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Operation Stack is likely to continue until Sunday at the earliest, says Kent Police.
Junction 9 and 10 coastbound are now closed to allow a traffic backlog to clear.
Freight traffic must join queues at junction 8.
This has caused traffic chaos around Dover with lorries blocking Jubilee Way and Duke of Yorks roundabout and the A258 to Deal.
The port is expected to have a busy weekend and bad weather conditions.
It remains open for business with both P&O Ferries and DFDS Seaways operating a full service.
Tourist traffic heading for the Port of Dover is, however, strongly advised to use the A2/M2 route, but please still allow plenty of time for your journey.
A spokesman for the port said: "The Port and the ferry operators are together working hard to serve all customers travelling to and from the continent. We apologise for any delays or inconvenience as a result of traffic congestion."
This comes after there has been unrest among MyFerryLink workers at the port of Calais who last week blockaded the port.
The Transport Ministry in France has now unveiled proposals for Eurotunnel to establish a ferry charter contract that will save 150 jobs. DFDS will also hire 230 MyFerryLink workers and the ministry said it would seek local employment options for remaining workers within three months, according to Reuters.