The relation between the bottom-hole pressure and surface flow rate
during the stabilised formation flow (see the reference to original paper of Gilbert):
p_{wf} = p_{wf}(q) |
which may be linear (Fig. 1) or non-linear (Fig. 2).
One of the two key concepts of Well Flow Performance analysis along with Vertical Lift Performance (VLP).
The word "Inflow" is misnomer as Inflow Performance Relationship (IPR) analysis is applicable for both producers and injectors.
The Inflow Performance Relationship (IPR) analysis is closely related to well Productivity Index (PI) which is defined as below:
| for oil producer with oil flowrate | |
| for gas producer with gas flowrate | |
| for gas injector with injection rate | |
| for water injector with injection rate |
where
field-average formation pressure estimate within the drainage area |
Based on above defintions the aribitrary IPR can be wirtten in a general form:
p_{wf} = p_r - \frac{q}{J_s} |
providing that has a specific meaning and sign as per the table below:
| for producer | |
| for injector | |
| for oil producer | |
| for gas producer or injector | |
| for water injector or water producer or water production from oil producer |
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. 1) or dependent on bottomhole pressure or equivalently on flowrate
(showing a curved line on IPR like on Fig. 2) .
In general case of multiphase flow the PI features a complex dependance on bottom-hole pressure
(or equivalently on flowrate
) which can be etstablished based on numerical simulations of multiphase formation flow.
For undersaturated reservoir the numerically-simulated Inflow Performance Relationship (IPR)s have been approximated by analytical models and some of them are brought below.
These correlations are usually expressed in terms of as alternative to
.
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.
For a single layer formation with low-compressibility fluid (water or dead oil) the PI does not depend on drawdown (or flowrate) and IPR plot is reperented by a straight line (Fig. 1)
|
| Fig.1. IPR plot for constant productivity (water and dead oil) |
This is a typical IPR plot for water supply wells, water injectors and dead oil producers.
The PI can be estimated using the Darcy equation:
J_s = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + \epsilon+ S} |
where – water-based or water-oil-based transmissbility above bubble point
,
for steady-state SS flow and
for pseudo-steady state PSS flow.
The alternative form of the constant Productivity Index IPR is given by:
\frac{q}{q_{max}} = 1 -\frac{p_{wf}}{p_r} |
where is the maximum reservoir deliverability when the bottom-hole is at atmospheric pressure and also called Absolute Open Flow (AOF).
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. 2).
|
Fig. 2. IPR for dry gas producer or gas injector into a gas formation |
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 is the turbulent flow exponent, equal to 0.5 for fully turbulent flow and equal to 1 for laminar flow.
The more accurate approximation is given by LIT (Laminar Inertial Turbulent) IPR model:
a \, q + b \, q^2 = \Psi(p_r) - \Psi(p_{wf}) |
where – is pseudo-pressure function specific to a certain gas PVT model,
is laminar flow coefficient and
is turbulent flow coefficient.
It needs two well tests at two different rates to assess or
.
But obviously more tests will make assessment more accruate.
For saturated oil reservoir the free gas flow inflict the downward trend of IPR plot similar to dry gas (Fig. 3).
|
Fig. 3. IPR for 2-phase oil+gas production below and above bubble point |
The analytical correlation for saturated oil reservoir flow is given by Vogel IPR @ model:
For undersaturated oil reservoir the behavior of IPR model will vary on whether the bottom-hole pressure is above or below bubble point.
When it is higher than bubble point then formation flow will be single-phase oil and production will follow the constant Inflow Performance Relationship (IPR).
When bottom-hole pressure goes below bubble point the near-reservoir zone free gas slippage also inflicts the downward trend at the right side of IPR plot (Fig. 3).
It can be interpreted as deterioration of near-reservoir zone permeability when the fluid velocity is high and approximated by rate-dependant skin-factor.
|
Fig. 3. IPR for 2-phase oil+gas production below and above bubble point |
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 related to bubble point flowrate
via following correlation:
q_{max} = q_b \, \Big[1 + \frac{1}{1.8} \frac{p_b}{(p_r - p_b)} \Big] |
For saturated 3-phase water-oil-gas reservoir the IPR analysis is represented by oil and water components separately (see Fig. 4.1 and Fig. 4.2).
|
|
Fig. 4.1. Oil IPR for saturated 3-phase (water + oil + gas) formation flow | Fig. 4.2. Water IPR for saturated 3-phase (water + oil + gas) formation flow |
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 |
For undersaturated 3-phase water-oil-gas reservoir the IPR analysis is represented by oil and water components separately (see Fig. 4.1 and Fig. 4.2).
|
|
Fig. 4.1. Oil IPR for udersaturated 3-phase (water + oil + gas) formation flow | Fig. 4.2. Water IPR for undersaturated 3-phase (water + oil + gas) formation flow |
The analytical correlation for saturated 3-phase oil flow is given by Wiggins model:
|
Petroleum Industry / Upstream / Production / Subsurface Production / Well & Reservoir Management
Subsurface E&P Disciplines / Production Technology / Well Flow Performance
[ Vogel IPR @model ] [ Richardson and Shaw IPR @ model ] [ Wiggins IPR @ model ]
Gilbert, W.E.: "Flowing and Gas-Lift Well Performance," Drill. and Prod. Prac., API (1954) 126.
