Flow and reactive transport problems in engineering, medical and environmental applications often involve complex geometries. Grid based methods (e.g. finite volume, finite element, etc.) are a vital tool for studying such problems. Cartesian grids are one of the most attractive options as they possess simple discretization stencils and are usually straightforward to generate at roughly no computational cost. The Immersed Boundary Method, a Cartesian based methodology, maintains most of the useful features of structured grids, while it exhibits a great resilience in dealing with complex geometries. These features make it increasingly more attractive to model transport in evolving porous media as the cost of grid generation reduces greatly. Yet, stability issues due to the geometry of the interpolation stencil combined with limited studies on the implementation of Neumann (constant flux) and linear Robin (e.g. reaction) boundary conditions have significantly limited its applicability to transport in complex topologies. We develop a high-order compact Cartesian model based on ghost cell immersed boundary method for incompressible flow and scalar transport subject to different boundary conditions. The accuracy test shows at least second order of accuracy in L-1, L-2 and L-infinity norms of error. The proposed method is capable of accurately capturing the transport physics near the boundaries for Dirichlet, Neumann and Robin boundary conditions. We tested the method for several transport and flow scenarios, including heat transfer close to an immersed object and mass transport over reactive surfaces. (C) 2019 Elsevier Ltd. All rights reserved.