From 4baff996f441dd359fbfa62d9173f013fc256d64 Mon Sep 17 00:00:00 2001
From: Lars Bilke <lars.bilke@ufz.de>
Date: Tue, 17 Oct 2023 14:18:26 +0200
Subject: [PATCH 01/11] [web] Deleted old static page.

---
 .../T/BHE_1P/figures}/pipe_flow_3d_model.png    | Bin
 .../Analytical_wellbore_heat_transport.zip      | Bin 1358 -> 0 bytes
 .../pipe_flow_EBHE/T_out_comparison.png         | Bin 33699 -> 0 bytes
 .../absolute_error_fluid_T_30d.png              | Bin 56395 -> 0 bytes
 4 files changed, 0 insertions(+), 0 deletions(-)
 rename {web/content/docs/benchmarks/heat-transport-bhe/pipe_flow_EBHE => Tests/Data/Parabolic/T/BHE_1P/figures}/pipe_flow_3d_model.png (100%)
 delete mode 100644 web/content/docs/benchmarks/heat-transport-bhe/pipe_flow_EBHE/Analytical_wellbore_heat_transport.zip
 delete mode 100644 web/content/docs/benchmarks/heat-transport-bhe/pipe_flow_EBHE/T_out_comparison.png
 delete mode 100644 web/content/docs/benchmarks/heat-transport-bhe/pipe_flow_EBHE/absolute_error_fluid_T_30d.png

diff --git a/web/content/docs/benchmarks/heat-transport-bhe/pipe_flow_EBHE/pipe_flow_3d_model.png b/Tests/Data/Parabolic/T/BHE_1P/figures/pipe_flow_3d_model.png
similarity index 100%
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diff --git a/web/content/docs/benchmarks/heat-transport-bhe/pipe_flow_EBHE/Analytical_wellbore_heat_transport.zip b/web/content/docs/benchmarks/heat-transport-bhe/pipe_flow_EBHE/Analytical_wellbore_heat_transport.zip
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From 35fb223a947b75dbb96ffe091fa497e73a7c2006 Mon Sep 17 00:00:00 2001
From: Frieder Loer <frieder.loer@ufz.de>
Date: Tue, 17 Oct 2023 15:13:15 +0200
Subject: [PATCH 02/11] [web,T] Added pipe_flow_ebhe notebook.

---
 ProcessLib/HeatTransportBHE/Tests.cmake       |   4 +
 Tests/Data/Parabolic/T/BHE_1P/BHE_1P.prj      |  18 +-
 .../Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md | 423 ++++++++++++++++++
 3 files changed, 444 insertions(+), 1 deletion(-)
 create mode 100644 Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md

diff --git a/ProcessLib/HeatTransportBHE/Tests.cmake b/ProcessLib/HeatTransportBHE/Tests.cmake
index 4434c63ec81..a2c60ddefd0 100644
--- a/ProcessLib/HeatTransportBHE/Tests.cmake
+++ b/ProcessLib/HeatTransportBHE/Tests.cmake
@@ -160,3 +160,7 @@ AddTest(
     BHE_1P_newton_ts_10_t_600.000000.vtu BHE_1P_newton_ts_10_t_600.000000.vtu temperature_BHE1 temperature_BHE1 1e-12 1e-14
     BHE_1P_newton_ts_10_t_600.000000.vtu BHE_1P_newton_ts_10_t_600.000000.vtu temperature_soil temperature_soil 1e-12 1e-13
 )
+
+if(NOT OGS_USE_PETSC)
+    NotebookTest(NOTEBOOKFILE Parabolic/T/BHE_1P/pipe_flow_ebhe.md RUNTIME 600)
+endif()
diff --git a/Tests/Data/Parabolic/T/BHE_1P/BHE_1P.prj b/Tests/Data/Parabolic/T/BHE_1P/BHE_1P.prj
index 5f0776157f8..11bc83042ff 100644
--- a/Tests/Data/Parabolic/T/BHE_1P/BHE_1P.prj
+++ b/Tests/Data/Parabolic/T/BHE_1P/BHE_1P.prj
@@ -144,9 +144,25 @@
                     <t_end> 600 </t_end>
                     <timesteps>
                         <pair>
-                            <repeat>10</repeat>
+                            <repeat>60</repeat>
                             <delta_t>60</delta_t>
                         </pair>
+						<pair>
+                            <repeat>60</repeat>
+                            <delta_t>360</delta_t>
+                        </pair>
+                        <pair>
+						    <repeat>41</repeat>
+						    <delta_t>3600</delta_t>
+						</pair>
+						<pair>
+						    <repeat>16</repeat>
+						    <delta_t>43200</delta_t>
+						</pair>
+                        <pair>
+						    <repeat>28</repeat>
+						    <delta_t>86400</delta_t>
+						</pair>
                     </timesteps>
                 </time_stepping>
             </process>
diff --git a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
new file mode 100644
index 00000000000..4a591274623
--- /dev/null
+++ b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
@@ -0,0 +1,423 @@
++++
+title = "Wellbore Heat Transport - EUBHE"
+date = "2023-08-03"
+author = "Chaofan Chen, Haibing Shao, Frieder Loer"
+image = "figures/pipe_flow_3d_model.png"
+web_subsection = "heat-transport-bhe"
+project = ["Parabolic/T/BHE_1P/BHE_1P.prj"]
++++
+
+
+## Problem description
+
+Ramey et al. (1962) proposed the analytical solution concerning the wellbore heat transmission, which can be used to quantify the fluid temperature change in the wellbore.  
+In order to verify the single pipe flow model in the OGS, the numerical results was compared with Ramey's analytical solution (Ramey et al. 1962).  
+The detailed calculation of the Ramey's analytical solution is given below.  
+A detailed analysis of an enhanced U.tube borehole heat exchanger (EUBHE) can be found in (Chen, C. et al. 2021).
+
+## Model Setup
+
+In this benchmark, the length of the wellbore is 30 m as shown in Figure 1 and the cold water is injected into the inlet point of the wellbore with temperature of 20°C.  
+The initial temperature of the fluid and grout in the wellbore is 20 $^{\circ}$C, and temperature of the surrounding rock is 55 $^{\circ}$C.  
+The wellbore and pipe diameter are 0.28 m and 0.25826 m, respectively.  
+And the flow rate is 0.0002 $m^3/s$.
+
+![Single pipe flow model](./figures/pipe_flow_3d_model.png)
+
+Figure 1: Single pipe flow model
+
+```python
+# Import required libraries
+
+import numpy as np
+import matplotlib.pyplot as plt
+import pandas as pd
+```
+
+## Ramey's analytical solution
+
+The equations are followed by their implementation in python.
+
+```python
+# Define constants
+
+T_d = 55 # Temperature of undistributed formation
+T_i = 20 # Temperature of inlet fluid
+Z = np.insert(np.linspace(0.3,30,100), 0, 1e-10) # Depth below surface, start with very small value to prevent division by zero
+q = 0.0002 # Flow rate of fluid in wellbore, m3/s
+rho_f = 1000 # Density of fluid, unit kg/m3
+c_p_f = 4190 # Specific heat capacity of fluid in wellbore
+mu_f = 1.14e-3  # Unit kg/m/s;
+lambda_f = 0.59  # Unit W/m/K
+rho_re = 1800 # Density of geothermal reservoir
+c_p_re = 1778 # Specific heat capacity of geothermal reservoir
+lambda_re = 2.78018 # Heat conductivity coefficient of geothermal reservoir
+lambda_g = 0.73 # Heat conductivity coefficient of grout and pipe
+lambda_pi = 1.3
+r_pi = 0.12913 # Inner radius of pipe and wellbore
+r_b = 0.14
+t_pi = 0.00587 # Thickness of the pipe
+t = 86400 * 30 # Operation time
+```
+
+In Ramey's analytical solution (Ramey et al. (1962)), the outlet temperature of the pipe inside the wellbore can be calculated by
+
+$$
+    T_o(t) = T_{s} + (T_i(t) - T_{s})\exp(-\Delta z/X)
+$$
+
+```python
+def temp(T_d, T_i, delta_z, X):
+    return T_d + (T_i - T_d) * np.exp(-delta_z / X)
+```
+
+The coefficient $X$ is determined by
+
+$$
+    X = \frac{q\rho_fc_{p,f}(\lambda_{s}+r_pUf(t))}{2\pi r_pU \lambda_{s}}
+$$
+
+where $q$ is the flow rate of the fluid in the wellbore.
+
+```python
+def coefficient_x(q, rho_f, c_p_f, lambda_re, r_pi, U, f_t):
+    return (q * rho_f * c_p_f) * (lambda_re + r_pi * U * f_t) / (2 * np.pi * r_pi * U * lambda_re)
+```
+
+With dimensionless time
+
+$$
+    t_D = \frac{\lambda_{s}t}{(\rho_{s}c_{p,s}r_b^{2})}
+$$
+
+```python
+def dimless_t(lambda_re, delta_t, rho_re, c_p_re, r_b):
+    return lambda_re * delta_t / (rho_re * c_p_re * r_b * r_b)
+```
+
+the time function $f(t)$ can be calculated by
+
+<!-- markdownlint-disable reference-links-images -->
+
+$$
+    f(t) = [0.4063+0.5\ln(t_D)][1+\frac{0.6}{t_D}], t_D > 1.5
+$$
+
+<!-- markdownlint-enable reference-links-images -->
+
+$$
+    f(t) = 1.1281\sqrt{t_D}(1-0.3\sqrt{t_D}), t_D \leqslant 1.5
+$$
+
+```python
+def time_function(t_D):
+    if t_D > 1.5:
+                f_t = (0.4063 + 0.5 * np.log(t_D)) * (1 + 0.6 / t_D)
+    else:
+        f_t = 1.1281 * np.sqrt(t_D) * (1 - 0.3 * np.sqrt(t_D))
+    
+    return f_t
+
+```
+
+The Prandtl and Reynolds number can be calculated as follows
+
+$$
+    Pr = \frac{\mu_f c_{p,f}}{\lambda_f}
+$$
+
+$$
+    Re = \frac{\rho_f v d_{pi}}{\mu_f}
+$$
+
+where, $\mu_f, \rho_f$ and $\lambda_f$ is the fluid viscosity, density and thermal conductivity.
+
+```python
+# Calculate the convective film resistance inside pipe
+v = q / (np.pi * r_pi * r_pi)
+
+# Reynolds number
+Pr = mu_f * c_p_f / lambda_f
+Re = rho_f * v * (2 * r_pi) / mu_f
+
+```
+
+The Nusselt number can be determined by the following equation (Diersch 2013):
+
+$$
+    Nu = 4.364,\  Re < 2300
+$$
+
+$$
+    Nu = \frac{(\xi_{k}/8)\  \mathrm{Re}_{k}\  \mathrm{Pr}}{1+12.7\sqrt{\xi_{k}/8}(\mathrm{Pr}^{2/3}-1)} \left[ 1+\left(\frac{d_{k}^{i}}{L}\right)^{2/3} \right],\   Re \geq 10^4
+$$
+
+$$
+    Nu = (1-\gamma_{k})\  4.364 + \gamma_{k} \Bigg\{ \frac{(0.0308/8)10^{4}\mathrm{Pr}}{1+12.7\  \sqrt{0.0308/8}(\mathrm{Pr}^{2/3}-1)} \left[ 1+\left(\frac{d_{k}^{i}}{L}\right)^{2/3} \right] \Bigg\}, 2 300 \leq Re < 10^{4}
+$$
+
+with
+
+$$
+    \xi_{k}=(1.8\  \log_{10}\mathrm{Re}_{k}-1.5)^{-2}
+$$
+
+and
+
+$$
+    \gamma_{k} = \frac{\mathrm{Re}_{k}-2300}{10^{4}-2300}
+$$
+
+```python
+# Estimate convective film coefficient
+xi = (1.8 * np.log(Re) - 1.5)**(-2)
+Nu_l = (xi / 8.0 * Re * Pr) / (1.0 + 12.7 * np.sqrt(xi / 8.0) * ((Pr)**(2.0 / 3.0) - 1.0)) * (1.0 + (r_pi / Z)**(2.0 / 3.0))
+Nu_lam = 4.364
+gamma = (Re - 2300) / (10000 - 2300)
+
+Nu_m = (1.0 - gamma) * 4.364 + gamma * ((0.0308 / 8.0 * 1.0e4 * Pr) / (1.0 + 12.7 * np.sqrt(0.0308 / 8.0) * ((Pr)**(2.0 / 3.0) - 1.0)) * (1.0 + (r_pi / Z)**(2.0 / 3.0)))
+
+# Evaluate Nusselt number; based on the value of Re choose appropriate correlation
+if Re < 2300:
+    Nu_p = Nu_lam
+elif Re > 10000:
+    Nu_p = Nu_l
+else:
+    Nu_p = Nu_m
+```
+
+and the overall heat transfer coefficient $U$ is written as follows,
+
+$$
+    U = [\frac{r_{pi}+t_{pi}}{r_{pi}h}+(r_{pi}+t_{pi})(\frac{\ln{\frac{r_{pi}+t_{pi}}{r_{pi}}}}{\lambda_{pi}}+\frac{\ln{\frac{r_b}{r_{pi}+t_{pi}}}}{\lambda_{grout}})]^{-1}
+$$
+
+with
+
+$$
+    h = \frac{\lambda_f Nu}{2r_{pi}}
+$$
+
+```python
+h = lambda_f * Nu_p / (2 * r_pi)  # Unit: W/m2/K
+
+U = 1 / (((r_pi + t_pi) / (r_pi * h)) + (r_pi + t_pi) * (np.log((r_pi + t_pi) / r_pi) / lambda_pi + np.log(r_b / (r_pi + t_pi)) / lambda_g))
+```
+
+The friction factor $f$, is evaluated by Churchill correlation (Churchill et al. (1977)),
+
+$$
+    f = \frac{1}{(\frac{1}{[((\frac{8}{Re})^{10}+(\frac{Re}{36500})^{20})]^{1/2}}+[2.21(\ln{\frac{Re}{7}})]^{10})^{1/5}}
+$$
+
+```python
+# Churchill correlation for friction factor
+f = 1 / ((1 / (np.sqrt((8 / Re) ** 10 + (Re / 36500) ** 20))) + (2.21 * np.log(Re / 7)) ** 10) ** (1 / 5)
+```
+
+With these equations the analytical solution can be computed:
+
+```python
+# Compute solution
+
+time_range = range(0, t+1, 43200)
+
+# Create a dataframe to store results for different z- and t-steps
+row_names = Z
+column_names = time_range
+
+
+df = pd.DataFrame(index=row_names, columns=column_names) 
+
+
+for delta_z in Z:
+    T_0 = []
+    for delta_t in time_range:
+        
+        # Compute dimensionless time
+        t_D = dimless_t(lambda_re, delta_t, rho_re, c_p_re, r_b)
+
+        # Compute time function
+        f_t = time_function(t_D)
+
+        # Compute X
+        X = coefficient_x(q, rho_f, c_p_f, lambda_re, r_pi, U, f_t)
+
+        # Compute outlet temperature
+        T_0.append(temp(T_d, T_i, delta_z, X))
+
+    df.loc[delta_z] = T_0
+```
+
+## Numerical solution
+
+```python
+# Create output path 
+import os
+
+out_dir = os.environ.get('OGS_TESTRUNNER_OUT_DIR', '_out')
+if not os.path.exists(out_dir):
+    os.makedirs(out_dir)
+```
+
+```python
+from ogs6py.ogs import OGS
+
+# Create an OGS-object
+# Pass the project file and set an output file path
+model=OGS(INPUT_FILE="BHE_1P.prj", PROJECT_FILE=f"{out_dir}/BHE_1P.prj")
+
+# set end time to 30 days (in seconds)
+model.replace_text(30 * 24 * 60 * 60, xpath="./time_stepping/t_end/value")
+
+# Write to the output file 
+model.write_input()
+
+# Run OGS
+model.run_model(logfile=os.path.join(out_dir, "log.txt"), args=f"-o {out_dir} -m .")
+```
+
+## Results and discussion
+
+The outlet temperature change over time was compared against analytical solution and presented in Figure 2.  
+After 30 days, the fluid temperature distribution in the wellbore is shown in Figure 3.  
+The maximum relative error between the numerical model and Ramey's analytical solution is less than 0.15%.
+
+In numerical model, the outlet temperature at beginning stage is affected by the initial temperature in the pipe inside the wellbore.  
+The initial fluid temperature set in the benchmark means there is water with 20°C filled in the wellbore already before injecting water into the wellbore.  
+But in the analytical solution, no initial temperature is set and the temperature keeps equilibrium state at every moment.  
+The impact of initial temperature condition in numerical model is decreasing with the increasing of the operational time as shown in Figure 2.
+
+```python
+import vtuIO
+import numpy as np
+import os
+
+out_dir = os.environ.get('OGS_TESTRUNNER_OUT_DIR', '_out')
+
+# Load the result
+pvdfile = vtuIO.PVDIO(f"{out_dir}/BHE_1P.pvd", dim=3)
+
+# Get point field names
+fields = pvdfile.get_point_field_names()
+print(fields)
+
+# Get all written timesteps
+time = pvdfile.timesteps
+
+# Define observation points
+observation_points = {'outlet':(0.0,10.0,-30.0)}
+
+# Read the time series of point field 'temperature_BHE1'
+temperature_BHE1 = pvdfile.read_time_series('temperature_BHE1', observation_points)
+```
+
+```python
+# Compute analytical solution again but at timesteps of numerical solution for error calculation
+for delta_z in Z:
+    T_0 = []
+    for delta_t in time:
+        
+        # Compute dimensionless time
+        t_D = dimless_t(lambda_re, delta_t, rho_re, c_p_re, r_b)
+
+        # Compute time function
+        f_t = time_function(t_D)
+
+        # Compute X
+        X = coefficient_x(q, rho_f, c_p_f, lambda_re, r_pi, U, f_t)
+
+        # Compute outlet temperature
+        T_0.append(temp(T_d, T_i, delta_z, X))
+```
+
+```python
+# Plot numerical vs. analytical result
+fig, ax1 = plt.subplots(figsize=(10, 8))
+
+x_pos = np.arange(0, 2592000+60*60*24*5, 60*60*24*5)
+x_ticks = np.arange(0,35,5)
+
+
+ax1.plot(time_range, df.iloc[-1, :], 'k.', markersize=10, markerfacecolor='none', label = 'Ramey\'s analytical solution')
+ax1.plot(time, temperature_BHE1['outlet'][:,0], 'r', label = 'OGS')
+ax1.set_ylabel('Outlet temperature (°C)', fontsize=20)
+ax1.set_xlabel('Time (d)', fontsize=20)
+ax1.set_xticks(x_pos, x_ticks)
+ax1.spines['left'].set_linewidth(2)
+ax1.spines['bottom'].set_linewidth(2)
+ax1.tick_params(axis='both', labelsize=16)
+
+color = 'blue'
+ax2 = ax1.twinx()
+ax2.plot(time ,temperature_BHE1['outlet'][:,0]-T_0, color=color, label = 'Absolute error')
+ax2.set_ylabel('Absolute error (°C)', color=color, fontsize = 20)
+ax2.tick_params(axis='y', labelcolor=color)
+ax2.spines['right'].set_color('blue')
+ax2.spines['right'].set_linewidth(2)
+ax2.tick_params(axis='y', labelsize=16)
+
+plt.figlegend(fontsize=18, loc = (0.4, 0.1), frameon=False)
+fig.tight_layout()
+
+plt.show()
+```
+
+Figure 2: Comparison with analytical solution results
+
+```python
+# Extract values along a line in the domain
+
+# Space axis for plotting
+length = np.linspace(0, -30, 101)
+
+# Draws a line through the domain for sampling results
+x_axis=[(0,10,i) for i in length]
+
+# At timestep i
+i = -1
+temp = pvdfile.read_set_data(time[i], "temperature_BHE1", pointsetarray=x_axis)
+
+# Plot
+fig, ax1 = plt.subplots(figsize=(10, 8))
+ax1.set_xlabel('Distance (m)', fontsize = 20)
+ax1.set_ylabel('Temperature (°C)', fontsize = 20)
+ax1.plot(np.linspace(0,30,101), temp[:,0], 'r', linewidth = 2, label = 'OGS')
+ax1.plot(Z,df.iloc[:, -1],'k.', markersize=10, markerfacecolor='none', label = 'Ramey\'s analytical solution')
+ax1.set_ylim(19,25)
+ax1.set_xlim(0,30)
+ax1.spines['left'].set_linewidth(2)
+ax1.spines['bottom'].set_linewidth(2)
+ax1.tick_params(axis='both', labelsize=16)
+
+color = 'blue'
+ax2 = ax1.twinx()
+ax2.set_ylabel('Absolute error (°C)', color=color, fontsize = 20)
+ax2.plot(np.linspace(0,30,101), temp[:,0]-df.iloc[:, -1], linewidth = 2, color=color, label = 'Absolute error')
+ax2.tick_params(axis='y', labelcolor=color)
+ax2.set_ylim(0,0.04)
+ax2.spines['right'].set_color('blue')
+ax2.spines['right'].set_linewidth(2)
+ax2.tick_params(axis='y', labelsize=16)
+
+fig.tight_layout()
+plt.figlegend(fontsize=18, loc = (0.4, 0.1), frameon=False)
+plt.show()
+
+```
+
+Figure 3: Distributed temperature of fluid and absolute error.
+
+## References
+
+<!-- vale off -->
+
+[1] Ramey Jr, H. J. (1962). Wellbore heat transmission. Journal of petroleum Technology, 14(04), 427-435.
+
+[2] Chaofan Chen, Wanlong Cai, Dmitri Naumov, Kun Tu, Hongwei Zhou, Yuping Zhang, Olaf Kolditz, Haibing Shao (2021). Numerical investigation on the capacity and efficiency of a deep enhanced U-tube borehole heat exchanger system for building heating. Renewable Energy, 169, 557-572.
+
+[3] Gnielinski, V. (1975). New equations for heat and mass transfer in the turbulent flow in pipes and channels. NASA STI/recon technical report A, 75, 8-16.
+
+[4] Churchill, S. W. (1977). Comprehensive correlating equations for heat, mass and momentum transfer in fully developed flow in smooth tubes. Industrial & Engineering Chemistry Fundamentals, 16(1), 109-116.
+
+<!-- vale on -->

From 54c87fe299bef66327b425d54ef663e105c5c621 Mon Sep 17 00:00:00 2001
From: Chaofan Chen <chaofan.chen@ufz.de>
Date: Mon, 30 Oct 2023 15:23:18 +0000
Subject: [PATCH 03/11] suggestion applied

---
 Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
index 4a591274623..7a2527c13a0 100644
--- a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
+++ b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
@@ -11,7 +11,7 @@ project = ["Parabolic/T/BHE_1P/BHE_1P.prj"]
 ## Problem description
 
 Ramey et al. (1962) proposed the analytical solution concerning the wellbore heat transmission, which can be used to quantify the fluid temperature change in the wellbore.  
-In order to verify the single pipe flow model in the OGS, the numerical results was compared with Ramey's analytical solution (Ramey et al. 1962).  
+In order to verify the single pipe flow model in the OGS, the numerical results were compared with Ramey's analytical solution (Ramey et al. 1962).  
 The detailed calculation of the Ramey's analytical solution is given below.  
 A detailed analysis of an enhanced U.tube borehole heat exchanger (EUBHE) can be found in (Chen, C. et al. 2021).
 

From 9e96077df186e30618ac0eec7b0990e8e396dde7 Mon Sep 17 00:00:00 2001
From: Chaofan Chen <chaofan.chen@ufz.de>
Date: Mon, 30 Oct 2023 15:23:36 +0000
Subject: [PATCH 04/11] suggestion applied

---
 Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
index 7a2527c13a0..a9124e2e659 100644
--- a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
+++ b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
@@ -13,7 +13,7 @@ project = ["Parabolic/T/BHE_1P/BHE_1P.prj"]
 Ramey et al. (1962) proposed the analytical solution concerning the wellbore heat transmission, which can be used to quantify the fluid temperature change in the wellbore.  
 In order to verify the single pipe flow model in the OGS, the numerical results were compared with Ramey's analytical solution (Ramey et al. 1962).  
 The detailed calculation of the Ramey's analytical solution is given below.  
-A detailed analysis of an enhanced U.tube borehole heat exchanger (EUBHE) can be found in (Chen, C. et al. 2021).
+A detailed analysis of an enhanced U-tube borehole heat exchanger (EUBHE) can be found in Chen, C. et al. (2021).
 
 ## Model Setup
 

From e190617a64f9c79596e1a8dd9c9a187126188d9e Mon Sep 17 00:00:00 2001
From: Chaofan Chen <chaofan.chen@ufz.de>
Date: Mon, 30 Oct 2023 15:23:58 +0000
Subject: [PATCH 05/11] Apply 1 suggestion(s) to 1 file(s)

---
 Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
index a9124e2e659..d7b67c3a0c5 100644
--- a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
+++ b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
@@ -17,7 +17,7 @@ A detailed analysis of an enhanced U-tube borehole heat exchanger (EUBHE) can be
 
 ## Model Setup
 
-In this benchmark, the length of the wellbore is 30 m as shown in Figure 1 and the cold water is injected into the inlet point of the wellbore with temperature of 20°C.  
+In this benchmark, the length of the wellbore is 30 m as shown in Figure 1 and the cold water is injected into the inlet point of the wellbore with the constant temperature of 20°C.  
 The initial temperature of the fluid and grout in the wellbore is 20 $^{\circ}$C, and temperature of the surrounding rock is 55 $^{\circ}$C.  
 The wellbore and pipe diameter are 0.28 m and 0.25826 m, respectively.  
 And the flow rate is 0.0002 $m^3/s$.

From de4a2e93c894926f07d20f16b48e3e4c44aeb09b Mon Sep 17 00:00:00 2001
From: Chaofan Chen <chaofan.chen@ufz.de>
Date: Mon, 30 Oct 2023 15:24:35 +0000
Subject: [PATCH 06/11] Apply 1 suggestion(s) to 1 file(s)

---
 Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
index d7b67c3a0c5..45d9e2ef2e4 100644
--- a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
+++ b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
@@ -279,7 +279,7 @@ model.run_model(logfile=os.path.join(out_dir, "log.txt"), args=f"-o {out_dir} -m
 
 ## Results and discussion
 
-The outlet temperature change over time was compared against analytical solution and presented in Figure 2.  
+The outlet temperature change over time was compared against the analytical solution and presented in Figure 2.  
 After 30 days, the fluid temperature distribution in the wellbore is shown in Figure 3.  
 The maximum relative error between the numerical model and Ramey's analytical solution is less than 0.15%.
 

From 3196ff861b9681f410af30aa64131c5cdafe8162 Mon Sep 17 00:00:00 2001
From: Frieder Loer <frieder.loer@ufz.de>
Date: Mon, 30 Oct 2023 16:40:39 +0100
Subject: [PATCH 07/11] removed reference

---
 Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md | 4 +---
 1 file changed, 1 insertion(+), 3 deletions(-)

diff --git a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
index 45d9e2ef2e4..e28b52f7fd2 100644
--- a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
+++ b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
@@ -416,8 +416,6 @@ Figure 3: Distributed temperature of fluid and absolute error.
 
 [2] Chaofan Chen, Wanlong Cai, Dmitri Naumov, Kun Tu, Hongwei Zhou, Yuping Zhang, Olaf Kolditz, Haibing Shao (2021). Numerical investigation on the capacity and efficiency of a deep enhanced U-tube borehole heat exchanger system for building heating. Renewable Energy, 169, 557-572.
 
-[3] Gnielinski, V. (1975). New equations for heat and mass transfer in the turbulent flow in pipes and channels. NASA STI/recon technical report A, 75, 8-16.
-
-[4] Churchill, S. W. (1977). Comprehensive correlating equations for heat, mass and momentum transfer in fully developed flow in smooth tubes. Industrial & Engineering Chemistry Fundamentals, 16(1), 109-116.
+[3] Churchill, S. W. (1977). Comprehensive correlating equations for heat, mass and momentum transfer in fully developed flow in smooth tubes. Industrial & Engineering Chemistry Fundamentals, 16(1), 109-116.
 
 <!-- vale on -->

From aa83d16f5979faf763dccb985e2d08705046f997 Mon Sep 17 00:00:00 2001
From: "Feliks K. Kiszkurno" <feliks-kuba.kiszkurno@ufz.de>
Date: Tue, 28 Nov 2023 15:48:04 +0100
Subject: [PATCH 08/11] Small fixes

---
 Tests/Data/Parabolic/T/BHE_1P/BHE_1P.prj      | 22 +++++++++----------
 .../Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md |  4 ++--
 2 files changed, 13 insertions(+), 13 deletions(-)

diff --git a/Tests/Data/Parabolic/T/BHE_1P/BHE_1P.prj b/Tests/Data/Parabolic/T/BHE_1P/BHE_1P.prj
index 11bc83042ff..c0a197093dd 100644
--- a/Tests/Data/Parabolic/T/BHE_1P/BHE_1P.prj
+++ b/Tests/Data/Parabolic/T/BHE_1P/BHE_1P.prj
@@ -147,22 +147,22 @@
                             <repeat>60</repeat>
                             <delta_t>60</delta_t>
                         </pair>
-						<pair>
+                        <pair>
                             <repeat>60</repeat>
                             <delta_t>360</delta_t>
                         </pair>
                         <pair>
-						    <repeat>41</repeat>
-						    <delta_t>3600</delta_t>
-						</pair>
-						<pair>
-						    <repeat>16</repeat>
-						    <delta_t>43200</delta_t>
-						</pair>
+                            <repeat>41</repeat>
+                            <delta_t>3600</delta_t>
+                        </pair>
                         <pair>
-						    <repeat>28</repeat>
-						    <delta_t>86400</delta_t>
-						</pair>
+                            <repeat>16</repeat>
+                            <delta_t>43200</delta_t>
+                        </pair>
+                        <pair>
+                            <repeat>28</repeat>
+                            <delta_t>86400</delta_t>
+                        </pair>
                     </timesteps>
                 </time_stepping>
             </process>
diff --git a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
index e28b52f7fd2..ac42c565b66 100644
--- a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
+++ b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
@@ -142,7 +142,7 @@ Re = rho_f * v * (2 * r_pi) / mu_f
 
 ```
 
-The Nusselt number can be determined by the following equation (Diersch 2013):
+The Nusselt number can be determined by the following equation (Diersch, 2013):
 
 $$
     Nu = 4.364,\  Re < 2300
@@ -153,7 +153,7 @@ $$
 $$
 
 $$
-    Nu = (1-\gamma_{k})\  4.364 + \gamma_{k} \Bigg\{ \frac{(0.0308/8)10^{4}\mathrm{Pr}}{1+12.7\  \sqrt{0.0308/8}(\mathrm{Pr}^{2/3}-1)} \left[ 1+\left(\frac{d_{k}^{i}}{L}\right)^{2/3} \right] \Bigg\}, 2 300 \leq Re < 10^{4}
+    Nu = (1-\gamma_{k})\  4.364 + \gamma_{k} \Big\{ \frac{(0.0308/8)10^{4}\mathrm{Pr}}{1+12.7\  \sqrt{0.0308/8}(\mathrm{Pr}^{2/3}-1)} \left[ 1+\left(\frac{d_{k}^{i}}{L}\right)^{2/3} \right] \Big\}, 2 300 \leq Re < 10^{4}
 $$
 
 with

From ea754d3dc44998045540a4ddd9d424e3848b4a8d Mon Sep 17 00:00:00 2001
From: "Feliks K. Kiszkurno" <feliks-kuba.kiszkurno@ufz.de>
Date: Wed, 29 Nov 2023 08:27:24 +0100
Subject: [PATCH 09/11] Fixing brackets issue

---
 Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
index ac42c565b66..8ef34ae9e78 100644
--- a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
+++ b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
@@ -153,7 +153,7 @@ $$
 $$
 
 $$
-    Nu = (1-\gamma_{k})\  4.364 + \gamma_{k} \Big\{ \frac{(0.0308/8)10^{4}\mathrm{Pr}}{1+12.7\  \sqrt{0.0308/8}(\mathrm{Pr}^{2/3}-1)} \left[ 1+\left(\frac{d_{k}^{i}}{L}\right)^{2/3} \right] \Big\}, 2 300 \leq Re < 10^{4}
+    Nu = (1-\gamma_{k})\  4.364 + \gamma_{k} ( \frac{(0.0308/8)10^{4}\mathrm{Pr}}{1+12.7\  \sqrt{0.0308/8}(\mathrm{Pr}^{2/3}-1)} \left[ 1+\left(\frac{d_{k}^{i}}{L}\right)^{2/3} \right] ), 2 300 \leq Re < 10^{4}
 $$
 
 with

From b13d1b03c280a791aa90b1604b55f5dd187feef8 Mon Sep 17 00:00:00 2001
From: "Feliks K. Kiszkurno" <feliks-kuba.kiszkurno@ufz.de>
Date: Wed, 29 Nov 2023 16:35:19 +0100
Subject: [PATCH 10/11] Fixing latex equation and formatting

---
 Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md | 12 +++++++-----
 1 file changed, 7 insertions(+), 5 deletions(-)

diff --git a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
index 8ef34ae9e78..ac656a0b8d3 100644
--- a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
+++ b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
@@ -123,11 +123,11 @@ def time_function(t_D):
 The Prandtl and Reynolds number can be calculated as follows
 
 $$
-    Pr = \frac{\mu_f c_{p,f}}{\lambda_f}
+    \mathrm{Pr} = \frac{\mu_f c_{p,f}}{\lambda_f}
 $$
 
 $$
-    Re = \frac{\rho_f v d_{pi}}{\mu_f}
+    \mathrm{Re} = \frac{\rho_f v d_{pi}}{\mu_f}
 $$
 
 where, $\mu_f, \rho_f$ and $\lambda_f$ is the fluid viscosity, density and thermal conductivity.
@@ -145,15 +145,17 @@ Re = rho_f * v * (2 * r_pi) / mu_f
 The Nusselt number can be determined by the following equation (Diersch, 2013):
 
 $$
-    Nu = 4.364,\  Re < 2300
+    \mathrm{Nu} = 4.364,\  \mathrm{Re} < 2300
 $$
 
 $$
-    Nu = \frac{(\xi_{k}/8)\  \mathrm{Re}_{k}\  \mathrm{Pr}}{1+12.7\sqrt{\xi_{k}/8}(\mathrm{Pr}^{2/3}-1)} \left[ 1+\left(\frac{d_{k}^{i}}{L}\right)^{2/3} \right],\   Re \geq 10^4
+    \mathrm{Nu} = \frac{(\xi_{k}/8)\  \mathrm{Re}_{k}\  \mathrm{Pr}}{1+12.7\sqrt{{\xi_k}/8}(\mathrm{Pr}^{2/3}-1)} [ 1+(\frac{{d_k}^{i}}{L})^{2/3}], Re \geq 10^4
 $$
 
 $$
-    Nu = (1-\gamma_{k})\  4.364 + \gamma_{k} ( \frac{(0.0308/8)10^{4}\mathrm{Pr}}{1+12.7\  \sqrt{0.0308/8}(\mathrm{Pr}^{2/3}-1)} \left[ 1+\left(\frac{d_{k}^{i}}{L}\right)^{2/3} \right] ), 2 300 \leq Re < 10^{4}
+
+$$
+    \mathrm{Nu} = (1-\gamma_{k})\  4.364 + \gamma_{k} ( \frac{(0.0308/8)10^{4}\mathrm{Pr}}{1+12.7\  \sqrt{0.0308/8}(\mathrm{Pr}^{2/3}-1)} \left[ 1+\left(\frac{d_{k}^{i}}{L}\right)^{2/3} \right] ), 2 300 \leq \mathrm{Re} < 10^{4}
 $$
 
 with

From deb6ecb83a1c05cc143b049bae8c648941c37c62 Mon Sep 17 00:00:00 2001
From: "Feliks K. Kiszkurno" <feliks-kuba.kiszkurno@ufz.de>
Date: Thu, 30 Nov 2023 11:46:15 +0100
Subject: [PATCH 11/11] Added missing citation and improved consistency

---
 Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md | 14 ++++++++------
 1 file changed, 8 insertions(+), 6 deletions(-)

diff --git a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
index ac656a0b8d3..bfd9bba7c02 100644
--- a/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
+++ b/Tests/Data/Parabolic/T/BHE_1P/pipe_flow_ebhe.md
@@ -10,10 +10,10 @@ project = ["Parabolic/T/BHE_1P/BHE_1P.prj"]
 
 ## Problem description
 
-Ramey et al. (1962) proposed the analytical solution concerning the wellbore heat transmission, which can be used to quantify the fluid temperature change in the wellbore.  
-In order to verify the single pipe flow model in the OGS, the numerical results were compared with Ramey's analytical solution (Ramey et al. 1962).  
+(Ramey et al. (1962)) proposed the analytical solution concerning the wellbore heat transmission, which can be used to quantify the fluid temperature change in the wellbore.  
+In order to verify the single pipe flow model in the OGS, the numerical results were compared with Ramey's analytical solution (Ramey et al. (1962)).  
 The detailed calculation of the Ramey's analytical solution is given below.  
-A detailed analysis of an enhanced U-tube borehole heat exchanger (EUBHE) can be found in Chen, C. et al. (2021).
+A detailed analysis of an enhanced U-tube borehole heat exchanger (EUBHE) can be found in (Chen, C. et al. (2021)).
 
 ## Model Setup
 
@@ -142,7 +142,7 @@ Re = rho_f * v * (2 * r_pi) / mu_f
 
 ```
 
-The Nusselt number can be determined by the following equation (Diersch, 2013):
+The Nusselt number can be determined by the following equation (Diersch, (2011)):
 
 $$
     \mathrm{Nu} = 4.364,\  \mathrm{Re} < 2300
@@ -416,8 +416,10 @@ Figure 3: Distributed temperature of fluid and absolute error.
 
 [1] Ramey Jr, H. J. (1962). Wellbore heat transmission. Journal of petroleum Technology, 14(04), 427-435.
 
-[2] Chaofan Chen, Wanlong Cai, Dmitri Naumov, Kun Tu, Hongwei Zhou, Yuping Zhang, Olaf Kolditz, Haibing Shao (2021). Numerical investigation on the capacity and efficiency of a deep enhanced U-tube borehole heat exchanger system for building heating. Renewable Energy, 169, 557-572.
+[2] Diersch, H-JG, et al. (2011). "Finite element modeling of borehole heat exchanger systems: Part 1. Fundamentals." Computers & Geosciences 37.8, 1122-1135.
 
-[3] Churchill, S. W. (1977). Comprehensive correlating equations for heat, mass and momentum transfer in fully developed flow in smooth tubes. Industrial & Engineering Chemistry Fundamentals, 16(1), 109-116.
+[3] Chaofan Chen, Wanlong Cai, Dmitri Naumov, Kun Tu, Hongwei Zhou, Yuping Zhang, Olaf Kolditz, Haibing Shao (2021). Numerical investigation on the capacity and efficiency of a deep enhanced U-tube borehole heat exchanger system for building heating. Renewable Energy, 169, 557-572.
+
+[4] Churchill, S. W. (1977). Comprehensive correlating equations for heat, mass and momentum transfer in fully developed flow in smooth tubes. Industrial & Engineering Chemistry Fundamentals, 16(1), 109-116.
 
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