WFP – Well Flow Performance analysis is a comparative analysis between the formation deliverability and wellbore deliverability.
It is based on correlation between surface flowrate and bottomhole pressure as a function of tubing-head pressure and formation pressure.
Most reservoir engineers exploit material balance thinking which is based on long-term well-by-well flow rates targets (where producers or injectors).
In practice, the flow rate targets are closely related to bottomhole pressure and associated limitations and require a specialised analysis to set up the optimal lifting parameters.
This is primary domain of WFP analysis.
WFP 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.
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| Steady state regime (SS) | Pseudo-steady state (PSS) |
|---|
| Boundary |
| Full pressure support | No pressure support |
| Productivity index | | LaTeX Math Inline |
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| body | J(t) = \frac{q}{\Delta p} |
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|
| | |
| Flow rate | | | |
| | LaTeX Math Inline |
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| body | \Delta p(t) = p_e(t) - p_{wf}(t) |
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|
| | |
| Botom-hole pressure | | | |
| Formation pressure | | | |
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
| LaTeX Math Inline |
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| body | \{ p_{wf} \ {\rm vs} \ q \} |
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cross-plot with two model curves:
IPR – Inflow Performance Relation represents the relation between the bottom-hole pressure
and surface flow rate
during the
stabilised formation flow:
| LaTeX Math Block |
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|
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:
| LaTeX Math Block |
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| J_s(q_{\rm liq}) = \frac{q_{\rm liq}}{p_R-p_{wf}} |
|
for oil producer with surface liquid production (water and oil) |
| LaTeX Math Block |
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| J_s(q_g) = \frac{q_g}{p_R-p_{wf}} |
|
for gas producer |
| LaTeX Math Block |
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| J_s(q_g) = \frac{q_g}{p_{wf}-p_R} |
|
for gas injector |
| LaTeX Math Block |
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| J_s(q_w) = \frac{q_w}{p_R-p_{wf}} |
|
for water injector |
where
| water, oil, gas flow rates at separator |
| field-average formation pressure withing the drainage area of a given well |
Based on these notions the general WFP – Well Flow Performance can be wirtten in univseral form:
| LaTeX Math Block |
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p_{wf} = p_R - \frac{q}{J_s(q)} |
providing that
has a specific meaning and sign as per the table below:
| for producer |
| for injector |
| LaTeX Math Inline |
|---|
| body | q=q_{\rm liq}=q_o+q_w |
|---|
|
| for oil producer |
| for gas producer or injector |
| for water injector or water-supply producer |
For a single-layer formation with low-compressibility fluid the PI does not depend on flowrate
and
WFP – Well Flow Performance plot is reperented by a straight line (Fig. 1)
This is a typical WFP – Well Flow Performance plot for water supply wells, water injectors and oil producers above bubble point.
The PI can be estimated using the Darcy equation:
| LaTeX Math Block |
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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 or water-oil 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.
For gas wells, condensate producers, light-oil producers, and oil producers below bubble point
the fluid compressibility is high, formation flow in well vicinity becomes non-linear (deviating from Darcy) and free gas slippage effects inflict the downward trend on
WFP – Well Flow Performance plot (Fig. 2).
It can be interpreted as deterioration of near-reservoir zone permeability with fluid velocity growth.
The alternative way to define the
vs
correlation is to present
as a function of
:
| Note |
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Despite of terminological similarity there is a big difference in the way WFP and Well Testing deal with formation pressure and flowrates which results in a big difference in productivity index definition and corresponding analysis. This difference is summarized in the table below:
| WFP | Well Testing |
|---|
| Formation pressure | – field-average pressure within the drainage area
| – pressure value at boudary of the drainage area
| | Flow rate | | | LaTeX Math Inline |
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| body | q_t = B_w \, q_W + \frac{B_o - R_s B_g}{1 - R_v R_s} \, q_O + \frac{B_g - R_v B_o}{1 - R_v R_s} \, q_G |
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|
– total flowrate at sandface | | Prroducivity Index | | LaTeX Math Inline |
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| body | J = \frac{q_{\rm liq}}{p_R - p_{wf}} |
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|
| LaTeX Math Inline |
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| body | J_g = \frac{q_g}{p_R - p_{wf}} |
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|
| | LaTeX Math Inline |
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| body | J = \frac{q_t}{p_e - p_{wf}} |
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|
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VLP – Vertical Lift Performance also called Outflow Performance Relation or Tubing Performance Relation represents the relation between the bottom-hole pressure
and surface flow rate
during the
stabilised wellbore flow under a constant Tubing Head Pressure (THP):
| LaTeX Math Block |
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|
p_{wf} = p_{wf}(q) |
which may be non-linear.
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| Fig 3. VLP for low-compressible fluid |
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| Fig 4. VLP for compressible fluid |
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| Fig. 5. WFP for stairated oil |
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| Fig. 6. WFP for stairated oil |
