Modelling of Hot Water Flooding

Transcription

1 Unversty of Readng Modellng of Hot Water Floodng as an Enhanced Ol Recovery Method by Zenab Zargar August 013 Department of Mathematcs Submtted to the Department of Mathematcs, Unversty of Readng, n Partal fulflment of the requrements for the Degree of Master of Scence n Mathematcs of Scentfc and Industral Computaton

2 Acknowledgment I would lke to thank my supervsor Professor Mke Banes for hs help and support. I also would lke to acknowledge Alson Morton and Paul Chlds for ther help and advce. Fnally specal thanks to my husband and famly for ther encouragement and support.

3 Declaraton I confrm that ths s my own work and the use of all materals from other sources has been properly and fully acknowledged. Sgned...

4 Abstract Ths dssertaton descrbes and compares two numercal technques that smulate one dmensonal hot water njecton. In total four equatons are ntroduced n order to model hot water njecton; the Buckley-Leverett equaton, two mass balance equatons for water and ol phases and an energy balance equaton, all of whch are hghly non-lnear. The objectve of the mathematcal model s to solve these equatons under the approprate ntal and boundary condtons. Ths soluton provdes space and tme dstrbutons of water and ol pressures, saturatons and temperature. One of the major dffcultes wth numercal modellng of ths process s the dependence of the flud propertes on the pressure and temperature. In the frst technque, the Buckley-Leverett equaton s used to calculate ol and water saturaton dstrbutons whch s a nonlnear hyperbolc equaton. The second order Lax-Wendroff scheme s used to solve ths equaton. The results of the saturatons are used n the mass balance equaton, whch s a nonlnear equaton snce ts coeffcents depend on temperature and pressure. A fully mplct central scheme s used n order to dscretze the equaton and then the Newton-Raphson method s used to solve ths nonlnear system n order to fnd the pressure dstrbuton. Fnally, the pressure results are used n the nonlnear energy equaton to obtan the temperature profle. In the second model, the mplct pressure/explct saturaton IM- PES technque s used for the mass balance equatons of water and ol phases n order to fnd the pressure and saturaton dstrbutons, then the results are used n the energy equaton to get temperature profles. Snce all these equatons are nonlnear and depend on each other, the energy equaton needs to be coupled wth materal balance equatons. Results show that saturaton front n the frst model lag behnd that of the second model whch can be a result of ncompressblty assumpton used n t. The second model has to be appled wth some care as t can be easly become unstable, but f t s used n ts stablty doman the results are more relable.

5 Contents 1 Introducton 1 Characterstcs of the Model 4.1 Assumptons Rock and Flud Propertes Descrpton Darcy s Law Porosty Saturaton Permeabltes Hydrocarbon Vscosty Phase Mass Densty Introducng Model Equatons Buckley-Leverett Mass Balance Equaton Energy Balance Equaton Intal and Boundary Condtons Heat losses Frst Model Buckley-Leverett Dscretzaton Effect of boundary condtons The CFL condton Dscretzaton of the Mass balance equaton Newton s Method for Nonlnear Systems of Equatons Jacoban Matrx Defnton for Mass Balance Equaton Well Couplng Energy Balance Equaton Dscretzaton Dscretzaton of Rght Hand Sde of the Energy Equaton Dscretzaton of Left Hand Sde of Energy Equaton 3.57 for Mddle Cells =,...,N x Calculatons for the Left Boundary Cell = Calculatons for the Rght Boundary Cell =N x Summary of The Frst Model Second Model IMPES Technque

6 4. Jacoban Calculatons for the Pressure Equaton Saturaton Calculatons Summary of the Second Model Results Frst Model Results Second Model Results Comparng Two Models Senstvty Analyss Concluson Future work

7 1 Introducton A hydrocarbon reservor s an underground volume comprsed of porous rock contanng a mxture of water and hydrocarbon fluds n the form of ol and gas, occupyng the vod space of the pores n the rock. Ols can be dvded nto two categores, lght ols and heavy ols. Lght ols have a low vscosty whle heavy ols have a hgh vscosty. The vscosty of a flud s a measure of how easly that flud wll flow, for nstance, water has a very low vscosty whle honey has a hgh vscosty. When ol recovery s hgh due to hgh natural reservor pressure. The rate of natural ol producton wll dmnsh wth tme, but there are some ol recovery methods to mprove the producton rate. Ol recovery processes nvolve the njecton of flud or a combnaton of flud and chemcals nto the ol reservor va njecton wells to force as much ol as possble towards and, hence, out of the producton wells. Lght ols are extracted under prmary and secondary recovery methods whch nvolve allowng the flud to flow out under the natural pressure of ts surroundng. These methods cannot be appled to the extracton of heavy ols, whose vscosty s far too hgh for such methods to be effectve; ther vscosty needs to be reduced. Ths s acheved by varous thermal stmulaton technques lke hot water floodng, steam njecton, n-stu combuston and so far whch rase the temperature of the ol, effectvely reducng ts vscosty. The approach whch s under consderaton here s hot water njecton modelng. It s necessary to model and smulate ths process n order to provde nformaton about producton and the future of the reservor to get the best recovery. All thermal recovery processes tend to rase the temperature of the crude n a reservor to reduce the reservor flow resstance by reducng the vscosty of the crude [1]. It s desrable to heat the reservor effcently, but nevtably some of the heat n the reservor s lost through produced fluds, and some s lost to the adjacent overburden and underburden formatons. The heat loss to the adjacent formatons s controlled by conducton heat transfer whch t can be readly estmated. In hot water floodng, as can be seen n fgure 1.1 many reservor equvalent volumes of hot water are njected nto a number of wells n order to reduce the vscosty and subsequently dsplace the ol n place more easly towards ol producton wells. Hot water njecton may be preferred n shallow reservors contanng ols n the vscosty range of cp [4]. 1

8 Fgure 1.1: Schematc dagram of hot water njecton process The mathematcal model representng the physcal process of hot water njecton requres rock and flud propertes n order to descrbe the flud flow and heat transfer wth a set of partal dfferental equatons and algebrac equatons, whch are derved from physcal prncpals. Ths set of equatons s derved from four man prncples: Conservaton of mass of phases water and ol; Darcy s Law for volumetrc flow rates whch descrbes how the flud phases flow through the reservor; volume balance equaton, a condton whch states that the flud flls the rock pore volume; conservaton of energy of phases. Snce the resultng equatons are too complex for more realstc models to be solved usng analytc technques, here s focused on numercal technques. In ths dssertaton, two dfferent models are appled and analyzed for the hot water njecton process. Chapter contans the problem defnton and characterstcs of the model. Introducng some necessary concepts about rock and flud propertes, and requred equatons. Intal and boundary condtons and heat loss n our model are also ncluded n ths part. The frst model s ntroduced n chapter 3, where n order to fnd the saturaton dstrbuton the Buckley-Leverett equaton s used. The nonlnear mass balance equaton s solved by a fully mplct central technque by usng the results of ol and water saturatons from the Buckley- Levertt equaton. Subsequently, the saturaton and pressure results are appled to a nonlnear energy equaton dscretzed by a fully mplct method. Fnally, the mass balance equaton

9 pressure equaton and energy balance equaton temperature equaton are coupled to fnd the best result for pressure and temperature dstrbuton, snce these equatons are hghly nonlnear. In chapter 4 the second model s presented. In ths model, mplct pressure explct saturaton IMPES technque s appled to our hot water model. Durng one tme step, the results of IMPES are used n the temperature equaton whch s solved fully mplctly, and fnally there s a couplng between IMPES technque and fully mplct temperature equaton n order to fnd the fnal pressure, saturaton and temperature dstrbuton results. In both approaches, bottom hole pressures at the boundares for the two model are also calculated usng a well couplng method. Because of the complexty of the models we have tred to gve easer understandng of the models by summarzng the models n flowchart dagrams at the end of each chapter. Chapter 5 shows and compares the results of the two models and some senstvty analyss are presented as well. Fnally chapter 6 outlnes the conclusons whch are drawn from the results. 3

10 Characterstcs of the Model In ths project, we have tred to model the hot water floodng process n a reservor whch s ntally saturated wth ol and water. The reservor s consdered to be one-dmensonal between an njecton and producton wells. A schematc dagram of the model s gven n fgure.1. Hot water s njected wth a constant rate and temperature nto the porous meda whch s flled wth cold and heavy ol. In such a system flud flow, heat transfer and heat losses are modeled n order to gve a better understandng of the process and ts effect on ol recovery. Fgure.1: Schematc dagram of the problem.1 Assumptons The followng assumptons are made to model the process; 1. In all reservor processes, every pont wthn the reservor s n thermodynamcs equlbrum.. The njected flud reaches thermal equlbrum nstantaneously wth the reservor fluds and sand, meanng that all phases and rock n the same locaton have the same temperature. 4

11 3. The model smulates one-dmensonal flud flow and heat convecton but twodmensonal heat conducton throughout the underburden reservor overburden system. 4. There s a two-phase water and ol system whch s mmscble. 5. There n no capllary pressure P O = P w = P. 6. Gravty effects are neglected.. Rock and Flud Propertes Descrpton The data of rock and flud propertes are requred to understand the concept of the model. Among these, Darcy s Law, porosty, saturaton, permeabltes and phase vscostes and denstes are ntroduced brefly below...1 Darcy s Law Darcy s Law descrbes the flow of a flud through a porous medum. It determnes how fast the phases flow through the reservor and gves the phase veloctes [1]. For one dmensonal flow, the Darcy s phase veloctes can be wrtten Pα V α = C α K abs x αg d x α = Ol, W ater.1 where C α = K rα µ α. denotes phase mobltes whch are phase relatve permeabltes dvded by phase vscostes denotes phase mass denstes; d x represents the depth gradent; and K abs s the absolute permeablty of the reservor; P α s the pressure of each phase. The flud flow s therefore due to a pressure gradent and a gravtatonal potental, g. In ths project, by assumptons 5 and 6 Darcy law s smplfed to P V α = C α K abs x α = Ol, W ater.3 A fuller descrpton of some of the terms n Darcy law s now gven n more detal... Porosty Ol s contaned n rocks whch are a type of porous meda. Porosty s the rato of vod space over the bulk volume of the rock [1], 5

12 ϕ = P ore V olume V p Bulk V olume V b.4..3 Saturaton The pore volume space s not always flled wth a sngle flud. Saturaton of each flud phase s defned as the rato of ts volume over the total pore volume occuped by all phases [1], S = P hase V olume V P ore V olume V p.5 By defnton, the saturatons are all non-negatve, and sum to one...4 Permeabltes One of the man propertes of porous rock s ts capablty to allow flud flow through ts connected pores whch s known as permeablty. There are two defntons of permeablty n the ol ndustry; absolute and relatve permeabltes. Under the condton of sngle phase flow, ths capablty s named absolute permeablty. But when the porous meda s flled by more than one phase, due to varous ways the phases can occupy the pore volume, the phases adversely affect the flow of each other n a complcated manner [1]. Ths effect s descrbed usng phase relatve permeabltes, K ro and K rw. The dependence of the relatve permeabltes on the rock and flud propertes s very complcated []; the K ro and K rw consdered here are non-negatve functons of the saturaton, S. In ths case, t s necessary that relatve permeabltes must tend to zero as ts saturatons approaches zero. There are dfferent methods used to fnd relatve permeabltes. In ths project the Corey-type, whch s a power law n the water saturaton, S w, s chosen [3]. K row S w = K max ro 1 S wn no n o = 3 K rw S w = Krw max.swn nw n w = 3.6 where S wn S w = S w S w 1 S w S orw.7 K row S w = K max ro, K row 1 S or = 0 K rw S w = 0, K rw 1 S or = K max rw.8 6

13 S w : Irreducble water saturaton S wc : Connate water saturaton S orw : Resdual water saturaton water-ol system S w : Water saturaton S wn : Normalzed water saturaton Fgure. shows the results of water and ol relatve permeabltes versus water saturaton n the system by applyng the Corey correlaton.in descrbng two-phase flow mathematcally, t s always the relatve permeablty rato, system that enters the equatons. K ro K rw, versus water saturatons for ol and water Fgure.: Ol and water relatve permeabltes usng Corey correlaton..5 Hydrocarbon Vscosty Phase vscosty represents the resstance of a phase to flow under the nfluence of a pressure gradent. The most obvous effect of thermal recovery on a reservor fluds s the reducton of ol vscosty. In fgure.3 two ponts are evdent. Frst, the rate of vscosty mprovement s greatest as the ntal temperature ncreases. Lttle vscosty beneft s ganed after reachng a certan temperature. Second, greater vscosty reductons are experenced n the more vscous low API gravty crudes API s a degree of measurement for ol densty than n hgher API gravty crudes. Heatng from 100 o F to 00 o F reduces the vscosty, 98% for 10 o API crudes but only 73% for 30 o API ols. These observatons show that the greatest vscosty reducton occurs wth the more vscous ols at the ntal temperature ncreases [4]. 7

14 Fgure.3: Effect of temperature on vscosty Vscosty s a functon of temperature and pressure, but water and ol vscostes are stronger functons of temperature n a thermal process rather than pressure. Snce a thermal ol recovery method s modeled n ths project, the effect of pressure s neglected [5]. µ o = T µ w = T T Phase Mass Densty Phase densty s defned as mass per unt volume for each phase. Water and ol denstes n ths context are consdered to be a functon of temperature and pressure. In the absence of expermental data, emprcal relatons are used to express denstes of ol and water as functons of both temperature and pressure [5]: w = 63 exp T exp P o = 59 exp T exp P Fgure.4 shows the effect of temperature and pressure ncrease on the denstes of both phases. 8

15 Fgure.4: Effect of temperature and pressure on ol and water denstes.3 Introducng Model Equatons In the two hot water models presented n ths dssertaton, four equatons are requred; the Buckley-Leverett equaton, mass balance equatons for water and ol, and an energy Balance equaton..3.1 Buckley-Leverett The Buckley-Leverett BLequaton s used n ol recovery n order to fnd the saturaton dstrbuton n 1D reservor. In the BL mechansm ol s dsplaced by water from a rock n a smlar as flud s dsplaced from a cylnder by a leaky pston. In order to have better understandng of the Buckley-Leverett equaton, t s frst necessary to ntroduce the fractonal flow equaton Dervaton of Fractonal Flow for the Model When ol s dsplaced by water n the system, from Darcy s equaton we have K rw P q w = 1.17 K abs A x µ w x.13 9

16 K ro P q o = 1.17 K abs A x µ o x By addng the two equatons Krw q w + q o = 1.17 K abs A x + K ro P µ w µ o x Substtutng for q = q w + q o.16 and f w = q w q.17 and solvng for the fracton of water flowng, we obtan snce f w = Kro µ o. µw K rw K r S w f w S w.19 Now, the Buckley-Leveret equaton s derved for a 1D sample based on mass conservaton and some assumptons [6], namely flow s lnear and steady state, the flud s ncompressble, capllary pressure P c s just a functon of the saturaton and pressure gradent for two phases s equal dpc ds = 0, where P c = P o P w. By applyng mass balance of water around a control volume see fgure.5 of length x we get the followng system for a tme perod of t : Fgure.5: Mass Balance Element for Fractonal Flow Equaton The materal balance may be wrtten: ] [ [ w q w x w q w x+ x t = A xϕ w S w t+ t w S w t].0 whch, when x 0 and t 0, reduces to the contnuty equaton: x wq w = Aϕ t ws w.1 10

17 by assumng an ncompressble flud w = constant and we have that q w = f w q, Therefore f w x = Aϕ q S w t. Snce f w S w, equaton. may be rewrtten as [6] f w S w S w where fractonal water flow s defned as f s S cv = 1 1+ Kro Kw. µw µo S n = 1 = 1+ Kmax ro Krw max x = Aϕ q. 1 Sn Sn 3. µw µo S w t Sw Swc 1 S wc S or S wc < S w < 1 S or.3.4 f w S w = 3 µw µ o K max ro Krw max 1 S n S or S wc.sn Kro K rw. µw µ o. 1 Sn S n f w = f Kro max, Krw max, µ o, µ w, S wc, S or, S w Fgure.6 shows the fractonal water flow functon and ts dervatves as a functon of water saturaton Fgure.6: Fractonal water functon and ts dervatve versus water saturaton Equaton. s known as the Buckley-Leverett equaton whch s a frst order hyperbolc equaton. The equaton can be solved analytcally by the method of characterstcs and graphcally [7]. In ths project a second order numercal scheme, the Lax-Wendroff scheme, s used to solve t. The method s explaned n more detal n the followng chapters. 11

18 .3. Mass Balance Equaton The man flow equaton n reservor engneerng can smply be derved by applyng materal balance to a control volume, as shown n fgure.7. Mass accumulaton nsde a control volume s the dfference between nput and generated mass and output and consumed mass as below: m ṁ o ṁ cons ṁ gen = M t, ṁ = Mass F lux =. q.6 whereq s flow rate, s densty, M s mass and t s tme. Tme Based on what we have n equaton.6 for 1D flow, the nput mass rate for x drecton shown n fgure.7 wll be: ṁ x =. u x.da x.7 da x, u x and are the normal cross sectonal area n x drecton, velocty and densty respectvely. Fgure.7: Materal Balance Control Volume Usng the Euler approxmaton for the mass rate ṁ x+dx = ṁ o = ṁ x + ṁ x x. dx.8 The generaton and consumpton terms n reservor engneerng are producton and njecton n wells and can be specfed as: ṁ cons =. q prod, q prod = P roducton Rate ṁ gen =. q nj, q nj = Injecton Rate.9 By substtutng equatons.7,.8 and.9 n equaton.6: x.u x.dy.dz.dx.q p.q = t Usng V b = dx. dy. dz and dvdng both sdes to V b gves: x.u x 1 V b.q prod.q nj = t.ϕ. dx.dy.dz.30.ϕ.31 1

19 Equaton.31 s the most general type of the mass conservaton law n ts one dmensonal form [8]. To make t more usable n reservor engneerng, Darcy law equaton.3 s used to substtute the veloctes. Hence, the result for mult-phase flow n porous meda wll be α P K abs K rα α q prod q nj = x µ α x V b t α.s α.ϕ.3 In ths dssertaton, α denotes water and ol phases..3.3 Energy Balance Equaton Thermal smulaton s all about energy balance and temperature calculatons. The energy conservaton law s very smlar to the mass conservaton law and can be wrtten as: ė ė o ė cons ė gen = E t, ė = Energy F lux.33 In hot water modelng, the energy consumpton and generaton are related to njecton or producton streams. There are two man heat transfer equatons that are wdely used for energy balance, conducton and convecton whch are defned as, Conducton heat transfer; q = k. T x Convecton heat transfer; q =. u.h where k s thermal conductvty and H s enthalpy. Therefore, the energy at each pont can be wrtten as: ė = k. T x +. u.h Cross secton area.34 Smlar to mass balance, energy balance can also be drven by applyng equaton.33 and equaton.34 on a sngle element lke the one shown n fgure.8. Fgure.8: Energy balance element ė x = ė x = k. T x +.u x.h dydz.35 13

20 [ ė x+dx = ė ox = k. T x +.u x.h + k. T ] x x +.u x.h.dx dydz.36 ė x ė ox = k T x x.u x.h dxdydz = k T x dxdydz + x µ.k.h. P dxdydz x.37 For wells, the heat transfer can dvded nto conducton and convecton based on bottom hole temperature and flud enthalpy, as below: ė w = ė cons ė gen = q nj q prod H πkhr w T r r=r w.38 For the accumulaton term n equaton.33, both the rock and flud must be taken nto account as they both have heat capactes and are able to store energy n themselves. Therefore the accumulaton term wll be: E t = t o U os o + w U w S w.ϕ.dxdydz + r U r 1 ϕ dxdydz.39 Now all the above equatons must be combned to obtan an energy balance equaton for one dmensonal flow n an ol and water system: o,w,r α k α. T x + o,w α.k α.h α. P + ėw = o,w ϕ α.u α.s α + 1 ϕ r U r x µ α α x V b t α.40 where U α, U r, H α and r are nternal energy for each phase, rock nternal energy, enthalpy for each phase and Rock densty [8]. As t s mentoned before there s a condton whch states that the flud flls the rock pore volume. Ths condton gves the very helpful relaton S o + S w = 1.41 Equatons.3,.3 for ol and water phases and.40 are non-lnear partal dfferental equatons wth coeffcents that are complex functons of temperature, pressure and saturaton. No technques exst to solve these types of equatons analytcally. In ths dssertaton two numercal technques are chosen to model these equatons n order to fnd pressure, temperature and saturaton dstrbutons whch are explaned n detal n the followng chapters..4 Intal and Boundary Condtons In ths one dmensonal reservor, t s supposed that ntally the reservor rock s fled wth ol and rreducble water under unform and constant pressure and temperature. It s also 14

21 supposed that two wells are located at two sdes or boundares of the reservor. An njecton well wth constant rate or total rate and njecton temperature T nj s placed at the left boundary and water saturaton at ths boundary equals to 1 S or. By usng Darcy s equaton n ol feld unts refer to nomenclature. A constant total rate provdes the pressure gradent Neumann boundary condton q t = u x A x.4 K rw P q t = q wnj = 1.17 K abs A x µ w x.43 At the rght boundary, ol and water are produced from a producton well. It s consdered that the total rate s constant from the producer, so the pressure gradent can be calculated at ths boundary usng Darcy s equaton n followng way; Krw q t = q oprod + q wprod = 1.17K abs A x + K ro P µ w µ o x bl br.44.5 Heat losses The heat losses n a system begn at the thermal unt or heat source, wth subsequent heat losses ocurrng n the surface njecton lnes, the njecton wellbore, the wellbore and the formaton tself and the adjacent strata see fgure.9. Theoretcal and laboratory studes have shown that the rate of heat loss to adjacent strata s the most mportant factor whch determnes the economc feasblty of a heat njecton project[5]. The heated area of the reservor rock s qute large, and the heat must be sustaned for a long perod of tme. Therefore, the cumulatve heat loss to adjacent strata s also large, n spte of the fact that the thermal conductvty of earth materal s very small [5]. In ths project, heat losses to wellbore and surface facltes are neglected and heat loss to adjacent layers whch s the most sgnfcant one s modelled by thermal conducton. The other way of heat loss whch s consdered s through the producng fluds from the producton well whch s modeled by thermal convecton. 15

22 Fgure.9: Illustraton of heat losses whch occur n a heat njecton system 16

23 3 Frst Model In ths model, the Buckley-Leverett equaton s used to fnd saturaton profles and then ts results are used n the pressure equaton whch s solved by a fully mplct numercal technque. In order to fnd the temperature profle the saturaton and pressure results are appled to a fully mplct energy equaton. Fnally, the mass balance equaton pressure equaton and the energy balance equaton temperature equaton are coupled to fnd the optmal result for pressure and temperature dstrbutons, snce these equatons are hghly nonlnear. 3.1 Buckley-Leverett Dscretzaton In ths model, the Buckley- Leverett equaton s used to fnd the saturaton dstrbuton. The numercal scheme used to solve ths hyperbolc equaton s the Lax-Wendroff scheme. The scheme s a second order fnte dfference method where the dervatves are approxmated by dfferences of dscrete values. An mportant requrement of numercal methods for such nonlnear hyperbolc equatons s to be n conservatve form to mantan the conservaton property of the equaton. To derve the numercal method n conservatve form we use standard fnte dfference dscretzaton of the conservatve form of the partal dfferental saturaton equaton, not the quaslnear form of the equaton. For a numercal scheme to be n conservaton form [9] t must have the form U j = U n j t X 1 n+ {F U j+ 1 F U n+ 1 }, 3.1 j 1 where U n j s an approxmaton to the cell average of the analytc functon, FU n+ 1 s the j+ 1 numercal flux functon, t =t t n, and x =x j+ 1 x j 1. By usng ol feld unts refer to nomenclature, the Buckley-Leverett equaton wll be, Aϕ q t By choosng, S t + f x =

24 X = x L dx = 1... Ldx t D = x = 1... L X qt A.ϕ.L t dt D = A.ϕ.L dt t = 5.615qt A.ϕ.L... t D the BL equaton converts nto the dmensonless equaton, S + f t D X = In order to drve the Lax-Wendroff scheme appled to above equaton to be n conservatve form, for all ntermedate blocks =,..., N x 1, we start wth; S + f t D X = 0 S = f t D X 3.5 and usng Taylor-seres expanson about t D S X, t D + t D = S X, t D + t D S X, t D t D + t D S X, t D t D + O t D S t D S t D = f X = f s td = f td X = f S. S t = f D X S f X = f X S. f X X By substtutng central dfferences for space dervatves 3.7 S S = S n t f+1 f 1 D = S n t D x By comparson: h + 1 where ν + 1 [ h + 1 X = 1 f +1 + f t D = t D X t D X f+1 f S +1 S f S h 1 + t D ] f S. f X + 1 [ ] f S. f X + 1 f S. S X 1 X : Conservatve f orm f S S = 1 f +1 + f 1 ν + 1. f +1 f 3.9 f S = S

25 Smlarly h 1 = 1 f + f 1 ν 1 f f 1 t D f f 1 X S S 1 f S S 1 ν 1 = 3.11 t D f X S f S = S Effect of boundary condtons For the frst cell = 1, t s supposed that there s a known value of S at the boundary S = 1 S or. At ths pont the Lax-Wendroff scheme s derved based on the unequal spacng see fgure 3.1 Fgure 3.1: Unequal spacng for the left boundary cell after ntroducng an magnary node f X 1 + X = f X 1 + X. f X 1 X b = f X 1 X b. f X X 1 f X X1 f = X X 1 S1 = S1 n f f b t f S t D + D. f X 1+ X X + X b f f b X + X b 3.1 f S. f X 1 X b 3.13 X/ + X b / = S n 1 t D X + X b { f f b t D [ f S. f X S 1 = S n 1 t D X+ X b. h 1+ X h 1+ X h 1 X b = f + f 1 ν 1+ X. f f 1 1+ X f S. f ] X 1 X b h 1 X b = f 1 + f b ν 1 X b f 1 f b 19

26 where ν 1+ X ν 1 X b = = t D X. f f 1 S S 1 S S 1 S = S 1 t D X. f S 1 t D X b. f 1 f b t D X b. f S S 1 S b S b S 1 S b = S 1 b 3.16 All the fractonal water values and ts dervatves n above formulas are calculated from equatons.4 and.5 For last cell =N x, the equaton 3.4 s dscretzed based on forward n tme and backward n space as follows, S t D S n + f Sn f S n 1 X = S = S n td [f S n f S n ] 1 X 3.18 Fnally, the result for all grd ponts s a lnear system, whch we denote t by A.S = b 3.19 where A s a trdagonal matrx and S s a vector of unknowns saturatons, whch s solved to fnd the saturaton profle The CFL condton In order to have stablty when usng explct numercal schemes, we are requred to apply the necessary condton known as the Courant-Fredrchs-Lewy condton. It s often referred to as the CFL or Courant condton, [9] and [10], and s µ= a t x µ max 3.0 Where n ths context, a = a S = f S. Here t and x are the tme and space steps, respectvely. The value of µ max changes wth the method used to solve the dscretzed equaton. Ths condton s not suffcent for stablty, as t s only a necessary condton for scheme to be stable. The Lax-Wendroff scheme s known to be stable for the regon µ= a t 1. The pcture on the rght of fgure 3. shows the doman of dependence for ths numercal scheme. If a t x s the slope of AB then the CFL condton s satsfed because AB les n 0 x

27 the stencl of the scheme, whlst the lne AC volates the CFL condton, by lyng outsde the doman of dependence. Fgure 3.: Stencls for the Lax-Wendroff scheme 3. Dscretzaton of the Mass balance equaton The equaton.3 n ol feld unts refer to nomenclature, for ol phase and substtutng S o = 1 S results n o p K abs K ro o q p q = x µ o x 1.17V b t o.1 S.ϕ 3.1 In order to dscretze ths equaton, a fully mplct scheme s used. As mentoned before, the coeffcents of ths equaton depend on pressure, temperature and saturaton. The results of the saturaton from the Buckley-leverett equaton are used drectly and ndrectly through the relatve permeablty n ths equaton. In order to expand the rght hand sde of equaton 3.1, we need to remember that densty s a functon of pressure and temperature = p t, T t, so we have o t = o p. p t + o p T. T t = op. + ot t T t Hence, expanson of the equaton 3.1 results n K o o + oσq o = ϕ [ 1 S x µ o x 1.17 V b 6.38 op. p t + 1 S ot T t 0 ] S t

28 Fgure 3.3 shows the grd ndexng scheme used for the method. dscretzed as, The above equaton s Fgure 3.3: Grd ndexng scheme n materal balance equaton K o p o µ o x + 1 = ϕ 6.38 t K o p o µ o x 1 x [ 1 S + oσq o 1.17 V b op p p n + 1 S ot T T n o S S n ] 3.4 By usng frst order central dfference scheme wth equal spacng for the pressure gradent at nterfaces, 1 x = K o P o µ o + 1 ϕ 6.38 t [ 1 S +1 P x op p 1 x p n o K o µ o S P P 1 x ot T + oσq o 1.17 V b T n o S S n 3.5 Ol propertes, densty and vscosty, are calculated from equatons.9 and.1 at the nterfaces, based on averages of pressure and temperature of two neghbourng blocks. Ol relatve permeablty s defned from average ol saturatons. By defnng D o = 1 x K 0 0 µ 0 1, = 1,..., N x Equaton 3.5 can be rewrtten n a smpler way for nternal cells supposng no generaton and no consumpton n these cells, ]

29 Do+1 p = ϕ 6.38 t +1 p [ 1 S D o p p 1 op p p n + 1 S ot T T n o S S n ] 3.7 So, general ol pressure equaton for each mddle cell =,...,N x 1 s defned as a functon of three varable p +1, p and p 1, D o.p 1 D o + D ϕ 6.38 t T n 1 S ϕ 6.38 t o o+1 + ϕ 6.38 t op. p n ϕ t S S n 1 S op 1 S p + D o+1 p ot T +1 = 3.8 Treatng the equaton for boundary cells s slghtly dfferent. For the frst cell = 1, the equaton 3.3 s dscretzed as K o P K o µ o x o P o + 1 µ o x bl x = ϕ 6.38 t [ 1 S op p + oσq o 1.17 V b p n + 1 S ot T T n o S S n ] 3.9 Usng equaton.43 and boundary condtons help to fnd water propertes and boundary pressure gradent n above equaton, S b = 1 S or, T b = T nj, p b = p bl K wbl = K abs.k rw 1 S or = K abs.k max rw µ wb = µ w P bl, T nj P P bl P q t.µ wbl = = = GP I x/ x 1.17 K wbl A n bl 3.30 Note that the njecton well s located on the boundary, so the effect of generaton terms s consdered n the boundary pressure gradent. Hence, the dscretzed ol pressure equaton for the frst cell = 1 s Do+1 + ϕ 1 S op 6.38 t ϕ 1 S op P n + ϕ 1 S ot 6.38 t 6.38 t p + D o+1. p T +1 = D o T n ϕ o 6.38 t. x I S S n

30 Condton at the rght boundary cell s dfferent. parameter, λ tb, the pressure gradent for ths boundary s calculated as Usng equaton.44 and defnng a new λ tb = K rob µ ob + K rwb µ wb P x b q t = = GP O K abs A n λ tb Boundary condtons = Nx S b = 3 S Nx 1 S Nx 1 p b = p br T b = 3 T Nx 1 T Nx 1 x 1 K rob = K ro 1 S b K rwb = K rw S b µ ob = µ o p b, T b µ wb = µ w p b, T b By substtutng ths term nto the equaton below dscretzed pressure equaton for the last cell, K o P K o µ o x o P o br µ o x 1 x = ϕ 6.38 t [ 1 S op p p n + 1 S ot T T n o S S n ] 3.33 And notng that the generaton and consumpton term n the man equaton 3.1 s replaced by the effect of the boundary condton, we have the equaton below for = N x, Do P 1 D ϕ 1 S op 6.38 t o + ϕ 1 S op P = Do. x.gp O 6.38 t P n + ϕ 1 S ot T T n ϕ o 6.38 t 6.38 t S S n 3.34 Fnally, wrtng these equatons for all grd blocks, we obtan the nonlnear system of AS = b where A s a trdagonal matrx n whch all ts elements depend on the unknowns.such a system s not easy to solve, but n the followng secton we explaned how to deal wth ths dffculty. 3.3 Newton s Method for Nonlnear Systems of Equatons To fnd the soluton of a system Ax = b of N x nonlnear equatons n N x unknowns [13], the system can be wrtten n the homogenous form 4

31 F x = Ax B = Consder the Taylor-Seres expanson of F x about x = x 0. Usng only the frst two terms of the expanson, a frst approxmaton to the root of F x can be obtaned from F x = F x 0 + x = F x 0 + x x 0 F x x 0 Let F x = J x 0 gvng F x = F x 0 + J x 0 x x 0 = 0 x 0 Ax 0 B + J x 0. x x 0 = 0 Ax 0 B + J x 0. x J x 0. x 0 = 0 J x 0. x = J x 0.x 0 Ax 0 B If A C = J x 0, C = J x 0. x 0 Ax 0 B and x = x 0 a vector represents the frst guess of the soluton, successve approxmaton to the soluton are obtaned from A C.x = C 3.36 Ths s the Newton Newton-Raphson method for solvng the system. It requres the evaluaton of the Jacoban matrx of the system whch s defned as: F x x = J x = F 1 F 1 x 1 x F x 1 F x.. F N x 1 F N x.. F 1 x N F x N Dfferent convergence crtera can be appled to the system, to fnd the soluton. F N x N 3.37 In ths project, the maxmum of the modulus dfference of the between consecutve vectors s used to be less than a certan tolerance ε. In mathematcal terms ths s expressed as max x x n < ε 3.38 The man complcaton wth usng Newton-Raphson to solve such a system of non-lnear equatons s havng to defne all the functons 5 F x j, for, j = 1,,..., N x, ncluded n the

32 Jacoban. As the number of equatons and unknowns, N x, ncreases, so do the number of elements n the Jacoban. The convergence of Newton s method s quadratc when the Jacoban matrx s non-sngular and the ntal guess s close enough. 3.4 Jacoban Matrx Defnton for Mass Balance Equaton In order to solve the nonlnear system resulted from applyng fully mplct method to ol mass conservaton equaton ol pressure equaton, the Jacoban s defned as follow, For nternal equatons =,..., N x 1 F P 1, P, P +1 = D o.p 1 ϕ t + ϕ 6.38 t o 1 S S D op. p n S n = 0 o + D ϕ 6.38 t o+1 + ϕ 6.38 t 1 S 1 S op ot T T n p + D o+1 p F = 1 P D Do ηop o ηop P 1 + D o D o+1 η op+1 P ϕ 6.38 t 1 S D o + D η op ϕ 6.38 t op T o+1 + ϕ 6.38 t T ϕ 6.38 t 1 S 1 S opp 1 S opp P n ϕ + T n 6.38 t op op η P S S n 3.40 F = Do + 1 P 1 D o ηop P 1 1 [ D o ηop P = Do η o P 1 P ] 3.41 [ F = Do P +1 η op+1 P +1 P ] 3.4 For the frst equaton = 1 F P = o+1 + ϕ 1 S op 6.38 t D + 1 η op+1 D o+1 P + ϕ 1 S 6.38 t opp η op+1 D GP I. x η op Do ϕ 1 S 6.38 t P n 6 opt o+1 + ϕ 1 S 6.38 t + 3 T opp GP I. x.d o. µ wpb µ wb P T n ϕ op + S S n 6.38 t

33 3.43 F = 1 P +1 η op+1 D o+1 P +1 P + D o+1 1 GP I. x. η op D o 1 For the last equaton = N x GP I. x.d o. µ wpb µ wb 3.44 F = 1 P D o ηop P 1 Do + ϕ 1 S op 6.38 t 1 D o ηop + ϕ 1 S opp P 6.38 t + ϕ 1 S opp P n ϕ 1 S opt 6.38 t 6.38 t + ϕ op S S n 6.38 t η op P 1 P + 1 D F = Do P 1 where ηop op = µ op o µ o 1 3 D o ηop x.gp O o ηop T T n 3.45 x.gp O 3.46 = 1,..., N x Dervatves of ol and water denstes and vscostes to pressure and temperature are defned usng the equatons Well Couplng In order to fnd bottom hole pressures at the left and rght boundares, couplng between the block and well pressures s performed. It starts by takng ntal guesses for boundary well pressures. Grd pressures are then solved based on these guesses. Because block and well pressures are correlated, new well pressures can be calculated from equatons 3.48 and 3.49 and these new values used n a new teraton to calculate new grd pressures. Iteratons contnue to fnd fxed block and well pressures. These two equatons are nonlnear for the well pressures, the Newton-Raphson method s used to fnd the roots. The left boundary well pressure equaton s q t = 1.17K abskrw max A n P B P wl µ w P wl x/

34 The rght boundary well pressure equaton s q t = 1.17K absa n x/ { Krw µ w P wr + K } ro P wr P B 3.49 µ o P wr 3.6 Energy Balance Equaton Dscretzaton The energy equaton.40 n ol feld unts refer to nomenclature s rewrtten as x.4k T H x x.{ K o K w o H o + w H w P µ o µ w x } + Σe V b = t {ϕ os o U o + w S w U w + 1 ϕ r U r } 3.50 H α = H ref α + C P α T T ref α = o, w U α = U ref α + C V α T T ref α = o, w, r 3.51 A fully mplct central fnte dfference scheme s used to dscretze the equaton. Frst, the equaton s expanded by substtutng equaton 3.51 and then new equaton s shortened by defnng some parameters, x.4k T H x x.{ K o o Ho ref + C P o T T ref P µ o x } x { K w w Hw ref + C P w T T ref P µ w x } + Σe V b = t {ϕ os o Uo ref + C V o T T ref + w S w Uw ref + C V w T T ref + 1 ϕ r U ref r + C V r T T ref } 3.5 By defnng, K o HR = o H ref K w o C P o T ref + w Hw ref C P w T ref 3.53 µ o µ w HB = o K o µ o C P o + w K w µ w C P w 3.54 UR = ϕ o S o U ref o C V o T ref + w S w U ref w C V w T ref + 1 ϕ r U ref r C V r T ref 3.55 UB = ϕ o S o C V o + w S w C V w + 1 ϕ r C V r 3.56 Equaton 3.5 s smplfed to 8

35 x.4k T H x e HR + HB.T P x x + = UR + UB T 3.57 V b t Dscretzaton of Rght Hand Sde of the Energy Equaton 3.57 Replace the dervatves as follow t UR + UB.T = 1 t {UR + UB.T UR + UB.T n } = UR UR n t + UB.T UBn t t T n Dscretzaton of Left Hand Sde of Energy Equaton 3.57 for Mddle Cells =,...,N x Ths sde of the equaton s dvded nto two terms; conductve and convectve heat transfer. Conducton s the transfer of heat energy by dffuson due to the temperature gradent. In ths project, conducton takes place n both rock and fluds. Whle convectve heat transfer takes place through advecton mostly, n whch heat s transferred by the moton of currents n the flud Conducton Term In ths dssertaton, conductve heat s consdered to transfer n two dmensons n order to model heat loss to adjacent strata [11]. Fgure 3.4 shows the schematc dagram of the model. 9

36 Fgure 3.4: Heat transfer n the x and y drecton by conducton Usng a central dfference scheme wth equal spacng n the x drecton and unequal spacng n the y drecton results n x 4K T H x + y 4K T r Y = T 4K H x T 4K + 1 H x 1 + 4k r T x 4K H x T 1 48K H x where y b = y + y. + 48k r y b y UP 4K r T y Down y + y T + 4K H x T = K r T y b Convecton Term Dscretzaton Usng a frst order central scheme n space, ths term wll be dscretzed as, 6.38 HR+HB.T P x x = 6.38 P {HR + HB.T x x } {HR + HB.T P x x } and by defnng GP pressure gradents on the nterfaces of blocks and takng an average temperature on nterfaces, between two neghbourng blocks, P x = GP 1 T + 1 = T + T +1 /, T 1 = T + T 1 / 30 = 1,..., N x + 1

37 Convecton term s sorted out as 6.38 x 6.38 HR + HB.T P x = x HR x HB HR 6.38 x HB T + +1 HB T x HB T 3.61 By combnng equatons 3.57, 3.58, 3.59 and 3.61 together, the general energy equaton for mddle cells s obtaned as =,..., N x ; { 4K H x 6.38 x HB + { 48K H x 48k r y b + { 4K H x UBn t T n x HB +1 } T x HB +1 48K r y T 6.38 b +1 } T { 4K H 6.38 x x HB } T x HR HB +1 = UR UR n t +1 HR UB } T t Calculatons for the Left Boundary Cell =1 For ths cell, the conducton and convecton terms are treated dfferently because of the effect of boundary condtons Conducton Term Applyng a central dfference scheme to the conducton term n the x drecton wth unequal spacng gves; x 4K T H x = 4K H and n the y drecton T +1 T n x 4K H T T nj x/ 3 x / 3.63 y 4K T r y = 4k r T T y 4K + y/ r T y + y/ T y + y/ 3.64 By addng these two together and rearrangng the terms, the conducton term s wrtten as 31

38 x 4K T H x + y 4K T r y = x 4K H T x + y 4K r T y = 96K H x 48K r y b Convecton Term T + 3K H x T+1 +64K H x T nj + 48K 3.65 r T By expandng the convecton term on the frst cell and substtutng the njecton temperature, y b 6.38 x 6.38 x HR + HB.T P x = {HR.GP + HB.GP.T } {HR.GP + HB.GP.T } b = x x HB T + T x {HR HR } 6.38 x HB n T nj 3.66 Hence, the dscretzed energy equaton s rearranged based on man varables temperatures as below; { 96K H x + UB K r y b t t UR x {HR +1 x HB +1 }. T +{ 3K H x UR n UBn t T n +1 x HB }. T +1 = + 4K r y T nj T 64K H b x T nj 48K r T y b +1 HR } x HB T nj Calculatons for the Rght Boundary Cell =N x Consderng the rght boundary condtons defned n the prevous chapter, the conducton term n the x drecton s zero for ths cell. In the y drecton s defned as n the other blocks from equaton The convecton term s derved n a smlar way; the dfference beng to substtute T b = 1 3T T 1 for T +1 so, t wll be 3

39 6.38 HR + HB.T P x x = x x HR HB HB. T x HB +1 Heat loss s ncluded n the energy term, +1 + HB. T 1 +1 HR 3.68 e V b = 4K H x. x 3 T 1 T 1 T 3.69 Hence, the energy equaton n ts dscretzed form for the last cell = N x s { x HB HB + { x 6.38 x UBn t T n 3 HB HB {HR HR 1K r + } T 1 x. x 4K r T 48K r x. x y T b 36K r 48K r x. x } + 1 t UR UR n y b UB } T = t 3.70 Wrtng these equatons for all grd blocks wll result n a nonlnear system. To solve ths system, t s necessary to defne the Jacoban. For smplcty, before calculatng the Jacoban, new parameters are defned as the dervatves of HB, HR, UB and UR. All these terms can be redefned usng D o equaton 3.6 so, DHB = D o η ot C P o + D w η wt C P w 3.71 DHR = D o η ot H ref o C P o.t ref + D w η wt H ref w C P w.t ref 3.7 DUB = ϕ ot S o.c V o + ϕ wt S w C V w 3.73 DUR = ϕ ot S o.u ref o where η αt = αt α µ αt µ α 1 C V o T ref + ϕ wt S w U ref w C V w T ref 3.74 = 1,..., N x + 1 and α = ol and water 3.75 Note that C P α and C V α are consdered to be ndependent of temperature constant and the dervatves of densty and vscosty wth respect to temperature and pressure are defned usng the equatons

40 Based on above defntons and equaton 3.6, the Jacoban calculatons for =,..., N x 1 gve F = 4K H T 1 x x HB x. DHR x DHB.T + T 1 F = x. DHB.T + T 1 T x + { 48K H x. DHB T + T 48K r y b x DHR +1 F T +1 = 4K H x K H x x HB +1 x HB +1 x HB DUB t +1 HB +1 DHR T x. DHR x. DHR DUR t UB } t +1 F T x. DHB T + T+1 In the case of the frst cell, the Jacoban elements for the energy equaton 3.67 are obtaned as F = { 96K H T x x + DUB t + 48K r y b x HB UB } t. DHB T + T+1. T F T +1 = +{ 3K H x DUR t x HB x.dhr } x. DHB T + T x. DHR and fnally, the Jacoban calculatons for the last cell = N x wll be F = T 1 x x x HB HB DHB +1 1K r + x. x +1 DHB T 1 3 DHB DHB. T x.dhr DHR

41 F T = {3.164 x 3HB x + {3.164 x +1 HB 3 DHB DHB. T 1 9 DHB DHB x.3 DHR DHR 36K r 48K r x. x y UB } b t DUB }. T t DUR t

42 3.7 Summary of The Frst Model Fgure 3.5: The Frst Model Calculatons Flow Chart 36

43 4 Second Model In ths model, the pressure wll be solved mplctly and after fndng the pressure soluton, saturaton values can be determned explctly. Ths technque s called IMPES and s much used n the ol ndustry [8]. Durng one tme step the results of IMPES are used n the temperature equaton whch s solved fully mplctly, and fnally there wll a couplng between the IMPES technque and the fully mplct temperature equaton n order to fnd the fnal pressure, saturaton and temperature dstrbuton results. 4.1 IMPES Technque In the ol and water system, the general 3D equatons for ol and water are K o o µ Po o + oσqo 1.17V b = 1 K w w µ Pw w + wσqw 1.17V b = t oϕ S o t wϕ S w 4.1 By consderng the assumptons made n secton.1, and consderng that generaton and consumpton terms are replaced by boundary condtons and expanson of rght hand sde of the equatons, we have x x K o P o µ o x w K w µ w P x = ϕ 6.38 = ϕ 6.38 [ 1 S P op t + 1 S ot T t o S ] t [ S P wp t + S wt T t w S ] 4. t Dvde ol pressure equaton by o and water pressure equaton by w, then add them together, and rearrange the fnal equaton to fnd the dscretzed pressure equaton for nternal cells =,..., N x 1, gvng 1 o x o K o ϕ w P µ o x [ 1 S op + S wp o w K w P w x µ w x ] P t + = ϕ 6.38 [ 1 S ot o + S wt w ] T t

44 For smplcty we defne the equatons parameters and terms to be smlar to those defned n the frst model refer to secton 3.. The above equaton 4.3 s dscretzed usng a fully mplct scheme and by consderng pressures as the man varables as follows, D o o + D w + D o+1 o P 1 w ϕ1 S n 6.38 t ot o + {D o + D o+1 o + D w+1 P+1 = {ϕ1 S n w 6.38 t op + o ϕ S n 6.38 t wt + D w + Dw+1 + ϕ1 S n w 6.38 t op + o }T w T n ϕ S n 6.38 t wp w } P n + ϕ S n 6.38 t wp } P w 4.4 Dfferent treatments are requred to obtan the pressure equaton for the left boundary cell = 1, by consderng ts boundary condtons refer to secton.4. 1 o. x { K o o = ϕ 6.38 o K o µ o P P µ o x + 1 [ 1 S op + S wp o w x bl} + 1 w x { w ] P t + ϕ 6.38 K w P µ w x + 1 [ 1 S ot o K w P w µ w x bl} ] T t + S wt w 4.5 By usng equaton 3.30 for the pressure gradent on ths boundary, { D o+1 o + D w+1 w + D w+1 + ϕ1 S n w 6.38 t op + o + D o+1 P+1 o { ϕ1 S n 6.38 t op o + o = GP I. x.d o ϕ S n 6.38 t wp } P n + w { ϕ1 S n 6.38 t ot o + ϕ S n 6.38 t wt ϕ S n 6.38 t wp } P w }T w + D w w T n The nvestgaton for the rght boundary cell = N x shows that o x { K o P o µ o ϕ 6.38 x br o K o µ o P [ 1 S op + S wp o w x 1 ] P t + } + 1 w x { w ϕ 6.38 [ 1 S ot o K w P µ w x br + S wt w K w P w µ w ] T t x 1 = 4.7 By usng equaton 3.3 as the pressure gradent on ths boundary and the ol and water propertes at boundary whch are calculated from extrapolated temperature and saturaton, and usng all these defntons, the dscretzed pressure equaton for = N x s 38

45 D o o + D w P w GP O. x. D o+1 o { D o o { ϕ1 S n 6.38 t ot o + + D w + ϕ1 S n w 6.38 t op + ϕ S n o 6.38 t wp } P = w + D w+1 { ϕ1 S n w 6.38 t op + o ϕ S n 6.38 t wt { ϕ1 S n 6.38 t ot + ϕ S n o 6.38 t wt }T w }T w T n T n ϕ S n 6.38 t wp } P n w 4.8 The result of wrtng the pressure equaton for all blocks s also a nonlnear system to solve, so the defnton of the Jacoban for ths system s requred. 4. Jacoban Calculatons for the Pressure Equaton The general equaton 4.4 s a functon of three varables; P 1, Jacoban matrx s defned as follow P and P +1, therefore the F = 1 D w η P 1 w wp + D o η op P 1 P o + D o o + D w 4.9 w F P = {D o+1 1 η op+1 o { D o + Do+1 o op + Dw+1 1 o η wp+1 w wp } P+1 w + D w + Dw+1 + ϕ1 S n w 6.38 t. op + ϕ S n o 6.38 t. wp } w { D o ηop + Do+1 η op+1 + D w ηwp + Dw+1 η wp+1 o w Do + Do+1 op Dw + Dw+1 wp + ϕ1 S n o w 6.38 t. opp o op o + ϕ S n 6.38 t. wpp w wp w }P + {D o 1 η op o + { ϕ1 S n 6.38 t. opp o op + ϕ S n o 6.38 t. { ϕ1 S n 6.38 t. opt o op ot + ϕ S n o 6.38 t. op + Dw 1 o wpp w wp w } P n wpt w wp wt w η wp w }T T n wp } P 1 w 4.10 F = 1 D w+1 η P +1 w wp+1 + D o+1 η op+1 P+1 P o 39 + D o+1 o + D w w

46 The frst cell = 1 pressure equaton depends on two varables; p and p +1, so F P = {D o+1 1 η op+1 o op + Dw+1 1 o η wp+1 w wp w { ϕ1 S n 6.38 t. opp o op + ϕ S n o 6.38 t. { D o+1 o + D w+1 + ϕ1 S n w 6.38 t. op + ϕ S n o 6.38 t. wp } w + GP I. x.{do. 3 ηop o op + Dw. 3 o }P+1 P wpp w wp η wp w w wp } w }P P n o.gp I. x.d o + D w µ wpb w µ wb { ϕ1 S n 6.38 t. opt o op ot + ϕ S n o 6.38 t. wpt w wp wt w }T T n F = 1 D w+1 η P +1 w 1 GP I. x.{ D o wp+1 + D o+1. η op η op+1 P+1 P o + Dw o. η wp 1 w + D o+1 o o.gp I. x.d o + D w+1 w + D w µ wpb w µ wb 4.13 and fnally, for the last cell = N x pressure equaton whch s functon of p and p 1, we have; F = 1 D w η P 1 w + 1 wp + D o w+1 GP O. xd η w η op P 1 P o wp+1 + D o+1 + D o o + D w w 4.14 η op+1 o F P = {D o 1 η op o op + Dw 1 o η wp w ϕ1 S n 6.38 t. opp o op + ϕ S n o 6.38 t. { D o o wp w } P 1 P wpp w wp w + D w + ϕ1 S n w 6.38 t. op + ϕ S n o 6.38 t. wp } w GP O. x.{do+1.3 ηop+1 o op + Dw+1.3 o { ϕ1 S n 6.38 t. opt o op ot + ϕ S n o 6.38 t. η wp+1 w }P P n wp } w wpt w wp wt w }T T n

47 4.3 Saturaton Calculatons Ol and water saturatons are evaluated explctly by usng the results of the fully mplct pressure equaton. The followng equatons are appled to fnd the saturatons x K o P o µ o x = t ϕ os o x K w P w µ w x = t ϕ ws w 4.16 These equatons are dscretzed as Do+1. P+1 P Do P P = ϕ 6.38 { o. So n o So} n Dw+1. P+1 P Dw P P = ϕ 6.38 { w. Sw n w Sw} n Hence, ol and water saturatons are calculated for nternal cells =,..., N x from 4.17 S o S w = 6.38 t ϕ {Do+1. P+1 Do + Do+1 P + Do. P o = 6.38 t ϕ {Dw+1. P+1 Dw + Dw+1 P + Dw. P w For the frst cell = 1, } + n o o } + n w w S n o S n w 4.18 S o S w = 6.38 t ϕ {Do+1 P+1 + P + GP I. x. Do o = 6.38 t ϕ {Dw+1 P+1 + P + GP I. x. Dw w and fnally for the rght boundary cell = N x } + n o o } + n w w S n o S n w 4.19 S o S w = 6.38 t ϕ {Do P P 1 + GP O. x. Do+1 } + n o o o = 6.38 t ϕ {Dw P P 1 + GP O. x. Dw+1 } + n w w w Ths s the whole procedure of the IMPES method for ths model. S n o S n w 4.0 In summary, n ths model pressure and saturatons are calculated from the IMPES technque, but a smlar method fully mplct method wth the frst model s used to fnd the temperature dstrbuton. 41