WATERRESOURCES RESEARCH, VOL. 30,NO. 11,PAGES 2945-2957, NOVEMBER 1994
Slugtests in partially penetratingwells ZafarHyder, JamesJ. Butler, Jr., Carl D. McElwee, and Wenzhi Liu KansasGeological Survey, University of Kansas, Lawrence
Abstract.A semianalyticalsolution is presentedto a mathematicalmodel describing theflow of groundwaterin responseto a slugtest in a confinedor unconfinedporous formation.The modelincorporates the effectsof partialpenetration, anisotropy, finite- radiuswell skins, and upper and lower boundariesof either a constant-heador an impermeableform. This modelis employedto investigatethe error that is introduced intohydraulic conductivity estimates through use of currentlyaccepted practices (i.e., Hvorslev,1951; Cooper et al., 1967)for the analysisof slug-testresponse data. The magnitudeof the error arisingin a varietyof commonlyfaced field configurationsis the basisfor practicalguidelines for the analysisof slug-testdata that can be utilizedby fieldpractitioners.
Introduction that the parameter estimatesobtained using this approach must be viewed with considerableskepticism owing to an Theslug test is one of the mostcommonly used techniques error in the analytical solution upon which the model is by hydrogeologistsfor estimatinghydraulic conductivityin based. the field [Kruseman and de Ridder, 1989]. This technique, In terms of slug tests in unconfined aquifers, solutionsto whichis quite simple in practice, consistsof measuringthe the mathematicalmodel describingflow in responseto the recoveryof head in a well after a near instantaneouschange induced disturbance are difficult to obtain because of the in waterlevel at that well. Approachesfor the analysisof the nonlinear nature of the model in its most general form. recovery data collected during a slug test are based on Currently, most field practitioners use the technique of analyticalsolutions to mathematical models describingthe Bouwer and Rice [1976; Bouwer, 1989], which employs flow of groundwater to/from the test well. Over the last 30 empirical relationshipsdeveloped from steady state simula- years,solutions have been developed for a number of test tions usingan electrical analog model, for the analysis of slug configurationscommonly found in the field. Chirlin [1990] tests in unconfinedflow systems.Dagan [1978] presentsan summarizesmuch of this past work. analytical solution based on assumptionssimilar to those of In terms of slug tests in confined aquifers, one of the Bouwer and Rice [1976]. Amoozegar and Warrick [!986] earliestproposed solutions was that of Hvorslev [1951], summarizerelated methods employed by agricultural engi- whichis based on a series of simplifyingassumptions con- neers. All of these techniquesresult from the application of cerningthe slug-inducedflow system (e.g., negligiblespe- severalsimplifying assumptions to the mathematicaldescrip- cific storage, finite effective radius). Much of the work tion of flow to a well in an unconfinedaquifer (e.g., negligible followingHvorslev has been directed at removing one or specificstorage, finite effectiveradius, representationof the moreof these simplifyingassumptions. Cooper et al. [1967] water table as a constant-headboundary). As with the developeda fully transientsolution for the caseof a slugtest confinedcase, the ramificationsof these assumptionshave in a well fully screenedacross a confinedaquifer. Moench not yet been fully evaluated. andHsieh [1985] extendedthe solutionof Cooper et al. to In this paper a semianalyticalsolution is presented to a thecase of a fully penetratingwell with a finite radiuswell mathematicalmodel describingthe flow of groundwater in skin.A numberof workers[e.g., Dougherty and Babu, 1984; responseto an instantaneouschange in water level at a well Hayashiet al., 1987]have developedsolutions for slugtests screenedin a porousformation. The model incorporatesthe in wellspartially penetrating isotropic, confined aquifers. effectsof partial penetration,anisotropy, finite-radius well Butlerand McElwee [ 1990]presented a solutionfor slugtests skinsof either higheror lower permeabilitythan the forma- inwells partially penetrating confined aquifers that incorpo- tion as a whole, and upper and lower boundariesOf either a ratesthe effectsof anisotropyand a finite-radiusskin at the constant-heador an impermeableform. This model can be testwell. In mostfield applications, the methodsof Hvorslev employedfor the analysisof data from slug tests in a wide [1951]or Cooperet al. [1967]are used.The error that is variety of commonlymet field configurationsin both con- introducedinto hydraulic conductivity estimates by employ- fined and unconfinedformations. Although packers are not ingthese models in conditionswhere their assumptionsare explicitlyincluded in the formulation,earlier numerical work inappropriatehas not yet beenfully evaluated.Note that has shown that such a model can also be used for the Nguyenand Pinder [1984] proposed a methodfor the anal- analysisof multilevel slug-testdata when packers of moder- ysisof data from slugtests in wells partiallypenetrating ate length(0.75 m or longer) are employed[e.g., Bliss and confinedaquifers that has receiveda fair amountof use. Rushton, !984; Butler et al., !994a]. Recently,however, Butler and Hyder [1993]have shown The major purposeof this paper is to use this solutionto Copyright!994 by theAmerican Geophysical Union. quantifythe errorthat is introducedinto parameterestimates Papernumber 94WR01670. as a result of using currently accepted practices for the 0043-1397/94/94WR-01670505.00 analysisof responsedata from slugtests. The magnitudeof
2945 2946 HYDER ET AL.: SLUG TESTS IN PARTIALLY PENETRATING WELLS
H(0) = H 0 (3) whererw is the screen radius ILl, B isaquifer thickness [L], H islevel of water in well [L], andH0 isheight of initial slug, equalto level of water in well at t = 0 (H(0)), [L]. Aquitard The boundaryconditions are the following:
Legend h2(o•, z, t) = 0 t > 0 0 -< z -
Aquifer • screenedinterval Ohi(r,0, t) Ohi(r,B, t) =0 rw
J rskJ -1 fd+b h l(rw, z, t) dz = H(t) t >0 (6) Aquitard bald Figure 1. Cross-sectionalview of a hypothetical confined aquifer (notation explained in text). Oh•(rw, z, t) •rr c2 dH(t) 2•rr•Kr• Or b dt l-l(z) t > 0 (7)
whered is distancefrom the top of the aquiferto the topof the error arising in a variety of commonly met field config- the screen[L]; b is screenlength [L]; r c is radius of well urationsserves as the basisfor practicalguidelines that can casing(casing and screendo not have to be of equal radius) be utilized by field practitioners. Although such an investi- [L]; andVl(z) is the boxcarfunction, equalto zero at z < d, gationof parameter error could be carded out usingeither a z > b + d, and equal to 1 otherwise. numerical or analytical model, the analytical model de- In order to ensure continuity of flow between the skin and scribedin the previousparagraph is employedhere in order the formation, auxiliary conditions at the skin-formation to provide a convenient alternative for data analysis when boundary (r = rsk) must also be met: the error introducedby conventionalapproaches is deemed too large for a particular application. hl(rsk, z, t)= h2(rsk,z, t) O-
Ohl(rsk, Z, t) Oh2(rsk,Z, t) Statement of Problem Krl -' Kr2 (9) Or Or The problem of interest here is that of the head response, as a function of r, z, and t, producedby the instantaneous O-< z0 introductionof a pressuredisturbance into the screenedor open section of a well. For the purposesof this initial Equations (1)-(9) approximate the flow conditions of in- development, the well will be assumedto be located in the terest here. Appendix A provides the details of the solution confinedaquifer shown in Figure 1. Note that, as shownon derivation.In summary,the approachemploys a seriesof Figure 1, there is a well skin of radiusrsk that extends integraltransforms (a Laplace transformin time and a finite through the full thickness of the aquifer. The skin has Fourier cosinetransform in the z direction) to obtain func- transmissiveand storageproperties that may differ from the tions in transform space that satisfy the transform-space formationas a whole. Flow propertiesare assumeduniform analogsof (1)-(9). The transform-spacefunction that is within both the skin and formation, although the vertical obtainedfor the head in a partially penetratingwell witha (K z) and radial (Kr) componentsof hydraulicconductivity finite-radiuswell skinin an anisotropicconfined aquifer can may differ. be written in a nondimensional form as The partial differential equation representingthe flow of groundwaterin responseto an instantaneouschange in water (y/2)fI tI)(p) -- (10) level at a central well screenedin a porousformation is the [1 + (3•/2)pfI] samefor both the skin and the aquifer and can be written as where(I)(p) is the Laplacetransform of H(t)/Ho, p is the i 1 Ohi Kzi 02hi Ohi Laplace-transformvariable, a = (2r•2Ss2b)/rc2, 3/ TMKr2[ 82hOr2 +-•-t-r Or '•r/), 8'•'= (S•il•Kri]"•' (1) Krl , and wherehi is the headin zonei [L]; Ssi is the specificstorage of zonei [I/L]; Kzi, Kri are the verticaland radial compo- D,= f(+l{Fj'•[Fc(w)f•]} drl nents, respectively, of the hydraulic conductivityof zone i [L/T]; t is time [T]; r is radial direction [L]; z is vertical direction, z = 0 at the top of the aquifer and increases whereto is the Fourier-transform variable, Fc(tO) is the finite downward[L]; i is the zonedesignator, for r -< rsk, i = 1, Fouriercosine transform of El(z), Fj -• is theinverse finite Fourier cosinetransform, andfor rsk--< r, i = 2; and rskis the outerradius of skin[L]. The initial conditions can be written as [A2K0(vl) -- Allo(Vl) ] hi(r, z, O)= h2(r,z, O)= 0 rw < r < c• 0 < z < B (2) I•I[A2K!(Vl) + Alll(Vl) ] ' HYDER ET AL.: SLUG TESTS IN PARTIALLY PENETRATING WELLS 2947
•1= z/b;•' = d/b;vi = (½/2to2q_Rip)O.5. ½i = (Ai/a2)ø'5' 5.0 - Ai -- Kzi/Kri;a = b/rw;R1 = 7a/2A;R 2 = a/2; A = Ss2/Ssl; 4..0 ...... a = 0.1 .... at = 0.001 at = 1.0e-5
Al=KO(Vl•sk)Kl(V2•sk)-(TN•)Ko(v2•:sk)Kl(Vl•sk);3.0 A2=Io(Vl•sk)Kl(v2•:sk)+(•)Ko(V2•sk)II(Vl•sk);---- a= 1.0e-7 N = Vl/V 2 •:sk= rsk/rw.
For the unconfined case the upper no-flow boundary 1.0 conditionin (5) is changed into a constant-headboundary condition,so the upper and lower boundary conditionsare
rewritten as 0.0 i ill' I 0.0()01 0.001 o.1 hi(r, O, t) = O r w < r < oo t>0 (11)
Ohi(r, B, t) Figure2. Plot of conductivityratio (Cooperet al. estimate O• = 0 rw< r < oo t > 0 (12) (Kest)over actual conductivity (Kr))• versus ½((Kz/K r)1/2/ (b/rw))as a functionof a((2rw2Ssb)/r•)for the case of a well AppendixA alsoprovides the detailsof the solutionderiva- screenednear the center of a very thick aquifer (/3 = 64, • = tion for the unconfined case. The transform-spacefunction 32). that is obtained for the head in a partially penetratingwell with a finite-radius well skin in an anisotropic unconfined [1967]for the skinand no-skin cases, respectively. Similarly, aquifercan be writtenin a nondimensionalform as if the test well is assumedto be partially screenedacross an (•/2)a* isotropic,confined aquifer, (10) reducesto the Laplace- ß uo(p) = (13) spaceform of the solutionof Doughertyand Babu [1984, [1 + (y/2)pll*] equation(62)]. Butler et al. [1993b] describeadditional where•uc(P) is the Laplacetransform of H(t)/Ho for the checksperformed with a numericalmodel to verify the unconfinedcase; solutionsproposed here.
n* = {Ff•[Fs(to*)f•]} Ramifications for Data Analysis f••+l As discussedin the introduction,the primary purposeof this paperis to evaluatethe error that is introducedinto F,(to*)is the modifiedfinite Fourier sine transform of[2(z); parameterestimates through use of currentlyaccepted prac- and to* is the Fourier transform variable for the modified sine ticesto analyzeresponse data from slugtests performed in transform. conditionscommonly faced in the field. This evaluationis For expressionsof the complexityof (10) and (13), the carriedout by using(10) and (!3) to simulatea seriesof slug analyticalback transformationfrom transformspace to real tests.The simulatedresponse data are analyzedusing con- spaceis only readilyperformed under quite limited condi- ventionalapproaches. The parameterestimates are then tions.in the generalcase the transformationis best per- comparedwith the parametersemployed in the original formednumerically. Numerical evaluationof the Fourier simulationsto assessthe magnitudeof the error introduced transformsand their inversions was done here using discrete into the estimatesthrough use of a particular approachfor Fouriertransforms [Brigham, 1974], thereby allowing com- the data analysis.The simulationand analysisof slugtests putationallyefficient fast Fourier transformtechniques were performedin this work usingSUPRPUMP, an auto- [Cooleyand Tukey, 1965]to be utilized.This approach, matedwell-test analysispackage developed at the Kansas whichis brieflyoutlined in AppendixB, did not introduce GeologicalSurvey [Bohling and McEtwee, 1992]. significanterror into the inversionprocedure. An algorithm Partial Penetration Effects developedby Stehfest[!970], which has been found to be of greatuse in hydrologicapplications [Moench and Ogata, The first factor examined here was the effect of partial 1984],was employedto performthe numericalLaplace penetrationon parameter estimates in a homogeneousaqui- inversion. fer (no-skincase). Figure 2 displaysa plot of the hydraulic Severalchecks were performed in orderto verifythat (10) conductivityratio (K½st/K r) versus½, where ½ is the square and(13) are solutionsto the mathematicalmodel outlined rootof the anisotropy ratio (Kz/Kr) •/2 over the aspect ratio here.Substitution of (!0) and(13) intothe respectivetrans- (b/rw), for a configurationin whichthe upperand lower form-spaceanalogs of (1)-(9) and (11)-(12)demonstrated boundaries are at sucha large distancefrom the screened thatthe proposedsolutions honor the governingequation intervalthat they have no effect(/3 = B/b = 64, •r= d/b • andauxiliary conditions in all cases.In addition,if the test 32). In this casethe hydraulicconductivity estimates are wellis assumedto be fully screenedacross an isotropic, obtainedusing the solutionof Cooperet al. [1967],which confinedaquifer, (10) reduces to theLaplace-space form of assumesthat the well is fully screenedacross the aquifer thesolutions of Moenchand Hsieh [!985] and Cooperet al. (i.e., flow is purelyradial). Figure 2 showsthat the error 2948 HYDER ET AL.' SLUG TESTS IN PARTIALLY PENETRATING WELLS
arising from the radial flow assumption diminisheswith decreasesin $. This is as expected, since $ reflects the 5.0 - proportionof verticalto radialflow in the slug-inducedflow -- o• = 0.1 system. Decreasesin $ correspondto decreasesin the 0.001 4.0 1.0e-5 anisotropyratio or increasesin the aspectratio, the effectof 1.0e- 7 both of which is to constrain the slug-induced flow to the interval bounded by the top and bottom of the well screen 3.0 (i.e., the proportionof radial flow increases).In addition, Figure 2 showsthat the error in the conductivity estimates decreasesgreatly with increases in a, the dimensionless storageparameter. This is in keeping with the resultsof Hayashi et al. [1987], who noted that, for a constantaspect
1.0 ratio, vertical flow decreaseswith increasesin the storage parameter.Based on Figure 2, it is evidentthat application of the Cooper et al. solutionto data from slug tests per- 0.0 formedin conditionswhere $ is less than about 0.003 should O.Ol o,1 0.001 introduce little error into the conductivity estimates. For isotropicto slightlyanisotropic systems, this • range corre-
2.5 spondsto aspectratios greaterthan about 250. Only in the case of a very low dimensionless storage parameter will significanterror (>25%) be introduced into the estimates. 2.0 -- -- •[ = 0.020.005 Note that Figure 2 should be consideredan extension of the -½,'= o.'•o findingsof Hayashi et al. [!987] to the case of slug testsin open wells, a more commonconfiguration for groundwater 1.5 applicationsthan the shut-inpressurized slug test configura- tion that they examined. ..-.'"' Currently, the most common method for analysis of slug / .....,""' ..., testsin partially penetratingwells in confinedaquifers is that proposedby Hvorslev [1951]. Hvorslev developeda model that can be used for the analysisof slugtests performedin a screenedinterval of finite length in a uniform, anisotropic, b vertically unbounded medium. Figure 3a displays a plot o.oo• 'o'.6', 6.", analogousto Figure 2 for the case of the Hvorslev model being used to obtain the conductivity estimates. Note that the Hvorslev model requires the use of a "shape factor," which is related to the geometry of the well intake region. 4.0 The shapefactor used in Figure 3a is that for case8 describedby Hvorslev [1951] and results in the following expressionfor the radial componentof hydraulicconductiv- ity:
Fc 2 In {1/(2½)+ [1 + (1/(2½))2]1/2} KHv= 2bTo (14) v ,-.2.0 ..- whereKHv is the estimatefor the radial componentof hydraulicconductivity obtained using the Hvorslevmodel 1.0 ...... c• = 0,1 o: = 0.001 andTo is thebasic time lag, the time at whicha normalized • = 1.0e-5 headof 0.37is reached.As the aspectratio gets large (1/2½ c• = 1.0e-7 getslarge), (14) will reduceto Hvorslev's expressionfor a 0,0 ...... , , , , , .... fully penetratingwell (case9) if the effectiveradius (distance 0.()01 0.01 0.'1 beyondwhich the slug-induceddisturbance has no effecton heads)is setequal to the screenlength in case9. Note that Figure 3. Plot of conductivity ratio (Hvorslev estimate the anisotropyratio, whichappears in the ½term, andKHV (Kest)over actual conductivity (Kr)) versus•(Kz/Kr)•/2/ are perfectlycorrelated in (14), so theseparameters cannot (b/rw)) for the case of a well screenednear the center of a be estimatedindependently. very thick aquifer (/3 = 64, •' = 32). (a) Hvorslev estimates obtainedwith (14) (anisotropyratio known) as a functionof In Figure3a, all analyseswere performed using (14) while t•((2r•2Ssb)/rc2).(b) Hvorslev estimates obtained with (14) assumingthat the anisotropyratio was known.Given •e (anisotropyratio unknown) as a function of & (• term with difficultyof reliablyestimating the degreeof anisotropyin an assumedanisotropy ratio) for a = 1.0 x 10-5. (c) naturalsystems, this assumption must be consideredrather Hvorslev estimatesobtained with the fully penetratingwell unrealistic.Therefore, the analyses were repeated assuming variant of the Hvors!ev model (assumingan effective radius that the degreeof anisotropywas not known. However, 9 2 of 200rw) as a functionof a((2r•,Ssb)/rc). sincethe anisotropy ratio and KHV cannot be estimated independently,some value for theanisotropy ratio must be HYDER ET AL.: SLUG TESTS IN PARTIALLY PENETRATING WELLS 2949 assumedfor the analysis.This assumptionof an arbitrary 4.0 - anisotropyratio will give rise to an apparent0 (•) value, tt whichis the squareroot of the assumedanisotropy ratio over ¾, = 0.1 /" . theaspect ratio. Figure3b displaysresults obtained for slug -- -- • = 0.02 testsanalyzed using different •* values. When consideredin 3.0 orderof decreasingmagnitude, the • curvescorrespond to - ¾,= 0.001 .,/'/// aspectratios of 10,50, and200, respectively, for thecase of anassumed anisotropy ratio of 1 (a commonassumption in '-2.0 field applications).These curves will apply to different ",e' aspectratios when an anisotropyratio other than 1 is assumed.
Often,field analysesare performedusing the fully pene- 1.0 tratingwell model of Hvorslev (case9). For this approach, someassumption must be made concerning the effective radiusof the slugtest. In a frequentlycited publication,the U.S. Department of the Navy [1961] recommendsthat an 1 -7 10-6 10-5 10-4 10 -• 10-2 10-• effectiveradius of 200 times the well radius be employed. Figure3c displaysthe error that is introducedinto conduc- tivity estimateswhen that recommendationis adopted. Figure 4. Plot of conductivity ratio (Hvorslev estimate Figure3a indicatesthat the estimatesprovided by (14) will (Kest)over actual conductivity (Kr)) versusa((2rw2Ssb)/rc 2) be reasonable for moderate to small values of dimensionless asa functionof •(Kz/K r) 1/2/(b/rw))for thecase of a well storageif the anisotropy ratio is known. At larger a, how- screenednear the center of a very thick aquifer (/3 • 64; •' • ever, the error introduced into the parameter estimates 32; • = 0.005). increasesbeyond the limit of what is consideredreasonable for this investigation (+-25%). Note that in Figure 3a, as in the remaining figures of this paper, the smallest • value [e.g., Freeze and Cherry, 1979] recommend use of the plottedis 0.001. This is a result of the relationshipsshown in isotropic form of (14), these results indicate that such an Figure 2, which indicate that, except in the case of very approachis only appropriatein isotropic to slightly aniso- small values of dimensionless storage, the Cooper et al. tropic systems.This recommendationwill result in a consis- modelis the appropriate tool for analysisfor ½ values less tent underpredictionof hydraulicconductivity in moderately than 0.001. to strongly anisotropic systems. Figure 3b indicates that the quality of the estimates Figure3c indicatesthat the fully penetratingwell model of providedby (14) will be dependenton the assumedapparent Hvorslev (using an effective radius of 200 times the well ½ (•) value for the case of an unknown anisotropyratio. radius) is appropriate in conditions where •t is less than This figure demonstratesthat for each • value there is a about 0.01 for moderate to small values of dimension!ess rangeof actual •for whichthe Hvorslevmethod will provide storage.This ½range corresponds to an aspectratio greater reasonableestimates. Although it is difficult to summarize than 100 for isotropic systems. For strongly anisotropic the results of Figure 3b succinctly, it is clear that, if the systems(Kz/K r considerablyless than !), the aspectratios assumedanisotropy is moderatelyclose to the actualaniso- at which the fully penetratingwell model becomesappropri- tropy (within a factor of 2-3), the Hvors!ev estimate will ate are much smaller. Given the form of the fully penetrating meetthe criterionof reasonabilityemployed here (_+25%).It well model of Hvorslev employed here and the earlier canbe readily shownthat the • curves of Figure 3b are discussedrelationship between the fully and partially pene- relatedto one anotherby a simplemultiplicative factor. This tratingvariants of the Hvorslev model, it shouldbe clearthat relationshipenables curves for • values other than those the curves on Figure 3c correspondto a q/* value of 0.005. consideredhere to be generatedby multiplyingthe Kest/Kr Usingthe relationshipsdiscussed in the previousparagraph, ratio for one of the curves given in Figure 3b by a factor the curvesplotted in Figure 3c can be employedto generate consistingof the ratio of the naturallogarithm term from (14) all needed• curvesfor commonvalues of the dimension- for the curve to be generatedover the same term for the less storageparameter. Table 1 presents the results from curvein Figure 3b. Although several standardreferences Figure3c in a tabularform, so that the reader can generate the curve neededfor a particular application. Since the ½* curves can be readily related to one another, the results Table 1. Tabulated Values of the Conductivity Ratio for presentedin the remainderof this paper will be for one thePlots of Figure3c particular½* value (½* = 0.005),which as statedabove, also a= 0.1 a=0.001 a= 1.0x !0 -5 a= !.0 x !0 -7 correspondsto thefully penetratingwell modelof Hvorslev. , The tabulated values for all the curves presented here are 1.00x 10-3 3.196 1.249 0.867 0.803 given by Hyder [ !994]. 2.23 x 10-3 3.198 1.275 0.950 0.909 Figures 3a-3c show that the quality of the Hvorslev 3.16 x 10-3 3.203 1.293 1.00! 0.964 7.07 x 10-3 3.221 1.374 1.150 1.125 estimatesdeteriorates rapidly as dimensionlessstorage in- !.00 x 10-2 3.244 1.429 1.233 1.2!0 creasesabove 0.001. Figure 4 graphically displaysthe large 2.22 x 10-2 3.330 1.641 1.49! 1.470 errors that are introduced into parameter estimates as a 3.20 x 10-2 3.399 1.774 1.638 1.615 approaches1 for the sameconditions as shownin Figure3c. 7.10 x 10-2 3.693 2.225 2.108 2.076 Clearly, the Hvorslev model must be used with extreme 1.00x 10-! 3.920 2.508 2.388 2.347 cautionat large values of the dimensionlessstorage param- 2950 HYDER ET AL.: SLUG TESTSIN PARTIALLY PENETRATINGWELLS
and specificstorage. If some reasonableestimates can be made about these parameters, Figures 2--4 and Table 1 can be used to assesswhich method is most appropriate for that Constont-Heod Boundory Impermeoble Boundory specificapplication. Note that the model of Cooperet al. 3.0 shouldbe employedat all aspectratios when the dimension- lessstorage parameter is large. '½, = 0.1 BoundaryEffects The previous discussionhas focused on the effectsof partialpenetration in a verticallyinfinite system.Although one mightsuspect that most natural systemscan be consid- 1.0 !/' = 0.001 ered as vertically infinite for the purposesof the analysisof responsedata from slug tests, there may be situationsin which the upper and/or lower boundaries of the system influencethe responsedata. Thus the next factor examined 0.0 i i 'l i • ', i i ' i i • I t"-i i i i i i 'i' '1 I here was the effect of impermeable and constant-heM 0.00 1•'.00 ...... 20.00 30.00 boundaries in the vertical plane on parameter estimates. Figure5 displaysa plot of the hydraulicconductivity ratio Figure 5. Plot of conductivity ratio (Hvorslev estimate versus the normalized distance to a boundary (sr = d/b). (Kest) over actual conductivity (Kr)) versus •(d/b) as a Results are shown for both impermeable and constant-heM functionof •(Kz/K r) i/2/(b/rw)).Solid lines designate im- boundaries.In all cases,an apparent ½ (½*) value of 0.005is permeableupper boundary; dashedlines designateconstant- used to obtain the conductivity estimates. It is clear from head upper boundary; circles and triangles designateesti- Figure 5 that a boundary will only have a significanteffect mates obtained using the semi-infinite variant of the (>25%) on parameter estimates when the screen is very Hvorslev model for • values of 0.1 and 0.001, respectively; closeto the boundary (i.e., • < 1-2) and • is relatively large. /3= 64;a = 1.0 x 10-s; ½ = 0.005. If there is any degree of anisotropy in hydraulic conductiv- ity, the influence of the boundary will be considerably lessened.Note that Hvorslev [1951] also proposed a semi- eter. Since Figure 2 indicates that the Cooper eta!. model infinite,partially penetratingwell model (singleimpermeable provides excellent conductivity estimatesat large values of boundarywith screenextending to boundary) for slugtests. dimensionlessstorage, the Cooper et al. model should al- The equationfor estimationof hydraulic conductivityin this ways be employed when such dimensionlessstorage values caseis the sameas (!4) except • is used instead of 25 in the are expected. As shown by Chirlin [1989], large values of logarithmicterm. The circles and triangles in Figure 5 show dimensionlessstorage will often be reflected in a distinct the estimatesthat would be obtained usingthis model for the concave upward curvature in a log head versus time plot. confined case. Clearly, the semi-infinite variant of the Note that the ½curves on Figure 4 become nearly horizontal Hvorslevmodel is only necessaryat large • values (wellsof as a decreases. Therefore the results that are discussed in small aspectratios in isotropicaquifers). As the proportion this paper concerningthe viability of the Hvorslev model at of vertical flow decreases(• gets small), the semi-infinite a valuesof 10-5-10-7 shouldbe verygood approximations model becomes slightly inferior to the vertically infiniteform for conditionswhere a values are smallerthan 10-7. of the Hvorslevmodel. Althoughall of the parameteresti- An importantgoal of this paper is to defineguidelines for matesin Figure5 were obtainedusing the Hvorslev model, the field practitioner. Since in actual field applicationsthe the method of Bouwer and Rice [1976] would normally be aspectratio shouldbe a known quantity, guidelinesbased on employedif an unconfinedboundary is suspected.Hyder the magnitudeof the aspect ratio would be preferred. Al- and Butler [1994] provide a detailed discussionof the error though the general lack of information concerninganisot- introducedinto parameterestimates using the Bouwerand ropy and specificstorage introduces uncertainty, the results Rice model. of this section can be used to roughly define aspect ratio The above discussion focuses on results when only a guidelinesfor the analysisof responsedata from slugtests in singleboundary is influencingthe responsedata. In thin partially penetrating wells. Clearly, at large aspect ratios formations, one may face conditions when both the upper (greaterthan 250), the Cooper et al. [1967]model is the most andlower boundaries are closeenough to the screento be appropriatetool for data analysis. In strongly anisotropic affectingthe slug-test responses. Figure 6 displaysa plotof systems(Kz/K r considerablyless than 1), the Cooperet al. the hydraulicconductivity ratio versusnormalized aquifer model will be applicable at much smaller aspect ratios. thickness(/3 = B/b) for the caseof a well screenlocated at Although it is difficult to accurately estimate the degree of thecenter of theunit. Clearly,in thin confinedsystems, the anisotropyfrom slug-testresponse data, Butler eta!. [ 1993a] pair of impermeableboundaries will havea significanteffect present a simple approach that can be used to assessif onHvors!ev estimates for relativelylarge values of •. In thin significantanisotropy is present. In the general case, the unconfinedsystems, the lower impermeableboundary acts fully penetratingwell model of Hvorslev [ 1951] would be the in an oppositemanner to the upperconstant-head boundary, best approachfor analyzing responsedata from wells of so that the estimates are more reasonable than in the aspect ratios between 100 and 250. At smaller aspect ratios single-boundarycase. the partially penetrating model of Hvorslev is best in the Theresults of thissection indicate that, except in casesof mostgeneral case. However, the most appropriatemodel for very thin formations(/3 < 10), screenslocated very closeto any particularapplication will dependon the anisotropyratio a boundary(g < 5), and large valuesof • (>0.05), the HYDERET AL.' SLUGTESTS IN PARTIALLYPENETRATING WELLS 2951 assumptionof a verticallyinfinite system introduces a very smallamount of error into the parameterestimates obtained 0.1 usingthe Hvors!ev model. Thus the relationships presented inFigures 3 and4 canbe consideredappropriate for the vast majorityof field applications.Note that no analyseswere performedin this sectionusing the Cooperet al. [1967] model.In the previoussection, a rangeof aspectratios (>250)was defined for whichthe Cooperet al. modelwould providereasonable estimates. Since boundaries in the verti- calplane will only introducesizable errors into parameter estimateswhen there is a considerablecomponent of vertical flow, the effects of boundarieswill be very small if the 0.1 Cooperet al. modelis only appliedover the previously lOO.O definedrange.
Well-Skin Effects ,,, 0.001 '0'.01 0.'1 The resultsof the previoussections pertain to the caseof slugtests performedin homogeneousformations. Often, however,as illustratedin Figure 1, well drilling and devel- opmentcreate a disturbed,near-well zone (well skin) that 1 - b may differ in hydraulic conductivity from the formation in 8- whichthe well is screened.It is importantto understandthe 7- effectof well skins on conductivity estimatesin order to avoidusing estimates representative of skin propertiesto •- •,• = 2.0• characterize the formation as a whole. Figure 7a illustrates the effect of a well skin on conduc- 4- tivity estimatesobtained using the Hvorslev model(• -- 0.005)for a broad range of contrastsbetween the conductiv- ity of the skin and that of the formation. Clearly, the existence of a well skin can have a dramatic effect on the Hvorslevestimates. In the caseof a skinless permeable than ,, t/,k=10• the formation, a conductivity estimate differing from the actualformation value by over an order of magnitudecan easilybe obtained.Figure 7a displaysresults for the caseof a skin whose outer radius is twice that of the well screen 10 -1 , , , , • , , , • , , , ,"',',n il (•sk= rsk/rw= 2). Figure7b showshow the resultsdepend 0.001 0.01 0.1 on the thickness of the skin for the case of a skin I order of magnitudeless conductive than the formation (3' = 10.0). Figure 7. Plot of conductivity ratio (Hvorslev estimate (Kest) over actual formation conductivity (Kr)) versus ½i((Kzl/Krl)l/2/(b/rw))for thecase of highand low con- ductivitywell skins (/• • 64, •r.• 32, a = 1.0 x 10-5 6' = 3.0 - Impermeable Upper Boundary 0.005, qq = ½2,X = 1). (a) Hvorslev estimatesas a function Constant-Head Upper Boundary of •'(Kr2/Kri) for •sk = 2. (b) Hvorslev estimates as a functionof •sk(rsk/rw)for ?,= 10.0.
Note that when the skin radius equals the effective radius 2.0•• tP=0.1 assumedin the Hvorslevfully penetratingwell model(•sk = 200), the estimated conductivity will approach that of the skinfor small values of ½. Figure 8 displays a plot of a simulated slug test and the best fit Hvors!ev model, which is representative of all the 'v' 1.0 low-conductivity skin cases shown in Figures 7a and 7b. As ½ = 0.001 can be seen from Figure 8, the Hvorslev model matches the simulated data extremely well. In fact, a large number of additional simulations have shown that the Hvors!ev fit for the low-conductivity skin case is almost always better than 0.0 10.0 20.0 30.0 40.0 50.0 60.0 that for the homogeneouscase. This is especially true at small ½ values(moderate to large aspectratios), where the Figure6. Plot of conductivityratio (Hvorslevestimate response data for the homogeneouscase generally will (Kest) over actualconductivity (Kr)) versus13(B/b) as a display a distinct concave upward curvature [e.g., ½hirlin, functionof •(Kz/Kr)i/2/(b/rw)).Solid lines designate im- 19891. permeableupper boundary; dashed lines :designate constant- At moderateto small ½ values, an underlying assumption headupper boundary; a = 1.0x 10-5;½ = 0.005. of the Hvorslev model is that there is an effective radius 2952 HYDER ET AL.: SLUG TESTS IN PARTIALLY PENETRATING WELLS 1• .....Simulated slugtest
lO.O__...
0.1 ?, = lOO.O
0.01 ...., , , , , , , , , I ] ] '• ' ; ] ' [ • I "]'"' '• • • [ [ [ ] I I I I I I I I I I I [ I , I I l 0.0 ,50.0 100.0 1•56.0 0.001 0.01 O.tl Tinge (secs) Figure 8. Normalized head versus time plot of simulated slug test and the best fit Hvorslev model for the case of a skin Figure 9. Plot of conductivityratio (Cooperet al. estimate (Kest) over actual formation conductivity (Kr)) versus 2 orders in magnitude less conductive than the formation ½l((Kzl/Krl)l/2/(b/rw))as a functionof •/(Kr2/Krl)for (b/r1.0wx = 10 50 -] rskm- •, 0.10Kr2 m,= Kz2rw == rc 0.001= 0.05 m/s, m,KrlSst = = Kz•Ss2 = •sk=2(/3•64,•r•32, a= 1.0X 10-5 ½1= ½2A= 1). 0.00001 m/s).
proposedan empiricalequation that relatesthe Cooperet al. beyond which the slug-induced disturbance has no effect on conductivityestimate to skin and formation properties. The aquifer heads. In the low-conductivity skin case, this as- practical use of this equation is limited, however, since sumptionis a very close approximation of reality, for almost estimationof formation conductivity from the Cooper et al. all of the head drop occurs across the skin; heads in the estimaterequires knowledge of skin conductivity and thick- formation are essentially unaffected by the slug test [e.g., ness. Faust and Mercer, 1984]. Another major assumptionof this In the high-conductivity skin case, as shown in Figures 7a model is that the specific storage has no influence on the and 9, conductivity estimateswill be greater than the forma- response data. In most cases the thickness of the skin is tion conductivity as a result of a considerable amount of relatively small, so the influenceof the specificstorage of the vertical flow along the more conductive skin. The difference skin on slug-testresponses is essentially negligible.Thus the will be greatest at large ½tvalues because of the larger assumptionsof the Hvorslev model actually appear to be proportionof vertical flow under those conditions. Note that more reasonable in the low-conductivity skin case than in the differencebetween the two high-conductivity skin cases the homogeneous case. So, if one assumes an effective (y = 0.01 and 0.1) decreasesat small ½valuesbecause ofthe radiusequal to the skin radius(e.g., •sk = 200 in Figure 7b), lesseningimportance of verticalflow. If the radiusof thewell the estimated conductivity will be a reasonableapproxima- tion of the conductivity of the skin at moderate to small ½ values. Hyder and Butler [1994] show that a low- 1.00 - conductivity skin has a similar effect on parameter estimates obtained using the Bouwer and Rice [1976] method. Figure 9 illustrates the effect of a well skin on conductivity estimatesobtained usingthe Cooper et al. model. In general, 0.80: '•\ the effect of a skin on the Cooper et al. model estimatesis similar to that seen with the Hvorslev model. Again, the effect of a low-conductivity skin is quite pronounced. If the o0,60 specificstorage is assumedknown or constrainedto physi- cally realisticvalues, applicationof the Cooperet al. model '•/0.40 to data from a well with a low-conductivity skin will produce an estimatethat is heavily weighted toward the conductivity of the skin. In addition, there will always be a considerable • - '- Simulatedslug test deviation between the best fit Cooper et al. model and the responsedata in a manner similar to that shownin Figure !0. At small ½tvalues (moderate to large aspect ratios), the 0.2o•.--Cooper etal.--model '\• combination of an excellent Hvorslev fit and a systematic o.1 I lO lOO deviationbetween the Cooper et al. model and the test data Time (secs) appears to be a very good indication of a low-conductivity Figure10. Normalizedhead versus time plot of simulat.'ed skin. At larger ½values (lower aspectratios), however, such slugtest and the best fit Cooperet al. modelfor thecase of a combinationis also an indicationof a strongcomponent of a skin 2 orders in magnitudeless conductivethan the vertical flow. Note that McElwee and Butler [1992] have formation(parameters as in Figure8). HYDER ET AL.: SLUG TESTS IN PARTIALLY PENETRATING WELLS 2953 screenis set to the nominalscreen radius in the analysis, 250 (much smaller aspect ratios for strongly anisotropic therewill alwaysbe an offsetbetween the high-conductivityformations). In systemswith a large dimensionlessstorage skincases and the homogeneouscase at small ½ values,as (a > 0.01) the Cooperet al. model shouldprovide reasonable shownin Figures 7a and 9. estimatesat virtually all commonly used aspect ratios. The Sincethere is a very small head drop in the radial direction viability of this model at ½ < 0.003 is only in questionfor acrossa high-conductivityskin, one might expectthat pa- configurationswith very small values of dimensionlessstor- rameterestimates for the high-conductivityskin casecould age(a < 10-6). beconsiderably improved by assumingthe radius of the well 2. In a homogeneousformation the Hvorslev model screenequals the radius of the high-conductivityskin. Al- (case 8) will provide reasonableestimates of the radial thoughsuch an approachwill decreasethe offsetat small½ componentof hydraulicconductivity at moderate to small valuesdisplayed in Figures7a and 9, additionalsimulations valuesof dimensionlessstorage (a < 10-4) for a broadrange haveshown that the gains obtainedthrough this approach of ½values if the magnitudeof the anisotropyratio is known. are quite modest0ess than 10%). The reasonfor these If the anisotropyratio is not known, which is the situation smallerthan might have been expected gains is that an commonlyfaced in the field, the Hvorslev model will pro- increasein the well radius only influencesa and ½. As has vide reasonableestimates if the assumedanisotropy ratio is been shown in plots in the previous sections, hydraulic within a factor of 2-3 of the actual ratio. Table 1 allows the conductivityestimates are not stronglyaffected by moderate error introducedby the anisotropyratio assumptionto be changesin these dimensionlessvariables. The major cause readily assessedfor any value of the assumedanisotropy. If of the differences between the high-conductivity skin and the effective radius is assigneda value 200 times the well homogeneouscases shown in Figures 7a and 9 is the radius,the fully penetratingvariant of the Hvors!ev model uncertainty concerning the screen length. Since screen (case9) will provide reasonableconductivity estimates for ½ lengthis a term in the dimensionlesstime variable (•- = values less than 0.01. (tbKr2)/rc2),an errorin the screenlength estimate of a 3. Except in casesof large valuesof ½(>0.05), and very certainmagnitude directly translates into an error in the thinformations (/3 < 10)or well screenslocated very closeto estimatedhydraulic conductivity of the same magnitude. a boundary(• < 5), upperor lowerboundaries will have little Thusuncertainty about the value to use for the screenlength influence on parameter estimates obtained using conven- canintroduce considerable error into the conductivity esti- tionalapproaches. If the formationhas any degreeof anisot- mates. In the case of a partially penetrating well, the ropy in hydraulicconductivity, the rangeof conditionsover high-conductivityskin (e.g., the gravel pack) will normally which boundary effectsare significantwill be quite limited. be of greaterlength than the well screen. In this situation, In general, the assumptionof a vertically infinite system thelength of the high-conductivityskin, and not the nominal introduces a very small amount of error into parameter lengthof the well screen, is the quantity of interest. This estimates.Relationships developed for vertically infinite larger-than-the-nominalscreen length can be termed the systemsshould thus be appropriatefor most field applica- "effectivescreen length" for the purposesof this discussion. tions. In Figures7a and 9 the high-conductivityskin caseswere 4. In the case of a low-conductivity skin, neither the analyzedassuming that the nominalscreen length was the Hvorslev nor the Cooperet al. model provides reasonable appropriatescreen length for the analysis.At large½ values, estimatesof hydraulic conductivityof the formation. Both suchan approachis clearly incorrect.A more appropriate approacheswill yield estimatesthat are heavily weighted approachwould have beento attemptto estimatethe actual towardthe conductivityof the skin. The underlyingassump- effectivescreen length. If there is an adequateseal in the tions of the Hvorslev model actually appear to be more annulus,the effectivescreen length should be the lengthof reasonablein the low-conductivity skin case than in the thegravel pack up to that seal.However, in caseswhere the homogeneouscase. At small ½ values (moderateto large lengthof the high-conductivityskin is considerablylonger aspectratios) the combinationof an excellentfit of the thanthe nominalscreen length, such as in Figures7a and 9, Hvorslev model to the test data and a systematic deviation wherethe skinextends to the upperboundary of the forma- between the test data and the best fit Cooper et al. model tion,the effective screenlength will be dependenton the appearsto be a verygood indication of a low-conductivity conductivitycontrast between the formationand the skin. sldn. Furtherwork is requiredto developapproaches for estima- 5. In the case of a high-conductivityskin the Cooper et tionof the effectivescreen length in suchsituations. al. modelwill providereasonable estimates of formation conductivityat small ½ values. The fully penetratingwell variant of the Hvorslev model (effective radius 200 times the Summaryand Conclusions well radius)will provideviable estimatesat ½ valuesless A semianalyticalsolution to a modeldescribing the flow of thanabout 0.01. The qualityof the estimatesfor both models groundwaterin response to a slugtest in a porousformation can be slightlyimproved if the radius of the screenis set hasbeen presented. The primarypurpose for the develop- equalto anapproximate skin radius. At ½> 0.01the viability mentof thismodel was to assessthe viabilityof conventional of Hvors!ev conductivityestimates will strongly dependon methodsfor the analysisof responsedata from slugtests. the qualityof estimatesfor the effectivescreen length. In Theresults of this assessmentcan be summarizedas follows. such conditionsthe length and radius of the gravel pack 1. In a homogeneousformation the Cooper et al. model shouldbe used in place of the nominal screen length and willprovide reasonable estimates (within 25%) of the radial radius,respectively, for the analysisof the responsedata. componentof hydraulic conductivity for •Ovalues less than The restfits of this assessmentindicate that there are many about0.003. For isotropicto slightlyanisotropic systems, commonlyfaced field conditionsin which the convention.al this½ range corresponds to aspect ratios greater than about methodologyfor theanalysis of responsedata from slugtests 2954 HYDER ET AL.: SLUG TESTS IN PARTIALLY PENETRATING WELLS appears viable. Since the definition of what constitutesa gravelpack that extendsabove the watertable. Ongoing reasonableparameter estimate will be applicationdepen- numericaland field investigationsare currentlybeing under- dent, the user can consultthe figures of this paper to assess taken to assessthe error that is introduced throughthis if the introducederror is acceptablefor a specificapplica- assumptionand to suggestapproaches for data analysis tion. If it appears that conventional approacheswill not whenthat error is deemedunaceeptab!y large [Butler et al., provide acceptable parameter estimates for a test in a 1994b]. particular configuration,the model developedhere can be used to analyze the response data. Butler et al. [1993a] describe a series of slug tests in both consolidatedand AppendixA unconsolidated formations in which the model described in In this section the mathematical derivations of the soh. this article is employedfor the data analysis.Considerable tions discussedin the main body of the text are presented. experienceis required, however, for successfulapplication For the sake of generality, the solutions are obtained in a in configurationswith low-conductivity skins or a moderate dimensionlessform. The solutionswill be presentedhere as degreeof anisotropy owing to uncertaintiesintroduced by a transform-spaceexpressions. Details concerningthe scheme high degree of parameter correlation. usedto numericallyinvert these expressionsto real space Note that the results of this study must be consideredin are given in Appendix B. Note that the expressionsgiven light of the three major assumptionsused in the mathemat- here are only for the head within the stressedwell. Solutions ical definitionof the slug-testmodel employedin this work. for heads outside the stressed well are available from the First, in (7) we adoptedthe commonlyemployed assumption authors upon request. of a uniform radial hydraulic gradient along the well screen as a mathematical convenience. In actuality, one would ConfinedAquifer Solution suspectthat the gradient would be larger at either end of the screen, producinga U-shaped profile in the vertical plane. Equations(1)-(9) describe the flow conditionsof interest Butler et al. [1993b], however, have performed detailed here. To work with the most general form of the solution, simulations with a numerical model to show that the use of this derivation is performed using dimensionlessforms of this mathematical convenience introduces a negligible de- (1)-(9). The dimensionlessanalogs of (1)-(9) are as follows: gree of error to the results reported here and virtually all practical applications. (A1) Second, in (8) and (9) we assumed that the skin fully penetratesthe formation beingtested. Althoughthis assump- •(g, n, O)=0 g> 1 O< n < t• (A2) tion is appropriate for the case of multilevel slug tests performed in a well'fully screenedacross the formation, it is 0(0)= ] (^3) clearly not representative of reality in the general case. For tests in wells with a low-conductivity skin, however, this 02(oo, ,/, r)= 0 . > 0 0-< n -1 r>O flow in response to a slug-induced disturbance will be Or/ primarily constrainedto an interval bounded by the top and bottom of the well screen. In the case of a high-conductivity skin, this assumptionwill produce considerablymore verti- r>O (A6) cal flow in the skin than would actually occur. Butler et al. [1993b], however, have shown through numerical simulation that a slugtest performed in a partially penetratingwell with ao•(1, r/, r) ? dO(,-) a high-conductivity skin that extends to the bottom of the ------[2(•) r>O (• a[ 2 dr screenis indistinguishablefrom a slug test performed in a similar configuration in which the well screen terminates o_
• -- H/Ho 'Y= Kr2/Krl a = (2r2•bSs2)/rc2 ¾ tI)(p)-- •-[ 1 - p(I)(p)]l• (A13) = o < •>f+l [2](r/) = 1 elsewhere where •' = d/b •:sk= rsk/rw. I), = f•+l{F•-l[Fc(w) f l]) A solutioncan be obtainedfor (A1)--(A9) throughthe use ofintegral transforms [Churchill, 1972]. A Laplacetransform Fg • isthe inverse finite Fourier cosine transform. intime followed by a finiteFourier cosine transform in the ./ Solvingfor •(p) yields directionproduces a Fourier-Laplacespace analog to (A1) of thefollowing form: (•,/2)fl t!>(p) = (A14) o24,• 1 o4• [1 + (y/2)pll] (A10) 2 (q,2o,2+ =o Appendix B providesdetails of the fast Fourier transform schemeused to invert the expressionin the 11term. The where•i is the Fourier-Laplacetransform of6i,f(•, •o,p); algorithmof Stehfest[1970] was usedto performthe numer- •0is the Fourier-transformvariable, equalto (n•r)//5, n = 0, ical Laplace inversionof (A14). 1, 2, --'; and p is the Laplace-transformwadable. The Fourier-Laplace space solution to (P•10) is quite UnconfinedAquifer Solution straightforward,as (A10) is simply a form of the modified For the unconfinedease, (A5) is replaced by the dimen- Besselequation [Haberman, 1987]. A solutioncan therefore sionlessanalogs of (11) and (12): be proposedin the form •bi(•:, 0, •) = 0 • > 1 z > 0 (A15) •i-' CiKo(vi•) q-Dilo( (All) *) wherevi = (½/2to2+ Rip)ø'5;Ci, Di axeconstants; Ki is a = 0 • > 1 r > 0 (A16) modifiedBessel function of the secondkind of order i; andli O• is a modified Bessel function of the first kind of order i. A solution for (A1)-(A,4), (A6)-(A9), and (A15)-(A16) is Usingthe transform-spaceanalogs of auxziliaryconditions obtainedusing the same approachas in the confined case. (A4) and (A6)-(A9), the constantsin (A11) can be evaluated. The Fourier-Laplace expressionfor head at a radial distance Sincethe focus of interest in mostslug tests is responsesin of •: = 1 in the tanconfinedcase can be written as the stressedwell, only the transform-spaceexpression for headat a radial distance of • = 1 is givenhere: 01u½(1,to*,p) =y [1- p•u½(p)]Fs(w*)fl(A17) (A12) qbl(1, to, p)= •- [1- p(I>(p)JFc(o)f1 where4•l•c is the Fourier-Laplacetransform of 6h,, the nondimensionalf0rrn of hi for the unconfinedcase; •uc(P) whereCb(p) is the Laplacetransform of •(t), the nondimen- is the Laplacetransf'onn of the nondimensionalform of H(t) sionalform of H(t); Fc(oo) is the finite Fourier cosine for the unconfinedcase; F•(•o*) is the modifiedfinite Fourier transformof Vl(z), equal to sine transformof El(z), equal to
-- sin cos ß 0• = n •r//3 to 2 •0' sin 2 '
n=l, 2,3'--, and oJ*is the Fourier transform variable for the modified sine transform,equal to (nrr)/2/3, n = 1, 3, 5, '- -. andequal to 1, to = 0; The applicationof an inversemodified finite Fourier sine transformto (A 17) for ./within the screen and rewriting in [A2K0(v 1) - AI!0(•'l)] . termsof •uc(P) producesthe followingexpression: Vl[A2Ki(vl) + Alll(Vl)-l' (y/2)fl* = +
Al'=Ko(Vl•sk)Kl(V2•sk)-(•.).Ko(v2fsk)Kl(Vl•sk);where A2=Io(Vl•sk)Kl(V2•sk)+(•)KO(V2•$k)II(Vl•sk);11' = f•+l{Ff•[F•(w*)fl]} drl.
N = Vl/V2. The modifiedfinite Fourier sinetransform employed in the Theapplication of an inversefinite Fourier cosinetrans- unconfinedcase requiresa bit of discussion.The standard formto (A12)for ,/withinthe screen and •atilization of the finite Fourier sinetransform is quite useful when a constant Laplace-spaceanalog of (A6)produces the folloxving expres- head is maimairier at both boundaries. In the unconfined sionfor head in the stressedwell: casethe upperboundary (• = 0) is definedas a constant- 2956 HYDER ET AL.: SLUG TESTS IN PARTIALLY PENETRATING WELLS headcondition, while the lowerboundary (r/- b) is defined as a no-flow condition.Churchill [1972] presentsthe modi- fied finite Fourier sine transform Fs(n)= t•fuc(r/) sink2/• ' ' (B3) n=1,3,5,'.' whereF s is the modifiedfinite Fourier sinetransform and Fs(n)= •f(r/) sin•, 2bJdr/ fuc(r/) = Fs(w*)fi. Equation (B3)is only definedfor (A19) odd-numberedn. For ready implementationwith standard FFT algorithms,(B3) is rewritten in terms of a continuous as an example of a Sturm-Liouville transformation. When sequenceof n: this modifiedsine transformis appliedto the second-order derivativewith respectto r/, integrationby partsyields F•(n)=[1 +(-1)n+•]2" I•fuc(r/) sin• 2/S ] a,/ (B4)
0'-•'sin • 2• d•/= --O•*2qbi q-O•*•i(0) n=l, 2, 3,...
Equation(B4) is now approximatedusing a discreteFourier - (- 1)n (A20) transform: where
2 w*= (n•')/2fl n= 1, 3, 5,... Fs(n)[1+(1) +l]A k-1• fuc(Ak) sin•2N] (B5) For the boundaryconditions employed here, (A20) reduces n=l, 2,---,N-1 to Equations(B2) and (B5) can be directly implementedin --0•*2•// (A21) standardFFT algorithms.In this work an FFT algorithm givenby Presset al. [1992]was employed.The total number of samplingpoints (N) in r/was constrained,such that there Appendix B would always be at least 10 points within the screened interval. In this section,details are presentedof the procedures In order to check on the approach outlined above, an employedto numericallyinvert the transform-spaceexpres- additionalseries of simulationswas performedin whichthe sionsderived in AppendixA. As discussedin the maintext, continuous forms of the finite Fourier transforms were thefast Fourier transform (FFT) procedurewas employed to employed for the required transforms/inversions.The 11 perform the required Fourier transforms/inversionsin this term that is employed in (A14) can be written in the work. In order to demonstrate that the discrete Fourier continuous form as transformsintroduced negligible error into the numerically inverted solution,a comparisonbetween the discretesolu- tion and the continuous form is discussed. = fl + • Z n2 sin22fl cø$2 2fl ' For the confinedcase (cL (A14)), a finite Fourier cosine flfl(n=O)8t• n=l•ø f•(n)nw (n•r(l+2•)) transformwas employed. The continuousform of this trans- form can be written as (B6) The fl* that is employedin (A18) can be written in contin- Fc(n)=flsf(•)cos (n-•) d• (B1)uous form as 16/3 • whereF c is the finite Fouriercosine transform and f(r/) = '2 n=l In order to utilize the FFT procedure,(B1) is approxi- mated using a discrete Fourier transform: ß[1+ (-1)n+l] fl(n)• sin2•'•n,r sin2 (nw(.1...+.k 4• 2•')) (B7) rc() a f(ak) cos (B2) In all casesthe inversionof (B6) and(B7) produced results k--0 thatwere virtually indistinguishable from thosefound using an FFT algorithmwith (B2) and (B5). The computational n=0, 1,2,...,N-1 time,however, was significantly greater. The inverseLaplace transform,the final step of the whereN, the numberof equallyspaced points between 0 and numericalinversion procedure, was performedhere us• /•, must be an integer power of 2; and A is the interval the algorithmof $tehfest[1970]. Sixteen terms were used in betweenequally spaced points, equal to I3/N. the summationof the Stehfestalgorithm for all the cases For the unconfined case (cf. (A18)) a modified finite examinedin thiswork. Note that the procedures discusse•d Fourier sinetransform was employed.The continuousform hereare implemented in a seriesof Fortranprograms fo•d of this transform can be written as in thework by Hyderet al. [1993]. HYDER ET AL.: SLUG TESTSIN PARTIALLY PENETRATINGWELLS 2957
Admowledgments.This research was sponsored in partby the Dagan,G., A note on packer,slug, and recovery tests in unconfined Air ForceOffice of ScientificResearch, Air Force SystemsCom- aquifers,Water Resour.Res., I4(5), 929-934, 1978. mand,USAF, under grant or cooperative agreementnumber Dougherty,D. E., and D. K. Babu, Flow to a partially penetrating AFOSR91-0298. This researchwas also supported in partby the well in a double-porosityreservoir, Water Resour. Res., 20(8), Departmentof the Interior,U.S. GeologicalSurvey, through the 1116-1122, 1984. KansasWater ResourcesResearch Institute. The views and conclu- Faust, C. R., and J. W. Mercer, Evaluation of slug tests in wells sionscontained in this documentare thoseof the authorsand should containing a finite-thicknessskin, Water Resour. Res., 20(4), notbe interpretedas necessarilyrepresenting the officialpolicies, 504-506, 1984. eitherexpressed or implied, of the U.S. government. Freeze, R. A., and J. A. 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