Giansilvio Ponzini

Empirical relation between electrical transverse resistance and hydraulic transmissivity

Abstract An empirical function between electrical transverse resistances of an aquifer, corrected for the effect of their porewater resistivities and corresponding hydraulic transmissivities, has been found. This function is not a linear one but a direct one in the sense that by increasing the transmissivity even the corrected transverse resistance increases. In order to obtain a reliable and useful relationship between these two parameters the field test sites should be suitably chosen covering the largest possible transmissivity range, since extrapolations of the obtained function are often doubtful. The porewater resistivities of the aquifer should be broadly constant, well known and representative of the whole aquifer thickness. The pumping tests for computing hydraulic transmissivities should be carried out utilizing fully penetrating wells screened for a significant percentage of the aquifer thickness. These kinds of relationships between geoelectrical and hydraulic parameters, where they exist and where they are detectable are sufficiently reliable for aquifers characterized by relatively simple hydrogeological schemes and corresponding electric stratigraphies. For more complicated aquifer systems, such as multilayers, or phreatic aquifers, or aquifers with anisotropic reservoir rocks, the relationships are often difficult if not impossible to obtain. Finally the empirical function obtained, between corrected transverse resistance and transmissivity, is utilizable for most of the surveyed aquifer; however, it is not directly applicable to other aquifers. The methodology, however, can be suitable for finding analogous relationships characterizing other aquifer systems.

The comparison model method: A new arithmetic approach to the discrete inverse problem of groundwater hydrology: 1. One-dimensional flow

Let the steady-state pressure z(·) of a fluid in a one-dimensional domain be governed by the equation dx(a dxz) = f subject to Dirichlet boundary conditions. We consider the identification of the transmissivity a (·), given f(·), and measured pressure z(·) by the comparison model method, a direct method which has been known and applied for some time but lacked theoretical background. With reference to a domain in one spatial dimension, we examine both the infinite-(‘continuous’) and finite-(discrete) dimensional cases. In the former, the method is based on the solution p(·) of an auxiliary flow equation, where f(·) and the two-point boundary conditions are unchanged with respect to the original or z(·) equation, whereas a tentative constant value b is assigned to the auxiliary transmissivity. The ratio of the first derivatives of p(·) and z(·) multiplied by b yields a solution ã(·) to the inverse problem. We examine in detail the nonuniqueness of ã(·) as a function of b, some cases where existence implies uniqueness, the role of positivity constraints, and a special feature: self-identifiability. We then translate all available results into the discrete case, where the good unknowns for the inverse problem are the internode coefficients. Several algebraic and numerical examples are presented.

Identification of Thermal-Conductivities by Temperature-Gradient Profiles. One-Dimensional Steady Flow

The comparison model method (CMM) is applied to the identification of spatially varying thermal conductivity in a one‐dimensional domain. This method deals with the discretized steady‐state heat equation written at the nodes of a lattice, a lattice which models a stack of plane parallel layers. The required data are temperature gradient and heat source (or sink) values. The unknowns of this inverse problem are not nodal values but internode thermal conductivities, which appear in the node heat balance equation. The conductivities, e.g., the solutions to the inverse problem obtained by the CMM, are a one‐parameter family. In order to achieve uniqueness (which coincides with identifiability in this case), a suitable value of this parameter must be found. To this end we consider (1) parameterization, i.e., introducing equality constraints between the unknown coefficients, (2) the use of two data sets at least in a subdomain, and (3) self‐identifiability. Each of these items formally translates the available ...

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.

Modeling water resources of a highly irrigated alluvial plain (Italy): calibrating soil and groundwater models

Modern and effective water management in large alluvial plains that have intensive agricultural activity requires the integrated modeling of soil and groundwater. The models should be complex enough to properly simulate several, often non-linear, processes, but simple enough to be effectively calibrated with the available data. An operative, practical approach to calibration is proposed, based on three main aspects. First, the coupling of two models built on well-validated algorithms, to simulate (1) the irrigation system and the soil water balance in the unsaturated zone and (2) the groundwater flow. Second, the solution of the inverse problem of groundwater hydrology with the comparison model method to calibrate the model. Third, the use of appropriate criteria and cross-checks (comparison of the calibration results and of the model outputs with hydraulic and hydrogeological data) to choose the final parameter sets that warrant the physical coherence of the model. The approach has been tested by application to a large and intensively irrigated alluvial basin in northern Italy.RésuméLa gestion moderne et efficace de l’eau dans les grandes plaines alluviales siège d’une agriculture intensive requiert la modélisation intégrée du sol et des eaux souterraines. Les modèles doivent être suffisamment complexes pour simuler correctement plusieurs processus souvent non linéaires, mais suffisamment simples pour être calibrés efficacement avec les données disponibles. Une approche de calibration pratique et opérationnelle est proposée, basée sur trois aspects principaux. Premièrement, le couplage de deux modèles construits sur des algorithmes bien validés, pour simuler (1) le dispositif d’irrigation et le bilan en eau du sol au sein de la zone non saturée et (2) l’écoulement d’eau souterraine. Deuxièmement, la résolution du problème hydrogéologique inverse avec la Méthode de Comparaison de Modèle pour caler le modèle. Troisièmement, l’utilisation de critères appropriés et de vérifications croisées (comparaison des résultats du calage et des sorties du modèle avec des données hydrologiques et hydrogéologiques) pour choisir les jeux de paramètres finaux qui garantissent la cohérence physique du modèle. L’approche a été testée par application à un vaste bassin alluvial intensément irrigué du Nord de l’Italie.ResumenEl manejo moderno y efectivo del agua en grandes planicies aluviales que tienen una intensa actividad agrícola requiere el modelado integrado del suelo y del agua subterránea. Los modelos deber ser lo suficientemente complejos como para simular correctamente varios procesos, a menudo no lineales, pero lo suficientemente simple para ser efectivamente calibrado con los datos disponibles. Se propone un enfoque práctico y operativo de la calibración, basada en tres aspectos principales. Primero, el acoplamiento de dos modelos construidos sobre algoritmos bien validados, para simular (1) el sistema de irrigación y el balance de agua en el suelo en la zona no saturada y (2) el flujo de agua subterránea. Segundo, la solución del problema inverso de hidrología de agua subterránea con el Método de comparación de modelos para calibrar el modelo. Tercero, el uso de criterios apropiados y controles cruzados (comparación entre los resultados de la calibración y las salidas del modelo con los datos hidrogeológicos e hidráulicos) para elegir el conjunto de parámetros finales que garanticen la coherencia física del modelo. El enfoque ha sido probado por su aplicación a una gran cuenca aluvial, intensamente irrigada en el norte de Italia.摘要在有密集型农业活动的大型冲积平原进行有效的现代水资源管理需要对土壤和地下水进行综合模拟。模型需要足够复杂,以模拟几种常见的非线性的过程,同时又要足够简单,能够采用现有数据进行校准。本文主要基于以下三个方面,提出了一个具备可操作性且符合实际的校准方法:1. 基于有效算法的两种模型的耦合,以模拟(1)非饱和带灌溉系统和土壤水平衡和(2)地下水流。2. 针对地下水文学的反演问题,采用比较法去校正模型。3.使用合适的准则和交互检验(根据水力和水文地质数据,比较模型的校准结果和模型输出结果)去选择最后的参数设置以确保模型的物理一致性。将该方法已经应用在意大利北部的一个大型密集灌溉冲积平原中进行ResumoUma moderna e eficaz gestão da água nas grandes planícies aluviais com atividade agrícola intensiva requer uma modelação integrada do solo e das águas subterrâneas. Os modelos devem ser suficientemente complexos para simular adequadamente vários processos, muitas vezes de caraterísticas não-lineares, mas suficientemente simples para serem adequadamente calibrados com os dados disponíveis. Neste artigo é proposta uma abordagem para calibração baseada em três aspetos principais. Primeiro, o acoplamento de dois modelos construídos sobre algoritmos bem validados, para simular (1) o sistema de rega e balanço hídrico do solo na zona não saturada e (2) o fluxo de águas subterrâneas. Segundo, uma solução do problema inverso da hidrologia subterrânea, baseada no Método de Comparação do Modelo para calibrar o modelo. Terceiro, o uso de critérios adequados e validações cruzadas (comparação dos resultados de calibração com as saídas do modelo com dados hidráulicos e hidrogeológicos), de modo a selecionar o conjunto final de parâmetros que garantem a coerência física do modelo. A abordagem tem sido testada numa bacia aluvial extensa e intensamente sujeita a rega, localizada no norte da Itália.

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.

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.

A technique to test finite difference schemes to model some geophysical processes in a geological structure

A preliminary problem to solve in the set-up of a mathematical model simulating a geophysical process is the choice of a suitable discrete scheme to approximate the governing differential equations. This paper presents a simple technique to test finite difference schemes used in the modeling of geophysical processes occurring in a geological structure. This technique consists in generating analytical solutions similar to the ones characterizing a geophysical process, given general information on some relevant parameters. Useful information for the choice of the discrete scheme to employ in the mathematical model simulating the original geophysical process can be obtained from the comparison between the analytical solution and the approximated numerical solutions generated by means of different discrete schemes. Two classes of numerical examples approximating the differential equation that governs the steady state earths heat flow have been treated using three different finite differences schemes. The first class of examples deals with media whose phenomenological parameters vary as continuous space functions; the second class, instead, deals with media whose phenomenological parameters vary as discontinuous space functions. The finite difference schemes that have been utilized are: Centered Finite Difference Scheme (CDS), Arithmetic Mean Scheme (AMS), and Harmonic Mean Scheme (HMS).The numerical simulations showed that: the CDS may yield physically inconsistent solutions if the lattice internodal distance is too large, but in case of phenomenological parameters varying as a continuous function, this pitfall can be avoided increasing the lattice node refinement. In case of phenomenological parameters varying as a discontinuous function, instead, the CDS may yield physically inconsistent solutions for any lattice-node refinement. The HMS produced good results for both classes of examples showing to be a scheme suitable to model situations like these.