Understanding the long-term effect of nonpoint source (NPS) pollution on groundwater of agricultural regions is an increasing challenge of global importance. A novel groundwater modeling framework is developed to assess and evaluate the dynamic, spatio-temporally distributed linkages between nonpoint sources above a groundwater basin and groundwater discharges to wells, streams, or other compliance discharge surfaces (CDSs) within a groundwater basin. The modeling framework allows for efficient evaluation of NPS pollution scenarios and of their short- and long-term effects on pollutant exceedance probabilities in CDSs. Using the model, we investigate the effect of aquifer heterogeneity, well design variability, and spatio-temporal nitrate source variability on nitrate in domestic and large production wells of a semiarid, irrigated agricultural region. Results show that the timing of nitrate breakthrough in wells is significantly controlled by aquifer recharge and pumping rates in NPS areas and by the effective porosity of the aquifer system. Results further show that mixing within a domestic or large production well is a dominant source of dispersive behavior in pollutant breakthrough. In production wells with shorter screens, macrodispersivity due to aquifer heterogeneity accelerates the earliest breakthrough. Variability in well construction and spatio-temporal variability of nitrate sources most strongly control the temporal dynamics of the nitrate exceedance probability and the variability of nitrate between wells, regardless of the degree of aquifer heterogeneity. Hence, characterization of the heterogeneity of external sources and sinks is critical to understand variability and uncertainty about nonpoint source pollution in groundwater discharge locations across basins.
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2. Developing a Rationale for a NPS Groundwater Modeling Framework
2.1. Nonpoint Source Pollution: A Global Groundwater Quality Threat
 Groundwater nitrate is largely derived from fertilizer nitrogen and animal nitrogen applied in agriculture [United Nations World Water Assessment Program (UN/WWAP), 2006, p. 117], where nitrogen is a vital nutrient for plant growth. The increasingly intensive use of nitrogen-based fertilizers in agriculture has allowed global food production to stay ahead of rapid population growth [Laftouhi et al., 2003], but at a potentially significant cost to current and future water quality [Corwin et al., 1999; Dubrovsky et al., 2010; Sutton et al., 2011; USEPA, 2011]. With further growing world population and higher standards of living, food consumption is estimated to increase 70% over the next four decades, while global land and water resources have limited growth reserves. Intensification of agriculture will therefore continue [Molden, 2007]. Long-term salinization of groundwater basins from nonpoint sources, particularly in semiarid and arid irrigated agricultural regions, has also been recognized as a critical threat to groundwater quality around the globe [Burkhalter and Gates, 2005; Martín-Queller et al., 2010]. Besides agriculture, other significant nonpoint sources of groundwater nitrate and salt include urban wastewater discharge, septic systems, wastewater holding ponds, irrigation water, and atmospheric deposition. The degradation of groundwater resources affects ecosystems worldwide via return flow of groundwater to surface water [Bouwman et al., 2009; Darracq et al., 2010]; and it affects the quality of both irrigation water (salinity) and drinking water (nitrate, salinity, pesticides, pathogens) pumped from aquifers [Sutton et al., 2011; USEPA, 2011]. Approximately half of the global population depends on groundwater as a drinking water source [United Nations World Water Assessment Program (UN/WWAP), 2003; Giordano, 2009]. In contrast, most of the global population in intensively farmed agricultural regions such as the California Central Valley, the North-American High Plains and Floridan aquifers, central Europe's unconsolidated aquifers, the Indo-Gangetic aquifer complex, and the North China plains, relies almost exclusively on, often shallow, groundwater [e.g., Power and Schepers, 1989; Chakraborti et al., 2011]. Globally, 43% of consumptive water used in irrigation is groundwater [Siebert et al., 2010]. Furthermore, the need to protect drinking water quality for large population sectors has driven and continues to drive NPS policy, particularly in Europe and North America [e.g., Sonnevald and Bouma, 2003; Dowd et al., 2008].
 Sound nonpoint source management and control policy requires thorough scientific understanding of nonpoint sources and of the linkage between nonpoint sources and groundwater discharges to users or affected ecosystems via wells, springs, or inflow to streams. Significant scientific effort has been dedicated to understand, manage, and monitor potential sources, to understand the dynamics of NPS pollutants in the vadose zone and in groundwater, and to assess the environmental and public health consequences of NPS pollution of groundwater [Addiscott and Wagenet, 1985; Corwin et al., 1999; Pavlis et. al., 2010]. The spatio-temporal and process complexity of NPS pollution of groundwater on one hand and the number and large diversity of affected stakeholders on the other hand (Figure 1) requires management of large data sets, the bridging of possibly huge data gaps, upscaling, the use of potentially complex models, and, most importantly, that science effectively communicates with policy and decision makers [King and Corwin, 1999; National Research Council, 1993].
 For assessment, planning, and regulation purposes, nonpoint source pollution, particularly from nitrate and salt sources, is a distinctly different problem set compared to point sources. Point sources commonly are of limited spatial extent (<0.1 km2), limited temporal duration (hours to months), do not contribute significantly to basin recharge [Freeze and Cherry, 1979; Bower, 2000; Domenico and Schwartz, 2008], and the associated point source concentrations are locally very intensive, often many orders of magnitude above regulatory limits. In contrast, nonpoint source pollution typically occurs repeatedly over long time periods across entire groundwater basins as part of recharge, particularly in agricultural regions [UN/WWAP, 2006; Burow et al., 2010; GWSP Digital Water Atlas, Map 48: Nitrogen Load (Mobilizable) (V1.0), 2008, available at http://atlas.gwsp.org]. For salts and nitrogen, polluted waters are typically less than one order of magnitude above regulatory limits (relatively low intensity), while background concentration levels are often less than one order of magnitude below regulatory limits.
 Stringent control measures for point sources have been in development for four decades (e.g., U.S. Code Title 42, chapter 103 U.S. Comprehensive Environmental Response, Compensation and Liability Act of 1980), while control and monitoring measures for nonpoint sources of groundwater are only beginning to be developed over the last one to two decades (e.g., EU Nitrate Directive, 91/676/EEC, available at http://ec.europa.eu/environment/water/water-nitrates/index_en.html, California Salt and Nutrient Basin Plan development, available at http://www.swrcb.ca.gov/centralvalley/water_issues/salinity/index.shtml). These differences between point source pollution and nonpoint source pollution of groundwater require that assessment methods, monitoring approaches, and regulatory frameworks for nonpoint source control do not simply copy the approaches taken in the point source arena, but that methods be developed specifically for nonpoint sources of groundwater.
 The implementation of a fully three-dimensional flow and transport model for nonpoint source assessment is largely limited by computational resources. Numerical flow and transport models are designed with at most 105–108 degrees of freedom (particle lines, finite difference cells, finite elements), allowing for 102–103 discretization points per dimension. At typical (point source) contamination sites to which these are applied, the resulting spatial discretization is on the order of 10−1–102 meters [Carle et al., 2006]. At similar spatial resolution, simulation of entire groundwater basins affected by nonpoint source pollution and being tens to hundreds of kilometers across (e.g., Floridan aquifer system, High Plains aquifer system, Central Valley aquifer system, North China Plain aquifer system, Indo-Gangetic aquifer system), would require spatial grids that are four to six orders of magnitude larger (109–1014 degrees of freedom). High-spatial resolution is necessary to properly capture individual sources (e.g., crop fields, lagoons, septic leach fields) and the impact to individual compliance discharge surfaces (CDSs) including domestic wells, municipal wells, irrigation wells, and stream reaches, sometimes referred to as receptors [Bloomfield et al., 2006]. There may be tens of thousands of individual sources and affected wells across a single groundwater basin (Figure 1).
 To render the computational burden tractable, alternative methods have been proposed. For example, Lin et al.  developed a simplified numerical model where the governing equations of 3-D groundwater flow and contaminant transport are replaced by a 2-D finite element approximation in the x-y direction and 1-D finite difference approach in the vertical direction. Almasri and Kaluarachchi  used surrogate models such as modular neural networks in order to predict nitrate contamination in the Sumas-Blaine aquifer in Washington state, but performance was found inferior to the classical fate and transport model.
 A widely used alternative technique is the streamline simulation model, where a multidimensional simulation problem is decoupled into multiple one-dimensional problems [Martin and Wegner, 1979]. Streamline models have been used extensively in petroleum engineering [Blunt et al., 1996; Baker et al., 2002]. Jang et al.  utilized the streamline model to simulate solute transport in fractures, and found that the breakthrough curves from simulations matched excellently with experimental data. Bandilla et al.  combined an analytic element-based solution of groundwater flow with the streamline method neglecting transverse dispersivity effects. McMahon et al. [2008a, 2008b] used the streamline model considering advection only and applied it to four study areas in the United States, while Green et al.  utilized random walk particle tracking in a highly heterogeneous system to examine the effects of heterogeneity on the reaction parameters. Recently, Herrera et al.  proposed an improved version of the method for simulating reactive solute transport in porous media. Streamline methods have also been used for model calibration [Jang, 2007; Jang and Choe, 2004], where the flow domain is decomposed into streamlines and the calibration parameters are adjusted along the streamlines. The efficiency of the streamline method stems from neglecting transverse dispersion and focusing on specific source-CDS relations. Depending on the modeling objective, the method is computationally far less demanding than a fully three-dimensional solution at equivalent high resolution.
 In this paper we present an efficient physically based groundwater modeling framework for nonpoint source pollution transport in groundwater, here referred to as the NonPoint Source Assessment Tool (NPSAT). The approach is designed to simulate fully three-dimensional groundwater flow and streamline-based transport at high-spatial resolution, yielding pollutant breakthrough curves for each of a large number of spatially distributed compliance discharge surfaces (CDSs) within a groundwater basin. Individual breakthrough curves are then assembled to compute time-dependent pollutant exceedance probability functions for groups of CDSs as a function of spatio-temporally distributed nonpoint source loading histories from past, current, and future land use.
3.1. Conceptual Approach
 Groundwater flow is governed by Darcy's Law and the conservation of mass [Bear, 1979]:
subject to appropriate initial and boundary conditions. Here is the Darcy flux, is the hydraulic head, is the hydraulic conductivity tensor, represents a vector of sources and/or sinks, is the storage coefficient, and represents time. The governing equation of contaminant transport in groundwater is [Bear, 1979]:
where is the concentration of the contaminant, is the velocity field, is the porosity of the porous medium, is the dispersion tensor, represents time, represents sources and sinks (e.g., via recharge, wells), and is the retardation factor [Putti et al., 1990]. Pollutant concentrations (2) in a groundwater basin are controlled by spatio-temporally variable, dynamic sources and sinks of water and associated (dissolved) pollutants (Figure 1), and by spatially distributed (heterogeneous) aquifer properties.
 Of particular interest to groundwater quality protection and management is the pollutant concentration (historically, current, and in the future) in water discharged from a (finite) set of individual wells or gaining stream reaches (CDSs) within a groundwater basin. The pollutant concentration history of the well water or stream reach discharge is referred to here as the breakthrough curve (BTC). The BTC at the CDS is controlled by the pollutant loading history in the source area of the CDS and by the solute reactions and dispersion along the groundwater flow paths between the source area and the CDS. The CDS source area is defined as the recharge area associated with all groundwater flow discharging into the CDS. Generally, recharge and pollutant loading within the source area may be spatially and temporally variable, and not all locations within the source area contribute to the associated CDS at all times due to transient changes in groundwater flow direction.
 To yield the solution of (1) and (2) tractable for nonpoint source pollution at the basin scale, yet with sufficiently high resolution, we make three critical simplifications: first, we assume that groundwater flow is steady state:
 Second, we assume that transverse dispersion in (2) is negligible (longitudinal dispersion only), and third, pollutant reactions are limited to first order degradation, linear sorption, or a combination thereof.
 We support the first assumption with the following heuristic consideration: a nonpoint source pollutant entering an aquifer at a specific location, but continuously over a period of time may discharge at multiple proximate CDSs at different times if groundwater flow is sufficiently transient. However, our focus here is on exceedance probabilities and hence on the ensemble set of BTCs across a group of CDSs (and hence, a group of source areas), which are much less sensitive to transient changes in the source area. Source areas of CDSs may partially overlap due to transient flow conditions. Hence, a steady state flow approximation still allows for capturing both central tendencies (mean travel time) and the degree of variability of travel time within a CDS and between CDSs.
 The steady state flow problem (3) is separable from the transport problem (2) and here is solved subject to the appropriate aquifer domain and boundary conditions using a finite elements methods (FEM). The grid resolution is chosen to capture the spatial pollutant loading variability as well as the flow dynamics around individual CDSs with sufficient detail. For example, the average size of individual sources in a typical agricultural region (California, Central High Plains, North China Plains, central Europe) varies from 102 to 106 m2, hence, the maximum size of a side of an element is in the range from 10 m to 1000 m. Near the CDSs, resolution is on the order of 10 m to provide appropriate flow field resolution near the well.
3.2. Transport Simulation: Streamlines
 Neglecting transverse dispersion, the transport equation (2) is solved through an ensemble of one-dimensional streamline-based solutions focused on the CDSs rather than a fully three-dimensional solution. Obtaining a quasi-3-D solution with the streamline approach specifically for CDS locations can be significantly more efficient than computing fully three-dimensional transient solutions over the entire groundwater nonpoint source contamination domain for time-horizons spanning decades to centuries [Thiele, 2001].
 Here we define streamlines by their exit points on the CDS and use backtracking to the source area (Figure 2). Exit points of streamlines are distributed across each CDS such that each streamline represents a known fraction of flow into the CDS. Streamline exit points around a well screen CDS are organized in multiple horizontal layers (Figure 2), each with a finite number of exit points per layer: where is the number of layers, is the number of exit points per layer, and is the total number of streamlines exiting on the well screen. Streamline contributions to the CDS BTC are weighted based on the exit velocity at the CDS.
 The accuracy of the breakthrough curve simulation at the CDS is determined by the number of streamlines used relative to the (spatio-temporally variable) pollutant loading across the source area (source loading). For computational efficiency, a balance must be sought between accuracy and numerical efficiency. The choice of the number of streamlines used is application-specific and depends on the desired accuracy. Given typical uncertainties and inaccuracies associated with estimating groundwater flow parameters and source loading, a practical simulation goal is to obtain discharge (well) concentrations that have a numerical accuracy within 5% of the true mathematical solution or, alternatively, at the 5%–10% level of a problem-specific contaminant concentration level of interest (e.g., drinking water limits for nitrate, salinity), whichever is larger. Consider a nonpoint source concentration that varies between 10% (background) and 1000% (intensive source) of the regulatory control level while recharge is relatively uniform. This is typical for nitrate and salinity pollution from agricultural landscapes [Harter et al., 2002; Burow et al., 2010]. In this case, the scenario requiring the highest resolution occurs if 1% of the CDS source area has the maximum concentration of 1000% of the regulatory control level, while the remainder of the source area recharges at a background concentration of only 10% of the regulatory control level. Hence, the high polluter adds ( =) 10% of the regulatory concentration level to the background concentration at the CDS. The number of streamlines must be sufficiently large to ascertain that the procedure captures the 1% of the source area with high concentration. Generally, from 102 to 103 streamlines are therefore needed to properly simulate the BTC at such a CDS.
3.3. Streamline Computation
 For each streamline exit point on the CDS , where is the number of streamlines per CDS, a backward particle tracking is performed until the particle intersects the water table, yielding a stream line . In groundwater, streamlines describe the time for a particle to travel a certain distance within the groundwater velocity field. The streamlines are defined as,
where is the integration variable and is the pore velocity vector. The equation (4) is an ordinary differential equation which can be solved analytically [Pollock, 1994] or numerically. Here we used an explicit fifth order Runge-Kutta-Fehleberg method [Fehlberg, 1969], which uses an adaptive step approach. Backward tracking of streamlines has two distinct advantages: Transport is computed only for the part of the aquifer that is of interest to the simulation outcome (concentration hydrographs at contaminant sinks). By using backward streamline simulation, with sufficient resolution around sources and sinks we avoid the so-called “weak-sink” problem in numerical solutions of (1) [Zheng and Wang, 1999].
 The result of the backward tracking is a positional vector and velocity vector for each streamline. The positional vector contains the distance along the streamline measured from the initial point , and the velocity vector contains the velocity magnitude that corresponds to positions of vector . Note that the last element of vector is the key link of the streamline with the nonpoint-source loading function , where is the number of land uses, corresponds to the identification of each land use and eventually links each exit point on the CDS with a contamination source .
 For each streamline, a one-dimensional transport model is applied to compute the URF. Generally, any transport model may be applied within the NPSAT framework to compute the URF provided that the superposition principle can be applied, e.g., continuous time random walk [Berkowitz et al., 2006], fractional advection dispersion equation (ADE) [Meerschaert et al., 1999], the tempered one-sided stable residence time density [Cvetkovic, 2011], and others. Here we use the one-dimensional ADE [Jury and Roth, 1990]:
where represents the effective macrodispersion, given by . The macrodispersivity, intrinsically accounts for the effects of aquifer heterogeneity within the streamtube represented by the 1-D streamline. Consistent with field experiments [Gelhar et al., 1992] and numerical experiments [Green et al., 2010], the longitudinal macrodispersivity, is scaled relative to the length of the streamline,
where is the streamline length. The velocity varies along the streamline, and is calculated from the flow solution by the norm . The solution to (5) is obtained numerically for each streamline. The streamline URF is computed from the resulting solution
 This ensures that the area of the URF is always equal to 1 (Figure 3).
3.6. Unit Response Function Parameterization
 Streamline URFs are archived for retrieval during the forecasting phase of the NPSAT. We found that the shape of the URFs is similar to that of common probability distribution functions (pdfs) found in the statistical literature and in statistical software. Both functions intrinsically integrate to a unit area. We can therefore readily fit the empirically determined URFs to a library of pdf functions using existing software. Pdfs are typically defined by two to three parameters and the function itself. The fitting procedure used consists of a gradient-based optimization method aiming at minimization of the error between the empirical and fitted URF. Rather than archiving on the order of 104–105 bytes of data per CDS (102 or more data points per each of 102–103 streamlines), this procedure reduces the archive to 3–4 numbers (2–3 parameters and one index to identify the function) per streamline or between 102–103 bytes of data per CDS. This allows for efficient data storage and forward modeling in applications to large groundwater regions, where the number of CDSs may range from 104–106 or even higher.
 Use of the URF approach computationally decouples the transport process from the nonpoint source loading process. URFs can be computed a priori without knowledge of the actual nonpoint source concentration history. We call this the NPSAT construction phase. The NPSAT construction phase requires the following steps (Figure 4):
 (1) Geospatial mapping of the individual land use parcels and their (nontransient, average) recharge of the CDSs (e.g., wells) and their (nontransient, average) discharge, and of other boundary conditions.
 (2) Computation of a detailed three-dimensional steady state groundwater velocity field using a high-resolution numerical solution to (3) with distributed recharge, groundwater pumping, and groundwater discharge to streams.
 (3) Geospatial mapping of the desired distribution of streamline exit points on the set of CDSs.
 (4) The backward computation of streamlines from their CDS exit points to the water table (source area).
 (5) Computation of a URF as the one-dimensional solution of (2) separately for each streamline, given a unit input step function, and fitting a parametric function to this empirically obtained URF at each streamline, a step that drastically reduces the data storage requirements.
 The construction phase yields a geospatial database of the location of CDSs (wells, drains, springs, stream reaches), an identification of associated streamlines, the parameters and a code identifying the form-function of the URF for each streamline, and each streamline's recharge and discharge (beginning and end) location.
3.7. Implementation Phase: Computing Breakthrough Curves at CDSs
 In the implementation phase of the NPSAT, the BTC of a CDS is computed by convoluting each streamline-specific unit response function with the actual, location-specific nonpoint source loading function, then performing a flux-weighted integration of streamline-output concentrations at time over all streamlines exiting in a specific CDS (model prediction phase). Let us suppose that streamlines and associated unit response functions were computed for the compliance discharge surface. The source loading functions associated with the are denoted as . For each streamline, the concentration history is obtained by convolution:
where increases in the summation at time step intervals and is the total runtime of the transport model. Equation (9) is the numerical approximation of the general convolution operator between two functions and expressed as . After the calculation of the concentration history at each streamline, the BTC of the CDS is computed from:
where is the weight that represents the amount of flow that corresponds to each streamline.
 Unlike in watershed NPS modeling, where the CDS output of individual stream reaches or tributaries is effectively integrated at the watershed outlet [Basso et al., 2010], the solute output at individual groundwater well CDSs does not further mix among CDSs (similarly in the case where a large number of low-order stream reaches are considered separately). Instead, the BTCs provide the basis for constructing time-dependent pollutant exceedance probability distribution functions (pdfs) across user-specified specific population sets of CDSs (e.g., domestic wells, irrigation wells, drinking water wells, stream reaches) within the modeling domain (Figure 5). This stochastic analysis is the final step in the NPSAT process.
4. Dynamics of Nitrate Breakthrough in Wells of a Semiarid Agricultural Alluvial Aquifer Systems
4.1. Alluvial Aquifer Simulation
 We demonstrate the utility of the NPSAT by investigating the temporal dynamics of long-term nitrate BTCs across a representative set of production wells (domestic, irrigation, or public drinking water supply wells) in an alluvial, semiarid aquifer system within an intensively farmed, irrigated region. Two major aquifer stresses in such systems are spatially distributed irrigation water recharge across the top of the aquifer and distributed groundwater pumping for irrigation and drinking water supply within the aquifer [e.g., Ruud et al., 2004; Faunt, 2009]. We use the NPSAT to determine travel time distributions and to elucidate the sensitivity of the nitrate exceedance pdf (the main simulation results of the NPSAT) to aquifer spatial variability and to spatio-temporal variability of the nitrate source loading (N loading) to the aquifer. For this, we apply the NPSAT to a conceptual representation of the Tule River groundwater sub-basin of the Central Valley aquifer system (Figure 1), an ∼600 m deep alluvial aquifer system of which the upper 300 m serve as the primary production unit [Ruud et al., 2004].
4.2. Simulation Scenario
 Without loss of generality, here we consider the aquifer to consist of a repeated (quasi-infinite) pattern of individual fields and groundwater wells (domestic and irrigation wells). The pattern is represented by a random, 8 km by 4 km landscape sample from the Tule River groundwater basin (Figures 1 and 6). The aquifer depth, effective horizontal hydraulic conductivity, and effective vertical hydraulic conductivity are uniform and equal to 300 m, 20 m d−1, and 1 m d−1, respectively. The top and bottom elevation of the aquifer are 30 m and −270 above mean sea level, and porosity is set to 20%. The northern (top) and southern (bottom) boundaries are pattern image boundaries (no flow). The eastern (right) and western (left) boundaries are constant head boundaries with hydraulic head equal to 20 and 30 m, respectively.
 A total of 40 wells are randomly distributed throughout the landscape pattern (Figure 6). It is assumed that 20 wells are used for irrigation and 20 wells for domestic use. Irrigation wells are screened deeper with large production rates, while domestic wells are shallow with low extraction rates. Depths and pumping rates were assigned randomly from a normal distribution based on typical well depths and pumping rates in the alluvial aquifer system of the Central Valley, California (Table 1).
Table 1. Distributions Regarding Well Depths, Well Screens, and Pumping Rates
expresses the normal distribution with μ mean and σ SD.
Screen length (m)
Normal, N(100, 50)
Top screen (m below water table)
Pumping rate (m3 d−1)
 The mean irrigation pumping rate was computed after generating random domestic well pumping rates:
where is the recharge in the land use polygon and is the individual domestic well pumping rate at well . Pumping rates, were obtained from a set of 20 randomly generated pumping rates, using:
 The mesh structure shown in Figure 6 is identical in all 30 layers resulting in a highly detailed flow model with 49,590 degrees of freedom. For each well (CDS), 100 streamline endpoints, distributed in 25 layers, are generated and a backward streamline calculation is performed. Some streamlines may originate upgradient of the inflow boundary. Here we use an iterative backward streamline simulation procedure: streamlines that intersect the upgradient boundary during the backward tracking procedure are reinserted at the downgradient boundary of the same flow domain at the same elevation and lateral position where they exited at the upgradient boundary.
 This procedure effectively simulates a semi-infinite aquifer flow domain with a periodic recharge and pumping stress pattern, where the recharge and pumping pattern of Figure 6 is repeated in the upgradient direction of the flow field. However, we randomly vary the N-loading distribution across the top of the simulation domain with each upgradient repetition (reinsertion iteration) of the aquifer domain. As a result, two streamlines that intercept the water table within the same land parcel, but after a different number of reinsertion iterations, have different N-loading histories under the stochastic forcing approach described below. In our aquifer example, the longest streamline requires 21 reinsertions before the particle exits from the water table of the aquifer. In effect, the streamline-transport simulation domain is therefore 176 km long. It accounts for nonuniform flow field conditions created by 880 wells and N loading across a 704 km2 nonpoint-source pollution area.
 For each streamline, the URF is computed and then fitted using the lognormal distribution function [Darracq et al., 2010]. Each URF consists of 2000 discrete double-precision numbers, requiring 16 Mb of storage in an archive. Using fitted URFs, the archive consists of only 36 kb (two location IDs and three URF parameters for each of 4000 URFs in single-precision real numbers).
4.3. Computational Efficiency
 Steady state flow, streamline, URF, BTC, and statistical simulations are programmed and executed using MATLAB and COMSOL with spatio-temporal input information stored in a geographic information system (GIS). Note that the NPSAT framework outlined in section 2 is meant to be general and not tied to a particular modeling software. With some additional programming to handle data transfer and processing, the framework can principally be implemented with a variety of existing flow and streamline transport models. On a Xeon E5507, 2.27GHz, 24GB RAM PC, the CPU time for computing the 4000 streamlines is 20 min, for computing the numerical solution of (2) along all of the 4000 streamlines is 33 min, and for the fitting procedure for all 4000 URFs is 2.1 min. Hence, the construction phase of the NPSAT uses ∼53 min of CPU time. The total runtime for calculating 1000 yr of nitrate transport to the 40 wells (prediction phase of NPSAT: 40 BTCs from 4000 URFs), and repeated for 100 Monte Carlo realizations of a 1000-yr spatio-temporally varying N loading (see discussion below), is on the order of 15 min. In comparison, the runtime for a 20-yr simulation period, for one N-loading scenario based on the full 3-D ADE transport model was ∼25 min. The high efficiency of the prediction phase makes this approach particularly suitable for the evaluation of N-loading scenarios and associated best management practices.
4.4. Effect of Aquifer Heterogeneity on Nitrate Transport to Wells
 Alluvial aquifer systems exhibit heterogeneity at scales varying from centimeters to meters (vertically) to hundreds of meters (laterally) [Weissmann and Fogg, 1999; Zhang et al., 2006; Phillips et al., 2007; Faunt, 2009]. A significant body of literature has investigated prediction uncertainty and contaminant flux variability in spatially heterogeneous aquifers [cf. Govindaraju, 2002]. The effects of aquifer heterogeneity on pollutant transport can be modeled directly using the appropriate high-resolution 3-D flow field [e.g., Carle et al., 2006; Gotavac et al., 2009; Zhang et al., 2006; Green et al., 2010]. Here without loss of generality of the NPSAT framework, we intrinsically account for heterogeneity by applying (5) to a homogenized, upscaled macroscopic flow field at sufficiently large scale [Cvetkovic et al., 1996]. The transport solution for each streamline represents solute transport along an infinite set of stream filaments within a heterogeneous aquifer subdomain [Ginn, 2002]. The length scale of aquifer heterogeneities are assumed to be smaller than the dimensions of the hypothetical streamtube domain associated with each streamline. Large-scale aquifer hetereogeneities would be explicitly built into the groundwater flow model (1).
where is the length of streamtube , is the effective aquifer porosity within the streamtube, is the geometric mean hydraulic conductivity of the aquifer material, and is the average hydraulic gradient. Highly heterogeneous alluvial aquifers may include interbedded layers and inclusions of fine-textured, extremely low permeable materials, leading to accelerated transport in the higher permeable materials. Here we account for the accelerated transport that leads to potentially early arrival of pollutants at the CDS by reducing the effective porosity, , used to compute in (5):
where is the volume fraction of aquifer sediments responsible for the most solute transport and is the effective porosity within those aquifer materials. For example, Faunt  report the fraction of coarser-textured aquifer material in the California Central Valley alluvial aquifer to vary from 20% to 60% with corresponding specific yields or macroscopic effective porosity varying from 9% to 40%. Field investigations such as those summarized by Gelhar et al.  and numerical investigations [e.g., Zhang et al., 2006; Green et al., 2010] find that longitudinal macrodispersivity varies with travel distance and is on the order of 2%–10% of the distance of the CDS from the pollutant source. The use of a streamline transport solution (in contrast to a fully 3-D transport solution) of (2) allows for scaling of macrodispersivity as a function of each streamline's length between the source and the CDS exit point. Hence, varies from streamline to streamline.
 We compare results for three effective porosity values and three macrodispersivities, including one scaling function . The macrodispersivity scaling function was suggested by Neuman  and, for the length scales considered here, yields longitudinal macrodispersivities that are approximately one order of magnitude smaller than the streamline lengths. We compare results to those for constant macrodispersivity of 100 m, and, as a control case, to a purely advective model with zero dispersivity. For N loading, here we assume spatially uniform concentration at > 0 (no source variability). Comparison of the BTC at four wells of different depths shows that the results are highly sensitive to the effective, large-scale porosity, which linearly scales the mean travel time associated with each streamline and, hence, with each BTC (Figure 7). Results are much less sensitive to the choice of macrodispersivity.
 Earliest arrival and largest BTC spreading occurs with the distance scaling macrodispersivity function suggested by Neuman . With the distance scaling function, macrodispersivities on long streamlines may exceed 1 km leading to an earlier onset of breakthrough, especially in deeper wells, when compared to the other two scenarios. Using a constant macrodispersivity of 100 m (typical for solute transport across 1–10 km distance) results in BTCs that are nearly identical to that for the Neuman scaling function in the shallow-most well, but with less spreading than that for the Neuman scaling function in the average depth domestic well. In deep wells, the constant macrodispersivity case yields identical results to the zero dispersivity case (Figure 7).
 For the shallower domestic wells, the purely advective streamline solution yields BTCs with less spreading over time than those BTCs that include macrodispersion. Hence, for the shallower domestic wells, a purely advective model predicts first arrival times of nitrate, e.g., for c/c0 = 0.1, that occur ∼10%–20% later than predicted by the macrodispersive solutions. For the same reason, the advection-only case also exhibits less tailing in the domestic (shallow) well BTC than in the heterogeneous aquifer case with macrodispersion.
 Importantly, much of the overall effective spreading observed in the macrodispersive breakthrough curves is also captured by the advection-only simulation, where the spreading is solely caused by the mixing process within the CDS (here the well screen). In other words, the CDS mixing generates most of the observed, effective BTC dispersion, while aquifer heterogeneity, represented by macrodispersion, contributes significantly less to the spreading of the BTC. Streamline lengths across deep well CDSs with intermediate and long screens vary widely yielding a large spread in age (travel time), which is larger in long-screened high-production wells than in shorter-screened domestic wells. In contrast, the shape of the breakthrough curve in extremely short monitoring well screens (<1 m), representing a very small CDS with nearly uniform travel time distribution across its surface, is mostly affected by macrodispersion in the heterogeneous aquifer [Green et al., 2010].
 The dimensionless, advection-only BTC (Figure 7) represents the travel time distribution across the set of streamlines associated with each well. Higher travel time variance across the CDS yields larger apparent dispersive behavior in the nitrate breakthrough curve [Kreft and Zuber, 1978]:
where is the mean travel time, is the travel time variance, and is the average streamline length. The arithmetic mean travel time of the ensemble of streamlines representing groundwater flow to domestic wells is 80.7 yr at 20% effective porosity. Arithmetic mean travel time to irrigation wells is an order of magnitude longer (Table 2). The average standard deviation of the travel time across domestic wells and irrigation wells are 31 and 520 yr (Table 2). Hence, the dimensionless apparent dispersivity (15) is for domestic wells and for irrigation wells. These apparent dispersivities, due to the nonstationary streamline travel time across individual CDSs, are on the same order (domestic wells) or larger (irrigation wells) than the macrodispersivity typically found at sites even with very high aquifer heterogeneity [Zhang et al., 2006; Green et al., 2010]. In typical domestic, public supply, and irrigation well settings, but also in many gaining stream settings, the wide range of advective mean travel times across individual CDSs are therefore the key control mechanism for the dispersive nature of the BTC.
Table 2. Statistical Moments of Streamline Lengths and Travel Times for Domestic Wells and for Irrigation Wells for 20% Porositya
Statistical Moments, Streamline Lengths, Travel times
Set of 20 Domestic Wells
Set of 20 Irrigation Wells
Flow to each well is represented by 100 streamlines. See Table 1 for well design parameters.
Arithmetic mean (geometric mean), advective travel time of 2000 streamlines (years)
Standard deviation (SD), advective travel time of 2000 streamlines (years)
Arithmetic mean (geometric mean), length of 2000 streamlines (m)
SD, length of 2000 streamlines (m)
Average SD of travel time of 20 wells (years)
SD of arithmetic mean (SD of geometric mean) travel time of 20 wells (years)
 Aquifer heterogeneity is known to affect both, first arrival time and the late tailing of a BTC in ways not described by the macrodispersivity solution to (5). This has been shown to be particularly important for instantaneous (or time-limited) pollution, where non-Fickian long-term tailing in the BTC has been observed [e.g., Willman et al., 2008]. However, the long-term continuity of the salt and nitrate loading (as opposed to accidental occurrence), which typically varies interannually within a concentration range of one order of magnitude or less, limits the effect of nonideal transport on the salt and nitrate BTC [cf., e.g., Carle et al., 2006; Zhang et al., 2006; Gotovac et al., 2009]. Where nonideal tailing is considered important, the general framework of the NPSAT still applies, as appropriate upscaled 1-D transport equations [e.g., Cortis et al., 2007; Willmann et al., 2008; Cvetkovic, 2011] may replace (5) when solving streamline transport.
 Importantly, statistics of the travel time distribution among streamlines and wells indicate significant variability between breakthrough curves across the entire set of 40 wells that is due to the spatial variability in recharge rates and due to the variability in screen length, screen depth, and pumping rate between individual wells. For each of the two sets of wells (domestic and irrigation), the variability in mean travel time to each well is of similar magnitude as the average variability of the streamline travel time associated with each individual well (Table 2). The large amount of variability in mean travel time between wells (within the domestic well set or within the irrigation well set) and the large difference between the overall mean travel time to domestic (shallow) wells and to irrigation (deep) wells has significant implications for regional planning and assessment of well water quality (or water quality in groundwater discharge to stream reaches). Significant implications between CDS variability in the breakthrough behavior of nitrate (and, e.g., salt) are likely, even if source loading conditions were homogeneous and effects of aquifer heterogeneity were negligible. Because of the decadal to centuries-long transfer time within the groundwater basin, high variability between CDSs should be expected to be observed over very long periods of time.
 The large variability in BTCs between wells, under conditions of uniform input concentration and realistic macrodispersion, also demonstrates that it is important for a nonpoint source assessment to properly capture the wide range of spatial arrangements between nonpoint sources and CDSs, thus providing further rationale for using high-resolution flow and transport models in nonpoint source assessments. Our results demonstrate that the spatial patterns of the distribution of recharge across the aquifer surface, and of well screens and pumping within the aquifer (or, by extension, of stream reaches connected to groundwater) need to be properly captured to assess the potential variability of pollutant concentration between wells, irrespective of aquifer heterogeneity or pollutant loading variability. This aspect has been unexplored in the hydrogeologic literature.
4.5. Effect of Stochastic Forcing by Spatio-Temporally Variable Nitrate Source Loading
 Next, we investigate the effect of spatio-temporally variable pollutant loading across the aquifer's water table, within the source area of each of the 40 wells considered here. The N loading within a source area is highly uncertain. Two main sources of uncertainty are the lack of measurements of actual N loading either across space or over time, and, from a planners perspective, uncertainty about the source area and its land use composition. Here the uncertainty in N loading is treated as a stochastic forcing problem, whereby the boundary conditions of (5) at the source side is a probability distribution function rather than a deterministic value [e.g., Botter et al., 2006]:
where is the probability density function of loading history.
 To solve the stochastic forcing problem, we apply the Monte Carlo method to the NPSAT, using the previous example with and Neuman's macrodispersivity scaling function, . Land use parcels are divided into five categories following Figure 6, which represents typical land use distribution in the intensively farmed region of the southeastern Central Valley, California (Figure 1). For each category, we assign a pollutant loading distribution with mean mg L−1. These rates represent a hypothetical, but typical range of nitrate-nitrogen (N) loading in agricultural recharge [VanderSchans et al., 2009; Burow et al., 2007, 2010]. The standard deviation is assumed constant for all categories and equal to mg L−1 representing the combined, unknown (stochastic) farm-to-farm, field-to-field, and year-to-year variations in recharge nitrate-N concentration. No spatial correlations between individual sources (parcels) are considered. Random, autocorrelated loading histories are generated for each parcel by employing a first order autoregressive model , where is the loading concentration at time , is a model parameter expressing the correlation between and , is Gaussian noise where is related to through , and is a constant expressed as . In our application we selected .
 For the stochastic forcing analysis, 100 realizations of a 1000-yr N-loading history to each land use parcel are generated with a temporal discretization of 1 yr. The generated loading functions were subsequently used to calculate the actual breakthrough curves for the 40 wells. The fitted unit response functions, previously computed, were convoluted with each N-loading history realization resulting in 100 breakthrough curves for each well (4000 BTCs in total).
 For comparison to the stochastic forcing analysis with NPSAT, we calculate an alternative, deterministic BTC using a simple instantaneous mixing model [Mercado, 1976; Lee, 2007]: one for the aquifer volume between water table and the bottom of the domestic well zone, and a second one for the aquifer volume between the water table and the bottom of the irrigation well zone. The concentration for the instantaneous mixing model is obtained as the ratio of the cumulative mass of N recharged through time t and the total groundwater volume considered.
 For further comparison, we also compute a 1-D vertical plug flow model solution [Cho and Mostaghimi, 2009] for a domestic and for an irrigation well at average well depth and with average well length, respectively. Concentration is zero until where is the depth to the top of the well screen and is the downward velocity. Concentration then increases linearly up to the recharge-weighted average concentration. The maximum concentration is reached at , where is the depth to the bottom of the well screen.
 The results show dramatic differences in BTC patterns between domestic and irrigation (and other large production) wells (Figure 8), consistent with the homogeneous pollutant loading example (Figure 7). In the shallower, low-production domestic wells, nitrate concentrations increase within less than one to a few decades and attain a relatively steady maximum after 30–200 yr (note: here we have neglected vadose zone travel times). Deeper, large production (irrigation) wells respond very slowly (at timescales of multiple decades to centuries) to nitrate loading at the water table. In this specific example, many irrigation wells do not reach their maximum concentration even after a millennium.
 In contrast, the simple mixing model would predict an instantaneous onset of nitrate breakthrough, followed by a linear increase until . The equivalent homogeneous plug flow model predicts a delayed onset of breakthrough that is nearly consistent with the median (50% of the exceedance probability) onset of nitrate pollution in the NPSAT simulations, it then predicts a rapid increase to the full concentration at . For both, the domestic and irrigation well set, of these simple models is significantly earlier than the time, when the median exceedance probability of the NPSAT solution reaches a maximum, steady concentration (Figure 8). The large difference between of the analytical models and the much longer time to quasi-steady state behavior in the exceedance probability distributions is due to the heterogeneity in recharge rates and, more importantly, due to the nonuniform velocity field, with recharge at the top boundary matching pumping rates from the aquifer, average vertical velocities decrease with depth and are zero at the bottom boundary of the aquifer. In particular, deep wells or wells with long screens extending to large depth discharge a disproportionate amount of old, deep groundwater, while the younger, polluted groundwater is siphoned off by shallower large production wells, and by the upper screen portions of long-screened, deep wells. The nonuniformity of the average velocity field with depth therefore significantly delays breakthrough in wells, when compared to the simple analytical models.
 The total variability of pollutant concentration, which is not captured by the simple analytical models, is found to be consistent with results from fully three-dimensional simulations of a heterogeneous aquifer system [Carle et al., 2006]. This is particularly evident during the second half of the simulation period, when most domestic wells have reached a quasi-steady concentration level. The results reflect two sources of variability: the land use composition in the source area of each well varies among wells. It determines the long-term average pollutant concentration in the well (reflected by the difference between bundles of colored lines in Figure 8). And second, the spatio-temporal variability of pollutant loading between fields within the same land use category is reflected in the variability of concentration at any one well across multiple realizations of land use loading (reflected by the many lines of one single color for each well in Figure 8).
 Notably, the concentration contrast between wells is similar for irrigation and domestic wells: At the end of the simulation period (t = 1000 yr), the coefficient of variation of the mean BTCs (20 samples, each being the mean concentration across 100 BTC realizations at a given well) is 0.35 for domestic wells and 0.33 for irrigation wells. This high variability reflects the contrast in the total mean loading rate between each well's source area.
 In contrast, at any given well, the uncertainty about its concentration due to uncertain loading alone is much smaller than between-well variability. More importantly, it is much smaller for irrigation wells than for domestic wells (compare colored bundles in Figure 8, top panels, which are tighter for irrigation wells than for domestic wells). The average coefficient of variation across multiple realization BTCs at each well is 0.075 for domestic wells and 0.013 for irrigation wells. The difference between the two groups is due to the much larger size of the source area of irrigation wells compared to that of domestic wells. The uncertainty about loading in individual fields (land use parcels) tends to cancel out across these larger source areas. From a regional planning or assessment perspective, the results imply that the spatio-temporal variability of loading within a particular land use category affects the uncertainty about a particular well BTC significantly less than the uncertainty about the total mean loading within the well source area, which is determined by the areal fraction and long-term mean loading of individual land use categories within a source area.
 In this context, variability in the early breakthrough can be further quantified by considering the average time to exceedance of 10 mg L−1 (the regulatory limit for nitrate-N). increases with the depth of the well screen, at a rate of ∼5 yr m−1 in domestic wells, but significantly faster, at 3.5 yr m−1 in the deeper irrigation wells due to the effect of macrodispersion and longer screens (Figure 9). The mean and standard deviation of is 41 and 22 yr for all domestic wells, respectively, and 386 and 177 yr for all irrigation wells (all loading realizations), respectively. The standard deviation of the expected (mean) time to reach 10 mg L−1 at each well is 2.7 and 3.7 yr for the domestic and irrigation wells, respectively, which reflects the uncertainty caused by the unknown, variable N loading.
 The total variability in (or uncertainty about) well concentration is a function of the variability of mean loading among land uses, as shown above, but also a function of the spatio-temporal variability within land use categories. Repeating the analysis above with various source-loading variabilities reveals that BTC concentration variations between wells increase monotonically with source loading variability (Figure 10). Domestic wells not only exhibit a higher variability (higher prediction uncertainty at individual wells), but the increase in that variability with loading variability is much larger than in irrigation wells due to the lack of source mixing across multiple land use parcels.
 We developed a high-resolution, highly efficient, quasi-3-D framework capable of simulating the transport of nonpoint source pollution within large groundwater basins (hundreds of kilometers across) while also accounting for detailed variability in flow field dynamics due to spatially variable fluxes across external and internal aquifer boundaries (recharge, well pumping or, equivalently, stream discharge variability at the decameter scale). The model produces individual CDS BTCs from which exceedance probability distributions across CDSs or subgroups of CDSs (by depth, by hydrogeologic subregion, or by land use subregion) can be generated. The approach provides broad flexibility in accommodating linear reactive transport, as well as the effects of aquifer heterogeneity either implicitly or explicitly.
 The NPSAT framework can be applied to existing steady state aquifer models with relatively coarse resolution using any number of domain decomposition methods or mesh refinement techniques [Mehl and Hill, 2002; Li et al., 2006] as a basis for computing the steady state flow and velocity field at sufficiently high resolution for the subsequent streamline and URF computation. Streamline start- and end-point coordinates and associated URF parameters can readily be stored in standard geospatial databases at densities of 102–103 data points per CDS, a manageable database even at the large basin scale (105 km2), considering that the typical number of CDSs is on the order of 1–10 per square km. BTCs at a large number of CDSs across large groundwater basins can be computed efficiently via convolution for arbitrary, user-defined loading functions (∼100,000 200-yr well-BTCs with 100 streamlines per BTC, per hour CPU time on a standard year 2010 personal computer), e.g., for optimization within a hydro-economic or policy scenario analysis. By separating the modeling process into a construction phase and a highly efficient simulation phase, the NPSAT framework lends itself for a wide range of land use planning and nonpoint source assessment applications.
 Using the NPSAT framework to investigate nitrate transport to wells in a representative alluvial aquifer system in an agricultural landscape with spatially varying recharge and pumping rates, we make some key novel observations that fundamentally define the effect of nonpoint source pollution on a set of CDSs (e.g., domestic wells, public supply wells, irrigation wells, gaining stream reaches) within a groundwater basin and nonpoint source region of interest:
 (1) The large variability in the spatial arrangement of nonpoint source pollution recharge sources across the top of the aquifer and of CDSs (sinks) within the aquifer introduces large variability in pollutant breakthrough between CDSs within a hydrogeologic and landscape region, irrespective of the degree of spatio-temporal nitrate loading variability or aquifer heterogeneity.
 (2) The location of production well CDSs and their spatial arrangement relative to recharge sources dictate that the main source of the apparent dispersive behavior in CDS pollutant breakthrough is the mixing of water collected within the CDS and the large variability of the (mean) travel time across the surface of the CDS. For large production wells, mixing is by far the dominant source of the apparent dispersive behavior in the CDS breakthrough, even in highly heterogeneous aquifer systems. For smaller domestic wells, mixing is still dominant, but aquifer heterogeneity and resulting macrodispersion may have a significant effect on early concentrations of the BTC. Only in the BTC of very short monitoring wells (<1 m), the apparent dispersion of the BTC is primarily controlled by aquifer heterogeneity (macrodispersion).
 (3) While concentrations are highly variable between wells, the analysis shows that the overall response dynamics across a large set of wells is driven by and consistent with the regional average vertical transport rate and the typical range of vertical positions of well screens. Equivalently, the range of transport times to a stream reach would be expected to be dictated by the range of travel times to the stream reach.
 (4) Simulated BTC dynamics in wells are highly sensitive to regional recharge rates imposed by the flow model and to effective aquifer porosity.
 (5) Consistent with the latter finding, nitrate CDFs for shallow, low production domestic wells increase much more rapidly than those in deeper high production irrigation/municipal wells. In a typical large alluvial aquifer system, response times in domestic wells are on the order of one to few decades, whereas it can be on the order of centuries for irrigation wells or large, deep production wells for drinking water.
 (6) In contrast to aquifer heterogeneity, which has only limited influence on BTCs in domestic and large production wells, stochastic forcing due to uncertainty about N loading in individual land use units results in significant uncertainty about the BTC at a well and adds significantly to the variability of concentrations between wells.
 (7) Much higher year-to-year variability of concentration histories are found in domestic wells than in deeper large production wells.
 (8) Large production wells exhibit significant between-well variability due to differences in mean long-term nitrate loading between source areas. But loading uncertainty within a large source area does not result in high uncertainty about concentration in individual wells.
 (9) Understanding long-term average nitrate loading associated with specific land use classes is therefore more critical to predicting long-term water quality in large CDSs than understanding field-to-field or year-to-year variations in nitrate loading for a specific land use class.
 A potentially significant limitation of the NPSAT framework is the assumption of steady state flow condition. While seasonal variations are likely to have limited impact on the analysis [Stuart et al., 2006], longer term changes, particularly in recharge dynamics, cannot be accounted for in the NPSAT. Such long-term changes in groundwater dynamics are intrinsically associated with many semiarid and arid groundwater basins, which experienced a significant change in recharge pattern during the 20th century, as they converted from natural landscapes or extensive agriculture to irrigated landscapes with intensive agriculture. Future policy options ideally include large-scale changes in irrigation efficiency, thus reducing future groundwater recharge from irrigated fields. A further limitation to note is that the response function approach, which takes advantage of the superposition principle, requires linearity in the reaction equations. Nonlinear reactions, precipitation, and chemical dissolution processes could not be directly accommodated within the response function approach.
 Future work with the NPSAT framework includes full application to regional aquifer systems, application to reactive pollutant systems, validation of CDFs against means and trends in real groundwater quality data sets, and further evaluation of the role of CDS-mixing versus aquifer heterogeneity effects, using various upscaling schemes including explicit representation of heterogeneity, e.g., Carle et al.  and implicit representation, e.g., the transfer function approach [Jury and Roth, 1990] or a continuous time random walk approach [Cortis et al., 2007].
Appendix A:: Comparison to Fully 3-D Transport Model With Transverse Dispersivity
 To assess the error due to neglecting transverse dispersivity, we compare the streamline-based, quasi-three-dimensional NPSAT approach against a fully three-dimensional solution with fixed longitudinal and transverse (macro-) dispersivity. The 3-D solution to (2) is obtained using the COMSOL numerical finite element solver. The comparison is based on a hypothetical aquifer with a single extraction well that has a source area ∼140-m wide (Figure A1).
 Three cases of spatial loading are considered: one that is not sensitive to transverse dispersion, and two that are particularly sensitive to transverse dispersion (Figures A1a, A1b, A1c) (uniformly distributed loading over the entire surface of the aquifer [Figure A1a] loading over the source area of the well only, and loading over the entire aquifer except within the well source zone). Dispersivity coefficients are 25 m, 2.5 m, and 0.25 m in the longitudinal, transverse horizontal, and vertical direction.
 For scenario 1 (uniform source loading), the two solutions (NPSAT with 100 streamlines and 3-D) are in close agreement (Figure A2) with differences as much as 10%–15% in the tail of the BTC. In the second case, the 3-D transverse dispersion model predicts smaller concentrations than in the first scenario due to pollutant dispersion into the adjacent, clean groundwater. The NPSAT overestimates the maximum concentration of the 3-D dispersion case by 1 × 10−3 kg m−3 (1 mg L−1) or 25% of the peak concentration. As expected, the third scenario provides the opposite solution to the second scenario, with NPSAT predicting zero concentration (no streamlines originating in areas with pollution) and the full 3-D model predicting fluxes that peak at 1 mg L−1. At higher dispersivities, transverse fluxes increase and the accuracy of the streamline solution further decreases. Therefore, while the NPSAT is applicable to most common cases, where concentration contrasts at the CDS source area boundary are not significant, the errors in peak concentration due to negligence of transverse dispersion are expected to be significantly smaller than shown in this special case. Yet there are cases where agricultural land uses are adjacent to nonirrigated lands, such as near riparian zones or other nonagricultural land uses, where the transverse dispersion cannot be neglected and alternative methods such as random walk particle tracking may apply to address the issue of transverse dispersion with nonpoint source solutes [Green et al., 2010], but those methods require substantially longer runtimes.
 We are immensely grateful for the extensive and very helpful comments by Christopher T. Green, USGS Menlo Park, and the two anonymous reviewers, who helped shape this manuscript and focus its message. We gratefully acknowledge funding for this work by the CA State Water Resources Control Board under agreements 04-184-555-3 and 09-122-250.