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# Boundary effects in diagnostic plots

Reservoir boundaries have significant influences on the shape of the diagnostic plot. The effects of boundaries appear following the middle-time region (infinite-acting radial flow) in a test.

## Recognizing boundaries

Recognizing the influence of boundaries can be as important as analyzing the test quantitatively. However, a problem in recognition is that many reservoir models may produce similar pressure responses. The model selected to interpret the test quantitatively must be consistent with geological and geophysical interpretations. Once the proper reservoir model has been determined, test analysis may be relatively straight-forward type-curve matching or regression analysis using modern well-test analysis software.

The shapes of the diagnostic plots for a buildup test and a drawdown test are essentially identical during the early- and middle-time regions for most tests. However, boundary effects can cause quite different shapes for a given reservoir model at late times in buildup and drawdown tests. This problem is augmented by the common use of "equivalent time" functions to analyze buildup tests on drawdown type curves. (There are different equivalent time functions for radial flow, linear flow, and bilinear flow, as discussed in more detail in Fluid flow in hydraulically fractured wells.)

Basically, equivalent time functions apply rigorously only to situations where either the producing time and the shut-in time both lie within the middle-time region or, as is commonly the case, the shut-in time is much less than the producing time before shut in.

To further complicate matters for buildup test analysis, the shape of the derivative curve depends on how the derivative is calculated and plotted. The derivative of pressure change may be taken with respect to the logarithm of either shut-in time or equivalent time. The derivative may then be plotted vs. either of these time functions, and the shape differs for each plotting function. Some pressure transient test analysis software allows the user a choice in the time function used in taking the derivative and another choice in time plotting function; for other software, the time functions used are "hard-wired." The results can be bewildering.

## Well in an infinite-acting reservoir

Infinite-acting, radial flow reservoirs are described on this page. **Figs. 1 and 2** show their diagnostic curves. For these plots, the derivative was taken with respect to shut-in time and derivative and pressure change curves are plotted vs. shut-in time. Both pressure and time are in terms of dimensionless variables. Wellbore storage distortion is not included in any of the diagnostic plots in this section.

Notice the significant difference in the shapes of both the derivative and pressure change curves for buildup and drawdown tests, with the pressure change curves flattening for buildup tests and the derivatives moving downward with an ultimate slope of –1. The time at which the flattening of the pressure change curve (and corresponding downward movement of the derivative) becomes apparent is a function of the producing time before shut-in. The longer the producing time, the longer the flattening is delayed and the longer the time the buildup diagnostic plot is essentially identical to the drawdown diagnostic plot.

**Fig. 3** is the diagnostic plot that results when the derivative is taken with respect to radial equivalent time and the time-plotting function is radial equivalent time. The drawdown and buildup curves appear to be identical for all times. However, the radial equivalent time has a maximum value of the producing time before shut-in, so, for the buildup plots, the curves terminate at these maximum values of the time plotting function, and all points "stack up" at these limiting values of the plotting function. Our conclusion is that radial equivalent time is more satisfactory as a variable for taking the derivative and as a plotting function for an infinite-acting reservoir because the shape of the diagnostic plot is the same as for a constant-rate drawdown test.

## Linear no-flow boundary

When a well is near a single no-flow boundary (**Fig. 4**) or, as a practical matter, much closer to one boundary than to any other, and when sufficient time has elapsed for the boundary to have an influence on the pressure response during the test, the characteristic diagnostic plot, as **Fig. 5** shows, results for a constant-rate drawdown test. (Wellbore storage may distort some of the earlier data on this diagnostic plot.) The derivative will double in value over approximately 1 2/3 log cycles (from 0.5 to 1.0 on a plot of dimensionless variables). Similar responses occur in naturally fractured reservoirs with transient flow from the matrix to the fractures.

**Fig. 6** is the diagnostic plot for a buildup test with the derivative taken with respect to shut-in time and plotted vs. shut-in time. (Wellbore storage may distort some of the earlier data on this plot.) The longer the producing time before shut-in, the more nearly the shape of the diagnostic plot for a buildup test resembles the diagnostic plot for a drawdown test. The derivative has a slope of –1 for shut-in times much longer than producing time before shut-in.

**Fig. 7** is the diagnostic plot for a buildup test with derivative taken with respect to radial equivalent time and plotted vs. equivalent time. Derivatives double over a small fraction of a log cycle for short producing times and, in general, the shapes of the diagnostic plots for buildup tests are similar to drawdown diagnostic plots only for longer producing times before shut-in.

**Fig. 8** is the diagnostic plot for a buildup test, with derivative taken with respect to radial equivalent time and plotted vs. shut-in time. In this case, the diagnostic plot is similar to the drawdown response, but the plots are not identical. Notice that the derivative doubles over approximately 1 2/3 log cycle. This procedure for taking the derivative and preparing the diagnostic plot is the most satisfactory of the alternatives considered.

## Linear constant-pressure boundary

When a well is much nearer a single boundary (similar to **Fig. 4**) but with a constant-pressure at that boundary and boundary effects are encountered during the test, the diagnostic plot shown in **Fig. 9** will result in a constant-rate drawdown test. (Wellbore storage effects could also occur early in the test.) The derivative has a slope of –1 at late times on the diagnostic plot.

**Fig. 10** is the diagnostic plot for a buildup test, with derivative taken with respect to shut-in time and plotted vs. shut-in time. This diagnostic plot is identical to the drawdown plot if steady state was achieved during the flow period preceding the buildup test. For other cases, with shorter producing times, the derivative has a slope steeper than the drawdown slope of –1.

**Fig. 11** is the diagnostic plot for a buildup test, with derivative taken with respect to radial equivalent time and plotted vs. equivalent time. For short producing times, the derivative falls precipitously.

**Fig. 12** is the diagnostic plot for a buildup test, with derivative taken with respect to radial equivalent time and plotted vs. shut-in time. The shapes of the diagnostic plots are similar to, but not identical to, the drawdown diagnostic plot for all producing times before shut-in. The diagnostic plot prepared in this way is the most satisfactory of the alternatives considered.

## Well in a channel

When a well is between two parallel no-flow boundaries and the pressure transient encounters both during a test long before the ends of the reservoir influence the test data, the diagnostic plot in **Fig. 13** results for a constant-rate drawdown test. Before the boundary effects, with characteristic derivative slope of 1/2, wellbore storage, radial flow (or hemiradial flow if the well is much nearer one boundary than the other) will usually appear on the diagnostic plot. Diagnostic plots with similar shapes occur for a well between two sealing faults, a hydraulically fractured well with a high-conductivity fracture, and a horizontal well during early linear flow.

**Fig. 14** is the diagnostic plot for a buildup test, with derivative taken with respect to shut-in time and plotted vs. shut-in time. The longer the producing time before shut-in, the more similar the curve shape is to the drawdown-test diagnostic plot. The derivative has a slope of –1/2 when shut-in time is much larger than producing time.

**Fig. 15** is the diagnostic plot for a buildup test with derivative taken with respect to radial equivalent time and plotted vs. equivalent time. This plot is not particularly useful for test analysis. However, linear equivalent time produces a more useful diagnostic plot as long as channel ends do not affect the pressure response.

**Fig. 16** is the diagnostic plot for a buildup test with derivative taken with respect to radial equivalent time and plotted vs. shut-in time. The derivative is similar to, but not identical to, the drawdown response. This method is the most useful for test analysis among the alternatives discussed.

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## See also

Fluid flow through permeable media