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Probabilistic Forecast of Single‐Phase Flow in Porous Media With Uncertain Properties
Author(s) -
Jun Yang Hyung,
Boso Francesca,
Tchelepi Hamdi A.,
Tartakovsky Daniel M.
Publication year - 2019
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/2019wr026090
Subject(s) - hydraulic conductivity , monte carlo method , hydraulic head , cumulative distribution function , probabilistic logic , flow (mathematics) , uncertainty quantification , probability distribution , variance (accounting) , moment (physics) , mathematics , computer science , mathematical optimization , statistical physics , probability density function , statistics , geology , mechanics , geotechnical engineering , physics , soil science , accounting , classical mechanics , business , soil water
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).