A new approach for uncertainty assessment in porous media is devised. The goal is to study tracer or phase transport while assuming that the multi-point velocity statistics is known. The method depends on solving a transport equation for the joint probability density function (PDF) of velocity and concentration (or phase saturation). Similar PDF methods have been developed for turbulent flow simulations and proved extremely successful in providing joint statistics of species concentration and velocity required for turbulent combustion modeling. As in the case of turbulent flows, the joint PDF equation for uncertainty assessment of transport in porous media is defined over a high dimensional space (physical, velocity, concentration or phase, and time). This results in severe computational limitations, if a finite-volume, finite-difference or finite-element method is employed, which is the motivation for the use of a particle method. The crucial advantage of the PDF modeling approach is its flexibility and high level of closure. For example, no modeling is required for macro-dispersion (ensemble mean of the product of velocity-concentration fluctuations). Moreover, no assumptions are made regarding correlation length scales and unlike perturbation-based Statistical Moment Equation (SME) methods, the PDF method is not restricted to small input variance. Here, the PDF methodology is presented for incompressible single-phase tracer transport in heterogeneous porous media, but more general applications involving multi-phase transport are discussed. A detailed description of the PDF method is given and its accuracy is demonstrated by comparison with published Monte Carlo results.


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