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This is the published version of a paper presented at European Geothermal Congress 2016,Strasbourg, France, 19-24 Sept 2016.

Citation for the original published paper:

Lazzarotto, A., Acuña, J., Monzó, P. (2016)Analysis and modeling of a large borehole system in Sweden.In:

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European Geothermal Congress 2016Strasbourg, France, 19-24 Sept 2016

Analysis and modeling of a large borehole system in Sweden

Alberto Lazzarotto1, José Acuña1, Patricia Monzó11 KTH - Royal Institute of Technology, Department of Energy Technology, Brinelvägen 68, S-100 40, Stockholm

alberto.lazzarotto@energy.kth.se

Keywords: Bore field, modeling, finite line source.

ABSTRACTThis paper presents a study on the thermal simulationof a large existing borehole thermal energy storage(BTES) system located in Stockholm, Sweden. Thebore field investigated presents an uneven pattern,which comprises vertical and inclined boreholes, for atotal of 130 units. Such complex bore field geometrycannot be modeled with the current availablecommercial design tools. The test case presented isutilized to explore the influence of boundaryconditions and level of detail utilized for representingthe model geometry on the output of the simulation.Two boundary conditions and three geometricalconfigurations were studied. The results show that, inthe considered case, the results obtained with thetested models give a marginal difference, hence alsothe greatest level of simplification can be utilizedwithout loosing accuracy in the analysis.

1. INTRODUCTION The design of bore field systems for shallowgeothermal application requires the use ofmathematical models for the long term prediction ofthe thermal behavior of these systems. Many of theavailable modeling tools for the thermal simulation ofbore fields (Blomberg et al., 2015; Spitler, 2000) arebased on the so-called Eskilson’s g-function approach(Eskilson 1987). In this method the ground is modeledas a semi-infinite solid with uniform and isotropicthermal properties. Under these conditions, the heattransfer process is governed by a linear partialequation and superposition of the effects can beapplied. As a result the so-called boreholetemperature, a key variable that influences the overallperformance of bore field systems, can be computedby using convolution methods which are fast from acomputational point of view, hence suitable fordesign.

An important novelty introduced in the work ofEskilson was the idea of determining a suitableborehole temperature response for an entire boreholefield. In particular he focused on the evaluation of theborehole response temperature to a unitary step heatinjection for bore fields in parallel arrangement. This

condition was approximated by imposing uniformtemperature along the boreholes and among theboreholes as heat is injected in the bore field(Cimmino et al., 2013). Eskilson utilized a numericalcode (Eskilson, 1986) to calculate these temperatureresponses in non-dimensional form and built adatabase containing responses for a large number ofbore field configurations (the g-functions). Thisdatabase constitutes the foundation of state of the artdesign tools.

A parallel path to the studies performed by Eskilsonon numerical modeling has been taken by severalauthors who worked on the development of modelsbased on analytical solutions given theircomputational efficiency, robustness and flexibility.These models were firstly explored by Ingersoll, Plass(1948) and by Carslaw, Jaeger (1959) who introducedrespectively the infinite line source and the cylindricalline source models. Claesson and Eskilson addressedthe problem of the three dimensional nature of the heattransfer in the surrounding of a borehole heatexchanger and proposed a finite line source model inthe unpublished journal article “Conductive HeatExtraction by a Deep Borehole. Analytical Studies.”contained in Eskilson’s PhD thesis (Eskilson, 1987).In the last decades many researchers (Claesson andJaved, 2011; Lamarche and Beauchamp, 2007; Zenget al., 2002), further developed this analytical solutionin order to make it usable for calculation regardingmultiple borehole systems. Cui et al. (2006), Marcotte,Pasquier (2009)and Lamarche (2011) utilized finiteline sources to simulate borehole systems withinclined boreholes showing the variety of conditionswhere these kind of methods can be applied.

In a recent development by Cimmino, Bernier (2014)a semi-analytical method based on finite line sourceshas been employed to reproduce the boundaryconditions used by Eskilson for the generation of g-functions. The authors compared their results with g-functions from Eskilson for a large set ofconfigurations and found good agreement between thetwo methods. A similar methodology has been thenutilized by Lazzarotto, Björk (2016); Lazzarotto(2016) to study the case of arbitrarily inclinedborehole systems.

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mailto:alberto.lazzarotto@energy.kth.se

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As it is clear from the literature review given so far,the modeling technique within the field of boreholesystems has been evolving over the years to providefast computational tools capable of a greater level ofdetail both in regards to the physical description of thesystem as well as in regards to its geometricalrepresentation. In this paper we present a study wherea few selected modeling strategies with different levelof complexity are applied to simulate the thermalbehavior of a large existing bore field systemcharacterized by a relatively articulated geometricalconfiguration. The objective is to evaluate the effectthat the choice of boundary conditions and the detailutilize for the geometrical representation of the systemhave on the simulated results for the given study case.

2. METHODOLOGYThis section presents the system object of study andthe set of modeling strategies that have beeninvestigated within this work. The conditionsreviewed comprises two sets of boundary conditionsand three different geometrical configurations.

The analysis of the dynamic behavior of the bore fieldhas been performed using the g-functionsmethodology. g-functions have been calculated foreach of the modeling strategy investigated using themethodology reviewed by Lazzarotto (2016). Thecomputed g-functions along with the design loadingcondition for the actual borehole system have beenthen utilized for the calculation of the temperature ofthe brine circulating within the borehole heatexchangers after a life-time horizon of 25 years.

2.1 The bore fieldThe bore field analyzed is an existing system that hasbeen recently built at the Stockholm UniversityCampus to supply heating and cooling to a group ofoffices and laboratories. The analysis provided in thispaper is relative to design condition..

The geometrical configuration of the system consistsof 130 boreholes unevenly distributed within theavailable ground volume. Two of these boreholes are270 meters long while the remaining ones are 230meters long. A plan view of the configuration isshown in Figure 1a: the blue dots represents thedrilling point at the surface while the blue segmentsrepresents the projection of the boreholes into thesurface for those boreholes that are inclined incomparison with the vertical direction.

2.2 Simplified geometrical configurationsThe design configuration described in section 2.1 isrelatively complex and the calculation of the g-function for such a system is a challenging problemthat can be tackled only using advanced research tools.It is therefore of interest studying the g-function of theoriginal configuration, but also of simplifiedgeometrical configuration which are easier tocompute.

The design configuration described in section 2.1 isrelatively complex and the calculation of the g-function for such a system is a challenging problemthat can be tackled only using advanced research tools.It is therefore of interest studying the g-function of theoriginal configuration, but also of simplifiedgeometrical configuration which are easier to handlefrom a modeling stand point.

“Mid point geometry”. The first simplificationintroduced consists in eliminating the difficulty ofstudying arbitrarily inclined boreholes. Each non-vertical borehole has been substituted by a vertical oneplaced at the mid point of its projection on the planview. An illustration of the plan view of this system isgiven in figure 1b. The boreholes in this case arevertical and are marked with red triangles. Theoriginal borehole field is plot in transparency forreference.

“Hexagonal pattern”. The second simplificationintroduced consists in placing the boreholes uniformlyin the storage area. The 130 boreholes of the systemare placed in hexagonal pattern as illustrated in figure1c (green squares). The distance between neighboringboreholes in this case is uniform. The choice of thisdistance has been made in order to get a good matchbetween the green area in figure 1b and the red area infigure 1c. Both areas are the ones delimited by theconvex hull relative to the set of points representingthe boreholes for the considered bore fieldconfiguration. The main idea behind this choice isstudying two system that occupies roughly the samevolume but have a slightly different boreholedistribution.

2.3 Boundary conditionsA key aspect in the thermal analysis of bore fieldsystem is the set of boundary conditions utilized forthe calculation of the g-function. In this study twoclassical boundary conditions are investigated. Thenotation utilized is coherent with the one introducedby Eskilson (1987)

BC1. Boreholes are modeled as finite line sources.The heat flux injected along the boreholes is uniform.The temperature along the boreholes is not uniformand the borehole temperature Tb is computed as themean temperature along the boreholes.

BC3. Borehole are modeled as finite lines, but in thiscase the distribution of heat in not uniform. Instead,heat is distributed so that the temperature along theborehole and among the boreholes is uniform andequal to the borehole temperature Tb.

BC3 is the boundary condition utilized by Eskilson forthe calculation of g-functions and is regarded as thereference condition. An in depth review on boundaryconditions for the calculation of g-functions has beenrecently done by Cimmino, Bernier (2014).

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2.4 Fluid temperature calculationGiven the g-function g(t) for the system object ofstudy, the fluid temperature is calculated as follow.Time is discretized into hourly steps and for each stepj, a correspondent value for the borehole temperatureTb[j], a value for the g-function g[j] and a value for theheat per meter injected into the ground q0[j] aredefined. The unknown borehole temperature iscalculated by applying the superposition of effectsaccording to the following expression:

T b [ j ]−T 0=1

2πk g∑i=2

j

(q ' [i ]−q ' [i−1])g(t [ j ]−t [i ]) [1]

Where T0 is the undisturbed ground temperature andkg is the thermal conductivity of the ground. Since theload q’[i] with i = 1,···,j is given, this calculation hasbeen performed using an equivalent method based onFFT (Marcotte and Pasquier, 2008) that ensure greatcomputational performance.

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Figure 1: Plan view of the bore field geometrical configurations considered within this study.

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Once the temperature Tb[j] is given, the fluidtemperature is then calculated using the followingexpression.

T f [ j ]−T b[ j]=q ' [ j]Rb [2]

where Rb* is the effective borehole thermal resistance.The value of the resistance Rb* and of thermalconductivity utilized for the calculations within thispaper are respectively 0.06 m K/W and 3.9 W/m K.

3. RESULTS AND DISCUSSION3.1 g-function analysisThe results of the calculation of the g-functions for thesix cases investigated are given in figure 2. Purplecurves are relative to calculations performed using theBC1 boundary condition while light blue curves arerelative to calculations performed using the BC3boundary condition. Solid lines, dash-dot lines anddashed lines are relative respectively to the designgeometry, to the ‘mid point” configuration and to the“hexagonal pattern” configuration. Finally the blacksolid line with triangle markers is the infinite linesource solution for a single borehole system which isdisplayed for reference. The g-functions are plotted inthe classic fashion as a function of ln(t/ts) where tsequal to H2/9αg is the time when the system approachsteady state conditions.

Figure 2 shows that the six systems analyzed can beclustered into three groups with distinguishable g-function:

• group 1: BC1 - design configuration, BC1 - “midpoint” configuration, BC1 - “hexagonal pattern”.

• group 2: BC3 - “mid point” configuration, BC3-“hexagonal pattern”.

• group 3: BC3 - design configuration.

Moreover, three time regions can be distinguishedwithin the plot:

• −8 ≤ ln(t/ts) < −6: All the g-functions are coincidentand equal to the infinite line source solution. At thisstage there is no mutual thermal influence betweenneighboring boreholes.

• −6 ≤ ln(t/ts) < −3: The g-functions are still coincidentbut they deviate from the infinite line source solution.In this region the thermal influence effect is noticeablebut the set of configuration and boundary conditionsinvestigated yield negligible difference in the thermalresponse.

• ln(t/ts)≥−3: The three groups of g-functionsintroduced in this section start to deviate. The effect ofboundary conditions and geometry plays a role in thethermal behavior of the system.

The results show that the major factor that affects theresponse temperature for the systems investigated is

the choice of boundary condition. For boundarycondition BC1 all the three geometrical patternsstudied yield roughly the same g-functions showingthat in this case the simplifications introduced in thegeometrical pattern do not affect the thermalcalculation. For boundary condition BC3 there is adeviation between the g-function of the originalconfiguration and the ones relative to “mid point” and“hexagonal pattern” configurations: the originalconfiguration with inclined boreholes yieldssignificantly greater g-function values for ln(t/ts) > −2suggesting that the heat distribution in the ground forthis case is noticeably different in comparison with theone relative to the simplified geometrical patterns.

3.2 Time frame of interestTraditionally g-functions are represented as a functionof the non-dimensional time ln(t/ts). This is a usefulrepresentation that enables the description of thetemperature response for families of configurationswith a given set of common characteristics. Thedrawback of this representation is that it is arguablynot so intuitive from a physical perspective. For thisreason an extra axis with a time scale in years hasbeen added to Figure 2. The time scale is relative tothe system object of study where αg = 1.875e −6 m2/sand H = 230 m.

The introduction of the new time scale is functional toanalyze the g-functions considering that by design theinstallation life-time horizon is typically set at around25 years. For the case study considered the 25 yearstime threshold corresponds to ln(t/ts) equal to -1.38.Although this value of ln(t/ts) is within the regionwhere the three groups of g-functions consideredmutually diverge, their difference is still relativelysmall and therefore the three groups of systems areexpected to behave in a similar fashion.

3.2 Bore field system simulationsIn order to quantify the effect of the choice of modelfor a time-horizon of 25 years, an hourly simulation ofthe dynamic variation of the temperature of thesecondary fluid circulating within the ground loop hasbeen performed. The thermal loading conditionconsidered for this test (Figure 3) is the load that hasbeen utilized for the design of the installation.

Figure 4 displays the results of the simulation. Theyearly mean borehole temperature for the systemsinvestigated increases with time since the yearlyinjection load is greater than the yearly extractionload. Even though a discrepancy between the fluidtemperatures obtained with the different models ispresent, this difference is relatively small: themaximum difference between group 1 and group 3 is1°C and the maximum temperature difference betweengroup 2 and group 3 is 0.45 °C.

The divergence between the models increases withtime as the effect of the yearly unbalanced becomesmore relevant.

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Figure 3: Design ground loading condition

Figure 2: g-functions computed using BC1 and BC3 boundary conditions for the three geometrical configurationsinvestigated. The g-functions are plotted as functions of the non-dimensional time ln(t/ts) with ts equal to H2/9αg. Anadditional axis has been added to visualize time in years for αg = 1.875e −6 m2/s and H = 230 m.

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Figure 4: Fluid temperature resulting from a 25 years hourly simulations. The three curves are relative to the threegroups of bore field systems considered yielding distinct g-functions (section 3.1).

Figure 5: Fluid temperature during the 25th year of operation.

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Figure 5 shows the fluid temperature for the threegroups of systems investigated during the 25th year ofoperation which is the period when the models yieldsthe greatest differences within the design time frame.The divergence between the models will tend toincrease also after the 25th year as it can be easilyderived from figure 2.

4. CONCLUSIONSIn this paper we investigated the effect of the choiceof boundary condition and geometrical configurationon the modeling of an existing large bore field system.The configuration studied presented a relativelycomplex geometrical pattern with both vertical andinclined boreholes. Such system cannot be studiedwith standard simulation tools commercially availableand requires advanced research tools. Two simplifiedconfigurations were proposed as possible substitute tothe original configuration for modeling purposes.Concerning the physical modeling of the system, bothuniform heat flux boundary condition (BC1) anduniform temperature boundary condition (BC3) wereinvestigated.

The six models studied showed similar thermalbehaviors and negligible temperature differences froma bore field design perspective. In conclusion for thisparticular study case the proposed approximations ofthe geometrical pattern and of the boundary conditionare acceptable and yield results that are comparablewith the ones obtained with more sophisticated modelscapable of a more detailed description of the system.A major reason for this result is the fact that the 25years life-time horizon utilized for bore field design isa relatively short time in comparison with the dynamicof the bore field configuration analyzed within thisstudy.

4. ACKNOWLEDGMENTS

The work in this publication was financed by theSwedish Energy Agency. Their support is kindlyacknowledged.

5. REFERENCESBlomberg, T., Claesson, J., Eskilson, P., Hellström, G., and

Sanner, B. (2015). Earth Energy Designer.

Carslaw, H.S., and Jaeger, J.C. (1959). Conduction of Heatin Solids (Oxford Oxfordshire : New York: OxfordUniversity Press).

Cimmino, M., and Bernier, M. (2014). A semi-analyticalmethod to generate g-functions for geothermal borefields. International Journal of Heat and Mass Transfer70, 641–650.

Cimmino, M., Bernier, M., and Adams, F. (2013). Acontribution towards the determination of g-functions

using the finite line source. Applied ThermalEngineering 51, 401–412.

Claesson, J., and Javed, S. (2011). An Analytical Method toCalculate Borehole Fluid Temperatures for Time-scalesfrom Minutes to Decades. ASHRAE Transactions 117,279–288.

Cui, P., Yang, H., and Fang, Z. (2006). Heat transferanalysis of ground heat exchangers with inclinedboreholes. Applied Thermal Engineering 26, 1169–1175.

Eskilson, P. (1986). Superposition Borehole Model - Manualfor Computer Code. (Lund, Sweden: Department ofMathematical Physics, Lund Institute of Technology).

Eskilson, P. (1987). Thermal analysis of heat extractionboreholes. Ph.D Thesis.

Ingersoll, L., and Plass, H. (1948). Theory of the groundheat pipe heat source for the heatpump. Transactions ofthe American Society of Heating and VentilatingEngineers.

Lamarche, L. (2011). Analytical g-function for inclinedboreholes in ground-source heat pump systems.Geothermics 40, 241–249.

Lamarche, L., and Beauchamp, B. (2007). A newcontribution to the finite line-source model forgeothermal boreholes. Energy and Buildings 39, 188–198.

Lazzarotto, A. (2016). A methodology for the calculation ofresponse functions for geothermal fields with arbitrarilyoriented boreholes – Part 1. Renewable Energy 86,1380–1393.

Lazzarotto, A., and Björk, F. (2016). A methodology for thecalculation of response functions for geothermal fieldswith arbitrarily oriented boreholes – Part 2. RenewableEnergy 86, 1353–1361.

Marcotte, D., and Pasquier, P. (2008). Fast fluid and groundtemperature computation for geothermal ground-loopheat exchanger systems. Geothermics 37, 651–665.

Marcotte, D., and Pasquier, P. (2009). The effect of boreholeinclination on fluid and ground temperature for GLHEsystems. Geothermics 38, 392–398.

Spitler, J.D. (2000). Glhepro – a Design Tool forCommercial Building Ground Loop Heat Exchangers.In Proceedings of the Fourth International Heat Pumpsin Cold Climates Conference, (Aylmer, Québec).

Zeng, H.Y., Diao, N.R., and Fang, Z.H. (2002). A finiteline-source model for boreholes in geothermal heatexchangers. Heat Trans. Asian Res. 31, 558–567.

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Analysis and modeling of a large borehole system in SwedenAlberto Lazzarotto1, José Acuña1, Patricia Monzó1Abstract1. Introduction2. Methodology2.1 The bore field2.2 Simplified geometrical configurations2.3 Boundary conditions2.4 Fluid temperature calculation

3. RESULTS AND DISCUSSION3.1 g-function analysis3.2 Time frame of interest3.2 Bore field system simulations

4. CONCLUsions4. Acknowledgments5. REFERENCES