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Analytical solutions for transient temperature distribution in a geothermal reservoir due to cold water injection

Solutions analytiques pour la distribution transitoire de la température due à l’injection d’eau froide dans un réservoir géothermal

Soluciones analíticas para una distribución transitoria de temperatura en un reservorio geotérmico debido a la inyección de agua fría

地热储由于冷水注入造成瞬时温度分布的解析解

Soluções analíticas para a distribuição de temperatura em regime transitório num reservatório geotérmico em resposta à injeção de água fria

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Abstract

An analytical solution to describe the transient temperature distribution in a geothermal reservoir in response to injection of cold water is presented. The reservoir is composed of a confined aquifer, sandwiched between rocks of different thermo-geological properties. The heat transport processes considered are advection, longitudinal conduction in the geothermal aquifer, and the conductive heat transfer to the underlying and overlying rocks of different geological properties. The one-dimensional heat transfer equation has been solved using the Laplace transform with the assumption of constant density and thermal properties of both rock and fluid. Two simple solutions are derived afterwards, first neglecting the longitudinal conductive heat transport and then heat transport to confining rocks. Results show that heat loss to the confining rock layers plays a vital role in slowing down the cooling of the reservoir. The influence of some parameters, e.g. the volumetric injection rate, the longitudinal thermal conductivity and the porosity of the porous media, on the transient heat transport phenomenon is judged by observing the variation of the transient temperature distribution with different values of the parameters. The effects of injection rate and thermal conductivity have been found to be profound on the results.

Résumé

Une solution analytique pour décrire la distribution transitoire de la température suite à une injection d’eau froide dans un réservoir géothermal est présentée. Le réservoir est composé d’un aquifère captif pris en sandwich entre des roches de propriétés thermo-géologiques différentes Les processus de transport de chaleur considérés sont l’advection, la conduction longitudinale dans l’aquifère géothermal et le transfert conductif de chaleur vers les roches sous et sus-jacentes aux propriétés géologiques différentes. L’équation de transfert de chaleur à une dimension est résolue en utilisant la transformée de Laplace sous l’hypothèse d’une densité constante et des propriétés thermiques à la fois de la roche et du fluide. Deux solutions simples sont ensuite dérivées, en négligeant d’abord la conductivité longitudinale de transport de chaleur et ensuite le transport de chaleur vers les épontes. Les résultats montrent que la perte de chaleur dans les épontes joue un rôle majeur en ralentissant le refroidissement du réservoir. L’influence de certains paramètres, tel que le débit d’injection, la conductivité thermique longitudinale et la porosité du milieu poreux, sur le phénomène transitoire du transport de chaleur est appréhendé en observant la variation de la distribution des températures dans le temps avec différentes valeurs des paramètres. Il apparait que débit d’injection et la conductivité thermique ont un impact important sur les résultats.

Resumen

Se presenta una solución analítica para describir la distribución transitoria de la temperatura en un reservorio geotérmico en respuesta a una inyección de agua fría. El reservorio está compuesto de un acuífero confinado, intercalado entre rocas de diferentes propiedades termo geológicas. Los procesos de transporte de calor considerados son advección, conducción longitudinal en el acuífero geotérmico, y la transferencia conductiva del calor a rocas subyacentes y suprayacente de diferentes propiedades geológicas. La ecuación de transferencia de calor unidimensional ha sido resuelta usando la transformada de Laplace con la suposición de densidad y propiedades térmicas constantes tanto de rocas como de fluidos. Se extrajeron dos soluciones simples, la primera despreciando la conductividad longitudinal del transporte de calor y por lo tanto del transporte de calor a las rocas confinantes. Los resultados muestran que la pérdida de calor hacia las capas de rocas confinantes juega un rol vital en retardar el enfriamiento del reservorio. La influencia de algunos parámetros, por ejemplo la tasa volumétrica de la inyección, la conductividad térmica longitudinal y la porosidad del medio poroso, sobre el fenómeno de transporte transitorio de calor son juzgados observando la variación de la distribución transitoria de la temperatura con diferentes valores de los parámetros. Se han encontrado los profundos efectos de la velocidad de inyección y de la conductividad térmica en los resultados.

摘要

研究展示了地热储由于冷水注入造成瞬时温度分布的解析解。热储由一个夹在不同地热特性岩层之间的承压含水层组成。考虑到的热传输过程有对流、地热含水层中的纵向传导、向上覆及下伏的具有不同地质特性的岩石的热传导。假定岩石和液体恒定密度和热特性并采用Laplace变换求解决一维热传导方程。随后推导出两个简单的 解决方法,首先忽略纵向传导的热传输,然后忽略传导到承压岩石的热传输。结果显示对承压岩石层的热损耗在减速热储的冷却上发挥至关重要的作用。有些参数如容积注入率、纵向人传导及多孔介质的孔隙度对瞬时热传输现象的影响根据观测瞬时温度分布的变化的不同参数值来判断。注入速率和热传导对结果有深远影响。

Resumo

É apresentada uma solução analítica para descrever a distribuição da temperatura em regime transitório num reservatório geotérmico em resposta à injeção de água fria. O reservatório é formado por um aquífero confinado, localizado entre rochas de diferentes propriedades térmicas e geológicas. Os processos de transporte de calor são a adveção, a condução longitudinal do aquífero geotérmico e a transferência de calor por condução para as rochas sub- e sobrejacentes de diferentes propriedades geológicas. A equação de transferência de calor unidimensional tem sido resolvida utilizando a transformada de Laplace, com a assunção das hipóteses de densidade e propriedades térmicas constantes das rochas e do fluido. São derivadas duas soluções simples, primeiro negligenciando a condutividade longitudinal do transporte de calor e, em seguida, o transporte de calor para as rochas confinantes. Os resultados mostram que a perda de calor para as camadas de rocha confinantes desempenha um papel vital no abrandamento do arrefecimento do reservatório. A influência de alguns parâmetros, como a taxa de injeção volumétrica, a condutividade térmica longitudinal e a porosidade do meio no fenómeno de transporte de calor em regime transitório, é avaliada observando a variação da distribuição de temperatura em transitório com diferentes valores dos parâmetros. Os efeitos da taxa de injeção e da condutividade térmica mostram ser muito grandes nos resultados obtidos.

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Correspondence to M. S. Mohan Kumar.

Appendices

Appendix 1

The inverse transformation in the second term of Eq. (34) is done by Carslaw and Jaeger (1959) according to whom

$$ {L}^{-1}\left[ \exp \left\{\frac{-\frac{x^2}{4\lambda {\zeta}^2}\left(\alpha {s}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+ Cs\right)}{s}\right\}\right]= erfc\left(\frac{\alpha {x}^2}{8\lambda {\zeta}^2\sqrt{t-\frac{C{x}^2}{4\lambda {\zeta}^2}}}\right)U\left(t-\frac{C{x}^2}{4\lambda {\zeta}^2}\right) $$
(60)

where U is the unit step function given by

$$ U\left(t-\frac{C{x}^2}{4\lambda {\zeta}^2}\right)=\left\{1\kern0.5em ;\kern1.1em t>\frac{C{x}^2}{4\lambda {\zeta}^2}0\kern0.5em ;\kern0.7em t<\frac{C{x}^2}{4\lambda {\zeta}^2}\right. $$
(61)

Another result (Oberhettinger and Badii 1973) is invoked here to facilitate the inverse transform in the third term of Eq. (34)

$$ {L}^{-1}\left\{\frac{ \exp \left(-a{s}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}{s\left({s}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\beta \right)}\right\}= erfc\left(\frac{1}{2}a{t}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)- \exp \left( a\beta +{\beta}^2t\right) erfc\left(\frac{1}{2}a{t}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\beta {t}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right) $$
(62)

The preceding result is subjected to the fact that

$$ {L}^{-1}\left\{ \exp \left(- cs\right)F(s)\right\}=U\left(t-c\right)f\left(t-c\right) $$
(63)

where

$$ {L}^{-1}F(s)=f(t) $$
(64)

and leads to

$$ \begin{array}{l}{L}^{-1}\left[\frac{ \exp \left\{-\frac{x^2}{4\lambda {\zeta}^2}\left(\alpha {s}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+ Cs\right)\right\}}{s\left({s}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$C$}\right.\right)}\right]=\left[ erfc\left\{\frac{\alpha {x}^2}{8{\lambda}^2{\zeta}^2{\left(t-\frac{C{x}^2}{4{\lambda}^2{\zeta}^2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right\}\right.\\ {}\left.- \exp \left\{\frac{\alpha^2{x}^2}{4{\lambda}^2{\zeta}^2C}+\frac{\alpha^2}{C^2}\left(t-\frac{C{x}^2}{4{\lambda}^2{\zeta}^2}\right)\right\}\cdot erfc\left\{\frac{\alpha {x}^2}{8{\lambda}^2{\zeta}^2{\left(t-\frac{C{x}^2}{4{\lambda}^2{\zeta}^2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}+\frac{\alpha }{C}{\left(t-\frac{C{x}^2}{4{\lambda}^2{\zeta}^2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}\right]\end{array} $$
(65)

The Laplace inverse of the fourth term in Eq. (34) can be given according to Oberhettinger and Badii (1973) as

$$ {L}^{-1}\left\{\frac{1}{s\left({s}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$C$}\right.\right)}\right\}=\frac{C}{\alpha}\left[1- \exp \left\{{\left(\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$C$}\right.\right)}^2\cdot t\right\}\cdot erfc\left(\frac{\alpha }{C}{t}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)\right] $$
(66)

Substituting these results in Eq. (34), the final form of the solution is derived as Eq. (35).

Appendix 2

The solution of the heat transfer Eq. (58) in the Laplace domain for negligible heat flux to the confining rock media is given by

$$ \overline{T}= \exp \left(\frac{ Ux}{2\lambda}\right)\left({T}_0-{T}_{in}\right)\left[\frac{ \exp \left\{\frac{ Ux}{2\lambda }{\left(1+\mu s\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}}{s\left[ \exp \left\{\frac{ UL}{2\lambda }{\left(1+\mu s\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}-1\right]}\kern0.5em -\kern0.5em \frac{ \exp \left\{\frac{U\left(2L-x\right)}{2\lambda }{\left(1+\mu s\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}}{s\left[ \exp \left\{\frac{ UL}{2\lambda }{\left(1+\mu s\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}-1\right]}\right]+\frac{T_0}{s} $$
(67)
$$ \mathrm{where}\kern6em \mu =\frac{4 C\lambda}{U^2} $$

Considering the first term in the bracket

$$ {\overline{v}}_1(s)=\frac{ \exp \left\{\frac{ Ux}{2\lambda }{\left(1+\mu s\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}}{s\left[ \exp \left\{\frac{ UL}{2\lambda }{\left(1+\mu s\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}-1\right]} $$
(68)

The inverse Laplace transform of the aforementioned, by the complex inversion formula, is given by

$$ {v}_1(t)=\frac{1}{2\pi i}{\displaystyle \underset{c- i\mathit{\infty}}{\overset{c+ i\mathit{\infty}}{\int }}\frac{ \exp (st)\cdot \exp \left(\frac{ Ux}{2\lambda}\sqrt{1+\mu s}\right)}{s\left\{ \exp \left(\frac{ UL}{\lambda}\sqrt{1+\mu s}\right)-1\right\}}} ds $$
(69)

Now, by the residue theorem, v 1(t) equals the sum of the residues of the preceding integrand. Hence, to evaluate the residues, the poles, or the values of s are to be located, at which the aforementioned integrand fails to be analytic.

The denominator \( s\left\{ \exp \left(\frac{ UL}{\lambda}\sqrt{1+\mu s}\right)-1\right\}=0 \) whenever s = 0 or \( s=-\frac{1}{\mu}\left(\frac{4{n}^2{\pi}^2{\lambda}^2}{U^2{L}^2}+1\right) \) n = 0, 1, 2, 3....

Hence a simple pole exists of order one at s = 0. The residue at s = 0 is

$$ \begin{array}{c}\hfill ={}_s\underrightarrow{ \lim}_0\left[\frac{s \exp \left( st+\frac{ Ux}{2\lambda}\sqrt{1+\mu s}\right)}{s\left\{ \exp \left(\frac{ UL}{\lambda}\sqrt{1+\mu s}\right)-1\right\}}\right]\hfill \\ {}\hfill =\left[\frac{ \exp \left(\frac{ Ux}{2\lambda}\right)}{\left\{ \exp \left(\frac{ UL}{\lambda}\right)-1\right\}}\right]\kern6em \hfill \end{array} $$

The residue at \( s=-\frac{1}{\mu}\left(\frac{4{n}^2{\pi}^2{\lambda}^2}{U^2{L}^2}+1\right) \) is evaluated as

$$ \begin{array}{c}\hfill {}_s\underrightarrow{ \lim}_{-\frac{1}{\mu}\left(\frac{4{n}^2{\pi}^2{\lambda}^2}{U^2{L}^2}+1\right)}\left[\frac{\left\{s+\frac{1}{\mu}\left(\frac{4{n}^2{\pi}^2{\lambda}^2}{U^2{L}^2}+1\right)\right\} \exp \left( st+\frac{ Ux}{2\lambda}\sqrt{1+\mu s}\right)}{s\left\{ \exp \left(\frac{ UL}{\lambda}\sqrt{1+\mu s}\right)-1\right\}}\right]\kern6em \hfill \\ {}\hfill ={}_s\underrightarrow{ \lim}_{-\frac{1}{\mu}\left(\frac{4{n}^2{\pi}^2{\lambda}^2}{U^2{L}^2}+1\right)}\left\{\frac{ \exp \left( st+\frac{ Ux}{2\lambda}\sqrt{1+\mu s}\right)}{s}\right\}{}_s\underrightarrow{ \lim}_{-\frac{1}{\mu}\left(\frac{4{n}^2{\pi}^2{\lambda}^2}{U^2{L}^2}+1\right)}\frac{\left\{s+\frac{1}{\mu}\left(\frac{4{n}^2{\pi}^2{\lambda}^2}{U^2{L}^2}+1\right)\right\}}{\left\{ \exp \left(\frac{ UL}{\lambda}\sqrt{1+\mu s}\right)-1\right\}}\hfill \end{array} $$

Applying L’Hospital’s rule for the second limit, the residue becomes

$$ =-\frac{4 n\pi i{\lambda}^2}{4{n}^2{\pi}^2{\lambda}^2+{U}^2{L}^2} \exp \left\{-\frac{t}{\mu}\left(\frac{4{n}^2{\pi}^2{\lambda}^2}{U^2{L}^2}+1\right)\right\} \exp \left( n\pi i\frac{x}{L}\right) $$

Thus the inverted first term in the bracket in Eq. (67) becomes

$$ {v}_1(t)=\frac{ \exp \left(\frac{ Ux}{2\lambda}\right)}{\left\{ \exp \left(\frac{ UL}{\lambda}\right)-1\right\}}-{\displaystyle \sum_{n=0}^{\infty}\frac{4 n\pi i{\lambda}^2}{4{n}^2{\pi}^2{\lambda}^2+{U}^2{L}^2} \exp \left\{-\frac{t}{\mu}\left(\frac{4{n}^2{\pi}^2{\lambda}^2}{U^2{L}^2}+1\right)\right\} \exp \left( n\pi i\frac{x}{L}\right)} $$

In a similar manner the inverse of the second term in the bracket in Eq. (67) can be found as

$$ =\frac{ \exp \left\{\frac{U\left(2L-x\right)}{2\lambda}\right\}}{\left\{ \exp \left(\frac{ UL}{\lambda}\right)-1\right\}}-{\displaystyle \sum_{n=0}^{\infty}\frac{4 n\pi i{\lambda}^2}{4{n}^2{\pi}^2{\lambda}^2+{U}^2{L}^2} \exp \left\{-\frac{t}{\mu}\left(\frac{4{n}^2{\pi}^2{\lambda}^2}{U^2{L}^2}+1\right)\right\} \exp \left(-\kern0.5em n\pi i\frac{x}{L}\right)} $$

Adding the terms, the solution can be written as

$$ \begin{array}{l}T={T}_0+ \exp \left(\frac{ Ux}{2\lambda}\right)\left({T}_0-{T}_{in}\right)\left[\frac{ \exp \left(\frac{ Ux}{2\lambda}\right)}{\left\{ \exp \left(\frac{ UL}{\lambda}\right)-1\right\}}-{\displaystyle \sum_{n=0}^{\infty}\frac{4 n\pi i{\lambda}^2}{4{n}^2{\pi}^2{\lambda}^2+{U}^2{L}^2} \exp \left\{-\frac{t}{\mu}\left(\frac{4{n}^2{\pi}^2{\lambda}^2}{U^2{L}^2}+1\right)\right\} \exp \left( n\pi i\frac{x}{L}\right)}\right.\\ {}\left.-\kern1em \frac{ \exp \left\{\frac{U\left(2L-x\right)}{2\lambda}\right\}}{\left\{ \exp \left(\frac{ UL}{\lambda}\right)-1\right\}}+{\displaystyle \sum_{n=0}^{\infty}\frac{4 n\pi i{\lambda}^2}{4{n}^2{\pi}^2{\lambda}^2+{U}^2{L}^2} \exp \left\{-\frac{t}{\mu}\left(\frac{4{n}^2{\pi}^2{\lambda}^2}{U^2{L}^2}+1\right)\right\} \exp \left(- n\pi i\frac{x}{L}\right)}\right]\end{array} $$
(70)

which, in turn, can be simplified to the final form of the solution given in Eq. (59) by using the fact that

$$ \exp \left( i\theta \right)= \cos \theta +i \sin \theta $$
(71)
$$ \mathrm{and}\kern.5em \exp \left(- i\theta \right)= \cos \theta -i \sin \theta $$
(72)

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Ganguly, S., Mohan Kumar, M.S. Analytical solutions for transient temperature distribution in a geothermal reservoir due to cold water injection. Hydrogeol J 22, 351–369 (2014). https://doi.org/10.1007/s10040-013-1048-2

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Keywords

  • Geothermal reservoirs
  • Analytical solutions
  • Heat transport
  • Thermal conditions