Productivity Index, IPR & Vogel
How much rate does a single psi of drawdown buy? The productivity index answers in one number, and the inflow performance relationship traces how that rate behaves as you lower the wellbore pressure - straight above the bubble point, curved below it.
How much rate does a single psi of drawdown buy? The productivity index answers in one number - and the IPR curve traces how that rate behaves as you lower the wellbore pressure.
The Productivity Index
The radial inflow equation ties rate to drawdown, and engineers compress that relationship into a single number: the productivity index, $J = \dfrac{q}{\bar p - p_{wf}}$, in stock-tank barrels per day per psi (STB/d/psi). It asks a simple question - how much rate does one psi of drawdown buy? A high-$J$ well is a good well: it delivers a lot of oil for little drawdown. A low-$J$ well is a poor one, needing a large drawdown to make a modest rate. Because $J$ folds permeability, pay thickness, viscosity, skin, and the geometry all into one number, it is the everyday yardstick for comparing wells and for watching a single well decline (or improve after stimulation) over time.
The IPR Straight Line and AOF
Plotting rate $q$ against flowing wellbore pressure $p_{wf}$ traces the well's inflow performance relationship (IPR). Above the bubble point - where the oil is undersaturated and no free gas has come out of solution - the IPR is a straight line with slope $-J$: halve the drawdown and you halve the rate. Extrapolating that line all the way down to $p_{wf} = 0$ gives the absolute open flow (AOF), $q_{max} = J\,\bar p$ - the theoretical rate if the well were open to the atmosphere. AOF is a convenient upper bound, even though no well is ever actually produced at zero bottomhole pressure. The straight-line IPR is the engineer's first, simplest model of what a well can deliver.
Below the bubble point, dissolved gas breaks out of the oil and free gas flows preferentially, so the IPR bends downward - rate rises more slowly as you draw the pressure down. Vogel's dimensionless relation captures this curvature: $\dfrac{q_o}{q_{o,max}} = 1 - 0.2\dfrac{p_{wf}}{\bar p} - 0.8\left(\dfrac{p_{wf}}{\bar p}\right)^2$. Vogel normalizes both rate (by the maximum $q_{o,max}$) and pressure (by the average $\bar p$), so the same curve describes many undersaturated-to-saturated oil wells. At $p_{wf} = 0$ it gives 1 (the AOF); at $p_{wf} = \bar p$ it gives 0 (no drawdown, no flow); and in between it always predicts less rate than the straight line would, because free gas steals mobility from the oil. The lesson: use the straight line above the bubble point, and switch to Vogel once you cross it.
- (a) Drawdown: $\bar p - p_{wf} = 3000 - 2000 = 1000$ psi.
- Productivity index: $J = 932/1000 = 0.932$ STB/d/psi.
- (b) Straight-line AOF: $q_{max} = J \times \bar p = 0.932 \times 3000 = 2{,}796$ STB/d.
- (c) At $p_{wf}/\bar p = 0.5$: $q_o/q_{o,max} = 1 - 0.2(0.5) - 0.8(0.5)^2$.
- $= 1 - 0.1 - 0.2 = 0.70$.
Check your understanding
- The productivity index J = q / (p-bar - p_wf) is rate per unit drawdown (STB/d/psi), the everyday yardstick for well performance.
- The IPR plots q versus p_wf as a straight line above the bubble point (constant J), with the absolute open flow AOF = J times p-bar at p_wf = 0.
- Below the bubble point, free gas bends the IPR, described by Vogel's q_o/q_o,max = 1 - 0.2(p_wf/p-bar) - 0.8(p_wf/p-bar)^2; at p_wf/p-bar = 0.5 it gives 0.70.