This abandoned railway was once the Central Indiana Railroad, or "The Midland Route" — basically a stepchild of the Pennsylvania Railroad (Panhandle) and New York Central (Big Four) for much of its life. It once ran from Brazil to Muncie, Indiana with plans to extend to Paris, Illinois in order to connect with St. Louis that never materialized. The line saw much financial hardship and was deemed unwanted by its parent companies who jointly owned the property. While intended as a bypass around Indianapolis, studies deemed the track unsatisfactory and it wasn't long before abandonment was started. After struggles, the line was finally left intact from Lebanon to Anderson. After the inception of Penn Central Railroad, and then Conrail later, the line was absorbed into the parent companies and abandoned except for the Anderson-to-Lapel segment which is still used today by shortline Central Indiana and Western Railroad.
Thanks to James Norwood for contributing information.
The section from Waveland to Lebanon was abandoned before 1954, the rest was abandoned in the 1980s. Large bridges remain in the Westfield to Noblesville section, part of which is to become a rail-trail.
Much work to do with the large bridges mentioned in Mike's post, as approaches have been removed from one or both bridges but will happen. Of note, Indiana Transportation Museum once used the Westfield to Noblesville section for at least one year to shuttle their "Fair Train" over to the former Monon Tracks in the early 1980's before both tracks were removed. Great pictures of this line can be found on Indiana Railroad Bull Session discussion list or Roger Hensley's "Railroads of Madison County" page.
It is my understanding, that Waveland, to Advance was abandoned in 1928.