the file path should be physical not virtual. Use
if (File.Exists(Server.MapPath(wordDocName)))
File.Exists() and probably everything else you want to do with the file will need a real Path.
File.Exists()
Your wordDocName is a relative URL.
wordDocName
Simply use
string fileName = Server.MapPath(wordDocName);
Use
Server.MapPath("~/specifications/" + Convert.ToInt32(ViewState["projectSelected"]) + ".doc")
to get the fully-qualified path. That should do the trick for ya.