# Solute transport including kinetic reaction

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## 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. 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.

## Problem Description

1d scenario: 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. Respective material properties, initial and boundary conditions are listed in the tables below.

2d scenario: The 2d–scenario only differs in the domain geometry and assignement of the boundary conditions. The horizontal domain is 0.5 m in x and 0.5 m in y direction, and, discretized into 10374 quadratic elemtents with an edge size of 0.0025 m. Parameter Description Value Unit
$\phi$ Porosity 1.0
$\kappa$ Permeability 1.157e-12 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^
$\mu_w$ 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~

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~
$g_{D,upstream}^p$ 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~

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. 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~. This verifies the implementation of OGS-6–IPhreeqc. Simulated component concentrations over domain length for different time steps (1d scenario) .

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