Multiscale heterogeneity and insufficient characterization data for the specific subsurface formation of interest render predictions of multi-phase fluid flow in geologic formations highly uncertain. Quantification of the propagation uncertainty from the geomodel to the fluid-flow response is typically done within a probabilistic framework. This task is computationally demanding due to, e.g., the slow convergence of Monte Carlo simulations (MCS), especially when computing the tails of a distribution that will be used 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 still 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 the 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.

Probabilistic models of subsurface flow and transport are required for risk assessment and reliable decision making under uncertainty. These applications require accurate estimates of confidence intervals, which generally cannot be ascertained with such statistical moments as mean (unbiased estimate) and variance (a measure of uncertainty) of a quantity of interest (QoI). The method of distributions provides this information by computing either the probability density function or the cumulative distribution functions (CDF) of the QoI. The method can be orders of magnitude faster than Monte Carlo simulations (MCS), but is applicable to stationary, mildly-to-moderately heterogeneous porous media in which the coefficient of variation of input parameters (e.g., log-conductivity) is below three. Our CDF-RDD framework alleviates these limitations by combining the method of distributions and the random domain decomposition (RDD); it also accounts for uncertainty in the geologic makeup of a subsurface environment. For a given realization of the geological map, we derive a deterministic equation for the conditional CDF of hydraulic head of steady single-phase flow. The solutions of this equation are then averaged over realizations of the geological maps to compute the hydraulic head CDF. Our numerical experiments reveal that the CDF-RDD remains accurate for two-dimensional flow in a porous material composed of two heterogeneous hydrofacies, a setting in which the original CDF method fails. For the same accuracy, the CDF-RDD is an order of magnitude faster than MCS.