Method of distributions for quantification of geologic uncertainty in
flow simulations
Abstract
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.