Definition


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


and


It is based on correlation between surface flowrate  and bottomhole pressure  as a function of tubing-head pressure  and formation pressure  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


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.



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 supportNo pressure support
Productivity index

constant

constant

Flow rate

constant

constant

Drawdown

constant

constant

Botom-hole pressure

constant

varying

Formation pressure

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  cross-plot with two model curves:

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  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



  1. Check the current production rate against the production target from FDP

  2. If the diffference is big enough to justify the cost of production optimization (see point 8 below) then proceed to the step 3 below

  3. Assess formation pressure based on well tests

  4. Simulate IPR/OPR based on the current WOR/GOR

  5. Calculate the stabilized flow bottom-hole pressure

  6. Gather the current bottom-hole pressure

  7. Check up the calculation aganst the actual 

  8. Recommend the production optimisation activities to adjust bottom-hole pressure :

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   and surface flow rate    during the stabilised formation flow:

p_{wf} = p_{wf}(q)

  which may be non-linear. 


The IPR analysis is closely related to well PI – Productivity Index   which is defined as below:

J_s(q_{\rm liq}) = \frac{q_{\rm liq}}{p_R-p_{wf}}


for oil producer with liquid flowrate (water and oil at surface conditions)

J_s(q_G) = \frac{q_G}{p_R-p_{wf}}


for gas producer with gas flowrate at surface conditions

J_s(q_g) = \frac{q_{GI}}{p_{wf}-p_R}


for gas injector with gas flowrate at surface conditions

J_s(q_w) = \frac{q_{WI}}{p_R-p_{wf}}


for water injector with water flowrate at surface conditions

where

field-average formation pressure within the drainage area of a given well:




Based on above defintions the general WFP – Well Flow Performance 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-supply 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 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  or   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 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)


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:

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  WFP – Well Flow Performance 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 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:

\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,   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 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:

\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.


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:

\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}



Multiphase IPR


For 3-phase water-oil-gas flow the IPR analysis is perfomed on oil and watr components (see Fig. 4.1 and Fig. 4.2).


Fig. 4.1. Oil WFP – Well Flow Performance for 3-phase (water + oil + gas) formation flow

Fig. 4.2. Water WFP – Well Flow Performance for 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   and surface flow rate    during the stabilised wellbore flow under a constant Tubing Head Pressure (THP):

p_{wf} = p_{wf}(q)

  which may be non-linear. 


Fig 3. OPR for low-compressible fluid

Fig 4. OPR for compressible fluid



Sample Case 1 –  Oil Producer Analysis



Fig. 5. WFP for stairated oil



Fig. 6. WFP for stairated oil


Sample Case 2 – Water Injector Analysis



Sample Case 3 – Gas Producer Analysis



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