A Quadratic Cumulative Production Model for the Material Balance of Abnormally-Pressured as Reservoirs F.E. onale, M.S. Thesis Defense 7 October 2003 Deartment of Petroleum Engineering Texas A&M University College Station, TX 77843-36 A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide
Executive Summary "/- 2 " Relation (/4) The rigorous relation for the material balance of a dry gas reservoir system is given by Fetkovich, et al. as: ce ( )( i i i ) inj i i W Rsw 5. 65 ( WBw Bg W inj Bw W e ) Eliminating the water influx, water roduction/injection, and gas injection terms; defining =c e ()( i -) and assuming that <, then rearranging gives the following result: i ( ) 2 i A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 2
Executive Summary "/- 2 " Relation (2/4) Simulated Dry as Reservoir Case Abnormal Pressure: Volumetric, dry gas reservoir with c f () (from Fetkovich). Note extraolation to the "aarent" gas-in-lace (revious aroaches). Note comarison of data and the new "Quadratic Cumulative Production" model. A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 3
Executive Summary "/- 2 " Relation (3/4) Anderson L Reservoir Case Abnormal Pressure: South Texas (USA) gas reservoir with abnormal ressure. Benchmark literature case. Note erformance of the new "Quadratic Cumulative Production" model. A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 4
Presentation Outline Executive Summary Objectives and Rationale Rigorous technique for abnormal ressure analysis. Develoment of the /- 2 model Derivation from the rigorous material balance. Validation Field Examles Case Dry gas simulation (c f () from Fetkovich). Case 3 Anderson L (South Texas, USA). Demonstration (MS Excel Anderson L case) Summary Recommendations for Future Work A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 5
Objectives and Rationale Objectives: Develo a rigorous functional form (i.e., a model) for the / vs. behavior demonstrated by a tyical abnormally ressured gas reservoir. Develo a sequence of lotting functions for the analysis of / data (multile lots). Provide an exhaustive validation of this new model using field data. Rationale: The analysis of / data for abnormally ressured gas reservoirs has evolved from emirical models and idealied assumtions (e.g., c f ()= constant). We would like to establish a rigorous aroach one where any aroximation is based on the observation of some characteristic behavior, not simly a mathematical/grahical convenience. A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 6
Develoment of the /- 2 model Concet: Use the rigorous material balance relation given by Fetkovich, et al. for the case of a reservoir where c f () is NOT resumed constant. Use some observed limiting behavior to construct a semi-analytical relation for / behavior. Imlementation: Develo and aly a series of data lotting functions which clearly exhibit unique behavior relative to the / data. Use a "multilot" aroach which is based on the dynamic udating of the model solution on each data lot. Develo a "dimensionless" tye curve aroach that can be used to validate the model and estimate. A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 7
/-Cumulative Model: (/3) The rigorous relation for the material balance of a dry gas reservoir system is given by Fetkovich, et al. as: ce ( )( i i i ) inj i i W Rsw 5. 65 ( WBw Bg W inj Bw W e ) Eliminating the water influx, water roduction/injection, and gas injection terms, then rearranging gives the following definition: i / i / [where ce ( )( i )] ( ) A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 8
/-Cumulative Model: (2/3) Considering the condition where: D Then we can use a geometric series to reresent the D term in the governing material balance. The aroriate geometric series is given by: / 2 x x x x 3... ( x ) or, for our roblem, we have: ( ) ( i ( ) 2 i ) Substituting this result into the material balance relation, we obtain: A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 9
A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 0 A more convenient form of the /-cumulative model is: We note that these arameters resume that is constant. Presuming that is linear with, we can derive the following form: /-Cumulative Model: (3/3) i i ) ( i i 3 2 ) ( ) ( i i i i i i i i b b a a wherea b Obviously, one of our objectives will be the study of the behavior of vs. (based on a rescribed value of ). 2 i i
- Performance (Case ) (/2) a. Case : Simulated Performance Case Plot of versus (requires an estimate of gas-inlace). Note the aarent linear trend of the data. Recall that =c e ( -). b. Case : Simulated Performance Case Plot of / versus. The constant and linear trends match well with the data essentially a confirmation of both models. Simulated Dry as Reservoir Case Abnormal Pressure: A linear trend of vs. is reasonable and should yield an accurate model. is aroximated by a constant value within the trend. A hysical definition of is elusive =c e ()( i -) imlies that has units of /volume, which suggests is a scaling variable for. A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide
- Performance (Case 3) (2/2) a. Case 3: Anderson L Reservoir Case (South Texas, USA) Plot of versus (requires an estimate of gas-in-lace). Some data scatter exists, but a linear trend is evident (recall that =c e ( )( i -)). b. Case 3: Anderson L Reservoir Case (South Texas, USA) Plot of / versus. Both models are in strong agreement. Anderson L Reservoir Case Abnormal Pressure: Field data will exhibit some scatter, method is relatively tolerant of data scatter. Constant aroximation is based on the "best fit" of several data functions. The linear aroximation for is reasonable (should favor later data). A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 2
Validation of the /- 2 model: Orientation Methodology: All analyses are "dynamically" linked in a sreadsheet rogram (MS Excel). Therefore, all analyses are consistent should note that some functions/ lots erform better than others but the model results are the same for every analysis lot. Validation: Illustrative Analyses /- 2 lotting functions based on the roosed material balance model. - erformance lots used to calibrate analysis. an analysis resumes 2-straight line trends on a /- lot for an abnormally ressured reservoir. D - D tye curve aroach use /- 2 material balance model to develo tye curve solution this aroach is most useful for data validation. A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 3
/- 2 Plotting Functions: Case (/5) a. i Δ( / ) vs. b. Δ( / ) vs. c. Δ( / ) d vs. 0 i d. Δ( / ) d vs. e. 2 0 Δ( / ) Δ( / ) d vs. f. Δ( / ) Δ( / ) d vs. 0 0 A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 4
- Plotting Functions: Case (2/5) a. Case : Simulated Performance Case Plot of c e ()( i -) versus (requires estimate of ). b. Case : Simulated Performance Case Plot of /c e ()( i -) versus (requires estimate of ). c. Case : Simulated Performance Case Plot of versus (requires estimate of ). d. Case : Simulated Performance Case Plot of versus / (requires estimate of ). A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 5
- Plotting Functions: Case (3/5) Simulated Dry as Reservoir Case Abnormal Pressure: Summary / lot for =constant and =linear cases. ood comarison of trends, =linear trend aears slightly conservative as it emerges from data trend but both solutions aear to yield same estimate. A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 6
an-blasingame Analysis (200): Case (4/5) a. Case : Simulated Performance Case an Plot c e ()( i -) versus (/)/( i / i ) (requires est. of ). c. Case : Simulated Performance Case an Plot 3 (/) versus (results lot). b. Case : Simulated Performance Case an Plot 2 (/)/( i / i ) versus ( /) (requires est. of ). an-blasingame Analysis: Aroach considers the "match" of the c e ()( i -) (/)/( i / i ) data and "tye curves." Assumes that both abnormal and normal ressure / trends exist. Straight-line extraolation of the "normal" / trend used for. A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 7
D - D Tye Curve Aroach: Case (5/5) a. D - D Tye curve solution based on the /- 2 model. D = [( i / i )-(/)]/( i / i ) and D = / both D and Di functions are lotted. b. Case : Simulated Performance Case Tye curve analysis of (/)- data, this case is "force matched" to the same results as all of the other lotting functions. A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 8
/- 2 Plotting Fcns: Case 3 (Anderson L) (/5) a. i Δ( / ) vs. b. Δ( / ) vs. c. Δ( / ) d vs. 0 i d. Δ( / ) d vs. e. 2 0 Δ( / ) Δ( / ) d vs. f. Δ( / ) Δ( / ) d vs. 0 0 A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 9
- Plotting Functions: Case 3 (2/5) a. Case 3: Anderson L (South Texas) Plot of c e ()( i -) versus (requires estimate of ). b. Case 3: Anderson L (South Texas) Plot of /c e ()( i -) versus (requires estimate of ). c. Case 3: Anderson L (South Texas) Plot of versus (requires estimate of ). d. Case 3: Anderson L (South Texas) Plot of versus / (requires estimate of ). A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 20
- Plotting Functions: Case 3 (3/5) Case 3 Anderson L Reservoir (South Texas (USA)) Summary / lot for =constant and =linear cases. ood comarison of trends, =constant and =linear cases in good agreement. Data trend is very consistent. A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 2
an-blasingame Analysis (200): Case 3 (4/5) a. Case 3: Anderson L Reservoir an Plot c e ()( i -) versus (/)/( i / i ) (requires est. of ). c. Case 3: Anderson L Reservoir an Plot 3 (/) versus (results lot). b. Case 3: Anderson L Reservoir an Plot 2 (/)/( i / i ) versus ( /) (requires est. of ). an-blasingame Analysis: We note an excellent "match" of the c e ()( i -) (/)/( i / i ) data and the "tye curves." Both the abnormal and normal ressure / trends aear accurate and consistent. Straight-line extraolation of the "normal" / trend used for. A New /- 2 Material Balance for Abnormally-Pressured Reservoirs Slide 22
D-D Tye Curve Aroach: Case 3 (5/5) Case 3 Anderson L Reservoir (South Texas (USA)) D-D tye curve solution matched using field data. Note the "tail" in the D trend for small values of D (common field data event). "Force matched" to the same results as each of the other lotting functions. A New /-2 Material Balance for Abnormally-Pressured Reservoirs Slide 23
Examle Analysis Using MS Excel: Case 3 Case 3 Anderson L (South Texas (USA)) Literature standard case. A 3-well reservoir, delimited by faults. ood quality data. Evidence of overressure from static ressure tests. Analysis: (Imlemented using MS Excel) /-2 lotting functions. - erformance lots. an analysis (2-straight line trends on a /- lot). D-D tye curve aroach. A New /-2 Material Balance for Abnormally-Pressured Reservoirs Slide 24
Summary: (/3) Develoed a new /-2 material balance model for the analysis of abnormally ressured gas reservoirs: i ( ) 2 i where: ce ( )( i ) The -function is resumed (based on grahical comarisons) to be either constant, or aroximately linear with. For the =constant case, we have: i 2 i ( ) i i i i A New /-2 Material Balance for Abnormally-Pressured Reservoirs Slide 25
Summary: (2/3) Base relation: /-2 form of the gas material balance i 2 i a. Plotting Function : ( ) i i i i d. Plotting Function 4 : (quadratic) (linear) Δ( / ) i vs. i b. Plotting Function 2: 2 c. Plotting Function 3: (quadratic) Δ( / ) 0 Δ( / ) d vs. Δ( / ) d vs. 0 f. Plotting Function 6: (quadratic) Δ( / ) d vs. 0 e. Plotting Function 5 : (linear) Δ( / ) vs. (linear) Δ( / ) 0 Δ( / ) d vs. A New /-2 Material Balance for Abnormally-Pressured Reservoirs Slide 26
Summary: (3/3) The lotting functions develoed in this work have been validated as tools for the analysis reservoir erformance data from abnormally ressured gas reservoirs. Although the straight-line functions (PF2, PF4, and PF6) could be used indeendently, but we recommend a combined/simultaneous analysis. The - lots are useful for checking data consistency and for guiding the selection of the -value. These lots reresent a vivid and dynamic visual balance of all of the other analyses. The an analysis sequence is also useful for orienting the overall analysis articularly the ce()(i-) versus (/)/(i/i) lot. The D-D tye curve is useful for orientation articularly for estimating the or ( D ) value. A New /-2 Material Balance for Abnormally-Pressured Reservoirs Slide 27
Recommendations for Future Work: Consider the extension of this methodology for cases of external drive energy (e.g., water influx, gas injection, etc.). Continue the validation of this aroach by alying the methodology to additional field cases. Imlementation into a stand alone software. A New /-2 Material Balance for Abnormally-Pressured Reservoirs Slide 28
A Quadratic Cumulative Production Model for the Material Balance of Abnormally-Pressured as Reservoirs F.E. onale, M.S. Thesis Defense 7 October 2003 Deartment of Petroleum Engineering Texas A&M University College Station, TX 77843-36 A New /-2 Material Balance for Abnormally-Pressured Reservoirs Slide 29