Storage of Fresh Water in Saline Aquifers Using a Well Field.

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LSU Historical Dissertations and Theses Graduate School

1974

Storage of Fresh Water in Saline Aquifers Using aWell Field.Walter Richard WhiteheadLouisiana State University and Agricultural & Mechanical College

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75-1965

WHITEHEAD, Walter Richard, 1937- STORAGE OF FRESH WATER IN SALINE AQUIFERS USING A WELL FIELD.

The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1974 Engineering, civi l

Xerox University Microfilms , Ann Arbor, Michigan 48106

STORAGE OF FRESH WATER IN SALINE AQUIFERS USING A WELL FIELD

A D i s s e r t a t i o n

Submitted to the Graduate Facul ty of the Louisiana Sta te U n iv e r s i t y and

A g r i c u l t u r a l and Mechanical Col lege in p a r t i a l f u l f i l l m e n t of the

requirements f o r the degree of Doctor o f Phi losophy

1 n

The Department o f C i v i l Engineering

byWal ter R. Whitehead

B . S . , U n iv e r s i t y of Southwestern Lou is ia na , 1960 M .S . , Louisiana St a te U n i v e r s i t y , 1964

August, 1974

ACKNOWLEDGMENTS

The author is indebted to Professor Raphael G. Kaz-

mann, Professor of C i v i l Engineer ing, and Dr. Oscar K.

Kimbler , Professor of Petroleum Engineer ing, under whose

guidance and superv is ion th i s work was accomplished.

The author would l i k e to express his g r a t i t u d e to

Messrs. Fred Dedon, Thomas P a i n t e r , Jamie Lewis, Conrad

Pearson, and many others who helped 1n the con st r uc t ion of

the exper imental equipment. To Mr. Tony Owens, who spent

many s leepless nights c o l l e c t i n g exper imental da ta , a

spec ia l note of thanks is due. The author extends his s i n

cere a pp re c i a t i o n to Mary H. A l s t o n , who typed th i s manu

s c r i p t , and to Norma B. Du f fy , who d r a f te d the f i g u r e s .

F inanc ia l support f o r th i s I n v e s t i g a t i o n was through

funds made a v a i l a b l e by the O f f i c e of Water Resources Re

search, U.S. Department of the I n t e r i o r , under P.L. 88 -279 ,

and administered by the Louisiana Water Resources Research

I n s t i t u t e as p r o j e c t A-022-LA.

This d i s s e r t a t i o n is dedicated to my w i f e , M a r j o r i e ,

whose as s is ta n ce , encouragement and s a c r i f i c e s are noted

wi th the g r e a te s t measure of g r a t i t u d e and a f f e c t i o n .

1 i

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ...................................................................................... i i

LIST OF T A B L E S ...................................................................................... v

LIST OF FIGURES...................................................................................... v1

A B S T R A C T ........................................................................................................v l i i

ChapterI . INTRODUCTION ......................................................... 1

I I . DISCUSSION OF THEORY ................................................ 6

General ................................................................................. 6Mixing Due to Molecular D i f f u s i o n and

Convect ive Dispersion ........................................... 7G r a v i t a t i o n a l Segregat ion Due to Densi ty

D i f f e r e n c e ........................................................................ 13

I I I . DEVELOPMENT OF COMPUTATIONAL PROCEDURE FORCALCULATING RECOVERY EFFICIENCIES ................... 18

General ................................................................................. 18C a l c u l a t i o n of Mixing Due to Molecular

D i f f u s i o n and Longi tudina l Convect iveD i s p e r s i o n ........................................................................ 18

C a l c u l a t i o n of G r a v i t a t i o n a l Segregat ionDue to Densi ty D i f f e r e n c e .................................. 20

C a l c u l a t i o n of Recovery E f f i c i e n c y ................... 27M u l t i p l e Well S y s t e m s ................................................ 29Computer Programs f o r Computing Recovery

E f f i c i e n c i e s ................................................................... 31

IV. EXPERIMENTAL PROCEDURE AND RESULTS .............................. 32

General ................................................................................. 32De scr ip t ion of the M i n i a q u i f e r ............................. 33The E f f e c t of B o u n d a r i e s ........................................... 35Fluids Used 1n E x p e r i m e n t s ...................................... 38Determinat ion of M1niaqu1fer P e r m e a b i l i t y . 45Determinat ion of M in i a q u i f e r Homogeneity

and P o r o s i t y ................................................................... 48

i i i

Determinat ion of the Longi tudina l D1s- p e r s i v l t y C o e f f i c i e n t f o r theM i n i a q u i f e r ................................................................... 51

Comparison o f Experimental and PredictedR e s u l t s ............................................................................ 53

V. THE ECONOMICS OF FRESH-WATER STORAGE INSALINE AQUIFERS .............................................................. 57

General ................................................................................. 57Storage 1n Steel Tanks ................................................ 58Storage 1n Sa l in e Aq ui fe r ...................................... 59Comparison of the Two Storage Methods . . . 65

V I . CONCLUSIONS AND RECOMMENDATIONS ............................. 66

Conclusions ....................................................................... 66Recommendations .............................................................. 67

NOMENCLATURE .......................................................................................... 68

BIBLIOGRAPHY .......................................................................................... 70

APPENDICES .......................................................................................... 73

V I T A ............................................................................................................. 141

1 v

LIST OF TABLES

Table Page

4.1 Prope r t ies of Pure Fluids Used 1nExperimental R u n s .......................................................... 40

4 . 2 Comparison of Observed and Predic ted RecoveryE f f i c i e n c i e s fo r S ingle Well Operat ion . . . 54

4 .3 Comparison of Observed and Predic ted RecoveryE f f i c i e n c i e s f o r M u l t i p l e Well Operat ion . . 55

5.1 Storage Aq ui fe r C h a r a c t e r i s t i c s .................................. 59

5 .2 Ca p i ta l Costs of I n s t a l l i n g Well F ie ld . . . . 64

5 .3 Summary of Annual Costs f o r Tank Storage andSa l ine Aq ui f e r Storage ............................................... 65

v

LIST OF FIGURES

Figure Page

2.1 D1mens1onless C o r r e l a t i o n Used to ComputeG r a v i t a t i o n a l Segregat ion ...................................... 17

3.1 Schematic Representat ion of the DisplacementProcess During an I n j e c t i o n H a l f - C y c l e . . 19

3 .2 Schematic Representat ion of the DisplacementProcess During an I n j e c t i o n H a l f - C y c l e . . 21

3 .3 Schematic Representat ion to I l l u s t r a t e theC a l c u l a t i o n of G r a v i t a t i o n a l Segregat ion Before Approximation to Radial Geometry . . 23

3 .4 Schematic Representat ion to I l l u s t r a t e theApproximation of G r a v i t a t i o n a l Segregat ion Ca lc u l a t i on s to Radial Geometry ........................ 26

3.5 Schematic Diagram to I l l u s t r a t e the Calcul a t i o n of Recovery E f f i c i e n c y ............................. 28

3 .6 Some Possible Well F i e ld P a t t e r n s ...............................30

4.1 System Represented by M1n1aqu1fer Used 1nThis S t u d y ..............................................................................34

4 .2 View of M l n i a q u i f e r , Pumps, and I n s t r u mentat ion .................................................................................. 36

4 .3 View of Three Large-Bar re l and Three Smal l -Barre l Pumps Used 1n Experiments ................... 37

4 .4 Comparison of Frontal Pos i t ions f o r aF i n i t e System and an I n f i n i t e System . . . 39

4 . 5 View of Capaci tance Cel l Used to DetectConcent rat ion Changes ........................................... 42

4 .6 P l o t of Recorder R e f l e c t i o n Versus Concent r a t i o n of Nat ive F lu id 1n ProducedS t r e a m ....................................................................... 43

4 .7 Schematic o f Experimental Apparatus .................... 44

v 1

4 . 8 P l o t of Pressure Versus Radius Used toDetermine M i n i a q u i f e r P e r m e a b i l i ty . . . . 47

4 . 9 Photograph Showing the C i r c u l a r Advance ofthe In j e c t e d F lu id Front ...................................... 49

4 .10 P l o t of Volume In j e c t e d Versus Area Sweptf o r Run No. 1 .........................................................................50

4.11 Comparisons of Computed and Observed Concent r a t i o n P r o f i l e s Used 1n Computing the Longi tudinal D1spers1vi ty ...................................... 52

5.1 Assumed Well F i e ld Conf igu ra t ion f o r CostC o m p a r i s o n ............................................................................. 60

v i l

ABSTRACT

The computat ional procedure presented 1n th is d i s s e r

t a t i o n should enable the p r a c t i c i n g engineer to design wel l

f i e l d s f o r the storage of f resh water 1n h o r i z o n t a l s a l i n e

a qu i fe rs in which there is no p r e - e x i s t i n g ground-water

movement. The recovery e f f i c i e n c y of the I n j e c t i o n / s t o r a g e /

r e t r i e v a l process can now be r e l i a b l y computed, thus making

possible an economic ana lysis of the process 1n any s p e c i

f i e d a rea . An economic comparison of the storage of ap

proximate ly one b i l l i o n gal lons In a s a l i n e a q u i f e r th a t

un der l ies the New Orleans area was made wi th the present

most f e a s i b l e a l t e r n a t e - - s t e e l tanks. The r e s u l t s favored

the s a l i n e a q u i f e r storage p r o j e c t by a f a c t o r of more

than 50 to 1 .

The v a l i d i t y of the computat ional procedure was de

termined by comparing recovery e f f i c i e n c i e s obtained from

a l a b o r a t o r y - s i z e m in ia tu re a q u i f e r ( m l n i a q u i f e r ) wi th r e

covery e f f i c i e n c i e s predic ted by the computat ional pro

cedure. The computat ional procedure predic ted the e x p e r i

mental data w i t h i n 10 percent f o r m u l t i p l e we l l systems.

The pred ic ted recovery e f f i c i e n c i e s were i n v a r i a b l y lower

than the ex pe r i m e n ta l l y determined recovery e f f i c i e n c i e s .

The a q u i f e r parameters th a t must be determined before

v i i i

the computat ional procedure can be used are th ickness ,

p e r m e a b i l i t y , p o r o s i t y , s t o r a t i v i t y , l o n g i t u d i n a l d i s p e r -

s i v i t y c o e f f i c i e n t , and v i s c o s i t y and dens i ty of the na t i v e

f l u i d . Of the parameters mentioned above the l o n g i t u d i n a l

d i s p e r s i v i t y c o e f f i c i e n t is the most d i f f i c u l t to ob ta in .

The procedure used in th is i n v e s t i g a t i o n to determine the

l o n g i t u d i n a l d i s p e r s i v i t y c o e f f i c i e n t f o r the m i n i a q u i f e r

can be r e a d i l y adapted to f i e l d use.

In a d d i t io n to the a q u i f e r parameters, the wel l f i e l d

c o n f i g u r a t i o n , the opera t ion schedule of the f i e l d , the

volume of f resh water to be i n j e c t e d , i n j e c t i o n r a t e s ,

probable dura t ion of s torage , product ion r a t e s , and f r a c

ture pressure of the upper conf in ing bed must be known in

order to make an economic a na l y s i s .

When f resh water is i n j e c t e d i n t o a h o r i z o n t a l ,

homogeneous, s a l i n e a q u i f e r which has no p r e - e x i s t i n g

ground-water movement, the two most impor tant fa c to rs which

determine the amount of usable water t h a t can be recovered

are: (1 ) mixing of the two f l u i d s due to molecular d i f f u

sion and convect ive d i s p e rs i o n , and (2) g r a v i t a t i o n a l seg

reg at io n of the two f l u i d s due to den s i ty d i f f e r e n c e .

CHAPTER I

INTRODUCTION

One of the most important fa c t o rs f o s t e r i n g the con

t inued existence and growth of any community is an adequate

and r e l i a b l e supply of f resh water t h a t is a v a i l a b l e on a

d a i l y b a s i s , each and every day of the y e a r . Many centers

of populat ion have to depend on sur face r u n - o f f f o r t h e i r

primary water supply, and since the a v a i l a b l e q u a n t i ty of

sur face r u n - o f f is sub ject to wide v a r i a t i o n , such popula

t io n centers must b u i l d r es e rv o i rs to s t ore surplus f resh

water in times of p lenty f o r use in times of low r u n - o f f or

actual drought.

Rivers or streams t h a t are sub jec t to p o l l u t i o n by

chemical s p i l l s , or na tura l p o l l u t i o n such as s a l t water

i n t r u s i o n from the ocean, are the sources of water fo r many

important communit ies. Here, again , there 1s need to store

an adequate supply of potable f resh water .

The storage of water in times of p len ty f o r use in

times of s c a r c i t y 1s not a no v e l ty . In many areas of the

world the topography 1s s u i t a b l e f o r the const ruc t ion of

dams and the c r e a t i o n of r e s e r v o i r s . However, in o ther

areas of the v o r l d , such as the coastal zones, the land 1s

too f l a t fo r the construc t ion of dams and the c re a t i on of

2

Impoundments, al though f resh water may be a v a i l a b l e 1n s u r

plus during c e r t a i n times of the y e a r . Also, 1n many I n

d u s t r i a l i z e d areas , al though the topography may be s u i t a b l e ,

the land may have been preempted f o r use by i n d u s t r y , com

merce, or residences and the problem of water storage s i t e s

ar ise s in a s l i g h t l y d i f f e r e n t guise. Nonetheless, water

must be stored where water sources are I n t e r m i t t e n t , as a

water supply t h a t 1s not a v a i l a b l e on a d a i l y b a s is , each

and every day, w i l l not support a popula t ion .

The usual answers to the problem of s t or in g surplus

water where p o t e n t i a l r e s e r v o i r s i t e s are not r e a d i l y

a v a i l a b l e have been:

1. The const ruc t ion of r e s e r v o i r s a t some distance

from the po int of use and conveying the water

from the impoundments to the populated areas

through la rg e pipe l i n e s .

2. In areas of f l a t topography, the c on st r uc t ion of

r e s e r v o i r s w i t h i n r ing levees .

3. The const ruc t ion of s torage tanks.

Some of the disadvantages of the f i r s t two methods

( 1 n a d d i t io n to i n i t i a l con st r uc t ion costs) a re : (1 ) sus

c e p t i b i l i t y to p o l l u t i o n , (2 ) water loss through evapora

t io n and/or seepage, and (3 ) the temporary or permanent

submergence of la rge areas of land t h a t might be used f o r

other purposes. Surface storage in tanks e l im in a te s the

p o l l u t i o n and w ate r - l os s problems but when very large v o l

umes are involved the cost of th is method of storage is

3

usu a l ly p r o h i b i t i v e . Based on 1973 f i g u r e s , i t is e s t i

mated t h a t the cost of tank storage in a ty p i c a l urbanized

area might be ten cents per ga l lon of storage c a p a c i t y , in

ad d i t io n to the cost of land (Klmbler et a l . , 1973) .

Many of the areas where sur face storage is a problem

are under la in by s a l i n e aqu i fe rs a t depths as shal low as

500 f e e t (Kohout, 1970) . As an a l t e r n a t e to sur face s t o r

age o f f resh w a te r , many authors have suggested underground

storage 1n these s a l i n e aqu i fe rs (Cederstrom, 1947; Moulder

and Fr a zo r , 1957; Esmail and Klmbler , 1967; Green and Cox,

1968; Kimbler , 1970; Moulder, 1970; Kumar and Kimbler ,

1970; Kimbler et a l . , 1973) . In b r i e f , the storage of

potable water 1n s a l i n e aqu i fe rs can be descr ibed as f o l -

1 ows :

1. When a surplus of water e x i s ts i t 1s In j e c t e d

i n t o a s u i t a b l e s a l i n e a q u i f e r , misc ib ly d i s

p lac ing the na t i v e s a l t water away from the 1n-

j e c t i o n w e l 1.

2. A f t e r the desi red q u a n t i t y has been I n j e c t e d ,

the stored water is al lowed to stand u n t i l

needed.

3. The stored water is produced through the same

well used f o r i n j e c t i o n u n t i l e i t h e r the s a l i n i t y

of the produced stream reaches an u n s a t is f a c t o r y

l eve l or u n t i l the demand is met.

These three steps c o n s t i t u t e what is c a l l e d one cyc le .

When there is again a surplus of potable water the process

4

1s repeated. In step 3 i t was noted t h a t product ion was

stopped when the s a l i n i t y of the produced stream reached

an u n s a t i s f a c t o r y l e v e l . T e c h n i c a l l y , the term "break

through" 1s used to descr ibe th i s c o n d i t i o n . Throughout

th i s i n v e s t i g a t i o n the term breakthrough is used to de

scr ibe the con di t ion when 3 percent of the produced stream

consists of na t i v e f l u i d . The r a t i o of the volume of

usable water recovered to the t o t a l volume I n j e c t e d is de

f ined as "recovery e f f i c i e n c y . " Several f i e l d tes ts of the

process have been or are 1n the process of being conducted

(Cederstrom, 1947; Moulder and Fr azor , 1957; Green and Cox,

1968; T i b b a l s , 1970; Kimbler , 1971; Brown and S l l v e y ,

1973) . Although the publ ished data from these t e s t s are

r a t h e r l i m i t e d I t appears t h a t the process is f e a s i b l e .

A few I n v e s t i g a t o r s (Kumar and Klmbler , 1970; Gelhar

et al . , 1972) have developed mathematical models f o r pre

d i c t i n g the recovery e f f i c i e n c y of the underground storage

technique. A l l of these models have been f o r a s i n g l e -

w e l 1 , r a d i a l system.

In a f i e l d a p p l i c a t i o n of the process, however, I t 1s

u n l i k e l y t h a t a s in g le wel l w i l l s u f f i c e when the quest ion

of del 1v e r a b i 1i t y Is considered. Although the surplus

water might be recharged at a slow r a t e over a long period

of time (4 to 6 months) , the water demand 1s l i k e l y to be

high over a shor t er I n t e r v a l of t ime (2 to 3 months). So

i t 1s l i k e l y th a t a number of we l ls would have to be i n

s t a l l e d to improve water del 1v e r a b i 1i t y . In f a c t , should a

5

number o ' such wel l f i e l d s be es tab l ished as pa r t of a

municipal water supply system, they might be used to supply

the sh o r t - te rm peak demands as wel l as long- term water sup

ply when the sur face source 1s tem por ar i ly shut down.

In t h is I n v e s t i g a t i o n a mathematical model to p r e d i c t

the recovery e f f i c i e n c y o f a m u l t i p l e wel l system has been

developed. The v a l i d i t y of the mathematical model was

v e r i f i e d under a wide v a r i e t y of parameters using e x p e r i

mental r e s u l t s obtained from a la bo ra to ry s ize min i a tur e

a q u i f e r ( m i n i a q u i f e r ) . The computat ional procedure used in

the mathematical model is an extension o f th a t used by

Kumar (1 9 6 8 ) .

Since t h i s i n v e s t i g a t i o n 1s concerned wi th a mlsc ib le

displacement process 1n a porous medium, the r e s u l t s are

d i r e c t l y a p p l i c a b l e to other mlsc ib le displacement pro

cesses in porous media such as the underground disposal of

water so luble wastes, the leaching of ores, the storage of

nat ur a l gas in aqu i fe rs when an I n e r t cushion gas is used,

secondary and t e r t i a r y recovery of petroleum, and the

c y c l i ng of re t rograde gas-condensate r e s e r v o i r s .

CHAPTER I I

DISCUSSION OF THEORY

2.1 General

In the storage of f resh water in s a l i n e aqu i fe rs the

fa c t o rs which can i n f lu e n c e the recovery e f f i c i e n c y are:

(1) mixing of the two f l u i d s due to molecular d i f f u s i o n and

convect l ve d ispersion, (2) g r a v i t a t i o n a l segregat ion of the

two f l u i d s due to dens i ty d i f f e r e n c e , (3) viscous f i n g e r

ing due to a d i f f e r e n c e in v i s c o s i t i e s between the i n

je c te d and na t i v e f l u i d s , (4) a q u i f e r h e t e r o g e n e i t i e s ,

(5 ) a q u i f e r d i p , and (6) p r e - e x i s t i n g ground-water move

ment in the storage a q u i f e r . Throughout th i s i n v e s t i g a t i o n

the f o l lo w i n g assumptions are made:

1. A h o r i z o n t a l , homogeneous, i s o t r o p i c storage

a q u i f e r of I n f i n i t e a rea l e x t e n t .

2. The v i s c o s i t i e s o f the i n j e c t e d f l u i d and the

n a t i v e f l u i d are equal .

3. No p r e - e x i s t i n g ground-water movement in the

storage a q u i f e r .

Hence, the fa c t o rs in f lu e n c i n g recovery e f f i c i e n c y which

are studied In t h is I n v e s t i g a t i o n are: (1) mixing of the

two f l u i d s due to molecular d i f f u s i o n and convect ive d i s

persion and (2 ) g r a v i t a t i o n a l segregat ion of the two

f l u i d s due to dens i ty d i f f e r e n c e .

6

7

2.2 Mixing Due to Molecular D i f f u s i o n and Convective PispersforT

I f two mlsc ib le f l u i d s o f d i f f e r e n t composit ion are

in c o n ta c t , a t r a n s f e r of molecules w i l l r e s u l t . As time

passes, the random movement of molecules creates a mixed

zone between the two f l u i d s ; t h a t i s , the two f l u i d s d i f

fuse I n t o one another . This process 1s c a l l e d molecular

di f f u s i o n .

When one f l u i d m isc ib ly d isplaces another f l u i d in a

porous medium the mixing between the two f l u i d s w i l l be

g r e a te r than t h a t due to molecular d i f f u s i o n a lone. The

a d d i t i o n a l mixing is p r i m a r i l y dependent on pore geometry

and is a r e s u l t of the varying v e l o c i t y f i e l d and constant

In t e r m i n g l i n g of f low paths as the displacement process

progresses. This a d d i t i o n a l mixing is c a l l e d convect ive

di s p e rs i o n . Convective d ispers ion is c l a s s i f i e d as l o n g i

tud in a l or t ra nsverse . Longi tudina l d ispers ion is in the

d i r e c t i o n o'f gross f l u i d movement, wh i l e t ransverse d i s

persion is in a d i r e c t i o n perpendicu lar to gross f l u i d

movement. Previous i n v e s t i g a t i o n s have shown tha t l o n g i

tud ina l d ispers ion is 6 to 20 times g r e a t e r than t ransverse

dispers ion (de Josse l in de Jong, 1958; Pozzi and B l ackwel l ,

1963) . In th is i n v e s t i g a t i o n t ransverse d ispers ion is

n e g l e c t e d .

Taking i n t o con s i der a t i on mixing due to molecular

d i f f u s i o n and l o n g i t u d i n a l d is p e r s i o n , Raimondi e t a l .

(1959) and Hoopes and Harleman (1967) s t a t e t h a t i f a

f l u i d moves r a d i a l l y outward from a l i n e source through a

8

homogeneous, I s o t r o p i c , porous medium, d i s p la c in g a f l u i d

of the same v i s c o s i t y and d e n s i t y , the mixing of the two

f l u i d s 1s descr ibed by the fo l l o w in g equat ion:

Where: c = concentrat ion of I n j e c t e d f l u i d a t r a d i u s , r ,

and t ime , t .

r = rad ius , (cm)

t = t ime, (seconds)

Q = q/(2frh<1>) (cm^/sec)

q = volumetr ic f low r a t e , (cc / se c )

h * a q u i fe r th ickness , (cm)

<J> = a q u i f e r p o r o s i t y , ( f r a c t i o n )

D = c o e f f i c i e n t of molecular d i f f u s i o n of f l u i d s

in porous medium, (cm^/sec)

a = l o n g i t u d i n a l d i s p e r s i v i t y c o e f f i c i e n t of

porous medium, (cm)

In the d e r i v a t i o n of Equation 2.1 i t is assumed th a t

the c o e f f i c i e n t of molecular d i f f u s i o n , D, is a constant .

This is not completely accurate because the value of D is

dependent on concentrat ion ( J o s t , 1960) . However, th is

dependence is small and f o r p r a c t i c a l purposes the value

of D may be considered constant (Perkins and Jonston,

1963) . S t oesse l l (1974) has shown e x p e r im e n ta l l y t h a t the

value of D in a porous medium s i m i l a r to t h a t used in the

exper imental work of th is i n v e s t i g a t i o n is about

( 2 . 1 )

9

10"6 cm^/sec.

Also assumed 1n the d e r i v a t i o n of Equation 2.1 1s

tha t lo n g i t u d i n a l d ispers ion is propor t iona l to the f i r s t

power of the average v e l o c i t y of f l u i d movement through the

porous mediurn, the p r o p o r t i o n a l 1ty constant being the l o n g i

tud ina l d i s p e r s i v i t y c o e f f i c i e n t , a. Ar is and Amundson

(1957) have shown t h a t th i s assumption is v a l i d i f the

v e l o c i t y is slow enough to a l low d i f f u s i o n to equa l i ze the

concent ra t ion w i t h i n each pore space. I f the v e l o c i t y is

high enough so t h a t concentrat ion e q u a l i z a t i o n does not

occur d ispers ion becomes propor t iona l to a higher power of

v e l o c i t y . Most exper imental data i n d ic a te s th a t th is

higher power has an upper l i m i t of about 1 .2 (Brigham et

a l . , 1961; Perkins and Johnston, 1963) . I t is f e l t th a t

t h is d e v i a t i c n from the assumption on which Equation 2.1 1s

based is not la rge enough to i n v a l i d a t e the equat ion when

viewed from an engineer ing s tandpoin t .

The value of the lo n g i t u d i n a l d i s p e r s i v i t y c o e f f i

c i e n t , a , f o r a porous medium is a c h a r a c t e r i s t i c s of the

porous medium and increases wi th increasing u n i f o r m i t y co

e f f i c i e n t (as the ma te r ia l contains a g r e a te r range of

p a r t i c l e s i z e ) and/or increasing i n t r i n s i c pe rm e a b i l i t y

(Raimondi e t al . , 1959) . Experiments by Brigham e t a l .

(1961) and Bentsen and Nielsen (1965) have shown t h a t the

value of a 1s also a fun ct ion of the r a t i o of the v i s

c os i t y of the displaced f l u i d to the v i s c o s i t y of the d i s

plac ing f l u i d . The l a r g e r the value of the r a t i o the

10

l a r g e r the value of a . Throughout t h i s i n v e s t i g a t i o n a

v i s c o s i t y r a t i o of un i ty was mainta ined. The value of a

fo r the m in i a q u i f e r used in the exper imental work 1n th is

I n v e s t i g a t i o n was found to be 0 .02 cm. The computational

procedure used to obtain th i s value of a is discussed l a t e r .

For the continuous i n j e c t i o n of a f l u i d at a steady

ra t e wi th a concent ra t ion cQ a t r = 0 , Raimondi e t a l . (1959)

and Hoopes and Harleman (1967) proposed the fo l lo w in g s o l u

t io n to Equation 2 .1 :

Where: R * radius of i n j e c t e d f l u i d at t ime t , assuming

no mixing or g r a v i t a t i o n a l segregat ion , (cm)

e r f c (?) * complementary e r r o r funct ion of £.

Equation 2 . 2 s a t i s f i e s the boundary condi t ions c ^ ( r = 0,

t > 0) = cQ and c ^ ( r = <®, t = 0) = 0; however, i t does not

s a t i s f y the i n i t i a l c o n d i t io n , c ^ ( r , t = 0) * 0. This is

due to the f a c t t h a t in obta in ing Equation 2 . 2 , i t was as

sumed t h a t 3 c / 3 t = 0 a t t = 0. Hoopes and Harleman (1967)

s t a t e t h a t th i s assumption 1s approximately t rue away from

the Immediate v i c i n i t y of the source; however, i t is not

t rue w i t h i n 10-20 p a r t i c l e diameters of the source. The

v a l i d i t y of the s o lu t i o n given by Equat ion 2.2 has been

• • ( 2 . 2 )

e r f c

11

demonstrated e x pe r i me n ta l ly by Bentsen and Nielsen ( 1 9 6 5 ) ,

Hoopes and Harleman (1967) and Esmall and Klmbler (1 9 6 7 ) .

I t can be shown (see Appendix A) t h a t Equation 2.2

can be wr1t ten as :

In terms of concent ra t ion of na t i v e f l u i d , c , i t cannbe shown (see Appendix A) t h a t Equation 2 .3 can be w r i t t e n

as :

Gardner e t a l . (1962) have extended the s o lu t i o n

given by Equation 2 .4 to apply to successive i n j e c t i o n and

product ion h a l f - c y c l e s . The f o l l o w in g equations are es

s e n t i a l l y those given by Gardner e t al . (1962) except t h a t

they have been rearranged, s i m p l i f i e d and subscr ip ted .

The concent ra t ion a t any r a d i u s , r , and f o r any i n j e c t i o n

or product ion h a l f - c y c l e can be computed by using the ap

p r o p r ia t e form of the s o l u t i o n :

( 2 . 3 )

2Where: f ( t ) » 4 /3 a ( 2 - Q * t ) 3 / 2 + D

( 2 . 4 )

cc

0n e r f c ( 2 . 5 )

Where

Where:

F i r s t I n j e c t i o n H a l f -C y c le

ONOMj, • 2Cf x ( t , ) 3 1/ 2

F i r s t Product ion Ha l f -C y c l e

DNOMp>2 . 2 [ fp 2 ( t 2 ) - f p >2( t , ) ♦

Second I n j e c t i o n H a l f -C y c le

DN0MI , 3 = 2 ^f I , 3 ^ t 3 " f I , 3 ^ t 2 + f P ,2^ t 2

" f P ,2^ t l + f I ,1 ^ 1 ^ 1/2

Second Product ion Ha l f -C y c le

DN0MP , 4 = 2 ^ f P , 4 ^ 4 ^ " f P , 4 ^ t 3^ + f I , 3 ^ t 3^

f I , 3 ( 1 2) + f P, 2^t 2 ’ f P, 2^t l^

etc

3 / 2 C 2 QrI t j ( t k ) = 4 / 3 a l Z Q , ^ ) 3' 2 * D h i k

* * J

( 2 Q o < t k ) 2f P j j ( t k ) = 4 / 3 a ( 2 Q p > j t k ) 3 / 2 ♦ D - T J U i -

" » J

( I or P) , j

r 2 r 2f 1 nal " i n i t i a l

T aF2

(cm /s ec )

i t = t ime fo r f l u i d to t r a v e l from r ^n^ - j a ] to

r f 1 n a l * ( s e c >

13

t ] , t £ , tg» . . . « t ime measured from s t a r t of f i r s t

I n j e c t i o n h a l f - c y c l e , (sec)

I , P = subscr ipts f o r I n j e c t i o n and product ion

respect i v e l y .

j , k * In tegers

Equation 2 .5 has been v e r i f i e d by Esmail and Klmbler (1967)

fo r two complete cycles.

2. 2 G r a v i t a t i o n a l Segregat ion Due to Density P i f f e r e n c T

When f l u i d s of unequal d e n s i t i e s are in contact in a

porous medium g r a v i t a t i o n a l forces cause the less dense

f l u i d to r i s e r e l a t i v e to the more dense. The i n t e r f a c e

w i l l assume a progress ive ly g r e a te r angle wi th respect to

the v e r t i c a l . Gardner et a l . (1962) have shown mathemati

c a l l y and Esmail (1966) has shown e x pe r i m e n ta l l y t h a t fo r

a v i s c o s i t y r a t i o of un i ty the i n t e r f a c e may be t re a te d as

a plane surface in l i n e a r systems. Kumar (1968) has shown

e x pe r i me n ta l ly th a t th is also appears to hold fo r r a d i a l

systems.

G r a v i t a t i o n a l segregat ion in porous media may be

separated in to two cases. One case is the s o - c a l l e d

" s t a t i c " case where there is no bulk f low of f l u i d s except

t h a t a r i s i n g from convect ive cur rents a t t r i b u t a b l e to

g r a v i t y . The second case is c a l l e d dynamic g r a v i t a t i o n a l

se gre gat ion , since i t occurs 1n the presence of bulk f low

( Esmai1 and K1mbler, 1967) . An example of the l a t t e r would

14

be the g r a v i t a t i o n a l segregat ion t h a t occurs during the

displacement of a f l u i d by an i n j e c t e d f l u i d of d i f f e r e n t

d e n s i t y . Esmail (1966) assumed and Kumar (1968) l a t e r

v e r i f i e d t h a t f o r a v i s c o s i t y r a t i o of one, dynamic and

s t a t i c g r a v i t a t i o n a l segregat ion a r e , f o r p r a c t i c a l pur

poses, equal and t h a t r e s u l ts obtained from l i n e a r systems

may be d i r e c t l y r e l a t e d to r a d i a l systems by c o r r e c t i n g

the l i n e a r r e s u l t s to r ad ia l geometry.

Gardner e t a l . (1962a) studied s t a t i c g r a v i t a t i o n a l

segregat ion of m isc ib le f l u i d s in l i n e a r , h o r i z o n t a l sys

tems. They r e p o r t t h a t , f o r p r a c t i c a l purposes, the pro

j e c t i o n of the i n t e r f a c e on the h o r iz o nt a l can be a p p ro x i

mated by the equat ion:

2/ 2 X L \ 2 16 r 2 kH ' t / t o* .................................( 2 . 6 )

- T F * 7 (1 ♦ t / t „ )

Where:

2XL = p r o j e c t i o n of the i n t e r f a c e on the h o r i z o n t a l ,

(cm)

h = a q u i f e r th ickness . (cm)p

= h o r i z o n t a l i n t r i n s i c p e r m e a b i l i t y , (cm )2

ky = v e r t i c a l i n t r i n s i c p e r m e a b i l i t y , (cm )

F = a dimensionless f a c t o r dependent on v is c o s i t y

r a t i o . (F = 1 .0 f o r a r a t i o of one)

t = t ime. (sec)

* _ 4 d> • h - u • F , ,*o ■ J Cy.g.ap <sec>

<t> = p o r o s i t y , ( f r a c t i o n )

15

y ■ average v i s c o s i t y of the two f l u i d s , (po ise )

g = a c c e le r a t i o n due to g r a v i t y , (cm/sec^)

Ap = dens i ty d i f f e r e n c e between the f l u i d s , (gm/cc)

Equation 2 .6 is based on the assumption of a sharp

I n t e r f a c e between the two f l u i d s . Esmall (1966) contended

t h a t in p r a c t i c e , a sharp i n t e r f a c e would not r e s u l t and a

mixed zone would be present due to d i f f u s i o n and d i sp ers io n .

He f u r t h e r reasoned tha t th i s mixed zone would r e t a r d the

e f f e c t of g r a v i t a t i o n a l segregat ion and y i e l d a smal ler

r a t e of i n t e r f a c e laydown than the sharp i n t e r f a c e theory

would p r e d i c t .

Esmail (1966) introduced a term f o r the dens i ty gra

d i e n t , S, which he def ined as the qu o t i e n t of the dens i ty

d i f f e r e n c e and the mixed zone length . Assuming a hor izon

t a l , homogeneous, i s o t r o p i c porous medium, he used dimen

s ional ana lys is and exper imental data from several l i n e a r

systems to obta in the f o l l ow in g equat ion t h a t describes

g r a v i t a t i o n a l segregat ion in l i n e a r systems:

f ( ^ ) = some fun ct io n of </;

In each of Esmai l 's exper imental runs he s t a r t e d with a

known mixed zone length which remained constant

- f U )2XL ( 2 . 7 )

Where:

16

throughout the exper iment . He also s t a r t e d each e x p e r i

ment wi th the i n t e r f a c e v e r t i c a l a t t ime ■ 0 and then r e

corded values of 2XL/h as t ime progressed.

To f in d the f un ct i on a l r e l a t i o n s h i p between 2XL/h

and ip the exper imental data obtained by Esmail (1966) were

p l o t t e d (see Fig. 2 . 1 ) . The data were d iv ided i n t o two

sect ions and a curve f i t t e d to each p a r t . The equat ion of

the curve f o r the f i r s t par t was:

p p = 20.0ip ; (0<ip<0.1)........ ...................................... ( 2 . 8 )

and f o r the second par t

p p = 0 . 8 + 1 2. 5ip - 4.8ip2 ; ( 0 .1 « p < 1 .0 ) . . ( 2 . 9 )

Since there was no data beyond ip = 1.0 the f un ct io n a l r e l a

t io ns h ip was r a t h e r a r b i t r a r i l y taken to be:

p p = 6 . 5 + 2. Oip; ( i p > 1 . 0 ) .......................................... ( 2 . 1 0 )

I t w i l l be shown l a t e r th a t th i s r e l a t i o n s h i p y i e l d s r e

s u l ts of recovery e f f i c i e n c y on the safe side ( lower than

those a c t u a l l y observed) .

In computing the values of ip p lo t t e d in Figure 2 . 1 ,

Esmail (1966) c a l c u l a t e d the dens i ty gr ad i en t on the basis

th a t the mixed zone length was the distance between the

r a d i i where the concent rat ions of na t i v e f l u i d were 3 pe r

cent and 97 percent . This same basis was used throughout

t h is i n v e s t i g a t i o n when mixed zone lengths were computed.

10.0

9.0

8.0

7.0

6.0

5.0

4.0

3.0

2.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0 .7 0.8 0.9 1.0 1.21.1 1.3 1.4 1.5

♦Figure 2.1 Dimensionless C o r r e l a t i o n Used to Compute G r a v i t a t i o n a l Segregation

CHAPTER I I I

DEVELOPMENT OF COMPUTATIONAL PROCEDURE

FOR CALCULATING RECOVERY EFFICIENCIES

3.1 General

The f o l l ow in g t reatment assumes a h o r i z o n t a l , homo

geneous, i s o t r o p i c , storage a q u i f e r of i n f i n i t e area ex

t e n t , which has no p r e - e x i s t i n g ground-water movement.

Ad di t io na l assumptions are t h a t the r a t i o of the v i s c o s i

t i e s of the i n j e c t e d and na t i v e f l u i d s is uni ty and t h a t

the f low geometry is r a d i a l . The fa c to rs which i n f lue nc e

recovery e f f i c i e n c y th a t are considered are: (1 ) mixing

due to molecular d i f f u s i o n and l o n g i t u d i n a l convect ive

d i s p e r s i o n , and (2) g r a v i t a t i o n a l segregat ion due to

dens i ty d i f f e r e n c e .

3 .2 C a l c u l a t i o n of Mixing Due to Molecular D i f f u s io n and Longi tud in a l Convective Dispersion

Consider Figure 3.1 which i d e a l i z e s the f low system

during an i n j e c t i o n h a l f - c y c l e and in which g r a v i t a t i o n a l

segregat ion is ignored. The i n j e c t e d f resh water d i s

places the na t i v e s a l t water away from the source. As the

i n t e r f a c e between the f resh water and s a l t water moves in

the a q u i f e r , the mixing between the two f l u i d s w i l l

18

'POINT SOURCE5 0 % CONCENTRATION

LINE

SALT WATER

Flow

FRESHWATER

MIXED ZONE

FRESHWATER

MIXED ZONE-

5 0 % CONCENTRATION LINE

SALTWATER

Flow

Figure 3.1 Schematic Representat ion of the Displacement Process During an I n j e c t i o n H a l f - C y c l e . G r a v i t a t i o n a l segregat ion is ignored.

VO

20

generate a t r a n s i t i o n or mixed zone 1n which the composi

t io n of e i t h e r f l u i d w i l l vary from 100 percent to 0 per

cent . The length of th i s mixed zone as 1t moves in the

a q u i f e r is dependent on the t o t a l d istance t r a v e le d by the

I n t e r f a c e , the v e l o c i t y at which the i n t e r f a c e moves, the

t o t a l t ime of contact between the l i q u i d s , the pro pe r t i es

of the l i q u i d s , and the proper t ies of the porous medium.

In Figure 3 . 1 , R is the radius of the i n j e c t e d f l u i d

at any time t , assuming no mixing. Th e re fo re , the average

value of c ^ / c Q or cn/ c Q a t radius R is 0 . 5 . In t h is com

p u t a t io n a l procedure the mixed zone length a t any time t ,

and about any radius R, is computed using the app ropr ia te

form of Equation 2 . 5 .

3 .3 C a lc u l a t i o n of G r a v i t a t i o n a lSegregat ion Due to Density Di f fe r enc e

The dens i ty d i f f e r e n c e between the i n j e c t e d f resh

water and the n a t i v e s a l t water w i l l cause the mixed zone

between the two l i q u i d s to i n c l i n e wi th respect to the

v e r t i c a l (see Fig. 3 . 2 ) . The less dense f resh water w i l l

r i s e over the more dense s a l t water . The g r a v i t a t i o n a l

segregat ion between the two f l u i d s at any time can be rep

resented by the tangent of the ang le , 0, th a t the 50 per

cent concent ra t ion l i n e makes with respect to the v e r t i c a l

and is given by the dimensionless group 2XR/h.

Equations 2 . 8 , 2 . 9 , and 2.10 cannot be used d i r e c t l y

to c a l c u l a t e g r a v i t a t i o n a l segregat ion in a r a d i a l system

POINT SOURCE


SALTWATER

MIXED ZONE

FRESHWATER


FRESHWATER

SALTWATER

MIXED ZONE

1 2XR L R L50 1_■ RL50 |. 2XR

Figure 3 .2 Schematic Representat ion of the Displacement Process During an I n j e c t i o n H a l f - C y c l e . Both mixing and g r a v i t a t i o n a l segregat ion are included.

22

f o r two reasons. These are: (1 ) they descr ibe g r a v i t a

t i o n a l segregat ion in l i n e a r systems o n l y , and (2 ) the

den s i ty g r a d i e n t , S, is not a constant value but var ies

cont inuously as the i n j e c t i o n , s t orage , and r e t r i e v a l

process progresses. In order to use Equations 2 . 8 , 2 . 9 ,

and 2.10 to compute g r a v i t a t i o n a l segregat ion in a r a d i a l

system, i t is necessary to f i r s t use a stepwise procedure

to a l low f o r the cont inuously changing value of S, and

second, to apply a co r re c t io n to approximate the computed

values of g r a v i t a t i o n a l segregat ion to r a d i a l geometry.

The method by which the equations are app l ied is ou t l in e d

below.

Consider Figure 3 .3 which i l l u s t r a t e s the stepwise

procedure f o r c a l c u l a t i n g the value o f g r a v i t a t i o n a l seg

regat ion before approximat ion to r a d i a l geometry. Note

t h a t only the 50 percent concent rat ion l i ne s are consid

ered. Let the f resh water be i n j e c t e d to a radius of

Divide the dis tance R in to equal i n t e r v a l s such max maxas 0 Rj , Rj Rg, e t c . The length of the mixed zone is c a l

cula ted at the c e n t e r , , of the f i r s t i n t e r v a l using the

ap p ro pr ia te form of Equation 2 . 5 . This value of mixed

zone length at is used to compute the value of dens i ty

g r a d i e n t , , a t C^. I t is assumed t h a t the dens i ty gra

d i e n t has had a constant value of over the i n t e r v a l

0 R| . Using the value of S-j and the rea l t ime of t r a v e l ,

t-j , from 0 to R.| the h o r i zo n t a l p r o j e c t i o n (2 X L ) a t the

end of the f i r s t i n t e r v a l can be c a lc u la t ed using

POINT SOURCE(2 XL) (2XL)

0 C, P, R, C 7 P2 R2

Figure 3 .3 Schematic Representat ion to I l l u s t r a t e the Ca lc u l a t i on of G r a v i t a t i o n a l Segregation before Approximation to Radial Geometry.

24

Equation 2 . 8 , 2 . 9 , or 2 . 1 0 . For the second i n t e r v a l ,

Rl Rg, the mixed zone length is computed a t the c e n t e r ,

Cg, o f the i n t e r v a l . The mixed zone length a t C2 w i l l be

g r e a te r than at C1 , t h e r e f o r e S2 w i l l be less than . I t

is now assumed t h a t the dens i ty g ra d i e n t has had a con

s t a n t value of S2 over the i n t e r v a l 0 R2 . Using S2 , a

pseudo-t ime, t^ , is c a l c u l a t e d which w i l l give a ho r iz o n

t a l p r o j e c t i o n a t the end of the f i r s t i n t e r v a l t h a t is

equal to (2XL)^. The pseudo-t ime, t-j , w i l l be gr e a te r

than t j since S^ is less than . The rea l t ime of t r a v e l

from R.] to R2 is added to th i s pseudo-t ime to give t o t a l

t ime , t 2 » which is then employed to c a l c u l a t e the ho r iz o n

t a l p r o j e c t i o n , (2 XL) 2 » at the end of the second i n t e r v a l .

The assumption t h a t the dens i ty gr a d i e n t remains constant

from 0 to the end of the i n t e r v a l in quest ion and the i n

t ro du ct i on of pseudo-t ime 1s necessary due to the manner

1n which the c o r r e l a t i o n given in Figure 2.1 and Equations

2 . 8 , 2 . 9 , and 2.10 were d e r iv e d . This has been previous ly

discussed in Chapter I I . The c a l c u l a t io n s f o r subsequent

i n t e r v a l s are c a r r i e d out in a s i m i l a r fash ion . For the

l a s t i n t e r v a l of the i n j e c t i o n h a l f - c y c l e , the t ime of

s t a t i c storage before product ion begins Is included in the

t o t a l t ime before making c a l c u l a t io n s of g r a v i t a t i o n a l

segregat i on.

In the preceding t rea tm ent , the p r o j e c t i o n of the

i n t e r f a c e was ca lc u l a te d on the basis of a l i n e a r geometry.

In r a d i a l geometry, the p r o j e c t i o n of the i n t e r f a c e is

25

less than t h a t f o r l i n e a r geometry f o r i n j e c t i o n h a l f

cycles and gr ea ter f o r product ion h a l f - c y c l e s . The c a lc u

la ted pr o jec t ions shown in Figure 3 .3 must be converted to

r a d i a l geometry. Consider Figure 3 .4 which I l l u s t r a t e s the

approximat ion of g r a v i t a t i o n a l segregat ion c a l c u l a t io n s to

rad i a l geometry. Lines p-jpj and P2 P2 represent the 50 p e r

cent concentrat ion l i ne s a t the ends of the f i r s t and

second i n t e r v a l s r e s p e c t i v e l y , before the approximat ion to

r a d i a l geometry. Note th a t p^p-j and P2 P2 F^9ure 3*4

are the same as p^pj and P2 P2 Figure 3 . 3 . For the f i r s t

i n t e r v a l i t is assumed th a t no c o r r e c t i o n is necessary,

t h a t i s , ( 2X L) -j * (2 X R) . The cor rec ted p r o j e c t i o n ,

( 2 XR) 2 > at the end of the second i n t e r v a l is obtained in

the fo l lo w in g manner: Compute the radius to po in t (a )2 2such t h a t the annular a rea , ttC ~ a 3* equal to the

2 2annual a re a , tt[Rj - p^ ] . Compute the radius to po int ( a ' )

2 2such tha t the annual a rea , 7 r [ (a ' ) - Rg], is equal to the2 2annual a re a , tt[ ( p-J) - R-j]. This gives the l i n e aa' which

is the l i n e p-jp.j converted to r a d i a l geometry at the end

of the second i n t e r v a l . The a d d i t i o n a l t i l t i n g of the

i n t e r f a c e in t r a v e l i n g from R to R2 is given by the d i f

ference , [ ( 2 X L ) 2 - ( 2XL) j ] . This d i f f e r e n c e is equa l ly

d i s t r i b u t e d on each side of the l i n e a a ' . Hence:

P ( 2 X L ) p - ( 2 X L ) , - ] ba = a ' b ' = I --------- ^ ------------ U

The ho r i zo n t a l p r o j e c t i o n , (2 X R) 2 » of the l i n e bb ' is the

(2XL)POINT SOURCE(2XR) (2XR)j

0 P.

Figure 3 .4 Schematic Representat ion to I l l u s t r a t e the Approximation of G r a v i t a t io n a l Segregat ion Ca lc u la t i ons to Radial Geometry.

27

p r o j e c t i o n of the 50 percent concentrat ion l i n e 1n r a d i a l

geometry a t the end of the second I n t e r v a l . The c a l c u l a

t ions fo r subsequent i n t e r v a l s are c a r r i e d out 1n a s i m i l a r

manner.

3 .4 C a l c u l a t i o n of Recovery E f f i c i e n c y

Stored f resh water is produced u n t i l the leading

edge of the mixed zone reaches the breakthrough r a d i u s ,

RBT (see F ig . 3 . 5 ) . The breakthrough radius would be the

we l lbore radius for a s in g l e wel l system. For a m u l t i p l e

wel l system, the breakthrough radius would be the radius

from the center of the wel l pa t t e rn to the outer r ing of

w e l l s . The volume of water contained in the f rustrum of

the cone having a h e i g h t , h, and upper and lower r a d i i of

RU50 and RL50 r e s p e c t i v e l y , is the volume of unrecovered

f resh water (see F ig . 3 . 5 ) .

The cumulat ive recovery e f f i c i e n c y is c a l c u l a t e d by

a computat ional program from the equat ion:

Cum. Recovery E f f i c i e n c y

The recovery e f f i c i e n c y f o r a p a r t i c u l a r cycle is computed

from the equat ion:

Recovery E f f i c i e n c y

Cum. volume of f resh water i n j e c t e d

Volume of unrecovered f resh water

Cum. volume of f resh water i n j e c t e d

Vol. of fresh water recovered in previous ______ cycles_______

Cum. vol. of fresh water injected

Vol. of unrecovered fresh water

Volume of f resh water i n je c t e d during cycle

POINT SINK


RU 50 R U 50 5 0 % CONCENTRATION LINE

FRESH FRESHWATER

SALTWATER

SALTWATER WATER

Flow

MIXED ZONE MIXED ZONE

R

RBT RBT

0L 5 0 RL5

Figure 3 .5 Schematic Diagram to I l l u s t r a t e the Ca lc u l a t i on of Recovery E f f i c i e n c y

rooo

3 .5 M u l t i p l e Well Systems

The preceding has d e a l t p r i m a r i l y with s i n g le wel l

systems. In a f i e l d a p p l i c a t i o n of the storage process i t

is most l i k e l y t h a t a wel l f i e l d w i l l be used instead of a

s in g le w e l l . In th i s i n v e s t i g a t i o n i t was assumed th a t

these wel l f i e l d s would be symmetrical and would have a

wel l a t the center of the pa t t e rn (see Fig. 3 . 6 ) . I t was

also assumed t h a t the f i e l d s would be operated so t h a t the

i n j e c t e d bubble of f resh water would remain e s s e n t i a l l y

ci r c u l a r .

The operat ing procedure f o r a wel l f i e l d c o n f i g u r a

t ion such as t h a t shown in Figure 3.6c would be as fol lows

(1) I n j e c t i n t o the center wel l u n t i l the lagging edge of

the mixed zone passes the inner r ing of w e l l s . (2 ) S t a r t

i n j e c t i o n in these we l ls (wi th i n j e c t i o n cont inuing in the

center w e l l ) u n t i l the lagging edge of the mixed zone

passes the outer r ing of w e l l s . (3 ) I n j e c t in to a l l nine

wel ls u n t i l the des ired q u a n t i t y is i n j e c t e d . (4 ) Al low

the i n j e c t e d water to stand u n t i l needed. (5) produce a l l

nine we l ls u n t i l breakthrough occurs a t the outer r ing of

w e l l s , a t which time product ion from the wel l f i e l d is

stopped. Subsequent cycles are made wi th i n j e c t i o n beg in

ning in a l l nine wel ls s imul taneous ly . The water requi red

to i n i t i a l l y "sweep out" the p a t t e rn is termed "cushion

water" and is water tha t w i l l , fo r p r a c t i c a l purposes,

never be recovered.

The mathematical procedures f o r computing recovery

30

d ( T y p )

a.) FOUR WELL PATTERN

b) FIVE WELL PATTERN

c) NINE WELL PATTERN

Figure 3.6 Some Possible Well F ie ld Pat terns

31

e f f i c i e n c i e s t h a t have been proposed thus f a r have been

f o r s in g l e we l l systems. To use these procedures f o r

m u l t i p l e we l l systems, i t must be assumed th a t a l l i n j e c

t ion and product ion takes place through the center wel l of

the m u l t i p l e wel l p a t t e r n . The exper imental r e s u l t s t h a t

w i l l be presented in the next chapter i n d i c a t e t h a t th is

is a v a l i d assumption as long as the wel l f i e l d is symmet

r i c a l and is operated so t h a t the i n j e c t e d bubble of f resh

water remains e s s e n t i a l l y c i r c u l a r .

3 .6 Computer Programs f o r Computing Recovery E f f i c i e n c i e s

D et a i l ed computer programs have been developed f o r

computing recovery e f f i c i e n c i e s f o r one, two, and three

cycle opera t ion of the storage process. A d e s c r i p t i o n and

l i s t i n g of these program are presented in Appendix B. The

programs presented are f o r s i ng le wel l systems on l y , but

complete i n s t r u c t i o n s are given on how to modify the pro

grams f o r m u l t i p l e we l l use. The programs are in FORTRAN

IV language and are w r i t t e n f o r use on an IBM 360/65

sys tern.

CHAPTER IV

EXPERIMENTAL PROCEDURE AND RESULTS

4.1 General

The exper imental work was conducted in three stages.

The f i r s t stage consisted of the const ruc t ion of a syn

t h e t i c sandstone m i n i a q u i f e r . The second stage involved

the de terminat ion of the physical p r op e r t ie s of the m i n i

a q u i f e r such as homogeneity, p o r o s i t y , p e r m e a b i l i t y , and

lo n g i t u d i n a l d i s p e r s i v i t y c o e f f i c i e n t . With t h is data the

performance of we l ls and wel l f i e l d s in the m i n i a q u i f e r

could be used as a standard aga ins t which the c a l c u l a t i o n

procedure could be t e s t e d . In the t h i r d stage several

i n j e c t i o n - p r o d u c t i o n runs were made wi th a wide v a r i a t i o n

of the parameters t h a t a f f e c t recovery e f f i c i e n c y . The

recovery e f f i c i e n c i e s obtained from these exper imental

runs were compared wi th those pred ic ted by the computa

t i o n a l procedure in order to a s ce r ta in the v a l i d i t y of the

procedure.

I t should be noted t h a t a m i n i a q u i f e r is not a model

of some prototype but is an actual independent , physical

system. Consequently, i f a mathematical model (a computa

t i o n a l procedure) w i l l p r e d i c t such things as mixed zone

l engths , g r a v i t a t i o n a l se gre gat ion , movement of in j e c t e d

32

33

f l u i d , e t c . , in the m ln i a q u i f e r there 1s every reason to

be l i e v e t h a t i t w i l l do the same fo r a f i e l d s i t u a t i o n :

both the m i n i a q u i f e r and a water -bea r ing sand or sandstone

invo lve laminar f low through porous media and the equations

take cognizance of the s ize or thickness of an a q u i f e r only

through the magnitudes of the hydrologic constants . That

i s , whether a sandstone is h a l f an inch t h i c k or 500 f t

t h i c k , the same Darcy equation expresses the r e l a t i o n s h i p

between discharge and change in p o t e n t i a l , only the numeri

cal c o e f f i c i e n t s being d i f f e r e n t .

4 .2 De sc r ip t ion of the M in i a q u i f e r

The technique used fo r the const ruc t ion of the m i n i

a q u i f e r is descr ibed in d e t a i l in Appendix C and was s i m i

l a r to th a t employed by Caudle ( 1 9 6 3 ) , Esmail ( 1 9 6 6 ) ,

Kumar (1 9 6 8 ) , and P a in te r (1 9 7 1 ) . The m i n i a q u i f e r is h a l f

of a rec ta ng ul a r system whose dimensions are 305 cm by

292 cm by 3.81 cm t h i c k , bordered completely by an i s o

p o t e n t i a l and conta in ing a n in e -w e l l a r ray at the center

(see F ig . 4 . 1 ) . Note th a t i f the wel l f i e l d is operated

such t h a t the r a t e of wel l 3 1s the same as the r a t e of

wel l 9 , wel l 4 the same as wel l 8 and wel l 5 the same as

wel l 7, a l i n e of symmetry (a lso a no- f low boundary)

e x is ts as shown in Figure 4 . 1 . Hence, the i n t e r f a c e p o s i

t ions on one s ide of the l i n e are m i r ro r images of the

i n t e r f a c e pos i t ions on the other side o f the l i n e . Taking

advantage of t h i s symmetry, i t was necessary to const ruc t

i

34

3 0 5 cm

152.5 cm 5 2 .5 cm

Isopotential

Line of Symmetry

Figure 4.1 System Represented by M i n i a q u i f e r Used in This Study.

146

cm

only h a l f the system (305 cm by 146 cm by 3.81 cm t h i c k ) .

In the exper imental procedure wel ls 1, 2, and 6 were

t r e a t e d as h a l f w e l l s . Figure 4 .2 is a photograph of the

m i n i a q u i f e r together with the moni tor ing equipment used

during the exper iments. F lu id was i n j e c t e d i n t o , or w i t h

drawn from, each we l l by means of constant-speed p o s l t l v e -

displacement pumps (see Fig. 4 . 3 ) , each of which had been

p r e c i s e ly machined and c a l i b r a t e d . Each pump consists of

a c y l i n d e r and a p iston powered by a synchronous motor.

By means of proper ly se lec ted gear r a t i o s , the r a t e of i n

j e c t i o n or product ion can be set a t a predetermined value.

The a v a i l a b l e i n j e c t i o n and product ion rates range from

8.046 cc/min to 0 .334 x 10"^ cc/min.

4 .3 The E f f e c t of Boundaries

One of the assumptions made in t h i s i n v e s t i g a t i o n

(Chapter I I ) was t h a t the f low systems were of i n f i n i t e

area l e x t e n t . The m i n i a q u i f e r pr ev ious ly descr ibed v i o

la tes th i s assumption. I t was t h e r e f o r e necessary to de

termine the maximum radius to which a bubble of f l u i d

could be i n j e c t e d in the m i n i a q u i f e r before boundary e f

fects became a pp re c ia b le . To determine t h i s maximum

ra d i u s , i t was assumed th a t i n j e c t i o n would take place at

a constant r a t e in wel l 1 of the system shown in Figure

4 . 1 . Using the image we l l technique to take in to account

the i s o p o t e n t i a l boundar ies, the f r o n t a l posi t ions of the

i n j e c t e d bubble were computed f o r var ious i n j e c t i o n t imes.

36

Figure 4 .2 - -V1ew of m i n i a q u i f e r , pumps, and Ins t rumen ta t i o n . Note camera on support f o r photographing f r o n t a l p o s i t i o n of I n j e c t e d f l u i d . In l e f t foreground are thre e l a r g e - b a r r e l pumps and one s m a l l - b a r r e l pump. Chemical o s c i l lometer and recorder on on stand j u s t above wel l f i e l d . To l e f t o f o s c i l l o m e t e r are I n d i v idual c on t ro ls f o r each w e l l .

37

Figure 4 . 3 - -V1ew of th re e l a r g e - b a r r e l and three s ma l l - b a r re l pumps used 1n exper iments.

38

I t was then assumed t h a t the system was I n f i n i t e (no

boundaries) and f r o n t a l pos i t ions f o r the i n j e c t e d bubble

were computed a t the same i n j e c t i o n times as fo r the

f i n i t e system. Figure 4 . 4 is a comparison of these f r o n

t a l p o s i t i o n s . I t can be seen t h a t there is no no t ic ea b l e

d i f f e r e n c e u n t i l a radius of 60 cm is reached and even at

a radius of 100 cm, the d i f f e r e n c e is smal l . In the ex

per imental runs made in th is i n v e s t i g a t i o n the r a d i i of

the i n j e c t e d bubbles seldom exceeded 60 cm, hence, the

assumption of a system wi th i n f i n i t e areal e x te n t is

reasonable.

4 . 4 Fluids Used in Experiments

Instead of s a l t water and f resh wate r , analog f l u i d s

were used in a l l of the exper iments. The analog f l u i d s

were made of various mixtures of naphtha, S o l t r o l 170,

carbon t e t r a c h l o r i d e , and iodobenzene. These f l u i d s are

m isc ib le in a l l p r opo r t ion s . The pr ope r t ie s of the pure

f l u i d s are given in Table 4 . 1 . The d e n s i t i e s were meas

ured using a Chainomatic G ra v i t o m e te r . The v i s c o s i t i e s

were measured using an Ostwald Viscometer . The same equ ip

ment was used to determine the pro pe r t i es of the f l u i d

mi x t u r e s .

Analog f l u i d s were used f o r two reasons: (1 ) such

f l u i d s are more d e s i r a b l e f o r exper imental use than fresh

water and s a l t water in t h a t much b e t t e r control of p roper

t i e s is po s s ib le , f o r example, a wide range of dens i ty

100

FINITE SYSTEM INFINITE SYSTEM

80

£ 60

</>3O<tz 40

20

20 40 80 10060Well no. IRADIUS, cm

Figure 4 . 4 Comparison of Frontal Posi t ions f o r a F i n i t e System and an I n f i n i t e System.

COto

40

d i f f e r e n c e s between na t i v e f l u i d and I n j e c t e d f l u i d s can

be obtained wi thout changing the v i s c o s i t y r a t i o , (2) i n

t rodu ct i on of water i n t o a s y nth et ic sandstone mat r ix

s i m i l a r to that used in the const ruct ion of the m i n i a q u i f e r ,

sometimes causes the mat r ix to i n e x p l i c a b l y d e t e r i o r a t e .

TABLE 4 . 1 - - P r o p e r t i e s of Pure Fluids Used in Experimental Runs

Fluid Density a t 22°C (gm/cc)

V is co s i t y a t 22°C (cp)

Naphtha 0.747 0.570

2S o l t r o l 170 0.771 2.5043

Carbon T e t r a c h lo r i d e 1 .590 0.9924

Iodobenzene 1 .832 1 .573

^Naphtha; V.M. & P . ; Humble Oi l and Ref ining Co.; Baton Rouge, La.

2S o l t r o l 170; A l i p h a t i c Hydrocarbon; P h i l l i p s Petroleum Co.; B a r t l e s v i l l e , Okla.

3Carbon T e t r a c h l o r i d e ; Technical Grade; F. H. Ross &

Co . ; Baton Rouge, L a .4

Iodobenzene; Matheson, Coleman & Bel l Mfg. Chemists; Norwood, Ohio.

P a in te r (1971) found t h a t a mixture of 45 percent

S o l t r o l and 55 percent naphtha by volume has a v is c o s i t y

equal to t h a t of carbon t e t r a c h l o r i d e . Thus by adding

carbon t e t r a c h l o r i d e to the mixture of S o l t r o l and naphtha,

a more dense f l u i d can be obtained wi tho ut any change in

vi scos i t y .

The m i n i a q u i f e r was i n i t i a l l y sa tura ted wi th a f l u i d

mixture t h a t contained 43.5 percent S o l t r o l , 53 .2 percent

naphtha, and 3 .3 percent carbon t e t r a c h 1or ide (percents

are by volume). The i n j e c t e d f l u i d s were prepared s t a r t

ing wi th the basic 45-55 percent S o l t r o l - n a p h t h a mix ture .

To th is was added 2 .5 percent iodobenzene, enough carbon

t e t r a c h l o r i d e to obtain the requi red dens i ty d i f f e r e n c e ,

and an o i l - s o l u b l e red dye. The purpose of the dye was to

make the progress of the i n j e c t e d f l u i d v i s i b l e . The pur

pose of the iodobenzene was to make the d i e l e c t r i c con

s tan t of the i n j e c t e d f l u i d d i f f e r e n t from t h a t of the

nat iv e f l u i d . During a product ion h a l f - c y c l e , when na t i v e

f l u i d appeared in the produced stream, the change in d i

e l e c t r i c constant of the produced f l u i d was sensed by a

capaci tance c e l l (see Fig. 4 . 5 ) which was connected to a

chemical o s c i l l o m e t e r . When the o s c i l l o m e t e r de tected a

change in ca pac i tance , i t generated a m i l l i v o l t s ignal

which caused the pen on a m i l l i v o l t recorder to move. The

response of the recorder and the concent ra t ion of na t i v e

f l u i d in the produced stream were l i n e a r l y r e l a t e d (see

Fig. 4 . 6 ) . When the concent ra t ion of na t i v e f l u i d in the

produced stream reached 3 per cent , breakthrough (as def ined

in th i s study) had occurred. A schematic of the e x p e r i

mental apparatus is shown in Figure 4 . 7 . For c l a r i t y , a

f low l i n e and coaxia l cable are shown connected to one

wel l on ly . In r e a l i t y there is a f low l i n e to each wel l

and a coaxia l cable running from each wel l to the r o t a r y

coaxial swi tch.

42

i i'!P *'i_i _ -r^

Male Adapter1/8" Tube to 1/8” Pipe Thrd Swagelok Cat. No. 2 0 I-A -2

Ground Connection

1/8 O.D. Brass Rod

Series 83 UHF Coaxial Connector Amphenol Type 8 3 * IR

Soldered Joint

l/ 4 ,,X | l/ 4 ,,X l" Phenolic Block

1/4 O.O. Stainless Steel Tube

J L

M iniaquifer

0 Seal S tra ighti(Thread Connector, 1 /4 " Tube Swagelok Cat. No. 4 0 0 - I -0 R

I,/4 " X I I/4 " X I" Phenolic Block Epoxied to Top Surface of M iniaquifer

-l!'4"xij'4"xi" Phenolic Block Epoxied to Bottom Surface of M iniaquifer

Male Adapter1/8" Tube to 1/8" Pipe Thrd. Swagelok Cat No 20I-A -2

Figure 4 .5 View of Capacitance Cel l Used to Detect Concentrat ion Changes.

NATI

VE

FLU

ID

IN PR

ODU

CED

STR

EA

M

(Vol

ume

Frac

tion

)

43

0.8

0.6

0.4

0.2

15050 100 200 2500RECORDER DEFLECTION (mm)

Figure 4 .6 P lo t of Recorder R e f l e c t i o n Versus Concentrat ion of Nat ive F lu id in Produced Stream.

FluidReservoir

Filter Filter

Synthetic Sandstone MiniaquiferIsopotential

WellField

Flush LineRecorder

Adapter

—Pump

Recorder ChemicalOscillometer

Figure 4 .7 Schematic of Experimental Apparatus.

45

I t should be noted t h a t 1n a l l the exper imental runs,

the more dense f l u i d was i n j e c t e d . This does not a f f e c t

the exper imental r e s u l t s in any way since the recovery e f

f i c i e n c i e s obtained would be the same f o r a given set of

condi t ions regardless of whether the i n j e c t e d f l u i d was

more dense or less dense than the na t i v e f l u i d . The more

dense f l u i d was i n j e c t e d because i t was more convenient to

do so in the exper imental work.

4 .5 Determinat ion of M in i a q u i f e r Permeab^1\ t v

To determine the p e r m e a b i l i t y of the m i n i a q u i f e r , a

pressure gauge was at tached to the i s o p o t e n t i a l . A pres

sure t ransducer wi th a readout was at tached to the wel l

f i e l d contro ls so t h a t the pressure a t any wel l could be

monitored. Well 1 was opened to f low under the hy d ro s ta t ic

head produced by the f l u i d in the f l u i d r e s e r v o i r (see

Fig . 4 . 7 ) . The f low r a t e was determined by observing the

t ime 1t took f o r a measured volume of f l u i d to f low from

wel l 1. During th i s f low pe r io d , the pressures a t the i s o

p o t e n t i a l and a t we l ls 2, 3, 4 , 5, and 6 were recorded.

The pressures f o r we l ls 2, 4 , and 6 were 1.32 psig and

t h e i r d istances from wel l 1 are the same--31.12 cm. The

pressures f o r we l ls 3 and 5 were 1.27 psig and t h e i r d i s

tances from wel l 1, 22.00 cm. The pressure a t the isopo

t e n t i a l was 1.62 psig and the minimum d istance from wel l 1

to the i s o p o t e n t i a l is 142.00 cm. The f low r a t e was

0 .943 cc/sec and the f l u i d v i s c o s i t y was 0 .946 cp.

46

Darcy's Law f o r steady s t a t e r a d i a l f low of an i n

compressible f l u i d 1s:

. Zjrhk [ P2 ' Pl> ............................................ ( 4 . 1 )q \x )n ( r 2/ r ] )

Where: q = volumet r ic f low r a t e (c c / s e c )

h = a q u i f e r thickness (cm)

k ■ i n t r i n s i c p e r m e a b i l i t y of a q u i f e r ( d a r c i e s )

y = absolute v i s c o s i t y of f l u i d (cp)

P2 = pressure a t radius r 2 (atm)

p = pressure at radius r^ (atm)

r 2 = radius (cm)

r^ = radius (cm)

From Equation 4 . 1 , a semi- log p l o t of pressure versus

radius should be a s t r a i g h t l i n e . The pressures fo r the

f low t e s t were p l o t t e d as shown in Figure 4 . 8 . A s t r a i g h t

l i n e was drawn through the point p l o t t e d f o r we l ls 2 , 4,

and 6 and the poin t p l o t t e d f o r the i s o p o t e n t i a l since

these pressures would most c lo s e ly approximate those f o r a

t r u l y r a d i a l system. Using Equation 4.1 and a f low r a t e

of 0 .943 x 2 or 1.886 cc/sec to a l low f o r the f a c t th a t

wel l 1 is a h a l f w e l l , the p e r m e a b i l i t y of the m i n i a q u i f e r

was c a lc u l a t e d to be 5.57 da rc ie s .

PRES

SURE

(p

sig)

47

2.0

Isopotential

Wells 2 ,4 &6

Wells 3 & 5

a .

1.0

0.5100010 100

RADIUS (cm)

Figure 4 . 8 P lo t of Pressure Versus Radius Used to Determine M i n i a q u i f e r P e r m e a b i l i t y .

48

4 .6 Determinat ion of Mini a q u i f e r Homogeneity and Porosi ty

The m in i a q u i f e r homogeneity and po ros i ty were d e t e r

mined during Run No. 1. The complete opera t ing procedure

and parameters f o r Run No. 1 are given 1n Appendix E. The

dens i ty d i f f e r e n c e between the na t i v e f l u i d and i n j e c t e d

f l u i d was made very low (0 .002 gm/cc) f o r th i s run so t h a t

g r a v i t a t i o n a l segregat ion would not be a s i g n i f i c a n t f a c

t o r .

As i n j e c t i o n progressed, the f r o n t of i n j e c t e d f l u i d

was observed to advance in e s s e n t i a l l y r a d i a l arcs (see

Fig. 4 . 9 ) . The advancement of the f r o n t 1n r a d i a l arcs

was considered s u f f i c i e n t evidence t h a t the m in i a q u i f e r

was homogeneous and i s o t r o p i c .

As the p o s i t i o n of each f r o n t was marked (see

F1g. 4 . 9 ) , the t ime of i n j e c t i o n corresponding to tha t

f r o n t was recorded and the volume of f l u i d In j e c t e d up to

t h a t t ime computed. The area w i t h i n each f r o n t a l po s i t io n

was computed and a p l o t of area swept versus volume i n

j e c te d was made (see F ig . 4 . 1 0 ) . For any i n j e c t i o n t ime:

Volume I n j e c t e d ■ (Area Swept) • h • <J>

o r :

h ‘ ♦ * V° A r ea Swept* 8*1 ' S,ope of P' ot ( F 1 9 ' 4 ' 10)

Using the above equat ion and Figure 4 . 1 0 , the poro s i ty of

the m i n i a q u i f e r was computed to be 0 . 2 5 .

49

Figure 4 . 9 - -Photograph showing the c i r c u l a r advance o f the i n j e c t e d f l u i d f r o n t ( i n j e c t i o n is tak ing pi ace i n w e l 1 1 ) .

VOLU

ME

INJE

CTE

D

(cm

3)50

6 0 0 0

4 0 0 0

SLOPE = 0 9 5 cm

2000

6 0 0 05 0 0 01000 3 0 0 02000AREA SWEPT (cm2)

4 0 0 0

Figure 4 .10 P lo t o f Volume I n j e c t e d Versus Area Swept f o r Run No. 1.

51

4 . 7 Determinat i on of t h e L o n g i t u d i n a l O l s p e r s i v i t y C o e f f i c i e n t For the Mln iaqu i^er

To determine the lo n g i t u d i n a l d i s p e r s i v i t y c o e f f i

c i e n t fo r the m i n i a q u i f e r , two exper imental runs (Run No.

D-l and Run No. D-2) were made. In both runs, the de n s i

t i e s and v i s c o s i t i e s of the I n j e c t e d and na t i v e f l u i d s

were e x a c t l y matched so t h a t there would be no g r a v i t a

t i o n a l segregat ion . F lu id was i n j e c t e d in wel l 1 a t a

constant r a t e ( 2 4 . 1 3 8 cc/min f o r both runs) f o r a prede

termined length of t ime (7090 seconds f o r Run No. D - l ,

7441 seconds f o r Run No. D - 2 ) . The i n j e c t e d f l u i d was

then produced through wel l 1 a t a constant r a t e (2 4 .138

cc/min f o r both runs) and a complete concent rat ion p r o f i l e

of na t i v e f l u i d in produced stream versus t o t a l t ime since

s t a r t of i n j e c t i o n was obtained for both runs. Points

from these concent ra t ion p r o f i l e s are p l o t t e d on

F1gure 4 . 1 1 .

Using the computer program descr ibed 1n Appendix D,

t h e o r e t i c a l concent ra t ion p r o f i l e s were computed f o r the

above condi t ions and f o r a range of l o n g i t u d i n a l d i s p e r

s i v i t y c o e f f i c i e n t s . In both cases (Run No. D-l and Run

No. D-2) the concent ra t ion p r o f i l e s computed using a d i s

p e r s i v i t y c o e f f i c i e n t of 0 .0 2 cm most c lo s e ly matched the

e x pe r i m e n ta l l y determined concent ra t ion p r o f i l e s . On th i s

b a s is , the l o n g i t u d i n a l d i s p e r s i v i t y c o e f f i c i e n t of the

m i n i a q u i f e r was taken as 0 .02 cm.

52

-? 100

< 90 l i i

H 80 CO

O 70 Ulo3 60oo? 5°CL

? 4 0O5 30_Ju. 20 COMPUTED (« = 0.02 cm)

EXPERIMENTAL DATAUJ><z

14400 14800 1520012800 13200 13600 14000 15600

TIME SINCE START OF INJECTION (S tconds)

8 100 3E

2 90

80CO

S 70Oo 60 oa. so

- 40O3 3 0- IU.

20UJ> COMPUTED ( a - 0 0 2 cm)

O EXPERIMENTAL DATA»-<Z

13600 14000 14400 14800 15200 15600 16000 16400

TIME SINCE START OF INJECTION (Stcondt)

Figure 4.11 Comparisons of Computed and Observed Concent r a t i o n P r o f i l e s Used in Computing the Longi tud ina l D i s p e r s i v i t y C o e f f i c i e n t , (a ) Comparisons f o r Run D - l ; (b) Comparisons fo r Run D-2.

53

4 .8 Comparison of Experimental and Predicted Results

Seventeen exper imental runs were made 1n the m1n1-

a q u l f e r using a wide range of the parameters t h a t a f f e c t

recovery e f f i c i e n c y . Complete de sc r i p t i ons and r e s u l t s of

16 of these runs are presented in Appendix E. Run No. 7

was of no q u a n t i t a t i v e va lue , hence, no re s u l ts f o r th i s

run are presented.

Table 4 .2 compares the observed and predic ted recov

ery e f f i c i e n c i e s f o r s ing le wel l o p e ra t i on . For s in g le

wel l opera t ion a l l the predicted values of recovery e f f i

ciency were less than the observed values. On the average

the d i f f e r e n c e between the predic ted and observed values

f o r the f i r s t cycle was 13 percent wi th a range of 8 to 19

percent . For the second c y c le , the average d i f f e r e n c e was

7 percent wi th a range of 4 to 10 percent . The t h i r d

cycle had an average d i f f e r e n c e of 7 percent wi th a range

of 5 to 8 percent . In a l l runs t h a t were c a r r i e d out f o r

more than one c y c l e , the recovery e f f i c i e n c y improved wi th

each cycle and the r es u l t s of computat ion more near ly ap

proached the observed recovery e f f i c i e n c y .

Table 4 . 3 compares the observed and pred ic ted r e

covery e f f i c i e n c i e s for m u l t i p l e wel l op era t i on . For mul

t i p l e wel l operat ion a l l the pred ic ted values of recovery

e f f i c i e n c y were equal to or less than the observed values

except f o r the f i r s t cycle of Run No. 15. On the average,

the pred ic ted values of recovery e f f i c i e n c y f o r the f i r s t

TABLE 4 . 2- -Compar ison o f Observed and P re d i c te d Recovery E f f i c i e n c i e s f o r S in g le WellOperat ion

Rlin F i r s t Cycle Second Cycle Third CycleObserved Rredi cted Observed Predicted Observed predi cted

1 93* 85*

2 94 86 95% 90% 97% 92%

3 91 77 94 85 __

6 74 61 87 77 92 85

8 93 78 94 86 96 88

11 77 64

12 77 64 89 85 92 87

14 68 58 83 79 89 81

16 39 20 - -

17 17 0 - -

cn

TABLE 4 . 3 - -Comparison o f Observed and P re d i c te d Recovery E f f i c i e n c i e s f o r M u l t i p l eWell Opera t ion

Run No. Fi rs t Cycl e Second Cycl e Third Cycl eObserved Predi cted Observed Predi cted Observed Predi cted

4 86% 82%

5 90 76 90% 83% 98% 88%

9 90 76 93 85 99 89

10 91 78 97 86 100 89

13 80 78 88 85 91 86

15 71 72 78 78 89 80

cncn

56

cyc le were 8 percent lower than the observed values. The

range of d i f f e r e n c e s ranged from -1 to 14 percent . For the

second cycle the average d i f f e r e n c e was 6 percent wi th a

range o f 0 to 11 percent . The t h i r d cycle had an average

d i f f e r e n c e of 9 percent wi th a range of 5 to 11 percent .

In a l l runs t h a t were c a r r i e d out f o r more than one c y c l e ,

the recovery e f f i c i e n c y improved wi th each cycle .

CHAPTER V

THE ECONOMICS OF FRESH-WATER STORAGE

IN SALINE AQUIFERS

5.1 General

To i l l u s t r a t e the r e l a t i v e economy of s t or in g f resh

water in s a l i n e a q u i f e r s , i t w i l l be assumed t h a t the mu

n i c i p a l water department of an urbanized, coastal area

(such as New Orleans) has excess f i l t e r capac i ty and r e

qui res a d d i t i o n a l storage c a p a c i t y . The a d d i t i o n a l storage

f a c i l i t i e s should have the c a p a b i l i t y of supplying potable

water a t the r a t e of 10 ,800 ,000 apd from November 1 through

January 31 (92 days) . This per iod is the normal dura t ion

of low f low f o r the r i v e r which is the pr imary water source

of the area and the per iod of the poorest q u a l i t y wate r .

The requi red storage capac i ty needed is approximately one

b i l l i o n g a l l o n s . For purposes of c a l c u l a t i o n the "exact"

number, 92 x 10 ,800,000 or 993 ,600 ,000 g a l l o n s , w i l l be

used. Less m in e ra l i z e d w a te r , s u i t a b l e fo r s t o r a g e , is

normal ly a v a i l a b l e from February 1 through July 15.

In such a densely populated urban area the most

f e a s i b l e way ( from a purely physical s tandpoin t ) of o b t a i n

ing surface storage is through the use of closed s tee l

tanks, when topography, poss ib le contaminat ion during

57

58

hu rr i c a n e s , and evapora t ion during droughts are taken In to

c o n s id e r a t i o n . An a l t e r n a t e method of storage would be

the i n j e c t i o n of the excess f resh water I n t o a s a l i n e

a q u i f e r 1f a s u i t a b l e a q u i f e r 1s a v a i l a b l e . Assuming t h a t

a s u i t a b l e storage a q u i f e r e x i s t s , an economic comparison

between the two methods of storage f o l l o w s .

5 .2 Storage in Steel Tanks

According to Klmbler e t a l . (1973) a good es t imat ing

f i g u r e f o r tank cost is ten cents per ga l l on of storage

c a p a c i t y . Hence, the cost o f tanks to s tor e 99 3 , 600 ,000

gal lons w i l l be 9 9 ,3 60 , 00 0 d o l l a r s . The cost of tank farm

piping 1s est imated to be 200,000 d o l l a r s . Engineering

and legal fees are est imated to be 1 , 000 ,000 d o l l a r s .

Al lowing a 5 percent cont ingency, th is gives a t o t a l con

s t r u c t i o n cost of ( 9 9 , 3 6 0 , 0 0 0 + 200,000 + 1 , 0 0 0 ,0 0 0 ) x 1.05

or 105 ,588 ,000 d o l l a r s . On the basis of an 8 percent i n

t e r e s t r a t e and a 50 -year l i f e f o r the tank farm, the

annual cost of i n t e r e s t and c a p i t a l recovery w i l l be

0.0817 x 105 ,588 ,000 or 8 , 6 2 7 ,0 0 0 d o l l a r s . Assuming a

tank s i z e o f 160 f e e t 1n diameter by 50 f e e t high

( 7 ,5 0 0 , 0 0 0 g a l l o n s ) , 132 tanks w i l l be needed to s tore

993 ,600 ,000 ga l l o n s . I f a tank spacing of 200 f e e t 1s

assumed, the land area requi red f o r the tank farm w i l l be

122 acres. Assuming land a c q u i s i t i o n a t 10,000 d o l l a r s

59

per acre , the annual i n t e r e s t charge on bonds to purchase

land w i l l be 122 x 10,000 x 0 .0 8 or 98 ,000 d o l l a r s .

Annual opera t ing and maintenance costs are est imated to be

1 percent of the const ruc t ion cost , 105 ,588 ,000 x 0.01 or

1 . 05 6 .0 00 d o l l a r s . This gives a t o t a l annual cost of

8 .62 7 .0 0 0 + 98,000 + 1 ,0 56 ,00 0 or 9 ,7 8 1 ,0 0 0 d o l l a r s to

st ore 9 9 3 ,6 00 , 00 0 gal lons of water in sur face tanks.

5. 3 Storage in Sa l ine Aqui fe r

As an a l t e r n a t e to tanks, consider underground s t o r

age in a s a l i n e a q u i f e r having the p r op e r t ie s given in

Table 5 . 1 .

Assume a we l l f i e l d c o n f i g u r a t i o n as shown in Figure 5 . 1 .

The requi red product ion r a t e of each we l l w i l l be

10 ,800,000 t (1440 x 5) or 1500 gpm. The i n j e c t i o n r a te

f o r each wel l w i l l be 1000 gpm. The w e l ls w i l l have

TABLE 5 . 1 - - S t o r a g e Aq ui fe r C h a r a c t e r i s t i c s

Aq ui f e r S t o r a t l v i t y :Aq u i f e r Poros i ty :Longi tudinal D i s p e r s i v i t y C o e f f i c i e n t : C o e f f i c i e n t of Molecular D i f f u s io n : S t a t l c Water L e v e l :Depth to Roof of Aqui f e r :Nat ive Water Densi ty:

Aqui fe r Thickness: Aqui fe r Pe r m e a b i l i t y :

100 f t400 g p d / f t 2 (20 d ar c i es )10-4

Nat ive Water V i s c o s i t y :

0 .30 1 .0 cm10"® cm^/sec land surface 800 f t1.0087 (gm/cc) (8 ,0 0 0 ppm TDS)1.0 cp

60

2 5 0 '2 5 0 '

2 5 0

250*

Figure 5.1 Assumed Well F ie ld Co nf i gu ra t ion f o r Cost Compari son.

61

l 8 -1nch diameter casing wi th 100 f t of 12- inch s t a in l e s s

st ee l screen t h a t is surrounded by a 3- inch gravel w a l l .

The t o t a l depth of each wel l from ground surface to the

bottom of the screen w i l l be 900 f t . In add i t i on to the

pump, motor, p i p i n g , va lv es , e t c . , normal ly associated

wi th a water w e l l , each wel l w i l l be equipped wi th the

f o l l o w in g : (1) a p ip ing mani fold t h a t w i l l a l low bypass

ing of the product ion s t r i n g so the wel l can be used f o r

i n j e c t i o n , (2) f low control regu la tors t h a t w i l l accu

r a t e l y control the i n j e c t i o n and product ion r a t e s , and

(3) ins t rumenta t ion to monitor and record the r a t e and

pressure f l u c t u a t i o n s and the changes in water q u a l i t y as

i n j e c t i o n or product ion proceeds. The water to be i n

je c te d w i l l a r r i v e a t the wel l f i e l d a t a pressure of

40 p s i , have a dens i ty of 1 .0 gm/cc, and a v i s c o s i t y of

1.0 cp. The a d d i t i o n a l pressure needed to i n j e c t the

water at the s p e c i f i e d rates w i l l be produced by a s in g le

booster pump f o r the e n t i r e we l l f i e l d . In a d d i t i o n to

a l l the pre v ious ly mentioned equipment, i t w i l l be neces

sary to have a w a t e r - t r e a t i n g f a c i l i t y f o r c h l o r i n a t i o n

and pH adjustment of the produced water before i t 1s r e

introduced in to the d i s t r i b u t i o n system. The wel l f i e l d

w i l l r eq u i r e a land area of 8 acres.

Using the computat ional procedures descr ibed in

Chapter I I I , i t was found t h a t i t w i l l be necessary to i n

j e c t in to wel l No. 1 f o r 71 days a t 1000 gpm before the

lagging edge of the mixed zone passes the outer r ing of

62

w e l l s . This w i l l give a cushion water volume of 1000 x

1440 x 71 or 102,240 ,000 g a l l o n s . The cost of th i s volume

of water is considered as pa r t of the c a p i t a l cost of the

system. At the end of 71 days, i n j e c t i o n w i l l begin in

a l l f i v e w e l ls a t 1000 gpm per w e l l . Again using the com

p u t a t i o n a l procedure descr ibed in Chapter I I I , i t was found

t h a t i f f resh water is i n j e c t e d in a l l f i v e wel ls at 1000

gpm per wel l f o r 156 days, al lowed to stand f o r 117 days,

and then produced a t 1500 gpm per we l l u n t i l breakthrough

occurs a t the outer r ing of w e l l s , the volume produced a t

breakthrough w i l l be 995 ,944 ,000 ga l l o n s . This exceeds

the requi red product ion of 993 ,600 ,00 0 g a l l o n s , th e r e f o r e

the procedure should be adequate to supply the requi red

water since subsequent cycles w i l l have b e t t e r re c o v e r i e s .

Hence, the maximum water loss per cyc le w i l l be

(156 x 1440 x 1000 x 5) - 99 3 , 600 ,000 or 129 ,600 ,000 g a l

lons. The cost of th i s water w i l l be considered as an

opera t ing cost .

Using the Theis equat ion ( T h e i s , 1935) to determine

the pressure changes t h a t w i l l be encountered during I n

j e c t i o n , i t was found t h a t a head of 140 f t w i l l have to

be produced by the booster pump in order to i n j e c t a t the

s p e c i f i e d r a t e s . The motor horsepower requi red by the

booster pump w i l l be (5 x 1000 x 140) t (3960 x 0 . 8 0 ) or

220 horsepower. The drawdown during product ion w i l l be

approximately 325 f t . The horsepower requi red to l i f t the

water 325 f t a t a r a t e of 1500 gpm and produce a surface

63

pressure of 40 ps1 w i l l be (1500 x [325 + 92 ] ) f (3960 x

0 . 8 0 ) or 200 horsepower. Each wel l w i l l req u i re a 200

horsepower motor.

The e l e c t r i c a l power requi red to I n j e c t the cushion

water w i l l be ( [2 20 i 5] x 0 .746 x 71 x 24) t 0 .90 or

63.000 Kw-Hr. The cost of th i s power is considered as

p a r t of the c a p i t a l cost of i n s t a l l i n g the system. The

power requi red during a normal i n j e c t i o n h a l f - c y c l e w i l l

be (220 x 0 .746 x 156 x 24) t 0 .9 0 or 683,000 Kw-Hr. The

cost of t h i s power is an op era t ing expense. The power r e

qui red during a normal product ion h a l f - c y c l e w i l l be

(200 x 5 x 0 .746 x 92 x 24) * 0 .90 or 1 , 830 ,000 Kw-Hr. The

cost of th i s power is an opera t ing expense.

The t o t a l c a p i t a l cost of i n s t a l l i n g the wel l f i e l d

1s shown in Table 5 .2 On the basis of an 8 percent i n t e r

est r a t e and a 25 -year l i f e f o r the we l l f i e l d the annual

cost of i n t e r e s t and c a p i t a l recovery w i l l be 0 .0937 x

( 1 , 2 3 6 , 0 0 0 - 80 ,0 00 ) or 108,000 d o l l a r s . The annual cost

of i n t e r e s t f o r land purchase w i l l be 80,000 x 0 . 08 or

6 .000 d o l l a r s . The annual cost of e l e c t r i c a l power w i l l

be (683 ,000 + 1 , 8 3 0 ,0 0 0 ) x 0 .015 or 38,000 d o l l a r s . The

annual cost f o r water losses w i l l be 129,600 x 0 .05 or

7.000 d o l l a r s . Other annual opera t ing and maintenance

costs are est imated to be 1 percent of the t o t a l c a p i t a l

cost less the costs of land and the hydrogeological s u r

vey, hence, these costs w i l l be ( 1 , 2 3 6 , 0 0 0 - 80,000 -

250,000) x 0.01 or 9 ,000 d o l l a r s . The t o t a l annual cost

64

TABLE 5 . 2 - - C a p i t a l Costs of I n s t a l l i n g Well

a) Hydrogeological Survey *

b) Land (8 acres 0 $ 1 0 , 0 0 0 / a c r e ) =

c) Wells (5 wel ls 0 $75,000 each) *

d) Motor and pump f o r we l ls =($20 ,000 per w e l l )

e) Accessor ies, f low r e g u l a t o r s , v a l v e s , =in s t ru m e n ta t i o n , e t c . , f o r we l ls($10 ,000 per w e l 1)

f ) I n j e c t i o n booster pump =

g) Water t reatment f a c i l i t y *

h) Cushion water =( 1 02 ,2 4 0 ,0 0 0 gal @ 5$ per 1000 ga l )

i ) E l e c t r i c power to i n j e c t cushion water =(63 ,000 Kw-Hr @ 1.5$ per Kw-Hr)

j ) Engineering and lega l fees =(25% of items c through g)

k) Contingency =(20% of items c, d, e , f , g, and j )

Total Capi ta l Cost $1

F ie ld

$250,000

80,000

375.000

100.000

50.000

25.000

50.000

5,000

1 ,000

150.000

150.000

, 236,000

65

of s t o r in g 993 ,600 ,00 0 gal lons of f resh water in the

s a l i n e a q u i f e r is 108,000 + 6 ,000 + 38,000 + 7 ,000 + 9 ,000

or 168,000 d o l l a r s .

5 .4 Comparison of the Two Storage Methods

Table 5 .3 summarizes the annual costs of the two

methods of s torage . I t can be seen th a t storage in the

s a l i n e a q u i f e r has an overwhelming economic advantage

(58 to 1) when compared to sur face storage in s tee l tanks.

Also, sur face storage uses 122 acres of land t h a t can sub

sequent ly be used f o r no o ther purpose than a tank farm,

whi le underground storage in the s a l i n e a q u i f e r requires

an area of only 8 acres and th i s area can simul taneously

be used f o r o ther purposes such as r e c r e a t i o n a l f a c i l i t i e s

or a parking l o t , since most of the s t r u c t u re s are beneath

the land sur f ace .

TABLE 5.3--Summary of Annual Costs f o r Tank Storage and S a l i n e Aq ui fe r Storage

Annual Cost T^ | £ L a ^ _

I n t e r e s t & Ca p i ta l Recovery $8 ,6 27 ,000 $108,000I n t e r e s t on Land 98,000 6,000Power - - 38,000Water Losses - - 7 ,000Operat ion and Maintenance 1 ,056 ,000 9,000

Totals $9 ,7 81 ,000 $168,000

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

6.1 Cone!uslons

On the basis of the r es u l t s obtained from th i s I n

v e s t i g a t i o n , the f o l l ow in g conclusions are drawn:

1. The recovery of f resh water stored In a s a l i n e

a q u i f e r using a wel l f i e l d can be predic ted using the com

p u t a t l o n a l procedure presented 1n t h is d i s s e r t a t i o n . The

wel l f i e l d must by symmetrical and must be operated so

t h a t the I n j e c t e d bubble of f resh water remains essen

t i a l l y c i r c u l a r .

2. The computat ional procedure pre d ic ts the exper l

menta l ly determined recovery e f f i c i e n c i e s f o r m u l t i p l e

wel l systems w i t h i n 10 percent .

3. The predic ted recovery e f f i c i e n c i e s are I n

v a r i a b l y smal le r than the observed recovery e f f i c i e n c i e s .

4. The exper imental work showed: (a ) the recovery

e f f i c i e n c y of the storage process Improves as the l er

of cycles of opera t ion increases , and (b) the l a r g e r the

den s i ty d i f f e r e n c e between the i n j e c t e d and na t i v e f l u i d s

the lower the recovery e f f i c i e n c y .

5. Storage of f resh water in a s a l i n e a q u i f e r by

means of a wel l f i e l d has a tremendous economic advantage

66

84

67

when compared to sur face storage of an equal volume of

water in tanks.

6. The method used to determine the l o n g i t u d in a l

d i s p e r s i v i t y c o e f f i c i e n t of the m in i a q u i f e r can be seadi ly

adapted to f i e l d use.

6 .2 Recommendati ons

Recommendations f o r f u r t h e r study on the storage of

f resh water in s a l i n e aqu i fe rs are:

1. The computat ional procedure proposed in th is

d i s s e r t a t i o n should be expanded by inc lud ing the e f f e c t s

of the f o l l o w in g f i e l d condi t ions on the recovery e f f i

ciency of the storage process:

a. Aq u i f e r dip

b. Unequal v i s c o s i t y r a t i o s between the i n j e c t e d and nat i ve f l u i d s .

c. P r e - e x i s t i n g ground-water movement in the storage a q u i f e r .

2. Experimental and t h e o r e t i c a l i n v e s t i g a t i o n s

should be undertaken to:

a. Determine the r a t e of g r a v i t a t i o n a l segregat io n between misc lb le f l u i d s f o r condi t ions where the dimensionless group, is gr e a te r than 1 (see Fig . 2 . 1 ) .

b. Develop a more e legant mathematical d e s c r i p t io n of g r a v i t a t i o n a l segregat ion for a r a d i a l system than t h a t presented in th i s d i s s e r t a t i o n .

NOMENCLATURE

c- = concent rat ion of i n j e c t e d f l u i d a t any radius and t ime.

cQ = i n i t i a l concent ra t ion of i n j e c t e d f l u i d .

c = concent rat ion of na t i v e f l u i d a t any radius and t i m e .

D = c o e f f i c i e n t of molecular d i f f u s i o n of f l u i d s in porous medium. (cm2/sec)

e r f c (?) = complementary e r r o r funct ion of ?

e r f c (?) = —/~tr

2e~w dw

F = a dimensionless f a c t o r dependent on v i s c o s i t y r a t i o . (F = 1.0 f o r a r a t i o of one)

g = a c c e l e r a t io n due to g r a v i t y , (cm/sec^)

h = a q u i f e r th ickness. (cm)

k = i n t r i n s i c p e r m e a b i l i t y of a q u i f e r , (cm^ in Equation 2 . 7 , darc ies in Equation 4 . 1 )

k^ = h o r i z o n t a l i n t r i n s i c p e r m e a b i l i t y , (cm^)

ky = v e r t i c a l i n t r i n s i c permeabi1i t y . (cm^)

P = pressure, (atm)

Q = q/ (2nh<p) . (cm^/sec)

q = volumetr ic f low r a t e . (cm^/sec)

R * radius of i n j e c t e d f l u i d a t any time t , assuming no mixing or g r a v i t a t i o n a l segregat ion , (cm)

r * r a d i u s . (cm)

S * dens i ty g r a d i e n t , (gm/cm^)

68

69

t = t ime , (sec)

2XL = p r o j e c t i o n of the 50 percent concent ra t ion l i n e on the h o r i z o n t a l f o r a l i n e a r system, (cm)

2XR = p r o j e c t i o n of the 50 percent concent ra t ion l i n e on the h o r i zo n t a l f o r a r a d i a l system, (cm)

a = l o n g i t u d i n a l d i s p e r s i v i t y c o e f f i c i e n t of porous medium, (cm)

Ap = den s i ty d i f f e r e n c e between i n j e c t e d and nat ive f l u i d s , (gm/cc)

0 = the angle t h a t the 50 percent concent rat ion l i n e makes wi th respect to the v e r t i c a l .

p = v i s c o s i t y of the na t i v e f l u i d , ( c e n t l p o i s e )

y- = average v i s c o s i t y of the i n j e c t e d and na t i v e f l u i d s , (po ise )

<J> = a q u i f e r p o ro s i t y , ( f r a c t i o n )

BIBLIOGRAPHY

Ar1s, R . , and N. R. Amundsen, 1957, Some remarks on l o n g i tud ina l mixing or d i f f u s i o n in f ixe d beds, AIChE Jou rna l , v o l . 3, no. 2, June 1957, pp. 280-282.

Bentsen, R. G . , and R. F. N ie ls e n , 1965, A study of plane r a d i a l m ls c lb le displacement in a consol idated porous medium, Transact ions , SPE of AIME, v o l . 234, pt . I I , March 1965, pp. 1 -5 .

Brigham, W. E . , P. W. Reed, and J. N. Dew, 1961, Ex p e r i ments on mixing during mlsc lb le displacement in porous media, T r ans act ions , SPE o f AIME, v o l . 222, pt . I I , March 1961, pp. 1 -8 .

Brown, D. L . , and W. 0. S i l v e y , 1973, Underground storage and r e t r i e v a l of f resh water from a bra ck is h-w ate r a q u i f e r , v o l . 1 of P r e p r i n t s , Second I n t e r n a t i o n a l Symposium on Underground Waste Management and A r t i f i c i a l Recharge, New Orleans , Sept . 26 -3 0 , 1973, pp. 379-419.

Caudle, B. H . , 1963, Laboratory models of o i l r es er vo i rsproduced by na t ura l water d r i v e , Ph.D. D i s s e r t a t i o n , Dept, of Petroleum Engineer ing, U n iv e r s i t y of Texas, Aus t i n .

Cederstrom, D. J . , 1947, A r t i f i c i a l recharge of a br ack lsh- water w e l l , The Commonwealth, Dec. 1947, pp. 31, 71-73.

de Josse l in de Jong, G . , 1958, Longi tudina l and t ransverse d i f f u s i o n in gr anu la r de pos i t s , T ransact ions , AGU, v o l . 39, no. 1, Feb. 1958, pp. 67 -74.

Esmai l , 0. J . , and 0. K. Kimbler , 1967, I n v e s t i g a t i o n of the technica l f e a s i b i l i t y of s t o r in g f resh water in s a l i n e a q u i f e r s , Water Resources Research, v o l . 3, no. 3, 1967, pp. 683 -95.

Esmai l , 0. J . , 1966, I n v e s t i g a t i o n of the techn ica l f e a s i b i l i t y of s t o r in g f resh water in s a l i n e a q u i f e r s ,M.S. Thesis , Dept, of Petroleum Engineer ing, L o u i s i ana State U n i v e r s i t y , Baton Rouge.

70

71

Gardner, G. H. F . , J. Downle, and M. R. J. W y l U e , 1962, Problems 1n the recovery of gas from aqu i fe r s used f o r gas s t o r ag e , Journal of the I n s t i t u t e of Pe t ro leum, v o l . 48, no. 457, Jan. 1962, pp. 1 -6 .

Gardner, G. H. F . , J. Downie, and H. A. Ke nda l l , 1962a, Grav i ty segregat ion of mlsc lb le f l u i d s in l i n e a r models, T r ans ac t i on s , SPE of AIME, v o l . 225, p t . I I , June 1962, pp. 95-104 .

Gelhar , L. W., J. L. Wi lson, J. S. M i l l e r , and J. M. Hamr i c k , 1972, Densi ty induced mixing in confined a q u i f e r s , EPA Water P o l l u t i o n Control Research Ser ies 16060 ELJ 0 3 / 7 2 , March 1972.

Green, D. W., and R. L. Cox, 1968, Storage of f resh water in underground r e s e r v o i r s conta in ing s a l i n e water , Phase I I , Co nt r ib ut io n No. 36, Kansas Water Resources Research I n s t i t u t e , Manhattan, Kansas.

Hoopes, J. A . , and D. R. F. Harleman, 1967, Dispersion in r a d i a l f low from a recharge w e l l , Journal of Geophysical Research, v o l . 72, no. 14, July 1967, pp. 3595-3607.

J os t , W., 1960, D i f f u s i o n in s o l i d s , l i q u i d s , gases, Academic Press I n c . , New York.

Klmbler , 0. K . , R. G. Kazmann, and W. R. Whitehead, 1973, Sal ine a q u i f e r s - - f u t u r e storage r e s e r v o i r s f o r f resh water?, v o l . 1 of P r e p r i n t s , Second I n t e r n a t i o n a l Symposium on Underground Waste Management and A r t i f i c i a l Recharge, New Or leans , Sept . 2 6 -3 0 , 1973, pp. 192-206.

Kimbler , 0. K . , 1971, Personal communication regarding the status of an underground f r e s h - w a te r storage p r o j e c t at Empire, La.

Kimbler , 0. K . , 1970, F lu id model s tudies of the storage of f resh water in s a l i n e a q u i f e r s , Water Resources Research, v o l . 6 , no. 5, Oct. 1970, pp. 1522-1527.

Kohout, F. A . , 1970, R e o r i e n t a t i o n of our s a l i n e water r e sources t h i n k i n g , Water Resources Research, v o l . 6 , no. 5, Oct. 1970, pp. 1442-1448.

Kumar, A . , and 0. K. Kimbler , 1970, E f f e c t of d i s p e r s i o n , g r a v i t a t i o n a l se gr e ga t i on , and format ion s t r a t i f i c a t io n on the recovery of f resh water stored in s a l i n e a q u i f e r s , Water Resources Research, v o l . 6, no. 6, Dec. 1970, pp. 1689-1700.

72

Kumar, A . , 1968, Dispersion and g r a v i t y segregat ion ofmisc lb le f l u i d s 1n porous media f o r s t r a t i f i e d r a d i a l f low systems, M.S. Thesis , Dept, of Petroleum Engi neer ing, Louisiana Sta te U n i v e r s i t y , Baton Rouge.

Moulder, E. A . , 1970, Fresh-water bubbles: a p o s s i b i l i t yf o r using s a l i n e aqu i fe r s to s tore w a te r , Water Resources Research, v o l . 6 , no. 5, Oct . 1970, pp. 1528-1531.

Moulder, E. A . , and D. R. F r a zo r , 1957, A r t i f 1 d a l - r e c h a r g e experiments a t McDonald Well F i e l d , A m a r i l l o , Texas, Texas Board of Water Engineers, B u l l e t i n 5701, Jan. 1957.

P a i n t e r , T. R . , 1971, Unequal dens i ty m is c l b l e d i s p l a c e ments 1n th in homogeneous t i l t e d beds, M.S. Thesis, Dept, of Petroleum Engineer ing, Louisiana Sta te Uni v e r s i t y , Baton Rouge.

Perk ins , T. K . , and 0. C. Johnston, 1963, A review of d i f fusion and d ispers ion 1n porous media, T ransact ions , SPE of AIME, v o l . 228, p t . I I , March 1963, pp. 70 -84.

P o z z i , A. L . , and R. J. B l a c k w e l l , 1963, Design of l a b o r a tory models f o r study of m lsc ib le d isplacement , Transact ions , SPE of AIME, v o l . 228, p t . I I , March 1963, pp. 28-40.

Raimondi, P . , G. H. F. Gardner, and C. B. P e t r i c k , 1959, E f f e c t of pore s t r u c t u r e and molecular d i f f u s i o n on the mixing of m isc ib le l i q u i d s f lowing 1n porous media, P r e p r i n t 43, AIChE-SPE J o in t Symposium on Fundamental Concepts of M is c ib le F lu id Displacement: Par t I I , F i f ty-Second Annual Meet ing, San Francisco, Dec. 6 - 9 , 1959.

Stoessel , R. K . , 1974, Experimental study of m u l t l - c a t i o n d i f f u s i o n 1n an a r t i f i c i a l quar tz sandstone, M.S. Thes is , Dept, of Geology, Louisiana Sta te U n i v e r s i t y , Baton Rouge.

The ls , C. V . , 1935, The r e l a t i o n between the lowering ofthe piezometr ic sur face and the r a t e and dura t ion of a wel l using ground-water s to ra g e , T ransact ions , AGU, v o l . 16, August 1935, pp. 319-524.

T i b b a l s , C. H . , 1970, Temporary storage o f f resh water ina s a l i n e a q u i f e r by use of w e l l s - - a f i e l d exper iment , Report prepared by the USGS in cooperat ion with the Ci ty of Cocoa, F l o r i d a .

APPENDICES

APPENDIX A

DERIVATION OF EQUATIONS

A. 1 De r iv a t i o n of Equation 2 .3

S t a r t i n g wi th Equation 2 .2 :

c i 1 - - 1 e r f c r 2/ 2 - Ot “ V " n/n( 4 / 3 a R + D/Q RH)

( 2 . 2 )

Where:

c. = concent ra t ion of i n j e c t e d f l u i d at r ad iu s , r ,

and t ime, t . (volume f r a c t i o n )

r = rad ius , (cm)

R = radius of i n je c t e d f l u i d at t ime t , assuming no

mixing or g r a v i t a t i o n a l segregat ion , (cm)

Q = q/(2Trh<f>) (cm^/sec)

q = volumet r ic f low r a t e , (cc / se c ) h = a q u i f e r th ickness , (cm)<t> = a q u i f e r p o r o s i t y , ( f r a c t i o n )

D = c o e f f i c i e n t of molecular d i f f u s i o n , (cm /s ec )

a = l o n g i t u d i n a l d i s p e r s i v i t y c o e f f i c i e n t of

porous medium, (cm)

e r f c = complementary e r r o r fu n c t i o n .

Note t h a t :

q • t * nR2h$ ............................................................. ( A . l )

74

75

Hence:

Sh? * r2 ................................................................................. <A- 2 >

D i v i d i n g both sides by 2:

*_a_ . t = 55 .................................................. <A-3>

T h e r e f o r e :

q . t - £ < A - 4 >

Solving for R:

R = / 2 - Q * t ................................................................... ( A . 5)

S u b s t i t u t i n g Equations A . 4 and A . 5 i n t o Equation 2 .2;

c i e r f c r 2/ 2 - R2/ 2

[ 4 / 3 a ( 2 - Q - t ) 3 /2 + DT77 ( A . 6)

Or:

f t ■ * e r f c

r 3 - R3

_ 2 / f T t T( A . 7)

Where:

f ( t ) = 4 / 3 a ( 2 « Q - t ) 3 /2 + D ( 2 l ^ ‘ t )'

Note t h a t Equation A . 7 is i d e n t i c a l to Equation 2 .3

A . 2 D e r iv a t i o n of Equation 2 .4

S t a r t i n g wi th Equation 2 .3 :

76

i 1 e r f c R

2 / T T t T

( 2 . 3 )

Note t h a t :

cn

Hence:

2 / T T t T( A . 8)

M u l t i p l y i n g through by 2:

2c.= 2 - e r f c

2 / n t T( A . 9)

Rearrangi ng:

2c. t= 1 + { 1 - e r f c r 2 - R2_ 2 / T T t T _

( A . 10)

Recal l the i d e n t i t i e s :

e r f c (?) = 1 - e r f (?)

e r f (?) = 1 - e r f c (?)

e r f ( - ? ) = - e r f (?)

( A . 11)

( A . 12)

( A . 13)

77

Using Equation A . 12, note t h a t Equation A . 10 can be w r i t

ten as :

2c n _= 1 + e r f r 2 - R2' ( A . 14)


ten as:

2c.= 1 - e r f R2 - r 2'

2 / r r tT( A . 15)


ten as:

2c.= e r fc R2 - r 2'

_ 2/fTtT_( A . 16)

D iv id ing through by 2:

= i e r f c c o 1

R2 - r 2

L 2 /TTtTJ( A . 17)

Note t h a t Equation A . 17 is i d e n t i c a l to Equation 2 . 4 .

APPENDIX B

COMPUTER PROGRAMS FOR CALCULATING RECOVERY EFFICIENCIES

The three computer programs l i s t e d In the f o l l ow in g

pages are f o r p r e d i c t i n g the recovery e f f i c i e n c i e s of the

storage process f o r one, two, and three cycle operat ion of

a s i n g l e wel l system. The programs are in FORTRAN IV l a n

guage and are w r i t t e n f o r use on an IBM 360/65 system.

A l i s t of the requi red input data is presented a t the

beginning of each program. Fol lowing th i s is a complete

l i s t of a l l the v a r i a b l e names used in the program along

with t h e i r d e f i n i t i o n s .

To i l l u s t r a t e the manner in which the programs have

to be modi f ied f o r use wi th a m u l t i p l e wel l system, a wel l

f i e l d p a t t e rn s i m i l a r to t h a t shown in Figure 7b w i l l be

assumed. The opera t ing procedure would be as fo l lo ws :

(1 ) I n j e c t in to the center wel l u n t i l the lagging edge of

the mixed zone passes the outer r in g of w e l l s . (2) I n j e c t

i n t o a l l f i v e wel ls u n t i l the desi red q u a n t i ty 1s i n j e c t e d .

(2) Al low the i n j e c t e d water to stand u n t i l needed.

(4 ) Produce a l l f i v e wel ls u n t i l breakthrough occurs at

the outer r ing of w e l l s , at which time product ion from the

wel l f i e l d is stopped. Subsequent cycles are made with

i n j e c t i o n beginning 1n a l l f i v e we l ls s imul taneously.

78

79

As s ta ted In Chapter I I I 1t 1s assumed t h a t a l l I n

j e c t i o n and product ion takes place through the center we l l

of the p a t t e r n . The programs l i s t e d 1n the fo l l o w in g pages

are f o r s i ng le wel l systems and 1t 1s assumed t h a t during

any h a l f - c y c l e the r a t e during t h a t h a l f - c y c l e remains

constant . To use the programs f o r the operat ing procedure

o u t l i n e d above prov is ions would have to be made to a l low a

r a t e change during the f i r s t i n j e c t i o n h a l f - c y c l e . This

can be accomplished as fo l low s:

1) Change statement 21 to read:

21 READ(5 , 1 3000)QR1GM,QR1AGM,QR2GM

Where:

QR1GM * I n j e c t i o n r a t e in center wel l u n t i l mixed zone passes outer r ing of w e l l s , (gpm)

QR1AGM = Combined i n j e c t i o n r a te of a l l f i v e w e l l s , (gpm)

QR2GM = Combined product ion r a t e of a l l f i v e w e l l s , (gpm)

2) Immediately fo l l o w in g statement 22 add the statement:

READ(5,15100)TIMDAY

Where:

TIMDAY = Time of i n j e c t i o n in center we l l u n t i l the mixed zone passes the outer r ing of we l ls and i n j e c t i o n begins in a l l f i v e w e l l s , (days)

3) Change statement 24 to read:

24 WRITE( 6 , 1 9 0 0 0 )QR1GM.QRlAGM.QR2GM

4) Immediately f o l l ow in g statement 51 add the statement:

QR1A=QR1AGM*CGMCCS

80

Where:

QR1A = Combined i n j e c t i o n r a te of a l l f i v e w e l l s , (cc /s e c )

5) Immediately f o l 1owing statement 57 add the statement:

TIMSEC=TIMDAY*CFDSEC

Where:

TIMSEC = Time of i n j e c t i o n in center wel l u n t i lmixed zone passes outer r in g of we l ls and i n j e c t i o n begins in a l l f i v e w e l l s , (sec)

6) Immediately fo l l o w in g statements 151 and 191 add the the statements:

TCHECK=TRT-TIMSEC

IF(TCHECK.GE.0.0)QR1=QR1A

Where:

TCHECK = A t ime check to see i f t ime of i n j e c t i o n has reached or exceeded TIMSEC. (sec)

7) Change format statement 13000 to read:

13000 F0RMAT(3F12 .0 )

8) Immediately f o l l o w in g format statement 15000 add the the format statement:

15100 FORMAT(1 FI 2 .0 )

9) Make the fo l l o w i n g changes in the con t in ua t i on s t a t e ments of format statement 19000:

3 11X , ' RATE FOR FIRST STEP', 3 5 X . F 1 4 . 8 / ,

4 11X , 1 RATE FOR SECOND STEP' ,34X , FI 4 . 8 / ,

5 9X,'PRODUCTION RATE FOR FIRST PRODUCTIONHALF-CYCLE' . 9 X . F 1 4 . 8 / )

The above has shown how the programs can be modi f ied

to take care of one r a t e change in the f i r s t i n j e c t i o n

h a l f - c y c l e . I f there is more than one r a t e change in the

81

f i r s t i n j e c t i o n h a l f - c y c l e or 1f the re are r a t e changes in

any subsequent h a l f - c y c l e , e i t h e r i n j e c t i o n or product ion,

they can be handled in a s i m i l a r manner.

I t should be noted t h a t in the s in g l e wel l case and

the m u l t i p l e we l l case i t is necessary to run the one cycle

program to obtain the f l u i d produced dur ing the f i r s t pro

duct ion h a l f - c y c l e and the t o t a l number of computation i n

t e r v a l s through the end of the f i r s t product ion h a l f - c y c l e

before the two-cycle program can be run. This procedure is

necessary since these two values are input data fo r the

two-cycle program. S i m i l a r l y the two-cycle program must

be run before the t h r e e - c y c l e program can be run.

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P R O G R A M T O C A L C U w A T E T H E R E C O V E R Y E F F I C I E N C Y O F T H E P R O C E S S O P S T O R I N G F R E S H W A T E R I N S A L I N E A Q U I F E R S *

P R O G R A M • C Y C L E | A * I S I N G L E W E L L - O N E C Y C L E )

D A T A T O B E R E A D I N

F I R S T C A R O - F O R M A T ! G F 1 2 * 0 )R S T F T ■ R A D I U S A T W H I C H B R E A K T H R O U G H I S C O M P U T E O * I F T )C B T • A L L O W A B L E C O N C E N T R A T I O N O F N A T I V E S A L T W A T E R I N

P R O D U C E D S T R E A M . I V O L U M E F R A C T I O N )

T I L I P T ■ I N T E R V A L L E N G T H F O R C A L C U L A T I O N S D U R I N G I N J E C T I O N H A L F ' - C Y C L E S * I F T )

T I L P F T ■ I N T E R V A L L E N S T H F O R C A L C U L A T I O N S 3 U R I N S P R O D U C T I O N H A L F - C Y C L E S . I F T )

R I N C P T » L E N G T H O F I N C R E M E N T F O R C A L C U L A T I O N O F M I X E D Z O N E L E N G T H S * ( F T )

T I N C F T ■ I N C R E M E N T B Y W H I C H T I L I F T - I S I N C R E A S E D I F M I X E O Z O N E I N T E R S E C T S T H E L I N E S O U R C E D U R I N G C A L C U L A T I O N S F O R F I R S T I N T E R V A L O F F I R S T I N J E C T I O N H A L F - C Y C L E * ( F T )

S E C O N O C A R O - F O R M A T ( S F 1 2 . 0 >H F T - A Q U I F E R T H I C K N E S S . ( F T )P L Y D A R ■ A Q U I F E R P E R M E A B I L I T Y . ( D A R C V S )P R - P O R O S I T Y . ( F R A C T I O N )A L F * L O N G i r j D I N A u D 1 S P E R S I V I T Y C O E F F I C I E N T * ( C M )D I F M O L • C O E F F I C I E N T O F M O L E C U L A R D I F F U S I O N . ( ( S Q C M ) / S E C )

T H I R D C A R D - F O R M A T ! S F 1 2 * 0 )V I S C P t » V I S C O S I T Y O F T H E I N J E C T E D F R E S H W A T E R . ( C P ) V I S C P 2 • V I S C O S I T Y O F T H E N A T I V E S A L T W A T E R * ( C P ) O E N S I • O E N S I T V O F T H E I N J E C T E D F R E S H W A T E R * ( G M / C C ) D E N S 2 - D E N S I T Y O F T H E N A T I V E S A L T W A T E R . ( G M / C C )A C N G ■ A C C E L E R A T I O N D U E T O G R A V I T Y . ( ( C N / S E C I / S E C I

F O U R T H C A R D - F O R M A T ! 2 F 1 2 . 0 1 0 R 1 G M » I N J E C T I O N R A T E F O R F I R S T I N J E C T I O N N A L P —C Y C L E * I G P M ) 0 R 2 G M ■ P R O O U C T I O N R A T E F O R F I R S T P R O D U C T I O N H A L F - C Y C L E * ( G P M )

F I F T H C A R O - F O R M A T I I F 1 2 . 0 )F L I N G I ■ F L U I O I N J E C T E D I N F I R S T I N J E C T I O N H A L F - C Y C L E * ( S A L )

S I X T H C A R D - F O R M A T ( 1 F I 2 * 0 )T S T 1 D » S T A T I C S T O R A G E T I M E A T T H E E N D O F T H E F I R S T I N J E C T I O N

H A L F - C Y C L E * ( O A V S )

D E F I N I T I O N O F V A R I A B L E N A M E S U S E D I N P R O G R A M

A t ■ I N T E R M E D I A T E V A L U E U S E D I N C O M P U T I N G V A L U E S F I R S T I N J E C T I O N H A L F - C Y C L E * ( S O C M )

O F R U S O F O R

A 2 ■ I N T E R M E D I A T E V A L U E U S S O I N C O M P U T I N G V A L U E SF I R S T P R O D U C T I O N H A L F - C Y C L E * ( S Q C M )

O F R U S O F O R

at ■ I N T E R M E D I A T E V A L U E U S E D I N C O M P U T I N G V A L U E S F I R S T I N J E C T I O N H A L F - C Y C L E . ( S Q C M )

O F A L S O F O R

B 2 ■ I N T E R M E D I A T E V A L U E U S E D I N C O M P U T I N G V A L U E S F I R S T P R O D U C T I O N H A L F - C Y C L E . ( S O C M )

O F R L S O N O ■

C F C P P ■ C O N V E R S I O N F A C T O R * ( P O I S E / C E N T I P O I S E )C F D S C M ■ C O N V E R S I O N F A C T O R . ( I S O C M ) / D A R C V )C F O S E C ■ C O N V E R S I O N F A C T O R . ( S E C / O A V )C F F T C M * C O N V E R S I O N F A C T O R . ( C M / F T )C F G L C C ■ C O N V E R S I O N F A C T O R . ( C C / G A L )

83

C C M C C S ■ C O W E M I O N F A C T M t ( ( C C / S E C ) / ( S A L / M I N ) I C O N S T I ■ V A L U I U S t O I N C H E C K I N S F O R M I A K T H R O M N OW N I N S A

P R O D U C T I O N H A L F - C Y C L E .C O N S T S - V A L U E U S E O I N C H E C K I N S F O R B A E A K T H A O O S H 9 U N I N S A

P R O D U C T I O N H A L F - C Y C L E .C R C E P P a C U M U L A T I V E R E C O V E R Y E P P I C I E N C Y . ( F R A C T I O N )C V L I N 6 * C U M U L A T I V E V O L U M E O P P L U I O I N J E C T E D . ( S A L )C V L R O S a C U M U L A T I V E V 3 L U M E O P I N J E C T E D F L U I D N S C O V C R E O . ( S A L )C V O L I N a C U M U L A T I V E V O L U M E O P P L U I O I N J E C T E D . ( C C )C V O L R O a C U M U L A T I V E V O L U M E O P I N J E C T E D P L U I D R E C O V E R E D . ( C C )C l I a C O M P U T E D C O N C E N T R A T I O N A T T H E R A D I U S A N O A T T H E T I M E

B E I N G C O N S I D E R E D . ( V O L U M E P R A C T I O N )D C a O E N S I T Y G R A D I E N T . ( ( G M / C C I / C M )O M 1 a A C O N S T A N T U S E O I N T H E C O M P U T A T I O N O P T H E 0 1 M E N S I O N L E S S

P A R A M E T E R G I V E N B Y E O U A T I O N 2 . T A .O N 2 a O I M E N S I O N L E S S C R O U P . S E C O N D G R O U P O N R I G H T S I D E OP

E O U A T I O N 2 . T A .0 M 2 A a A C O N S T A N T U S E D I N T H E C O M P U T A T I O N OP T H E D l M E N S I O N L E S S

P A R A M E T E R G I V E N B Y E O U A T I O N 2 . 7 A .O N O M I I a D E N O M I N A T O R O P A R G U M E N T O P C O M P L E M E N T A R Y E R R O R F U N C T I O N

I N E O U A T I O N 2 . S F O R F I R S T I N J E C T I O N H A L F - C Y C L E .O N O M P 2 a D E N O M I N A T O R O P A R G U M E N T O P C O M P L E M E N T A R Y E R R O R F U N C T I O N

I N E O U A T I O N 2 . S F O R F I R S T P R O O U C T I O N H A L F - C Y C L E .O S O P a O E N S I T Y D I F F E R E N C E B E T W E E N I N J E C T E D A N D N A T I V E

F L U I D S . ( G M / C C )0 S C 2 a I N T E R M E D I A T E V A L U E U S E D I N C O M P U T I N G V A L U E S O P R L S O F O R

F I R S T P R O D U C T I O N H A L F - C Y C L E . ( S O C M )P L I N J I a F L U I D I N J E C T E D I N F I R S T I N J E C T I O N H A L F - C Y C L E . ( C C ) P L P R G 2 a F L U I D P R O O U C E D I N F I R S T P R O O U C T I O N H A L F - C Y C L E . ( G A L ) P L P R N 2 a F L U I D P R O O U C E D I N F I R S T P R O D U C T I O N H A L F - C Y C L E . ( C C )H a A O U I F E R T H I C K N E S S . ( C M )I a S U B S C R I P T D E S I G N A T I N G C O M P U T A T I O N I N T E R V A L .N I N T 1 a N U M B E R O P C O M P U T A T I O N I N T E R V A L S T H R U T H E E N O O P T H E

F I R S T I N J E C T I O N H A L F - C Y C L E .H I N T 2 a N U M B E R O F C O M P U T A T I O N I N T E R V A L S T H R U T H E F I R S T

P R O O U C T I O N H A L F - C Y C L E .P L Y a A O U I F E R » E R M E A B I L I T V . ( S O C M )p p p a P R O D U C T O F P I . P O R O S I T Y . A N D T H I C K N E S S . ( C M )P P P I a 2 P P P P . ( C M )O i l a T W O D I M E N S I O N A L F L O W R A T E F O R F I R S T I N J E C T I O N

H A L F - C Y C L E . ( I S O C M ) / S E C )0 P 2 a T W O D I M E N S I O N A L P L O W R A T E F O R F I R S T P R O O U C T I O N

H A L F - C Y C - E . ( ( S O C M ) / S E C )0 R 1 a F L O W R A T E F O R F I R S T I N J E C T I O N H A L F - C Y C L E . ( C C / S E C )0 R 2 a F L O W R A T E F O R F I R S T P R O D U C T I O N H A L F - C Y C L E . ( C C / S E C )O S 1 1 a T W O D I M E N S I O N A L P S E U O O P L O W R A T E F O R F I R S T I N J E C T I O N

H A L F - C Y C L E . ( ( S O C M ) / S E C )0 9 P 2 a T W O D I M E N S I O N A L P S E U O O P L O W R A T E F O R F I R S T P R O O U C T I O N

H A L F —C Y C L E U P T O C O M P U T A T I O N I N T E R V A L A T W H I C H B R E A K T H R O U G H C H E C K I S B E I N G M A D E . ( ( S O C M ) / S E C )

R ( ! ) a R A D I U S O P I N J E C T E D P L U I D A T T H E I T H C O M P U T A T I O NI N T E R V A L . A S S U M I N G N O M I X I N G O R G R A V I T A T I O N A L S E G R E G A T I O N . ( C M )

R B T a R A D I U S A T W H I C H B R E A K T H R O U G H I S C O M P U T E D . ( C M )R C B T ( I ) a L E A S T R A O I U S T O A V A L U E O F C O N C E N T R A T I O N O P C B T F O R T H E

l a s t c o m p u t a t i o n i n t e r v a l O F T H E L A S T P R O O U C T I O N H A L F - C Y C L E . ( C M )

R C B T F T a L E A S T R A O I U S T O A V A L U E O F C O N C E N T R A T I O N O F C B T F O R T H EL A S T C O M P U T A T I O N I N T E R V A L O F T H E L A S T P R O D U C T I O N H A L F - C V C u E . ( F T )

R C E F F a C Y C L E R E C O V E R Y E F F I C I E N C Y . ( F R A C T I O N )R F T a R A O I U S O F I N J E C T E O F L U I D A T T H E I T H C O M P U T A T I O N

I N T E R V A L A S S U M I N G N O M I X I N G OR G R A V I T A T I O N A L S E G R E G A T I O N . ( F T )

R I N C a L E N G T H O F I N C R E M E N T U S E D F O R C A L C U L A T I O N O P M I X E D I O N E L E N G T H S . ( C M )

R I N J I a R A D I U S O F I N J E C T E D F L U I D A T T H E E N D O F T H E F I R S T I N J E C T I O N H A L F - C Y C L E A S S U M I N G N O M I X I N G A N D N O G R A V I T A T I O N A L S E G R E G A T I O N . ( C M )

R L S O a R A D I U S T O L O W E R E N D O F S O P E R C E N T C O N C E N T R A T I O N L I N E . ( C M )

84

M L S O F T ■ R A O I U S T O L O N E R S N O O F S O P E R C E N T C O N C E N T R A T I O N L I N E . ( F T )

R L S O I 1 ■ R A O I U S T O L O N E R E N D O F S O P E R C E N T C O N C E N T R A T I O N A T T H E S T A R T O F T H E F I R S T P R O D U C T I O N H A L F - C Y C L E

L I N E • ( C M )

R U S O ■ R A D I U S T O U P P E R E N D O F S O P E R C E N T C O N C E N T R A T I O N L I N E . ( C M )

R U S O F T ■ R A O I U S T O U P P E R E N D O F S O P E R C E N T C O N C E N T R A T I O N L I N E . ( F T )

R l ■ R A O I U S A T M H I C H C O N C E N T R A T I O N I S B E I N G C O M P U T E D . I N N E R R A O I U S O F N I X E D Z O N E . ( C M )

A L S O

R I F T ■ I N N E R R A D I U S O F M I R E D Z O N E . ( F T )R 2 ( I ) ■ O U T E R R A D I U S O F M I X E D Z O N E . ( C M )R 2 F T M O U T E R R A D I U S O F M I X E D Z O N E . ( F T )R 3 a L E N G T H O F M I X E D Z O N E . ( C M )R 3 F T a M I X E D Z O N E L E N G T H . ( F T )R 4 a T I L I / 2 . ( C M )R S a R A O I U S TO T H E M I O P O I N T B E T M E E N R ( t ) A N D R ( l - I ) . ( C M )R S S O m R S S O U A R E O . ( S Q C M )R » a T I L P / 2 . ( C M )S O S N O M a D E N O M I N A T O R O F A R G U M E N T O F C O M P L E M E N T A R Y E R R O R F U N C T I O N

I N E Q U A T I O N 2 . 8 U S I N G P S E U D O N A T E S T O C H E C K F O N B R E A K T H R O U G H D U R I N G A P R O O U C T I O N M A L E - C Y C L E *

T I L I ■ I N T E R V A L L E N G T H F O R C O M P U T A T I O N S D U R I N G I N J E C T I O N H A L F - C Y C l E S . ( C M )

T I L P ■ I N T E R V A L L E N G T H F O R C O M P U T A T I O N S D U R I N G P R O D U C T I O N H A L F - C Y C L E S . ( C M )

T R T ■ C U M U L A T I V E T R A V E L T I M E O F F R E S H M A T E R - S A L T M A T E R I N T E R F A C E . ( S E C )

T R T I ■ C U M U L A T I V E T R A V E L T I M E O F I N T E R F A C E T H R U E N D O F F I R S T I N J E C T I O N H A L F - C Y C L E . ( S E C )

T R T 2 ■ C U M U L A T I V E T R A V E L T I M E O F I N T E R F A C E T H L U C O M P U T A T I O N I N T E R V A L A T M H 1 C H B R E A K T H R O U G H C H E C K I S B E I N G M A D E . ( S E C )

T S T t ■ S T A T I C S T O R A G E T I R E A T T H E E N D O F T H E F I R S T I N J E C T I O N

H A L F - C Y C L E . ( S E C )T S P » P S E U D O T I N E U S E O I N G R A V I T A T I O N A L S E G R E G A T I O N

C A L C U L A T I O N S . ( S E C )T S P D ■ P S E U D O T I M E U S E D I N G R A V I T A T I O N A L S E G R E B A T I O N

C A L C U L A T I O N S . ( D A Y S )

T T - T I M E O F T R A V E L A C R O S S A N Y C O M P U T A T I O N I N T E R V A L P L U S T H EP S E U D O T I N E F O R T H A T I N T E R V A L . ( S E C )

T T D ■ T I M E O F . T R A V E L A C R O S S A N Y C O M P U T A T I O N I N T E R V A L P L U S T H EP S E U O O T I M E F O R T H A T I N T E R V A L . ( O A V S I

T I ■ T I M E A T M H I C H C O N C E N T R A T I O N I S B E I N G C O M P U T E D D U R I N GF I R S T I N J E C T I O N H A L F - C Y C L E . A L S O C U M U L A T I V E T R A V E L T I M E O F I N T E R F A C E T H R U E N D O F F I R S T I N J E C T I O N H A L F - C Y C L E . ( S E C )

T 2 a T I M E A T M H I C H C O N C E N T R A T I O N I S B E I N G C O M P U T E D D U R I N GF I R S T P R O D U C T I O N H A L F - C Y C L E . ( S E C )

T i l - T I M E O F T R A V E L A C R O S S A N Y C O M P U T A T I O N I N T E R V A L . ( S E C )T 1 1 0 ■ T I M E O F T R A V E L A C R O S S A N Y C O M P U T A T I O N I N T E R V A L . ( D A Y S )V I S ■ M E A N V I S C O S I T Y O F I N J E C T E D A N O N A T I V E F L U I D S . ( P O I S E )V I S C P - M E A N V I S C O S I T Y O F I N J E C T E D A N O N A T I V E F L U I D S . ( C P )V O L N R ■ T O T A L V O L U M E O F I N J E C T E O F L U I D N O T R E C O V E R E D . I C C )X a V A L U E O F O I M E N S I O N L E S S P A R A M E T E R G I V E N B Y

E Q U A T I O N 2 . 7 A .X L I I } - H O R I Z O N T A L P R O J E C T I O N O F B O P E R C E N T C O N C E N T R A T I O N

L I N E F O R . I N E A R G E O M E T R Y . ( C M )X L F T • H O R I Z O N T A L P R O J E C T I O N O F S O P E R C E N T C O N C E N T R A T I O N

L I N E F O R L I N E A R G E O M E T R Y . ( F T )X R ( I ) - H O R I Z O N T A L P R O J E C T I O N O F S O P E R C E N T C O N C E N T R A T I O N

L I N E F O R R A D I A L G E O M E T R Y . ( C M )X R F T ■ H O R I Z O N T A L P R O J E C T I O N O F S O P E R C E N T C O N C E N T R A T I O N

L I N E F O R R A D I A L G E O M E T R Y . ( F T )X X a A R G U M E N T O F C O M P L E M E N T A R Y E R R O R F U N C T I O N F O R

E Q U A T I O N 2 . S .Y L ( I ) - R A T I O O F H O R I Z O N T A L P R O J E C T I O N O F S O P E R C E N T

C O N C E N T R A T I O N L I N E T O A O U I F E R T H I C K N E S S F O R L I N E A R G E O M E T R Y .

85

C V R I I ) ■ H A T 1 0 O F H O R I Z O N T A L P R O J E C T I O N O F SO P E R C E N TC C O N C E N T R A T O R L I N E T O A O U I F E R T H I C K N E S S F O R R A D I A LC 6 E O N E T R T •Cc 4 4 4 4 4 4 4 4 4 4 4 4 4 6 4 4 * » * 4 4 4 4 * 4 * * 4 4 * * « 4 * * 4 * 6 * 4 4 4 * 4 * * * 6 * * * » » * * * » » 4 4 4 4 * 6 <cCc c

0 1 M E N S I O N R ( t 0 0 0 ) • R 2 I l O O O J . Y L l I 0 0 0 I . X L I 1 0 0 0 1 * X R ( 1 0 0 0 1 • R C S T t l O O O l

c cC P A R T 1 - R E A D I N C D A T A

CC

2 0 R E A O I S . 1 0 0 0 0 * E N D * I 3 0 0 I R B T F T * C B T * T I L I F T * T I L P F T * R 1 N C F T * T I N C F T R E A D 1 S . 1 t O O O ) H F T * P L V D A R * P R * A L F . D I F N O LR E A O I S • 1 2 0 0 0 ) V I S C P 1 • V I S C P 2 * O E N S 1 * D E N S E • A C N G

2 1 R E A O I S . I 3 0 0 0 I Q R I C R . O R 2 C N R E A 0 1 S . 1 4 0 0 0 ) F L I N C I

2 2 R E A D ( 5 * I S 0 0 0 1 T S T I D

CC C A L C U L A T I O N O F D E N S I T T D I F F E R E N C E A N D N E A R V I S C O S I T Y O F F L U I D S

CD S O F - A B S I D E N S l - O E N S S l V I S C P * I V I S C P I 4 V I S C P 2 1 / 2 . 0

P A R T 2 * P R I N T I N S D A T A

V R I T E I 6 * 1 6 0 0 0 I H F T • P . V O A R * P R * A L P * 0 I F M O L ■ R I T E ( 6 . 1 7 0 0 0 ) V I S C P 1 • V I S C P 2 » V I S C P W R I T E 1 6 * 1 0 0 0 0 1 D E N S I * D E N S 2 * O S O F * A C N C

2 4 M R I T E I 6 * 1 0 0 0 0 ) Q R I C N . 0 R 2 C M M R I T E I 6 * 2 0 0 0 0 1 F L I N C 1 M R I T E ( A * 2 I 0 0 0 I T S T I D

C

M R I T E I 6 * 2 2 0 0 0 )C

C O T O S O S O T I L I F T * T I L I P T 4 T I N C P T

C O T O S Oc cC P A R T S - C O N S T A N T S A N O C O N V E R S I O N F A C T O R S

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I N T E R S E C T S T H E L I N E S O U R C E D U R I N G C A L C U L A T I O N S F O R F I R S T I N T E R V A L O F F I R S T I N J E C T I O N H A L F - C Y C L I . ( F T )

S E C O N D C A R O - F O R M A T I S F 1 2 . 0 )H F T a A Q U I F E R T H I C i N E S S . ( F T )P L Y O A R a A Q U I F E R P E R M E A B I L I T Y . ( O A R C V S )P R a P O R O S I T Y . ( F R A C T I O N )A L F a L O N G I T U D I N A L O I S P E R S I V I T V C O E F F I C I E N T . ( C M )D 1 F M 0 L a C O E F F I C I E N T 3 F M O L E C U L A R O I F F U S I O N . ( ( S O C M ) / S E C )

T H I R D C A R O - F O R M A T I S F I 2 . 0 )V I S C P 1 a V I S C O S I T Y O F T H E I N J E C T E D F R E S H W A T E R . ( C P )V I S C P 2 a V I S C O S I T Y O F T H E N A T I V E S A L T W A T E R . ( C P )O E N S I a O E N S I T V O F T H E I N J E C T E D F R E S H W A T E R . ( G M / C C )O E N S 2 a D E N S I T Y O F T H E N A T I V E S A L T W A T E R . ( G M / C C )A C N G a A C C E L E R A T I O N D U E T O G R A V I T Y . ( ( C M / S E C ) / S E C )

F O U R T H C A R D - F O R M A T ! 4 F 1 2 . 0 )Q R I G M a I N J E C T I O N R A T E F O R F I R S T I N J E C T I O N H A L F - C Y C L E . ( G P M ) Q R 2 G M a P R O D U C T I O N R A T E F O R F I R S T P R O D U C T I O N H A L F - C Y C L E . ( 6 P M ) 0 R 3 G M a I N J E C T I O N R A T E F O R S E C O N D I N J E C T I O N H A L F - C Y C L E . ( G P H I O R A CM a P R O D U C T I O N R A T E F O R S E C O N D P R O D U C T I O N H A L F - C Y C L E . ( G P M )

F I F T H C A R O - F O R M A T ( 3 F I 2 . 0 )F L I N G I - F L U I O I N J E C T E D I N F I R S T I N J E C T I O N H A L F —C Y C L E . ( G A L ) F L P R G 2 a F L U I D P R O D U C E D I N F I R S T P R O D U C T I O N H A L F - C Y C L E . ( G A L ) F L I N G 3 a F L U I D I N J e C T E O I N S E C O N O I N J E C T I O N H A L F - C Y C L E . ( G A L )

S I X T H C A R O - F O R M A T ( 2 F I 2 . 0 )T S T I O • S T A T I C S T O R A G E T I M E A T T H E E N D O F T H E F I R S T I N J E C T I O N

H A L F - C Y C L E . ( O A V S )T S T S O a S T A T I C S T O R A G E T I M E A T T H E E N D O F T H E S E C O N O I N J E C T I O N

H A L F - C Y C L E . ( D A Y S )

S E V E N T H C A R D - F O R M A T ! I I S )

N I N T 2 a N U M B E R O F C O M P U T A T I O N I N T E R V A L S T H R U T H E F I R S T

P R O D U C T I O N H A L F - C Y C L E .

D E F I N I T I O N O F V A R I A B L E N A M E S U S E D I N P R O G R A M

A I a I N T E R M E D I A T E V A . U E U S E D I N C O M P U T I N G V A L U E S O F R U 5 0 F O RF I R S T I N J E C T I O N H A L F - C Y C L E . ( S O C M )

A 2 a I N T E R M E D I A T E V A L U E U S E D I N C O M P U T I N G V A L U E S OF R U S O F O RF I R S T P R O D U C T I O N H A L F - C Y C L E . ( S O C M )

92

A 3 ■ i n t e r m e d i a t e v a l u e u s e d i n c o m p u t i n g v a l u e s o p N U B S P O NS E C O N O I N J E C T I O N H A L F - C Y C L E . I SO C N )

A 4 a I N T E R M E D I A T E V A L U E U S E D I N C O M M U T I N G V A L U E S OP R U S O P O RS E C O N O P R O D U C T I O N H A L F —C Y C L E * I S O CM>

S t ■ i n t e r m e d i a t e v a l u e u s e d i n c o m p u t i n g v a l u e s o p R L S O P O RF I R S T I N J E C T I O N H A L F - C Y C L E . I S O C M )

B 2 a I N T E R M E D I A T E V A L U E U S E D I N C O M M U T I N G V A L U E S O F R L S O P O RF I R S T P R O D U C T I O N H A L F - C Y C L E . I S O C M )

S 3 ■ I N T E R M E D I A T E V A L U E U S E O I N C O M P U T I N G V A L U E S O F R L S O P O NS E C O N ? I N J E C T I O N H A L F - C Y C L E . I S O C M )

S A ■ I N T E R M E D I A T E V A L U E U S E O I N C O M M U T I N G V A L U E S O F R L S O P O RS E C O N O P R O D U C T I O N H A L F - C Y C L E . I S O C M )

C F C P P a C O N V E R S I O N F A C T O R . I P O I S E / C E N T I P O I S E )C F O S C M a C O N V E R S I O N F A C T O R . I I S O C M ) / D A R C Y )C F D S E C ■ C O N V E R S I O N F A C T O R . I S E C / O A Y )C F F T C M ■ C O N V E R S I O N F A C T O R . I C M / F T )C F G L C C a C O N V E R S I O N F A C T O R . I C C / G A L )C G M C C S a C O N V E R S I O N F A C T O R . I I C C / S E C ) / I G A L / M I N ) )C O N S T ! a V A L U E U S E O I N C H E C K I N G F O R B R E A K T H R O U G H D U R I N G A

P R O O J C T I O N H A L F - C Y C L E .C O N S T 2 a V A L U E U S E O I N C H E C K I N G F O R B R E A K T H R O U G H D U R I N G A

P R O D U C T I O N H A L F - C Y C L E .C R C E F F a C U M U L A T I V E R E C O V E R Y E F F I C I E N C Y . { F R A C T I O N )C V L I N G a C U M U L A T I V E V O L U M E O F F L U I D I N J E C T E D . I G A L )C V L R O G a C U M U L A T I V E V O L U M E O F I N J E C T E D F L U I O R E C O V E R E D . I G A L ) C V O L I N a C U M U L A T I V E V O L U M E O F F L U I D I N J E C T E D . I C C )C V O L R O a C U M U L A T I V E V O L U M E O F I N J E C T E D F L U I O R E C O V E R E D . I C C )C l l a C O M P U T E D C O N C E N T R A T I O N A T T H E R A D I U S A N D A T T H E T I M E

B E I N G C O N S I D E R E D . I V O L U M E F R A C T I O N )OG a D E N S I T Y G R A D I E N T . ( I G M / C C I / C N lD M I a A C O N S T A N T U S E O I N T H E C O M P U T A T I O N O F T H E D I M E N S t O N L E S S

P A R A M E T E R G I V E N B Y E Q U A T I O N 2 . 7 A .D M 2 a O I M E N S I O N L E S S G R O U P . S E C O N D G R O U P O N R I G H T S I D E O F

E O U A T I O N 2 . 7 A •D M 2 A a A C O N S T A N T U S E O I N T H E C O M P U T A T I O N O F T H E D I M E N S 1 O N L E S S

P A R A M E T E R G I V E N B Y E O U A T I O N 2 . 7 A .D N O M I I a D E N O M I N A T O R O F A R G U M E N T O F C O M P L E M E N T A R Y E R R O R F U N C T I O N

I N E O U A T I O N 2 . S F O R F I R S T I N J E C T I O N H A L F - C Y C L E .O N O M I 3 a D E N O M I N A T O R 3 F A R G U M E N T O F C O M P L E M E N T A R Y E R R O R F U N C T I O N

I N E Q U A T I O N 2 . 3 F O R S E C O N O I N J E C T I O N H A L F - C Y C L E .O N O N P S a D E N O M I N A T O R O F A R G U M E N T O F C O M P L E M E N T A R Y E R R O R F U N C T I O N

I N E O U A T I O N 2 . S F O R F I R S T P R O D U C T I O N H A L F - C Y C L E .O N O M P 4 B D E N O M I N A T O R O F A R G U M E N T O F C O M P L E M E N T A R Y E R R O R F U N C T I O N

I N E O U A T I O N 2 . 8 F O R S E C O N O P R O D U C T I O N H A L F - C Y C L E .O S O F a O E N S I T Y D I F F E R E N C E B E T M E E N I N J E C T E D A N D N A T I V E

F L U I D S . I G M / C C )D S C A a I N T E R M E D I A T E V A L U E U S E D I N C O M P U T I N G V A L U E S O F R L S O F O R

S E C O N O P R O D U C T I O N H A L P - C Y C L E . I S O C M )P L I N J I a F L U I O I N J E C T E D I N F I R S T I N J E C T I O N H A L F - C Y C L E . I C C )

F L I N J 3 a F L U I O I N J E C T E D I N S E C O N D I N J E C T I O N H A L F - C Y C L E . I C C )F L P R G 4 a F L U I O P R O D U C E D I N S E C O N D P R O D U C T I O N H A L F - C Y C L E . I G A L )F L P R N 2 a F L U I D P R O O U C E D I N F I R S T P R O D U C T I O N H A L F - C Y C L E . I C C )F L P R N 4 a F L U I D P R O O U C E O I N S E C O N O P R O D U C T I O N H A L F - C Y C L E . I C C )H a A Q U I F E R T H I C K N E S S . I C M )I a S U B S C R I P T D E S I G N A T I N G C O M P U T A T I O N I N T E R V A L *N I N T a I N T E R M E D I A T E V A L U E U S E D I N C A L C U L A T I N G T N E N U M B E R OP

C O M P U T A T I O N I N T E R V A L S .N I N T I a N J M B E R O F C O M P U T A T I O N I N T E R V A L S T H R U T H E S N O OF T H E

F I R S T I N J E C T I O N H A L F - C Y C L E .N I N T 3 a N U M B E R O F C O M P U T A T I O N I N T E R V A L S T H R U T H E S E C O N O

I N J E C T I O N H A L F C Y C L E .N I N T 4 a N U M B E R O F C O M P U T A T I O N I N T E R V A L S T H R U T H E S E C O N O

P R O D U C T I O N H A L F - C Y C L E .P L Y a A Q U I F E R P E R M E A B I L I T Y . I S O C M )P P P a P R O O U C T O F P | . P O R O S I T Y . A N D T H I C K N E S S . I C M )P P P I a 2 4 P P P . I C M )O i l a T W O D I M E N S I O N A L F L O W R A T E F O R F I R S T I N J E C T I O N

H A L F - C Y C L E . I I SO C M J / S E C )0 1 3 a T W O O I M E N S 1 O N A L F L O W R A T E F O R S E C O N O I N J E C T I O N

H A L F - C Y C - E . ( I S O C M I / S E C )

93

O P 2 a T W O D I M E N S I O N A L P L O W M A T E P O M P I M S T P M O O U C T I O NM A L P ~ C Y C L E * ( I S O C M l / S E C I

0 P 4 ■ T W O D I M E N S I O N A L P L O W M A T E P O M S E C O N O P M O O U C T I O NH A L F - C Y C - E . ( ( S O C M I / S E C I

O M I - P L O W M A T E P O M P I M S T I N J E C T I O N H A L F - C Y C L E . ( C C / S E C I0 M 2 ■ P L O W M A T E P O T P I M S T P M O O U C T I O N H A L P - C Y C L E . ( C C / S E C IO R 3 - P L O W H A T E P O T S E C O N O I N J E C T I O N H A L P - C Y C L E . ( C C / S E C IO M A - P L O W R A T E P O T S E C O N O P R O D U C T I O N H A L P - C Y C L E . ( C C / S E C IO S I I • T W O O I M E N S I O N A L P S E U O O P L O W R A T E P O M P I M S T I N J E C T I O N

H A L P - C Y C L E . ( ( S O C M I / S E C I O S 1 3 • T W O D I M E N S I O N A L P S E U O O P L O W R A T E P O R S E C O N O I N J E C T I O N

H A L F —C Y C - E * ( ( S O C M I / S E C I O S P 2 • T W O 0 1 M E N S I O N A L P S E U O O P L O W R A T E P O R F I R S T P M O O U C T I O N

H A L P - C Y C L E . ( I S O C M I / S E C I 0 S P 4 a T W O D I M E N S I O N A L P S E U O O P L O W R A T E P O R S E C O N O P M O O U C T I O N

H A L P - C V C . E u p T O C O M P U T A T I O N I N T E R V A L A T W H I C H B R E A K T H R O U G H C H E C K I S B E I N G M A D E . ( ( S O C M I / S E C I

R ( I I - R A D I U S O P I N J E C T E O P L U I O A T T H E I T H C O M P U T A T I O NI N T E R V A L A S S U M I N G N O M I X I N G OR G R A V I T A T I O N A LS E G R E G A T I O N . ( CM I

R B T • M A O I U S A T W H I C H B R E A K T H R O U G H I S C O M P U T E D . ( C M )R C B T ( I ! ■ L E A S T R A D I U S T O A V A L U E O P C O N C E N T R A T I O N O P C B T P O R T H E

L A S T C O M P U T A T I O N I N T E R V A L O P T H E L A S T P R O D U C T I O NH A L F —C Y C t - E . ( CM I

R C 0 T P T a L E A S T R A D I U S T O A V A L U E O P C O N C E N T R A T I O N O P C B T P O R T H EL A S T C O M P U T A T I O N I N T E R V A L OP T H E L A S T P M O O U C T I O NH A L P - C Y C L E . ( P T 1

R C E P P • C Y C L E R E C O V E R Y E F F I C I E N C Y . ( P R A C T I O N I R P T a M A O I U S O P I N J E C T E D F L U I D A T T H E I T H C O M P U T A T I O N

I N T E R V A L A S S U M I N G N O M I X I N G O R G R A V I T A T I O N A LS E G R E G A T I O N . ( F T )

R I N C a L E N G T H O P I N C R E M E N T U S E D P O R C A L C U L A T I O N O P M I X E O Z O N E L E N G T H S . ( CM I

R I N J I a M A O I U S O P I N J E C T E D F L U I D A T T H E E N O O P T H E F I R S T I N J E C T I O N H A L P - C Y C L E A S S U M I N G N O M I X I N G A N D N O G R A V I T A T I O N A L S E G R E G A T I O N . ( C M |

R I N J 3 a R A O I U S O P I N J E C T E D P L U I O A T T H E E N O O P T H E S E C O N O I N J E C T I O N H A L P - C Y C L E A S S U M I N G N O M I X I N G A N O N O G R A V I T A T I O N A L S E G R E G A T I O N . ( C M )

R L S O • R A D I U S T O L O W E R E N D O P S O P E R C E N T C O N C E N T R A T I O N L I N E . ( C M )

R L S O P T a R A D I U S T O L O W E R E N D O P S O P E R C E N T C O N C E N T R A T I O N L I N E . ( F T )

R L B O I I a R A D I U S T O L O W E R E N O O P S O P E R C E N T C O N C E N T R A T I O N L I N EA T T H E S T A R T O P T H E F I R S T P R O D U C T I O N H A L P - C Y C L E . ( C M )

R L B O I t a R A O I U S T O L O W E R E N O 3 F S O P E R C E N T C O N C E N T R A T I O N L I N EA T T H E S T A R T O P T H E S E C O N O P R O D U C T I O N H A L P - C Y C L E . ( C M )

R L S O P I a R A O I U S T O L O W E R E N O O P S O P E R C E N T C O N C E N T R A T I O N L I N EA T T H E S T A R T O P T H E S E C O N O I N J E C T I O N H A L P - C Y C L E . ( C M )

R U S O a R A D I U S T O U P P E R E N D O P B O P E R C E N T C O N C E N T R A T I O N L I N E . ( C M )

R U S O P T a R A D I U S T O U P P E R E N O O P B O P E R C E N T C O N C E N T R A T I O N L I N E . ( F T )

R l a R A D I U S A T W H I C H C O N C E N T R A T I O N I S B E I N G C O M P U T E D . A L S OI N N E R R A O I U S O P M I X E O Z O N E . ( C M )

R I F T a I N N E R R A O I U S O P M I X E O Z O N E . ( F T )R 2 ( 1 1 a O U T E R R A D I U S O P M I X E D Z O N E . ( C M )R 2 P T a O U T E R R A D I U S O P N I X E D Z O N E . ( F T )R 3 a L E N G T H O P M I X E D Z O N E . ( C M )R 3 P T a M I X E D Z O N E L E N G T H . ( F T )R 4 a T I L 1 / 2 . ( C M )

R S a R A O I U S T O T H E M I D P O I N T B E T W E E N R ( I ) A N D R ( l - I ) . ( C M )R S S O a R S S Q U A R E D . ( S O C M )M « a T I L P / 2 . ( C M )S O E N O M a D E N O M I N A T O R O P A R G U M E N T O P C O M P L E M E N T A R Y E R R O R F U N C T I O N

I N E O U A T I O N 2 . S U S I N G P S E U D O R A T E S T O C H E C K P O R B R E A K T H R O U G H D U R I N G A P R O D U C T I O N H A L P - C Y C L E .

T I L I a I N T E R V A L L E N G T H F O R C O M P U T A T I O N S D U R I N G I N J E C T I O N H A L F - C Y C L E S . ( C M )

94

T I L P

T R T

T R T 1

T R T 2

T R T 3

T R T *

T S T 1

T S T 3

T S P

T S P D

T T

T T O

T1

T 2

T 3

T *

T H

T 1 I OV I SV I S C PV O L N RX

X L ( I I

X L F T

X R ( I )

X R P T

X X

V L ( I I

V R I I )

I N T E R V A L L E N G T H P O R C O M P U T A T I O N S 3 U R I N O P M O O U C T I O N H A L F - C Y C L E S . ( C M I

C U M U L A T I V E T R A V E L T I M E O P P R E S H M A T E R - S A L T H A T E R I N T E R P A C E * ( S E C )

C U M U L A T I V E T R A V E L T I M E O P I N T E R F A C E T H R U E N O O P P I R S T I N J E C T I O N H A L P - C T C L E . ( S E C )

C U M U L A T I V E T R A V E L T I M E O P I N T E R F A C E T H R U E N O OP P I R S T P R O O U C T I O N H A L P - C Y C L E * ( S E C )

C U M U L A T I V E T R A V E L T I M E O P I N T E R P A C I T H R U E N O O P S E C O N D I N J E C T I O N H A L P - C Y C L E . ( S E C )

C U M U L A T I V E T R A V E L T I M E O P I N T E R F A C E T H R U C O M P U T A T I O N I N T E R V A L A T H H I C H B R E A K T H R O U G H C H E C K I S B E I N G M A D E . ( S E C )

S T A T I C S T O R A G E T I M E A T T H E E N O O P T H E P I R S T I N J E C T I O N H A L P - C Y C L E . ( S E C )

S T A T I C S T O R A G E T I M E A T T H E E N O O P T H E S E C O N O I N J E C T I O N H A L P - C Y C L E . ( S E C )

P S E U D O T I M E U S E O I N G R A V I T A T I O N A L S E G R E G A T I O N C A L C J l A T I Q N S . ( S E C )

P S E U O O T I M E U S E O I N G R A V I T A T I O N A L S E G R E G A T I O N C A L C U L A T I O N S . ( D A Y S )

T I M E O P T R A V E L A C R O S S A N Y C O M P U T A T I O N I N T E R V A L P L U S T H EP S E U O O T I M E P O R T H A T I N T E R V A L . ( S E C )

T I M E O P T R A V E L A C R O S S A N Y C O M P U T A T I O N I N T E R V A L P L U S T H EP S E U O O T I M E P O R T H A T I N T E R V A L . ( D A Y S )

T I M E A T H H I C H C O N C E N T R A T I O N I S B E I N G C O M P U T E D D U R I N G F I R S T I N J E C T I O N H A L P - C Y C L E . A L S O C U M U L A T I V E T R A V E L T I M E O P I N T E R F A C E T H R U E N O O P P I R S T I N J E C T I O N H A L P - C Y C L E . ( S E C )

T I M E A T H H I C H C O N C E N T R A T I O N I S B E I N G C O M P U T E O D U R I N G

P I R S T P R O O U C T I O N H A L F - C Y C L E . A L S O C U M U L A T I V E T R A V E L T I M E O P I N T E R F A C E T H R U E N O O P F I R S T P R O O U C T I O N H A L P - C Y C L E . ( S E C )

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1 0 0 0 0 F O R M A T ( O X . • D E N S I T Y O F T H E F L U I D S ( G M / C C ) * / .1 O X . * O E N S I T Y O F T H E I N J E C T E O F L U I D * . 2 0 X . F I 2 . S / •2 O X , • O E N S I T Y O F T H E N A T I V E F L U I D • . 3 I X . F I 2 . 0 / •3 O X , * O E N S I T V D I F F . B E T V E E N T H E F L U I D S * . 2 6 X . F I 2 . 0 / / / / *4 6 X . « A C C E L E R A T I O N D U E T O G R A V I T Y I I C M / S E C > / S E C ) • • I 9 X . F 1 2 . S / / / I

1 9 0 0 0 F O R M A T I 6 X • * O P E R A T I N G C O N D I T I O N S * / / .1 S X , * I N J E C T I O N A N D P R O D U C T I O N R A T E S ( G A L / M I N ) * / .2 O X , • I N J E C T I O N R A T E F O R F I R S T I N J E C T I O N H A L F - C Y C L E * . S X . F I 4 . S / •3 O X . ' P R O D U C T I O N R A T E F O R F I R S T P R O D U C T I O N H A L F - C Y C L E • • S X . F 1 4 . S / *4 O X , * I N J E C T I O N R A T E F O R S E C O N O I N J E C T I O N H A L F - C Y C L E * . S X . F I 4 . 0 / •5 O X . * P R O D U C T I O N R A T E F O R S E C O N O P R O D U C T I O N H A L F - C Y C L E • . B X . F I 4 . 0 / »

2 0 0 0 0 F O R M A T ! B X . * V O L U M E O F F L U I O I N J E C T E O O R P R O D U C E D ( G A L L O N S ) * / .1 O X . ' F L U I O I N J E C T E O I N F I R S T I N J E C T I O N H A L F - C Y C L E * . 4 X . F 2 O . 0 / .2 O X . ' F L U I D P R O D U C E D I N F I R S T P R O D U C T I O N H A L F - C Y C L E • . 4 X . F 2 O . 0 / •3 O X , * F L U I O I N J E C T E O I N S E C O N D I N J E C T I O N H A L F - C Y C L E • , 4 X . F 2 0 . B / )

2 1 0 0 0 F O R M A T ( S X , ' T I M E O F S T A T I C S T O R A G E ( D A Y S ) * / .1 O X . * A T T H E E N O O F F I R S T I N J E C T I O N H A L F - C Y C L E * . 1 6 X . F I 3 . 0 / .2 O X . * A T T H E E N O 3 F S E C O N O I N J E C T I O N H A L F - C V C L E • . 1 0 X . F I 3 . S / / / I

2 2 0 0 0 F O R M A T ! I H 1 . 4 I X . ^ C A L C U L A T I O N S F O R F I R S T I N J E C T I O N H A L F - C Y C L E * / .I 4 2 X . * — ---------------------------------------------— --------------------------------— • / / / )

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2 4 0 0 0 F O R M A T ! I X . • I - * . 1 3 . 6 X . * O M I - • , E I S . 6 . 6 X . * D M 2 - • , E I 8 . B . S X , * D G - * . E I S . S .I 4 X , * X - * . E I S . S )

2 9 0 0 0 F O R M A T ( I X . • I - * . 1 3 . S X . • T l I D — • • E l 8 . 0 . S X . • T S P D —• . E I S . 0 . 4 X . • T T D - * •1 C I 5 . B . 3 K . * V L - * . E 1 S . B . 3 X . * X L F T - • . E I S . S )

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P R O O R A M • C Y C L E 3 A * ( S I N G L E N E L L - T H R E E C Y C L E S )

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P R O D U C E D S T R E A M . ( V O L U M E F R A C T I O N )T I L I F T • I N T E R V A L L E N G T H F O R C A L C U L A T I O N S 3 U R I N G I N J E C T I O N

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H A L F - C Y C L E S . ( F T )R I N C F T a L E N G T H O F I N C R E M E N T F O R C A L C U L A T I O N O F M I X E O Z O N E

L E N G T H S . ( F T )T I N C F T a I N C R E M E N T B Y W H I C H T I L I F T I S I N C R E A S E D I F M I X E O Z O N E

I N T E R S E C T S T H E L I N E S O U R C E D U R I N G C A L C U L A T I O N S F O R F I R S T I N T E R V A L O F F I R S T I N J E C T I O N H A L F - C Y C L E . ( F T )

S E C O N O C A R O - F O R M A T ! S F I 2 . 0 )H F T ■ A Q U I F E R T H I C K N E S S . ( F T )P L Y O A R B A Q U I F E R P E R M E A B I L I T Y . ( D A R C V S )P R a P O R O S I T Y . ( F R A C T I O N )A L F a L O N G I T U D I N A L D I S P E R S I V I T Y C O E F F I C I E N T . ( C M )O I F M O L a C O E F F I C I E N T O F M O L E C U L A R D I F F U S I O N . ( ( S O C M I / S C C )

T H I R D C A R O - F O R M A T ! 8 F 1 2 . 0 )V I S C P ! a V I S C O S I T Y O F T H E I N J E C T E O F R E S H W A T E R . ( C P )V I S C P 2 a V I S C O S I T Y O F T H E N A T I V E S A L T W A T E R . ( C P )O E N S I a O E N S I T Y O F T H E I N J E C T E D F R E S H W A T E R . ( G M / C C )D E N S 2 a D E N S I T Y O F T H E N A T I V E S A L T W A T E R . ( G M / C C )A C N G a A C C E L E R A T I O N D U E T O G R A V I T Y . ( ( C M / S E C ) / S E C )

F O U R T H C A R O - F O R M A T I 6 F 1 2 . 0 )O R I G M a I N J E C T I O N R A T E F O R F I R S T I N J E C T I O N H A L F - C Y C L E * ( G P M ) O R 2 G M b P R O O U C T I O N R A T E F O R F I R S T P R O O U C T I O N H A L F - C Y C L E . ( G P M ) 0 R 3 G M a I N J E C T I O N R A T E F O R S E C O N O I N J E C T I O N H A L F - C Y C L E . ( G P M ) Q R 4 G M 8 P R O O U C T I O N R A T E F O R S E C O N D P R O O U C T I O N H A L F - C Y C L E . ( G P M I Q R S G M a I N J E C T I O N R A T E F O R T H I R D I N J E C T I O N H A l F - C Y C L E . ( G P M ) Q R 6 G M B P R O O U C T I O N R A T E F O R T H I R D P R O O U C T I O N H A L F - C Y C L E . ( G P M )

F I F T H C A R D - F O R M A T I S F 1 2 . 0 )F L I N G I B F L U I O I N J E C T E O I N F I R S T I N J E C T I O N H A L F - C Y C L E . ( G A L )F L P R G 2 B F L U I D P R O D U C E D I N F I R S T P R O D U C T I O N H A L P - C V C L Z . ( G A L )F L I N G 3 a F L U I D I N J E C T E D I N S E C O N O I N J E C T I O N H A L F - C Y C L E . ( G A L )F L P R G 4 a F L U I D P R O D U C E O I N S E C O N O P R O O U C T I O N H A L F - C Y C L E . ( G A L )F L I N G S B F L U I O I N J E C T E O I N T H I R D I N J E C T I O N H A L F - C Y C L E . ( G A L )

S I X T H C A R O - F O R M A T I 3 F I 2 . 0 )T S T I O a S T A T I C S T O R A G E T I N E A T T H E E N D O F T H E F I R S T I N J E C T I O N

H A L F - C Y C L E . ( D A Y S )

T S T 3 0 a S T A T I C S T O R A G E T I M E A T T H E E N O O F T H E S E C O N O I N J E C T I O N H A L F - C Y C L E . ( D A Y S )

T S T S O B S T A T I C S T O R A G E T I M E A T T H E E N O O F T H E T H I R O I N J E C T I O N H A L F - C Y C L E . ( O A V S )

S E V E N T H C A R O - F O R M A T I 2 1 3 )N I N T 2 a N U M B E R O F C O M P U T A T I O N I N T E R V A L S T H R U T H E F I R S T

P R O O J C T I O N H A L F - C Y C L E .H I N T A B n u m b e r O F C O M P U T A T I O N I N T E R V A L S T H R U T H E S E C O N O

P R O O U C T I O N H A L F - C Y C L E .

105

O S P I N I T I O M

A t ■

A 2 ■

A 3 •

A 4 ■

A S ■

A A ■

6 1 •

6 2 ■

63 »6 4 •

es •

66 ■C F C P P • C P D S C M ■ C F D S E C » C F F T C M • C F G L C C • C G M C C S • C O N S T I •

C O N S T 2 •

C R C E P P • C V L I N G ■

C V L R O G » C V O L I N ■ C V O L R O *

C l l •

00 - O N I •

D M A •

. D M 2 A ■

O N O M I 1 ■

O N O N 1 3 ■

O N O M I S ■

O N O M P 2 ■

D N 0 M P 4 »

O N O M P A ■

O S O F ■

O S C A ■

F L 1 N J I m P L I N J 3 ■

O F V A N I A 6 L E N A M E S U S E O I N P N O O N A M

I N T E N M E O I A T E V A L U E U S E D I N C O M P U T I N O V A L U E S O P H U E S P O NF I R S T I N J E C T I O N H A L F - C V C L E . ( S O C M )

I N T E R M E D I A T E V A L U E U S E O I N C O M P U T I N G V A L U E S O P R U S O P O R F I R S T P R O O U C T I O N H A L F - C V C L E . ( S O C M )

I N T E R M E D I A T E V A L U E U S E O I N C O N P U T I N O V A L U E S O F R U S O F O R S E C O N O I N J E C T I O N H A L F - C V C L E . ( S O C M )

I N T E R M E D I A T E V A L U E U S E O I N C O M P U T I N O V A L U E S O F R U S O P O R S E C O N O P R O O U C T I O N H A L F - C V C L E . ( S O C M )

I N T E R N E D I A T E V A L U E U S E O I N C O M P U T I N S V A L U E S O F R U S O F O R T H I R D I N J E C T I O N H A L F - C V C L E . I S O C M )

I N T E R M E D I A T E V A L U E U S E D I N C O M P U T I N G V A L U E S O P R U S O P O R T H I R O P R O O U C T I O N H A L F - C V C L E . I S O C M )

I N T E R M E D I A T E V A L U E U S E O I N C O M P U T I N G V A L U E S O P R L S O P O R F I R S T I N J E C T I O N H A L F - C V C L E . ( S O C M )

I N T E R M E D I A T E V A L U E U S E O I N C O M P U T I N G V A L U E S O P R L S O P O R F I R S T P R O O U C T I O N H A L F - C V C L E . ( S O C M )

I N T E R M E D I A T E V A L U E U S E D I N C O M P U T I N G V A L U E S O F R L S O F O R S E C O N O I N J E C T I O N H A L F - C V C L E . ( S O C M )

I N T E R M E D I A T E V A L U E U S E O I N C O M P U T I N G V A L U E S O P R L S O P O R S E C O N O P R O D U C T I O N H A L F - C V C L E . ( S O C M )

I N T E R M E D I A T E V A L U E U S E O I N C O M P U T I N S V A L U E S O F R L S O P O R T H I R D I N J E C T I O N H A L F - C V C L E . ( S O C M )

I N T E R M E D I A T E V A L U E U S E D I N C O M P U T I N G V A L U E S O P R L S O P O R T H I R O P R O O U C T I O N H A L F - C V C L E . ( S O C M )

C O N V E R S I O N F A C T O R . ( P O I S E / C E N T I P O I S E )C O N V E R S I O N F A C T O R . ( ( S O C M ) / D A R C V >C O N V E R S I O N F A C T O R . ( S E C / D A Y )C O N V E R S I O N F A C T O R . ( C M / F T )C O N V E R S I O N F A C T O R . ( C C / G A L )C O N V E R S I O N F A C T O R . ( ( C C / S E C ) / ( G A L / M I N ) )V A L U E U S E O I N C H E C K I N G F O R S R E A K T M R O U G H D U R I N G A

P R O O U C T I O N H A L F - C V C L E .V A L U E U S E O I N C H E C K I N G F O R B R E A K T H R O U G H D U R I N G A

P R O O J C T I O N H A L F - C V C L E .C U M U L A T I V E R E C O V E R Y E F F I C I E N C Y . ( F R A C T I O N )C U M U L A T I V E V O L U M E O F F L U I O I N J E C T E D . ( G A L )C U M U L A T I V E V O L U M E O P I N J E C T E D F L U I D R E C O V E R E D . ( G A L )C U M U L A T I V E V O L U M E O F F L U I O I N J E C T E O . ( C C )C U M U L A T I V E V O L U M E O F I N J E C T E O F L U I O R E C O V E R E D . ( C C )C O M P U T E O C O N C E N T R A T I O N A T T H E R A O I U S A N D A T T H E T I M E

B E I N G C O N S I D E R E D . ( V O L U M E F R A C T I O N )O E N S I T V G R A D I E N T . ( ( G M / C C ) / C M )A C O N S T A N T U S E O I N T H E C O M P U T A T I O N O F T H E 0 1 M E N S I O N L E S S

P A R A M E T E R G I V E N B V E O U A T I O N 2 . 7 A .O I M E N S I O N L E S S G R O U P . S E C O N O G R O U P O N R I G H T S I D E O P

E O U A T I O N 2 . T A .A C O N S T A N T U S E O I N T H E C O M P U T A T I O N O F T H E D t M E N S I O N L E S S

P A R A M E T E R G I V E N B V E O U A T I O N 2 . 7 A .D E N O M I N A T O R O F A R G U M E N T O F C O M P L E M E N T A R Y E R R O R F U N C T I O N

I N E O U A T I O N 2 . S F O R F I R S T I N J E C T I O N H A L F - C V C L E . D E N O M I N A T O R O F A R G U M E N T O F C O M P L E M E N T A R Y E R R O R F U N C T I O N

I N E Q U A T I O N 2 . 8 F O R S E C O N O I N J E C T I O N H A L F - C V C L E . D E N O M I N A T O R O F A R G U M E N T O F C O M P L E M E N T A R Y E R R O R F U N C T I O N

I N E O U A T I O N 2 . 8 F O R T H I R O I N J E C T I O N H A L F - C V C L E . D E N O M I N A T O R O P A R G U M E N T O P C 0 M P L 5 M S N T A R V E R R O R P U N C T I O N

I N E Q U A T I O N 2 . S F O R F I R S T P R O O U C T I O N H A L F - C V C L I . D E N O M I N A T O R O P A R G U M E N T O F C O M P L E M E N T A R Y E R R O R F U N C T I O N

I N E O U A T I O N 2 . S F O R S E C O N O P R O O U C T I O N H A L F - C V C L E . D E N O M I N A T O R O P A R G U M E N T O F C O M P L E M * N T A R V E R R O R F U N C T I O N

I N E O U A T I O N 2 . 8 F O R T H I R O P R O D U C T I O N H A L F - C V C L E . D E N S I T Y D I F F E R E N C E B E T W E E N I N J E C T E O A N D N A T I V E

F L U I O S . ( G M / C C )I N T E R M E D I A T E V A L U E U S E O I N C O M P U T I N G V A L U E S O P R L S O P O R

T H I R O P 4 0 0 U C T I O N H A L F - C V C L E . ( S O C M )F L U I O I N J E C T E D I N F I R S T I N J E C T I O N H A L F - C V C L E . ( C C )F L U I O I N J E C T E D I N S E C O N O I N J E C T I O N H A L F - C V C L E . ( , C C )

106

F L I N J S ■ F L U I D I N J E C T E O I N T H M O I N J E C T I O N H A L F - C V C L E * I C C IF L P R G G a F L U I O P R O O U C E O I N T H I R O P R O D U C T I O N H A L F - C V C L E * ( G A L )P L P R N 2 a F L U I O F R O O U C C O I N F I R O T P R O D U C T I O H H A L F - C V C L E * I C C IF L P A N 4 ■ F L U I D P R O D U C E D I N S E C O N O P R O O U C T I O N H A L F - C V C L E * I C C IF L P R N A a F L U I D P R O D U C E D I N T H I R D P R O D U C T I O N H A L F - C V C L E * I C C IH ■ A O U I F E R T H I C K N E S S * I C N >I ■ S U B S C R I P T D C S I G N A T I N S C O N F U T A T I O N I N T E R V A L *N I N T a I N T E R M E D I A T E V A L U E U S E D I N C A L C U L A T I N G T H E N U N O E R O F

C O M P U T A T I O N I N T E R V A L S *N I N T I a N U M B E R O F C O M P U T A T I O N I N T E R V A L S T H R U T H E E N D O F T H E

F I R S T I N J E C T I O N H A L F - C V C L E *N I N T 3 a N U M B E R O F C O M P U T A T I O N I N T E R V A L S T H R U T H E S E C O N O

I N J E C T I O N H A L F C Y C L E *N I N T S a N U M B E R O F C O M P U T A T I O N I N T E R V A L S T H R U T H E T H I R O

I N J E C T I O N H A L F - C V C L E *H I N T S a N U M B E R O F C O M P U T A T I O N I N T E R V A L S T H R U T H E T H I R O

P R O O U C T I O N H A L F - C V C L E .P L V a A O U I F E R P E R M E A B I L I T Y . I S O C M IP P P a P R O O U C T O F P I * P O R O S I T Y . A N D T H I C K N E S S * I C M IP P P I a 2 * P P P . I C M IO i l a T W O 0 1 M E N S I O N A L F . O M R A T E F O R F I R S T I N J E C T I O N

H A L F - C V C L E * U S O C M I / S E C I 0 1 3 ■ T W O D I M E N S I O N A L F L O * R A T E F O R S E C O N O I N J E C T I O N

H A L F - C V C L E * 1 1 S O C M I / S E C I O I S a T W O D I M E N S I O N A L F . O M R A T E F O R T H I R O I N J E C T I O N

H A L F - C V C L E . 1 1 S O C M I / S E C I O P E a T W O D I M E N S I O N A L F L O M R A T E F O R F I R S T P R O O U C T I O N

H A L F - C V C L E . 1 1 S O C M I / S E C I 0 P 4 a T V O 0 1 m e n s I O N A L F L O M R A T E F O R S E C O N O P R O D U C T I O N

H A L F - C V C L E * 1 1 S O C M I / S E C I O P S a T M O D I M E N S I O N A L F L O M R A T E F O R T H I R O P R O O U C T I O N

H A L F - C V C L E . 1 1 S O C M I / S E C I O R 1 a P L O W R A T E F O R F I R S T I N J E C T I O N H A L F - C V C L E . ( C C / S E C IO R E a P L O W R A T E F O R F I R S T P R O O U C T I O N H A l F - C V C L E * ( C C / S E C )O R 3 a P L O W R A T E F O R S E C O N O I N J E C T I O N H A L F - C V C L E * ( C C / S E C IO R A a F . O M R A T E F 0 « S E C O N O P R O O U C T I O N H A L F - C V C L E * ( C C / S E C )O R B a F L O M R A T E F O R T H I R O I N J E C T I O N H A L F - C V C L E . ( C C / S E C )O R S a F L O M R A T E F O R T H I R O P R O O U C T I O N H A L F - C V C L E * ( C C / S E C )O S 1 1 a T M O O I M E N S I O N A L P S E U O O F L O M R A T E F O R F I R S T I N J E C T I O N

H A L F - C V C L E . ( ( S Q C M ) / S E C )O S 1 3 a T W O D I M E N S I O N A L P S E U O O F L O M R A T E F O R S E C O N O I N J E C T I O N

H A L F - V C L E . ( I S O C M I / S E C I

O S I S a T M O D i E N S I O N A L P S E U D O F L O M R A T E F O R T H I R O I N J E C T I O N H A L F - C V C L E . ( ( S O C M ) / S E C )

0 S P 2 a T W O D I M E N S I O N A L P S E U O O F L O M R A T E F O R F I R S T P M O O U C T I O N H A L F - C V C L E * ( ( S O C N ) / S E C )

O S P A a T M O D I M E N S I O N A L P S E U O O F L O M R A T E F O R S E C O N O P R O O U C T I O NH A i r - C V C L L . ( ( S O C M ) / S E C )

0 S P 6 a T W O 0 1 M E M S I O N A L P S E U O O F L O M N A T E F O R T H I R O P R O O U C T I O N H A L F - C V C L E U P T O C O M P U T A T I O N I N T E R V A L A T H H I C H B R E A K T H R O U G H C H E C K I S B E I N G M A D E . ( I S O C M ) / S B C )

R ( I I a R A O I U S O F I N J E C T E O F L U I O A T T H E 1 T H C O M P U T A T I O NI N T E R V A L A S S U M I N G MO M I X I N G O R G R A V I T A T I O N A LS E G R E G A T I O N . ( C M )

R B T a R A D I U S A T H H I C H B R E A K T H R O U G H I S C O M P U T E D * ( C M )R C B T ( I ) a L E A S T R A O I U S T O A V A L U E O F C O N C E N T R A T I O N O F C B T F O R T H E

L A S T C O M P U T A T I O N I N T E R V A L O F T H E L A S T P R O O U C T I O NH A L F - C V C L E * ( C M )

R C B T F T a ( . E A S T R A O I U S T O A V A L U E O F C O N C E N T R A T I O N O F C O T F O R T H EL A S T C O M P U T A T I O N I N T E R V A L O F T H E L A S T P R O O U C T I O NH A L F - C V C L E * ( F T )

R C E F F a C Y C L E R E C O V E R Y E F F I C I E N C Y * ( F R A C T I O N )R F T a R A D I U S O F I N J E C T E O F L U I O A T T H E I T N C O M P U T A T I O N

I N T E R V A L A S S U M I N G N O M I X I N G O R G R A V I T A T I O N A LS E G R E G A T I O N . ( F T )

R I N C a L E N G T H O F I N C R E M E N T U S E O F O R C A L C U L A T I O N O F M I X E O Z O N E . E N G T H S * ( C M )

R I N J I a R A O I U S O F I N J E C T E D F L U I O A T T H E E N O O F T H E F I R S T I N J E C T I O N H A L F - C V C L E A S S U M I N G N O M I X I N G A N D N O G R A V I T A T I O N A L S E G R E G A T I O N . ( C M )

u u

107

R I N J 3 ■ R A O I U S O F I N J E C T E O F L U I O A T TH ® E N D O F T H E S 1 C O N O I N J E C T I O N H A L F - C Y C L E A S S U N I N 6 N O N I X I N G A N O N O

6 R A V I T A T I O N A L S E G R E G A T I O N . ( C N )R 1 N J S • R A O I U S O F I N J E C T E O F L U I O A T T H E E N D O F T H E T H I R D

I N J E C T I O N H A L F - C V C L E A S S U R I N G N O N I X I N G A N O N O G R A V I T A T I O N A L S E G R E G A T I O N . I C R }

R L S O a R A O I U S T O L O V E R E N O O F 3 0 P E R C E N T C O N C E N T R A T I O N L I N E . ( C R )

R L S O F T • R A D I U S T O L O V E R E N O O F S O P E R C E N T C O N C E N T R A T I O N L I N E . I F T )

R L S O I I a R A O I U S T O L O V E R E N O O F 8 0 P E R C E N T C O N C E N T R A T I O N L I N EA T T H E S T A R T O F T H E F I R S T P R O O U C T I O N H A L F - C V C L E . ( C N )

R L S O 1 2 a R A O I U S T O _ O V E R C N O O F S O P E R C E N T C O N C E N T R A T I O N L I N EA T T H E S T A R T O F T H E S E C O N O P R O O U C T I O N H A L F - C V C L E . I C R )

R L S 0 1 S a R A O I U S T O L O V E R E N D O F S O P E R C E N T C O N C E N T R A T I O N L I N EA T T H E S T A R T O F T H E T H I R O P R O O U C T I O N H A L F - C V C L E . ( C R )

R L S O P I a R A O I U S T O L O V E R C N O O F S O P E R C E N T C O N C E N T R A T I O N L I N EA T T H E S T A R T O F T H E S E C O N O I N J E C T I O N H A L F - C V C L E . ( C R )

R L 8 0 P 1 a R A O I U S T O L O V E R C N O O F S O P E R C E N T C O N C E N T R A T I O N L I N E A T T H E S T A R T O F T H E T H I R O I N J E C T I O N H A L F - C V C L E . ( C N )

R U S O a R A D I U S T O U P P E R E N O O F S O P E R C E N T C O N C E N T R A T I O N L I N E . ( C R )

R U S O F T a R A O I U S T O U P P E R C N O O F S O P E R C E N T C O N C E N T R A T I O N L I N E . ( F T )

R I a R A O I U S A T V H I C H C O N C E N T R A T I O N I S B E I N G C O N F U T E D . A L S OI N N E R R A O I U S O F N I X E D Z O N E . ( C R )

R I F T a I N N E R R A O I U S O F N I X C O Z O N E . ( F T )R 2 ( I ) a O U T E R R A O I U S O F R I X C O Z O N E . ( C N )R Z F T a O U T E R R A O I U S O F H I K E D Z O N E . ( F T )

R 3 a L E N G T H O F R I X C O Z O N E . ( C N )R 3 F T a R I X C O Z O N E L E N G T H . ( F T )R A a T I L 1 / 2 . ( C R )R S a R A O I U S T O T H E R I O P O I N T B E T V E E N R ( ! ) A N O R ( t - I ) . ( C N )R S S O a R S S Q U A R E D . I S O C R )R 6 a T I L P / 2 . ( C N )S D C N O N a O E N O R I N A T O R } F A R G U H E N T O F C O R P L E R E N T A R V E R R O R F U N C T I O N

I N E O U A T I O N 2 . 5 U S I N G P S E U D O R A T E S T O C H E C K F O R B R E A K T H R O U G H D U R I N G A P R O D U C T I O N H A L F - C V C L E .

T I L I a I N T E R V A L L E N G T H F O R C O N F U T A T I O N S D U R I N G I N J E C T I O N H A L F —C Y C L E S . ( C N )

T 1 L P a I N T E R V A L L E N G T H F O R C O N F U T A T I O N S D U R I N G P R O O U C T I O N

H A L F - C Y C L E S . ( C N )T R T a C U H U L A T I V C T R A V E L T I R E O F F R E S H V A T E R - S A L T V A T E R

I N T E R F A C E . ( S E C )T R T 1 a C U M U L A T I V E T R A V E L T I M E O F I N T E R F A C E

I N J E C T I O N H A L F - C Y C L E . ( S E C )T H R U E N O O F F I R S T

T R T 2 a C U M U L A T I V E T R A V E L T I M E O F I N T E R F A C E P R O O U C T I O N H A L F - C V C L E . ( S E C )

T H R U E N O O F F I R S T

T R T 3 a C U M U L A T I V E T R A V E L T I M E O F I N T E R F A C E I N J E C T I O N H A L F - C V C L E . ( S E C )

T H R U E N O O F S E C O N O

T R T A a C U M U L A T I V E T R A V E L T I M E O F I N T E R F A C E P R O O U C T I O N H A L F - C V C L E . ( S E C )

T H R U E N O O F S E C O N D

T R T 8 a C U M U L A T I V E T R A V E L T I K E O F I N T E R F A C E I N J E C T I O N H A L F - C V C L E . ( S E C )

T H R U E N O O F T H I R O

T R T * a C U M U L A T I V E T R A V E L T I M E O F I N T E R F A C E T H R U I N T E R V A L A T V H I C H B R E A K T H R O U G H C H E C K I S M A D E . ( S E C )

C O M P U T A T I O NB E I N G

T S T 1 a S T A T I C S T O R A G E T I N E A T T H E E N O O F T H E F I R S T I N J E C T I O N H A L F - C V C L E . ( S E C )

T S T 3 a S T A T I C S T O R A G E T I R E A T T H E E N O O F T H E S E C O N O I N J E C T I O N H A L F - C V C L E . ( S E C )

T S T S a S T A T I C S T O R A G E T I R E A T T H E C N O O F T H E T H I R O I N J E C T I O N H A L F - C Y C L E . ( S E C )

T S P a P S E U O O T I M E U S E O I N G R A V I T A T I O N A L S E G R E G A T I O NC A L C J L A T I O N S . ( S E C )

T S P O a P S E U O O T I M E U S E D I N G R A V I T A T I O N A L S E G R E G A T I O NC A L C U L A T I O N S . ( D A Y S )

T T a T I R E O F T R A V E L A C R O S S A N Y C O M P U T A T I O N I N T E R V A L P L U S T H EP S E U O O T I M E F O R T H A T I N T E R V A L . ( S E C )

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T T O ■ T I M E O F T R A V E L A C R O S S A N Y C O M P U T A T I O N I N T E R V A L P C U S T H E P S E U O O T I M E F O R T H A T I N T E R V A L * I O A V S I

T l ■ T I M E A T W H I C H C O N C E N T R A T I O N I S B E I N G C O M P U T E D D U R I N SF I R S T I N J E C T I O N H A L F - C Y C L E . A L S O C U M U L A T I V E T R A V E L T I M E O F I N T E R F A C E T H R U C N O O F F I R S T t N J E C T I O N H A L F - C V C L E . ( S E C )

T * ■ T I M E A T W H I C H C O N C E N T R A T I O N I S B E I N G C O M P U T E O D U R I N GF I R S T P R O O U C T I O N H A L F - C V C L E . A L S O C U M U L A T I V E T R A V E L T I M E O F I N T E R F A C E T H R U C N O O F F I R S T P R O O U C T I O N H A L F - C V C L E . ( S E C )

T 3 ■ T I M E A T W H I C H C O N C E N T R A T I O N I S B E I N G C O M P U T E D D U R I N GS E C O N O I N J E C T I O N H A L F - C V C L E . A L S O C U M U L A T I V E T R A V E LT I M E O F I N T E R F A C E T H R U C N O O F S E C O N O I N J E C T I O N H A L F - C V C L E . ( S E C )

T 4 - T I M E A T W H I C H C O N C E N T R A T I O N I S B E I N G C O M P U T E D D U R I N GS E C O N D P R O O U C T I O N H A L F - C V C L E . A L S O C U M U L A T I V E T R A V E LT I M E O F I N T E R F A C E T H R U E N O O F S E C O N O P R O O U C T I O N H A L F - C V C L E . ( S E C )

T S - T I M E A T W H I C H C O N C E N T R A T I O N I S B E I N G C O M P U T E O D U R I N GT H I R O I N J E C T I O N H A L F - C V C L E . A L S O C U M U L A T I V E T R A V E L

T I M E O F I N T E R F A C E T H R U E N O O F T H I R O I N J E C T I O N H A L F - C V C L E . ( S E C )

T 6 ■ T I M E A T W H I C H C O N C E N T R A T I O N I S B E I N G C O M P U T E D D U R I N GT H I R O P R O O U C T I O N H A L F - C V C L E . ( S E C )

T i l ■ T I M E O F T R A V E L A C R O S S A N V C O M P U T A T I O N I N T E R V A L . ( S E C )T I I O ■ T I M E O F T R A V E L A C R O S S A N V C O M P U T A T I O N I N T E R V A L . ( O A V S )V I S ■ M E A N V I S C O S I T Y O F I N J E C T E D A N D N A T I V E F L U I O S . ( P O I S E )V I S C P ■ M E A N V I S C O S I T Y O F I N J E C T E O A N O N A T I V E F L U I D S . ( C P )V O L N R ■ T O T A L V O L U M E O F I N J E C T E D F L U I D N O T R E C O V E R E O . ( C C )X » V A L U E O F O I M E N S I O N L E S S P A R A M E T E R G I V E N B V

E O U A T I O N 2 . 7 A .X L I I ) ■ H O R I Z O N T A L P R O J E C T I O N O F S O P E R C E N T C O N C E N T R A T I O N

L I N E F O R L I N E A R G E O M E T R Y . ( C M )X L F T • H O R I Z O N T A L P R O J E C T I O N O F S O P E R C E N T C O N C E N T R A T I O N

L I N E F O R L I N E A R G E O M E T R Y . ( F T )X R ( I ) ■ H O R I Z O N T A L P R O J E C T I O N O F S O P E R C E N T C O N C E N T R A T I O N

L I N E F O R R A O I A L G E O M E T R Y . ( C M )X R F T ■ H O R I Z O N T A L P R O J E C T I O N O F S O P E R C E N T C O N C E N T R A T I O N

L I N E F O R R A O I A L G E O M E T R Y . ( F T )X X ■ A R G U M E N T O F C O M P L E M E N T A R Y E R R O R F U N C T I O N F O R

E O U A T | O N 2 . S .Y L I I ) ■ R A T I O O F H O R I Z O N T A L P R O J E C T I O N O F S O P E R C E N T

C O N C E N T R A T I O N L I M E T O A Q U I F E R T H I C K N E S S F O R L I N E A RG E O M E T R Y .

V R < 1 ) m R A T I O O F H O R I Z O N T A L P R O J E C T I O N O P S O P E R C E N TC O N C E N T R A T I O N L I N E T O A O U I F E R T H I C K N E S S F O R R A O I A LG E O M E T R Y .

0 1 M E N S I O N R ( I 0 0 0 I . R K I O O O ) . Y L ( 1 0 0 0 1 . X L I 1 0 0 0 ) * X R ( I B O O ) . R C B T I 1 0 0 0 )

P A R T I - R E A D I N G O A T A

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P A R T 2 - P R I N T I N G O A T A

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C

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H « H F T » C P P T C M P P P * 3 * 1 4 I 6 9 P R 9 H P P P l » 2 * 0 * P P P P L Y > P L Y O A R * C P O S C M P L I N J I - P L I N G I 6 C F G L C C P L P R N 2 - F L P R G 2 4 C P G L C C P L I N J J - F L I N G 3 4 C F G L C C F L P R N 4 - P L P R G 4 4 C P G L C C F L I N J 5 » P L I N G S 6 C F G L C C

5 1 O R I * Q R I G M 6 C G M C C S5 2 0 R 2 - 0 R 2 G N 4 C G M C C S5 3 0 R 3 a Q R 3 G M « C G M C C S 8 4 O R 4 * Q R 4 G N 6 C G M C C S S S O R S * O R S G M » C G M C C S8 6 0 R 6 « 0 R 6 G M « C G M C C S8 7 T S T I > T S T I D * C P D S E C 8 6 T S T 3 - T S T 3 0 « C P O S e C 8 9 T S T S > T S T S O * C P D S E C

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6 0 T I L I » T I L I P T * C P P T C R R 4 - T I L 1 / 2 . 0

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C A L C U L A T I O N S O P I N T C R V A L S A N O R A O I U S O P I N J C C T I O N

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R I F T * R 1 / C F F T C N R 2 F T * R 2 I I 5 / C F F T C N R 3 F T * R 3 / C F F T C N

C A L C U L A T I O N O F I N T C R P A C t P R O J K C T I O N O U K T O G R A V I T Y S K S R K 6 A T I 0 N

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C O T O I S O 1 7 5 Y L I I l - 6 . 5 4 2 . 0 4 X I S O X L I D - V L I I ) 6 M

X L F T . X L t D / C F P T C N T l 1 D - T 1 l / C F O S E C T S P O - O . O T T O * T I I O C O T O 2 9 0

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1 9 1 T R T * T R T 4 7 1 1I F I V L I l - D - 2 . 0 ) 1 9 5 . 1 9 5 . 2 0 0

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2 0 0 I P I V L I l - l > - 6 . 5 ) 2 0 5 . 2 1 0 . 2 1 02 0 5 X * 1 . 2 9 9 3 - S O R T 1 1 7 2 . I S 7 3 - 1 9 . 2 7 6 4 6 V H I - 1 ) 1 / 9 . 6 3 9 2

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I F ! I - N I N T I 1 2 4 0 . 2 3 0 . 3 3 0 2 3 0 T T * T T 4 T S T I 2 4 0 X * D M I 4 T T 4 0 N 2 4 6 0 . 5

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3 1 0 X R I 1 ) ■ R U S O - R L S O Y R * X R ( t l / H R U S O F T * R U S O / C F F T C M R L S O F T * R L S 0 / C F F TC M X R F T - X R I I > / C F F T C M R F T * R < I l / C F F T CM

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1 3 1 0 RUSOaSORT1 ( 3 . 1 4 2 6 R I I ) 6 4 2 6 A 6 1 / 3 . I 4 2 1 ♦ ( X L 1 1 l - X L I t - 1 1 1 / 2 . 0CC C H E C K I N G P O R O R E A X T H R O U G H

CR L 5 0 S O a R L S 0 6 R L 0 0Q 5 P 6 a ( R L S O IS 4 R L 5 0 I 3—RLSOSO) / ( 2 . 06| T R T O - T R T S ) 15 OENONa2. 0 6SORTI 1 . 3 3 3 4 A .P 4 1 1 2 .0 6 Q O P 6 6 TRT61661. 5 —1 2 . 0 6 0 S P 6 4 T R T S 1 6 4 |

1 . S + C O N S T 1 1 . 0 I P M 0 L 6 I ( 2 .0 6 O S P G 4 T R T 6 ) 4 6 2 /Q S P 6 —( 2 . 0 4 O 3 P 6 6 T R T S ) 42 4 2 / 0 3 P 6 6 C 0 N S T 2 ) 1

R la R L S O -R IN CI P I R l - R B T ) 1 3 3 0 . 1 3 2 0 . 1 3 2 0

1 3 2 0 X X a ( RLSOSO—R 1 6 R I l /S O E N O M C l l a E R P C I R X 1 / 2 . 0I P I C l I -C O T 1 1 3 2 2 . 1 3 2 2 . 1 3 2 1

1 3 21 R | a R t - R | N CI P I R 1 - R O T ) 1 3 3 0 . 1 3 2 0 . 1 3 2 0

1 3 2 2 R C B T I I ) a R |X R I 1 1 ■ R U 3 0 -R L 5 0 V R -X R I D / M RU 50P TaRU50/CPPTCN RLSOPT aR L5 0 /C P P T C M X R P T a X R I I l /C P P T C M R P T a R | I l / C P P T CM

119

cC M I N T I M S C O M P U T A T IO N S PON T N I NO PRODUCT IO N H A L F -C Y C L EC

MR I T E I A . ( 3 0 0 0 ) I . R I P T . R 2 P T . R 3 P T M R I T E ! A . ( A 0 0 0 ) I * D M 1 * D M 2 . 0 G « X M R 1 T E I A . 2 B 0 0 0 ) 1 . T 1 1 0 . T S P O . T T D . Y L 1 1 1 . X L P T M R 1 T E I A . 2 6 0 0 0 ) 1 . R U S O P T . R L S O P T . X R P T . Y R . R P T

c1 3 ( 3 1 - 1 * 1

G O T O I I S O 1 3 3 0 I - I - I

N I N T 6 - tCcC P A R T 1 0 - C A L C U L A T I O N O P R E C O V E R Y C P P 1 C I C N C Y

Cc

1 3 S 0 M R I T E t 6 * 3 ( 0 0 0 )C

C V O L I N - P L I N J 1 * F L I N J 3 + F L I N J 9 I P I I . O T . N I H T S I O O T O I S A O

P L P R N 6 - 0 . 0C V O L R D - P L P R N 2 « P L P R N 4 * P L P R N A G 0 T O 1 3 7 0

1 3 6 0 A 6 - 3 . I 4 2 * ! I R 1 l - D + X R I 1 - 1 ) / 2 . 0 ) * * 2 - R I I - I > • * ( >B A - 3 . 1 4 2 * 1 R l l - l > * * 2 - 1 R l l - l > - X R * * 2 . A A ) / 3 . 1 4 2 ) * 1 X L 1 1 I - X L I I - I ) ) / ( . 0 R L 5 0 —S O R T I I 3 . 1 4 2 * R - I X L 1 1 ) - X L I I - 1 1 l / ( . 0 V O L N R - P P P * I I R U 5 0 * R U 5 0 * R U 5 0 * R L 9 O * R L B 0 * R L 9 O > / 3 . O - I R C B T 1 1 1 * R C S T -

I R B T * R B T ) )P L P R N 6 - C V O L I N —V O L M R —P L P R N 2 —P L P R N 4 C V 0 L R D - F L P R N 2 + F L P R N 4 * F L P R N A R C B T P T - R C B T I I l / C P P T C M M R I T C I A . 3 3 0 0 0 I I * R B T P T . C B T . R C B T P T

1 3 7 0 R C E F P - P L P R N A / P L I N J S

C R C E P P - C V O L R O / C V O L I N P L P R G A —P L P R N A / C F C L C C C V L I N G - C V O L I N / C P G L C C C V L R D G - C V D L R D / C P G l C CM R I T E I A . 3 4 0 0 0 1 1 . P L P R G A * R C E P P . C M L I N G . C V L R O G . C R C E P P

CG O T 0 2 0

Ccc P A R T I I - P O R M A T S T A T C M E N T S

Cc1 0 0 0 0 P O R M A T I A P 1 2 . 0 1 1 1 0 0 0 P O R M A T I S P K . 0 1 1 2 0 0 0 P O R M A T ( S P 1 2 . 0 )1 3 0 0 0 P O R M A T I A P I 2 . 0 )1 4 0 0 0 P O R M A T I 5 P 1 2 . 0 )1 9 0 0 0 P O R M A T I 3 P 1 2 . 0 )1 9 2 0 0 P O R M A T 1 2 1 3 )1 6 0 0 0 P O R M A T I I H t . 3 S X . ' O A T A ' / . S A X . • — — » ✓ / / / . A X . ' P O R O U S N E O I U M * / / .

1 9 X . ' T H I C K N E S S O P T H E M E D I U M I P T ) • . 3 0 X * P 1 2 . 0 / .2 O X . • P E R M E A B I L I T Y O P T H E M E D I U M t O A R C I E S ) • . 2 2 X . P 1 2 . A / .3 O X . ' P O R O S I T Y O P T H E M E D I U M I P R A C T I O N ) ' • 2 9 X . P 1 2 . 6 / .4 O X . ' L O N G I T J 0 1 N A L O I S P E R S I V I T V O P T H E M E D I U M I C M ) • • 1 4 X . P 1 2 . 0 / .9 O X . ' C O E P P I C I E N T O P M O L E C U L A R D I P P U S I O N I S O C M / S E C ) * . 1 2 K . P 1 2 . S / / / A )

1 7 0 0 0 P O R M A T I A X . ' P L U I O P R O P E R T I C S ' / / . A X . • V I S C O S I T Y O P T H E P L U I D S I C P ) ' / .1 O X . ' V I S C O S I T Y O P T H E I N J E C T E D P L U I 0 ' . ( 7 X . P 1 2 . S / .2 O X . ' V I S C O S I T Y O P T H E N A T I V E P L U I O ' . 2 0 X . P 1 2 . 0 / .3 O X . ' M E A N V I S C O S I T Y O P T H E T M O F L U I D S ' . 2 6 X . P 1 2 . S / 1

1 S 0 0 0 P O R M A T I 8 X . ' D E N S I T V O P T H E P L U I D S I G M / C C ) ' / .1 O X . ' D E N S I T Y O P T H E I N J E C T E D P L U I O • . 2 0 X , P I 2 . • / .2 O X . ' D E N S I T Y O P T H E N A T I V E P L U I D * • 3 1 X . P t 2 . 8 / .3 O X . ' D E N S I T Y D I P P . B E T M E E N T H E P L U I D S ' . 2 6 X . F 1 2 . S / / / / »4 A X . ' A C C E L E R A T I O N D U E T O G R A V I T Y I I C M / S E C > / S E C ) • . I O X . P 1 2 . A / / / )

120

1 9 0 0 0 P O R M A T ! 8 X . * O P E R A T I N C C O N D I T I O N S * / / .1 O X . • I N J E C T I O N A N D P R O D U C T I O N R A T E S ( G A L / M I N ) * / .2 O X . * I N J E C T I O N R A T E F O R F I R S T I N J E C T I O N H A L P - C Y C L S * . 8 X . P I 4 . S / .3 O X » • P R O D U C T I O N R A T E F O R F I R S T P R O D U C T I O N H A L F - C V C L E • . 0 X . P I 4 . 0 / .♦ O X . * I N J E C T I O N R A T E F O R S E C O N D I N J E C T I O N H A L F - C V C L E * . O X . F 1 4 . 0 / .9 O X , * P R O D U C T I O N R A T E F O R S E C O N D P R O D U C T I O N H A L F —C Y C L E * . O X . P I 4 . S / .9 O X . • I N J E C T I O N R A T E F O R T H I R O I N J E C T I O N H A L F - C Y C L E * . S X . F 1 4 * 8 / .7 O X . ' P R O D U C T I O N R A T E F O R T H I R O P R O D U C T I O N H A L F - C Y C L E * . S X . F I 4 . 8 / 1

2 0 0 0 0 F O R M A T ! O X . • V O L U M E O F F L U I D I N J E C T E D OR P R O D U C E D ( G A L L O N S ) * / .1 O X . • F L U I D I N J E C T E D I N F I R S T I N J E C T I O N H A L F - C Y C L E * . 4 X . F 2 0 . 0 / •2 O X . ' F L U I D P R O D U C E D I N F I R S T P R O D U C T I O N H A L F - C Y C L E * . 4 X . F 2 0 . 8 / .3 O X , • F L U l O I N J E C T E D I N S E C O N D I N J E C T I O N H A L F - C Y C L E * . 4 X . F 2 0 . 0 / •4 O X . • F L U I D P R O D U C E D I N S E C O N D P R O D U C T I O N H A L F - C V C L E • . 4 X . F 2 0 . 0 / •5 O X . ' F L U I D I N J E C T E D I N T H I R D I N J E C T I O N H A L F - C Y C L E • « 4 X « F 2 0 . 0 / )

2 1 0 0 0 F O R M A T ( O X . • T I M E O F S T A T I C S T O R A G E ( D A Y S ) * / .1 O X • • A T T H E E N D O F F I R S T I N J E C T I O N H A L F - C V C L E • • I O X . F 1 3 . 0 / .2 O X . * A T T H E E N D O F S E C O N D I N J E C T I O N H A L F - C V C L E • • I O X . F I 3 . 8 / •3 O X . • A T T H E E N D O F T H I R D I N J E C T I O N H A L F - C V C L E • . I 0 X . F I 3 . 8 / / / )

2 2 0 0 0 F O R M A T ( 1 H I . 4 1 X . * C A L C U L A T I O N S F O R F I R S T I N J E C T I O N H A L F - C V C L S • / .I 4 2 X . *---------------------------------------------------------------— ---------- — • / / / )

2 3 0 0 0 F O R M A T ! I X . • ! • • • 1 3 . S X . • R I F T - ' . E 1 9 . 0 . 5 X . * R 2 P T « * . E l 9 . 8 . 3 X . * R 3 F T « * •1 E I 9 . 8 )

2 4 0 0 0 F O R M A T d X . * I * - ' . 1 3 . O X . • O M l » • . E I S . 8 . 0 X . * 0 M 2 « * . E i S . S . S X . * O G » * . E 1 9 . 8 .1 4 X . * X - * . E I 9 * 8 >

2 8 0 0 0 F O R M A T ! I X . • ! ■ * . 1 3 . S X . * T 1 I 0 » * . E l 8 . 8 . 5 X . • T S P O a * . E I 5 . 8 . 4 X . * T T D > * .I E I S . 8 . 3 X . * V L - * . C I 9 . 8 . 3 X . * X L F T** • • E 1 9 . 8 )

2 0 0 0 0 F O R M A T ! I X • • 1 3 . 3 X . • R U S O F T - • . E 1 9 . 8 . 3 X . • R L S O F T a * . E l 9 . S . 3 X . * X R F T > * 1 . E I S . S . 3 X . * V R - * . E t S . 8 . 4 X . * R F T * * . E 1 9 . 0 / / »

2 7 0 0 0 F O R M A T ! I N I . 4 I X . * C A L C U L A T I O N S F O R F I R S T P R O D U C T I O N H A L F —C Y C L E * / •

2 8 0 0 0 F O R M A T 1 1 H I * 4 1 X . • C A L C U L A T I O N S F O R S E C O N O I N J E C T I O N H A L F - C Y C L E * / .1 4 2 X , * ---------------------------- — -------- ----------------------------------------------------— - — - • / / / )

2 9 0 0 0 F O R M A T ! I H I . 4 1 X . * C A L C U L A T I O N S F O R S E C O N O P R O D U C T I O N H A L F - C Y C L E * / •I --------------------------------- 4 2 X . -* ------------- • ✓ / / )

3 0 0 0 0 F O R M A T ! 1 H I . 4 1 X . ' C A L C U L A T I O N S F O R T H I R O I N J E C T I O N H A L F - C Y C L E • / •

3 1 0 0 0 F O R M A T ! I H I * 4 1 X * * C A L C U L A T I O N S F O R T H I R O P R O O U C T I O N H A L F - C Y C L E * / *

3 2 0 0 0 F O R M A T ! I H I . 4 0 X . * C A L C U L A T I O N O F R E C O V E R Y E F F I C I E N C V • / •1---------------------------------4 7 X . -* -------------------------- — --------------------------------------------- • / / / »

I • R C B T F T « * . E 1 S . U / I 3 4 0 0 0 F O R M A T I I X . • ! ■ • • 1 3 . 3 X . • F L P R 6 4 - * * E 1 8 . 8 . 3 2 . * R C E F F » * . E I S . 8 . 3 X .

I * C V L t N G - * . C I S . 3 . 3 X . * C V L R 0 G > * . E I S . 8 . 3 X . * C R C E F F « * . E I 8 . 8 IC

1 3 8 0 S T O P E N O

APPENDIX C

DESCRIPTION OF MINIAQUIFER CONSTRUCTION

The sand used f o r the con st r uc t ion of the m i n i a q u i f e r

was a f i n e - g r a i n e d commercial grade b l a s t i n g sand, com

monly c a l l e d "sugar sand," obtained from I n d u s t r i a l Sand

and Abrasives Corpora t ion , Port A l l e n , Louis iana . Three

sieve analyses of the sand are shown in Table C - l .

The un i f o rm i t y c o e f f i c i e n t is def ined as the r a t i o

of the 40 percent grain s ize to the 90 percent grain s i z e .

The 40 percent s i ze was 0 .0068 inches and the 90 percent

s iz e was 0 .0038 inches. Hence, the u n i fo r m i t y c o e f f i c i e n t

was 0 .0068 t 0 .0038 or 1 .7 9 . The low value of u n i fo rm i ty

c o e f f i c i e n t i nd i ca tes a very uni form sand, one 1n which

the bulk of the grains are of s i m i l a r s i z e . The mean pa r

t i c l e diameter of the sand (50 percent gra in s i z e ) was

found to be 0.0063 inches.

The epoxy adhesive used to conso l ida te the sand was

Armstrong C-7 resin wi th 8 percent by weight of A c t i v a t o r A

(Armstrong Products Co . , Warsaw, I n d i a n a ) . Mixing of the

adhesive and sand was accomplished by thoroughly hand

kneading the mixture fo r 10 to 15 minutes. The maximum

batch s ize t h a t could be proper ly mixed in the requi red

t ime was approximately 3000 grams. The amount of adhesive

to sand was 6 percent by we ight . The working l i f e of a

121

TABLE C - 1 - - Sieve Analyses of Sand Used in M in ia q u i f e r Construct ion

SieveNo.

(US)

ScreenOpening

( in . )

Analysis No. 1 Analysis No. 2 Analysis No. 3Weight

Retained(gm)

PercentRetained

Cum. % Retained

WeightRetained

(gm)

PercentRetained

Cum. % Retained

WeightRetained

(gm)Percent

RetalnedCum. %

Retained

30 0.0328 0 0 0 0 0 0 0 0 0

40 0.0164 0 .4 0.1 0.1 0 .4 0.1 0.1 0 .4 0.1 0.1

50 0.0116 2.0 0 .5 0 .6 2.3 0.6 0.7 2.4 0.6 0.7

70 0.0082 42.5 10.7 11 .3 52.0 13.1 13.8 47.9 12.0 12.7

100 0.0058 184.5 46.5 57.8 185.4 46.7 60.5 186.6 47.0 59.7

140 0.0041 104.6 26.3 84.1 96.6 24.3 84.8 99.4 25.0 84.7

200 0.0029 57.9 14.6 98.7 53.9 13.6 98.4 53.4 13.5 98.2

Pan

Totals

5.3

397.2

1.3 100.0 6.5

397.1

1 .6 100.0 7.1

397.2

1 .8 100.0

122

123

batch a f t e r mixing was between 90 and 120 minutes. With

such a short working l i f e , i t was a physical i m p o s s i b i l i t y

to mix the requi red amount of sand-epoxy m ix t u r e , a p p r o x i

mately 120 batches, and compact i t in the mold be fore the

epoxy began to s e t . I t was found t h a t i f the sand-epoxy

mixture was quick f rozen immediately a f t e r mix ing , i t could

be stored f o r as long as two weeks wi th out any de t r im en ta l

e f f e c t s . When thawed, the mixture was s t i l l workable and

had a working l i f e between 60 and 90 minutes.

A f t e r each batch of sand and epoxy was thoroughly

mixed, i t was placed in a p l a s t i c bag and quick f rozen

using "dry ice" in a chest - type food f r e e z e r . The batches

were placed so t h a t there were a l t e r n a t i n g layers of dry

i ce and sand-epoxy mix ture . Over a per iod o f 5 days, 130

batches were mixed and f r o z e n . On the s i x th day, the

f rozen bags of sand-epoxy mixture were removed from the

f r e e z e r . As they thawed, they were placed in a mold t h a t

was 10 '3" by 5 '0" by 1.5" deep. The mold consisted of a

3 /4" plywood base t h a t had been r e i n f o r c e d wi th 2 x 4 ' s .

The base of the mold had been covered by successive layers

of aluminum f o i l and 6 mil t h i c k polye thy lene f i l m . The

purpose of the aluminum f o i l was to serve as a heat r e

f l e c t o r during the cur ing process. The po lyethy lene f i l m

served as a mold r e le a s e .

The sand-epoxy mixture was packed i n t o the mold using

hand tampers, p lac ing a po lye thy lene f i l m over the mold and

running in place on i t , and by r o l l i n g the mixture w i th a

124

4-1nch diameter by 64 - inch long s o l i d s t a i n l e s s s t ee l bar .

The f i n a l r o l l i n g wi th the bar insured a smooth, uniform

f i n i s h . The elapsed time from thawing of the mixture to

f i n a l r o l l i n g was approximately 75 minutes.

A f t e r the sand-epoxy mixture had been placed in the

mold, 1t was al lowed to cure a t room temperature f o r about

48 hours. A l a rg e in su la ted wooden curing oven was then

placed over the m i n i a q u i f e r and the temperature was

gra dua l l y increased over a per iod of 24 hours to a p p ro x i

mately 200°F. The temperature was maintained a t 200°F f o r

24 hours and then gr ad u a l ly decreased to room temperature .

Temperature changes were gradual to avoid shr inkage cracks.

Even though care was taken to avoid shr inkage cracks

during c ur i n g , some cracking did occur at the edges of the

m i n i a q u i f e r . These were removed by trimming the edges

using a c i r c u l a r , e l e c t r i c handsaw t h a t was equipped wi th

a diamond t ipped blade. The dimensions of the m i n i a q u l f e r

a f t e r tr imming were lO'O" by 4 * 9 . 5 " by 1.5" t h i c k .

A l l surfaces of the m i n i a q u i f e r were sealed using

f i v e coats of Armstrong C-7/A epoxy. The f i r s t three

coats were app l ied using a 7" f l a t / l a t e x pa i n t r o l l e r in

order to obtain a th in coat . Use of th is technique r e

su l ted in a c o n t r o l l e d i m b i b i t i o n depth as the th in c oa t

ing al lowed the epoxy to q u ic k ly a t t a i n c a p i l l a r y e q u i l i b

rium w i t h in the sur face pores. The f i n a l two coats were

much t h i c k e r than the f i r s t three and were appl ied using a

12" mortar t rowel to evenly spread the epoxy in order to

125

obtain a smooth, glossy f i n i s h . Each of the f i v e coats was

al lowed to cure a t room temperature f o r a t l e a s t 24 hours

before the next coat was a pp l i e d .

A 3 /8" by 3 /8" s l o t was cut In to the m i n i a q u l f e r

along three of the four edges In order to produce an Is o -

p o t e n t i a l a t the boundaries (see F ig . 4 . 7 ) . A l i n e n - b a s e ,

laminated phenol ic m at e r i a l was used to cover the s l o t and

to serve as base blocks f o r the I n s t a l l a t i o n o f valves and

f i t t i n g s . Three f u l l we l ls and three h a l f w e l ls were I n

s t a l l e d 1n the m i n i a q u l f e r (see F1g . 4 . 7 ) . Each we l l was

equipped wi th a capaci tance c e l l s i m i l a r to t h a t shown 1n

Figure 4 . 5 .

The m in i a q u l f e r was then tes ted f o r leaks by pr es sur

ing 1t up to 5 ps1 wi th f re o n . An e l e c t r o n i c halogen leak

d e t e c to r was then used to loca t e any le ak s . The leaks were

patched wi th epoxy and the epoxy al lowed to cure. The

m i n i a q u l f e r was then pressured up to 10 ps1 and shut-1n.

A f t e r 48 hours there was no apprec iab le change In pressure,

hence, i t was concluded t h a t a l l leaks had been located

and patched.

In order to assure complete s a t u r a t i o n , the pore

spaces of the m in i a q u l f e r were evacuated by a t ta c h in g a

vacuum pump to wel l 1 (see F1g. 4 . 7 ) and p u l l i n g a vacuum

f o r 72 hours. A f l u i d mixture conta in ing 43 .5 percent

s o l t r o l , 53.2 percent naphtha, and 3.3 percent carbon

t e t r a c h l o r i d e (percents are by volume) was then al lowed to

en te r the m in i a q u i f e r from the i s o p o t e n t i a l . The f l u i d

126

f r o n t moved through the m in ia q u l f e r f i l l i n g the evacuated

pore spaces and converged to wel l 1. When product ion of

the f l u i d s t a r t e d a t wel l 1, the wel l was shut-1n and the

m i n i a q u i f e r al lowed to come to e q u i l i b r i u m . The min i -

a q u i f e r was then f lushed wi th approximately two pore

volumes of f l u i d to d isso lve any gases t h a t came out of

s o l u t i o n during the i n i t i a l s a t u r a t i o n .

APPENDIX D

COMPUTER PROGRAM USED IN COMPUTING THE LONGITUDINAL

DISPERSIVITY COEFFICIENT OF THE MINIAQUIFER

The computer program l i s t e d in the fo l l o w in g pages

is used to c a l c u l a t e the concent ra t ion of na t iv e f l u i d in

the produced stream as a funct ion of cumulat ive t ime since

i n j e c t i o n s t a r t e d . The program is f o r one cyc le only and

i t is assumed th a t no dens i ty or v i s c o s i t y d i f f e r e n c e s

e x i s t between the i n j e c t e d and na t i v e f l u i d s . The program

is in FORTRAN IV language and is w r i t t e n f o r use on an IBM

360/65 system.

A l i s t of the requi red input data is presented at

the beginning of the program. Fol lowing th i s is a com

p l e t e l i s t of a l l the v a r i a b l e names used in the program

along wi th t h e i r d e f i n i t i o n s .

128

PROGRAM TO CALCULATE CONCENTRATION AT THE WELL BORE AS A FUNC TIO N OF CU M ULATIVE T IM E S IN C E IN J E C T IO N STARTED* T H IS FROSRAN I S FOR ONE CYCLE ONLY AND ASSUMES NO O B N S IT V OR V IS C O S IT Y D IFFER ENCES BETWEEN IN JE C T E D AND N A T IV E F L U ID S *

PROGRAM • CONCENT*

OATA TO BE REAO IN

F IR S T CARO - FORM A T I A P I S . 0 )ORISM ■ FLOW RATE FOR IN J E C T IO N H A L F -C V C L I* 16PM )OREGN ■ FLOW RATE FOR PRODUCTION H A L F -C Y C L E * IG P R I T IO » TOTAL IN J E C T IO N T IM E * IO A Y S )TSO * CU M ULATIVE T IM E AT WHICH CONCENTRETION C ALC U LATIO N S

ARC STARTCO . IO A Y S )HFT ■ AQUIFER T H IC K N E S S * I F T )PR » AQ U IFER P O R O S IT Y * IF R A C T IO N )

SECONO CARO - FORMAT! A P IC .O )ALP ■ LO N G IT U D IN A L O IS P C R S IV IT V C O E F F IC IE N T * IC M )DIPM O L » C O E F F IC IE N T OF MOLECULAR D IF F U S IO N * I IS O C M )/S E C )

• DEL TO - T IM E INCREMENT FOR CONCENTRATION C A LC U LA T IO N S . IO A Y S ) RVFT ■ WELL BORE R A D IU S . I F T )

D E F IN IT IO N OF V A R IA B L E NAMES USED IN PROGRAM

CFFTCM ■ CONVERSION FACTO R . IC M /F T )CGMCCS ■ CONVERSION FA C TO R . I IC C / S C O / IG A L /M I N ) )CFOSCC ■ CONVERSION FACTOR* IS E C /O A Y )OR I ■ FLOW RATE FOR IN J E C T IO N H A L F -C Y C L E . IC C /S C C )ORE ■ FLOW RATE FOR PRODUCTION H A L F -C Y C L E * IC C /S S C )H - AQ UIFER T H IC K N E S S * IC M )OELT ■ T IM E INCREMENT FOR CO N C IM TR ATlO N C A LC U LA T IO N S . IS E C ) RW - WELL BORE R A D IU S * IC M )T l ■ TOTAL IN J E C T IO N T IM E . IS C C )TS ■ CU M ULATIVE T IM E AT WHICH CONCENTRATION IS

COMPUTED. I S E C )T 2D ■ CU M U LA TIVE T IM E AT WHICH CONCENTRATION IS

COMPJTEO* IO A Y S )PPP - PRODUCT OF P I . P O R O S IT Y * ANO TH IC K N E S S * IC M )PPP1 ■ E R P P P I. IC M )O i l ■ TWO O IM E N S IO N A L FLOW RATE FOR IN J E C T IO N

H A L F -C Y C L E * 11 SO C M I/S C C )OPS - TWO D IM E N S IO N A L FLOW RATE FOR PRODUCTION

H A L F -C Y C L E * 11 SO C M I/S B C )c i i ■ c o M P u re o c o n c e n t r a t io n * i v o l u m e f r a c t i o n )FPE • SOUARC OF DENOMINATOR OF ARGUMENT OF COMPLEMENTARY

ERROR F U N C TIO N *XX ■ ARGUMENT OF COMPLEMENTARY ERROR FU N C TIO N *

R EAD IN G OATA

10 R B A O IS .1 0 0 0 * E N O -S S )O R IQ M .B R B S M .T IB .T S O .H F T .P R R E A O IS • BOO• ) A L P .B IF R O L • D B L T O .R W T

129

ccC P R IN T IN G OATA

CMR I r e I 6 • 3 0 0 0 ) ALP• O IFM OL • RM FT. T 10 M R IT E t 6 . 4 0 0 0 )

ccC C O N STAN TS AND C O N V E R S IO N FA C T O R S I T I I L O U N IT S TO C . S . S * U M T S !C -------------------------------- — — ------- — -----------— — — ----------------c

c p o s e c * s « « o o * oC F F T C M « 3 0 .4 S 0 IC G M C C S » 6 3 .0 9 0 60 E L T -0 E L T 0 4 C P 0 S E CH « H F T * C F P T C NP P P > 1 . U I 6 « P R « HP P P |» 2 .0 * P P POR 1 «O R IC M *C G M C C S01 l * O R l / P P P lQ R 2>0R 2C M *C O M C C S0 P 2 « 0 R 2 /P P P |R M *R M P T *C F P T C M T l » T t 0 *C P D S E C

C A L C U L A T IO N OP C O N C E N TR A T IO N VERSUS C U M U L A T IV E T IM E

T 2 - T S D * C P 0 S E C20 F P 2 » I . 3 3 3 4 A L P 4 I < 2.0*0P2*T 2)**t.0 - 1 2 .0*0P2*TI)•• ! .8 * 1 2 .0 * 0 1 1 ATI I M l

t . S ) * 0 I F M O L 4 I I 2 »0*QP2*T2 l**2/0PS «l2»0*O P2*T l)**2/0P2*12 .0*01l*T2 1 1 * * 2 / 0 1 1 )

X X - I - O P 2 * ! T 2 - T I ) - R M * « 2 / 2 . 0 * O t I * T 1 ) / P P 2 * * 0 . SC U - E R P C I X X ) / 2 . 0T 2 D - T 2 / C P D S E C

P R IN T IN G COMPUTED V A LU E S OP C O N C E N T R A T IO N VE R SU S C U M U L A T IV E T IM E

M A I T E t 8 . SOOO) T S O .C l1C

I P I C l 1 . o r . 0 . 9 * ) COTO10T 2 *T 2 * 0 E L TCOT 0 2 0

CcC PORMAT STATEMENTSC - - - -------------------------— -

IS O S P O R M A TIS P 1 2 *0 I 2SSS F O R M A T I4 F !2 * 0 )3 0 0 0 P O R M A T II M l. 9 2 • •L O N G IT U D IN A L D IS P S R S IV IT V C O C P P IC IS M T I C M ) * .O X .P IS *

1 8 / . IO X . • C O E F F IC IE N T OP MOLECULAR D IF F U S IO N I IS O C M l/S E C ) • . 2 X . P I S2 * 8 / * I O X . • MELL BORE R A O IU S I F T ) * . 2 9 X .P 1 5 . 8 / . I O X .3 • I N J E C T I ON T IM E IO A V S )* . 2 9 X . P I S . 8 / / / )

4 0 0 0 FORMAT I3 2 X .*C O N C E N T R A T IO N P R 0 P I L S * / / . 2 9 X . ‘ T IM E • . 1 2 X .I 'C O N C E N T R A T IO N * /.2 8 X .* IO A V S )* .O X .*< V O L U M E F R A C T IO N ) * / / )

SOOO F O R M A T I 2 S X . P I 2 . S . I I X . F 7 .S )C

SO STOP EMO

APPENDIX E

RESULTS OF EXPERIMENTAL RUNS MADE TO VERIFY THE MATHEMATICAL MODEL

The f o l l o w in g tables summarize the la b o ra t o r y data

t h a t have been obtained during th i s i n v e s t i g a t i o n and in a

previous i n v e s t i g a t i o n (Kumar, 1968) on the f e a s i b i l i t y of

s t o r in g f resh water in s a l i n e a q u i f e r s . Also tabu lated

are the recovery e f f i c i e n c i e s predic ted f o r the various

exper imental runs using the computat ional procedure pro

posed 1n th i s paper. In t h i s i n v e s t i g a t i o n and t h a t by

Kumar, breakthrough was when the produced stream contained

3 percent na t i v e f l u i d .

Tables E-1 through E-4 summarize the data obtained

during th i s i n v e s t i g a t i o n . Since the m in i a q u l f e r used 1n

t h i s i n v e s t i g a t i o n is o n e - h a l f of a f u l l system, the tabu

la te d values of rates and volumes are o n e - h a l f o f those

f o r a f u l l system.

Tables E-5 and E-6 summarize the data Kumar obtained

using a 45° se c tor of a r a d i a l system. The physical prop

e r t i e s of the system were: thickness = 7.65 cm,

po ros i ty = 0 . 2 5 5 , p e r m e a b i l i t y « 6 .90 d a r c i e s , l o n g i t u d i n a l

d i s p e r s i v i t y c o e f f i c i e n t = 0 .04 cm, and w e l lbore

radius = 0 . 4 cm. Note t h a t since th i s system was one-

eighth of a f u l l system, the tabu la ted values of rates and

130

131

volumes are one-e1ghth of those f o r a f u l l system.

The de s c r i p t io n s of the Run Types l i s t e d 1n the

tab les are:

Run Type 1A: F lu id was i n j e c t e d I n t o the center wel l

of the system f o r a predetermined length of t ime. This was

the I n j e c t i o n h a l f - c y c l e . The I n j e c t e d f l u i d was then pro

duced through the center we l l u n t i l breakthrough occurred.

This was the product ion h a l f - c y c l e . I f s t a t i c storage of

the i n j e c t e d bubble was to be Included 1n the run, th is

took place between the end of the I n j e c t i o n h a l f - c y c l e and

the beginning of the product ion h a l f - c y c l e . The steps o u t

l i ned above c o n s t i t u t e d one complete c y c le .

Run Type I B : Same as Type 1A except cont inued f o r

two complete cyc les .

Run Type 1C: Same as Type 1A except cont inued f o r

three complete cycles.

Run Type 2A: F lu id was i n j e c t e d i n t o we l l 1 (see

F ig . 4 . 7 for we l l l o c a t i o n s ) f o r a predetermined length of

t ime. This was the I n j e c t i o n h a l f - c y c l e . During the pro

duct ion h a l f - c y c l e f l u i d was withdrawn through w e l ls 1, 3,

and 5 u n t i l breakthrough occurred In e i t h e r we l l 3 or 5.

At t h a t t ime wel ls 3 and 5 were shut -1 n . This concluded

the f i r s t step of the product ion h a l f - c y c l e . Product ion

through wel l 1 continued u n t i l breakthrough occurred. This

concluded the product ion h a l f - c y c l e . I f s t a t i c storage of

the I n j e c t e d bubble was to be Included 1n the run t h i s took

place between the end of the i n j e c t i o n h a l f - c y c l e and the

132

beginning of the product ion h a l f - c y c l e . The steps ou t l in e d

above c o n s t i t u t e d one complete cy c l e .

Run Type 2C: Same as Type 2A except continued f o r

three complete cy c les .

Run Type 3C: F lu id was I n j e c t e d In to we l l 1 u n t i l

the f r o n t passed we l ls 3 and 5. This concluded the f i r s t

step of the i n j e c t i o n h a l f - c y c l e . I n j e c t i o n was then

s t a r t e d 1n we l ls 3 and 5, wi th i n j e c t i o n cont inuing In

wel l 1, and cont inued u n t i l the desi red volume had been I n

j e c t e d . This completed the i n j e c t i o n h a l f - c y c l e . During

the product ion h a l f - c y c l e f l u i d was withdrawn through

wel ls 1, 3, and 5 u n t i l breakthrough occurred 1n e i t h e r

wel l 3 or 5. At t h a t t ime we l ls 3 and 5 were shut -1n .

This concluded the f i r s t step of the product ion h a l f - c y c l e .

Product ion through we l l 1 continued u n t i l breakthrough oc

curred. This concluded the product ion h a l f - c y c l e . I f

s t a t i c storage of the I n j e c t e d bubble was to be Included

in the run, th is took place between the end of the I n j e c

t io n h a l f - c y c l e and the beginning of the product ion h a l f

cyc le . The steps o u t l i n e d above c o n s t i t u t e d the f i r s t

cyc le . The run was continued f o r three complete cycles .

Run Type 4C: F lu id was I n j e c t e d in to we l l 1 u n t i l

the f r o n t passed we l ls 3 and 5. This concluded the f i r s t

step of the f i r s t I n j e c t i o n h a l f - c y c l e . The water I n j e c t e d

1n th is step is termed "cushion w a t e r . " I n j e c t i o n was then

s t a r t e d in w e l ls 3 and 5, wi th I n j e c t i o n cont inuing in

wel l 1, and continued u n t i l the desi red volume had been

133

I n j e c t e d . This completed the I n j e c t i o n h a l f - c y c l e . During

the product ion h a l f - c y c l e , f l u i d was withdrawn through

wel ls 1, 3 and 5 u n t i l breakthrough occurred 1n e i t h e r

wel l 3 or 5. This concluded the product ion h a l f - c y c l e . I f

s t a t i c storage of the I n j e c t e d bubble was to be Included In

the run, th i s took place between the end of the I n j e c t i o n

h a l f - c y c l e and the beginning of the product ion h a l f - c y c l e .

Two a d d i t i o n a l cycles were c a r r i e d out 1n a manner s i m i l a r

to the steps descr ibed above except t h a t the f i r s t step of

the f i r s t i n j e c t i o n h a l f - c y c l e was excluded 1n the two

a d d i t i o n a l cyc les .

Run Type 5B: F lu id was I n j e c t e d I n t o the center wel l

of the system f o r a predetermined length o f t ime. This was

the f i r s t I n j e c t i o n h a l f - c y c l e . The In j e c t e d f l u i d was then

produced through the center we l l but product ion was stopped

short of breakthrough. This concluded the f i r s t product ion

h a l f - c y c l e and the f i r s t f u l l c y c le . The process was r e

peated f o r a second cycle except t h a t during the second

product ion h a l f - c y c l e product ion continued u n t i l b reak

through occurred. I f s t a t i c storage was to be Included 1n

the run, i t took place between the end o f an I n j e c t i o n h a l f

cycle and the beginning of a product ion h a l f - c y c l e . Kumar

u t i l i z e d Run Type 5B f o r his two-cyc le exper iments.

TABLE E - l Data f o r S i n g l e Wel l I n j e c t i o n and S in g le Well P roduct ion

R U N N U M B E R1 2 3 6 8 11 12 l<t 16 17

R u n T y p e1A 1C 11 1C - i r 1» tc "TC ' T l - ” " 1A

0.942 0.966 0.967 0.900 0.950 0.924 0.924 0.923 0.917 0.9170.954 0.967 0.961 0.901 0.956 0.926 0.926 0.925 0.919 0.9190.946 0.967 0.964 0.901 0.953 0.925 0.925 0.924 0.918 0.9181 0.783 0.783 0.784 0.854 0.791 0.861 0.861 0.859 0.854 0.854I 0.781 0.781 0.782 0.776 0.782 0.785 0.785 0.782 0.778 0.7780.002 0.002 0.002 0.077 0.009 0 076 0.076 0.077 0.076 0.076

24.138 24.138 6.705 17.880 20.115 20.115 20.115 20.115 1.609 0.3665170. 5170. 5174. 4300. 5029. 5029. 5029. 5029. 5064. 5029.

0. 0. 0. 0. 0. 0. 0. 250. 0. 0.24.138 24.138 6.705 17.880 20.115 20.115 20.115 20.115 1.609 0.366

4797. 4867. 4707. 3194. 4687. 3660. 3887. 3436. ; ; ? i . 852.

93 94 91 74 93 77 77 68 31 1785 66 77 61 78 64 64 58 20 0

93 94 91 74 93 77 77 68 39 1785 86 77 61 78 64 64 58 20 0

0.074 0.063 0.284 0.970 0.241 0.964 0.964 1.537 9.927 - -

24.138 6.705 17.880 20.115 20.115 20.1155170. 5174. 4800. 5029. 5029. 5029.

0. 0. 0. 0. 0. 250.24.136 6.705 17.880 20.115 20.115 20.115

4932. 4866. 3760. 4747. 4484. 4194.

95 94 87 94 89 8390 85 77 86 85 79

95 93 81 94 83 7688 81 74 82 75 68

0.103 0.482 1.646 0.405 1.711 2.770

FLUIO PROPERTIESVise, o f I n j . F lu id (cp)Vise, o f Rat. F lu id (cp)Mean V iscos ity (cp)Oenslty o f I n j . F lu id (ga/cc) Density o f Rat. F lu id (ga/cc) Density D iffe rence (ga/cc)

FIRST CYCLEIn je c t io n Rate (cc /a ln )Voluae In jec ted (cc)S ta t ic Storage (a ln )Production Rate (c c /a ln ) Voluae Produced (cc)Cycle Recovery E f f . (S)

Experlaental Coaputed

Cua. Recovery E f f . (1) Experlaental Coaputed

Value o f D 'less Croup f R.T.

SECONO CYCLEIn je c t io n Rate (c c /a ln )Voluae In jec ted (cc)S ta t ic Storage (a ln )Production Rate (c c /a ln ) Voluae Produced (cc)Cycle Recovery E f f . ( I )


Cua. Recovery E f f . (S) Experlaental Coaputed

Value o f O 'less Croup # 8.T.

TABLE E- l ( con t inu ed)

THIRO CYCLEIn je c t io n Roto (c c /a ln ) VoI u m In joc tod (cc)S ta t ic Storage (a ln )Production Rato (c c /a ln ) Yoluao Produced (cc)Cycle Recovery E f f . (X)


Cua. Recovery E f f . (X) Experlaental Coaputed

R U N N U M B E R1 2 3 6 8

R u nTJ ------------ K ' I I 1C T l

24.138 17.880 205170. 4300. 5029

0 . 0 . 024.138 17.880 20

5036. 3937. 4823

97 92 9692 85 88

96 84 9589 78 84

0.141 2.316 0.S50

11 12 M 16 17T y p e _________________________________

lA 1C 1C ~TA 1*

20.I IS 20 .I IS5029. 5029

0. 25020.115 20

4643. 4472

92 8987 81

86 8079 72

2.355 3.827

TABLE E-2 Data f o r S in g le Wel l I n j e c t i o n and M u l t i p l eWel l P roduc t ion

R U N N U M B E R R U N N U M B E R«» 5 « s

__ ..H u T i l t - . . . { » TJfW

FLUI4 FROFERTIES SEC0R4 CYCLE (cootlaooO)Vise. or I n j . M a l l (cp) V ltc . o f Rat. F1a10 (cp)

0 .* Ft 4.PP4 Caa. Racavar? E f f . (4)R.POS 4.*44 Enperlaaatal 44

Moon V tscaslt? (cp) 0.PJ2 4.441 CaapataO 73Dansl t? o f I n j . M a H (pa/cc) O.FPJ 4.742 Valaa a f 4 ' lass 4ro«p 4 4.T . 4.444Dens I t? o f Rot. M a lJ (pa/cc) Oeaslt? S lfforonco (pa/cc)

0 .FI1t . M t

0.74)0 .0 1 1 SecoaO FraOactlon Rata (c c /a ln )

Valaaa FraOacaO (cc)14.444

1442.FIRST CYCLE

In je c tio n Rato (c c /a ln ) iR .ta s 14.444Stop Racavar? E f f . (4)

Eaperlaeatal 4TVotoaa InJoetoO (cc) 4/14. 4141. CaapataO 13S ta tic Storapa (a ln ) 1.744 4.S4S C?c1a Racavar? E f f . ( I )F ir s t FroOoctlon Rata (cc /a ta ) n . m 14.244 Enperlaaatal 44Valaaa FraOacaO (cc) 3134. 4340. CoapateO 43Stop Recover? ( f f . ( I ) Caa. Racavar? E ff . (4)

Enperla ta ta l 44 ts Esperlaaatal 44CaapataO 47 42 CaapataO 44

Can. Recover? I f f . ( I ) Valaa a f 4 'laaa Rroep 4 R.T. 4.431( ip e r la e a ta l 44 4S THIRO CYCLE

In je c tio n Rata (c c /a ln )CaapataO 47 44

14.444Valaa o f P 'lass troop P P.T. 4.444 4.444 Valaaa InjactaO (cc) 4141.SacanO FraOactlan Rato (cc /a ta ) Yalwaa FroOacaO (cc)

1.447444.

3.4471444.

S ta tic Storapa (a ln) 7.444F ir s t PreOactloi. Rata (c c /a la ) 14.444Stop Racavar? I f f . (S)

Experlaaatat CaapataO

44IS

4S13

Valaaa FroOacaO (cc) 3444.Stop Racavar? E ff . (4)

C?e)e Racavar? E ff . ( I ) Enperlaaatal 44 40

EnperlaentstCaapataO

7174

CaapataO 44 74 Caa. Racavar? E f f . (4)Caa. Racavar? E ff . (4 ) . E iporlaon ta l

CaapataO4474

Enperlaaatal 44 44CaapataO 44 74 Valaa a f 4*lass Rroop 4 R.T. 4.744

fataa e f 0‘ lasa Proap 4 R.T. 4.144 4.344 SacanO FroOoctlon Rato (c c /a ln ) 14.444Volaaa FroOacaO (cc) 1443.

SECRRP CYCLEIn je c tio n Rato (c c /a ln ) 14.444

Stop Racavar? E f f . (4) E iporlaaa ta l 47Valaaa InJoctoO (cc) 4141. CaapataO 14

S ta tic Storapa (a ln ) 4.417 C?clo Racavar? E f f . (4)F irs t FroOoctlon Rato (c c /a la ) 14.444 la po rlaa a ta l 44Yoloaa FroOacaO (cc) 3444. CaapataO 44Stop Racavar? E ff . (S) Caa. Racavar? I f f . (4)

44Eaporlaoatal 44 EnparlaantalCoapotoO 74 CaapataO

Valoo o f 4 ' lo ta Rroop 4 R.T.444.444

13

6

137

TABLE E-3 Data f o r M u l t i p l e Well Well Product ion

I n j e c t i o n and M u l t i p l e

RUN NUMBER 9

R u n T y p •3C

FLUID PROPERTIESVise, o f I n j . F lu id (cp)Vise, o f Nat. F lu id (cp)Mean V isco s ity (cp)Oonslty o f I n j . F lu id (ga/cc) Density o f Nat. F lu id (ga/cc) Oonslty D lf fa ranca (ga/cc)

FIRST CYCLEF i r s t In je c t io n Rate (c c /a ln ) Voluao In jec ted (cc)Second In je c t io n Rata (cc/a1n) Voluae In jec ted (cc)S ta t ic Storage (a ln )F i r s t Production Rate (cc /a ln ) Voluae Producod (cc)

(S)Step Recovery E f f . Experlaental Coaputed

(S)Cua. Racovary E f f .Experlaental Coaputed

Value o f C la s s Group 6 8.T.Second Production Rato (c c /a ln )Voluae Produced (cc)Step Recovery E f f . (S)


Cycle Recovery E f f . (S) Experlaental Coaputed

Cua. Recovery E f f . (S) Experlaental Coaputed

Value o f O 'less Group 9 B.T.

SECONO CYCLEF i r s t In je c t io n Rate (c c /a ln )Voluae In jec ted (cc)Second In je c t io n Rate (c c /a ln )Voluae In jec ted (cc)S ta t ic Storage (a ln )F i r s t Production Rate (cc/a1n)Voluae Produced (cc)Step Recovery E f f . ( t )


0.9440.9460.9450.7920.7830.009

4.0231006.

20.115 4023.

0.20.115

3755.

7563

75630.3984.023

768.

1512

9076

90760.472

4.0231006.

20.115 4023.

0.20.115

3984.

7971

SECONO CYCLE (continued)Cua. Recovery E f f . (S)


Value o f O 'less Group 8 B.T.Second Production Rate ( c c /a ln )Voluae Produced (cc)Step Recovery E f f . (X)


Cycle Recovery E f f . (S) Experlaental Coaputed

Cua. Recovery E f f . ( t ) Experlaental Coaputed

Value o f O 'less Group 8 B.T.

THIRO CYCLEF i r s t In je c t io n Rate (c c /a ln )Voluae In jec ted (cc)Second In je c t io n Rate (c c /a ln )Voluae In jec ted (cc)S ta t ic Storage (a ln )F i r s t Production Rata (c c /a ln )Voluae Produced (cc)Step Recovery E f f .


(*)

( * )

88740.7164.013

698.

1414

9385

9281

0.768

.023

Cua. Recovery E f f .Experlaental Coaputed

Value o f O'less Group 6 B.T.Second Production Rate (c c /a ln )Voluae Produced (cc)Step Recovery E f f . (X)


Cycle Recovery E f f . (X) Experlaental Coaputed

Cua. Recovery E f f . (X) Experlaental Coaputed

Value o f O'less Group 9 B.T.

41006

20.115 4023.

0.20.115

4036.

8075

88790.9734.023

973.

1913

9989

9484

1.015

138

TABLE E-4 Data f o r M u l t i p l e Well I n j e c t i o n and M u l t i p l e Well Product ion

RUN NUMBER 10 13 15

FLUID PROPERTIES Vise, o f I n j . F lu id (cp)V1*c. o f Hot. F lu id (cp)Noon V isco s ity (cp)Density o f I n j . F lu id (ga/cc) Density of Hot. F lu id (ga/cc) Oonslty D if fe rence (ge/cc)

FIRST CYCLEF i r s t In je c t io n Rote (cc/a1n) VoIusm In jec ted (cc)Second In je c t io n Rote (c c /e ln ) Voluae In jec ted (cc)S to t lc Storage (a ln )Production Rote (c c /a ln )Voluae Produced (cc)Cycle Recovery E f f . (S)


•Cua. Recovery E f f . (i) Experlaental Coaputed

Volue o f O 'less Group • B.T.

SECOHO CYCLEIn je c t io n Roto (c c /a ln )Voluae In jec ted (cc)S to t lc Storoge (a ln )Production Rote (c c /a ln )Voluae Produced (cc)Cycle Recovery E f f . It)


•Cua. Recovery E f f . (S) Experlaental Coaputed

Volue o f D 'less Group D B.T.

THIRD CYCLEIn je c t io n Rote (c c /a ln )Voluae In jec ted (cc)S to t lc Storoge (a ln )Production Rote (c c /a ln )Voluae Produced (cc)Cycle Recovery E f f . (S)


•Cua. Recovery E f f . (S) Experlaental Coaputed

Volue o f D 'less Group D B.T.

0.94S 0.922 0.9260.942 0.92S 0.9270.945 0.923 0.9270.793 0.860 0.8600.783 0.784 0.7840.010 0.076 0.076

4.023 4.023 4.0231006. 1341. 1341.

20.115 20.115 20.1155029. 6029. 5029.

0. 0. 280.20.115 20.115 20.115

4600. 4020. 3566.

91 80 7178 78 72

76 63 S665 62 870.463 2.045 2.592

20.115 20.115 20.1155029. 5029. 5029.

0. 0. 250.20.115 20.115 20.115

4861. 4429. 3913.

97 88 7886 85 78

86 74 6675 72 660.645 2.766 3.769

20.115 20.115 20.1555029. 5029. 5029.

0. 0. 250.20.115 20.115 20.115

5015. 4568. 4476.

100 91 8989 86 80

9079

0.810

7976

3.404

73704.806

• A l l cuau lo t lve recovery e f f ic ie n c ie s were coaputed w ith the "cushion water" voluae Included.

139

TABLE E-5 Data f o r S ing le Well Operat ion (Kumar, 1968)

RUN2

R u1A

NUMBER 3 6

n T y p ei A .... i r

FLUID PROPERTIESVise, o f I n j . F lu id (cp) Vise , o f Nat. F lu id (cp)

0.833 0.801 0.6560.781 0.647 0.571

Mean V is c o s i t y (cp) 0.807 0.724 0.614Densi ty o f I n j . F lu id (gm/cc) 0.765 0.765 0.751Densi ty o f Nat. F lu id (gm/cc) 1.129 0.957 0.823Densi ty D i f fe rence (gm/cc) 0.364 0.192 0.072

FIRST CYCLEI n j e c t i o n Rate (cc /mln) 74.843 22.857 6.000Volume In jec te d (cc) 4528. 3200. 1974.S t a t i c Storage (mln) 14.500 13.333 121.000Product ion Rate (cc/m1n) 95.124 50.455 6.000Volume Produced (cc) 2933. 1850. 168.Cycle Recovery E f f . ( 1 )

Experimental 65 58 9Computed 62 52 31

Cum. Recovery E f f . (X)Experimental 65 58 9Computed 62 52 31

Value o f D ' l ess Group 9 B.T. 0.388 0.526 0.844

140

TABLE E-6 Data f o r Sint (Kumar, 1968]

)1 e Well Operat i on

RUN NUMBER4 5

R u n ■■ W — L H r -

FLUID PROPERTIESVise, o f I n j . F lu id (cp) Vise, o f Nat. F lu id (cp)

0.799 0.7290.656 0.592

Moan V is c o s i t y (cp) 0.728 0.661Oonsl ty o f I n j . F lu id (gn /cc ) 0.764 0.761Densi ty o f Nat. F lu id (gn /cc) 0.870 0.805Densi ty D i f fe rence (gn/cc) 0.106 0.044

FIRST CYCLEI n j e c t i o n Rate ( c c /n ln ) 63.000 89.714Volune In jec te d (cc ) 3150. 3140.S t a t i c Storage (n1n) 20.167 20.000Product ion Rate (cc/m1n) 66.250 70.667Volune Produced (cc) 1325. 2120.

SECOND CYCLEI n j e c t i o n Rate (cc/n1n) 72.422 86.333Volune In je c te d (cc) 1545. 2590.S t a t i c Storage (n ln ) 20.167 13.333Product ion Rate ( c c /n ln ) 73.489 78.912Volune Produced (cc) 2250. 2900.Cun. Recovery E f f . (X)

Exper lnenta l 76 88Conputed 70 78

Value o f D ' less Group • B.T. 0.253 0.139

VITA

Water Richard Whitehead, the son of Velma D. and

Richard A. Whitehead, was born August 29, 1937, 1n Lake

Char les , Louis iana. He was graduated from Forest H i l l High

School , Forest H111 , Lou is iana , in June 1955. In September

of t h a t y e a r , he entered Southwestern Louisiana I n s t i t u t e

(now the U n iv e r s i t y of Southwestern L o u i s i a n a ) , L a f a y e t t e ,

Louis iana , from which he received the Bachelor of Science

degree in C i v i l Engineering in May of 1960. He was imme

d i a t e l y employed by the U.S. Forest Service as a C i v i l

Engineer . A f t e r four months wi th the U.S. Forest Service

he entered a c t i v e m i l i t a r y s e rv i ce 1n September of 1960.

Upon re lease from a c t i v e m i l i t a r y serv ice in March of 1961,

he accepted a p o s i t i o n wi th the Louisiana Department of

Highways as an A s s i s ta n t D i s t r i c t Laboratory Engineer. In

September o f 1962 he resigned from the Department and

entered the Graduate School of Louisiana Sta te U n i v e r s i t y

in Baton Rouge, from which he received the Master of

Science degree 1n C i v i l Engineering 1n May of 1964. In

June 1964, he jo ined the General Engineering D iv is io n of

the Ethyl Corporat ion where he held pos i t ions from Design

Engineer to Head of the C i v i l Engineering Group. In June

of 1970, he resigned from the Ethyl Corporat ion and r e

turned to Louisiana State U n iv e r s i t y to begin work toward

141

142

the Doctor of Phi losophy degree in C i v i l Engineer ing.

He is married to the former M a r j o r i e Smith of McCal l ,

M i s s i s s i p p i . They are the parents of four daughters:

Cheryl ( 1 6 ) , Kathi ( 1 4 ) , Cynthia ( 9 ) , Lynet te ( 7 ) ; and one

son, Daryl ( 1 2 ) .

EXAMINATION AND THESIS REPO RT

Candidate: W a lte r R. W hitehead

Major Field: C i v i l E n g in eerin g

Title of Thesis: S to rage o f Fresh W ater in S a lin e A q u ife rs Using a W e ll F ie ld .

Approved:

M ajor Professor a: lairman

Dean of the Graauate School

IN IN G M IT T E E :

Date of Examination:

J u ly 2 , 1974

Storage of Fresh Water in Saline Aquifers Using a Well Field.

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