Guido Parravicini

Minimal a priori assignment in a direct method for determining phenomenological coefficients uniquely

We identify the coefficients of the transport equation in N dimensions grad c.grad h+c Delta h=d delta h/ delta t+f by solving a differential system of the form grad c+ca=b. The assignment of c at one point only yields a unique solution, found by integration along arbitrary paths. This arbitrariness guarantees a good control of the error, notwithstanding the ill-posedness of the problem. For N=2, the hypotheses allowing for this identification are satisfied when one knows two stationary potentials with non-overlapping equipotential lines and a third non-stationary one-this last needed only for determining d. The theory is applied to a numerical synthetic example, for various grid sizes or for noisy data. Notwithstanding the minimal a priori information required for the coefficients, we are able to compute these at a large number of nodes with good precision. For the sake of completeness, we give other results on identification.

The differential system method for the identification of transmissivity and storativity

The differential system (DS) method for the identification of transmissivity and storativity is applied to a confined isotropic aquifer in transient conditions. The data that are required for the identification are the piezometric heads and the source terms, together with the value of transmissivity at a single point only, which is the only parameter value needed a priori. In particular, no a priori knowledge of storativity is needed and, moreover, the identification of transmissivity does not depend upon storativity. The DS method yields the internode transmissivities necessary for the conservative finite differences models in a natural way, because it identifies transmissivities along the internodal segments, so that a well-known formula can be applied that bypasses the difficulty of finding an equivalent cell transmissivity and an averaging scheme. In addition, the DS method takes into account several different flows all over the aquifer, so that the identified parameters are to a certain degree ‘global’ and‘flow independent’. Moreover, the method allows for a piecemeal identification of the parameters, thus keeping away from the regions where wells are pumping so that a two-dimensional model can be used throughout. We test the applicability of the DS method with noisy data by means of numerical synthetic examples and compare the identified internode transmissivities with the reference values. We use the identified parameters to forecast the behaviour of the aquifer under different exploitation and boundary conditions and we compare the forecast piezometric heads, their gradients and the associated fluxes with those computed with the reference parameters.

A numerical comparison between two upscaling techniques: non-local inverse based scaling and simplified renormalization

Abstract In this paper, we face the problem of upscaling transmissivity from the macroscopic to the megascopic scale; here the macroscopic scale is that of the continuous flow equations, whereas the megascopic scale is that of the flow models on a coarse grid. In this paper, we introduce the non-local inverse based scaling (NIBS) and compare it with the simplified renormalization (SR). The latter is a classical technique that we adapt to compute internode transmissivities for a finite differences flow model in a direct way. NIBS is implemented in three steps: in the first step, the macroscopic transmissivity, together with arbitrarily chosen auxiliary boundary conditions and sources, is used to solve forward problems (FPs) at the macroscopic scale; in the second step, the resulting heads are sampled at the megascopic scale; in the third step, the upscaled internode transmissivities are obtained by solving an inverse problem with the differential system method (DS) for which the heads resulting from the second step are used. NIBS is a non-local technique, because the computation of the internode transmissivities relies upon the whole transmissivity field at the macroscopic scale. We test NIBS against SR in the case of synthetic, isotropic, confined aquifers under the assumptions of two-dimensional (2D) and steady-state flow; the aquifers differ for the degree of heterogeneity, which is represented by a normally distributed uncorrelated component of ln T . For the comparison, the reference heads and fluxes at the megascopic scale are computed from the solution of FPs at the macroscopic scale. These reference values are compared with the heads and the fluxes predicted from models at the megascopic scale using the upscaled parameters of SR and NIBS. For the class of aquifers considered in this paper, the results of SR are better than those of NIBS, which hints that non-local effects can be disregarded at the megascopic scale. The two techniques provide comparable results when the heterogeneity increases, when the megascopic scale is large with respect to the heterogeneity length scale, or when the source terms are relevant.

Single- and Dual-domain Models of Solute Transport in Alluvial Sediments: the Effects of Heterogeneity Structure and Spatial Scale

Fine-scale heterogeneity of alluvial aquifers controls solute transport in groundwater at the scales relevant for practical applications: the architecture of sedimentary structures might create preferential flow paths (PFPs) or hydraulic barriers, which affect the breakthrough curves (BTCs). Objective of this paper was the assessment of the relevance of single- and dual-domain models for different heterogeneity patterns and scale lengths in alluvial sediments. Three case studies have been analysed with a classical single-domain model (SDM) and with three dual-domain models (DDMs): a dual-porosity model (DPorM) and two dual-permeability models (DPerM), which differ for the presence or the absence of solute exchange between the two domains. The first case study includes numerical tracer tests in metre-scale blocks of alluvial sediments; the second is a laboratory experiment of tracer injection in a decimetre-scale column of homogeneous sand; the third is a field tracer test performed at hectometre scale at the Cape Cod site. The relevance of the solute exchange in the DDMs is analysed with the characteristic advection and exchange times and with the Péclet and Damköhler numbers. The SDM is satisfactory for alluvial sediments with unstructured heterogeneity. The uncoupled DPerM is shown to be a better approach than the DPorM in sediments with PFPs; in this case, the coupled DPerM does not improve significantly the results of the uncoupled DPerM. A minor difference between the results of the three DDMs is observed for sediments in which the non-Fickian behaviour is not clearly determined by the presence of PFPs.

Why are very short times so long and very long times so short in elastic waves

In a first study of thermoelastic waves, such as in the textbook of Landau and Lifshitz, one might at first glance understand that when the given period is very short, waves are isentropic because heat conduction does not set in, while if the given period is very long, waves are isothermal because there is enough time for thermalization to be thoroughly accomplished. When one pursues the study of these waves further, by the mathematical inspection of the complete thermoelastic wave equation one finds that if the period is very short, much shorter than a characteristic time of the material, the wave is isothermal, while if it is very long, much longer than the characteristic time, the wave is isentropic. One also learns that this fact is supported by experiments: at low frequencies the elastic waves are isentropic, while they are isothermal when the frequencies are so high that can be attained in few cases. The authors show that there is no contradiction between first-glance understanding and the mathematical treatment of the elastic wave equation: for thermal effects very long periods are so short and very short periods are so long.

Identification of Aquifer Transmissivity with Multiple Sets of Data Using the Differential System Method

The mass balance equation for stationary flow in a confined aquifer and the phenomenological Darcy’s law lead to a classical elliptic PDE, whose phenomenological coefficient is transmissivity, T, whereas the unknown function is the piezometric head. The differential system method (DSM) allows the computation of T when two “independent” data sets are available, i.e., a couple of piezometric heads and the related source or sink terms corresponding to different flow situations such that the hydraulic gradients are not parallel at any point. The value of T at only one point of the domain, x0, is required. The T field is obtained at any point by integrating a first order partial differential system in normal form along an arbitrary path starting from x0. In this presentation the advantages of this method with respect to the classical integration along characteristic lines are discussed and the DSM is modified in order to cope with multiple sets of data. Numerical tests show that the proposed procedure is effective and reduces some drawbacks for the application of the DSM.

SOME CONSIDERATIONS ABOUT UNIQUENESS IN THE IDENTIFICATION OF DISTRIBUTED TRANSMISSIVITIES OF A CONFINED AQUIFER

The identification of the transmissivity of a confined aquifer can be achieved by the solution of a generally ill-posed inverse problem when measurements of piezometric head and source term are available. Herewith some classical results are considered; the most promising approach consists of the simultaneous utilisation of several sets of data, namely piezometric heads and source terms relative to different steady hydraulic conditions of the aquifer. The main advantage of this approach is that the required data are the easiest to measure in hydrogeological field applications.