This file serves as supplementary material, including additional content not included in the main manuscript for the sake of brevity. The manuscript is titled "Physics-Informed Deep Operator Learning for Computational Hydraulics Modeling". Its abstract is as follows. Traditional 2D hydraulic models face significant computational challenges that limit their applications that are time-sensitive or require many model evaluations. This study presents a physics-informed Deep Operator Network (DeepONet) framework for computational hydraulics modeling that learns the solution operator of the 2D shallow water equations (SWEs) to create fast surrogate models. The framework can operate in two modes: a purely data-driven SWE-DeepONet that learns from numerical solver such as SRH-2D, and a physics-informed PI-SWE-DeepONet that additionally incorporates the continuous SWEs as constraints during training. Based on a real-world case, steady flows in a reach of the Sacramento River in California, it is demonstrated that PI-SWE-DeepONet possesses much enhanced prediction capability than SWE-DeepONet when applied to out-of-distribution scenarios. The physics-informed model is shown to exhibit slower error growth and larger breakdown distances in comparison with SWE-DeepONet. The gain of the physics-informed training, however, comes with costs, chief among which are the simulated results have slightly higher errors for in-distribution cases. It reflects the existence of a tension between the two competing training objectives: fitting the results from the traditional hydraulic model and satisfying the continuous governing equations. In this study, guidelines are developed for selecting the appropriate approach based on a real-world case: PI-SWE-DeepONet is preferred for out-of-distribution predictions, uncertain training data, or when physical consistency is a priority, while SWE-DeepONet is recommended if the modeling objective is to replicate faithfully the traditional hydraulic model results within the training distribution. Other challenges are also discussed, such as the loss weighting approach.