ASSESSMENT OF UNCERTAINTY IN ESTIMATION OF STORED AND RECOVERABLE THERMAL ENERGY IN GEOTHERMAL RESERVOIRS BY VOLUMETRIC METHODS

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1 PROCEEDINGS, Thrty-Fourth Workshop on Geothermal Reservor Engneerng Stanord Unversty, Stanord, Calorna, February 9-11, 009 SGP-TR-187 ASSESSMENT OF UNCERTAINTY IN ESTIMATION OF STORED AND RECOVERABLE THERMAL ENERGY IN GEOTHERMAL RESERVOIRS BY VOLUMETRIC METHODS Hülya Sarak, Ö. İnanç Türeyen and Mustaa Onur Istanbul Tehnal Unversty ITU Petrol ve Dogal Gaz Muhendslg. Bolumu. Maslak, Istanbul, 4469 TURKEY e-mal: ABSTRACT In ths paper, we nvestgate the propagaton o unertantes n the nput varables (used n the volumetr method) on to stored and reoverable thermal energy estmates alulated rom volumetr methods. Eets o the derent types o nput dstrbutons, orrelaton among nput varables and ogntve bases are also nvestgated. Both Monte Carlo (MC) and the analyt unertanty propagaton (AUP) methods are onsdered and ompared or unertanty haraterzaton. Analyt unertanty propagaton equatons (AUPEs) are derved based on a Taylor-seres expanson around the mean values o the nput varables. The AUPEs are general n that orrelaton among the nput varables, t exsts, an also be aounted or on the resultng unertanty. Monte Carlo methods (MCMs) were used to very the results obtaned rom the AUPEs. A omparatve study that we have onduted shows that the AUPM s as aurate as the MCM or the problem o nterest. We show that AUPM an be used as a ast tool, wthout resortng to the MCM beause the resultng dstrbutons o stored and reoverable heat are always log-normal, whh makes t possble or the results o the AUPM to aurately haraterze the unertanty. It s also shown that t s norret a ommonly made mstake to add the proved and probable (whh orresponds to P 10 and P 90 perentles o the umulatve dstrbuton unton, respetvely) thermal energy reserves rom ndvdual wells (or elds) to get proved eld (or ountry) reserves. Applatons on synthet and eld data ases are presented to demonstrate the methodology onsdered n ths work. INTRODUCTION Unertanty s nherent and ubqutous n estmaton o any type o reserves (ol, gas or heat) rom underground energy systems. The thermal energy or power reserve o a gven geothermal reservor s no exepton. Unortunately, ths s also true regardless o any method used or estmaton, e.g., volumetr, delne urve, or reservor smulaton methods beause the nput varables requred or the reserve estmaton problem always ontan unertantes to some degree that propagate nto reserve estmates. Thereore, to make good desons, one must be able to aurately assess and manage the unertantes and rsks. In ths work, we lmt our study to the assessment o unertanty n estmated thermal energy reserve (n the orm o stored or reoverable) by the volumetr methods (or example, see Muler and Catald, 1978). Volumetr methods are usually used to estmate stored heat and reoverable power reserves n the early le o geothermal reservors. Estmaton o the thermal energy requres geologal, geophysal, and petro-physal data nludng reservor temperature, reservor area, thkness, porosty, rok and lud spe heats, et. The values o these nput varables have usually large unertantes assoated wth them, and hene t s very mportant to propagate these unertantes on to the estmaton o the thermal energy reserves. From the vew ponts o a eld nvestment, an aurate assessment o unertanty n stored and reoverable heat s a rual task rom whh to make desons that wll reate value and/or mtgate loss n value (rsk). In the past, varous authors have onsdered assessng unertanty n estmated stored and reoverable power rom the volumetr method by usng the Monte Carlo (MC) method (Brook et al., 1979; Serpen, 001; Lovekn, 004; Arkan and Parlaktuna, 005; Serpen et al., 008). However, none o these works provde a deep nvestgaton o and nsght to the unertanty assessment rom the volumetr method. These works smply apply the MCM to haraterze unertanty n estmated stored heat or reoverable power or the geothermal reservors nterest.

2 It s no doubt that the MCM used n the prevous reerenes ted s a general approah or assessng the unertanty. In ths work, we show that there s a smple and ast alternatve method whh we reer to t as the analyt unertanty propagaton method (AUPM) to the MCM method or haraterzng unertanty. The valdty o the AUPM or aurately haraterzng unertanty results rom the at the dstrbutons o stored heat and power or a zone, well, or eld to be omputed rom the volumetr method are almost always log-normal. Ths result smply ollows rom the undamental theorem o statsts and probablty the Central Lmt Theorem (CLT) (e.g., see Parzen, 196). The CLT states that the sum o a suently large number o ndependent random varables eah wth nte mean and varane wll be approxmately normally dstrbuted. As a onsequene o ths theorem and the untonal relatonshp o the volumetr method whh nvolves a produt/quotent o several ndependent random varables or omputng thermal energy reserves, one should expet that the resultng dstrbuton o the thermal energy wll tend to be log-normal as the number o nput random varables nreases. Ths result s n at vald no matter what orm o unertanty the nput varables assume. The same ndngs have been reported prevously or the assessment o unertanty n ol or gas reserves omputed rom the volumetr methods by Capen (1996, 001). Other obetves o ths work are as ollows: We would lke to show how the derent types o nput dstrbutons, orrelaton among nput varables, and ogntve bases regardng the nput dstrbutons (Capen, 1976; Capen, 001; Welsh et al., 007) propagate nto the unertanty o the estmated thermal energy or power. In addton, we show that t s norret a ommon mstake to add the proved thermal energy reserves rom ndvdual wells (or elds) to get proved eld (or ountry) reserves, whh has long been establshed by Capen (1996, 001). VOLUMETRIC METHOD Here, we ntrodue the volumetr method used to ompute the stored heat and reoverable thermal energy (n terms o power) or a gven geothermal eld. Throughout, we wll assume a geothermal reservor ontanng hot sold rok and sngle-phase lqud water. Based on the volumetr method, the total stored heat n the reservor s equal to the sum o the stored heat n the (sold) rok and water: Ht Hs + Hw, (1) where H s and H w are gven, respetvely, by and ( 1 φ) ( ) H Ah T T, () s s s s r ( ) H φ Ah T T. () w w w w r In Eqs. 1-, the subsrpt r, s, t, and w denote reerene, sold rok, total, and water, respetvely. The varables used n Eqs. 1- and ther unts are gven below: A area n m spe heat n kj/(kg o C) h net thkness n m H stored heat n kj T temperature n o C φ porosty (raton) densty n kg/m A ew remarks are n order or Eqs. and : Eqs. and are gven by Muler and Catald (1978) who onsder derent temperatures or the sold rok and lud omponents o the reservor. In Eqs. and, however, we wll assume that a loal thermal equlbrum s always vald n the reservor so that sold rok and lud temperatures are dental n Eqs. and ;.e., T s T w T. Hene, usng Eqs. and wth ths assumpton n Eq. 1 yelds: ( 1 φ) φ ( ) Ht s s + w w Ah T Tr. (4) H t dened by Eq. 4 s usually reerred to as the aessble resoure base and t an be onverted to reoverable power (n MW) by the ollowng equaton (Serpen, 001; Arkan and Parlaktuna, 005): HtRY F. (5) 10 LFt p The varables used n Eq. 5 and ther unts are gven below: total stored heat n kj load ator (raton) reoverable power n MW R F reovery ator (raton) Y transormaton yeld (raton) proet le n seonds (s.) H t L F t p Here, L F, the load ator, represents the raton o the total tme n whh the dret heatng or power generaton s n operaton, and Y, the transormaton yeld, represents the eeny o transerrng thermal energy rom the geothermal lud to a seondary lud.

3 It s worth notng that the stored heat n the sold rok s typally an order o magntude larger than the stored heat n the water. In other words, the rato o stored heat n the sold rok to the total stored heat n the sold rok plus water,.e., H s /H t, s nearly 0.80 to 0.90 (or n perentage 80 to 90%). Note that n the ase o hot dry rok, ths rato s unty beause φ s zero or hot dry rok applatons. In other words, or most o the applatons, Ht H s. Ths observaton ndates that gnorng the stored heat n water n omputng H t and rom Eqs. 4 and 5 may not ntrodue sgnant underestmaton n H t and. It s also worth notng that Eqs. 4 and 5 are vald or predtng the reoverable power or both dretheatng and power (eletrty) generaton applatons provded that the varables suh as T, T r, R F, Y, L F, and t p are adequately hosen or the spe applaton o nterest. For example, or most o the dret-heatng applatons, T r usually ranges rom 15 to 60 o C, whereas or eletrty generaton, T r usually ranges between 70 to 100 o C. Although t s mportant how to hoose the varables and ther ranges or aurately assessng unertanty n H t (Eq. 4) and (Eq. 5), our purpose here s not to get nto a detaled dsusson o approprate soures o data and how one ould determne the approprate values o the varables and the assoated unertantes. Assessment o unertantes n the nput varables tsel s a notorously dult problem, beause, to our knowledge, there s no standard rule or haraterzng unertanty n the nput varables. On ths aspet, we reer the readers to the works o Capen (1976) and Welsh et al. (007). Unertanty (n the stored heat and the reoverable power) results rom our lak o knowledge n most o the nput varables n Eqs Quantaton o unertanty s nevtably subetve beause knowledge about the nput varables s dependent on avalable data and personal experene o the nterpreter. As well stated by Welsh et al. (007), t s qute possble or two people to have derent probablty estmates or the same nput varable, based on ther derng knowledge. Thus, there s no sngle orret probablty dstrbuton, unless all people have dental experene and normaton, and proess t n the same way (Welsh et al., 007). Based on the dsusson gven n the prevous paragraph, there s no reason to lam that any partular type o probablty dstrbuton (e.g., unorm, normal, log-normal, trangular, et.) or our nput varables be preerable. As we wll show later based on the CLT, the resultng dstrbutons o H t and are log-normal regardless o the types o probablty dstrbutons hosen or the nput varables. When omputng H t rom Eq. 4, we treat H t as a unton o eght random, nput varables; namely, A, h, φ, s, w, s, w, (T-T r ). When omputng rom Eq. 5, we treat as a unton o eleven random nput varables; the same eght nput varables gven prevously or the H t plus three more nput varables; namely R F, L F, and Y. In our applatons, we x T r at a onstant value, but the varable (T-T r ) n Eq. 4 wll be treated as a random varable beause T n Eq. 4 s treated as a random varable. I the mean and varane o T are T and T, then the mean and varane o the (T-T r ) are ( r ) T T T T r and T, respetvely. Furthermore, we assume that there s no unertanty assoated wth the varable t p n Eq. 5. Some nput varables nvolved n Eqs. 4 and 5 an be expeted to be statstally orrelated. For example, w, the densty o water, s expeted to be dependent on the value o T, reservor temperature n Eq. 4. We would expet that nreasng T derease w, whh ndates that these two varables are negatvely orrelated. Hene, ths ndates that w and T may not be treated as two ndependent varables n Eq. 4. We may also expet that s, the sold rok spe heat be negatvely orrelated wth s, the sold rok densty, and s be postvely orrelated wth temperature. In addton, reservor area may be postvely orrelated wth the net thkness (Murtha, 1994). Our pont s that gnorng exstng orrelatons between nput varable pars may lead to an norret haraterzaton o unertanty n H t and. I data and avalable normaton permt, one should make satter plots o nput varable pars to denty the orrelaton between them, any, and then nlude these orrelatons nto the unertanty assessment proedure. QUANTIFICATON OF UNCERTAINTY In ths seton, we revew some bas equatons and methods used or quantaton o unertanty n the estmaton o H t and by the volumetr method (Eq. 1-5). We rst onsder the Monte Carlo method (MCM) and then the analyt unertanty propagaton method (AUPM). However, beore gettng nto detals o these methods, we rst note that Eq. 4 or estmatng H t nvolves a produt o our random varables; ( 1 φ) ss + φww, A, h, and (T-T r), whereas Eq. 5 or estmatng nvolves a produt o three random varables; H t, R F and Y, and a quotent o the random varable L F. As H t nvolves a produt o our random varables, then s a produt o seven random varables and a quotent o one random varable.

4 I we take the natural logarthm o gven by Eq. 5 and rearrangng the resultng equaton so that the rght-hand sde o the resultng equaton an only be expressed n terms o the nput random varables, we obtan: ln + ln(10 tp ) lnht + lnrf + lny lnlf, (6) where lnh t, whh ollows rom E. 4, s gven by: ( ) ln ( ) ln Ht ln 1 φ ss + φww + ln A. (7) + ln h+ T T r Eqs. 6 and 7 learly ndate that ln and lnh t an be wrtten as a sum o the natural logarthms o all ndependent random varables. I all these random varables are treated as ndependent, then t ollows rom the CLT, dsussed prevously, that the resultng dstrbuton o lnh t or ln wll tend to be normal. It s mportant to note that ths s true no matter what type o dstrbuton the nput random varables assume. I lnh t and ln wll tend to be normal, then the orrespondng dstrbutons o H t and wll tend to be log-normal. We would expet the dstrbuton o to be more lke log-normal than that o H t beause nvolves more produts/quotents than does H t. Note that the CLT promses that lnh t and ln be normal all the random varables are ndependent. Hene, we may not lam that lnh t and ln be stll normal some o the nput random varables are not ndependent. However, our results to be shown later ndate that the resultng dstrbutons o lnh t and ln stll tend to be normal even some o the nput varables are treated as dependent. Fnally, we state our dentons to be used or haraterzng unertanty n H t or. For ths haraterzaton, we adopt the onventon proposed by Capen (001). We wll reer to P 10 as proved, P 50 as probable, and P 90 as possble, where P 10, P 50 and P 90 orrespond to 10th, 50th and 90th perentles o the umulatve dstrbuton unton, respetvely, or H t or. Monte Carlo Method (MCM) The MCM reles on a speed probablty dstrbuton o eah o the nput varables and generates an estmate o the overall unertanty n the predton due to all unertantes n the varables (Kalos and Wthlok, 008). As t does not requre a lnearzaton o the unton and a ontnuty o the random varables, t s a more general approah or haraterzng the unertanty or any gven nonlnear random unton. In our ase, represents H t gven by Eq. 4 or gven by Eq. 5. In the applatons to be gven, we perorm Monte Carlo smulatons by TM spreadsheet-based sotware (004). Analytal Unertanty Propagaton Method (AUPM) Here, we derve an unertanty propagaton equaton or a unton (suh as H t n Eq. 4 or n Eq. 5 or ther natural logarthms gven by Eqs. 6 and 7) where t s treated as a ontnuous random unton due to unertantes n the nput varables. We assume that all unertantes are due to the random unertantes n the nput varables and gnore the systemat errors n the nput varables and modelng errors. The error propagaton equaton we present s based on a Taylor seres approxmaton o the unton around the mean values o the varables up to ts rst dervatves wth respet to eah o the nput varables. As a onsequene o ths approxmaton, the unertanty propagaton equaton provdes a lnearzaton o the unton n terms o ts nput random varables (Barlow, 1989; Coleman and Steele, 1999; Zeybek et al., 009). The AUPM provdes a smple approah or estmatng the varane o a unton dened by several random varables partularly so, o a unton dened by produts and quotents o random varables, whether they are ndependent or orrelated. The method does not assume a spe type o dstrbuton or the nput varables and all needed to use the AUPM are the statstal propertes o the dstrbuton o eah random varable; speally the mean, varane (or std. dev.), and the ovarane (or orrelaton oeent) among varable pars the random varables are orrelated. Beore we present the dervaton o the AUPM, t s worth notng that the AUPM provdes an exat result or the mean and varane o a random unton s lnear wth respet to the nput random varables. Otherwse,.e., s nonlnear, then the AUPM provdes only approxmate estmates o the mean and varane o. The approxmaton gets better nonlnear an be well approxmated by a lnear unton near the means o the nput random varables. For the problem o nterest n ths work, we wsh to estmate the varanes o H t and gven by Eqs. 4 and 5. As an be seen rom these equatons, H t s, n general, a nonlnear unton o the varables A, h, φ, s, w, s, w, (T-T r ). Beause H t s nonlnear, then gven by Eq. 5 s also a nonlnear unton o the same nput varables plus the varables R F, L F and Y. As noted beore, we work, however, wth the natural logarthms o H t and (Eqs. 7 and 6), then we obtan nearly lnearzed equatons or H t and

5 . We say nearly lnearzed beause lnh t and ln s stll nonlnear, but now are weakly nonlnear. For example, lnh t (Eq. 7) s more lnear ompared to H t (Eq. 4) beause lnh t s only nonlnear wth respet to the varables; φ, s, w, s, w, but s lnear wth respet to lna, lnh, ln(t-t r ), whereas H t s nonlnear wth all o these eght random varables. Smlar arguments are vald or and ln untons. Thereore, as to be shown later, we obtan more aurate estmates o the varanes o lnh t and ln we apply the AUPM dretly to lnh t (Eq. 7) and ln (Eq. 6) nstead o applyng t to H t (Eq. 4) and (Eq. 5). One nal remark s that the porosty s nearly zero (e.g., n the ase o hot dry rok), then lnh t gven by Eq. 7 an be wrtten as: ln H ln ln ln A ln h ln ( T T ) (8) t s s r So, n ths ase, lnh t s a lnear unton o the natural logarthms o the nput varables; s, s, A, h, and T- T r. Usng Eq. 8 n Eq. 6 yelds: ln + ln(10 tp) ln s + ln s + ln A+ ln h + ln ( T Tr) + ln RF + lny, (9) ln L whh ndates that ln s a lnear unton o the natural logarthms o the nput varables; s, s, A, h, T-T r, R F, Y, and L F. AUP Equatons or H t (Eq. 4) and (Eq. 5) Let s onsder a random unton o M varables,, 1,,,M,.e., ( 1,, K, M ). Then, expandng around the mean (or true) values o s (denoted by, 1,, K, M ) by usng a Taylor seres up to rst dervatves, we obtan: ( 1,, K M ) (,, K, ) F 1 M + ( ) 1, 1,..., M M. (10) It an be shown that the mean ( ) and varane o ( ) are approxmated by: and (,,, ) K, (11) 1 M M M 1 M + ov , (1) θ θθ (, ) where ov(, ) represents the ovarane between the varable pars and, and we use the relaton between ovarane and orrelaton oeent,,, then we an express Eq. 1 n terms o the orrelaton oeent as: M M 1 M +, (1) θ θθ In Eqs. 1 and 1, θ s the dervatve o wth respet to the varable,.e., θ, 1,..., M. (14) Note that θ represents the senstvty o to the varable evaluated at the mean values o all the varables. It an be noted (rom Eqs. 1-14) that the unertanty propagaton nto s determned not only by the varanes o the varables and orrelaton among them, but also the senstvty o to eah varable n the volumetr method (Eqs. 4-7). We an derve several useul orms o AUPEs. For example, we dvde both sdes o Eq. 1 by, we obtan: M θ 1, M 1 M θ θ +, (15) where / s alled the relatve unertanty n and / represents the relatve unertanty or the varable, and smply denoted by RU. It s worth notng that the term θ / n Eq. 15 represents the normalzed (or dmensonless) senstvty o wth respet to the varable and s expltly omputed rom: θ, 1,..., M. (16) Another useul measure o the total unertanty an be obtaned by multplyng both sdes o Eq. 15 by ( / ) and then rearrangng the resultng equaton:

6 θ M 1. θ θ M 1 M +, (17) Eq. 17 an also be smply rewrtten as: M M 1 M UPC + UPC, 1, (18) where we dene UPC as a measure o the ndvdual ontrbuton (n raton) o eah varable s unertanty to the total unertanty (or varane) n : θ UPC θ, (19) and dene UPC, as a measure o the ontrbuton o the orrelated pars and (n raton) to the total unertanty (or varane) n : the varables ontrbute more to the total unertanty n. I the varables are orrelated, then one should ompute UPC s n addton to UPC to denty whh o the varables and the orrelated pars ontrbute more to the total unertanty n. It may not be obvous rom Eqs. 1-0 to see how orrelaton among varables propagates nto unertanty n beause n the ase o orrelaton, the unertanty n (Eq. 1) depends on () the number o orrelated pars, () magntude and sgn o the orrelaton oeent between the orrelated pars, and () magntude and sgn o the senstvty o to the orrelated varables. So, the unertanty n an nrease or derease wth orrelaton. However, how orrelatons among varables propagate nto unertanty n an be easly dented by alulatng and then nspetng the UPC and UPC gven by Eqs. 19 and 0, derved rom the error propagaton equaton aountng or orrelaton among the varables. As mentoned prevously, or the problem o nterest, n Eqs ould represent H t gven by Eq. 4 or gven by Eq. 5. The senstvtes o H t and (.e., θ s) requred n Eqs. 1-0 an be obtaned by analytal derentaton o Eqs. 4 and 5. For example, the senstvty o H t to a gven varable n Eq. 4 s obtaned by derentatng Eq. 4 wth respet to that varable. These senstvtes are tabulated n Table 1. To ompute unertanty propagaton n gven by Eq. 5 rom Eqs. 1-0, then we rst ompute the senstvtes rom the ormulas gven n Table. UPC θ θ,, θ θ,. (0) Note that UPC (Eq. 19) s always postve, but UPC, (Eq. 0) an be postve or negatve dependng on the sgns o the normalzed senstvtes (Eq. 16) and the orrelaton oeent. I, and only, all varables are unorrelated, all UPC, are zero. A ew mportant remarks (rom Eqs. 1-0) are n order: I the varables are ompletely ndependent, the total unertanty n wll be domnated more by the varables havng the larger produts o normalzed senstvtes and relatve unertantes. So, or ths ase, one only needs to nspet the produt o normalzed senstvty and relatve unertanty, or UPC or eah o the varables to denty whh o Table 1: Senstvty o H t (Eq. 4) wth respet to a gven varable n Eq. 4. * Varable θ H t / φ ( + ) A h ( T T ) s w w r s ( 1 φ) ( ) s A h T Tr s ( 1 φ ) ( ) s A h T Tr w φ w A h ( T Tr ) w φ w A h ( T Tr ) A ( 1 φ) + φ h ( T T ) s s w w r h ( 1 φ) + φ A ( T T ) T-T r ( 1 ) s s w w r φ s φ + w w A h *evaluated at the mean values o the varables s

7 Table : Senstvty o (Eq. 5) wth respet to a gven varable n Eq. 5. * Varable θ / φ ( ) + s w w A h ( T Tr) R F Y 10 t LF p s ( 1 φ) ( ) 10 s A h T Tr RF Y L t F p s ( 1 φ ) ( ) s A h T Tr RF Y 10 L t F p w φ w A h ( T Tr) R F Y 10 L t F p w φ w A h ( T Tr) R F Y 10 Lt F p 1 φ s φ + w w h T Tr R F Y 10 t A ( ) ( ) LF p h ( 1 φ) + φ ( T T ) T-T r ( 1 ) s s w w A r RF Y 10 L t F p + φ s φ w w A h RF Y 10 L t F p R F ( 1 φ) + φ ( T T ) s s w w A h r Y 10 L t F p Y ( 1 φ) + φ ( T T ) s s w w A h r RF 10 L t F p L F ( 1 φ) + φ ( T T ) s s w w A h r RF Y 10 t p L F *evaluated at the mean values o the varables s AUP Equatons or lnh t (Eq. 7) and ln (Eq. 6) In the ase where we onsder unertanty propagaton n ln (e.g, ln lnh t gven by Eq. 7 or ln ln gven by Eq. 6) nstead o, then we may use two derent approahes to derve AUP equatons. These approahes yeld two derent AUP equatons or estmatng the varane o lnh t or ln. Approah 1: The rst approah s based on the Taylor seres expanson o lnh T (or ln) around the mean values o the nput varables. For ths ase, s the AUP equatons are gven by Eqs wth replaed by ln. Hene, the senstvtes θ s requred n Eqs. 1-0 are gven by: ln θ, 1,..., M. (1) I we onsder unertanty propagaton n ln lnh t, gven by Eq. 7, then the senstvty o lnh t wth respet to s smply obtaned by usng the han rule as: ln Ht 1 H H t, 1,..., M t, () where and H t / are tabulated n the rst and seond olumns o Table 1. Smlarly, we onsder unertanty propagaton n ln gven by Eq. 6, then the senstvty o ln wth respet to s obtaned by usng the han rule as: ln 1, 1,..., M, () where and / are tabulated n the rst and seond olumns o Table. Table and 4 present the ormulas or the dervatves or lnh t and ln wth respet to, respetvely. Table : Senstvty o lnh t (Eq. 7) wth respet to a gven varable n Eq. 7. * Varable θ lnh t / φ ( ) + s w w ( 1 ) + s ( 1 φ) s ( 1 ) + φ s φ w w s ( 1 φ ) ( 1 ) w w φ s φ w w s φ s φ + w w φ w ( 1 φ) s φ + w w φ w ( 1 φ) s φ + w w A 1/ A h 1/ h T-T r 1/ ( T T r ) *evaluated at the mean values o the varables s

8 Table 4: Senstvty o ln (Eq. 6) wth respet to a gven varable n Eq. 6. * Varable θ ln/ φ ( ) + s w w ( 1 ) + φ s φ w w s φ + s φ w w s φ s φ + w w φ w ( 1 φ) s φ + w w φ w ( 1 φ) s φ + w w s ( 1 φ) ( 1 ) s ( 1 φ ) ( 1 ) w w A 1/ A h 1/ h T-T r 1/ ( T T r ) R F 1/ Y R F 1/ Y L F 1/ LF *evaluated at the mean values o the varables s Approah : The seond approah s based on the Taylor seres expanson o lnh t (or ln) around the mean values o natural log o the nput varables;.e., s. For ths ase, the AUP equatons are gven by ln Eqs wth replaed by ln, s by ln s, and by ln s. Hene, or ths approah, the s senstvtes θ s requred n Eqs. 1-0 are to be omputed rom: ln θ ln, 1,..., M. (4) I we onsder unertanty propagaton n ln lnh t, then the senstvty o lnh t wth respet to ln s smply obtaned by usng the han rule as: ln H H ln H t t t, 1,..., M. (5) Smlarly, we onsder unertanty propagaton n ln ln gven by Eq. 6, then the senstvty o ln wth respet to ln s obtaned by usng the han rule as: ln ln, 1,..., M. (6) The dervatves (Eqs. 5 and 6) are smply omputed by multplyng the dervatves gven n the seond olumns o Table and 4 by the mean value o the varable nterest,. Our numeral results ndate that Approah provdes slghtly better estmate o the varane than Approah 1. However, unlke the Approah 1, the Approah or estmatng the varanes o lnh t and ln wll requre us to work wth the means and varanes o the natural-log o the nput model varables,.e., ln and ln. In the orrelated ase, we wll also need to onvert to orrelaton oeent between two pars, say, to ln,ln. All these requre an addtonal eort to ompute the mean ln and varane ln rom a gven dstrbuton o the nput varable. I the dstrbuton o s log-normal wth mean and varane, then ln s normal wth the mean ln and varane, gven by: and ln 1 ln ln ln 1 + ln ln 1+, (7). (8) I the hosen dstrbuton or the nput varable s not log-normal, then we an use desrptve statsts on the avalable data to ompute ln and ln. I suh exhaustve data are not avalable, then we may generate samples rom a known dstrbuton and use desrptve statsts on these samples to ompute and. ln ln Estmaton o 10th, 50th and 90th perentles o H t,, lnh t and ln As dsussed prevously and to be shown wth numeral results later, the resultng dstrbutons o H t and are nearly log-normal due to the CLT. Consequently, the dstrbutons o lnh t and ln are nearly normal. Hene, usng ths observaton, we an

9 ompute other statstal measures suh as P 10, P 50, and P 90 or haraterzng unertanty n H t or n addton to the mean and varane determned rom the AUPM. When usng the AUPM, we rst ompute natural logs o P 10, P 50, and P 90 ;.e., lnp 10, lnp 50, and lnp 90 rom the ollowng ormulas gven or a normal dstrbuton (also see Capen, 001): and ln P 1.8, (9) 10 ln ln ln P + 1.8, (0) ln P 90 ln ln ln P + ln P , (1) where ln represents ether lnh t or ln. The 1.8 n Eqs. 9 and 0 stems rom the at that or a normal dstrbuton, 10 or 90 perentle les 1.8 standard devaton unts away rom the mean. It s mportant to note that how we ompute the natural log o perentles rom Eqs. 9-1, whh requre the values o ln and ln (square root o the varane yelds the standard devaton ln ) rom the AUPM. I we dretly apply AUPM to H t (Eq. 4) or (Eq. 5), then we an estmate the mean and varane o H t or ;.e., and, where represents ether H t or. Then, usng these values n Eq. 7 and 8 (where s replaed by ) beause s log-normal due to the CLT, we an estmate ln and ln. As mentoned prevously, we do not reommend usng ths approah beause applyng the AUPM dretly to H t or does not provde as aurate estmates o and as applyng t to lnh t or ln due to nonlnearty n H t or. Thereore, we estmate dretly the mean and varane o lnh t or ln;.e., and, where ln ln represents ether H t or, by usng the AUPM method based on ether Approah 1 or, dsussed prevously. Then, we ompute lnp 10, lnp 50, and lnp 90 by usng Eqs We nally exponentate the omputed values o lnp 10, lnp 50 and lnp 90 to obtan P 10, P 50 and P 90, respetvely, or H t and. RESULTS Here, we present our results obtaned by onsderng a ew examples to make our ponts. Unorrelated Case Frst, we onsder a ase where all nput varables are ndependent n H t and gven by Eqs. 4 and 5. For the purpose o ths example, we hoose to model all dstrbutons wth a trangular dstrbuton. As dsussed prevously, n at, there s no reason to nsst upon any partular probablty dstrbuton or our nput varables. The mnmum, most lkely (mode), and maxmum values o the nput varables are gven n Table 5. The values o mean and varane gven n Table 5 were omputed rom the well-known ormulas or a trangular dstrbuton: and Mn + Max + Mode, () ( Mn) + ( Max) + ( Mode) 18 ( Mn Max + Mn Mode + Max Mode) 18 () The data gven n Table 5 pertan to Izmr Balçova- Narlıdere geothermal eld n Turkey and were taken rom Satman et al. (001). For ths applaton, T r 60 o C. Table 5: Dstrbutons o the nput varables. Varable Mn Mode Max Mean + Varane +. φ s, kj/(kg o C) s, kg/m w, kj/(kg o C) w, kg/m A, m h, m T-T r, o C R F Y L F t p, s mean and varane were omputed rom the known ormulas gven or a trangular dstrbuton, see Eqs. and.

10 Fgures 1-4 show hstograms o H s (Eq. ), H w (Eq. ), H t (Eq. 4), and (Eq. 5) generated rom the MCM by usng the dstrbutons gven n Table 5 The statstal varables (e.g., mean, varane, P 10, P 50, and P 90 ) or eah hstogram are gven n the nsets o Fgures 1-4. As s expeted rom the CLT, all hstograms o Fgures 1-4 are nearly log-normal. Ths an be urther vered by smply sortng the omputed values n asendng order and plottng the resultng values on a log-normal probablty paper. I the ponts all lose to a straght lne, then the dstrbuton an be onsdered as log-normal. Fgure 5 shows a plot o on a log-normal probablty paper. The result o Fgure 5 veres that s reasonably log-normal. Although not shown here, plots o H s, H w, and H t on a log-normal probablty paper ndate that these varables are also reasonably log-normal. Frequeny Mean : 9.761x10 1 Varane :.08x10 7 P 10 : 0.486x10 14 P 50 : 0.884x10 14 P 90 : 1.589x H t (x10 14 ), kj Fgure : Hstogram o total stored heat n sold rok plus water, H t, generated rom the MCM Mean : 8.876x10 1 Varane : 1.694x10 7 P 10 : 0.441x10 14 P 50 : 0.805x10 14 P 90 : 1.445x Mean : Varane : P 10 : P 50 : 4.16 P 90 : Frqueny 600 Frequeny H s (x10 14 ), kj Fgure 1: Hstogram o stored heat n sold rok, H s, generated rom the MCM , MW Fgure 4: Hstogram o reoverable power,, generated rom the MCM Frequeny Mean : 8.849x10 1 Varane :.41x10 5 P 10 : 0.07x10 14 P 50 : 0.077x10 14 P 90 : 0.154x10 14 Smulated values o, MW H w (x10 14 ), kj Fgure : Hstogram o stored heat n water, H w, generated rom the MCM Probablty, % less than Fgure 5: Cumulatve plot or reoverable power,, generated rom the MCM. It s worth notng that the total stored H t s the sum o stored heat n the sold rok and the stored heat n water oupyng the pore spaes. Inspetng the hstograms o H s (Fg. 1), H w (Fg. ) and H t (Fg. ),

11 we see that the ontrbuton o H w to H t s small, and H t s domnated manly by H s, stored heat n the sold matrx. In at, as dsussed prevously, ths s not surprsng beause most o the heat s normally stored n the sold rok n geothermal reservors. It may be also worth notng that as expeted rom a statstal pont o vew, the sum o the means o the dstrbutons o H w and H s s exatly equal to the mean o the dstrbuton o H t. However, we see that the sum o ndvdual varanes o H w and H s dstrbutons, though they are lose and they agree wthn 15%, s not exatly equal to the varane o the dstrbuton o H t. It s atually less than the varane o the dstrbuton o H t. The reason or ths s due to the at that H w and H s are somewhat (postvely) orrelated wth eah other due to the ommon varables (φ, A, h, and T-T r ) nvolved n samplng the dstrbutons o H w and H s. Hene the varane o the dstrbuton s derent (greater) rom (than) the sum o the ndvdual varanes o H w and H s dstrbutons. Ths ndates that we an smply add the means, but not the varanes o ndvdual dstrbutons or estmatng the mean and varane o a random unton nvolvng the sum o two ndvdual random dstrbutons unless they are unorrelated. Now, we ompare the estmates o means, varanes, P 10, P 50, and P 90 omputed or H t,, lnh t, ln rom the MCM and AUPM. As mentoned prevously, when we use AUPM, we an onsder two derent approahes or estmatng the varanes o lnh t and ln untons. The seond approah (Approah ) requres that we should work wth the mean and varanes o ln s nstead o s. Table 6 gves ln and ln values omputed rom desrptve statsts ater applyng natural log transormaton to eah or the nput varables lsted n Table 5. Table 6: Mean and Varane o and ln. Varable ln ln φ s, kj/(kg o C) s, kg/m w, kj/(kg o C) w, kg/m A, m h, m T-T r, o C R F Y L F t p, s Table 7 ompares the values o means and varanes omputed rom MCM and AUPM or H t,, lnh t and ln. The omputed values o mean and varane rom the MCM and AUPM or H t,, lnh t, and ln untons agree well (Table 7). We also note that the values o mean and varane or lnh t and ln omputed rom the AUPM based on Approah 1 and are nearly dental to those estmated rom the MCM. The means and varanes omputed rom the MCM and AUPM or H t and untons, however, do not agree as good as those omputed or lnh t and ln untons. Ths s not surprsng though, and s an expeted result beause as mentoned prevously, the AUPM provdes a lnear approxmaton to a nonlnear random unton around the mean values o the nput varables and an provde exat results or the mean and varane or a unton that unton s lnear wth the nput random varables. In our ase, both H t (Eq. 4) and (Eq. 5) are n at nonlnear untons o ther nput varables. On the other hand, lnh t (Eq. 7) and ln (Eq. 6) are almost lnear (or weakly nonlnear) untons o the nput varables. Consequently, the AUPM (based on ether Approah 1 or ) provdes estmates o means and varanes or lnh t and ln untons that agree very well wth those omputed rom the MCM. Note that AUPM based on Approah provdes estmates essentally dental to those rom the MCM. Table 7: A omparson o means and varanes rom the MCM and AUPM or H t,, lnh t, and ln untons. Mean Varane MCM AUPM MCM AUPM H t, kj , MW lnh t ln AUPM based on Approah 1 AUPM based on Approah Next, we ompute the 10 th, 50 th and 90 th perentles or H t,, lnh t and ln untons. As mentoned prevously, to ompute these perentles when usng the AUPM, we apply the AUPM (based on ether Approah 1 or ) dretly to lnh t or ln to estmate and ln ln, where represents ether H t or. Then we use these values o mean and varane n Eqs. 9-1 to ompute lnp 10, lnp 50, and lnp 90. Exponentatng the values o lnp 10, lnp 50, and lnp 50 yelds P 10, P 50, and P 90, respetvely, or the H t or

12 unton. Table 8 ompares the values o P 10, P 50, and P 90 omputed rom the MCM and AUPM (based on Approah 1 and ). The values o P 10, P 50, and P 90 omputed rom the AUPM or H t and (Table 8) were omputed wth the proedure ust mentoned above. The values o P 10, P 50 and P 90 gven n the 5 th and 6 th rowa o Table 8 or lnh t and ln represent the values lnp 10, lnp 50 and lnp 90 omputed rom Eqs The values o P 10, P 50, and P 90 perentles omputed rom the MCM and AUPM agree well (Table 8). The maxmum relatve derene or the MCM and AUPM based on Approah 1 s around 10%, whereas the maxmum relatve derene or the MCM and AUPM based on Approah s around %. Another ne eature o the AUPM s that we an denty and quanty whh o the nput varables ontrbute more to the total unertanty n H t, lnh t,, or ln. Beause lnh t and ln are nearly lnear untons o the nput varables as dsussed prevously, we onsder lnh t and ln to denty whh o the nput varables ontrbute more to the total unertanty n lnh t and ln. For ths purpose, as the nput varables are treated as ndependent random varables n ths example, we an smply use UPC gven by Eq. 19, the relatve measure o the ndvdual (ratonal) ontrbuton o eah varable s unertanty to the total unertanty (or varane) n lnh t or ln. Table 9 tabulates the values o UPC along wth the relatve unertanty (RU ) and the produts o normalzed senstvty (NS ) and relatve unertanty or the nput varables nvolved n lnh t. Table 10 present the results or the nput varables nvolved n ln. In Tables 9 and 10, RU, NS, and UPC are dened as: and ln RU ln NS, (4) ln ln ln ln, (5) ln ln ln ln ln UPC. (6) ln ln ln In Eqs. 4-6, represents H t or. Table 8: A omparson o the values o 10th, 50th and 90th perentles omputed rom the MCM and AUPM or H t,, lnh t, and ln untons. P 10 P 50 P 90 MCM AUPM MCM AUPM MCM AUPM H t 10 1, kj 4.95 * 8.8 * 15.8 *, MW 17.6 * 4. * 66. * lnh t *.1 *.7 * ln *.5 * 4.19 * AUPM based on Approah 1 * AUPM based on Approah Table 9: The values o RU, UPC and normalzed senstvtes (NS) or eah varable n lnh t. Varable RU NS RU NS UPC φ s, kj/(kg o C) s, kg/m w, kj/(kg o C) w, kg/m A, m h, m T-T r, o C Table 10: The values o RU, UPC and normalzed senstvtes (NS) or eah varable n ln. Varable RU NS RU NS UPC φ s, kj/(kg o C) s, kg/m w, kj/(kg o C) w, kg/m A, m h, m T-T r, o C R F Y L F

13 The UPC values gven n Table 9 ndate that or the example onsdered, manly three varables are ontrbutng to the total unertanty n lnh t. These are, n the order o, thkness (h), area (A), and temperature (T-T r ). Smlarly, rom Table 10, we see that manly our varables are ontrbutng to the total unertanty n ln, whh are, n the order o, thkness (h), area (A), reovery ator (R F ), and temperature (T-T r ). Another mportant pont that we wsh to make rom the results o Tables 9 and 10 s that the varables that are more nluental on the total unertanty n lnh t or ln an be dented by nspetng the absolute value o the produts o RU and NS, but annot be dented by nspetng solely ther ndvdual values. In other words, nspetng RU or NS alone s not suent to denty the varables that are more nluental on the total unertanty n lnh t or ln. Note that suh dentaton an be made beore applyng the MCM or AUPM by omputng the values o normalzed senstvtes rom the ormulae presented (Tables 1-4) and the gven values o relatve unertanty or eah nput parameter. Correlated Case Here, we nvestgate the propagaton o orrelatons among nput varables nto the total unertanty n lnh t and ln. As mentoned prevously, t s possble that varous nput varables n stored heat and reoverable power an be orrelated wth eah other. For example, the sold rok spe heat may be negatvely orrelated wth the densty o the sold rok, the densty o water may be negatvely orrelated wth temperature, and the sold rok spe heat an be postvely orrelated wth temperature. In addton, we may expet that area (A) and thkness (h) are postvely orrelated (Murtha, 1994). For ths nvestgaton, we use the same nput dstrbutons gven n Table 5, but assume orrelaton between the ve orrelated pars and the orrelaton oeents gven n Table 11. Table 11: Correlated varable pars and orrelaton oeents. Correlated Varable Pars (, ) Correlaton Coeent, ( s, T Tr ) ( s, s ) 0.44 ( w, T Tr ) 0.6 ( w, w ) 0.4 ( A, h ) Fgures 6 and 7 show hstograms o H t (Eq. 4), and (Eq. 5) generated rom the MCM by usng the dstrbutons gven n Table 5 and the orrelaton oeents gven n Table 11 Frequeny Mean : 1.004x10 14 Varane :.666x10 7 P 10 : 0.456x10 14 P 50 : 0.889x10 14 P 90 : 1.718x H t (x10 14 ), kj Fgure 6: Hstogram o total stored heat H t, generated rom the MCM, orrelated ase. Frequeny Mean : Varane : P 10 : P 50 : P 90 : , MW Fgure 7: Hstogram o reoverable power,, generated rom the MCM, orrelated ase. It an be seen that orrelaton dd not hange the shape o the H t and dstrbutons. When the results o Fgs. 6 and 7 or the orrelated ase are ompared wth the results o Fgs. and 4 or the unorrelated ase, we see that orrelaton nreased the varane sgnantly (about 0%). Correlaton nreased the 10 th, 50 th and 90 th perentles o H t and slghtly (about 6%) ompared to the orrespondng results or the unorrelated ase. Table 1 ompares the values o means and varanes omputed rom MCM and AUPM or H t,, lnh t and ln. We used the AUPM based on Approah to ompute means and varanes o lnh t and ln untons. Ths approah requres that we work wth the orrelaton oeent between the pars n terms o the natural-log o nput random varables;.e., ln,ln. Our results ndate ln,ln,. That s; one an use the orrelaton oeents based

14 , when usng the AUPM method based on Approah. Note that the means and varanes obtaned rom MCM and AUPM based on Approah or lnh t and ln are essentally dental. Table 1 ompares the values o P 10, P 50, and P 90 omputed rom the MCM and AUPM (based on Approah ). Agan, there s a very good agreement n the 10 th, 50 th and 90 th perentles omputed rom both methods. Table 1: A omparson o means and varanes rom the MCM and AUPM or H t,, lnh t, and ln untons, orrelated ase. Mean Varane MCM AUPM MCM AUPM H t, kj , MW lnh t ln n ln and lnh t untons even or the orrelated nput varables. Table 1: A omparson o the values o 10th, 50th and 90th perentles omputed rom the MC and AUP methods or H t,, lnh t, and ln untons, orrelated ase. P 10 P 50 P 90 MCM AUPM MCM AUPM MCM AUPM H t 10 1, kj, MW lnh t ln Table 14: The values o UPC, and UPC or ln. Varable UPC UPC φ s, kj/(kg o C) s, kg/m Table 14 tabulates the values o UPC (Eq. 6) and UPC (Eq. 7) values omputed rom the AUPM method or ln unton. UPC s s omputed rom: ln ln ln UPC, ln,ln ln ln ln ln (7) The results o Table 14 ndate that orrelatons n ve orrelated pars ontrbute postvely about 16% to the total unertanty. The remanng 84% n the total unertanty s due to ndvdual varables ontrbuton (UPC s) to the total unertanty. In at, these results explan why the varane o ln (or lnh t ) or the orrelated ase nreased about 17% (or 4%), ompared to that o ln (or lnh t ) or the unorrelated ase. From Table 14, we also note that the orrelaton between area (A) and thkness (h) ontrbute the most (about 1%), among the other our orrelated pars, to the total o UPC s. In summary, our results show that orrelaton among varables, partularly between A and h, they exst and that data avalable permts one to denty orrelaton among varables, should be aounted or aurate haraterzaton o unertanty n H t,, lnh t, and ln. The results also ndate that the AUPM works as good as the MCM to estmate unertanty (varane, 10 th, 50 th, and 90 th perentles) w, kj/(kg o C) w, kg/m A, m h, m T-T r, o C R F Y L F Correlated Pars, (, ) (, T T ) s (, ) s s r (, T T ) w (, ) w w r ( A, h )

15 Cogntve Bases It has long been known e.g., Tversky and Kahneman (1974), Capen (1976), and Welsh et al. (005, 007) that ogntve bases mpat desons taken under ondtons o unertanty. Cogntve bases results rom dsrepanes between alulated, optmal desons and those made usng ntuton (Welsh et al., 005). There are three derent spe ogntve bases o nterest; overondene, trust heurst (or anhorng), and avalablty. Overondene results rom people s tendeny to understate unertanty due to havng no good quanttatve dea o unertanty. Thus, they overestmate the preson o ther own knowledge (Capen, 1976, Welsh et al., 005 and 007). For example, 80% ondene ranges are ommonly used when nterpreters are asked to estmate varables suh as average porosty and thkness o reservor. Data olleted rom nterpreters however, ndate that suh ranges on average nlude the atual values less than 50% o the tme rather than 80% as the ondene range should ndate (Welsh et al., 007). In other words, ommonly stated or used ranges or the varables by nterpreters are too narrow. In ths work, we lmt our nvestgaton only to how ogntve bas due to overondene propagates nto the total unertanty n lnh t and ln. We wll not onsder ogntve bases due to trust heurst results rom managers tendeny to rely on the udgments o ndvduals that they have learnt to trust, rather than norporatng the opnons o everyone workng on an nterpretaton task and avalablty results rom people s tendeny to gnore the total number o possble events and to use the number o events that an be realled (Welsh et al., 005 and 007). We reer the nterested readers to Welsh et al. (007) or detals regardng modelng trust heurst and avalablty. To model ogntve bas due to overondene, we use the same approah used by Welsh et al. (007). We onsder the nput varable dstrbutons gven n Table 5 as the base dstrbutons or the overondene model. To model the mpat o overondene on the total unertanty, the 10 th and 90 th perentles o eah o the base nput dstrbutons gven Table 5 were alulated. Then, or eah o the base dstrbutons, a trangular dstrbuton wth the same mode and mean was ound by adustng the mnmum and maxmum values by equal amounts so that the 10 th and 90 th perentles o the base dstrbuton orresponds to 0 th and 80 th perentles o the real dstrbuton. Hene, the base dstrbutons gven n Table 5 s 0% overondent ompared to the transormed dstrbutons (reerred to as real ). For example, Fgure 8 ompares the real (wder) dstrbuton o porosty (φ) aganst the 0% overondent (narrow) dstrbuton o φ. Frequeny overondent dstrbuton "real" (or "true") dstrbuton Porosty (φ) Fgure 8: A omparson o overondent φ dstrbuton wth real φ dstrbuton. To assess the eet o overondene, transormatons desrbed above were appled to all o the nput varables (gven n Table 5) used n alulatng the H t,, lnh t and ln. We assume that all nput varables are ndependent. Here, we only present our results obtaned or. Smlar results obtaned or H t. Table 15 ompares the mean, varane and 10 th, 50 th, and 90 th perentles omputed or based on the 0% overondent nput varables and the real nput varables by usng the AUPM (based on Approah ). The mean and varane gven or n Table 15 were obtaned ater omputng the mean and varane or the ln and then usng the ollowng transormaton rom normal to log-normal: and exp ln + ln, (8) ( ) ln ln ( ln ) exp + exp 1. (9) As an be seen rom Table 15, although the mean s dental or both the 0% overondent model and the real model, the unertanty markers; varane and perentles, are sgnantly derent, as expeted. The varane or the real ase s about twe the varane o the 0% overondene ase. Hene, the 10 th and 90 th perentles o the pertanng to these two ases are derent. As expeted, the 10 th and 50 th perentles or the 0% overondene ase are

16 larger than the orrespondng ones or the real ase, and 90 th perentle or the 0% overondene ase s smaller than that or the real ase. Hene, 0% overondene model predts sgnantly too low unertanty n the estmates o stored heat and reoverable power. Table 15: The values o mean, varane, 10th, 50th and 90th perentles or the real and %0 overondene models. P 10 P 50 P 90 or the 0% overondene or the real Addng Indvdual Well (or Feld) Reserves to Determne Feld (or Country) Reserves Here, we show that t s norret a ommonly made mstake to add the proved and probable (P 10 and P 90 perentles) thermal energy reserves rom ndvdual wells (or elds) to get proved eld (or ountry) reserves, whh has long been establshed by Capen (1996, 001) n the ontext o stohast estmaton o ol and gas reserves. For the purpose o demonstraton, we assume a ountry Z onsstng o 10 pseudo geothermal elds, eah havng a log-normal dstrbuton or wth means, varanes and P 10, P 50, P 90 perentles as gven n Table 16. I we let to denote the reoverable power or eld, 1,,,N, where N denotes the total number o elds, the total reoverable power (denoted by C ) or the ountry Z s smply gven by C N 1. (40) Beause we treat eah eld s as ndependently dstrbuted random untons, the mean and varane o the C gven by Eq. 40 are smply the sums o eah eld s mean and varane, respetvely. These values are gven n the last row o the seond and thrd olumns n Table 16;.e., C MW and 110,40 MW (or.95 MW). C C Based on the CLT, we expet that addng s (Eq. 40) leads asymptotally to the normal dstrbuton as N beomes large. Then, assumng that C s normal, we an ompute the P 10, P 50 and P 90 perentles o the C dstrbuton by smply usng the ormulas gven by: and P , (41) C C P , (4) 50 C C C P + P P. (4) The values o 10 th, 50 th and 90 th perentles omputed rom Eqs or the C or the 10-eld example ase onsdered are gven n Table 17. Table 16: The values o mean, varane, 10th, 50th and 90th perentles or eah pseudo eld s. P 50 P 90 P 10 MW MW MW MW MW Feld Feld Feld Feld Feld Feld Feld Feld Feld Feld Total Table 17: Comparson o 10th, 50th and 90th perentles omputed rom derent approahes or the Country Z. P 10 MW P 50 MW P 90 MW 1 C C C C perentles omputed based on normal assumpton perentles obtaned rom the MC smulaton perentles omputed based on log-normal assumpton 4 perentles alulated as the smple sum o the ndvdual eld s orrespondng perentles. To hek the valdty o the perentles omputed (see nd row o Table 17) or C based on the normalty assumpton, we perormed Monte Carlo smulaton based on Eq. 40 wth eah eld s,