We address an important component of risk mitigation for ultra-deep sea drilling in the Gulf of Mexico (GoM), namely the probabilistic characterization of fluid fluxes at the seafloor from future drilling operations. In the process, we develop a stochastic representation of functions defined on a high-dimensional space conditional on their marginal statistics and their global correlation structure. The representation leverages a particular structure of the functional dependence of interest which exhibits scale separation. Specifically, we construct a polynomial chaos representation for scalar quantities of interest whose coefficients are themselves random. The intrinsic randomness of the polynomial chaos expansion (PCE) reflects local uncertainty and captures dependence on a subset of the parameters, while randomness in the PCE coefficients captures a global structure of the uncertainty and dependence on the remaining parameters in the high-dimensional space. This construction is demonstrated by predicting wellbore signatures in the GoM where a 120-dimensional table is populated at several thousand wellbore locations throughout the GoM. Physics-based models of multiphase flow in porous media are used to calculate the PCE representations at the sites where data is available. In this context, random parameters describing the subsurface define the parameter set with respect to which PCE is constructed. A Gaussian process in parameter space is then developed for each coefficient in these representations. The combined probabilistic representation permits the delineation of separate stochastic influences on predictions of interest.