Summary. Introduction. Basic equations. Simulation of gas and oil flow in hydrocarbon reservoirs. P. Cole & P. A. Collins

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1 Quarterly Jurnal f ngineering Gelgy, Lndn, 1986, Vl. 19, pp Printed in Nrthern Ireland Simulatin f gas and il flw in hydrcarbn reservirs P. Cle & P. A. Cllins xplratin Department, British Gas Crpratin, 59 Bryanstn Street, Lndn W1 Summary The nn-specialist is intrduced t the uses made f numerical simulatin in mdelling the behaviur and perfrmance f hydrcarbn reservirs. Hydrcarbn reservir mdelling can be cmpared with grundwater mdelling, but is generally mre cmplex, cmmnly dealing with multiphase fluid systems in tw r three dimensins. Bth finite difference and finite element techniques are used t slve numerically the equatins f mtin. The applicatin f tw pwerful mdels, PORS and PROGRSS, illustrates the use f simulatin during the varius stages f develpment f the Mrecambe and Rugh gas fields by British Gas. Intrductin Hydrcarbn accumulatins frm when il and gas, prduced frm buried rganic material, cllect in prus rck frm which they are unable t escape. Three prerequisites are essential t their frmatin. 1. A prus rck capable f hlding hydrcarbns. 2. A 'surce' rck cntaining the material frm which hydrcarbns are frmed, at suitable cnditins f temperature and pressure. 3. A trapping mechanism, i.e., a cmbinatin f an impermeable layer verlaying the reservir rck and a structural r stratigraphic feature in which the hydrcarbns accumulate. xamples f impermeable cap rcks are layers f anhydrite, halite r shales. Typical large Nrth Sea reservirs cver tens f square kilmetres and may cmprise reservir rcks ver 300 m in thickness. Reservirs may cntain bth gas and il, r nly ne f these fluids. Beneath the hydrcarbns, the reservir rck will generally be saturated with water. The initial pressure in the reservir will usually be determined by the hydrstatic pressure f the water at the reservir depth. Wells drilled t recver the hydrcarbns will usually be several hundred metres apart and are ften several thusand metres frm the mre remte parts f the reservir it is hped they will drain. The ability t estimate the effectiveness f a given well pattern in draining the reservir is essential t the Petrleum ngineer when planning the mst ecnmic way f develping a hydrcarbn reservir. The use f cmputer simulatin prgrams t make these esti- mates is nw an accepted part f develpment planning. The simulatin mdels aim nt nly t describe the flw f hydrcarbns and water in the reservir, but als the flw f these fluids thrugh the wells t the surface. In the case f ffshre gas fields it is cmmn practice als t incrprate in the simulatin mdel a descriptin f the gas flw thrugh cmpressrs and pipelines t the shre. In the early stages f develpment f a hydrcarbn reservir, infrmatin n the distributin f rck prperties (prsity, permeability, etc.) will ften be limited t that btained frm a small number f explratin wells cmbined with an initial gelgical cncept f the spatial distributin f rcks, including thse which frm the reservir. Cnsequently, early simulatin mdels may incrprate versimplified reservir descriptins. Hwever, as wells are drilled and prductin data (flws frm individual wells and the pressures in them) cllected, it is pssible t refine the descriptin f the reservir. The equatins gverning the flw f fluids in gas and il reservirs are intrduced and cmpared with the equatins used t mdel the flw f grundwater. Mdels used by British Gas in cnnectin with the develpment f its Mrecambe and Rugh gas fields are used t illustrate different numerical appraches t the slutin f these equatins. Basic equatins The mvement f hydrcarbn fluids thrugh prus media is gverned by a cmbinatin f Darcy's equatin and a mass cnservatin cnditin. The cmplexity f the equatins increases with the number f phases present in the system, althugh the physical principles are unchanged. Additinal relatinships are required t describe fluid and rck prperties, fr example, the change in fluid density and viscsity with pressure, and the effect f pressure n rck cmpressibility. (The latter effect can be ignred in sme hydrcarbn reservirs). This verall system f equatins is the basis fr all mdelling f hydrcarbn fluid mvements in prus media. The precise frm f the equatins depends n the number f dimensins that are thught t be required t mdel the prcess f interest adequately, as well as the number f fluid phases present. In the cases cnsidered belw the three-dimensinal frm f the equatins will be given.

2 122 P. COL & P. A. COLLINS Single phase systems One f the simplest reservir systems t describe is the mvement f gas in a reservir where n ther hydrcarbn phases are present. In many such reservirs the interface between the gas and the mveable water in the underlying aquifer* (the gas-water cntact) remains statinary thrughut the prductive lifetime f the reservir. In this case the aquifer is excluded frm the system and a single phase mdel is sufficient t predict the withdrawal f the gas and subsequent pressure decline. In practice the gasbearing rck will cntain a water phase, but this is effectively immbile; it simply reduces the strage capacity (effective prsity), and ability t transmit gas (effective permeability) f the reservir rck. The cmplexity f this system is similar t that f grundwater flw in a cnfined aquifer. Single phase equatins When frmulating the system equatins, grundwater mdellers tend t use hydraulic ptential (~) as the dependent variable, where reservir, it is custmary in this case t break dwn K int the frm K=Pg.k It where It is the fluid density and k is the effective permeability tensr (k depends n the fractin f the pre space ccupied by each phase present, as well as the rck permeability). Thus, replacing ~ by p as dependent variable leads t an alternative frm f Darcy's law: k q=--[vp+pgvz] (3) It Cnservatin f mass. This cnservatin equatin, ften called the material balance equatin, simply states that, ver any time interval, the difference between the mass f fluid entering and leaving any vlumetric element f the prus medium is equal t the increase in fluid mass in that element. If, in additin, fluid surces and/r sinks are included, the full equatin becmes S St (pn) + V. (pq) + Q = 0 (4) r -- (1) Pg Here, z is the elevatin abve sme datum, P0 is the reference pressure and p is the fluid density. It is, hwever, mre cnvenient t wrk in terms f pressure (p) when mdelling single phase gas systems. The equatin f cnservatin f mass and Darcy's law are expressed belw in terms f each f these variables. It is als custmary, in bth applicatins, t cmbine the Darcy and mass balance equatins t leave a single diffusin type equatin. The assumptins made in deriving each equatin are utlined belw. Darcy's Law. This empirical law, which is mst naturally expressed as a relatinship between vlumetric rate and ptential gradient, may be written as q=-k.vo (2) where q is the Darcy velcity, K is the hydraulic cnductivity tensr and V-(O/Ox, S/Sy, S/9z). It is usual t assume that the directins f anistrpy f K are c-linear with the crdinate system, s that nly three nn-zer cmpnents Kxx, Kyy, Kzz need be cnsidered. In grundwater mdelling, equatin (2) is the standard frm f Darcy's law. Hwever, it is clear that the hydraulic cnductivity is a functin f fluid as well as rck prperties. Since there may be hydrcarbn fluids with very different prperties within the same * Aquifer is used t describe a water-saturated permeable stratum. where n is the prsity and Q is the vlumetric withdrawal (injectin) rate. Here n in equatin (4) refers t the true rck prsity when applied t grundwater flw, but in the single phase gas reservir n is that fractin f the reservir vlume ccupied by gas, the presence f immbile water reducing the reservir strage capacity. In grund water mdels q wuld be eliminated between equatins (2) and (4) t give 9 St (pn) = V. (pk. VO) - Q (5) Fr flw in an uncnfined aquifer it is assumed that water is effectively incmpressible (s that p--p, a cnstant), but that the aquifer des defrm with pressure, s that changes in prsity can be related t changes in hydraulic ptential. The cncept f strage cefficient is intrduced t give S as= V. (K. VO)- Q" (6) s Ot P where Ss is the specific strage. On the ther hand, with mdels f hydrcarbn systems q wuld be eliminated between equatins (3)and (4) t give Ot (pn)= V. (Vp + pgvz) - Q (7) Althugh frmatin cmpressibility is imprtant in il reservirs, generally it can be ignred in gas reservirs, as it is several rders f magnitude smaller than the cmpressibility f gas. In this case the Quarterly Jurnal f ngineering Gelgy, Lndn, 1986, Vl. 19

3 diffusin equatin becmes npc-~- = V. (Vp + pgvz) - e (S) quatins expressing fluid cmpressibility (c) and density in terms f pressure are nw needed t clse the system. These cme directly frm the real gas equatin f state, making equatin (8) highly nnlinear. In an attempt t reduce this nn-linearity it is cmmn practice, when mdelling gas flw, t rewrite equatin (8) in terms f a variable knwn as the 'real gas ptential', defined as HYDROCARBON RSRVOIR MODLLING 2(Pp dp m(p) = # ~ (9) where Z is the real gas deviatin factr in the equatin f state: Me P - ZRT (l) Here M is the mlecular weight, R is the universal gas cnstant and T is the abslute temperature. Multiphase systems In mst hydrcarbn reservirs it is necessary t accunt fr the mvement f mre than ne fluid phase. Oil reservirs at a pressure belw the 'bubble pint'* generally have a 'cap' f assciated gas abve the il zne; almst all reservirs have sme gas in slutin in the il phase. In additin, at sme stage during the prductin histry f an il field, water is ften injected int the reservir as a secndary recvery mechanism. In these cases the interactins f the fluid phases and their interactins with the reservir rck must be accunted fr by additinal physics in the reservir system. Tw f the mst imprtant cncepts which nw need t be cnsidered are thse f capillary pressure and relative permeability. Bth f these quantities are defined in terms f fluid saturatins. In any system in which the pre space is ccupied by immiscible fluids, the saturatin f any ne fluid is defined as the prprtin f the ttal pre vlume ccupied by that fluid. Saturatins are therefre fractins whse sum is unity. xperimental evidence suggests that it is a gd apprximatin t cnsider bth capillary pressure and relative permeability as functins f saturatins nly. Capillary pressure 123 At any pint in the reservir where immiscible phases cexist, the lcal average pressure in each phase will differ. This is because a pressure difference exists acrss the curved interface between any tw immiscible fluids. T maintain equilibrium in a prus medium cntaining tw such fluids, the average pressure in the nn-wetting phase must exceed that in the wetting phase. Fr a tw phase system, if Pw (Pnw) is the lcal average pressure in the wetting (nn-wetting) phase, then the capillary pressure is defined as pc(sw) = Pnw - Pw 9 The capillary pressure defined in this way is a cnvenient variable fr the reservir engineer t use. It has t be appreciated, hwever, that it is being used t describe, in average terms, differences which vary rapidly n the scale f the actual pres in the frmatin. mpirically, the capillary pressure (Pc) is generally fund t be a functin f the wetting phase saturatin (Sw) nly. Hwever, the capillary pressure/ saturatin relatinship is nt unique in any given system; it depends als upn the displacement mechanism. If a nn-wetting fluid displaces a wetting fluid frm a prus medium initially saturated with the wetting fluid, the change in Pc as Sw decreases might be as illustrated in curve 1 f Fig. 1. This is a drainage v s _ (2~ Curve (I) " drainage Curve (2) 9 imbibiti0n * Many reservirs cntain il which is undersaturated with gas at initial reservir pressure. When il is prduced frm such reservirs, the pressure declines until the 'bubble pint', at which the first bubble f gas escapes frm slutin, is reached. i, 0 I Wetting phase saturatin (Sw) FIG. 1. Capillary pressure versus wetting phase saturatin. Quarterly Jurnal f ngineering Gelgy, Lndn, 1986, Vl. 19

4 124 P. COL & P. A. COLLINS prcess. The frmatin f il and gas reservirs is such a prcess. Here the nn-wetting phase is il (gas) and the reservir rck is initially saturated with water, the wetting phase. The displacement f a nn-wetting phase by a wetting phase is knwn as an imbibitin prcess. The secndary recvery f il by water injectin is such a prcess. As the wetting phase saturatin increases, the capillary pressure fllws a different curve t that fr drainage, as illustrated, fr example in curve 2 f Fig. 1. This hysteresis effect is bserved in all tw phase systems. Usually, the nature f any displacement prcess t be mdelled is knwn a priri, s that an apprpriate capillary pressure/saturatin relatin can be chsen. The gradient f the Pc versus Sw curve becmes very large as Sw decreases, s that eventually a very large change in capillary pressure prduces nly an infinitesimal change in wetting phase saturatin. The limiting saturatin is knwn as the irreducible saturatin r, in the special case f water, the cnnate water saturatin. Capillary pressure is therefre nly effective in regins f the reservir in which there is mre than ne phase present at ptentially mveable saturatins. One such regin is the transitin zne, which ccurs between the hydrcarbn-bearing frmatin at cnnate water saturatin and the underlying water-saturated rck in a reservir in hydrstatic equilibrium. Relative permeability Darcy's law in its riginal frm describes the effect f rck permeability n single phase flw in prus media. In multiphase systems, Darcy's law is applied t each phase m turn, the permeability f the system t any phase (effective permeability) being a functin f the phase saturatins, as well as the rck prperties (abslute permeability). It is custmary t define relative permeability as the rati f the effective and abslute permeability. mpirically, it is fund that, t a gd apprximatin, the relative permeability is a functin f the phase saturatins nly. Figure 2 shws typical relative permeability curves fr a (water-wet) il/water system. The value f Sw at which water starts t flw is the critical saturatin, Swc. The saturatin at which il ceases t flw is called the residual saturatin, St. The wetting state f the rck has a majr effect n the shape f the relative permeability curves and in particular n the psitin f the end pint values. The wetting phase fills the crevices f the pre spaces and therefre its presence des nt greatly reduce the permeability t the nn-wetting phase. Hwever, when the nn-wetting phase is at irreducible saturatin, it frms drplets which ccupy the pre spaces and may bstruct the pre thrats, increasing the resistance t the flw f the wetting phase. Hence, S,,c I-S, FIG. 2. Tw phase relative permeabilities fr a water-wet il/water system. in water-wet, il/water systems the relative permeability t il at cnnate water saturatin is typically 0.6 t 0.9, whereas the relative permeability t water at irreducible il saturatin is nly 0.05 t 0.3. (Hagrt 1984). Multiphase equatins The derivatin f the partial differential equatins gverning the flw f each phase in multiphase hydrcarbn reservirs is similar t that fr single phase flw. Hwever, capillary pressure and relative permeability nw have t be taken int accunt. In additin, the equatin gverning gas mvements must reflect the fact that gas can appear bth as a free phase and in slutin in the il. Gas is generally assumed nt t have significant slubility in water at typical reservir cnditins. Thermdynamic prperties f the fluids at reservir cnditins are needed in the frm f density and viscsity versus pressure relatins. Similarly, the frmatin cmpressibility must be accunted fr by a prsity/pressure relatin. In rder t simplify material balance calculatins it is custmary in reservir simulatin t frmulate all flw rates at standard cnditins f temperature and pressure. Cnversin factrs relating reservir vlumes t surface vlumes are therefre required in the frm f frmatin vlume factrs (B), which are Quarterly Jurnal f ngineering Gelgy, Lndn, 1986, Vl. 19

5 HYDROCARBON RSRVOIR MODLLING 125 functins f reservir pressure. Fr il, water and gas these are, respectively, [V + vd~]~c B= [V]sc [Vw]~c Bw= [Vw]~c i , :k Bg = [Vg] Rc 02 [Vg]sC 0-02 where RC implies reservir cnditins, SC implies standard cnditins, and V, Vg, Vw and Vdg are vlumes f il, free gas, water and gas disslved in il, respectively. The quantity f gas disslved in the il phase at a given pressure (the slutin gas-il rati) is defined as Gg g L.. 0 ( BQ ~, (.9 This describes the mass transfer between the il and gas phases. All f the abve infrmatin must be inferred frm reservir fluid and rck samples by labratry bservatin and experimentatin. The pressure dependence f sme f these quantities is shwn in Figs 3 and 4. Assuming the abve infrmatin, the dependent variables in the equatins f mtin are the phase pressures and saturatins. Hwever, the pressures in the il and water (gas) phases are linked thrugh the il-water (gas) capillary pressure relatins. Hence a I 3 '6 I-2 g "6,,,i ~ / _.m..q r~ B I I [ I I 1"14"0"0 6 8 I0 12 Pressure (MP) FIG. 3. Oil frmatin vlume factr, il viscsity and slutin gas-il rati as functins f pressure, 60 a: i 40.~_ g._ 20 = -6 O') 3 14 ; ;,6,i 14 Pressure (MPa) FIG. 4. Gas frmatin vlume factr and gas viscsity as functins f pressure. system f fur equatins fr the il-phase pressure and the saturatins in each phase is all that is required. The relatin S+Sw+Sg= 1 (12) tgether with the equatin f mtin fr each fluid phase make up the required system. Cmbining Darcy's Law and the material balance equatin fr each phase leads t the fllwing equatins (see, fr example, Odeh 1982): 0 (ns~ (13) V'(k'V*)-q=~\B / V-(k,- V~g) + V. (nsk 9 V~) - qg =O---[n(Sg+~~ (14) at [_ \Bg 8 (nsw) (15) V.(Kw'V~w)-qw=~ -B-~- 9 Here the phase ptentials are defined such that and 7~ = Vp + gpvz V~g = Vp + gpgvz + Vpcg 70w = Vp + gpwvz - VPcw where p is the il phase pressure, Pcw is the capillary pressure between il and water, Pcg is the capillary pressure between gas and il, q is the prductin rate, p is the density, and the subscripts, g, and w refer t Quarterly Jurnal f ngineering Gelgy, Lndn, 1986, Vl. 19

6 126 P. COL & P. A. COLLINS il, gas and water. The phase transmissibilities ~. are defined by where k is the abslute permeability tensr and k~i the relative permeability t phase i. Many f the cefficients in equatins (13) t (15) are functins f p r S. In additin, the gas equatin cntains 'il' cefficients ~, and S. Hence, the system f multiphase equatins is cupled and nn-linear. The rle f the reservir simulatin mdel is t slve these equatins accurately and efficiently. The equatins are discretised using either the finite difference r finite element methd and are generally slved using ne f tw principal methds--- simultaneus slutin (fully implicit methd) r sequential slutin (implicit pressure, explicit saturatin methd). In the simultaneus prcedure all flw (and well) equatins are slved simultaneusly fr the unknwns at each grid cell. In the sequential prcedure, equatins (13) t (15) are cmbined t give the il phase pressure equatin kri ap (16) Z BiV" (/~iv(i)i) -- q, =nct -~ where ct is ttal cmpressibility f the system and qt is ttal prductin rate. This equatin is first slved implicitly fr the pressure at each grid pint. The phase saturatins are then btained by the explicit slutin f the discretised frms f equatins (13) t (15). The slutin f the matrix equatins is by either a direct methd (nly practical in tw-dimensins) r iterative methds (tw and three dimensins). The equatins abve assume that all hydrcarbn species present in a reservir can be represented by a 'black il' in which the gas and il are assumed always t have the same hydrcarbn cmpsitins, but with the 'gas' partly disslved in the il and als existing as a free phase. This is nt a valid assumptin fr all reservir fluids. In particular, in vlatile il and gas cndensate systems it is necessary t mdel precisely the equilibrium between vapur and liquid phases. This is because the hydrcarbn cmpsitins f the gas and liquid phases can vary substantially during the reservir's prductin lifetime. The fluids may als exhibit a cunter-intuitive behaviur knwn as isthermal retrgrade cndensatin, whereby a reservir vapur phase partly cndenses t liquid as the reservir pressure is reduced, fllwed by revaprizatin f the liquid as the pressure falls even further. In such systems it is necessary t mdel the mvements f several hydrcarbn species individually, greatly increasing the cmplexity f the system equatins. Quarterly Jurnal f ngineering Gelgy, Lndn, 1986, Vl. 19 Chice f mdel Simulatin mdels The reservir engineer may chse t apprximate the three-dimensinal system represented by a hydrcarbn reservir using a simulatin mdel having anything frm zer t three dimensins, depending upn the prevailing circumstances. Such factrs as reservir gemetry and hetergeneity, rck prperties, the number f fluid phases, their distributin and mbility, the availability and quality f data, and the availability f cmputing facilities must all be cnsidered carefully. Zer-dimensinal (tank) mdels can, fr example, give satisfactry results when mdelling the pressure decline in dry gas reservirs having very high permeability (and in which the underlying aquifer is nt active). In this case gas can mve rapidly under very small pressure gradients. Darcy's law therefre becmes redundant, and it is assumed that the average reservir pressure drps at a rate that is related t the field prductin rate thrugh the material balance equatin. One-dimensinal simulatrs are ccasinally used, fr example, t estimate the displacement efficiency f injected water as an il recvery mechanism. Hwever, increases in cmputer speed and capacity in recent years have favured the use f tw- and three-dimensinal mdels. Depth averaged, twdimensinal mdels are ften justified because mst reservirs are rders f magnitude bigger in the hrizntal directin than they are in the vertical. It can ften be assumed that the reservir remains in vertical equilibrium with capillarity frces balancing gravity effects in any transitin zne. If, in additin, rck prperties d nt vary greatly thrugh the vertical, the tw-dimensinal areal mdel will give satisfactry results. In such mdels pseud-reservir prperties can be used t accunt fr vertical variatins, and these prperties are ften derived frm a preliminary tw-dimensinal crss-sectinal mdel. If nne f the abve simplifying effects are in evidence a fully three-dimensinal mdel may be required. The vertical dimensin will always be necessary when mdelling the effects f water influx frm an underlying aquifer. Slutin techniques and types f simulatin prgrams The finite difference methd is used predminantly thrughut the petrleum engineering industry t slve the equatins f mtin f reservir fluids. Here the reservir vlume is divided int a netwrk f cells, simple material balance fr each phase in each cell

7 HYDROCARBON RSRVOIR MODLLING 127 being bserved. Darcy's law then describes fluid mvements frm ne cell t any f its immediate neighburs. The number f cells with which any given cell interacts relates directly t the band width f the matrix which needs t be inverted t slve the system f simultaneus equatins representing the cmplete system at each time step. This increases the cmputer time required t slve three-dimensinal prblems as cmpared t ne- r tw-dimensinal prblems, in additin t the bvius increase in the number f cells required. British Gas uses a multipurpse finite difference simulatr, PORS (Pnting et al. 1983), which can be used t slve prblems in ne, tw r three dimensins and/r fluid phases, and which uses a fully implicit slutin algrithm. In additin, a twdimensinal, single phase finite element simulatr, PROGRSS (Gldwater et al. 1978), is used fr mdelling dry gas reservirs. PORS is an extremely sphisticated prgram written by UKAA Harwell n behalf f the Department f nergy, British Gas Crpratin and Britil PLC. It cmbines pwerful, efficient numerical techniques and rigrus cnvergence criteria with a wide range f 'engineering ptins' in the frm f a well and prductin cntrls. A variety f methds, bth numerical and algebraic, are available fr mdelling the interactin f the aquifer with the re.servir. The prgram als allws grid refinement (r carsening) by allwing nn-matching grids t be jined tgether in a way in which the transmissibility acrss the interface is crrectly described. It is als pssible t mdel gelgical faults s that the displacement f strata is prperly described. Althugh less sphisticated in its numerical techniques, PROGRSS, allws many f the advantages f the finite element methd t be explited when mdelling hydrcarbn reservirs. It is easy t fllw irregular bundaries, faulting within the reservir (these may be sealed r partially transmitting), and rapidly changing reservir thickness, by grid refinement in the apprpriate regins. Cnversely, it may be pssible t use a carser grid in regins away frm wells where little infrmatin in the frm f reservir data is available. PROGRSS has the facility t lcate many wells within a single element at distinct lcatins, rather than lumped at the cell centre as in the finite difference frmulatin. If vertical variatins within a reservir are t great t be ignred, a multilayer versin f PROGRSS is available in which vertical gas mvements can be mdelled. An imprtant feature f PROGRSS is the availability f a pipeline and cmpressr mdel, integrated with the reservir mdel. This enables British Gas t mdel, nt nly the mvement f gas in the reservir, but als its transmissin t shre, where it is delivered at a cntractually agreed minimum pressure. Sme practical aspects f mdelling gas fields In general terms the reservir simulatr is used t mdel gas fields in tw distinct kinds f studies. The first kind ccurs befre the field ges int prductin, at which stage the reservir engineer has available nly the initial gelgical mdel f the reservir, built up frm infrmatin gained frm early explratin and appraisal wells. In such a study the predictive capability f the simulatr is clearly limited. Valuable infrmatin can, hwever, be btained frm cmparative runs. The secnd kind f study ccurs well int the prductive lifetime f a field. At this stage the perfrmance f the simulatr can be matched against recrded well prductin rates and pressures. In this way reservir parameters can be amended t imprve the agreement between mdel and field perfrmance. This calibratin prcess clearly enhances the predictive capability f the mdel. Befre any gas field ges int prductin many pssible develpment strategies must be evaluated. The number and lcatin f wells used t deplete the field must be decided upn. Here, bth the permeability f the structure and the lcatin f gelgical faults have a large affect. The rate f return n capital expenditure is als very imprtant; clearly (up t a pint) the mre wells available, the higher the rate f prductin the field can meet and the greater the revenue generated. Hwever, rapid prductin will entail greater capital expenditure; a balance has t be fund taking int accunt a wide range f cmmercial and technical factrs. Once all planned wells are available, mst gas fields are initially prduced at a 'plateau' rate. Typically, fr larger fields, a sufficiently high number f wells will be drilled t allw the plateau rate t be maintained until arund 60% f the estimated recverable reserves have been prduced. Usually, this can nly be achieved by the installatin f cmpressin n ne f the platfrms r nshre. This allws field prductin t cntinue when the well-head pressures are cnsiderably less than the cntractually agreed delivery pressure nshre (usually abut 6.9MPa). Once a gas field cmes ff plateau, prductin rates will decline until they are s lw that the field is n lnger ecnmically viable. At this stage the field will be abandned. British Gas' Mrecambe field may be taken as an example f the use made f simulatrs in planning the develpment f a gas field. This field, having reserves f abut 150 billin standard cubic metres f gas, is situated apprximately 30 km ffshre, in the Irish Sea (Fig. 5). It is currently under develpment and will be used initially t meet a need fr additinal supplies t satisfy peak demand fr gas during the winter. Bth PORS and PROGRSS have been used t mdel the Mrecambe field. A detailed three- Quarterly Jurnal f ngineering Gelgy, Lndn, 1986, Vl. 19

8 128 e. COL & P. A. COLLINS FIG. 5. Suth Mrecambe develpment scheme. FIG. 6. Suth Mrecambe: vertical crss-sectin thrugh PORS mdel grid. Quarterly Jurnal f ngineering Gelgy, Lndn, 1986, Vl. 19

9 HYDROCARBON RSRVOIR MODLLING 129 dimensinal mdel has been set up using PORS and, as mre infrmatin is btained frm the prductin wells nw being drilled, the mdel will be cnstantly revised t becme eventually ne f the main 'tls' used in the management f the reservir. A crss-sectin thrugh the PORS mdel fr the reservir is shwn in Fig. 6. Fr the initial planning f the numbers f platfrms and wells required t meet the desired prductin rates, a mdel has been cnstructed using PROG- RSS. The finite element grid used is shwn in Fig. 7. In rder t represent crrectly the gelgical characteristics f the reservir a three-layer mdel has been used. Mst wells will penetrate mre than ne f these layers, but it is expected that prductin will cme principally frm the highest permeability layer, with gas gradually migrating frm prer quality rck int the better material and frm there int the wells. One prblem faced in mdelling reservirs is that f determining the values f parameters, such as.7. 9 Pssible well lcatins Faults FIG. 7. Mrecambe: PROGRSS finite element triangulatin. Quarterly Jurnal f ngineering Gelgy, Lndn, 1986, Vl. 19

10 130 P. C O L & P. A. C O L L I N S permeability and prsity t be used in mdels f the full field. In these mdels a single grid blck r element may represent an area f the reservir up t a kilmetre acrss r mre. The actual reservir material will cmmnly shw wide variatins in characteristics within such a blck r element. Therefre, in rder t estimate the average parameters fr use in a mdel, subsidiary mdels are ften set up t represent in mre detail the smaller scale variatins in the reservir. These details ften have t be inferred frm well data and rely n the gelgists' cnceptual understanding f the reservir. Figure 8 illustrates ne f the PORS mdels, representing a sectin thrugh a high permeability channel embedded in a system f prer quality sands, which was used in estimating average effective permeabilities in Mrecambe. The Rugh field, lcated in the suthern Nrth Sea, like Mrecambe, will be used t meet seasnal demands. In this partly depleted reservir, a strage prgramme is being implemented whereby gas will be injected during the summer mnths t replenish that prduced during winter (Hllis, 1984). Previusly, Rugh had been prduced cnventinally using six wells. A further 18 wells are being drilled t achieve the greater peak prductin nw required. Over a perid f several years mre gas will be reinjected than is prduced s as t build up the pressure f the reservir t near its riginal level. This will increase the strage capacity in additin t the wells' ability t deliver. The riginal pressure will nt be exceeded s as t ensure that gas will nt expand int a part f the aquifer where it wuld be n lnger 'trapped'. Because Rugh has been in prductin frm 1975 it has been pssible t refine the mdel fr the reservir by matching measured well pressures t thse predicted by the mdel. Figure 9 illustrates the match btained fr ne f the wells. The cyclic injectin and prductin f gas will result mmmmmmmmmmiiiii iiiiiiiiiii mm mmmm mmmmm ii!ii!s mii!!!i!ii II Immmmmmm IIIIIii~iiiii)ii)i@ii@i!ii!i@ii!ilIIII@ I iiii!il))i)))))))))@)/))))) mm ~i!~i~@!~i~m i~iii~m i~i!il m R m i R Immmm@ssmswR lm mmmmmmmmmbm= minim iiiiiiimmili iiiiiiiiiii~ii~i!~iim!~iim~i)i 1, ~ ~ 0-1 md ~ 1-10 md OmD : mmmmmmm i im imiii) : I! 1 m... li 5000' FIG. 8. Mrecambe: PORS crss-sectin f channel sand. Quarterly Jurnal f t~gineering Gelgy, Lndn, 1986, Vl. 19 ~ md ~ > 100 md

11 32- x9 9 9 Mdel value X Recrded value x x ~,~ex x 9 x Ol 09 --OxX 9 x x QX x 00 x 9 ~.Xx x %x 18 i XxX X I I i I i Time FIG. 9. Rugh: typical well histry match. 50- ~.g_ ~ '6 9 g 40 5O 2O I0 " -I0 - Number f wells -~--~- 6 Deliverability (injectivity) " I 24-~ 25r- 20 I- i 15 L FIG. 10. Rugh: prductin and injectin prgramme.

12 132 P. COL & P. A. COLLINS in rises and falls in the average reservir pressure with cnsequent variatins in the maximum rates f gas that can be prduced r injected using the available facilities. The simulatin mdels are used t predict these maximum rates in rder t calculate the length f time fr which a given prductin rate culd be achieved, r the vlume f gas which needs t be injected t meet a given peak requirement. Figure 10 illustrates a typical prductin/injectin cycle, alng with the resulting variatins f reservir pressure. It is clear frm the abve applicatins that reservir simulatrs are able t play a valuable rle in predicting the behaviur and planning the develpment f hydrcarbn reservirs. It is, hwever, imprtant fr the engineer always t have in mind the fact that the mdel can never be better than the input data, and that the nature f reservir hetergeneities can never be knwn with cmplete precisin. References GOLDWATR, M. H., COLLINS, P. A. & TAYLOR, B. A The use f a finite element prgram t mdel faulted gas reservirs f the Suthern Nrth Sea Basin. Prc. ur. Offshre Petrl Cnf. Paper UR 87, HAGOORT, J Measurement f relative permeability fr cmputer mdelling/reservir simulatin. Oil Gas J. 82(8), HOLLIS, A. P Sme petrleum engineering cnsideratins in the change-ver f the Rugh Gas Field t strage mde. J. Petrl. Technl. 36, 79%804. ODH, A. S An verview f mathematical mdelling f the behaviur f hydrcarbn reservirs. S.LA.M. Review, 24, PONTING, D. K., FOSTR, B. A., NACCACH, P. F., NICHOLAS, M. O., POLLARD, R. K., RA, J., BANKS, D. & WALSH, S. K An efficient fully implicit simulatr. Sc. Petrl. ng. J. 23, Quarterly Jurnal f ngineering Gelgy, Lndn, 1986, Vl. 19