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Identification of unknown physical processes and parameters of groundwater contaminant sources is a challenging task due to their ill-posed and non-unique nature. Numerous works have focused on determining nonlinear physical processes through model selection methods. However, identifying corresponding nonlinear systems for different physical phenomena using numerical methods can be computationally prohibitive. With the advent of machine learning (ML) algorithms, more efficient surrogate models based on neural networks (NNs) have been developed in various disciplines. In this work, a theory-guided U-net (TgU-net) framework is proposed for surrogate modeling of three-dimensional (3D) groundwater contaminant problems in order to efficiently elucidate their involved processes and unknown parameters. In TgU-net, the underlying governing equations are embedded into the loss function of U-net as soft constraints. For the considered groundwater contaminant problem, sorption is considered to be a potential process of an uncertain type, and three equilibrium sorption isotherm types (i.e., linear, Freundlich, and Langmuir) are considered. Different from traditional approaches in which one model corresponds to one equation, these three sorption types are modeled through only one TgU-net surrogate. The three mentioned sorption terms are integrated into one equation by assigning indicators. Accurate predictions illustrate the satisfactory generalizability and extrapolability of the constructed TgU-net. Furthermore, based on the constructed TgU-net surrogate, a data assimilation method is employed to identify the physical process and parameters simultaneously. This work shows the possibility of governing equation discovery of physical problems that contain multiple and even uncertain processes by using deep learning and data assimilation methods.