Reservoir simulation and geomechanical modeling in Leming Lake, Alberta

Transcription

1 Reervoir imulation Reervoir imulation and geomechanical modeling in Leming Lake, Alberta John J. Zhang, Tony Settari and Laurence R. Bentley ABSTRACT Thi paper deal with reervoir imulation and geomechanical modeling. The phyic o luid low and geomechanical ytem i expreed a a et o conervation equation and imulation i their numerical olution. GEOSIM i ued to carry out imulation o a coupled reervoir and geomechanical model created uing well log, core meaurement and eimic data. Reervoir imulation how a patial pattern o preure, temperature and aturation change around well, which can be ued to predict the change in eimic repone. Geomechanical modeling how igniicant diplacement in the reervoir and in area ar away rom the reervoir, but igniicant change in tre and train ield are mainly limited to the reervoir. Elatic moduli decreae due to exolved ga releaed by lowering preure and high temperature ater a period o production. The moduli will be urther reduced due to horizontal racture predicted by geomechanical modeling. The cumulative eect i expected to be trong enough to be detected by time-lape eimic data. INTRODUCTION Heavy oil ha been produced uing cyclic team timulation rom the Clearwater ormation in Leming Lake, Alberta, Canada. Typically the proce conit o cheduled cycle o injection and production rom vertical, deviated or horizontal well. High injection preure i ued to generate racture, which equentially introduce high temperature team into the reervoir in hope o improving injectivity and productivity. The team chamber and temperature and preure zone are conventionally orecated with reervoir imulation. But the prediction i ar rom complete due to heterogeneity. Recently Imperial Oil hot time-lape three-d eimic urvey over a ew production pad in an attempt to monitor luid low and reervoir condition. Seimic interpretation or change in aturation, preure and temperature in reervoir add another contraint to reervoir imulation in addition to production hitory matching. Reervoir management baed on reervoir imulation optimized by both production perormance and time-lape eimic would enhance heavy oil recovery. In thi paper the author ocu on reervoir imulation and geomechanical modeling with an aim to evaluate how eaible time-lape eimic can be in monitoring reervoir recovery. UNDERLYING PRINCIPLES Fluid injection or/and production rom a reervoir diturb the tatic equilibrium o pore luid and olid ramework. A natural tendency to reach a new balance initiate luid low and deormation. During the dynamic proce conervation law can be applied to aected. Equation (1) i the mathematical ormula or luid conervation (notation ound at end o paper). CREWES Reearch Report Volume 16 (2004) 1

2 Zhang and Bentley ρ φv d = ( ρ φdv) Q t +, (1) where Q i the diturbance (amount o injected or produced luid) leading to luid preure change and luid movement within the reervoir, which in turn caue change o eective tre and deorm the olid ramework. Similarly the conervation o grain olid ma i applicable, a hown in Equation (2): v v ρ (1 φ) v d = [ ρ (1 φ)dv] + Q t. (2) The conervation o momentum or both luid and grain olid are expreed in a vector orm in Equation (3), which can alo be derived rom Newton econd law: S d + [ ρ (1 φ) + ρ φ] gdv = { [ρ (1 φ) v + ρ φv ]dv}. (3) t A econdary outcome o luid low and deormation i energy traner in the media, which i expreed a an energy conervation Equation: [ρ φv H + ρ (1 φ) v H + CT T] d = { [ρ U φ + ρu (1 φ)]dv} + Q t (4) Applying the Green-Gau theorem and removing the integral over the repreentative volume v, we obtain the ollowing equation: v v heat and ( ρ φv ) = (ρ φ) + Q, (5) t [ ρ (1 φ) v ] = [ρ (1 φ)] + Q, (6) t S + [(1 φ)ρ + φρ ] g = [ρ (1 φ) v + ρ φv ], (7) t [ρ φv H + ρ (1 φ) v H + CT T] = [ρ U φ + ρu (1 φ)] + Q heat. (8) t The luid velocity i related to Darcy velocity, ν=(ν -ν )φ, and Equation (5) or inglephae luid low can be reormulated a: [ ρ k/µ ( P - ρ g ) + ρ φv ] = (ρ φ) + Q. (9) t For a multi-component, multi-phae ytem, the equation i decompoed into a number o equation that correpond to each component. 2 CREWES Reearch Report Volume 16 (2004)

3 Reervoir imulation Conidering that the time cale in day during hydrocarbon recovery i very large compared with the magnitude o diplacement in millimeter or maller, ν i negligibly mall. Conequently, all the term multiplied by ν can be eliminated. Likewie Equation (6) i inigniicant and can be removed rom the group o equation. In addition, ν i typically around 10 m/day and ν / t i much maller. To the irt-order approximation, the term related to ν on the right-hand ide in Equation (7) can be omitted. A inal et o equation reult a ollow: and [ ρ k/µ ( P - ρ + t g)] = (ρ φ) Q, (10) [ρ k/µ P H + CT T] = [ρ U φ + ρu (1 φ)] + Q heat, (11) t S + ρ (1 φ) + ρ φ] g 0. (12) [ = Total tre S i reolved into the um o luid preure and eective tre, which relate to the diplacement through the contitutive equation. A a reult, Equation (12) ha two unknown (preure and diplacement vector). All together, Equation (10), (11) and (12) have three unknown and the olution i unique. A numerical olution to Equation (10) and (11) i traditionally called reervoir imulation, which ind luid preure, temperature and aturation (i it i a multi-phae low). There are many commercial otware package or thi purpoe uch a STARTS, EXOTHERM, TETRED, ECLIPSE, VIP, GEOSIM etc., which can handle multi-phae multi-component luid low and heat traner. A limitation o traditional reervoir imulation i that poroity and permeability are aumed to be contant or are artiicially et a a unction o luid preure. For well conolidated rock, thi treatment may not generate ubtantial error. For unconolidated and, however, deormation i igniicant epecially when racture are generated. Poroity and permeability evolution during injection and recovery need to be accurately calculated in order to have accurate reervoir imulation. On the other hand, the tre ield diturbed by production and/or injection i not limited to the reervoir and the urrounding area may undergo igniicant tre change and deormation. The uplit in Cold Lake ater team injection appear to reult rom coniderable thermal expanion ar beyond the production zone. The geomechanical modeling thereore mut be coupled with traditional reervoir imulation. Reervoir imulation and geomechanical modeling can be ully coupled, i.e., Equation (10), (11) and (12) olved imultaneouly, but we rarely do that. Reervoir imulation and geomechanical modeling have been developed eparately in the lat decade.. The commonly ued olution method i iteration or partial coupling between the two independent application. I iterated to ull convergence, it olve the problem a rigorouly a a ully coupled (imultaneou) olution (Settari, 1998). The link i luid preure and poroity and permeability. The preure computed rom reervoir imulation CREWES Reearch Report Volume 16 (2004) 3

4 Zhang and Bentley in a time tep pae a an input to geomemechanical modeling, which compute diplacement, train and tree. The new poroity and permeability computed rom train are then ued by the reervoir imulator, which re-compute the preure and other unknown. Thi proce i repeated until convergence. RESERVOIR CHARACTERIZATION MODEL The reervoir characterization model deine the geometry and phyical propertie o the reervoir, which are the input or the reervoir imulation. In the companion paper, the author decribe a reervoir characterization model developed rom core meaurement, well log and three-dimenional eimic data. Thi ection dicue boundary condition, gridding and how to cale up the phyical propertie rom the mall cale or eimic data to the coarer grid ued or reervoir imulation. A hown in Figure 1, 3-D eimic urvey were repeated in March 1997, February 1998 and February 1999 in an attempt to monitor the ive horizontal well or cyclic team timulation and production. Steam injection tarted in September 1997 and the 1997 eimic urvey i then viewed a the baeline or thee ive horizontal well. The adjacent deviated well in the urvey area were alo injected and produced at the ame period and at much earlier time a well. Although they aect the 3-D urvey area, their interaction i neglected or the moment and will be detailed later in the uture tudy. Since the imulator boundarie are no-low boundary, the area or reervoir imulation wa extended on three ide beyond the eimic urvey area, a hown in Figure 1. The three-dimenional eimic urvey cover an area o 1200 m * 960 m with 120*96 tacked trace. The bin ize i 10 m * 10 m. The reervoir zone extend vertically or 50 m rom 410 m to 460 m, ampled at an interval o 1 m. I thi grid wa or reervoir imulation, the total number o block exceed hal a million. A coarer grid i required in order to carry out the tak o reervoir imulation. By experiment, a grid o 40*36*20 wa inally elected or the area with eimic data and then an additional grid beyond that wa three by our cell, a een in Figure 2. Thi i a tradeo between horizontal and vertical reolution. Horizontally, each reervoir imulation cell contain ix eimic trace. Vertically, 50 time ample correpond to 20 cell and averaging can not be achieved or each cell. It i noted rom the companion paper that the mot important thing rom the eimic ection i location o peak, which are attributed to the exitence o tight rock with zero poroity and zero permeability. Well log and core meaurement indicate that the thickne o tight rock i approximately 2 meter. We re-ampled the time window o 50 m at an interval o 0.5 m. The new total number o ample i 100 and each ample repreent 0.6 meter. Each aumed tight rock peak and the two immediately adjacent ample are aumed to be tight rock. So each peak correpond to 1.8 meter o tight rock. Now each reervoir imulation cell contain 3*3*5 eimic ub-cell. Averaging can be done to ind it poroity and permeability. The poroity i imply an arithmetic average o 45 eimic ub-cell. Permeability, however, i more complicated. We aumed that oil and have a horizontal and vertical permeability o 2000 md and 650 md, repectively. The permeability or tight rock i zero. The averaging methodology i 4 CREWES Reearch Report Volume 16 (2004)

5 Reervoir imulation a ollow: averaging irt along z direction or each o nine vertical column o ub-cell, arithmetic or Kx and Ky and harmonic or Kz. Second along y direction or each o three row, arithmetic or Kx and Kz and harmonic or Ky. Finally along x direction to average the lat remaining ub-cell, harmonic or Kx and arithmetic or Ky and Kz. The inal reult i a hown in Figure 3 and 4. FIG. 1 Time-lape 3-D eimic urvey area (blue) and reervoir imulation area. The goal o timelape eimic urvey wa to monitor the ive horizontal well or cyclic team timulation. GEOMECHANICAL MODEL One o the goal o geomechanical modeling i to ind change in tre and train ar beyond well and the reervoir. The geomechanical model i et to include the reervoir and urrounding ormation. Retricted by computer memory and peed, the number o block outide o the reervoir mut be limited. A een in Figure 5, we choe a number o important eimic relection above the reervoir a block boundarie and extended the CREWES Reearch Report Volume 16 (2004) 5

6 Zhang and Bentley model to the urace. The block below the reervoir extend downward over one thouand meter. The horizontal boundary deined in reervoir imulation i applied to geomechanical modeling and it i deined a a no-diplacement boundary condition. The whole grid or geomechanical model i een in Figure 6. The tatic moduli or geomechanical modeling are oten acquired rom extenive lab teting and modeling, which are limited to a ew ample rom target zone. Depite their validity at coring location, a large area i let without data and interpolation and extrapolation rom a ew point i not reliable. On the other hand, well log and eimic data are abundant and dynamic moduli can be calculated rom them and ued to populate the whole area. The key i how to convert dynamic moduli, which meaure tre-train relationhip at low train amplitude and at a hort period o time, to tatic moduli, which decribe ull long-term tre-train. The dynamic and tatic moduli are dierent becaue material are elatic or the ormer and undergo partially irrecoverable deormation or the latter. In thi tudy, we blocked onic and denity log according to the vertical grid o the geomechanical model and computed the dynamic moduli. An empirical relationhip between dynamic Young modulu and tatic Young modulu wa applied to calculate the tatic moduli. The equation i E=a Ed b (a=0.097 and b=1.485, Wang and Nur, 2000). E i tatic Young modulu and Ed i dynamic Young modulu. Mot value were ound to be cloe to thoe obtained rom lab teting (Walter and Settari, 2002). FIG. 2. Three-dimenional viualization o reervoir imulation grid (43*36*20). The eimic urvey area ha a dimenion o 40*36. 6 CREWES Reearch Report Volume 16 (2004)

7 Reervoir imulation FIG. 3. Three-dimenional viualization o poroity in the area with eimic data FIG. 4. Three-dimenional viualization o permeability (horizontal) in the area with eimic urvey CREWES Reearch Report Volume 16 (2004) 7

8 Zhang and Bentley Glacier Drit Colorado Shale Colny-Vik GrandRapid L. GrandRapid Clearwater Reervoir FIG. 5. Seimic ection in the north-outh direction. It how the major trata above the reervoir. FIG. 6. Three-dimenional viualization o geomechanical modeling (43*36*30). The eimic urvey area ha a dimenion o 40*36. 8 CREWES Reearch Report Volume 16 (2004)

9 SIMULATION PARAMETERS Reervoir imulation In addition to poroity, permeability and tatic moduli, many other parameter are needed to be deined. We ued the mode o light oil to imulate cyclic team injection into the Clearwater ormation in Leming lake. The mode i a three-component and threephae ormulation. The three component are water, light oil and heavy oil and the three phae are water, ga and oil. Depending on preure and temperature, light oil could be in ga phae or in oil phae or in both. Similarly water wa allowed to exit in both ga phae and water phae. But heavy oil i not allowed to vaporize. The three-phae relative permeability wa computed rom two relative permeability table or a water-oil ytem and a liquid-ga ytem uing a modiication o Stone II equation (Walter and Settari, 2000). Other rock phyical propertie were provided by Tauru Reervoir Solution. Settari and Raibeck (1978) meaured the vertical and horizontal tree at a number o depth in poorly conolidated and in Cold Lake. They ound the vertical tre gradient wa 21.3 kpa/m and the horizontal tre gradient wa 20.0 kpa/m. The number wa ued a input or imulation. The initial ditribution o tre implie that vertical racture would be opened i team wa injected. However, the horizontal racture were created when team wa injected becaue team injection changed the tre ield. SOFTWARE TOOL The coupled reervoir imulation-geomechanical modeling wa perormed with GEOSIM, a otware package by Tauru Reervoir Solution. GEOSIM i a modular otware ytem combining a 3-D, 3-phae thermal reervoir imulation with a general 3- D inite element tre-train imulation (Walter and Settari, 2000). TERASIM-THERM model the low o oil, team, water and non condening (NC) ga in an oil reervoir with heat traner in the reervoir and over/under-burden. Steam injection and production can be modeled under thermal condition. FEM3D analye coupled poro- and thermoelaticity or tree and deormation. A mentioned previouly, the two component are partially coupled or iterated until convergence. RESULTS AND DISCUSSION The time-lape eimic urvey were conducted to monitor the team chamber and the ditribution o preure, temperature and aturation or ive horizontal well. The inluence rom other adjacent well probably need to be accounted or, but they have been neglected or the time being. Steam injection tarted in September, 1997, or three o the wetern horizontal well. It lated or two month to October or two well and wa ollowed by production in November. The third well wa injected or three month to November, and began to produce in January The other two well on the eat ide were injected in October until November and production tarted in January In February 1998, when irt time-lape eimic urvey wa done, all well were on production. Our imulation tarted rom September at the beginning o injection and continued to the end o February, Figure 7 how the 3-D viualization o oil aturation ater thirty-eight-day injection or three well. The aturation decreae i immediately around the injection well and thi CREWES Reearch Report Volume 16 (2004) 9

10 Zhang and Bentley zone could be viewed a the team chamber. Away rom the well, the aturation goe back to normal quickly and it appear that the eect i limited to around 30 meter. The zone o high temperature, a een in Figure 8, i larger than the team chamber, panning three cell around 100 metre. The temperature anomaly i alo obervable above the well in the middle. A indicated in Figure 9, luid preure propagate ater than temperature and aturation change and orm a uniied high-preure regime, which may be ueul in generating racture conducive to luid low and heat traner. The cope o inluence rom luid preure exceed 200 meter. Thee three picture how that three zone are ormed: team chamber, temperature anomaly and luid preure ront. Thee three zone would have dierent velocitie and denitie and would repond dierently eimically. In October, the other two well were irt injected and injection continued to November. The two well on the wet ide tarted production in November ater twomonth injection. The remaining well in the middle wa injected until November. The ollowing three picture how condition at the end o November. In Figure 10, the oil aturation around two well on the wet ide decreae becaue o production. Two team chamber can be ound or two well on the eat ide. In Figure 11, the temperature zone or two well on the wet ide ha expanded even though they are on production. The temperature zone increae or the well in the middle due to continued injection. In Figure 12, the luid preure drop or the two well on the wet ide. Once again, luid preure i the mot rapidly propagating parameter and it decline promptly with production. I the velocity i aected by luid preure igniicantly becaue o ga diolution or ex-olution, the ront can be captured by eimic ignal. In February 1998, the irt time-lape eimic urvey wa hot in an attempt to capture change in reervoir condition. At thi time, all well are already on production or two month and we expect preure decline. Temperature may not change much and oil aturation around the borehole may increae compared to team injection period a heated oil low into the. In Figure 13, oil aturation doe increae lightly around all the well. In Figure 14, temperature decreae a little. In Figure 15, the high preure zone hrink and low preure zone expand away rom well. Time-lape eimic repone in the region around well may be trong ince high temperature and exolved ga will decreae the bulk modulu, however low luid preure will increae rame dry modulu. Geomechanical modeling erve two purpoe, a mentioned previouly. One i to couple with reervoir imulation or more accurate reult, which ha been done in the aoregoing part. The other i to ind tre, train and diplacement in area ar beyond the reervoir zone. They may caue collective eimic repone larger than thoe due directly to change within the reervoir. Kenter etc. (2004) ound that time hit in overburden ormation due to hydrocarbon recovery can be an order o magnitude larger than change ound in the reervoir. In the ollowing, we will how a ew picture and examine how thee geomechanical parameter repond to team injection and production in the reervoir zone. On thirty-eighth day ater injection in the three well on the wet ide, the vertical tre doe not change appreciably in mot o the overburden ormation a hown in Figure 16, but it become negative within the reervoir, a ubtantial deviation rom initial tre. Note that Sz i eective tre, which indicate horizontal racture are 10 CREWES Reearch Report Volume 16 (2004)

11 Reervoir imulation generated by team injection. Horizontal tree have a imilar ditribution. No tre change implie that eimic data may not be able to detect team injection rom overburden ormation a Kenter uggeted (2004). Figure 17 exhibit volumetric train, which i limited to the reervoir zone and ha little inluence on the overburden ormation a expected rom tre. However, the vertical diplacement i igniicant and in exce o a ew millimeter on ground in Figure 18. Relatively large diplacement accompanied by inigniicant tre i attributed to mall moduli in unconolidated and and hale in overburden ormation. At the end o November 1997 (eventy-ixth day), the two well on the eat ide have been injected or two month. Injection in the well in the middle ha continued or three month. The two well on the wet ide are on production. In Figure 19, the high magnitude o negative vertical tree are ituated in the newly injected well, while they have become relatively mall, even poitive, in two well put on production in November. Dierent rom the cenario in Figure 17, the volumetric train ha reached the ground urace a een in Figure 20. Thi i caued by prolonged injection in the underlying reervoir. The expanion o the overlying ormation may decreae the elatic moduli and may be eimically detectable. The vertical diplacement o the overburden are urther increaed and expanded to a larger area a hown in Figure 21. The uplit at the ground urace i on the order o centimetre. In February 1998, all well were on production. It i expected that vertical tre decreae and could be poitive within the reervoir, depending on the duration and quantitie o oil production. The volumetric train i expected to drop and become negative. The vertical diplacement hould decreae. Figure 22 indicate an increae in vertical tree in the reervoir zone compared to injection time. A a matter o act, vertical tree have become poitive, which may caue racture created during team injection to cloe. In Figure 23, the volumetric train remain poitive and the initial tate ha not been retored. In Figure 24, the vertical diplacement till maintain high value. The implication i that the horizontal racture exited at the time o the irt time-lape eimic urvey and we expect low rame moduli. Combined with the previouly mentioned luid actor, the elatic moduli in the reervoir zone i expected to be much lower than the initial moduli. We expect it i eaible to monitor thee zone eimically. The change o elatic moduli in the overburden appear not to reult in ubtantial time hit and it may not be eaible to ee the change eimically.. CREWES Reearch Report Volume 16 (2004) 11

12 Zhang and Bentley O il S aturation D itribution ater 38 D ay o S team Injection Y Z X So D FIG. 7. Ditribution o oil aturation ater one-month team injection. T e m pe ra ture D itribution ater 3 8 D ay o S team Injection Y Z X Temp D FIG. 8. Ditribution o temperature ater one-month team injection. 12 CREWES Reearch Report Volume 16 (2004)

13 Reervoir imulation F lu id P re u re D itrib u tio n a te r 3 8 D a y o S te a m Inje ction Y Z X Pre D FIG. 9. Ditribution o luid preure ater one-month team injection. O il S a tura tion D itribution ater 78 Day o Steam Injection Y Z X So D FIG. 10. Ditribution o oil aturation ater more than two-month team injection. CREWES Reearch Report Volume 16 (2004) 13

14 Zhang and Bentley Temperature Ditribution ater 78 Day o Steam Injection Y Z X Temp D FIG. 11. Ditribution o temperature ater more than two-month team injection. Fluid Preure Ditribution ater 7 8 D ay o S team Injection Y Z X Pre D FIG. 12. Ditribution o luid preure ater more than two-month team injection. 14 CREWES Reearch Report Volume 16 (2004)

15 Reervoir imulation O il S aturation D itribution ater 60 D ay o O il P roduction Y Z X So D FIG. 13. Ditribution o oil aturation ater two-month oil production. Temperature Ditribution ater 60 Day o Oil Production Y Z X Temp D FIG. 14. Ditribution o temperature ater two-month oil production. CREWES Reearch Report Volume 16 (2004) 15

16 Zhang and Bentley Fluid Preure Ditribution ater 60 Day o Oil Production Y Z X Pre D FIG.15. Ditribution o luid preure ater two-month oil production. FIG. 16. Ditribution o vertical tree in the reervoir and overburden ormation on thirtyeighth day ater injection (the unit i in KPa). 16 CREWES Reearch Report Volume 16 (2004)

17 Reervoir imulation FIG. 17. Ditribution o volumetric train in the reervoir and overburden ormation on thirty-eighth day ater injection (the unit i in %). FIG. 18. Ditribution o vertical diplacement in the reervoir and overburden ormation on thirty-eighth day ater injection (the unit i in meter). CREWES Reearch Report Volume 16 (2004) 17

18 Zhang and Bentley FIG. 19. Ditribution o vertical tree in the reervoir and overburden ormation on eventy-eighth day ater injection (the unit i in KPa). FIG. 20. Ditribution o volumetric train in the reervoir and overburden ormation on eventy-eighth day ater injection (the unit i in %). 18 CREWES Reearch Report Volume 16 (2004)

19 Reervoir imulation FIG. 21. Ditribution o vertical diplacement in the reervoir and overburden ormation on eventy-eighth day ater injection (the unit i in meter). FIG. 22. Ditribution o vertical tree in the reervoir and overburden ormation ater two-month production (the unit i in KPa). CREWES Reearch Report Volume 16 (2004) 19

20 Zhang and Bentley FIG. 23. Ditribution o volumetric train in the reervoir and overburden ormation ater two-month production (the unit i in %). FIG. 24. Ditribution o vertical diplacement in the reervoir and overburden ormation ater two-month production (the unit i in meter). 20 CREWES Reearch Report Volume 16 (2004)

21 CONCLUSIONS Reervoir imulation The principle o coupling reervoir imulation with geomechanical modeling i to olve the equation o conervation (ma, energy and momentum) by iteration between low imulation and geomechanical imulation. A reervoir characterization model uitable or reervoir and geomechanical imulation wa created or a portion o the Leming Lake reervoir. Reervoir imulation how zone o changing aturation, preure and temperature around injection well. Reervoir imulation how the ater a period o production low preure exolve ga rom the oil. The ga plu the high temperature contribute to lowering the bulk modul. However, the lower preure will caue and increae in eective tre tending to increae the dry rame moduli. Geomechanical modeling reveal tre, train and diplacement in the reervoir and in area ar away rom the reervoir. Change in tre and train happen mainly within the reervoir and they are not igniicant in other area. Geomechanical modeling predict igniicant change in vertical diplacement both in the reervoir and on ground even ater a period o production. It remain to be determined what magnitude o change in eimic repone will be produced by thee complex competing procee. ACKNOWLEDGMENTS We would like to expre our appreciation to the CREWES ponor or their upport o thi reearch. We alo grateully acknowledge Imperial Oil or their permiion to ue their eimic data. REFERENCES Kenter, C. J., Van den Beukel, A. C., Hatchell, P. J., Maron, K. P., Molenaar, M. M., 2004, Evaluation o reervoir characteritic rom timehit in the overburden, American Rock Mechanic Aociation. ARMA/NARMS Settari, Antonin and F. M. Mourit, 1998, A coupled reervoir and geomechanical imulation ytem, SPE Journal, Settari, Antonin and Raibeck, J. M., 1978, Fracture mechanic analyi in in-itu oil and recovery. Paper , 29 th Annual Technical Meeting o the Petroleum Society o the Canadian Intitute o Mining and Metallurgy, Calgary, June 13-16, Walter, D. A. and A. Settari, 2000, Poreelatic eect o cyclic team timulation in the Cold Lake Reervoir, SPE Wang, Zhijing and A. Nur, 2000, Seimic and acoutic velocitie in reervoir rock, recent development, Geophyic reprint erie. CREWES Reearch Report Volume 16 (2004) 21

22 Zhang and Bentley C T : thermal conductivity (E/mT) g: gravity accelerator (m/ 2 ) H : enthalpy o luid (E/kg) H : enthalpy o olid (E/kg) k: permeability (m 2 ) P: preure (kg/m 2 ) Q : ource or ink o luid (kg/) Q : ource or ink o olid (kg/) Q heat : ource or ink o heat (E/) S: total tre (kg/m 2 ) T: temperature (T) t: time (t) U : internal energy o luid (E/kg) U : internal energy o olid (E/kg) v : luid velocity (m/) v : olid velocity (m/) µ : vicoity o luid (kg/m) ρ : luid denity (kg/m 3 ) ρ : luid denity (kg/m 3 ) φ: poroity E: energy kg: ma m: length : time T: temperature V: volume NOTATIONS 22 CREWES Reearch Report Volume 16 (2004)