[web] update latex formulars in benchmark docs

    Hydro-component/contracer
    Reactive Transport/kineticreactant_allascomponents
    Reactive Transport/wetland

web/content/docs/benchmarks/hydro-component/contracer/ConTracer.md

@@ -20,8 +20,8 @@ Additionally, simulations have been compared with experimental data obtained fro
 
 The experimental system consists of a box of 4.7 m in length, 1.2 m in width and 1.05 m in depth (Nivala *et al.* 2013).
 The box was filled with gravel giving an average porosity of 0.38. The initial water level was at 1.0 meter and the outflow was realized by an overflow pipe of 1.0 m height.
-Hydraulic water influx was 0.768 meter^3^ d^-1^ at the left side.
-The tracer (40.26 g of Br^-^) was diluted in 12 L of waste water and added as a single impulse event at $t=0$.
+Hydraulic water influx was 0.768 $\textrm{m}^3~\textrm{d}^{-1}$ at the left side.
+The tracer (40.26 g of $\textrm{Br}^-$) was diluted in 12 L of waste water and added as a single impulse event at $t=0$.
 Note, that only 89\% of the tracer was recovered at the outlet.
 
 ![Top: Schematic representation of the experiment. Middle and bottom: Simulated domain and input parameters in 1D and 2D, respectively. Modified with permission from Boog *et al.* (2019).](../ConTracer_domain.png)
@@ -40,14 +40,14 @@ The boundary condition of the tracer ($g_{D,left}^{c_{tracer}}$) at the inlet wa
 |Parameter | Description | Value | Unit  |
 |:-------  | :-------- |-----:|:------:|
 |$\phi$     | Porosity   | 0.38  |  |
-|$\kappa$   | Permeability | 1.00E-08 | m^2^ |
+|$\kappa$   | Permeability | 1.00E-08 | $\textrm{m}^2$ |
 |$S$       | Storage     | 0     |  |
 |$a_L$     | long. Dispersion length | 0.7   | m |
 |$a_T$     | transv. Dispersion length | 0.07  | m |
-|$\rho_w$   | Fluid density | 1.00E+03 | kg m^-3^ |
+|$\rho_w$   | Fluid density | 1.00E+03 | $\textrm{kg m}^{-3}$ |
 |$\mu_w$    | Fluid viscosity | 1.00E-03 | Pa s |
-|$D_{tracer}$ | Tracer diffusion coef. | 0  | m^2^ s^-1^ |
-|$g$       | Gravity acceleration in $y$ direction | 9.81 | m s^-2^ |
+|$D_{tracer}$ | Tracer diffusion coef. | 0  | $\textrm{m}^2~\textrm{s}^{-1}$ |
+|$g$       | Gravity acceleration in $y$ direction | 9.81 | $\textrm{m s}^{-2}$ |
 
 ---------------------- ---------- ----------- ------------
 
@@ -57,15 +57,11 @@ Table 1: Material Properties
 
 | Parameter | Description | Value | Unit  |
 |:--------- |:----------  | -----:|------:|
-|$g_{N,left}^p$ | Influent mass influx | 6.55093e-03 | kg s^-1^ |
+|$g_{N,left}^p$ | Influent mass influx | 6.55093e-03 | $\textrm{kg s}^{-1}$ |
 |$g_{D,outlet}^p$ | Pressure at outlet | 9810 | Pa |
-|$g_{D,left}^{c_{tracer}}$ | Tracer concentration |  $$ c(t)=
-                                                            \begin{cases}
-                                                                1.19438 & \text{for } t<=3600 \\
-                                                                0 & \text{for } t>3600
-                                                            \end{cases}$$ | g L^-1^|
+|$g_{D,left}^{c_{tracer}}$ | Tracer concentration | $1.19438~\textrm{for}~t\leq 3600,~0~\textrm{for}~t>3600$ | $\textrm{g L}^{-1}$|
 |$p(t=0)$   | Initial pressure | 9810  | Pa |
-|$c_{tracer}(t=0)$ | Initial tracer concentration | 0  | g L^-1^ |
+|$c_{tracer}(t=0)$ | Initial tracer concentration | 0  | $\textrm{g L}^{-1}$ |
 
 ---------------------- ---------- ----------- ------------
 
@@ -81,15 +77,11 @@ Material properties, initial and boundary conditions are presented in the table
 
 | Parameter | Description | Value | Unit  |
 |:--------- | :---------- | -----:| -----:|
-| $g_{N,left}^p$ | Influent mass influx | 3.27546e-02 | kg s^-1^ |
+| $g_{N,left}^p$ | Influent mass influx | 3.27546e-02 | $\textrm{kg s}^{-1}$ |
 | $g_{D,outlet}^p$ | Pressure at outlet | 9320 | Pa |
-| $g_{D,left}^{c_{tracer}}$ | Tracer concentration | $$ c(t)=
-                                                            \begin{cases}
-                                                                1.19438 & \text{for } t<=3600 \\
-                                                                0 & \text{for } t>3600
-                                                            \end{cases}$$ | g L^-1^ |
-| $p(t=0)$  | Initial pressure | $9810-9810*y$ | Pa |
-| $c_{tracer}(t=0)$ | Initial tracer concentration | 0  | g L^-1^ |
+| $g_{D,left}^{c_{tracer}}$ | Tracer concentration | $1.19438~\textrm{for}~t\leq 3600,~0~\textrm{for}~t>3600$ | $\textrm{g L}^{-1}$ |
+| $p(t=0)$  | Initial pressure | $9810-9810y$ | Pa |
+| $c_{tracer}(t=0)$ | Initial tracer concentration | 0  | $\textrm{g L}^{-1}$ |
 
 ---------------------- ---------- ----------- ------------
 
@@ -111,4 +103,4 @@ Boog, 2013. Effect of the Aeration Scheme on the Treatment Performance of Intens
 
 Boog, J., Kalbacher, T., Nivala, J., Forquet, N., van Afferden, M., Müller, R.A., 2019. Modeling the relationship of aeration, oxygen transfer and treatment performance in aerated horizontal flow treatment wetlands. Water Res. 157 , 321 - 334
 
-Nivala, J., Headley, T., Wallace, S., Bernhard, K., Brix, H., van Afferden, M., Müller, R.A, 2013. Comparative analysis of constructed wetlands: the design and construction of the ecotechnology research facility in Langenreichenbach, Germany. Ecol. Eng., 61, 527-543
+Nivala, J., Headley, T., Wallace, S., Bernhard, K., Brix, H., van Afferden, M., Müller, R.A, 2013. Comparive analysis of constructed wetlands: the design and construction of the ecotechnology research facility in Langenreichenbach, Germany. Ecol. Eng., 61, 527-543

web/content/docs/benchmarks/reactive-transport/kineticreactant_allascomponents/KineticReactant2.md

@@ -16,9 +16,9 @@ title = "Solute transport including kinetic reaction"
 ## Overview
 
 This scenario describes the transport of two solutes (Snythetica and Syntheticb) through a saturated media.
-Both solutes react to Productd according to $\text{Productd}=\text{Synthetica}+0.5~\text{Syntheticb}$.
-The speed of the reaction is described with a first--order relationship $\frac{dc}{dt}=U(\frac{c_{\text{Synthetica}}}{K_m+c_{\text{Syntheticb}}})$.
-The coupling of OGS-6 and IPhreeqc used for simulation requires to simulate the transport of H^+^--ions, additionally.
+Both solutes react to Productd according to $\text{Product d}=\text{Synthetic a}+0.5~\text{Synthetic b}$.
+The speed of the reaction is described with a first--order relationship $\frac{dc}{dt}=U(\frac{c_{\text{Synthetic a}}}{K_m+c_{\text{Synthetic b}}})$.
+The coupling of OGS-6 and IPhreeqc used for simulation requires to simulate the transport of $H^+$--ions, additionally.
 This is required to adjust the compulsory charge balance computation executed by Phreeqc.
 The solution by OGS-6--IPhreeqc will be compared to the solution by a coupling of OGS-5--IPhreeqc.
 
@@ -28,9 +28,9 @@ The solution by OGS-6--IPhreeqc will be compared to the solution by a coupling o
 The 1d--model domain is 0.5 m long and discretized into 200 line elements.
 The domain is saturated at start--up ($p(t=0)=$ 1.0e-5 Pa).
 A constant pressure is defined at the left side boundary ($g_{D,\text{upstream}}^p$) and a Neumann BC for the water mass out-flux at the right side ($g_{N,\text{downstream}}^p$).
-Both solutes, Synthetica and Syntheticb are present at simulation start--up at concentrations of $c_{\text{Synthetica}}(t=0)=c_{\text{Syntheticb}}(t=0)= 0.5$ mol kg^-1^~water~, the influent concentration is 0.5 mol kg^-1^~water~ as well.
-Productd is not present at start--up ($c_{\text{Productd}}(t=0)=0$); neither in the influent.
-The initial concentration of $\text{H}^+$--ions is 1.0e-7 mol kg^-1^~water~; the concentration at the influent point is the same.
+Both solutes, Synthetic a and Synthetic b are present at simulation start--up at concentrations of $c_{\text{Synthetic a}}(t=0)=c_{\text{Synthetic b}}(t=0)= 0.5~\textrm{mol kg}^{-1}~\textrm{water}$, the influent concentration is $0.5~\textrm{mol kg}^{-1}~\textrm{water}$ as well.
+Product d is not present at start--up ($c_{\text{Productd}}(t=0)=0$); neither in the influent.
+The initial concentration of $\text{H}^+$--ions is $1.0e\textrm{-}7~\textrm{mol kg}^{-1}~\textrm{water}$; the concentration at the influent point is the same.
 Respective material properties, initial and boundary conditions are listed in the tables below.
 
 **2d scenario:**
@@ -44,16 +44,16 @@ The horizontal domain is 0.5 m in x and 0.5 m in y direction, and,  discretized
 |Parameter | Description | Value | Unit |
 |:-------- | :---------- | :---- | :--- |
 | $\phi$   | Porosity    | 1.0   |  |
-| $\kappa$ | Permeability | 1.157e-12 | m^2^ |
+| $\kappa$ | Permeability | 1.157e-12 | $\textrm{m}^2$ |
 | $S$   | Storage | 0.0     |  |
 | $a_L$ | long. Dispersion length | 0.0   | m |
 | $a_T$ | transv. Dispersion length | 0.0  | m |
-| $\rho_w$ | Fluid density | 1.0e+3 | kg m^-3^ |
+| $\rho_w$ | Fluid density | 1.0e+3 | $\textrm{kg m}^{-3}$ |
 | $\mu_w$ | Fluid viscosity | 1.0e-3 | Pa s |
-| $D_{\text{H}^+}$ | Diffusion coef. for $\text{H}^+$ | 1.0e-7     | m^2^ s |
-| $D_{solutes}$ | Diffusion coef. for Synthetica, Syntheticb and Productd | 1.0e-12 | m^2^ s |
-| $U$ | Reaction speed constant | 1.0e-3 | h^-1^ |
-| $K_m$ | Half--saturation constant | 10 | mol kg^-1^~water~ |
+| $D_{\text{H}^+}$ | Diffusion coef. for $\text{H}^+$ | 1.0e-7     | m$^2$ s |
+| $D_{solutes}$ | Diffusion coef. for Synthetica, Syntheticb and Productd | 1.0e-12 | m$^2$ s |
+| $U$ | Reaction speed constant | 1.0e-3 | h$^{-1}$ |
+| $K_m$ | Half--saturation constant | 10 | mol kg$^{-1}$ water |
 
 Table: Media, material and component properties
 
@@ -62,25 +62,25 @@ Table: Media, material and component properties
 | Parameter | Description | Value | Unit |
 |:--------- | :---------- | :---- | :--- |
 | $p(t=0)$  | Initial pressure | 1.0e+5  | Pa |
-| $g_{N,downstream}^p$ | Water outflow mass flux | -1.685e-02 | mol kg^-1^~water~ |
+| $g_{N,downstream}^p$ | Water outflow mass flux | -1.685e-02 | mol kg$^{-1}$ water |
 | $g_{D,upstream}^p$ | Pressure at inlet | 1.0e+5 | Pa |
-| $c_{Synthetica}(t=0)$  | Initial concentration of Synthetica | 0.5     | mol kg^-1^~water~ |
-| $c_{Syntheticb}(t=0)$  | Initial concentration of Syntheticb | 0.5     | mol kg^-1^~water~ |
-| $c_{Productd}(t=0)$  | Initial concentration of Productd | 0     | mol kg^-1^~water~ |
-| $c_{\text{H}^+}(t=0)$  | Initial concentration of $\text{H}^+$ | 1.0e-7     |  mol kg^-1^~water~ |
-|  $g_{D,upstream}^{Synthetica_c}$ | Concentration of Synthetica |  0.5 | mol kg^-1^~water~ |
-|  $g_{D,upstream}^{Syntheticb_c}$ | Concentration of Syntheticb |  0.5 | mol kg^-1^~water~ |
-|  $g_{D,upstream}^{Productd}$ | Concentration of Productd |  0.0 | mol kg^-1^~water~ |
-|  $g_{D,upstream}^{\text{H}^+}$ | Concentration of $\text{H}^+$ |  1.0e-7 | mol kg^-1^~water~ |
+| $c_{Synthetica}(t=0)$  | Initial concentration of Synthetica | 0.5     | mol kg$^{-1}$ water |
+| $c_{Syntheticb}(t=0)$  | Initial concentration of Syntheticb | 0.5     | mol kg$^{-1}$ water |
+| $c_{Productd}(t=0)$  | Initial concentration of Productd | 0     | mol kg${^-1}$ water |
+| $c_{\text{H}^+}(t=0)$  | Initial concentration of $\text{H}^+$ | 1.0e-7     |  mol kg$^{-1}$ water |
+|  $g_{D,upstream}^{Synthetica_c}$ | Concentration of Synthetica |  0.5 | mol kg$^{-1}$ water |
+|  $g_{D,upstream}^{Syntheticb_c}$ | Concentration of Syntheticb |  0.5 | mol kg$^{-1}$ water |
+|  $g_{D,upstream}^{Productd}$ | Concentration of Productd |  0.0 | mol kg$^{-1}$ water |
+|  $g_{D,upstream}^{\text{H}^+}$ | Concentration of $\text{H}^+$ |  1.0e-7 | mol kg$^{-1}$ water |
 
 Table: Initial and boundary conditions
 
 ## Results
 
-The kinetic reaction results in the expected decline of the concentration of Synthetica and Syntheticb, which is super-positioned by the influx of these two educts through the left side.
-By contrast, the concentration of Productd increases in the domain.
-Over time, opposed concentration fronts for educts and Productd evolve.
+The kinetic reaction results in the expected decline of the concentration of Synthetic a and Synthetic b, which is super-positioned by the influx of these two educts through the left side.
+By contrast, the concentration of Product d increases in the domain.
+Over time, opposed concentration fronts for educts and Product d evolve.
 Both, OGS-6 and OGS-5 simulations yield the same results in the 1d as well as 2d scenario.
-For instance, the difference between the OGS-6 and the OGS-5 computation for the concentration of Productd expressed as root mean squared error is 1.76e-7 mol kg^-1^~water~ (over all time steps and mesh nodes, 1d scenario); the corresponding median absolute error is 1.0e-7 mol kg^-1^~water~.
+For instance, the difference between the OGS-6 and the OGS-5 computation for the concentration of Product d expressed as root mean squared error is 1.76e-7 mol kg$^{-1}$ water (over all time steps and mesh nodes, 1d scenario); the corresponding median absolute error is 1.0e-7 mol kg$^{-1}$ water.
 This verifies the implementation of OGS-6--IPhreeqc.
 {{< img src="../KineticReactant2.gif" title="Simulated component concentrations over domain length for different time steps (1d scenario) .">}}

web/content/docs/benchmarks/reactive-transport/wetland/Wetland.md

@@ -23,7 +23,7 @@ The scenario presented here is a modification of a case already described in Boo
 The experimental system consists of a basin of 4.7 m in length, 1.2 m in width and 0.9 m in depth (Figure 1) filled with gravel and saturated with water.
 The domestic wastewater enters the system at a constant flow rate on the left side and leaves it via an overflow at the right side.
 
-The coupling of OGS-6 and IPhreeqc used in the simulation requires to include the transport of H^+^--ions to adjust the compulsory charge balance computated by Phreeqc.
+The coupling of OGS-6 and IPhreeqc used in the simulation requires to include the transport of H$^+$--ions to adjust the compulsory charge balance computated by Phreeqc.
 The results obtained by OGS-6--IPhreeqc will be compared to the ones of OGS-5--IPhreeqc.
 
 ## Problem Description
@@ -36,9 +36,9 @@ For the water efflux, a constant pressure is defined as boundary ($g_{D,\text{ou
 
 ![Schematic representation of the experimental system and 1D model domain used in the simulation. The influent and effluent zone in the 1D model are represented by solid lines.](../Wetland_domain.png)
 
-The microbiological processes are modeled by a complex network of kinetic reactions based on the Constructed Wetland Model No. 1 (CWM1) described in Langergraber, (2009).
+The microbiological processes are modeled by a complex network of kinetic reactions based on the Constructed Wetland Model No. 1 (CWM1) described in Langergraber (2009).
 The network includes dissolved oxygen ($So$) and nine different soluble and particulated components ("pollutants") that some of them can be metabolized by six bacterial groups resulting in 17 kinetic reactions (Figure 2).
-A "clean" system is assumed at start-up in the basin, therefore, initial concentrations of all components (oxygen + "pollutants") and bacteria are set to 1.0e-4 and 1.0e-3 mg L^-1^, respectively.
+A "clean" system is assumed at start-up in the basin, therefore, initial concentrations of all components (oxygen + "pollutants") and bacteria are set to 1.0e-4 and 1.0e-3 mg L$^-1$, respectively.
 For the wastewater components ("pollutants" and oxygen) entering the system, time-dependent Dirichlet BC are defined at the influx point.
 Respective material properties, initial and boundary conditions are listed in Table 1--2.
 
@@ -50,14 +50,14 @@ Respective material properties, initial and boundary conditions are listed in Ta
 |:-------- | :---------- | :---- | :--- |
 | Influent & effluent zone |||
 | $\phi$   | Porosity    | 0.38  | - |
-| $\kappa$ | Permeability | 1.0e-7 | $m^2$ |
+| $\kappa$ | Permeability | 1.0e-7 | m$^2$ |
 | $S$   | Storage | 0.0     | - |
-| $a_L$ | long. Dispersion length | 0.45   | $m$ |
+| $a_L$ | long. Dispersion length | 0.45   | m |
 | Treatment zone |||
 | $\phi$   | Porosity    | 0.38  | - |
-| $\kappa$ | Permeability | 1.0e-8 | $m^2$ |
+| $\kappa$ | Permeability | 1.0e-8 | m$^2$ |
 | $S$   | Storage | 0.0     | - |
-| $a_L$ | long. Dispersion length | 0.40   | $m$ |
+| $a_L$ | long. Dispersion length | 0.40   | m |
 
 -----------------------------------------
 
@@ -67,14 +67,14 @@ Table 1: Media, material and component properties
 
 | Parameter | Description | Value | Unit |
 |:--------- | :---------- | :---- | :--- |
-| $p(t=0)$  | Initial pressure | 8829  | $Pa$ |
-| $g_{N,in}^p$ | Water influx | 5.555e-3 | kg s^-1^ |
+| $p(t=0)$  | Initial pressure | 8829  | Pa |
+| $g_{N,in}^p$ | Water influx | 5.555e-3 | kg s$^{-1}$ |
 | $g_{D,out}^p$ | Pressure at outlet | 8829 | Pa |
-| $c_{components}(t=0)$  | Initial component concentrations | 1.0e-4     | g kg^-1^~water~ |
-| $c_{bacteria}(t=0)$  | Initial bacteria concentrations | 1.0e-3  | g kg^-1^~water~ |
-| $c_{\text{H}^+}(t=0)$  | Initial concentration of $\text{H}^+$ | 1.0e-7     |  mol kg^-1^~water~ |
-|  $g_{D,in}^{components_c}$ | Influent component concentrations |  $f(t)$ | g kg^-1^~water~ |
-|  $g_{D,in}^{\text{H}^+}$ | Influent concentration of $\text{H}^+$ |  1.0e-7 | mol kg^-1^~water~ |
+| $c_{components}(t=0)$  | Initial component concentrations | 1.0e-4     | g kg$^{-1}$ water |
+| $c_{bacteria}(t=0)$  | Initial bacteria concentrations | 1.0e-3  | g kg${^-1}$ water |
+| $c_{\text{H}^+}(t=0)$  | Initial concentration of $\text{H}^+$ | 1.0e-7     |  mol kg${^-1}$ water |
+|  $g_{D,in}^{components_c}$ | Influent component concentrations |  $f(t)$ | g kg${^-1}$ water |
+|  $g_{D,in}^{\text{H}^+}$ | Influent concentration of $\text{H}^+$ |  1.0e-7 | mol kg${^-1}$ water |
 
 -----------------------------------------
 
@@ -94,7 +94,7 @@ After 20 days (1.7286e+6 seconds) the microbial reaction network in the wetland
 The biochemical reactions are now governing the system behaviour.
 
 Both, OGS-6 and OGS-5 simulations yield the same results.
-For instance, the difference between the OGS-6 and the OGS-5 computation for the concentration of $S_A$  expressed as root mean squared error is 1.11e-4 g L^-1^ (over all time steps and mesh nodes); the corresponding relative mean squared error is 0.37%.
+For instance, the difference between the OGS-6 and the OGS-5 computation for the concentration of $S_A$  expressed as root mean squared error is 1.11e-4 g L$^{-1}$ (over all time steps and mesh nodes); the corresponding relative mean squared error is 0.37%.
 The relatively high error may be associated with the missing transport or charge in the OGS-6 simulation, which affects computations by Phreeqc.
 Please note that due to the long computation time of the simulation (~13 h), the corresponding test (Wetland_1d.prj) is reduced to the first four time steps (28800 s).