# Heterogeneous Saturated Mass Transport

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## Overview

This example involves the usage of a cell-based heterogeneous parameterization for the parameter “intrinsic permeability”. We show a 2D and a 3D setup and compared the results to those from OGS5; the original examples are described in chapter “5.2 Groundwater flow in a heterogeneous medium” in Kolditz et al. (2012).

We extended the setup to show mass transport in the heterogeneous medium for testing.

## Problem description

The setups are steady-state for flow, with an extent of a $100$ m x $100$ m horizontal plane for the 2D setup and a $100$ m x $100$ m x $50$ m cube for the 3D setup. Mesh elements have side lengths of $1$ m. The initial conditions are hydrostatic and concentration $c=0$. The boundary conditions are translated into equivalent hydrostatic pressure values from hydraulic heads $h_{left}=10$ m and $h_{right}=9$ m and for concentration $c_{left}=1$, $c_{right}=0$ for left and right sides, respectively. All other sides are defined as no-flow (Zero-Neumann).

Porosity is $0.01$, specific storage is $0$, fluid density is $1000$ kg$\cdot$m$^3$, dynamic viscosity is $10^{-3}$ Pa$\cdot$s, molecular diffusion coefficient is $2\cdot 10^{-9}$ m$\cdot$s$^{-2}$, dispersivities are longitudinal $1$ m and transverse $0.1$ m. The heterogeneous parameter fields of intrinsic permeability are shown in the figures below; creation of the tensor field is documented here.

### Model results

The comparison of velocity and hydraulic head are shown below. The numerical results of OGS6 fit very well to the OGS5 results with relative differences for velocity in the order of ca $10^{-2}$ and for hydraulic head in $10^{-4}$. Relative differences of 2D simulation results between OGS5 and OGS6. On the top left figure, white lines represent hydraulic head values of OGS5, blue lines of OGS6. Relative differences of 3D simulation results between OGS5 and OGS6. On the top left figure, grey dots represent hydraulic head values of OGS6.

The mass transport simulation results (figures below) show an expected heterogeneous mass front moving through the domain. Concentration distribution at simulation time $1e8$ s for the 2D setup. Concentration distribution at simulation time $1e8$ s for the 3D setup.

## Literature

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.

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