Method of Distributions for Two‐Phase Flow in Heterogeneous Porous Media

Multiscale heterogeneity and insufficient characterization data for a specific subsurface formation of interest render predictions of multi‐phase fluid flow in geologic formations highly uncertain. Quantification of the uncertainty propagation from the geomodel to the fluid‐flow response is typically done within a probabilistic framework. This task is computationally demanding due to, for example, the slow convergence of Monte Carlo simulations (MCS), especially when computing the tails of a distribution that are necessary for risk assessment and decision‐making under uncertainty. The frozen streamlines method (FROST) accelerates probabilistic predictions of immiscible two‐phase fluid flow problems; however, FROST relies on MCS to compute the travel‐time distribution, which is then used to perform the transport (phase saturation) computations. To alleviate this computational bottleneck, we replace MCS with a deterministic equation for the cumulative distribution function (CDF) of travel time. The resulting CDF‐FROST approach yields the CDF of the saturation field without resorting to sampling‐based strategies. Our numerical experiments demonstrate the high accuracy of CDF‐FROST in computing the CDFs of both saturation and travel time. For the same accuracy, it is about 5 and 10 times faster than FROST and MCS, respectively.