From f4af587fd5fac224d4a287dc1a2238a4b2619796 Mon Sep 17 00:00:00 2001
From: Marc Walther <marc.walther@ufz.de>
Date: Thu, 2 May 2019 14:29:50 +0200
Subject: [PATCH] Add Theis Benchmark for ComponentTransport

---
 ProcessLib/ComponentTransport/Tests.cmake     |  44 +++
 .../ComponentTransport/HC_Theis/HC_theis.prj  | 302 ++++++++++++++++++
 .../HC_Theis/abstraction_whole_surf.vtu       |   3 +
 .../HC_Theis/analytical_theis.xlsx            | Bin 0 -> 332354 bytes
 .../HC_Theis/drawdown_s-30.txt                |  25 ++
 .../HC_Theis/drawdown_t-5e3.txt               | 202 ++++++++++++
 .../HC_Theis/inflow_surf.vtu                  |   3 +
 .../out_HC_theis_pcs_0_ts_0_t_0.000000.vtu    |   3 +
 .../out_HC_theis_pcs_0_ts_10_t_100.000000.vtu |   3 +
 .../out_HC_theis_pcs_0_ts_20_t_400.000000.vtu |   3 +
 ...out_HC_theis_pcs_0_ts_30_t_1000.000000.vtu |   3 +
 ...out_HC_theis_pcs_0_ts_40_t_2000.000000.vtu |   3 +
 ...out_HC_theis_pcs_0_ts_50_t_5000.000000.vtu |   3 +
 ...ut_HC_theis_pcs_0_ts_60_t_20000.000000.vtu |   3 +
 ...ut_HC_theis_pcs_0_ts_70_t_70000.000000.vtu |   3 +
 ...t_HC_theis_pcs_0_ts_73_t_100000.000000.vtu |   3 +
 .../HC_Theis/slice_reorder.vtu                |   3 +
 .../benchmarks/hydro-component/theis/BCs.png  |   3 +
 .../hydro-component/theis/HC_Theis.pandoc     |  65 ++++
 .../hydro-component/theis/comparison.png      |   3 +
 20 files changed, 680 insertions(+)
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/HC_theis.prj
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/abstraction_whole_surf.vtu
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/analytical_theis.xlsx
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/drawdown_s-30.txt
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/drawdown_t-5e3.txt
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/inflow_surf.vtu
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/out_HC_theis_pcs_0_ts_0_t_0.000000.vtu
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/out_HC_theis_pcs_0_ts_10_t_100.000000.vtu
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/out_HC_theis_pcs_0_ts_20_t_400.000000.vtu
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/out_HC_theis_pcs_0_ts_30_t_1000.000000.vtu
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/out_HC_theis_pcs_0_ts_40_t_2000.000000.vtu
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/out_HC_theis_pcs_0_ts_50_t_5000.000000.vtu
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/out_HC_theis_pcs_0_ts_60_t_20000.000000.vtu
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/out_HC_theis_pcs_0_ts_70_t_70000.000000.vtu
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/out_HC_theis_pcs_0_ts_73_t_100000.000000.vtu
 create mode 100644 Tests/Data/Parabolic/ComponentTransport/HC_Theis/slice_reorder.vtu
 create mode 100644 web/content/docs/benchmarks/hydro-component/theis/BCs.png
 create mode 100644 web/content/docs/benchmarks/hydro-component/theis/HC_Theis.pandoc
 create mode 100644 web/content/docs/benchmarks/hydro-component/theis/comparison.png

diff --git a/ProcessLib/ComponentTransport/Tests.cmake b/ProcessLib/ComponentTransport/Tests.cmake
index b7e977d61dc..91dd68099a6 100644
--- a/ProcessLib/ComponentTransport/Tests.cmake
+++ b/ProcessLib/ComponentTransport/Tests.cmake
@@ -853,3 +853,47 @@ AddTest(
     vdbc_pcs_0_ts_21990_t_210000.000000_expected.vtu vdbc_pcs_0_ts_21990_t_210000.000000.vtu concentration Si 1e-5 1e-4
     vdbc_pcs_0_ts_25990_t_250000.000000_expected.vtu vdbc_pcs_0_ts_25990_t_250000.000000.vtu concentration Si 1e-5 1e-4
 )
+
+AddTest(
+    NAME HC_Theis
+    PATH Parabolic/ComponentTransport/HC_Theis
+    EXECUTABLE ogs
+    EXECUTABLE_ARGS HC_theis.prj
+    WRAPPER time
+    TESTER vtkdiff
+    REQUIREMENTS NOT OGS_USE_MPI
+    RUNTIME 90
+    DIFF_DATA
+    out_HC_theis_pcs_0_ts_0_t_0.000000.vtu
+    out_HC_theis_pcs_0_ts_0_t_0.000000.vtu pressure pressure 1e-3 1e-6
+    out_HC_theis_pcs_0_ts_0_t_0.000000.vtu
+    out_HC_theis_pcs_0_ts_0_t_0.000000.vtu darcy_velocity darcy_velocity 1e-10 1e-6
+    out_HC_theis_pcs_0_ts_10_t_100.000000.vtu
+    out_HC_theis_pcs_0_ts_10_t_100.000000.vtu pressure pressure 1e-3 1e-6
+    out_HC_theis_pcs_0_ts_10_t_100.000000.vtu
+    out_HC_theis_pcs_0_ts_10_t_100.000000.vtu darcy_velocity darcy_velocity 1e-10 1e-6
+    out_HC_theis_pcs_0_ts_20_t_400.000000.vtu
+    out_HC_theis_pcs_0_ts_20_t_400.000000.vtu pressure pressure 1e-3 1e-6
+    out_HC_theis_pcs_0_ts_20_t_400.000000.vtu
+    out_HC_theis_pcs_0_ts_20_t_400.000000.vtu darcy_velocity darcy_velocity 1e-10 1e-6
+    out_HC_theis_pcs_0_ts_30_t_1000.000000.vtu
+    out_HC_theis_pcs_0_ts_30_t_1000.000000.vtu pressure pressure 1e-3 1e-6
+    out_HC_theis_pcs_0_ts_30_t_1000.000000.vtu
+    out_HC_theis_pcs_0_ts_30_t_1000.000000.vtu darcy_velocity darcy_velocity 1e-10 1e-6
+    out_HC_theis_pcs_0_ts_40_t_2000.000000.vtu
+    out_HC_theis_pcs_0_ts_40_t_2000.000000.vtu pressure pressure 1e-3 1e-6
+    out_HC_theis_pcs_0_ts_40_t_2000.000000.vtu
+    out_HC_theis_pcs_0_ts_40_t_2000.000000.vtu darcy_velocity darcy_velocity 1e-10 1e-6
+    out_HC_theis_pcs_0_ts_60_t_20000.000000.vtu
+    out_HC_theis_pcs_0_ts_60_t_20000.000000.vtu pressure pressure 1e-3 1e-6
+    out_HC_theis_pcs_0_ts_60_t_20000.000000.vtu
+    out_HC_theis_pcs_0_ts_60_t_20000.000000.vtu darcy_velocity darcy_velocity 1e-10 1e-6
+    out_HC_theis_pcs_0_ts_70_t_70000.000000.vtu
+    out_HC_theis_pcs_0_ts_70_t_70000.000000.vtu pressure pressure 1e-3 1e-6
+    out_HC_theis_pcs_0_ts_70_t_70000.000000.vtu
+    out_HC_theis_pcs_0_ts_70_t_70000.000000.vtu darcy_velocity darcy_velocity 1e-10 1e-6
+    out_HC_theis_pcs_0_ts_73_t_100000.000000.vtu
+    out_HC_theis_pcs_0_ts_73_t_100000.000000.vtu pressure pressure 1e-3 1e-6
+    out_HC_theis_pcs_0_ts_73_t_100000.000000.vtu
+    out_HC_theis_pcs_0_ts_73_t_100000.000000.vtu darcy_velocity darcy_velocity 1e-10 1e-6
+)
\ No newline at end of file
diff --git a/Tests/Data/Parabolic/ComponentTransport/HC_Theis/HC_theis.prj b/Tests/Data/Parabolic/ComponentTransport/HC_Theis/HC_theis.prj
new file mode 100644
index 00000000000..f5ec6c80e0c
--- /dev/null
+++ b/Tests/Data/Parabolic/ComponentTransport/HC_Theis/HC_theis.prj
@@ -0,0 +1,302 @@
+<?xml version="1.0" encoding="ISO-8859-1"?>
+<OpenGeoSysProject>
+    <meshes>
+        <mesh>slice_reorder.vtu</mesh>
+        <mesh>inflow_surf.vtu</mesh>
+        <mesh>abstraction_whole_surf.vtu</mesh>
+    </meshes>
+
+    <processes>
+        <process>
+            <name>ComponentTransport</name>
+            <type>ComponentTransport</type>
+            <integration_order>2</integration_order>
+            <process_variables>
+				<concentration>concentration</concentration>
+                <pressure>pressure</pressure>
+            </process_variables>
+            <secondary_variables>
+                <secondary_variable type="static" internal_name="darcy_velocity" output_name="darcy_velocity"/>
+            </secondary_variables>
+            <!-- <calculatesurfaceflux>
+                <mesh>abstraction_whole_surf</mesh>
+                <property_name>flux</property_name>
+                <output_mesh>out_abstraction_whole_surf_flux.vtu</output_mesh>
+            </calculatesurfaceflux> -->
+            <fluid>
+                <density>
+                    <type>ConcentrationAndPressureDependent</type>
+				    <fluid_density_concentration_difference_ratio>0</fluid_density_concentration_difference_ratio>
+                    <fluid_density_pressure_difference_ratio>5.0968E-08</fluid_density_pressure_difference_ratio>
+                    <reference_concentration>0</reference_concentration>
+				    <reference_pressure>0</reference_pressure>
+				    <reference_density>1000</reference_density>
+                </density>
+                <viscosity>
+                    <type>Constant</type>
+                    <value>1.0e-3</value>
+                </viscosity>
+            </fluid>
+            <porous_medium>
+                <porous_medium id="0">
+                    <permeability>
+                        <type>Constant</type>
+                        <permeability_tensor_entries>kappa1</permeability_tensor_entries>
+                    </permeability>
+                    <porosity>
+                        <type>Constant</type>
+                        <porosity_parameter>porosity1</porosity_parameter>
+                    </porosity>
+                    <!-- The implementation of the ComponentTransport process 
+                    is without Boussinesq approximations, thus, 'storage' is not 
+                    used here, but should be set through S = thickness * gravity * porosity * d\rho / dp / rho.  -->
+                    <storage>
+                        <type>Constant</type>
+                        <value>0</value>
+                    </storage>
+                </porous_medium>
+            </porous_medium>
+            <fluid_reference_density>rho_fluid</fluid_reference_density>
+            <retardation_factor>retardation</retardation_factor>
+            <decay_rate>decay</decay_rate>
+            <specific_body_force>0 0 -9.81</specific_body_force>
+        </process>
+    </processes>
+    <media>
+        <medium id="0">
+            <phases>
+                <phase>
+                    <type>AqueousLiquid</type>
+                    <components>
+                        <component>
+                            <name>concentration</name>
+                            <properties>
+                                <property>
+                                    <name>molecular_diffusion</name>
+                                    <type>Constant</type>
+                                    <value>2e-9</value>
+                                </property>
+                            </properties>
+                        </component>
+                    </components>
+                </phase>
+            </phases>
+            <properties>
+                <property>
+                    <name>longitudinal_dispersivity</name>
+                    <type>Constant</type>
+                    <value>0</value>
+                </property>
+                <property>
+                    <name>transversal_dispersivity</name>
+                    <type>Constant</type>
+                    <value>0</value>
+                </property>
+            </properties>
+        </medium>
+    </media>
+    <time_loop>
+        <processes>
+            <process ref="ComponentTransport">
+                <nonlinear_solver>basic_picard</nonlinear_solver>
+                <convergence_criterion>
+                    <type>PerComponentDeltaX</type>
+                    <norm_type>NORM2</norm_type>
+                    <reltols>1e-5 5e-6</reltols>
+                </convergence_criterion>
+                <time_discretization>
+                    <type>BackwardEuler</type>
+                </time_discretization>
+                <time_stepping>
+                    <type>FixedTimeStepping</type>
+                    <t_initial> 0.0 </t_initial>
+                    <t_end> 1e5 </t_end>
+                    <timesteps>
+                        <pair>
+                            <repeat>10</repeat>
+                            <delta_t>10</delta_t>
+                        </pair>
+                        <pair>
+                            <repeat>5</repeat>
+                            <delta_t>20</delta_t>
+                        </pair>
+                        <pair>
+                            <repeat>10</repeat>
+                            <delta_t>40</delta_t>
+                        </pair>
+                        <pair>
+                            <repeat>5</repeat>
+                            <delta_t>80</delta_t>
+                        </pair>
+                        <pair>
+                            <repeat>10</repeat>
+                            <delta_t>100</delta_t>
+                        </pair>
+                        <pair>
+                            <repeat>5</repeat>
+                            <delta_t>200</delta_t>
+                        </pair>
+                        <pair>
+                            <repeat>5</repeat>
+                            <delta_t>400</delta_t>
+                        </pair>
+                        <pair>
+                            <repeat>5</repeat>
+                            <delta_t>1000</delta_t>
+                        </pair>
+                        <pair>
+                            <repeat>5</repeat>
+                            <delta_t>2000</delta_t>
+                        </pair>
+                        <pair>
+                            <repeat>10</repeat>
+                            <delta_t>5000</delta_t>
+                        </pair>
+                        <pair>
+                            <repeat>5</repeat>
+                            <delta_t>10000</delta_t>
+                        </pair>
+                        <pair>
+                            <repeat>5</repeat>
+                            <delta_t>20000</delta_t>
+                        </pair>
+                    </timesteps>
+                </time_stepping>
+            </process>
+        </processes>
+        <output>
+            <type>VTK</type>
+            <prefix>out_HC_theis</prefix>
+            <timesteps>
+                <pair>
+                    <repeat> 1 </repeat>
+                    <each_steps> 10 </each_steps>
+                </pair>
+            </timesteps>
+            <variables>
+                <variable> pressure </variable>
+                <variable> concentration </variable>
+                <variable> darcy_velocity </variable>
+            </variables>
+        </output>
+    </time_loop>
+
+    <parameters>
+    	<parameter>
+            <name>rho_fluid</name>
+            <type>Constant</type>
+            <value>1000</value>
+        </parameter>
+        <parameter>
+            <name>retardation</name>
+            <type>Constant</type>
+            <value>1</value>
+        </parameter>
+        <parameter>
+            <name>decay</name>
+            <type>Constant</type>
+            <value>0</value>
+        </parameter>
+        
+        <parameter>
+            <name>porosity1</name>
+            <type>Constant</type>
+            <value>0.2</value>
+        </parameter>
+        <parameter>
+            <name>kappa1</name>
+            <type>Constant</type>
+            <values>7.6453E-13 0 0 0 7.6453E-13 0 0 0 7.6453E-13</values>
+        </parameter>
+
+        <parameter>
+            <name>c_ini</name>
+            <type>MeshNode</type>
+            <mesh>slice_reorder</mesh>
+            <field_name>c_ini</field_name>
+        </parameter>
+        <parameter>
+            <name>p_ini</name>
+            <type>MeshNode</type>
+            <mesh>slice_reorder</mesh>
+            <field_name>p_ini</field_name>
+        </parameter>
+        <parameter>
+            <name>c_0</name>
+            <type>Constant</type>
+            <value>0</value>
+        </parameter>
+        <parameter>
+            <name>p_BC_Dirichlet</name>
+            <type>MeshNode</type>
+            <mesh>inflow_surf</mesh>
+            <field_name>p_BC_Dirichlet</field_name>
+        </parameter>
+        <parameter>
+            <name>p_BC_Neumann_c</name>
+            <type>Constant</type>
+            <value>-2.83542E-03</value>
+        </parameter>
+
+
+    </parameters>
+
+    <process_variables>
+        <process_variable>
+            <name>concentration</name>
+            <components>1</components>
+            <order>1</order>
+            <initial_condition>c_ini</initial_condition>
+            <boundary_conditions>
+                <boundary_condition>
+                    <type>Dirichlet</type>
+                    <mesh>inflow_surf</mesh>
+                    <parameter>c_0</parameter>
+                </boundary_condition>
+            </boundary_conditions>
+        </process_variable>
+        <process_variable>
+            <name>pressure</name>
+            <components>1</components>
+            <order>1</order>
+            <initial_condition>p_ini</initial_condition>
+            <boundary_conditions>
+                <boundary_condition>
+                    <type>Dirichlet</type>
+                    <mesh>inflow_surf</mesh>
+                    <parameter>p_BC_Dirichlet</parameter>
+                </boundary_condition>
+                <boundary_condition>
+                    <type>Neumann</type>
+                    <mesh>abstraction_whole_surf</mesh>
+                    <parameter>p_BC_Neumann_c</parameter>
+                </boundary_condition>
+            </boundary_conditions>
+        </process_variable>
+    </process_variables>
+
+    <nonlinear_solvers>
+        <nonlinear_solver>
+            <name>basic_picard</name>
+            <type>Picard</type>
+            <max_iter>30</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-10 -maxiter 10000</lis>
+            <eigen>
+                <solver_type>CG</solver_type>
+                <precon_type>DIAGONAL</precon_type>
+                <max_iteration_step>10000</max_iteration_step>
+                <error_tolerance>1e-10</error_tolerance>
+            </eigen>
+            <petsc>
+                <prefix>lf</prefix>
+                <parameters>-lf_ksp_type cg -lf_pc_type bjacobi -lf_ksp_rtol 1e-16 -lf_ksp_max_it 10000</parameters>
+            </petsc>
+        </linear_solver>
+    </linear_solvers>
+</OpenGeoSysProject>
diff --git a/Tests/Data/Parabolic/ComponentTransport/HC_Theis/abstraction_whole_surf.vtu b/Tests/Data/Parabolic/ComponentTransport/HC_Theis/abstraction_whole_surf.vtu
new file mode 100644
index 00000000000..79c0ff13078
--- /dev/null
+++ b/Tests/Data/Parabolic/ComponentTransport/HC_Theis/abstraction_whole_surf.vtu
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4c73a8b4660971a61bd186fb837e306e6590d635a7f1054cf086a22098c8b70b
+size 9506
diff --git a/Tests/Data/Parabolic/ComponentTransport/HC_Theis/analytical_theis.xlsx b/Tests/Data/Parabolic/ComponentTransport/HC_Theis/analytical_theis.xlsx
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diff --git a/Tests/Data/Parabolic/ComponentTransport/HC_Theis/drawdown_s-30.txt b/Tests/Data/Parabolic/ComponentTransport/HC_Theis/drawdown_s-30.txt
new file mode 100644
index 00000000000..82a5fc7ab59
--- /dev/null
+++ b/Tests/Data/Parabolic/ComponentTransport/HC_Theis/drawdown_s-30.txt
@@ -0,0 +1,25 @@
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\ No newline at end of file
diff --git a/Tests/Data/Parabolic/ComponentTransport/HC_Theis/drawdown_t-5e3.txt b/Tests/Data/Parabolic/ComponentTransport/HC_Theis/drawdown_t-5e3.txt
new file mode 100644
index 00000000000..76739684204
--- /dev/null
+++ b/Tests/Data/Parabolic/ComponentTransport/HC_Theis/drawdown_t-5e3.txt
@@ -0,0 +1,202 @@
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diff --git a/web/content/docs/benchmarks/hydro-component/theis/HC_Theis.pandoc b/web/content/docs/benchmarks/hydro-component/theis/HC_Theis.pandoc
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++++
+author = "Marc Walther"
+weight = 142
+project = "Parabolic/ComponentTransport/Theis/HC_theis.prj"
+date = "2019-05-02T14:41:09+01:00"
+title = "Theis solution for well pumping"
+
+[menu]
+  [menu.benchmarks]
+    parent = "hydro-component"
+
++++
+
+{{< data-link >}}
+
+
+## Overview
+
+Abstraction of water from an aquifer is a common procedure, as groundwater is usually better protected from contamination and steadily available even in dry seasons. For a sustainable water usage, the evaluation of the spatio-temporal behaviour of the water level for a specified abstraction rate is of high importance for water management. 
+
+This benchmark verifies the groundwater level drawdown for an aquifer that is subjected to pumping.
+
+
+## Problem description
+
+For pumping from a well, Theis (1935) proposed an analytical solution to calculate the water level drawdown over time and distance from the well. A short summary of the solution can be seen in 
+web/content/docs/benchmarks/liquid-flow/liquid-flow-theis-problem.pandoc
+
+Here, we verify pumping abstraction with Theis for the `ComponentTransport` process using a 3D setup. The setup is adapted from Walther (2014), section 3.2.
+
+
+### Model setup
+
+The setup comprises a 1/8th slice of a full circle (see figure 1). 
+
+{{< img src="../BCs.png" title="Mesh and boundary conditions (BC); blue = outer Dirichlet pressure and concentration BC, red = inner Neumann abstraction BC.">}}
+
+The outer boundary condition is set as Dirichlet with a hydrostatic pressure along the shell surface of the slice equivalent to a head of $h = 0 m$ (i.e. water level equals top of domain). For mass transport, a Dirichlet boundary conditions with concentration $c = 0$ is set at the outer shell. The inner boundary condition is equivalent to the eighth of a total abstraction rate of $Q_t = 15 m^3/d$ for a full cylinder. *NB: In the `ComponentTransport` process, the Neumann BC is given as mass flux and has to be calculated per area, such that the value for the project file is $Q = Q_t / 8 / A \cdot \rho = 2.83542E-03 m^3/s/m^2 \cdot kg/m^3$ with fluid density $\rho = 1000 kg/m^3$ and abstraction area $A = 7.65 m^2$.* 
+
+
+The homogeneous, isotropic domain is defined for the radius $1 < r < 100 m$ and a thickness $b = 10 m$. Saturated intrinsic permeability is $\kappa = 7.6453E-13 m^2$ yielding a transmissivity of $T = 7.5E-05 m^2/s$; porosity is $\phi = 0.2$; specific storage is $S_s = 1.0E-03$ and defined through compressibility $\gamma = 5.0968E-08 s^2/m/kg$ (input tag fluid_density_pressure_difference_ratio is $\frac{\partial \rho}{\partial p} = \frac{S_s}{b \phi g \rho}$ with gravitational acceleration $g = 9.81 m^2/s$).
+
+Mass transport properties are irrelevant as no transport processes are calculated.
+
+Initial conditions are $c = 0$ and hydrostatic pressure conditions.
+
+{{< data-link "Link to the HC_theis project file" "Parabolic/ComponentTransport/HC_Theis/HC_theis.prj" >}}
+
+
+### Results
+
+The figure below compares the analytical Theis solution vs. the simulated values from OGS6.
+
+{{< img src="../comparison.png" title="Comparison between numerical (crosses) and analytical (lines) values.">}}
+
+The top figure shows drawdown (i.e. the difference in water level compared to an initial state) over time at distance $r = 30 m$: for a simulation time $t < 40000 s$, the differences between analytical and numerical solutions are marginal; at later simulation times, the drawdown shows lower values than predicted from the analytical solution, as it is influenced by the outer Dirichlet pressure boundary condition. 
+
+The lower figure shows drawdown over radius at time $t = 5000 s$; the top figure also signifies this with a green vertical: over the whole domain, the analytical and numerical solutions are coinciding very well. *NB: With higher values of $S_s$, one has to take care to convert the simulated water level to equivalent freshwater head; in this case, fluid density variation is very low such that a conversion is not required.*
+
+
+## References
+
+Theis, C.V., 1935. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage. Trans. Amer. Geophys. Union 16, 513–514.
+
+Walther, M., 2014. Variable-Density Flow Processes in Porous Media On Small, Medium and Regional Scales (Dissertation). Technische Universität Dresden.
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-- 
GitLab