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 -<B (4) Aquifer • screenedinterval Ohi(r,0, t) Ohi(r,B, t) =0 rw<r<o• t>0 (5) ß Kr Oz Oz J rwJ 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-<z<B t>0 (8) 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-< z<B t>0 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 transform in 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.