We present a combined experimental and numerical modeling study that addresses two principal ques... more We present a combined experimental and numerical modeling study that addresses two principal questions: (i) is any particular Eulerian-based method used to solve the classical advection-dispersion equation (ADE) clearly superior (relative to the others), in terms of yielding solutions that reproduce BTCs of the kind that are typically sampled at the outlet of a laboratory cell? and (ii) in the presence of matches of comparable quality against such BTCs, do any of these methods render different (or similar) numerical BTCs at locations within the domain? To address these questions, we obtained measurements from carefully controlled laboratory experiments, and employ them as a reference against which numerical results are benchmarked and compared. The experiments measure solute transport breakthrough curves (BTCs) through a square domain containing various configurations of coarse, medium, and fine quartz sand. The approaches to solve the ADE involve Eulerian-Lagrangian and Eulerian (f...
... Water Resour Res., 23, 293-312, 1987. 11. Ackerer, Ph. , A. Younes and R. MosS, Modeling vari... more ... Water Resour Res., 23, 293-312, 1987. 11. Ackerer, Ph. , A. Younes and R. MosS, Modeling variable density flow and solute transport in porous medium: 1. Numerical model and verification, Transport in Porous Media, 35(3), 345-373, 1999. 12. ...
In this work, we show how the use of global sensitivity analysis (GSA) in conjunction with the po... more In this work, we show how the use of global sensitivity analysis (GSA) in conjunction with the polynomial chaos expansion (PCE) methodology can provide relevant information for the interpretation of transport experiments in laboratory‐scale heterogeneous porous media. We perform GSA by calculating the Sobol indices, which provide a variance‐based importance measure of the effects of uncertain parameters on the output of a chosen interpretive transport model. The choice of PCE has the following two benefits: (1) it provides the global sensitivity indices in a straightforward manner, and (2) PCE can serve as a surrogate model for the calibration of parameters. The coefficients of the PCE are computed by probabilistic collocation. The methodology is applied to two nonreactive transport experiments available in the literature, while considering both transient and pseudo steady state transport regimes. This method allows a rigorous investigation of the relative effects and importance of ...
International Journal for Numerical Methods in Fluids, 2008
In this work, we develop a new model to solve the advection–dispersion transport equation on unst... more In this work, we develop a new model to solve the advection–dispersion transport equation on unstructured triangular meshes. The model combines numerical methods that are specifically suited to achieve high accuracy for each type of equation without using the time splitting procedure. It is based on a combination of the upwind discontinuous Galerkin (DG) method for advection and the multipoint flux approximation (MPFA) method for dispersion.In contrast to mixed finite elements, MPFA methods provide fluxes at element interfaces explicitly by weighted sums of discrete element concentrations. Therefore, the combination of DG and MPFA methods allows taking into account total flux boundary conditions while using different numerical techniques.A theta‐scheme time discretization is developed for advection and an implicit scheme for dispersion. Accuracy of the numerical model is shown by simulating (i) the transport of a tracer in a simplified bidimensional problem with highly unstructured ...
The Henry saltwater intrusion problem provides a semi-analytical solution that is largely used fo... more The Henry saltwater intrusion problem provides a semi-analytical solution that is largely used for benchmarking density-dependent groundwater flow models. The major drawback of this problem arises from the high dispersion value used by Henry (represented by the dimensionless parameter b = 0.1). Finding a stable semi-analytical solution for small values of b is challenging due to the low convergence of the corresponding nonlinear system. In this work, an accurate semi-analytical solution is developed in the case of a very narrow transition zone corresponding to b = 0.005. About 6,330 terms are used in the Fourier series to accurately represent the solution. The resolution of the corresponding highly nonlinear system is made possible by the modified Powell hybrid algorithm due to the analytical evaluation of the Jacobian, which drastically reduces the computational time. The new test problem is also investigated numerically using different numerical methods and different mesh sizes to show its high worthiness, compared to the standard Henry problem, for benchmarking density driven flow codes.RésuméLe problème d’Henry de l’intrusion saline fournit une solution semi-analytique qui est largement utilisée pour des analyses comparatives de modèles d’écoulement d’eau souterraine avec prise en compte de la densité. L’inconvénient majeur de ce problème découle de la valeur utilisée par Henry caractérisée par une forte dispersion (représentée par le paramètre sans dimension b = 0.1). Trouver une solution semi analytique stable pour des faibles valeurs de b est un défi à cause de la faible convergence du système non linéaire correspondant. Dans ce travail, une solution semi-analytique précise a été développée pour le cas de zone de transition très étroite correspondant à une valeur de b = 0.005. Environ 6330 termes sont utilisés dans des séries de Fourier pour représenter de manière précise la solution. La résolution du système fortement non linéaire correspondant est rendue possible en ayant recours à l’algorithme hybride modifié de Powell dû à l’évaluation analytique de l’équation de Jacob, ce qui réduit de manière drastique le temps de calcul. Le nouveau problème de test est également étudié du point de vue numérique en utilisant différentes méthodes numériques et différentes dimensions de maillage afin de montrer sa grande rigueur, en comparaison au problème standard d’Henry, dans le cadre d’analyses comparatives de codes d’écoulement avec prise en compte de la densité.ResumenEl problema de Henry para la intrusión de agua salada proporciona una solución semianalítica que es grandemente usada para la evaluación comparativa de modelos de flujo de agua subterránea dependiente de la densidad. El mayor inconveniente de este problema se presenta a partir de los valores altos de dispersión usados por Henry (representados por los parámetros sin dimensiones b = 0.1). Encontrar una solución semianalítica estable para pequeños valores de b es un desafío debido a la baja convergencia correspondiente a los sistemas no lineales. En este trabajo se desarrolla una solución semianalítica exacta para el caso de una muy estrecha zona de transición correspondiente a b = 0.005. Se usan alrededor de 6330 términos en las series de Fourier para representar exactamente la solución. La resolución correspondiente al sistema altamente no lineal se hace posible por el algoritmo híbrido modificado de Powell debido a la evaluación analítica del Jacobiano, lo cual reduce drásticamente el tiempo computacional. El nuevo problema de ensayo es también investigado numéricamente usando diferentes métodos numéricos y diferentes dimensiones de malla para mostrar su alta solvencia, comparado con el problema estándar de Henry, para la evaluación comparativa de códigos de flujo dependientes de la densidad.摘要亨利海水入侵问题提供了一个主要用于标记密度制约的地下水流模型半解析法。这个问题的主要缺点起因于亨利所用的高色散值(由无量纲参数b = 0.1表示)。由于相应的非线性系统内敛很低,为小的b值找到一个稳定的半解析法是一个挑战。在本研究中,开发了b = 0.005过渡带非常窄的情况下精确的半解析法。大约6330条术语用于傅里叶序列以精确描述该解析法。由于雅克比矩阵解析评价,改进的鲍威尔混合算法使解决相应的高非线性系统问题成为可能,雅克比矩阵大大降低了计算时间。与标准的亨利问题相比,还用不同的数值方法和不同的网格调查了新的实验问题以展示对于标记密度驱动的水流编码来说其高的价值。ResumoO problema da intrusão de água salina de Henry dá uma solução semi-analítica largamente utilizada para análise comparativa de modelos de escoamento subterrâneo dependentes da densidade. A maior desvantagem deste problema surge do elevado valor de dispersão utilizado por Henry (representado pelo parâmetro adimensional b = 0.1). É desafiante encontrar uma solução semi-analítica estável para valores pequenos de b, devido à baixa convergência do sistema não linear correspondente. Neste trabalho desenvolve-se uma solução semi-analítica precisa para o caso de uma zona de transição muito estreita correspondente a b = 0.005. São utilizados cerca de 6300 termos da série de Fourier para representar a solução de forma precisa. A resolução do sistema altamente não linear correspondente é possível com o algoritmo híbrido modificado de Powell devido à avaliação analítica do…
Computer Methods in Applied Mechanics and Engineering, 2003
A new resolution of parabolic and elliptic partial differential equations (PDEs) based on the mix... more 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.
... This approach cannot be extended to 2 or 3 dimensions. Russell and Binning [10] proposed a se... more ... This approach cannot be extended to 2 or 3 dimensions. Russell and Binning [10] proposed a selective lumping scheme to avoid excessive numerical diffusion. The basic idea is to add the minimum numerical diffusion required to avoid oscillations. ...
In this work, a numerical model is developed to investigate the influence of fluid flow and heat ... more In this work, a numerical model is developed to investigate the influence of fluid flow and heat transfer on the thermo-mechanical response of a cracked porous media. The fluid flow, governed by the Darcy’s law, is discretized with the nonconforming finite element method. Time splitting is used with the energy conservation equation to solve the fluid and the solid phases separately. A combination of Discontinuous Galerkin (DG) and multi-point flux approximation methods is used to solve the advection-diffusion heat transfer equation in the fluid phase. While the conductive heat transfers equation in the solid phase is solved using the eXtended finite element method (XFEM) to better handle the temperature discontinuities and singularities caused by the cracks. Further, the resulted temperature is used as body force to solve the thermo-mechanical problem using the XFEM. In the post processing stage, the thermal stress intensity factor is computed using the interaction integral technique at each time step and used to validate the obtained results. A good agreement was found when the results were compared with the existing ones in the literature.
We present a combined experimental and numerical modeling study that addresses two principal ques... more We present a combined experimental and numerical modeling study that addresses two principal questions: (i) is any particular Eulerian-based method used to solve the classical advection-dispersion equation (ADE) clearly superior (relative to the others), in terms of yielding solutions that reproduce BTCs of the kind that are typically sampled at the outlet of a laboratory cell? and (ii) in the presence of matches of comparable quality against such BTCs, do any of these methods render different (or similar) numerical BTCs at locations within the domain? To address these questions, we obtained measurements from carefully controlled laboratory experiments, and employ them as a reference against which numerical results are benchmarked and compared. The experiments measure solute transport breakthrough curves (BTCs) through a square domain containing various configurations of coarse, medium, and fine quartz sand. The approaches to solve the ADE involve Eulerian-Lagrangian and Eulerian (f...
... Water Resour Res., 23, 293-312, 1987. 11. Ackerer, Ph. , A. Younes and R. MosS, Modeling vari... more ... Water Resour Res., 23, 293-312, 1987. 11. Ackerer, Ph. , A. Younes and R. MosS, Modeling variable density flow and solute transport in porous medium: 1. Numerical model and verification, Transport in Porous Media, 35(3), 345-373, 1999. 12. ...
In this work, we show how the use of global sensitivity analysis (GSA) in conjunction with the po... more In this work, we show how the use of global sensitivity analysis (GSA) in conjunction with the polynomial chaos expansion (PCE) methodology can provide relevant information for the interpretation of transport experiments in laboratory‐scale heterogeneous porous media. We perform GSA by calculating the Sobol indices, which provide a variance‐based importance measure of the effects of uncertain parameters on the output of a chosen interpretive transport model. The choice of PCE has the following two benefits: (1) it provides the global sensitivity indices in a straightforward manner, and (2) PCE can serve as a surrogate model for the calibration of parameters. The coefficients of the PCE are computed by probabilistic collocation. The methodology is applied to two nonreactive transport experiments available in the literature, while considering both transient and pseudo steady state transport regimes. This method allows a rigorous investigation of the relative effects and importance of ...
International Journal for Numerical Methods in Fluids, 2008
In this work, we develop a new model to solve the advection–dispersion transport equation on unst... more In this work, we develop a new model to solve the advection–dispersion transport equation on unstructured triangular meshes. The model combines numerical methods that are specifically suited to achieve high accuracy for each type of equation without using the time splitting procedure. It is based on a combination of the upwind discontinuous Galerkin (DG) method for advection and the multipoint flux approximation (MPFA) method for dispersion.In contrast to mixed finite elements, MPFA methods provide fluxes at element interfaces explicitly by weighted sums of discrete element concentrations. Therefore, the combination of DG and MPFA methods allows taking into account total flux boundary conditions while using different numerical techniques.A theta‐scheme time discretization is developed for advection and an implicit scheme for dispersion. Accuracy of the numerical model is shown by simulating (i) the transport of a tracer in a simplified bidimensional problem with highly unstructured ...
The Henry saltwater intrusion problem provides a semi-analytical solution that is largely used fo... more The Henry saltwater intrusion problem provides a semi-analytical solution that is largely used for benchmarking density-dependent groundwater flow models. The major drawback of this problem arises from the high dispersion value used by Henry (represented by the dimensionless parameter b = 0.1). Finding a stable semi-analytical solution for small values of b is challenging due to the low convergence of the corresponding nonlinear system. In this work, an accurate semi-analytical solution is developed in the case of a very narrow transition zone corresponding to b = 0.005. About 6,330 terms are used in the Fourier series to accurately represent the solution. The resolution of the corresponding highly nonlinear system is made possible by the modified Powell hybrid algorithm due to the analytical evaluation of the Jacobian, which drastically reduces the computational time. The new test problem is also investigated numerically using different numerical methods and different mesh sizes to show its high worthiness, compared to the standard Henry problem, for benchmarking density driven flow codes.RésuméLe problème d’Henry de l’intrusion saline fournit une solution semi-analytique qui est largement utilisée pour des analyses comparatives de modèles d’écoulement d’eau souterraine avec prise en compte de la densité. L’inconvénient majeur de ce problème découle de la valeur utilisée par Henry caractérisée par une forte dispersion (représentée par le paramètre sans dimension b = 0.1). Trouver une solution semi analytique stable pour des faibles valeurs de b est un défi à cause de la faible convergence du système non linéaire correspondant. Dans ce travail, une solution semi-analytique précise a été développée pour le cas de zone de transition très étroite correspondant à une valeur de b = 0.005. Environ 6330 termes sont utilisés dans des séries de Fourier pour représenter de manière précise la solution. La résolution du système fortement non linéaire correspondant est rendue possible en ayant recours à l’algorithme hybride modifié de Powell dû à l’évaluation analytique de l’équation de Jacob, ce qui réduit de manière drastique le temps de calcul. Le nouveau problème de test est également étudié du point de vue numérique en utilisant différentes méthodes numériques et différentes dimensions de maillage afin de montrer sa grande rigueur, en comparaison au problème standard d’Henry, dans le cadre d’analyses comparatives de codes d’écoulement avec prise en compte de la densité.ResumenEl problema de Henry para la intrusión de agua salada proporciona una solución semianalítica que es grandemente usada para la evaluación comparativa de modelos de flujo de agua subterránea dependiente de la densidad. El mayor inconveniente de este problema se presenta a partir de los valores altos de dispersión usados por Henry (representados por los parámetros sin dimensiones b = 0.1). Encontrar una solución semianalítica estable para pequeños valores de b es un desafío debido a la baja convergencia correspondiente a los sistemas no lineales. En este trabajo se desarrolla una solución semianalítica exacta para el caso de una muy estrecha zona de transición correspondiente a b = 0.005. Se usan alrededor de 6330 términos en las series de Fourier para representar exactamente la solución. La resolución correspondiente al sistema altamente no lineal se hace posible por el algoritmo híbrido modificado de Powell debido a la evaluación analítica del Jacobiano, lo cual reduce drásticamente el tiempo computacional. El nuevo problema de ensayo es también investigado numéricamente usando diferentes métodos numéricos y diferentes dimensiones de malla para mostrar su alta solvencia, comparado con el problema estándar de Henry, para la evaluación comparativa de códigos de flujo dependientes de la densidad.摘要亨利海水入侵问题提供了一个主要用于标记密度制约的地下水流模型半解析法。这个问题的主要缺点起因于亨利所用的高色散值(由无量纲参数b = 0.1表示)。由于相应的非线性系统内敛很低,为小的b值找到一个稳定的半解析法是一个挑战。在本研究中,开发了b = 0.005过渡带非常窄的情况下精确的半解析法。大约6330条术语用于傅里叶序列以精确描述该解析法。由于雅克比矩阵解析评价,改进的鲍威尔混合算法使解决相应的高非线性系统问题成为可能,雅克比矩阵大大降低了计算时间。与标准的亨利问题相比,还用不同的数值方法和不同的网格调查了新的实验问题以展示对于标记密度驱动的水流编码来说其高的价值。ResumoO problema da intrusão de água salina de Henry dá uma solução semi-analítica largamente utilizada para análise comparativa de modelos de escoamento subterrâneo dependentes da densidade. A maior desvantagem deste problema surge do elevado valor de dispersão utilizado por Henry (representado pelo parâmetro adimensional b = 0.1). É desafiante encontrar uma solução semi-analítica estável para valores pequenos de b, devido à baixa convergência do sistema não linear correspondente. Neste trabalho desenvolve-se uma solução semi-analítica precisa para o caso de uma zona de transição muito estreita correspondente a b = 0.005. São utilizados cerca de 6300 termos da série de Fourier para representar a solução de forma precisa. A resolução do sistema altamente não linear correspondente é possível com o algoritmo híbrido modificado de Powell devido à avaliação analítica do…
Computer Methods in Applied Mechanics and Engineering, 2003
A new resolution of parabolic and elliptic partial differential equations (PDEs) based on the mix... more 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.
... This approach cannot be extended to 2 or 3 dimensions. Russell and Binning [10] proposed a se... more ... This approach cannot be extended to 2 or 3 dimensions. Russell and Binning [10] proposed a selective lumping scheme to avoid excessive numerical diffusion. The basic idea is to add the minimum numerical diffusion required to avoid oscillations. ...
In this work, a numerical model is developed to investigate the influence of fluid flow and heat ... more In this work, a numerical model is developed to investigate the influence of fluid flow and heat transfer on the thermo-mechanical response of a cracked porous media. The fluid flow, governed by the Darcy’s law, is discretized with the nonconforming finite element method. Time splitting is used with the energy conservation equation to solve the fluid and the solid phases separately. A combination of Discontinuous Galerkin (DG) and multi-point flux approximation methods is used to solve the advection-diffusion heat transfer equation in the fluid phase. While the conductive heat transfers equation in the solid phase is solved using the eXtended finite element method (XFEM) to better handle the temperature discontinuities and singularities caused by the cracks. Further, the resulted temperature is used as body force to solve the thermo-mechanical problem using the XFEM. In the post processing stage, the thermal stress intensity factor is computed using the interaction integral technique at each time step and used to validate the obtained results. A good agreement was found when the results were compared with the existing ones in the literature.
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