Dynamic Data Integration and Quantification of Prediction Uncertainty Using Statistical-Moment Equations

Water Resources Research

2019

Uncertainty about geologic makeup and properties of the subsurface renders inadequate a unique quantitative prediction of flow and transport. Instead, multiple alternative scenarios have to be explored within the probabilistic framework, typically by means of Monte Carlo simulations (MCS). These can be computationally expensive, and often prohibitively so, especially when the goal is to compute the tails of a distribution, that is, probabilities of rare events, which are necessary for risk assessment and decision making under uncertainty. We deploy the method of distributions to derive a deterministic equation for the cumulative distribution function (CDF) of hydraulic head in an aquifer with uncertain (random) hydraulic conductivity. The CDF equation relies on a self‐consistent closure approximation, which ensures that the resulting CDF of hydraulic head has the same mean and variance as those computed with statistical moment equations. We conduct a series of numerical experiments dealing with steady‐state two‐dimensional flow driven by either a natural hydraulic head gradient or a pumping well. These experiments reveal that the CDF method remains accurate and robust for highly heterogeneous formations with the variance of log conductivity as large as five. For the same accuracy, it is also up to four orders of magnitude faster than MCS in computing hydraulic head with a required degree of confidence (probability).

Section: Accuracy Of the Cdf Methodssupporting

confidence: 88%

Section: 1029/2019wr026090mentioning

confidence: 99%

Section: Appendix A: Derivation Of Moment Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Probabilistic Forecast of Single‐Phase Flow in Porous Media With Uncertain Properties

Water Resources Research

2019

“…The first step involves finite-volume solutions of the moment equations (B4)-(B13), that is, provides numerical approximations of the mean and variance of the hydraulic head, hðxÞ and σ 2 h ðxÞ. This step relies on the research code developed by Likanapaisal et al (2012).…”

Section: Numerical Implementationmentioning

confidence: 99%

“…The presence of multiple hydrofacies Ω i manifests itself in a histogram of the measurement set

{false{K false({boldx}_{n} false) false}}_{n = 1}^{N_{meas}}

(an estimate of the PDF of K ) that exhibits multimodal behavior and large standard deviation σ K ‐. This typical setting would increase the computational cost of MCS and invalidate the perturbation‐based moment differential equations (Likanapaisal et al., 2012) and PDF/CDF equations (Yang et al., 2019), both of which require the perturbation parameter

σ_{Y}^{2}

(the variance of log‐conductivity

Y = normalln K

) to be relatively small.…”

Section: Problem Formulation and Its Probabilistic Solutionmentioning

confidence: 99%

Method of Distributions for Quantification of Geologic Uncertainty in Flow Simulations

Water Resources Research

2020

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 statistical moments such 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 function (CDF‐) of the QoI. The standard 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 four. Our CDF‐random domain decomposition (RDD) framework alleviates these limitations by combining the method of distributions and 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 map‐ to compute the hydraulic head CDF. Our numerical experiments reveal that the CDF‐RDD method remains accurate for two‐dimensional flow in a porous medium composed of two heterogeneous hydrofacies, a setting in which the original CDF method fails. For the same accuracy, the CDF‐RDD method is an order of magnitude faster than MCS.

Method of distributions for quantification of geologic uncertainty in flow simulations

2022

Preprint

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.