Estimation of groundwater flow from temperature monitoring in a borehole heat exchanger during a thermal response test

Abstract

To estimate the groundwater flow around a borehole heat exchanger (BHE), thermal properties of geological core samples were measured and a thermal response test (TRT) was performed in the Tsukuba upland, Japan. The thermal properties were measured at 57 points along a 50-m-long geological core, consisting predominantly of sand, silt, and clay, drilled near the BHE. In this TRT, the vertical temperature in the BHE was also monitored during and after the test. Results for the thermal properties of the core samples and from the monitoring indicated that groundwater flow enhanced thermal transfers, especially at shallow depths. The groundwater velocities around the BHE were estimated using a two-dimensional numerical model with monitoring data on temperature changes. According to the results, the estimated groundwater velocity was generally consistent with hydrogeological data from previous studies, except for the data collected at shallow depths consisting of a clay layer. The reasons for this discrepancy at shallow depths were predicted to be preferential flow and the occurrence of vertical flow through the BHE grout, induced by the hydrogeological conditions.

Résumé

Pour estimer les écoulements d’eau souterraine autour d’une sonde géothermique verticale (SGV), les propriétés thermiques d’échantillons géologiques ont été mesurées et un test de réponse thermique (TRT) a été mis en œuvre sur le plateau de Tsukuba, au Japon. Les propriétés thermiques ont été mesurées en 57 points répartis le long d’un profil géologique de 50 m, essentiellement constitué de sables, de silts et d’argiles, forés près de la SGV. Lors de ce TRT, la température verticale dans la SGV a aussi été suivie pendant et après le test. Les résultats issus des propriétés thermiques des échantillons et du suivi lors des tests indiquent que l’écoulement d’eau souterraine augmente les transferts thermiques, en particulier aux faibles profondeurs. Les vitesses d’écoulement autour de la SGV ont été estimées en utilisant un modèle numérique à deux dimensions, avec les données de suivi des variations de température. Sur la base de ces résultats, les vitesses d’écoulement estimées des eaux souterraines étaient globalement cohérentes avec les données hydrogéologiques d’études antérieures, en dehors des données collectées aux faibles profondeurs correspondant aux formations argileuses. Les raisons de cette différence aux faibles profondeurs seraient liées à des écoulements préférentiels et à l’existence d’écoulements verticaux via la SGV, induits par les conditions hydrogéologiques.

Resumen

Para estimar el flujo de agua subterránea alrededor de un intercambiador de calor de pozo (BHE), se midieron las propiedades térmicas de las muestras de testigos geológicos y se realizó un ensayo de respuesta térmica (TRT) en el altiplano de Tsukuba, Japón. Las propiedades térmicas se midieron en 57 puntos a lo largo de testigos geológicos de 50 m de longitud, que consistían principalmente de arena, limo y arcilla, perforados cerca del BHE. En este TRT, la temperatura vertical en el BHE también se controló durante y después del ensayo. Los resultados de las propiedades térmicas de las muestras del testigo y del monitoreo indicaron que el flujo del agua subterránea mejoró las transferencias térmicas, especialmente a poca profundidad. Las velocidades del agua subterránea alrededor del BHE se estimaron utilizando un modelo numérico bidimensional con datos del monitoreo de los cambios de temperatura. De acuerdo con los resultados, la velocidad estimada del agua subterránea fue generalmente consistente con los datos hidrogeológicos de los estudios previos, a excepción de los datos recolectados a poca profundidad que consisten en una capa de arcilla. Se pronosticó que las razones de esta discrepancia a poca profundidad son el flujo preferencial y la ocurrencia de flujo vertical a través del relleno del BHE, inducido por las condiciones hidrogeológicas.

摘要

为了估算钻孔热量交换器周边的地下水流,在日本Tsukuba高地测量了地质岩心样品的热特性,进行了热响应试验。沿50米长的地质岩心50个点测量了热特性,岩心主要包括砂、粉砂和粘土,钻孔地点位于钻孔热交换器附近。在这个热响应试验中,试验期间以及试验后还监测了钻孔热交换器中的垂直温度。岩心样品的热特性结果及监测结果显示,地下水流提高了热传输,特别是在浅部更是如此。利用温度变化监测数据采用二维数值模型估算了钻孔热交换器周围的地下水速度。根据结果来看,估算的地下水速度通常与先前研究中的水文地质数据一致,包含粘土层的浅部处收集的数据除外。预测了浅部这种不符现象的原因,原因就是水文地质条件导致出现的优先流和穿过钻孔热交换器泥浆的垂直流。

Resumo

Para estimar o fluxo de águas subterrâneas no entorno de um tubo permutador de calor (borehole heat exchanger – BHE), foram medidas as propriedades térmicas em testemunhos geológicos e realizado um teste de resposta térmica (TRT) na região do monte Tsukuba, Japão. As propriedades térmicas foram medidas em 57 pontos ao longo de um testemunho de 50 metros de comprimento, com predomínio de areia, silte e argila, amostrado próximo ao BHE. No TRT, a temperatura vertical no BHE também foi monitorada durante e após a realização do teste. Os resultados das propriedades térmicas das amostras de testemunho e do monitoramento indicaram que o fluxo de águas subterrâneas elevou as transferências térmicas, sobretudo em profundidades rasas. As velocidades das águas subterrâneas em torno do BHE foram estimadas utilizando um modelo numérico bidimensional com dados de monitoramento das mudanças de temperatura. Conforme os resultados, a velocidade estimada para as águas subterrâneas está consistente com dados hidrogeológicos de estudos anteriores, exceto para os dados coletados em profundidades rasas em uma camada de argila. Os motivos para esta discrepância em profundidades rasas foram preditas como sendo um fluxo preferencial e a ocorrência de fluxo vertical através do cimento do BHE, induzida pelas condições hidrogeológicas.

Appendices

Appendix 1

Analysis of the thermal response test (TRT)

The effective thermal conductivity was estimated from the averaged inlet and outlet temperature during the TRT heating stage based on Kelvin’s line-source theory. In this analysis method, the borehole heat exchanger (BHE) is assumed to be a fine infinite line source and the subsurface is assumed to be homogeneous with a constant temperature over a finite radial distance. Temperature change at a distance r from the line source is given by the following equations.

$$ T-{T}_{\mathrm{i}}=\frac{Q}{4\uppi {\lambda}_{\mathrm{s}}}E(X) $$

(5)

$$ X=\frac{r^2}{4{\alpha}_{\mathrm{s}}t} $$

(6)

$$ E(X)={\int}_X^{\infty}\frac{e^{-u}}{u}\mathrm{d}u $$

(7)

where T is the temperature [K], T _i is the initial temperature [K], Q is the heat output [W/m], λ _s is the effective thermal conductivity [W/(mK)], α _s is the thermal diffusivity [m²/s], t is the time [s], E(X) is the exponential integral and u is the variable of integration [-]. E(X) can be approximated for a small range of X (<0.05, which means, for example, in the case where α _s is 5.0 × 10⁻⁷ m²/s, r is 0.08 m, and t is larger than ~18 h) as shown in the following equation:

$$ E(X)=-\ln (X)-0.5772 $$

(8)

Thus, the average temperature is approximated as

$$ \mathrm{T}-{\mathrm{T}}_{\mathrm{i}}=\frac{Q}{4\uppi {\lambda}_{\mathrm{s}}}\left(\ln \frac{4{\alpha}_{\mathrm{s}}t}{r^2}-0.5772\right) $$

(9)

Eq. (5) can be rewritten as

$$ T-{T}_{\mathrm{i}}=m\ \ln (t)+b $$

(10)

$$ m=\frac{Q}{4\uppi {\lambda}_{\mathrm{s}}} $$

(11)

Equation (6) means that the effective thermal conductivity can be estimated from the gradient of the relationship between the average temperature and the logarithmic time scale. Generally, T is given by the average of the inlet and outlet temperature. In practical situations, the gradient several hours after heating during the TRT will be non-linear because of the influence of heat exchange pipes and grout material in the well. Therefore, the heating should be monitored until steady-state conditions are obtained (IEA ECES ANNEX 21 2013).

Appendix 2

Treatment of thermal properties in FEFLOW

In FEFLOW, the energy conservation equation is expressed in the following forms (equations of mass conservation are referred to as described in Diersch 2005):

$$ \frac{\partial }{\partial t}\left[\varepsilon {\rho}^{\mathrm{f}}{E}^{\mathrm{f}}+\left(1-\varepsilon \right){\rho}^{\mathrm{s}}{E}^{\mathrm{s}}\right]+\frac{\partial }{\partial {x}_i}\left({\rho}^f{q}_i^{\mathrm{f}}{E}^{\mathrm{f}}\right)-\frac{\partial }{\partial {x}_i}\left({\lambda}_{ij}\frac{\partial T}{\partial {x}_j}\right)={Q}_T $$

(12)

$$ {\displaystyle \begin{array}{l}{\lambda}_{ij}={\lambda}_{ij}^{{\mathrm{cond}}_{\mathrm{f}}}+{\lambda}_{ij}^{{\mathrm{cond}}_{\mathrm{s}}}+{\lambda}_{ij}^{{\mathrm{disp}}_{\mathrm{f}}}\\ {}=\left[{\varepsilon \lambda}^{\mathrm{f}}+\left(1-\varepsilon \right){\lambda}^s\right]{\delta}_{ij}+{\rho}^{\mathrm{f}}{c}^{\mathrm{f}}\left[{\alpha}_T{V}_q^{\mathrm{f}}{\delta}_{ij}+\left({\alpha}_L-{\alpha}_T\right)\frac{q_i^{\mathrm{f}}{q}_j^{\mathrm{f}}}{V_q^{\mathrm{f}}}\right]\end{array}} $$

(13)

$$ {Q}_{\mathrm{T}}=\varepsilon {\rho}^{\mathrm{f}}{Q}_T^{\mathrm{f}}+\left(1-\varepsilon \right){\rho}^s{Q}_T^s $$

(14)

where E refers to the enthalpy, ε is the porosity, T is the temperature, ρ is the bulk density, q is the Darcy velocity, t is the time, λ is the thermal conductivity, c is the specific heat, α _L and α _T are the longitudinal and transverse dispersivities, respectively, Q is the energy, and ${V}_q^{\mathrm{f}}=\sqrt{q_i^{\mathrm{f}}{q}_j^{\mathrm{f}}}$ indicates the absolute Darcy flux. Subscripts f and s represent the fluid phase and solid phase, respectively, and superscripts “cond” and “disp” represent the conduction and dispersion, respectively. Directly measured thermal conductivity and heat capacity (shown in Table 3) are bulk properties, i.e., these values are the averages of solid, liquid, and partly air. The required soil properties in this model are the thermal conductivity and heat capacity of the “solid phase”. Therefore, the thermal conductivity and heat capacity of solids were determined by using these equations, and the porosity was assumed to correspond to the volumetric water content.