The fuel system can be considered in two parts, fuel supply and fuel delivery with the fuel delivery system being the throttle body and injector. The throttle body and components are largely the same on all the year models.
Component Part No info for all year models & 650 variants is in the FuelSys Parts pdf
There are 3 variations on the fuel injector, the basic connection & operation is the same on all the models. Injector details & the alternatives are in the pdf. Fuel Injector Info pdf
Lamp operation is covered in the 00-09 Dash info pdf to assist repairs. 00-09 Dash Info pdf
Fuel Level Sensor info for all year models is in the pdf to assist repairs. Fuel Sensor Info pdf
The "mystery connector" on top of the fuel tank is the connector for the US only Tank Vent Valve. The pdf here has information on the valve & fault finding. Vent Valve Info pdf
On 2013+ BMS-E bikes the fuel pump operates via a relay under the seat near the fuse box
The resistance of the BMW fuel pump is difficult to measure, readings between 1.5 to 9.0 Ohms may be seen.
Current draw is 2.6 amps @ 12.5v (filter regulator disconnected), 4.6 amps @ 12.5v (filter regulator connected)
The GS911 has diagnostics routines to operate the fuel pump and the fuel injector to aid in fault finding.
Fuel system parts are available from HiFlow, EuroMotoElectrics and others, info is here:
Fuel Sys Info pdf
SFMike at f650.com has done a write up on fitting the Hiflow pump, a pdf is here: Fuel Pump Replacement pdf
A number of owners have reported fuel pump failures. It seems the three supports sag with age leaving the pump supported by the wiring. The power wire to the fuel pump can fracture and fail. The failures have been documented at f650.com and the repair procedure below is from the threads. A pdf of the repair procedure. Repair pdf
Remove the seat, disconnect the plugs to the fuel tank cap.
Unscrew the big plastic ring on the top of the fuel tank.
Pull the pump/gasket out of the tank.
Use a multimeter to check the wiring is connected.
Reconnect wiring & use a length of wire to extend wiring if needed.
Wire colors Red is Power, Brown is Earth
Carefully place the fuel pump into the fuel tank.
Take care with the rubber gasket, it can get caught on the big threaded ring and then does not seal well.
Run a smear of grease between the gasket and the fuel tank plus between the gasket and the ring to ensure it seals.
Reconnect the external connectors to the fuel pump & check the bike starts and runs. Put the seat on, all finished.
Full credit to Workerant and Richard who posted the details of the problem and the repair at f650.com
Bad connections in the fuel pump connector on top of the fuel tank can melt the plastic and cause fuel leaks. This is a known problem on other types of vehicles with in-tank fuel pumps. Check connections periodically.
The metal neck of the fuel filler can rust, particularly where the machine
has been parked for an extended period. The rust can fall into the bottom
of the fuel tank and clog the filter. Cleaning the tank is then hard.
The Filler Cap screws thread into plastic and brass inserts which can release and spin preventing removal of the filler cap. Once removed by either drilling the head or spinning out by rotating anti clockwise with a drill to generate some heat, they can be epoxied back into place.
The seal on the fuel tank locking ring under the seat can perish or not seat correctly and cause fuel leaks.
The Sertao in particular has had problems with the seal and has been subject to warranty remedial action.
The problem of cracking of the top of the fuel pump mounting has been reported on both US and EU fuel tanks. A recall was done on the F650CS models in the US only but not on the GS. The symptoms are fuel odour and fuel leaks when on the side stand. Check the mount during maintenance when the seat is off and if cracks are found replace the assembly. Note. The mount is only available new as part of the complete fuel pump assembly
Filters and Regulators are a source of annoyance with owners, the GS combined unit cost, and the lack of BMW Part No's for the 650X parts. Alternatives are not well documented, what is known is in the Filter Regulator Info pdf
The regulator specification is for between 3.3 and 3.7 bar. This results in significant variations in fuel injector flow rates (341 to 361 cc/min). When replacing a regulator reset BMS adaptions with a GS911 to adjust for the change.
The G650X has an in tank fuel filter eliminating the fuel return line freeing up space under the cowlings. There are no Part No's for the filter/regulator. The F800 also has an in tank filter. It may be possible on the 650GS, more investigation is required. G650X FuelSys Info pdf
The TR650 has an in-tank fuel system and is only available as a complete assembly. The information collected by owners so far is in the pdf. TR650 FuelSys Info pdf