Radial flow equation Outer boundary: CR Inner boundary: CP Inner boundary: Initial: pi 1/2/2019 ENGI6324: A-2-1
Dimensionless Radial (cylindrical source) initial Outer boundary: no flow Inner boundary CR CP 1/2/2019 ENGI6324: A-2-1
Intuitive Concept of Productivity Index boundary well re Production Rate Drawdown (Driving Force) 1/2/2019 ENGI6324: A-2-1
Pseudo-steady State IPR: Impact of Reservoir Pressure AOF 1/2/2019 ENGI6324: A-2-1
Irregular Drainage Area and/or Location Dietz shape factor Euler’s constant CA = 31.6 CA = 30.9 1/2/2019 ENGI6324: A-2-1
How to be smart? Pss: p is defined with average pressure 1/2/2019 ENGI6324: A-2-1
How to be smart? Assume rw = 0.3 ft A/acre 40 re /ft JDpss = 120 1/2/2019 ENGI6324: A-2-1
Pseudosteady-State Performance with Skin (Undersaturated Oil) Average reservoir press There is –3/4 1/2/2019 ENGI6324: A-2-1
Single Phase Pss IPR Productivity Index and Skin Average reservoir pressure (NOT average between reservoir and wellbore !!!) Slope: 1/J NOT J ! 1/2/2019 ENGI6324: A-2-1
Effective Wellbore Radius Definition of r’w E.g. Steady-state 1/2/2019 ENGI6324: A-2-1
Effect of Stimulation Represented as a Folds of Increase in PI Spot acidizing: 0 carbonate acidizing: -2 hydraulic fracturing: -5 1/2/2019 ENGI6324: A-2-1
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) 1/2/2019 ENGI6324: A-2-1
Boundary-dominated State Constant wellbore pressure, time elapsed is enough to stabilize the "shape" of the pressure distribution No flow boundary 1/2/2019 ENGI6324: A-2-1
Steady-state Constant wellbore pressure*, time elapsed is enough to stabilize the "shape" of the pressure distribution Constant outer pressure *or constant rate 1/2/2019 ENGI6324: A-2-1
Steady-State Performance with Skin (Undersaturated Oil) Press at outer boundary There is no –3/4 1/2/2019 ENGI6324: A-2-1
Akif Ibragimov and Peter Valkó Theory http://www.isc.tamu.edu/iscpubs/0005.pdf Akif Ibragimov and Peter Valkó 1/2/2019 ENGI6324: A-2-1
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 1/2/2019 ENGI6324: A-2-1
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 1/2/2019 ENGI6324: A-2-1
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") 1/2/2019 ENGI6324: A-2-1
Circular drainage area reD 1 1/2/2019 ENGI6324: A-2-1
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Pseudo-steady State Def 1 (CR PI) Def 2 (PSS) Def: Auxiliary Problem 1 (sol: pD1 with zero average) 1/2/2019 ENGI6324: A-2-1
General Results: Specific Results: 1/2/2019 ENGI6324: A-2-1
Boundary-dominated Def 1 (CP PI) Def 2 (BD) Def: Auxiliary Problem 3 Def: Auxiliary Problem 4 (zero-zero transient) 1/2/2019 ENGI6324: A-2-1
General Results: Specific Results: 1/2/2019 ENGI6324: A-2-1
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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 1/2/2019 ENGI6324: A-2-1
PSS 1/2/2019 ENGI6324: A-2-1
BD 1/2/2019 ENGI6324: A-2-1
Difference: PPSS-PBD 1/2/2019 ENGI6324: A-2-1