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The radius of investigation is one of the most useful, yet least understood, concepts in well test interpretation.

2.6.1 Radius of Investigation—Drawdown. If we set the argument x of the Ei function to 1, then solve for t as a function of distance from the wellbore r_i, we have

t c r

k

=948φµ t i². . . . (2.39)

Similarly, solving for r_i, we have

r kt

i c

t

= 948φµ . . . . (2.40)

Here we have used r_i to denote the distance at which the argument of the Ei function is 1 for a given time t. This distance is referred to as the radius of investigation. The radius of investigation is a simple, yet powerful, tool for understanding the movement of a pressure transient through the reservoir.

In Fig. 2.12, the pressure distribution is shown at several different points in time. As time increases, the pres-sure transient moves further into the reservoir. The radius of investigation provides a convenient dividing line

0.01 hr 0.1 hr 1 hr 10 hr 100 hr

Distance From Axis of Wellbore 0

1000

Fig. 2.12—Radius of investigation and movement of a pressure transient through the reservoir, drawdown.

for distinguishing the region distant from the well, where the pressure is essentially unaffected by the pressure transient, from the near-well region influenced by the pressure transient.

2.6.2 Radius of Investigation—Buildup. Every change in rate at the wellbore creates a new pressure transient that propagates into the reservoir independently of other pressure transients. Fig. 2.13 shows the pressure profile in the reservoir at several points in time during a buildup, with the radius of investigation for the buildup transient calculated using the shut-in time Δt. Meanwhile, the drawdown transient continues to move into the reservoir. For example, at 1000 hours into the buildup, the drawdown transient has moved to the point labeled “r_i @ 2000 hour.”

In spite of the name, the effects of a pressure transient do not end abruptly at the radius of investigation, as can be seen from Fig. 2.7 and Fig. 2.8. In Fig. 2.7, at t = 10 hours, the pressure has fallen slightly below the initial pressure at distances shortly beyond the radius of investigation of 400 ft; in Fig. 2.8, the pressure falls below the initial pressure shortly before the radius of investigation reaches 400 ft at time t = 10 hours.

Every change in rate at the wellbore creates a new pressure transient that propagates into the reservoir indepen-dently of any other pressure transients.

2.6.3 Discussion. Although the radius of investigation was developed for single-phase flow of a slightly com-pressible liquid, we may use the radius of investigation for gas wells by evaluating the viscosity and compressibility at the initial pressure for a new well, or at the current average drainage area pressure for a producing well.

One common misconception is that a higher flow rate will cause the pressure transient to move further into the reservoir in a given time. However, as Fig. 2.14 shows, changing the flow rate has no effect on the radius of investigation.

Distance From Axis of Wellbore

Fig. 2.13—Radius of investigation and movement of a pressure transient through the reservoir, buildup.

6000

Distance From Axis of Wellbore Fig. 2.14—Effect of flow rate on radius of investigation.

As may be seen from Eq. 2.39, the only parameters that affect the radius of investigation are the for-mation permeability, k, the fluid viscosity, μ, the total compressibility, c_t, and the porosity, f. Except for the porosity, each of these parameters may vary over several orders of magnitude. As a result, for a given time t, the radius of investigation may be thousands of feet for a high permeability light oil reservoir above its bubblepoint, while that for a moderate permeability heavy oil reservoir below its original bubblepoint pressure may be only a few dozen feet. This fact will have a major influence on how pressure transient tests are conducted and how much information can be obtained from a test of a specified duration in different types of reservoirs.

Table 2.1 shows the radius of investigation after 24 hours and the time required to reach a radius of investigation of 880 ft, for an oil reservoir with typical rock and fluid properties and different permeabilities.

2.6.4 Derivation of Radius of Investigation. Why did we define the radius of investigation by setting the argument of the Ei function to 1? Other choices are certainly possible. For example, we could have chosen the radius of investigation, ˆr_i, to be the distance at which the logarithmic approximation to the Ei function is zero.

The logarithmic approximation to the Ei function is zero when the argument of the logarithm is 1, or

1 781. x=1. . . . (2.41) Thus, we have

x c r

kt

=948 t i = 1 1 781 φµ ˆ2

. , . . . (2.42) which leads to

t c r

k

=1688φµ ˆt i². . . . (2.43)

So, exactly what is special about setting the argument of the Ei function to 1? Consider the following. Taking the time derivative of Eq. 2.23, we have

∂

∂ = − −

 

 p

t

qB kht

c r kt

70 6. µexp 948φµ t ² . . . . (2.44) It is impossible to increase the radius of investigation at a given time by producing at a higher flow rate.

TABLE 2.1—RADIUS OF INVESTIGATION EXAMPLE, OIL RESERVOIR Reservoir and Fluid Properties

m ^{1.7 cp} c_o 12.0 × 10^–6 psi^–1

f ^18% c_w 3.0 × 10^–6 psi^–1

S_w 30% c_f 4.0 × 10^–6 psi^–1

c_t 13.3 × 10^–6 psi^–1

k r_i @ t = 24 hour Time required to reach r_i = 880 ft

md ft hr days

1000 2,494 3

100 789 29.8 1.25

10 249 299.0 12.5

1 79 2,990.0 125

At any fixed distance r, the magnitude of the rate of change of pressure initially increases, reaches a maximum, then decreases asymptotically toward 0. For the example drawdown test given above, the time derivative of pressure at a radius of 400 ft is shown in Fig. 2.15.

To determine the time at which the maximum occurs at given distance, we set the second derivative of the pressure to zero and solve for t as a function of r:

∂

For Eq. 2.46 to be satisfied, one of the three terms on the left side must be zero. It is easy to show that the first term decreases monotonically, approaching zero at large times:

lim .

The second term increases monotonically as t decreases, approaching zero as t approaches zero:

lim exp

Neither Eq. 2.47 nor Eq. 2.48 describes an extremum in the derivative. On the other hand, setting the third term to zero, and replacing r by r_i, gives which leads to the desired extremum. Thus, at a given distance r_i, the magnitude of the rate of change of pressure will reach a maximum at a time t given by

t c r

k

=948φµ t i², . . . (2.50)

which is the same as Eq. 2.39.

12

Fig. 2.15—Magnitude of time rate of change of pressure as a function of time, at a distance r = 400 ft from the axis of the wellbore.

Thus, we may define the radius of investigation r_i for a given time t as that distance at which the magnitude of the time derivative of pressure reaches a maximum value (in time) at the given time t.

We will revisit the radius of investigation in Section 2.8.5, after we have introduced the pressure response for PSSF from a closed circular reservoir.