A new kind of crisis was observed in a system where a transition from conservative toquasi-dissipative can be observed. The crisis signifies a sudden and intrinsic change of a stochasticweb, which is formed by the end-results of the images of the discontinuous borderlines of the systemfunction. In the crisis, a strange quasi-repeller can be defined. When changing the controllingparameter, the variation of the fractile dimension of the quasi-repeller obeys a logarithmic rule.