Production optimization is of significance for carbonate reservoirs, directly affecting the sustainability and profitability of reservoir development. Traditional physics-based numerical simulations suffer from insufficient calculation accuracy and excessive time consumption when performing production optimization. We establish an ensemble proxy-model-assisted optimization framework combining the Bayesian random forest (BRF) with the particle swarm optimization algorithm (PSO). The BRF method is implemented to construct a proxy model of the injection–production system that can accurately predict the dynamic parameters of producers based on injection data and production measures. With the help of proxy model, PSO is applied to search the optimal injection pattern integrating Pareto front analysis. After experimental testing, the proxy model not only boasts higher prediction accuracy compared to deep learning, but it also requires 8 times less time for training. In addition, the injection mode adjusted by the PSO algorithm can effectively reduce the gas–oil ratio and increase the oil production by more than 10% for carbonate reservoirs. The proposed proxy-model-assisted optimization protocol brings new perspectives on the multi-objective optimization problems in the petroleum industry, which can provide more options for the project decision-makers to balance the oil production and the gas–oil ratio considering physical and operational constraints.

The advancement in big data and artificial intelligence has enabled a novel exploration mode for the study of petroleum engineering. Unlike theory-based solution methods,the data-driven intelligent approaches demonstrate superior flexibility,computational efficiency and accuracy for dealing with complex multi-scale,and multi-physics problems. However,these intelligent models often disregard physical laws in pursuit of error minimization,which leads to certain uncertainties. Therefore,physics-informed machine learning approaches have been developed based on data,guided by physics,and supported by machine learning models. This study summarizes four embedding mechanisms for introducing physical information into machine learning models,including input data-based embedding,model architecture-based embedding,loss function-based embedding,and model optimization-based embedding mechanism. These “data + physics” dual-driven intelligent models not only exhibit higher prediction accuracy while adhering to physic laws,but also accelerate the convergence to improve computational efficiency. This paradigm will facilitate the guide developments in solving petroleum engineering problems toward a more comprehensive and efficient direction.