The relation between the bottom-hole pressure
and surface flow rate
during the
stabilised formation flow:
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p_{wf} = p_{wf}(q) |
which may be linear (Fig. 1) or non-linear (Fig. 2).
Widely used in Well Flow Performance analysis.
The IPR (Inflow Performance Relation) analysis is closely related to well Productivity Index (PI)
which is defined as below:
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| J_{sO} = \frac{q_O}{p_r-p_{wf}} |
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for oil producer with oil flowrate at surface conditions |
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| J_s(q_G) = \frac{q_G}{p_r-p_{wf}} |
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for gas producer with gas flowrate at surface conditions |
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| J_s(q_g) = \frac{q_{GI}}{p_{wf}-p_r} |
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for gas injector with injection rate at surface conditions |
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| J_s(q_w) = \frac{q_{WI}}{p_r-p_{wf}} |
|
for water injector with injection rate at surface conditions |
where
| field-average formation pressure within the drainage area of a given well: LaTeX Math Inline |
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body | p_r = \frac{1}{V_e} \, \int_{V_e} \, p(t, {\bf r}) \, dV |
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Based on above defintions the aribitrary IPR can be wirtten in a general form:
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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 bottom-hole 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 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
LaTeX Math Block Reference |
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|
.
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)
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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:
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J_s = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + \epsilon+ S} |
where
LaTeX Math Inline |
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body | \sigma = \Big \langle \frac{k} {\mu} \Big \rangle \, h = k \, h\, \Big[ \frac{k_{rw}}{\mu_w} + \frac{k_{ro}}{\mu_o} \Big] |
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|
– water-based or water-oil-based transmissbility above bubble point
LaTeX Math Block Reference |
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anchor | Perrine2phase_alpha |
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page | Linear Perrine multi-phase diffusion (model) |
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,
for steady-state
SS flow and
for pseudo-steady state
PSS flow.
The alternative form of the constant Productivity Index IPR is given by:
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\frac{q}{q_{max}} = 1 -\frac{p_{wf}}{p_Rr} |
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).
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Fig. 2. IPR for dry gas producer or gas injector into a gas formation |
The popular dry gas IPR correlation is Rawlins and Shellhardt:
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anchor | IPRGas |
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alignment | left |
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\frac{q}{q_{max}} = \Bigg[ \, 1- \Bigg( \frac{p_{wf}}{p_Rr} \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:
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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
LaTeX Math Inline |
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body | \{ q_{max} \, , \, n \} |
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|
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).
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Fig. 3. IPR for 2-phase oil+gas production below and above bubble point |
The analytical correlation for saturted oil flow is given by Vogel model:
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\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} |
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
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 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:
LaTeX Math Block |
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\frac{q}{q_b} = \frac{p_r - p_{wf}}{p_r - p_b} \quad , \quad p_r > p_{wf} > p_b |
LaTeX Math Block |
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anchor | ModifiedVogel |
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alignment | left |
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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:
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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:
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\frac{q_o}{q_{o, \, max}} = 1 - 0.52 \, \frac{p_{wf}}{p_r} - 0.48 \Bigg(\frac{p_{wf}}{p_r} \Bigg)^2 |
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\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 |