Influence of regional groundwater flow on ground temperature around heat extraction boreholes

Geothermics 56 (2015) 119–127

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Influence of regional groundwater flow on ground temperature around heat extraction boreholes Alberto Liuzzo-Scorpo a,b,∗ , Bo Nordell b , Signhild Gehlin c a b c

Dipartimento di Ingegneria Civile e Architettura, Università degli Studi di Trieste, via A. Valerio 10, 34127 Trieste, Italy Division of Architecture and Water, Dept. Civil, Mining and Natural Resources Engineering, Luleå University of Technology, SE-97187 Luleå, Sweden Swedish Centre for Shallow Geothermal Energy, P.O. Box 1127, SE-22104 Lund, Sweden

a r t i c l e

i n f o

Article history: Received 25 June 2014 Accepted 13 April 2015 Available online 16 May 2015 Keywords: Vertical borehole heat exchanger TRT Groundwater flow Influence Length Numerical model

a b s t r a c t The increasing popularity of ground-coupled heat pumps has resulted in almost 20% of all Swedish family houses being heated this way. To avoid undesirable interactions between neighboring boreholes and disturbance of the ground temperature, the general rule and recommendation of Swedish authorities is that the distance between two neighboring boreholes must be ≥20 m. However, according to previous studies, relatively low groundwater flow rates may significantly reduce the borehole excess temperature compared to the case of pure heat conduction. In this work the Influence Length is defined and its relations with flow rate, real thermal conductivity of the ground and effective thermal conductivity obtained by thermal response analysis are investigated. The aim of this study was to find a way to use the thermal response test as a means to determine the groundwater flow influence in order to reduce the borehole spacing perpendicular to groundwater flow direction. The results confirm that very low groundwater flow rates are enough to significantly reduce the Influence Length, hence this is a crucial parameter which should be considered. Moreover, a first estimation, even before the thermal response test analysis, of the Influence Length is possible if the knowledge of hydrogeological conditions of the site allows good predictions about real thermal conductivity of the ground and flow rate. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Increased energy cost and environmental concerns have made ground-coupled heat pumps (GCHP) increasingly popular in Europe and North America. Almost 20% of all Swedish single family houses are heated this way i.e. by extracting heat from boreholes in the ground. Such heat extraction systems mean that the surrounding ground is slowly cooled until a steady-state is reached after some years. Borehole systems that are placed too close together will therefore influence each other, by further lowering the ground temperature. To avoid conflicts between neighbors that also might want to use the ground in a similar way, the Swedish authorities have set up some rules to consider before the required drilling permit is granted. The general rule or recommendation (for single family

∗ Corresponding author at: Dipartimento di Ingegneria Civile e Architettura, Università degli Studi di Trieste, via A. Valerio 10, 34127 Trieste, Italy. E-mail address: [email protected] (A. Liuzzo-Scorpo). http://dx.doi.org/10.1016/j.geothermics.2015.04.002 0375-6505/© 2015 Elsevier Ltd. All rights reserved.

houses) is that the distance between two neighboring boreholes must be ≥20 m to avoid significant thermal interaction. For a typical heat extraction borehole system (see Table 1) the steady-state ground temperature is ≈1.4◦ C lower than the undisturbed temperature at a distance of 10 m from the borehole, assuming that the heat is transported away from the borehole by conduction only, see Fig. 1. Unlike the case of a heat source of finite length, pure conduction will never result in steady-state for a heat source of infinite length. In current work, a two-dimensional numerical model is used (see

Table 1 Typical Swedish heat borehole system average values; (*) according to Statens Energimyndighet (2006); (**) according to Swedish Energy Agency (2008). Total borehole depth: 132 m(*) Active borehole depth: 117 m(*) Annual heat extraction: 22,000 kW h(**) Bedrock mean thermal conductivity: 3.5 W (m K)−1 Bedrock mean undisturbed temperature: 5 ◦ C

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Nomenclature thermal diffusivity, Eq. (7) (m2 s−1 ) p porosity (–)  thermal conductivity (W m−1 K−1 ) eff effective thermal conductivity calculated by the TRT data analysis (W m−1 K−1 )  dynamic viscosity (Pa s)  density (kg m−3 ) time (s)   end duration of the simulation (s) n outer unit normal from the borehole surface to the ground U volumetric flow rate per unit of cross-sectional area, Eq. (1) (m s−1 ) c specific heat (J kg−1 K−1 ) h hydraulic head (m) hydraulic gradient, Eq. (2) (–) i k hydraulic permeability (m2 ) hydraulic conductivity (m s−1 ) K p static pressure (Pa) q heat transfer rate (Wy m−1 ) q uniform heat flux released from the borehole surface (W m−2 ) r radial distance from the borehole axis (m) T temperature (K) Tf heat carrier mean fluid temperature (◦ C) unperturbed ground temperature (=278.15 K) T0 x Cartesian coordinate Pe Peclét number, Eq. (12) Linf05 , Linf1 Influence Lengths as defined in Section 3.2, considering the temperatures T0 ± 0.5◦ C and T0 ± 1◦ C, respectively for all ∀ ˛

Subscripts b denotes quantities, defined on the borehole surface denotes thermophysical properties of soils and r rocks w denotes thermophysical properties of water denotes thermophysical properties of the porous gr media (soil/rocks + water), Eq. (6a) TRT denotes value related to Thermal Response Test (TRT) analysis denotes steady state conditions SS

Section 2.2) and the state after 40 years of heat injection/extraction is assumed as steady-state condition. However, groundwater flow through the ground will change the ground temperature disturbance so that the temperature plume around the borehole will be distorted, as shown in Fig. 2. The radial distance from the borehole, perpendicular to the flow direction of the groundwater, shows that any given temperature disturbance is closer to the borehole than in the case of no groundwater flow (Figs. 1 and 2). The distance to the same steady-state temperature disturbance as that at 10 m distance for conductive flow perpendicular to the groundwater flow direction (Figs. 1 and 2) is called Influence Length, Linf . This length decreases with increasing groundwater flow rate. Since the steady-state temperature change of the ground is the basis for the recommended minimum borehole spacing it would be possible to reduce the borehole spacing perpendicular to the groundwater flow, provided that this temperature influence could

Fig. 1. Section through the ground. Radial steady-state ground temperature difference (T = | T − T0 |) in the ground outside a typical Swedish heat extraction borehole. (a) The T at 10 m distance from the borehole is 1.4 ◦ C without any groundwater movements. (b) The T at 10 m distance from the borehole, perpendicular to the groundwater flow, is considerably lower with increasing groundwater flow.

be determined in advance. Here the idea is to use a thermal response test as a means of determining the Influence Length. 1.1. Thermal response test and groundwater flow Mobile thermal response test (TRT) is an in situ measurement method developed by Eklöf and Gehlin (1996) and Austin (1998). It is used to determine the effective thermal conductivity of the ground and the thermal resistance within the borehole. These data are necessary to design and to predict the thermal performance of a borehole heat exchanger (BHE) system (Nordell, 2011). Most commonly these tests are performed with heat injection. The evaluated effective thermal conductivity of the bedrock must be greater or equal than the thermal conductivity that would be obtained by laboratory testing of rock cores. The reason is that the heat transfer from the borehole is not only by heat conduction but also carried by occurring groundwater movements and in fact, according to Sanner et al. (2000), groundwater movements influence the test results. Chiasson et al. (2000) developed a finite-element groundwater flow and heat transfer model to simulate forced convection of heat in various geologic materials and concluded that regional groundwater flow only influenced the heat transfer in BHEs for certain geohydrological conditions. Results suggest that in these cases, the advection of heat by groundwater flow significantly enhances heat transfer due to the high hydraulic conductivity of the geologic materials, such as sands, gravels, and formations showing secondary porosity (fractures and solution channels). The phenomena was further studied by Witte (2001) who performed experimental analysis combined with a numerical study of a clayey cover layer and a water bearing formation consisting mainly of sand. His results show that even small groundwater flows cause higher estimated value of the effective thermal conductivity.

Fig. 2. Plan. Radial steady-state ground T in the ground outside a typical Swedish heat extraction borehole. (a) The T at 10 m distance from the borehole is 1.4 ◦ C without any groundwater movements. (b) The T at 10 m distance from the borehole, perpendicular to the groundwater flow, is considerably lower with increasing groundwater flow.

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Gehlin and Hellström (2003) compared three different model approaches to investigate the influence from regional groundwater flow by two-dimensional, finite-difference simulations of groundwater flow and heat transfer around a single borehole. The soil was modeled as equivalent and continuous porous media, fracture zone with a homogeneous porosity and vertical fracture, respectively. The results showed that in all cases relatively low flow rates may significantly enhance the heat transfer, which means that the borehole temperature change is considerably lower than in the case of pure heat conduction even through a model simulating the ground as continuous porous media using Darcy’s equation. Several analytical and numerical models have been developed over the years to compute the ground-coupled heat exchanger (GCHE) transient heat transfer. Most of them are based on the analytical solutions for the conduction heat transfer from a line source (Ingersoll et al., 1954) or a cylindrical source (Carslaw and Jaeger, 1959). The effect of groundwater flow on the heat transfer efficiency of vertical GCHEs was investigated, among others, by Eskilson (1987), Diao et al. (2004), and Fan et al. (2007). Eskilson (1987) considered a homogeneous, isotropic, cracked, rocky soil filtered by groundwater and investigated the effects at the steady-state for a single vertical borehole. Eskilson (1987) claimed that the influence of natural ground-water advection is negligible. A way to describe the heat transfer between the borehole and the surrounding ground is based on the resistance analogy: it considers a thermal resistance between the working fluid temperature (inside the U-tube) and the far-field temperature of the ground, with respect to the borehole (Sutton, 1998; Sutton et al., 2003). Design algorithms typically subdivide the overall thermal resistance into two parts: a resistance associate with the borehole interior (including ground, tubing material and flow resistance) and a resistance associated to the ground (Hellström, 1991; Sutton, 1998; Sutton et al., 2003). Sutton et al. (2003) developed a ground resistance to integrate the existing BHE design algorithms. Starting from the moving line heat source solution by Carslaw and Jaeger (1959) and Sutton et al. (2003) developed a ground resistance which accounts for the convective contribution to the heat transfer provided by the groundwater flow. The authors presented the combined ground resistance in terms of the dimensionless Fourier and Péclet parameters, obtaining significant differences in the resulting convection ground resistance with respect to the conduction-only ground resistance, when Pe >0.01 and for a particular geological regime and groundwater flow scenario. Diao et al. (2004) derived a transient, two-dimensional, analytical solution for convective heat transfer around vertical borehole heat exchangers. The analytical solution by Diao et al. (2004) reveals several features of the considered heat transfer process. The temperature loses the axial symmetry and is affected towards the flow direction. In addition, steady-state conditions are attained asymptotically when groundwater advection occurs. Diao et al. (2004) also computed the time-rate of change of the borehole average surface temperature, which is of great importance in the design of GCHE systems. The analytical solution by Diao et al. (2004) was investigated by Piller and Liuzzo-Scorpo (2012, 2013), considering a wide range of Peclét number. The conclusions of these studies are that the analytical solution proposed by Diao et al. (2004) fits very well with the numerical solution for Peclét number, Pe  0.2 and Fourier number, Fo  1, despite the geometric approximation of the analytical solution, which is based on the infinite line source model. The analytical solution, obtained by the superposition principle, is in excellent agreement with the numerical solution also in the cases of borehole fields, if typical values of spacing between boreholes encountered in practical GCHE application are considered.

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Fan et al. (2007) developed a two-dimensional, numerical model for the simulation of a field of vertical geothermal boreholes, with significant groundwater advection. An equivalent single pipe was used to represent the actual U-shaped pipe (Yian and O’Neal, 1998). A unidirectional groundwater field is assumed over the entire numerical domain. The proposed model also took into account freezing and thawing of the soil moisture. The results are in good agreement with the analytical solution by Diao et al. (2004). The model suggested by Fan et al. (2007) is used to simulate the heat transfer between a borehole field and the surrounding soil during both winter and summer operating conditions, with different groundwater velocities. It is found that the presence of groundwater flow has a variable, through always significant, impact on the heat transfer between the borehole field and the surrounding soil, depending on the operating conditions. Molina-Giraldo et al. (2011) proposed a new analytical solution for the flow and heat transfer around a vertical, finite line source (MFLS model) that accounts for groundwater flow. The MFLS model was compared with existing analytical solutions such as the purely conductive finite line source model (FLS) (Eskilson, 1987; Zeng et al., 2002; Marcotte et al., 2010; Lamarche and Beauchamp, 2007), the moving infinite line source model (MILS) (Diao et al., 2004; Sutton et al., 2003) and with a numerical solution. The proposed analytical approach can be applied to all groundwater flow conditions and borehole lengths. The authors claimed that FLS model can be reliably applied for Péclet numbers lower than 1.2, while the MILS approach yields accurate results for Péclet numbers greater than 10; for a Péclet number range between 1.2 and 10, the use of the MFLS model is suggested (Molina-Giraldo et al., 2011). 2. Methodology 2.1. Mathematical model A two-dimensional model is used, thus neglecting any interaction with the ground surface and any variation with depth. A uniform flow of incompressible, constant-property fluids flows through a saturated, homogeneous, isotropic porous matrix and is deflected by the borehole surface, which is assumed as impervious. No hydraulic boundary layer develops on the borehole surface as the fluid is assumed inviscid, according to the classical Darcy model. Liquid flow in a porous medium is usually described by Darcys law, which can be expressed as U = −K · i

(1)

where K is the hydraulic conductivity and i the hydraulic gradient defined as i=

dh dx

(2)

The volumetric flow rate per unit of cross-sectional area, U, is equal to the average velocity of the liquid over the cross-section. The flow is assumed horizontal, then gravitational effects are neglected in Eq. (1). Under the aforementioned assumptions, the steady-state continuity equation reads (Bear, 1972):

∇ 2 pw = 0;

r ≥ rb

(3)

The following boundary conditions hold −

k ∂pw −→ U; w ∂x

∇ pw · n = 0;

r −→ +∞

(4a)

r = rb

(4b) q

is released from Starting at time  = 0 a constant heat flux the borehole surface. An unsteady thermal disturbance results and propagates through the soil by advection and diffusion. The soil

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Fig. 3. Plain-section computational domain, with indication of the boundary conditions.

is assumed thermally homogeneous and isotropic. The thermophysical properties of both the soil and the fluid are considered as constant and thermal equilibrium between fluid and surrounding soil is assumed (Bear, 1972). Under these assumptions the energy equation, with initial and boundary conditions, may be expressed as:

∂T + (c)w U · ∇ T = gr ∇ 2 T ; ∂ T = T0 ;  = 0, ∀ r ≥ rb ( c)gr

T = T0 ; −gr

 > 0, r > rb

 > 0, r → +∞

∂T  =q ; ∂n

 > 0, r = rb

(5a) (5b) (5c) (5d)

The real quantities are calculated by ( c)gr = p ( c)w + (1 − p ) ( c)r

(6a)

gr = p w + (1 − p ) r

(6b)

2.2. Numerical model The mass and energy conservation equations ((3) and (2.1)), with the appropriate initial and boundary conditions, are solved numerically by the commercial finite element package COMSOLMultiphysics< ce : sup > ® . The computational domain, shown in Fig. 3 with boundary conditions indicated, is 2560 × 1280 m, and it takes advantage of the symmetry of the case. A common BHE model approximation is annular geometry instead of the more complex U-pipe geometry (Gustafsson, 2010). This approach allows 1D or 2D calculations that diminish the computational time, and therefore in this study the borehole is modeled considering an equivalent radius equal to 0.0400 m and boreholewall radius equal to 0.0518 m (Gustafsson, 2010; see Fig. 4). The gap between the borehole walls is considered filled with water. All computations are carried out in a mesh encompassing 1.26 × 106 quadrilateral and 1.48 × 104 triangular (total of 1.28 × 106 ) fifth order Lagrange elements. In order to verify the grid and the far-field independence of the numerical solutions, the simulations for highest and lowest considered K, for each gr value, have been repeated on a finer mesh (3.29 × 106 elements) and on a larger domain (6000 × 3000 m). The relative maximum errors (in terms of temperature differences) are 1.7 × 10−8 and 1.3 × 10−6 , respectively. 2.2.1. The reference values for the simulations The values shown in Table 2 are chosen because they are typical mean values for practical application. In detail: the range of K, gr , p and (c)gr are chosen according to the mean hydraulic and thermal

Variable

Value

U (m s−1 )

0, 1 × 10−7 , 1 × 10−6 , 5 × 10−6 , 1 × 10−5 , 2 × 10−5 , 3 × 10−5 , 4 × 10−5 , 5 × 10−5 , 6 × 10−5 , 7 × 10−5 , 8 × 10−5 , 9 × 10−5 , 1 × 10−4 , 1.1 × 10−4 , 1.2 × 10−4 , 1.3 × 10−4 , 1.4 × 10−4 , 1.5 × 10−4 , 1.6 × 10−4 , 1.7 × 10−4 , 1.8 × 10−4 , 1.9 × 10−4 , 2 × 10−4 0.01 0.25 3 × 106 1.5, 2.5, 3.5 0.50 × 10−6 , 0.83 × 10−6 , 1.17 × 10−6 0.0400

i (m m−1 ) p (–) (c)gr (J m−3 K−1 ) gr (W (m K)−1 ) ˛gr (m2 s−1 ) rb (equivalent radius) (m) Borehole-wall radius (m) q TRT (W m−2 ) q SS (W m−2 ) T0 (K)  end,TRT (s (h))  end,SS (s (yr))

0.0518 240 70 278.15 2.592 × 105 (72) 1.26144 × 109 (40)

properties of soils and rocks proposed in Chiasson et al. (2000). The resulting values of thermal diffusivity of the ground ˛gr , considering the equation ˛gr =

gr (c)gr

(7)

are also reported in Table 2. According to Eq. (7), the direct proportionality between gr and ˛gr allows to notice that all the relations studied changing the thermal conductivity gr (Section 3) have the same behavior if the thermal diffusivity ˛gr is considered instead. ˚ According to Aberg and Johansson (1988), hydraulic gradients in Sweden are usually within the range 0.01–0.001. In current work a value of i = 0.01 is chosen. A wider range of flow rates U (Eq. (1)) is encompassed.  The heat flux used in the TRT models, qTRT , is derived from the typical heat power injected during a TRT analysis (9 kW for a 150 m deep borehole, as suggested in Gustafsson, 2010), as well as  end,TRT is chosen in a precautionary way considering the duration of a TRT suggested from Gehlin (1998) and Gustafsson (2010), i.e. between 60 and 72 h. As a result, the eff values obtained using these settings can be considered more than satisfactorily accurate even when groundwater flow does not occur. On the other hand in the  long-time analysis models, the heat flux, qSS , is derived assuming a consumption of energy equal to 22,000 kW h per year (i.e. 8760 h), according to the data reported in Statens Energimyndighet (2006), and considering the mean power of ∼2.5 kW distributed on 150 m deep borehole (i.e. ≈17.5 W/m) with the chosen equivalent radius equal to 0.04 m. 2.3. Data analysis The analysis of the response test data is based on a description of the heat as being injected from a line source (Morgensen, 1983; Eskilson, 1987; Gehlin, 1998). When heat is injected into a borehole a transient process starts that is approximated by: Tf =

q ln  + 4 eff 2/a

for  ≥ 5rb

  q

1 4 eff



ln

4a rb2



−



− Rb + T0

and with

Tf = heat carrier mean fluid temperature)◦ C). q = heat transfer rate (W/m). a = thermal diffusivity (m2 s−1 ).



(8)

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123

Fig. 4. Detail of the mesh.

= Euler’s constant. Rb = borehole thermal resistance (K/(W/m)). The equation can be simplified to a linear relation between Tf and ln : Tf = m ln  + h

(9)

where m and h are constants. According to (8) and (9), eff can be calculated from the inclination of the graph as eff =

q 4 m

(10)

3. Results 3.1. Thermal response data analysis The eff values calculated by the thermal response data analysis are shown in Table 3, while in Fig. 5 the relation between eff and U is displayed. Fig. 5 shows the influence of the gr on the eff values obtained by TRT analysis, as well as the effect of U on the eff which, as expected, is not linear. The introduction of the value , defined as





 = eff − gr 

Fig. 5. The effective thermal conductivity eff values with respect to the volumetric flow rate U. The dotted lines indicate the gr values while the solid lines indicate the eff values. Cross-markers denote the eff values related to gr = 1.5, X-markers denote the eff values related to gr = 1.5, circle-markers denote the eff values related to gr = 3.5.

(11)

allows to draw Fig. 6, where an effect described by Liebel (2012) is highlighted: by fixing flow rate U,  is higher for lower gr values, while considering gr as constant,  increases with increasing U. These responses are expected because the conductive contribution to the heat transfer increases if gr increases, while the convective contribution is constant (with constant flow rate). With respect to this, Fig. 7 gives a supplemental clue. In fact, considering U as a constant parameter, the ratio eff /gr is closer to 1 for higher gr values. The explanation of this behaviour is that

Fig. 6. , as defined in (11), with respect to U. Cross-markers denote the  values related to gr = 1.5, X-markers denote the  values related to gr = 2.5, circle-markers denote the  values related to gr = 3.5.

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Table 3 The calculated eff values. U (m s−1 )

gr (W m−1 K−1 )

0.0 1.0 × 10−09 1.0 × 10−08 5.0 × 10−08 1.0 × 10−07 2.0 × 10−07 3.0 × 10−07 4.0 × 10−07 5.0 × 10−07 6.0 × 10−07 7.0 × 10−07 8.0 × 10−07 9.0 × 10−07 1.0 × 10−06 1.1 × 10−06 1.2 × 10−06 1.3 × 10−06 1.4 × 10−06 1.5 × 10−06 1.6 × 10−06 1.7 × 10−06 1.8 × 10−06 1.9 × 10−06 2.0 × 10−06

1.5

2.5

3.5

1.508 1.508 1.508 1.509 1.509 1.513 1.518 1.526 1.536 1.548 1.563 1.580 1.599 1.621 1.646 1.673 1.703 1.736 1.771 1.810 1.853 1.898 1.947 2.000

2.507 2.507 2.507 2.507 2.507 2.510 2.514 2.520 2.528 2.538 2.549 2.562 2.577 2.594 2.613 2.633 2.656 2.680 2.707 2.735 2.766 2.798 2.833 2.870

3.502 3.502 3.502 3.502 3.502 3.505 3.509 3.515 3.523 3.532 3.543 3.556 3.570 3.586 3.604 3.624 3.645 3.669 3.694 3.721 3.750 3.780 3.813 3.848

since eff changes with respect to both the convective and the conductive heat transfer rates, a higher gr value implies a greater share of the conduction on the total heat transfer (described by eff ). In addition to that, Fig. 7 shows that the spacing between the curves related to different gr are not as regular as expectations would suggest. This means that the influence of the advection on the total heat transfer decreases less-than-linearly while the ground conductivity (gr ) increases. In fact, according to the TRT simulations of this study, comparing Tf in the case of no flow rate and Tf for U = 2 ·10−6 m s−1 , at  =  end , the resulting temperature decreases are equal to ≈1.35 ◦ C, ≈0.53 ◦ C and ≈0.28 ◦ C considering gr = 1.5, gr = 2.5 and gr = 3.5 W m−1 K−1 , respectively. This behaviour is due to the larger differences between Tf and the groundwater temperature (equal to T0 ) when lower gr are considered. Introducing the non-dimensional Peclét number, defined as Pe =

Urb (c)w gr

Fig. 8. Cross-markers denote the values related to gr = 1.5, X-markers denote the values related to gr = 2.5, circle-markers denote the values related to gr = 3.5; the solid lines denote the best fit-correlations (Eqs. (13) and (14) in (a) and (b), respectively). (a) The variation of the Pe value, as defined in Eq. (12), with respect to the ratio U/eff ; (b) log10 (Pe) with respect to log10

U

/eff .

it is possible to relate it to the ratio between U and eff , as shown in Fig. 8a. The polynomial regression function displayed in the figure is given by

 U = aj (Pe)j eff 3

(13)

j=0

with a0 a2

= =

−5.7378 × 10−10 −4.5689 × 10−6

a1 a3

= =

+6.1049 × 10−6 −1.1921 × 10−5

(12) The resulting correlation coefficient is 0.9941; it means that the polynomial regression described by Eq. (13) fits the numerical data for every Pe value included in the range considered in this study. Fig. 8b shows the relation between the logarithms of both the parameters already considered in Fig. 8a and Eq. (13). In this way it is possible to appreciate the behaviour of U/eff for lowest Pe values. The correlation function is given by log

U = 0.9916 log Pe − 5.2468 eff

(14)

Eq. (14) describes very well the relation for log10 (Pe) ≤ −0.90 (Pe ≤ 0.123), i.e. considering low values of U, but does not fit the data very well when Pe is higher than the suggested value, which means for relatively high U. 3.2. Steady-state analysis eff

Fig. 7. The ratio /gr with respect to U. Cross-markers denote the ratio values related to gr = 1.5, X-markers denote the ratio values related to gr = 2.5, circlemarkers denote the ratio values related to gr = 3.5.

As previously mentioned, in this study the state after 40 years of heat injection/extraction is assumed as steady-state condition.

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Fig. 9. A graphical representation of the Influence Length; T0 denotes the undisturbed ground temperature.

In order to investigate the effects of the heat transfer at the steady-state, the Influence Length is defined as the distance from the borehole axis where the temperature is 0.5 ◦ C (Linf05 ) and 1 ◦ C (Linf1 ) higher (or lower) than the undisturbed ground temperature T0 , considering the perpendicular with respect to the groundwater flow direction (see Fig. 9). Fig. 10 shows the variation of the Influence Length, Linf , related to the different values of eff . The highest value of Linf1 is reached for eff ≤ 1.508 W m−1 K−1 , which means lowest gr (= 1.5 W m−1 K−1 ) and very limited flow rate (U < 5 ×10−8 m3 /s). On the other hand, if the value of T0 ± 0.5 ◦ C is taken into account, it is possible to see that Linf05 is obviously larger for all considered eff , but in particular if low U values are considered. Fig. 11 gives the possibility to predict the Influence Length when hydrogeological conditions of the site are known and therefore a good estimation of gr is possible. From Fig. 11 can be inferred that a very low convective contribution is enough to significantly decrease the Influence Length at steady-state. In fact all the values of eff /gr ≥ 1.01 result in a Linf05 ≤ 5 m, which means less than half the distance required by the Swedish authorities. This substantial

decrement of the Influence Lengths in presence of very low flow rate displayed in Figs. 10 and 11 is fully in agreement with Gehlin and Hellström (2003). This aspect is even more emphasized in Fig. 12, which displays the variations of the Influence Lengths Linf05 and Linf1 consequent to the increase of the flow rate U. If we consider a typical heat extraction borehole system, as defined in Section 1 (see Table 1), Fig. 12 shows that when U ≥ 0.9 × 10−7 m s−1 the temperature at the distance of 10 m is ≤T0 ± 1 ◦ C, while in presence of flow rate ≥1.8 × 10−7 m s−1 the temperature will be ≤T0 ± 0.5 ◦ C, instead of ≈T0 ± 1.4 ◦ C as in case of pure conduction. In addition, it is interesting to consider that the Influence Lengths Linf1 and Linf05 became lower than 5 m if U ≥ 1.9 × 10−7 m s−1 and U ≥ 3.5 × 10−7 m s−1 , respectively. Besides, it is noticeable that in the case gr is lower than 3.5 W (m K)−1 , for the same U values, Linf1 and Linf05 become even shorter. Also Fig. 13 could be very useful for a first estimation, even before the TRT analysis, of the Influence Length. In fact, if hydrogeological conditions of the site are known and therefore it is possible to make a good prediction about gr and flow rate U, the

Fig. 10. The Influence Lengths Linf1 and Linf05 with respect to the effective thermal conductivity eff . Cross-markers denote the values related to gr = 1.5, X-markers denote the values related to gr = 2.5, circle-markers denote the values related to gr = 3.5; solid and dotted lines are for Linf1 and Linf05 , respectively; dash-dotted line highlights the distance of 5 m from the borehole center.

Fig. 11. The Influence Lengths Linf1 and Linf05 with respect to the ratio eff /gr . Crossmarkers denote the values related to gr = 1.5, X-markers denote the values related to gr = 2.5, circle-markers denote the values related to gr = 3.5; solid and dotted lines are for Linf1 and Linf05 , respectively; dash-dotted line highlights the distance of 5 m from the borehole center.

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Fig. 12. The Influence Lengths Linf1 and Linf05 with respect to the flow rate U. Crossmarkers denote the values related to gr = 1.5, X-markers denote the values related to gr = 2.5, circle-markers denote the values related to gr = 3.5; solid and dotted lines are for Linf1 and Linf05 , respectively; dash-dotted line highlights the distance of 5 m from the borehole center.

correlated Pe number could be calculated and applied according to Eq. (12). The Pe number tends to increase when lower gr values are considered, while higher Pe values, at the same gr , imply lower Influence Lengths. According to Fig. 13, Linf1 is ≤10 m for Pe values ≥1.0 × 10−2 , while Linf1 is ≤5 m if Pe values are ≥1.9 × 10−2 ; on the other hand Linf05 turns out to be ≤10 m for Pe ≥1.6 × 10−2 and ≤5 m when Pe values are ≥3.1 × 10−2 . Taking into account the same Pe number, it happens that lower gr imply higher Influence Lengths; for these reasons, the above mentioned indications are related to the case where gr = 1.5 W m−1 K−1 . If gr is higher, the indicated Influence Lengths could be considered precautionary, while in case of lower gr the Pe number has to be higher to obtain the same Influence Lengths.

4. Discussion In this study 2D models are used in order to investigate the relation between gr , eff and U. Moreover, the Influence Length perpendicular to the groundwater flow direction has been defined as the radial distance from the borehole axis where the temperature disturbance is equal to a default value (in this study: T0 ± 0.5 ◦ C and T0 ± 1 ◦ C). The examination of the Influence Length has been useful to analyze the effects of the heat transfer around a BHE at steadystate and in particular it has been configured in order to reduce the borehole spacing perpendicular to the groundwater flow with respect to the currently recommended minimum (20 m) based on the steady-state temperature variations.

Fig. 13. The Influence Lengths Linf1 and Linf05 with respect to Pe (Eq. (12)). Crossmarkers denote the values related to gr = 1.5, X-markers denote the values related to gr = 2.5, circle-markers denote the values related to gr = 3.5; solid and dotted lines are for Linf1 and Linf05 , respectively; dash-dotted line highlights the distance of 5 m from the borehole center.

The relation between eff and U gives good indication about the conductive and the convective heat transfer contribution to the eff values calculated by thermal response analysis. The effect, described by Liebel (2012), of the increase of the difference  (Eq. (11)) and of the ratio eff /gr connected to relatively lower gr values, has been confirmed by the results of this study. Two regression functions for a first estimation of eff have been given: the first one (Eq. (13)), which relates the Pe number (Eq. (12)) with the ratio U /gr , is a third-degree polynomial and it fits the data on the whole considered range of values; the second proposed correlation (Eq. (14)) considers the logarithm of both Pe and U /gr and it is a linear equation which allows a sufficient accuracy only for Pe ≤ 0.123. It is evident that a good knowledge of the hydrogeological characteristics (gr and U) of the site is necessary in order to evaluate Pe and subsequently use it in the estimation of eff . In order to investigate the effects of the heat transfer at the steady-state, the Influence Lengths Linf05 (= T0 ± 0.5 ◦ C) and Linf1 (= T0 ± 1 ◦ C) have been defined. The behaviour of these Influence Lengths has been studied taking into account different values of eff and considering the ratio eff /gr . Both the comparisons indicate that a very low flow rate is enough to reduce Linf , though gr is a crucial parameter which has to be considered. In fact, considering a soil with gr = 1.5 W m−1 K−1 , will result Linf1 > 20 m in case of pure conduction (i.e. eff ≈ gr and eff /gr ≈ 1), while considering a groundwater flow rate such that eff ≈ 1.65 W m−1 K−1 (i.e. eff /gr ≈ 1.1) will result Linf1 < 1 m and Linf05 ≈ 1.2 m. If we compare these results with the currently recommended minimum borehole spacing, equal to 10 m and based on the steady-state temperature variations, the potential advantage in terms of perpendicular spacing between boreholes is evident. Moreover, while the eff value is measured with TRT analysis, the value of gr can be easily estimated to a first approximation using a geological map to determine the nature of the bedrock (and therefore its thermo-physical features). Even the relation between the flow rate U and Linf suggests that a very low value of U is sufficient to determine a significantly decrease of the required spacing between neighboring boreholes. Common hydraulic gradients are in the interval 0.01–0.001 ˚ (Aberg and Johansson, 1988), and typical hydraulic conductivities for sands are 2 × 10−7 to 6 × 10−3 m s−1 , for fractured crystalline rock 8 × 10−9 to 3 × 10−4 m s−1 (Chiasson et al., 2000). Clay and sandstone have even lower hydraulic conductivities. That means U for fractured rock would be in the order of 8 × 10−11 to 3 × 10−6 m s−1 , and sands would be 2 × 10−9 to 6 × 10−5 m s−1 . Hence in all sands the Influence Length could be expected to be considerably shorter than for pure conductive conditions. In fractured rock the Influence Length will be affected when U values exceed 2 × 10−7 m s−1 . Finally, if the hydrogeological conditions of the site are known and it is possible to make a good prediction about gr and flow rate U, the Peclét number could be calculated. Due to this, a first estimation, even before the TRT analysis, of the Influence Length is possible, always taking into account that the value of gr is a decisive parameter also in this case. Two-dimensional models do not take into account convective heat transfer which may occur due to buoyancy forces: when this phenomenon takes place, the convective heat transfer plays an important role for the thermal behavior of groundwater filled BHEs (Gehlin, 2002). Also the interactions between the borehole and the ground surface, together with any variation with depth (including end effects), are neglected in the study, resulting in an overestimation of the Influence Length. However, these assumptions imply that the results are obtained in a precautionary way. Moreover, the study has been carried out modeling a BHE filled with groundwater, which is the case in almost 100% of all Swedish

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BHEs. Therefore all the conclusions deriving from current work, as well as the obtained Influence Lengths, are valid for these specific conditions. Another important aspect it that the maximum Influence Length, perpendicularly to the groundwater flow direction, is located downstream of the borehole, as seen in Fig. 9. The effects of this behavior on the borehole performances will be investigated in future studies. The choice to use numerical models, instead of already existing analytical solutions, was made in prospective of continued studies on the Influence Length by including fractures in the numerical model. 5. Conclusions In this paper, the Influence Length, perpendicularly to the groundwater flow direction and from the borehole location has been defined. Knowing this value is of interest for dissipative borehole systems where it is important that the boreholes are as unaffected as possible by other boreholes. The boreholes must be placed in a line perpendicular to the groundwater flow to gain from the decreased Influence Length. Considering that the Influence Length is also of interest in areas with many single-borehole systems, e.g. in a housing area, where boreholes should not affect each other. In that case, a similar Influence Length, but along the flow direction, would be of interest to estimate, so that boreholes are not affecting each other downstream. However, the results discussed in this study could be taken into account only for multiple boreholes arranged perpendicular to the groundwater flow direction, because of the given definition of Influence Length. For storage systems with boreholes drilled in a compact configuration, Influence Length is of less importance: in storage systems the interaction between boreholes is desired, while the study on the Influence Length is made in order to find the minimum distance between boreholes where the thermal influence is negligible. References ˚ Aberg, B., Johansson, S., 1988. Vattenströmning till och från borrhålsvärmelager. (Water flow to and from a borehole heat store). Tech. Rep. R6: 1988. Swedish Council for Building Design (in Swedish). Austin, W., 1998. Development of an In-Situ System for Measuring Ground Thermal Properties (Master’s thesis). Oklahoma State University, Oklahoma, OK, USA. Bear, J., 1972. Dynamics of Fluids in Porous Media. Dover Publications, Inc., New York, USA. Carslaw, H., Jaeger, J., 1959. Conduction of Heat in Solids, 2nd ed. Oxford Press, Oxford, England. Chiasson, A., Rees, S., Spitler, J., 2000. A preliminary assessment of the effects of groundwater flow on closed-loop ground-source heat pump systems. ASHRAE Trans. 106, 380–393.

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