Use of the Pressure Derivative for Diagnosing Pressure-Transient Behavior
Authors: Ehlig-Economides C. 
Published: December 2007

Technology Today Series articles provide useful summary informationon both classic and emerging concepts in petroleum engineering. Purpose:To provide the general reader with a basic understanding of a significantconcept, technique, or development within a specific area of technology.

The combined plot of log pressure change and log derivative ofpressure change with respect to superposition time as a function of log elapsedtime was first introduced by Bourdet et al. as an aid to type-curve matching. Features that are hardly visible on the Homer plot or are hard to distinguishbecause of similarities between one plot or are hard to distinguish because ofsimilarities between one reservoir system and another are easier to recognizeon the pressure-derivative plot. Once the patterns have been diagnosed on thepressure-derivative plot. Once the patterns have been diagnosed on the log-logplot. specialized plots can be used to compute reservoir parameters or the datacan be matched to a type curve.

The Homer plot has been the most widely accepted means for analyzingpressure-buildup data since its introduction in 1951. The slope of the lineobtained by plotting pressure vs. log Horner time is used to compute thereservoir permeability. (Homer time is the log of production time plus shut-intime divided by shut-in time.) The extension of this line to the time 1 hourafter the start of the buildup provides a means for calculating the skinfactor. The extension of this line to when the Homer time equals 1 is theextrapolated pressure used to determine the average reservoir pressure.pressure. Another widely used aid to pressure-transient analysis is the plot oflog pressure. change vs. log elapsed (shut-in) time. This plot serves twopurposes. First, the data can be matched to type curves, which are plots ofanalytically generated reservoir response patterns for specified reservoirmodels. Second, the type curves can illustrate the expected trends inpressure-transient data for a large variety of well and reservoir systems,

The visual impression afforded by the log-log presentation has been greatlyenhanced by the introduction of the pressure derivative. In practice, thederivative of the pressure change is taken with respect to the superpositiontime function, which corrects for variations in the surface flow rate thatoccurred before the flow period being analyzed. As such, it represents theslope of the generalized Homer plot for buildup data. When the data produce astraight line on a semilog plot, the pressure-derivative produce a straightline on a semilog plot, the pressure-derivative will, therefore, be constant. That is, the log-log pressure-derivative plot will be flat for that portion ofthe data that can be correctly plot will be flat for that portion of the datathat can be correctly analyzed as a straight line on the Homer plot.

Many analysts rely on the plot of log-log pressure vs. pressure derivativeto diagnose which reservoir model can pressure derivative to diagnose whichreservoir model can represent a given pressure-transient data set. Patternsvisible in the log-log diagnostic and Homer plots for five frequentlyencountered reservoir systems are shown in Fig. 1. The simulated curves in Fig.1 were generated from analytical models. For each case, the log-log plotillustrates the features typically seen in real data. The curves on the leftrepresent buildup responses; the derivatives were computed with respect to theHomer time function. The curves on the right show what the same examples looklike on a plot of pressure vs. log Homer time.

For each log-log plot, the upper curve is the pressure change, delta p, vs.the shut-in time, delta t, and the lower curve is the pressure changederivative, (delta p)'delta t. Patterns in the pressure derivative that arecharacteristic of a particular pressure derivative that are characteristic of aparticular reservoir model are shown in a different type of line that isreproduced on the Homer plot. The portions of the derivative curves that appearflat determined where to draw the lines on the Homer plots, which weredetermined from a least-squares fit using the points between the arrows on theplot. When the Homer plot line has been diagnosed from the derivative response, the values computed for permeability, skin, and extrapolated pressure will bebased on the radial flow response required for the Homer analyst

The Homer plots were drawn with Homer time increasing on the horizontal plotaxis. This means that the earliest data points appear to the right of the plotand the last data point points appear to the right of the plot and the lastdata point appears farthest to the left. For this reason, the flow regimesrepresented by different line types appear in reverse order on the Homerplots.

Using common response patterns like those shown in Fig. 1 as a reference, even the novice can begin to spot trends in actual data that characterizecertain well/reservoir systems. Once the system has been diagnosed, variousportions of the data can be replotted in specialized plots that produce a linefor points within a specific range of values identified on the log-logpressure/pressure-derivative diagnostic plot.

The following examples should help the reader to discern what to look for inthe log-log diagnostic plots shown in Fig. 1

Example A illustrates the most common response-that of a homogeneousreservoir with wellbore storage and skin. Wellbore-storage derivativetransients are recognized as a "hump" in early time. The flat derivativeportion in late time is easily analyzed as the Homer semilog straight line.

Example B shows behavior of an infinite conductivity, which ischaracteristic of a well that penetrates a natural fracture. The half slopes inboth the pressure change and its derivative result in two parallel lines duringthe flow regime, representing linear flow to the fracture.

Example C shows the homogeneous reservoir with a single vertical planarbarrier to flow or a fault. The level of the second-derivative plateau is twicethe value of the level of the first-derivative plateau, and the Horner plotshows the familiar slope-doubling effect.

Example D illustrates the effect of a closed drainage volume. Unlike thedrawdown pressure transient, which has a unit-slope line in late time that isindicative of pseudosteady-state flow, the buildup pressure derivative dropspseudosteady-state flow, the buildup pressure derivative drops to zero. Thepermeability and skin cannot be determined from the Homer plot because noportion of the data exhibits a flat derivative for plot because no portion ofthe data exhibits a flat derivative for this example. When transient dataresemble Example D, the only way to determine the reservoir parameters is witha typecurve match.

Example E exhibits a valley in the pressure derivative that is indicative ofreservoir heterogeneity. In this case, the feature

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