PROCEEDINGS, Thirty-First Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, January 3-February 1, 6 SGP-TR-179 DEVELOPMENT OF MULTI-FEED P-T WELLBORE MODEL FOR GEOTHERML WELLS Murat Cinar, Mustafa Onur and bdurrahman Satman Petroleum and Natural Gas Engineering Department Istanbul Tehnial University Maslak Istanbul 34469 TURKEY e-mails: inarmura@itu.edu.tr, onur@itu.edu.tr, mdsatman@itu.edu.tr BSTRCT wellbore pressure temperature (p-t) model for geothermal wells with multiple feed zones is presented. The flow rate, temperature and pressure may vary depending on the feed zones. The fluid onsidered is water ontaining low amount of CO. Espeially in Turkey, geothermal waters ontain onsiderable amounts of CO dissolved in water. The presene of CO leads to two phase flow at higher pressure onditions than the vapor pressure of water. The wellbore model onsiders all these onditions. The model assumes that the flow is in steady state and Henry s law is valid for CO s partial pressure alulations. The main inputs of the model are reservoir properties and the depths of the feed points. The main outputs are pressure and temperature profiles. If measured profiles of temperature and pressure are available then the model an be used to estimate the mass fration of CO. Dynami temperature profiles measured at some wells in fyon Omer-Geek geothermal field in Turkey showed multi-feed wellbore behavior. The multi-feed p-t wellbore model developed in this study was applied to analyze the p-t profiles of one of the wells, the well F-1. There are two feed points in the wellbore, one of whih is ooler. The field data from the F-1 well was analyzed with the model and a good math between the alulated and measured data was obtained. lso the CO ontent was estimated from the math. This paper disusses the analysis proedure as well as the results obtained from the modeling approah. INTRODUCTION Prediting flowing pressure and temperature profiles in geothermal wells is a very important issue to exploit geothermal resoure in a rational way. High temperatures and flow rates are ommonly enountered in geothermal wells. These onditions inrease the errors in the measurements. In addition, the presene of CO, frequently found in geothermal fluids auses sale deposition of alium arbonate in the wellbore where the fluid is flashing. In order to take the neessary preautions, the depth of flashing has to be known (Satman and lkan, 1989). Compliated geologial strutures of geothermal fields make wellbore flow a muh more ompliated problem. For example, feeds with different temperatures at different depths may be enountered. In the following work, a pressure-temperature (p-t) model for geothermal wells with multiple feed points is presented. Based on the assumptions made, the model disussed is valid for shallow wells of water dominated systems ontaining low amount of dissolved CO. DESCRIPTION OF THE MODEL Many of the geothermal systems are water dominated systems produing a two-phase mixture of liquid and gas. In Turkey the majority of the geothermal systems is water-dominated systems inluding CO dissolved in water. These systems produe a two phase mixture, vapor phase of whih is formed mostly by CO, at the wellbore. The alulation of pressure and temperature profiles for those systems has to onsider the following effets: 1. Flashing point of CO and mass transfer between phases.. Potential and fritional pressure losses for both single and two phase onditions 3. Heat loss to the surrounding formations. SINGLE PHSE FLOW In this setion, the equations for pressure and temperature drop for flow of single-phase water are presented.
Pressure Profile Calulations General pressure drop equation for fluid flow is given by Equation 1. dp dp dp dp = + + dz dz dz dz F E (1) Here the first term on the right hand side represents pressure drop due to frition, seond term represents pressure drop due to elevation hanges, and the last term represents pressure drop due to aeleration. Pressure loss due to aeleration is muh smaller than pressure loss due to elevation hange and frition and so an be ignored. Therefore Equation 1 beomes, dp dp dp = + dz dz dz Here, F dp f = dz F g d υ ρ dp ρg sinθ = dz E g ssuming the well is vertial (sinθ =1), dp fυ ρ ρg = dz g d g E () (3) (4) (5) By using Equation 5, pressure drop for single phase water flow an be alulated. Temperature Profile Calulations For prediting temperature profile in the wellbore in single phase onditions, Ramey s lassial method is used (Ramey, 1961). Ramey s method for alulating temperature distribution in the wellbore is based on the following two simplifying assumptions: 1. Heat flows radially away from the wellbore. Heat flow through various thermal resistanes in the immediate viinity of the wellbore is rapid ompared to heat flow in the formation and an be represented by steady state solution. Ramey s approah was investigated by many researhers and validity of the model was examined (Hagoort, 4; lves et al., 199; Wu and Pruess, 199). In Hagoort s reent work, it is observed that Ramey s solution is an exellent approximation exept for an early transient period. For geothermal wells the equation for temperature distribution along the well is given by, y ( ) ( 1 ) ( ) T = T ay + a e + T T e (6) bh / y/ bh Here is the diffusion depth and for geothermal wells is defined as, () t () = wf t π k (7) In Equation 7, f(t) is the dimensionless time funtion representing the transient heat transfer to the formation. In literature there are different orrelations to alulate this term (Ramey, 1961; Hagoort,4; Hasan and Kabir, 1994). The orrelation proposed here is found by urve fitting the data given by Ramey et al. (see Ramey Table 6.1, 1961). The dimensionless time funtion an be defined as, () =.7165 +.3947 ( D ) x Log( t ) x Log( t ) Log f t x Log t 3.487 D +.3574 D 4 for 1 td 1 (8) where, αt td = r Log( f(t) ).8.4 -.4 -.8 ( ) ' garwal - Ramey Hagoort Proposed (Eq.8) (9) - 4 Log(t D ) Figure. 1. Comparison of different orrelations in the literature for f(t).
The dimensionless time funtion may be found from Figure 1. By using Equations 6, 7, 8 and 9 temperature distribution along the well for the flow of single-phase water an be estimated. MULTI FEED WELLBORE MODEL Figure.. Representation of the multiple feed (Tokita and Itoi, 4). Representation of a multi-feed ase is shown in Figure. In the model it is assumed that the flow from the deepest feed zone ours first and forms a pressure gradient in the well. That pressure gradient stimulates the other shallower feed zones to flow onseutively. Here flow rate will inrease after eah feed zone, and the temperature of the fluid flowing to the wellbore may be different from the fluid flowing in the well. ssuming that thermal equilibrium is established instantaneously, new temperature at whih fluids are mixed at the feed point an be alulated by the following formula. Qtotal = Qt + Q f (1) T mix = q ρt + q ρ T t t t f f f ( qf + qt) ρ f ( Tmix) (11) Here subsript f stands for the feed properties, and t stands for the properties of the fluid in the tubing. TWO PHSE FLOW The fluid flowing in the well forms a pressure gradient. Pressure dereases while the fluid rises in the well. s a result of pressure drop the dissolved CO in the fluid leaves the water and forms a seond phase. Developing the model it is assumed that the gas phase is only omposed of CO. Furthermore, Henry s law is assumed to be valid for partial pressure alulations of CO. Under those assumptions, the following method is used for alulating the two-phase pressure and temperature profiles. Pressure Profile Calulations Two phase alulations are ompliated by the fat that two phases exist simultaneously. The interfae between phases an be formed in many different patterns whih is alled flow pattern. To make aurate alulations, flow pattern and the riteria for flow pattern transitions have to be determined orretly. In the model it is assumed that flow is in the bubble flow regime. The harateristi of this flow pattern is the flowing of gas in the liquid as disrete bubbles. In two-phase flow, phases flow with different veloities. The gas phase, having lower density relatively, flows faster than the liquid phase. s a result the faster flowing gas phase holds up the liquid phase. This phenomenon is alled liquid hold up and the differene in the veloity is alled slip (Hasan and Kabir, ). Using homogenous flow equation for pressure drop alulations in bubble flow redues omplexity without introduing any signifiant inauray (Hasan and Kabir, ). dp f g dv = dz g d g g dz mυmρm ρm υmρm m (1) Fluid aeleration during bubble flow is very small so the last term in Equation 1 is negleted. Hene the equation beomes, dp fmυmρm ρmg = dz g d g (13) The density of the mixture an be alulated by the following equations, ( 1 ) ρ = ρ f + ρ f (14) f m L L g L L vsg = 1 C υ + m υ ( L g) g ρ ρ σ υ = 1.53 ρ L 1/ 4 (15) (16) The flashing point pressure for a given temperature an be alulated by the help of Henry s law that is given by,
pi = KH ni (17) For CO, 18 p = x K 44 p = p mpa CO CO HW CO CO where, ( ) ( ) ( ) ln 4.51748673.554534 1 1.13x1 xt + 9.3689x1 xt T = T C KHW = + x xt 4 8 3 The flashing point pressure an be estimated by, ( 1 ) fp CO CO s (18) (19) p = p + n p () For the vapor pressure alulations of water ntoine equation is used (Ohe, 1976). Temperature Profile Calulations Temperature profile alulations are governed by the equation proposed by lves (lves et al., 199) whih redues to Ramey's method under appropriate assumptions and is given by, z ( ) ( 1 ) ( ) T = T az + a e + T T e bh 1 g + ηρ g e Jg gjp / z/ bh z/ ( 1 ) (6) For the single-phase onditions the above equation redues to Equation 6. For flow rates higher than 5 lb m /se, ρηg term an be taken as (Sagar et al., 1991). Hene the equation redues to the following one: L/ ( bh ) ( 1 ) L / g L/ ( T Tbh ) e ( 1 e ) T = T al + a e + gj P (7) By using Equation 7 temperature distribution along the well for two phase onditions an be estimated. P-T MODEL Two governing equations (temperature and pressure equations) are oupled through the dependene of the fluid density on temperature for single-phase and on both temperature and pressure for two-phase. z/ ( bh θ) θ( 1 ) T = T azsin + asin e + Here, 1 ρ J z / z/ ( T Tbh ) e + Φ( 1 e ) P dp dz (1) The wellbore depth profile is broken up into a number of segments and the pressure and temperature equations applied suessfully to eah segment. The first segment is hosen as the bottom of the well and from that referene point the alulations are performed up to the wellhead for eah segment. dp g d dp ρ p Sin ρυ υ Φ= ρη θ dz g g dz dz 1 T z η = yg yl p z + T p () (3) Pressure loss due to elevation hange is the most important omponent of pressure drop equation of vertial flow and pressure losses due to aeleration and frition are muh more smaller ompared to pressure loss due to elevation hange so we an take, dp g Sin dz g ρ θ (4) ρυ dυ g dz (5) By assuming the well is vertial Equation 1 is redued to the following form, MODEL VLIDTION Dynami temperature profiles measured at some wells in the fyon Omer-Geek geothermal field in Turkey showed multi-feed wellbore behavior. The multi-feed p-t wellbore model developed in this study was applied to analyze the p-t profiles of one of the wells, the well F-1. There are two feed points in the wellbore, one of whih is ooler. The F-1 well is 695 ft deep. ording to the measurements the reservoir water ontains.4 % CO by mass. The field data from the F-1 well was analyzed using the model developed in this study and a good math between the alulated and measured data was obtained. By mathing the field data with the alulated ones, the properties of the feed zones (temperature and flow rate) and CO ontent were estimated. The temperatures of feed zones are found by the analysis of a stati p-t test onduted in the well.
Figure 3 shows the stati and dynami temperature profiles. In the stati temperature profile there is a sharp transition between 5 F and 13.8 F whih is an indiation of the upper and ooler feed point. Here the temperature of the seond feed is found to be 13.8 F. To alulate the flow rate of the seond feed Equation 11 is used. Hene the only unknown in Equation 11 is the flow rate of the feed and is estimated to be.53 ft 3 /se. Temperature, F 1 3 4 Calulated values Measured values The CO ontent was found to be.4 % from the math of alulated and dynami p-t test data. lso the flashing point was found to be approximately 38 ft. The results are presented in Figures 4 and 5. Depth, ft 4 CONCLUSIONS In this paper, a wellbore p-t model for geothermal wells with multiple feed points is presented. The model has the ability to alulate pressure and temperature profiles along the well for both single and two phase onditions in the presene of CO. Model an be used to alulate CO ontent by using field data. In addition a orrelation for prediting dimensionless time funtion is proposed. good math between alulated and measured data is obtained. 6 Figure. 4. Comparison of alulated and measured temperature data. Pressure, psia 1 3 Temperature, F 17 18 19 1 3 Calulated values Measured values Depth, ft 4 Depth, ft 4 6 6 8 Stati Temperature Profile Dynami Temperature Profile 8 Figure. 5. Comparison of alulated and measured pressure data. Figure. 3. Stati and dynami temperature profiles. NOMECLTURE a = geothermal gradient, F / ft = diffusion depth, ft p = fluid heat apaity (onstant pressure), Btu / (lb m - F) d = diameter, ft f = frition fator f g = void fration f L = liquid hold up
f(t) = dimensionless time funtion g = aeleration of gravity, ft / se g = unit onversion fator,3.17(lb m - ft) / lb f / se J = mehanial equivalent of heat, 778 ft-lb fore / BTU k = thermal ondutivity of formation, 33.6 Btu / (ft-day- F) K H = Henry s oeffiient. n = mole fration p = pressure, psia p fp = flashing point pressure p s = vapor pressure q = flow rate, ft 3 / se Q = Heat, Btu / lb m r = outside radius of asing, ft t = time, days t D = Fourier time, dimensionless T = temperature, F T bh = bottom hole temperature T = temperature of fluid in tubing w = mass flow rate, lb m / hr x = mass fration. y = no slip hold up. z = Length, ft. z = gas ompressibility fator α = thermal diffusivity of formation,.96 ft /gün ρ = density, lb m / ft 3 υ = veloity of the fluid, ft / se υ sg = superfiial in-situ veloity, ft / se υ = terminal rise veloity, ft / se θ = inlination angle with horizontal Φ = dimensionless orretion parameter η = Joule Thomson oeffiient σ = surfae tension, lb m / se SUBSCRIPTS m = mixture g = gaseous L = liquid W = water Hasan,.R., Kabir, C.S. (1994), spets of Wellbore Heat Transfer During Two-Phase Flow, SPE Prodution & Failities, ugust, 11-16. Hasan,.R., Kabir, C.S. (), Fluid Flow and Heat Transfer in Wellbores, Soiety of Petroleum Engineers, Texas. Ohe, S. (1976), Computer ided Data Book of Vapor Pressure, Data Book Publishing Co., Tokyo. Ramey, H.J., Jr. (1981) Reservoir Engineering ssessment of Geothermal Systems, Stanford University, Stanford, C. Ramey, H.J., Jr. (1961) Wellbore Heat Transmission, IME Transations, 47-435. Sagar, R.K., Dotty, D.R., and Shmidt, Z. (1991) Prediting Temperatures in a Flowing Oil Well, SPE Prodution Engineering, 1991, 441-448. Satman,., lkan, H. (1989) Modeling of Wellbore Flow and Calite Deposition for Geothermal Wells in the Presene of CO, SPE Paper No: 1935. Tokita, H., Itoi, R., (4) Development of the MULFEWS Multi-Feed Wellbore Simulator, 9th Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, California, 6-8 January. Yu-Shu Wu and Pruess, K. (199) n nalytial Solution for Wellbore Heat Transmission in Layered Formations, SPE Reservoir Engineering, November, 531-538. SUPERSCRIPTS = average REFERENCES lves, I.N., lhanati, F.J., Shoham, O. (199), Unified Model for Prediting Flowing Temperature Distribution in Wellbores and Pipelines, SPE Prodution Engineering, November, 363-367. Hagoort, J. (4), Ramey s Wellbore Heat Transmission Revisited, SPE Journal, Deember, 465-474.