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Predictions of solute transport in subsurface environments are notoriously unreliable due to aquifer heterogeneity and uncertainty about the values of hydraulic parameters. Probabilistic framework, which treats the relevant parameters and solute concentrations as random fields, allows for quantification of this predictive uncertainty. By providing deterministic equations for either probability density function or cumulative distribution function (CDF) of predicted concentrations, the method of distributions enables one to estimate, e.g., the probability of a contaminant's concentration exceeding a safe dose. We derive a deterministic equation for the CDF of solute concentration, which accounts for uncertainty in flow velocity and initial conditions. The coefficients in this equation are expressed in terms of the mean and variance of concentration. The accuracy and robustness of the CDF equations are analyzed by comparing their predictions with those obtained with Monte Carlo simulations and an assumed beta CDF.

We derive deterministic cumulative distribution function (CDF) equations that govern the evolution of CDFs of state variables whose dynamics are described by the first-order hyperbolic conservation laws with uncertain coefficients that parametrize the advective flux and reactive terms. The CDF equations are subjected to uniquely specified boundary conditions in the phase space, thus obviating one of the major challenges encountered by more commonly used probability density function equations. The computational burden of solving CDF equations is insensitive to the magnitude of the correlation lengths of random input parameters. This is in contrast to both Monte Carlo simulations (MCSs) and direct numerical algorithms, whose computational cost increases as correlation lengths of the input parameters decrease. The CDF equations are, however, not exact because they require a closure approximation. To verify the accuracy and robustness of the large-eddydiffusivity closure, we conduct a set of numerical experiments which compare the CDFs computed with the CDF equations with those obtained via MCSs. This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties of the two input parameters, such as their correlation lengths and variance of the coefficient that parametrizes the advective flux.