Axial Effects of Borehole Design

7/29/2019 Axial Effects of Borehole Design

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The importance of axial effects for borehole design of geothermalheat-pump systems

D. Marcotte a,b,c,*, P. Pasquier a, F. Sheriffb, M. Bernier c

a Golder Associates, 9200 lAcadie, Montreal, (Qc), H4N 2T2 Canadab CANMET Energy Technology Centre-Varennes, 1615 Lionel-Boulet Blvd., P.O. Box 4800, Varennes, (QC), J3X 1S6 Canadac Departement des genies civil, Geologique et des mines, Ecole Polytechnique de Montreal, C.P. 6079 Succ. Centre-ville, Montreal, (Qc), H3C 3A7 Canada

a r t i c l e i n f o

Article history:

Received 13 May 2008

Accepted 18 September 2009

Available online 23 October 2009

Keywords:

Infinite line source

Finite line source

Ground loop heat exchangers

Hybrid systems

Underground water freezing

a b s t r a c t

This paper studies the effects of axial heat conduction in boreholes used in geothermal heat pump

systems. The axial effects are examined by comparing the results obtained using the finite and infinite

line source methods. Using various practical design problems, it is shown that axial effects are relatively

important. Unsurprisingly, short boreholes and unbalanced yearly ground loads lead to stronger axial

effects. In one example considered, it is shown that the borehole length is 15% shorter when axial

conduction effects are considered. In another example dealing with underground water freezing, the

amount of energy that has to be removed to freeze the ground is three times higher when axial effects

are considered.

2009 Elsevier Ltd. All rights reserved.

1. Introduction

Geothermal systems using ground-coupled closed-loop heat

exchangers (GLHE) are becoming increasingly popular due to

growing energy costs. Such a system is presented in Fig. 1.

The operation of the system is relatively simple: a pump circu-

lates a heat transfer fluid in a closed circuit from the GLHE to a heat

pump (or a series of heat pumps). Typically, GLHE consists

of boreholes that are 100150 m deep and have a diameter of

1015 cm. The number of boreholes in the borefield can range from

one, for a residence, to several dozens, in commercial applications.

Furthermore, several borehole configurations (square, rectangular,

L-shaped) are possible. Typically, a borehole consists of two pipes

forming a U-tube (Fig. 1). The volume between these pipes and the

borehole wall is usually filled with grout to enhance heat transferfrom the fluid to the ground. In some situations it is advantageous

to design so-called hybrid systems in which a supplementary heat

rejecter or extractor is used at peak conditions to reduce the length

of the ground heat exchanger.

Given the relatively high cost of GLHE, it is important to design

them properly. Among the number of parameters that can be

varied, the length and configuration of the borefield are important.

There are basically two ways to design a borefield. The first method

involves using successive thermal pulses (typically 10-years 1month 6 h) to determine the length based on a given configura-tion and minimum/maximum heat pump entering water temper-

ature [8,3]. There are design software programs that perform these

calculations. Some use the concept of the g-functions developed by

Eskilson [5]. The g-functions are derived from a numerical model

that, by construction, includes the axial effects. The other approach

is to perform hourly simulation. This last approach is essential for

design of hybrid systems in which supplemental heat rejection/

injection is used. There are several software packages that can

perform hourly borehole simulations. For example, TRNSYS [9] andEnergyPlus [4] use the DST [6] and the short-time step model [5],

respectively. Even though these packages account for axial effects,

they necessitate a high level of expertise. Furthermore, it is not

easily possible to obtain ground temperature distributions like the

ones shown later in this paper. In this paper hourly simulations are

performed using the so-called finite and infinite line source

approximations where the borehole is approximated by a line with

a constant heat transfer rate per unit length. These approximations

present, in a convenient analytical form, the solution to the tran-

sient 2-D heat conduction problem. Despite their advantages,

hourly simulations based on the line source approximation are

* Corresponding author. Departement des genies civil, geologique et des mines,

Ecole Polytechnique de Montreal, C.P. 6079 Succ. Centre-ville, Montreal, (Qc),

H3C 3A7 Canada. Tel.: 1 514 340 4711x4620; fax: 1 514 340 3970.E-mail address: [email protected] (D. Marcotte).

Contents lists available at ScienceDirect

Renewable Energy

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / r e n e n e

0960-1481/$ see front matter 2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.renene.2009.09.015

Renewable Energy 35 (2010) 763770
mailto:[email protected]://www.sciencedirect.com/science/journal/09601481http://www.elsevier.com/locate/renenehttp://www.elsevier.com/locate/renenehttp://www.sciencedirect.com/science/journal/09601481mailto:[email protected]


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rarely used in routine design due to the perceived computational

burden.

The major difference between the finite and infinite line source

lies in the treatment of axial conduction (at the bottom and top of

the borehole) which is only accounted for in the former. Thetheoretical basis of the finite line source, although more involved

than for the infinite line source, was first established by Ingersoll

et al. [7]. It has been rediscovered recently by Zeng et al. [15] who

improved the model by imposing a constant temperature at the

ground surface. Lamarche and Beauchamp [11] have made a useful

contribution to speed up the computation of Zengs model. Finally,

Sheriff[13] extended Zengs model by permitting the borehole top

to be located at some distance below the ground surface. She also

did a detailed comparison of the finite and infinite line source

responses, but did not examine the repercussion on borefield

design.

At first glance, the axial heat-diffusion is likely to decrease

(increase) the borehole wall temperature in cooling (heating)

modes respectively. Therefore, designing without consideringaxial effects appears to provide a safety factor for the design. But,

is it really always the case? Moreover, are the borehole designs

incorporating axial effects significantly different from those

neglecting it? Under which circumstances are we expected to

have significant design differences? These are the main questions

we seek to answer. The main contribution of this research is to

describe, using synthetic case studies, the impact of considering

axial effects on the GLHE design. Our main finding is that for

many realistic circumstances the axial effects cannot be neglec-

ted. Therefore, design practices should be revised accordingly to

include the axial effects.

We first review briefly the theory for infinite and finite linesource models. Then, we present three different design situations.

The first two situations involve the sizing of geothermal systems

with and without the hybrid option, under three different hourly

ground load scenarios. The last design problem examines the

energy required and ground temperature evolution in the context

of ground freezing for environmental purposes.

2. Theoretical background

The basic building block of both infinite and finite line source

models is the change in temperature felt at a given location and

time due to the effect of a constant point source releasing q0 units ofheat per second [7]:

DTr; t q0

4pksrerfc

r

2ffiffiffiffiffiat

p

(1)

where erfc is the complementary error function, r the distance to

the point heat source, and a is the ground thermal diffusivity.

The line is then represented as a series of points equally spaced.

In the limit, when the distance between point sources goes to zero,

Fig. 1. Sketch of a GLHE system.

Nomenclature

a Thermal diffusivity (m2 s1)A, B, C, D Synthetic load model parameters (kW)

b r/H

Cs Ground volumetric heat capacity (Jm3 K1)

erfc (x) Complementary error function

(erfcx 12ffiffiffip

pRN

x et2 dt

EWT Temperature of fluid entering the heat pump (K or C)Fo Fourier number, Fo at/r2ks Volumetric ground thermal conductivity (Wm

1 K1)H Borehole length (m)

HP Heat Pump

q0 Radial heat transfer rate (W)q Radial heat transfer rate per unit length (Wm1)

S Borehole spacing (m)

r Distance to borehole (m)

rb Borehole radius (m)

Rb Borehole effective thermal resistance (Km W1)

t Time

DT(r, t) Ground temperature variation at time tand distance r

from the borehole (K or

C)

Tf Fluid temperature (K orC)

Tg Undisturbed ground temperature (K orC)

Tw Temperature at borehole wall (K orC)

u H2ffiffiffiffiat

px, y Spatial coordinates (m)

z Elevation (m)

D. Marcotte et al. / Renewable Energy 35 (2010) 763770764


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the combined effect felt at distance rfrom the source is obtained by

integration along the line.

2.1. Infinite line source

In an infinite medium, the line-integration gives the so-called

(infinite) line source model [7]:

DTr; t q4pks

ZNr2=4at

eu

udu (2)

2.2. Finite line source

In the case of a finite line source, the upper boundary is

considered at constant temperature, taken as the undisturbed

ground temperature [15]. This condition is represented by adding

a mirror image finite line source with the same load, but opposite

sign, as the real finite line. Then, integrating between the limits of

the real and image line, one obtains [15],[13]:

DTr; t;z q4pks

ZH

0

0@erfc du2ffiffiffiffiatp du

erfc

d0u2ffiffiffiffiatp d0u

1Adu (3)where du

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 z u2

qand d0u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 z u2

q, z is the

elevation of the point where the computation is done. The left part

of the integrand in Equation (3) represents the contribution by the

real finite line, the right part, the contribution of the image line.

Fig. 2 shows the vertical temperature profile obtained with

Equation (3) at radial distance r 2 m, after 200 days, and atr 1 m, after 2000 days of heat injection. The correspondinginfinite lines-source temperature is indicated as a reference. In this

example,the borehole is 50 m long, the groundthermal parameters

are ks 2.1 Wm1 K1 and Cs 2e06 Jm3 K1. The ground is inti-tially at 10 oC. The applied load is 60 W per m for a total heating

power of 3000 W. As expected, the importance of axial effects and

the discrepancy between infinite and finite models increases with

the Fourier number (at/r2 4.54 and 181.4 for these two cases).

In hourly simulations, the fluid temperature (Tf in Fig. 1) is

required. This necessitates knowledge of the borehole thermal

resistance Rb (i.e. from the fluid to the borehole wall), and of the

borehole wall temperature (Tw in Fig. 1) [2]. The average borehole

wall temperature it obtained by integrating Equation (3) along z.

However, this is computationally intensive due to the double

integration. Lamarche and Beauchamp [11] have shown, using an

appropriate change of variables, how to simplify Equation (3) to

a single integration. Accounting for small typos in [11] and [15] as

noted by Sheriff[13], the average temperature difference, between

a point located at distance rfrom the borehole and the undisturbed

ground temperature, is given by:

DTr; t q2pks

0BBB@

Zffiffiffiffiffiffiffiffiffib21p

b

erfcuzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 b2

q dz DA

Zffiffiffiffiffiffiffiffiffib24p

ffiffiffiffiffiffiffiffiffib21perfcuz

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 b2

qdz DB

1CCCA

(4)

where b r/H, r is the radial distance from the borehole center,u H

2ffiffiffiffiat

p and DA, and DB are given by:

DA ffiffiffiffiffiffiffiffiffiffiffiffiffiffib

2 1q

erfc

u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffib

2 1q

b erfcub

eu2b21 eu2b2

uffiffiffip

p!

and

DB ffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 1

qerfc

u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 1

q 0:5

b erfcub

ffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 4q erfcu ffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 4q

eu2b

21

0:5

eu2b2 eu2

b

24

uffiffiffip

p!

10 12 14 16 18 20 22 24

0

10

20

30

40

50

60

Temperature (oC)

Dept

h(m)

Vertical temperature profile

Infline, r=2, t=200 d

Fline, r=2, t=200 d

Fline average, r=2, t=200 d

Infline, r=1, t=2000 dFline, r=1, t=2000 d

Fline average, r=1, t=2000 d

Fig. 2. Vertical ground temperature profile at radial distances r1 m and r2 m after

respectively 2000 days and 200 days, Fo (r1, t2000)181.4 and Fo (r2,

t200)4.54.Constantheat injectionof 3000 W.Thermal parameters:ks2.1 Wm1 K1,

Cs

2e06 Jm3

K1

.

0 1000 2000 3000 4000 50000

2

4

6

8

10

12

Days

T(oC)

Infinite

Finite

FEM

Fig. 3. Comparison of Finite and Infinite line source model with finite element model

(FEM) for a 30 m borehole. Average temperature variation computed at 0.5 m from the

borehole axis, over the borehole length. Constant heat transfer rate of 1000 W.

Thermal parameters: ks

2.1 Wm1

K1

, Cs

2e06 Jm3

K1

.

D. Marcotte et al. / Renewable Energy 35 (2010) 763770 765


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The particular case r rb in Equation (4) gives the borehole walltemperature.

2.3. Numerical validation

Fig. 3 compares the variation in temperature over time

computed with finite and infinite line source to the numerical

results of a finite element model (FEM) constructed within

COMSOL. The finite element model is 2-D with axial symmetry

around the borehole axis. The ground is represented by a 50 m long

and 50 m radius cylinder. The borehole is represented by a 30 m

long and 0.075 m radius cylinder delivering 1000 W. The axis ofrevolution is located at the borehole center and constitutes

a thermal insulation boundary whereas all external boundaries are

set to the undisturbed ground temperature. Over 6000 triangular

elements equipped with quadratic interpolating functions are used

to discretize the model. The agreement between the FEM model

and the finite line source is almost perfect, the maximum absolute

difference in temperature over the 5000 days period being only

0.019 oC.

Fig. 4 compares the temperature obtained with the infinite

and finite line source models, at r 1 m and r 0.075 m (atypical value for rb), with the thermal parameters specified

above. A 1 oC temperature difference between the infinite and

finite models is obtained after 2.5 y and 2 y, at 1 m and 0.075 m

respectively. Note that the temperature reaches a plateau for the

finite line source model indicating that a steady-state condition

has been reached. In contrast, the infinite line source model

exhibits a linear behavior.

Fig. 5 shows the ground temperature, computed at a distance of1 m from the borehole, for increasing values of the borehole length.

As expected, the finite line source solution reaches the infinite line

source solution for long boreholes.

0.001 0.01 0.1 1 10 100 100010

20

30

40

50

60

70

80

r=0.07

5m

r=1m

Ground temperature

Time (y)

Temperature(oC)

Fig. 4. Comparison of Finite (solid) and Infinite (broken) line source model, computed

at distance 1 m and 0.075 m from the borehole. Constant heat transfer rate per unitlength of 100 W/m. Thermal parameters: ks2.1 Wm

1 K1, Cs2e06 Jm3 K1.

0 100 200 300 400 500 600 700 800 900 100012

12.5

13

13.5

Borehole length (m)

Temperature

(oC)

Average temperature vs borehole length

Infinite linesourceFinite linesource

Fig. 5. Infinite vs finite line source average temperature along a vertical profile. The load

is 20 W/m, thermal parameters: ks2.1 Wm1 K1, Cs2e06 Jm

3 K1. Temperature

computed after one year at r

1 m from the borehole.

1 2 3 4 5 6

100

0

100

Cooling (+) Heating () load

Time (h)

Load(kw)

1 2 3 4 5 6200

100

0

100

Load decomposition

Load(kw)

Fig. 6. Principle of temporal superposition for variable loads.

0 5 10 15 20 25 30 353

3.5

4

4.5

5

5.5

6

6.5

7

7.5

COP vs EWT

EWT

COP

Cooling

Heating

Fig. 7. COP as a function of EWT.



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3. Design of complete geothermal systems

In this section we compare the design length of borefields

obtained with the finite and infinite line source models for given

hourly ground load scenarios. These calculations imply that single

borehole solutions will need to be superimposed spatially. We have

already seen an instance of this principle of superposition while

computing the line source solution from a series of constant point

sources along a line [7], see Equations (1 and 2). The additivity of

effects (variation in temperature) stems from the linear relationbetween q and DT, and the fact that energy is an extensive and

additive variable. The temporal superposition also follows the same

general principle of addition of effects as described by Yavuzturk

and Spitler [14] and illustrated by Fig. 6. When the load is varying

hourly, a new pulse is applied each hour. It is simply the difference

between the load for two consecutive hours. More formally, for the

infinite line source as an example, with a single borehole, we have:

DTr; t X

i; tit

qi

4pk

ZNr2=4atti

eu

udu (5)

where: q*1 q1, and q*i qiqi1, i 2.I, tI t, is the incrementalload between two successive hours. With multiple boreholes,

DTx0; t Xn

j 1

Xi; tit

q0i

4pk

ZNkxjx0k2=4atti

eu

udu (6)

where: n is the number of boreholes, xj and x0 are the coordinate

vectors of borehole j and point where temperature is computed,

respectively. Note that for long simulation periods, the computa-

tional burden becomes important.

In the test cases that follow we assume that all of the buildingheating and cooling loads are to be provided by the GLHE system,

i.e. there is no supplementary heat rejection/injection. Synthetic

building loads are used to enhance the reproducibility of our

results. These building loads are simulated using:

Qt A B cos

t

87602p

C cos

t

242p

D cos

t

242p

cos

2t

87602p

(7)

In Equation (7), t is in hours, A controls the annual load

unbalance, B the half-amplitude of annual load variation, C and D

40 30 20 10 0 10 20 30 40

40

30

20

10

0

10

20

30

40

1

2

3 4

5

6 7

8 9

10

11 12

13

14 15

16 17

18 19

20 21

22 23

24 25

26

27 28

29

30 31

32 33

34 35

36 37

38 39

40 41

42 43

44 45

46

47 48

49

50 51

52 53

54 55

56 57

58 59

60 61

62 63

64 65

66 67

68 69

70

71 72

73 74

75 76

77 78

79 80

81

82 83

84 85

86 87

88 89

90 91

92 93

94 95

96 97

98 99

100 101

102 103

104 105

106 107

108 109

110

111 112

113

114 115

116 117

118 119

120 121

122 123

124 125

126 127

128 129

130 131

132 133

134 135

136 137

138 139

140 141

142 143

144 145

146

147 148

149

150 151

152 153

154 155

156 157

158 159

160 161

162 163

164 165

166 167

168 169

170 171

172 173

174 175

176 177

178 179

180 181

182 183

184 185

186 187

188 189

190 191

192 193

194 195

196 197

198 199

200 201

202 203

204 205

206 207

208 209

210 211

212 213

214 215

216 217

218 219

220 221

222 223

224 225

Borehole location and priority number

Coord. x (m)

Coord.y(m)

Fig. 8. Borehole grid and priority number. Number indicates order of inclusion in the design when required.

Table 1

Number of boreholes required, complete geothermal system. Constant T assumes

a constant ground surface temperature of 10 oC, Periodic T assumes a periodic

ground surface temperature with an amplitude of 20 oC in phase with the heatload.

Scenar io Borehole length Infinite li ne Fini te l in e

Constant T Periodic T

Balanced (A 17) 100 m 33 33 34Balanced 50 m 76 74 80

Cooling dominant (A 17) 100 m 39 36 37Cooling dominant 50 m 93 79 81

Heating dominant

(A 30)100 m 57 53 56

Heating dominant 50 m 134 115 124

Table 2

Number of boreholes required, hybrid system. HP capacity represents 40% of

maximum building load. The last two column represent the percentage of the

building load supplied by the HP for each mode.

Scenario Borehole

length

Number of

boreholes

% Energy

Infinite Finite Cooling Inf.

(Fin.)

Heating Inf.

(Fin.)

Balanced 100 m 19 19 69 (69) 77 (78)

Balanced 50 m 37 37 67 (67) 72 (73)

Cooling dominant 100 m 24 24 69 (69) 86 (86)

Cooling dominant 50 m 41 39 70 (69) 90 (88)

Heating dominant 100 m 37 37 67 (67) 83 (86)

Heating dominant 50 m 55 53 72 (70) 70 (73)



6/8

the half-amplitude of daily load fluctuations. D/C controls the

relative importance of the damped component used to simulate

larger daily fluctuations in winter and summer. Coefficients A to D

are in kW.

We consider three different load scenarios, each with B 100,C 50, and D 25. One is approximately balanced (A 17), one isa cooling dominated load (A 17) and the other is a heating domi-natedload(A 30).Conversionof buildingloads to ground loads isdone with: qground qbuilding(1 1/COP). The heat pump COP variesas a function of entering water temperature as depicted in Fig. 7.

For all scenarios, we consider a unique set of possible locations

for the boreholes. The locations are at the nodes of a regular grid of

mesh S

6 m. The boreholes are assigned a priority number (lower

number / highest priority), moving excentrically from the grid

center to the fringes (see Fig. 8). The system is simulated with

100 m and then, 50 m long boreholes. The borehole heads are

located at the ground surface.

For each scenario and given number of boreholes, we simulate

the hourly fluid temperature for a 10-year period. We repeat this

simulation for different number of boreholes and finally keep the

lowest number of boreholes ensuring that the fluid temperature at

the HP entrance never exceeds the HP limit specifications. We do

this computation using both the infinite and finite line source

models. Thermalparameters of the ground are: thermal conductivity

ks 2 Wm1 K1, volumetric heat capacity Cs 3.4e06 Jm3 K1and borehole thermal resistance Rb 0.1 m KW1. The ground isinitially at 10 oC and all the boreholes have a radius of rb

0.075 m.

Table 1 gives the number of boreholes required for a specificscenario and model. The choice of model does not have a strong

impact on the balanced load scenario. However, for cooling and

heating dominant scenarios, the finite line source model indicates

a reduction in the number of boreholes of approximately 7% and

15% for the 100 m and 50 m borehole length. As expected, the

shorter the borehole length, the greater are the discrepancies

between both models.

3.1. Seasonal effects

The finite line source model assumes a constant temperature at

the ground surface equal to the undisturbed ground temperature.

This hypothesis would more or less correspond to the case of

geothermal boreholes located under a building slab. However, in

many cases, the boreholes are located outside the building area

where the ground surface temperature varies in phase with the

heat load. One can expect the axial effect at the ground surface will

be less than under the constant temperature assumption. The

influence of ground surface temperature variation on the ground

temperature, at any depth and time, can be computed for a periodic

signal [7]. Using the principle of superposition, the influence on the

design can be assessed.

Last column of Table 1 gives the number of boreholes required

when the groundsurface temperature is periodic, in phase with the

heat load, and shows a yearly variation of 20 oC. The boreholeheads are located at the ground surface, a feature that maximizes

the seasonal effects on the design. As expected, the designs with

periodic ground surface temperature require more boreholes than

with constant ground surface temperature. For unbalanced

scenarios, the periodic ground surface temperature solutions

obtained are intermediate between the infinite and the finite line

source designs.

0 5 10 15 20 25 306

4

2

0

2

4

6

8

Time (year)

o

Temperature(

C)

Rock temperature at x=(13,5,12.0)

Infinite linesource

Finite linesource

15 10 5 0 5 10 1515

10

5

0

5

10

15Control point

x (m)

y(m)

Fig. 9. Vertical average rock temperature at point x (13.5, 12.0). Thermal parameters: ks2 Wm1 K1, Cs3.4e06 Jm

3 K1.

0 5 10 15 20 25 301000

900

800

700

600

500

400

300

200

100

0

Time (year)

Load(W)

Ground cooling load / borehole

Infinite linesource

Finite linesource

Fig. 10. Evolution of average load per borehole.



7/8

4. Design of hybrid geothermal systems

The design situation borrows its essential features (borehole

grid, thermal parameters, heat load scenarios) from the previous

complete geothermal system example. The main difference is that

the HP and auxiliary system power are respectively selected at 40%

and 60% of the maximum building load. The simulation works thesame way as previously with the following modification. Each hour,

the fluid temperature at the HP entrance (EWT) is computed. If

EWT, at any hour, exceeds the HP specification limit, it is assumed

that the HP cycles to keep the EWT at its limit value, the excess load

being taken by the auxiliary system. We keep track of all the loads

provided by the auxiliary system. At the end of the 10-year simu-

lation, we verify that the load to the auxiliary system never exceeds

its capacity. If exceeded, we add more boreholes. The final design is

obtained with the smallest number of boreholes compatible with

the auxiliary system capacity.

Table 2 shows the number of boreholes for a specific scenario

and computation model. The proportion of energy provided by the

geothermal system, relative to the total building energy required

for the 10-year period, is also given. Unbalanced scenarios with

short boreholes display reduction of only 6% for the number of

boreholes required. However, comparison of Tables 1 and 2 reveals

striking reduction in numberof boreholes. The hybrid systems need

only between 46% and 70% of the number of boreholes for the

complete geothermal system. Yet, the geothermal components of

hybrid systems are able to supply between 67% and 70% of the

cooling load and between 73% and 88% of the heating load for thevarious scenarios examined. As the borehole construction cost

represents the main investment in a geothermal system, it indi-

cates that hybrid systems can be economically advantageous.

Even though hourly simulations are computationally more

involved, efficient methods exist to speed up the computations

[1,10,12]. For example, a hybrid system 10-years hourly simulation

for a 45 boreholes field is computed in less than 2 min on a stan-

dard laptop.

5. Comparing performances for a rock freezing problem

In some environmental problems, like those involving DNAPL

(dense non-aqueous phase liquid), the contaminants reach the

bedrock and then can propagate through the rock fractures. In such

X coordinate (m)

Ycoordinate(m

)

Infinite line: Temperature after 5 years

15 10 5 0 5 10 15

15

10

5

0

5

10

15

12

10

8

6

4

2

0

2

X coordinate (m)

Ycoordinate(m

)

Finite line: Temperature after 5 years

15 10 5 0 5 10 15

15

10

5

0

5

10

15

12

10

8

6

4

2

0

2

X coordinate (m)

Ycoordinate(m)

Infinite line: Temperature after 20 years

15

10

5

0

5

10

15

12

10

8

6

4

2

0

2

X coordinate (m)

Ycoordinate(m)

Finite line: Temperature after 20 years

15 10 5 0 5 10 15 15 10 5 0 5 10 1515

10

5

0

5

10

15

12

10

8

6

4

2

0

2

Fig. 11. Ground temperature after 5 years (top) and 20 years (bottom). Infinite line source model (left) and finite line source model (right).



8/8

circumstances, remediation can be extremely difficult and expen-

sive. One solution is to catch and treat all the contaminants flowing

through the fractures. This requires designing a series of wells

forming an hydraulic trap. The pumping andtreatment is expensive

and it has to be maintained typically for hundreds of years. More-

over, the flow of water to treat could be relatively important. An

alternative is to permanently freeze the liquid within the fractures

to avoid any movement of the contaminant.

In the following example, we consider the problem of freezing

water inside rock fractures, in a domain 30 m by 30 m by 30 m, for

the purpose of long term confinement of toxic DNAPLs. A 10 10borefield is considered (see Fig. 9). Thermal parameters of the rock

are: thermal conductivity ks 2 Wm1 K1 and volumetric heatcapacity Cs 3.4e06 Jm3 K1. The borehole thermal resistance isRb 0.1 m KW1. The rock is initially at 10 oC. The rock porosity issmall enough so as to neglect the latent heat of water and the

increase of rock thermal conductivity occurring when water

freezes. The total HP capacity is 100 kW (one kW per borehole)

and the EWTlimitis 8 oC. Themass flow of the circulating fluid is19 l/s (or 0.19 l/s for each borehole). The simulation is run for 30

years. The HP capacity is kept at its maximum until the EWT has

reached the limiting value of

8 oC. Then, the HP cycles to

maintain the EWT at this temperature. The performance of thesystem is monitored by computing rock temperature at a control

point located at the borefield periphery at (x, y) (13.5, 12) m (seeFig. 9).

Fig. 9 shows the rock temperature obtained at this control point

using the infinite and finite line source models. Fig. 10 shows the

corresponding average hourly load per borehole. In both figures,

the rugged nature of the load curves is due to intermittent opera-

tion of the HP. As shown in Fig. 9, the rock becomes permanently

frozen after 10 months and 2.2 years for the infinite and finite

models, respectively. The temperature difference between both

models increases steadily with time, to reach approximately 2 oC

after 30 years. The finite line source model predicts that about

150 W per borehole are required compared to only 50 W per

borehole for the infinite line source model (Fig. 10). Thus, despitea higher heat transfer rate per borehole, the finite line source model

predicts a higher rock temperature at the control point.

In this next example, the same 10 10 borefield is cooled ata rate of 10 W/m (300 W/borehole) for the first year and then

3.33 W/m (100 W/borehole) permanently. The temperature distri-

bution in the ground is computed using the infinite and finite line

source models after 5 and 20 years. Results are presented in Fig. 11.

The proportion of the domain with temperature below freezing is

79% and 100% after 5 and 20 years for the infinite line source

calculation whereas it is only 10% and 52% with the finite line

source model. Therefore, a design based on the use of the infinite

line source would have wrongly predicted ground freezing. Thus,

contrary to the two previous examples, the infinite model does not

provide a conservative design.

6. Conclusion

This paper studies the effect of axial heat conduction in bore-

holes used in geothermal heat pump systems. The axial effects are

examined by comparing the results obtained using the finite and

infinite line source methods. Using various practical design prob-

lems it is shown that axial effects are relatively important. Even

with a borehole spacing / borehole length (S/H) as lowas 0.06, the

finite line source model indicated that 7% less boreholes are

required,for a complete geothermal system, when the yearly load is

unbalanced either in cooling or in heating. The reduction in the

number of boreholes due to axial effects increases to 15% for 50 m

long boreholes (ratio S/H 0.12). Differences between the finite

and infinite line source models were noticeably smaller for the

hybrid system but still indicate that less boreholes are generally

required with the finite line source model. As sizing of geothermal

system is a determining factor for the profitability of a project, axial

effects should certainly be considered in similar circumstances.

Also, the value of using hybrid systems to substantially reduce the

number of boreholes, yet preserving most energy savings, was

confirmed.

In another example dealing with underground water freezing,

the amount of energy that has to be removed to freezethe ground is

three times higher when axial effects are considered. Furthermore,

for this particular example, a design based on the infinite line

source would not achieve the ground freezing goal. Indeed, while

the infinite line source calculations show that 100% of the ground

would be frozen, the finite line source model predicts that only 52%

of the ground would be frozen.

Together, these examples clearly indicate that axial effects are

problem dependent. The magnitude of these effects is difficult to

anticipate and cannot generally be neglected. Moreover, neglecting

the axial effects (i.e. using the infinite line source model) does not

always provide a conservative design. Since the computing cost of

finite line source model, when properly implemented, is only

slightly superior to the computing cost of the infinite line sourcemodel, there is no reason to overlook axial effects in any borefield

design.

Acknowledgments

Part of this research was financed by the National Science and

Engineering Research Council of Canada. We thank Jeremie

Gaucher for English editing. Helpful suggestions from anonymous

reviewers are acknowledged.

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Axial Effects of Borehole Design