Hybrid Forecasting Methods for Multi-Fractured Horizontal Wells: EUR Sensitivities

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1 Advances in Peroleum Exploraion and Developmen Vol. 3, No., 0, pp. -0 DOI:0.3968/j.aped ISSN 95-54X [Prin] ISSN [Online] Hybrid Forecasing Mehods for Muli-Fracured Horizonal Wells: EUR Sensiiviies Moreza Nobakh,* ; Chrisopher Clarkson Universiy of Calgary, 500 Universiy Dr. NW, Calgary, Albera, TN N4, Canada. *Corresponding auhor. Received 6 January 0; acceped 8 February 0 Absrac In his paper, he sensiiviy of expeced ulimae recovery (EUR) for horizonal wells wih muliple fracures o decline exponen is sudied using he simplified forecasing mehod inroduced by Nobakh e al. []. This is very imporan from he reserves evaluaion perspecive due o uncerainy in decline exponen, b. This uncerainy is caused by many facors like desorpion and reservoir/ compleion heerogeneiy. I is found ha in case of imebased forecas (duraion of forecas is specified), he raio of EURs for wo differen specified values of decline exponen depends on he raio of economic life ime of a well o he duraion of linear flow. On he oher hand, his EUR raio depends on he raio of rae a he end of linear flow o economic rae limi for economic limi-based forecas (economic rae limi is specified). Key words: EUR sensiiviies; Muli-fracured horizonal wells; Hybrid forecasing mehods Nobakh, M., & Clarkson, C.R. (0). Hybrid Forecasing Mehods for Muli-Fracured Horizonal Wells: EUR Sensiiviies. Advances in Peroleum Exploraion and Developmen, 3(), -0. Available from: URL: hp:// ne/index.php/aped/aricle/view/j.aped DOI: hp://dx.doi.org/0.3968/j.aped Nomenclaure A Drainage area, f b Hyperbolic decline exponen, dimensionless b Inercep of inverse gas rae versus suare roo of ime plo, /(Mscf/D) B g Gas formaion volume facor, f 3 /scf c g Gas compressibiliy, psi - c Toal compressibiliy, psi - D Decline rae a he end of linear flow, /day D i Decline rae a he sar of hyperbolic forecas, / day EUR Expeced ulimae recovery, Mscf h Ne pay hickness, f k Permeabiliy, md m Slope of inverse gas rae versus suare roo of ime plo, day / /Mscf n The raio of he rae a he end of linear flow o he economic rae limi OGIP Original gas-in-place, Mscf p Pressure, psi p i Iniial pressure, psi p pi Pseudopressure a iniial pressure, psi /cp p pwf Pseudopressure a flowing pressure, psi /cp p wf Flowing pressure, psi Gas rae, Mscf/D i Gas rae a he sar of hyperbolic forecas, Mscf/D D Dimensionless rae, dimensionless Dd Dimensionless rae, dimensionless ( Dye ) Dimensionless rae a he end of linear flow, dimensionless Gas rae a he end of linear flow, Mscf/D Q Gas cumulaive producion, Mscf Q Gas cumulaive producion a he end of linear flow, Mscf Q Volume of gas produced beween he end of linear flow and end of forecas, Mscf S g Gas sauraion, fracion Economic life of a well, days Dye Dimensionless ime, dimensionless ( Dye ) Dimensionless ime a he end of linear flow, dimensionless Duraion of linear flow, days T Reservoir emperaure, R Copyrigh Canadian Research & Developmen Cener of Sciences and Culures

2 Hybrid Forecasing Mehods for Muli-Fracured Horizonal Wells: EUR Sensiiviies x e Reservoir widh, f x f Fracure half-lengh, f y e Reservoir lengh, f Z Gas compressibiliy facor Greek Symbols γ g Reservoir gas specific graviy (air) φ Porosiy, fracion µ g Gas viscosiy, cp INTRODUCTION Shale gas reservoirs have become a significan source of gas supply in Norh America owing o he advancemen of drilling and simulaion echniues o enable commercial developmen. Producion analysis and forecasing of shale gas wells is difficul due o complex reservoir behavior (ex. ulra-low marix permeabiliy, dual porosiy or dual permeabiliy behavior, heerogeneiies a all scales, sressdependen porosiy and permeabiliy ec.) and complex wellbore archiecures and simulaion and compleion mehods. The mos popular mehod for exploiing shale gas reservoirs oday is he use of long horizonal wells compleed wih muliple-fracuring sages; he resuling, ofen complex, hydraulic fracure nework can impar furher complexiy for producion analysis and forecasing. Alhough rigorous mehods o accoun for reservoir and hydraulic fracure nework complexiies in producion daa analysis and forecasing using numerical modeling approaches have been proposed [], i is desirable o seek simpler mehods for analysis and forecasing ha can be used more rouinely by he reservoir engineer. Simple empirical mehods such as Arps decline curve mehod [3] have been used hisorically o forecas mulifracured horizonal wells (MFHW). The Arps mehod, where he rae profile is described by a consan b-value, is limied o cases where operaing condiions are no changing and he well is in boundary-dominaed flow. For igh formaions, he ransien flow may coninue for a long ime and herefore, Arps decline may no be applicable o hese formaions. Srigh and Gordon [4] sudied he producion decline curves of hree lowpermeabiliy gas wells in Piceance basin and found ha linear flow euaions can be used o approximae long erm producion in hese wells. To forecas raes during boundary-dominaed flow, hey suggesed swiching from he forecas based on linear flow projecion o exponenial decline when decline rae reaches a pre-deermined value. Maley [5] sudied he linear flow in igh formaions and concluded ha linear flow can be modeled using Arps decline using b. Hale [6] analyzed he monhly producion raes of more han 6000 hydraulically fracured gas wells locaed in low-permeabiliy reservoirs. He compared exponenial forecasing, logarihmic forecasing and linear flow forecasing. However, no mehod was defined on how o forecas raes during boundarydominaed flow. Waenbarger e al. [7] also focused on linear flow as he flow regime ha lass for a long ime in igh gas reservoirs. They used he reservoir geomery shown in Fig. and developed general soluions for linear flow. They proposed o use he consan produciviy index during boundary-dominaed flow. Kupchenko e al. [8] proposed using b o forecas during linear flow and swich o b 0.5 during boundary-dominaed flow. They deermined he end of linear flow for a hydraulicallyfracured (verical) well, locaed in suare reservoir geomery, based on assumed value for porosiy and some assumpion for gas properies. The power-law and he sreched exponenial decline are new empirical mehods inroduced by Ilk e al. [9] and Valkó [0], respecively. These mehods accoun for changing b-value wih ime. Due o number of parameers in hese echniues, hey produce non-uniue forecass []. Recenly, a semi-empirical mehod was developed ha models changing b-value wih ime in muli-fracured horizonal wells wih uneual fracure half-lenghs and/or non-uniform fracure spacing along horizonal well []. Recognizing ha he flow regime mos ofen observed in muli-fracured horizonal wells is linear flow, Nobakh e al. [] developed a simplified forecasing mehod for horizonal wells wih muliple fracures in igh/shale gas reservoirs. This semi-empirical mehod was developed assuming ha he ransien linear flow is he dominan flow-regime in horizonal wells wih muliple fracures, which is a reasonable assumpion. This flow regime may coninue for several years, and will ulimaely become boundary-dominaed flow a much laer imes. This mehod combines he linear flow ransien period wih hyperbolic decline during boundary-dominaed flow. The slope of he inverse gas rae versus suare roo of ime plo is used o yield a forecas for ransien linear flow and hen applied he Arps hyperbolic decline for boundarydominaed flow. A mehod for esimaing he sar of boundary-dominaed flow was provided ha does no rely on an esimae of marix permeabiliy (ofen elusive for igh formaions because of he difficuly in measuremen) or fracure half-lengh. In his paper, firs, he above-menioned simplified forecasing mehod is briefly reviewed. Secondly, i is shown ha for gas producion under consan flowing pressure in a volumeric reservoir, a some poin during depleion he decline exponen sars o deviae from 0.5. The ime of his ransiion is correlaed o permeabiliy, porosiy, gas properies and reservoir widh. Thirdly, i is shown ha ignoring his ransiion from b 0.5 for forecasing is no affecing he forecas pracically. Finally, he sensiiviy of expeced ulimae recovery o b-value is sudied for boh ime-based and economic-limi based forecass. Copyrigh Canadian Research & Developmen Cener of Sciences and Culures

3 Moreza Nobakh; Chrisopher Clarkson (0). Advances in Peroleum Exploraion and Developmen, 3(), -0. REVIEW OF SIMPLIFIED FORECASTING METHOD The base reservoir geomery, for which he simplified forecasing mehod was developed, is shown in Fig.. The well is a he cener of a recangular reservoir and he fracure exends all he way o he laeral boundaries of he reservoir. This geomery was firs used by Waenbarger e al. [7] for modeling linear flow in igh gas reservoirs. This base geomery was chosen as i is reasonable o assume ha drainage beyond he simulaed region is insignifican for ulra-low marix permeabiliy reservoirs. The consan flowing pressure soluion was used by Nobakh e al. [] for linear flow analysis, as in pracice, igh gas and shale gas wells are produced under high drawdown o maximize producion. The forecasing procedure using his simplified mehod of Nobakh e al. [] follows he 7 seps below: Fracure y e Sep 4. Calculae he producion rae a he end of he linear flow period,. () m + b Sep 5. Calculae he decline rae a he end of he linear flow period, D. m D (3) m + b Sep 6. Forecas raes for using he following euaion: m + b' Sep 7. Assume a value for decline exponen, b, and use E. (5) o forecas raes for >. + bd ( ) /b In he preceding euaions, b and b are in no way relaed. b is he inercep of inverse gas rae versus suare roo of ime plo whereas b is he hyperbolic decline exponen. Alhough his forecasing procedure was developed for he reservoir geomery shown in Fig., i can be used o forecas raes for horizonal well wih muliple fracure sysem shown in Fig. using he area of simulaed reservoir volume (SRV) as he inpu o he simplified mehod insead of drainage area []. (4) (5) x e Figure A Hydraulically Fracured Verical Well in he Cener of a Recangular Reservoir Sep. Plo versus, where is gas rae and is ime, on Caresian coordinaes and place a line hrough he daa corresponding o linear flow. Deermine he slope, m, of his line and is inercep, b. Linear flow can be recognized independenly using semi-log derivaive analysis. Sep. Specify a value for drainage area. Sep 3. Calculae he duraion of linear flow,. Ah zngc im ( ppi - ppwf) ; ^ h E () 00.6T In his euaion, A is he drainage area, h is he ne pay hickness, φ is he reservoir porosiy, µ g is gas viscosiy, c is oal compressibiliy, subscrip i refers o iniial reservoir condiions, p pi and p pwf are pseudopressures a iniial pressure and flowing pressure, respecively and T is he reservoir emperaure. I should be noed ha E. () is derived for field unis. Figure Schemaic of a Homogeneous Muli-Fracured Horizonal Well Compleion. DECLINE EXPONENT (b-value) This forecasing procedure reuires a decline exponen o forecas raes during boundary-dominaed flow (Sep 7). I is well documened in he lieraure ha for volumeric gas reservoirs b 0.5 can be used in hyperbolic decline. Therefore, b 0.5 is a good number o be used for igh gas. However, for shale gas reservoirs, depending on operaing condiions and he shape of desorpion isoherm, desorpion can cause he b value o be above 0.5. b value larger han 0.5 is also expeced in muli-layer reservoirs 3 Copyrigh Canadian Research & Developmen Cener of Sciences and Culures

4 Hybrid Forecasing Mehods for Muli-Fracured Horizonal Wells: EUR Sensiiviies wih no cross flow. This forecasing mehod wih b 0.5 is only applicable o horizonal wells wih homogeneous compleions (i.e., he fracures have he same lengh) []. Ambrose e al. [] proposed ha for a heerogeneous compleion, where he fracure lenghs are no he same (Fig. 3), he sysem needs o be divided ino sub-componens and hen for each subcomponen, he simplified forecasing mehod of Nobakh e al. [] can be applied. I was shown ha for a volumeric gas reservoir afer he firs wo fracures sar o inerfere, he decline exponen value sars o deviae from b and when he whole sysem is in boundary-dominaed flow he decline exponen value reaches b 0.5. Using some real daa, Ambrose e al. [] showed ha Nobakh e al. mehod [] wih a decline exponen beween 0.5 and (afer he end of linear flow) can be used o forecas raes from heerogeneous compleions. The decline exponen ha should be used depends on he compleion geomery. This shows ha when using he simplified forecasing mehod for shale gas reservoirs, here migh be uncerainies in decline exponen and herefore, i is imporan o sudy he relaive impac of b-value used afer he end of linear flow on long-erm producion forecas for linear-flow dominaed wells. Figure 3 Schemaic of a Heerogeneous Muli-Fracured Horizonal Well Compleion. Evoluion of Decline Exponen During Depleion I is observed in he lieraure ha for consan flowing pressure producion from a single layer volumeric gas reservoir, he decline exponen during boundarydominaed flow is going o deviae from b 0.5 and evenually reduces o b 0 [,3]. I is imporan o find he ime ha his deviaion sars and deermine if i pracically affecs he expeced ulimae recovery (EUR) calculaed from Arps decline. In his secion, his ime is deermined for he reservoir geomery shown in Fig.. Firs, he consan-pressure ype curve for his reservoir geomery was developed using he linear flow soluion for a slighly compressible fluid during linear flow [7] and hyperbolic decline wih b 0.5 during boundarydominaed flow. The ploing forma for his ype curve, shown in Fig. 4, is ye Dd agains [7] Dye, where: D xe 4.Bn D (6) kh^pi- pwfh Dye k (7) zn cy e The manner in which he ype curve is generaed is presened in Appendix A. This ploing forma gives only one ype curve for geomery shown in Fig. raher han xf families of ype curves wih differen values of [7]. As ye shown in Appendix A, Dye a he end of he linear flow period (i.e., a ). To calculae he ime ha he decline exponen sars o deviae from b 0.5, a oal number of 0 simulaion cases were generaed for consan flowing pressure condiion. The inpu daa for he numerical simulaion cases are given in Table. The blank cells in his able indicae ha he value for ha parameer is he same as ha of Case. The rae versus ime daa poins for hese cases was ransformed o Dd versus Dye forma and ploed on he ype curve. Fig. 4 also shows his daa ploed on he ype curve for Cases 5. This figure shows ha for all cases, he b value sars o gradually decrease from b 0.5 around Dye 0.4. As Dye a, i can be concluded ha he b value sars o deviae from b 0.5 around 6. The simulaed daa during linear flow do no fall on he half-slope par of he ype curve in Fig. 4. This is because we simply used ime insead of pseudoime for ploing he daa on he ype curve [4,5]. Copyrigh Canadian Research & Developmen Cener of Sciences and Culures 4

5 Moreza Nobakh; Chrisopher Clarkson (0). Advances in Peroleum Exploraion and Developmen, 3(), -0 Table Inpu Parameers Used for Numerical Simulaion for Differen Cases, he blank cells in his able indicae ha he value for ha parameer is he same as ha of Case Case p i (psi),000,000 4,000 T ( F) 0 h (f) 00 φ (%) 0 S g (%) 00 γ g 0.65 p wf (psi) ,000 x f (f) 50 x e (f) 500 y e (f) 5,000 0,000,500 k (md) Type Curve Case Case Case 3 Case 4 Case 5 Dd Dye Figure 4 The Rae Versus Time Daa Poins for Cases 5 Ploed on he Dd Versus Dye Type Curve To invesigae if he gradual deviaion of decline exponen from b 0.5 affecs he EUR, he following assumpions are made o simplify he problem:. The gas is ideal (Z ). Using he definiion of gas compressibiliy, his assumpion leads o: dz cg (8) p Z dp p. Gas viscosiy is no changing wih pressure. Using he definiion of pseudopressure and ideal gas assumpion, p pi (9) ppi # dp # pdp ngz n n gi 3. Toal compressibiliy is dominaed by gas compressibiliy, i.e., c Sgcg (0) 4. Pseudopressure a he consan flowing pressure is negligible compared o ha a he iniial pressure. In oher gi words, ppi - ppwf. ppi () 5. The inercep of inverse gas rae versus suare roo of ime plo is ignored. In oher words, he rae can be calculaed using he following euaion during linear flow period: () m Using E. (), he cumulaive producion a he end of linear flow (i.e., ) is: Q d d # # 0 m m (3) 0 Combining Es. (), (8) () and (3) and using he definiion of gas formaion volume 5 Copyrigh Canadian Research & Developmen Cener of Sciences and Culures

6 Hybrid Forecasing Mehods for Muli-Fracured Horizonal Wells: EUR Sensiiviies facor 0.08ZT i Bgi wih Z i (ideal gas assumpion), pi Q 0. 8OGIP (4) where OGIP is he original gas-in-place in Mscf. This means ha under assumpions menioned above, 8% of he oal gas in he reservoir geomery shown in Fig. is produced during linear flow. The relaionship beween cumulaive producion and ime for hyperbolic decline is as follows: i - Q bd b 7 - ] + i g A (5) ] - bgdi where i is he producion rae a he sar of he hyperbolic forecas period, b is he hyperbolic decline exponen, D i is decline rae corresponding o i and is ime since he sar of he hyperbolic forecas. Since hyperbolic forecas sars afer he end of linear flow, we will use he hyperbolic decline euaion in he form shown in E. (6), which is obained from E. (5) by reiniializing he ime a : T Q -^+ bd] - ] - bgd - b 7 gh A (6) In his euaion, Q is he volume produced during boundary-dominaed flow using hyperbolic decline. Using and D from E. () and E. (3) respecively (wih b 0), he volume produced beween and 6 (i.e., when he decline exponen sars o deviae from b 0.5) using b 0.5 becomes:. D Q ( b 0.5) (7) m Combining Es. (7) and (B-4), Q ( b 0. 5) 0. 33OGIP (8) This shows ha 3% of he original gas-in-place is produced beween and 6 using hyperbolic decline wih b 0.5. In oher words, almos 60% of he oal gas will be produced by he ime he decline exponen sars o deviae from b 0.5. As shown in Appendix B, 5% of he oal gas will be recovered afer 6 using b 0.5 o forecas for infinie ime, whereas % of he oal gas will be produced afer 6 using exponenial decline. Therefore, he volume of gas produced afer 6 is beween % and 5% of he original gas-in-place, as here is a ransiion from b 0.5 o b 0. This shows ha for all pracical purposes, we can ignore he change in decline exponen wih ime for 6. Noes: Alhough he calculaions presened above and in Appendix B are based on ideal gas assumpion (Z ), hey are valid for cases where Z is no changing drasically wih pressure. The cumulaive producions a differen imes presened above and in Appendix B are obained based on he duraion of linear flow from E. (). Alhough i is repored in he lieraure ha E. () is an approximaion for he duraion of linear flow [6 8], i is used for simpliciy in his sudy. 3. SENSITIVITY OF FORECAST TO b-value As menioned in he previous secion, afer he fracures sar o inerfere in a muli-fracured horizonal well, he decline exponen can vary beween 0.5 and due o differen fracure lenghs and/or uneual fracure spacing along he horizonal well (heerogeneous compleions). The simplified forecasing procedure can be used o invesigae he sensiiviy of EUR o decline exponen, b, which is being used for forecasing during boundarydominaed flow. 3. EUR Based upon Time E. (6) shows he volume produced afer he end of linear flow according o a hyperbolic decline forecas. To obain he expeced ulimae recovery (EUR) a he end of he forecas (when > ), he volume produced using E. (6) can be added o he cumulaive producion a he end of linear flow calculaed from E. (3). Combining Es. (), (3), (3) and (6): - b EUR < + < - b b + b m - b -ll FF (9) Here, is he economic life of he well. Using E. (9), i can be shown ha he raio beween EUR values obained for wo differen values of decline exponen depends on he value of, given ha he wo decline exponens are specified. EUR( b.3) and EUR( b 0.5) EUR( b 0.8) versus EUR( b 0.5) are shown in Fig. 5 and Fig. 6, respecively. These plos can be used o sudy he sensiiviy of forecas o he value of decline exponen. For example, i can be seen from Fig. 6 ha for 30, EUR for b 0.8 is almos 0% higher han EUR for b 0.5. I should be noed ha b 0.5 is for homogeneous compleion, b 0.8 is for slighly heerogeneous compleion and b.3 is for very heerogeneous compleion []. Copyrigh Canadian Research & Developmen Cener of Sciences and Culures 6

7 Moreza Nobakh; Chrisopher Clarkson (0). Advances in Peroleum Exploraion and Developmen, 3(), -0 EUR (b.3) / EUR (b 0.5) / Figure 5 EUR( b. 3) Plo of EUR( b 0. 5 ) Versus EUR (b 0.8) / EUR (b 0.5) / Figure 6 Plo of EUR( b 0. 8) Versus EUR( b 0. 5) 3. EUR Based upon Economic Limi When economic limi, f, is known, he volume produced using hyperbolic decline is calculaed using he following euaion: Q b i - b -b 7i - f A (0) - b Di ] g Therefore, he volume produced during boundarydominaed flow Q for hyperbolic decline is: b -b b T Q f A () ] - bgd Using and D from E. () and E. (3) respecively (wih b 0) and assuming n is he raio of he rae a he end of linear flow o he economic rae limi (i.e., n f wih n ), he EUR for hyperbolic decline becomes: b- EUR : + ] - n gd () m - b Using E. (), i can be shown ha he raio beween EUR values obained for wo differen values of decline exponen depends on he value of n, given ha he wo decline exponens are specified. When EUR is being calculaed based upon economic limi, E. () can be used o invesigae he sensiiviy of EUR o decline exponen. CONCLUSIONS This paper applied he simplified forecasing mehod of Nobakh e al. [] o sudy he sensiiviy of expeced ulimae recovery (EUR) o decline exponen used afer he end of linear flow. This is imporan for reserve evaluaion because of uncerainy in decline exponen due o facors like desorpion and heerogeneiy in compleion. I was shown ha for gas producion under consan flowing pressure in a volumeric reservoir, a some poin during depleion he decline exponen sars o deviae from 0.5. The ime a which his deviaion happens is almos 6 for he reservoir geomery shown in Fig., where is he duraion of linear flow. Using some assumpions, i was shown ha ignoring he gradual decrease in b-value afer 6 will no pracically affec he EUR. Finally, he sensiiviy of EUR o b-value is sudied for boh imebased and economic limi-based forecass. I is found ha for wo differen specified values of decline exponen, he raio beween heir EURs depends on he raio of economic life of he well o he duraion of linear flow for ime-based forecas and he raio of he rae a he end of linear flow o he economic rae limi for economic limibased forecas. ACKNOWLEDGEMENTS The auhors would like o hank ConocoPhillips for heir suppor of his research. Chris Clarkson would like o acknowledge Encana for suppor of his Chair posiion in Unconvenional Gas a he Universiy of Calgary, Deparmen of Geoscience. Finally, boh auhors would like o hank Fekee Associaes Inc., paricularly Louis Maar, for fruiful discussions on he subjec of raeransien analysis. REFERENCES [] Nobakh, M., Maar, L., Moghadam, S., & Anderson, D. M. (00). Simplified Ye Rigorous Forecasing of Tigh/ Shale Gas Producion in Linear Flow. Paper SPE 3365 presened a he SPE Wesern Regional Meeing, Anaheim, California, 7 9 May. DOI: 0.8/3365-MS. [] Cipolla, C. L., Williams, M. J., Weng, X., & Maxwell, S. (00). Hydraulic Fracure Monioring o Reservoir Simulaion: Maximizing Value. Paper SPE presened a he SPE Annual Technical Conference and Exhibiion, Florence, Ialy, 9 Sepember. DOI: 7 Copyrigh Canadian Research & Developmen Cener of Sciences and Culures

8 Hybrid Forecasing Mehods for Muli-Fracured Horizonal Wells: EUR Sensiiviies 0.8/33877-MS. [3] Arps, J. J. (945). Analysis of Decline Curves. Trans., AIME, 60, [4] Srigh, D. H., & Gordon, J. I. (983). Decline Curve Analysis in fracured Low Permeabiliy Gas Wells in he Piceance Basin. Paper SPE 640 presened a he SPE/ DOE Low Permeabiliy Gas Reservoirs Symposium, Denver, Colorado, 4 6 March. DOI: 0.8/640-MS. [5] Maley, S. (985). The Use of Convenional Decline Curve Analysis in Tigh Gas Well Applicaions. Paper SPE 3898 presened a he SPE/DOE Low Permeabiliy Gas Reservoirs Symposium, Denver, Colorado, 9 March. DOI: 0.8/3898-MS. [6] Hale, B. W. (986). Analysis of Tigh Gas Well Producion Hisories in he Rocky Mounains. SPE Producion Engineering, (4), SPE-639-PA. DOI: 0.8/639-PA. [7] Waenbarger, R. A., El-Banbi, A. H., Villegas, M. E., & Maggard, J. B. (998). Producion Analysis of Linear Flow ino Fracured Tigh Gas Wells. Paper SPE 3993 presened a SPE Rocky Mounain Regional/Low Permeabiliy Reservoirs Symposium and Exhibiion, Denver, Colorado, 5 8 April. DOI: 0.8/3993-MS. [8] Kupchenko, C. L., Gaul, B. W., & Maar, L. (009). Tigh Gas Producion Performance Using Decline Curves. Paper SPE 499 presened a he CIPC/SPE Gas Technology Symposium, Calgary, Albera, 6 9 June. DOI: 0.8/499-MS. [9] Ilk, D., Rushing, J. A., Perego, A. D., & Blasingame, T. A. (008). Exponenial vs. Hyperbolic Decline in Tigh Gas Sands Undersanding he Origin and Implicaions for Reserve Esimaes Using Arps' Decline Curves. Paper SPE 673, presened a he Annual Technical Conference and Exhibiion, Denver, Colorado, 4 Sepember. DOI: 0.8/673-MS. [0] Valkó, P. P. (009). Assigning Value o Simulaion in he Barne Shale: A Simulaneous Analysis of 7000 Plus Producion Hisories and Well Compleion Records. Paper SPE 9369 presened a SPE Hydraulic Fracuring Technology Conference, The Woodlands, Texas, 9 January. DOI: 0.8/9369-MS. [] Ambrose, R. J., Clarkson, C. R., Youngblood, J., Adams, R., Nguyen, P., Nobakh, M., & Biseda, B. (0). Life-Cycle Decline Curve Esimaion for Tigh/Shale Gas Reservoirs. Paper SPE 4059 presened a he SPE Hydraulic Fracuring Technology Conference and Exhibiion, The Woodlands, Texas, 4 6 January. DOI: 0.8/4059- MS. [] Nobakh, M., Ambrose, R. J., & Clarkson, C. R. (0). Effec of Heerogeneiy in a Horizonal Well Wih Muliple Fracures on he Long-Term Forecas in Shale Gas Reservoirs. Paper CSUG/SPE presened a he Canadian Unconvenional Resources Conference, Calgary, Albera, Canada, 5 7 November. DOI: 0.8/ MS. [3] Okuszko, K. E., Gaul, B. W., & Maar, L. (007). Producion Decline Performance of CBM Wells. Paper CIPC , presened a Canadian Inernaional Peroleum Conference, Calgary, Albera, 4 June. DOI: 0.8/ [4] Nobakh, M., & Clarkson, C. R. (0). A New Analyical Mehod for Analyzing Producion Daa from Shale Gas Reservoirs Exhibiing Linear Flow: Consan Pressure Producion. Paper SPE presened a he Norh American Unconvenional Gas Conference, The Woodlands, Texas, 6 June. DOI: 0.8/43989-MS. [5] Nobakh, M., & Clarkson, C. R. (0). A New Analyical Mehod for Analyzing Producion Daa from Shale Gas Reservoirs Exhibiing Linear Flow: Consan Rae Producion. Paper SPE presened a he Norh American Unconvenional Gas Conference, The Woodlands, Texas, 6 June. DOI: 0.8/43990-MS. [6] Ibrahim, M., & Waenbarger R. A. (005). Rae Dependence of Transien Linear Flow in Tigh Gas Wells. Paper CIPC presened a Canadian Inernaional Peroleum Conference, Calgary, Albera, 7 9 June. DOI: 0.8/ [7] Ibrahim, M., & Waenbarger R. A. (006). Analysis of Rae Dependence in Transien Linear Flow in Tigh Gas Wells. Paper SPE presened a he Abu Dhabi Inernaional Peroleum Exhibiion and Conference, Abu Dhabi, UAE, 5 8 November. DOI: 0.8/00836-MS. [8] Nobakh, M., & Clarkson, C. R. (0). Esimaion of Conaced and Original Gas-in-Place for Low Permeabiliy Reservoirs Exhibiing Linear Flow. Paper CSUG/SPE presened a he he Canadian Unconvenional Resources Conference, Calgary, Albera, Canada, 5 7 November. DOI: 0.8/49398-MS. Copyrigh Canadian Research & Developmen Cener of Sciences and Culures 8

9 Moreza Nobakh; Chrisopher Clarkson (0). Advances in Peroleum Exploraion and Developmen, 3(), -0 APPENDIX A: Duraion of Linear Flow and Developmen of Figure 4 When he well is producing under consan flowing pressure in he reservoir geomery shown in Fig., he disance of invesigaion, y, can be obained from he following euaion during he linear flow period [7] : y k (A-) 0.59 ( zngc) i According o his euaion, he end of linear flow is given by: ye k 0.59, (A-) ] zngcgi where y e is reservoir lengh and is he duraion of linear flow. E. (A-) can be rewrien as follows: d i n (A-3) y e ( zngc) # 0.59 k Using calculaed from E. (A-3) in E. (7), Dye a he end of linear, ( Dye ), becomes: ( ) Dye k zncy e (A-4) In oher words, Dye a he end of linear flow. Noe ha his value is differen from Dye 0.5 Waenbarger e al. [7] derived simply because in heir work y e was half of reservoir lengh, whereas in his sudy y e is he reservoir lengh. To generae he ype curve shown in Fig. 4 for he reservoir geomery in Fig., we used he following euaion, which is linear flow soluion for consan flowing pressure, for Dye (i.e., duraion of linear flow). r r (A-5) Dye Dd For Dye (i.e., boundary-dominaed flow), we used he following euaion (wih b 0.5), which is he exension of Nobakh e al. [] mehod in dimensionless form: Dd + bd ( Dd) ( - ( ) ) (A-6) 6 /b Dye Here, ( Dd ) is value of Dd a Dye ( Dye ) (( Dd ) 0.78) and D is as follows: D 8 ( ) Dye (A-7) APPENDIX B: Calculaion of Volume Produced During Differen Time Inervals For he reservoir geomery shown in Fig., he end of linear flow,, is [,7] : ^ h Ah zngc m ( p p ) i pi - pwf ; E (B-) 00.6T Combining Es. (B-), (8) () resuls in: Ahz ngi Sg pi AhzSg pi m m (B-) 00.6T p n 00.6 T i gi Using he definiion of gas formaion volume facor 0.08ZiT Bgi wih Z i (ideal gas assumpion), p i pi 0.08 T B (B-3) gi Combining Es. (B-) and (B-3) and using he definiion of AhzSg OGIP 0.00 (OGIP is in Mscf), Bgi Ahz Sg OGIP (B-4) m 00.6 B Using he value of m gi from E. (B-4) ino E. (3), Q 0.8OGIP (B-5) The rae a 6, can be calculaed by combing Es. (3) and (5) (wih b 0, i.e., zero inercep on inverse gas rae versus suare roo of ime plo): (B-6) Using hyperbolic decline wih b 0.5 beween and 6, he decline rae a 6 can be calculaed using: D D c m (B-7) Combining Es. (B-6) and (B-7), D D (B-8) The volume produced afer 6 using hyperbolic decline wih b 0.5 and exponenial decline can be calculaed using Es. (B-9) and (B-0), respecively: b 6 b T Q 6 - ( - bd ) 6 ^ h (B-9) - -b f 6 f Q (B-0) D 6 Using f 0, Es. (B-9) and (B-0) will change o:. 784 (B-) 6 Q D 6 D (B-) 6 Q D 6 D 9 Copyrigh Canadian Research & Developmen Cener of Sciences and Culures

10 Hybrid Forecasing Mehods for Muli-Fracured Horizonal Wells: EUR Sensiiviies Using Es. (), (3) and (B-4), Q 0. 46OGIP (B-3) Q 0. 3OGIP (B-4) I can be concluded from Es. (B-3) and (B-4) ha 5% of he oal original gas-in-place is produced afer 6 using hyperbolic decline wih b 0.5, whereas % of he oal original gas-in-place is produced afer 6 using exponenial decline. Copyrigh Canadian Research & Developmen Cener of Sciences and Culures 0