14thNew Zealad Workshop 1992 NON-LNEAR MODELLNG OF A GEOTHERMAL STEAM PPE Y. Huag ad D. H. Freesto Geothermal stitute, Uiversity of Aucklad SUMMARY Recet work o developig a o-liear model for a geothermal pipe etwork system is preseted i this paper. Previous methods of modellig are reviewed. Usig the mass ad eergy balace at each ode ad loop of a pipe etwork, a o-liear equatio set ca be established. The equatio set is formulated from a coceptual model of the pipe etwork, which is composed of steam wells ad coected pipelies. The well characteristics curves, pipelie frictioal characteristic ad the differet combiatios of the coected compoets all cotribute to the pressure ad mass flow rate distributios i the system. Some recet umerical experimets o the o-liear model are compared with talia experiece. The results coverged satisfactorily. Usig this model, the user ca chage the coectio ad the characteristic of each idividual demad i a etwork ad chose a desired maifold pressure; the simulator will the calculate the balaced mass flow rate ad pressure distributio alog the defied pipe etwork. 1 NTRODUCTON Computer modellig of geothermal pipe etworks for both desig ad simulatio purposes has bee reported. Most of these are liear computig models, which ca work well o a tree-like liked simple pipe etwork. However with more complex practical systems, it is easy to fid that both loops of pipelies ad a series of oliear well characteristics have to be cosidered. These two factors cotribute to the difficulty of the modellig work. Loops i a pipe etwork ca ormally be solved usig a umber of established methods (Stepheso, 1989). However if the pipe etwork is coected to a umber of wells with parabolic like characteristic curves rather tha liear curves, eve the calculatio of a simple etwork ca be a cumbersome process. Covergece of the solutio is ofte very slow because of the trial ad error methods used. This paper presets recet developmet work o a umerical model which uses a o-liear method to simulate a geothermal steam pipelie etwork. The mass ad pressure balace at each ode ad loop of a pipe etwork are cosidered to be uder the cotrol of both the coected well ad the maifold workig poit. After settig up a o-liear equatio set for the above balace, umerical methods are used to solve for the mass flow ad pressure distributio. Covergece of the solutio has bee satisfactory for the umerical experimets coducted. this paper, a recet simulatio o oe of the logest etworks at Larderello, taly, is preseted. The simulatio results are compared with field measuremets ad a earlier simulatio by Marcocii ad Neri (1979). Sesitivity of the simulatio to the pipelie loss factor ad well characteristics are discussed. Sice the o-liear model solves the set of equatios describig a etwork simultaeously, it has good flexibility, makig it particularly applicable to looped etworks which are much more difficult to solve tha a tree-like etwork. 2 NON-LNEAR MODELLNG STRATEGY A o-liear model solves the equatio set simultaeously usig a umerical method formulated from the coceptual model of the pipe etwork. A computer code has bee developed based o this o-liear modellig strategy. The simulatio results for a series of test models have show a satisfactory covergece of the solutio. 2.1 Review The pressure ad mass flow rate distributio through a pipe etwork are geerally cotrolled by the pressure differece betwee the iput ad output poits of the system. a steady-state operatio, the mass flow ad pressure drop at each ode ad loop should be balaced that is: the et flow towards ayjuctio or odes is ZERO, ad the et head loss aroud ay closed loop is ZERO. Head loss alog a pipe lie are usually assumed to be of the form Ah= where Ah is the head loss, k the factor, L is the pipe legth, Q the volume flow rate ad D the iteral diameter of the pipe. Most methods of etwork aalysis are based o the above equatio (Stepheso, 1989). Two early approachs to pipe etwork aalysis were the Loop Flow Correctio Method ad the Node Head Correctio Method. Both methods use successive correctors speeded by a mathematical techique developed by Hardy Cross (1936). The developmet of micro computers has made it much easier to perform a etwork aalysis by umerical methods. This ivolves the simultaeous solutio of equatios describig flow ad pressure balace. Whe the iputs to a etwork system are steam wells with parabolic characteristics, a oliear umerical solutio is a requiremet of the model.
106 2.2 No-liear Modellig A coceptual model which ca represet the real pipe etwork is the fust requiremet for settig up a o-liear equatio set. A effective umerical method is ecessary for a accurate solutio with quick covergece. 2.2.1 Coceptual Model Before pressure ad mass balaces are applied at each loop ad ode, a coceptual model is required. This model should reflect all the iterrelatios betwee each part of the etwork ad should be cocise ad easy to use. Fig. 1 shows a typical coceptual model of a etwork where ad idicate the characteristic curves of the three steam wells, idicates the required steam pressure at the maifold,,, ad are the stable-state mass flows i the etwork, ad the odes are umbered as 1, 2, 3, 4, 5 ad 6. Dummy lies, represeted by dashed lie are used betwee each iput ad output poit of the system. With the help of these dummy lies, the etwork is liked by a umber of eclosed loops o which the pressure balace rule of a loop ca be applied. The expected flow directios are marked o each pipelie. The followig covetios are the applied to establish a equatio set for the etwork. At a balaced ode the iput flow is positive, output flow is egative. a balaced loop, a arbitrary calculatio directio of the loop is assumed. f a well output has a same directio as tlie calculatio directio the well head pressure is positive, otherwise it is egative; the opposite rule is applied for output pressure i steam maifold. f a pipe flow has a same directio as the calculatio directio, tlie frictioal pressure drop is positive. otherwise it is egative. tlie dummy pipe, the mass flow rate is defied as zero. 2.2.2 Establishig the equatio the coceptual model of Fig. 1, well head pressure ca be expressed as: + + where A, B ad C are regressio coefficiets of the well characteristic curve ad m is mass flow of the well. Frictioal pressure loss alog the pipelies is give as: where K is the loss factor o a sectio of a pipelie. A system costrait is the requiremet of a fixed output pressure at ode 6, so the mass ad pressure balace equatio set ca be writte as: - - = - - - = - - - - = - - - - = From equatio (1) ad (2) it is obvious that the equatio set (3) is a o-liear set ad should have a simultaeous solutio for, ad,if they exist. Numerical Solutio For a oliear equatio set such as i = 1, 2, We ca defie a objective fuctio 6 m 50 m 50m m Fig. 1 A typical cocept model of a etwork i Whe we have the set (4). 1 are roots of the oliear equatio Programmig for the above process is performed as follows: Startig from a iitial guess of o-zero roots suppose tlie iteratio has progressed to the step, the we have 2. Calculate the objective fuctio
107 3. f c E, the is take as a solutio, otherwise go to ext step; 4. Calculate: well 1 well 3 where or take c = 5. Calculate: i = 1, 2, Fig. 2 Basic test model j=l the repeat from step 2 util covergece is obtaied at step 3. A computer code, writte i TURBO PASCAL, for the solutio process has bee developed (Huag, 1991). 2.3 Test Model order to test the computer program, well data from the geothermal field has bee used i a test model. The operatio of the program ad the iterrelatios betwee parameters (Huag, 1992) have bee ivestigated. Differet coectios, usig the same basic model were simulated i order to compare results for differet operatio coditios. Test Network The basic model is show i Fig. 2. The model is composed of four productio wells, a sigle-looped brach lie etwork betwee the wells, ad a mai lie coectig the etwork to a power house (steam cosumer). With this model, it is show how a oliear looped etwork is solved, although such coectios may ot be ecessary i practice. the layout of the coceptual model of this etwork, m2, m3, ad represet the steam mass flow from wells 1, well 2, well 3 ad well 4 respectively. All these mass flows are cotrolled by the well ad separator characteristics ad However, the frictioal pressure loss alog each lie also make a cotributio to the cotrol of all the mass flow rates from to mg. This frictioal pressure loss is described by = The frictio loss factor is defied as: 32 f where f is Faig frictio factor (Perry, R. H., ad D are the pipe legth ad iside diameter, ad desity of the steam. is the The turbie ilet steam pressure is set at = 13.5 bara.. The pipe etwork has ie ukow variables, to ad ie oliear equatios ca be established. A costat value is used for this test model. 2.3.2 Test Model Results Usig the model described above, a series of test rus were made with differet combiatios of the workig wells used to simulate the possible differet workig coditios. Each combiatio followed the law that if oe well is shut off, the correspodig likig brach lies are also shut off. All the simulatios coverged satisfactorily. Results are show i Table 1. The etwork output steam is used as the iput for Turbie ad the system output is give i terms of Turbie output i MW. This umerical experimet o the o-liear model illustrates that a pipe etwork ca be simulated for ay combiatio of wells o-lie ad differet pipe frictioal characteristics.
Table 1 System performace with differet wells o-lie Well MAN D 2-6 2-7 14.059 14.063 14.014 35.682 5.46 12.565 8.199 1.820 1.533 1.000 13.747 13.127 23.597 3.61 15.231 8.366 3-- TURBONE 2 2 - t Fig. 3 A steam pipe etwork layout Note: RMFC - Relative Mass Flow Cotributio The results of these simulatios ca be used to predict the geeral performace of a o-liear pipe etwork system whe (a) the umber ad order of the wells o-lie have bee chaged (some wells shut off); (b) the frictioal characteristics of ay pipelie have bee chaged (valvig); the characteristic curve of ay productio well chages due to log term operatio. 3. SMULATON OF A STEAM PPE NETWORK The o-liear model was applied o oe of the logest steam pipe etworks operatig at Larderello, taly ad the results compared to published data. The coceptual model of the etwork ad the simulatio results are preseted. 11 Maifold Productiowells 1 8 ' 7 6 8' 7' 4' 3.1 Network Layout ad Coceptual Model This etwork carries fluid from 1, Querciola 2, Capriola, Grottitaa ad VC 2 wells to the Serrazzao power plat. Fig. 3 illustrates the layout of the pipe system. Several codesate dischargers are placed alog tlie lie. The characteristic curves of the differet wells were calculated from published data. The loss factor, K, is based o the geometry of each pipelie ad the correspodig steam state. A coceptual model for the etwork is show i Fig. 4 Coceptual model
109 3.2 Results A o-liear umerical simulatio of the Larderello pipelie etwork was successfully performed with three differet sets of iput data i TEST 4, TEST 5 ad TEST 6. The iterative procedure takes about 5 CPU miutes to coverge o a BM 386 computer. The results were prited as the mass flow rate from to The required maifold pressure was take from published data (Marcocii ad Neri, 1979). TEST 5, the K value for pipelies betwee well Capriola ad VC 2 has bee modified by a icrease show as follows TEST4 0.00024 TEST 5 0.00037 K7-8 0.00020 0.00030 TEST 6, a costat Faig frictio factor which is a average value for geothermal pipig (Huag, was used i the calculatio of all the K values throughout the etwork. The mass flow rate for each well from the results is used i the well characteristic curves to give the operatig wellhead pressures. Pressure profiles are the plotted over the measured data alog the pipelie etwork, as show i Fig. 5. The resultat curves represetig TEST 4, TEST 5 ad TEST 6 fall withi a close rage of the measured data irrespective of the differece i iput data. Cosiderig the simplificatio made, the result is good eough to show a agreemet betwee the umerical simulatio ad the measuremets. 4. DSCUSSON Oe of the sesitive parameters i the simulatio is the loss factor K. mproperly used K values i the system may lead to a o-coverget simulatio. Fortuately, K value foud i practice is depedet o the chage of Faig frictio factor f, which happes to fall i a arrow rage. Differet tests have show that the simulatio results for pressure drop are ot too sesitive to Faig frictio factor f. the compariso betwee the simulatio ad measuremet is illustrated. The arrow gap betwee the two measuremets may idicate measuremet errors. Most of the simulatio results fall withi the rage of measuremet error, which idicate a good agreemet with measuremet. All three simulatios, TEST 4, TEST 5 ad TEST 6 have a slightly flatter pressure profile alog the pipelie tha the measured data. This is because localized frictioal loss has bee eglected o the simulatio at this stage. For the results for TEST 4, the pressure profile has a similar slope to the measuremets except for the pipelies from ode 4 to 8. This might be due to some additio localized frictioal loss. TEST 5, the K value for all pipelies has bee modified by a small icrease. The simulatio result shows that the slope of the pressure profile betwee ode 4 ad 8 is closer to the measured oe, demostratig the sesitivity to the K value. TEST 6 is based o a costat Faig frictio factor f of 0.0033. The objective of this test was to ivestigate the sesitivity to f. t is iterestig to ote that the resultat pressure profile of this test is very close to that of TEST 4. This is a idicatio that the f value, if withi a resoable rage, is ot a sesitive parameter i this umerical simulatio. Amog the variables ivolved i evaluatig K, pipe diameter is the most sesitive oe for estimatig pressure drop. Fortuately it is oe of most well specified parameters. Of the mai parameters, productio well characteristic curves are of special importace. These cotrol the geeral performace of a coected pipelie etwork, though the chage of the frictioal characteristic of the etwork ca brig a chage i the geeral performace to some extet. t is the well curve which cotrols how much mass flow the well cotributes to the etwork. Sice each well may have its ow characteristic curve, ay pressure chage alog pipelie ca cause a correspodig chage of the workig poit of the well alog its productio curve. As a result, the well productio rate is chaged which the leads to a cosequet chage i pressure drop o the etwork. Needless to say, this ca also brig a ifluece o the workig poits of other wells ad their productio rates. 9 9 8 Simulatio TEST4 Simulatio TEST5 Simulatio Measuremet-1 Measuremet-2 TEST6 VAPSTAT a 6 0 3000 5000 6000 7000 Progressive distace Fig. 5 Results compared with measuremets 0 2000 3000 6000 7000 Progressive distace Fig. 6 compared with other simulatio
110 The ifluece o the other well operatig poits of chagig a well characteristic curve is very complex. t is a fuctio of their curve shape, the frictioal characteristic of each pipelie ad the mass ad eergy balace of the whole etwork. some cases, it ca cause a icrease i the total system output while i some other opposite effect occurs. Geerally speakig, a cluster of wells havig similar characteristic curves ted to give a simpler etwork which is easy to cotrol. With the help of the simulatio, well head cotrol of the workig poit ca be used as a systematic cotrol to a pipe etwork, especially after there has bee a chage to the etwork system. To examie more detailed iterrelatios betwee wells, a study of differet combiatios of wells ad the correspodig simulatio results are eeded. At this stage, the calculatio i the mathematical model is based o saturated steam. Heat loss alog the pipelie has ot yet bee take ito accout. The results from simulatio TEST 6 have show little differece from the published results of a other simulatio VAPSTAT 1 (Marcocii ad Neri, 1979) as illustrated i Fig. 6. The similarity of the two simulatio results idicates that the superheat of the steam ad the heat loss alog the pipelie play little part i the umerical simulatio. VAPSTAT 1 about 80 C superheat of the steam is cosidered. Heat losses through the pipelie surface are also icluded. From steam tables, it ca be see that 80 C super-heatig causes a 18% chage i desity of steam at a pressure of 8 This small chage seems to cause very little chage to the f value ad the cosequet pressure drop. Sice the pipelie is isulated, the temperature drop alog the pipelie is small. This small reductio of the superheat has little effect o the pressure drop. 5. CONCLUSONS 1. A o-liear model for the umerical simulatio of a geothermal steam pipe etwork has bee developed. A mass flow ad pressure balace at each ode ad loop i the etwork is used to establish a o-liear equatio set. Well characteristic curves ormally domiate the distributio of the mass flow withi a pipe etwork ad total output. alog the pipelie is caused by eglectig the localized frictioal loss. 3. Differet simulatio results have show that they are ot sesitive to the Faig frictio factor This may idicate the possibility of usig a costat or liear (agaist pipe roughess f value for a simplified calculatio of pressure drop. The correlatio used for frictioal pressure drop is proportioal to the pipe diameter to the mius fifth power, the diameter D becomes the most sesitive factor i the calculatio. 4. With the o-liear model, the simulatio of a pipelie etwork ca be applied ot oly to a tree-like system, as for VAPSTAT 1, but also to a looped oe. The o-liear model allows chages of coditios i the etwork system ad chages to the maifold pressure to be made at the discretio of the user. 6. ACKNOWLEDGEMENTS The authors would like to ackowledge Desigpower New Zealad limited for the fiacial support ad their special advice to this research project. 7. REFERENCES Cross, H. (1936): Aalysis of flow i etworks of codiuts or coductors, Uiversity of lliois Bulleti, 286 Huag, Y.C. (1991): Computer Modellig o A Multipump-statio Geothermal Pipe Network System (Part A), Research Report (4) to Desigpower, 1991, Huag, Y.C. (1992): Computer Modellig o A Multipump-statio Geothermal Pipe Network System (Part B): A Numerical Experimet, Research Report (5) to Desigpower, March 1992, pp. 39 Marcocii, R. ad Neri, G. (1979): Numerical Simulatio of a Steam Pipelie Network, Geothermics, Vol. 7 pp. 17-27, 1979 2. With the successful umerical simulatio of oe Perry, R.H. ad Gree, D.M. (1987): Perry's Chemical logest steam pipe etwork i Larderello, the oliear model has bee validated. Superheatig of the egieerig series. Egieers' Hadbook, McGraw-Hill chemical steam ad the heat loss through pipelie surface are ot calculated at this Stage. The coverged, With results close to the measured data withi 5 miutes CPU Elsevier Sciece Publishers B. V., The time. The flatter slope of the simulated pressure profile Netherlads