A Study of Water and Carbonated Water Injection with Constant Pressure. Boundaries. Huan Yang

Transcription

1 A Study f Water and Carbnated Water Injectin ith Cnstant Pressure Bundaries by Huan Yang A Thesis submitted t the Schl f Graduate Studies in partial fulfillment f the requirements fr the degree f Master f Engineering Faculty f Engineering and Applied Science Memrial University f Nefundland May 2014 St. Jhn s Nefundland

2 Abstract The Buckley-Leverett thery fr ne-dimensinal cnstant fluid velcity is idely used in the il and gas industry. Hever, given a changing fluid velcity ith fixed pressure bundary cnditins, limitatins arise. This rk is based n an existing extensin f the Buckley-Leverett thery in a ater-il system ith fixed pressure bundary cnditins. This alls the Buckley-Leverett thery t be applied t situatins f injecting ater at a cnstant bttm-hle pressure and prducing il at a fixed bttm-hle pressure. Based n mass cnservatin, numerical simulatin is perfrmed in Matlab using the Implicit Pressure Explicit Satuatin (IMPES) methd fr t-phase fl. The numerical slutin is cmpared t the recently develped analytical slutin fr different case studies. The cmparisn is als used t illustrate the effect f numerical dispersin and rund-ff errrs. This extensin f the Buckley-Leverett thery has significant cnsequences in its applicability t mre realistic perating scenaris and cmputatinal savings thrugh analytical slutins. Carbnated ater injectin is studied numerically based n the validated ater injectin mdel. In carbnated ater injectin, CO 2 is disslved in ater phase befre injectin. After injectin, the prperties f reservir fluids ill change due t the partitining f CO 2 beteen bth the ater and il phases. Therefre, the reductin f il viscsity and il-ater interfacial tensin uld be the main factrs affecting the il recvery. Hever, there is minimal research n carbnated ater flding cmbining bth thermdynamics and reservir simulatin mdels. This research aims t study the effect ii

3 fr il recvery in carbnated ater injectin based n bth physical and numerical perspectives. iii

4 Acknledgements I uld like t take this pprtunity t express my sincere appreciatin t my supervisrs, Dr. Lesley James and Dr. Thrmd Jhansen. Thank yu fr yur utstanding technical guidance that inspires me t rk n this thesis. Yur patience takes me thrugh every difficulty that I have encuntered during my study. I am grateful fr yur supprt and encuragement that mtivate me t accmplish this prject. I als ant t thank Nrah Hyndman. Thank yu fr helping ith my riting. Thanks als t my research clleagues: Hashem Nekuie, Jie Ca, Nan Zhang, Xialng Liu, Ali Suri Laki, Xiayan Tang, Mahsa Mayed, Eskandari Saeed. Thanks fr their suggestins and help during my research. I really enjyed the time that e ere rking tgether. My thanks als g t my dear friend Weiyun Lin. Thanks t the friendship that makes me feel lucky and happy. Last but nt least, I uld like t express my deeply thanks t my parents and my fiancé Yi Li. T my parents, I culd nt cmplete my study ithut all the sacrifice that yu have made. Yur selfless giving is my best treasure in my life. Thanks t my fiancé Yi Li. Because f his caring and cncerning I alays feel arm and secure. Thanks t my little cat Tiger, fr his cmpany in every rking night. iv

5 Table f Cntents Abstract... ii Acknledgements... iv Table f Cntents... v List f Tables... ix List f Figures... x Nmenclature... xii Abbreviatin... xv Chapter 1 Intrductin The Overvie f Glbal Oil Prductin and Cnsumptin The Backgrund f Oil Recvery Primary Oil Recvery Secndary Oil Recvery Tertiary Oil Recvery Current and Invested CO 2 EOR Prject Scpe f Thesis... 6 Chapter 2 Literature Revie f CO 2 Oil Recvery Prcess Revie f Buckley-Leverett Thery Limitatin f The Buckley-Leverett thery fr a Cnstant Fl Rate Extensin f the Buckley-Leverett Thery ith Cnstant Pressure Bundaries Summary f CO 2 EOR Prcess CO 2 Gas Injectin Water Alternative Gas Injectin Carbnated Water Injectin Literature Revie f Carbnated Water Injectin (CWI) Experimental Investigatins f Carbnated Water Injectin v

6 2.3.2 Numerical Mdeling f Carbnated Water Injectin Variatin f Fluid Prperties ith Disslved CO Change in Fluid Viscsity Change in Fluid Density CO 2 Oil Selling Oil-Water Interfacial Tensin Chapter 3 Water Injectin under Cnstant Pressure Bundary Cnditin Mathematical Mdel The Cntinuity Equatin Darcy s La fr a Single Phase One-Dimensinal Hrizntal Water Flding Intrductin The Black Oil Mdel Numerical Mdel Implicit pressure and Explicit Saturatin (IMPES) Saturatin Prfile Numerical Simulatin and Analysis Data Preparatin Fluid Prperties Reservir Prperties Bundary Cnditins Cmputatin f Finite Difference Equatin Pressure Distributin Saturatin Prfile and Ttal Fluid Velcity Case Study under the Cnstant Pressure Bundary Cnditins Input Parameters Case Studies Results Saturatin Prfile and Pressure Distributin vi

7 Ttal Fluid Velcity Cmparisn beteen Numerical and Analytical Slutins Numerical and Analytical Cmparisn Discussin f Numerical Errrs Summary Chapter 4 Carbnated Water Injectin Intrductin Mathematical Mdel Partitining f CO 2 in a Three-cmpnent T-phase System CO 2 Slubility CO 2 Slubility in Water CO 2 Slubility in Oil Fluid Characterizatin Oil Phase Prperties CO 2 -Oil Viscsity Change f CO 2 -Oil Density Carbnated Water Prperties Carbnated Water Viscsity The Change f Carbnated Water Density IFT f Water-il Numerical Simulatins Case Study Initial Cnditins Oil Initial Cmpsitin Initial Reservir and Fluid Prperties Case 1: Different Injecting Pressures Discussin Case 2: Different Reservir Temperatures Case 2a: Higher IFT (ler reservir temperature) vii

8 Case 2b: Ler IFT (higher reservir temperature) Discussin Viscsity Effect Cumulative Oil Prductin and Recvery Factr f CWI and WI Chapter 5 Cnclusins and Recmmendatins Cnclusins Recmmendatins Bibligraphy Appendix A Unit cnversin factrs Appendix B Pre-print: Slutins f Multi-Cmpnent, T-Phase Riemann Prblems ith Cnstant Pressure Bundaries viii

9 List f Tables Table 2.1 Cefficients used t calculate il viscsity fr bth live and dead ils Table 3.1 Parameters used in case study Table 4.1 Initial Oil Cmpsitin Table 4.2 Initial Infrmatin regarding Reservir and Fluid Prperties Table 4.3 Summary f CWI Case Studies Table 5.1 Summary f CWI ith Different Cnditins ix

10 List f Figures Figure 1.1 Number f CO 2 EOR Prjects (Melzer and Midland, 2012)... 6 Figure 1.2 Summarized Cncept Map f Thesis... 8 Figure 2.1 The Typical Water Saturatin Prfile Figure 2.2 Fractin Fl Curve Figure 2.3 Develped Miscibility Figure 2.4 Vaprizing Gas Drive Figure 2.5 Cndensing Gas Drive Figure 2.6 Cmparisn Results beteen Emera and Sarma (2006) and Simn and Graue (1965) Oil Selling Factr (due t CO 2 ) Crrelatins Predictin Results (Jarba and Anazi, 2009) Figure 2.7 Oil-Water Interfacial Tensin as Functin f CO 2 Injected (Zekri et al., 2007) Figure 3.1 Blck-Centered Finite Difference Mdel Figure 3.2 Typical relative permeability curves f ater and il Figure 3.3 Summarized by The Fl Chart fr Slving One-Dimensin Water Flding Figure 3.4 Saturatin Prfile at Water Breakthrugh Time 0.25, 0.5, 0.75, Figure 3.5 Fractinal fl curve f case 1 (μ > μ ) and case 2 (μ < μ ) Figure 3.6 Pressure Distributin at Water Breakthrugh Time 0.25, Figure 3.7 Pressure Distributin vs. Water Saturatin at 0.25 Breakthrugh Time in Case 2 (μ < μ ) Figure 3.8 Water Saturatin Prfile at 0.5 Breakthrugh Time f Each Case Figure 3.9 Ttal Fluid Velcity x

11 Figure 3.10 F vs. Water Saturatin Figure 3.11 Cmparisn beteen Numerical and Analytical Slutins Figure 3.12 Cmparisn beteen Numerical and Analytical Slutins f Case 1 after Minimizing Rund-ff Errr Figure 3.13 Water Saturatin Prfile fr Case 1 (μ >μ ) under Different Numbers f Grid Blcks Figure 4.1 Water Saturatin Prfiles f CWI and WI under 33 MPa Injecting Pressure after 25 Days Figure 4.2 Water Saturatin Prfiles f CWI and WI under 32 MPa Injecting Pressure after 38 Days Figure 4.3 Water Saturatin Prfile after 38 Days Figure 4.4 CO 2 Mass Cncentratin vs. Residual Oil Saturatin ver 200 Days Figure 4.5 Water Saturatin Prfile after 200 Days Figure 4.6 CO 2 Mass Cncentratin vs. Oil Viscsity ver 200 Days f Case 2a Figure 4.7 Cumulative Oil Prductin after 200 Days xi

12 Nmenclature A Crss sectin area, m 2 c c c, c 2 c2 CO 2 mass cncentratins in il, ater phases c Cmpressibility c Rck cmpressibility pv E t Ttal seep efficiency E a Areal seep efficiency E v Vertical seep efficiency E m Micrscpe seep efficiency f F Fractin fl functin Mass flux, kg/(m 3 s ) K Frmatin permeability, m 2 K, K Permeability f il, ater, m 2 k r, k Relative permeability f il, ater r k, k Equilibrium cefficients f CO 2 mle fractin in il, ater phases c2 c2 k c, 2 k Equilibrium cefficients f CO c 2 mass cncentratin in il, ater 2 phases L Length, m MW, MW Mlecular eight f il, gas c 2 xii

13 m Mlality f CO c 2 (ml/kg) 2 p p c Pressure, Pa Capillary pressure, Pa p b Bubble pint pressure, Pa Q Vlumetric fl rate, m 3 /s Q Mass fl rate, kg/s q Mass fl rate per unit vlume, kg/(m 3 s ) R S S Universal gas cnstant Saturatin Average saturatin S Residual il saturatin r S Irreducible ater saturatin c S Frnt ater saturatin f t Time f breakthrugh, s BT t Time, s T Temperature, u t Ttal fluid velcity, m/s u, u Velcity f ater, il, m/s V Vlume, m 3 V Bulk vlume, m 3 b xiii

14 v f Frnt velcity, m/s x, x CO 2 mle fractin in il, ater phases c2 c2 y CO c 2 mle fractin in the gas phase 2 Activity cefficient c 2, Frmatin vlume factr f il, ater Partitin cefficient Oil specific gravity CO c 2 fugacity cefficient 2 Initial il density, kg/m 3 i Prsity Mbility Mass cncentratin in bulk vlume, kg/m 3, Viscsity f il, ater phases, Pa s, l(0) c2 Standard liquid and gas chemical ptential in ideal cnditins, J/kg v(0) c2 Density, kg/m 3, Density rati beteen stck tank cnditin and reservir cnditin * * Water-il interfacial tensin, N/m xiv

15 Abbreviatin EOR IFT IMPES WI CWI CFL MMP WAG NGL GA RF Enhance Oil Recvery Interfacial Tensin Implicit Pressure, Explicit Saturatin Water Injectin Carbnated Water Injectin Friedrichs-Ley Cnditin Minimal Miscible Pressure Water Alternating ith Gas Natural Gas Liquid Genetic Algrithm Recvery Factr xv

16 Chapter 1 Intrductin 1.1 The Overvie f Glbal Oil Prductin and Cnsumptin Energy demand has grn significantly rldide since the late 1970s and early 1980s. Hydrcarbn fluids are cnsidered as ne f the majr surces f energy. As a fundamental resurce, il has been prviding heat, light and per as ell as nn-energy prducts like chemicals and lubricants (Rrda, 1979). The demand r cnsumptin f il is affected by many aspects such as ppulatin, ecnmic activity, gvernment plicies, internatinal il prices and technlgical advances etc. (Jrdan, 1998). East Asia has been rapidly expanding il markets due t the ppulatin and ecnmic develpment in the 1990s (Davies, 1994). Hever, a rapid decline f il prductin has been bserved frm many il fields rldide ith many mre fields transitining int decline each year. The ttal rld il prductin stpped expanding in apprximately mid-2004 (Höök et al., 2009). Accrding t Andre Guld, CEO f Schlumberger, althugh an accurate average decline rate is hard t estimate, an verall figure f 8% is nt an unreasnable assumptin (Schlumberger, 2005). In rder t meet the gring il demand, expliting ne il fields cntinues. Hever, it is difficult t justify the develpment f ne il reservirs that are generally lcated in remte, islated r harsh ff-shre areas (Stahl et al., 1987). Efficient recvery frm already discvered il fields becmes a matter f imprved and enhanced il recvery ecnmics factring in capital csts f directinal and multilateral ells, advanced ells, and including perating csts 1

17 fr different injectin strategies such as plymer, CO 2, ater-alternating-gas (WAG), etc. alng ith the need fr an additinal separatin and treatment capacities. 1.2 The Backgrund f Oil Recvery Typically, nly a fractin f the ttal resurces in place can be recvered frm a reservir. The fractin f il ultimately prduced frm a given field depends n the gelgy f the field, the recvery mechanisms, and ecnmic cnditins (Lake, 1989) Primary Oil Recvery During the primary recvery stage, initial prductin fl by the pressure difference beteen the reservir and the ell fling pressure. Operating ith a cnstant reservir and ell fling pressure, albeit fr a finite perid f time, can be mathematically cnsidered as cnstant pressure bundary cnditins. This ill be discussed in Sectin 3.5. Artificial lift can be used if the pressure is nt sufficient (Tzimas et al., 2005). Hever, due t the pressure depletin during prductin the reservir il eventually ceases t fl leaving cnsiderable il trapped in the pres f reservir rck (Lake, 1989) Secndary Oil Recvery Secndary il recvery is hen gas r ater is injected int the frmatin in rder t maintain the reservir pressure and frcing il t fl tards prductin ells. Althugh a great amunt f il can be prduced after applying a secndary il recvery 2

18 strategy, in mst reservirs, 50-80% f the il remains in the reservir after the aterfld since ater is immiscible ith il (Dullien, 1991) Tertiary Oil Recvery The purpse f tertiary r enhanced il recvery (EOR) is t increase the extractin by changing the il mbility. Examples include: gas injectin, carbn dixide flding, plymer injectin, ht ater r steam injectin, in-situ cmbustin, etc. The gal f thermal methds is t achieve a mre mbile il phase thermally by reducing the il viscsity. In-situ cmbustin intends t crack the heavy il mlecules int a light (mbile) fractins and a heavy fuel fractin by cmbusting the il in-situ ith injected air. During the prcess the cmbustin frnt prpagates thrugh the reservir. The carbn-rich prduct is frmed by the thermal cracking and distillatin f the residual il near the cmbustin frnt hich sustains the in-situ cmbustin (Mahinpey et al., 2007). Immiscible slvent injectin is anther EOR prcess here chemicals can be injected int the reservir in rder t reduce il viscsity and interfacial tensin (IFT) beteen the ater and il. Natural gas, CO 2, and air can be injected immiscibly depending n the reservir cnditins and il and gas cmpsitins (Tuni et al., 2011). Plymer flding is the mst cmmn ay in immiscible injectin ith the gal f increasing the viscsity f the ater phase, decreasing the mbility rati and therefre increasing the seep efficiency (Needham and De, 1987). Gas injectin r miscible tending flding is presently the mst cmmnly used apprach in EOR. The miscible displacement prcess is adpted t maintain the reservir 3

19 pressure and imprve il displacement by lering IFT beteen gas and il. Miscibility can develp beteen injectin gas and il phases depending n their injectin cmpsitin, pressure and temperature (Ra and Lee, 2003). Gases used in the miscible prcess include: methane under high pressures, natural gas enriched ith intermediate hydrcarbn, nitrgen under high pressures, and carbn dixide (CO 2 ) under suitable temperature and pressure cnditins etc. The mst cmmnly used fluid during miscible flding is CO 2 due t the ler cst, available supply and ability t achieve miscibility at ler pressure Current and Invested CO 2 EOR Prject Accrding t a summary dcument recently released by The U.S. Department f Energy there are hundred CO 2 -driven enhanced il recvery prjects peratinal in the United States (Dley et al., 2009). The CO 2 used in the majrity f existing EOR prjects is captured frm natural gelgic depsits. Hever, in recent years a number f large industrial surces have als cntributed t the use f CO 2 in EOR (Peter, 2010). The successful applicatin f CO 2 in EOR ver the last 35 years suggests that ver the next ten years the incremental il prduced by CO 2 injectin and the number f CO 2 fld prjects ill gr steadily (Melzer and Midland, 2012). Figure 1.1 shs the number f prjects f CO 2 in EOR frm 1984 t As indicated in the figure, the number f CO 2 related prjects in EOR has grn frm 1986 t CO 2 is a idely used injectin agent, in either free frm r as a slvent fr bth secndary and tertiary il recvery prcesses. Even under immiscible fld cnditins il 4

20 recvery is increased by changing the physical prperties f the il phase. Pr seep efficiency has been reprted (Patel et al., 1987) due t the high mbility f gas and gravity driven gas smetimes verride leading t premature CO 2 breakthrugh. The impact f pr seep significantly reduces the cntact beteen the CO 2 and il. Carbnated ater injectin in here CO 2 is disslved in ater prir t injectin may help alleviate the l seep efficiency during CO 2 injectin, as the CO 2 partitins t the il phase upn ater-il cntact. The physical prperties f il phase are changed by primarily altering the il phase cmpsitin as a result f mixing ith CO 2. Carbnated Water Injectin (CWI) has three main advantages: 1) the CO 2 disslved in the il phase changes the il viscsity and hence the mbility rati; 2) experimentally the interfacial tensin (IFT) beteen the ater and il phases is reduced (Mungan, 1964) resulting in an imprved verall perfrmance f CWI cmpared t ater injectin (Dng et al., 2011), and 3) significant selling f il as bserved during CWI due t CO 2 disslutin in the il phase. The discnnected il ganglia left behind after cnventinal ater flding may recnnect as il sells. The trapped il then becme rembilized and recvered due t the recnnectin f the il (Mungan, 1981). Carbn dixide slvent flding prcesses ere initially investigated in labratry flding experiments here additinal il as recvered by the carbn dixide slutin drive (Hlm, 1961). Accrding t the results f experimental research (Siregar et al., 1999), an ptimum il recvery can be achieved by maximizing the CO 2 cncentratin in the ater phase. A calculatin methd has been develped fr CWI (Nel, 1964). The slutin revealed the crucial effect f viscsity reductin and il selling using CWI. Cmpared t ater injectin (WI), the 5

21 experimental results shed a better il recvery in bth cases thrugh the t different mechanisms discussed abve (Shrabi et al., 2009a). Figure 1.1 Number f CO 2 EOR Prjects (Melzer and Midland, 2012) 1.3 Scpe f Thesis The varius CO 2 EOR prcesses are revieed in chapter 2. Bth the secndary and tertiary (EOR) il recvery prcesses have been studied regarding the fluid behavir undergrund and the efficiency f il recvery. As the mst cmmnly used gas in EOR, CO 2 has been injected in either free frm r as a slvent int il reservirs. In rder t evaluate the perfrmance f carbnated ater injectin (CWI) numerically, the scpe f this thesis starts ith a simple ater injectin prcess. In Chapter 3 the mathematical mdel f hrizntal t-cmpnent t-phase 6

22 ater flding is frmulated and discretized fr cnstant pressure bundaries. This is the first study here the mdel is develped under cnstant pressure bundary cnditins reflecting the mre realistic perating scenari. As the fundamental study fr a hrizntal three-cmpnent t-phase CWI, this nedimensinal numerical simulatin fr t-cmpnent t-phase WI is validated by the cmparisn f numerical slutin ith analytical slutin. The numerical errrs have been minimized. As a third cmpnent, CO 2 is added t the simple ater injectin in Chapter 4. By cmbining bth mathematical mdels and thermdynamics the perfrmance f CWI is evaluated numerically in this chapter. Since WI is validated in Chapter 3 the CWI mdel here is als partly validated as mathematically it is a reasnably mdest extensin f validated WI mdel. The gal f CWI study is t: 1. Study the effects f il viscsity and interfacial tensin reductin n CWI. 2. Evaluate the perfrmance f CWI by cmparing it ith WI. The cnclusins and further suggestins fr a deeper study are given in Chapter 5. Figure 1.2 shs a summarized cncept map regarding this thesis. 7

23 Water injectin Black il Mass cnservatin IMPES Finite difference methd Analytical slutin f WI fr cnstant Numerical simulatin f WI fr cnstant pressure bundary Cmpare Numerical errrs Carbnated Water injectin Cmpsitinal mdel GA-based mdels Fluid thermaldynamics Linear crrelatin f IFT Cmpare Numerical simulatin f CWI fr cnstant pressure bundary Cnclusin References used in the thesis Accmplishment f the thesis Figure 1.2 Summarized Cncept Map f Thesis 8

24 Chapter 2 Literature Revie f CO 2 Oil Recvery Prcess 2.1 Revie f Buckley-Leverett Thery The fractinal fl thery, as develped by Buckley and Leverett (1942) is revieed here bth in its riginal frm fr cnstant fl rate and its extended frm fr cnstant pressure bundaries (Jhansen and James, 2012). In this study the displacement is nly t cmpnents il and ater, ith negligible capillary pressure and incmpressible fluids. The fractinal fl (f ) functin can be ritten as (Buckley and Leverett, 1942): Equatin Sectin 2 f u u t = k 1 1 ( S ) k ( S ) r r. (2.1) As shn in equatin (2.1), since the fluids are incmpressible and therefre the viscsities are cnstant, it is clear that f is a unique functin f ater saturatin S hich can be expressed as f (S ). The cntinuus equatins fr ater and il can be ritten as: S t S t u x u x 0, (2.2) 0. (2.3) 9

25 The summatin f ater and il saturatin equals t ne, therefre, by adding equatin (2.2) and equatin (2.3) the expressin f ttal fluid velcity is: ut ( ux u ) x x 0. (2.4) The equatin (2.2) then can be ritten as: S u f S ut 0. (2.5) t x t x Due t the cnstant ttal fl rate alng distance hich is shn in equatin (2.4), equatin (2.5) can be reritten as: S t u t f ( S ) x 0. (2.6) The fractinal fl functin, f depends nly n S, therefre equatin (2.6) can be als expressed by: df S S ds x u t. (2.7) t The abve equatin (2.7) is knn as the Buckley-Leverett equatin. The ttal fluid velcity here is treated as cnstant fr any psitin at any time. The frnt velcity (v f ) is given by: v f u df ds t. (2.8) S Sf Clearly, this is directly related t the slpe f tangent t the fractinal fl curve. In a ater injectin prcess ith a cntinuus injectin the ater frnt travels alng the reservir frmatin frm the injecting pint t the utlet bundary and the il phase is partly displaced by the mving ater. In rder t btain the uniqueness in ater 10

26 saturatin values at any lcatin ith prpagatin f time, a shck frnt hich is als called ater frnt is intrduced t indicate an abrupt changing frm frnt ater saturatin t the cnnate ater saturatin shn by Figure 2.1. Figure 2.2 shs a typical fractinal fl curve. As shn frm the figure the fractinal fl curve starts frm zer at irreducible ater saturatin and ends at ne ith the maximum ater saturatin (1-S r ) hich is 70% in this thesis. The determinatin f the frnt velcity must fll bth the entrpy cnditin and velcity cnstraint. Entrpy cnditins state that the shck velcity ( df ds S Sf ) has t df be larger than (r equal t) the dnstream ave velcity ( ds df equal t) the upstream ave velcity ( ds ) but smaller than (r ). Based n the velcity cnstraint the ave velcity shuld decrease mntnically frm dnstream t upstream. Accrding t these cnditins the shck velcity can be determined as shn in Figure 2.2. In Figure 2.2 the straight line is the tangent f fractinal fl curve and the ater saturatin at the tangent pint is crrespnding t the frnt saturatin shn by Figure 2.1. Since, the term df ds S Sf in equatin (2.8) is the slpe f the tangent t the curve at the frnt saturatin, and the cnstant ttal fluid velcity is assumed knn, the frnt velcity can be calculated analytically as depicted in Figure 2.2 fr a cnstant ttal fluid velcity. The average ater saturatin in the reservir ( S ) can be btained graphically by extending the tangent frm the pint f f (S f ) t the upper limit here the y-value is 11

27 equal t ne. The average ater saturatin then can be fund by reading the value n the x-axis at the intersectin pint (Welge, 1952) shn in Figure Water saturatin S Length (m) S f f Figure 2.1 The Typical Water Saturatin Prfile f (S f ) S f ater saturatin S S Figure 2.2 Fractin Fl Curve 12

28 2.1.1 Limitatin f The Buckley-Leverett thery fr a Cnstant Fl Rate The previus discussin f the Buckley-Leverett thery is fr a ne-dimensinal tphase displacement ith a cnstant ttal fluid velcity. Accrding t equatin (2.8), the cnstant ttal fl rate and the unique shape f the fractinal fl curve result in the ater frnt mving ith a cnstant velcity. Hence the ater breakthrugh time can be easily determined by dividing the ttal length f the frmatin by the frnt velcity. Hever, in the fixed pressure bundary cnditins case, Buckley-Leverett thery (under a cnstant fl rate) is n lnger applicable (Jhansen and James, 2012), since the fl rate is nt cnstant (see Appendix B). Due t this limitatin, ithut an analytical slutin the valuable infrmatin f ater injectin under the fixed pressure bundary cnditins can nly be generated by numerical simulatin. Hever, a numerical simulatin cnsumes a lt f cmputatin time and creates numerical errrs that may lead t a misunderstanding f the reservir r result in a physically unrealistic situatin. Mrever, in industry ater is mre likely t be injected at a cnstant pressure t displace il at a fixed bttm-hle pressure. Therefre, fr the sake f accuracy and time efficiency, an analytical slutin is desirable. 13

29 2.1.2 Extensin f the Buckley-Leverett Thery ith Cnstant Pressure Bundaries An analytical slutin has been recently described (Jhansen and James, 2012) hich is an extensin f the classical Buckley-Leverett thery. This analytical methd is applicable fr multi-cmpnent fl under the cnstant pressure bundary cnditins. Using this analytical methd the pressure at any lcatin alng the frmatin can be predicted as ell as the ater breakthrugh time. The ttal fl rate can be calculated at any time befre r after breakthrugh during the displacement. The analytical methd is prvided in Appendix B. In terms f time and accuracy, the analytical extensin f Buckley-Leverett thery fr cnstant pressure bundaries prvides an efficient and reliable slutin fr multicmpnent prblems ith fixed pressure bundary cnditins. In this sectin e fcus n the aterflding prblem. Based n this analytical methd, the displacement time has been divided int t sectins hich are defined as befre and after ater breakthrugh. Fr each sectin, the ttal velcity has different mathematical expressins. Time less than r equal t the breakthrugh time (t tbt) Equatin (2.9) predicts the ttal velcity prir and up t ater breakthrugh. The frnt psitin, breakthrugh time, and pressure at the frnt psitin are described in equatins (2.10) thrugh (2.12). Defining = * S SL f ''( S) ds and letting S* and λ be the frnt ater saturatin and the il ( S) t 14

30 phase mbility, respectively, the ttal fluid velcity is given as: u t B 2 p 4ACt, (2.9) here, A 0.5 f 1, '( Sf ) L B, f '( S f ) C p. The slutin f frnt psitin (x f ) is: x f 2 B B 4ACt. (2.10) 2A The ater breakthrugh time ( t BT ) can be calculated by: t BT 2 AL BL. (2.11) C The equatin f frnt pressure can be calculated as: u p p L x. (2.12) t f ut f The abve calculatins are applied hen the time is less than r equal t the breakthrugh time. After breakthrugh different expressins apply t calculate the ttal fluid velcity and the pressure at the specific ater saturatin. Time after ater breakthrugh (t > tbt) The ater saturatin at any lcatin behind the frnt (alng the frmatin) is larger than the frnt saturatin. Assuming a specific ater saturatin that is larger than frnt saturatin and smaller than injecting ater saturatin, i.e. S inj > S > S f. Let t BT dente the ater breakthrugh time and t s is the time hen this specific S reaches the uter bundary. 15

31 The psitin f S at the breakthrugh time can be calculated by: here, ( t) u ( t) dt. t 0 t f '( S) x( S, tbt )= ( tbt ), (2.13) The time fr ater saturatin S t break thrugh, t s, can then be expressed as: t s t BT (, ) ( ). (2.14) 2 2 pf '( S) 2 L x S tbt S Applying equatin (2.14) the crrespnding value f ttal fluid velcity at t s finally can be calculated as: u S t L x( S, t ) 2 Lf '( S) t t 2 2 BT s BT. (2.15) The summarized prcedure f this calculatin is as flls: 1) Calculate t s fr the ater saturatin S breakthrugh frm equatin (2.14), 2) and btain the ttal velcity by equatin (2.15). Based n the previus discussin, this analytical slutin is applicable t bth befre and after ater breakthrugh, under cnstant pressure bundary cnditins. In Chapter 3 the numerical slutins f WI is cmpared ith this analytical slutins in rder t validate the numerical apprach used in this thesis. 16

32 2.2 Summary f CO 2 EOR Prcess Carbn dixide (CO 2 ) has been used fr injectin in enhanced il recvery (EOR) since the 1970s. Carbn dixide (CO 2 ) can be injected int a reservir under different cnditins. In this sectin the main strategies f CO 2 EOR are discussed CO 2 Gas Injectin In first cntact miscible gas injectin, the injectin gas and reservir il frm t a single phase under a sufficiently high pressure. It is nt pssible t frm first-cntact miscibility at all reservir cnditins r injected gas cmpsitins, thus, multi-cntact miscibility strategies are adpted in gas injectin prcesses depending n reservir cnditins and prperties. Vaprizing gas drive is ne f multi-cntact miscible fld methds. In this prcess the crude il (I 2 ) is lcated in the right hand side f critical tangent, as ppsite t injected gas (J 1 ) as shn in Figure 2.3. Lean injectin gas vaprizes the intermediate cmpnents frm the il phase and creates a miscible transitin zne. The gas frnt mves thrughut the reservir. By cntacting the riginal reservir il, gas is enriched in intermediate cmpnents and eventually the cmpsitin f enriched gas is sufficiently rich in intermediate cmpnents that the enriched gas phase becmes miscible ith the il as shn in Figure 2.4. Cndensing gas drive, the ther mechanism, is hen the crude il (I 1 ) is n the left hand side f critical tangent line and the injected gas (J 2 ) is n the right hand side f this line shn in Figure 2.3. In cndensing gas drive the intermediate cmpnents (C 2 -C 6 ) transfer frm the displacing gas t the il phase. After multiple cntacts, the il and injected gas becmes miscible by enriching the 17

33 transitin il phase ith C 2 -C 6 generating a critical mixture at the displacing frnt shn in Figure 2.5. In cndensing-vaprizing gas drive the miscibility is develped by the cmbined cndensing/vaprizing mechanism. In this cndensing-vaprizing gas drive, gas is initially enriched by vaprizatin. This enriched gas is nt rich enugh t be miscible ith the il but it cntains intermediate cmpnents (C 2 -C 6 ) hich generate the cndensing gas drive mechanism ith the riginal il. The intermediate cmpnents then cndense hen the gas encunters the fresh il hich is very similar t the cndensing gas drive mechanism. Figure 2.3 Develped Miscibility 18

34 Miscible frnt Injected gas Miscible transitin zne Oil Figure 2.4 Vaprizing Gas Drive Miscible frnt Injected gas Miscible transitin zne Oil Figure 2.5 Cndensing Gas Drive CO 2 injectin is a ell-established technlgy fr EOR. The main factr that affects the efficiency f gas injectin (CO 2 ) EOR is the miscibility f CO 2 in the il phase after injectin. Once the pressure exceeds the minimum miscible pressure (MMP), CO 2 is miscible ith the il phase reducing the interfacial tensin t zer, reducing il viscsity and causing il selling. The il trapped in the pre space can therefre be mbilized and fl thrugh the rck, hence enhancing the il recvery (Dullien, 1991). The study f CO 2 injectin in the Dulang field in Malaysia (Zain et al., 2001) at a reservir temperature f 101 and a reservir f pressure MPa indicates that althugh the miscibility cannt be achieved under perating cnditins, additinal il recvery is pssible. Mrever, accrding t their test, CO 2 as capable f extracting the hydrcarbn cmpnents heavier than C 7. 19

35 In multi-cntact miscible CO 2 gas injectin the intermediate and high mlecular eight hydrcarbns are extracted int the CO 2 -rich phase. Under certain cnditins, this CO 2 - rich phase can reach a cmpsitin hich is miscible ith the riginal reservir il. Once this pint is achieved, miscible r near-miscible cnditins are established at the displacement frnt. A miscible r near-miscible CO 2 injectin can result in a cnsiderable il recvery. Hever, l seep efficiency has been reprted due t high mbility f CO 2. Hence, different injectin strategies have been prpsed and used t alleviate this prblem (Riazi, 2011) Water Alternative Gas Injectin Water alternating gas (WAG) injectin as develped t imprve the mbility efficiency f high mbility gas verriding ler mbility reservir fluid. In this prcess ater injectin and gas injectin are cnducted alternately fr perids f time. As the evlutinary step in gas based EOR, WAG has been applied in bth immiscible and miscible gas injectin EOR (Rgers and Grigg, 2000). The injectin gases in a WAG prcess are usually CO 2 mixed ith natural gas liquid (NGL). The perfrmance f WAG is affected by many factrs, such as reservir cnditins, fluid prperties, injectin techniques and WAG parameters hich include the WAG rati and the slug size (Jiang et al., 2012). A number f cre fld experiments revealed that the timing f cyclic injectins has direct impact n WAG perfrmance. An untimely WAG injectin ill lead t l seep efficiency (Nuryaningsih et al., 2010). The ptimal time t inject CO 2 WAG is hen the 20

36 fld frnt passes the middle f the cre (Jiang et al., 2012). In ther rds, CO 2 WAG shuld be injected after half f the il is prduced by secndary ater flding. The evaluatin f CO 2 WAG il recvery in hetergeneus prus media has been dne using a cmpsitinal simulatr. (Ghmian et al., 2008) listed the mst t least influential factrs regarding il recvery as reservir hetergeneity characteristics, cmbinatin f WAG rati and slug size, and the slug size itself. This research has als pinted ut that a higher il recvery is expected under l hetergeneity ith high WAG rati and large CO 2 slug size. Carbn dixide (CO 2 ) WAG prjects have been applied in the il industry fr the interest f additinal il recvery, hever, the pr perfrmance caused by the presence f ater layers beteen islated il ganglia has been bserved. This s-called ater blcking effect prevents the cntact beteen il and CO 2 reducing the CO 2 slutin in il (Lin and Huang, 1990). Due t the pr seep efficiency f direct CO 2 injectin and reductin f il recvery by ater-blcking in WAG, ther alternative CO 2 injectin appraches are being cnsidered Carbnated Water Injectin Cnventinal CO 2 gas injectins such as cntinuus CO 2 gas flding and WAG injectin require a large amunt f CO 2 and may nt achieve the desirable results due t pr seep efficiency and ater-blcking. In carbnated ater injectin (CWI), CO 2 is disslved in the ater r brine prir t injectin. Due t the high slubility in il, CO 2 ill transfer frm the ater t the il phase changing the reservir fluid prperties. 21

37 CWI as first intrduced as an imprved secndary il recvery prcess. The first implementatin f CWI as t enhance ater flding in the K&S prject in Oklahma. Abut 43% mre il as recvered cmpared t the riginal frecast and imprved mbility rati as als reprted (Kechut et al., 2011). Carbnated ater injectin can cntact mre reservir il since CO 2 is disslved int ater befre injectin rather than presenting as a free gas. As an EOR technique, CWI nt nly can reduce the il viscsity and imprve the mbility rati, it can als reduce the IFT beteen the ater and il. The selling effect caused by the disslved CO 2 in the il phase results in a recnnectin f discnnected residual il leading t additinal il recvery (Riazi et al., 2009). 2.3 Literature Revie f Carbnated Water Injectin (CWI) Experimental Investigatins f Carbnated Water Injectin CWI as first cnsidered in early 1950s as a ptential EOR strategy by cnducting crefld experiments. CWI as reprted t reduce the initial il saturatin frm 30% t 22%. In 1959, experimental cre displacement tests ere cnducted t study the impact f CO 2 slvent flding n il recvery (Hlm, 1959). They fund that after slvent flding, carbnated ater flding recvered mre il than using cnventinal ater flding. Carbnated ater flding shed a higher il recvery cmpared t CO 2 slug injectin driven by plain ater. 22

38 High-pressure (pressure range frm 600 t 2500 psig) direct fl micrmdel experiments have been implemented t reveal the mechanisms f CWI frm a pre-scale perspective (Shrabi et al., 2009b). The experimental results shed additinal il recvery in bth heavy and light il via different mechanisms. The experiments revealed that the dminant mechanism f il recvery by CWI in light il is the il selling due t the recnnectin f discnnected il trapped in the pre space, hereas the reductin in the il viscsity as the main cntributr t increased il recvery fr mre viscus il resulting in imprved mbility. Kechut et al. (2011) carried ut cre flding experiments t investigate the il recvery and benefit f CO 2 strage. The experiments ere carried ut under MPa, The il samples used in their tests ere pure n-decane and stck tank il ith viscsity Pa s. The experimental results demnstrated an increased il recvery in CWI under bth secndary and tertiary il recvery prcesses. Mre il can be prduced by CWI cmpared ith WI. In terms f CO 2 strage, the study revealed that mre than 46% f ttal vlume f injected CO 2 as stred after CWI Numerical Mdeling f Carbnated Water Injectin Nel (1964) intrduced a mathematical mdel fr carbnated ater flding based n a Buckley-Leverett linear fl mdel ithut dispersin and cnsidering an incmpressible fluid system. This methd cnsidered the effect f reductin f il viscsity and the il selling due t carbn dixide mixed ith reservir il. The sample calculatins shed that the increased il recvery is mainly due t the decrease f il 23

39 viscsity. Cntrary t previus rk, this calculatin methd as used t discuss a slug injectin f carbnated ater rather than a cntinuus injectin ith a cnstant fluid cmpsitin. A fe cnclusins can be summarized as flls: 1. Oil viscsity reductin due t disslved CO 2 leads t higher il recvery frm carbnated ater flding than pure ater flding. 2. Imprved seep efficiency als cntributes t higher il recvery. 3. The CO 2 slug size and the cncentratin f CO 2 in the ater injected als affect il recvery. (Ramesh and Dixn, 1972) develped a mdel t predict the perfrmance f carbnated ater flding in a hetergeneus il reservir fr a three-phase fluid system here the CO 2 is alled t exist in the gas state. The mathematical mdel as based n the transprt equatin ith t dimensins and a three-phase simultaneus fl system in a hetergeneus reservir. The numerical slutin predicted the transfer f ne phase t the adjacent blcks t sn cmpared ith experimental results. T minimize the errr, an arbitrary cut-ff saturatin as specified at hich the transmissibility f the displacement phase as set t zer. An adequate applicatin f this mdel as validated by experimental results. The abve numerical studies nly incrprate the reservir numerical mdels and ignred the change f fluid prperties such as IFT. In this thesis carbnated ater injectin (CWI) is studied by cmbining bth thermdynamics and reservir simulatin mdels ith the gal t better understand the effects n il recvery frm bth physical and numerical perspectives. 24

40 2.4 Variatin f Fluid Prperties ith Disslved CO 2 Due t the presence f CO 2 in the liquid phase, the prperties f il and ater are variable during carbnated ater flding. The reductin f il viscsity and il selling are majr mechanisms that affect the il recvery in CWI; hence, a better understanding f the change in fluid prperties is essential in CWI. In recent years, the Genetic Algrithm-based (GA-based) technique (Emera and Sarma, 2006) has been used t prvide the crrelatins t predict CO 2 slubility, CO 2 -il sell factr, CO 2 -il density and CO 2 -il viscsity fr bth dead and live ils. The mdels ere develped and tested based n the experimental data (Jarba and Anazi, 2009). Unlike ther crrelatins hich are applicable in a limited range f cnditins, the GA-based crrelatins can be applied ver a ide range f cnditins. These crrelatins have been validated ith published experimental data. In this study, the GA-based crrelatins are used t calculate the il fluid prperties ith disslved CO 2 shn in the next sectins. The disslved CO 2 in the ater phase als results in a variatin f ater prperties. Increasing the amunt f disslved CO 2 in the ater and il phase is expected t reduce the IFT during the CO 2 flding prcess Change in Fluid Viscsity Oil viscsity decreases significantly due t increasing amunt f CO 2 in the il phase. A graphical crrelatin as built t predict the il viscsity after CWI as a functin f pressure and initial il viscsity, (Welker, 1963). Hever, this crrelatin as 25

41 established based n dead il at a temperature f and a pressure f 800 psia thus; failing t be applicable ver a ide range f reservir cnditins. A crrelatin (Lhrenz et al., 1964) as prpsed t calculate the decrease in il viscsity hen gases are disslved in il phases. The calculatin as based n the cmpsitin f the fluid; hence the results ere greatly affected by the fluid density. Since the crrelatin as mainly applied t light il samples; it is nt applicable fr the case f heavy il recvery. Beggs and Rbinsn (1975) develped a CO 2 -il viscsity calculatin methd based n reservir temperature and CO 2 slubility. In this methd, the impact f pressure n il viscsity as neglected. (Chang et al., 1998) fund that the disslved CO 2 in the ater phase has minimal effect n ater viscsity. The ater viscsity as calculated at the reservir cnditins (temperature, pressure, and the salinity f ater). Hever, the CO 2 slubility as neglected (Kestin et al., 1978). A GA-based CO 2 -il viscsity crrelatin (Emera and Sarma, 2006) as develped based n CO 2 slubility, initial il viscsity (Pa s), pressure (MPa), temperature ( ) and il specific gravity under SI unit. This crrelatin is applicable fr bth dead and live il fr pressures in the range frm 0.1 MPa t 34.5 MPa, and temperature in the range frm 21.1 t 140. Cmpared ith ther crrelatins mre accurate results have been bserved by the cmparisn ith experimental data. The crrelatin fr il viscsity used here is the GA-based mdel (Emera and Sarma, 2006): 26

42 x c2 yi + A, (2.16) i here y= X B, p X = Ci 1.8T 32 D xc2 here A, B, C and D are dimensinless cnstants prvided in Table 2.1., Table 2.1 Cefficients used t calculate il viscsity fr bth live and dead ils Oil type A B C D Live Dead Change in Fluid Density The CO 2 -il density fr bth dead and live il can be calculated based n the Genetic Algrithm-based (GA-based) crrelatins (Emera and Sarma, 2006), hich accunt fr the reservir pressure (MPa) and temperature ( ), il specific gravity ( ) and the initial il density ( i, g/cm 3 ) at bubble pint pressure (MPa) under the defined temperature ( ) and a specific il cmpsitin. The density crrelatin in the GA-based mdel (Emera and Sarma, 2006) fr SI unit is: here i y y , (2.17) 27

43 p p 1.25 i b y 1.8T 32 The density f carbnated ater can be updated by applying an existing crrelatin based n the relatin f pressure, vlume, temperature and cncentratin f slvent (Re Jr and Chu, 1970).. In this study, the mdel fr ater density ver a range f pressures and given reservir temperature is generated by PVTsim Calcep simulatr CO 2 Oil Selling The il selling in CO 2 flding is cnsidered as ne f the imprtant factrs directly affecting the recvery f efficiency. In the 1960s, the relatinship beteen selling factr, mle fractin f disslved CO 2 and mlecular size as studied (Simn and Graue, 1965). The GA-based crrelatin fr the il selling factr (SF) as a functin f CO 2 slubilty (Sl, mle fractin), il mlecular eight (MW) and il specific gravity ( ) has been prpsed by Emera and Sarma (2006) : SF y y y y y y , (2.18) here 2 y 1000 Sl MW exp MW. 28

44 Accrding t Figure 2.6, (Jarba and Anazi, 2009) the GA-based crrelatin gives a mre accurate result cmpared t previus rk. Figure 2.6 Cmparisn Results beteen Emera and Sarma (2006) and Simn and Graue (1965) Oil Selling Factr (due t CO 2 ) Crrelatins Predictin Results (Jarba and Anazi, 2009) Oil-Water Interfacial Tensin As mentined previusly, additinal il recvery in carbnated ater flding can be attributed t the reductin f il viscsity, il selling as ell as the decrease f interfacial tensin beteen ater and il hen sufficient CO 2 has been disslved in il phase. Accrding t experimental CO 2 flding results at ambient temperature fr effluent il cuts (Zekri et al., 2007), after injectin f 0.25 pre vlume f CO 2 the maximum drp in 29

45 IFT beteen il and ater as 85%. Figure 2.7 (Zekri et al., 2007) shs the relatinship f IFT beteen crude il and brine as a functin f CO 2 pre vlume injected. The reduced IFT is expected t reduce the residual il saturatin. Hever, based n the previus rk f (Trabzadey, 1984), the residual il saturatin as nt affected under high IFT (IFT > 20 mn/m); but a significant drp f residual il as beserved under l IFT fluid system (IFT < 0.2 mn/m) IFT, dyne/cm Cre#144-C4-4D Pre vlume injected Figure 2.7 Oil-Water Interfacial Tensin as Functin f CO 2 Injected (Zekri et al., 2007) 30

46 Chapter 3 Water Injectin under Cnstant Pressure Bundary Cnditin In any injectin prcess the breakthrugh time is ne f the mst imprtant variables that need t be estimated by reservir engineers. An early breakthrugh time usually indicates pr seep efficiency and an unecnmic recvery strategy. The ater breakthrugh time under cnstant pressure bundaries is, therefre, predicted in this study bth numerically and analytically. The numerical mdel is validated by the cmparing the numerical and analytical slutins. The numerical errrs are minimized in this study. 3.1 Mathematical Mdel The mvement f fluids thrugh prus media in the subsurface is gverned by the cnservatin f mass, and mmentum and energy. The gverning equatins mdel a physical system. The behavir f the hle system is cmplex; hence, the primary task in mdeling is t chse a set f equatins that can accurately describe the cmplex fluid system. The mst idely used equatins in reservir simulatin are built upn the las f cnservatin, hich fr isthermal system cnsist f 1) the mass balance, and 2) the mmentum balance (Darcy s la). Equatin Chapter (Next) Sectin The Cntinuity Equatin Given a cntrl vlume (V) that fluid can fl thrugh, the cnservatin f mass states that the rate f change f mass in V is equal t the mass flux acrss the bundary f V 31

47 plus any mass injected (surce) r remved (sink). The change f mass in this cntrl vlume V, in a unit time can be ritten as: Mass f cmpnent inv dv t. (3.1) Bulk vlume f V V The cnservatin f mass in the vlume V can be described using the Gauss therem: t mass f cmpnent in =- + Bulk vlume f V V dv F dv qdv, (3.2) V V V here F is the mass flux int the medium and q is the mass fl rate per unit vlume injected r prduced ithin V. If e define Mass f cmpnent in V =, (3.3) Bulk vlume f V then, since V is an arbitrary cntrl vlume, the flling equatin hlds at any pint + F q =0. (3.4) t Darcy s La fr a Single Phase Based n Darcy s la, fr a single phase hrizntal fl, the vlumetric fl rate Q thrugh a hrizntal prus medium ith length L and crss sectinal area A can be ritten as: Q = KA p, (3.5) L here K is permeability describing the ability f the rck t transfer the amunt f fluid, μ is the fluid viscsity and p is the pressure difference beteen inlet and utlet. Fr a fl in nly ne directin (x) Darcy s la can als be expressed in the flling 32

48 differential frm: Q K p u A x. (3.6) Applying Darcy s La fr ne dimensinal hrizntal fl, equatin (3.4) becmes: 1 K p q x x. (3.7) t 3.2 One-Dimensinal Hrizntal Water Flding Intrductin The displacement f il by either ater r gas is cmmnly investigated using reservir simulatin. In a ater-il system, e assume that there is n mass transfer beteen the t phases. The vid vlume f the prus medium is ccupied by the t phases (il and ater), i.e. S S 1. (3.8) In a ater-il system, ater is usually the etting phase hich ets the prus medium mre than il. Due t the curvature f the interface beteen ater and il in the micrscpic pres, the pressure in the il ( ( p ). This pressure difference is the capillary pressure, p c : p ) is higher than the pressure in the ater pc p p. (3.9) When ater and il fl simultaneusly thrugh a prus medium, each f the t phases interferes ith the ther. Hence, the permeability f each phase is less than r 33

49 equal t the permeability fr single phase fl. If the permeability fr il and ater are K and K respectively, then the relative permeability f these t phases ( k r, k r ) can be defined as: k k r r K K K K 1, 1. (3.10) The relative permeabilities are unique functins f the phase saturatins The Black Oil Mdel A simplified mdel, the black il mdel, as intrduced fr describing the equilibrium f a hydrcarbn system at temperature ell bel critical temperature. In this mdel the assumptin is made that n mass transfer ccurs beteen ater/gas and ater/il. In a ater-il system, bth cmpnents (il and ater) are defined at the standard cnditin n the surface called stck tank il and stck tank ater, hever, the mass balance f these cmpnents is perfrmed at reservir cnditins. A parameter is intrduced called frmatin vlume factr (FVF) dented as β: here, RC V =, (3.11) ST V RC V is the vlume f a phase under reservir cnditin and the same phase under stck tank cnditin. ST V is the vlume f In a ater-il system, the mass f ater cmpnent per bulk vlume is defined as: ST ST ST RC ST ST Mass f cmpnent in V V V SVbulk S = =, (3.12) Bulk vlume f V V V V ( p ) bulk bulk bulk 34

50 STC STC STC RC STC STC Mass f cmpnent V V V SVbulk S = =, (3.13) Bulk vlume V V V V ( p ) bulk bulk bulk here, S is phase saturatin and is the prsity. The mass influx (F) can be expressed as: ST RC F u= u. (3.14) ( p ) Equatin (3.4) can then be reritten fr ater and il cmpnents: Water: S u q q ST. (3.15) t ( p ) ( p ) Oil: S u q q ST, (3.16) t ( p ) ( p ) here, u u K p Kkr p x x K p Kkr p x x., (3.17) Using equatin (3.8) and equatin (3.9) fr hrizntal ne-dimensinal ater flding case, the gverning system f equatins can be ritten as: 35

51 S u q t ( p) x ( p), S u q t ( p ) x ( p ), u Kkr( S) p, x Kkr( S) p u, x (3.18) S S 1, p p p c. The system f equatins can be simplified by assuming that there is n surce r sink terms in the 1-D reservir and that capillary pressure can be neglected ( p p p ). The system f equatins (3.18) then becmes: S u 0, t ( p) x ( p) S u 0, t ( p) x ( p) u Kkr( S) p ( ), x (3.19) u Kkr( S) p x, S S 1. 36

52 In the abve system f equatins there are five unknns ( S, S, u, u, p ) and five equatins. Therefre, by applying certain bundary cnditins this system can be slved at any lcatin at any pint in time Numerical Mdel In this sectin, the numerical slutin fr ne-dimensinal hrizntal ater injectin is develped. Defining ater mbility as: Kk ( S ), (3.20) r = the ater cntinuity equatin becmes: S p. (3.21) t ( p) x x After expansin the equatin can be rerganized as: p S p S cpv c = t t x x, (3.22) here, 1 d c and dp c pv 1 d are fluid and rck cmpressibility, respectively. dp By the same prcedure, the il phase cntinuity equatin can be btained: p S p S cpv c = t t x x. (3.23) As as previusly discussed, the summatin f il and ater saturatin is equal t ne, hence by adding equatin (3.22) and equatin (3.23) the il saturatin S cancels ut: 37

53 p p p cpv Sc c Sc t x x x x. (3.24) Implicit pressure and Explicit Saturatin (IMPES) Equatin (3.24) indicates that a system f t-phase immiscible fl thrugh 1D prus media is gverned by a nnlinear time-dependent partial differential equatin. T types f discretizatin numerical schemes can be applied t slve this type f equatin, the fully implicit and the implicit-explicit. In this case, the Implicit Pressure Explicit Saturatin (IMPES) apprach has been adpted. The IMPES methd as intrduced t slve a partial differential cupled system fr t-phase fl in a prus medium (Sheldn et al., 1959; Stne and Garder Jr, 1961). The main idea f this classical methd is t separate the cmputatin f pressure frm that f saturatin. Using this methd, the cupled system is split int t separate equatins fr pressure and saturatin. The saturatin and pressure equatins are slved using explicit and implicit time apprximatin appraches, respectively. This methd can be set up easily and efficiently implemented fr t-phase immiscible fl. Hever, the IMPES methd is cnditinally stable and cnverges if and nly if the time step is selected carefully accrding t the classical Curant-Friedrichs-Ley cnditin (CFL) (Curant et al., 1928). In the current case, the CFL cnditin expresses that the time step must ensure that change f mass in each cntrl vlume is less than the mass f ne pre vlume f the cell per time step. In ther rds, L t, (3.25) u t 38

54 here L is the length f the grid blck. Discretizatin f the fl equatins The blck-centered finite difference methd ith cnstant size f grid blcks is used t slve the numerical prblem in this rk. Figure 3.1 indicates three blcks (x i 1,x i and x i+1), each ith a cnstant length Δx. The value ( y i ) in psitin x i can be apprximated by: y y (3.26) 2 i1/2 i1/2 yi Figure 3.1 Blck-Centered Finite Difference Mdel Fr a general differential equatin as: p p, (3.27) x x t the discretizatin can be expressed as: p 1 p p x x x x x i1/2 i1/2 i i1/2 i1/2 1 p p p p x x x i1 i i i1 i1/2 i1/2 i1/2 i1/2. (3.28) Applying IMPES, equatin (3.28) becmes: 39

55 n1 n1 p 1 n p n p = i1/2 i1/2 x x x x x i1/2 i1/2 1 p p p p n x x x n1 n1 n1 n1 i1 i n i i1 i1/2 i1/2 i1/2 i1/2. (3.29) n In the abve equatin nly pressure is slved implicitly and is the cefficient frm the last time step n, i.e. t n t. Therefre, the scheme fr slving a general partial differential equatin (refer t equatin (3.27)) can be ritten as: 1 p p p p p p = i1/2 i1/2 x xi 1/2 x i1/2 t n1 n1 n1 n1 n1 n n i1 i n i i1 i i. (3.30) Pressure distributin in a ater-il system Based n equatin (3.28), the ater term n the right hand side f the equatin (3.24) becmes: n n1 n1 n1 n1 p p p p p i n i1 i n i i1 i1/2. (3.31) i1/2 x x x xi 1/2 x i1/2 Leting i1/2, (3.32) i1/2 x i 1/2 and i1/2, (3.33) i1/2 x i 1/2 and defining, A i, (3.34) i i 1/2 x i 40

56 B i, (3.35) i i 1/2 x i C A B, (3.36) i i i equatin (3.31) can be reritten as: p A p C p B p x x n n1 n n1 n n1 i i1 i i i i1. (3.37) Repeating the same prcedure fr the il term n the right hand side f equatin (3.24): p A p C p B p x x n n1 n n1 n n1 i i1 i i i i1 The expansin f the left hand side f equatin (3.24) can be btained by:. (3.38) p t cpv Sc c Sc cpv Sc c Sc n i p n 1 i p t n i. (3.39) Leting cpv Sc n c Sc Ei = t n i, (3.40) equatin (3.39) becmes: p t n 1 n c n pv Sc c Sc Ei pi pi. (3.41) Substituting equatins (3.37), (3.38) and (3.41) int equatin (3.24), the final expressin cnsisting f bth ater and il appears as: n n1 n n1 n n1 n n i i1 i i i i1 i i A A p C C E p B B p E p. (3.42) The abve linear equatin can be slved fr n 1 p under certain bundary and initial reservir cnditins. Based n the blck centered methd, the cefficients shn in 41

57 equatin (3.42) are apprximated by the value in the center f the grid blck and the flux is apprximated n the edge f each grid blck Saturatin Prfile Once the pressure difference is slved implicitly, the saturatin value can then be updated explicitly at each pint in the medium. By applying the explicit finite difference apprach equatin (3.22) is arranged as: n 1 n n S n i Si i n p p S i cpv c i t x x t. (3.43) Applying the pressure distributin that as slved implicitly, equatin (3.43) becmes: S n1 n1 i n1 n1 p 1 p p p 1. (3.44) n n1 n S cpv c pi p i i i 1/2 1/2 n i i i i n 1 n i t xi xi i S i The ater saturatin prfile can then be updated at time step n 1 frm time step n and pressure n 1 p i. 3.3 Numerical Simulatin and Analysis The numerical mdel as develped and executed in Matlab. The numerical mdel is easily mdified fr different reservir prperties, cnditins r extra calculatins. The simulatin prcedures cnsist f fur majr steps: 1. Cmputatin f the pressure distributin 2. Updating f the saturatin prfile 3. Calculatin f the ttal fluid velcity 4. Generatin f the main graphics (figures) 42

58 The fluid prperties are specified prir t simulatin; hence the pressure distributin is a unique functin f time and psitin under the given bundary cnditins. The saturatin prfile and ttal fluid velcity can then be calculated based n the distributin f pressure at any time and psitin. Graphical representatins f the results are presented in the last step fr better bservatin and analysis Data Preparatin The aim f the simulatin study is t better understand the fluid behavir in the reservir. The data selectin is imprtant fr a realistic cmprehensin f fluid perfrmance in the reservir. The three prperty sets that need t be defined are the fluid, reservir, and rck prperties plus the reservir bundary cnditins Fluid Prperties The initial fluid prperties can be selected frm varius resurces. The reservir fluid prperties are functins f pressure, fluid cmpsitin and temperature; hence the fluid prperties ill vary accrding t the change f the fluid itself and the reservir cnditins. The simulatins in this study are based n the black il mdel and the reservir temperature is cnstant, therefre, the fluid prperties are cnsidered t be functins nly f pressure. The pressure distributin is functin f time; hence the fluid prperties need t be recalculated and updated at each cmputatinal time step during ater flding. Frmatin vlume factr ( ) The vlume f bth ater and il in the reservir is inversely prprtinal t pressure ( p -1 ). In many simulatin studies, a linear crrelatin is adpted as a simplificatin 43

59 t estimate the cefficients that are used during the cmputatin. In the case study e assume the fluid is incmpressible and is assumed t be 1. Fluid cmpressibility ( c, c ) The cmpressibility cefficient f il ( c ) and ater ( c ) is defined as the abslute rati beteen the amunt f vlume change per unit change in pressure and initial vlume. In general, the vlume f single phase petrleum fluid decreases ith increasing pressure under cnstant reservir temperature. Since the fluid is assumed incmpressible the fluid cmpressibility equals t zer in this study. Viscsity (, ) Pressure has been shn t have an insignificant effect n the viscsity f a liquid, except under extremely high pressure cnditins. In this study the impact f pressure n fluid viscsity is neglected Reservir Prperties Reservir permeability (K) Permeability is a parameter hich quantifies the ability f rck t pass and receive fling fluid. In the petrleum industry, permeability (K) is measured in Darcy. Hever, t maintain the cnsistency during the calculatin all the units are cnverted int SI unit. Relative permeability (k r, k r ) Crey s mdel is intrduced fr calculating the relative permeability in this rk. The expressins fr ater and il in this study are chsen as: 44

60 k r 2 S S c 0.2, (3.45) 1 Sc Sr k r 2 1 S S r 0.8, (3.46) 1 Sc Sr here, S c is reservir irreducible ater saturatin and S r is the residual il saturatin. The typical relative permeability curves are shn by Figure 3.2. Relative Permeability, k r S c kr kr 1-S r Water Saturatin - S Figure 3.2 Typical relative permeability curves f ater and il Prsity (ϕ) Prsity is defined as the fractin f vid vlume f ver the ttal bulk vlume. In this study a cnstant prsity, =0.18, is used. Initial frmatin ater and il saturatin (S 0, S 0 ) Initial frmatin saturatin has t be defined in rder t start the calculatin. The initial ater saturatin is 0.25 in this study. 45

61 Pre vlume cmpressibility (c pv ) Due t the prsity the pre space in the frmatin tends t change under different pressure cnditins. The cnstant prsity is adpted in this study i.e. effect n change f pre vlume is neglected, i.e. cpv Bundary Cnditins In rder t slve the pressure equatin shn in equatin (3.42) the specified bundary cnditins have t be applied. Fr a secndary il recvery prcess the ater is injected t drive the il t the surface. The injectin pressure (p in ), 21 MPa, ill be fixed as ell as the utlet pressure (p ut ), 17 MPa. Since mst f vid space is ccupied by the ater phase at the pint f injectin the injectin ater saturatin (S in ) is equal t 1 S r hich is 0.7 in this case Cmputatin f Finite Difference Equatin Pressure Distributin Having defined the reservir prperties, fluid prperties and bundary cnditins; the pressure distributin can be btained by slving the finite difference equatin (3.42): n n1 n n1 n n1 n n i i1 i i i i1 i i A A p C C E p B B p E p. (3.42) Fr a system ith M grid blcks, M equatins can be develped crrespnding t each blck hich are shn bel: 46

62 + n n1 n n1 n n1 n n 1 in B B p C C E p A A p E p + n n1 n n1 n n1 n n B B p C C E p A A p E p + n n1 n n1 n n1 n n B B p C C E p A A p E p... n n 1 n n 1 n n 1 n n m m1 m m m ut m m B B p C C E p + A A p E p. (3.47) The abve system f equatins (3.47) can be rganized int matrix frm as: n Epi n n n n n1 C C E A 1 A Ep B 1 1 B p 1 in n1 p i n n n n n 1 B B C C E Ep A m m m A p m ut (3.48). In this equatin, the cnstant matrix cnsists f three diagnals. The main diagnal is sitting n the diagnal f the matrix adjacent t an upper diagnal and a ler diagnal. The rest f the elements f this matrix are zer. The pressure in each grid blck can be easily calculated by cmputing the matrix slutin in Matlab Saturatin Prfile and Ttal Fluid Velcity Saturatin prfile As discussed previusly, equatin (3.44), the saturatin prfile alng the cell, can be updated by applying the values f p n+1 that ere slved frm pressure prfile (linear system in equatin (3.48)). Ttal fluid velcity In rder t assess the il prductin frm the ecnmic perspective the estimatin f ater breakthrugh time becmes crucial in the prcess. T achieve a better understanding f ater breakthrugh, the ttal fluid velcity is necessary. This value is 47

63 calculated as flls: The ttal fluid velcity can be ritten as: Applying Darcy s La, equatin (3.49) becmes: u ut u u, (3.49) Kk Kk p p r r i i1 t, i1/2 ( ) ( ) x. (3.50) The saturatin distributin is used t determine the relative permeability calculated by the Crey s mdel (equatins (3.45) and (3.46)) in equatin (3.50). Therefre, nce saturatin and pressure are slved the ttal fluid velcity can be easily calculated alng the distance fr each time level. The prgramming prcedures can be summarized by Figure 3.3: 48

64 Time Lp Data Calculatin f parameters Frmulating the finite differential equatin fr slving the next time step pressure Slve the three diagnals matrix system Calculate next time step saturatin based n ne pressure distributin Calculate u t Figure 3.3 Summarized by The Fl Chart fr Slving One-Dimensin Water Flding 49

65 3.4 Case Study under the Cnstant Pressure Bundary Cnditins In this sectin e cmpare the results f t cases develped fr ater flding including breakthrugh time, saturatin prfile, and pressure distributin. The numerical slutins ill be presented graphically and discussed frm a physical pint f vie. The cnditins used t generate the numerical simulatin are frm the text bk PVT and Phase Behaviur f Petrleum Reservir Fluids by Danesh (1998) Assumptins Assumptins f the mdel include: 1) The mdel is fr ne-dimensin hrizntal fl. 2) Capillary pressure is negligible. 3) The fluids are incmpressible. 4) N mass transfer ccurres beteen the ater and il phases. 5) Viscsities f bth liquids are assumed cnstant and independent f pressure. 6) Reservir frmatin ill nt be defrmed during the depletin, cpv Input Parameters Water is cntinuusly injected int the reservir at cnstant injecting pressure (21 MPa) during ater flding. The utlet pressure is fixed t 17 MPa hich is equal t the initial reservir pressure. At the injectin pint, the frmatin rck is saturated by 70% ater and 30% residual il. Initially, the reservir ater saturatin is 25% and the il saturatin is 75%. The reservir frmatin is 100 meters lng ith a prsity f 0.18 and a 50

66 permeability f 1 Darcy. All values are shn in Table Case Studies 1) In the first case il is the least mbile phase cmpared ith ater. The viscsity f il is 20 cp hich is 20 times larger than ater, 1cP. 2) In the secnd case ater is assumed t be much mre viscus than il, having a viscsity f 20 cp ith the il viscsity is equal t 1cP. Table 3.1 Parameters used in case study Data Unit Case 1 Case 2 μ Pa s μ Pa s μ / μ L m 100 ϕ P in Pa P res Pa P ut Pa S r S c K m β - 1 β - 1 k r - S Sc ( ) 1 S S k r - c r 1 S Sr 0.8 ( ) 1 S S c r 2 51

67 3.4.3 Results Saturatin Prfile and Pressure Distributin Under cntinuus injectin, the ater frnt travels alng the reservir frmatin frm the injectin pint t the utlet bundary displacing the il phase. The ater breakthrugh time is calculated numerically fr case 1 (μ / μ = 20) and case 2 ((μ / μ = 0.05) and is fund t be 27 and 120 days, respectively. Figure 3.4 shs the ater frnt prpagatin crrespnding t fur different times (0.25, 0.5, 0.75, 1f breakthrugh time) fr the t cases. As illustrated by Figure 3.4 the frnt is lcated at different psitins alng the frmatin at different times. As shn in Figure 3.5, the shape f fractinal fl curve behaves differently accrding t different fluid systems. The values f ater saturatin at the frnt frm Figure 3.4 are apprximately 0.43 and 0.68 fr case 1 (μ / μ = 20) and case 2 ((μ / μ = 0.05), respectively and this can be als bserved frm Figure 3.5. Under cnstant pressure injectin the reservir pressure ill increasing ith time. The distributin f pressure, therefre, is different at each time step. Figure 3.6 shs the pressure prfiles at 0.25 and 0.75 f breakthrugh time in bth cases. The higher injectin pressure inlet is respnsible fr the fluid fl and the pressure prfile mves frm left t right (injectin t prductin) ith time. Since discntinuity appears in shck frnt, the pressure des nt vary smthly alng the reservir frmatin. As shn by Figure 3.6, the break pint has been bserved in the pressure prfile. The shape f pressure prfile changes at the same lcatin pint as the il-ater shck as illustrated in Figure

68 0.25 BT 0.5 BT 0.75 BT 1 BT Water saturatin Length (m) ater saturatin Length (m) 3.4a (case 1 μ > μ ) 3.4b (case 2 μ < μ ) Figure 3.4 Saturatin Prfile at Water Breakthrugh Time 0.25, 0.5, 0.75, 1 f f S f 0.20 S f Water Saturatin Water Saturatin 3.5a (case 1 μ > μ ) 3.5b (case 2 μ < μ ) Figure 3.5 Fractinal fl curve f case 1 (μ > μ ) and case 2 (μ < μ ) 53

69 Pressure (MPa) Length (m) 3.6a (case 1 μ > μ ) Pressure (MPa) BT 0.75 BT Length (m) 3.6b (case 2 μ < μ ) Figure 3.6 Pressure Distributin at Water Breakthrugh Time 0.25,

70 Water saturatin ater saturatin Length (m) pressure distributin Pressure (MPa) Figure 3.7 Pressure Distributin vs. Water Saturatin at 0.25 Breakthrugh Time in Case 2 (μ < μ ) Cmparisn f saturatin prfiles As shn in the figures abve the saturatin prfiles in the t cases sh the same trend. The shck frnt is mving frm the injectr t the prducer and eventually ater breaks thrugh. The average reservir pressure keeps increasing ith time. The break pint is bserved in pressure prfile at the lcatin here the ater frnt is passing thrugh. The differences are shn by Figure 3.8. As shn, the frnt saturatin is different in case 1(μ / μ = 20) and case 2 (μ / μ = 0.05). The frnt saturatin keeps cnstant arund 1-S r in case 2 (μ / μ = 0.05) hile in case 1 (μ / μ = 20) it is nly arund 0.43 resulting cntinuusly increasing ater saturatin after ater breakthrugh. Due t the different mbility rati ater breaks thrugh at different times. When the ater is less dense cmpared ith il, ater travels faster than il leading t an early breakthrugh 55

71 time. This can be prved by the cmparisn beteen these t cases. In case 1, the ater viscsity is 20 times less than the il viscsity, hever, in the secnd case ater viscsity is 20 times larger than il viscsity. The crrespnding breakthrugh time f case 1 and case 2 is 27 days, and 120 days, respectively. 0.8 Water saturatin case 1 case Length (m) Figure 3.8 Water Saturatin Prfile at 0.5 Breakthrugh Time f Each Case Ttal Fluid Velcity Recall the equatin (2.4) ut ( ux u ) 0. (2.4) x x Equatin (2.4) indicates a cnstant ttal fl rate alng the length. Althugh the fl rate remains cnstant alng the distance it ill change ith time. Cmparisn f velcity distributin 56

72 The velcity distributin varies ith time. The results f the simulatin sh that the ttal fluid velcity increases during the injectin hen the ater viscsity is less than il viscsity (μ > μ ), i.e. case 1. The frnt saturatin f case 1 is less than the maximum ater saturatin f 70%; hence after breakthrugh the ater saturatin cntinues t increase as des the ttal velcity. In the case hen il is less viscus than ater (i.e. case 2, μ < μ ), the ttal fluid velcity deceased then remained almst cnstant. The frnt saturatin f this case is arund 70% therefre after breakthrugh the ttal velcity remains cnstant. Figure 3.9 shs that the ttal fluid velcity in case 1 (μ / μ = 20) gradually increases frm m/s t m/s ith time, in the meanhile in case 2 (μ / μ = 0.05) the ttal fluid velcity starts ith m/s then drps t m/s and after that stays almst cnstant during ater flding. Ttal fluid velcity 10 6 (m/s) Time (day) BT 3.8a (case 1 μ > μ ) 57

73 Ttal fluid velcity 10 6 (m/s) Time (day) 3.8b (case 2 μ < μ ) Figure 3.9 Ttal Fluid Velcity The time dependent ttal velcity can be als bserved by defining F as: F ut f, (3.51) the change f F ith ater saturatin in case 1 (μ > μ ) at different times (0.25BT, 0.5BT, 0.75BT and BT) are shn in Figure In this figure the graph f F shifts at different time steps. Since the fractinal fl functin f is a functin nly f ater saturatin, F changes ith the varying ttal fluid velcity at different times. The ttal fluid velcity is nt cnstant during ater flding hen the pressure bundary cnditins are kept cnstant. In fact, it is a functin f time nly. 58

74 u t f Water saturatin Figure 3.10 F vs. Water Saturatin Since the velcity des nt remain cnstant, the shck frnt velcity is varying ith time. The breakthrugh time cannt be calculated by simply using length divided by frnt velcity. The Buckley-Leverett thery fr a cnstant fl rate therefre fails t determine ater breakthrugh time under fixed cnstant pressure bundaries. 3.5 Cmparisn beteen Numerical and Analytical Slutins In Sectin 2.1.2, the analytical slutin (Jhansen and James, 2012) as intrduced as an extensin f the Buckley-Leverett thery t cnstant pressure bundary cnditins. This methd is applicable fr varying ttal fluid velcity befre and after ater breakthrugh. Imprtant infrmatin, such as the ttal velcity, the pressure distributin and ater breakthrugh time, can be calculated frm the equatins prvided by this analytical methd. The results f bth the numerical slutin and the analytical slutin ill be 59

75 cmpared in this sectin. The numerical slutin as cmpleted using t different cases in Sectin 3.4. It is necessary t test the validity f the numerical slutin by applying the analytical methd under the same cnditins. In this sectin, the cmparisn beteen the numerical and the analytical slutins is illustrated and discussed Numerical and Analytical Cmparisn Fr the purpse f cnvenience, the ttal fluid velcity values calculated by bth the numerical and the analytical methds are presented in the same figure. The t case cmparisns are shn and the results sh a gd agreement beteen the numerical and analytical slutins. As shn in Figure 3.11, the numerical slutin agrees ell ith the analytical slutin befre and after ater breakthrugh fr bth cases. The numerical and analytical slutins in case 2 are almst identical (Figure 3.11b). The ater breakthrugh time in each case, generated by numerical simulatin is slightly earlier cmpared t the analytical slutin. The breakthrugh time fr the first case (μ > μ ) is calculated as 28.5 days by analytical slutin cmpared t 27 days generated frm the numerical slutin. In the secnd case (μ < μ ) the analytical breakthrugh time is days, apprximately 3 days later cmpared ith 120 days calculated by the numerical methd. This is primarily due t the numerical smearing effect n the ater displacement frnt hich makes a unique definitin f breakthrugh time impssible. The cmparisn f the numerical and analytical rk validates the finite-difference methd presented in previus sectins. Figure 3.11a shs a break pint in the analytical slutin hen the ater frnt appraches the utlet (time is clse t the breakthrugh time) due t rund-ff errr. By substituting equatin (2.10) int equatin (2.9), the ttal 60

76 fluid velcity befre breakthrugh time can be reritten as: u t p. (3.52) x f x f L f '( S ) f Equatin (3.52) indicates that hen the ater frnt appraches the utlet the frnt psitin x f is infinitely clse t the ttal length L causing the rund-ff numerical errr during the calculatin. The errr can be minimized by using ttal length (L) instead f actual frnt psitin ( x f ) hen the distance beteen ater frnt and ttal length is less than ne meter. The result after minimizing rund-ff errr is shn in Figure As shn, the substitutin f the ttal length reduces the rund-ff errr. Ttal fluid velcity 10^6 (m/s) analytical slutin numerical slutin Time (day) 3.10a (case 1 μ >μ ) 61

77 Ttal fluid velcity 10^6 (m/s) numerical slutin analytical slutin Time (day) 3.10b (case 2 μ <μ ) Figure 3.11 Cmparisn beteen Numerical and Analytical Slutins Ttal fluid velcity 10^6 (m/s) numerical slutin analytical slutin Time (day) Figure 3.12 Cmparisn beteen Numerical and Analytical Slutins f Case 1 after Minimizing Rund-ff Errr Discussin f Numerical Errrs The stability f the numerical methd is imprtant as it directly affects the accuracy f 62

78 the numerical slutins. If cmputatinal errrs are intrduced in any step, they may be amplified during the cmputing prcess and btaining reasnable infrmatin frm the simulatin becmes impssible. Therefre, the selectin f an apprpriate frmulatin fr the reservir simulatin becmes a key factr. In the current study, a single pint upstream eighting methd has been applied t btain the mbility beteen t blcks. It is ell knn that single pint upstream apprach is a first rder apprximatin in hich mbility beteen t blcks is assumed t be equal t the upstream blck s mbility, and als that the apprpriate stability criterin fr this scheme is the CFL cnditin discussed in Sectin equatin (3.25). In the numerical simulatin, the slutins are btained by applying the upstream methd in hich the average fluid mbility is calculated frm the ater saturatin frm the previus grid blck. Hever, if the dnstream ater saturatin is applied, the ater ill nt be able t fl. The upstream methd is cnditinally stable hile the dnstream is uncnditinally unstable. The accuracy f simulatin results als depends n the number f grid blcks. A better agreement beteen the numerical and analytical slutins can be achieved by refining the number f grid blcks during simulatin. Figure 3.13 shs the ater saturatin prfiles at different numbers f grid blcks cmpared t analytical slutin hich is calculated by using equatin (2.10). As indicated by this figure, by refining grids frm 100 t 1000 the numerical dispersin is alleviated by reducing the truncatin errr (first rder in bth time and space). 63

79 Water saturatin prfile 0.80 Water saturatin grids=1000 grids=100 analytical slutin Length (m) Figure 3.13 Water Saturatin Prfile fr Case 1 (μ >μ ) under Different Numbers f Grid Blcks 3.6 Summary In this chapter the simple ater injectin ith cnstant pressure bundary cnditins has been studied numerically using IMEPS methd. The numerical slutin has been validated by the great agreement beteen numerical and analytical slutins. Therefre, the numerical IMPES scheme is adpted fr study f carbnated ater injectin (CWI) hich is discussed in Chapter 4. 64

80 Chapter 4 Carbnated Water Injectin 4.1 Intrductin The simple ater injectin, discussed in chapter 3, is a secndary il recvery strategy fcused n il displacement and the pressure maintenance. After ater injectin, tertiary il recvery (EOR) can be applied t increase incremental il recvery. Flling n chapter 3, here e ill develp the mathematical mdel fr injecting CO 2 in carbnated ater injectin (CWI) under cnstant pressure bundaries t predict il recvery, saturatin distributin and the pressure distributin frm injectr t prducer. Althugh much theretical rk has been dne regarding CO 2 injectin, there is limited rk fcusing n CWI. In this chapter a mathematical mdel is develped based n mass cnservatin t study the perfrmance f CWI. The effects f viscsity and interfacial tensin are cnsidered in this mdel, as imprtant factrs in enhanced il recvery (EOR). Hever, the il selling effect is nt included in the mdel. Since the WI mdel has been validated by cmparing the analytical and numerical slutins, a mdest extensin frm ater injectin t carbnated ater injectin is deemed valid. The numerical apprach used in Chapter 3, therefre, als is used here. 4.2 Mathematical Mdel The mathematical mdel is based upn the mass cnservatin equatins that describe ater-il t-phase simultaneus fl. The free CO 2 gas phase is nt present, and the slubility f CO 2 under high reservir pressure is ignred, meaning that physically the 65

81 CO 2 nly exists as a cmpnent f the liquid ater r liquid il phases. Differing frm the ater injectin that is cnsidered as a t-cmpnent t-phase fluid system, the CWI prcess is a three-cmpnent, t-phase prblem. The black il mdel is nt applicable in this case since the mass f each phase (ater r il) is nt cnserved due t the transfer f CO 2 beteen the ater and il phases. In this sectin, a ne-dimensinal cmpsitinal mdel is develped t study the reservir behavir and the perfrmance f CWI. The flling assumptins are made in this numerical simulatin: 1. Fl is ne dimensinal hrizntal, i.e., the effect f gravity is neglected. 2. There is n surce r sink term beteen injectr and prducer. 3. The capillary pressure effect is negligible. 4. The reservir pressure is sufficiently high that n free CO 2 r hydrcarbn gases exist in the reservir. 5. Water and il fl simultaneusly and CO 2 is present in slutin in these t phases. 6. CO 2 transfers frm the ater t the il phase but there is n mass transfer f il and ater cmpnents beteen phases. 7. The diffusin f CO 2 is ignred. The advance f CO 2 is nly due t the mvement f il and ater. 8. In an il-co 2 -ater fluid system, the equilibrium beteen il and ater saturated ith CO 2 is reached instantaneusly. The maximum slutin f CO 2 in bth il and ater phases is btained at any pint in time. 9. The frmatin prperties such as prsity and permeability are cnstant during 66

82 the CWI prcess. The cmpsitinal mdel is based n the mass cnversatin fr each cmpnent ater, il and CO 2, respectively: RC RC S 1 c c 1, 2 u c c2 t x RC RC S 1 cc 1, 2 u c c2 t x RC RC RC RC Scc +. 2 Scc 2 ucc 2 ucc 2 t x (3.53) Since the densities f the ater and il phases vary ith pressure and CO 2 slubility at different lcatins and times, fr simplicity, the stck tank cnditin is chsen t be the reference state in hich the densities f the ater and il phases are cnstant. Therefre, the abve system f equatins (3.53) can be reritten as: u S * 1 cc 2 1 c, * c2 t x u S * 1 cc 1, 2 c * c2 t x ST ST u ST u ST S * cc 2 S * cc 2 c * c 2 c * c 2 t x. (3.54) here, * and * is the density rati beteen stck tank cnditins and reservir cnditins fr the ater and il, respectively. The saturatin and CO 2 mass cncentratin equatins are incrprated t slve the numerical mdel: S S 1, (3.55) c c, (3.56) c2 c2 67

83 here, is the partitin cefficient f CO 2 beteen the ater and il phases. Equatins (3.54), (3.55) and (3.56) describe the three-cmpnent t-phase fl, hich cntains five equatins ith five unknns. The five unknns are the saturatin f il and ater ( S, S ), CO 2 mass cncentratins in il and ater ( c, c c2 c2 ) and pressure (p). 4.3 Partitining f CO 2 in a Three-cmpnent T-phase System The slubility f CO 2 in il is t t ten times greater than its slubility in ater in mst cases. During carbnated ater flding, CO 2 transfers frm the ater t the il resulting in a change in the il s prperties. An il viscsity reductin ith increasing CO 2 slubility in il phase as bserved by many experimental researchers (Enick and Klara, 1990). Accrding t the results f the experiments, the il selling factr caused by disslved CO 2 cntributes t the additinal il recvery, especially in nn-heavy il reservirs. Mrever, the residual il saturatin decreases ith sufficiently l IFT due t the high CO 2 cncentratin in the il phase. Therefre, determining the amunt f CO 2 in the il phase becmes a key step fr carbnated ater flding simulatin. Hever, due t the lack f infrmatin regarding the CO 2 distributin beteen il and ater after injectin, a simple scheme is applied. In this calculatin ater and il are assumed t be saturated by CO 2 simultaneusly. In three-phase fl, hen the system reaches equilibrium, the CO 2 mle fractins in gas, ater and il satisfy the relatin: k x k x y, (3.57) c2 c2 c2 c2 c2 68

84 here k c2 and k c2 are equilibrium cefficients regarding CO 2 mle fractin; x c2 and x c 2 are CO 2 mle fractin in ater and il phases and phase. yc 2 is the CO 2 mle fractin in the gas In this study t-phase fl (ater and il) is assumed. Therefre, the gas phase is n lnger present in the system. The ne relatinship f CO 2 mle fractins in liquid phase is: k x k x. (3.58) c2 c2 c2 c2 The cmpsitinal mdel is based n the mass cnservatin; hence the mle fractin needs t be cnverted t mass cncentratin during the numerical calculatin. By cnverting t mass cncentratin the relatinship f CO 2 mass cncentratin in liquid phase becmes: c2 c c 2 2 c2 k c k c, (3.59) here, k c and 2 k c are equilibrium cefficients regarding CO 2 2 mass cncentratin in separated systems (ater-co 2 and il-co 2 ). Equatin (3.59) can be als ritten as: k c2 c2 c 2 c2 k c2 c c c. (3.60) The partitin cefficient ( ) f CO 2 in three-cmpnent t-phase fluid system is defined as the rati beteen mass equilibrium cefficients: c2 c2 k. (3.61) k 69

85 In the lack f infrmatin as h CO 2 partitins beteen ater and il hen CO 2 is insufficient t saturate bth phases, a simple scheme is adpted. Since in this study e assume (1) the equilibrium beteen il and ater saturated ith CO 2 is reached instantaneusly, and (2) there is n interacting beteen ater and il cmpnents, the CO 2 slubility in ater and il as the functin f pressure can be calculated frm separate systems f ater-co 2 and il-co 2 (Ramesh and Dixn, 1972) CO 2 Slubility As discussed in previus sectin the partitin cefficient is calculated frm the mass equilibrium cefficients f CO 2 in ater phase ( k c ) and CO 2 2 in il phase ( k c ). T 2 determine these t cefficients t fluid systems, ater-co 2 and il-co 2, are applied. In each system the CO 2 is present as bth slvent in liquid phase and gas in vapur phase. The mass cncentratin f CO 2 in gas phase is assumed t be ne CO 2 Slubility in Water The cmpsitinal mdels used t simulate the enhanced il recvery prcesses usually neglect the slubility f hydrcarbn in ater. Hever, as an exceptin, CO 2 has much higher slubility in ater cmpared t hydrcarbn cmpnents. Due t the slubility in ater, CO 2 -assisted ater flding is n applied fr mbility cntrl in il recvery prcesses. In general, t increase the CO 2 slubility in the aqueus phase higher pressure and ler temperature cnditins are required. The slubility f CO 2 in ater has been studied by many researchers. Based n Henry s La (Li and Nghiem, 1986) a mdel as 70

86 develped t predict the CO 2 slubility in liquid phase under the equilibrium fluid system. Using the same thery, Enick and Klara (1990) estimated the CO 2 slubility in distilled ater. A cmpsitinal mdel as presented by Chang et al. (1998) t describe CO 2 flding including the CO 2 slubility in ater. In this study, the mdels develped by Duan and Sun (2003) ere selected. The mdel predicts the CO 2 slubility in bth pure and salt ater under a ide range f pressures and temperatures. The slubility mdel in their rk is based n the balance f CO 2 chemical ptential beteen the liquid and gas phases at equilibrium, i.e. y p l (0) v(0) c2 c2 c2 ln lnc ln 2 c 2 m c RT 2, (3.62) here l(0) c2 and v(0) c2 are the standard liquid and gas chemical ptential at ideal cnditins, yc 2 is the mle factin f CO 2 in the gas phase (hich is assumed t be ne during the simulatin), c 2 is CO 2 fugacity cefficient, c 2 is activity cefficient, respectively, and mc 2 is mlality f CO 2 (ml/kg) in the liquid phase. Since CO 2 is assumed t ccupy the hle gas phase and v(0) c2 is set t zer, equatin (4.9), regarding mlality f CO 2 in the ater phase, can be reritten as ln m l (0) c 2 ln c p, (3.63) RT c2 2 c 2 here, the term RT l(0) and c 2 can be calculated frm the crrelatins prvided. The CO 2 mass cncentratin in the ater-co 2 fluid system can be ritten as: 71

87 MW m, c2 cc = c2 c2. (3.64) Since CO 2 is assumed t take ver the hle gas phase, the mass equilibrium cefficient ( k c2 ) can be calculated frm flling equatin: k c2 = c 1, c2 c2. (3.65) CO 2 Slubility in Oil The majr parameter that affects the results f CO 2 flding in an il reservir is the CO 2 slubility in the il phase. A higher slubility results in a less viscus il, thus increasing the il mbility. Mrever, ith disslved CO 2, il selling ccurs during CO 2 flding. Oil selling helps t increase il recvery under unchanged residual il saturatin. Mungan (1964) experimentally shed that the reductin f il ater IFT enhances the efficiency f ater injectin. In CWI, the IFT can be further decreased due t the disslved CO 2 in the ater and il phases, resulting in an imprved verall perfrmance. Due t these imprtant effects n the il recvery prcess, CO 2 slubility in il has been studied by many researches. A graphical crrelatin (Welker, 1963) f CO 2 slubility as develped as a functin f pressure and il API gravity at a cnstant temperature. Simn and Graue (1965) presented slubility data fr dead ils at a temperatures range f C t C and pressures up t MPa. Later n, Mehrtra and Svrcek (1986) calculated the CO 2 slubility at pressures up t 6.38 MPa and temperatures frm t Emera and Sarma (2006) develped a genetic algrithm based n experimental data t predict the CO 2 slubility and il phase prperties as a functin f 72

88 disslved CO 2. The results f these crrelatins have been validated by published experimental data ith a ler errr cmpared t ther crrelatins. In this study the apprach f Emera and Sarma (2006) using GA-based crrelatins are adpted t calculate CO 2 slubility as ell as ther il prperties. In a CO 2 -il fluid system t phases (CO 2 and il) are assumed t present. Since CO 2 is in the gaseus state the flling crrelatins are applied t calculate the mle fractin, 2 f CO 2 in il phase ( c ): x c 2 x Y 3.273Y 4.3Y, (3.66), c c2 here, Y T p pb 1 exp MW. The abve equatins sh that the CO 2 slubility in this GA-based mdels depends n the il specific gravity ( ), the il bubble pint pressure ( p b ), temperature (T ) and the il mlecular eight ( MW ). The CO 2 mass cncentratin can be then calculated by cnverting frm the mle fractin: c M MW x, c2, c c 2 2 CO2 c2 c 2, c2, c2 M c M (1 ) 2 MWCO x 2 c x 2 c MW 2 MW x MW x MW MW, c2 CO2 c2, c2 c 2 CO2, (3.67) here M and M c 2 are mass f il and CO 2, respectively. Accrding t the assumptin made previusly (CO 2 ccupies the entire gas phase) the mass equilibrium cefficient ( k c ) is expressed as: 2 73

89 k c2 1 =. (3.68) c, c2 c2 All the infrmatin used t determine the partitin cefficient ( ) f CO 2 in a CWI fluid system are knn frm the abve crrelatins under reservir cnditin fr a specific il cmpsitin, hence, n can be calculated using equatin (3.61). 4.4 Fluid Characterizatin Due t the presence f CO 2 in the liquid phase, the prperties f il and ater vary during carbnated ater flding. Oil selling and the reductin f il viscsity are majr mechanisms that affect the il recvery in CWI; hence, these need t be evaluated during the calculatin ith the change f ttal pressure and CO 2 mass cncentratin in each phase. The disslved CO 2 in the ater phase als results in a variatin f ater prperties Oil Phase Prperties CO 2 -Oil Viscsity Oil viscsity decreases significantly ith increasing f CO 2 slubility. In the isthermal reservir cnditin, il viscsity varies mainly due t the change f CO 2 mass cncentratin and ttal pressure. The crrelatin fr il viscsity used here is the GA-based mdel (Emera and Sarma, 2006) shn in Sectin 2.4.1: x c2 yi + A, (2.16) i 74

90 here y = B X, p X = Ci 1.8T 32 D xc2 here A, B, C and D are cnstants prvided in Table 2.1., Change f CO 2 -Oil Density The vlume f reservir il expands ith increasing amunts f disslved CO 2, but is reduced under higher pressures. Therefre, the reservir il density is changed by the effect f bth ttal pressure and CO 2 mass cncentratin. The il density increased ith an increase f CO 2 slubility hich results frm the higher pressure (DeRuiter et al., 1994). As mentined in Sectin the density crrelatin (equatin (2.17)) in GAbased mdel (Emera and Sarma, 2006) is adpted in this study. i y y here i p pb y 1.8T Carbnated Water Prperties , (2.17) Carbnated Water Viscsity A crrelatin has been prpsed (Kestin et al., 1978) t calculate the ater viscsity here the ater viscsity is a functin f temperature, pressure and salt cncentratin but nt CO 2 cncentratin. Since ater viscsity is minimally affected by the disslved CO 2, 75

91 e assume that the carbnated ater viscsity remains cnstant irrespective f CO 2 cncentratin The Change f Carbnated Water Density The density f carbnated ater varies as a functin f pressure. Cmpared t pressure the effect f disslved CO 2 n ater density is very small hich is assumed t be negligible this study. The mdel f ater density under a certain pressure range (34 MPa t 30 MPa) and given reservir temperature (80 t 250 ) is generated by PVTsim Calcep simulatr. By regressin, the expressin f ater density in SI unit is: RC (3.69). ST p IFT f Water-il Carbn dixide (CO 2 ) is miscible in bth the il and ater phases and it is assumed that the interfacial tensin beteen these t liquids decreased ith increasing CO 2 cncentratin. A significant remval f residual il after simple ater flding has been bserved ith decreasing IFT (Abrams, 1975). Accrding t the experiment (Shen et al., 2006), the residual il saturatin nly decreases by reducing IFT t a certain range. In the l interfacial tensin regin ( N/m < σ < N/m), the residual il is reduced, thus increasing the il relative permeability (Kumar et al., 1985). Hever, the interfacial tensin beteen the t liquid phases cannt be measured in this numerical study. Due t limited available literature, the interfacial tensin is assumed t be a simple 76

92 linear crrelatin based n the IFT data f ater-ctane prvided by the bk (Danesh, 1998). The ater-il IFT (N/m) changes ith reservir temperature ( ) and CO 2 mass cncentratin: cc ( T 80). (3.70) 2 T simplify, e assume that the residual il saturatin in this study changes linearly as a functin f IFT, ith IFT under a l interfacial tensin regin ( N/m < σ < N/m): S S , (3.71) 0 1 r r here CWI). 0 S r is the reference residual il saturatin (the residual il saturatin befre 4.5 Numerical Simulatins The prblem t be slved is a three-cmpnent t-phase hrizntal ne-dimensinal fluid system. Carbnated ater is injected n ne side and il is prduced frm the ther side f reservir under cnstant pressure bundary cnditins. The prcedures f slutin are similar t ater flding simulatin hich has been discussed in Chapter 3. The IMPES methd as used fr numerical calculatin in rder t evaluate the perfrmance f CWI. By substituting equatins (3.55) and (3.56) int equatin (3.54), the pressure distributin is first slved by applying the knn bundary cnditins. Water saturatin and CO 2 slubility in the ater phase are then updated by pressure hich as slved previusly. The slutin frm the IMPES methd is cnditinally stable depending n 77

93 the size f time step ( t ), therefre, the apprpriate time step need t be selected. As discussed in Sectin , the time step selectin here flls the same cnstraint as the ne e adpted in ater flding called CFL (Curant et al., 1928). As mentined previusly, the carbnated ater injectin mdel here is an extensin f ater injectin mdel. The same numerical scheme has been applied in bth plain ater flding and carbnated ater flding. Since the numerical slutins f WI have been validated in Chapter 3 the numerical mdel f CWI is als validated. 4.6 Case Study Several cases have been studied t validate the simulatin results. In rder t investigate the factrs that affect il recvery, different scenaris (such as different injectin pressures and reservir temperatures) are included. Since an expected IFT reductin ccurs hile increasing reservir temperature, the effect f IFT is als evaluated in this study. Cases are develped under the cnstant pressure bundary cnditins and cmpared ith ater injectin. The summary cnditins applied in different cases studies are shn in Table 4.3. The infrmatin used t prcessed the numerical simulatin is frm textbk (Danesh, 1998) Initial Cnditins Oil Initial Cmpsitin The initial il cmpsitin is required in rder t calculate and update the CO 2 -il prperties. Table 4.1 prvides the infrmatin f initial il cmpsitin in mle fractin: 78

94 Table 4.1 Initial Oil Cmpsitin Cmpnent Ml % N CO2 1 C C C ic nc ic nc C C Initial Reservir and Fluid Prperties The initial reservir and fluid prperties are required befre prceeding t the cmputing prcess. The initial infrmatin t slve the prblem is listed in Table 4.2: 79

95 Table 4.2 Initial Infrmatin regarding Reservir and Fluid Prperties Data Units (SI) CWI WI p res MPa 31 p ut MPa 30 μ i Pa s μ Pa s c _ in c2 c _ in c2 c _ res c2 c _ res c2 ST kg/m ST kg/m S r S c L m 100 ϕ K m k r -- k r -- k k r r 1 S Sr 0.8( ) 1 S S c r S Sr 0.2( ) 1 S S c r 2 2 Table 4.3 Summary f CWI Case Studies Case Cnditin 1 (different injectin pressures) Injectin pressure: 1) 33MPa 2) 32MPa 2a (high IFT, N/m) 80 2b (l IFT, N/m)

96 4.6.2 Case 1: Different Injecting Pressures In the first case CWI is cnducted under the same injectin pressure as the ater injectin (33 MPa) initially. A slightly ler injectin pressure, 32 MPa, is then used in subsequent cases in rder t btain the same ater breakthrugh time as the pure ater injectin. The reservir temperature is kept cnstant, at 80 fr bth injectin cnditins Discussin The saturatin prfile shn in Figure 4.1 shs that after 25 days the ater frnt f CWI breaks thrugh, hever, the ater frnt f WI is nly three quarters f the ay frm injectr t prducer, i.e. 72 m. Cmpared t CWI, the ater frnt in pure ater injectin mves mre slly indicating a later ater breakthrugh in the WI prcess. Due t il viscsity reductin (frm 9 cp t 1 cp) the displacement frnt mves faster leading t an earlier ater breakthrugh in the case f CWI. The average ater saturatin behind the frnt ( S ) (Welge, 1952) in CWI and WI can be calculated using the integratin f ater saturatin divided by the ttal length hich equals t 0.53 and 0.5 in CWI and WI, respectively. The il recvery factr (RF) then can be calculated by: 1 1 Sc Vb S Vb S S RF= = 1 S V 1 S c b c c. (3.72) In CWI, the recvery factr is hich is greater than in WI indicating better il recvery using CWI even thugh the breakthrugh time is less. 81

97 Since the average ater saturatin f bth CWI and WI are knn, the il prductin per pre vlume injected (J) can be calculated by: J 1 c 1 Ttal il prductin S L S L, (3.73) Ttal pre vlume injected S L f here L f is frnt lcatin. Accrding t equatin (3.73) the il prductin per pre vlume injected in CWI is Althugh a better il recvery is achieved using CWI, WI has higher il prductin ith ne pre vlume injectin, hich is cmpared t in CWI. The ther factr t evaluate the perfrmance f an EOR is knn as ttal r verall seep efficiency. This factr can be divided int three different seep efficiencies. In the 2-D area the sept regin by the displacing fluid uld never equal the entire reservir area because f ecnmic cnstraints. The rati f the sept area ver the reservir area is the areal seep efficiency (E a ). The areal seep efficiency is primarily a functin f the mbility rati, reservir hetergeneity, cumulative vlume f ater injected and aterfld pattern cnfiguratin. Due t the vertical hetergeneities ithin the reservir, sme parts f the reservir ill nt be reached by the displacing fluid. A vertical seep efficiency (E v ) is intrduced t accunt fr the vertical hetergeneity. In additin t these t factrs, micrscpic displacement efficiency (E m ) describes the displacement efficiency at the pre scale. Micrscpic displacement efficiency is cntrlled by the balance f gravity, capillary and viscus frces and als pre size distributin. Based n these three seep efficiencies, the ttal seep efficiency (E t ) can be estimated as: 82

98 Et Ea Ev Em. (3.74) Hever, in this numerical study CWI is perfrmed hrizntally ith cnstant bundary pressure cnditins. Therefre, the results fail t evaluate the ttal seep efficiency hich requires the infrmatin frm the 3-D mathematical mdel. A later ater breakthrugh can be reached by lering the injectin pressure. Figure 4.2 shs the ater saturatin prfile after 38 days ith a change f injectin pressure t 32 MPa fr the CWI case. After 38 days, the ater frnts f bth CWI and WI break thrugh at the same time (as designed). ater saturatin prfile ater saturatin CWI WI length (m) Figure 4.1 Water Saturatin Prfiles f CWI and WI under 33 MPa Injecting Pressure after 25 Days 83

99 ater saturatin prfile ater saturatin WI CWI Length (m) Figure 4.2 Water Saturatin Prfiles f CWI and WI under 32 MPa Injecting Pressure after 38 Days Case 2: Different Reservir Temperatures In the secnd case t scenaris are applied t study the effect f IFT n additinal il recvery. As shn by (Okye et al., 1988) IFT decreases ith increasing temperature. T scenaris are presented in this sectin based upn t different reservir temperature cnditins. The cnstant pressure bundary cnditins are still applicable in this sectin. A cmparisn beteen CWI and WI is als prvided by simulatin slutins. In rder t reach the same breakthrugh time carbnated ater is injected under a pressure f 32 MPa fr bth scenaris hile the ater is injected at a pressure f 33 MPa. Since carbnated ater is injected under the sufficiently high pressure (32 MPa), the initial CO 2 cncentratin and CO 2 slubility in ater and il phases are nt affected by the change f temperature beteen t scenaris. 84

100 Case 2a: Higher IFT (ler reservir temperature) In the ler reservir temperature carbnated ater is injected at 80. The IFT varies based n equatin (3.70). The decrease in IFT ccurs due t the increasing slubility f CO 2 in the il phase. Hever, ith a maximum CO 2 slubility in the il phase the IFT still des nt reach t l IFT regin ( N/m< σ <0.002 N/m) alling fr effective reductin in residual il saturatin (Trabzadey, 1984). The residual il saturatin, therefre, remains cnstant, at 0.25, during the flding Case 2b: Ler IFT (higher reservir temperature) In rder t investigate the impact f l IFT n il recvery prcess a higher reservir temperature, 250, is examined t reduce the IFT, thus decrease the residual il saturatin Discussin The saturatin prfiles at a breakthrugh time f 38 days are shn in Figure 4.3 fr bth the l IFT (IFT = N/m), and high IFT (IFT= N/m) and the pure WI. Because the amunt f residual il is nt affected in the high IFT range, (larger than N/m in this case), the maximum ater saturatin (1-S r ) is the same as the ne in WI. In the high temperature system (l IFT), the IFT decreases ith increasing CO 2 cncentratin; hence, mre il is recvered under ler IFT cnditins. This can be verified by examining the ater saturatin prfile fr the l IFT (high temperature) case. The ater saturatin in this case is much higher clse t the injectin pint. Hever, due t insufficient CO 2 slubility, after a sharp reductin the curve flls the same saturatin prfile as it des in case 2a (high IFT). The verlapping ater saturatin 85

101 prfile f the three prcesses are shn in Figure 4.3 implies the same fluid behavir. This cnfirms the calculatin results frm previus rk (Nel, 1964) hich stated that due t the cntact ith il the initial carbnated ater injected lses its CO 2 and then prceeds as plain ater. Thus, the CO 2 mves behind the pure ater in CWI. Since the temperature in case 2b (l IFT) is much higher than in case 2a (high IFT), the IFT is decreased t ithin the range hereby the residual il saturatin is decreased (equatin (3.71)) and recvery factr is increased. Figure 4.4 shs the change in residual il saturatin and CO 2 cncentratin in the il phase alng the length f the cre. Once the CO 2 mass cncentratin is larger than 56%, a sufficiently l IFT is reached leading t ler residual il saturatin. Water saturatin prfile ater saturatin WI case 2b (l IFT) case 2a (high IFT) Length (m) Figure 4.3 Water Saturatin Prfile after 38 Days 86

102 CO 2 Mass Cncentratin CO2 mass cncentratin residual il saturatin Length (m) Residual Oil Saturatin Figure 4.4 CO 2 Mass Cncentratin vs. Residual Oil Saturatin ver 200 Days Viscsity Effect Althugh the residual il saturatin stays cnstant ith bth CWI at high IFT (case 2a) and WI increased il recvery is bserved ith CWI. Figure 4.5 shs a cmparisn beteen ater saturatin prfiles f case 2a (high IFT) and simple ater injectin after 200 days f injectin. The ater saturatin (ler il saturatin) hen injected ith carbnated ater cmpared t pure ater injectin. In ther rds, mre il ill be prduced during the CWI prcess. This is mainly because f the deceasing il viscsity ith increasing CO 2 cncentratin. This is illustrated in Figure

103 0.75 Water saturatin prfile ater saturatin WI case 2a (high IFT) Length (m) Figure 4.5 Water Saturatin Prfile after 200 Days CO 2 Mass Cncentratin CO2 mass cncertratin il viscsity Length (m) Oil Viscsity (cp) Figure 4.6 CO 2 Mass Cncentratin vs. Oil Viscsity ver 200 Days f Case 2a Cumulative Oil Prductin and Recvery Factr f CWI and WI The cumulative amunt f il prduced in case 2a (IFT = N/m), case 2b (IFT = N/m) and ater flding are pltted in Figure 4.7. The CWI, ith a l IFT, has the best result flled by CWI at a high IFT cnditin. Cmpared t CWI, less il can be recvered by pure ater flding. 88

104 After 200 days, the recvery factr (RF) fr l IFT CWI is 0.68, 6% higher than the recvery factr fr a high IFT in CWI hich is The WI has the lest recvery factr at 0.55 in this case. Based n the numbers shn abve the mst desirable il recvery perfrmance is achieved by lering interfacial tensin (case 2b). Due t the sufficiently l interfacial tensin beteen ater and il ( N/m), the residual il saturatin has been reduced during CWI prcess. Mrever, the il viscsity is als reduced in this l interfacial tensin fluid system by the slutin f CO 2 in the il phase. The cmbinatin f effects in reductin f il viscsity and residual il saturatin results in the mst desirable scenari in case 2b shn by Figure 4.7. With the high interfacial tensin ( N/m) in case 2a, the residual il saturatin stays cnstant during the CWI prcess. Hever, the il viscsity is decreasing ith increasing CO 2 cncentratin in il phase. As indicated by Figure 4.7 mre il is recvered in case 2a due t the il viscsity reductin cmpared ith plain ater injectin (WI). In cnclusin, additinal il recvery using carbnated ater injectin is due t the il viscsity reductin by disslved CO 2. In additin, if the fluid system is clse t miscibility and l IFT can be btained, the significant increase in il recvery ill be bserved due t reductin in residual il saturatin. Hever, in the case hen the fluid system is far frm miscibility the il viscsity reductin plays the main rle in il recvery enhancement. 89

105 cumulative il prductin il prductin (m 3 ) 4, , , , , , , case 2a (high IFT) WI case 2b (l IFT) Time (days) Figure 4.7 Cumulative Oil Prductin after 200 Days 90

106 Chapter 5 Cnclusins and Recmmendatins 5.1 Cnclusins The thesis mainly fcuses n the il recvery prcesses in bth ater injectin (WI) and carbnated ater injectin (CWI). These il recvery prcesses are cnsidered under cnstant pressure bundary cnditins. Numerical slutins are calculated by an IMPES blck-centered finite-difference methd using upstream evaluatin f mbilities. In WI, due t varying ttal fluid velcity under fixed pressure bundaries, the classical Buckley Leverett thery fr a cnstant ttal fluid velcity is nt applicable. Based n mass cnservatin, the numerical simulatin is perfrmed in Matlab. The numerical slutin is then cmpared t an existing analytical extensin f the classical Buckley Leverett thery fr a cnstant pressure bundary and the WI simulatin mdel is validated. As an extensin f WI mdel the mdel f CWI, therefre, is validated. Results fr different case studies are shn. The cmparisn is als used t illustrate the impact f numerical errrs by shing h the numerical slutin appraches the analytical slutin hen the number f grid blcks is refined. In carbnated ater injectin, CO 2 is disslved in the ater phase prir t injectin. After injectin, CO 2 ill partitin in bth the ater and il phases. The fractins f CO 2 in each phase are the main variables that affect the recvery factr. This rk presents the results f CWI by cmbining bth thermdynamics and reservir simulatin mdels. The effects f il recvery in CWI are als discussed in this rk. 91

107 The findings are presented next. The cnclusins f WI can be summarized as: 1. The ttal fluid velcity changes ith time fr cnstant pressure bundary cnditins. 2. We successful applied the cnstant pressure bundary cnditins f the Buckley- Leverett thery extensin. 3. A numerical slutin as cmpared t the analytical extensin ith gd agreement fr different fluid systems. Carbnated ater injectin as studied under different scenaris: 1) different injectin pressures (33 MPa and 32 MPa), and 2) different reservir temperatures: 80 (high IFT) and 250 (l IFT) resulting in the flling cnclusins: 1. An early breakthrugh has been fund ith a higher injectin pressure. The pstpned breakthrugh time can be realized by decreasing injectin pressure. 2. IFT is decreasing ith increasing temperature. With maximum CO 2 slubility l IFT can be reached. Cntrary t the high temperature, in a l temperature reservir (80 ) high IFT fluid system is present. 3. A reductin f residual il saturatin in l IFT under the higher temperature (250 in this case) is bserved resulting in a higher cumulative il prductin. As EOR, CWI have been als cmpared ith WI (secndary il recvery) under the same breakthrugh time. The main cnclusins are summarized as flls: 1. In l temperature fluid system the viscsity is the main mechanism f 92

108 enhancing il recvery. 2. In high temperature fluid system bth reductin f viscsity and IFT cntribute t the additinal il recvery. 3. The CWI, ith a l IFT, has the best result flled by CWI in a high IFT cnditin. Cmpared t CWI, less il can be recvered by ater flding. Table 5.1 Summary f CWI ith Different Cnditins Case Cnditin Result 1 (different injectin pressures) 2a (high IFT, N/m) 2b (l IFT, N/m) Variable injectin pressure High pressure led an early breakthrugh 2. Decreasing the injectin pressure prlnged breakthrugh 1. High IFT fluid system existed 2. N change f residual il as bserved 1. L IFT fluid system as present 2. A reductin f residual as bserved 3. High cumulative il prductin as shn 5.2 Recmmendatins During this study, sme aspects f CWI fr il recvery ere theretically investigated. Due t the limitatin f this theretical research the imprtant infrmatin, such as IFT, cannt be measured experimentally. The numerical slutins have nt been cmpared ith r validated against experimental data. Therefre, a further experimental investigatin is recmmended fr a deeper understanding f CWI in EOR. The effects f 93

109 CO 2 diffusin in the fluid system as neglected in this rk. Further research can be carried n by accunting fr the diffusin f CO 2 in bth ater and il phases. This research is cnducted fr a 1-D mdel. T have full evaluatin f CWI a 3-D mathematical study is recmmended t the deeper cmprehensin. 94

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112 Okye, C., Oungpasuk, P., and Wang, P. (1988). Elevated temperature multiphase rckfluid prperties fr high and l tensin systems. Paper presented at the Internatinal Meeting n Petrleum Engineering. Patel, P., Christman, P., and Gardner, J. (1987). Investigatin f Unexpectedly L Field-Observed Fluid Mbilities During Sme CO2 Tertiary Flds. SPE Reservir Engineering, 2(4), Ramesh, A., and Dixn, T. (1972). Numerical simulatin f carbnated aterflding in a hetergeneus reservir. Paper presented at the Fall Meeting f the Sciety f Petrleum Engineers f AIME. Ra, D. N., and Lee, J. I. (2003). Determinatin f gas il miscibility cnditins by interfacial tensin measurements. Jurnal f cllid and interface science, 262(2), Riazi, M. (2011). Pre scale mechanisms f carbnated ater injectin in il reservirs. Herit-Watt University. Riazi, M., Shrabi, M., Jamilahmady, M., Ireland, S., and brn, c. (2009). Oil recvery imprvement using CO2-enriched ater injectin. Paper presented at the EUROPEC/EAGE Cnference and Exhibitin. Rgers, J., and Grigg, R. (2000). A literature analysis f the WAG injectivity abnrmalities in the CO2 prcess. Paper presented at the SPE/DOE Imprved Oil Recvery Sympsium. Rrda, J. n. (1979). RTD 1 (2) Petrleum in Wrld Energy Balances t the Year Paper presented at the 10th Wrld Petrleum Cngress. Re Jr, A. M., and Chu, J. C. (1970). Pressure-vlume-temperature-cncentratin relatin f aqueus sdium chlride slutins. Jurnal f Chemical and Engineering Data, 15(1), Sheldn, J., Zndek, B., and Cardell, W. (1959). One-dimensinal, incmpressible, nncapillary, t-phase fluid fl in a prus medium. Trans. SPE AIME, 216, Shen, P., Zhu, B., Li, X.-B., and Wu, Y.-S. (2006). The influence f interfacial tensin n ater-il t-phase relative permeability. Paper presented at the SPE/DOE Sympsium n Imprved Oil Recvery. Simn, R., and Graue, D. (1965). Generalized crrelatins fr predicting slubility, selling and viscsity behavir f CO2-crude il systems. Jurnal f Petrleum Technlgy, 17(1), Siregar, S., Mardisej, P., Kristant, D., and Tjahyadi, R. (1999). Dynamic Interactin Beteen CO2 Gas and Crude Oil in Prus Medium. Paper presented at the SPE Asia Pacific Imprved Oil Recvery Cnference. Shrabi, M., Riazi, M., Jamilahmady, M., Ireland, S., and Brn, C. (2009a). Enhanced Oil Recvery and CO2 Strage by Carbnated Water Injectin. Paper presented at the Internatinal Petrleum Technlgy Cnference. Shrabi, M., Riazi, M., Jamilahmady, M., Ireland, S., and Brn, C. (2009b). Mechanisms f Oil Recvery by Carbnated Water Injectin. Paper presented at the Internatinal Sympsium f the Sciety f Cre Analysts held in Nrdijk aan Zee, The Netherlands. 97

113 Stahl, C. R., Gibsn, M. A., and Knudsen, C. W. (1987). Thermally-enhanced il recvery methd and apparatus: Ggle Patents. Stne, H., and Garder Jr, A. (1961) G-Analysis f Gas-Cap r Disslved-Gas Drive Reservirs. Old SPE Jurnal, 1(2), Trabzadey, S. (1984). The effect f temperature and interfacial tensin n ater/il relative permeabilities f cnslidated sands. Paper presented at the SPE Enhanced Oil Recvery Sympsium. Tuni, S. Q., Tuni, A. H., Ghiran, N. A., and El Aday, Z. M. (2011). Cmparisn f different enhanced il recvery techniques fr better il prductivity. Internatinal Jurnal f Applied Science and Technlgy, 1. Tzimas, E., Gergakaki, A., Garcia Crtes, C., and Peteves, S. (2005). Enhanced il recvery using carbn dixide in the Eurpean energy system. Eurpean Cmmissin Jint Research Centre, Reprt EUR, Welge, H. (1952). A simplified methd fr cmputing il recvery by gas r ater drive. Jurnal f Petrleum Technlgy, 4(4), Welker, J. (1963). Physical prperties f carbnated ils. Jurnal f Petrleum Technlgy, 15(8), Zain, Z., Kechut, N., Ganesan, N., Nraini, A., and Raja, D. (2001). Evaluatin f CO2 Gas Injectin fr Majr Oil Prductin Fields in Malaysia-Experimental Apprach Case Study: Dulang Field. Paper presented at the SPE Asia Pacific Imprved Oil Recvery Cnference. Zekri, A., Shedid, S., and Almehaideb, R. (2007). Pssible Alteratin f Tight Limestne Rcks Prperties and the Effect f Watershelding n the Perfrmance f SC CO2 Flding fr Carbnate Frmatin. Paper presented at the SPE Middle East Oil and Gas Sh and Cnference. 98

114 Appendix A Unit cnversin factrs Cvert frm T Multiply by Inverse Area acre m E E-4 ft 2 m E E+1 Density pund/ft 3 kg/m E E-2 Mass pund kg E E+0 Pressure psi Pa 6.895E E-4 atm Pa 1.013E E-6 bar Pa E E-5 Permeability md m E E+15 Time day s 8.64 E E-5 hur s E E-4 Viscsity cp Pas E E+3 Vlume ft 3 m E E+1 barrel m E E+0 Length ft m E E+0 inch m E E+1 Interfacial tensin dyn/cm N/cm E E+3 99

115 Appendix B Pre-print: Slutins f Multi- Cmpnent, T-Phase Riemann Prblems ith Cnstant Pressure Bundaries 100

116 Transprt in Prus Media Fractinal Fl Analysis fr Multi-Cmpnent Prblems ith Cnstant Pressure Bundaries --Manuscript Draft-- Manuscript Number: Full Title: Article Type: Keyrds: Crrespnding Authr: Fractinal Fl Analysis fr Multi-Cmpnent Prblems ith Cnstant Pressure Bundaries Original Research Paper Buckley-Leverett; fractinal fl thery; cnstant pressure bundaries; Reimann prblems Thrmd E Jhansen, PhD Memrial University f Nefundland St. Jhn's, Nefundland CANADA Crrespnding Authr Secndary Infrmatin: Crrespnding Authr's Institutin: Memrial University f Nefundland Crrespnding Authr's Secndary Institutin: First Authr: Thrmd E. Jhansen, Ph.D. First Authr Secndary Infrmatin: Order f Authrs: Thrmd E. Jhansen, Ph.D. Lesley James, Ph.D. Order f Authrs Secndary Infrmatin: Abstract: Fractinal fl thery has been used t describe the linear, incmpressible, fluid displacement in a prus medium under cnstant flux cnditins ithut dispersin. The hyperblic prblem is usually assciated ith mving shck frnts mathematically describing the changing lcatin f the abrupt change in fluid cmpsitin. Realistically, reservirs are ften prduced under cnstant pressure bundaries ith a cnstant injectin pressure and a cnstant ell fling pressure. Here, e extend the Buckley-Leverett's classic fractinal fl thery t multicmpnent prblems ith cnstant pressure bundaries. We revie Riemann prblems and derive expressins fr the vlumetric flux, bth befre and after breakthrugh f multiple cmpnent aves. Expressins fr predicting the breakthrugh time and calculating the pressure prfile frm utlet t inlet as ell as generalised frmulas are presented. Finally, t aterflding cases and a plymer case are develped fr time dependent pressure bundaries t demnstrate and exemplify the cnstant pressure bundary slutin t fractinal fl thery. Pered by Editrial Manager and Preprint Manager frm Aries Systems Crpratin

117 Manuscript Click here t dnlad Manuscript: Cnstant Pressure Bundaries Fractinal Fl Thery (vsubmitted).dcx Click here t vie linked References Fractinal Fl Analysis fr Multi-Cmpnent Prblems ith Cnstant Pressure Bundaries Thrmd E. Jhansen 1 and Lesley A. James 2 Faculty f Engineering & Applied Science Memrial University f Nefundland St. Jhn s, Nefundland A1B 3X5 Canada ; thrmdj@mun.ca 1 ; ljames@mun.ca 2 Fractinal fl thery has been used t describe the linear, incmpressible, fluid displacement in a prus medium under cnstant flux cnditins ithut dispersin. The hyperblic prblem is usually assciated ith mving shck frnts mathematically describing the changing lcatin f the abrupt change in fluid cmpsitin. Realistically, reservirs are ften prduced under cnstant pressure bundaries ith a cnstant injectin pressure and a cnstant ell fling pressure. Here, e extend the Buckley-Leverett s classic fractinal fl thery t multicmpnent prblems ith cnstant pressure bundaries. We revie Riemann prblems and derive expressins fr the vlumetric flux, bth befre and after breakthrugh f multiple cmpnent aves. Expressins fr predicting the breakthrugh time and calculating the pressure prfile frm utlet t inlet as ell as generalised frmulas are presented. Finally, t aterflding cases and a plymer case are develped fr time dependent pressure bundaries t demnstrate and exemplify the cnstant pressure bundary slutin t fractinal fl thery. Key rds: Buckley-Leverett; fractinal fl thery; cnstant pressure bundaries; Reimann prblems; EOR: Enhanced Oil Recvery 1

118 1. Intrductin The Buckley-Leverett slutin (1941) is synnymus ith fractinal fl thery here an immiscible fluid displaces anther in ne-dimensinal fl in a prus medium. Physically, fractinal fl thery describes the linear displacement f ne phase by anther immiscible phase here there is a frnt described by a shck r sudden change in cncentratin. In its simplest frm it describes ne cmpnent displacing anther immiscible cmpnent in ne dimensin in the absence f diffusive and cmpressible fl, i.e. ater displacing il (Buckley and Leverett 1941, Welge 1952). Mathematically, the Buckley-Leverett equatin is a first rder hyperblic partial differential cnservatin equatin in time and space. This cncentratin shck (il-ater interface) travels frm the start (injectin) t the end (prductin) as depicted in Figures 1 and 2. Fig. 1 One-dimensinal Riemann prblem Instead f a shck, the interface beteen the t phases may exhibit a gradual change in cncentratin indicated as a rarefactin ave, ν i r ν i-1 in Fig. 2. Fractinal fl prblems are mathematically knn as Riemann prblems that can be slved using the methd f characteristics. Riemann prblems are hyperblic first rder partial differential equatins ith a cnstant initial value and a cnstant injected value. The methd f characteristics finds a characteristic curve f the Riemann prblem here the partial differential equatin becmes an rdinary differential equatin and here an analytical slutin can be fund. The bjective f this rk is t extend the Buckley-Leverett thery frm the cnstant flux cnditin t cnstant pressure bundaries fr multicmpnent systems. The mathematical frmulatin is derived fr time befre the first ave breaks thrugh, time after the first ave breaks thrugh but befre the next ave, 2

119 time after the subsequent ave and then generally. Three cases are used t demnstrate the cnstant pressure bundary multicmpnent extensin t the Buckley-Leverett slutin; case 1a) aterflding ith µ µ = 0.2, case 1b) aterflding ith µ µ = 20 and case 2) plymer flding ith a single plymer cmpnent residing in the aqueus phase (Jhansen and Winther, 1988). This cnstant pressure multicmpnent extensin t the Buckley-Leverett equatin is particularly imprtant as many actual fields are perated under cnstant pressure bundaries and being a generalised analytical slutin it can be readily adapted and used fr better predicting prductin rates. 2. Riemann Prblems Given a hyperblic system f cnservatin las such as an n-cmpnent t phase mdel fr ne dimensinal fl in prus media subject t standard fractinal fl assumptins (1D cnstant vlume fl ith negligible dispersin). If Fi Fi u1 un 1 = (,..., ) is the fractinal flux functin fr cmpnent i, and u = [ u,..., ] represents the verall vlume fractin f the fluid cmpnent(s) 1 un 1 here the sum f the individual cmpnents must be ne ( ui = 1), the cnservatin f mass mdel under cnsideratin may be ritten as Fi φ [ ui + ai ( u )] + ut = 0 ; i = 1,..., n 1 (2.1) t x here a( u ) is vlume fractin f the stagnant part f cmpnent i, e.g. caused by adsrptin. Furthermre, φ is prsity and u T is the cnstant vlumetric flux. If e have t phases, Fi = fui1 + (1 f ) ui 2 ; ui = Sui1 + (1 S ) u here u i 2 ij is vlume fractin f cmpnent i in phase j, S is saturatin f phase 1 and f is the fractinal fl functin f phase 1, e assume the mdel can be refrmulated as n i= 1 u u T u + A( u ) = 0 (2.2) t φ x 3

120 here A( u) is an ( n 1) ( n 1) matrix ith real eigenvalues λ1,..., λn (since 1 e assume the system in equatin (2.2) is hyperblic). We assume the slutin fr the multi cmpnent Riemann prblem described by equatin (2.2) is knn fr the case hen the vlumetric flux u is cnstant bth T in x and t. A Riemann prblem is an initial/bundary value prblem ith cnstant states L u(0, t) = u ; t 0 R u( x,0) = u ; 0 x L (2.3) here L is the length f the 1D medium. As explained in the intrductin, in this paper the cnstant flux slutin is used t determine the slutin f the assciated prblem ith cnstant pressure bundaries, p = p(0, t) ; p = p( L, t). Fr such cnstant pressure bundaries, the vlumetric flux ill be cnstant as a functin f x because f the incmpressibility assumptin, hever u = u ( t) ill be time dependent. The cnstant vlumetric flux slutin cnsists f a sequence f self - similar aves (i.e. aves that can be described as a functin f ξ = x / t ) L R cnnecting the t states u, u, in such a ay that the verall ave velcity increases frm L u t in R u. Each f these elementary aves belngs t ne f the eigenvalues λ1,..., λn 1 = x / t either as a Rarefactin ave (smth) r a Shck ave (including cntact discntinuity). Any t adjacent aves are separated by a cnstant state. The slutin f the assciated prblem ith cnstant pressure bundaries and the cnstant fl rate slutin are cngruent in the sense that either slutin at a given time can be btained frm the ther by stretching the x- axis. The sequence f elementary aves is illustrated in Fig. 2, here als the nmenclature used in this paper is defined. ut T T 4

121 ν i ν N S R = S N ν 1 S i-1 S i S L = S 0 ν i-1 S i-1 ν i S i x i-1 y i x i Fig. 2 Elementary Waves We assume that each ave can be defined by the parameter S (e.g. phase saturatin). The leading edge f the ave ν i is xi and the trailing edge is y i shcks xi = yi. Als, it is pssible t have xi 1 = yi, such as in the classic Buckley Leverett slutin, here a shck has the same velcity as the leading edge f the trailing rarefactin ave. Any t aves ν, i 1 ν are separated by a cnstant state i S. i 1 fr Cnsider the case here ν i is a rarefactin ave, parameterized by S. We d nt assume that the system is strictly hyperblic, s λi λk may change sign fr any pair f eigenvalues. Hence, e cannt assume that the elementary aves crrespnd ne by ne t a sequence f increasing eigenvalues. Instead, ν i ( S) = λk ( S ) fr sme k. If ν i is a shck, it must satisfy the Rankine-Hugnit cnditin (shck mass cnservatin) fr each cmpnent, hich in particular means it ill satisfy [ Fk ] ν i = ; k = 1,..., n 1 (2.4) [ u ] k here [-] represents a jump frm ne side f the shck t the ther. This equatin gives rise t n 1 elementary shck aves crrespnding t each f the eigenvalues λ1, λ2,..., λn. 1 5

122 ut In ur ntatin, the prpagatin velcity f a ave ν is i Vi = ν i. φ ut If ν i ; i = 1,..., N represents the slutin f the cnstant flux Riemann prblem φ L R cnnecting u, u, the slutin f the cnstant pressure bundary slutin is represented by ut ( t) ν i ; i = 1,..., N. φ In brief, this paper assumes e kn the slutin (unique r nt) f a multicmpnent Riemann prblem subject t the assumptin f cnstant vlumetric flux u. The main result f the paper is t determine the functin T ut ( t ) fr the case f cnstant pressure bundaries fr the same Riemann prblem. In this derivatin e als btain clsed expressins fr the time hen a given state is breaking thrugh at the utlet end. Furthermre, e determine the pressure distributin at any time in 0 x L. 3. Determinatin f the vlumetric flux u ( t) fr cnstant pressure bundaries We ill ithut ambiguity, since eigenvalues ( λ ) d nt appear in this sectin, 2 let λ = K ( k / µ ) dente ttal mbility, here K is permeability, k rj phase T rj j j= 1 relative permeability and µ j phase viscsity. We assume, in this sectin, cnstant pressure bundaries; pin = p(0, t) ; put = p( L, t), and u is cnstant as a functin T f x but nt t. We btain T p dx u = λ p = u ; p = p p (3.1) T T T in ut x λ 0 T L Let t BT, i be the time hen the leading edge f a ave ν is breaking thrugh at i the utlet end x = L. 6

123 t BT,N 3.1. The Case We first derive explicit expressins fr the velcity, u ( t ), befre the fastest ave breaks thrugh at the utlet end, and the time hen this breakthrugh ccurs, t BT, N. Assuming e kn u ( τ ) at any time τ t BT, N and letting 0 T t Ψ ( t) = u ( τ ) dτ, e first use integratin by parts fr a rarefactin ave ν i as flls: T T x x i i Si Si dx x λ T xi yi Ψ( t) ν iλ ut = ut + x( s) ds u 2 T ds 2 λ ( ) ( 1 1) i T λ = + y T λ λ y S i i T T Si λt Si φ λ Si 1 T Si Si Si xi yi Ψ( t) ν i ν i Ψ( t) ν = ut + ( + ds) ut ds λt ( Si ) λt ( Si 1) φ λ = T λ S S i 1 i 1 T φ λ Si 1 T (3.2) Obviusly, if the ave is a shck, this integral is zer. We, therefre, define 0 if ave i is a shck Si ' I i = ν i (3.3) ds if ave i is a rarefactin λ Si 1 T We can n rite equatin (3.1) as N yi xi 1 Ψ( t) L x N p = ut [ + I i ] +, (3.4) i= 1 λt ( Si 1) φ λt ( SR ) ut and define the flling here Vi ( S) = ν i ( S) : φ ν i (S) ν i (S) = [F i (S)] [S] if i is rarefactin if i is shck (3.5) Given the leading edge f the ave, x i and the trailing edge, y i, e can relate the velcity f the leading edge ( dx dt ) and the velcity f the trailing edge f the shck ( dy dt ) t the prpagatin velcity f the ave, V( S ). dyi dxi 1 = Vi ( Si 1) ; = Vi 1( Si 1) ; i = 1,... N 1 (3.6) dt dt 7

124 ν i ( Si 1) ν i( Si ) If e define cnstants, βi = ; αi = here subscript R is the ν ( S ) ν ( S ) N R N R saturatin at the right hand side (the exit), equatin (3.6) implies that dyi dx N dx = β = α = α =. (3.7) i i 1 ; i 1 ; i 1,..., N 1 ; 0 0 dxn Since x = y = x = 0 at t = 0, Hence, N i i y = β x ; x = α x. (3.8) i i N i 1 i 1 N y i x i 1 λ T (S i 1 ) = r i x N ; r i = β i α i 1 λ T (S i 1 ) (3.9) Substituting this int equatin (3.4), e btain N N Ψ( t) L x N p = ut ri xn + I i + i= 1 φ i= 1 λt ( SR ) (3.10) The leading edge f the ave, at breakthrugh, is difference is defined as x Ψ( t) = ν ( S ), the pressure φ N N R T [ ] p = u Ax + B (3.11) N here N N 1 1 L A = r + ; B = ( ) ( ) ( ) Using equatin (3.11) and integratin gives I (3.12) i i i= 1 ν N SR i= 1 λt SR λt SR dxn ut pν N ( SR ) = ν N ( SR ) =, (3.13) dt φ φ( Ax + B) N here Ax + 2Bx = Ct (3.14) 2 N N C = 2 pν ( S ) / φ. (3.15) N R 8

125 Accepting nly the psitive rt in equatin (3.14), the lcatin f the leading edge f the fastest ave is given by ( ) xn t = A 2 B B ACt (3.16) Furthermre, e can find an explicit expressin fr the break thrugh time f ave ν by substituting N xn = L in equatin (3.14), i.e. t BT, N = 2 AL + 2BL C (3.17) Finally, the pressure at the leading edge f the fastest ave, befre this ave breaks thrugh at time t BT, N is calculated as SR ds ut N = ut + T = ut + N λ ( ) ( ) SN 1 T S λt S R p ( t) p u p ( L x ( t)). (3.18) The pressure at any lcatin can then be calculated backards (tards the inlet end) using equatin (3.10). The abve applies t t t BT, N. We next describe h u ( t ) is calculated fr t < t t, i.e. after the break thrugh f the first ave. BT, N BT, N 1 T 3.2. The Case t < t BT,N BT,N -1 If the fastest ave is a shck ν N ith a cnstant saturatin state, S N, separating 1 ν frm ν, the velcity, u ( t ), fr t, < t t, 1 is calculated exactly as N 1 N T abve, simply by remving ν and putting N S R = S N. This is because e already 1 kn Ψ ( t) fr t t BT, N. If the first ave is a rarefactin, the calculatin f u ( t ) is as described bel. BT N BT N T Let S beteen S N = S and R S N be arbitrary but fixed. Let x( S, t 1 BT, N ) be the lcatin f S at time t BT, N edge at x in Fig. 3., i.e. the time hen ν breaks thrugh ith its leading N = L. Als, let t S be the time hen S arrives at x = L. This is illustrated 9

126 Fig. 3 Example f a Rarefactin Wave at Breakthrugh Let ˆt be a time beteen t BT, N and t, and let S t = tˆ. Assuming e kn u ( t) ; t tˆ, then and T Ŝ be the value f S at x L = at ˆ ν N ( ) x( S; t ) = S Ψ ( tˆ ) (3.19) φ giving r ˆ N ˆ N 1 S ˆ ˆ yi xi 1 Ψ( t ) Ψ( t ) ν N ( s) ds p = ut ( t ) + I i + i= 1 λt ( Si 1) φ i= 1 φ, (3.20) λ ( ) SN 1 T s u ( tˆ ) = T p y x ˆ ( s) ds ( S ) ( S) ( s) N N 1 Sˆ i i 1 x( S, t ) ν N 1 + I i + i 1 λt i 1 ν = N i= 1 λ S T N 1 (3.21) 10

127 u ( tˆ ) = T x( S, tˆ ) N p 1 ˆ 1 N S ν N ( s) ds r + I i + ( ) λ ( ) SN 1 T s i i= 1 ν N S i= 1 here r i is given by equatin (3.9). We als have ( ˆ, ˆ) ( ) (3.22) dx S t vn S = u ( ˆ T t ). (3.23) dt φ Cmbining equatin (3.22) ith equatin (3.23), e get (, ˆ dx x S t ) dt pν ( S) N = ˆ N N 1 S 1 ν N ( s) ds φ ri + I i + i= 1 ν N ( S) i= 1 λt ( s) S N 1 hich, hen integrated beteen t BT, N and ˆt letting tˆ t s can be ritten as (3.24) φ x( S, t ) 2 pν ( S) t t. ( s) ds ( s) N S BT, N BT, N L = ˆ N N 1 S ν N ri ν N ( S) + I i + i= 1 i= 1 λ SN 1 T (3.25) Here, t BT, N is knn frm equatin (3.17) and Hence, t S can be calculated frm ν N ( S) x( S, tbt, N ) = Ψ ( tbt, N ). (3.26) φ S φ x( S, t, ) N N BT N L ν N ( s) ds ts = tbt, N + ( ) 2 ri ν N S + I i + 2 pν N ( S). (3.27) i= 1 i= 1 λ ( ) SN 1 T s The crrespnding value fr u ( t ) is given by T S 2 2 φ [ x( S, tbt, N ) L ] ut ( ts ) = 2 Lν ( S)( t t ) N S BT, N (3.28) fr ts > tbt, N. Fr t = t BT, N it is easy t see that u in equatin (3.28) appraches T the value f u T given by equatin (3.10), i.e. u T is cntinuus, hever, nt differentiable at t = t BT, N. The prcedure then can be summarized as flls: We can calculate the time t S hen S breaks thrugh at x = L frm equatin (3.27) fr any S n the rarefactin 11

128 ave ν. Once this time is knn, the crrespnding value f N ut ( ts ) is given by equatin (3.28). t > t BT,N The Case When the entire leading ave ν has passed x = L, as described in sectins 3.1 N and 3.2, the prcedure can be repeated by remving ν frm the ave train and N starting ver again ith S R = S N. The cmputatinal prcedure is, therefre, 1 cmplete fr the case hen p is fixed. The special case hen fractinal fl thery) can be treated by using in equatin (3.11), i.e. u is cnstant in bth x and t (as in the classical T ut xn = ν N ( SR ) t (3.29) φ * * ut p( t) = ut [ A t + B] ; A = A ν N ( SR ). (3.30) φ Equatin (3.30), f curse, reduces t u T p(0) = λ T. (3.31) L The prcedure fr calculating p( t) fr the ther cases is straightfrard Generalisatin The abve derivatin fr a fixed p can easily be generalized t the situatin here p( t) is given as a functin f time. Denting it is easily seen that (as in sectin 3.2), e get t D( t) = p( τ ) dτ, (3.32) 0 and 2 ( ) B + B + 4 ACD( t) xn t = (3.33) A 12

129 u ( t) = T p( t) Ax ( t) + B N (3.34) The time t break thrugh f ν ( S ) is then fund frm N R 2 AL + 2BL D( tbt, N ) = (3.35) C and similarly fr ther cases. 4. Cnstant Pressure Bundary Case Studies T case studies are develped in this sectin illustrating the use and effectiveness f the generalized cnstant pressure fixed bundary Reimann prblem. The first illustratin is a simple aterflding case here in a) the viscsity f the ater is greater than that f il, ith µ µ = 0.2 and in b) the il viscsity is greater than the ater viscsity ith µ µ = 20. The secnd case is a plymer flding case here the viscsity f the ater phase in linearly dependent n the cncentratin f plymer added. The parameters used in the case studies are utlined in the flling table. The cre is ne meter lng ith 18% prsity and a permeability f ne Darcy. There is a 500 psi pressure drp acrss the cre that is initially 25% ater saturatin as cnnate ater and 75% il saturatin. The displacing ater saturates t 70% leaving 30% residual il saturatin. We use nrmalized saturatins, i.e. S Sc S =. (4.1) 1 S S r c 13

130 Table 1 Parameters used in the Cnstant Pressure Bundary Cases Waterflding Plymer Flding Case 1a Case 1b Case 2 Parameter µ >> µ µ << µ = 1 µ (cp) 10 1 µ ( c) = µ + 200c µ µ (cp) µ /µ φ 0.18 L (m) 1 P in 2.1x10 7 Pa (3000 psi) P f 1.7x10 7 Pa (2500 psi) S r 0.30 S c 0.25 K (m 2 ) 1 x r k ( ) 2 r k r 1 S S = 0.8 = a 1 S 1 Sc Sr c 2 k r kr = 0.2 = as 1 S c Sr S S Waterflding The Riemann prblem fr aterflding is defined as flls fr the simple system illustrated in Fig. 1 here either in case 1a) a mre viscus ater displaces a less viscus il r in case 1b) a less viscus ater displaces a mre viscus il. The viscsity rati f il t ater varies 100x beteen the t cases. The Riemann prblem is S µ T f ( s) + = 0 t φ x (4.2) L S = 1 S r = 0.7 (4.3) R S = S c = 0.25 (4.4) The fractinal fl f a phase is defined frm the mbility f the phase (λ) ith respect t the ttal mbility as: λ f ( s) = λ + λ (4.5) 14

131 Kk λ = r (4.6) µ We use the illustratin in Fig. 4 t depict t aves (N=2) Fig. 4 Depictin f a t ave Reimann prblem The rarefactin ave is dented by ν and the shck ave, 1 ν. The prpagatin 2 velcities are dented V and 1 V fr the rarefactin and shck aves, 2 respectively. Table 2 Wave Descriptins Wave Prpagatin Velcity Rarefactin µ T ν 1 = f ( S) (4.7) V1 = ν 1 φ (4.8) Shck * R f ( S ) f ( S ) µ T ν 2 = (4.9) V * R 2 = ν 2 S S φ (4.10) 15 Case 1b μ /μ = 20 F (x 10 4 ) 10 5 Cnnate Water Saturatin Case 1a μ /μ = 0.2 Residual Oil Saturatin S Fig. 5 Fractinal Fl Functins fr the Waterflding Cases 1a) µ µ = 0.2 and 1b) µ µ = 20 at Breakthrugh Times 0.3 ( ), 0.5 ( ), 0.7 ( ), and 0.9 ( ) 15

132 Using the data given abve, e get: * S f ( S) 11 I 1 = ds = x 10 (4.11) λ ( ) S T S I 2 = 0 (4.12) r ( S ) ( ) ( ) β α β α ν ν ν r2 = + = + = 0 (4.13) λt ( S ) λt ( S1 ) λt S λt S 2 p C = ν 2 ; p = 500 psi (4.14) φ L B = λ ( S ) (4.15) T 2 R 1 1 A = I 1 (4.16) ν λ ( S ) T R The crrespnding ater saturatin prfiles are shn in Figure 6. The high ater saturatin fr case 1a is physically realistic here a much ler mbility rati (represented by µ µ ) ill result in better seep efficiency, i.e. higher ater saturatin behind the fld frnt. 0.8 Residual Oil Saturatin Water Saturatin (S ) Case 1b μ /μ = 20 Case 1a μ /μ = 0.2 Cnnate Water Saturatin Length (m) Fig. 6 Saturatin Prfiles fr the Waterflding Cases 1a) µ µ = 0.2 and 1b) µ µ = 20 at Breakthrugh Times 0.3 ( ), 0.5 ( ), 0.7 ( ), and 0.9 ( ) 16

133 Likeise, the time t breakthrugh can be determined frm equatin (3.35). The time fr ater t breakthrugh in the mbility cntrlled case 1a, is 523 s hereas it is nly 228 s fr the mbility unstable case 1b here the ater (displacing) viscsity is much less than the il viscsity (displaced). First, calculate the integral numerically using u ( t ) befre breakthrugh, using eqtn. (3.2). The time after breakthrugh and the vlumetric flux r ttal velcity can be then calculated frm the flling equatins: t si BT ( ) φ = t + (4.17) 2 2 L x S, s i i tbt f ( S) ds 2 2 pν 2 * λt ( S) s (, ) 2 2 φ L x si t ut ( tsi ) = [ tsi tbt ] = 2 f ( s ) i BT T (4.18) The ttal velcity prfiles are shn fr bth aterflding cases in Fig. 7 (the flling explanatin als makes reference t the ater saturatin prfiles depicted in Fig. 6). The ttal velcity decreases nn-linearly fr case 1a as expected due t the increasing high viscsity ater saturatin. After breakthrugh, the ttal velcity is almst cnstant ing t the unifrm 70% ater saturatin. The ppsite is bserved fr case 1b here the much ler viscsity displacing ater saturates less pre vlume at breakthrugh. As the l viscsity ater saturatin des cntinue t increase after breakthrugh s des the ttal velcity. Fig. 7 Ttal Velcities fr Waterflding Cases 1a) µ µ = 0.2 ( ) & 1b) µ µ = 20 ( ) 17

134 The frnt velcity, in general, des nt advance linearly as a functin f time, ith the applicatin f cnstant pressure bundaries. It is calculated using equatin (3.34) kning the frnt psitin. Befre breakthrugh, the frnt psitin is calculated using equatin (3.19) r mre explicitly, as shn in the flling equatin and is shn in Figure 8 fr bth aterflding cases. t ν BT 2 ( i, BT ) T ( ) φ 0 x S t = u t dt (4.19) Fig. 8 Psitin f the Fld Frnt (Shck Wave) befre Breakthrugh fr Waterflding Cases 1a) µ µ = 0.2 ( ) and 1b) µ µ = 20 ( ) 4.2. Plymer Flding The plymer flding case illustrates the fact that the cnstant pressure bundary slutin rks fr multi-cmpnents, i.e. multiple aves. Physically, plymer may be added t the ater t increase its viscsity t vercme an adverse mbility rati ith respect t the mre viscus il. The parameters used fr the plymer case are shn in Table 1. If c is plymer cncentratin in ater, e chse a linear dependence f ater slutin viscsity n plymer cncentratin, µ ( c) = µ + 200c (4.20) 18

135 hich gives, i.e., f ( S, c) = S 2 S µ a +. (1 S) µ a S f ( S, c) = S + ( c)(1 S) 2 2 (4.21) (4.22) Fr the Riemann prblem, e chse: S L = S = 1.0 ; S R = 0.0 c L = 0.01 ; c R = 0.0 An adsrptin istherm f the frm shn in equatin (4.23), is used t describe the effect f the plymer cncentratin. The additin f the plymer results in the creatin f t shcks and a rarefactin as shn in Fig. 10. This Riemann prblem as analysed by Jhansen and Winther (1988). 0.2c a( c) = (4.23) c 3 12 Water Phase Viscsity (cp) 2 1 Viscsity Adsrptin Adsrptin Istherm x C R Plymer Cncentratin C L Fig. 9 Water Phase Viscsity ( ) and Adsrptin Istherm ( ) as a Functin f Plymer Cncentratin 19

136 Calculated using Netn-Raphsn, the cnstant saturatins f the shcks are fund t be S 1 = and S 2 = 0.514, and are shn in Figure 10. The fractinal fl functin is shn Fig. 10 Fractinal Fl Functin fr the Plymer Case 2 The saturatin prfiles are shn in Figure 11 shing the t shcks and the rarefactin aves. Fig. 11 Saturatin Prfiles fr the Plymer Case 2) at Breakthrugh Times 0.3 ( ), 0.5 ( ), 0.7 ( ), and 0.9 ( ) 20

137 The fractinal fl functins shn in Figure 12 are functins f the plymer cncentratin and saturatin as shn in equatins (4.24) t (4.26). df ν 1 = ( S, cl) (rarefactin) (4.24) ds ν f ( S, c ) (shck) (4.25) 1 L 2 = =1.01 S1 + hlr f ( S, c R ) ν (shck) (4.26) 2 3 = =1.37 S2 Cmparing the right and left fractinal fl curves and the slpe f the tangents at the pint f inflectin fr the shck aves in Figure 12, e see that the right shck is travelling mre quickly than the left shck, i.e. the ater-il shck is advancing thrugh the prus media faster than the increased viscsity plymer ater. This is cnfirmed hen cmpared t the initial higher ttal flux shn in Figure 13 that decreases as the ater-il interface advances thrugh the prus media. Bth, shck ave ν 3 and shck ave ν 2 decelerate as they mve thrugh the prus medium (Figure 12) as des the ttal flux (Figure 13). The ttal flux prfile (in Figure 13) shs that there is a discrete change in shck velcity frm shck ν 3 t shck ν 2. Fig. 12 Fractinal Fl Functins fr the Plymer Case 2) at Saturatin Prfiles fr the Plymer Case 2) at Breakthrugh Times 0.3 ( ), 0.5 ( ), 0.7 ( ), and 0.9 ( ) fr c L = 0.01 and c R = 0 21