The relation between the bottom-hole pressure and surface flow rate
during the stabilised formation flow:
p_{wf} = p_{wf}(q) |
which may be non-linear.
The IPR (Inflow Performance Relation) analysis is closely related to well PI – Productivity Index 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 within the drainage area |
Based on above defintions the aribitrary IPR (Inflow Performance Relation) 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 |
The Productivity Index can be constant or dependent on bottom-hole pressure or equivalently on flowrate
.
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 IPR (Inflow Performance Relation)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 customized IPR (Inflow Performance Relation) 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) and IPR (Inflow Performance Relation) plot is reperented by a straight line (Fig. 1)
|
| Fig.1. IPR (Inflow Performance Relation) plot for constant productivity (water and dead oil) |
This is a typical IPR (Inflow Performance Relation) 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 (Inflow Performance Relation) 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 atmosperic pressure and also called AOF – Absolute Open Flow.
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 (Inflow Performance Relation) plot (Fig. 2).
|
Fig. 2. IPR (Inflow Performance Relation) for dry gas producer or gas injector into a gas formation |
The popular dry gas IPR (Inflow Performance Relation) 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.
Saturated Oil IPR
For saturated oil reservoir the free gas flow inflict the downward trend of IPR (Inflow Performance Relation) plot similar to dry gas (Fig. 3).
|
Fig. 3. IPR (Inflow Performance Relation) for 2-phase oil+gas production below and above bubble point |
The analytical correlation for saturted oil flow is given by Vogel model:
\frac{q}{q_{max}} = 1 - 0.2 \, \frac{p_{wf}}{p_R} - 0.8 \Bigg(\frac{p_{wf}}{p_R} \Bigg)^2 \quad , \quad p_b > p_R > p_{wf} |
Undersaturated Oil IPR
For undersaturated oil reservoir the behavior of IPR (Inflow Performance Relation) 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 IPR (Inflow Performance Relation).
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 (Inflow Performance Relation) 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 (Inflow Performance Relation) 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] |
Saturated Multiphase IPR
For saturated 3-phase water-oil-gas reservoir the IPR (Inflow Performance Relation) analysis is represented by oil and water components separately (see Fig. 4.1 and Fig. 4.2).
|
|
Fig. 4.1. Oil IPR (Inflow Performance Relation) for saturated 3-phase (water + oil + gas) formation flow | Fig. 4.2. Water IPR (Inflow Performance Relation) 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 |
Undersaturated Multiphase IPR
For undersaturated 3-phase water-oil-gas reservoir the IPR (Inflow Performance Relation) analysis is represented by oil and water components separately (see Fig. 4.1 and Fig. 4.2).
|
|
Fig. 4.1. Oil IPR (Inflow Performance Relation) for udersaturated 3-phase (water + oil + gas) formation flow | Fig. 4.2. Water IPR (Inflow Performance Relation) for undersaturated 3-phase (water + oil + gas) formation flow |
The analytical correlation for saturated 3-phase oil flow is given by Wiggins model:
|