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 q and bottomhole pressure p_{wf} as a function of tubing-head pressure p_s and formation pressure p_R 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.
The conventional WFP – Well Performance Analysis is perfomed as the
\{ p_{wf} \ {\rm vs} \ q \} 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 p_s 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
p_{wf}
- Check up the calculation aganst the actual
p_{wf}
- Recommend the production optimisation activities to adjust bottom-hole pressure
p_{wf}:
- adjusting the choke at surface
- adjusting the pump settings from surface
- changing the pump depth
- changing the tubing size
- changing the pump
- adjusting the choke at surface
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 p_{wf} and surface flow rate q during the stabilised formation flow:
(1) | p_{wf} = p_{wf}(q) |
which may be non-linear.
The IPR analysis is closely related to well PI – Productivity Index J_s which is defined as below:
| for oil producer with liquid flowrate q_{liq} = q_O + q_W (water and oil at surface conditions) | ||
| for gas producer with gas flowrate
q_G at surface conditions | ||
| for gas injector with gas flowrate
q_{GI} at surface conditions | ||
| for water injector with water flowrate q_{WI} at surface conditions |
where
p_R | field-average formation pressure within the drainage area V_e of a given well: 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:
(6) | p_{wf} = p_R - \frac{q}{J_s} |
providing that q has a specific meaning and sign as per the table below:
- | for producer |
+ | for injector |
q=q_{\rm liq}=q_o+q_w | for oil producer |
q=q_g | for gas producer or injector |
q=q_w | for water injector or water-supply producer |
The Productivity Index can be constant or dependent on bottom-hole pressure p_{wf} or equivalently on flowrate q.
In general case of multiphase flow the PI J_s features a complex dependance on bottom-hole pressure p_{wf} (or equivalently on flowrate q) 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 q = q (p_{wf}) 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) J_s = \rm const 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) | J_s = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + \epsilon+ S} |
where \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
,\epsilon = 0.5 for steady-state SS flow and \epsilon = 0.75 for pseudo-steady state PSS flow.
The alternative form of the constant Productivity Index WFP – Well Flow Performance is given by:
(8) | \frac{q}{q_{max}} = 1 -\frac{p_{wf}}{p_R} |
where q_{max} = J \, p_R 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) | \frac{q}{q_{max}} = \Bigg[ \, 1- \Bigg( \frac{p_{wf}}{p_R} \Bigg)^2 \, \Bigg]^n |
where n 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) | a \, q + b \, q^2 = \Psi(p_R) - \Psi(p_{wf}) |
where \Psi – is pseudo-pressure function specific to a certain gas, a is laminar flow coefficient and b is turbulent flow coefficient.
It needs two well tests at two different rates to assess \{ q_{max} \, , \, n \} or \{ a \, , \, b \}.
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) | \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 p_R > p_b 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 p_{wf} > p_b 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 p_{wf} < p_b 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) | \frac{q}{q_b} = \frac{p_R - p_{wf}}{p_R - p_b} \quad , \quad p_R > p_{wf} > p_b |
(13) | 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} |
where AOF q_{max} is defined by the following correlation:
(14) | q_{max} = q_b \, \Big[1 + \frac{p_b}{1.8 \, (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).
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) | \frac{q_o}{q_{o, \, max}} = 1 - 0.52 \, \frac{p_{wf}}{p_R} - 0.48 \Bigg(\frac{p_{wf}}{p_R} \Bigg)^2 |
(16) | \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).
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 p_{wf} and surface flow rate q during the stabilised wellbore flow under a constant Tubing Head Pressure (THP):
(17) | p_{wf} = p_{wf}(q) |
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