Recession Characteristics of Groundwater Outflow and Base Flow from Mountainous Watersheds

WATER RESOURCES RESEARCH, VOL. 24, NO. 10, PAGES 1651-1658, OCTOBER 1988

Recession Characteristics of Groundwater Outflow and Base Flow From Mountainous Watersheds

YEMANE B. ZECHARIAS AND WILFRIED BRUTSAERT

Schoolqf Civil and EnvironmentalEnqineering, Cornell University, Ithaca, New York

The relationshipbetween base flow recessioncharacteristics in steepwatersheds and geomorphologic and soil parametersis investigated.The formulationfor the groundwateroutflow was obtained by means of a hydraulicapproach applied to a simpleconceptual model for a hillslope.Long-term flow data of 19 representativebasins in the AlleghenyMountain sectionof the AppalachianPlateaus were analyzedon the basis of this formulation. Results showed that the reaction factor, which is a time scale of base flow recession,is dependenton the meanland slope,the drainagedensity, and the ratio (K/f) of the hydraulic conductivity and the drainable porosity. On account mainly of the nonuniformdistribution of the physicalcharacteristics within a basin,the reaction factor for a givenwatershed is somewhatvariable with time, but the adoption of a constantvalue is usefulto representaverage conditions for a recession period.Analysis of the (K/f) dependencyshowed that macroporesand otherstructural features may greatlyaffect the watershed base flow. Evaporation from groundwater appears to constituteonly a minor portionof overallbasin evaporation.

INTRODUCTION Mountain sectionof the AppalachianPlateau physiographic Duringrainless periods, the flow in a basin'sstream system, province.The sectionconsists of a sequenceof near-horizontal whichis often referred to as baseflow is sustainedby ground- paleozoicsedimentary rocks mainly conglomerate,sandstone, waterdischarge. But evenduring wet periods,base flow con- and shale[Thornbury, 1965,p. 130)]. The rock formationsare stitutesan importantand sometimesdominant streamflow only mildy deformedand form broad, open anticlinaland componentparticularly in humidforested areas. Therefore an synclinalstructures with a gentleregional dip; consequently, understandingof the behaviorof groundwateroutflow is es- the evolutionof drainagesystems in the area has not been sentialin studiesof water budgetsand the responseof catch- stronglyaffected by structuralcontrols. Throughout the sec- mentsto varioushydrologic and climaticinputs. tion, whichis unglaciated,fluvial erosionhas created a deeply The natural groundwaterdischarge in a catchmentas a and irregularlydissected topography that is characterizedby functionof timeis largelycontrolled by the physicaland hy- steepslopes and V-shapedvalleys. Average elevations of drologicproperties of theaquifer materials, many of whichare mountainsummits range from 700 to 830 m and, depending reflectedin the morphology of the basin.Since geomorpholog- on their size,stream valleys may be up to 300-600 m deep ic characteristicscan be readilyobtained from mapsand air [e.g.,Morisawa, 1959]. As a result,the sectionconstitutes one photos,it is of interestto establishreliable relationships be- of the highestand mostrugged parts of the AppalachianPla- tweenthe groundwater outflow rate andthe controlling geo- teauprovince fAtwood, 1940, p. 115;Hunt, 1967, p. 175].The climate is humid continentalwith an averageannual precipi- morphologicparameters of a basin.As notedelsewhere [Ze- tation of about 1000-1500 mm. Most of the precipitationis &ariasand Brutsaert, this issue], previous hydrogeomorphic studieshave been primarily statistical in nature.On theother deliveredbetween early spring and late summerand supports hand,in groundwaterhydrology, most investigations that deal a densevegetative cover 75% of which is forestand 10% withthe problem of drainagefrom hillslopes have focused on cropland[Baker and Dill, 1971]. Nineteen watersheds,within the Allegheny Mountain sec- thehydraulics and almostno attemptshave been made to relatethe outflowto basin-widegeomorphologic or aquifer tion wereselected as studybasins. To avoidthe needof using parameters.The objectiveof thispaper is to investigatethe channelrouting techniques, basin area was restricted to about 200 km2. All the basinswere at one time or anothergaged by effectof geomorphologicand soil characteristics on base flow recessionby means of a simpleconceptual model and on the theU.S. Geological Survey and have therefore long-term flow records.These records indicate that the streamflowdata are basisof basinscale parameters. The parametersin question aremainly land slope and drainage density of watersheds,but classifiedas goodand that the flowswere not subjectedto thehydraulic conductivity and the drainableporosity of the regulationor diversion.The basins are we!l-distributed within aquifermaterials are also considered.The approachis then theAllegheny Mountain section and each contains and/or is appliedto dataobtained for somebasins located in the Appa- locatedin the vicinityof a numberof meteorologicalstations lachianPlateaus. whoserecords are contemporaneous with the streamflowdata. A notabletopographic feature of thebasins is the absence of floodplains in theirupland regions. First- and second-order DESCRIPTION OF STUDY AREA streamsoccupy the bottom of V-shapedvalleys and a signifi- Thisstudy makes use of hydrologicand geomorphologic cant fraction of the narrowflood plains associated with third- dataof somerepresentative watersheds in the Alleghenyand fourth-orderstreams is occupiedby the streamchannels themselves.Soil Survey reports of the Countiesin whichthe Copyright!988 by theAmerican Geophysical Union. basinsare locatedshow that the bedrockis mantledby a Papernumber 88WR03084. weatheredlayer which has a nearlyuniform thickness ofabout 0043-1397/88/88WR-03084505.00 1.5-2.0m. Theremarkable uniformity of soilthickness in the

1651 1652 ZECHARIASAND BRUTSAERT:GROUNDWATER OUTFLOW AND BASE FLOW

NN $ Y L V A N I A

2• 7

•4 R Y L A N D KEY ]

' • studybasin '"• populationcenter • stream W E $ T V I R G I .... /' state boundary •12 •J• ! 80'Kin'

Fig. 1. Study basinsand associateddrainage systems: A. R., AlleghenyRiver; M. R., MonongahelaRiver; O. R., Ohio River; P. R., PotomacRiver; W. B., West Branch SusquehannaRiver. study watershedsas well as this averagesoil thicknesscan be a number of (not fewer than four) weather stationslocated in observedalong stream banks and road cuts at various topo- or around each basin was identified.Daily averagevalues of graphic levels.In all the watersheds,this layer forms an un- temperature,rainfall, and snowfallof thesestations for periods confined aquifer. The size, location, and geographicdistri- coveredby the flow records of the associatedbasins were then bution of the study basins are shown in Figure 1 (see also obtained from published records of the National Oceanicand Zechariasand Brutsaert[1985]). Atmospheric Administration.

DATA GeomorphologicData StreamflowData The geomorphologicparameters that affect groundwater The long-term streamflow data of the basins,from which outflow were generatedfrom recent 1:24,000 U.S. Geological their baseflow hydrographswere derived,were obtainedfrom Surveytopographic maps. The methodsor formulasof com- the appropriate Water Supply Papers publishedby the U.S. putingthem havebeen described elsewhere [see Zecharias and GeologicalSurvey. Daily temperaturerecords of meteorologi- Brutsaert, 1985, this issue]. The drainage densityvalues are cal stationsthat are locatedin and/or near the studybasins presentedin Table 1 and the land slopevalues are givenby and other relatedinformation [see Baker and Dill, 1971] indi- Zechariasand Brutsaert[1985]. cate that frozen ground conditionsmay exist in the area Soil Data duringthe period Octoberto May inclusive.The shallowness of the unconfinedaquifers in the area and the effectthat freez- The U.S. Departmentof AgricultureCounty Soil Survey ing may haveon the naturalgroundwater flow regimemade it reportsprovide data on depthto bedrock(i.e., soil thickness}• necessaryto excludethe parts of the streamflowrecords that soilhydraulic conductivity, and ground surface slope all of correspondto thesemonths of the year. On the basisof rain- whichare givenas averagevalues for eachsoil unitin anarea. fall recordsof weatherstations associated with each study In addition,the reportsinclude information on the texture basin(see Meteorological Data below),all daily streamflows andstratification of soil unitsas well as mapsshowing all on rainy dayswere next excludedto obtainthe dry periodor mappableunits. For the presentstudy, the outline of each low flow data for the basin.The data thusscreened may still basin was transferredfrom topographicmaps on to corre- containa "direct"runoff' component from precipitationthat spondingsoil maps, and the fraction of basinarea that each occurredin precedingdays. Therefore in addition,the flow for unitcomprises was measured. These values and horizon thick- the first day of every dry period was excludedand the re- nessesof soil unitswere then usedas weightingfactors to sultingdata were assumedto constitutea base flow record for obtainbasin averages of hydraulic conductivity and depth to the basin. bedrock.Furthermore, it was possible to make a qualitative assessmentofthe dominant grain size and texture of the soils MeteorologicalData ineach study basin. This information together with the "grain Throughthe conjunctiveuse of the locationmaps of the size--specificyieldgraphs" developed byJohnson [!967] al- studybasins and thoseof meteorologicalstations in the area, lowedthe estimation of the average drainable porosity for ZECHARIASAND BRUTS^ERT'GROUNDWATER OUTFLOW AND BASEFLOW 1653

TABLE 1. DrainageDensity and Average Parameters of Soilsin the StudyBasins

Soil Parameters

Drainage H yd raulic Drainable Density, Conductivity Soil Depth Porosity No. Drainage Basin km-'• K, cm/hour D, m f

1 Grassy Run 1.6871 3.37 1.04 0.20 2 Green Lick Run 1.2727 5.50 1.70 0.15 3 Clear Run 1.2152 1.52 4 Lick Run 1.2574 13.50 I. 10 0.25 5 Bradley Run 1.0430 1.45 6 Little Yellow Creek 1.3879 7.96 1.38 0.20 7 Poplar Run 1.1172 5.22 1.62 0.25 8 South Fork Beech Creek 0.8450 9. I1 1.65 0.15 9 Corey Creek 0.9492 4.80 1.22 0.15 10 Buffalo Creek 1.2038 5.00 1.31 0.15 11 Muncy Creek 1.0018 1.25 12 Roaring Creek 1.6081 4.50 1.52 0.20 13 Blockhouse Creek 1.0857 7.20 1.45 0.20 14 Young Womans Creek 0.9007 4.10 1.40 0.15 15 SavageRiver 1.1510 6.90 1.50 0.20 16 Yellow Creek 1.2702 7.41 1.46 0.20 17 Casselman River 0.7517 5.60 1.45 0.15 18 Deckers Creek 1.2591 5.20 1.39 0.15 19 Moshannon Creek 0.9442 3.90 1.50 0.15 eachbasin. The resultingbasin averagevalues are presented term in (1) is neglected,for any horizontal aquifer this ap- Table 1. proximationproduces a zero flow. As a result,the kinematic wave approachis unsuitablewhen a wide rangeof aquifer slopes,including very small ones, has to be considered. ANALYSIS In the presentstudy, (1) and the physicalmodel shown in Figure2 wereused to derivea simpleoutflow equation by Formulationof Outflow Rate meansof the quasisteady state approach. This approachwas The geologicand geomorphologicsetting of the study pioneeredaround 1886 by K. E. Lembke (cited by basins,their soil characteristicsand reconnaissancefield ob- Polubarinova-Kochina[1962, p. 573]) for an infinitely long servationswere used as a basis to establish a simple con- aquiferand it has been applied successfullyby DeZeeuw ceptualmodel. The crosssection shown in Figure2 represents [1979;also Kraijenho. ff, 1979,p. 315] for a horizontalaquifer a shallow,unconfined aquifer which has a small depth-to- of finitelength. Interestingly, this same approach was applied lengthratio D cosO/B which rests on an inclinedimpermeable by Landahl[1953] in the solutionof lineardiffusion and later layer{IL) representingthe underlyingbedrock. The two ends by Macey[1959] to a moregeneral class of nonlineardiffu- of the aquifermodel correspond to the groundwaterdivide sionproblems. Parlange [1971] usedit in the studyof hori- andthe bank of a fully penetratingstream. zontal infiltration.The accuracyof the quasisteady state ap- Drainagefrom an inclinedslab of porousmaterial involving proachfor sorptionproblems has been examined by Brutsaert both saturated and unsaturated flow can be described by [1976]. Richards'sequation [e.g., Brutsaert and El-Kadi, 1984, 1986]. As shownin the appendix,the applicationof the approach Butthis equation can only be solvednumerically, and as a to slopingaquifers results in theequation result,the approach cannot easily be implementedto derivea q = (qo+ EB)e-"t- EB (2) parametricequation suitable for the presentpurpose. More appropriateis the hydraulicapproach based on the following whereq is the outflowrate per lengthof streamchannel at equation x ---0,qo its initial value, E themean (i.e., constant) evapora- q = --Kh[(dh/dx) cos0 + sin 0] (•) whereq[L2/T] is theflow rate per unit widthof aquifer,K the hydraulicconductivity, h the thicknessof the saturatedzone measuredperpendicular to the IL, and0 theslope angle of the IL. Manystudies devoted to the problemof hillslopedrainage havemade use of the hydraulicapproach by meansof (1) whichwas first developed by Boussinesq[!877; Childs,1971]. However,in mostof thesestudies [e.g., Henderson and Wood~ it•o,1964; Betten, 1981] the approach was further simplified by thekinematic wave approximation. Also, Sloan and Moore r [19'84]and Stagnitti et al. [1986]used this approximation to Fig.2. Anidealized physical model of a hillslope.B,horizontal d,.•ribehillslope drainage as partly saturated flow. Because aquifer length' D, aquifer thickness' IL, impermeable layer' 0, incli- thehydraulic gradient is assumedto besin 0 andthe (dh/dx) nationangle of impermeablelayer. 1654 ZECHARIASAND BRUTSAERT'GROUNDWATER OUTFLOW AND BASE FLOW

tion rate, and B the horizontal projectionof the aquiferlength. straightlineupper and lower envelopes ofthe dQ/dt versus Q The parameter:z is the aquifer reactionfactor and is givenby datapoints through the origin; to exclude possible anomalous • = 2K(pDcos-' 0 + B sin 0)/(fB2) (3) outliers,upto 10%of the points were allowed tofall outside theupper and lower envelopes. Thesevalues aredenoted as• in which.fis the drainableporosity, D the mean aquiferdepth, in Table 2. and p is a constantarising in the linearization. It should be noted that the aquiferreaction factor is a con- DISCUSSIONOF RESULTS venientmeasure of the groundwater outflow. This resultsfrom the fact that the total groundwater outflow volume (hence Theresults given in Table2 areobtained by means ofstan- total aquiferstorage) is the integral of the outflow rate namely, dardstatistical and graphical analyses of (5).However, the •: q dt = qo/•,where q0, the initialoutflow rate, is dependentform of (5) is derived on the basis of strictly physical consider. only on precipitation input. Therefore for a given input, ations.These results can provide insight into severalissues. groundwaterdischarge volume is solelya functionof the aquifer reaction factor •z. Put differently, cz-• is a characteristic Linearity of LumpedSystem time scalefor duration of groundwateroutflow. Thevalues of • (i.e.,%, %0, and %) andE givenin Table2 H ydrologicApplication arebasin scale, time average values obtained from the applicationof theexponential decay base flow equation (5). As an The base flow recession characteristics of the basins can be illustration,the datapoints for SavageRiver are shown in determinedby meansof a methodproposed by Brutsaertand Figure3. The plot displays considerable scatter indicating that Nieber [1977]. The method is basedon the generalform in fora givenvalue of Q,the value of •zis not a uniqueconstant, whichmost groundwateroutflow equations can be expressed, but probablya variabledepending on hydrologicconditions namely, in the basin.Undoubtedly, part of thisscatter is dueto noise dQ/dt =f(Q) (4) in the flow data,but part of it mustbe dueto limitationsin the analysiswhich was basedon conceptualizingthe water- where Q is the outflow rate from the basin at the gaging shedas a singlelumped and linearstorage unit. station.The functionalrelationship f(Q) may be determined To analyzethe variation of • with time,several plots of the from a plot of ldQ/dtl versusQ. For actual flow data this baseflow data of a givenbasln were prepared each time by consistsof plotting I(Qi- Q,-x)/Atl against(Qi + Qi_•)/2, usingonly those flows that occurred after a successivelylarger whereQ, is the flowrate at time t and Q•_• is that at t- At. numberof daysfollowing the beginningof everyrecession One advantageof this methodis that it eliminatesthe prob- period.In otherwords, the IdQ/dtlversus Q data,correspond- lem of determining the time referencet--0 after each inter- ingto a givenstate of therecession process, were plotted after ruption of baseflow recessionby precipitation.Moreover, as theywere isolated from those of thepreceding days. This way one is only concernedwith the rate of changeof discharge it became possible to examine how the recessionbehavior from one day to the next, the length of a givenrecession changed2, 4, 6, and7 daysinto the dry period.This graphical periodis not of primary importance.If czand E can be taken analysiswas performedby meansof the upperand lower en- asaverage or characteristicbasin scale values, q canbe readily velopesof the points.Clearly, the upperand lowerenvelopes relatedto Q by integratingit alongall baseflow contributing representthe maximal and minimal observedrates, respec- stream channels in the basin. The characteristic basin scale tively,of the recessionof the groundwateroutflow. Note that value of B can be taken as [see Horton, 1932; Brutsaertand thisgraphical analysis was carriedout as indicatedearlier for Nieber, 1977] one half the ratio of the watershed area to the %; thusfor conveniencethe envelopeswere takenthrough the total length of streams(A/L), i.e., one half the inverse of the origin, and, to allow for possibleerror in the data,outliers drainagedensity. Then (2) can be put in the form of (4) as wereexcluded by letting up to 10% of the pointsfall outside follows: the upperand lower envelopes. dQ/dt= -cz(Q + EA) (5) To illustrate the procedure,the successiveplots for Savage River have been condensedinto a singlecomposite plot in whereA isthe watershed area. With tkr as 1 day,average daily Figure 3. For the plots of flow valuesthat occurredafter 2, 4, flowsof recessionperiods were plotted for eachstudy basin. 6, and 7 daysfollowing a rainfall event,the slopesof theupper Accordingto (5) the aquifer reaction factor for the watershed envelopesare 0.33, 0.23, 0.19, and 0.15 and thoseof the lower is theslope of theregression line, and the evaporation rate can envelopes0.062, 0.062, 0.063, and 0.066,respectively. Thus the be computedby usingboth the slopeand the interceptof the slopes,i.e., •, of the lower envelopesremain approximately line.The values of • (denotedby %)and of E, determinedby constant,while those for the upper envelopesdecrease con- applyingthe methodof leastsquares to (5),are givenin Table tinuouslywith time.The valueof Q dependsmainly on the 2. storageof water in the watershed.But the upperenvelope The valuesof E A, obtainedthis way, are very small,and providesinformation on the groundwateroutflow regime in probably on the order of the Q data, or even smaller.There- theearly stages of a dry periodwhen the rates of recession, i.e., forevalues of • werealso determined by assumingE --0 in IdQ/dtlare high.The successivevalues of czshow that there are (5), that is, by forcingthe bestfit straightline for eachbasin aquifersin the basinwhose reaction factors are initiallylarge, throughthe origin.The fit wascarried out in two ways.The but decreasesensibly as the rainlessperiod continues. Ad- firstwas determined by the methodof leastsquares; the re- vancedstates of theoutflow process, which are accompanied sulting valuesof the reaction factor are shown in Table 2 as bysmall recession rates, are represented bythe lower envelope •ro. The secondmethod was graphical.For each basin the whoseslope in the successivescatter diagrams remains essen- reactionfactor was taken as the averageof the slopesof the tiallythe same. This suggests the presence of aquiferswhose •, ZECHARIA$AND BRUTSAERT'GROUNDWATER OUTFLOW AND BASEFLOW 1655

TABLE 2. ReactionFactor a and EvapotranspirationE Values Determined for the StudyBasins

a, (day-t) From a,.,, From Flow Flow Data EvaporationE (mm/day) Data Regression Predicted%, .• RegressionWith From Flow Data With (5) for a,. From Flow Data 102 by Meansof No. DrainageBasin RegressionWith (5) E=O Average Envelopes Slopes (3} I GrassyRun 0.1159 0.1602 0.1054 0.1145 0.466 2 GreenLick Run 0.2388 0.0326 0.2304 0.2132 0.860 3 ClearRun 0.1693 0.0961 0.1512 0.1207 4 Lick Run 0.2995 0.0896 0.2717 0.2452 2.355 5 BradleyRun 0.1608 0.3099 0. t 206 0.1117 6 Little Yellow 0.3061 0.1387 0.2602 0.1961 0.858 Creek 7 PoplarRun 0.3022 0.0655 0.2723 0.2121 0.427 8 SouthFork Beech 0.1762 0.2592 0.1299 0.1265 0. 679 Creek 9 CoreyCreek 0.1658 0.0153 0.1584 0.1581 0.721 10 BuffaloCreek 0.3388 0.0614 0.3096 0.2168 1.292 I I MuncyCreek 0.2200 0.1802 0.1789 0. ! 562 12 RoaringCreek 0.2799 0.0467 0.2593 0.2487 1.812 !3 BlockhouseCreek 0.1559 0.0517 0.1392 0.1327 0.995 14 YoungWomans 0.1343 0.0876 0.1195 0.1123 0.471 Creek 15 SavageRiver 0.2498 0.0631 0.2238 0.1959 1.060 16 Yellow Creek 0.1324 0.0068 0.1308 0.1043 0.995 !7 CasselmanRiver 0.1777 0.0479 0.1649 0.1379 0.421 18 DeckersCreek 0.2939 0.0464 0.2700 0.2542 1.319 19 MoshannonCreek 0.1347 0.2455 0.1032 0.1023 0.612

althoughrelatively small, remain nearly constant throughout sensitivityof theseresults, partial correlationswere also com- mostof the recessionperiod. putedby usingslightly different sets of variables,e.g., 0, JB 2, It is likelythat the variationof • with time is largelythe and K. In addition, the tests were repeated after successive resultof the nonuniformdistribution of the physicalcharacter- eliminationof larger watersheds.(It was suspectedthat the isticswithin a watershed.Equation (3) showsthat the steeper conceptualmodel might perhapsbe more applicablefor partsof a basin,where 0 is largeand whereB tendsto be smaller(and perhapsK/f larger),must have fast depletion rates:such areas are generallylocated in the headwatersec- tions of a basin. In contrast, the downstream regions of a ,, successiveupper envelopes /+ basinhave smaller inclinations, hence relatively lower ratesof •- _._.average lowerenvelope / depletion.As indicatedearlier, the layerof weatheredmaterial in thestudy area is fairly uniform;therefore D doesnot seem tobe a factorcausing substantial • variability. Theseresults show that while the characterization of a basin asa singlelumped unit withbasin scale parameters is a useful approximation,it can have certain limitations.The total groundwaterdischarge is the sum of flow contributionsof aquifersections which have unequal reactionfactors. This + 4- totalflow is initiallydominated by the dischargesof the steeper partswhich contribute a large fractionof the total flow duringthe first few days of a recessionperiod. As therecession progresses,however, their storagedecreases rapidly and the gentlerparts of theaquifer, now being the major contributors, determinethe outflow.

œ•,ctof BasinScale Characteristics Thelumped approach (see the appendix)indicated that the reactionfactor • is a function of the basin scale parameters thatappear on theright-hand side of (3).Three different sets of• valuesof thestudy basins (i.e., %, •ro, and •e) havealready beendetermined from the flow data by meansof (5), and the otherparameters in (3) are knownquantities. Therefore to assessthe relativestrengths of the effectsof the parameters, 0,50 1.0,fl l.Sl• 5,00 •.51• • •0 o rEMS] theirrespective partial correlations with these three sets of • valueswere computed. Because the highestcorrelations were Fig.3. A plotof the base flow data showing envelopes that corre- obtainedwith %, only these results are given here (see Table spondto successivestages of recessionperiods tSarage River basin, 3•.However, the othercorrelations were similar. To testthe M aryland }. 1656 ZECHARIASAND BRUTSAERT: GROUNDWATER OUTFLOW AND BASE FLOW

TABLE 3. Partial CorrelationsBetween Variables Shown and probablyindicates that as alreadyintimated earlier, the vari- the Reactions Factor ationsin thesmall values of E are,indeed, unreliable, and that Partial Correlation ConfidenceLevel morerobust estimates of•z can be obtained by means of(5) by Variable Coefficient Significantat simplyconstraining the interceptat E = 0. Thisissue will be , , discussedfurther in the following section. K/f 0.32 73% Besidethe correlationcoefficients, a second measure of the B -0.37 92% T=D cos20/B 0.52 98% qualityof thepredicted values of czis theirmagnitude. A direct sin 0 comparisonof the columnsof czvalues given in Table2 shows thatthe predicted values czv are roughly two orders of mag-

nitudeß smaller than the actualones. In additionto deficiencies in the modelleading to (3), the underpredictionof thereaction smallerbasins). However, the resultswere not significantlyfactor by (3) can also be due to the fact that any of thevari. differentfrom those given and are therefore not presented. The ablesK/f, l/B, or 0 mayhave been taken too small.However, resultsshown in Table 3 indicate that the basin ground sur- B doesnot vary much within a basin; also, the 0 valueswere faceslope term T exertsthe strongest effect on the magnitude determined by a reliablemethod [see Zecharias and Brutsaert, of •. A little smaller,but practicallyjust as significant,is the 1985], and even large discrepanciesin 0 would not produce effectof the basinaquifer width B (or streamdensity). The major variations in the slope term T. Therefore the causefor effectof thesoil parameter K/f is smallestand not very signifi- theunderestimation of •zmay be the parameterK/f; actually, cant. as shown in Table 3 it is also the parameter which is least Thebasin scale morphologic/soil characteristics can also be correlated with •. usedto predict•z by meansof (3).The • valuespredicted this The hydraulic conductivitiesgiven in the Soil Surveyre. way(with p = 1) are givenin Table2 as %. To assessthe ports are the result of laboratory permeabilitytests of soil reliabilityof (3) asa predictorof cz,the predicted % valuesand samples[Olson, 1976, p. 32] and are thereforelocal parameter the valuesobtained directly from the flow data (i.e., values.As such they probably representonly the effectof in- weresubjected to correlationtests after the individualvalues tersticeson the water-transmittingproperties of the aquifer werefirst log transformed.The bestcorrelation was obtained material. Basin scale or overall hydraulic conductivities,on between% and•e; the correlation coefficient, Rw was found the other hand, include the effect of both micro- and macro- to be 0.68(significant at the 99% confidencelevel and with a pores (root holes, worm holes, cracks, and fissures).Conse. 95% confidenceinterval 0.27 < Rv,< 0.88).The correlationquently, they are much larger than the correspondinglocal coefficient,Rv,o, between % andCZro was somewhat smaller, scale or laboratory values. Since the latter were usedto corn. namely,0.58 (significantat the 95% confidencelevel, with a pute 0•from (3), they may be the causefor most of theunder- 95%confidence interval 0.11 <_ Rv, o < 0.83).The lowest corre- predictionof the reactionfactor values.In contrast,drainable lationwas found between •, andcz,, namely, Rv, -- 0.56,with a porosityhas a much smaller range of possiblevalues and is similarconfidence level as Rv,o. These three correlation coef- therefore not significantlyaffected by the scale of measure- ficients,namely, 0.68, 0.58, and 0.56 are all quite high with ment. However, the use in (3) of possibly overestimatedf goodconfidence levels, for this type of data. In light of the values(as obtained from Johnson's[1971] curves)may als0 limitationsof the lumped approachon which (3) is based, have,to a lesserextent, contributed to the underpredictionof these correlations can be consideredsignificant. Therefore 0•. even for hilly watersheds,the adoptionof the lumped approach(i.e., the use of basinscale parameter values to estimate Groundwater Evaporation an average•) can produceuseful results. The questionremains now why the correlationcoefficient The applicationof the leastsquares method to thedata between% and0•e is higherthan between % andthe other pointsresulted in E values(see Table 2) whichvary widely two, or why the morphological-soilparameters are bettercor- frombasin to basinand whoseaverage value is considerably relatedwith •e (seeTable 3) thanwith cz,or 0•ro.The reaction smallerthan the known averagevalue for the region[e.g. factors0•, were calculated as the averagesof theslopes of the Brutsaert,1982]. Differences in land use, such as proportion of upperand lower envelopes.As foundin the sectionLinearity forestedto clearedor cultivatedareas, would be expectedto of LumpedSystem, (see also Figure 3) the slopeof the lower be reflectedin differencesof evaporationrates of thebasins. enveloperemained fairly constantwith durationof the dry However,most of thevariation in thecomputed rates may be period,whereas the slopeof the upperenvelope decreased apparentrather than real. As mentioned,these E valuesare continuouslyas the length of the dry period increased.Hence probablywithin the rangeof error of the dischargemeasure- the upperenvelope of all the data pointsreflects the recession ments.Therefore it is difficultto makea meaningfulanalysis behaviorin the early stagesof a dry period. This meansthat of the variation in computedE. thecze, which depend on the slope of this upper envelope, are Thegenerally small magnitude ofthe individual E values, alsomore heavily influenced bythe early drainage behavior of onthe other hand, seems to suggestthat very little of the thecatchment, than the other two reaction factors 0•o and 0•; overallwatershed evaporation isfurnished bygroundwater. In thesewere obtained byleast squares, sothat they characterize any given watershed, thewatertable tends to approachthe morethe average long-term drainage behavior. Since the % landsurface only in riparianzones along stream channels, werederived from morphologic andsoil characteristics, the which usually occupy a small fraction ofbasin area. Over the factthat R•,, is the largest suggests thatthese characteristics remainder ofthe watershed, theremay be a decoupling ofthe affectoutflow behavior more strongly inthe early phases ofa upperunsaturated layers from the lower saturated zoneof the dryperiod and that their effect decreases asthe rainless period unconfined aquifers. Under such conditions evapotranspira' becomeslonger. Thefact that Rvr ois the smallest ofthe three tion is drawn mostly from upper zone soil moisture, and ZECHARIASANDBRUTSAERT' GROUNDWATER OUTFLOW AND BASE FLOW 1657 groundwaterstorage is affectedmuch less by thisdepletion be taken as a true free surface-it is also assumedthat the mechanism.Note that in a studyof 23 km2 basinin Alabama, Dupuit(-Forchheimer) assumptions can be adaptedto the Daniel[1979] foundthat evaporationfrom groundwaterwas slopingaquifer. In addition,in mostapplications the aquifer onlyabout 0.16 mm/day.Even smaller values (about 0.04 materialis taken as homogeneous and isotropic. ram/day)may be derived from the base flow data of Tschinkel In the quasisteady state approach it is furtherassumed that [1963]obtained in southernCalifornia if E is referredto the the shapeof the movingwater table is the sameas that calcu- entirewatershed area and not just to the riparian woodlands. latedfor steadyflow conditions. Under steady conditions the Thesefindings obtained by different methodsare consistent outflowfrom a slabis equalto the recharge.If i [LIT] is the withthe E valuesderived in the presentwork. rechargerate, Figure 2 showsthat at anyx onehas to a good CONCLUDING REMARKS approximation(since D < B) by virtue of {11 In an earlierstudy [Zechariasand Brutsaert,this issue]it Kh(dh/'dx+ tan 0} = (/(B/cos0)- x) (AI) wasdetermined that drainagedensity, average basin slope, Uponlinearization by putting• -- h, with the boundarycon- andlength of perennialstreams are the mostimportant geo- ditionsh = h• for x = 0, andh = he for x = B/cos0, integra- morphologiccontrols of groundwaterdischarge. These find- tion of (A 1) yields ingsare confirmed herein, since the firsttwo of theseparame- tersappear in the expressionfor • (3). The physicalmodel i = 2K•[(h2 - ht) cos2 0 + B sinO]/'B 2 (A2) usedhere representsgroundwater outflow q from an individ- For steadyflow conditions (lB/cos O) = q[L2/T] is the outflow ual cross section of a hillslope into the adjacent stream ratefrom the aquifer per unit length of channel.In thepresent channel.Therefore the third parameter (the total length of derivationit is assumedthat the channelis empty,so that perennialstreams) enters in the computationthrough the inte- h• = 0. Also,to keepthe flow systemlinear it is assumedthat grationprocess of the elementaloutflows to derivethe total h2 is a constantfraction p of the aquiferdepth D suchthat 0 outflowQ at the outlet of the basin. < p _< 1. Thus (A2) can be written as The outflow equation (2) used in this study is of the ex- ponentialdecay form and is therefore equivalent to the q ----:zS (A3) lumpedstorage model formalizedby Sugawaraand Maruyama where• is the aquiferresponse constant given by (3}' $ [L2] is [1956],Nash [1958], Dooge[1973] and others.Evidently, the the amountof water stored(per unit lengthof channel)in the notion that groundwater outflow from a watershed can be aquifer, given by characterizedby an exponential decay function can be traced to the work of Boussinesq[1877], and over the years it has S = f/•B/cos0 (A4) beenapplied widely in hydrology with considerablesuccess. In in whichf is the drainableporosity. The unsteadyflow prob- thepresent paper this outflow equation is derived on the basis lem can now be solvedby meansof the lumpedequation of 'ofa simpleconceptual model by meansof hydraulictheory of continuity groundwaterand in terms of basin morphologic and soil characteristics. q + EB = --dS/dr (A5) Becausethe physicalcharacteristics (mainly slope and, to a where ElL/T] is the mean evapotranspirationrate. Integra- lesserextent, drainage density and hydraulicconductivity) are tion of (A5) with (A3) producesthe desiredresult (2). not uniformlydistributed within the basinsin the region,a distributedmodel approach may perhapsyield better results. ComparisonWith Other Studies However,the primaryadvantage of adoptingan averagevalue of thereaction factor •z, that is the lumpedlinear approach The parameterizationused in the presentstudy was also withbasin scale parameters, is its convenienceand practical compared with the experimental results of Hewlett and Hib- applicability. bert [1963]. Their data includeqo = 0.692 m3/day,K = 168 The valuesof the geomorphologicvariables derived from mm/hour, 0 = 21.8•', B/cos 0 = 13.72 m, D = 0.91 m, total out- maps,namely the slope 0 and the drainage density (1/2B) flow[•,, = 1.26m 3, and total volume of soildrained V• = 10.58 appearto be suitablefor the parameterizationof baseflow. In m3. The drainableporosity f is determinedby dividingl• by contrast,the values of the soil variable (K/f) obtained from • which givesa value of 0.12. In the presentstudy, some of CountySoil Survey reports are poorly coordinated with cz and these parametershad to be adjusted to ensure that the as- probablyaround two ordersof magnitudetoo small.This sumptions made in developing the physical model are ap- discrepancysuggests that thelocal or small-scaleestimation of proximatelysatisfied by the experimentalsetup. someparameters does not fully capture all features of the To account for the small wedge of soil at the lower end of phenomenainvolved at largerscales. In thecase of hydraulic the trough, which remainedundrained at the end of the exper- conductivity,this is probablydue to thepresence of meso-or iment, the effectiveheight of the trough was taken as 0.86 m. macroporesin the aquifers.More researchis requiredto bring For the same reason, the flow direction had an inclination of clarityin thisproblem. 21.8" only in the upper part of the trough; the general direction of flow at the lower end was horizontal. Therefore to APPENDIX:CONCISE PARAMETERIZATION OFHILLSLOPE obtain an "average" inclination that is applicable over the DRAINAGE entire lengthof the trough,the given angle was reducedby a third as suggestedby the geometry of the structure. It is QuasiSteady State Approach known that fitting free surfacemodels to field drainageprob- Equation(I), firstderived by Boussinesq[1877], embodies lems requires an adjustmentof the hydraulicconductivity. The t.Mso-ca!led hydraulic theory of groundwater.Thecapillary results of E!-Kadi and Brutsaert [1986] for outflow from un- flowabove the water table is neglected, sothat the latter can confinedaquitErs in which B/D is of the order of 10, suggest 1658 ZECHARIASAND BRUTSAERT: GROUNDWATER OUTFLOW AND BASE FLOW

Daniel,J. F., Estimatinggroundwater evapotranspiration from streamflowrecords, Water Resour.Res., 12, 360-364,1976. DeZeeuw, J. W.,Hydrograph analysis for areas with mainly ground. 10 ...... ': waterrunoff, pp. 321-357, Publ. 16, Int. Inst.for Land Reclamation and Improvement,The Netherlands, 1979. E1-Kadi,A.I., andW. Brutsaert, Can unsaturated flowduring gravity drainagebe represented by Boulton's formulation?, Water Resour. Res.,22, 1361-1366, 1986. Dooge,J. C. I., Lineartheory of hydrologic systems, Tech. Bull. 1468, Agric.Res. Serv., U.S. Dep. of Agric.,Washington, D.C., 1973. Henderson,F. M., andR. A. Wooding,Overland flow and ground. waterflow from a steadyrainfall of finite duration, J. Geophys.Res., 69, 1531-1540, 1964. Hewlett,J. D., and A. R. Hibbert,Moisture and energyconditions withina slopingsoil during drainage, J. Geophys.Res., 68, 1081- 1087, 1963. • 0.5 Horton,R. E., Drainagebasin characteristics, Eos Trans. AGU, l& 350-361, 1932.

. Hunt,C. B., Physiographyof the UnitedStates, Freeman, Cooper, San Francisco, Calif., 1967. Johnson,A. I., Specificyield--Compilation of specificyield for variousmaterials, U.S. Geol. Surv. Water SupplyPap., 1622-/), 0•1 0.5 I 5 10 1967. time since beginning of drainage [daysl Kraijenhoffvan de Leur,D. A., Rainfall-runoffrelations and compu- tational methods, pp. 245-320, Publ. 16, Int. Inst. Land Recla- Fig. 4. Comparisonof outflowvolumes computed with (:2) and (3) mation and Improvement, The Netherlands, 1979. of quasisteady state approach (dashed curve) with experimentalre- Landahl, H. D., An approximation method for the solutionof diffu- suitsof Hewlett and Hibbert [1963] (solidcurve). sion and related problems,Bull. Math. Biophys.,15, 49-61, 1953. Macey, R. I., A quasi-steady-stateapproximation methodfor diffu. that a reductionfactor of 0.5 is probably appropriate;accord- sion problems, I, Concentration dependent diffusion coefficien• ingly,a valueof'K = 84 mm/hourwas taken.The lin- Bull. Math. Biophys.,21, 19-32, 1959. earizationparameter p was taken as 1/3; this valueis suggest- Morisawa, M. E., Relation of quantitative geomorphologyto stream ed by comparisonof Polubarinova-Kochina's[1962, p. 507] flow in representativewatersheds in the AppalachianPlateau province, Tech. Rep. 20, Off. of Naval Res.,Columbia Univ., NewYork, exact solution of the nonlinear Boussinesqequation for drain- 1959. age from an infinitelylong aquiferwith Edelman'sapproxi- Nash, J. E., The form of the instantaneousunit hydrograph,General mate solution obtainableby linearization(as quotedby Krai- Assembly Toronto, Publ. 45, Int. Assoc. of Sci. Hydrol., Gen- jenhoffvan de Leur 1'1979]and Brutsaertand Nieber, [1977'!). tbrugge,pp. 114-118, 1958. The use of theseparameters in (3) gave an • value of 0.67, Olson, G. W., Landusecontributions of soil surveywith geomorphol- ogy and engineering,in Geomorphologyand Engineering,edited by which was, in turn, used in (2) to compute the cumulative D. R. Coates, pp. 23-41, State University of New York, Bing. outflow volumes for E = 0. The resultsare presentedin Figure hamton, 1976. 4, where they can be compared with the experimentalresults Parlange, J.-Y., Theory of water-movement in soils, I, One- of Hewlett and Hibbert [1963]. The agreementis better than, dimensionalabsorption, Soil Sci., 111, 134-137, I971. Polubarinova-Kochina, P. Y-A., Theory of GroundsrarerMovement, or at least as good as, that for any of the other modelsana- translatedfrom Russianby R. J. M. De Wiest, p. 613, Princeton lyzedby Sloanand Moore [1984] and Stagnittiet el. [1986-1. University Press,Princeton, N.J., 1962. Sloan, P. G., and I.D. Moore, Modeling subsurfacestormflow on REFERENCES steeplysloping forested watersheds, Water Resour.Res., 20, 1815- Atwood, W. W., The PhysiographicProvinces of North America,Ginn 1822, 1984. and Company,Boston, Mass., 1940. Stagnini,F., M. B. Pafiange,T. S. Steenhuis,and J.-Y.Parlange, Baker, S., and H. W. Dill, The Look of our Land, U.S. Department of Drainagefrom a uniformsoil layer on a hillslope,Water Resour. Agriculture,Washington, D.C., 1971. Res., 22, 631-634, 1986. Beven, K., Kinematic subsurface storm flow, Water Resour. Res., 17, Sugawara,M., andF. Maruyama,A methodof previsionof thefiver I419-I424, 1981. dischargeby meansof a rainfallmodel, Symposia Darcy, Dijon, Brutsaert,W., The conciseformulation of diffusivesorption of water Publ.42, pp. 71-76 Int. Assoc.Sci. Hydrol., Gentbrugge, 1956. in a dry soil, Water Resour.Res., 12, 1118-1124, 1976. Thornbury,W. D., RegionalGeomorphology ofthe United States, John Brutsaert,W., Evaporationinto the Atmosphere,D. Reidel,Hingham, Wiley, New York, 1965. Mass., 1982. Tschinkel,H. M., Short-termfluctuation in streamflowasrelated to Brutsaert, W., and A. I. EI-Kadi, The relative importance of com- evaporationand transpiration, J. Geophys. Res., 68, 6459-6469, pressibilityand partial saturationin unconfinedgroundwater flow, 1963. Water Resour. Res., 20, 400-408, 1984. Zecharias,Y. B., and W. Brutsaert,Ground-surface slope as a basin Brutsaert, W., and A. I. E1-Kadi, Interpretation of an unconfined scaleparameter, Water Resour. Res., 21, 1895-1902,1985. groundwater flow experiment, Water Resour. Res., 22, 419-422, Zecharias,Y. B., and W. Brutsaert,The influence ofbasin morpholo- 1986. gyon groundwateroutflow, Water Resour. Res., this issue. Brutsaert, W., and J. L. Nieber, Regionalizeddrought flow hydro- graphsfrom a mature glaciatedplateau, Water Resour.Res., 13, W. Brutsaertand Y. B. Zecharius,School of Civiland Environmen- 637-643, 1977. talEngineering, Hollister Hall, Cornell University, Ithaca, NY 14853. Boussinesq,J., Essai sur la theorie des eaux courantes,Mere. Pres. Divers Savants l'Acad. Sci. l'Inst. France, 23, 252-260, 1877. (ReceivedOctober 20, 1986; Childs, E. C., Drainage of groundwater resting on a sloping bed, revised June 6, I988; Water Resour. Res., 7, 1256-1263, 1971. acceptedJune 15, 1988.)