Uncertainty quantification for flow in highly heterogeneous porous media

doi:10.1016/S0167-5648(04)80092-3

Developments in Water Science

Volume 55, Part 1, 2004, Pages 695-703

https://doi.org/10.1016/S0167-5648(04)80092-3 Get rights and content

Natural porous media are highly heterogeneous and characterized by parameters that are often uncertain due to the lack of sufficient data. This uncertainty (randomness) occurs on a multiplicity of scales. We focus on geologic formations with the two dominant scales of uncertainty: a large-scale uncertainty in the spatial arrangement of geologic facies and a small-scale uncertainty in the parameters within each facies. We propose an approach that combines random domain decompositions (RDD) and polynomial chaos expansions (PCE) to account for the large- and small-scales of uncertainty, respectively. We present a general fremework and use a one-dimensional flow example to demonstrate that our combined approach provides robust, non-perturbative approximations for the statistics of the system states.

References (22)

A. Chorin
Hermite expansion in Monte-Carlo simulations
J. Comput. Phys.
(1971)
H. Matthies et al.
Finite elements for stochastic media problems
Comput. Methods Appl. Mech. Engrg.
(1999)
D. Xiu et al.
Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos
Comput. Methods Appl. Math. Engrg.
(2002)
D. Xiu et al.
Modeling uncertainty in flow simulations via generalized polynomial chaos
J. Comput. Phys.
(2003)
D. Xiu et al.
A new stochastic approach to transient heat conduction modeling with uncertainty
Inter. J. Heat Mass Trans.
(2003)
G. Dagan
Flow and transport in porous formations
(1989)
J.H. Cushman
The physics of fluids in hierarchical porous media: Angstroms to miles
(1997)
Y. Rubin
Applied stochastic hydrogeology
(2003)
D.M. Tartakovsky et al.
Stochastic averaging of nonlinear flows in heterogeneous porous media
J. Fluid Mech.
(2003)

K. D. Jarman, T. F. Russell, Moment equations for stochastic immiscible flow, SIAM J. on Multiscale Modeling and...

Cited by (9)

Streamflow, stomata, and soil pits: Sources of inference for complex models with fast, robust uncertainty quantification
2019, Advances in Water Resources
Citation Excerpt :
More broadly, one needs a holistic approach to UQ to seamlessly encapsulate all uncertainties of computer simulations within their specific contexts. This is a valuable pursuit for many computational sciences, not just hydrology, and as such, UQ has emerged in the last two decades as an active research field, which has incorporated applied mathematics, engineering, and physical sciences (e.g., Ghanem and Doostan, 2006; Gilbert et al., 2016; Knio and Le Maître, 2006; Najm, 2009; Sargsyan et al., 2014; Xiu and Tartakovsky, 2004). The overarching goal of UQ is to provide improvements in predictions and understanding of key sources and magnitudes of uncertainty, which can inform decision making and control for management of natural and engineered systems (Ascough Ii et al., 2008; da Cruz et al., 1999; García et al., 2015; Morss et al., 2005).
The scale and complexity of environmental and earth systems introduce an array of uncertainties that need to be systematically addressed. In numerical modeling, the ever-increasing complexity of representation of these systems confounds our ability to resolve relevant uncertainties. Specifically, the numerical representation of the governing processes involve many inputs and parameters that have been traditionally treated as deterministic. Considering them as uncertain introduces a large computational burden, stemming from the requirement of a prohibitive number of model simulations. Furthermore, within hydrology, most catchments are sparsely monitored, and there are limited, heterogeneous types of data available to confirm the model’s behavior. Here we present a blueprint of a general approach to uncertainty quantification for complex hydrologic models, taking advantage of recent methodological developments. We rely on polynomial chaos machinery to construct accurate surrogates that can be efficiently sampled for the ecohydrologic model tRIBS-VEGGIE to mimic its behavior with respect to a selected set of quantities of interest. The use of the Bayesian compressive sensing technique allows for fewer evaluations of the computationally expensive tRIBS-VEGGIE. The approach enables inference of model parameters using a set of observed hydrologic quantities including stream discharge, water table depth, evapotranspiration, and soil moisture from the Asu experimental catchment near Manaus, Brazil. The results demonstrate the flexibility of the framework for hydrologic inference in watersheds with sparse, irregular observations of varying accuracy. Significant computational savings imply that problems of greater computational complexity and dimension can be addressed using accurate, computationally cheap surrogates for complex hydrologic models. This will ultimately yield probabilistic representation of model behavior, robust parameter inference, and sensitivity analysis without the need for greater investment in computational resources.
Uncertainty quantification and inference of Manning's friction coefficients using DART buoy data during the Tōhoku tsunami
2014, Ocean Modelling
Citation Excerpt :
Here, we rely on Polynomial Chaos expansions to build a surrogate model, which, in addition, can efficiently provide statistical properties, such as the mean, variance and sensitivities. Polynomial Chaos (PC) is a probabilistic methodology that expresses the dependencies of model outputs on the uncertain model inputs as a truncated polynomial expansion (Villegas et al., 2012; Lin et al., 2009; Xiu and Tartakovsky, 2004). We briefly describe the PC method below; for more details the reader is referred to Le Maıˇtre and Knio (2010).
Tsunami computational models are employed to explore multiple flooding scenarios and to predict water elevations. However, accurate estimation of water elevations requires accurate estimation of many model parameters including the Manning’s n friction parameterization. Our objective is to develop an efficient approach for the uncertainty quantification and inference of the Manning’s n coefficient which we characterize here by three different parameters set to be constant in the on-shore, near-shore and deep-water regions as defined using iso-baths. We use Polynomial Chaos (PC) to build an inexpensive surrogate for the GeoClaw model and employ Bayesian inference to estimate and quantify uncertainties related to relevant parameters using the DART buoy data collected during the Tōhoku tsunami. The surrogate model significantly reduces the computational burden of the Markov Chain Monte-Carlo (MCMC) sampling of the Bayesian inference. The PC surrogate is also used to perform a sensitivity analysis.
Robust nonlinear MPC based on Volterra series and polynomial chaos expansions
2014, Journal of Process Control
A robust nonlinear model predictive controller (NMPC) based on a Volterra series is proposed. Polynomial chaos expansions (PCE) are used to represent the uncertainty in the Volterra series coefficients and this uncertainty is then propagated onto the output predictions. The key advantage of the PCE is that it provides an analytical expression to compute the L²-norm of the output prediction error resulting in computational savings, compared to previously proposed techniques, which are essential for real time implementation. Terminal and input constraints based on Structured Singular Value based-norms are used to ensure convergence to a set-point and compliance with constraints in manipulated variables. The algorithm is applied to a multivariable pH neutralization system. A comparative study shows superior closed loop performance and computational efficiency of the proposed technique as compared to previously proposed algorithms.
An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems
2007, Journal of Computational Physics
Citation Excerpt :
Variations of the method have been used to study diffusion [3,45] and fluid dynamics [47,51]. There is also a growing literature on applications of PC methods to steady state flow in porous media and related elliptic equations [4,5,22,41,64,77,79,80]. As opposed to MC methods, which require many domain-sized solves, PC methods typically produce large coupled systems.
We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method.
Quantification of Approaching Wind Uncertainty in Flow over Realistic Plant Canopies
2024, Boundary-Layer Meteorology
Breaking Down the Computational Barriers to Real-Time Urban Flood Forecasting
2021, Geophysical Research Letters

View all citing articles on Scopus

^*: This research was performed under the auspices of the U.S. Department of Energy, under contract W-7405-ENG-36. This work was supported in part by the U.S. Department of Energy under the DOE/BES Program in the Applied Mathematical Sciences, Contract KC-07-01-01, and in part by the LDRD Program at Los Alamos National Laboratory. This work made use of shared facilities supported by SAHRA (Sustainability of semi-Arid Hydrology and Riparian Areas) under the STC Program of the National Science Foudation under agreement EAR-9876800.

View full text

Uncertainty quantification for flow in highly heterogeneous porous media

J. Comput. Phys.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Math. Engrg.

J. Comput. Phys.

Inter. J. Heat Mass Trans.

Flow and transport in porous formations

The physics of fluids in hierarchical porous media: Angstroms to miles

Applied stochastic hydrogeology

Stochastic averaging of nonlinear flows in heterogeneous porous media

J. Fluid Mech.