I am modeling a hypothetical backup protection system for a power transmission grid and need to know if a directional relay can detect a fault if there are multiple lines and buses between the relay and the fault.
No a directional relay cannot detect a fault on line 13-14 (between buses 13 & 14).
Your relay has several buses between the fault location and the protective device in question. A directional relay only uses information from the circuit breakers that it is connected to (red boxes in your grid).
So in this case the relay does not know what is happening at the other relays/buses outside of its zone(s) of protection. You don’t have those zones shown but they are typically something like the following image.
The fault would most likely be detected by a relay on bus 13 and/or 14.