I am no battery expert, but I can help some.
I am going to assume you are using a lead acid battery since you did not specify what type you are using. Different types of batteries will discharge differently. At 11.5 Vdc you have about 10% of the capacity left in the battery and it is probably going to start getting damaged soon. Looking around the internet you generally do not want to take a normal battery to less than 20% of charge (11.66V). You can find charts for 12V battery voltage vs state of charge charts using google.
I you want your battery to last a specified amount of time you need to find out how much power you are using and how long you want the battery to last. Here is a site that will give you more information about lead acid batteries.
http://all-about-lead-acid-batteries.capnfatz.com/all-about-lead-acid-batteries/lead-acid-battery-fu...
You can look up the differences in output between lead acid and lithium ion. Lithium Ion batteries hold a more constant voltage and then drop off.
