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NumericalsimulationoftheCerroPrietosteampipelinenetwork

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AlfonsoGarcia-Gutierrez

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RosemberOvando

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Applied Thermal Engineering 75 (2015) 1229e1243

Contents lists avai

Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

Hydraulic model and steam flow numerical simulation of the CerroPrieto geothermal field, Mexico, pipeline network

A. García-Guti�errez a, *, A.F. Hern�andez a, J.I. Martínez a, M. Cece~nas a, R. Ovando a,I. Canchola b

a Instituto de Investigaciones El�ectricas, Ave. Reforma 113, Col. Palmira, Cuernavaca, Mor. 62490, Mexicob Comisi�on Federal de Electricidad, Gerencia de Proyectos Geotermoel�ectricos, Campo Geot�ermico de Cerro Prieto, Carretera Pascualitos-Pescadores,Km. 26.5, Mexicali, BC 21700, Mexico

h i g h l i g h t s

� Extensive literature review of flow models of geothermal steam gathering networks.� Hydraulic model of the Cerro Prieto geothermal field steam network.� Selection and validation of the employed pressure-drop model.� Numerical flow simulation of the world's largest geothermal steam gathering network.� Detailed network pressure drop analysis and mapping of steam flow distribution.

a r t i c l e i n f o

Article history:Received 8 January 2014Received in revised form3 September 2014Accepted 6 September 2014Available online 4 November 2014

Keywords:Steam pipeline networkNumerical simulationCerro PrietoHydraulic modelPressure lossesHeat lossesModel documentation

* Corresponding author. Tel.: þ52 777 362 3811x73E-mail address: aggarcia@iie.org.mx (A. García-Gu

http://dx.doi.org/10.1016/j.applthermaleng.2014.09.081359-4311/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

The development of a hydraulic model and numerical simulation results of the Cerro Prieto geothermalfield (CPGF) steam pipeline network are presented. Cerro Prieto is the largest water-dominantgeothermal field in the world and its transportation network has 162 producing wells, connectedthrough a network of pipelines that feeds 13 power-generating plants with an installed capacity of720 MWe. The network is about 125 km long and has parallel high- and low-pressure networks. Prior tothis study, it was suspected that steam flow stagnated or reversed from its planned direction in somesegments of the network. Yet, the network complexity and extension complicated the analysis of steamtransport for adequate delivery to the power plants. Thus, a hydraulic model of the steam transportationsystem was developed and implemented numerically using an existing simulator, which allowed theoverall analysis of the network in order to quantify the pressure and energy losses as well as the steamflow direction in every part of the network. Numerical results of the high-pressure network were ob-tained which show that the mean relative differences between measured and simulated pressures andflowrates are less than 10%, which is considered satisfactory. Analysis of results led to the detection ofareas of opportunity and to the recommendation of changes for improving steam transport. A maincontribution of the present work is having simulated satisfactorily the longest (to our knowledge), andprobably the most complex, steam pipeline network in the world.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In geothermal fields, the steam from producing wells is usuallytransported through a network of steam pipelines to the powerplants which may be sited several hundred of meters or even somekilometers away. The network geometry becomes rather

06; fax: þ52 777 362 3804.ti�errez).

8

complicated and the pressure and temperature drop along thenetwork may become quite high. Thus, it becomes quite difficult topredict the pressure or flowrate changes due to normal operationor events like the opening, or regulating or closing of the valves ofwells, pipelines or power plants; the integration of newwells; shut-down of existing wells, and start-up or shut down of power plants.Therefore, numerical modeling and simulation of the steamtransportation network become essential to obtain a detailedknowledge of the operating conditions, pressures, temperaturesand flowrates at almost any position in the network; information

Nomenclature

A4 constant ¼ f (dimensionless numbers), 0 � A4 � 1B4 constant ¼ 1 � A4

d internal pipe diameterD differencef friction factorg acceleration due to gravitygc gravitational constantG mass velocityH hold-up, a function of several parameters and

constants for different flow regimesK resistance coefficientL distancep pressureQ hat flux per unit lengthRe Reynolds numberS exponent in Eq. (7)T temperatureU overall heat transfer coefficientn fluid velocityy parameter in Eq. (8)

Subscriptsac acceleration

cd conduction heat transfercv convection heat transferD Darcyf frictionG gas, gravityi internalint intermittentL liquidM mixturen subscript used in Eq. (8) to denote the Moody friction

factor for smooth pipesr radiation heat transfersat saturatedseg segregated

Greek symbolsD differencef phaseF two-phase pressure drop multiplierl input liquid contentm viscosityr densityq pipe angle from the horizontal∞ infinity

A. García-Guti�errez et al. / Applied Thermal Engineering 75 (2015) 1229e12431230

which otherwise is very difficult to obtain experimentally sincesteam pipelines are regarded as high-pressure vessels.

A literature search shows that flow modeling and simulation ofgeothermal steam networks has received some degree of attentionin the past. In one of the first works on geothermal steam pipelinenetwork simulation [1] the computer program VAPSTAT-1 wasdeveloped to simulate a four-well steam pipeline network of thesame type used in the Larderello geothermal field. Pressure, flow-rate and temperature were calculated at a given number of points.Later, a method for the calculation of two-phase flow in a singlesteam pipeline subjected to variations in elevation, pipe diameter,etc. was developed and incorporated in the FLUDOF computer code[2]. For vertical pipelines, FLUDOF uses the Griffith method [3] forbubbly flow, the Orkiszewski method [4] for intermittent flow, andthe Duns and Ros method [5] for transient and annular dispersedflow. For horizontal pipelines, FLUDOF uses the Dukler semi-theoretical method [6] along with the Hughmark criteria [7] forcalculating void fraction. For inclined pipes, FLUDOF uses the driftmodel [8] in conjunction with the TaiteleDukler diagram [9] todetermine the flow pattern. A model of a geothermal supply andreinjection pipe network was developed [10] and applied tosimulate the Ohaki geothermal pipe network, which had 24 pro-duction wells. This network is of the satellite-system type where afew sets of separators are located strategically in the field, eachreceiving the full flow from a number of production wells, and re-quires significant lengths of two-phase flow pipelines between thewells and the satellite stations. The basic method used in Ref. [10] isthe “transport” of the backpressure curve of a well, through thepipeline and separator mixer in the system. The reinjection systemand pump are also included, so that the modeled system includesthe geothermal supply system, the reinjection pump and itsoperation.

A numerical model based on a non-linear method to simulate ageothermal steam pipeline network is described in Ref. [11]. Thismodel was used to simulate one of the longest networks fromLarderello, Italy, whose total length was 6.9 km and incorporatedfive wells. Results compared satisfactorily with field measurements

and with an earlier simulation [1]. The effect of changes in piperoughness on network pipe flows and pressures was studied [12] bysimulating the reinjection pipe network of the Ohaki power station.It was found that the effects on the performance of the network dueto surface roughness changes from 4.57 � 10�5 m to 5.0 � 10�4 mwere small.

An energy-exergy study of the geothermal fluid network in theLarderello-Farinello-Valle Secolo area (Tuscany, Italy) was per-formed [13]. This network had 32 wells and 3 turbines. The simu-lation results showed that the network exergy losses were muchlower than the production plant losses, where the main exergylosses were concentrated in the turbine, the low-pressurecondenser and the cooling tower. The PowerPipe numerical codewas developed [14] to study the behavior of steam pipeline net-works including the characteristic curves of the network compo-nents and the network off-design operation. The code was appliedto analyze a geothermal field with four wells and a power gener-ation plant. Design and off-design analyses were performed byaccounting for the exergy losses of the system components. It wasfound that modifications in pipe geometry and insulation thicknessto reduce thermal losses could improve the system thermodynamicefficiency. Analysis of the off design operation led to determinationof some operating condition limits like steam condensation.

A numerical hydraulic model of the Los Azufres GeothermalField (LAGF), M�exico steam pipeline network was developed anddocumented in detail [15]. Flow simulations were performed todetermine pressure and heat losses, flow directions, and velocitiesin that network using one- and two-phase numerical simulators.This pipeline network has a length of some 28 km and transportsthe steam produced by 41 wells to 14 power plants.

The Northern California Power Agency (NCPA) developed anintegrated model that combines reservoir simulation with mathe-matical modeling of the wellbores, pipelines, and power plants.This has proven very useful for evaluating the most cost-effectiveimprovements to the combination of wells and surface facilitiesas well as for studying the benefit of increasing the volume ofaugmented injection at The Geysers Geothermal Field, USA [16].

Fig. 1. Cerro Prieto steam pipeline network.

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Verma [17] wrote the computer program, GeoSteamNet, forthe numerical simulation of steam transport in geothermalpipeline networks using Visual Studio.NET. The program con-siders (a) internally consistent thermodynamic properties ofwater, and (b) a numerical algorithm based on the principles ofconservation of mass, linear momentum (Newton's second law),and energy (the first and second laws of thermodynamics). Anexample of application of this program includes steam flow in ahypothetical geothermal power plant with two wells and oneproduction unit.

Specific studies on CPGF steam network include a modeldeveloped in the 800s based on the polytropic expansion of steamas it flows down a horizontal network of thermally insulatedpipelines [18,19]. Later, an adiabatic flow model in which only themomentum equation is solved [20] was developed. These modelswere not fully tested and their real scope is unknown. More recentstudies include a preliminary hydraulic model of the high-pressurenetwork [21], and the development of the high- and low-pressuresub-networks hydraulic models and some numerical simulationsof only the high-pressure network [22].

Pressure drop in single-phase pipelines is generally carried outfrom Moody friction factors [23] while pressure drop in two-phaseflow pipelines have traditionally followed empirically derivedmethods [3e9] that are limited to the range of conditions underwhich they were developed. Methods that are more recent have atheoretical basis and they have been under development for anumber of years. Ansari et al. [24] proposed a mechanistic modelfor upward two-phase flow in wellbores, while the TACITEcompositional transient, multiphase flow simulation tool wasincorporated into the Pipephase two-phase simulator as an add-onprogram [25]. More recent models include a study of the voidfraction propagation in a bubbly two-phase flow [26] and a two-region hydraulic averaging model for cuttings transport duringdrilling of horizontal wells [27], among others.

The present work is the result of an applied investigation on theCPGF network operation to address a real problem of the powerplant: The steam flow produced by geothermal wells appeared notto arrive entirely to the generating units. Thus, one of our mainobjectives was to locate the problematic network segments wheresteam flow stagnates or reverses from its planned direction. Properidentification of those areas can lead to recommending appropriatechanges to the operation and/or design of the network to assureand improve steam flow delivery to the power plants. Herein, thedevelopment of a more detailed, fully documented and validatedhydraulic model of the complete CPGF steam network is described.This network is perhaps, the most complex steam-gatheringnetwork in the world. The hydraulic model was implemented inthe Pipephase v9.1 numerical simulator [25], which allows for thedetermination of pressure drops, steam flowrates, heat losses, andcondensate quantification. An analysis for the selection and vali-dation of a pressure drop model for the flow simulations isincluded. Simulation results are presented on the behavior of thecomplete high-pressure network (divided into North and Southblocks). The objective is to obtain a global view of the operatingperformance of the network. Mapping of the steam flow distribu-tion in the complete high-pressure network, which constitutes animportant result, is also included.

2. The CPGF steam pipeline network

The CPGF is one of the largest liquid dominant geothermal fieldsin the world and its present installed capacity is 720 MWe. It iscomposed of four field areas named progressively from Cerro PrietoOne (CP1) to Cerro Prieto Four (CP4). The installed power plantsinclude four-37.5MWe units and one-30MWe unit in CP1; two-

110MWe units in CP2 and in CP3; and four-25MWe units in CP4.All units are of the condensing type [28].

The Cerro Prieto pipe network is essentially of the distributedsystem type where most of the separators are located immediatelyadjacent to each production well and individual pipelines trans-port steam to the main collecting ducts, called branches. Thenetwork is highly complex and has several arrangements forsteam separation. CP1 has high-pressure steam separation onlywhereas CP2, CP3, and CP4 have high- and low-pressure separa-tion. In CP2 and CP3, there also exist several “sites” for steamseparation. In a “site”, the high-pressure steam is separated firstand the separated water is sent to the low-pressure separatortogether with the brine from other neighboring wells. In CP4,there are “separation islands” which are squared areas, dividedinto four modules. Each module has four high-pressure separators,each receiving the two-phase flow from a well to separate thehigh-pressure steam. Then, the separated water of the fourstreams is mixed and fed to a single separator to obtain low-pressure steam. There also exist some auxiliary wells, which donot actually produce water or steam, but their facilities are used toseparate the steam from the mixture produced by a neighbor well.The large majority of the separated water is finally sent to theevaporative pond via open channels, although some water isinjected back to the reservoir.

The separated steam is transported to the power plants in apipeline network 125 km long approximately. The pipe diametersrange from 800 to 4600. The pipes are thermally insulated withmineral wool or glass fiber and an exterior layer of aluminum orwrought iron. The network has 183 connected wells of which 162are producing wells, the rest are wells for future integration; yet, asingle branch may be fed with steam from 1 to 36 wells. CP1 haseight high-pressure branches, while CP2, CP3, and CP4 have bothhigh- and low-pressure parallel branches, two per field area. Thesebranches feed steam to the power plants and thus, the presentmodel terminates at the end of the branches where steam isdelivered to the power plants.

The network also has several interconnections among thedifferent field areas in order to assure an adequate steam flow tothe power plants. Fig. 1 shows the steam pipeline network of theentire geothermal field. Thus, the complexity of the steam pipelinenetwork, the changes in the operating conditions, andmaintenanceand integration of new wells difficult the estimation of pressure orflowrates under normal operation conditions or under abnormal

A. García-Guti�errez et al. / Applied Thermal Engineering 75 (2015) 1229e12431232

events making steam transportation and supply to the powerplants a rather complex task, even for the experienced engineers.

3. Development of the pipeline network hydraulic model

In viewof the size and complexity of the network, it was decidedto develop a carefully documented hydraulic model of the pipelinenetwork such that every single piece of information of the networkdata could be traced to its origin and was therefore referenced inthe model documentation. It was also decided to use Pipephase, anexisting pipeline flow simulator, to analyze pressure distributionand steam flow in the network. The model was designed to includesteam flow from the orifice plate of each well into the trans-portation network and delivery to each of the four power gener-ating stations, and since there are no important differences intopographic height in the field, the whole network was regarded asa set of horizontal pipes.

The methodology employed included: (a) Gathering of networkdata; (b) documentation of the geometric model; (c) gathering ofoperative data; (d) selection of a numerical flow simulator; (e) se-lection and validation of a pressure drop model, and (f) calculationof heat losses.

3.1. Network data gathering

The development of the hydraulic model began with the gath-ering of information related to the pipeline network. This includedextensive revision of all existing information on the design char-acteristics of the network available in hardcopy or electronic files:e.g., pipe geometry, pipe materials, valves, fittings, insulation, aswell as information on producing wells, power plants, etc. Otherinformation such as historical well production records and statis-tical environmental data, like ambient temperatures, humidity, andwind velocity, were also collected. Visits to the field were made inorder to be acquainted with all flow devices and operations of thenetwork.

Additional fieldwork was done in order to verify or complete allrequired data. This included visual inspection of pipes, flow devicesand insulation, direct verification of dimensions (diameter, thick-ness) and measurement of temperatures in pipes, fittings, valves,and thermal insulations. All observations and measurements weresupported with photographic evidence in order to document thenetwork information in sufficient detail using formats compatiblewith those required by the numerical simulator.

3.2. Documentation of the geometric model

In view of the huge amount of the network data, a specificnomenclature was designed in order to facilitate their manage-ment, particularly to identify each pipeline segment (link), and toenter data into the Pipephase simulator. This nomenclature uses a12-character code that identifies each link in the followingmanner:

For example, the code M116 S01 R1 DA indicates that this linkstarts at the well M-116 site; that it is the first segment of the link;that the link is connected to Branch 1 of the CP2 field area, and thatit belongs to the high-pressure network system.

All data related to pipes and components of the steam pipelinenetwork were documented in MS Word formats specificallydesigned for this purpose. These formats are tables containingtypes, dimensions, schedules, of all flow devices, such as pipes,valves, fittings, etc. in a single link. Formats also contain thephotographic evidence of every link. Fig. 2 shows an example ofinformation documentation, where the upper left part shows thegeometry of the steam duct (link) M116 S01 R1 DA, which lies be-tween the orifice plate of well M-116 and the high-pressure Branch1 of CP2. The upper right part shows the photographic evidence ofthe corresponding duct. The filled format below the upper figurescontains the device data for link M116 S01 R1 DA. This format fol-lows the Pipephase GUI's link device data window, which was usedto ease entering data into the numerical simulator by simplycopying, pasting, and mouse-clicking the information contained inthe Word formats.

With the information properly classified, verified, and docu-mented in its respective format, the implementation of the hydraulicmodel was performed by integrating the four field areas and thehigh- and low-pressure pipeline network systems. The implementedmodel allows determination of the network performance via esti-mation of pressure drops, steam flowrates and heat losses, andquantification of mass condensate. It may also be used for assessingthe impact of changes in operating conditions, steam flowvariations,maintenance operations, and design changes such as integration ofnew wells. The model does not make decisions regarding thenetwork operationmode but its results provide decision elements tooperate the network and improve its performance.

3.3. Gathering of operative information

Operative data required for the simulations included pressureand flowrate at each well and at each delivery point of the powergeneration stations CP1 through CP4. Data were taken from dailyfield reports. The listing of data for the 162 producing wells for atypical day is too long to be included here but will be shown in thegraphical results later on. Table 1 is a summary of the wells con-nected to each steam transportation branch. An analysis of histor-ical data was performed in order to verify if the data wereconsistent and within expected trends, and to detect abnormaldata. Then simulations were carried out for the operations of agiven date, selected by the operating personnel of the geothermalfield.

3.4. Selection of a numerical flow simulator

Based on a search for two-phase geothermal water-steam flowsimulators in internet, in the open literature, by contact with in-ternational colleagues, and from preliminary steam flow simula-tions of the CPGF [21,22] and the LAGF [15] steam pipelinenetworks, it was decided that the Pipephase v9.1 [25] numericalsimulator was the best alternative for the present study. As ourmain objectivewas to determine pressure, energy losses, and steamflow throughout the network, our search included simulators onlyfor steady state.

Pipephase is a powerful simulation tool that accounts forsteady-state multiphase flow and heat transfer in wells, pipes andpipeline networks transporting oil and gas or steam and itscondensate, and has been extensively tested in the oil industry. Ithas a friendly Graphical User Interface (GUI) that facilitates theconstruction of hydraulic models and viewing of results, and has anoptimization module for pipeline networks. Its advanced userinterface enormously facilitated the construction and imple-mentation of the highly complex and detailed model of the CerroPrieto network as well as the analysis of results.

Fig. 2. An example of information documentation of the Cerro Prieto geothermal field steam pipeline network.

Table 1Summary of wells connected to each steam transportation branch.

Field area Branch Number of wells Number of sites Integrated wells Integrated sites Future wells Future sites

CP1 1 5 0 4 0 1 0CP1 2 2 0 2 0 0 0CP1 3 2 0 2 0 0 0CP1 4 2 0 0 0 2 0CP1 5 1 0 1 0 0 0CP1 6 0 0 0 0 0 0CP1 7 2 0 2 0 0 0CP1 8 7 0 5 0 2 0

CP2 1 37 0 36 0 1 0CP2 2 37 4 32 4 5 0

CP3 1 28 4 24 3 4 1CP3 2 35 3 33 3 2 0

CP4 1 13 2 10 1 3 1CP4 2 12 2 11 2 1 0

Totals 183 15 162 13 21 2

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A. García-Guti�errez et al. / Applied Thermal Engineering 75 (2015) 1229e12431234

3.5. Selection and validation of a pressure drop model

The Pipephase flow simulator has several pressure drop modelsthat can be used for the present case. Selection of one such modelinvolved measuring pressure profiles in the high- and low-pressurenetworks of Branch 1 of CP2 and matching these measurementswith results obtained from different pressure drop models. Pres-sure measurements were performed with two calibrated manom-eters and one digital pressure transducer. From the variouspressure drop models tested, it was found that the Beggs & Brill[29] method was the one that best compared with measured data,and on this basis, it was chosen for the simulation of the wholenetwork. An advantage of this model is that the correlations forliquid holdup and friction factor degenerate to single-phase con-ditions as the flow approaches all liquid or all gas [29]. Fig. 3 showsa comparison of measured and computed pressures using twopressure drop models for the case of the high-pressure Branch 1 ofCP2. As seen from the graph, the Beggs& Brill model matches betterthe measured pressures while the Beggs & Brill-Moody modeldeparts from the measured data at shorter distances.

3.6. Calculation of heat transfer coefficients

In order to reduce computation load on numerical simulations,overall heat transfer coefficients were calculated for each field areaof Cerro Prieto, including both the high- and low-pressure pipe-lines. A wide range of flow and pressure operating conditions,temperature-dependent thermophysical properties of pipes, insu-lation materials and covering metallic layers, average ambienttemperatures, and wind speed and direction were used in thecomputations. The physical condition of the insulation materialswas mapped using the software ArcGis and considered in thecalculations.

Pipe diameters varied from 800 to 4600, and steam flowrates var-ied between 15 and 70 t/h at the wells and up to 875 t/h at thepower plant delivery points. Pressure ranged between 8 and 16 kg/

Fig. 3. Comparison of measured and computed pressures using the Beggs & Brill-Moody and Beggs & Brill pressure drop models in the high-pressure Branch 1 ofCerro Prieto 2.

cm2 in the high-pressure network and between 4 and 7 kg/cm2 inthe low-pressure network. Steam temperatures were taken as thesaturated values corresponding to the operating pressures.Ambient conditions (temperature and wind velocity) were takenfrom the field records as the average for the month of this study.Calculation of the overall heat transfer coefficients included con-duction, internal and external convection, and pipe surface radia-tion with appropriate material emissivities. An MS Excelspreadsheet was specifically developed for this purpose. Validationof pipe external surface temperatures estimated with this spread-sheet was effected by comparing against field measured tempera-tures using infrared technology.

It was found that overall heat transfer coefficients are practi-cally independent of flowrates for most operating conditions butrather more dependent on pipe diameter. For well insulated pipes,the coefficients reach values of 1.0e1.2 W/(m2 �C) for pipe di-ameters from 12.7500 to 2000, respectively, and for a flowrate of 20 t/h of steam at 7 barg; however they can be as high as 21.0 for barepipes.

Useful values of the heat transfer coefficients for the simulationswere determined from an overall inspection of the physical con-ditions of the insulating materials and their metallic protectioncover as well as from comparisons of infrared thermographicmeasurements on pipes, insulating materials and metallic coversversus temperatures computed for different values of the overallheat transfer coefficient. From these analyses, an overall heattransfer coefficient of 1.5 W/m2-K was selected, which proved to bea good choice during the simulation work.

4. Theoretical aspects

4.1. Calculation of pressure losses in pipelines

The pressure drop in a pipe of length L and diameter D understeady-state conditions is given by:�dpdL

�¼�dpdL

�fþ�dpdL

�gþ�dpdL

�ac

(1)

The term on the left is the total pressure loss in the pipe, the firstterm on the right is the pressure loss due to friction, the secondterm is the pressure loss due to elevation, and the third term rep-resents pressure loss due to acceleration. The frictional losses relatethe shear stress at the wall and the DarcyeWeisbach (or Moody)friction factor. Pipephase uses this friction factor defined as fourtimes the Fanning friction factor.With this and the usual definitionsfor the other terms, Eq. (1) can be rewritten as.

�dpdL

�¼�fDrv2

2gcd

�þ�rgsen q

gc

�þ�rvdvgcdL

�(2)

Two-phase flow occurs when steam condensation occurs in apipe and thus, in Eq. (2) the friction factor and the physical prop-erties are replaced by the two-phase flow equivalent resulting in:

�dpdL

�2f

¼ fMrMv2M2gcd

!þ�rMg sen q

gc

�þ�rMvMdvM

gcdL

�(3)

where M stands for mixture. The definitions of density and frictionfactor for the mixture are specific to each two-phase flow correla-tion or pressure drop model. In the present study, the Beggs & Brillmodel was selected from a comparisonwith experimental data anda description of this model follows from the Pipephase User'sManual [25]. In this model, pressure drop due to friction forsegregated, intermittent, and distributed flow is given by:

A. García-Guti�errez et al. / Applied Thermal Engineering 75 (2015) 1229e1243 1235

�dpdL

�f¼ f2frMv2M2gcd 144

!(4)

where 2f denotes two-phase flow. The Reynolds number is definedby:

Re ¼�1488rMvMd

mM

�(5)

The mixture density is given by:

rM ¼ rLHL þ rGð1� HGÞ (6)

where HL and HG are the liquid and gas hold-up, respectively, andthe two-phase friction factor is defined as:

f2f ¼ eSfn (7)

In Eq. (7), fn is the Moody single-phase friction factor and S is givenby:

S ¼�

y�0:0523þ 3:318y� 0:8725y2 � 0:01863y4

�(8)

where S ¼ ln(2.2ey � 1.2) for (1 < ey < 1.2) and y ¼ ln(lL/nL2).The pressure loss due to elevation is given by:�

dpdL

�g¼ rMg senðqÞ

144gc(9)

The pressure loss due to acceleration is given by:�dpdL

�ac

¼ vMvSGrn144gcp

�dpdL

�T

(10)

For transitional flow, the friction losses are given by:�dpdL

�f¼ A4

�dpdL

�f ;seg

þ B4

�dpdL

�f ;int

(11)

where A4 is a function of diverse dimensionless numbers and takeson values such that 0� A4 � 1, and B¼ (1 � A4). The pressure lossesdue to elevation are given by the same expression of segregatedflow.

4.2. Calculation of pressure losses in fittings

Pressure losses in fittings are computed from.

Dp ¼ KG2F

2rg(12)

where Dp is the pressure loss in a fitting, K is resistance coefficient,G is the mass velocity (mass flowrate/flow area), g is the accelera-tion due to gravity, and r is the fluid density.

4.3. Calculation of heat losses

The heat exchange between the steam and the environment iscontrolled by forced convection between the steam and theinternal wall of the pipe; heat conduction through the pipe wall,insulating material and metallic lining; and natural convectionand radiation from the external wall. Thus, heat losses are givenby:

Q ¼ UADT (13)

where Q is the heat loss per unit length of pipe, U is the overallheat transfer coefficient and DT is the temperature differencebetween the bulk of the fluid and the environment. Typically, thethermal resistance due to convection inside the pipe is negligible.Thus for an insulated pipe the heat loss per unit length to theenvironment is the sum of heat transfer by convection Qcv andradiation Qr, and these are balanced with the heat conductedthrough the pipe wall, insulating material and metallic lining Qcd.This is given by:

Qcd ¼ Qcv þ Qr ¼ UpDiðTsat � T∞Þ (14)

where Tsat is the temperature of saturated steam and T∞ is themeanambient temperature.

As mentioned above, an overall heat transfer coefficient of1.5 W/m2-K was selected and fed to the numerical simulator forcarrying out the simulations.

5. Analysis and discussion of results

5.1. Cerro Prieto high-pressure steam pipeline network

The Cerro Prieto geothermal field is composed of four main fieldareas, CP1 thru CP4, and except for CP1, which has only a high-pressure network, all of them have low- and high-pressure net-works. CP1 has eight high-pressure branches, while CP2, CP3, andCP4 have both high- and low-pressure parallel branches, two perfield area. The high-pressure network has a number of in-terconnections between field areas in order to provide an adequatesteam flow to the power plants, while the low-pressure networkhas fewer interconnections and wells. Hence, the high-pressurenetwork includes all wells from the four field areas that generatesteam from a primary separator.

Based on the operative information of the date selected forsimulations, this network is composed by two main blocks whichare joined, or separated, at the south end of InterconnectionCP2eCP3, by a pipe equipped with a closed valve and a bypass witha 1200 butterfly valve opened at 5% of full aperture. As a firstapproximation to solve the high-pressure networkmodel, the valvewas assumed to be fully closed so that the model could be split intotwo independent blocks called the North Block and the South Block.The North Block is composed of Branches 1 and 2 of CP3; Branches 1and 2 of CP4, and some wells from CP2. It has three main in-terconnections: Interconnection B that conducts steam from CP3towards CP1; Interconnection C that collects steam from someadditional wells of CP3 and discharges to Interconnection B; andInterconnection CP2eCP3 that links CP2 with CP3, see Fig. 4. TheSouth block is composed of Branches 1 and 2 of CP2 and all eightCP1 Branches, except for Interconnection B, which is included aspart of the North Block.

A steam balance on the two blocks yielded an excess of 30 t/h ofthe flow expected at the CP3 power plants and a deficiency of about30 t/h at the CP2 power plants. To achieve a correct mass balance,the valve connecting the two blocks was modeled as a 30 t/h sinkfor the North Block and a 30 t/h source for the South Block. Thesevalues represent the flow at an aperture of 5% of the bypass valve.The pressure and enthalpy are tuned to the same values for both thesink and source.

Once the model was set-up according to the field data, it wasthen tested by comparing simulation results with pressure and flowmeasurements obtained from the field reports, and by analyzingpressure profiles in selected lines and steam flow distribution in the

Fig. 4. View of the high-pressure steam transportation network of Cerro Prieto as implemented in the numerical simulator.

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network. Fig. 4 shows a diagram of the high-pressure networkmodel as implemented in the numerical simulator, where theNorth and South Blocks are delimited by dotted lines.

5.2. North Block of the high-pressure network

The North Block transports steam from the CP3 and CP4 wellsand from the CP2 wells that feed Interconnection CP2eCP3. Each ofthe two main branches of CP3 and CP4 transports steam to their

own power plants while Interconnection B transports steam to theCP1 power plants. Each well was considered as a source boundarycondition since flowrate at the orifice plate of each well is known.Similarly, the pressures at the arrival of all power plants wereoriginally fixed as boundary conditions. However, due to instabilityof the numerical solution and misdistribution of steam flows, onlythe CP3 power plants were modeled with pressure as boundaryconditions while the power plants of CP1 and CP4 were modeledwith flowrate as boundary condition.

Fig. 6. Comparison of absolute measured and computed pressures of the wells con-nected to Branch 1 of Cerro Prieto 3. The dotted line indicates a linear regression.

Fig. 7. Comparison of absolute measured and computed pressures of the wells con-nected to Branch 2 of Cerro Prieto 3. The dotted line indicates a linear regression.

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Preliminary analysis of field data showed that a fairly steadytrend was characteristic for all wells one month before and onemonth after the date selected for simulation. However, severalCP3 wells presented either a peak or a valley at the date of in-terest. Simulations performed with the original data showed thatthese wells presented relative deviations well in excess of 10%with respect to measured values. Hence, the field data of suchwells were replaced by an average value over the month of in-terest, and this approach yielded much better results. Fig. 5shows an example of a well that presented this problem. Thevalue of September 17 is considerably far from the mean value ofthe trend.

Fig. 6 shows a comparison of the measured and computedpressures of the CP3wells connected to Branch 1. A tendency can beseen to follow a 45� line, indicative of a good prediction by thenumerical model. Also shown is a regression line of computed andmeasured pressures at thewells. The regressed line is parallel to the45� line and above it, indicating that the computed values arehigher than the measured ones; however the maximum relativedifference is 4.8% while the average of relative differences is 1.74%and the standard deviation is 2.04. Similarly, Fig. 7 shows a com-parison of measured and computed pressures of the CP3 wellsconnected to Branch 2. Once again, a well-defined trend to follow a45� line can be seen which again is indicative of a good predictionby the numerical model. Also shown is a regression line ofcomputed and measured pressures that is parallel to the 45� lineand above it. In this case, the average of relative differences is 1.19%with a standard deviation of 2.23 while the maximum relativedifference is 5.25%.

Fig. 8 shows the computed and measured pressures of the wellsthat supply steam to the CP4 power plants. The computed wellpressures were higher than measured data, as shown by the 45�

line and the regression line above it, with a maximum relativedifference of 5.15%. Fig. 9 shows a comparison of measured andcomputed pressures of the wells that feed InterconnectionCP2eCP3. The maximum relative difference is 6.6% while theaverage of relative differences is 3.5% with a standard deviation of1.49. In this case, computed well pressures are also higher thanmeasured pressures, as shown by the 45� line and the regressionline above it; however, the differences are quite reasonable.

Figs. 10 and 11 show the pressure profiles of the two in-terconnections between Branches 1 and 2 of CP3. In Fig. 10, it isobserved that the pressure drop of Interconnection CP2eCP3 be-tween Branches 1 and 2 is quite small, 0.04 kg/cm2, along the nearly440-m length of this duct. This small pressure difference indicates arelatively small steam flowrate in this part of the interconnectionwith the steam flowing from Branch 2 to Branch 1.

Fig. 5. Time series of line pressure of a well showing a peak at the date selected forevaluation of the network performance.

Fig. 11 shows the pressure profile of Interconnection B betweenBranches 1 and 2 of CP3. In this case, the pressure drop along thenearly 440-m length is 0.14 kg/cm2, a value that is higher than thatof Interconnection CP2eCP3 between the two branches. Hence a

Fig. 8. Comparison of absolute measured and computed pressures of the wells con-nected to Branches 1 and 2 of Cerro Prieto 4. The dotted line indicates a linearregression.

Fig. 9. Comparison of absolute measured and computed pressures of the CP2 wellsthat supply steam to the Cerro Prieto 3 power plants via Interconnection CP2eCP3. Thedotted line indicates a linear regression.

Fig. 10. Absolute pressure profile of Interconnection CP2eCP3 between Branches 1 and2 of CP3.

Fig. 12. Absolute pressure profile of Interconnection CP2eCP3.

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higher flowrate is expected in this segment of the interconnection;however, steam in this line flows from Branch 1 to Branch 2.

Fig. 12 shows the pressure profile of the interconnection be-tween CP2 and CP3. This steam duct is nearly 1600m long and has apressure drop of about 0.35 kg/cm2. The profile has a small gradientduring the first 100 m approximately and then the pressure drops

Fig. 11. Absolute pressure profile of Interconnection B between Branches 1 and 2 ofCP3.

more rapidly due to the growing number of wells that feed this line.Fig. 13 shows the pressure profiles of Branches 1 and 2 of CP3.Starting from the left, at 0 m, the pressure of Branch 1 is greaterthan that of Branch 2 up to around 1600 m where the pressure ofBranch 2 becomes greater and at about 4000 m the profiles crosseach other again. These crossing points roughly correspond to thepoints where the two branches intersect with InterconnectionsCP2eCP3 and B. The point at which the pressures are equal isdetermined mainly by Interconnection B, which transports steamto CP1.

The model considers fixed flows at the arrival to the CP4 plantsand at the flow measuring point corresponding to the arrival ofInterconnection B to CP1. On the other hand, the model considersfixed pressures at both CP3 plant arrivals, which are tuned todistribute uniformly the flow relative differences between the twoplants.

Table 2 summarizes the pressure results. It should be noted thatCP4 showed a larger relative difference because aperture of thepressure-reducing valves, which distribute the flow towards bothCP3 and CP4, located 60 m upstream of the CP4 plants, was notmodeled. The location of these valves is in a highly complicatedinterconnection between CP3 and CP4, and in practice, these valvesoperate partially open creating a large pressure drop locally (chokepoint) since well pressures in CP4 are near 16 kg/cm2 and powerplant pressures in CP4 are about 12 kg/cm2. In the model, regula-tion of these valves led to numerical instability and convergence

Fig. 13. Pressure profiles of Branches 1 and 2 of CP3.

Table 2Comparison of computed and measured steam pressures of the network branchesfeeding the power plant flows (the asterisk * indicates a boundary condition value).

Measuredpressure, kg/cm2

Computedpressure, kg/cm2

Relativedifference, %

CP3-Branch 1 12.86 12.96* 0.76CP3-Branch 2 13.23 13.07* �1.02CP4-Branch 1 11.74 15.35 30.75CP4-Branch 2 11.43 15.57 36.17CP1-IntB 11.33 10.74 �5.22

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problems. During simulation this problem was solved by leavingthe valves fully open, which established a free path towards CP4;however, the plant pressures had to be increased in order to matchthe steam flowrates specified as boundary condition. Thus, theestimated pressures of CP4 shown in Table 2 actually represent thepressure at the inlet of the pressure-reducing valves. The largerelative pressure differences in the two CP4 branches are mainlyexplained by the fact we are comparing pressure values that arephysically separated by a distance of about 60 m, between thepower plant delivery points and the upstream point of thepressure-reducing valves. The relative difference in the pressures ofthe other plants is less than 5.5%. Table 3 summarizes the flowrateresults, for which the relative difference is less than 1%.

Fig. 14 shows a schematic diagram of the flow distribution of themain steam pipelines that comprise the high-pressure North Blockof Cerro Prieto. In this figure, Branch 1 of CP3 runs horizontally fromleft to right and Branch 2 runs in parallel. The interconnectionsbetween these two branches and other interconnections are shownin the figure. The quantities shown on the diagram correspond tothe total mass flowrate (t/h) while the values in parenthesis indi-cate the steam quality (%). The results shown in Table 3 correspondto the steam flowrate (total mass flowrate times quality).

5.3. South Block of the high-pressure network

The South Block transports steam from thewells of CP1 and CP2,except for the wells of CP2 that supply steam to the North Blockthrough Interconnection CP2eCP3, as already mentioned. In CP1,there exist eight branches that are located at both the north andsouth sides of the CP1 power plant. In CP2, there are two mainbranches, each of them transporting steam to their own powerplants. Two short interconnections permit an adequate steam dis-tribution between branches 1 and 2 of CP2. Even though CP1 haseight branches, there are very few wells connected to themwhereas most of the wells from the South Block are connected tothe CP2 branches, as shown in Fig. 4. Interconnection CP2eCP1allows the transfer of steam from Branch 1 of CP2 to the CP1branches located at the south side of the CP1 power plant. Aregulated valve at the end of Interconnection CP2eCP1 controls thesteam flow by reducing the flowing pressure.

All wells of the South Block were set as a source boundarycondition since flowrates at the orifice plates are known. Similarly,

Table 3Comparison of computed and measured steam flowrates of the network branchesfeeding the power plant flows (the asterisk * indicates a boundary condition value).

Measuredflowrate, t/h

Computedflowrate, t/h

Relativedifference, %

CP3-Branch 1 737 743.18 0.84CP3-Branch 2 865 868.15 0.36CP4-Branch 1 421.6 421.94* 0.08CP4-Branch 2 338.1 338.28* 0.05CP1-IntB 663 760.22* 0.08

the pressure at the arrival of all power plants was originally set asboundary condition. Nevertheless, under this scheme, numericalinstability and convergence problems arose, and thus the sinkscorresponding to the end (flow delivery) points of Branch 7 of CP1and Branch 2 of CP2 were modeled with flowrate as boundarycondition, while pressure was maintained as boundary conditionfor the remaining sinks. As in the North Block simulation, un-certainties and mismatches in input data from the South Blockwells were eliminated through preliminary analysis and selectionof appropriate data.

Fig. 15 shows a comparison of the measured and computedpressures for all wells included in the South Block. A trend maybe observed for all well pressures to follow a 45� line, indicativeof a good prediction by the numerical model. Also shown is aregression line of computed and measured pressures. Theregressed line is parallel to the 45� line and slightly below it,indicating that the computed values are lower than the measuredones; however, the maximum relative difference is �8.4% whilethe average of relative differences is �0.3% and the standarddeviation is 3.6.

Three main groups of wells can be identified in this plot. Group1, with pressures of about 8e9 kg/cm2, which corresponds to thewells connected to Branches 1e8 of CP1. Group 2, with pressuresranging from 10 to 12 kg/cm2 corresponding to the wells connectedalong Interconnection CP2eCP1. Group 3, with pressures rangingfrom 12 to 16 kg/cm2, which corresponds to the wells connected toBranches 1 and 2 of CP2. The wells of Group 2 showed themaximum relative differences, which can be explained by thestrong pressure differences between the wells of CP2 and CP1 andby the regulation of the valve located at the end of InterconnectionCP2eCP1 that produces an additional pressure drop, as will beshown later with regard to Fig. 20.

Fig. 16 shows the pressure profiles of Branches 1, 2, and 3 of CP1,while Fig.17 shows the pressure profiles of Branches 5 thru 8 of CP1.Branch 4 was not operating at the date of simulation. In Fig. 16,Branch 1 shows a pressure drop of 0.4 kg/cm2 while the pressuredrop along Branches 2 and 3 is smaller. This is because Branches 2and 3 currently transport much less steam than their design valueas some wells have ended their productive life.

Fig. 17 shows a comparison of the pressure profiles for CP1Branches 5 thru 8. Starting from the points farthest from the plant(left side on the plot), it can be seen that Branch 6 initially operatesat higher pressures than the other branches since it operates as asecond interconnection between CP2 and CP1. The slope change inthe pressure profile of Branch 6 at 1600 m may be explained by anincrease in pipe diameter from there onwards. The pressure profileof Branch 7 also presents a notorious change in slope around 725m,and this change is related to the junction of InterconnectionCP2eCP1 with this branch.

Pressure drop becomes greater from this point on due to theincrease in steam flow being transported. It can also be noted thatat the end of the four branches, pressure tends to be equal. Thepoint at which the pressures are equal is determined mainly by amanifold located 80m upstream from the arrival points of branches5, 6, and 7 to the CP1 power plant. The steam from Branch 8 iscollected in this manifold and then delivered to the CP1 powerplant through Branches 5, 6 and 7.

Fig.18 shows a comparison of the pressure profiles of Branches 1and 2 of CP2. It can be seen that Branch 1 initially operates at higherpressure than Branch 2 up to about 1400 m where the pressure ofBranch 2 becomes greater. This crossing point roughly correspondsto the interconnection between both branches where Branch 1supplies steam to Branch 2 as shown in Fig. 19. The larger pressuredrop in Branch 1 indicates that this branch transports more steamthan Branch 2.

Fig. 14. Flow distribution in the high-pressure North Block network of the CPGF.

Fig. 15. Comparison of absolute measured and computed pressures of the CP1 and CP2wells included in the South Block. The dotted line indicates a linear regression.

Fig. 16. Pressure profiles of Branches 1, 2, and 3 of CP1.

Fig. 17. Pressure profiles of Branches 5, 6, 7, and 8 of CP1.

Fig. 18. Pressure profiles of Branches 1 and 2 of CP2.

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Fig. 19. Absolute pressure profile of the interconnection between Branches 1 and 2 ofCP2.

Table 4Comparison of computed andmeasured pressures in the network branches that feedthe power plant flows (the asterisk * indicates a boundary condition value).

Measuredpressure, kg/cm2

Computedpressure, kg/cm2

Relativedifference, %

CP1-Branch 1 7.76* 7.76 0.01CP1-Branch 2 7.38* 7.38 0.00CP1-Branch 3 7.73* 7.73 0.00CP1-Branch 5 8.31 7.78* �6.37CP1-Branch 6 7.72 7.80* 1.05CP1-Branch 7 7.61 7.72 1.42CP2-Branch 1 12.34 12.30* �0.34CP2-Branch 2 12.39 12.32 �0.58

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Fig. 19 shows the pressure profile of the interconnection be-tween Branches 1 and 2 of CP2. This pressure profile has a verysmall gradient along the interconnection whereas a slightly higherpressure-drop occurs at the valves located towards the end of theinterconnection that allows steam to flow from Branch 1 to Branch2. Fig. 20 shows the pressure profile of Interconnection CP2eCP1.This steam duct is nearly 2300 m long and has a pressure drop ofabout 2.5 kg/cm2 between inlet and the regulated valve locatednear the end of the duct. The valve produces an additional pressuredrop of about 1 kg/cm2, so that steam can flow into Branch 7 of CP1.

As previously mentioned, the South Block model consideredfixed flows at the sinks corresponding to Branch 7 of CP1 andBranch 2 of CP2, while the rest of the branches were modeled withpressure as boundary condition. In some cases, the fixed pressureshad to be tuned to distribute uniformly the flow relative differencesbetween the CP1 power plants (Branches 5, 6, and 7) and the CP2power plants. Table 4 summarizes the computed pressure results ofthe power plants of the South Block. The relative differences areless than 1.5% with the only exception of Branch 5 of CP1. It wasassumed that this difference was due to a simple uncertainty in theinput data since Branch 5 is fed with the steam coming from anindividual well and from the manifold, which collects and tends toequalize the pressures of the steam coming from Branches 6, 7, and8. Therefore, the measured pressure in Branch 5 seems to besomewhat large when compared to those of Branches 6 and 7.

Fig. 20. Absolute pressure profile of Interconnection CP2eCP3.

A comparison of computed and measured steam flowrates ofeach branch at the arrival to the CP1 and CP2 power plants is shownin Table 5. It can be seen that except for Branch 2 of CP1, all relativedifferences are less than 5% in absolute value. The largest relativedifference in Branch 2 of CP1 can be explained, again, as amismatchsince Branch 5 of CP1 received the steam of only two wells and thesum of their computed flowrates is less than the measuredflowrates.

Figs. 21 and 22 show the flow distribution in the high-pressureSouth Block network. Fig. 21 includes Branches 1 and 2 of CP2. Fromthis figure, it may be seen that Branch 2 received steam fromBranch1 through the interconnection between both branches, and fromsome of the CP2 wells that feed Interconnection CP2eCP3 (seeNorth Block description and Fig. 4). Just before the arrival at the CP2power plants, the steam from well 630 and some steam fromBranch 2 flowed back into Branch 1, as shown in the figure. At thesame time, Branch 1 supplied steam to Interconnection CP2eCP1.Fig. 22 shows the flow distribution in Branches 5 thru 8 of CP1 andInterconnection CP2eCP1. The steam coming from CP2 is distrib-uted through both Interconnection CP2eCP1 and Branch 6 of CP1.All the steam coming from Branches 5, 6, 7, and 8 of CP1 enters themanifold and is then delivered to the CP1 power plant throughBranches 5, 6, and 7. The mapping of steam flow distribution in thenetwork, shown in Figs. 14, 21 and 22, constitutes an importantresult from the present study since it would have been very difficultto achieve this result without numerical simulation.

6. Conclusions

A hydraulic model of the Cerro Prieto geothermal field steampipeline network has been developed. The model was carefullydocumented in graphical and tabular form and every piece of infor-mation is traceable to its origin. Thus, the model information is nowcompiled into a single document, in a single place and under a singleadministrator, and can be implemented numerically usingpracticallyin anyexisting two-phase simulator. This result is important since the

Table 5Comparison of computed and measured steam flowrates in the network branchesthat feed the CP1 and CP2 power plant flows (the asterisk * indicates a boundarycondition value).

Measuredflowrate, t/h

Computedflowrate, t/h

Relativedifference, %

CP1-Branch 1 58.71 61.47 4.69CP1-Branch 2 34.79 31.85 �8.46CP1-Branch 3 21.29 20.55 �3.48CP1-Branch 5 187.99 195.91 4.21CP1-Branch 6 197.67 205.29 3.86CP1-Branch 7 244* 243.41 �0.24CP2-Branch 1 794 774.72 �2.43CP2-Branch 2 754* 748.23 �0.76

Fig. 21. Flow distributions in Branches 1 and 2 of CP2.

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transportation network is highly complex as it transports steam from162 producing wells, of a total of 183 connected wells, to 13 powerplants in the four field areas using parallel high- and low-pressurenetworks with several interconnections between them.

The numerically implemented model was used to performcomputations and evaluate the network performance. Results arepresented only for the field-wide high-pressure network due tospace limitations. In general, well pressures compare withmeasured pressures with relative differences under 8%. Similardifferences were obtained when comparing computed flowratesand pressures with measured data at the inlet of the power plants,except for the CP4 power plants where convergence problemsoccurred due to the large pressure drop at the control valves locatedsome 60 m upstream of the power plants. This was solved by

Fig. 22. Flow distributions in Branches 5e8

modeling the valves as fully open and increasing pressure at theinlet of the power plants until the flowrate set as boundary con-dition was matched.

The pressure profiles of the interconnections between Branches1 and 2 of the CP3 field area indicate very low pressure-drops (lessthan 0.14 kg/cm2) and steam flowing in opposite directions. Asimilar result was obtained for the interconnections betweenBranches 1 and 2 of the CP2 field area. Conversely, the pressureprofiles of the interconnections between CP2 and CP1 field areasshow large pressure drops, of about 3.5 kg/cm2. Such differences inoperating pressures between different field areas represented adifficult task to solve during network simulation.

The implemented model allowed determination of the overallnetwork performance and the quantification of the steam flow

of CP1 and Interconnection CP2eCP1.

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characteristics such as flowrate, quality, direction and velocity,pressure drops, and heat losses through the various networkcomponents. It also permitted identification of some areas of op-portunity for improving network performance, and at present,most of the recommended changes have been implemented;however, these findings are out of the scope of the present workand will be the subject of a future paper.

Furthermore, the model can be used for the analysis of theimpact of changes in operating conditions, network changes due toremoval or addition or removal of new wells and pipelines, as wellas of maintenance activities. A main overall contribution of thepresent work is having simulated satisfactorily the longest (to ourknowledge), and probably the most complex, steam pipelinenetwork in the world. Our work also shows also that it is feasible tosimulate reliably large, complex steam networks via up-to-datenumerical simulators. The mapping of steam flow distribution inthe network clearly shows a result that would have been verydifficult to achieve without numerical simulation.

Acknowledgements

Thanks are due to the authorities of the Cerro Prieto geothermalfield and Instituto de Investigaciones El�ectricas for permission andencouragement to publish this work. We also thank the paper re-viewers who helped us greatly to improve this paper.

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