Philippe Ackerer

On the finite volume reformulation of the mixed finite element method for elliptic and parabolic PDE on triangles

Abstract A new resolution of parabolic and elliptic partial differential equations (PDEs) based on the mixed finite element approximation on triangles has been recently developed [24] , [25] . This new approach reduces the number of unknowns from fluxes or Lagrange multiplier defined on edges to a single unknown per element. In this paper, we analyze this transformation mathematically, and describe in details how to handle singular elements and singular edges. For these singular elements, the standard mixed method on triangles can always be made equivalent to a finite volume formulation, where the finite volumes are obtained by aggregation of finite elements across singular edges. The positive definiteness of the system matrix obtained with the new formulation is analyzed in details. A criterion is given concerning the property of this matrix which show that its conditioning is related to the shape of the triangle and the contrast in parameters from one element to the adjacent ones. Numerical experiments are performed for elliptic and parabolic PDEs. The comparisons between an iterative solver (PCG) and a direct solver (unifrontal/multifrontal) show that the direct solver is more efficient. Moreover, its performance is not correlated with the system matrix conditioning. It appears that the new formulation requires significantly less CPU time for elliptic PDEs and is competitive for parabolic PDEs. The new formulation remains also accurate enough even in nearly singular situations.

Quantification of nitrate removal by a flooded alluvial zone in the Ill floodplain (Eastern France)

The nitrate reducing capacity of a flooded system in the Ill floodplain (Eastern France) was investigated for a period of 2 years. The methodology used consisted of a spatio-temporal monitoring of stream flow and nitrate concentrations in the groundwater and surface water, calculation of input and output fluxes and modelling of groundwater fluxes and nitrate transfer through the alluvial area. A comparison of chloride flux (used as hydrological tracer) and nitrate flux was done to determine a floodplain effect on the retention of nitrate. We show that up to 95% of the nitrate load in the groundwater is retained by the system, whereas the retention in the stream network is very low. Ammonium fluxes increased from inputs to outputs in the stream and in the groundwater. The chloride input in the groundwater is higher than the output, whereas in the surface water the output is higher than the input, the amount evacuated in streams corresponding to the losses from groundwater. The nitrate removal rate calculated for the whole modelized surface area (40 km2) represented 559 t N yr-1 or 1397.7 kg N ha-1 yr-1. The ammonium fluxes exported by the system represented 102 kg N ha-1 yr-1. A part of nitrate is reduced and exported by the groundwater and stream network in the form of ammonium. These results can be explained by the duration of floods which controls the equilibrium between the various forms of nitrogen. Thus, long watering periods favour nitrogen removal (denitrification and plant uptake) and limit nitrate production which compensates elimination during the dry period.

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.

Modeling the effects of water velocity on TiO2 nanoparticles transport in saturated porous media.

The transport of manufactured titanium dioxide (TiO2, rutile) nanoparticles (NP) in porous media was investigated under saturated conditions. Experiments were carried out with different fluid velocities, with values in the range of observed velocities in alluvial aquifers. As reported on the literature for different kinds of NPs, the amount of retained NPs decreased when the water velocity increased. Moreover, no retention was observed for ionic strength values smaller than 5mM. A transport model coupling convective-dispersive transport with a Langmuirian kinetic deposition was used to fit the BTCs. Empirical linear equations were developed to estimate the attachment rate ka and the maximal solid phase concentration smax. Both parameters were found to be linearly depending on the collector efficiency (η0). It was also observed that attachment efficiency (α) did not change with increase of water velocity under the given experimental conditions and that the model had a low sensitivity to α. Based on these estimates of the retention parameters, the classical dispersion-convection model coupled with a Langmuir type adsorption model was able to reproduce quite well the observed TiO2 breakthrough curves for every fluid velocity used in the experiments.

An Adaptive Subdivision Algorithm for the Identification of the Diffusion Coefficient in Two-dimensional Elliptic Problems

In this work, we consider the identification problem of the diffusion coef-ficient in two-dimensional elliptic equations. For parameterization, we use the zonation method: the diffusion coefficient is assumed to be a piecewise constant space function and unknowns are both the diffusion coefficient values and the geometry of the zones. An algorithm based on geometric principles is developed in order to determine the boundaries between the zones. This algorithm uses the refinement indicators which are easily computed from the gradient of the objective function. The efficiency of the algorithm is proved by testing it in some simple cases with and without noise on the data.

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.

Acid/base front propagation in saturated porous media: 2D laboratory experiments and modeling

We perform laboratory scale reactive transport experiments involving acid-basic reactions between nitric acid and sodium hydroxide. A two-dimensional experimental setup is designed to provide continuous on-line measurements of physico-chemical parameters such as pH, redox potential (Eh) and electrical conductivity (EC) inside the system under saturated flow through conditions. The electrodes provide reliable values of pH and EC, while sharp fronts associated with redox potential dynamics could not be captured. Care should be taken to properly incorporate within a numerical model the mixing processes occurring inside the electrodes. The available observations are modeled through a numerical code based on the advection-dispersion equation. In this framework, EC is considered as a variable behaving as a conservative tracer and pH and Eh require solving the advection dispersion equation only once. The agreement between the computed and measured pH and EC is good even without recurring to parameters calibration on the basis of the experiments. Our findings suggest that the classical advection-dispersion equation can be used to interpret these kinds of experiments if mixing inside the electrodes is adequately considered.

Modélisation du transport d'un soluté réactif en milieux poreux non saturés

Abstract Solute transfer in soils may create sharp concentration fronts. The numerical modeling of these fronts remains a difficult problem. The proposed one-dimensional model (Wamos-T) is robust and well adapted to these problems. It is based on operator splitting: discontinuous finite elements combined with a slope limiting procedure is used to discretize the advective term, the dispersive and reactive terms of the transport equation being solved by an implicit finite differences scheme. Compared to a widely used finite element method (Hydrus-1D), Wamos-T provides more accurate results when sharp fronts occur.

Accuracy and efficiency of time integration methods for 1D diffusive wave equation

The diffusive wave approximation of the Saint-Venant equations is commonly used in hydrological models to describe surface flow processes. Numerous numerical approaches can be used to solve this highly nonlinear equation. Nonlinear time integration schemes—also called methods of lines (MOL)—were proven very efficient to solve other nonlinear problems in geosciences but were never considered to deal with surface flow modeling with the diffusive wave equation. In this paper, we study the relative performance of different time and space integration schemes by comparing the results obtained with classical approaches and with nonlinear time integration approaches. The results show that (i) the integration method with a higher order in space shows high accuracy regarding an integrated indicator such as the global mass balance error but is less accurate regarding local indicators, and (ii) nonlinear time integration techniques perform better than classical ones. Overall, it seems that integration techniques combining nonlinear time integration and a low spatial order need to be considered when developing hydrological modeling tools owing to their simplicity of implementation and very good performance.

Characterization of reciprocity gaps from interference tests in fractured media through a dual porosity model

We analyze drawdown reciprocity gaps emerging in interference tests performed in a confined fissured karstic formation. Modeling the system as a dual porosity continuum allows characterizing the dynamics of the relative contribution of the connected fractures and the rock matrix to the total flow rate extracted at the pumping wells. Observed lack of reciprocity of drawdowns can then be linked to the occurrence of processes that are not accounted for in the classical flow models based on a single-continuum representation of the system through flow equations grounded on Darcys law only. We show that interpreting the system as a dual porosity continuum can cause drawdown reciprocity gaps to emerge as a consequence of local effects associated with an identifiable contribution of the matrix to the total fluid extracted at the well location during pumping. These theoretical results are then employed to identify the contribution to the flow being supplied to the pumping well by the low conductivity matrix constituting the host rock formation, in contrast to that provided by the fractures. An application to data from two interference tests performed at the Hydrogeological Experimental Site (HES) in Poitiers, France, illustrates the approach. We show that, whenever the matrix is assumed to provide a contribution to the total flow rate extracted, nonreciprocity is expected, the latter being linked to the occurrence of a differential drawdown between fracture and matrix at the pumping well. This difference decreases with time in the example presented, displaying a power law late time behavior, with nonreciprocity effects persisting up to remarkably long times.