An Improved Model for Predicting Fluid Temperature in Deep Wells

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1 Mthemticl Modelling nd Applictions 016; 1(1): doi: /j.mm An Improved Model for Predicting Fluid Temperture in Deep Wells Boyun Guo, Jinze Song Petroleum Engineering Deprtment, University of Louisin t Lfyette, Louisin, USA Emil ddress: guo.boyun@gmil.com (Boyun Guo), pete.songjinze@gmil.com (Jinze Song) To cite this rticle: Boyun Guo, Jinze Song. An Improved Model for Predicting Fluid Temperture in Deep Wells. Mthemticl Modelling nd Applictions. Vol. 1, No. 1, 016, pp doi: /j.mm Received: July 17, 016; Accepted: October 14, 016; Published: October 1, 016 Abstrct: The objective of this study ws to develop n improved method to predict fluid temperture profiles in hightemperture wells for designing production string in deep-wter development. The method ws developed on the bsis of het trnsfer involves het convection nd conduction inside the production string nd in the nnulr spce. The governing equtions were solved using the method of chrcteristics, resulting in two simple closed-form equtions. The method ws coded in spredsheet for esy pplictions. Dt from three wells were employed to check the ccurcy of the new method. Comprisons of results from Hsn's method, Gilbertson et l.'s method, nd the new method with temperture dt mesured in two gs-lift wells show tht the new method best predicts well tempertures in trend. A comprison of results given by Mo's method nd the new method with tempertures observed in deep-wter gs well testing indictes tht the new method better predicts well tempertures with errors less thn 4%. This work provides petroleum engineers simple nd ccurte method for predicting temperture profiles in oil nd gs production opertions, especilly deep-wter opertions. It elimintes the need for sophisticted nlyticl nd numericl models in fluid temperture nlysis. Keywords: Fluid Temperture, Deep Wells, Gs-Lift Wells, Het Trnsfer 1. Introduction Prediction of fluid temperture profile is vitlly importnt for designing test string in deep-wter gs wells. A literture survey shows tht severl reserchers hve proposed their theoreticl models for fluid temperture profiles in oil wells. Rmey (196) presented theoreticl model to estimte fluid temperture s function of well depth nd production time. An pproximte nlyticl solution to the trnsient hetconduction problem involved in movement of hot fluids through wellbore ws derived. This model ws modified by lter reserchers. Sgr extended Rmey s model to multiphse flow in wellbore by considering kinetic energy nd Joule-Thompson expnsion effect [1], []. A simplified model suitble for hnd clcultions ws proposed on the bsis of the generl model in which the Joule-Thomson nd kinetic-energy terms were replced with correltions. In ddition, his contribution ws to introduce the Coulter-Brdon eqution into gs lift wells. Alves developed generl model for predicting flowing temperture in devited wellbores nd pipelines [3]. Also, pproximte methods for determining two-phse het cpcity nd Joule-Thomson coefficient were proposed. Hsn presented n pproch to estimte wellbore fluid temperture during stedy-stte two phse flow. It llows for wellbore het trnsfer by conduction, convection, nd rdition [4]. King showed n nlyticl solution for trnsient temperture field round csed nd cemented wellbore [5]. Guo developed simple model for predicting het loss nd temperture profiles in insulted pipelines [6]. Spindler derived nlyticl models for wellbore-temperture distribution [7]. Some investigtions were performed on het losses in stem injection in the wellbore. Investigtors include Stter, Huygen, Bck, Durrnt, nd Pcheco [8] ~ [1]. Chiu nd Thkur presented het losses in directionl wells considering the chnge of injection conditions [13]. The initil investigtion of gs temperture t injection depth of gs-lift wells ws presented by Kirkptrick [14]. His simple model presented flowing temperture grdient to clculte gs temperture t depth of injection vlves. Winkler presented lgorithm for more ccurtely predicting nitrogen-chrged gs-lift vlve opertion t different

2 Mthemticl Modelling nd Applictions 016; 1(1): tempertures [15]. Lgerlef climed gs-lift-vlve test rck opening design methodology for extreme kickoff temperture conditions [16]. Hsn developed mechnistic model for the flowing temperture of nnulr nd tubing in gs lift wells bsed on energy blnce eqution [17]. The uthor ssumes stedy stte flow nd stedy het trnsfer between tubing nd csing. However, kinetic nd potentil energy terms re neglected in the energy blnce eqution. Hernndez performed downhole temperture survey nlysis for wells on intermittent gs lift [18]. Yu modeled the prediction of wellbore temperture profiles during hevy oil production ssisted with light oil lift [19]. Severl reserchers, including Gilbertson et l, hve designed thermlly ctuted sfety vlves for gs lift wells [0]. Gilbertson modeled stedy-stte temperture profile in gs lift wells nd verified with experimentl dt. However, Joule-Thompson effect ws not considered when clculting the mixed temperture in tubing. So the ccurte prediction of temperture profiles becme the limits for their design. Hn presented itertion lgorithms for multi-interfce het trnsfer in pipe flow bsed on mss- nd momentum conservtion [1]. Wooley computed downhole tempertures in circultion, injection, nd production wells with numericl model []. Other numericl models include those developed by Leutwyler, Trgesser, nd Nelson [3] ~ [5]. Although these numericl models hve removed severl unrelistic ssumptions mde for deriving those nlyticl models, their pplictions hve not been populr due to their very limited ccess by most engineers. In summry, number of therml models, both nlyticl nd numericl, hve been developed for predicting fluid temperture profiles in oil wells. These models cover nturl flow, gs-lift, nd therml-recovery oil wells. Among these models, Hsn s mechnistic model hs gined most pplictions in gs lift wells nd Gilbertson et l s model hs been widely ccepted for nturlly flowing oil wells [17] [0]. These two models re compred with new model developed for deep-wter gs wells nd field dt in this work.. New Anlyticl Model A new nlyticl solution ws derived in this study for predicting temperture profiles inside work string (test string, tubing, or drill string) nd in the nnulus, ssuming upwrd flow in the string nd down-wrd flow in the nnulus. Resultnt equtions in the new model re summrized in this section. Derivtion of the model is vilble upon request. The derivtion of the mthemticl models ws bsed on the following ssumptions: (1) The therml conductivity of csing is infinite. () The geotherml grdient is not ffected by the fluids in the wellbore. (3) Het cpcity of fluid is constnt. (4) Friction-induced het is negligible. The fluid temperture profiles inside the string T t nd in the nnulus T re expressed s: C1 rl C 1 rl T t e e EL D q + q + + (1) 1 r1 rl r 1 rl h C1 rl C 1 rl T Ce 1 + C e + E e + e + EL + D f q q f q q () where c βg (10) R( A J) SG C1 (3) FR KG C K( A J) SF (4) GK FR p + p 4q r1 (5) p p 4q r (6) γ β (7) πρd ' 4mɺ t t πdk t t b Cmɺ D d t ( ) t t t t (8) (9) d βt go (11) dfq Q D (1) q F G cf E (13) q r + h 1 (14) fq f b (15) r + h (16) fq h b (17)

3 Boyun Guo nd Jinze Song: An Improved Model for Predicting Fluid Temperture in Deep Wells Eh I (18) f C mɺ T -43.7C mɺ σ C mɺ + C mɺ o o oil o o (31) E + hd J (19) f K θf e q 1 rl 1 mx (0) m γ (1) N ( h mf ) M + ( + h) cf () p + h (3) q h mf (4) R θg e q 1 rlmx (5) S EL + D θil θj σ (6) πρ α ( d D ) c t 4mɺ πdckc β C mɺ D D ( ) w c πdk t t γ C mɺ D d ( ) t t 0.84 C mɺ θ C mɺ + C mɺ o o (7) (8) (9) (30) where A Cross section re of the nnulus, m. A t Cross section re of string, m. C Het cpcity of fluid in nnulus, J/kg- C. C o Het cpcity of fluid from formtion, J/kg- C. C t Het cpcity of fluid in string, J/kg-C. D c Outer dimeter of cement sheth, m. d t Inner dimeter of string, m. D t Outer dimeter of string, m. D w D c, wellbore dimeter of csed hole, m. K c Therml conductivity of cement, W/m- C. K t Therml conductivity of string, W/m- C. L mx Totl depth, m mɺ Mss flow rte of fluid in nnulus, kg/s. mɺ t Mss flow rte of fluid in string, kg/s. Q Flow rte in nnulus, m 3 /s. T,0 Temperture in nnulus t surfce L, C. T,L Temperture in nnulus in point L, C. T oil Temperture of formtion fluid, C. T t,l Temperture in string t point L, C. ρ Density of fluid in nnulus, kg/m 3. ρ t Density of fluid in string, kg/m Model Comprison The new nlyticl model ws compred with Hsn s model, Gilbertson et l. s model, nd dt mesured in the ctul wells reported by these uthors [17], [0]. Figure 1 presents comprison of results given by Hsn s model nd the new model using the bsic well dt presented by Hsn s pper [17]. Figure 1. Comprison between the new model nd Hsn s model. Figure 1 indictes tht, in generl, there is good greement between the temperture profiles in nnulus nd tubing from Hsn s model nd the new model. The clculted tempertures of fluid in tubing from these two

4 Mthemticl Modelling nd Applictions 016; 1(1): models re both close to the ctul dt. The temperture of fluid inside the tubing given by the new model is slightly higher t shllow depth nd lower t deep depth thn tht by Hsn s model. Temperture profile given by the new model is more ccurte thn tht by Hsn s model. The new model tht considers Joule-Thomson cooling rigorously gives tempertures in tubing nd in nnulus tht re lower thn the geotherml temperture t the bottom. However these three tempertures re identicl ccording to Hsn s model. This is becuse in Hsn s model, the Joule-Thomson effect is ccounted by using the theoreticl pproch developed by Alves et l. where the mss frction of nnulr fluid is neglected [3]. In ddition, the kinetic nd potentil energy terms re neglected in the energy blnce eqution in Hsn s model. Figure illustrtes comprison of results given by Gilbertson et l s model nd the new model using the bsic well dt provided by Gilbertson et l s pper [0]. It shows tht Gilbertson et l s (013) model underestimtes tubing temperture t shllow depth nd significntly over-estimtes tubing temperture t deep depth. This is due to the fct tht Gilbertson s model does not consider Joule-Thomson cooling effect. In ddition, Gilbertson s model does not hve the cpbility of clculting the nnulr temperture, which limits its pplictions. In contrst, the new model overestimtes tubing temperture t shllow depth nd underestimtes tubing temperture t deep depth. This is consistent with the result shown in Figure 1. The reson is due to the fct tht the new model considers sonic flow of fluid when it enters the string through restriction (gs lift vlve in this cse), which genertes the upper bound of Joule-Thomson cooling. Figure. Comprison between the new model nd Gilbertson s model. 4. Model Appliction Tble 1. Bsic prmeters of the deep-wter well in South Chin Se. The new nlyticl model ws derived for generl pplictions including oil nd gs production through tubing (nnulr flow is set zero in the model), oil production through gs lift, wter nd gs injection (pipe size nd in-pipe flow re set zero in the model), nd well drilling nd work over with reverse circultion. It ws first used for designing test string in deep-wter gs well where the nnulr fluid is the sttionry drilling fluid both in the wellbore nd drilling riser sections. The well ws drilled in South Chin Se. The bsic well prmeter vlues of this deep wter well re presented in Tble 1. Estimted mteril properties re given in Tble. Well depth Wter depth Outer dimeter of testing pipe Inner dimeter of testing pipe Outer dimeter of csing Inner dimeter of csing Outer dimeter of cement Inner dimeter of cement Outer dimeter of drilling riser Inner dimeter of drilling riser Geotherml grdient Surfce se wter temperture 3,00 m 1,380 m m m m m m m m m C/m 0 C

5 4 Boyun Guo nd Jinze Song: An Improved Model for Predicting Fluid Temperture in Deep Wells Tble. Estimted mteril properties for the deep-wter well in South Chin Se. Mteril Density Specific het Therml conductivity (kg/m 3 ) (J/kg-K) (J/(m-K) Gs 6.5, Se wter 1,05 4, Drilling fluid 1,00 1, Steel 7, Cement, Rock, Figure 3 presents the temperture profiles from bottom hole to the mudline clculted by the new model. As expected, the temperture of gs inside the test string increses with gs production rte. Bsed on the temperture t the mudline, the temperture profiles of the gs inside the drilling riser were clculted with the clssicl het trnsfer model presented by Guo et l. [6]. The predicted tempertures of gs t surfce re summrized in Tble 3. Also included in the tble re the tempertures mesured in the field test nd clculted by Mo s model [6]. It is seen tht the new model gives error less thn 4%. Although Mo s model gives error of less thn 6%, it is rgued tht the gs density of 650 kg/m 3 employed in Mo s model is too high. Figure 3. Temperture Profiles Clculted by the New Model. Tble 3. Comprison of Model-Clculted nd Observed Surfce Tempertures for the deep-wter well in South Chin Se. Gs Production Rte Temperture t Surfce (C) (m 3 /dy) Field Test Mo's (016) Model Error (%) New Model Error (%) 450, , ,500, Conclusions A new closed-form nlyticl model ws developed in this study for predicting fluid temperture profiles in deep-wter wells. The following conclusions re drwn: (1) Comprisons of results from Hsn s (1996) model, Gilbertson et l. s (013) model, nd the new model with temperture dt mesured in two gs-lift wells show tht the new model best predicts well tempertures in trend. () A comprison of results given by Mo s (016) model nd the new model with tempertures observed in deep-wter gs well testing indictes tht the new model better predicts well tempertures with errors less thn 4%. (3) The new model ws derived for generl pplictions including oil nd gs production through tubing, oil production through gs lift, wter nd gs injection, nd well drilling nd work over with reverse circultion. Accurcy of the model in these pplictions needs further investigtions. Acknowledgements This reserch ws sponsored by the Chin Ntionl Nturl Science Foundtion Founding No The uthor is grteful to the support from the Southwest Petroleum University through the Stte 1000-Scholr Progrm. References [1] Sgr, R., Doty, D. R., nd Schmidt, Z Predicting Temperture Profiles in Flowing Well. SPE Production Engineering 6 (04): SPE-1970-PA. doi: /1970-PA. [] Rmey, H. J Wellbore Het Trnsmission. Journl of Petroleum Technology 14 (04): SPE-96-PA. doi: /96-PA.

6 Mthemticl Modelling nd Applictions 016; 1(1): [3] Alves, I. N., Alhnti, F. J. S., & Shohm, O A Unified Model for Predicting Flowing Temperture Distribution in Wellbores nd Pipelines. Presented t SPE Annul Technicl Conference nd Exhibition, New Orlens, Louisin, USA, 3-6 September. SPE MS. doi: /063-PA. [4] Hsn, A. R., nd Kbir, C. S Aspects of Wellbore Het Trnsfer During Two-Phse Flow (includes ssocited ppers 306 nd 30970). SPE Production & Fcilities 9 (03): SPE-948-PA. doi: /948-PA. [5] King, V. P. S., Coelho, L. C., Guigon, J., Cunh, G., nd Lndu, L Anlyticl Solution for Trnsient Temperture Field Around Csed nd Cemented Wellbore. Presented t SPE Ltin Americn nd Cribben Petroleum Engineering Conference, Rio de Jneiro, Brzil, 0-3 June. SPE MS. doi: /94870-MS. [6] Guo, B., Dun, S., nd Ghlmbor, A A Simple Model for Predicting Het Loss nd Temperture Profiles in Insulted Pipelines. SPE Production & Opertions 1 (01): doi: /86983-PA. [7] Spindler, R. P Anlyticl Models for Wellbore- Temperture Distribution. SPE Journl 16 (01): SPE PA. doi: / PA [8] Stter, A Het Losses during Flow of Stem down Wellbore. Journl of Petroleum Technology 17 (07): SPE-1071-PA. doi: /1071-PA. [9] Huygen, H. H. A., & Huitt, J. L Wellbore Het Losses nd Csing Tempertures during Stem Injection. Presented t Drilling nd Production Prctice, New York, New York, USA, 1 Jnury. API [10] Bck, L. H., nd Cuffel, R. F Anlysis of Het Losses nd Csing Tempertures of Stem Injection Wells with Annulr Coolnt Wter Flow. Presented t SPE Cliforni Regionl Meeting, Sn Frncisco, Cliforni, USA, 1-14 April. SPE MS. doi: /7148-MS. [11] Durrnt, A. J., nd Thmbynygm, R. K. M Wellbore Het Trnsmission nd Pressure Drop for Stem/Wter Injection nd Geotherml Production: A Simple Solution Technique. SPE Reservoir Engineering 1 (0): SPE PA. doi: /1939-PA. [1] Pcheco, E. F., & Ali, S. M. F Wellbore Het Losses nd Pressure Drop In Stem Injection. Journl of Petroleum Technology 4 (0): SPE-348-PA. doi: /348-PA. [13] Chiu, K., nd Thkur, S. C Modeling of Wellbore Het Losses in Directionl Wells under Chnging Injection Conditions. Presented t SPE Annul Technicl Conference nd Exhibition, Dlls, Texs, USA, 6-9 October. SPE- 870-MS. doi: /870-MS. [14] Kirkptrick, C. V Advnces in Gs-lift Technology. Presented t Drilling nd Production Prctice, New York, New York, USA, 1 Jnury. API [15] Winkler, H. W., nd Eds, P. T Algorithm for More Accurtely Predicting Nitrogen-Chrged Gs-Lift Vlve Opertion t High Pressures nd Tempertures. Presented t SPE Production Opertions Symposium, Oklhom City, Oklhom, USA, Mrch. SPE MS. doi: /18871-MS. [16] Lgerlef, D. L., Smlstig, W. H., nd Erwin, M. D Gs- Lift-Vlve Test Rck Opening Design Methodology for Extreme Kickoff Temperture Conditions. Presented t SPE Western Regionl Meeting, Bkersfield, Cliforni, USA, 30 Mrch-1 April. SPE MS. doi: /4065-MS. [17] Hsn, A. R., nd Kbir, C. S A Mechnistic Model for Computing Fluid Temperture Profiles in Gs-Lift Wells. SPE Production & Fcilities 11 (03): SPE-6098-PA. doi: /6098-PA. [18] Hernndez, A., Grci, G., Concho, A. M., Grci, R., nd Nvrro, U Downhole Pressure nd Temperture Survey Anlysis for Wells on Intermittent Gs Lift. Society of Petroleum Engineers. doi: /39853-MS. [19] Yu, Y., Lin, T., Xie, H., Gun, Y., nd Li, K Prediction of Wellbore Temperture Profiles During Hevy Oil Production Assisted With Light Oil Lift. Presented t SPE Production nd Opertions Symposium, Oklhom, USA, 4-8 April. SPE MS. doi: /11956-MS. [0] Gilbertson, E., Hover, F., nd Freemn, B A Thermlly Actuted Gs-Lift Sfety Vlve. SPE Production & Opertions 8 (01): SPE PA. doi: / PA. [1] Hn, G., Ling, K., nd Zhng, Z A Trnsient Two-Phse Fluid- nd Het-Flow Model for Gs-Lift-Assisted Wxy- Crude Wells with Periodicl Electric Heting. Presented t SPE Hevy Oil Conference-Cnd, Clgry, Albert, Cnd, June. SPE MS. doi: / MS. [] Wooley, G. R Computing Downhole Tempertures in Circultion, Injection, nd Production Wells. Journl of Petroleum Technology 3 (09): SPE-8441-PA. doi: /8441-PA. [3] Leutwyler, K Csing Temperture Studies in Stem Injection Wells. Journl of Petroleum Technology 18 (09): 1,157-1,16. SPE- 164-PA. doi: /164-PA. [4] Trgesser, A. F., Crwford, P. B., & Crwford, H. R A Method for Clculting Circulting Tempertures. Journl of Petroleum Technology 19 (11): 1,507-1,51. SPE-1484-PA. doi: /1484-PA. [5] Nelson, W. C Circulting Tempertures Existing Prior To Cementing Csing In Prudhoe By Wells. Presented t SPE Annul Fll Technicl Conference nd Exhibition, Denver, Colordo, USA, 9-1 October. SPE- 680-MS. doi: /680-MS. [6] Mo, J. nd Liu, Q. (016). Temperture prediction model of gs wells for deep-wter testing in South Chin Se. Personl communiction, JNGSE-D

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Entropy 006, 8[], 50-6 50 Entropy ISSN 099-4300 www.mdpi.org/entropy/ ENTROPY GENERATION IN PRESSURE GRADIENT ASSISTED COUETTE FLOW WITH DIFFERENT THERMAL BOUNDARY CONDITIONS Abdul Aziz Deprtment of Mechnicl