Vectorized simulation of groundwater flow and ...

Environmental Modelling & Software 52 (2014) 207e221

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Environmental Modelling & Software journal homepage: www.elsevier.com/locate/envsoft

Vectorized simulation of groundwater flow and streamline transport George Kourakos, Thomas Harter* Dept. of Land Air and Water Resources, University of California Davis, Davis, 95616 CA, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 11 April 2013 Received in revised form 21 October 2013 Accepted 26 October 2013 Available online 27 November 2013

We describe a modeling suite of Matlab functions for simulating nonpoint source (NPS) pollution in groundwater aquifers. The NPS model simulates groundwater flow and contaminant transport from a large array (order of 102 e 107) of spatially distributed sources with time-varying pollution strength to a similarly large array of spatially distributed production wells (receptors) using the streamline transport approach. The code solves three equations: steady-state groundwater flow, particle tracking, and transient advection dispersion contaminant transport. The code performs convolution integration in its predictive step. Written in highly efficient vectorized form to avoid time consuming “for/while” loops, the code is also suitable for other groundwater flow and transport problems. The code is verified against analytical solutions and finite element software Comsol. An application illustrates 200 years of transient nitrate transport in the 2000 km2 Tule River aquifer sub-basin of the Central Valley, California, with 9000 individual nitrate sources and 1920 wells. 2013 Elsevier Ltd. All rights reserved.

Keywords: Non-point source pollution Finite element method Matlab Streamline transport Groundwater modeling

Software availability Name of Software: mSim Developers: George Kourakos, Thomas Harter First available year: 2013 Program Language: Matlab, C/Cþþ Availability: http://groundwater.ucdavis.edu/mSim/ Primary contacts: George Kourakos, Thomas Harter E-mail: [email protected], [email protected], thharter@ ucdavis.edu 1. Introduction Numerical simulation models have been established as standard methods for studying a vast variety of physical phenomena and environmental processes including water resources (Moriasi et al., 2012; Bobba, 2012; Bennett et al., 2013). In many countries, water management decisions are based, to a large extent, on simulation with numerical models (Refsgaard and Henriksen, 2004). Hence, numerical models have found broad applicability in research and education as well as in consulting, decision making, and industry. Following this trend, groundwater hydrologists also have a long history of developing and using numerical models for simulating

* Corresponding author. E-mail addresses: [email protected], [email protected] (G. Kourakos), [email protected] (T. Harter). 1364-8152/$ e see front matter 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.envsoft.2013.10.029

groundwater flow and contaminant transport i.e. Modflow (Harbaugh et al., 2000), Parflow (Ashby and Falgout, 1996), HydroGeoSphere (Therrien et al., 2007), Feflow (Trefry and Muffels, 2007), SUTRA (Voss, 1984), HYDRUS (Simunek et al., 2012), IWFM (Integrated Water Flow Model, 2012), FEMWATER (Lin et al., 1997) etc. In general, numerical models can be divided into two broad categories: Open source, where the code is available to the user under certain license types (e.g. Modflow, IWFM, FEMWATER) and closed source (e.g. HydroGeoSphere, Feflow, Comsol (COMSOL, 2008)) where the user has no access to the original code. Typically, closed source software products are commercial products supported through license sales (Feflow, Comsol). Neither type is necessarily preferable. The choice depends on user experience and needs and on budget constraints. Closed source models, especially those available commercially, are typically accompanied by user-friendly graphical user interfaces, which attempt to assist and protect the user from dealing with many of the details involved in a numerical simulation, i.e. grid generation, assembly process, preconditioners, solvers, parallel implementation, etc. While this is desirable in many cases, it is also very important, especially in research, that users have access and be able to deepen and intervene with the numerical codes. In addition, open source codes allow widespread reproduction of results, which is fundamental to science and useful to a wider audience than closed source codes. Hence, open source codes are more common in education and research, while commercial products are more popular in consulting and industry. Open source codes, while transparent to the user, are not always user friendly. For example,

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G. Kourakos, T. Harter / Environmental Modelling & Software 52 (2014) 207e221

open source codes such as MODFLOW and FEMWATER, which simulate the groundwater flow equation and saturated/unsaturated, density driven, flow and transport respectively, require a tedious input data preparation process, even for relatively simple applications. For the aforementioned codes, pre- and post-processors have been developed that assist in the data input and output processing, e.g. GroundwaterVista, Visual Modflow, GMS, ModelMuse (Winston, 2009), etc. The majority of open source codes is written in FORTRAN (Modflow, IWFM, Hydrogeosphere, SUTRA, FEMWATER) or C/Cþþ (Comsol, OpenFoam (OpenFoam, 2013), Dune (Dedner et al., 2011)), which provide strong computational efficiency. However reading and understanding languages such as FORTRAN and C/Cþþ requires significant experience and knowledge of the programming language itself. Implementing and testing research ideas that require modifications to an existing code are potentially difficult and error prone tasks. In addition, most recent codes are likely to use an object oriented programming style e.g. Cþþ, which is convenient for the experienced developer, yet adds another level of difficulty to users that are not programmers. In addition to products based on high performance languages such as Fortran and C/Cþþ, there is a number of high-level scripting languages that promise to increase productivity without compromising performance on high performance computing systems (Chaves et al., 2006) such as Python, Matlab, Octave etc. In addition these languages are thought to have less steep learning curves when compared to Fortran or C/Cþþ. In particular, Matlab is a proprietary platform that is very popular among engineering and environmental sciences students. It provides an interactive environment for numerical computation, visualization, and programming, yet the code is visible to the user and available for further modifications. Octave on the other hand is an open source programming language under the GNU license, which is quite similar to Matlab, so that most programs are easily portable to Matlab. But Octave currently lacks an interactive and user friendly debugging environment. Matlab and Octave codes are not considered as efficient as codes written in compiled programming languages, yet there is a growing number of examples where Matlab is used to carry out computationally intensive tasks, such as mesh generation (Persson and Strang, 2004; Talischi et al., 2012), model emulation (Tych and Young, 2012), remote sensing (Teza et al., 2012), numerical simulations (Lee et al., 2004; Kattan, 2008; Louwyck et al., 2012; Lie et al., 2012; Kumar and Dodagoudar, 2010), optimization (Kourakos and Mantoglou, 2008; Kourakos and Mantoglou, 2012a), neural networks (Kourakos and Mantoglou, 2012b), parallel computing (Kepner, 2009) etc. The major drawback of Matlab and Octave is that the execution of repeated tasks in the form of for and while loop becomes very time consuming when compared to compiled languages. To alleviate this shortcoming, Matlab and Octave provide a “vectorized” formulation of loops. According to Matlab’s definition, vectorization is the conversion of “for” and “while” loops to equivalent vector or matrix operations which can be performed in Matlab at a speed comparable to FORTRAN and C/ Cþþ. The efficiency of vectorized implementation for assembling system matrices that arise from the numerical solution of partial differential equation has been reported by various studies (Higham, 2002; Koko, 2007; Funken et al., 2011; Andreassen et al., 2011), while Aslam and Hendren (2012) developed a framework to optimize Matlab code. In this paper we develop a Matlab program, which we will refer to as mSim. The code is intended for the simulation of Non-Point Source (NPS) pollution in groundwater aquifers. The majority of the Matlab code is portable to Octave without any modification. Yet there is a small number of preprocessing functions, which are used to convert the geometrical description of a study area to a

constructive solid geometry object, that are written in the new style of Matlab classes introduced in the 2008a edition, which is currently not supported by Octave. Generally, tools applied to non-point source assessment can be grouped into three categories (NRC, 1993): 1) Index-based methods which combine spatial properties of the study area such as soil type, slope, climate, depth to groundwater, etc. and provide vulnerability maps (Aller et al., 1987); 2) statistical methods such as regression models (Nolan and Hitt, 2006), neural networks (AlMahallawi et al., 2012), etc. 3) physically based methods attempt to simulate the fate of contaminants in groundwater by solving the flow and transport equations. Due to the large extent of NPS pollution in groundwater basins, full 3D transport simulation is limited by the available computational resources to simplified cases (Gallardo et al., 2005; Jiang and Somers, 2009). A promising alternative to full 3D transport simulation is the application of streamline transport methods (Fogg et al., 1999; McMahon et al., 2008; Starn et al., 2012). To simulate NPS we utilize the streamline modeling framework proposed by Kourakos et al. (2012) (NPSAT e NonPoint Source Assessment Toolbox). The NPSAT consists of three major processes: i) simulation of steady state groundwater flow, ii) backward particle tracking, and iii) 1D transient transport simulation along the streamlines. Note that the governing equations used in the NPSAT also describe a variety of environmental processes, therefore the code can be used for simulations other than NPS modeling that are governed by the same equations. In our implementation we take advantage of the capabilities of Matlab and develop a highly vectorized code, minimizing the number of loops as much as possible. Hence we are able to solve problems up to several million degrees of freedom, while keeping the assembly and solution processes on the order of minutes. Matlab/Octave-provided solvers are only used for the solution of relatively small sparse systems. For large systems, a variety of existing solvers can seamlessly be combined with mSim, for example, pyAMG (Bell et al., 2011), which is a python implementation of the Algebraic Multigrid solvers, and HYPRE (HYPRE, 2012) and Trilinos (Heroux et al., 2003), which provide rich libraries for solving large sparse linear systems of equations on massively parallel computers. This paper is divided into six sections. The second section presents an overview of mSim. The third section briefly describes the governing equations employed by NPSAT and its finite element formulation. The fourth section discusses the vectorized implementation. We then describe a validation of mSim against analytical and numerical codes and apply mSim to a real case study in south-central California, USA. The last section summarizes the key points of this study. 2. Overview of mSim The mSim code is a collection of Matlab functions that are primarily intended for the simulation of non-point source (NPS) pollution in agricultural groundwater basins. The modeling approach, assumptions and justifications are described in detailed in Kourakos et al. (2012). In this paper we focus on efficient code development based on the high-level language Matlab/Octave. The processes that are involved with the NPS pollution modeling are the simulation of groundwater flow, of particle tracking and of solute transport along particle streamlines based on the solution of the advectionedispersion equation. The Matlab/Octave functions are organized into five groups: i) mFlow, ii) mPart, iii) mTrans, iv) mUtil and v) mNPSAT, where each group is actually a directory that contains the required files for the simulation of flow, the particle tracking, the simulation of transport,


various utility functions, and functions related to non-point source assessment toolbox (NPSAT), respectively. A detailed description of the functions provided is beyond the scope of this paper. Instead, details of the implementation can be found in the help files of mSim and the website http://groundwater.ucdavis.edu/mSim. The workflow of mSim is similar to any finite element method (FEM)-based models. First the simulation domain is defined through a set of primitive objects (e.g. points, lines etc) and then a mesh is generated. For the mesh generation we use the Gmsh software (Geuzaine and Remacle, 2009), which is an open source code for mesh generation. Although Gmsh has its own GUI, all of its options can be scripted. This allows us to write suitable Matlab functions that prepare the input files for mesh generation using Gmsh. In the current version of mSim, Gmsh is used to generate a 2D mesh only, and the 3D mesh is obtained by extruding the 2D mesh in the vertical direction within mSim. The mSim is a scripting toolbox, (i.e. without GUI). Therefore, the assignment of boundary conditions, which involves the selection of appropriate element faces, lines or points, and the assignment of properties to these, may become challenging in complex geometries. But based on our experience, the lack of a GUI provides an advantage in terms of modeling flexibility as GUIs can be useful when geometries are relatively simple and users have to interactively select only few planes, which correspond to the boundary conditions. However in realistic aquifer simulation, where boundaries may be consist of hundreds of different planes, interactive selection becomes very inefficient. Typically, GUIs therefore support linking to GIS databases to assist with the assignment of complex boundary conditions. With mSim, few lines of Matlab code allow the user to obtain and process data from almost any available database format. After the assignment of boundary conditions, the matrices are assembled and solved using three main commands Assemble_LHS, Assemble_RHS and solve_system, which are used to assemble the left-hand side matrix of the linear system, then the right hand side of the system, and finally solve the system, respectively. The final simulation step is the interpretation of the result. Matlab and Octave provide a large variety of toolboxes for postprocessing analysis such as statistics, fitting, optimization. It also has a large number of built-in visualization functions. However, few of these are suitable for visualization of very complex domains or for large scale visualizations which involve simulations with many degrees of freedom. We therefore use ParaView, which is an open source visualization software. mSim provides a set of functions that assist the data preparation for visualization using Paraview. Although mSim was initially developed for the simulation of NPS pollution using a very specific modeling framework (Kourakos et al., 2012), the code is intended also as an educational and research tool for studying and experimenting with highly efficient finite element methods in the broader context of groundwater hydrology. Since it is written in Matlab/Octave, the code is portable to any platform and can be readily run on different architectures. In addition it provides immediate access to linear system matrices, therefore it can be combined with many solver libraries to increase the solution efficiency. The mSim Code is available from http://groundwater.ucdavis. edu/mSim/. The code is provided free of charge and can be redistributed and modified under the terms of the GNU General Public License version 2 (GPL-2.0). 3. Mathematical framework of NPSAT In this section we will briefly discuss the main components of the NPSAT approach, which are simulated by mSim. The same set of

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governing equations can be used to simulate a variety of phenomena. Therefore the simulation of diffuse pollution will be the starting point of the mSim development but the methods and subsequently the code can be extended to simulate other, similar phenomena. Details and the rationale behind the NPSAT approach can be found in Kourakos et al. (2012). The NPSAT is implemented in two phases a) The construction phase which solves the governing equations for user-specified flow boundary conditions and a unit solute pulse boundary condition to obtain a unit response function and b) the implementation or forward phase where the model is used for predictions using convolution of the unit response functions with complex NPS pollution boundary conditions. The first phase involves three main steps i) the groundwater flow equation is solved under the steady state assumption; ii) based on the computed velocity field, backward particle tracking is performed with the starting points of the particles distributed around Compliance Discharge Surfaces (CDS) of interest (e.g. wells, gaining streams etc.) and the contributing recharge and pollution sources to the CDS are identified; iii) a 1D transport simulation is performed along each streamline to compute the travel time and magnitude of the pollution at the CDS. In the transport model we assume a unit Heaviside step loading function at the recharge boundary, thus the output of the 1D transport simulation is an ensemble of unit response functions (URFs) at the CDS. During the implementation phase the URFs are convoluted with actual loading scenarios (space-time maps of loading concentration). Since convolution involves only analytical computation, NPSAT can be used for rapid evaluation of many alternative loading scenarios (Kourakos et al., 2011, 2012). 3.1. Groundwater flow The governing equation that describes groundwater flow in steady state is expressed as:

v vh v vh v vh Kxx þ Kyy þ Kzz Q ¼ 0 vx vx vy vy vz vz

(1)

where h(x, y, z) is the unknown hydraulic head field, Kxx, Kyy, Kzz are the values of the hydraulic conductivity tensor along the major anisotropy x, y, z axes, and Q represents the various sources and sinks. Note that the same equation can also be used to simulate heat transfer, diffusion etc. Eq. (1) can be solved analytically only in very special cases. To obtain a general solution we solve Eq. (1)with numerical methods after discretizing the physical domain. The mSim utilizes a finite element discretization method, where the continuous hydraulic head field h(x, y, z) is approximated as the weighted average of n discrete points hðx; y; zÞ ¼ N1 h1 þ N2 h2 þ . þ Nn hn . Here hi is the approximate head at the discretization node and Ni is an interpolation function also known as shape function. After the domain discretization, the partial differential equations are simplified into a system of n linear equations (see details in Istok, 1989; Pepper and Heinrich, 2006; Fish and Belytschko, 2007) in the form of

AH ¼ F

(2)

where A is the conductance matrix with dimensions n n, H is a vector of the unknown head n 1, and F is the flux vector n 1 that contains the stresses and boundary conditions. Based on the Galerkin finite element discretization (Fairweather, 1978) the matrix A is expressed as:

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Z A ¼

BT KBdU

(3)

U

and the flux vector F is

Z Z F ¼ N T qB dG þ NT qI dU G

(4)

3.3. Advectionedispersion The last step of the construction phase of the NPSAT is the quasi3D transport simulation, which involves the solution of the 1D advection dispersion equation along each streamline. The governing equation that describes the contaminant transport assuming advection and dispersion is expressed as:

U

where B is the derivative matrix of the shape functions, K is the hydraulic conductivity tensor, N denotes the shape functions, qB and qI represents the sources and sinks from the boundaries and internal to domain (e.g. diffuse groundwater recharge, stream interactions pumping etc.) respectively. Hence, to transform Eq. (1) to a system of linear equations, the domain is first discretized into elements. Next, for each element we compute the element contributions of the conductance and flux matrices Aei ; Fie i˛½1; .; n via Eqs. (3) and (4) to global matrices A and F and finally the element matrices are assembled to formulate the linear system. 3.2. Streamlines and particle tracking After the computation of a steady state velocity field, backward particle tracking is performed which links the polluting sources with the discharge surfaces within the flow domain (e.g. wells) based on streamline computation. Streamlines are a family of curves that are instantaneously tangent to the velocity vector. The streamlines are computed via particle tracking by introducing a particle into the velocity field and then tracking its trajectory through the domain. Particle tracking has been used extensively in groundwater hydrology, e.g. for well protection zone delineation (Moutsopoulos et al., 2008), simulation of contaminant transport (Perianez, 2005; Morio et al., 2010), computation of travel times (Green et al., 2010; Eberts et al., 2012) and visualization (Chen et al., 2011). Mathematically, particle tracking is an initial value problem of an ordinary differential equation:

dvy dvx dvz ¼ Vx ðxs ; ys ; zs ; tÞ; ¼ Vy ðxs ; ys ; zs ; tÞ; ¼ Vz ðxs ; ys ; zs ; tÞ ds ds ds (5) where xs, ys, zs are known for some initial time t ¼ 0. xs, ys, zs are the particles coordinates, vx, vy, vz are the components of the velocity vectors at each point s and Vx, Vy, Vz are functions which define the velocity as functions of time and 3D space. To solve this problem, a variety of numerical methods exist. The simplest method is the Euler’s method (Butcher, 2008), whereby the position of the next point is a function of the velocity of the previous point. More advanced techniques such as RungeeKutta (Butcher, 2008) methods take multiple samples of the velocity space, and the actual velocity is a weighted average of the sampled velocities. Hence all methods require the interpolation of the velocity from hydraulic head field. The velocity within a groundwater flow field is expressed as v ¼ KVh/q where q is the porosity of the porous medium. However in a discretized solution obtained by FEM the continuous hydraulic head field is approximated ashðx; y; zÞ ¼ N1 h1 þ . þ Nn hn , hence the gradient of the head can be computed using the shape function derivatives multiplied by the approximated nodal heads vhðx; y; zÞ ¼ vN1 h1 þ . þ vNn hn . Note that this scheme, where velocity is directly computed from the derivative of the head field is known to produce a discontinuous velocity field along the boundaries of the elements (Cordes and Kinzelbach, 1992).

R

vc ¼ VðDVcÞ VðvcÞ þ G vt

(6)

where c is the concentration of the contaminant, v is the velocity which is computed by Eq. (5), and particle tracking, D is the dispersion tensor, R is the retardation factor, G represents the additional contaminants sources or sinks and t represents time. Eq. (6) is generic in terms of dimensionality, however in NPSAT we solve Eq. (6) in 1D assuming a constant unit input from the loading (recharge) side of the streamline and compute the unit breakthrough curve at the discharge side of a streamline. The unit breakthrough curve is subsequently converted to a unit response function. Similarly to the solution of the groundwater flow equation, the ADE domain is discretized into a number of elements and the ADE is simplified to a linear system of the form AC ¼ F. In this case matrix A is expressed as:

Z A ¼

BT DBdU þ vNB

(7)

U

Note that the first part of the right hand side of Eq. (7) is identical to groundwater flow integral (Eq. (3)). In addition it can be seen that the contribution of the advection term leads to a nonsymmetric matrix as opposed to pure diffusion problem. The expression for the vector F is practically identical to Eq. (4):

Z Z F ¼ NT cB dG þ N T ci dU G

(8)

U

where cB and cI represent the contributions to concentration from external sources and sinks from the boundaries and domain respectively. To formulate the linear system AC ¼ F we first compute the element matrices which define dispersivity Ae and forcing Fe for each element. Then the matrices are assembled into a global system of equations.

3.4. Implementation phase The implementation phase of NPSAT involves only analytical computations. The output of the construction phase is an ensemble of URFs that describe the response of the groundwater aquifer system to a unit input pulse. To compute the breakthrough curves for actual diffuse pollution we convolute the URFs with the actual loading scenarios (a transient, spatially variable signal at the recharge boundary). Convolution is an analytical computation which mathematically is expressed as:

cðtÞ ¼

t X

lðt dÞ$UðdÞ

(9)

d¼0

where c(t) is the streamline breakthrough curve, l is the loading function, U is the unit response function, and d increases in the summation at time step intervals. Finally the streamline breakthrough curves associated with an individual CDS (e.g. a well) are averaged according to their flow contribution to the


discharge surface, to obtain the final discharge surface breakthrough curve. Among the three aforementioned processes (e.g. simulation of flow, particle tracking and 1D simulation of the ADE) the most computationally demanding are those associated with the solution of the partial differential equations, i.e. the simulation of flow. This requires the assembly of a system of matrices and their solution. The next section describes a highly vectorized Matlab implementation of the matrix assembly process. It also describes the similarly vectorized Matlab implementation of the particle tracking algorithm. 4. Vectorized implementation 4.1. Matrix assembly Using Eqs. (3) and (4), the individual element matrices Ae and Fe, are computed and assembled to final matrices of the linear system PNel eT e e Eq. (2) A and F based on the formulas A ¼ e ¼ 1 L A L and PNel eT e F ¼ e ¼ 1 L F where the matrix L is a gather matrix operator which consist of “0”s and “1”s and actually maps the elements of

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matrix Ae to the global matrix A. Fig. 1(left) illustrates the flowchart for the serial assembly by iterating through each element. The figure shows the assembly of 2D isoparametric triangular quadratic element using Gaussian quadrature for the evaluation of the integrals of Eqs. (3) and (4). The code consists of two nested loops. The outer loops iterates through Nel elements. Initially the element coordinates are extracted and the local element matrix is initialized to a zero matrix with dimensions Nsh Nsh, where Nsh is the number of shape functions that are used for the element interpolation. In the case of a 2D quadratic triangular element Nsh ¼ 6. The integrals in Eq. (3) and Eq. (4) can be evaluated analytically only for simple linear elements, thus Gauss quadrature is typically used, where the integrals are estimated by the weighted average of point evaluations of the function to be integrated. Therefore, an additional inner loop is needed to iterate through Nq sampling points of the Gauss quadrature. Every loop requires the computation of the shape function derivatives vN with respect to the parametric coordinates, the Jacobian J and the inverse Jacobian J1 of the transformation between parametric and physical coordinates. If the hydraulic conductivity is variable within the element, then one needs to

Fig. 1. Flowchart of Serial (left) and vectorized (right) assembly of matrix A.

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compute the value at the sampled point and finally add the weighted elements on the local matrix Ael. After the code exits the inner loop, the local element contributions are added to the global matrix. This serial method becomes very inefficient in Matlab or Octave as the problem of the size grows, due to the fact that the iterations of the outer loop increase linearly with the size of the problem. To alleviate this inefficiency we employed a vectorized implementation of the assembly process. The flowchart of the vectorized assembly is illustrated in Fig. 1 (right side). In the vectorized implementation we split the process into two loops. First we iterate through the Gauss quadrature sampling points and compute the elements of the local matrices. A second independent loop is used for assembling the global matrix. The code starts by extracting the element coordinates outside of the loop. In the vectorized implementation the matrices X and Y are vectors with dimensions Nel Nsh, where Nel is the number of elements, therefore the element coordinates need to be read only once. Next for each Gauss quadrature point (x, h) the computations are performed in a vectorized (parallel) manner. For example, in serial implementation, the dimensions of the Jacobian matrix J are Ndim Ndim where Ndim is the dimension of the problem (e.g. 1D, 2D or 3D). In the vectorized implementation however, the dimensions 2 of the Jacobian matrix are Nel Ndim where for each element the values of the Jacobian are stored in a row-major order in a single row and each column contains the values of Jacobian for a different element. Similarly the determinant and the inverse Jacobian matrix are computed in an element wise manner. The shape function derivatives are then multiplied with the elements of the inverse Jacobian matrix, and the outcome is stored into a matrix variable (B) with dimensions Nel (Ndim Nsh) The hydraulic conductivity is also stored into a matrix Nel m where m depends on the hydraulic conductivity anisotropy. The vectorized implementation is limited to anisotropy along the main diagonal Kxx s Kyy s Kzz and off diagonal terms are always zero. However, it is important that hydraulic conductivity K is always stored in a row-major order to allow for efficient vectorized computations. Finally the contribution of each local element Ae to the global matrix A is computed simultaneously for all elements and stored into a matrix in row major order, i.e. the dimensions of Ae 2 . In the based on the vectorized implementation are Nel Ndim vectorized implementation, we first compute the product of Eq. (3) BTKB and store it temporarily for each quadrature point and then we add the weighted contribution of the integrating point to matrix Ae. At the end of the first loop matrix, Ae contains the final values that need to be gathered to the global matrix A. To gather the values of matrix Ae, simultaneously for all elements, into the global matrix we iterate through the columns of the 2 iterations. During the assembly of the global matrix Ae e.g. Nsh matrix we take advantage of the additive property of the “sparse” function (i.e. when two different rows map a local value at the same entrie of the global matrix, then these values are added). Although the vectorized assembly is significantly faster, it requires more storage than the serial implementation. Fig. 2 shows the number of floating numbers that need to be stored as a function of the number of elements. For all 2D elements and 3D linear elements the required extra storage is practically negligible, given the memory that commodity computers offer. However in 3D hexahedral quadratic elements the storage requirement increases significantly with the number of elements. For example for 9000 elements we need to store 9000 272 ¼ 6.5$106 floating point numbers, while for quadratic prism elements we would need 9000 182 ¼ 2.9$106. A possible remedy to this problem is the use of serendipity (incomplete) elements which use less control points

Fig. 2. Memory requirements for the vectorized assembly.

(e.g. 20 instead of 27 for quadratic hexahedral) per elements at the expense of less flexible elements. An alternative approach is to use a semi-vectorized assembly which we refer to as nested assembly. In nested implementation the assembly is performed in three nested loops, where the two outer loops iterate Nsh times and the inner loop iterates through the integration points. Therefore the nested implementation requires Nsh Nsh Nint, where Nint is the number of integration points, as opposed to vectorized approach which requires Nint independent iterations and Nsh Nsh iterations. Despite the additional storage, the assembly time using the vectorized method is at least 1e2 orders of magnitude less than the nested and serial assembly. The efficiency increases with the number of elements and as their complexity increases. Fig. 3 compares the assembly time for several common element types as a function of the number of elements. The family of red, green, and blue lines correspond to serial, nested and vectorized implementation. In all cases, vectorization significantly reduces the assembly time. The solid lines correspond to 3D elements. By comparing the green solid lines (nested assembly for 3D elements) with the red solid lines (serial assembly of 3D elements) we observe that the serial assembly was faster than the nested in the case of relatively few elements. Yet, in every case the vectorized approach is the preferred method. If the number of elements is very large, then it is preferable to use parallel vectorized implementation (under development). The assembly process for the transport ADE is practically identical with the process for the groundwater flow equation. The only difference is that an additional matrix needs to be assembled that contains the entries of the advective term. Therefore the analysis presented is identical for the assembly of the ADE. 4.2. Particle tracking The particle tracking is a strictly serial process as the next position of the particle is a function of the previous position. However, if the particles do not interact among each other, as is the case here, it is possible to vectorize the particle tracking so that streamlines can be computed simultaneously. In our implementation we provide both a serial and a vectorized version of the code. The total simulation time in the vectorized implementation is dictated by the time of the particle with the largest trajectory. Particle


213

Fig. 3. Comparison between the different assembly methods for various types of elements. The blue, green and red lines correspond to vectorized, nested and serial assembly respectively. The solid lines correspond to 3D elements and the dotted lines to 2D elements. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

vectorization is a relatively complex process and we repeatedly utilize similar techniques as the ones described in the previous section. The code is approximately 700 lines and a detailed explanation is omitted from this paper. Currently, four options are available for particle tracking. The simplest is the Euler method, but the code also offers the option of second order and fourth order Runge Kutta schemes. To obtain good streamline approximations, the streamlines must be computed in very small steps. As a fourth option we therefore provide an adaptive step approach based on the Runge Kutta Fehlberg scheme or Runge Kutta 45. Although the latter requires six interpolations of the velocity for each time step it can be significantly faster and more accurate compared to the other integration methods. The efficiency of the vectorized particle tracking is shown in Fig. 4. In a hypothetical flow field we release particles, which are

tracked until they exit the flow field. Note that special care is taken to ensure that the paths of the particles are approximately equal (i.e. the particles were released from the same elevation). All scenarios are performed with the adaptive Runge Kutta 45 scheme. As expected, the CPU time for the serial implementation (Fig. 4, blue line (in web version)) increases linearly with the number of particles. The CPU time for the vectorized particle tracking (Fig. 4, green line) is consistently 3e4 times faster than the serial implementation. Due to the fact that particle tracking is in essence a serial process and that the main while loop cannot be avoided, it is advantageous to convert this part of the code into a C/Cþþ version that can be executed within Matlab. The red line in Fig. 4 shows the particle tracking time using the Cþþ implementation of the code. It can be seen that it is consistently faster than the vectorized implementation. In summary, mSim offers three modes of particle tracking. i) Serial in matlab, which is relatively slow, but very useful for debugging and experimenting. ii) Vectorized in matlab, which is recommended approach if debugging is necessary. iii) A serial Cþþ implementation called from matlab, which is the most efficient approach, but also most difficult in terms of debugging and of adding code to investigate and understand the flow field and simulated particle tracks (e.g. multi-dimensional visualization of particle tracks and velocity vectors).

5. Application 5.1. Model validation

Fig. 4. Comparison of the different modes of particle tracking. Serial and vectorized modes are implemented in Matlab, Cþþ is written in Cþþ and runs from Matlab.

The mSim suite contains a large number of functions which have been thoroughly tested in several applications. In this section we are presenting two verification examples. The code is first compared with an analytical solution and then against the wellestablished numerical software COMSOL Inc. Since mSim is primarily intended to be used for NPS modeling it is very important to accurately capture the drawdown caused by pumping wells. In the first example we simulate the water table

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around a fully penetrating well. The analytical solution of drawdown due to constant pumping is given by

Sr ¼

Q R ln 2pT r

(10)

where Sr is the drawdown at a distance r form the well, Q is the pumping rate, T is the transmissivity and R is the radius of influence of the well. To simulate the head loss as function of pumping rate we set up a hypothetical aquifer of rectangular shape with size 10 km on each side. We assume that a fully penetrated well with pumping rate equal to Qw ¼ 5000 m3/day is located at the center of the domain. The hydraulic head at the boundaries of the aquifer is assumed constant and equal to 100 m. We also assumed that the hydraulic conductivity of the aquifer is K ¼ 10 m/day, while the depth of the aquifer is equal to b ¼ 50 m. For the discretization we used a variable size mesh where the maximum length of the elements near the well is approximately 0.1 m and the maximum length of the elements is 500 m near the boundaries. The elements within the aquifer vary between these two values. Fig. 5 shows the difference between the drawdown computed with the mSim code and the analytical expression Eq. (10). As expected the error is large near the well and drops to zero as it approaches the prescribed head boundaries. The maximum relative error of the model prediction is 0.12% which is typical for finite element solutions (Donea and Huerta, 2003). Next we compare the mSim code with the COMSOL software, a proprietary multiphysics simulation package often used for science and engineering applications. The domain of the problem is described in Fig. 6. The hypothetical aquifer is a rectangular domain with dimensions 5 2.5 km and it is divided into two zones of hydraulic conductivity with K1 ¼ 10 m/day and K2 ¼ 2 m/day. The depth of the aquifer is equal to 50 m. The top and bottom boundaries of the aquifer are considered impervious, and a constant head boundary is assigned at the left and right boundaries with 40 m and 20 m hydraulic head, respectively. The recharge is uniform and equal to 0.0004 m/day. In addition there are 10 wells (Fig. 6) that pump 500 m3/day each. The aquifer was discretized using triangular linear elements and the final model consists of 811 degrees of freedom. However the actual number of unknowns is 777 as 34 nodes are located on the constant head boundaries. The aquifer was simulated using COMSOL, which has the option of extracting the actual conductance matrix and forcing vector of the linear system. Therefore by comparing the conductance matrix we are able to compare not only the final outcome of the model (e.g. the solution) but also the numerical methods used during the assembly process such as the Gaussian quadrature and the computation of shape function

Fig. 6. Hypothetical aquifer for validation with COMSOL inc.

derivatives. By comparing the conductance matrix obtained from COMSOL with the one obtained from mSim we observe that they are practically identical as the maximum relative error of the nonzero values is in the order of 1012 (Fig. 7 green line (in web version)). Similarly we observe that the maximum discrepancy of the forcing vector (Fig. 7 blue line) between COMSOL and mSim is in the order of 1015. Since both matrices are practically identical the solution (Fig. 7 red line) returns practically identical solutions (maximum error is approximately 1013 m). Last we verify the solution of the one-dimensional advectione dispersion equation. Note that the streamline model requires the computation of multiple 1D ADE problems in order to derive unit response functions for each streamline. For simple geometric boundary conditions, the ADE can be solved analytically (Ogata and Banks, 1961). The analytical solution of the 1D ADE for a semi-infinite homogeneous isotropic porous medium with constant input concentration at the boundary can be expressed as:

xv C 1 x vt x þ vt erfc pffiffiffiffiffiffi þ exp erfc pffiffiffiffiffiffi ¼ Co 2 D 2 Dt 2 Dt

(11)

where C(x, t) corresponds to the breakthrough curve as a result to the constant input Co, v is the velocity, and D is the dispersion coefficient. In the example below we compare this analytical solution against the solution obtained from mSim and Comsol for a 1D problem with length L ¼ 10 km, longitudinal dispersivity aL ¼ 1000 m and velocity v ¼ 0.3 m/day. In the numerical model, the inlet boundary is set to constant unit concentration, the outlet

Fig. 5. Relative error between model predictions and analytical solution.


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5.2. Large scale application

Fig. 7. Comparison of the conductance matrices, forcing vectors, and solution vectors obtained from COMSOL and mSim.

boundary is defined by a zero concentration gradient, and the initial condition is set to zero. The diffusion matrix, forcing matrix and the mass matrix are identical for mSim and Comsol. Fig. 8aec shows the resulting concentration along the 1D domain for 200 years. The discrepancy between the two numerical solutions (mSim and Comsol) is negligible. The relative difference between analytical and numerical solutions from both numerical codes (mSim and COMSOL) is generally well below 1%, with highest relative error at the outlet (up to 10%) due to the finite size of the numerical domain, consistent with other studies (Yeh et al., 1993). Relative error is also higher at early times immediately in front of the solute plume where absolute concentrations are near zero.

The motivation for the development of the mSim is the simulation of regional non-point source pollution. In this section we illustrate an application of mSim to simulate 200 years of nitrate contaminant transport in the Tule River aquifer, which is located in the southern Central Valley, California, US. The Tule River basin covers 2000 km2. It is an intensively farmed agricultural area that relies heavily on groundwater for domestic and irrigation purposes. Fig. 9 illustrates the outline of the simulated domain. The area is divided into approximately 9000 land use polygons that belong to nearly 60 different land use categories. Groundwater recharge and groundwater pumping is polygon specific (Ruud et al., 2004). Known pumping is distributed across 1920 virtual wells (Boyle et al., 2012), which represent mostly irrigation wells and some municipal supply wells, and w150 line segments that correspond to recharging (losing) streams here considered to be free of pollution. This setting creates a highly complicated domain, which is very challenging for mesh generation. The mSim functions were used to prepare the input files for the mesh generation using Gmsh. First we created a 2D triangular mesh, which consists of 309,258 triangles with edge length that varies between 50 m near the wells and 500 m, in void of interest areas (e.g. see southwest area in Figs. 9 and 10), which is mostly native vegetation. Although Gmsh was instructed to generate a constrained mesh, due to complexity of the land use surface, it was found that Gmsh sometimes failed to follow the constraints at the benefit of generating a better quality mesh (e.g. mesh with triangles without acute angles). The mesh was subsequently extruded in the vertical direction to formulate 11 layers of vertical prism elements (Fig. 10) e.g. (3,401,838 prism elements and 1,861,560 degrees of freedom). The hydraulic conductivity was estimated based on a previous study that employed a pilot point optimization method and varies between 1 and 150 m/

Fig. 8. Concentration over time, along the 1D domain. (a) and (b) show the concentration distribution based on mSim and Comsol simulation. (c) shows the analytical solution and the (d) the relative to maximum concentration error between numerical and analytical simulation.

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Fig. 9. Domain and land use cover of Tule river basin, California (modified from Ruud et al. (2004)).

day with median and mean equal to 5 m/day and 24 m/day respectively (Larry Walker and Associates, 2010). In addition, there is a confining layer known as Corcoran Clay member that extends eastward from the western edge of the basin for about 20 km. To account for the confining layer the mesh elevation was adapted in the area above and below the Corcoran Clay layer so that the bottom of the 5th layer and the top of the 9th layer follow the elevation of Corcoran Clay confining unit (see Fig. 10 the area inside the red rectangle).

The total amount of diffuse groundwater recharge estimated by Ruud et al. (2004) is approximately 6.4 105 m3/day, and the estimated groundwater recharge from streams is 3.6 105 m3/ day. Therefore 36% of the total recharge occurs from streams and 64% is diffuse recharge. There are also 1920 virtual wells with a mean and median pumping rate of 526 m3/day and 300 m3/day, respectively. The upper portion of the Tule river basin is an unconfined aquifer. To simulate the water table elevation with an adaptive

Fig. 10. Left: Top view of finite element discretization. Right Top: Zoom to the green rectangle. Right bottom: cross section of the finite element mesh. The red rectangle shows the area where the mesh was adapted to honor the Corcoran Clay member. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)


Fig. 11. Distribution of hydraulic head of the top layer. Due to iterative solution the elevation of the top nodes is equal to their hydraulic head. Note that the lighting source is from the north-northeast. Recharging rivers show as (elevated) ridges, while pumping center show as cones of depression.

mesh, we utilize an iterative solution scheme to match the top node elevation with the final hydraulic head. A total of four iterations are adequate for the unconfined problem to converge. During each iteration, the mesh elevation is modified to match the previous head solution, then the conductance matrix is reassembled. Here we choose to update the entire matrix, because the elevations of all layers are modified as a function of the elevations of the top layer. The flux vector is assembled only once and remains constant during the iterations. The assembly time for the 3.4 106 element model was approximately 94 s. The time to solve the system with 1.8 106 dofs was approximately 8 min on a Xeon 2.27 GHz, with 24 GB RAM. This time corresponds to default Matlab solver (e.g. slash \ operator, where Matlab identifies the most appropriate solver based on the properties of the matrix). However we found that pyAMG (Bell et al., 2011) was significantly faster. Fig. 11 shows the head distribution of the top layer, which also correspond to the elevation of the top layer nodes. Note that the streams with relatively high recharge rates appear slightly extruded as a result of the iterative scheme for the solution of the nonlinear unconfined problem.

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Note also that the mesh created noticeable cones of depression around the wells with relatively large pumping rates. Next we apply the streamline transport model based on the flow field computed in the previous step. At each well, 100 particle endpoints are uniformly distributed around the well screen. Backward particle tracking is employed until streamlines exit from the flow field at the point of recharge. Because the aquifer is surrounded by no flow boundary conditions, all particles originate from the water table. Based on the sediment analysis by Faunt (2009), porosity values were estimated to be between f ¼ 0:07 0:15. In streamline modeling, the porosity becomes a very important scaling factor (Kourakos et al., 2012; Starn et al., 2013). As sensitivity analysis, the transport model was implemented for two cases, f ¼ 0:07 and f ¼ 0:15. The particle travel time scales linearly with the porosity. Fig. 12 shows the full histogram of the particle’s age. Note that the x axis represents the logarithm of the travel time, therefore the age distribution in the aquifer is approximately log-normal distributed. In both cases, the particle’s travel time spans from less than a year to tens of thousands years. In general, very old water is extracted from wells which are located in areas with relatively low recharge rate and from deep parts of well screens. For example, wells that are located in the western-most and southeastern part of the study area extract very old water that originates either from streams or from remote high recharge areas. Streamline age patterns (Fig. 13) follow the recharge pattern across the simulation domain. The northern part of the aquifer has the youngest water as it receives most of the stream recharge. The central and southern part of the aquifer, which consists of vineyards and other crops receives primarily water from diffuse recharge. Next, for each streamline we solved the 1D advection dispersion equation assuming a Heaviside unit loading function at the water table and a free advective flux (zero concentration gradient) boundary at the well side. In the current implementation of the streamline transport simulation, the transverse dispersivity is neglected. The longitudinal dispersivity was set equal to aL ¼ 0:32$L0:83 (Neuman, 1990), which defines the dispersivity coefficient to be approximately equal to 10% of the streamline length. For each well, each particle’s x, y, z position is different. Depending on well depth and screen length, streamline length and travel time across the well screen may vary within a narrow range (shallow wells with short screens lengths) or have a very broad range (deep wells with long screens). The response for shallow wells is significantly faster compared to the response for deep wells. At the same time the peak concentrations for deep wells are much lower compared to shallower wells (Fig. 14). Results are highly sensitive to the porosity chosen (Fig. 14).

Fig. 12. Histogram of the particle’s travel time into the groundwater aquifer for each case. The green line corresponds to median travel time. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 13. Simulated streamlines. Blue to red colors correspond to logarithmic travel time from the point of recharge onward. The gray anaglyph background corresponds to the bottom of the aquifer. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

We also compute the age distribution of water age that enters a well (Woolfenden and Ginn, 2009). In shallower wells with shorter screen, pumped water is a mix of relatively young water (20e40 years) and some older water (more than 100 years). In deep wells, water age along the intake screen ranges from 100 to 1000 years. The final phase of the NPSAT implementation involves convolution of the unit response functions with the actual pollutant loading functions. Here, we illustrate the outcome of the final phase by convoluting the unit response functions with uniform continuous loading equal to 50 kg/ha/year. Note that this number represents the loading mass distributed across the entire surface of the aquifer. During the convolution the mass is converted to

concentration by dividing the mass with the recharge that corresponds to the streamline point at the recharge boundary. Therefore, although we assumed uniform loading, the actual concentration will be variable as a function of the recharge rate. We assumed zero loading for streamlines that originate from streams. The implementation phase involves convolution of the loading functions with each unit response function. Hence for each well we obtain 100 breakthrough curves, where each corresponds to a different streamline. Note that by using URFs we are able to quickly recalculate the concentration for alternative, time-varying source loading, without again solving the entire 1D ADE problem. To obtain the actual mixed well breakthrough curve, we calculate the weighted average of the individual streamline breakthrough curves. Weights reflect the relative flow contribution of each streamline to the total pumped water volume at a well. In Fig. 15, each thin colored line corresponds to an individual, aggregated well breakthrough curve (e.g. each plots consist of 1920 breakthrough curves), which form the basis for statistical analyses of regional, subregional, or land use-specific impacts. Statistical processing also allows for quantification of uncertainty or prediction reliability. Based on well breakthrough curves we computed exceedance probabilities for the study area for the two porosity scenarios that were considered previously with f ¼ 0:07 and f ¼ 0:15 (shown with yellow thick lines in Fig. 15). With 50 kg/ha/year uniform nitrate loading, it appears that 10% of the wells will exceed the drinking water limit (10 mg NO3eN/L, shown in thick green line) after a period of approximately 4 (Fig. 15 Left) to 8 (Fig. 15 Right) years. One quarter of the wells in the study area are expected to exceed the drinking water limit after 15e30 years. Approximately half of the wells will exceed the drinking water limit after 110 years if f ¼ 0:07 and after more than 200 years at 15% porosity. It becomes apparent from the above analysis that prediction uncertainty driven by imprecise porosity estimates increases significantly at higher design exceedance thresholds. For example the uncertainty in estimating the exceedance probability for the most immediately impacted 5% of wells is in the order of few years. On the other hand the uncertainty in estimating the time at which

Fig. 14. Bottom panels: Unit response functions for an average and deep well and different values of porosity. Top panels: Age distribution for the same wells.


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Fig. 15. Left: Exceedance probability for f ¼ 0:07. Right: Exceedance probability for f ¼ 0:15. The thin colored lines correspond to individual well BTCs. The yellow lines correspond to calculated exceedance probability based on the individual BTCs (MCL: Maximum Contaminant Level). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

40% of the wells exceed the drinking water threshold is in the order of several decades. These uncertainties also depend on the exceedance threshold (here: drinking water limit) chosen, relative to the source loading. The uncertainty increases exponentially as the well exceedance concentration increase (Fig. 16). 6. Conclusions In this paper we present an efficient numerical code for the simulation of non-point source contamination based on concepts proposed in our earlier work (Kourakos et al., 2012), but extendable to more general groundwater flow and transport simulation problems using the finite element method. The code is written exclusively in Matlab and is available for free download from http://groundwater.ucdavis.edu/msim. The equations that mSim solves in the present version are steady state groundwater flow, particle tracking, and 1D transient advection dispersion with up to first order degradation and decay. mSim is based on the finite element method using Galerkin approximation of the partial differential equations. mSim relies on Gmsh code for the mesh generation, and on ParaView for large scale post-processing and visualization. Note that both codes are open source and free.

The code is implemented in a highly vectorized manner which allows it to be used for large scale applications. Currently the system of linear equations is solved using the Matlab solvers (i.e. backslash operator). Several verification problems are implemented and we illustrate the potential application of mSim with a model that estimates long-term nitrate contamination in the southern Central valley, California. The simulated model was discretized into a grid representing 1.8 millions degrees of freedom and was solved on a Xeon 2.27 GHz, with 24 GB RAM. Although the example is based on hypothetical loading we draw important conclusions regarding the uncertainty effect of porosity. For example by comparing two porosity scenarios with 4 ¼ 0:07 and 4 ¼ 0:15 we found w100 years difference in the estimation of the median travel time in the aquifer. The uncertainty due to porosity has also significant impact on predictions. We found that the uncertainty is less significant for predicting concentration at wells impacted early. Furthermore, prediction uncertainty due to unknown effective porosity increases exponentially as the exceedance probability of interest increases. The uncertainty about the duration until a threshold concentration is exceeded also increases as the threshold of interest becomes higher. In this paper we used uniform loading, and future work will examine the effects of actual loading scenarios. mSim in its current version is potentially the starting point for adding other capabilities. Future work will involve the intergration of domain decomposition methods and algebraic multigrid solvers in order to overcome the limitations imposed by the default matlab solvers. We are also working on parallel assembly options. Acknowledgments We gratefully acknowledge the constructive comments provided by six anonymous reviewers. Their thorough analysis allowed us to provide a much improved manuscript. Funding for this project was provided through the CA State Water Resources Control Board under agreements 04-184-555-3 and 09-122-250. References

Fig. 16. Uncertainty in exceedance probability as a function of the uncertainty of porosity.

Al-Mahallawi, K., Mania, J., Hani, A., Shahrour, I., 2012. Using of neural networks for the prediction of nitrate groundwater contamination in rural and agricultural areas. Environ. Earth Sci. 2012 (65), 917e928. Aller, L., Bennett, T., Lehr, J.H., Petty, R.J., Hackett, G., 1987. DRASTIC: a Standardized System for Evaluating Groundwater Pollution Potential Using Hydrogeologic Settings. Rep. EPA-600/2-87-035. U.S. Environ. Prot. Agency, Ada, Okla, p. 641.

220


Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O., 2011. Efficient topology optimization in MATLAB using 88 lines of code. Struct. Multidiscip. Optim. 43, 1e16. http://dx.doi.org/10.1007/s00158-010-0594-7. Ashby, S.F., Falgout, R.D., 1996. A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations. Nucl. Sci. Eng. 124 (1), 145e159. Aslam, A., Hendren, L., 2012. McFLAT: a profile-based framework for MATLAB loop analysis and transformations, languages and compilers for parallel computing. In: 23rd International Workshop, LCPC 2010, vol. 6548, pp. 1e15. Revised Selected Papers. Bell, W.N., Olson, L.N., Schroder, J.B., 2011. PyAMG: Algebraic Multigrid Solvers in Python v2.0. http://www.pyamg.org. Bennett, N.D., Croke, B.F.W., Guariso, G., Guillaume, J.H.A., Hamilton, S.H., Jakeman, A.J., Marsili-Libelli, S., Newham, L.T.H., Norton, J.P., Perrin, C., Pierce, S.A., Robson, B., Seppelt, R., Voinov, A.A., Fath, B.D., Andreassian, V., 2013. Characterising performance of environmental models. Environ. Model. Softw. 40, 1e20. http://dx.doi.org/10.1016/j.envsoft.2012.09.011. Bobba, A.G., 2012. Ground Water-Surface Water Interface (GWSWI) modeling: recent advances and future challenges. Water Resour. Manag. 26 (14), 4105e 4131. http://dx.doi.org/10.1007/s11269-012-0134-x. Boyle, D., King, A., Kourakos, G., Lockhart, K., Mayzelle, M., Fogg, G.E., Harter, T., 2012. Groundwater nitrate occurrence. Technical Report 4. In: Addressing Nitrate in California’s Drinking Water with a Focus on Tulare Lake Basin and Salinas Valley Groundwater. Center for Watershed Sciences, University of California, Davis. Report for the State Water Resources Control Board Report to the Legislature. Butcher, J.C., 2008. Numerical Methods for Ordinary Differential Equations. second ed. J. Wiley, Chichester. Chaves, J.C., Nehrbass, Guilfoos, J.B., Gardiner, J., Ahalt, Krishnamurthy, S.A., Unpingco, J., Chalker, A., Warnock, A., Samsi, S., 2006. Octave and python: highlevel scripting languages productivity and performance evaluation. In: HPCMP Users Group Conference, pp. 429e434. http://dx.doi.org/10.1109/HPCMPUGC.2006.55. Chen, C.-K., Yan, S., Yu, H., Max, N., Ma, K.-L., 2011. An Illustrative visualization framework for 3D vector fields. Comput. Graph. Forum 30 (7), 1941e1951. http://dx.doi.org/10.1111/j.1467-8659.2011.02064.x. COMSOL Multiphysics, 2008. COMSOL Multiphysics Modeling Guide: Version 3.5. COMSOL AB, Stockholm, Sweden. Cordes, C., Kinzelbach, W., 1992. Continuous groundwater velocity fields and path lines in linear, bilinear, and trilinear finite elements. Water Resour. Res. 28 (11), 2903e2911. http://dx.doi.org/10.1029/92WR01686. Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M., 2011. A generic interface for parallel and adaptive scientific computing: abstraction principles and the DUNE-FEM module. Computing 90 (3), 165e196. Donea, J., Huerta, A., 2003. Finite Element Methods for Flow Problems. John Wiley & Sons Ltd., West Sussex, UK. Eberts, S.M., Böhlke, J.K., Kauffman, L.J., Jurgens, B.C., 2012. Comparison of particletracking and lumped-parameter age-distribution models for evaluating vulnerability of production wells to contamination. Hydrogeol. J. 20 (2), 263e 282. http://dx.doi.org/10.1007/s10040-011-0810-6. Fairweather, G., 1978. Finite Element Galerkin Methods for Differential Equations. In: Lecture Notes in Pure and Applied Mathematics, vol. 34. CRC Press, New York, p. 263. Faunt, C.C. (Ed.), 2009. Groundwater Availability of the Central Valley Aquifer, California. U.S.Geological Survey Professional Paper 1766, p. 225. Fish, J., Belytschko, T., 2007. A First Course in Finite Elements. John Wilay & Sons Ltd, NJ. Fogg, G.E., Labolle, E.M., Weissmann, G.S., 1999. Groundwater vulnerability assessment: hydrogeologic perspective and example from Salinas Valley, California. In: Corwin, D.L., Loague, K., Elisworth, T.R. (Eds.), Assessment of Non-Point Source Pollution in the Vadose Zone, pp. 45e61. http://dx.doi.org/10.1029/ GM108. Funken, S., Praetorius, D., Wissgott, P., 2011. Efficient implementation of adaptive P1-FEM in matlab. Comput. Methods Appl. Math. 11 (4), 460e490. http:// dx.doi.org/10.2478/cmam-2011-0026. Gallardo, A.H., Reyes-Borja, W., Tase, N., 2005. Flow and patterns of nitrate pollution in groundwater: a case study of an agricultural area in Tsukuba City, Japan. Environ. Geol. 48, 908e919. http://dx.doi.org/10.1007/s00254-005-0029-8. Geuzaine, C., Remacle, J.-F., 2009. Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Meth. Eng. 79 (11), 1309e1331. Green, C.G., Böhlke, J.K., Bekins, B.A., Phillips, S.P., 2010. Mixing effects on apparent reaction rates and isotope fractionation during denitrification in a heterogeneous aquifer. Water Resour. Res. 46 (8), W08525. http://dx.doi.org/10.1029/ 2009WR008903. Harbaugh, A.W., Banta, E.R., Hill, M.C., McDonald, M.G., 2000. MODFLOW-2000, the U.S. Geological Survey Modular Ground-water ModeldUser Guide to Modularization Concepts and the Ground-water Flow Process. Open-File Report 00e92. U.S. Geological Survey, p. 121. Heroux, M., Bartlett, R., Howle, V., Hoekstra, R., Hu, J., Kolda, T., Lehoucq, R., Long, K., Pawlowski, R., Phipps, E., Salinger, A., Thornquist, H., Tuminaro, R., Willenbring, J., Williams, A., 2003. An Overview of Trilinos. Sandia National Laboratories. SAND2003e2927. Higham, D.J., 2002. Nine ways to implement the binomial method for option valuation in MATLAB. SIAM Rev. 44 (4), 661e677.

HYPRE, 2012. User’s Manual. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory. Integrated Water Flow Model, 2012. Theoretical Documentation. Integrated Hydrological Models Development Unit, Modeling Support Branch, Bay-Delta Office, Department of Water Resources, Bay-Delta. Istok, J.D., 1989. Groundwater Modeling by the Finite Element Method. In: Water Resources Monograph, vol. 13. AGU, Washington, DC. Jiang, Y., Somers, G., 2009. Modeling effects of nitrate from non-point sources on groundwater quality in an agricultural watershed in Prince Edward Island, Canada. Hydrogeol. J. 17 (3), 707e724. http://dx.doi.org/10.1007/s10040-0080390-2. Kattan, P.I., 2008. MATLAB Guide to Finite Elements, an Interactive Approach. Springer, Berlin Heidelberg. http://dx.doi.org/10.1007/978-3-540-70698-4. Kepner, J., 2009. In: Dongarra, J.J. (Ed.), Parallel MATLAB for Multicore and Multinode Computers. Society for Industrial and Applied Mathematics. Koko, J., 2007. Vectorized Matlab codes for linear two-dimensional elasticity. Sci. Program. 15, 157e172. Kourakos, G., Mantoglou, A., 2008. Remediation of heterogeneous aquifers based on multiobjective optimization and adaptive determination of critical realizations. Water Resour. Res. 44, W12408. http://dx.doi.org/10.1029/ 2008WR007108. Kourakos, G., Mantoglou, A., 2012a. Inverse groundwater modeling with emphasis on model parameterization. Water Resour. Res. 48, W05540. http://dx.doi.org/ 10.1029/2011WR011068. Kourakos, G., Mantoglou, A., 2012b. Development of a multi-objective optimization algorithm using surrogate models for coastal aquifer management. J. Hydrol. 479, 13e23. http://dx.doi.org/10.1016/j.jhydrol.2012.10.050. Kourakos, G., Klein, F., Harter, T., 2011. A GIS-linked unit response function approach to stochastic groundwater nonpoint source pollution modelling. In: Proceedings of ModelCARE 2011 Models-repositories of Knowledge, vol. 355. IAHS Publ., pp. 106e112, 2012. Kourakos, G., Klein, F., Cortis, A., Harter, T., 2012. A groundwater nonpoint source pollution modeling framework to evaluate long-term dynamics of pollutant exceedance probabilities in wells and other discharge locations. Water Resour. Res. 48, W00L13. http://dx.doi.org/10.1029/2011WR010813. Kumar, R.P., Dodagoudar, G.R., 2010. Two-dimensional meshfree modelling of contaminant transport through saturated porous media using RPIM. Environ. Earth Sci. 61 (2), 341e353. http://dx.doi.org/10.1007/s12665-009-0346-4. Larry Walker and Associates, February 2010. Salt and Nitrate Sources Pilot Implementation Study Report. Final Report to the Central Valley Salinity Coalition (CV-SALTS), p. 247. Lee, H.S., Matthews, C.J., Braddock, R.D., Sander, G.C., Gandola, F., 2004. A MATLAB method of lines template for transport equations. Environ. Modell. Softw. 19 (6), 603e614. http://dx.doi.org/10.1016/j.envsoft.2003.08.017. Lie, K.eA., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B., 2012. Open-source MATLAB implementation of consistent discretisations on complex grids. Comput. Geosci. 16 (2), 297e322. http://dx.doi.org/10.1007/s10596-0119244-4. Lin, H.-C.J., Richards, D.R., Talbot, C.A., Yeh, G.-T., Cheng, J.-R., Cheng, H.-P., Jones, N.L., 1997. A Three-dimensional Finite Element Computer Model for Simulating Density-dependent Flow and Transport in Variable Saturated Media: Version 3.0. U.S Army Engineer Research and Development Center, Vicksburg, Miss. Louwyck, A., Vandenbohede, A., Bakker, M., Lebbe, L., 2012. Simulation of axisymmetric flow towards wells: a finite-difference approach. Comput. Geosci. 44, 136e145. http://dx.doi.org/10.1016/j.cageo.2011.09.004. McMahon, P.B., Burow, K.R., Kauffman, L.J., Eberts, S.M., Böhlke, J.K., Gurdak, J.J., 2008. Simulated response of water quality in public supply wells to land use change. Water Resour. Res. 44, W00A06. http://dx.doi.org/10.1029/ 2007WR006731. Moriasi, D.N., Wilson, B.N., Douglas-Mankin, K.R., Arnold, J.G., Gowda, P.H., 2012. Hydrologic and water quality models: use, calibration, and validation. T ASABE 55 (4), 1241e1247. Morio, M., Finkel, M., Martac, E., 2010. Flow guided interpolation e a GIS-based method to represent contaminant concentration distributions in groundwater. Environ. Model. Softw. 25 (12), 1769e1780. http://dx.doi.org/10.1016/ j.envsoft.2010.05.018. Moutsopoulos, K.N., Gemitzi, A., Tsihrintzis, V.A., 2008. Delineation of groundwater protection zones by the backward particle tracking method: theoretical background and GIS-based stochastic analysis. Environ. Geol. 54, 1081e1090. http:// dx.doi.org/10.1007/s00254-007-0879-3. National Research Council, 1993. Groundwater Vulnerability Assessment: Predicting Relative Contamination Potential Under Conditions of Uncertainty. National Academy Press, Washington, D. C, p. 204. Neuman, S.P., 1990. Universal scaling of hydraulic conductivities and dispersivities in geologic media. Water Resour. Res. 26 (8), 1749e1758. http://dx.doi.org/ 10.1029/WR026i008p01749. Nolan, B., Hitt, K.J., 2006. Vulnerability of shallow groundwater and drinkingwater wells to nitrate in the United States. Environ. Sci. Technol. 40 (24), 7834e7840. Ogata, A., Banks, R.B., 1961. A Solution of the Differential Equation of Longitudinal Dispersion in Porous Media. In: US Geological Survey Professional Papers, No. 34, p. 411-A. OpenFoam, 2013. The Open Source CFD Tool: User Guide. VErsion 2.2.1. OpenFoam Foundation.

G. Kourakos, T. Harter / Environmental Modelling & Software 52 (2014) 207e221 Pepper, D.W., Heinrich, J.C., 2006. The Finite Element Method: Basic Concepts and Applications. Taylor & Francis Group, FL. Perianez, R., 2005. GISPART: a numerical model to simulate the dispersion of contaminants in the Strait of Gibraltar. Environ. Model. Softw. 20 (6), 797e802. http://dx.doi.org/10.1016/j.envsoft.2004.06.008. Persson, P.O., Strang, G., 2004. A simple mesh generator in MATLAB. SIAM Rev. 46 (2), 329e345. http://dx.doi.org/10.1137/S0036144503429121. Refsgaard, J.C., Henriksen, H.J., 2004. Modelling guidelines - terminology and guiding principles. Adv. Water Resour. 27 (1), 71e82. http://dx.doi.org/10.1016/ j.advwatres.2003.08.006. Ruud, N.C., Harter, T., Naugle, A.W., 2004. Estimation of groundwater pumping as closure to the water balance of a semi-arid irrigated agricultural basin. J. Hydrol. 297, 51e73. Starn, J.J., Bagtzoglou, A.C., Robbins, G.A., 2012. Methods for simulating solute breakthrough curves in pumping groundwater wells. Comput. Geosci. 48, 244e 255. http://dx.doi.org/10.1016/j.cageo.2012.01.011. Starn, J.J., Bagtzoglou, A.C., Robbins, G.A., 2013. Uncertainty in simulated groundwater quality trends in transient flow. Hydrogeol. J. 21 (4), 813e827. Talischi, C., Paulino, G.H., Pereira, A., Menezes, I.F.M., 2012. PolyMesher: a generalpurpose mesh generator for polygonal elements written in Matlab. Struct. Multidiscip. Optim. 45 (3), 309e328. http://dx.doi.org/10.1007/s00158-0110706-z. Teza, G., Marcato, G., Castelli, E., Galgaro, A., 2012. IRTROCK: a MATLAB toolbox for contactless recognition of surface and shallow weakness of a rock cliff by

221

infrared thermography. Comput. Geosci. 45, 109e118. http://dx.doi.org/10.1016/ j.cageo.2011.10.022. Therrien, R., McLaren, R.G., Sudicky, E.A., 2007. HydroGeoSphereda Three-dimensional Numerical Model Describing Fully Integrated Subsurface and Surface Flow and Solute Transport. Groundwater Simulations Group, University of Waterloo. Trefry, M.G., Muffels, C., 2007. Feflow: a finite-element ground water flow and transport modeling tool. Ground Water 45 (5), 525e528. http://dx.doi.org/ 10.1111/j.1745-6584.2007.00358.x. Tych, W., Young, P.C., 2012. A Matlab software framework for dynamic model emulation. Environ. Model. Softw. 34, 19e29. http://dx.doi.org/10.1016/ j.envsoft.2011.08.008. Voss, C.I., 1984. A Finite Element Simulation Model for Saturated-unsaturated, Fluid Density-dependent Groundwater Flow with Energy Transport or Chemically Reactive Single-species Solute Transport. US Geological Survey, pp. 84e4369. Winston, R.B., 2009. ModelMuse-A Graphical User Interface for MODFLOW-2005 and PHAST. In: U.S. Geological Survey Techniques and Methods 6eA29, p. 52. Woolfenden, L.R., Ginn, T.R., 2009. Modeled ground water age distributions. Ground Water 47, 547e557. http://dx.doi.org/10.1111/j.1745-6584.2008.00550.x. Yeh, T.C.J., Srivastava, R., Guzman, A., Harter, T., 1993. A numerical model for water flow and chemical transport in variably saturated porous media. Groundwater 31 (4), 634e644. http://dx.doi.org/10.1111/j.1745-6584.1993.tb00597.x. Simunek, J., van Genuchten, M.Th., Sejna, M., 2012. Hydrus: model use, calibration, and validation. T ASABE 55 (4), 1261e1274.