The Superposition Plot
If we assume the following multiplerate production profile:
From t= t_{P }onwards, the well is shutin to perform a PBU test of duration Δt
The PBU pressure data P_{ws}(Δt) or [Pi P_{ws}(Δt)] are plotted versus the superposition time (a monstruous equation…):
or a similar superposition function, with q_{n1 }the rate previous to the PBU test, and Pi the initial pressure.
This is a semilog plot (only the time axis is in log scale)
I heard about the famous Horner Plot…
Isn’t the Horner Plot the same as the Superposition Plot?
Both plots are slightly different.
The Horner Plot shows PBU pressure data versus the Horner time:
and not the superposition time.
The Horner plot is only used for the first initial PBU (the first shutin period following an initial constant rate production period). For any other shutin period, following a multiple rate history, the superposition time needs to be used.
However, both the Horner and Superposition Plots follow the same principles and rules. For example:

for production wells, the gauge pressure values at late times are indicated on the leftend side of the semilog plot (for injectors, it is the inverse).

During radial flow regime, the plot should follow a straight line of slope inversely proportional to the mobility KH/u (K:permeability, H: net reservoir thickness and u: fluid viscosity). The slope does not depend on rate.
The total skin value can also be calculated from this straight line.
Practical Use of the Superposition Plot:

Calculation of Derivative:
The conventional derivative is calculated by deriving pressure with respect to the superposition time. Thus, the idea is to use the slope of the superposition plot and a data window to define two points around the point of interest to calculate the derivative. The size of the window is controlled by some smoothing coefficient.
If the derivative curve is too noisy, the smoothing coefficient is increased until the derivative response is smooth enough. However, it is worth mentioning that oversmoothing (more than 0.2, 0.3) could introduce some erroneous features. In case of too much noise, it is therefore recommended to use the superposition plot to define the radial flow regime and extract KH and total skin.
At the end of the shutin period, the point of interest where we want the derivative to be calculated becomes closer to the last recorded point. Smoothing may not be possible, and some errors may be created at the end of the derivative response. It is recommended to be cautious regarding any downward or upward shortterm derivative trend at late times.

Calculation of KH, Skin and P* values:
The Superposition Plot is first used to identify radial flow regime by overlaying the actual data with a straight line.
The start and end of the overlay with the straight line indicate the start and end of the radial flow regime. It should coincide with the identification of radial flow regime on the derivative plot, as shown below.
If not, it probably means that the selected portion of data does not represent radial flow regime.
In the superposition plot, the intercept of the straight line with the pressure axis will give a mathematical value called P*. Under some assumptions, such as a same production time before each PBU (ideally the time to reach pseudosteady state), the evolution of P* over time could reflect the change in reservoir pressure over time.

Indication of presence of boundaries
In the above example, the straightline in the superposition plot defines the radial flow regime from 1 to 20 hours. At late times (leftend side of the Superposition plot), the deviation from the straightline is explained by the presence of boundaries.
By plotting data from different PBU tests in the Superposition Plot, a downward shift in the straight lines during radial flow regime could be used to support the presence of boundaries. However, if the multiple rate history is used with the total production time, this shift would also be expected.

Verification tool:
The Superposition plot is also a verification tool. It will compare the simulated pressure data from the interpretation model with the actual gauge pressure values.
The data in the superposition plot should be matched. If not, reasons should be understood. This could due to an incorrect initial pressure value Pi, permeability or skin value in the model.
As a quick guide, the plots below suggest the impact of erroneous model parameters:
– Error on initial pressure or distances to the boundaries:
As shown in the figure above, this error could be recognized by a shift in the data.
– Error on kh value:
As shown in the figure below, an error in Kh value gives the impression of a “rotation” in the data.
– Error on Skin value:
An error in skin value does not affect the match of the radial flow regime. The data at early times (right end side of the superposition plot) will not be matched if the skin value is erroneous.
For more information or for a discussion on this topic, please don’t hesitate to contact us.
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