If the ground is so wet that the grass is declining rushes may come to the fore, as mowing will be less frequent and enough regrowth can occur to maintain the plant. After cultivation the large reservoir of seed will continue to germinate for a few years, but they will not develop under regular mowing conditions.
Weedkillers to use:-
Glyphosate or Ammonium Sulphamate, systemic action killing the whole plant. A few drops of detergent in the mix will help adhesion to the shiny vertical surface. Keep a lookout for new seedlings which germinate

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