application of the method of orthogonal collocation on finite elements

8/9/2019 Application of the Method of Orthogonal Collocation on Finite Elements

1/194

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! *

l v u f eJ '

■1)c


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3/194

THE UNIVERSITY OF ALBERTA

) : APPLICATION OF THE METHOD OF ORTHOGONAL COLLOCATION ON FINITE.

ELEMENTS TO ENGINEERING .PROBLEMS

V

by

' { p. ' Dilee p Kumar

A,THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES AND RESEARCH

TNr PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE

OF MASTER OF SCIENCE

TN ‘ '

• . ■ CHEMICAL ENGINEERING, .( ' l

' DEPARTMENT OF CHEMICAL ENGINEERING

EDMONTON, ALBERTA

Spring, 1979

r odu ced with permission of the copyright owner. Furth er reproduction prohibited without permission.


4/194

■ ) ' VTHE UNIVERSITY OF ALBERTA

, \

FACULTY OF GRADUATE STUDIES ANQy RESEARCHf*

The undersigned c e r t i f y th a t t hey have read, and recommend

to the ~F acu lty of Graduate Studies and Research, fo r acceptance, a ■

, Ap pl ica tio n o f the Method of Orthogonal Collo cati onth es is . e n t i t l e d . . . L . ....................... . . .

on Finite Elements to Engineering Problems.

Dileep Kumar submitted by .................... ;.......................................

in p a r t i a l fu l f i lm e n t o f the requi r emen ts fo r the degree o f v

Master of Science.

(Superv isor )

Date. n

n . 7 7

rodu ced with permission of the copyright owner. Furth er reproduction prohibited without permission.


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Abst r a c t ■¥

The method of orthogonal co llo ca ti on on fi n i t e elements

(OCFE) was ap pl ied to two en gi neer in g problems. One of. the problems

considered is tha t of the f low of a Newtonian f l u i d . i n an in te rn a ll y

finn ed%ube . C yl in d ri ca l coordi nates were employed and Legendre

sh ift ed orthogonal polynomials were used as the t r i a l fun cti on s. An

al te rn at in g di re ct io n im p li c it (ADI) method was used to' solve the

re su lt in g set of equations. Bette r accuracy was achieved by increa sing

the number of collocation points per element rath|er than increasing the

number of elements for a given number of interior collocation points.i

Al th ough, in general, f o r a given ' to ta l number o f c o l lo ca t io n p o in ts , the

OCFE was found sup er io r tq the f f n i t e d if fe re n ce method in terms of /

accuracy, the computational time requirement was much higher for the

method o f orthogonal col lq®5-tion on f i n i t e elements.

The second problem considered deals with the simulation of

two-dimensional m isc ibl e displacement of o i l by a solven t inp or ou s

media. -A d ir e c t method of so lu tio n was used. Soluti on of the

c o n t in u it y equadjon provided ex ce lle nt mass conserva tion. However,*

realistic concentration profiles could only be obtained from the

.convection d iff us io n equation for a very high value of the di ffu si onc o e f f i c i e n t . T

pr oduc ed with permission of the copyright owner. Furthe r reproduction prohibited without permission.


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A C K N O W L E D G E M E N T S

- VThe .author wishes to acknowledge the help and guidance,*• - '

■ rec ei ved 'fr om . Dr . J. H. Masliyah througho ut the cour.se of t hi s st ud y.

The au th or is, indebted to h i£ f r ie n d , Rajeev D. -Deshmukh,

fo r valuable suggestions and useful c ri ti c is m s . , • v /

The aut hor Wishes • to thank Mrs. Audrey M^yes fo r c a r e f u l l y •.0 . -1 » . . ■. ’

typ ing the the sis and meeting the deadl ine. - •

Finally, the f inancial assistance provided by the Universityt ' ^of Alberta is gratefull,y acknowledged. 1 '



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Chapter

12 >

3

~4T

Table of Contents

• ’ Page*

I n tr o d uc t io n . . . . ............ f T .. ........................................ 1« * *

Li te ra tu re Rev.iew .......................... 1 .......... 3

Method of Weighted Residuals and OrthogonalCollocation on Finite Elements ................................ 6

3.1 General Trea tment : . . . . ........... 6

(a) Subdomain Method ............................... . ........ ...................... .7(b) Least Squares Method .......................................................... 8

(c) .Galerkin Method- ............................. •' ................................. 8

(d) Method of Moments .................. 8(e) Collocation Method ............................................................... 9

3.2 Choice of Tr ia l Funct ions .................. 9

3.3 The: Method of Orthogonal C o l lo c a t io n ........................ .- ...10

3.4 OVthogonal .Collocation for Two-DimensionalProblems ............................................... 13

3.5 Orthogonal Collocation on Finite Elements. (OCFE) . . . . . . . ^ ........ . . . . . . . .................... ............. ■.............. 17

. 3.5.1 ' Approximation E rr o rs .............. 18

Ap p lica tion o f OCFE to Finned T u b es . ................. ’.............. 19

4.1 Statement of the Problem ........................... - ........... -19

4.2 OCFE For mulat ion o f the P ro bl em .............................. . . . . . . 2 1

4."3 Solution of the Equations .............. 25

4 .3 .1 Constant, r s o lu t io n . • • • • . . . . . . . . . . . . . . . . . . . - - -25

4.3.2 Constant.O s o lu ti o n ’. .......................... -26

4.4 -Computat ional Scheme \* • - .. ........................ 28

4.5 C alcu la t ion o f Average V e l o c i t y ............. 32

4.6 Other Techniques .............. 354.6.1 Least Square~-Madching Technique .............................. 35

4.6.2 Fi ni te- Di ffe re nc e Method,.F.D. 36

4.6.3 Soliman and Feingold Approach .................... . . . . . . . 3 6

4.7 Discussion of Results. ................. 38

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Table of Contents - cont inued -

Cha pter ' ' . ,Page

5 A pp li cat io n of OCFE to. Porous M ed ia ......................................... 46

5.1 Governing Equations .......... I . . . . . . ................... 46

, 5.2 Numbering Scheme ................... . 1 .................... .50

5.3 OCFE For mulati on of the. Governing Equati ons .. 50

5.3.1 Cont inuit y Equation . . . . . . ,J. ... . . ....... 50

5.3.2 Co^ivection-Di ffus^on Equation . . ......................... 58

5. 3. 2. 1 Runge-Kutta Method-, R.K. . . * . ......................... 59

5.3.-2.2 Total Im p li c it Method . ................................ 60

5.4 Determination of Source. Term .................. . . . . . . 6 1

5.5 Computational Scheme ..................................... 61

5.6 Results and Discussion ................... -.............. .62* •r

< 5.6.1 Velo city Results ................................. 625.6.1.1 Configur ation One . . . ........ . . . 6 4

5.6 .1. 2 C onfi gura tion Two .......................... 69

5.6.2 Concentration Results ........... ,................................... 70

6 Conc lus ions and Recommendations . . ; .................................... . . . . 7 6

6.1 Conclusions .................................................... j . . . . 7 6

6.1.1 Finned Tube Problem ................ 76

6.1.2 Porous Media Problem ..................................... ;76

6.2 Recommendations ........ > ............................ .. ................ J.76’

Bibliography . . . . 77

Appendices A Computer Program to Generate Matrices, A, B and w. .. . . .79

B Tabulation of Matrices , A, B and w ................................... . . . '84

C S ol ut io n o f a One-Dimensional P r ob le m ..................................... 97

D F i ni t e Dif fe ren ce Formulat ion o f the FinnedTube Problem ..................... 108

E Pressure and Ve lo c it y Resul ts from C on ti nu it yEquation ................ 110

F Concentrat ion Results from Convection D if fu si onEquation ....................................................... .-148

G Hand c a lc u la t io n f o r Concentr at ion Results ...... ...................... 170

H Physical Data fo r Porous Media Problem ................................

vn

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■ ’l— . C, '

List of TablesTable \ ' Page

■ A1 Summary of computations-for the Orthogonal Colloca tionon Finite Element Method ................................. t ............... 33

2 Summary of Computations fo r the F in it e Dif fer enc e Method. . .. .3 7

3 Summary o f Resul ts f o r the OCFE and FD ............... . . . 3 9

4 Comparison of Evaluated and In je ct ed Q .......................................... ^. . .65

5 Co ll oc at io n Point number of the Point locat ed at the . .Cent re o f Each Element f o r N=3 and 5. ....................... 66

& '

\ 6 Comparison o f the Pressures at the Centre o f Each\ Element f o r N=3 and 5 . ' ...................... 67-

\\

C . l \ Comparison of the An aly ti ca l and Numerical Results fo r \ a One-Dimensional Problem ................................................................ 104

\ \

E.l Con figu rat ion One: Pressure and Ve lo ci ty Results fo r ■100 cp and N=2. . . , ....................... . . .................................. 112

E.2 Co nf igu ra tio n One: Pressure and Ve lo ci ty Results fo r 4 ' u = 10,000 cp and N=2........................ t .......... .............................. 113

• .'E.3 Co nfi gu rat ion One: Pressure and V el oc it y Results fo r

y = 100 cp \ a n d N=3 .............. .114

E.4 Co nf igu ra tio n One: Pressure and Ve lo ci ty Results fo r y = 10,000 cp add N=3. ' . 116 -

•j ; \ •E.5 Con fig ur ati on One: Pressurexand Ve lo ci ty Results fo r

y '= 100 cp and N=4. \ .................................................................. 118 \ _ ■ ' ■ '

E.6 Con fig ura ti on One: Pressure and^ Vel oci ty Results fo r y = 10,000 cp and N=4. . . . \ . ................................ 121

xyj\iE. 7 Con fig urat ion' One: Pressure and Ve lo ci ty Results f or

y = 100 cp and N=5 ............ ■................... 124

, E.8 Co nf igu ra tio n One: Pressure and Ve lo ci ty Results fo r y = 10,000 cp and N=5 ............ -................................ ' .......... .127

E.9 Conf igur ati on Two: Pressure and Ve lo ci ty Re su lts rfo r y = 100 cp and N=2 .............................. . . r . ............................ . . . . 1 3 0

E.10 Co nf igu ra ti on Two: Pressure and Ve lo ci ty Results fo r y = 10,000 > cp and N=2. ....................................................... .131..

* . . \

• ■ »■' V l l l

od uced with permission of the copyright owner. Further reproduction prohibited without permission.


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*♦ ,List of Tables - continued.

Tabl e

E .l l Co

■ k

E.12 Con fi gur at ion Two: Pressure and Ve lo ci ty Results f or v = .10,000 cp and N=3. . . .......................... 134V

E.13 Con fig ura tio n Two: Pressure and Ve lo ci ty Results fo r u = , 100 cp and N=4. • ..................................................................... 136

E.14 Confi gurati on Two : Pressure and Veloci ty Results f or u = 10,000 cp and N=4. • 139 .

jE.15 Confi gu rati on Two : Pressure and Veloci t y Results f or /

u = 100 cp and N=5 ........... ........ 142

E.16 Co nf ig ur at io n Two: Pressure and V el oc it y Resul.'ts fo r

v = 10,000 cp and N=5 ....................... 145F.l Conf igur ati on One: Concentration Results fo r

Kp = .0001075 cm2/ s and N=3 ....................................... 150

F.2 Conf igur atio n One: Concentration Results fo r Kp = .01075 cm2/s and N=3 .................................... ". ..................... 152

F.3 Conf igur ati on One: Concentration Results fo r . Kq = .0001075 cm2/s and N=5 ..................................................... . .154

F.4 Co nf igu ra ti on One: Concentr ation Results fo r Kd = .01075 cm2/s and N=5 ........................ .. .................... , ........... 157

F.5 Con figu rati on Two: Concentration Resdlts fo r Kd = .0001075 cm2/s and N=3 ............................... 160

F.6 Con fig ura tio n Two:- Concent rati on Results fo r . Kit-= .01075 cm2/s - and N=3 .................................................. . . . . . 1 6 2

I , oF.7=- Configuration Two: Concentration Results for

Kp = .0001075 cm2/s and N=5.. ...................................... .164

F.8 Con fig ura ti on Two: Concen trat ion R^sul ts f or • Kp = .01075 cm2/s and N-5. ................... ' . . . . . 1 6 7

Page

n f i ^ u r a t i o n\ t = 10 0

Two: Pressure and Velocity Results forcp and N=3. ................................. ............ ,132

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' L is t of Figures

Figure. _ - ’ ' Page

1 Flow Geometry o f a Fanned Tube 1 ......... _ ............. '.20i

2 F in it e Elements and Col lo ca ti on 'P oi nts ...................... ..22

3 Col lo ca ti on Poi nts near the Fin Tip .............. f. .................. 23

4 Flow char t f o r the Computational Scheme f o r

5 l o ^ ^ P ^ o f the Poi nt s f o r F i n i t e ! D if fe re nc e

6 Va ri at ion of Centre' and Average.Ve lo ci tie sfor L=0.5 with number of Collocation Points

' f o r GOC ‘ ................. ; ................................ “. .41

7 Va ria tio n of Centre ve lo ci ty with number of Col loc ati on P oi nt s. fo r OCFE . . . . j ...................................... 44

8 . Va ri at io n of Average Ve lo ci ty wi th number of


12/194

List of Figures - continued

( ‘Figure ' Page

• \C.3 Block Diagonal Ma tr ix -as a Band St ru ct ur ed

M a t r i x . . . . . . . ........ . \ ......... . - • • • . ................................. 102

G.l Col loc atio n Points fo r F ir s t and Second Deri vati vesfor N=3..... ...................................................... ' .............................. 1710

G.2 Co ll oc ati on Points fo r Fi r s t and- Second Der iva tiv es

f o r N=5 : ........................... ! ..................... i .............. 174

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Nomenclature ' -

A F i r s t d e r iva t iv e representation in the or th og on al^ co l loca t ion mathod

Ap . Dimensior iless cross-sectional flow area (Ap/R2 )t B Second d e r iv a t ive representation in the orthogonal

co ll oc at io n method. -

C Concentration o f the solvent in the o i l - s o lv e n t mixture

Cp Dimensioniess wett ed pe rimete r (Cp/R)

C. Source concentration * . ' *in '

dV Di f fe ren t ia l . -vol ume, cm3o ' ■

f Fanning f ra ct i on fact or

f.'Re ' , Product o f Fanning f r i c t i o n fa ct or and Reynolds number'

Axk • ' VF = Z— - , a dimens ion ies s quan t i ty" Ay? . ■ • Xj . ta

^ - Dimensioniess va ri ab le

Co l loca t ion po in t a t f in t ip*

Kg Di sper sion C o e f f i c ie n t , cm2/ s ‘ (Kg= Kg.p).

Kgj 'Total Dispersion C oe ff ic ie nt , cm2/s

Kp P er me ab il it y, darcy \ '

k 1,k2 ,k 3 and. k4 k-value s of the fo ur th ord er Runge-Kutta method

L 1 Dimensional length , cm • i '* *.

L Fin len gth , dimensioniess (Chapter 4)

Length o f the fo rm at io n, cm (Chapter 5)

£ p ■ F in t i p element "

N Number.of i n t e r i o r c o l lo c a t io n points in one directionper element

NE Number o f elements ' ^

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NR, NX' and Ne Total number o f co l lo ca t io n po in ts (i nc lud in gboundary c ol lo ca ti o n po ints f o r GOC in r , x and e -d i r ec t ions , r espec t ive ly

NEe, NER, NEX and NEY Number o f elements in e, r , x and y di re c t io n srespec t ive ly

NPe, NPR,-NPX and NPY Number of co l lo ca t io n po in ts •per. eleme nt 'i n-0 , r , x and y di re ct io ns re sp ec tiv el y (=N+2)

P, p ' , j - . Pressure, psi

P- Orth ogo nal ’ po lyn omi al, Legendre sh if te d polynomial

k £ ■ thP-’ -' - Pressure at co llo ca tio n pov t (x . , y .) in the k *■

’’ element 1

q, q(x( , y ' } ' Source term, cm-3/cm3fo rm ati o n^Sc v

Q Source term, cm3/s> ® ’

v Calculated-Q using a quadrature approach

R ■.' . Tube ira di us , cm

r ' Radial coo rdin ate

r Dimensioniess radi al coordinate ( r :/R) ■ ,» 1

r ■ Value.of r a t c o l lo c a t i o n - p o in t ( e . . r . ) - .3 . . i ' j

Re Reynolds, number vs> ,* S

S ' Thickness o f the fo rm at io n, cm

t Time, s

At Increment in t , s ‘ " • ' • . v

u. ‘ Darcy Jveloci ty , cm/s

ux ,Uy Component o f the Darcy v e lo c it y in x and y di re ct io ns ',cm/sk , £ k £

•u x! . or Ux* .. (x i ’ y i) ^-component of the ve lo ci ty o f coll oc ati on1>J poi n- t/ (x^ .j 'y .) in the k^ th element"

» - .

xi i i

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w, ,w. Weighting fa c to r, wei ghti ng fa ct or in the i*"*1 row

W' j Width o f the fo rm at io n, cm

W' Dimensional v e lo c it y , cm/s* u

k i W ’ ( o , r \ Dimensioniess ve lo ci tyv l - 'i

Wc Centre v e lo ci ty , dimensioniess

Average v e lo c i ty , dimensi onies s

a-f- rn l l n r a t i n n nm* n+ /ft T* i in "t"h P1.J

Vel oci ty a t co l l oca t ion poin t (e . . , r. ) in the k?,element, dimens ioni ess . ^

WF A weighting fa c to r

x 1 . Ax ial d i rec tion, . -cm


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Subscri p.ts

i , j , k , £ and n ind ices ,

0 ' Oi l V ,

s So lve n t \

f Fin

x,y In x and y di re ct io ns

Vector

Superscri pts

s+

t , t + A t

Vector

Matrix

Denotes a dimensioniess quantity

k,£ Denotes k t ^ element

Denotes in i t ia l value, value a f te r one hal f i te ra t io nand value after one comple/te iteration, respectively

Denotes value after t and t+At time levels respectively

V

xv



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+ The A p p l i c a b i l i t y o f t he OCFE method to porous media i s• • ‘

discussed in Chapter 5. The problem deals with the, unsteady st at e

miscib le d isplacement of o i l by so lvent in an o i l re se rv oi r. Unlike

the finned tube problem, a direct method of solution was used. .The'

d ir e c t method solved a ll the equations s imultan eousl y using the LU

decomposition technique with i te ra ti v e refinement. The d ir e c t method

was employed I because the ADI techn ique was found t o be f a i r l y expensive

fo r the-tinner) tube problem. The two d if fe re n t loc at io ns f o r the

pr od uc tio n we l \ were consi dere d. In one o f the schemes, the geometry

repre sen ted a qu ar te r of a f i v e spo t. The second scheme which has no

practical importance was used mainly to check the numerical results.,'

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1 / CHAPTER 2

.’ . LITERATURE REVIEW'

The method o f weight ed r e si dua ls eric ipasses several methods

(Subdomain, Co llo ca tio n, Galerki n e tc . ) . These methods were f i r s tun if ie d by Crandall (1956) as the method of we ig ht ed T^ si du al s (MWR).

A comprehensive rev iew o f the l i t e r a t u r e on MWR is a va i la b le in Finjays on

(1972, 1974).

■".'■V The method o f weighted re si du al s was ap pl ie d to a ,i dc

variety of engineering problems by Clymer and Braun (1973), rinl_ayson

and Scriven (1966) and VichneV’etsk y (1969). Ap pl ic at io n of the Galerki ni '


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(1968) s uc ce ss fu lly appl ied a le as t square c ol lo ca ti on method to steady

st ate 'hea t conduction in ar bi tr ar y bodies . / The f i r s t known app lic ati on

o f a boundary c o ll o c a ti on method is due to Sparrow and L o e ff le r (195 9).

The method of orthogonal c ol lo ca ti on was f i r s t appl ied by

Lanczos (1938, 1956). I t has since been ap pl ie d by Cleanshaw and

Norton (1963)\ Norton (1964), and Wright (1964) to solve ordinary

d if fe re n ti a l equations. Villadsen and Stewart (1967) applied the ort ho -."v

gonal co ll o ca t ion method to boundary value problems. The method o f

^orthogonal collocation has been shown to be very effective for certain

non-linear chemical engineering problems and has been highly advocated

by Finlayson (1971), Young and Finlayson (1973).,

Sincovec (1977) described the development of a generalized

collocation method for the solution of coupled non-linear parabolic

pa rt ia l d if f e r e n ti a l equations. He showed th at the co llo ca ti on method

with Gaussian co ll oc at io n po ints was more e ff e c ti v e than the conventional

f i n i t e d if fe re nc e/ s ol ut io n. He also showed tha t fo r problems wit h a smoothsolution, one would obtain more accuracy per unit time by increasing

the order of the co llo ca ti on method. "

The method of orthogonal co ll oc at io n on f i n i t e elements

(OCFE)- which is, the su bj ec t of. th is th es is is a r at he r new tec hniq ue.

The area is not well explored and not much work has been done on this

method. Douglas and Dupont (1973) stud ied t h e o r e t ic a ll y a f i n i t e

element co llo ca ti on method fo r parabolic equations. Bladie r (1973)

'u sed OCFE to solve a di e swel l problem un su cc es sf ul ly . Anderman (1974)

used th i s method to sol ve a two dimensional f l u i d fl ow around a sphere •

a t a ver y low Reynolds number. He found th a t the comput ati onal time

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req ui remen t was ver y la rg e . Carey and Fi nlayson (1975) used the OCFE

method to solve a one dimensional effectiveness factor problem

c a ta ly s t p e l l e t and th ey h ig h ly recommended the use o f OCFE-. Chang

and Fi nl ay so n (1977) appl ied the OCFE method to a two dimens ional

problem and used the al te rn a ti ng di re ct io n im p l i c it (ADI) method to

solve the resulting algebraic equations.



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CHAPTER 3

THE METHODS OF WEIGHTED RESIDUALS AND ORTHOGONAL COLLOCATION ON

FINITE ELEMENTS

3.1 General Treatment :

The method o f weight ed residuals (MWR) i s a -ganeral method of

obtaini ng solutions to d if fe r e n ti a l equations. The solu tion to be

determined is expanded in a se t o f spec if ied t r i a l fun cti ons .' The

constants of the t r i a l fun cti ons are obtained using MWR. The f i r s t

appr oxima tio n gives a so lu ti on w it hi n 20%. However, more accurates ol ut io ns can b'e obtained using hi gher approx imat ions .

For i l l u s t r a t i v e purposes, a boundary value problem is fe‘ . 5

considered Finlayson (1972).(

v2T = Txx + Tyy = 0 in V(x »y)- ' (3 -1 )

, -T = T q on the boundary of V ^ ^ ( 3 . 2 )

Assuming^ t r i a l function o f t-ljse form

nT = T + 7 C.T. ' ( 3 . 3 )

0 i= i 1 1 -■where functions T^ satisfy the boundary conditions (T^=0 on the

boundary). Su bs ti tut e Equatioriy (3 .3) i n Equation (3. 1) to form the

residual (The residual is zero everywhere in V when the t r i a l f u n c t io n js\

the exac t so lut ion) .N

. R = V2(T + I C.T.) ,o ^‘ i 1 l v , (3.4)

0r NR = v 2T +T C. v 2T.

0 i -1 1 1

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In the method of weighted re si d u al s, are chosen in such.a

way th at the residu al is forced to be.zero in-a n average sense.

Equating the weighted integrals of the residual to zero yields

(WFj , R) = O j = 1 , 2 n ' (.3 -5)

where (WF . , R) =

, V

WFj R dV - (o 6)

and WF i s a we igh ti ng fa c to r . When WF and R are orthogona l then

WF R dV = 0

From'Equations (3.4)and (3.5) one obtains,I '

I C (WF., v2Tf ) = - (WF V2T ) (3.7)i = l 3

Equation (3.7) can be written as

j , V l ’ V ' (3.8 )

where G^ = (WF ,̂ v2T \ )

H. = - (WF . ,V2T )J J ■ o

Here T and T. are known. Ther efor e G.. and H. can be evalua te d i f WF.o t J i J J

i s known. The methods o f choosing WF. are described below. Once G.. and. J Ji

H. are known; can be eva lua ted using Equation (3 .8 ) . can then be

substituted in Equation (3.3) to obtain the approximate solution.

There are vari ous ways o f choosing the weig ht in g fu nc ti on s WF.

Each choice pr ovides a d if fe r e n t method of weighted re si du al s. Some of

the important methods are considered below.

a) Subdomain method: Di vi di ng the domain -V in to n^ sma ll er subdomli

V . and de fi ni ngJ

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1 X 'in V.wf . =< J

J O X no t in' V.J ’ '

One observes th at the residual R, of the d if fe r e n t ia l equation

when integrated'over the subdomain, V j , is zero as given by Equation (3 .5 ).

As the number of subdomains inc rea se ^t he d i f f e r e n t i a l equa tion is

satisfied in more and more subdomains and the residual approaches zero

everywhere in the li m i t as nr

ng

or

d ; •

b) Least squares method: In the le as t squares method th e. we ig ht i

dR fu nc ti on is . Equating the res idua l to zero one ob ta in s,

i

f : R dv = 0 • ( 3. 9)

R2 dV = 0 f o r i = 1 , 2 . .. n . ^ ]g )3C.1

Hence the in te gr al is minimized wit h resp ect to . Sol ut ioh of

Equation (3.1 0) provides the C.. c o e ff ic ie n t s . The algebra invol ved

using this method is usually rather tedious.

c) Galerkin- Method: fn the Galer kin method, the weigh ting fu nc ti on s

are also the t r ia l func tions , i . e . , WF.. = T . . The t r i a l functio ns

must be pa rt of a complete set of func tio ns so that the tr i a l ,a

solution is capable of representing the exact solution provided

enough terms are used. The Galerkin method forces the residual to bet

zero by making i t ort hogonal to each member o f a complete se t of

funct ions .

d) Method of .Moments: In th is method the we ig ht in g fu nc ti on s f o r

the one dimensional case are 1, x,x 2 ,x3 , . . .Therefore successively(

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high er moments o f the re sid u als are forc ed to be zero . I t is

evident that fo r the f i r s t approximation this method is

J identical to the subdomain method,

e) Co llo cat ion Method: In the co ll oc at io n method the weighti ng

func tion s are' the displaced Dirac delt a fun cti on P

WF = 6(x - x . )J J

which has the property that

WF. R dV = R|J

(3.12)

V

Thus the residual is . zer o at the N co lloc ati on points and i t

approaches zero everywhere in the l i m i t as N °°.

I t has been shown th at in the c o ll o c a ti o n method the sol ut io n

is set to ze ro, Finl ayson (1972). In order to reduce such dependence

one can apply the le a s t squares , col lo cat io n method. In th i s method the

res idua l is evaluated at more point s than there are c o e ff ic ie n ts and theover-determined set of alg ebr aic equations are solved by a le as t ..

squares method. . -

The orthogonal collocation method which is a special case of

the collocation method ts discussed in Section 3.3.

2 \ 2 Choice of T r ia l Functions':

choice of the t r i a l fun cti ons . Such a choice is very important fo r low

order approximations but fo r higher order approximations i t is not as

c r i t i c a l since the rate o f convergence becomes the prime c r it e ri o n ,

Finlayson (1972).

depends upon the choice of the collocation points at which the residual

. One of t he most im por tan t consi der at ion s i n usi ng MWR is tfife ,

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The t r i a l fu nc ti on s must be complete so th at they represent

the exact solu t ion i f enough terms are used. Further the tr ia l functi on

should be as simple as possible and should not complicate the analysis

unn eces sar ily . For a problem wit h a boundary co nd it io n of the type

y(x,z) = F(x,z) one may choose

Ny( x, z) = F(x,z) + I a y. (x, z)

i = l ■where i t is s pe ci fi ed t h a t y. = 0 on the boundary. Thus the choice

s a t is fi e s the boundary co nd it io n. One may s ta r t wi th a general

polynomial and can obtain a reasonable tr ia l fun ctio n afte r ' 'a pply ing/

boundary con dit ion s and symmetry co nd iti on s. Orthogonal polynomials

were found to be ex ce ll en t t r i a l fun cti on s (Fin layso n 1972) and can be*

con stru cted to sa ti s fy some of the boundary co nd it io ns . This approach

is usual ,y used in the orthogonal co llo ca ti on method.

3.3 Method o f Orthogonal Co ll oc at io n:

The method of orthogonal collocation has been well covered by'•'v •

* '.T

Finlayson (1972). However, a b ri e f de sc rip tio n is presented in thi s

sec tio n in order to f a c i l i t a t e the understanding- of the method which’ wase>

applied to the two dimensional problems discussec) in this thesis.

The advantage of the method of orthogonal collocation is the

rapid convergence to the solution as the number of collocation points is

inc rea sed . Ferguson and Fin lays on (1972) showed th a t fo r an or di na ry

d i f fe re n t i a l equat ion the er ro r was p ropor t ional to ( j j j - ) ^ '^ where N isthe number of in te r io r co llo ca ti on p oin ts. As N changes from 5 to 6

the e rr or decreases by a fa ct or of 100. In the f i n i t e

d i f f e ren ce ca l cu l a t i o n o f 0 ( a x 2 ) , a change o f N ( = | —) from 5 to 6„ tAX

with permission of the copyright owner. Furth er reproduction prohibited without permission.


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decreases the er ro r by only a fa ct or of 1.4.

In t .,e orthogonal co l loc at i on me th o ^ th e-c oll oc at i on points

are taken as the root s to. orthogonal pol ynomi als. Vi llc ds en andStewart (1967) c.iose U \ e t r ial functions to be the sets of orthogonal

polynomial which ilsc satisfied the boundary cotTditions and the roots to

the polynomials gave „he collocation points.t 9 • '

A b r i e f d esc r ip t io n o f th e o r th og on a l^co llocat ion method i s

presented below.

Let a fun cti on t be approximated by a tr ia ' l fu nc tio n,

NX-2 't ( x ) = b1 + b 9 x + x (1 - x ) I a.P. , (x) (3.13)i z . i = 1 i - i

where the pol ynom ial s, , are define d by

to tal number of col l oc at i on points ( in clu din g boundary po int s) . Since

both even and odd powers o f x are incl uded in the il f un ct io ns , i t is '

clear that the orthogonal polynomials have no special symmetry pro

pert ies. Equation (3.‘T3) can be subst i tuted in the differential equation

whose solution is required and the residuals are set to zero at the given

co l loca t ion po in ts in the in ter val (0 ,1 ) . Consequently the co eff ic ie n t s

o f Equation (3.13) can be evaluated. The co llo ca tio n po ints are the roots

to the polynomial Pp(x)- Finlayson1s approach is not to solve for the

WF'(x) Pn(x) Pm(x) dx = 0 3.13a

0

and m f n

Thus the successive polynomials are orthogonal to all polynomials of

orde r less than m with:some weig htin g f un ct io n WF(x) > 0. NX is the

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constants but fo r t at the co llo ca tio n poin t. Equation ( 3 . 1 3 ) can be

wr i t t en as ,

NX • ■ ^t (x ) = I d • x1" 1 (3.14)

• " 1 = 1 - I - - - ■ ■ ■ ■

*Taking .f i r s t and second der ivat ives a t the co l locat ion poin ts , one

\ .* •obtains '

i ' NX . .

• t ( x . ) = I d . ( x . ) 1~ . ’ (3.15)i = 1 .

.. NX . •' , , NX . _ a r * , ' 1 x . di - . J , d - n ( x j ) 1' 2 d,

J J 1=1 J

'NX . NX • ■ 'd2t

dx2X ; > Z1 f l x1" i : x , d t = b l ( ' - I X i - Z X x ^ 1 ' 3 dt (3.1*7)

J J

wfiere NX=N+2. N- is the to ta l number o f in t e r i o r co ll o ca ti on p oi nt s.

Wri tin g the above equations in the mat rix notat ion y ie ld s , I

t = ^ d . „ , " - (3 .! c ' *dt = -j" . ^dx :■= c d . . - ■ (3.19)

and j—T-r _ .. ± 1 = d d ' ' ' (3 .20)

dx2 v 2 - ,

where 0 • = x ^: j i - j .(3.-21).

Cji - ( i -1) xj 2 - (3.22)* •

' ^ "

Dj i = ( i - l ) ( l - 2 ) x ] - 3 : - ' " ; " n . 23)



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1 3

From equation (3.18' .

d = ! f 11 t \ .3-24;

Su bs ti tu ti ng d in Equat ions( 3.19) and (3.20) yi el ds

S ' . c r 1 t l A t where A = C C _ ( 3 . 25 )

— =! D Q _1 t = B t where S = 0 q _1 ,■ .dx 2 .

As the c o l l o c a t i o n points x . are known, ma tr ices Q, C and d can beJ

eva luated and hence A and B can be determined. A general computer

program to c a lcu l a te mat ri ces Q>, C, d, A and B i s pr ov ided in Appendix

A. The ma trices ar e t abulated in Appendix B f o r 3 £ NX l 10. Tabulation

o f matr ice s A and B is also given in F inl ays on (1972) f o r NX = 3 and 4. '

* . The de riv ati ve s in a d if fe re n ti a l equation whose so lut ion

is to be found can be rep lac ed by Equati ons ( 3.25 ) and (3 .2 6 ). The

resulting set of algebraic equations can be solved subject to thepre vai l ing boundary condit io ns . For a d if fe re n t i a l equat ion of the

type ^ .

d 2t , d t , , n ". ' d x 2 dx X t ~ 0 . (3 .27)

ne obtains

NX NXy b . . t . + y a . . t. + x . t .

i= i J ’ 1 1 f=i j ’ 1 1 j jo

f o r e a c h i n t e r i o r c o l l o c a t i o n p o in t j .

(3.28)

3.4 Orthogonal Co ll oc at io n fo r Two-Dimensional Problems:

For two-dimensional problems a t r j a l fun cti on analogous to



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Fquat'on (3. 13) i s given by, Fin lays on (1 974).

NX-2• T(x ,y) = < b-j+b^x ,+ x( l- x ) I ' akP ^ _ i (x)

k- 1x

NY-2c1+c2y + y d - y ) ^ (3.29)

Of NX

T( x, y) = [ d k -1

k=l x,k

NYy d y

i £ = l y ’ *

'£••1(3.30)

where

x , k

y»£

(3.31)

(3.32)

For any particular point ( i , j) one can write

NX k - 1

k=r

NY £-1y d y .

£=1 y ’ £ J

(3.33)

D e f i n i n g mat ri x T by T. . = T (x ., y .) E q u a t i o n . (3.33) becomes"1 »J "J J

TuQ-

*•

(3.34)

where Q and Q are Q in x and y d ire ct ion s re spec t ively and th ei r valuesx y i*

are given by Equation (3.21) or in the matrix form by

^ / y ] (3.35)T - Q dX X

or T = Q d d 1 Q 1x x y vyx (3.36)



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16hence

Z. . = T. .i .J J

Equation (3 .2 °) gives

3Z. • . . 1 >J = ' V a 7sy fe i , k . k , j - ( 3 . 46 )

Interchanging i and j in Equatio n(3.46) yie lds

3Z.

cy or

£ Aj , k Zk , i . ' (3 .4 7)

S ± i * ? * 3 .lc h , k (3-48)

Similar relations hold for the second derivatives.

..*>VThus for affi$wo dimensional problem of the type

- i f l + = f (x>y) ( 3 . 4 9 ), x : ay 2

one obtains

NX NYJ B. T . + y B. T. = f ( x . , y . ) /•, r n \

n =1 i»n n , j ^ J ,n i ,n i ( 3.50 )

f o r a n y i n t e r i o r c o l l o c a t i o n p o in t ( i , j )

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1 7/ '

3.5 The Method o f Qrthogonal Co ll oc at io n on F in i te Elements (OCFE)

The main difference between the method of weighted residuals

(MWR) and t\ie tr a d it i o n a l f i n i t e element method is i n the choice of the

tr i a l func tion s. Normally MWR uses tr ia l functio ns defined over theI

en tire domain whereas in the tr ad iti on al f i n it e element method t r i a l

or shape fun ct io ns are def ine d over each element. The advantage of the

f i n i t e element method i s th at the elements can be changed in shape or

size to f i t the physi cal boundaries. The method of orthogonal

co ll oc at io n on fi n i t e elements is an, attempt to combine the featu res o f

both the orthogonal co ll oc a ti on method and the f i n i t e element method.

The main feature of'OCFE is that the domain of interest is

divid ed i nto subdomains and tha t the t r i a l fun cti on is appl ied over the

domain in a piecewise fashion element by element. Using such

d is c r e ti z a ti o n OCFE should then be able to handle sol ut io ns which have

steep gradient s s the t r i a l fu ncti ons are orthogonal in the region

(0,T) i t become., ,-cessary to have the independent va ri ab le (s ) to l i e

between (0 ,1 ). For a given element, the value of the re si d ua ls gi ve an

indication as to whether more elements need to be added in a given

reg ion . The to ta l number o f it e ra ti o n s can be reduced by using a' '

solution obtained with a lesser number of collocation points as the

i n i t i a l guess fo r a higher number of co llo ca tio n poin ts. Irre spe ctiv e

of the nature of the solution, (symmetric or unsymmetric) , ^general

polynomial a ppr oxi mati on should be used in OCFE. A give n d i f f e r e n t i a lI

equation is sa t is f ie d a t each in te r i o r co l loc at io n poin t . At the^

element interboundar ies , c on t inu i ty of the f i r s t de ivat ive is sought .

For a one-di mensiona l problem, one ob ta in s a blo ck diagona l m a tr ix . Two-

dimensional problems can be solved as one-dimentional problems using

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« 18

the a l t ern at i ng d ir ec t io n im p li c i t method (ADI) . . /

Appendix C i l l u s t r a t e s the so lu t ion o f a simple one

dimensional problem using OCFE. The computer program is also

incl ude d. A s im il a r approach was used to solve 'the two-dimensional f i n

problem discussed in ChaptePM.

3.5.1 . Approximati on Err ors :

Douglas and Dupont (1973) showed t h a t , f o r pa ra bo li c

d iff e re n t ia l equat ions the d is cre t iz at i on er r or was propor t ional to h 14

(h=l/NE) fo r N= 2 , when the c o ll o c a ti o n po in ts were the Gaussian

quadratur e po in ts. However, when the c ol lo ca ti on po in ts were un ifo rm ly

d is tr ib u te d in each element, the er ro r was pro por tio nal to h2. ^ Thus

changing the collocation points to Gaussian quadrature points reduces

the er ro r dra ma ti ca ll y. More ge ne ra lly , Deboor and Swartz (1973)

showed- tha t fo r a d if fe re n ti a l equation of the type D2y = f( y ) , the

er ro r f o r the OCFE could, be given by the fo ll ow in g re la ti o n .r.

i 1 nN+ 2e r r o r a ( ^ )

Douglas ( 1973) showed t h a t fo r 1i.riear . problems ,when the

t r i a l poly nomia ls were degree (N +l ), convergence proceeded as Axn+^ g lo b a l l y and Ax^n a t the c o l lo ca t io n poin ts .



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C H A P T E R 4

APPLICATION OF THE-METHOD OF ORTHOGONAL COLLOCATION

ON FINITE ELEMENTS TO FINNED TUBES

me method of orthogonal collocation was applied to a

d i f f i c u l t f lu id flow problem. The selected physical j>roblem is that

of an incompressible Newtonian f lu id flow in the in te rn a ll y finned js

tube shown in Figure 1. The governing equation i s an e l l i p t i c type of

parti a-l d if f e r e n t ia l equati on. This problem was chosen mainly

because tra di t io na l f in it e d iffe renc e techniques did not provide anadequate solution unless a large number of grid points were

used and other techniques which are usually used to solve these type

of equations fa i le d to produce a sat isf act ory sol uti on.

4.1 Statement o f the Problem:

The momentum equa tion governing the f l u i d fl ow is given by.

'

1


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21

W = 0 at r = 1 and 0 £ 0


38/194

2 2

-f

4 . 1 /.----- ---

4, 2 J 4, 3 4, 4

r'c/,3. A

-------- -Ukt'-th

element3, 2 1

- AX£

13, 3 3, 4

J

A

A2 , 2 r o C O 2. 4

1. 1 !1, 2 1, 3 1. 4

J W 1r = 0 r = 1

76

5

4

3

2 =1

k+ 1, C

- ("

k, C-1. kJ K)z+ 1

k -1 , e

\V

FIGU RE 2. Finlte ElemenCd and collo catio n Points

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23

cII

(a)NEC = NER = 5NP0 = NPR = 5

L = 0.3

'(b)

j c j

NE0 = #JER - 2NP0 = NPR = 5

L = 0.3

FIGURE 3 Collocation Points Near the Fin Tip

r odu ced with permission of the copyright owne r. Furth er reproduction prohibited without permission.


40/194

where (4.8)

and 2 , NPe-1

-h r 2 , NPR-1

k(4.9)

, NER

For a given element k i , NP0 and NPR are the to ta l number o f c o ll o c a ti o n

points ( including boundary points) in 0 and r d irec t ion s , resp ect ive ly.

NEo and NER are the number o f elements in 6 and r d i r e c t io ns sr^s pe c t iv e ly .

Figure 2 shows the elements on which Equation (4.7) is applied.

Es se nt ia lly Equation (4 .7) is appl ied to each element of the' flow f i e ld .

On the element int erbo und arie s co n ti n u it y of the fun cti on ai>d' i t s

f i r s t der iv at iv e is assumed. For the element boundaries th at coincide

with the tube boundaries, the physi cal boundary con di tio ns as given by

Equation (4.3 ) are app lie d. The alg ebr ai c Equations (4. 7) toge the r

with the boundary conditions are solved using the ADI method, Peaceman

and Rachford (1955). /

produced with permission of the copyright owner. Further reproduction prohibited without permission.


41/194

25

4.3 Solution-of the Equations:

Equation (4.7) can be written as

2Jr +v£ Ar i NPR l, „ f r +v £ Ar 1 NPR ,

L' V B . Wk ’ £ H 1 - I A. W. ’ £ Ar£ ! j , n i , n \ J | Ar^ ! j , n i , n

t N P e k o o „ ■

+— I Bi n V i = ‘ (r"+v̂ Aro} (4J0)\e? n=l 1,n n ’ J SL 1 SL A0 £ n= I ' J

1 & solve the system o f equations using ADI, Equation (4.10) is wr it te n

in two different forms.

■4.3.1 Constant r solution:

For consta nt r , Equation (4.10 ) can be w ri tt e n as.

M , s+h __J_ Ny9 B wk,A,s.+Js = ^ wk,£ ,si , j a 0 2 n= 1 T»n n >3 T' J

'k

r +v - Ar I 2 NPRSL J SL r

k NPR , r , t i NPR^ - 4 y b . wk F 5 + V v j AM y , wk .£ ,s

£ j n=l J ' n 1-n ---------- ------- I n b j >n i ,n

+ ( r £+' ,j Ar£)2 ' , ( 4 J 1 )

where w is an it e r a t i o n parameter and (s+ 4 ) denotes the ve lo ci tie s at

the end of a constant r sweep.

At the element interboundaries, the fo l lo w ing condi tions are imposed

( i ) Con t inu i ty o f the func t ion :

wnp « ’ S+'"= Wk+1,£,S+is f o r k = l , ......... NE0-1 . (4.12) N r 0 , j 1 , J

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26

( i i ) C o n t in u i t y o f the f i r s t d e r i v a t i v e : -

NPa • NPe ' -1— y a ' _ _L . y A W*


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Hk-t.s+.l . ' " P B wk .« . s +li , j 1 Ar& n L h j , n i , n

2 7

l.r +v^Ar I NPR, — - -J-— L \ V a wk,£, s+1 = k,£,s+b

Ari M L n=l J ’ n 150 - 1 _

i N P6 , j

■ ^ ^ J , Bi ,n “ n j ’ 5 i + ( V ^ ^ t ) r «•>«>

where (s+ 1 )! denotes . the v e lo c i t i e s - a t .end of a con sta nt e sweep.

At the'element interboundaries the fo l lo wing^hon d it io ns ar e imposed:

i Conti nu tty o f , the-, fun cti on :

Wi ’,NPR+1= Wi j ’ S+1 f o r ^=1»- -NER-1 ...f -■ ( A . 17)

i i . C o n ti n u it y o f the f i r s t d e r i v a t iv e :

NPRJ L y k . £ ' S+1 i NPR k ,£ + l , s+1 = 0 Ar . A m d d W.K,£’ S 1 - i - y A, W.• £ n=l NPR,n i , n - Ar,,,- , l , n i>n

£+ 1 n=l

f o r £ = 1 , NER-1 (4. 18 )

In addition, the following boundary conditions are applied,

B.C. 1:

1 NPR

4r.l nilI A. wk - ! >s+1 = o a t . r = 0 fo r k = 1... .NEO _i I . n i 9n x

(4.19)

l.C. 2:,k,NER,s+1i ,NPRWi npr = 0 a t r = 1 f o r k = 1, NEe _ (r4.20) '

Equation (4. 16) tog eth er wi th Equations (4. 17) to (4.2Q) w§«#rj, solved

l ine by l ine a t constan t 6 [ i . e . fo r i = 2 , . . .N Pe=l , and k .= l , . . .N Ee ] ,

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28

4.4 Computational Scheme:

I t has already been stated th at the 'Al ter na tin g D irec tion

I m p l i c i t ’ ADI method was used to solve the re su lt in g al geb rai c equat ions.

The system of equations described in Section 4.3.1 was solvpd line by

1ine at constant r ( i . e . fo r j=2, ...NPR=1 and £=1,.. .NER). Since theI

right hand side of the Equation (4.11) is known, the system of equations

in Sect ion 4 .3 .1 can be solved as a one-dimen sion al problem by the method

described in Appendix C. To s ta rt the it e r a ti v e procedure an i n i t i a l

so lu ti on was assumed f o r the en ti re domain. Due to the nature of the

boundary condition along e=a, two different matrices were obtained.

Therefor^ for part 1, one needs to invert only two matrices reqardless

of the number o f ite ra ti o n s . As discussed in'Appendi x C, both the le f t%

hand sid e mat ri ces were block di agonal and were conv erted to a band

s tr u ct u re p r io r to ent er in g the su brout in e GELB. One may use LU

decomposit ion equal ly ef fe ct iv el y. Af t er one ha lf i te ra t io n the

so lu ti on is known fo r j = 2, ...NPR- l but n ot fo r J=1 and NPR.The s o lu ti on

at these points'may be obtained by smoothing, Chang and Finlayson (197.7).

However in this work, old values were used at these points for the

second ha l f of the i t e r a t i o n scheme and no smoothing was performed.

The second half of the iteration scheme (for the system'of equations

described in Sect ion 4 .3 .2) is s imi lar to the f i r s t ha lf except that

onl y one matr ix is in ver ted regardl ess o f the number, of

i te ra ti o n s . The computational scheme is shown in Figure 4. A ft e r the

completion of the second half of the iteration scheme, the velocities

at the point s marked wit h so lid c ir cl e s in Figure 5 are s t i l l not known.

The sol uti on a t these points was obtained using a f i n i t e di ffe re nc e

pr oduc ed with permission of the copyright owner. Furthe r reproduction prohibited without permission.


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29

FIGURE 4.

START

READ (x WR IT E

DATA

SWEEP IN 0 DIRECTION. (CONSTANT r )

SOLVE BLOCK DIAGONAL MA TR IX

I----SWEEP IN

r DIRECTION (CONSTANT 0)

YES

^CONVERGENCE

CALCULATE AVE. VELOCI TY

USE F.D. TO EVALUATE

CORNOR POINTS

SOLVE BLOCK DIAGONAL

MA TR IX

Flow Chart for the Computational Scheme for Finned Tube

Reprodu ced with permission of the copyright owner. Further reproduction prohibited without permission.


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e

6 =

/

FIGURE 5.

f

L

r=0 r=l

I ’

POINTS WHERE FINITE DIFFERENCEMETHOD WAS APPLIED

Location of the Points for Finite Difference Method



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technique. The f i n i t e dif fer en ce development of Equation (4.7 ) is

given in Appendix D.

Once the solution at all the points was known, the procedure

was repea ted t i l l convergence was achi eved . Convergence was

assumed to have been reached when the average v e lo c i ty di d no t change

-4by more than 10 in 50 i t e r a t i o n s . A quad rat ure approach was used to

calculate the average f low velocity.

The ra te o f convergence was found to be s. ver y s tr ong

fun cti on of the it e ra ti o n fa ct or w. The it e ra ti v e procedure becomes

uns tabl e when w is la rge and the rat e o f convergence is ver y slow when

w is sma ll. The optimal or, near optimal value of w was found by

t r ia l and er ro r. In general, the appropri ate value of w increased with■ i < • '

the number of interior collocation points and with the number of

elements. Tabl£ 1 shows tha t w varie d from 2 (f o r NEg = NER = 5,

NPe = NPR = 3) to 1150 ( fo r NEe = NER = 2, NPe, NPR = 8 ). The

optimal valu e o f w was also a fun ct io n o f the number o f f i n s and the

f i n 1 ength. *•

As the i t e r a t i o n parameter i s the re cr ip roca l o f a ti me

^ in te rv al when solvi ng si m il ar .problems with the time de ri va ti ve of

W inc lu ded , a large u> means th at the time step is small and the

number o f the time in te rv al s to reach "steady sta te" is l arg e. Table

1 shows tha t as w is increased the number o f i te ra ti o n s to convergenceo

als o in creas ed . The CPU time per 100 i t e r a t i o n s i s 'shown in Table 1.

The computational time was found to i ncrease fa i r l y ra pi d ly wit h the

number o f co ll o ca ti on p oin ts. For example, fo r two elements in the

e and r di r e c ti o n s , the CPU time per 100 it e r a ti o n s was 1.7 and 22.4

odu ced with permission of the copyright owner. Furth er reproduction prohibited without permission.


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32

seconds fo r NPe•= NPR = 5 and NPe = NPR = 10, r espect ive ly . For t h i s

' $ty pe -o f problem, the ADI method becomes f a i r l y expensive i n terms of

CPU demand f o r cases using a la rge number of c o l lo c a t ion p o in ts . '

4 .5 Calculati on of t he Average V el oc ity :

The average velocity in finned tubes is obtained using the

follo win g expression: >

or

/ “ / Wr dr de j j = 0 r = 0 _____ _

/ “ j r d r dee = 0 r = 0

W > = — j a | y r dr doa 0 0

(4.21

(4.22)

Equation (22) was eval uate d using a quad ratu re approach. The fo ll o w in g

ormula was used to evaluate the average velocity

< W‘ NEe

Ik=l

NP0 . NERI wi I

i= l £= 1

NPP \ k "y ( w. Wk ’ T r.

j = ] J - J (4.23)

The l is t in g o f the weig hting f ac to r w is given in Appendix. B. In te gr at io n

was performed f i r s t in the r- di re ct io n. The res ulti ng\ val ud s were«* \ \

integr ated again in th£ e-d irec tion to complete the integr ation, over

\ xinterboundaries do not affect the average velocity as the quadrature\at

these points is zero.

rodu ced with permission of the copyright owne r. Furth er reproduction prohibited without permission.


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3 3

Ta b l e 1

Summary of Computa t ions f o r theO r t h o g o n a l C o l l o c a t i o n o n F i n i t e E l e m e n t s M e t h o d

L NF NEeNER

NP eNPf

0 . 3 3 2 50 . 3 3 2 60 . 3 3 2 70 . 3 3 2 8

0 . 5 3 2 50 . 5 3 2 60 . 5 3 n£ 70 . 5 3 2 8

0 . 7 3 2 50 .7 3 2 6

"D.7 3 2 70 . 7 3 2 80 .7 3 2 10

0 .4 8 2 50 .4 8 2 60 .4 8 2 70 .4 8 2 .8

0 . 3 3 5 30 . 3 3 5 5

0 . 5 3 5 30 . 5

■63 5 5

0 .7 3 . 5 30 . 7 3 5 5

U CPU( s e c o n d s )

To t a l No .o f

I t e r a t i o n s

40 1 .7 25 070 2 .2 "250

120 6 .3 750185 10 1100

40 1 . 7 - 25070 2 .2 30 0

n o . 6 .3 60 0170 10 50 0

40 1.7' 30080 2 .2 50 0

120 6 . 3 80 0180 10 75 0385 2 2 .4 1600

20 0 1 . 7 25 0500 . 2 . 2 ' 60 0780 6 . 3

1150 10 1000

2 1 .7 10 0250 6 .1 . , 1000

2 1 . 7 10 0300 6.1 • 250

2 1 . 7 10 0300 6.1 ' 200

roduced with pernrission of the copyright owner. Fodder reproduction prohibited without p e n s io n .


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3 4

One qu an tit y which is qu ite useful in' the study of fl u id flow

in f inned tubes is the product o f . f r i c t i o n fa ct or and Reynolds number

(f .R e) . I t is given by the fo ll ow in g expression, Nandakumar and"v

Mas1i ya h (1 9 7 5).

8 A2

f -Re = ■ (4 -24>>

where is the average v e lo c i ty , Ap is the cr o ss -s ec ti o na l area, and

Cp is' the wetted pe rimet er . Here Re is based on- the e qu iv al en t

d ia me te r D .eThe representative flow area and the wetted perimeter are

given by, respectively,-

AjI = tt R2 ^ ) = | R2 ■ (4 .2 5 )

andC' = (2 tt R)(£-)■ + L' = ccR+L ‘ ' (4.26;t" L TT

where Ap = Ap/R 2 (4.27)

Cp = Cp/R ■ (4.28)

andL = L' /R (4 .29 )

Using Equations (4.25) to (4.28) Equation (4.24) becomes

■f-Re = '(»H 7R )a (4'30)

In a case when there is no fin^present (L=0), Equation (4.30) becomes

f.Re = ~ , ■ (4.31)

Equation (4131) is used to check the numerical results presented

in the Section 4.7



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35

4 . 6 Oth er Techniques

4.6 .1 Least Square Matching Technique

The general solution of the Poisson equation representing themomentum equa ti on is wel l known and i s given by

W = bo + + tW n r '+ ^ (a kr_k + bkr k ) cos 0k

-k k 1 - '+ I (c ^ r + d^ r ) si n ek . - jk

U ti li z in g the boundary cond iti ons (4.3c) and the fac't th a t the sol uti on

must be f i n i t e at r= 0 . a more specific solution is,

r 2 M kW = ~~A> + I 3 kr C0S 6k

4 k= 0

The coefficients a^ can be determined by choosing N(=M+1)

poi nts along the flow duct boundaries. Each boundary co ll oc a ti on p oin t

provides one alg ebra ic equation. The re su lt in g N simultaneous

equations can be solved fo r the N co e ff ic ie n ts . However, by. considering more boundary co llo ca tio n points ( f t - y N) than co ef fi ci en ts ,

the over-determined set of algebraic equations can be reduced to a set

"o f (M+l) e quations by a lease square approach wit h a wei ght ing fa ct or

o f un i ty.>

Al tho ugh th i s method was found to c/' very su cce ss ful in the

so lu tio n of fl ow in a r b i t r a r i l y shaped ducts, Ratkowsky and Epstein

(1968) and other complicated flows, Bowen and MasTiyah (1973), the

method fa il ed to give any meaningful f low f ie ld fo r f in lengths greater

than 0.2 . The number.of co e ff ic ie n ts va rie d between 5 and 20. Si mi la r

conclusions were also reached by Soliman and Feingold (1977))

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4.6 .2 F i n ite D ifferen ce Method, F.D.

'M a s i iya h (1 975) has used a F.D. method to solv e Equat ion ( 4 .2 ). '

The momentum equat io n was di scr e ti ze'd 1 using a thr ee -p oin t central

di ff er en ce module. The de ri va ti ve s at the flo w boundaries were

approximated by Newton forward and backward th re e -p o in t formulae . A

successive over-relaxation'method was used with a relaxation factor of

1.7. Sol uti ons f o r grids, o f (11x11), (2^x21) and (41x41) were

obtained.

Convergence for the (11x11) grid was fast and the rate of

convergence was found to decrease rapidly as the number of the grid

poi nts was incre ased. For a gr id of (41x41), the rate of convergence

was so slow f o r the case o f a f i n le n gt h , L = 0.4 and number of fi n s

NF = 8 th at i t was not pos sibl e to as ce rta in whether convergence had

occurred a ft e r a tot al of 7000 it er at io ns .

Table 2 shows the time requirements and the total number

of it e ra ti o n s needed to achieve convergence. In gene ral, convergence

was assumed to have been reached when the average v e lo c i t y d id n o t

-5change by more than 1 0 i n - 50 i t e r a t i o n s .

4. 6. 3 Soliman and Fein gold Approach.

Due to the fa i lu r e of the le as t square matching approach, ,

and to overcome the mixed-type boundary conditions, the flow domain

was divided into two regions separated by a circular arc of

radius (1- L ), Soliman and Feingold (19 77) .General t r i a l functi ons fo r

each region were evaluated . These fun ctio ns s a ti s fi e d the resp ecti ve

region boundary con diti ons . Using the co nt in ui ty of vel oc ity and it s* ,

derivative at the boundary of the two regions at equi-distant collo

cation p oin ts, the constants contained in the tr ia l f low functio ns were



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Table 2

Summary of Comoutations for the Finite Difference Method

CPU time in seconds Total number of it e ra ti o nsper 1 0 0 i te ra tio ns to convergence

NF L 1 1 x11 2 1 x21 41x41 1 1 x11 2 1 x21 41x41

3 0.3 0 . 1 2 0.46 2 . 0 1 0 0 0 3^00 7000

3 0. 5 0 . 1 2 0.46 2 . 0 800 2400 2500

3 0.7 0 . 1 2 0.46 2 . 0 700 3400 . 2600

8 0.4 0 . 1 2 0.46 2 . 0 800 2500 (7000)

N/C no conver gence,

GJ


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evaluated. The number of inter -boun dary co ll oc at io n poi nts varied

between 10 and 20. Soliman and Fe in gold found th a t the average

ve lo ci ty of the flow was w it hi n 1% fo r 10 and 20 co e ff ic ie n ts . The

results using 20 coefficients are given in Table 3.

4.7 Dis cussi on of Results \£/~'yt

In order to compare the re su lt s for. the f l u i d - f l o w in .an-

internally f inned tube, the central velocity and the average velocity■♦

w i ll be used fo r comparison. In orde r to gain confide nce in -t he

numerical r es u lt s, a li m it in g case is considered. When the fi n leng th

is zero the exact value of f. R e .i s 16. 0CFE also- gives a value o f 16.

This shows th at the numerical re su lt s are i n pe rfe ct agreement wi th the

exact solution for the limiting case considered.

As the purpose o f th i s work is no t to st udy the flow in

finned tubess but rath er to study the general a p p li c a b il it y of 0CFE to

obta in so lu tio ns to th is type of problem, only a few flo w cases are . :

considered?and these cases are p rim ar ily dicta ted by the a v a il a b il it y

of r es ul ts from oth er workers. The flo w cases considered are

■ fo r TTF = 3*8 wi th f in leng ths of 0.3, 0.4, 0.5 and 0.7 . NF is the

number o f f i n s . ”

H. Kan (1978) has presented some' re su l t s fo r G0C. Due

to the presence of a di sc on ti n ui ty along 0 = “ (presence of the f i n ) ,

g lobal or thogonal co l loc at ion is o f l im ited app l icat ion . I t is notpossib le to a r b i t r a r i ly se lec t a f in leng th , s ince the t ip o f the f in

must l i e on a co l lo ca ti o n p o in t. When NR is an odd number, the

middle co llo ca tio n point is always at r = 0.5. I t is f or th is

reason t ha t the res ul ts fo r G0C are only given fo r one f i n len gth ,

o duce d with permission o f the copyright owner. Furth er reproduction prohibited without permission.


55/194

39


56/194


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41

21 x 21 I’ DGOC0 .0 3 0 0,09

0 . 0 2 5 0.08

0.07 W< W > 0.020

L = 0.50.015 0.06NF

0 . 0 1 0 0.05

0.140.07

0.13 i/V 0.06< w >

- 0 .50.05

0.04 0.114 5 76 8 9 121 1

NR or N d

FIG URE 6. Variati on of Centre and Average Velocities for L“0.5 wi th Number of Co ll oc at io n Po in ts for GO C



58/194

■ ' ' 2In other words, the g-ap'between the 4th and the 5th collocation points

i s f a i r ly l a rge and the re fo re the ve lo c i ty re so lu t io n i s f a i r ly poor

near the f in t ip . As the f i n t ip pos i t ion i s very c r i t i c a l in

dete rmin ing the ve lo c i ty f i e ld , i t i s not su rp r i s ing tha t the re su l t s

as given by the GOC are poor, although a to ta l of 49 i n t e r i o r

collocation points are used.

The r e s u l t s f o r the QCFE f o r the case of NPe = NPR = 3 and )

5 with.. NEn = NER = 5 ar e given in columns 6 and 7 o f Tatfle 3. As the I

total number of collocation points is increased from 3 to 5, theagreement wi th the " tr u e" so lu ti on improved. The maximum di ff er en ce i s

about 7% ( f o r the case of-W , L = 0. 7) . For NPR = 5, the co ll o c a ti o n

poi nt s are 0, 0.1127, 0.5 , 0.8873 and 1.0 . With Ar 0. 2, t h is means*

th at the i nt er va l "between the second col lo ca ti on po in t and the. th ir d

col loc a t io n point ( j f ') i s 0 .2 (0 .5-0 .1127)=0.07746. A f in i t e

difference method with a 14x14 grid would produce a uniform Ar similar

to th a t near t he f i n t i p f o r the case of NEe = NER = 5 and NPe = NPR = 5.

Comparison o f column 7 wit h columns 8 and 9 of Tabl e 3 shows th a t the

re su lt s o f 0CFE do not f a l l between those given by the f i n i t e

di ff er en ce method fo r the (11x11)' and the (21x21) gr id s. In fa c t the

0CFE re su lt s fa l l below those fo r the (11x11) gr id . This ind ic ate s

th a t the OCFE having uni for m element s ize wi th NEe = NER = 5 and

NPe = NPR = 5 (a t o ta l o f 25x9 c ol lo c a ti o n p oi nt s) is not s ui ta bl e f o r this type of a problem.

Further numerical experimentation was conducted with two

elements of unequal size in r-direction and two elements of equal') ..

size in e -d ir ec ti on , i . e . , NEe = NER = 2. The number o f col lo ca ti on



59/194

points was varied , v iz , NPe = NPR = 5 , 6 , 7 , 8 and 10. *~As the number t.

o.f c o ll o c a ti o n po int s was in cre ase d, the values o f Wc and

approached those given by the fi n i t e di ffe re nc e method wit h a (41x41)

g ri d . Figu res 7 and 8 show the variation of the centre velocity and

the average veloci ty with the number of col locat ion points, respectively.

The val ues o f W and f o r the NF = 3 and L = 0.3 f o r the case o fcNPe = NPR = 8 ( to ta l o f 144 in ter io r co l locat ion po in ts) are c lose to

those give n by a gr id o f about 2 1 x21 using the f in i t e diff ere nce method.

S i m il a r ly , using W and as basi s fo r comparison, fo r NF = 3 and

L = 0.5 and 0.7 , the NPe = N P R = 8 case was found to be equivalent to

a g r id o'f about (15x 15). For the case o f a more number of f i n s , NF = 8 ,

a NPe = NPR = 6 ( to ta l o f 64 in te ri o r co l lo ca t io n points ) was found to

be equiv alen t to the f in i t e diff ere nce scheme of ( l . lx l l ) grid and a

NPe = NPR = 8 was found to be eq uiv ale nt to at le a st a (21x21) gr id .

The re su lt s of Soliman and Feing old are shown in Figures ,7

and 8 fo r compari son. For the., case o f L = 0.7 (NF=3) and L = 0.4 (NF- 8 )

t h e i r r e s u lt s are in good agreement wi th those f o r NPe = NPR = 81s

However, as the fin length is--decreased, Soliman and Feingold results

.become equivalent to those of lower order grid points.



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w

vv.

w .

vv

0.210„L = 0.-3

0.205 - NEC = NE R = 5 NP0 = NPR = 5

- NEC = NE R = 2- ooliman and Feingold- Finite Difference, FD

0 . 1 9 5

41 x 41 gD21 x 21 fD

0.155FD

NF = 30 . 1 4 5

0 . 1 4 0

0 . 1 3 5

21 x 21 FD

0 . 1 3 0

L = 0.4 NF = 8

0 . 1 2 5

0.120

41 x 41 F D f 21 x 21 FD

0 . 0 8 0

0 . 0 7 0 L “ 0.7 NF = 3

X0 . 0 6 0

NPR or NP0-

FIGU RE 7. Variatio n of Centre Velocity with Number* of Coll oca tio n Poin ts for OCF E .



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< W > X

1 0

'

< W > X

1 0

< W > X

1 0

< W > X

1 0

45

NPO = NPR = 5

Soliman and Feingold Finite Difference, FD

41 x 41 FD

0 . 9 9

71 x 21 FD0.98

0 . 9 7

L = 0.3 N F - 3

0 . 9 6

0 . 9 5

0 . 9 4

0 .6 8 41 x 41 FD

0 . 6 7

0.6 6

T1 x 11 FD0 . 6 5

L = 0.5 NF = 30 6 4

0 . 6 3

0 . 6 2

0 . 4 5

21 x 21 FD0 . 4 4

0 . 4 3

L = 0.4 NF = 8

0 . 4 2

0 . 4 0

0 . 4 5

41 x 41 F D21 x 21 FD

0 . 4 3

0 . 4 2L *= 0.7

NF = 3

0 . 4 0

NP0 or NPR

FIGU RE 8. Variation of Average Velocity with Number of Collocation Pointo for 0CFE



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CHAPTER 5

APPLICATION OF THE METHOD OF ORTHOGONAL COLLOCATION ON FINITE ELEMENTS

TO_POROUS MEDIA

« $This chapterihd&nonstrates the applicability of the method of

orthogonal c oll oc ati on on f i n i t e • ements (OCFE) to a f low problem in1

porous media. The equations sim ula tin g mi sc ib le displacement in

porous media were solved usii ig OCFE. Unl ik e the f in ned tube problem,

a di ne ct method of so lu ti on was used to solve the r es ul ti ng alg ebr ai ct

eq uat ions. , •_

The phys ical process considere d here is th a t of a homogenous

porous medium having a very viscous and p r a c t i c a l l y immobile o i l . A

solvent of low- vi sc os ity is in jected into the res erv oir through an

in je ct io n well in order to reduce the vi sc os ity of the o i l . The o il

is assumed to be hig hl y misci ble w it h' th e solve nt and the o il -s o lv e n t

mixture v is co si ty is taken to be a strong fun cti on o f the solve nt

conce ntra tion. As solve nt in je ct io n proceeds, a mixture of oiT and

s( Ive nt is produced through the pro duct ion w e ll ."vV>

5.1 gover nin g Equati ons: .

| The two dimensional incomp ressib le mi sc ib le displacement i n '

a porous media can be described by the following equations, Settari

e t a l . ( l V 6 ) •

' - \ . u = q (5.1 )

and

• V .U Kdt V C) - V .(u C) = 4, | y + q C. n - (5 .2 )

where


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4 7

ve lo ci ty and K_T is the tot al disp ersio n co ef fi ce nt . C. is the sourceDT m

concentration and C is the concentration of the solvent in the o i l -

solve nt mixture , q is the amount of fl u id inj ect ed or produced in

cubic centimeters per cubic centimeter of the formation per second.

T(he vis co si ty of the .o il -s ol ve nt' mi xtu re is assumed to be a

func tion of c onc entr atio n and is given bysS ettar i et al '. (1976).\

\ ':° ' s ( 5. 3)[(i-c\^1/e + C pQ1/e] e

where u and w ' ar e‘ the vi s co s it ie s of the o il and the solven t,Mo ^V \

re sp ec ti ve ly and e is a mi\ in g parameter.\ - V

The two-dimensional\form of'Equations (5.1) and (5.2) can be

wr. i t te n \ respec t ively, as\

3 u.. - A u = q (5.4)ax/. x ay y

and

ax' D ax' x ay' D ay' y

= ♦ ■ + ^ x’’ Cin (5;5)

where Kd Kdt ^

Using Darcy's law the components of the fl u id ve lo ci ty in Eq ua tio n (5.4)

can be replaced by

> „ * - £ I t - ' ( 5 -6)

and v 1u = _ I E . . (5 .7 ) '

y u ay

o duce d with permission of the copyright owner. Furthe r reproduction prohibited without permission.


64/194

where is the pe rm ea bi li ty of the porous medium. The

48

disp ersi on c o e ff ic ie n t Kn is taken as a constant in both dir ec tio ns and

is assumed to be independent of the f l u i d ve lo ci ty .

M ult ip lyi ng equation (5.4 ) by C and using e lation (5.5)

yi elds

3 r iy 3C i 3 r i/ 3C -| 3C 3C _ 3x' -D sx 3y' D. 3y . ux 3x 1 y 3y'

f ^ + q ( x ' , y ’ ) (C ,C) W (5 .8)d i 1 n j

/ •where C. is the source con cen tra tio n. Thi s®conc en tra tio n is•s . in

equal to the input concentration for an injection well and to C(x, y)

fo r a producti on we ll . Consequently the la s t term in Equation (5.8)

disappears for all production wells.

The boundary con diti ons considered are fo r H f^ is ol at e d system.

The geometry o f the porous medium is shown in Figu re 9. The porous medium

is assumed to be a re ct an gl e. Equations (5 .4) and (5. 8) were solved

using the foll ow ing boundary con diti on s, '

| ^ = 0 f o r ( x , y ) e dA ( 5. 9a )d r.i ,

3C 0 ' f o r (x ,y ) E dA (5.9b )v) ■ 3n

m u { ux = Uy = 0 fo r (x ,y ) e dA (5. 9c )

C( x, y, 0) = 0 fo r (x ,y ) e dA (5.9 d)

where dA is the boundary o f A and n is the outward normal to

dA.



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4 9

COLLOCATIONPOINT NUMBER

10 FOR N=>214 FOR N=318 FO R N=“422 FO R N=5

y=0.5

COLLOCATION POINT NUMBER

• 35 Ft)R - n =2

M F°R j

' ^ 2 9 . '

*52 0

* i .v6.

o

r COLLOCATION POINT NUMBER

6 FOR N°2 8 FOR N=*3

10 FOR N=4 I 12 FO R N=5

x*=0. 5

0

COLLOCATION POINT NUMBER

31 FOR N=2 5 9 F OR N g3 1 ^

095139

FOR FOR

N=4 N=5

CONFIGURATION ONE

CONFIGURATION TWO

• REFERENCE WELL

® PR OD UC TI ON WE LL

O INJECTION WELL

FIG UR E 9*‘ Geo met ry of the Porous Mediu m

oduced with permission o f the copyright owner. Furth er reproduction prohibited without permission.


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50

5.2 -

For computational purposes a quarter of a five-spot was

cons ider ed. Only two elements of equal siz e were used in each d ir e c t io n .The number of interior collocation points was varied from 2 to 5.

Five in t e r io r c ol lo ca ti on points correspond to a to ta l of 160 unknowns.

The numbering scheme f o r N=2,3,4 and 5 is shown in Figure s 10 to 13,

number,since they did not appear i n OCFE fo rm ul at io n o f the equati ons.

ide n t i f i e d as being the f i r s t in t e r io r co l loc a t ion po in t in the inc reasing

di re ct io n of both x and y. Two ■diff erent-' l oc at io ns of the production

wel l were consid ere d. In one scheme the pr od uc ti on wel l was loca ted

di ag on al ly opposite to the in je c ti on w el l. s!£n the oth er scheme, thev

production well was such that the injection and the production wells were

symmetric to the li ne x = 0 .5 . (In th is si tu at io n the geometry does not

repr esent a fi v e - s p o t ) . At any time, onl y one prod ucti on and one

injection well were considered, however, the computer program can

handle any number of production and injection wells.

5.3 OCFE For mulat ion o f the Governing Equa tions:

5.3.1 Continuity Equation:

For a porous medium of con stan t pe rm ea bi li ty , Equation (5. 10)

becomes

re sp ec tiv ely . The co rn er, poi nts o f the blocks were not assigned any

As shown in Fi gures 10 to 13, the i n j e c t i o n we ll can always be

Consi deri ng Equation (5. 4) and using Darcy' s law, one obt ain s

(5.10)

K x 14.696764P

3 . (5.11)

od uced with permission of the copyright owner. Further reproduction prohibited without permission.


67/194

5 1

a

ny-1

26

20

24

23x - 0 x -1X

# REFERENCE WELL

® PRODUCTI ON WE LL

* O INJECTION w e l l

FIG URE 10. Numbe ring of Unkn owns for N**2

o duced with permission of the copyright owner. Further reproduction prohibited without permission.


68/194

52 j *

•y-i

14

22

30 5 4 6 3

20y

10 43

60

41 .

4 9y- 0x*>0 X“ 1

^ i • REFERENCE WELL^ 2 ; ® PRODUCTION WE LL

O INJECTION WELL

0 : • - ■$ . '■ _

.FIGURE IT. Numb erin g of Unknown s for N“3



69/194

53

y-i

y=0

8- < ___ __ - 2 9 .4 0 111----Lo_

is_

i3

37

70

o :

81

wj 10/f 0 7 in i. A

7 1 7 28 39 10 69 80 91 1n?

6 16 27 38 19 68 79 on i o i"

5 15 26 37 18 67 78 89 100

14 25 36 17 66 77 88 904 13 24 35 16 36 65 76 87 98

3 12 ‘ 2 3 • 34 15_

44

3J5_

34

64 75 86 97

2 11 22 33 63 1L 85 96

1 r U P. . . V :'f a p .

m j r . 32 , 43 33 62 73

t.

84 95 .

JL _ 20 31 +2 61 72

--------------- 6

83 9ii‘ — -

• REFERENCE WELL

® PRODUCTION WELL

O INJECTION WELL

112

111

110

109

108

107

106

105

FIG URE 12. .Numbering of Unknowns' for N“4

pro duc ed with permission of the copyright owner. Furthe r reproduction prohibited without permission.


70/194

54

y»0 11112 3 p. 36 60

48 15960 - 73 122109

.4 6 108.; 121 47 158

94 107 110 . 157

44 1195 7- 70 156118117 155

.11 115ULQ2 153

26 101 ~ -4C 152

100 11324

x-0

• reference well

© production well

O ■ i n j e c t i o n WELL

Cl

FIGU RE 13. 'Numbering of'Unkn owns for N“5

p rodu ced with permission of the copyright owner. Furthe r reproduction prohibited without permission.


71/194

The constant 14.696764 is for dimensional consistency

Let x 1 - xL (5.1 2)

and y ' = yW - (5.13)

Here L and Ware the length and the width of the formation4

und^r consideration and x and y are the dimension!ess coordinates.

| Su bs titu tin g the dimensionless qu an titi es in Equation (5.10)

and rea rran gin g,o ne obta ins . •I

li e . _ ik. M + Z 2 r alp _ ip IE.] = 9L k V ( 5 14 vu 3x2 3x 9x - LP 9 y 2 9y 9yJ Kp* 1 4 . 696764

where Z = L/W • ' (5.15)

Ir> Equatio n (5 .14 ) on ly x, y and Z are dimension l es s.

Two new variables g and v are defined such that

x-x,I

■

^ = ax T (5-16a)kand

y ~ y i V =

Ay£(5.16b)

where Axfc = x ^ - x ^

and Ay£ = y t +r y £

Ap plyi ng the method o f orthog on al c o l lo c a t io n on f i n i t eelements to Equation (5.1), and using the new independent-variables g and

v as defined by Equation (5.16), one obtains,



72/194

■a * 5 6

f t •v

Ax

• NPXr ' Ik n=l

!. Pk , f 'i . n n , j

+ V - p Ay 2

NPY

fc n=lT b . p*l . i n t

AX

k,£

NPXI A

k n=l

k, £pi . n n , j I AX.L k

NPY

J . n i ,n Ay 2£

y ’ a . pL _ . t r n=l

k, £J.n "i ,n

q (x i , y j) L2 p 2K * 14.696764

P

N"X pk , , . i

" 4 ■n“ 1 ,

f NP YI A. P?

n=l J . n i .n j k . £

(5.17)

where

l

jk£

2 ,.

2 ,.1 ,

1 ,

. . , NPX-1

.. , NPY-1

. . ,NEX

. . ,NEY

(5.18)

NEX and NE

application of the method of orthogonal collocation on finite elements

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