drain between fields, irrigation
畎 centers on a small water channel cut through fields: a drain or irrigation ditch that moves water between plots.
畎 combines 田 (field) with 犬 (dog), but the historical reason for 犬 is uncertain. It may have originally represented a phonetic element or a different component that simplified to 犬.
Picture a field 田 with a dog 犬 running along a narrow irrigation ditch between the crops. The dog follows the water channel, showing where the drain cuts through the field.
For ケン, imagine a keen-eyed dog patrolling the irrigation ditch: keen -> ケン, and the dog keeps the water flowing between the fields.