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Radial flow equation

Radial flow equation. Outer boundary:. CR Inner boundary:. CP Inner boundary:. Initial: p i. Dimensionless Radial (cylindrical source). initial. Outer boundary: no flow. Inner boundary CR CP. Intuitive Concept of Productivity Index. boundary.

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Radial flow equation

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  1. Radial flow equation Outer boundary: CR Inner boundary: CP Inner boundary: Initial: pi ENGI6324: A-2-1

  2. Dimensionless Radial(cylindrical source) initial Outer boundary: no flow Inner boundary CR CP ENGI6324: A-2-1

  3. Intuitive Concept of Productivity Index boundary well re Production Rate Drawdown (Driving Force) ENGI6324: A-2-1

  4. Pseudo-steady State IPR: Impact of Reservoir Pressure AOF ENGI6324: A-2-1

  5. Irregular Drainage Area and/or LocationDietz shape factor Euler’s constant CA = 31.6 CA = 30.9 ENGI6324: A-2-1

  6. How to be smart? Pss: p is defined with average pressure ENGI6324: A-2-1

  7. Assume rw = 0.3 ft How to be smart? ENGI6324: A-2-1

  8. Pseudosteady-State Performance with Skin (Undersaturated Oil) Average reservoir press There is –3/4 ENGI6324: A-2-1

  9. Single Phase Pss IPRProductivity Index and Skin Average reservoir pressure (NOT average between reservoir and wellbore !!!) Slope: 1/J NOT J ! ENGI6324: A-2-1

  10. Effective Wellbore Radius Definition of r’w E.g. Steady-state ENGI6324: A-2-1

  11. Effect of Stimulation Represented as a Folds of Increase in PI Spot acidizing:  0 carbonate acidizing:  -2 hydraulic fracturing:  -5 ENGI6324: A-2-1

  12. Elements of Skin Effect(Accounting for well geometry, perforation, etc) • Skin Components • damage • penetration+slant • perforation • pseudo (e.g. non-Darcy, condensate) • plus reservoir shape and well location:(later: stimulation) the “original” The easiest (will be always plus) ENGI6324: A-2-1

  13. Boundary-dominated State Constant wellbore pressure, time elapsed is enough to stabilize the "shape" of the pressure distribution No flow boundary ENGI6324: A-2-1

  14. Steady-state Constant wellbore pressure*, time elapsed is enough to stabilize the "shape" of the pressure distribution Constant outer pressure *or constant rate ENGI6324: A-2-1

  15. Steady-State Performance with Skin (Undersaturated Oil) Press at outer boundary There is no –3/4 ENGI6324: A-2-1

  16. Theory http://www.isc.tamu.edu/iscpubs/0005.pdf Akif Ibragimov and Peter Valkó ENGI6324: A-2-1

  17. Productivity Index • Intuitive: • for a given reservoir-well geometry, the ratio of production rate to some pressure difference between the reservoir and the well is basically independent from production history or even from actual operating conditions, once the well production is "stabilized" • Math: Key concept is invariance • From time • From production rate or pressure difference ENGI6324: A-2-1

  18. 3 basic flow regimes with invariance properties • Steady-State: Constant pressure outer boundary and constant pressure (or flowrate) at the well • Pseudo-Steady State: No flow outer boundary and constant flowrate at the well • Boundary-Dominated State: No flow outer boundary and constant pressure at the well ENGI6324: A-2-1

  19. Unified View • Driving force is average reservoir pressure minus wellbore pressure • We are looking for an initial distribution of pressure PROVIDING time invariance at once • In the case of Pseudo-steady and Boundary-dominated states time invariance applies only to PI and some characteristics of the pressure distribution ("shape") ENGI6324: A-2-1

  20. Circular drainage area reD 1 ENGI6324: A-2-1

  21. ENGI6324: A-2-1

  22. Pseudo-steady State • Def 1 (CR PI) • Def 2 (PSS) • Def: Auxiliary Problem 1 (sol: pD1 with zero average) ENGI6324: A-2-1

  23. General Results: Specific Results: ENGI6324: A-2-1

  24. Boundary-dominated • Def 1 (CP PI) • Def 2 (BD) • Def: Auxiliary Problem 3 • Def: Auxiliary Problem 4 (zero-zero transient) ENGI6324: A-2-1

  25. General Results: Specific Results: ENGI6324: A-2-1

  26. ENGI6324: A-2-1

  27. R = 1000 PPSS = -6.15776809 - 5.0000050e-7 r2 + ln(r) PBD = 1.227462 J0(0.000568798 r ) + 0.254100Y0(0.000568798 r) - 1 When both average pressures (0) and wellbore pressures (-1) are the same ENGI6324: A-2-1

  28. PSS ENGI6324: A-2-1

  29. BD ENGI6324: A-2-1

  30. Difference: PPSS-PBD ENGI6324: A-2-1

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