Water Quality Model

1. Overview

HydroPol2D simulates pollutant transport using a build-up and wash-off formulation coupled to the hydrodynamic model.

The model tracks:

• pollutant mass stored on the surface,
• pollutant removal due to flow (wash-off),
• redistribution between neighboring cells,
• export through outlet boundaries.

The governing variable is:

$$B_t$$

where:

• $B_t$ = pollutant mass per cell $[\mathrm{M}]$

The model is fully mass conservative and dynamically coupled to flow routing.

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2. Pollutant storage

Each cell stores a pollutant mass:

$$B_t \quad [\mathrm{kg}]$$

The corresponding areal concentration is:

$$b = \frac{B_t}{A_{\mathrm{cell}}} \quad [\mathrm{M}\,\mathrm{L}^{-2}]$$

where:

• $A_{\mathrm{cell}}$ = cell area $[\mathrm{L}^2]$

Minimum and maximum thresholds are defined:

• $B_{\min}$ → minimum active pollutant mass
• $B_{\max}$ → saturation limit

Below a threshold:

$$b < b_{\min} \Rightarrow \text{no wash-off}$$

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3. Hydrodynamic coupling

The water quality model uses hydrodynamic outflows:

$$q_{\mathrm{out}} = \{ q_{\mathrm{left}}, q_{\mathrm{right}}, q_{\mathrm{up}}, q_{\mathrm{down}}, q_{\mathrm{outlet}} \}$$

with units:

$$[\mathrm{L}\,\mathrm{T}^{-1}] \quad (\mathrm{mm/h})$$

These are converted internally to volumetric flow:

$$Q = \frac{q_{\mathrm{out}}}{1000 \cdot 3600} \cdot A_{\mathrm{cell}} \quad [\mathrm{L}^3\,\mathrm{T}^{-1}]$$

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4. Wash-off formulation

Two wash-off formulations are implemented.

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4.1 Rating-curve wash-off model

When flag_wq_model = 1, the wash-off is:

$$W_{\mathrm{out}} = C_3 \, Q^{C_4} \, f(B_t)$$

where:

• $W_{\mathrm{out}}$ = pollutant mass flux $[\mathrm{M}\,\mathrm{T}^{-1}]$
• $C_3, C_4$ = empirical coefficients
• $Q$ = flow rate $[\mathrm{L}^3\,\mathrm{T}^{-1}]$

The storage correction factor is:

$$f(B_t) = 1 + \max(B_t - B_r, 0)$$

where:

• $B_r$ = residual pollutant mass threshold $[\mathrm{M}]$

This formulation ensures that:

• wash-off increases with flow,
• wash-off increases with available pollutant mass.

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4.2 Mass-based wash-off model

When flag_wq_model = 0, the formulation becomes:

$$W_{\mathrm{out}} = C_3 \, Q^{C_4} \, B_t$$

This formulation assumes:

• wash-off is proportional to available pollutant mass,
• transport increases with flow magnitude.

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5. Directional pollutant fluxes

Pollutant fluxes are computed for each direction:

• left
• right
• up
• down
• outlet

The total pollutant outflow from a cell is:

$$W_{\mathrm{out,tot}} = \sum_k W_{\mathrm{out},k}$$

Incoming pollutant fluxes are computed from neighboring cells:

$$W_{\mathrm{in}} = \sum_k W_{\mathrm{in},k}$$

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6. Net pollutant balance

The net pollutant rate is:

$$dW = W_{\mathrm{out,tot}} - W_{\mathrm{in}} \quad [\mathrm{M}\,\mathrm{T}^{-1}]$$

The mass balance equation is:

$$B_t^{t+\Delta t} = B_t^t - dW \cdot \Delta t$$

This formulation accounts for:

• pollutant removal due to wash-off,
• pollutant redistribution across the grid.

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7. Adaptive time stepping (critical feature)

To ensure stability and avoid negative mass, HydroPol2D computes a minimum admissible time step:

$$\Delta t_{\min} = \min \left( \frac{B_t}{|dW|} \right)$$

where:

• only cells losing mass are considered,
• very small values are excluded.

If:

$$\Delta t_{\min} < \Delta t$$

the model splits the time step into sub-steps:

$$\Delta t = \sum \Delta t_i$$

and updates pollutant mass iteratively:

$$B_t^{i+1} = B_t^i - dW \cdot \Delta t_i$$

This ensures:

• no negative pollutant mass,
• numerical stability,
• accurate mass conservation.

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8. Mass conservation and corrections

After updating:

• negative values are removed:

$$B_t = \max(B_t, 0)$$

• rounding is applied to avoid numerical noise.

Mass lost due to numerical corrections is tracked:

$$\mathrm{mass}_{\mathrm{lost}}$$

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9. Pollutant concentration

Pollutant concentration is computed as:

$$C = \frac{W_{\mathrm{out,tot}}}{Q_{\mathrm{tot}} \cdot A_{\mathrm{cell}}} \cdot 10^6 \quad [\mathrm{mg/L}]$$

where:

• $Q_{\mathrm{tot}}$ = total water outflow $[\mathrm{L}\,\mathrm{T}^{-1}]$

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10. Outlet concentration

The outlet concentration is computed as:

$$C_{\mathrm{out}} = \frac{ \sum W_{\mathrm{out, outlet}} }{ \sum Q_{\mathrm{outlet}} \cdot A_{\mathrm{cell}} } \cdot 1000 \quad [\mathrm{mg/L}]$$

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11. Total washed mass

The cumulative washed pollutant mass is:

$$M_{\mathrm{washed}} = \sum W_{\mathrm{out,tot}} \cdot \Delta t \quad [\mathrm{kg}]$$

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12. Numerical constraints

The model enforces:

• minimum pollutant threshold:

$$b < b_{\min} \Rightarrow W_{\mathrm{out}} = 0$$

• minimum flow threshold:

$$q_{\mathrm{out}} \ge 10 \quad [\mathrm{mm/h}]$$

to avoid numerical instability.

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Summary

The HydroPol2D water quality model:

• tracks pollutant mass per cell,
• computes wash-off as a function of flow and storage,
• redistributes pollutants across the domain,
• ensures strict mass conservation,
• uses adaptive time stepping to maintain stability.

It provides a robust and flexible framework for simulating pollutant transport coupled with hydrologic and hydrodynamic processes.