A method for reducing or completely eliminating water influx in an oil or natural gas borehole includes introducing in the borehole an aqueous solution of a copolymer comprising: A) 40 98 wt. % of structural units of formula (I), wherein: R.sup.1 is hydrogen or methyl; R.sup.2 represents C.sub.2 C.sub.10 alkylene; and Me.sup.+ represents an ammonium metal ion and an alkali metal ion; B) 0.1 to 58 wt. % of structural units of formula (II); C) 0.1 to 10 wt. % of structural units of formula (III), wherein: R.sup.3 and R.sup.4 independently of each other represent hydrogen, methyl or ethyl, or R.sup.3 and R.sup.4 represent together a propylene group, which, with inclusion of a radical (A), form a pyrrolidon radical or, with inclusion of a pentamethylene group, forms a caprolactam radical; D) 0.10 to 10 wt % of structural units of formula (IV). The method also includes introducing in the ground or deposit, simultaneously with the copolymer or subsequently, an agent for crosslinking the copolymer.