Mechanisms of heat exchange between water and rock in karst conduits

Mechanisms of heat exchange between water and rock inkarst conduits

M. D. Covington,1 A. J. Luhmann,2 F. Gabrov�sek,1 M. O. Saar,2 and C. M. Wicks3

Received 18 March 2011; revised 19 July 2011; accepted 5 September 2011; published 15 October 2011.

[1] Previous studies, motivated by understanding water quality, have explored themechanisms for heat transport and heat exchange in surface streams. In karst aquifers,temperature signals play an additional important role since they carry information aboutinternal aquifer structures. Models for heat transport in karst conduits have previously beendeveloped; however, these models make different, sometimes contradictory, assumptions.Additionally, previous models of heat transport in karst conduits have not been validatedusing field data from conduits with known geometries. Here we use analytical solutions ofheat transfer to examine the relative importance of heat exchange mechanisms and thevalidity of the assumptions made by previous models. The relative importance ofconvection, conduction, and radiation is a function of time. Using a characteristic timescale,we show that models neglecting rock conduction produce spurious results in realistic cases.In contrast to the behavior of surface streams, where conduction is often negligible,conduction through the rock surrounding a conduit determines heat flux at timescales ofweeks and longer. In open channel conduits, radiative heat flux can be significant. Incontrast, convective heat exchange through the conduit air is often negligible. Using therules derived from our analytical analysis, we develop a numerical model for heat transportin a karst conduit. Our model compares favorably to thermal responses observed in twodifferent karst settings: a cave stream fed via autogenic recharge during a snowmelt event,and an allogenically recharged cave stream that experiences continuous temperaturefluctuations on many timescales.

Citation: Covington, M. D., A. J. Luhmann, F. Gabrov�sek, M. O. Saar, and C. M. Wicks (2011), Mechanisms of heat exchange

between water and rock in karst conduits, Water Resour. Res., 47, W10514, doi:10.1029/2011WR010683.

1. Introduction[2] Water temperature is a crucial parameter in determin-

ing water quality and can serve as a natural groundwaterflow tracer [e.g., Anderson, 2005; Saar, 2011]. Temperaturecan be an indicator of anthropogenic influences on surfacestreams, and many aquatic species are sensitive to tempera-ture perturbations [Caissie, 2006]. Cave streams typicallyprovide a more stable thermal environment than surfacestreams, though many cave streams also experience signifi-cant deviations in water temperature as a result of precipita-tion events, snow-melt, or seasonal changes in rechargetemperatures [e.g., Luhmann et al., 2011]. Cave-adaptedspecies may be sensitive to the magnitude, frequency, or pe-riodicity of these temperature changes [Poulson and White,1969; Jegla and Poulson, 1970]. Water and streambedtemperatures are frequently used in surface streams to quan-tify surface water–groundwater interactions, and can also

influence those interactions via the temperature-dependentviscosity of water [Silliman and Booth, 1993; Sinokrot andStefan, 1993; Constantz, 1998, 2008; Hatch et al., 2006,2010; Dogwiler et al., 2007; Vogt et al., 2010]. Similarly, itmay be possible to quantify conduit–matrix exchange, orhyporheic exchange with sediments in cave streams usinganalysis of the longitudinal propagation of temperaturepulses within the streams [Dogwiler and Wicks, 2005].

[3] Variations in spring temperature and chemistry playan important role in the field of karst hydrology as they areone of the few pieces of information that can be easilyobtained from many karst aquifers, in hopes of constrainingthe properties of the conduit system. Lags between dischargeresponses and conductivity or temperature responses havebeen used to estimate the volume of conduit systems[Ashton, 1966; Atkinson, 1977; Sauter, 1992; Ryan andMeiman, 1996; Birk et al., 2004]. Thermal signals at karstsprings have also been suggested to be a function of conduitgeometry, recharge mode, aquifer depth, and conduit–matrixexchange flow [Benderitter et al., 1993; Bundschuh, 1993;Liedl et al., 1998; Liedl and Sauter, 2000; Birk et al., 2006;Long and Gilcrease, 2009; Luhmann et al., 2011]; however,the exact information content of thermal signals is not fullyunderstood.

[4] One barrier to an improved understanding of heattransport in cave streams is that a variety of models havebeen used to calculate heat exchange between the water in

1Karst Research Institute, Znanstvenoraziskovalni Center SlovenskeAkademije Znanosti in Umetnosti (ZRC SAZU), Postojna, Slovenia.

2Department of Earth Sciences, University of Minnesota, Twin Cities,Minneapolis, Minnesota, USA.

3Department of Geology and Geophysics, Louisiana State University,Baton Rouge, Louisiana, USA.

Copyright 2011 by the American Geophysical Union.0043-1397/11/2011WR010683

W10514 1 of 18

WATER RESOURCES RESEARCH, VOL. 47, W10514, doi:10.1029/2011WR010683, 2011

http://dx.doi.org/10.1029/2011WR010683

a conduit and the surrounding rock. Some of these modelsrely on assumptions that have not been examined in detailfrom a theoretical perspective or confirmed using fieldobservations of thermal signal propagation through con-duits with known properties. Conduction, convection, evap-oration, flow rate, and radiation have all been shown tohave important effects on the water temperatures of surfacestreams [Raphael, 1962; Brown, 1969; Sinokrot andStefan, 1993; Hondzo and Stefan, 1994; Gu and Li, 2002];however, some of these mechanisms are either absent or in-significant in karst conduits, particularly under full-pipeconditions.

[5] Some models of heat exchange in karst conduitsassume that the rate of heat transfer between the water (orair) and the conduit wall is controlled by convective heattransport within the water (or air), effectively assuming aconstant rock temperature at the conduit wall [Wigley andBrown, 1971; Long and Gilcrease, 2009]. In contrast, mod-els of surface stream temperature frequently assume thatheat exchange with the streambed is controlled by conduc-tion of heat within the rock or sediment, such that thestreambed temperature is equal to the water temperature[Sinokrot and Stefan, 1993]. An alternative model of karstconduit heat exchange includes both conduction in the rockand convection within the water [Birk et al., 2006]. In somemodels of surface stream temperatures, heat exchange withthe streambed is neglected altogether [e.g., Caissie et al.,2007], as, over long timescales, it is small in comparison tometeorological heat exchanges. Here we examine if andwhen each of these approaches is realistic in the calculationof heat transfer within karst conduits, using both heat trans-fer theory and observations of thermal signals at multiplelocations in two cave streams that experience differenttypes of thermal forcing. Previous models of water temper-ature in cave streams have only considered exchange underfull-pipe flow conditions. We further examine whetherradiative and air-convective heat exchange between waterand rock are important in karst conduits with open channelstreams.

2. Theoretical Analysis of the Mechanisms ofHeat Transport2.1. Basic Mathematical Framework

[6] The longitudinal propagation of heat within surfacestreams is often calculated using an unsteady heat advec-tion-dispersion equation containing source and sink termsthat represent heat exchange with the surrounding environ-ment [e.g., Sinokrot and Stefan, 1993; Younus et al., 2000;Caissie et al., 2007]. For surface streams, heat exchangeresults from incident short wave radiation, emitted longwave radiation, evaporation, convection through the air,and conduction into the streambed. However, in karstconduits that are completely filled with water, the majorityof these mechanisms are inactive and conductive heatexchange with the rock is the only remaining mechanism.Strictly speaking, heat exchange with the rock is a functionof both convective transfer rates between the water and theconduit walls and conductive transfer rates through therock surrounding the conduit, with these two processesacting in series (Figure 1). Taking this into account, heattransport along a karst conduit and exchange with the

surrounding rock may be represented by coupling a tran-sient heat advection-dispersion equation

@Tw

@t¼ DL

@2Tw

@x2� V

@Tw

@xþ 4hconv

�wcp;wDHðTs � TwÞ; ð1Þ

and a two-dimensional heat conduction equation, in cylin-drical coordinates,

1r@

@rr@Tr

@r

� �þ @

2Tr

@x2 ¼1�r

@Tr

@t; ð2Þ

where Tw, Ts, and Tr are the water, conduit wall, and rocktemperatures, respectively, x is the longitudinal distance

Figure 1. The model for heat exchange between a full-pipe karst conduit and the surrounding rock. Heat passesfrom the bulk, mixed water at T�w through a convectiveboundary layer into the conduit wall at temperature T�s .Heat then conducts through the rock body (initially atT�r ¼ 1), influencing an increasing volume of rock as theduration of a thermal pulse increases.

W10514 COVINGTON ET AL.: HEAT TRANSPORT IN KARST W10514

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along the conduit, r is the radial distance from the conduitcenter, t is time, DL is the longitudinal dispersivity, V(t) isthe flow velocity, hconv is the convective heat transfer coef-ficient, �w is the density of water, cp,w is the specific heatcapacity of water, DH is the conduit hydraulic diameter,and �r ¼ kr=ð�rcp;rÞ is the rock thermal diffusivity, wherekr is the rock thermal conductivity, �r is the rock density,and cp,r is the specific heat capacity of the rock.

[7] For the following theoretical analysis, it is useful toreduce the equations into a dimensionless form. Equation(1) becomes

@T�w@t�¼ 1

Pe@2T�w@x�2

� V�V@T�w@x�þ StðT�s � T�wÞ; ð3Þ

and equation (2) becomes

1r�

@

@r�ðr� @T�r

@r�Þ þ R2

L2

@2T�r@x�2

¼ �@T�r@t�

; ð4Þ

where T�w ¼ Tw=Tr;0 is the dimensionless water tempera-ture, T�s ¼ Ts=Tr;0 is the dimensionless conduit wall tem-perature, Tr,0 is the initial rock temperature (or rocktemperature at infinity), �V is a time-averaged or referenceflow velocity, t� ¼ t �V=L is the dimensionless conduit flowthrough time, where L is conduit length, and x� ¼ x=L isthe dimensionless longitudinal distance. In equation (4),r� ¼ r=R is the dimensionless radial coordinate, R is theconduit radius, T�r ¼ Tr=Tr;0 is the dimensionless rock tem-perature, and

� ¼ R2 �VL�r

ð5Þ

is a dimensionless ratio of conduction and advectiontimescales.

[8] Equation (3) contains two dimensionless numbers, Stand Pe. Pe is the Peclet number, which quantifies the rela-tive importance of advection and longitudinal dispersion,

Pe ¼ L�VDL

: ð6Þ

St is the Stanton number, which represents the ratio of heatflux into the conduit wall to the heat flux along the conduitand is given by

St ¼ 4hconvL�wcp;wDH �V

: ð7Þ

The convective heat transfer coefficient, hconv, is

hconv ¼kwNuDH

; ð8Þ

where kw is the thermal conductivity of water and Nu is theNusselt number, defined as the ratio of convective to pureconductive heat transfer through the convective boundarylayer in the water. For turbulent flow within a conduit, Nu

is given by the empirically derived Gnielinski correlation[equation (8.62) Incropera et al., 2007],

Nu ¼ ðf =8ÞðRe� 1000ÞPr

1þ 12:7ðf =8Þ1=2ðPr2=3 � 1Þ; ð9Þ

where f is the Darcy-Weisbach friction factor, Re ¼�wVDH=�w is the Reynolds number, Pr ¼ cp;w�w=kw is thePrandtl number, and �w is the dynamic viscosity of water.

[9] Equations (3) and (4) are subject to the boundaryconditions (Figure 1)

T�wðx� ¼ 0; tÞ ¼ f ðtÞ; ð10Þ

@T�w@x�

��x�¼1¼ 0 ð11Þ

@T�r@r�! 0 as r� ! 1; ð12Þ

@T�r@r�

��r�¼1¼ 1

2��StðT�s � T�wÞ; ð13Þ

where � ¼ �wcp;w=ð�rcp;rÞ is a ratio of the volumetric heatcapacities of water and rock, and we have defined R ¼ DH/2.As initial conditions we set the rock and water temperaturesequal to the rock temperature at infinity, T�wðx�; 0Þ ¼T�r ðr�; 0Þ ¼ 1:

2.2. Numerical Methods[10] In order to simulate the observed thermal signals (sec-

tion 2.6.1), and to examine the validity of the assumption ofplanar symmetry commonly used when calculating heat fluxthrough the stream beds of surface streams (section 2.5), wesolve the coupled heat advection-dispersion and conductionequations using the finite element package COMSOL Multi-physicsVR , version 3.5. Equation (4) is solved using the con-duction heat transfer application mode with either a 2-Dsymmetric or 2-D axisymmetric box geometry to simulateplanar or cylindrical configurations, respectively. Equation(3) is solved along a 1-D line using the coefficient form PDEmode. The two geometries are coupled to each other at oneof the box boundaries using the extrusion coupling variablesfeature in COMSOL.

[11] For all simulations, both the conduit and the rock atthe conduit wall were discretized into 1000 finite elementsalong the conduit length. Within the rock, elements gradu-ally coarsen moving away from the conduit wall toward theopposite boundary, which is always set far enough from theconduit that it does not influence the solution. The conduc-tion simulations typically employ around 22,500 elementswithin the rock. COMSOL solves the equations using animplicit method, and adjusts the time steps during the simu-lation by comparing the estimated error against relative andabsolute tolerances. For the simulations in section 2, and allpurely convective simulations, relative and absolute toleran-ces were set to 10�6 and 10�7, respectively. To reduce thecomputational time for the simulations in section 3, whichsimulate longer periods using field data, we employedhigher relative and absolute tolerances of 10�4 and 10�5,


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respectively. Several cases of simulations of field data werealso run with higher resolution, and the differences werenegligible, suggesting that the lower resolution was suffi-cient. Additional assumptions and methods used to simulateobserved signals can be found in section 3.2.

2.3. Analytical Solutions Using SimplifyingAssumptions

[12] In section 2.6.1 we numerically solve the precedingequations in order to reproduce observed thermal signals inkarst conduits. However, analysis of analytical solutionsusing a variety of simplifying assumptions allows moregeneral insights concerning the relative importance of dif-ferent heat exchange mechanisms and the range of applic-ability of common approximations.2.3.1. Convection-Limited Heat Exchange

[13] Since convection and conduction act in series, if oneof the two processes is significantly slower at exchangingheat than the other, then the slow process will be rate-limit-ing, and exchange rates can be approximated by only con-sidering the slow (limiting) process. The first analyticalsolution we consider is derived by assuming constant-temperature conduit walls, where heat conduction in therock is assumed to be very fast so that heat exchange is lim-ited by convective heat transport rates within the water(convection-limited exchange). This constant wall tempera-ture assumption is used by Long and Gilcrease [2009] tomodel temperature variations in a well in a karst aquifer. Asimilar approach was employed by Wigley and Brown[1971] to estimate heat exchange between air and conduitwalls and by Luhmann et al. [2011] to make a roughapproximation of heat exchange effectiveness. The approxi-mation is expected to hold for sufficiently short timescales,when the thermal boundary layer in the rock is very thinand conduction rates are correspondingly large. In section2.3.1 we quantify the timescale after which the convection-limited heat-exchange approximation breaks down.

[14] If a constant conduit wall temperature is assumed,then heat transport is governed solely by the heat advec-tion-dispersion equation (3) with T�s held constant. If weneglect dispersion and examine the constant-velocitysteady state behavior for a constant input temperature, T�w;0,the equation can be integrated resulting in

FTðx�Þ ¼T�w � T�s

T�w;0 � T�s¼ e�St x� ; ð14Þ

where FT is the fractional temperature variation. The tem-perature decays exponentially along the conduit. Figure 2adisplays an example of a convection-controlled temperatureprofile for a conduit with fairly typical parameters of DH ¼0.5 m and V ¼ 0.1 m s�1.2.3.2. Conduction-Limited Heat Exchange

[15] If we assume that convective heat exchange is veryfast, such that heat fluxes are controlled by conduction inthe rock, and again assume that longitudinal dispersion canbe neglected, then we can derive another analytical approx-imation for the longitudinal temperature profile. The solu-tion is aided by additionally assuming planar rather thancylindrical symmetry to calculate rock conduction. The va-lidity of a planar approximation is examined in furtherdetail in section 2.5. If the boundary condition in equation

(13) is substituted into the exchange term in equation (3),then this gives

@T 0w@t�¼ �V

�V@T 0w@x�þ 2

��

@T 0r@y�

��y�¼0

; ð15Þ

where the position coordinate within the rock is nowy� ¼ y=R instead of r�, the conduit wall is at y� ¼ 0, andthe substitutions T 0r ¼ T�r � 1 and T 0w ¼ T�w � 1 are appliedto simplify the boundary conditions in the Laplace trans-form domain. The heat conduction equation becomes

@2T 0r@y�2

¼ �@T 0r@t�

; ð16Þ

where we have presumed that the effects of longitudinalvariations in rock temperature are negligible, or R2ð@2T 0r=@x�2Þ � L2ð@2T 0r=@y�2Þ. For the conduction-limited solutionwe assume that the rock temperature at the conduit is equal

Figure 2. Longitudinal temperature profiles for the con-vection-limited (a) and conduction-limited (b) solutions.The conductive solution is shown for one flow-throughtime (t1) and five flow-through times (t2). Note that the con-ductive solution allows significantly deeper penetration oftemperature pulses (x-axis in km).


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to the water temperature, T�w. The conduction boundary con-dition becomes

T 0r jy�¼0 ¼ T 0wðx�Þ: ð17Þ

We solve these equations assuming that the initial rock andwater temperatures are equal everywhere. As for the con-vection-limited solution, we calculate the temperature pro-file for a constant temperature input T 0w;0 6¼ T 0r for t� > 0.Unlike the convection-limited solution, the temperatureprofile evolves over time.

[16] The set of equations and boundary conditionsdescribed here is mathematically identical to a model previ-ously used to calculate solute transport in a river with hypo-rheic exchange [Wörman, 1998; De Smedt, 2007], wherethe solution in the latter study includes the effects of disper-sion. The solution for the water temperature can beobtained by taking the Laplace transform of the equations.The resulting temperature profile is

T 0wðx�; t�Þ ¼ T 0w;0Hðt� � x�Þerfcx�

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ðt� � x�Þ

p" #

; ð18Þ

where H is the Heaviside function and erfc is the comple-mentary error function. As an illustration of the conduc-tion-controlled temperature profile, we plot the temperatureprofile for an example conduit with the same parametersused for the convection-limited profile above, choosingtimes of one and five flow-through times (Figure 2b). Thedistance that a thermal pulse can penetrate into a conduit issignificantly enhanced in the case of conduction-limitedheat exchange. The significant differences between the twoprofiles in Figure 2 illustrate the disparity between themechanistic assumptions made in previous karst conduitheat exchange models.

[17] Despite neglecting convection, and the assumptionof planar rather than cylindrical symmetry, the conduction-limited analytical solution (equation (18)) closely matchesresults produced by the numerical solution of equations (3)and (4) using COMSOL so long as � has a large-enoughvalue (the effect of � on error produced by the planarapproximation is discussed in detail in section 2.5). To dem-onstrate this, we compare the conduction-limited planar-symmetric analytical solution with the cylindrically sym-metric numerical solution from COMSOL calculated byapplying a sudden change in water temperature at theupstream boundary at t� ¼ 0, and with a value of � ¼ 100,which roughly corresponds to a kilometer-long conduit witha radius of 1 m and a flow velocity of 0.1 m s�1 (Figure 3).

2.4. Convective Versus Conductive Control of HeatExchange

[18] If we consider the nature of heat exchange inresponse to a sudden change in water temperature, then, atearly times, when the conduit wall is near the rock tempera-ture at infinity, Tr,0, the convection-limited approximation(section 2.3.1) is accurate. At late times, the temperature ofthe conduit wall approaches the water temperature, and theconduction-limited approximation (section 2.3.1) is accu-rate. The success of the example case simulated using theconduction-limited approximation (Figure 3) suggests that

the timescale over which the convection-limited is valid isshort in at least some realistic cases. However, since bothof these approximations have been used in models of heatexchange in surface streams and karst conduits, we derivethe timescale that roughly divides the regions of applicabil-ity of each approximation.

[19] We simplify the problem by assuming planar symme-try, as in the conduction-limited solution above. This approx-imation holds for any geometry for small Fourier number,Fo ¼ �rt=R2 � 1, when there is only a thin skin of rock sur-rounding the conduit that has changed temperature. The so-lution for the surface temperature of a semi-infinite solidwith a convective boundary condition along a planar surfaceis well known [equation (3.87) Rohsenow et al., 1998],

T�s ¼ T�w � ðT�w � 1Þ expðBi2Þ erfcðBiÞ; ð19Þ

where

Bi ¼ hconv

kr

ffiffiffiffiffiffi�rtp

¼ kwNukrDH

ffiffiffiffiffiffi�rtp

ð20Þ

is the Biot number. For Bi � 1 the surface temperatureapproaches the water temperature ðT�s � T�wÞ. By the timewhen Bi ¼ 1, that is

tconv ¼k2

r D2H

k2w�rNu2

; ð21Þ

the convection-limited approximation has broken down.

Figure 3. Dimensionless temperature profiles along aconduit at three different times. The analytical solution(dashed), which assumes planar symmetry and neglects theconvective boundary layer within the water, and the numer-ical solution (solid), which includes the effects of cylindri-cal geometry and convection within the water, are nearlyidentical for this choice of model parameters ð� ¼ 100Þ,which roughly corresponds to a conduit length of 1 km, ra-dius of 1 m, and a flow velocity of 0.1 m s�1. x� ¼ 1 is thedownstream end of the conduit, T� ¼ 1 is the initial rockand water temperature, T� ¼ 2 is the input temperature fort� > 0, and t� ¼ 0 is one flow-through time.


5 of 18

[20] We calculate this convection timescale as a functionof DH assuming a variety of head gradients within the rangeof those expected in karst aquifers (Figure 4). The convec-tion-limited approximation would only hold for timesmuch less than tconv. For karst conduits carrying turbulentflow, tconv is typically on the order of a fraction of a secondto a few tens of seconds. Therefore, any calculation of heatexchange in a conduit carrying turbulent flow shouldinclude conduction in the surrounding rock. Since our deri-vation of tconv assumes a planar symmetry, the derivedtimescale is only accurate if tconv � tFo, where tFo is thetime at which Fo ¼ 1. Therefore, in Figure 4 we also depicttFo as a function of conduit diameter. For turbulent flowcases, tconv � tFo, and the effects of cylindrical geometryon tconv are insignificant. For laminar flow, tconv � tFo, andthe cylindrical geometry will have an effect, increasing thevalue of tconv; however, conduits with such small diameterswould also be more likely to occur within fractures andthus have planar shapes.

2.5. Cylindrical Versus Planar Symmetry[21] For some of the analytical solutions above we

ignored the cylindrical geometry of the rock surroundingthe conduit and used planar symmetry instead. Further-more, models that calculate water temperatures in streams,if they include rock conduction, typically assume planarsymmetry. If the depth of temperature changes in the wallis small compared to the conduit radius then planar symme-try is a good approximation. A semi-infinite solid with aplanar surface and an internally bounded semi-infinitecylindrical solid maintain nearly equal surface heat fluxes

so long as the Fourier number is very small, Fo � 1 [seee.g., Figure 5.10 in Incropera et al., 2007]. For limestone,�r � 10�6 m2 s�1, meaning that for a conduit with a radiusR ¼ 1 m, the cylindrical and planar solutions will begin todeviate after a few hours (Fo � 0.01), and will have devi-ated significantly after a week (Fo � 1). Therefore, onemight expect that neglecting the cylindrical geometry sur-rounding the conduit could produce errors, at least forlong-term simulations.

[22] To characterize this error, we numerically solve thecylindrical model using COMSOL and calculate two springthermographs with � ¼ 10 and � ¼ 1. Then we comparethese numerical cylindrical solutions to thermographs cal-culated using the analytical conduction-limited planar-symmetry solution and a second COMSOL numericalmodel that employs planar symmetry (Figure 5). The pla-nar-symmetry COMSOL model is in excellent agreementwith the analytical conduction-limited planar-symmetry so-lution for both choices of �, because tconv is very short. Asexpected from the examination of Fo above, the cylindricalsolution diverges from the planar solutions at early times.However, after sufficient time, the ratio between the cylin-drical and planar solutions approaches a constant value. Theratio between the cylindrical and planar solutions at latetimes is shown as a function of � in Figure 6. For �[10the error in assuming planar symmetry becomes significant.� is most strongly dependent on conduit radius, with smallconduits producing a more significant error if planar sym-metry is assumed. � is depicted as a function of radius forfive different flow-through times in Figure 7.

2.6. Additional Heat Exchange Mechanisms in OpenChannel Conduits

[23] When a conduit is full of water, convection in thewater and conduction in the rock are the primary processes

Figure 4. The timescale over which the convection-lim-ited (or constant rock temperature) approximation breaksdown, tconv, as a function of DH. For conduits carrying tur-bulent flow, tconv is on the order of seconds, and neglectingrock conduction around the conduit will produce significanterrors after that time. Each line depicts values of tconv for agiven choice of hydraulic head gradient. Dotted lines markthe transition from laminar to turbulent flow. The dashedline depicts the time tFo as a function of conduit diameter.The approximation used to calculate tconv is only validwhen tconv� tFo, which holds for the turbulent flow regime.

Figure 5. Thermographs at the outlet of conduits whereconduction is calculated using planar and cylindrical sym-metry for two different choices of �. The cylindrical solu-tion is numerical, whereas both analytical and numericalsolutions are depicted for planar symmetry. The error pro-duced by an assumption of planar symmetry approaches aconstant value at late times.


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by which heat is exchanged with the surrounding rock,greatly reducing the number of heat exchange terms incomparison to models for surface streams. However, manycave streams flow through conduits that are partially filledwith air. In this case, the free surface of the water alsoexchanges heat with the rock via both radiation and con-vection through the air (Figure 8).2.6.1. Radiative Heat Exchange

[24] The radiative heat flux at the surface of the dry rockwall is approximated by the equation for blackbody radia-tion, since the emissivities of water and rock are close to 1,

q00rad ¼�

AdðT4

w � T4s;dÞ; ð22Þ

where q00rad is the radiative heat flux at the rock surface,� ¼ 5:67� 10�8 W m�2 K�4 is the Stefan-Boltzmann con-stant, Ad ¼ Pd/Wfs is a ratio of dry rock wall surface towater free surface areas, Pd is the dry conduit perimeter,Wfs is the width of the water-free surface, and Ts,d is thetemperature of the dry conduit wall. Temperatures here arein Kelvin. Since radiative heat exchange between water androck acts in series with conduction through the rock, at longtime-scales heat flux will be governed by rock conduction.

[25] In analogy to the convective boundary case (section2.3.1), we can estimate the timescale, trad, over which thedry rock wall reaches the water temperature by radiativeexchange using the planar solution from equation (19) andsubstituting a temperature-dependent radiative heat transfercoefficient, hrad, into Bi, where

hrad ¼q00rad

ðTw � Ts;dÞ¼ �

AdðTw þ Ts;dÞðT2

w þ T2s;dÞ: ð23Þ

This gives a timescale, at Bi ¼ 1, of

trad ¼A2

dk2r

�r�2ðTw þ Ts;dÞ2ðT2w þ T2

s;dÞ2 : ð24Þ

Since the water and rock temperatures in most karst systemsonly vary within a narrow range of 273 K [ T [ 310 K,most of the variation in trad results from Ad and not tem-perature. trad is typically on the order of days to weeks(Figure 9). trad may be somewhat longer than calculatedby equation (24) for conduits with D[1 m because of theeffects of cylindrical geometry. Tall, narrow conduits, andother cross-sectional geometries where Pd � Wfs, willhave the longest equilibration times and also the largestradiative heat fluxes at t � trad, when exchange rates areconduction-limited.

Figure 6. The ratio between conduit output temperaturesfor planar and cylindrical solutions at late times (see Figure 5for solutions as a function of time). The planar approximationprovides good results for conduits with values of �Z10.

Figure 7. The dimensionless parameter � as a functionof conduit radius for five flow-through times (h). If �Z10then the thermographs produced by planar and cylindricalgeometries are nearly identical.

Figure 8. Water in open-channel karst conduits exchangesheat with the rock via radiation and convection through theair, in addition to the processes that occur in full-pipe con-duits (Figure 1).


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[26] An additional question of interest is if and whenradiative heat exchange is a significant percentage of thetotal exchange. For times longer than seconds, we can ap-proximate the heat flux at the water–wall interface usingthe solution for heat flux at the surface of a semi-infinitemedium with a constant temperature boundary

q00cond ¼krðTw � Tr;0Þffiffiffiffiffiffiffiffiffi

��rtp ð25Þ

[equation (3.81) Rohsenow et al., 1998]. The radiative heatflux into the dry conduit wall is given by

q00rad ¼ hradðTw � Tr;0Þ expðBi2Þ erfcðBiÞ ð26Þ

[equation (3.88) Rohsenow et al., 1998]. Figure 10 depictsthe ratio of these two fluxes as a function of time, using thesame temperature range as above to calculate hrad, andassuming three different values for Ad. Radiative flux issmall, but nonnegligible, at short timescales (hours) andcomparable to heat flux at the water–rock interface overlong time-scales (days to weeks). The overall heatexchange resulting from each mechanism will be a functionof both flux and the surface area for exchange. The impor-tance of radiative exchange is reduced for cross-sectionalgeometries where Pw � Wfs or Pw � Pd, where Pw is thewetted perimeter.2.6.2. Convective Heat Exchange Through the Air

[27] Convective heat transport in the air is more chal-lenging to calculate than the other mechanisms of heatexchange because of the complex set of processes at work.In particular, both free [Wilson and Dwivedi, 2009] andforced convection occur within the conduits, and variationsin above-ground air temperature, barometric pressure, andhumidity are first-order influences on the heat exchangesnear entrances. However, since the direct influence of

surface conditions will typically be localized near entrances[Wigley and Brown, 1971; Dreybrodt et al., 2005], heatexchange between the cave and the surface via air-convec-tive processes is unlikely to have significant influences onstream temperatures across an entire system. Here we onlyconsider convective heat exchange deep inside of caveswhere the effects of exchange of air between the surface andsubsurface can be neglected and the partial pressure of watervapor is near equilibrium. In this case, the air simply acts as amedium for heat exchange between water and rock, andweather conditions at the surface only affect convective heatexchange by influencing the air velocity via thermal (chim-ney effect) and barometric forcing of airflow. Convectiveheat exchange will only be enhanced by free convectionwhen the water is warmer than the rock. Additionally, underforced turbulent airflow conditions, which are typical in air-filled caves, enhancements of heat exchange due to free con-vection are generally small [Osborne and Incropera, 1985].Therefore, we can estimate heat exchange via air convectionby calculating the exchange due to only forced convection.

[28] Heat flux through the air must traverse two bound-ary layers, one at the water–air interface and one at the air–rock interface (Figure 8). Therefore, we can approximatethe total heat transfer coefficient by using equation (8) anddividing by two. To estimate Nu, we use the Colburn equa-tion [equation (8.59) Incropera et al., 2007],

Nu ¼ 0:023Re4=5Pr1=3; ð27Þ

which is a simpler replacement for the Gnielinski correla-tion (equation (9)). In both equations we substitute the

Figure 9. The timescale over which radiative exchangebetween the free water surface and dry conduit wall bringsthe conduit wall to the water temperature for a range of val-ues of the surface area ratio, Ad. The solid line representsaquifers with water temperatures near 273 K (0�C) and thedashed line represents 310 K (37�C). Labels depict appro-priate values of Ad for different cross-sectional shapes.

Figure 10. The ratio of conductive heat flux at the water-rock boundary, q00cond, to the radiative heat flux into the dryconduit perimeter, q00rad, as a function of time for threechoices of the ratio between surface areas of dry rock andthe water free surface, Ad. At short timescales (hours) radi-ative flux is small in comparison to the water-rock flux. Byday to week timescales, wetted conduit and dry conduitheat fluxes are approximately equal, since they are bothcontrolled by the conductive heat fluxes in the rock. Totalheat exchange via each mechanism will also be a functionof the total wet and dry rock surface areas.


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appropriate constants for air, ka ¼ 0.026 W m�1 K�1,�a ¼ �a=�a ¼ 1:5� 10�5 m2 s�1, and Pr ¼ 0.7. The result-ing heat transfer coefficient is

hair ¼0:023kaPr1=3V 4=5

a

�4=5a D1=5

H

; ð28Þ

where Va is the airflow velocity. To examine the relative im-portance of air-convective heat transport, we compare theair-convective to radiative heat fluxes as a function of airvelocity (Figure 11). To calculate radiative flux, we haveassumed a wall temperature of 15�C, though results are notvery sensitive to this choice. For typical air velocities of0.01 m s�1 < Va < 0.1 m s�1, air-convective fluxes are afraction of a percent to a few percent of the radiative flux.Even for high air velocities of up to 1 m s�1, air-convectivefluxes are only 20% of the radiative flux. Therefore, formost cases it appears to be reasonable to neglect air-convec-tive heat transport when calculating water temperatures,particularly when one considers that water-rock fluxes arestill larger than radiative fluxes over short timescales, andthat, on its own, radiation effectively couples water and dryrock surface temperatures at time-scales of days to weeks.

3. Reproduction of Observed Signals[29] To test the applicability of our mathematical model

to karst conduits we calculate downstream thermographs intwo cave streams using observed upstream thermographsand conduit geometry as model inputs. We then compare thecalculated thermographs with the thermographs observed ateach site.

3.1. Description of the Field Sites and Methods3.1.1. Tyson Spring Cave, Minnesota

[30] Tyson Spring Cave is located in Fillmore County insoutheastern Minnesota (Figure 12) and has developed inthe Upper Ordovician Prosser and Cummingsville Forma-tions of the Galena Group [Alexander Jr. et al., 2009]. The

Prosser Formation consists of very fine-grained limestonewith thin shale partings, and the Cummingsville Formationis composed of interbedded very fine-grained limestoneand calcareous shale [Mossler, 2008]. The Galena Grouphas very low to low matrix porosity and permeability, butdissolution features have been noted around the Cum-mingsville-Prosser contact, even in deep bedrock condi-tions [Runkel et al., 2003]. Furthermore, dendritic streamcaves are common in the lower Cummingsville Formation[Alexander Jr. et al., 2009]. Tyson Spring Cave is a dendri-tic branchwork cave, but within the segment studied it pri-marily consists of one large passage with an active streamand several, minor infeeders. The cave is recharged auto-genically, via a sinkhole plain. Mean monthly surfacetemperatures (1971–2000) for nearby Preston, MN rangefrom �11�C in January to 21�C in July, with an averageannual precipitation of 870 mm according to the MidwesternRegional Climate Center (http://mcc.sws.uiuc.edu/climate_midwest/mwclimate_data_summaries.htm).

[31] We installed four Schlumberger CTD-divers inTyson Spring Cave. The data loggers recorded water spe-cific conductivity, temperature, and depth over a 15-monthperiod (Figure 12). Three loggers were deployed at differentlocations along the mainstream channel within the cave,and one logger was deployed at the spring. A SchlumbergerBarologger was installed near the spring to allow compensa-tion for atmospheric pressure changes. Measurements weremade and recorded every 5 or 10 min, and the temperatureresolution was 0.01�C, with an accuracy of 0.1�C from�20�C to 80�C.

[32] The spring snowmelt in 2010 caused the largestobserved variation in water temperature. Surface air tem-peratures rose and remained above freezing in mid-March2010, providing a continuous snowmelt pulse that producedfive daily temperature minimums from 11 to 15 March.Temperature data from this event recorded by data loggerT1 was used as an input to attempt to match data recordedat data logger T2. This portion of the cave was used in thesimulation because relatively little additional water entersthe stream between these points, and we have the best setof discharge measurements in this segment of the subsur-face stream. The average hydraulic diameter along thisreach is DH ¼ 1.4 m, and the distance between data loggersT1 and T2 is 845 m.

[33] We developed a rating curve using salt trace dis-charge measurements to convert level measurementsrecorded by data logger T1 to discharge Q. However, we didnot measure discharge during highest flows, when much ofthe cave is inaccessible, and therefore the rating curve is ex-trapolated beyond the highest observed discharges. Theresulting uncertainty in discharge is a likely source of errorin modeling heat transport during the highest flows.3.1.2. Postojna Cave, Slovenia

[34] The Postojna Cave system is a famous show cave atthe contact between Eocene flysch and Upper Cretaceouslimestone near Postojna, Slovenia. With a length of morethan 20 km, it is currently the longest known cave systemin Slovenia. The cave is formed within both flanks of thePostojna anticline, and the Pivka River sinks in Senonianlimestone and then passes through Turonian and into UpperCenomanian limestone [�Sebela, 1998]. The main rechargeto the system is allogenic, via the Pivka River, which flows

Figure 11. The ratio of air-convective, q00air, to radiative,q00rad, heat fluxes as a function of airflow velocity for twochoices of air-filled hydraulic diameter. For typical air veloc-ities of 0:1 m s�1, air-convective heat exchange is negligible.


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from the adjacent Pivka Basin. The surface flow of the riveris about 15 km long. The basin extends from 750 to 500 mabove sea level and is surrounded by high karst plateaus(up to 1200 m). The water re-emerges in Planina Cave andbecomes part of the Unica River, which flows across Pla-ninsko polje. On the west and north rims of the polje theriver sinks again, emerging finally at the edge of the Ljubl-jana Basin as the Ljubljanica River.

[35] The climate of Postojna is transitional between conti-nental and Mediterranean. Mean annual temperature (1961–1990) is 8.4�C. Mean annual precipitation is 1578 mm[Gospodari�c and Habi�c, 1976]. Precipitation maxima occurin spring and autumn.

[36] Measurements were taken at two stations in PostojnaCave (Figure 13). The upstream station (P1) was located inVelika dvorana (Great Chamber), just downstream from thesink point. The downstream site (P2) was positioned nearSpodnji Tartar. The stream distance between the two meas-uring points is 700 m. At low and mean discharge, as duringthe period modeled in this work, open channel flow prevailsalong the length of the channel. However, several channelsegments experience transitions to pressurized flow duringlarge floods.

[37] Water temperature and pressure were measuredusing Schlumberger CT-Divers, with a temperature resolu-tion of 0.01�C and an accuracy of 0.1�C. Data wererecorded at 15-min intervals, and atmospheric pressure var-iations were compensated with barometric data from thesurface. Discharge was determined using a gaging stationin the Pivka River near the sink point into Postojna Cave.

[38] Since Postojna Cave receives allogenic recharge fromthe Pivka River, the stream in the cave undergoes consistentdiurnal variations. Additionally, the stream temperatureresponds strongly to individual recharge events. We analyzea period during the summer of 2008, when the cave stream

experiences a variety of conditions. First, a cool precipitationevent lowers the stream temperature in the cave, with lowtemperatures around 12�C. During this period, little dampingoccurs between the two temperature stations because thetemperature is close to that of the conduit wall. The coolevent is followed by both a recession in discharge and awarming of surface temperatures, during which temperaturesat P1 rise to nearly 18�C. During this period, there is morethermal damping between P1 and P2. The superposition oftemperature variations on differing timescales provides auseful test for the heat transport model.

3.2. Simulations of Observed Thermal Signals3.2.1. Modeling Methods

[39] The physical constants and parameters used in thesimulations are displayed in Tables 1, 2, and 3. The modelrequires values for a reference velocity, �V , and instantane-ous velocities, V(t). For both field sites, we determine �V bydividing the stream distance between the two loggers bythe time difference between two sharply defined features inthe thermographs. For Tyson Spring Cave, we use the watertemperature maximum recorded by both loggers after theinitial water temperature minimum of the 2010 snowmeltperiod. For Postojna Cave, we use the water temperaturemaximum recorded by both loggers on 16 May 2008. Theresults of the simulations do not depend critically on thechoice of a reference velocity. The reference velocity issimply needed to nondimensionalize the equation. How-ever, since the flow-through timescale is defined via thischoice in velocity, using a value that is typical for eachcave produces more easily interpretable results. To calcu-late V(t) we use

V ðtÞ ¼ Q�V�Q; ð29Þ

Figure 12. A line plot of the mapped portion of Tyson Spring Cave, Minnesota, depicting the locationof the two data loggers used in the study, T1 and T2. The spring is just downstream of T2, and the hill-shade created from LIDAR data depicts the sinkhole plain drained by the cave.


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Figure 13. The location of data loggers, P1 and P2, within Postojna Cave, Slovenia.


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where �Q is discharge from the same time that was used tocalculate �V . This equation assumes that the cross-sectionalarea of the flow is constant in time, and therefore will nothold for large variations in Q. Choosing a value for �V nearthe average velocity during the simulated period reducesthe error from this assumption, and we find that equation(29) produces reasonable results, with changes in lag timethroughout the simulated periods being properly matchedby the model outputs. For the simulations of both cavestreams we chose a large value of Pe ¼ 1000, essentiallyturning off dispersion, because there is very little dispersionin the observed temperature and conductivity signals.Lower values of Pe are ruled out because they would smoothout the sharp features observed at the downstream stations.The surface area ratios, Ad ¼ Pd/Wfs and Aw ¼ Pw/Wfs, werematched to known values from cave surveys, and for eachcave DH was coarsely adjusted to produce a good fit.

[40] Using the rules obtained above concerning the va-lidity of model assumptions and the relative importance ofheat exchange mechanisms, we derive an appropriate set ofmodel assumptions to apply to our field sites. The dry con-duit radius roughly determines the curvature relevant forsolving the heat conduction within the rock. For both fieldsites, dry conduit radii are typically larger than 2 m, withflow-through times of a few hours. Therefore, a planar-symmetric solution will produce accurate results (Figure7). Furthermore, both cave streams are open channels dur-ing the simulated periods. Since the timescales simulatedare on the order of a week or longer, we must account forradiative exchange between water and rock (Figure 10).

[41] The substitution of equation (29) into equation (3)and the addition of radiative heat exchange leads to the fol-lowing heat advection-dispersion equation:

@T�w@t�¼ 1

Pe@2T�w@x�2

�Q�Q@T�w@x�þStðT�s;w�T�wÞþAd�ðT�s;d�T�wÞ; ð30Þ

where T�s;w is the surface temperature of the wet conduitwall, T�s;d is the surface temperature of the dry conduit wall,and

�¼ hradLWfs

�wcp;wAc �V¼ 4hradL�wcp;wAwDH �V

ð31Þ

is a dimensionless number describing radiative heatexchange, where Aw ¼ Pw/Wfs.

[42] We calculate conduction through the wet and dryrock perimeters separately, using the planar-symmetricheat conduction equation for each,

@2T�r@y�2

þ R2

L2

@2T�r@x�2

¼ �@T�r@t�

: ð32Þ

Conduction through the wet perimeter is coupled to theadvection-dispersion equation (30) using the boundary con-dition given in equation (13) with T�r ¼ T�r;w, T�s ¼ T�s;w,and changing variables to y�. The boundary condition forconduction at the dry perimeter is

@T�r;d@y�

��y�¼1¼ 1

2��AwðT�s;d � T�wÞ: ð33Þ

The boundaries at large y� are placed sufficiently far fromthe conduit so that equation (12) holds without influencingthe heat exchange at the conduit wall during the duration ofthe simulations. The rock boundaries at x� ¼ 0 and x� ¼ 1are insulated, and the upstream water temperature (equation(10)) is set to the values recorded at the upstream data log-ger as a function of time.3.2.2. Simulation Results: Tyson Spring Cave

[43] Figure 14a shows the result of our simulation oftemperature signal at T2 during the snowmelt event atTyson Spring Cave as well as the water temperature datafrom T1 and T2 and the discharge at T1. The simulationincludes conduction into the surrounding rock, as well asradiative heat exchange with the dry conduit wall, and pro-duces a reasonably good fit to the observed temperature atT2 using DH ¼ 0.3 m, Aw ¼ 2, and Aw ¼ 4. The first tem-perature minimum is closely matched, whereas the modelproduces insufficient heat exchange during the second tem-perature minimum. The error in the second minimum likelyresults from poorly constrained discharge values during thehighest flows, where the peak discharge exceeds the highestobserved discharge by a factor of two. From the secondminimum on, the model slightly underpredicts the observedtemperature, which may be a residual effect from over-shooting the temperature minimum. For both the simulatedand observed data, the temperature difference between T1and T2 decreases over time as the cold temperature pulse

Table 1. The Values of Constants and Parameters Used in theSimulations of Both Tyson Spring and Postojna Caves

Parameter Value Units

cp,w 4200 (J kg�1 K�1)kw 0.58 (W m�1 K�1)�w 1.3 � 10�3 (kg s�1 m�1)�w 1000 (kg m�3)cp,r 810 (J kg�1 K�1)kr 2.15 (W m�1 K�1)�r 2320 (kg m�3)f 0.05 Unitless

Pe 1000 UnitlessPr 9.5 Unitless� 5.67 � 10�8 (W m�2 K�4)

Table 2. The Values of the Geometrical Parameters Used in theSimulation of Tyson Spring Cave


L 845 (m)DH 0.3 (m)Ad 4 UnitlessAw 2 Unitless

Table 3. The Values of the Geometrical Parameters Used in theSimulation of Postojna Cave


L 700 (m)DH 1.5 (m)Ad 6.67 UnitlessAw 1.33 Unitless


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propagates further into the conduit wall, cooling more ofthe surrounding rock. This behavior is expected from theanalytical, conduction-limited solution in section 2.3.1. Todemonstrate the difference between the full model and aconvection-limited model, we also depict a simulation inFigure 14a that assumes a constant conduit wall tempera-ture, using the same hydraulic diameter. In this simulationthe thermal signal is entirely damped when it reaches T2,producing a flat line that is nothing like the observed signal.3.2.3. Simulation Results: Postojna Cave

[44] Figure 14b shows the result of our simulation ofthermal signals at P2 in Postojna Cave along with the watertemperature data measured at P1 and P2 and the dischargedata recorded at the gaging station near the entrance of thecave. This simulation also includes conduction into thesurrounding rock and radiative exchange with the dry

conduit wall, and produces a close fit to the observed tem-perature at P2 using DH ¼ 1.5 m, Aw ¼ 1.33, and Aw ¼6.67. The simulated temperatures at P2 closely match theobserved temperatures during the 18-d duration of the sim-ulation. Early on, before the cool storm, there are slighttemperature offsets between P1 and P2. During the storm,P1 and P2 record nearly identical temperatures, a behaviorthat is matched by the model. As surface temperatureswarm, the water temperature at P1 rises to nearly 18�C andthe damping between P1 and P2 increases. The model alsoprovides a good match to the observed temperature duringthis period. Again, we run an additional convection-limitedsimulation using the same hydraulic diameter. The temper-ature variations in this model at P2 are small (0.2�C),though they are not entirely damped as they were for TysonSpring Cave.3.2.4. Simulations Using Different Diameters

[45] While the observed data cannot be reproduced usingthe purely convective model and conduit hydraulic diame-ters that are close to the observed diameters for the casesabove, here we adjust the diameters within the convectivemodel until we obtain the best fit to observations (Figure15). A simulation for Tyson Spring Cave with DH ¼ 30 mprovides a reasonably good fit to the observed tempera-tures. For this simulation, as above, we have focused on thefit of the first minimum, since discharge values are less cer-tain during the second minimum. The convection-limitedmodel actually produces a better fit during the high flowsof the second minimum than the full model above. This is aresult of the relative insensitivity of convective heatexchange to changes in flow velocity. In the turbulent flowregime, decreases in heat exchange because of higher flowvelocities (shorter residence times), are almost exactly off-set by increases in the heat exchange because of increasedturbulent mixing (smaller convective boundary layer). As aresult, errors in discharge values are less significant for theconvection-limited model. A relatively good fit to theobserved temperature in Postojna Cave is obtained usingDH ¼ 250 m. However, it is not possible to obtain an exactfit for the entire simulated period using a single value ofDH for either simulation. Near the end of the simulatedperiods for both caves there are minor offsets between theobserved and simulated relations.

[46] While a full sensitivity analysis is beyond the scopeof this work, we simulate a few cases using the full heatexchange model with diameters different from the best fitvalues. For each cave, we display cases where DH has beenincreased or decreased by a factor of two from the best fit(Figure 16). These results provide some idea of how themodel responds to changes in diameter. For Postojna Cave,we have focused the plot on the region with the largestchanges in thermal response.

4. Discussion[47] Typically, the properties of the conduits within a

karst aquifer are a critical unknown. The conduits deter-mine the paths of flow and transport, but only a fraction ofthe existing conduits can be traversed and mapped. Conse-quently, it is important to develop tools to derive informa-tion about the conduit system using external observations,such as the recording of variations in the physical and

Figure 14. Time series of the simulated (solid-black) andobserved (thick solid gray) temperature signals from TysonSpring Cave (a) and Postojna Cave (b). For each case werun an additional model using constant conduit wall tem-perature, i.e., a convection-limited model that neglects con-duction (dashed-dotted line), showing that this assumptionproduces significantly more heat exchange than observed.Locations T1 and T2 are shown in Figure 12, and locationsP1 and P2 are shown in Figure 13.


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chemical properties of water discharged at karst springs.Unfortunately, discharge hydrographs are frequently con-trolled by conduit recharge, in which case they carry littleinformation about the conduits [Covington et al., 2009];however, temperature signals potentially carry significantinformation about both the mode of recharge and the sizesof conduits traversed [Liedl et al., 1998; Birk et al., 2006;Luhmann et al., 2011]. Additionally, the frequency, dura-tion, and magnitude of temperature variations in cavestreams likely influence the ecosystems within caves andsurface streams fed by karst springs [Poulson and White,1969; Jegla and Poulson, 1970; Caissie, 2006; Alexanderet al., 2008]. Finally, groundwater temperature can serve asa natural groundwater flow tracer [e.g., Saar, 2011]. In orderto understand the relationships between thermal variations incave streams, aquifer properties, and external forcing, reli-able models for heat transport in karst conduits are needed.

4.1. The Relative Importance of Heat ExchangeMechanisms

[48] Previous models of heat transport in karst conduitsand surface streams have employed a variety of assump-tions. Most crucially, in karst, some models have assumedthat the walls of a conduit are at constant temperature, suchthat heat exchange between water and rock is limited byconvective rates. Models of surface streams have often pre-sumed that convection is so efficient that one can assumethat the streambed is at the water temperature, so that heatexchange is limited by conductive rates. Using analyticalsolutions of conduction in semi-infinite solids with convec-tive boundary conditions, we demonstrate that, under turbu-lent flow conditions, conduction rates limit overall heatexchange rates except at extremely short time-scales, onthe order of seconds (Figure 4), or for conduits/fracturesthat are in the laminar flow regime.

[49] This conclusion is confirmed by the failure of purelyconvective models to reproduce the temperature signalsobserved in Tyson Spring Cave and Postojna Cave using re-alistic conduit diameters (Figure 14). Convective models canreproduce the observed signals relatively well (Figure 15);

Figure 16. Sensitivity of the full heat exchange model tochanges of a factor of two from the best fit values of hy-draulic diameter, DH.Figure 15. Time series of the simulated (solid-black) and

observed (thick solid gray) temperature signals from TysonSpring Cave (a) and Postojna Cave (b), using the purelyconvective model, with constant temperature conduit walls.The hydraulic diameters, DH, used are much larger thanobserved.


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however, this requires hydraulic diameters that are one totwo orders of magnitude larger than the physical conduitdiameters. When conductive effects are included, heatexchange is a function of the timescale and history of tem-perature changes. Therefore, the convective models aremore likely to fail when simulating longer periods with awider range of timescales of temperature changes. While thefits of the convective models using large diameters are fairlygood, both cases show systematic offsets during the finalperiods of the simulations. This effect is larger for PostojnaCave where the timescales of temperature changes span alarger range. In both cases it is not possible to fit the entirecurve with a single choice of DH.

[50] A simplification frequently made in models for sur-face stream temperatures is that of planar symmetry. Whilethis is typically reasonable in the limit of wide, shallowstreams, many full-pipe karst conduits have roughly circu-lar cross-sections, so that calculating conduction througha cylindrical body [as in the work of Birk et al., 2006]may be more appropriate. However, we find that planar-symmetric models closely match the results of cylindricallysymmetric models when the dimensionless number �Z10,where � is a function of conduit radius, flow-through time,and the thermal diffusivity of rock. Specifically, � / R2,such that the planar approximation is better for larger con-duits (Figure 7). Therefore, in many cases, such as the cavestreams modeled in this work, the additional complexity ofcylindrical geometry can be neglected.

[51] When conduits contain open channels, additionalmechanisms for heat exchange arise that have not beenconsidered in previous models of heat transport in karst(Figure 8). A rough estimation suggests that, in most cases,air-convection does not play an important role in determin-ing heat exchanges between water and rock (Figure 11). Onthe other hand, the exchange of long-wave radiationbetween the free surface of the water and the dry portion ofthe conduit wall can have a significant effect. Radiativeexchange typically will bring the water and dry conduitwall into thermal equilibrium on the timescale of days toweeks (Figure 9). Models that neglect radiative exchangein open channel conduits may grossly underestimate thetotal exchange over longer timescales. Additionally, thecoupling of water and rock temperatures via convectionover a timescale of seconds, and via radiation over a time-scale of days to weeks, suggests that long term temperaturevariations in a cave stream could be modeled by simplycalculating conductive heat transport away from the entireconduit cross-section, wet and dry, without a significantloss of accuracy. Figure 17 provides a schematic summaryof the ranges of applicability for each assumption concern-ing heat exchange in karst conduits.

[52] One final mechanism that is not considered here isthe exchange of water between the conduit and the sur-rounding rock or sediments. While inflow from the porousmatrix or fracture networks will certainly act to ‘‘dilute’’temperature signals, it seems unlikely that exchange flow

Figure 17. A summary of the applicability of various approximations in heat transport models for karstconduits.


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will significantly perturb the conduction through the rock,except in extremely porous or fractured rocks where thepropagation of thermal pulses due to flow into the conduitwalls exceeds the rate of heat diffusion via rock conduc-tion. Hyporheic exchange with sediments in a conduit mayalso affect temperature, via either dilution or the introduc-tion of a thicker boundary layer within the sediment at theconduit wall, where some combination of convection andconduction controls heat exchange with the rock [Dogwilerand Wicks, 2005]. Such exchanges are a potentially usefularea of future research, particularly concerning the influen-ces on stream biota.

4.2. Simulation of Observed Signals[53] For both field sites, our mathematical model is able

to produce thermographs that are similar to those observed.The two field sites encompass a range of thermal behaviors.Tyson Spring Cave in Minnesota, is recharged autogeni-cally via a sinkhole plain (Figure 12). At base flow therecharge water is primarily coming from draining soil andthe fracture and pore space within the rock. Consequently,the cave stream is at constant temperature for much of theyear, only responding to large storms or snowmelts that gen-erate runoff from the surface into sinkholes that feed con-duits within the system. In contrast, Postojna Cave, inSlovenia, receives the majority of its recharge from a singlesource, the Pivka River, which has flowed across the surfacefor many kilometers (Figure 13). Therefore, the stream inPostojna Cave undergoes constant temperature fluctuations,with diurnal changes frequently exceeding 1�C. During thesimulated period, diurnal fluctuations are superimposedonto short-term events as well as larger (5�C) week-scalechanges in surface temperature. Our theoretical analysissuggests that the function of various heat exchange mecha-nisms is sensitive to the timescale of temperature variations.Despite this fact, the model closely matches the observedbehavior on all timescales (Figure 14b). It is also importantto note that the constant rock temperature models that weuse for comparison cannot match the observed behaviorwith reasonable choices for conduit properties (Figures 14and 15).

[54] While the mathematical models produce encourag-ing results, the exact hydraulic diameters in our best-fittingmodels are less than the average surveyed diameters. Forthe simulation of Tyson Spring Cave we use DH ¼ 0.3 m,whereas the average surveyed value is DH ¼ 1.4 m. Thediscrepancy for Postojna Cave is smaller with model andsurvey diameters of 1.5 and 2.0 m, respectively. One sourceof error is that the real cave systems have hydraulic diame-ters that vary every few meters over a wide range of values,yet we are representing DH with a single effective value.Since heat exchange is strongly dependent upon DH, the seg-ments with smaller DH may weigh more heavily in deter-mining the effective DH for each simulated reach. Thisseems a likely explanation for the discrepancy at PostojnaCave, as the model value is well within the range of varia-tion in DH in the modeled reach. An additional source ofuncertainty is the thermal properties of the rock. For bothcases we have used properties representative of Salem Lime-stone [Incropera et al., 2007], but these values can only beconsidered approximate since the exact thermal propertiesof the rock at the two field sites were not measured. The

thermal properties of the sediments deposited within eachstreambed are also unknown.

[55] The discrepancy between surveyed and modeled DHfor Tyson Spring Cave is more significant than that of Post-ojna Cave, since the modeled value for DH is near the lowend of the surveyed values along the reach. However, if weconsider the geological setting of Tyson Spring Cave thenanother possible explanation arises. Because of the auto-genic recharge, the stream within Tyson Spring Cave grad-ually gains flow along its length, with base flowapproximately doubling over a 2.25 km length, of whichthe simulated section is roughly a third. This flow is addedvia both discrete infeeding streams and as diffuse rechargethrough the fractured rock matrix. The modeled reach waschosen because it has few visible infeeding streams com-pared to other portions of the cave. However, one signifi-cant infeeder enters the stream with a flux of 5%–10% ofthe mainstream during low flows. The quantity of diffuseinput into the stream along the simulated reach is not well-constrained due to the difficulty of discharge measurementsat T2. Water recharged via a distributed network of smallerflow paths is likely to be near the equilibrium rock tempera-ture. Therefore, if neglected in the model, such inflowswould lead to low values of DH to compensate for the addi-tional thermal damping. Diffuse recharge along the conduitlength could also be a factor in creating the small tempera-ture discrepancy during the recession period, particularlysince diffuse flow will compose a higher portion of the totalflow during recession periods than during peak flow. Sincethe recharge into Postojna Cave is dominated by the PivkaRiver, any contribution from distributed recharge in themodeled reach would have negligible effects on streamtemperature, leading to an accurate value for DH. This rea-soning suggests that cave stream temperature may providea useful means of constraining diffuse recharge compo-nents in cave streams, much as temperature is used in sur-face streams to quantify groundwater exchange. However,it also suggests an uncertainty in estimating conduit diame-ters using temperature signals unless diffuse input can beindependently constrained.

5. Conclusions[56] Using analytical solutions, we have demonstrated

the relative importance of the primary mechanisms for heatexchange in karst conduits, showing that conduction typi-cally dominates under turbulent flow conditions over longtimescales ( t � 1 wk) and that radiative exchange is animportant effect in open channel conduits. Additionally, wehave explored the validity of a number of common assump-tions in models of cave and surface stream temperatures.While many of the same mechanisms influence surfacestream and cave stream temperatures, the dominance ofmechanisms in each setting is in striking contrast. Often,conduction into the streambed is ignored in surface streamtemperature models, because it becomes slow over longtimescales and there are many other paths for heatexchange with the surrounding environment [Caissie et al.,2007]. In karst conduits, rock conduction arises as the pri-mary control on heat exchange over long timescales. Thisresults, again, because heat conduction becomes graduallyless effective over time. Only, in the case of karst conduits,


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essentially all of the heat exchanged must ultimately passthrough the rock via conduction.

[57] Our results can serve as a guide to future models ofkarst heat transport, illuminating proper and improper sim-plifications for a given setting. Using assumptions moti-vated by our theoretical results, we constructed successfulheat transport models in two significantly different cavesettings, illustrating the applicability of the theory. Theresults also suggest that diffuse inflow is an important con-trol on karst conduit water temperatures in some situations.In future work, accurate models of heat transport will allowcontinued examination of the relationships between theconfiguration of preferred flow paths within an aquifer andthe observed thermal signals.

Notation

�r thermal diffusivity of rock (m2 s�1)Ac conduit wet cross-sectional area (m2)Ad Pd/Wfs (unitless)Aw Pw/Wfs (unitless)Bi Biot number (unitless)

cp,r specific heat capacity of rock (J kg�1 K�1)cp,w specific heat capacity of water (J kg�1 K�1)DH conduit hydraulic diameter (m)DL longitudinal dispersivity (m2 s�1)Fo Fourier number (unitless)

f Darcy-Weisbach friction factor (unitless)H Heaviside (step) function

hair air convection heat transfer coefficient (W m�2 K�1)hconv water convection heat transfer coefficient (W m�2

K�1)hrad radiation heat transfer coefficient (W m�2 K�1)

ka thermal conductivity of air (W m�1 K�1)kr thermal conductivity of rock (W m�1 K�1)

kw thermal conductivity of water (W m�1 K�1)L conduit length (m)�w dynamic viscosity of water (kg m�1 s�1)�w dynamic viscosity of air (kg m�1 s�1)va kinematic viscosity of air (m2 s�1)

Nu Nusselt number (unitless)� dimensionless radiation number (unitless)� �wcp;w=ð�rcp;rÞ (unitless)Pd conduit dry perimeter (m)Pw conduit wetted perimeter (m)Pe longitudinal dispersion Peclet number (unitless)Pr Prandtl number (unitless)Q volumetric discharge (m3 s�1)�Q time-averaged or reference discharge (m3 s�1)

q00rad radiative heat flux at rock surface (W m�2)q00cond conductive heat flux at rock surface (W m�2)�a density of air (kg m�3)�r density of rock (kg m�3)�w density of water (kg m�3)R conduit radius (m)

Re Reynolds number (unitless)r� dimensionless radial distance (unitless)� Stefan-Boltzmann constant (J s�1 m�2 K�4)

St stanton number (unitless)� advection and conduction time ratio (unitless)Tr rock temperature (C or K)

Tr,0 initial rock temperature (C or K)

Ts conduit surface temperature (C or K)Ts,d conduit surface temperature (C or K)Tw water temperature (C or K)T�r dimensionless rock temperature (unitless)T 0r T�r � 1 (unitless)T�s dimensionless conduit wall emperature (unitless)

T�s;w dimensionless wetted wall temperature (unitless)T�s;d dimensionless dry wall temperature (unitless)T�w dimensionless water temperature (unitless)

T�w;0 initial dimensionless water temperature (unitless)T 0w T�w � 1 (unitless)

t time (s)t� dimensionless time (unitless)

tconv convection-limited timescale (s)tFo time when Fo ¼ 1 (s)trad radiation-limited timescale (s)

V flow velocity in conduit (m s�1)�V average or reference flow velocity (m s�1)

Va airflow velocity (m s�1)Wfs width of the water free surface (m)

x� dimensionless longitudinal position (unitless)y� dimensionless distance from conduit (unitless)

[58] Acknowledgments. We are grateful for the access provided toTyson Spring Cave by John Ackerman and the MN Cave Preserve. Thework of M. D. C. was supported by National Science Foundation (NSF)Earth Sciences Postdoctoral Fellowship 081647 as well as an NSF Interna-tional Research Fellowship 0754495. A. J. L. is supported by a Doctoral Dis-sertation Fellowship from the University of MN Graduate School. M. O. S.thanks the George and Orpha Gibson endowment and a McKnight Land-grant Professorship for their support of the Hydrogeology and GeofluidsResearch Group, and also acknowledges support from NSF grant EAR-0941666. We also express our gratitude to three anonymous reviewers,whose comments improved the clarity and precision of this manuscript.Any opinions, findings, and conclusions or recommendations expressed inthis material are those of the authors and do not necessarily reflect theviews of the NSF.

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Mechanisms of heat exchange between water and rock in karst conduits

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