A Quadratic Cumulative Production Model for the Material Balance of Abnormally-Pressured Gas Reservoirs F.E. Gonzalez M.S.

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1 Natural as Engneerng A Quadratc Cumulatve Producton Model for the Materal Balance of Abnormally-Pressured as Reservors F.E. onale M.S. Thess (2003) T.A. Blasngame, Texas A&M U. Deartment of Petroleum Engneerng Texas A&M Unversty College Staton, TX t-blasngame@tamu.edu Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde

2 Executve Summary: "/- 2 " Relaton (/3) The rgorous relaton for the materal balance of a dry gas reservor system s gven by Fetkovch, et al. as: ce( )( ) nj W R sw ( W B g B w W nj B w W Elmnatng the water nflux, water roducton/njecton, and gas njecton terms; defnng =c e ()( -) and assumng that <, then rearrangng gves the follow-ng result: e ) 2 ( ) Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 2

3 Executve Summary: "/- 2 " Relaton (2/3) Smulated Dry as Reservor Case: Abnormal Pressure: Volumetrc, dry gas reservor wth c f () (from Fetkovch). Note extraolaton to the "aarent" gas-n-lace (revous aroaches). Note comarson of data and the new "Quadratc Cumulatve Producton" ("/- 2 ") model. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 3

4 Executve Summary: "/- 2 " Relaton (3/3) Anderson L Reservor Case: Abnormal Pressure: South Texas (USA) gas reservor wth abnormal ressure. Benchmark lterature case. Note erformance of the new "Quadratc Cumulatve Producton" model. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 4

5 Presentaton Outlne: Executve Summary Objectves and Ratonale Rgorous technque for abnormal ressure analyss. Develoment of the /- 2 model Dervaton from the rgorous materal balance. Valdaton Feld Examles Case Dry gas smulaton (c f () from Fetkovch). Case 3 Anderson L (South Texas, USA). Demonstraton (MS Excel Anderson L case) Summary Recommendatons for Future Work Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 5

6 Objectves and Ratonale: Objectves: Develo a rgorous functonal form (.e., a model) for the / vs. behavor demonstrated by a tycal abnormally ressured gas reservor. Develo a sequence of lottng functons for the analyss of / data (multle lots). Provde an exhaustve valdaton of ths new model usng feld data. Ratonale: The analyss of / data for abnormally ressured gas reservors has evolved from emrcal models and dealed assumtons (e.g., c f ()= constant). We would lke to establsh a rgorous aroach one where any aroxmaton s based on the observaton of some characterstc behavor, not smly a mathematcal/grahcal convenence. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 6

7 Develoment: The /- 2 Model Concet: Use the rgorous materal balance relaton gven by Fetkovch, et al. for the case of a reservor where c f () s NOT resumed constant. Use some observed lmtng behavor to construct a sem-analytcal relaton for / behavor. Imlementaton: Develo and aly a seres of data lottng functons whch clearly exhbt unque behavor relatve to the / data. Use a "mult-lot" aroach whch s based on the dynamc udatng of the model soluton on each data lot. Develo a "dmensonless" tye curve aroach that can be used to valdate the model and estmate. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 7

8 Base as Materal Balance Model: (/3) The rgorous relaton for the materal balance of a dry gas reservor system s gven by Fetkovch, et al. as: ce( )( ) nj W R sw ( W B g B w W nj B w W Elmnatng the water nflux, water roducton/njecton, and gas njecton terms, then rearrangng gves the followng defnton: / / [where ce ( )( )] ( ) e ) Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 8

9 Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Consderng the condton where: Then we can use a geometrc seres to reresent the D term n the governng materal balance. The arorate geometrc seres s gven by: D ) ( 3 2 x... x x x x / ) ( ) ( or, for our roblem, we have: 2 ) ( Substtutng ths result nto the materal balance relaton, we obtan: Base as Materal Balance Model: (2/3) Slde 9

10 Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. A more convenent form of the /-cumulatve model s: We note that these arameters resume that s constant. Presumng that s lnear wth, we can derve the followng form: ) ( 3 2 ) ( ) ( b b a a where a b Obvously, one of our objectves wll be the study of the behavor of vs. (based on a rescrbed value of ). 2 Base as Materal Balance Model: (3/3) Slde 0

11 Alcaton: - Performance (Case ) (/2) a. Case : Smulated Performance Case Plot of versus (requres an estmate of gas-n-lace). Note the aarent lnear trend of the data. Recall that =c e ( -). b. Case : Smulated Performance Case Plot of / versus. The constant and lnear trends match well wth the data essentally a confrmaton of both models. Smulated Dry as Reservor Case: Abnormal Pressure A lnear trend of vs. s reasonable and should yeld an accurate model. s aroxmated by a constant value wthn the trend. A hyscal defnton of s elusve =c e ()( -) mles that has unts of /volume, whch suggests s a scalng varable for. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde

12 Alcaton: - Performance (Case 3) (2/2) a. Case 3: Anderson L Reservor Case (South Texas, USA) Plot of versus (requres an estmate of gas-n-lace). Some data scatter exsts, but a lnear trend s evdent (recall that =c e ( )( -)). b. Case 3: Anderson L Reservor Case (South Texas, USA) Plot of / versus. Both models are n strong agreement. Anderson L Reservor Case: Abnormal Pressure Feld data exhbt some scatter, method s relatvely tolerant of data scatter. Constant aroxmaton based on the "best ft" of several data functons. The lnear aroxmaton for s reasonable (should favor later data). Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 2

13 Valdaton: The /- 2 Model Methodology: All analyses are "dynamcally" lnked n a sreadsheet rogram (MS Excel). Therefore, all analyses are consstent should note that some functons/ lots erform better than others but the model results are the same for every analyss lot. Valdaton: Illustratve Analyses /- 2 lottng functons based on the roosed materal balance model. - erformance lots used to calbrate analyss. an analyss resumes 2-straght lne trends on a /- lot for an abnormally ressured reservor. D - D tye curve aroach use /- 2 materal balance model to develo tye curve soluton ths aroach s most useful for data valdaton. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 3

14 Plottng Functons: /- 2 Case (/5) Δ( / ) vs. Δ( / ) vs. a. b. c. 0 Δ( / ) d vs. 2 Δ( / ) d vs. d. e. 0 Δ( / ) Δ( / ) d vs. f. Δ( / ) Δ( / ) d vs. 0 0 Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 4

15 Plottng Functons: - Case (2/5) a. Case : Smulated Performance Case Plot of c e ()( -) versus (requres estmate of ). b. Case : Smulated Performance Case Plot of /c e ()( -) versus (requres estmate of ). c. Case : Smulated Performance Case Plot of versus (requres estmate of ). d. Case : Smulated Performance Case Plot of versus / (requres estmate of ). Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 5

16 Plottng Functons: - Case (3/5) Smulated Dry as Reservor Case: Abnormal Pressure: Summary / lot for =constant and =lnear cases. ood comarson of trends, =lnear trend aears slghtly conservatve as t emerges from data trend but both solutons aear to yeld same estmate. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 6

17 an-blasngame Analyss (200): Case (4/5) a. Case : Smulated Performance Case an Plot c e ()( -) versus (/)/( / ) (requres est. of ). c. Case : Smulated Performance Case an Plot 3 (/) versus (results lot). b. Case : Smulated Performance Case an Plot 2 (/)/( / ) versus ( /) (requres est. of ). an-blasngame Analyss: Aroach consders the "match" of the c e ()( -) (/)/( / ) data and "tye curves." Assumes that both abnormal and normal ressure / trends exst. Straght-lne extraolaton of the "normal" / trend used for. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 7

18 Tye Curve Aroach: D - D Case (5/5) a. D - D Tye curve soluton based on the /- 2 model. D = [( / )-(/)]/( / ) and D = / both D and D functons are lotted. b. Case : Smulated Performance Case Tye curve analyss of (/)- data, ths case s "force matched" to the same results as all of the other lottng functons. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 8

19 Plottng Functons: /- 2 Case 3 (Anderson L) (/5) Δ( / ) vs. Δ( / ) vs. a. b. c. 0 Δ( / ) d vs. 2 Δ( / ) d vs. d. e. 0 Δ( / ) Δ( / ) d vs. f. Δ( / ) Δ( / ) d vs. 0 0 Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 9

20 Plottng Functons: - Case 3 (Anderson L) (2/5) a. Case 3: Anderson L (South Texas) Plot of c e ()( -) versus (requres estmate of ). b. Case 3: Anderson L (South Texas) Plot of /c e ()( -) versus (requres estmate of ). c. Case 3: Anderson L (South Texas) Plot of versus (requres estmate of ). d. Case 3: Anderson L (South Texas) Plot of versus / (requres estmate of ). Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 20

21 Plottng Functons: - Case 3 (Anderson L) (2/5) Case 3: Anderson L Reservor (South Texas (USA)) Summary / lot for =constant and =lnear cases. ood comarson, =constant and =lnear cases n good agreement. Data trend s very consstent. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 2

22 an-blasngame Analyss (200): Case 3 (Anderson L) (4/5) a. Case 3: Anderson L Reservor an Plot c e ()( -) versus (/)/( / ) (requres est. of ). c. Case 3: Anderson L Reservor an Plot 3 (/) versus (results lot). b. Case 3: Anderson L Reservor an Plot 2 (/)/( / ) versus ( /) (requres est. of ). an-blasngame Analyss: We note an excellent "match" of the c e ()( -) (/)/( / ) data and the "tye curves." Both the abnormal and normal ressure / trends aear accurate and consstent. Straght-lne extraolaton of the "normal" / trend used for. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 22

23 Tye Curve Aroach: D - D Case 3 (Anderson L) (5/5) Case 3: Anderson L Reservor (South Texas (USA)) D - D tye curve soluton matched usng feld data. Note the "tal" n the D trend for small values of D (common data event). "Force matched" to the same results as the other lottng functons. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 23

24 Examle Analyss Usng MS Excel: Case 3 Case 3 Anderson L (South Texas (USA)) Lterature standard case. A 3-well reservor, delmted by faults. ood qualty data. Evdence of overressure from statc ressure tests. Analyss: (Imlemented usng MS Excel) /- 2 lottng functons. - erformance lots. an analyss (2-straght lne trends on a /- lot). D - D tye curve aroach. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 24

25 Summary: (/3) Develoed a new /- 2 materal balance model for the analyss of abnormally ressured gas reservors: 2 ( ) where: c e ( )( The -functon s resumed (based on grahcal comarsons) to be ether constant, or aroxmately lnear wth. For the =constant case, we have: ) 2 ( ) Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 25

26 Summary: (2/3) Base relaton: /- 2 form of the gas materal balance a. Plottng Functon : (quadratc) Δ( / ) b. Plottng Functon 2: (lnear) vs. Δ( / ) vs. c. Plottng Functon 3: (quadratc) 2 ( ) 2 0 Δ( / ) 0 d. Plottng Functon 4 : (lnear) Δ( / ) d vs. e. Plottng Functon 5 : (quadratc) Δ( / ) d vs. f. Plottng Functon 6: (lnear) 0 Δ( / ) d vs. Δ( / ) 0 Δ( / ) d vs. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 26

27 Summary: (3/3) The lottng functons develoed n ths work have been valdated as tools for the analyss reservor erformance data from abnormally ressured gas reservors. Although the straght-lne functons (PF 2, PF 4, and PF 6 ) could be used ndeendently, but we recommend a combned/smultaneous analyss. The - lots are useful for checkng data consstency and for gudng the selecton of the -value. These lots reresent a vvd and dynamc vsual balance of all of the other analyses. The an analyss sequence s also useful for orentng the overall analyss artcularly the c e ()( -) versus (/)/( / ) lot. The D - D tye curve s useful for orentaton artcularly for estmatng the or ( D ) value. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 27

28 Recommendatons for Future Work: Consder the extenson of ths methodology for cases of external drve energy (e.g., water nflux, gas njecton, etc.). Contnue the valdaton of ths aroach by alyng the methodology to addtonal feld cases. Imlementaton nto a stand-alone software. Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 28

29 Natural as Engneerng A Quadratc Cumulatve Producton Model for the Materal Balance of Abnormally-Pressured as Reservors F.E. onale M.S. Thess (2003) (End of Presentaton) T.A. Blasngame, Texas A&M U. Deartment of Petroleum Engneerng Texas A&M Unversty College Staton, TX t-blasngame@tamu.edu Natural as Engneerng 26 May - 30 May 204 (U. Kavala/REECE) Tom BLASINAME t-blasngame@tamu.edu Texas A&M U. Slde 29

A Quadratic Cumulative Production Model for the Material Balance of Abnormally-Pressured Gas Reservoirs F.E. Gonzalez M.S.

A Quadratic Cumulative Production Model for the Material Balance of Abnormally-Pressured Gas Reservoirs F.E. Gonzalez M.S. Formaton Evaluaton and the Analyss of Reservor Performance A Quadratc Cumulatve Producton Model for the Materal Balance of Abnormally-Pressured as Reservors F.E. onale M.S. Thess (2003) T.A. Blasngame,