William W.-G. Yeh

A stochastic inverse solution for transient groundwater flow: Parameter identification and reliability analysis

In this paper, a stochastic approach is developed for solving the inverse problem of parameter identification for transient groundwater flow. Adjoint state equations are derived for the stochastic partial differential equation (SPDE) relating transient head and log hydraulic conductivity perturbations. The derived equations can be used to calculate the covariance matrix of head observations and the cross-covariance matrix between head observations and log hydraulic conductivity measurements for each observation time, as well as the covariance and cross-covariance matrices between different observation times. The number of simulation runs required for obtaining these matrices is equal to the number of head observations. The reliability of model prediction is evaluated through the variance estimate method using adjoint sensitivity analysis and the cokriging estimate. The reliability evaluation of model prediction can be used as an aid in the optimization of experimental design for sampling strategies. A numerical example is given to illustrate the stochastic inverse procedure for a general transient flow problem. The example shows that using head observations of all observation times simultaneously produces much better results than using them sequentially (commonly known as the quasi-steady approach). The example also shows that it is possible to obtain a reliable stochastic inverse solution for meeting a given prediction objective based on a limited number of observations.

Impacts of the 2004 tsunami on groundwater resources in Sri Lanka

The 26 December 2004 tsunami caused widespread destruction and contamination of coastal aquifers across southern Asia. Seawater filled domestic open dug wells and also entered the aquifers via direct infiltration during the first flooding waves and later as ponded seawater infiltrated through the permeable sands that are typical of coastal aquifers. In Sri Lanka alone, it is estimated that over 40,000 drinking water wells were either destroyed or contaminated. From February through September 2005, a team of United States, Sri Lankan, and Danish water resource scientists and engineers surveyed the coastal groundwater resources of Sri Lanka to develop an understanding of the impacts of the tsunami and to provide recommendations for the future of coastal water resources in south Asia. In the tsunami-affected areas, seawater was found to have infiltrated and mixed with fresh groundwater lenses as indicated by the elevated groundwater salinity levels. Seawater infiltrated through the shallow vadose zone as well as entered aquifers directly through flooded open wells. Our preliminary transport analysis demonstrates that the intruded seawater has vertically mixed in the aquifers because of both forced and free convection. Widespread pumping of wells to remove seawater was effective in some areas, but overpumping has led to upconing of the saltwater interface and rising salinity. We estimate that groundwater recharge from several monsoon seasons will reduce salinity of many sandy Sri Lankan coastal aquifers. However, the continued sustainability of these small and fragile aquifers for potable water will be difficult because of the rapid growth of human activities that results in more intensive groundwater pumping and increased pollution. Long-term sustainability of coastal aquifers is also impacted by the decrease in sand replenishment of the beaches due to sand mining and erosion.

Mathematical simulation of soil vapor extraction systems: Model development and numerical examples

This paper describes the development of a numerical model for prediction of soil vapor extraction processes. The major emphasis is placed on field-scale predictions with the objective to advance development of planning tools for design and operation of venting systems. The numerical model solves two-dimensional flow and transport equations for general n-component contaminant mixtures. Flow is limited to the gas phase and local equilibrium partitioning is assumed in tracking contaminants in the immiscible fluid, water, gas, and solid phase. Model predictions compared favorably with analytical solutions and multicomponent column venting experiments. Sensitivity analysis indicates equilibrium phase partitioning is a good assumption in modeling organic liquid volatilization occurring in field venting operations. Mass transfer rates in volatilization from the water phase and contaminant desorption are potentially rate limiting. Simulations of hypothetical field-scale problems show efficiency of venting operations is most sensitive to vapor pressure and the magnitude and distribution of soil permeability.

A proposed stepwise regression method for model structure identification

This paper proposes a new methodology for constructing groundwater models. The proposed methodology, which determines simultaneously both model structure and model parameters, is based on the following ideas: (1) When solving the inverse problem, different model structures always produce different model parameters; (2) since the number of possible model structures of an aquifer is infinite, the number of possible representative parameters is also infinite; (3) to obtain a set of appropriate representative model parameters, we must have an appropriate model structure; and (4) an appropriate model structure should be determined not only by observation data and prior information but also by the accuracy requirements of model applications. In this proposed methodology we start with a homogeneous model structure and, step by step, gradually increase the complexity of the model structure. At each level of complexity we calculate not only the fitting residual of parameter identification but also the error of model structure to determine if a more complex model structure is needed. The model structure error of using one model structure to replace another model structure is defined by a maximum-minimum (max-min) problem that is based on the distance between the two models and is measured in parameter, observation, and prediction (or decision) spaces. This proposed methodology is used to solve a hypothetical remediation design problem in which the true hydraulic conductivity is a random field with a certain trend. We have found that for the example problem, virtually identical pumping policy is obtained when a five-zone model with an optimized zonation pattern is used to represent the nonstationary random field. We have also found that observation errors have minimum impact on management solution in comparison with structure errors. To calculate the model structure error for this example, the inverse solution is coupled with a management problem. We have also developed an effective iteration method to handle nonlinear water quality constraints.

Hydro Unit Commitment via Mixed Integer Linear Programming: A Case Study of the Three Gorges Project, China

This paper develops a methodology for optimizing the hydro unit commitment (HUC) for the Three Gorges Project (TGP) in China. The TGP is the worlds largest and most complex hydropower system in operation. The objective is to minimize the total operational cost. The decision variables are the startup or shutdown of each of the available units in the system and the power releases from the online units. The mathematical formulation must take into account the head variation over the operation periods as the net head changes from hour to hour and affects power generation. Additionally, the formulation must consider the operation of 32 heterogeneous generating units and the nonlinear power generation function of each unit. A three-dimensional interpolation technique is used to accurately represent the nonlinear power generation function of each individual unit, taking into account the time-varying head as well as the non-smooth limitations for power output and power release. With the aid of integer variables that represent the on/off and operation partition statuses of a unit, the developed HUC model for the TGP conforms to a standard mixed integer linear programming (MILP) formulation. We demonstrate the performance and utility of the model by analyzing the results from several scenarios for the TGP.

Inverse modeling for locating dense nonaqueous pools in groundwater under steady flow conditions

In this work we develop an inverse modeling procedure to identify the location and the dimensions of a single-component dense nonaqueous phase liquid (DNAPL) pool in a saturated porous medium under steady flow conditions. The inverse problem is formulated as a least squares minimization problem and solved by a search procedure based on the Levenberg-Marquardt method. Model output is calculated by an existing three-dimensional analytical model describing the transport of solute from a dissolving distributed noise upon the forward model-generated concentration field. We further test the algorithms ability to predict the location and size of a DNAPL pool placed in a controlled three-dimensional bench-scale experiment. In this case we apply the Levenberg-Marquardt algorithm to the minimization of three types of residuals: ordinary residuals, weighted residuals with weights equal to the square of the inverse of the observations, and weighted residuals with weights obtained by adding a constant term to the observed concentrations. The results are sensitive to the location of the observation wells and to the type of residuals minimized. In general, better results in terms of pool location and dimensions were obtained by the minimization of weighted residuals with weights obtained by adding a constant term to the observed concentrations. The results also indicate that the inverse problem is nonunique and nonconvex even in the absence of observation errors. Finally, the sensitivity of the inverse modeling scheme to transport parameter uncertainty was addressed. The inverse solution was found to be extremely sensitive to errors in the dispersion coefficients and relatively insensitive to errors in the mass transfer coefficient.

Optimization of Large-Scale Hydrothermal System Operation

This paper presents the development of a mathematical model to optimize the management and operation of the Brazilian hydrothermal system. The system consists of a large set of individual hydropower plants and a set of aggregated thermal plants. The energy generated in the system is interconnected by a transmission network so it can be transmitted to centers of consumption throughout the country. The optimization model offered is capable of handling different types of constraints, such as interbasin water transfers, water supply for various purposes, and environmental requirements. Its overall objective is to produce energy to meet the country’s demand at a minimum cost. Called HIDROTERM, the model integrates a database with basic hydrological and technical information to run the optimization model, and provides an interface to manage the input and output data. The optimization model uses the General Algebraic Modeling System (GAMS) package and can invoke different linear as well as nonlinear programming ...

A Proposed Geological Parameterization Method for Parameter Identification in Three‐Dimensional Groundwater Modeling

A new parameterization method for parameter identification is presented. This method allows us to incorporate all well log data and any other geological information into the inverse solution procedure for three-dimensional groundwater modeling. In the proposed method, unknown parameters, such as hydraulic conductivity and storage coefficient, are directly related to the geological materials. Existing well logs of an aquifer can provide information of local geological structures along the vertical direction. By using these data, as well as any other geological information available, the three-dimensional structure of the aquifer can be estimated by means of the geostatistical method. Then, art inverse problem can be formulated, leaving fewer unknown parameters to be identified. The advantages of the proposed method are the following: (1) all existing geological and hydrogeological information available is used for parameter identification, (2) the identified parameters are independent of the complexity of the simulation model, (3( the ill-posedness of the inverse solution is mitigated, and (4) the identified parameters are distributed and physically meaningful. In a hypothetical example, the three-dimensional distribution of hydraulic conductivity is easily identified using the proposed method. The results indicate that the identified distributed parameter vector is very close to the “true” distribution, and the inverse solution is highly stable with respect to observation errors. The proposed methodology has enabled us to determine the three-dimensional distribution of hydraulic conductivity and storativity in the Hemet basin, Riverside, California.

Parameter estimation in rainfall-runoff models

Abstract This paper presents techniques for the estimation of parameters in rainfall-runoff models. In the practical application of models a number of catchment parameters are not directly measureable and it is desirable to make the best possible estimate from known rainfall-runoff data. Three techniques are presented. Linear programming (LP) is used to minimize the sum of the absolute errors (MSAE) of the computed hydrograph. Quadratic programming (QP) is used in the two other techniques, namely ordinary least squares (OLS) and generalized least squares (GLS). OLS uses the traditional regression objective of minimizing the square of the deviation while GLS uses a weighted form of the OLS objective which can eliminate the effect of serially correlated errors (noise). The techniques are demonstrated using a hypothetical catchment for which rainfall-runoff series are generated using a conceptual model. Four parameters necessary for the models operation are then estimated, with varying levels of noise superimposed on the generated series, using the three techniques and then finally the techniques are applied to a real catchment. The covariance and the correlation matrices of the estimated parameters are computed.

Review: Optimization methods for groundwater modeling and management

Optimization methods have been used in groundwater modeling as well as for the planning and management of groundwater systems. This paper reviews and evaluates the various optimization methods that have been used for solving the inverse problem of parameter identification (estimation), experimental design, and groundwater planning and management. Various model selection criteria are discussed, as well as criteria used for model discrimination. The inverse problem of parameter identification concerns the optimal determination of model parameters using water-level observations. In general, the optimal experimental design seeks to find sampling strategies for the purpose of estimating the unknown model parameters. A typical objective of optimal conjunctive-use planning of surface water and groundwater is to minimize the operational costs of meeting water demand. The optimization methods include mathematical programming techniques such as linear programming, quadratic programming, dynamic programming, stochastic programming, nonlinear programming, and the global search algorithms such as genetic algorithms, simulated annealing, and tabu search. Emphasis is placed on groundwater flow problems as opposed to contaminant transport problems. A typical two-dimensional groundwater flow problem is used to explain the basic formulations and algorithms that have been used to solve the formulated optimization problems.RésuméLes méthodes d’optimisation ont été utilisées pour la modélisation des eaux souterraines ainsi que pour la planification et la gestion de ces systèmes. Le présent article passe en revue et évalue les méthodes d’optimisation variées qui ont été utilisées pour apporter une solution au problème inverse d’identification (estimation) des paramètres, de démarche expérimentale et de planification et gestion des eaux souterraines. Le problème inverse d’identification des paramètres concerne la détermination optimale des paramètres du modèle faisant appel aux observations sur le niveau de l’eau. En général, la démarche expérimentale optimale vise à atteindre des stratégies d’échantillonnage qui permette d’estimer les paramètres non connus du modèle. Un objectif classique de la planification d’une utilisation conjuguée optimale des eaux de surface et des eaux souterraines est de minimiser les coûts opérationnels de la réponse à la demande en eau. Les méthodes d’optimisation incluent des techniques de programmation mathématique, telles que la programmation linéaire, la programmation quadratique, la programmation dynamique, la programmation stochastique, la programmation non linéaire et des algorithmes de recherche globale comme les algorithmes génétiques, le recuit simulé et la recherche avec tabou. L’accent est mis sur les problèmes d’écoulement souterrain par opposition aux problèmes de transfert de contaminants. Le problème-type d’écoulement souterrain bi-dimensionnel est utilisé pour expliciter les formulations de base et les algorithmes employés pour résoudre les problèmes d’optimisation formulés.ResumenLos métodos de optimización se han utilizado en la modelación, planificación y manejo de los sistemas de agua subterránea. Este trabajo revisa y evalúa los distintos métodos de optimización que han sido usados para resolver el problema inverso de la identificación de parámetros (estimación), diseño experimental y planificación y manejo del agua subterránea. Se discuten varios criterios de selección de modelos, así como los criterios usados para la discriminación del modelo. El problema inverso de la identificación de parámetros se refiere a la determinación óptima de los parámetros del modelo usando observaciones de niveles de agua. En general, el diseño óptimo experimental busca encontrar estrategias de muestreo con el fin de estimar los parámetros desconocidos del modelo. Un objetivo típico de óptima planificación de uso conjuntivo de agua superficial y agua subterránea es minimizar los costos operativos de la demanda de agua. Los métodos de optimización incluyen técnicas de programación matemática, tales como programación lineal, programación cuadrática, programación dinámica, programación estocástica, programación no lineal, y la búsqueda global de algoritmos, tales como algoritmos genéticos, de recocidos simulados y de búsqueda tabú. Se hace hincapié sobre los problemas de flujo de las aguas subterráneas en contraposición a los problemas del transporte de contaminantes. Se utiliza un típico problema de flujo bidimensional de agua subterránea para explicar las formulaciones básicas y los algoritmos que han sido usados para resolver los problemas de optimización formulados.摘要最优化方法用于地下水模拟以及用于地下水系统的规划和管理。本文综述和评估了用于解决参数识别(估算)、试验设计和地下水挂会和管理逆问题的各种最优化方法。探讨了各种模型选择标准,以及探讨了用于模型识别标准。参数识别的逆问题采用水文观测数据关注模型参数的最优化确定。总的来说,最优化试验设计寻求找到采样策略,以估算未知的模型参数。地表水和地下水最优化联合利用规划中一个典型的目标就是在满足供水需求的情况下尽量减少运行费用。最优化方法包括数学编程技术,诸如线性编程、二次方编程、动态编程、随机编程、非线性编程及全局搜寻算法,诸如遗传算法、模拟的处理及禁忌算法。重点强调了与污染物运移问题对立的地下水流问题。利用典型的二维地下水流问题解释用于解决所阐述的最优化问题的基本构想和算法。ResumoOs métodos de otimização têm sido utilizados tanto para a modelagem de águas subterrâneas quanto para o planejamento e gerenciamento desses sistemas. Esse artigo revisa e avalia diversos métodos de otimização que têm sido utilizados para resolver o problema inverso da identificação (estimação) de parâmetros, delineamento experimental e planejamento e gerenciamento de águas subterrâneas. São discutidos vários critérios de seleção de modelos, assim como critérios usados para o descarte de modelos. O problema inverso de identificação de parâmetros consiste na determinação de parâmetros ótimos do modelo por intermédio de observações de níveis de água. O planejamento ótimo de experimentos, por sua vez, busca estratégias de amostragem necessárias para tal estimação de parâmetros desconhecidos do modelo. No planejamento integrado ótimo entre águas superficiais e subterrâneas, o objetivo típico é minimizar os custos operacionais de atendimento à demanda. Os métodos de otimização incluem técnicas de programação matemática, como programação linear, programação quadrática, programação dinâmica, programação estocástica, programação não-linear e os algoritmos de busca global, como algoritmos genéticos, recozimento simulado (simulated anneling) e busca tabu. É dada ênfase em problemas de escoamento de águas superficiais, diferentemente dos problemas de transporte de contaminantes. As formulações básicas dos métodos e seus algoritmos, que tem sido utilizados para resolver os problemas de otimização formulados, são discutidos a partir de um mesmo problema típico de escoamento bi-dimensional de águas subterrâneas.