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

Abstract

This work presents an efficient mathematical/numerical model to compute the sensitivity coefficients of a predefined performance measure to model parameters for one‐dimensional steady‐state sequentially coupled radionuclide transport in a finite heterogeneous porous medium. The model is based on the adjoint sensitivity approach that offers an elegant and computationally efficient alternative way to compute the sensitivity coefficients. The transport parameters include the radionuclide retardation factors due to sorption, the Darcy velocity, and the effective diffusion/dispersion coefficients. Both continuous and discrete adjoint approaches are considered. The partial differential equations associated with the adjoint system are derived based on the adjoint state theory for coupled problems. Physical interpretations of the adjoint states are given in analogy to results obtained in the theory of groundwater flow. For the homogeneous case, analytical solutions for primary and adjoint systems are derived and presented in closed forms. Numerically calculated solutions are compared to the analytical results and show excellent agreements. Insights from sensitivity analysis are discussed to get a better understanding of the values of sensitivity coefficients. The sensitivity coefficients are also computed numerically by finite differences. The numerical sensitivity coefficients successfully reproduce the analytically derived sensitivities based on adjoint states. A derivative‐based global sensitivity method coupled with the adjoint state method is presented and applied to a real field case represented by a site currently being considered for underground nuclear storage in Northern Switzerland, “Zürich Nordost”, to demonstrate the proposed method. The results show the advantage of the adjoint state method compared to other methods in term of computational effort.

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

“…The dimension of the adjoint states are those of the performance measure J times (TL 3 /M). A similar interpretation was recently obtained in the case of steady-state sequentially coupled radionuclide transport (Hayek et al, 2019).…”

“…Both continuous and discrete approaches can provide very similar results (Sykes et al, 1985;Samper & Neuman, 1986;Delay et al, 2017;Hayek et al, 2019). The discrete approach is easy to implement and computationally efficient because the resulting adjoint system involves the transposes of the matrices associated with the forward system.…”

“…This appendix presents the key steps needed to obtain the final adjoint system of PDEs 13 starting from 8. The gradient matrix operators ∇ C L and ∇ p L are defined as (Hayek et al, 2019)…”

“…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).…”

“…In a recent work, Hayek et al (2019) proposed an adjoint sensitivity model applied to RN transport problem coupled with first-order reactions in a steady-state regime. In the present work, the adjoint methodology is extended to the more realistic transient case.…”