We can recover average reservoir pressure from shut-in periods for producers and injectors using well test analysis. Plotting reservoir pressure versus time for different wells will help to monitor the recovery process in the reservoir.

Calculate reservoir pressure with well test analysis

There are several ways to calculate an average reservoir pressure from a PBU/PFO test.

• Using a mathematical value P*:

In the superposition plot, the intercept of the straight line defining radial flow regime with the pressure axis will give a mathematical value called P*.

If the same production time t_P is used for every PBU test (ideally the time to reach pseudo-steady state or longer), the evolution of P* over time will reflect the evolution of reservoir pressure over time.

Radial flow regime needs to be highlighted with a straight line on the superposition plot. Then P* is obtained with its intercept with the pressure axis. Please note that the straight line is the radial flow regime straight line, which defines permeability and total skin. It is not an extrapolation from the final data points on the superposition plot.

The same process can be repeated for a different PBU test, keeping a constant production time. The difference in P* over the two PBU tests would indicate reservoir depletion.

Comment on P* method:

The P* method is a simple extrapolation of the radial flow straight line on a superposition plot, without the need to define a drainage area. For a single well with small cumulative production volume in a reservoir of infinite extent, the P* value should be equal to initial pressure. For a long term production in a reservoir with multiple producers and injectors, the evolution of P* over time would approximately represent the evolution of average reservoir pressure around a well.

However the P* method has a number of limitations. First, P* does not represent the reservoir pressure of any defined drainage area.  It might over-estimate the reservoir pressure around a production well, which could have some consequences for the voidage replacement strategy. Also, it is difficult to use these pressure values for calibration of reservoir simulation models, since those models require input of pressures in specified volumes. Then, the P* method requires some data handling before each PBU to ensure that the same production time is used.

Finally, the difference between the true reservoir pressure and P* depends on the rate before the PBU. So if the rate is not the same before each PBU, the evolution of P* over time won’t reflect the evolution of reservoir pressure over time.

This is why the second method below is preferred.

• Using a fixed drainage area and the MBH method:

The MBH (Matthews, Brons, Hazebroek) method estimates average reservoir pressure for a PBU test given the reservoir drainage area and well location.

 A reservoir drainage area has to be defined around the well and kept constant for any evaluation of average reservoir pressure. The drainage area may not be correct but the evolution of the average reservoir pressure over time should be. To help define the drainage area, we should use the nearest boundaries that are identified in well test analysis and mid-distances between neighbouring wells.

As the MBH method assumes a constant rate during the production time, the multirate production history before the PBU of interest then needs to be replaced by a constant rate production period of rate q and duration T_Pe .

 T_Pe is the equivalent Horner time:

T_Pe= Q/q with Q the cumulative production before the test and q the final flow rate before the PBU of interest.

In general, well test programs will handle this conversion to a constant rate production period.

By plotting the Horner plot (semi-log plot showing pressure versus Horner time) and defining radial flow regime with a straight line, P* value can be obtained. Then the user can select the MBH analysis and define the drainage area to get access to the calculated average reservoir pressure in this area.

The same process can be repeated for another PBU test. The multiple rate history previous to the test needs to be converted into a constant rate production period of duration equal to the Horner equivalent time t_Pe, then the Horner plot is used to define P* and the MBH method to calculate an average pressure in the specified area.

Comment on the MBH method:

The MBH method requires the input of a drainage area and as a result the method gives an average reservoir pressure for a definite region.  It is a rigorous approach for production wells with no-flow boundaries and without pressure support from injection for example. It is therefore a more appropriate input for reservoir simulation models. Also, the MBH method does not require the user to fiddle with the amount of rate history to use (unlike the P* method with the use of the same production Tp).

However,  the MBH method is not going to be exactly correct when applied to volumes which may not have strict no-flow boundaries, especially with the presence of injection wells or an aquifer.  In addition, the drainage area might be uncertain. It is therefore recommended to use an inner part of the drainage area around a given well. It may have some flow across the “fictitious” boundaries but with low pressure gradient far away from the well, this could result in small under-estimation of reservoir pressure. In this case, the pressure values obtained from this method could be seen as approximations (and a different average reservoir pressure would have been derived from a different reservoir size). However, the evolution of reservoir pressure versus time should be representative.

• Analytical Simulation

The analytical simulation is the preferred way to derive average reservoir pressure since the rate history does not need to be modified and no straight line analysis needs to be performed.

When a well model in a closed reservoir is matched to the pressure data, average reservoir pressure can be obtained for a particular PBU test.

Monitor recovery process

Average reservoir pressure values could be obtained for each PBU test and each well. The results could be shown versus time.

This will give a quick and efficient way to monitor recovery process over time, assess the strategy and update accordingly the well guidelines. For example, Well 2 might see the influence of a new water injection well while Well 3 might be impacted by the new start of production from the nearby Well 4.

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