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~ ) Pergam onInt. J. Rock Mech. Min. Sci. Vol. 34, No. 8, pp. 1201-1211, 1997 199 8 Elsevier Science Ltd. All r ights reservedP I I : S 0 1 4 8 - 9 0 6 2 ( 9 7 ) 0 0 3 0 8 - 2 P r in te d i n G r ea t B rit ai n0148-9062/97 $17.00 + 0.00

A N u m e r i c a l M o d e l f o rT h e r m o - H y d r o - M e c h a n i c a l C o u p l i n g i nF r a c t u r e d R o c kK . M . B O W E R tG . Z Y V O L O S K I ~

Coupled thermo-hydro-mechan ica l codes w i th the ab i li t y to m odel f rac tu redmater ia l s are u sed for pred ic t ing groundwater f low behavior in f rac tu redaqu ifers containing therm al sources. The po ten tial applications o f such a codeinclude the analysis o f groun dw ater behavior wi thin a geotherm al reservoir .The capab i l it y o f model ing hydro- thermo sys tems wi th a dua l poros i t y ,

f rac ture f low mode l has been prev ious ly deve loped in the f in i t e e l emen t code ,F EH M . F E H M has been modi f ied to inc lude s tress coup ling wi th the dua lporos i t y f ea ture . F E H M has been fur the r deve loped to couple the dependenceof frac ture hydraul ic conduct ivi ty on ef fective s tress wi thin two-dimension al ,sa tura ted aqu i fer s con ta in ing f rac ture sys t ems . The cub ic law or f low be tweenpara l l e l p la tes w as used to mode l f rac ture permeab i l it y . The Ba r ton-B and i srelat ion was used to determine the frac ture aperture w i thin the cubic law . Thecode used a Ne w ton -R ap hso n i terat ion to impl ici tly solve fo r s ix unk now ns ateach node.

Resu l t s f ro m a model o f hea t f low f rom a reservo ir to the moving f lu id ina s ingle fracture compared wel l wi th analyt ic resul ts Resul ts of a modelshow ing the increase in frac ture f lo w due to a single frac ture opening underf lu id pressure com pared wel l wi th ana ly ti c resul ts . A ho t dry rock , geo therma lreservoir was modeled wi th real is t ic t ime s teps indicat ing that the modi f iedF E H M code does successfu l ly mo del coup led f low prob lems wi th noconvergence problems. 1 9 9 8 Elsevier Science Ltd

I N T RODUC T I O NFlu id f low in f rac tu re d reservo i rs is o f in t e res t inapp l i ca t ions where the f rac tu re permeab i l i ty i s in f lu -enced by chang ing t em pera tu re , f lu id p ressu re , o r s t res sc o n d i t i o n s o f t h e m e d i u m. A p p l i c a t i o n s i n c l u d e a n a l y si so f g e o t h e r ma l h o t d r y r o c k r e s er v o ir s , mo d e l i n g f l u idf low in a h igh l eve l nuc lear was te repos i to ry , pe t ro leump r o d u c t i o n , a n d g e o t h e r ma l s u b s i d e n c e . A l l t h e s ea p p l i c a t i o n s ma y i n v o l v e l a r g e t i me s o v e r ma n y y e a r s .D u a l p o r o s i t y mo d e l s h a v e b e e n d e v e l o p e d t o d e s c r i b ef lu id f low th rough f rac tu re sys tems wi th f lu id mass ande n e r g y s t o r a g e w i t h i n t h e r o c k ma t r i x . Su c h mo d e l sa l l o w f r a c t u r e s t o b e mo d e l e d w i t h u n i f o r m s i z e de lement s l ead ing to be t t e r so lu t ion convergence . In af rac tu red , sa tu ra te d aqu i fe r the bu lk o f the f lu id f low isgenera l ly th rough the f rac tu re sys tem. The f rac tu rec o n d u c t i v i t y i s s t r o n g l y d e p e n d e n t o n t h e f r a c t u r eaper tu re which i s in f luenced in tu rn by the e f fec t ive s t ressacross the f rac tu re . I t i s impor tan t to model th i st L o s A l a m o s N a t i o n a l L a b o r a t o r y , E a r t h a n d E n v i r o n m e n t a l S c ie n ce ,L o s A l a m o s , N e w M e x i c o , U . S .A .

beha v io r whe n de te rmin ing the coup led e f fec t s in af r a c t u r e d g r o u n d w a t e r s y s t e m.

I n o r d e r t o mo d e l l a rg e t ime s , t h e d u a l p o r o s i t y c o d es h o u l d b e c o u p l e d a n d h i g h l y i mp l i c i t ( t h e N e w t o n -Ra p h s o n s o l u t io n s h o u l d i n c lu d e a l l t e rms) . Such a codecould eff icient ly model the energy, f luid mass , and s t ressc o u p l i n g w i t h i n t h e f r a c t u r e d p o r o u s me d i a . T h i sresearch modi f i es FEHM, an ex i s t ing f in i t e e l ementwhich so lves fo r hea t and f lu id ma ss t rans fer , t o f i l l t h i sn e e d . FE H M h a d t h e c a p a b i l it y o f mo d e l i n g f r a c tu r ef l ow u si n g t h e d u a l p o r o s i t y me t h o d . T h e c o d e h a d b e e nf u r t h e r d e v e l o p e d t o p r o v i d e c a p a b i li ti e s o f c o u p l e ds t ress wi th hea t and f lu id mass t rans fer though no t wi tht h e d u a l p o r o s i t y f e a t u re [ 1 ] . FE H M d i d n o t a l l o w t h ef rac tu re hyd rau l i c cond uct iv i ty to change as the e f fec tives t res s across the f rac tu re changed . Th i s paper descr ibesmo d i f i c a ti o n s o f FE H M t h a t e x t e n d t h e s t re s s s o l u t io no f FE H M t o t h e d u a l p o r o s i t y c a p a b i l i ty a n d t h a t u s ethe e f fec t ive s t res s across f rac tu res to de te rmine theh y d r a u l i c c o n d u c t i v i t y o f t h e f r a c t u r e s y s t e m. T h emo d i f i c a ti o n s c o u p l e h y d r o lo g i c , t h e rma l , a n d me c h a n -ica l behav io r in f rac tu re sys tems . To the au thors '

1201

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1 2 0 2M a t r i x

B O W E R a n d Z Y V O L O S K I:

F r a c t u re S y s t e mF i g . 1 . D u a l p o r o s i t y m o d e l s h o w i n g i s o l a t io n o f m a t r i x c e ll s .

knowledge, th is i s the f i rs t ful ly coupled, highly impl ici tcoup led f lu id s t res s model tha t uses a dua l poros i tyapproa ch . The m odi f i ed code is ver i fi ed by com par ing i t sresu l t s wi th known so lu t ions . I t i s a l so used to model ah o t d r y r o c k g e o t h e r ma l r e s e r v o i r .

B A C K G R O U N DT h e r e a r e s e v e r a l a p p r o a c h e s t o t h e s o l u t i o n o f

c o u p l e d h y d r o l o g i c , t h e r ma l a n d me c h a n i c a l p r o b l e ms .The f i rs t i s t he fu l ly exp l i c it method . In th i s app roac h a l lthe conserva t ion equat ions a re t rea t ed exp l i c i t l y in t ime.The non- l inear t e rms , be ing eva lua ted a t t he p rev ioust ime s t ep , a re eas i ly accoun ted fo r . The t ime s t eps a rel im i t e d b y t he s p e e d o f s o u n d i n th e r o c k a n d a r eex t remely smal l . The nex t approach so lves eachconserva t ion equat ion impl i c i t l y , bu t updates thecoup l ing t e rms ( fo r example , t he e f fec t o f d i sp lacement son rock permeab i l i t y ) in a sequen t i a l manner . Th i sm etho d i s no t l imi t ed to the smal l t ime s t eps o f the fu l lyexp l i c i t method , bu t the convergence ra t e i s dependen to n t h e m a g n i t u d e o f t h e c h a n g e o f th e c o u p l i n g t e rms .O u r e x p e ri e n ce f o r t h e t y p e o f p r o b l e ms g i ve n b e l o wind ica tes tha t very smal l t ime s t eps a re requ i red fo rconvergence . Fo r app l i ca t ions wh ere f rac tu re f low i simpor tan t the fu l ly impl i c i t bu t sequen t i a l ly coup leda p p r o a c h w a s a b a n d o n e d i n f a v o r o f a f u l l y i mp l i c i t ,fu l ly coup led approach . In th i s approach the fu l l s e t o fe q u a t i o n s is s ol v e d w i t h a N e w t o n - R a p h s o n i t er a t io n

T H M N U M E R I C A L M O D E LT a b l e 1 . E x a m p l e 1 - - P a r a m e t e r s u s e d in m o d e l

P a r a m e t e r V a l u eL e n g t h 1 0 0 0 .0 mW i d t h 5 0 . 0 mY o u n g ' s m o d u l u s 1 x 103 M P aP o i s s o n ' s r a t io 0 . 0R o c k s p e c i fi c h e a t 1 0 4 6 J k g - ~ C - ~R o c k d e n s i t y 2 6 5 0 k g m - 3R o c k t h e r m a l c o n d u c t i v i t y 2 . 59 W m -~ C -~F r a c t u r e p o r o s i t y 1 .0I n i t i a l a p e r t u r e 0 . 0 01 mA p e r t u r e p o w e r ( f o r c u b i c l a w ) 3A p e r t u r e f a c t o r 8 . 3 3 x 1 0 ~M a x i m u m a p e r tu r e c l o s u re 0 .0 01 mF r a c t u r e s t i f f n e s s 1 x 1 03 0 M P a m - ~M a t r i x p e r m e a b i l i t y 1 x 1 0 -2 0 m 2M a t r i x p o r o s i t y 0 .1V o l u m e t r i c f l o w 6 . 1 5 2 x 1 0 - 3 k g s e c - ~

me t h o d . T h i s h a s n o f o r ma l t i me s t e p l i m i t a t i o n b u trequ i res the so lu t ion o f l a rge sys t ems o f equa t ions w i thup to s ix unknowns per g r id b lock . Impl i c i t so lu t ionme t h o d s a r e p r e d o m i n a n t l y u s e d t o s o l v e c o u p l e dprob lem s in the o il i ndus t ry ; exam ples o f which a refoun d in Bo berg [2 ] and Co at s [3 ] . The a u thors be l i evew i t h t h e a d v a n c e s i n p r e c o n d i t i o n e d c o n j u g a t e g r a d i e n tme thods fo r the so lu t ion o f l i near equa t ion s , t h i sa p p r o a c h h a s p r o mi s e .

N o o r i s h a d a n d T s a n g [ 4] h a v e d e v e lo p e d R O C M A SII , a f in i t e e l ement p rogram wi th hydro-mechan ica lc o u p l i n g f o r f r a c t u r e d p o r o u s me d i a . T h e f r a c tu r e s mu s tbe m odele d exp l i c i tl y as l inear e l em ent s . Sw enson [5 ] hasd e v e lo p e d a s im i la r c od e , G E O C R A C K , w h i ch m o d e l si mp l ic i t c o u p li n g o f me c h a n i c a l a n d h y d r o l o g i c p h e n o m-e n o n w i t h e x p li c it c o u p l in g o f th e r ma l p h e n o m e n o n .G E O C R A C K u s e d a d i s c r e t e f r a c t u r e n e t w o r k a n dignores f lu id s to rage in the rock m at r ix [6 ] . TheD E CO V A L E X p r o j e c t i s a n i n t e r n a t i o n a l c o - o p e r a t i v eresearch p ro jec t fo r s tud ies o f coup led p rocesses in hardrocks . Wi th in the p ro jec t , numer ica l code resu l t s a rec o mp a r e d i n mo d e l i n g h y d r o - t h e r mo - me c h a n i c a lc o u p l e d p r o c e s se s [ 7 ] . N o n e o f t h e c o d e s u s e d f o r t h i sp r o j e c t u s e a d u a l p o r o s i t y c a p a b i l i t y t o mo d e l t h e s e

T o = 1 0 0 CP o = 2 5 M P a

Tinjeetion = 9 0 CQ = 6 .125 x 10"3 k g s e e "1

I 0 0 0 . 0 k . . . _ r a c tu r e b o t t o m i n e f o d e s )

P = 2 5 . 0 M P a "

vertical xagger ation 2F i g . 2. C o n c e p t u a l m o d e l o f E x a m p l e 1 .

- f5 0 . 0l

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B O W E R a n d Z Y V O L O S K I : T H M N U M E R I C A L M O D E L 1 20 3Table 2 . Exam ple 1 - -Bo unda ry cond i t ions used in m ode l (1 -- -l ef t,r = r igh t , t = top , b = bo t tom boundary)T y p e B o u n d a r y C o n d i t i o nStress 1, L b No norm al displacem entFlu id f low r , 1o t , b No norm al f lowFrac ture node, 1 Q = 6.125 10 3 kg sec ~Frac tu re node , r P = 25 MP aHea t t r ans fe r r , L t , b No norm al t r ans fe r

Frac tu re node , 1 T = 90CFrac tu re node , r T = as numer ica l ly ca lcu la ted

coupled thermal, hydrologic and force balance processesin fractured systems.The dual porosity method has been developed tomodel fracture flow on the scale where discretefractures cannot be treated effectively. Douglas andArbogast [8] and Showalter [9] give descriptions of thismethod. The method models the fracture and matrixsystems as separate but connected systems as shown inFig. 1. The fracture system has fluid flow along thelength of the continuum. The matrix only has fluidflow to and from its corresponding node in thefracture system. Matrix nodes cannot communicatewith each other. This method models fracture fluidflow and matrix fluid storage. It attempts to capturethe important features of each system without usingcomplex geometry and large numbers of nodes tomodel discreet fractures.FEHM has the energy transfer in the porous mediaformulated in a similar fashion. Energy flows fromelement to element in the fracture system. The matrixsystem only allows energy to flow from matrix node toits corresponding fracture node. This does notcorrespond well with energy transfer in an actual rockmass if the thermal conduction in the rock massdominates the thermal system. Energy transfer in a dualporosity system will be a topic of a future paper.If a single fracture is considered as formed by twoparallel plates, the flow through it can be described by

I00.00

98.00 - - An a ly t ic

90.00 I I0 100000 200000Time, days

Fig . 3 . Example 1 - - -Com par i son o f ana ly t i c and FE HM resu l ts .

O~ 96.00

~ 94.00

~ 92.00

100OO

( . - , ) 9 8 . 0 0/ Time step = 99 days and 936 days

96.00 ~b ~ / / T im e s tep = 5429 days~ ~ ~ y s~ 94.00 ~ l0 9 2 . 0 0

I0 . 0 0 lI O O O O 2 0 0 0 0 0Time, days

Fig . 4. Example l - -Co mp ar i s on o f ou t le t t empera tu res us ing va ry ingtime step size.

the cubic law [10] : w b 3 d PO - 12/~ dx (1)

where w is the width of the reservoir, b is the fractureaperture, P is the pressure, and /~ is the dynamicviscosity. Equating the flow through the fracture withflow through a reservoir, the intrinsic permeability of afractured reservoir is:b 2 nkr = 1--2L (2)

where n is the number of fractures and L is the heightof the reservoir. Equation (2) assumes b is the hydraulicaperture described by Tsang [11]. Barton e t a l . [12] usethe mechanical aperture as the actual aperture anddescribe the relation between the hydraulic andmechanical aperture. Zhao and Brown [13] also giverelations between the two.The aperture, b, is a function of the effective normalstress and the shear stress across the fracture. Bandise t a l . [14] have described the change in aperture due tothe normal stress. Assuming the normal stress is theeffective normal stress, then the Barton-Bandis relationcan be expressed as:

Ab - Aa~Aa~' (3)g n i - - -I 'mwhere Atr~ is the change in the effective stress from some

.2 87 nodes, NR = 4438~.00 // / / / 1 4 7 n od es , N R = 4 41 1o X N N / / . 77 nodes N R = 4363o8.oo ~ 4 2 9 196.00

O

0 92.oo9 0 . 0 0 [ 1 I

0 IOOOOO 2OOOOOTime, daysFig . 5 . Example 1 - -Com par i son o f ou t le t t empera tu res us ing va ry ingmesh size.

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1 20 4 B O W E R a n d Z Y V O L O S K I :1.011

0 . 8 0

[,.0 , 4 0

0 . 2 0?50 . ~

/ A T = 9 0 C , N R = 5 8 5 7/ A T = 7 0 C , N R = 5 8 0 4

I / . / ~ T = 5 0 C , N R = 5 6 8 7< ~ Ar = t0 c, NR = 5301/ ~ C , N R = 4 7 1 7

I I1 0 1 1 0 0 0 2 0 0 0 0 0

Tim e , da ysF i g . 6 . E x a m p l e 1 - - C o m p a r i s o n o f o u t l e t te m p e r a t u r e f o r d i f fe r en c e sof in je c t ion t e m pe ra ture a n d in i t i a l roc k t e m pe ra ture . T he ve r t i c a l a x i si s d im e ns ionle s s t e m pe ra ture

T" - T o u t l e t - T i n j ~t i onT i n i t i a l - T i n j e c t i o n "

in i t ia l s ta te , gn i i s the in i t ia l f r ac tu re s t i f fness , and Vm ist h e m a x i m u m f r a c t u r e c l o s u r e . N o t e t h a t t h e d e n o m i -n a to r r e p r e se n t s t h e f r a c tu r e s t if f ne s s w h ic h i n c r ea se s i nr e sp o n se t o a n i n c r e a s in g c o mp r e s s iv e s t r e s s ( t e n s i l estress is posit ive) .

TH E C O U P L E D M O D E LT h e f in i te e le m e n t m e t h o d i s b a s e d u p o n t h e i d e a t h a t

a c o n t i n u u m m a y b e m o d e l e d a s a c o n f i g u r a t i o n o fd i s c r e t e e l e me n t s . F o r e a c h e l e me n t , e q u a t i o n s a r ew r i t t e n t h a t d e sc r ib e t h e i n t e r a c t i o n o f t h e e l e me n t w i thi t s n e ig h b o r s . T h e se e q u a t io n s d e sc r ib e t h e h y d r o lo g i c ,t h e r m a l , a n d m e c h a n i c a l b e h a v i o r o f th e e l e m e n ts . T h ef in i t e e l e me n t me th o d l e a d s t o a s e t o f n o n - l i n e a re q u a t i o n s w h i c h a r e t h e n s o l v e d w i t h i n F E H M . F o r ad e t a i l e d p r e se n t a t i o n o f t h e f i n i t e e l e me n t me th o d , r e f e rt o Z i e n k i e w ic z [1 5 ]. F o r a d e t a i l e d d e sc r ip t i o n o f F E H M ,r e fe r t o Z y v o l o s k i e t a l . [16].

I t i s a s su m e d in t h i s w o r k t h a t t h e f r a c tu r e d r e se r v o i r sa r e tw o - d ime n s io n a l , s a tu r a t e d , a n d p o s se s s u p t o tw op e r p e n d ic u l a r f r a c tu r e s e t s . T h e mo d e l p l a n e s a r eh o r i z o n t a l so t h a t g r a v i t y h a s n o e f f e c t . T h e e q u a t io n s

T H M N U M E R I C A L M O D E LT a b l e 3 . E x a m p l e 2 - - P a r a m e t e r s u s e d i n m o d e l

P a r a m e t e r V a l u eL e n g t h 2 5 mW i d t h 1 mRoc k de ns i ty 2716 kg m -3Ro ck specif ic hea t 803 J kg -~ KRoc k the rm a l c ondu c t iv i ty 2 .57 W m -~ KY o u n g ' s m o d u l u s 1 x 1 03 M P aPois so n ' s r a t io 0 .0I n i t i a l a p e r t u r e 1 x 1 0 - S mFra c tu re poros i ty 1 .0A p e r t u r e p o w e r ( f o r c u b i c l a w ) 3Ap e r ture f a c tor ( for c ubic l a w) 8333M i n i m u m a p e r t u r e 1 x 1 0 - 3 mFra ctu re s t if fness 1 x 105 M Pa m -~M a t r ix pe rm e a b i l i ty 1 x 10 20 m 2M a t r i x p o r o s i t y 0 .1

s h o w n a r e f o r a n i s o t r o p l c m e d i u m u n d e r p l a n e s t r a i n ,t h o u g h t h e s e r e s tr i ct i o n s d o n o t e x i st i n F E H M . I n t h ed u a l p o r o s i t y sy s t e m, t h e ma t r i x ( n o t a t e d w i th su b sc r ip t ,m ) r e p r e s e n t s t h e p o r o u s r o c k a n d c o n t a i n s b o t h p o r e sa n d so l i d r o c k ma te r i a l . T h e f r a c tu r e s ( n o t a t e d w i thsu b sc r ip t , f ) a r e mo d e l e d a s a s e p a r a t e e f f e c t i v ec o n t i n u u m c o n n e c t e d t o t h e m a t r i x s y s t e m . T h ec o n se r v a t i o n o f fl u id ma ss w i th in t h e f r a c tu r e sy s t e m i s:

8 A r ~.. . + A-fr. . . . + qr, . . . + q . . . . . = 0 ( 4 )where Af .~a~s s the f lu id mass pe r un i t vo lume g iven by:

A f , . . . = f p f ( 5 )E q u a t io n ( 5 ) imp l i e s t h a t w i th in t h e f r a c tu r e sy s t e m th ef lu id v o lu m e c h a n g e w i l l e f fe c t th e ma ss s t o r a g e , f f ,m~ ist h e f l u id ma ss f l u x g iv e n b y :

J ~ , . . . = P r # r (6 )~br i s the po ros i ty in th e f ra c ture sys tem (genera l ly equ a lto 1 .0), p r i s the f lu id den s i ty in the f rac tur e sy s tem , qr, . . .i s the f lu id ma ss s ource in the f rac ture sy s tem , and qm . .. .i s t h e f l u id ma ss f l o w f r o m th e ma t r i x sy s t e m. T h ev e lo c i t y o f t h e f l u id i n a n e q u iv a l e n t f r a c tu r e sy s t e m c a nb e e x p r e s se d b y D a r c y ' s L a w :

~f = _ kr VP f (7),uf

T o = 20 CPo = 21 M P a

P = 2 1 . 9 M P a

f r a c tu r e ( c e n t e r l i n e n f n n d e ,q ~

v e r t i c a l ex a g g era t i o n = 2F i g . 7 . C o n c e p t u a l m o d e l o f E x a m p l e 2 .

l . m

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B O W E R a n d Z Y V O L O S K I: T H M N U M E R I C A L M O D E L 1 2 05T a b l e 4 . E x a m p l e 2 - - B o u n d a r y c o n d i t i o n s u s e d i n m o d e l ( I = le f t,r -- - r i g h t , t = t o p , b = b o t t o m b o u n d a r y )T y p e B o u n d a r y C o n d i t i o nS t r e s s 1, t , b N o n o r m a l d i s p l a c e m e n tF l u i d f l o w r , 1 , t , b N o n o r m a l f l o wF r a c t u r e n o d e , 1 P = 2 1 . 9 M P aF r a c t u r e n o d e , r P = 2 5 M P aH e a t t r a n s f e r r , 1, t , b N o n o r m a l t r a n s f e r

wh ere P i s t he : fl uid pre ssu re an d k i s t he in t r ins i cp e r m e a b i l i t y . C o n s e r v a t i o n o f f lu i d - r o c k e n e r g y isg i v e n b y :

d A f ,nergYdt + V - ~ , e n e r g y "+"qf, nergy+ q .... .~y = 0 ( 8 )w h e r e A r . , ~ y i s t h e e n e r g y p e r u n i t v o l u m e g i v e n b y :

A f , energy = ( 1 - ~ ) f ) p r U r " ] - ~ b f p f ua n d f r ,on~rgy s the energy f lux given by:

.)~energy p f h ~ f - K V T f

(9)

(10)u is t h e sp e c if i c i n t e r n a l e n e r g y a n d c a n b e e x p r e s se d a su= cT f w h e r e c i s th e sp e c i fi c h e a t a n d T f i s th et e m p e r a t u r e i n t h e f r a c t u r e , tc i s t h e t h e r m a l c o n d u c -t iv i ty , qr,~,o~gy s t he ene rgy sou rce fo r t h e f ra c tu re sys t em ,a n d q . . .. ~ i s th e , e n e rg y so u r c e f r o m t h e m a t r i x sy s t e mt o t h e f r a c t u r e s y s t em . T h e s u b s c r i p t , r , r e f er s t o t h e r o c km a t e r i a l . T h e d e n s i t y , p , v i s c o s i t y , # , a n d sp e c if i ce n t h a l p y , h a r e fu n c t i o n s o f p r e s s u r e a n d t e m p e r a t u r ea n d a r e a p p r o x i m a t e d b y r a t i o n a l f u n c t i o n s w i t h i nF E H M .T h e c o n s e r v a t i o n o f fl u id m a s s i n t h e m a t r i x s y s t e m i s:

d Am . m a s sd t q . . . . . = 0 (11)w h e r e t h e s u b s c r i p t , m , r e f e r s t o t h e m a t r i x s y s t e m .q . . . . . r e p r e s e n t s t h e m a t r i x s o u r c e w h i c h i s t h e f lo wf r o m t h e m a t r i x t o t h e f r a c t u r e s . A . . . s s i s t h e m a ss p e ru n i t v o l u m e w i t h i n t h e m a t r i x . I n F E H M , c h a n g e s i nm a t r i x v o l u m e a r e c o n s i d e r e d n e g l i g i b l e a n d d o n o ta f f e c t t h e m a t r i x m a ss s t o r a g e , q .. . . .. is t h e su m o f t h e

LOOE-05 - - A na ly t i c F E H M, S.OOE-O6o

~ 6.00E-06.~_

2 . 0 0 E 4 R I

O . O O E + O 000.00E+00 5.00E+00 t .00E +01 1.50E+01 2.00E+01 2.50E+01D i s t a n c e , m

F i g . 8. E x a m p l e 2 - - - C o m p a r i s o n o f a n a l y ti c a n d n u m e r i c a p e r t u r e s a t5 0 0 a n d 2 0 0 0 d a y s .

T o = 230 CPo = 62 .7 MP a

! I I ' l ' l l l ' ' l ' ' l l l ' l ' l l l t ' l l ]I I I I [ l l T I t l l l l l l l l l l l l l l lI I I I I I I I I ] I [ I I I

I [ I l l l l l l r l t l l EI I / l l l l l l l l l l l [l l l / r p l l l l l l ~ l l l l l l l l l l l l l l l l l l l l lt

i n jec t ion w e l l / 600 m \ p roduction wel l1 4

P = 62 .7 MPa P = 45 .0 MP aT = 7 0 CF i g . 9. C o n c e p t u a l m o d e l o f E x a m p l e 3 .

1 5 0 m

I I I II I I

I I I

n o r m a l c o m p o n e n t s o f fl o w f r o m m a t r i x c el l t h r o u g h t h ec e l l b o u n d a r y :

= - - f V " km p m V P m , f 'n d x (12)J~ l b o u n d a r y ] . AT h e c o n s e r v a t io n o f e n e r gy e q u a t i o n f o r t h e m a t r i x c a nb e w r i t t e n a s :

dAm,energydt qe,matnx= 0 ( 1 3 )w h e r e :

= - - | Vm,enorgyl l b o u n d a r y

' ' 4 '

I n c o n t r a s t t o t h e f lu i d m a s s a n d e n e r g y e q u a t io n s , t h ee q u i l i b r i u m e q u a t i o n s f o r s t r e s s a r e s t e a d y - s t a t e b a s e do n t h e o b s e r v a t i o n t h a t s t r e s s f r o n t s t r a v e l m u c h f a s t e rt h a n p r e s s u r e a n d t e m p e r a t u r e f r o n t s i n t h e f l u i d o rs o li d . T h e s t a t i c f o r c e b a l a n c e i n s i d e t h e p o r o u s m a t r i xa s su m i n g sm a l l d e f o r m a t i o n s i s [ 1 7 ] :

~ j + b i = 0 (15)w h e r e ~ r~ i s th e s t r e ss t e n so r i n t h e so l i d a n d b i i s t h e b o d yf o r c e (s u m m a t i o n i n t e nd e d ) . T h e c o m p o n e n t s o f s tr es s

T a b l e 5 . E x a m p l e 3 - - P a r a m e t e r s u s e d in m o d e lP a r a m e t e r V a l u eL e n g t h 6 0 0 . 0 m '~W i d t h 1 5 0. 0 m tY o u n g ' s m o d u l u s 8 . 5 12 x 1 04 M P a ~P o i s s o n ' s r a t i o 0 . 2 5 tR o c k d e n s it y 2 7 1 6 k g m - 3 tR o c k s p e c if ic h e a t 8 0 3 J k g - ' C - ~ tR o c k t h e r m a l c o n d u c t i v i t y 2 . 5 7 W m - t C- 1" 1"F r a c t u r e p o r o s i t y 1 .0I n i t i a l a p e r t u r e 9 . 3 x 1 0 - S mA p e r t u r e p o w e r ( f o r c u b i c l a w ) 3A p e r t u r e f a c t o r ( f o r c u b i c l a w ) 8 9 6M i n i m u m a p e r t u r e 9 . 3 x 1 0 - S mF r a c t u r e s t i ff n e s s 1 . 0 x 1 08 M P a m - ~M a t r i x p e r m e a b i l i t y 1 . 0 x 1 0 - 20 m 2M a t r i x p o r o s i t y 0 . 01R o c k t h e r m a l e x p a n s i o n c o e f f i c i e n t 7 . 5 x 1 0 - 6 C -~ t"t F r o m S w e n s o n et a l . ( 1 9 9 5 ) .

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1 20 6 B O W E R a n d Z Y V O L O S K I :T a b l e 6 . E x a m p l e 3 - - B o u n d a r y c o n d i t i o n s u s e d i n m o d e l ( 1 = l e ft ,r = r i g h t , t = t o p , b = b o t t o m b o u n d a r y )T y p e B o u n d a r y C o n d i t i o nS t r e s s r , 1, t , b N o n o r m a l d i s p l a c e m e n tF l u i d f l o w r , 1, t , b N o n o r m a l f l o wI n j e c t i o n w e l l P = 6 2 .7 M P aP r o d u c t i o n w e l l P = 4 5 .0 M P aH e a t t r a n s f e r r , 1, t , b N o n o r m a l t r a n s f e r

I n j e c t i o n w e l l T = 7 0 CP r o d u c t i o n w e l l T = a s n u m e r i c a l l y c a l c u l a t e d

w i t h i n t h e m a t r i x m a t e r i a l ( a s s u m i n g t e n s i o n i s p o s it i v e)a r e d e s c r i b e d b y :

e~ = ~-~ a 0 -- l + v ~rkk60 + O O L -~ + otATm (16)

w h e r e E0 i s t h e s t r a i n , B l o t ' s c o n s t a n t , H , i s t h e i n v e r s ep r o p o r t i o n a l i t y c o n s t a n t o f t h e p o r e p r e s s u r e e ff e ct , a i st h e c o e f f i c i e n t o f t h e r m a l e x p a n s i o n , G i s t h e s h e a rm o d u l u s , a n d v i s P o i s s o n ' s r a t i o .

I n F E H M t h e b a l a n c e o f f o r c e s w i t h i n th e m a t r i x i sa c t u a l l y f o r m u l a t e d i n t e r m s o f d e f o r m a t i o n s a t t h en o d e s , u i . T h e s t r a i n m a y b e w r i t t e n a s :

~ i j : l ( u i f l J [ - U j , ) ( 1 7 )E q u a t i o n s ( 1 5 - 1 7 ) m a y b e c o m b i n e d r e s u l t i n g i n :

T H M N U M E R I C A L M O D E L

N o t e t h a t e q u a t i o n ( 1 8 ) a p p l i e s t o t h e m a t r i xs u r r o u n d i n g t h e f r a c t u r e s . T h e s t r e s s a n d s t r a i n i n t h ef r a c t u r e s th e m s e l v e s a r e h a n d l e d i n a d i f f e r e n t m a n n e r .T h e i n t ri n s ic p e r m e a b i l i t y o f t h e f r a c t u r e s y s t e ma p p e a r s i n e q u a t i o n ( 7) . T h i s t e r m i s f o r m u l a t e d f o r t w op e r p e n d i c u l a r f r a c t u r e s e t s a n d i s a l l o w e d t o v a r y a s af u n c t i o n o f t h e n o r m a l e f f ec t iv e s tr e ss a c r o s s e a c hf r a c t u r e s e t u s i n g e q u a t i o n s ( 2 ) a n d ( 3 ) w h e n t h e s t r e s si s c o m p r e s s i v e . T h e e f f e c ti v e s t r es s o f a f r a c t u r e i s t h en o r m a l s t r e s s w i t h i n t h e s u r r o u n d i n g m a t r i x m i n u s t h ef l u i d p r e s s u r e w i t h i n t h e f r a c t u r e . T h e f r a c t u r e s t i f f n e s si s h e l d c o n s t a n t i n t e n s i o n . T h e d i l a t i o n d u e t o t h es h e a r s t r e s s a c r o s s t h e f r a c t u r e i s n o t m o d e l e d . T h u sk = f (P r , Pm, T in , U l , U 2 ) . S i n c e a n i n c r e a s i n g a p e r t u r ec a u s e s a n i n c r e a s e i n t h e c o m p r e s s i v e s t r e s s , t h e s t r a i nd u e t o t h e c h a n g i n g a p e r t u r e i s a d d e d t o t h ep r e - e x i s t i n g s t r a i n i n t h e r o c k . F l u i d s t o r a g e i n t h ef r a c t u r e s y s t e m a ls o c h a n g e s w i t h t h e c h a n g i n g f r a c t u r ea p e r t u r e .

T h e q u a d r a t u r e r u l e s u s e d f o r t h e a c c u m u l a t i o n t e r m si n e q u a t i o n s ( 5 ) a n d ( 9 ) u s e w e i g h ts a t t h e n o d e s r a t h e rt h a n t h e u s u a l G a u s s q u a d r a t u r e w e i g h t s f o r t h e i n t e r i o ro f t h e e l em e n t . T h e n o d e p o i n t i n t e g r a t i o n i s e q u i v a l e n tt o n o d e l u m p i n g f o r t h e t r a n s i e n t c a p a c i t a n c e t e r m . I t is ,h o w e v e r , m o r e r i g o r o u s a n d i s a l s o u s e d f o r t h e f l o wt e r m s w h e r e n o d a l l u m p i n g i s n o t u s e d . T h e a c c u r a c y isl e s s t h a n G a u s s q u a d r a t u r e , b u t s t i l l s e c o n d o r d e r .

~ 2 2 Ia331 2G(1 -L ~ , ~ j

v 01 1 - vv 1 01 - - vv v 0

1 - v 1 - v

1 - 2v0 0 2(1 - v )

~ u t

~U 2~3x2

( ~ U l ~ / 2

( 1 8 )

I i ]50/

F i g . 1 0 . E x a m p l e 3 - - P r e s s u r e d i s t r i b u t i o n o f h o t d r y r o c k r e s e r v o i r a t 7 2 0 d a y s ( M P a ) .

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B O W E R a n d Z Y V O L O S K I : T H M N U M E R I C A L M O D E L 1 20 72 3 0Fig . 11. Example 3 - -T emp era tu re d i s t r ibu t ion o f ho t d ry rock rese rvo i r a t 720 days (C) .

A control volume finite element approach is used inFEHM. Here, quadrilaterals and triangles in twodimensions and hexahedra and tetrahedra in threedimensions are divided into volumes associated withnodes and areas associated with internodal distances (forthe problems in this paper, rectangular elements areused). These geometric terms are used to form controlvolume difference equations for the conservationequations. This method is not traditional becauseequation parameters are defined by node, not element.In FEHM the non-linear and purely geometric parts

are separated. This separation is explained in detail inFung et al. [18] and is valid over lower order elements.The non-linear part uses average transmissibility,D = k p / # , between two nodes. This is usually taken tobe the upstream nodal value. The method thus requiresparameters to be input at nodes rather than elements.The result is a much more stable code for solvingnon-linear problems while still retaining much of thegeometric flexibility of finite elements. The method hasbeen used in FEHM since 1983 [19] and has beenextensively verified [20].A 6 x 6 system of equations is developed defining all

six unknowns implicitly; Pf, Tr, Pro, Tin, u~, and u2:

in order to couple hydrologic, mechanical, and thermalphenomenum in a dual porosity system. Equation (19)is iterated until the norm of the residuals decreases by aprescribed factor.E X A M P L E S

The coupled system described can be used to solvecoupled groundwater problems in frac tured reservoirs. Acomprehensive verification of FEHM is beyond thescope of this paper but is described in Dash et al. [20]and Bower [21]. The following three examples aredescribed as verification of the fracture system coupledflow version of FEHM. Example 1 tests the thermal andhydrologic coupling for a single fracture with a constantaperture within a dual porosity system. Example 2 teststhe hydrologic and mechanical coupling for a singlefracture with a variable aperture. Example 3 test theability of FEHM to model realistic problems withreasonable time steps.Exa mple 1: Single fractu re, constant aperture

Example I consists of a 2-dimensional dual porositysystem with a single horizontal fracture. The reservoir is

OMf 3Mr OMf ~Mf 3Mr OMf~Pf ~ ~Pm O ~ ~Ul ~U2OPt O~ OPm ~ OUl Ou2dMm OMm OMm BMm OMm ~Mm~Pf OTr OPm OTm C3Ut OU2OEm OEm OEm OEm OEm OEmOP t OTf OPm OTto OUl ~U2~ S l O S l O S l ~ S l ~ S 1 O SIOP'-~f Of?~ OPm OTto OUl Ou2OS2 0S2 OS2 ~$2 ~$2 OS2Oef OTr OPm OTm ~3Ui 3U2

- Ep f(-TfEPr~E-Tm(-u1Eu2

- 9~ef~ r f~ e m= 9trm9~u2

(19)

where Me represents equation (4), Ef represents equation(8), Mm represents equation (11), Em represents equation(13), and S~ and Sz represent the static force balanceequations in the x- and y-directions. In equation (19), 9lrepresents the residual of the equations and E representsthe solution vector for each of the unknowns.Newton-Raphson iteration is used to solve equation (19)

50 m high and 1000 m long with a unit thickness (Fig.2). Initially the reservoir has a tempera ture of 100C anda pressure of 25 MPa. The left boundary of the fracturehas a constant inflow of 6.152 10 -3 kg see -1 per unitthickness of the reservoir. The pressure at the rightboundary is constant at 25 MPa. At t = 0 the fractureinjection temperature is set to 90C. The finite element

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1208 BOWER and ZYVOLO SKI: THM NUM ERICAL MOD ELr e s u l t s u s in g c o n s t a n t f r a c tu r e a p e r tu r e a r e c o mp a r e dw i t h t h e a n a l y t i c s o l u t io n p r e s e n t e d b y G r i n g a r t e n et al.[221.I n F E H M w i t h d u a l p o r o s i t y c a p ab i li ti e s, t h e r e a retw o o v e r l a p p in g e l e me n t c o n f ig u r a t i o n s ; t h e ma t r i xe l e me n t s a n d t h e f r a c tu r e s y s t e m e l e me n t s . T h u s t h e r ea r e r e a l ly tw ic e t h e n u m b e r o f i n p u t n o d e s u s e d f o rs o lu t i o n . I n E x a m p le l , t h e r e i s a s i n g le l in e o f n o d e sr e p r e s e n t i n g t h e f r a c t u r e b u t c o n t a i n e d w i t h i n t h ef r a c tu r e s y s t e m c o n f ig u r a t i o n . I n o r d e r t o m o d e l a s i n g l ef r a c tu r e , t h e ma te r i a l p r o p e r t i e s o f t h e f r a c tu r e s y s t e m a tt h e f r a c tu r e mu s t h a v e d i f f e r e n t p r o p e r t i e s t h a ne lsewhere .

T h e F E H M m e s h c o n s i st s o f 28 7 n o d e s ( F ig . 2 ). T h ee l e me n t s i ze n e a r t h e f r a c tu r e i s s ma l l to b e t t e r mo d e l t h ee f fe c t s a r o u n d t h e f r a c tu r e . T h i s l e a d s t o a p o t e n t i a lp r o b l e m d u e t o t h e a s p e c t r a t i o o f t h e e le me n t s .H o w e v e r , a s i g n i f i c a n t i n c r e a s e i n t h e n u m b e r o f n o d e su s e d i n o r d e r t o i m p r o v e t h e a s p e c t r a t i o c o u l d n o t b ei m p l e m e n t e d d u e t o m e m o r y l i m i t a t i o n s o n c o m p u t e rsys tem s ava i lab le fo r th is r e sea rch . The in i t ia l s ti f fness ofthe f r ac tur e i s se t to 1030 M Pa m -~ so th a t the f r ac tu rea p e r tu r e r e ma in s c o n s t a n t . T h e i n j e c t i o n t e mp e r a tu r e i scha ng ed to 90C a t t = 0 . S ince th is i s a t r an s ien tp r o b l e m , t h e t e m p e r a t u r e s a n d d i s p l a c e m e n t s a r emo d e l e d t o 6 21 y e a r s t o s h o w t r a n s i e n t b e h a v io r . T a b l e 1s h o w s p a r a m e te r s u s e d i n E x a m p le 1 . T a b l e 2 s h o w s t h eb o u n d a r y c o n d i t i o n s u s e d .

T h e i m p e r m e a b l e m a s s o f r o c k s u r r o u n d i n g t h ef r a c tu r e t r a n s f e r s h e a t t o t h e w a t e r . H o w e v e r , i n a d u a lp o r o s i t y s y s t e m , th e e n e r g y t r a n s fe r t h r o u g h t h e m a t r i xc a n n o t b e t r a n s f e r r e d f r o m m a t r i x n o d e t o m a t r i x n o d ed i r e ct l y b u t m u s t p a s s t h r o u g h t h e c o r r e s p o n d i n gf r a c tu r e n o d e s ( F ig . 1 ) . T h e d u a l p o r o s i t y mo d e l w o u ldn o t u s u a l l y b e c h o s e n w h e r e c o n d u c t i o n o f h e a t i s t h ed o m i n a n t m e t h o d o f e n e rg y t ra n s f e r. I n o r d e r t o u s e t h em o d i f ie d F E H M o n t h i s p ro b l e m t h e v o l u m e f r ac t io n o ft h e f r a c tu r e s y s t e m n o t i n t h e f r a c tu r e n e e d e d t o b e s e tv e r y h ig h ( 0 . 9 9 ) i n o r d e r t o a l l o w e n e r g y t r a n s f e r t o b es imu la t e d i n t h e r o c k ma s s . T h i s i n c r e a s e i n v o lu mef r a c t io n i s c o m p e n s a t e d f o r b y m u l t i p l y i n g t he a p e r t u r ef a c to r f o r t h e f r a c tu r e s y s t e m n o t i n t h e f r a c tu r e , b y w h a tw o u l d h a v e b e e n t h e v o l u m e f r a c ti o n . N o t e t h a t t h i s w a yo f m o d e l i n g t h e e n e r g y t r a n s fe r w o u l d n o r m a l l y c a u se a ne r r o r i n t h e f l u id s t o r a g e . T h i s i s n o t t h e c a s e i n t h i sm o d e l b e c a u se o f t h e l o w p e r m e a b i l it y o f t h e f r a c t u r es y s t e m n o d e s w h ic h a r e n o t i n t h e f r a c tu r e . V e r y l i t t l ef l u id g e t s i n to t h e s e n o d e s t o b e s t o r e d .

T h e n u m e r i c r es u l ts o f F E H M a r e c o m p a r e d t o t h ea n a ly t i c r e s u lt s o f G r in g a r t e n i n F ig . 3 . A s c a n b e s e e nin F ig . 3 , t h e r e s u l ts o f F E H M a r e i n e x c e l le n t a g r e e m e n tw i th t h e a n a ly t i c r e s u l t s .

W h e n a n a ly z in g a g e o lo g i c s y s t e m, i t i s r a r e t h a t a l lt h e p a r a me te r s n e e d e d t o c l e a r l y d e f i n e a g r o u n d w a te rm o d e l a r e k n o w n . S o m e p a r a m e t e r s m a y b e k n o w n .O t h e r s m a y b e b o u n d e d w i t h i n a r a n g e o r o n l ye s t ima t e d . T h e r e f o r e , k n o w le d g e o f t h e s e n s i t i v i t y o f as y s t e m t o c h a n g e s i n p a r a m e t e r s i s i m p o r t a n t . F o r t h i ss e n s i t iv i t y a n a ly s i s , a mo d i f i e d E x a m p le 1 w a s u s e d a st h e b a s e c a s e b y c h a n g i n g t h e f l o w i n p u t b o u n d a r y

c o n d i t i o n t o a f i x e d p r e s s u r e b o u n d a r y c o n d i t i o n o f2 5 .0 48 M P a . T h e i n i t ia l f r a c tu r e s t if f n es s o f E x a m p le 1w a s c h a n g e d f r o m 1 x 1 0 3 M P a m -~ ( a c o n s t a n ta p e r tu r e c o n d i t i o n ) t o 1 x 105 M P a m - ~. T h e s e c h a n g e sd o n o t a l t e r t h e p r e v io u s r es u l t s o f E x a m p le 1 a n d a l l o wth e f r a c tu r e p e r me a b i l i t y t o c h a n g e a s t h e f o l l o w in gp a r a me te r s a r e v a r i e d . N o te t h a t , w i th t h e e x c e p t i o n o ft h e a b o v e c h a n g e s , t h e i n p u t p a r a me te r s w e r e r e tu r n e dto t h o s e o f E x a m p le 1 b e f o r e e a c h o f t h e f o l l o w in g ru n s .

T h e c o n v e r g e n c e c a p a b i l i t y o f t h e m o d e l m a y b ein f l u e n c e d b y t h e v a lu e o f p a r a me te r s . I n t h e f o l l o w in gs e n s i t iv i t y r u n s t h e a v e r a g e t ime s t e p w a s a c o n s t a n t , 9 9d a y s , e x c e pt w h e r e n o t e d . T h e n u m b e r o f N e w t o n -R a p h s o n i t e r a t i o n s p e r t i m e s t e p ( N R ) i s s h o w n i n t h ef i g u r e s t o i n d i c a t e t h e a b i l i t y o f t h e mo d e l t o c o n v e r g e .L o w e r N e w t o n - R a p h s o n i t e r a t i o n s i n d i c a t e f a s t e rc o n v e r g e n c e .

F in i t e e l e me n t r e s u l t s a r e o f t e n s e n s i t i v e t o t h e t imes t e p u s e d f o r t h e c a l c u l a t i o n s . T o o s ma l l a t ime s t e p w i llc a u s e l o n g e r c o m p u t a t i o n t i m e s . E x t r e m e l y s m a l l t i m es teps cause ins tab i l i ty . Too la rge a t ime s tep wi l l r e su l ti n e r ro r s i n t h e o u tp u t o r e v e n i n a l a c k o f c o n v e rg e n c e .T h e u s e r d o e s n o t u s u a l l y s p e c i f y t h e t ime s t e p u s e d i nF E H M . T h e i n it i al ti m e s te p a n d t h e m a x i m u m t i m e s t epa re spec i f ied . The t ime s tep a t eve ry t ime i s ca lcu la tedw i t h i n F E H M b a s e d o n c o n v e r g e n c e c r it e ri a . T h e t i m es teps spec i f ied in F ig . 4 a re the ave rage t ime s teps ove rth e e n t i r e r u n . F ig u r e 4 i n d i c a te s t h a t r e s u l t s d o d e p e n du p o n t h e t ime s t e p b u t l i t t l e c h a n g e o c c u r s w i th a t imes t e p s ma l l e r t h a n 9 3 6 d a y s . U s in g t h i s t ime s t e p r e s u l t si n a r e a s o n a b l e c o m p u t a t i o n t i m e f o r t h e p r o b l e m .

I t i s w e l l k n o w n th a t i n c r e a s in g t h e n o d a l s p a c in g w i l ld e c r e a s e t h e a c c u r a c y o f r es u l ts . T h e s e n s i ti v i t y o fE x a m p le 1 t o h o r i z o n t a l e l e m e n t s iz e w a s t e s t e d b yi n c re a s in g t h e n u m b e r o f n o d e s i n t h e x - d i r e c t io nr e s u l t i n g i n a t o t a l n u m b e r o f 2 8 7 , 1 4 7, 7 7, a n d 4 2 n o d e s .T h e r e s u l t s a r e s h o w n in F ig . 5 . F ig u r e 5 i n d i c a t e s t h a tt h e r e s u l ts o f E x a m p le 1 a r e n o t p a r t i c u l a r l y s e n s i ti v e tot h e n u m b e r o f n o d e s i n t h e h o r i z o n t a l d i r e c ti o n w i t h i nth e r a n g e t e s t e d .

T h e v e r t i c a l n o d a l s p a c in g w a s v a r i e d t o d e t e r m in e t h es e n s i ti v i t y o f t h e p r o b l e m to t h e v e r t i ca l s iz e. F o r t h i sp r o b l e m r e s u l t s w e r e n e a r ly i d e n t i c a l . H o w e v e r , t h ev e r t i c a l n o d a l s p a c in g h a s b e e n k n o w n to a f f e c t r e s u l t sf o r o th e r p r o b l e ms .

M a t r i x p e r m e a b i l i t y w a s v a r i e d f r o m 1 x 1 0 - 8 m 2 t o1 x 1 0 - 2 m 2. T h e r e w a s n o c h a n g e i n t h e o u t l e tt e m p e r a tu r e d i s t r i b u t i o n i n d i c a t i n g t h a t t h e r e i s v i r t u a l l yn o f l o w to t h e ma t r i x .

T h e d i f f e r e n c e i n t e mp e r a tu r e w a s f o u n d t o a f f e c t t h ec o n v e r g en c e c a p a b i l it y o f F E H M d u r i n g t h e m o d e l i n g o fE x a mp le 1 . L a r g e d i f f e r e n c e s b e tw e e n t h e i n j e c t i o nt e mp e r a tu r e a n d t h e i n i t i a l t e mp e r a tu r e c a u s e s l o w e rc o n v e r g e n c e a n d ma y r e q u i r e s ma l l e r t ime s t e p s . T h i sp r o b l e m i s w o r s e w h e n u s i n g e x t r e m e t h e r m a l p a r -a me te r s s u c h a s l a r g e v a lu e s o f t h e t h e r ma l e x p a n s io nc o e f fi c ie n t . F ig u r e 6 s h o w s t h e r e s u l ts o f E x a m p le 1 w h e nth e d i f f e r e n c e b e tw e e n t h e i n i t i a l t e mp e r a tu r e a n d t h ein j e c t i o n t e mp e r a tu r e i s i n c r e a s e d . F o r t h e s e r u n s acoe f f ic ien t o f th e rm a l e xpa ns io n of 1 .0 x 10-4C - I i s

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used. Note tha t 1Lhe vertical axis is dimensionless. Thenumber of Newton-Raphson iterations increases as thetemperature difference increases. This indicates that thecode has more trouble converging with larger tempera-ture differences. If the temperature difference is largeenough the problem may not continue to run with areasonable time step size. The poorer convergence withlarger temperature difference is probably made worse bythe fact that this problem is of a single fracture in animpermeable rock (see previous discussion on energytransfer in dual porosity systems).It has been found that, while changes in eachindividual parameter may not have a large effect,combinations of parameters can result in instability ofthe problem. Large temperature differences will increaseinstability. However, the large temperature differenceswill have more effect on stability if there is a largecoefficient of thermal expansion and a small initialfracture stiffness. Large differences will have more effectwhen using the modified FEHM to model singlefractures in impermeable rock masses as the dualporosity system is designed to model fracture systems. Ingeneral, the following make a difference in the ability ofFEHM to converge when under the influence of largetemperature differences:

Problems with larger sized rock blocks surroundinga single fracture will converge better than problemsinvolving smaller rock blocks when the thermalexpansion coefficient is large and the boundaryconditions are fixed displacement. Under unfavorableconditions the thermal expansion or contraction effectsextend to the boundary. Problems with smaller initial fracture stiffness andlarge thermal expansion coefficients will converge poorlyunder large temperature differences. The large amountof expansion or contraction in the rock causes largerchanges in the fracture aperture which can result ininstability. This problem is made worse by very smallinitial apertures because the change in aperture is a largefraction of the initial state. It can be made worse by largepressure changes.The user of FEHM should be aware of theselimitations as they may affect the convergence propertiesof models.E x a m p le 2 : S in g le f ra c tu re , f l u id p re s su re

Example 2 is taken from the semi-analytical similarityanalysis of fluid flow through a single joint developed byWijesinghe [23]. This problem has been used by Kelkarand Zyvoloski [1], Noorishad e t a l . [24], and Swensone t a l . [25] to test coupled fracture codes. This test caseconsists of fluid mass and force coupling in a variableaperture fracture. Wijesinghe's problem is a 1-dimen-sional flow coupled with deformation in a single jointsurrounded by a solid. The solid extends to infinitynormal to the fracture. The fracture is subjected to auniform in situ stress normal to the fracture which isassumed large enough so that the fracture sides areprevented from losing contact with each other under theRMM S 34/8--F

influence of the injected fluid pressure. Wijesinghe usedthe cubic law to model the joint permeability. Jointstiffness was constant.The reservoir is 25 m long and 1 m high with a singlehorizontal fracture down the center. Initially the fracturehas an aperture of 1.0 x 10 -5 m. The fracture stiffness is1.0 x 105 MPa m -l. Init ially there is a constant pressureof 21 MPa throughout the fracture and matrix system.At t = 0, the fluid is injected into the fracture a t apressure of 21.9 MPa. This forces the fracture apertureto increase as the pressure front moves down thefracture. The conceptual model o f the problem is shownin Fig. 7.In FEHM with dual porosity capabilities, there aretwo overlapping element configurations; the matrixelements and the fracture system elements. Thus thereare really twice the number of input nodes used forsolution. In Example 2, there is a single line of nodesrepresenting the fracture but contained within thefracture system configuration. In order to model a singlefracture, the material properties of the fracture system atthe fracture must have different properties thanelsewhere.There are 287 nodes in the fracture system. In orderto model the fracture as a single line of nodes, everyother node in the fracture system uses a very highfracture stiffness and very low fracture aperture. Theinput parameters are given in Table 3. The boundaryconditions are given in Table 4.In order to allow the area of fracture flow to changeas the aperture changes, the aperture power used is 3 andthe aperture factor of 1/12 is divided by the originalaperture of the fracture. In order to prevent closing ofthe fracture, the minimum aperture closure is set to1 x 1 0 - 3 m .

A comparison o f Wijesinghe's solution with the resultsof FEHM is shown in Fig. 8. The results are comparedat 500 and 2000 sec. Figure 8 indicates that there is goodagreement between the analytic and numeric results.E x a m p le 3 : H o t d ry ro c k r e serv oi r

Example 3 consists of a 2-dimensional, horizontalmodel of the Fenton Hill hot dry rock reservoir. TheFenton Hill hot dry rock reservoir has been previouslydescribed and modeled by Swenson e t a l . [25]. The hotdry rock reservoir is 300 m long, 600 m wide (Fig. 9), andit is estimated to be 30 m high. The initial temperatureof the reservoir is 230C. The fracture system controlsthe fluid flow and before hydrofracturing was essentiallyimpermeable due to its great depth. The wells weresubsequently pressurized to 62.7 MPa and shut off.During geothermal production the fluid was forcedthrough the fracture system to the production well (at45.0 MPa) which cools the rock and causes thermalcontraction. This tends to slowly increase the fracturesystem permeability. The decreasing fluid pressurearound the production well tends to decrease thepermeability.The FEHM model consists of 370 nodes. Only half ofthe reservoir need be modeled due to symmetry. The

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1210 BOW ER and ZYVOLO SKI: THM NUM ERICAL MOD ELp a r a m e t e r s u s e d i n t h e m o d e l a r e g i v e n i n T a b l e 5 . T h eb o u n d a r y c o n d i t i o n s a r e g i v e n in T a b l e 6 . T h e i n i t ia lp r e s s u r e o f th e m o d e l i s s e t a t 6 2 .7 M P a a n d t h e i n it i a lt e m p e r a t u r e i s s e t t o 2 3 0 C . T h e s e a r e c o n d i t i o n s a f t e rt h e i n i t ia l p r e s s u r i z a t i o n o f t h e r e s e r v o i r . A t t = 0 , t h ei n j e c t i o n w e l l t e m p e r a t u r e i s r e d u c e d t o 7 0 C . T h ep r o d u c t i o n w e l l p r e s s u re i s c h a n g e d t o 4 5 .0 M P a . T h es i m u l a t i o n i s r u n t o 7 2 0 d a y s , o v e r t w i c e t h e t i m e t h eF e n t o n H i l l re s e r v o i r w a s o p e r a t e d .

I n o r d e r t o a l l o w t h e a r e a o f f r a c t u r e f lo w t o c h a n g ea s th e a p e r t u r e c h a n g e s , t h e a p e r t u r e p o w e r u s e d i s 3 a n dt h e a p e r t u r e f a c t o r o f 1 / 1 2 i s d i v i d e d b y t h e o r i g i n a la p e r t u r e o f t h e f r a c tu r e .

S o m e p a r a m e t e r s o f t h e F e n t o n H i l l h o t d r y r o c kr e s e r v o i r a re n o t w e l l k n o w n . I n p a r t i c u l a r t h e f r a c t u r es p a c i n g , i n it i a l a p e r t u r e , a n d f r a c t u r e s t i f fn e s s a r e o n l yk n o w n i n s o m e r a n g e . T h e m o d e l w a s r u n t o s t e a d y s t a tew i t h o u t t h e r m a l d e f o r m a t i o n a n d t h o s e v a l ue s o f in i ti ala p e r t u r e a n d f r a c t u r e s t if f n es s w h i c h r e s u l t ed i n ap r o d u c t i o n w e l l f l o w o f 5 .7 k g se c -1 ( a s s u m i n g a 3 0 mt h i c k r e s e r v o i r w i t h s y m m e t r y ) a r e u s e d i n t h e m o d e l .T h e m o d e l a s s u m e s t w o p e r p e n d i c u l a r f r a c t u r e s e t sp a r a l l e l t o t h e m e s h b o u n d a r i e s . A f r a c t u r e s p a c i n g o f1 0 m i s u s e d . H o w e v e r , t h e m a t r i x t o f r a c t u r e l e n g t h u s e di n t h e m o d e l i s o n l y 1 m t o a l l o w g r e a t e r h e a t t r a n s f e rt o t h e m a t r i x s y s t e m ( a n d c o n s e q u e n t l y g r e a t e r h e a ts t o r a g e ) .

T h e r e su l ts o f E x a m p l e 3 a t 7 2 0 d a y s a r e s h o w n i n F i g s1 0 a n d 1 1. T h e p r o d u c t i o n w e l l f l o w a f t e r a d j u s t m e n t o ft h e f r a c t u r e p r o p e r t i e s i s 5. 7 k g / s e c w h i c h i s t h e a c t u a lf i e l d r e s u l t . T h e t e m p e r a t u r e a t t h e p r o d u c t i o n w e l l i s2 2 9 C a t 2 7 0 d a y s w h i l e t h e a c t u a l F e n t o n H i l l r e s e r v o i rs h o w e d a p r o d u c t i o n w e l l t e m p e r a t u r e o f 2 3 0 C a t 27 0d a y s o f f ie l d te s t in g . T h i s i n d i c a t e s e x c e l l e n t a g r e e m e n tb e t w e e n F E H M r e s u l ts a n d f i el d d a t a f o r th i s s i ng l ep o i n t a t 2 7 0 d a y s . T h i s i s t h e o n l y t e m p e r a t u r e d a t ap o i n t k n o w n f o r c o m p a r i s o n .

T h e i n it ia l p e r m e a b i l i t y o f t h e m o d e l e d f r a c t u r es y s t e m i s 6 .7 x 1 0 -1 5 m 2. A f t e r 7 2 0 d a y s t h e p e r m e a b i l i t yc h a n g e s n e a r t h e in j e c ti o n w e ll t o t h e m a x i m u m a m o u n to f 6 . 9 x 1 0 -1 5 m 2 a n d n e a r t h e t o p o f t h e r e s e r v o i r t o t h em i n i m u m a m o u n t o f 6 .6 x 1 0 -15 m 2. T h u s t h e f r a c t u r ep e r m e a b i l i t y c h a n g e s l i t t l e i n t h e f r a c t u r e s y s t e m .

T h e a v e r a g e t i m e s t e p s i z e i s 1 0 d a y s w i t h a n a v e r a g eo f 4 .0 N e w t o n - R a p h s o n i t e ra t i o n s p e r ti m e s te p . T h is isa r e a s o n a b l e t i m e s t e p s i z e f o r r u n n i n g a p r o b l e m o v e rl o n g p e r i o d s o f t im e .

C O N C L U S I O NN u m e r i c a l c o u p l i n g o f h y d r o l o g ic , t h e r m a l , a n d

m e c h a n i ca l p h e n o m e n u m is i m p o r t a n t i n m o d e li n gf r a c t u r e d r e s e r v o i r s i t u a t i o n s s u c h a s h o t d r y r o c kg e o t h e r m a l r e s e r v o i r s , h i g h l e v el n u c l e a r w a s t e r e p o s i t o -r ie s, p e t r o l e u m p r o d u c t i o n , a n d g e o t h e r m a l s u b s id e n c e.N u m e r i c a l c o d e s m u s t b e s t a b l e so t h a t r e a s o n a b l e t i m es t e p s i z e s a r e u s e d i n p r o b l e m s i n v o l v i n g l a r g e t i m es c a l e s . T h i s c a n b e a c h i e v e d b y i m p l i c i t l y c o u p l i n g t h ep h e n o m e n u m i n v o l v e d .

A d u a l p o r o s i t y , f in it e e l e m e n t co d e , F E H M , h a s b e e nm o d i f i e d t o c o u p l e h y d r o l o g ic , t h e r m a l , a n d m e c h a n i c a lp h e n o m e n u m i n a f r a c tu r e d r e s e r v o i r b y e x t e n d in g t h es t r es s c a l c u l a t i o n s t o a d u a l p o r o s i t y s y s t e m a n d b yl e tt in g t h e p e r m e a b i l i ty o f u p t o t w o p e r p e n d i c u l a rf r a c t u r e s y s t e m s c h a n g e a s t h e e f fe c t iv e n o r m a l s t r e s sa c r o s s t h e f r a c t u r e c h a n g e s . T h i s w a s a c c o m p l i s h e d b yu s i n g a c o m b i n a t i o n o f t h e B a r t o n - B a n d i s r e l a ti o n f o rf r a c t u r e a p e r t u r e s a n d t h e c u b i c l a w f o r f r a c t u r e f l o w .T h e r e s u lt in g c o d e w o r k s w e l l i n a v a r i e t y o f p r o b l e m si n v o l v i n g c o u p l i n g . T h e t h r e e e x a m p l e s d e s c r i b e d h e r ei n v o l v e h y d r o l o g i c a n d t h e r m a l c o u p l i n g ( E x a m p l e 1 ),h y d r o l o g i c a n d m e c h a n i c a l c o u p l i n g ( E x a m p l e 2 ) , a n dh y d r o l o g i c , th e r m a l , a n d m e c h a n i c a l c o u p l i n g ( E x a m p l e3 ) . T h e r e q u i r e d t i m e s t e p f o r c o n v e r g e n c e i n a l l t h r e ee x a m p l e s i s r e a s o n a b l e . T h u s r e a l i s t i c s i t u a t i o n s c a n b ee f f e ct i v el y a n d e f f ic i en t ly m o d e l e d . F E H M i s a n e f f e c t iv ea n d r o b u s t t o o l i n m o d e l i n g c o u p l e d f l o w s i t u a t i o n s i nf r a c t u r e d r e s e r v o i r s .

F u t u r e m o d i f i c a t i o n s t o F E H M a r e p l a n n e d . T h e s em o d i f i c a t i o n s i n c l u d e a d d i n g t h e e f f ec t s o f f r a c t u r ed i l a t i o n d u e t o s h e a r s t r e s s o n t h e f r a c t u r e a p e r t u r e ,e x t e n d i n g t h e c u r r e n t w o r k t o 3 - d i m e n s i o n s , a n dc o u p l i n g t h e h y d r o l o g i c b e h a v i o r t o s o l u t e t r a n s p o r t .T h e s e p l a n n e d m o d i f i c a t i o n s w i l l e x t e n d t h e a p p l i c a b i l i t yo f F E H M i n m o d e l i n g f ra c t u re d g r o u n d w a t e r s y s te m s .Acknowledgements --This work was supported by the US Depart-ment of E nergy at the Los Alamos National Laboratory.

Accepted for publication 8 July 1997.

REFERENCES1. K elkar S. and Zyvolosk iG. Hydro-thermo-mechanical esponse ofa fractured rock blo ck. Rock Mechanics Contributions andChallenges (Edited by Hustrulid W. A. and Joh nson G. A.),pp. 337-344. Balkema, Rotterdam (1990).2. Bobe rg T. C. Thermal Methods o f Oil Recovery. Wiley, New York(1988).3. Coats K. H. A highly implicit steamflood model. Soc. Pet. Eng.J. 369-383 (October 1978).4. Noorishad J. and Tsan g C. F. D evelopment and verification of anumerical technique for coupled hydromechanical phenomena in

rocks. Rock Joints (Edited by Barton N. and Stephansson O.),pp. 673-679. Balkema, Rotterdam (1990).5. Swenson D. User's Manual for GEOCRACK: A Coupled FluidFlow~Heat Transfer~Rock Deformation Program for Analysis o fFluid Flow in Jointed Rock, Manual Release 3.0. Kansas StateUniversity (1994).6. Sprecker T. A . A finite elemen t heat transfer m odel of fluid flowin joint roc k. Masters thesis, Kan sas State Un iversity (1994).7. Jing L., Tsan g C. F. and Stephansson O. DEC OV ALEX --Aninternational co-operative research proje ct on mathematicalmodels of couple d THM processes for safet y analys is ofradioactive was te repositories. Int. J. Rock Mech. Min. Sei. &Geomech. Abstr. 32, 389-398 (1995).8. Douglas J. Jr. and Arbogast T. D ual porosity models for flow innaturally fractured reservoirs. Dynamics of Fluids in HierarchicalPorous Media (Edited by Cushman J. H.), pp. 177-221. AcademicPress, New York (1990).9. Show alter R. E. Diffusion models wi th microstructure. Math-ematical Modeling for Flow and Transport Through Porous Media(Edited by Dagan G., H ornung U . and Knabner P.), pp. 567-580.Kluw er Acade mic Publishers, Nethe rlands (1991).10. Kundu P. K. Fluid Mechanics, p. 269. A cademic Press, New York(1990).

7/28/2019 1-s2.0-S1365160997800718-main

11/11

B O W E R a nd Z Y V O L O S K I: T H M N U M E R I C A L M O D E L 121111. Tsang Y. W . Usag e o f "equ iv a len t aper tu res" fo r rock f rac tu resas de r ived f rom hydrau l ic and t race r te s t s . Wate r Re s . Re s . 28 ,1451-1455 (199211.12 . Bar ton N. , Band is S . and Bakh ta r K. S t reng th , de fo rmat ion andconduc t iv i ty coup l ing o f rock jo in ts . I n t . J. R o c k M e c h . M i n . S c i .& Geomech. Ab str . 22, 121-140 (1985).13 . Zhao J . and Brown E . T . Hydro- the rmo-mechan ica l p roper t ie s o fjo in ts in the C arnmene l l i s g ran i te . Q. J . o f Engng Ge ol. 25, 279-290(1992).14 . Band is S . C . , Lumsden A. C . and Bar ton N. R . Fundamenta ls o fr o c k j o i n t d e f o r m a t i o n . In t . J . Roc k M e c h. M in . Sc i . & Geome ch.Abs t r . 20, 249-268 (1983).15. Zienk iewicz O. C. The F in i t e E le me nt M e thod. M c G r a w - H i l l B o o kCompany , London (1977) .16. Z yvo losk i G. , Dash Z . and K e lkar S . F E H M N 1 .0 : F i n i te E l e m e n tH e a t a n d M a s s T r a n s fe r C o d e, L o s A l a m o s N a t i o n a l L a b o r a t o r y ,LA - 12062-MS (1991).17 . B io t M. A. G enera l theory o f th ree -d imens iona l conso l ida t ion . J .A p p l . P h y . 12, 155-164 (1941).1 8 . F u n g L . S . , B u c h a n a n L . a n d S h a r m a R . H y b r i d _ C V F E m e t h o dfor f lexible-grid reservoir simulation. S. Pe t . Engs J . 19, 188-199(1994).19 . Zyvo losk i G. F in i te e lemen t methods fo r geo the rmal rese rvo i r

s imula t ion . In t . J . Nume r . Anal . Me thods Ge ome c h. 7 , 75 -86(1983).20 . Dash Z . V., Rob inson B . A. and Zyvo losk i G. A. V & V R e p o r t f o rt h e F E H M N A p p l i c a t i o n , L o s A l a m o s N a t i o n a l L a b o r a t o r y ,LA-UR-95-2063 (1995) .2 1 . B o w e r K . M . A n u m e r i c a l m o d e l o f h y d r o - th e r m o - m e c h a n i c a lcoup l ing in a f rac tu red rock mass . Ph .D . thes is , U n ivers i ty o f NewMex ico (1996).22 . Gr ingar ten A. C . , Withe rspoon P . A. and Ohnish i Y. Theory o fhea t ex t rac t ion f rom f rac tu red ho t d ry rock . J . Ge ophy s ic a l Re s .80, 1120-1124 (1975).23. Wijesinghe A. M . A S imi lar i t y Solu t ion fo r Cou ple d De format ionand F lu id F low in D isc re te Frac ture s , L a w r e n c e L i v e r m o r eNat iona l Labora to ry , UCRL-95316 (1986) .24 . Noo r ishad J . , Tsang , C . F . and W ithe rspoon P . A. Theore t ica l andf ie ld studies o f coup led hydrom echan ica l beh av io r o f f rac tu redrocks - -1 . D eve lopm ent and ve r i f ica t ion o f a numer ica l s imula to r .In t . J . Roc k M e c h. M in . Sc i . & Geome ch. A bs t r . 29, 401-409(1992).25 . Swenson D. , Du Teau R . and Sp recker T . Mode l ing f low in ajo in ted geo the rmal rese rvo i r . P r o c ee d i ng s o f t h e W o r m G e o t h e r m a lCongress, 1995 (Ed i ted by Barb ie r E . , F rye G. , Ig les ias E . andPa lmason G. ) . In te rna t iona l Geo thermal Assoc ia t ion (1995) .