Flow Geometries

7/26/2019 Flow Geometries

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FLOW GEOMETRIES & DIMENSIONS

1.RECTANGULAR FLOW GEOMETRY

2. RADIAL CYLINDRICAL FLOW

GEOMETRY

3. ELLIPTICAL CYLINDRICAL FLOW

GEOMETRY

4. SPHERICAL FLOW GEOMETRY

5.CURVILINEAR FLOW GEOMETRY


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Flow Geometries and Dimensions

The same factors that dctate o!r choce ofcoord"ate s#stems $%a# a dom"a"t ro%e "o!r decd"& ho' ma"# dme"so"s to ass&"to a $ro(%em)

E"&"eer"&* to a %ar&e e+te"t* is a marriagebetween pure science and practical

reality) I" other 'ords* se%ect"& a h&her"!m(er of dme"so"s to re$rese"t a s#stemma# (e scientificallycorrect* (!t 'e ma# %ac,the "formato" or the com$!tato"a%o-erhead "eeded to ass&" ths ma"#

dme"so"s) So* 'e ass&" fe'er dme"so"sa"d sett%e for a %ess.tha".dea% $ro(%emdef"to") S!ch com$romse ma# seemdrastc) /!t " rea%t#* for most e"&"eer"&$ro(%ems* 'e ca" &e"erate a" ade0!ateamo!"t of "formato" e-e" 'th" these

%mtato"s)


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123RECTANGULAR FLOW GEOMETRY

Rectangular geometrys the o"e that s mostfam%ar to !s* as e"&"eers* dat"& (ac, too!r h&h schoo% ca%c!%!s) In reservoirmodeling we often use this familiarity toour advantage, since most field-scalemulti-well studies are done in this co-

ordinate system.

AsF&!re (e%o' %%!strates* 'e ca" co"sder

the reser-or to (e a recta"&!%ar (o+ 'th thef%!d $artc%es mo-"& " stra&ht %"es*

perhaps at different speeds in differentdirections and locations.
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Figure 1

I" ths case* the stream%"es are $ara%%e% tothe three $r"c$a% a+es 1+* #* a"d 43* 'hchare ortho&o"a%)


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F&!re a(o-ea%so sho's the $artto" of the(o+ "to ma"# sma%%er (o+es* 'hch arerecta"&!%ar $rsms) Each of these recta"&!%ar

$rsms re$rese"ts a certa" $orto" of thereser-or* a(o!t 'hch 'e ca" $roc!re"formato" thro!&h sm!%ato" st!des) We

use this smaller element of dimensions, (x,

y, z) as a control volume to set up and

discretize the governing equations.

NOTE5

We should emphasize that a fluid particleentering an elemental volume in onedirection does not necessarily exit in thesame direction; by the same token, thefluid particle leaving the elementalvolume in one direction did notnecessarily enter it in the same direction.


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Formato" characterstcs* 1s!ch ashetero&e"et#* $ermea(%t# co"trasts a"d theforce fe%ds m$osed (# the co"dto"s at the

(o!"dares3 dictate the flow path once theelement enters the control volume)Ths s theesse"ce of f%o' m!%t.dme"so"a%t#)

F&!re (e%o' %%!strates the co"ce$t of o"e.

dme"so"a% f%o' a%o"& the +.drecto")A%tho!&h t s dffc!%t to f"d rea%.'or%de+am$%es of tr!%# o"e.dme"so"a% f%o'* ma"#t#$es of a"a%#ses a"d s#stems do %e"dthemse%-es to descr$to" as o"e.dme"so"a%)


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The f%o' str!ct!re sho'" " F&!re a(o-e

$rec%!des 1for(ds3 f%o' " a"# other drecto"*which implies there is no propertyvariation along the y and z directions.

Therefore* f 'e ta,e a secto" $er$e"dc!%ar

to the "dcated f%o' drecto"* there '%% "ot(e a"# $ro$ert# -arato" across the $%a"e)

Sm%ar%#* a"# cross secto" ta,e" " the +.4 or+.# $%a"es 1$ara%%e% to the stream%"es3 '%%

re-ea% the !"formt# of the f%o' str!ct!re)

More e+$%ct%#* the $ress!re $rof%es of f%o'$aths '%% (e sm%ar) A %o"&* s,""# reser-orthat s co"f"ed (et'ee" t'o c%ose%# s$aced*

$ara%%e% fa!%ts fts ths descr$to")


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The "e+t %e-e% of descr$to" !sed " reser-or

sm!%ato" s t'o.dme"so"a% f%o') Ma"#reser-or sm!%ato" st!des em$%o# t'o.dme"so"a% Cartesa" coord"ate s#stems)Ths ma,es se"se 'he" 'e co"sder the %ar&e%atera% e+te"t of most reser-ors com$ared'th ther thc,"esses) F&!re (e%o'%%!stratesa t'o.dme"so"a% f%o' str!ct!re a%o"& the +a"d # drecto"s)

Ths $rec%!des f%o' " the 4.drecto")


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Therefore* a"# s%ce ta,e" $ara%%e% to the +.#$%a"e '%% "ot sho' a"# -arato" " terms of$ro$ert# a"d f%!d dstr(!to" s!ch as

$orost#* $ermea(%t# a"d sat!rato"s)

The "trod!cto" of the seco"d dme"so"a%%o's !s to descr(e a 'de -aret# of$ro(%ems) We ca"* for "sta"ce* acco!"t for

drecto"a% $ermea(%t# 1HT* 6ef"e 73-arato" a"d %atera% 1drected to a sde3 'e%%dstr(!to"s)

Moreo-er* a t'o.dme"so"a% a$$roach a%%o's

!s to re$rese"t a -aret# of 'e%% com$%eto"strate&es 1e)&)* -ertca% 'e%%s* hor4o"ta%'e%%s* stm!%ated 'e%%s3) Th"* (%a",et sa"dsthat te"d to ds$%a# %ar&e area% co-era&e aredea%%# s!ted for descr$to" (# a t'o.

dme"so"a% mode%)


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The (est re$rese"tato" of f%o' s the three.dme"so"a% mode%* (eca!se t a%%o's !s to$roc!re the most "formato" a(o!t the

reser-or)

U"fort!"ate%#* t a%so re0!res the %ar&estamo!"t of "$!t "formato" a"d a h&her%e-e% of com$!tato"a% $o'er a"d o-erhead)

St%%* "cor$orat"& a thrd dme"so" &-es !sthe latitude 'e "eed to "c%!de a%% the$ro$ert# -arato"s " a%% three s$ata%drecto"s)

Ths mea"s that f 'e ta,e t'o $ara%%e% s%ces$er$e"dc!%ar to the thrd dme"so"* the# '%%e+h(t $ro$ert# a"d f%o' dffere"ces) F&!re

(e%o' %%!strates a three.dme"so"a% f%o'str!ct!re)
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Figure 4

A three.dme"so"a% re$rese"tato" a%%o's !sto accommodate a 'de -aret# of $ro(%ems of$ractca% "terest* s!ch as %a#ered reser-ors1'th or 'tho!t crossf%o'3* $arta%%#


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$e"etrat"& 'e%%s* m!%t.%a#ered $rod!cto"schemes* a"d thc, reser-ors 'here&ra-tato"a% forces co!%d (e s&"fca"t)

A three.dme"so"a% mode% ma,es t $oss(%efor !s to come !$ 'th more rea%stcre$rese"tato"s of dr-e mecha"sms 1or a"#com("ato" thereof3* s!ch as &as ca$

e+$a"so" 173* (ottom 'ater dr-e 173* a"d soforth)16ef) of Gas ca$ e+$a"so"3Process of reservoir-liquids displacement by the natural expansion of the reservoir gas cap to ll thevoids vacated by recovered liquids.

I" s$te of a three.dme"so"a% mode%8s ma"#ad-a"ta&es* t s %ess ofte" !sed " $ractcetha" 'e m&ht e+$ect)

Ths s (eca!se we have to weigh suchfactors as cost, data availability andmarginal utility)


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RECTANGULAR FLOW GEOMETRY AN6 VOLUME ELEMENT

6ETAILS 5 1a3 26 FLOW* + 6IRECTION ONLY

1(3 :6 FLOW* + AN6 # 6IRECTION ONLY

1c3 ;6 FLOW* +* # AN6 4 6IRECTIONS


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1:3 RA6IAL < CYLIN6RICAL FLOW GEOMETRY

The rada%.c#%"drca% coord"ate s#stem s$artc!%ar%# a$$ea%"& for descr("& s"&%e.

'e%% $ro(%ems)F&!re (e%o'sho's the $r"c$a% drecto"s ofths f%o' &eometr# a"d ts e%eme"ta% -o%!me)

Figure


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The three $r"c$a% f%o' drecto"s are rada%

1r3* -ertca% 143 a"d ta"&e"ta% 13)

To -s!a%4e ths f%o' str!ct!re* ma&"e as"&%e 'e%% %ocated " the ce"ter of a crc!%arreser-or* s!ch that the wellbore and the

reservoir boundary are two concentriccircles)

If 'e ass!me a reser-or of !"formthc,"ess* the" the system becomes two

concentric cylinders of the same height)

A $artc%e mo-"& " a three.dme"so"a%rada%.c#%"drca% f%o' &eometr# ca" (e%%!strated as " F&!re (e%o')


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Figure !

A typical onedimensional, radialcylindricalflow model s the c%assca% re$rese"tato"


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!sed " 'e%% test a"a%#ss) I" ths case* f%o' sco"stra"ed to the r.drecto" s!ch thatstream%"es are ra#s co"-er&"& to'ards the

ce"ter of the 'e%% 1F&!re (e%o'3)

Figure "

St!d#"& the $ro(%em a%o"& o"e tra=ector# s


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s!ffce"t (eca!se of s#mmetr#) A"# $artc%e%ocated o" a"# of the tra=ectores '%%e+$ere"ce sm%ar forces)

/# "e&%ect"& the f%o' " a"&!%ar 1>3 a"d a+a%143 drecto"s* 'e "trod!ce a seres ofass!m$to"s* s!ch as "o $ermea(%t#&radato" a%o"& the >.drecto" a"d "o

&ra-tato"a% effect a%o"& the 4.drecto")

As 'e ca" ma&"e from %oo,"& at F&!re *o"e.dme"so"a% f%o' re$rese"tato"s " the>. a"d 4.drecto"s have no practical

significance in reservoir studies)

The t'o.dme"so"a% 1r.43 re$rese"tato" sappealing for single-well problems where

gravity andor layering effects ares&"fca"t 1F&!re (e%o'3)

Ths r.4 $%a"e ca" (e ta,e" at a"# > %ocato"'tho!t cha"&"& the $ro(%em (eca!se of tsa+s.s#mmetrc "at!re)
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Figure #

The three.dme"so"a% f%o' str!ct!re "rada%.c#%"drca% coord"ate s#stem admits

property variation in all three directions)F&!re (e%o'sho's ths s#stem)


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Figure $


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CYLIN6RICAL FLOW GEOMETRY AN6 VOLUME ELEMENTS 1a3 26 FLOW* r6IR) ONLY 1(3 :6 FLOW* r AN6 THETA 6IR) ONLY 1C3 ;6 FLOW* r* THETA ? @

6IRS)


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ELLIPTICAL < CYLIN6RICAL FLOW GEOMETRYWe sometmes !se e%%$tca%.c#%"drca% f%o'&eometr# " s"&%e.'e%% st!des 'he" a stro"&

$ermea(%t# co"trast e+sts " t'o $r"c$a%drecto"s o" the %atera% $%a"e)

A"other common application of thscoord"ate s#stem s when a vertical well isintercepted by a vertical, high-conductivity fracture 1theoretca%%#$res!med to (e of "f"te co"d!ct-t#3)

U"der these co"dto"s* the "orma%%#co"ce"trc equipotential contoursdegenerate into confocal ellipses)Sm%ar%#* the streamlines become

distorted into confocal hyperbolas)

F&!re (e%o' de$cts ths f%o' str!ct!re)


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Figure 1%


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!"#$%&'() *)+W $+$%/

A%tho!&h "ot commo"%# !sed for &e"era%sm!%ato"* the s$herca% coord"ate s#stem$ro-des a &ood re$rese"tato" of somes$ecfc reser-or e"&"eer"& $ro(%ems)

T'o e+am$%es are $arta% $e"etrato" to a

thc, formato" (# a $rod!cto" 'e%%* a"d f%o'aro!"d $erforato"s)

The $r"c$a% f%o' drecto"s " s$herca%

coord"ates are rada% 1r3* ta"&e"ta% 13 a"d

a4m!tha% 13* as sho'" " F&!re (e%o' )


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Figure 11


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'0%1&)&2$(% *)+W $+$%/

The most &e"era%4ed coord"ate s#stem sc!r-%"ear) I" fact* a%% of the coord"ate

s#stems 1$re-o!s%# dsc!ssed3 co"stt!te as!(set of the c!r-%"ear s#stem)

A c!r-%"ear coord"ate s#stem a%%o's a(etter re$rese"tato" of the f%o' &eometr#* as

'e%% as the (o!"dar# &eometr# where thelatter dictates the former)

Wth the f%o' &eometr# more acc!rate%#re$rese"ted* the results obtained with acurvilinear coordinate system do not getdistorted by grid orientation effects* asofte" ha$$e"s 'th other coord"ate s#stems)


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A"other ad-a"ta&e s that c!r-%"ear s#stemsma# he%$ red!ce the "!m(er of &rd (%oc,s"eeded for the same %e-e% of acc!rac#)

F&!re (e%o'sho's the area% m$%eme"tato"of c!r-%"ear coord"ates to a f-e.s$ot

"=ecto"$rod!cto" $atter")

Figure 1


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Note that the stream%"es a"d e0!$ote"ta%co"to!rs def"e the c!r-%"ear e%eme"ta%-o%!me)

Although curvilinear coordinate systems offerattractive advantages, their use is limited!ecause of the added mathematical andinterpretational complexity they introduce)

Choos"& the a$$ro$rate coord"ate s#stema"d "!m(er of dme"so"s s "ot o"%#$aramo!"t to a sm!%ato" st!d#8s s!ccess*(!t a%so to ts re%at-e sm$%ct#) It s th!s

esse"ta% that 'e !se so!"d e"&"eer"&=!d&me"t a"d $erform thoro!&h a"a%#sesthro!&ho!t ths $rocess)


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We m!st a"s'er 0!esto"s $erta""& to thereser-or8s a$$ro+mate &eometr#* $oss(%edr-e mecha"sms* 'e%% a"d com$%eto"

co"f&!rato"s* %e-e% of deta% re0!red* t#$ea"d amo!"t of data a-a%a(%e* a"d so o")

As far as reser-or sm!%ato" s co"cer"ed*(&&er s "ot "ecessar%# (etter) We m!st

e+ercse &ood e"&"eer"& =!d&me"t "esta(%sh"& the sco$e of o!r st!d#)

We "eed to a-od o-er,%%* (!t at the sametme* !"dersta"d that !"der.re$rese"t"& the"eeded deta%s s da"&ero!s) Sm$%# $!t* 'em!st str,e a (a%a"ce)

33333333333333333333


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Comparison between Cartesian and

Curvilinear grids

Comparison between Cartesian and Curvilinear

grids shows that in Cartesian grid cells are

wasted in dealing with objects. The distribution

of function is very fine in curvilinear grid. Theresources required in curvilinear grids are less

as compared to Cartesian grids thus saving lots

of memory. Therefore we can say that coarse

grids are able to capture flow details efficiently.

Disadvantages of Curvilinear grids

Difficulties associated with the curvilinear gridsare related to equations.

While in Cartesian system the equation can be

solved easily with less difficulty but in


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curvilinear coordinate system it is difficult to

solve the complex equations. Difference

between various techniques lies in the fact thatwhat type of grid arrangement is required and

the dependent variable that is required in

momentum equation. To generate meshes so that

it includes all the geometrical features mapping

is very important. In mapping hysicalgeometry is mapped with computational

geometry. There are difficulties which we face

in generating the body fitted grids in geometries

li!e IC engine combustion chamber. "or

example the #alve mapping in Internal

Combustion $ngine is done very carefully so

that the region of one type is mapped carefully

with another type of regions. There are regions

where dense mesh is done deliberately to

accommodate complex features. %ut this results

in unnecessary grid resolution which leads to

local variation of solution domain.
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Flow Geometries