3. INTRODUCCIÓN
3.10 FPIES EN SITUACIONES ESPECIALES
When the dimensionless pressure pDis differentiated with respect to the natural logarithm of dimension-less time tD, then
( 2-8) where pD′is the dimensionless pressure derivative with respect to dimensionless time tD.
The use of this particular form of pressure deriva-tive represents a major advancement in pressure tran-sient analysis. It was first presented to the petroleum literature by Bourdet et al. (1983). Figure 2-3 repre-sents the complete solution of Gringarten et al.’s (1979) work for an infinite-acting reservoir, comple-mented by the pressure derivative as developed by Bourdet et al. (1983).
During wellbore storage effects, the dimensionless pressure is related to dimensionless time and
dimen-p t
sionless wellbore storage by Eq. 2-7, which, when differentiated and combined with Eq. 2-8, yields
(2-9) On log-log paper, this shows a unit straight line exactly as does the dimensionless pressure.
During the radial flow period and when the semi-logarithmic approximation is in effect (Eq. 1-19),
(2-10) and, thus, the dimensionless derivative curve at late time approaches a constant value equal to 0.5.
In general, if
(2-11) where m is equal to 1.0 for wellbore storage, 0.5 for linear flow and 0.25 for bilinear flow, then
(2-12) which on log-log coordinates implies that the deriva-tive curve is parallel to the pressure curve departed vertically by log m.
The derivative is useful in pressure transient analysis, because not only the pressure curve but also the pressure derivative curve must match the
analytical solution. More importantly, the derivative is invaluable for definitive diagnosis of the test response. Although pressure trends can be confusing at “middle” and “late” times, and thus subject to multiple interpretations, the pressure derivative val-ues are much more definitive. (The terms early, mid-dle and late time are pejorative expressions for early-, midway- and late-appearing phenomena. For exam-ple, wellbore storage effects are early, fracture behavior is middle, and infinite-acting radial flow or boundary effects are late.) Many analysts have come to rely on the log-log pressure/pressure deriva-tive plot for diagnosing what reservoir model is rep-resented in a given pressure transient data set. To apply this method of analysis, the derivative of the actual pressure data must be calculated. A variety of algorithms is available. The simplest is to calculate the slope for each segment, using at least three time intervals. More sophisticated techniques also may be contemplated.
Patterns visible in the log-log and semilog plots for several common reservoir systems are shown in Fig. 2-4. The simulated curves in Fig. 2-4 were gen-erated from analytical models. In each case, the buildup response was computed using superposition.
The curves on the left represent buildup responses, and the derivatives were computed with respect to the Horner time function.
dpD d
( )
lntD =t pD D′ =tD CD.dpD d
( )
lntD =t pD D′ =0 5. ,pD ~tDm,
dpD d
( )
lntD =t dpD(
D dtD)
~mtDm,Figure 2-3. Dimensionless type curves for pressure drawdown and derivative for an infinite-acting reservoir with wellbore storage and skin effect (see discussion of type-curve use in Bourdet et al., 1983).
Dimensionless time, tD/CD pD and p′D (tD/CD)
0.1 1 10 100 1000 10,000 100
10
1
0.1
CDe2s
103 1030 1020
1015
10 1023
108 104 1010 106
0.3 0.1
Figure 2-4. Log-log and semilog plots for common reservoir systems.
Log-Log
Diagnostic Horner Plot Specialized Plot
A
B
C
D
E
F
G
H
C
Wellbore storage Infinite-acting radial flow From specialized plot
kb2
Wellbore storage Linear channel flow
From specialized plot
Wellbore storage Partial penetration Infinite-acting radial flow
Wellbore storage
Sealing fault Infinite-acting radial flow
Wellbore storage No-flow boundary
Wellbore storage Dual-porosity matrix to fissure flow (pseudosteady state)
Linear flow to an infinite-conductivity vertical fracture From specialized plot
kxf2
Bilinear flow to a finite-conductivity vertical fracture From specialized plot
kfw
Patterns in the pressure derivative that are charac-teristic of a particular reservoir model are shown with a different line, which is also reproduced on the Horner plot. In cases where the diagnosed behavior can be analyzed as a straight line with a suitable change in the time axis, the curves are shown as spe-cialized plots in the third column. Determination of the lines drawn on the Horner plots for each example was based on the diagnosis of radial flow using the derivative.
Example A illustrates the most common response, a homogeneous reservoir with wellbore storage and skin effect. The derivative of wellbore storage tran-sients is recognized as a hump in early time (Bourdet et al., 1983). The flat derivative portion in late time is easily analyzed as the Horner semilog straight line. In example B, the wellbore storage hump leads into a near plateau in the derivative, followed by a drop in the derivative curve to a final flat portion. A plateau followed by a transition to a lower plateau is an indication of partial penetration (Bilhartz and Ramey, 1977). The early-time plateau represents radial flow in an effective thickness equal to that of the interval open to flow into the partially penetrat-ing wellbore. Later, radial flow streamlines emanate from the entire formation thickness. The effects of partial penetration are rarely seen, except in tests that employ a downhole shut-in device or the convolu-tion of measured downhole flow rates with pressure (Ehlig-Economides et al., 1986).
Examples C and D show the behavior of vertical fractures (see Chapter 12). The half-slope in both the pressure change and its derivative results in two par-allel lines during the flow regime representing linear flow to the fracture. The quarter-sloping parallel lines in example D are indicative of bilinear flow. During linear flow, the data can be plotted as pressure versus the square root of ∆t, and the slope of the line in the specialized plot is inversely proportional to √kxf2, where xfis the vertical fracture half-length in ft.
During bilinear flow, a plot of pressure versus the fourth root of ∆t gives a line with the slope inversely proportional to 4√k(kfw), where kfis the fracture per-meability in md and w is the fracture width in ft.
Example E shows a homogeneous reservoir with a single vertical planar barrier to flow or a fault. The level of the second derivative plateau is twice the value of the level of the first derivative plateau, and the Horner plot shows the familiar slope-doubling effect (Horner, 1951). Example F illustrates the
effect of a closed drainage volume. Unlike the draw-down pressure transient, which sees the unit slope in late time as indicative of pseudosteady-state flow, the buildup pressure derivative drops to zero (Proano and Lilley, 1986).
When the pressure and its derivative are parallel with a slope of 1⁄2in late time, the response may be that of a well in a channel-shaped reservoir (Ehlig-Economides and (Ehlig-Economides, 1985), as in example G. The specialized plot of pressure versus the square root of time is proportional to kb2, where b is the width of the channel.
Finally, in example H the valley in the pressure derivative is an indication of reservoir heterogeneity.
In this case, the feature is due to dual-porosity behavior (Bourdet et al., 1984).
Figure 2-4 clearly shows the value of the pressure/
pressure derivative presentation. An important advantage of the log-log presentation is that the tran-sient patterns have a standard appearance as long as the data are plotted with square log cycles. The vis-ual patterns in semilog plots are enabled by adjusting the range of the vertical axis. Without adjustment, much or all of the data may appear to lie on one line, and subtle changes can be overlooked.
Some of the pressure derivative patterns shown are similar to those characteristic of other models. For example, the pressure derivative doubling associated with a fault (example E) can also be an indication of transient interporosity flow in a dual-porosity system (Bourdet et al., 1984). The abrupt drop in the pressure derivative in the buildup data can indicate either a closed outer boundary or a constant-pressure outer boundary resulting from a gas cap, aquifer or pattern injection wells (Proano and Lilley, 1986). The valley in the pressure derivative (example H) could be an indication of a layered system instead of dual porosity (Bourdet, 1985). For these cases and others, the ana-lyst should consult geological, seismic or core analy-sis data to decide which model to use for interpreta-tion. With additional data, there may be a more con-clusive interpretation for a given transient data set.
An important use for pressure/pressure derivative diagnosis is at the wellsite. The log-log plot drawn during transient data acquisition can be used to determine when sufficient data have been collected to adequately define the infinite-acting radial flow trend. If the objective of the test is to determine per-meability and skin effect, the test can be terminated once the derivative plateau is identified. If
hetero-geneities or boundary effects are detected in the tran-sient, the test can be run longer to record the entire pressure/pressure derivative response pattern required for analysis.