Radius of Investigation

The radius of investigation concept is one of the most important concepts in pressure transient analysis. We will go through a bit of theory, define the radius of investigation and see how we can derive it from well test analysis.

Theory

Let’s assume a fully completed vertical well producing at constant rate q during Δt in an infinite homogeneous isotropic reservoir with constant properties, bounded above and below by impermeable planes. At the start of production, the reservoir is at equilibrium state with a uniform pressure equal to the initial pressure Pi.

In these conditions, a solution of the diffusivity equation is given by:

Hence, we have:

Fortunately, there is a log approximation of the Ei function:

As a result, the solution of the diffusivity equation could be simplified as:

At the wellbore (r = r_w), we have:

This solution defines the fundamental flow regime in well testing: radial flow regime. In the bedding plane, the flow will have the following cylindrical path:

When the reservoir production is established, the flow-lines converge towards the well with a radial geometry.

Within the reservoir, the pressure around the well decreases due to production, with the following pressure change:

This pressure change ∆P increases with time and decreases as the radius r (distance from the well) increases. This defines the transient effect. It is also worth noticing that the pressure disturbance is proportional to the well rate q.

The figure below shows the pressure profile in the reservoir versus the radius r, at an instant t, which is created by the well producing at a constant rate q.

The minimum pressure is at the wellbore with p(r_w,∆t). Pressure increases with the distance from the well and tends towards the initial reservoir pressure Pi further away from the well.

The figure below shows the evolution of the pressure profile as the production time increases.

At a time ∆t₁ and at a reservoir radius R₁, we have:

At a time ∆t₂, the pressure at the point R₁ and time ∆t₁ will move to a distance R₂ in the reservoir:

So we have:

As a result, the pressure that was in the reservoir at the point R₁ and time ∆t₁ will move by the time ∆t₂ to: 

On a log scale, the pressure profile is shifted by 0.5 log(∆t₂)-0.5 log(∆t₁), as shown below.

.

As it can be observed on the figure above, a larger part of the reservoir is affected by this pressure disturbance as the production time increases. The transient response evolves away from the well.

The radius from the well to the affected reservoir region is called the radius of investigation Ri. As production time increases, the radius of investigation also increases.

Definition of the Radius of Investigation Ri

Let’s first use a very simple definition of the radius of investigation at the time ∆t as p(Ri,∆t)= Pi. At this stage, a more complex definition of Ri could be argued.

So we have:

And

In this case, the radius of investigation is defined as:

Here we used a simplistic approach to define the radius of investigation. In addition, we used the log approximation of the Ei function, but this is only valid for small values of r²/(4ƞ∆t). When using the exponential integral function, the definition of ΔP= 0 will not result in a finite Ri value and therefore will not work.

We could define the radius of investigation as the point where the pressure variation is the fastest.

The pressure variation is defined as the derivative of pressure with respect to time:

(dp/dt=dp/du.du/dt)

The maximum of the pressure variation is defined as:

In this case, the radius of investigation is defined as:

In the oilfield units, this gives:

(Muskat, Van Poolen, Matthews & Russell, Lee, Streltsova, Bourdarot)

Below is another definition of the radius of investigation, in the oilfield units, from Earlougher:

More definitions of the radius of investigation are available in SPE 120515.

For all these definitions, the radius of investigation increases with time with the square root of ηΔt. This is our third golden rule in the following post: Conventional Well Test Derivative.

We can also observe that the radius of investigation doesn’t directly depend on the rate q. However, the rate will affect the amplitude of the pressure signal.

So far, we have seen that the theory is applicable to a constant-rate production period, with a vertical well in a homogeneous reservoir of infinite extent. In this particular case, the flow is radial/circular towards the well, following the radial flow regime.

In practice, it is difficult to define the radius of investigation from a production period.

The theory above is applicable to a constant-rate production period, when radial flow regime or the infinite-acting transient period is visible.

It is quite difficult in practice to keep a production rate constant for a long time period. The time ∆t from the equations above cannot be defined as the longest production time period. It has to be the time when radial flow regime is present. The radius of investigation can then be estimated from the time when the radial flow regime ends.

One way to achieve a constant-rate period over a long time interval is to shut the well in and to look at the PBU test.

PBU radius of investigation

When the PBU and the drawdown responses are consistent, the radius of investigation can be derived from the PBU test.

To do so, the stabilization line is selected on the derivative plot. This defines the radial flow regime and results in a KH value. Then the radius of investigation is calculated from the formula above.

The radius of investigation can be estimated from the time when radial flow regime ends. In the PIE software, this is marked by a “cross” on the last point of the derivative stabilization line. In this example, Ri= 255 ft.

Whenever possible, the derivative plot should be used to estimate the radius of investigation. If the derivative is noisy, the superposition plot could be used.

In this case, the radius of investigation Ri depends only on the duration of the PBU test. While it is in theory independent of the previous production rate and duration, the PBU radius of investigation is limited by the accuracy of the pressure gauge. We can imagine that for a very long PBU test, the pressure changes recorded at the end of the PBU may become too small to be detected.

Nowadays with high quality gauges, this is no longer a common issue. However, some care must be taken when designing the PBU test. The company may have to increase the duration of the drawdown period or increase the flow rate.

Extend the radius of investigation with Deconvolution

The above method to extract the radius of investigation from the conventional derivative is limited by the PBU duration, i.e. by the data from a single flow period.

The Deconvolution technique could then be used to extend the radius of investigation. Indeed, Deconvolution results in the equivalent initial constant-rate production period, of duration equal to the entire duration of the test sequence. More information on the Deconvolution technique is available here: Deconvolution.

Using the example above, the deconvolution in red is defined over a longer time interval. By using the same technique with the derivative stabilization line, the radius of investigation is tripled for this particular case. In this example, Ri= 759 ft.

Does it work in reservoirs with boundaries ?

Strictly speaking, the radius of investigation is only applicable during the radial flow regime, or the infinite-acting transient period, when the flow is radial towards the well.

In case of boundaries, then the user can define the radius of investigation by using the last points in the derivative stabilization, as such:

In this derivative plot, the increase in derivative marks the presence of boundaries.

For this case, we can define the minimum connected volume. This is the minimum hydrocarbon volume that the test proves. More info on this is available here: How to define the reserves associated with a well.

An example

Let’s assume the reservoir below, with an increase in reservoir net thickness on one direction, and with the presence of boundaries.

The following test was performed: (note that this test sequence is not optimized…)

What should we do to calculate the radius of investigation ?

And which time ∆t should we use ? The production time, the longest drawdown period, the entire PBU time ?

The first thing we need to do is to plot all the PBU tests in the log-log plot (or derivative plot).

This helps us to gain some confidence in the data and it is easier to identify the radial flow regime. We then draw the stabilization line to highlight the radial flow regime, which then results in a permeability-thickness KH value.

The ∆t to use for the radius of investigation is the last point of the derivative stabilization indicative of radial flow regime. In PIE, it is automatically calculated, based on the “cross” marker in the stabilization line. As shown in the plot, this gives a radius of investigation Ri= 285 ft.

We derived below the deconvolution response in red and ensured its validity.

The deconvolved derivative in red is consistent with the conventional derivative in blue until 3 hours. Then we can note a discrepancy between deconvolution and the conventional derivative. This is because of a distortion on the conventional derivative. While deconvolution is derived with respect to the log of time, the superposition time is used for the conventional derivative. This brings some error in the conventional derivative from 3 hours onwards.

From deconvolution, we can extract a larger radius of investigation, Ri= 449 ft.

The log-log plot below shows one of the models that matches the deconvolution data. We also verified that all the other plots are matched. (click on the plot to enlarge it)

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with the following legend:

To obtain a minimum connected volume, we can close the interpretation model by adding some fictitious boundaries. Then we can reduce the distances from the well to these fictitious boundaries until a discrepancy is noted on the derivative at late times (SPE 102483). This is shown below.

The minimum pore-volume is 136.3 million cubic feet.

As a result, this test proves a minimum connected oil volume of 8.3 million barrels of oil at surface.

Another technique can be used, which is to place a unit-slope straight line on the final point of the deconvolved derivative (SPE 116575). This results in a conservative minimum connected volume of 7.2 million barrels of oil at surface.