Fabien Cornaton

Groundwater age, life expectancy and transit time distributions in advective–dispersive systems: 1. Generalized reservoir theory

Abstract We present a methodology for determining reservoir groundwater age and transit time probability distributions in a deterministic manner, considering advective–dispersive transport in steady velocity fields. In a first step, we propose to model the statistical distribution of groundwater age at aquifer scale by means of the classical advection–dispersion equation for a conservative and non-reactive tracer, associated to proper boundary conditions. The evaluated function corresponds to the density of probability of the random variable age, age being defined as the time elapsed since the water particles entered the aquifer. An adjoint backward model is introduced to characterize the life expectancy distribution, life expectancy being the time remaining before leaving the aquifer. By convolution of these two distributions, groundwater transit time distributions, from inlet to outlet, are fully defined for the entire aquifer domain. In a second step, an accurate and efficient method is introduced to simulate the transit time distribution at discharge zones. By applying the reservoir theory to advective–dispersive aquifer systems, we demonstrate that the discharge zone transit time distribution can be evaluated if the internal age probability distribution is known. The reservoir theory also applies to internal life expectancy probabilities yielding the recharge zone life expectancy distribution. Internal groundwater volumes are finally identified with respect to age and transit time. One- and two-dimensional theoretical examples are presented to illustrate the proposed mathematical models, and make inferences on the effect of aquifer structure and macro-dispersion on the distributions of age, life expectancy and transit time.

Analytical 1D dual-porosity equivalent solutions to 3D discrete single-continuum models. Application to karstic spring hydrograph modelling

Abstract One-dimensional analytical porosity-weighted solutions of the dual-porosity model are derived, providing insights on how to relate exchange and storage coefficients to the volumetric density of the high-permeability medium. It is shown that porosity-weighted storage and exchange coefficients are needed when handling highly heterogeneous systems—such as karstic aquifers—using equivalent dual-porosity models. The sensitivity of these coefficients is illustrated by means of numerical experiments with theoretical karst systems. The presented 1D dual-porosity analytical model is used to reproduce the hydraulic responses of reference 3D karst aquifers, modelled by a discrete single-continuum approach. Under various stress conditions, simulation results show the relations between the dual-porosity model coefficients and the structural features of the discrete single-continuum model. The calibration of the equivalent 1D analytical dual-porosity model on reference hydraulic responses confirms the dependence of the exchange coefficient with the karstic network density. The use of the analytical model could also point out some fundamental structural properties of the karstic network that rule the shape of the hydraulic responses, such as density and connectivity.

Flow behavior in a dual fracture network

A model that incorporates a pseudo-random process controlled by mechanical rules of fracturing is used to generate 3D orthogonal joint networks in tabular stratified aquifers. The results presented here assume that two sets of fractures, each with different conductivities, coexist. This is the case in many aquifers or petroleum reservoirs that contain sets of fractures with distinct hydraulic properties related to each direction of fracturing. Constant rate pump-tests from partially penetrating wells are simulated in synthetic networks. The transient head response is analyzed using the type curve approach and plots, as a function of time, of pressure propagation in the synthetic network are shown. The hydrodynamic response can result in a pressure transient that is similar to a dual-porosity behavior, even though such an assumption was not made a priori. We show in this paper that this dual porosity like flow behavior is, in fact, related to the major role of the network connectivity, especially around the well, and to the aperture contrast between the different families of fractures that especially affects the earlier hydrodynamic response. Flow characteristics that may be interpreted as a dual porosity flow behavior are thus related to a lateral heterogeneity (large fracture or small fault). Accordingly, when a dual porosity model matches well test data, the resulting reservoir parameters can be erroneous because of the model assumptions basis that are not necessarily verified. Finally, it is shown both on simulated data and well test data that such confusion in the interpretation of the flow behavior can easily occur. Well test data from a single well must therefore be used cautiously to assess the flow properties of fractured reservoirs with lateral heterogeneities such as large fractures or small faults.

Stochastic forecasts of seawater intrusion towards sustainable groundwater management: application to the Korba aquifer (Tunisia)

A stochastic study of long-term forecasts of seawater intrusion with an application to the Korba aquifer (Tunisia) is presented. Firstly, a geostatistical model of the exploitation rates was constructed, based on a multi-linear regression model combining incomplete direct data and exhaustive secondary information. Then, a new method was designed and used to construct a geostatistical model of the hydraulic conductivity field by combining lithological information and data from hydraulic tests. Secondly, the effects of the uncertainties associated with the pumping rates and the hydraulic conductivity field on the 3D density-dependent transient model were analysed separately and then jointly. The forecasts of the impacts of two different management scenarios on seawater intrusion in the year 2048 were performed by means of Monte Carlo simulations, accounting for uncertainties in the input parameters as well as possible changes of the boundary conditions. Combining primary and secondary data allowed maps of pumping rates and the hydraulic conductivity field to be constructed, despite a lack of direct data. The results of the stochastic long-term forecasts showed that, most probably, the Korba aquifer will be subject to important losses in terms of regional groundwater resources.RésuméOn présente ici une étude stochastique des prévisions à long terme de l’intrusion marine, avec son application à l’aquifère de Korba (Tunisie). En premier lieu, un modèle géostatistique des débits d’exploitation a été construit sur la base d’un modèle de régression multilinéaire combinant des données directes incomplètes et des informations secondaires exhaustives. Ensuite, on a élaboré et utilisé une nouvelle méthodologie de construction d’un modèle géostatistique du champ de conductivité hydraulique en combinant des données lithologiques et des données issues des tests hydrauliques. Dans un second temps, on a analysé séparément puis conjointement les effets des incertitudes associées aux débits de pompage et au champ de conductivité hydraulique sur le modèle transitoire 3D avec effet de densité. Les prévisions des impacts sur l’intrusion saline de deux scénarios différents de gestion à l’horizon 2048 ont été calculés au moyen de simulations de Monte Carlo, tenant compte à la fois de l’incertitude sur les paramètres d’entrée et de possibles changements dans les conditions aux limites. La combinaison de données principales et secondaires a permis la construction de cartes de débits d’exploitation et de champ de conductivité hydraulique malgré le manque d’information directe. Les résultats des prévisions stochastiques à long terme montrent que l’aquifère de Korba subira très probablement d’importantes pertes des ressources en eaux souterraines.ResumenSe presenta un estudio estocástico a largo plazo de pronóstico de la intrusión de agua de mar con una aplicación al acuífero de Korba (Túnez). En primer lugar, se construyó un modelo geoestadístico de los caudales de explotación, basado en un modelo de regresión multi linear combinando datos directos incompletos y información secundaria exhaustiva. Luego, se diseñó y usó un método para construir un nuevo modelo geoestadístico del campo de conductividad hidráulica combinando información litológica y datos de ensayos de bombeo. En segundo lugar, se analizaron separada y conjuntamente, los efectos de las incertezas asociadas con los caudales de bombeo y el campo de conductividad hidráulica en un modelo transitorio tridimensional dependiente de la densidad. Se realizaron pronósticos de los impactos de dos escenarios diferentes de gestión sobre la intrusión de agua de mar en el año 2048 por medio de simulaciones de Monte Carlo, teniendo en cuenta las incertezas en los parámetros de entrada así como posibles cambios de las condiciones de borde. La combinación de datos primarios y secundarios permitió construir mapas de los caudales de bombeo y de los campos de conductividad hidráulica, a pesar de una falta de datos directos. Los resultados de los pronósticos estocásticos a largo plazo mostraron que, muy probablemente, el acuífero de Korba estará sujeto a pérdidas importantes en términos de los recursos regionales de agua subterránea.摘要本文研究了对海水入侵的长期预测随机模型在Korba含水层(突尼斯)的应用。首先,本文在结合了不完全的直接资料和详尽的次级资料的多线性回归模型的基础上,建立了开采率的地质统计模型。然后,文中设计了一种新方法,通过与岩性资料和水力试验得到数据相结合来建立渗透系数场的地质统计模型。第二,文中单独并联合分析了与抽水率、在密度有关的3D传导模型中的渗透系数场有关的不确定性的影响。利用蒙特卡罗模拟,本文预测了到2048年时两种不同的管理情景对海水入侵的影响,解释了输入参数的不确定性和边界条件可能发生的变化。虽然缺乏直接资料,通过综合考虑主要资料和次级资料,文中建立了抽水率和渗透系数场图。随机长期预测的结果显示,就区域地下水资源而言,Korba含水层的水量很有可能会大幅度减少。ResumoÉ apresentado um estudo estocástico de previsões a longo prazo de intrusão marinha com aplicação ao aquífero Korba (Tunísia). Em primeiro lugar foi construído um modelo geoestatístico das taxas de extração, baseado num modelo de regressão multi-linear que combina dados diretos incompletos com informação secundária exaustiva. Depois, foi desenvolvido e utilizado um novo método para construir um modelo geoestatístico do campo de condutividade hidráulica, através da combinação de informação litológica com dados de ensaios hidráulicos. Em segundo lugar, foram analisados, separadamente e depois conjuntamente, os efeitos das incertezas associadas às taxas de bombagem e ao campo de condutividade hidráulica no modelo 3-D transitório dependente da densidade. Foram realizadas as previsões dos impactes de dois cenários distintos de gestão na intrusão marinha no ano de 2048, através de simulações de Monte Carlo, tendo em conta as incertezas nos parâmetros de entrada, assim como eventuais alterações das condições de fronteira. Não obstante a escassez de dados diretos, a combinação de dados primários e secundários permitiu construir mapas de taxas de bombagem e do campo da condutividade hidráulica. Os resultados das previsões estocásticas a longo prazo mostraram que, muito provavelmente, o aquífero Korba vai estar sujeito a perdas importantes em termos de recursos hídricos subterrâneos regionais.

Analytical modelling of stable isotope fractionation of volatile organic compounds in the unsaturated zone

Analytical models were developed that simulate stable isotope ratios of volatile organic compounds (VOCs) near a point source contamination in the unsaturated zone. The models describe diffusive transport of VOCs, biodegradation and source ageing. The mass transport is governed by Ficks law for diffusion. The equation for reactive transport of VOCs in the soil gas phase was solved for different source geometries and for different boundary conditions. Model results were compared to experimental data from a one-dimensional laboratory column and a radial-symmetric field experiment. The comparison yielded a satisfying agreement. The model results clearly illustrate the significant isotope fractionation by gas phase diffusion under transient state conditions. This leads to an initial depletion of heavy isotopes with increasing distance from the source. The isotope evolution of the source is governed by the combined effects of isotope fractionation due to vaporisation, diffusion and biodegradation. The net effect can lead to an enrichment or depletion of the heavy isotope in the remaining organic phase, depending on the compound and element considered. Finally, the isotope evolution of molecules migrating away from the source and undergoing degradation is governed by a combined degradation and diffusion isotope effect. This suggests that, in the unsaturated zone, the interpretation of biodegradation of VOC based on isotopic data must always be based on a model combining gas phase diffusion and degradation.

A finite element formulation of the outlet gradient boundary condition for convective–diffusive transport problems

A simple finite element formulation of the outlet gradient boundary condition is presented in the general context of convective–diffusive transport processes. Basically, the method is based on an upstream evaluation of the dependent variable gradient along open boundaries. Boundary normal unit vectors and gradient operators are evaluated using covariant bases and metric tensors, which allow handling finite elements of mixed dimensions. Even though the presented method has implications for many fields where diffusion processes are involved, discussion and illustrative examples address more particularly the framework of contaminant transport in porous media, in which the outlet gradient concentration is classically, but wrongly assumed to be zero. Copyright

Regional Flow and Deformation Analysis of Basin‐Fill Aquifer Systems Using Stress‐Dependent Parameters

Changes in effective stress due to water pressure variations modify the intrinsic hydrodynamic properties of aquifers and aquitards. Overexploited groundwater systems, such as basins with heavy pumping, are subject to nonrecoverable modifications. This results in loss of permeability, porosity, and specific storage due to system consolidation. This paper presents (1) the analytical development of model functions relating effective stress to hydrodynamic parameters for aquifers and aquitards constituted of unconsolidated granular sediments, and (2) a modeling approach for the analysis of aquifer systems affected by effective stress variations, taking into account the aforementioned dependency. The stress-dependent functions were fit to laboratory data, and used in the suggested modeling approach. Based on only few unknowns, this approach is computationally simple, efficiently captures the hydromechanical processes that are active in regional aquifer systems under stress, and readily provides an estimate of their consolidation.

Regional Flow Simulation in Fractured Aquifers Using Stress‐Dependent Parameters

A model function relating effective stress to fracture permeability is developed from Hookes law, implemented in the tensorial form of Darcys law, and used to evaluate discharge rates and pressure distributions at regional scales. The model takes into account elastic and statistical fracture parameters, and is able to simulate real stress-dependent permeabilities from laboratory to field studies. This modeling approach gains in phenomenology in comparison to the classical ones because the permeability tensors may vary in both strength and principal directions according to effective stresses. Moreover this method allows evaluation of the fracture porosity changes, which are then translated into consolidation of the medium.