Dynamic optimal ground water remediation including fixed and operation costs

In time-varying ground water remediation, the lack of an optimal control algorithm to simultaneously consider fixed costs and time-varying operating costs makes it nearly impossible to obtain an optimal solution. This study presents a novel algorithm that integrates a genetic algorithm (GA) and constrained differential dynamic programming (CDDP) to solve this time-varying ground water remediation problem. A GA can easily incorporate the fixed costs associated with the installation of wells. However, using a GA to solve for time-varying policies would dramatically increase the computational resources required. Therefore, the CDDP is used to handle the subproblems associated with time-varying operating costs. A hypothetical case study that incorporates fixed and time-varying operating costs is presented to demonstrate the effectiveness of the proposed algorithm. Simulation results indicate that the fixed costs can significantly influence the number and locations of wells, and a notable total cost savings can be realized by applying the novel algorithm herein.     

A Hybridized GA with LP‐LP Model for the Management of Confined Groundwater

A novel hybrid methodology is introduced in this paper for the optimal solution of the groundwater management problem. The problem to be addressed is the optimal determination and operation of a predefined number of wells out of a priori known set of potential wells with fixed locations to minimize the pumping cost of utilizing a two‐dimensional (2D) confined aquifer under steady‐state flow condition. The solution to this problem should satisfy a downstream demand, a lower/upper bound on the pumping rates, and a lower/upper bound on the water level drawdown in the wells. The problem is solved by hybridizing a genetic algorithm (GA) which suggests the candidate configurations for the operational wells and a hybrid linear programming (LP‐LP) approach with the duty of finding the optimal operation policy of the candidate wells defined by their pumping rates. Two different codings, namely binary and integer codings, are used for the GA and their performances are compared. The ability of the proposed hybrid method is tested against two benchmark problems: (1) finding the optimal configuration and pumping rates of a predefined number of wells out of potential wells and (2) finding the optimal number, configuration and pumping rates of the operating wells out of potential wells and the results are presented and compared with the available ones showing superior efficiency and effectiveness of the proposed method.     

A general solution for groundwater flow in an estuarine's leaky aquifer system after considering aquifer anisotropy

This paper presents an analytical model for describing the tidal effects in a two‐dimensional leaky confined aquifer system in an estuarine delta where ocean and river meet. This system has an unconfined aquifer on top and a confined aquifer on the bottom with an aquitard in between the two. The unconfined and confined aquifers interact with each other through leakage. It was assumed that the aquitard storage was negligible and that the leakage was linearly proportional to the head difference between the unconfined and confined aquifers. This model's solution was based on the separation of variables method. Two existing solutions that deal with the head fluctuation in one‐dimensional or two‐dimensional leaky confined aquifers are shown as special cases in the present solution. Based on this new solution, the dynamic effect of the water table's fluctuations can be clearly explored, as well as the influence of leakage on the behaviour of fluctuations in groundwater levels in the leaky aquifer system. Copyright © 2015 John Wiley & Sons, Ltd.     

Unconfined linear flow to a horizontal well

The validity of a previously proposed but untested modification to equations for flow to a horizontal well is assessed using a specially developed finite-difference model. This modification extends confined flow equations to allow the head in the well and the saturated depth at the well to be estimated in unconfined conditions. The study is limited to the case of two-dimensional flow with no flow in the direction parallel to the line of the well. The results show that the modified equations for both a finite unconfined aquifer and, by inference, an infinite unconfined aquifer are adequately accurate for practical application.     

Optimal control algorithm and neural network for dynamic groundwater management

Researchers have found that obtaining optimal solutions for groundwater resource‐planning problems, while simultaneously considering time‐varying pumping rates, is a challenging task. This study integrates an artificial neural network (ANN) and constrained differential dynamic programming (CDDP) as simulation‐optimization model, called ANN‐CDDP. Optimal solutions for a groundwater resource‐planning problem are determined while simultaneously considering time‐varying pumping rates. A trained ANN is used as the transition function to predict ground water table under variable pumping conditions. The results show that the ANN‐CDDP reduces computational time by as much as 94·5% when compared to the time required by the conventional model. The proposed optimization model saves a considerable amount of computational time for solving large‐scale problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Identifying Aquifer Type in Fractured Rock Aquifers using Harmonic Analysis

Determining aquifer type, unconfined, semi‐confined, or confined, by drilling or performing pumping tests has inherent problems (i.e., cost and complex field issues) while sometimes yielding inconclusive results. An improved method to cost‐effectively determine aquifer type would be beneficial for hydraulic mapping of complex aquifer systems like fractured rock aquifers. Earth tides are known to influence water levels in wells penetrating confined aquifers or unconfined thick, low‐porosity aquifers. Water‐level fluctuations in wells tapping confined and unconfined aquifers are also influenced by changes in barometric pressure. Harmonic analyses of water‐level fluctuations of a thick (～1000 m) carbonate aquifer located in south‐central Oklahoma (Arbuckle‐Simpson aquifer) were utilized in nine wells to identify aquifer type by evaluating the influence of earth tides and barometric‐pressure variations using signal identification. On the basis of the results, portions of the aquifer responded hydraulically as each type of aquifer even though there was no significant variation in lithostratigraphy. The aquifer type was depth dependent with confined conditions becoming more prevalent with depth. The results demonstrate that harmonic analysis is an accurate and low‐cost method to determine aquifer type.     

Analytical methods for transient flow to a well in a confined-unconfined aquifer

Concurrent existence of confined and unconfined zones of an aquifer can arise owing to ground water withdrawal by pumping. Using Girinskii's potential function, Chen (1974, 1983) developed an approximate analytical solution to analyze transient ground water flow to a pumping well in an aquifer that changes from an initially confined system to a system with both unconfined and confined regimes. This article presents the details of the Chen model and then compares it with the analytical model developed by Moench and Prickett (1972) for the same problem. Hypothetical pumping test examples in which the aquifer undergoes conversion from confined to water table conditions are solved by the two analytical models and also a numerical model based on MODFLOW. Comparison of the results suggests that the solutions of the Chen model give better results than the Moench and Prickett model except when the radial distance is very large or aquifer thickness is large compared with drawdown.     

Determination of hydraulic diffusivity of aquifers by spectral analysis

This study uses the cyclical frequency to develop the mathematical relationship between hydraulic diffusivity and spectral density functions calculated from groundwater level variation. Such relationship can be applied to (1) unsteady state, one-dimensional confined aquifer with time-dependent water level on both end boundaries, and (2) linearized unconfined aquifer with or without vertical recharge. The spectral density functions of groundwater fluctuations are largely affected by the spectral density functions obtained from time-dependent end boundaries and their cross-spectral density functions. Hydraulic diffusivity of an aquifer can be solved by type-curve matching technique at a specified frequency band under the conditions of (1) confined aquifer having equal time-dependent boundaries on both ends, (2) unconfined aquifer having equal time-dependent boundaries on both ends with surface recharge, and (3) unconfined aquifer subjected to surface recharge but neglecting the water table fluctuations on both end boundaries.     

Information effect on remediation design of contaminated aquifers using the pump and treat method

We examine the effect of uncertainty due to limited information on the remediation design of a contaminated aquifer using the pump and treat method. The hydraulic conductivity and contaminant concentration distributions for a fictitious contaminated aquifer are generated assuming a limited number of sampling locations. Stochastic optimization with multiple realizations is used to account for aquifer uncertainty. The optimization process involves a genetic algorithm (GA). As the number of realizations increases, a greater extraction rate and more wells are needed. There was a total cost increase, but the optimal remediation designs became more reliable. Stochastic optimization analysis also determines the locations for extraction wells, the variation in extraction rates as a function of the change of well locations, and the reliability of the optimal designs. The number of realizations (stack number) that caused the design factors to converge could be determined. Effective stochastic optimization may be achieved by reducing computational resources. An increase in the variability of the conductivity distribution requires more extraction wells. Information about potential extraction wells can be used to prevent failure of the remediation task.     

A new method for aquifer system identification and parameter estimation

The standard practice for assessing aquifer parameters is to match groundwater drawdown data obtained during pumping tests against theoretical well function curves specific to the aquifer system being tested. The shape of the curve derived from the logarithmic time derivative of the drawdown data is also very frequently used as a diagnostic tool to identify the aquifer system in which the pumping test is being conducted. The present study investigates the incremental area method (IAM) to serve as an alternative diagnostic tool for the aquifer system identification as well as a supplement to the aquifer parameter estimation procedure. The IAM based diagnostic curves for ideal confined, leaky, bounded and unconfined aquifers have been derived as part of this study, and individual features of the plots have been identified. These features were noted to be unique to each aquifer setting, which could be used for rapid evaluation of the aquifer system. The effectiveness of the IAM methodology was investigated by analyzing field data for various aquifer settings including leaky, unconfined, bounded and heterogeneous conditions. The results showed that the proposed approach is a viable method for use as a diagnostic tool to identify the aquifer system characteristics as well as to support the estimation of the hydraulic parameters obtained from standard curve matching procedures. Copyright © 2012 John Wiley & Sons, Ltd.     

SIMPLE ANALYTICAL SOLUTIONS FOR THE HP41CV PROGRAMMABLE CALCULATOR

Abstract. Two useful programs have been developed for the Hewlett Packard HP41CV programmable calculator. The THEIS program is designed to simulate a well pumping from a confined or unconfined aquifer. Drawdown, residual drawdown, t/t₁ and t/r² are calculated. The BOUN program is designed to solve for drawdown in a well pumping from an aquifer bounded by two parallel impermeable barriers. The programs can be used in aquifer pumping test design, pumping test analysis, and aquifer response predictions.     

Tide-induced groundwater head fluctuation in coastal multi-layered aquifer systems with a submarine outlet-capping

This paper considered the tide-induced head fluctuations in two coastal multi-layered aquifer systems. Model I comprises two semi-permeable layers and a confined aquifer between them. Model II is a four-layered aquifer system including an unconfined aquifer, an upper semi-permeable layer, a confined aquifer and a lower semi-permeable layer. In each model, the submarine outlet of the confined aquifer is covered with a skin layer (“outlet-capping”). Analytical solutions of the two models are derived. In both models, leakages of the semi-permeable layers decrease the tidal head fluctuations. The outlet-capping reduces the aquifer’s head fluctuation by a constant factor and shifts the phase by a positive constant. The solution to Model II explains the inconsistency between the relatively small lag time and the strong amplitude damping effect of the tidal head fluctuations reported by Trefry and Johnston [Ground Water 1998;36:427–33] near the Port Adelaide River, Australia.     

Using principal component analysis in the investigation of groundwater hydrochemistry of Upper Jezireh Basin,Syria

Principal component analysis (PCA) was applied to hydrochemical and isotopic data of 34 groundwater samples. This allowed the reduction of 20 variables to four significant PCs that explain 81.9% of the total variance; F1 (47.1%) explains the groundwater mineralization, whereas F2 (17%) shows isotopic enrichment and nitrate pollution. Based on an iso-factor scores map of F1, three water zones were delineated: Zone A (F1 < ?1), with fresh groundwater from the unconfined aquifer; Zone B (1 > F1 > ?1), with moderate mineralization from the confined–unconfined aquifer boundary; and Zone C (F1 > 1), with the most mineralized hot water from the confined aquifer. The iso-factor scores map of F2 delineates positive values representing samples from the unconfined aquifer, with freshwater and nitrate contamination associated with stable isotope enrichment, whereas negative values represent samples from the confined aquifer. The results clearly demonstrate the usefulness of PCA in groundwater hydrochemistry investigations.     

A Finite Layer Formulation for Groundwater Flow to Horizontal Wells

A finite layer approach for the general problem of three‐dimensional (3D) flow to horizontal wells in multilayered aquifer systems is presented, in which the unconfined flow can be taken into account. The flow is approximated by an integration of the standard finite element method in vertical direction and the analytical techniques in the other spatial directions. Because only the vertical discretization is involved, the horizontal wells can be completely contained in one specific nodal plane without discretization. Moreover, due to the analytical eigenfunctions introduced in the formulation, the weighted residual equations can be decoupled, and the formulas for the global matrices and flow vector corresponding to horizontal wells can be obtained explicitly. Consequently, the bandwidth of the global matrices and computational cost rising from 3D analysis can be significantly reduced. Two comparisons to the existing solutions are made to verify the validity of the formulation, including transient flow to horizontal wells in confined and unconfined aquifers. Furthermore, an additional numerical application to horizontal wells in three‐layered systems is presented to demonstrate the applicability of the present method in modeling flow in more complex aquifer systems.     

Analysis of steady ground water flow toward wells in a confined-unconfined aquifer

A confined aquifer may become unconfined near the pumping wells when the water level falls below the confining unit in the case where the pumping rate is great and the excess hydraulic head over the top of the aquifer is small. Girinskii's potential function is applied to analyze the steady ground water flow induced by pumping wells with a constant-head boundary in a mixed confined-unconfined aquifer. The solution of the single-well problem is derived, and the critical radial distance at which the flow changes from confined to unconfined condition is obtained. Using image wells and the superposition method, an analytic solution is presented to study steady ground water flow induced by a group of pumping wells in an aquifer bounded by a river with constant head. A dimensionless function is introduced to determine whether a water table condition exists or not near the pumping wells. An example with three pumping wells is used to demonstrate the patterns of potentiometric surface and development of water table around the wells.     

A two‐dimensional analytical solution for groundwater flow in a leaky confined aquifer system near open tidal water

Groundwater in coastal areas is commonly disturbed by tidal fluctuations. A two‐dimensional analytical solution is derived to describe the groundwater fluctuation in a leaky confined aquifer system near open tidal water under the assumption that the groundwater head in the confined aquifer fluctuates in response to sea tide whereas that of the overlying unconfined aquifer remains constant. The analytical solution presented here is an extension of the solution by Sun for two‐dimensional groundwater flow in a confined aquifer and the solution by Jiao and Tang for one‐dimensional groundwater flow in a leaky confined aquifer. The analytical solution is compared with a two‐dimensional finite difference solution. On the basis of the analytical solution, the groundwater head distribution in a leaky confined aquifer in response to tidal boundaries is examined and the influence of leakage on groundwater fluctuation is discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Interpretation of Water Level Changes in the High Plains Aquifer in Western Kansas

Water level changes in wells provide a direct measure of the impact of groundwater development at a scale of relevance for management activities. Important information about aquifer dynamics and an aquifer's future is thus often embedded in hydrographs from continuously monitored wells. Interpretation of those hydrographs using methods developed for pumping‐test analyses can provide insights that are difficult to obtain via other means. These insights are demonstrated at two sites in the High Plains aquifer in western Kansas. One site has thin unconfined and confined intervals separated by a thick aquitard. Pumping‐induced responses in the unconfined interval indicate a closed (surrounded by units of relatively low permeability) system that is vulnerable to rapid depletion with continued development. Responses in the confined interval indicate that withdrawals are largely supported by leakage. Given the potential for rapid depletion of the unconfined interval, the probable source of that leakage, it is likely that large‐scale irrigation withdrawals will not be sustainable in the confined interval beyond a decade. A second site has a relatively thick unconfined aquifer with responses that again indicate a closed system. However, unlike the first site, previously unrecognized vertical inflow can be discerned in data from the recovery periods. In years of relatively low withdrawals, this inflow can produce year‐on‐year increases in water levels, an unexpected occurrence in western Kansas. The prevalence of bounded‐aquifer responses at both sites has important ramifications for modeling studies; transmissivity values from pumping tests, for example, must be used cautiously in regional models of such systems.     

Geochemical factors controlling radium activity in a sandstone aquifer

Geochemical processes behind the occurrence of radium activities in excess of the U.S. EPA's drinking water limit of 5 pCi/L combined radium were investigated in a regional sandstone aquifer located in southeastern Wisconsin. Geochemical speciation modeling (PHREEQC 2.7) combined with a detailed understanding of the regional flow system provided by recent flow modeling efforts was used to determine that radium coprecipitation into barite controls radium activity in the unconfined portion of the aquifer. As the aquifer transitions from unconfined to confined conditions, radium levels rise and the water becomes more sulfate rich yet the aquifer remains at saturation with barite throughout. Calculations based on published distribution coefficients and the observed Ra:Ba atomic ratios indicate that barite contains approximately 12 mug/kg coprecipitated radium. Confined portions of the aquifer have high concentrations of sulfate, and barium concentrations become too low to be an effective control on radium activity. Additional, as yet undefined, controls on radium are operative in the downgradient, confined portion of the aquifer.     

Modelling of water table variation in response to time-varying recharge from multiple basins using the linearised Boussinesq equation

A mathematical model is presented to describe the variations of the water table in an unconfined aquifer due to time-varying recharge applied from four rectangular basins. The model is developed by solving the linearised Boussinesq equation using the extended finite Fourier cosine transform. The time-varying recharge rate is approximated by a number of piecewise linear elements of different lengths and slopes depending on the nature of the variation in recharge rate. Application of this model for the prediction of water table fluctuations and in the sensitivity analysis of various controlling parameters on the aquifer response is demonstrated in an example.     

Adjoint simulation of stream depletion due to aquifer pumping

If an aquifer is hydraulically connected to an adjacent stream, a pumping well operating in the aquifer will draw some water from aquifer storage and some water from the stream, causing stream depletion. Several analytical, semi-analytical, and numerical approaches have been developed to estimate stream depletion due to pumping. These approaches are effective if the well location is known. If a new well is to be installed, it may be desirable to install the well at a location where stream depletion is minimal. If several possible locations are considered for the location of a new well, stream depletion would have to be estimated for all possible well locations, which can be computationally inefficient. The adjoint approach for estimating stream depletion is a more efficient alternative because with one simulation of the adjoint model, stream depletion can be estimated for pumping at a well at any location. We derive the adjoint equations for a coupled system with a confined aquifer, an overlying unconfined aquifer, and a river that is hydraulically connected to the unconfined aquifer. We assume that the stage in the river is known, and is independent of the stream depletion, consistent with the assumptions of the MODFLOW river package. We describe how the adjoint equations can be solved using MODFLOW. In an illustrative example, we show that for this scenario, the adjoint approach is as accurate as standard forward numerical simulation methods, and requires substantially less computational effort.