Clay Absorbent.
This stuff is available at most auto parts repair stores. Sometimes it is available at big box stores.
It is NOT the same as cat litter although it looks strikingly similar. The resemblance is so close that "kitty litter" is the 'street' name for this product.
However, clay absorbent is far superior in its ability to remove oil from concrete. For example, my father and I moved a broken 2.5 hp motor over a concrete walkway in a plastic tub. At some point the load became unstable and the engine tilted enough to force us to drop the tub. In doing so, we streaked the walkway with oil approximately 2" wide and 6" long.
Since I was away from home and he did not have any clay absorbent on hand, I promptly drove to purchase some and placed the product according to the manufacturers directions. After the second application was swept up, I made a third application and ground the product into the concrete with my shoe and left it to sit until the next rain washed it away (not recommended, but hey the walkway was pristine beforehand and my dad has a low carbon footprint).
To this day, there is a spot on the walkway that is cleaner than the rest. I am not exaggerating in the least when I say that the entire section now needs the ground-in-and-washed-off treatment to help minimize the white spot left behind where there was a dark black oil spot before.
Two applications should do for what you need. Skip the ground in trick unless you promise to recover the waste water ;)
Edit:// Just read where you said spots are old and kitty litter will not work.
Try the clay absorbent. It is about 100% more effective than cat litter on any day. Also, if it does not come up, leave it for a night or two.
Other than that, (using a test spot!!!) dampen the concrete with fresh oil (detergents good...) then place the oil dry over it and leave for at least a day. If all else fails, grind it in!
Your eavestroughs (or gutters, in some parts of the world) collect the rain from your roof and direct it through a downspout to the ground level. It sounds like you have concrete splash blocks below your downspouts - like this:
Image source
(Besides preventing soil erosion, the splash blocks keep the dirt from splashing up on your house.)
Drainage is easy - water goes downhill. Just make sure you have a constant downhill path for the water to flow and you're golden. It won't go uphill into a raised planter, unless you have some special capillary mat or plans with long roots. Plantings are good to slow down the flow of water and also absorb it.
You could put any old soil in to fill in the erosion. If you don't have a splashblock, then you should get some -- flat rocks will work also and are often more aesthetically pleasing.
I agree that rain barrels are unsightly, so don't really like them myself. I once put in a french drain, but that was a lot of work. (Dug a trench, put in geotextile, then some gravel on top of the fabric, then 4" weeping tile, then some more gravel, covered the geotextile over top, then backfilled. The geotextile fabric is to prevent dirt from washing into the gravel and eventually blocking it from working effectively for drainage.)
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