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The estimation of the wave-induced instantaneous liquefaction is particularly important for the design of foundations of offshore structures. Regarding the occurrence of liquefaction in a non-cohesive seabed, most existing studies using constant permeability were found to cause fallacious tensile stresses in the liquefied zone and further pollute the overall pore pressure distribution. A dynamic permeability model was previously presented to mitigate the shortcoming but posed difficulties in the nonlinear convergence. To overcome the shortcoming of the previous studies, this study proposes the concept of modelling the liquefaction-involved wave-seabed interactions as a nonlinear complementarity problem, wherein a Karush–Kuhn–Tucker condition is constructed, based on revisiting the liquefaction criterion most widely applied in ocean engineering. The Lagrange multiplier method and the primal–dual active set strategy are employed to numerically deal with the nonlinear complementarity problem. The performance of the chosen multiplier space is investigated by theoretical analyzing and numerical modelling. Compared with the previous dynamic permeability model, the present model is totally free of extra parameters and precisely fulfills the no-tension requirement. Moreover, the difficulties of dynamic permeability in the nonlinear convergence are overcome and no divergence is observed in the numerical tests. [ABSTRACT FROM AUTHOR]