Excerpt frm the Prceedings f the COMSOL Cnference 9 Bstn w-imensinal COMSOL Simulatin f Heavy-Oil Recvery by Electrmagnetic Heating M. Carrizales *, and Larry W. Lake he University f exas at Austin, *Crrespnding authr: he University f exas at Austin, University Statin, C3, Austin, X 787, carrizalesmaylin@mail.utexas.edu Abstract: Intrducing heat t the frmatin has prven t be an effective way f lwering the il viscsity f heavy ils by raising the temperature in the frmatin. he applicatin f electrical energy has gained mre interest during the last decade because it ffers fewer restrictins fr its successful applicatin cmpared t the cnventinal steam flding methds [-]. Althugh this recvery technique has been studied befre [-8], there are n cmmercial reservir simulatrs yet available t mdel the respnse f a reservir when it underges heating. his paper presents the use f COMSOL Multiphysics t simulate single-phase flw in a reservir when an surce is applied. Starting frm mass and energy balance, and arcy s law we mdeled the effect n heating n temperature, pressure and flw rate fr a axis-symmetric system (r and z crdinates). Numerical results frm COMSOL Multiphysics [9] were validated with analytical slutins fr simplified cases develped earlier []. We determined temperatures, pressures and the ultimate il prductin btained frm a reservir when heating is applied. Keywrds: Heavy-il recvery, Electrmagnetic heating, single-phase flw.. Intrductin hermal il recvery methds add heat t a reservir t reduce il viscsity and make il mre mbile. hermal recvery invlves different well-knwn prcesses such as steam injectin, in situ cmbustin, steam assisted gravity drainage (SAG), and a mre recent technique that cnsists f heating the reservir with electrical energy [, 5, 8]. Steam flding leads in develpment and applicatin by far; hwever, the use f electric heating fr heavy-il reservirs can be especially beneficial where cnventinal methds can nt be used because f large depth, thin frmatins, frmatin discntinuity, n water available, reservir hetergeneity, r excessive heat lsses. Chakma and Jha [8] shwed that heating is an effective way t intrduce energy t the reservir in a cntrlled manner and that this energy can be directed int a specific regin. Hence, the applicatin f electrical energy has gained mre interest lately. In this study, heating refers t highfrequency heating, radi frequencies (RF) and micrwave (MW) are examples, that is prduced by the absrptin f electrmagnetic energy in the frmatin. he amunt f heat absrbed will depend n the absrptin cefficient f the medium, which in turn, will depend n the electrical prperties that vary with temperature and water saturatin. In this wrk, water saturatin is very lw and assumed t be immbile; therefre, electrical prperties vary nly with temperature. Althugh several authrs have dealt with the pssibility f using heating t enhance recvery frm heavy il reservirs [-8], there are n cmprehensive mdels r cmmercial tls yet available that cuples heating t reservir simulatin. his study was carried ut with COMSOL t slve an heating mdel that cuples fluid flw and the thermal respnse f a reservir when an surce is applied at a vertical wellbre. he mdel cnsists f tw nn-linear partial differential equatins (PE s) derived frm an energy balance, where the energy frm the antenna is added as a surce term, and a mass balance in which fluid flw is described by arcy s law. hese equatins are cupled thrugh the dependency n the flw velcity t slve fr temperature as well as the dependence n temperature t calculate the flw velcity thrugh the viscsity in arcy s law. In slving this mdel, we used COMSOL because f its flexibility when cupling Multiphysics. Numerical results were validated with analytical slutins fr a ne-dimensinal heating mdel previusly develped [].
. Use f COMSOL Multiphysics COMSOL ffers tw ptins fr the slutin f the prpsed mdel. Fr a single-phase flw mdel, fluid flw and heat transfer can be taken frm the Earth Science Mdule r mdeled using the PE fr time dependent prblems applicatin [9]. Althugh using the Earth Science Mdule can be simpler and faster than inputting specific PE s int COMSOL Multiphysics, the latter seems t be mre cnvenient fr ur future gal f mdeling multiphase fluid flw cupled t heating using COMSOL Multiphysics. Symmetry is cnventinally assumed in a reservir at the wellbre fr single-well numerical reservir simulatin. aking advantage f this cnditin, we mdeled nly half f the reservir in the radial directin assuming the well with the surce is lcated at the center f the reservir (See Figure ). he mdel cnsists f three layers (z-directin); the tp and bttm are nn-reservir layers used merely t accunt fr heat transfer by cnductin (heat lsses) thrugh the interir bundary frmed with the middle layer. he middle layer crrespnds t the reservir f interest, where the energy surce is applied, and fluid flw ccurs. energy flw is cunter-current, which means it flws ppsite t the fluids t be prduced.. Gverning equatins he mdel is derived frm a mass balance n the il phase where fluid flw is described by arcy s law, and a ttal energy cnservatin equatin that includes heat transprt by cnvectin, cnductin and the energy as a surce term. he verall energy cnservatin equatin is btained frm an energy balance dne n tw phases, a s-called "phtn" phase that transprts the energy, and the cnventinal "material" phase where the reservir and the fluids reside [] Assuming there is nly an il phase flwing with il as a single cmpnent in the reservir, with n gas disslved in it; the mass balance fr il can be written as t φρ ( ) + ( ρ ) = i (.) u Figure. Schematic view f heating fr cuntercurrent flw. An antenna is placed at the center f the prducing well in frnt f the target zne cnfined by the adjacent layers. Here, u represents the vlumetric il rate, which can be expressed by arcy s law as kk r u = ( p + ρ g z) (.) μ where φ and k dente the prsity and abslute permeability tensr f the prus system, while ρ, μ, k r, p represent the density, dynamic viscsity, relative permeability, and pressure f the mbile phase (il), and g is the gravitatinal vectr (pinting dwnward). he medium is istrpic, k = ki, and gravity effects are ignred. Expanding the derivative, and replacing arcy s velcity in equatin (.), we btain what is usually called the pressure equatin as ( φc ) ( p ) where Pwer Cnfining Layer Fluid Flw Cnfining Layer p k i = (.3) t μ c = B B p Fluid Flw flw Antenna h = hickness and represents the il cmpressibility, and B is the il frmatin vlume factr. Fluid prperties are cnstant; except fr the il viscsity, which is determined accrding t the fllwing relatinship F/ μ = e (.4)
where and F are empirical cnstants determined frm tw measured viscsities at knwn temperatures (abslute). Equatin (.3) mdels the il flw in the reservir. Since the idea f heating is t intrduce heat t the reservir, we need the cnservatin f ttal energy equatin t cmplete the mdel. Fr the single flw f il, the ttal energy in the system made f the cntributin f energy transprt by cnductin, cnvectin, and heating is given by M + Mu i + H i u t i k = iq ( ) eff ( ρ ) Where M = ( φms + ( φ) Ms) (.5) Here, M is the vlumetric heat capacity, H is the enthalpy, and k eff is the effective reservir thermal cnductivity that cmprises the rck (index s ) and the il (index ). he term n the right side f equatin (.5) represents the heating surce, and its expressin is derived in the fllwing sectin.. Electrmagnetic () Heating erm q he term n the right side f equatin (.5) is the gain in heat cntent because f the pwer applied thrugh the "phtn" phase as discussed by Bird et al. []. his term can be btained frm a separate energy balance n the phtn phase assuming steady-state since the mass f the phtns is negligible [4, ]. he mathematical frmulatin fr this term can be als derived frm the slutin f Maxwell s equatins r frm the applicatin f Lambert s law [6]. he gain in heat cntent prvided frm the surce can be mathematically expressed in multiple ways; hwever, which f these is the mst accurate expressin, especially in a multidimensinal flw, is still unknwn. Fr this wrk, the energy balance n the phtn phase is expressed as: i q = q (.6) where the term q is the magnitude f the flux vectr. In a radial system, assuming that energy flws frm the surce in the hrizntal directin nly, equatin (.6) can be written as: ( rq ) =q (.7) r r Integratin f equatin (.7) gives: rq r = Ce (.8) ( ) r where C is an integratin cnstant that can be evaluated with the fllwing bundary cnditin: q ( rw ) = P (.9) Using the abve bundary cnditin, equatin (.8) can be rewritten as: ( r r w ) Pe q ( r) = (.) r where P is the incident pwer radiated at the wellbre, α is the absrptin cefficient, r is the radial distance, and r w is the wellbre radius. hen, the energy cntributin beacause f the surce applied in a radial system can be expressed as: ( r r w ) Pe i q = (.) r his expressin represents the surce term in the energy balance fr a radial system. he absrptin cefficient is derived frm Maxwell s equatins [5] and has the fllwing expressin εμ ' σ α = ω + εω (.) where ω is π times the frequency, ε is the real part f the cmplex permittivity, μ ' is the real part f the cmplex magnetic permeability, and σ is the dielectric cnductivity f the medium, which is a functin f temperature.
.3 Initial and Bundary Cnditins he heating mdel described by the equatins abve wuld nt be cmplete withut a descriptin f the initial and bundary cnditins used t slve the system. he primary variables slved fr are pressure and temperature assuming single-phase flw. Cnstant temperature and pressure thrughut the reservir are taken as the initial state. In slving the pressure equatin, the pressure at the external bundary (r e ) is kept cnstant. At the wellbre, a cnstant flwing bttmhle pressure (p wf ) was used. At the interir bundaries (z=h, and z=) between the reservir and the adjacent frmatins, a n-flw bundary cnditin was impsed fr the slutin f the mass balances, s n crssflw is allwed. Fr the slutin f the energy equatin, temperature is kept cnstant and equal t the initial temperature at the external bundary f the reservir, at the tp f the verburden, and at the bttm f the underburden. At the wellbre, cnvective flux was used as the cnditin t btain the temperature distributin. Cnvectin heat lss ccurs nly in the radial directin. Cnductin heat lss thrugh the adjacent frmatins is included by setting the cntinuity f heat as a bundary cnditin between the reservir and the tp and bttm frmatins. 3. Numerical Simulatins he numerical implementatin f the mdel previusly derived was accmplished by using the PE applicatin in general frm prvided by COMSOL Multiphysics. We first validated the implementatin f the numerical mdel in COMSOL with analytical slutins fr transient flw fr a special simplified case (See Appendix). hen, we used the numerical mdel t study the effect f heating n recvery, with sensitivities n the input pwer and the frequency f the surce. he dmain is a, three layer system with a radial extent f 5 ft, and a ttal reservir vertical extensin f 46 ft. Fluid, rck, and electrical prperties used were cllected frm varius published papers. able summarizes the basic data used fr a hypthetical reservir under cnsideratin. 4. Results and iscussin 4. Validatin Figure shws a cmparisn between the numerical slutin fr temperature btained frm COMSOL and the analytical slutin derived fr a radial heating mdel neglecting cnductin, and using a cnstant prductin rate cnditin at the wellbre f bbl/day. Numerical and analytical results are in gd agreement. Using the same prperties input in the mdel slved by COMSOL with a dmain f 64 ft f length, a sealed external bundary, and assuming n heat is intrduced t the reservir, we carried ut a simulatin using the reservir simulatr SARS [] t cmpare the results fr pressures, and il rate btained during cld prductin frm bth slutins. Figures 3-4 shw a cmparisn f the pressure with distance, and the il rate btained. A n flw cnditin at the external bundary was impsed fr a prper cmparisn with the results frm SARS. A reasnable agreement is shwn, which allws cnfirming the validity f the mdel implemented in COMSOL fr the simulatin f the heavy il recvery by using heating. able : Basic data f a hypthetical reservir used fr the validatin f the heating mdel. Prperty Value Oil density, lbm/ ft 3 6.4 Permeability, md, Prsity, fractin.38 Well radius, ft.3 Initial pressure, psi 3 Initial temperature, F Wellbre pressure, psi 7 Oil cmpressibility, /psi 5E-6 hermal cnductivity, lbf/s.f.38 Oil vlumetric heat capacity, lbf.ft/ft 3.F.9E4 Empirical cnstant fr viscsity crrelatin, cp.e-6 Empirical cnstant F fr viscsity crrelatin, F.4E4 Initial viscsity, cp 3,78 Pwer input, Watt 63, Absrptin cefficient @ 95 MHz, /ft.4
emperature, F 7 6 5 4 3 5 days_analytical days_analytical 5 days_comsol days_comsol 3 4 5 istance frm wellbre radius, ft Figure. Cmparisn f transient temperature prfiles fr cunter-current radial flw btained with COMSOL vs. analytical slutins fr the special case f n cnductin. Pressure, psia 35 3 5 5 5 SARS_initial SARS_days SARS_36days SARS_3days COMSOL_days COMSOL_36days COMSOL_3days 4 8 6 istance frm wellbre, ft Figure 3. Cmparisn f transient pressure prfiles fr cunter-current radial flw btained with COMSOL vs. SARS fr cld prductin (N heating). Oil rate, bbl/day 8 6 4 6 8 4 3 ime, days SARS COMSOL Figure 4. Cmparisn f il prductin btained with COMSOL vs. SARS fr the cld case (N heating). 4. w-dimensinal Heating Case Our main bjective here is t use COMSOL t simulate a reservir underging heating. Once the implementatin in COMSOL was validated fr the n heating case, we added the surce term t study the effect f heating n temperature, pressure, and il rate prduced. ata fr this prblem is shwn in able, a radial extent f m (38 ft) was used. Figures 5 and 6 are surface plts f temperature and pressure; the distance crdinates (r, z) are displayed in meters. Figure 5 shws the temperature distributin fr a reservir after 3 years f heating. he temperature at the wellbre reaches a maximum f 45. F, and abut 4 F at m frm the wellbre, which means a cnsiderable area f the reservir is heated t an effective temperature in terms f viscsity reductin. Since vertical heat lss by cnductin is allwed, the temperature in the reservir (middle layer) clse t the cnfining layers, where n heating is cnducted, is lwer than at the center f the reservir. his result shws the ability f fcusing the heat intrduced t the reservir with heating aviding excessive heat lsses as is ften the case f steam injectin. Figures 6 and 7 shw the pressure prfile btained fr the heating case and the prductin rate frm heating cmpared t cld prductin rate. Since there is n flw in the cnfining layers, nly the prducing layer is shwn. Figure 8 shws an imprvement in cumulative il prductin frm heating f abut 5.4 times cld prductin. able : Basic data f a hypthetical reservir used fr the study f heating fr heavy-il recvery Prperty Value Permeability, md, hickness, m 3 Prsity, fractin.38 Well radius, m. External radius, m Initial pressure, psi 77 Initial temperature, F Wellbre pressure, psi 5 Initial viscsity, cp 3,78 Pwer input, Watts 7, Absrptin cefficient @ 95 MHz, /m.33
8 Oil rate, bbl/day 6 4 N heating heating 4 6 8 ime, day Figure 7. Oil rate cmparisn fr the heating case and cld prductin (n heating). Figure 5. emperature ( F) prfile fr a twdimensinal reservir after 3 years f heating btained with COMSOL. Cumulative Oil, bbl/day 4 3 N-heating Heating 4 6 8 ime, days Figure 8. Cumulative il prductin cmparisn fr the heating case vs. cld prductin (n heating). 5. Cnclusins A - numerical simulatin f heavy il recvery by heating using COMSOL, has been successfully cnducted and validated with analytical slutins fr a - case, and with the reservir simulatr SARS fr a - case f cld prductin. Results frm this wrk are encuraging t the use f COMSOL fr simulating heating fr heavy il recvery, and they will be extended t study multiphase flw and phase changes when an heating surce is applied. Figure 6. Pressure (psi) prfile fr a tw-dimensinal reservir after 3 years f heating btained with COMSOL. 6. References. Hiebert, A.., Vermeulen, F.E., Chute, F.S. and Capjack, C.E. Numerical Simulatin Results
fr the Electrical Heating f Atabasca Oil-Sand Frmatins. SPERE (): 76-84, (986).. Sahni, A., Kumar, M., Knapp, R.B. and Livermre, L. Electrmagnetic Heating Methds fr Heavy Oil Reservirs. Paper SPE 655, (). 3. Abernethy, E.R. Prductin Increase f Heavy Oil by Electrmagnetic Heating. J. Cdn. Pet. ech 5 (3): 9-97 (976) 4. Araque, A. and Lake, L.W. Aspects Relevantes Sbre Fluj de Fluids Baj Calentamient Electrmagnétic. PVSA- Intevep, IN 9388. () 5. Kim, E.S. Reservir Simulatin f in situ Electrmagnetic Heating f Heavy Oils. Ph dissertatin, exas A & M U., Cllege Statin, exas. (987). 6. Ovalles, C., Fnseca, A., Lara, A., et al.. Opprtunities f wnhle ielectric Heating in Venezuela: hree Case Studies Invlving Medium, Heavy, and Extra-Heavy Crude Oil Reservirs. Paper SPE 7898, (). 7. McPhersn, R.G., Chute, F.S. and Vermeulen, F.E. Recvery f Atabasca Bitumen with the Electrmagnetic Fld Prcess. J. Cdn. Pet. ech 4 (): 44-5, (985). 8. Chakma, A. and Jha, K.N. 99. Heavy Oil Recvery Frm hin Pay Znes by Electrmagnetic Heating. Paper SPE 487, (99). 9. COMSOL Multiphysics, COMSOL Multiphysics Mdeling Guide Versin 3.5a, COMSOL AB (7). Carrizales, M., Lake, L.W., and Jhns, R. Prductin Imprvement f Heavy-Oil Recvery by Using Electrmagnetic Heating. Paper SPE 573, (8). Bird, R.B., Steward, W.E. and Lightft, E.N. ransprt Phenmena, 56-57. New Yrk City: Jhn Wiley & Sns, Inc. (). Cmputer Mdeling Grup, Ltd., SARS Users Guide, Versin 8, Calgary, Canada. 7. Acknwledgements Larry W. Lake hlds the W. A. (Mnty) Mncrief Centennial Chair at he University f exas. 8. Appendix A. ransient emperature. N Cnductin Fr cunter-current radial flw, neglecting cnductin, and intrducing the surce term, the energy balance reduces t ( r r ) Mq w α Pe M = + (A.) t π h r r π rh where the ttal vlumetric heat capacity (M ) is given by ( φ ) M r M = φm + simplify Eq. A., we defined the variable ξ = r, then d ξ = rdr. Substitutin f this int A. gives ( ξ ξ ) Mq α Pe M = + t h π ξ πhξ (A.) In dimensinless frm Eq. A. can be written as w α ξ ξ = e (A.3) t ξ ξ with BC s ( ξ, ) = (,t ) = where: M q t = t πhm ξ, α = αξ e, M u = P e ( ) ξ, and ξ =. ξ e Applying Laplace transfrms, Eq. A.3 can be transfrmed int: w d α ξ ξ s = e dξ sξ (A.4) with BC { (,t )} (,s ) L = =. Slutin f Eq. A.4 using the given BC, gives: e = e ( t+ ξ) α ξ ξ ξ e + e α ξ ξ ξ ξ < t ξ > t (A.5)