APPLICATIONS OF LUMPED PARAMETER MODELS FOR SIMULATION OF LOW-TEMPERATURE GEOTHERMAL RESERVOIRS

Transcription

1 PROCEEDINGS, Twenty-Eighth Wokshop on Geothemal Resevoi Engineeing Stanfod Univesity, Stanfod, Califonia, Januay 27-29, 23 SGP-TR-173 APPLICATIONS OF LUMPED PARAMETER MODELS FOR SIMULATION OF LOW-TEMPERATURE GEOTHERMAL RESERVOIRS Hulya Saak, Mustafa Onu, and Abduahman Satman Petoleum and Natual Gas Engineeing Depatment Istanbul Technical Univesity Maslak, 8626, Istanbul, Tukey s: ABSTRACT A companion pape titled New Lumped Paamete Models fo Simulation of Low-Tempeatue Geothemal Resevois pesents pactical analytical models fo simulation of low-tempeatue geothemal esevois. This pape deals with the applications of the new lumped-paamete models to field cases. The models ae used to match the long-tem obseved wate level o pessue esponse to a given poduction histoy. Fo histoy matching puposes, we use an optimization algoithm based on the Levenbeg-Maquadt method to minimize an objective function based on weighted least-squaes fo estimating elevant aquife/esevoi paametes. In addition, we constain the paametes duing nonlinea minimization pocess to keep them physically meaningful and compute statistics (e.g, standad 95% confidence intevals) to assess uncetainty in the estimated paametes. Thee field examples taken fom the liteatue ae consideed to show the use of the models and optimization algoithm. The obseved and simulated wate level changes obtained fom the models ae discussed. Results show a vey good ageement between the obseved field data and simulated data fom the lumped models given in this wok, in spite of long data sets. INTRODUCTION The modeling appoach used in this pape is descibed as lumped-paamete. A companion pape (Saak et al. 23) pesents new lumped paamete models fo simulation of low-tempeatue geothemal esevois. In lumped-paamete modeling aveage popeties ae assigned to the two components of the geothemal system, the esevoi and the aquife. The changes of these popeties ae monitoed and pedicted. Saak et al. (23) epot and discuss the lumped paamete models in the liteatue and pesent the new ones (Figue 1). In this pape, ou objective is to apply these new lumped paamete models within the context of automatic histoy matching to liquiddominated fields and to show thei validities. The models wee evaluated using poduction data fom thee fields as examples. p i w α a1, a1 p i α a1 p i w, α a a Resevoi, p V, φ, ρ, c a) 1-tank model Aquife, p 1 V a, φa, ρa, cta wa 2, αa2 b) 2-tank model Aquife 1 p 1, κ a 1 α a2 c) 3-tank model Figue 1. Schematics of lumped models consideed in simulations. Poduction w p Aquife 2 p 2, κ a 2 t Poduction w p Resevoi, p 2 φ V,, ρ, ct α a3 Poduction w p Resevoi p 3, κ

2 OPTIMIZATION PROCEDURE As is well known, the poduction causes the pessue in the geothemal systems to decline, which is eflected in the loweing of the wate level in boeholes. The ate of pessue decline is detemined by the ate of poduction, the size and popeties of the geothemal system, and the echage chaacteistics of the system. The echage wate invades the system in esponse to the loweed pessue o wate level. Afte a geothemal esevoi has been poduced fo a peiod of time, a lumped paamete model can be matched to obseved pessue (o wate level) data with the available poduction/einjection ate histoy to obtain optimum paametes of a paticula lumped model. As moe data become available, moe infomation can be obtained about the esevoi and the system. With time thee ae data available which may be used to impove the undestanding the behavio of the esevoi. Theefoe, in modeling, data must be collected as the esevoi is poduced. The model is limited to the data used, so all the pessue (o wate level) esponses must be included fo honoing all the data available. In matching obseved poduction data, in geneal, moe and moe poduction data ae desied. This is quite impotant in educing the uncetainty in pefomance pedictions as well as in futhe development of the system unde consideation. Fitting model paametes to the obseved data equies accuate and fast appoaches. The method of least squaes fitting is a convenient one to apply. Fo example, Axelsson (1989) used the standad (unweighted) least squaes fitting wheeas Olsen (1984) used the standad least squaes fitting and also gaphical techniques fo histoy matching. As is well known, the taditional (unweighted) least squaes estimation is often unsatisfactoy when some obsevations ae less eliable than othes and/o vaious measuements having dispaate odes of magnitude ae simultaneously used in estimation. In the fome case, we want to make sue that ou paamete estimates will be moe influenced by the moe eliable obsevations than by the less eliable ones. In the latte case, we wish to make sue that any infomation contained in the data with small magnitudes is not lost because of summing togethe squaes of numbes of such dispaate odes of magnitude. Theefoe, in this wok, we conside weighted least-squaes fitting so that the above mentioned disadvantages associated with the standad least squaes fitting can be ovecome. The details of ou optimization algoithm ae given in the following subsection. Paamete Estimation The invese poblem of estimating unknown paametes fom vaious lumped models deived in Saak et al. (23) can be fomulated as a nonlinea optimization poblem. We pefom nonlinea paamete estimation by minimizing a weighted leastsquaes (LS) objective function (J) fo which the weights (invese of the vaiances of measuement eos assumed to be nomal and independent) ae assumed to be known. In geneal, we minimize a weighted least-squaes objective function: J [ f ( t χ ) y ( t ] 2 M n ( χ ) = w j, i j i, j i ) j= 1 i= 1 (1) whee M epesents the total numbe of model function f and, (t i, y j (t i )), i=1,,n is a set of n obsevations of the model function f j, j=1,,m. χ is an l-dimensional column vecto whose elements ae unknown paametes fo a chosen lumped model. In Equation 1, the positive weights w j,i ae the invese of vaiance of measuement eos coesponding to measued value y j at time t i. In ou applications, y j could epesent pessue (o wate-level) data measued as a function of time fom wells in esevois o aquifes. One can constuct weighted LS objective functions based on Equation 1, depending on the pessue data available and the lumped model chosen fo egession, and conside matching of a single pessue data set (M=1 in Equation 1) as well as simultaneous matching of diffeent pessue data sets (M>1 in Equation 1) to optimize χ. Suppose we conside a two-tank lumped model (Figue 1(b)), whee the system is assumed to be consisted of one aquife and one esevoi and assume that we have a set of n measued pessue data fom a well in the esevoi and a set of n measued pessue data fom a well in the aquife. Then, M = 2 in Equation 1, and one can choose y 1 and f 1 to epesent the measued and model pessue data fo the esevoi, wheeas y 2 and f 2 to epesent measued and model pessue data fo the aquife, espectively. In this case, the positive weights w 1,i, i=1,,n, will epesent the invese of vaiance of the measuement eo fo the ith measued pessue fo the esevoi, wheeas, w 2,i, i=1,,n, will epesent the invese of vaiance of the measuement eo fo the ith measued pessue fo the aquife. It means that fo this case, we constuct a weighted-ls objective function based on two diffeent sets of pessue data and simultaneously match both sets to optimize χ, whee, fo this lumped model, in geneal, χ can be epesented as χ = [ κ, κ, α α p ] T a a 1, a 2, i (2)

3 whee T denotes the tanspose. See Saak et al. (23) fo the definition of paametes in Equation 2. In ou applications, we minimize the objective function given by Equation 1 by using the Levenbeg-Maquadt method with a esticted step pocedue as descibed by Fletche (1987) and constain the unknown paametes in nonlinea egession by using the so-called imaging method of Cavalho et al. (1996). In addition, we compute 95% confidence intevals and coelation coefficients by using the standad definitions (Dogu et al. 1977). As is well known, computing and inspecting such statistics in egession analysis is vey useful fo identifying which paametes can be eliably detemined fom available data. As is well known, in nonlinea egession, paamete estimation fom lumped models stats with a set of initial guess fo the paametes, and then the paametes ae updated by the method discussed above until a successful match of data with the model esponse can be obtained. We use the standad teminating citeia given by Gill et al. (1993). At temination, fo each data set matched, we also compute the standad deviation of eos as well as the oot mean squae eos (RMS). Hee, we use the standad definition of RMS given by RMS j = whee χo 1 n n [ f j ( ti, o ) y j ( ti )] i= 1 2 χ, (3) epesents the optimized paamete vecto. Befoe closing this section, we should note that choosing good initial guesses and constaints fo paametes plays an impotant ole in nonlinea egession analysis because nonlinea egession algoithms could often become tapped at unacceptable local minima. Paticulaly, this would be valid in cases whee the models with a lage numbe of unknown paametes ae chosen fo the data to be matched and/o the obseved data contain lage measuement eos. As shown in the next section, the analytical equations and the asymptotic expessions given in ou companion pape (Saak et al. 23) ae useful to obtain a good set of initial guesses fo the paametes pio to pefoming nonlinea egession analysis. FIELD APPLICATIONS Fo the field applications discussed below, all obseved data obtained fom liteatue wee given in tems of wate levels. As we pefe fomulating ou models in tems of pessue, all the obseved wate level data fist conveted to pessue equivalence and then used in egession algoithm. Thus, all paamete estimates ae given in pessue units. Howeve, all gaphical esults ae pesented in tems of wate levels to be consistent with the published field data. Svatsengi Field The Svatsengi field in Iceland is a liquid-dominated esevoi with fluids of nealy constant tempeatue at 235 o C. Fluid poduction fom the esevoi stated in The composition of the fluids poduced is about twothids seawate and one-thid ainwate. Fluid extaction and esevoi dawdown in Svatsengi wee monitoed. The dawdown was measued as wate level in monitoing wells. The wate level was measued in wells 4, 5 and 6. The esistivity measuements indicated a esevoi suface aea of 5 km 2 at 2 m depth, and 7 km 2 at 6 m below sea level. Olsen (1984) and Gudmundsson and Olsen (1987) studied the poduction data of the Svatsengi field. Thei objective was to study the use of wate influx methods in geothemal esevoi evaluation. They found that the steady state Schilthuis (1936) method gave a easonable match and the Hust (1958) simplified unsteady-state method gave the best match of the models they tied. Poduction esponse data of the Svatsengi geothemal esevoi consist of a seven-yea continuous ecod fom neaby obsevation wells in the field. These data ae pesented in Figue 2. Wate Level, m Wate Level Poduction Time, days Figue 2. Poduction esponse data of the Svatsengi geothemal esevoi (Olsen, 1984). Figue 3 shows the Schilthuis steady-state match and the Hust (simplified) unsteady-state match obtained by Olsen (1984). The Schilthuis match is bette fo the ealy pat of the data than fo the late data. Olsen obtained the following values fom the Schilthuis match: κ =5.34x1 8 kg/ba and α a = kg/(bas). Among the models that he tied to match the data, Olsen found that the Hust simplified model Poduction, kg/s

4 assuming an infinite adial aquife best matched to the data. Wate Level, m Measued Schilthuis Hust Simplified (see Figue 5 and RMS values in Table 1) fo both models. The 69% confidence inteval fo κ a in the 2-tank closed model gives an indication that this paamete is not well detemined compaed to the othe paametes of the model fom the data available. Because esults of ou 1-tank and 2-tank closed model simulations did not exhibit any significant diffeences, futhe infomation and detailed analysis ae equied to identify the most appopiate model fo the system Time, days Figue 3. Schilthuis steady-state and Hust (simplified) unsteady-state matches (Olsen, 1984). Wate Level, m Measued Schilthuis 1-Tank Model As seen fom Fig. 2, the poduction ate data can be consideed almost constant in the time inteval fom 15 to 1 days. In addition, we note that the wate level data in this time inteval incease linealy with time. Based on the asymptotic expession of Eq. 9 given in Saak et al. (23), we pefomed staightline analysis of wate level data in this time inteval 9 and found κ = 2x1 kg/ba. We also note that wate-level data do not show any stabilized value. So, we cannot obtain an estimate of α a fom the asymptotic expession of Eq. 1 in Saak et al. 9 (23), Then, using κ = 2x1 kg/ba and α a = 3.44 kg/(ba-s) as given by Olsen (1984) as initial guesses, we pefomed nonlinea egession of the Svatsengi poduction data using ou 1-tank model. Figue 4 shows the match between the obseved and simulated wate level. The match obtained by Olsen using the Schilthuis model is also shown fo compaison puposes. Notice that ou match based on the 1-tank model fits the measued wate level data bette than the Olsen s match based on the Schilthuis model. We then used ou 1- and 2-tank models. Table 1 summaizes the paametes of the best fitting lumped models. Hee and thoughout, the pecentages given in paentheses epesent the 95% confidence inteval in tems of pecentages (Hone 1995). As seen fom Table 1, the confidence pecantages computed fo the paametes of the 2-tank open model ae quite high, (paticulaly see confidence intevals fo κ and α a2 ) compaed to those of the 1-tank and 2-tank closed models. This gives an indication that the 2- tank open model is inappopiate fo the data. Howeve, 1-tank and 2-tank models appea to be appopiate fo the data as the fits between the obseved and simulated data ae quite satisfactoy a Time, days Figue 4. Compaison of obseved and calculated wate level changes in the Svatsengi field. Wate Level, m Measued Wate Level 1-Tank Model 2-Tank Open Model 2-Tank Closed Model Time, days Figue 5. Simulation esults of 1- and 2-tank models. Lauganes Field The Lauganes field in SW-Iceland is a consideably lage field. The majo feed zones ae between depths of 7 and 13 m and the wate tempeatue is between 115 and 135 o C. A continuous wate level ecod was available fom one well. The Lauganes field is discussed in Axelsson (1989), and Axelsson and Gunnlaugsson (2). Axelsson (1989) used this data to simulate the pessue esponse of the field and to estimate its poduction capacity.

5 Table 1. Paametes of the best fitting lumped paametes Olsen This This This Study Study Study Schilthuis 2-Tank 2-Tank 1-Tank Open Closed κ a, kg/ba κ, kg/ba α a1, kg/(ba-s) α a2, kg/(ba-s) x x19 (3%) (6%) 6.9x1 8 (63%) 6.1x1 8 (38%) 28.3 (27%) 144 (15%) 1.x1 1 (69%) 9.1x1 8 (1%) (16%) RMS, ba Pio to exploitation the hydostatic pessue at the suface in the geothemal field was 6-7 bas coesponding to a fee wate level 6-7 m above the land suface. Exploitation caused pessue dop in the field and wate level fell. Figue 6 shows the wate level changes and poduction histoy of the Lauganes system. Wate Level, m Measued Wate Level Axelsson's Match Ou Model With Axelsson's Paametes Yea Figue 7. Compaison of obseved and calculated wate level changes in the Lauganes field. Next, using the paamete estimates of Axelsson (1989) as initial guesses fo the paametes, we pefomed nonlinea egession analysis based on ou model to estimate the paametes and obtained the best fit with the paametes given in Table 2. The RMS value fo the fit was.581 ba. We note that the pecantage confidence fo the paamete a2 κ is quite high, indicating that this paamete is not well detemined compaed to the othe paametes fom the data. The Axelsson s paametes ae also given fo compaison puposes. Axelsson did not computed the confidence intevals fo the paametes. Wate Level, m Poduction Wate Level Poduction, l/s Figue 8 shows the obseved and simulated wate level changes in the Lauganes field. Thee is a vey good ageement between the obseved and simulated data, in spite of long data sets, eflecting the flexibility of this method of lumped modeling and optimization algoithm Yea Figue 6. The wate level changes and poduction histoy of the Lauganes field (Axelsson, 1989; Axelsson, Gunnlaugsson, 2). Axelsson (1989) used a closed thee capacito lumped model (a thee-tank with closed oute bounday model) fo simulation. The simulations wee caied out automatically by a compute. He teated the modeling as an invese poblem. He obtained quite a satisfactoy match between the obseved and calculated data (Figue 7). Results of ou 3-tank closed model assuming the values of paametes given by Axelsson ae also plotted fo compaison. Axelsson s match and ou match look almost identical. Table 2. Paametes of the best fitting lumped paametes-(3-tank with closed oute bounday model) Axelsson This Study κ a1, kg/ba 3.64x x1 1 (14%) κ a2, kg/ba 2.9x x1 9 (17%) κ, kg/ba 7.73x x1 7 (2%) α a2, kg/(ba-s) 61.8 α a3, kg/(ba-s) (21%) 33.5 (9%)

6 Figue 9 pesents the simulation esults obtained fom ou 1-tank, 2-tank open, and 3-tank closed models. The 1-tank model does not give an acceptable match. Although the 3-tank closed lumped model gave the best match with the lowest RMS of the models tied, howeve, the 2-tank open model also gave a easonable match as well as lowe confidence intevals fo the paametes (not shown hee). This indicates that a 3-tank closed o 2-tank open model can be used to epesent the system detemine the esevoi and aquife paametes and as well as the initial wate level. Ou optimization study of the field data yielded an initial wate level of -38 m. Wate Level, m Wate Level (Well 7) Poduction Poduction, l/s Wate Level, m Measued Wate Level Axelsson's Match 3 Tank Closed Model Yea Figue 1. The wate level data and data on the poduction in the Gleadalu field (Axelsson, 1989) Yea Figue 8. Compaison of Axelsson s match with ou match Measued Wate Level 75 1-Tank Model 2-Tank Open Model 1 3-Tank Closed Model Wate Level, m Yea Figue 9. Simulation esults of 1-, 2-, and 3-tank models. Gleadalu Field The wate level and the poduction ate data in the Gleadalu low-tempeatue geothemal field in N- Iceland ae pesented in Figue 1 (Axelsson, 1989). This field has been utilized since The esevoi tempeatue at Gleadalu is about 61 o C. The main feed zone is at 45 m depth. Most of the wells dilled ae shallow (1-3 m) exploation wells. One poblem involved in simulating the Gleadalu field was the absence of the initial esevoi pessue o the initial wate level data. Lumped paamete modeling equies the initial wate level to be known a pioi. Hence, the simulations wee caied out to The obseved wate level behavio shown in Fig. 1 fo the Gleadalu field esembles the behavio of a system with constant pessue oute bounday. Fo a constant poduction ate, the esevoi pessue of a constant pessue oute bounday system declines shaply at ealy times and then eaches to a constant value at late times. Equations 9, 1 and 16 given in Saak et al. (23) descibe the ealy time and late time behavios of the 1- and 2-tank systems with constant pessue souce. The ealy time decline of the wate level eflects the esevoi popeties and its slope is equal to w p / κ as given by Equation 9. In fact this slope elationship is valid fo all systems since it is not a function of the aquife popeties and the oute bounday conditions. The late time steadystate esevoi pessue dop, howeve, is a function of the hamonic aveage of esevoi and aquife poductivities ( α a1 and α a2 ) and the poduction ate (w p ). Figue 1 exhibits a elatively constant poduction ate and a stabilized steady-state pessue dop. An equilibium between poduction and echage is eventually eached duing long-tem poduction, causing the esevoi pessue (o wate level) dawdown to stabilize. Such a behavio is valid fo systems with constant pessue oute bounday. Theefoe we consideed the 2-tank model with constant pessue oute bounday besides the 3-tank closed model as suggested by Axelsson (1989) fo simulating the esponse of the Gleadalu field. Pefoming gaphical analysis on the data by using the asymptotic equations given in ou companion pape (Saak et al. 23), we estimated 7 κ = 2x1 kg/ba and α a = kg/(ba-s). Hee, α a epesents the hamonic aveage. Based on these values, we chose ou initial guesses of the paametes and pefomed nonlinea egession. Results of ou simulation study ae given in Table 3.

7 Table 3. Paametes of the best fitting lumped paametes Axelsson This This Study Study 3-Tank 3-Tank 2-Tank Closed Closed Open κ a1, kg/ba 6.8x x19 (19%) κ a2,kg/ba 6.66x x17 (23%) κ, kg/ba 5.9x x16 (36%) α a2, kg/(ba-s) 1.89 α a3, kg/(ba-s) (11%) - 8.7x1 7 (12%) 8.17x1 6 (29%) 1.42 (5%) 3.75 (21%) RMS, ba (15%) Figue 11 shows the compaison of the obseved data, calculated wate level changes obtained by Axelsson and calculated by ou 3-tank closed model using Axelsson s paametes of the best fitting models. Note that thee ae some diffeences between Axelsson s esults and ou esults at ealy time data points. The eason fo such diffeence could be due to the accuacy of the poduction data used in ou model o due to the diffeence in initial wate level values used in models. Axelsson used the measued values wheeas we used the poduction data eadings obtained fom Figue 5 of Axelsson (1989). the thid column of Table 3 with a RMS value of.6 ba. The esults of the simulation, that is the compaison between obseved and calculated wate level, ae pesented in Figue 12. Wate Level, m Measued Wate Level Axelsson's Match 3-Tank Closed Model Yea Figue 12. Compaison of obseved and calculated wate level changes. Futhemoe, we applied 1-tank and 2-tank open lumped models. Simulation esults of those models as well as esults of the 3-tank closed model ae given in Figue 13 fo compaison puposes. As is clea fom Fig. 13, the wate level computed fom 1-tank model does not fit well the obseved data. Howeve, the 2- tank open and 3-tank closed models yield almost identical matches. This indicates that a 2-tank model with constant pessue oute bounday could also epesent the Gleadalu field esponse Wate Level, m 1 2 Measued Wate Level Ou Model With Axelsson's Paametes Axelsson's Match Wate Level, m 1 Measued Wate Level 1-Tank Model 2-Tank Open Model 3-Tank Closed Model Yea Figue 11. Compaison of obseved and calculated wate level changes (3-tank closed lumped model case) Yea Figue 13. Simulation esults of 1-, 2-, and 3-tank models. As a next step, ou model was used to detemine the 3-tank closed lumped model paametes. Simulation esults of ou best fit yielded the paametes given in Although Axelsson (1989) used a 3-tank closed model to match the data, a compaison of confidence intevals given in Table 3 fo the paametes of both models indicates that a 2-tank open model is moe

8 appopiate fo the data. Note fom Table 3 that the confidence intevals of the estimated paametes fo the 2-tank open model ae naowe than the confidence intevals of the estimated paametes fo the 3-tank closed model. Howeve, the geological and geophysical conditions in the aea should be consideed in choosing the most appopiate model. This is not the scope of this study and no futhe analysis was conducted. DISCUSSION The pessue measued in the obsevation well is not necessaily epesentative of the aveage pessue in the esevoi. Thee may be intefeence fom the poducing wells aound the obsevation well, causing the pessue to appea lowe. To get the tue aveage pessue, the esevoi should be shut in and allowing the pessue to stabilize. This is aely accomplished since the esevoi is continually poducing. To include the effects fom each well, a supeposition of the effects fom all the wells would be necessay. Howeve, in all the field cases discussed in this pape, the measued pessue is assumed to be epesentative fo the esevoi. Because the lumped models ae based on many simplifying assumptions, thei eliability is sometimes doubted. The pedictions and simulations ae made based on the data available. Amount (fequency) and duation (time length) of the data definitely affect the eliability of the simulations. The diffeence between the calculated and measued pessue o wate level changes, if occus, does not demonstate uneliability of lumped modeling, but the uncetainty in such pedictions. The amount and quality of the data and the type of model (1-, 2-, 3- tanks, with closed oute bounday o constant pessue bounday) chosen fo modeling ae the impotant paametes involved in uncetainty. Geologic and geophysical data, wheneve available, should be consideed in peliminay estimates of esevoi and aquife popeties. Such data ae paticulaly useful fo ealy evaluation of κ = Vφρc fo confined systems o κ = A φ / g fo unconfined systems. Confined esevoi is the one poduced by expansion. The wate expands because of its compessibility. Unconfined esevoi is poduced because of a fall in liquid level. We should emphasize that the fomulation of the lumped model is not affected by whethe the system is assumed to be confined o unconfined. The κ value obtained fom the simulation is analyzed to estimate the bulk volume V fo the confined case o the lateal aea A fo the unconfined case. Checking the values of V o A with the those estimated fom othe souces can indicate the dominant poducing mechanism of the system, whethe it is confined o unconfined (Olsen, 1984; Gudmundsson and Olsen, 1987). Effect of Injection In ode to maintain pessue in a esevoi, einjection may be consideed. The injection fluid will be colde and will cool down the esevoi. When the volume injected is known, an estimate of heat depletion in the esevoi can be made. In lumped paamete modeling the injected fluid must be included in the mass balance: W = W W W + W + W (5) c i p l e whee W i =initial mass, W p =mass poduced, W l =mass loss, W e =wate influx(echage), W in =mass injected. The natual mass loss due to natual dischage o evapoation, W l, has been assumed negligible in lumped paamete models. This may not be a good appoximation. If the ate of mass loss is constant, this eo is most ponounced fo ealy time, since that is when the ate was low. Assuming that the injection of fluid will not change the compessibility o total density of the system consideably, the injection and poduction tems can be lumped in a net poduction tem: W p, net = W p Win (6) which in tems of mass ate becomes: w p, net = w p win (7) Using this, the dawdown fo a vaiety of poduction schedules can be pedicted. It should be noted that no tansient effects in the esevoi and changes in tempeatue, density and compessibility as a esult of injecting cold wate have been consideed. CONCLUSIONS The main conclusions of ou wok discussed in this pape ae: 1. Lumped-paamete models used in this pape adequately match pessue (o wate level) dawdown-poduction data. 2. Although up to thee o fou tanks can be consideed in modeling the aquife-esevoi system, howeve, the two tank models, one esevoi and one aquife, seem to epesent the low-tempeatue geothemal esevois sufficiently. Additionally, educing the in

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