Computerized solutions for field management optimization often require reduced-order models to be computationally tractable. The purpose of this paper is to compare two different graph-based approaches for building such models. The first approach represents the reservoir as a graph of 1D numerical flow models that each connects an injector to a producer. One thus builds a network in which the topology is primarily determined by “well nodes” to which “non-well nodes” can be connected if need be. The second approach aims at building richer models so that the connectivity graph mimics the intercell connections in a conventional, coarse 3D grid model. One thus builds a network with topology defined by a mesh-like placement of “non-well nodes”, to which wells can be subsequently connected. The two approaches thus can be seen as graph-based analogues of traditional streamline and finite-volume simulation models. Both model types can be trained to match well responses obtained from underlying fine-scale simulations using standard misfit minimization methods; herein we rely on adjoint-based gradient optimization. Our comparisons show that graph models having a connectivity graph that mimics the intercell connectivity in coarse 3D models can represent a wider range of fluid connections and are generally more robust and easier to train than graph models built upon 1D subgridded interwell connections between injectors and producers only.