It seems that the last batch of heavy rain has eroded the soil beside your concrete driveway. In order to shore it up, I suggest building a small retaining wall out of sleepers to prevent soil erosion in that part of the driveway. I also recommend putting in Vinidex 100mm x 20m Socked Slotted Draincoil inside the retaining wall to re-direct the water falling into it.
I've placed a sample sketch below to give you an idea of what it looks like and how you can build it.
If you have any questions about it, please let me know.