As has been discussed in the preceding five sections, a well test response may have different behavior at different times.
The earliest time response is usually wellbore storage. Somewhat later, the response of fractures or primary porosity may be evident. At intermediate times, infinite acting radial flow may appear.
Finally, late time responses may show the effects of reservoir boundaries. This combination of responses may give rise to an overall transient such as in Figure 2.27. Clearly there are many possible combinations of effects, as can be imagined by examination of Table 2.1.
Figure 2.27
Table 2.1
Recognition and appropriate analysis of these different responses is the key to proper interpretation of a well test. Data from a real well test can start and end at any time, so one or more of the responses can be missing. Also, depending on parameter values, one response may overlap and hide another.
For example, in a dual porosity reservoir, the first semilog straight line characteristic of the secondary porosity may be completely hidden by wellbore storage (Figure 2.28).
Figure 2.28
2.13 SUPERPOSITION
One of the most powerful techniques in reservoir engineering is superposition.
This approach makes it possible to construct reservoir response functions in complex situations, using only simple basic models. Superposition is especially useful in well test analysis, since we can use it to represent the response due to several wells by adding up the individual well responses. By appropriate choice of flow rate and well location, we can also represent various reservoir boundaries.
In addition, we can use superposition in time to determine the reservoir response to a well flowing at variable rate, by using only constant rate solutions.
The principle of superposition is very simple. It says that the response of the system to a number of perturbations is exactly equal to the sum of the responses to each of the perturbations as if they were present by themselves. It should be noted in passing that the principle of superposition only holds for linear systems (in the mathematical sense), however these include most of
the standard response functions used in well test analysis, such as the constant rate radial flow, dual porosity, fractured and bounded well solutions described earlier.
To begin to understand use of superposition, consider the pressure drop in the reservoir at point A due to the production of two wells at B and C (Figure 2.29).
Figure 2.29
Assuming that wells B and C are both line source wells without wellbore storage, the pressure drop at A due to the production of both wells is (assuming tD
> 10):
(2.56)
It can be confirmed from Eq. 2.56 that the total pressure drop is equal to the sum of the individual pressure drops.
This is true for any number of wells.
Another interesting observation can be made if both wells produce at identical rate, and the point A is exactly midway between them. In this case the pressure gradient towards the other wells, thus the net flux towards either well, is zero.
Hence any point midway between the wells is a no flux point, and we can replace all such points by an impermeable barrier without affecting
the flow distribution or the pressure field (Figure 2.30).
Figure 2.30
Alternatively, if the wells are equidistant, but have equivalent flow rates opposite in sign, then the pressure drop at the midpoint will be exactly zero, since the pressure drop due to one well will be exactly canceled by the pressure rise due to the other. The net result is that all such midpoints remain at constant pressure, and the effect is identical to the situation in which a linear constant pressure boundary is present (Figure 2.31).
Figure 2.31
Thus we have discovered an important method of representing the effect of a boundary, using only very simple
pressure drop solutions for wells in an infinite reservoir. The effect of an impermeable boundary can be replicated exactly by placing an "image" well at a distance from the original well that is exactly twice the distance of the boundary from the original well. Also, we can see from Eq. 2.56 that at late time the effect of two identical wells, measured at the original well, will be given by Eq. 2.57.
(2.57)
where re is the distance to the impermeable boundary. Hence we can understand why the slope of the semilog straight line doubles as the boundary effect becomes significant (as was described earlier in Section 2.8.2, and is shown in Figure 2.32).
Why is the semilog slope not doubled for all time? Since re2 is so much larger than rw2, the second well effect does not become significant until time t becomes large (recall that the logarithmic approximation to the exponential integral function is only valid for tD/rD2 >
10).
Using the image well concept, it is very simple to create the effect of quite complex boundary shapes, including mixtures of impermeable and constant pressure boundaries, even in three dimensions. Consider for example the infinite array of images that can be used to create the effect of a rectangular boundary (Fig. 2.33).
Figure 2.32
Figure 2.33
2.14 Time Superposition -- Multirate Tests
A second important use of superposition is to add together the effects of wells at different times. Consider for example the case of Fig. 2.28 where A, B and C are all at the same point. The pressure drop of two wells at the same location, one with flow rate qB and the other with flow rate qC, is identical to the pressure drop due to a single well with flow rate qB + qC. We can imagine replacing two small pumps by one big pump. However, let us go on to consider the case where the first well starts flowing at time zero, and the second well does not start flowing until time tp (Fig. 2.34).
Figure 2.34
The net effect is of a single well flowing at rate qB for time tp, and at rate qB + qC
thereafter (Fig. 2.35).
Figure 2.35
This same superposition can be used for any number of "wells", each with constant flow rates starting at different times, thus it is possible to generate the reservoir response to a single well flowing at variable rate, using only the same constant rate solutions described already. The variable rate is approximated by a series of "stairsteps,"
as in Fig. 2.36.
Figure 2.36
When a well test contains a series of different flow rates, or a continuously
varying flow rate, the combination of the pressure transients due to the varying flow rate is called convolution.
Convolution can be understood physically in terms of the principles of superposition just described, or mathematically in terms of Duhamel's Principle, which can be stated in the context of pressure transient computation as:
(2.58) or,
(2.59) Where qD is the flow rate relative to some reference value qref. It is important to note that qref will be the flow rate used to define pwD. The prime designates the derivative with respect to time t. These equations originate from the paper by van Everdingen and Hurst (1949).
The complement of convolution is deconvolution, in which the pressure response for constant rate production can be computed from the (measured) pressure response due to the actual (multirate) flow history. The rigorous mathematical treatment of both convolution and deconvolution usually require the application of computer-based methods, so full discussion will be deferred until later (see 3.7 Desuperposition).
There are several traditional methods of handling convolution approximately in cases where the well flows in infinite acting radial flow (semilog pressure
behavior). The following description follows that of Earlougher (1977). For a series of N constant step changes in flow rate, such as those shown in Fig. 2.36, the integral in Eq. 2.58 can be rewritten:
(2.60) Where qj is the flow step between tj-1 and tj. In cases where the response can be approximated by semilog behavior (infinite acting radial flow), the pressure from Eq. 2.60 can be written:
(2.61) This equation is more convenient to use if rewritten as:
(2.62) To use this equation correctly, it is important to notice than the value of N depends on the time t at which the pressure is determined. The flow rate qN is the flow rate at the time t in Eq. 2.62.
The same consideration regarding N and qN also applies to Eq. 2.61.
Based on Eq. 2.62, a plot of
. should show a straight line with slope
162.6Bµ/kh during infinite acting radial flow behavior. Such a plot is sometimes known as a rate-normalized plot or a multirate superposition plot.
As an example of the use of rate-normalization, Fig. 2.38 shows a normal semilog plot of the drawdown test with variable flow rate shown in Fig. 2.37.
The rate-normalized plot in Fig. 2.39 shows the expected straight line behavior, whereas the semilog plot in Fig. 2.38 shows a continuously changing curve.
Figure 2.37
Figure 2.38
Figure 2.39
A similar but more accurate form of the rate-normalization idea was described by Meunier, Wittmann, and Stewart (1985), who improved the procedure by approximating continuously varying flow rate data as a series of linear "ramp"
segments instead of as the series of constant steps used in Eq. 2.60 and depicted in Fig. 2.36. This sandface rate convolution method as described by Meunier, Wittmann, and Stewart (1985) still depends on the assumption that the reservoir is responding with semilog behavior (infinite acting radial flow). In the sandface rate convolution method, Eq. 2.58 is approximated by replacing the flow rate function by a series of linear
"ramps" between adjacent flow rate measurements, and the pressure function is approximated using the semilog function described by Eq. 2.27 or 2.28, such that:
(2.63) where:
(2.64)
(2.65)
It should be noted that this procedure will fail if any of the individual flow rates are zero.