From ab1f59fd31d04c88774530a3b71996883ccb8c9e Mon Sep 17 00:00:00 2001
From: HBShaoUFZ <haibing.shao@ufz.de>
Date: Wed, 21 Feb 2018 13:49:35 +0100
Subject: [PATCH] add a new benchmark "BHE Array" to the web content.
---
ProcessLib/HeatConduction/Tests.cmake | 13 +
Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.gml | 3 +
Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.prj | 302 ++++++++++++++++
Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.vtu | 3 +
...n_bhe2d_pcs_0_ts_840_t_72576000.000000.vtu | 3 +
.../heatconduction/BHE_array_benchmark.pandoc | 115 ++++++
.../BHE_array_benchmark_figures/figure_1.png | 3 +
.../BHE_array_benchmark_figures/figure_2.png | 3 +
.../BHE_array_benchmark_figures/figure_3.png | 3 +
.../BHE_array_benchmark_figures/figure_4.png | 3 +
.../BHE_array_benchmark_figures/figure_5.png | 3 +
.../bhe_array_analytical_solver.py | 332 ++++++++++++++++++
.../heatconduction/bhe_array_benchmark.bib | 41 +++
13 files changed, 827 insertions(+)
create mode 100644 Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.gml
create mode 100644 Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.prj
create mode 100644 Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.vtu
create mode 100644 Tests/Data/Parabolic/T/2D_BHE_array/standard_solution_bhe2d_pcs_0_ts_840_t_72576000.000000.vtu
create mode 100644 web/content/docs/benchmarks/heatconduction/BHE_array_benchmark.pandoc
create mode 100644 web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_1.png
create mode 100644 web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_2.png
create mode 100644 web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_3.png
create mode 100644 web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_4.png
create mode 100644 web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_5.png
create mode 100644 web/content/docs/benchmarks/heatconduction/bhe_array_analytical_solver.py
create mode 100644 web/content/docs/benchmarks/heatconduction/bhe_array_benchmark.bib
diff --git a/ProcessLib/HeatConduction/Tests.cmake b/ProcessLib/HeatConduction/Tests.cmake
index 36f273dd8f0..f7a13ba7e8c 100644
--- a/ProcessLib/HeatConduction/Tests.cmake
+++ b/ProcessLib/HeatConduction/Tests.cmake
@@ -42,3 +42,16 @@ AddTest(
# EXECUTABLE ogs
# EXECUTABLE_ARGS wedge_1e2_axi_ang_0.02.prj
# )
+
+# The 25 BHE array benchmark
+# test results are compared to 2D simulation result
+AddTest(
+ NAME BHE_Array_2D
+ PATH Parabolic/T/2D_BHE_array
+ EXECUTABLE ogs
+ EXECUTABLE_ARGS bhe2d.prj
+ TESTER vtkdiff
+ DIFF_DATA
+ standard_solution_bhe2d_pcs_0_ts_840_t_72576000.000000.vtu bhe2d_pcs_0_ts_840_t_72576000.000000.vtu temperature temperature 1e-12 0.0
+ REQUIREMENTS NOT OGS_USE_MPI
+)
diff --git a/Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.gml b/Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.gml
new file mode 100644
index 00000000000..ef457454166
--- /dev/null
+++ b/Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.gml
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:0f628c00748a3dd14aff03d777db2900f6652da5a287f903971754bb2a265745
+size 2500
diff --git a/Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.prj b/Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.prj
new file mode 100644
index 00000000000..4e0bf0361b2
--- /dev/null
+++ b/Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.prj
@@ -0,0 +1,302 @@
+<?xml version="1.0" encoding="ISO-8859-1"?>
+<OpenGeoSysProject>
+ <mesh>bhe2d.vtu</mesh>
+ <geometry>bhe2d.gml</geometry>
+ <processes>
+ <process>
+ <name>HeatConduction</name>
+ <type>HEAT_CONDUCTION</type>
+ <integration_order>2</integration_order>
+ <thermal_conductivity>lambda</thermal_conductivity>
+ <heat_capacity>c_p</heat_capacity>
+ <density>rho</density>
+ <process_variables>
+ <process_variable>temperature</process_variable>
+ </process_variables>
+ <secondary_variables>
+ <secondary_variable type="static" internal_name="heat_flux_x" output_name="heat_flux_x"/>
+ </secondary_variables>
+ </process>
+ </processes>
+ <time_loop>
+ <processes>
+ <process ref="HeatConduction">
+ <nonlinear_solver>basic_picard</nonlinear_solver>
+ <convergence_criterion>
+ <type>DeltaX</type>
+ <norm_type>NORM2</norm_type>
+ <abstol>1.e-6</abstol>
+ </convergence_criterion>
+ <time_discretization>
+ <type>BackwardEuler</type>
+ </time_discretization>
+ <output>
+ <variables>
+ <variable> temperature </variable>
+ <variable> heat_flux_x </variable>
+ </variables>
+ </output>
+ <time_stepping>
+ <type>FixedTimeStepping</type>
+ <t_initial> 0.0 </t_initial>
+ <t_end> 93312000 </t_end>
+ <timesteps>
+ <pair>
+ <repeat>1</repeat>
+ <delta_t>86400</delta_t>
+ </pair>
+ </timesteps>
+ </time_stepping>
+ </process>
+ </processes>
+ <output>
+ <type>VTK</type>
+ <prefix>bhe2d</prefix>
+ <timesteps>
+ <pair>
+ <repeat> 8 </repeat>
+ <each_steps> 120 </each_steps>
+ </pair>
+ </timesteps>
+ </output>
+ </time_loop>
+ <parameters>
+ <parameter>
+ <name>lambda</name>
+ <type>Constant</type>
+ <value>2.0</value>
+ </parameter>
+ <parameter>
+ <name>c_p</name>
+ <type>Constant</type>
+ <value>1500</value>
+ </parameter>
+ <parameter>
+ <name>rho</name>
+ <type>Constant</type>
+ <value>1950.0</value>
+ </parameter>
+ <parameter>
+ <name>T0</name>
+ <type>Constant</type>
+ <value>10</value>
+ </parameter>
+ <parameter>
+ <name>p_spatial</name>
+ <type>Constant</type>
+ <value>1</value>
+ </parameter>
+ <parameter>
+ <name>sourceterm_heat_transfer_coefficient</name>
+ <type>CurveScaled</type>
+ <curve>point_Neumann</curve>
+ <parameter>p_spatial</parameter>
+ </parameter>
+ <parameter>
+ <name>ambient_temperature</name>
+ <type>Constant</type>
+ <value>10</value>
+ </parameter>
+ </parameters>
+ <curves>
+ <curve>
+ <name>point_Neumann</name>
+ <coords>0 10368000 10368000.01 31104000 31104000.01 41472000 41472000.01 62208000 62208000.01 72576000 72576000.01 93312000</coords>
+ <values>
+ -35 -35 0 0 -35 -35 0 0 -35 -35 0 0
+ </values>
+ </curve>
+ </curves>
+ <process_variables>
+ <process_variable>
+ <name>temperature</name>
+ <components>1</components>
+ <order>1</order>
+ <initial_condition>T0</initial_condition>
+ <boundary_conditions>
+ <boundary_condition>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>bottom_left</geometry>
+ <type>Dirichlet</type>
+ <parameter>ambient_temperature</parameter>
+ </boundary_condition>
+ </boundary_conditions>
+ <source_terms>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po0</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po1</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po2</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po3</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po4</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po5</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po6</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po7</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po8</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po9</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po10</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po11</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po12</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po13</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po14</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po15</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po16</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po17</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po18</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po19</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po20</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po21</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po22</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po23</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ <source_term>
+ <geometrical_set>geometry</geometrical_set>
+ <geometry>po24</geometry>
+ <type>Nodal</type>
+ <parameter>sourceterm_heat_transfer_coefficient</parameter>
+ </source_term>
+ </source_terms>
+ </process_variable>
+ </process_variables>
+ <nonlinear_solvers>
+ <nonlinear_solver>
+ <name>basic_picard</name>
+ <type>Picard</type>
+ <max_iter>10</max_iter>
+ <linear_solver>general_linear_solver</linear_solver>
+ </nonlinear_solver>
+ </nonlinear_solvers>
+ <linear_solvers>
+ <linear_solver>
+ <name>general_linear_solver</name>
+ <lis>-i cg -p jacobi -tol 1e-16 -maxiter 10000</lis>
+ <eigen>
+ <solver_type>CG</solver_type>
+ <precon_type>DIAGONAL</precon_type>
+ <max_iteration_step>10000</max_iteration_step>
+ <error_tolerance>1e-16</error_tolerance>
+ </eigen>
+ <petsc>
+ <prefix>gw</prefix>
+ <parameters>-gw_ksp_type cg -gw_pc_type bjacobi -gw_ksp_rtol 1e-16 -gw_ksp_max_it 10000</parameters>
+ </petsc>
+ </linear_solver>
+ </linear_solvers>
+</OpenGeoSysProject>
\ No newline at end of file
diff --git a/Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.vtu b/Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.vtu
new file mode 100644
index 00000000000..505f6564901
--- /dev/null
+++ b/Tests/Data/Parabolic/T/2D_BHE_array/bhe2d.vtu
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:3a6e0e794cbeb6777e29d7a362b7e4184476eaa9c7d9505d9952f6c20d2ac180
+size 3724838
diff --git a/Tests/Data/Parabolic/T/2D_BHE_array/standard_solution_bhe2d_pcs_0_ts_840_t_72576000.000000.vtu b/Tests/Data/Parabolic/T/2D_BHE_array/standard_solution_bhe2d_pcs_0_ts_840_t_72576000.000000.vtu
new file mode 100644
index 00000000000..50531da935b
--- /dev/null
+++ b/Tests/Data/Parabolic/T/2D_BHE_array/standard_solution_bhe2d_pcs_0_ts_840_t_72576000.000000.vtu
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:77367cbf00ccecb5f6e1970e5ac0eac59523fb13e43a59f0768e032d231f4f64
+size 1372484
diff --git a/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark.pandoc b/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark.pandoc
new file mode 100644
index 00000000000..d5145334b7a
--- /dev/null
+++ b/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark.pandoc
@@ -0,0 +1,115 @@
++++
+author = "Shuang Chen and Haibing Shao"
+date = "2018-02-21T13:44:00+01:00"
+title = "BHE Array 2D"
+weight = 123
+project = "Parabolic/T/2D_BHE_array/bhe2d.prj"
+
+[menu]
+ [menu.benchmarks]
+ parent = "heatconduction"
+
++++
+
+{{< data-link >}}
+
+## Problem description
+
+When shallow geothermal energy is extracted by using Borehole Heat Exchanger (BHE) for heating of buildings, it causes the decrease of subsurface temperature in the vicinity of BHE. In this benchmark, a 2D numerical model has been constructed to model the above temperature variation. The model is validated against the super-imposed analytical solution. Additionally the impact of mesh density to the accuracy of numerical result is also discussed.
+
+## Analytical Solution
+
+For the temperature change in an infinite homogeneous subsurface caused by one single BHE, it can be calculated by the line-source analytical solution (cf. Stauffer et al. (2013)Â , section 3.1.3), with the assumption that thermal conduction is the only process and no groundwater flow is present. In this case, the ground temperature $T$ is subject to the following equation.
+
+\begin{equation}
+T-T_0 = \frac{q_b}{4\pi \lambda}E_1 \frac{r^2}{4\alpha t}
+\end{equation}
+
+where $q_b$ is the heat extraction rate on the BHE and $E_1$ denotes the exponential integral function. $T_0$ refers to the initial ground Temperature, and $r$ is the distance between the observation point and the BHE.
+
+In case multiple BHEs are present, Bayer et al. (2014) proposed to calculate the temperature change by super-impose the line-source model of equation (1). The overall temperature change at an observation point with a local coordinate (i, j) can be then calculated as
+
+\begin{equation}
+\Delta \mathop T\nolimits_{i,j} \left( {t,\mathop q\nolimits_{k = 1,...,n} } \right) = \sum\limits_{k = 1}^n {\Delta \mathop T\nolimits_{i,j,k} } \left( {t,\mathop q\nolimits_k } \right).
+\end{equation}
+
+where ${\mathop q\nolimits_k }$ is a sequence of heat extraction pulses at t =1, ... ,m time steps. In this benchmark, the time step size of heat extraction pulses is set to 120 days, reflecting a 4-month long heat period and the 8-month long recovery interval every year. Within one time step the heat extraction rate on each BHE remains constant. By combining equation (1) and equation (2), the super-imposed temperature change due to the thermal load ${\mathop q\nolimits_k }$ imposed on each BHE can be calculated as
+
+\begin{equation}\begin{split}
+ \Delta \mathop T\nolimits_{i,j} \left( {t,\mathop q\nolimits_{k,l = 1,...,m} } \right)= \sum\limits_{k = 1}^m {\frac{{\mathop q\nolimits_l - \mathop q\nolimits_{l - 1} }}{{4\pi L\lambda }}} E_1\left[ {\frac{{{{\left( {i - \mathop x\nolimits_k } \right)}^2} + {{\left( {j - \mathop y\nolimits_k } \right)}^2}}}{{4\alpha \left( {\mathop t\nolimits_m - \mathop t\nolimits_l } \right)}}} \right] \\
+ = \sum\limits_{l = 1}^m {\sum\limits_{k = l}^n {\frac{{\mathop q\nolimits_{k,l} }}{{4\pi L\lambda }}} } \left( {E_1\left[ {\frac{{{{\left( {i - \mathop x\nolimits_k } \right)}^2} + {{\left( {j - \mathop y\nolimits_k } \right)}^2}}}{{4\alpha \left( {\mathop t\nolimits_m - \mathop t\nolimits_{l - 1} } \right)}}} \right] - E_1\left[ {\frac{{{{\left( {i - \mathop x\nolimits_k } \right)}^2} + {{\left( {j - \mathop y\nolimits_k } \right)}^2}}}{{4\alpha \left( {\mathop t\nolimits_m - \mathop t\nolimits_l } \right)}}} \right]} \right).
+ \end{split}\end{equation}
+
+where ${\mathop q\nolimits_{k,l} }$ is the heat extraction of the k-*th* BHE at time step *l*. The equation (3) will be used to calculate the analytical solution of the overall temperature change in this model for validating the numerical results. It is written in python code and can be found [here](../bhe_array_analytical_solver.py).
+
+## Numerical model setup
+
+A 2D numerical model was constructed and simulated with the Finite element code OpenGeoSys (OGS). The subsurface was represented by a $100 \times 100~m$ square-shaped domain, inside of which 25 BHEs were installed (cf. Figure 1). The distance between adjacent BHEs is kept at 5 m. The ground temperature variation caused by the operation of the BHE array was simulated over a three years’ period. In the model, a 4-month heating period is assumed every year from January to April, with a constant heat extraction rate of 35 W/m on each BHE. In the rest months, the BHE system is shut down and no heat extraction was imposed. The parameters applied in the numerical model can be found in the table below.
+
+In this model, the quad element was adopted to compose the mesh. The initial temperature of the model domain is set to 10 $^{\circ}$C. For the need of modelling a fixed 10 $^{\circ}$C temperature boundary condition was imposed at coordinate (0 m, 0 m). In the model domain, the locations of the BHEs are identified as red dots as in Figure 1. On each of these points, a sink term was specified in the numerical model, with the specific heat extraction rate as listed in the following table.
+
+| Parameter | Symbol | Value | Unit |
+| -------------------------------- |:------------ | -------------------:| ----------------:|
+| Soil thermal conductivity | $\lambda$ | 1.720 | $Wm^{-1}K^{-1}$ |
+| Soil heat capacity | $\rho c$ | $2.925\times10^{6}$ | $J^{-3} mK^{-1}$ |
+| Ground thermal diffusivity | $\alpha$ | $5.7\times10^{-7}$ | $Wm^{-1}K^{-1}$ |
+| Initial subsurface temperature | $T_0$ | 10 | $^{\circ}C$ |
+| Heat extraction rate of the BHE | $q$ | 35 | $W/m$ |
+| Diameter of the BHE | $D$ | 0.15 | $m$ |
+
+{{< img src="../BHE_array_benchmark_figures/figure_1.png" >}}
+
+
+Figure 1: Model geometry, BHE location, and the observation profile
+
+Different meshes were adopted to analyse the impact of mesh density on the numerical results. According to Diersch et al. (2011) the different element size can affect the accuracy of the numerical result significantly for such type of BHE simulation. The optimal element size $\triangle$ in a 2D model around the BHE node should have the following relationship with respect to the BHE diameter:
+
+\begin{equation}
+\begin{split}
+ \Delta = {\rm{ }}a{r_b}\ \hspace{6mm}
+ a = \left\{ \begin{array}{l}
+ 4.81 \hspace{2mm} for\hspace{2mm} n=4\\
+ 6.13 \hspace{2mm} for\hspace{2mm} n=6\\
+ 6.16 \hspace{2mm} for\hspace{2mm} n=8
+ \end{array}\right.
+ \label{eq_4}
+\end{split}
+\end{equation}
+
+where $r_b$ is the BHE radius. n denotes the number of surrounding nodes. n =Â 8 is typical for a squared grid meshes. In this study, the BHE diameter is assumed to be 0.15Â m. Based on equation (4) the optimal element size should be set to approximately 0.5Â m.
+
+## Numerical modelling results
+
+Figure 2 and 3 show the comparison of the temperature distribution along the observation profile (position see Figure 1) using analytical solution with the numerical results from OGS5 and OGS6 for every 4 months in the whole simulated time. It shows the numerical solution has a very good agreement with the analytical solution.
+
+
+{{< img src="../BHE_array_benchmark_figures/figure_2.png" width="200">}}
+
+Figure 2: The temperature evolution of the BHEs field along the observation profile
+
+{{< img src="../BHE_array_benchmark_figures/figure_3.png" width="200">}}
+
+Figure 3: The temperature evolution of the BHEs field along the observation profile
+
+In order to investigate the impact of mesh density on the accuracy of numerical result, the simulated temperature profile at the observation point A (53 m, 52.5 m) was plotted and compared against the analytical solution. Figure 3 shows the relative difference of the computed temperature between the analytical and numerical solution by using different mesh size (2.5 m, 1 m, 0.5 m, 0.25 m and 0.2 m). The results show that the difference becomes smaller when the mesh size is approaching 0.5 m, which is expected as the optimal mesh size mentioned in Diersch et al. (2011). From Figure 4, it can be found that the absolute error of temperature values at point A should be less than 2.5e-3 if the mesh size is kept denser than 0.5m.
+
+{{< img src="../BHE_array_benchmark_figures/figure_4.png" width="200">}}
+
+Figure 4: The relative difference of computed temperature at point A between the analytical and numerical solution using different mesh size
+
+{{< img src="../BHE_array_benchmark_figures/figure_5.png" width="200">}}
+
+Figure 5: The absolute difference of computed temperature along the diagonal profile between the analytical and numerical solution using different mesh size
+
+## Summary
+
+In this benchmark, a 2D numerical model has been constructed to simulate the ground temperature variation caused by heat extraction by BHE. The results show a very good agreement against the analytical solution. Additionally the impact of mesh density on the accuracy of numerical result was also investigated, and the optimal mesh size was found to be 0.5 m. In future studies, the pipe-line network system will be implemented into the model and coupled with the BHE.
+
+## References
+
+[1] Bayer, P., de Paly, M., Beck, M., 2014. Strategic optimization of borehole heat exchanger field for seasonal geothermal heating and cooling. Applied Energy 136, 445-453.
+URL http://dx.doi.org/10.1016/j.apenergy.2014.09.029
+
+[2] Diersch, H.-J., Bauer, D., Heidemann, W., Ruehaak, W., Sch¨atzl, P., 2011. Finite element modeling of borehole heat exchanger systems: Part 2. Numerical simulation. Computers & Geosciences 37, 1136-1147.
+
+[3] Stauffer, F., Bayer, P., Blum, P., Giraldo, N., Kinzelbach, W., 2013. Thermal Use of Shallow Groundwater. CRC Press, 290.
diff --git a/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_1.png b/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_1.png
new file mode 100644
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+++ b/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_1.png
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+version https://git-lfs.github.com/spec/v1
+oid sha256:e2195a70425617d6cb553fd4ddd5c6983889ad1962561897e31dea369de4d419
+size 124921
diff --git a/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_2.png b/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_2.png
new file mode 100644
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+version https://git-lfs.github.com/spec/v1
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diff --git a/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_3.png b/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_3.png
new file mode 100644
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+version https://git-lfs.github.com/spec/v1
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+size 259956
diff --git a/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_4.png b/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_4.png
new file mode 100644
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+version https://git-lfs.github.com/spec/v1
+oid sha256:b3970b1aed528183571182b22a8aac7dee75b8b8e3baf8c4081d7071070357b2
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diff --git a/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_5.png b/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_5.png
new file mode 100644
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+++ b/web/content/docs/benchmarks/heatconduction/BHE_array_benchmark_figures/figure_5.png
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e65a5d57d0a0525354368e140e055cf9a8361d69df54f6c2776b3d1fc40f8961
+size 121517
diff --git a/web/content/docs/benchmarks/heatconduction/bhe_array_analytical_solver.py b/web/content/docs/benchmarks/heatconduction/bhe_array_analytical_solver.py
new file mode 100644
index 00000000000..a51015bd1ae
--- /dev/null
+++ b/web/content/docs/benchmarks/heatconduction/bhe_array_analytical_solver.py
@@ -0,0 +1,332 @@
+# -*- coding: utf-8 -*- {}
+import matplotlib.pyplot as plt
+import numpy as np
+import scipy
+import os
+import math
+from mpmath import *
+mp.dps = 25; mp.pretty = True
+
+#part 1:gif_splitfile to txt
+
+#新建文件夹txt，用于å˜å‚¨åŽŸæ–‡ä»¶çš„分解文件集。
+os.mkdir("C://george//PhD//papers//BHE_sc//result//txt")
+#os.chdir(path) 方法用于改å˜å½“å‰å·¥ä½œç›®å½•åˆ°æŒ‡å®šçš„路径。æ¤å¤„创建路径到新建文件夹txt
+os.chdir("C://george//PhD//papers//BHE_sc//result//txt")
+txtfiles = os.getcwd() #获得当å‰è¿è¡Œè„šæœ¬æ‰€åœ¨ç›®å½•ä½œä¸ºæ–‡ä»¶å‚¨å˜åœ°å€
+
+# define the path of the data file
+str_tec_file_path = "C://george//PhD//papers//BHE_sc//result//tec_0_5m.tec"
+
+data_num =0
+with open(str_tec_file_path, 'r') as f_in:
+ i = -1 #æ¤å¤„写-1æ˜¯å› ä¸ºè¦ä½¿è¾“出文件å第一个必须是以0结尾，ä¸ç„¶å’Œpart2的文件读入循环对ä¸ä¸Šã€‚å¿…é¡»è¦å¯¹ä¸Šæ˜¯å› 为\n
+ #part2数组值从0ä½å¼€å§‹å‚¨å˜çš„。
+ for line in f_in:
+ if '=' not in line:
+ txtfiles.write(line)
+ #读出å•ä½æ–‡ä»¶ä¸æœ‰å¤šå°‘行数æ®ï¼Œå˜å…¥å˜é‡data_num
+ data_num = data_num + 1
+ elif 'ZONE' in line: #从关键è¯ZONE开始读å–è¡Œ
+ data_num = 0
+ i = i + 1
+ txtfiles = open('txtfile{}.txt'.format(i), 'w')#æ ¼å¼åŒ–命å文件åå˜å‚¨
+
+#记录总共的分出文件数目，用于part 2
+numfile = i + 1
+#å…³é—并完整文件释放内å˜
+txtfiles.close()
+
+
+# plotting T_in T_out curves,建立作图文件夹åŠåŠæ”¹å˜å·¥ä½œè·¯å¾„
+os.mkdir("C://george//PhD//papers//BHE_sc//result//png")
+os.chdir("C://george//PhD//papers//BHE_sc//result//png")
+pngfiles = os.getcwd()
+
+#建立source termåæ ‡çŸ©é˜µï¼Œç”¨ä»¥è®¡ç®—å„çƒæºå¯¹referece point的温度影å“矩阵
+po_x = np.array([40,40,40,40,40],dtype=float).reshape(-1,1) + np.arange(0,25,5)
+po_y = np.arange(60,35,-5).reshape(-1,1) + np.zeros(5)
+
+po_dist_to_referencepo = np.zeros([5,5])
+Temp_po_to_referencepo = np.zeros([5,5])
+#解æžè§£1基础数æ®é¡¹,q是timestepå˜åŒ–é‡
+q1 = -35
+q2 = 35
+q3 = 0
+
+time_trans = 120*24*60*60 #3个月一输出数æ®
+T0 = 10
+T = T0
+
+
+poro = 10e-20
+lamda_f = 0.5984
+density_f = 998.2032
+cp_f = 4182.0
+lamda_sp = 2.0
+density_sp = 1950.0
+cp_sp = 1500.0
+
+alpha_f = lamda_f/(density_f*cp_f)
+alpha_sp = lamda_sp/(density_sp*cp_sp)
+
+lamda_med = (1 - poro)*lamda_sp + poro*lamda_f
+alpha_med = (1 - poro)*alpha_sp + poro*alpha_f
+
+#解æžè§£2基础数æ®é¡¹ï¼š
+qq = np.array([-35,0,0,-35,0,0,-35,0,0,-35,0]).reshape(1,-1)
+qq_all = np.repeat(qq,data_num,axis=0)
+expandedloads= np.array(qq)
+numtimesteps = 11
+
+numbhe = 25
+ppo_x = np.array([40,40,40,40,40],dtype=float).reshape(-1,1) + np.arange(0,25,5)
+ppo_y = np.arange(60,35,-5).reshape(-1,1) + np.zeros(5)
+ppo_x_re = np.reshape(po_x, (-1,1))
+ppo_y_re = np.reshape(po_y, (-1,1))
+
+# 建立动æ€å˜é‡å
+createVar = locals()
+
+# 建立月份åå—è¡¨ï¼Œå› ä¸ºæ•°æ®è®°å½•ä»Ž1月1日有一组数æ®ï¼Œæ‰€ä»¥å¤šå‡ºä¸€ä¸ªæœˆæ•°æ®ã€‚
+month = [" Temperature after 0 months ", " Temperature after 4 months", " Temperature after 8 months", " Temperature after 12 months", " Temperature after 16 months", " Temperature after 20 months",
+ " Temperature after 24 months", " Temperature after 28 months", " Temperature after 32 months", " Temperature after 36 months"," Temperature after 40 months"," Temperature after 44 months"]
+
+#解æžè§£2项：
+txtpath2 = f"C://george//PhD//papers//BHE_sc//result//txt//txtfile0.txt"#fæ ¼å¼åŒ–å—符串
+dist_arrayy = 0.0
+with open(txtpath2, 'r') as f_in_txt:
+ for line in f_in_txt:
+ # split line
+ data_line = line.split(" ") #maxsplit
+ # time_double = 0.0
+ dist_double2 = float(data_line[0])
+ cordinate2 = math.sqrt(dist_double2**2/2)
+
+ # put into the list
+ dist_arrayy= np.vstack((dist_arrayy, cordinate2))
+
+ dist_arrayy_new = np.delete(dist_arrayy, 0, 0)
+
+ numtemppoints = len(dist_arrayy_new)
+ T2=np.zeros([numtemppoints,numtimesteps])
+
+ coeff_all = np.zeros([numtemppoints,numtimesteps])
+
+ for currstep in range(0,numtimesteps):
+ Temp_po_to_referencepo= np.zeros([numtemppoints,numbhe])
+ po_dist_to_referencepo= np.zeros([numtemppoints,numbhe])
+ localcoeff_all= np.zeros([numtemppoints,1])
+ localcoeff= np.zeros([numtemppoints,numbhe])
+ localcoeff1= np.zeros([numtemppoints,numbhe])
+ for i in range(0,numbhe):
+ if(time_trans*(currstep+1)-time_trans*0>0):
+ for j in range(0,numtemppoints):
+ po_dist_to_referencepo[j,i] = abs(ppo_x_re[i] - (dist_arrayy_new[j]+1))**2 + abs(ppo_y_re[i] - dist_arrayy_new[j])**2
+ exp = po_dist_to_referencepo[j,i]/(4*alpha_med*time_trans*(currstep+1))
+ n = e1(exp)
+ localcoeff[j,i] = 1/(4*math.pi*lamda_med)*n
+ if(time_trans*(currstep+1)-time_trans*1>0):
+ for j in range(0,numtemppoints):
+ po_dist_to_referencepo[j,i] = abs(ppo_x_re[i] - (dist_arrayy_new[j]+1))**2 + abs(ppo_y_re[i] - dist_arrayy_new[j])**2
+ exp1 = po_dist_to_referencepo[j,i]/(4*alpha_med*time_trans*currstep)
+ n1 = e1(exp1)
+ localcoeff[j,i] = localcoeff[j,i] - 1/(4*math.pi*lamda_med)*n1
+
+ localcoeff_all= np.sum(localcoeff,axis=1).reshape(-1,1)
+ coeff_all[:,1:]=coeff_all[:,:numtimesteps-1]
+ coeff_all[:,:1]=localcoeff_all
+
+ for currstep in range(0,numtimesteps):
+ T2[:,currstep] = np.sum(coeff_all[:,numtimesteps-1-currstep:]*qq_all[:,:currstep+1],axis=1) +10
+
+
+
+# loop over
+for m in range(0 , numfile):
+ #文件读å–路径
+ txtpath = f"C://george//PhD//papers//BHE_sc//result//txt//txtfile{m}.txt"#fæ ¼å¼åŒ–å—符串
+ txtpath_ogs6 = f"C://george//PhD//papers//BHE_sc//result//txt_ogs6//txtfile{m}.txt"
+
+ #动æ€å˜é‡åæ¯ç»„赋åˆå€¼0.0
+ createVar['dist_array'+ str(m)] = 0.0
+ createVar['temperature_array_ogs5'+ str(m)] = 0.0
+ createVar['temperature_array_ana'+ str(m)] = 0.0
+ createVar['dist_array2'+ str(m)] = 0.0
+ createVar['temperature_array_ana2'+ str(m)] = 0.0
+
+ createVar['dist_array_ogs6'+ str(m)] = 0.0
+ createVar['temperature_array_ogs6'+ str(m)] = 0.0
+
+ #解æžè§£1：
+# if m == 0 or m ==2 or m ==3 or m ==5 or m ==6 or m ==8 or m ==9 or m ==10:
+# q = q2
+# elif m == 1 or m ==4 or m ==7:
+# q = q1
+ if m == 0:
+ with open(txtpath, 'r') as f_in_txt:
+ for line in f_in_txt:
+ # split line
+ data_line = line.split(" ") #maxsplit
+ # time_double = 0.0
+ dist_double = float(data_line[0])
+ cordinate = math.sqrt(dist_double**2/2)
+ temperature = float(data_line[1])
+
+ #解æžè§£æ•°æ®é¡¹
+ time = time_trans + 10e-6
+ for i in range(0,5):
+ for j in range(0,5):
+ po_dist_to_referencepo[i,j] = abs(po_x[i,j]-(cordinate+1))**2 + abs( po_y[i,j]-cordinate)**2
+ for i in range(0,5):
+ for j in range(0,5):
+ exp = po_dist_to_referencepo[i,j]/(4*alpha_med*time)
+ n1 = e1(exp)
+ Temp_po_to_referencepo[i,j] = q3/(4*math.pi*lamda_med)*n1
+
+ T = np.sum(Temp_po_to_referencepo) + T0
+
+ # put into the list
+ createVar['dist_array'+ str(m)] = np.vstack((createVar['dist_array'+ str(m)], cordinate))
+ createVar['temperature_array_ogs5'+ str(m)] = np.vstack((createVar['temperature_array_ogs5'+ str(m)], temperature))
+ createVar['temperature_array_ana'+ str(m)] = np.vstack((createVar['temperature_array_ana'+ str(m)], T))
+
+ f_in_txt.close()
+ # end of loop
+
+ #去除21å’Œ22行的0.0å€¼ï¼Œå› ä¸ºæ¤å€¼è¢«å¸¦å…¥äº†dist_arrayçš„list第一行，å¯æŸ¥çœ‹variable explorer temperature_array观察。
+ #目的是去除åšå›¾æ—¶çš„第一个数æ®0值。
+ dist_array_new = np.delete(createVar['dist_array'+ str(m)], 0, 0)
+ temperature_array_ogs5_new = np.delete(createVar['temperature_array_ogs5'+ str(m)], 0, 0)
+ temperature_array_ana_new = np.delete(createVar['temperature_array_ana'+ str(m)], 0, 0)
+
+ with open(txtpath_ogs6, 'r') as f_in_txt:
+ for line in f_in_txt:
+ # split line
+ data_line = line.split(" ") #maxsplit
+ # time_double = 0.0
+ dist_double = float(data_line[0])
+ cordinate = math.sqrt(dist_double**2/2)
+ temperature = float(data_line[1])
+ createVar['dist_array_ogs6'+ str(m)] = np.vstack((createVar['dist_array_ogs6'+ str(m)], cordinate))
+ createVar['temperature_array_ogs6'+ str(m)] = np.vstack((createVar['temperature_array_ogs6'+ str(m)], temperature))
+ f_in_txt.close()
+ dist_array_new_ogs6 = np.delete(createVar['dist_array_ogs6'+ str(m)], 0, 0)
+ temperature_array_ogs6_new = np.delete(createVar['temperature_array_ogs6'+ str(m)], 0, 0)
+
+ #plotting
+ plt.figure()
+ plt.plot(scipy.dot(dist_array_new,1.0),scipy.dot(temperature_array_ogs5_new,1.0),color = 'r',ls=':',lw=1, marker='^',markersize=1.5, label= 'OGS5')
+ plt.plot(scipy.dot(dist_array_new_ogs6,1.0),scipy.dot(temperature_array_ogs6_new,1.0),c='g', ls=':',lw=1, marker='o',markersize=1, label= 'OGS6')
+# plt.plot(scipy.dot(dist_array_new,1.0),scipy.dot(temperature_array_ana_new,1.0), color = 'g',label= 'ana1' + months[m])
+ plt.plot(scipy.dot(dist_array_new,1.0),scipy.dot(temperature_array_ana_new,1.0), "b",label= 'Analytical')
+ plt.xlim([0,100])
+ plt.ylim([-10,20])
+ plt.ylabel('Temperature [$^\circ$C]')
+ plt.xlabel('x [m]')
+ plt.legend(loc='best',fontsize=8)
+ plt.title(month[m],fontsize=12)
+ # plt.show()
+ plt.savefig('pngfile{}.png'.format(m),dpi = 300, transparent = False)
+
+
+ else:
+ #解æžè§£æ•°æ®é¡¹,实际是温度场斜三角矩阵相åŠ
+ T = np.zeros([12])
+ with open(txtpath, 'r') as f_in_txt:
+ for line in f_in_txt:
+ # split line
+ data_line = line.split(" ") #maxsplit
+ # time_double = 0.0
+ dist_double = float(data_line[0])
+ cordinate = math.sqrt(dist_double**2/2)
+
+ i1 = m +1
+ for m1 in range(1, m + 1 ):
+ i1 = i1 - 1
+ #时间å˜é‡ï¼Œtimestep温度éšæ—¶é—´å åŠ çŸ©é˜µæ¯è¡Œé‡æ–°è®¾ä¸º0
+ time = time_trans * i1 + 10e-6
+ if m1 == 2 or m1 == 5 or m1 == 8 or m1 == 11:
+ q = q2
+ elif m1 == 1 or m1 ==4 or m1 ==7 or m1 ==10:
+ q = q1
+ elif m1 == 3 or m1 == 6 or m1 == 9:
+ q = q3
+
+ for i in range(0,5):
+ for j in range(0,5):
+ po_dist_to_referencepo[i,j] = abs(po_x[i,j]-(cordinate+1))**2 + abs( po_y[i,j]-cordinate)**2
+ for i in range(0,5):
+ for j in range(0,5):
+ exp = po_dist_to_referencepo[i,j]/(4*alpha_med*time)
+ n1 = e1(exp)
+ Temp_po_to_referencepo[i,j] = q/(4*math.pi*lamda_med)*n1
+
+ T[m1] = np.sum(Temp_po_to_referencepo)
+
+ T_sum = np.sum(T) + T0
+
+ createVar['temperature_array_ana'+ str(m)] = np.vstack((createVar['temperature_array_ana'+ str(m)], T_sum))
+
+ f_in_txt.close()
+
+
+ #ogs5项：
+ with open(txtpath, 'r') as f_in_txt:
+ for line in f_in_txt:
+ # split line
+ data_line = line.split(" ") #maxsplit
+ # time_double = 0.0
+ dist_double = float(data_line[0])
+ cordinate = math.sqrt(dist_double**2/2)
+ temperature = float(data_line[1])
+
+ # put into the list
+ createVar['dist_array'+ str(m)] = np.vstack((createVar['dist_array'+ str(m)], cordinate))
+ createVar['temperature_array_ogs5'+ str(m)] = np.vstack((createVar['temperature_array_ogs5'+ str(m)], temperature))
+
+ f_in_txt.close()
+ #去除21å’Œ22行的0.0å€¼ï¼Œå› ä¸ºæ¤å€¼è¢«å¸¦å…¥äº†dist_arrayçš„list第一行，å¯æŸ¥çœ‹variable explorer temperature_array观察。
+ #目的是去除åšå›¾æ—¶çš„第一个数æ®0值。
+ dist_array_new = np.delete(createVar['dist_array'+ str(m)], 0, 0)
+ temperature_array_ogs5_new = np.delete(createVar['temperature_array_ogs5'+ str(m)], 0, 0)
+ temperature_array_ana_new = np.delete(createVar['temperature_array_ana'+ str(m)], 0, 0)
+
+
+ #ogs6项
+ with open(txtpath_ogs6, 'r') as f_in_txt:
+ for line in f_in_txt:
+ # split line
+ data_line = line.split(" ") #maxsplit
+ # time_double = 0.0
+ dist_double = float(data_line[0])
+ cordinate = math.sqrt(dist_double**2/2)
+ temperature = float(data_line[1])
+
+ # put into the list
+ createVar['dist_array_ogs6'+ str(m)] = np.vstack((createVar['dist_array_ogs6'+ str(m)], cordinate))
+ createVar['temperature_array_ogs6'+ str(m)] = np.vstack((createVar['temperature_array_ogs6'+ str(m)], temperature))
+
+ f_in_txt.close()
+ dist_array_new_ogs6 = np.delete(createVar['dist_array_ogs6'+ str(m)], 0, 0)
+ temperature_array_ogs6_new = np.delete(createVar['temperature_array_ogs6'+ str(m)], 0, 0)
+
+ #plotting
+ plt.figure()
+ plt.plot(scipy.dot(dist_array_new,1.0),scipy.dot(temperature_array_ogs5_new,1.0),color = 'r',ls=':',lw=1, marker='^',markersize=1.5, label= 'OGS5')
+ plt.plot(scipy.dot(dist_array_new_ogs6,1.0),scipy.dot(temperature_array_ogs6_new,1.0),c='g', ls=':',lw=1, marker='o',markersize=1, label= 'OGS6')
+
+# plt.plot(scipy.dot(dist_array_new,1.0),scipy.dot(temperature_array_ana_new,1.0), color = 'g',label= 'ana' + month[m])
+ plt.plot(scipy.dot(dist_array_new,1.0),T2[:,m-1],"b",label= 'Analytical')
+ plt.xlim([0,100])
+ plt.ylim([-10,20])
+ plt.ylabel('Temperature [$^\circ$C]')
+ plt.xlabel('x [m]')
+ plt.legend(loc='best',fontsize=8)
+ plt.title(month[m],fontsize=12)
+ # plt.show()
+ plt.savefig('pngfile{}.png'.format(m), dpi = 300, transparent = False)
+
+ # end of loop
+txtpath.close()
+pngfiles.close()
\ No newline at end of file
diff --git a/web/content/docs/benchmarks/heatconduction/bhe_array_benchmark.bib b/web/content/docs/benchmarks/heatconduction/bhe_array_benchmark.bib
new file mode 100644
index 00000000000..8b78ab42718
--- /dev/null
+++ b/web/content/docs/benchmarks/heatconduction/bhe_array_benchmark.bib
@@ -0,0 +1,41 @@
+@article{Bayer,
+author = {Bayer, Peter and de Paly, Michael and Beck, Markus},
+doi = {10.1016/j.apenergy.2014.09.029},
+file = {:C$\backslash$:/george/PhD/Lieteratur/Bayer{\_}AE2014.pdf:pdf},
+issn = {03062619},
+journal = {Applied Energy},
+keywords = {Closed systems,Finite line source,Low-enthalpy geothermics,Seasonal use,Simulation},
+mendeley-groups = {PhD/artikel/BHE{\_}benchmark},
+pages = {445--453},
+publisher = {Elsevier Ltd},
+title = {{Strategic optimization of borehole heat exchanger field for seasonal geothermal heating and cooling}},
+url = {http://dx.doi.org/10.1016/j.apenergy.2014.09.029},
+volume = {136},
+year = {2014}
+}
+
+
+@article{Diersch,
+author = {Diersch, H.-J.G. and Bauer, D and Heidemann, W and R{\"{u}}haak, W and Sch{\"{a}}tzl, P},
+doi = {10.1016/j.cageo.2010.08.002},
+file = {:D$\backslash$:/Dropbox/Literature/geothermal{\_}heat{\_}pump/Diersch{\_}FEFLOW{\_}2{\_}2011.pdf:pdf},
+issn = {0098-3004},
+journal = {Computers {\&} Geosciences},
+keywords = {Borehole heat exchanger},
+mendeley-groups = {Automatically Imported},
+pages = {1136--1147},
+title = {{Finite element modeling of borehole heat exchanger systems: Part 2. Numerical simulation}},
+volume = {37},
+year = {2011}
+}
+
+@article{Stauffer,
+author = {Stauffer, Fritz and Bayer, Peter and Blum, Philipp and Giraldo, Nelson and Kinzelbach, Wolfgang},
+doi = {10.1201/b16239},
+file = {:C$\backslash$:/george/PhD/Lieteratur/K15881{\_}CoverDraft{\_}01.pdf.pdf:pdf},
+isbn = {978-1-4665-6019-2},
+journal = {CRC Press},
+pages = {290},
+title = {{Thermal Use of Shallow Groundwater}},
+year = {2013}
+}
--
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