Chapter 5 Generalized Material Conservation Equations and Special Cases

Transcription

1 Chapter 5 Generalized Material Cnservatin Euatins and Special Cases The basic cncepts discussed earlier will be eneralized t the multiphase and multicmpnent flw systems that ccur in reservir recvery prcesses. The mass cnservatin euatins will be derived as a mre eneral material cnservatin euatins. The frm f the euatins will be illustrated fr the varius material uantities bein cnserved. A Generalized Material Cnservatin Euatin The fluid flw euatins in mathematical reservir simulatrs cnsists f tw basic types f euatins: the euatins fr the cnservatin f mmentum which is just a eneralizatin f Darcy's law and the euatins fr the cnservatin f mass r material. The material uantity bein cnserved is nt necessarily in mass units. Fr example, the uantity bein cnserved may be stck tank barrels (STB) f liuid hydrcarbns at stck tank cnditins (14.7 psia and 60 F) prduced frm the separatr. A barrel is a unit f vlume eual t ft 3, 42 U.S. allns, r m 3. The mass f ne STB f il will nt chane unless the specific ravity r API ravity chanes durin the depletin f the reservir. Other cnserved uantities may include STB f water, MCF f as, r mles f individual cmpnents r rups f cmpnents. These uantities will be euivalent t an unit f mass if the specific ravity r averae mlecular weiht des nt chane. [The "ravity" f an il is enerally expressed in "API derees," cmputed frm the specific ravity at 60 F, ρ, by the relatin API = (141.5/ρ) ] A eneral cncentratin will be dented as C. The cncentratin, C, is defined as the amunt f the cnserved uantity per unit vlume (bulk) f the system. Fr example, the cncentratin f stck tank il per unit f bulk vlume f the reservir is STB Cil = b S φ bbl f bulk reservir vlume where S φ b il saturatin frmatin prsity reciprcal il frmatin vlume factr, STB/RB A eneral surce term will be dented as. It is defined as the injectin rate r the neative f the prductin rate per unit f bulk reservir vlume. It may be expressed as the amunt f the uantity injected r prduced per unit vlume per unit time. The surce term will be used t represent wells in reservirs in the differential frm f the cnservatin euatins. 5-1

2 A eneral flux term will be dented as the vectr, f. The flux can be interpreted as the rate f transprt r flw rate f a uantity per unit area. Since the rate f transprt per unit area depends n the rientatin f the surface, the value f f in any directin may be interpreted as the rate f transprt acrss a unit f area with its nrmal in that directin. Thus, if we have a Cartesian crdinate system, the cmpnents f f, f x, f y, and f z can be interpreted as the rate f transprt per unit f area in the x, y, and z directins respectively. An example f a flux is the mass flux vectr iven by f mass = ρ u where ρ u f fluid density, mass/vlume superficial fluid velcity, lenth/time r vlume/(area time) mass flux, mass/(area time). T illustrate a derivatin f the eneralized cnservatin euatin, cnsider a small element f vlume in Cartesian crdinates. The element is a parallelepiped with a vlume eual t x y z. A material balance ver this element f vlume ver a time interval t can be expressed as ACCUMULATION = IN OUT + SOURCE The ACCUMULATION term represents the accumulatin r ain f the cnserved uantity in the element f vlume ver the time interval frm t t t+ t. ( + ) ACCUMULATION = C t t C t x y z The IN-OUT terms represent the net influx f the material thruh the surfaces bundin the element f vlume durin the time interval, t. This may be expressed as ( x x+ x) IN OUT = fx fx y z t ( y y y y+ y) + f f x z t ( z z z z+ z) + f f x y t. 5-2

3 If f x is psitive, the first term f the sum may be interpreted as the transprt f material int the vlume acrss the face at x minus the transprt f the material ut f the vlume at the face at x + x. The SOURCE term represents the additin f material int the vlume ver the time interval, t, due t the surces. This may be expressed as SOURCE = x y z t. Substitutin f the abve euatins int the cnservatin euatin and divisin by x y z t results in C C fx f f t t t x x x x y f y y y f y z f z+ z z z = +. t x y z Takin the limit as x, y, z,and t es t zer results in C f f f t x y z x y z = +. The sum f derivatives f the cmpnents f f can be represented as the diverence f f. C = f +. t Since the diverence peratr is independent f the crdinate system, the euatin in vectr ntatin is applicable t any ther crdinate system. Fr the case f tw dimensinal (x-y) flw in a reservir f variable thickness, h(x,y), a similar derivatin will result in where ( hfx) ( hfy) C h = + h. t x y 5-3

4 C = f f x y = = 0 h h 0 h 0 h Cdz h f dz x h f dz y h dz 0 = h. 5-4

5 ydrcarbn Phase Behavir Crude ils are enerally a cmplex mixture f hundreds f cmpnents. wever, the material that is prduced at the surface facilities and marketed is a "stck tank" il and a separatr as. It will simplify matters reatly if the crude il culd be treated as if it cnsisted f these tw cmpnents and the material cnservatin euatins be expressed in terms f these tw cmpnents. Since bth fluids are cmpressible and the cmpnents distribute between the liuid and vapr phases at the reservir temperature and pressure, sme means are needed t describe the vlumetric behavir f these fluids. The vlumetric behavir f a typical mixture f hydrcarbns with decrease in pressure is illustrated in the fllwin fiure. Suppse the pressure, p 1 represents the initial reservir pressure where this mixture is under-saturated, ne phase liuid (il) system. As the pressure drps t p 2, the vlume increases as the liuid expands. This expansin is characterized by the il cmpressibility. c 1 V = V p T At pressure p 3 the first bubble f a vapr phase appears. This pressure is called the bubble pint pressure, p b. As the pressure is decreased further, additinal as is evlved and expands. Even thuh the pressure is decreasin, the il vlume may decrease if the il vlume decrease due t the lss f as is reater than the il vlume increase due t cmpressibility. The chane in vlume with pressure depletin needs t be uantified fr the purpse f simulatin. We d this by expressin the vlumes per unit f "stck tank il" and "separatr as" at standard temperature (60 F) and pressure (14.7 psia). The separatr as will enerally be mstly methane with small amunts f ethane and carbn dixide. The fllwin terms are emplyed: (a) Frmatin Vlume Factr (FVF) B : the vlume f reservir liuid phase that wuld yield a unit vlume f il at stck tank cnditins. (b) Slutin Gas Oil Rati (GOR) R s : the standard vlume f as that the reservir liuid phase culd yield with a unit vlume f stck tank il. (c) Gas Frmatin Vlume Factr B : the vlume f reservir vapr phase that wuld yield a unit vlume as at standard cnditins. 5-5

6 Fi. 5.1 Variatin in hydrcarbn phases and vlumes with decreasin pressure. (Csse,1993) 5-6

7 Fi. 5.2 Cnditins fr measurin the frmatin vlume factr and slutin as il rati (Csse, 1993). The il frmatin vlume factr is expressed in reservir barrels per stck tank barrel (in US) r cubic meter per cubic meter. The cnditins fr the frmatin vlume factrs are illustrated in fiure 5.2. The dependence f the prperties n the pressure is illustrated in fiure 3 fr a liht il and a mderate ravity il. There is a break in the slpe f the curves at the bubble pint pressure. Abve p b the il expands n reductin f pressure and the slutin GOR remains cnstant. Belw p b there is shrinkae f the il n reductin f pressure due t lss f slutin as. Suppse yu had an ptin t first prduce by primary depletin, i.e., il expansin and slutin as drive, befre water fldin. Als, suppse that the same averae il saturatin wuld remain after water fldin. What wuld be the effect n recvery f depletin t a lw pressure befre water fldin cmpared t startin the water fld at the bubble pint pressure? Fi. 5.3 Dependence f il prperties n pressure (Csse, 1993) 5-7

8 Anther factr that culd have a sinificant dependence n pressure is the il viscsity. The liht il has a lw viscsity cmpared t water and has little dependence n pressure. The medium ravity il increases in viscsity as the pressure is depleted because f the lss f the lw mlecular weiht as cmpnents and the remainin cmpnents have a much hiher averae mlecular weiht. The increase in viscsity will be much mre sinificant with lwer ravity (heavy) ils. Fi. 5.4 Dependence f the il viscsity n pressure depletin (Csse, 1993) If repressurizatin by water injectin ccurs after pressure depletin, the curve fr the slutin GOR may appear as in fiure 5.5. Suppse the riinal cnditins are at pint A. A break in the curve ccurs at p b. Suppse that pressure depletin and as evlutin ccurred until at pint a water injectin reversed the pressure trend and increased the pressure. As the pressure increases, the free as will disslve back int slutin aln the same curve. As a result f as miratin and prductin, a iven lcatin may nt have a much as as initially present and the as Fi. 5.5 R s as a functin f pressure, pressure depletin frm prductin fllwed by repressurizatin frm water injectin (Kederitz, arvey, narpur, 1989). saturatin may t zer at sme pressure which is called the saturatin pressure. (The bubble pint pressure is the saturatin pressure f the riinal il.) As pressure is increased further R s will nt chane since there is n mre as t int slutin. Als, the il frmatin vlume factr will chane nly by the cmpressibility effect abve this pressure since there is n as in int slutin. This saturatin pressure upn repressurizatin can be different fr each rid blck. 5-8

9 Cnservatin Euatins fr Particular Fluid Systems Readin Assinment: Chapter 4 and 11 f Reservir Simulatin The eneral frm f the material cnservatin was shwn earlier. The euatins will nw be expressed fr particular reservir fluid systems. Unless there is a sinificant amunt f disslved hydrcarbns, the water cnservatin euatins will be the same fr the varius hydrcarbn fluid mdels. wever, the hydrcarbn fluids can be mdeled as black il and as, incmpressible fluids, dry as, wet as, cndensate as, vlatile il, cmpsitinal, and miscible. In all f the fluid systems there will be at mst three fluid phases, the aueus phase, hydrcarbn liuid phase, and hydrcarbn vapr phase. These phases will enerally be dented with the subscripts w,, and r a,, and v. The phase saturatins Sw, S,and S are defined as the vlume fractin f the pre space ccupied by each phase. Thus, we have Sw + S + S = 1. The relative permeability f each phase is a functin f the saturatins. In a water-wet frmatin the water and as relative permeabilities can be apprximated as functins f the respective saturatins. wever, the il relative permeability is a functin f tw f the saturatins in a three-phase system. k = k S rw rw w 1 k = k S = k S + S k = k S S r r r w (, ) r r w At a iven pint in the reservir, the phase pressures will be different due t the interfacial tensin and the curvature f the interfaces. This phase pressure difference can be expressed as a capillary pressure. p p = P S w cw w 1 p p = P S = P S + S c c w where P cw is a functin f S w in a water-wet frmatin and P c is always a functin f the as saturatin. (a) Water. The cncentratin and flux f water can be expressed as 5-9

10 C f = φ S b = b u w w w w w w where C w f w φ b w u w water cncentratin, STB/(bulk reservir vlume) water flux, STB/(area time) frmatin prsity reciprcal water frmatin vlume factr, STB/RB water superficial velcity, vlume/(area time) The superficial fluid velcity fr water in multiphase flw is k k u = p D ( ρ ) rw w w w w where k rw water relative permeability, functin f water saturatin p w water phase pressure. The cnservatin euatin fr water can be expressed as φbwsw k krw b w = ( pw ρw D) + w. t w (b) Black il. Crude il with a ravity less than 40 API and a reservir temperature less than 150 F can be represented with a black il mdel. The black il mdel assumes that all f the stck tank il is prduced frm the reservir liuid phase, i.e., the reservir vapr phase yields n stck tank il. In the black il fluid systems there is very little difference between the labratry euilibrium flash vaprizatin and the differential vaprizatin data. Lihter crude (40-45 API) with hiher reservir temperature ( F) will shw a mre sinificant difference between the flash and differential data. Pssible appraches t mdelin these crudes with a black il mdel is t either cnduct the labratry experiment such as t differentially vaprize the il sample and flash t surface cnditins r t use the differential data fr the reservir fluid flw and expansin and use the flash data t cnvert the fluids t surface cnditins. Crudes with hiher ravity (45-55 ) and hiher reservir temperature (abve 250 F) cannt be mdeled as a black il as much f the stck tank il is prduced frm the reservir vapr phase. These type f fluid systems must be mdeled as a vlatile il with ne f the ther mdels discussed later. 5-10

11 The cnserved uantities in the black il mdel are stck tank barrels (STB) f il and thusands f cubic feet (MCF) f separatr as at standard cnditins f 14.7 psia and 60 F. The as is mdeled t be sluble in the reservir liuid hydrcarbn phase. The cncentratin f the stck tank il in the liuid phase and separatr as in the vapr phase are represented by the reciprcal il frmatin vlume factr b (STB/RB) and the reciprcal as frmatin vlume factr b (MCF/RB) respectively. The cncentratin f the separatr as in the liuid phase is represented by b R s (MCF/RB) where the slutin as/il rati R s, has the units f MCF/STB. The cncentratin f stck tank il and separatr as in the black il mdel is where CSTB = φ b S C = φ b S + b R S Mcf s S S saturatin f hydrcarbn liuid (il) phase, fractin f pre vlume saturatin f hydrcarbn vapr (as) phase, fractin f pre vlume. The flux f stck tank il in the black il mdel is f = b u STB k k b ( p ρ D) r = and the flux f separatr as is. f = b u + b R u Mcf s k kr b k kr b Rs = ( p ρ D) ( p ρ D). 5-11

12 The surce terms fr the black il mdel are = b STB v = b + b R * Mcf v s v * where R s is the slutin as/il rati existin in the reservir fr a prducin well and is a specified slutin as/il rati if il is bein injected. v and v are the vlumetric surce terms fr the hydrcarbn liuid and hydrcarbn vapr phases. T simplify the ntatin, and will just be dented as and. STB The stck tank il and separatr as cnservatin euatins fr the black il mdel are φbs k kr b = ( p ρ D) + t Mcf ( b S + φb S Rs) φ t = k k b k k b R ( ρ ) + ( ρ ) + r r s p D p D. (c) Incmpressible fluid flw. In water r as injectin pressure maintenance prjects with neliible chane in averae pressure, the fluids and frmatin can be mdeled t be incmpressible. T mdel the reservir fluids as bein incmpressible des nt reuire that the fluid have neliible cmpressibility, but-rather that the chane in the pressure dependent prperties with time and psitin in the reservir has neliible effect n the reservir perfrmance. Fr example, the cmpressibility f as may be lare and is imprtant in recvery by primary depletin, but in pressure maintenance by crestal as injectin, the as can be mdeled as incmpressible. In incmpressible fluid flw, the slutin as/il rati is cnstant. Since the flw and prductin f slutin as will always be prprtinal t that f il, it is nt necessary t mdel the slutin as. Thus, the cnservatin euatin fr as will nly need t include the free as. As was discussed earlier, the expressin fr the Darcy's law can be simplified in the case f incmpressible fluids by expressin the pressure and buyancy terms as a flw ptential. 5-12

13 u u u kk rw w = Φw w kk r = Φ kk r = Φ where Φ = p ρ D+ C w w w Φ = p ρ D+ C Φ = p ρ D+ C where C 1, C 2, C 3 are cnstants determined by the pressure and capillary pressure at specified depths, e.. zer capillary pressures at the free water level and free liuid level. With the incmpressible fluid assumptin, the cnservatin euatin can be expressed as fllws. φ S k k = Φ + t w rw w w w bw φ S k k = Φ + t r b φ S k k = Φ + t r b. A restrictin n the incmpressible mdel is that if the reservir has nflw bundaries, then the net sum f + + dv must eual t zer. w b V w b b That is t say that the net vlumetric injectin and prductin must be zer. (d) Dry as. The cnservatin euatins fr a water drive, dry as reservir can be represented by substitutin as prperties fr il prperties, and excludin the slutin as terms. If the slutin as in water is imprtant, such as in extremely hih pressure as reservirs, then the water cnservatin euatin can be represented substitutin water prperties fr il and usin R s, as the slutin as/water rati. 5-13

14 Sinle-phase dry as flw can be mre accurately cmputed by transfrmin the as cnservatin euatin in terms f the real as pseudpressure discussed earlier. (e) Wet as. If the pressure in the reservir stays abve the dew pint pressure but cndensate is recvered frm the separatr, the reservir is dented as a wet as reservir. Since the cndensate/as rati remains cnstant, a wet as reservir can be mdeled as a dry as reservir and the cndensate prductin be cmputed as the prduct f the cndensate/as rati and the as prductin. (f) Cndensate as. A as reservir in which liuid hydrcarbn cndenses ut f the vapr phase in the reservir as the pressure declines belw the dew pint pressure is dented as a cndensate as reservir. Nrmally the liuid hydrcarbn phase is immbile but if the liuid saturatin becmes sufficiently lare, it can becme mbile. If the liuid hydrcarbn phase is swept by a dry as, it can be revaprized int the vapr phase. In eneral, a cndensate as system can be mdeled as a multicmpnent cmpsitinal system. wever, the mdel that will be discussed is a much simpler frmulatin presented by A. Spivak. The frmulatin is analus t the black il mdel. The hydrcarbn system is mdeled as a pseud tw-cmpnent system f the STB f hydrcarbn liuid and MCF f as prduced frm the separatr as in the black il mdel. The basic assumptin f the mdel is that the euilibrium cndensate cntent f the vapr is a sinle valued functin f pressure durin bth the depletin and revaprizatin. The mdel als assumes that the separatr as cntained in the liuid hydrcarbn phase can be nelected. The STB f cndensate and MCF f as recvered frm the separatr frm ne reservir barrel f the liuid hydrcarbn and vapr phase, respectively are represented by the reciprcal frmatin vlume factrs, b and b. The STB f cndensate recvered frm ne reservir barrel f the vapr phase is represented by r s b, where r s has units f STB/MCF. The reciprcal cndensate frmatin vlume factr, b, is a functin nly f the pressure since the liuid hydrcarbn phase is assumed t always be in euilibrium with the vapr phase. The values f b and r s are sinle valued functins f pressure when the vapr and liuid are in euilibrium. wever, when n liuid phase is present (e.., when abve the initial dew pint r when all f the liuid has been vaprized by the cycled dry as), the value f b is a functin f bth pressure and r s and r s, is cmputed frm the cnservatin euatins. 5-14

15 The cncentratins, fluxes, and surce terms fr the STB f cndensate and MCF f separatr as can be expressed as C = φ S b + S b r STB s CMCF = φ S b f = b u + b r u f STB s k k * STB v s v r s ( p ρ D) ( p ρ D) r = + MCF ( p ρ D) r = MCF v b = b u k k b = b + b r = b k k * where r s is the cndensate/as rati existin in the reservir if the well is a prducer and is a specified value if the well is an injectr. The terms v are vlumetric surce terms. T simplify the ntatin, STB and MCF will just be dented as and. The cnservatin euatins fr the cndensate and separatr as may be expressed as ( φ ) b r φsb + S b rs k kr b = ( p ρ D) t k kr b r s + ( p ρ D) + ( φsb) k kr b = ( p ρ D) + t When n liuid hydrcarbn phase exists in the reservir, then r s is nt a functin f pressure and must be cmputed. In the case f zer liuid hydrcarbn phase, the euatin fr STB liuids reduces t φsbs rs k kr b r s = ( p ρ D) + t Nw r s, is a dependent variable which can be cmputed frm abve euatin. 5-15

16 (h) Vlatile il. The cncepts f the black il mdel and the cndensate as mdel previusly discussed can be cmbined t represent either a vlatile il r a cndensate as system. The separatr as cntent in the reservir liuid hydrcarbn phase can be represented by b R s and the stck tank hydrcarbn liuid in the reservir vapr phase can be represented by b r s. The cncentratin f the stck tank liuid hydrcarbn and separatr as can then be represented as C = φ S b + S b r STB s C = φ S b R + S b MCF s The term partial density r partial specific mass has been used t dente the cncentratins f the separatr prducts in the reservir liuid and vapr phases. Fr example, the cncentratin f the surface prducts culd have been expressed as where ρ = b ρv = b rs ρ = b R ρ = b ( ) ( ) C = φ S ρ + S ρ STB v C = φ S ρ + S ρ MCF v s v The partial densities are determined as a functin f pressure frm experimental depletin data. In ne case the cmpsitinal chanes were apprximated by usin the averae mlecular weiht as a crrelatin parameter. (i) Cmpsitinal mdel. All f the previusly mentined hydrcarbn system culd have been represented with a cmpsitinal mdel. The cmpsitinal mdel represents the hydrcarbn system as a mixture f tw r mre hydrcarbn cmpnents r pseudcmpnents that have their cmpsitin in the vapr and liuid phases determined by an euilibrium rati, Ki = yi / xi, which may be mdeled as cnstant, functin f pressure, r a functin f pressure and cmpsitin. The number f cnservatin euatins is eual t the number f cmpnents. The cmpsitinal mdel enerally takes mre cmputin time than the black il r cndensate as mdel because f the additinal cnservatin euatins and the vapr-liuid euilibrium calculatins. Thus, the simpler mdels shuld be used when pssible. wever, in recvery prcess with injected fluids with a cmpsitin very different frm the reservir fluids, it may 5-16

17 be necessary t use a cmpsitinal mdel. Examples are the injectin f enriched as, lean as, carbn dixide, r flue as at cnditins where miscibility is nt attained. (j) Miscible displacement. The permeability, relative permeability, and viscsity determine the mbility f a phase in immiscible displacement. wever, in miscible displacement f tw fluids, bth fluids have the same phase relative permeability and have a viscsity that may be a blend with the ther cmpnent. Fr example, the mbility f tw miscible fluids in multiphase flw may be expressed as λ λ kk S rp 1 1= 1e S1+ S2 kk S rp 2 2= 2e S1+ S2 where λ 1, λ 2 mbility f fluid cmpnents 1 and 2 k abslute permeability k rp relative permeability f phase S 1,S 2 vlume fractin f pre space ccupied by fluid cmpnent 1 and 2, effective viscsity f fluid cmpnent 1 and 2 1e 2e It was assumed that the phase under cnsideratin cntained nly the tw cmpnents. A methd f mdelin the effective viscsity will be discussed with the miscible displacement prcesses. With the mbilities described as the abve euatins and the cncentratins described with the vlume fractins, S 1 and S 2, and frmatin vlume factrs, the cnservatin euatins can be expressed and slved similar t the black il mdel. 5-17

18 References 1. Jacby, R.., and Berry, V. J., Jr. (1957), A Methd fr Predictin Depletin Perfrmance f a Reservir Prducin Vlatile Crude Oil, Trans. AIME, v. 210, Standin, M. B. (1952), Vlumetric and Phase Behavir f Oil Field ydrcarbn Systems, Rienhld Publishin Crp., New Yrk. 3. Ddsn, C. R., Gdwill, P., and Mayer, E.. (1953), Applicatin f Labratry PVT Data t Reservir Enineerin Prblems, Trans. AIME, v. 198, Reudelhuber, F. O., and inds, R. F. (1957), A Cmpsitinal Material Balance Methd fr Predictins f Recvery frm Vlatile Oil Depletin Drive Reservirs, Trans. AIME, v. 210, Spivak, A. (1971), Mathematical Simulatin f Cndensate-Gas Reservirs, Ph.D. Thesis, University f Texas at Austin. 6. Kniazeff, V. J., and Naville, S. A. (1965), Tw-Phase Flw f Vlatile ydrcarbns, Sc. Petr. Enr. J., p. 37, March. 7. McCrd, D. R., and Assciates (1968), Users Reference Manual fr Pint, Tw-Dimensinal, Tw-Phase Gas, Tw-Phase Liuid Unsteady-State Reservir Simulatr with Dynamic Fluid Crrelatin Parameters. 8. Price,. S, and Dnhue, D. A. T. (1967), Isthermal Displacement Prcess with Interphase Mass Transfer, Sc. Petr. Enr. J., p 205, June. 9. Culham, W. E., Faru Ali, S. M., and Stahl, C. D. (1969), Experimental and Numerical Simulatin f Tw-Phase Flw with Interphase Mass Transfer in One and Tw-Dimensins, Sc. Petr. Enr. J., p. 323, September. 10. Rebuck, I. F., Jr., endersn, G. E., Dulas, J., Jr., and Frd, W. T. (1969), The Cmpsitin Reservir Simulatr: Case I - The Linear Mdel, Sc. Petr. Enr. J., p. 115, March. 11. Van-Quy, N., Simandux, P., and Crteville, J. (1970), A Numerical Study f Diphasic Multicmpnent Flw, SPE Preprint n. 3006, presented at the 45th Annual Fall Meetin f SPE in ustn, Octber 4-7,