Marwan Fahs

Short communication: An easy and efficient combination of the Mixed Finite Element Method and the Method of Lines for the resolution of Richards' Equation

In this work, the Mixed Hybrid Finite Element (MHFE) method is combined with the Method Of Lines (MOL) for an accurate resolution of the Richards Equation (RE). The combination of these methods is often complicated since hybridization requires a discrete approximation of the time derivative whereas with the MOL, it should remain continuous. In this paper, we use the new mass lumping technique developed in Younes et al. [Younes, A., Ackerer, P., Lehmann, F., 2006. A new mass lumping scheme for the mixed hybrid finite element method. International Journal for Numerical Methods in Engineering 67, pp. 89-107.] for the MHFE method. With this formulation, the MOL is easily implemented and sophisticated time integration packages can be used without significant amount of work. Numerical simulations are performed on both homogeneous and heterogeneous porous media to show the efficiency and robustness of the developed scheme.

Variable-density flow in heterogeneous porous media - laboratory experiments and numerical simulations.

Konz, M., Ackerer, P., Younes, A., Huggenberger, P., Zechner, E., 2009a. 2D Stable Layered Laboratory-scale Experiments for Testing Density-coupled Flow Models. Water Resources Research, 45. doi:10.1029/2008WR007118., a series of laboratory-scale 2D tank experiments were conducted and accurately simulated for density driven flow problems on homogeneous porous media. In the present work, we extended the numerical and experimental studies to heterogeneous problems. The heterogeneous porous medium was constructed with a low permeability zone in the centre of the tank and had well-defined parameters and boundary conditions. Concentration distributions were measured in high resolution using a photometric method and an image analysis technique. The numerical model used for the simulations was based on efficient advanced approximations for both spatial and temporal discretizations. The Method Of Lines (MOL) was used to allow higher-order temporal discretization. Three different boundary conditions, corresponding to different localizations of the inflow and the outflow openings at the opposite edges of the tank, were applied to investigate different flow scenarios in the heterogeneous porous medium flow tank. Simulation results of all three density coupled experiments revealed a density-dependent behavior of dispersion. Thus, a reduction of dispersivites was required to obtain a good matching of the experimental data. The high quality of the experiments enabled a detailed testing of numerical variable-density flow codes under heterogeneous conditions. Therefore, the experiments were considered to be reliable benchmark tests.

Modelling variable density flow problems in heterogeneous porous media using the method of lines and advanced spatial discretization methods

Modelling variable density flow problems under heterogeneous porous media conditions requires very long computation time and high performance equipments. In this work, the DASPK solver for temporal resolution is combined with advanced spatial discretization schemes in order to improve the computational efficiency while maintaining accuracy. The spatial discretization is based on a combination of Mixed Finite Element (MFE), Discontinuous Galerkin (DG) and Multi-point Flux Approximation methods (MPFA). The obtained non-linear ODE/DAE system is solved with the Method of Lines (MOL) using the DASPK time solver. DASPK uses the preconditioned Krylov iterative method to solve linear systems arising at each time step. Precise laboratory-scale 2D experiments were conducted in a heterogeneously packed porous medium flow tank and the measured concentration contour lines are used to evaluate the numerical model. Simulations show the high efficiency and accuracy of the code and the sensitivity analysis confirms the density dependence of dispersion.

A Reference Benchmark Solution for Free Convection in A Square Cavity Filled with A Heterogeneous Porous Medium

The Fourier-Galerkin (FG) method is used to produce a highly accurate solution for free convection in a square cavity filled with heterogeneous porous medium. To this end, the governing equations are reformulated in terms of the temperature and the stream function. These unknowns are then expanded in infinite Fourier series truncated at given orders. The accuracy of the FG solution is investigated for different truncation orders and compared to the results of an advanced finite-element numerical model using fine-mesh discretization. The obtained results represent a set of high-quality data that can be used for benchmarking numerical models.

The Henry problem: New semianalytical solution for velocity‐dependent dispersion

A new semi-analytical solution is developed for the velocity-dependent dispersion Henry problem using the Fourier-Galerkin method (FG). The integral arising from the velocity-dependent dispersion term is evaluated numerically using an accurate technique based on an adaptive scheme. Numerical integration and nonlinear dependence of the dispersion on the velocity render the semi-analytical solution impractical. To alleviate this issue, and to obtain the solution at affordable computational cost, a robust implementation for solving the nonlinear system arising from the FG method is developed. It allows for reducing the number of attempts of the iterative procedure and the computational cost by iteration. The accuracy of the semi-analytical solution is assessed in terms of the truncation orders of the Fourier series. An appropriate algorithm based on the sensitivity of the solution to the number of Fourier modes is used to obtain the required truncation levels. The resulting Fourier series are used to analytically evaluate the position of the principal isochlors and metrics characterizing the saltwater wedge. They are also used to calculate longitudinal and transverse dispersive fluxes and to provide physical insight into the dispersion mechanisms within the mixing zone. The developed semi-analytical solutions are compared against numerical solutions obtained using an in house code based on variant techniques for both space and time discretization. The comparison provides better confidence on the accuracy of both numerical and semi-analytical results. It shows that the new solutions are highly sensitive to the approximation techniques used in the numerical code which highlights their benefits for code benchmarking. This article is protected by copyright. All rights reserved.

A high-accurate solution for Darcy-Brinkman double-diffusive convection in saturated porous media

ABSTRACT The main purpose of this article is to present a high-accurate reference solution for double-diffusive convection in a confined saturated porous medium. The solution is developed using the Fourier-Galerkin spectral method for the coupled flow, heat, and mass transfer equations. The accuracy of the obtained solution is investigated in terms of the truncation orders of the Fourier series and by comparison against a newly developed finite-element model. Results of simulations highlight the accuracy of the proposed reference solution and show its worthiness for benchmarking numerical models dealing with Darcy-Brinkman double-diffusive convection.

Modeling 2D Multispecies Reactive Transport in Saturated/Unsaturated Porous Media with the Eulerian–Lagrangian Localized Adjoint Method

In the present paper, the Eulerian–Lagrangian localized adjoint method (ELLAM) formulation developed by Younes et al. (Advances in Water Resources 29:1056–1074, 2006) is combined with the sequential noniterative approach to accurately simulate 2D multicomponent reactive transport in saturated/unsaturated porous media. The performance and accuracy of the developed model, named ELLAM_REACT, are compared against those of an existing numerical model based on a combination of discontinuous Galerkin and multipoint flux approximation methods (DGMPFA_REACT). Three studied test cases, dealing with reactive transport in saturated and unsaturated porous media and involving chemical reactions with only aqueous species or both fixed and aqueous species, show the superiority of the ELLAM_REACT model compared to the DGMPFA_REACT model.

A High-Accurate Fourier-Galerkin Solution for Buoyancy-Driven Flow in a Square Cavity

In this work, a high-accurate solution is proposed for the verification of coupled fluid flow and heat transfer codes. The solution, based on the Fourier-Galerkin (FG) method, is developed for free convection in a square cavity with Rayleigh numbers ranging from 103 to 108. The Fourier coefficients are calculated by solving a highly nonlinear system of algebraic equations using Powells method. The FG results are then used as reference solutions to study the performance of a finite-element model and to illustrate the benefit of the developed solutions in benchmarking buoyancy-driven flows.

A new benchmark reference solution for double-diffusive convection in a heterogeneous porous medium

ABSTRACT A new benchmark with a high accurate solution is proposed for the verification of numerical codes dealing with double-diffusive convection in a heterogeneous porous medium. The new benchmark is inspired by the popular problem of square porous cavity by assuming a stratified porous medium. A high accurate steady state solution is developed using the Fourier–Galerkin method. To this aim, the unknowns are expanded in double infinite Fourier series. The accuracy of the developed solution is assessed in terms of the truncation orders of the Fourier series. Comparison against finite element solutions highlights the worthiness of the proposed benchmark for numerical code validation.

Global sensitivity analysis and Bayesian parameter inference for solute transport in porous media colonized by biofilms

The concept of dual flowing continuum is a promising approach for modeling solute transport in porous media that includes biofilm phases. The highly dispersed transit time distributions often generated by these media are taken into consideration by simply stipulating that advection-dispersion transport occurs through both the porous and the biofilm phases. Both phases are coupled but assigned with contrasting hydrodynamic properties. However, the dual flowing continuum suffers from intrinsic equifinality in the sense that the outlet solute concentration can be the result of several parameter sets of the two flowing phases. To assess the applicability of the dual flowing continuum, we investigate how the model behaves with respect to its parameters. For the purpose of this study, a Global Sensitivity Analysis (GSA) and a Statistical Calibration (SC) of model parameters are performed for two transport scenarios that differ by the strength of interaction between the flowing phases. The GSA is shown to be a valuable tool to understand how the complex system behaves. The results indicate that the rate of mass transfer between the two phases is a key parameter of the model behavior and influences the identifiability of the other parameters. For weak mass exchanges, the output concentration is mainly controlled by the velocity in the porous medium and by the porosity of both flowing phases. In the case of large mass exchanges, the kinetics of this exchange also controls the output concentration. The SC results show that transport with large mass exchange between the flowing phases is more likely affected by equifinality than transport with weak exchange. The SC also indicates that weakly sensitive parameters, such as the dispersion in each phase, can be accurately identified. Removing them from calibration procedures is not recommended because it might result in biased estimations of the highly sensitive parameters.