How can I repair a bad differential?

it could be center bearing, if its going bad it can also cause a vibration, make sure you have sufficient fluid in the diff as well, you can also get a noise and vibration from collapsed motor or trans mounts

It should have been thoroughly checked. Noises like these on four wheel drives could several different things. A bearing, CV axle, or the differential, or bearings in the differential. A wheel bearing will make different sounds when making a turn. A CV axle is a constant sound. Differential bearings can come and go, depending on if it is a bearing for the axle or a pinion bearing. Axle side bearings usually make a constant noise. The differential can also come and go. Depends on acceleration or slowing down. All these sounds are also with a vibration. Slight to harsh.

You havwe worn gears in the differential. You will need to rebuild it.

Assuming that you have already ruled out the brakes, severely worn engine mounts can transmit vibration and noises from the engine which are not normally heard. You don't say in what I see but is the vehicle two or four wheel drive. Since you mention CV half shafts I have to assume it is at least front wheel drive. If the vehicle is four wheel drive then there is a front differential with bearings in it that may be at fault. Have you done both sides when you say "hub assembly". Defective hub bearings will make grinding noises which will change tone or intensity when turning, accelerating or braking however the noise is usually fairly constant unless they are only just starting to fail. CV joints don't normally "grind" unless they are really bad. They usually make progressively worse "clicking" sounds or "clunking" sounds especially when turning hard, changing gears or direction of travel (forward to reverse and back again). Do you have non-stock tires/wheels? Larger/wider than factory spec tire sizes can rub the fender wells when the car is moving or turning. Another thing it could possibly be is the air dam below the front bumper. If it is loose it will push down in the airstream while the car is moving and rub on the ground.

If the noise is coming from below the trunk area(by the rear wheels), It may be that the differential is probably going bad and the grinding is coming from inside the differential. Inside the differential are gears and they will grind and cluck if they are going bad or bad. The car may or may not move depending on the damage inside the differential.

indicates a fuel pump problem as that is the only thing in that area that is ignition controlled or possibly the EVAP system solenoid/ valve if it is in that area
the frequency will be constant as the pump rpm is constant
the noise could be a venting noise from the EVAP in in that area
the distance of 15 km indicates that the problem is heat related as it take that long for something to over heat and cause the noise
you may get an indicator by running the fault code

GET IT TO A MECHANIC!! Something is loose or about to collapse! Remember that vibration and noise can travel away from the source so get the mechanic to start in that corner.
You need to get the following checked!
-Prop shaft mountings at transfer box.
-Prop shaft mountings at differential.
-Prop shaft bearings.
-Differential condition and oil level.
-Drive shafts at differential end.
-Drive shafts at the wheel end.
-All flexing joints in the system including UV joints and CV joints.

I'd be looking a the driveshaft ujoints.. Push and pull the driveshaft looking for any play at the cup areas.

carrier bearings are in the rear differential,and when they go bad,you will have a VERY noticable grind noise come from the rear and some vibration. They are reltively easy easy to replace yourself,though a good shop will charge around $200 for parts AND labor.

Check the front drive axles and boots. A bad cv (constant velocity) joint can cause this.