We address uncertainty quantification of diffusion processes in layered media, where the properties of the layers and their locations are uncertain and therefore described as stochastic quantities. The method of generalized polynomial chaos expansions is attractive for uncertain quantification in computational mechanics because of its fast convergence. However, the jumps associated with layered media slow down the convergence significantly. In this paper we describe a new technique to circumvent the convergence problem, however this technique requires an extension of the classical polynomial chaos method to handle dependent random variables. We develop a convenient computational way of incorporating dependent random variables in polynomial chaos expansions. Numerical investigations involving porous media flow through two layers demonstrate that the new method can effectively deal with stochastic permeabilities and stochastic location of the permeability jumps. Comparisons with existing methods show a significant increase in the convergence rate.