Analytical sensitivity analysis of transient groundwater flow in a bounded model domain using the adjoint method

Abstract

Sensitivity analyses are an important component of any modeling exercise. We have developed an analytical methodology based on the adjoint method to compute sensitivities of a state variable (hydraulic head) to model parameters (hydraulic conductivity and storage coefficient) for transient groundwater flow in a confined and randomly heterogeneous aquifer under ambient and pumping conditions. For a special case of two-dimensional rectangular domains, these sensitivities are represented in terms of the problem configuration (the domain size, boundary configuration, medium properties, pumping schedules and rates, and observation locations and times), and there is no need to actually solve the adjoint equations. As an example, we present analyses of the obtained solution for typical groundwater flow conditions. Analytical solutions allow us to calculate sensitivities efficiently, which can be useful for model-based analyses such as parameter estimation, data-worth evaluation, and optimal experimental design related to sampling frequency and locations of observation wells. The analytical approach is not limited to groundwater applications but can be extended to any other mathematical problem with similar governing equations and under similar conceptual conditions.

Characterizing subsurface hydraulic heterogeneity of alluvial fan using riverstage fluctuations

“…where ( ) r fh J (n×m) is the sensitivity of head with respect to the change of parameter. The sensitivity matrix is evaluated using the adjoint state approach [Li and Yeh, 1998;Li and Yeh, 1999;Lu and Vesselinov, 2015]. After completion of the linear estimation, the conditional residual covariance of ( 1) r + f is updated subsequently by…”

An Adjoint Sensitivity Model for Steady‐State Sequentially Coupled Radionuclide Transport in Porous Media

“…These coefficients reflect the sensitivities of user-defined scalar measures of system behavior or model performance, called performance measures, to various model parameters. For instance, they are used to determine the relative sensitivity of performance measures with respect to the parameters of the system (Lu & Vesselinov, 2015;Sykes et al, 1985;Wilson & Metcalfe, 1985), to guide gradient search in nonlinear optimization (Hayek et al, 2008), or to perform firstand second-order, second-moment uncertainty analyses (Li & Yeh, 1998).…”

“…The adjoint state method has been used successfully in a wide range of disciplines such as mathematical physics, geophysics, systems engineering, economics, constrained optimization, nuclear engineering, electrical engineering, meteorology, oceanography, hydrogeology, petroleum engineering, and seismology. In hydrogeology, the adjoint state method has been employed in many applications including interpretation of interference tests using geostastistical techniques (de Marsily et al, 1984), steady-state groundwater flow (Sykes et al, 1985;Wilson & Metcalfe, 1985), groundwater travel time uncertainty analysis (LaVenue et al, 1989), automated calibration of transmissivity fields (LaVenue et al, 1995;RamaRao et al, 1995), coupled nonlinear multiphase multicomponent flow (RamaRao & Mishra, 1996), modeling multidimensional groundwater flow (Clemo, 2007), adaptive multiscale parameterization of flow in unsaturated porous media (Hayek et al, 2008), transient groundwater flow in a bounded model domain (Lu & Vesselinov, 2015), fractured dual-porosity media (Delay et al, 2017;Fahs et al, 2014), and coupled surface water-groundwater modeling (RamaRao et al, 2017). However, few applications of the adjoint state method have been devoted to solute transport (e.g., Larbkich et al, 2014;Michalak & Kitanidis, 2004;Neupauer & Wilson, 1999, 2001Piasecki & Katopodes, 1997).…”

An Adjoint Sensitivity Model for Transient Sequentially Coupled Radionuclide Transport in Porous Media

“…The adjoint state method offers an elegant and computationally efficient alternative to the parameter perturbation method and the sensitivity equation method. It has already been successfully applied in a broad range of fields including mathematical physics, electrical engineering, systems engineering, nuclear reactor assessment, hydrology, petroleum engineering, surface-water hydrology, meteorology, oceanography, and geophysics (Cacuci, 1981;Carrera & Neuman, 1986;Chavent et al, 1975;de Marsily et al, 1984;Delay et al, 2017Delay et al, , 2019Director & Rohrer, 1969;Hayek et al, 2008Hayek et al, , 2019Lavenue et al, 1989Lavenue et al, , 1995LaVenue & de Marsily, 2001;Losch & Heimbach, 2007;Lu & Vesselinov, 2015;Plessix, 2006;RamaRao et al, 1995;Sykes et al, 1985;Wilson & Metcalfe, 1985).…”