An analytical solution for modeling thermal energy transfer in a confined aquifer system

Abstract

A mathematical model is developed for simulating the thermal energy transfer in a confined aquifer with different geological properties in the underlying and overlying rocks. The solutions for temperature distributions in the aquifer, underlying rock, and overlying rock are derived by the Laplace transforms and their corresponding time-domain solutions are evaluated by the modified Crump method. Field data adopted from the literature are used as examples to demonstrate the applicability of the solutions in modeling the heat transfer in an aquifer thermal energy storage (ATES) system. The results show that the aquifer temperature increases with time, injection flow rate, and water temperature. However, the temperature decreases with increasing radial and vertical distances. The heat transfer in the rocks is slow and has an effect on the aquifer temperature only after a long period of injection time. The influence distance depends on the aquifer physical and thermal properties, injection flow rate, and injected water temperature. A larger value of thermal diffusivity or injection flow rate will result in a longer influence distance. The present solution can be used as a tool for designing the heat injection facilities for an ATES system.

Résumé

Un modèle mathématique est développé pour simuler le transfert d’énergie thermique dans un aquifère captif avec des caractéristique géologiques différentes dans les roches sous-jacentes et sus-jacentes. Les solutions pour la distribution de la température dans l’aquifère, les roches sous-jacentes et les roches sus-jacentes sont dérivées des transformées de Laplace et leurs solutions fonction du temps correspondantes sont évaluées par la méthode de Crump modifiée. Les données de terrain choisies dans la littérature sont utilisées comme exemples pour démontrer la pertinence des solutions de modélisation du transfert de chaleur dans un système aquifère de stockage d’énergie thermique (ATES). Les résultats montrent que la température de l’aquifère croît avec le temps, le débit d’injection, et la température de l’eau. Toutefois, la température décroît avec des distances radiales et verticales croissantes. Le transfert de chaleur dans les roches est lent et a une influence sur la température de l’aquifère après une longue période d’injection seulement. La distance sous influence dépend des propriétés physiques et thermiques de l’aquifère, du débit d’injection, et de la température de l’eau injectée. Une valeur plus importante de la diffusion thermique et du débit d’injection aura pour résultat une distance sous influence plus étendue. La solution obtenue peut être utilisée comme outil pour élaborer les installations d’injection de chaleur dans un système ATES.

Resumen

Se ha desarrollado un modelo matemático para simular la transferencia de energía térmica en un acuífero confinado con diferentes propiedades geológicas de las rocas suprayacentes y subyacentes. Las soluciones para la distribución de la temperatura en el acuífero, las rocas suprayacentes y subyacentes se derivan de la transformada de Laplace y sus correspondientes soluciones en el dominio temporal se evalúan con el método de Crump modificado. Se utilizan datos de campo extraídos de la literatura como ejemplos de la aplicabilidad de las soluciones para modelar la transferencia de calor en el sistema de almacenamiento de energía térmica de un acuífero (AETA). Los resultados indican que la temperatura del acuífero se incrementa con el tiempo, con los caudales inyectados, y con la temperatura del agua. Sin embargo, la temperatura disminuye a medida que se incrementa las distancias radial y vertical. La transferencia de calor en las rocas es lenta y tiene efectos en la temperatura del acuífero luego de un largo período de inyección. La distancia de la influencia depende de las propiedades físicas y térmicas del acuífero, del caudal de inyección y de la temperatura del agua de inyección. Valores mayores de difusividad térmica o de caudales de inyección resultarán en mayores distancias de influencia. La solución presentada puede usarse como herramienta para el diseño de instalaciones de inyección de calor en sistemas AETA.

Chinese

摘要考虑上覆和下伏岩层的地质特性的差异, 研究了承压含水层中的热传递问题, 建立了数学模型。应用Laplace变换求解了含水层、上覆和下伏岩层中温度分布, 并应用改进的Crump方法评估了它们对应的时空解。以文献中的现场数据为例, 演示了该解法对于模拟含水层储能 (ATES)系统热传递过程的适用性。结果显示, 含水层温度随时间、注入速率和水温的增加而升高, 随径向和垂向距离的增大而降低, 而岩石中的热传递过程缓慢, 在注入很长时间后才会对含水层温度产生影响。影响半径取决于含水层的物理和热力学性质、注入速率以及注入水温度。热扩散系数或注入速率越大, 影响半径越大。该解法的提出为ATES系统注热设备设计提供了工具。

Resumo

É desenvolvido um modelo matemático para simular a transferência de energia térmica num aquífero confinado em que as rochas sobrejacentes e subjacentes apresentam diferentes propriedades geológicas. As soluções obtidas para as distribuições de temperatura no aquífero e nas rochas confinantes são obtidas a partir de transformadas de Laplace e as correspondentes soluções no domínio temporal avaliadas pelo método modificado de Crump. Para demonstrar a aplicabilidade do modelo, são utilizados como exemplos dados de campo retirados da literatura, em casos de modelação de transporte de calor num sistema aquífero para armazenamento de energia térmica. Os resultados mostram que a temperatura do aquífero aumenta com o tempo, o caudal de injecção e a temperatura da água, embora esta última diminua à medida que as distâncias radial e vertical aumentam. A transferência de calor nas rochas é baixa e apenas tem um efeito na temperatura do aquífero após um longo período de injecção. A distância de influência depende das propriedades físicas e termais do aquífero e do caudal e da temperatura da água de injecção. Quanto maior for a difusividade termal ou o caudal de injecção maior será a distância de influência. Este modelo pode ser utilizado no desenho de sistemas de injecção de calor em sistemas aquíferos para armazenamento de energia térmica.

Appendix

The coupled boundary value problem represented by Eqs. (1)–(11) can be expressed in a dimensionless form based on the dimensionless groupings of Eq. (12). For the aquifer, the governing equation for the steady-state heat conduction is

$$\left( {b\rho _{\text{A}} c_{\text{A}} } \right)\left( {\frac{Q}{{2\pi rb}}} \right)\frac{{\partial T_{{\text{AD}}} \left( {r,t} \right)}}{{\partial r}} = \left. { - K_{{\text{R}}1} \frac{{\partial T_{{\text{R}}1{\text{D}}} \left( {r,z,t} \right)}}{{\partial z}}} \right|_{{\text{z}} = 0} \left. { + K_{{\text{R}}2} \frac{{\partial T_{{\text{R}}2{\text{D}}} \left( {r,z,t} \right)}}{{\partial z}}} \right|_{{\text{z}} = {\text{b}}} $$

(30)

The initial condition is

$$T_{{\text{AD}}} \left( {r,0} \right) = 0$$

(31)

The boundary condition at the rim of wellbore is

$$T_{{\text{AD}}} \left( {r_{\text{w}} ,t} \right) = 1$$

(32)

For the underlying rock, the heat conduction equation is

$$\frac{{\partial ^2 T_{{\text{R}}1{\text{D}}}^\prime \left( {r,z,t} \right)}}{{\partial z^2 }} = \frac{1}{{\alpha _{{\text{R}}1} }}\frac{{\partial T_{{\text{R}}1{\text{D}}}^\prime \left( {r,z,t} \right)}}{{\partial t}},\alpha _{{\text{R}}1} = \frac{{K_{{\text{R}}1} }}{{\rho _{{\text{R1}}} c_{{\text{R}}1} }}$$

(33)

The initial condition is

$$T_{{\text{R}}1{\text{D}}}^\prime \left( {r,z,0} \right) = 0$$

(34)

The boundary conditions are

$$T_{{\text{R}}1{\text{D}}}^\prime \left( {r,0,t} \right) = T_{{\text{AD}}} \left( {r,t} \right) - T_{{\text{R}}10{\text{D}}} $$

(35)

and

$$T_{{\text{R}}1{\text{D}}}^\prime \left( {r, - b_1 ,t} \right) = 0$$

(36)

For the overlying rock, the heat conduction equation for the temperature distribution can be written as

$$\frac{{\partial ^2 T_{{\text{R}}2{\text{D}}}^\prime \left( {r,z,t} \right)}}{{\partial z^2 }} = \frac{1}{{\alpha _{{\text{R}}2} }}\frac{{\partial T_{{\text{R}}2{\text{D}}}^\prime \left( {r,z,t} \right)}}{{\partial t}},\alpha _{{\text{R}}2} = \frac{{K_{{\text{R}}2} }}{{\rho _{{\text{R}}2} c_{{\text{R}}2} }}$$

(37)

The initial condition is

$$T_{{\text{R}}2{\text{D}}}^\prime \left( {r,z,0} \right) = 0$$

(38)

The boundary conditions are

$$T_{{\text{R}}2{\text{D}}}^\prime \left( {r,b,t} \right) = T_{{\text{AD}}} \left( {r,t} \right) - T_{{\text{R}}20{\text{D}}} $$

(39)

and

$$T_{{\text{R}}2{\text{D}}}^\prime \left( {r,b + b_2 ,t} \right) = 0$$

(40)

Applying Laplace transforms to Eqs. (30)–(40), the governing equation for the aquifer becomes

$$\left( {\frac{{\rho _{\text{A}} c_{\text{A}} Q}}{{2\pi }}} \right)\left( {\frac{1}{r}\frac{{d\bar T_{{\text{AD}}} \left( {r,s} \right)}}{{dr}}} \right) = \left. { - K_{{\text{R}}1} \frac{{d\bar T_{{\text{R}}1{\text{D}}} \left( {r,z,s} \right)}}{{dz}}} \right|_{{\text{z}} = 0} \left. { + K_{{\text{R}}2} \frac{{d\bar T_{{\text{R}}2{\text{D}}} \left( {r,z,s} \right)}}{{dz}}} \right|_{{\text{z}} = {\text{b}}} $$

(41)

and the boundary condition becomes

$$\bar T_{{\text{AD}}} \left( {r_{\text{w}} ,s} \right) = \frac{1}{s}$$

(42)

The heat conduction equation for the underlying rock results in

$$\frac{{d^2 \bar T_{{\text{R}}1{\text{D}}}^\prime \left( {r,z,s} \right)}}{{dz^2 }} = q_1^2 \bar T_{{\text{R}}1{\text{D}}}^\prime \left( {r,z,s} \right),q_1^2 = \frac{s}{{\alpha _{{\text{R}}1} }}$$

(43)

and the related boundary conditions are

$$\bar T_{{\text{R}}1{\text{D}}}^\prime \left( {r,0,s} \right) = \bar T_{{\text{AD}}} \left( {r,s} \right) - \frac{{T_{{\text{R}}10{\text{D}}} }}{s}$$

(44)

and

$$\bar T_{{\text{R}}1{\text{D}}}^\prime \left( {r, - b_1 ,s} \right) = 0$$

(45)

The heat conduction equation for the overlying rock becomes

$$\frac{{d^2 \bar T_{{\text{R}}2{\text{D}}}^\prime \left( {r,z,s} \right)}}{{dz^2 }} = q_1^2 \bar T_{{\text{R}}2{\text{D}}}^\prime \left( {r,z,s} \right),q_2^2 = \frac{s}{{\alpha _{{\text{R}}2} }}$$

(46)

and the related boundary conditions are

$$\bar T_{{\text{R}}2{\text{D}}}^\prime \left( {r,b,s} \right) = \bar T_{{\text{AD}}} \left( {r,s} \right) - \frac{{T_{{\text{R}}20{\text{D}}} }}{s}$$

(47)

and

$$\bar T_{{\text{R}}2{\text{D}}}^\prime \left( {r,b + b_2 ,s} \right) = 0$$

(48)

Substituting Eqs. (44) and (45) into Eq. (43) yields

$$\bar T_{{\text{R}}1{\text{D}}}^\prime \left( {r,z,s} \right) = \frac{{\sinh \left( {q_1 \left( {b_1 + z} \right)} \right)}}{{\sinh \left( {q_1 b_1 } \right)}}\left( {\bar T_{{\text{AD}}} \left( {r,s} \right) - \frac{{T_{{\text{R}}10{\text{D}}} }}{s}} \right)$$

(49)

Based on Eq. (12), Eq. (49) can be rewritten as

$$\bar T_{{\text{R}}1{\text{D}}} \left( {r,z,s} \right) = \frac{{\sinh \left( {q_1 \left( {b_1 + z} \right)} \right)}}{{\sinh \left( {q_1 b_1 } \right)}}\left( {\bar T_{{\text{AD}}} \left( {r,s} \right) - \frac{{T_{{\text{R}}10{\text{D}}} }}{s}} \right) + \frac{{T_{{\text{R}}10{\text{D}}} }}{s}$$

(50)

Similarly, the Laplace-domain solution for the underlying rock can be obtained by substituting Eqs. (47) and (48) into Eq. (46) as

$$\bar T_{{\text{R}}2{\text{D}}}^\prime \left( {r,z,s} \right) = \frac{{\sinh \left( {q_2 \left( {b + b_2 - z} \right)} \right)}}{{\sinh \left( {q_2 b_2 } \right)}}\left( {\bar T_{{\text{AD}}} \left( {r,s} \right) - \frac{{T_{{\text{R}}0{\text{D}}} }}{s}} \right)$$

(51)

The above equation can be rewritten by the use of Eq. (12) as

$$\bar T_{{\text{R}}2{\text{D}}} \left( {r,z,s} \right) = \frac{{\sinh \left( {q_2 \left( {b + b_2 - z} \right)} \right)}}{{\sinh \left( {q_2 b_2 } \right)}}\left( {\bar T_{{\text{AD}}} \left( {r,s} \right) - \frac{{T_{{\text{R}}0{\text{D}}} }}{s}} \right) + \frac{{T_{{\text{R}}0{\text{D}}} }}{s}$$

(52)

The substitution of Eqs. (50) and (52) into Eq. (41) yields

$$\begin{array}{*{20}l} {\left( {\frac{{\rho _{\text{A}} c_{\text{A}} Q}}{{2\pi }}} \right)\left( {\frac{1}{r}\frac{{d\bar T_{{\text{AD}}} \left( {r,s} \right)}}{{dr}}} \right) = } \hfill {\begin{array}{*{20}l} { - K_{{\text{R}}1} q_1 \coth \left( {q_1 b_1 } \right)\left( {\bar T_{{\text{AD}}} \left( {r,s} \right) - \frac{{T_{{\text{R}}10{\text{D}}} }}{s}} \right)} \hfill \\ { - K_{{\text{R}}2} q_2 \coth \left( {q_2 b_2 } \right)\left( {\bar T_{{\text{AD}}} \left( {r,s} \right) - \frac{{T_{{\text{R}}20{\text{D}}} }}{s}} \right)} \hfill \\ \end{array} } \hfill \\ \end{array} $$

(53)

After substituting Eq. (42) into Eq. (53), the Laplace-domain solution, Eq. (13), can then be obtained.