In Chen et al. (2013), the fundamental concepts of the modular UQ methodology have been introduced for general multiphysics applications in which each physics module can be independently embedded with its internal UQ method (intrusive or nonintrusive) without losing the global uncertainty propagation property. In the current paper, we extend the modular UQ methodology to subsurface flow and reactive transport applications, which are characterized by high dimensionality in the stochastic space due to spatially random velocity field in randomly heterogeneous porous media. Specifically, we develop a scheme to reduce the dimension of the stochastic space. This is achieved via a doubly nested dimension reduction by applying Karhunen-Loève expansion to the logarithmic hydraulic conductivity field, followed by Proper Orthogonal Decomposition to the velocity field. This scheme enables the modular UQ framework to handle spatially random models efficiently while maintaining solution accuracy. When compared against sampling-based nonintrusive UQ methods, the modular UQ method demonstrates a similar accuracy at a fraction of computational cost on designed numerical experiments.