498 n a horzontal wellbore n a heavy ol reservor L Mngzhong, Wang Ypng and Wang Weyang Abstract: A novel model for dynamc temperature dstrbuton n heavy ol reservors s derved from and axal dfference equatons and then ntegrated. Takng nto account the couplng of temperature and pressure n the reservor and wellbore, models for calculatng dstrbutons of the reservor temperature, reservor pressure, and water saturaton are also developed. The steam njected nto the wellbore has a that the reservor temperature and pressure decrease exponentally wth ncreasng dstance from the as that of the temperature feld, and both varaton ranges decrease from the wellbore heel to the toe. varaton ranges of the reservor temperature and pressure ncrease wth steam njecton tme, but rate of ncrease dmnshes gradually. Key words: 1 Introducton Heavy ol reservors are common worldwde. Most of the heavy ol contans a sgnfcant amount of asphaltenes and the presence of asphaltene s the man reason for the thermal recovery methods have been wdely used as the calculated accurately by consderng the couplng processes, but prevous studes have mostly concentrated on pressure couplng or temperature couplng separately because of the wellbore pressure couplng was studed wthout consderng changes of the reservor temperature (Zhou and Guo, 009). couplng not n a steady state has been conducted (Ouyang of varable mass flow n the wellbore (Huang et al, 010). Research nto temperature couplng was theoretcally and numercally nvestgated. Reservor and wellbore temperature dstrbuton models were studed (Hasan et al, 007; Lu et al, 007). Methods of establshng steady state temperature Receved December, 011 coupled models were studed (Lvescu et al, 008a; 008b), Hgh-accuracy models were obtaned by correctng porosty the above temperature models gnored changes of reservor numercal models for steam flow n wellbore and reservor were obtaned (Wu et al, 004). The numercal models consdered the couplng of temperature and pressure, but the temperature dstrbuton n the model was statc. s deduced accordng to energy conservaton. The model s solved through dfference processng. A block-centered grd system s appled to the entre reservor doman. Axal and radal dfference equatons are bult separately at frst, and then are ntegrated. Models for the reservor temperature profle, reservor pressure profle and water saturaton dstrbuton are all developed by consderaton of couplng procedures for the coupled model are gven, and calculaton Physcal model
physcal model s a two-dmensonal axally symmetrc model for a thermal drve. The followng assumptons are made for the reservor obeys Darcy s law; 3) The physcal propertes of temperature; 4) The temperature of the reservor outsde the Wellbore heel sp L Wellbore toe Fg. 1 3 Mathematc formulatons 3.1 Dynamc temperature dstrbuton n the reservor ncrement of the mcro-unt ncludes three parts: the heat unt and the heat njected nto the mcro-unt from the outsde. Energy conservaton can be expressed as follows: 1 T T ecpet e r e r r r x x where r s the radus of the mcro-unt, m; x s the length of the mcro-unt, m; e T s temperature, ºC; r e 3 ; c pe 3 t s tme, s. The product of the effectve densty and the effectve heat c 1 c S c S c e pe s ps o o po w w pw Accordng to the assumptons mentoned above,, s, o, w, c ps, c po, and c pw n Eq. () are all ndependent of tme. Based on the mass conservaton equaton, l r l (l represents o, w) n Eq. () can be expressed as a functon of pressure P: S 1 kk P kk P r o o o ro o ro r r o r x o x wsw 1 wkkrw P wkkrw P r r r r x x w w (1) () (3) (4) 499 1 T 1 okkro P e r cpot r r r r r r o r 1 wkkrw P T cpwt r e (5) r r w r x x okkro P wkkrw P T cpot cpwt ecpe x o x x w x Eq. (5) s an energy balance equaton for reservors n two-dmensonal cylndrcal coordnates. The dynamc temperature feld wthn the reservor can be obtaned by solvng Eq. (5). Eq. (5) can be solved wth dfference processng. A blockcentered grd system s appled to the entre reservor doman. Axal and radal dfference equatons are bult separately, and and j. An equal-step dfference x and the radal step r. The frst term on the left-hand sde of Eq. (5) through radal dfference processng can be expressed as follows: 1 T e r r r r 1 1 1 T j1t j e j r j r jr r 1 1 T jt j1 e j r j r The second term on the left-hand sde of Eq. (5) through radal dfference processng can be expressed as follows: P c j T j ct r r r r r j r P jp j j j r j r P jp j j j r j r The thrd term on the left-hand sde of Eq. (5) through radal dfference processng can be expressed as follows: P c j T j ct r r r r r j r P jp j j j r j r P jp j j j r j r
500 The fourth term on the left-hand sde of Eq. (5) through axal dfference processng s shown as follows: T T T T T x x x x x P c T P P P P ct x x x x x The sxth term on the left-hand sde of Eq. (5) through axal dfference processng s shown as follows: 1 1 1 1 1 1 P cpw T P P P P cpwt w w w w w w x x x x x T j T j jc j AT j A T j A A A A A A T j A T j A T j (6) wth j r j A, r j r j r j A r j r k A c j j k j r j P jp j j r j r c j j k j r j P j P j j c j j k j r j P j P j j c j j k j r j P jp j j j A, x j A x c j jk j P j P j j c j jk jp jp j j c j jk jp jp j j
501 effectve thermal conductvty, relatve permeablty to ol and water are all represented as a functon of water saturaton. The effectve thermal conductvty can be wrtten as: 1/ 1 1/ 1 1 k S S ro w w The relatve permeablty to ol and water are gven as follows (Cheng et al, 009): 1/ 1 1/ 1 1 kro Sw Sw (8) rw 5 w k S The relatonshp between heavy ol vscosty and temperature s expressed as follows (Chen, 1996): T (7) (9) (10) where s the reservor porosty; P s pressure, MPa; k s ; ndex; k ro, k rw are the relatve permeablty to ol and water; S o, S w are the ol saturaton and water saturaton, respectvely; o, w are the ol vscosty and water vscosty, respectvely, s, o, w denote the denstes of rock, ol and water, 3 ; c ps, c po, c pw denote the heat capacty of rock, ol and 3 l s, l o, l w denote the thermal conductvty of T s s the reservor temperature xx=l N. The temperature at the perforaton tunnel T w. Let the radal seral number j=1, and the borehole wall temperature T(,1) s obtaned, that s T w. N equatons ncludng T w, P w, S w are obtaned from Eq. (6). The equatons nclude 3N unknown varables: T w, P w, S w (N), and can be expressed as follows: F T, P, S 0 TF w w w (11) Eq. (11) shows the nfluence of formaton pressure and water saturaton on the formaton temperature at the borehole wall at segment. 3. Wellbore temperature dstrbuton The wellbore s dvded nto N segments accordng to the number of perforatons. The model for temperature change of each segment s obtaned by establshng control equatons (Wang et al, 010): 1 vm v r dt dqdw dsp spcs sp 1 dt 1 dsp spvm d P mah T dp P mah (1) where dqw s the work sp s the steam absorpton v m s the steam flow rate n v r s the steam rate whch flows nto the m s the steam densty n the current segment, 3 ; A h s the cross sectonal area of the segment, m ; P s the steam njecton pressure, MPa; T s the steam temperature, ºC; C s 3 When the steam njecton pressure and the steam T w at perforaton tunnel can be obtaned from Eq. (1). T w nto the followng equaton: 1 Tw Tw, 1 Tw, 1 Tw N Tw0 Twf Tw0 0 1,,, We obtan N equatons ncludng sp, T w, P w : F, T, P 0 TW sp w w (13) (14) where T wf s the temperature of the wellbore heel, ºC; sp s the steam absorpton at perforaton tunnel absorpton on the steam temperature at segment. The equatons nclude 3N unknown varables: T w, P w, sp (N). 3.3 Pressure dstrbuton n the reservor at perforaton tunnel s sp. The pressure at perforaton tunnel s P w. Accordng to potental superposton prncple, the seepage equaton of the two-phase ol-water mxture whch follows Darcy s law can be derved, and can be wth AI B T I sp1, sp, sp3,, spn 11 e1 1 e 13 e3 1 N e N 1 e1 e 3 e3 N en A 31 e1 3 e 33 e3 3N en N1 e1 N e N3 e3 NN en 4Lp1 Hw1 He 4Lp Hw He B 4Lp3 Hw3 He 4Lp N Hw N He (15)
50 H N r L ln r L C L p p 1 Pw Pw, 1 Pw, 1 Pw N Pw0 Pwf Pw0 0 1,,, (18) sn cos r x x rw y z rw z w p x x r L sn y z rw Lp cos z where A s the coeffcent matrx assocated wth the calculaton poston; B s the matrx assocated wth pressure; I s the column matrx composed of the mcro-unt steam absorpton; H e s the potental at the reservor boundary; H w s the potental at perforaton tunnel ; L p s the length of perforaton tunnel ; j s the value of at segment j; e s the value of at the reservor boundary; C s the ntegral constant; x s the abscssa of perforaton tunnel ; z s the coordnate of perforaton tunnel n the z drecton; r w s the, m; q, rad; (x, y, z) s the coordnates of the calculaton poston n the reservor. Pressure P w, temperature T w, and water saturaton S w are all mpled n the potental functon H. Therefore Eq. (15) ncludes 4N unknown varables: sp, T w, P w, S w ( N), and can be expressed as follows: PF T, P, S 0 sp, w w w F (16) Eq. (16) shows the nfluence of formaton temperature, steam absorpton and water saturaton on the formaton pressure of the borehole wall at segment. 3.4 Wellbore pressure dstrbuton A pressure drop model for the wellbore can be derved by dvdng the wellbore nto N segments and establshng control equatons (Wang et al, 010): d P A h vmdsp c 1dT 1 v T d P P m sp (17) where t c When the steam temperature and steam absorpton are P w at perforaton tunnel can P wf at the P w nto Eq. (18), we can an N-dmensonal vector, and then the equaton ncludes 3N unknown varables and can be expressed as follows: F, P, T 0 PW sp w w (19) Eq. (19) shows the nfluence of steam temperature and absorpton on the steam pressure at segment. 3.5 Coupled thermo-hydro model Based on the couplng relatonshp between temperature contnuty and pressure contnuty, both the reservor temperature and wellbore temperature and the reservor pressure and wellbore pressure are the same at the borehole wall. A model for thermo-hydro couplng n the reservor and temperature Eq. (11), wellbore temperature Eq. (14), reservor varables n the coupled model nclude sp, P w, T w, and S w, whose number s 4N, equalng the number of equatons. Therefore, the coupled model has a unque soluton. The boundary and ntal condtons of the coupled model are lsted below: r r, P P, T T r r, P Pe, T Te rw r r Sw So 1 w w w (0) where P w s the wellbore pressure, MPa; T w s the wellbore temperature, ºC; P e s the ntal reservor pressure, MPa; T e s the ntal reservor temperature, ºC. 4 Soluton procedures for the coupled model parameters, the coupled model s a complex nonlnear equaton system, whch should be solved by an teratve method. As can be seen from the coupled model, n each tmng-cycle, the temperature nfluences the flud vscosty, the coupled model are as follows: t, temperature T w and water saturaton S w ) Accordng to the three-dmensonal coordnates of the j n Eq. (15) can be calculated, and then matrx A can be obtaned;
503 P w. P w =P wf can be assumed before calculaton; 4) Calculate the potental functon H w n Eq. (15), and then calculate matrx B; and then obtan sp ; sp pressure model, update P w and record as P' w ; 7) Wth pre-gven p, f max P' w P w p, let P w = P' w, and P' w cannot meet the accuracy requrement, let P' w sp and P w nto the wellbore temperature equaton, and calculate the temperature dstrbuton T' w n the wellbore; P w and T' w nto Eq. (6), and calculate the water saturaton dstrbuton S' w at current tme; 10) Wth pre-gven T and, f max T' w T w < T and max S' w S w, let T w = T ' w and S w =S' w T' w or S' w cannot meet the accuracy requrements, let S' w and S' w 11) Compare wth t max < max, let = and go to are lsted n Table 1. Table 1 Basc data for the example Durng the calculaton process, the tme step s 1 h, the = 96 h, the dstrbuton of steam absorpton along the reservor temperature, reservor pressure and water saturaton along the radal drecton are shown n Fgs. 3-5. When =1,, 4, 6, 8 and 10 d, the varatons of reservor temperature and pressure at the poston of x= 60 m are shown n Fgs. 6 and 7, respectvely. Steam absorpton, t/h 0.431 0.430 0.49 0.48 0 50 100 150 00 Dstance from the wellbore heel, m Fg. Fg. ndcates that steam absorpton decreases gradually pressure decreases from heel to toe (Wang et al, 010b), and the pressure dfference between reservor and wellbore becomes smaller, resultng n a reducton n steam absorpton ablty. Reservor parameters Wellbore parameters Heavy ol parameters parameters Depth, m 750 Thckness, m 1 Orgnal temperature, C 48.5 Orgnal pressure, MPa 7.5 Porosty, % 31. Permeablty, μm 1 Orgnal ol saturaton, % 60 Orgnal water saturaton, % 40 4 3. C).0 10 3. C) 1.7 Wellbore dameter, mm 159.4 00 3. C).1 10 3. C) 1.35 Vscosty at 50 C, mpa. s 3.4 10 4 Temperature at the wellbore heel, C 337 Pressure at the wellbore heel, MPa 14.1 Dryness at the wellbore heel, % 56 9 Reservor temperature, 300 Dstance from the wellbore heel: 60 m Dstance from the wellbore heel: 10 m Dstance from the wellbore heel: 180 m 00 100 0 0 1 3 4 5 Radal dstance from the wellbore, m Fg. 3 Radal dstrbuton curves of the reservor temperature Reservor pressure, MPa 14 Dstance from the wellbore heel: 60 m Dstance from the wellbore heel: 10 m 1 Dstance from the wellbore heel: 180 m 10 8 0 4 6 8 10 Radal dstance from the wellbore, m Fg. 4 Radal dstrbuton curves of the reservor pressure
504 Water saturaton, fracton 1.0 0.8 0.6 0.4 Dstance from the wellbore heel: 60 m Dstance from the wellbore heel: 10 m Dstance from the wellbore heel: 180 m 0 1 3 Radal dstance from the wellbore, m Fg. 5 Radal dstrbuton curves of the water saturaton The dstrbutons of the reservor temperature and pressure n the radal drecton (llustrated n Fgs. 3 and 4) ndcate by steam. Maxmum values of the reservor temperature and pressure appear at the borehole wall. Along the radal drecton, the reservor temperature and pressure decrease exponentally wth ncreasng dstance from the wellbore. The njecton s twce as wde as that of the temperature feld, and both varaton ranges decrease from the wellbore heel to the toe. Ths s because the reservor pressure and heavy ol Reservor pressure, MPa 300 00 100 Injecton tme: 1 d Injecton tme: d Injecton tme: 4 d Injecton tme: 6 d Injecton tme: 8 d Injecton tme: 10 d 0 0 4 6 8 Radal dstance from the wellbore, m Fg. 6 Radal dstrbuton curves of the reservor temperature 14 1 10 8 Injecton tme: 1 d Injecton tme: d Injecton tme: 4 d Injecton tme: 6 d Injecton tme: 8 d Injecton tme: 10 d 0 4 6 8 10 Radal dstance from the wellbore, m Fg. 7 Radal dstrbuton curves of the reservor pressure vscosty are both hgh under orgnal reservor condtons. s heated by steam and the reservor temperature and pressure decrease to ther ntal values at and 4 meters from the borehole wall. The steam njected nto the wellbore has a temperature, whch s manly caused by thermal convecton pressure decrease wth the dstance from the wellbore heel. Fg. 5 shows that the water saturaton n the vcnty of the wellbore decreases rapdly from about 1.0 to 0.4 (orgnal value), exhbtng the same varaton as the temperature and s smaller than that on the reservor temperature and pressure, As shown n Fgs. 6 and 7, the steam njecton tme has a sgnfcant effect on the reservor pressure and temperature, and the radus of the heavy ol reservor affected by steam vares from 1 to 5 m when the njecton tme ranges from 1 days to 10 days. However, the ncrement decreases gradually. Ths s because the temperature dfference between steam and reservor s hgh at the ntal stage of steam njecton, and the heat transfer rate (.e., the quantty per unt tme) s hgh, so wth steam njecton tme. After a perod of steam njecton, the temperature dfference between the steam and the formaton near the wellbore becomes low, so the rate of heat transfer from the steam to the formaton reduces substantally and the rates of temperature and pressure changes dmnsh n the radal drecton. Moreover, the ncreasng radus of the reducton n the effect of the njected steam. 6 Conclusons 1) A model s derved for calculatng the dynamc temperature dstrbuton n heavy ol reservors. By combnng dynamc temperature dstrbuton and pressure dstrbuton equatons for the reservor and wellbore, a coupled thermohydro model s establshed. The model consders the couplng of temperature and pressure, and can be used to calculate of reservors. steam njecton. The reservor temperature and pressure decrease exponentally wth ncreasng dstance from the wellbore. Maxmum values of the reservor temperature and pressure appear at the borehole wall. The radal varaton range of pressure feld nduced by steam s twce as wde as that of temperature feld, whle, from heel to toe, both saturaton has the same varaton as the temperature and 3) The steam njecton tme has a sgnfcant effect on the reservor pressure and temperature, and the radus of the heavy ol reservor affected by steam vares from 1 to 5 m when the steam njecton tme ranges from 1 days to 10 days. However, the ncrement decreases gradually.
505 References water flow model for predcton of sand face condton n steam of a fshbone well consderng couplng among mult-segment profles durng soak perods n steam-stmulated wells. Petroleum 1999. 0(3): 8-86 (n Chnese) multphase wellbore-flow model for use n reservor smulaton. Chnese) (n Chnese) applcatons to steam njecton n heavy ol reservors. Chnese (n Chnese) reservor modelng nvestgatng poroelastc effects of cyclc steam 17-13 (n Chnese) wellbore. Petroleum Exploraton and Development. 004. 31(1): 88-90 (n Chnese) 13755) and Gas West Asa, 11-13 Aprl 010a, Muscat, Oman wells wth reservor flow. Petroleum Drllng Technques. 009. 37(4): 84-87 (n Chnese) (Edted by Sun Yanhua)