In subsurface flow settings, deep-learning-based surrogate modeling has been shown to be an effective approach for dealing with cases that require a substantial amount of model simulations. However, a large number of high-fidelity training simulations are usually required to construct these deep-learning-based surrogate models. For large-scale models, it can be computationally prohibitive to perform these training simulations. To address this limitation, in this work, we develop a new approach to construct surrogate models using multifidelity training data. The model is based on a U-Net deep-learning architecture. In the process of model construction: firstly, the reservoir model is coarsened by grid-based upscaling, which allows the generation of a large amount of low-fidelity training data for the construction of a pre-trained deep-learning model; Subsequently, a small amount of high-fidelity training data is used to fine-tune the network parameters in order to accurately predict fluid flow in fine-scale reservoirs. The results show that our proposed surrogate models trained using multifidelity data provide predicted dynamic pressure and saturation fields that are in close agreement with the corresponding results calculated by reservoir numerical simulation, while achieving near 75% reduction in terms of the required fine-scale simulations.