JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 94, NO. B7, PAGES 9439-9451, JULY 10, 1989
The Influence of GroundwaterFlow on Thermal Regimes
in MountainousTerrain' A Model Study
CRAIG FORSTER
Departmentof Geology,Utah StateUniversity,Logan
LESLm SMITH
Departmentof GeologicalSciences,Universityof British Columbia,Vancouver
Groundwaterflow in high-relief mountainousterrain is modeled to examine how geology, surface
topography,climate, and regionalheat flow influencethe patternand magnitudeof an advectivethermal
disturbance.A numericalprocedureis usedto estimatethe positionof the water table within constraints
providedby the availableinfiltrationrate and the permeabilityof the mountainmassif. Resultsshowthat
wherethe watertableis locatedat depthwithin the mountainmassif,the rate of groundwater
recharge,rather
thanpermeability,is the appropriate
factorto characterize
the potentialfor an advectivedisturbance.Thermal
disturbances
shouldbe assessed
within the frameworkof the two-dimensionalcharacterof the flow system
becauseinterpretations
basedon one-dimensional
modelsare proneto significanterror. Modeling of sitespecificsystems
canexploitthe existenceof a permeability"window,"where temperatures
in dischargeareas
andin springsdischarging
from fracturezonesreachpeakvalues. Temperatures
of thermalspringsreflectthe
complexinteractionbetweenflow within the mountainmassifand flow throughpermeablefracturezones.
Consequently,simple calculationsrelating the geothermalgradient, depth of maximum groundwater
circulation,anddischarge
temperature
improperlyrepresent
thephysicsof theprocess.
INTRODUCTION
Ongoinginterestin characterizing
theEarth'sthermalstate,
investigating
the natureof certaingeologicprocesseswithin
the uppercrust,and exploringfor geothermalresources
has
led to thecollectionof heatflow datain mountainous
regions.
A subsetof thesedata indicatesthat groundwaterflow can
causea significantadvectivedisturbanceof thermal regimes
[Steeleand Blackwell, 1982; Mase et al., 1982; Black et al.,
1983; Reader and Fairbank, 1983]. Although advective
disturbance
of thermalregimeshas long beenrecognized(see
review by Lachenbruchand Sass[1977]), the interactionof
groundwaterflow and heat transferwithin a mountainmassif
hasreceivedlittle study.
Schematicflow systemsshownin Figure 1 illustratethe
amplitudeand wavelengthof groundwaterflow systemsthat
might develop in both high-relief and low-relief terrain.
Greater
relief
found
in
mountainous
terrain
causes
an
enhancedvertical componentof groundwaterflow that, in
turn, may enhance advective disturbance of the thermal
regime. Furthermore,closelyspacedridges and valleys in
high-relief terrain cause an advectivedisturbancethat has a
restricted lateral extent when compared to the longwavelengthdisturbances
typical of low-relief terrain. As a
consequence,
groundwaterflow within a mountainmassifcan
causelarge-amplitude,short-wavelength
perturbationsof the
thermalregime that may distortour view of broad regionalscalevariationsin heatflow. With few exceptions
[Steeleand
sparseand selectively distributeddata may provide little
insight into regional heat flow in mountainousterrain. In
contrast,groundwaterflow systemsin low-relief terraincan
causeperturbationsthat are relatively easy to characterize
usingheat flow data collectedfrom boreholesseparatedby
distancesexceedingseveral tens of kilometers [Willet and
Chapman,1987]. Suchdatasetshavehelpedconstrainmodel
studiesthat illustratehow basin-scalethermalregimesin lowrelief terraincan be perturbedby regionalgroundwaterflow
[Brott et al., 1981; Smith and Chapman, 1983; Garven and
Freeze, 1984; Woodbury and Smith, 1985; Gosnold, 1985;
Majorowicz et al., 1985).
Numericalmodelsprovide a quantitativeframeworkfor
interpreting subsurface temperature data collected in
mountainousterrain. Hanoaka [1980] examinesthe influence
of surface topography on the interaction between
topographicallydriven fluid flow and free convectionby
modelingidealized,steadystatehydrothermalsystems.While
a nonuniformdistributionof regionalheat flow is implicitly
specified;Hanoaka [1980] did not addressthe issuesof
climate,differentconditionsof regionalheatflow, and depth
to the watertable.lngebritsenand Sorey[1985] incorporated
the influence of high-relief topographyin their transient
numerical
model
of advection-dominated
heat transfer
at
Mount Lassen,Oregon. Becausea basal sourceof heated
fluid is used to representthe circulation and heating of
groundwater recharged on the mountain flanks, the
relationship
betweenthe magnitudeof the fluid sourceand
Blackwell, 1982; Reader and Fairbank, 1983], most thermal
rates
of
groundwater
rechargeat the groundsurfaceis poorly
data sets in mountainous terrain are obtained from isolated
defined. Recenthydrologicmodelingperformedby Forster
boreholes
locatedat the valleyfloor or at lowerelevationson
and Smith, 1988b illustratespatternsof groundwaterflow
the mountain flank. In addition, distances separating
expectedwithin a mountain massif under a variety of
boreholes often exceed 5 km.
Inferences based on these
geologic,climatic,andtopographic
conditions.
Our approachdiffers from thoseof Hanoaka [1980] and
Copyright1989by the AmericanGeophysical
Union.
lngebritsenand Sorey [1985] in two key respects.First, we
explorethe influenceof climateon advectiveheattransferby
Papernumber89JB00297.
0148-0027/89/89 JB-00297 $05.00
specifyingrechargeto the flow systemas an input variable
9439
9440
FORSTERA.• SMrr}l: Motn, rr•
GROUNDWATER
AND THERMALREOIMES
•_ (a)
2 -
-2
z
9
2
•
1
w
W
I
0
•.
I
I
I
8
1•2
8
12
I
I
I
16
20
(b)
o
-1
0
4
16
20
1
20
(c)
0.5
ß
-0.5
•
0
•
4
8
DISTANCE
-- Water table
•
,
12
(kin)
Hypothetical
groundwater
pathline
Fig. 1. Schematic
groundwater
flow systems
for (a) CoastMountains
of BritishColumbia,(b) RockyMountainsof British
Columbia-Alberta,and (c) conventionallow-reliefterrainafterFreezeand Witherspoon[1967].
rather than assumingan arbitrary water table configuration.
This is accomplished
usinga free-surfacemodelingtechnique.
Second,we incorporatepermeablefracturezonesas discrete
entitiesthat play an importantrole in the developmentof
thermalsprings.This approach
hasenabledus to characterize
the nature of groundwater flow systems in mountainous
terrain and to demonstratethe important influence of the
thermalregime on mountain-scaleflow systems[Forster and
Smith, 1988b]. This paper expandsour work to examinethe
influence of fluid flow on thermal regimes.The modeling
resultshave implicationsfor assessing
the way that advective
heat transfermodifiesthe near-surfaceexpression
of regional
heat flow, controls the genesis of landform-controlledore
deposits,and influences the developmentof geothermal
systems.Topics to be consideredinclude(1) the natureand
magnitudeof advectiveheat transfer, (2) the interactionof
thermallydriven free convectionand topographicallydriven
forced convection within a mountain massif, (3) the transfer
of heat within fracture zones, and (4) the nature of thermal
Mountainsof Canada and the United States(Figure lb), and
the central Cascades of the Pacific Northwest. The horizontal
basal boundary is assumedimpervious to fluid flow.
Regionalconductiveheat flow is represented
by a uniform
heatflux (Ha) appliedacrossthebasalboundary.
The upperboundaryof the domainis the bedrocksurface.
The thin cover of discontinuous
surficialdepositstypical of
uplandareasof mountainslopesis viewedas a thin skin of
variablethickness
thatis not explicitlyincludedin the model.
Resistanceto heat transferin the thin surficial depositsis
assumednegligibleso temperaturesat the bedrocksurface
match those at the ground surface. Surface temperature
conditions
are defined
in
terms
of
a reference
surface
temperature
(Tr) and a thermallapserate (G). Exceptions
occur where fracturesoutcropto producesprings. In such
cases,temperatures
at the bedrocksurfaceare assumedto
reflectthe temperature
of groundwater
flowing in thefracture
zoneratherthanthe ambientair temperature.
In earliernumericalstudiesof coupledfluid flow andheat
flow systems,it is assumedthat the
springs. Following the example of Thompson[1964], transferin regional-scale
mountainousterrain is defined as rugged topographywith watertableis the upperboundaryof the flow domain[e.g.,
local relief in excess of 600 m.
Smithand Chapman,1985]. An importantlimitationof this
approachis that the magnitudeand spatialdistributionof
groundwaterrechargeare imposedimplicitly when a water
APPROACH
table configurationis specifiedexplicitly.Modeling results
Our conceptualmodel for groundwaterflow and heat indicatethat it is preferableto estimatethe water table
of groundwater
rechargerate,
transfer is shown in Figure 2. Vertical boundaries are positionwithin the constraints
[ForsterandSmith,1988b].
insulatedand impermeable,reflectinga periodicrepetitionin basalheatflow, andpermeability
surfacetopographywith wavelengthof 12 km andamplitude If thisalternativeapproachis not adopted,varyingvaluesof
or heatflow in a sensitivityanalysismay yield
of 2 km. This topographyresemblesconditionsfoundin the permeability
CoastMountainsof British Columbia(Figure la), the Rocky unrealisticvalues for groundwaterrecharge. Theseresults
FORSTERAND SMITH: MotrNTAIN GROUNDWATERAND THERMAL,REGIMES
Mean
Annual
Precipitation
Available
POD
•
Recharge
rate I s, is the verticalvolumetricflux of fluid acrossa
horizontal
planeofunitarea((m3 s4) m'2).In theabsence
of
Infiltration
i
-'-<_t
--.•
9441
T T
detailedfield data,I sis bestthoughtof as a percentage
of the
mean annualprecipitationrate (Figure 2). By providing
explicit control on groundwaterrecharge,this modeling
approachallows us to examinethe influenceof various
climaticenvironments(e.g., rangingfrom arid to humid) on
the natureand magnitudeof fluid flux and advectiveheat
transfer.
Discharge
The availableinfiltrationrate represents
an upperlimit for
rechargeto the groundwaterflow system. Where the water
tablecoincideswith the bedrocksurface,only a portionof the
I
,,
2
available
infiltration
is transmitted
to the bedrock.
The
remainderis contributedto surfacerunoff. The ability of the
flow systemto transmitthe incomingfluid dependsupon a
variety of factors including topographic relief, rock
permeability,andbasalheatflow [Forsterand Smith,1988b].
0J
Where the water table lies below the bedrock surface, the
:
/
entire
/
Z
available
infiltration
rate
is assumed to cross the
bedrocksurfaceto be transmitteddirectly to the water table
via one-dimensional flow through an unsaturated zone.
Adoptingthis approachenablesus to controlthe patternand
b.•.•
kf••
magnitudeof groundwaterrechargewithout specifying,a
.:.'T.:.:•:.:.'.--t'...:..•:..•.:...'•....:.•:.........,...........-.,
....
ß
........................
•.• ...............
..... priori,thepositionof thewatertable.
-2
Heat is transferredby both conductionand advection
throughoutthe domainshownin Figure 2. Advectiveheat
::::::::::::::::::::::::::::::::::::::::::::::::::::::
,.'i:i::
:i:!:i:::
3::::::::::::::::
:'• -::::i:!:::
3:::::::::::::::::::::::
:i:!:!:::::
•' transfer above the water table is representedby onedimensionalmovementof fluid, in the liquid phase,at a rate
determined
by the availableinfiltrationrate.By assuming
that
.:.:.:...:.:.:.:.:
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::::::.
::::::
•
':. :::::::::::::::::::::::::::::::::::::::::::::::::
iiii!!i:i!i:iiii!i:ii!iiiiii:!:!:iii:i:!:!:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i.i:i:i:i:i:::i:i:i:i:i:i:i:i:i:i:i:!:i:!:)i:i:i:!:!::/•
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?
::
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::
::
::
::
i::::::::::::
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7' ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
'22'):•'2"2"i"7"'2')
.......
2'7-') ....::::::::::::::::::::::::::::::::::::::::::
I / / / / /
/":?:
-4
0
X:X
2
4
6
DISTANCE(km)
0
x=x
Heat
Flow
2 x 10'8m s'• or0.6m yr4 [Forster
andSmith,1988b].
Free-surface
Insulated
and impermeable
Impermeable
Basal
boundary
boundary
low-permeability
Unsaturated zones with infiltration rates less than this amount
are likely found only in the most arid of climates. A
reasonableupperlimit for availableinfiltrationratesis about
o
Basal
the lowerlimit for infiltration
rate is 10'•2 m s4, vapor
movement can be taken to be negligible [Ross, 1984].
unit
Fig. 2. Conceptualmodel for groundwater
flow and heat transferin
mountainous
terrain. POD (pointof detachment)
and HP (hingepoint)
are termsdefinedto describethe pointswherethe watertableintersects
the groundsurfaceand the transitionbetweenrechargeand discharge,
respectively.
Z0wandZ• definetheelevation
of thewatertableatX0 and
X•;. Z0sandZff definetheelevationof the groundsurfaceat X0 andXL. A
more detaileddescription
of the boundaryvalue problemis given by
Forster and Smith, 1988a.
A basallow-permeability
unit occupiesthe lower 2 km of
the domainto providea regionof conduction-dominated
heat
transfer(Figure 2). The remainderof the domaincontainsa
higher-permeability
unit where advectiveheat transfermay
dominate. Althoughvaluesof bulk permeabilityare likely
controlledby the degreeof fracturingwithin the rock mass,
we treat the rock mass as an equivalentporousmedium.
Exceptionsto this assumptionare representedas discrete
permeable
fracture
zones
witha homogeneous
permeability
kf
andwidth b (Figure 2).
The mathematicalmodel describingthe systemshownin
Figure2 is expressed
by coupledpartialdifferentialequations
for heattransferandfluid flow, by equationsof statefor the
fluid properties,
andby boundaryconditions
for the thermal
andfluid flow problems.Readersinterested
in detailsof the
mathematical
can, in turn, have an importantimpact on the pattern and
magnitude of advective heat transfer. Therefore, when
simulatinglargerangesin the valuesof permeabilityandheat
flow, it is importantto explicitly constrainfluid flux across
the upper boundary. This constraintcan be imposedby
assumingthat the watertableneednot coincidewith the upper
boundary,which in our studyis assumedto be the bedrock
surface.
Fluid moving within the thin skin of surficial depositsis
lumpedwith overlandflow as a runoff term. The remaining
fluid availablefor recharge,termed an available infiltration
model
are referred
to Forster
and Smith
[1988a]. Key assumptions
include(1) steadystatefluid flow
and heat transferin a saturatedporousmedium with onedimensionalfluid flow throughthe unsaturatedzone at the
availableinfiltrationrate, (2) no internalheat sourcesor sinks,
(3) thermalspringspresumedto dischargeonly from major
through-going
fracturesintersectingthe bedrocksurface,(4)
thermalequilibriumexistingbetweenfluid andsolid,(5) the
fluid being pure liquid water with densityand viscositya
functiononly of temperatureand pressure,(6) the heat
capacityandthermalconductivity
of thefluid beingconstant,
and (7) porosity, permeability,and thermal conductivity
9442
Fo;:sTERANDSMITH:MoLr•r•
TABLE
Parameter
GROUNDWATER
ANDTHERM• REGIMES
1. TYPlc.nL SIMUlaTION PARAMm'rms
Definition
Value
Fluid Flow Parameters
permeabilityof basalunit
permeabilityof upperunit
vertical infiltration
rate
Thermal
b *
s
1.0 x 10-22m2
1.0 x 104• m2
2.0 x 10-9m s4
Parameters
basal heat flow
60.0 mWm -2
surfacetemperature
gradient
referencesurfacetemperature
porosityof basalunit
porosityof upperunit
solidthermalconductivity
fluid thermalconductivity
vaporthermalconductivity
specificheatcapacityof water
5 øCkm4
10øC
0.01
0.10
2.50W m4 øK4
0.58W m4 øK4
0.024 W m4 øK4
4186.0
Jkg40c 4
saturation above water table
0.0
longitudinalthermaldispersivity
transverse
thermaldispersivity
100.0 m
* Parameters varied in the simulation
10.0m
series.
varyingin spacebut not as a functionof temperature
or
pressure.
permeabilityand porosity (e.g., decreasingk and n with
increasingdepth) have been simulated with the model;
In adoptingthisapproach,
we neglecta varietyof factors
(slope aspect, transientsin
surface temperatureand
groundwaterflow, and complex heterogeneities
in thermal
and hydraulic properties)in order to concentrateon the
however, the overall character of the results differs little from
andhenceon advective
heattransferhasbeenassessed
by
constant for all simulations.
thatobtainedwith thetwo homogeneous
units. Althoughrock
thermalconductivity(•/) is uniformthroughout
the system,
varyingporosityandsaturationproducecontrasts
in effective
character of advective heat transfer within a mountain massif.
thermal conductivity (•.e). Longitudinal and transverse
The influenceof thesefactorson groundwater
flow systems thermal dispersivities(tz•, %) are uniform and are held
Forster and Smith [1988b]. Factorsother than long-term
In the followingparagraphs,
frequentreferenceis madeto
hydrologicandthermaltransients
causedby climatechanges a referencecasesimulatedusingtheparameter
valueslistedin
or igneous
activityarejudgedto be of secondary
importance Table 1. Resultsfor the referencecase,shownin Figure3c,
[Forsterand Smith, 1988b]. We are in the processof are used as a standardfor comparisonwith subsequent
modifying the model to accommodatethermal transients simulation
results. The reference
I z of 2 x 10'9 m s4
causedby igneousactivity. The equationsdescribingfluid representsthe rate of infiltration that might reasonablybe
flow and heat transferare solved using a Galerkin finite available for recharge in a climate transitional between
element technique. The water table elevationis estimated
semiarid
andhumid.Theupper
unitpermeability
(k•)of 1045
usingan iterativefree-surface
methodthatis constrained
by m2 represents
relativelypermeable
conditions
typicalof
the availableinfiltrationrate appliedat the bedrocksurface. moderatelyfracturedcrystallineor argillaceousrock [Freeze
Two-dimensionalvertical sectionswith planar or radial and Cherry, 1979; Neuzil, 1986]. The referenceheat flux
symmetryare representedby linear triangularelements.Thin,
(H•) of 60 mWm'2is a typicalconductive
regional
heatflux.
high-permeability
fracturezonesare includedby embedding Thermal
conductivity
(•/) is uniformlyfixed at 2.5 W m'•
one-dimensional
line elementsin the field of triangular øK4(typical
of values
reported
forcrystalline
rocks).Barry
elements.Details of the computational
procedureare given [1981]suggests
thatatmospheric
thermallapseramsmayvary
by Forster and Smith [1988a].
RESULTSOF NUMERICAL SIMU•TIONS
Simulations
areperformedby assigning
a setof fluid flow
andthermalparameters
(Table1) withina geometry
similarto
fromlessthan2 øCkm'• to greater
than8 øCkm4. In this
study,temperatures
at the bedrocksurfaceare definedusinga
median
lapserate(G) of 5 øCkm4 anda valleyreference
temperature
(T•)of 10øC.
that of Figure 2.
Characteristic variations in surface
topographyare incorporatedby consideringtwo extremesin Patternsof AdvectiveHeat Transfer
slopeprofile(Figure3): a convextopographic
profiletypical
Patternsof groundwaterflow are depictedin Figure3 by
of glaciated
crystalline
terrainanda concave
profilecommon path lines (dotted lines) representingthe track of a fluid
in volcanicterrain.In mostsimulations,
bothupperandlower particleenteringthe flow systemat a specifiedpointon the
zoneshavehomogeneous
andisotropic
permeability
(ku, kb) bedrocksurface.Pathline spacingis inverselyproportional
to
anduniformporosity(%, %). More complicated
variations
in the flux of fluid (specific discharge)through flow robes
FORSTER
AI'4D
SMITH:Motm'r•
GROUb/DWATER
ANDTHERMAL
REGIMES
9443
2
ao
2
d.
...........
•,. Groundwater
pathline
/•
•-
•
Heatline
5---- Isotherm (øC)
Watertable
Regionnot penetrated
by basal heat flux
"'"""•
"':'"'" Basallow-permeability
unit
0
2
4
6
DISTANCE (km)
0
2
4
6
DISTANCE (km)
Fig. 3. Patternsof groundwaterflow and heat transferin convexand concavetopographysimulatedwith the reference
conditions
ofTable1 (except
where
noted);
(a)k, = 1048m2,(b)k, = 1046m2, (c)/q,= 1045m2(reference
case),
(d)k, = 1048
m2andHi,= 120mWm-2,(e)ku= 1045
m2andHi,= 120mWrn-2,(f)k•= 104•m2,(g)k•= 1045
m2.
boundedby eachpair of pathlines. For example,path line
spacingin the referencecase(Figure3c) varies becausefluid
flux in the convexdomainincreases
approximately
fivefold
from a valueof 2 x 10-9m s'1 at the free surfaceto about10-8
intersectthe free surfacewith equal spacing(recall that a
uniformfluid flux is explicitlyappliedat thefreesurface).
Patternsof heattransferareshownin Figure3 by heatlines
(thick dashedlines), while subsurfacetemperaturesare
m s'1atthedischarge
area.Fluidfluxismoreuniform
in the representedby isotherms(solid lines). Kimura and Bdjan
concave
domain
(Figure3g),decreasing
from2 x 10-9m s'1at [1983] suggestthat heat lines are usefulfor mappingthe
thefreesurface
toabout1.6x 10-9m s-1atthedischarge
area. transferof thermalenergyby conductionand advection.By
Althoughsimilar,pathlinesthataredefinedby trackingthe definition,any heat line is locallyparallelto the directionof
trajectoriesof fluid particlesshouldnot be confusedwith net energyflow throughthe domain. Note that this yields
streamlines.These approaches
differ becauseflow of fluid heat lines normal to the isothermsonly in regionswhere
through each stream tube (bounded by each pair of advectiveheat transferby groundwaterflow is negligibleor
streamlines)is fixed throughoutthe domain, while fluid flow
throughflow tubesdefinedon the basisof path lines may
where the direction of fluid flow is normal to the isotherms.
Heat lines are eveNwhere normal to isotherms in the
differ from flow tube to flow tube. Flow tubes shown in
conduction-dominated
thermalregimesshownin Figures3a,
Figure3 carry the sametotal flow only when the flow tubes 3d, and3f. In this study,heat linesare identifiedby tracking
9444
FORSTERAND SMITH: Motnvr•
GROUNDWATERAND THF.JLMALREalMES
the advectiveand conductivetransportof thermal energy
originatingat the basal boundary. Becausea uniform heat
flow is appliedat thebasalboundary,eachheattube(bounded
by two heatlines)carriesthe sametotal thermalenergy.
Heat lines and isothermsshown in Figure 3 illustrate the
increasing influence of advective heat transfer as rock
permeabilityis increasedby three ordersof magnitude. In
each case, the available infiltration rate is fixed
at
the
>
referencevalue (I, = 2 x 10-9m s4). Threeseries
of
w
simulation results are shown; the convex domain with
w
referenceheat flow (Figures3a-3c), the convexdomainwith
doubled heat flow (Figures 3d and 3e) and the concave
domain with reference heat flow (Figures 3f and 3g).
Conductive
temperature
fields,obtained
fork• equalto 1048
m2, areshownin theleft-hand
panelof eachseries
(Figures
3a, 3d, and 3f). An advectivedisturbance,characterized
by
heat lines at an oblique angle to the isotherms,is evident
DISTANCE (km)
Fig. 4. Patternsof groundwater
flow andheattransferwhenI, is reduced
fivefoldfromthatofthereference
case(Figure
3c)to4 x 1040m s4.
whenk,, exceeds
about1046m2 (Figure3b). A strong
disturbance,
shownin the right-handpanel of eachseries,is
identifiedby heat lines subparallelto isothermswhen
Magnitudeof AdvectiveThermalDisturbance
exceeds
about1045m2 (Figures
3d,3e,and3g). Woodbury The magnitudeof an advectivethermaldisturbancecanbe
characterized
by notingthemagnitude
andspatialvariationof
two effects:(1) the degreethat conductivethermalregimes
arecooled,or heated,by groundwater
circulation,and(2) the
impact of advectiveheat transferon vertical temperature
gradients.These effects are illustratedin contourplots of
residualsandtemperature
gradientratios(Figure
the bedrocksurfacebecausethe basalheat flow is effectively temperature
maskedby downwardflowinggroundwater.Elsewhere,
heat 5). Temperatureresidualsare calculatedby subtracting
obtained
for a givensimulation
fromthoseof the
line patternsindicate that the entire basal heat flow is temperatures
conductivecase, at each node in the f'mite
absorbed
by thecirculatinggroundwater
andredirected
to exit corresponding
and Smith [1988] use the dependenceof the thermalregime
on bulk permeabilityin developingan inverseapproach
for
estimating
thebulkpermeabilityof a mountainmassif.
Temperatures
withintheshadedareasshownin Figures3c,
3e, and3g reflectthe temperature
of groundwater
rechargeat
in the dischargearea.
Changes in permeability or basal heat flow
are
accompanied
by little changein patternsof groundwater
flow
(Figure3). Becauseratesof fluid flow increaseask,,or H a is
increased,reducedwater table elevationsare found in higher-
permeabilityor higherheat flow cases(Figures3c, 3e, and
3g). A reducedthermaldisturbanceis foundfor the concave
slope profile (compareFigure 3c to Figure 3g) because
smaller volumes of groundwaterare transmittedat slower
ratesthroughtheconcavedomain[ForsterandSmith,1988b].
In systemswith a deep water table (greaterthan 50 m
deep),permeability
is not thefactorthatbestcharacterizes
the
potentialfor a thermal disturbance. Rather, the rate of
groundwater
rechargebestcharacterizes
the rate of fluid flux
throughthe domain.For example,reducingthevalueof 1, by
a factorof 5, from2 x 10'9 to 4 x 10-20m s'• (reflecting
a
transition to a more arid climate than that of Figure 3c),
causes a fivefold decrease in groundwaterflux and a
significant decline in water table elevation (Figure 4).
Reduced fluid flux creates a weaker thermal disturbance and a
warmer thermal regime than that found for the higherinfiltration case of Figure 3c. This result could not be
predictedon the basisof permeabilityalonebecausethe rate
of fluid flux is controlledby the infiltrationrate.Furthermore,
implicitinfiltrationratesresultingfrom arbitraryestimates
of
water table configurations may be inconsistentwith the
presumedconditionsof climateand permeability.Adopting
the free-surfaceapproachreducesthe dependence
of model
resultson the difficult-to-estimatewater table positionand
providesan additionalconstrainton the resultsby providing
explicit controlof estimatedinfiltrationrates [Forster and
Smith, 1988b].
elementmesh. Figures5a-5c illustratetemperatureresiduals
obtainedfor the advectivelydisturbedthermalregimesshown
in Figures3b, 3c, and 3g, respectively.Temperaturegradient
ratios are calculatedby normalizing vertical temperature
gradients
computed
for eachelementin themeshwithrespect
to vertical temperature gradients calculated in the
corresponding
conduction-dominated
case. Figures 5d-5f
illustratetemperature
gradientratioscalculatedfor thethermal
regimesof Figures3b, 3c, and3g.
The plots of temperaturegradient ratios indicate the
possiblemagnitudeof errors in heat flow estimatescaused
only by advectiveheat transfer. Note that additionalerrors
may be causedby temporalvariationin surfacetemperature
and spatial variation in surface topographyand thermal
conductivity.Shadedregions indicate where estimatesof
uncorrectedheat flow might be made with an error of less
than 25%.
Inferencesbasedon one-dimensional
analysesof fluid flow
and heat transfer suggest that the maximum advective
disturbance
occursin regionsof maximumfluid flux. Because
temperature
residualsand temperaturegradientratiosreflect
the disturbanceof subsurfacetemperatures
by advectiveheat
transfer,maximum values of each might be expected in
regionsof maximum fluid flux. This is not the case in the
two-dimensionaldomainspresentedhere. For example,the
absolutevalue of the temperatureresidualis greatestin a
region of cooling located beneath the mountain summit
(Figures5a, 5b, and 5c) whereratesof fluid flux within the
upper permeableunit approacha minimum (indicatedby
widely spaced flow lines in Figures 3b, 3c, and 3g).
Temperaturegradientratios in this region, however,differ
only slightly from a value of 1.0. Therefore, althougha
FORSTE••-D SMrm: Motrs'r,•r
GROUNDWATER
AND THERMAL REGIMES
9445
2 -
-- -2
+1o
•
---40•
Heating
/.0
-60
o•
0•
-4
fø
Oo
1•0
0.5
F-
?.4
0.5
--•
LIJ
LIJ
-4
0
2
DISTANCE
4
(km)
6
0
2
4
DISTANCE
6
0
(km)
!
I
2
4
DISTANCE
6
(km)
0.75 <-temperature
gradientratio<-1.25
Temperature
gradientratio < 0.0
Fig.5. Contour
plots
oftemperature
residual
(øC)forconvex
and
concave
topography'
(a) ku = 1046
m2,(b)k,= 10-15
m2,
(c)k.u= 10-lsm2andoftemperature
gradient
ratio,(d)/q,= 1046m2,(e)/q,= 10-lsm2,(f)k•= 10-15m2.
significant disturbanceis indicated by the temperature
residual,temperaturegradientsare relatively unaffected.In
contrast, maximum values of temperature gradient ratio
locatedin a region of maximum fluid flux near the valley
floor indicate a significant disturbance,while the absolute
value of temperatureresidualprovidesonly limited insight
into themagnitudeof the advectivedisturbance.Theseresults
suggestthat advective thermal disturbancein mountainous
terrain is best assessed
using a methodthat incorporatesthe
two-dimensional
characterof the nonuniformflow systemsin
an explicitmannerandpermitsbothtemperature
residualsand
temperature
gradientratiosto be calculated.
Figures5a and 5b illustratethe greatercoolingthat occurs
similarin both convexand concavedomains(the sameI z is
appliedat the free surface),the longerfree-surfacesegmentof
the convex domain yields a total flow of groundwaterabout
twice that of the concavedomain. This providesa greater
volume of fluid to absorbthe basal heat flux, leads to greater
cooling throughout the convex domain, and produces a
smallerregionof heating(Figure 5b).
Plots of temperaturegradient ratio provide the greatest
insightinto the possibleinfluenceof advectivedisturbance
on
heat flow studiesin mountainousterrain. For example, the
maximumtemperature
gradientratioexceeds2.0 at thevalley
floor in eachcaseshownin Figure 5. Additionalsimulations
(not shown)indicatemaximumtemperature
gradientratiosas
whenk, is increased
tenfold
from1046m2 to 104sm2. highas3.6in thedoubled
heatflowcase(Hb= 120mWm'2)
Increasedgroundwaterflux absorbsthermalenergysupplied whenk, equals
10'is m2 (Figure
3e).Theseresults
suggest
at the basalboundarywith lessheatingof the domain. As a that hydrologiceffectsalonecouldcausevaluesof regional
consequence,the higher-permeabilitydomain contains a heatflow to be overestimated
by at least200% wherethermal
smallerregionof heatingwith reducedvaluesof temperature data are collectedfrom boreholeslocatedat the valley floor in
residual.
Ask, is decreased
below10']6m2,advective
heat terrain
withpermeability
inexcess
of 10'17m2.A permeability
transferis reduced,a greaterpercentage
of the basalheatflux of 1047 m2 is typicalof slightly
fractured
crystalline
or
is transmitted
by conduction,and a largerregionof heating
develops.
Comparingternperature
residualsin Figure 5b with those
of Figure 5c highlightsthe greatercoolingthat occursin the
argillaceousrocks [Freeze and Cherry, 1979]. Therefore
greatervaluesof permeabilitymight masonablybe expected
to causea significantadvectivedisturbancein fracturedrocks
domainwith convexslopeprofile. Althoughfluid flux is
Wheninterpreting
heatflow data,it is oftenassumed
thata
located at lower elevations on a mountain flank.
9446
FORSTER
ANDSMrm: MOUNTAINGROUNDWATER
ANDTHERMAL
REGIMES
linear relationshipbetweentemperatureand depth indicates
the absenceof a hydrologicdisturbance.In low-relief terrain
with relativelysimplepatremsof groundwaterflow this is
likely a reasonable assumption.Using the linearity of
temperature-depth
profiles to test for the presenceof an
advectivedisturbancein high-relief terrainmay be misleading
for two reasons. First, vertical variation in the spacingof
isothermsshown in Figures 3a, 3d, and 3f confirms that
nonlineartemperature-depth
profiles shouldbe expectedin
conductive thermal regimes in mountainousterrain. Thus a
nonlinear temperature-depthprofile does not necessarily
indicatethe presenceof an advectivedisturbance.Second,
temperature-depth
profilesmay be obtainedin regionswhere
temperature
gradientsappearconductive,despitea significant
advective disturbance. For example, regions with nearverticalcontoursof temperature
gradientratio(Figures5d, 5e,
and5f) indicatewhereperturbedtemperature
gradientswould
differ from thoseof thecorresponding
unperturbed
conductive
casesonly by a constantmultiple (the value of the gradient
ratio). Analysisof temperature-depth
profilesobtainedin
theseregionswould incorrectlysuggesta conductivethermal
regimeunperturbed
by groundwater
flow. The likelihoodfor
confusionis reducedin higher-permeability
mountainmassifs
(Figures5e and 5f) where fewer regions with near-vertical
contoursof temperature
gradientratio may be found.
Crosshatchedareas shown in Figures 5e and 5f indicate
wheretemperatureinversionscan be expectedin temperaturedepthboreholelogs (temperaturegradientratiosare lessthan
0.0). Such conditionsare predictedto occuronly when ku
2 ..................
__ ....................
Groundwater
...•
Io
i Dividei,
ii
>
.-.-.-.. 5 :'.:.:.:.:.:
::::::::::::::::::::::::::::::::::::
..'.===========================================
.:.:.:.:..,..:.:.:.:.:.:.:.:.:,:..•.:.:.:.:.:.:.:.:.:.:.:..:.:........
............ -...::::1;...7...................7........
::i:i:i:i.!.i.i.!.!.i.•.i.•.i.i.i.
::::::::::::::::::::::::::::::::::::
=========================
:::::
.............
'i•
.......
•:::il:!:i:i:i:i:i:i:•:i:Eii!!:!:!:i:E:
....
'.........
:::::::::::::::::::::::::
.i:i:i:!:!:i:i:!:i:i:E:!:!:!:ii•
_J
- 4 ::::::::::::::::::::::::::::::::::::
ß'....
0
2
4
::::::::::::::
--,..'.'.'.'"'-...'.t.-'-•...::.....'.:•:',
..:...::....:.:.:.:.:,.:;:::::2::::.:.:.:.:.:.:.:.:.:.:
6
DISTANCE
8
10
12
(km)
Fig. 6, Thezmal regime near an asymmetricmountainvalley with
opposingsummitsof equalrelief, modeledwith the referenceconditions
of Table 1.
the convexprofile. This imbalancein flow causesboth the
groundwater
divide and the regionof upwarpedisothermsto
be displacedto the right. The regionof upwarpedisotherms
is displacedfurtherto theright as therelief of the linearslope
is reduced.
Figure6 indicatesthat the groundwater
flow dividemay
intersectthe bedrocksurfaceupslopeof the valley floor in
regions of asymmetricsurface topography. Therefore
chemical or thermal signaturesof a subsurfaceheat source
locatedbeneaththe left-handconvexslopemay be expressed
in springsdischargingupslopefrom the valley floor on the
opposinglinear slope. In addition,groundwatersamples
exceeds
10-•5m2. Isotherms
foundwithintheshaded
regions collectedat the valley floor may provide little information
of Figures3c, 3e, and3g illustratethe temperatureinversions regardingconditionsbeneaththe linear slope profile. As
that might be expectedin terrainwith relief of 2 km over 6 permeabilityis reduced,the warpingof isothermsbecomes
km. Basedon thesemodelingresults,significanttemperature less pronounced. The position of the groundwaterflow
inversions observed in boreholes located on the flank of
divide,however,is unchanged.
Mount Hood, Oregon [Steeleand Blackwell, 1982], suggest Free Convection in Mountainous Terrain
that the bulk permeabilityof the volcanicpile likely exceeds
10-15m2. Thisinferred
valueis withintheestimated
rangeof
Conceptualmodels developedto explain the origin of
permeability
for fractured
igneous
rockof 1046to 10-• m2 geothermalsystemsand the genesisof volcanogenicore
[Freezeand Cherry, 1979].
depositsofteninvokethe processof free convectionwithin a
mountainmassifto provideelevatedtemperatures
at shallow
Influenceof MountainTopography
depth [e.g., Henley and Ellis, 1983]. Although thesemodels
The high topographicrelief of mountainousterrain usuallyincorporatetheinfluenceof a localizedheatsource,it
amplifiesthe influenceof surfacetopography
on patternsof is of interest to examine the interaction between
groundwater flow and the resulting advective thermal topographically
driven and thermally driven fluid flow in
disturbance.For example,asymmetryin surfacetopography regionswith uniform heat flow.
cancausea strongwarpingof isotherms
nearthevalley floor
Figure 7a illustratesthe thermalregime and patremsof
(Figure6) that might be mistakenfor the thermalsignatureof fluid flow associatedwith developmentof free-convection
a free-convection
cell or a permeablefracturezone. Figure6 cells in mountainousterrain, given the geologicconditions
represents
a mountainvalley with a convexslopeprofile on shownin Figure7d. A thin low-permeability
horizon(100 m
the left and a linear slope profile on the right. Mirror thickwith permeabilityki) separates
two permeable
units;an
symmetry is assumed at vertical boundarieslocated at upper
unitwithk. equal
to 104•m2anda lowerunitwithk•
opposingsummitsof equalelevation. The stronglydisturbed equal
to5 x 104•;
m2. Path
lines
shown
intheright-hand
thermalregimeshownin Figure6 is obtainedby assigningthe panel of Figure 7a depict two free-convectioncells that
referenceconditionsgiven in Table 1.
developin thelowerpermeableunit whena basalheatflow of
The pronounced
warpingof isotherms
shownin Figure6 is 120mWm-2is applied
andtheupperandlowerpermeable
centered on the groundwaterflow divide located within a units
areisolated
bysetting
ki to1049m2.Usingthedefinition
regionof closelyspacedsubverticalheat lines. The position of Cheng [1978], the Rayleigh number for the lower
of the divide reflects a balance between
the total flow
of
groundwaterbeneatheach slope profile. In the symmetric
casesshownin Figure 3, the divide is vertical and locatedat
the valley floor. In the asymmetriccaseof Figure 6, total
flow throughthe linear slopeprofile is about60% of that of
permeableunit is computedto be approximately170.
Upwarpedisothermsand convergentheat lines shownin
Figure7a highlightthe regionof upwellingfluid in the freeconvection cells. Heat lines pass through the lowpermeabilityhorizon to be swept in the direction of
Fol•s•
nh'D S]vlrm: Motrffr•
GROIJ'NDWA• AND THF_.RMAL
REGIMES
9447
:•:•:..-:..'•fRegion
not penetrated
ter
table
Do
•
I ;'-.-'".
'"'"'•-..
•r"Grøundwater
pathline
Heatline
I •
..
2
4
DISTANCE
6
0
(km)
horizon,
ki
_[ku=10-15m2
•
-.-.•:.:.:.:.:.:.:.:.:.:.:
:.:.::•:•••:•:•:•:•:•:•:•:•:•:•:•:•:•••••••••:•••••••:•:•:•:•:•:•:•:•:•:•:•:•:•:•:•:•:•:•••••••••••••
..
0
Low-permeability
Permeable layer, k•
2
4
DISTANCE
6
0
(km)
2
DISTANCE
4
6
(km)
Fig.7. Thermalregimes
andpatterns
of groundwater
flowin mixedfreeandforcedconvection
scenarios
withHt, = 120mW
m-2,I, = 5 x 10'9m s4' and/q,
= 10-15me;(a)k/= 1049
meandkl = 5 x 1045
me,(b)ki = 1047meandk•= 5 x 1045me,(c)k/=
1049m2andk•= 5 x 1044m2,(d)geologic
conditions
forsimulations.
topographicallydriven flow in the upper permeablezone.
the permeable
lowerunit (3 x 1040m s-], on average).
Therefore
Clearly,free-convection
cellsare lesslikely to be disturbedas
the
basal
heat
flux
has
little
influence
on
temperaturesdirectly beneaththe mountainsummit (shaded
regionof Figure7a).
Previous studies of mixed free and forced convection
specifya nonuniformheat flux at the basal boundaryto
representa localizedheat sourcethat inducesfree convection
[Elder, 1967; Hanoaka, 1980]. Modeling resultsshownin
Figure 7, however, indicate that high-relief surface
topographycan causelateralvariationsin the thermalregime
that, in turn,causefree convectionto developin regionswith
uniform basal heat flux.
either k, or topographicrelief are reduced,or as H b is
increased.
Increasing
k]by oneorderof magnitude
to 5 x 10-14m2
increasesthe Rayleighnumberto approximately520, causes
more vigorousfree convectionwithin the lower permeable
layer (Figure 7c), and produces fluid flux similar in
magnitude
tothatof theupper
permeable
layer(5 x 10-9m s4,
on average). Becausethere is little movementof fluid across
the low-permeability horizon, the volume rate of fluid
fluxin thepermeable
upper
unit(5 x 10-9m s4, onaverage)
is
enteringand exiting the domain is little changedfrom that
found for the weakly convectingcase (Figure 7a). The
computedincreasein buoyancy-driven
fluid flux in the lower
permeablelayer, however, suggeststhat the free-convection
cells will be better able to resist displacementby
topographically
driven fluid flow as the permeabilityof the
interveninghorizonis increased. In the vigorousconvection
case (Figure 7c), enhancedrates of buoyancy-driven
fluid
flow yield significamcoolingthroughoutthe lowerpermeable
layer, except at the top of the region of upwelling, where
temperatures
are similar to thoseof the weakly convecting
case(Figure7a). Temperatures
in the upperpermeableunit
differ little from thosefoundin the weakly convectingcase,
exceptin the vicinity of the valley floor. These modeling
resultssuggestthatit will be difficult to definethepresenceof
free convectionwithin thepermeablelowerunit usingthermal
about150timesgreaterthanthebuoyancy-driven
fluidflux of
data collected in shallow boreholes.
Increasingthe permeabilityof the interveningunit ki
to 10-17m2increases
thegroundwater
fluxbetween
theupper
and lower permeable units and increasesthe total flow
throughthe system. Increasedlateral flow in the lower
permeableunit displacesthe regionof upwellingto the right,
increasescooling beneaththe mountainsummit, and causes
the region of warmer temperatures
to migrate beneaththe
valleyfloor(Figure
7b). Further
increasing
ki to 1046m2
completelyobliteratesthe free-convectioncells to producea
strongly disturbed thermal regime and patterns of
groundwaterflow similar to thoseof Figure 3f. Under the
conditions simulated here, the locations of free-convection
cellsandregionsof elevatedtemperature
are easilydisturbed
by topographicallydriven fluid flow becausegroundwater
9448
FORST• riND SMrm: MotnvrA•
GROUNDWA'IER
AND THERMALREGIMES
ao
PermeableFracture Zonesand Thermal Springs
Fracture
As topographicrelief increases,permeablefracturezones
have a greaterimpact on the developmentof groundwater
flow systems and thermal regimes [Forster and Smith,
1988b]. The interrelationshipbetween permeablefracture
zonesand regionalflow systemshas receivedlittle attention,
yet it has importantimplicationsfor understanding
the origin
of fault-controlledthermalspringsandassessing
theeffectsof
chemicalalterationon the mechanicalpropertiesof faulted
rocks.In contrastto previousmodel studies[Lowell, 1975;
Sorey, 1978; Kilty et al., 1979; Goyal and Kassoy, 1980;
-4
bo
Bodvarssonet al., 1982], fluid flux within the fracture zone in
this study is dictatedby conditionscontrollingthe regional
flow system,ratherthanby a specifiedfluid sourceat depthor
by a specifieduniform fluid flux within the fracture zone.
This differencereflectsthe fact that our approachallowsfluid
to enter, or leave, the fracture zone at any point along its
length. Previousworkersoftenassumean imperviousfracture
wall or specifya uniformfluid flux within thefracture.
We representfracturezonesin our model with a seriesof
line elements having uniform thickness (b) and a
homogeneous
equivalent
porousmediumpermeability
(kf).
Becausean approximatelinear relationshipexists between
permeability
andfluidflux,various
combinations
of b andkf
Z•
"•
-2
-4
0
2
4
6
DISTANCE
8
10
12
(km)
producethe same patternsof groundwaterflow in systems Fig.8. Influence
of a steeply
dipping
fracture
zonewithuniform
k/- b =
(m2.m)
onthermal
regimes
withintheasymmetric
topography
of
withmatching
watertableconfigurations
andmatching
kf. b 10n.ku
6;(a)ku= 10-25m2,(b)k,= 10-26m•.
[Forster
andSmith,1988b].Theproduct
kf. b (expressed
in Figure
unitsof m2-m)is termedthetransmissivity
of thefracture
zone
and
is used
to
characterize
the
simulation
results.
Figures 8a and 8b include a steeply dipping fracture zone, the surrounding
rock massto yield a warmerthermalregime
extending from the valley floor to the basal boundary. (Figure
8b) anda warmerspringtemperature
(24øC).The
Fracture
widthb andpermeability
kfare constant
everywherepattern of upwarpedisothermsshown in Figure 8b differs
along the fracture zone. The transmissivityof the fracture from those of Sorey [1978] and Kilty et al. [1979], who
zonekf. b is assumed
toequal
104.k,,
(m2.m)
andmight predictlarge temperaturegradientsat shallowdepthsin the
reasonably
correspond
to several
combinations
of b andkf;b fracture zone and almost isothermalconditionsat greater
of1mand
kfof104.k,,,
bof10mandkfof103.k,,,
orbofi00 depth becausethey specify uniform fluid flux within the
fracturezone. Reducedfluid flux at depth in the fracture
mand
kfof102.k,,.
Figure 8a illustrates the influence of the fracture zone zones studied here causes reduced advective heat transfer at
described
above,
withkuof 10-]5m2,ongroundwater
flowand depth and producesa more uniform temperaturegradient
heat transferwithin the asymmetrictopographyof Figure 6. throughoutthe fracturezone.
Patternsof fluid flow and thermalregimesshownin Figure
The effect of the permeablefracturezone is to increasethe
totalflow of groundwaterby 75%. This changein fluid flux 8 provide a quantitativeframework for studyingprocesses
yields a region of cooling (temperaturesare depressed
by at suchas chemicalalterationof faulted rocks or depositionof
least10øC)that is concentrated
near the fault zonebut ore minerals in fault zones. Resultspresentedin Figure 8
extends,to a lesser degree, throughoutthe domain. Patterns show how the permeabilityof the fault zone may influence
of fluid flow and heat transferare greatlymodifiedbecause the developmentof regional-scalegroundwaterflow systems
80% of the total fluid flow, and the entire basal heat flow, is
andthermalregimes. The regionalsystemmust,in turn,have
capturedby the fracture zone. Furthermore,the basal heat a strongimpact on the detailed characterof fluid flow and
flow is effectively masked everywhere along the upper temperaturewithin the fault zone that controlsthe processes
boundary (shaded region in Figure 8a). Inflow from the of chemical alteration and mineral deposition.Our results
surroundingrock masscausesfluid flux in the fracturezone to provide support for the growing opinion that local-scale
increasealmostlinearly, aboutone order of magnitude(from processesare best examinedin the context of the regional
10-6to 10-5m s4),in moving
upward
alongthefracture
zone hydrogeologicsystem.
Figure9 illustratesthe variationof springtemperature
as a
from the top of the low-permeabilityunit to the bedrock
surface. Rapid fluid flow in the fracture zone, fed by the functionof upperunit permeability(k•) for boththereference
surrounding
rock mass,yields almostisothermalconditions anddoubledheatflow cases. Maxima in springtemperature
whenrockmass
permeability
equals
about
10-]6m2.
along the fracture zone and a spring temperatureelevated arefound
slightly
above
theambient
surface
temperature
of 10øC.
Decreasingthe permeabilityof the surrounding
rock mass
to 10-]6 m2, while retainingthe fracturetransmissivity
of
10a-k,,
(m2-m),
reduces
fluidfluxinboththefracture
zoneand
Recallthatthisvalueof k, alsoproduces
maximumadvective
heating in the unfracturedconvex slope profile (compare
temperature
residualsshownin Figures5a and5b), reflecting
the way that the overall characterof the thermal regime is
FORSTER
ASl) SMITH: Motrs'rAn• GROUNDWATER
AND THERMALRY_.GIME$
9449
46
o
42-
0
2
o
38
n'
34-
Hb=120mW/m
•.•..•//
okf-b
=104-ku
105-ku
(m2-m)
/e
/ •
ß
kf-b=
(m2-m)
/
--
<•
30-
•
26-
•
22-
\
• kf-b= 103-ku
(m2-m)
/
/
/
Hb 60mW/m
2•/
/
ßkf-b= 104-ku
(m2-m)/
z
13_
/e•
\
\
%/'
,
/
', ,
\
\\
ß
18-
14-
•._
.......
10
-19
10
..............,&---
\•,
i
i
i
-18
-17
-16
10
10
10
i
10
-15
-14
10
ku (m2)
Fig.9. Spring
temperature
asa function
of upperzonepemaeability/q,,
transmissivity
of thefracture
zonekf. b, andbasalheat
flow H•,.
controlledby regionalgroundwaterflow throughthe rock
mass. On a more local scale, the fracture zone providesa
conduitfor more rapid fluid flow that servesto concentrate
the regionof warntingshownin Figure 5a within a smaller
area and to producea warmerregimenear the valley floor
zonekf. b andthepermeability
of theupperunitk,,dictates
the degree of thermal disturbance.In the systemshown in
Figure 8b, the fracture zone exerts its maximum influence
when
kf.bequals
about
104.k.
(m2-m).
Atvalues
ofkf.bin
excess
of 104.k.(m2.m),fluidflux to thefracture
zoneis
rockmassand
(Figure
8b). Increasing
k. above10'26m2 causes
reduceddictatedby thepermeabilityof the surrounding
springtemperatures
becausethe larger volmetric flux of
groundwater
absorbsall thermalenergyappliedat the baseof
the systemwith little increasein temperature.Decreasingk.
not by the transmissivityof the fracture zone. Thus,
fluid flow to the fractureis reducedand a greaterportionof
the basalheat flow is transferredby conductionthroughthe
ofkf.bless
than
104.k•
(m2.m),
fluidfluxtothefracture
zone
increasing
kf.bto10S.k,,
(m2.m)
has
little
additional
effect
on
theregional
flowsystem
andcauses
onlya IøC increase
in
below10'26m2 causes
reduced
spring
temperatures
becausespringtemperature(opencircleplottedin Figure9). At values
is dictated
by thetransmissivity
of thefracture
zone(kf. b).
As a consequence,
changingthe transmissivity
of the fracture
zone
influences
the
rates
and
patterns
of
groundwater
flow
The obviousdependence
of springtemperature
on thebulk
permeabilityof therockmasssuggests
thatmisleadingresults throughoutthe domainandhasan importantimpacton spring
may be obtainedwhenusingspringtemperatures
to estimate temperature.
Forexample,
reducing
kf.b to103.k,,
(m2.m)
adecrease
inspring
temperature
of7øC(Figure
9).
the depth of groundwatercirculationin a domain with a causes
Resultsshownin Figure 9 are representative
only of the
specifiedgeothermalgradient.Changingthe effectivelength
Spring
of the fracture zone (for example, by intersectingother idealized geometries and conditions tested.
are controlledby a numberof factorsthat are
permeablefracture zones) modifies its influence on the temperatures
thermalregime. For example,addinga secondfracturezone difficult to map and quantify, includingfracture position,
length,and intersection
with other
that extendsfrom the uplandrechargearea to intersectthe• orientation,transmissivity,
first fracturezoneat a depthof 2 km causesa greaterportion fractures. In addition, the three-dimensional nature of
of infiltrating groundwaterand a greaterpercentageof the intersectingfracture zones and rugged mountainousterrain
basal heat flow to be transmittedthrough the first fracture makes detailed extrapolation from two-dimensional
zone.The resultingincreasein fluid flux and advectiveheat simulationsdifficult. Results shown in Figure 9 suggest,
transfer through the fracture zone causes warmer spring however, that there is a permeability"window" wherein
spring
temperatures
aremostprobable
(k,,= 1047m2
temperature
anda coolerrechargearea.Conversely,
reducing elevated
be notedthatlocalized
shallow
heat
the lengthof the fracturezone to 1 km causesonly a slight to 104s m2). It should
reductionin springtemperature
becauseeliminatinginflow to sources,not consideredin this study, could promote the
deeperregionsof the fracturezone has little e•lect
re, on hhe development
of high-temperature
springsevenwhen k. lies
outsidethisrangeof values.
nonuniform
patternof fluid flux in thefracturezone.
Insightgainedfrom our modelingresultsenablesus to use
The contrastbetween the transmissivityof the fracture
surroundingrock mass.
9450
FORSTER
ANDSMITH:Motm'rAn,•GROUNDWATER
ANDTHERMAL
REGIMES
hotspringdatato inferregional-scale
permeability
withinthe
"window"for the mountainmassifwherethe temperatureof a
thermalspringreachesa peak value. Simplecalculations
that high-temperature
phenomenasuchas boiling springs, relating the geothermal gradient, depth of maximum
fumaroles,
andmudpotsareabsentin theCoastMountainsof circulation,and dischargetemperaturedo not reflect the
BritishColumbia,despiteelevatedheat flow [Lewiset al., properphysicsof theprocess.
1985]. Sixteenthermalsprings,with temperatures
seldomin
Coast Mountains of British Columbia. Souther [1975] notes
excess
of65øC,arescattered
throughout
aregion
of 1.3x 104
km2 [Souther
andHalstead,1973;Souther,1975]. Because
thesespringsinvariablyissuefrom fracturesin crystalline
rock, the associatedthermal regimes are presumedto be
stronglycontrolledby permeablefracturezones. The sparse
distributionof thermal springs suggeststhat conditions
favoringelevatedspringtemperatures
arefoundonlyin a few
localizedregions. In the absenceof local heat sources,the
Acknowledgments.
This work was fundedby grantsfrom the Natural
Scienceand EngineeringResearchCouncil of Canada(NSERC) and a
GraduateResearchin Engineeringand Technology (GREAT) award
providedby the BritishColumbiaScienceCouncil(in cooperation
with
Nevin Sadlief-Brown Goodbrand Company Ltd.).
Numerous
discussionswith David Chapman are gratefully acknowledged.
Computations
were carriedout on an FPS 164/MAX array processor
supportedby an NSERC major installationgram to the Universityof
British Columbia.
location of the observed thermal springs might indicate
localized
regions
wherebulkpermeability
approaches
10-16
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likelyfallsoutside
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(ReceivedAugust14, 1987;
revisedJanuary31, 1989;
acceptedNovember29, 1988)