Synergistic security relationships and risk measurement of water resources-social economy-ecological environment in Beijing-Tianjin-Hebei region

China’s productivity progress has promoted good socio-economic functioning, but has brought serious negative impacts on natural resources and eco-environment. Promoting synergistic security of water resources (W), social economy (S) and ecological environment (E) provides technical support for promoting regional economic growth, social progress and ecological harmony. This study constructed an index system based on stability (S), coordination (C) and resilience (R). Then the entropy weight-CRITIC method, coupled coordination degree model, and obstacle degree model were used to comprehensively evaluate the level of synergistic security of WSE-SCR in the Beijing-Tianjin-Hebei (BTH) region. Finally, the two-dimensional and three-dimensional joint risk probability distributions of different provincial regions were explored by combining the Copula function. The results showed that the level of synergistic security in the BTH region increased from 2006 to 2022, and the multi-year average value was within the security range, specifically Beijing (0.69) > Hebei (0.66) > Tianjin (0.65). The key factors affecting the WSE-SCR are per capita water resources (S1), per capita water consumption (S3), proportion of secondary industry (S5), ecological water use ratio (C4), and investment in infrastructure construction (R8). In terms of two-dimensional joint risk probability, the C-R of Beijing and the S-R of Tianjin and Hebei had the highest probability of security risk, which were 0.5598, 0.5308 and 0.5488, respectively. The three-dimensional joint risk probability of the S-C-R of Beijing, Tianjin and Hebei (S ≤ 0.6, C ≤ 0.6, and R ≤ 0.6) were greater than 30 %, which were 0.389, 0.341 and 0.352, and increased with the increase of single dimension risk. This study can provide scientific information for advancing the synergistic security and sustainable improvement of WSE in arid zones.