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Reinforcement learning (RL) is a promising tool to solve robust optimal well control problems where the model parameters are highly uncertain, and the system is partially observable in practice. However, RL of robust control policies often relies on performing a large number of simulations. This could easily become computationally intractable for cases with computationally intensive simulations. To address this bottleneck, an adaptive multi-grid RL framework is introduced which is inspired by principles of geometric multi-grid methods used in iterative numerical algorithms. RL control policies are initially learned using computationally efficient low fidelity simulations using coarse grid discretization of the underlying partial differential equations (PDEs). Subsequently, the simulation fidelity is increased in an adaptive manner towards the highest fidelity simulation that correspond to finest discretization of the model domain. The proposed framework is demonstrated using a state-of-the-art, model-free policy-based RL algorithm, namely the Proximal Policy Optimisation (PPO) algorithm. Results are shown for two case studies of robust optimal well control problems which are inspired from SPE-10 model 2 benchmark case studies. Prominent gains in the computational efficiency is observed using the proposed framework saving around 60-70% of computational cost of its single fine-grid counterpart.