Convergence of deterministic and stochastic approaches in optimal remediation design of a contaminated aquifer

The comparison between two series of optimal remediation designs using deterministic and stochastic approaches showed a number of converging features. Limited sampling measurements in a supposed contaminated aquifer formed the hydraulic conductivity field and the initial concentration distribution used in the optimization process. The deterministic and stochastic approaches employed a single simulation–optimization method and a multiple realization approach, respectively. For both approaches, the optimization model made use of a genetic algorithm. In the deterministic approach, the total cost, extraction rate, and the number of wells used increase when the design must satisfy the intensified concentration constraint. Growing the stack size in the stochastic approach also brings about same effects. In particular, the change in the selection frequency of the used extraction wells, with increasing stack size, for the stochastic approach can indicate the locations of required additional wells in the deterministic approach due to the intensified constraints. These converging features between the two approaches reveal that a deterministic optimization approach with controlled constraints is achievable enough to design reliable remediation strategies, and the results of a stochastic optimization approach are readily available to real contaminated sites.     

Effects of compound-specific dilution on transient transport and solute breakthrough: A pore-scale analysis

This pore-scale modeling study in saturated porous media shows that compound-specific effects are important not only at steady-state and for the lateral displacement of solutes with different diffusivities but also for transient transport and solute breakthrough. We performed flow and transport simulations in two-dimensional pore-scale domains with different arrangement of the solid grains leading to distinct characteristics of flow variability and connectivity, representing mildly and highly heterogeneous porous media, respectively. The results obtained for a range of average velocities representative of groundwater flow (0.1–10 m/day), show significant effects of aqueous diffusion on solute breakthrough curves. However, the magnitude of such effects can be masked by the flux-averaging approach used to measure solute breakthrough and can hinder the correct interpretation of the true dilution of different solutes. We propose, as a metric of mixing, a transient flux-related dilution index that allows quantifying the evolution of solute dilution at a given position along the main flow direction. For the different solute transport scenarios we obtained dilution breakthrough curves that complement and add important information to traditional solute breakthrough curves. Such dilution breakthrough curves allow capturing the compound-specific mixing of the different solutes and provide useful insights on the interplay between advective and diffusive processes, mass transfer limitations, and incomplete mixing in the heterogeneous pore-scale domains. The quantification of dilution for conservative solutes is in good agreement with the outcomes of mixing-controlled reactive transport simulations, in which the mass and concentration breakthrough curves of the product of an instantaneous transformation of two initially segregated reactants were used as measures of reactive mixing.     

Fitting a stochastic partial differential equation to aquifer data

The steady state two dimensional groundwater flow equation with constant transmissivities was studied by Whittle in 1954 as a stochastic Laplace equation. He showed that the correlation function consisted of a modified Bessel function of the second kind, order 1, multiplied by its argument. This paper uses this pioneering work of Whittle to fit an aquifer head field to unequally spaced observations by maximum likelihood. Observational error is also included in the model. Both the isotropic and anisotropic cases are considered. The fitted field is then calculated on a two dimensional grid together with its standard deviation. The method is closely related to the use of two-dimensional splines for fitting surfaces to irregularly spaced observations.     

A stochastic interpretation of the tailing effect in solute transport

 The advection-dispersion equation (ADE) is inadequate for describing tails in solute breakthrough curves. Re-examination of solute breakthrough curves from one-dimensional experiments in porous media and channel flow literature shows a consistent discrepancy compared with solutions to the ADE. The leading tail of breakthrough curves is sharper, and the trailing tail is longer and smoother, than best fitting, least-squares ADE solutions. A random particle simulation exercise shows that the ADE may firstly be erroneous because of the assumption of time steps over which random solute movements are considered independent. Definition of such time steps hinges upon the slowest random movements, such as those predominantly by molecular diffusion. A second potential source of error is the highly skewed nature of the inverse distribution of underlying, micro-scale velocities, which causes slow convergence to normality under the central limit theorem.     

沉积模型和储层随机建模

沉积模型是地层分析的重要工具，可以分为比例尺模型、概念模型和数学模型三大类型，其中数学模型又可分为确定性模型和随机模型。在实际地层分析及模拟工作中，特别是在小尺度问题的研究中，采用随机模型(或称统计学模型)往往更为有利。储层随机建模技术，作为这方面研究的典范，近年来成为储层预测和风险评价的一项较为有效的手段。然而，由于研究目标的复杂性，不同沉积模型之间的嵌套制约关系亦应引起重视。     

Recharge process and aquifer models of a small watershed

Abstract

Kanchanapally watershed covering an area of about 11 km² in Nalgonda district, Andhra Pradesh, India is located in granitic terrain. Groundwater recharge has been estimated from a water balance model using hydrometeorological data from 1978–1994. The monthly recharge estimates obtained from the water balance model formed input for the groundwater flow model during transient model testing. The groundwater flow model has been prepared to simulate steady state groundwater conditions of 1977 using the nested squares finite difference method. The transient groundwater flow model has been tested during 1977–1994 using the estimated recharge values. The present study helped verify the usefulness of monthly recharge estimates for accounting dynamic variations in recharge as reflected in water level fluctuations in hydrographs.     

One-dimensional uniform and time varying solute dispersion along transient groundwater flow in a semi-infinite aquifer

An analytical solution for the space-time variation of contaminant concentration in one-dimensional transient groundwater flow in a homogenous semi-infinite aquifer, subjected to time-dependent source contamination, is derived. The uniform and time varying dispersion along transient groundwater flow is investigated under two conditions. First, the flow velocity distribution in the aquifer is considered as a sinusoidally varying function, and second, the flow velocity distribution is treated as an exponentially increasing function of time. It is assumed that initially the aquifer is not solute free, so the initial background concentration is considered as an exponentially decreasing function of the space variable which is tending to zero at infinity. It is assumed that dispersion is directly proportional to the square of the velocity, noting that experimental observations indicate that dispersion is directly proportional to the velocity with a power ranging from 1 to 2. The analytical solution is illustrated using an example and may help benchmark numerical codes and solutions.     

Data-based modelling approach for variable density flow and solute transport simulation in a coastal aquifer

Data-based models, namely artificial neural network (ANN), support vector machine (SVM), genetic programming (GP) and extreme learning machine (ELM), were developed to approximate three-dimensional, density-dependent flow and transport processes in a coastal aquifer. A simulation model, SEAWAT, was used to generate data required for the training and testing of the data-based models. Statistical analysis of the simulation results obtained by the four models show that the data-based models could simulate the complex salt water intrusion process successfully. The selected models were also compared based on their computational ability, and the results show that the ELM is the fastest technique, taking just 0.5 s to simulate the dataset; however, the SVM is the most accurate, with a Nash-Sutcliffe efficiency (NSE) ≥ 0.95 and correlation coefficient R ≥ 0.92 for all the wells. The root mean square error (RMSE) for the SVM is also significantly less, ranging from 12.28 to 77.61 mg/L.     

Worth of head data in well-capture zone design: deterministic and stochastic analysis

In this study, we examine the effects of conditioning spatially variable transmissivity fields using head and/or transmissivity measurements on well-capture zones. In order to address the challenge posed by conditioning a flow model with spatially varying parameters, an innovative inverse algorithm, the Representers method, is employed. The method explicitly considers this spatial variability.

A number of uniform measurement grids with different densities are used to condition transmissivity fields using the Representers method. Deterministic and stochastic analysis of well-capture zones are then examined. The deterministic study focuses on comparison between reference well-capture zones and their estimated mean conditioned on head data. It shows that model performance due to head conditioning on well-capture zone estimation is related to pumping rate. At moderate pumping rates transmissivity observations are more crucial to identify effects arising from small-scale variations in pore water velocity. However, with more aggressive pumping these effects are reduced, consequently model performance, through incorporating head observations, markedly improves. In the stochastic study, the effect of conditioning using head and/or transmissivity data on well-capture zone uncertainty is examined. The Representers method is coupled with the Monte Carlo method to propagate uncertainty in transmissivity fields to well-capture zones. For the scenario studied, the results showed that a combination of 48 head and transmissivity data could reduce the area of uncertainty (95% confidence interval) in well-capture zone location by over 50%, compared to a 40% reduction using either head or transmissivity data. This performance was comparable to that obtained through calibrating on three and a half times the number of head observations alone.     

Drought forecasting using stochastic models

Drought is a global phenomenon that occurs virtually in all landscapes causing significant damage both in natural environment and in human lives. Due to the random nature of contributing factors, occurrence and severity of droughts can be treated as stochastic in nature. Early indication of possible drought can help to set out drought mitigation strategies and measures in advance. Therefore drought forecasting plays an important role in the planning and management of water resource systems. In this study, linear stochastic models known as ARIMA and multiplicative Seasonal Autoregressive Integrated Moving Average (SARIMA) models were used to forecast droughts based on the procedure of model development. The models were applied to forecast droughts using standardized precipitation index (SPI) series in the Kansabati river basin in India, which lies in the Purulia district of West Bengal state in eastern India. The predicted results using the best models were compared with the observed data. The predicted results show reasonably good agreement with the actual data, 1–2 months ahead. The predicted value decreases with increase in lead-time. So the models can be used to forecast droughts up to 2 months of lead-time with reasonably accuracy.     

A stochastic optimization framework for the restoration of an over-exploited aquifer

ABSTRACT

This study investigates the impact of hydraulic conductivity uncertainty on the sustainable management of the aquifer of Lake Karla, Greece, using the stochastic optimization approach. The lack of surface water resources in combination with the sharp increase in irrigation needs in the basin over the last 30 years have led to an unprecedented degradation of the aquifer. In addition, the lack of data regarding hydraulic conductivity in a heterogeneous aquifer leads to hydrogeologic uncertainty. This uncertainty has to be taken into consideration when developing the optimization procedure in order to achieve the aquifer’s sustainable management. Multiple Monte Carlo realizations of this spatially-distributed parameter are generated and groundwater flow is simulated for each one of them. The main goal of the sustainable management of the ‘depleted’ aquifer of Lake Karla is two-fold: to determine the optimum volume of renewable groundwater that can be extracted, while, at the same time, restoring its water table to a historic high level. A stochastic optimization problem is therefore formulated, based on the application of the optimization method for each of the aquifer’s multiple stochastic realizations in a future period. In order to carry out this stochastic optimization procedure, a modelling system consisting of a series of interlinked models was developed. The results show that the proposed stochastic optimization framework can be a very useful tool for estimating the impact of hydraulic conductivity uncertainty on the management strategies of a depleted aquifer restoration. They also prove that the optimization process is affected more by hydraulic conductivity uncertainty than the simulation process.

Editor Z.W. Kundzewicz; Guest editor S. Weijs     

Investigations into critical earthquake load models within deterministic and probabilistic frameworks

This paper deals with the determination of critical earthquake load models for linear structures subjected to single‐point seismic inputs. The primary objective of this study is to examine the realism in critical excitations and critical responses vis a vis the framework adopted for the study and constraints that these excitations are taken to satisfy. Two alternative approaches are investigated. In the first approach, the critical earthquake is expressed in terms of a Fourier series that is modulated by an enveloping function that imparts transient nature to the inputs. The Fourier coefficients are taken to be deterministic and are constrained to satisfy specified upper and lower bounds. Estimates on these bounds, for a given site, are obtained by analysing past earthquake records from the same site or similar sites. The unknown Fourier coefficients are determined such that the response of a given structure is maximized subjected to these bounds and additional constraints on intensity, peak ground acceleration, peak ground velocity and peak ground displacement. In the second approach, the critical earthquake is modelled as a partially specified non‐stationary Gaussian random process which is defined in terms of a stationary random process of unknown power spectral density (psd) function modulated by a deterministic envelope function. The input is constrained to possess specified variance and average zero crossing rate. Additionally, a new constraint in terms of entropy rate representing the expected level of disorder in the excitation is also imposed. The unknown psd function of the stationary part of the input is determined so that the response of a given structure is maximized. The optimization problem in both these approaches is solved by using sequential quadratic programming method. The procedures developed are illustrated by considering the seismic response of a tall chimney and an earth dam. It is concluded that the imposition of lower and upper bounds on Fourier coefficients in the first approach and constraints on amount of disorder in the second approach are crucial in arriving at realistic critical excitations. Copyright © 2002 John Wiley & Sons, Ltd.     

Tracing effects of decalcification on solute sources in a shallow groundwater aquifer, NW Germany

δ⁸⁷Sr values and Ca/Sr ratios were employed to quantify solute inputs from atmospheric and lithogenic sources to a catchment in NW Germany. The aquifer consists primarily of unconsolidated Pleistocene eolian and fluviatile deposits predominated by >90% quartz sand. Accessory minerals include feldspar, glauconite, and mica, as well as disperse calcium carbonate in deeper levels. Decalcification of near-surface sediment induces groundwater pH values up to 4.4 that lead to enhanced silicate weathering. Consequently, low mineralized Ca–Na–Cl- and Ca–Cl-groundwater types are common in shallow depths, while in deeper located calcareous sediment Ca–HCO₃-type groundwater prevails. δ⁸⁷Sr values and Ca/Sr ratios of the dissolved pool range from 7.3 to −2.6 and 88 to 493, respectively. Positive δ⁸⁷Sr values and low Ca/Sr ratios indicate enhanced feldspar dissolution in shallow depths of less than 20 m below soil surface (BSS), while equilibrium with calcite governs negative δ⁸⁷Sr values and elevated Ca/Sr ratios in deep groundwater (>30 m BSS). Both positive and negative δ⁸⁷Sr values are evolved in intermediate depths (20–30 m BSS). For groundwater that is undersaturated with respect to calcite, atmospheric supplies range from 4% to 20%, while feldspar-weathering accounts for 8–26% and calcium carbonate for 62–90% of dissolved Sr²⁺. In contrast, more than 95% of Sr²⁺ is derived by calcium carbonate and less than 5% by feldspar dissolution in Ca–HCO₃-type groundwater. The surprisingly high content of carbonate-derived Sr²⁺ in groundwater of the decalcified portion of the aquifer may account for considerable contributions from Ca-containing fertilizers. Complementary tritium analyses show that equilibrium with calcite is restricted to old groundwater sources.     

A numerical Lagrangian stochastic approach to upscaling of dispersivity in solute transport

The upscaling of dispersivity in solute transport in heterogeneous aquifers is addressed with a numerical stochastic formulation. This work represents progress toward converting theory into scalable numerical models that can be compared to laboratory experiments. Traditional global assumptions of low variance, ergodicity, single correlation scale, stationarity, and the like are avoided through the use of a flexible Lagrangian numerical, not analytical, framework, which allows assumptions to be local. A method of calculating grid-block upscaled dispersivities is presented. Computational results are obtained for a heterogeneous tank experiment, with reasonable behavior.     

Metaphors and models: the ASR bubble in the Floridan aquifer

Studies at the intersection of cognitive science and linguistics have revealed the crucial role that metaphors play in shaping our thoughts about phenomena we cannot see. According to the domains interaction theory of cognition, a metaphoric expression sets up mappings between a target domain that we wish to understand and a familiar source domain. The source domain contains elements ("commonplaces") that we manipulate mentally, like parts of an analogue model, to illuminate the target domain. This paper applies the structure of domains interaction theory to analyze the dynamics of a metaphor in hydrogeology: the so-called bubble formed by water injected into an aquifer during aquifer storage and recovery (ASR). Of the four commonplaces of bubbles--(1) they are discrete; (2) they are geometrically simple; (3) they rise; and (4) they burst--we focus on the first two using both displacement and dispersion (tracer) models for both homogeneous and heterogeneous storage zones patterned from geological studies of the Suwannee Limestone of Sarasota County, Florida. The displacement model easily shows that "bottle brush" better represents the geometric complexity predicted from the known and inferred heterogeneity. There is virtually no difference, however, in the prediction of recovery efficiency using the dispersion model for a bubble (homogeneous flow zone) vs. bottle brush (heterogeneous flow zone). On the other hand, only the bottle brush reveals that unrecovered tracer is located preferentially in the low-permeability layers that lie adjacent to high-permeability channels in the flow zones.     

Nitrate fluctuations at the water table: implications for recharge processes and solute transport in the Chalk aquifer

This study investigates fluctuations in nitrate concentration at the water table to improve understanding of unsaturated zone processes in the Chalk aquifer. Sampling was conducted using a novel multi‐level sampler during periods of water table rise over 5 years at a vertical resolution of 0.05 m. Nitrate concentration increased as the water table seasonally recovered, with similar inter‐annual trends with depth. The rising water table activated horizontal fractures facilitating the delivery of water elevated by up to 10 mg/l of nitrate with respect to the adjacent groundwater below. These fractures are considered to activate via piston displacement of water from the adjoining matrix. Hydrograph analysis identified 16 events which significantly perturbed the water table within 24–48 h of rainfall. Consistent nitrate concentrations indicate recharge through persistent fracture flow from the surface was not generally the primary driver of the rapid water table response during these events. Instead, the response was attributed to the piston displacement of porewater immediately above the water table. However, a single event in November 2012 delivered relatively dilute recharge indicating rapid persistent fracture flow following rainfall was possible to a depth of 14–15 m. Decreases in porewater nitrate concentration around fracture horizons and the dilution of many groundwater samples with respect to porewaters indicate a fresher source of water at depth. This was considered most likely to be a result of near surface water bypassing the matrix because of widespread mineralization on fracture surfaces, which retard water and solute exchange. Therefore, persistent fracture flow maybe considered a frequent process, operating independently of the matrix, and is not necessarily event driven. Copyright © 2015 John Wiley & Sons, Ltd.