Anis Younes

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.

Addressing factors fixing setting from given data

This paper deals with global sensitivity analysis of computer model output. Given an independent input sample and associated model output vector with possibly the vector of output derivatives with respect to the input variables, we show that it is possible to evaluate the following global sensitivity measures: (i) the Sobol indices, (ii) the Borgonovos density-based sensitivity measure, and (iii) the derivative-based global sensitivity measure of Sobol and Kucherenko. We compare the efficiency of the different methods to address factors fixing setting, an important issue in global sensitivity analysis. First, global sensitivity analysis of the Ishigami function is performed with the different methods. Then, they are applied to two different responses of a soil drainage model. The results show that the polynomial chaos expansion for estimating Sobol indices is the most efficient approach. We compare several methods for global sensitivity analysis of model output.All the methods only require one single sample with input variables independently distributed.We study their efficiency to address factors fixing setting.The different methods are tested on the Ishigami function and a soil drainage model.Sobol indices via sparse PCE are found to be the most efficient approach.

Analytical solution and Bayesian inference for interference pumping tests in fractal dual-porosity media

A new analytical solution is developed for interference hydraulic pumping tests in fractal fractured porous media using the dual-porosity concept. Heterogeneous fractured reservoirs are considered with hydrodynamic parameters assumed to follow power-law functions in radial distance. The developed analytical solution is verified by comparison against a finite volume numerical solution. The comparison shows that the numerical solution converges toward the analytical one when the size of the time step decreases. The applicability of the fractal dual-porosity model is then assessed by investigating the identifiability of the parameters from a synthetic interference pumping test with a set of noisy data using Bayesian parameter inference. The results show that if the storage coefficient in the matrix is fixed, the rest of the parameters can be appropriately inferred; otherwise, the identification of the parameters is faced with convergence problems because of equifinality issues.

Behaviour of mixed finite element and multipoint point flux approximations for flow with discontinuous coefficients

We study the numerical behaviour and the relationships between some numerical methods derived from the Mixed Finite Element (MFE) formulation and from the Control Volume formulation using the Multi-Point Flux Approximation (MPFA) of Aavatsmark et al. (1996). All methods are locally mass conservative and are well suited to solve the flow problem for both steady state (elliptic case) an transient (parabolic case) on a general irregular grid with anisotropic and heterogeneous discontinuous coefficients. Hybridization of the standard MFE method allows to obtain a symmetric positive definite system which is not, in general, an M-matrix. Moreover, the mixed hybrid finite element method uses more unknowns (one per edge or face) than the finite volume formulation (one per element). On the other hand, for a general element grid, the MPFA method does not give a symmetric matrix of coefficients and monotonicity issues are known to arise for high aspect ratios combined with skewed of computational grids. In this work, we propose two variants of the MFE method: (i) the lumped-MFE method, based on the equivalence between MHFE method and the P1-nonconforming Galerkin method for the Laplace equation and (ii) the multipoint-MFE method based on Multipoint finite volume discretization using the framework of the modified mixed finite element space. We discuss connections between the different approaches and perform numerical experiments for both steady state and transient cases to compare (number of unknowns, matrix properties, condition number) and study the numerical behaviour of these methods (unphysical oscillations, CPU time).

A New Mass Lumping Scheme for the Mixed Hybrid Finite Element Method: Application to unsaturated water flow modelling

Abstract: Groundwater flow modelling is of interest in many sciences and engineering applications for scientific understanding and/or technological management. Accurate numerical simulation of infiltration in the vadose zone remains a challenge, especially when very sharp fronts are present. This study is focused principally on an alternatively numerical approaches referred to in the literature as the mixed hybrid finite element (MHFE) method. MHFE schemes simultaneously approximate both the pressure head and its gradient. For some problems of unsaturated water flow, the MHFE solutions contain oscillations. Various authors ( see [1]) suggest the use of a mass lumping procedure to avoid this unphysical phenomenon. An analyse of the resulting matrix system shows that the recommended technique differs from the standard mass-lumping well-established for Galerkin finite element methods. A “new” effective mass-lumping scheme adapted from [2] has been specially developed for the MHFE method. Its ability for eliminating oscillations have been tested in unsaturated conditions. Various test cases in a 2D domain, for homogeneous and heterogeneous dry porous media and subject to different boundary conditions are presented. References: [1] Farthing, M. W., C. E. Kees, and C. T. Miller. 2003. Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow. Adv. Water Resour. 26:373-394. [2] Younes A., Ackerer P. and Lehmann F., 2005.A new mass lumping scheme for the mixed hybrid finite element method, Int. J. Numer. Meth. Engng. (submitted).