They make a perforated plastic dry well (dry bed) liner to put in the hole you dug. An alternative is bricks laid not without mortar with a prefabricated concrete or steel lid. Then just about anything (PVC pipe suggested) can be used to re-route the washing machine drain line to that.
Some city codes may prohibit the use of a dry well for anything other than plain water such as rain water.
You really should not need to re-route your washing machine drainage from your septic tank unless you are experiencing septic system problems.
(One morning on WRKO Radio) The tub's not clean! Let's remodel the bathroom.
(One afternoon on WRKO Radio) It's time for an oil change. I'm going to trade in the car for next year's model.