Definition

WFP – Well Performance Analysis analysis is a comparative analysis between:

the formation fluid deliverability (the ability of reservoir to produce or take-in the fluid) which is called WFP – Well Flow Performance

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)

It is based on correlation between surface flowrate $\begin{array}{l}q\end{array}$ and bottomhole pressure $\begin{array}{l}p_{wf}\end{array}$ as a function of tubing-head pressure $\begin{array}{l}p_s\end{array}$ and formation pressure $\begin{array}{l}p_R\end{array}$ and current reservoir saturation.

Ideally, the well flow model for WFP – Well Flow Performance analysis should be performed individually for each well but even typical for a given asset can.

Application

Setting up the required production or injection regime for each well upon the current formation pressure, reservoir saturation and production target specified by FDP
Generating WFP – Well Flow Performance tables as input for 3D simulations

Technology

Most reservoir engineers exploit material balance thinking which is based on long-term well-by-well surface flowrate targets (whether producers or injectors).

In practice, the flowrate targets are closely related to bottomhole pressure and associated limitations and require a specialised analysis to set up the optimal lifting (completion, pump, chocke) parameters.

This is primary domain of WFP – Well Flow Performance analysis.

WFP – Well Flow Performance is performed on stabilised wellbore and reservoir flow and does not cover transient behavior which is one of the primary subjects of Well Testing domain.

More on stabilised reservoir and wellbore flow

The wellbore flow is called stabilised if the delta pressure across wellbore is not changing over time.

The formation flow is called stabilised if the well productivity index is not changing over time.

It's important to remember the difference between constant rate formation flow and stabilised formation flow.

The stabilised formation flow may go through a gradually changing flow rate due to formation pressure change, while the productivity index stays constant.

On the other hand, the constant rate formation flow may not represent a stabilised formation flow as the bottom-hole pressure and productivity index maybe still in transition after the last rate change.

The WFP methods are not applicable if the well flow is not stabilised even if the flow rate is maintained constant.

There are two special reservoir flow regimes which are both stabilised and maintain constant flow rate: steady state regime (SS) and pseudo-steady state regime (PSS).

The steady state regime (SS) regime is reached when the flow is stabilised with the full pressure support at the external boundary.

The pseudo-steady state (PSS) regime is reached when the flow is stabilised with no pressure support at the external boundary.

In both above cases, the drawdown and flow rate will stay constant upon productivity stabilisation.

As for formation and bottom-hole pressure in PSS they will be synchronously varying while in SS they will be staying constant.

The table below is summarizing the major differences between SS and PSS regimes.

		Steady state regime (SS)	Pseudo-steady state (PSS)
Boundary		Full pressure support	No pressure support
Productivity index	$\begin{array}{l}J(t) = \frac{q}{\Delta p}\end{array}$	constant	constant
Flow rate	$\begin{array}{l}q(t)\end{array}$	constant	constant
Drawdown	$\begin{array}{l}\Delta p(t) = p_e(t) - p_{wf}(t)\end{array}$	constant	constant
Botom-hole pressure	$\begin{array}{l}p_{wf}(t)\end{array}$	constant	varying
Formation pressure	$\begin{array}{l}p_e(t)\end{array}$	constant	varying

It's again important to avoid confusion between the termines stationary conditions (which mean that refered properties are not chaning in time) and stabilised flow conditions which may admit pressure and rate vraition.

In practice, the productivity index is usually not known at all times as there is no routine procedure to assess it.

It is usually accepted that a given formation takes the same time to stabilise the flow after any change in well flow conditions and the stabilisation time is assessed based on the well tests analysis.

Although, this is not strictly true and the flow stabilisation time depends on well-formation contact and reservoir property variation around a given well.

This is also compromised in multi-layer formations with cross-layer communication.

The conventional WFP – Well Performance Analysis is perfomed as the $\begin{array}{l}\{ p_{wf} \ {\rm vs} \ q \}\end{array}$ cross-plot with two model curves:

IPR – Inflow Performance Relation – responsible for resevroir deliverability (see below)

OPR – Outflow Performance Relation ( also called TPR – Tubing Performance Relation and VFP – Vertical Lift Performance ) – responsible for well deliverability (see below )

The intersection of WFP – Well Flow Performance and OPR curves represent the stabilized flow (see Fig. 1)


Fig. 1. The stablised flow rate is represnted as junction point of WFP – Well Flow Performance and OPR curves.	Fig. 2. The dead well scenario.

Given a tubing head pressure $\begin{array}{l}p_s\end{array}$ the WFP Junction Point will be dynamic in time depending on current formation pressure (see Fig. 2) and formation saturation (see Fig. 3).


Fig. 2. A sample case of stablised flow rate as function of formation pressure.	Fig. 3. A sample case of stablised flow rate as function of formation water saturation and corresponding production water-cut.

Workflow

Check the current production rate against the production target from FDP
If the diffference is big enough to justify the cost of production optimization (see point 8 below) then proceed to the step 3 below
Assess formation pressure based on well tests
Simulate IPR/OPR based on the current WOR/GOR
Calculate the stabilized flow bottom-hole pressure
Gather the current bottom-hole pressure $\begin{array}{l}p_{wf}\end{array}$
Check up the calculation aganst the actual $\begin{array}{l}p_{wf}\end{array}$
Recommend the production optimisation activities to adjust bottom-hole pressure $\begin{array}{l}p_{wf}\end{array}$ :
- adjusting the choke at surface
- adjusting the pump settings from surface
- changing the pump depth
- changing the tubing size
- changing the pump

The above workflow is very simplistic and assumes single-layer formation with no cross-flow complications.

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 $\begin{array}{l}p_{wf}\end{array}$ and surface flow rate $\begin{array}{l}q\end{array}$ during the stabilised formation flow:

(1)	$\begin{array}{l}\displaystyle p_{wf} = p_{wf}(q)\end{array}$

which may be non-linear.

The IPR analysis is closely related to well PI – Productivity Index $\begin{array}{l}J_s\end{array}$ which is defined as below:

(2)	$\begin{array}{l}\displaystyle J_s(q_o) = \frac{q_o}{p_R-p_{wf}}\end{array}$

for oil producer with liquid flowrate $\begin{array}{l}q_{liq} = q_O + q_W\end{array}$ (water and oil at surface conditions)

(3)	$\begin{array}{l}\displaystyle J_s(q_G) = \frac{q_G}{p_R-p_{wf}}\end{array}$

for gas producer with gas flowrate $\begin{array}{l}q_G\end{array}$ at surface conditions

(4)	$\begin{array}{l}\displaystyle J_s(q_g) = \frac{q_{GI}}{p_{wf}-p_R}\end{array}$

for gas injector with gas flowrate $\begin{array}{l}q_{GI}\end{array}$ at surface conditions

(5)	$\begin{array}{l}\displaystyle J_s(q_w) = \frac{q_{WI}}{p_R-p_{wf}}\end{array}$

for water injector with water flowrate $\begin{array}{l}q_{WI}\end{array}$ at surface conditions

where

$\begin{array}{l}p_R\end{array}$

field-average formation pressure within the drainage area $\begin{array}{l}V_e\end{array}$ of a given well: $\begin{array}{l}p_R = \frac{1}{V_e} \, \int_{V_e} \, p(t, {\bf r}) \, dV\end{array}$

Based on above defintions the aribitrary WFP – Well Flow Performance can be wirtten in a general form:

(6)	$\begin{array}{l}\displaystyle p_{wf} = p_R - \frac{q}{J_s}\end{array}$

providing that $\begin{array}{l}q\end{array}$ has a specific meaning and sign as per the table below:

$\begin{array}{l}-\end{array}$	for producer
$\begin{array}{l}+\end{array}$	for injector
$\begin{array}{l}q=q_o\end{array}$	for oil producer
$\begin{array}{l}q=q_g\end{array}$	for gas producer or injector
$\begin{array}{l}q=q_w\end{array}$	for water injector or water producer or water production from oil producer

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The Productivity Index can be constant or dependent on bottom-hole pressure $\begin{array}{l}p_{wf}\end{array}$ or equivalently on flowrate $\begin{array}{l}q\end{array}$ .

In general case of multiphase flow the PI $\begin{array}{l}J_s\end{array}$ features a complex dependance on bottom-hole pressure $\begin{array}{l}p_{wf}\end{array}$ (or equivalently on flowrate $\begin{array}{l}q\end{array}$ ) which can be etstablished based on numerical simulations of multiphase formation flow.

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 $\begin{array}{l}q = q (p_{wf})\end{array}$ as alternative to (6).

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) $\begin{array}{l}J_s = \rm const\end{array}$ and WFP – Well Flow Performance plot is reperented by a straight line (Fig. 1)

Fig.1. WFP – Well Flow Performance plot for constant productivity (water and dead oil)

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:

(7)	$\begin{array}{l}\displaystyle J_s = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + \epsilon+ S}\end{array}$

where $\begin{array}{l}\sigma = \Big \langle \frac{k} {\mu} \Big \rangle \, h = k \, h\, \Big[ \frac{k_{rw}}{\mu_w} + \frac{k_{ro}}{\mu_o} \Big]\end{array}$ – water-based or water-oil-based transmissbility above bubble point

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,

$\begin{array}{l}\epsilon = 0.5\end{array}$ for steady-state SS flow and $\begin{array}{l}\epsilon = 0.75\end{array}$ for pseudo-steady state PSS flow.

The alternative form of the constant Productivity Index WFP – Well Flow Performance is given by:

(8)	$\begin{array}{l}\displaystyle \frac{q}{q_{max}} = 1 -\frac{p_{wf}}{p_R}\end{array}$

where $\begin{array}{l}q_{max} = J \, p_R\end{array}$ 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).

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:

(9)	$\begin{array}{l}\displaystyle \frac{q}{q_{max}} = \Bigg[ \, 1- \Bigg( \frac{p_{wf}}{p_R} \Bigg)^2 \, \Bigg]^n\end{array}$

where $\begin{array}{l}n\end{array}$ 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:

(10)	$\begin{array}{l}\displaystyle a \, q + b \, q^2 = \Psi(p_R) - \Psi(p_{wf})\end{array}$

where $\begin{array}{l}\Psi\end{array}$ – is pseudo-pressure function specific to a certain gas PVT model, $\begin{array}{l}a\end{array}$ is laminar flow coefficient and $\begin{array}{l}b\end{array}$ is turbulent flow coefficient.

It needs two well tests at two different rates to assess $\begin{array}{l}\{ q_{max} \, , \, n \}\end{array}$ or $\begin{array}{l}\{ a \, , \, b \}\end{array}$ .

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).

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:

(11)	$\begin{array}{l}\displaystyle \frac{q}{q_{max}} = 1 - 0.2 \, \frac{p_{wf}}{p_R} - 0.8 \Bigg(\frac{p_{wf}}{p_R} \Bigg)^2 \quad , \quad p_b > p_R > p_{wf}\end{array}$

Undersaturated Oil IPR

For undersaturated oil reservoir $\begin{array}{l}p_R > p_b\end{array}$ 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 $\begin{array}{l}p_{wf} > p_b\end{array}$ 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 $\begin{array}{l}p_{wf} < p_b\end{array}$ 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.

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:

(12)	$\begin{array}{l}\displaystyle \frac{q}{q_b} = \frac{p_R - p_{wf}}{p_R - p_b} \quad , \quad p_R > p_{wf} > p_b\end{array}$

(13)	$\begin{array}{l}\displaystyle q = (q_{max} - q_b ) \Bigg[ 1 - 0.2 \, \frac{p_{wf}}{p_b} - 0.8 \Bigg(\frac{p_{wf}}{p_b} \Bigg)^2 \Bigg] + q_b \quad , \quad p_R > p_b > p_{wf}\end{array}$

with AOF $\begin{array}{l}q_{max}\end{array}$ related to bubble point flowrate $\begin{array}{l}q_b\end{array}$ via following correlation:

(14)	$\begin{array}{l}\displaystyle q_{max} = q_b \, \Big[1 + \frac{1}{1.8} \frac{p_b}{(p_r - p_b)} \Big]\end{array}$

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).


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:

(15)	$\begin{array}{l}\displaystyle \frac{q_o}{q_{o, \, max}} = 1 - 0.52 \, \frac{p_{wf}}{p_R} - 0.48 \Bigg(\frac{p_{wf}}{p_R} \Bigg)^2\end{array}$

(16)	$\begin{array}{l}\displaystyle \frac{q_w}{q_{w, \, max}} = 1 - 0.72 \, \frac{p_{wf}}{p_R} - 0.28 \Bigg(\frac{p_{wf}}{p_R} \Bigg)^2\end{array}$

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).


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

OPR – Outflow Performance Relation

OPR – Outflow Performance Relation also called TPR – Tubing Performance Relation and VLP – Vertical Lift Performance represents the relation between the bottom-hole pressure $\begin{array}{l}p_{wf}\end{array}$ and surface flow rate $\begin{array}{l}q\end{array}$ during the stabilised wellbore flow under a constant Tubing Head Pressure (THP):

(17)	$\begin{array}{l}\displaystyle p_{wf} = p_{wf}(q)\end{array}$

which may be non-linear.

Fig 3. OPR for low-compressible fluid

Fig 4. OPR for compressible fluid

References

Joe Dunn Clegg, Petroleum Engineering Handbook, Vol. IV – Production Operations Engineering, SPE, 2007

Michael Golan, Curtis H. Whitson, Well Performance, Tapir Edition, 1996

William Lyons, Working Guide to Petroleum and Natural Gas production Engineering, Elsevier Inc., First Edition, 2010

Shlumberge, Well Performance Manual

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