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1 SPE 757 Decline Curve Analysis Using Type Curves Evaluaion of Well Performance Behavior in a Muliwell Reservoir Sysem T. Marhaendrajana, Schlumberger, and T. A. Blasingame, Texas A&M Universiy Copyrigh, Sociey of Peroleum Engineers Inc. This paper was prepared for presenaion a he SPE Annual Technical Conference and Exhibiion held in New Orleans, Louisiana, 3 Sepember 3 Ocober. This paper was seleced for presenaion by an SPE Program Commiee following review of informaion conained in an absrac submied by he auhor(s). Conens of he paper, as presened, have no been reviewed by he Sociey of Peroleum Engineers and are subjec o correcion by he auhor(s). The maerial, as presened, does no necessarily reflec any posiion of he Sociey of Peroleum Engineers, is officers, or members. Papers presened a SPE meeings are subjec o publicaion review by Ediorial Commiees of he Sociey of Peroleum Engineers. Elecronic reproducion, disribuion, or sorage of any par of his paper for commercial purposes wihou he wrien consen of he Sociey of Peroleum Engineers is prohibied. Permission o reproduce in prin is resriced o an absrac of no more han 3 words; illusraions may no be copied. The absrac mus conain conspicuous acknowledgmen of where and by whom he paper was presened. Wrie Librarian, SPE, P.O. Box , Richardson, TX , U.S.A., fax Absrac In his paper we presen a new muliwell reservoir soluion and an associaed analysis mehodology o analyze single well performance daa in a muliwell reservoir sysem. The key o his approach is he use of field cumulaive producion daa and individual well flow rae and pressure daa. Our new soluion and analysis mehodology couples he single well and muliwell reservoir models and enables he esimaion of oal reservoir volume and flow properies wihin he drainage area of an individual well wih he analysis performed using a single well reservoir model (ype curve). This muliwell analysis using a single well model is made possible by a coupling of he single well and muliwell soluions based on a oal maerial balance of he sysem. The daa required for his approach are readily available in pracice: basic reservoir properies, fluid properies, well compleion daa, and well rae (and pressure) daa and cumulaive producion daa for he enire field. Currenly, all exising decline ype curve analyses assume a single well in closed sysem (or single well wih consan pressure or prescribed influx a he ouer boundary). In many cases a well produces in associaion wih oher wells in he same reservoir and unless all wells are produced a he same consan rae or he same consan boomhole flowing pressure, nonuniform drainage sysems will form during boundary-dominaed flow condiions. Furhermore, i is well esablished ha new wells seal reserves from older wells, and his behavior is commonly observed in he producion behavior. Our new approach accouns for he enire producion hisory of he well and he reservoir and eliminaes he influence of well inerference effecs. This approach provides much beer esimaes of he in-place fluids in a muliwell sysem, and he mehodology also provides a consisen and sraighforward analysis of producion daa where well inerference effecs are observed. This work provides he following deliverables:. A new muliwell reservoir soluion for which he formulaion yields a simplified form for an arbirary (individual) well during boundary-dominaed flow condiions.. A complee analysis mehodology for oil and gas reservoir sysems based on convenional producion daa (on a perwell basis) as well as he cumulaive producion of he enire field. 3. A sysemaic validaion of his approach using a numerical reservoir simulaor for cases of homogeneous, regionally heerogeneous, and randomly heerogeneous reservoirs. 4. An applicaion o a large gas field (Arun field, Indonesia). This approach provides very consisen esimaes of inplace fluids and reservoir properies. All analyses (simulaed and field daa) clearly demonsrae ha he effecs of well inerference on individual wells were eliminaed as a resul of his analysis mehodology. Inroducion The single well model has been widely used o forecas he producion decline of reservoir and wells sysems. Alhough he analyical soluions for a single well in circular reservoir came as early as in 934, his effor was pioneered by Arps, who presened a suie of (empirical) exponenial and hyperbolic models for his purpose. Fekovich 3 presened he heoreical basis for Arps s producion decline models using he pseudoseady-sae flow equaion. He also developed decline ype curves ha no only enable us o forecas well performance bu also o esimae reservoir properies (i.e., flow capaciy kh) as well as original oil-in-place (OOIP). This classic work by Fekovich laid he foundaion for all he work ha followed regarding decline ype curves. McCray 4 (99) developed a ime funcion ha ransformed producion daa for sysems exhibiing variable rae or pressure drop performance ino an equivalen sysem produced

2 T. Marhaendrajana and T. A. Blasingame SPE 757 a a consan boomhole pressure (his work was laer exended by Blasingame e al. 5 (99) o an equivalen consan rae analysis approach). In 993, Palacio and Blasingame 6 developed a soluion for he general case of variable rae/variable pressure drop for he flow of eiher single-phase liquid or gas. Rodriguez and Cinco-Ley 7 (993) developed a model for producion decline in a bounded muliwell sysem. The primary assumpions in heir model are ha he pseudoseady-sae flow condiion exiss a all poins in he reservoir and ha all wells produce a a consan boomhole pressure. They concluded ha he producion performance of he reservoir was shown o be exponenial in all cases, as long as he boomhole pressures in individual wells are mainained consan. Camacho e al. 8 (996) subsequenly improved he Rodriguez Cinco-Ley model by allowing individual wells o produce a differen imes. However, Camacho e al. also assumed he exisence of he pseudoseady-sae condiion and ha all wells produce a consan boomhole pressures. Valko e al. 9 () presened he concep of a muliwell produciviy index for an arbirary number of wells in a bounded reservoir sysem. These auhors also assumed he exisence of pseudoseady-sae flow, bu proved ha he concep was valid for consan rae, consan pressure, or variable rae/variable pressure producion. The limiaions of he available muliwell models are summarized as follows: Consan boomhole pressure producion (excep for Valko e al. 9 ). This is rarely he case in pracice. The assumpion of pseudoseady-sae may be violaed, especially for condiions where he producion schedule (rae/pressure) changes dramaically, he reservoir permeabiliy is very low, and/or he well spacing changes (because of infill wells). None of he available muliwell mehods provides mechanisms for rigorous producion daa analysis. In his work, we have developed a general muliwell soluion ha is valid for all flow regimes (ransien, ransiion, and boundary-dominaed flow). This new soluion is rigorous for any rae/pressure profile (consan rae, consan pressure, or variable rae/variable pressure). I also provides a mechanism for he analysis of producion daa based on a maerial balance for he enire reservoir sysem. Decline Curve Analysis in a Muliwell Reservoir Sysem. Moivaion. Fig. is a ypical p/z plo of a gas well producing from Arun field (Indonesia). The daa ploed in his figure are from Syah. This plo is characerized by concave downward behavior ha could easily be inerpreed as an abnormal pressure sysem. However, applicaion of he maerial balance ha accouns for abnormal pressure mechanisms does no validae ha assumpion. Fig. illusraes an aemp o mach he well performance daa funcions wih he single well decline ype curve as proposed by Palacio and Blasingame. 6 The well performance daa funcions deviae from he b = sem during boundary dominaed flow condiion (he b = sem represens he maerial balance model for he reservoir sysem). This behavior (i.e., daa funcions deviaing from he maerial balance rend) has been consisenly observed for he analysis of well performance daa from Arun field. I is ineresing ha he p/z versus G p plo for he oal field performance a Arun field shows a sraigh-line rend as expeced for a volumeric gas reservoir (Fig. 3). This observaion suggess ha he behavior observed in Figs. and is perfecly correc individual wells compee for reserves, while he cumulaive (or aggregae) performance of he sysem is represened by a oal balance of pressure and producion. In oher words, if we inend o consider he local performance of an individual well, hen he effecs of oher nearby wells mus also be considered. To prove his poin we canno rely solely on oal field analyses (such as shown in Fig. 3) because he compuaion of he average pressure from he oal field is based on an averaging of he available local pressure measuremens. This averaging echnique iself may yield numerical arifacs, and he issue of he accuracy and relevance of he local pressures becomes quie imporan. How accurae and represenaive are locally averaged pressures? The developmen of an analysis and inerpreaion approach ha is rigorous, ye does no rely on an average reservoir pressure scheme, was our moivaion. Our sraegy was o develop a muliwell daa analysis mehod using a general muliwell model bu o develop his model in such a form ha i uses individual well performance daa in he esimaion of oal reserves and he permeabiliy in he drainage region for a paricular well. Muliwell Soluion. The general soluion for he well performance in a bounded muliwell reservoir sysem is given by (see Appendix A for deails) : p D ([x wd,k + ε],[y wd,k + ε], DA )= DA q D,i (τ) dp D,cr ( DA τ) dτ dτ k,i + q Dk ( DA )s k... () The physical model used o develop Eq. is shown in Fig. 4. This model assumes a closed recangular reservoir wih a consan hickness, which is fully peneraed by muliple verical wells (he well locaions are arbirary). The reservoir is assumed o be homogeneous, and we also assume he singlephase flow of a slighly compressible liquid. The soluion for a single well produced a a consan rae in a bounded recangular reservoir is given by Eq. A- (or Eq. A-). The

3 SPE 757 Decline Curve Analysis Using Type Curves Evaluaion of Well Performance Behavior in a Muliwell Reservoir Sysem 3 consan rae soluion inside he inegral in Eq. is compued a a paricular well (well k ) and includes he effecs of each well in he reservoir sysem. The accuracy of Eq. is validaed using numerical reservoir simulaion. We use a homogeneous square reservoir of consan hickness and include nine wells in he sysem in a regular well paern. Each well is assigned an arbirary boomhole flowing pressure ha can vary wih ime (Fig. 5). This soluion can also include an arbirary flow rae scheme. The reservoir and wells configuraion is shown in Fig. 6, and he reservoir and fluid propery daa for his case are lised in Table. The compuaion of he oil flow rae from boh he analyical soluion and numerical reservoir simulaion are ploed in Fig. 7. Noe ha he analyical soluion is in close agreemen wih numerical soluion. Decline Type Curve Analysis for a Muliwell Reservoir Sysem. Having developed and validaed our muliwell soluion, we proceeded wih he developmen of a daa analysis mehodology ha could be derived from he muliwell soluion. In Appendix B we show ha Eq. can be wrien as q k () ( p i p wf,k ()) =... () Nc o +f() where o,k = q k () q i (τ) dτ = N p,o... (3) q k () The variable f() is obviously ime-dependen (see Appendix B) however, his variable becomes consan during boundary-dominaed flow condiions. Eq.. represens a general formulaion of Arp s harmonic decline equaion and should be recognized as a maerial balance relaion. Noe ha his formulaion accouns for he variaions in producion schedules ha occur in pracice in paricular, he approach is valid for consan rae, consan pressure, or variable rae/variable pressure behavior during boundary-dominaed flow condiions. This equaion suggess ha if we plo q k ()/(p i -p wf,k ()) versus he oal maerial balance ime funcion, hen we can esimae he original oil-in-place (OOIP) for he enire reservoir using decline ype curves. For boundary-dominaed flow, Eq. can be wrien in erms of he dimensionless decline variables as q Dde =... (4) Dde + where 4. Bμ q q Dde = k () kh (p i p wf,k ()) ln r / β D... (5) Dde =.633 k o φμc A π ln r / β D... (6) This resul saes ha he performance of an individual well in a muliwell sysem behaves as a single well in a closed sysem provided ha he oal maerial balance ime funcion is used. Furhermore, his observaion implies ha he Fekovich/McCray ype curves for a single well which include boh ransien and boundary-dominaed flow can also be used o analyze daa from a muliwell reservoir sysem, provided ha properly defined dimensionless variables are used (Eqs. 5 and 6). To validae our concep we use Eq. as a mechanism o generae he behavior of a muliwell reservoir sysem. This is a subsanial deparure from he work of Fekovich 3 (and ohers), where a well cenered in a closed circular reservoir is used as he reservoir model. In paricular, we use a square reservoir wih nine wells on a regular well spacing and producing a he same consan rae. Using he case described above we found ha we can produce resuls ha are essenially idenical o he Fekovich/McCray ype curve developed for a single well. For he muliwell reservoir case we plo q Dde versus Dde on log-log scale and use r / β D as he family parameer (as shown in Fig. 8). We define an ineracion coefficien β D,which is used o represen he oher wells in he muliwell sysem. An ineracion coefficien of is he single well case and, consequenly, is a special case of he muliwell model. For gas reservoirs, Eqs. hrough 6 are valid provided ha we use he appropriae pseudopressure, pseudoime, and oal maerial balance pseudoime funcions. These funcions replace pressure, ime, and he oal maerial balance ime, respecively. The pseudopressure and pseudoime funcions are defined by 6,3 p p = μz p p p i μzdp... (7) p base a = μc i μc pavg dτ... (8) and he oal maerial balance pseudoime is expressed as 6 a,o = μc i q g q g,o μc pavg dτ... (9) Applicaion of Muliwell Model o Simulaed Cases Homogeneous Reservoir Example. In his paricular case we can direcly validae he muliwell soluion proposed in he previous secion. The reservoir and well configuraion is shown in Fig. 5 and he well performance behavior is shown in Figs. 5 and 7. The imporan issue for his case is ha he analyical soluion (Eq. ) and he numerical simulaion model represen exacly he same case of a homogeneous, bounded recangular reservoir. Validaion of he analyical soluion for his case implies (as would any reservoir engineering soluion) ha his resul can be used for he analysis and inerpreaion of performance daa.

4 4 T. Marhaendrajana and T. A. Blasingame SPE 757 The well performance daa for all wells are ploed on a loglog scale in Fig. 9. Noe ha all he daa rends from he nine wells (each wih a differen producion schedule) overlie one anoher. The behavior a early ime (all rends overlie) confirms he homogeneous naure of he reservoir, whereas he alignmen of all he daa a lae ime confirms our oal maerial balance on he enire reservoir sysem. An imporan noe is ha we have no modified any daa or daa rend shown in Fig. 9 he excellen agreemen of hese daa is solely due o he accuracy of he new soluion. The value of his example is he confirmaion ha he performance of a single well can be used o esablish he reservoir volume. I appears ha he early ime (ran-sien flow) daa can be used o esimae he permeabiliy in he local drainage area for each well. We con-firm his hypohesis using he locally homogeneous and heerogeneous reservoir cases considered in he fol-lowing secions. Anoher advanage of using our muliwell approach (i.e., he oal maerial balance ime funcion) over he single well approach (i.e., he maerial balance ime funcion for a single well) can be observed in Fig.. This figure shows he performance of well [3,] he daa are ploed on a log-log scale using boh he single and muliwell approaches. The daa for he muliwell approach (denoed by open symbols) clearly bes mach he ype curve model for all flow regimes (ransien, ransiion, and boundary-dominaed). The daa for he single well approach (denoed by solid symbols) deviae sysemaically from he decline ype curve model. The deviaion is significan during boundarydominaed flow. Any analyses based on his mach of he daa could easily yield erroneous resuls. To exend his approach for he analysis of gas well performance, we mus exend he concep of our maerial balance ime funcion o include he oal gas producion. This requires modificaion of he pseudoime formulaion for he gas reservoir case and combinaion wih he maerial balance ime concep. This is relaively sraighforward (see Eq. 9). The remainder of his secion is devoed o he validaion of he gas well performance case. Figs. hrough 3 show he applicaion of our muliwell decline ype curve mehod o simulaed performance daa from a homogeneous, dry gas reservoir. The reservoir and wells configuraion is he same as for he oil case (Fig. 6). The fluid and reservoir properies are lised in Table. Fig. 3 indicaes ha all he well performance daa imply a paricular reservoir volume (i.e., a unique original gas-in-place, OGIP) as prediced by our muliwell analysis echnique. This behavior is denoed by he convergence of he boundary-dominaed daa (i.e., he lae ime daa) for all wells ino a single maerial balance rend. Locally Homogeneous Reservoir Example. In his example, he reservoir and fluid properies are he same as in Table for he homogeneous bounded reservoir. The primary difference in his case is ha he reservoir permeabiliy disribuion is no homogeneous, bu is considered locally homogeneous (Fig. 4). Similar o he previous case, a numerical simulaion is performed where each well is produced under variable boomhole flowing pressure condiions. The boomhole pressure profile for each well is shown in Fig. 5, and he oil flow rae response for each well is shown in Fig. 6. The well performance daa for all wells are ploed on a loglog scale in Fig. 7. All he daa rends converge o a single maerial balance rend a lae ime, which corresponds o a unique reservoir volume. Also noe ha he differen responses a early ime correspond o he differen average permeabiliies for he drainage areas defined by individual wells. The daa in Figs. 7 and 8 clearly show he abiliy of our muliwell approach o model he enire sysem based on he well performance daa for in-dividual wells. Simulaneous maching of he daa for all wells using he Fekovich/McCray decline ype curve is shown on Fig. 8. The daa for all wells mach he ype curve very well for all flow regimes (ransien, ransiion, and boundary-dominaed). The daa are a lile scaered in he ransiion flow regime, where his behavior is due o a sudden change of he well flowing condiion gradual, raher han sudden, changes in raes and pressures are more likely in pracice. The resuls for his example are lised in Table 3. Inpu and calculaed values of OOIP are in excellen agreemen bu his is somewha expec-ed because he oal maerial balance ime correlaes wih he oal (in-place) volume. The differences in esimaed permeabiliy values occur because of he frame of reference. The inpu permeabiliy is he value assigned o he well spacing for a paricular well; whereas he calculaed permeabiliy is he harmonic average of he permeabiliies ha occur in he drainage area of a paricular well. The issue of a drainage area for an individual well in a muliwell reservoir sysem is somewha problemaic because he drainage areas change wih ime, corresponding o changes in he producion schedule for all he wells in he reservoir. Heerogeneous Reservoir Example. This case differs from subsanially from he wo previous cases in ha he reservoir permeabiliy disribuion is random (see Fig. 9) he permeabiliy values are assigned o grid blocks arbirarily from. o md and vary hroughou he well spacing and he reservoir. This case is inended o demonsrae how well our muliwell analysis/inerpreaion approach works for a randomly heerogeneous reservoir. The reservoir and fluid properies are as lised in Table. The specified boomhole flowing pressure profile for each well is also differen from he wo previous cases (Fig. ), and he oil flow rae response for each well is shown in Fig.. The well performance daa for all wells are shown on log-log scale in Fig.. Again, all well responses con-verge o a single maerial balance rend a lae ime, which again corresponds o a unique reservoir volume. As we noed for he

5 SPE 757 Decline Curve Analysis Using Type Curves Evaluaion of Well Performance Behavior in a Muliwell Reservoir Sysem 5 locally homogeneous reservoir, he differen rends a early ime are due o he differen average permeabiliies wihin an individual well drainage areas. In Fig. 3 all he daa rends mach he correc (maerial balance) soluion a lae ime (i.e., boundary-domi-naed flow). The variaions in he early ime (ransien flow) behavior correspond o differen permeabiliies. The sca-ered daa wihin he ransiion region are due o severe rae changes ha affec he derivaive compuaion. In pracice, such severe rae changes are unlikely o occur. If hey do occur, screening he bad daa provides a smooher derivaive funcion. The resuls of our analysis are provided in Table 4. Again, he inpu and calculaed values ofooip are in excellen agreemen. The OOIP is a unique propery of he reservoir, and using our muliwell approach preserves his uniqueness (based on he oal maerial balance ime). Alhough i is somewha unclear as o how o compare he inpu and calculaed permeabiliy values for each well, we chose o compare he harmonic average permeabiliy wihin a paricular well spacing o he calculaed permeabiliy from he decline ype curve mach. The resuls in Table 4 confirm our proposiion ha his approach can be used o esimae permeabiliies in a heerogeneous muliwell reservoir sysem. Field Applicaion To demonsrae he applicaion of our mehod o field daa, we analyzed several cases of well performance daa from Arun field (Indonesia). Arun field has wells (79 producers, injecors, 4 observaios, and 7 abandoned wells). The layou of Arun field is shown in Fig. 4 would cerainly be considered a muliwell reser-voir sysem. Arun field is a supergian gas condensae reservoir wih a maximum liquid dropou of approximaely.5% a he dewpoin (alhough mos daa sugges ha he maximum liquid producion should be less han %). In our analysis, he variaion of fluid properies wih pressure is incorporaed by he use of pseudopressure and pseudoime. In addiion, we use he oal (molar) gas rae. Using his procedure we expeced o esimae he correc gas-in-place volume for he enire field, as well as correcly esimae he local (per well) effecive permeabiliies o gas. We analyzed seleced cases of well performance daa from he following Arun field wells: Well C-II- (A-37) Well C-III-4 (A-6) Well C-II-3 (A-3) Well C-III-5 (A-35) Well C-II-4 (A-4) Well C-III-6 (A-7) Well C-II-6 (A-9) Well C-III-9 (A-8) Well C-III- (A-5) Well C-III-5 (A-4) Well C-III-3 (A-34) We discuss in deail he analysis resuls obained using he producion daa from Well C-III- (A-5). The produc-ion hisory of Well C-III- (A-5) (wellhead pressure and gas rae versus ime) is ploed in Fig. 5. The producion hisory includes boh wellhead flow raes and flowing wellhead pressure daa. In his example, we use boh single well (i.e., single well maerial balance pseudoime) and our proposed muliwell decline ype curve analysis (i.e., oal maerial balance pseudoime) echniques. The decline ype curve maches for boh he single and muliwell approaches are shown in Fig. 6. Our muliwell analysis approach maches he producion daa funcions (solid symbols) o he ype curve very well (we used pseudopressure and pseudoime funcions o accoun for he dependency of fluid properies on pressure). The single well approach (based only on he rae and pressure daa for a single well) fails o mach he lae ime maerial balance rend, where he boundary-dominaed flow daa deviae sysemaically from he ype curve (Fig. 6, open symbols). We recognize ha his behavior is due o well inerference effecs caused by compeing producing wells, bu he single well approach has no mechanism o correc or accoun for well inerference behavior. Our analysis using he muliwell approach yields an esimae of he OGIP for Arun field of approximaely 9.8 TCF. The esimae of he effecive flow capaciy (o gas) for his well is,79 md-f, which is based on he mach of he early ime (ransien flow) daa. Figs. 7 and 8 show he log-log plos of he rae/pressure drop and decline ype curve mach, respecively, for all wells ha we considered for our combined analysis. All he curves converge o he unique maerial balance rend a lae ime. This region (i.e., he boundary-dominaed flow daa) will be used o esablish an esimae of he oal (in-place) gas reservoirs for Arun field. The resuls of our analysis for he wells seleced from Arun field are summarized in Table 5. The OGIP compued using our approach is consisen ha is, each of he well analyses yields he same esimae of OGIP for he enire Arun field. Our mehodology assumes ha he OGIP is consan; herefore, we should be able o force all analyses o a single value of gas-in-place, which is wha we obained. Conclusions The following conclusions are derived from his work. We have developed a general muliwell soluion ha is accurae and provides a mechanism for he analysis of producion daa from a single well in a muliwell reservoir sysem. We have developed a mehodology for he analysis of producion daa from an individual well in a muliwell sysem. Using his mehod we can esimae he original fluid-in-place for he enire reservoir, as well as he local permeabiliy. Our mehodology can be applied for boh oil and gas reservoirs. Our approach uses he single well decline ype curve (i.e., he Fekovich/McCray ype curve) coupled wih he appropriae daa ransforms for he muliwell

6 6 T. Marhaendrajana and T. A. Blasingame SPE 757 reservoir sysem. We developed a oal maerial balance ime ploing funcion, which includes he performance from all he producing wells in he muliwell reservoir sysem. Our mehod honors he volumeric balance of he enire reservoir and preserves he uniqueness of he reservoir volume. Furhermore, he esimaes of flow capaciy (or permeabiliy) obained from our numerical simulaion sudies indicae ha our approach provides esimaes ha are boh accurae and represenaive for homogeneous and heerogeneous reservoir sysems. Nomenclaure A = area, f B = formaion volume facor, RB/STB c = oal compressibiliy, psi G p = cumulaive gas producion, MMscf h = ne pay hickness, f k = permeabiliy, md N = original oil-in-place, STB N p = cumulaive oil producion, STB = number of wells p = pressure, psia p i = iniial pressure, psia p p = pseudopressure funcion, psia p wf = well flowing pressure, psia q = flow rae, STB/D q g = gas flow rae, MSCF/D q g,o = oal gas flow rae (all wells), Mscf/D q o = oal flow rae (all wells), STB/D r e = reservoir radius, f r w = wellbore radius, f s = near-well skin facor, dimensionless = ime, day a = pseudoime, day a,o = oal maerial balance pseudoime, day o = oal maerial balance ime, day x = x coordinae from origin, f y = y coordinae from origin, f x e = reservoir size in he x direcion, f y e = reservoir size in he y direcion, f x w = x coordinae of well from origin, f y w = y coordinae of well from origin, f z = gas z-facor β D = muliwell ineracion coeeficien, dimensionless ε = small sep, dimensionless μ = fluid viscosiy, cp τ = dummy variable φ = porosiy, fracion Subscrips A =area is used as he reference avg = average bar = evaluaion is performed a average pressure base = arbirary reference cr = consan rae D = dimensionless k,i = well index MP = mach poin mw = muliwell ref = reference Acknowledgemens The auhors hank he former Mobil E&P Technology Co. (MEPTEC, now ExxonMobil) in Dallas, Texas, for financial and compuing services suppor provided during his work. The firs auhor also hanks Ms. Kahy Harman, Mr. Norman Kaczorowski, and Mr. Ravi Vaidya of ExxonMobil for virually unlimied access o daa and specifically for he personnel suppor. References. Hurs, W.: Unseady Flow of Fluids in Oil Reservoirs, Physics (Jan. 934) 5,.. Arps, J.J.: Analysis of Decline Curves, Trans., AIME (Dec. 945) 6, Fekovich, M.J.: Decline Curve Analysis Using Type Curves JPT (June 98) McCray, T.L.: Reservoir Analysis Using Producion Decline Daa and Adjused Time, MS hesis, Texas A&M Universiy, College Saion, TX (99). 5. Blasingame, T.A., McCray, T.C. and Lee, W.J.: Decline Curve Analysis for Variable Pressure Drop/Variable Flowrae Sysems, paper SPE 53 presened a he 99 SPE Gas Technology Symposium, Houson, Jan Palacio, J.C. and Blasingame, T.A.: Decline Curve Analysis Using Type Curves: Analysis of Gas Well Producion Daa, paper SPE 599 presened a he 993 SPE Rocky Mounain Regional/Low Permeabiliy Reservoirs Symposium, Denver, April Rodriguez, F. and Cinco-Ley, H.: A New Model for Producion Decline, paper SPE 548 presened a he 993 Producion Operaions Symposium, Oklahoma Ciy, March Camacho-V, R., Rodriguez, F., Galindo-N, A. and Pras, M.: Opimum Posiion for Wells Producing a Consan Wellbore Pressure, SPEJ (June 996) Valko, P.P., Double, L.E. and Blasingame, T.A.: Developmen and Applicaion of he Muliwell Produciviy Index (MPI), SPEJ (Mar. ).. Syah, I.: Modeling of Well Inerference Effecs on The Well Producion Performance in A Gas-Condensae Reservoir: A Case Sudy of Arun Field, PhD disseraion, Texas A&M Universiy, College Saion, TX, Marhaendrajana, T.: Modeling and Analysis of Flow Behavior in Single and Muliwell Bounded Reservoir, PhD disseraion, Texas A&M Universiy, College Saion, TX,.. Marhaendrajana, T. and Blasingame, T.A.: Rigorous and Semi- Rigorous Approaches for he Evaluaion of Average Reservoir Pressure from Pressure Transien Tess, paper SPE 3875 presened a he 997 SPE Annual Technical Conference and Exhibiion, San Anonio, Oc Fraim, M.L. and Waenbarger, R.A.: Gas Reservoir Decline Analysis Using Type Curves wih Real Gas Pseudopressure and Normalized Time, SPEFE (Dec. 987) 6.

7 SPE 757 Decline Curve Analysis Using Type Curves Evaluaion of Well Performance Behavior in a Muliwell Reservoir Sysem 7 Appendix A General Soluion for Muliwell Sysem The mahemaical model describing he pressure behavior in a bounded recangular reservoir wih muliple wells is in ref.. In his model each well produces a an arbirary consan rae and any well can be locaed a an arbirary posiion in he reservoir (as shown in Fig. 4). This soluion is given as p x + p y q i ()B Ah(k/μ) δ(x x w,i,y y w,i Σi ) = φμc p (A-) = k Eq. A- can be wrien in erms of he radiional dimensionless variables as follows: p D x D + p D y +π q D,i ( DA ) δ(x D x wd,i,y D y wd,i ) D = p D DA...(A-) where p D = πkh( p i p(x,y,)) ; q ref Bμ DA = φμc k (Darcy unis) A q D ( DA )= q() q ; x ref D = x A ; y D = y A In field unis, he dimensionless variables are defined as p D = kh( p i p(x,y,)) 4.q ref Bμ ; DA =.633 k φμc A Employing of Duhamel s principle for variable rae/variable pressure sysems, we obain he following soluion for Eq. A- subjec o he presumed no-flow ouer boundary condiion: p D (x D,y D, DA )=π q D,i (τ) ψ i (x D,y D, DA τ,x wd,i,y wd,i ) dτ DA...(A-3) where ψ(x D,y D DA,x wd,y wd ) is an insananeous line source soluion wih uni srengh locaed a (x w,y w ). From Eq. A-3, we can wrie he consan rae soluion for a single well as DA p D,cr (x D,y D, DA )=π ψ(x D,y D, DA τ,x wd,y wd ) dτ...(a-4) We define ξ= DA -τ and use his definiion in Eq. A-4 o obain DA p D,cr (x D,y D, DA )=π ψ(x D,y D,ξ,x wd,y wd ) dξ...(a-5) Taking he derivaive of Eq. A-5 wih respec o DA, we obain p D,cr (x D,y D, DA ) =πψ(x D,y D, DA,x wd,y wd )...(A-6) DA Subsiuing Eq. A-6 ino Eq. A-3, we obain he convoluion inegral formulaion for he pressure response a any locaion in an arbirary muliwell reservoir sysem. p D (x D,y D, DA )= DA q D,i (τ) p D,cr (x D,y D, DA τ,x wd,i,y wd,i ) DA... (A-7) If all wells are produced a individual (consan) flow raes, Eq. A-7 can be simplified o yield p D (x D,y D, DA )= q D,i p D,i (x D,y D, DA,x wd,i,y wd,i )... (A-8) Using Eq. A-7, he pressure soluion for well k is given by: p D ([x wd,k +ε],[y wd,k +ε], DA )= i Σ = DA q D,i (τ) dp D,cr( DA τ ) dτ k,i dτ dτ... (A-9) To accoun for he effec of he near-well skin facor s a well k we use p D ([x wd,k +ε],[y wd,k +ε], DA )= i Σ = DA q D,i (τ) dp D,cr( DA τ ) dτ dτ k,i + q Dk ( DA )s k... (A-) The consan rae soluion for an arbirary locaion in a bounded recangular reservoir is given by, p D,cr (x D,y D, DA )=π DA +4π +4π +8π Σn = Σn = exp n π n π x x DA exp n π y DA Σm =Σn = n π y exp n π + m π x cos nπ x x D cos nπ x x wd cos nπ y y D cos nπ y y wd y n π x + m π y DA cos nπ x x D cos nπ x x wd cos mπ y y D cos mπ y y wd... (A-) From ref. we noe ha Eq. A- can also be wrien in he form of an exponenial inegral series: p D,cr (x D,y D, DA )=

8 8 T. Marhaendrajana and T. A. Blasingame SPE 757 Σ Σ m = n = E (x D + x wd +nx ) +(y D + y wd +my ) 4 DA + E (x D x wd +nx ) +(y D + y wd +my ) 4 DA + E (x D + x wd +nx ) +(y D y wd +my ) 4 DA (x + E D x wd +nx ) +(y D y wd +my )...(A-) 4 DA The pressure response for an individual well produced a a consan rae is given by p D,cr ([x wd,k + ε],[y wd,k + ε], DA )=π DA +4π +4π +8π Σn = Σn = Σm =Σn = exp n π n π x x DA exp n π y DA n π y exp n π + m π x cos nπ x (x wd + ε) cos nπ x x wd cos nπ y (y wd + ε) cos nπ y y wd y n π x + m π y DA cos nπ x (x wd + ε) cos nπ x x wd cos y mπ (y wd + ε) cos mπ y y wd...(a-3) Subsiuing Eq. A- ino Eq. A-3 we obain p D,cr ([x wd + ε],[y wd + ε], DA )= Σ Σ m = n = E (x wd + ε +nx ) +(y wd + ε +my ) 4 DA + E (ε +nx ) +(y wd + ε +my ) 4 DA + E (x wd + ε +nx ) +(ε +my ) 4 DA + E (ε +nx ) +(ε +my ) 4 DA...(A-4) Appendix B Developmen of Producion Daa Analysis Technique in Muliwell Sysem In his Appendix we develop he ploing funcions ha serve as he basis for our proposed decline ype curve analysis of well and field producion performance daa from a bounded muliwell reservoir sysem. We begin by subsiuing Eq. A-3 ino Eq. A-: p D (x wd,k + ε,y wd,k + ε, DA )=π DA q Di (τ) dτ +π q Di (τ) F([x wd,k + ε],[y wd,k + ε],[ DA τ],x wd,i,y wd,i ) dτ DA + q Dk ( DA )s k... (B-) where F([x wd,k + ε],[y wd,k + ε],[ DA τ],x wd,i,y wd,i )= + exp n π ( DA τ) cos x nπ (x wd,k + ε) cos nπ x x wd,i Σn = x + exp n π Σn = y ( DA τ) cos nπ y (y wd,k + ε) cos nπ y y wd,i + 4 exp n π x + m π y ( DA τ) Σm =Σn = cos x nπ (x wd,k + ε) cos nπ x x wd,i cos y mπ (y wd,k + ε) cos mπ y y wd,i... (B-) Wriing Eq. B- in field unis and muliplying boh sides by q ref /q k (), we obain kh (p i p wf,k ()) =π 4.Bμ q k ().633 φμc k A q k () +π.633 k φμc A q k () q i (τ) dτ q(τ) F([x w,k + ε],[y w,k + ε],[ τ],x w,i,y w,i )dτ + s... (B-3) We immediaely recognize ha N p,o = q i (τ) dτ... (B-4)

9 SPE 757 Decline Curve Analysis Using Type Curves Evaluaion of Well Performance Behavior in a Muliwell Reservoir Sysem 9 where N p,o is he cumulaive oil producion for he enire field. Furhermore, we define a oal maerial balance ime as o,k = q k () q i (τ) dτ = N p,o... (B-5) q k () Subsiuing Eq. B-5 ino Eq. B-3 and muliplying boh sides by 4.Bμ/(kh), we obain (p i p wf,k ()) q k () + Nc q k () = Nc o q(τ) F([x w,k + ε],[y w,k + ε],[ τ],x w,i,y w,i )dτ 4. Bμ + s kh... (B-6) For simpliciy, we can wrie Eq. B-6 as (p i p wf,k ()) q k () = Nc o + f ()... (B-7) Taking reciprocal of Eq. B-7, we obain q k () ( p i p wf,k ()) =... (B-8) Nc o +f() The variable f() is obviously ime-dependen however, his variable becomes consan during boundary-dominaed flow condiions. Eq. B-8 is he general formulaion of Arp s harmonic decline equaion. This is an elegan relaion, considering ha i is rigorous and ye simple. Specifically, his resul akes ino accoun he complexiy of he producion schedule (consan rae, consan pressure, or variable rae/variable pressure). The formulaions given by Eqs. B-7 and B-8 are convenien for daa analysis excep ha f() is ime-dependen. Neverheless, during boundary-dominaed-flow condiions his erm becomes consan and we can rea he analysis of muliwell performance daa in he same manner as he single well case. Our purpose is o use he radiional single well decline ype curve analysis echniques o esimae he (oal) volume and (near-well) flow properies simulaneously. Recalling he boundary-dominaed flow (or pseudoseady-sae flow) soluion for a single well, we have q() ( p i p wf,k ()) =... (B-9) +b Nc pss where b pss = 4. Bμ kh e 4 A γ... (B-) C A r wa For he muliwell case, he Diez shape facor is deermined no only by reservoir shape and well posiion bu also by he sae of he oher wells (number, posiion, and rae/pressure). The apparen drainage area of a well in muliwell sysem depends on he raio of producing rae o oal field producing rae. We call his raio he ineracion coefficien β D. For boundary-dominaed flow, Eq. B-8 becomes q k () ( p i p wf,k ()) =... (B-) Nc o + b pss,mw where b pss,mw = 4. Bμ kh e 4 A/β D γ... (B-) C A r wa Fekovich 3 used a modified definiion of he b pss variable defined as b pss = 4. Bμ kh lnr... (B-3) Eq. B-3 has been used as he defining ransform variable for all he decline ype curves presened for he case of a single well cenered in a bounded circular reservoir. Accordingly, we presen a similar expression for he muliwell sysem: b pss,mw = 4. Bμ kh ln r / β D... (B-4) Subsiuing Eq. B-4 ino Eq. B- and muliplying boh sides by 4.Bμ/(kh), we obain 4. Bμ q k () kh ( p i p wf,k ()) = π.633 k o φμc A + ln r / β D... (B-5) Rearranging Eq. B-5 slighly, we finally arrive a he following formulaion: 4. Bμ q k () kh (p i p wf,k ()) ln r / β D = π.633 k... (B-6) o φμc A ln r / β D + The appropriae dimensionless decline variables are defined as 4. Bμ q q Dde = k () kh (p i p wf,k ()) ln r / β D... (B-7) Dde =.633 k o π φμc A ln r / β D... (B-8) Hence we can wrie Eq. B-6 as q Dde =... (B-9) Dde + We immediaely recognize ha Eq. B-9 is he Arp s harmonic decline relaion. This resul verifies ha he producion decline characer of an individual well in a muliwell reservoir sysem has he same behavior as a single well in

10 T. Marhaendrajana and T. A. Blasingame SPE 757 a closed reservoir if we use he oal maerial balance ime. Furhermore, he Fekovich/McCray ype curves for a single well sysem can be used for he analysis and inerpreaion of he performance of a muliwell reservoir sysem provided ha we use he appropriae definiions of he dimensionless variables (Eqs. B-8 and B-9). Table Reservoir and Fluid Properies for Synheic Example, Oil Reservoir. Reservoir Properies: Iniial Pressure, p i = 5, psia Reservoir Thickness, h = 5 f Toal Reservoir Area, A = acres Original-Oil-In-Place, OOIP = 4,78 MMSTB Permeabiliy, k = 5 md Wellbore radius, r w =.5 f Porosiy, φ =. (fracion) Fluid Properies: Toal Compressibiliy, c = 3 6 psia Oil Viscosiy, μ =.8 cp Oil Formaion Volume Facor, B =.84 RB/STB Table Reservoir and Fluid Properies for Synheic Example, Gas Reservoir. Reservoir Properies: Iniial Pressure, p i = 5, psia Reservoir Thickness, h = 5 f Toal Reservoir Area, A = acres Original-Gas-In-Place, OGIP= 6.34 Tscf Permeabiliy, k = 5 md Well radius, r w =.5 f Porosiy, φ =. (fracion) Fluid Properies: Pressure z-facor Gas FVF Viscosiy Compressibiliy (psia) (bbl/scf) (cp) (/psi) E E E E E E E E E E E E E E E-4 Table 3 Resuls of Muliwell Analysis (Locally Homogeneous Example). Well Permeabiliy, k (md) Absolue (calculaed) (inpu) Relaive Error (%) [,] [,] [,3].. [,] [,] [,3] [3,] [3,] [3,3] Original Oil-In-Place (inpu) : 4,78 MMSTB Original Oil-In-Place (calculaed) : 4,78 MMSTB Table 4 Resuls of Muliwell Analysis (Heerogeneous Example). Table 5 Well Permeabiliy, k (md) Absolue (calculaed) (inpu) Relaive Error (%) [,] [,] [,3] [,] [,] [,3] [3,] [3,] [3,3] Original Oil-In-Place (inpu) : 4,78 MMSTB Original Oil-In-Place (calculaed) : 4,78 MMSTB Summary of he Decline Type Curve Analysis Resuls for Arun Field, Indonesia (Muliwell Approach). Well Name [Δ/ Dde ] MP [q/δp]/ [q Dde ] MP r / β D OGIP (Tcf) kh (md-f) C-II- 8,44 95, 9.8,946 C-II-3 8, ,33 C-II-4, ,76 C-II-6, C-III- 9,433 9, 9.8,79 C-III-3 5,979 5, 9.8 3,56 C-III-4 5, ,4 C-III-5 9, C-III-6 9, 9, 9.8 5,893 C-III-9 5,4 5, 9.8 3,567 C-III-5 3, ,6

11 SPE 757 Decline Curve Analysis Using Type Curves Evaluaion of Well Performance Behavior in a Muliwell Reservoir Sysem (,) y e y w,i x w,i x e Figure Typical p/z plo for a well in Arun field (Well A-5). Figure 4 Bounded recangular reservoir wih muliple wells locaed a arbirary posiions wihin he reservoir. Figure Decline ype curve mach using single well approach (Well A-5). Figure 5 Boomhole flowing pressure profiles (homogeneous reservoir example). Y-Direcion, f [3,] [3,] [3,3] [,] [,] [,3] [,] [,] [,3] Figure 3 p/z plo for Arun field (oal field performance) X-Direcion, f Figure 6 Homogeneous bounded square reservoir wih nine producing wells (homogeneous reservoir example).

12 T. Marhaendrajana and T. A. Blasingame SPE 757 Figure 7 Oil rae versus ime profiles (homogeneous reservoir example). Figure Log-log plo of rae/pressure drop funcions as a funcion of oal maerial balance ime (homogeneous reservoir example all cases). Figure 8 Plo of Dimensionless decline variables for single well and muliwell performance cases simulaed performance was used o validae he muliwell concep. Figure Boomhole flowing pressure profiles for individual wells (gas, homogeneous reservoir example). Figure 9 Log-log plo of rae/pressure drop funcions as a funcion of oal maerial balance ime (homogeneous reservoir example). Figure Gas flow rae profiles for individual wells (gas, homogeneous reservoir example).

13 SPE 757 Decline Curve Analysis Using Type Curves Evaluaion of Well Performance Behavior in a Muliwell Reservoir Sysem 3 Figure 3 Log-log plo of rae/pseudopressure drop funcions versus oal maerial balance pseudoime (gas, homogeneous reservoir example). Figure 6 Oil rae versus ime profiles for individual wells (locally homogeneous reservoir example). Figure 4 Locally homogeneous bounded square reservoir wih nine producing wells (locally homogeneous reservoir example). Figure 7 Log-log plo of he rae/pressure drop versus oal maerial balance ime (locally homogeneous example). Figure 5 Boomhole flowing pressure profiles for individual wells (locally homogeneous reservoir example). Figure 8 Decline ype curve mach using he muliwell approach (oal maerial balance ime) (locally homogeneous example).

14 4 T. Marhaendrajana and T. A. Blasingame SPE 757 Figure Log-log plo of he rae/pressure drop versus oal maerial balance ime (heerogeneous reservoir example). Figure 9 Random permeabiliy case, bounded square reservoir wih nine producing wells (heerogeneous reservoir example). Figure 3 Decline ype curve mach using he muliwell approach (oal maerial balance ime) (heerogeneous reservoir example). Figure Boomhole flowing pressure profiles for individual wells (heerogeneous reservoir example). Figure 4 Layou of he Arun field, Indonesia. Figure Oil rae versus ime profiles for individual wells (heerogeneous reservoir example).

15 SPE 757 Decline Curve Analysis Using Type Curves Evaluaion of Well Performance Behavior in a Muliwell Reservoir Sysem 5 Figure 5 Producion hisory of Well C-III- (A-5) Arun Gas field, Indonesia. Figure 8 Decline ype curve mach for wells of Arun field, Indonesia. Figure 6 Decline ype curve mach of Well C-III- (A-5) single well and muliwell approaches. Figure 7 Log-log plo of rae/pressure drop funcions versus oal maerial balance pseudoime for wells of Arun Field, Indonesia noe ha all curves converge o a unique maerial balance rend.