[Web] Updated description.

web/content/docs/benchmarks/reactive-transport/calcite.pandoc

@@ -2,8 +2,8 @@
 author = "Vanessa Montoya, Renchao Lu"
 weight = 142
 project = "Parabolic/ComponentTransport/ReactiveTransport/EquilibriumPhase/calcite.prj"
-date = "2017-09-07T14:41:09+01:00"
-title = "Advective flow and precipitation/dissolution equilibrium reactions"
+date = "2019-07-08T14:41:09+01:00"
+title = "Precipitation/dissolution equilibrium reactions in a saturated column"
 
 [menu]
   [menu.benchmarks]
@@ -16,22 +16,60 @@ title = "Advective flow and precipitation/dissolution equilibrium reactions"
 
 ## Overview
 
-This benchmark was originally described in Kolditz et al. (2012) as a reactive transport benchmark including precipitation and dissolution reactions along a saturated column of calcite which is fluxed with a diluted solution of $\mathrm{MgCl}_2$. Three different codes (OGS.5-ChemApp, OGS.5-iPHREEQC and OGS.5-GEMS) were used in that exercise.
-Kolditz et al. (2012) considered that all reactions occurred at thermodynamic equilibrium. Later on, in He et al. (2015) and Jang et al. (2018), the incorporation of kinetic rates for reaction precipitations in OGS.5-iPHREEQC was investigated and compared with the modelling results obtained with OGS.5-ChemApp and PHREEQC.
-Very briefly, the studied system consists on a column of 0.5 m, initially containing crushed calcite ($\mathrm{CaCO}_3$(s)), which is continuously fluxed with a 1 millimolar (mmol/L) magnesium chloride ($\mathrm{MgCl}_2$) solution at pH = 7 and 25°C during 350 min (~ 5.8h) at a constant flow rate of 0.18 L/min.
-Previous to the experiment, the column had been packed giving a mean bulk density and porosity of 1.80 Kg/L and 0.32, respectively. The column was saturated and equilibrated by injecting pure water giving a saturated solution with respect calcite and pH = 9.9.
+This benchmark was firstly described in Kolditz *et al.* (2012) as a reactive transport benchmark including precipitation and dissolution reactions along a saturated column of calcite which is fluxed with a diluted solution of MgCl$_2$. Three different codes (OGS-5#ChemApp, OGS-5#IPhreeqc and OGS-5#GEMS) were used in that exercise.
 
+Kolditz *et al.* (2012) considered that all reactions occurred at thermodynamic equilibrium. Later on, in He *et al.* (2015) and Jang *et al.* (2018), the incorporation of kinetic rates for reaction precipitations in OGS-5#IPhreeqc was investigated and compared with the modelling results obtained with OGS-5#ChemApp and Phreeqc.
+
+Very briefly, the studied system consists on a column of 0.5 m, initially containing crushed calcite (CaCO$_3$(s)), which is continuously fluxed with a 1 millimolar (mmol/L) magnesium chloride (MgCl$_2$) solution at pH = 7 and 25°C during 350 min (~ 5.8h) at a constant flow rate of 0.18 L (or a Darcy velocity (q) of $3\times10^{-6}$ m/s, considering a cross-sectional area of 1 m$^2$).
+
+Previously, the column had been packed giving a mean bulk density and porosity of 1.80 kg/L and 0.32, respectively. Additionally, the column was saturated and equilibrated by injecting pure water giving a saturated solution with respect calcite and pH = 9.9.
+
+## Numerical approach
+
+Reactive component$^{(*)}$ transport processes in saturated porous media illustrated in this benchmark have been possible by integrating IPhreeqc v.3.5.0 module (Parkhurst and Appelo,2013) in OGS-6. The operator-splitting (OS) model formulation (Strang, 1968) with a sequential non-iterative approach (SNIA) has been used for the implementation.
+
+In a first step, the mass balance equation describing the component transport within the fluid is solved by a parabolic operator (see ../hydro-component/HC-Process.pdf). Then, in a second step, the nonlinear algebraic equations describing the equilibrium chemical reactions between the aqueous components and the solid (i.e. mass action law) are solved with a modified Newton-Raphson method implemented in IPhreeqc.
+
+($^*$*Note: component denotes a chemical species that belongs to the minimum number of independent chemical species necessary to completely describe their mass*).
+
+In this way, a conservative mass transport equation for component $i$ is solved
+$$
+\begin{equation}
+\frac{\partial C_i}{\partial t} = - \nabla\left(v C_i\right) + \nabla\left(D_i \nabla C_i \right) + Q_i,
+\end{equation}
+$$
+$$
+\begin{equation}
+\frac{\partial S C_i}{\partial t} = \Gamma_i \left(C_i ... C_m\right),
+\end{equation}
+$$
+with $C_i$ (mol m$^{-3}$) standing for the molar concentration of component $i$, $v$ (m s$^{-1}$) for the pore velocity in the fluid phase, $D_h$ (m$^2$ s$^{-1}$) for the dispersion tensor of component $i$, $Q_i$ (mol m$^{-3}$ s$^{-1}$) for a source/sink term, and $\Gamma_i$ (C$_1$ ...C$_m$) (mol m$^{-3}$ s$^{-1}$) being a source/sink term for component $i$ due to chemical reactions with $m$ other components. The Scheidegger dispersion tensor is implemented in two dimensions as
+$$
+\begin{equation}
+D_{kl} = \alpha_T |v| \delta_{kl} + \left( \alpha_L - \alpha_T \right) \frac{v_k v_l}{|v|} + D_e,
+\end{equation}
+$$
+where $\alpha_L$ (m) and $\alpha_T$ (m) are the longitudinal and transversal dispersion length, respectively. $\delta_{kl}$ (–) is the Kronecker symbol, $v_{k,l}$ (m s$^{-1}$) is the fluid pore velocity in direction $k$,$l$, and $D_e$ (m$^2$ s$^{-1}$) is the molecular diffusion coefficient.
 
 ## Model setup
 
-The operator-splitting (OS) model formulation (Strang, 1968) with a sequential non-iterative approach (SNIA) has been used to implement the reactive component transport processes in saturated porous media illustrated in this benchmark. In a first step, the mass balance equations describing each component transport present in the fluid is solved according to
+A one-dimensional (1D) model domain of 0.5 m discretized into 100 uniform elements has been selected for the spatial discretization of the system. Dirichlet (constant concentration) and Neumannn (no flux) boundary condition are defined for the upstream inflow and the downstream, respectively. A longitudinal dispersivity of 0.0067 m and a time step size of 100 s have been taken into account in the simulation. See Figure below:
+
+{{< img src="../calcite-Figures/Scheme.png" title="Schematic representation of the model setup and parameters.">}}
 
+Thermodynamic data for hydrolysis, aqueous speciation, and dissolution/precipitation reactions between Mg, Ca, Cl, and carbonate were selected from version 12/07 of the PSI/NAGRA chemical thermodynamic database (Thoenen *et al.* 2014). Although several other minerals containing Mg and Ca were available in the PSI/NAGRA database (*i.e.* magnesite), only two solids were allowed to precipitate or dissolve in the studied system (calcite and dolomite (CaMg(CO$_3$)$_2$)).
 
-### Model results
+## Model results
+
+A comparison of the results obtained with OGS-6#IPhreeqc and OGS-5#IPhreeqc at the end of the simulation is shown in the Figures below.
+
+At the simulated time, it can be clearly seen that the MgCl$_2$ solution front has penetrated ~0.3m of the column resulting in the dissolution of calcite and dolomite precipitation. Total aqueous concentration and solid profiles obtained of OGS-6 are in good agreement with those of OGS-5. The absolute error in terms of component concentrations is $2.15\times10^{-5}$ (Cl), $1.13\times10^{-5}$ (Mg), and $4.57\times10^{-6}$ (Ca). Additionally, pH profiles calculated with both codes are in good agreement.
+
+{{< img src="../calcite-Figures/ResultComparison.png" title="Total aqueous concentration and solid profiles obtained with OGS-6#IPhreeqc (empty triangle symbol) and OGS-5#IPhreeqc (empty circle symbol) at 350 min. (C(4) = total carbonate)">}}
+
+{{< img src="../calcite-Figures/ResultComparisonPH.png" title="pH value profiles obtained with OGS-6#IPhreeqc (empty triangle symbol) and OGS-5#IPhreeqc (empty circle symbol) at 350 min.">}}
 
-A comparison of the numerical data is shown in the figure below. The numerical results of OGS-6 are in good agreement with those of OGS-5. The maximum number of absolute error in terms of varying element concentrations is $2.15\times10^{-5}$ ($\mathrm{Cl}$), $1.13\times10^{-5}$ ($\mathrm{Mg}$), and $4.57\times10^{-6}$ ($\mathrm{Ca}$).
 
-{{< img src="../ResultComparison.png" width="80" title="Results for numerical (OGS-5 - with empty citrcle marker, OGS-6 - with empty triangle marker) results together with concentration distribution in the domain and mesh resolution for different time steps.">}}
 
 {{< data-link >}}
 
@@ -40,8 +78,10 @@ A comparison of the numerical data is shown in the figure below. The numerical r
 
 He, W., Beyer, C., Fleckenstein, J.H., Jang, E., Kolditz, O., Naumov, D., Kalbacher, T., 2015. A parallelization scheme to simulate reactive transport in the subsurface environment with OGS#IPhreeqc 5.5.7-3.1.2. Geosci. Model Dev. 8 (10), 3333 - 3348
 
-Hummel,W., Berner, U., Curti, E., Thoenen, T., Pearson, F.J., 2002. Nagra/PSI Chemical Thermodynamic Data Base Version 01/01 (Nagra/PSI TDB 01/01) NAPSI_290502.DAT. NAGRA Technical Report 02-16, Wettingen, Switzerland.
-
 Jang, E., Boog, J., He, W., Kalbacher, T., 2018. OpenGeoSys Tutorial. Computational hydrology III: OGS#IPhreeqc coupled reactive transport modeling SpringerBriefs in Earth System Sciences. Springer International Publishing, Cham, 103 pp.
 
 Kolditz, O., Görke, U.-J., Shao, H., Wang, W., 2012. Thermo-Hydro-Mechanical-Chemical Processes in Porous Media: Benchmarks and Examples, Lecture notes in computational science and engineering. Springer. ISBN: 3642271766.
+
+Parkhurst, D.L., Appelo, C.A.J., 2013. Description of Input and Examples for PHREEQC Version 3 - a Computer Program for Speciation, Batch-reaction, One-dimensional Transport, and Inverse Geochemical Calculations.
+
+Thoenen, T., Hummel,W., Berner, U., Curti, E., 2014. The PSI/Nagra Chemical Thermodynamic Data Base 12/07. PSI Report 14-04, Villigen, Switzerland.