Quinn River, Cultus River, Browns Creek, and Fall River ..... Discharge in spring-dominated streams for Quinn River, Browns Creek, ..... AGU, 77, Fall Meet.
WATER RESOURCES RESEARCH, VOL. 33, NO. 8, PAGES 1813-1822, AUGUST 1997
A model for discharge in spring-dominated streams and implications for the transmissivity and recharge of quaternary volcanics in the Oregon Cascades Michael Manga Departmentof GeologicalScience,Universityof Oregon,Eugene
Abstract. A model for dischargein spring-fedstreamsis describedand appliedto streamsin the OregonCascades. Thesestreamsare assumedto be fed by an unconfined aquifercomposed predominantly of quaternarybasaltsandbasalticandesites. The model is basedon the unsteadyBoussinesq equationand is characterizedby a singleparameter, a normalizedlengthscaleof the aquifer.Flow in a runoff-dominated streamis usedas a proxyfor the time-dependent groundwater recharge.Four spring-dominated streamswith 50 yearsof dailydischarge recordsare studied,and modelingefficiencies are in the range of 0.76-0.89. The effectivetransmissivity is found to be approximatelyproportionalto the lengthscaleof the aquifers.Groundwaterrechargeratesare in the rangeof 66-127 cm/yr and 40-73% of the mean annual precipitation. tailed and completehydrologicmodel for the entire Upper DeschutesRiver Basinis currentlybeingdevelopedby the U.S. A centralproblemin hydrologyis understanding and char- GeologicalSurvey(USGS) [Gannettet at., 1996].Our modelis acterizingthe relationshipbetweenthe responseof hydrologic physically basedin that it followsfrom Darcy'sequationand systems (suchasstreamflow), variousinputs(suchasrainfall), conservationof mass,but it greatlysimplifiesbasingeometry and the physicalattributesof the system.Determiningsuch and groundwaterrechargepatterns.Despitetheseapproximahydrologicrelationships at the regionalscaleinvolvesdevelop- tions, the model allows us to determine an effective transmising modelsand fitting parameters,for example,developing sivity(seeworkbySanchez-Vita etat. [1996]for a definition)of modelsthat relate rainfall/snowmeltand runoff. Owing to the the aquifersthat feed the springsand providesconstraintson rates. geologicand hydrologiccomplexityof most natural systems, groundwaterrechargeand evapotranspiration modelsthat are physicallybasedare typicallycharacterizedby a large number of physicaland geometricparameters[e.g., Beven,1989].Alternatively,conceptualmodelscan be devel- 2. Geologic and Hydrologic Setting opedandcan oftensuccessfully accountfor measuredstreamWe focuson a set of four spring-dominated streamsin the flow evenwhen characterizedby a smallnumberof parameters Upper DeschutesRiver basinin the High OregonCascades: [e.g.,Jakemanand Hornberger,1993]. Here we showthat a Quinn River, Cultus River, Browns Creek, and Fall River certainclassof streamhydrographs, namelythoseof spring-fed (Table 1 and Figure 1). Daily dischargemeasurements were streams,cansometimesbe explainedby physically basedmod- made by the USGS from 1939 to 1991. Dischargein these spring-dominated streamstypicallyvariesby lessthan a factor els that are characterizedby a small number of parameters. are shownlater in Unlike "typical"runoff-dominatedstreams,those that are of 2 over the courseof a year (hydrographs spring-dominated often have simpledischargecharacteristics Figure 5). Peak dischargetypicallyoccursin the summeror (terminologyafter Whitingand Stamm [1995]). The hydro- fall, even thoughmost of the precipitationfalls as snow,and graphsof spring-dominated streamsoften exhibitvery little peak snowmelt,as observedin runoff-dominatedstreams,ocvariation,evenin regionswith extendeddry seasonsor follow- curs in the spring [Whitingand Stamm, 1995]. By contrast, ing a period of significantprecipitationor snowmelt.In such runoff-dominatedstreams,for example, Cultus Creek, Deer systems,subsurface flow originatingfrom an aquifer (as op- Creek, and ChadtonCreek (Table 1 and Figure 1), haverelwith little or no baseflow for much posedto directrunoff,whichconsists of both surfaceflow and ativelyflashyhydrographs of the year (e.g., the hydrograph for Deer Creek, shownlater interflow)accounts for mostof the streamflow. Previousstudin Figure 5). ies haveshownthat it is possibleto explainmanyof the charThe local geologyconsistsprimarily of Quaternarybasalts acteristics of suchspring-fedstreamswith simplifiedphysically 1.
Introduction
and basaltic andesites. The Mount Bachelor volcanic chain,
basedmodels[e.g.Manga, 1996;Leonardiet at., 1996]. abouthalf the rechargeareafor the Fall River Here we describea one-parametermodel for dischargein whichcomprises (shown later in Figure 10), consistsof basalt and basaltic spring-dominated streamsthat is based on the Boussinesq andesitesthat eruptedfrom cinderconesand shieldvolcanos equationfor unsteadyflowin an unconfinedaquiferand apply from about18 to 8 ka [Scottand Gardner,1992].The recharge the model to six streams and rivers near the headwaters of the area for the other streamsis underlain primarily by basaltic DeschutesRiver in the High Oregon Cascades.A more deandesites with well preservedsurfacefeatures;a K-At date of Copyright1997by the AmericanGeophysical Union. this unit givesan age of 0.73 Ma [Shetrod,1991]. The entire regionwascoveredat 7 ka by Mazamaash;the thickness of the Paper number 97WR01339. 0043-1397/97/97WR-01339509.00 ashlayervariesfrom about0.5 m in the southernpartsshown 1813
1814
Table
MANGA:
1.
Stream
Name
MODEL
FOR
Locations
Number
Elevation, m
in Figure1 to lessthan 10 cmin the northernpartsof the study area shownin Figure 1 [MacLeodand Sherrod,1992]. Annual precipitationin the (assumed)rechargearea varies from about 20 inches(50 cm) in the east to more than 100 inches(250 cm) near the top of the SouthSisterand Broken Top mountains(see insetof Figure 1). The rechargearea is forestedby Ponderosaand Lodgepolepines at lower elevations.At higherelevations,andalongthe crestof the Cascades, forestsare dominatedby Hemlock and mixed conifers.As we will showlater, groundwaterrechargeratesare veryhigh,presumablya result of the high permeabilityof the younglava flowsand the layer of Mazama ash. The four spring-dominated streamsdischargemeteoricwa-
Latitude, Longitude
Spring-DominatedStreams Cultus River Quinn River Browns Creek Fall River
14050500 14052500 14054500 14057500
Cultus Creek Deer Creek Char!ton Creek
14051000 14052000 14053000
Deschutes River Snow Creek
14051000 14074900
1356 1354 1332 1286
43049'06", 43ø47'0Y', 43ø42'57", 43ø47'48",
121047'40" 121050'06" 121ø48'10" 121ø34'18"
Runoff-DominatedStreams 1385 1378 1359
43ø49'17", 121049'22" 43ø48'48", 121ø50'18" 43ø46'51", 121050'06"
Other Streams Studied Here
1385 1378
DISCHARGE
43ø49'17", 121049'22" 44ø06'59", 121039'34"
ter.In Figure2 weplotthemeasured 8•80 and8D values(in partsper thousandrelativeto standardmeanoceanwater)for
Precipitationin inches
,,,,,•x,s
20 km ß ß
Sparks Lake '"i•i• Bachelor Mountain
Elk Lake
NI 4km I spring gagingstation
Cultus Lake
7• Cultus Cultus/
'
/
',Cultus
(43ø45 'N,
+
Figure 1. Map showingthe locationof streams,gagingstations,and elevation(in feet). The dashedcurve showsthe crestof the Cascades. Onlystreamsstudiedin thispaperare shown.Insetshowsthe isohyetalmap of meanannualprecipitationin inchesper year of rainfall equivalent(basedon recordsfrom 1961to 1991 compiledby G. Taylor, stateclimatologist, OregonClimateService).
MANGA:
MODEL
FOR DISCHARGE
1815
waterssampledfrom the four spring-dominated streamson September 28, 1996(soliddisks).For comparison we alsoplot
theglobalmeteoric waterline,&D = 8&•80+ 10.The close agreementbetweenthe meteoricwaterline andthe measurementsis probablythe resultof integratingthe compositions of rainfall and snowmeltfrom at least severaldecadesof precip-
itation.For comparison with the (large)springsstudiedhere,
in Figure2 weshow&•80and&Dvalues(fromIngebritsen etal. [1994]) for other springson the east and west side of the Cascades. Deviations
from the meteoric water line of these
L
x=O
Figure 3.
otherspringsare attributedto localweatherpatternvariability andvaryingamountsof evaporation[Ingebritsen et al., 1994].
x=L
Geometryof the one-dimensionalmodel.
wherek, p, /x, #, Sy,t, andx arepermeability, waterdensity, waterviscosity,gravitationalacceleration,effectiveporosityor We assumethat the spring-dominated streamsare fed by specificyield [Freezeand Cherry,1979,p. 61], time, and horiunconfined aquifersthat are replenished by rainfallandsnow- zontal position, respectively,and N describesthe timemelt (becausemeteoricwater is beingdischarged). Here we dependentwaterinput.Equation(1) for saturatedflowis subdescribea one-parametermodel,illustratedin Figure 3, for ject to the standardDupuit approximations,ignores the faces,andassumes the aquifer subsurfaceflow and dischargethat can accountfor the two capillaryfringeandanyseepage is isotropic and homogeneous and that the aquifer hasa hordistinctivefeaturesof the hydrographsof spring-dominated izontal and relativelyimpermeablebase. streams:(1) peakdischarge occursin the summeror fall even The Boussinesq equationis frequentlylinearizedby setting thoughsnowmelttypicallyoccursin April or May, and (2) h (x, t) = ho + h • (x, t), whereho is a constant;assuming dischargevaries by lessthan a factor of about 2 over the h•/ho is smallsothat quadratictermsin h• canneglected,(1)
3.
Model
50-yeartimeperiodfor whichwe havestreamgagingrecords. Below we describethe governingequations,boundaryconditions,and the numericalmethodsemployedto determinethe numericalvalue of the one parameterwhichcharacterizes the
becomes
Oh•
kp#hoO2h • N(x, t)
O• -= ladyOx 2+ Sv
model.
(2)
The coefficient kp#ho/txSy canbe interpreted asa diffusivity that describesthe rate at which variationsin the height of the The evolutionof a water table in an unconfinedaquifer,with water table decay. Manga[1996]showedthat a modelbasedon the Boussinesq heighth, can be describedby the Boussinesqequationfor unsteadyflowwith accretion,whichfollowsfrom conservation equationcan quantitativelyexplainthe observedtime lag be-
3.1.
BoussinesqEquation
of massandDarcy'sequation[e.g.,Bear,1969,section8.2]. In one horizontaldimension,the Boussinesq equationis givenby
h + Oh 0(Oh) at kp# txSy Ox •xx N(x,t) Sy
-- =
(1)
tweensnowmeltand peak dischargeat the springs,aswell as the magnitudeof annualdischargevariations.Leonardiet al. [1996]successfully applieda similarmodelbasedon the diffusion equationto springsin Armenia which dischargefrom
basalticaquifers. 3.2.
Recharge
Because runoff is related to the same events that result in
-70-
-- Global meteoric water line o Springs east ofCascade crest
-80 - /•Springs west of Cascade crest•x •5D
-90
Browns Creek oo Cultus River•ø•eø • o^• Q•inn o
-100 -
o
o River o ø o
o
o o
o
o
o
-120 -
• -15
• -14
• -13
-12
(51•0
-11
-10
groundwaterrecharge,namely,rainfall and snowmelt,we use flow in a runoff-dominatedstream as a proxy for the timedependentrecharge,N(t). Groundwaterrechargemay also vary in space.In-this casewe expectN(x) to increasewith increasingdistancefrom the springbecausethe mean annual precipitation increases upslope, which,for all the streamsstudied here,corresponds to increasing distancefrom the mouthof the springs(seeFigure 10, shownlater). Here we assumeN increaseslinearlywith increasingdistancefrom the spring. The model describedhere is similarto that of Manga [1996] exceptfor the assumedspatialdistributionof recharge:Manga [1996]assumed that rechargeoccurredat a uniformrate over the region0 < x < L' andwaszerooverthe regionL' < x < L. 3.3.
Boundary Conditions and Geometry
Figure2. Relationship between &•80and&Dvalues (inparts We assumethat flow is primarilyin one directionsothat we per thousand relativeto standard meanoceanwater)for the model,(2). Thisapproximation fourspring-dominated streams studiedhere(solidcircles).For canapplythe one-dimensional becausethe rechargeareas(seeFigure 10) comparison, valuesfor othersprings on the eastandwestside is not unreasonable are long and narrow. In addition,estimatedwatertableheights of the crestof the cascades are shownwith open circlesand in this region are also compatiblewith nearlyunidirectional opentriangles, respectively [fromIngebritsen et al., 1994].The solidlineistheglobalmeteoric waterline&D= 8&•80+ 10. flowbeneaththe rechargeareatowardsthe springs[Gannettet
1816
MANGA:
MODEL
FOR
DISCHARGE
0.20
i Quinn River
E=••
0.18
N t=l
q•-q/ø,
(6)
and maximizethe modelingefficiency
0.16
•=• (q•_ q/O)2
0.14
ME= 1- •_• (qy_?::/•4)2, (7)
0.12
where q is discharge,the superscriptM denotesthe model prediction,the superscriptO denotesthe observedvalues,and /::/isthe mean of q. In all casesthe choiceof d that minimizes E is similarto the value that maximizesME. An exampleof E and ME for different valuesof d is shownin Figure 4 for the
0.10 0.9
Quinn River.
4. 0.6
Here we considerfour spring-dominatedstreamsfor which reliabledaily dischargerecordsexistover a period of 50 years:
0.5
Quinn River, Cultus River, Browns Creek, and Fall River
0.4
0.3 0.8
1.0
1.2
1.4
1.6
Figure 4. Normalized mean absoluteerror (top), equation (6), and modelingefficiency(bottom),equation(7), as a function of the diffusionlengthscaled, equation(5), for the Quinn River. The best model parameter is d = 1.27.
al., 1996].We let L denotethe lengthof the aquifer,and atx = L we apply the boundaryconditionh = h o. Discharge,q, is proportionalto Oh/Ox at x = L. 3.4.
Modeling Procedure
In order to nondimensionalizethe linearized Boussinesq equation,we choosecoto be the characteristicrechargefrequency(1 year) and the characteristic lengthscaleXc to be the "diffusive"length scale(see work by Manga [1996] for a detailed discussion),
Xc=(2kpgho• •ot•S• ?•/2 '
(3)
Thus settingt' - cotandx' = x/x c, the linearizedBoussinesq equation becomes
Oh•
Results
1 02h• N(x', t')
Ot'- 20x '2+
cosy
h l = 0 atx' = L/xc = d.
(4)
We solvethe linearizedBoussinesq equationnumericallyusing a Crank-Nicholson
finite-difference
method.
Our model, (4), is characterizedby a singleparameter,the normalizedlength of the aquifer, d = 6/Xc.
(5)
The averageannual rechargeis determinedby requiringthat the mean streamdischargeequalsthe mean groundwaterrecharge. In order to determine mean-absolute
error
(Figure 1 andTable 1). As discussed in the previoussection,we use dischargein a runoff-dominatedstream as a proxy for recharge.For the resultspresentedhere we use flow in Deer Creek. We have also carried out the same analysisusingthe hydrographsof two other nearby runoff-dominatedstreams (CultusCreek and ChafftonCreek;seeFigure 1 and Table 1) and obtainednearly identicalresults. Before presentingand discussing the results,it is important to highlightcertain key model approximationsand assumptions so that results are not misinterpreted:the model is a one-dimensionalapproximationof a two-dimensionalproblem; the parameter d describesan effective (aquifer-scale) transmissivity; the aquifer is assumedto be homogeneousand isotropic;and rechargein the entire region is assumedto be proportionalto dischargein the runoff-dominatedstreams.We also note that there is some uncertaintyand error associated with the streamflowmeasurements(determined by relating measuredwater stageto discharge).In particular,the highest peak flowsand the low baseflowsin Deer Creek,whichwe use as a proxyfor recharge,are noted asbeingestimatedvaluesin the WaterResources Data, Oregon[1992] books. For all four spring-dominated streamsthere is a well-defined minimumE and maximumME (see Figure 4 for the Quinn River). Our bestvaluesfor d and the corresponding modeling efficiencies(basedon the entire time series)are listedin Table 2. Modeling efficienciesare in the range of 0.76-0.89 and are comparableto those of the IHACRES lumped conceptual rainfall-runoffmodel [e.g.,Schreideret al., 1996]. In Figure 5 we showobservedhydrographs (thin curves)and modelhydrographs(bold curves)for all four streamsover the time period 1947-1987. In general, the model describesboth the annual and decadaldischargevariations.
the best value of d we minimize
the
Table 2. Model Parametersand Modelling Efficienciesfor Spring-DominatedStreams Name
d
ME
Quinn River
1.27
0.89
Cultus River Brown Creek Fall River
1.54 1.78 2.50
0.76 0.80 0.78
MANGA:
3.0 •.,• 0
MODEL
FOR
DISCHARGE
1817
Deer CreekI
I
'
1.6 0.8 0
, •
19521953 195••••'•-
3.0 •
.
1.5
_ I ,_ I
Deer Creek
,
I
1.6 0.8 0
6 3
1948
1950
1952
1954
1956
1958
3.0•
1960
1962
1964
1966
DeerCree•
1.5 0
.
Cultus Rive•
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
year
Figure 5. Discharge in spring-dominated streamsfor QuinnRiver,BrownsCreek,CultusRiver,andFall Riverfor the period1947-1986(timein calendaryears).Observed discharge is shownwith a thin curve,and the modeldischarge is shownwith a boldcurve.Arrowsindicatepossible surfacerunoffcontributions.
of dischargevariations.In Figure 6 we plot the difference betweenobservedand modeleddischargefor the Quinn and Fall Rivers, the streamswith the smallestand largestvalue of d, respectively. The largestdifferencesare generallyat the of the riseof thehydrograph. For bothstreamsthese flow variationsoccuron the timescaleof severalyearsand are beginning relatedto droughtyears.In particular,wateryears(October- differencesappear as negativespikesand indicatethat the streamflow. September)1973 and 1977had almostno precipitation(see modelpredictsa slightlyearlieronsetof increasing time differencebetweenthe Deer Creekhydrograph, Figure5), andall four springshaveno In particular,the smallresponse model and observation is most clear for the Fall River. This new pulseof dischargein the summeror fall. couldbe dueto a slightdelaybetweenrechargeof We notethat the modelsuccessfully reproducesthe slopesof discrepancy both the risingand fallinglimbsof the hydrograph(seeinset Fall River'saquiferrelativeto that inferredfrom Deer Creek; withmicroclimate variability. for the QuinnRiver) andin generalreproduces the amplitude suchvariationsmightbe associated The rangeof d for thesefour streamsis 1.27-2.50.For the Quinn River (d - 1.27) the annualdischargevariationsare large.At the otherextreme,for the Fall River (d -- 2.50) the annualdischargevariationsare small,and the largeststream-
1818
MANGA:
MODEL
FOR
DISCHARGE
6
3• .
.
0
model -obs.ervat; • .....
1968
1972
•
1976
1980
1984
year
Figure 6. Model predictionanddifferencebetweenmodelpredictionandobservation for Quinn River (top) and Fall River (bottom), streamswith the smallestand largestvaluesof d, respectively.
In order to determined for the resultsshownin Figure 5 and Table 2, we used the entire time series.In Figure 7 we show valuesof d obtainedusingshortertime seriesof 1 year (open circles)and 7 years(solidcircles)centeredaroundthe symbol (i.e., the value of d for the 7-yeartime interval1947-1953is shownby a solid circle at 1950). The greater variabilityof inferred d for Fall River basedon 1 year of data is due to the relatively small signal which remains after subtractingthe mean. For example,between1978 and 1982the hydrographis nearlyflat sothat determiningd is problematic.For the Quinn River, annualdischargevariationsare largeenoughthat 1 year of flow measurementsis sufficient provide reasonableestimatesof d. In fact, usingdischargemeasurementsat 1-month intervalsover a time periodof 1 year is sufficientto determine
dominatedstream(indicatedwith a blue-dashedcurveon 7.5min USGS topographicmaps)whichsuggests that thesespikes are due to a runoff component.Indeed, thesespikesare correlated in time with peaksin the runoff-dominatedstreams.
5.
Application to Other Streams
Here we apply the model to two other streamsystemsfor whichdailydischargerecordsare available:SnowCreek (north of Broken Top Mountain) and the DeschutesRiver above CranePrairie Reservoir(Figure 1 and Table 1). Both streams have no diversionor regulationupstream,and both have dischargecharacteristics similar to the spring-dominatedstreams studiedin the previoussection.
d for the Quinn River. 5.1.
For the CultusRiver there are a few yearswith a spikein the springtime(indicatedwith arrowsin Figure 5). Betweenthe spring source and gaging station there is a small runoff-
Snow Creek
The model describedabovewasappliedto SnowCreek, and the resultingobservedandpredictedhydrographsare shownin
10
0 1year fit
0
ß 7 yearfit
Fall River o
-
o
-
0
ß
0
•
O
Oo
0
OO
•
O
Oøo•eeee øe..ooee•e•e .. •
oO
0 1 yearfit
2.0
ß 7 yearfit
Quin0 River
1.6
1.2
-
• OoO• • Oo o•
O
Oege•••eß•eeee•eooø OOO
0.8
0.4
I 1940
. I
I
I
1950
1960
1970
I 1980
year
Figure 7. Model valuesof the diffusionlength scale,d, usingonly selectedlengthsof time seriesfor Fall River (top) andQuinnRiver (bottom).The opencirclesarevaluesof d basedon 1 yearof datacorresponding to the date on the x axis.The solidcirclesare valuesof d basedon 7 yearsof data centeredaroundthe date on the x axis.
MANGA:
MODEL
FOR
DISCHARGE
1819
0.5 for this secondmodel. Model and observedhydrographs are shownin Figure 9. Although this secondmodel resultsin a better modeling efficiency,it still underestimatespeak discharge,at least for years with peak dischargesgreater than about 10 m3/s. The ability of this secondmodel to accountbetter for dis-
2.0
Deer Creek
charge variations mayprovide information about thesubsurI
1988
face flow pathwaysfor the DeschutesRiver's aquifer. The model suggeststhat most of the rechargeoccursover a small area at somedistancefrom the springbecauseL '/L is sosmall. The basindrainedby the DeschutesRiver is a largebasin(area
I
1989
1990
1991
•340 km2) surrounded by the crestof the Cascades to the
year
west,SouthSisterandBrokenTop mountainsto the north, and Mount Bachelor to the east.A number of springsand runoffdominated streams dischargeinto Sparks, Homer, and Elk
Figure 8. Model discharge(bold curve) and observeddischarge(thin curve)for SnowCreek.
lakes from which there is no outlet. The small values of L'/L
Figure8, againusingthe Deer Creek hydrographasa proxyfor recharge;the best fit occursfor d = 0.90. The Snow Creek hydrographclearly containstwo components.The first is a short-periodand large-amplitudecomponent(big spikesin Figure8) associated with runoff and interflowfrom snowmelt in the springtimeandrainfall.The modelsuccessfully describes a secondcomponentwhichhas a long period and small amplitude, and in our model correspondsto flow associatedwith groundwaterflow. The modelingefficiencyof the bestmodelis 0.057(a smallvaluebecausethe modeldoesnot reproducethe large and frequentrunoff-dominateddischarges). 5.2.
Deschutes
River
The hydrographof the DeschutesRiver, just above the Crane Prairie Reservoir,alsohasfeaturessimilarto the springdominated
streams studied in section 4. The Deschutes
River
drains the spring-fedLava Lake (Figure 1), and the gaging station record for the Deschutes River also contains a smaller
contributionfrom spring-dominated SnowCreek (a different SnowCreek from the one studiedin section5.1). The model described here results in ME
- 0.69 and under-
estimatespeak discharges by severalcubicmetersper second. Becausethismodelingefficiencyis low,we alsousedthe model studied by Manga [1996]. Briefly, rather than assumingrechargewas proportional to distancefrom the spring,we assumethat the rechargeoccursUniformlyin the region0 < x < L' (and that the aquiferextendsfrom 0 < x < L); thustwo parametersneed to be estimated,d and L'/L. For the DeschutesRiver we find d = 1.07 and L'/L: 0.24, resultingin ME = 0.82. For comparison, the four spring-dominated streamsstudiedin the previoussectionare bestfit by L'/L •
and d (comparedwith the Fall River) suggestthat much of rechargefor the aquifer which feedsthe Lava Lakes and DeschutesRiver occursnear Sparks,Elk, and Homer lakes. 6.
Discussion
The modelingresultsindicate that the dischargecharacteristics of the spring-dominatedstreamsin any given year are governedprimarily by the previouswinter'ssnowmeltand rainfall. For example, in the 1972-1973 and 1976-1977 winters there was relativelylittle precipitation(see Figure 5), and the hydrographsfor all the streamsshowa smoothdecreaseover a 2-yearperiod. However, as discussed in section6.2, the "age" of the groundwaterbeingdischargedis probablyin the rangeof decadesto centuries.Clearly there are two timescalesoperating in the unconfinedaquifer: the first is the timescaleover whichvariationsin the heightof the water table propagate(a diffusivetimescale),and the secondis related to the actual velocity of water in pore spaces,which moves much more slowly. The time lag betweenpeak runoff and peak flow in springdominatedstreamsis related to the time required for a "pulse" of groundwaterto propagate and diffuse the length of the aquifer.As the pulsemoves,its amplitudedecreases.Thus, as the time lag increases,dischargevariations decrease.Such a relationshipis characteristicof diffusiveprocessesand is con-
sistentwith our model, (4), which is of the sameform as a diffusionequation. 6.1.
Length Scale of the Aquifers
In Figure 10 we show estimated recharge areas for the spring-dominated streamsQuinn River, BrownsCreek, Cultus
12
6
1948
1950
1952
1954
1956
1958
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
year
Figure 9. Model discharge(boldcurve)and observeddischarge(thin curve)for the DeschutesRiver above Crane
Prairie
Reservoir.
1820
MANGA:
MODEL
FOR
L = 24.2 km
DISCHARGE
1994], we find that the groundwaterwarms up at the rate of about iøC per 30 years.We note that borehole temperature measurements in the High Cascadesshowthat temperaturein manyareasis nearlyconstantto depthsof hundredsof meters [e.g.,Blackwellet al., 1990],an observationconsistent with the large volumes of circulatinggroundwatersuggestedby our model [Ingebritsen et al., 1996].On the basisof (8), the age of the water dischargedat Quinn River, Cultus River, and Browns Creek should be about 50-100 years, and that discharged at the Fall River should be about 100-200 years. These numbersare compatiblewith age estimatesfrom residencetime arguments[e.g.Roseet al., 1996]:assuminga rechargerate of 1 m/year, an aquifer thicknessof 500 m, and
Sy - 0.2 leadsto a residence timeof 100years. The tritium concentrationof water samplescollectedon September28, 1996, are 8.1, 12.8, 6.2, and 9.9 tritium units (TU) for the Quinn River, BrownsCreek, Cultus River, and Fall River, respectively,with an uncertaintyof +_2.7TU on eachmeasurement.Theseconcentrationscannotbe unambiguouslyrelated to an apparentgroundwaterage.If the groundwater beingdischargedconsistedentirelyof pre-1952water,we ? L= 13.3km would expecttritium concentrationslower than about 3 TU. However,mixingof water of all agesoccursalongthe lengthof the aquiferso that tritium concentrations in the rangeof 6-12 Figure 10. Estimatedregions(areasboundedby the dashed TU may indicate that somepost-1952water has been mixed curves) of groundwaterrechargefor Quinn River, Browns with older water.
reek; )t• [,. ....... '"""
Creek, Cultus River, and Fall River. L is the characteristic
length scalefor groundwaterflow.
6.3.
Transmissivity
Our model allows us to determine
the effective
transmissiv-
ity (and equivalentpermeabilityif we knowho) of the young River, and Fall River, aswell as the characteristiclengthscale basaltsand basalticandesiteswhichcomprisethe aquifers.The L of the aquifers(10.6, 13.3, 11.6, and 24.2 km, respectively). effectivetransmissivityis related to d by The boundariesof theseregionsare basedon surfacetopography.However,as notedin the USGS WaterResources Data, SytoL 2 r= 2d2 . (9) Oregon[1992] publications,the actual rechargearea is uncertain becauseof interbasinexchange.We assumehereafter an Shownin Figure 12 are estimatesof T basedon the length uncertaintyof 15% in our estimatedlength scaleL. scalesL shownin Figure 10; for Quinn River, BrownsCreek, 6.2. Determining Groundwater Age From Temperature Cultus River, and Fall River aquiferswe find T = 6.9, 5.6,
In orderto extendtherangeof In order to estimatethe age of the groundwaterbeing dis- 5.6, and9.4m2/s,respectively. L we also plot the transmissivity obtainedfor SnowCreek (see chargedat the springs,we recordedwater temperaturesat the mouth of the springs.Measurementswere made once per section5.1). We did not includea point for the DeschutesRiver because, month for 5 months, and the temperature was found to be constantto within the accuracyof the thermometer(+_0.2øC). For Quinn River, Browns Creek, Cultus River, and Fall River
we measured3.4, 3.6, 3.6 and 5.9øC,respectively.Temperatures are shownin Figure 11 as a functionof the length scale 6 -
L of the aquifer(seeFigure10).
Fall River
We can relate spring temperatureto groundwaterage as
follows. Because mostofthewaterenters theground assnowmelt, it shouldbe closeto freezing(y intercepton the graph). As the water flowstowardsthe spring,it is slowlywarmed by
inputof geothermal heat.Nowconsider a verticalcolumnof water with height h. Assumingthe direction of fluid flow is perpendicularto the temperature gradient (an assumption consistent with the Dupuit approximationusedin the model), the rate of temperatureincreaseis givenby dr
Q
d•-=pChSy
5-
4
-
3
-
Cultus River
Quinn River •• .
. .qq?' '
2ß
1 -
0
(8)
where Q is the heat flux into the bottom of the aquifer and C
0
5
10
15
20
25
30
lengthscale L (km)
isthespecific heat.Choosing Q = 100 mW/m2 [e.g.,Blackwell Figure 11. Spring temperature as a function of the length andPriest,1996],h = 500 m, andSy = 0.2 [Ingebritsen etal., scaleof the aquifer,L (see Figure 10).
MANGA:
T
Quinn River J•T 6
MODEL
c•.s •i•/ .
'"'
. ß
ß
ß ß ß
Snow Creek 10
15
20
25
1821
of theCascades foundbyIngebritsen etal. [1992]is O(102 103) timessmallerthanthenear-surface permeability [Manga,
k
. ß
ß
5
DISCHARGE
dominatedstreams,G can be estimatedfrom the dischargein the spring-dominated streams(assumingthat the springsare fed by groundwater),and ET can be calculatedby ET - P G - R. We are implicitlyassumingthat all the groundwateris dischargedat springs,which is clearly not the casebecauseat least a small amount of groundwater circulates to greater depthsand is dischargedat lower elevations[Ingebritsen et al., 1992].However,the effectivepermeabilityof the upper 1-2 km
I.•wns i. Creek
-
FOR
30
lengthscale L (km)
Figure 12. Transmissivitydividedby specificyield as a function of the lengthscaleof the aquifer,L (seeFigure 10).
as discussed in section5.2, the hydrologyof the systemfeeding the DeschutesRiver is relatively complicatedand required modifications to the model. However, if we chooseL to be the
distancebetweenthe Lava LakesandSparksLake (seeFigure
1),wefindL • 10 km andT • 4.4 m2/s,andthuswouldplot
1996] sothat only a smallfractionof groundwatercirculatesto greater depths.Surface runoff can be calculatedonly for the Quinn River rechargeareawhere surfacerunoff is measuredin Charlton Creek (see Figure 1). However, for the Fall River there are no runoff-dominatedstreamsin the recharge area. For Browns Creek and the Cultus River, runoff-dominated
streamsenter the spring-dominatedstreamsupstreamof the gagingstation,and the absenceof large and clearrunoff signals in the hydrographsshown in Figure 5 indicates that direct runoff makesa small contributionto the overall water budget. In Table 3 we summarize estimated values for P, R, G, and
ET on the basisof basinsizesshownin Figure 10 and the mean annual precipitation map shown in the inset of Figure 1. Groundwaterrechargerates are in the range of 66-127 cm/yr
(between40 and73% of meanannualprecipitation),andET is in the rangeof 43-98 cm/year(27-60% of mean annualpreDespitethe large uncertainties,there is evidencefor a scale- cipitation).Thesehigh ratesof groundwaterrechargeare simet al. [1994] for the Oak dependenteffective transmissivity, with T increasingby about ilar to thoseestimatedby Ingebritsen Grove Fork Basin, which lies on the western side of the crest a factor of 10 as L increasesby a factor of 10. The scale dependenceof transmissivity is a well-knownphenomenondue of the Cascades,and is composedof
