SYDNEY—Forecasters expect heavy rain in parts of Australia’s fire-ravaged southeast in the coming days, offering relief to firefighters, but raising worries that harmful ash could be washed into rivers and contaminate the country’s water supply.

At a dam that provides most of the drinking water for Sydney, the country’s largest city, authorities placed silt curtains and booms to stop ash and other debris from getting into the city’s water system. The area, or catchment, around the dam has burned extensively, and without vegetation…

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