I have been struggling for the past few years in trying to remove green goutweed that has infested one quarter of my lawn and garden. I have been digging it out, covering the soil with newspapers and black liners but it still escapes and sends out its runners. I have kept the newspapers and black liners covering the infested area for two years but now the goutweed is spreading in other areas of the lawn.
I have wrestled with goutweed, too. Because of its ability to spread far and wide and regrow from every bit of root left behind, it is one of few weeds where herbicide is necessary, in my experience. Another option is solarization which takes up to an entire growing season to heat the soil to a temp that kills plant roots. The problem is extreme soil heat also kills earthworms and beneficial bacteria. Another option is to dig up all the soil, sift it to get out as much root as possible and then spray or use a paintbrush to apply herbicide to new sprouts. Round-Up works and is not active and not persistent in the soil so it is the least toxic herbicide. It is going to require vigilance for a year or two to make sure it is gone. Good luck.