Optimization of perforation distribution for horizontal wells based on genetic algorithms

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1 3 Pet.Sci.()7:3-38 DOI.7/s Optimiztion of perfortion distribution for orizontl ells bsed on genetic lgoritms Wng Ziming, Wei Jingung, Zng Jin,, Gong Bin 3 nd Yn Hiyun Scool of Petroleum Engineering, Cin University of Petroleum, Beijing 49, Cin Cin United Colbed Metne Co., Ltd., Beijing, Cin 3 Fculty of Engineering, Peking University, Beijing 84, Cin Cin University of Petroleum (Beijing) nd Springer-Verlg Berlin Heidelberg Abstrct: Erly ter brektroug nd rpid increse in ter cut re lys observed in igpermebility completion intervls en perfortions re uniformly distributed in te ellbore in eterogeneous reservoirs. Optimiztion of perforting prmeters in prtitioned sections in orizontl intervls elps omogenize te inflo from te reservoir nd tus is criticlly importnt for ennced oil recovery. Tis pper derives coupled reservoir-ellbore flo model bsed on inflo controlling teory. Genetic lgoritms re pplied to solving te model s tey excel in obtining te globl optimum of discrete functions. Te optimized perforting strtegy pplies lo perfortion density in igpermebility intervls nd ig perfortion density in lo-permebility intervls. As result, te inflo profile is omogenized nd idelized. Key ords: Well completion, perfortion optimiztion, genetic lgoritms, prtition, orizontl ell Introduction In te conventionl completions for orizontl ells, perfortions re uniformly distributed long te orizontl ellbores. Permebility eterogeneity long orizontl ellbore ill cuse ig rte of fluid inflo from te reservoir into te ellbore in ig-permebility intervls, tus n erly ter brektroug ill occur in tese intervls. On te oter nd, te pressure drop sos ig vlue t te eel of orizontl ellbores (Dikken, 99; Sric et l, 994). Tis lso leds to iger inflo rte t te eel nd n erlier ter brektroug. In 99, by modeling perfortions s finite, Lndmn nd Goldtorp (99) estblised perfortion optimiztion model for orizontl ells nd nlyzed te perfortion distribution en te fluid flo from te reservoir to te ell is uniform. Hoever, tey did not tke into ccount te effect of formtion dmge nd crused zone dmge. In 993, tking into considertion te formtion dmge, perforting dmge, ell inclintion, nd deposit boundry, Mrett nd Lndmn (993) estblised n optimiztion model of perforting prmeters for orizontl ells by introducing te perfortion skin fctor into te coupled model. Teir model is cpble of computing perfortion density in presence of ellbore inclintion nd permebility eterogeneity. In *Corresponding utor. emil: ngzm@cup.edu.cn, ellcompletion@6.com Received ovember 8, 8 997, Aseim nd Oudemn clssified te floing pressure drop into tree prts inflo pressure drop, perfortion pressure drop, nd ellbore pressure drop. Tey modeled te perfortions s microellbores. In consequence, penetrtion lengt, dimeter, nd spcing re modeled s ell lengt, dimeter, nd dringe rdius, respectively. By clculting every floing pressure drop, n nlyticl model of perfortion optimiztion for orizontl ells s estblised bsed on pressure continuity. Te optiml distribution of perfortion density long te ellbore nd te effect of optimized perfortion on te orizontl ell flo efficiency ere nlyzed. Zou et l (; ; 7) studied te perfortion density long te ellbore using te sme metod s Aseim nd Oudemn. Wng et l (5) studied te optimiztion of perfortion distribution for orizontl ells in omogeneous reservoirs. Wng et l (7) studied te effect of perfortion distribution on inflo profiles in orizontl ells. Teir reserc illustrted tt te completion prmeters re te min fctors influencing inflo profiles in orizontl ells. Hoever, permebility eterogeneity long te ellbore s not considered nd optimiztion of perforting prmeters in ec prtitioned segments s not been conducted ccording to te distribution of permebility long orizontl ellbore. In tese studies, perfortion distribution long te ellbore s only qulittively nlyzed, nd engineering prctices ve not been ell implemented in order to provide relistic boundry conditions for perfortion optimiztion.

2 Pet.Sci.()7: Coupled model of ellbore nd eterogeneous reservoir flo As son in Fig., te inflo profile ill be non-uniform becuse te permebility is different long orizontl ell. An ide of optimiztion of perforting prmeters for orizontl ell is to prtition te orizontl ellbore into number of sections nd optimize te perforting prmeters in ec section ccording to flo resistnce in te reservoir nd ellbore. As result, te inflo profile of ec section is optimized to be s close s possible to n idel inflo profile ic is only vlid for infinite-conductivity, omogeneous reservoirs. Heel r Idel inflo profile Actul inflo profile Pressure curve long te ellbore Fig. Inflo profiles long orizontl ell. Reservoir flo model L K r K z, i K K z, i z, i K S t, i L K v 3 L K v e, i exp.75 L Kv v ln sin Toe Te folloing ssumptions re necessry to te flo model: ) Only single-pse incompressible etonin fluid is involved; ) Isoterml condition; 3) Te reservoir is of uniform tickness; 4) Te reservoir formtion fr y from te ellbore is omogeneous, nd its permebility is te verge reservoir permebility; 5) Te permebility is uniform in ec section long te ellbore. As son in Fig., te orizontl ell is divided into sections, ec it lengt of ΔL nd ell inclintion of θ. Te distnce beteen ec section nd te reservoir bottom is z,j. For uniform inflo from te reservoir into te ellbore under pseudo stedy-stte conditions, ec section cn be looked upon s verticl ell, nd te equivlent rdius of ec verticl ell section is: ere r e,i is te equivlent rdius of section i, m; ΔL is te section lengt, m; is te reservoir tickness, m; r is te rdius of te orizontl ellbore, m; z, i is te distnce beteen section i nd te bottom of te reservoir formtion, () m; K is te orizontl permebility, -3 μm ; K v is te verticl permebility, -3 μm ; S t, i is te totl skin of section i, dimensionless. As son in Fig. 3, bsed on te superposition principle, te potentil t n rbitrry point is expressed s follos: M qiln ri C ().5487 i Tking positions on te supply boundry nd ellbore of ec equivlent verticl ell, te reltionsip beteen bottom-ole pressure nd flo rte t different positions of orizontl ells cn be derived s: p p Lq r e e fr, i s, nln.5487k n r n,i n,i z, M M θ Fig. Cross-sectionl geometric prmeters for conventionl orizontl ell re, i ( i n) ( xn xi) ( yn yi) ( i n) ere μ is te fluid viscosity, mp.s; p e is te reservoir boundry pressure, MP; r e is te supply rdius, m; p fr,i is te pressure t te ll of section i, MP; q s,i is te inflo rte of section i, m 3 /(s m); r n,i is te distnce from te centre of section n to te center of section i, m; x i nd y i re te coordintes of te center of section i, m. M r r r i y L M i r M - M r - r x q q q i q - q Fig. 3 Potentil superposition principle for ell sections. er ellbore flo model Te totl resistnce to flo from te reservoir to te eel of te orizontl ell consists of te resistnce to flo in te reservoir fr y from te ellbore, te inflo resistnce nerby nd troug te ell, nd te complicted flo resistnce in te ellbore (Fig. 4). As result of permebility eterogeneity t different positions long te ellbore, te inflo rte is different long te orizontl ell. Tis ill speed up te inflo in ig-permebility regions nd slo don te inflo in lopermebility regions. Te inflo profile is tus fluctuting long te ole ellbore. In order to investigte te influence of permebility eterogeneity on orizontl ell inflo (3) (4)

3 34 Pet.Sci.()7:3-38 d profiles nd te optimiztion result of completion prmeters, te equivlent dimeter or investigtion dimeter in te eterogeneous region is ssumed to be d. According to te definition of te skin fctor, te eterogeneous model of te pseudo skin fctor ner te ellbore cn be defined s: K d S ( xi ) ln (5) K( xi ) d Te totl skin of ec infinitesiml ell section cn be represented s: S ( x ) S ( x ) S ( x ) (5b) t i p i i ere K (x i ) is te verge permebility of infinitesiml ell section i, -3 μm ; K is te verge permebility long te ole ellbore, -3 μm ; d is te dimeter of te region exibiting permebility eterogeneity, m; d is te ell dimeter, m; S t (x i ) is te totl skin of infinitesiml ell section i, dimensionless; S p (x i ) is te perfortion skin fctor of infinitesiml ell section i, dimensionless, clculted from te model proposed by Furui (4); S (x i ) is te pseudo skin fctor of ell infinitesiml section i, dimensionless..3 Wellbore flo model Te pressure drop cross section i in te orizontl ellbore is derived ccording to te conservtion of momentum nd mss. It is te sum of ccelertion pressure drop, friction pressure drop, nd grvity pressure drop: p p p p p f, i f, i cc, i ll, i g, i Te ccelertion pressure drop cn be expressed s: 3qs, i pcc, i q 4, i L d Te friction pressure drop cn be expressed s: 3 ft, i pll, i q 5, i L d Te grvity pressure drop cn be expressed s: pg, i gcosi L Substituting Eq. (7) into Eq. (6) gives: pf, i pf, i 3qs, i 3 ft, i q 4, i q 5, i gcosil d d Te cross-sectionl flo rte is: q q L, i s, j ji P WS d W P WS P r Inflo region q W P f Reservoir flo L Wellbore flo Fig. 4 er-ellbore flo q WS q WS (6) (7) (7b) (7c) (8) (9) ere p f,i is te ellbore pressure in te center of infinitesiml ell section i, MP; p cc,i is te ccelertion pressure drop in infinitesiml section i, MP; p ll,i is te friction pressure drop in infinitesiml section i, MP; p g,i is te grvity pressure drop in infinitesiml section i, MP; q,i is te cross-sectionl flo rte of infinitesiml section i, m 3 /s; f t,i is te totl vrible mss inflo friction coefficient of infinitesiml section i, dimensionless, clculted from te model proposed by Ouyng et l (996); θ i is te devition ngle of infinitesiml section i, degrees..4 Derivtion of coupled flo model At ny point on te ell ll, te reservoir fluid pressure is equl to te ellbore fluid pressure. Tis nturl boundry condition yields: P f,i = p fr,i () Wen te ell produces t constnt bottom-ole floing pressure, te boundry condition is: pf, pf () Combining Eq. (3) it Eq. () gives A X b () Combining Eq. (8) it Eq. () gives: A X b it X q q q s, s, s, X p p p b f, f, f, L r e ni, ln.5487k r n,i pe p pe p p p pe p f, f, e f, f, T T (b) 3qs, 3 ft, q 4, q 5, gcos L pf d d 3qs, 3 ft, q 4, q 5, gcos L d d b 3qs, 3 ft, q 4, q 5, gcos L d d 3qs, 3 ft, q 4, q 5, gcos L d d

4 Pet.Sci.()7: A,,,,,,,, A 3 Optimiztion of perforting prmeters 3. Optimiztion procedure Te optimiztion procedure for perforting prmeters in ec section is given s belo: ) According to te permebility long te ellbore in eterogeneous reservoirs, te permebility function K is developed it te interpoltion nd ten te verge permebility K is clculted. ) Idel inflo rtes q id (x ), q id (x ),..., q id (x n ) re clculted from te model ic couples orizontl ellbore flo it inflo from te eterogeneous reservoir into te ellbore under te condition of omogeneous permebility K in n infinite-conductivity reservoir. 3) Idel skin fctors S id (x ), S id (x ),..., S id (x n ) re clculted from te coupled model under te condition of omogeneous permebility K in n infinite-conductivity reservoir. 4) According to permebility function K, eterogeneous model of totl skin fctor, nd te inflo profile velocity control teory, te perforting prmeters sould stisfy min Sid ( x) St ( x) s muc s possible. 5) According to te ctul field limittions nd specifictions, te optimiztion of perforting prmeters in ec section is cieved using genetic lgoritms. 3. Appliction of genetic lgoritms 3.. Encoding of te perforting prmeters Te optimiztion of perforting prmeters for ec section in orizontl ell is to minimize Sid ( x) St ( x) under some constrints. Te perforting prmeters include te bullet type, perfortion density, perfortion psing, nd perfortion loction. Te binry code is selected s son in Fig. 5. Te solution spce, i.e., te lengt of binry code string cn be cnged to blnce te ccurcy nd efficiency. 3.. Determintion of te fitness function Te fitness of individuls is evluted using te objective function, min S id -S t. Te evlution function cn be given by expressing te totl skin s function of coordintes nd perforting prmeters: min S ( x ) S ( x, c, n,, ) (3) id i t i pi pi pi pi ere c pi is te bullet type, dimensionless; n pi is te perfortion density, m - ; θ pi is te perfortion psing, degrees; α pi is te perfortion zimut, degrees. Constrints for optimiztion of perforting prmeters for ec section in te orizontl ell include: () mximum/ minimum perfortion density vlues exist for ec type of perforting bullet, () te lloed rnge of pse ngle, nd (3) optimized prmeters sould be integers, suc s bullet type, perfortion density, perforting psing, nd perfortion position. 4 Computtionl exmples 4. Prmeters Tble summrizes te input prmeters in te exmple. Permebility vlues long te orizontl ell re son in Tble. Four cses, it different permebility distributions, re modeled respectively, nd te verge permebility of ec cse is 5 md. Te dimeter of te eterogeneous region is tree times te ell dimeter. Tble Vlues of relted prmeters Supply rdius, m 5. Oil viscosity, mp. s 5. Reservoir tickness, m. Distnce beteen te ellbore to te formtion bottom, m Oil formtion volume fctor.5 5. Degree of dmge.4 Outer csing dimeter, m.397 Dmged dept, m. Csing ll tickness, m.97 Fluid density, kg/m 3 95 Reltive rougness of csing ll. Liquid production, m 3 /d Well lengt, m 4. Perforting gun - Mximum perfortion density, sots/m 6 Perfortion lengt, m.55 Perfortion dimeter, m. umber of prtitioned sections Crused zone tickness, m. Crused zone permebility, -3 μm. min S ( x ) S ( x, c, n,, ) id i t i pi pi pi pi Fig. 5 Binry code of perforting prmeters 4. Interprettion of results In order to study te effect of permebility eterogeneity long te orizontl ell on te optimiztion of perforting prmeters, optimized perforting prmeters ere nlyzed for four cses. In tese cses, only te optimiztion of perfortion density s crried out to equlize te specific inflo into ec section of tenty -meter ell sections. Te optimized perfortion density vlues for ec section re listed in Tble 3. Te optimized perfortion density in ec

5 36 Pet.Sci.()7:3-38 Tble Permebility vlues long te orizontl ell Tble 3 Optimized perfortion density long te orizontl ell Distnce to te toe of te orizontl ell, m Permebility, -3 μm Cse Cse Cse 3 Cse Rtio of mximum to verge permebilities..5. Rtio of mximum to minimum permebilities section is different long te orizontl ell. Te perfortion density is lo in ig-permebility regions nd ig in lopermebility regions. Te inflo crcteristics of te modified nd conventionl completions re plotted in Figs. 6 nd 7. For te uniformly perforted cse, te specific inflo rte cnges severely s te permebility eterogeneity increses (see Fig. 6). For te optimized perforted cse, te specific inflo rte from ig permebility regions reduces nd te specific inflo rte from lo permebility regions increses. Te inflo profile from te reservoir to te ellbore is close to idel, tus delying ter brektroug nd voiding rpid increse in ter cut. Comprisons of pressure drops, s son in Fig. 7, indicte te pressure drop fluctution is more noticeble long te orizontl ell it more eterogeneity. Hoever, te vrition in perfortion density s little influence on pressure Distnce to Optimized perfortion density, sots/m te toe of te orizontl ell, m Cse Cse Cse 3 Cse otes: Perforting gun: -; Penetrting crge: DP44RDX- drop cross te entire orizontl ell, i.e. te pressure drop long te entire orizontl ell lmost remins te sme fter optimiztion. Tble 4 sos tt te productivity index of te ell reduces sligtly fter te perfortion density is optimized in ec section in te orizontl ell, nd te reduction in te productivity index is more obvious s te level of permebility eterogeneity is greter. Cse Tble 4 A comprison of productivity indexes before nd fter perfortion optimiztion Productivity index m 3 /(d. MP) Before optimiztion After optimiztion Reduction in productivity index m 3 /(d. MP) Loss rtio of productivity index % Cse Cse Cse Cse

6 Pet.Sci.()7: Inflo rte per unit ell lengt, m 3 / (d. m) Idel () Cse Inflo rte per unit ell lengt, m 3 / (d. m) Idel (b) Cse Inflo rte per unit ell lengt, m 3 / (d. m) Idel (c) Cse 3 Inflo rte per unit ell lengt, m 3 / (d. m) Idel (d) Cse 4 Fig. 6 Optimiztion of te perfortion density to cieve more uniform specific inflo Pressure drop long te orizontl ell, P () Cse Pressure drop long te orizontl ell, P (b) Cse Pressure drop long te orizontl ell, P (c) Cse 3 Pressure drop long te orizontl ell, P (d) Cse 4 Fig. 7 Comprison of pressure drops before nd fter optimiztion of te perfortion density

7 38 Pet.Sci.()7:3-38 In generl, te optimiztion of perfortion distribution in ec section does not ve significnt effect on te productivity index of te orizontl ell. 5 Conclusions ) A model is developed ic incorportes orizontl ellbore flo it flo from te eterogeneous reservoir. Te optimiztion of perfortion density is cieved using genetic lgoritms. ) of te perfortion distribution for ec section in te orizontl ell, te inflo rte fluctutes severely s te reservoir permebility cnges srply. After optimiztion of perfortion distribution in ec section, te inflo from ig permebility regions reduces nd te inflo from lo permebility regions improves. Terefore, te inflo from te reservoir to te ellbore is close to idel, tus reducing te risk of premture ter brektroug nd voiding rpid increse in ter cut. 3) Perfortion distribution is not uniform from te toe to te eel of te orizontl ell. Te perfortion density is lo in ig-permebility regions nd ig in lopermebility regions. Te productivity index reduces sligtly fter perfortion distribution is optimized in ec section of te orizontl ell nd te reduction in productivity index is sligtly more obvious s te level of permebility eterogeneity is greter. Generlly speking, te optimiztion of perfortion distribution for orizontl ells does not ve significnt effect on te productivity index. Acknoledgements Tis reserc is supported by tionl Scientific Project (o. 8ZX54-3). References As eim H nd Oudemn P. Determintion of perfortion scemes to control production nd injection profiles long orizontl ells. SPE Drilling & Completion (): 3-8 (Pper SPE 975) Dik ken B J. Pressure drop in orizontl ells nd its effect on production performnce. Journl of Petroleum Tecnology. 99. : Furui K. A Compreensive Skin Fctor Model for Well Completions Bsed on Finite Element Simultions. P.D Tesis. Te University of Texs t Austin. 4 Ln dmn M J nd Goldtorpe W H. Optimiztion of perfortion distribution for orizontl ells. Pper SPE 35 presented t SPE Asi-Pcific Conference, 4-7 ovember 99, Pert, Austrli Mr ett B P nd Lndmn M J. Optimiztion of perfortion distribution for orizontl ells in reservoirs it boundries. Pper SPE 5366 presented t SPE Asi Pcific Oil nd Gs Conference, 8- Februry 993, Singpore Ouy ng L B, Arbbi S nd Aziz K. Generl ellbore flo model for orizontl, verticl, nd slnted ell completions. Pper SPE 3668 presented t te 996 SPE Annul Tecnicl Conference nd Exibition, 6-9 October 996, Denver; lso publised t SPE Journl (): 4-33 Sr ic C, Hciislmoglu M, Rgvn R, et l. Influence of ellbore ydrulics on pressure bevior nd productivity of orizontl gs ells. Pper SPE 8486 presented t SPE Annul Tecnicl Conference nd Exibition, 5-8 September 994, e Orlens, Louisin Wn g S P, Li Z P, Luo Y, et l. Anlysis of perfortion tunnel distribution mode in orizontl ells. Drilling & Production Tecnology. 7. 3(): 39-4 (in Cinese) Wn g Z M, Xu J, Wng X Q, et l. Study of vrible density perforting model of to-pse flo in orizontl ells. Journl of te University of Petroleum, Cin. 5. 9(3): (in Cinese) Zo u S T nd Su X Y. Specific properties of orizontl ellbores nd optiml nlysis of te perfortion density. Journl of te University of Petroleum, Cin.. 5(6): (in Cinese) Zo u S T, M D Q nd Liu M. Optimiztion of perfortion tunnel distribution in perforted orizontl ells. Journl of te University of Petroleum, Cin.. 6(3): 5-54 (in Cinese) Zou S T. Anlysis of perfortion density optimiztion in perforted orizontl ells. Petroleum Drilling Tecniques. 7. 5(5): (in Cinese) (Edited by Sun Ynu)

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