/SPETecluwlogyToday SERIES

Use of the Pressure Derivative forDiagnosing Pressure-Transient Behaviorc. Ehllg-Economldes, SPE, Schlumberger

Summary. The combined plot of log pressure change and log derivative of pressure change with respect to superposition time asa function of log elapsed time was first introduced by Bourdet et al. I as an aid to type-curve matching. Features that are hardlyvisible on the Horner plot or are hard to distinguish because of similarities between one reservoir system and another are easier torecognize on the pressure-derivative plot. Once the patterns have been diagnosed on the log-log plot, specialized plots can be usedto compute reservoir parameters or the data can be matched to a type curve.

The Horner plot has been the most widely accepted means foranalyzing pressure-buildup data since its introduction in 1951. 2

The slope of the line obtained by plotting pressure vs. log Hornertime is used to compute the reservoir permeability. (Horner timeis the log of production time plus shut-in time divided by shut-intime.) The extension of this line to the time 1 hour after the startof the buildup provides a means for calculating the skin factor. Theextension of this line to when the Horner time equals 1 is the ex-trapolated pressure used to determine the average reservoirpressure. 3

Another widely used aid to pressure-transient analysis is the plotof log pressure change vs. log elapsed (shut-in) time. This plot servestwo purposes. First, the data can be matched to type curves.v>which are plots of analytically generated reservoir response patternsfor specified reservoir models. Second, the type curves can illus-trate the expected trends in pressure-transient data for a large varietyof well and reservoir systems.

The visual impression afforded by the log-log presentation hasbeen greatly enhanced by the introduction of the pressurederivative. 1.6,7 In practice, the derivative of the pressure changeis taken with respect to the superposition time function.s whichcorrects for variations in the surface flow rate that occurred beforethe flow period being analyzed. As such, it represents the slopeof the generalized Horner plot for buildup data. When the dataproduce a straight line on a semilog plot, the pressure derivativewill, therefore, be constant. That is, the log-log pressure-derivativeplot will be flat for that portion of the data that can be correctlyanalyzed as a straight line on the Horner plot.

Many analysts rely on the plot of log-log pressure vs.pressure derivative to diagnose which reservoir model canrepresent a given pressure-transient data set. Patterns visible inthe log-log diagnostic and Horner plots for five frequentlyencountered reservoir systems are shown in Fig. 1. Thesimulated curves in Fig. 1 were generated from analyticalmodels. For each case, the log-log plot illustrates the featurestypically seen in real data. The curves on the left representbuildup responses; the derivatives were computed with respectto the Horner time function. The curves on the right showwhat the same examples look like on a plot of pressure vs. logHorner time.

For each log-log plot, the upper curve is the pressurechange, ap, vs. the shut-in time, ill, and the lower curve isthe pressure change derivative, (ilp)' ill. Patterns in thepressure derivative that are characteristic of a particularreservoir model are shown in a different type of line that isreproduced on the Horner plot. The portions of the derivative

Copyright t966 Society 01 Petroleum Engineers

1280

curves that appear flat determined where to draw the lines onthe Horner plots, which were determined from a least-squaresfit using the points between the arrows on the plot. When theHorner plot line has been diagnosed from the derivativeresponse, the values computed for permeability, skin, andextrapolated pressure will be based on the radial flow responserequired for the Horner analysis. I

The Horner plots were drawn with Horner time increasingon the horizontal plot axis. This means that the earliest datapoints appear to the right of the plot and the last data pointappears farthest to the left. For this reason, the flow regimesrepresented by different line types appear in reverse order onthe Horner plots.

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

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

Example A illustrates the most common response-that of ahomogeneous reservoir with wellbore storage and skin.Well bore-storage derivative transients are recognized as a"hump" in early time. I The flat derivative portion in late timeis easily analyzed as the Horner semilog straight line.

Example B shows behavior of an infinite conductivity,which is characteristic of a well that penetrates a naturalfracture.f The half slopes in both the pressure change and itsderivative result in two parallel lines during the flow regime,representing linear flow to the fracture.

Example C shows the homogeneous reservoir with a singlevertical planar barrier to flow or a fault. The level of thesecond-derivative plateau is twice the value of the level of thefirst-derivative plateau, and the Horner plot shows the familiarslope-doubling effect. 2

Example D illustrates the effect of a closed drainagevolume. Unlike the drawdown pressure transient, which has aunit-slope line in .late time that is indicative of pseudosteady-state flow, the buildup pressure derivative drops to zero.? Thepermeability and skin cannot be determined from the Hornerplot because no portion of the data exhibits a flat derivative forthis example. When transient data resemble Example D, theonly way to determine the reservoir parameters is with a type-curve match.> .

Example E exhibits a valley in the pressure derivative thatis indicative of reservoir heterogeneity. In this case, the feature

Journal of Petroleum Technology. October 1988

10' ~p --(~p)'~t 2.0

.............. " /e,

b10

.......I RA01AL Well with Infinite••••• FLOW

.~TRANSITION .... Conductivity Vertical"",,- ....~RFLOW 0.0 ....-------_. Fracture in a

10.1 " Homogeneous Reservoir10· to') tOl 10" 10' 10' 10' 10' 10' 10' 10 ' 10'

~t (tp+~t)/At

10' 10.0

10'N"J

~p

.~~(~p)' ~ t Q. S.O C

10' lL:1············/ ,"···7 ---,0- ••••• ~ ___ •• Well with Wellbore

.:::...- WELLBORE""""'--- 'RADIAL -SEALING -, Storage and Skin in aSTORAGE FLOW FAULT

Homogeneous Reservoir10" ".0.0 with One Sealing Fault

10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10'~t (tp+~t)/~t

10' 10.0 ------------ ........-,-,~p

10'(~p)'~t d'./ "-10' >..~ 5.0 Well with Wellbore

WELLBORE '" NO • FLOW Storage and Skin in aT(i I ~ STORAGE '\ BjNDARY Homogeneous Reservoir

with Closed Outer\

" . Boundary.,10 0.0

10" 10'· Hi) 10.2 10.,

10' 10' 10' 10' 10 10' 10'

~t (tp+~I)/~110.0

10'

~~~p10' , ~ ..'<, e

(~p)'tdWell with Wellbore••.••• / PSEUDO·STEADY STATE C. 5.0

10' I~I'" •••• FLOW FROM MATRIX Storage and Skin in ac- ' / \- TO FISSURES---,.~Y ~~~~8A~Rl / •••~, ,<#"/ / '. Dual Porosity System10

RAOIAL FLOW •.•_,' /,/ '. with Pseudo-Steady( IN FISSURES) RAOIAl FLO'N(TOTAL SYSTEM) State Flow from Matrix to

10 0.0 Fractures10'

,10'

,10'

,10' 10'

.10 10 10 10 10 10 10 10 10

~t (tp+~t)/~1r:

Fig. 1-Examples A through E. adapted from Ref. 12.

Log - LogDiagnostic Plot

10

" "••••• WELLBORE ----- \- ----oi• STORAGE \ /

RADIAL FLOW

(~p)' ~I..•• I

l°'lF~ ~ ~ -. ~10

10

Journal of Petroleum Technology, October 1988

Horner Plot10.0 ,-------------------------,

aWell with WellboreStorage and Skin in aHomogeneous Reservoir

Q. 5.0

.....<.,

0.0 +----r----,.---r--r----..,--"'''''''r ....•.10° 10' 101 10' 10· 10$ 10

4.0"T""-------------------------,

1281

results from dual-porosity behavior, for the case ofpscudosteady flow from matrix to fractures. 10

Fig. I clearly shows the value of the pressure/pressure-derivative presentation. An important advantage of the log-logpresentation is that the transient patterns have a standardappearance as long as the data are plotted with square logcycles. The visual patterns in semi log plots are amplified byadjusting the range of the vertical axis. Without adjustment,many or all of the data may appear to lie on one line andsubtle changes can be overlooked.

Some of the pressure-derivative patterns shown are similar tothose characteristic of other models. For example, thepressure-derivative doubling associated with a fault (ExampleC) can also indicate transient interporosity flow in a dual-porosity system. 10 The sudden drop in the pressure derivativein buildup data can indicate either a closed outer boundary or aconstant-pressure outer boundary resulting from a gas cap, anaquifer, or pattern injection wells.? The valley in the pressurederivative (Example E) could indicate a layered system insteadof dual porosity. II For these cases and others, the analystshould consult geological, seismic, or core-analysis data todecide which model to use in an interpretation. With additionaldata, a more conclusive interpretation for a given transient dataset may be found.

An important place to use the pressure/pressure-derivativediagnosis is on the wellsite. If the objective of the test is todetermine permeability and skin, the test can be terminatedonce the derivative plateau is identified. If heterogeneities orboundary effects are detected in the transient, the test can berun longer to record the entire pressure/pressure-derivativeresponse pattern needed for the analysis.

Ref. 6 provides a method for computing the pressurederivative. Modern electronic gauges typically produce datathat are readily differentiable and, often, data from amechanical gauge produce an adequate derivative presentation.Hence, to avoid errors caused by analyzing the "wrong"straight line on a Horner plot, a look at the log-log plot ofpressure and its derivative is always recommended. With someexperience, the analyst can readily recognize the most commontransient-behavior patterns on this plot and can learn muchmore from each data set.

1282 Journal of Petroleum Technology. October 1988

Acknowledgments[ would like to acknowledge Schlumberger for permission topublish this paper and to thank Joe Martin of the SchlumbergerEducational Services staff for his painstaking effort in draftingthe illustration. .

ReferencesI. Bourdet. D. et al.: "A New Set of Type Curves Simplifies Well Test

Analysis." World Oil (May 1983) 95-106.2. Horner. D.R.: "Pressure Build-up in Wells," Proc .. Third World Pet.

Cong., The Hague (1951) Sec. 11,503-23; Pressure Analysis Methods,Reprint Series. SPE. Richardson, TX (1967) No.9, 25-43.

3. Matthews, C.S., Brons, F., and Hazebroek, P.: "A Method for De-termination of Average Pressure in a Bounded Reservoir," Trans.,AIME (1954) 201, 182-91.

4. Earlougher, R.C. Jr.: Advances in Well Test Analysis, MonographSeries, SPE, Richardson, TX (1977) 5.

5. Gringarten, A.C.: "Type-Curve Analysis:What It Can and CannotDo,"JPT (Jan. 1987) 11-13.

6. Bourdet, D., Ayoub, J.A., and Pirard, Y.M.: "Use of the PressureDerivative in Well Test Interpretation," paper SPE 12777 presentedat the 1984 SPE California Regional Meeting, Long Beach, March27-29.

7. Pirard, Y.M. and Bocock, A.: "Pressure Derivative Enhances Use ofType Curves for the Analysis of Well Tests," paper SPE 14101presented at the 1986 SPE International Meeting on Petroleum Engi-neering, Beijing, March 17-20.

8. Economides, M.J. and Nolte, K.G.: Reservoir Stimulation, Schlurn-berger Educational Services, Houston (1987) Chap. II.

9. Proano, E.A. and Lilley, I.J.: "Derivative of Pressure: Applicationto Bounded Reservoir Interpretation," paper SPE 15861 presented atthe 1986 SPE European Petroleum Conference, London, Oct. 20-22.

10. Bourdet, D. et al.: "New Type Curves Aid Analysis of Fissured ZoneWell Tests," World Oil (April 1984).

II. Bourdet, D.: "Pressure Behavior of Layered Reservoirs WithCrossflow," paper SPE 13628 presented at the 1985 SPE CaliforniaRegional Meeting, Bakersfield, March 27-29.

12. Matthews, C.S. and Russell, D.G.: Pressure Buildup and Flow Testsin Wells, Monograph Series, SPE, Richardson, TX (1967) 1, 123.

JPTThis paper is SPE 18594. Technology Today Series articles provide useful summary in-formation on both classic and emerging concepts in petroleum engineering. Purpose: Toprovide the general reader with a basic understanding of a significant concept, technique,or development within a specific area of technology.