ADCIRC is both a 2-dimensional (2D) and 3D model, and the formulations of these are somewhat different. The best documentation of the base model physics for both is currently in the ADCIRC Theory Report.
2D ADCIRC Physics
2D ADCIRC is based on a form of the shallow water equations, which includes the continuity equation and the horizontal momentum equations. The model is depth-integrated, meaning that although vertical flow is not solved for in the model, the effects of such vertical variations can still be included to some degree, such as the 2D baroclinic model.