The Conventional Well Test Derivative in the log-log plot

Rate-normalized pressure and derivative data from the PBU test are plotted versus the shut-in duration in a log-log scale.
The conventional derivative data are calculated from the derivative of pressure with respect to the superposition time.
Remember the Superposition Time ?
This is the monstrous function that we saw in the note about The Superposition Plot
Why do we derive in such a way ?
to highlight the radial flow regime, but more on this in the Well Test Theory and Practice course.


Main Statements

The main statements can be summarized by the figure below.
There are three main statements to understand the derivative and ∆P plot:


1. A stabilization in the derivative plot could be indicative of radial flow regime. If so, the level of that stabilization is inversely proportional to KH/u (K: permeability, H net reservoir thickness and u: fluid viscosity). As a result, a lower stabilization level would indicate a higher KH, and a higher stabilization level a lower KH.
During the radial flow regime across the entire net reservoir thickness, all PBU derivative plots should share the same stabilization level. If not, the stabilization might not represent radial flow regime, or well performance might be changing, or the flow rate data must be checked. A higher level of derivative stabilization could indicate an under-estimated rate previous to the test, a lower level an over-estimated rate. It is worth noticing that any adjustment in the flow rate should be limited to the rate uncertainty (10-20%).
Other typical flow regimes can also be recognized on the derivative plot, as shown in the video training course.


2. During radial flow regime, the vertical separation between the derivative and DP plots is representative of the total skin. Higher separation will mean a higher skin, smaller separation smaller skin.


3. The shut-in duration or Delta-T is linked to the radius from the well, it reflects the distances from the wellbore. Thus at small ∆t, we are looking near the wellbore, and in general we will see the “wellbore storage and skin”: a unit-slope straight line followed by a hump (if the well is damaged). When you start to produce the well, production at surface is due to the expansion of the fluid in the wellbore. When you shut a well in, flow at sand face will not stop immediately.
As ∆t increases, we see further away from the well. At large Dt, far away from the well, we will see some boundaries, and this is represented by an increase in the derivative.


Use of the derivative plot


  • Evolution of the derivative shape over time: identify typical flow regimes and changing flow conditions.
A derivative plot can be used to identify radial flow regime and other typical flow regimes over time (linear, bilinear, spherical flow regimes, etc…). The evolution of the derivative shape by overlaying the data from different PBU tests on the same log-log plot will help to identify these regimes and to notice any changing flow conditions in the near wellbore region.


  • Calculation of well and reservoir parameters with an interpretation model
Once confidence in the derivative shape is obtained and typical flow regimes identified, an interpretation model (or type curve model) can be selected. This represents a possible well and reservoir response. Flow regimes are then confirmed with a straight-line analysis: derivative data can be fit with straight lines of different slopes. This will result in first estimates of the well and reservoir parameters, later refined by matching the ΔP and derivative plots possibly by using a non-linear regression tool.
Much more info and details are available in the Well Test Theory and Practice course.


Distortions on the conventional derivative

The derivative is not measured but calculated. While the initial constant-rate derivative is obtained with respect to the log of time, the PBU derivative is calculated with respect to the superposition time. This may add some distortions on the PBU derivative, depending on the previous production history or for example when the end of the previous flow period is not in radial flow.
A “notorious” distortion in the conventional derivative (in blue) is shown below, with a channel response. The red plot shows the theoretical initial constant-rate drawdown response (deconvolved response).


  • Channel response:
From 100 hours, the conventional derivative drops below the radial flow regime. If the test stops here, the channel response could potentially be interpreted as a region of higher mobility and/or storativity away from the well or as a closed reservoir.
The derivative follows then a half-unit slope straight line at later times. The channel response seems delayed.
Much more info and details will be available in the training video called Deconvolution.


Other effects to consider

The shape of the derivative can be affected by a number of factors such as noise, data quality, pressure or time errors at start and end of flow period, wellbore phase redistribution which could create a minimum or a discontinuity, etc… It is recommended to be cautious with late time fast changing trend and to overlay different derivative responses to build some confidence.
A general rule of thumb states that any effect which lasts less than half a log cycle may not be caused by the reservoir.
These effects could be removed from the derivative data by removing the corresponding analysis pressure points from the analysis.
Otherwise, these effects will have to be discarded if the regression tool is used to match the pressure data.
For more information or for a discussion on this topic, please don’t hesitate to contact us.

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