A finite volume, shock-capturing scheme is used to solve the shallow water equations on unstructured triangular meshes. The conditions are characterised by: slow flow velocities (up to 1m/s), long time scale (around 10 days), and large domains (50-100km across). Systematic verification is carried out by comparing numerical with analytical results, and by comparing parameter variation in the numerical scheme with perturbation analysis, and good agreement is found. It is the first time a shock-capturing scheme has been applied to slow flows in Moreton Bay. The scheme is used to simulate transport of a pollutant in Moreton Bay, to the east of the city of Brisbane, Australia. Tidal effects are simulated using a sinusoidal time-dependent boundary condition. An advection equation is solved to model the path of a contaminant that is released in the bay, and the effect of tide and wind on the contaminant is studied. Calibration is done by comparing numerical results with measurements made at a study site in Moreton Bay. It is found that variation in the wind speed and bed friction coefficients changes the solution in the way predicted by the asymptotics. These results vary according to the shape of the bathymetry of the domain: in shallower areas, flow is more subject to shear and hence changes in wind speed or bed friction had a greater effect in adding energy to the system. The results also show that the time-dependent boundary condition reproduces the tidal effects that are found on the Queensland coast, i.e. semi-diurnal with amplitude of about 1 metre, to a reasonable degree. It is also found that the simulated path of a pollutant agrees with field measurements. The computer model means different wind speeds and directions can be tested which allows management decisions to be made about which conditions have the least damaging effect on the area.