Browsing by Author "Huang, Ching-Sheng"

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A general analytical model for head response to oscillatory pumping in unconfined aquifers: effects of delayed gravity drainage and initial condition
(2019-03-08)
Huang, Ching-Sheng
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Tsai, Ya-Hsin
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Yeh, Hund-Der
;
Yang, Tao
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環境工程研究所
Oscillatory pumping tests (OPTs) provide an alternative to constant-head and constant-rate pumping tests for determining aquifer hydraulic parameters when OPT data are analyzed based on an associated analytical model coupled with an optimization approach. There are a large number of analytical models presented for the analysis of the OPT. The combined effects of delayed gravity drainage (DGD) and the initial condition regarding the hydraulic head are commonly neglected in the existing models. This study aims to develop a new model for describing the hydraulic head fluctuation induced by the OPT in an unconfined aquifer. The model contains a groundwater flow equation with the initial condition of a static water table, Neumann boundary condition specified at the rim of a partially screened well, and a free surface equation describing water table motion with the DGD effect. The solution is derived using the Laplace, finite-integral, and Weber transforms. Sensitivity analysis is carried out for exploring head response to the change in each hydraulic parameter. Results suggest that the DGD reduces to instantaneous gravity drainage in predicting transient head fluctuation when the dimensionless parameter a(1) = epsilon S(y)b/K-z exceeds 500 with empirical constant epsilon, specific yield S-y, aquifer thickness b, and vertical hydraulic conductivity K-z. The water table can be regarded as a no-flow boundary when a(1) < 10(-2) and P < 10(4) s, with P being the period of the oscillatory pumping rate. A pseudo-steady-state model without the initial condition causes a time-shift from the actual transient model in predicting simple harmonic motion of head fluctuation during a late pumping period. In addition, the present solution agrees well with head fluctuation data observed at the Savannah River site. Highlights. An analytical model of the hydraulic head due to oscillatory pumping in unconfined aquifers is presented. Head fluctuations affected by instantaneous and delayed gravity drainages are discussed. The effect of the initial condition on the phase of head fluctuation is analyzed. The present solution agrees well with head fluctuation data taken from field oscillatory pumping.
A general analytical solution for flow to a single horizontal well by Fourier and Laplace transforms
(2011-05-01)
Huang, Ching-Sheng
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Chen, Yu-Lin
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Yeh, Hund-Der
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環境工程研究所
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Institute of Environmental Engineering
The objective of this paper is to present an analytical solution for describing the head distribution in an unconfined aquifer with a single pumping horizontal well parallel to a fully penetrating stream. The Laplace-domain solution is developed by applying Fourier sine, Fourier and Laplace transforms to the governing equation as well as the associated initial and boundary conditions. The time-domain solution is obtained after taking the inverse Laplace transform along with the Bromwich integral method and inverse Fourier and Fourier sine transforms. The upper boundary condition of the aquifer is represented by the free surface equation in which the second-order slope terms are neglected. Based on the solution and Darcy's law, the equation representing the stream depletion rate is then derived. The solution can simulate head distributions in an aquifer infinitely extending in horizontal direction if the well is located far away from the stream. In addition, the solution can also simulate head distributions in confined aquifers if specific yield is set zero. It is shown that the solution can be applied practically to evaluate flow to a horizontal well. (C) 2011 Elsevier Ltd. All rights reserved.
A general analytical solution for groundwater fluctuations due to dual tide in long but narrow islands
(2012-05-04)
Huang, Ching-Sheng
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Yeh, Hund-Der
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Chang, Chia-Hao
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環境工程研究所
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Institute of Environmental Engineering
This paper develops a general mathematical model for describing head fluctuations in an aquifer of long but narrow islands subject to a dual tide effect. The upper boundary condition of the aquifer is represented by an equation combining the simplified free surface equation with a leakage term. Such an equation is considered as a general expression representing the upper boundary condition of a confined, unconfined, or leaky confined aquifer. The closed-form solution of the model represented by two series terms is developed by the direct Fourier method and finite Fourier sine transform. One of the series can reduce to a closed-form expression by means of contour integral and residue theorem. If the width of the island is very large, this solution gives the predicted head almost the same as that of the solution for an aquifer subject to a single tide effect. It is found that the presence of an upper aquitard produces significant vertical flow in the lower leaky confined aquifer even if the aquitard permeability is low. Neglecting such vertical flow may result in an overestimate of hydraulic head in the leaky confined aquifer. The attenuation factor and phase lag predicted from the present solution subject to the dual tide effect agree well with those estimated from 57 day head fluctuation data observed at Garden Island, Australia.
An analytical approach for the simulation of flow in a heterogeneous confined aquifer with a parameter zonation structure
(2016-11)
Huang, Ching-Sheng
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Yeh, Hund-Der
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環境工程研究所
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Institute of Environmental Engineering
This study introduces an analytical approach to estimate drawdown induced by well extraction in a heterogeneous confined aquifer with an irregular outer boundary. The aquifer domain is divided into a number of zones according to the zonation method for representing the spatial distribution of a hydraulic parameter field. The lateral boundary of the aquifer can be considered under the Dirichlet, Neumann or Robin condition at different parts of the boundary. Flow across the interface between two zones satisfies the continuities of drawdown and flux. Source points, each of which has an unknown volumetric rate representing the boundary effect on the drawdown, are allocated around the boundary of each zone. The solution of drawdown in each zone is expressed as a series in terms of the Theis equation with unknown volumetric rates from the source points. The rates are then determined based on the aquifer boundary conditions and the continuity requirements. The estimated aquifer drawdown by the present approach agrees well with a finite element solution developed based on the Mathematica function NDSolve. As compared with the existing numerical approaches, the present approach has a merit of directly computing the drawdown at any given location and time and therefore takes much less computing time to obtain the required results in engineering applications.
An analytical approach for the simulation of flow in a heterogeneous confined aquifer with a parameter zonation structure
(2016-11)
Huang, Ching-Sheng
;
Yeh, Hund-Der
;
環境工程研究所
;
Institute of Environmental Engineering
This study introduces an analytical approach to estimate drawdown induced by well extraction in a heterogeneous confined aquifer with an irregular outer boundary. The aquifer domain is divided into a number of zones according to the zonation method for representing the spatial distribution of a hydraulic parameter field. The lateral boundary of the aquifer can be considered under the Dirichlet, Neumann or Robin condition at different parts of the boundary. Flow across the interface between two zones satisfies the continuities of drawdown and flux. Source points, each of which has an unknown volumetric rate representing the boundary effect on the drawdown, are allocated around the boundary of each zone. The solution of drawdown in each zone is expressed as a series in terms of the Theis equation with unknown volumetric rates from the source points. The rates are then determined based on the aquifer boundary conditions and the continuity requirements. The estimated aquifer drawdown by the present approach agrees well with a finite element solution developed based on the Mathematica function NDSolve. As compared with the existing numerical approaches, the present approach has a merit of directly computing the drawdown at any given location and time and therefore takes much less computing time to obtain the required results in engineering applications.
An analytical solution for a radial collector well near a stream with a low-permeability streambed
(2012-06-26)
Huang, Ching-Sheng
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Tsou, Pei-Rong
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Yeh, Hund-Der
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環境工程研究所
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Institute of Environmental Engineering
Radial collector wells are often constructed near a stream to obtain more water and produce smaller drawdown in comparison with traditional vertical wells. This paper aims at developing a mathematical model for describing head distributions induced by pumping in a radial collector well near a stream in an unconfined aquifer. A low permeable streambed between the stream and the aquifer is considered. The first-order free surface equation is used to describe the movement of water table. A point-sink solution of the model is developed first by a general transform combining Fourier sine and cosine transforms and then by Fourier transform and Laplace transform. The transient solution of head distributions for different well types such as the horizontal well or radial collector well can be obtained by integrating the point-sink solution along the well. Based on Darcy's law and the developed solution, an equation for temporal stream depletion rate describing filtration from the stream is then obtained. The steady-state solutions for filtration can be obtained from the transient solution when neglecting the exponential term associated with time. It is found that steady-state filtration depends only on the ratio of streambed permeability over aquifer permeability. Steady-state filtration equals water extraction from a well when the ratio is larger than 10(-2). The streambed is regarded as completely impermeable when the ratio is less than 10(-7). Additionally, the lowest water table happens near the well before occurrence of filtration. The lowest water table moves landward and away from the well after filtration occurs. (C) 2012 Elsevier B.V. All rights reserved.
An analytical solution for a radial collector well near a stream with a low-permeability streambed (vol 446, pg 48, 2012)
(2016-11)
Huang, Ching-Sheng
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Tsou, Pei-Rong
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Yeh, Hund-Der
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環境工程研究所
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Institute of Environmental Engineering
Analysis of radially convergent tracer test in a two-zone confined aquifer with vertical dispersion effect: Asymmetrical and symmetrical transports
(2019-09-05)
Huang, Ching-Sheng
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Tong, Chenchen
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Hu, Wei-Shan
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Yeh, Hund-Der
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Yang, Tao
Radially convergent tracer test (RCT) with an extraction well and a tracer injection well is commonly conducted for determining aquifer dispersivity associated with the spreading of contamination plume. A variety of analytical models for RCT have been proposed, but the effects of skin zone and vertical dispersion are rarely explored. In addition, little attention is paid to the validity of those analytical models assuming asymmetrical tracer transport (ATT) as radially symmetrical transport (RST) toward the pumping well. This study develops a new analytical model for RCT subject to the above-mentioned effects in a confined aquifer with a skin zone around the wellbore. The Laplace-domain solution of the model for a continuous or instantaneous input is developed. A finite element solution (FES) for AFT is also developed to verify the Laplace-domain solution based on RST. Results suggest ATT can be regarded as RST in predicting the breakthrough curve (BTC) at the pumping well when four quantitative conditions are met. A lumped dimensionless parameter dominates those two effects on the BTC. Both the FES and Laplace-domain solution agree with monitored concentration data from a field RCT.
Analysis of three-dimensional unsaturated-saturated flow induced by localized recharge in unconfined aquifers
(2018-07-20)
Chang, Chia-Hao
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Huang, Ching-Sheng
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Yeh, Hund-Der
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環境工程研究所
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Institute of Environmental Engineering
In the process of groundwater recharge, surface water usually enters an aquifer by passing an overlying unsaturated zone. Little attention has been given to the development of analytical solutions to a coupled unsaturated-saturated flow model due to localized recharge up to now. This paper develops a mathematical model to depict three-dimensional transient unsaturated-saturated flow in an unconfined aquifer with localized recharge on the ground surface. The model contains Richards' equation for unsaturated flow, a flow equation for saturated formation, and the Gardner constitutive model describing the behavior of unsaturated soil properties. Both flow equations are coupled through the continuity conditions of the head and flux at the water table. The semi-analytical solution to the coupled flow model is derived by the methods of Laplace transform and Fourier cosine transform. A sensitivity analysis is performed to explore the head response to the change in each of the aquifer parameters. A quantitative tool is presented to assess the recharge efficiency signifying the percentage of the water from the recharge to the aquifer. We found that the effect of unsaturated flow on the saturated hydraulic head is negligible if two criteria associated with the unsaturated soil properties and initial aquifer thickness are satisfied. The head distributions predicted from the present solution match well with those from finite-difference simulations. The predictions of the present solution also agree well with the observed data from a field experiment at an artificial recharge pond in Fresno County, California.
Analysis of Unconfined Flow Induced by Constant Rate Pumping Based on the Lagging Theory
(2019-05-01)
Lin, Ye-Chen
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Huang, Ching-Sheng
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Yeh, Hund-Der
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環境工程研究所
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Institute of Environmental Engineering
This study proposes a new mathematical model for describing the drawdown distribution due to a constant rate pumping (CRP) in an unconfined aquifer considering the lagging theory. We introduce two lag times in Darcy's law and in turn in a free surface equation to reflect the effects of the capillary fringe and the capillary suction on the water table motion. The present free surface equation can reduce to those used in previous studies. The Laplace and Weber transform methods are used to derive the semianalytical solution to the model including the effect of wellbore storage. The algorithm of numerical Laplace inversion is applied to obtain the transient solution of the model. We find that the delay index, commonly used in the literature, is equivalent to the lag time associated with the effect of the capillary suction. The sensitivity analysis is performed to assess the drawdown behavior in response to the change in each aquifer parameter. The drawdown distributions predicted by the present solution agree fairly well to the field data taken from CRP tests at Cape Cod, Massachusetts; the Canadian Forces Base Borden, Ontario; and Saint Pardon de Conques, Gironde (France). The lag times determined by both CRP tests seem to decrease linearly with increasing distance on the logarithmic scale from the pumping well. The consideration of two lag times considerably improves the accuracy in estimated specific yields.
Approximate Solution for a Transient Hydraulic Head Distribution Induced by a Constant-Head Test at a Partially Penetrating Well in a Two-Zone Confined Aquifer
(2014-07-01)
Yang, Shaw-Yang
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Huang, Ching-Sheng
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Liu, Chih-Hsuan
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Yeh, Hund-Der
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交大名義發表
A mathematical model describing the transient hydraulic head distribution induced by constant-head pumping/injection at a partially penetrating well in a radial two-zone confined aquifer is a mixed-type boundary value problem. The analytical solution of the model is in terms of an improper integral with an integrand having a singularity at the origin. The solution should rely on numerical methods to evaluate the integral and handle the problems of convergence and singularity. This study aims at developing a new approximate solution describing the transient hydraulic head distribution for a constant-head test (CHT) at a partially penetrating well in the aquifer. This approximate solution is acquired based on a time-dependent diffusion layer approximation proposed in the field of electrochemistry. The diffusion layer can be analogous to the radius of influence in the area of well hydraulics. The approximate solution is in terms of modified Bessel functions for aquifers with a partially penetrating well and can reduce to a simpler form in terms of a natural logarithmic function for the case of well full penetration. The predicted hydraulic heads from the present approximate solution are compared with those estimated by the Laplace-domain solution of the model. The result shows that the predicted spatial head distributions are accurate in the formation zone and fairly good in the skin zone. In addition, the present solution gives an accurate temporal head distribution at a specific location when the radius of influence is far away from the observation wells. This newly developed approximate solution has advantages of easy computing and good accuracy from practical viewpoint, and thus is a handy tool to evaluate temporal and spatial hydraulic head distributions for the CHT. (C) 2014 American Society of Civil Engineers.
Discussion of "Integral and Closed-Form Analytical Solutions to the Transport Contaminant Equation Considering 3D Advection and Dispersion" by Luan Carlos de S. M. Ozelim and Andre Luis Brasil Cavalcante DOI: 10.1061/(ASCE)GM.1943-5622.0000245
(2014-10-01)
Huang, Ching-Sheng
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Yeh, Hund-Der
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交大名義發表
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環境工程研究所
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National Chiao Tung University
Estimating stream filtration from a meandering stream under the Robin condition
(2015-06-01)
Huang, Ching-Sheng
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Yeh, Hund-Der
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環境工程研究所
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Institute of Environmental Engineering
This study applies image well theory to estimate the stream depletion rate (SDR) due to pumping near a meandering stream with a clogged streambed treated as the Robin condition. The stream is considered as an irregular boundary represented by discrete nodes. Image wells are arranged along the stream and near those nodes. On the basis of the Theis (1935) solution and the principle of superposition, the solution for the aquifer drawdown subject to the stream can then be expressed as the sum of the Theis solution and a simple series representing the effect of those image wells. The discharge rates of the image wells are determined by solving a system of equations obtained by substituting the drawdown solution into the Robin condition. Quantitative criteria for assessing the applicability of the image well theory are provided. On the basis of the drawdown solution and Darcy\'s law, the analytical solution for SDR can then be obtained. A finite element solution is also developed to verify the SDR solution. Temporal SDR distributions predicted by both the analytical solution and finite element solution agree well over the entire period except at late time when the stream filtration rate approaches the pumping rate (i.e., SDR 1). It is found that a meandering stream has a significant effect on SDR compared with a rectilinear one and the effect should be taken into account in estimating SDR.
Groundwater flow to a pumping well in a sloping fault zone unconfined aquifer
(2014-05-01)
Huang, Ching-Sheng
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Yang, Shaw-Yang
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Yeh, Hund-Der
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環境工程研究所
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Institute of Environmental Engineering
This study develops a mathematical model for simulating the hydraulic head distribution in response to pumping in a sloping fault zone aquifer under a water table boundary condition. A two-dimensional equation with a sink term representing the pumping is used for describing the head distribution in the aquifer. In addition, a first-order free surface equation is adopted to represent the change in water table at the outcrop. The analytical solution of the model, derived by the Laplace and finite Fourier cosine transforms, is expressed in terms of a double series. A finite difference solution within a deformable grid framework is developed to assess the solution obtained by specifying the free surface equation at the outcrop. Based on the analytical solution, we have found that the model\'s prediction tends to overestimate drawdown in a late pumping period. The temporal head distribution is independent of the aquifer slope if the water table change is small, and exhibits a double-humped shape due to the effect of the free surface. The temporal drawdown predicted from the analytical solution is further compared with those measured from a pumping test conducted in northern Portugal.
New Analytical Models for Flow Induced by Pumping in a Stream-Aquifer System: A New Robin Boundary Condition Reflecting Joint Effect of Streambed Width and Storage
(2020-04-01)
Huang, Ching-Sheng
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Wang, Zicheng
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Lin, Ye-Chen
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Yeh, Hund-Der
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Yang, Tao
A Robin type boundary condition (BC), commonly adopted at stream-aquifer interface, excludes a term associated with streambed accounting for the effects of streambed storage and width. This study presents two new analytical models for describing confined flow induced by pumping in a stream-aquifer system. One model considers a single-zone aquifer and treats the streambed as a lagging Robin BC with a time lag parameter related to the effects. The other considers a two-zone aquifer consisting of aquifer and streambed zones. A Dirichlet BC for stream water level is specified at the edge of the streambed. The time-domain solutions of both models are developed to describe spatiotemporal drawdown and temporal stream filtration/depletion rate (SDR). The finite element solutions (FESs) of both models are also built. Results suggest the lag time equals the half of the squared streambed width divided by the streambed hydraulic diffusivity. The effects of streambed width and storage on SDR should be considered when their lumped parameter exceeds 0.1. Neglecting their effects causes 25% difference in SDR when the lumped parameter equals 10. Based on the FESs, the use of the lagging Robin BC takes nearly a tenth of computing time of obtaining accurate steady-state SDR for the simulation of the two-zone aquifer. In addition, the present solutions agree to a field SDR experiment conducted by Hunt et al. (2001, ). To conclude, this study presents two new models for describing groundwater flow in a stream-aquifer system and explores the joint effect of streambed width and storage on SDR. Key Points Two new analytical solutions are developed to estimate aquifer drawdown and stream depletion rate in a pumped stream-aquifer system One treats streambed as a Robin type boundary based on the lagging theory while the other considers streambed as a finite-width zone The lag time in the Robin boundary equals the half of squared streambed width divided by the streambed hydraulic diffusivity
Review of analytical models to stream depletion induced by pumping: Guide to model selection
(2018-06-01)
Huang, Ching-Sheng
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Yang, Tao
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Yeh, Hund-Der
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交大名義發表
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環境工程研究所
Stream depletion due to groundwater extraction by wells may cause impact on aquatic ecosystem in streams, conflict over water rights, and contamination of water from irrigation wells near polluted streams. A variety of studies have been devoted to addressing the issue of stream depletion, but a fundamental framework for analytical modeling developed from aquifer viewpoint has not yet been found. This review shows key differences in existing models regarding the stream depletion problem and provides some guidelines for choosing a proper analytical model in solving the problem of concern. We introduce commonly used models composed of flow equations, boundary conditions, well representations and stream treatments for confined, unconfined, and leaky aquifers. They are briefly evaluated and classified according to six categories of aquifer type, flow dimension, aquifer domain, stream representation, stream channel geometry, and well type. Finally, we recommend promising analytical approaches that can solve stream depletion problem in reality with aquifer heterogeneity and irregular geometry of stream channel. Several unsolved stream depletion problems are also recommended.
Stream filtration induced by pumping in a confined, unconfined or leaky aquifer bounded by two parallel streams or by a stream and an impervious stratum
(2014-05-26)
Huang, Ching-Sheng
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Lin, Wen-Sheng
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Yeh, Hund-Der
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環境工程研究所
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Institute of Environmental Engineering
A mathematical model is developed for describing three-dimensional groundwater flow induced by a fully-penetrating vertical well in aquifers between two parallel streams. A general equation is adopted to represent the top boundary condition which is applicable to either a confined, unconfined or leaky aquifer. The Robin (third-type) boundary condition is employed to represent the low-permeability streambeds. The Laplace-domain head solution of the model is derived by the double-integral and Laplace transforms. The Laplace-domain solution for a stream depletion rate (SDR) describing filtration from the streams is developed based on Darcy\'s law and the head solution and inverted to the timedomain result by the Crump method. In addition, the time-domain solution of SDR for the confined aquifer is developed analytically after taking the inverse Laplace transform and the time-domain solutions of SDR for the leaky and unconfined aquifers are developed using the Fade approximation. Both approximate solutions of SDR are expressed in terms of simple series and give fairly good match with the Laplace-domain SDR solution and measured data from a field experiment in New Zealand. The uncertainties in SDR predictions for the aquifers are assessed by performing the sensitivity analysis and Monte Carlo simulation. With the aid of the time-domain solutions, we have found that the effect of the vertical groundwater flow on the temporal SDR for a leaky aquifer is dominated by two lumped parameters: it = K (v)x(0)(2)/(KhD2) and k\'=k\'(B\'K-v) where D is the aquifer thickness, x(o) is a distance between the well and nearer stream, K-h and K are the aquifer horizontal and vertical hydraulic conductivities, respectively, and K\' and B\' are the aquitard hydraulic conductivity and thickness, respectively. When it < 10, neglecting the vertical flow underestimates the SDR. When it >= 10, the effect of vertical flow is negligible. When K\'<= 10(-4), the aquitard can be regarded as impermeable, and the leaky aquifer behaves as a confined one. (C) 2014 Elsevier B.V. All rights reserved.
Technical Note: Three-dimensional transient groundwater flow due to localized recharge with an arbitrary transient rate in unconfined aquifers
(2016-01-01)
Chang, Chia-Hao
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Huang, Ching-Sheng
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Yeh, Hund-Der
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環境工程研究所
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Institute of Environmental Engineering
Most previous solutions for groundwater flow induced by localized recharge assumed either aquifer incompressibility or two-dimensional flow in the absence of the vertical flow. This paper develops a new three-dimensional flow model for hydraulic head variation due to localized recharge in a rectangular unconfined aquifer with four boundaries under the Robin condition. A governing equation describing spatiotemporal head distributions is employed. The first-order free-surface equation with a source term defining a constant recharge rate over a rectangular area is used to depict water table movement. The solution to the model for the head is developed with the methods of Laplace transform and double-integral transform. Based on Duhamel's theorem, the present solution is applicable to flow problems accounting for arbitrary time-dependent recharge rates. The solution to depth-average head can then be obtained by integrating the head solution to elevation and dividing the result by the aquifer thickness. The use of a rectangular aquifer domain has two merits. One is that the integration for estimating the depth-average head can be analytically achieved. The other is that existing solutions based on aquifers of infinite extent can be considered as special cases of the present solution before the time when the aquifer boundary had an effect on head predictions. With the help of the present solution, the assumption of neglecting the vertical flow effect on the temporal head distribution at an observation point outside a recharge region can be assessed by a dimensionless parameter related to the aquifer horizontal and vertical hydraulic conductivities, initial aquifer thickness, and the shortest distance between the observation point and the edge of the recharge region. The validity of assuming aquifer incompressibility is dominated by the ratio of the aquifer specific yield to its storage coefficient. In addition, a sensitivity analysis is performed to investigate the head response to the change in each of the aquifer parameters.
Transient analysis for fluid injection into a dome reservoir
(2011-12-01)
Kuo, Chia-Chen
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Huang, Ching-Sheng
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Yeh, Hund-Der
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環境工程研究所
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Institute of Environmental Engineering
A dome-shaped layer can be selected as a storage site for fluid injection. In this study, we develop a mathematical model for simulating transient head distribution in a heterogeneous and anisotropic dome-shaped layer due to a constant-head injection in a fully penetrating well. In the model, a form of step change is adopted to approximate the upper and lower boundaries of the dome and then the layer is split into two regions. The Laplace-domain solution of the model is developed using the Laplace transform and method of separation of variables. The transient injection rate at wellbore can then be obtained based on Darcy's law and Bromwich integral method. The predicted head contours from the head solution show significant vertical flow components near the location of step change in the dome reservoir. The results of sensitivity analysis indicate that the hydraulic conductivity is the most sensitive parameter and the specific storage is the least sensitive one to the injection rate after a short period of injection time. In addition, the injection rate for a dome reservoir is also very sensitive to the change of the height for the reservoir near the injection well (first region) at a very early injection time. In contrast, the injection rate is more sensitive to the change of the height of the second region than that of the first region at late time. This analytical solution may be used as a primary tool to assess the capacity of fluid injection to various dome reservoirs. (C) 2011 Elsevier Ltd. All rights reserved.