One of the Productivity Diagnostics methods  based on relation between the bottom-hole pressure 

p_{wf} = p_{wf}(q)

One of the two key concepts of Well Flow Performance analysis along with Vertical Lift Performance (VLP).

The word  "Inflow" is misnomer as IPR analysis is applicable for both producers and injectors.

The most general proxy-model is given by LIT (Laminar Inertial Turbulent) IPR model:

a \, q + b \, q^2 = \Psi(p_r) - \Psi(p_{wf})

where

It needs well tests at least three different rates to assess  

The IPR analysis is closely related to well Productivity Index (PI)  

J_s(q_O) = \frac{q_O}{p_r-p_{wf}}

J_s(q_G) = \frac{q_G}{p_r-p_{wf}}

J_s(q_{GI}) = \frac{q_{GI}}{p_{wf}-p_r}

J_s(q_{WI}) = \frac{q_{WI}}{p_r-p_{wf}}

where

Based on above defintions the aribitrary IPR can be wirtten in a general form:

p_{wf} = p_r - \frac{q}{J_s}

providing that  

See more on the variations of PI definition between Dynamic Modelling,  Well Flow Performance and Well Testing. 

The  Productivity Index can be constant (showing a straight line on IPR like on  Fig. 2) or dependent on bottomhole pressure 

In general case of multiphase flow the PI 

For undersaturated reservoir the numerically-simulated IPRs have been approximated by analytical models and some of them are brought below. 

These correlations are usually expressed in terms of 

They are very helpful in practise to design a proper well flow optimization procedure.

These correaltions should be calibrated to the available well test data to set a up a customised IPR model for a given formation.

Water and Dead Oil IPR

For a single layer formation with low-compressibility fluid (water or dead oil) the PI does not depend on drawdown (or flowrate) 

This is a typical IPR plot for water supply wells, water injectors and dead oil producers.

For the oil/water production the PI can be estimated using the Dupuit PI @model:

J_s = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + \epsilon+ S}

where 

The alternative form of the constant Productivity Index  IPR is given by:

\frac{q}{q_{max}} = 1 -\frac{p_{wf}}{p_r}

where 

Dry Gas IPR

For gas producers, the fluid compressibility is high and formation flow is essentially non-linear, inflicting the downward trend on the whole IPR plot (Fig. 3).

The popular dry gas IPR correlation is Rawlins and Shellhardt:

\frac{q}{q_{max}} = \Bigg[  \, 1- \Bigg(  \frac{p_{wf}}{p_r} \Bigg)^2  \, \Bigg]^n

where 

Saturated Oil IPR

For saturated oil reservoir the free gas flow inflict the downward trend of IPR plot  similar to dry gas (Fig. 4).

The analytical correlation for saturated oil reservoir flow is given by Vogel IPR @ model:

Undersaturated Oil IPR

For undersaturated oil reservoir 

When it is higher than bubble point 

When bottom-hole pressure goes below bubble point 

It can be interpreted as deterioration of near-reservoir zone permeability when the fluid velocity is high and approximated by rate-dependant skin-factor.

The analytical correlation for undersaturated oil flow is given by modified Vogel model:

\frac{q}{q_b} = \frac{p_r - p_{wf}}{p_r - p_b} \quad , \quad p_r > p_{wf} > p_b 

q = (q_{max} - q_b ) \Bigg[ 1 - 0.2 \, \frac{p_{wf}}{p_b} - 0.8 \Bigg(\frac{p_{wf}}{p_b} \Bigg)^2  \Bigg] + q_b \quad , \quad p_r > p_b > p_{wf}

with AOF 

q_{max} = q_b \, \Big[1 + \frac{1}{1.8} \frac{p_b}{(p_r - p_b)}  \Big]

Saturated Multiphase IPR

For saturated 3-phase water-oil-gas reservoir the IPR analysis is represented by oil and water components separately (see Fig. 6.1 and Fig. 6.2).

The analytical correlation for saturated 3-phase oil flow is given by Wiggins model:

\frac{q_o}{q_{o, \, max}} = 1 - 0.52 \, \frac{p_{wf}}{p_r} - 0.48 \Bigg(\frac{p_{wf}}{p_r} \Bigg)^2  

\frac{q_w}{q_{w, \, max}} = 1 - 0.72 \, \frac{p_{wf}}{p_r} - 0.28 \Bigg(\frac{p_{wf}}{p_r} \Bigg)^2 

Undersaturated Multiphase IPR

For undersaturated 3-phase water-oil-gas reservoir the IPR analysis is represented by oil and water components separately (see Fig. 7.1 and Fig. 7.2).

\frac{q_o}{q_{o, \, max}} = 1 - 0.52 \, \frac{p_{wf}}{p_r} - 0.48 \Bigg(\frac{p_{wf}}{p_r} \Bigg)^2  

\frac{q_w}{q_{w, \, max}} = 1 - 0.72 \, \frac{p_{wf}}{p_r} - 0.28 \Bigg(\frac{p_{wf}}{p_r} \Bigg)^2 

See Also

Petroleum Industry / Upstream /  Production / Subsurface Production / Subsurface E&P Disciplines / Field Study & Modelling / Production Analysis / Productivity Diagnostics

[ Production Technology / Well Flow Performance ]

[ Vogel IPR @model ] [ Richardson and Shaw IPR @ model ] [ Wiggins IPR @ model ]

[ Dual-layer IPR][ Multi-layer IPR ] [ Dual-layer IPR with dynamic fracture ]

Reference

Archer, R. A., Del Castillo, Y., & Blasingame, T. A. (2003, January 1). New Perspectives on Vogel Type IPR Models for Gas Condensate and Solution-Gas Drive Systems. Society of Petroleum Engineers. doi:10.2118/80907-MS