Word 2016, 2013 and 2010 allows users to add filename to the footer section using Quick Parts. For example, if you want add filename in a three column footer then you can do the following.
In the Word document, navigate to Insert menu, click the Footer option and pick Blank (Three Columns) from drop down list.
Insert filename in Footer
After selecting the column where you want to insert filename, navigate to Quick Parts under Text section and click the Filed menu option.
In the Filed window, select Filename under Field names and choose your preferred format i.e. Uppercase, Lowercase, First capital, Title case and None.
If you want to add path to the filename then navigate to Fields options section and tick the check box with label as Add path to filename. Click OK button to confirm and save the changes. Now your Footer section will reflect the filename with the path.
Update filename in Footer
In future if you change the name of the file then you can update the footer with the new filename by right clicking on the filename section and selecting Update Filed menu option.