1501.Applied Parameter Estimation for Chemical Engineers (Chemical Industries) by Peter Englezos...

8/22/2019 1501.Applied Parameter Estimation for Chemical Engineers (Chemical Industries) by Peter Englezos [Cap18]

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18Parameter Estimation in PetroleumEngineering

Parameter est imat ion is r o u t i n e ly used in many areas of petroleum engi-neering. In this chapter , we present several such applicat ions. First , w e d e m o n -strate how m u l t i p l e l inear regression i s employed to est imate parameters for adri l l ing penetrat ion rate mode l . Second, w e u s e kinet ic data to est imate parametersfrom s i mp le mod els tha t are used to describe the complex kinet ics of b i t u m e n lowtemperature oxidat ion and h igh temperature cracking react ions of Alberta oi lsands . F in a l l y , w e descr ibe an appl ica t ion of the G a u s s - N e w t o n m e t h o d t o PDEsystems. In par t icu lar we present the development of an eff ic ient automat ic his torym a tc h in g s i m u l a t o r fo r reservoir e n g i n e e r i n g ana l y s i s .

18.1 MODELING O F D R IL L IN G R A T E USING C A N A D I A NO F F S H O R E W E L L DATAOffshore dr i l l ing costs m ay exceed s imi lar land operat ions by 30 to 40 t imesan d hence it is impor tant to be able to m i n i m i z e the overall dr i l l ing t ime . This isaccompl i shed th rough mathemat ica l model ing of the dr i l l i ng penetrat ion rate an d

operation (Wee and K alogerakis , 1989).T h e processes i nvo l ved in rotary dr i l l ing are complex and our cur rent under -s tanding is far from complete . Nonethe less , a bas ic unders tanding has c o m e fromfield an d laboratory exper ience over the years . The most comprehens ive model i st he one deve loped by Bour goyne and Young (1986) that relates th e penetrat ionrate (dD/dt) to eight process variables . The model i s t rans formably l inear with

353

Copyright 2001 by Taylor & Francis Group, LLC


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354 Chap ter 18

respect to i ts eight ad jus tab le parameters. The model for the penetra t ion rate isg iven b y th e e xp o n e n t i a l re la t ionsh ip ,d Ddt = exp Xa.ixJ.1=2

w h i c h can be t ransformed to a l inear m o d e l b y taking natura l logar i thms from bothsides to y i e ld

T he explanat ion o f each dr i l l ing parameter (aj) re lated to the correspondingdri l l ing variable ( X j ) is given b e l o w :a\ fo rmat ion s t rength constanta2 normal compact ion t rend constanta3 un d e rc o mp a c t i o n constanta4 pressure differential constanta s b it we i g h t constanta $ rotary speed cons tan ta 7 tooth wear constanta8 hydrau l i c s cons tan t

T h e a b o v e m ode l is referred to as the Bourgoyne-Young model. After carefulm a n ip u la t i o n of a set of raw dr i l l ing data for a given format ion type , a set of p e n e -tration rate data is u s u a l l y obta ined . T h e dr i l l ing variables are also measured an dth e measu r ement s b e c o m e part of the raw data set.T he object ive of the regressionis then to estimate the parameters a } b y match ing th e m o d e l to the dr i l l ing penetra-t ion data.Parameter ai in Equat ion 18.1 accounts for the l u m p e d effects o f factorsother than those descr ibed by the dri l l ing variables x 2 , x 3 , . . . ,x 8 . Hence, i t s value isexpected to be different from w e l l to w e l l whereas th e parameter va l ues fora2 ,a 3 ,..,a8 are expected to be s imi la r . T h u s , w h e n data from tw o wel l s (wel l A a n dw e l l B) are s im u l t a n e o u s ly analyzed, th e m o d e l takes th e form

+ a9x9 +a, 0x, 0 (18.3)

wh e re th e f o l l o w i n g addi t ional def in i t ions a p p l y .Copyright 2001 by Taylor & Francis Group, LLC


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Parameter Estimation in Petroleum Engineering 355

a 9 fo r ma t i o n strength c o n s ta n t ( pa ramete r a { ) fo r w e l l Aa 10 format ion s t rength cons tan t ( pa ramete r a , ) fo r w e l l Bx 9 =1 for data from wel l A and 0 for all other datax ] 0 =1 for data from w e l l B and 0 for a l l other dataT he above model is referred to as the Extended Bourgoyne-Young model.Simi la r ly , ana l y s i s o f combined data from mor e than two wel ls i s s t ra ightforwardand it can be d o n e by adding a new variable and parameter for each addi t ionalwe l l . A large a m o u n t o f field data from th e dr i l l ing process are needed to re l iablyestimate all m o d e l parameters .W e e a n d K al ogerak i s (1989) h a v e also cons idered th e s i m p l e three-parame t e r model g i v e n n e x t

dD , . = exp(a\ + a 5 x 5 + a 6 x 6 Jdt (1 8 .4 )

18.1.1 Application to Canadian Offshore Well DataWee and Kalogerakis (1989) tested th e above models us ing Canadian off-shore w e l l penetra t ion data (o f f shore dri l l operated b y H u s k y Oil). Co n s id e r a b le

effort w as re q u i re d to conver t the raw data into a set of data su i tab le for regress ion .The comple te dataset is g iven in the above reference.3.0

0.00.0 0,5 1.0 1.5 2.0 2.5 3.0ln(Predicted Rate)

Figure 18.1 Observed versus calculated penetration rate and 95% confidence inter-vals for we ll A using the Bourgoyne-Young model [reprinted from th eJournal of Canadian Petroleum Technology with pe rmission].



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356 Chapter 18

3.0

0.00.0 0.5 1.0 1,5 2.0 2.5 3.0

ln{Predicted Rate)Figure 18.2 Observed versus calculated penetrat ion rate and 95% confidence in -tervals for well B using th e Bourgoyne-Young model [reprintedfrom

the Journal of Canadian Petroleum Technology with permission].

Figure 18.3

0.00.0 0.5 1.0 1.5 2.0 2.5 3.0in(Predicted Rate)

Observed versus calculated penetration rate and 95 % confidence in-tervals fo r well for well A using th e 5-parameter model [reprinted fromthe Journal of Canadian Petroleum Technology with permission].



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3.0

Figure 18.4

0.00.0 0.5 1 .0 1.5 2.0 2.5 3.0ln{Predicted Rate)

Observed versus calculated penetra t ion rate and 95 % confidence inter-vals fo r well fo r well B using th e 5-parameter model [reprinted from th eJournal of Canadian Petroleum Technology with pe rmission].

A s expected, the authors found that the eight-parameter mode l w as suff i -cient to m o d e l th e data; how ever , they ques t ioned th e need to have eight parame-ters. F i g ure 18.1 s h o w s a p lo t of the logari thm of the observed penetrat ion rateversus th e l ogar i thm of the calculated rate us ing the Bour goyne-Young mode l .The 95 % conf idence intervals are a lso show n . The resul ts for w e l l B are s h o w n inF i g u r e 18.2.R e s u l t s w i t h th e th r e e p a r a m e te r m o d e l s h o w e d th a t the f i t was p o o re r t h a nthat obtained b y t h e B o u r g o y n e - Y o u n g mod el . In addi t ion th e dispersion abou t th e45 degree l ine w as mor e signif icant . The au thors conc luded tha t even though theB o u r g o y n e - Y o u n g m o d e l gave good resul ts i t was w o r t h w h i l e to e l imina te possi -b le re d un d a n t parameters . Th i s w o u l d reduce th e data requi rements . Indeed , byus ing appropriate statistical procedures i t was demonstrated that for the data ex-amined f ive out of the eight parameters were adequate to calculate the penetrat ionrate and match th e data suff ic ient ly well . The f ive parameters were a \, a2 , a4 , a f t anda 7 and the correspondingfive-parameter model is g iv e n b y

dD~ d T = exp(a l + a 2 X 2 + a 4 x 4 + a 6 x 6 + a 6 x 6 ) (18.5)

Figures 18.3 and 1 8 . 4 show th e observed versus calculated penetrat ion ratesfo r w el l s A and B us ing the five-parameter model . As seen the resul ts have n o t



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358 Chapter 18

c h a n g e d s i g n i f i c a n t l y b y e l i m i n a t i n g th ree of the parameters (a3, a 5 an d a8). T h ee l im in a t i o n o f each p a r a m e te r w as d o n e s e q u e n t i a l l y t h ro u g h h y p o t h e s i s tes t ing .O b v i o u s l y , th e fact tha t o n l y t h e a b o v e f ive var iab le s affect s ignif icant ly th e p e n e -trat ion rate means that th e three-parameter model i s ac tua l ly i nadequa te eventhough i t might be able to fit the data to some extent .

18.2 MOD EL IN G OF B I T U M E N OXI DATI ON AND C R A C K I N GKINETICS USING DATA FROM ALBE R TA OIL SANDSIn th e labora tory o f Pr o fe s so r R.G. M o o r e at the U n i v e r s i t y o f C al gary , k i-ne t ic data were ob ta ined u s i n g b i t u m e n s a m p l e s of the "North B o d o an d A th a b a s c ao il sands of nor thern Alberta. Low temperature oxidation data were taken at 50,75, 100, 125 and 150"C w h e r e a s th e h i g h t e m p e r a tu r e t h e r m a l crack ing data at

360, 39 7 a n d 420"C.P re l i m i n a ry w o rk s h o w e d th a t f i rs t order react ion m o d e l s are a d e q u a te fo rth e descr ipt ion of these p h e n o m e n a even t h o ug h th e actual reaction me c h a n i s msare ext remely complex and hence di f f icu l t to determine . Thi s s impli f ica t ion is adesired feature of the mo dels s ince such s impl e m o d e l s are to be used in numer ica ls imula tors of in situ c o mb us t i o n processes . The bi tumen is d iv ided in to five maj orpseudo-componen t s : coke (COK) , aspha l tene (ASP) , heavy o i l (HO), l ight o il(LO) and gas (GAS). These pseudo-componen t s w e r e l u m p e d together as neededto produce two, three and f o u r c o m p o n e n t models . Two, three an d fo ur -c o mp o n e n tmo d e l s were considered to descr ibe these compl ica ted react ions (Ha n s o n and K a-logerakis , 1984).

18.2 .1 Two-Component ModelsI n t h i s class o f models , th e f ive b i tumen pseudo-componen t s are l u m p e d

into two in an effor t to descr ibe th e f o l l o w i n g react ion

(18.6)w her e R (reactant) and P (product ) are the two l u m p e d pseudo-componen t s . Of a l lposs ib le combina t ions , the four mathemat ica l models that are of interest here areshown in Tab le 18.1. A n A r r h e n i u s t ype t empera ture dependence of the u n k n o w nrate cons tan t s is a lways assu med.



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Table 18.1 Bitumen Oxidation a nd Cracking: Formulation of Two-ComponentModels

Modelr\

D

E

Reactant

r\\.j\\^\j

R-HO+LO+ASP

R-HO+LO+ASP

Producti V _.Wr v i .r~YiJi

I V^-wrv ' n-ji

UUK

Model OD Ed C p ( i c R ) k A x / ~ E A ldt K A ( RT j

d C p = ( l - C R ) NBkBe J - EBldt K U ( R T JdCp-(l CR)kDe;J~ED)dt ^ RT J

d C p = ( l - C R ) NH kB/~EEldt I R T J

In Tab le 18.1 th e f o l l o w i n g variables are used in the m odel equat ions :C P Product concentrat ion (weight %)C R Reac tan t c o n c e n t r a t i o n (weight %)k j React ion rate cons tan t in m o d e l j = A , B , D , EE J Energy of activation in mode l j =A,B,D,E .N B E xp o n e n t used in mode l BN E Ex ponen t u sed in model E

18.2.2 Three-Component ModelsB y l u m p i n g pseudo-components , we can formulate five t h r ee -componen tmode l s o f interest. Pseudo-componen t s show n together in a circle are treated aso n e p s e u d o - c o m p o n e n t for the cor r espond ing kinet ic mo d e l .

Figure 18.5 Schematic of reaction network for model F.Copyright 2001 by Taylor & Francis Group, LLC


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360 Chapter 18

Model F is depicted schemat ica l ly in F i g u r e 18.5 and the correspondingmathemat ica l model is g iven by the f o l lo w in g tw o O D E s :dC--

dt-CASP -CC OK)-k2CA S P (18.7a)

dt = k3 (l-CA S P -CC OK) + k2CASP (18.7b)

Figure 18.6 Schematic of reaction nehvorkfor model G.

Model G is depicted schematical ly in F igu re 18.6 and the corresponding mathe-matical mode l i s given b y the f o l lo w in g tw o ODE s:dC A S P

dt = - k , CASP (18.8a)

dCr,O Kdt -k2 ( l -C A S P - (18.8b)

Figure 18.7 Schematic of reaction nehvorkfor model I.



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Model I is depicted schemat ica l ly in F igure 18.7 and the corresponding ma t h e -mat ica l mod el i s g iven by the f o l l o w i n g tw o O D E s :dC A S P

dt ~kl ( l ~C A S P ~C COK)~k2 C A S P (18.9a)

dC C O Kdt 'ASP (18.9b)

Figure 18.8 Schematic of reaction network fo r model K .

Model K is depicted schemat ica l ly in Figure 18.8 and the corresponding mathe-mat ica l mo del i s g iven by the f o l lo w in g tw o O D E s :

dt = k ] ( I - C A S P + H O ~ ' A S P + H O - k 3 C A S p + H o ( 1 8 . 1 0 a )

dC r,O Kdt ( 18 . 10 b)

Figure 18.9 Schematic of reaction network fo r model J.



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362 Chapter 18

Model J is depicted schematical ly in F igure 18.9 and the corresponding mathe-matica l m o d e l is given by the fo l lowing tw o O D E s :- A S Pdt l~

C ASP ~C COK.)~k2C ASP ~k3CA SP ( 1 8 . H a )

dCr,C O Kdt ( I S . l l b )

In all the above t h re e -c o mp o n e n t models as wel l as in the fo ur -c o mp o n e n tmo dels presented next , an Arrhenius - type temperature dependence is assumed forall th e kinet ic parameters. N a m e l y each parameter k ; is of the form Ajer/7(-Ej/RT).

18.2.3 Four-Component ModelsW e cons ider th e fo l lowing f o u r - c o m p o n e n t mode l s . Model N is depictedschemat ica l ly in Figure 18.10 and the corresponding mathemat ica l model is givenb y t h e fo l lowing three O D E s :

Figure 18.10 Schematic of reaction network for model N.

dC H Odt

dC

- - k CH OASP

dt ~kC2 A S PdC C O Kdt =k2CASP

(18.12a)

(18 .12b)

( 18 . 12c)



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Figure 18.11 Schematic of reaction network for model M.

M o d e l M is depic ted s c h e m a t i c a l l y in Fig u r e 18.11 and the cor respond ing m a th e -matica l model is given by the f o l lo w in g three O D E s :dC H O

dt - k](l-C H O -CASP -C C OK)-k2C H O -k3CH OdcA S P

dC C O Kdt

(18 .13a)

(1 8 .1 3b )

(18.13c)

Figure 18.12 Schematic of reaction network fo r model O.

Model O is depicted schematical ly in Figure 18.12 and the corresponding mathe-matical m o d e l is g iven by the f o l lo w in g th ree O D E s :



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364 Ch apter 18

dC ASP = kdt !dCHO

=k2

CA S p -

k3

CH O -

k4CHO (18.14b)dt

LO _\,r^ _ i_ L - r1 (\St \Ar\ 5 A S P 4 H O v ^^)dt

18.2.4 Results and DiscussionT he t w o - c o m p o n e n t mo d e l s are "too s i mp le " to be able to descr ibe th e

complex react ions t ak ing place. O n l y model D was f o u n d to descr ibe early coke( C O K ) product ion adequa te ly . For Low Temperature Oxidat ion (LTO) condi t ionsth e mod el was adequ ate o n l y up to 45 h and for cracking cond i t ions up to 25 h.T he t h re e -c o mp o n e n t mode l s were f o u n d to fit the exper imenta l data betterthan th e t w o - c o m p o n e n t ones. M odel I was found to be able to f i t both LTO andc r a c k in g data very w e l l . Th i s m o d e l w a s cons ide red the bes t of all models eventhough it is unable to calculate th e HO/LO split (H a n s o n a n d Kalogerakis , 1984).F o u r c o m p o n e n t m o d e l s w e r e found very d i f f i cu l t o r i mp o ss i b l e to con-verge. Model s K , M and O are mor e compl ica ted and have m o r e reaction pathscompared to m odels I or N. W h e n e v e r th e parameter with th e h i g h e s t variance w asel iminated in any of these three models , it w o u l d revert back to the s impler ones :Model I or N. Model N was the o n ly four p s e u d o - c o m p o n e n t mo d e l that con-verged. Th i s model a l so provides an es t imate of the HO/LO spl i t . Th i s mo d e l to -gether with mo d e l I were r ecommended for use in s i tu combus t ion s imula tors(H anson and Kalogerak i s , 1984). Typica l resu l t s are presented next for model I .Figures 18.13, through 18.17 show the experimental data and the calcula-t i o n s b a s e d on model I for the low t e mp e ra t u re o x i d a ti o n at 50, 75, 100, 125 and150 C of a Nor th Bodo oi l sands b i t ume n with a 5% oxygen gas. A s seen, there isgeneral ly good agreement between the exper imenta l data and the resul ts obtainedby th e s impl e t h ree pseudo-compo nen t mode l at all temperatures except the run at125^. T h e on l y drawback of the model i s that i t cannot calculate t h e H O / L Ospl i t . The est imated parameter values fo r mo d e l 1 and N are s h o w n in Table 18.2 .T h e observed large standard deviat ions in the parameter est imates is rather t yp ica lfor A r r h e n i u s t y p e e xp r e s s io n s .

In F igu res 18.18, 18.19 an d 18.20 th e ex per imenta l da ta and the ca lcu la -t i o n s based on m ode l 1 a re show n fo r the h i g h t e m p e r a tu r e c r a c k i n g at 360, 3 97and 420 C of an Athabasca oil sands b i t ume n (D rum 20).Simi la r resul ts are seenin F igures 18.21, 18.22 and 18.23 fo r another Athabasca oil sands b i t u m e n (Drum433). The estimated parameter values for model I are shown in Tab le 18.3 forD rums 20 and 433 .



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1UU -/sIE o-oL . 8 o -0>a*. 7 0 ~.c5" 60-n*, 50-0 40-|30-0 20-u0 10-o

n -

1 1 1 I I I I I I i i i i i iT = 50 C

~ O O D D-- ---_I _---Q----__------JO-~a~0---------------------------

4] A I I I | i _J_A_L.. _ . ! . . L-J 5 10 1 5 20 25 30 35 40 45 50 55 60 65 70 75 80T i m e (h)

Figure 18.13 Exp erimental a nd calculated concentrations of Coke (COK) " A " ,Asphaltene (ASP) "o" and Heavy Oil + Light Oil (HO+LO) "a"at 50 "C for the low temperature oxidation of North Bodo oi lsands bitumen using model I.

IUU-,* s" 90n0)Ou 80-0}a.-t- 70"JC.? 60-o*, 50-_0 40-g 30-*-a 20-u0 10-0

D -

T = 75 C,H........................a a -----D-----.-.-..............0

---_

n _a.--- -o-Q---9----O------~ o- -t*,4-4.1 A 1 .........-4t-J L. U-U15 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80T ime (h)

Figure 18.14 Experimental and calculated concentrations of Coke (COK) "A" ,Asphaltene (ASP) "o " a nd Heavy Oil + Light Oil (HO-LO) "n "at 75 "C for the low temperature oxidation of North Bodo oilsands bitumen using model I.



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366 Chapter 18

^90-" 80-01

70-.? 60-s*. 50 -0 40-

120-uo 1 0 -CJ0 -

I I 1 1 1 1 1 1 1 '"! 1 T = 100C

"--........D D " "

e- - - - - - . . a . . D " " " ............p

-

..9o _ _ _ _ _ _ _ _ _ _ - e ~ a--"""

*l u I I t vrf-=i-3i=^3=jrriT "I ~ &0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80Time (h)

Figure 18.15 Experimental a nd calculated concentrations of Coke (COK) " A " ,Asphaltene (ASP) "o" a nd Heavy Oil + Light Oil (HO+LO) "a "at 100 "C for the low temperature oxidation of North Bodo oilsands bitumen using model I.

CooIDOLJC

o

Coacocoo

1UU -

90-80 -70-60-50-40-30-20 -10 -i0-C

1 1 1

T = 125C." * . """..

'"'"""D """--.......

"""""----..... "o _0

^-"^-0-" ."^~~'"~^~ ' ^

^~-~~' ~^^^^ ~**~~^ f} \h*)5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 8!

Time (h)

Figure 18.16 Experimental a nd calculated concentrations of Coke (COK) " A " ,Asphaltene (ASP) "o" a nd Heavy Oil + Light Oil (HO-LO) "a"at 125 "C for the low temperature oxidation of North Bodo oilsands bitumen using model I .



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1UU -

~c 9-o" e o -0Q 70 --C.? 60 -Oi50-0 40 -10 30 -i20-0o 10 -o n -

IT = 150C

_ -fi ***''- . . . ^^-' i

- D D '"-, , 'A''.. x*"-....i ^DXX'--.D~ o / '"--../ ' - . . . n/' """""--

- - - - - oX' '""""""""---o/' o o " o""""" -/* l I I I

0 5 10 15 20 25 JO 35 40 45 50 55 60 65 70 75 80T i m e (h )

Figure 18.17 Experimental and calculated concentrations of Coke (COK) "A" ,Asphaltene (ASP) "o " a nd Heavy Oil + Light Oil (HO+LO) "n "at 150 "C for the low temperature oxidation of North Bodo oi lsands bitumen using model I.

Table 18.2 Est imated Parameter Values fo r Models I and N for the LowTemperature Oxidation of North Bodo Oil Sands BitumenModel

I

N

ParameterA ,E ,A 2E 2A !EiA 2E,A 3C j " - ,

Parameter Va lue s5 . 4 9 9 x l 0 4

6 25 53 . 0 7 5 x 1 0 "

121501 .885x l0 67866

4 . 9 3 4 x l 0 1 213298

3 . 4 5 0 x l 0 1 417706

StandardDevia t ion (%)976.62 729.21176.44 3 513.5

321890

Source: Ha n s o n and Kalogerakis (1984).



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368 Chapter 18

Table 18.3 Estimated Parameter Values for M ode l I for High TemperatureCracking of Athabasca Oil Sands BitumenBi tu m e nC od e

Drum2 0

D r u m/I "") 1433

ParameterA ,E ,A 2E 2A!E ,A 2E 2

Parameter Values3 . 5 4 x l 0 4

593208 . 7 6 x 1 0 "

194851 . 2 3 x l 0 3 1

5 23 4 09 . 5 1 x l 0

1 52 6 2 3 2

StandardDeviat ion (% )87701022 558.8

4 2 9 757

2 3 56.1

Source: Ha n s o n an d Kalogerak i s (1984).

C M+u 0--ca_ 7 0 - -j:5> 6 0 - -)* 50-1-

P J O - -

couOH0

T = 360 C

- 3 - - -10T i m e (h)

Figure 18.18 Exp erimental and calculated concentrations of Coke (COK) "A ",Asphaltene (ASP) " o " a nd H ea vy Oil + Light O il ( HO+ LO) "a' :at 360 "Cfor the high temperature cracking of Athabasca oilsands bitumen (Drum 20) using model I.



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r-sC0UUQ.E0c

cUco0

1UU -

90-80-70-60-50-40-SO-20-10-n ^

l >-do. n o o

--

T = 397 C--.

** "* " - - ~"- ~~~->-^f"~ 6 ^4> ..10Time (h) 20

Figure 18.19 Exp erimental and calculated concentrations of Coke ( C O K ) "A ",Asphal tene ( A S P ) " o " a nd Heavy Oil + Light O il (HO+LO) "a "at 397 "C for the high temperature cracking of Athabasca oi lsands bitumen (Drum 20) using model I ,

COUQ


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370 Chapter 18

cook.Q.IjCok.cooco0

li/w -80-

80-70-60 -50-40 -30-20-10-0 ^C

1 1 1 -

a. pD D D

--

T = 3 6 0 C--

< S o o - -5_

^ _ .___._._5 15 2iT i m e (h)Figure 18.21 Experimental a nd calculated concentrations of Coke (COK) "A",Asphaltene (ASP) "o" a nd Heavy Oil + L ight Oil (HO+LO) "n"

a t 360 "C for the high temperature cracking of Athabasca oi lsands bitumen (Drum 433) using model I.

I 90+0" 8 0 - - . . . H . . . !

70--.? 60 - -oA 50--

o 10+O

T = 397 C

0 -.10T ime (h)

Figure 18.22 Experimental a nd calculated concentrations of Coke (COK) "A" ,Asphaltene (ASP) "o" a nd Heavy Oil + Light O il (HO- LO) "a "at 397 "C for the high temperature cracking of Athabasca oi lsands bitumen (Drum 433) using model I.



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100c M+oS t-D . . .a0

_ _ _ 70--.S 60 - -o*, 5 0 - -

P 30--fe 20UO TO --O

T = 420 C

10T i m e (h)

18.3

Figure 18.23 Experimental and calculated concentrations of Coke (COK) " A " ,Asphal tene (ASP) "o" a nd Heavy Oil + Light Oil (HO+LO) "a "at 4 20 "C for the high temperature cracking of Athabasca oi lsands bitumen (Drum 433) using model I.

A U T O M A T I C H I S T O R Y M A T C H I N G IN R ESER V OIRE N G I N E E R I N GHistory matching in reservoir engineering refers to the process of es t imat inghydrocarbon reservoir parameters (l ike porosity an d permeabi l i ty dis t r ibut ions) sothat th e reservoi r s imula tor matches the observed f ield data in s o m e o p t i ma l fash-ion. T he intention is to use the his tory matched-mode l to forecast future behav io rof the reservoi r un d e r different deple t ion p lans and thu s opt imize product ion .

18.3.1 A Fully Implicit, Three Dimensional, Three-Phase Simulator withAutomatic History-Matching CapabilityT h e mathemat ica l mo d e l for a hydrocarbon reservoi r consists o f a n u m b e r o fpart ial dif ferent ia l equa t ions (PDEs) as w e l l as a lgeb ra ic equat ions . T h e n u m b e r o fequa t ions depends on the scope/capabi l i t ies of the m o d e l . The set of PDEs is oftenreduced to a set of O D E S b y grid discret izat ion. The estimation of the reservoirparameters of each grid cell is the essence of history matching.T he discret ized reservoir model can be writ ten in the general form presented

in Section 10.3. T he state variables are the pressure and the o i l , water and gas satu-Copyright 2001 by Taylor & Francis Group, LLC


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rat ions at each grid ce l l in a "b lack o i l " r e s er v o i r m o d e l . T h e vector of ca lcu la tedvar iables y is related to the state variables t h rough a nonl inear re lat ionship. Fore xa mp le , th e c o mp ut e d water to o i l ratio from a product ion o il we l l is a n o n l in e a rfunct ion o f pressure, saturat ion leve ls an d f l u i d compos i t ion of the grid cells th ew e l l i s com pleted in . In s o m e cases we ma y also observe a very l im i te d n u m b e r o fth e state var iables . N o rma l l y me a s ure me n t s are ava i l ab l e from product ion or ob-servation we l l s placed t h r o u g h o u t th e reservoir . In general , func t ion evaluat ionsare t ime c o n s u m i n g a n d c o n s e q u e n t ly t h e n u m b e r o f func t ion eva l ua t ions requ iredby the i terative a lg o r it h m s h o u ld b e m i n i m i z e d ( T a n an d Kalogerakis , 1991).T h e parameters are est imated by m i n i m i z i n g th e u s u a l L S object ive func t ionwhere th e w e i g h t i n g matr ix is often chosen as a diagonal matrix that normal izesth e data an d m a k e s a l l measurements be of the same order o f magni tude. S ta t i s t i -ca l ly , t h i s is the correct approach if the error in the m e a s u r e m e n t s i s p ropor t iona lto the magni tude of the measured var iable .Tan and K al ogerak i s (1991) modif ied a t h ree - d imens iona l , th ree-phase ( o i l ,water , g a s ) m u l t i - c o m p o n e n t (N c c o mp o n e n t s ) industr ia l ly avai lab le s imula tor .T h e reservoir s i m u la t i o n m o d e l consisted of a set of O D E s that described th ec o m p o n e n t mater ia l balances subject to constraints for the saturations, S, and themole fract ions, x ip , i = l , . . . , N c , i n each of the three phases ( o i l , g a s , water) . Eachc o m p o n e n t c o u l d b e found in any one o f the three phases . T h e state variables werepressure , sa tura t ions (S g as , S water, S0u ) and the master mo le fractions at each gridcell of the reservoir . The master m o l e fract ions are related to the actual m o l e frac-t i ons t h ro ug h the e q u i l i b r i u m dis t r ibu t ion ratios also k n o w n as "K-va lues . " It isk n o w n from phase e q u i l i b r i u m t h e rmo d yn a mi c s tha t th e "K - v a lu e s " are a c o n -ven ien t way to describe phase behavior . T he parameters to be est imated were po-rosity ( c p ) and hor izonta l and vert ical permeabi l i ty (kh and k v) in each grid cel l . Theobserved var iables that c o u l d b e matched were th e pressures of the grid cel ls , w a-ter-oil ratios an d gas-oi l ratios of the i n d i v i d u a l product ion wel l s , and f l o w i n gbot tom ho l e pressures . Th e G a us s -N e wt o n a lgor i thm w as imp l emented wi th anop t ima l step size pol icy to avoid overs tepping. An automat ic t imestep selectoradjusted th e t imes teps in order to ensure that th e s imula tor per forms th e ca lcu la -t ions of the pressures and the produc ing ra t ios at the corresponding observat iont imes .Fu rthermo re , the implementat ion of the Gauss -New ton m ethod also incorpo-rated the use of the pseudo- inverse method to avoid instabi l i t ies caused by the i l l -c o n d i t i o n i n g o f matr ix A as discussed in C h a p t e r 8. In reservo i r s i mu l a t i o n th ism ay o c c ur fo r ex ampl e when a parameter z o n e i s ou ts ide the dra inage rad ius o f aw e l l and is therefore n ot observab le from th e w e l l data. Most impor tant ly , in orderto real ize subs tan t ia l sav ings in computa t ion t i me , th e se q u e n t i a l c o m p u ta t i o n o fth e sensi t ivi ty coefficients discussed in detai l in Section 10.3 .1 was implemented.Fi n a l l y , th e n u m e r ic a l integration procedure that w as used was a f u l l y imp l ic i t o n eto ensure stabil i ty and convergence over a w i d e range o f parameter est imates .

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Parameter Estimation in Petroleum Engineering 3 73

18.3.2 Applicat ion to a Radia l Coning Problem (Second SPE Com p arativeSolution Problem)F o r i l lustrat ion purposes , Tan and Kalogerak i s (1991) appl ied their historymatching approach to the Second SPE Comparat ive Solu t ion Problem. T he prob-

lem i s described in detai l b y Chappelear an d N o l e n (1986). T h e reservoir consistso f t e n concentr ic r ings an d fifteen l ayers . A gas cap and oi l zone and a water zoneare all present. Starting wi th an arbitrary initial guess of the reservoi r porosity an dpermeabi l i ty dis t r ibut ion i t was attempted to match: (a) the observed reservoirpressure; (b) the water-oi l ratio; (c) the gas-oi l ratio; (d) the bottom hole pressure;and (e) combina t ions o f observed data. It is noted that th e s imu l a to r wi th its b u i l tin p a r a m e t e r e s t i ma t i o n c a p a b i l i t y c a n m a t c h rese rvo i r pressures, w a t e r - o i l ratios,gas-oil ratios and f l o w i n g bottom hole pressures either ind iv idua l ly or all of thems i m u l t a n e o u s l y . T h e reservo i r m o d e l w a s u s e d t o generate ar t i f ic ia l o b s e rv a t i o n susing the or iginal v a lue s of the reservoir parameters and by adding a random noiseterm. Subsequent ly , s ta r t ing with an arbitrary init ial guess of the reservoir pa-rameters, i t was attempted to recover the or ig ina l reservoir parameters by match-ing th e "observed" data.T h e var ious s imu l a t ion runs revealed that the Gauss-Newton i mp le me n t a -tion by Tan and Kalogerakis (1991) was ex t remely efficient compared to otherreservoir h i s to ry match ing me t h o d s reported earl ier in the l i terature.

18.3.2.1 Matching Reservoir PressureT he fifteen layers of cons tant permeabi l i ty and porosity were taken as thereservoir zones fo r w h i c h these parameters w o u l d be est imated. T h e reservoirpressure is a state var iab le an d h e n c e in th i s case th e relat ionship between th e out-pu t vector (observed variable s) and the state variables is of the form y(t,)=Cx(ti).T a n a n d K a l o g e r a k i s ( 1 9 9 1 ) perfo rmed four d i f f e ren t r u n s . I n t h e 1 s t run,layers 3 to 12 were selected as the zones w h o s e hor i zon ta l permeabi l i t ies were th eu n k n o w n parameters . T h e ini t ial guess for a l l the permeabi l i t ies was 200 md. I tw as found that th e or ig ina l permeabi l i ty va lues were obtained in n i n e i terations o f

th e Gauss -Newton method b y match ing th e pressure profi le. In Figures 18.24a an d18.24b the parameter va l ues and the LS object ive function are s h o wn d ur i n g th ecourse of the i terat ions. The cpu t i me required for the n i n e i terat ions was e q u i v a -lent to 19.51 mo d e l - run s by the s imu l a to r . Therefore, the average cpu t i m e for oneiteration was 2.17 mode l - r uns . Th i s is the t ime requi red for one model -run (com-putat ion of a l l state variables) and for the computat ion of the sensit ivi ty coeffi-cients of the ten paramete r s w h i c h w as o b v i o u s l y 1.17 m o d e l - r u n s . T h i s c o m p a r e sexcept ional ly wel l to 10 m o d e l - r u n s that w ou ld have normal l y been requi red byth e standard formulat ion of the Gauss-Netwon method . These sav ings i n compu-tation t ime are so le ly due to the efficient computat ion of the sensit ivity coefficientsas described in Section 10.3.1. Thi s resul t represents an o rder of magn i tude reduc-t ion in computa t iona l requi rements and makes automat ic h i s tory match ing through



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n o n l i n e a r regress ion prac t ica l an d e c o n o m ic a l l y feas ib le for the first t i m e in reser-voir s imula t ion (Tan an d Kalogerak i s , 1991).Ne xt , in the 2 n d run the hor izon ta l p e r me a b i l i t i e s of a l l 15 l ayers were est i -mated by us i n g th e v a lu e of 200 md as in i t ia l guess . I t required 12 i terat ions toconverge to the op t ima l permeabi l i ty va lues .I n th e 3 l d run the porosi ty of the ten zones w as est imated by u s ing an in i t ia lg ue s s of 0.1 . Final ly , in the 4 th run the porosi ty of al l fifteen zones w as estimatedb y us ing th e same init ial guess (0.1) as above . In this case, matr ix A w a s found tobe extremely i l l -condi t ioned and the pseud o- inverse opt ion had to be used .U n l i k e the permeabi l i ty runs , th e resu l t s show ed that, the observed datawere n o t su f f ic ien t to d i s t i n g u i sh al l fi fteen va lues o f the p o r o s i t y . T h e i l l -c o n d i t i o n in g of the p rob l em was m a in ly due to the l imi ted observabi l i ty and i tcould be over come b y s u p p l y i n g mor e in format ion such as addi t ional data or by are -parameter iza t ion of the reservo i r m o d e l i t se l f ( r e z o n in g th e reservoir) .

18.3.2.2 Matching Water-Oil Ratio, Gas-Oil Ratio or Bottom Hole PressureThe water-oi l ratio i s a complex t ime-dependent func t ion of the state vari-ables s ince a w e l l can p roduce o i l from several grid cel ls at the same time. In thiscase th e re la t ionship of the outpu t vector and the state variables is non l inea r of the

form y(ti)=h(x(ti)).T he central ly located w e l l i s completed in layers 7 and 8. The set of obser-vat ions consis ted of water-oi l ratios at 16 t imesteps that were obtained from a baser u n . These data were then used in the next three r u n s to be matched with calcu-lated valu es. In the 1 s t run the hor izonta l permeabi l i t ies o f layers 7 and 8 were es-t imated u s i n g an ini t ial g u e s s o f 200 md. Th e o p t i mum wa s reached w i th in fivei terat ions of the Gauss -Newton method . In the 2 n d run the object ive was to esti-mate th e unknown permeab i l i t i e s in layers 6 to 9 ( four zones) us i n g a n ini t ialguess of 200 md. In spi te of the fact that the calcu lated water-oi l ra t ios agreed withth e observed va lues very wel l , th e calculated permeabi l i t i es were n o t f o u n d to beclose to the reservoir values . Fina l ly , th e porosi ty of layers 6 to 9 was est imated inthe 3 rd run in five Gauss-Newton i terations and the est imated va lues were f o u n d tobe in good agreement with th e reservoir porosity va lues .Sim i la r f indings were observed for the gas-oi l ratio or the bottom h o l e pres-sure of each wel l wh ich is also a state variable when the w e l l product ion rate iscapacity restricted (Tan and Kalogerakis , 1991) .

18.3.2.3 Matching All Observed DataI n this case th e observed data consisted of the water-oi l ratios, gas-oil ratios,

f l o w in g bottom hole pressure measu r ement s and the reservoir pressures at twolocat ions of the wel l (layers 7 and 8). In the first run , the hor izonta l permeabi l i t iesof layers 6 to 9 were est imated by u s i n g th e v a lu e of 200 md as the init ial guess .Copyright 2001 by Taylor & Francis Group, LLC


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As expec ted , the es t imated v a lu e s w e r e f o u n d to be closer to the correct onesc o mp a re d wi th th e est imated v a lu e s w h e n t h e wate r-o i l rat ios are o n ly matched . Inthe 2 n d run, th e h o r i z o n ta l p e r m e a b i l i t ie s o f layers 5 to 10 (6 z o n e s ) w e r e es t imatedu s i n g th e v a l u e o f 2 0 0 m d as in i t ia l g ue s s . I t was f o u n d necessa ry to use thepseudo- inverse op t ion in th is case to ensure conver gence of the c o mp ut a t i o n s . T h einit ial and converged profi les generated by the m o d e l are compared to the ob-served data in F igures 18.25a and 18.25b.I t w as also attempted to est imate permeabi l i ty va lues for e igh t zones but i twas no t success fu l . It wa s c o n c lud e d that in order to extent the reservoir that canbe identified from measu r ement s on e needs observation data over a longer history.Final ly , in another run, i t was show n tha t th e poros i t ies of layers 5 to 10 cou ld bereadi ly est imated wi th in 10 i terat ions. However, i t was not p ossible to est imate th eporosi ty values for eigh t layers due to the same reason as the permeabi l i t i es .Ha v i n g performed all the above computer run s , a s imple l inear correlationw as f o u n d re la ting the requi red cpu t ime as mode l e q u i v a l e n t runs (MER) wi th then u m b e r o f u n k n o w n parameters regressed. T h e correlat ion w as

M E R = 0.319+0.07x/? ( 1 8 . 1 5 )This indicates that after an i n i t i a l overhead o f 0.319 m o d e l runs to set up thea lg o r i t h m , an addi t iona l 0.07 o f a m o d e l - r u n w as re q u i re d f o r t h e c o m p u ta t io n o fthe sensit ivity coeff ic ients for each addit ional parameter . Thi s is abou t 14 t i me sless compared to the one addit ional mode l - run requ i red by the standard i m p le -

mentat ion of the Gauss -New ton method . O b v i o u s l y these numbers serve on l y as aguide l ine how ever , the computa t iona l savings real ized t h ro ug h t h e efficient inte-grat ion of the sensi t ivi ty O D E s are expected to be very s ign i f ican t w henever anim p l i c i t o r s e m i - i m p l i c i t r e se rvo i r s im u la to r is i n v o l v e d .

107

10s

10"10'

10 ' 1010'1ID '210'3

2000

2 4 6 8ITERATION

10 3 5 7ITERATION

Figure 18.24 2 SPE problem with 15 permeabil ity zones and using all measure-ments: (a ) Reduction of LS objective function and (b) Parametervalues during th e course of th e iterations [reprinted with permiss ionfrom the Society of Petroleum Engineers].Copyright 2001 by Taylor & Francis Group, LLC


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4000 3700

3500-

uic c3< n< nu iIT0.

Pressure data Layer #7o Layer #8n BHP

2900200 400 600 800 1000

TIME (days)200 400 600 800 1000

T I M E (days)

Figure 18.25 Observed data and model calculations fo r initial a nd convergedparameter values for the 2" SPE problem, (a ) Ma tch of gas-oil ratioa nd wa ter-oil rat io, (b ) Match of bottom-hole pressure a nd reservoirpressures a t layers 7 and 8 [reprinted with permission from the So-ciety of Petroleum E ngineers].

18.3.3 A Three-Dimensional, Three-Phase Automatic History-Match ingModel: Reliability of Parameter EstimatesA n impor tant benef i t o f u s i n g a t h ree - d imens iona l th ree-phase au tomat ichis tory matching s imula tor is that besides th e est imat ion of the u n k n o w n parame-ters, it is poss ib le to a n a lyz e th e qual i ty of the fi t that h as been achieved betweent h e m o de l a n d t h e field data. In the h i s t o r y m a t c h i n g p h a s e of a re se rvo i r s i m u l a -

t ion s t u d y , th e p r a c t i c i n g e n g i n e e r w o u l d l i k e to k n o w h o w accura te ly the pa -rameters h a v e been est imated, whether a par t icu lar parameter has a s ign i f i can tin f luence on the his tory match and to w h a t degree. T h e tradit ional procedure ofh i s to ry match ing by va ry ing different reservoir parameters based on eng ineer ingj u d g m e n t m ay prov ide an adequate match; however , th e eng ineer has no me a s ureof the rel iabi l i ty of the fitted parameters.As already discussed in Chapter 11, matr ix A ca lcu la ted dur ing each itera-t ion of the Gauss -Newton m e t h o d can be used to de te rmine th e covariance matrixof the estimated parameters , w h i c h in turn provides a measure of the accuracy ofthe parameter est imates (Tan and Kalogerak i s , 1992).The covar iance matr ix , C O f ^ k * ) , of the parameters is given b y E q u a t io n11.1, i.e.,C0 F(k*)=[A*]~ 'SL s(k*)/(d.f .) and the var iances of the est imated p a-rameters are s i m p l y obtained as the diagona l e l ement s o f COK(k *) . I t i s remin dedthat the square root of the variances are the standard deviations of the parameter



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Parameter Estimation in Petroleum Eng ineering 37 7

est imates (a lso k n o w n as the s tandard es t imat ion error) . T h e m a g n i t u d e of thestandard devia t ion o f a par t icu la r parameter est imate indicates the degree o f conf i -dence tha t s h o u l d be p laced on that parameter va l ue .The covar iances between the parameters are the off -d iagonal e lements of thecovariance matrix. The covar iance indicates ho w closely tw o parameters are cor-related. A large v a lue for the covariance between tw o parameter est imates indi-cates a very close correlat ion. Pract ical ly , this m e a n s that these tw o parametersmay not be poss ib le to be est imated separately. This i s shown better t h ro ug h th ecorrelation matrix. The correlat ion matr ix , R, is obtained by t r ans fo rming the co-variance matr ix as f o l lo w s

w h e r e D is a d i a g o n a l m a t r i x w i t h e le m e n t s th e s q u a r e roo t of the d ia g o n a l ele-m e n t s o f COV(k*) . T h e d iagona l e lements o f the cor re la t ion matr ix R w i l l a l l bee q u a l t o o n e w h e r e a s th e o f f - d i a g o n a l o n e s w i l l h a v e v a lu e s b e tw e e n - 1 a n d 1. Ifan o f f - d i a g o n a l e le m e n t h a s a n abso l u te v a l u e very c lo s e t o o n e then th e corre-sponding parameters are h i g h l y correlated. Hence, the of f -d iagona l elements ofmatrix R p ro v i d e a direct indication of two-parameter correlat ion.Corre la t ion b e t w e e n th ree o r more pa ramete r s is very d i f f i cu l t to detect u n -less an e i g e n v a lu e d e c o m p o s i t i o n of matr ix A* i s p e r f o r m e d . As a l ready d i scussedin Chapter 8, matr ix A* i s symmetr ic and hence an eigenvalue decomposition isalso an or thogonal d e c o mp o s i t i o n

A * = V A V T (18.17)where A i s a d iagonal matrix wi th pos i t ive e l e me n t s that are the eigenvalues ofmatr ix A *, V is an or thogonal matr ix w h o s e c o l u m n s are the normalized eigen-vectors o f matr ix A * a n d hence, V T V = I . Fu r th e r m o r e , a s s h o w n in C h a p t e r 11, th e(1-a) 1 0 0 % joint confidence region for the paramete r vector k i s descr ibed by theel l ipsoid

(18.18)

I f this el l ipsoid is h i g h l y elongated in one direction then the uncertainty inth e parameter va l ues in that direction is signif icant. The/? e igenvectors o f A formthe direction of the p pr incipa l axes of the el l ipsoid wh ere each eigenvalue is equa lto the inverse of the length of the cor responding pr incipa l axis . The longest axis,corresponds to the smallest e igenva lue whereas th e shortest axis to the largest ei-genva lue . The rat io of the largest to the smal les t eigenvalue i s the condi t ion n u m -ber of matrix A. As discussed in Chapter 8, the condition n u m b e r can be calcu-

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lated d u r i n g th e m i n i m i z a t i o n steps to p r o v id e an i nd ica t ion of the i l l - cond i t ion ingo f matr ix A. If s o m e parameters are corre la ted then th e v a l u e of the condi t ionn u m b e r wil l b e very large. On e s h o u ld b e careful , however , to note that the e igen-values are not in a one- to-one correspondence with th e parameters. It requ ires in -spection of the e igenvectors cor responding to the sma l l e s t e igenva lues to k n o wwhich parameters are h ig h ly correlated.T he e lements of each eigenvector are the cosines of the angles the eigen-vector m a k e s with th e axes corresponding to the p parameters. I f s o m e of the pa-rameters are h ig h ly correlated then thei r axes w i l l l ie in the same direction and theangle be tween them w i l l be very smal l . Hence, the e igenvector corresponding toth e smallest e igenva l ue w i l l have s igni f icant contr ibu t ions from th e correlatedparameters and the cosines of the angles w i l l tend to 1 since the angles tend to 0.Hence , the l a rge r e le m e n t s of the e igenvec to r w i l l e n a b le ident if ica t ion of the pa-rameters w h i c h are correlated.

18.3.3.1 Implementation and Numerical ResultsT a n a n d K a lo g e r a k i s ( 1992) i l lus t ra ted th e a b o v e p o in t s u s i n g tw o s i m p le

reservoir p r o b le m s r e p r e se n t in g a u n i f o r m areal 3 x 3 m o d e l with a p r o d u c i n g w e l la t g r id b lock (1 ,1 ) . In th i s sect ion the i r re su l t s from the SPE second compara t iveso l u t i o n prob lem are o n ly repor ted . This i s the w e l l - k n o w n t h r e e - p h a s e radia lcon ing prob lem descr ibed by Chappe lea r an d N o le n (1986). T he reservoir has 15layers each of a cons tan t hor izonta l permeabi l i ty . The w e l l i s completed at layers 7and 8. The object ive here is to estimate th e permeabi l i ty of each layer us ing th eme a s ure me n t s made at the wel l s . The measu r ement s inc lude water-oi l ra t io , g a s -oil ratio, f l o w in g b o t t o m - h o l e pressure and reservoir pressures of the w e l l loca-t ions . Observa t ions were gen erated r u n n i n g th e mo de l wi th th e or ig ina l descript ionand by adding no i se to the m o d e l calculated va l ues as f o l l o w s

(18.19)w her e y j ( t j ) are the n o i s y observat ions o f var iable j a t t i m e t,, o e i s the normal-ized standard me a s ure me n t error and z y is a ra n d o m var iable distr ibuted normal l ywith zero me a n and standard devia t ion o n e . A v a l u e of 0 . 0 0 1 w as used fo r o e.In th e first a t tempt to character ize th e reservoir , all 15 layers w ere t reated asseparate parameter zones . A n ini t ial guess o f 300 md w as used for the permeabi l i tyo f each z o n e . A diagona l we igh t ing mat r ix Q , w i th e l ement s y j ( t j ) " 2 w as used.After 10 iterations of the Gauss-Newton method the LS object ive funct ion w asreduced to 0.0147. T h e est imat ion prob lem as def ined, is severely i l l -condi t ioned.A l t h o u g h the algori thm did not converged, th e est imation part of the program p r o -vided est imates of the standard deviat ion in the parameter values obtained t h us f a r .



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Table 18.4 2m SPE Problem: Permeabili ty Est imates of Original Zonation

Layer12Oj456789101 112131415

TruePermeab i l i ty

354 7 . 514 82 0290

4 1 8 . 5775606824 7 21253 0 0

1 3 7 . 5191350

Est imatedPe r me a b i l i t y23 7 . 5329.22 2 . 4

289.2240 . 8324.6676.2

5 2 . 6636

4 1 1 . 48 2 1 . 1323 . 71 5 9 . 9

2 4 6 . 1256.4

StandardDeviat ion(% )

867.71 4 1 6725 .24 0 . 76 9 . 14 0 . 35 . 3 320.9

1 1 6 . 81 2 1 . 0

2 6 1 . 41 2 8 433921 2 4 11 2 5 6

Eigenvectorof SmallestEigenva lue0.1280-0 .25010.1037-0 .00410.0108-0.00560.00030.0001-0 .0194-0.0074-0.0428-0.28480.86130.1693-0.2373

Source: Tan and Kalog erakis (1992) .

The es t imated permeabi l i t ies together with th e t rue permeabi l i t ies of eachlayer are show n in Tab le 18.4. I n Tab le 18.4, the eigenvector corresponding to thesmallest e i g e n v a lue o f matrix A is a lso s h o w n . T h e largest e l ement s of the eigen-vector correspond to z o n e s 1,2,3,12,13,14 an d 15 sugges t ing that these zones arecorrelated. Based on the above and the fact that the eigenvector of the secondsmal les t e igenva lu e sugges ts that layers 9,10 ,11 are also correlated, i t was decidedto re z o n e the reservo i r in the f o l l o w i n g m a n n e r : L a y e r s 1,2 a n d 3 a r e l u m p e d intoo n e z o n e , layers 4,5,6,7 and 8 r e m a in as dis t inc t z o n e s an d l aye rs 9 ,1 0 ,1 1 ,1 2 ,1 3 ,1 4and 15 are lumped in to one zone . With th e revised zonation there are on l y 7 un-k n o w n parameters .With th e revised zonation t he L S object ive funct ion w as reduced to 0 .00392 .The f inal match is s h o w n in Figures 18.26 and 18.27. Th e est imated parametersand their s tandard deviat ion are shown in Tab le 18.5. A s seen, th e est imated per-meabi l i ty fo r layer 7 has the lowest standard error. Thi s is s impl y due to the facttha t th e w e l l is c o m p l e t e d in layers 7 and 8. I t i s in teres t ing to no te tha t cont rary toexpectat ion, the es t imated permeab i l i ty corresponding to l ayer 5 has a h i g h e r u n -certainty compared to zones even further away from i t . The true permeab i l i t y o flayer 5 is low and mos t l ikely it s effects have been missed in the t i m i n g of the 16observat ions taken .



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380 Chapter 18

Table 18.5 2nd SPE Problem: Permeability Estimates ofRezoned Reservoir

L a y e r123456789101112131415

T r u ePe r me a b i l i t y354 7 . 51482 0290

4 1 8 . 5775606824 7 21253 00

1 3 7 . 51913 50

Z o n e

123456

7

Est imatedPermeabi l i ty

246.2248.01 2 8 . 1478.6696.86 7 . 5

480.5

StandardDeviation(% )1 4 . 4

2 1 . 43 8 . 11 0 . 11.98.7

1 2 . 1

Source: T a n a n d K a lo g e r a k i s (1 9 9 2 ) .

18.3.4 Imp roved Reservoir Ch aracterization Through Automatic HistoryMatching

I n th e his tory m a t c h i n g phase o f a reservoir s im u la t i o n s tudy , th e reservoirengineer is phased with tw o pr ob lems : First, a grid cell mode l that represents thegeo log ica l s t ruc ture of the underground reservo i r mus t be deve loped an d secondth e porosi ty and permeabi l i ty dis t r ibu t ions across th e reservoir m u s t be determinedso that the reservoir s imulat ion model matches satisfactori ly the f ie ld data. In anactual f ield s tudy , the postulated grid cell mo d e l may not accurately represent th ereservoir descript ion s ince th e geologica l interpretat ion o f se ismic data can easilybe in error. Furthermore, the variat ion in rock propert ies could be such that thepostulated grid cell m o d e l ma y n o t have e n o u g h detail. Unfor tunate ly , even withcurrent c o mp ut e r hardware one cannot use very fine grids with h u n d r e d o f t hou -sands o f cel ls to model reservo i r heterogeneity in poros i ty or permeabi l i ty distri-bu t ion . T h e s imples t way to address the reservoir descr ipt ion prob lem is to use azonation approach w h e r e b y th e re se rvo i r is d iv ide d i n to a re l a t ive ly s m a l l n u m b e rof zones of cons tant poros i ty and permeabi l i ty (each zone m ay have many gridcells dictated by accuracy an d stabili ty considerat ion) . Chances are that the sizeand shape of these zones in the postu la ted model w i l l b e qu i te different than th etrue distr ibution in the reservoir . In this section it is s h o wn how automat ic his torymatching can be used to g u i d e the pract ic ing engineer towards a reservoir charac-



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ter iza t ion that is m o r e representa t ive of the ac tua l u n d e r g r o u n d reservoir ( T a n an dKalogerak i s , 1993).In practice w h e n reservoir parameters such as porosi t ies and permeabi l i t iesare estimated by ma t c h i n g reservoi r m o d e l calcu la ted va lues to f ield data, one hassome pr ior in format ion about the parameter values . For e xa mp le , poros i ty andpermeabi l i ty va l ues ma y b e avai lable from core data ana lys i s and w e l l test analy-s i s . In addict ion, th e parameter v a lue s are k n o w n to be with in certain b o u n d s for apart icular area. A ll t h i s informat ion can be incorporated in the est imat ion methodof the s imula tor by in t roduc ing pr ior parameter dis t r ibu t ions and by i mp o s i n g c o n -straints on the parameters ( T a n and Kalogerakis , 1993) .

3800

3700 -

3300

3200 -

400 600T i m e ( d a y s )

Figure 18.26 Observed a nd calculated bottom-hole pressure and reservoir pres-sures a t layers 7 and 8 for the 2"d SPE problem using 7 perme-ability zones [reprinted from th e Journal of the Canadian Petro-leum Technology with permission].



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382 Chapter 18

7000

200 400 600Time (days)

800 1000

Figure 18.27 Observed and calculated water-oil ratio a nd gas-oil ratio for the2" d SP E problem using 7 permeabil ity zones [reprinted from th eJournal of th e Canadian Petroleum Technology with permission].

18.3.4.1 Incorporation of Prior Information and Constraints on theParameters

It is r e a s o n a b le t o a s s u m e tha t th e m o s t p r o b a b l e v a lu e s of the parametershave n o r m a l dis t r ibu t ions wi th m e a n s equa l to the va lues that were obta ined fromwel l test an d core data analyses. These are the pr ior estimates. Each one o f thesem o s t probab le parameter va l ues (k B j, j = l , . . . , p ) a lso has a corresponding standarddeviation a k j w h i c h i s a m easure o f the uncer ta in ty of the prior parameter est imate.A s already discussed in Ch a p t e r 8 (Section 8.5) u s i n g m a x i m u m l ike l ihood argu-ments th e pr ior in fo rmat ion i s in t roduced by au gm ent ing the L S object ive func t ionto i n c l u d e

S p r i e r = [k-kBlr VB1 [k-kB] (18.20)

T h e prior covariance matrix of the parameters (VB) i s a diagonal matr ix an dsince in the solu t ion of the p ro b le m o n ly th e inverse o f V B is used, i t is preferableCopyright 2001 by Taylor & Francis Group, LLC
http://dk5985_ch08.pdf/http://dk5985_ch08.pdf/http://dk5985_ch08.pdf/http://dk5985_ch08.pdf/


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Parameter Estimation in Petroleum Engineering 3 83

to use as i n p u t to the program th e inverse i tself . N a m e l y , th e i n fo r ma t i o n is enteredinto th e program as VB" ' = diag(a u~ 2, o k 2 " 2 , . . . , o kp "2).Severa l au thors ha ve suggested the use of a weig ht ing factor fo r S pr io i in theoveral l object ive func t ion . A s there is really n o r igorous way to assign an opt i -mized weight ing factor, i t i s better to be left up to the user ( T a n an d Kalogerakis ,1993). If there is a s igni f icant a m o u n t o f his tor ica l observa t ion data avai lable , th eobject ive func t ion wi l l b e domina ted by the least squares term whereas the s ignif i -cance of the prior term increases as the a m o u n t of such data decreases. If the esti-mated parameter va lues differ s ign i f ican t ly from th e prior estimates then the users h o u ld e i ther revise th e pr io r in fo rmat ion o r ass ign it a greater s ign i f i cance b y i n -creasing th e v a lue of the we i g h t i n g factor o r e q u iv a le n t ly decrease th e va l ue of thep r i o r es t imates o kj . As po in ted ou t in Sect ion 8 . 5 , th e cruc ia l ro le o f the pr io r terms h o u ld b e e m p h a s i z e d w h e n th e es t imat ion p ro b le m is severely i l l -condi t ioned. Insuch cases , th e parameters tha t a re no t affected b y th e data main ta in essen t ia l lytheir prior est imates (kj and o k j) and at the same t ime, the condi t ion n u m b e r o fmatrix A does no t b e c o m e proh ib i t ive l y large.While pr ior in format ion may be used to in f luence th e parameter est imatestowards realistic va lues , there i s no guarantee that th e f inal est imates w i l l n o t reachextreme va lues par t icu lar ly when the postulated grid cell m o d e l is incorrect an dthere is a large a m o u n t of data avai lable . A s i mp le way to i mp o s e inequal i ty c o n -straints on the parameters i s through the incorporat ion of a penalty funct ion asalready discussed in C hap ter 9 (Section 9.2.1.2) . B y this approach extra terms areadded in the objective funct ion that tend to explode when the parameters approachnear th e bound ary an d become negl ig ib le w h e n th e parameters are f a r . One caneasily construct such penal ty func t ions . F o r e xa mp le a s i mp le and ye t very effec-t ive penalty funct ion tha t keeps th e parameters in the interval (k m m j, k m ax j) is

], r , r , t f- K "K ^ " " . . / I O - M \; 1 = 1,...,P (18.21),...,]f , __ \r \r , __ I; . M ^mm,! ^max,! " MTh e s e f u n c t i o n s are also m u l t i p l i e d b y a u s e r s u p p l i e d w e i g h t i n g cons tan t , to

(>0) w hich s h o u ld have a large value d u r in g the early i terations of the Gauss-Newton m ethod w h e n th e parameters are away from t he i r opt imal va l ues . In g e n -eral, c o s h o u ld be reduced as the parameters approach th e o p t i mum so that th econtribut ion of the penal ty func t ion is essent ia l ly negl ig ib le (so that no bias is in-troduced in the parameter est imates). I f / ? pena l ty funct ions are incorporated thenthe overal l ob ject ive funct ion b e c o m e sStotal (k) = SLS + Spr ior + Spe na l t y (18.22)

w her e



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384 Chapter 18

Spe n a hy =c o >i ( k i ) (18.23)

A s seen above , the objec t ive func t ion is m o d i f i e d in a way tha t i t increasesqui te s ignif icant ly as the solut ion approaches th e constraints but remains pract i -cally un c h a n g e d ins ide the feasible reg ion . Inside the feasible region, ;(kj) issmall an d resul ts in a s m a l l contribut ion to the main d iagona l e l e m e n t of matrix A.C o n s e q u e n t l y , Ak^1' wi l l not be affected s ignif icant ly . I f on the other hand, th eparameter v a lue s are near th e b o un d a ry ,(ki) becomes large and dominates th ediagonal e l e me n t of the parameter to w h i c h the constraint i s appl ied . As a resul t ,the va lue of A k ^ + 1 ) is very sma l l an d k ( is not a l l owed to cross that b o u n d a r y .

18.3.4.2 Reservoir Characterization Using Au tomatic History MatchingAs i t was pointed o u t earlier, there a re two major prob lems associated withthe history m a t c h i n g of a reservoir . The first is the correct representat ion of thereservoir with a grid cell mode l with a l imited n u m b e r o f zones , and the second isthe regression ana lys i s to obtain th e op t ima l parameter va lues . W h i l e the secondp r o b le m h as been addressed in th i s chapter , there i s no r igorous m e t h o d ava i lab lethat someone m ay fo l low to address th e first pr ob lem. Engineers rely o n in forma-t ion from other sources such as se i smic mapping an d geological data. However,

Tan and Kalogerak i s (1993) h a v e s h o w n that automat ic history ma t c h i n g can alsob e use fu l in de te rmin ing the existence of impermeable boundar ies , assessing th eposs ib i l i ty of reservoir extensions, prov id ing est imates of the vo lumes of oi l , g asan d water in p l a c e w h i l e at the same t ime l imi t ing th e parameter est imates to stayw i t h i n rea l is t ic l imi t s .They performed an extensive case s tudy to demonstrate the use of automatichistory m a tc h in g to r e s e r v o i r character iza t ion . F o r ex ampl e , i f the est imated per-meabi l i ty o f a part icular zone i s unrea l i s t ica l ly s m a l l compared to geologica l in -format ion, there is a good chance tha t an i mp e rme a b le barrier is present . Similar lyif th e est imated porosi ty o f a z o n e approaches unrea l i s t ica l ly h igh values , chancesare the zone of the reservoir s h o u l d be expanded bey ond i ts curren t bou ndary .Based on the analys is of the case s tudy, Tan and Kalogerak i s (1993) rec-o m m e n d e d th e f o l lo w in g practical g u id e l i n e s .Step 1 . Const ruct a postulated gr id cell model of the reservoir b y u s i n g all theava i lab le in fo rmat ion .Step 2 . S ubd iv ide the model into zones b y f o l lo w in g an y geologica l zo nat ionas closely as poss ib le . T h e zones fo r poros i ty and permeabi l i ty needno t b e identical and the cel ls al located to a zone need not be cont igu-

ous.



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Step 3 . Pr o v id e est imates of the mos t p ro b a b le v a lu e s of the parameters an dthe i r s tandard devia t ion (kj, o kj , j = l , . . . , p ) .Step 4. Provide the b o u n d a r y ( m i n i m u m an d m a x i m u m v a lu e s ) that each pa-rameter s h o u l d b e cons t ra ined w i t h .Step 5. Ru n the auto ma tic history ma t c h i n g mo d e l to obta in estimates of theu n k n o w n parameters .Step 6. W h e n a converged so lu t ion has been achieved or at the m i n i m u m ofthe object ive function check the variances of the parameters and thee igenvec tors cor respond ing to the s m a l l e s t e ig e n v a lu e s to ident i fy an y

h i g h l y correlated z o n e s . C o m b i n e an y adjacent z o n e s o f h ig h var iance .Step 7. In order to m o d i f y and improve the postulated gr id cell representat ionof the reservoir , analyze any zones wi th values close to these con-straints.Step 8. Go to Step 3 if yo u have made any changes to the model . Other w i seproceed with the s tudy to predict fu ture reservoir performance, etc.

I t s h o u l d be noted that th e nature of the problem is such that it is practical lyimpossible to obtain a pos tu la ted model w h i c h is able to uniquely represent th ereservoir . As a result, i t is required to c o n t i n uo us ly update th e match w h e n addi-t ional information becomes ava i lab le and poss ib ly also change the grid cell de-scription of the reservoir .B y u s i n g au tomat ic his tory m a t c h i n g , th e reservoir engineer i s no t facedw i t h th e u s u a l d i l e m m a w h e t h e r to reject a par t icu la r grid cell m o d e l because it isnot a good approximat ion to the reservoir or to proceed with the param eter searchbecause the best set of parameters has not been de termined yet.

18.3.5 Reliability of Predicted Well Performance Through AutomaticHistory MatchingIt is of interest in a reservoi r s imulat ion s tudy to c o m p u t e fu ture pr oduc-tion levels of the his tory matched reservoir under al ternat ive deplet ion plans . Inaddit ion, th e sensit ivity of the anticipated performance to different reservoir de-scr ipt ions is also evaluated. Such s tudies contr ibute towards assess ing the risk

associated with a part icular deplet ion p l a n .Kalogerakis and T o m o s (1995) presented an efficient procedure for thedeterminat ion of the 95 percent conf idence intervals of forecasted horizontal wel lCopyright 2001 by Taylor & Francis Group, LLC


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386 Chapter 18

p ro du c t i o n rates. S u c h c o m p u t a ti o n s are u s u a l l y th e f ina l task of a reservo i r s i m u -la t ion s tudy . T h e p ro c e d ure is based on the G a us s -N e wt o n m e th o d for the est ima-t ion of the reservoir parameters as was descr ibed earl ier in t h i s chapter. The est i -mat ion of the 95 % c o n f i d e n c e in te rva ls o f w e l l p r o d u c t io n rates w a s d o n e a s fol-l ows . The product ion rates for oil (Q O J), water (Q W J) and gas (Qa) from a verticalo r hor izonta l wel l completed in layer j are given b y t h e f o l l o w i n g e q ua t i o n sQ o j = W p i M 0 ( P b h +H-P g r id)/B0 (18.24a)

Q W J = Wp iMw (P b h +H-P g r i d )/Bw (1 8 .2 4b )0 & j=Wp iMg( P bh+H-P g r i d )Eg+R sQ 0 (18.24c)

where W p; is the wel l product iv i ty index g iven by

2 7 i h , / k t , k vWpi =" s k i n

and M is defined for each phase as the rat io of the re lat ive perm eabi l i ty to the vis-cosity of the oil,water or gas, Pbh is the f l o w i n g bot tom hole pressure, H is thehydrostat ic pressure (assumed zero for the top layer), B 0 and B w are the format ionv o l u m e factors for oil and water , R s is the so lu t ion gas-oil ratio, Eg is the gas ex-p a n s io n factor, kh and k v are the hor izonta l and vertical permeabi l i t ies and h is theb l ock th ickness . T h e determinat ion o f r0 , th e effect ive w e l l rad ius an d F sk in , th eskin factor , for horizontal wel ls has been th e subject of cons iderable discuss ion inth e l i terature.Th e total o i l p roduc t ion rate is obtained by the sum of a l l p ro d uc i n g layersof the w e l l , i.e.,

N LQ0 = Q0(x,k) = Q o (18.26)

Simi lar express ions are obtained for the total g as (Q g) an d water (Q w) pro-duct ion rates by the su m of the Q g j and Q w j terms over all N L layers.I n order to der ive est imates of the uncertainty of the computed w e l l pro-duct ion rates, the express ion for the w e l l product ion rate Q c (c=o,w, g) is l ine-



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arized a r o u n d th e o p t i m a l r e s e r v o i r paramete r s to arr ive a t an express ion for thed e p e n d e n c e o f Q c o n th e paramete r s (k) at any p o i n t in t i m e t, n a m e l y ,T

(k-k*) (18 .27)

wh ere k* represents the con verged va lues of the parameter vector k .T he overall sensitivity of the well production rate can now be est imatedt h ro ug h i m p l i c i t different iat ion from th e sensi t ivi ty coefficients already compu tedas part of the Gauss -New ton method c o mp ut a t i o n s as fo l lows

(18.28)1Id kw h e r e G(t) is the paramete r sensi t iv i ty matr ix descr ibed in Ch a p t e r 6. T h e e v a lu a -t ion of the part ial derivat ives in the above equat ion is performed at k=k* and att ime t .

Once , (dQ c/ dk) has been compu ted , the variance O Q C of the w e l l pr oduc-t ion rate can be readi ly obta ined b y t ak ing variances from both s ides of Equat ion18.27 to yie ld ,

H a v i n g est imated O Q , the (l-a)% c o n f i d e n c e in terval fo r Q c a t anypo in t in t i m e is g iv e n b y

Qc(k* t) - t* /2aQ < Q c ( k , t ) < Q c ( k * , t ) + C2oQ (18.30)where t^ / 2 is obta ined from th e statistical tables for the t-distribution for v de-grees o f freedom wh i c h a re e q u a l to the total n u m b e r o f observat ions m i n u s th en u m b e r o f u n k n o w n parameters .

If the re a re N w p r o d u c t io n w e l l s th e n th e (l-a)% conf idence in te rva l fo rth e total production from th e reservo i r is given by

c t o t < Q c t o t ( k , t ) < Qctot(k* t) +C2o- (18.31)c t o tCopyright 2001 by Taylor & Francis Group, LLC


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388 Chapter 18

wh e re

and tot

(18.32)

is th e standard error of est imate of the total reservoir produc t ion ratewhich i s g iven by

Q c l t o t d k C O K ( k ) dk (18.33)

It is noted that the part ial derivat ives in the above variance express ionsdepend o n t ime t and therefore the variances s h o u l d b e compu ted s imu l taneous l ywith the state variables and sensit ivi ty coeff icients. Final ly , th e confidence inter-vals of the cumulative production of each w e l l and of the total reservoir are calcu-la ted by in tegrat ion w i t h respect to t i m e (Ka lo g e ra k i s an d T o m o s , 1995).

18.3.5.1 Quantification of RiskO n c e the standard error of est imate of the mean forecasted response hasbeen est imated, i.e., the uncertainty in the total product ion rate, one can computethe probabi l i ty level , a, for w h i c h th e m i n i m u m total product ion rate is b e lo wsome pre-determined va l ue based on a previous ly conducted e c o n o mi c analys is .Such ca lcu la t ions can be performed as part of the post-processing ca lcu la t ions .

18.3.5.2 Multiple Reservoir DescriptionsWith the he lp of au tomat ic his tory matching, the reservoir engineer canarrive at several p l aus ib l e history matched descript ions of the reservoir . Thesedescr ip t ions m ay differ in the grid block representat ion of the reservoir, exis tenceo f sea l ing and non-sea l ing faul ts , o r s impl y dif ferent zonat ions of constant poros-ity or permeabi l i ty (Kalogerakis an d Tomos , 1995). In addi t ion , we ma y assign toeach one of these reservoir mo d e l s a probabi l i ty o f b e i n g th e correct one.Th i sprobabi l i ty can be based o n addi t ional geological in format ion about the reservoiras wel l as the plaus ib i l i ty of the va l ues of the est imated reservoir parameters.Thus , for the r th descript ion of the reservoir , based on the material presentedearlier, one can c o m p u t e expected total oi l p roduct ion rate as w e l l as the m i n i m u ma n d ma x i mu m p ro d uc t io n rates corresponding to a desired conf idence level (1-a).T h e m i n i m u m an d m a x i m u m total o il product ion rates for the r th reservoirdescript ion are given b y



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andQ(r )

Qx(r)

r (k*t) - t v / , oo t o t v ' a/2 Q0 tot

( k , t ) = Q(r )

( k * , t ) + tv,,0^' x v ' ,,o t o t a / 2 otot

(18.34)>

(18 .35)'Next , the probabi l i ty , P b(r) that th e r th m o d e l is the correct one, can be usedto compu te th e expected overall field production rate based on the product ion data

from N m dif ferent reservoi r m odels , n a m e l y

(18.36)

Si mi l a r l y th e m i n i m u m a n d m a x i m u m b o u n d s a t (l-a)100% c o n f i d e n c elevel are comp uted as

and( 18 . 3 7)

r= lAgain, the above l imi t s can be used to compute the r isk level to meet adesired product ion rate from th e reservoir .

18.3.5.3 Case Study - Reliability of a Horizontal Well PerformanceKalogerakis and T o m o s (1995) demo nst ra ted the above methodo logy fo ra hor izonta l w e l l b y adapt ing th e e xa mp le used b y C ol l in s et al . (1992) . They as-s ume d a per iod of 1,250 days fo r his tory match ing purposes and pos tu la ted three

different descript ions of the reservoir . A ll three models matched the his tory of thereservoir pract ical ly with the same success . However, the forecasted performancefrom 1,250 to 3,000 days w as q u i t e different. Thi s is s h o w n in Figures 18.28a an d18.28b wh ere the pred ic t ions by Models B and C are compared to the actual reser-voir performance (g iven by Model A). In addit ion, the 95% confidence in terval sfo r Model A are shown in Figures 18.29a and 18.29b. Form these plots it appearsthat the uncertainty is not very h i g h ; how ever , it shou ld b e kept in mind that thesecompu ta t i ons were m a d e under the n u l l h yp o t h e s i s that al l other parameters in the



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390 Chapter 18

reservoir mo d e l (i.e., relative permeabi l i t ies) and the grid cel l descript ion of theare a l l k n o w n prec i se ly .

o5= Vteter0 - EJXOCCCO: :a:0 1000

onfin

6000-

4000-

2000-

n

Historylmatched| horeCt

I'1 2x10s-0

0x10-[

Oil~---

,.-/.../

---/'' ,-'\Gas

D.UXIUS"s . 4.5x105-tjuDO^j J.UX1UI :o>S 1.5x105-30

Wrter^ \

/'

Hf /1M/

fC

) 1000 2000 3000 0 1000 2000 300Q(days) The(days)

1501.Applied Parameter Estimation for Chemical Engineers (Chemical Industries) by Peter Englezos...

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Transcript of 1501.Applied Parameter Estimation for Chemical Engineers (Chemical Industries) by Peter Englezos...