WATER RESOURCES RESEARCH, VOL. 24, NO. 7, PAGES 999-1010, JULY I988
Groundwater Flow Systemsin Mountainous Terrain 1. Numerical Modeling Technique
CRAIG FORSTER
Departmentof Geology,Utah State University,Logan
LESLIE SMITH
Departmentof GeologicalSciences, University of British Columbia,Vancouver, British Columbia
A coupledmodel of fluid flow and heattransfer is developedto characterizesteady groundwater flow withina mountainmassif. A coupledmodel is necessarybecause high-relief terrain can enhance ground- water flow to depthswhere elevated temperatures are encountered.A widerange in watertable form and elevationexpected in high-reliefterrain is accommodatedusing a free-surfacemethod. This approach allowsus to examinethe influenceof thermalconditions on the patternsand ratesof groundwaterflow and the position of the water table. Vertical fluid flow is assumedto occur within the unsaturatedzone to providea simplebasis for modelingadvective heat transfer above the watertable. This approach ensures that temperaturesat the water table, and throughoutthe domain, are consistentwith temperature conditionssl•ecified at the bedrocksurface. Conventional free-surface methods provide poor estimatesof the water table configurationin high-relief terrain. A modified free-surfaceapproach is introducedto accommodaterecharge at upper elevationson the seepageface, in addition to rechargeat the free surface.
INTRODUCTION draulic conductivity that might be found within Meager Mountainousterrain occupies20% of the Earth's land sur- Mountain, British Columbia. Ingebritsen and Sorey used a face[Barry, 1981] yet little is known of the details of ground- coupled model to simulate the transient development of a waterflow at depth within a mountain massif. Upper regions parasitic steam field in a liquid-dominated geothermal system of flow have been explored, to a limited extent, in field studies at Mount Lassen, California. Although topographic relief was that emphasizethe interface between surface hydrology and assumedto drive the flow system,recharge was representedas shallowgroundwater flow [Halstead, 1969; Sklash and Farvol- a basal source of heated groundwater. den,1979; Bortolami et al., 1979; Martinec et al., 1982; Smart, The nature of deep groundwater flow is of interest in studies 1985].Although these studies provide insight into the hydrol- of geothermal systems in mountainous terrain. Groundwater ogyof alpine watershedsand the relationships between water samples obtained from springs and boreholes during geother- tablefluctuations and seasonal snowmelt, they yield little in- mal exploration often provide geochemical indications of a formationon deep flow systems. resource at depth. Identifying the source of a chemical signa- The characterof permeablezones within a mountain massif ture, however, requires an understandingof the rates and pat- are describedin reports describing inflows to alpine tunnels terns of groundwater flow. Efforts to identify a geothermal [$chardt,1905; Fox, 1907; Hennings, 1910; Keays, 1928; resourcealso rely on temperature data collectedin shallow Meats,1932]; however,measurements of fluid pressurethat boreholes. Advective disturbance of conductive thermal re- couldaid in definingthe nature of mountain flow systemsare gimes by groundwater flow can complicate the interpretation generallylacking. Jamier [1975] assessed the hydraulic of borehole temperature logs and may mask the thermal sig- characteristicsof fractured crystalline rock deepwithin Mount nature of an underlying resource. Geochemical and thermal Blanc(France) on the basisof geochemicaland hydraulicdata data have been used by several workers [Lahsen and Trujillo, obtainedduring construction of a highwaytunnel, but an inte- 1975; Blackwell and Steele, 1983; Adams et al., 1985; Sorey, grateddescription of the flow system within the mountain 1985] to form generalizationson the nature of mountain hy- massifwas not attempted.Water table and hydraulic head drothermal systems.A comprehensive,quantitative analysisof dataare rarely availableat mountain summitsbecause most groundwaterflow systemsin mountainousterrain has yet to wellsand boreholes are located on the lower flanks of moun- be reported in the literature. tain slopes.Two summit water level measurementsare noted The objectiveof this study is to investigatethe characterof in theliterature- at a depthof 30 m in fracturedcrystalline groundwater flow and thermal regimesin mountainous ter- rockat Mt. Kobau,British Columbia [Halstead, 1969] and at rain. This study is presentedin two parts. Paper 1 describes a depthof 488m in the basaltsof Mt. Kilauea,Hawaii [Za- the conceptualmodel, mathematical formulation, and numeri- blockiet al.,1974]. cal method developedto simulate the fluid flow and thermal Numericalstudies of mountainscale flow systems have been regimes.In paper 2 [Forster and Smith,this issue]this model is used to examine factors controlling patterns and mag- presentedby Jamiesonand Freeze [1983] and lngebritsenand Sorey[1985]. Jamieson and Freezeused a free-surfacemodel nitudesof groundwaterflow in mountainousterrain. anda waterbudget approach to estimatethe rangeof hy- CONCEPTUAL MODEL FOR GROUNDWATER FLOW IN MOUNTAINS Copyright1988 by the American Geophysical Union. Numerical modeling provides a quantitative basis for exam- Papernumber 7W5017. ining the influenceof topography,climate, thermal regime, 0043-1397/88/007W_5017505.00 and permeability on the rates and patterns of groundwater 999 1000 FORSTERAND SMITH: GROUNDWATER ];'LOW SYSTEMS,
(a)
o
-1
-2 0 4 8 12 16 2o
z (b) • 2 < 1
0 4 8 12 16 20
(c)
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o 4 8 16 20 DISTANCE (km)
Water table • Hypothetical groundwater pathline
Fig. 1. Hypothetical groundwater flow systemsfor homogeneouspermeability. (a) Coast Mountains of British Colum- bia. (b) Rocky Mountains of British Columbia/Alberta.(c) Conventionallow-relief terrain after Freeze and Witherspoon [1967].
flow. In this study, idealized mountain flow systemsare mod- encountered. Spatial variation in temperature has a strong eled for a range of conditions representative of the Western effect on fluid density and viscosity that, in turn, have an Cordillera in North America. Mountainous terrain is defined important influence on the rates and patterns of groundwater as rugged topography with local relief in excess of 600 m flow. Thermally induced differencesin fluid density producea [Thompson, 1964]. In the Coast Mountains of British Colum- buoyancy-driven component of fluid flow that enhancesverti- bia and the central Cascades of the Pacific Northwest, topo- cal movement of groundwater. In addition, reduced fluid vis- graphic relief of 2 km over a horizontal distance of 6 km is cosity in regions of elevated temperature contributes to in- typical. In the Rocky Mountains of Canada and the United creasedrates of groundwater flow. States, a more subdued relief of 1 km over 6 km is not uncom- In addition to the above differences,it is important to note mon. Vertical sectionsand schematicflow lines representative that rocks found in mountainous terrain have significantl)' of the Coast Mountains of British Columbia and the Rocky lower permeability than materials commonly encounteredin Mountains at the Alberta-British Columbia border are shown aquifer simulation studies.This reduced permeability means in Figures la and lb. For comparison,Figure lc shows flow that considerably longer time scales are encountered when systemsin a low-relief topography similar to those described examining processesoperating in mountain flow systems. by Freeze and Witherspoon [1967]. An idealized flow systemis shown, without vertical exagger- Mountain flow systemsdiffer from low-relief systemsin two ation, in Figure 2. Vertical no-flow boundaries are definedto important respects. reflecttopographic symmetry at valley floor and ridgetop.The 1. For a given set of conditions, with greater topographic domainshown in Figure 2 representsthe regionlying beneath relief, a greater range in water table elevation and form is a singleridge-valley segment of the topographicprofile shown possible.In low-relief terrain, water table configurationscan in Figure la. The basal boundary is presumedto be imperme- be defined with reasonable accuracy using water level eleva- able.The upperboundary of the groundwaterflow systemis tions and hydraulic head data obtained from boreholesand the bedrock surface.Erosional processesoperating in moun- wells located acrossthe region of interest. In many instances, tainousterrain often promote development of a thin coverof estimated water table elevations are used in defining the upper discontinuoussurficial deposits, often lessthan 10 m thick. boundary of regional flow systems.In mountainousterrain over upland areasof mountainslopes. In our concept• measured water table elevations and hydraulic head data are model,these deposits are thoughtof as a thin skinof variable sparseand, whereavailable, usually concentrated on the lower thicknessthat is not explicitly included in the model. Subsur. flanks of mountain slopes.This restricteddistribution of data faceflow within this skin,in additionto overlandflow and leads to considerableuncertainty in defining water table con- evapotranspiration,is lumpedin a singlerunoff term. These figurationsbeneath mountain summits. processesare strongly affected by spatialvariations in precipi' 2. High-reliefterrain enhancesgroundwater circulation to tation, slope angle, and soil permeability as well as by tempo' depthswhere elevated temperatures (in excessof 50øC)may be ral variationsin precipitationevents. Field observationsofthe Foast•a ArqnSM•T•' GROUNDWATERFrow S¾s•:œMs,1 10131
Mean Annual Precipitation complexnature and interaction ofthese factors are lacking for individualmountain slopes. Therefore as a firstapproxi- <• •z Available-•'a•Infiltration ...... mation,a lumped steady state approach is adopted and an availableinfiltration rate is defined. Available infiltration rep- PO resentsthe maximum rate of recharge possible at thebedrock C3 - surfacefor specified climatic, geologic, and topographic con- ,, i ditions.In the absence ofdetailed site data, the available infil- Recharge • øI - I trationrate is bestthought of as a percentageof the mean annualprecipitation rate. Rechargeto the flow system reflects the magnitude of the availableinfiltration rate, the capacity for fluidflow through ) thesystem and the nature of thethermal regime. In high- permeabilityterrain, groundwater flowsystems may accept all theavailable infiltration and producea watertable that lies v• 1.• belowthe bedrock surface (Figure 2). In lower-permeability o b terrain,where recharge accepted by the flow systemis less z• kf ...... thanavailable infiltration, the water table will be foundclose to the bedrocksurface. Conventionalapproaches to modelingregional ground- waterflow recognize that a transitionfrom groundwaterre- chargeto groundwaterdischarge occurs at a specifiedpoint onthe upper boundary. In thisstudy, this point is termedthe hingepoint (HP) (shown in Figure2). Because the unsaturated 0 2 4 6 zoneis includedin this conceptualmodel, it is also necessary x = xO DISTANCE(km) x = xL toidentify the point where the water table meets the bedrock surface.This point is definedas the point of detachment (POD)(shown in Figure2). In developingnumerical models to Basal Heat Flow analyzeseepage through earthen dams, an exitpoint is com- • _v.• Free-surface monlydefined that hasthe propertiesof boththe POD and • Insulatedan••mpermeable boundary theHP. In high-reliefterrain, the usualexit point cannotbe • Impermeable boundary identifiedbecause the POD and HP do not coincide.In the :"•:• Basallow-permeability unit regionbetween the point of detachmentand the hingepoint, the water table coincideswith the bedrock surface and the Fig. 2. Conceptualmodel for groundwaterflow in mountainouster- saturatedzone is rechargeddirectly from overlyingsurficial rain. deposits.This surprising result implies that rechargecan occur on whatis usuallyconsidered to be the seepageface. This behavioris discussedin more detail in a subsequent section temperature.Given the assumed presence of only a thinskin andin the appendix. of surficialdeposits on mountainslopes, temperatures at the Upslopeof the pointof detachment,the watertable lies bedrocksurface are presumedto matchtemperatures at the belowthe bedrock surfaceand infiltration is transferred to the groundsurface. watertable by unsaturatedflow. Significant lateral flow in the Heat transferoccurs by conductionand advectionboth unsaturatedzone could cause patterns of rechargeat the water aboveand below the water table; thereforethe influenceof tableto differfrom patterns of infiltrationat the bedrocksur- fluid flow on the thermal regimein the unsaturatedzone must face.The nature and magnitude of lateralflow in unsaturated be considered.In the model presentedhere, we considerthe regionsof mountainflow systems,however, is poorlyunder- thermaleffects of moisturemovement in the liquidphase and stood.As a first approximation,a one-dimensionalmodel of heat conductionthrough the solid-vapor-fluidcomposite. verticalflow from the bedrock surface to the water table is Thermal effectsof condensation,evaporation, and heat trans- adopted. fer by vapormovement within the unsaturated zone are ne- Boundariesof the thermalregime coincide with thoseof the glected.In developing a conceptual model of deep unsaturated fluidflow system. A basalconductive heat flow suppliesther- zones,Ross [1984] suggests that moisture transport by vapor malenergy to the mountainflow system.The basalheat flow, movementbecomes negligible when recharge rates exceed whichrepresents heat transfer from deeperlevels in the Earth's about10-•2 m/s.Unsaturated zones with recharge rates less crust,is a characteristicof the tectonic environment in which thanthis amount are likely found only in themost mountain- themountain is located.Chapman and Rybach[1985] report ousterrain of theWestern Cordillera. Thus in developingthis that representativeheat flow values range from 30 to 150 conceptualmodel, the influence of vapormovement in the mW/m2 with a medianof 61 mW/m2. Followingthe example unsaturatedzone is neglectedand a minimuminfiltration rate ofBirch [1950], it is assumedthat temperaturesat the ground of 10-t2 m/s is assumed. surfacereflect an altitudinal gradient in surfacetemperature While fractureslikely providethe primarypathways for {thermallapse rate) with mean annualtemperatures at the groundwaterflow through a mountain massif, an equivalent groundsurface a few degreeswarmer than meanannual air porousmedia approach is adopted. This approach isreason- temperature.Exceptions occur where fracture zones outcrop able whenfracture densities are sufficientlyhigh, and the to producegroundwater springs. In suchcases, temperature at lengthand spacing of individualfractures are much smaller the groundsurface is assumedto reflect the temperatureof than the scaleof the mountainmassif. In keepingwith this groundwaterflowing in the fracturezone, rather than the air lumpedapproach, groundwater discharge is represented as 1002 FORSTSR^ND SMITH: GrtOUNDWATSRFLOW S¾STSMS,1
seepagedistributed acrossthe dischargearea (downslopeof The crossterms for the permeability tensor (k,=, k=x) are in. the HP in Figure 2), rather than at isolated groundwater cludedto preservethe generalityof the subsequentnumerical springs. Local variations in surfacetopography, rock per- formulation. The analagousequation in one-dimensionallocal meability, and thicknessof surficial depositscontrolling the coordinates for a fracture zone is distribution of springsare assumedto have little effect on the overall pattern and magnitude of groundwater flow. Excep- tions to this assumptionare causedby major throughgoing 7; :0 fracturezones represented as discretepermeable fracture zones witha homogeneouspermeability ks andwidth b (Figure2). Boundary conditions for the fluid flow system shownin Figure 2 are as follows: MATHEMATICAL MODEL The mathematicalmodel for groundwaterflow in moun- Oh for --4km c9-•Epsq•] + •zzEp fq•] = 0 (1) wherex o equals0.0 km, L is the horizontal length of thefl0w system(6.0 km), and z,•• is the water table elevation at the specifiedx coordinate. wherePt = pœ(T,p) is fluid densityand q,•and q=are horizon- In Figure 2, the free surface is that portion of the water tal and vertical components of fluid flux (specific discharge), table that lies below the bedrock surface. Equation (9c) reflects respectively. Fluid flow is driven by gradients in fluid pressure the fact that freshwater head must equal the water table eleva. and thermally induced density contrasts. Fluid potential h is tion everywhere at the upper boundary of the saturated region defined in terms of an equivalent freshwater head: of flow. On the free-surface segment an additional constraint (2) is required to limit the infiltration available for rechargeto the bedrockflow system.Fluid flux q•,, directedalong the unit where P0 is a referencefluid density at a specifiedtemperature, normal n• to the free surface,is driven by potential gradients g is acceleration due to gravity, p is fluid pressure,and z is the and thermal buoyancy where elevation where the freshwater head is calculated. Frind [1982] advocates the use of an equivalent freshwater head to (10) describe potential gradients exclusiveof static fluid pressures. q•= - ku• +P• -- n• = (I_.cos O)n• Fluid flux through the saturated porous matrix is given by The available infiltration rate I= is applied directly on thefree surface to represent one-dimensional flow from the bedrock q'= --V PogO•j + (p•' --Po)g (3) surface to a water table that slopes at an angle 0 from hori- wherep = p(T, p) is the dynamicviscosity of the fluid and kU zontal. This boundary condition differs from that usedbs, is the permeability tensor for the porous matrix. This equation Neuman and Witherspoon[1970] in two respects.First, fluid is simplifiedby defininga relative density flux and freshwaterhead conditions are specifiedon the free- surfacesegment while only freshwaterhead is specifiedon the seepage face (Figure 2). This approach allows rechargeto to obtain occur on the seepageface between HP and POD. Belowthe HP, only dischargeoccurs. Second, a buoyancyterm statedin terms of the relative buoyancy p• is included inside the square q•=--k u• +p, (5) bracketsof (I 0) becausethermal effects are incorporatedin the In thin fracture zones and discrete fractures, fluid flux in formulation. The appendix contains a discussionof the con- directions parallelto the fracture-matrixboundary is givenby ditions that promote separationof the POD and HP and outlinesuse of a hodographto demonstratethe theoretical basisfor such a separation. qs:--kz •poq [c3h•ss+ p" •ss•z] (6) The steadystate balance of thermalenergy in a variab!y saturated porous medium with no internal heat sourcesor wherekz is thepermeability of materialwithin a discretefrac- sinks is describedby ture or fracture zone. For open fractures, a parallel plate model is usedto definethe fracturepermeability in terms of an effectiveaperture b wherekz = b2/12. The equationdescribing the distributionof freshwaterhead c3[/1.xx *cST c3T]c•[ 8T in the porousmatrix is obtainedby substituting(5) into (1): --Piti q,•xxOT+ q='•z'z =0 (11) where =o (7) T temperature; FORSTERAND SMITH: GROUNDWATER FLOW SYSTEMS, 1 1003 Cs specificheat capacity offluid; expandedfor an isotropicporous medium as follows: 2•f thermalconductivity tensor for solid-fluid-vapor composite. nDxx= psCs(atqx2/•q- atq:2/tj)q- (13a) The last term on the left side of (11) describesthe advective nD....-- pfCf(atqx2/•-b atq:2/cj)+ n2œ (13b) transferof heat by fluid flow both above and below the water nD,½..= nD:,, = pœCœ(at - at)qxq../c7 (13c) table.In (11),thermal equilibrium is assumedto exist between fluid,solid, and vapor. Here,a t and at are longitudinaland transversedispersivities, Modelingheat transfer in the unsaturatedzone ensuresa 3f is thethermal conductivity of the fluid, and • is themag- water table temperature and thermal regime consistent with nitude of the fluid flux. temperatureconditions specified at the bedrocksurface. By The analagousenergy balance equation for a fullysaturated adoptinga free-surfacemethod, the rate of verticalfluid flow fracturezone is written using one-dimensionallocal coordi- in the unsaturated zone is defined to be the available infiltra- nates as tion rate (applied directly on the free surface).In solving the equationsof energytransport, one-dimensional advective heat transferabove the water table is modeled by setting the hori- •ssn'cDs +(I -- n f)2 s '•-J's-c•T PsC'rq• •c?T = 0 (14) zontalfluid flux q•,to zero and the verticalfluid flux q..to the availableinfiltration rate I.. in (11). where Heat conductionand thermal dispersionby fluid flow in the nsDs--' pj.Cj.(atqs + nf,tf) (15) solid-liquid-vaporcomposite is representedby the first two termsof (11) usinga thermal conductivitytensor definedby In (14)and (15),nœ is the porosityof the fracture,where 0 < nœ< 1 forfractures filled with porous material and n s equals1 ,•if -- SnDi•+ (1 -- S)n2•' + (1 -- n)J,us (12) for openfractures. Where n s is lessthan 1, an isotropicther- mal conductivity,U is assumedfor the fracturefilling. where Boundaryconditions for the heat transferproblem are n porosityof the porous matrix; •T for --4km to adopt this approachwhen studying groundwater flow in mountainousterrain can producesolutions with unrealistic watertable configurations. Although this approach toviewing thenature of seepagefaces has been revealed through study of mountainousterrain, it is also applicableto flow systemsin low-relief terrain. Solution Procedure -2 The coupledequations of fluid flow and heat transferare solvedusing an iterativeprocedure. The freshwaterhead and temperaturedistributions are calculatedusing the following steps -4 1. Individual finite elementmeshes are constructedfor the 0 2 4 6 fluid flow and thermal problems. bo 2. Mesh accuracy is tested and meshesare reconstructed as necessary. 3. A conductivetemperature field and a hydrostaticpres. sure distribution are computed to obtain initial valuesfor the temperature and pressure-dependentproperties of water. 4. Iterations are performed to solve for freshwaterhead. estimate a new water table configuration, deform the mesh used in solving for freshwater head, reconstruct the meshfor the thermal problem, and solve for an updated temperature ILl -2 fie!& 5. Iterations terminate when successive water table elena. tions differ by no more than a specifiedtolerance. The number of iterations required typically ranges from 7 to 15. -4 6. The mass flux of fluid and heat across boundaries of the 0 2 4 6 domain is calculatedto evaluatemass and energybalances. DISTANCE (km) l•'lesh Construction Fig. 3. Typicalfinite element meshes: (a) fluid flowproblem (1137 nodes,2139 elements)and (b) heat transferproblem (1230 nodes,2201 Finite element meshes that can deform in a vertical direc- elements). tion are constructedfor the fluid flow and thermal problems usingan internalmesh generator. The fluid flow meshis gen- erated only within the saturated region of flow. The upper Free-Smfiwe Models boundary of the initial mesh is defined using a reasonable guessof the water table configuration.In subsequentiter- Free-surface methods have been developed to solve un- ations, the current estimateof the position of the water tableis confinedgroundwater flow problemsin which the positionof used. An example of a mesh used in the solution for freshwater the water table is initially unknown. Most free-surface meth- head,obtained at the final water table configuration,is shown ods involve successive estimation and correction of water in Figure 3a. Throughoutthe solutionprocedure, the structure table positions in an iterative procedure (see, for example, of the mesh is unchangedbelow the elevation of the valle.• Neuman and Witherspoon [-1970]. The majority of free-surface floor. Although the upper region of the mesh deformsto con- codes have been applied to relatively small-scaleproblems of form to successiveestimates of the water table configuration. isothermal flow through porous media in systemswith low the number of nodes(1137) and elements(2139) in thismesl: topographic relief. Although Jamiesonand Freeze [1983] ap- remainsunchanged during the iterative procedure.Checks are plied the steady FREESURF model of Neuman and Wither- performed before and after each simulation to ensure that spoon to the rugged topography of Meager Mountain, their acceptable aspect ratios are maintained for elements located solutions suggestthat the seepageface is poorly represented. within the deformingregion of the mesh. Nonisothermal free-surface models are described by Horne Coupling between heat transfer and fluid flow is facilitatea and O'Sullivan [1978] and Bodvarssonand Pruess [1983]. by ensuringa one-to-onecorrespondence between nodes 10- These authors applied their models to regionsof flat topogra- cated below the water table in each mesh. Heat transfer ab0•e phy near producing geothermal fields. the water table is modeledby extendingthe meshfor the An important difference between the numerical method thermalproblem to the bedrocksurface (Figure 3b). Because used in this study and other methods lies in the approach used the meshfor the fluid flow problemdoes not extendabove the to solve the free-surfaceproblem in terrain with high topo- water table, fluid flow in the unsaturatedzone is implicitl.• graphic reliefi Conventional techniques assume that ground- definedby the availableinfiltration rate appliedat thefree water recharge occurs only above the POD that is implicitly surface.Although the boundariesof the thermal problem assumedto coincide with the HP. In developing our modeling remain fixed, the thermal mesh must be deformed internall.• approach, it was found that the POD need not coincide with {or reconstructed)to match changesin the fluid mesh. the HP in steep terrain with rocks of relatively low per- A simpledeformation of the meshfor the thermalproblem meability (appendix). Therefore recharge is allowed to occur is unableto accountfor lateral pinchingout of the unsatu- downslope of the POD on the seepageface (Figure 2). Failure ratedzone as the lengthof the seepageface increases and the FORSTERAND SMITH:GROUNDWATER FLOW SYSTEMS, 1 1005 PODmoves upslope on the bedrocksurface during the itera- alongthe top surfaceto maintaina balancebetween recharge tiveprocedure. Therefore the thermalmesh must be recon- and discharge.As the positionof the POD changes,the length structedin its entiretyat eachiteration to accomodateupslope of the free-surfacechanges and the regionwhere the available or downslopemovement of the POD. Each time the meshis infiltration rate is applied varies. reconstructed,the numberof nodesand elementsmay change In this modified free-surfacemethod, it is assumedthat the dependingupon the degree of POD movement.A finalther- POD marks the uppermostpoint where freshwaterheads are malmesh, which corresponds to the fluid meshof Figure 3a, is known to equal the elevation of the bedrock surface,rather shownin Figure 3b and contains 1230 nodes and 2201 ele- than the uppermostpoint of groundwaterdischarge. Although ments.The numericalformulation doesnot includea method rechargerates are initially unknown for the region between for maintainingthe position of fracture zones and geologic the POD and the HP, they can be computedfrom the fresh- boundarieswithin deformingregions of the mesh.As a conse- water head solution.After eachiteration, recharge rates com- quence,fracture zones and heterogeneitiesmust be restricted puted usingfreshwater heads in the vicinity of the water table to nondeformingregions of the mesh. are compared to available infiltration rates. If rechargeex- The accuracyof each initial mesh is evaluated by solving ceedsavailable infiltration near the POD, additional iterations the isothermalfluid flow problem and computing a total bal- are performed to refine further the water table configuration. anceof fluid mass crossingall system boundaries and a fluid This approach differs from that of Neurnanand Witherspoon massbalance acrossthe top surface. Becausevertical grid lines [1970•] in two respects.First, fluid fluxes calculated at the in the deforming mesh are rarely orthogonal to the sloping seepageface are not explicitly used in solving (7). Rather, top boundary,computing fluid flux normal to the surface freshwater head is specifiedwhere the water table coincides usingfreshwater heads can be inaccurateif the mesh is too with the bedrock surfaceand fluid flux is specifiedonly on the coarse. free surface.This minimizesthe dependenceof the solution on Flux inaccuracies are minimized in a two-step process. calculated fluid fluxes and allows the systemof equations to First,the free-surfacemethod is usedin an isothermal mode to be solved only once for each iteration, rather than the two- solvefor freshwater head and to obtain the water table config- step process used by Neuman and Witherspoon. Second, be- uration.Fluid fluxes are calculated at each boundary using the cause only vertical mesh deformation is allowed, near-vertical freshwaterhead solution. Second, the problem is reformulated topographic slopes are poorly represented in this modified usingstream functions, with the previouslydefined water table formulation. configurationforming the upper boundary of flow. This In developing this approach, it is assumed that available streamfunction solution provides an alternative method for infiltration is controlled by processesacting at the bedrock calculatingfluid fluxes normal to the water table. surface (evapotranspiration, subsurface stormflow, surface Large differencesbetween boundary fluxes obtained using runoff) rather than by the position of the water table or by the two solution methods often indicate regions where the processes acting in the unsaturated zone. Therefore when water table configuration may be poorly estimated and the varying parameters such as permeability in a numerical sensi- finite element mesh must be refined. Acceptable grids are de- tivity analysis, the available infiltration rate is assumed to be fined when mass flux balances for flow across the water table unaffected by varying depth to water table. Where the flow differ by less than 1% and total mass balances differ by less system acceptsthe entire available infiltration, the water table than 5%. In addition to obtaining a good match in mass flux will be predicted to lie below the bedrock surface and recharge balance,it is also important to ensure that patterns of fluid to the saturated region of flow is exactly equal to the available flux on the upper boundary are the same for the two different infiltration rate. Where only a portion of the available infiltra- formulations.Therefore an acceptablegrid must also provide tion rate is accepted,the predicted water table will coincide hingepoint positionsthat match for each solution method. with the bedrock surfaceand recharge to the flow system will be less than the available infiltration rate. Iterative Procedure Solutions to the thermal transport equation ((equation 11)) A coupledsolution is obtained by solving (7) and (11) in an may be poorly approximated by conventional finite element iterativeprocedure controlled by the free-surfacemethod. At techniqueswhen advective heat transfer dominates conduc- theupper boundary of the fluid flow problema mixedbound- tion. Such conditions are defined on the basis of a grid Peclet ary conditionis defined.An available infiltration rate is ap- number plied where the water table lies below the bedrock surface and freshwaterhead is specifiedwhere the water table coincides Pe= CfP ft•Lc (17) withthe bedrock surface. Prior to initiatingthe iterations,(11) issolved for the conductivecase to obtainthe initial temper- where Lc is a characteristiclength for an individualelement. aturefield. These initial temperaturesare used to compute Patankar [1980] and Huyakorn and Pinder [1983] report that fluidproperties required in the firstfree-surface iteration. The reliable solutions to both one- and two-dimensional transport numericalsolution proceeds by updatingthe fluid properties problemsare possiblewhen Pe _<2. In this study,an accept- usingthe latest estimate of the temperaturefield, solving (7) able triangular elementmesh is constructedby ensuringthat forfreshwater head, obtaining a newestimate of the water the grid Petlet numbersare lessthan 2 for eachelement. tableconfiguration to calculate specific discharge, and updat- In representingfracture zones as line elementswithin the ingthe temperature field by solving (11). triangularmesh, the fracturepermeability must exceedthat of Thesolution begins by solving(7) for freshwaterhead and the adjacentrock massby at least four ordersof magnitude extrapolatinga new water table positionusing the method [Baca et al., 1984]. Such large permeabilitycontrasts imply describedbyNeuman and Witherspoon [1970]. The finite ele- that largefluid flux contrastsare possible.In suchcases, solu- mentmesh is then deformedto conformto the shapeof the tions with triangular elementPe lessthan 2 may also have line newupper boundary. At each iteration, the POD is moved elementPe much greater than 2. This problem is handled by 1006 FORSTER^Nr) SM•TI-•' GROUSr>w^TER FLOW SYSTEMS, 1 TABLE 1. Typical Simulation Parameters availableinfiltration rate. Heat and moisturetransport by vapormovement in the unsaturated zone are assumed negligi. Values ble. Fluid Flow Parameters 9. Fluid densityand viscosityvary as a functionof tern. kt,, permeability of basal unit 1.0 x 10-22 m2 peratureand pressure while thermal conductivity and specific k,,, permeability of upper unit 1.0 x 10-15 m2 heatcapacity of thefluid are assumed constant. I•, available infiltration rate 2.0 x 10-9 m/s IMPLEMENTATIONOF THE MODEL Thermal Parameters Ha, basal heat flow 60.0 mW/m2 The numericalmodel describedin the precedingsection is Gt, thermal lapse rate 5 øC/km usedin paper2 [Forsterand Smith, this issue] to examinethe Tr, reference surface temperature 10 øC factorscontrolling groundwater flow through a mountain n•,, porosity of basal unit 0.01 massif.To demonstratethe basicform of the results,an exam. n,,, porosity of upper unit 0.10 A•', solid thermal conductivity 2.50 W/møC pleis presentedhere that characterizes thethermal regime in if, fluid thermal conductivity 0.58 W/møC mountainousterrain. Simulations are performedby assigning X*', vapor thermal conductivity 0.024 W/møC a set of fluid flow and thermal parameters(Table 1) withina C•,;specific heat capacityof 4186.0 J/kgøC geometrysimilar to thatof Figure2. Theimportance ofsur. water S, saturation above water table 0.0 facetopography is examinedby consideringtwo extremesin a•, longitudinal thermal 100.0 m slopeprofile: one convex and one concave (Figure 4). Convex dispersivity profilesare typical of glaciatedcrystalline terrain while con. a t, transversethermal 10.0 m caveprofiles are often found in foldedmountain belts and at dispersivity volcanic cones. A basalzone of low permeabilityoccupies the lower2 kmof thesystem to providea regionof conduction-dominatedheat definingan exponentialbasis function whose form depends transfer.The remainderof the systemis occupiedby a higher. uponPe at the previousiteration. This function is incorpor- permeabilityunit where advective heat transfer may dominate. ated in the finite elementequations and usedfor calculating This configurationallows advective thermal disturbances in averagefluid properties within each line element. The method theupper unit to propagateinto the basal conductive regime, adoptedis similarto upstreamweighting methods described ensuring a reasonabletransition between advection-and for finitedifference grids by Spalding['1972], Raithby and Tor- conduction-dominatedthermal regimes. The basal 10w- rance[1974], and Patankar [1980'1. permeabilityunit is sufficientlythick that the majority of sim. The iterativeprocedure is terminatedwhen water table ele- ulation resultsshow isothermsnear the basal boundaryto be vationsare resolvedwithin a toleranceof 1 m and calculated subparallelto theboundary. As a consequence,theconductive rechargeis lessthan available infiltration near the POD. In heatflux applied at thebasal boundary is transferredby verti- flowsystems with topographic relief of 2 km,this tolerance of cal conductionaway from the boundary. 1 m is 0.05% of the maximumpossible water table relief. Once Bothupper and lowerzones have homogeneous and iso- the solution is terminated,a final checkis performedto ensure tropicpermeability (ku, k b) and uniform porosity (n,,, nb). Ther- that nodal temperaturesobtained at final and penultimate mal conductivityof the solid2 s is uniformthroughout the iterationsdiffer by no more than 0.1øC. system;however, varying porosity and saturationproduce contrastsin rock thermalconductivity 2 e.Because we consider SUMMARY OF ASSUMPTIONS thesteady state problem, porosity only has an indirectinflu- Major assumptionsinvoked in developingthe numericalence on the flow systemthrough its impacton rock thermal method are summarized below. conductivity.A uniform heat flow H• is appliedat thebase 0f 1. Two-dimensionalcoupled fluid flow and heat transfer in the flow systemand surfacetemperature conditions are de verticalsections with planaror axisymmetricsymmetry. fined in terms of a referencesurface temperature T, anda 2. Steadyfluid flow and heat transfer within a stablegeo- thermallapse rate Gv Longitudinaland transversethermal logic environment. dispersivities(at, a t) areuniform throughout the system and 3. Anequivalent porous medium is assumedexcept where held constant for all simulations. majorthroughgoing fracture zones are represented asdiscrete Temperaturefields, reflecting the interaction between advec- entitieswith permeabilityat least10 '• timesthat of the sur- tive andconductive heat transfer, are influencedby bothfluid roundingrock mass. Permeability and thermal conductivity flow and thermal parameters.Figure 4 showstemperature maybe heterogeneous and anisotropic. fieldsobtained using the values of thermalparameters speci- 4. Thermalequilibrium exists between fluid, solid, and fiedin Table i and valuesof I z requiredto maintainthe same watertable configurations within each system as ku varies vapor. 5. Verticalboundaries for fluid flow and heattransfer are from !0- • to 10- x• m2 (Table 2). By maintainingthe same symmetryboundaries (impermeable andinsulated). watertable position, the thermal conductivity structure isun- 6. Thebasal boundary is horizontal and impermeable with changedand temperature fields shown in Figure4 differonly a conductiveheat flux appliedalong the boundary. in theway that advective heat transfer, asinfluenced byfluid 7. Theupper boundary of thedomain is the bedrock sur- flowparameters alone, affects the thermalregime. facewhere temperatures are specified using a thermallapse Pathlines represent the track of a particleentering the ft0w rate. systemat a specifiedpoint on the bedrock surface, while the 8. One-dimensionalvertical flow of fluidthrough the un- spacingof thepath lines indicates relative changes in themag- saturatedzone causes one-dimensional advective heat transfer nitudeof fluidflux in the directionof flow.Maintaining the (combinedwith two-dimensional heatconduction) above the samewater table position produces patterns of fluidflow that watertable. Vertical fluid flux above the water table equals the arevirtually identical for a givenslope profile, despite a wide FORSTER^ND SMITH' GROUNDWATERFLOW SYS•MS, 1 1007 Run C1 Run C2 Run B2 ,>,, i ...... i Run C3 Run C4 Run B7 .. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0 2 4 6 0 2 4 6 0 2 4 6 DISTANCE (km) DISTANCE (km) DISTANCE (km) .... -•- Groundwater --5--.• Isotherm (øC) Water table pathline Fig.4. Temperaturefields and groundwater flow patterns as a functionof upperzone permeability k, for fixed water tableconfigurations in concave and convex slope profiles: (a) k, = 10-x8 m 2 (I== 2 x 10-x2m/s for convex run C1 and 5 x 10-•2 m/sfor concaverun C3); (b) k u = 10-t6 m2 (I•.= 2 x 10-•ø m/sfor convex run C2 and5 x 10-tø m/sfor concaverun C4); and (c) k,, = I0- • m2 (I= = 2 x 10-9 m/sfor convex run B2 and 5 x 10-9 m/sfor concave run B7). variationin the temperaturefields. Despite their obvioussimi- Figures4a to 4c show the transitionfrom conduction-to larry, path lines shouldnot be confusedwith the streamlines advection-dominatedthermal regimesas a function of increas- generatedfrom contour plots of a suitably defined stream ing permeability.Upper zone permeabilities less than 10- • functionor velocitypotential. These approachesdiffer because m2 producea purely conductivethermal regime with iso- fluidflow through each streamtube(bounded by each pair of thermssubparallel to the surfacetopography (runs C! and C3 path lines) is fixed throughout the domain while fluid flow in Figure4a). An increaseof two ordersof magnitudein k,,to throughflow tubes defined on the basis of path lines may 10- •6 m• producesa weakadvective disturbance as evidenced differ from flow tube to flow tube. by thewarping of isotherms(runs C2 andC4 in Figure4b). An Water table configurationsare maintained by increasingI• additionalone order of magnitudeincrease in k,, to 10- • m2, by the sameincrement that k, is increased.Maintaining the createsa stronglydisturbed thermal regime (runs B2 and B7 watertable position as permeabilityis increasedimplies that in Figure4c). Smith and ChapmanE1983] found that ground- increasinglyhumid climatessupport the increasedinfiltration water flux is sufficientlylarge to cause a transition from rates.The relationshipbetween infiltration rate, permeability, conduction-dominated heat transfer to a weakly disturbed and water table elevation is examined in detail in paper 2 thermalregime when permeability is about10- • v m•' in low- [Forsterand Smith, this issue]. reliefsedimentary basins (1 km over 40 km). In regionsof high-reliefmountainous topography, fluid flux is sufficiently enhancedby thegreater hydraulic gradients to causethis tran- TABLE 2. Influenceof Permeabilityk,, and InfiltrationI, on Maximum Temperatures in Flow System Tma,• sitionto occurat a lowerpermeability of about10- • m2. The maximumtemperature attained along the baseof each Run Profile k,,,m 2 I z, m/s Tm..x,øC flowsystem Tm• • is tabulated in Table2 foreach run shown in CI X 10-•8 2 X 10-12 138 Figure4. Ask. increases,Tmax decreases from about 130øC to C2 X 10-]6 2 x !0 -•ø 130 about85øC. Higher permeabilities enhance groundwater flow, B2 X 10-]5 2 X 10-9 81 cool the subsurface,and reduceconductive thermal gradients C3 V 10-!8 5 X 10-12 122 in regionsof strongdownward fluid flow. Reduced thermal C4 V 10-]6 5 x 10-•ø 117 gradientsreflect the ability of groundwaterto absorb and re- B7 V 10-•5 5 X 10-9 87 distributeheat that would otherwisecause increased rock tem- X, convex:V, concave.Thermal conductivity of thesolid A'" = 2.5 peratures.The temperature fields shown in Figure4 illustrate W /rnoC. Basalheat flow H•, 60.0mW/m 2. the rangeof thermalconditions that are encounteredin the 1008 FOmSTœRAND SMITH: Gt sensitivityanalyses carried out in paper2 [Forsterand Smith, trolledas an inputvariable rather than implicitly calculated in this issue]. the solution procedure. SUMMARY APPENDIX:DISTINCTION BETWEEN POINT OF DETACHMI•NT 1. Groundwater flow systems in mountainous terrain AND HINGE POINT ON A SEEPAGE FACE differfrom thosein low-reliefterrain in two key respects:(1) for a givenset of hydrogeologicconditions, a greater range in Conventional approachesto solving free-surfaceground. water table elevationand form is possible;(2) high-reliefter- waterflow problems assume that a singlepoint, the exit point, rainenhances groundwater circulation to depthswhere signifi- marksthe boundary between the free-surface and the seepage cant heatingcan occur,implying that thermaleffects influence face. Two conditionsoccur at this point: the free-surfacede. the patternsand rates of groundwaterflow. viatesfrom the seepage face, and the upperlimit of discharge 2. A conceptual model has been outlined to describe on the seepageface is defined.These conditionsoccur at a groundwater flow and thermal regimesin mountainouster- singlepoint for free-surfaceproblems similar to the triangular rain. The uppermostboundary of the domain is the bedrock earthendam shown in Figure5a. In thisisothermal example, surface. Where the water table coincides with the bedrock a homogeneousisotropic hydraulic conductivity K is assumed surface,freshwater head equalsthe bedrockelevation. Where for all panels shown in Figure 5. The horizontal baseCD is the water table lies below the bedrock surface,a uniform impermeable while segmentsBC and DE are constant head availableinfiltration rate is appliedthat is transmitteddirectly boundarieswith head differential•Sh = h•- h2. The formof to the water table by one-dimensionalflow throughthe un- free-surfaceAB, the length of the seepageface AE and the saturated zone. Heat supplied by a regional heat flux is trans- position of the exit point A are controlled by this head differ. ferred through the system by advection and conductionin ential,the geometryof the dam,the hydraulicconductivity K, both saturated and unsaturatedregions of flow. Fluid flow and the pattern of infiltration applied on the upper surface0f and heat transfer through a thin cover of discontinuoussur- the dam. ficial depositson upland areasof mountainslopes is not in- In a simplifiedmountain groundwater flow problem,the cluded in the model. Fluid moving within thesedeposits is verticalleft-hand boundary of Figure 5a becomesa symmetry lumpedwith overlandflow as a runoff term. The remaining boundary(Figure 5b). At a highuniform infiltration rate fluid available for recharge is termed an available infiltration the water table coincides with the ground surface acrossthe rate that in the absenceof detailed field data, is best thought flow system. In this case, a free surface is absent and the entire of as a percentageof the mean annualprecipitation rate. upper surface can be considered a seepage face. The HP at 3. Traditional free-surfaceapproaches incorrectly assume point E on the seepage face marks the boundary between that rechargecannot occur downslopeof the point where the recharge and discharge and fulfills one condition of an exit water table meets the bedrock surface (commonly defined as point by defining the upper limit of discharge.The point the exit point marking the upper limit of the seepageface). In detachment A is undefined in Figure 5b becausea free surface developingthe approach usedin this study,it was necessaryto is absentunder conditions where the water table is everywhere identify two separatepoints on the upper boundary; the point at the bedrock surface.Reducing the infiltration rate to I., where the water table meets the bedrock surface (POD) and causesa free surface to develop (Figure 5c) with POD at A the point that marks the transition between rechargeand dis- and a HP at E. In this case,a single point cannot be defined charge (HP). Between these points the water table coincides that fulfills the definition of an exit point and rechargemay with the bedrock surface. In this region, the boundary con- occur on the seepageface between the POD and HP. Further dition is specifiedby setting hydraulic head equal to the eleva- reducingthe infiltration rate decreasesthe separationbetween tion of the upper boundary and allowing rechargeto occur at POD and HP (A and E). In low-relief topography,the separa- a rate defined by the nature of the groundwater flow system. tion becomesnegligible and the usual exit point definitionis In steep terrain, with rocks of low permeability, rechargecan valid. in high-reliefmountainous topography, the separation occur downslope of the POD in a region that might normally between POD and HP can be substantial and must be con- be considered part of the seepageface. In regions of reduced sidered in the numerical formulation. topographic relief, the separation between HP and POD be- Bear [1972, pg. 272] notes that mapping free-surface comes smaller and these points merge at the commonly de- groundwater flow problems into the hodograph plane is a fined exit point. useful method for examining flow conditions at the bound- 4. The fluid and thermal regimes are modeled using a Ga- aries. While the boundary between the free surface and the lerkin finite element technique in conjunction with a free- seepageface is initially unknown in the physicalplane, in the surface approach to estimate the position of the water table. hodograph plane it is completely defined.This mappingpro- Two finite element grids are required in solving this problem. cedureprovides an analyticalargument that supportsthe con- The mesh for fluid flow is generated only within the saturated cept of separatedPOD and HP on seepagefaces in mountain- zone. The mesh for heat transfer extends from the basal ousterrain. Details on methodsof mappingfrom the physical boundary to the bedrock surface, incorporating both satu- to hodographplanes can be foundin the work by Bear[1972] rated and unsaturated regions of flow. Coupling between heat and Verruijt [1970]. transfer and fluid flow is facilitated by ensuring a one-to-one Figure 5d is the hodographrepresentation of the physical correspondencebetween nodes located below the water table. systemof Figure 5c. In the hodographplane, verticaland 5. The numerical method developedin this paper provides horizontalcomponents of fluid flux (qz,q,•) at each pointon the means to examine the factors that control groundwater the boundaryof the physicalsystem form the hodograph.The flow and heat transfer in mountainous terrain: geology, sur- outline of the hodograph is defined as the locus of points face topography, climate, and regional heat flow. It is advanta- marking the distal ends of specificdischarge vectors originat- geous to adopt the flee-surface approach when performing ing at pointC (seevectors F•, F 2, and F 3 shownin Figures5c sensitivityanalyses because recharge to the flow systemis con- and 5d). In the mappingprocess, however, the spatialrelation- FO]•STE]•AND SMITH: GROUNDWATERFLOW SYSTEMS,1 1009 Iz0 ao b / E / ½ D D [z C, / / •%• Iz / / J / / c D qz e. z c • Iz / ß' x2 c•' D qz Fig. 5. Mappingwater table configurationsand flow patternsfrom physicalto hodographplanes: (a) triangulardam, (b) idealizedfully saturatedmountain profile, (c) mountainfree-surface problem with infiltrationrate I_.,(d) hodographof Figure5c, (e) mountain free-surface problem with infiltrationrates I:z and I=2,and (f) hodographof Figure5e. shipsbetween adjacent points in the physical plane are no Increasingthe vertical infiltration rate from Iz2 to I..• causes longerdefined in the hodograph plane. Along the horizontal the HP to move upslopefrom E 2 to E• and the POD to move impermeableboundary CD, q.. is zero and qx increaseswith upslopefrom A 2 to A 1 in the physicalplane of Figure5e. The increasingdistance from the origin to a maximum value equal HP remains fixed in the hodograph plane (points E x and E2), to the hydraulic conductivity K at point D. Similarly, along becausethis point alwaysmarks the point of zero flux normal the vertical impermeableboundary CB, qx is zero and the to the upper boundary in the hodographrepresentation magnitudeof q.. increasesto a maximum with absolutevalue (Figure5f). Separationbetween POD and HP increaseswith equalto the verticalinfiltration rate I s at B. increasinginfiltration in the physicalplane (Figure 5e). Al- In the hodograph plane, the free surface is describedby a thoughthe physicaldistance between POD and HP cannotbe circulararc with radius(K- I•)/2 [Bear, 1972]. The highest defined in the hodograph plane, it seemslogical to assume point on the free-surface occurs at the intersection with the that increasingseparation in the hodograph plane corre- verticalflux axis (point B), while the lowestpoint occursat the spondsto increasingseparation in the physicalplane (Figure POD (point A) where the free surfaceintersects the upper 5e). boundaryof the domain. In the simplesystems shown in Figures5c and 5e, the hori- Threefluid flux vectorsFx, F 2, and F 3 are shownin Figure zontal and verticalcomponents of fluid flux are explicitlyde- 5c with their correspondinghodograph representationin fined at points A, B, C, D, and E in the hodographplane Figure5d. In thephysical plane, vector F 3 is directedoutward (Figures5d and 5f). Hydraulichead solutions obtained with indicatingdischarge across the seepageface, while vector F x is the finite element method are used to estimate the free-surface directedinward and indicatesrecharge. At the hingepoint, configurationand calculate boundary fluxes. Boundary fluxes vectorF 2 is parallel to the upper boundary in the physical calculated with the finite element model correspondwell to planeand perpendicular to the correspondingline AD in the those defined in the hodograph plane. hodographplane (Figure 5d). Because F 2 is parallelto the upperphysical boundary, fluid flux normal to the boundaryat Acknowledgments.This work was fundedby grantsfrom the Na- pointE is zero.Therefore point E is the hingepoint that tural Sciencesand EngineeringResearch Council of Canada(NSERC) marksthe boundarybetween recharge and dischargeon the and a GraduateResearch in Engine,ering and Technologyaward pro- seepageface. videdby the BritishColumbia Science Council {in cooperationwith 1010 FORST•-RAND SMITH: GROUNDWATER FX.OW SYSTEMS, 1 Nevin Sadlief-BrownGoodbrand Company Limited). Discussions Lahsen, A., andP. Trujillo,The geothermal field of El Tatio,Chile, in with David Chapmanare gratefullyacknowledged. Thanks are also Proc.2nd. U.N. Syrup.on Developmentand Use of GeothermalRe. extendedto Allan Freezefor reviewingthe manuscriptand sharing sources,vol. 1, pp. 157-175,U.S. Government Printing Office, San hisviews on the developmentof seepagefaces in mountainousterrain. Francisco, Calif., 1975. GedeonDagan is to be thankedfor providingan illuminatingdis- Martinec,J., H. Oeschager,U. Schotterer,and U. Siegenthaler,Snow. cussionof hodographmethods. Computations were carried out on an melt and groundwaterstorage in an alpinebasin, Hydrological FPS 164 MAX Array Processorsupported by an NSERC Major Aspectsof Alpineand High MountainousTerrain, IAH$ Publ., InstallationGrant to theUniversity of BritishColumbia. 138, 169-175, 1982. Mears,F., Theeight mile Cascade Tunnel, Great Northern Railway• REFERENCES I!, Surveys,construction methods and a comparisonof routes, Adams,M. C., J. N. Moore, and C. 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