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The development of the adjoint state method for the system of partial differential equations (PDEs) describing radionuclide transport in heterogeneous discrete fractured porous media in steady‐state regime is presented in this study for both continuous and discrete adjoint approaches. The continuous adjoint approach has its own advantage in the sense that the adjoint system can be derived in terms of PDEs which can be solved analytically for specific performance measures. These PDEs can then be used to obtain a physical interpretation of the adjoint state functions. The discrete adjoint approach permits a simpler mathematical treatment based on the use of the same matrix (or its transpose) of the linear system of the discretized transport equations referring to from hereon as the forward problem. Analytical solutions for both the forward problem and its associated adjoint problem have been derived for radionuclide transport in layered formations including a fracture using an analytical inversion of tridiagonal matrices. A set of 14 formulas of sensitivity coefficients with respect to pertinent system parameters has been derived and presented in integral forms. Analytical results have been compared with numerical results and have shown excellent agreements. The discrete adjoint approach has been coupled with the differential sensitivity analysis method and applied to a real field case involving the performance assessment of a site currently being considered for underground nuclear storage in northern Switzerland, Zürich Nordost. The results show that when a fracture is present in the vicinity of waste repository, the state parameter controlling flow through the fracture, referred to herein as fracture transmissivity, is among the important parameters which should be considered in sensitivity analysis studies. Key Points: A full mathematical/numerical adjoint sensitivity model for steady‐state RN transport in heterogeneous discrete fractured media is presentedAnalytical solutions for concentrations and adjoint states have been derived for the case of layered formations including a fractureThe developed adjoint approach has been coupled with the differential sensitivity analysis method and applied to a real field case [ABSTRACT FROM AUTHOR]

We present an efficient mathematical/numerical adjoint sensitivity analysis model for transient radionuclide transport through heterogeneous porous media. This work extends our previous research on radionuclide transport in a steady‐state regime. Both continuous and discrete adjoint approaches have been developed in this current research. The mathematical equations associated with the adjoint system and required for the continuous approach have been derived. The methodology for deriving the discrete adjoint solution is also presented. For the homogeneous case, the adjoint system for radionuclide decay chain is solved analytically. The four‐member decay chain (238Pu → 234U → 230Th → 226Ra) is considered for validation. The validity of the analytical adjoint sensitivity model has been shown using two illustrative examples associated with two performance measures. The analytical adjoint states are compared with calculated numerical results and show excellent agreements. The sensitivity coefficients are also computed numerically using the perturbation method. The numerical sensitivity coefficients successfully reproduce the analytically derived sensitivities based on adjoint states. The adjoint model has been coupled with a derivative‐based global sensitivity method and applied to a real field case involving the performance assessment of a site currently being considered for underground nuclear storage in northern Switzerland, Zürich Nordost. An illustrative case study of spent fuel and high‐level waste (SF/HLW) repository involving 6 radionuclides and 48 parameters has been considered for identifying the most important parameters with respect to the maximum radionuclide dose at the interface geosphere/biosphere. Based on the results of sensitivity analysis, the uncertainty of the maximum radionuclide dose has been performed using Monte Carlo simulations. Key Points: An adjoint sensitivity model for transient sequentially coupled radionuclide transport in heterogeneous media is presentedAnalytical expressions including adjoint state functions and sensitivity coefficients are derived in closed forms for the homogeneous caseA derivative‐based global sensitivity method coupled with the adjoint approach is presented and applied to a real field case [ABSTRACT FROM AUTHOR]

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. Key Points: A full mathematical/numerical adjoint sensitivity approach for steady‐state sequentially coupled radionuclide transport in heterogeneous media is presentedAnalytical expressions including adjoint state functions and sensitivity coefficients are derived in closed forms for the homogeneous caseA derivative‐based global sensitivity method coupled with the adjoint approach is presented and applied to a real field case [ABSTRACT FROM AUTHOR]

Despite its popularity in many engineering fields, the nonlinear diffusion–convection equation has no general analytical solutions. This work presents a family of closed-form analytical traveling wave solutions for the nonlinear diffusion–convection equation with power law nonlinearities. This kind of equations typically appears in nonlinear problems of flow and transport in porous media. The solutions that are addressed are simple and fully analytical. Three classes of analytical solutions are presented depending on the type of the nonlinear diffusion coefficient (increasing, decreasing or constant). It has shown that the structure of the traveling wave solution is strongly related to the diffusion term. The main advantage of the proposed solutions is that they are presented in a unified form contrary to existing solutions in the literature where the derivation of each solution depends on the specific values of the diffusion and convection parameters. The proposed closed-form solutions are simple to use, do not require any numerical implementation, and may be implemented in a simple spreadsheet. The analytical expressions are also useful to mathematically analyze the structure and properties of the solutions. [ABSTRACT FROM AUTHOR]

The subsurface spill of hydrocarbon fluids (from an underground leaking tank or from pipelines) is a widespread problem in the oil industry and hydrology. Under the assumption of low oil saturation, the migration of oil pollutant may be modeled by a nonlinear convection-diffusion equation with power law nonlinearities. The saturation-dependent diffusion coefficient is proportional to a power of saturation for which the exponent may be negative or positive depending on the porous medium and the pollutant characteristics. This paper presents general analytical solutions for the one-dimensional transient saturation distribution in immiscible zone. The solutions describe the longtime behavior of a contaminant oil pollutant caused by a continuous source of contaminant (e.g., leakage from pipelines, disposal dumps or storage tanks). Two general forms corresponding to positive and negative power exponents of the diffusion coefficient are presented. The general forms express implicitly the position as function of oil saturation and time. An alternative explicit closed-form solution near the front position is also proposed. An exact explicit solution is obtained for the particular case of constant diffusion coefficient. The proposed solutions are fully analytical in the sense that they do not require any numerical implementation. The solutions may be used to study the porous medium and pollutant parameters. They can be applied to aquifer contamination problems, and they are also important for the verification of numerical solutions. [ABSTRACT FROM AUTHOR]

Attempts have been made to find exact solutions for the one-dimensional transient gas flow equation in porous media. By introducing a traveling wave variable, a traveling wave solution of the gas flow equation has been found. The traveling wave solution is presented in an explicit form of the space and time variables, and it takes into account both gravity and Klinkenberg effects (pressure-dependent permeability). We investigated the properties of the traveling wave solution and the effect of some parameters such as the Klinkenberg coefficient. A numerical study has been carried out, which confirms the stability of the traveling wave solution. The traveling wave solution is then used to derive two benchmark solutions defined over the semi-infinite domain. The first one assumes uniform initial gas pressure and non-uniform boundary condition, and the second assumes uniform boundary condition and non-uniform initial distribution of the gas pressure. The benchmark solutions are easy to use and are useful for validating numerical solutions. Two illustrative examples are presented in order to compare the benchmark solutions with the numerical solutions. The results show good agreements between the solutions. [ABSTRACT FROM AUTHOR]

Summary The paper presents certain exact solutions describing the vertical movement of a water pulse through a semi-infinite unsaturated porous column. The saturation-based form of the Richards’ equation is used with special power law relative-permeability functions. Both capillary and gravity effects are taken into account. Three exact solutions are derived corresponding to three relative-permeability functions, linear, quadratic and cubic. The Richards’ equation is nonlinear for the three cases. The solutions are obtained by applying a general similarity transformation. They are explicit in space and time variables and do not contain any approximation. They describe the evolution of the water saturation in the vertical column and they can be used to predict the post-infiltration movement of a finite quantity of water. Exact expressions of the masses of water leaving a given depth are also derived for the three cases. We analyze the effect of relative-permeability and capillary pressure. The proposed solutions are also useful for checking numerical schemes. One of the exact solutions is used to validate numerical solution obtained from an arbitrary initial condition. Results show that the numerical solution converges to the exact solution for large times. [ABSTRACT FROM AUTHOR]

Highlights: [•] We develop a complete algorithm for the parameterization in 2D fractured dual media. [•] The parameterization algorithm is based on the refinement indicators. [•] We use adjoint state method for the calculation of the local gradients. [•] We study robustness of the developed algorithm using several numerical experiments. [•] Algorithm provides a good parameterization even in critical cases. [ABSTRACT FROM AUTHOR]

Abstract: We present an upscaled model for the vertical migration of a CO2 plume through a vertical column filled with a periodic layered porous medium. This model may describe the vertical migration of a CO2 plume in a perfectly layered horizontal aquifer. Capillarity and buoyancy are taken into account and semi-explicit upscaled flux functions are proposed in the two following cases: (i) capillarity is the main driving force and (ii) buoyancy is the only driving force. In both cases, we show that the upscaled buoyant flux is a bell-shaped function of the saturation, as in the case of a homogeneous porous medium. In the capillary-dominant case, we show that the upscaled buoyant flux is the harmonic mean of the buoyant fluxes in each layer. The upscaled saturation is governed by the continuity of the capillary pressure at the interface between layers. In the capillary-free case, the upscaled buoyant flux and upscaled saturation are determined by the flux continuity condition at the interface. As the flux is not continuous over the entire range of saturation, the upscaled saturation is only defined where continuity is verified, i.e. in two saturation domains. As a consequence, the upscaled buoyant flux is described by a piecewise continuous function. Two analytical approximations of this flux are proposed and this capillary-free upscaled model is validated for two cases of heterogeneity. Upscaled and cell averaged saturations are in good agreement. Furthermore, the proposed analytical upscaled fluxes provide satisfactory approximations as long as the saturation set at the inlet of the column is in a range where analytical and numerical upscaled fluxes are close. [ABSTRACT FROM AUTHOR]

Abstract: In the analysis of the unsaturated zone, one of the most challenging problems is to use inverse theory in the search for an optimal parameterization of the porous media. Adaptative multi-scale parameterization consists in solving the problem through successive approximations by refining the parameter at the next finer scale all over the domain and stopping the process when the refinement does not induce significant decrease of the objective function any more. In this context, the refinement indicators algorithm provides an adaptive parameterization technique that opens the degrees of freedom in an iterative way driven at first order by the model to locate the discontinuities of the sought parameters. We present a refinement indicators algorithm for adaptive multi-scale parameterization that is applicable to the estimation of multi-dimensional hydraulic parameters in unsaturated soil water flow. Numerical examples are presented which show the efficiency of the algorithm in case of noisy data and missing data. [Copyright &y& Elsevier]