Inferring Immobile and In-situ Water Saturation from Laboratory and Field Measurements

Transcription

1 SGP-TR-167 Inferring Immobile and In-itu Water Saturation from Laboratory and Field Meaurement Rodolfo P. Belen Jr. June 2000 Financial upport a provided through the Stanford Geothermal Program under Department of Energy Grant No. DE-FG07-95ID13370 and No. DE-FG07-99ID13763, and by the Department of Petroleum Engineering, Stanford Univerity

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3 Abtract Analyi of experimental data and numerical imulation reult of dynamic boiling experiment revealed that there i an apparent correlation beteen the immobile ater aturation and the hape of the team aturation profile. An elbo in the team aturation profile indicate the udden drop in team aturation that mark the tranition from team to to-phae condition inide the core during boiling. The immobile ater aturation can be inferred from thi elbo in the team aturation profile. Baed on experimental reult obtained by Satik (1997), the inferred immobile ater aturation of Berea andtone a found to be about 0.25, hich i conitent ith reult of relative permeability experiment reported by Mahiya (1999). Hoever, thi technique may not be ueful in inferring the immobile ater aturation of le permeable geothermal rock becaue the elbo in the team aturation profile i le prominent. Model of vapor and liquid-dominated geothermal reervoir that ere developed baed on Darcy la and material and energy conervation equation proved to be ueful in inferring the in-itu and immobile ater aturation from field meaurement of cumulative ma production, dicharge enthalpy, and donhole temperature. Knoing rock and fluid propertie, and the difference beteen the table initial, T o, and dry-out, T d, donhole temperature, the in-itu and immobile ater aturation of vapor-dominated reervoir can be etimated. On the other hand, the in-itu and immobile ater aturation, and the change in mobile ater content of liquid-dominated reervoir can be inferred from the cumulative ma production, m, and enthalpy, h, data. Comparion ith to-phae, radial flo, numerical imulation reult confirmed the validity and uefulne of thee model. v

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5 Acknoledgment Thi reearch a conducted ith financial upport through the Stanford Geothermal Program under the US Department of Energy Grant No. DE-FG07-95ID To Prof. Roland Horne for hi guidance, encouragement, upport and great ene of humor. To Stefan Finterle and Malou Guerrero for their patience ith me in learning TOUGH2 and itough2. To Huda Naori and Keen Li for their patience ith me in the laboratory. vii

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7 Content Abtract.. v Acknoledgment vii Content ix Lit of Table.xi Lit of Figure. xiii 1. Introduction 1 2. Inferring Immobile Water Saturation from Laboratory Meaurement Experimental Deign Reult of Boiling Experiment Senitivity Analyi Inferred Immobile Water Saturation of Berea Sandtone Rock Modeling Reult of Boiling Experiment Uing Geothermal Rock Concluion and Recommendation Inferring Immobile and In-Situ Water Saturation from Field Meaurement Inferring In-itu Saturation from Variation in Ga Content Zero-Dimenional Model TOUGH2 To-Phae Radial Flo Model Comparion of TOUGH2 To-Phae Radial Flo Model Concluion and Recommendation Concluion Nomenclature 39 Reference 41 A. TOUGH2 and itough2 File for Berea Sandtone. 43 B. TOUGH2 File for Geyer Geothermal Rock. 65 C. TOUGH2 and itough2 File for To-Phae Radial Flo Simulation...67 ix

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9 Lit of Table Table 2-1: Propertie of Berea andtone core... 2 Table 2-2: Geyer geothermal rock propertie Table 3-1: TOUGH2 reervoir model and ellbore parameter.. 20 Table 3-2: TOUGH2 imulation reult and fluid propertie: i = 0.4, r = Table 3-3: TOUGH2 imulation reult and fluid propertie: i = 0.4, r = xi

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11 Lit of Figure Figure 2-1: Experimental et-up of boiling experiment... 2 Figure 2-2: Steam aturation profile of vertical boiling experiment.. 3 Figure 2-3: Temperature profile of vertical boiling experiment 4 Figure 2-4: Preure profile of vertical boiling experiment... 4 Figure 2-5: Steam aturation profile: contant heat rate, r = 0.2, Berea..6 Figure 2-6: Heat rate a a tep function of time. 6 Figure 2-7: Steam aturation profile: variable heat rate, r = 0.1, Berea 7 Figure 2-8: Steam aturation profile: variable heat rate, r = 0.2, Berea 7 Figure 2-9: Steam aturation profile: variable heat rate, r = 0.5, Berea.8 Figure 2-10: Steam aturation profile of vertical boiling experiment 9 Figure 2-11: Steam aturation profile of horizontal boiling experiment 9 Figure 2-12: Steam aturation profile of top-heating vertical boiling experiment Figure 2-13: Steam aturation profile: contant heat rate, r = 0.2, Geyer. 11 Figure 2-14: Steam aturation profile: variable heat rate, r = 0.1, Geyer 12 Figure 2-15: Steam aturation profile: variable heat rate: r = 0.2, Geyer. 13 Figure 2-16: Steam aturation profile: variable heat rate: r = 0.5, Geyer. 13 Figure 3-1: Log-log plot of CO2-H2S concentration for KMJ11 (Grant 1979)...17 Figure 3-2: Vapor and liquid-dominated geothermal reervoir.. 18 Figure 3-3: Effect of grid refinement on imulation reult. 20 Figure 3-4: Water aturation profile: i = 0.3, r = Figure 3-5: Temperature profile: i = 0.3, r = Figure 3-6: Preure profile: i = 0.3, r = Figure 3-7: Production enthalpy and reervoir temperature: i = 0.2, r = Figure 3-8: Production enthalpy and reervoir temperature: i = 0.3, r = Figure 3-9: Production enthalpy and production temperature: i = 0.3, r = Figure 3-10: Production enthalpy and reervoir temperature: i = 0.3, r = Figure 3-11: Production enthalpy and reervoir temperature: i = 0.4, r = Figure 3-12: Production enthalpy and reervoir temperature: i = 0.5, r = Figure 3-13: TOUGH2 enthalpy and temperature: varying i, r = Figure 3-14: Zero-dimenional enthalpy and temperature: varying i, r = Figure 3-15: TOUGH2 enthalpy and temperature: varying r, i = Figure 3-16: Zero-dimenional enthalpy and temperature: varying r, i = Figure 3-17: TOUGH2 cumulative ma and enthalpy: i - r = Figure 3-18: Zero-dimenional cumulative ma and enthalpy: i - r = xiii

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13 Chapter 1 1. Introduction Relative permeability i important in decribing the flo of to-phae team in geothermal reervoir and in performing geothermal reervoir engineering calculation. Preently, hoever, relative permeability relation for team and liquid ater are not completely undertood. Permeability relation are normally adopted from field data or from nitrogen and ater flo experiment. The experimental determination of team and liquid ater relative permeabilitie i a central target of the Stanford Geothermal Program. Flo-through experiment on Berea andtone ere performed by Ambuo (1996), Satik (1998) and Mahiya (1999) uing X-ray computer tomography to determine the team aturation profile. In a different approach, numerical imulation a ued by Guerrero et al. (1998) to infer relative permeabilitie of Berea andtone, baed on temperature, preure, and team aturation data obtained from dynamic boiling experiment performed by Satik (1997). All of thee earlier tudie ued Berea andtone in order to capitalize on it higher permeability relative to geothermal rock, hich enabled the experiment to be performed in reaonable time. Thi tudy aimed to extend the undertanding to lo permeability geothermal rock by determining only the endpoint aturation of the relative permeability curve. The endpoint or irreducible or immobile aturation of a certain phae i the aturation at hich that phae become mobile in multiphae flo. Combining information about the endpoint aturation from the lo geothermal rock experiment ith information about the general hape of the relative permeability curve from the fater andtone rock experiment could then define the team-liquid ater relative permeability behavior completely. Furthermore, knoledge of the immobile and in-itu ater aturation ill provide a better undertanding of the adorption characteritic and fluid torage capacitie of geothermal rock. Thi ill be valuable in etimating the ize of the available geothermal reource accurately and in evaluating geothermal reervoir performance. The objective of thi tudy a to determine the endpoint aturation of the team and liquid ater relative permeability curve by inference from preure, temperature and aturation data obtained from pat dynamic boiling experiment conducted by Satik (1997) uing Berea andtone rock. 1

14 In addition, the tudy developed model of both vapor and liquid-dominated geothermal reervoir to allo inference of the in-itu and immobile ater aturation from field meaurement of cumulative ma production, dicharge enthalpy, and donhole temperature. 2

15 Chapter 2 2. Inferring Immobile Water Saturation from Laboratory Meaurement Thi tudy analyzed reult of pat dynamic boiling experiment performed by Satik (1997) uing Berea andtone rock. The intent a to invetigate the feaibility of inferring the immobile ater aturation from experimental preure, temperature and aturation data. Uing numerical imulation, thi tudy alo evaluated the feaibility of extending the technique to geothermal rock Experimental Deign In 1996 and 1997, Satik performed a erie of boiling experiment uing Berea andtone core. The objective of the tudy a to further the undertanding of the boiling proce in porou media and ultimately to obtain capillary function and relative permeability relation for team and liquid ater. The dynamic boiling experiment involved the heating of a rock aturated ith liquid ater and oberving the boiling proce by continuou meaurement of preure, temperature, heat flux and team aturation ithin the core. The X-ray CT canner a ued to viualize the boiling proce and to determine the three-dimenional fluid ditribution ithin the rock. Thee experiment are analogou to drainage experiment in oil and ater ytem, in hich oil, the nonetting fluid, i injected into a rock aturated ith ater, the etting fluid, to diplace the ater from the rock. Hoever, in the cae of the boiling experiment, team produced by heating the ater-aturated rock diplace the liquid ater from the rock. The experimental apparatu conited of a core holder houing the Berea andtone core, a data acquiition ytem, a vacuum pump, a ater pump and a balance (Figure 2-1). The core a inulated ith a fiber blanket to minimize heat loe. The heater a attached to the end of the core that a cloed to fluid flo. The other end a connected to a ater reervoir placed on a balance that a ued to monitor the amount of ater diplaced from the core during the boiling proce. Preure, temperature and heat flux ere meaured in the core uing preure tranducer, thermocouple and heat flux enor repectively and ere recorded automatically in a data acquiition ytem. 1

16 The core a firt dried and then vacuumed to remove air inide the pore pace. The core a then canned at 1-cm interval to obtain dry-core CT value to be ued in computing the team aturation. The core a then aturated ith deaerated ater and then canned again at the ame location to obtain et-core CT value. The heater a then turned on and preure, temperature and heat flux ere meaured continuouly during each heating tep until teady-tate condition ere reached. Steady tate had been reached hen no more ater floed out of the core and hen preure, temperature and heat flux meaurement had tabilized. At the ame time, the core a canned again to obtain CT value that ere ued to calculate the team aturation ditribution. The heating rate a increaed and the procedure a repeated. Different experiment ere performed in hich the core a mounted vertically and horizontally. ater pump core P, T, heat flux meaurement balance vacuum pump data acquiition ytem Figure 2-1: Experimental et-up of boiling experiment (Satik 1997). Propertie of the Berea andtone core ued in the boiling experiment are ummarized in Table 2-1. Table 2-1: Propertie of Berea andtone core. Poroity 22% Permeability 600 md Length 43 cm Diameter 5.04 cm 2.2. Reult of Boiling Experiment In all the boiling experiment, teady-tate team aturation, temperature and preure profile indicated a progreive boiling proce ith the formation of ditinct region of team, to-phae and liquid ater. The profile at earlier time and loer heat flux have the characteritic of a to-phae region folloed by a ingle-phae liquid zone hile at later time and higher heat flux the profile indicate a to-phae region beteen ingle 2

17 phae team and liquid ater region. Figure 2-2 to 2-4 illutrate the team aturation, temperature and preure profile of a vertical boiling experiment performed by Satik (1997) hoing the formation of team, to-phae and liquid region ithin the core a the heat flux a increaed. The udden drop in the team aturation near the heater end of the core mark the tranition from team to to-phae condition. Thi drop i deignated in thi tudy a the "elbo" in the team aturation profile after 5 day, heat rate = 8 W after 6 day, heta rate = 9 W after 7 day, heat rate = 10 W Steam Saturation Ditance from heater, cm Figure 2-2: Steam aturation profile of vertical boiling experiment (Satik, Spring 1997). 3

18 after 5 day, heta rate = 8W after 6 day, heat rate = 9W after 7 day, heta rate = 10W Tat 80 Temperature, C Ditance from heater, cm Figure 2-3: Temperature profile of vertical boiling experiment (Satik, Spring 1997). 5 4 after 5 day, heat rate = 8W after 6 day, heat rate = 9W after 7 day, heat rate - 10W Preure, pig Ditance from heater, cm Figure 2-4: Preure profile of vertical boiling experiment (Satik, Spring 1997). 4

19 2.3. Senitivity Analyi It a hypotheized that the irreducible ater aturation can be correlated ith the oberved elbo in the team aturation profile. Analyzing the enitivity of the boiling proce to the irreducible ater aturation through numerical modeling a ued to tet the validity of thi hypothei. The boiling proce a imulated uing different value of immobile ater aturation. The preure, temperature and aturation profile ere predicted to invetigate if the elbo in the aturation profile could be correlated ith the irreducible ater aturation. In 1998, Guerrero et al. developed a to-dimenional radial itough2 model to infer relative permeability relation from the reult of the boiling experiment. The ame model a ued in the enitivity analyi but the grid ere refined to give a better reolution of the variation in the team aturation near the heater end of the core. The TOUGH2 and itough2 input file are included in thi report a Appendix A. Forard calculation in itough2 ere performed to imulate the preure, temperature, and team aturation profile along the core. It a aumed that linear team-liquid ater relative permeability function (X-curve) and Leverett capillary function govern the flo of to-phae team in the andtone core. The nonadiabatic boiling proce a imulated uing contant and variable heating rate. Figure 2-5 ho the imulated team aturation profile for the contant heating rate cae. The plot correpond to the profile after one day of heating the core continuouly at a contant rate. A the heating rate a raied, boiling commenced and the team and to-phae region formed and expanded. The boiling proce a then imulated uing progreively increaing heat rate. Figure 2-6 ho the heat rate a a tep function of time. Three different value of immobile ater aturation, 0.1, 0.2, and 0.5, ere ued in the enitivity analyi. Steam aturation a then plotted at 0.2-cm. interval and up to a ditance 8 cm. aay from the heater end. The aturation profile are hon in Figure 2-7 to 2-9. The imulation reult indicated an apparent correlation beteen the elbo in the team aturation profile and the immobile ater aturation. In all three cae, the team region extend to a ditance one centimeter from the heater end. An abrupt drop in the team aturation mark the tranition to to-phae condition, hich correpond to the elbo in the profile. The team aturation tay cloe to thi value behind the elbo and then goe don further ith ditance aay from the heater end. 5

20 W 9 W 11 W 13 W 15 W 17 W Steam Saturation Ditance from Heater, cm Figure 2-5: Steady tate team aturation profile ith ditance and contant heating rate: immobile ater aturation = 0.2, Berea andtone rock Heating Rate in W Time in day Figure 2-6: Heat rate a a tep function of time. 6

21 after 4 day after 5 day after 6 day after 7 day after 8 day Steam Saturation Ditance from Heater, cm Figure 2-7: Steam aturation profile ith ditance and time uing variable heating rate: immobile ater aturation = 0.1, Berea andtone rock after 4 day after 5 day after 6 day after 7 day after 8 day Steam Saturation Ditance from Heater, cm Figure 2-8: Steam aturation profile ith ditance and time uing variable heating rate: immobile ater aturation = 0.2, Berea andtone rock. 7

22 after 4 day after 5 day after 6 day after 7 day after 8 day Steam Saturation Ditance from Heater, cm Figure 2-9: Steam aturation profile ith ditance and time uing variable heating rate: immobile ater aturation = 0.5, Berea andtone rock. It i important to note that teady-tate condition ere reached after about ten day of heating the core. Steady tate mean that the imulated team aturation, preure and temperature profile remained invariant ith time. A a conequence, the imulated team aturation profile indicate team condition in the region tarting from the edge of the core here the heater i attached to a ditance one centimeter aay. Beyond thi region, the imulated team aturation profile indicate to-phae and liquid condition Inferred Immobile Water Saturation in Berea Sandtone Figure 2-2, and 2-10 to 2-12 ho the team aturation profile obtained from the boiling experiment that ere conducted by Satik in Spring and Summer of The figure ho the team aturation profile ith ditance along the core and a a function of the heating rate. It i important to note that team aturation, preure and temperature ere meaured only at 1-cm interval along the core. 8

23 after 22 day, heat rate = 8 W after 21 day, heat rate = 10 W after 20 day, heat rate = 11 W Steam Saturation Ditance from heater Figure 2-10: Steam aturation profile of vertical boiling experiment (Satik, Spring 1997) after 3 day, heat rate = 6 W after 5 day, heat rate = 8 W after 8 day, heat rate = 10 W after 10 day, heat rate = 11 W Steam Saturation Ditance from heater Figure 2-11: Steam aturation profile of horizontal boiling experiment (Satik, Summer 1997). 9

24 after 4 day, heat rate = 6 W after 5 day, heat rate = 8 W after 6 day, heat rate = 9 W after 7 day, heat rate = 11 W Steam Saturation Ditance from heater Figure 2-12: Steam aturation profile of top-heating vertical boiling experiment (Satik, Summer 1997). Baed on the location of the elbo in the team aturation profile obtained from the actual experimental reult, it can be inferred that the irreducible ater aturation of Berea andtone i about Thi value i conitent ith the experimental reult reported by Mahiya (1999) in relative permeability experiment conducted uing imilar Berea andtone core Modeling Reult of Boiling Experiment Uing Geyer Geothermal Rock Boiling experiment uing Geyer geothermal rock ere then imulated to determine the feaibility of thi technique in inferring the immobile ater aturation of lo poroity and lo permeability geothermal rock. The propertie of the geothermal rock ued in the imulation run are tabulated in Table 2-2. A previouly in the Berea andtone cae, the nonadiabatic boiling proce a imulated uing contant and variable heating rate. An excerpt of the TOUGH2 input file hoing the rock propertie i included in thi report a Appendix B. Figure 2-13 ho the team aturation profile ith ditance and heating rate. The plot correpond to the profile after one day of heating the core continuouly at a contant rate. 10

25 Table 2-2: Geyer geothermal rock propertie. Poroity 5% Permeability 1 x m 2 Rock denity 2600 kg/m 3 Rock pecific heat 485 J/kg C Rock heat conductivity 2.43 W/m C W 9 W 11 W 13 W 15 W 0.7 Steam Saturation Ditance from Heater, cm Figure 2-13: Steady tate team aturation profile ith ditance and contant heating rate: immobile ater aturation = 0.2, Geyer geothermal reervoir rock. It i evident from the imulation reult that the team and to-phae region expand a the heating rate i increaed. Hoever, it i important to note that in the cae of the loer-poroity and loer-permeability geothermal rock, the to-phae region i much horter and the boiling front i much harper in comparion to the Berea andtone cae. Conequently, the elbo in the team aturation profile i le conpicuou in the geothermal rock cae. Figure 2-14 to 2-16 ho the imulation reult hen variable heating rate are ued. The ame heating rate profile a ued in the imulation run, a hon in Figure 2-6. Likeie, team and to-phae region formed and expanded a the boiling proce progreed. A expected, the to-phae region covered only a horter ditance and the boiling front a harper. A a reult, it i difficult to identify an elbo in the team 11

26 aturation profile. Thi very important imulation reult mut be taken into conideration in planning and deigning boiling experiment in tighter and le permeable geothermal reervoir rock. The difficulty in identifying the elbo in the team aturation profile may limit the uefulne of thi technique in determining the immobile ater aturation of geothermal rock from laboratory data obtained from boiling experiment Concluion and Recommendation Baed on experimental data and TOUGH2 numerical imulation reult of dynamic boiling experiment, there i an apparent correlation beteen the immobile ater aturation and the hape of the team aturation profile. The inferred immobile ater aturation of Berea andtone rock i about 0.25, hich i conitent ith experimental reult from pat relative permeability experiment reported by Mahiya (1999). Hoever, thi technique may not be ueful in inferring the immobile ater aturation for lo poroity and lo permeability geothermal rock becaue of the difficulty in locating the elbo in the team aturation profile after 4 day after 5 day after 6 day after 7 day after 8 day Steam Saturation Ditance from Heater, cm Figure 2-14: Steam aturation profile ith ditance and time uing variable heating rate: immobile ater aturation = 0.1, Geyer geothermal reervoir rock. 12

27 after 4 day after 5 day after 6 day after 7 day after 8 day Steam Saturation Ditance from Heater, cm Figure 2-15: Steam aturation profile ith ditance and time uing variable heating rate: immobile ater aturation = 0.2, Geyer geothermal reervoir rock after 4 day after 5 day after 6 day after 7 day after 8 day Steam Saturation Ditance from Heater, cm Figure 2-16: Steam aturation profile ith ditance and time uing variable heating rate: immobile ater aturation = 0.5, Geyer geothermal reervoir rock. 13

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29 Chapter 3 3. Inferring Immobile and In-Situ Water Saturation from Field Meaurement The dicharge of aturated or uperheated team during the exploitation of vapordominated geothermal reervoir greatly exceed the amount that can be tored a vapor. Therefore, vapor-dominated reervoir mut contain ubtantial amount of liquid ater to utain production (Jame, 1968; Nathenon 1975; Grant, 1979). In decribing the repone of vapor-dominated reervoir to exploitation, it i valid to aume that the liquid ater i completely immobile. Although ater may be lightly mobile in the natural tate of the reervoir, it oon become immobile becaue the ater aturation drop a fluid are produced (Grant, 1979). The liquid ater i adorbed in the pore of the reervoir matrix and i able to vaporize, but i not able to flo a liquid ater. Grant (1979) etimated the in-place ater aturation of the Kaah Kamojang geothermal field, Indoneia, baed on variation in the ga content of the production fluid. Changing the flo rate at the ellhead produce a repone in the reervoir preure and ga content, hich allo for the etimation of the in-place ater aturation or the immobile ater aturation of the reervoir rock. In contrat, thi tudy aim to infer the in-itu and immobile ater aturation from field meaurement of change in the floing enthalpie of producing ell a ell a donhole temperature. Zero-dimenional model can be ued to decribe the preure, temperature and aturation profile accompanying production. Thi tudy developed model of both vapor and liquid-dominated geothermal reervoir that can be ued to infer the in-itu and immobile ater aturation uing field meaurement of cumulative ma production, dicharge enthalpy, and donhole temperature Inferring In-itu Saturation from Variation in Ga Content Variation in preure and aturation in the reervoir during exploitation reult in the tranfer of ma beteen phae and, conequently, variation in the ga content of the liquid and vapor phae. The change are determined by the equation for the conervation of ga (Grant, 1977) given by n [ ( ) ] φ ρ + = n 1 ρ n k p (3-1) t υ 15

30 Uing a flo model, Grant (1979) hoed that the variation in ga content ith flo rate could be decribed uing a parameter, ζ, hich depend on ga, team and reervoir propertie. The tranient repone in the ga content of the team produced hen a ell i opened at time t = 0 to a flo rate W kg/ i given by n ln n o = ζ exp( ξ ) 0 ξ + ζ exp( ξ ) dξ (3-2) c t W µ ζ = (3-3) 4πkh γ γ = ( 1 ) ρ + Aρ (3-4) n A = (3-5) The in-place ater aturation can then be inferred from ga data by conidering the difference in the repone of to gae ith production. Hoever, thi method i valid only if the total amount of ga i mall and each ga obey Equation 3-2 independently ith a different ζ parameter. After further implification, Grant (1979) hoed that the logarithm of the concentration of ga 1 in team i a linear function of the logarithm of the concentration of ga 2 in team. The lope of the traight line give the in-itu ater aturation of the unditurbed reervoir. n lope of (1 ) ρ + A ρ ) plot 1 (1 ) ρ + A ρ ln( n ) v ln( n2 2 = (3-6) Figure 3-1 i a log-log plot of CO 2 and H 2 S concentration for production ell KMJ11 in the Kaah Kamojang geothermal field. Production data ho that Kaah Kamojang i a claic vapor-dominated ytem (Grant 1979). The lope of the traight line that fit the ga concentration data give an in-itu ater aturation of about 0.35 uing Grant model and approach. 1 16

31 %CO % H 2 S Figure 3-1: Log-log plot of CO2-H2S concentration for KMJ11 (Grant 1979) Zero-Dimenional Model Darcy la and the differential material and energy conervation equation that decribe the repone of geothermal reervoir to exploitation can be combined to form imple zero-dimenional model. For the cae of vapor-dominated geothermal ytem, the only mobile phae i team (Figure 3-2). The flo of dry team can be decribed by Equation 3-7 to 3-9. It i aumed that the immobile ater doe not impede the flo of team and that Darcy la decribe the team flo. t φ t { ρ + ( ) ρ } = ( u ρ ) 1 (3-7) {( φ ) ρ c T + φρ h + φ( 1 ) ρ h } = ( u ρ h ) r r 1 (3-8) u kk = µ r p (3-9) Under reervoir condition, the enthalpy of aturated team i nearly contant ith temperature. Thi approximation reult in a implified relation beteen preure and aturation at any point in the reervoir. ( 1 ) ρ C T + φρ ( h h ) = contant φ (3-10) r r 17

32 Reduction in the reervoir preure reulting from production i accompanied by a reduction in reervoir temperature to maintain aturation condition. Heat mut then be mined from the rock to cool it, hich i achieved by vaporizing ome of the liquid ater to team. Therefore, a decline in the reervoir preure reult in a decline in the reervoir aturation. Thi mining of heat and conequent decline in aturation continue a long a aturation condition exit. The aturation fall to zero at dry-out condition and the inplace ater aturation can then be etimated uing the initial and dry-out reervoir condition (Grant 1979). o = ( 1 φ ) ρrcr ( To Td ) φ ρ ( h h ) T o (3-11) Thi zero-dimenional model allo the calculation of the in-itu ater aturation uing rock and fluid propertie and the initial, T o, and dry-out, T d, donhole or reervoir temperature. In thi cae, the dry-out temperature i that temperature at hich the reervoir ha completely dried out and ha tarted to produce uperheated team. dry aturated team to-phae team mobile team mobile ater and team immobile ater Vapor Dominated mobile ater Liquid Dominated Figure 3-2: Vapor- and liquid-dominated geothermal reervoir. On the other hand, for the liquid-dominated reervoir cae (Figure 3-2), the differential material and energy balance equation (Equation 3-12 and 3-13) and the implified equation decribing the zero-dimenional model (Equation 3-14 and 3-15) are much more complicated becaue both ater and team phae are mobile. φ t { ρ + ( ) ρ } = ( u ρ + u ρ ) 1 (3-12) 18

33 t 1 (3-13) {( φ ) ρ c T + φρ h + φ( 1 ) ρ h } = ( u ρ h + u ρ h ) r r ( ρ ) + φv { ( 1 ) ρ } + m + m = 0 φ V (3-14) φv ( ρ h ) + φv { ( 1 ) ρ h } + ( 1 φ ) Vρ C T + m h + m h = 0 r r (3-15) The compoition and enthalpy of the production fluid are determined by the mobility of team and liquid ater in the porou rock given by the relative permeabilitie and vicoitie of the to phae a hon in Equation 3-16 and m k r ρ µ = m (3-16) k r k r ρ + ρ µ µ k r ρ µ m = m (3-17) k r k r ρ + ρ µ µ 3.3. TOUGH2 To-Phae Radial Flo Model The zero-dimenional model aume that the preure, temperature and aturation are uniform throughout the reervoir. Thee model do not take into account the tranient and patial effect of to-phae team flo. In order to verify the validity and uefulne of thee model in determining the in-itu and immobile aturation, to-phae radial flo a modeled uing the numerical imulator TOUGH2. The TOUGH2 imulation reult ere then compared ith thoe predicted by the zero-dimenional model. A cylindrical reervoir model a ued in the TOUGH2 imulation run. A ingle ell maintained at contant donhole preure a placed in the middle of the reervoir. Table 3-1 ummarize the reervoir and ellbore parameter ued in the imulation run. The reervoir model conited of 100 grid block. The grid ize increae logarithmically from the center to the boundary of the reervoir. The effect of the grid block ize and number on the imulation reult i illutrated in Figure 3-3. Grid refinement minimize the dicretization error that are particularly large during the early tranient production period. 19

34 Table 3-1: TOUGH2 reervoir model and ellbore parameter. Poroity 5% Permeability 1 x m 2 Rock denity 2600 kg/m 3 Rock pecific heat 485 J/kg C Reervoir radiu 1000 m Reervoir thickne 10 m Initial reervoir temperature 280 C Contant donhole ellbore preure 200 pia i = 0.3 Enthalpy, kj/kg i = block 30 block 100 block E +0 1E +1 1E+2 1E +3 1E +4 1E +5 1E +6 1E +7 1E +8 1E +9 Cumulative Ma Production, kg Figure 3-3: Effect of grid refinement on imulation reult. Figure 3-4 to 3-6 are emilog plot of the aturation, temperature and preure profile ith time and radial ditance from the ell. Initially, ater aturation throughout the reervoir in thi particular imulation run i 0.3. A a repone to production, reervoir preure, temperature and aturation drop and boiling near the ellbore commence. The boiling front move farther into the reervoir aay from the ell a to-phae team i produced. Hoever, ater aturation tay above and never goe belo the immobile ater aturation. The reervoir eventually produce dry aturated team. During thi time, reervoir aturation ha dropped belo the immobile ater aturation. Water remaining 20

35 in the reervoir during thi period i immobile ater, hich i able to vaporize but not able to flo. At ome point during production, the reervoir completely drie out and tart to produce uperheated team. Water aturation ha dropped to zero throughout the reervoir. The equence of event in the production hitory of the reervoir i labeled in Figure To-phae 1 ec 10 ec 1 min 10 min 1 hour 1 day 10 day 100 day 1000 day Water Saturation Dry Saturated Steam 5000 day 7500 day day Superheated Steam day Wellbore Radial Ditance from Well, meter Figure 3-4: Water aturation profile ith ditance and time: initial ater aturation = 0.3, immobile ater aturation =

36 ec 10 ec Temperature, degree C min 10 min 1 hr 1 d 10 d 100 d 1000 d 5000 d 7500 d d d Wellbore Radial Ditance from Well, meter Figure 3-5: Temperature profile ith ditance and time: initial ater aturation = 0.3, immobile ater aturation = ec 10 ec 1 min Preure, MPa min 1 hr 1 d 10 d 100 d d 5000 d 7500 d d d Wellbore Radial Ditance from Well, meter Figure 3-6: Preure profile ith ditance and time: initial ater aturation = 0.3, immobile ater aturation =

37 3.4. Comparion of TOUGH2 To-Phae Radial Flo Model ith Zero- Dimenional Model Vapor-Dominated Geothermal Reervoir Figure 3-7 compare the production enthalpie and reervoir temperature imulated by TOUGH2 ith thoe predicted by the zero-dimenional model for a vapor-dominated reervoir cae ith in-itu ater aturation equal to 0.2. There appear to be a very good agreement beteen the imulation and the modeling reult. The modeling reult ere then ued to calculate the in-itu ater aturation through Equation 3-11 a hon belo. o = ( 1 φ ) ρ rcr ( To Td ) φ ρ ( h h ) To = kg ( )( 2600 )( kj o o kgc)( C C) m kg ( kj kj ) m kg kg = 0.20 The zero-dimenional model gave the correct in-itu ater aturation, hich in thi cae, i equal to the immobile ater aturation. Thi reult confirm that the zero-dimenional model can be ued to infer both the in-place and immobile ater aturation of vapordominated geothermal reervoir by knoing the initial and dry-out reervoir temperature T o Production Enthalpy, kj/kg T d Temperature, degree C E+0 5.0E+7 1.0E+8 1.5E+8 2.0E+8 2.5E+8 3.0E+8 Cumulative Ma Production, kg TOUGH2 Enthalpy Zero-D Model Enthalpy TOUGH2 Temperature Zero-D Model Temperature Figure 3-7: Production enthalpy and reervoir temperature profile: vapor-dominated cae, initial ater aturation = 0.2, immobile ater aturation =

38 Figure 3-8 ho another vapor-dominated cae but ith a higher in-itu ater aturation equal to 0.3. Equation 3-11 give the correct in-itu ater aturation a hon belo. o = ( 1 φ ) ρ rcr ( To Td ) φ ρ ( h h ) To = kg ( )( 2600 )( kj o o kgc)( C C) m kg ( kj kj ) m kg kg = T o Production Enthalpy, kj/kg T d Temperature, degree C E+0 5.0E+7 1.0E+8 1.5E+8 2.0E+8 2.5E+8 3.0E+8 3.5E+8 4.0E+8 4.5E+8 Cumulative Ma Production, kg TOUGH2 Enthalpy Zero-D Model Enthalpy TOUGH2 Temperature Zero-D Model Temperature Figure 3-8: Production enthalpy and reervoir temperature profile: vapor-dominated cae, initial ater aturation = 0.3, immobile ater aturation = 0.3. In the lat to cae, TOUGH2 imulated reervoir temperature agreed atifactorily ith the modeled temperature. Hoever, reervoir temperature are not normally meaured in the field. Intead, urface and donhole temperature are routinely meaured in the production ell. Figure 3-9 i a plot comparing TOUGH2 imulated donhole temperature and modeled reervoir temperature. The initial udden drop in the donhole ellbore temperature a a repone to production i clearly evident from Figure 3-9. After thi early tranient period, donhole temperature decline at the ame rate a the reervoir temperature. TOUGH2 imulated donhole temperature ere then ued to etimate the in-itu ater aturation uing Equation The initial temperature, T o, a taken to be the table temperature after the early tranient period. The zero-dimenional model till gave a very cloe approximation of the correct in-itu ater aturation a hon in the folloing calculation. 24

39 o = ( 1 φ ) ρ rcr ( To Td ) φ ρ ( h h ) To = kg ( )( 2600 )( kj o o kgc)( C C) m kg ( kj kj ) m kg kg = Production Enthalpy, kj/kg T o Temperature, degree C 500 T d E+0 5.0E+7 1.0E+8 1.5E+8 2.0E+8 2.5E+8 3.0E+8 3.5E+8 4.0E+8 4.5E+8 Cumulative Ma Production, kg TOUGH2 Enthalpy Zero-D Model Enthalpy TOUGH2 Temperature Zero-D Model Temperature Figure 3-9: Production enthalpy and production temperature profile: vapor-dominated cae, initial ater aturation = 0.3, immobile ater aturation = 0.3. Therefore, either donhole ellbore or reervoir temperature can be ued to etimate the in-place ater aturation uing the zero-dimenional model. Hoever, it i important that the appropriate initial, T o, and dry-out, T d, temperature be ued in Equation o = ( 1 φ ) ρrcr ( To Td ) φ ρ ( h h ) T o (3-11) The initial reervoir temperature mut be ued in evaluating the fluid propertie hile the table donhole ellbore temperature mut be ued in evaluating the temperature difference. The table donhole ellbore temperature i the temperature meaured donhole hen flo condition have already tabilized. 25

40 Liquid-Dominated Geothermal Reervoir The repone of liquid-dominated geothermal reervoir to exploitation can be decribed by a equence of to-phae, dry aturated and uperheated team production. TOUGH2 and the zero-dimenional model predict imilar trend in production enthalpy and temperature a illutrated in Figure 3-10 to The enthalpy of produced to-phae team increae and reervoir temperature decline a the reervoir i depleted of it mobile ater. The production enthalpy then plateau a the reervoir continue to dry out and produce dry aturated team hile the temperature continue to fall almot linearly. Finally, the enthalpy increae further a the reervoir completely drie out of immobile ater and produce uperheated team. Hoever, the zero-dimenional model, hich doe not take into account tranient or patial variation in temperature, preure and aturation, fail to match the early tranient to-phae production period predicted by TOUGH2. The zero-dimenional model predict a continuou increae in to-phae team enthalpy ith cumulative ma production. On the other hand, TOUGH2 predict a production period of apparently table to-phae team enthalpy immediately after the early tranient period. Then the enthalpy increae ith production until dry aturated team i produced Production Enthalpy, kj/kg Temperature, degree C E+0 5.0E+7 1.0E+8 1.5E+8 2.0E+8 2.5E+8 3.0E+8 3.5E+8 4.0E+8 Cumulative Ma Production, kg TOUGH2 Enthalpy Zero-D Model Enthalpy TOUGH2 Temperature Zero-D Model Temperature Figure 3-10: Production enthalpy and production temperature profile: liquid-dominated cae, initial ater aturation = 0.3, immobile ater aturation =

41 Production Enthalpy, kj/kg Temperature, degree C E+0 5.0E+7 1.0E+8 1.5E+8 2.0E+8 2.5E+8 3.0E+8 3.5E+8 4.0E+8 4.5E+8 5.0E+8 Cumulative Ma Production, kg TOUGH2 Enthalpy Zero-D Model Enthalpy TOUGH2 Temperature Zero-D Model Temperature Figure 3-11: Production enthalpy and production temperature profile: liquid-dominated cae, initial ater aturation = 0.4, immobile ater aturation = Production Enthalpy, kj/kg Temperature, degree C E+0 1.0E+8 2.0E+8 3.0E+8 4.0E+8 5.0E+8 6.0E+8 Cumulative Ma Production, kg TOUGH2 Enthalpy Zero-D Model Enthalpy TOUGH2 Temperature Zero-D Model Temperature Figure 3-12: Production enthalpy and production temperature profile: liquid-dominated cae, initial ater aturation = 0.5, immobile ater aturation =

42 Intereting obervation can be made from the modeling reult. Figure 3-13 and 3-14 are plot of production enthalpy and temperature for different value of in-itu ater aturation and a fixed immobile ater aturation. On the other hand, Figure 3-15 and 3-16 are plot for different value of immobile ater aturation and a fixed in-itu ater aturation. Baed on the modeling reult, to-phae team enthalpie are a function of the mobile ater content of the reervoir, that i, the difference beteen the initial and immobile ater aturation. Reervoir ith higher mobile ater aturation produce to-phae team ith loer enthalpie. Thi i clearly evident from Figure 3-17 and 3-18, hich ho production enthalpie for cae of contant mobile ater content but varying in-itu and immobile ater aturation. Likeie, dry-out temperature follo the ame trend. Reervoir ith higher mobile ater content dry-out at loer temperature. Furthermore, dry aturated team production commence at a later time in the life of a reervoir containing greater amount of mobile ater. The mobile ater content and the immobile ater aturation of liquid-dominated geothermal reervoir can then be inferred from the to-phae team production enthalpy and the cumulative ma production uing the zero-dimenional model. The dicharge enthalpy of to-phae team can be expreed a a eighted average of the individual team and liquid ater enthalpie given by Equation h ρ k h k + ρ h r r µ µ = (3-18) kr k r ρ + ρ µ µ Auming that linear team-liquid ater relative permeability relation are valid, the above equation can be rearranged to obtain an expreion for the mobile ater content in the reervoir, r 1 r r = ρ µ ρ µ ( h h ) ( h h ) + ( h h ) ρ µ (3-19) Equation 3-19 can be ued to calculate the ratio of the mobile ater in the reervoir to the maximum poible mobile ater. Thi ratio give an indication of ho much mobile ater i preent in the reervoir and ho fat the reervoir i drying out. Similar expreion can be obtained by auming different relative permeability relation. For example, if Corey-type relative permeability relation govern the flo of to-phae team in the reervoir, the reulting expreion for the mobile ater content i given by Equation

43 Production Enthalpy, kj/kg Temperature, degree C E+0 1E+8 2E+8 3E+8 4E+8 5E+8 6E+8 Cumulative Ma Production, kg i = 0.3, Enthalpy i = 0.4 Enthalpy i = 0.5, Enthalpy i = 0.3, Temp i = 0.4, Temp i = 0.5, Temp Figure 3-13: TOUGH2 imulated production enthalpy and reervoir temperature profile: varying initial ater aturation, immobile ater aturation = Production Enthalpy, kj/kg Temperature, degree C E+0 1E+8 2E+8 3E+8 4E+8 5E+8 6E+8 Cumulative Ma Production, kg i = 0.3, Enthalpy i = 0.4, Enthalpy i = 0.5, Enthalpy i = 0.3, Temp i = 0.4, Temp i = 0.5, Temp Figure 3-14: Zero-dimenional model production enthalpy and reervoir temperature profile: varying initial ater aturation, immobile ater aturation =

44 Production Enthalpy, kj/kg Temperature, degree C E+0 1E+8 2E+8 3E+8 4E+8 5E+8 Cumulative Ma Production, kg r = 0.2, Enthalpy r = 0.3 Enthalpy r = 0.2, Temp r = 0.3, Temp Figure 3-15: TOUGH2 imulated production enthalpy and reervoir temperature profile: varying immobile ater aturation, initial ater aturation = Production Enthalpy, kj/kg Temperature, degree C E+0 1E+8 2E+8 3E+8 4E+8 5E+8 Cumulative Ma Production, kg r = 0.2, Enthalpy r = 0.3 Enthalpy r = 0.2, Temp r = 0.3, Temp Figure 3-16: Zero-dimenional model production enthalpy and reervoir temperature profile: varying immobile ater aturation, initial ater aturation =

45 Production Enthalpy, kj/kg E+00 1E+08 2E+08 3E+08 4E+08 5E+08 6E+08 7E+08 Cumulative Ma Production, kg i = 0.4; r = 0.2 i = 0.5; r = 0.3 i = 0.6; r = 0.4 Figure 3-17: TOUGH2 imulated cumulative ma production and enthalpy profile: mobile ater content = Production Enthalpy, kj/kg E+00 1E+08 2E+08 3E+08 4E+08 5E+08 6E+08 7E+08 Cumulative Ma Production, kg i = 0.4; r = 0.2 i = 0.5; r = 0.3 i = 0.6; r = 0.4 Figure 3-18: Zero-dimenional model cumulative ma production and enthalpy profile: mobile ater content =

46 r 1 r r 4 = ρ µ ρ µ ( h h ) ( h h ) + ( h h ) ρ µ (3-20) The change in the reervoir aturation can be etimated uing cumulative ma production data. The material balance equation decribing to-phae team production can be implified to Equation 3-21 by auming that the denitie of team and liquid ater are approximately invariant ith preure and temperature. Thi equation give the change in reervoir aturation a a function of the cumulative ma production, m, knoing the reervoir pore volume, φv. m = 1 = (3-21) φv 2 ( ρ ρ ) The immobile ater aturation can then be etimated baed on the change in mobile ater content from Equation 3-19 and the change in reervoir aturation from Equation Thi technique i demontrated belo uing fluid propertie, and enthalpy and cumulative ma production value imulated by TOUGH2, hich are ummarized in Table 3-2 and 3-3. In both cae, the immobile team aturation, r, a aumed to be equal to 0.1. Table 3-2: TOUGH2 imulation reult and fluid propertie: initial ater aturation = 0.4, immobile ater aturation = 0.2. T C h kj/kg ρ kg/m 3 ρ kg/m 3 µ cp µ cp h kj/kg h kj/kg Calculating the mobile ater content at the initial tate uing Equation 3-19: 1 r 1 r r = ρ µ ρ µ ( h h ) ( h h ) + ( h h ) ρ µ = Calculating the mobile ater content at the final tate uing Equation 3-19: r 1 ρ ( h h ) 2 µ = = ρ ρ r r ( h h ) + ( h h ) µ µ

47 Calculating the difference in mobile ater beteen the initial and final tate: 1 r r r = 1 2 r = 1 1 = r r r r r r Calculating the change in reervoir aturation uing Equation 3-21: = m kg = = = 2 3 kg kg φv ( ρ ρ ) 0.05( π m )( ) m m Calculating the immobile ater aturation baed on the change in mobile ater content and the change in reervoir aturation: r r = = = r r r = r 0.1, therefore, = 0.2 Table 3-3: TOUGH2 imulation reult and fluid propertie: initial ater aturation = 0.4, immobile ater aturation = 0.3. T C h kj/kg ρ kg/m 3 ρ kg/m 3 µ cp µ cp h kj/kg h kj/kg Calculating the mobile ater content at the initial tate uing Equation 3-19: r 1 ρ ( h h ) 1 µ = = ρ ρ r r ( h h ) + ( h h ) µ µ

48 Calculating the mobile ater content at the final tate uing Equation 3-19: r 1 ρ ( h h ) 2 µ = = ρ ρ r r ( h h ) + ( h h ) µ µ Calculating the difference in mobile ater beteen the initial and final tate: 1 r r r = 1 2 r = 1 1 = r r r r r r Calculating the change in reervoir aturation uing Equation 3-21: = m kg = = = 2 3 kg kg φv ( ρ ρ ) 0.05( π m )( ) m m Calculating the immobile ater aturation baed on the change in mobile ater content and the change in reervoir aturation: r r = = = r r r = r 0.1, therefore, = Baed on the calculation reult hon above, the zero-dimenional model prove to be ueful in inferring the immobile ater aturation from field production data. The model gave fairly good etimate of the immobile ater aturation uing "field" data from the TOUGH2 imulation reult. Having etimated the immobile ater aturation, the in-itu ater aturation can be inferred from the initial dicharge enthalpy uing Equation Concluion and Recommendation The in-itu and immobile ater aturation can be inferred from field meaurement of cumulative ma production, dicharge enthalpy and donhole temperature uing imple zero-dimenional model developed baed on material and energy conervation equation and Darcy la. For vapor-dominated reervoir, the in-itu and immobile ater aturation can be etimated knoing rock and fluid propertie, and the difference beteen the table initial and the dry-out donhole temperature and uing Equation

49 o = ( 1 φ ) ρrcr ( To Td ) φ ρ ( h h ) T o (3-11) On the other hand, in liquid-dominated reervoir the in-itu and immobile ater aturation can be inferred from cumulative ma production, dicharge enthalpy, and donhole temperature data uing Equation 3-19 and r 1 r r = ρ µ ρ µ ( h h ) ( h h ) + ( h h ) ρ µ (3-19) m = 1 = (3-21) φv 2 ( ρ ρ ) The validity and uefulne of thee zero-dimenional model ere verified by comparing the modeling reult ith TOUGH2 to-phae radial flo imulation reult. It i recommended that actual field production data be ued to further verify the validity and uefulne of thee model. 35

50

51 Chapter 4 4. Concluion The immobile ater aturation can be inferred from laboratory meaurement of preure, temperature and team aturation in dynamic boiling experiment. There i an apparent correlation beteen the immobile ater aturation and the hape of the team aturation profile, baed on experimental data and TOUGH2 numerical imulation reult of dynamic boiling experiment. The inferred immobile ater aturation of Berea andtone rock i about 0.25 baed on experimental reult obtained by Satik (1997). Thi i conitent ith experimental reult from pat relative permeability experiment reported by Mahiya (1999). Hoever, thi technique may not be ueful in inferring the immobile ater aturation for lo poroity and lo permeability geothermal rock becaue the elbo in the team aturation profile i le prominent. The in-itu and immobile ater aturation can be inferred from field meaurement of cumulative ma production, dicharge enthalpy and donhole temperature uing imple model developed baed on Darcy la and material and energy conervation equation. For vapor-dominated reervoir, the in-itu and immobile ater aturation can be etimated knoing rock and fluid propertie, and the difference beteen the table initial, T o, and the dry-out, T d, donhole temperature. On the other hand, in liquid-dominated reervoir the in-itu and immobile ater aturation can be inferred from cumulative ma production, m, enthalpy, h, data. The validity and uefulne of thee zero-dimenional model ere verified by comparing the modeling reult ith TOUGH2 to-phae radial flo imulation reult. It i recommended that actual field production data be ued to further verify the validity and uefulne of thee model. 37