D.10 Mass balance - Groundwater
The reporting of both water volume and pollutant mass fluxes into groundwater layers from above allows for subsurface mass balance analysis to be undertaken.
D.10.1 Concept
The concept is to compare the time evolution of total water volume and pollutant mass in a TUFLOW CATCH groundwater layer via two independent methods that use:
- An initial layer volume / mass and flux output timeseries
- Porosity and groundwater level and concentration output timeseries
These are described following.
D.10.1.1 Flux calculations
D.10.1.1.1 Volume
The volume of water \(V^{F,inst}_t\) at time \(t\) in a given groundwater layer can be computed by summing the previous volume \(V^{F,inst}_{t-1}\) and the reported instantaneous volumetric fluxes \(F\) via Equation (D.3):
\[\begin{equation} V^{F,inst}_t = V^{F,inst}_{t-1} + (F^{V,inst}_{t,in} - F^{V,inst}_{t,out}) \times \Delta t \tag{D.3} \end{equation}\]
where
- \(F^{V,inst}_{t,in}\) is the instantaneous volumetric flux reported by TUFLOW CATCH for the considered layer
- \(F^{V,inst}_{t,out}\) is the instantaneous volumetric flux reported by TUFLOW CATCH for the layer below the considered layer (the flux entering the layer below is the same as the flux exiting the layer considered)
- \(\Delta t\) is the output timestep
- \(V^{F,inst}_{t-1}\) is the previous computed volume and is the initial volume when \(t\) equals one
A similar computation for \(V^{F,tint}_t\) applies if time integrated fluxes are used:
\[\begin{equation} V^{F,tint}_t = V^{F,tint}_{t-1} + (F^{V,tint}_{t,in} - F^{V,tint}_{t,out}) \tag{D.4} \end{equation}\]
where
- \(F^{V,tint}_{t,in}\) is the time integrated volumetric flux reported by TUFLOW CATCH for the considered layer
- \(F^{V,tint}_{t,out}\) is the time integrated volumetric flux reported by TUFLOW CATCH for the layer below the considered layer (the flux entering the layer below is the same as the flux exiting the layer considered)
D.10.1.1.2 Mass
The mass of pollutant \(M^{F,inst}_t\) at time \(t\) in a given groundwater layer can be computed by summing the previously computed mass \(M^{F,inst}_{t-1}\) and the reported instantaneous mass fluxes \(F\) via Equation (D.5):
\[\begin{equation} M^{F,inst}_t = M^{F,inst}_{t-1} + (F^{M,inst}_{t,in} - F^{M,inst}_{t,out}) \times \Delta t \tag{D.5} \end{equation}\]
where
- \(F^{M,inst}_{t,in}\) is the instantaneous mass flux reported by TUFLOW CATCH for the considered layer
- \(F^{M,inst}_{t,out}\) is the instantaneous mass flux reported by TUFLOW CATCH for the layer below the considered layer (the flux entering the layer below is the same as the flux exiting the layer considered)
- \(\Delta t\) is the output timestep
- \(M^{F,inst}_{t-1}\) is the previously computed mass and is the initial mass when \(t\) equals one
A similar computation applies if time integrated fluxes are used:
\[\begin{equation} M^{F,tint}_t = M^{F,tint}_{t-1} + (F^{M,tint}_{t,in} - F^{M,tint}_{t,out}) \tag{D.6} \end{equation}\]
where
- \(F^{M,tint}_{t,in}\) is the time integrated mass flux reported by TUFLOW CATCH for the considered layer
- \(F^{M,tint}_{t,out}\) is the time integrated mass flux reported by TUFLOW CATCH for the layer below the considered layer (the flux entering the layer below is the same as the flux exiting the layer considered)
D.10.1.2 Porosity calculations
D.10.1.2.1 Volume
The volume of water \(V^{P}_t\) at time \(t\) in a given groundwater layer can be computed by multiplying the reported groundwater depth, cell area and porosity in each cell and summing across the domain via Equation (D.7):
\[\begin{equation} V^{P}_t = \Delta x^2 \times \sum_{i=1}^{n_{cells}} (GWd)^i_t \times \phi^i \tag{D.7} \end{equation}\]
where
- \(\Delta x\) is the model cell size
- \((GWd)^i_t\) is the reported groundwater depth in cell \(i\) at time \(t\)
- \(\phi^i\) is the soil porosity in cell \(i\)
D.10.1.2.2 Mass
The mass of pollutant \(M^{P}_t\) at time \(t\) in a given groundwater layer can be computed by multiplying the reported groundwater depth, cell area, pollutant concentration and porosity in each cell and summing across the domain via Equation (D.8):
\[\begin{equation} M^{P}_t = \Delta x^2 \times \sum_{i=1}^{n_{cells}} (GWd)^i_t \times C_t^i \times \phi^i \tag{D.8} \end{equation}\]
where
- \(C_t^i\) is the pollutant concentration in cell \(i\) at time \(t\)
D.10.2 Configuration
A TUFLOW CATCH model with 6 groundwater layers was used to undertake the analysis above. It included two sediment fractions. A description of the model is presented in this webinar: Whole of system simulation of catchment water quality treatment devices.
D.10.3 Method
Mass balance was assessed for water volume and one sediment fraction and all six layers. Only one layer’s results are presented here for clarity. Mass balance performance is presented in scatter plots that compare:
- \(V^{F,inst}\) to \(V^{P}\)
- \(V^{F,tint}\) to \(V^{P}\)
- \(M^{F,inst}\) to \(M^{P}\)
- \(M^{F,tint}\) to \(M^{P}\)
Mass balance requires that these scatter plots have all points lying on the 1:1 line, i.e. that computation of flux and porosity based quantities at a given timestep are equal.
D.10.4 Results
D.10.4.1 Volume
The comparison between \(V^{F,inst}\) and \(V^{P}\) is presented in Figure D.13. The correspondence between the two computed quantities is 1:1.
Figure D.13: Groundwater volume in a single layer: comparison of volumes computed from instaneous fluxes and porosity
The comparison between \(V^{F,tint}\) and \(V^{P}\) is presented in Figure D.14. The correspondence between the two computed quantities is 1:1.
Figure D.14: Groundwater volume in a single layer: comparison of volumes computed from time integrated fluxes and porosity
D.10.4.2 Mass
The comparison between \(M^{F,inst}\) and \(M^{P}\) is presented in Figure D.15. The correspondence between the two computed quantities is 1:1.
Figure D.15: Groundwater mass in a single layer: comparison of masses computed from instaneous fluxes and porosity
The comparison between \(M^{F,tint}\) and \(M^{P}\) is presented in Figure D.16. The correspondence between the two computed quantities is 1:1.
Figure D.16: Groundwater mass in a single layer: comparison of masses computed from time integrated fluxes and porosity
