Temperature Behavior of Geothermal Wells During Production, Injection and Shut-in Operations

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1 PROCEEDINGS, Thrty-Nnth Workshop on Geothermal Reservor Engneerng Stanford Unversty, Stanford, Calforna, February 24-26, 2014 SGP-TR-202 Temperature Behavor of Geothermal Wells Durng Producton, Injecton and Shut-n Operatons Kaan Kutun, Omer Inanc Tureyen and Abdurrahman Satman ITU Maden Fakultes, Petrol ve Dogal Gaz Muh. Bol., Maslak Istanbul, TURKEY Keyords: Temperature profle, statc condtons, dynamc condtons, numercal modelng ABSTRACT Statc and dynamc measurements of pressure and temperature along geothermal ells are commonplace practces for characterzaton purposes n the geothermal ndustry. Such profles provde good nsght regardng the delverablty of the ell, locaton of the upper and loer boundares of the reservor / reservors and etc. The statc temperature profles, for example, can be used for modelng the natural state of the system through hstory matchng. As these profles are taken, t s mportant that the actual statc and dynamc condtons are reached. In other ords, to take a dynamc temperature profle, one ould have to at untl the temperature profle stablzes n the ell after producton has started and vce versa for the statc profle. Hence t becomes crucal to kno beforehand ho long the stablzaton tme takes. In ths study a numercal model s developed to study the parameters effectng the stablzaton tme of statc and dynamc condtons n the ells for a sngle phase ater system. The model s based on mass and energy balance equatons and couples the reservor th the ell and takes nto account the heat losses to the surroundngs of the ell. The model s valdated usng varous analytcal models n the lterature. A synthetc applcaton s provded to dentfy key parameters that effect temperature dstrbutons. In the applcaton e model temperature behavor along the ell for the transton from statc to dynamc and from dynamc to statc. 1. INTRODUCTION Statc and dynamc temperature profles taken along geothermal ells provde very useful nformaton regardng the flud and petrophyscal propertes of the reservor. For example, dynamc temperature profles could help dentfy dfferent zones of ater entry (at dfferent temperatures) nto the ell. Dynamc temperature profles taken at dfferent tmes could gve ho the bottomhole temperature changes th tme. Or they could provde nsght nto ho much heat s lost to the surroundngs of the ell as the flud s movng n the ell. Statc profles on the other hand could be used to dentfy the top and bottom ponts of the reservor tself. In fact multple entry ponts may be determned from the statc profles. When nterpretng ether the statc or dynamc temperature profles along the ells, to ponts are very crucal to consder. The frst s that a mathematcal model hch properly descrbes the physcs of the phenomenon must exsts so that nference of varous flud and/or petrophyscal propertes can be made. The second s that durng the measurements, a long enough tme must be consdered for ether the statc or dynamc condtons to be establshed. Falng to do so could result n a mscharacterzaton of the system. The necessary tme requred to reach statc or dynamc condtons can be assessed through the use of an approprate mathematcal model. Many authors have developed models that descrbe temperature behavor n geothermal ells for varous cases. Perhaps the earlest and the most cted ork s gven by Ramey (1962). In Ramey s ork, an approxmate soluton to the ellbore heat transmsson problem for the njecton of hot or cold fluds s provded. Ths soluton assumes steady state flo n the ellbore and heat transfer nto the surroundng formatons s treated to be unsteady radal conducton. Wooley (1980) presents a numercal model hch models the heat conducton effects n the horzontal and vertcal drectons as ell as modelng the convectve nature of the flo. Another numercal model s gven by Farouq Al (1981) hch presents a comprehensve numercal model that can deal th dfferent ell operaton condtons. Ths model s capable of handlng steam/ater mxtures and s based on solvng the momentum, mass balance and energy balance equatons n the ellbore. Durrant and Thambynayagam (1986) gve a straghtforard teratve procedure for the upard and donard movement of a steam/ater mxture. Wu and Pruess (1990) have presented an analytcal soluton that consders heat losses to an arbtrary number of layers th dfferent propertes. The smplfyng assumptons of Ramey (1962) ere not ncluded n ths study. Hagoort (2004) assess Ramey s (1962) method for the calculaton of temperatures n njecton and producton ells. Ths study states that although Ramey s approxmaton and a rgorous soluton are n good agreement at later tmes, Ramey s model overestmates the temperatures at early transent perods. Hasan and Kabr (2010) gve a model for to phase flo usng the drft-flux approach. Lvescu et. al. (2010) gve a sem analytcal approach here the extenson of sothermal ellbore-flo models to non-sothermal cases are presented. 2. THE MATHEMATICAL MODEL The mathematcal model developed n ths study s based on solvng the mass balance and energy balance equatons smultaneously for a gven set of grd blocks. The structure of the model s very smlar to that gven by Tureyen and Akyap (2011). Lets consder any grd gven n Fgure 1. We assume that the grd s composed of ater and rock components. Grd can make an arbtrary number of connectons th any other grd n the system. The total number of connected grds s termed N c. Energy and mass transfer s alloed for beteen grd and the connected grds represented by the ndex j. The mass balance appled to grd s gven n Eq. 1. 1

2 V b, d dt N c, l 1, p p z z 0,, p, nj, (1) Here, V b denotes the bulk volume, the densty, the porosty, t tme, α the transmssblty, the pressure gradent, p the pressure and the mass rate. The subscrpts represents ater and b represents bulk. The frst term of Eq.1 represents the mass accumulaton. The second term represents the sum of the mass transfer beteen grd and the connected grds and takes nto account the flo component due to gravty. The thrd term gves the mass producton rate and the fourth term gves the mass njecton rate. Here t s mportant to note that the mass njecton rate s performed at a specfed temperature T nj. The mass transfer beteen the grds s based on the pressure dfference beteen the grds and the transmssblty. The transmssblty beteen the grd block and any neghborng grd block j l can be defned as gven n Eq. 2. ka, (2) d, Here k s the permeablty, A s the cross-sectonal area, d s the dstance beteen the grd ponts of grds and j l and µ s the vscosty. Equaton 2 gves the transmssblty term for to neghborng grd blocks n Cartesan coordnates. Dfferent expressons for the transmssblty may be obtaned for dfferent coordnate systems such as a radal coordnate system or a sphercal coordnate system., j 1 Grd : j 1 W nj, Injecton T nj, W p, Producton T, j 2 Grd : j 2, j 3 Grd : j 3 Grd : Water + rock Volume : V b Porosty : Temperature : T Pressure : p, j l, j N c 1 Grd : j l Grd : j Nc-1, j N c Grd : j Nc Fgure 1: Illustraton of any grd and the neghborng grds. The energy balance equaton used n the model s gven n Eq. 3. V 1 b, N c d dt, C T u h h h m m c, p p z z T T,,, l 1 l 1 N nj (3) Here C s the heat capacty, u s the nternal energy, h s the enthalpy and s the heat conducton transmssblty. The subscrpt m represents the matrx. The frst term n Eq. 3 s the accumulaton of energy n grd block. As t s clear, the accumulaton of energy takes place n both components of the grd block; both n the rock and n the ater. The second term represents the energy contrbuton due to njecton, the thrd term represents the energy contrbuton due to producton, the fourth term represents the convectve heat transport from and to the neghborng grd blocks and the fnal term represents the conductve heat transfer to and from the neghborng grd blocks. Here t s mportant to note that an upndng scheme s appled to the convectve heat transfer beteen grd blocks. The subscrpt denotes the drecton of upndng and s defned n Eq. 4. 2

3 α=0 =computed z f p p z,, (4) j f p p z z l,, The heat conducton transmssblty for Cartesan coordnates s defned as gven n Eq. 5. Just as n evaluatng the flo transmssblty term gven n Eq. 2, the heat conducton transmssblty term gven n Eq. 5, can also be rtten for dfferent coordnate systems. A d, j l, (5) Here represents the thermal conductvty. In evaluatng the thermal conductvty and the permeablty for the nterface beteen grd blocks and j l n Eq s 2 and 5, harmonc averages are used. For the other parameters evaluated at the nterface (such as the densty or vscosty) arthmetc averagng s used. The mass and energy balance equatons are treated n a fully mplct manner causng them to become hghly non-lnear. Hence a Neton Raphson procedure s used to solve Eq s 1 and 3 smultaneously. For constructng the Jacoban matrx n the Neton Raphson procedure, numercal dervatves are used. A forard dfference scheme s used to handle the dervatves th respect to tme. The above mathematcal method has been used prevously by Palabyk et. al. (2013) and later agan by Palabyk (2013) for a detaled senstvty analyss of factors effectng pressure and temperature behavors n geothermal reservors. In ths study e extend ths model to model the ellbore temperature behavor. The schematcs of the grd blocks used n ths model are gven n Fgure 2. Specfed Producton or njecton Strata grd blocks =0 Well grd blocks =1 α=0 =computed Reservor grd blocks Fgure 2: Schematcs of the grd blocks used n the study. As can be seen n Fgure 2, the ell s also dscretzed n the z drecton usng the same dscretzaton as the reservor and the overburden strata. Afterards, the mass and energy balance equatons are solved n the ellbore grds just as e do so n the reservor and strata grds. The only dfference s that a much hgher permeablty s used for the ellbore grds. Throughout ths study n all our runs, the permeablty of the ellbore grd blocks are taken to be m 2. Ths knd of an approach here e use Darcy type of a flo for the ellbore allos us to couple the reservor flo th flo n the ellbore thout solvng the momentum balance equaton n the ellbore. To elmnate the rock component from the ellbore grd blocks, the porosty of the ellbore s taken to be 1. The thck lnes separatng the reservor grds from the strata grds represent the reservor upper boundary. At ths boundary the flo transmssblty s set to zero for avodng flo nto the strata grds. The heat conducton transmssblty hoever s set to a computed value to allo for heat transfer from the reservor nto the strata by ay of conducton. The same approach s taken for the boundary beteen the ell grds and the strata grds. Ths ay heat loss to the formatons by ay of 3

4 conducton can be modelled hle preventng flud movement nto the strata grds. Mass producton or njecton rates are specfed only at the very top grd of the ellbore. No other mass rate s specfed for any other grd block. 3. VERIFICATION OF THE MODEL In ths secton e provde a verfcaton of the model usng to of the common analytcal approaches for modelng the temperature behavor of ells; the approach gven by Ramey (1962) and the approach provded by Hagoort (2004). Table 1 lsts the propertes used n the example for the verfcaton case. Fgure 3 gves the grd system used n the verfcaton case. The ntal dstrbuton of temperature has been obtaned usng a temperature gradent of C /m. Intally the same temperature gradent s assumed to have establshed n the ell. Only one grd layer n the z drecton s used to represent the reservor. The rest of the grds n the z drecton are used for the strata. For the verfcaton, e consder only a producton scenaro. Table 1: Well and reservor propertes for the verfcaton example. Well radus, m 0.1 Reservor radus, m 1000 Reservor permeablty, m Reservor porosty, fracton 0.2 Rock compressblty, bar Rock thermal expanson coeffcent, C -1 0 Rock densty, kg/m Rock specfc heat capacty, J/kg-C 1000 Thermal conductvty of ater, J/m-s-C 0.67 Thermal conductvty of rock, J/m-s-C 2.92 Depth of reservor, m 1000 Heght of reservor, m 100 Intal pressure (@ 1050 m), bar Intal temperature (@ 1050 m), C Number of grd blocks n r drecton 11 Number of grd blocks n z drecton 15 Producton rate, kg/s 4.39 Well Strata Reservor Fgure 3: The structure of the grds used n the verfcaton example. The results of the verfcaton example are gven n Fgure 4. In Fgure 4 e provde a comparson of ell head temperature beteen our model, Ramey s (1962) model and the model proposed by Hagoort (2004). To perform the comparson, the solutons presented n Fgure 5 of Hagoort (2004) are used. The Ramey number and the Graetz number for these solutons are one. Hence, to do the comparson, the values presented n Table 1 gve the same Ramey and Graetz numbers. The solutons have been dgtzed and compared th the model presented n ths ork n Fgure 4. As can be seen n Fgure 4, the rgorous soluton gven by Hagoort and the model presented n ths study match rather ell both for early tmes and for late tmes. Matches for early tmes cannot be 4

5 Temperature, C obtaned th the Ramey model snce the Ramey model tends to overestmate the temperatures for early tmes as stated by Hagoort (2004) Ramey Hagoort Model Fgure 4: Comparson of varous models. Tme, Days 4. SYNTHETIC APPLICATIONS In thıs secton e provde varous synthetc applcatons of the model specfcally geared toards analyzng the parameters affectng the stablzaton tme of the temperature. We consder a producton case and a shut n case and pont out the most mportant parameters that effect the stablzaton tme. In all the synthetc applcatons, presented n ths secton, the parameters gven n Table 1 are used unless otherse stated. Producton Case In ths subsecton, e analyze th more detal the synthetc applcaton for the verfcaton example gven n the prevous secton. We frst provde ho the temperature profle nsde the ell evolves th tme. Fgure 5 gves the temperature profles for varous tme slces of 110-6, 0.01, 0.1, 1, 10, 100 and 1000 days. It s mportant to note that, most of the change occurs durng the frst 10 days. Then the changes n profle th tme become very small. At days, there s practcally no change n the temperature profle. The dstrbuton s almost dentcal to the ntal temperature dstrbuton n the ell. As tme progresses, the lnear behavor of the profle s dstorted and the profle starts to shft. To man mechansms of heat transfer play a role n the changng ell temperature profle. The frst s the convectve heat transfer from the bottom of the ell to the top va producton of the flud. The second s the conductve heat loss to the surroundngs of the ell. In the fnal profle at 1000 days, the ellhead temperature reaches to about 133 C dfferent from the ntal reservor temperature hch as at around 144 C. Ths dfference beteen the ellhead temperature and the bottomhole temperature s because of the heat losses to the surroundngs of the ell. Next e consder the effect of the producton rate on the stablzaton tme of the temperature. Hoever, t s frst mportant to dscuss brefly the physcs of the problem. From a purely mathematcal pont of ve, t s never possble to reach a stablzed temperature n the ell smply because of the conductve heat losses to the surroundngs of the ell. Hoever, from a practcal pont of ve, at late tmes durng the producton, the temperature does not change much th tme as e have seen n Fgure 5. In order to analyze better the change of temperature th tme, e look at the dervatve of the ellhead temperature th respect to tme. Fgure 6 gves the behavor of ths dervatve th tme. As can be seen from Fgure 6, the dervatve ntally dsplays a constant behavor. Then, t decreases more or less lnearly. Ths s ndcatve of to dfferent behavors. The constant dervatve porton of the curve reflects the tme perod here convectve flo s domnatng the temperature change. The lnearly decreasng porton of the data reflects the perod here the conductve heat losses no domnate the temperature change. In order to compare the stablzaton tmes, an arbtrary dervatve value of 0.1 C/s for the ellhead temperature s chosen as the pont here stablzaton s sad to have occurred. In other ords, t s assumed that the temperature has stablzed f the dervatve decreases belo ths cut off durng producton. 5

6 dt/dt, C/D Depth, m Temperature, C t= Days t=0.01 Days t=0.1 Days t=1 Days t=10 Days t=100 Days t=1000 Days Fgure 5: Temperature profles of a producng ell Fgure 6: The behavor of the temperature dervatve th tme. Tme, Days We frst compare the stablzaton tmes for varous mass flo rates. The results are gven n Fgure 7. As t s clear, the stablzaton tmes decrease hyperbolcally th ncreasng mass flo rate. There are to reasons for ths. The frst reason s because at loer mass flo rates t takes a longer tme for the hotter flud to travel to the ellhead. The second reason s that at 6

7 Stablzaton Tme, Days Stablzaton Tme, Days loer mass flo rates, as the flud s movng toards the ellhead, more heat s lost to the surroundngs of the ell, causng the flud to arrve at the ellhead th loer temperatures compared to hat ould have been th hgher mass flo rates. Next e look at the effect of the ell radus on the stablzaton tme. Fgure 8 llustrates the results. An ncrease n the ell radus causes an ncrease n the stablzaton tme. Ths s because, at a constant mass flo rate an ncrease n the radus results n a decrease n the velocty of the flud. Hence t takes a longer tme for the hotter flud to arrve at the ellhead Mass Flo Rate, kg/s Fgure 7: Wellhead temperature stablzaton tmes for varous mass flo rates Well Radus, m Fgure 8: Wellhead temperature stablzaton tmes for varous ell rad. 7

8 Stablzaton Tme, Days Shut In Case In ths secton e only consder the effects of producton tme on the stablzaton tme of temperature. For ths purpose e perform producton for varous duratons of tme. Then the ell s shut n and the stablzaton tmes are observed. The same cut-off dervatve value of 0.1 s used for the shut-n perod. Fgure 9 gves the results. Accordng to Fgure 9 as the producton ncreases the stablzaton tme durng shut n ncreases as ell. After a certan pont, the stablzaton tme becomes constant. As mentoned earler, durng producton the surroundngs of the ell are heated due to the hot flud flong n the ell. Larger producton tmes causes the heated regon around the ell to become der. Hence more heat s stored n the surroundngs of the ell. Ths leads to longer stablzaton tmes durng shut n snce t takes a longer tme for the surroundngs of the ell cool th more heat stored durng producton Fgure 9: Wellhead temperature stablzaton tmes for varous producton duratons. 5. CONCLUSIONS The follong conclusons have been obtaned from ths study: Producton Tme, Days A model capable of modelng temperature profles n the ell s developed. The developed model s also coupled to the reservor allong for realstc profles to be computed. The developed model has been verfed by to of the common methods used n the lterature; the method of Ramey (1962) and the rgorous soluton gven by Hagoort (2004). The focus of ths study are the stablzaton tmes of ellhead temperature for varous flo rates, ell rad and producng tmes for shut n. It s found that th ncreasng flo rates, the stablzaton tmes for temperature durng producton are decreased. An ncrease n the ell radus causes an ncrease n the ellhead temperature stablzaton tmes durng producton. Increasng the producton tme causes an ncrease n the stablzaton tme of the ellhead temperature for shut n. REFERENCES Durrant A. J. and Thambynayagam, R. K. M.: Wellbore Heat Transmsson and Pressure Drop for Steam/Water Injecton and Geothermal Producton: A Smple Soluton Technque, SPE Reservor Engneerng, 1, (1986), Farouq Al, S. M.: A Comprehensve Wellbore Steam/Water Flo Model for Steam Injecton and Geothermal Applcatons, SPE Journal, 21, (1981), Hagoort, J.: Ramey s Wellbore Heat Transmsson Revsted, SPE Journal, 9, (2004), Hasan, A.R. and Kabr, C. S.: Modelng To-Phase Flud and Heat Flos n Geothermal Wells, Journal of Petroleum Scence and Engneerng, 71, (2010), Lvescu, S., Durlofsky, L. J. and Azz, K.: A Semanalytcal Thermal Multphase Wellbore-Flo Model for Use n Reservor Smulaton, SPE Journal, 15, (2010), Palabyk, Y: A Study on Pressure and Temperature Behavors of Geothermal Wells n Sngle-Phase Lqud Reservors, PhD Dssertaton, Istanbul Techncal Unversty Graduate School of Scence Engneerng and Technology, Istanbul, Turkey (2013). 8

9 Palabyk, Y., Tureyen, O. I., Onur, M. and Paker Denz, M: Pressure and Temperature Behavors of Sngle-Phase Water Geothermal Reservors Under Varous Producton/Injecton Schemes, Proceedngs, 38 th Workshop on Geothermal Reservor Engneerng, Stanford Unversty, Stanford, CA, USA (2013). Ramey, H.J.: Wellbore Heat Transmsson, Journal of Petroleum Technology, 14, (1962), Tureyen, O.I., and Akyap, E.: A Generalzed Non-Isothermal Tank Model for Lqud Domnated Geothermal Reservors, Geothermcs, 40, (2011), Wu, Y.S. and Pruess, K.: An Analytcal Soluton for Wellbore Heat Transmsson n Layered Formatons, SPE Reservor Engneerng, 5, (1990),