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In this study, we develop a probabilistic approach to map the parametricuncertainty to the output state uncertainty in first-order hyperbolicconservation laws. We analyze this problem for nonlinear immiscible two-phasetransport in heterogeneous porous media in the presence of a stochasticvelocity field. The uncertainty in the velocity field can arise from theincomplete description of either porosity field, injection flux, or both. Theuncertainty in the total-velocity field leads to the spatiotemporal uncertaintyin the saturation field. Given information about the spatial/temporalstatistics of the correlated heterogeneity, we leverage method of distributionsto derive deterministic equations that govern the evolution of single-point CDFof saturation. Unlike Buckley Leverett equation, the equation for the raw CDFfunction is linear in space and time. Hereby, we give routes to circumventingthe computational cost of Monte Carlo scheme while obtaining the fullstatistical description of saturation. We conduct a set of numericalexperiments and compare statistics of saturation computed with the method ofdistributions, against those obtained using the statistical moment equationsapproach and kernel density estimation post-processing of high-resolution MonteCarlo simulations. This comparison demonstrates that the CDF equations remainaccurate over a wide range of statistical properties, i.e. standard deviationand correlation length of the underlying random fields, while the correspondinglow-order statistical moment equations significantly deviate from Monte Carloresults, unless for very small values of standard deviation and correlationlength.

Probabilistic forecast of multiphase transport under viscous and buoyancy forces in heterogeneous porous media