How To Recognize "Double-Porosity"Systems From Well TestsAlain. C. Gringarten, SPE, Scientific Software-Intercomp

IntroductionWell tests are performed to acquire qualitative andquantitative knowledge of the well and the reservoirbeing tested. Typically, a well test involvesmodification of the rate or the pressure at one ormore wells in the reservoir and observation of theresultant reservoir response (a change in pressure orrate, respectively) at the perturbed well and/oradjacent wells. The reservoir response is then used toconstruct a well test interpretation model from whichwell and reservoir parameters, such as permeabilityand skin, can be calculated.

The well test interpretation model describes thepressure or rate behavior of the actual well/reservoirsystem during the test and must be identified from theshape of the reservoir response. This information isdifferent from, but complementary to, the informationprovided by other interpretation models that can bederived, for example, from log measurements orgeologic observations. A well test interpretationmodel indicates primarily how many media withsignificantly different permeabilities and porosities areinvolved in the flow process and how these mediainteract.

One possible well test interpretation model is thedouble-porosity model, which describes double-porosity behavior. Double-porosity behavior isobtained when two different media are involved in theflow process: a higher-permeability medium thatproduces fluid into the well and a lower-permeabilitymedium that recharges the higher-permeabilitymedium. Double-porosity behavior is typical offissured reservoirs and multilayered reservoirs withhigh permeability contrast between layers.

Different double-porosity behaviors are possible,depending on the degree of interaction, orinterporosity flow, between the two constitutivemedia. The two extremes are (I) restricted, or"pseudosteady-state," interporosity flow, obtainedwhen there is a significant impediment to flow, orinterporosity skin, between the most-permeable andthe least-permeable media, and (2) unrestricted, or"transient, .• interporosity flow, obtained when thereis no interporosity skin.

The following describes the various techniquesavailable for identifying double-porosity behaviorfrom well test pressure data.

Conventional AnalYSisConventional analysis involves plotting test pressuredata vs. some function of time on a semilog plot(Figs. la and lb).

Copyright 1987 Society of Petroleum Engineers

Journal of Petroleum Technology. June 1987

In theory, double-porosity behavior yields twoparallel straight lines on a semilog plot, providedthere are no near-wellbore or outer-boundary effects.Such a semilog plot is schematically represented inFig. la. The first semilog straight line represents thehomogeneous behavior of the most-permeable mediumbefore the least-permeable medium starts recharging.As Fig. la indicates, this first straight line lastslonger for restricted interporosity flow than forunrestricted interporosity flow. The second semilogstraight line represents the homogeneous behavior ofboth media when recharge from the least-permeablemedium is fully established. The two parallel straightlines are separated by a transition zone thatcorresponds to the onset of interporosity flow. Thetransition can be a straight line in the case ofunrestricted interporosity flow. The slope of such atransition straight line is equal to half that of the twoparallel straight lines.

In practice, however, the two parallel straight linesmayor may not be present. This depends on thecondition of the well, the composition of the reservoirfluid, and the duration of the test. As a result, thesame well may yield different responses in differenttests. Fig. lb illustrates a case of double-porositybehavior where only the last semilog straight lineexists. This straight line represents the homogeneousbehavior of the total system and is not characteristicof double-porosity behavior.

Thus a semilog plot is not an efficient tool foridentifying double-porosity behavior. More generally,straight-line analysis techniques are not valid asdiagnostic tools, because an apparent straight linethrough a range of data does not necessarily provethe existence of a specific flow regime.

Log-Log AnalysisLog-log analysis involves a log-log plot of pressurechange vs. elapsed time (Fig. 2).

Double-porosity behavior yields an S-shaped log-logpressure curve on a log-log plot, as illustrated in Fig.2. The initial portion of the curve represents ahomogeneous behavior resulting from depletion inonly the most-permeable medium. This correspondsto the region labeled "homogeneous behavior (most-permeable system)" in Figs. la and lb. A transitionfollows, corresponding to interporosity flow, during ~which pressure in the two media tends to equilibrate.Finally, homogeneous behavior resumes again, as aresult of depletion in both constitutive media at thesame time. This corresponds to the region labeled"homogeneous behavior (total system)" in Figs. la

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and lb. As Fig. 2 indicates, transition may start atvery early times in the case of unrestrictedinterporosity flow; in such a case, the firsthomogeneous behavior may not be seen in practice.

Log-log analysis represents a significantimprovement over conventional semilog analysis foridentifying double-porosity behavior. It is not fullyreliable, however. For instance, the S shape isusually difficult to see in highly damaged wells; thewell behavior can then be erroneously diagnosed as

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homogeneous. Furthermore, a similar S shape mayalso be found in semi-infinite reservoirs withhomogeneous behavior.

Pressure-Derivative AnalysisPressure-derivative analysis involves a log-log plot ofthe derivative of the pressure with respect to somefunction of elapsed time vs. elapsed time (Figs. 3aand 3b).

Double-porosity behavior is characterized by theexistence of a minimum on the pressure derivative.For a test of adequate duration, this minimum can beeither preceded and followed by a stabilization, as inFig. 3a, or only followed by a stabilization, as inFig. 3b. In addition, there could be a maximum atearly times if the well is damaged (as shown in Figs.3a and 3b) or no maximum if the well is nondamagedor is stimulated. There could also be an upward ordownward trend at late times if the reservoir isbounded.

The first stabilization shown in Fig. 3a representsthe homogeneous behavior of the most-permeablemedium and corresponds to the first semilog straightline in Fig. la. The second stabilization occurs at thesame pressure-derivative value as the first one andrepresents the homogeneous behavior of the totalsystem. It corresponds to the second parallel semilog

Journal of Petroleum Technology. June 1987

straight line in Fig. Ia. The stabilization in Fig. 3brepresents the homogeneous behavior of the totalsystem and corresponds to the semilog straight line inFig. lb.

The shape of the minimum depends on the type ofdouble-porosity behavior. As shown in Figs. 3a and3b. restricted interporosity flow yields a V-shapedminimum. whereas unrestricted interporosity yields anopen U-shaped minimum. The lower part of theunrestricted interporosity flow minimum correspondsto a pressure-derivative value that is always greaterthan or equal to half the pressure-derivative value forthe stabilization level.

Pressure derivatives provide the most efficientmeans for identifying double-porosity systems fromwell test data if a suitable pressure-derivative curvecan be obtained. The main limitation comes from thequality of the pressure data available and. moreimportantly. from the algorithm used for calculatingpressure derivatives. Reliability is also greatly

Journal or Petroleum Technology. June 1987 633

improved; the only significant ambiguity is betweendouble-porosity behavior with unrestrictedinterporosity flow and homogeneous behavior with asingle sealing fault. because both exhibit similarpressure-derivative shapes.

ConclusionsDouble-porosity systems can be identified from well

-tmaata, and the best method is to use pressurederivatives. Note, however, that although it ispossible to recognize the type of double-porositybehavior exhibited by the system, it is usuallydifficult to decide whether the reservoir is naturallyfissured or multilayered. This requires additionalinformation from sources other than well testing.

JPTThis paper is SPE 16437. Technology TodaySe,le. articles provide useful summaryinformation on both classic and emerging concepts in petroleum engineering. Pur-pese: To provide the general reader w~h a basic understanding of a significant con-cept. technique. or development within a specifiC area of lechnology.