### 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:

##### With the diffusivity coefficient:

##### and the exponential integral function:

##### 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:

_{w})

##### 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 ∆t_{1} and at a reservoir radius R_{1}, we have:

##### At a time ∆t_{2}, the pressure at the point R_{1} and time ∆t_{1} will move to a distance R_{2 } in the reservoir:

##### So we have:

##### As a result, the pressure that was in the reservoir at the point R_{1} and time ∆t_{1} will move by the time ∆t_{2 }to:_{ }

##### On a log scale, the pressure profile is shifted by 0.5 log(∆t_{2})-0.5 log(∆t_{1}), 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 r^{2}/(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:

##### Hence:

##### 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.

thank you very much for this gift in this end of year.

Hi hadi_ansari238, thanks very much for this kind comment!

We are pleased to be able to help!

The TestWells team

Hi,

Thanks a lot for sharing this.

What would be the Rinv in an injection well? when dealing with heavy viscous oil?

cheers,

Hi hhendizadeh,

Thanks for your comment. With a (thermal) fracture and multiphase flow in the reservoir, the radius of investigation should -strictly speaking- not be applied to injection wells. However, it could give you some ideas, in particular about the distance to the water/oil interface or water front. This could be supported by a volume calculation. More information is available in the training course called “Pressure Transient Analysis in Producing Fields” (https://www.testwells.com/pta-in-producing-fields-content).

We hope this helps.

Best Regards,

The TestWells team

Thanks so much, really helpful article…

Appreciate your efforts

Thanks OmarBittar for your comment, much appreciated.

Thank you for this.

Since Radius of investigation is not applicable for the boundaries then, how is it possible that Saphir and PIE calculate the distance to the second boundary or even the third given that these boundaries are not equally spaced to the well location???

Are these wrong practices???

Hi Ali,

Thanks for your comment. Distances to these boundaries are obtained from the interpretation model. The radius of investigation formula can be used to get some “idea” about the distances to the boundaries or the size of the reservoir, but strictly speaking, the equation should be applied only to the radial flow regime (circular flow geometry) in a homogeneous reservoir.

We hope this helps,

Best Regards,

The TestWells team

Appreciate your quick reply.

Thank you this is very helpful.