s between Ramey and Dreher (Yara-

thon O i l C o . , Denver) r e v e a l e d t h a t o c t a n o l o f t e n behaves more

d l i k e an o i l t h a n a n aqueous w e t t i n g phase i n s ands tones .

r e t r o s p e c t , it might have been b e t t e r t o h a v e selected b o -

p r o p y l a l c o h o l (IPA) as t h e second p o l a r s u b s t a n c e . It i s

recommended t h a t IPA be cons idered f o r f u t u r e s t u d i e s .

I n

J

6 . 5 D i s c u s s i o n

J I

have been one r ea son for t l h e effect of t e m p e r a t u r e l e v e l upon I 1 Cass6’ concluded t h a t c lay- water i n t e r a c t i o n s may

i I ~

a b s o l u t e p e r m e a b i l i t y t o w a t e r f o r s a n d s t o n e s h e observed.

G r i m 3 ’ 31 p r e s e n t e d a comprehensive rev iew of the mechanism

involved i n c lay- water i n t e r a c t i o n s . Because t h e r o c k s used J

i n t h i s work w e r e t o t a l l y or n e a r l y c l a y- f r e e , c lay- water ~

1 i n t e r a c t i o n i s cons ide red r e s p o n s i b l e f o r o n l y a minor role

i n the phenomenon observed i n t h i s s tudy . .d

It i s almost c e r t a i n t h a t water-cilica i n t e r a c t i o n s

a re r e s p o n s i b l e f o r t h e mafior e f fec ts obse rved w i t h w a t e r . I n ! 13

I Glass F i l t e r s of low p e r m e a b i l i t y ( 1 6 0 rrd a t room t e m p e r a t u r e ) ’ \ I

a d d i t i o n t o the r e s u l t s of t h i s s t u d y , Grunberg and Nissan

used aqueous s o l u t i o n s ( i n c l u d i n g Amyl Alcohol) w i t h J ena

and found a l i n e a r dec rease i n a b s o l u t e p e r n e a b i l i t y wi th an 1

3

J

i n c r e a s e i n t e m p e r a t u r e . 3 2 F l u i d f l o w mechanisms o f t e n fit t w o c a t e g o r i e s :

3

(1) mechanisms which are e s s e n t i a l l y mechanical i n n a t u r e , t h e 1 1

- 62-

f l o w depending on t h e bulk p r o p e r t i e s of t h e f l u i d s concerned

..

!-

and upon the mechanical f o r c e s e x e r t e d upon the f l u i d b o d i e s ;

and ( 2 mechanisms which are e s s e n t i a l l y m o l e c u l a r i n char&-

t e r , t h e f l o w depending l a r g e l y on t h e motion of t h e i n d i v i d u a l

molecules, t h e moleculair, weight , t h e c o l l i s i o n cross-section,

t h e mean f ree p a t h , r o c k- f l u i d a t t r a c t i v e forces, etc., ra tder

than upon t h e d e n s i t y , p r e s s u r e v i s c o s i t y , e t c . , of t h e f l u i d

i n bulk .

of t h e e f fec t of t empera tu re upon water f l o w i n s ands tones

under p r e s s u r e .

The l a t t e r mechanisms may be dominant i n t h e case

S u r f a c e a t t r a c t i v e forces between s i l i c a and water

molecules may be l a r g e enough t o l e a d t o chemi- sorp t ion . I n

such a case, t h e a d s o r b a t e molecules become p r a c t i c a l l y a

p a r t o f t h e solid s u r f a c e and, compared with o the r rnolecules

i n the l i q u i d , such chemfi-sorbed molecu les may be l a r g e l y

immobilized. The e f f e c t i v e c r o s s- s e c t i o n under v i s c o u s f l o w I

could t h e n be d i f f e r e n t f o r water from t h e o t h e r f l u i d s test&.

I n c r e a s e s i n t e m p e r a t ~ r e ~ ~ seem t o i n c r e a s e t h i s e f f e c t from

t h e r e s u l t s of t h i s s t u d y . An exhaus t ive l i t e r a t u r e su rvey

i n d i c a t e d no known phenomna capable of e x p l a i n i n g t h e magni-

t u d e of t h e ef fects observed i n t h i s s tudy .

t h a t a major, and h e r e t o f o r e unsuspected a t t r a c t i o n between

water and s i l i c a h a s been d i scovered . It is p o s s i b l e t h a t

t h i s a t t r a c t i o n i s e s s e n t i a l l y r e s p o n s i b l e for t h e l a r g e

tempera ture e f fec t s on p r a c t i c a l i r r e d u c i b l e water s a t u r a t i o n

and r e l a t i v e p e r m e a b i l i t i e s 34916,35, and on c a p i l l a r y p r e s -

s u r e s 36-38 and r e s i s t i v i t y p rev ious ly r e p o r t e d .

It is b e l i e v e d 1

- 6 3-

If a s t r o n g a t t r a c t i o n between water and s i l i c a i s _.

t h e major t e m p e r a t u r e e f f ec t , s e v e r a l new exper iments w i t h

g a s- o i l flow i n sands tones w i l l show no t e m p e r a t u r e e f fec t , I

~

whi le gas- water f l o w i n sands tone w i l l depend on t empera tu re .

I n a d d i t i o n , exper iments of any s o r t i n l i m e s t o n e s shou ld

no t depend s t r o n g l y on t e n p e r a t w e .

- 64 -

J

v,

J

J

d

J

J

J

J

J

J

i

7. CONCLUSI3NS AND RECOY!!!TYIkTIONS

Experimental r e s u l t s i n d i c a t e t h a t t h e a b s o l u t e per-

meabi l i ty t o water f o r conf ined sands tones i s s t r o n g l y t e m -

p e r a t u r e dependent. The a b s o l u t e p e r m e a b i l i t y of s a n d s t o n e s

t o o t h e r f l u i d s ( n i t r o g e n , mine ra l o i l , o c t a n o l ) i s e i t h e r ,

una f fec ted o r on ly s l i g h t l y a f f e c t e d by t e m p e r a t u r e l e v e l .

Temperature increase has t h e e f f e c t of d e c r e a s i n g t h e abso-

l u t e p e r m e a b i l i t y t o w a t e r f o r sands tones remarkably.

Temperature change h a s l i t t l e o r no effect on t h e abeo-

l u t e p e r m e a b i l i t y o f s a n d s t o n e s t o n i t r o g e n , m i n e r a l oil, qr

o c t a n o l . I n t h e case o f water f l O t 7 , p e r m e a b i l i t y r e d u c t i o h s

o f up t o 60% were o b s e w e d o v e r a t empera tu re r a n g e of 70-$OO0F. ~

Regardless of t h e t y p e of f lowing f l u i d , t h e l e v e l of

c o n f i n i n g p r e s s u r e a f f e c t e d a b s o l u t e p e r m e a b i l i t y i n the s@ne

manner , i .e . , p e r m e a b i l i t y decreased wi th i n c r e a s i n g confizhing

p r e s s u r e . For water f:Low exper iments , i n c r e a s i n g t h e c o n f i n i n g

p r e s s u r e had t h e a d d i t i o n a l effect of i n t e n s i f y i n g t h e tembera-

t u r e dependence o f a b s o l u t e p e r m e a b i l i t y i n many cases (but

n o t a l l ) .

I n t h e l i g h t of t h e r e s u l t s o b t a i n e d , it seems t h a t wn-

suspec ted f l u i d - s o l i d s u r f a c e a t t r a c t i v e f o r c e s between w a t e r

molecules and s i l i ca were l a r g e l y r e s p o n s i b l e for t h e m a j o r

e f f e c t s observed. The f o l l o w i n g recomnendations a p p e a r

,

p e r t i n e n t .

-65 -

J' - 66 -

It i s recornended t h a t t h e r e s u l t s o b t a i n e d by G r u n - 13 b e r g and Ni s san be rechecked for t h e same aaueous s c l u t i o n s

and for c o n f i n e d porous media. It i s also recornended t h a t

exper iments b e r e p e a t e d f o r h i g h e r t e m p e r a t u r e s and conf inin ,g

p r e s s u r e s .

J

S i n c e o c t a n o l behaves more l i k e an o i l t h a n a n aq-ueous J

w e t t i n g phase i n s ands tones , it i s recommended that i sop ropy l

alcohol b e used as the second polar s u b s t a n c e for f u t u r e

s t u d i e s .

t h e d e c r e a s e i n water pe rmeab i l i t y w i t h i n c r e a s i n p temperatqhe

is due t o t h e p o l a r i t y o f water.

R e s u l t s from such s t u d i e s may h e l p t o e x p l a i n whether A

d I

I f a s t r o n g a t t r a c t i o n between w a t e r and s i l i c a causes '

t h e t e m p e r a t u r e e f f ec t , experiments des igned t o i so la te watap

a n d / o r s i l i ca w i l l be e n l i g h t e n i n g . Exper iments w i t h water I f l o w i n l i m e s t o n e s should no t depend s t r o n F l y on tempera tura

I n a d d i t i o n , exper iments wi th g a s- o i l f l o w i n s ands tones may

show no t e m p e r a t u r e e f f ec t whi le gas-water f l o w i n sandstond

w i l l depend on t empera tu re .

J

i

J

J

-66-

8 . REFERENCES

1.

2 .

3.

4.

5.

6 .

7.

8.

9 .

10.

11.

1 2 .

Cassg , F. J . : "The Effect of Temperature and Confining P re s su re on F l u i d Flow P r o p e r t i e s of Consol ida ted Xocks ," Ph.D. D i s s e r t a t i o n , S t a n f o r d U n i v e r s i t y ( 1 9 7 4 ) .

A r i h a r a , N . : "A Stud17 of Non- Isothermal S i n g l e and Two- Phase Flow th rough Consol ida ted Sandstones ,!' Ph.D. D i s - s e r t a t i o n , S t an fo rd U n i v e r s i t y ( 1 9 7 4 ) .

Greenberg, D .B . , Cresap, R.S:, and Malone, T.A.: " I n t r i n - s i c Pe rmeab i l i t y of Hydro log ica l Porous Mediums: V a r i a t i o n wi th Temperature," Water Resources Research (Aug. 19681, - 4 , No. 4, 7 9 1 .

Muskat, M. : Flow of Homogeneous F l u i d s through Porous Media, M c G r a w - H i l l Book Company, I n c . , 1937, C h . 11.

Fancher , G.H. ? L e w i s , J . A . , and Barnes , K.: "Some Physi- , ca l C h a r a c t e r i s t i c s of O i l Sands,"Ind. Eng. C h e m . , 2 5 : 1 1 3 9 ( 1 9 3 3 ) .

Geertma, J. : " Est imat ing t h e C o e f f i c i e n t of t h e I n e r t i a l ' Res i s t ance i n F l u i d Flow th rough Porous Media," SOC. Pe t . Eng. Jou r . ( O c t . 1 9 7 4 1 , 465.

- -

Amyx, J . W . , Bass, D.M., Jr., and Whiting, R.L.: Pet roleul Rese rvo i r Engineer in ,&, M c G r a w - H i l l Book Company, Inc . , ' 1 9 6 0 , Ch . . 2

Kundt, A . , and Warbu:rg, E. : Pogpendorf ' s Ann. Phys ik , pp 155, 337, 525.

Klinkenberg , L. J . : "'The P e r m e a b i l i t y of Porous Media t o Liquids and Gases," D r i l l . and Prod. P rac . , API (19411, 200 .

Forchheimer, P.: "Hydraulik," Le ipz ig a n d . B e r l i n , C h . 1s S e c t i o n s 1 1 6- 1 1 8 , Druck and Ver l ag von B.G. Teubner ( 1 9 1 4 ) .

C o r n e l l , D . , and Katz, D.L. : "Flow of Gases through Consol ida ted Porous Media," Ind . Eng. Chem. (19531, Vol. 4 5 , p . 2145.

Dranchuk, P.H., and Kolada, L . J . : " I n t e r p r e t a t i o n of Steady L i n e a r V i s c o- I n e r t i a l Gas F ~ O W Data," Jour . ' Can. P e t . Tech. ( 1 9 6 8 1 , -. 7 , No. 1, p. 36.

- 6 7 -

13.

14.

1 5 .

1 6 .

1 7 .

18.

1 9 .

2 0 .

2 1 .

2 2 .

23.

24 .

25.

2 6 .

i

J

J

J

kJ

d

J

4

J

.J

I

.J

Grunberg, L . , and Nissan, A . H . : "The Permeabi l i tv of Porous S o l i d s t o Gases and Liquids ," Jou r . of . I n k . of P e t . (19531, 2 9 , No. 2 3 6 , p . 1 9 3 . - - Calhoun, J . C . , Jr . , and Yus te r , S.T.: "A Study of t h e Flow of Homogeneous F l u i d s th rouph I d e a l Porous Media," D r i l l and 'Pro'd. Prac., API ( 1 9 4 6 1 , 335.

API RP 2 7 : "Recommended P r a c t i c e for Determining Permea- b i l i t y of Porous Media," Am. P e t . I n s t . (Aug. 1956).

Weinbrandt , R.M. : "The Effect of Temperature on Relat ive P e r m e a b i l i t y , " Ph .D. D i s s e r t a t i o n , S t a n f o r d U n i v e r s i t y (1972).

W i l h e l m i , B . , and Somerton, W.H.: "Simul taneous Measure- ment of Po re and Elas t ic P r o p e r t i e s of Rocks under T r i - ax i a l S t r e s s Cond i t i ons , " SOC. P e t . Eng. J o u r . (Sep t . 1 9 6 7 1 , - 240, 283.

w i t h Overburden Pressure," Trans . , AIME (1952) , - 3 1 9 5 329.

F a t t , I. , and Davis , : D . H . : "Reduction in P e r m e a b i l i t y I I

Gray, D.H. , F a t t , I:, and Bergamini, G. : "The Effect of I

I S t r e s s on P e r m e a b i l i t y of Sandstone Cores," SOC. Pe t . Eng. Jour . ( J u n e 1 9 6 3 ) , 9 5 .

Zoback, M . D . , and Byeiqlee, J . D . : " P e r m e a b i l i t y and Ef- f e c t i v e S t r e s s , " A m e r . Assoc. P e t . Geol. B u l l . , 59 (1) (1975a1, 1 5 4 .

Somerton, W.H. , Mehta,, M.M. , and Dean, G.W. : "Thermal A l t e r a t i o n of Sandstones ," J. P e t . Tech. (May 19651, 589.

Wyble, D.O.: "Effect of Applied P r e s s u r e on t h e Conduc-

i 1

t i v i t y , P o r o s i t y , and Pe rmeab i l i t y of Sands tones ," Trans . , AIME (19581. 213. 430.

Meyer, C.A. , McClintock, R.B. , S i l v e s t r i , G . J . , and Spencer , R . C . , Jr.: 19167 ASME S t eam Tab le s , Am, SOC. I

Mech. Engrs . , 2nd e d . , New York, N . Y . , 1968.

ASTM-IP-Petroleum Measurement T a b l e s , ASTM, P h i l a d e l p h i a I

(19521 . P e r r y , J . H. : Chemical Engineer ' s Handbook, M c G r a w - H i l l Book Company, Inc . (19691, pp. 23- 10.

Larman, H . J . : " V a r i a t i o n s i n P e r m e a b i l i t y and P o r o s i t y of S y n t h e t i c O i l Rese rvo i r Rock--!<ethods of Control," SOC. of P e t . Eng. J o u r . (Dec. 19651, 329.

I

-6 8-

2 7 .

2 8 .

29.

30.

31.

32,

33.

34.

35.

36.

37.

38.

39 .

Katz, D . L . , and C o a t s , K . H . : Underground S t o r a g e of F l u i d s , U l r i c h s Books, Inc . , Ann Arbor, Michigan c19 6 8 ) . Jones , S .C . : "A Rapid , Accurate Unsteady- State Kl inken- berg Permeameter," SOC. P e t . Eng. J o u r . ( O c t . 1 9 7 2 1 , 483.

Sax, N. I. : Dangerdus P r o p e r t i e s of I n d u s t r i a l Materials , Reinhold P u b l i s h i n g Corpora t ion (1963) . G r i m , R.E. : Mineralogy, M c G r a w - H i l l Book Company, I n c

G r i m , R.E. : Clay Mineralogy, M c G r a w - H i l l Book Company, Inc . (1968) .

Flood, E .A. : " I n f l u e n c e o f Su r face Fo rces o n F l o w of F l u i d s t h rough C a p i l l a r y Systems ,I' Highway Research Board, Spec. Rep. N o . 40 (19581,

Bar te l l , F . E . , Thomas, T .L . , and Fu, Y.: "Thermdynam$cs of Adsorpt ion from S o l u t i o n , i v . Temperature Dependenae of Adsorp t ion ," J o u r . - of Phys. Chem. (19511, - 5 5 , 1 4 5 6 ,

Poston, S . W . , Ysrael, S. , Hossain , A.K.M.S. , Montgomery, E.F. , 111, and Rarney, H.F. , Jr.: "The Effec t of Tempeva- t u r e on I r r e d u c i b l e Water S a t u r a t i o n and R e l a t i v e Permed- b i l i t y of Unconsol-idated Sands," SOC. of P e t . Eng. Joqr. ( June 1 9 7 0 1 , 1 7 1 .

Weinbrandt, R.M., Casse, F , J . , and Ramey, H . J . , Jr.: , "The Effect of Temperature on R e l a t i v e and Abso lu t e ' Permeab i l i t y of Sands tones ," SOC. of P e t . Eng. J o u r . ( O c t . 1 9 7 5 1 , 376.

S innokro t , A.A. , Ramey, Hi; J., Jr, , and Marsden, S.S , , ~ Jr, : "Effect of Temperature Level upon C a p i l l a r y P r e s s u r e ' Curves," SOC. o f P e t . Eng. Jou r . (March 19711, 13 .

Okandan, E . , Marsden, S .S., Jr. , and Ramey, H.J., Jr. E " C a p i l l a r y P r e s s u ~ e and Contact Angle Measurements at Eleva ted Temperatures , I 1 presen ted a t AIChE Meet ing, FLlsa, Oklahoma, March 1:2, 1 9 7 4 .

Sanya l , S .K . , Ramey, H . J . , J r . , and Marsden, S . S . , Jri. : "The Ef f ec t of Teinperature on C a p i l l a r y P r e s s u r e Propkr- t i e s of Rocks," S:PWLA 1 4 t h Annual Logging Symposium (May 1973) .

Na t iona l Research Counci l of U. S .A. : " I n t e r n a t i o n a l Cr i t ica l Tables of Numerical Data, P h y s i c s , Chemis t ry and Technology," 'Vol. I-1111, 1 9 3 3 ,

-69-

J

9. APPENDICES

J

3

c

9.1 L:st of Tabulated ! lata

Table 1. Dens i ty and Vi . scos i ty of Water vs. Temperature

(from Steam

P r e s s u r e = 2 0 0 p s i g

Temperature . ' OF

6 0

1 0 0

1 5 0

2 0 0

250

3 0 0

350

Dens i ty gm/cc

0.9990

0.9931

0.9803

0 . 9 6 3 2

0 .9423

0.9180

0.8904

V i s c o s i t y cp

1.100

0.679

0 .426

0 .300

0.228

0 .183

0 .152

J

r'

J

J

4

J

4

4

3

4

-72 -

c Table 2. V i s c o s i t y of Nitrogen vs. Temperature

I

Pr-essure = 14.7 p s i a

Temperature Temperature Viscosity OF ' OK c p

60 288.7 0.01739

100

150

200

2 50

300

310.9

338.7

366.5

394.3

0.01839

0.01959

0.02074

0.02185 I

422.0 0.02291

OK = T(°F)-32 + 273.16 1.8

-73-

n f U

a 0

5 u 0 rl k si Q,

rl F m 0 0

0 .

rl Ln f co 0 0

0

m - 03 03 0 0

0 .

0 3 3. f rl 0

0

CD 03 Lo CD m 0

0

3 Ln W a 0

0 9

(D f * 0 rl

0

- I

E W

v) I 0 rl X b a hl + ri U 0 t-i ri v) 0

I I

E b

: Q) c 3

VlD.413 0 0 0 0 0 0

. I

- 74 -

J

J

J

J

J

J

J

J

3

J

J

b-4

0 E: rd

0 I

c 0, n

cv I

I cv $ 1

+J a Llc4

W N O

ti

-75-

Table 5.

Temp. OF

66

100

150

200

250

300

Viscosity of 2-Octanol vs. Temperature

“T cpTsec ps1-cc

0.05112

0.05117

0.05124

0.05131

0.05138

0.05144

Pressure = 200 psig

A P

p s i

1.50

1.25

1.00

1.00

0.75

0.75

Q cc/sec

0.00725

0.01266

0.02222

0.04000

0.04494

0.0625

where aT. = .05113 1+2.67~10-~ (T-7O0F‘I]

J

I

!

J lJ

+ 5.0

2.306 .J

0.8 7 J

0.6L4

3

J

-76-

'1 I

I t I X I E N

b2 G ' - E .rl u r n m acv L-

m m o c v c o 0 . ' h l m j

II II I t

o Q , C D o ~ m o C D o o o o c o o ~ m o u l o m m ~ c - ( V O ( V m ~ N O f c n ~ ~ 0 h l m 0 c v 0 c v ~ a , C D f m h l ~ c n f m c v d 0 7 ~ m ( V ~ ~ ~ m c v ~

O O O O O O O O O O O O Q O O O O O O O O . . . . . . . . . . . . . . . . . . . . .

Q ) a < 5 d 5

$ Z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000000000000000000000 000000000000000000000 r l ~ d ~ l d d ( V c v N ~ c v m 0 m m m ~ ~ ~ j f

Q i r n

E - l m 2-g a & 3

Q) c 0 +J rn a c a v)

c 0 rl

Fc 0

44

a +J

rd n 3 0 d Ll

W

Q) d A a b

c - m f f c o m f m a c - c n m f f b c n m d c v u l m b m r ( c ' ~ o ~ ~ c o u 3 j a , o d o c - ~ c n j ~ m ~ o m ~ I ~ c n d c v c v r l c v d ( V c v ( V d ~ ~ ~ r l 0 0000 '0d000000000000000

000000000000000000000 . . . . . . . . . . . . . . . . . . . . . .

- 7 7-

n i b CD a303

o r l c n 0 . . m m f . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 0 0 CI 0 0 0 0 0 0 0 0 0 0 d rl 0 0 0 0 0 0 0 0 I I I I II

Q ) J < & 3 [I] 0000000000000000000000000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c> 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 l - i l - i l - i r l r i r l l - i ~ ~ m ~ ~ ~ v ~ ~ ~ ~ m o m m o m m o ~

v)

P-!

z l "I

E

+ -P c 0 V 4

J

J

0 0 0 0 ~ 0 ~ 0 ~ f r l r l l - i f ~ 0 0 f W f h l ~ f O ~

Q[I] . . . . . . . . . . . . . . . . . . . . . . . . . . 0000000000000000000000000

..A ~ O N ~ ~ I m f ~ ~ N ~ N N ~ m ~ m m W h l W N W o ~

J

~ 0 0 0 0 0 ~ m 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 W,l b m 0 LD 0 m r- - UJ 0 Ln 0 UJ 0 m u3 u3 UJ 0 Lo 0 Ln 0 In + P O r(mmCvr4 l - i m m m h l h l r l d N m m m h l d

3

J

c

31n X ;1

m 0 I-4

I1 x

0 co CD co 0 aJ CI) CD 0 0 CD 0 OD 0 03 0 a 0 0 hl N N a3 N CI) N u3 CD N 0 N OD N 0 01 0 d f zt N r l w m f c n m f m ~ d h l m f m

I-4 0 0 0 0 0 c> 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . .

0 0 0 0 0 0 c> 0 0 0 0 0 0 0'0 0 0 0 0 0 0 0 0 0 c> 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c> 0 0 0 0 0 0 0 0 0 0 0 m m m m m 0 cr) m m f f f f f f f f f

0 m m CJ 0 r\l cu m 0 0 m N N 0 N hl m f a, o o m o cr) rr) o co o o cr) m co m m o w mODC0 f 43 (D 00 0 Q, 00 j 1-43 CD 03 CT, a, 0 0) a, 0 3 cn cn 0 0 , m a, 0, cn m Q, m o m

0 0 0 0 0 0 (3 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . .

0 CD CD a3 0 co (9 b rn b f f 0, d Ln m m 6, b a, 0, co 0743 fr) cn co (3, b fi co 0 CD CD a3 m rl rl I-4 rl rl 4 I+ li l-i d li d I-4 cu rl rl rl rt 0 0 0 0 0 0 (2 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 (3 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . .

0 0 0 N 0 ru .j. 0 00 hl OD m N f hl m f 0 Lo m m N hl 6JC-J m 3 m 04 c\1 CJ N 04 N m w

0 0 0 0 0 0 '3 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . .

a

c .?I

3 u c 0 0

21 -79 -

m 0 rl

J m m f co m m b a, m 0 rl 0 rlf N 00 0 a, CD a, i-l LO i-l Lo 0 a, (v rl co r l c o CJ v) 0 b Lo m m 0 cv (v CJ m m m m m m hi N I 3 m m m X

- I

0 co 0 co m 00 0 co 0 0 co 0 co cr) OD 0 co Lo Lo CJO cv a0 cv 0 N c- m CJ 0 CJ OD N 0 hl m c n f mcu r l C J mrfcoa, f m h i r l N m f c n

0 0 0 c3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . .

bl 0 4

a m P

0 0 0 c3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c> 0 0 0 0 0 0 0 0 0 0 0 0 0 0 rl ri d r-i d rl rl rl r i hl CJ hi cv N c\1 hi (v

( u a + J F c R l & 3

a, " F c 3 0

& 0 0

Q\ tu

0 m el cu 0 01 N m cn co m hl CJ 0 hl h i m 00 00 0 m YO a, 07 m 0 co co 0 m m QD 0 m 0 b cn a, co 3 rl f m a, a, 0, a, co f rl f UJ a, cn a, cn a, cn cn a, cn cn a, cn cn cn cn cn cn cn a, a, 0 0 0 c3 0 0 0 0 00 0000000 0 . . . . . . . . . . . . . . . . . .

(u 0 b 3 r-l rl h l f N b f 0 Q, a3 0 rl OD cr) 0 0 0 c3 0 0, cn cn a 0 0 co b * r- a3 CO cn CJ cv N c.4 (v d rl rl rl CJ hl rl rl i-l rl d rl rl 0 0 0 c> 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 c3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . .

c, 0 +J VI a c, Rl cn

J

F: 0 rl rl a d

f m (v c3 cn 0, rl f hi f (D 0 CD f CD 0 f CJ Lo m (* 0') CJ (v m m L n m m Cr) hi CJ 0 4 m m v)

0 0 0 c3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . .

Lo v) rd x

3

E l 0

Q i

rd c, a c7 J

I I

j

03

a) rl A a

E-c

J

. . . -80 -

-- , I

3 0 rl F4

c a, M 0 & +J -d 2 n

Lo

0 z k” 0 u a, c 0 +J rn a c a cn c 0 rl rl 4 rn rn a c & 0 ccr a C’ rd 0

$ d L

cn Q) rl A a E-c

‘cl E

I ii x

0 0 0 c l 0 f f ~ c n ~ ~ L o m L o a a f f f f c o c o C D ~ b c o c o c o a L o m L o c o w ~ c o c o c o c o c o c o b F F ~ . F P F c n m ~ m d r l r l r l ~ F b t - F r l r l r l ~ l r l r l r l r l r l r l r l ~ c v h l c v c v r l r l r l r l o o o c l o o o o o o o o o o o o o o o o

. . . ? I . . . . . . . . . . . . . . . . - 4 o o o c b o o o o o o o o o o o o o o o o

Lo Lo cn cr) m t- m rl m co m co (u cv b m G co a ii C D f m ~ l c v r l r l c o L o c v r l f m c v r l r l 0 0 0 0

f a . . . . . . . . . . . . . . . . . . . . o o o o o c o o o o o o o o o o o o o o

51 rl

a0 0 cv U) c v o rl f 0 0 f m CD b In F) 0 f O c v 3 0 L o m F 0 f m c o L o m c o f a 0 F ) I n o r l ~ ~ i c v ~ c v o ~ c v ~ ~ r l c v h l m ~ ~ c o c o

0 0 0 C) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . . . .

a rl *d

p c m a

-81- & I

J

J

.J

+J

c 0 0

bl 0 4

a m P

; 1 P U

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000000000000000000000000 0000000000000000000000000 d d d r l r l r i ~ r l r l r l r l r l m o m m o m o o m m o o o

J

u E x 0

m E

x 0

U Q) 00000000000000000000000c0

O O O O O c > o O O O O O O O O o O O O O o o o o o O O O O O C ~ O O O O O O O O O O O 0 O O O O O O O

N N N N N ~ ~ N N N N N N N N N N N C V N N N N N ~ ~ . . . . . . . . . . . . . . . . . . . . . . . . . m

d

d I E c, m

J

rd E 9-l

a m P

4 a m a o !

fd rl -4

c 4 r o P

3

J

d z c 3 p: - 8 2 -

I li s

0 0 0 0 0 0 0 0 0 0 0 0 0 o o o o o o o c o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 f f j f f f 3 f j f f f f

r - r - f i m m m m m m m m m b a a a m m m m m a 3 m a 3 a 3 a r - r - r - m c n m m c n d r - i d r - i b d r - i l - i d d d d d m m m m d 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . .

u a,

a m u u

o o o o o o o o c o o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

m N N m m m N N m m N m N . . . . . . . . . . . . .

d o m o l - l F o m m ~ m w m m m m m o m m m a d m o a 3 f m N d 3 N m d d f N m r - i

0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . .

m o w r l l n m a 3 a o o o d o m m l n N m f a , m f m N a l - 4 Nddr lC9C'Jddd2tmC'4N

o o o o o o o o a 3 o o o o . . . . . . . . . . . . .

Q)

I c .rl

n

0 d

0 a, z A c a 3 I3 d

d

-83-

a, Fc 0 a F: rd a, t:

I Cr)

0

x I

s c 3 Iz

o o c ~ o o c o o o o o o o o o o o o o o o o o 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o c ' o o o o o o o o o o o o o o o o o o o o r l r l ~ l r l r l ~ r i r i r l N r n C J N C J ~ C J f f f ~ f f ~

. . . . . . . . . . . . . . . . . . . . . . . 00,000000000000000000000

f c n L o r . 0 c o 0 m 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o m c n m o c n o m o o o o o o o o o o o o o o o . . . . . . . . . . . . . . . . . . . e . . . f o m m f m f m f f ~ ~ f f f f f f f f f f f

- 8 4 -

a r k 0 u+ a IJ a a

tu rl

a, -4 A a I3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d d r l r l 4 d m o m C Q o m

f w c n L n W ( r 1 0 03 m 3 c n m c o m c n c V C l m c n o m = l - c o b c o b c D m = t - m c n m O m b m b O o m b P J m o 0 0 0 d 0 0 0 0 0 d 0 0

0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . .

o o o o o o r l o o o o o 0 0 0 0 0 0 0 0 0 0 0 0

f f f f f f f f f f f f . . . . . . . . . . . .

-85-

3 0 rl r.4

Fc a, v a 5 n

3- rl =k

a c: a cn a u *#-I rl *ti m a 3 a c, c, 0

a 0 c, a a .ti rl 0 [I)

F: 0 0 r= 3 Fc 0 %

rd v a

3 0 rl EL(

n

m rl

a, rl A a I3

0, a m 0 CD o m 0 . . c\I v) Loo

II II I I II

c d rd 4 a, a r:

X m

li X I

o w o m o w c m c a o m o c \ I w m b N a m L o @ J u l w d f ~ r l 0 f c J r l O f C u l - l

r l 0 0 0 4 0 0 0 I - i 0 0 0 . . . . . . . . . . . .

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

p c u l L o L o v ) L o o o o o ~ m L o m *I r l l - l r l r l l - l l - i r l r l

3 b m m v ) c D m 3 m m r r ) m l - i a m L o f m ( v f f b ? l 3 L o m d a m N a m c D a m 0 0 N 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . .

0 0 r - L o m ~ O ( v m r l f a f m O O L o ( v d m m m m h l

pc 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . .

.I

z O1

J

3

J

J

I

- 8 6 -

d tc Fc Q) c, id 3

0 z

z 0 u a E: a m a u *ti d -rl m

a c, c, C

a Q) c, a a

0 . 4

rl 0 m E: 0 u E: 5 k 0 Ict

a +J a a 3 0

f r i

Q) d c1 a E-c

O O O O O O O O O O Q O O O O O O O O O O O O O O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o a o o ~ L o m L o ~ m ~ a ~ ~ ~ ~ ~ 0 0 0 0 0 0 0 0 0 ~ 0 0

Lo r l r l r l l - i r l c v c v m w w m c v

, cv

co o m 0 a3 a,- 0307 L n f O r n a , f r t O 00 m O c v ~ N ~ b O O L o Q m c v c - ~ c - f ~ m c o b m a , ~ ~ m c o 0 3 m W m c v f m 0 a m m c v m O c v N m f m ~ c v l ~ C u 0 0 0 0 0 b 0 0 0 0 0 f 0 0 a 0-0 0 0 0 0 0 0110 0 . . . . . . . . . . . . . . . . . . . . . . . . . 0000000000000000000000000

0 W a , m ~ 0 3 m a , o o o ~ ~ c v ~ ~ ~ c v o o b o ~ ~ F c o c o ~ . m ~ ~ ( D c o r n f ~ ~ ~ ~ ~ O m c o o m ~ a , f ~ ~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 a . . . . . . . . . . . . . . . . . . . . . . . . . 0000000000000000000000000

-87-

0 z

7 CG

f F : d

3 0 rl r.4 & a, c, a 3

a c a G a, M

c, .rl z

E

h

CD rl

0 z a, h 0 u a

cn rd 0

*rl d *r l cn a

c, c, 0

a a, c, a a *ti d 0 v) F: 0 0 c 3 s4 0 (H

a c1 a 0

2

5

I+

L4

m rl

Q) d A a

€+

EmE 0 0

0000000000000 0000000000000 0000000000000 m m m m t w m m m m m m m m

rn Q q 0

I

3 4 g 0

m d a o m C D m m 3 ~ m m b d m m m m m m m 3 m N m 3 0 0 0 0 0 0 0 0 0 0 0 0 0

O O O O O O O O O O O O O . . . . . . . . . . . . .

0

N ..

. bU0"I *

Ea0Y s: I

-88-

30 ri b-l

ri 0 c a +J u 0

I hl

n

c- r i

0 z a Fc 0 0

a c a cn a 0

*rl ri -rl cn a

c, CJ 0

a a P a a -rl rl 0 (I) c 0 0 c 3 SI 0 +I

a c, fd

3

n

8 T I F4

LD ri

QJ d A a w

u a,

M

3,12

2 Lo Lo

c *d

c a al a

k c5

W co co W W W ~ O f r i f 0 ~ m 0 f r i f 0 0 0 0 f r i ~ 0 O m r n o c o o o t - ~ o o c o o o c o w ~ o L D o ~ r l 0 N r i 0 ~ ~ 0 0 N r i 0 r i N ~ ~ c l r i O d N ~ . . . . . . . . . . . . . . . . . . . . .

r l r l

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000000000000000000000 m L O m m m m m 0 0 0 0 0 0 0 0 0 0 0 0 0 0

r l r idr l r idr lc l@lCJNhlNN 2 -1

u u a \

bc

m ~ ~ r n N o ~ m ~ O ~ N m r i L O t - f ~ t - ~ O c o r i ~ f W O C D ~ t - ~ t - ~ ~ t - O f L O L o ~ ~ r n O r n d f ~ f d d ~ r i r n t - O ~ C J N O c - m N N C J 000000000000000000000

000000000000000000000 . . . . . . . . . . . . . . . . . . . . .

t - 0 0 0 m 0 0 0 m ~ Q 0 0 0 0 0 ~ 0 0 ~ 0 0 N ~ ~ m 0 0 ~ ~ 0 0 f N N c l 0 ~ c - h l

04 ~ r l ~ ~ 0 0 f ~ d 0 ~ ~ ~ ~ W ~ N ~ 0 ~ ~ Q . . . . . . . . . . . . . . . . . . . . .

000000000000000000000

- 89 -

9.2 Core Data

-90-

J

.d

i

J

J

.J

J

J

3

J

4

N l l

c-4 N

ri rl

N c4

0 rl

CJ N

N hl

o m m o C I ( D m D I

L D N O O O a,

. . . . . a

N N

C3 W

3 m

cv

m N

c 0

*rl f

hl CJ

+, .rf [I) 0

0 0

m N N

N I

r i I

N

rl I

c-4 cn

- 9 1 -

J

9.2- 1-3 D e s c r i p t i o n : The Mass i l lon sands tone i s found

i n Holmes County, Ohio, and h a s been e x t e n s i v e l y q u a r r i e d

for a v a r i e t y of purposes . The s t o n e is medium-to-coarse .J

gra ined and i s composed m s t l y of q u a r t z g r a i n s , which are

subangular t o subrounded i n shape. The cementing materials

are i r o n o x i d e , c l a y m i n e r a l s , and secondary s i l ica . d'

Tests made for t h e Briar Hill Stone Company show t h a t

t h e s t o n e h a s a n a b s o r p t i o n of water of approximate ly 6 % b y

weight .

5000 p s i being abou t ave rage .

.J

The c r u s h i n g s t r e n g t h r a n g e s from 4000 t o 6000 p s i ,

The s t o n e weighs 135 pounds

p e r c u b i c foot . The average p e r m e a b i l i t y of t h e samples J

used i n t h i s s t u d y w a s 8 0 0 md, and t h e average p o r o s i t y about

2 2 % .

9 .2- 2 Uncons 'ol idated O t t a w a S i l i c a

9 . 2- 2- 1 P h y s i c a l P r o p e r t i e s

Core No. 1 4 1 5 1 6 1 7 1

Length, c m 5 .696 5.590 5.657 5 .894 J

Cross- Sec t iona l Area, c m 5 .067 5.067 5.067 5 . 0 6 7

mesh 1 mesh) m e s h ) mesh) Average Grain S i z e , mm ( 3 5- 4 5 (80- 100 (80- 100 (80- 100

3

9 .2 .2- 2 M i n e r a l o g i c a l Composition

S i l i c o n d i o x i d e ( S i O , )

Aluminum ox ide (A1203) I r o n oxide ( F e 2 0 3 )

T i tan ium d i o x i d e ( T i 0 2 ) Calcium oxide ( C a O )

Magnesium oxide (MgO)

Loss on I g n i t i o n (LO11

Weight %

99.806 0.047

0 .019

0.018 <o. 01 <0,01

0.09

3

4

- 92-

J

9.2- 2- 3 Desc r ip t ion : The O t t a w a sand used i n

t h i s s t u d y i s t h e No. 1 7 s i l i c a g rade and i s commercial ly

a v a i l a b l e from t h e O t t a w a D i v i s i o n , P a r k e r I n d u s t r i a l and

Foundry Supply , Rurlingame, C a l i f o r n i a . It i s more t h a n 9 9 %

weight q u a r t z and t h e g r a i n s are f a i r l y w e l l rounded. The

d i s t r i b u t i o n of g r a i n s i z e s i s as fo l lows:

U.S. Sieve N o .

40

50

7 0

1 0 0

1 4 0

2 0 0

2 7 0

Opening, !li l l imeter

0.420

0 .297

0 .210

0 . 1 4 9

0.105

0 .074

0 . 0 5 3

P e r c e n t Weight Re ta ined

8 . 1

44.5

28 .8

12 .7

4.4

1.1

0.2

The sand w a s s i e v e d and o n l y un i formly s i z e d g r a i n s w e m

used t o p r e p a r e a core f o r t h i s s tudy .

- 93 -

9.3 D e r i v a t i o n of Equat ions

-94 -

9 .3- 1 C a p i l l a r y Tube Viscometer: Because l i q u i d v i s -

c o s i t y d a t a ob ta ined from s u p p l i e r s are ave rage p roduc t q u a n t i -

t i e s and t h o s e from c o n v e n t i o n a l r e f e r e n c e s were n o t given at

high working t e m p e r a t u r e s , it was n e c e s s a r y t o de t e rmine v i s -

c o s i t y ve r sus t e m p e r a t u r e a t o p e r a t i n g pressures.

A c a p i l l a r y t ube v i scome te r , whose o v e r a l l l e n g t h w a s

62.875 inches and whose i n t e r n a l d i ame te r was 0.033 i n c h e s ,

w a s c o n s t r u c t e d f rom a 316

c o e f f i c i e n t of t h e r m a l expans ion w a s 8 .9 x

s t a i n l e s s s t e e l tube . The l i n e a r

in/in-OF.

The laminar f low o f l i q u i d s th rough c i r c u l a r c o n d u i t s

obeys P o i s e u i l l e ' s l a w :

4 rr Ap ' = 8pR

where q = f l o w r a t e , cc/sec; r = i n s i d e r a d i u s , cm; Ap =

p r e s s u r e drop , dynes/cm 2 ; ?J = l i q u i d v i s c o s i t y , p o i s e s , 2 =

l e n g t h of c a p i l l a r y , c m .

For r and R i n i n c h e s , Ap i n p s i , and i n cp u n i t s , Eq.

9- 1 becones:

and

4 7 r A 2 RP

q = 4 . 4 3 7 x 1 0

?J = 0 . 5 2 3 0 !& 9

( 9 - 2 )

(9-3)

Knowing t h e a c t u a l v a l u e of v i s c o s i t y a t 7OoF, and f r o m

measurements of p and q , t h e m u l t i p l y i n g c o n s t a n t w a s found

t o be 0.5113. The re fo re :

= 0 . 0 5 1 1 3 % ( 9 - 4 ) '70

-95-

J p, Ap (l+--) b

T h e r e f o r e , a graph of Y = Dm vs . x a (1+- b 1 ; & Pa 9, prn

shou ld y i e l d a s t r a i g h t l i n e of s lope rj and i n t e r c e p t E 1 , a

d

provided t h a t t h e va lue of b is known.

J

J

3

J

J

J

J

J

Eq . 9-10 was t h e working equa t ion for t h i s a p p a r a t u s .

The r e s u l t s a g r e e c l o s e l y w i t h t h o s e found i n t h e I n t e r n a t i o n a l

C r i t i c a l Tab le s . 3 9

9 . 3 - 2 V i s c o - I n e r t i a l Flow of G a s : The q u a d r a t i c equatbh

as proposed by Forchheimer and modified by C o r n e l l and Katz

(see re f s . 1 0 and 11) i s :

(9-11) 2 -8 = avq + BPq

for t h e case of v i s c o - i n e r t i a l flow o f gas . For v i s c o u s flea,

t h e q u a d r a t i c t e r m o f 9- 1 1 i s small enough t o be n e g l e c t e d ,

and a h a s t o be t h e r e c i p r o c a l o f p e r m e a b i l i t y , i . e . , Q = l / k .

The i n t e g r a t e d form of E q . 9- 1 1 f o r h o r i z o n t a l l i n e a r

flow i n t h e absence of s l i p p a g e i s :

1 b ' When t h e r e i s gas s l i p p a g e , a becomes

_I k (l+:) and s i m p l i f y i n g , b

Pm k(l+-

Replac ing a by

or Y = E + fix

- 97 -

A t t e m p e r a t u r e s , T , o t h e r t h a n room t e m p e r a t u r e , t h e

l e n g t h of t h e t u b e i s :

and

o r

rT 4 = r 7 0 ‘[l + B ( ~ - 7 0 1 1 ~

Because B , t h e l i n e a r c o e f f i c i e n t of t h e r m a l expans ion , i s

small, Eq. 9-7 can be approximated as:

4 4

‘T ‘70 ’T - = - r70 6 + 3 B ( T - 7 0 1 7

For t h e a p p a r a t u s :

= 0,05113 l l + 2 . 6 7 x (T-70)3 UT Q

There fo re , t h e v i s c o s i t y a t any t empera tu re is :

Where : aT = 0.05113ll t 2 . 6 7 x 10’’ ( T - 7 0 1 1

(9 -5 )

(9-6)

( 9 - 7 )

(9-9)

(9 -10)

-96-

9 . 4 L i s t of Manufacturers

- 9 9 -

LIST OF YANUFACTURERS

The fo l lowing i s a l i s t of t h e v a r i o u s p i e c e s of equip-

ment or s u p p l i e s t h a t were used i n t h i s work, t o g e t h e r w i t h

t h e corresponding names of manufac ture rs and /o r s u p p l i e r s

from whom t h e y were a c q u i r e d .

Chevron White Minera l O i l N o . 3 - Van Waters E Rogers, I n c . San Franc isco .

P re s su re Transducers - Dynasciences Corp. , Model KP 1 5 , c/o Gaco Inst rument S a l e s , 655 Castro S t r e e t , S u i t e 2 , Mountain V i e w , C a . , 9 4 0 4 0 ( 9 6 1 - 2 2 2 2 ) .

B a r n e t t I n d u s t r i a l Dead Weight Tester - c/o Gaco Ins t rumen t Sales, 655 C a s t r o S t r e e t , S u i t e 2 , Mountain V i e w , C a . , 94040 ( 9 6 1 - 2 2 2 2 ) .

P r e s s u r e I n d i c a t o r - Pace Node1 CD 25, c / o Gaco Ins t rumen t Sales, 655 Castro S t r e e t , S u i t e 2 , Mountain V i e w , C a . , 9 4 0 4 0 ( 9 6 1 - 2 2 2 2 ) .

P r e s su re Regu la to r - Volumet r ics , Model VRC- 400 , 1 0 2 5 Arbor Vitae S t r e e t , Inglewood, C a . , 90301 ( 2 1 3 ) 6 4 1- 3 7 4 7 , o r c /o Gaco Ins t rument S a l e s , 6 5 5 Castro S t r e e t , S u i t e 2 , Foun ta in V i e w , C a . , 94040 ( 9 6 1 - 2 2 2 2 ) .

High P re s su re Regu la to r - Model 4-580, Matheson Gas P r o d u c t s , Newark, C a . (793-2559).

Va r i ab l e R a t e O i l Pump - Model PC, Whitey Tool E D i e Company, 3 6 7 9 Landregan Emvl., Oakland, C a , (653-51001, loan from USBM.

Hydraul ic Hand Pump - Enerpac, Model P-39, Pau l Monroc Hydrau l i c s , I nc . , 1 5 7 0 G i l b r e t h Road, Burlinganre, C a . , 9 4 0 1 0 (697-2950).

Constant R a t e Pump - Hand-made e q u i v a l e n t t o Ruska 2 2 0 0 series. Uses Viton 0 r i n g w i t h machined t e f l o n back -up . r ing .

Accumulators ( G r e e r o l a t o r ) , Model No. 20- 30 TMR-S-% WS, Hy- d r a u l i c C o n t r o l s , I nc . , 1330 6th S t . , Emervvi l l e , C a . , 94608 ( 6 5 8- 8 3 0 0 ) .

Recording Po ten t iome te r (for t empera tu re ) 3- Model Speedo Max v, Leeds E Northrup, 1 0 9 5 Market S t . , S a n Francisco, C a , (349-6656).

-100-

Thermocouples - I r o n- C o n s t a n t a n , Conax, c /o I n s t r u m e n t Lab- o r a t o r y , 6 4 4 Emerson S t . , P a l o A l t o , C a . , 94303 (328-1040) .

Temperature C o n t r o l l e r - API Model 228 with 4010 Power P a c k , A P I I n s t r u m e n t s , 2339 Charleston Rd . , Mountain V i e w , Ca., 94040 (964-0512) .

L a b o r a t o r y F lowra tor K i t - Model No. 10A3565ALK2, F i s h e r E P o r t e r C o . , 1 3 4 1 North Main S t . , Walnut Creek, C a . , 94596 (933-8880) .

Core S l e e v e - V i t o n A t u b i n g , \?‘est American Rubber Co., 750 North Main S t . , Orange , Ca., 92668 (714-532-3355)-

0 !ings - Viton A w i t h t e f l o n back- up r i n g s , ABSCOA I n d u s- t r i e s , 880 B u r l i n g a m e Ave . , Redwood C i t y , C a . (369-4897) OR 1 0 7 1 W. Arbor Vitae S t . , Znglewood, C a . , (213- 7764561).

T u b i n - T u b e s a l e s , 500 Sansome S t . , S a n F r a n c i s c o , Ca. d l 9 1 9 ) .

F i t t i n g s E Valves - Van Dyke E F i t t i n g C o . , 5525 M a r s h a l St., O a k l a n d , C a . (658-1700) .

Gas Analyzer - Gas Master, Laboratory Model, GOW-MAC Ins t r tk - ment C o . , 1 0 0 Kings F.d., Mad i son , N . J . 07940 (201-377-34501 o r Appl ied Ins t rument C o . , 1 9 9 1s t S t r e e t , Los A l t o s (941-592k3).

Recording P o t e n t i o m e t e r ( f o r p r e s s u r e ) - Two pen m u l t i - r a n g e recorder , L e e d s and N o r t h r u p , BD-9, P.O. Box 634 , B e l m n t , C a . (593-8392) .

Alundum Core - N o r t o n Co., I C D , 2555 L a y a f e t t e S t . , S a n t a Clara, C a . 95050 (234-7710) .

Lapp P u l s a f e e d e r Pump - Model LS-20, 316SS, 1 / 6 hp, 94 Natoma S t . , S a n F r a n c i s c o , Ca. 94105 (391-7650) .

O t t a w a S i l i c a , No. 1 7 S i l i c a , P a r k e r I n d u s t r i a l and Foundry S u p p l y , 1 8 8 1 R o l l i n g Rd. , Burlingame, C a . , 94010 (697-8865) .

Massillon S a n d s t o n e - The Briar H i l l S t o n e C o . , Glenrnont , Ohio, 44628 (216-2764011) .

-101-

10. NOMENCLATURE

E n g l i s h

A = area, c m 2

k = p e r m e a b i l i t y , md

L = l e n g t h of core , c m

p = p r e s s u r e , p s i

q = flow r a t e , cc/sec

w = flow r a t e , gm/sec

T = t e m p e r a t u r e , 0 K or OF

r = r a d i u s , c m

v = flow v e l o c i t y , cm/sec

G r e e k

p = d e n s i t y , gm/cc

u = v i s c o s i t y , c p

4 = p o r o s i t y , f r a c t i o n

A = i n c r e m e n t

Subscr i D t s

a = a t m o s p h e r i c c o n d i t i o n

c = c o r e or c o n f i n i n g

m = mean v a l u e

T = t e m p e r a t u r e

-102-