One of the primary advantages of Slurry over other types of fertilizer is that it can be stored right near the field. This is done by placing a Slurry Field Tank near a field that's currently being worked by a Slurry Spreader, and filling that field tank using a Slurry Tanker.
By placing such a Field Tank near the worked field, a Slurry Spreader that runs out of Slurry does not need to be taken all the way back to a Slurry refilling point, such as the Biogas Plant or the Cow Pasture. Instead, it can be taken a short distance to the Slurry Field Tank, where it can be refilled as normal.
The Field Tank cannot fill itself from the stationary Slurry refilling points, so instead you will need a Slurry Tanker to make the trips between those points and the Field Tank. The only Slurry Tanker in the game (the Kotte TSA 30000) is quite large, and is capable of filling a Field Tank in two runs.
Due to a bug/feature in the game design, this entire process can be skipped simply by allowing a Hired Worker to use the Slurry Spreaders, without refilling at all. The Hired Worker will automatically purchase Slurry, at a fractional price. If you do not mind using this apparent exploit, and don't mind allowing a Worker to do all the work for you, there is no real need to use Slurry Field Tanks or Slurry Tank