Sensitivity Methods for Time‐Continuous, Spatially Discrete Groundwater Contaminant Transport Models

Abstract

Existing sensitivity methods for spatially discrete groundwater contaminant transport models have been developed for time-stepping numerical algorithms and cannot be readily used with time-continuous approaches to transport simulation, such as the Laplace transform Galerkin technique. We develop direct and adjoint sensitivity methods in which sensitivity coefficients are computed in the Laplace domain and inverted numerically to the time domain. The methods are computationally efficient when used in conjunction with time-continuous transport equations. The relative efficiency of the two methods depends on the number of model parameters, number of performance measures, and number of spatial discretization nodes. The adjoint method is favored when the number of performance measures is much smaller than the number of model parameters. The adjoint method is limited in that performance measures are restricted to being linear functions of state variables. A two-dimensional transport example is developed in detail, and sensitivities with respect to nodal hydraulic conductivities are computed. In the problems analyzed, the direct and adjoint methods are 9 to 156 times faster than the perturbation method, with the computational savings increasing as the size of the problem is increased. IntroductionMany facets of groundwater contaminant transport modeling require or benefit from sensitivity analysis, i.e., a systematic evaluation of how a model responds to changes in input parameters. Among other things, sensitivity analysis is used to estimate model parameters, design experiments, analyze groundwater management problems, and evaluate model robustness. The most important quantity in sensitivity analysis is the sensitivity coefficient, defined in the context of transport modeling as the first derivative of concentration (a state variable) with respect to a model parameter. Sensitivity coefficients, which are also called state sensitivities or simply sensitivities, provide a quantitative measure of the direction and magnitude of a model's response to small changes in input parameters.Three standard approaches to computing sensitivity coefficients are (1) the perturbation method, (2) the direct method, and (3) the adjoint method. The perturbation method, also known as the influence coefficient or divided difference method, computes a numerical approximation to the sensitivity coefficients using state variables calculated during multiple simulation runs. Following an initial base run of the model, further runs are made in which parameters are slightly perturbed one by one. Using state variables from the base run and perturbed runs, divided difference approximations to the sensitivity coefficients are computed. Thus if sensitivities for Np parameters are sought and a two-point divided difference approximation is used, Np + 1 simulation runs are required. The advantage of the perturbation method is that an existing com-1Now at U.S. Salinity Laboratory, Riverside, California. puter code can be used with only minimal additional p...

Multidataset Study of Optimal Parameter and Uncertainty Estimation of a Land Surface Model with Bayesian Stochastic Inversion and Multicriteria Method

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“…There are many methods to infer parameter sensitivity that have been applied in meteorological contexts, including the traditional perturbation (one parameter at a time) technique (e.g., Wilson et al 1987;Bonan et al 1993;Pitman 1994;Alapaty et al 1997), variational (adjoint) methods (e.g., Skaggs and Barry 1996;Margulis and Entekhabi 2001), factorial methods (Henderson-Sellers 1993;Niyogi et al 1999), Fourier amplitude sensitivity tests (e.g., Collins and Avissar 1994), muticriteria methods (e.g., Bastidas et al 1996Bastidas et al , 1999, reduced form model (Beringer et al 2002), and response surface methods (Niyogi et al 1998(Niyogi et al , 2002. Each of these techniques has benefits and drawbacks.…”

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“…Then, the ODE system is analytically solved to yield the final closed-form solution. The semidiscrete approach has been applied to contaminant transport modeling in porous media (Guymon, 1970;Guymon et al, 1970;Nalluswami et al, 1972;Willis, 1979;Hwang et al, 1984;Umari and Gorelick, 1986;Kamra et al, 1991a,b;Skaggs and Barry, 1996). This solution method overcomes the limitations to the analytical solution engendered by heterogeneous media and irregular boundary geometry (Hwang et al, 1984).…”

Modelling the fate of oxidisable organic contaminants in groundwater

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“…Thus, the same rate expressions apply as in Case 1b, with an additional rate expression as given by Eq. (33). The actual species that are produced will depend on the chemical composition of the aqueous phase at the time of release.…”