...
and
- wellbore fluid deliverability (the ability of well to lift up or lift down the fluid) and which is called OPR or TPR or VFP (equally popular throughout the literature)
...
In practise, the WFP – Well Flow Performance analysis is often very tentative and production technologists spend some time experimenting with well regimes on well-by-well basis.
...
IPR – Inflow Performance Relation
IPR – Inflow Performance Relation represents the relation between the bottom-hole pressure
and surface flow rate during the stabilised formation flow:| LaTeX Math Block |
|---|
|
p_{wf} = p_{wf}(q) |
which may be non-linear.
...
The WFP – Well Flow Performance analysis is closely related to well PI – Productivity Index
which is defined as below:| LaTeX Math Block |
|---|
|
J_{sO} = \frac{q_O}{p_R-p_{wf}} |
for oil producer with oil flowrate
at surface conditions| LaTeX Math Block |
|---|
|
J_s(q_G) = \frac{q_G}{p_R-p_{wf}} |
for gas producer with gas flowrate
at surface conditions
| LaTeX Math Block |
|---|
|
J_s(q_g) = \frac{q_{GI}}{p_{wf}-p_R} |
for gas injector with injection rate at surface conditions
| LaTeX Math Block |
|---|
|
J_s(q_w) = \frac{q_{WI}}{p_R-p_{wf}} |
for water injector with injection rate
at surface conditionswhere
...
...
field-average formation pressure within the drainage area
of a given well: | LaTeX Math Inline |
|---|
| body | p_R = \frac{1}{V_e} \, \int_{V_e} \, p(t, {\bf r}) \, dV |
|---|
|
Based on above defintions the aribitrary WFP – Well Flow Performance can be wirtten in a general form:
| LaTeX Math Block |
|---|
|
p_{wf} = p_R - \frac{q}{J_s} |
providing that
has a specific meaning and sign as per the table below:...
...
...
...
...
...
The Productivity Index can be constant or dependent on bottom-hole pressure
or equivalently on flowrate .
...
...
...
...
For undersaturated reservoir the numerically-simulated WFP – Well Flow Performances 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 |
|---|
|
.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 WFP – Well Flow Performance 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 WFP – Well Flow Performance plot is reperented by a straight line (Fig. 1)...
Image Removed
...
This is a typical WFP – Well Flow Performance plot for water supply wells, water injectors and dead oil producers.
The PI can be estimated using the Darcy equation:
| LaTeX Math Block |
|---|
|
J_s = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + \epsilon+ S} |
where
| LaTeX Math Inline |
|---|
| body | \sigma = \Big \langle \frac{k} {\mu} \Big \rangle \, h = k \, h\, \Big[ \frac{k_{rw}}{\mu_w} + \frac{k_{ro}}{\mu_o} \Big] |
|---|
|
– water-based or water-oil-based transmissbility above bubble point | LaTeX Math Block Reference |
|---|
| anchor | Perrine2phase_alpha |
|---|
| page | Linear Perrine multi-phase diffusion (model) |
|---|
|
,
for steady-state SS flow and for pseudo-steady state PSS flow....
The alternative form of the constant Productivity Index WFP – Well Flow Performance is given by:
| LaTeX Math Block |
|---|
|
\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 WFP – Well Flow Performance plot (Fig. 2).
...
Image Removed
...
Fig. 2. WFP – Well Flow Performance for dry gas producer or gas injector into a gas formation
The popular dry gas WFP – Well Flow Performance correlation is Rawlins and Shellhardt:
| LaTeX Math Block |
|---|
| anchor | IPRGas |
|---|
| alignment | left |
|---|
|
\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:
| LaTeX Math Block |
|---|
|
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 |
|---|
| body | \{ q_{max} \, , \, n \} |
|---|
|
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 WFP – Well Flow Performance plot similar to dry gas (Fig. 3).
...
Image Removed
...
Fig. 3. WFP – Well Flow Performance for 2-phase oil+gas production below and above bubble point
The analytical correlation for saturted oil flow is given by Vogel model:
| LaTeX Math Block |
|---|
|
\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 WFP – Well Flow Performance 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 WFP – Well Flow Performance. 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 WFP – Well Flow Performance 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.
...
Image Removed
...
Fig. 3. WFP – Well Flow Performance 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 |
|---|
|
\frac{q}{q_b} = \frac{p_R - p_{wf}}{p_R - p_b} \quad , \quad p_R > p_{wf} > p_b |
| LaTeX Math Block |
|---|
| anchor | ModifiedVogel |
|---|
| alignment | left |
|---|
|
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:| LaTeX Math Block |
|---|
|
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 WFP – Well Flow Performance analysis is represented by oil and water components separately (see Fig. 4.1 and Fig. 4.2).
...
Image Removed
...
Image Removed
...
Fig. 4.1. Oil WFP – Well Flow Performance for saturated 3-phase (water + oil + gas) formation flow
...
Fig. 4.2. Water WFP – Well Flow Performance for saturated 3-phase (water + oil + gas) formation flow
The analytical correlation for saturated 3-phase oil flow is given by Wiggins model:
| LaTeX Math Block |
|---|
|
\frac{q_o}{q_{o, \, max}} = 1 - 0.52 \, \frac{p_{wf}}{p_R} - 0.48 \Bigg(\frac{p_{wf}}{p_R} \Bigg)^2 |
| LaTeX Math Block |
|---|
|
\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 WFP – Well Flow Performance analysis is represented by oil and water components separately (see Fig. 4.1 and Fig. 4.2).
...
Image Removed
...
Image Removed
...
Fig. 4.1. Oil WFP – Well Flow Performance for udersaturated 3-phase (water + oil + gas) formation flow
...
Fig. 4.2. Water WFP – Well Flow Performance for undersaturated 3-phase (water + oil + gas) formation flow
...
The analytical correlation for saturated 3-phase oil flow is given by Wiggins model:
| LaTeX Math Block |
|---|
|
\frac{q_o}{q_{o, \, max}} = 1 - 0.52 \, \frac{p_{wf}}{p_R} - 0.48 \Bigg(\frac{p_{wf}}{p_R} \Bigg)^2 |
...
...
OPR – Outflow Performance Relation
...
