Protocol: 1. The problems in this exam are to be worked completely and independently. 2. The exam is "CLOSED NOTES," students are limited to the following resources: You are permitted the use of 3 (three) standard 8.5"x11" pieces of paper to serve as your personal "resource pages" for this exam these pages can contain any notes, equations, plots, etc. that you may wish to utilize on the exam. You may use both sides of your "resource pages." You are also permitted the use of the "Practice Final Exam" distributed via e-mail. Your own calculator, pencil, straight-edge, etc. 3. Students are specifically forbidden to work together in any capacity. 4. Any and all questions should be directed to the instructor. 5. Exam duration will be 2 hours. General Rules: 1. You must show all work for credit no credit for unsupported work. 2. Be as neat and organized as possible no credit for illegible or incomplete work. 3. Carefully organize and attach all of your work (even work on scratch paper) when you reassemble the exam to turn in. You are to only write on the front portion of any particular page and you must number all pages in some logical order. Rules for Analysis Problems: 1. Graphical Analysis: a. Identify the model and provide appropriate analysis relations. b. Identify and label every pertinent feature clearly note all features on a particular plot (slopes, intercepts, trends, etc), and comment (in detail) as needed. c. Proceed with calculations you must show all details of your calculations. 2. Compare the results of your various analyses, making sure to note that some techniques should yield similar results (if applicable). 3. You are responsible for using the appropriate analysis relations (equations). If you use the wrong relation(s) for analysis, no credit will be given for your work. Rules for Theory/Development/Application Problems: 1. Identify all pertinent relations and assumptions before you begin your solution you are encouraged to provide a brief outline of the proposed solution before you start any work. 2. Proceed carefully in a logical and consistent manner through the solution, showing all pertinent details "magic steps" will be given no credit. 3. Carefully check the solution, if possible, by comparison of your result(s) to other similar solutions. For numerical results, you should attempt to check your values against other solutions (if avail-able/applicable) as a means of ensuring accuracy. Academic Honesty Statement: Aggie Code of Honor: Aggies do not lie, cheat, or steal, nor do they tolerate those who do. I have neither given nor received help on this exam. (your signature)
2 Description: Orientation Pressure Transient Data Analysis: This examination problem considers the analysis of pressure buildup test performed in a gas well that has been hydraulically fractured. The pseudopressure and pseudotime transformations are applied to the data functions you can simply analyze these well test data in the same manner as you would for data from an oil well test. The governing equations for these analyses are attached. Production Data Analysis: This examination problem also considers the analysis of the long-term production rate and pressure data (provided on a daily basis) for the same well as mentioned above. The pseudopressure and material balance pseudotime transformations are applied to the data functions so that you can perform decline type curve analysis of these data directly (i.e., by using the provided hand analysis plots). The governing equations for these analyses are attached. Required: (All relevant plots are provided) The following results are required: Pressure Transient Analysis Preliminary Log-log Analysis: Wellbore storage coefficient, C s. Dimensionless wellbore storage coefficient, C D or C Df. Formation permeability, k. Cartesian Analysis: Early Time Data Pressure drop at the start of the test, p pws ( t=0). Wellbore storage coefficient, C s. Dimensionless wellbore storage coefficient, C D or C Df. Semilog Analysis: ( t a (MDH) analysis is sufficient for this case) Formation permeability, k. Near well skin factor, s. Log-Log Type Curve Analysis: ( t a analysis is sufficient for this case) Dimensionless fracture conductivity, C fd. Formation permeability, k. Fracture half-length, x f. Wellbore storage coefficient, C s. Dimensionless wellbore storage coefficient, C D or C Df. Pseudoradial flow skin factor, s pr. Production Data Analysis Semilog Analysis: (Model: q g q gi exp(-d i t), Plot: log(q g ) versus t) Initial rate, q i. Decline constant, D i. Gas-in-place, G. Cartesian Analysis: (Model: q g q gi - D i G p, Plot: q g versus G p ) Initial rate, q i. Decline constant, D i. Gas-in-place, G. Log-Log Type Curve Analysis: (Fetkovich-McCray-style decline type curves) Dimensionless fracture conductivity, C fd (or F cd (old name)). Reservoir drainage area, A. Formation permeability, k. Fracture half-length, x f. Pseudoradial flow skin factor, s pr.
3 Governing Equations for Well Test Analysis Well Test Analysis Pseudopressure-Pseudotime Relations: Pseudopressure: µ p gizi p p = dp p pi p µ g z base Pseudotime: t t = gi c 1 µ ti d t a 0 µ g c t Well Test Analysis Early-Time Cartesian Analysis: (pseudopressure-pseudotime form) qg Bgi qg Bgi p pws = p pwf ( t = 0) + mwbs ta (or tae), where mwbs =, which yields Cs =. 24Cs 24mwbs C The dimensionless wellbore storage coefficients are, 0.894 s C C and 0.894 s D = C. 2 Df = 2 φhctirw φhctix f Well Test Analysis Semilog Analysis: (pseudopressure-pseudotime form) Horner: t p + ta qg Bgiµ gi qg Bgiµ gi ppws = p pi msl ln, where msl = 162.6, which gives k = 162.6. ta kh mslh and the skin factor, s, is given by: ( p pws ( t a = 1hr) p pwf ( t a = 0)) t p k s = 1.1513 log log + 3. 2275 m sl t p + 1 2 φµ gi c ti r w MDH: k qg Bgiµ gi ppws = p pwf ( ta = 0) + msl ln( ta) + m sl log 3.2275 0.8686s +, where m 162.6. 2 kh gi c ti r sl = φµ w where the formation permeability, k, and the skin factor, s, are given by: qg Bgiµ gi k = 162.6 and ( p pws ( t a = 1hr) p pwf ( t a = 0)) k s = 1.1513 mslh log + 3. 2275 m 2 sl φµ gi c ti r w Well Test Analysis Type Curve Analysis Relations: (pseudopressure-pseudotime form) "Bourdet -Gringarten" Type Curves: (Unfractured Vertical Well with Wellbore Storage and Skin Effects) Family Parameter: C D e 2s Formation Permeability: q ' g Bgiµ gi [ p or ] 141.2 wd p k = wd MP h ' [ p p or p p ] MP Dimensionless Wellbore Storage Coefficient: "Economides" Type Curves: (Fractured Vertical Well with Wellbore Storage Effects) Family Parameter: C Df Formation Permeability: q ' g Bgiµ gi [ p or ] 141.2 wd p k = wd MP h ' [ p p or p p ] MP Fracture Half-Length: k [ t or ] 0.0002637 a tae MP 2 k 1 [ t or ] CD = 0.0002637 a t x ae MP 2 φµ [ td / CD] f = gictirw MP φµ gicti CDf [ tdxf / CDf ] MP Skin Factor: 1 [ C 2s ] = ln De s MP 2 CD
4 Governing Equations for Production Data Analysis Production Data Analysis Pseudopressure-Pseudotime Relations: Pseudopressure: µ p gizi p p = dp p pi p µ g z base Pseudotime: (base definition) t t = gi c 1 µ dt a ti 0 µ g ( p) c t ( p) Production Data Analysis Rate-Time and Rate Cumulative Analysis: q g q gi exp(-d i t) (plot of log(q g ) versus t) q g q gi - D i G p (plot of q g versus G p ) Material Balance Pseudotime: (variable-rate) µ c t gi ti q g ( t) t = dt a q g µ g ( p) c t ( p) 0 (These results are derived for the liquid case assumptions include a volumetric reservoir produced at a constant bottomhole pressure. These relations are "conservative" approximations for the gas case.) Production Data Analysis Type Curve Analysis Relations: (pseudopressure-material balance pseudotime form) (from Pratikno (2002), Decline Curve Analysis Using Type Curves Fractured Wells) Type Curve Matching Procedure: 1. Assemble the production (gas MSCF/D) and bottomhole pressure (psia) versus time (in days). Compute the "material balance pseudotime" function given by: µ c t gi ti q g ( t) t = dt a q g µ g ( p) c t ( p) 0 2. Compute the pressure drop normalized rate and rate integral functions. The pseudopressure drop normalized rate function is given by: ( qg qg qg p p ) = = ( p pi pwf ) p p For gas, the pressure drop normalized rate integral function is given as: 1 t a ( qg p p ) i = ( qg p ) dτ t p a 0 For gas, the pressure drop normalized rate integral-derivative function is given by: ( qg p p ) id [( q p ) ] d[ ( q p ) ] d g p i = d ln ( ta ) = t a g dt a p i 3. The following data functions are plotted on a scaled log-log grid for type curve matching using the Fetkovich- McCray format type curve: a. ( qg p p ) versus t a b. ( q g p p ) i versus t a c. ( q g p p ) id versus t a
5 Governing Equations for Production Data Analysis Production Data Analysis Pseudopressure-Pseudotime Relations: (continued) 4. We now "force" match the depletion data trends onto the Arps b = 1 (harmonic) stems for each of the Fetkovich- McCray style type curves being used (i.e., q Dd, q Ddi, and q Ddid ). Once a "match" is obtained, we record the "time" and "rate" axis match points as well as the r ed transient flow stem. Recall that for this case, r ed = r e /x f. a. Rate-axis Match Point: Any ( q p) MP ( qdd ) MP pair b. Time-axis Match Point: Any ( t ) MP ( t Dd ) MP pair c r ed transient flow stem: look for the ( q p), ( q p) i, ( q p) id functions that best match the transient data stems. d. Lastly, we calculate the b Dpss value (using Eq. 4.5 below): b 2 Dpss = ln ( red ) 0.049298 + 0.43464 red a1 + a + 1+ b 1 2 u + a u + b 2 3 u u 2 2 + a + b 3 4 u u 3 3 + a + b where the following constants are defined: u = ln ( FcD ) a 1 = 0.93626800 b 1 = -0.38553900 a 2 = -1.00489000 b 2 = -0.06988650 a 3 = 0.31973300 b 3 = -0.04846530 a 4 = -0.04235320 b 4 = -0.00813558 a 5 = 0.00221799 (For the case of an unfractured well the b Dpss parameter is defined as: b Dpss = ln(r ed ) - 1/2) Estimation of Reservoir Properties: Using the "match point" results we can estimate the following reservoir properties: Original Gas-In-Place: 1 G = cgi ( t ) ( q ) MP g p a p MP ( t Dd ) MP ( qdd ) MP Reservoir Drainage Area: GBgi A = 5.6148 φtot h(1 S wirr ) "Effective" Drainage Radius: r e = A π Formation Permeability: (effective permeability to gas) k g Bgi µ = 141.2 h gi Fracture Half-Length: re x f = red b Dpss ( q g p p ) ( qdd ) MP MP 4 5 u u 4 4
6 Required Results Required Results: Pressure Transient Analysis (perform analyses as applicable) Preliminary Log-log Analysis: Wellbore storage coefficient, C s = RB/psi Dimensionless wellbore storage coefficient, C D or C Df = Formation permeability, k = md Cartesian Analysis: Early Time Data Pressure drop at the start of the test, p pws ( t=0) = psi Wellbore storage coefficient, C s = RB/psi Dimensionless wellbore storage coefficient, C D or C Df = Semilog Analysis: ( t a (MDH) analysis is sufficient for this case) Formation permeability, k = md Near well skin factor, s = Log-Log Type Curve Analysis: ( t a analysis is sufficient for this case) Dimensionless fracture conductivity, C fd = Formation permeability, k = md Fracture half-length, x f = ft Wellbore storage coefficient, C s = RB/psi Dimensionless wellbore storage coefficient, C D or C Df = Pseudoradial flow skin factor, s pr = Required Results: Production Data Analysis (perform analyses as applicable) Semilog Analysis: (Model: q g q gi exp(-d i t), Plot: ln(q g ) versus t) Initial rate, q i = MSCF/D Decline constant, D i = 1/D Gas-in-place, G = MSCF Cartesian Analysis: (Model: q g q gi - D i G p, Plot: q g versus G p ) Initial rate, q i = MSCF/D Decline constant, D i = 1/D Gas-in-place, G = MSCF Log-Log Type Curve Analysis: (Fetkovich-McCray-style decline type curves) Dimensionless fracture conductivity, C fd (or F cd (old name)) = Reservoir drainage area, A = acres Formation permeability, k = md Fracture half-length, x f = ft Pseudoradial flow skin factor, s pr =
"Cinco-Samaniego" Type Curve "Cinco-Samaniego" Type Curve: p wd and p wd ' vs. t Dxf Various C fd Values (NO WELLBORE STORAGE EFFECTS) (1"x1" format) 7
"Cinco-Samaniego" Skin Factor Correlation "Cinco-Samaniego" Skin Factor Correlation: (used to relate the fractured well case to the Pseudoradial flow skin factor) 8
Bourdet-Gringarten Type Curve (Unfractured Well) Bourdet-Gringarten Type Curve: p wd and p wd ' vs. t D /C D Various C D Values (Radial Flow Case Includes Wellbore Storage and Skin Effects) (1"x1" format) 9
"Economides" Type Curve (C fd =1, C Df =various) "Economides" Type Curve: p wd and p wd ' vs. t Dxf /C Df C fd =1 (Fractured Well Case Includes Wellbore Storage Effects) (1"x1" format) 10
"Economides" Type Curve (C fd =2, C Df =various) "Economides" Type Curve: p wd and p wd ' vs. t Dxf /C Df C fd =2 (Fractured Well Case Includes Wellbore Storage Effects) (1"x1" format) 11
"Economides" Type Curve (C fd =2, C Df =various) "Economides" Type Curve: p wd and p wd ' vs. t Dxf /C Df C fd =5 (Fractured Well Case Includes Wellbore Storage Effects) (1"x1" format) 12
"Economides" Type Curve (C fd =10, C Df =various) "Economides" Type Curve: p wd and p wd ' vs. t Dxf /C Df C fd =10 (Fractured Well Case Includes Wellbore Storage Effects) (1"x1" format) 13
"Economides" Type Curve (C fd =1x10-3, C Df =various) "Economides" Type Curve: p wd and p wd ' vs. t Dxf /C Df C fd =1x10 3 (Fractured Well Case Includes Wellbore Storage Effects) (1"x1" format) 14
Fetkovich-McCray Decline Type Curve: Unfractured Well (Radial Flow System) Fetkovich-McCray Decline Type Curve Unfractured Well (Radial Flow System) 15
Correlation Plot for the Pseudosteady-State Constant Fractured Wells (Various C fd (or F cd ) Cases Correlation Plot for the Pseudosteady-State Constant: Fractured Wells (Various C fd (or F cd ) Cases 16
Fetkovich-McCray Decline Type Curve: Fractured Well Case C fd (or F cd ) = 0.1 Fetkovich-McCray Decline Type Curve Fractured Well Case C fd (or F cd ) = 0.1 17
Fetkovich-McCray Decline Type Curve: Fractured Well Case C fd (or F cd ) = 1 Fetkovich-McCray Decline Type Curve Fractured Well Case C fd (or F cd ) = 1 18
Fetkovich-McCray Decline Type Curve: Fractured Well Case C fd (or F cd ) = 10 Fetkovich-McCray Decline Type Curve Fractured Well Case C fd (or F cd ) = 10 19
Fetkovich-McCray Decline Type Curve: Fractured Well Case C fd (or F cd ) = 1x10 3 Fetkovich-McCray Decline Type Curve Fractured Well Case C fd (or F cd ) = 1x10 3 20
21 Data Inventory: Well "B" Data Inventory Well "B" Reservoir properties: φ = 0.088 S wirr = 0.131 (fraction) h = 170 ft r w = 0.292 ft Gas properties: (at the initial reservoir pressure (p i ) of 9330 psia) γ g = 0.61 (air=1) CO 2 = 2.5 mole percent T= 300 deg F B gi = 0.5442 RB/MSCF µ gi = 0.0322 cp c t = c gi = 5.58x10-5 psia -1 Production parameters: p pwf ( t=0) = 228.48 psia q g = 900 MSCF/D (constant) t p = 14784 hr
Production Summary Plot: Production Summary Plot 22
Cartesian Plot: Early-Time Pseudopressure versus Pseudotime Data Cartesian Plot Early-Time Pseudopressure versus Pseudotime Data 23
Semilog Plot: "MDH" Plot Pseudopressure versus Pseudotime Data Semilog Plot "MDH" Plot Pseudopressure versus Pseudotime Data 24
Log-log Plot Pseudopressure Drop and Pseudopressure Drop Derivative Data Versus Shut-In Pseudotime (1 inch x 1 inch) Log-log Plot: Pseudopressure Drop and Pseudopressure Drop Derivative Data Versus Shut-In Pseudotime (1 inch x 1 inch) 25
Semilog Plot: Rate-Time Plot (Production Data Analysis) Semilog Plot Rate-Time Plot (Production Data Analysis) 26
Cartesian Plot: Rate-Cumulative Plot (Production Data Analysis) Cartesian Plot Rate-Cumulative Plot (Production Data Analysis) 27
Log-log Plot Pseudopressure Drop Normalized Rate Versus Material Balance Pseudotime (1 inch x 1 inch) Log-log Plot: Pseudopressure Drop Normalized Rate Versus Material Balance Pseudotime (1 inch x 1 inch) 28