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Petroleum Engineering 613 Natural Gas Engineering Texas A&M University Lecture 05: Gas Material Balance T.A. Blasingame, Texas A&M U. Department of Petroleum Engineering Texas A&M University College Station, TX 77843-3116 +1.979.845.2292 — t-blasingame@tamu.edu PETE 613 Gas Material Slide — 1 (2005A) Balance Material Balance "Accounting" Concept of Material Balance: Require all inflows/outflows/generations. (Average) reservoir pressure profile is REQUIRED. Require rock, fluid, and rock-fluid properties (at some scale). Oil Material Balance: Less common than gas material balance (pressure required). Gas Material Balance: Volumetric dry gas reservoir (p/z versus Gp (straight-line)). Abnormally-pressured gas reservoirs (various techniques). Waterdrive/water influx cases (always problematic) (i.e., we don't know the influx, so we use a model). Material Balance yields RESERVOIR VOLUME! PETE 613 Gas Material Slide — 2 (2005A) Balance Material Balance of a Petroleum Reservoir General Concept of Material Balance... a. Initial reservoir conditions. b. Conditions after producing Np STB of oil, and Gp SCF of gas, and Wp STB of water. From: Petroleum Reservoir Engineering — Amyx, Bass, and Whiting (1960). Material Balance: Key Issues Must have accurate production measurements (oil, water, gas). Estimates of average reservoir pressure (from pressure tests). Suites of PVT data (oil, gas, water). Reservoir properties: saturations, formation compressibility, etc. PETE 613 Gas Material Slide — 3 (2005A) Balance Average Reservoir Pressure for Material Balance Average Reservoir Pressure From: Engineering Features of the Schuler Field and Unit Operation — Kaveler (SPE-AIME, 1944). Average Reservoir Pressure: Key Issues Must "average" pressures over volume or area (approximation). Pressure tests must be representative (pavg extrapolation valid). Can average using cumulative production (surrogate for volume). PETE 613 Gas Material Slide — 4 (2005A) Balance Gas Material Balance Case (1/3) General Gas Material Balance: p 1 ce ( p )( pi p ) z pi pi 1 zi zi G Gp Ginj Wp Rsw 5.615 1 (Wp Winj ) Bw We Bg "Dry Gas" Material Balance: (no reservoir liquids ) p pi 1 1 Gp G z zi PETE 613 Gas Material Slide — 5 (2005A) Balance Gas Material Balance Case (2/3) General Gas Material Balance: p 1 ce ( p )( pi p ) z pi pi 1 zi zi G Gp Ginj Wp Rsw 5.615 1 (Wp Winj ) Bw We Bg "Abnormal Pressure" Material Balance: (cf=f(p)) p pi 1 Gp 1 z zi 1 ce ( p )( pi p ) G 1 VpNNP VpAQ ce ( p ) S wi cw c f (cw c f ) (1 S wi ) VpR VpR "Quadratic Cumulative" Approximation: p pi 1 1 ( ) Gp G 2 zi p z G G PETE 613 Gas Material Slide — 6 (2005A) Balance Gas Material Balance Case (3/3) General Gas Material Balance: p 1 ce ( p )( pi p ) z pi pi 1 zi zi G Gp Ginj Wp Rsw 5.615 1 (Wp Winj ) Bw We Bg "Water Influx" Material Balance: 1 Gp p/z pi /zi 1 W B G 1 e w GB gi "Cubic Cumulative" Approximation: (Current Research) 2 3 p pi 1 (1 ) Gp Gp Gp ( ) z zi G G G PETE 613 Gas Material Slide — 7 (2005A) Balance Volumetric Gas Material Balance "Dry Gas" Material Balance: Normally Pressured Reservoir Example Volumetric reservoir — no external energy (gas expansion only). p/z versus Gp yields unique straight-line trend. Linear extrapolation yield gas-in-place (G). PETE 613 Gas Material Slide — 8 (2005A) Balance Gas MBE Abnormally-Pressured Reservoir "Dry Gas" Material Balance: Abnormally Pressured Reservoir Example Volumetric reservoir — no water influx or leakage. p/z versus Gp yields unique quadratic trend (from approximated MBE). Quadratic extrapolation yield gas-in-place (G). PETE 613 Gas Material Slide — 9 (2005A) Balance Gas MBE "Water Influx" Case a. Gas Material Balance Plot: p/z vs. Gp — simulated b. Gas Material Balance Plot: p/z vs. Gp — simulated performance. Note effect of aquifer permeability on performance. Note effect of displacement field performance. efficiency (Ep). Gas Material Balance: Water Drive Gas Reservoir Pressure (hence p/z) is maintained during production via communication with an unsteady-state aquifer (this study). From: Unsteady-State Performance of Water Drive Gas Reservoirs, Agarwal (Texas A&M Ph.D., 1967). PETE 613 Gas Material Slide — 10 (2005A) Balance Concept: p/z vs. Gp — Water Influx Case Simulated Performance: Agarwal Dissertation (1967) Recovery is a function of production rate, Ep, and kaquifer. p/z vs. Gp performance appears to be cubic (i.e., f(Gp3)). PETE 613 Gas Material Slide — 11 (2005A) Balance Petroleum Engineering 613 Natural Gas Engineering Texas A&M University Lecture 05: Gas Material Balance (End of Lecture) T.A. Blasingame, Texas A&M U. Department of Petroleum Engineering Texas A&M University College Station, TX 77843-3116 +1.979.845.2292 — t-blasingame@tamu.edu PETE 613 Gas Material Slide — 12 (2005A) Balance Petroleum Engineering 613 Natural Gas Engineering Texas A&M University A Quadratic Cumulative Production Model for the Material Balance of Abnormally-Pressured Gas Reservoirs F.E. Gonzalez M.S. Thesis (2003) Department of Petroleum Engineering Texas A&M University College Station, TX 77843-3116 PETE 613 Gas Material Slide — 13 (2005A) Balance Executive Summary — "p/z-Gp2" Relation (1/4) The rigorous relation for the material balance of a dry gas reservoir system is given by Fetkovich, et al. as: p 1 ce ( p)( pi p) pi z zi pi 1 5.615 Gp Ginj W p Rsw (W p Bw Winj Bw We ) zi G Bg Eliminating the water influx, water production/injection, and gas injection terms; defining Gp=ce(p)(pi-p) and assuming that Gp<1, then rearranging gives the follow- ing result: p pi 1 1 ( ) Gp G 2 p z zi G G PETE 613 Gas Material Slide — 14 (2005A) Balance Executive Summary — "p/z-Gp2" Relation (2/4) Simulated Dry Gas Reservoir Case — Abnormal Pressure: Volumetric, dry gas reservoir — with cf(p) (from Fetkovich). Note extrapolation to the "apparent" gas-in-place (previous approaches). Note comparison of data and the new "Quadratic Cumulative Production" model. PETE 613 Gas Material Slide — 15 (2005A) Balance Executive Summary — "p/z-Gp2" Relation (3/4) Anderson L Reservoir Case — Abnormal Pressure: South Texas (USA) gas reservoir with abnormal pressure. Benchmark literature case. Note performance of the new "Quadratic Cumulative Production" model. PETE 613 Gas Material Slide — 16 (2005A) Balance Presentation Outline Executive Summary Objectives and Rationale Rigorous technique for abnormal pressure analysis. Development of the p/z-Gp2 model Derivation from the rigorous material balance. Validation — Field Examples Case 1 — Dry gas simulation (cf(p) from Fetkovich). Case 3 — Anderson L (South Texas, USA). Demonstration (MS Excel — Anderson L case) Summary Recommendations for Future Work PETE 613 Gas Material Slide — 17 (2005A) Balance Objectives and Rationale Objectives: Develop a rigorous functional form (i.e., a model) for the p/z vs. Gp behavior demonstrated by a typical abnormally pressured gas reservoir. Develop a sequence of plotting functions for the analysis of p/z—Gp data (multiple plots). Provide an exhaustive validation of this new model using field data. Rationale: The analysis of p/z—Gp data for abnorm- ally pressured gas reservoirs has evolved from empi- rical models and idealized assumptions (e.g., cf(p)= constant). We would like to establish a rigorous ap- proach — one where any approximation is based on the observation of some characteristic behavior, not simply a mathematical/graphical convenience. PETE 613 Gas Material Slide — 18 (2005A) Balance Development of the p/z-Gp2 model Concept: Use the rigorous material balance relation given by Fetkovich, et al. for the case of a reservoir where cf(p) is NOT presumed constant. Use some observed limiting behavior to construct a semi-analytical relation for p/z—Gp behavior. Implementation: Develop and apply a series of data plotting functions which clearly exhibit unique behavior relative to the p/z—Gp data. Use a "multiplot" approach which is based on the dynamic updating of the model solution on each data plot. Develop a "dimensionless" type curve approach that can be used to validate the model and estimate G. PETE 613 Gas Material Slide — 19 (2005A) Balance p/z-Cumulative Model: (1/3) The rigorous relation for the material balance of a dry gas reservoir system is given by Fetkovich, et al. as: p 1 ce ( p)( pi p) pi z zi pi 1 5.615 Gp Ginj W p Rsw (W p Bw Winj Bw We ) zi G Bg Eliminating the water influx, water production/injection, and gas injection terms, then rearranging gives the following definition: p i /z i Gp p/z 1 [where Gp c e ( p )( p i p )] (1 Gp ) G PETE 613 Gas Material Slide — 20 (2005A) Balance p/z-Cumulative Model: (2/3) Considering the condition where: D Gp 1 Then we can use a geometric series to represent the D term in the governing material balance. The appropriate geometric series is given by: 1 / 1 x 1 x x 2 x3 ... (1 x 1) or, for our problem, we have: 1 1 Gp (1 Gp 1) (1 Gp ) Substituting this result into the material balance relation, we obtain: p pi 1 1 ( ) Gp G 2 zi p z G G PETE 613 Gas Material Slide — 21 (2005A) Balance p/z-Cumulative Model: (3/3) A more convenient form of the p/z-cumulative model is: p pi Gp G 2 p z zi 1 pi pi ( ) G zi G zi We note that these parameters presume that is con- stant. Presuming that is linear with Gp, we can derive the following form: p pi 1 pi a pi 2 b pi 3 ( a) Gp ( b) Gp Gp z zi G zi G zi G zi where a bGp Obviously, one of our objectives will be the study of the behavior of vs. Gp (based on a prescribed value of G). PETE 613 Gas Material Slide — 22 (2005A) Balance -Gp Performance (Case 1) (1/2) a. Case 1: Simulated Performance Case — Plot of b. Case 1: Simulated Performance Case — Plot of versus Gp (requires an estimate of gas-in- p/z versus Gp. The constant and linear trends place). Note the apparent linear trend of the match well with the data — essentially a con- data. Recall that Gp=ce(pp-p). firmation of both models. Simulated Dry Gas Reservoir Case — Abnormal Pressure: A linear trend of vs. Gp is reasonable and should yield an accurate model. is approximated by a constant value within the trend. A physical definition of is elusive — Gp=ce(p)(pi-p) implies that has units of 1/volume, which suggests is a scaling variable for G. PETE 613 Gas Material Slide — 23 (2005A) Balance -Gp Performance (Case 3) (2/2) a. Case 3: Anderson L Reservoir Case (South b. Case 3: Anderson L Reservoir Case (South Texas, USA) — Plot of versus Gp (requires an Texas, USA) — Plot of p/z versus Gp. Both estimate of gas-in-place). Some data scatter models are in strong agreement. exists, but a linear trend is evident (recall that Gp=ce(p )(pi-p)). Anderson L Reservoir Case — Abnormal Pressure: Field data will exhibit some scatter, method is relatively tolerant of data scatter. Constant approximation is based on the "best fit" of several data functions. The linear approximation for is reasonable (should favor later data). PETE 613 Gas Material Slide — 24 (2005A) Balance Validation of the p/z-Gp2 model: Orientation Methodology: All analyses are "dynamically" linked in a spread- sheet program (MS Excel). Therefore, all analyses are consistent — should note that some functions/ plots perform better than others — but the model results are the same for every analysis plot. Validation: Illustrative Analyses p/z-Gp2 plotting functions — based on the proposed material balance model. -Gp performance plots — used to calibrate analysis. Gan analysis — presumes 2-straight line trends on a p/z-Gp plot for an abnormally pressured reservoir. pD-GpD type curve approach — use p/z-Gp2 material balance model to develop type curve solution — this approach is most useful for data validation. PETE 613 Gas Material Slide — 25 (2005A) Balance p/z-Gp2 Plotting Functions: Case 1 (1/5) pi p Gp 0 1 1 a. Δ( p/z ) vs. Gp b. Δ( p/z ) vs. Gp c. Δ( p/z ) dGp vs. Gp zi z Gp Gp Gp Gp 0 0 1 Gp 1 1 0 1 Δ( p/z ) Δ( p/z ) dGp vs. Gp d. Δ( p/z ) dGp vs. Gp e. Δ( p/z ) Δ( p/z ) dGp vs. Gp f. G2 Gp Gp Gp p PETE 613 Gas Material Slide — 26 (2005A) Balance -Gp Plotting Functions: Case 1 (2/5) a. Case 1: Simulated Performance Case — Plot of b. Case 1: Simulated Performance Case — Plot of ce(p)(pi-p) versus Gp (requires estimate of G). 1/ce(p)(pi-p) versus Gp (requires estimate of G). c. Case 1: Simulated Performance Case — Plot of d. Case 1: Simulated Performance Case — Plot of versus Gp (requires estimate of G). versus Gp/G (requires estimate of G). PETE 613 Gas Material Slide — 27 (2005A) Balance -Gp Plotting Functions: Case 1 (3/5) Simulated Dry Gas Reservoir Case — Abnormal Pressure: Summary p/z—Gp plot for =constant and =linear cases. Good comparison of trends, =linear trend appears slightly conservative as it emerges from data trend — but both solutions appear to yield same G estimate. PETE 613 Gas Material Slide — 28 (2005A) Balance Gan-Blasingame Analysis (2001): Case 1 (4/5) a. Case 1: Simulated Performance Case — Gan Plot 1 b. Case 1: Simulated Performance Case — Gan Plot 2 ce(p)(pi-p) versus (p/z)/(pi/zi) (requires est. of G). (p/z)/(pi /zi ) versus (Gp/G) (requires est. of G). Gan-Blasingame Analysis: Approach considers the "match" of the ce(p)(pi-p) — (p/z)/(pi/zi) data and "type curves." Assumes that both abnormal and normal pressure p/z trends exist. Straight-line extrapolation of the c. Case 1: Simulated Performance Case — Gan Plot 3 "normal" p/z trend used for G. (p/z) versus Gp (results plot). PETE 613 Gas Material Slide — 29 (2005A) Balance pD-GpD Type Curve Approach: Case 1 (5/5) a. pD-GpD Type curve solution based on the p/z-Gp2 b. Case 1: Simulated Performance Case — Type model. pD= [(pi/zi)-(p/z)]/(pi/zi) and GpD=Gp/G — curve analysis of (p/z)-Gp data, this case is both pD and pDi functions are plotted. "force matched" to the same results as all of the other plotting functions. PETE 613 Gas Material Slide — 30 (2005A) Balance p/z-Gp2 Plotting Fcns: Case 3 (Anderson L) (1/5) pi p Gp 0 1 1 a. Δ( p/z ) vs. Gp b. Δ( p/z ) vs. Gp c. Δ( p/z ) dGp vs. Gp zi z Gp Gp Gp Gp 0 0 1 Gp 1 1 0 1 Δ( p/z ) Δ( p/z ) dGp vs. Gp d. Δ( p/z ) dGp vs. Gp e. Δ( p/z ) Δ( p/z ) dGp vs. Gp f. G2 Gp Gp Gp p PETE 613 Gas Material Slide — 31 (2005A) Balance -Gp Plotting Functions: Case 3 (2/5) a. Case 3: Anderson L (South Texas) — Plot of b. Case 3: Anderson L (South Texas) — Plot of ce(p)(pi-p) versus Gp (requires estimate of G). 1/ce(p)(pi-p) versus Gp (requires estimate of G). c. Case 3: Anderson L (South Texas) — Plot of d. Case 3: Anderson L (South Texas) — Plot of versus Gp (requires estimate of G). versus Gp/G (requires estimate of G). PETE 613 Gas Material Slide — 32 (2005A) Balance -Gp Plotting Functions: Case 3 (3/5) Case 3 — Anderson L Reservoir (South Texas (USA)) Summary p/z—Gp plot for =constant and =linear cases. Good comparison of trends, =constant and =linear cases in good agreement. Data trend is very consistent. PETE 613 Gas Material Slide — 33 (2005A) Balance Gan-Blasingame Analysis (2001): Case 3 (4/5) a. Case 3: Anderson L Reservoir — Gan Plot 1 ce(p)(pi-p) b. Case 3: Anderson L Reservoir — Gan Plot 2 (p/z)/(pi /zi ) versus (p/z)/(pi/zi) (requires est. of G). versus (Gp/G) (requires est. of G). Gan-Blasingame Analysis: We note an excellent "match" of the ce(p)(pi-p) — (p/z)/(pi/zi) data and the "type curves." Both the abnormal and normal pressure p/z trends appear ac- curate and consistent. Straight-line extrapolation of the c. Case 3: Anderson L Reservoir — Gan Plot 3 (p/z) "normal" p/z trend used for G. versus Gp (results plot). PETE 613 Gas Material Slide — 34 (2005A) Balance pD-GpD Type Curve Approach: Case 3 (5/5) Case 3 — Anderson L Reservoir (South Texas (USA)) pD-GpD type curve solution matched using field data. Note the "tail" in the pD trend for small values of GpD (common field data event). "Force matched" to the same results as each of the other plotting functions. PETE 613 Gas Material Slide — 35 (2005A) Balance Example Analysis Using MS Excel: Case 3 Case 3 — Anderson L (South Texas (USA)) Literature standard case. A 3-well reservoir, delimited by faults. Good quality data. Evidence of overpressure from static pressure tests. Analysis: (Implemented using MS Excel) p/z-Gp2 plotting functions. -Gp performance plots. Gan analysis (2-straight line trends on a p/z-Gp plot). pD-GpD type curve approach. PETE 613 Gas Material Slide — 36 (2005A) Balance Summary: (1/3) Developed a new p/z-Gp2 material balance model for the analysis of abnormally pressured gas reservoirs: p pi 1 where: 1 1 ( ) Gp G 2 ce ( p)( pi p) zi p z G G Gp The -function is presumed (based on graphical comparisons) to be either constant, or approximately linear with Gp. For the =constant case, we have: p pi Gp G 2 p z zi 1 pi pi ( ) G zi G zi PETE 613 Gas Material Slide — 37 (2005A) Balance Summary: (2/3) Base relation: p/z-Gp2 form of the gas material balance p pi 1 pi pi Gp G 2 p ( ) z zi G zi G zi a. Plotting Function 1: d. Plotting Function 4 : (quadratic) (linear) p p Gp 0 1 Δ( p/z ) i vs. Gp Δ( p/z ) dGp vs. Gp zi z G2 p b. Plotting Function 2: e. Plotting Function 5 : (linear) (quadratic) Gp 0 1 1 Δ( p/z ) vs. Gp Δ( p/z ) Δ( p/z ) dGp vs. Gp Gp Gp c. Plotting Function 3: f. Plotting Function 6: (quadratic) (linear) Gp Gp 0 0 1 1 1 Δ( p/z ) dGp vs. Gp Δ( p/z ) Δ( p/z ) dGp vs. Gp Gp Gp Gp PETE 613 Gas Material Slide — 38 (2005A) Balance Summary: (3/3) The plotting functions developed in this work have been validated as tools for the analysis reservoir performance data from abnormally pressured gas reservoirs. Although the straight-line functions (PF2, PF4, and PF6) could be used independently, but we recommend a combined/simultaneous analysis. The -Gp plots are useful for checking data con- sistency and for guiding the selection of the -value. These plots represent a vivid and dynamic visual balance of all of the other analyses. The Gan analysis sequence is also useful for orient- ing the overall analysis — particularly the ce(p)(pi-p) versus (p/z)/(pi/zi) plot. The pD-GpD type curve is useful for orientation — particularly for estimating the or (D ) value. PETE 613 Gas Material Slide — 39 (2005A) Balance 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 approach by applying the methodology to additional field cases. Implementation into a stand alone software. PETE 613 Gas Material Slide — 40 (2005A) Balance Petroleum Engineering 613 Natural Gas Engineering Texas A&M University A Quadratic Cumulative Production Model for the Material Balance of Abnormally-Pressured Gas Reservoirs (End of Presentation) F.E. Gonzalez M.S. Thesis (2003) Department of Petroleum Engineering Texas A&M University College Station, TX 77843-3116 PETE 613 Gas Material Slide — 41 (2005A) Balance