The well performance equations that we have discussed to this point; **Equation 4.18**, **Equation 4.34**, and **Equation 4.40**; are known collectively as the well * Inflow Performance Relationships, IPR* (relationship between the flowing well pressure, ${p}_{wf}$, and the production rate, $q$, from the well).

**Table 4.02**summarizes the flow regimes and the IPR equations.

Steady-State Flow Regime | Pseudo Steady-State Flow Regime | |
---|---|---|

Pressure Distribution | $$p={p}_{wf}+\frac{141.22\text{}\mu B\text{}q}{kh}lo{g}_{e}\left(\frac{r}{{r}_{w}}\right)$$ | $$p={p}_{wf}+\frac{141.22\text{}\mu B\text{}q}{kh}\left[lo{g}_{e}\left(\frac{r}{{r}_{w}}\right)-\left(\frac{{r}^{2}}{{r}_{e}^{2}}\right)\right]$$ |

In terms of ${p}_{e}$ at the external radius, ${r}_{e}$ , of the drainage volume | ||

Drawdown | $${p}_{e}-{p}_{wf}$$ | $${p}_{e}-{p}_{wf}$$ |

Productivity Index |
$$\frac{kh}{141.22\text{}\mu B\left[lo{g}_{e}\left(\frac{{r}_{e}}{{r}_{w}}\right)+S\right]}$$ | $$\frac{kh}{141.22\text{}\mu B\left[lo{g}_{e}\left(\frac{{r}_{e}}{{r}_{w}}\right)-\frac{1}{2}+S\right]}$$ |

IPR | $$q=\frac{kh\left({p}_{e}-{p}_{wf}\right)}{141.22\text{}\mu B\left[lo{g}_{e}\left(\frac{{r}_{e}}{{r}_{w}}\right)+S\right]}$$ | $$q=\frac{kh\left({p}_{e}-{p}_{wf}\right)}{141.22\text{}\mu B\left[lo{g}_{e}\left(\frac{{r}_{e}}{{r}_{w}}\right)-\frac{1}{2}+S\right]}$$ |

In terms of $\overline{p}$ in the interior of the drainage volume | ||

Drawdown | $$\overline{p}-{p}_{wf}$$ | $$\overline{p}-{p}_{wf}$$ |

Productivity Index |
$$\frac{kh}{141.22\text{}\mu B\left[lo{g}_{e}\left(\frac{{r}_{e}}{{r}_{w}}\right)-\frac{1}{2}+S\right]}$$ | $$\frac{kh}{141.22\text{}\mu B\left[lo{g}_{e}\left(\frac{{r}_{e}}{{r}_{w}}\right)-\frac{3}{4}+S\right]}$$ |

IPR | $$q=\frac{kh\left(\overline{p}-{p}_{wf}\right)}{141.22\text{}\mu B\left[lo{g}_{e}\left(\frac{{r}_{e}}{{r}_{w}}\right)-\frac{1}{2}+S\right]}$$ | $$q=\frac{kh\left(\overline{p}-{p}_{wf}\right)}{141.22\text{}\mu B\left[lo{g}_{e}\left(\frac{{r}_{e}}{{r}_{w}}\right)-\frac{3}{4}+S\right]}$$ |

[A] Note, we did derive the equations in the shaded cells, but they are included for future reference. |

In **Table 4.02**, I introduced some new terminology. The pressure drop in these equations, $\Delta p$, is referred to as the * Drawdown* or the

*; while the term multiplying the drawdown is the referred to as*

**Drawdown Pressure***(sometimes the productivity index is also referred to by the symbol $J$ ). The oilfield units of the drawdown are psi; while the oilfield units of the productivity index are STB/(day psi).*

**Productivity Index, PI**Using this terminology, we can simplify the Inflow Performance Relationship to:

Where the appropriate definitions for the drawdown and productivity index are selected from **Table 4.02** given the prevailing reservoir conditions (flow regime). **Equation 4.41** is very useful for practical applications. In the field, we can change the well flowing pressure, ${p}_{wf}$ , by changing the choke size on the well and measure the resulting stabilized production rate, $q$ . By doing this several times, we can estimate the productivity index from:

To do this, we must have some knowledge of the field pressure, typically measured by shutting in the well of interest or by shutting in * Offset Wells* (adjacent wells). Using this approach, the productivity of the well can be established without the need of knowing the individual well properties ($k$, $h$, $S$, etc.) or the flow regime (steady-state or pseudo steady-state). The measured drawdowns and production rates provide the appropriate productivity index to allow for future well calculations.

In fact, we can use the field measured productivity indices even in cases with mixed flow boundaries/regimes. For example, we may have a situation where a strong aquifer is located south of a production well which keeps the southern external boundary of the drainage area nearly constant (steady-state). While north of this production well, pressure depletion may be occurring due to production from offset wells. In this situation, we do not need to make any assumptions regarding the flow regime, which best describes the well or what definitions of drawdown or productivity index we need to use. If we do the field measurement, then the test will provide the correct results for that specific well.