Photographs of the region's airport showed planes grounded by flood water and as of Thursday morning, some 250 passengers were stranded.
Emotional images of dramatic rescues in inflatable boats and civilians hoisting people over muddy flood waters have been posted on social media under the hashtag Chennaifloods.
One video showed residents forming a human chain through perilously fast moving flood waters to rescue a stranded pedestrian.
Indian Prime Minister Narendra Modi visited the region on Thursday and said he was "pained" by the devastation he witnessed from the air and on the ground.
Laxman Singh Rathore, director of the Indian Meteorological Department told CNN that the downpours were unlikely to cease this week, with heavy rain forecast for the next 48 hours.
Chennai, a city of nine million, received more 350 mm of rain on Wednesday more than double the amount it usually receives in December, according to CNN Meteorologist Pedram Javaheri.
