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WRC RESEARCH REPORT NO 77

ANALYSIS OF LIQUID-WASTE INJECTION WELLSI N ILLINOIS BY MATHEMATICAL MODELS

Manoutchehr Heidar iand

Keros Car twr igh tILLINOIS STATE GEOLOGICAL SURVEY

andP a u l E Sa y l o r

DEPARTMENT OF COMPUTER SCIENCEUNIVERSITY OF ILLINOIS AT URBANA-CHAIIPAIGN

U rbana I l l i n o i s

F I N A L R E P O R TProject A-058-ILL

The work upon which t h i s pu b l i ca t io n s based was suppo rted by fund sp r o v i d e d b y t h e U S . Dep ar tm en t of t h e I n t e r i o r a s a u t h o r i z e d u n de r

t h e Water Resourc es Research Act of 19 64 P.L. 880379Agreem ent No. 14-31-0001-3813

UNIVERSITY OF ILLINOISWATER RESOURCES CENTER

2535 Hydrosystems Labora toryU rb an a I l l i n o i s 6 18 01

February 1974


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ABSTRACT

ANALYSIS OF LIQUID-WASTE INJECTION WELLS I N ILLINOIS BY MATHEMATICAL MODELS

T h i s r e p o r t c o n t a i n s t h e r e s u l t s o f a p r e li m i na r y t h e o r e t i c a l s t ud y of t h e f a t eof l i q u i d i n d u s t r i a l wastes i n j e c t ed i n t o deep g eo l o g i c fo rm a t i o n s . The J o n esand Laughlin Cor pora t ion w el l was used as a model and th e geo logy o f th e a re a w a sid ea l i ze d i n t o a 15- layered homogeneous and an is o t ro p i c mathemat ica l model . Thef i n i t e e l em ent m et ho d w a s t e s t e d and pr ov ed t o b e a n e f f e c t i v e ma t he m at i ca l t o o li n t h e s o l u t i o n of t h e eq u a t i o n o f f l o w. The f l o w and p re s s u re bu i ld -up show t h a tt h e r o c k s are cap ab l e of r ece i v i n g g r ea t e r vo lu mes of waste t h a n are now beingi n j e c t e d w i th o u t e nd an ge ri ng t h e i n t e g r i t y o f t h e a q u i f e r o r t h e c o n f i ni n g l a y e r .The mass - t rans por t equa t ion fo r l a r ge and complex g round-water re se rv o i r systemsw a s i n v e s t i g a t e d a n d t w a s c on cl ud ed t h a t t h e d i s p e r s i o n a nd d i f f u s i o n p a r t s o ft h e e q u at i o n are r e l a t i v e l y i n s i g n i f i c a n t a nd u nd er e xt re me c o n d it i o n s t h ed i s p e r s ed zo n e w i l l no t be more than a f ew f e e t w i de . T h e re fo re t w a s concludedt h a t a more p r ac t i ca l ap pro ach t o t h e p ro b l em wo u l d b e t h e s o l u t i o n o f a sys temw i t h a moving i n t e r f ac e boundary i n which m a s s t r a n s p o r t r e s u l t s ma in ly fr omconvect ion .To overcome d i f f i c u l t i e s encountered wi t h computer t ime and memory i n th e so lu t i ono f t h e mas s - t r an s p o r t eq u a t i o n f o r l a rg e com pl ex s y s tem s an i t e r a t i v e m et hod sproposed f o r t h e s o l u t i o n of t h e e q u a t io n s whic h s u b s t a n t i a l l y r e du c es t h e s ed i f f i c u l t i e s .

H e i d a r i M . S a y l o r P . and Cartwright KANALYSIS OF LIQUID-WASTE INJECTION WELLS I N ILLINOIS BY MATHEMATICAL MODELSUn ive rs i ty o f I l l i n o i s Water Resources Cen ter Report No. 77KEYWORDS ma ss- tra nsp ort eq ua ti on / fl ow eq ua ti on / ma the ma tic al mode ling /f i n i t e e l em e nt m et ho d/ m u lt i - l ay e r ed s y st e ms / s o l u t i o n o f l a r g e s e t of l i n e a re q u a t i o n s / adaptive-Chebyshev-factor izat ion method


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pagei i Three -Layered System wi t h Cons tan t In je c t io nRat- Boundary Condition ........................ 9

c. A ppl i c a t i on o f t h e F i n i t e El em en t Method t o aField Problem ....................................... 48

i Geology ........................................ 48i i S o l u t i on by F i n i t e E le me nt Method 6

AN ADAPTIVE CHEBYSHEV FACTORIZATION METHOD FOR THE SOLUTION OF LINEAREQUATIONS ARISING FROM THE NUMERICAL SOLUTION OF FLOW EQUATION 7

D. Approximat- Fac tor i za t ions ...................................... 8E. The Fini te-D iffe renc e Case ...................................... 82F. A pprox im at e F a c t o r i z a t i o n o f F i n i t e E le me nt M a t r i c e s 88


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v

LIST OF FIGURES

pageF igur e 1. Schematic column s et up and t y p i ca l measurements f o r an

ins t a n ta ne ous po in t sou r c e i n je c t i o n ....................... 5Figure 2. Schemat ic rep res ent a t i on of th e zone of mixed f l u id s

o r gradual boundary ........................................ 6F igur e 3 C o n ce n tr a ti o n d i s t r i b u t i o n o b t a i n e d w i t h t h e a bo ve d a t a

(s ee Fi gu re 3) and Hoopes and Harleman s approx imates o l u t i o n w i t ho u t m o le c ul a r d i f f u s i o n ....................... I

F igur e 4. C onc e nt r a tion d i s t r i bu t i on ob ta ine d wi th th e a bove da t a(s ee Fi gu re 4) and Hoopes and Harleman s approx imates o l u t i o n w i t h m ol e cu l ar d i f f u s i o n .......................... 14

F igur e 5. C o n ce n tr a ti o n d i s t r i b u t i o n o b t a in e d w i t h t h e abo ve d a t a(s ee Fi gu re 5) and Hoopes and Harleman s approx imates o l u t i o n ................................................... 1 7

Figure 6-a . Divis io n of two-dimensional regio n in to t r ia ng ul a rl m n t s.................................................. 23

Figure 6-b. An a x isym me t ri c e l em e n t w i th c ons ta n t t r i a n gu la r.............................................. 23

Figure 7 . Schemat ic rep res ent a t i on of f low through node n 28F igur e 8 . S chem a ti c r e p r e se n ta t i on of th e nodes which c on t r i bu te t o

t h e e le m en t s a a and dnm, where m = i, i 1 n -1 n, n 1,nm-1 and ................................................ 31

F i g u r e 9 S chem at ic r e p r e se n ta t i on of i r r e gu la r e l em e n tsF igure 10-a. Resu l ts obta i ned by f i n i t e e lement method as compared wi th

a n a l y t i c a l s o l u t i o n s f o r im pe rm ea bl e and c o n s t a n t p o t e n t i a l


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ou te r boundary system, wi th we l l of ze ro rad ius and

F igur e lob . Res ults obt ain ed by f i n i t e element method as compared witha n a l y t i c a l s o l u t i o n s f o r i n f i n i t e r a d i a l s y st em , wi t h w e l lof r a d i u s 0. 25 f t . and c o n s t a n t r a t e o f i n j e c t i o n 37

F igur e 11 R e s u l t s of t h e f i n i t e el em en t s o l u t i o n f o r a t wo -l ay er edsys tem adja cent t o a pond wi th con s tan t head andr a d i u s r = 80.0 f t . r /H = 10.0 , r r = 5 .0 , H2/H1 =w w e w1.0, and K /K = 2 O) as compared wi th t he an a l y t ic a l2 1

Figure 12 . Schemat ic rep res ent a t i on of a three - lay ered sys tem 41Figure 13 . Resu l ts obta ine d by f i n i t e e lement method as compared

w i t h a n a l y t i c a l s o l u t i o n s f o r a t h r ee - l ay e r ed l e a kysys tem wi th

B 2@32= 0 . 1 an d c o n s t a n t i n j e c t i o n

F igur e 1 4 Resu l ts obta ine d by f i n i t e e lement method as comparedw i t h a n a l y t i c a l s o l u t i o n s f o r a th r e e -l a y e re d l e a ky sy s te m

12 632 = 1 . 0 w i th c o n s t an t i n j e c t i o n r a t ei th B 46F igur e 15. Resu l ts obta ine d by f i n i t e e lement method as compared

w i t h a n a l y t i c a l s o l u t i o n s f o r a t h r ee - l ay e r ed l e a kysys tem wi th 12 = 32 = 1 . 0 and c o n s t a n t i n j e c t i o n

F igur e 16 . R e su l t s ob ta ine d by f i n i t e e l em e nt method f o r a th r e e -l a ye r e d syst e m wi th 6 = 32 = 0 .01 , c ons ta n t12in je c t io n r a t e , and de ca y pe r iod t im e d c on t inuous ly


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Fi g u re 1 7 .

F i g u re 1 8 .

F i g u re 1 9 .

F igure 20 .F igure 21 .F igure 22 .

F igure 23 .

F igure 24 .

Figure 25.

F igure 26 .

Figure 27.

Figure 28.

App ro xi ma te co n t o u r s o f eq u a l Ah fo r t h e t h ree -l ay e red l eak y s y s t em a t t h e en d o f b u i ld -u p p e r i o dApp ro xi ma te co n t o u r s of eq u a l Ah fo r t h e t h ree -l ay e red l eak y s y s tem a t t h e en d of d ecay p e r i o dP o s s i b l e d i s p o s a l r e s e r v o i r s o f l i q u i d w a s te s , andw a t e r q u a l i t y o f d e ep s a n d s t o n e s ...........................Thickness o f th e M t Simon sandstone .......................St ru c t u r e on t o p o f M t Simon Formation ....................Geolog ic c ros s -se c t io n th rough th e Hennep in Reg ion,s ho wi ng t h e s e c t i o n mo de le d i n t h i s s t u d yGeol o gi c s e c t i o n o f t h e J o nes and L aug h l in C o rp o r a t i o n sw a s te d i s p o s a l w e l l , H e nn epi n, I l l i n o i s ....................Det a i l s o f t h e m ode led zo ne i n t h e J o n es an d L au gh l i nC o rp o ra t i o n s we l l a t H e n n e p i n , I l l i n o i sS ch em at ic r e p r e s e n t a t i o n o f t h e i d e a l i z e d l a y e r e d s y s te mu s ed t o an a l y ze t h e J o n es and L au g hl i n C o rp o ra t i o n si n j e c t i o n w e l l a t H e n n e p i n , I l l i n o i s .......................R e s u l t s of f i n i t e el em en t method f o r t h e g r i d s p a c i n gu s ed i n t h e J o n es a nd L au g hl i n C o r p o r a ti o n s i n j e c t i o nwe l l w i t h h omogeneous and i s o t ro p i c p r o p e r t i e s , asc ompar ed w i t h a n a l y t i c a l s o l u t i o n ..........................Change of pr es su re vs . t i m c a l c u l a t e d a t t h e i n t e r f a c ebetween Iro nto n-G ale svi l le and Eau Claire f o r d i f f e r e n td i s t an ces f ro m t h e cen t e r o f m o de l .........................Change of pr es su re vs . t i m c a l c u l a t e d a t t h e i n t e r f a cebetween Eau Claire and M t Simon f o r d i f f e r e n t d i s t a n c e sfrom the ce nt er of model ...................................


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v i i i

F i g u re 2 9 . C h a n g e o f p r e s s u r e v s . t i m e c a l c u l a t e d a t t h e m id d l eof M t Simon f o r d i f f e r e n t d i s t a n c e s f ro m t h e c e n t e rof model 67

Fig ure 30. Approximate cont ours of eq ua l Ap (p si ) f o r th e modelo f Jone s and La ugh lin C or pora t ion ' s i n j e c t io n we l l a tth e end of th e bui ldeup per iod 71

Fig ure 31. Approximate cont ours of equ al Ap (p si ) f o r th e modelo f Jone s and La ugh lin C or pora t ion ' s i n j e c t io n we l l a tt h e end of th e decay perio d 72

Figur e 32. f r i ang ula r iz a t i on of domain 105Fig ure 33. Non-zero e lements of sp ar se ma tr ix 106Figur e 34. Magni tude of e r ro r vs . number of i t e ra t i o n s f o r 570,

1600, 8732, and 10660 107


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x

LIST OF TABLES

pageTable 1 Analysis of data o f F igure 3 ................................... 16Table 2 . Analy sis of data o f F igure ................................... 1 6Table 3 Ph y s i c a l an d g eo metr i ca l p r o p e r t i e s o f t h e mod el of

J o n es and L au gh l in Co r p o r a t io n s i n j e c t i o n w e l l 62


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Acknowledgments

The au thors would l i k e t o express t h e i r g r a t i tu d e t o D r RobertBergstrom, M r Peter Sarapuka , D r Edward Holley, and D r P a u l H ei go ld f o r t h e i rreview of th e or ig in al manuscript and th e i r comments. D r H ol le y p a r t i c i p a t e d i nt h i s p r o j e c t a s a c o n s u lt a n t . H i s s t i m u l a t i n g s u g g e s ti o n s h e l pe d t h e a u t h o r s i n

t h e s t u dy of t h e t o p i c s s t u d i e d i n t h e co u r se of t h i s i n v e s t i g a t i o n .The au tho rs would a l so l i k e t o acknowledge wi th thanks t he resea rch funds

prov ided by t he Water Resources Cen ter a t th e Univers i ty o f I l l i n o i s underAgreement 14-31-0001-3813.

We a r e g r a t e f u l t o M r s Edna Yeargin and M r s M ar ga re t G i d el f o r t h e i rp a i n s ta k i n g c a r e i n t y p i n g t h e m a nu sc r ip t.


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I INTRODUCTION

A c t i v e sThis i s t h e co mp le t i o n r ep o r t o n Ph ase I of a p r o j ec t w hich h ad

o r ig in a l l y b een p l an n ed f o r two y ea r s . The b a s i c o b j ec t i v e o f t h e tw o- yearp r o j e c t was t o i n v e s t i g a t e n um e ri c al and f i e l d m ethods t o p r e d i c t t h e f a t e o fl i q u i d w a s t e i n j e c t e d i n t o un de rg ro un d r oc k fo r m a ti o n s i n I l l i n o i s . From t h e s es t u d i e s i t was ho ped to d ev e lo p en g in ee r in g an d h y d r o g eo lo g ica l c r i t e r i a t h a tw ould h e l p t h e I l l i n o i s S t a t e G e ol o g ic a l Su rv ey a s s i s t t h e S t a t e E nv ir on m en ta lP r o t e c t i o n Agency i n t h e e v a l u a t i o n of p e rm i t a p p l i c a t i o n s by i n d u s t r i e st h ro u gh o ut t h e S t a t e t o i n j e c t l i q u i d w a s t e u nd er gr ou nd .

From t h e be g in n in g of t h e p r o j e c t two c r i t e r i a w e re c o ns i d e re d v i t a lt o t h i s s t u d y : p r a c t i c a l i t y a nd economy. One c a nn o t d e f i n i t e l y p r e d i c t t h e

prob lems which may be encountered i n a model based on a c tu a l f i e l d da ta bys tu d y in g o n ly a sma l l and h y p o th e t i c a l mo de l. T h e r e f o r e i t had been p lanned t ob a s e t h e b u l k of t h e c o n c l us i o ns i n t h i s s t u d y o n t e s t r un s w i t h f i e l d d a t a . I nt h i s r eg a rd w e f a c e d a b ud g et a ry l i m i t a t i o n ; i . e . t h e a n a l y s i s o f f i e l d pro bl em sby numer ica l methods r eq u i re s s u bs ta n t i a l amounts o f computer t i m e and computermemory. Thus co ns id er ab le ef f o r t was expended on t h e development of th e newtech n iq u es t h a t w ould r edu ce t h e co s t o f an a l y s i s .

Because th e p ro je c t was no t funded fo r th e second ye ar many of the seo b j e c t i v e s w e r e no t achieve d. What fol lo ws i s a r e p o r t o n t h e h i g h l i g h t s a ndi m p l i c a t i o ns o f t h e d i f f e r e n t s u b j e c t s s t u d i e d i n one y e a r and t h e i r e v a l u a t i o n.

B BackgroundThe p ro ces s o f i n j ec t i o n of l i q u i d w a s t e i n t o a s u b s u r f a c e fo r m a ti o n i s

u su a l ly accomp l ish ed b y th e d i sp l acemen t o f t h e n a t i v e f l u id s by some l i q u i d w as t e .


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I n t i m e, t h e l i q u i d w as t e moves i n acco rd ance w i t h t h e g r a d i en t d ev el op ed i nt h e r e c e i v i n g r e s e r v o i r . The p ro ces s of l i q u i d w ast e i n j e c t i o n h as b een u sedi n t h e o i l i n d u s t r y f o r many y e a r s 2 0, 000 b r i n e i n j e c t i o n w e l l s i n Texas a l o n eb y 19 66 ) w he re b r i n e s s ep a ra t ed f ro m pumped o i l a r e r e - i n j ec t ed i n t o t h e o i lp ro d u c i n g r e s e rv o i r s . An a dv an ta ge t h a t t h e o i l i n d u s t r i e s h a ve o v er o t h e ri n d u s t r i e s i n t h e d e si gn o f d i s p o s a l w e l l s s t h a t s u b s t a n t i a l p h y s ic a l a ndc he mi ca l d a t a a r e g e n e r a l l y a v a i l a b l e f o r o i l r e s e r v o i r s . By t h e u s e of t h e s ed a t a , t h e d e s ig n an d o p e ra t i o n o f a b r i n e i n j ec t i o n w e l l may b e s t u d i ed , an dmany i mp o r tan t f ac t o r s s u ch a s p r e s s u re h ead , r a t e o f i n j ec t i o n , d u ra t i o n ofcon t inuous in je c t io n , e t c . , may be dec ided i n advance . Other in du s t r i es whichs eek t o i n j e c t l i q u i d w ast e s commonly h ave l e s s d a t a av a i l ab l e .

The p roper s to r ag e o f was te l iq u i ds i n th e underground pore space sr eq u i r e s h y d ro l o g ic , g eo l o g i c , ma t h ema t ica l , and en g i n ee r i n g s t u d i e s . T hese

s t ud ie s mus t be under taken i n con junc t ion wi th one ano ther . The rev iew of t hel i t e r a t u r e r ev e a l s t h a t i n many i n s t an ces t h e d ev el op ment o f a h i g h l y s o p h i s -t i ca te d mathemat ica l model was based on t he smal l hy po the t i c a l model s. T h e re a r eg eo l o g i c r ep o r t s w hich e f f i c i en t l y d e s c r i b e o n ly t h e g eo l o g ic p a rame te r s need edf o r t h e s t o r a g e of l i q u i d w a s te i n t h e s u b s u rf a c e f o rm a ti on s. A tt em pt s t oi n t e g r a t e t h e g e o l o g i c a l , h y d r o l o g i ca l an d e n gi n ee r i ng s t u d i e s done i n w a s t edis po sa l management have been made i n th e re ce nt y ea rs , and wi th th e growingawareness of environm ental problems one would expect th a t th es e ef f o r t s w i l l b eexpanded.

C Previous WorksThe l i t e r a t u r e dea l ing wi th the movement of f l u i ds i n porous media s

v e ry ex t en s i v e . Many of t h e s e p u b l i ca t i o n s a r e r e l ev a n t t o s u b s u r f ace w as t ei n j e c t i o n d i r e c t l y o r i n d i r e c t l y . The U S Environmental Pr ot ec ti on Agency 1972)


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ha s c ompiled a n anno ta t e d b ib l iog r a phy o f 106 pub l i c a t ion s de a l ing d i r e c t ly w i ths u b s u r f a c e w a s te i n j e c t i o n . The p u rp o se o f t h i s s e c t i o n s n ot t o review a l lt h e s e p a pe rs b u t t o f oc us on s p e c i f i c s t u d i e s which c o n t r i b u t e d t o t h i si n v e s t i g a t i o n .

Van Everdingen and Freeze (1971) have compiled a l s t of 188 re fe r ence sd e a l i n g w i t h s u b s u r f ac e l i q u i d w a s t e s t o r a g e a s w e l l a s u nd er gr ou nd g a s s t o r a g eand hydrodynamic di sp er si on . They have ca te gor i zed th e problems p resented by t hes u b s u rf a c e i n j e c t i o n o f n a t u ra l l i q u i d s s uc h a s s a l i n e w a t e r, wa s te b r i n e , e t c .in to : (1) hydrodynamics , 2 ) s t r e s s m e ch an ic s, an d ( 3) f l u i d c o m p r e s s i b i l i t y .F or f o r e i g n l i q u i d w a s te s t h e p ro ble ms a r e n o t c o nf i ne d t o t h e s e t h r e e a r e a s ;r a t h e r t s r e a l i z ed t h a t ve ry l i t t l e s known about th e behavio r of the se l i qu id swhen the y come in c on ta c t w i th na t ive f l u i ds unde r r e se r vo i r c ond i t io ns ge ne r a te dby l o n g p e r i o d s of i n j e c t i o n .

The s u b j e c t o f e a r t h t r em o rs g e n e r a te d by i n j e c t i o n o f l i q u i d s i n t od i s p o s a l w e l l s h as be en br ou g ht t o t h e a t t e n t i o n of t h e i n v e s t i g a t o r s i n r e c e n tyea rs. The Rocky Mountain Arsena l di sp os al we ll s p erh ap s t h e f i r s t c a se f o rwhich enough geophysical data s a v a i l a b l e t o c o r r e l a t e i n j e c t i o n w it h e a r t htre mo rs . Van Po ol le n and Hoover (1970) have reviewe d some of t h e mechanisms pro-pose d t o e xp la in the s e t re m or s . Among th e se m echan ism s th e in t e r s t i t i a l - p r e s s u r emechanisms po st ul a t ed by Hubbert and Rubey (1959) , h yd ra ul ic f ra ct u r i n g ,r e d u ct i o n of c o e f f i c i e n t of f r i c t i o n , t h e rm al e f f e c t s , a n d c he mi ca l e f f e c t s a r ed i s c u s s e d b r i e f l y .

The movement of contam inants t hroug h porous media was i n v es ti g a te d asf a r b ack a s 1 90 5 by S l i c h t e r . I n t h i s s t u d y a s l u g of s a l t s o l u t i o n was i n j e c t e di n t o a w e l l a nd t was no t i c e d tha t t s a r r i v a l a t a n e ar by w e l l was g r a d ua l a ndnonuniform. Perhaps S l ic ht e r ' s observ a t ions formed th e ba s i s fo r th e deve lopment


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of t he t h eo re t i ca l m odel s t o desc r i be t he movement o f fo re ig n l i q u id s throughporous media. I n gener a l , th re e fa c t or s co nt r ibu te to the movement of anysubsta nce through porous media; disp er sio n, di ff us io n, and convect ion. Hoopesand Harleman (1965) have se t up a schemat ic repr ese nta t ion of d i s pe rs ion andd i f f u s io n , a s shown i n F ig. 1 -a . I n a s a t u ra t ed ve r t i c a l column wi th s t eady f l ow,a s l ug of dye i s i n j e c t e d a t a p o i n t s o ur c e. t i s o bs er ve d t h a t a s t h e s l u gt r a v e l s down t h e column, i t s s i z e i n c r e a s e s a nd m ix in g w i t h t h e n a t i v e f l u i dtak es p lace . This mix ing produces a concent ra t ion pr o f i l e a s shown i n F igs . 1-band 1-c a t t ime t as a func t i on o f d i s t a nce , z1 down th e column from t h e po in tsou rce , and d i s t an ce f rom the column ax i s . These concen t ra t i on p ro f i l e s a r eas soc i a t ed wi th d i sp e r s ion and d i f fu s ion . D i spe r sion i s caused by th e mechanicalm ixing o f t h e i n d i v i d u a l f o r e i g n f l u i d p a r t i c l e s w i t h t h e n a t i v e f l u i d s . Themixing i s a r e s u l t o f va r i ab l e ve lo c i t i e s t hrough t he po re spaces of t he medium.Di f fu s ion , on t he o the r hand , i s t h e r e su l t o f t he rm al o r mo lecu l ar mo tion o fi n d i v i d u a l f l u i d m o l ec u le s . The con t r i bu t i on of t he d i s pe r s ion p roces s t o aconcen t ra t i on p ro f i l e such as F ig s . 1-b and 1-c i n some pa r t s o f t he f l ow f i e l dw he re v e l o c i t i e s a r e h i g h , i s many t im es g rea t e r t han t ha t o f t he d i f fu s i onprocess . I n a porous medium being in je c t ed wi t h t racer , these two processescause t he boundary between t he t r ac e r and t h e na t i ve f l u i d s no t t o be a sha rp andwell-defined boundary. R a th er , t h e r a t i o of t h e c on c e nt r at i on o f t h e t r a c e r i n t h ez one of m ixed f l u i d s t o t h e c o n c e n t r at i o n o f t h e t r a c e r a t t h e f l u i d d e c r e a se s f ro m1 00 t o 0 a s one moves a cro ss t h i s zone of mixed f lu id s o r grad ual boundary a sshown i n Fi g. 2


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(p~.a~s

spumr


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- Zone of mixed f l u i d s witht r a c e r c o n ce n t ra t i on

I n j e c t e d f l u i d N at iv e f l u i dw i t hc onc e n t r a t ion o

Co

Figu re 2: Schematic re pr es en ta t i on of t he zone of mixedf lu ids o r g r a dua l bounda r y .

The ext ent of t he zone of mixed f lu id s , 6, depends on th e l o c a l ve lo c i ty ofth e f l u i ds a nd phys ic a l p r op e r t i e s o f th e po r ous m ed ia. I f no d i spe r s i on andd i f f us ion t a ke p l a c e i n th e medium, th e f o r e ign f lu id moves unde r t he in f lu e nc eof convec t ion only , and th e boundary between th e fo re ign f l u i d and th e na t i vef l u i d s w i l l be a sharp one , i . 6 = 0.

Many st u di es have been made on t h e sol ut io n of th e mass- transp orte qua t ion c on ta in ing d i sp e r s io n and d i f f u s ion . Hoopes and Harleman 1965) de ri ve dth e t r a n spo r t e qua t ion and , a s suming a s t e a dy flow, a na ly t i c a l so lu t io ns we refound f o r l o n g i t u d i n a l a s w e l l a s l a t e r a l d i s pe r s io n . I n c y l i n d r i c a l c o o r d i n a t e sth e i r c onve c t ive - d ispe r s ion e qua t ion f o r s t e a dy f low fr om a l i n e sou r c e per pe n-d i c u l a r t o t h e r - p la ne i s :

where: c = c o n ce n t ra t i on of t h e t r a c e r ; i . e . , f o r e i g n l i q u i dt = t imeA = Q / 2 n b


17/124

Q t o t a l flowb t h i ck n es s o f t h e g eo l o g ic fo rmat i on i n t o which i n j ec t i o n s

t a k i n g p l a c eI p o r o s i t y

an g u l a r co o r d i n a t e w i t h l i m i t s of t o 2mr r a d i a l d i st a nc ea D /q co n s t an t , a f u n c t i o n o f med ia s t r u c t u re1 1/Dl co e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n i n n on un iform f lo wq s eepag e v e l o c i t ya t D; /q co n s t a n t , a f u n c t i o n of med ia s t r u c t u re2D2 c o e f f i c i e n t of l a t e r a l d i s p e r s i o n i n no nu ni fo rm fl ow

T hen, a ss umin g a s y mmetr ica l co n cen t r a t i o n d i s t r i b u t i o n w i t h r e s p ec t t o an g u l a rco o rd i n a t e , i . e . , no l a t e r a l d i s p e r s i o n t ak es p l ace , eq u a t i o n 1 ) becomes:

T h i s would b e t h e eq u a t i on f o r t h e co n cen t r a t i o n d i s t r i b u t i o n r e s u l t i n g f ro m f lo wfrom a recharge we l l which f u l l y pen e t r a te s a homogeneous and i s o t ro p i c aqu i fe r .The so lu t i on of 2) may be somewhat s i mp l i f i ed by assuming th a t th e e f f e c t of th el o n g i t u d i n a l d i s p e r s i o n t er m, i . e . , t h e r ig ht -h an d s i d e , i n 2 ) s s m a l l i n cornpar iso n wi t h th e convect ive term i n 2) . Then 2) becomes:


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-cUsing 3) t o ca l cu l a t e on t he r igh t-hand s i de o f 2 ) , one obta ins :ar2

2ac A a c - rat- - - = aat r a r l E 2

The development of th es e equ atio ns i s more f u l l y disc us se d by Hoopes andHarleman 1965). Replacing th e approximate ly equal s ig n wi th t h e e q u al s i g n ,t he so lu t i o n o f 4 ) fo r t he case where a t r ac e r w i th concen t ra ion c i s i n j e c t e dc o n ti n uo u s ly a t t h e i n j e c t i o n s o u r c e , r 0 i s :

w he re : c o n c e n tr a t i o n o f t h e t r a c e r a t t h e o r i g i nc o n c e n tr a t i o n o f t h e t r a c e r a t a d i s t a n c e r f rom thei n j e c t i o n s o u r c e

r d i s t a n c e f rom t h e i n j e c t i o n s o u r c ee r f com plementa ry e r ro r func t i on o f t he quan t i t y i n b racke t

Equation 5) g i v e s a n ap pr ox im at e s o l u t i o n f o r t h e c o n c e n t r a ti o n o f t h e t r a c e r a sa f f ec t ed by convect i on and l on g i t u d in a l d i spe r s ion a t any d i s t ance r f rom theor ig i n . I n order t o inc lude the ef fe c t o f molecular d i f f us i on which becomes of somen q

m r 2 is i g n i f i c a n c e a t v e r y l a r g e d i s t a n c e s f rom t h e i n j e c t i o n s o u r c e , t h e t er m atmay be added t o th e r igh t -hand s i de of 4) . Then th e so lu t i on of th i s equat ionbecomes


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whe re : c o e f f i c i e n t o f m o le c u la r d i f f us io n , and r e s t o f th e t er m s ha vea l r e a dy be e n de f ine d .

I n t h e s ub s eq u en t s e c t i o n s , the s e two a ppr oxim a te so lu t io ns o f e qua t ion I ) ,i . e . , e q u a t i o n s 5) and 6) , w i l l b e us ed t o e v a l u a t e t h e e f f e c t s o f d i s p e r s i o na nd d i f f us ion on m ss t r a n s p o r t .

The so lu t i on of mass t ra ns po r t eq ua t ion by numer ica l methods hasencounte red some c r i t i c a l problems. Redde l l and Sunada 1970) used th e f i n i t ed i f f e r e n c e t e c h ni q ue t o s o l v e t h e f lo w e q u a ti o n f o r p r e s s u r e i n a n u n s t ea d y,n on un if or m fl o w f i e l d w i t h d e n s i t y and v i s c o s i t y v a r i a t i o n s b et we en t h e i n j e c t e dan d n a t i v e f l u i d s , a nd u se d t h i s p r e s s u r e d i s t r i b u t i o n i n c on j un c ti o n w i t h t h emethod of c ha ra c t e r is t i cs proposed by Garder, e t a l . 19 64) t o s o l v e t h e mass-t r a ns po r t e qua t ion . The m ethod o f c h a r a c te r i s t i c s i s employed t o avoid th eso - c a l l e d num e r ic a l d i s pe r s ion wh ic h i s u s u a l l y e n co u n te r ed when t h e f i n i t ed i f f e r e n c e t e c h n i q u e i s used t o so lve the m a ss - tr a nspor t e qua t ion .

Nal luswam i 1971) used t he f i n i t e el e m en t m ethod t o so lve t he mass-t r an sp or t equ a t io n i n a two-dimensional coo rdin a te sys tem. The second ord erl i n e a r p a r t i a l d i f f e r e n t i a l e q ua ti on f o r t h e ma ss -t ra ns po rt a s u se d i n t h a t s tu d yh a s mixed p a r t i a l d e r iv a t i ve s w h i ch r e s u l t from t r e a t i n g t h e d i s p e r s i o n c o e f f i c i e n t s

s se cond o r de r sym me tr ic t e ns o r s . Thus, a f un c t io na l ha d to be ob ta ine d t o ha nd le

t h e mixed p a r t i a l d e r i v a t i v e s . The r e s u l t s of t h e m i ni m i za t io n o f t h i s f u n c t i o n a lseem t o compare favorably wi t h t h e an a l y t ic so lu t i on s , and var i ous boundary con-d i t i on s may be in c o r por a t e d i n t o th e model.


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11 SOLUTION OF THE FLOW ND MASS-TRANSPORT EQUATIONS

The p re d i c t i on o f t he f a t e of l i qu id wa s t e s i n j e c t e d i n t o de ep sub-su rf ac e formations may be accomplished by th e fol lo wing st ep s. F i r s t , t h esu bs ur fa ce fo rma tio n must be id ea li ze d by a model wi th nonhomogeneous andan iso t ro pi c pr op er t ie s and wel l -de f ined boundar ies . Then, th e f low equa t ionshou ld be so lve d f o r t h i s mode l, wi th de ns i t y a nd v i sc os i t y va r i a t i o ns betwe ent h e i n j e c t e d w a st e and n a t i v e f l u i d s , t o o b t a i n p o t e n t i a l d i s t r i b u t i o n t hr ou gh ou tth e syst em. Th i s po t e n t i a l d i s t r i bu t i on may be used i n t h e ma ss- t ra nspor te q u at i on t o p r e d i c t t h e d i s t a n c e t h a t a p a r t i c l e o f l i q u i d w a st e would t r a v e l i nth e d i re c t i on o f de c re a s ing po t e n t i a l i n a given increme nt o f t ime . Based on th i sa pproa ch , f i r s t a syst e m o f e qua t ions c on s i s t i ng o f t he f l ow e qua t ions a nd t hemass-transport equ at io n was formulated i n which t h e sol ut io n of one equat io n wass u b s t i t u t e d i n o t h e r e q u a ti o ns t o obta in th e so l u t i on of th e second which was thens u b s t i t u t e d i n t h e f i r s t eq u at i on t o g e t a n ot h er s o l u t i o n , and e t c . T h is wasra th er s im i l a r t o Redde l l and Sunada s (1970) approach. However , i n order t o avoidth e problems asso c ia t ed wi th f i n i t e d i f f e r enc e technique and th e method ofc h a r a c t e r i s t i c s , t h e f i n i t e el emen t method was proposed a s t h e main t o o l f o r t heso lu ti on of th es e equ atio ns. Jav and el and Witherspoon (1968a-b and 1969) andNeuman and Witherspoon (1971) have demonstra ted th e ap pl ic ab i l i t y of th e f i n i t ee lement method t o t he so lu t i on of the flow equat ion i n two- and three- layeredsystems wit h hig h per mea bil i ty con tr as ts between la ye rs . Nal luswami (1971) hassolved th e mass- t ranspor t equa t ion by f i n i t e element method. Due t o t he l ac k ofthe p hys i c a l da t a , subsu r fa c e geo log i c format ions use d f o r l i qu id i n j e c t i on maybe s t be i de a l i z e d by a s t r a t i f i e d syst em wi th e ac h l a ye r b e ing homogeneous bu ta n i so t r op i c . Such an i d e a l i z a t i o n , t oge the r wi th t he re fe re nc e s c i t e d above


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s h o u ld pr o v id e t h e a ns we rs a p p r o p r i a t e t o t h e o b j e c t i v e s o f t h i s s t u d y .

However, b ef or e th e s t r t o f t h i s p r o c e s s , t h e fo l l ow i n g p o i n t s ha d t o b er e so lve d

A Impor tance of th e Dispers ion and Dif fus ion i n th e Mass Transpor t Equa tionI n o r d e r t o e v a l u a t e t h e i m po rt an ce of t h e te rm s r e l a t e d t o t h e

d i s p e r s i o n and d i f f u s i o n , e q u a t i o n (5) deve lop ed by Hoopes and Harleman (1965)was a p p l i e d t o a medium wi th t he f o l lowing c ha r a c te r i s t i c s :

a = d i s p e r s i o n c o e f f i c i e n t = .15 cm = 4 .9 2x 10 -~ f t .~ = c o n c e n t r at i o n o f l i q u i d i n j e c t e d = 1 .0 ( a r b i t r a r i ly may be c hose na s un it mass per u n i t volume)Q = r a t e o f i n j e c t i o n of l i q u i d w a s te = 200 GPMb = t h i c kne ss o f in j e c t i on z one = 1800 f t .

= po ro si ty of th e medium = 0.10

The se c o ns ta n t s a ppr ox ima te the c ond i t ions a t t he Jone s a nd La ughl in C or por a t ion si n j e c t i o n w e l l With thes e cons tan ts equa t ion (5) was so l ved f o r 128.0 days ofc o nt i nu o us i n j e c t i o n . The r e s u l t s f o r d ay s 1 2, 4 , 8 , 16 , 32 and 128, a rep l o t t e d i n F ig . 3 . Th i s f i gu r e shows th a t t h e wid th of th e z one o f mixed f lu id s , 6i n c r e a s e s w i t h t i m e

I n o r d e r t o e v a l u a te t h e e f f e c t o f t h e m ol ec ul ar d i f f u s i o n , e q u at i o n ( 6)was s o lve d f o r t h e same per iod of t i m e a nd th e same c ons t a n t s a s i n F ig . 3 bu tw i t h

D = m o le cu l ar d i f f u s i o n c o e f f i c i e n t = 1 . 0 ~ 1 0 -4 2m -5 m2 9 . 3 ~ 1 0 tsec day

The r e s u l t s a r e p l o t t e d i n F ig . 4 Visu a l examina tion of th es e curves lea ds ust o th e same conc lus ion as i n F ig . 3 . However, i n order t o (1) make any s ta te ment

a bou t t he c ompar ison of th e c u r ves i n F igu r e s 3 and 4 , ( 2) e va lu a te th e e f f e c t o f


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t ime on th e width of th e zone of mixed f lu i d s , 6 and (3) as se ss th eim por tance of t h e m o lecu l a r d i f fu s io n , a parameter such as t h e pe r ce n t i l eco ef f i c i en t of th e ku r to s i s g iven by Sp iege l (1961) may be used:

where : K = p e r c e n t i l e c o e f f i c i e n t o f k u r t o s i s1Q = (Q3 Ql)

Q1 and Q = t h e f i r s t and t h i r d q u a r t i l e s ( v al ue s of r i n F i gu r es 3and 4 which would give 0.25 and 0 .75 of t h e are as of th eres pec t ive f requency curves) where f requency curves ar ee q u i v a l e n t t o t h e d e r i v a t i v e s of t h e d i s t r i b u t i o n c ur ve si n F igures 3 and 4

Pgo and P = 90 and 10 pe rc en t i l e s (values of r i n F igures 3 and 4o which would giv e 0.90 and 0.10 of th e are as of th erespect ive f requency curves)

Using equat ion (7) , K was ca lcu la t ed f o r every curve i n Figures 3 and 4 . Thesec a l c u l a t i o n s a r e t a b u la t e d i n Ta bl es 1 and 2 t oge the r w i th t h e s i z e of t he zoneof mixed f lu ids , 6 and t h e r a t i o o f one ha l f o f 6 t o Q where Q2 i s t h e v a l ue2of r which would d iv id e the a re a under f requency curve i n to ha l f . In Tabl es 1 and2 , 6 i s measured from a point where C C = 99.0 1 t o a p o in t where C C = 1 .0These tab le s show tha t th e value s of K decrease a s t im e i nc re ases , i . t h ef l a t t e r t h e c u rv e t h e l ow er t h e K. The value of K fo r s t andard no rm al cu rveu = 0 U = 1 as g iven by Spie gel i s 0.263. Thus, i n Tables 1 and 2 curves of up

t o a b ou t 32 d ays a r e l e p t o k u r t i c ( a d i s t r i b u t i o n h av in g a r e l a t i v e l y h i gh p e ak ) ,and t h e c u r ve s a f t e r t h i s t im e a r e p l a t y k u r t i c ( a d i s t r i b u t i o n which i s f l a t - t opped ) .F u rtherm ore , one observes t ha t t he e f fe c t o f t h e m o lecu l a r d i f fu s ion i s shown i nthe sm a l l e r va lues o f K f o r the same per io d of time. However, t h i s ef fe c t i s


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C h La

I *7 . 8 . : VII

mI 4 oI00

I I8 . 2 . t = 1 . 0 day 1Crl 8 . 7 f. t = 2 . 0 dayslu 1

0PiKIxf :II /R I7 t = 4 . 0 daysKI r.JI

aC Pnrr(r.52 urg s 3 2 . 0

I75I7o U 3 3 . 1 t = 1 6 . 0 daysw r-

r 5n KIc a 3 4 . 1G 4r( r sI7 4 5 . 7L Z rr-rnn mr n 7 n t = 3 2 . 0 days

U] w. L 53 4 7 . 9

D

t = 6 4 . 0 days

9 1 . 8k


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= 8.0 d ys23.424.3 r

L

t = 16.0 d ys33.134.1c

10.9.11.7.1 2 3 y

r r rD q w15.7

fl 16.5P P.gs 17.3

46.7 t = 32.0 d ys

48.1

t = 2.0 d ys J

t = 4.0 d ys Jfl

Q u= 128.0 d ys G a


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i n s i g n i f i c a n t ( a r o u n d 3 of t he value of K a f t e r 128 days of cont inuousi n j e c t i o n ) . The e f f e c t o f m ol ec u la r d i f f u s i o n a t t h e e a r l y s t a g e s o f i n j e c t i o ni s e ve n l e s s s i g n i f i c a n t . T h is i s du e t o t h e h i gh v e l o c i t i e s which e x i s taround th e in je c t io n we l l , caus ing the d i sp ers ion t o be many t imes mores i g n i f i c a n t t h an t h e mo le c ul ar d i f f u s i o n .

The width of t he zone of mixed f l ui ds , 6 i n Tables and 2 i s

another measure of the impor tance of d i sp ers ion and d i f fus ion . For these curves6 in cr ea se s from a minimum of 0.95 f e e t ( a f t e r day o f i n j e c t i o n w i th o u tmolecular d if fu si on ) t o a maximum of 3.90 f e e t ( a f t e r 1 2 8 d ay s of i n j e c t i o n w i t hmolec ular dif fu si on ). Columns (10) and (11) of Tabl es and 2 show th e va lu es ofQ (median ) o r t he s econd qua r t i l e which r e l a t e s t o C /C 0.5 , and th e measureo f t h e w i d th o f t h e d i sp e r s e d zone r e l a t i v e t o t h e d i s t a n c e f rom t h e i n j e c t i o nsou rce , E I t was no t i ced t ha t E decr eased with t ime, and may be c onside redi n s i g n i f i c a n t f o r p r a c t i c a l s t ud i e s .

F igu re 5 shows th e r es u l t s o f the computa tions performed on a s e t o fdata which may be considered an extreme case. Af te r 512 days of cont inuousin j ec t i o n t h e wid th o f t h e zone of mixed f l u i d s , 6 i s expanded t o 15 f e e t w i thth e cen t e r of t h i s zone 295.5 f e e t f rom th e i n j e c t i on zone, i . e . , E 2.54 .


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T ab l e : Analysis of Data o f F i g u r e 3

1) 2) 3) 4) 5) 6) 7) 8) 9) 10 11)Q3 Q pg0 P K =Q P O Q2 E (- x100)q3-ql P g ~ - P 1 ~ Q /P Q2

days) f t ) f t ) f t ) f t ) 0 f t )

Tab le 2 : Analys i s o f Data o f F i g u r e 4


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n tI- 3R Dr 3R

R

3

I-aQnw

I- h

YI-

n g\ a


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B So lu ti on of Flow Equat ion f o r Multi-Layered Systemss was i nd ic at ed e a r l i e r , an ani so tr op ic and nonhomogeneous system may

be id ea l i ze d by a la yere d sys tem wi t h each la ye r be ing homogeneous but an iso-t r o p i c . T h i s i d e a l i z a t i o n may l e a d t o a s y s t em w i t h h i g h p e r m e a b i l i t y c o n t r a s tbetween th e la ye rs , and thus make the computa tion of t h e perm eab i l i t y der iv a t iv et er ms i n t h e e q u a t io n o f f lo w by t h e t r a d i t i o n a l f i n i t e d i f f e r e n c e t ec h ni q ues u b j e c t t o s i g n i f i c a n t e r r o r . The f i n i t e e le men t method i n t h i s r e ga r d seems t oh av e a d e f i n i t e a d va nt ag e o v e r t h e f i n i t e d i f f e r e n c e te c hn i qu e . Neuman andWitherspoon 1971) , i n exte ndi ng th e work of Javand el and Witherspoon 1969) t ol e ak y a q u i f e r s , c on cl ud ed t h a t t h e s o l u t i o n o f t h e e q u a t io n of f l ow f p r a t h r ee -l a ye r e d l e a ky sys t em wi th pe r m e a b i l i ty c on t r a s t s of 5000 by the f i n i t e el em e ntmethod was in e xc e l l e n t a gr ee me nt w i th the a n a l y t i c a l so lu t ion . Thus, a t t e n t io nwas focused on unders tanding t he f i n i t e e lement method.

1 F i n i t e Element Method: The development of t h e f i n i t e e lement methods t a r t e d i n t h e a i r c r a f t i n d u s t ry and s t r u c t u r a l e ng in ee ri ng . t has been adaptedt o t h e s o l u t i o n o f s e v e r a l t y p es of d i f f e r e n t i a l e q ua t io n s . Zienkiewicz andCheung 1970) were among th e f i r s t t o apply t h i s method t o th e f low equa t io n .The theory and formu la t ion of t he f i n i t e element method i s well documented

Zienk iewicz and Cheung, 19 70, Wilson and N ic k el l, 1966, Java nd el and Witherspoon,1969) . t s u f f i c e s h e r e t o men tio n t h a t i n t h e f i n i t e e le me nt method, t h ed i f f e r e n t i a l e qu at io n w i th t h e i n i t i a l c on d it i on i s f i r s t r e p l a ce d by a n e q u iv a l e ntfu nc ti on al which when minimized w i l l pr ov ide a so lu t io n wh ic h i s a v e r y c l o s ea pp ro x im at io n t o t h e s o l u t i o n o f t h e i n i t i a l bo un da ry v a l u e p ro bl em . I n t h efo l low ing paragraphs we s h a l l fo l low t he formul a t ion of Jav ande l and Witherspoon1969).

The i n i t i a l bounda ry va lue p r ob le m t o be so lve d i s th e equa t ion of f lowof a s l i g h t l y compress ib le f l u i d i n a nonhomogeneous and ani so t r op ic porous


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medium which s :

w i th i n i t i a l co n di ti on

and boundary condit ions :

B 1 , t = El B,) on bound ry B1

and

w here fo r ce p o t e n t i a l gh g [Z h]h h y d ra u l i c h eadg a c c e l e r a t i o n of g r a v i t y

e l e v a t i o n of t h e p o i n t whe re t h e f o r c e p o t e n t i a ls being measured

p r e s s ur e a t t h e p o i n t i n q u es t io nd e n s it y of t h e f l u i d a t t h e p o i n t i n q u e s ti o n


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x X . i j 1 ,2 ,3 ) t h e ax es of t h e s p ace co o rd i n a t e sJKi Perm eab i l i t y , a symmet ri c 3 x 3 mat r ix which usua l lyPgi s a f u nc t io n of s pa ce c o o rd i na t es , K ki j ii

k s p e c i f i c p e r m e a b i l i t y= v i s c o s i t y o f t h e f l u i d a t t h e p o in t i n q ue s ti o n

s s p e c i f i c s t o r a g e, i . e . , s t o r a ge p e r u n i t t h ic k n es s ofth e form at ion, +cpg

+ p o ro s i t y o f t h e fo rma t io nc e f f e c t i v e co mp res s i b i l i t y of t h e fo rma ti o nt t imen ou te r normal t o boundary B2I, El and E2 func t ions o r c ons tan t s depend ing on the

bo und ary an d i n i t i a l co n d i t i o n sB and B2 boundaries on which E and E m ust b e s a t i s f i e d2

By ob ta in ing the Lap lace t rans fo rm of the l e f t hand s i de of 8 ) and su bs t i tu t i ngf o r from 9 ) i n t o t h i s t ra n sf o rm , a f t e r i n v e r si o n one a r r i v e s a t :

a * - = s Sm m - m )E K i j a x Jwhere * s t a nds f o r convo lu tion as def ined by Gur t in 1964):

* G t - T ) T ) d TIEquation 11) i s equ iva len t t o equat ions 8) and 9) combined. This s te pe l i mi n a t e s o ne eq u a t i o n , namely t h e one f o r t h e i n i t i a l co n d i t io n e q ua t io n 9 ) .Therefo re , the p roblem i s reduced to th e so lu t io n of 11) s ub jec t to the boundarycond i t ions 10a) and lob) .


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The fun c t io na l r ep res en t ing t h i s prob lem i s :

where: re pr es en ts t he volume of th e medium, and i j 1 ,2 ,3 .The c o n d i ti o n s t h a t 13) h as t o s a t i s f y i n o r d e r t o b e t h e f u n c t i o n a l of 11 )

s u b j e ct t o 10a) and lo b) a r e given by Zienk iewic z and Cheung 1970).Equa tion 13), which i s t h e v a r i a t i o n a l p ro bl em o f 1 1) s u b j ec t t o 1 0a ) and

lob), when minimized w i l l prov ide a so lu t io n which i s t h e s o l u t i o n of 1 1) t o o.t represents a homogeneous but anisotropic medium. I n o r d e r t o a p pl y e q u at i on13) t o a heterogeneous and ani so t ro pi c sys tem, th e medium may f i r s t be divide d

i n t o sma ll ani so tr op ic and homogeneous eleme nts and equat ion 13) may bed i s c r e t i z ed , i . e . , r ep l aced by t h e summation of t h e fu n c t i o n a l s of e ach i n d i v i d u a le le me nt . I n t h i s fa s h i o n f u n c t i o n a l 1 3) f o r a s p e c i f i c e le me nt e i s :

Using equat ion 14) fo r every element , eq uat io n 13) may be repla ced by


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To minimize 15) , f o r each element must be approximated by a fun cti on. I fthe el e m en t s a r e su f f i c i e n t ly sm a l l , ove r e ac h e l em e nt f o r a two-d im ensionalsystem, may be approximated by a l i n e a r fu nc ti on such a s

whe re a ,b , c c o e f f i c i e n t s whic h a r e f unc t ions o f t im e , a nd r and z c oor d ina te s

o f a s p e c i f i c p o i n t w i t h i n el em en t e . The c o e f f i c i e n t s a , b , an d c f o r e ac helement must be chosen i n such a way t h a t s a t i s f y th e boundar y c ond i t ions o fe lement e , and be cont inuous wi th re sp ec t t o t he two-dimensional sys tem overelement e. The simplest e lement i s one w i t h a t r i a n g u l a r c r o s s s e c t i o n . F ora xi sy mm et ri c s y s te m s , r i n g s w i t h a t r i a n g u l a r c r o s s s e c t i o n c o n c e n t ri c w i t h t h ez-a xi s) may be con sid ere d. Fig ure s 6-a and 6-b r ep re se nt a two-dimensionalt r i a n g u l a r i z a t i o n a nd a r i n g wi t h t r i a n g u l a r c r o s s s ec t i o n . n element e andp o t e n t i a l w i t h i n i t may be t o t a l ly de f ine d by the th r e e node s 1 , 2 , and 3 a n d t h e i rr e s p e c t i v e p o t e n t i a l s l , 3 Thi s means th a t one may w r i t e equatio n 16)f o r t h e t h r e e n od es 1 , 2 , an d 3 and s o l ve f o r a , b , and c i n terms of the s r sand 2 s. Upon su bs t i tu t i on of the se va lue s o f a , b y a nd c in to e qua t ion 16) weo b t a i n :

where


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Figure 6-a: Division of two-dimensional region into triangular elements.After Zienkiewicz and Cheung, 1970)

Figure 6 b: An axisymmetric element with constant triangular cross-section.After Neuman and Witherspoon, 1971)


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(O*T

xEa

s

uu*aT

(zJZZ.AZ-z=ZT3


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Now, th e va lu es of Q i n 1 9) a r e t o b e c a l c u l at e d s o t h a t t he ynm in im iz e th e f un c t io na l 15 ). Th i s c a nno t be a c h ieve d by t a k ing the f i r s tv a r i a t i o n o f t 8) w i t h r e s pe c t t o Q~ , because Q i s a fu nc t i on ofboth s pace and t ime, and t Q ) c o n t a i n s a c o n v o l ut i o n , i . e . , i n t e g r a t i o n w i t hr e s p e c t t o t i m e , a nd i n t e g r a t i o n w it h r e s p e c t t o s p a c e c o o r d i n a t e s . F ol lo wi ng t h em ethod used i n c a l c u lu s of va r i a t i o n and pr ese nt ed by Neuman and Witherspoon 1971) ,Q t ) may b e replaced by t ) q t ) , where i s a n a r b i t r a r y c o n s t a n tn n

and q t ) i s a t ime dependen t c on t inuous ly d i f f e r e n t i a b le f unc t io n which va n i she sa t t = o and t = t Then 19) becomes:

- 2s ENn] Em T I TI d l mo l l d ~ d vS

which may be minimized now wi th re sp ec t t o by


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By assuming t h a t i s a n a r b i t r a r y f u n c t i o n , o ne may w r i t e :

E qua t i on 21) may be w r i t t e n f o r n = 1 2 , ... N where N i s t h e t o t a l number o fn od es i n t h e s y st em , t h u s c r e a t i n g N equ a t io ns wi t h t h e unknowns, m m = 1 , 2 , ...N). t m us t be no t i c e d t h a t when w r i t i n g 21) f o r node n , on l y e l e m e n t s w h ic hh av e no de n i n common w i l l c o n t r i b u t e t o t h i s e q u a t i o n . T h is w i l l p ro du ce as ym me t ri c s p a r s e m a t r i x whos e band w i d t h i s de te rmined by th e number ing sys tem of nodes .

To simp1 i f y t h e _ c om p u ta t io n s o f 2 1 ) l e t

Thus 20) f o r node n becomes


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The va lue s o f in t eg ra l 22) and 23) as fo rmula ted y Wilson and Ni ck el 1 1966)and proved y Neuman and Witherspoon 19 71) f o r e lemen ts w i t h t r i an g u l a r c ro s ss e c t i o n a r e :

where r + r2 + r3r = 1 3r r i a n ~ u l a r r os s s e c t io n

f o r p l a n ar t r i a n g l e

f o r c o n c en t r i c r i n gw i t h z - a x i s

1 2 ( K r r n m K r z n Yrn K z r Yn KZz Yn yro)d rdzJ A1A (Krr Bn Bm K Bn ym K Y B + KZz y n y m ) f o r p l a n a r t r i a n g lr z z r n m


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f o r c o n c e n tr i c r i n g .

The l a s t i n t e g r a l i n 23) i s a measure of th e ne t f low th rough t he e n t i r e boundaryB . T h is i n t e g r a l v an i s hes f o r i n t e rn a l bo u nd a r ie s be tw een e l emen t s, s i n c e fo r2the se boundaries t h e outf low from one element w i l l b e t h e i n f l ow t o a n o th e re l em e nt , a n d t h u s t h e n e t e f f e c t i s z e r o. T h e re f o re , t h i s i n t e g r a l n ee ds t o b eeva lu a ted on ly fo r t he ex te r na l boundar ies wi th the f low. Assuming th e flowthrough boundary Be of Figure i s Qe t ) , t h e f lo w a t t r i b u t e d t o n od e n f ro m

2 B2element e i s

and th e t o t a l f low th rough node n i s

Fi g u re : Schemat ic re pr es en tat ion of f low through node n .Le t :


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Then 2 4 ) becomes:

I n o r d e r t o e v a l u a t e t h e c o nv o lu ti on s i n 2 5 ) , l e t A t t h e t im e i n t e r v a l i nt h e t i me i n t e g r a l b e s ma l l enough s o t h a t @ and Q may be assumed t o var yl i n e a r l y w i th t im e :

and

I @ mm T) d r A t [mm t) @ t - A t ) ]rfl - A t

where - Qn t) Qn t - A t )Qn 2S u b s t i t u t i n g 2 6 ) and 2 7 ) i n t o 2 5 ) one ge t s :

- t D 0 t - a t )2 n n m mwhich by d efi nin g:

A t= D + - A An m 2 n m

A t 1@ t - a t ) nd Xm = [ m m ( t ) + @ t - a t ) ] ,r im = Qnl Dnm m2 7 ) becomes:


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The se t o f eq u a t i o n s r ep re s en t ed by 3 0) a r e N s im u lt an e ou s l i n e a r a l g e b r a i cequ atio ns wi th unknowns, X where rep res en t s th e t o t a l number o f noda lmp o i n t s i n t h e s y st em .

2 Boundary Condit io ns: Two typ es of boundary con di t io ns a r e ofp a r t i c u l a r i n t e r e s t i n g ro un d- wa te r h y dr ol og y: i ) c o n s t an t fl ow b ou nd ar y, i i )co n s t an t p o t e n t i a l b o u n d a ry . The t r ea t ment o f bo th o f the se boundar ies wi thequ at i on 30) has been demonstrated by Javan del and Witherspoon 1969). I n th eso lu t i on o f th e model s t o be p resen ted a cons tan t f low boundary was cons ideredmos t o f t en . However, a few runs were made wi t h a con sta nt p o te n t i a l boundary.

3 Impleme ntation of th e F i n i t e Element Method: Based on th e for-mula t ion of th e f i n i t e e l ement method and assumpt ions p rese n ted i n th e p rev ioussections, a computer program capable of handling any anisotropic and nonhomo-geneous conf ined sys tem, was wr i t t en and t e s t ed f o r var iou s hy po th e t i ca l modelswi th cons tan t p o t en t i a l boundary cond i t i on and cons tan t f low ra t e boundarycon di t i on. Then a f i e l d problem wi th consta nt f low boundary con di t i on was solvedwi th th i s p rogram. It must be ment ioned th a t mos t of t he r e su l t s p resen ted i n thefo l l o w in g s ec t i o n s a r e n o t u n iq u e t o t h i s s t u d y s ee Jav an d e l and W it he rs po on ,1968-a,b and 196 9, and Neuman and With erspo on, 19 71 ). What ha s been ac com plish edi n t h i s i nc o mp le te s t u dy i s t h e unders t and ing o f t he f i n i t e e lement method, i t sadvantages, and i t s d i sadvan tages .

a - Pro p e r t i e s o f Ma t r i ce s D and AA: The p roper t i es o f D and AA p l ay av e ry i m p or ta nt r o l e i n t h e f e a s i b i l i t y o f t h e s o l u t i o n of e q ua t io n 3 0) f o r l a r g es y st ems . I n t h e s e ma t r i ce s n , m = 1 ... N Thus, were the se mat r i ces dense , as t or ag e of 2 N x N) would have been requi red t o s t o r e elements of AA and D because

t i n 29) h a s t o b e v e ry s m a l l i n t h e e a r l y ti me s t e p s and i n c re a s e g r ad u a l l y ,bo th D and AA must b e acces s i b l e t h ro u g ho u t t h e t ime s t e p s ) . Bu t , s was


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men ti on ed e a r l i e r , t h e n on ze ro e l ement s i n AA and D a r e o n ly d ue t o t h enm nmnodes which sh ar e an element wi t h th e node whose a a and Dnm a r e b ei ng c a l c u l a t e d ,nmw here lo w er ca s e l e t t e r s r e f e r t o el emen ts of t h e co r r es p o nd i n g ma t r i ce s , i . e . , n

s ee F i g u re 8 ) . T h e re fo re, t h e s e ma t r i ce s a r e s p a r s e .

F igure 8 : Schemat ic rep res en t a t i on o f th e nodes whichco n t r i b u t e t o t h e e lement s aa and dnm nmwhere m i, i l, -1, n , n+ l, k-1, and k.

Another property of AA and D s t h a t t h ey a r e s ymmet ri c, i . e . , d dnm nm mnand aa aa This s because in . equat ions 2l )an d 22) th e terms and con sta ntsnm mnused f o r node n a r e th e same as f o r node m

T he se p r o p e r t i e s f a c i l i t a t e t h e com puter s t o r a g e and r e t r i e v a l of a anmand dnm gr ea t l y . The s t o r ag e requ i red f o r th ese ma t r i ces depends on the numberof nodes immediately surrounding th e node i n que st io n. I n Fi gu re 8 , when compu-t a t i o n s a r e b e i n g pe rformed f o r no de n , s t o r ag e must b e p ro v i ded a t most f o r s even


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co ef f i c i en t s i n th e mat r ix equa t io ns 24-a and b and, because of symmetry, as t o r a g e f o r f o u r c o e f f i c i e n t s w i l l b e s u f f i c i e n t . The r e t r i e v a l of t h e s e co-e f f i c i e n t s may be ach ieved by p rov id ing a po i n te r vec t o r , w w h i ch i n d i ca t e st h e node number, m whose con t r ibu t ion t o t he node n i s b ei ng s t o r e d i n l o c a t i o nj j = 1 2 , 3 , 4 , o f t h e fo u r l o ca t i o n s . T h i s method o f s t o r ag e an d r e t r i e v a li s somewhat c lo se t o t he George s (1971) Method 1 f o r th e compact s to ra ge schemef o r s p a r s e m a t r i c e s. The di f fe re nc e between th e method used her e and t h a t ofGeorge i s t h a t i n t h i s s tu dy t h e p o in t er v e c t o r, w co nta ins th e node whosec o n t ri b u ti o n t o n i s ca l c u l a t e d , w h i l e i n George s method t h i s p o i n t e r co n t a i n sthe d i f fe ren ce s be tween t he node numbers whose c on t r ibu t i ons t o n i s b e i n g ca l -cu lat ed and node n . Thus , fo l lowin g th e numbering sys tem of Figu re 8 , t he poin te rv e c t o r , w i n t h i s s t u d y f o r node n c o n t a in s :

w (n , 1 ) = n

w (n, 2) = n 1w (n , 3) = k - 1w (n , 4) = k

Any t i m e i t i s r e q u ir e d t o r e t r i e v e t h e c o n t ri b u ti o n o f , s a y , m t o n , i . e . , a anmand dnm, a se ar ch may be conducted i n vec to r w (n , j ) , j = 1 4 , t o f i n dwhich member of w co n t a i n s m. The elem ent denote d by j w i l l d i r e c t u s t o t h eloc a t i ons o f aa and d

m nmb - Mesh Cons t ruc t ion: The mesh cons t ruc t ion i n th i s s tudy s t a r t e d byd i v i d i n g t h e s ys te m i n t o e le m en ts w i t h s q u a r e o r t r i a n g u l a r c r o s s- s e c ti o n s. Theneach e lement wi th square c ross -sec t ions was t emporar i ly subd iv ided in to twoe l emen ts w i t h t r i an g u l a r c ro s s - s ec t i o n s , s uch a s shown i n F i g u re 8 t o c a l c u l a t eth e co ef f i c i en t s requ i red . Th i s type o f decompos i tion genera tes equa t ions which


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a r e i d en t i ca l w i th t hose de r i ved by t he we l l known f i n i t e d i f f e ren ce methodAllen, 1955). Sever a l o the r decomposi t ions were t r i e d , bu t t h i s scheme was

found t o be ea s i e r t o hand l e and requ i red l e s s s t o ra ge t han o the r s . F or exam ple,had th e decompos it ion of F igure 9 been used , th e co ef f i c i en ts of node n would haver e q u i r e d a s t o r a g e o f 2 x 5 10 t o s t o r e aa and d where m n , o , p , q , and inm nm

Figure 9 : S chem at ic r ep res en t a t i on o f i r r eg u l a r e l em ent sAf te r: Zien kiewi cz and Cheung, 1970)

A t th e same t ime the s tor ag e requi rement f o r node p s 2 x 6 The regularmesh system a s defin ed by Zienkiewicz and Cheung 1970) and a s prese nte d i nF igure 8 r e q u i r e s a s t o r a g e of 2 x 4 f o r every node i n th e system. Whereas th esums of the computer s tora ge f o r a l l nodes a r e th e same f o r th e two decompos i tions ,i n pr ac t i ce th e decomposi tion of F igure 9 requ i re s more s to rag e because of t h ei r r e g u l a r i t y i n t h e m eshes.


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c Change of Time Increment: I n t h e form ulat ion of t he method, i twas mentioned th a t because some of th e t ime dependent va r i ab le s such as and Qa r e approximated over an inc rement of t ime by a l i n e a r func t io n , th e t imeincrement , A t m ust be ve r y sm a l l a t t he e a r ly t im e s t e p s . However, be ca use o feconomy, A t m us t b e ch ang ed a s f a s t a s p o s s i b l e . T h e r e f o r e , i n t h i s s t u d y as p e c i f i c p o i n t , p , i n t h e s ys te m whose p o t e n t i a l i s e xp ec te d t o c ha ng e r a t h e r f a s tw i th t im e was c hosen a nd a s soon a s th e r e l a t i on sh ip 31) was sa t i s f i e d , A t wasm u l t i p l i e d by a c o n s t a n t g r e a t e r t h a n 1 I n t h i s s t ud y t h i s c o n s ta n t was a r b i t r a r i l ys e t e q ua l t o 1 .5 .

D ur in g t h e e a r l y t i m e s t e p s E was s e t a t 0.04 t o 0 .1 depend ing on the pr ob le m. Bu ta s t ime s t ep s cont inued t h i s con s tan t was g radu a l ly i nc re ased t o a maximum of 0 .2 .With t h i s scheme i t was p o s s i b l e t o i n c r e a s e A t f rom a f r a c t i on o f a sec ond t o10 t o 15 hour s i n a bout 40 t o 60 t im e s t e p s .

d S o lu t ion o f th e L ine a r Equa t ions : Th i s s t e p o f th e c om putat ionswas accomplished by t h e Gauss El im in ati on Method as pres en ted by Cran dal l 1956 ), andC r ou t 1941). Th i s t e c hn ique seem ed t o be e f f i c i e n t f o r a sm a l l syst e m o f e qua t ions ,s a y , N = 100. However, f o r a l a r g e sy st em of e q u a t i o n s , s a y , N = 1000, i t sde f i c i e nc ie s we re e v ide n t . sys t em o f 1000 e qua t ions took a p r o h ib i t i ve t im eof about 40 seconds pe r t ime s t e p on th e I M 360175. This ex ces siv e t imee nc oura ge d a more e f f i c i e n t so lu t io n o f l a r ge sys t em s o f l i n e a r e qua t ions . Ther e s u l t s o f t h i s s t u d y a r e p r es e nt e d i n s e c t i o n I of t h i s r e p o r t .

4. Te st of t h e Computer Program:a Homogeneous and Is o t r o p i c Systems: Two t e s t s were made on t h e homo-

geneous and i s o t ro p ic models . The r e su l t s a r e p lo t t e d i n F igu r e s 10 -a a nd 10-b


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i n t h e form o f :d im ens ion l es s po t en t i a l

dimensionless t ime

where: K perm eab i l i t yth ickn ess of th e format ion

Q t o t a l r a t e of i n j e ct i o ng cons t an t of g ra v i t y

A change of p o t e n t i a l w i th r e s p e ct t o t h e s t a r t of i n j e c t i o n , i . e . , twhere @ and @ a r e t h e p o t e n t i a l s a t t im es t and o r e s p e c t i v e l yt

t t imeS s p e c i f i c s t o r ag esr d i s t a n c e of t h e p o i n t from t h e c e n t e r o f t h e w e l l

r ou te r r ad ius of t he syst eme

The so l i d l i n e s i n th ese f ig ur es which a re copied f rom Javandel and Wi therspoon1968-b) a r e t h e ana ly t i c a l so l u t i o ns t o t h e r e spec t i v e p robl em s g iven by

Muskat 1946) and Mu lle r and Witherspoon 1965 ).Figure 10-a i s t h e s o l u t i o n of a f i n i t e r a d i a l s y s te m w i t h a n imper-

meable ou te r boundary and cons tan t po te n t ia l ou te r boundary , cons ta n t f low ra t e ,and we l l of ze ro r ad ius . The f i n i t e e lement so lu t i on m atches t h a t o f t he

ra n a l y t i c a l s o l u t i o n . F or a p o i n t w i t h r e where r i s t h e o u t e r r a d i u s, t h e8.9 e

a n a l y t i c a l s o l u t i o n was n o t r e a d i l y a v a i l a b l e , and t im e was no t spen t t o ca l cu l a t et h i s c u r v e .


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System with impermeable outerboundary

102

101

System with constanr100 potential outer

10-1

10-210-1 101 103 10

Dimensionless time K tSs r2Figure 10 at Results obtained by finite element method symbols) as compared with analyticalsolutions solid lines after Muskat, 1946) for impermeable and constant potential

outer boundary system, with well of zero radius and constant rate o f injection.


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I- r ur 4m JID m1 r n Fg v g.r . P , R

C mnrn RID DO M rkar 3m r rtR IDf l

P P nl gorPg 5s n r nU I 3 4R I D 3@ f l u3 rnR u r0 rn:G

I; o nm m ov l 3P.L.mI D O I Dn e eRF r Co 3 r .3 m~r 3?. PR 3ID P

T K HDimensionless potential = Er @


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Figure 10-b i s t h e s o l u t i o n of an i n f i n i t e r a d i a l s ys te m, r = wi thea w e l l of f i n i t e r a d i u s an d c o n s t an t fl ow . The d i f f i c u l t y w i th r e s p e c t t o r = aewas overcome by assuming a f i n i t e r which could be handled wit h t h e computerememory av ai la bl e , and temporal sol ut io n was cont inued u n t i l th e t ime wheredev ia t i ons f rom th e Theis Curve , such a s i n F igu re 10 -a , w e r e n o t i ce d i n t h e s o l u ti o n .Whereas t h e a n a l y t i c a l s o l u t i o n s f o r p o i n t s c l o s e t o t h e w e l l match t h o s e of t h ef i n i t e e lem en t me thod , t he re a re no t i cea b l e amounts o f v a r i a t i on between t here su l t s o f t he a na ly t i ca l method and t he f i n i t e el emen t method f o r po in t s w i thr 2.0 r o r l a r g e r , w h e r e r r ad ius of t h e we l l . Th i s may be a t t r i b u t ed t oW Wth e mesh s i z e and th e length of th e t ime increment . The length of th e t imeincrement , A t may a f f ec t t he accuracy because of t h e l i ne a r i t y a ssumed i n t heeva lua t ion of th e convolu t ions of equat ions 26) and 2 7 ) . Apparen t l y t h i s i s n o tthe cause of t he va r i a t i o ns m ent ioned above, o the rwise t h e r e su l t s f o r nodes withr = 2.0 r would show such va r i a t i ons t oo . The mesh s i z es , f o r t he a re as i n t he

Wsyst em , such as a rea s ad j acen t t o t h e we l l , w i th h igh and non - l inea r va r i a t i o nsof @ can a f fe c t t h e accuracy cons ide rab ly . Dur ing t h e ea r l y t im e s t ep s , t h eg rad i en t o f @ i n t h e s e a r e a s i s very high. Because @ i s approximated over ane lem en t by a l i ne a r func t i on , i . e . , equa t i on 16 ) , ca re must be exe rc i s ed t oass ur e tha t such an approximat ion i s v a l i d . One way t h a t t h i s may be a chiev edi s t o u se sm al l mesh s i ze s i n t hese a rea s . An exam ina ti on o f t he mesh s i z es o fthe model u sed t o ob t a in da t a i n Figu re 10-b r evea l ed t h a t t he mesh s i z e fo r nodesw i t h r r r w 2.0, i n Fi gu re 10-b where r i s a dim ens ion l ess qu an t i t y , i nc rea sesD Dby a f ac to r of 5 over th a t o f th e node wi th r r r w 1 . 0 w i t h i n a d i s t a n c eDless t han a foo t .

b - Layered Systems: I n t h i s sec t i on two models of co nfine d leakys ys te ms a r e t e s t e d .


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i ) Two-Layered System With Con stant P o t e n t i a l Boundary a t t h e Well:The f i r s t m odel t e s t e d was a two-laye re d c onf ine d , l e a ky a x i s ym e t r i c sys t e mwi th a pond of rad iu s r 80.0 f e e t l oc a t e d i n the c e n te r of th e m odel ando t h e r g e o m e tr i c al a nd p h y s i c a l p r o p e r t i e s g i ve n i n F i g u r e 11 Katz 1960, p. 86)has d er iv ed an expre ssi on f o r two-layered systems and Jav and el and Witherspoon

1968a ) have c a lc u la t e d and e va lua te d t h i s e xpr e s s ion f o r f ou r po in t s w i thr r r w 2 .0 , 3 .0 , 4 .0 , and 5 .0 and p r e se n te d them toge the r w i th the f i n i t ee le me n t so lu t io ns . I n F igu r e 11 t h e s e f o u r c u r v es a r e p r e se n t e d t o g e t h e r w i t ht h e f i n i t e el em en t s o l u t i o n s o b t a in e d i n t h i s s t u d y f o r t h e same p o i n t s an da d d i t i o n a l p o i n t s w i t h r 1 . 25 an d 1 .0 1 . The n u m er i ca l s o l u t i o n s f o r t h e f i r s tf o u r p o i n t s match v e ry c l o s e l y t h o s e of t h e a n a l y t i c a l s o l u t i o n s , and t h e p o i n twi th r 1.25 seems t o fo l low th e t ren d of th ese curves . However, t he so lu t i onf o r r 1 .0 1 e nc oun te r s o sc i l l a t i on s a s shown by the dashe d l i ne s . Th i so s c i l l a t i o n may b e a s s o c i at e d w i t h t h e mesh s i z e s on t h e v e r t i c a l s u r f a c e of t h es y st e m a d j a c e n t t o t h e pond. An a tt e m p t t o r e s o l v e t h i s p o i n t by u s i n g f i n e rmesh s i z e s o n t h e v e r t i c a l s u r f a c e d i d n o t c hange t h e r e s u l t s s u b s t a n t i a l l y .O t he r t e s t s w i t h a w e l l of r a d i u s r 0.25 f e e t rep lac i ng th e pond produced noo s c i l l a t i o n s and r e s u l t s matched t h e a n a l y t i c a l s o l u t i o n s .

i i ) Three-Layered System wit h Constant I n j ec ti o n Rate Boundary Condit ion:The ne x t model was a th r e e - l a ye r e d , c onf ine d sys t em wi th c o ns ta n t in j e c t io n r a t e a tt h e w e l l . Consider a sys tem such a s shown i n F igu re 12 , w i th t he a t t het o p of t h e i n j e c t i o n l a y e r l a y e r 1 ) and th e we l l bo r e.


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0 -aDimensionless potential = - 0I- I-

I-

I- I I I 1 I I I I I I

m rP , R3 7 00 . IDN l dm Oi i- Br- .D

9, c 1V) m r t1 m g

mP1 rIDa . 3 ID

m Im o m zr t 1 V I-=r 0R r tr * \9, Xe ,I&

I- n,p, ? ,Y0 op.P YI- IIv rnll-I- 1 0= *rtg

u.

- - -----[-zitI- 10%n

L


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Figure 12 : Schemat ic rep res en t a t io n of a th ree- l ayered sys temAf te r Neuman and With ersp oon, 1969a)

f

Assume in ject ion s t ak ing p lac e th rough a we l l o f rad ius r i n t o a homogeneousWand i s o t ro p i c l ay e r wh ich s i n f i n i t e i n h o ri z on t al ex t en t , i . e . , l a y e r 1 Thisl a y e r s bounded by an impermeable la ye r a t t he bot tom and by l ay e r 2 K2


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i n a l e a k y s y s t em i n w hich t h e e f f e c t of t h e s t o r a g e of l a y e r 2 was c o ns i de r ed .Neuman and Witherspoon 1969a,b) have dropped both of th e se ass um ptio ns, i . e . ,t h e i r mod el a l l o w s f o r t h e c ha ng e of p o t e n t i a l i n l a y e r 3 and d oe s n o t i g n o r es t o ra g e i n l a y e r 2 . They e v a l u a t e d t h e i r s o l u t i o n i n t e rm s o f t h e d i m en s io n le s sparameter s

and

where i 1 o r 3 i n d i c e s g iv en t o l a y e r s 1 and 3Ki o r Ss p e rm e a bi l it y o r s p e c i f i c s t o r a g e of l a y e r i r e s p e c t i v e l y

r a d i a l d i s t a n c e of t h e p o i n t whose r B and ai a r e b e in gic a l c u l a t e dHi th ic kne ss o f l a ye r i s ee F ig u r e 1 2 )

Equation 32) g i v e s a me as ur e of t h e r a t i o of t h e p e r n i e a b i l i t i e s of l a y e r 2 , t h ea q u i t a r d , an d t h e o t h e r two l a y e r s . Equat ion 33) give s a measure of t he same para-m e t er s and t h e e f f e c t s of s t o r a g e . Neuman and With erspoo n 196 9a, b) havee v a l u at e d t h e i r s o l u t i o n s f o r s e v e r a l c om bi na ti on s o f Bi and r B i 2 and comparedt h e i r a n a l y t i c a l r e s u l t s w it h t h o s e of t h e f i n i t e el em en t m ethod .

To t e s t t h e c a p a b i l i t i e s o f t h e c om pu te r prog ra m w r i t t e n f o r t h l s s t u d y ,a sys tem such as F igure 12 was s e t up wi th th e fo l lowin g parameter s :


53/124

r 30,000.0 fee teH 40.0 feet

T o t a l number of mesh po in ts 469T o ta l number of elemen ts 436

These ar e e s s e n t i a l l y th e same parameters t h a t Neuman and Witherspoon 1971) usedo e v a l u a t e t h e i r f i n i t e el em en t s o l u t i o n s . W ith t h e s e p ar a me t er s , when P i

r B i 2 0.01 , then r 10.0 f e e t ; when f r B i 2 0 .1 , t he n r 100 f e e t ; a ndiwhen r B i 2 1 .0 , t he n r 1000 fe e t . The in je c t io n per iod was d iv idedi 2in to a ) bu i ld - up pe r iod , a nd b ) de ca y- pe riod . I n the bu i ld - up pe r iod in j e c t io n

3c on ti nu ed a t a d e s i r e d r a t e o f 0 . 1 f t / s e c f o r 31.9 h o ur s. A t t h e en d of t h i sp e ri o d t h e f l ow r a t e c a l c u l a t e d a t r 0 . fro m t h e p o t e n t i a l d i s t r i b u t i o n was

3.09999 f t / s e c . A t t h i s t im e i n j e c t i o n was s to p p ed and t h e c a l c u l a t i o n f o r44.8 hours of the decay-per iod was s t a r te d . F igures 13 , 14 , and 15 pres ent

4 KI l K1 td im e nsion les s hea d in c r e a s e h A h vs . d imensionless time ts r21

obta ined by th e f i n i t e e lement method symbols) as compared wi th th e ana ly t i ca lso lu t i on s so l i d l i ne s) of Neuman and Witherspoon 1969a) fo r po in ts wi t h r 10f e e t f i g . 1 3 ) , r 100 f e e t f ig . 14 ) a nd r 1000 f e e t f i g . 1 5 ) a t e l ev a -t i o n s z -5.0 la ye r 1 i n j e c t io n z one ) , 4 .0 , 10 .0 , 16.0 l a y e r 2 , a qu i t a r d ) a nd25.0 f e e t l a y e r 3 ) f o r e ac h r Th ese f i g u r e s a l s o r e p r e s e n t t h e d a t a o b ta i n e d


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f rom th e decay pe r iod f o r t he s ame po in t s a s t he i n j ec t i o n pe r i od . Because of th ela ck of t ime and funds no atte mpt was made t o compare th e decay d at a wi th t h ea n a l y t i c a l s o l u t i o n s . I n F igur e 13 some d isc repa ncie s ar e observed between th ea n a l y t i c a l s o l ut i o rl s and t h e f i n i t e e le me nt s o l u t i o n s . t i s obse rved t ha t t hef i n i t e e le me nt s o l u t i o n s f o r s h o r t t im es d e v i a t e n o t i c e a b ly from t h e a n a l y t i c a lso l u t i on . Th i s may be due t o t h r ee f a c t o r s . One fa c t o r i s t h a t t h e a n a l y t i c a lso l u t i o ns fo r sh o r t t im es can be cons idered as - approxim at ions fo r t he a c tu a lchange of head i n th e system. Neuman and Witherspoon (1969a) de fi ne d th e l i m i tof t h e s h o r t i n j e c t i o n t i me s a s :

The o the r f ac to rs could be t he mesh s i z e used i n the model and th e leng th of th etime increment durin g th e e a rl y time st e ps . Neuman and Witherspoon (1969a) al s oobserved t h a t t h e i r num er i cal so lu t i on was on o r below t he a na ly t i c a l so lu t i on .T h is was s p e c i f i c a l l y n o t i c e d f o r r B i 2 0 . 01 i n l a y e r 2 (aqui tard) wherei 2the num eri ca l so l u t i o n was found t o be abou t 5 below th e a na ly t i c a l so lu t i on .The same typ e of e r r o r s may be observed i n Figur es 14 and 15.

The da t a f o r t he decay-pe riod c l ea r l y show the e f fe c t o f t h e e l a s t i c i t yo f t he l aye r s . S ince l ay e r 2 ( aqu i t a rd ) i s 8 t im es more e l a s t i c t han l aye r s o r 3 ,i t i s n o t i c e d t h a t Ah f o r n ode s w i t h i n t h i s l a y e r d oe s n o t s t a r t t o d e c r ea s eimm ediat ely a f t e r t he i n j e c t i o n s t ops . R a the r , a s expec t ed , t he da t a show a sm alli nc re ase i n Ah be fo re i t s t a r t s decreas ing . This i s more no t i c eab l e f o r po in t sc lo se r t o t h e i n j e c t i on we l l (F igu res 13 and 14 ) t han f o r t he nodes f a r away f romth e we l l (F igure 15) . The time coord inate of t he decay curves s t a r t s from t 1.0


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iur D C Tm Pr

OD3 1m aTi m un c d'Dl .s z m tlo r1g v.

O D 3 Rs r mn v.

3 n C Tb T R: : g v.R m Rr 5r o a IIti rtc u r n uCT I--


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T K1 H1Dimensionless increase in head = hD = h

D w7 r . r na s m(D Co m -1 n

ru mn m oD n bs m nn r ( Dr( z k@ ( D I Dn c a8 1 3 '7 YD Mr. 3 r.2 ?.5 :n nr 3 mo m +7 - ( Dm 3d (D

A z7 3mi* W X X O

W N X Ob * XO

* W N X O* W * X O* W N X O

W l r x e* W N I D* W X r n* W l r mW N P I. l r a unW mx Cw x 4, ,W N X, , w a x aw a x ID,,WOXX r(I t = ra m .Q 3 =a= IImpcn

q m


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1 K H2Dimensionlerr incre se n head = hD n


58/124

o r 0 . 0 1 r a t h e r t h a n c o n t i n ua t i o n of t h e t f o r t h e b u i l d- u p cu r v es . The r eas o ns t h a t when da ta of th e decay-per iod was p l o t te d wi th t a s t h e c o n t i n u a t i o n o f

t h e t o f t h e b u i ld - u p p e r io d , n o t r en d was v i s i b l e b ecaus e of t h e c lo s en es s o fd a t a p o i n t s . F ig ure 16 shows such a p lo t fo r r = 1 0. 0 f e e t . F i g ur e 1 7 r e p r e s e n t sth e app r ox imate co n to u r s of t h e eq u a l A h f o r t h i s mod el a t t h e end of t h e b u i ld - u pp e r io d . A f t e r 44 .8 h o u r s o f decay- pe r io d t h e s e co n to u r l i n e s l o ok ed a s p r e s en t edi n F i g u r e 1 8 .

c Appl ica t ion o f the F i n i t e E lement Method t o a F i e l d Problem: Up t ot h i s p o i nt w e h a ve d e s c r i b e d b r i e f l y t h e t h e o r y o f t h e f i n i t e e le m en t m eth od , a n de v a l u a t e d t s accuracy and some of th e p rob lems invo lve d i n t s implementa t ion .I n t h i s s e c t i o n t h e f i n i t e e le me nt method s a p pl i ed t o t h e M t Simon sandstonec u r r e n t l y u s ed f o r i n j e c t i o n of l i q u i d w a s t e s by J on e s and L aug h l in Co rp o r a t i o n a tHennepin I l l i n o i s .

i ) Geology : Deep permeable geo l og ic fo rmat ions appear t o have con-s i d e r a b l e p o t e n t i a l f o r w as te d i s po s a l. The d eep f o r ma t io n s co n t a in h ig h ly s a l i n eb r in es , w hich i n some a r e as ap pr o ach es 10 -t imes t h e s a l i n i t y o f s e a w a te r . Thei s o l a t i o n fro m s u r f a c e e nv ir on m en ts wh ic h ha ve r e s u l t e d i n t h e f o rm a t i on o f t h eb r i n e s s of p a r t i c u l a r i n t e r e s t i n deep w e l l d i s po s a l, a s t h i s s e x a c t l y t h ee nv ir on me nt d e s i r e d f o r i n j e c t i o n . B e rg st ro m (19 68) o u t l i n e d t h e p o s s i b l e d i s p o s a lr e s e r v o i r s ( F i g u r e 1 9 ) , a nd s u g g e st e d r e g i o n s w he re t h e r e w e re i n d i v i d u a l d i s p o s a lzones and a ss oc ia te d con f in in g fo rmat ions which may serve t o i s o l a t e t h e f or ma ti onf l u i d s a nd i n j e c t e d w a s t es .

E x i s t i n g hy d ro g eo l og i c and s t r u c t u r a l - g e o l o g i c d a t a a r e a v a i l a b l e s ot h a t t h e m at h e m a ti c a l m ode l d ev el op ed i n t h i s r es ea rc h w i l l r e p r e se n t a r e a l i s t i cp i c t u r e o f p o t e n t i a l d e ep w e l l d i s p o sa l . Of prim ary con ce rn s t h e co n t in u edis o l a t io n o f th e d i sp os a l zone from man s env i ronment. The p re ssu re bu i ld -up and


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K1 H1Dimensionless increase in head = D hQ- - - - - -

0 0 0 0I 0 ?. IU0- -V- 0---- +

00 S 0-

0 -----0 ;- 0- 0--- 0- 3

0w 3b 3 (3(3- r- wa-- 7_ b a 40 - 3 7w r

3- 3 a

II I I I I I I I I I,,


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61/124


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u l t i m a t e f a t e of w a s te s i n t h e d i s p o s a l f o r ma t io n s i s dependent on th e geolo gicand hydrogeologic paramete rs , and th e accuracy of th e re s u l t s i s dependent upont h e s e .

The e x i s t i n g d a t a on t h e p h y s i c a l p r o p e r t i e s o f t h e M t . Simon sandstonea r e f o r t h e up pe r z on es , g e n e r a l l y 400 f e e t o r l e s s , o f t h e fo r ma t io n . E x i s t i n gs o u r ce s o f d a t a a r e dee p w a t e r w e l l s i n t h e n o r t h e r n p a r t o f t h e s t a t e , u nd er -ground g a s s t o r a g e t e s t s , d eep s t r u c t u r a l t e s t s i n o i l pr od uc in g r e g i o ns , an d

e x i s t i n g d i s p o s a l w e l l s and t e s t s . Maps of t h e g e o l og i c s t r u c t u r e o f t h e I l l i n o i sB a sin ha ve bee n pub l i she d Bond, 1972) a nd a r e a de qua te f o r u se i n de f in in g thegeometry of the M t . Simon sandstone.

The M t . Simon sandstone i s t he d i spo sa l z one o f most in t e r e s t be c a useof i t s i s o l a t io n f rom su r f a c e e nv ir onm en t s a nd i t s gr e a t th i c kne ss a nd be c a usei t s n a t i v e waters a r e b r i n e s i n most of I l l i n o i s . I n t h i s re p o r t , M t . Simonr e f e r s t o t h e h y dr o lo g ic u n i t c o n s i s t i n g o f t h e M t . Simon Formation and thec o nt i gu o u s s a n d st o n e s i n t h e l ow er Eau C l a i r e F o rm at io n; t h e l a t t e r s a n d s t on emay be as th i ck as 400 fee t . ) The M t . Simon va r i e s i n th i c k ne ss f rom l e s s tha n1000 f e e t t o ove r 2500 f e e t , a nd i s g e n e r a l l y g r e a t e r t h a n 1500 f e e t t h i c k i n m ostof th e regio n where i t would be cons i dered a s a l i q u i d w as t e i n j e c t i o n zone

Figure 2Q) . However, th e sands ton e has been shown t o be absen t i n th e to p of th eM t . Simon; the M t . Simon var ies f rom l ss t h an 1000 f e e t b elow s e a l e v e l t o o v er12 QQO f e e t F igu r e 21 )

The M t . Simon i s dominantly a f ine - t o coarse-gra ined sands to ne wi thsome be ds of s ha le a nd s i l t s t o n e . The f o r m at ion i s moderately compact andf r i a b l e i n t h e n o r t h and n o rt h w e s t, w her e i t a l s o y i e l d s p o t a b l e wa t er . Deep i nth e ba s in a nd on i t s e a s t e r n m a rg in , t he sa nds tone i s w e l l cemented and quar tz i t ic .I n t h e s e r e g i o n s , t h e f o r m at i on h as a r e l a t i v e l y l ow p o r o s i t y a nd p er m e a bi l i t y.


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* ja-Uu .apaa


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. sdapua


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The geology i n th e Hennepin regi on was de scr ibe d by Cady 1919).McComas 1968) re la te d the geology of the ar ea t o p lanning and envi ronmenta lproblems . The fo l lowing br ie f d i sc uss i on s t aken f rom these pub l i ca t i ons .

The Jones Laughl in dis po sa l we l l s s i t u a t e d i n a s t r u c t u r a l l y lowt rough about 15 m i l es west of t he c r es t of t he La S a l l e an t i c l i n a l b e l t F igu re 22 ) .A b r i e f d e s c r i p t i o n and l i t h i c l o g of t h e w e l l s shown i n Fig ure 23. A t Hennepint h e M t Simon aquifer s about 1800 f e e t th ic k . The we l l s open through thee n t i r e aqu i fe r and through mos t of th e over ly ing Eau Cla i r e . Cores were takeno ve r se ve n d i f f e r e n t i n t e r v a l s i n t h e a q u i f e r , t o t a l i ng abou t 275 f e e t of co re.Measurements of h or iz on ta l perm ea bi li t y and po ro sit y were made by a commerciallab ora tor y on the core . Measured per mea bi l i t i es var ied f rom about 5 t o 500 m i l l idar cys ; po ro si ty measurements va ri ed between 8 and 22 per cen t. Measured po ro si tyco r re l a t e d we l l w ith geophys i ca l l ogs son i c and po ros i t y l ogs ) and t h e da t a

ANÁLISIS DE VERTIDOS EN POZOS DE INYECCIÓN

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