On the timescales characterizing groundwater discharge at springsseismo.berkeley.edu/~manga/paper32.pdf · 2001. 11. 25. · 1996; Manga, 1997). Discharge at Big Springs is based

On the timescales characterizing groundwater discharge at springs

Michael Manga*

Department of Geological Sciences, University of Oregon, Eugene, OR 97403, USA

Received 27 August 1998; accepted 22 March 1999

Abstract

In some regions, measurements made at springs can be used to study regional hydrogeologic processes, and determinehydraulic and transport properties of aquifers. Here, input–output models, spectral analysis, and time series analysis are used toidentify three different timescales characterizing discharge at springs. First, the “hydraulic time scale” depends on the trans-missivity of the aquifer and relates long term changes in discharge to long term changes in recharge. The hydraulic time scalewill describe, for example, the secular decrease in discharge during a period of drought. A second hydraulic time scale, referredto as the “time lag”, measures the time lag between groundwater recharge (in the studied examples, recharge is snowmelt duringthe springtime) and the time of peak discharge at the spring. Finally, the “age” or mean residence time of groundwaterrepresents the transport time of tracers and the volume of water in the aquifers. As examples, large discharge springs (typicallyseveral m3/s) in the Oregon and California Cascades are studied. These springs are associated with volcanic-hosted fracture-flow systems.q 1999 Elsevier Science B.V. All rights reserved.

Keywords:Springs; Tracers; Correlation; Spectral analysis; Time series analysis; Transmissivity; Unconfined aquifer

1. Introduction

In many hydrogeological environments, manage-ment decisions involving groundwater resourcesmust be based on data with limited spatial distribu-tion. For example, in some regions, ground-wateremerging at springs provides the only availablemeans for obtaining information about the regionalscale subsurface hydrology. Therefore, although thespatial distribution of aquifer data such as hydraulicconductivity and head might be unavailable, temporalvariations in properties of the springs such as tempera-ture (Bundschuh, 1993) or discharge (Mangin, 1984;Leonardi et al., 1996; Angelini, 1997) can be used tocharacterize the groundwater system as a whole.

Here I use a number of simplified models and time

series analysis to obtain characteristic timescalesrepresenting hydraulic and transport processes. Themodels employed are often termed “gray-box”, inte-gral balance, or lumped parameter models becausephysical properties are spatial averages, and thedetailed spatial variability of hydrogeologic variablesis ignored. Gelhar (1993), however, noted that“lumped-parameter models are often consistent withthe kind of limited data that is available for analyzingsuch problems”. Fortunately, in many cases, detailsnecessary to predict phenomena at local scales arenot necessary to understand the regional flow (Duffyand Lee, 1992).

As examples, I consider large springs in the Oregonand California Cascades. Despite the large size of therecharge areas (hundreds of square kilometers), few(and in some cases no) wells exist in the rechargeareas (Caldwell and Truini, 1997). Thus, inferencesabout the groundwater hydrogeology must be based

Journal of Hydrology 219 (1999) 56–69

0022-1694/99/$ - see front matterq 1999 Elsevier Science B.V. All rights reserved.PII: S0022-1694(99)00044-X

* Tel.: 1 1-541-346-5574; fax:1 1-541-346-4692.E-mail address:[email protected] (M. Manga)

primarily on measurements at the springs. Althoughthese springs are remote, they provide water forhydroelectric plants and irrigation, spawning groundsfor salmon and trout, and habitat for threatened andendangered aquatic species. As usual, one of the goalsfor the hydrogeologist is to describe quantitatively theproperties that characterize the springs and associatedaquifers in a manner that might be useful in resourcemanagement decisions. For the examples I studied,three timescales characterize the water discharged atthe springs, and are related to (1) the time of peakdischarge at the spring; (2) the long term responseof the spring to temporal changes in recharge; and(3) the mean residence time of water in the aquifer.Because these timescales are related to physicalproperties, the timescales can also be used to inferproperties of the aquifers such as theirtransmissivities.

2. Geological and hydrological setting

I consider a set of eight large springs in the Oregonand California Cascades (see Fig. 1) that dischargegroundwater from basalt and basaltic andesite aqui-fers (Sherrod, 1991; Rose et al., 1996): the DeschutesRiver, Quinn River, Cultus River, Browns Creek andFall River in central Oregon, and the Rising River,Big Springs and Crystal Lake Springs along HatCreek in northern California. Precipitation in theCascades occurs primarily during the winter monthsas snow, and groundwater recharge is dominated bysnowmelt during the springtime. Recharge areas forthe eight springs range from about 30 to 400 km2.Mean annual discharge ranges from 0.7 to 8.5 m3/s(Meinzer, 1927). All gauging data are based ondaily measurements collected by the U.S. GeologicalSurvey and published inWater Resources Databooks(Hubbard et al., 1992; Mullen et al., 1993). Table 1lists the mean discharge and the time period overwhich daily discharge measurements are available.

The geological setting of the springs in Oregon isdescribed in geologic maps by Sherrod (1991),MacLeod and Sherrod (1992), and Scott and Gardner(1992). Briefly, the Oregon springs lie in the HighCascades subprovince, and the aquifers consistprimarily of quaternary basaltic andesites. Becausethe individual lava flows rarely extend more than afew kilometers in the subsurface and are probably atmost a few tens of meters thick, the “aquifers” weconsider here are probably composed of many inter-bedded flows (Gannett et al., 1996). We expect thatmuch of the subsurface flow will occur in the rela-tively permeable and rubbly tops and bottoms of indi-vidual flows. The hydrogeology of the Hat Creekbasin, located at the southern end of the Cascaderange, is described by Rose et al. (1996). Again, inCalifornia, the aquifer rocks are of composed ofQuaternary volcanic rocks that range in compositionfrom basalt to andesite. Groundwater recharge ratesare typically one-half to two-thirds of the mean annualprecipitation (Ingebritsen et al., 1994; Rose et al.,1996; Manga, 1997).

Discharge at Big Springs is based on the measureddischarge in Hat Creek above Rising River, and thusincludes, in addition to the groundwater discharged atBig Springs, a runoff component and a contributionfrom other small springs (Rose et al., 1996).

M. Manga / Journal of Hydrology 219 (1999) 56–69 57

Fig. 1. Locations of large springs in the Oregon and CaliforniaCascades.

Following Rose et al. (1996), discharge at CrystalLake Springs is obtained by taking the difference indischarge measured upstream at the Hat #1 power-house (USGS station 11358800) and downstream atthe Hat #2 powerhouse (USGS station 11359300);similarly, the discharge from the largest springalong Hat Creek, Rising River, can be estimated bysubtracting the discharge measured upstream on HatCreek (USGS station 11355500) from the dischargemeasured at the Hat #1 powerhouse. Unfortunately,regulation by the powerhouses affects the hydrographfor the Rising River, so I only consider its meanannual discharge in this paper. For all the othersprings, the analysis is based on daily dischargemeasurements.

Some of the springs (Quinn River, Cultus River) are“classical” springs with all the discharge occurring ata single location, as shown in Fig. 2. Other “springs”,e.g. Browns Creek, represent the discharge of numer-ous smaller springs located close to each other. In allthese cases discharge is measured at gauging stationslocated on river reaches. Some of the discharge datalisted in Table 1 also include a component of diffuseaquifer discharge that occurs between the main springand the gauging station, such as the Fall River(Gannett et al., 1996). Finally, some of the gaugingdata also include a runoff component, in particularsome of California records shown in Fig. 4. Surfacerunoff and interflow contributions can be recognizedby the large amplitude and short duration of thepeaks in the hydrograph. In most cases, a runoffcontribution is not apparent in the hydrographs ofthe spring-fed streams in Oregon (see Fig. 5 in Manga,1997).

3. Hydraulic timescale,Th

In this section, I consider models that can be used toinfer a hydraulic timescale,Th, over which changes ininput to the system (recharge) are reflected in changesin output (discharge).Th characterizes, for example,long term changes in discharge related to droughts.Here I first follow the approach used by Manga(1997), because it was found to be the most straight-forward technique for determiningTh for the springsin the Cascades.

The evolution of a water table in an unconfinedaquifer, with heighth, can be described by the Bous-sinesq equation for unsteady flow with accretion(Bear, 1969, Section 8.2). In one horizontal dimen-sion, the Boussinesq equation is given by

2h2t� K

f

2

2xh2h2x

� �1

N�x; t�f

�1�

whereK, f , t andx are hydraulic conductivity, effec-tive porosity, time and horizontal position, respec-tively. N describes the time-dependent and possiblyspatially-variable recharge. Here,h is measured fromthe bottom of the aquifer, as illustrated in Fig. 3.

The Boussinesq equation is frequently linearized bysettingh�x; t� � �h 1 h1�x; t�; where�h is a constant andequal to the mean water height in the aquifer (see Fig.3). If h1= �h is small (i.e. p 1), terms quadratic inh1

can be neglected, and Eq. (1) becomes

2h2t� K �h

f

22h

2x2 1N�x; t�f

�2�

Discharge,q�t�, is proportional toh2h=2x at x� LwhereL denotes the length of the aquifer.

M. Manga / Journal of Hydrology 219 (1999) 56–6958

Table 1Properties of springs

Spring USGS station number Mean flow (m3/s) Period of recordavailable (month/day/year)

Deschutes River, OR 14050000 4.2 07/01/1938–09/30/1991Quinn River, OR 14052500 0.67 07/01/1938–09/30/1991Cultus River, OR 14050500 1.8 07/01/1938–09/30/1991Browns Creek, OR 14054500 1.1 07/01/1938–09/30/1991Fall River, OR 14057500 4.2 07/01/1938–09/30/1991Big Springs, CA 11355500 4.8 04/01/1930–09/30/1994Crystal Lake Springs, CA 11359300–11358800 3.2 10/01/1986–09/30/1996Rising River, CA 11355500–11359300 8.5 10/01/1986–09/30/1994

The time-dependent recharge is not known a priori,so I assume that the groundwater recharge isproportional to discharge measured at nearby runoff-dominated streams. This approximation is reasonable,because the same processes that produce recharge,namely rainfall and snowmelt, also produce runoff.The timing of snowmelt varies with elevation andlatitude, and the recharge elevation of the springsvaries by up to a couple of hundred meters. Never-theless, the “recharge proxy” seems to be successful atexplaining discharge variations over time scales of

months to years (see Figs. 5, 8 and 9 in Manga,1997). In the absence of any other information, therecharge proxy provides at least a starting point forestimating the time-dependent recharge. The runoff-dominated streams I use to estimate recharge are DeerCreek in Oregon (USGS station 14052000) and the PitRiver near Canby in California (USGS station11348500).

The coefficientK �h=f in Eq. (2) can be interpretedas a diffusivity that describes the rate at whichvariations in the height of the water table decay.


Fig. 2. Sources of the Quinn River (left) and the Cultus River (right). Mean discharges are 0.67 and 1.8 m3/s, respectively. Both rivers begin asfull-fledged rivers that emerge from springs. A person in each picture, shown for scale, is indicated by an arrow. The trees on the left areponderosa pine; the trees on the right are lodgepole pine.

▼▼

▼

recharge N(x,t)concentration c (t)in

discharge q(t)concentration c (t)out

x

h(x,t)

L

h

Fig. 3. Geometry and dimensions of the unconfined aquifers that discharge groundwater at the springs.cin andcout are the input and output tracerconcentrations.

The characteristic response time for a diffusive systemis the characteristic lengthscale squared and dividedby twice the diffusivity (Turcotte and Schubert, 1982,p. 157), in this case,

Th � f

2K �hL2 �3�

The timescaleTh defined by Eq. (3) differs only by afactor of 3/2 from the “hydraulic response time”fL2

=3T defined by Gelhar and Wilson (1974), whereT � K �h is the aquifer transmissivity.

In Fig. 4(a), calculated (bold curve) and measured(thin curve) hydrographs for Big Springs for the timeperiod 1945–1995 are shown. The calculated curve isthe “best-fit” solution to Eq. (2) forTh � 13 years. Iassume, in the absence of information about the spatialdistribution of recharge, that recharge of the aquiferoccurs uniformly over its length.Th of the “best”model is the one that maximizes the modeling effi-ciency (i.e. minimizes the difference between calcu-lated and measured hydrographs). Note thatTh is theonly model parameter for Eq. (2). In Eq. (1),


1945 1950 1955 1960 1965 1970

1970 1975 1980 1985 1990 1995

2

8

6

4

2

8

6

4

calender year

disc

harg

e (

m /

s)3

1987 1988 1989 1990 1991 1992 1993 1994

1987 1988 1989 1990 1991 1992 1993 1994

6

5

4

3

2

6

5

4

3

2

disc

harg

e (

m /

s)3

Big Springs

Crystal Lake Springs

Big Springs

Big Springs

a)

b)

Fig. 4. (a) Calculated (bold curve) and measured (thin curve) hydrographs for Big Springs, CA;Th� 13 years. (b) Calculated (bold curves) andmeasured (thin curves) hydrographs for Big Springs, CA and Crystal Lake Springs, CA during drought years.

however, I am assuming the aquifer has a horizontalbase. Eq. (1) can be modified in a straightforwardmanner to account for a finite slope (Bear, 1969,Section 8.2), though the model would then be char-acterized by two parameters rather than one. In thename of simplicity, for the present discussion I willassume that Eq. (1) is an adequate description of thegroundwater system. In fact, the optimal values ofTh

obtained in models that account for the finite slope aresimilar to those presented here, but have larger valuesof Th.

In Fig. 4(b), measured and calculated hydrographsfor Crystal Lake Springs for the time period 1987–1994 are shown, along with the Big Springs hydro-graph for the same time period; the best-fit for CrystalLake Springs is obtained forTh � 14 years. The late1980s and early 1990s was a period of drought inCalifornia, and discharge at both springs decreasedby about a factor of two between 1987 and 1993.The timescaleTh characterizes the rate of decrease.For the Rising River,Th is based on a comparison ofcalculated and measured mean annual discharge,rather than of daily discharge. In Fig. 5, the meanannual discharge for water years 1986 to 1992 alongwith model predictions forTh� 5, 15 and 25 years areshown. The calculated discharges forTh� 15 are mostsimilar to the measured discharges; shorter and longertimescales produce annual fluctuations that are toolarge and too small, respectively.

Manga (1997) applied the model described above

to the springs in the Oregon Cascades, thus measuredand calculated hydrographs will not be reproducedhere (partial time series for two springs are shownlater in Fig. 8). Th can be obtained from Manga(1997) usingTh � d2, where Th will be a time inyears andd was defined as a characteristic diffusionlengthscale. For the Deschutes River, Quinn River,Cultus River, Browns Creek and Fall River,Th �1.1, 1.6, 2.4, 3.2 and 6.3 years, respectively, wereobtained. In Table 2, I summarize values ofTh, L,and inferred transmissivityT � K �h based on an esti-mated effective porosityf � 0:1.

Other approaches can be used to determineTh forsprings, e.g. using three-dimensional groundwaterflow models (Angelini and Dragoni, 1997), applyinganalytical models (Leonardi et al., 1996; Bhar andMishra, 1997), using recession-flow analysis (Drogue,


Rising River

1986 1987 1988 1989 1990 1991 1992

3

4

5

6

7

measured

water year

mea

n an

nual

dis

char

ge (

m /

s)3

Fig. 5. Mean annual discharge for water years 1986–1992: measured discharge is shown by the solid line, and calculated discharge is shown bysymbols for various values ofTh. Water years begin October 1 and end September 30.

Table 2Hydraulic timescales and transmissivities

Spring Th (years) Tlag (days) L (km) T (m2/s)

Deschutes River 1.1 98 t 10 0.2Quinn River 1.6 64 11 0.1Cultus River 2.4 85 12 0.1Browns Creek 3.2 137 13 0.09Fall River 6.3 112 24 0.2Big Springs 13 47 18 0.04Crystal Lake Springs 14 – t 40 0.2Rising River t 15 – t 40 0.2

1972; Brutsaert and Nieber, 1977; Bonacci, 1993;Szilagi et al., 1998), and frequency–domain analysis(Gelhar, 1993). I attempted recession-slope analysis,following Brutsaert and Lopez (1998), and was unsuc-cessful becauseTh is always greater than the timeperiod without significant recharge (typically abouthalf a year). Next, frequency–domain analysis isconsidered.

3.1. Frequency–domain analysis

Gelhar and Wilson (1974) show that the water

balance for an unconfined aquifer can be simplifiedto a linear relationship between the average waterlevel and the rate of discharge if the change in waterlevel is small compared to the total thickness of theaquifer. For this so-called linear reservoir model, therelationship between the spectral density of discharge,Sout, and that of recharge,Sin, is

Sout � Sin

1 1 v2T2h

�4�

wherev is the frequency. Definitions of spectra aregiven by Duffy and Gelhar (1985).


Dimensionless frequency,

Deschutes River

Quinn River

Cultus River

Browns Creek

Fall River

Big Springs

10

1

0.1

10-4

10-2

10-3

0.01 0.1 1 10

10

1

0.1

10-4

10-2

10-3

0.01 0.1 1 10

10

1

0.1

10-4

10-2

10-3

0.01 0.1 1 10

10

1

0.1

10-4

10-2

10-3

0.01 0.1 1 10

10

1

0.1

10-4

10-2

10-3

0.01 0.1 1 10

10

1

0.1

10-4

10-2

10-3

0.01 0.1 1 10

Tra

nsfe

r fu

nctio

n,

Fig. 6. Transfer function (defined by Eq. (5)) as a function of dimensionless frequency (normalized byTh) for the Deschutes River, Quinn River,Cultus River, Browns Creek, Fall River, and Big Springs. The bold curve is a theoretical result for the so-called “linear reservoir model”(Gelhar, 1974); thin curves are based on discharge measurements using 214 measurements; filled discs are based on windowed dischargemeasurements (see text for details).

In Fig. 6, the transfer function is shown as

uH�v�u2 � Sout=Sin �5�for the six springs with long gauging records alongwith the theoretical transfer function (bold curve)given by Eq. (4). The thin black curve is the transferfunction based on one continuous time series of justunder 45 years of measurements (214 measurements).The filled discs are transfer functions based on datawindowed using a Parzen window and 8 segments of4096 discharge measurements. Spectral analysis wasperformed using the subroutines of Press et al. (1989).Frequency is normalized by the value ofTh

determined previously, rather than by following acurve matching procedure to determineTh from thetransfer function (Duffy and Gelhar, 1986).

In general, the value ofTh obtained previouslyproduces reasonable agreement with the linear reser-voir model (Eq. 4), at least for low frequencies. Myclaim of “reasonable” agreement is based on a visualcomparison with similar curves of Duffy and Gelhar(1986). The spectral approach is most successful (andmost useful), ifTh is small compared with the lengthof the discharge time series. For example, theDeschutes River and Quinn River haveTh < 1 year,and the spectral analysis of 45 years of data canresolve the bend in the transfer function; Big Springs,however, hasTh < 13 years, and spectral analysiscannot resolve the flat part of the transfer function.For short timescales, or for high normalized frequen-cies, the approximations made by the linear reservoirmodel (Eq. 4) will not be valid (Gelhar, 1974), and weshould thus expect worse agreement between themodel and measurements at high frequencies.

4. A second hydraulic timescale: the time lag,Tlag

Previous studies of the springs in the OregonCascades have noted that peak discharge occurs insummer and sometimes in fall even though peaksnowmelt occurs during the springtime (Whiting andStamm, 1995; Manga, 1996). In Fig. 7, the hydro-graphs for the Quinn River and runoff-dominatedDeer Creek for water year 1990 (October 1–Septem-ber 30) are shown. The gauging stations are locatedonly a few hundred meters apart and differ in eleva-tion by 24 m. Peak discharge at the spring (QuinnRiver), at least in 1991, is delayed from peak runoffby about 2 months.

The time lag shown in Fig. 7, hereafter denotedTlag,also represents hydraulic processes, but is muchshorter than the hydraulic time scaleTh determinedin Section 3. Whiting and Stamm (1995), during adiscussion over Browns Creek, suggested that “it isnot clear, if the fall runoff is associated with meltingof the snowpack from the previous winter or fromearlier years”. Whiting and Stamm (1995) furtherobserved that “at Browns Creek, the peak flows inthe late summer and fall are delayed by at least 4months, if not 1 year and 4 months, or more”. Idemonstrate next, thatTlag is in fact a few months,as indicated in Fig. 7.

In Fig. 8, measured and calculated hydrographs forthe Quinn River and Fall River along with the hydro-graph for Deer Creek (the recharge proxy) are shown.The calculated hydrographs are the “best-fits” to Eq.(2), using the entire discharge record for each spring.During the winter of 1972–1973, there was very littleprecipitation (see Deer Creek hydrograph) and the


0

0.5

1.0

2.0

1.5

Feb 1Oct 1 Dec 1 Apr 1 Jun 1 Aug 1

Tlag

Quinn River (spring-dominated)

Deer Creek (runoff-dominated)

Dis

char

ge (

m /

s)3

Oct 1

Fig. 7. Hydrographs for Deer Creek, a runoff-dominated stream, and Quinn River, a spring-dominated stream during water year 1990. Peakdischarge in Deer Creek is due to springtime snowmelt. Spring-fed streams are characterized by a nearly constant discharge, and a delay of peakdischarge relative to peak snowmelt.

hydrographs for the springs recede nearly continu-ously for 2 years with only small summer increases.Thus, the peak discharge that occurs each year iscorrelated with the previous snowmelt andTlag isless than 1 year (and less thanTh determinedpreviously).

Other investigations of springs have found thatcorrelation can be useful for characterizing the rela-tionship between recharge and discharge (Padilla andPulido-Bosch, 1995; Angelini, 1997). I show nextthat, for the springs studied here, cross-correlationscan be used to identify this second hydraulic timescale,Tlag.

In Fig. 9, the cross-correlation between discharge atthe spring and recharge is shown. Again, the proxy forrecharge is discharge measured in a runoff-dominatedstream (Deer Creek for the springs in Oregon and thePit River for the springs in California). For compar-ison, the auto-correlation for each spring is shown.The mathematical definitions of these correlationscan be found in Jenkins and Watts (1968). Briefly,the cross-correlation function for two time series illus-trates the degree to which the time series are corre-lated as a function of the time interval betweenindividual values in the series. The auto-correlationfunction is the cross-correlation function of a timeseries within itself. The correlation functions are

defined such that they have values between2 1(perfect negative correlation) and1 1 (perfect posi-tive correlation); a value of 0 indicates no correlation.

Tlag ranges from 47 to 137 days, i.e. peak dischargein the runoff-dominated stream is best correlated withdischarge at the springs a few months later. For all thesprings studied, the annual recharge by snowmeltproduces correlation functions with a period of 1year. The magnitudes ofTh andTlag are not correlated,as shown in Table 2. WhileTlag is clearly differentfrom Th, both timescales may still represent ground-water flow in the same aquifer. The Boussinesq equa-tion has both a short-time solution, when flow is notinfluenced by the no-flow boundary condition thatexists at one end of the aquifer, and a long time solu-tion (Polubarinova-Kochina, 1962). Both solutionscan be characterized by the recession-flow analysis(Brutsaert and Lopez, 1998; Szilagi et al., 1998) but,as discussed previously, the recession flow analysiscannot be successfully applied (at least in a straight-forward manner) to the spring hydrographs studiedhere.

Finally, Manga (1996) showed that for a subset ofthe springs in Oregon,Tlag increases asTh increases.The detailed relationship between the two time scales,however, also depends on the spatial distribution ofrecharge. For example, if most of the recharge occurs


Year

3

1.5

0

0

0.4

0.8

1.2

1,6

0

2

4

6

1969 1970 1971 1972 1973 1974

Fall River

Deer Creek (recharge proxy)

Quinn River

Dis

char

ge (

m /

s)3

computedmeasured

computedmeasured

dry year

Fig. 8. Hydrographs for Deer Creek, Quinn River and Fall River. Calculated hydrographs are shown with bold curves, and measuredhydrographs are shown with thin curves. In the winter of 1973, there was little precipitation (see hydrograph of runoff-dominated DeerCreek), and discharge in the spring-dominated streams decreases nearly continuously for a 2-year period.

at the distal end of the aquifer,Tlag will be relativelylong, whereas if recharge is uniformly distributed,Tlag

will be relatively short (Fig. 3 in Manga, 1996).

5. Groundwater age,Tage

The mean ageTage(also called mean transit time orturnover time) is related to the volume of mobilewater in the aquifer and the volume added per unittime,

Tage��hf�N

�6�

where �N is the mean recharge rate,�h the mean aquiferthickness, andf the effective porosity. While bothTage and Th depend on geometrical properties of theaquifer, such as its mean thickness,Tage does notdepend on hydraulic properties such as hydraulicconductivity,K.

An estimate of the reasonable range of possibleages can be made using approximate aquifer thick-nesses based on borehole temperature measurements.In quaternary volcanic rocks in the high Cascades,temperatures in many regions are nearly constantfrom the surface to depths of a few hundred meters,typically between 100 and 300 m (Blackwell et al.,1982; Swanberg et al., 1988; Ingebritsen et al.,1989). Such isothermal conditions are probably dueto high groundwater recharge rates and rapid ground-water flow (Ingebritsen et al., 1989). Nearly isother-mal geotherms are also found in parts of the SnakeRiver Plain in Idaho, and are again associated withhigh recharge rates and rapid groundwater flow (Brottet al., 1981). The depth of the nearly isothermal regionin the Oregon Cascades, unfortunately, is poorlyknown and probably highly variable (Ingebritsen etal., 1994). In our particular study areas, there arefew wells (and no deep wells), and most of thetemperature–depth data are from further north in theCascades. There is no obvious reason, however, toexpect our study area to be different from the geolo-gically similar regions further north. If we associatethe depth of this nearly isothermal region with ouraquifer thickness�h, assuming an effective porosityof 0.1, and a mean recharge of 1 m/year (Manga,1997),Tage should be about 10–30 years.

Measurements of environmental tracers Tritium,

3H, and tritogenic Helium,3He, can also be used toestimate Tage. Owing to the distributed nature ofrecharge, and mixing of waters recharged at differenttimes, models must be used to interpret such environ-mental tracers. Integral balance or lumped-parametermodels representing the relationship between inputand output can be described by a convolution integral.

cout�t� �Zt

2∞cin�t�g�t 2 t�e2l�t2t�dt �7�

Here,g(t) is the impulse response function (or weight-ing function),cin and cout (see Fig. 3) the input andoutput concentrations, respectively,t the time,t anintegration variable, andl the radioactive decayconstant. Particular models are defined in terms oftheir response functions,g(t).

The so-called “exponential model” introduced byEriksson (1958), assumes an exponential distributionof transit times, with

g�t� � 1Tage

e2t=Tage �8�

and is typically used for distributed sources (Duffyand Lee, 1992). The exponential model is appropriatefor unconfined aquifers withK and f decreasingexponentially with depth as well as constantK andf (Maloszewski and Zuber, 1982). The exponentialmodel (Eq. 8) is used for the streams in California,because I assumed previously that recharge wasuniformly distributed.

The input function for 3H is based on IAEAmeasurements made in Portland, Oregon, the closeststation with a long record. Measurements in Portlandbegan in 1963 and are extended back in time by scal-ing measurements made in Ottawa, Canada.3Hconcentrations are then reduced by 20% to approxi-mate the effect of the difference in latitude betweenthe springs in California and Portland, Oregon(Michel, 1989). In Fig. 10(a), the concentration of3H in discharged water is shown as a function ofTage assuming that3H recharge is proportional to3Hdeposition. The measured concentrations of3H at theRising River and Big Springs are 4.23̂ 0.5 and4.09^ 0.5 TU (Tritium Unit), respectively (Rose etal., 1995). To the extent that the input function andexponential model are reasonable approximations,ages for both springs of about 8̂ 2 years areobtained. Given a single3H measurement and some


chosen model forg(t), the inferred age is not unique,however: a much larger age. 100 years will alsoproduce concentrations of about 4 TU.

In principle, the measurement of3He (the decayproduct of 3H) allows “true” mean ages (referred tohereafter as the3He/3H age) to be obtained withoutrelying on the uncertain3H input function (Torgersonet al., 1978). Again, however, mixing will affect the3He/3H age, and a model of the form of Eq. (7) stillneeds to be applied for an aquifer with distributedrecharge and measurements made at the spring. Forexample, in Fig. 10(b), the3He/3H age is shown as afunction of Tage based on the exponential model (Eq.8) and the same3H input function used previously.

ForTageless than about 10 years, the3He/3H age over-estimatesTage whereas forTage greater than about 30years, the3He/3H age will remain close to about 30years. In both cases, the contribution of3He and3Hfrom < 30-year old water associated with the largestatomic bomb peaks biases the age inferred fromdischarged3He and 3H concentrations. Rose et al.(1995) also cite a3He/3H age of 20.5 years for theRising River, if no radiogenic or mantle He is present;the 4He concentration is that expected for the air satu-rated water. Because of the high recharge rates, about1 m/year (Rose et al., 1996), diffusive escape of3Heshould be negligible (Szabo et al., 1996). TheCascades is a volcanically active arc so that there


1.0

0.8

0.6

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0

0 200 400

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0 200 400

lag (days)

corr

elat

ion

Deschutes River

Quinn River

Cultus River

Browns Creek

Fall River

Big Springs

lag (days)

Fig. 9. Correlation functions for the Deschutes River, Quinn River, Cultus River, Browns Creek, Fall River, and Big Springs. Solid curves arecross-correlations between discharge at the spring and discharge in a nearby runoff-dominated stream. The dashed curves are auto-correlationsof spring discharge.

may indeed be a volcanic contribution of He, but I willassume for now that the3He/3H age cited by Rose etal. (1995) is appropriate. Based on Fig. 10(b), I thusinfer a true mean ageTageof about 8 years, identical tothe age inferred from the3H concentration.

Duffy and Lee (1992) used a set of numerical“experiments” to show that the exponential modelcan be successful, to within a few percent, at predict-ing discharged concentrations, especially for largeaspect ratio aquifers. The input–output behaviour iswell represented by “effective” parameters and isindependent of the second-spatial moments of proper-ties such as hydraulic conductivity and the magnitudeof local dispersive mixing. Thus, while the lumped-parameter models are in some ways “gray-box”models (Gelhar, 1993), they may in fact be adequateto obtain spatial averages such as estimates of meanresidence times.

6. Discussion and concluding remarks

The timescales that describe discharge variations atthe springs represent aquifer behaviour over largescales and long times. For the aquifers and springsstudied here, the only data available are interpolatedclimate data (Daly et al., 1994), temperature measure-ments (Manga, 1998), gauging measurements ofspring discharge, and, in a few cases, concentrationsof nonpoint source environmental tracers. The appli-cation of lumped-parameter models and the use oftime series analysis, indicate that three different time-scales characterize discharge at springs. Two of these,the hydraulic timescale,Th, and the age,Tage, areobtained by using input–output lumped-parametermodels in which physical properties are averagedover the entire aquifer. These timescales are basedonly on discharge measurements, spatially averagedrecharge estimates, and tracer concentrations at thesprings, and thus cannot predict the spatial distribu-tion of water levels in the aquifers. Although thedetailed groundwater flow cannot be determinedfrom measurements made at the springs, modelssuch as those applied here are “appropriate whenaddressing problems of overall policy and manage-ment decisions relating to the behaviour of the aquiferover extended periods of time” (Gelhar, 1993).

All three timescales describe features of the

discharged spring water, either the mean residencetime of the water or temporal variations of discharge,and are thus useful from a management point-of-view.For the springs studied here, the time lag is a fewmonths and is thus a useful timescale for issues relatedto the timing and volume of water use and reservoirregulation over similarly short periods of time. Thehydraulic time scale describes long term dischargevariations and thus describes the effect of, and therecovery from, droughts. The groundwater age, onthe other hand, characterizes mass transport and is


3

4

5

6

7

8

9

1 105 50

Rising River

H (

TU

)3

30

20

10

01 105 50

Rising RiverHe

/ H

age

(ye

ars)

33

T (years)age

a)

b)

4.23 ± 0.5 TU

20.5 years

Fig. 10. (a) Calculated3H concentration in spring water, measuredin Tritium units, as a function of groundwater age,Tage (dashedcurve). See text for details on model assumptions and the3Hinput function. The measured3H concentration at the RisingRiver springs is 4.23̂ 0.5 TU (Rose et al., 1995) which impliesa mean groundwater age of about 7–9 years. (b) Calculated relation-ship between the measured “true”3He/3H age and groundwater age.The measured3He/3H age at the Rising River springs is 20.5 years(Rose et al., 1995) which implies a groundwater age of about 8years.

thus relevant for assessing the impact of potentialsubsurface contamination associated with develop-ment (in an active volcanic arc such as the Cascades,this might be geothermal development).

While three distinct timescales can be identified,Th

and Tage are related by geometrical and subsurfacehydraulic parameters,

Th � f2L2

2K �NTage�9�

Using values for the Rising River aquifer ofTh < 15years andTage < 8 years (both values determined inthis paper), and an estimate forL of 40 km (Rose etal., 1996), a hydraulic conductivity is obtained

K < 0:002 m=s �10�This aquifer-averaged value ofK is large, but stillwithin the range of values for fractured basalts citedby Domenico and Schwartz (1998) and similar toother estimates for young basalts (Ingebritsen andScholl, 1993; Saar and Manga, 1999).

Finally, Manga (1998) observed that the tempera-ture of discharged groundwater at the springs inOregon, measured at approximately monthly inter-vals, varied by less than 0.1–0.28C over a 2 yearperiod. By contrast, some springs show seasonaltemperature variations of several degrees Celsius(Bundschuh, 1993). The nearly constant temperaturereflects the large volume of the aquifers and residencetime of groundwater (i.e.Tageon the order of a decade)which effectively averages annual temperature varia-tions, and will even damp climatic temperature varia-tions on the time scale of decades.

Acknowledgements

The author thanks E.R. James, M.O. Saar, T.P.Rose and an anonymous reviewer for thoughtfulcomments and helpful suggestions. B. Hudson, T.P.Rose and M.L. Davisson are thanked for providing the3H and 3He data. This work was supported by theNational Science Foundation, grant EAR9701768.

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Documents

On the timescales characterizing groundwater discharge at springsseismo.berkeley.edu/~manga/paper32.pdf · 2001. 11. 25. · 1996; Manga, 1997). Discharge at Big Springs is based