Experimental investigation of the thermal dispersivity term and its significance in the heat transport equation for flow in sediments


  • Gabriel C. Rau,

    1. Connected Waters Initiative Research Centre, Water Research Laboratory, School of Civil and Environmental Engineering,University of New South Wales, Sydney,Australia
    2. National Centre for Groundwater Research and Training, Water Research Laboratory, School of Civil and Environmental Engineering,University of New South Wales, Sydney,Australia
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  • Martin S. Andersen,

    1. Connected Waters Initiative Research Centre, Water Research Laboratory, School of Civil and Environmental Engineering,University of New South Wales, Sydney,Australia
    2. National Centre for Groundwater Research and Training, Water Research Laboratory, School of Civil and Environmental Engineering,University of New South Wales, Sydney,Australia
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  • R. Ian Acworth

    1. Connected Waters Initiative Research Centre, Water Research Laboratory, School of Civil and Environmental Engineering,University of New South Wales, Sydney,Australia
    2. National Centre for Groundwater Research and Training, Water Research Laboratory, School of Civil and Environmental Engineering,University of New South Wales, Sydney,Australia
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[1] A review of heat and solute transport in sediments demonstrates that the use of heat as a tracer has not been experimentally evaluated under the same experimental conditions as those used for the evaluation of solute as a tracer. Furthermore, there appears to be disagreement in the earth science literature over the significance of the thermal dispersivity term. To help resolve this disagreement, detailed experimentation with typical groundwater flow velocities (Darcy range, Re < 2.5) was conducted in a specifically designed hydraulic tank containing well-sorted saturated sand. The experiment enabled, for the first time, the precise monitoring of heat and solute tracer movement from a point source in separate runs under identical solid matrix and steady state flow conditions. Experimental results demonstrate that heat transport with natural groundwater flow velocities can reach a transition zone between conduction and convection (0.5 < Pet < 2.5). The thermal dispersion behavior can be described by using a thermal dispersivity coefficient and the square of the thermal front velocity. We propose an empirical formulation for thermal dispersion with Darcy flow in natural porous media and clarify the disagreement regarding its significance. Finally, it was observed that Darcy velocities independently derived from heat and solute experimentation show a systematic discrepancy of up to 20%, and that experimental thermal dispersion results contain significant scatter.

1. Introduction

[2] Heat has been used as a tracer in a wide range of applications. These have included the estimation of multidimensional transport characteristics and parameters in packed beds [e.g., Levec and Carbonell, 1985b], soil thermal parameters and water flux measurements [e.g., Bristow, 1998; Hopmans et al., 2002], near-surface vertical flow estimates [e.g., Suzuki, 1960; Stallman, 1963, 1965; Goto et al., 2005], subsurface heat flow [e.g., Bredehoeft and Papadopulos, 1965; Bodvarsson, 1972; Saar, 2011], and thermo-haline water movement behavior when coupled with solute transport [e.g., Rosenberg and Spera, 1992; Oldenburg and Pruess, 1998, 1999; Geiger et al., 2005]. In addition to these applications, the availability of reliable automated temperature measuring devices has supported an increase in the use of heat as a natural tracer for the investigation of surface water and groundwater connectivity [e.g., Silliman et al., 1995; Becker et al., 2004; Johnson et al., 2005; Schmidt et al., 2006; Hatch et al., 2006; Keery et al., 2007; Constantz, 2008; Rau et al., 2010; Jensen and Engesgaard, 2011].

[3] The fundamental description of heat propagation in a saturated porous matrix should take account of the fact that this propagation takes place by conduction, through both the matrix and the fluid and convection that occurs only through the fluid [e.g., Lagarde, 1965; Bear, 1972; de Marsily, 1986; Ingebritsen and Sanford, 1998; Anderson, 2005]. It is worth noting here that this combination of processes is different to that which occurs in solute transport, where propagation occurs by diffusion and advection, but only through the fluid phase. For this reason, conductive heat transport should occur more rapidly than diffusive solute transport when these processes occur in the same material and are subject to the same conditions [e.g., Bear, 1972; Ingebritsen and Sanford, 1998]. On the contrary, the advective transport of heat is slower than solute transport since the heat capacity of the solids will retard the advance of the thermal front [e.g., Bodvarsson, 1972; Oldenburg and Pruess, 1998; Schoofs et al., 1999; Geiger et al., 2006]. The implications of these differences on the advective transport require investigation.

[4] In this paper we first review the mathematical description of these processes with particular emphasis on the concept of thermal dispersivity. This analysis highlights the well-known fact that the solute Peclet number differs significantly from the thermal Peclet number for the same flow field conditions. The literature contains conflicting descriptions of the thermal dispersivity and the relationship between thermal dispersivity and fluid velocity with one group of authors suggesting a linear relationship [e.g., de Marsily, 1986; Anderson, 2005; Hatch et al., 2006; Keery et al., 2007; Vandenbohede et al., 2009; Vandenbohede and Lebbe, 2010; Rau et al., 2010], while others working in the chemical engineering area have identified the possibility of a nonlinear relationship [Green et al., 1964]. To better understand the mechanics of the thermal dispersivity and to resolve this significant uncertainty concerning linearity, we describe the results of a detailed laboratory experiment that measures heat and solute transport separately, but under the same conditions, representative of naturally occurring groundwater flow systems (Re < 3, according to Bear [1972]).

2. Methodology

2.1. Mathematical Description of Solute and Heat Transport

[5] Propagation of heat and solute in water saturated porous material can be described in a general sense using a differential transport equation (DTE) that takes into account both advective and dispersive transport [e.g., de Marsily, 1986]

display math

where inline image represents either the fluid solute concentration in Fick-type solute transport or the bulk temperature in Fourier-type heat transport, inline image represents solute dispersion or thermal dispersion, inline image is velocity of either the solute or the thermal front [m s−1], with inline image representing time [s]. The general conditions that apply [Stallman, 1963, 1965] are: (a) fluid flow is steady state and uniform in one direction and (b) characteristics of the fluid and medium are constant in space (homogeneity and isotropy) and time.

[6] The DTE thus represents a macroscopic description assuming a representative elementary volume (REV) large enough that the transport can be described with volume average parameters [Bear, 1972].

2.1.1. Solute Transport

[7] For solute transport in saturated sand, inline image is the concentration inline image [kg m−3]. The solute front velocity (pore water or interstitial velocity) is given by

display math

where the superscript inline image denotes solute, inline image represents specific discharge (Darcy velocity) [m s−1], and inline image effective porosity. The solute dispersion term ( inline image) is given by [e.g., de Marsily, 1986]

display math

where the subscript inline image represents solute dispersion in the longitudinal direction, inline image is the diffusion coefficient of the solute in water [m2 s−1], and inline image is the longitudinal hydrodynamic solute dispersivity coefficient [m].

[8] The contributions of both advective and diffusive transport can be characterized by the dimensionless solute Peclet number defined as [e.g., Bear, 1972; de Marsily, 1986]

display math

where inline image is a characteristic length [m]. Transport is advection dominated for inline image and diffusion dominated for inline image with regard to the characteristic length. In this study, inline image has been equated to the mean the grain size ( inline image) of the sand.

2.1.2. Heat Transport

[9] For heat transport, inline image is the temperature inline image [°C]. A further condition of the DTE is that temperatures at the boundary interface between solid and liquid need to be the same (local thermal equilibrium) [Stallman, 1965; Ingebritsen and Sanford, 1998]. The use of equation (1) in this form has been justified by Levec and Carbonell [1985a]. The thermal front velocity becomes

display math

where the superscript inline image denotes “thermal,” inline image is the specific volumetric heat capacity of water [J/m3/°C], inline image is the specific heat capacity of the bulk volume [J/m3/°C] defined as [Buntebarth and Schopper, 1998; Schärli and Rybach, 2001]

display math

[10] In this equation, inline image represents the specific volumetric heat capacity of the solids [J/m/°C]. Note that the Darcy velocity (q) in equation (5) is the same as that in equation (2). For this reason, the thermal front velocity (equation (5)) can also be expressed in terms of the solute front velocity (equation (2))

display math

where it is evident that the thermal front velocity is retarded compared to the solute front velocity [e.g., Bodvarsson, 1972; Oldenburg and Pruess, 1998; Geiger et al., 2006]. The thermal dispersion coefficient (or effective thermal diffusivity) inline image is given as [de Marsily, 1986]

display math

where subscript inline image denotes the longitudinal and subscript inline image the transverse effective thermal diffusivity [m2 s−1]. The parameter inline image represents the longitudinal and transverse thermal dispersivity. inline image is the bulk thermal conductivity calculated as [Woodside and Messmer, 1961]

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for a saturated porous matrix with random geometry, inline image is the thermal conductivity of water [W−1 m−1 °C], and inline image is thermal conductivity of the solids [W−1 m−1 °C]. Recently, Tarnawski et al. [2011] have experimentally evaluated and confirmed equation (9) for sands under saturated conditions.

[11] Analogous to solute transport, the thermal Peclet number is defined as [e.g., de Marsily, 1986; Anderson, 2005]

display math

[12] Transport is convection dominated for inline image and conduction dominated for inline image. It has previously been pointed out that solute diffusion is slower by several orders of magnitude compared to heat diffusion [e.g., Bear, 1972; de Marsily, 1986; Ingebritsen and Sanford, 1998; Anderson, 2005]. The values of the solute and thermal Peclet numbers therefore must differ by orders of magnitude for the same Darcy flow velocity [e.g., de Marsily, 1986]. This is of importance when solute and thermal dispersion are compared. As an example, for natural sediments and a typical groundwater flow condition, de Marsily [1986] reports a thermal Peclet number of 0.1 and a solute Peclet number of 50.

[13] It has been suggested that thermal dispersivity ( inline image) is important due to enhanced spreading of thermal energy caused by forced convection and thus should be part of the mathematical description, in analogy to solute dispersivity [e.g., de Marsily, 1986]. Accordingly, a number of authors have incorporated this term into their models in a similar manner to solute dispersivity [e.g., Smith and Chapman, 1983; Hopmans et al., 2002; Niswonger and Prudic, 2003]. Other authors claim that as heat propagates through both phases, the hydrodynamic dispersivity mechanism is inappropriate because the enhanced thermal spreading is either negligible or can be described simply by increasing the effective diffusivity [Bear, 1972; Ingebritsen and Sanford, 1998]. This divergence in opinion has previously been identified in the earth science community [e.g., de Marsily, 1986; Anderson, 2005; Hatch et al., 2006; Keery et al., 2007; Vandenbohede et al., 2009; Vandenbohede and Lebbe, 2010; Rau et al., 2010].

[14] Green et al. [1964] conducted laboratory experiments in ideal porous media (glass spheres) with advective flow to derive the effective thermal conductivity. They suggested the following empirical relationship for the thermal dispersion coefficient

display math

[15] In this equation, inline image is the thermal dispersion (or diffusivity) at zero flow. Both inline image and inline image are coefficients that were used to describe the nonlinear increase of the overall heat dispersion with flow velocity. The power law for thermal dispersion has been confirmed with experiments in ideal porous media (i.e., glass spheres) by Levec and Carbonell [1985b], and more recently by Metzger et al. [2004] and Testu et al. [2007].

[16] This finding has not been used for heat tracing in the earth sciences for natural materials. Furthermore, heat as a tracer has not been evaluated by independent means such as solute tracing under identical conditions. Inspection of the respective transport Peclet numbers reveals that the treatment of thermal dispersivity and its contribution to the overall thermal dispersion process in heat tracing requires clarification.

2.2. Hydraulic Laboratory Experiment

[17] To generate experimental data that could be used to verify the theoretical analysis, a laboratory experiment was designed in which the solute and thermal properties could both be evaluated in the same porous media subject to the same source term and boundary conditions. The core of the hydraulic experiment was composed of a rectangular Plexiglas tank (sand tank in Figure 1) with inner dimensions of 0.96 × 0.96 × 0.4 m (W × H × D). Its main function was to provide a variable but uniform flow field to a fully water-saturated sand mass. To achieve this, each of the four sides (top, right, bottom, left) featured 10 flow distribution chambers (Figure 1) where hydraulic access to the main manifold could be individually switched on or off using shut-off valves. Each of the 10 flow chambers allowed equal water pressure to be applied to the sand mass across the entire depth and width of the sand in the tank. However, in the results reported in this study, only a vertical flow field was applied to the sand (i.e., access to the side panels were switched off). In each chamber water could freely pass through a large number of perforations through the Plexiglas into the sediment (Figure 1). A permeable nylon mesh was placed on the inside of the Plexiglas to prevent the sand from leaking into the flow chambers. The front and the back of the tank were covered with 20 mm of thick, clear Plexiglas sheets. The back sheet was perforated to allow access for instrumentation. A system of steel bars and clamps on the outside of the tank provided mechanical support and stability when the tank was under pressure load. The final configuration can be seen in Figure 2a.

Figure 1.

Hydraulic setup of laboratory experiment. Flow through the tank is driven by difference in hydraulic head (ΔH) and regulated by the flow regulation valve.

Figure 2.

(a) Picture of experiment tank in preparation with water flushing from the bottom up. (b) Image of nine temperature (T) and seven EC sensors (EC) mounted on three carbon fiber tubes (dimension in mm) for horizontal installation in the tank.

[18] Figure 1 gives an overview of the hydraulics for the experimental setup. Water is fed through an inflow manifold to all top chambers on one side. The chambers that are at the bottom of the tank were collected in an identical outflow manifold. The inflow manifold was connected to a constant head tank located ∼4 m above the top of the sand tank that maintained a constant head at all times. For this investigation, all valves at the top and the bottom were opened and all side valves were closed, which forced water to flow vertically downward through the material in the tank. A pump continuously circulated water between a large storage tank and the constant head tank. The outflow manifold was hydraulically connected to a second constant head tank maintaining a head slightly above the top of the main tank to prevent the potential development of unsaturated conditions. The manifold was dimensioned with a sufficient diameter (200 mm) to prevent hydraulic head loss at fast flows between the connections to the different flow chambers, thus ensuring a uniform flow field within the tank.

[19] The effluent was collected in a vertical column made from a large diameter PVC pipe. This provided a means of assessing the bulk volumetric flow rate through the tank by monitoring the increase in water level over time (ΔL in Figure 1). The outflow could alternatively be discharged into the storage tank and recirculated through the system.

[20] Constant head tanks were installed at fixed vertical positions and connected to the inflow and outflow manifolds as shown in Figure 1. The fixed location of the tanks only allowed a single flow rate through the sands driven by the head difference (ΔH). To overcome this restriction, a pressure dissipation valve (Stübbe MV 310, Austria) was installed on the input side that allowed variable, but stable flow rates to be established.

2.2.1. Experiment Preparation and Setup

[21] The experiment chamber was filled with well-sorted sand that had a well constrained grain size distribution ( inline image, uniformity inline image, see image of material in Figure 2b). Laboratory XRD analysis of a sand sample indicated a quartz content of 98%. The temperature and fluid electric conductivity (EC) sensors were carefully inserted through the back of the Plexiglas sheet whenever the sand fill reached the level that corresponded to the vertical probe position (Figure 2b). The porosity of the material in the tank could be determined by carefully weighing all the dry material that was placed inside the known volume of the tank (Table 1).

Table 1. Summary of Parameters as Used for the Solute and Heat Modeling Procedures
Mean grain size inline image inline image2Grain size analysis
Sand uniformity inline image inline image1.68Grain size analysis
Hydraulic conductivity inline image(m s−1)1.3E−2Measurement
Effective porosity inline image inline image0.34Experiment setup
KCl diffusion coefficient inline image(m2 s−1)1.848E−9Lobo et al. [1998]
Water dynamic viscosity inline image inline image1.0016E−3at 20°C (NIST, 2011)
Water density inline image(kg m−3)998at 20°C (NIST, 2011)
Water thermal conductivity inline image inline image°C)0.598at 20°C (NIST, 2011)
Water heat capacity inline image inline image°C)4184at 20°C (NIST, 2011)
Solid density (quartz) inline image(kg m−3)2650Schön [1996]

[22] The sand filled tank was flushed with CO2 from the bottom up to displace the air in the pore space. Water was then forced upward through the sand and continuously replenished until all CO2 was dissolved and removed from the material. This avoided entrapped air and ensured 100% water saturation, supporting a homogeneous flow field and thermal parameters that are dependent on two phases only (liquid and solid).

[23] The average uniformity of the flow field within the sand was tested by color dye experiments. First, food color was mixed through the storage tank and inlet manifold until homogenous and then the vertical flow was switched on. The moving color front could be observed through the clear Plexiglas front panel and a time lapse of the moving color front was taken using a USB camera.

[24] The entire hydraulic experiment was located in a laboratory with a constant temperature of 20°C (±1.5°C) in order to minimize the impact of unwanted ambient thermal fluctuations on the experimental runs.

2.2.2. Design of Thermal and Solute Excitation and Sensor Equipment

[25] A laboratory PC with data acquisition hardware comprosed of analog inputs and outputs (NI-6225, National Instruments, USA) and software designed on the LabVIEW 2009 platform was deployed as the data acquisition system (DAS). The software contained functionality for excitation control and the collection of experimental data. The sampling rate could be adjusted for each experiment run. In order to ensure adequate signal capture and reduce unwanted high-frequency noise in the measurements, channels were scanned at 4 kHz and values were calculated as the median of 400 samples (except for fluid EC measurements). This approach resulted in a 0.1-s time average for each sample.

[26] Thermal excitation was achieved using a point heat source designed to produce a controlled amount of heat. A small axial resistor (wire wound, Ø 3 mm, of 48.34 Ω resistance) was waterproofed with thin, thermally durable Teflon heat shrink. The resistor was centered in the sand tank with its axis along the vertical flow path (Figure 3b). This ensured the best possible radial heat dissipation and minimum obstruction to the water flow. The power (heat) input could be precisely regulated with a constant current supply by the DAS output. The voltage across the resistor was monitored for performance and allowed for an accurate calculation of heat input. Furthermore, a resistance temperature detector (Platinum RTD) was mounted directly onto the point heat source in order to monitor its temperature during operation.

Figure 3.

(a) Frontal view showing numbered probe locations with temperature (dots) and EC (rings) sensors as well as solute and heat source (squares). Dimensions are in (mm). (b) Three-dimensional view of numbered temperature (dots) and EC (rings) sensors, solute source (S), and heat source (Q) in the experimental tank. The lines represent the carbon fiber tubes holding the sensors. The 3-D coordinate system used for the data evaluation is centered at the source (x-axis in the line of flow).

[27] Temperature measurements in the sands were achieved with sensors that were specifically designed for high resolution (detection of temperature changes as small as 0.0006°C) and minimal size. Thin film platinum resistors (Pt1000, 2 × 2 mm, compliant to IEC 60751, Omega Engineering, UK) were selected as resistance temperature detectors (RTD) due to their very high output linearity, excellent long-term measurement stability, and low noise. A resistance value of 1 kΩ was chosen for the high-resolution measurements. Because of its size and low thermal mass, this type of RTD also ensures a low response time. Self-heating effects caused by excitation of the platinum RTD's were minimized by using a low excitation current of 0.1 mA.

[28] A total of 30 RTD sensors were arranged to measure temperature in the following locations: the ambient laboratory air, the inflow manifold water, the point heat source, and 27 sensors distributed in the sand tank measuring the thermal plume spreading from the point heat source (for locations see Figure 3a). The sensors were mounted on a carbon fiber tube (Ø 4 mm, SKF Bearings, Australia) long enough to reach from the back to the front of the Plexiglas sheet. Each tube had between one and three RTD's arranged so that the exact positions in relation to the centerline of flow were known. Figure 3b shows the locations of these carbon fiber tubes. The platinum sensors were mounted on the outside of the tubes (Figure 2b) and sensor connection wires were fed through a small hole drilled into the tube near the intended sensor positions. All wires were then run through the hollow inside of the tube to the outside of the tank. Waterproofing was achieved by enclosing the entire tube with thin-walled clear heat-shrink tubing. One end was plugged with waterproof epoxy (Knead It Aqua, Selleys, Australia) while the other end was left open for connecting to the DAS.

[29] For the temperature data acquisition, a 30-channel high-resolution three-wire temperature control device was custom built. The electronic equipment was designed to translate temperature values between 15°C–35°C to the maximum input range for the data acquisition system (0–10 V and 16-bit DAS resolution). The temperature offset caused by self-heating was investigated theoretically and found to be negligible over the range of measurement. The Callendar-Van Dusen relationship for platinum thermometry was applied as a correction for the slight nonlinearity of the temperature resistance relationship. All temperature sensors were simultaneously calibrated using 20 different temperature values established in a well-mixed and insulated water bath. The platinum temperature relationship is linear enough that even large deviations from the absolute temperature scale only negligibly affect the nonlinearity correction, resulting in inaccuracies that were smaller than the detection limit. The RTD assembly and monitoring equipment was able to deliver a temperature resolution of 0.0006°C; this was sufficient for the precise detection of very small temperature differences.

[30] The solute tracer was injected through a small flexible silicone tube (Ø 1 mm) mounted next to the point heat source and pointing downward in the direction of flow. The tube was mounted on the carbon fiber probe and fed through a waterproof cable gland in the back panel of the experiment setup. Two three-way valves were connected in series to the tube at the outside of the back sheet and connected to two injection syringes: one for injection of solute tracer and one for flushing the tube after injection. This setup enabled a solute pulse to be injected at the same point as the heat source.

[31] The solute pulse was tracked using three microelectrodes (Microelectrodes, USA) that measured fluid EC. By placing the EC sensors above the RTDs, fluid EC was detected at the same spatial locations as three selected centerline temperature sensors (Figures 2b and 3a). A bipolar pulse technique was used for the EC measurement because it offers fast and accurate measurements [Johnson and Enke, 1970] and allows automated acquisition of high-resolution fluid EC time series [Papadopoulos and Limniou, 2001]. The three sensors were time-multiplexed to avoid artifacts through electrical cross-talk between different sensors in the sand. The best sampling conditions were found for pulses with ±10 V potential magnitude and a total duration of 8 ms (applied potential: 2 ms negative, 2 ms positive, and 4 ms zero) applied to each electrode and at every sampling cycle.

[32] The design response time of the Pt1000 resistors in water was ∼0.4 s. However, the response times for the temperature probes used in the experiment were expected to be somewhat larger due to the construction method used. To avoid using data sets that were possibly affected by response time delays, data from sensors that showed temperature slopes smaller than 0.01°C s−1 were selected. This could be achieved by selecting data from sensors at greater distances from the heat source. The EC sensors are not affected by a measurable response time so these measurements are considered instantaneous.

2.2.3. Tank Preparation and Experimental Conditions

[33] Experiment runs were completed for pure heat conduction (no convection) and 28 different Darcy velocities between ∼0.2 and 98 m d−1, all within the range of Darcy flow with respect to the sand. The sand tank was thoroughly flushed to remove any thermal or solute gradients. The pressure dissipation valve was then adjusted to achieve a specific flow rate, which could be estimated by monitoring the change in level (ΔL in Figure 1) in the effluent column over time.

[34] For each experiment the DAS software was programmed to continuously record temperatures, pressure levels (for flow stability control), and point source power dissipation, with a sampling rate selected to adequately measure the rates of change. A step change in heat input was achieved by instantaneously activating the heat source. Experiments were then run until sensors within the temperature plume achieved steady state. After allowing enough time for the thermal pulse to disappear, the DAS was switched to record fluid EC values and a single pulse of 1 mL of KCl solution was injected into the tank at the point source. The fluid electrical conductivity of the KCl solution was 2520 μS cm−1 providing a clear signal against the background feed water of ∼300 μS cm−1. All data recorded by the DAS was automatically saved as a text file for further processing. Table 2 summarizes the experiments conducted.

Table 2. Summary of Experiments Conducted and Methodologies Applied to the Data Setsa
ExperimentNo Flow (Heat)Vertical Flow (Solute)Vertical Flow (Heat)
  • a

    Note that solute and heat tracers were applied separately but for the same flow velocity (valve setting).

Runs128 different velocities28 different velocities
Darcy velocity range0 m d−1∼0.28–98 m d−1∼0.28–98 m d−1
Solute tracer (slug input)1 ml KCl slug 2520 μS cm−1
Heat tracer (step input)1.935 W1.935–5.023 W
ParametersSolid thermal parameters, vertical sensor locationsDarcy velocity, solute dispersionDarcy velocity, thermal dispersion
Equations usedEquations (20)(22) (with q = 0)Equations (16)(19)Equations (20)(22)
Data used12 sensors (41% of total data used)3 sensorsSelection from 29 sensors (40% of total data used)

2.2.4. Modeling of Experiment Results

[35] The differential transport equation (equation (1)) has been solved by Hunt [1978] for an infinite medium with uniform flow and the initial condition inline image for inline image, and the boundary conditions inline image and inline image, where inline image is a continuous source of mass flow inline image [kg s−1] in the case of solute transport, or a continuous heat flow source inline image [W] when the heat transport is modeled.

[36] With regard to solute transport, the analytical solution is stated as [Hunt, 1978]

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[37] For heat transport, equation (10) is slightly modified to

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[38] Both formulations include the central kernel

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with the spherical radius R [m] defined by

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[39] Equations (12) and (13) presume that the center of the source is at zero; the condition inline image applies. Because of the experiment geometry and the homogeneity of the material, both lateral dispersion components are assumed equal ( inline image. Note also that the Cartesian coordinate system has been used in these derivations and that longitudinal dispersion ( inline image) is assumed to occur in the x direction, and transverse dispersion ( inline image) is assumed to occur in a plane at right angles.

[40] In this experiment, heat was changed as a step function, while solute was injected as a pulse. However, a comparison of these two boundary conditions can still be achieved using the same analytical solution. The response to a Dirac Delta-type input (solute pulse) can be integrated with respect to time and is equivalent to the response of a step function (heat step) [Yu et al., 1999]. This equivalence is used in the analysis below.

2.2.5. Parameter Estimation

[41] Estimation of model parameters was performed by fitting high-resolution temperature or fluid EC time series, as measured by the DAS, to values calculated by solute and heat modeling (see section 2.2.6) using nonlinear least squares functions in MATLAB.

[42] The parameters to be estimated during each fitting procedure must be independent of each other as a prerequisite for correct matching. This offered two possible combinations for parameter estimation: (a) sensor position and dispersion coefficient, or (b) flow velocity and dispersion coefficient. The first approach was only applied to the conduction experiments ( inline image) in order to establish the sensor positions and the unknown solid thermal parameters ( inline image and inline image) from the dispersion coefficient (equation (3)). The second approach was applied to all experimental data with flow ( inline image). The quality of fit was evaluated with standard statistical parameters such as the coefficient of determination ( inline image) and the root-mean-square error ( inline image). For better comparison of fits, the normalized root-mean-square error ( inline image) was calculated, based on the temperature step ( inline image). A summary of a priori known parameters, which were used in the modeling procedures are listed in Table 1. Coordinates of each monitoring point in the 3-D space are depicted in Figure 3b.

2.2.6. Solute Concentration Breakthrough Curves

[43] The fluid EC probe calibrations were sensitive to the pressure change due to the hydraulic loading of the sand tank. Consequently, it was not possible to calibrate the fluid electric conductivity equipment in situ. As a result, only relative fluid EC values with linear dependence on solute concentration were obtained from the DAS. For this reason it was necessary to first integrate with respect to time and then normalize the solute breakthrough curves.

[44] Solute slug movement in saturated sand is a 3-D transport problem, but as only breakthrough curves along the flow centerline were monitored, it was only possible to determine velocity and longitudinal dispersion. The method of moments (MOM) [Wakao and Kagei, 1982] directly applied to the processed solute breakthrough curves offered a convenient way of estimating the experimental pore water (and also Darcy) velocity by finding the time at which 50% of the mass breakthrough occurred. Resulting velocities were considered robust because moment analysis does not require any underlying transport theory [Yu et al., 1999]. In this case, velocity results as calculated from the set of measurement points with the greatest distance (EC1 and EC3: 0.2 m) by the MOM were used as the baseline for all further comparison between solute and heat.

[45] To determine transport parameters according to the transport equation (1), the analytical solution (12) was normalized with respect to steady state values inline image

display math

[46] This allowed for solute forward modeling (SFM) of time integrated and normalized concentrations from any upstream position to a downstream position. This relationship can be expressed, in cylindrical coordinates, as (method adapted from Silliman et al. [1995])

display math
display math
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where inline image is the initial condition ( inline image), inline image is the circular radius [m] at location inline image perpendicular to the centerline, inline image is an error parameter [1 s−1] combining the possible slug mass offset from the centerline (i.e., deviation from vertical flow) and the unknown degree of lateral dispersion, and subscripts denote location.

[47] The center of the slug mass offset ( inline image) and lateral dispersion ( inline image) cannot both be determined individually in equations (19). One, or both, of them would have to be either known a priori or determined experimentally with a dense network of off-axis measurements. Combining them into one parameter ( inline image) minimizes the error in the computation of the longitudinal dispersion. In this way the lateral component can be excluded from the estimate of the longitudinal dispersion.

[48] Solute forward modeling can be used to predict breakthrough curves downstream. This seminumerical approach is versatile because it allows modeling of arbitrary boundary problems. Fitting the measured normalized EC values to predictions from the SFM by minimizing the inline image allowed for joint estimation of Darcy velocity, longitudinal dispersion coefficients, and the error parameter.

2.2.7. Temperature Breakthrough Curves

[49] Close inspection of the heat dissipation time series confirmed that a true heat step was only attained for flows at the lower end of the experimental range. At higher-flow velocities the clarity of the step function becomes more obscure and it is necessary to use a numerical model of the heat transfer process to fit the experimental data. The point source heat output, obtained as a time series from the DAS, was taken as input in order to predict the temperature response at any temperature probe (known location) according to (adapted to method from Silliman et al. [1995])

display math
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[50] The probes were installed in a predetermined and fixed position, however, it was observed that the sand settled as the tank was saturated and pressurized. This potentially caused a slight dislocation of the measurement locations and potentially introduces additional uncertainties to the fitting procedure. The dislocation was assumed to be significant only in the direction of subsidence (downward along the x-axis). Sensor locations ( inline image) and thermal parameters ( inline image and inline image) were jointly estimated using the heat forward modeling (HFM) procedure (equations (20) and (21) with inline image) and temperature data from the conduction only experiment. Correct vertical probe locations could then be obtained (from known inline image and inline image) using

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[51] These corrected locations were used in all subsequent convective heat transport evaluations.

[52] The HFM and fitting routine using equations (20) and (21) was applied to temperature data sets from all probes and experiment runs involving movement of heat (and water) through the sand. The appropriate starting point for each time series was determined from the DAS record. The end point was visually determined from the onset of steady state conditions. During the calculations, the fitting routine was allowed to adjust the parameters of interest, Darcy velocity ( inline image), longitudinal thermal dispersion ( inline image), and lateral thermal dispersion ( inline image) within realistic bounds. Ambient laboratory conditions posed a problem during daylight hours in summer and it was necessary to repeat many of the slower experiments to achieve the required degree of accuracy. To ensure that change in ambient conditions did not bias the interpretation, only data derived from the fitting routine was used if inline image. This model is applied to the experimental data sets (see Table 2).

3. Results

[53] The results of a range of experiments that measure the temperature response to a change in heat flux are presented below. It was necessary to measure the fluid EC response to a solute pulse in the same material and under the same flow conditions to provide an independent measure of the velocity and dispersion characteristics of the sand. The solute data is considered first.

3.1. Solute Transport and Derived Parameters

[54] An example of normalized solute concentration time series are shown for a low-flow velocity of 0.898 m d−1 in Figure 4a. The solute pulse gradually spreads as it is traveling down through the pore space, consistent with equation (1). The pulse passage was measured at very high-temporal resolution and shows very little noise, enabling very accurate time integration of the signal. A time-integrated breakthrough curve and best-fit solute forward model results for the same flow velocity are shown in Figure 4b for the same experiment. Data for a faster experiment with a velocity of 15 m d−1 are shown in Figure 4c. The mean breakthrough time (equivalent to the center of solute pulse mass) for each EC probe can be accurately determined from the time integrated graph at C/C0 = 0.5 (dashed line in Figures 4b and 4c). This method is considered to deliver robust results [Wakao and Kagei, 1982; Yu et al., 1999], and calculations were therefore performed for each combination of EC probes (EC1–2, EC2–3, and EC1–3) for every experiment run.

Figure 4.

(a) Normalized measurements of solute concentration at the three centerline EC probes (EC1–EC3: 200 mm) for a water flow velocity of 0.9 m d−1. (b) Time integrated breakthrough curves (only every fifth data point) and best fit solute forward model results for the same flow velocity (EC1–2: spacing 100 mm, R2 = 0.99992; EC1–3: spacing 200 mm, R2 = 0.99992). (c) Time-integrated breakthrough curves (only every fifth data point) and best fit solute forward model results for a velocity of 15.41 m d−1 (EC1–2: spacing 100 mm, R2 = 0.99997; EC1–3: spacing 200 mm, R2 = 0.99998). Both Figures 4b and 4c show the 50% breakthrough line for travel time determination. Note that the maximum deviation between measurement and model is 2% for Figure 4b and 0.05% for Figure 4c.

[55] The pore water velocities were also estimated by directly modeling solute concentrations using the DTE (equation (1)). Applying the solute forward model (equation (17)) to the three centerline observation points (EC1, EC2, and EC3) enables three different combinations for the determination of water flow velocity and dispersion. Best fits obtained from the solute forward model for EC2 and EC3 (using EC1 data as the starting point) are illustrated in Figures 4b and 4c. The solute model predicted concentrations at downstream points well for the low flow example (Figure 4b, maximum residual is 2%), and very well for the faster velocity (Figure 4c max. residual is 0.05%). All solute derived velocities (using the MOM and SFM) and dispersion coefficients (SFM only) are plotted in Figure 5 for each set of steady state velocity experiments.

Figure 5.

Results from solute transport: (a) velocities derived from solute forward model versus velocities from method of moments, and (b) longitudinal solute dispersion, including best model fit of the dispersivity coefficient, for all three probe combinations.

[56] Velocities derived from MOM and SFM for all possible probe combinations are compared in Figure 5a and show an excellent agreement ( inline image). Solute dispersion versus velocity for all possible probe combinations is given in Figure 5b. The longitudinal solute dispersivity coefficient, according to equation (3), was evaluated by the SFM and is plotted in Figure 5b. Note that the experimental Darcy velocity range for solute transport corresponds to inline image (see top x-axis in Figure 5b).

Figure 6.

(a) Relative temperature changes for the conduction-only experiment measured with centerline probes at different vertical positions (lines without symbols from top to bottom: T8, T17, T22) and off-center probes with the same distance to heat source R ∼ 60 mm (lines with open circles, top to bottom: T5, T7, T9, T11) and R ∼ 70 mm (lines with closed circles, top to bottom: T4, T6, T10, T12). (b) Measured (only showing every 20th measurement point) and modeled temperatures at points selected for increasing distance ( inline image) to the heat source (T8: 39 mm, T2: 46 mm, T11: 60 mm, T10: 75 mm, and T17: 137 mm), (c) Computed model residuals (including all measured samples) for the same probe locations as in Figure 6b.

3.2. Heat Transport Modeling

[57] Heat conduction results were established using the tank under zero flow conditions. The point heat source in the sand tank was instantaneously switched from zero to dissipate 1.935 W. This created a step heat input. The energy input was kept low enough for temperature responses not to exceed ∼3°C in order to avoid the free convection generated by density differences. Temperature responses at all observation points were simultaneously recorded by the DAS. The relative temperature time series for selected centerline and off-center locations are plotted in Figure 6a, grouped by distance from the heat source (by the spherical radius inline image).

[58] Temperature time series data obtained from individual temperature probes were used for matching results computed with the conduction HFM procedure (according to equation (20)). During this procedure, the vertical location ( inline image), solid thermal conductivity ( inline image), and solid heat capacity ( inline image) were varied in order to find the best fit. Figure 6 shows measurements (a) and modeling results (b) for selected measurement locations ordered by increasing spatial distance to the heat source (top to bottom: 0.039, 0.046, 0.06, 0.075, and 0.13 m) as well as all the residuals (c).

[59] It must be noted that determination of the vertical spatial probe location was limited to temperature probes with a distance of less than ∼0.2 m to the point source. This was because the thermal wave spreads spherically from the heat source and was expected to reach the tank's front and back plates 0.2 m from the source at the same time. The Plexiglas walls, with their very different thermal properties, caused artifacts in the heat propagation when reached by the heat front and clearly violate the validity of the analytical solution, which assumes an infinite porous media. For this reason, only results for probes within ∼0.2 m from the point source are shown. Thermal parameters for the solid phase were derived from the fitting procedure and are shown in Table 3. The averages and standard deviations of the data shown in Table 3 were: solid thermal conductivity ( inline image) 8.387 ± 0.578 W−1 m−1 °C, solid heat capacity ( inline image) 1078 ± 25 J−1 kg−1 °C and isotropic thermal diffusivity ( inline image 1.035 × 10−6 ± 5.46 × 10−8 m2 s−1. These results are used in the heat flow modeling with convection. Figure 6c shows the residuals between measured and modeled temperature data. The majority of residual temperatures are within ±0.002°C, except for T8 and T2 at the beginning of the recording, which show residual peaks of ±0.006°C.

Table 3. Solid Thermal Parameter Results Obtained From the Conduction Model Fitting Procedurea
Temperature ProbeT8T2T11T7T9T5T28T10T4T6T12T3
  • a

    Experiment details for all evaluated probes are: power dissipation: 1.934 W, total experiment time: 10,000 s, number of values: 2001.

  • b

    Parameters were varied to find best fit between data and model.

Source distance R (m)0.0390.0460.0600.0600.0600.0620.0670.0750.0760.0770.0770.099
X coordinate (m)b0.0390.0000.0350.0340.0340.037−0.0450.0260.0280.0310.0290.000
Y coordinate (m)0.0000.047−0.0490.0000.0000.0500.050−0.0500.0500.050−0.050−0.100
Z coordinate (m)0.0000.0000.0000.050−0.0500.0000.0000.0490.049−0.050−0.0500.000
Solid therm. cond. (W/m/°C)b9.1497.5118.5659.0458.5608.3897.2418.8498.7628.1428.3658.065
Solid Heat Cap. (J/kg/°C)b104610811057105610551052109910941098111911111071
Therm. dispersion (m2 s−1)1.12E–069.61E–071.06E–061.10E–061.06E–061.05E–069.29E–071.06E–061.05E–069.94E–071.02E–061.01E–06
Temperature step (°C)0.85500.78990.50480.48770.50250.48470.46240.35470.34980.35170.35080.2292
Max. temp. slope (°C s−1)8.80E–047.00E–044.60E–045.00E–045.00E–044.40E–048.60E–043.80E–045.20E–045.00E–044.00E–044.00E–04
Quality of fit R20.999930.999950.999980.999980.999910.999970.999920.999980.999970.999880.999880.99995

[60] Recorded temperatures from selected convection experiments with Darcy velocities derived from the solute experiment of 0.899, 6.382, and 15.533 m d−1 can be seen in Figure 7. These thermal responses are a result of step heating with 2.443 W (Figure 7a) and 3.468 W (for Figures 7b and 7c). It is clear that the temperature response measured by centerline probes (0.039, 0.13, and 0.23 m) increased as flow velocity increased, while the off-center probes decreased. The off-center response in Figure 7c was almost damped to zero as a result of the dominant convective heat transport in the direction of flow.

Figure 7.

(a) Relative temperature changes for the convection experiment with a solute derived flow rate of 0.899 m d−1, (b) 6.382 m d−1, and (c) 15.533 m d−1. Location of measuring points in each panel refer to centerline positions (lines top to bottom: T8: 39 mm, T17: 137 mm, T22: 230 mm) and off-center probes with distance R from heat source ∼60 mm (lines with open circles top to bottom: T5, T7, T9, T11) and ∼70 mm (lines with closed circles top to bottom: T4, T6, T10, T12).

[61] Parameter fitting was performed for temperature data sets from all 28 experiment runs and 27 sensors wherever the measurements permitted sensible output (response time criterion for sensors and a response detectable above the background noise). Representative modeling results are plotted in Figure 8 for the same flow velocities as shown in Figure 7. It is evident that very good replication of the temperature measurements were achieved by the heat model (equation (20)). The majority of residuals lie within the bounds of ±0.002°C; however, the model also produces values with larger residuals of ∼0.014°C at the beginning of the time series where temperature slopes are greatest.

Figure 8.

Recorded (gray) and modeled (black) temperatures for convection with (solute derived) flow rates: (a) flow rate of 0.9 m d−1 (only every 20th data point is shown for clarity), (b) flow rate 6.382 m d−1 (only every 15th data point shown), and (c) flow rate 15.533 m d−1 (only every 10th data point shown). Curves were recorded with selected centerline (T8: 39 mm, T17: 137 mm, T22: 230 mm, T25: 330 mm) and off-centerline (T2: 46 mm, T10: 75 mm, T11: 60 mm) probes.

[62] Finally, results derived from applying the HFM fitting procedure to all applicable temperature data sets can be viewed in Figure 9. Darcy velocities derived from heat tracing are plotted against the solute-derived MOM velocity values in Figure 9a. It is evident that heat-derived Darcy velocity ( inline image) results offer a reasonable velocity estimate when compared to solute-derived Darcy velocity ( inline image) results ( inline image). However, there is significant deviation between heat- and solute-derived values resulting in scattering around the 1:1 line. Heat-derived velocity results from all probes deviate with an average of 20% from solute derived velocities, with heat velocities generally overestimated.

Figure 9.

(a) Darcy velocities estimated from heat tracing and solute tracing compared (R2 = 0.97620). (b) Longitudinal thermal dispersion and best-fit thermal dispersivity model (R2 = 0.7553), and (c) transverse thermal dispersion and best-fit thermal dispersivity model plotted against solute MOM derived velocities (R2 = −0.0405).

[63] Longitudinal effective thermal dispersion ( inline image) versus solute velocity is plotted in Figure 9b. Results range between ∼5 × 10−7 and 3 × 10−6 m2 s−1 and show significant scatter in particular for velocities that are smaller than 30 m d−1. For larger velocities, however, an increasing trend can be observed. An increasing trend can also be seen for the transverse effective thermal dispersion term ( inline image, but to a lesser degree as can be seen in Figure 9c. Results scatter between 5 × 10−7 and 2 × 10−6 m2 s−1 with just a small trend over the entire range of flow velocities. Note that for the range of Darcy velocities in this work, the thermal Peclet numbers are inline image (see top x-axis in Figures 9b and 9c).

4. Discussion

4.1. Solute Transport

[64] Solute transport in porous media has been thoroughly investigated and the method of moments has been used for pore water flow quantification. Consequently, the solute results are not the main focus of this paper but require consideration as solute transport data are used to verify the heat data. Solute dispersivity coefficients are specific to the material and have to be established before a comparison between heat and solute transport can be carried out.

[65] Good fits (high R2) between data and model were achieved for Darcy velocities >15 m d−1 (Figures 4c and 4d), but increasingly poor fits were observed for the lower flow velocities (Figure 4b). This tailing effect can be attributed to non-Fickian or anomalous transport [Kennedy and Lennox, 2001] caused by local scale heterogeneities [Berkowitz et al., 2006]. Darcy velocities derived from MOM and SFM between all probes compare very well with each other (Figure 5a). The excellent agreement between MOM and SFM gives high confidence in both the validity of using the DTE and in the quality of dispersion results determined by the SFM.

[66] Previous experiments [e.g., Blackwell, 1962; Hiby, 1962; Pfannkuch, 1963; Han et al., 1985] show that the overall solute dispersion coefficient ( inline image) can be approximated using a dispersivity coefficient with linear dependency on the water flow velocity (see equation (3) and Figure 5b). This is in agreement with theoretical reviews on solute transport which assume that molecular diffusion and hydrodynamic dispersivity can be superimposed [e.g., Fried and Combarnous, 1971; Bear, 1972; Koch and Brady, 1985; de Marsily, 1986]. However, the linear velocity relationship is particularly accurate for advection-dominated flow with inline image. For slower flows, the slope is slightly steeper [e.g., Bijeljic et al., 2004] as is experimentally confirmed in Figure 10.

Figure 10.

Conceptual plot adapted from Bear [1972] showing the dependency of the thermal and solute dispersion coefficients on the thermal and solute Peclet numbers. Note that the plot contains two important messages: The solute and thermal dispersion regime exhibit similar behavior when compared using the respective Peclet number (bottom axis); the Darcy flow (q) velocities used for the current heat and solute results are equal (top axis). Thermal dispersion results are from centerline probes only.

[67] The longitudinal dispersivity coefficient of inline image m determined in this study is slightly lower than the values found by Klotz et al. [1980] for porous material with similar grain size and uniformity. However, they used a slightly different model with slightly nonlinear velocity dependency.

4.2. Heat Transport

4.2.1. Theoretical Model Fits to the Experimental Data

[68] The excellent quality of the model fits obtained from the analysis of the temperature data is reflected in the small average temperature residuals of ±2 m°C (measured minus modeled). The most prominent deviations in the residuals occur at early times. It is hypothesized that these are caused by different sensor response times that are not included in the modeled values. Sensors that are closest to the heat source show the most rapid temperature change and also the largest maximum residuals, an observation that strongly supports this hypothesis. Discarding data with temperature slopes >0.01°C s−1 ensured that errors caused by these response times were greatly minimized. As a result, excellent fits with residuals of no more than 2% with regard to the temperature step were obtained.

4.2.2. Thermal Parameters

[69] The solid thermal conductivity values presented in Table 3 were higher than the average for sand minerals in general, but well within the range of values found for quartz [Clauser and Huenges, 1995; Schön, 1996]. The sand sample XRD analysis results indicated a high-quartz content of 98%, which strongly supports the thermal analysis data. The high solid thermal conductivity of quartz means that the ratio of thermal conductivity between the solid and fluid phase inline image is ∼14. A similar observation can be made for the specific heat capacity. The values derived from the different probes were very similar and slightly higher than literature results for quartzite [Waples and Waples, 2004]. The excellent model fit (see Table 3 and Figure 6) gives confidence in the accuracy of the thermal parameter calculation.

4.2.3. Darcy Velocity and Thermal Dispersion

[70] Darcy flow velocities derived from the heat experiments show a clear linear correlation to the solute-derived Darcy velocities throughout the range of experimental velocities (Figure 9a). However, there is some scatter in the data and what appears to be a consistent offset from the 1:1 line with deviations in the two sets of velocity data of up to 20%. The mechanism behind this offset is not clear; but since the offset seems to depend on the location of individual measurement points some dependence on spatial heterogeneity could be a possible explanation.

[71] Figure 9b indicates that the effective longitudinal thermal dispersion increases with increasing velocity in a nonlinear way. Results within the approximate range inline image correspond to Bear's [1972] recommendation (derived from solute transport) that both conduction and convection have to be considered in the heat transport formulation. Even results for inline image exhibit a slight dependency on velocity, which corresponds to findings by Fried and Combarnous [1971] derived from solute transport experiments.

[72] The experimental results in this study, covering the groundwater flow regime, can be approximated adequately ( inline image) by a square velocity relationship

display math

with inline image and inline imageFigures 9b and 9c).

[73] Equation (23) is in conflict with equation (8), a linear formulation commonly found and applied in earth science literature [e.g., Bear, 1972; de Marsily, 1986; Anderson, 2005; Hatch et al., 2006; Keery et al., 2007; Vandenbohede et al., 2009; Vandenbohede and Lebbe, 2010; Rau et al., 2010]. However, the experimental results in this study are conceptually in agreement with the results by Green et al. [1964] as well as the theoretical derivations by Hsu and Cheng [1990] and Kaviany [1995] for heat transport in porous media with disordered grains and low flow ( inline image). An explanation for the slight increase in the longitudinal dispersion coefficient within the conduction dominant flow regime ( inline image) could be the high conductivity ratio between the solid and fluid, as this has been found to increase the longitudinal dispersion value for low inline image (Yuan et al. [1991], as illustrated by Kaviany [1995]).

[74] To the best of our knowledge, the only previous experimental investigations in porous media heat transport that include results for low flows comparable to this investigation ( inline image) have been conducted in one-dimension for artificial materials (glass beads) by Yagi et al. [1960] and Green et al. [1964], and for sand by Kunii and Smith [1961]. Yagi et al. [1960] found a linear increase in thermal dispersion with fluid velocity with substantial experimental scatter in the results.

[75] Results from Kunii and Smith [1961], as evaluated by Yuan et al. [1991], exhibit a slight increase in dimensionless dispersion for inline image, but this was considered an experimental error and not further addressed. However, the trend shown in data reported by Kunii and Smith [1961] did match the experimental data in this study which covers the inline image range from 0 to 2.5 and demonstrates that the longitudinal thermal dispersion is significantly increasing at higher inline image.

[76] The experimental results for longitudinal thermal dispersion reported by Green et al. [1964] for slightly higher flow rates and a wider flow range (∼ inline image) also clearly demonstrate a nonlinear relationship. Green et al.'s [1964] empirical power law (equation (11)) has been confirmed independently with experiments using ideal porous media and faster flow conditions ( inline image) [Levec and Carbonell, 1985b; Metzger et al., 2004; Testu et al., 2007]. Compared to these investigations, the current study utilizes natural porous media and focuses on lower thermal Peclet numbers. The current experimental range is representative of typical groundwater conditions ( inline image. It can be concluded that heat transport does not exceed beyond a transition zone where the thermal dispersion behavior changes from being independent of the flow velocity to showing a nonlinear increase with velocity. This transition zone can be approximated with a simple square relationship (equation (23)). The approximation corresponds to setting inline image in the power law (equation (11)). Our proposed model has the advantage that only one coefficient (rather than two) is required to adequately describe thermal dispersion in groundwater heat transport. The suggested thermal dispersion description (equation (23)) offers good approximation of the current experimental results (Figure 10).

[77] Interestingly, the results presented here show a more rapid increase in thermal dispersion with velocity (Figure 10) compared to Green et al.'s [1964] results, possibly due to the larger thermal conductivity of the matrix as mentioned earlier. In a theoretical context, Yuan et al. [1991] established a square relationship and concluded that the coefficient (in this case inline image is a function of porosity as well as fluid and solid thermal parameters. This all suggests that for flow in sediments, inline image is a function of both matrix geometry and thermal properties. Therefore, it is impacted by sedimentology and in particular inhomogeneity in the sediment properties.

[78] Although the experimental data is significantly poorer due to the data scatter, the transverse effective thermal dispersion also appears to exhibit a dependency on water flow velocity (Figure 9c). With the data available it is difficult to reliably quantify the dependency of the transverse thermal dispersivity coefficient on the velocity. However, the overall magnitude is significantly lower compared to that shown for the longitudinal direction. For consistency, fitting the same square model (equation (23)) as for the longitudinal case, to all experimental data resulted in a transverse thermal dispersivity coefficient of inline image (Figure 9c). This overall result is in good agreement with theoretical derivations by Kaviany [1995], illustrating that the transverse dispersion coefficient is expected to be smaller by a factor of four compared to the longitudinal dispersion coefficient.

4.3. Implications for the Use of Heat as Tracer

[79] The results of this study are limited to problems in natural porous media with temperature gradients that allow the Fourier description as inherent to the DTE (equation (1)). The results do not apply to systems with large temperature differences, i.e., when a change in thermal parameters with temperature cannot be neglected, and systems where heat gradients cause buoyancy-driven flow. These conditions can be found in hydrothermal subsurface systems as well as flow through fractured rock [e.g., Saar, 2011]. However, it has been shown theoretically that heat transport through fractured porous rocks can be described by a Fourier approach even for steep temperature gradients as long as the matrix blocks are well connected [Geiger et al., 2010].

[80] The experimental results for heat transport in saturated sand reported here indicate good agreement with theoretical derivations for heat transport [Hsu and Cheng, 1990; Kaviany, 1995] and the experimental results in artificial materials reported by Green et al. [1964]. The well-known fact that the relevant thermal Peclet number is more than two orders of magnitude lower than the corresponding solute Peclet number for the same flow velocity and grain size (Figures 5b, 9b, and 9c) can be consulted to explain this finding. This is illustrated in Figure 10, which links solute and thermal dispersion to the respective transport Peclet numbers (bottom x-axis) for the same Darcy flow velocities (top x-axis) within the framework established by Bear [1972]. Figure 10 shows the dependency of the solute and thermal dispersion coefficient on the transport ranges as represented by the appropriate Peclet number (bottom axis). The plot reveals a conceptual similarity between solute and thermal dispersion behavior.

[81] In the region that characterizes the heat transport range, conduction is initially dominant with no dependency on velocity (e.g., for inline image). However, as apparent from this study, the Darcy region also covers the transition zone between the dominance of conduction ( inline image) and the dominance of convection ( inline image) that occurs at higher velocities. The location of the transition zone in this space is a function of the thermal properties of the solid and the sedimentology as represented by inline image in equation (23). In addition, the thermal parameters are dependent on temperature and inline imagel will therefore change for large changes in temperature. The experimental data reported by Green et al. [1964] and included in Figure 10 show the location of the transition zone at a higher inline image as a result of different physical properties. For the range of velocities that likely occur in groundwater (Darcy range inline image see the experimental heat transport regime in Figure 10), the thermal dispersion transitions between not depending on the velocity and a nonlinear increase with velocity. This transition zone corresponds to the lower end of the power law previously established by Green et al. [1964] and Metzger et al. [2004] as expressed by equation (11). However, rather than using a true power law, the dispersion behavior can be adequately described by an empirical relationship with a square dependency on the thermal Peclet number (or Darcy velocity).

[82] In the region that characterizes the experimental solute transport range, advection is dominant and is approximated with a linear relationship to the velocity. In this experiment the difference between the heat and solute transport ranges is due to the fact that thermal conduction is much faster compared to solute diffusion. This is explained by the fact that the thermal conductivity of materials is usually much higher than water [e.g., Ingebritsen and Sanford, 1998], and the fact that heat propagates through both the solid and fluid phase [e.g., Bear, 1972; de Marsily, 1986; Anderson, 2005]. The exact opposite pertains to advective heat and solute transport, meaning that the thermal front velocity is slower than the solute front velocity for the same Darcy velocity [e.g., Bodvarsson, 1972; Oldenburg and Pruess, 1998; Geiger et al., 2006]. Consequently, the relative importance between advective and diffusive heat transport is different than that for solutes, as expressed by the respective Peclet numbers. The importance of the hydrodynamic thermal dispersion process occurring at the pore scale is consequently reduced.

[83] Because of the relatively faster thermal conduction compared to the solute diffusion mechanism, Fourier-type thermal transport likely applies to faster-flow velocities than Fick-type solute transport does. Therefore, heat transport modeling has an advantage over solute transport modeling because the classical advection-dispersion equation (equation (1)) has a larger range of validity [Geiger et al., 2010]. The excellent fits between heat model and temperature measurements for all flow velocities in this study, as compared to the solute case, strongly support this argument.

[84] An important corollary of the above is that thermal dispersivity is not significant at the low thermal Peclet numbers characteristic of many groundwater studies. However, it is important to note that as flow velocity increases a point will be reached at which at least longitudinal thermal dispersivity becomes significant. This will likely occur while velocities are still laminar and within the Darcy range, but depend on the thermal Peclet number, which is influenced by the thermal properties of the material. This finding is also significant for the use of heat as a tracer in hyporheic systems where flow velocities toward the top of the streambed can be significantly higher than typical groundwater flow velocities [e.g., Winter et al., 1998].

[85] If the systematic deviation between the Darcy velocities derived from the solute and the heat data, as mentioned previously (Figure 7a), is caused by inhomogeneity in the thermal transport mechanism, then it calls for caution in using heat data for calculating Darcy flow velocities in natural environments. The sand used in this experiment has a very well constrained grain-size distribution ( inline image), and for most purposes it would be characterized as homogeneous. Thus, for natural sediments, typically being much more heterogeneous, a much larger deviation between the solute (or real) Darcy velocity and the Darcy velocity estimated from the heat data are to be expected. However, the results presented here are inconclusive regarding this point and more research is needed. Further research is also required to test the validity of the volume average heat transport description (equation (1)) for flow through naturally occurring porous media, especially for less uniform grain-size distributions. Moreover, a possible impact of this on the thermal dispersion behavior is expected (i.e., similar to the findings for solute transport by Klotz et al. [1980]) but has not yet been investigated.

5. Conclusion

[86] The work reported in this paper compares for the first time heat and solute transport under experimental conditions with identical flow fields, boundary conditions, and fluid and solid properties for typical natural groundwater flow velocities (Re < 2.5) and temperature gradients. A theoretical analysis of the governing equations has brought renewed attention to the well-known dissimilarity between the mechanism of heat and solute transport through saturated porous media. The respective transport Peclet numbers for heat and solute differ by several orders of magnitude for the same Darcy flow velocities (Figure 10).

[87] Previous laboratory experiments investigating heat and water flow through ideal porous media have shown that thermal dispersion and flow velocity are related by a power law. Experimental results from the current work illustrate that the solute and thermal dispersion behavior exhibit a similar conceptual and nonlinear evolution when normalized and compared through the respective Peclet number. The thermal dispersion behavior for Darcy-related velocities in natural porous media does not exceed beyond a transition regime. The thermal dispersion can be approximated by a thermal dispersivity coefficient and a square dependency on the thermal front velocity of the form shown in equation (23). This result deviates from the linear description of thermal dispersion, which is assumed in analogy to solute transport. The difference can be explained with the different characteristics of heat and solute transport in porous media as expressed by the respective transport Peclet numbers (Figure 10). The results indicate that for relatively uniform coarse sand the thermal dispersivity term in the thermal dispersion equation can be neglected for Pet < 0.5.

[88] This finding clarifies the disagreement in the earth science community over the significance of the thermal dispersion term and proposes a more suitable formulation for the thermal dispersivity. Results also confirm and extend on earlier experimental work by Green et al. [1964], and the proposed relationship conceptually agrees with the theoretical analyses by Hsu and Cheng [1990] and Kaviany [1995]. Furthermore, it is shown that the thermal dispersivity coefficient is dependent on the properties of the porous media matrix.

[89] The experimental data showed a systematic discrepancy of ∼20% between the Darcy velocity estimates independently derived from separate heat and solute transport experiments and that the thermal dispersion results show significant scatter. We hypothesize that both of these observations can be attributed to the differences between the heat and solute transport mechanisms caused by the heterogeneity inherent to natural materials. The work reported here establishes a better understanding of the fundamental relationship between heat transport and sediment thermal characteristics, and its significance for calculation of Darcy flow velocities based on temperature measurements.


[90] This project was funded by the Australian Government through the National Water Commission's Raising National Water Standards Program and the partners of the National Program for Sustainable Irrigation (NPSI) research fund UNS5127. The practical aspects of the experimental setup are deeply in debt to the enormous contribution by the late John Hart, and he is warmly remembered. Mark Whelan is gratefully acknowledged for his invaluable efforts in supporting the design and construction of the electronic measurement equipment. Thanks also to Sebastian Geiger and two anonymous reviewers for substantially improving this manuscript with their suggestions and comments.