Application of Material Balance Equations of Multicompartment Gas Reservoirs in YC Gas Reservoir

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1 Internatonal Conference on Manufacturng Scence and Engneerng (ICMSE ) Applcaton of Materal Balance Equatons of Multcompartment as Reservors n YC as Reservor Quan-Hua Huang, a, Chong Chen, b, Lang Yn, c, Tong Lu,d and Tao Fang3,e State Key Laboratory of Ol & as Reservor eology and Explotaton, Southwest Petroleum Unversty, Chengdu, Schuan Provnce, People s Republc of Chna Research Insttute of Engneerng Technology, Snopec Southwest ol and gas feld Company, Deyang, Schuan Provnce, People s Republc of Chna 3 CNOOC Chna Lmted, Tanjn Branch, Tanjn Provnce, People s Republc of Chna a swpuhqh@6.com, bchenchenchong@6.com, c96783@63.com, d 487@qq.com, efangtao@cnooc.com.cn Keywords: compartmented gas reservor, materal balance, fault block, mathematcal model Abstract. A compartmented gas reservor s defned as a reservor nvolvng two or more dstnct blocks n hydraulc communcaton. A common one s a fault block gas reservor separated nto dfferent fault blocks by partally sealng faults. The fault plane n fault block gas reservor works as the mgraton pathway of ol and gas sometmes, whch makes the determnaton of reservor pressure and dynamc reserves as well as remanng reserves complcated. Hence, n order to accomplsh the goal of producng gas felds effcently, t s necessary to have further study on the determnaton of reservor pressure and evaluaton methods of dynamc reserves, whch can especally lay bass for the establshment of sequent development or adjustment program. Ths paper establshes the mathematcal model of multcompartment gas reservors and derves the correspondng materal balance equatons (MBEs) accordng to materal balance. Then Newton-Raphson teraton scheme s employed to solve these equatons. Ths model can be used not only for hstory matchng, but also for performance predcton. An example s presented to analyze how transmssblty, gas ntally n place (IIP), gas producton rate and commssonng tme of the nonproducng reservor affect the P/Z behavor. The practcal applcaton of ths method proves the method to be applcable. The change n nflux and nstantaneous flow rate between blocks can be quanttatvely expressed by ths method as well. Thus, producng degree of the nonproducng reservor can be determned, whch would be helpful to reduce costs and mprove economc effcency. Introducton Fault block gas reservors, as one knd of the man structural gas pools are rather commonly found n ol and gas area n East Chna and South Chna Sea where so many faults develop. The fault plane n fault block gas reservors works as the mgraton pathway of ol and gas sometmes, whle t performs as an obstacle at tmes[]. When t acts as the former one, ol or gas can flow laterally or vertcally,.e. the producng reserves between dfferent reservor compartments or layers durng the producton run are employed mutually, so the determnaton of reservor pressure, dynamc reserves and remanng reserves of gas reservor becomes complcated. There are many methods of performance analyss. Materal balance method s the most common one of those. Frstly employng materal balance method n 989, Hower and Collns[] establshed the model of compartmented gas reservors, and they proposed the analytcal solutons of that. They were adjustable to the condtons of constant gas producton rate, volumetrc depleton and constant gas propertes. However, the results could reflect the basc depleted characterstcs of that knd of gas reservor clearly. In 99, Lord and Collns[3] appled MBE to a multcompartment system. They formulated dfferental equatons of that and solved them by mathematcs. However, no specfc detals of. The authors - Publshed by Atlants Press 4

2 mplementaton of ths method were gven. Then Lord et al. extended the MBE method and appled t to the compartmented gas reservor n Fro Layer n South Texas. In 993, amed at the features of unbalanced explotaton n low-permeablty gas reservors, ao and Zhang et al.[4,,6,7] founded a new dynamc predcton model of constant volume gas reservors: materal balance of dual-compartment reservors. Ths model took the pressure dfference and gas nflux between hgh wthdrawal area and strpper area nto account, and t can be used to analyze producton performance and varaton of the suppled gas between blocks under dfferent well patterns and development ndexes. In 996, the multcompartment gas reservor model was appled to sngle tght gas reservors by Payne[8]. Payne solved the MBE by explct method; n the meanwhle, he gnored flud exchange between boundares and change n gas propertes durng a tme step. For the determnaton of crossflow between compartments, the calculaton procedure proposed by Payne s so easy that we can compute t drectly by utlzng EXCEL. However, adoptng explct procedure as well as the use of pressure-squared approxmaton may result n unacceptable errors. In 999, after summarzng and analyzng the pros and cons of MBEs by Hower, Lord and Payne, J.Hagoort and R.Hoogstra[9] put forward the ntegral form of MBEs of dual-compartment gas reservors and used the mplct teraton procedure to solve them. They also summarzed how crossflow between reservor compartments, gas reservor sze and gas producton rate nfluenced P/Z behavor. The numercal methods were so practcable because of ts smplcty and effcency. But they just gave MBEs of dual-compartment gas reservors. The MBEs of multcompartment gas reservors wth gas nflux and efflux reman to be studed n detals. Employng rgorous mathematcal methods, N.M.A. Rahman and A.K. Ambastha[] bult up 3D dfferental model of a compartmented gas reservor n, and ntegral transform method was used to resolve those equatons. Fnally, the detaled characterstcs of crossflow between reservor compartments and the rgorous soluton of reservor pressure n space and tme were obtaned. Lu, Lu and Wang[] proposed the calculaton of reserves of reservor compartments,whch are separated from compartmented gas reservors wth water n 7. They nvolved the effect of water aqufer nto the MBEs, and they utlzed apparent geologc reserve methods to calculate cumulatve gas nflux and the reserves of the separated reservor compartments after smply changng the form of those equatons. It was an easy method and could be completed n EXCEL. However, there are several unknowns, so multple solutons of the MBEs may lead to poor relable results. Accordng to materal balance, ths paper establshes a new model of dual-compartment gas reservors, consderng gas nflux and efflux between compartments, and presents MBEs of that. Then t s readly extended to multcompartment gas reservors wth gas nflux and efflux. Newton-Raphson teraton scheme s adopted to solve ths knd of model. Combned wth an example, the senstvty of several parameters n MBEs s also analyzed. After comparng the results of the example, we can fnd that ths model s applcable to such a complex and compartmented gas reservor. The gas nflux and efflux between dfferent reservor compartments can be determned accordng to ths model, and then further make sure whether we perform actual drllng and development n the strpper areas, whch contrbutng to mprovng development effectveness. Modelng of multcompartment gas reservors The physcal model of dual-compartment gas reservors s shown schematcally n Fg.. Ths model conssts of two porous and permeable reservor compartments, wth a permeable barrer dvdng them. They contan gas and water, and they were thought as storage tanks at unform pressure and temperature. Whatever the gas s dry gas or wet gas, the gas propertes n these two reservor compartments are approxmately the same. The assumpton that compartment and compartment cannot be exploted smultaneously s made.

3 PRODUCTION PRODUCTION INFLUX Fg. Schematc representaton of a typcal compartmented gas reservor consstng of two reservor compartments separated by a permeable boundary The MBEs of the dual-compartment gas reservor can be coupled to each other through cumulatve gas nflux or efflux. P P p p [ Ce ( P P )] = ( ) Z Z () P + P p p [ Ce ( P P )] = ( ) Z Z Where t TscZ sc p = qdt; q = Tr ( m m) () P T sc Smlarly, the general MBEs of multcompartment gas reservors wth n reservor compartments can be readly expressed as follows. P Z P Z LL P n Z n Pn Z n [ C ( P P )] e [ C ( P P )] e [ C ( P P )] ( n ) [ C ( P P )] e e n n P = Z P = Z n P = Z n n ( ( P( = Z p n ) ( n ) ( p pn p + ( + p pn p( n ) p3 + LL p3 + pn LL p( n ) + LL + 3 p( n ) p( n ) p( n ) n p( n ) pn LL ) pn ) p( n ) n p( n )( n ) Where t p p TscZ sc p p j pj = qjdt; qj = Trj ( m j m ) ; m = P T dp ; m j = u z u z sc In the above equatons, crossflow between reservor compartments ( q j ) s governed by Darcy s law n the form of pseudo-pressure, whch takes full account of the pressure dependence of gas propertes. Here we calculate the pseudo-pressure durng the explotaton by trapezodal quadrature formula. In case that gas nflux and efflux between dfferent reservor compartments exsts, the flow drecton of fluds s prmarly decded by pseudo formaton pressure. Here, we are provded that postve q j means that gas flows from reservor j nto reservor, and negatve q j means that gas flows from reservor nto reservor j. The ntal value condtons of these equatons are that the ntal reservor pressure of each gas reservor compartment s the same before the start of producton. We assume that the formaton temperature remans constant throughout the entre producton process,.e. sothermal decompresson producton. t = : P = P = LL = P n = P ; and T = T = L L = Tn = T Here, n means the numbers of reservor compartments. ) p( n ) n j j ) dp (3) 6

4 After producton, each reservor pressure and crossflow between reservor compartments are both functons of cumulatve gas producton and recoverng tme, so producton profles of dfferent reservor compartments wll present dfferent pressure drop and change n gas flow. t > : p = p( t) ; p = p( t) ; L L; pn = pn( t), andt = T = L L = Tn = T Here, n means the numbers of reservor compartments. Therefore, for the producng gas reservor, f the varaton of reservor pressure s known, we can substtute actual producton profles of each reservor nto the equatons above. Supposng that one or several parameters are unknown, such as geologcal reserves and transmssblty, gas nflux or efflux between reservor compartments can be determned by adjustng these parameters to ft the observed reservor pressures. Smultaneously, for the nonproducng gas reservor, we can predct reservor pressure varaton wth recoverng tme and when and how much natural gas supples wth each gas reservor compartment. There are several parameters referred to the MBEs above. They are Z-factor, gas vscosty, volume factor, pore compressblty and formaton water compressblty. DPR method and the correlaton of Lee et al. can be chose to calculate Z-factor and gas vscosty respectvely. Volume factor of natural gas can be obtaned accordng to pressure, temperature, Z-factor and specfed standard condton. Pore compressblty can be computed n the lght of emprcal formula descrbng the relatonshp of C versus φ. Formaton water compressblty can be determned by the expresson presented by p McCaB[]. Soluton of the model Newton-Raphson teraton scheme of nonlnear equatons s requred to solve the MBEs above. For an approprate teratve ntal value, the degree of convergence of ths scheme s second-order. Here the ntal value s ntal reservor pressure, often acqured by actual measurement. So t s relable and the equatons are generally convergent. Takng the nfluence of pressure on natural gas propertes nto account, then the updated pressure by teraton s employed to calculate the physcal parameters of natural gas. Durng an teraton cycle, the latest adjusted pressure s dfferentated. Therefore, ths scheme s an mplct method. Suppose that nstantaneous gas nflux durng a tme step s lnear to tme, then cumulatve nflux durng a tme step ( t ) can be calculated by the expresson below. pj ( t + t) = pj + [ qj ( t) + qj ( t + t) ] t (4) It can be appled to solve the equatons at successve tme step. The ntal estmate s ntal reservor pressure ( P ). The ntal estmate ( P B ) at a new tme step s regarded as the convergence pressure at the last tme step. When the corrected pressure s smaller than an allowable devaton value, such as.mpa, stop the teraton. The computer programmng demonstrated that Newton-Raphson teraton scheme s fast and steady. Pressure would converge wthn 3-8 teratons. The degree of devaton of the calculated reservor pressure manly depends on accuracy of the calculated crossflow between gas reservor compartments and the sze of tme step9. If there does not exst crossflow between gas reservor compartments, there s no constrant to the sze of tme step ether. It s needed to halve or double tme step sze to check the senstvty of these equatons to the sze of tme step. In the lght of J.Hagoort et al, change n pressure durng a tme step should be less than % of the ntal reservor pressure. As mentoned above, materal balance equaton method can be used for producton predcton as well as hstory matchng. In a certan teratve doman, computer programmng can automatcally adopt nonlnear regresson technque to complete hstory matchng accordng to method of nonlnear least squares. 7

5 Senstvty analyss of computatonl parameters enerally speakng, there are more potental unknowns than gven data n the MBEs. That s to say, there are too less equatons to derve unknowns, whch s a bg problem throughout the petroleum engneerng feld. Even numercal smulaton s not able to settle ths problem. It just added some potental unknowns to sparse equatons, such as geometry, porosty, permeablty and relatve permeablty. Just because of the multple solutons and uncertantes of the MBEs, t s sgnfcantly essental to analyze and determne the degree of the nfluence of parameters n MBEs on the results. Takng Case n SPE 76 for example here, called Base Case, we wll dscuss the man characterstcs of pressure depleton and gas nflux of the dual-compartment gas reservor. Table lsts the relevant reservor data and gas propertes of the Base Case. Research methods advanced by J.Hagoort were partally referred to. The results here are n accord wth that n SPE 76. We enlarged or reduced some mportant parameters n MBEs agan, nvolvng geologcal reserves, transmssblty and gas producton rate. Then the results were calculated under those dfferent condtons. In the meanwhle tme, we analyzed how these parameters nfluenced the results. Computer programmng was employed to solve the MBEs above. The smulated producton tme of ths typcal case s 9 days,.e. tme steps, and the computng tme was less than seconds. The results of the Base Case are presented n Fg.. The sold lnes n t are P/Z curves of the producng reservor and the nonproducng reservor. By comparson wth that, the dotted lnes, showng P/Z curves under two extreme condtons, are also ncluded n Fg.. One condton s the depleton development wthout gas flowng nto the producng reservor from the nonproducng reservor, whch means that transmssblty s. The other s the depleton development wth the assumpton that the two reservors are fully connected, whch means that transmssblty s nfntely great. The ntercept of P/Z curve and p-axs s equal to the geologcal reserves of the producng reservor for the former condton whle the ntercept of them s equal to the total geologcal reserves of the two reservors for the latter condton9. We can learn from Fg. that the slope of P/Z curve of the producng reservor n the early development perod s equal to the slope of the P/Z straght lne correspondng to transmssblty of. After a short tme, the P/Z curve nclnes to the P/Z straght lne correspondng to the nfnte transmssblty. Then the system enters pseudo-steady state before long. The rate of pressure declne of the two reservors s approxmately the same n ths stage. The P/Z curve correspondng to pseudo-steady state s nearly parallel to that correspondng to the nfnte transmssblty9. Fg.3 descrbed the relatonshp between cumulatve gas producton of the producng reservor and nstantaneous flow rate as well as cumulatve gas nflux from the nonproducng reservor. The nstantaneous flow rate enjoys ts peak soon and pseudo-steady state flow regme develops. Under pseudo-steady state flow, both reservors contrbute to the explotaton of the whole gas reservor, and the contrbutons s propostonal to ts geologcal reserves. Hence, cumulatve gas nflux from the nonproducng reservor s half of the total gas producton of the producng reservor n ths paper. 8

6 Besdes, the nstantaneous flow rate s propostonal to the pseudo pressure dfference of the two reservors on the bass of Darcy s law. 4 3 P/Z (MPa) Nonproducng Reservor Producng Reservor Cumulatve Producton ( 8 m 3 ) Fg. Base Case P/Z behavor of the producng and nonproducng reservor 3 Instantaneous Influx ( 4 m 3 /d) 3 Instantaneous Influx Cumulatve Influx Cumulatve Influx ( 8 m 3 ) Cumulatve Producton ( 8 m 3 ) Fg.3 Base Case nflux nto the producng reservor Influence of Transmssblty Transmssblty s an mportant parameter descrbng the connectvty between two reservors. The greater t s, the better percolaton ablty between reservors s and the more obvous the fault sealng s. enerally, t s hard to determne t by feld test. However, the MBEs could be a preferable choce to solve t. P/Z curves and nstantaneous flow rate curves under dfferent transmssblty were respectvely presented n Fgs.4 and. As shown n Fg.4, the P/Z curve correspondng to transmssblty of s close to that correspondng to transmssblty of. The formaton pressure of the two reservors keeps dvergent untl they are abandoned. It s apparent that pseudo-steady state flow regme cannot develop when transmssblty s small. When transmssblty s respectvely, and, the correspondng P/Z curves of both reservors nclne to a straght lne gradually. In other words, two fully connected reservors can be vewed as a sngle reservor to be exploted when transmssblty s nfnte. Accordng to Fg., t takes a relatvely long tme for the nstantaneous flow rate correspondng to transmssblty of to reach a maxmum gradually, whch s stll relatvely small. It would affect some parameters related to pressure, such as Z-factor, gas vscosty and volume factor, resultng n a sharp nonlnear relatonshp between flow rate and pressure drop fnally. Ths phenomenon s smlar to mddle-later secton of nflow performance curve of a gas well. The maxmal nstantaneous flow rate 9

7 ncreases wth the ncrement of transmssblty. However, the maxmum wll be 4 m 3 /d, half of the total flow rate, at the very most. 4 P/Z (MPa) 3 Tr= Tr= Nonproducng Reservor Producng Reservor Tr= Tr= Instantaneous Influx ( 4 m 3 /d) Cumulatve Producton ( 8 m 3 ) Fg.4 P/Z curves of the producng and nonproducng reservor under dfferent transmssblty Tr= Tr= 3 4 Cumulatve Producton ( 8 m 3 ) Tr= Tr= Fg. Instantaneous flow rate nto the producng reservor under dfferent transmssblty Influence of eologcal Reserves Fgs.6 and 7 show the results when geologcal reserves of the nonproducng reservor are twce of that of the Base Case. Consequently, the depleton characterstcs are dentcal wth these of the Base Case quanttatvely. The nonproducng reservor contrbutes more to the total producton as ts geologcal reserves ncrease. So does the nstantaneous flow rate. As a result of the ncrement of nstantaneous flow rate, the pressure dfference between the two reservors s larger than that of the Base Case. Fgs.8 and 9 present the results when geologcal reserves of the nonproducng reservor reduce by half of that of the Base Case. As a result of the reducton of nstantaneous flow rate, the pressure dfference between the two reservors s smaller than that of the Base Case. And the P/Z curve of the producng reservor s close to that correspondng to nfnte transmssblty. In concluson, under the same transmssblty and gas producton rate, the larger geologcal reserves of the nonproducng reservor are, the greater nstantaneous flow rate s. The more evdently the pressure of the two reservors dverges, the lower recovery percent s, and vce versa. 6

8 P/Z (MPa) 4 3 Producng Reservor Nonproducng Reservor Cumulatve Producton ( 8 m 3 ) Fg.6 P/Z curves of both reservors for twce of Base Case geologcal reserves of the nonproducng reservor 4 4 Instantaneous ( 4 m 3 /d) 3 Instantaneous Influx Cumulatve Influx 3 Cumulatve Influx ( 8 m 3 ) Cumulatve Producton ( 8 m 3 ) Fg.7 Influx nto the producng reservor for twce of Base Case geologcal reserves of the nonproducng reservor 4 3 P/Z (MPa) Producng Reservor Nonproducng Reservor Cumulatve Producton ( 8 m 3 ) Fg.8 P/Z curves of both reservors for half of Base Case geologcal reserves of the nonproducng reservor 6

9 4 Instantaneous Influx ( 4 m 3 /d) Instantaneous Influx Cumulatve Influx Cumulatve Influx ( 8 m 3 ) 3 4 Cumulatve Producton ( 8 m 3 ) Fg.9 Influx nto the producng reservor for half of Base Case geologcal reserves of the nonproducng reservor Influence of as Producton Rate Fg. shows the results when gas producton rate of the producng reservor s half or twce of that of the Base Case. In the crcumstance of gas producton rate reducton by half, the correspondng nstantaneous flow rate would reduce to half of that of the Base Case, because nstantaneous flow rate s lnear to nstantaneous flow rate under pseudo-steady state. Therefore, pressure dfference between the two reservors s half of that of the Base Case, leadng to the two curves beng close to each other. They both nclne to the P/Z curve correspondng to nfnte transmssblty. Ths mples that lowerng gas producton rate can enhance natural gas recovery for a gas reservor under a gven abandonment pressure. On the other hand, n the crcumstance of gas producton rate doublng, cumulatve gas producton s twce of that of the Base Case wthn the same producton tme. The end of P/Z curve of the producng reservor approaches the p-axs, so ths reservor can be abandoned nearly. However, there are stll some certan reserves remanng to be exploted n the nonproducng reservor. The follow-up research proves that contnuous development contrbutes to mprovng gas recovery. Consequently, ncreasng gas producton rate s conducve to develop gas reservor more quckly. Large gas producton rate and development of the nonproducng reservor n advance are effcent and economcal development methods for complex, offshore fault reservor gas reservors wth gas nflux or efflux. Fg. presents nstantaneous flow rate of the nonproducng reservor under dfferent gas producton rate. As showed, the lower gas producton rate s, the smaller the nstantaneous flow rate s, but the two reservors can enter pseudo-steady state ahead of tme. When gas producton rate s hgher, the nstantaneous flow rate s larger and the two reservors enter pseudo-steady state wth a delay of tme. What s more, nstantaneous flow rate tends to declne when recovery percent reaches a certan hgher value, smlar to the producton performance when transmssblty s small. That s because the pressure dfference between the two reservors s nclned to ncrease n producton tal and the nfluence of pressure on Z-factor, gas vscosty and volume factor becomes more and more sgnfcant, resultng n a nonlnear relatonshp between flow rate and pressure drop fnally. At that tme the fluds flow nearby the boundary does not follow Darcy s law. 6

10 P/Z (MPa) 4 3 qg= qg=7 qg=4 Nonproducng Reservor Producng Reservor Cumulatve Producton ( 8 m 3 ) Fg. P/Z curves of both reservors under dfferent gas producton rates 7 Instantaneous Influx ( 4 m 3 /d) qg= qg=7 qg= Cumulatve Producton ( 8 m 3 ) Fg. Instantaneous flow rate nto the producng reservor under dfferent gas producton rates Influence of Commssonng Tme of the Nonproducng Reservor Fg. shows P/Z curves of both reservors when the nonproducng reservor s put nto producton after to years development of the producng reservor, wth a tme step of years. In order to guarantee that the total producton capacty remans unchanged after commssonng of the nonproducng reservor, gas producton rate of both reservors are half of that of the Base Case, 4m3/d. t can be found that whenever the nonproducng reservor s put nto producton, gas recovery s approxmately the same. In concluson, the commssonng tme of the nonproducng reservor does not affect the gas recovery. Fg.3 presents the varaton of nstantaneous flow rate wth dfferent commssonng tme of the nonproducng reservor. Owng to the same geologcal reserves and gas producton rate of the two reservors, nstantaneous flow rate nto the producng reservor decreases unexpectedly and fnally t decreases to when formaton pressure of the two reservors strkes a balance. P/Z curves of both reservors draw near and concde wth each other at last. 63

11 4 P/Z (MPa) 3 years later years later years later years later Nonproducng Reservor Producng Reservor Cumulatve Producton ( 8 m 3 ) Fg. P/Z curves of both reservors under dfferent commssonng tme of the nonproducng reservor Instantaneous Influx ( 4 m 3 /d) 3 years later years later years later years later Cumulatve Producton ( 8 m 3 ) Fg.3 Curves of nstantaneous flow rate nto the producng reservor under dfferent commssonng tme of the nonproducng reservor Practcal applcaton and results A certan complex, offshore fault gas reservor wth total geologcal reserves of 93 8 m 3 s taken for example. The average porosty of core s 3%, wth the maxmum of 3%. And the average 3 3 permeablty of core s 8 µ m, wth the maxmum of 467 µ m, so the reservor s a knd of low porosty-medum permeablty~medum porosty-hgh permeablty reservor. The average daly gas producton rate s up to m 3. There are producton wells wth 9 wells openng at present. X well and X7 well were shut n respectvely n July, 6 and n June, due to waterfloodng. 64

12 396 Y X4 A X X X X6 X3 Y Y9 3 B X7 Y3b Y3a Y X D Y3a X3 Y3b C Y9 Fg.4 Wellarray map of a certan gas reservor As shown n Fg.4, the gas reservor s dvded nto four blocks. Block A s separated from block B by a fault. Block B, C and D are separated by a T-shaped fault. Block A was the frst one exploted, so the formaton pressure of the other three blocks has dropped dfferently to some extent when they were put nto producton. As presented n Table, reservor producng degree of Block A s over %, whle that of Block B s relatvely small and the average s 3.7%. Therefore, Block B maybe employed by Block A. In addton, reservor producng degree of Block AB s also more than %, meanng that Block CD may be employed. Well Block IIP, 8 m 3 Table Reservor producng degree among these blocks Flowng materal balance method Dynamc reserves ( 8 m 3 ) Producng degree(% ) as recovery method Dynamc reserves( 8 m 3 ) producng degree(% ) Ol reservor nfluence functon method Dynamc Producng reserves( degree(%) 8 m 3 ) X-X6 A X7 B X-X7 AB X3 C X D Sum After analyzng the estmated formaton pressure(table 3) and dynamc reserves, there manly exsted these cases as follows: () as flows nto Block A from Block B(B A); () as flows nto Block B or Block AB from Block C(C B or AB); (3) as flows nto Block B or Block AB from Block D(D B or AB); (4) as flows nto Block D from Block C(C D). 6

13 Table 3 The estmated formaton pressure data Well Date Estmated formaton pressure (MPa) Intal formaton pressure( MPa) Pressure drop( MPa) X X X Here, the MBEs of multcompartment gas reservors can be reduced to MBEs of four-compartment gas reservors. Then ths method was appled nto these four blocks mutually nfluenced to analyze the change n formaton pressure and producng reserves of these blocks. The man parameters to be referred to are n Table 4. Standard Condtons Table 4 Basc Data of the Reservor Reservor Data Standard pressure(mpa).3 Intal reservor pressure(mpa) 38. Standard temperature( ). Intal reservor temperature( ) 7 as Propertes Intal porosty, fracton.3 as gravty.684 Intal water saturaton, fracton.8 Pseudo-crtcal pressure(mpa) Pore compressblty(/mpa).69 Pseudo-crtcal temperature(k) 6.7 Water compressblty(/mpa).698 Informaton Of Numercal Control Tme step(d) 3 Pressure tolerance. Frst, we can fx geologcal reserves and ntal reservor pressure of the four blocks and make fttng analyss of the observed reservor pressures after settng the transmssblty as the fttng parameter. Then, fttng analyss s made after geologcal reserves of all the blocks are set as the fttng parameter. When the sum of resduals squared of fttng s the smallest, we can obtan the fttng geologcal reserves and transmssblty. The results are shown n Table. Block A does not connect wth block C and block D drectly, so cumulatve nflux and transmssblty between them are and they are not presented n the table below. Table Fttng results of all the blocks Block A Block B Block C Block D Fttng geologcal reserves, 8 m Remanng geologcal reserves, 8 m Block A - Block B Block B - Block C Block B - Block D Block C - Block D Cumulatve gas nflux, 8 m Transmssblty, md m Fg. presents the fttng curves of reservor pressure and t shows that the fttng results s rather satsfactory, demonstratng that ths method s applcable to such an open type fault block gas reservor. The sum of the resduals squared s 7.36MPa The fault sealng s poor between block A and block B accordng to the results of the parameter estmatons n Table. The hstory match resulted n a transmssblty of md m and a IIP of 66

14 block A of 3 8 m 3 and block B of 8 m 3. There was only X7 well n block B producng and t was shut n after producton of a short tme of fve years. The remanng reserves are m 3, ndcatng that the producng reserves s large. Fg.6 shows that nstantaneous flow rate nto block A decreased soon after commssonng of X7 well n block B n. So the apparent formaton pressure of both blocks tended to be equal. Then nstantaneous flow rate ncreased gradually n after t was shut n due to water breakthrough. The fault sealng s good between block B and block C as well as block D. The hstory match resulted n a small transmssblty of 6 md m and 4 md m respectvely. Because of the small transmssblty, pseudo-steady flow dd not develop between them after block B was put nto producton, so apparent formaton pressure between them kept dvergent. Fg.7 shows that nstantaneous flow rate nto block B decreased soon after commssonng of X3 well n block C n May, 7. The apparent formaton pressure of block C dropped rapdly and t was a lttle smaller than that of block B at the end of. Therefore, block B suppled block C wth gas. Fg.8 shows that nstantaneous flow rate nto block B decreased soon after commssonng of X well n block D n February,. The apparent formaton pressure of block D dropped rapdly. One year later, nstantaneous flow rate ncreased gradually after X was shut n because of producng water. The fault sealng s good between block C and block D. The hstory match resulted n a small transmssblty of 9 md m. Because of the small transmssblty, pseudo-steady flow dd not develop between them, so apparent formaton pressure between them kept dvergent. Fg.9 shows that nstantaneous flow rate nto block C decreased soon after commssonng of X3 well n block C n May, 6. The apparent formaton pressure of block C also dropped rapdly. However, X well n block D was shut n due to water breakthrough at the end of, and then formaton pressure of block D bult up, leadng to formaton pressure of block D beng larger than that of block C. Therefore, gas n block D flow nto block C. In summary, IIP of block B s almost employed so nfllng well s not recommended. Although X well n block D was shut n due to water breakthrough, the results of the parameter estmatons ndcate that IIP of block D s also employed. Formaton Pressure (MPa) Formaton pressure of block A Formaton pressure of block B Formaton pressure of block C Formaton pressure of block D Measured pressure of block A Measured pressure of block B Measured pressure of block C Measured pressure of block D 997// // 3// 6// 9// // // Date (yyyy/m/d) Fg. Hstory match of observed reservor pressures n a certan complex fault gas reservor 67

15 q AB ( 8 m 3 ) q AB pab /6/ /6/ 6/6/ /6/ 4/6/ Date (yyyy/m/d) Fg.6 Influx nto block A from block B pab ( 8 m 3 ) q BC ( 8 m 3 ) q BC pbc /6/ /6/ 6/6/ /6/ 4/6/ Date (yyyy/m/d) Fg.7 Influx nto block B from block C pbc ( 8 m 3 ) 3 8. q BD ( 8 m 3 ) q BD pbd pbd ( 8 m 3 ) q CD ( 8 m 3 ) q CD pcd pcd ( 8 m 3 ) 998/6/ /6/ 6/6/ /6/ 4/6/ Date (yyyy/m/d) Fg.8 Influx nto block B from block D /6/ /6/ 6/6/ /6/ 4/6/ Date (yyyy/m/d) Fg.9 Influx nto block C from block D Conclusons MBEs of multcompartment gas reservors were derved accordng to materal balance. The determnatons of the related physcal parameters of gas were ntroduced. Newton-Raphson teraton scheme was employed to solve the MBEs under successve tme step. Computer programmng proved ths scheme s fast and steady. Pressure would converge wthn 3-8 teratons. Amed at the exstence of nflux and efflux n a complex, offshore fault block gas reservor wth four blocks, the MBEs of four-compartment gas reservors was appled to analyze change n nflux and nstantaneous flow rate of these blocks. Hstory match of observed reservor pressures was good, demonstratng that ths method s applcable to such an open type fault block gas reservor. The results of the parameter estmatons ndcate that reserves of block B and block D were both employed, so there was no need to nfll well. Ths would be helpful to reduce costs and mprove economc effcency. Nomenclature pj cumulatve gas nflux from reservor j nto reservor, 8 m 3 q j nstantaneous gas flow rate from reservor j nto reservor, 8 m 3 m j pseudo pressure correspondng to the average formaton pressure of reservor j, MPa/mPa s T rj transmssblty of fault between reservor and reservor j, md m, K fj permeablty of fault between reservor and reservor j, md T = K p j formaton pressure of reservor j, MPa z j Z-factor correspondng to the average formaton pressure of reservor j, dmensonless rj fj A fj w fj 68

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