Thermal Dispersivity Research Articles

Introducing a novel method for determining the effective thermal conductivity at moderate and high Péclet numbers

AbstractFlow and heat transfer in porous materials is a common topic throughout many fields, including environmental and petroleum engineering. Using a coupled approach of experimental and simulation methods, this study presents a novel method for calculating thermal conductivity. Specifically, a new approach for the measurement of thermal dispersivity with both conduction and forced convection drives is proposed. In our experiments, temperature was monitored at different points within the porous medium, providing detailed spatial temperature distributions. These measurements allowed us to calculate and report effective thermal conductivity, enhancing the accuracy of our model. Experiments with various injection rates and temperatures were conducted on a sand pack. There is a relationship between the composition and connectivity of the solid in the geometry and heat transfer. However, in the case of forced convection, the key factor is the Péclet number which is important for optimal extraction of the heat inside the geothermal reservoir according to the cooling rate. When the Péclet number is high, the permeability of the porous medium plays a significant role. The velocity of the fluid can change the effective thermal conductivity up to four orders of magnitude. Due to the thermal resistance of solid and fluid, the temperature gradient between the boundary and the centre of the geometry was seen and temperature peaks were observed in the initial stages of the experiments. The size and number of peaks at the initial stage of the experiments are highly dependent on the matrix properties, such as thermal conductivity and surface area.

Quantifying vertical streambed fluxes and streambed thermal properties using heat as a tracer during extreme hydrologic events

As a natural tracer, heat has some remarkable advantages over competing test methods and has become a popular and widely used tool for quantifying streambed fluxes. In recent decades, extreme hydrological events have become intense and frequent, resulting in irregular or complex temperature changes at the stream-streambed interface. During extreme hydrological events, most existing analytical models with periodic changing top boundary conditions are inapplicable. In this study, we present a heat-based analytical model that accounts for the arbitrary top boundary and initial conditions to estimate time-variant vertical streambed fluxes (VSFs) and streambed effective thermal diffusivity using temperature–time series from streambed sediments during extreme hydrologic events. The particle swarm optimization is used to estimate the time-variant VSFs. The Morris global sensitivity analysis method is used to test the impacts of the input parameters on heat transport in the streambed. Results show that the output temperature data is most sensitive to VSFs, followed by the volumetric heat capacity of the streambed, thermal conductivity and thermal dispersivity. The performance of the new model is tested using a numerical model with time-variant VSFs associated with extreme hydraulic events such as hurricanes and dam-controlled river conditions. Results demonstrate that despite oscillations in the estimated VSFs and fluctuations being stronger when the VSFs begin to reverse and at the peak of the VSFs, the new model of this study captures changes in VSFs under extreme hydraulic events. In addition, the frequency domain model, LPMLEn, performs satisfactorily in estimating VSFs in hurricane conditions. Application to field data demonstrates that the temperature distribution can be effectively remodeled based on the estimated VSFs and streambed effective thermal diffusivity. Both the present model and LPMLEn model could be considered as alternatives to the traditional isotope-based method in estimating VSFs during extreme hydrologic events.

Analysis of heat transfer in an aquifer thermal energy storage system: On the role of two-dimensional thermal conduction

Aquifer thermal energy storage (ATES) system plays an important role in application of renewable energy, efficient energy utilization, and reduction of CO2 emission. To realize the successful operation of ATES systems, it is essential to understand the thermal behaviors thoroughly during heat injection. Serving as a powerful tool, analytical model is widely adopted while the existing analytical models usually simplify ATES systems by ignoring aquifer transverse thermal conduction and rock longitudinal thermal conduction. This study develops a novel analytical model for an ATES system, which accounts for thermal convection, dispersion, longitudinal and transverse thermal conduction in the aquifer, and longitudinal and transverse thermal conduction in the surrounding rocks. The Green's function method is applied to derive the semi-analytical solutions of temperature. A global sensitivity analysis is conducted to analyze the relative importance of each parameter. The results indicate that ignoring the aquifer transverse thermal conductivity will overestimate the temperature in the aquifer and ignoring the caprock longitudinal thermal conductivity may underestimate the heat loss in ATES systems. Similar conclusion can be made about the bedrock longitudinal thermal conductivity. The global sensitivity analysis indicates that temperature is sensitive to the changes in the flow velocity, thermal dispersivity and aquifer thickness.

Analyzing the relationship between doublet lifetime and thermal breakthrough at the Dogger geothermal exploitation site in the Paris basin using a coupled mixed-hybrid finite element model

Reinjection of produced water is a critical aspect of managing geothermal reservoirs. The sustainability of extracted thermal energy relies on the performance of well doublets, which in turn is linked to the hydrothermal properties of the reservoir, the thermal characteristics of confining layers, and operating conditions. This study presents an advanced mixed-hybrid finite element simulator that solves coupled hydrothermal equations on unstructured grids. The model is favorably compared with the conforming finite element method at a geothermal power plant site that exploits the Dogger reservoir in the Paris basin. The study also includes a parametric sensitivity analysis, using an ensemble simulation to assess doublet lifetime, thermal breakthrough, and their ratio as performance metrics. Our findings emphasize the significant influence of horizontal permeability anisotropy and production flow rate on doublet performance, with doublet orientation relative to the principal axes of the areal permeability tensor as a key control parameter. Other influential factors include the thermal conductivity of the confining beds, while longitudinal thermal dispersivity and the total productive thickness of the reservoir are less critical. We establish, for the first time, a simple relationship between doublet lifetime and thermal breakthrough, indicating that the former is twice as long as the latter. This underscores the importance of long-term monitoring to sustain geothermal exploitation. This relationship can inform enhanced management of the Dogger reservoir at the basin scale.

Heat transfer in a fracture embedded in a finite matrix: On the role of geometries and thermal dispersivity in the fracture

Understanding heat transfer in the discrete fracture embedded in a finite rock matrix is critical to the investigation of thermal energy transfer in the fracture network and the assessment of geothermal energy recovery in the enhanced geothermal system (EGS). In this study, an analytical model is proposed for transient and steady-state thermal energy transfer in a discrete fracture embedded in a finite rock matrix. The heat transfer model accounts for thermal convection, velocity-dependent thermal dispersion and longitudinal thermal conduction in the fracture, and thermal conduction in the rock matrix. The influence of the outer boundaries of the finite rock matrix will also be considered along with thermal exchange between the fracture and rock matrix. The thermal transfer processes in those two domains are coupled by assuming the continuity of temperatures and thermal fluxes along the fracture-rock matrix interfaces. The solutions are validated with the existing analytical solution and numerical model, and proved to be robust and accurate. The study implies that: 1) the geometries, such as the fracture aperture and thickness of the rock matrix, have an important influence on the temperature distribution. The fracture temperature will change in a greater range in the wider fracture and thicker rock matrix; 2) longitudinal thermal conduction in the fracture does not have a major influence on the temperature distribution in the system, and thus can be ignored; 3) thermal dispersivity in the fracture is an important mechanism to include. Ignoring thermal dispersion in the fracture will misestimate the temperature significantly. Also, thermal conductivity in the rock matrix is a key parameter for the thermal exchange between the rock matrix, surrounding environment and fracture. When the rock matrix has a greater thermal conductivity, the temperature in the fracture is observed to change less. The analytical model proposed in this study is implemented to analyze the thermal flow-through experiments in fractured core samples by Zhao (1987) and estimate the thermal dispersivity values in the fracture. The Morris global sensitivity analysis is conducted to identify the most influential input variables and estimate the interaction effects. The global sensitivity analysis is consistent with thermal breakthrough curve analysis and indicates that the temperature in the fracture is sensitive to fracture aperture, rock matrix radius, thermal dispersivity in the fracture and thermal conductivity in the rock matrix, and insensitive to longitudinal thermal conductivity in the fracture.

Stratum temperature recovery considering groundwater advection in periodic operations of deep borehole heat exchangers

• Effect of groundwater advection on stratum temperature recovery of DBHE is studied. • Thermal impact scope of long-term and periodic operations is involved. • Critical Darcy velocity conducive to DBHE durability is discovered. • Multiple low-temperature valleys occur when the critical Darcy velocity is reached. During periodic operations of the deep borehole heat exchanger (DBHE), stratum temperature recovery is significantly affected by groundwater advection, but it has not been fully addressed. In this study, the stratum temperature recovery of long-term and periodic operations of DBHE is quantitatively analyzed via an improved analytical model developed in our previous work. To characterize its dependence on Darcy velocity, recovery rate, quasi-equilibrium time, and thermal impact scope are accordingly defined. Sensitivity analyses of thermal conductivity and thermal dispersion are also conducted. The results show that the recovery rate increases with Darcy velocity but varies slightly with heat extraction power inside the borehole. There is a critical Darcy velocity in periodic operations. When the groundwater advection velocity reaches the critical value, stratum temperature at the borehole wall gets close to its initial state after every recovery period, and multiple low-temperature valleys downstream of the groundwater advection occur. The greater thermal conductivity of medium and thermal dispersivity can result in the higher critical Darcy velocity. The thermal impact scope of the DBHE decreases overall when Darcy velocity increases, while the downstream thermal impact radius has a sharp increase for low Darcy velocities before the decline.

Analytical Model for Heat Transfer in a Discrete Parallel Fracture-Rock Matrix System.

Mass transport and heat transfer in the single fracture situated in the rock matrix have been investigated extensively in the past decades. Extended from the single fracture, the model of parallel fractures in the rock matrix considers the interactions of multiple fractures and the ambient rock matrix. Heat transfer in such discrete fractures is important to understand thermal energy transfer in the fractured porous media. In this study, an analytical solution is developed for transient heat transfer in discrete parallel fractures in the rock matrix. The newly proposed model accounts for thermal convection, conduction, and dispersion in the fractures, transverse thermal conduction in the rock matrix, and the interactions between parallel fractures. The analytical solutions of the spatiotemporal temperature distributions in the fractures and rock matrix are derived in the Laplace domain and verified with the previous study. The results illustrate that: (1) the fracture aperture and spacing are important to the temperature evolutions in the system. Heat transfers faster when discrete parallel fractures are wide and closely spaced; (2) different roles of longitudinal thermal conduction are observed at high and low flow velocities in the fractures; (3) thermal dispersivity in the fractures is important for temperature evolution and should not be ignored; (4) when the fractures are closely spaced, transverse thermal conduction in the rock matrix has minor influence on fracture temperature. It becomes important when the fractures are sparsely distributed; and (5) the sensitivity analysis indicates that the parallel fracture-rock matrix is most sensitive to fracture thermal dispersivity.

On the role of rock matrix to heat transfer in a fracture-rock matrix system

In this study, a fully coupled analytical model is developed for thermal energy transfer in a single fracture-rock matrix system where the coupling implies that the governing equations of thermal transfer in the fracture and rock matrix are supplemented with the continuity conditions of temperature and thermal flux at the interfaces of the fracture-rock matrix. The proposed model accounts for thermal convection, longitudinal thermal conduction and thermal dispersion in the fracture, and transverse thermal conduction in the rock matrix. The fully coupled two-dimensional model is established to investigate the thermal energy transfer processes, assess the spatiotemporal temperature distribution in the fracture and rock matrix system and investigate the role of the rock matrix. The solutions are verified with the existing studies and proven to be accurate and robust. The present study demonstrates that: 1) thermal dispersion in the fracture plays an important role in the temperature distribution in the fracture and rock matrix domains, and longitudinal thermal conduction in the fracture has minor effects on the temperature distribution in the system; 2) transverse thermal conduction in the rock matrix is a critical parameter that determines the spatiotemporal temperature distribution in both the fracture and the rock matrix domains. Ignoring thermal conduction in the rock matrix will lead to a significant overestimation of temperature in the short and long terms; 3) the sensitivity analysis implies that thermal energy transfer in the system is sensitive to the fluid velocity in the fracture, thermal dispersivity in the fracture and thermal conductivity in the rock matrix, and less sensitive to thermal conductivity in the fracture.

The impact of porous medium heterogeneity on the thermal feedback of open-loop shallow geothermal systems

Groundwater has been increasingly used to provide low-carbon heating and cooling of buildings with open-loop shallow geothermal systems. Water is generally reinjected into the same aquifer after the heat exchange in order to avoid the aquifer depletion. However, this can result in the return of part of the injected water to the production well(s), causing a gradual thermal alteration known as thermal feedback. Thermal feedback is a major design issue of open-loop shallow systems but, so far, it has been mainly addressed neglecting the heterogeneity of the aquifer properties. This study investigates the impact of aquifer heterogeneity on two main metrics that characterize thermal feedback: thermal breakthrough time (i.e., the first arrival time of the thermal plume) and recirculating ratio (i.e., the fraction of water coming back to production well). A stochastic approach was adopted performing a large number of numerical simulations that cover a wide range of possible scenarios. The results highlight that conductivity heterogeneity plays a major influence on the temperature evolution at the production well. The breakthrough time alone might lead to misleading evaluations of the system efficiency, given that a few particles can reach the production well by traveling in the highly-conductive layers. Conversely, both the heterogeneity and the thermal dispersivity have a negligible impact on the recirculating ratio, which quantifies the long-term evolution of thermal feedback. As a consequence, the available approaches based on advection-only and homogeneous medium are a robust tool to predict the long-term behavior of shallow open-loop geothermal systems.

Evaluating anisotropy ratio of thermal dispersivity affecting geometry of plumes generated by aquifer thermal use

Studies on shallow geothermal applications have demonstrated that mechanical thermal dispersion is an important heat transport mechanism in saturated porous media. However, most previous studies have assumed the thermal dispersivity ratio, which has not been sufficiently examined. This study investigates the validity of the general assumption that the transverse dispersivity is one-tenth of the longitudinal dispersivity. For this purpose, heat transport experiments, specifically designed at the laboratory scale for estimating dispersivity, were conducted using two different heat sources at Darcy fluxes of 4.74 to 37.47 m/d (Reynolds number Re < 0.3). The results from the analytical models showed that the dispersivity ratios differed from the thermal dispersivity assumption, indicating that the ratio can vary depending on the properties of the porous media. Additionally, heat transport modeling was performed to confirm the influence of the dispersivity ratio on the temperature plumes that develop from the thermal use of shallow aquifers, such as groundwater heat pump systems. We found that, under steady-state conditions, the decrease in the dispersivity ratio led to smaller plume dimensions in all directions, owing to the increased heat dissipation in the transverse direction. Transient simulations for flow and heat transport indicated that the effect of the thermal dispersivity ratio was time-dependent and heterogeneous and increased with the injection rate. These results suggest that the thermal dispersivity ratio is crucial for assessing the environmental impacts of aquifer thermal use, especially for long-term and large-scale applications. Therefore, the magnitude and ratio of thermal dispersivity should be evaluated at the design stage for more efficient and sustainable use of shallow geothermal resources.

Temporal Temperature Distribution in Shallow Sediments of a Large Shallow Lake and Estimated Hyporheic Flux Using VFLUX 2 Model

Identifying and quantifying exchange flux across sediment-water interface is crucial when considering water and nutrient contributions to a eutrophic lake. In this study, observed temporal temperature distributions in shallow sediment of Lake Taihu (Eastern China) based on three-depth sensors at 14 sites throughout 2016 were used to assess temporal water exchange patterns. Results show that temporal temperature in shallow sediments differed with sampling sites and depths and the temperature amplitudes also clearly shrunk as the offshore distance increasing. Exchange fluxes estimated using the VFLUX 2 model based on temperature amplitude show that alternating-direction temporal flow exists in the eastern zone of Lake Taihu with averages of −13.0, −0.6, and 3.4 mm day−1 (negative represents discharging into the lake) at three nearshore sites (0.5, 2.0, and 6.0 km away from the shoreline, respectively). Whereas downwelling flow occurred throughout almost the entire year with averages of 37.7, 23.5, and 6.6 mm day−1 at the three southern nearshore sites, respectively. However, upwelling flow occurred throughout almost the entire year and varied widely in the western zone with averages of −74.8, 45.9, and −27.0 mm day–1 and in the northern zone with averages of −76.2, −55.3, and −51.1 mm day−1. The estimated fluxes in the central zone were relatively low and varied slightly during the entire year (−15.1 to 22.5 mm day−1 with an average of −0.7 mm day−1). Compared with the sub sensor pair (at 5 and 10 cm), the estimated hyporheic fluxes based on the top sensor pair (at 0 and 5 cm) varied within wider ranges and exhibited relatively larger values. Effects of upwelling flow at the western and northern zones need to be paid attention to on nearshore water quality particularly during winter and spring seasons. Estimated flow patterns at the four zones summarily reflect the seasonal water interaction near the sediment surface of Lake Taihu and are beneficial to improve its comprehensive management. Thermal dispersivity usually used for estimating the thermal diffusivity is more sensitive for upward hyporheic flux estimating even if with a low flux. Temperature amplitude ratio method can be used to estimate the exchange flux and suitable for low flux conditions (either upwelling or downwelling). A better evaluation of the exchange flux near inclined nearshore zones might need an optimized installation of temperature sensors along with the potential flow path and/or a vertical two-dimensional model in the future.

Experimental investigation of the thermal dispersion coefficient under forced groundwater flow for designing an optimal groundwater heat pump (GWHP) system

Mechanical thermal dispersion has often been neglected or underestimated in the simulation of heat transport in porous media, e.g., by using zero or the default value in simulators, or by using the scaling law for solute dispersivity as a thermal dispersivity value. However, large amounts of water usually injected in groundwater heat pump (GWHP) systems may increase the groundwater flow velocity much faster than natural flow and thus change the importance of mechanical dispersion in heat transport. In this study, to investigate the effects of water injection on the flow field, thermal dispersion coefficient, and associated heat transport process, a laboratory experiment using two different heat sources as tracers was performed at various background flow velocities (Re < 0.52). The analysis results from analytical and numerical models indicate that injected water increases both flow velocities and thermal dispersion coefficients, especially near the injection well, and thus makes the effect of mechanical dispersion on heat transport very important even at low background flow velocity. This result was also found in the field-based modeling results, but the radius of hydraulic and thermal effects was larger. In particular, thermal dispersivity on a field scale is known to increase depending on the scale of measurement and the degree of aquifer heterogeneity. Therefore, to ensure the efficiency and sustainability in field applications such as GWHP systems, it is necessary to evaluate site-specific thermal dispersivity through field experiments.

Identifying and assessing key parameters controlling heat transport in discrete rock fractures

Numerical modeling of heat transfer in low porosity fractured rock is challenging because of the complexity associated with the interactions between fractures, bedrock matrix, groundwater, and heat sources. Possible influential parameters include the heat source configuration, the thermal conductivity of the matrix, the velocity of the fluid, the thermal dispersivity in the fracture, and the aperture of the fracture. In this investigation we use factorial analysis (2K) to define which of the five parameters, or combinations thereof, significantly influence heat migration in a single fracture setting assuming a parallel plate condition. A 3-D numerical model based on the control-volume finite element method is used for the simulations. For the parameter ranges investigated, results indicate that the most influential factor controlling heat propagation in a single fracture setting is the velocity of the fluid in the fracture. The interaction between the thermal conductivity of the matrix and the velocity of the fluid, and between the thermal conductivity of the matrix and the aperture of the fracture, dominantly control the attenuation of the thermal front migration. Depending on the particular system, one or more of these parameters should be given better consideration during site characterization in fractured rock and in the compilation of site-specific models intended to predict heat propagation in fractured systems.

Importance of thermal dispersivity in designing groundwater heat pump (GWHP) system: Field and numerical study

Through field and numerical studies, this paper describes the importance of thermal dispersivity in designing groundwater heat pump (GWHP) systems. A push–pull test using heat as a tracer was performed to estimate the thermal dispersivity of the aquifer of this study at the Han River Environment Research Center in Korea. Measured data during the test were compared to the results of three-dimensional (3-D) groundwater flow and heat transport simulations. From the best fit between the measured data and the simulated results, thermal dispersivity was estimated to be 0.4 m. To evaluate the effects of the thermal properties on subsurface heat transport associated with GWHP systems, sensitivity analysis was also performed. The analysis confirmed that, despite small changes based on the estimated values, thermal dispersivity of the aquifer had a great influence on the subsurface temperature distribution as well as the extent of the thermal plume. Because groundwater pumping and injection can cause flow velocity around wells to be faster than natural groundwater velocity, thermal dispersion in this elevated velocity condition will have a considerable impact on the heat transport process with operation of the GWHP system.

Analysis and experimental study on thermal dispersion effect of small scale saturated porous aquifer

The coefficient model of small scale thermal mechanical dispersion of saturated porous aquifer is established and it is applied in the heat transfer process of convective dispersion. Step-by-step test is conducted for the two physical processes of one-dimensional unsteady heat conduction of semi-infinite medium and convection dispersion to obtain heat physical parameters, thus achieving the verification purpose of analytical solution. On this basis, the porous aquifer thermal dispersion effect is evaluated, the results show that if coefficient of thermal mechanical dispersion 1 × 10−2 W m−1 K−1 is selected as the critical point where thermal transport is affected, the distribution of thermal mechanical dispersion coefficients can be divided into non-ignorable triangular domain and ignorable polygon domain. However, the result shows that the maximum of longitudinal dispersivity is at centimeter order of magnitude, which is significantly different from the research result of thermal dispersivity under outdoors large scale conditions. This proves the existence of scale effect of thermal dispersion, and thus shows the direction of further research. At last, under condition that the thermal dispersion is ignored, the heat transfer method of thermal transport under conditions of different seepage velocities is also defined.

Effect of thermal-hydrogeological and borehole heat exchanger properties on performance and impact of vertical closed-loop geothermal heat pump systems

Ground-source geothermal systems are drawing increasing attention and popularity due to their efficiency, sustainability and being implementable worldwide. Consequently, design software and regulatory guidelines have been developed. Interaction with the subsurface significantly affects the thermal performance, sustainability, and impacts of such systems. Reviewing the related guidelines and the design software, room for improvement is evident, especially in regards to interaction with groundwater movement. In order to accurately evaluate the thermal effect of system and hydrogeological properties on a borehole heat exchanger, a fully discretized finite-element model is used. Sensitivity of the loop outlet temperatures and heat exchange rates to hydrogeological, system and meteorological factors (i.e. groundwater flux, thermal conductivity and volumetric heat capacity of solids, porosity, thermal dispersivity, grout thermal conductivity, background and inlet temperatures) are analyzed over 6-month and 25-year operation periods. Furthermore, thermal recovery during 25 years after system decommissioning has been modeled. The thermal plume development, transport and dissipation are also assessed. This study shows the importance of subsurface thermal conductivity, groundwater flow (flux > 10−7 m/s), and background and inlet temperature on system performance and impact. It also shows the importance of groundwater flow (flux > 10−8 m/s) on thermal recovery of the ground over other factors.

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

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 &lt; 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 &lt; Pet &lt; 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.

Experimental investigation of the thermal time‐series method for surface water‐groundwater interactions

Diel temperature fluctuations have been used to quantify vertical water flow through saturated sediments with 1‐D analytical heat models. The underlying transport equation relies on assumptions that could be violated using field temperature records. To test the capability of this method a hydraulic laboratory experiment was designed. A mass of fully water‐saturated homogeneous sand was exposed to 19 different uniform pressure gradients inducing steady state Darcy velocities 0 &lt; q &lt; 25 m d−1. An areal heating grid generated steady sinusoidal thermal forcing. Multipoint sediment temperature responses demonstrated increasing spatial variability for increasing velocities. This introduced horizontal temperature gradients. Velocities calculated from heat tracing were compared to velocities independently obtained from solute slugs. Heat‐ and solute‐derived velocities agreed for q &lt; 3 m d−1, but heat‐derived velocities were consistently larger at higher velocities. Temperature amplitude‐ and phase‐derived velocities revealed significant scatter when compared to solute velocities. This scatter reduced when amplitude‐ and phase‐derived velocities were compared for each sensor pair. The variability in heat‐derived velocities therefore represents a spatially variable flow field in the sand. However, amplitude‐ and phase‐derived velocities deviated from a 1:1 relationship at higher velocities. This can partly be explained by longitudinal thermal dispersivity, and partly by enhanced thermal spreading due to horizontal temperature gradients originating from nonuniform flow. This is surprising given the homogeneous sand and transition zone transport conditions (Pet &lt; 0.7). These findings have implications for the quantification of velocities from field records because field conditions are likely more heterogeneous, which would exacerbate the effects found in this study.

Numerical sensitivity study of thermal response tests

Thermal conductivity and thermal borehole resistance are basic parameters for the technical and sustainable design of closed ground source heat pump (GSHP) systems. One of the most common methods to determine these parameters is the thermal response test (TRT). The response data measured are typically evaluated by the Kelvin line source equation which does not consider all relevant processes of heat transfer in the subsurface. The approach only considers conductive heat transfer from the borehole heat exchanger (BHE) and all transport effects are combined in the parameters of effective thermal conductivity and thermal borehole resistance. In order to examine primary effects in more detail, a sensitivity study based on numerically generated TRT data sets is performed considering the effects of (1) the in-situ position of the U-shaped pipes of borehole heat exchangers (shank spacing), (2) a non-uniform initial thermal distribution (such as a geothermal gradient), and (3) thermal dispersivity. It will be demonstrated that the shank spacing and the non-uniform initial thermal distribution have minor effects (less than 10%) on the effective thermal conductivity and the determined borehole resistance. Constant groundwater velocity with varying thermal dispersivity values, however, has a significant influence on the thermal borehole resistance. These effects are even more pronounced for interpreted effective thermal conductivity which is overestimated by a factor of 1.2–2.9 compared to the real thermal conductivity of the saturated porous media.

Sand box experiments to evaluate the influence of subsurface temperature probe design on temperature based water flux calculation

Abstract. Quantification of subsurface water fluxes based on the one dimensional solution to the heat transport equation depends on the accuracy of measured subsurface temperatures. The influence of temperature probe setup on the accuracy of vertical water flux calculation was systematically evaluated in this experimental study. Four temperature probe setups were installed into a sand box experiment to measure temporal highly resolved vertical temperature profiles under controlled water fluxes in the range of ±1.3 m d−1. Pass band filtering provided amplitude differences and phase shifts of the diurnal temperature signal varying with depth depending on water flux. Amplitude ratios of setups directly installed into the saturated sediment significantly varied with sand box hydraulic gradients. Amplitude ratios provided an accurate basis for the analytical calculation of water flow velocities, which matched measured flow velocities. Calculated flow velocities were sensitive to thermal properties of saturated sediment and to temperature sensor spacing, but insensitive to thermal dispersivity equal to solute dispersivity. Amplitude ratios of temperature probe setups indirectly installed into piezometer pipes were influenced by thermal exchange processes within the pipes and significantly varied with water flux direction only. Temperature time lags of small sensor distances of all setups were found to be insensitive to vertical water flux.