Integrated, process-based numerical models in hydrology are rapidly evolving, spurred by novel theories in mathematical physics, advances in computational methods, insights from laboratory and field experiments, and the need to better understand and predict the potential impacts of population, land use, and climate change on our water resources. At the catchment scale, these simulation models are commonly based on conservation principles for surface and subsurface water flow and solute transport (e.g., the Richards, shallow water, and advection-dispersion equations), and they require robust numerical techniques for their resolution. Traditional (and still open) challenges in developing reliable and efficient models are associated with heterogeneity and variability in parameters and state variables; nonlinearities and scale effects in process dynamics; and complex or poorly known boundary conditions and initial system states. As catchment modeling enters a highly interdisciplinary era, new challenges arise from the need to maintain physical and numerical consistency in the description of multiple processes that interact over a range of scales and across different compartments of an overall system. This paper first gives an historical overview (past 50 years) of some of the key developments in physically based hydrological modeling, emphasizing how the interplay between theory, experiments, and modeling has contributed to advancing the state of the art. The second part of the paper examines some outstanding problems in integrated catchment modeling from the perspective of recent developments in mathematical and computational science.

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1002/2015WR017780

Physically based modeling in catchment hydrology at 50: Survey and outlook

[1] This study reports on two strategies for accelerating posterior inference of a highly parameterized and CPU-demanding groundwater flow model. Our method builds on previous stochastic collocation approaches, e.g., Marzouk and Xiu (2009) and Marzouk and Najm (2009), and uses generalized polynomial chaos (gPC) theory and dimensionality reduction to emulate the output of a large-scale groundwater flow model. The resulting surrogate model is CPU efficient and serves to explore the posterior distribution at a much lower computational cost using two-stage MCMC simulation. The case study reported in this paper demonstrates a two to five times speed-up in sampling efficiency.

/pdf/efficient-posterior-exploration-of-a-high-dimensional-1bav9vwse1.pdf

Efficient posterior exploration of a high‐dimensional groundwater model from two‐stage Markov chain Monte Carlo simulation and polynomial chaos expansion

Global sensitivity analysis is now established as a powerful approach for determining the key random input parameters that drive the uncertainty of model output predictions. Yet the classical computation of the so-called Sobol’ indices is based on Monte Carlo simulation, which is not af- fordable when computationally expensive models are used, as it is the case in most applications in engineering and applied sciences. In this respect metamodels such as polynomial chaos expansions (PCE) and Gaussian processes (GP) have received tremendous attention in the last few years, as they allow one to replace the original, taxing model by a surrogate which is built from an experimental design of limited size. Then the surrogate can be used to compute the sensitivity indices in negligible time. In this chapter an introduction to each technique is given, with an emphasis on their strengths and limitations in the context of global sensitivity analysis. In particular, Sobol’ (resp. total Sobol’) indices can be computed analytically from the PCE coefficients. In contrast, confidence intervals on sensitivity indices can be derived straightforwardly from the properties of GPs. The performance of the two techniques is finally compared on three well-known analytical benchmarks (Ishigami, G-Sobol and Morris functions) as well as on a realistic engineering application (deflection of a truss structure).

pdf/metamodel-based-sensitivity-analysis-polynomial-chaos-goxm01riqs.pdf

Metamodel-based sensitivity analysis: polynomial chaos expansions and Gaussian processes

https://hal.archives-ouvertes.fr/hal-01476649/document

Bayesian sparse polynomial chaos expansion for global sensitivity analysis

http://sharif.ir/%7Eataie/Printed-Articles/1JH_2015.pdf

Polynomial chaos expansions for uncertainty propagation and moment independent sensitivity analysis of seawater intrusion simulations

In this work, we show how the use of global sensitivity analysis (GSA) in conjunction with the polynomial chaos expansion (PCE) methodology can provide relevant information for the interpretation of transport experiments in laboratory-scale heterogeneous porous media. We perform GSA by calculating the Sobol indices, which provide a variance-based importance measure of the effects of uncertain parameters on the output of a chosen interpretive transport model. The choice of PCE has the following two benefits: (1) it provides the global sensitivity indices in a straightforward manner, and (2) PCE can serve as a surrogate model for the calibration of parameters. The coefficients of the PCE are computed by probabilistic collocation. The methodology is applied to two nonreactive transport experiments available in the literature, while considering both transient and pseudo steady state transport regimes. This method allows a rigorous investigation of the relative effects and importance of different uncertain quantities, which include boundary conditions as well as porous medium hydraulic and dispersive parameters. The parameters that are most relevant to depicting the system's behavior can then be evaluated. In addition, one can assess the space-time distribution of measurement points, which is the most influential factor for the identifiability of parameters. Our work indicates that these methods can be valuable tools in the proper design of model-based transport experiments.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2010WR009639

Use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory‐scale porous media

:

Groundwater flow modelling is of interest in many sciences and engineering 
applications for scientific understanding and/or technological management. Accurate 
numerical simulation of infiltration in the vadose zone remains a challenge, 
especially when very sharp fronts are present. 
This study is focused principally on an alternatively numerical approaches referred 
to in the literature as the mixed hybrid finite element (MHFE) method. MHFE schemes 
simultaneously approximate both the pressure head and its gradient. For some 
problems of unsaturated water flow, the MHFE solutions contain oscillations. Various 
authors ( see [1]) suggest the use of a mass lumping procedure to avoid this 
unphysical phenomenon. An analyse of the resulting matrix system shows that the 
recommended technique differs from the standard mass-lumping well-established for 
Galerkin finite element methods.
A “new” effective mass-lumping scheme adapted from [2] has been specially developed 
for the MHFE method. Its ability for eliminating oscillations have been tested in 
unsaturated conditions. Various test cases in a 2D domain, for homogeneous and 
heterogeneous dry porous media and subject to different boundary conditions are 
presented.

References:

[1] Farthing, M. W., C. E. Kees, and C. T. Miller. 2003. Mixed finite element 
methods and higher order temporal approximations for variably saturated groundwater 
flow. Adv. Water Resour. 26:373-394.
[2] Younes A., Ackerer P. and Lehmann F., 2005.A new mass lumping scheme for the 
mixed hybrid finite element method, Int. J. Numer. Meth. Engng. (submitted).

http://proceedings.dtu.dk/fedora/repository/dtu:1513/OBJ/article.pdf

A New Mass Lumping Scheme for the Mixed Hybrid Finite Element Method: Application to unsaturated water flow modelling

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Use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory‐scale porous media

A New Mass Lumping Scheme for the Mixed Hybrid Finite Element Method: Application to unsaturated water flow modelling