Earth-Science Reviews 129 (2014) 40–58

Contents lists available at ScienceDirect

Earth-Science Reviews

j ourna l homepage: www.e lsev ie r .com/ locate /earsc i rev

Heat as a tracer to quantify water flow in near-surface sediments

Gabriel C. Rau a,b,⁎, Martin S. Andersen a,b, Andrew M. McCallum a, Hamid Roshan a,b, R. Ian Acworth a,b

a Connected Waters Initiative Research Centre (CWI), Water Research Laboratory, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, Australiab National Centre for Groundwater Research and Training (NCGRT), Water Research Laboratory, School of Civil and Environmental Engineering, The University of New South Wales,Sydney, Australia

⁎ Corresponding author at: Water Research Laboratory,4188.

E-mail address: gabriel.rau@unsw.edu.au (G.C. Rau).

0012-8252/$ – see front matter © 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.earscirev.2013.10.015

a b s t r a c t

a r t i c l e i n f o

Article history:Received 18 October 2012Accepted 28 October 2013Available online 1 December 2013

Keywords:Heat as a tracerConduction heat transportConvection heat transportTemperature dataHydrogeologyGroundwater hydrology

The dynamic distribution of thermal conditions present in saturated near-surface sediments have been widelyutilised to quantify the flow of water. A rapidly increasing number of papers demonstrate that heat as a traceris becoming an integral part of the toolbox used to investigate water flow in the environment. We summarisethe existing body of research investigating natural and induced heat transport, and analyse the progression infundamental and natural process understanding through the qualitative and quantitative use of heat as a tracer.Heat transport research in engineering applications partly overlaps with heat tracing research in the earthsciences but is more advanced in the fundamental understanding. Combining the findings from both areas canenhance our knowledge of the heat transport processes and highlight where research is needed. Heat tracing re-lies upon the mathematical heat transport equation which is subject to certain assumptions that are oftenneglected. This review reveals that, despite the research efforts to date, the capability of the Fourier-model ap-plied to conductive–convective heat transport in water saturated natural materials has not yet been thoroughlytested. However, this is a prerequisite for accurate andmeaningful heat transport modelling with the purpose ofincreasing our understanding of flow processes at different scales. This review reveals several knowledgegaps that impose significant limitations on practical applications of heat as a tracer of water flow. The reviewcan be used as a guide for further research directions on the fundamental as well as the practical aspects ofheat transport on various scales from the lab to the field.

© 2013 Elsevier B.V. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412. Heat transport theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.1. Heat transport model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.1.1. Conductive heat transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.1.2. Convective heat transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2. Local Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.3. Thermal Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.1. Small-scale dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3.2. Field-scale Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4. Heat versus Hydrometric Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.5. Heat and solute tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.6. Non-Fourier heat transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3. The application of heat as a tracer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.1. Temperature as an indicator for water flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.1. Qualitative assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.1.2. Spatial variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2. Natural heat as a tracer to quantify water flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.1. Steady-state method with a constant temperature boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.2. Transient method with an arbitrary temperature boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.3. Transient method with a sinusoidal temperature boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

The University of New SouthWales, 110 King Street, Manly Vale NSW 2093, Australia. Tel.: +61 2 8071 9850; fax: +61 2 9949

ghts reserved.

41G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

3.3. Advances in natural heat tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.1. Hydrologic process understanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.2. Practical improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.3. Method limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4. Induced heat as a tracer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4.1. Saturated zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4.2. Vadose zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5. Heat tracing with numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1. Introduction

In this review the transport of heat through unconsolidated watersaturated porous materials is investigated with the focus on how touse it as a tracer for water flow in the environment. The theoreticalfoundation for conductive heat transfer calculations was given byFourier (1822) based onNewton's law of cooling. Mathematical descrip-tions of the temperature field caused by a moving fluid through porousmaterials were developed by Schumann (1929) and successfully testedwith different porous materials by Furnas (1930). The original heatconduction equation has been subject to comprehensive mathematicalexploration including the introduction of a moving heat source(Carslaw and Jaeger, 1959). The first analytical solutions for modellingheat transport under the influence of flowingwater in the environmen-tal realm were provided by Suzuki (1960), Stallman (1963, 1965) andBredehoeft and Papaopulos (1965).

The earth sciences research community did not immediately recog-nise the potential offered by heat tracingwhen applied towater saturatednear-surface environments. Instead, temperature was mainly consideredas a variable that primarily influences the water quality (e. g. Blakey,1966). Early tracing applications focused on the effect of heat transportacross the surface interface on subsurface temperatures (Birman, 1969),heat flow through fractures (Bodvarsson, 1969), the characteristics oftemperature depth profiles (Cartwright, 1970; Sorey, 1971), and the hor-izontal redistribution of geothermal heat (Cartwright, 1971). Somewhatlater, Smith and Chapman (1983) comprehensively investigated thecharacteristics of natural heat flow in the subsurface.

The usefulness of near-surface temperature data as an indicatorof natural heat flow were later rediscovered and linked quantitativelyto water flow through the soil zone (e. g. Cartwright, 1974; Lee, 1985;Taniguchi, 1993; Taniguchi and Sharma, 1993) and the streambed(e. g. Lapham, 1989; Silliman et al., 1995) using the heat transportmodel. Increasing research at the earth's surface, streams in particular,was sparked by the recognition that our surfacewater and groundwaterresources are connected and form a single resource (e. g. Winter et al.,1998), and that exchange flows have a crucial impact on the health ofour riparian ecosystems (e.g. Brunke and Gonser, 1997; Woessner,2000; Krause et al., 2011a). The use of natural heat as a tracer hassince become popular. Surface water temperature measurements havebeen used for detecting groundwater discharge zones (e.g. Andersenand Acworth, 2009; Baskaran et al., 2009). Sediment temperaturemeasurements were used for quantifying water fluxes through the sat-urated porous boundary (e.g. the sediments of any surface water body)(e.g. Stallman, 1965; Lapham, 1989; Hatch et al., 2006). The popularityof heat tracing can be attributed to temperature being a robust param-eter that can be measured cheaply and easily by using fully automateddevices (Johnson et al., 2005; Kalbus et al., 2006).

Introductions to forced heat convection in subsurface geologic pro-cesses are given in Ingebritsen and Sanford (1998) and Domenico andSchwartz (1998). Excellent reviews and summaries on heat as an envi-ronmental tracer can be found for groundwater in Anderson (2005),

arid zone recharge in (Blasch et al., 2007) and for streambed water ex-change in Constantz (2008). Furthermore, advanced heat transportmodelling techniques are comprehensively compiled in Nield andBejan (1992) and Kaviany (1995). These works are specific to their re-spective research disciplines, such as groundwater or engineering appli-cations. From this it becomes clear that the use of heat as a tracer in theearth sciences overlaps with the application of heat transport in engi-neering, which has been thoroughly investigated by researchers in therealm of chemical and process engineering. Consequently, additional in-sights into heat transport modelling can be gained from the literaturereporting on engineering research. The overlap between earth sciencesand engineering has been explored very little to date. A summary oftheoretical and laboratory research from the field of engineering hasthe potential for informing the use of heat as a tracer in the earthsciences.

Engineering research hasmainly focussed on heat transport throughbeds packed with porous materials (e.g. Green et al., 1964; Metzgeret al., 2004). The concept is equivalent to unconsolidated sedimentarymaterial studied in the earth sciences. However, porous packed bed ex-periments are better suited for fundamental investigations due to thegood control over the porous materials, fluid and boundary conditions.The main differences are the pure homogeneous materials with well-defined grain geometries (i.e. glass beadswith ideal shapes) in the engi-neering research versus heterogeneous natural materials with variablegrain shapes, sizes and mineralogy in earth sciences research.

In this paper heat as a tracer is reviewedwith the aim to (1) compileknowledge about the capability of the differential heat transport equa-tion and its parameters to model conductive–convective heat transportthrough unconsolidated natural porous materials, (2) summarise ad-vances to date in the development and applicability of methods thatquantify water flow from temperature data, and (3) reveal the gaps inour understanding of the heat transport process at different scales.The scope of this review comprises the quantification of heat andwater transport in unconsolidated near-surface sediments as wellas engineering research employing porous media heat laboratoryexperiments with the aim of understanding the heat transport modelunder different hydraulic conditions. This review paper extends thescope of the recent reviews by Anderson (2005) and Constantz (2008)by: (1) including heat transport research from engineering which pro-vides fundamental understanding useful for earth science research;(2) reporting on the significant progress in near-surface heat tracingthat has been achieved since Anderson (2005) identified this area asa “recent focus” and is now a rapidly growing area of research;(3) reporting on the outcome from “creative deployment of tempera-ture equipment” that Constantz (2008) envisioned as a future directionfor streambed research.

Literature reporting on heat transport in the deeper subsurface is ex-cluded in this review, unless significant contribution to the fundamentalunderstanding of heat transport can be gained. The primary reason forthis is the fact that typically groundwater flow velocities decreasewith depth from the surface (e.g. Winter et al., 1998) hence convective

42 G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

heat transport,which is central to this review, becomes less dominant atlarger depths. For a review of deeper subsurface heat tracing the readeris referred to Saar (2011).

The review first outlines the heat transport theory. A link is thenestablished to laboratory and theoretical investigations in engineering ap-plications. This ensures that research from different overlapping disci-plines (e.g. earth sciences and engineering) is examined in order toidentify the total body of knowledge,merge concepts and identify the po-tential areas for further research. Secondly, the discovery, evolution andusefulness of temperature as an independent physical state variable forthe quantification of water flow in near-surface sediments (e.g. hydrolo-gy, hydrogeology and soil science) are reviewed. The most significantliterature on near-surface heat tracing is listed in Table 1 categorised byresearch aim and further by specific focus.

2. Heat transport theory

Before the heat tracing literature can be analysed a thorough under-standing of the underlyingmathematical equations is required. The fol-lowing section reviews the Fourier–type heat transport model includingparameters which is widely applied to natural porous media.

2.1. Heat transport model

Propagation of heat in porous materials can be described using thedifferential heat transport equation (HTE) that takes into account bothconvective and conductive transport (e.g. De Marsily, 1986; Nield andBejan, 1992; Kaviany, 1995)

∇ � D∇Tð Þ−v �∇T ¼ ∂T∂t ; ð1Þ

where T represents the bulk temperature [°C]; D is the thermal disper-sion tensor [m2/s] (also referred to as effective thermal diffusivity); vis the thermal front velocity [m/s]; t represents the time [s]. Eq. (1) rep-resents a mathematical expression for the modelling of heat transporton the macroscopic scale (e.g. Bear, 1972). Hence, the HTE equation

Table 1Summary of significant and relevant literature reporting on heat as a tracer categorised by rese

Research aim Research focus Refere

Fundamental heat transport Thermal parameters BunteSchär2004a

Small-scale dispersion BonsMetzg

Field-scale dispersion Bear,Molin2009;

Thermal equilibrium De MaFlux quantification: method development Natural heat Brede

Keery1995;

Induced heat AngerLewan

Computer programs Vertical flux calculation GordoMethod comparison Heat and solute Batyc

TanigNatural variability in fluxes Spatial Anger

2011;Swan

Temporal HatchMcCa

Limitations: natural heat as a tracer Method related Carde2011;

Process related AnibaCuthb2010;2012;Schor

assumes a representative elementary volume (REV) large enough thatthe transport can be described with volume averaged parameters. Fur-thermore, Eq. (1) corresponds to the well-known heat conductionequation (Carslaw and Jaeger, 1959), but is extended to account for amoving heat source by the convection term. This is mathematicallyequivalent to a stationary heat source with water moving through a po-rous material assuming a uniform flow field. A further condition for thevalidity of the HTE equation is that temperatures at the boundary inter-face between all phases (solid, liquid and gas) need to be equal which isthe assumption of local thermal equilibrium (e.g. Stallman, 1965; Bear,1972; Nield and Bejan, 1992; Kaviany, 1995; Ingebritsen and Sanford,1998).

2.1.1. Conductive heat transportIn the heat transport Eq. (1) the thermal dispersion coefficient D is

given as (e.g. De Marsily, 1986)

D ¼ κ0ρc

ð2Þ

The thermal dispersion coefficient Eq. (2) is equivalent to ther-mal diffusivity. In this equation κ0 is the bulk thermal conductivity[W/m/°C], ρ is the density [kg/m3] and c is the specific heat capacity[J/kg/°C] of the bulk volume. The resulting specific volumetric heatcapacity is defined as (Buntebarth and Schopper, 1998; Schärli andRybach, 2001)

ρc ¼ nρwcw þ 1−nð Þρscs; ð3Þ

where n is the total porosity of thematerial, ρscs is the specific volumetricheat capacity of the solids [J/m3/°C] and ρwcw is the specific volumetricheat capacity of water [J/m3/°C]. Values for specific volumetric heat ca-pacities of different rocks and rock forming minerals can be obtainedfrom literature (e.g. Schön, 1996; Waples and Waples, 2004a,b).

The effective or bulk thermal conductivity κ0 is of particular impor-tance for heat transport calculations, and consequently empiricalmodels for the calculation of porous media thermal conductivity

arch aim and focus.

nce

barth and Schopper, 1998; Côté et al., 2011; Côté and Konrad, 2009; Revil, 2000;li and Rybach, 2001; Schön, 1996; Tarnawski et al., 2011; Waples and Waples,,b; Woodside and Messmer, 1961.et al., 2013; Green, 1962; Green et al., 1964; Kunii and Smith, 1961; Lu et al., 2009;er et al., 2004; Rau et al., 2012a,b; Yagi et al., 1960; Zeng-Guang et al., 1991.1972; Chang and Yeh, 2012; Dwyer and Eckstein, 1987; Hidalgo et al., 2009;a-Giraldo et al., 2011; Sauty, 1978; Sauty et al., 1982a,b; Vandenbohede et al.,Xue et al., 1990.rsily, 1986; Green, 1962; Kaviany, 1995; Levec and Carbonell, 1985a,b; Rau et al., 2010a.hoeft and Papaopulos, 1965; Conant, 2004; Goto et al., 2005; Hatch et al., 2006;et al., 2007; Lapham, 1989; Luce et al., 2013; McCallum et al., 2012; Silliman et al.,Stallman, 1965; Voytek et al., 2013; Wörman et al., 2012.mann et al., 2012b; Ballard, 1996; Byrne et al., 1967, 1968; Dudgeon et al., 1975;dowski et al., 2011; Ren et al., 2000.n et al., 2012; Swanson and Cardenas, 2011; Voytek et al., 2013.ky and Brenner, 1997; Constantz et al., 2003a; Cox et al., 2007; Rau et al., 2012a,b;uchi and Sharma, 1990; Vandenbohede et al., 2009.mann et al., 2012a,b; Briggs et al., 2012b; Conant, 2004; Jensen and Engesgaard,Lautz et al., 2010; Lautz and Ribaudo, 2012; Rau et al., 2010a; Schmidt et al., 2007;son and Cardenas, 2010; Vogt et al., 2010, 2012.et al., 2006, 2010; Jensen and Engesgaard, 2011; Keery et al., 2007; Lautz, 2012;llum et al., 2012; Rau et al., 2010a; Swanson and Cardenas, 2010; Vogt et al., 2010, 2012.nas, 2010; Hatch et al., 2006; Jensen and Engesgaard, 2011; Keery et al., 2007; Munz et al.,Rau et al., 2010b; Vogt et al., 2012.s et al., 2009; Bhaskar et al., 2012; Briggs et al., 2012a,b; Cardenas and Wilson, 2007;ert and Mackay, 2013; Cuthbert et al., 2010; Ferguson and Bense, 2011; Hatch et al.,Jensen and Engesgaard, 2011; Lautz, 2010, 2012; Lautz et al., 2010; Lautz and Ribaudo,Munz et al., 2011; Rau et al., 2010a, 2012a,b; Roshan et al., 2012; Schmidt et al., 2007;nberg et al., 2010; Shanafield et al., 2010, 2011; Vogt et al., 2010.

43G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

have been developed. For instance,Woodside andMessmer (1961) sug-gested a simplemodel for a two-phasemediumwith a randomdistribu-tion (i.e. water saturated sand), whereby the bulk thermal conductivityis the geometricmean of both phasesweighted by the volume fraction n(total porosity)

κ0 ¼ κnw � κ1−n

s ; ð4Þ

where κw is the thermal conductivity of water [W/m/°C] and κs is thethermal conductivity of the solids [W/m/°C].

General theoretical bounds of κ0 were defined for porous media(Hashin and Shtrikman, 1962; Carson et al., 2005) and the model wasfurther extended to account for various degrees of water saturation(Chen, 2008). Thermal conductivity values for different rock types androck forming minerals were summarised in Horai and Simmons(1969), Horai (1971) and Clauser and Huenges (1995).

Natural sediments are typically mixtures of different minerals andgrain sizes. A positive logarithmic correlation between grain size andthermal conductivity was found for sediment samples mixed using dif-ferent minerals (Midttomme and Roaldset, 1998). The isotropic bulkthermal conductivity could be predicted by taking into account a grainshape factor and a correlation between solid and fluid conductivity andporosity (Revil, 2000). Similarly sophisticated theoretical models havebeen developed more recently (e.g. Samantray et al., 2006; Reddy andKarthikeyan, 2009). In particular, Côté and Konrad (2009) investigatedthe effect of material structure on the thermal conductivity and foundthat effects of structure can be neglected for thermal conductivity ratiosof κs/κw b 15. This ratio is unlikely to be exceeded in hydrogeological ap-plications considering a solid thermal conductivity value of 8 W/m/°C forpure quartz as the highest value for natural materials.

The accuracy of the geometric mean model was confirmed experi-mentally for sands under saturated conditions (Tarnawski et al.,2011). Crushed granite mixtures featuring distinctly different grainsizes, as representative of natural grain size mixtures, did not changethe saturated bulk thermal conductivity appreciably (Côté et al.,2011). These researchworks suggest that the geometric mean equationis well suited for the calculation of thermal conductivity of water satu-rated natural sand, at least when its mineral content does not vary toomuch.

Thermal conductivity on scales from the pore to the sedimentarybasin was subject of discussion in Bachu (1991) with particular focuson the effect of heterogeneities. The work concluded that the thermalconductivity can best be described for all scales by using a generalisedweightedmeanmodel taking into account an estimate of fractional vol-umes occupied by differentmaterials (heterogeneity fraction) and theirconductivity contrast. Another interesting large scale investigation con-ducted byMarkle et al. (2006) quantified a two-dimensional conductiv-ity field in a sand and gravel aquifer and concluded that the thermalplume spreading in a heterogeneous versus a homogeneous (spatiallyaveraged) thermal conductivity field would cause spatial temperaturedifferences that are smaller than 1 °C.

2.1.2. Convective heat transportIn the HTE (Eq. (1)) the thermal front velocity becomes (e.g. De

Marsily, 1986; Batycky and Brenner, 1997)

v ¼ ρwcwρc

q ¼ ρwcwρc

nevs; ð5Þ

where q is theDarcy velocity (or specific discharge) of themovingwater[m/s]; ne is the effective porosity. Here it is important to point out thatthe effective porosity (ne) is used instead of the total porosity (n). Thisis because it quantifies the void ratio affected by flowing water and ex-cludes the dead end pores contrary to the total porosity. It has beenpointed out that there is competition between the conductive and con-vective mechanism in the pore channels (Bons et al., 2013), which

suggests that the pore space participating in active flow depends onthe flow magnitude. The heat transport literature does not distinguishbetween these two presumably because the effective porosity is verydifficult to quantify and depends on the material's pore structure. Inmost cases both values can be assumed equal.

Further to Eq. (5), we use the term convection instead of advectionto distinguish the heat transport process from solute transport. Aspointed out by Anderson (2005) both terms are used in the literatureto describe heat transport. The engineering literature is more stringentand refers to heat transport induced bywater flow due to external pres-sure gradients as forced convection. In contrast, heat transport driven bydensity differences is termed free convection. The latter is excludedfrom this review.

Note that the thermal front velocity (Eq. (5)) is retarded compared tothe pore water or interstitial velocity vs for sediments since neρwcw=ρc isgenerally smaller than 1 (e.g. Bodvarsson, 1972; Oldenburg and Pruess,1998; Geiger et al., 2006).

Relative contributions of conductive versus convective heat flow canbe characterised using the thermal Péclet number, which is defined as(e.g. De Marsily, 1986; Anderson, 2005)

Pe ¼ ρwcwκ0

qL ð6Þ

with characteristic length L [m] overwhich heat transport is considered.L is often set equal to average grain size diameter in order to generalisethe relative contributions of convective and conductive heat transport.Transport is convection dominated for Pe N 1 and conduction dominat-ed for Pe b 1.

2.2. Local Thermal Equilibrium

It is often assumed that there is no temperature difference betweensolid and liquid for earth science applications in sedimentary environ-ments. The condition is called local thermal equilibrium and it is inher-ent to the HTE (Eq. (1)). This assumption allows the separate transportequations for each phase to be expressed as one (Eq. (1)) and the ther-mal properties of the solid and the fluid phases can be averaged(Eqs. (3) and (4)). Alternatively, a physically more correct approach isto model simultaneous heat transport through the two phases. Thetwo separate equations for fluid and solids can then be coupled by aterm accounting for thermal transfer through the boundary betweenthe two phases (e.g. Green, 1962; Wakao and Kagei, 1982; Levec andCarbonell, 1985a; Kaviany, 1995). A comprehensive engineering per-spective on heat transport in porous media can be found in Kaviany(1995).

Justifying the simplification from two equations to one requires de-tailed analysis of the inherent limitations. Levec and Carbonell (1985a)theoretically investigated the two equation model along a flow path as-sumingmonodispersed particles and found that the propagation veloc-ity of a heat pulse initially differs for both phases until it becomes equalafter some time. As a result, the temperature responsesmeasured in thesolid and the fluid phases exhibit an increasing but finite time delay de-pending on the contrast in thermal properties and the particle diameter.These mathematical considerations were verified by laboratory experi-mentation measuring thermal breakthrough curves separately in thetwo phases (Levec and Carbonell, 1985b). Their work contains valuableconclusions justifying the one equation approach (Eq. (1)) and volumeaveraging of thermal properties; in particular for the slow flow veloci-ties encountered in earth sciences applications.

In natural environments heat transport occurs through materialswith variable characteristics such as mineral composition, grain sizedistribution and texture. The difference between a natural porousmedium with homogeneous and heterogeneous phase distribution isconceptualised in Fig. 1. Heat transport through both phases is simpli-fied by volume averaging the thermal properties of the individual

REV

heterogeneousmedium

scal

e

homogeneousmedium

REV

d50

d50

REV

system length

REV

system length, e.g. volume

REV

heterogeneousmedium

homogeneousmedium

Domain ofmicroscopic

effects

Domain ofporous medium

phys

ical

pro

pert

y, e

.g.p

oros

ity n

Fig. 1. Conceptual difference between natural porous media with homogeneous and heterogeneous phase distributions, and any physical property (e.g. porosity) in relation to the repre-sentative elementary volume (REV) (e.g. Bear, 1972).

44 G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

phases (for details refer to the Section 2.1). This assumes the existenceof an appropriate representative elementary volume (REV as explainedin Bear, 1972) which has not yet been clearly developed as a conceptfor heat transport in natural porous media.

For convective heat transport, when thefluid is inmotion, differentlysized grains will take different times to equilibrate thermally and conse-quently the suitability of the volume averaging approach to appropriate-ly describe the heat transfer process is complex. It is anticipated that theassumption of local thermal equilibrium is increasingly violated for in-creasing pore water velocities, to the extent that eventually the heattransport model (Eq. (1)) must fail as an accurate description of thephysical process. Whether or not this is likely to occur for flows innatural sediments within the Darcy range remains unknown and couldtrigger some interesting research questions.

2.3. Thermal Dispersion

The hydrodynamic thermal dispersion process has traditionallybeen assumed analogously to solute transport when applied to naturalsystems and a similar mathematical description is used (e.g. Bear,1972; De Marsily, 1986; Anderson, 2005). The thermal dispersion coef-ficient (also referred to as effective thermal diffusivity) D1,t used in theheat transport Eq. (1) is given as (e.g. Bear, 1972; De Marsily, 1986)

Dl;t ¼κ0

ρcþ βl;t

ρwcwρc

q����

����; ð7Þ

where subscripts l and t denote longitudinal and transverse directions,respectively. The first part of Eq. (7) is the thermal diffusivity, whilethe second part described thermal dispersion due to convection. The pa-rameter β1,t represents the longitudinal and transverse thermaldispersivity coefficient [m]. The significance of thermal dispersivityand its contribution to the overall process of heat spreading at differentspatial scales has long been disputed in the literature (e.g. Sauty, 1978;DeMarsily, 1986; Anderson, 2005; Hatch et al., 2006; Keery et al., 2007;Vandenbohede et al., 2009; Rau et al., 2010a). The following sections re-view the literature and summarise the findings regarding thermaldispersion on different scales.

2.3.1. Small-scale dispersionIt was recognised early that a flowing fluid through randomly ar-

ranged particles accelerates the spread of solute (e.g. Bear, 1961;Scheidegger, 1961; Saffman, 1959). In heat transport the analogousphenomenon of heat dispersion has received less attention. The ques-tion how dispersion depends on the flow velocity is addressed eithertheoretically or experimentally bydetermining an empirical correlation.The sum of conductive and convective dispersion (here D as in Eq. (7))

is often called effective thermal conductivity in the engineering researchliterature.

Yagi et al. (1960) used glass beads to quantify the thermal dispersionfor different flow rates. They found that thermal dispersion increaseslinearly with flow velocity, but their experimental dataset containedtoo much scatter for reliable interpretation. Similarly, Kunii and Smith(1961) experimented with sand at flow rates comparable to thosefound in natural settings, but the results were inconclusive regarding avelocity relationship. The first meaningful laboratory investigation wasconducted by Green (1962) for various flow rates through different di-ameter spheres. His experimental results clearly demonstrated that thethermal dispersion velocity relationship can be described with a powerlaw of the form

Dl

D¼ n·

vsd50D

� �m

; ð8Þ

where n and m are empirical coefficients (Green et al., 1964).The first sound theoretical expression for a thermal dispersion

model in ideal porous media was derived by Zeng-Guang et al. (1991).They concluded that for laminar flow the thermal dispersion reducesto thermal diffusivity (cf. first term in Eq. (7)) plus a term accountingfor additional spreading due to flow field variability on the pore scale.The latter term is complicated and was found to depend on a numberof parameters: the square of the Péclet number, the porosity of the ma-terial, the ratio of the thermal conductivity and heat capacity and thetemperature pattern. They also could not define a general dispersion-velocity relationship. The most interesting result of this study is theassessment of pore space heterogeneity, as is inherent to all naturalporous media, and the conclusion that the dispersion coefficient canvary significantly (Zeng-Guang et al., 1991).

The study formed a basis for much more detailed and complex ana-lytical and numerical investigations (e.g. Xue et al., 2010; Hsiao andAdvani, 1999; Yang and Nakayama, 2010). However, these are theoret-ical models and it is questionable whether the conclusions can be ex-tended to practical macroscopic transport models in natural sedimentswithout experimental verification.

More recently, the thermal dispersion velocity relationship wasexperimentally investigated by Metzger et al. (2004) for glass beads.The apparent longitudinal and lateral dispersion coefficients were cal-culated from fitting an analytical solution to measured temperaturebreakthrough curves for convection dominated transport conditions.However, their experimental conditions, such as flow velocities,exceeded those typically encountered in hydrogeology. Yet the empiri-cal power law, as determined by Green et al. (1964), was verified intheir work.

45G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

The only three-dimensional laboratory study using naturalmaterialswas conducted byKimet al. (2005). However, they used heat as an anal-ogy to solute transport in order to calculate the location of a contamina-tion source from temperature measurements at discrete spatial points.They used a procedure to find the best fit between measurements anda 3D analytical model including one longitudinal and two lateral ther-mal dispersion coefficients. They found the longitudinal to be largerthan both lateral dispersion coefficients (Kim et al., 2005). This shouldbe considered when flow velocity is calculated from temperature data.

Lu et al. (2009) used a heat pulse probe installed in a soil column toquantify the longitudinal thermal dispersion-velocity relationship forthree different soil types. Their results best matched a relationshipwhereby the dispersivity coefficient is related to the velocity by apower law with different coefficients for the different materials. Theirstudy highlights that thermal dispersion is a function of the materialtexture (i.e. grain size distribution).

In a comprehensive laboratory investigation, Rau et al. (2012a) com-pared the sub-metre scale dispersive behaviour of solute and heat forDarcian flows (Re b 2) through natural sand with a homogeneousgrain size distribution. They systematically investigated earlier specula-tions on the difference between solute and heat transport in porousma-terials, namely (1) that the magnitude of conductive heat transport isorders of magnitude faster than solute diffusion (e.g. Bear, 1972; DeMarsily, 1986; Ingebritsen and Sanford, 1998; Anderson, 2005), and(2) that the convective transport of heat is slower than that for solutesbecause the thermal capacity of the solids retard the advance of thethermal front (e.g. Bodvarsson, 1972; Oldenburg and Pruess, 1998;Geiger et al., 2006). This results in different Péclet numbers for heatand solute (e.g. De Marsily, 1986), which has the consequence that forthe same Darcy velocity solute and heat transport are within differenttransport regimes (see Fig. 2, published in Rau et al., 2012a). For exam-ple, at the Darcy velocity where advection dominates the solutetransport, conduction will dominate the heat transport mechanism.Consequently, thermal dispersion is much less significant than solutedispersion when heat transport is quantified. Most importantly, how-ever, the heat transport mechanism can be described by the samefundamental dispersion relationship previously established for solutedispersion.

10−2 10−1 10010−2

10−1

100

101

102

103

104

Nor

m. D

ispe

rsio

n C

oeffi

cien

t Dl/D

[−]

Thermal or Solu

Darcy Velocity [m/d]

100 101 102

Experimental Heat Transport Range

Darcy Range (Re < 2.5)

Fig. 2. Experimental dispersion data on the sub-meter scale for heat and solute transport in natconceptual graph adapted from Bear (1972).

As a result of their findings, Rau et al. (2012a) propose a new empir-ical dispersion formulation for transition zone heat transport on thesub-metre scale

Dl;t ¼κ0

ρcþ βl;t

ρwcwρc

q� �2

; ð9Þ

where β1,t is a dispersion coefficient [1/s] that must be determined ex-perimentally. The square relationship is only required for flows thatcan be considered fast in the realm of the earth sciences; however,this depends on the dominant grain size (e.g. q N 10 m/day for grainsize d50 = 2 mm). They further noted that thermal dispersion couldbe higher in materials with different grain size distributions and thatmore experiments are required to reveal the influence of materialtexture. Furthermore, a direct comparison between heat transport inideal and natural materials has not yet been conducted and could deliv-er interesting insight into the transport mechanism.

For the first time, Bons et al. (2013) unified both solute and heat dis-persionmodels into onemathematical framework under the hypothesisthat transport of both mechanisms (convection and conduction) com-pete in the pore channels. Their model defines the “critical Péclet num-ber” as the point at which convection starts to dominate overconduction as a material dependent parameter. The authors confirmthat further experimentation is required to elucidate the relationshipbetween material texture and dispersion. The above discussion rein-forces that thermal dispersion can be of significance and its role shouldbe investigated further to give a better basis for heat transport calcula-tions in applications relating to sediment heat tracing.

2.3.2. Field-scale DispersionA first mention of field-scale hydrodynamic thermal dispersion can

be found in Bear (1972). However, few field studies have consideredthermal dispersivity (β1,t in Eq. (7)) and reported actual values (Sauty,1978; Sauty et al., 1982a,b). De Marsily (1986) summarised the resultsfrom the different investigations and noted that thermal dispersivityshould be five times weaker than solute dispersivity, but also pointedout that there is a lack of field studies.

101 102 103 104

te Peclet Number

Darcy Velocity [m/d]

100 101 102

Darcy Range (Re < 2.5)

Experimental Solute Transport Range

Heat Experimentation (Rau et al., 2012a)Thermal Dispersion Model (Equation 9)Heat Experimentation (Green et al., 1964)Solute Experimentation (Rau et al., 2012a)Dispersion Concept (Bear, 1972)

ural homogeneous sand (d50 = 2 mm) as determined by Rau et al. (2012a) plotted on the

46 G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

Xue et al. (1990)modelled energy storage in an aquifer and conclud-ed that it is necessary to consider the hydrodynamic dispersion process.Molson et al. (1992) used dispersivity values determined from soluteexperiments but noted that appropriate thermal dispersivity valuesare an order of magnitude lower and attributed that to thermal trans-port occurring much faster compared to that of a leachate plume.Vandenbohede et al. (2009) performed borehole push–pull tests withheated water and noted that the thermal dispersivity is much smallerthan the solute dispersivity and that it seems independent from thescale of investigation. It was further noted that consideration of thermaldispersivity is important in particular under conditions of increasingflow velocity when convective transport becomes more important.

A number of theoretical studies involving hydrodynamic thermaldispersion have been conducted for different purposes. Dwyer andEckstein (1987) modelled thermal energy storage in aquifers andfound that the dispersivity has a significant impact on the energy recov-ery factor. The effect of a heterogeneous hydraulic conductivity field onheat dispersion in 3D was investigated numerically by Hidalgo et al.(2009). They concluded that transverse dispersion is of importancewhen the accurate geometrical extent of the thermal plume is of inter-est. They further suggested a simple dispersion model based on theirmodelling results. Molina-Giraldo et al. (2011) analysed how thermaldispersion, either scale dependent or expressed as a power law, can in-fluence calculated temperature plumes in the subsurface. They foundthat dispersion effects can be significant, but that they need to be eval-uated for each case based on the hydrogeological conditions, such asaquifer heterogeneity and ambient groundwater flow. Chang and Yeh(2012) stochastically analysed the influence of heterogeneity in the hy-draulic conductivity on the field-scale convective thermal transport.Subject to their assumptions they illustrated that the longitudinal andtransverse dispersivity coefficients increase with travel time beforereaching an asymptotic value. Their study also shows that the longitudi-nal dispersivity increases with rising Péclet numbers that indicate atransition between conduction and convection (Pe ≈ 1). Collectivelythese works highlight that further experimental work is required sothat the theoretical conclusions can be supported by real physical data.

To date many researchers still question the significance and behav-iour of field-scale hydrodynamic thermal dispersion (e.g. Anderson,2005; Vandenbohede et al., 2009; Rau et al., 2012a). From the discussionregarding the characteristics of naturalmaterials (i.e. different grain sizedistribution and texture as illustrated by Fig. 1 in Section 2.2) it can be

10-2

10-3

10-1

100

101

102

103

Hydraulic Conductivity K [cm/day]

Thermal Conductivityκ [W/m/°C]

Textural Range

Con

duct

ivity

: The

rmal

[W/m

/°C

] or

Hyd

raul

ic [m

/s]

Increasing Grain Size

Fourier Q = -κ·ΔT

Darcy q

= -KH

Fig. 3. The range in thermal and hydraulic conductivity (vertical axis) of natural sedimentsas a function of grain size (adapted from Stonestrom and Constantz, 2003).

perceived that heat dispersion in natural porousmedia becomes a ques-tion of the flux magnitude as well as sediment heterogeneity (Sautyet al., 1982a,b; Vandenbohede et al., 2009). On scales of several metresthe effect of material heterogeneities and the permeability field hasbeen found to significantly influence the temperature field calculatedfrom heat transport (e.g. Sauty et al., 1982b; Ferguson, 2007;Schornberg et al., 2010; Ferguson and Bense, 2011). Interestingly, theconclusions from small-scale findings (e.g. Rau et al., 2012a) are consis-tent with those from field scale investigations. However, there is a lackof field-scale investigations and further research is required to advanceour fundamental understanding of the significance and characteristicsof thermal dispersion under heterogeneous conditions (Ferguson,2007), and at different scales.

2.4. Heat versus Hydrometric Methods

Traditional hydrometric groundwater flow assessment methods re-quire water levels and hydraulic conductivity estimates. The sedimenthydraulic conductivity is generally a function of the sediment grainsize distribution (e.g. Kozeny, 1927; Bear, 1972; Côté et al., 2011), butalso depends on the temperature due to the change in water densityand viscosity (e.g. Muskat, 1937). Fig. 3 illustrates the advantage ofusing heat as tracer to quantify water fluxes. It can be seen that thedependence of thermal conductivity on sediment texture ismuch betterconstrained than the hydraulic conductivity. Furthermore, thermalconductivity does not depend on the grain size unlike the hydraulicconductivity (Côté et al., 2011). This promises to provide improved pre-dictive certainty andmore accurate flow calculations when heat is usedas a tracer compared to a hydrometric approach (Stonestrom andConstantz, 2003; Kalbus et al., 2006; Blasch et al., 2007; Constantz,2008). However, compared to Darcy's law the water flow derived fromtemperature data requires consideration of both the conductive andconvective thermal transport according to the HTE equation (Eq. (1)).

In order to solve for water flow using heat as a tracer the heat trans-port Eq. (1) requires physical variables specific to the sediment in additionto the thermal conductivity, such as porosity and specific volumetric heatcapacity. Although heat capacities of natural materials are also wellconstrained, their natural variability have the potential to introduce addi-tional uncertainty to velocity calculations derived from temperature data(refer to Eq. (5)). The impact of thermal parameter uncertainty on veloc-ity results depends on the analytical solution in use and has only beensparsely investigated. However, it has been recognised that uncertaintyin thermal parameters can cause large velocity errors especially underconduction dominated heat transport conditions (e.g. Constantz et al.,2003a; Vandenbohede and Lebbe, 2010a; Shanafield et al., 2011).

Stallman (1965) suggested that it is possible to detect water move-ment as low as 2 cm/day using his analytical solution with reasonableassumptions regarding thermal parameter uncertainty. However,many authors use literature values for thermal variables in which casethe uncertainties arising from the different variables must be combinedin order to calculate the uncertainty in the velocity quantification. It hasbeen illustrated that the uncertainty diverges as the velocity decreases,and that the limit of detection is a question of statistical confidence inthe results (Rau et al., 2010b). The detection limit is also influenced bythe fact that the relative contribution to temperature changes causedby convective transport is harder to distinguish from the conductivecontribution as the velocity decreases. The lower velocity detectionlimit is therefore also a function of temperature measurement accuracy(Shanafield et al., 2011) and discretisation in both space and time (Rauet al., 2010a; Soto-López et al., 2011). The above discussion suggeststhat it is important to use high quality temperature equipment withbest possible resolution in order to maximise the confidence in heattracing results at low velocities. Furthermore, it should be acknowl-edged that a velocity threshold exists below which the differentialheat transport Eq. (1) is incapable of quantifying a velocitywith reason-able certainty. Further effort is required to determine a conditional

47G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

limitation when heat as a tracer is used to quantify low flows and per-haps how this is affected by variable saturation.

2.5. Heat and solute tracing

Heat transport is fundamentally different from solute transport inporous materials. Combining both tracers can therefore be a powerfultool for increased process understanding. Both tracers can be reducedto the same mathematical equation assuming Fick-type transport(De Marsily, 1986). Batycky and Brenner (1997) have theoretically in-vestigated the similarity between solute and heat transport. Theycommented that solute transport is the equivalence of two phase heattransport, but with one phase muted for the conductive and convectivetransport mechanism. A direct comparison between both tracers underthe same conditions could be useful to improve our understanding ofthe behaviour of flow fields and transport mechanisms in porousmedia on the microscopic scale. If the microscopic tracer transport be-haviours are well known this knowledge could be used for a better de-scription of themacroscopic properties of the solid matrix. Surprisingly,little work has been done using heat and solute to characterise flow andtransport processes. However, a few investigations demonstrate thepower of combining heat and solute transport.

Taniguchi and Sharma (1990) investigated water percolationthrough two different natural materials under unsaturated conditions.Their laboratory column experiments demonstrated reasonable agree-ment between percolation rates calculated from the propagation oftemperature and solute concentration peaks recorded at differentdepths. The authors noticed the different speeds of percolation betweensolute and heat transport caused by the different transportmechanisms.

Constantz et al. (2003a) used temperature measurements to cali-brate a model for near stream sediments hydraulic conductivity values.These were then implemented in a solute transport model which suc-cessfully predicted concentrations of a solute tracer. The authors notedthat there is a substantial difference arising from the difference in trans-port mechanisms making interpretation of temperature data difficult,especially when preferential flow paths are present. They furthercommented that even though the thermal dispersivity is smaller thanthe solute dispersivity, it should not be neglected in the calculations.

Heat and chloride were used to investigate the dynamics of streamwater exchange with groundwater by Cox et al. (2007). They used nu-merical modelling to show that streambed hydraulic conductivitiesvary due to clogging and scouring as well as seasonal and annual tem-perature changes. An important conclusion from this research demon-strated that combining two independent tracers offered additionalinsight into the water exchange dynamics.

Vandenbohede et al. (2009) investigated the characteristics of soluteversus heat transport by subsurface push–pull tests. The study focusedon the determination of aquifer parameters, namely porosity anddispersivity for both solute and heat. The work was not able to resolvethe question whether the assumed mathematical analogy betweenthermal and solute dispersivity is appropriate. However, it did highlightthat aquifer heterogeneity has a dissimilar effect on solute and heattransport because of the differing propagation mechanisms.

The above research works indicate that more insight into transportmechanisms and its relations to the porous media characteristicscould be gained by simultaneously applying solute and heat tracers(i.e. identical conditions). As an example, the results by Rau et al.(2012a) point out that heat and solute tracers have their strengths in dif-ferent flow regimes. For example, the capability of heat as a tracer forflow is limited at the lower end of the velocity scale. This limit is reachedwhen the part of the temperature signature caused by convection is sosmall that it cannot be distinguished any more from conduction andthe velocity estimation as a consequence gradually becomes inaccurate(Rau et al., 2010b). This velocity limitmust be orders ofmagnitude fasterthan that for solute transport as heat conducts quicker than solutediffuses. This suggests that solute tracing is more robust at slow flows.

2.6. Non-Fourier heat transport

Some theoretical studies are of particular importance for heat trans-port applications in the earth sciences where the Fourier-type modelfails to represent the physical reality. These cases include heterogene-ities, for example fractured or double porosity materials. A few impor-tant studies investigating such cases are mentioned here as a furtherreference. Chevalier and Banton (1999) demonstrated that the randomwalk technique is capable of describing transport through fractured sys-tems. Emmanuel and Berkowitz (2007) adapted the continuous time ran-dom walk (CTRW) approach, originally designed for solute transport(Berkowitz et al., 2006), to heat transport in porous media. They notedthat this method is able to correctly account for thermal disequilibriumas well as material heterogeneity. Geiger and Emmanuel (2010) numer-ically analysed heat transport in a fractured porous medium using CTRWand concluded that the transport behaviour becomes Fourier-like whenthe fracture network is highly connected. This confirms that Fourier-like heat transport behaviour should apply to natural unconsolidatedsediments. The capabilities of the new CTRWmodel have yet to be testedby experimentation in the laboratory or the field.

3. The application of heat as a tracer

In this section the research on near-surface heat transport for calcu-lating water fluxes are summarised. The focus is on Fourier-type heattransport. Thus heat flow in fractures, high temperature gradients,non-Fourier transport, buoyancy effects and hydrothermal convectionare not included in this review.

3.1. Temperature as an indicator for water flow

3.1.1. Qualitative assessmentsFrom a hydrogeological point of view the streambed forms an impor-

tant part in the overall heat balance for shallow systems (Sinokrot andStefan, 1993) which can be quantified using sediment temperatures(Prats et al., 2011). Lee (1985) developed a tool that could identify pointsof groundwater discharge in lake beds using temperature anomalies. Theuse of sediment temperature time records to identify losing and gainingpoints along a river was qualitatively investigated by Silliman and Booth(1993). They concluded that temperature measurements at differentdepths provide an excellent indicator for water flow. Constantz (1998)found that diel variations in surface water and sediment temperaturesare greater for losing than for gaining reaches. Fig. 4 summarises thequalitative relationship between water and sediment temperatures forgaining and losing stream conditions. As an interesting feedbackmecha-nism thewater flow rate to the subsurfacewas found to be influenced bythis daily change in water temperature via its impact on water viscosityand density and consequently the hydraulic conductivity. This resultedin observable daily variations of the aquifer recharge and stream flow(Constantz et al., 1994; Ronan et al., 1998).

Similar investigations delivered evidence that variations in the an-nual or diel vertical temperature envelopes in sediments or streambedscan indicate up or downward flow and provide valuable informationabout temporal changes in the flux through near-surface sediment sys-tems (Cartwright, 1974; Lapham, 1989; Baskaran et al., 2009). It wasfurther observed that diel temperature fluctuations measured at differ-ent depths exhibit a distinct phase shift (Evans and Petts, 1997).

In arid and semi-arid environments temperature time series mea-sured at different depths of the sediment were successful in accuratelymonitoring ephemeral stream flow characteristics. These studies foundthat sediment temperature time series are qualitatively and quantitative-ly useful to describe percolation characteristics (Constantz and Thomas,1997), and that times with active channel flow as well as water percola-tion depth could accurately be determined (Constantz and Thomas,1996; Constantz et al., 2001). The percolation rate was estimated usingthepenetration rate of the thermal front andoffered only a crude estimate

GAINING REACH

TIME

TIME

TE

MP

ER

AT

UR

E

FLO

W

StreamGage

LOSING REACH

TIME

TIME

TE

MP

ER

AT

UR

E

FLO

W

StreamGage

Fig. 4. Illustration of the temporal temperature variations in the stream and streambed sediment as a response to diel temperature changes as a function of losing (recharge) or gaining(discharge) conditions in the stream (illustration adapted from Stonestrom and Constantz, 2003).

48 G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

compared to seepage measurements. The discrepancy was possibly duetomeasurement locations not being the same for the twomethods. How-ever, the methodology was successfully automated by applying a statisti-cal technique to thermal records offering a reliable method fordetermining stream flow timing and duration (Blasch et al., 2004). Anearly summary of qualitative heat field studies is given in Stonestromand Constantz (2003).

3.1.2. Spatial variabilityA comprehensive summary of surface water temperature research

with focus on spatial heterogeneity and impact on ecology is reportedin Webb et al. (2008). While Winter et al. (1998) provide an excellentgeneral overview of the spatial variability in river and groundwater in-teractions, more recent research shows that the temperature differencebetween the surface water and the streambed can indicate magnitudeand direction of exchange flow. For example, it was found that the allu-vium acts as storage body for thermal energy resulting in complex tem-perature patterns on both temporal and spatial scales (Arrigoni et al.,2008); that these patterns are highly influenced by groundwater surfacewater exchange (Malard et al., 2001); and that hyporheic temperaturepatterns are different for shallow alluvial and deeper groundwater dis-charge (Malard et al., 2002).

Spatially variable temperature patterns were also successfullycorrelated to independent flux measurements using seepage metres

940

Dis

tan

cem

940

950

Dis

tan

cem

1000 1010 1020 1030 1040 10

Distance m

A Streambed Temperatu

B Fluxes: Analytical Solu

Channel Boundary

River FlowDirection

930

950

Fig. 5. (A) Spatial streambed temperature variability over 60 m of a stream; (B) Estimated vertiSpatial variability in streambed water fluxes at the metre scale (Angermann et al., 2012a).

(Alexander and Caissie, 2003). Schmidt et al. (2006) investigated verti-cal streambed temperature profiles along a reach and concluded thattemperatures could well be correlated to vertical flow through thesediment. Conant (2004) and Schmidt et al. (2007) measured the hori-zontal distribution of sediment temperatures on a short section of ariver and found significant spatial variability. These temperature mea-surements are illustrated in Fig. 5A. More recent research investigatedthe sediment thermal dynamics for a regulated stream (Gerecht et al.,2011) and in pool rifle sequences (Hannah et al., 2009). Collectively itwas found that there is significant spatial heterogeneity caused by sur-face temperature fluctuations even for up-welling conditions (Krauseet al., 2011b). The velocity fields in studies of streambed sections atthe scales of 10s of metres and even down to 1 s of metres revealedlarge spatial variability (Fig. 6c, Angermann et al., 2012a,b; Lautz andRibaudo, 2012).

Advances inmeasurement technology enable high spatial resolutionsensing of temperature data either remotely or locally. This facilitatesspatially distributed quantification of groundwater discharge (Schuetzand Weiler, 2011) as well as large scale investigations to establishgood correlations between surface temperatures and water exchangeactivity (Atwell et al., 1971; Loheide and Gorelick, 2006; Pfister et al.,2010). Moreover, distributed temperature sensing (DTS) using fibreoptic cable technology opened up new possibilities regarding spatiallydistributed temperature monitoring (Selker et al., 2006a,b). DTS has

0

2

3

4

5

6

8

0

25

50

100

400

50

res

tion

T °C

Recharge

Discharge

Flux Lm d-2 -1

C Total Effective Flux

D Vertical Flux Component

cal exchange fluxes using a steady-state temperaturemodel (Schmidt et al., 2007). (C & D)

Annual (daily) streambed temperature range

Stream

Downward flux

Upward flux

Z=10 m (or 0.5 m)

DE

PT

H (

Z)

TEMPERATURE

Maximumannual (daily)temperature

Minimumannual (daily)temperature

Fig. 6. Concept of annual or daily sediment temperature depth envelopes with dependen-cy on vertical water flow (adapted from Stonestrom and Constantz, 2003).

49G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

been deployed to identify groundwater discharge zones (Westhoff et al.,2007); observe that zoneswith significant exchange flowdo not changeposition over time (Lowry et al., 2007); and investigate general heattransport characteristics in the stream environment (Westhoff et al.,2010, 2011a,b). Comparing the usefulness of DTS with conventionalmethods to determine groundwater discharge to streams, Briggs et al.(2012a) found that DTS employed in a tightly coiled arrangementprovides the highest spatial resolution. The capability of estimatinggroundwater discharge from distinct discharge points using DTS wassystematically investigated with the result that it is reliable when thecontrast between groundwater and surface water temperature exceedsa certain threshold (Lauer et al., 2013). Furthermore, caution should beapplied when using DTS to detect temperature changes at smaller spa-tial scales than the methods spatial sampling resolution as small-scaletemperature contrasts can deteriorate beyond the detection limit(Rose et al., 2013).While DTS is increasingly being used forfield studies,it appears that the interpretation of field data could benefit from furthertesting of the methods capabilities and limitations.

The above research works emphasise that there is generally largespatial temperature variability in the stream and streambed environ-ments, and that temperature datasets can improve our understandingof the spatial variability of groundwater surface water exchange pro-cesses. However, the reported spatial variability also clearly shows thata few spot measurements will not adequately capture the spatial flowcharacteristics. This calls for more research regarding meaningful up-scaling in order to characterise large scale exchange while minimisingthe deployment of monitoring resources.

Although it is often mentioned that higher infiltration rates influ-ence the propagation rate of heat by convection, most of the researchworks summarised in this section did not use their temperature datato quantify water flow. However, they provide detailed insight intothe spatial and temporal dynamics of the thermal regime in near-surface shallow hydrogeological systems.

3.2. Natural heat as a tracer to quantify water flow

Temperature data and the HTE (Eq. (1)) can be used to quantify thewater flow. When reduced to one spatial dimension (i.e. vertical) theequation allows the determination of simple analytical solutions forspecific boundary conditions representative of field problems. It mustbenoted that any analytical solution to Eq. (1) is subject to the followingassumptions (e.g. Stallman, 1965): (1) fluid flow is steady and uniform;(2) characteristics of the material are constant in space (homogeneousand isotropic) and time; (3) the temperature of thewater and the solidsat any particular distance from the surface are equal at all times (localthermal equilibrium).

3.2.1. Steady-state method with a constant temperature boundaryFor steady-state thermal and hydraulic conditions and long time-

scales a constant temperature boundary can be assumed. Bredehoeftand Papaopulos (1965) solved the differential equation with constanttemperature conditions at the upper and lower boundaries of a 1Ddomain. The resulting analytical solution describes the effect of waterflow distorting the temperature profile away from the purely conduc-tive vertical temperature profile. The authors developed dimensionlesstype curves to which normalised spatial temperature data can be ap-plied in order to find the water flow velocity responsible for the ob-served temperature profile.

The above steady-state solution was originally developed for the as-sessment of deeper vertical groundwater movements using geothermalgradients (e.g. Cartwright, 1970; Sorey, 1971). However, Lapham(1989) adopted the idea and applied it to near-surface systems inorder to quantify surface water groundwater exchange through thestreambed using seasonal temperature envelopes (Fig. 6). Land andPaull (2001) found that the temperature data in shallow coastal estua-rine sediments could not be explained by conduction alone and wereable to attribute the deviation to groundwater discharge. In an arid en-vironment, Constantz et al. (2003b) successfully assessed the shallowwater recharge rates in a stream and estimated percolation rates intothe deeper subsurface using temperature measurements. Taniguchiet al. (2003) applied the steady-state method to vertical borehole tem-peratures to estimate groundwater discharge.

Conant (2004) used an empirical method to correlate temperaturetime series to sediment flow on a streambed section. The potential forexchange fluxwas determined with water level measurements and hy-draulic conductivity estimates obtained from slug testing. This workforms a transition between qualitative interpretation and quantificationbased on temperature data. Moreover, the paper classifies five differentflow zones based on the magnitude and direction of sediment watervelocity measured. The same temperature dataset was re-evaluated bySchmidt et al. (2007) who used the steady-state temperature methodto estimate vertical exchange flows (see Fig. 5B). Both methods,Bredehoeft and Papaopulos (1965) and Schmidt et al. (2007)were incor-porated into a computational algorithmnamed Ex-Stream (Swanson andCardenas, 2011) which is capable of automated velocity calculation fromfield temperature records based on MATLAB®.

Schmidt et al. (2006) applied the steady-statemethod to estimate ex-change volumes on a reach scale. However, the validity of the steady-stateflow condition for the use with sediment temperature profiles was eval-uated with the conclusion that the method may not be able to quantifyvelocities correctly under low flow conditions because of the need forthe temperature boundary to be constant in time (Schmidt et al., 2007).The thermal conditions in near-surface sediments can cause significantdeviations in flow estimates because the steady-state assumption canbe inadequate during spring and autumn when the temperature differ-ence between surface and at depth is minimal (Anibas et al., 2009). How-ever, it is suggested that this method can serve as first pass mapping tool(Anibas et al., 2011). The validity of the steady-state vertical approachfor groundwater discharge was evaluated numerically for a heteroge-neous hydraulic permeability field on the streambed scale (Fergusonand Bense, 2011) as well as for transient temperature conditions(Schornberg et al., 2010). Both studies demonstrate that increasing spa-tial variability in the sediment permeability induces an increase in thehorizontal temperature gradients whichwould limit the quantitative ca-pability of the steady-state method in particular for low vertical flow ve-locities. The above summarised research suggests that the steady-statetemperature method has limited applicability in particular for realisticnatural permeability distributions and temperature dynamics found inshallow systems. However, Brookfield and Sudicky (2013) concludedthat qualitative patterns of flux can be described adequately despite hor-izontal components and heterogeneous streambed properties.

The influences of thermal diffusivity and vertical flow velocity ontemperature depth profiles were further tested by Vandenbohede and

50 G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

Lebbe (2010a). They concluded that velocity and diffusivity can be esti-mated for convection, but only the diffusivity can be determined whenconduction is dominating. In a different study, vertical temperature pro-files where analysed for the effects caused by temporally variable re-charge rates on the steady-state method (Vandenbohede and Lebbe,2010b). It was concluded that time variable recharge can cause asym-metric temperature depth profiles that are difficult to interpret. Insummary, the temperature-depth profiles contain the history of thetemporal temperature signal that the surface has been exposed to.

3.2.2. Transient method with an arbitrary temperature boundaryA transient near-surface heatmodelling approachwas developed by

Lapham (1989) who numerically solved the transport equation(Eq. (1)) to calculate vertical streambed water velocities and hydraulicconductivities. This advanced method utilised, for the first time,temperature-depth envelopes for a temporal estimation of water fluxesappreciating the seasonally varying temperatures at the surface.

Based on forward modelling of the heat transfer Silliman et al.(1995) devised another novel approach to quantify vertical streambedflow. This numerical technique is discretised in time and thus enablesthe calculation of the sediment thermal response at a selected depthbased on any temperature time series at the upper boundary. The sedi-ment temperature response depends on the water flow velocity whichcan be calculated using temperature data recorded at two differentdepths within the sediment. The model estimates the average flow ve-locity responsible for the thermal signature at the lower monitoringpoint by iterating the velocity value. Unfortunately, themethod requiresthat the velocitymust be constant throughout the entire duration of thetemperature–time series used for the calculation. Although Sillimanet al. (1995) commented that the method is only useful for calculatingthe streambed loss, Becker et al. (2004) successfully applied themethodto estimate upward flow.

Another explicit analytical solution for temperature predictions wasdeveloped by Shao et al. (1998) based on the Fourier transform. Themethod was applied to the soil zone and successfully approximatedtemperature variations at depth under convective influence caused bywater percolation from surface rainfall and irrigation. In contrast,Holzbecher (2005) published a study that made use of the differentialequation solver integrated into MATLAB® to calculate the inversion oftemperature time series for a combined estimation of vertical waterflow and sediment thermal diffusivity. However, the capabilities ofthis module have not yet been further evaluated. Streambed tempera-ture time series measured at different depths can now be processedusing the computer program 1DTempPro solving the coupled flowand heat transport equation to provide vertical fluxes (Voytek et al.,2013).

In a different approach,Wörman et al. (2012) analysed the sedimentdepth temperatures for their content of different frequencies by com-puting the amplitude and phase spectra in the frequency domain. Itwas found that the signal propagation depth is a function of signal fre-quency, with longer period signals propagating deeper. Further, thescaled spectrumcan beused to compute the thermal diffusivity and ver-tical flux between the two temperature pairs involved.

3.2.3. Transient method with a sinusoidal temperature boundarySuzuki (1960) first mentioned the HTE (Eq. (1)) for conductive and

convective heat flow. The equation was solved analytically for a one-dimensional vertical domain with a sinusoidal temperature functionas the upper boundary condition and a constant temperature at thelower boundary. The purpose of this method was to make use of thediel temperature signature observed underneath irrigated rice paddiesand estimate water percolation rates from temperature time series.Stallman (1965) noticed a mistake in the derivation and corrected theequation. He found a closed-form analytical solution describing thetemperature at any point in time and depth with sinusoidal perturba-tions as the thermal forcing at the surface.

According to the method, the sediment depth response depends onthe sediment thermal parameters (i.e. thermal conductivity and heatcapacity) and the velocity of vertical water flow. Stallman (1965) de-rived a method that allows the calculation of flow velocity from thedamping of temperature amplitudes and the phase shift within thedepth of heat penetration. The method requires two temperature timeseries recorded at different vertical positions. Velocities are calculatedwith special type curves that relate the attenuation of diel fluctuationsto the percolation rate. Velocities as low as 0.3 cm/day can be detectedaccording to the original publication, but only if the thermal propertiesof the sediment are known accurately. However, Stallman (1965) rec-ommended that field testing of the method reliability is required.

Potentially, the approach could be applied to all near-surface hydro-logic systems that contain diel temperature variations. An example oftemperature time series measured in shallow saturated sediments atfour discrete depths can be seen in Fig. 7. These data clearly show thatthe temperature amplitude decreases and that the phase is increasinglyshifted with depth of measurement. Stallman's method was an enor-mous scientific contribution in the field of hydrogeology (Weeks,2006). It further marked a historical milestone by raising the signifi-cance of temperature from being only a water quality parameter to aneasily measurable input for water flow quantification.

Stallman's method remained untouched for almost three decadesuntil Taniguchi (1993) successfully developed type curves from whichgroundwater flow rates in shallow aquifers can be estimated usingtemperature depth profiles. Neglecting short term temporal variationsin both heat and water flux, they focused on the annual temperatureenvelope and successfully calculated vertical flow both upwardsand downwards. Furthermore, groundwater recharge and dischargewere successfully estimated from relative temperature differencesover depth again making use of the near-surface temperature signal(Taniguchi and Sharma, 1993; Taniguchi et al., 2003). Similarly,Tabbagh et al. (1999) developed a method using the differences be-tween temperature amplitudes and phases obtained from a verticalpair of soil sensors fromameteorological station to estimate the averagerecharge. The methodology was later improved by incorporating thevertical soil profile heterogeneity (Cheviron et al., 2005), and itwas test-ed (Bendjoudi et al., 2005). The authors noted that themethod sensitiv-ity could be improved by deployingmulti-level subsurface temperaturemonitoring adapted to their specific purpose rather than using datafrom standard meteorological stations.

An important extension to Stallman's original analytical solutionwasdeveloped by Goto et al. (2005). They determined water fluxes througha hydrothermal mound at the ocean floor from temperature–timerecords. The study demonstrated that the propagation of arbitrary tem-perature–time series can bemodelled using Stallman's original solution.This is because the heat transport theory is linear by nature which al-lows the principle of superposition to be used for the decompositionof temperature records into a series of sinusoidal temperature functionswith different amplitudes and phases using the Fourier Theorem. Thetotal sediment temperature depth response is the summation of eachindividual frequency responseweightedwith Stallman's function.More-over, Goto et al. (2005) developed a dimensionless parameter relatingflow velocity to the periodicity of the surface fluctuation. Apart from es-tablishing three distinct thermal regimes in the sediment (convectiondominated, conduction dominated and a transition region), the verticalflow velocity and diffusivity of the sediment were successfully quanti-fied using their method.

Another important application derived from Stallman's method wasdeveloped by Hatch et al. (2006) who mathematically separated theexplicit temperature equation into two distinct expressions based on asinusoidal temperature (1) amplitude attenuation and (2) phase shift asa function of flow velocity. Fig. 8 define these parameters for tempera-ture records at two different depths. An additional mathematical termaccounting for hydrodynamic thermal dispersivity was included in themethod. The method first extracts the dominant diel temperature

15 16 17 18 19 20 21 2214

16

18

20

22

24

26

Tem

pera

ture

[°C

]

September 2008

Water0.15 m0.3 m0.45 m0.6 m

Fig. 7. Example of an unprocessedmulti-level streambed temperature dataset showing amplitude damping and phase shifting with increasing depth of measurement (Rau et al., 2010a).

51G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

fluctuations from sediment temperature datameasured at different ver-tical depths by forward–backward filtering (Hatch et al., 2006). Thisstep is important because temperature data used for the velocity calcu-lation must comply with the sinusoidal boundary condition as this is acondition for the analytical solution. Computation of the flow velocityis then performed by determining the temperature amplitude ratioand phase shift.

This method permits the quantification of the vertical sedimentwater flux with the two independent mathematical relationships, tem-perature amplitude damping and phase shifting (Hatch et al., 2006).Furthermore, the method is quasi–transient in regards to flow becauseit allows the determination of three velocity values per day, using the si-nusoidmaximum,minimum and zero crossings. The approach providesthe ability to determine whether flow is upwards or downwards. Onemajor advantage of the method is the fact that the calculation of watervelocity only requires the distance between sensor pairs and not theabsolute depth from the surface. Therefore, results are unaffected bynaturally occurring processes at the surface, such as streambed scouringor sediment deposition (as long as the probes are not exposed). Acomplication of themethod is the fact that fieldmeasurements requiresextensive data processing to extract the sinusoidal signal, with the par-ticular prerequisite of phase preservation, before the flow velocity canbe calculated. The impact of the temperature filtering process on the ve-locity results was only tentatively estimated and requires proper inves-tigation. As a further shortfall, the thermal properties of the sedimentmust be known a priori and are commonly derived from the literature(e.g. Lapham, 1989). Despite these shortfalls the method has already

A B

Fig. 8. (A) Example of a field installation measuring temperatures at multiple depths.(B) Illustration of temperature amplitude damping (ΔA) and phase shifting (Δϕ) withdepth of measurement (Hatch et al., 2006).

been used extensively to analyse field data (e.g. Fanelli and Lautz,2008; Rau et al., 2010a; Swanson and Cardenas, 2010; Jensen andEngesgaard, 2011; McCallum et al., 2012).

A similar methodwas developed by Keery et al. (2007)who derivedan explicit mathematical formulation which isolates the velocity fromStallman's original model as the parameter of interest. The equation re-lies on the phase shift between two temperature time series recorded atdifferent vertical depths of the sediment. Instead of zero-phase filtering,the method deploys Dynamic Harmonic Regression (DHR) (Young et al.,1999) to extract the diel component from the temperature time series.DHR is an extension to Fourier analysis applicable to non-stationary sig-nals and, in this context, is capable of extracting temperature amplitudeand phase information with a resolution of more than two values a day.This algorithm is implemented as a set of functions contained in theCaptain Toolbox (Young et al., 2007). Results from the analysis of fielddata indicated that water flow velocity varied considerably with time.The paper was the first to suggest that surface water groundwater ex-change monitoring using natural heat as a tracer could be automatedby implementing data processing functionality into temperaturelogging devices. As a further step towards the goal of automated fluxcalculations from thermal records, the Hatch and the Keery methodswere included in the code Ex-Stream (Swanson and Cardenas, 2011)and in the code VFLUX (Gordon et al., 2012), which included compre-hensive error and sensitivity analysis.

The mathematics of the Hatch method was recently revisited byMcCallum et al. (2012) and Luce et al. (2013). Independently fromeach other, McCallum et al. (2012) were able to extract a system oftwo equations from Stallman's analytical solution that explicitly solvefor the thermal front velocity and the sediment thermal diffusivityfrom consecutive pairs of temperature amplitude and phase shift data.The advantage of this newmethod is that the sediment's thermal diffu-sivity does not have to bemeasured or assumed, but can be calculated asa time series explicitly from streambed temperature time series. For agiven volume of sediment in the field the diffusivity should not changesignificantly over short to medium term time scales (days to months),unless scouring or deposition occurs. If the range of physically possiblediffusivity values is exceeded or if rapid changes in diffusivity areobserved, then one or more of the general method assumptions (seeabove)must be violated. The estimated diffusivity time series can there-fore be used as a temporal quality control parameter on the validity ofapplying themethod (McCallum et al., 2012). The analytical approachespresented by Luce et al. (2013) add significant flexibility to practicalapllications, such as the ability to estimate unknown sensor spacings,or even to estimate streambed changes due to scour or depositionwhen the computed sensor spacings become unrealistic.

Recently, wavelet analysis was applied to temperature time recordsin order to extract amplitudes and phase of the diel component. Theinformation provided flux estimates with high temporal resolution

52 G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

equalling that of the temperature time-series measurement. (Onderkaet al., 2013). However, reliability of these methods under transienthydraulic conditions has not yet been tested.

3.3. Advances in natural heat tracing

The development of techniques utilising diel temperature fluctua-tions has led to a significant amount of field deployments for variousapplications in the last decade. The main aims are to derive waterflow velocities and to advance near-surface hydrologic process under-standing. To date themethods have been applied to a range of differentfield situations. The following sections summarise this research.

3.3.1. Hydrologic process understandingAt this stage the quantitative methods using natural heat as a tracer

only deliver insight into a very small area around the monitoring loca-tion (point-in-space). In summary, the findings show spatially variableand non-uniform fluxes in the sediments across and along surfacewater bodies (e.g. Fanelli and Lautz, 2008; Rau et al., 2010a; Swansonand Cardenas, 2010; Jensen and Engesgaard, 2011) aswell as temporal-ly varying velocity results (e.g. Goto et al., 2005; Keery et al., 2007; Rauet al., 2010a; Vogt et al., 2010; Jensen and Engesgaard, 2011). Numericalstudies highlight that heat derived flux measurements should dependon the stream geometry, hydraulic conditions and the location of mea-surement within the channel (Shanafield et al., 2010; Roshan et al.,2012). Importantly, vertical flux estimates are mainly erroneous inareas where flow lines converge or diverge (Cuthbert and Mackay,2013).

A recent attempt to up-scale point-in-space heat tracing to a reachwith 10s of metres showed only reasonable success (Lautz andRibaudo, 2012). Judging by the hydrodynamic variability within near-surface aquatic environments it is difficult to infer exchange volumesfrom these point measurements by integration to larger spatial scales.

This also raises the issue that localised heat tracing has not been ableto quantify the relative contribution of hyporheic flows and exchangewith deeper groundwater. Further progress was achieved by Bhaskaret al. (2012) who illustrate that groundwater and hyporheic flow com-ponents can be distinguished by analysing shallow thermal signatures.They assume that an increased hyporheic flow component has the effectof increasing the shallow thermal dispersion. They also offer guidanceabout how to optimally place temperature sensors to detect deeper hor-izontal hyporheicflow and quantify the vertical streambedflow. It is an-ticipated that further research along these lines will strengthen the useof heat as a tracer with additional insight into the complexity of thestreambed hydrodynamics and heat transport.

3.3.2. Practical improvementsA new temperature profiler was developed by Vogt et al. (2010)

which significantly improves the spatial resolution of DTS by closelywrapping the fibre optic cable around a pipe. This new vertical“temperature profiler” was tested in river sediment and offered agreat spatio-temporal insight into the centimetre scale temperature dy-namics as well as a detailed quantification of water fluxes based onStallman's (1965)method in the hyporheic zone. In an attempt tomea-sure the loss of water from a river into the riparian environment (bankfiltration), a series of these profilers were installed (Vogt et al., 2012).However, the authors stress that flux quantification using the simple1D method is affected by horizontal flow and heat exchange with thenearby unsaturated zone. In a different study, a number of vertical tem-perature profilers were installed in a bar–pool–dam sequence and re-vealed very detailed insight into the vertical and temporal variabilityof vertical hyporheic fluxes (Briggs et al., 2012b).

A different advance has been to use heat derived fluxes and hydrau-lic head measurements in combination to estimate changes in stream-bed hydraulic conductivity with time (Hatch et al., 2010; Lautz et al.,2010; Rau et al., 2010a). Alternatively, temperature–depth variations

measured using coiled DTS were able to resolve centimetre-scaleflows that correlated well with depth profiles of hydraulic conductivi-ties (Liu et al., 2013). These advances are a logical consequence of suc-cessful heat tracing in the field. However, it must be noted that furtherresearch on the reliability of the derived fluxes is necessary before thisresearch can advance (refer to Section 3.3.3 for details).

3.3.3. Method limitationsThe abundance offield studies covering a broad range of thermal and

hydraulic conditions have collectively led to a significant advancementin the process understanding of heat transport research. Interestingly,laboratory experimentation on the fundamental characteristics of heattransport is remarkably rare in the hydrogeological community. This isdespite the obvious advantage of having the ability to control the initialand boundary conditions. For sub-metre spatial scales, the validity of theHTE (Eq. (1)) for heat transportmodelling in different types of sedimentwas questioned because it relies on the assumption of local thermalequilibrium between the solid and liquid phase (Rau et al., 2010a).

Rau et al. (2012b) experimentally tested the performance of the si-nusoidal temperature method utilising recorded amplitude and phaseshift data under conditions that comply with the assumptions for theanalytical model. They reported that even for a material with a narrowgrain size distribution (i.e. homogeneous sand for practical purposes)there is an increasingly non-uniform flow field with increasing flow ve-locity. This was for velocities well below groundwater Reynolds num-bers where turbulence should be expected (Re b 1). This non-uniformity seems to induce horizontal temperature gradients whichadd to the overall thermal dispersion process and thus result in increas-ingly erroneous velocity estimates for increasing velocities. They furtherspeculated that this effect could be enhanced when material heteroge-neity increases and that the impact of sediment heterogeneity (e.g. larg-er non-uniformity in the grain size distribution) on the thermaldispersion should be investigated (Rau et al., 2010a, 2012a).

Lautz (2012) used column experiments to test the capability of theanalytical heat model using temperature sinusoids for estimating tran-sient velocities on a shorter time scale than the period of the surfaceheat forcing. The study concluded that the 1D model is reliable andgives meaningful flux quantifications even for fast hydraulic changes.However, the work also stressed that filtering the temperature signalis a crucial requirement for accurate results. In the field this finding isrelevant for the confidence in the method for monitoring the effects ofstorm events, floods or dam releases on the surface water groundwaterexchange (McCallum et al., 2012). Investigations like these highlight theusefulness of laboratory experimentation to improve our general un-derstanding and confidence in the performance of the heat tracingmethods.

The rate of ongoing research highlights continuing interest in heat asa tracer and proves that it is regarded as a potentially powerful tool forthe determination of the localised vertical water flux in near-surfacesediments, albeit with several unresolved issues. Consequently a num-ber of factors were investigated and identified as having a significantimpact on the veracity of the flux quantifications. For a broad overviewof these factors refer to Table 1. For instance, it has widely been ac-knowledged that a horizontal flow component may distort the verticalflux estimated from 1D methods (Lautz, 2010; Rau et al., 2010a;Swanson and Cardenas, 2010; Jensen and Engesgaard, 2011; Briggset al., 2012b; Roshan et al., 2012; Vogt et al., 2012). Lautz (2010) there-fore proposed that velocities derived from the shift in the diel tempera-ture phase with depth should be disregarded due to a lack of reliabilitycompared to amplitude derived velocities. However, only Rau et al.(2010a) have presented flux calculations from both amplitude andphase derived solutions. They found these to deviate significantly fromeach other for two out of three locations investigated. They suggestedthat this artefact could be indicative of the method assumptions beingviolated, most probably by the deviation from vertical flow. A numericalinvestigation has confirmed this expectation and recommended that

Fig. 9.Conceptual principle of a heat pulse tool for themeasurement of directionalflowve-locities in saturated sediments. A heat pulse is emitted from the centre, and its response ismeasuredwith temperature probes in a 3D space. Thedirection andmagnitude of theflowcan be derived from a joint inversion of the temperature responses (Angermann et al.,2012a).

53G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

calculating and comparing both amplitude and phase derived velocitiescan offer important insight into the applicability of 1D methods to agiven field site (Roshan et al., 2012).

It was also found that uncertainties in thermal parameters and inac-curacy of temperature measurements can lead to large errors in the ve-locity estimates in particular for low flows (Shanafield et al., 2011;Roshan et al., 2012). For example the accumulation of biogenic gasesin the shallow sedimentsmay temporally alter the bulk thermal param-eters which can introduce significant errors in the exchange rates calcu-lated (Cuthbert et al., 2010). Both the sensor temperature resolution(Soto-López et al., 2011) and the accuracy of its spatial location(Shanafield et al., 2011) can significantly affect the accuracy of flow cal-culations. The above discussion calls for more fundamental research onthe validity of using the method under different field circumstances,such aswide grain size distributions, extreme flow velocities or violatedboundary conditions.

3.4. Induced heat as a tracer

3.4.1. Saturated zoneInduced heat has also been used to detect groundwater flow in bore-

holes in the saturated zone (Dudgeon et al., 1975; Hess, 1986). Althoughthese methods can identify hydraulically conductive zones by theirtemperature anomalies the flow velocities through these zones is notquantified. An instrument for the in-situ measurement of groundwaterflow direction and velocity under fully water saturated conditions wasdeveloped by Ballard (1996). This advanced technique relies on a cylin-drically shaped heater with 30 thermistors strategically placed on itssurface which is permanently buried in the unconsolidated saturatedsubsurface. Any groundwater movement around this cylinder leaves adistinct temperature field on the surface of the cylinder which can beanalysed for flow direction and magnitude using an advanced mathe-matical model. The probe was tested in an aquifer stressed artificiallyby pumping, and directional flow was successfully measured with alower detection limit of 0.8 m/day (Ballard et al., 1996). The impact ofsediment texture on the interpretation of temperature datawas numer-ically investigated concluding that there is a risk of misinterpretationdue to the spatial variability of thermal conductivity (Su et al., 2006).

A smaller probe, very similar to the HPP developed for soil science(Ren et al., 2000), with 6 sensors surrounding a heat emitter was ableto determine flow velocity in laboratory experiments (Sisodia andHelweg, 1998). However, later evaluations of this tool revealed large er-rors (Guaraglia and Pousa, 2006) and the methodology was refined byincreasing the amount of sensors around the heat pulse emitter(Guaraglia and Pousa, 2007; Guaraglia et al., 2008). Preliminary resultsof a revised rotary design indicated promising applications for themea-surement of fast velocities that may be associated with remediationtechniques for groundwater contamination (Guaraglia et al., 2009).

More recently, a tool using the heat pulse methodology was devel-oped (see illustration in Fig. 9) for the measurement of local scalegroundwater surface water exchange velocities and directions in theshallow sediments (Lewandowski et al., 2011). Angermann et al.(2012a) developed a 3D algorithm for the inversion of flux magnitudeand direction from temperature breakthrough curves. The toolwas test-ed by Angermann et al. (2012a,b) who, for the first time, quantified theflow variability with very high spatial resolution (~80measurements inan area of 2 x 2 m) in the sandy streambed of a lowland river in combi-nation with vertical head gradients. The results illustrated a surprisingdegree of flow heterogeneity with down-welling and upwelling zoneson the centimetre scale (Fig. 5C). The authors acknowledge that thetool is intrusive and that it has a range of practical limitations requiringconsideration and modification when it is applied to different types ofsaturated near-surface environments. However, the tool promises tobe a useful alternative to natural heat as a tracer as the point type heatinjection can be controlled and the flow velocities can be determinedin all directions. It should be noted that velocity quantification is subject

to theHTE (Eq. (1)) and the capabilities of the heat toolwill benefit fromthe research requirements mentioned in Section 2.

3.4.2. Vadose zoneThe vadose zone has the additional complexity of a gas phase which

can significantly alter the bulk thermal parameters compared to fullywater saturated conditions (Ochsner et al., 2001). The presence of agas phasemay also challenge the assumption of thermal equilibrium in-herent to the transport equation (Cuthbert et al., 2010). Despite thispossible limitation a number of studies have applied or adapted theheat transport model (Eq. (1)) for specific applications in the vadosezone.

In soil science the water flux was successfully determined by induc-ing heat and evaluating the steady-state temperature response (Byrneet al., 1967, 1968). The technique was subject to considerable practicalimprovements but still relied on an empirical calibration curve (Byrne,1971; Cary, 1973) rather than the heat transport theory given inSection 2.1. Based on this pioneering work, Heat Pulse Probes (HPP)were later developed to measure the specific heat capacity, thermalconductivity and water content of soil samples (e.g. Campbell et al.,1991; Bristow, 1998) using heat diffusion theory as outlined byCarslaw and Jaeger (1959) (Eq. (1) with v = 0).

The first HPP research work making use of the Suzuki (1960) heattransport model (Eq. (1) with v N 0) to calculate soil water fluxes waspresented by Ren et al. (2000). The probe induces a heat pulse of finiteduration and measures the thermal response at several locations. Thetechnique was tested on different soil materials, and it was noted thatmethod refinements are required in order to improve the reliability ofthe flux measurements. The technique was improved by simplifyingthe complex numerical model to an analytical solution (Kluitenbergand Warrick, 2001).

Hopmans et al. (2002) developed an inverse modelling technique,incorporated the effect of the probe geometry, and further noted thatthermal dispersivity, when expressed analogously to solute dispersivity(Eq. (2)), can have a significant effect on thewaterflux estimate derivedfrom the measurements. The mathematical solution was simplified(Wang et al., 2002), the method was tested in a laboratory study

54 G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

(Ochsner et al., 2005), and improved by increasing the points of mea-surement to three instead of two (Kluitenberg et al., 2007). Practical de-sign aspects and their possible influences, for example heat pulsecharacteristics and influence of possible vapour flow, have also beenevaluated numerically (Saito et al., 2007). A newly proposed designwith thicker needles can now measure water flux rates as low as1 cm/day (Kamai et al., 2008).

The HPP has been equipped with additional measurement tech-niques to improve flux quantification in the vadose zone (Ren et al.,2003; Mori et al., 2005) and provide multifunctional capabilities suchas the simultaneous measurement of solute concentrations (Moriet al., 2003;Mortensen et al., 2006). In summary soil science and vadosezone research demonstrates that heat is a powerful tool to quantifywater flow also under variably saturated conditions. Further researchis required to lower the flux detectability threshold under unsaturatedconditions (Kamai et al., 2008), and to determine a universal relation-ship between soil properties and their thermal dispersion behaviouras a further extension of Lu et al. (2009).

3.5. Heat tracing with numerical models

Most available numerical groundwater codes use a separate equa-tion for flow of water and one for transport of thermal energy. Theflow equation requires estimates of hydraulic heads and conductivities.If heat data is available, simultaneously solving for the transport ofthermal energy can serve to constrain the uncertainty in the hydraulicestimates. Accurately quantifying water fluxes thus inherently requiresdetailed hydraulic knowledge. This review aims to summarise progressin the quantification of water fluxes without any knowledge of the hy-draulic conditions based on simple analytical models derived from theheat transport equation and temperature data. However, a numericalapproach potentially offers more insight into the processes occurring,but at the same time requiresmore data as well as computational effort.The following section is a brief introduction into available numericalcodes as well as examples of heat tracing approaches. For a more de-tailed review of numerical applications in near-surface environmentsthe interested reader is referred to Constantz (2008).

With the increasing number of temperature field investigations aneed was recognised for flexible and complex temperature boundaryformulations to enable realistic and multi-dimensional simulations. Asearly as 1984, Voss developed a sophisticated numerical code forgroundwater simulations which incorporated the ability of modellingheat transport: SUTRA (Voss, 1984). To date a variety of numericalgroundwater software packages that can also solve coupled groundwa-ter and heat transport equations are available, for example VS2DH(Healy and Ronan, 1996), HydroGeoSphere (based on FRAC3DVS;Therrien, 1992; Therrien and Sudicky, 1996; Brookfield et al., 2009),FEFLOW (Diersch, 2005) and SEAWAT (Langevin et al., 2008).

Numerical models facilitate the use of heat as a tracer for differentpurposes, for example:

• Thematching of modelled and observed temperature data for naturalsystems in order to define flow paths, estimate residence times, flowvelocities or hydraulic conductivities (e.g. Constantz et al., 2002,2003a; Bense and Kooi, 2004; Su et al., 2004; Essaid et al., 2008;Kulongoski and Izbicki, 2008; Barlow and Coupe, 2009; Birkinshawand Webb, 2010; Duque et al., 2010; Naranjo et al., 2013).

• Temperature data can be used in combinationwith hydraulic heads ormeasured fluxes to help constrain models (e.g. Ronan et al., 1998;Bravo et al., 2002; Burow et al., 2005; Mutiti and Levy, 2010).

• Numerical heat transport models can be used to investigate the flowof water and its influence on the thermal signature in complex envi-ronments (e.g. Sauty et al., 1982a, b; Xue et al., 1990; Molson et al.,1992; Palmer et al., 1992; Cardenas and Wilson, 2007; Cardenas,2010; Vandenbohede and Lebbe, 2011; Constantz et al., 2013).

These research works demonstrate that application of numericalheat transport codes can advance our understanding of natural process-es. Numerical approaches may be much more flexible in regards toimplementing real world scenarios (i.e., boundary conditions, dimen-sionality and heterogeneity). However, as with analytical solutions nu-merical models rely on differential equations which are subject tofundamental assumptions, such as local thermal equilibrium, volumeaveraging of thermal parameters and empirical expressions for the ther-mal dispersion. The correctness of the transport equations underpin-ning the numerical codes should therefore be verified experimentally.

4. Concluding remarks

This review paper demonstrates that there is a large body of scientificresearch dedicated to the development and application ofmethods usingnatural and induced heat as a tracer in ideal and natural porousmaterials. The HTE (1) forms a basis for themajority of works and quan-tification of water flow and thermal dispersion are the parameters of in-terest. The review comprehensively summarises and discusses theFourier-type heat transportmodelling including parameters, and reviewsmodel and process understanding gained from research applying heattransport modelling in the different but overlapping fields of earthscience and engineering.

The summarised research shows that the characteristics of heattransport in natural materials under saturated conditions have not yetbeen fully established. For example, on small spatial scales the HTE (1)is subject to several assumptions such as local thermal equilibriumand volume averaging of material properties within a suitable REV.Research reported in the engineering literature contains clarificationfor some of these points, but for ideal porous materials (e.g. glassbeads). The heterogeneous phase distribution and spatial variability inhydraulic conductivity, inherent to natural near-surface systems, mayhave a significant impact on the thermal transport. For example, atrapid flow velocities the validity of the assumption of thermal equilibri-um between solid and liquid phase could be violated. In this case theapplicability of the standard HTE (1) would be compromised. It hasyet to be established whether this is likely for Darcian flows in differenttypes of naturalmaterials. Furthermore, the dependency of thermal dis-persion on sediment type and texture at different scales requiresinvestigation.

Engineering research exemplifies that controlled laboratory experi-mentation is integral to advancing our fundamental understanding ofconvective heat transport. This review reveals that there is a lack ofexperimentation in regards to solving fundamental questions aboutthe characteristics of heat transport in natural materials. In the earthsciences, the impact of these assumptions on the estimation of flowvelocities has rarely been acknowledged let alone thoroughly tested.Consequently, physical experiments at different scales (e.g. laboratoryand field) with natural materials of different grain size distributionsare recommended.

A growing body of research works has focussed on the earth's near-surface (e.g. notably streambeds)where hydraulic conditions and naturalheat variability are highly dynamic (e.g. diel temperature fluctuations).The publication timeline of studies summarised here demonstrate thatresearch into using heat as a tracer, in particular the stream environ-ment, is increasing. Heat tracingmethods have been applied under a va-riety of circumstances and algorithms for automated evaluation ofparameters, in particular water flow, have been developed. However,several aspects require further attention. For example, the relativemag-nitude of heat conduction, compared to heat convection, means thatwater flow quantification from temperature records becomes increas-ingly uncertain with decreasing flow velocities. Themeasurement reso-lution of temperature equipment currently available imposes a practicallimitation on the accurate estimation of low fluxes. Higher resolutiontemperature logging equipment is required to push the limit of resolv-ing convective influences in the thermal signature caused by flowing

55G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

water. Additionally, any uncertainty in thermal parameters further in-tensifies the uncertainty in flow calculations. These limitations havenot properly been appreciated and require further research.

The breadth of existing research demonstrates that heat tracing hasgreatly contributed to characterising and quantifying hydrologicalfluxes at the earth's near-surface. Furthermore, the characterisation ofshallow thermal regimes is useful beyond the determination of waterflow, as the temperature influences the biogeochemical processes andis significant for our understanding of changes to the water qualityand ecology (e.g. Brunke and Gonser, 1997; Krause et al., 2011a).

Acknowledgement

Funding for this research was provided by the National Centre forGroundwater Research and Training (NCGRT), an Australian Govern-ment initiative, supported by the Australian Research Council and theNational Water Commission. We thank Stephen Grasby and an anony-mous reviewer for their assessment which significantly improved thequality of this manuscript.

References

Alexander, M.D., Caissie, D., 2003. Variability and comparison of hyporheic water temper-atures and seepage fluxes in a small Atlantic salmon stream. Ground Water 41,72–82.

Andersen, M.S., Acworth, R.I., 2009. Stream–aquifer interactions in the Maules Creekcatchment, Namoi Valley, New South Wales, Australia. Hydrogeol. J. 17 (8),2005–2021.

Anderson, M.P., 2005. Heat as a ground water tracer. Ground Water 43, 951–968.Angermann, L., Lewandowski, J., Fleckenstein, J.H., Nützmann, G., 2012a. A 3D analysis

algorithm to improve interpretation of heat pulse sensor results for the determina-tion of small-scale flow directions and velocities in the hyporheic zone. J. Hydrol.475, 1–11.

Angermann, L., Krause, S., Lewandowski, J., 2012b. Application of heat pulse injections forinvestigating shallow hyporheic flow in a lowland river. Water Resour. Res. 48 (12),W00P02.

Anibas, C., Fleckenstein, J.H., Volze, N., Buis, K., Verhoeven, R., Meire, P., Batelaan, O., 2009.Transient or steady-state? Using vertical temperature profiles to quantifygroundwater-surface water exchange. Hydrol. Process. 23, 2165–2177.

Anibas, C., Buis, K., Verhoeven, R., Meire, P., Batelaan, O., 2011. A simple thermal mappingmethod for seasonal spatial patterns of groundwater–surface water interaction.J. Hydrol. 397 (1–2), 93–104.

Arrigoni, A.S., Poole, G.C., Mertes, L.A.K., O'Daniel, S.J., Woessner,W.W., Thomas, S.A., 2008.Buffered, lagged, or cooled? Disentangling hyporheic influences on temperaturecycles in stream channels. Water Resour. Res. 44, 1–13.

Atwell, B.H., MacDonald, R.B., Bartolucci, L.A., 1971. Thermal mapping of streams fromairborne radiometric scanning. J. Am. Water Resour. Assoc. 7, 228–243.

Bachu, S., 1991. On the effective thermal and hydraulic conductivity of binary heteroge-neous sediments. Tectonophysics 190, 299–314.

Ballard, S., 1996. The in situ permeable flow sensor: a ground-water flow velocity meter.Ground Water 34, 231–240.

Ballard, S., Barker, G.T., Nichols, R.L., 1996. A test of the in situ permeable flow sensor atSavannah River, SC. Ground Water 34, 389–396.

Barlow, J.R.B., Coupe, R.H., 2009. Use of heat to estimate streambed fluxes during extremehydrologic events. Water Resour. Res. 45, W01403.

Baskaran, S., Brodie, R.S., Ransley, T., Baker, P., 2009. Time-series measurements of streamand sediment temperature for understanding river–groundwater interactions:Border Rivers and Lower Richmond catchments, Australia. Aust. J. Earth Sci. 56,21–30.

Batycky, R.P., Brenner, H., 1997. Thermal macrotransport processes in porous media. Areview. Adv. Water Resour. 20, 95–110.

Bear, J., 1961. On the Tensor Form of Dispersion in Porous Media. Journal of GeophysicalResearch 66, 1185–1197.

Bear, J., 1972. Dynamics of fluids in porous media. 764.Becker, M.W., Georgian, T., Ambrose, H., Siniscalchi, J., Fredrick, K., 2004. Estimating flow

and flux of ground water discharge using water temperature and velocity. J. Hydrol.296, 221–233.

Bendjoudi, H., Cheviron, B., Guérin, R., Tabbagh, A., 2005. Determination of upward/down-ward groundwater fluxes using transient variations of soil profile temperature: testof the method with Voyons (Aube, France) experimental data. Hydrol. Process. 19,3735–3745.

Bense, V.F., Kooi, H., 2004. Temporal and spatial variations of shallow subsurface temper-ature as a record of lateral variations in groundwater flow. J. Geophys. Res. Solid Earth109 (B4), B04103.

Berkowitz, B., Cortis, A., Dentz, M., Scher, H., 2006. Modeling non-Fickian transport ingeological formations as a continuous time random walk. Rev. Geophys. 44, 1–49.

Bhaskar, A.S., Harvey, J.W., Henry, E.J., 2012. Resolving hyporheic and groundwater com-ponents of streambed water flux using heat as a tracer. Water Resour. Res. 48 (8),W08524.

Birkinshaw, S.J., Webb, B., 2010. Flow pathways in the Slapton Wood catchment usingtemperature as a tracer. J. Hydrol. 383, 269–279.

Birman, J.H., 1969. Geothermal exploration for ground water. Geol. Soc. Am. Bull. 80, 617.Blakey, J.F., 1966. Temperature of surface waters in the conterminous United States.Blasch, K.W., Ferre, T.P.A., Hoffmann, J.P., 2004. A statistical technique for interpreting

streamflow timing using streambed sediment thermographs. Vadose Zone J. 3,936–946.

Blasch, K.W., Constantz, J., Stonestrom, D.A., 2007. Appendix 1: thermal methods forinvestigating ground-water recharge. In: Stonestrom, et al. (Eds.), Ground-WaterRecharge in the Arid and Semiarid Southwestern United States.

Bodvarsson, G., 1969. On the temperature of water flowing through fractures. J. Geophys.Res. 74, 1987–1992.

Bodvarsson, G., 1972. Thermal problems in the siting of reinjection wells. Geothermics 1,63–66.

Bons, P.D., vanMilligen, B.P., Blum, P., 2013. A general unified expression for solute andheatdispersion in homogeneous porous media. Water Resour. Res. 49 (10), 6166–6178.http://dx.doi.org/10.1002/wrcr.20488.

Bravo, H.R., Jiang, F., Hunt, R.J., 2002. Using groundwater temperature data to constrainparameter estimation in a groundwater flow model of a wetland system. WaterResour. Res. 38, 1153–1162.

Bredehoeft, J.D., Papaopulos, I.S., 1965. Rates of vertical groundwater movement estimat-ed from the earth's thermal profile. Water Resour. Res. 1, 325.

Briggs, M.A., Lautz, L.K., McKenzie, J.M., 2012a. A comparison of fibre-optic distributedtemperature sensing to traditional methods of evaluating groundwater inflow tostreams. Hydrol. Process. 26, 1277–1290.

Briggs, M.A., Lautz, L.K., McKenzie, J.M., Gordon, R.P., Hare, D.K., 2012b. Using high-resolution distributed temperature sensing to quantify spatial and temporal variabil-ity in vertical hyporheic flux. Water Resour. Res. 48, W02527.

Bristow, K.L., 1998. Measurement of thermal properties and water content of unsaturatedsandy soil using dual-probe heat-pulse probes. Agric. For. Meteorol. 89, 75–84.

Brookfield, A., Sudicky, E., 2013. Implications of hyporheic flow on temperature-basedestimates of groundwater/surface water interactions. J. Hydrol. Eng. 18 (10),1250–1261.

Brookfield, A.E., Sudicky, E.A., Park, Y.J., Conant, B., 2009. Thermal transport modelling ina fully integrated surface/subsurface framework. Hydrol. Process. 23 (15),2150–2164.

Brunke, M., Gonser, T., 1997. The ecological significance of exchange processes betweenrivers and groundwater. Freshw. Biol. 37, 1–33.

Buntebarth, G., Schopper, J.R., 1998. Experimental and theoretical investigations on theinfluence of fluids, solids and interactions between them on thermal properties ofporous rocks. Phys. Chem. Earth 23, 1141–1146.

Burow, K.R., Constantz, J., Fujii, R., 2005. Heat as a tracer to estimate dissolved organiccarbon flux from a restored wetland. Ground Water 43, 545–556.

Byrne, G.F., 1971. An improved soil water flux sensor. Agric. Meteorol. 9, 101–104.Byrne, G.F., Drummond, J.E., Rose, C.W., 1967. A sensor for water flux in soil. “Point

source” instrument. Water Resour. Res. 3, 1073.Byrne, G.F., Drummond, J.E., Rose, C.W., 1968. A sensor for water flux in soil: 2. ‘Line

Source’ instrument. Water Resour. Res. 4, 607.Campbell, G.S., Calissendorff, C., Williams, J.H., 1991. Probe for measuring soil specific-

heat using a heat-pulse method. Soil Sci. Soc. Am. J. 55, 291–293.Cardenas, M.B., 2010. Thermal skin effect of pipes in streambeds and its implications on

groundwater flux estimation using diurnal temperature signals. Water Resour. Res.46, W03536.

Cardenas, M.B., Wilson, J.L., 2007. Effects of current–bed form induced fluid flow on thethermal regime of sediments. Water Resour. Res. 43, W08431.

Carslaw, H.S., Jaeger, J.C., 1959. Conduction of heat in solids. 510.Carson, J.K., Lovatt, S.J., Tanner, D.J., Cleland, A.C., 2005. Thermal conductivity bounds for

isotropic, porous materials. Int. J. Heat Mass Transf. 48, 2150–2158.Cartwright, K., 1970. Groundwater discharge in the Illinois basin as suggested by temper-

ature anomalies. Water Resour. Res. 6, 912.Cartwright, K., 1971. Redistribution of geothermal heat by a shallow aquifer. Geol. Soc.

Am. Bull. 82, 3197.Cartwright, K., 1974. Tracing shallow groundwater systems by soil temperatures. Water

Resour. Res. 10, 847.Cary, J.W., 1973. Soil water flowmeterswith thermocouple outputs. Soil Sci. Soc. Am. Proc.

176–181.Chang, C.M., Yeh, H.D., 2012. Stochastic analysis of field-scale heat advection in heteroge-

neous aquifers. Hydrol. Earth Syst. Sci. 16 (3), 641–648.Chen, S.X., 2008. Thermal conductivity of sands. Heat Mass Transf. 44, 1241–1246.Chevalier, S., Banton, O., 1999. Modelling of heat transfer with the random walk method.

Part 2. Application to thermal energy storage in fractured aquifers. J. Hydrol. 222(1–4), 140–151.

Cheviron, B., Guérin, R., Tabbagh, A., Bendjoudi, H., 2005. Determining long-term effectivegroundwater recharge by analyzing vertical soil temperature profiles at meteorolog-ical stations. Water Resour. Res. 41, W09501.

Clauser, C., Huenges, E., 1995. Thermal conductivity of rocks and minerals. Rock Physicsand Phase Relations: A Handbook of Physical Constants, 3, pp. 105–126.

Conant, B., 2004. Delineating and quantifying ground water discharge zones usingstreambed temperatures. Ground Water 42, 243–257.

Constantz, J., 1998. Interaction between stream temperature, streamflow, and groundwa-ter exchanges in alpine streams. Water Resour. Res. 34, 1609.

Constantz, J., 2008. Heat as a tracer to determine streambed water exchanges. WaterResour. Res. 44.

Constantz, J., Thomas, C.L., 1996. The use of streambed temperature profiles to estimatethe depth, duration, and rate of percolation beneath arroyos. Water Resour. Res. 32,3597.

56 G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

Constantz, J., Thomas, C.L., 1997. Stream bed temperature profiles as indicators of perco-lation characteristics beneath arroyos in the Middle Rio Grande Basin, USA. Hydrol.Process. 11, 1621–1634.

Constantz, J., Thomas, C.L., Zellweger, G., 1994. Influence of diurnal variations in streamtemperature on streamflow loss and groundwater recharge. Water Resour. Res. 30,3253.

Constantz, J., Stonestorm, D., Stewart, A.E., Niswonger, R., Smith, T.R., 2001. Analysis ofstreambed temperatures in ephemeral channels to determine streamflow frequencyand duration. Water Resour. Res. 37, 317.

Constantz, J., Stewart, A.E., Niswonger, R., Sarma, L., 2002. Analysis of temperature profilesfor investigating stream losses beneath ephemeral channels. Water Resour. Res. 38,1316.

Constantz, J., Cox, M.H., Su, G.W., 2003a. Comparison of heat and bromide as groundwater tracers near streams. Ground Water 41, 647–656.

Constantz, J., Tyler, S.W., Kwicklis, E., 2003b. Temperature-profile methods for estimatingpercolation rates in arid environments. Vadose Zone J. 2, 12–24.

Constantz, J., Eddy-Miller, C.A., Wheeler, J.D., Essaid, H.I., 2013. Streambed exchangesalong tributary streams in humid watersheds. Water Resour. Res. 49 (4),2197–2204.

Côté, J., Konrad, J.-M., 2009. Assessment of structure effects on the thermal conductivity oftwo-phase porous geomaterials. Int. J. Heat Mass Transf. 52, 796–804.

Côté, J., Fillion, M.-H., Konrad, J.-M., 2011. Estimating hydraulic and thermal conductivitiesof crushed granite using porosity and equivalent particle size. J. Geotech. Geoenviron.137, 834–842.

Cox, M.H., Su, G.W., Constantz, J., 2007. Heat, chloride, and specific conductance as groundwater tracers near streams. Ground Water 45, 187–195.

Cuthbert, M.O., Mackay, R., 2013. Impacts of nonuniform flow on estimates of verticalstreambed flux. Water Resour. Res. 49 (1), 19–28.

Cuthbert, M.O., Mackay, R., Durand, V., Aller, M.-F., Greswell, R.B., Rivett, M.O., 2010. Im-pacts of river bed gas on the hydraulic and thermal dynamics of the hyporheiczone. Adv. Water Resour. 33, 1347–1358.

De Marsily, G., 1986. Quantitative hydrogeology: groundwater hydrology for engineers,xix (440 pp.).

Diersch, H.-J.G., 2005. FEFLOW— Finite Element Subsurface Flow & Transport SimulationSystem — Reference Manual, Berlin, Germany.

Domenico, P.A., Schwartz, F.W., 1998. Physical and chemical hydrogeology.Dudgeon, C.R., Green, M.J., Smedmoor, W.J., 1975. Heat–pulse flowmeter for boreholes.Duque, C., Calvache, M.L., Engesgaard, P., 2010. Investigating river–aquifer relations using

water temperature in an anthropized environment (Motril–Salobreña aquifer).J. Hydrol. 381, 121–133.

Dwyer, T.E., Eckstein, Y., 1987. Finite-element simulation of low-temperature, heat-pump-coupled, aquifer thermal energy storage. J. Hydrol. 95 (1–2), 19–38.

Emmanuel, S., Berkowitz, B., 2007. Continuous time random walks and heat transfer inporous media. Transp. Porous Media 67, 413–430.

Essaid, H.I., Zamora, C.M., McCarthy, K.A., Vogel, J.R., Wilson, J.T., 2008. Using heat to char-acterize streambed water flux variability in four stream reaches. J. Environ. Qual. 37,1010–1023.

Evans, E.C., Petts, G.E., 1997. Hyporheic temperature patterns within riffles. Hydrol. Sci. J.42, 199–213.

Fanelli, R.M., Lautz, L.K., 2008. Patterns of water, heat, and solute flux through streambedsaround small dams. Ground Water 46, 671–687.

Ferguson, G., 2007. Heterogeneity and thermal modeling of ground water. GroundWater45, 485–490.

Ferguson, G., Bense, V., 2011. Uncertainty in 1D heat-flow analysis to estimate groundwa-ter discharge to a stream. Ground Water 49, 336–347.

Fourier, J.B.J., 1822. Théorie analytique de la chaleur. Didot.Furnas, C.C., 1930. Heat transfer from a gas stream to a bed of broken solids—II. Ind. Eng.

Chem. 22, 721–731.Geiger, S., Emmanuel, S., 2010. Non-Fourier thermal transport in fractured geological

media. Water Resour. Res. 46, W07504.Geiger, S., Driesner, T., Heinrich, C.A., Matthäi, S.K., 2006. Multiphase thermohaline con-

vection in the earth's crust: I. A new finite element–finite volume solution techniquecombined with a new equation of state for NaCl–H2O. Transp. Porous Media 63,399–434.

Gerecht, K.E., Cardenas, M.B., Guswa, A.J., Sawyer, A.H., Nowinski, J.D., Swanson, T.E., 2011.Dynamics of hyporheic flow and heat transport across a bed-to-bank continuum in alarge regulated river. Water Resour. Res. 47, W03524.

Gordon, R.P., Lautz, L.K., Briggs, M.A., McKenzie, J.M., 2012. Automated calculation of ver-tical pore-water flux from field temperature time series using the VFLUXmethod andcomputer program. J. Hydrol. 420–421, 142–158.

Goto, S., Yamano, M., Kinoshita, M., 2005. Thermal response of sediment with verticalfluid flow to periodic temperature variation at the surface. J. Geophys. Res. SolidEarth 110, B01106.

Green, D.W., 1962. Heat transfer with a flowing fluid through porous media.Green, D.W., Perry, R.H., Babcock, R.E., 1964. Longitudinal dispersion of thermal energy

through porous media with a flowing fluid. AIChE J 10, 645–651.Guaraglia, D.O., Pousa, J.L., 2006. Decreasing measuring errors of a thermal groundwater

flowmeter. J. Irrig. Drain. Eng. 132, 284–287.Guaraglia, D.O., Pousa, J.L., 2007. A method to improve flow-velocity measurements from

an array of partially cosine response sensors. IEEE Trans. Instrum. Meas. 56,1721–1724.

Guaraglia, D.O., Mayosky, M.A., Pousa, J.L., Kruse, E.E., 2008. Numerical and experimentalstudy of a thermal probe for measuring groundwater velocity. Rev. Sci. Instrum. 79,15102.

Guaraglia, D.O., Pousa, J.L., Kruse, E.E., Mayosky, M.A., 2009. A rotary thermal probe formeasuring groundwater velocity. Instrum. Sci. Technol. 37, 303–318.

Hannah, D.M., Malcolm, I.A., Bradley, C., 2009. Seasonal hyporheic temperature dynamicsover riffle bedforms. Hydrol. Process. 23, 2178–2194.

Hashin, Z., Shtrikman, S., 1962. A variational approach to the theory of the effective mag-netic permeability of multiphase materials. J. Appl. Phys. 33, 3125.

Hatch, C.E., Fisher, A.T., Revenaugh, J.S., Constantz, J., Ruehl, C., 2006. Quantifying surfacewater–groundwater interactions using time series analysis of streambed thermal re-cords: method development. Water Resour. Res. 42, W10410.

Hatch, C.E., Fisher, A.T., Ruehl, C.R., Stemler, G., 2010. Spatial and temporal variations instreambed hydraulic conductivity quantified with time-series thermal methods.J. Hydrol. 389, 276–288.

Healy, R.W., Ronan, A.D., 1996. Documentation of computer program VS2DH for simula-tion of energy transport in variably saturated porous media.

Hess, A.E., 1986. Identifying hydraulically conductive fractures with a slow-velocity bore-hole flowmeter. Can. Geotech. J. 23, 69–78.

Hidalgo, J.J., Carrera, J., Dentz, M., 2009. Steady state heat transport in 3D heterogeneousporous media. Adv. Water Resour. 32, 1206–1212.

Holzbecher, E., 2005. Inversion of temperature time series from near-surface porous sed-iments. J. Geophys. Eng. 2, 343–348.

Hopmans, J.W., Šimunek, J., Bristow, K., 2002. Indirect estimation of soil thermal proper-ties and water flux using heat pulse probe measurements: geometry and dispersioneffects. Water Resour. Res. 38.

Horai, K.-I., 1971. Thermal conductivity of rock-forming minerals. J. Geophys. Res. 76,1278–1308.

Horai, K.-I., Simmons, G., 1969. Thermal conductivity of rock-forming minerals. EarthPlanet. Sci. Lett. 6, 359–368.

Hsiao, K.-T., Advani, S.G., 1999. A theory to describe heat transfer during laminar incom-pressible flow of a fluid in periodic porous media. Phys. Fluids 11, 1738.

Ingebritsen, S.E., Sanford, W.E., 1998. Groundwater in geologic processes. 341.Jensen, J.K., Engesgaard, P., 2011. Nonuniform groundwater discharge across a streambed:

heat as a tracer. Vadose Zone J. 10, 98.Johnson, A.N., Boer, B.R., Woessner, W.W., Stanford, J.A., Poole, G.C., Thomas, S.A.,

O'Daniel, S.J., 2005. Evaluation of an inexpensive small-diameter temperaturelogger for documenting ground water–river interactions. Ground Water Monit.Rem. 25, 68–74.

Kalbus, E., Reinstorf, F., Schirmer, M., 2006. Measuring methods for groundwater–surfacewater interactions: a review. Hydrol. Earth Syst. Sci. 10, 873–887.

Kamai, T., Tuli, A., Kluitenberg, G.J., Hopmans, J.W., 2008. Soil water flux density measure-ments near 1 cm d − 1 using an improved heat pulse probe design. Water Resour.Res. 44, W00D14.

Kaviany, M., 1995. Principles of heat transfer in porous media, xxii (708 pp.).Keery, J., Binley, A., Crook, N., Smith, J.W.N., 2007. Temporal and spatial variability of

groundwater–surface water fluxes: development and application of an analyticalmethod using temperature time series. J. Hydrol. 336, 1–16.

Kim, J., Park, Y., Harmon, T.C., 2005. Real-time model parameter estimation for analyzingtransport in porous media. Ground Water Monit. Rem. 25, 78–86.

Kluitenberg, G.J., Warrick, A.W., 2001. Improved evaluation procedure for heat-pulse soilwater flux density method. Soil Sci. Soc. Am. J. 65, 320.

Kluitenberg, G.J., Ochsner, T.E., Horton, R., 2007. Improved Analysis of Heat Pulse Signalsfor Soil Water Flux Determination. Soil Sci. Soc. Am. J. 71, 53.

Kozeny, J., 1927. Über kapillare Leitung des Wassers im Boden. Sitzungsber. Akad. Wiss.,Wien, 136: 271-306.

Krause, S., Hannah, D.M., Blume, T., 2011a. Interstitial pore-water temperature dynamicsacross a pool-riffle-pool sequence. Ecohydrology 4, 549–563.

Krause, S., Hannah, D.M., Fleckenstein, J.H., Heppell, C.M., Kaeser, D., Pickup, R., Pinay, G.,Robertson, A.L., Wood, P.J., 2011b. Inter-disciplinary perspectives on processes in thehyporheic zone. Ecohydrology 4, 481–499.

Kulongoski, J.T., Izbicki, J.A., 2008. Simulation of fluid, heat transport to estimate desertstream infiltration. Ground Water 46, 462–474.

Kunii, D., Smith, J.M., 1961. Heat transfer characteristics of porous rocks: II. Thermal con-ductivities of unconsolidated particles with flowing fluids. AIChE J 7, 29–34.

Land, L.A., Paull, C.K., 2001. Thermal gradients as a tool for estimating groundwater advec-tive rates in a coastal estuary: White Oak River, North Carolina, USA. J. Hydrol. 248,198–215.

Langevin, C.D., Thorne, Daniel T.J., Dausman, A.M., Sukop, M.C., Guo, W., 2008. SEAWATVersion 4: A computer program for simulation ofmulti-species solute and heat trans-port. USGS Florida Integrated Science Center.

Lapham, W.W., 1989. Use of temperature profiles beneath streams to determine rates ofvertical ground-water flow and vertical hydraulic conductivity.

Lauer, F., Frede, H.-G., Breuer, L., 2013. Uncertainty assessment of quantifying spatiallyconcentrated groundwater discharge to small streams by distributed temperaturesensing. Water Resour. Res. 49 (1), 400–407.

Lautz, L.K., 2010. Impacts of nonideal field conditions on vertical water velocity estimatesfrom streambed temperature time series. Water Resour. Res. 46, W01509.

Lautz, L.K., 2012. Observing temporal patterns of vertical flux through streambed sedi-ments using time-series analysis of temperature records. J. Hydrol. 464–465,199–215.

Lautz, L.K., Ribaudo, R.E., 2012. Scaling up point-in-space heat tracing of seepage fluxusing bed temperatures as a quantitative proxy. Hydrogeol. J. 20 (7), 1223–1238.

Lautz, L.K., Kranes, N.T., Siegel, D.I., 2010. Heat tracing of heterogeneous hyporheic ex-change adjacent to in-stream geomorphic features. Hydrol. Process. 24, 3074–3086.

Lee, D.R., 1985. Method for locating sediment anomalies in lakebeds that can be caused bygroundwater flow. J. Hydrol. 79, 187–193.

Levec, J., Carbonell, R.G., 1985a. Longitudinal and lateral thermal dispersion in packedbeds. Part I: theory. AIChE J 31, 581–590.

Levec, J., Carbonell, R.G., 1985b. Longitudinal and lateral thermal dispersion in packedbeds. Part II: comparison between theory and experiment. AIChE J 31, 591–602.

57G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

Lewandowski, J., Angermann, L., Nützmann, G., Fleckenstein, J.H., 2011. A heat pulse tech-nique for the determination of small-scale flow directions and flow velocities in thestreambed of sand-bed streams. Hydrol. Process. 25, 3244–3255.

Liu, G., Knobbe, S., Butler, J.J., 2013. Resolving centimeter-scale flows in aquifers and theirhydrostratigraphic controls. Geophys. Res. Lett. 40 (6), 1098–1103.

Loheide, S.P., Gorelick, S.M., 2006. Quantifying stream–aquifer interactions through theanalysis of remotely sensed thermographic profiles and in situ temperature histories.Environ. Sci. Technol. 40, 3336–3341.

Lowry, C.S., Walker, J.F., Hunt, R.J., Anderson, M.P., 2007. Identifying spatial variability ofgroundwater discharge in a wetland stream using a distributed temperature sensor.Water Resour. Res. 43, W10408.

Lu, X., Ren, T., Gong, Y., 2009. Experimental investigation of thermal dispersion in saturatedsoils with one-dimensional water flow. Soil Sci. Soc. Am. J. 73 (6), 1912–1920.

Luce, C.H., Tonina, D., Gariglio, F., Applebee, R., 2013. Solutions for the diurnally forced ad-vection–diffusion equation to estimate bulk fluid velocity and diffusivity in stream-beds from temperature time series. Water Resour. Res. 49 (1), 488–506.

Malard, F., Mangin, A., Uehlinger, U., Ward, J.V., 2001. Thermal heterogeneity in thehyporheic zone of a glacial floodplain. Can. J. Fish. Aquat. Sci. 58, 1319–1335.

Malard, F., Tockner, K., Dole-olivier, M.-j., Ward, J.V., 2002. A landscape perspective of sur-face-subsurface hydrological exchanges in river corridors. Freshwater Biology 47,621–640.

Markle, J.M., Schincariol, R.A., Sass, J.H., Molson, J.W., 2006. Characterizing the two-dimen-sional thermal conductivity distribution in a sand and gravel aquifer. Soil Sci. Soc. Am.J. 70, 1281.

McCallum, A.M., Andersen, M.S., Rau, G.C., Acworth, R.I., 2012. A 1D analytical method forestimating surface water groundwater interactions and effective thermal diffusivityusing temperature time series. Water Resour. Res. 48, W11532.

Metzger, T., Didierjean, S., Maillet, D., 2004. Optimal experimental estimation of thermaldispersion coefficients in porous media. Int. J. Heat Mass Transf. 47, 3341–3353.

Midttomme, K., Roaldset, E., 1998. The effect of grain size on thermal conductivity ofquartz sands and silts. Pet. Geosci. 4, 165–172.

Molina-Giraldo, N., Bayer, P., Blum, P., 2011. Evaluating the influence of thermal disper-sion on temperature plumes from geothermal systems using analytical solutions.Int. J. Therm. Sci. 50, 1223–1231.

Molson, J.W., Frind, E.O., Palmer, C.D., 1992. Thermal energy storage in an unconfinedaquifer: 2. Model development, validation, and application. Water Resour. Res. 28(10), 2857–2867.

Mori, Y., Hopmans, J.W., Mortensen, A.P., Kluitenberg, G.J., 2003. Multi-functional heatpulse probe for the simultaneous measurement of soil water content, solute concen-tration, and heat transport parameters. Vadose Zone J. 2, 561–571.

Mori, Y., Hopmans, J.W., Mortensen, A.P., Kluitenberg, G.J., 2005. Estimation of vadosezone water flux from multi-functional heat pulse probe measurements. Soil Sci.Soc. Am. J. 69, 599.

Mortensen, A.P., Hopmans, J.W., Mori, Y., Šimůnek, J., 2006. Multi-functional heat pulseprobe measurements of coupled vadose zone flow and transport. Adv. Water Resour.29, 250–267.

Munz, M., Oswald, S.E., Schmidt, C., 2011. Sand box experiments to evaluate the influenceof subsurface temperature probe design on temperature based water flux calculation.Hydrol. Earth Syst. Sci. Discuss. 8, 6155–6197.

Muskat, M., 1937. The flow of fluids through porous media. J. Appl. Phys. 8, 274.Mutiti, S., Levy, J., 2010. Using temperature modeling to investigate the temporal variabil-

ity of riverbed hydraulic conductivity during storm events. J. Hydrol. 388, 321–334.Naranjo, R.C., Pohll, G., Niswonger, R.G., Stone, M., McKay, A., 2013. Using heat as a tracer

to estimate spatially distributed mean residence times in the hyporheic zone of ariffle-pool sequence. Water Resour. Res. 49 (6), 3697–3711.

Nield, D.A., Bejan, A., 1992. Convection in porous media.Ochsner, T.E., Horton, R., Ren, T., 2001. A new perspective on soil thermal properties. Soil

Sci. Soc. Am. J. 65, 1641.Ochsner, T.E., Horton, R., Kluitenberg, G.J., Wang, Q., 2005. Evaluation of the heat pulse

ratio method for measuring soil water flux. Soil Sci. Soc. Am. J. 69, 757.Oldenburg, C.M., Pruess, K., 1998. Layered thermohaline convection in hypersaline

geothermal systems. Transp. Porous Media 33, 29–63.Onderka, M., Banzhaf, S., Scheytt, T., Krein, A., 2013. Seepage velocities derived from

thermal records using wavelet analysis. J. Hydrol. 479, 64–74.Palmer, C.D., Blowes, D.W., Frind, E.O., Molson, J.W., 1992. Thermal energy storage in an

unconfined aquifer: 1. Field Injection Experiment. Water Resour. Res. 28, 2845.Pfister, L., McDonnell, J.J., Hissler, C., Hoffmann, L., 2010. Ground-based thermal imagery

as a simple, practical tool for mapping saturated area connectivity and dynamics.Hydrol. Process. 24, 3123–3132.

Prats, J., Ramos, A., Armengol, J., Dolz, J., 2011. Comparison of models for calculationof diel sediment-water heat flux from water temperatures. J. Hydraul. Eng. 137,1135–1147.

Rau, G.C., Andersen, M.S., McCallum, A.M., Acworth, R.I., 2010a. Analytical methodsthat use natural heat as a tracer to quantify surface water–groundwater exchange,evaluated using field temperature records. Hydrogeol. J. 18, 1093–1110.

Rau, G.C., Andersen, M.S., McCallum, A.M., Acworth, R.I., 2010b. Uncertainty of verticalstreambed seepage rates under realistic field conditions using diel temperature fluc-tuations. 38th IAH Congress, Krakow, Poland, 12–17 Sept 2010.

Rau, G.C., Andersen, M.S., Acworth, R.I., 2012a. Experimental investigation of the thermaldispersivity term and its significance in the heat transport equation for flow in sedi-ments. Water Resour. Res. 48, W03511.

Rau, G.C., Andersen, M.S., Acworth, R.I., 2012b. Experimental investigation of the thermaltime-series method for surface water-groundwater interactions. Water Resour. Res.48, W03530.

Reddy, K.S., Karthikeyan, P., 2009. Estimation of effective thermal conductivity of two-phase materials using collocated parameter model. Heat Transfer Eng. 30, 998–1011.

Ren, T., Kluitenberg, G.J., Horton, R., 2000. Determining soil water flux and pore watervelocity by a heat pulse technique. Soil Sci. Soc. Am. J. 64, 552.

Ren, T., Ochsner, T.E., Horton, R., 2003. Development of thermo-time domain reflectome-try for vadose zone measurements. Vadose Zone J. 2, 544–551.

Revil, A., 2000. Thermal conductivity of unconsolidated sediments with geophysical appli-cations. J. Geophys. Res. 105, 16749–16768.

Ronan, A.D., Prudic, D.E., Thodal, C.E., Constantz, J., 1998. Field study and simulation of di-urnal temperature effects on infiltration and variably saturated flow beneath anephemeral stream. Water Resour. Res. 34, 2137.

Rose, L., Krause, S., Cassidy, N.J., 2013. Capabilities and limitations of tracing spatial tem-perature patterns by fiber-optic distributed temperature sensing. Water Resour.Res. 49 (3), 1741–1745.

Roshan, H., Rau, G.C., Andersen, M.S., Acworth, I.R., 2012. Use of heat as tracer to quantifyvertical streambed flow in a 2-D flow field. Water Resour. Res. 48 (10), W10508.http://dx.doi.org/10.1029/2012WR011918.

Saar, M.O., 2011. Review: geothermal heat as a tracer of large-scale groundwater flow andas a means to determine permeability fields. Hydrogeol. J. 19, 31–52.

Saffman, P.G., 1959. A theory of dispersion in a porous medium. J. Fluid Mech. 6, 321.Saito, H., Šimůnek, J., Hopmans, J.W., Tuli, A., 2007. Numerical evaluation of alternative

heat pulse probe designs and analyses. Water Resour. Res. 43, W07408.Samantray, P.K., Karthikeyan, P., Reddy, K.S., 2006. Estimating effective thermal conduc-

tivity of two-phase materials. Int. J. Heat Mass Transf. 49, 4209–4219.Sauty, J.P., 1978. The effect of thermal dispersion on injection of hot water in aquifers.

Well testing Symposium, 2nd, Berkeley, California.Sauty, J.P., Gringarten, A.C., Menjoz, A., Landel, P.A., 1982a. Sensible energy storage in

aquifers: 1. Theoretical study. Water Resour. Res. 18, 245.Sauty, J.P., Gringarten, A.C., Fabris, H., Thiery, D., Menjoz, A., Landel, P.A., 1982b. Sensible

energy storage in aquifers: 2. Field experiments and comparison with theoretical re-sults. Water Resour. Res. 18, 253.

Schärli, U., Rybach, L., 2001. Determination of specific heat capacity on rock fragments.Geothermics 30, 93–110.

Scheidegger, A.E., 1961. General theory of dispersion in porous media. J. Geophys. Res. 66,3273–3278.

Schmidt, C., Bayer-Raich, M., Schirmer, M., 2006. Characterization of spatial hetero-geneity of groundwater-stream water interactions using multiple depth stream-bed temperature measurements at the reach scale. Hydrol. Earth Syst. Sci.Discuss. 3, 1419–1446.

Schmidt, C., Conant, B., Bayer-Raich, M., Schirmer, M., 2007. Evaluation and field-scale ap-plication of an analytical method to quantify groundwater discharge using mappedstreambed temperatures. J. Hydrol. 347, 292–307.

Schön, J.H., 1996. Physical properties of rocks: fundamentals and principles ofpetrophysics. 583.

Schornberg, C., Schmidt, C., Kalbus, E., Fleckenstein, J.H., 2010. Simulating the effects ofgeologic heterogeneity and transient boundary conditions on streambed tempera-tures — implications for temperature-based water flux calculations. Adv. WaterResour. 33, 1309–1319.

Schuetz, T., Weiler, M., 2011. Quantification of localized groundwater inflow into streamsusing ground-based infrared thermography. Geophys. Res. Lett. 38 (3), L03401.

Schumann, T.E.W., 1929. Heat transfer: a liquid flowing through a porous prism. J. Frankl.Inst. 208, 405–416.

Selker, J., van de Giesen, N., Westhoff, M., Luxemburg, W., Parlange, M.B., 2006a. Fiber op-tics opens window on stream dynamics. Geophys. Res. Lett. 33, 4.

Selker, J.S., Thévenaz, L., Huwald, H., Mallet, A., Luxemburg, W., van de Giesen, N., Stejskal,M., Zeman, J., Westhoff, M., Parlange,M.B., 2006b. Distributed fiber-optic temperaturesensing for hydrologic systems. Water Resour. Res. 42, 8.

Shanafield, M., Pohll, G., Susfalk, R., 2010. Use of heat-based vertical fluxes to approximatetotal flux in simple channels. Water Resour. Res. 46, W03508.

Shanafield, M., Hatch, C., Pohll, G., 2011. Uncertainty in thermal time series analysis esti-mates of streambed water flux. Water Resour. Res. 47, W03504.

Shao, M., Horton, R., Jaynes, D.B., 1998. Analytical solution for one-dimensional heatconduction-convection equation. Soil Sci. Soc. Am. J. 62, 123.

Silliman, S.E., Booth, D.F., 1993. Analysis of time-series measurements of sediment tem-perature for identification of gaining vs. losing portions of Juday Creek, Indiana.J. Hydrol. 146, 131–148.

Silliman, S.E., Ramirez, J., McCabe, R.L., 1995. Quantifying downflow through creek sedi-ments using temperature time series: one-dimensional solution incorporating mea-sured surface temperature. J. Hydrol. 167, 99–119.

Sinokrot, B.A., Stefan, H.G., 1993. Stream temperature dynamics: measurements andmodeling. Water Resour. Res. 29, 2299.

Sisodia, S., Helweg, O.J., 1998. Modeling the heat sense flowmeter. J. Irrig. Drain. Eng. 124,148–151.

Smith, L., Chapman, D.S., 1983. On the thermal effects of groundwater flow 1. Regionalscale systems. J. Geophys. Res. 88, 593–608.

Sorey, M.L., 1971. Measurement of vertical groundwater velocity from temperatureprofiles in wells. Water Resour. Res. 7, 963.

Soto-López, C.D., Meixner, T., Ferré, T.P.A., 2011. Effects of measurement resolution on theanalysis of temperature time series for stream-aquifer flux estimation. Water Resour.Res. 47, W12602.

Stallman, R.W., 1963. Computation of ground-water velocity from temperature data.Methods of collecting and interpreting ground-water data (Bentall, Ray).

Stallman, R.W., 1965. Steady one-dimensional fluid flow in a semi-infinite porous medi-um with sinusoidal surface temperature. J. Geophys. Res. 70, 2821–2827.

Stonestrom, D.A., Constantz, J., 2003. Heat as a tool for studying the movement of groundwater near streams. Circular 1260, US Geological Survey, Reston, Virginia.

Su, G.W., Jasperse, J., Seymour, D., Constantz, J., 2004. Estimation of hydraulic conductivityin an alluvial system using temperatures. Ground Water 42, 890–901.

58 G.C. Rau et al. / Earth-Science Reviews 129 (2014) 40–58

Su, G.W., Freifeld, B.M., Oldenburg, C.M., Jordan, P.D., Daley, P.F., 2006. Interpreting velocitiesfrom heat-based flow sensors by numerical simulation. Ground Water 44, 386–393.

Suzuki, S., 1960. Percolation measurements based on heat flow through soil with specialreference to paddy fields. J. Geophys. Res. 65, 2883.

Swanson, T.E., Cardenas, M.B., 2010. Diel heat transport within the hyporheic zone of apool-riffle-pool sequence of a losing stream and evaluation of models for fluid fluxestimation using heat. Limnol. Oceanogr. 55, 1741–1754.

Swanson, T.E., Cardenas, M.B., 2011. Ex-Stream: a MATLAB program for calculating fluidflux through sediment–water interfaces based on steady and transient temperatureprofiles. Comput. Geosci. 37, 1664–1669.

Tabbagh, A., Bendjoudi, H., Benderitter, Y., 1999. Determination of recharge in unsaturatedsoils using temperature monitoring. Water Resour. Res. 35, 2439.

Taniguchi, M., 1993. Evaluation of vertical groundwater fluxes and thermal proper-ties of aquifers based on transient temperature-depth profiles. Water Resour.Res. 29, 2021.

Taniguchi, M., Sharma, M.L., 1990. Solute and heat transport experiments for estimatingrecharge rate. J. Hydrol. 119, 57–69.

Taniguchi, M., Sharma, M.L., 1993. Determination of groundwater recharge using thechange in soil temperature. J. Hydrol. 148, 219–229.

Taniguchi, M., Turner, J.V., Smith, A.J., 2003. Evaluations of groundwater discharge ratesfrom subsurface temperature in Cockburn Sound, Western Australia. Biogeochemistry66, 111–124.

Tarnawski, V.R., Momose, T., Leong, W.H., 2011. Thermal conductivity of standard sands II.Saturated conditions. Int. J. Thermophys. 32, 984–1005.

Therrien, R., 1992. Three-dimensional analysis of variably-saturated flow and solutetransport in discretely-fractured porous media. PhD Thesis, University of Waterloo,Waterloo, Ontario, Canada.

Vandenbohede, A., Lebbe, L., 2010a. Recharge assessment by means of vertical tempera-ture profiles: analysis of possible influences. Hydrol. Sci. J. 55, 792–804.

Vandenbohede, A., Lebbe, L., 2010b. Parameter estimation based on vertical heat trans-port in the surficial zone. Hydrogeol. J. 18, 931–943.

Vandenbohede, A., Lebbe, L., 2011. Heat transport in a coastal groundwater flow systemnear De Panne, Belgium. Hydrogeol. J. 19, 1225–1238.

Vandenbohede, A., Louwyck, A., Lebbe, L., 2009. Conservative solute versus heat transportin porous media during push–pull tests. Transp. Porous Media 76, 265–287.

Vogt, T., Schneider, P., Hahn-Woernle, L., Cirpka, O.A., 2010. Estimation of seepage ratesin a losing stream by means of fiber-optic high-resolution vertical temperatureprofiling. J. Hydrol. 380, 154–164.

Vogt, T., Schirmer, M., Cirpka, O.A., 2012. Investigating riparian groundwater flow close toa losing river using diurnal temperature oscillations at high vertical resolution.Hydrol. Earth Syst. Sci. 16, 473–487.

Voss, C.I., 1984. A finite-element simulation model for saturated–unsaturated, fluid-density-dependent ground-water flow with energy transport or chemically-reactivesingle-species solute transport.

Voytek, E.B., Drenkelfuss, A., Day-Lewis, F.D., Healy, R., Lane, J.W., Werkema, D., 2013.1DTempPro: analyzing temperature profiles for groundwater/surface-water exchange.Ground Water.

Wakao, N., Kagei, S.O., 1982. Heat and mass transfer in packed beds, xxii 364 pp.

Wang, Q., Ochsner, T.E., Horton, R., 2002. Mathematical analysis of heat pulse signals forsoil water flux determination. Water Resour. Res. 38, 1091.

Waples, D.W., Waples, J.S., 2004a. A review and evaluation of specific heat capacities ofrocks, minerals, and subsurface fluids. Part 1: minerals and nonporous rocks. Nat.Resour. Res. 13, 97–122.

Waples, D.W., Waples, J.S., 2004b. A review and evaluation of specific heat capacities ofrocks, minerals, and subsurface fluids. Part 2: fluids and porous rocks. Nat. Resour.Res. 13, 123–130.

Webb, B.W., Hannah, D.M., Moore, R.D., Brown, L.E., Nobilis, F., 2008. Recent advances instream and river temperature research. Hydrol. Process. 22, 902–918.

Weeks, E.P., 2006. R.W. Stallman (1924–1977)—an underappreciated contributor tohydrogeology. Ground Water 44, 618–621.

Westhoff, M.C., Savenije, H.H.G., Luxemburg, W.M.J., Stelling, G.S., van de Giesen, N.C.,Selker, J.S., Pfister, L., Uhlenbrook, S., 2007. A distributed stream temperature modelusing high resolution temperature observations. Hydrol. Earth Syst. Sci. 11, 1469–1480.

Westhoff, M.C., Bogaard, T.A., Savenije, H.H.G., 2010. Quantifying the effect of in-stream rockclasts on the retardation of heat along a stream. Adv. Water Resour. 33, 1417–1425.

Westhoff, M.C., Bogaard, T.A., Savenije, H.H.G., 2011a. Quantifying spatial and temporaldischarge dynamics of an event in a first order stream, using distributed temperaturesensing. Hydrol. Earth Syst. Sci. 15, 1945–1957.

Westhoff, M.C., Gooseff, M.N., Bogaard, T.A., Savenije, H.H.G., 2011b. Quantifyinghyporheic exchange at high spatial resolution using natural temperature variationsalong a first-order stream. Water Resour. Res. 47, W10508.

Winter, T.C., Harvey, J.W., Franke, O.L., Alley, W.M., 1998. Ground water and surface water—a single resource. U.S. Geological Survey Circular 1139.

Woessner, W.W., 2000. Stream and fluvial plain ground water interactions: rescalinghydrogeologic thought. Ground Water 38, 423–429.

Woodside, W., Messmer, J.H., 1961. Thermal conductivity of porous media. I. Unconsoli-dated sands. J. Appl. Phys. 32, 1688.

Wörman, A., Riml, J., Schmadel, N., Neilson, B.T., Bottacin-Busolin, A., Heavilin, J.E., 2012.Spectral scaling of heat fluxes in streambed sediments. Geophys. Res. Lett. 39 (23),L23402.

Xu, J., Lu, T.-J., Hodson, H.P., Fleck, N.A., 2010. Analysis of thermal dispersion in an array ofparallel plates with fully-developed laminar flow. International Journal of Heat andFluid Flow 31, 57–69.

Xue, Y., Xie, C., Li, Q., 1990. Aquifer thermal energy storage: a numerical simulation of fieldexperiments in China. Water Resour. Res. 26 (10), 2365–2375.

Yagi, S., Kunii, D., Wakao, N., 1960. Studies on axial effective thermal conductivities inpacked beds. AIChE J 6, 543–546.

Yang, C., Nakayama, A., 2010. A synthesis of tortuosity and dispersion in effective thermalconductivity of porous media. Int. J. Heat Mass Transf. 53, 3222–3230.

Young, P.C., Pedregal, D.J., Tych, W., 1999. Dynamic harmonic regression. J. Forecast. 18,369–394.

Young, P.C., Taylor, C.J., Tych, W., Pedregal, D.J., 2007. The computer-aided program fortime-series analysis and identification of noisy systems. Lancaster University, UK(http://captaintoolbox.co.uk).

Zeng-Guang, Y., Somerton, W.H., Udell, K.S., 1991. Thermal dispersion in thick-walledtubes as a model of porous media. Int. J. Heat Mass Transf. 34 (11), 2715–2726.