watchOS 3.1.3 can be downloaded through the dedicated Apple Watch app on the iPhone by going to General --> Software Update. To install the update, the Apple Watch must have 50 percent battery, it must be placed on the charger, and it must be in range of the iPhone. An iPhone running iOS 10 or later is required to download the new software, but it is available for all Apple Watch models.
No significant changes or feature additions were discovered in watchOS 3.1.3, but presumably, it fixes the 3.1.1 installation issues that were bricking some Apple Watch Series 2 models, allowing customers who did not get a chance to download 3.1.1 to finally get the bug fixes included in that update.
watchOS 3.1.1 included several bug fixes for Messages, notifications, Activity, Maps, Calendar, and more.
There was no watchOS 3.1.2 update, as Apple appears to have gone straight to watchOS 3.1.3.