Assessing the impact of thermal feedback and recycling in open-loop groundwater heat pump (GWHP) systems: a complementary design tool

Abstract

Thermal feedback and thermal recycling in open-loop groundwater heat pump (GWHP) systems occurs when a fraction of the injected water in a well doublet returns to the production well. They reflect two different mathematical representations of the same physical process. Thermal feedback assumes a constant injection temperature, while thermal recycling couples the injection and production temperatures by a constant temperature difference. It is shown that thermal feedback, commonly used in GWHP design, and recycling reflect two thermal end-members. This work addresses the coupled problem of thermal recycling, which is, so far, the missing link for complete GWHP assessment. An analytical solution is presented to determine the return-flow fraction in a well doublet and is combined with a heat-balance calculation to determine the steady-state well temperatures in response to thermal feedback and recycling. This is then extended to advective-dispersive systems using transfer functions, revealing that the well temperatures in response to thermal feedback and recycling are functions of the capture probability. Conjunctive interpretation of thermal feedback and recycling yields a novel design approach with which major difficulties in the assessment of the sustainability of GWHP systems can be addressed.

Résumé

Le percement thermique et le recyclage thermique se produisent dans un doublet géothermique ouvert lorsqu’une fraction de l’eau injectée retourne au puits productif. Ils reflètent deux représentations mathématiques du même processus. Le percement thermique suppose une température d’injection constante, alors que le recyclage thermique relie la température d’injection et de production par une différence de température constante. On montre que le percement thermique, communément utilisé pour le dimensionnement de doublets géothermiques, et le recyclage définissent deux cas de sollicitation thermique extrêmes. Cet article traite du problème couplé du recyclage thermique, en général négligé lors de l’évaluation d’un doublet géothermique ouvert. Une solution analytique est établie pour déterminer la fraction du débit de retour, puis combinée avec un bilan thermique donnant les températures d'équilibre aux puits en réponse au percement thermique et au recyclage thermique. Cette nouvelle approche est ensuite étendue aux systèmes advectif-dispersifs au moyen des fonctions de transfert, révélant que ces températures d'équilibre sont entièrement déterminées par la probabilité de capture du puits productif. L'interprétation combinée des effets liés au percement thermique et au recyclage thermique fournit des critères plus réalistes pour le dimensionnement et l'évaluation des performances d'un doublet géothermique ouvert.

Resumen

La retroalimentación térmica y el reciclado térmico en sistemas de bomba de calor de agua subterránea en circuito abierto (GWHP) ocurren cuando una fracción del agua inyectada en pozos dobles retorna al pozo de producción. Ello configura dos representaciones matemáticas diferentes de un mismo proceso físico. La retroalimentación térmica supone una inyección a temperatura constante, mientras que el reciclado térmico la inyección se acopla a la temperatura de producción a través de una diferencia de temperatura constante. Se muestra que la retroalimentación térmica, comúnmente usada en el diseño GWHP, y el reciclado reflejan dos miembros térmicos extremos. Este trabajo aborda el problema del acoplamiento del reciclado térmico, que es, hasta aquí, el eslabón perdido para completar la evaluación de GWHP. Se presenta una solución analítica para determinar la fracción del flujo de retorno en un pozo doble y se combina con un cálculo de balance de calor para determinar el estado estacionario de temperaturas del pozo en respuesta a la retroalimentación y al reciclado térmico. Esto es luego extendido a sistemas dispersivos – advectivos usando funciones de transferencia, lo que revela que las temperaturas del pozo en respuesta a la retroalimentación y reciclado térmico son funciones de la probabilidad de captura. La interpretación conjunta de retroalimentación y reciclado térmico brinda un enfoque de nuevo diseño con el cual pueden ser abordadas las mayores dificultades en la evaluación de la sustentabilidad de los sistemas GWHP.

Resumo

Retorno térmico e reciclagem térmica em sistemas de bombas de calor baseados em água subterrânea em circuito aberto (open-loop groundwater heat pump – GWHP) ocorrem quando uma fração da água injetada no furo emparelhado retorna ao furo de produção. Estes termos correspondem a duas representações matemáticas do mesmo processo físico. O retorno térmico assume uma temperatura constante de injeção, enquanto a reciclagem térmica agrega as temperaturas de injeção e de produção através de uma diferença de temperatura constante. É mostrado que o retorno térmico, comummente usado nos projetos GWHP, e a reciclagem refletem dois membros térmicos finais. Este trabalho analisa o problema acoplado da reciclagem termal, o qual é, até agora, o elo perdido para avaliação completa do GWHP. É apresentada uma solução analítica para determinar a fração de retorno de fluxo num furo emparelhado e é combinada com um cálculo de balanço de calor para determinar o estado estável da temperatura nos furos em resposta ao retorno e reciclagem térmicas. Isto é então estendido a sistemas advetivo-dispersivos, recorrendo ao uso de funções de transferência, mostrando que as temperaturas no furo em resposta ao retorno e reciclagem térmicas são funções da probabilidade de captura. A interpretação conjuntiva do retorno e reciclagem térmicas leva a uma nova forma de aproximação que pode resolver situações que apresentavam maiores dificuldades de avaliação da sustentabilidade dos GWHP.

Appendix: Characteristic parameters of a well doublet with partial recirculation in an aquifer with uniform regional flow

The complex potential, Ω, resulting from the implementation of a well doublet parallel to a 2D uniform flow field in the (x, y)–plane, with the injecting well located downstream, is classically obtained (e.g. Strack 1989; Javandel and Tsang 1986; Luo and Kitanidis 2004) by the superposition

$$ \Omega = \frac{Q}{{2\pi \,b}}ln(z + d)-\frac{Q}{{2\pi \,b}}ln(z-d)-q\,z,z = x + i\,y $$

(19)

where b is the aquifer thickness and z is the coordinate in the complex plane. The first two terms correspond, respectively, to the flow rate Q extracted at z = −d and to the same flow rate injected (−Q) at z = d, while the third term accounts for the regional uniform Darcy flux q defined positive in the direction of the real axis (Fig. 4). The two wells are therefore centred about the origin along the x-axis and are separated by the distance L = 2d.

Fig. 4

Dimensionless complex plane representation with pumping (P) and injecting (I) wells located at z’ = ±1, equipotentials (dashed), streamlines (solid), stagnation points (dots) and flow separation lines (bold). Partial recirculation patterns for X = 3

The complex function above features the hydraulic potential h and the stream function ψ as real and imaginary parts, respectively, so that

$$ \Omega = k\,h + i\,\psi $$

(20)

where k is the hydraulic conductivity of the medium.

Using the dimensionless variables

$$ \Omega \prime = \frac{{b\,\Omega }}{Q},z\prime = \frac{z}{d},X = \frac{Q}{{\pi \,b\,q\,d}},h\prime = \frac{{b\,k\,h}}{Q},\psi \prime = \frac{{b\,\psi }}{Q} $$

(21)

the complex potential becomes

$$ \Omega \prime = h\prime + i\psi \prime = \frac{1}{{2\pi }}ln(\frac{{z\prime + 1}}{{z\prime-1}})-\frac{{z\prime}}{{\pi \,X}},z\prime = x\prime + i\,y\prime $$

(22)

and is illustrated in Fig. 4, where the dimensionless hydraulic potential h′ and stream function ψ′ are represented for the dimensionless flow rate X = 3.

According to the complex potential theory, the dimensionless specific discharge is obtained by the differentiation

$$ W\prime = -\frac{{d\,\Omega \prime}}{{dz\prime}} = \frac{1}{\pi }(\,\frac{1}{{{{z\prime}^2}-1}} + \frac{1}{X}\,) $$

(23)

This function has no complex components at either purely real or purely imaginary locations. It is singular at the well points and vanishes at the stagnation points ${z\prime_{\text{s}}}$.

Setting W′ (${z\prime_{\text{s}}}$) = 0 in (Eq. 23) yields

$$ {z\prime_{\text{s}}} = \pm \sqrt {{1-X}} , X < 1 $$

(24)

$$ {z\prime_{\text{s}}} = \pm i\sqrt {{X-1}} , X > 1 $$

(25)

Hence, as long as X < 1, two stagnation points are located on the real axis (y′ = 0) and there is no recirculation from the injection well to the pumping well. As X increases, the stagnation points shift towards the origin, merge at ${z\prime_{\text{s}}} = 0$ when X = 1, and then shift away along the imaginary axis (x′ = 0), as indicated by (Eq. 25) for X > 1. In true dimensions, the stagnation points are located at x = 0 and $y = \pm \,d\,\sqrt {{X-1}}$.

In this latter case, a recirculation cell develops and a fraction of the injected flow rate returns to the pumping well.

The shortest travel time of the recycled water particles is the hydraulic breakthrough-time, dictated by the inverse of the specific discharge along the straight flow-path connecting the two wells. Given that specific discharge has no complex components along this path, this characteristic time is easily obtained, in dimensionless form, by

$$ {t\prime_{{min}}} = \int\limits_1^{{-1}} {\frac{1}{{W\prime}}\,\,dz\prime} = 2\pi X(\frac{X}{{\sqrt {{X-1}} }}ta{n^{{-1}}}(\frac{1}{{\sqrt {{X-1}} }})-1) $$

(26)

with the limit value ${t\prime_{{{ \min }}}} = 4\pi / 3$ when X –– > ∞.

In true dimensions, ${t_{{{ \min }}}} = {t\prime_{{{ \min }}}}\phi \,b\,{d^2} / Q$ where $\phi$ is the porosity of the medium.

The fraction of the flow rate recycled in the well doublet is expected to increase with the flow rate itself. Under increasing recycling conditions, the stagnation points (Eq. 25) are always at purely imaginary locations and connected by a straight (equipotential) line intersecting all recirculating flow-paths. As specific discharge always remains real-valued along this line (Eq. 23), the fraction recycled is therefore straightforwardly obtained by

$$ \alpha = i\int\limits_{{-i\,\sqrt {{X-1}} }}^{{\,i\,\sqrt {{X-1}} }} {\,W\prime\,\,dz\prime} = \frac{2}{\pi }(ta{n^{{-1}}}(\sqrt {{X-1}} )-\frac{{\sqrt {{X-1}} }}{X}) $$

(27)