No abstract

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.

We study transport of a reactive solute in a chemically heterogeneous porous medium whose chemical properties are uncertain. The dissolved substance undergoes a heterogeneous chemical reaction with a solid phase in the presence of advection caused by extraction/injection from a point source. We present semi-analytical solutions for the probability density function of the solute concentration, which allows us to quantify predictive uncertainty associated with uncertain reaction rate constants for both linear and nonlinear reactions. This enables one to compute probabilities of rare events, which are required for quantitative risk analyses.