To calculate the annual Global Warming Potential (GWP) of the released greenhouse gases, we must account for the chemical transformation of methane during flaring and the individual GWPs of the resulting gases.

1. Account for Flaring of Methane ($CH_4$)

Flaring involves the combustion of methane, which converts it into carbon dioxide ($CO_2$) and water vapor ($H_2O$). The chemical reaction is: 
$CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O$

We use the molecular weights to find the mass of $CO_2$ produced from flaring:

• Molecular weight of $CH_4 \approx 16 \text{ g/mol}$
• Molecular weight of $CO_2 \approx 44 \text{ g/mol}$

Mass of $CO_2$ generated from $0.5 \text{ kg/day}$ of $CH_4$: 
$\text{Mass}_{CO_2, \text{ flared}} = 0.5 \text{ kg} \times \left( \frac{44}{16} \right) = 0.5 \times 2.75 = 1.375 \text{ kg/day}$

2. Calculate Total Daily $CO_2$ Equivalent Emissions

After flaring, the methane is gone, but we must add the newly created $CO_2$ to the existing $CO_2$ emissions. Then, we convert the $N_2O$ emissions into their $CO_2$ equivalents using its GWP.

• Total $CO_2$ released: $5 \text{ kg (original)} + 1.375 \text{ kg (from flare)} = 6.375 \text{ kg/day}$
• $N_2O$ in $CO_2$ equivalent: $0.1 \text{ kg} \times 310 = 31 \text{ kg/day}$

Total daily GWP: 
$\text{Total daily } CO_2\text{-eq} = 6.375 + 31 = 37.375 \text{ kg/day}$

3. Calculate Annual Emissions

To find the annual GWP, multiply the daily total by the number of days in a year (365): 
$\text{Annual GWP} = 37.375 \text{ kg/day} \times 365 \text{ days/year}$ 
$\text{Annual GWP} = 13641.875 \text{ kg } CO_2\text{-eq}$

Final Answer:

Rounding to the nearest integer, the annual GWP is 13642 kg $CO_2$ equivalent.