1. Radial flow equation Outer boundary: CR Inner boundary:
CP Inner boundary:
Initial: pi
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2. Dimensionless Radial (cylindrical source)
initial
Outer boundary: no flow
Inner boundary
CR CP
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3. Intuitive Concept of Productivity Index
boundary
well re
Production Rate
Drawdown (Driving Force)
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4. Pseudo-steady State IPR: Impact of Reservoir Pressure
AOF
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5. Irregular Drainage Area and/or Location Dietz shape factor
Euler’s constant
CA = 31.6
CA = 30.9
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6. How to be smart? Pss: p is defined with average pressure 1/2/2019
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7. How to be smart? Assume rw = 0.3 ft A/acre 40 re /ft JDpss = 120
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8. Pseudosteady-State Performance with Skin (Undersaturated Oil)
Average reservoir press
There is –3/4
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9. Single Phase Pss IPR Productivity Index and Skin
Average reservoir pressure
(NOT average between reservoir and wellbore !!!)
Slope: 1/J
NOT J !
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10. Effective Wellbore Radius
Definition of r’w
E.g. Steady-state
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11. Effect of Stimulation Represented as a Folds of Increase in PI
Spot
acidizing:  0
carbonate acidizing:  -2
hydraulic fracturing:  -5
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12. The easiest (will be always plus)
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)
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13. Boundary-dominated State
Constant wellbore pressure,
time elapsed is enough to stabilize
the "shape" of the pressure distribution
No flow boundary
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14. Steady-state Constant wellbore pressure*,
time elapsed is enough to stabilize
the "shape" of the pressure distribution
Constant outer pressure
*or constant rate
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15. Steady-State Performance with Skin (Undersaturated Oil)
Press at outer boundary
There is no –3/4
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16. Akif Ibragimov and Peter Valkó
Theory
Akif Ibragimov and Peter Valkó
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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
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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
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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")
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20. Circular drainage area
reD 1
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21. 1/2/2019
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22. Pseudo-steady State Def 1 (CR PI) Def 2 (PSS)
Def: Auxiliary Problem 1 (sol: pD1 with zero average)
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23. General Results:
Specific Results:
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24. Boundary-dominated Def 1 (CP PI) Def 2 (BD) Def: Auxiliary Problem 3
Def: Auxiliary Problem 4 (zero-zero transient)
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25. General Results:
Specific Results:
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26. 1/2/2019
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27. R = 1000
PPSS = e-7 r2 + ln(r)
PBD = J0( r ) Y0( r) - 1
When both
average pressures (0)
and wellbore pressures (-1)
are the same
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28. PSS
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29. BD
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30. Difference: PPSS-PBD
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