1. Semi-Analytical Rate Relations for Oil and Gas Flow
Petroleum Engineering 613
Natural Gas Engineering
Texas A&M University
Lecture 06:
Semi-Analytical Rate Relations
for Oil and Gas Flow
T.A. Blasingame, Texas A&M U.
Department of Petroleum Engineering
Texas A&M University
College Station, TX —
PETE 613
(2005A)

2. Rate Relations for Oil and Gas Flow
Historical Perspectives
"Backpressure" equation.
Arps relations (exponential, hyperbolic, and harmonic).
Derivation of Arps' exponential decline relation.
Validation of Arps' hyperbolic decline relation.
Specialized Gas Flow Relations:
Fetkovich Gas Flow Relation.
Ansah-Buba-Knowles Gas Flow Relations.
Specialized Oil Flow Relations:
Fetkovich Oil Flow Relation.
Inflow Performance Relations (IPR):
Early work (for rationale).
Oil IPR and Solution-Gas Drive IPR.
Gas Condensate IPR.
PETE 613
(2005A)

3. History: Deliverability/"Backpressure" Equation
Gas Well Deliverability:
The original well deliverability relation was completely empiri-cal (derived from observations), and is given as:
This relationship is rigorous for low pressure gas reservoirs, (n=1 for laminar flow).
From: Back-Pressure Data on Natural-Gas Wells and Their Application to Production Practices — Rawlins and Schellhardt (USBM Monograph, 1935).
PETE 613
(2005A)

4. History: p2 Diffusivity Equations
Diffusivity Equations for a "Dry Gas:" p2 Relations
p2 Form — Full Formulation:
p2 Form — Approximation:
PETE 613
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5. History: Gas p2 Condition (mgz vs. p, T=200 Deg F)
"Dry Gas" PVT Properties: (mgz vs. p)
Basis for the "pressure-squared" approximation (i.e., use of p2 variable).
Concept: (mgz) = constant, valid only for p<2000 psia.
PETE 613
(2005A)

6. History: Gas p2 Condition (mgz vs. p, T=200 Deg F)
"Dry Gas" PVT Properties: (mgz vs. p)
Concept: IF (mgz) = constant, THEN p2-variable valid.
(mgz)  constant for p<2000 psia.
Even with numerical solutions, p2 formulation would not be appropriate.
PETE 613
(2005A)

7. History: "Arps" Equations
Arps' (Empirical) Rate Relations:
Exponential decline case (conservative).
Harmonic decline case (liberal).
Hyperbolic decline case (everything in between).
Fetkovich (Radial Flow) Decline Type Curve:
Exponential, hyperbolic, harmonic decline cases.
Derivation of the Arps' Exponential Rate Relation:
Combination of liquid material balance and liquid pseudo-steady-state flow equation solved for pwf  constant.
Useful for deriving auxiliary relations (cumulative production functions, in particular).
Derivation of the Arps' Hyperbolic Rate Relation:
Interesting exercise, limited practical value.
PETE 613
(2005A)

8. Arps Relations: Summary (1/2)
Flowrate-Time Relations:
Exponential: (b=0)
Hyperbolic: (0<b<1)
Harmonic: (b=1)
Cumulative Production-Time Relations:
Exponential: (b=0)
Hyperbolic: (0<b<1)
Harmonic: (b=1)
PETE 613
(2005A)

9. Arps Relations: Summary (2/2)
Flowrate-Cumulative Production Relations:
Exponential: (b=0)
Hyperbolic: (0<b<1)
Harmonic: (b=1)
Plot of: q versus Np
Plot of: log(N-Np) versus log(q)
Plot of: log(q) versus Np
PETE 613
(2005A)

10. Arps Relations: Example 1 (1/2)
(Exponential)
(Hyperbolic)
a. Semilog "Rate-Time" Plot: Barnett Gas Field.
b. Cartesian "Rate-Cumulative" Plot: Barnett Gas Field (North Texas).
c. Log-Log "(G-Gp)-Rate" Plot: Barnett Gas Field (North Texas).
PETE 613
(2005A)

11. Arps Relations: Example 1 (2/2)
(Exponential)
(Hyperbolic)
EUR Analysis: Barnett Field (North Texas (USA))
Semilog "Rate-Time" Plot: Barnett Gas Field.
Note data scatter and apparent fit of hyperbolic function.
PETE 613
(2005A)

12. Arps Relations: Example 2 (1/2)
(Exponential)
(Hyperbolic)
a. Semilog "Rate-Time" Plot: SPE — East Texas Gas Well 1.
b. Cartesian "Rate-Cumulative" Plot: SPE — East Texas Gas Well 1.
c. Log-Log "(G-Gp)-Rate" Plot: SPE — East Texas Gas Well 1.
PETE 613
(2005A)

13. Arps Relations: Example 2 (2/2)
(Exponential)
(Hyperbolic)
EUR Analysis: SPE Well 1 (East Texas (USA))
Combination "Rate-Time" and "Pressure-Time" plot.
Note pressure buildup (used to check with PTA).
PETE 613
(2005A)

14. Fetkovich Decline Type Curve: Empirical
Fetkovich "Empirical" Decline Type Curve:
Log-log "type curve" for the Arps "decline curves" (Fetkovich, 1973).
Initially designed as a graphical solution of the Arps' relations.
PETE 613
(2005A)

15. Analytical Type Curves: Radial Flow
"Analytical" Rate Decline Curves:
Data from van Everdingen and Hurst (1949), replotted as a rate decline plot (Fetkovich, 1973).
This looks promising — but this is going to be one really big "type curve."
What can we do? Try to collapse all of the trends to a single trend during boundary-dominated flow (Fetkovich, 1973).
"Analytical" stems are another name for transient flow behavior, which can yield estimates of reservoir flow properties.
From: SPE — Fetkovich (1973).
From: SPE — Fetkovich (1973).
PETE 613
(2005A)

16. Fetkovich Decline Type Curve: Analytical
Fetkovich "Analytical" Decline Type Curve: (constant pwf)
Log-log "type curve" for transient flow behavior (Fetkovich, 1973).
First "tie" between pressure transient and production data analysis.
PETE 613
(2005A)

17. Fetkovich Decline Type Curve: Composite
Fetkovich "Composite" Decline Type Curve:
Assumes constant bottomhole pressure production.
Radial flow in a finite radial reservoir system (single well).
PETE 613
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18. Derivation: Arps' Exponential Decline Case
Oil Material Balance Relation:
Oil Pseudosteady-State Flow Relation:
Steps:
Differentiate both relations with respect to time.
Assume pwf = constant (eliminates d(pwf)/dt term).
Equate results, yields 1st order ordinary differential equation.
Integrate.
Exponentiate result.
PETE 613
(2005A)

19. Validation: Arps' Hyperbolic Decline Case
(Details of derivation are omitted, see paper SPE 19009, Camacho and Raghavan (1989)).
a. Hyperbolic flowrate relations for the case of constant pressure production from a solution gas drive reservoir (Camacho and Raghavan (1989)).
b. Hyperbolic decline type curve with data simulation performance data superimpos-ed (Camacho and Raghavan (1989)).
PETE 613
(2005A)

20. Specialized Gas Flow Relations
Fetkovich Gas Flow Relation (poor approximation):
Rate-time.
Characteristic behavior plot.
Results from Knowles-Ansah-Buba work:
Rate-cumulative.
PETE 613
(2005A)

21. Fetkovich Gas Flow Relation: Poor Approximation
Gas Material Balance Relation: (z=1 ! (ideal gas?))
Gas Pseudosteady-State Flow Relation: (Fetkovich)
Final Result: (Fetkovich)
PETE 613
(2005A)

22. Fetkovich Decline Type Curve: Gas
Fetkovich "Analytical" Gas Decline Type Curve: (pwf = 0)
Cheated (z=1) ... this is not a valid solution (Fetkovich, 1973).
Good intentions ... wanted to develop a "simple" gas solution.
PETE 613
(2005A)

23. Knowles — Gas Rate-Time Relation
"Knowles" rate-time relation for gas flow:
Models decline of gas flowrate versus time.
Better representation of rate-time behavior than the "Arps" hyperbolic decline relations.
Assumptions:
Volumetric, dry gas reservoir.
pi < 6000 psia.
Constant bottomhole flowing pressure.
PETE 613
(2005A)

24. Knowles-Buba — Gas Rate-Cumulative Relation
This work presents an analysis and interpretation se-quence for the estimation of reserves in a volumetric dry-gas reservoir. This is based on the "Knowles" rate-cumulative production relation for pseudosteady-state gas flow given as:
"Knowles" relations for gas flow:
qg — Gp follows quadratic "rate-cumulative" relation.
Approximation valid for pi<6000 psia.
Assumes pwf = constant.
PETE 613
(2005A)

25. Simplified Gas Flow: Validation of Knowles Eqs.
b. Simulated Performance Case: Gp versus t (pi= 5000 psia, pwf=1000 psia, Gquad=4.20 BSCF).
qg vs. t and Gp vs. t:
Base plots ― verify models by Ansah, et. al
Comparative trends of 0.9qgi , qgi and 1.1qgi . Comparison applied to all analysis plots.
Very good match on both plots, accuracy verifies model.
a. Simulated Performance Case: qg versus t (pi= 5000 psia, pwf=1000 psia, Gquad=4.20 BSCF).
PETE 613
(2005A)

26. Simplified Gas Flow: Validation of Buba Eq.
"Knowles-Buba" relations for gas flow:
Simulated performance case: qg-Gp (quadratic "rate-cumulative").
pi= 5000 psia, pwf=1000 psia, Gquad=4.20 BSCF.
Data function matches well with quadratic model function.
PETE 613
(2005A)

27. Specialized Oil Flow Relations
Fetkovich Oil Flow Relation:
Rate-time (Decline Type Curve Analysis).
Deliverability (Isochronal Testing of Oil Wells).
PETE 613
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28. Fetkovich Oil Flow Relation: (Approximation)
Oil Material Balance Relation: (p2 – formulation!)
Oil Pseudosteady-State Flow Relation: (Fetkovich)
Final Result: (Fetkovich)
PETE 613
(2005A)

29. Fetkovich Decline Type Curve: Solution Gas Drive
Fetkovich "Analytical" Oil Decline Type Curve: (pwf = 0)
Cheated (pressure-squared material balance relation?) ... this is not a valid solution (Fetkovich, 1973).
PETE 613
(2005A)

30. Oil "Backpressure" Relation: Fetkovich (1/2)
a. Deliverability ("backpressure") plot developed for Well 2/4-2X prior to matrix acidizing treatment. (Fetkovich [SPE (1973)]).
b. Deliverability ("backpressure") plot developed for Well 2/4-2X after matrix acidizing treatment. Note much higher flowrate performance and apparent non-linear (i.e., non-laminar) flow behavior (Fetkovich [SPE (1973)]).
PETE 613
(2005A)

31. Oil "Backpressure" Relation: Fetkovich (2/2)
a. Comparison of simulated and predicted IPR behaviors for solution-gas-drive case (Vogel [SPE (1968)]).
b. Deliverability ("backpressure") plot developed using Vogel data. Proof of concept for "backpressure" flow relation (Fetkovich [SPE (1973)]).
PETE 613
(2005A)

32. Inflow Performance Relations (IPR)
Early work (for rationale)
Oil IPR and Solution-Gas Drive IPR
Vogel IPR work (for familiarity with approach)
Other IPR work (for reference/orientation)
Gas Condensate IPR
Fevang and Whitson work (for reference)
PETE 613
(2005A)

33. History Lessons — Early Performance Relations
Early “Gas Deliverability Plot," note the straight-line trends for the data (circa 1935).
Early “Gas IPR Plot," note the quadratic relationship between wellhead pressure and flowrate (circa 1935).
Well deliverability analysis: (after Rawlins and Schellhardt)
These plots represent the earliest attempts to quantify behavior and to predict future performance.
PETE 613
(2005A)

34. History Lessons — "Backpressure" Equation
Gas Well Deliverability:
The original well deliverability relation was derived from observations:
The "inflow performance relation-ship" (or IPR) for this case is: (assuming n=1)
From: Back-Pressure Data on Natural-Gas Wells and Their Application to Production Practices — Rawlins and Schellhardt (USBM Monograph, 1935).
PETE 613
(2005A)

35. History Lessons — IPR Developments/Correlations
Early "Inflow Plot," an attempt to correlate well rate and pres-sure behavior — and to esta-blish the maximum flowrate, (after Gilbert (1954)).
IPR "comparison" — liquid (oil), gas, and "two-phase" (solution gas-drive) cases presented to illustrate comparative behavior (after Vogel (1968)).
Inflow Performance Relationship (IPR):
Correlate performance, estimate maximum flowrate.
Individual phases require, separate correlations.
PETE 613
(2005A)

36. Solution-Gas Drive Systems — Vogel IPR
Vogel Correlation: (Statistical)
The Vogel IPR correlation and its variations are well establish-ed as the primary performance prediction relations for produc-tion engineering applications. The original correlation is de-rived from reservoir simulation.
IPR behavior is dependent on the depletion stage (i.e., the level of reservoir depletion). No single correlation of IPR behavior is possible.
Vogel IPR Correlation: Solution Gas-Drive Behavior
Derived as a statistical correlation from simulation cases.
No "theoretical" basis — Intuitive correlation (qo,max and pavg).
PETE 613
(2005A)

37. Solution-Gas Drive Systems — Other Approaches
Fetkovich IPR: (Semi-Empirical)
Richardson, et al. IPR: (Empirical)
(x = phase (e.g., oil, gas, water))
Other IPR Correlations:
Fetkovich: Derived assuming linear mobility-pressure relationship.
Richardson, et al.: Empirical, generalized correlation.
PETE 613
(2005A)

38. Solution-Gas Drive Systems— Other Approaches
Wiggins, et al. IPR: (Semi-Rigorous)
Other IPR Correlations:
Wiggins, et al.: Used a polynomial expansion of the mobility function in order to yield a semi-rigorous IPR formulation.
Coefficients (a1, a2…) are determined based on the mobility function and its derivatives taken at the average reservoir pressure.
PETE 613
(2005A)

39. Solution-Gas Drive Systems— Other Approaches
Pseudopressure Formulation – Oil Phase
Mobility Function
Other IPR Correlations:
n strong function of pressure and saturation.
Semi-rigorous IPR formulation (derived for the solution-gas case) has the same form of the Richardson, et al. IPR (which is empirical).
PETE 613
(2005A)

40. Gas Condensate Systems — Pseudopressure
Three flow regions were characterized:
Region 1 — Main cause of productivity loss, oil and gas flow simultaneously.
Region 2 — Two phases coexist, but only gas is mobile.
Region 3 — single-phase gas.
Fevang and Whitson Correlation: Gas Condensate systems
Pressure and saturation functions need to be know in advance — GOR, PVT properties and relative permeabilities.
PETE 613
(2005A)

41. Gas Condensate IPR — Del Castillo 2003 (TAMU)
Model-Based Performance Study:
Radial, fully compositional, single well simulation model
Parameters/functions used in simulation:
Reservoir Temperature: T = 230, 260, 300 Deg F
Critical Oil Saturation: Soc = 0, 0.1, 0.3
Residual Gas Saturation: Sgr = 0, 0.15, 0.5
Relative Permeability: 7 sets of kro-krg data
Fluid Samples: 4 synthetic cases, 2 field samples
Assumptions used in simulation:
Interfacial tension effects are neglected
Non-Darcy flow effects are neglected
Capillary pressure effects are neglected
Refined simulation grid in the near-well region
Skin effect is neglected
Gravity and composition gradients are neglected
Simulations begun at the dew point pressure
Correlation of gas and gas-condensate performance using Richardson IPR model.
PETE 613
(2005A)

42. Gas Condensate — IPR Trends (Condensate)
Base IPR plot (condensate) — Case 16 (gas condensate sys-tem).
Dimensionless IPR plot (condensate) — Case 16 (gas condensate system)
Condensate IPR Correlations (gas condensate reservoirs)
All eight depletion stages regressed simultaneously.
Excellent correlation — all stages.
PETE 613
(2005A)

43. Gas Condensate — IPR Trends (Gas)
Base IPR plot (gas) — Case 16 (gas condensate system).
Dimensionless IPR plot (gas) — Case 16 (gas condensate system).
Gas IPR Correlations (gas condensate reservoirs)
All eight depletion stages regressed simultaneously.
Excellent correlation — even when there is a more pronounced curve overlap (gas).
PETE 613
(2005A)

44. Gas Condensate — Difference in IPR Trends
Base IPR plot (condensate) — Case 16 (Very rich gas condensate system).
Base IPR plot (condensate) — Case 1 (Lean gas condensate system).
Condensate IPR Shape (gas condensate reservoirs)
Remarkable difference in shape between a very rich gas condensate system and a lean one.
PETE 613
(2005A)

45. Gas Condensate — IPR Parameter (no or ng )
o,g
o,g
Dimensionless IPR plot.
Base IPR plot.
(x = phase (e.g., oil, gas, water))
Condensate or gas IPR parameter (gas condensate reservoirs)
Low no or ng values — IPR more concave.
Exact value of not crucial — similar curves for different no or ng values.
PETE 613
(2005A)

46. Semi-Analytical Rate Relations for Oil and Gas Flow
Petroleum Engineering 613
Natural Gas Engineering
Texas A&M University
Lecture 06:
Semi-Analytical Rate Relations
for Oil and Gas Flow
(End of Lecture)
T.A. Blasingame, Texas A&M U.
Department of Petroleum Engineering
Texas A&M University
College Station, TX —
PETE 613
(2005A)