Introduction

Numerical models have been widely used in recent decades to study groundwater flow in subsurface systems. However, developing groundwater models has never been an easy task for fluvial-in-origin subsurface systems owing to their inherent structural heterogeneity (Galloway 1977). The first challenge in developing a groundwater model is to adequately characterize the subsurface systems. The literature shows many methods to model hydrofacies for different heterogeneity scales using various input data sets (e.g., well logs, pumping test, and seismic data). Among them, widely used methods in hydrogeology are the two-point variogram statistics, such as indicator geostatistics (Journel 1983; Johnson and Dreiss 1989; Johnson 1995; Proce et al. 2004); transition probability-based indicator geostatistics (Carle and Fogg 1996; Lee et al. 2007; Koch et al. 2014); and multiple-point simulation (MPS) (Strebelle 2002; Journel 2005; dell’Arciprete et al. 2012). Reviews of these methods can be found in Koltermann and Gorelick (1996), de Marsily et al. (2005), and Hu and Chugunova (2008). While the applications of these methods are site specific and are subject to user preference and expertise, it has been well understood that different hydrofacies methods generate significantly different spatial distributions of hydraulic properties (Alabert and Modot 1993; Gómez-Hernández and Wen 1998; Western et al. 2001; Zinn and Harvey 2003; Zhang et al. 2006; Lee et al. 2007; Bianchi et al. 2011; Berg and Illman 2015) and consequent flow and solute transport responses.

The second challenge in developing a groundwater model, besides the estimation of spatially variable hydraulic parameters, is to construct better model grids that are consistent with the geometries of modelled hydrofacies. Errors from inaccurate model grids that fail to capture hydrofacies geometries may result in incorrect estimated hydraulic parameters during model calibration. The literature shows two approaches commonly used to construct a computational grid from well log data: the solid approach and the pre-defined grid approach. Before implementing either approach, one needs to obtain geological information (e.g., lithology, bed boundary elevation, formation dip, etc.) from well logs. Readers are referred to some classic books for well log interpretation techniques (Schlumberger 1972; Hilchie 1982; Bassiouni 1994), which were used to interpret well logs for this study.

Using the solid approach, one needs to manually correlate well logs and label distinct hydrofacies for each well log. Once the well log correlation is established, interpolation methods are applied to generating surfaces of the same types of hydrofacies. These surfaces represent the hydrofacies boundaries and result in a solid model. Jones et al. (2002) and Lemon and Jones (2003) developed a grid generator to generate computational grids for groundwater models from solid models. The beauty of this approach is the creation of non-uniform computational layers that match well the generated hydrofacies surfaces, including pinch-outs.

The biggest challenge in using this approach is performing manual correlation between well logs, which is subjective and can become laborious and impractical when dealing with a huge number of well logs in areas known to be highly complex. (e.g., fluvial depositional environments). Manually correlating well logs often results in inconsistencies with geological deposition, forces correlation of unrelated hydrofacies, and produces erroneous hydraulic connections of discontinuous hydrofacies.

The pre-defined grid approach usually generates uniform, relatively coarse layers directly to be used for flow and transport modeling. Examples of this approach include using T-PROGS (Carle 1999) and geostatistical tools—e.g., GSLIB (Deutsch and Journel 1997). This approach does not force generating surfaces of hydrofacies, and therefore, avoids the issues caused by manual correlation; however, the greatest concern of using pre-defined grids is to lose the vertical resolution of hydrofacies geometries if layers are not fine enough. Using very fine layers intuitively can improve vertical resolution of hydrofacies geometry, but will significantly increase computation time since pre-defined grids are directly used for flow simulation.

The third challenge when developing a groundwater model is model calibration; for complex aquifer systems, model calibration can be very time consuming. Without advanced search algorithms and computing resources, searching for better parameter values can end prematurely; moreover, grid error amplifies parameter estimation error. Model calibration may result in over-parameterization and unrealistic model parameter values in order to compensate model structure error due to an invalid grid.

This study presents a general procedure to develop complex groundwater models for siliciclastic aquifer systems and to overcome the challenges raised by grid generation and model calibration. The study first introduces a grid generation technique that maintains high vertical resolution of hydrofacies geometries with a reasonable number of non-uniform boundary-fitted layers. Second, the study adopts the covariance matrix adaptation-evolution strategy (CMA-ES; Hansen and Ostermeier 2001; Hansen et al. 2003) in parallel computing to calibrate complex groundwater models. Third, the grid generation and model calibration techniques are applied to developing a three-dimensional (3D) groundwater model for the fluvial-in-origin aquifer system underneath the Baton Rouge area, southeastern Louisiana.

Methodology

The flowchart in Fig. 1 presents major steps to develop a groundwater model for applications: data collection, model construction, model calibration, and model application. The first step is to collect and analyze pertinent data for the area of interest. The data include geological and geophysical data as well as hydrologic and hydrogeologic data, groundwater use data, etc. Data are always lacking for groundwater study and their quantity and quality directly affect the subsequent model development steps. The second step is to construct a conceptual groundwater model based on the collected data. Modelers often face decisions of which model components are fixed and which model components are to be adjusted in the model calibration phase. In model construction, hydrostratigraphy is the backbone of a groundwater model and directly controls the development of a computational grid for a numerical method. Hydrostratigraphy construction for an aquifer system is the key step to transfer known geological information from well logs to a groundwater model. This step is actually extremely difficult for a complex aquifer system such as the siliciclastic groundwater system in this study, but is often overlooked and overly simplified. The challenge leads to this study to address the question of how to use a large amount of well logs with highly irregular sand-clay sequences to generate a proper computational grid for groundwater modeling. Model calibration is a necessary step to ensure model integrity before a groundwater model can be applied to its applications. It involves tuning physical parameters (e.g., hydraulic conductivity, specific storage, etc.) and adjusting model components such that groundwater model output is close to observed data. For complex aquifer systems, model calibration can be time consuming.

Fig. 1
figure 1

Flow chart for developing a groundwater model

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This section focuses on two challenges in groundwater model development: grid generation and model calibration. The discussions in the following are mainly for siliciclastic sedimentary depositional environments, which are applicable to this case study demonstration.

Well log interpretation

The primary sources of information used to establish hydrofacies geometries are wire-line spontaneous potential (SP) and electrical resistivity logs for boreholes. Spontaneous potential and resistivity log responses are controlled largely by the ratio of sand to clay minerals and they have long been used to interpret sedimentary depositional environments. Galloway (1977) used SP and resistivity curve morphologies to identify fluvial facies for channel fill, levee, crevasse splay and floodplain, and established a meandering stream facies. Kerr and Jirik (1990) adapted Galloway’s (1977) facies model and provided examples of SP and resistivity responses that match known fluvial facies for the middle Frio formation, South Texas. Sands deposited by braided streams produce jagged, wedge-shaped curve morphologies (Miall 2010). Based on these established relationships between log responses and fluvial facies, Chamberlain et al. (2013) used SP and resistivity data to study depositional environments of siliciclastic sediments in the Baton Rouge area.

Following Chamberlain et al. (2013) and Elshall et al. (2013), this study uses SP, resistivity, and gamma ray (when available) to identify the location of sand facies at depth. Figure 2 shows a typical SP-resistivity log for saturated formations in the Baton Rouge area. Based on deviations from a visually estimated shale baseline, boundaries of sands can be drawn on inflection points of SP curves. A cutoff value that generally fell between 10 and 35 ohm-m for resistivity curves is assigned to determine boundaries of sands. Low long-normal resistivity generally indicates the occurrence of salty water. Low gamma ray response generally indicates a sand facies. Sand boundaries can be well identified by correlating SP, resistivity and gamma ray curves (Schlumberger 1972; Hilchie 1982; Bassiouni 1994)—for example, seven sand facies are picked and many thin sands are ignored as shown in Fig. 2. Non-sand intervals are assumed to be clay (shale or mudstone) facies.

Fig. 2
figure 2

Interpretation of fluvial facies for a well log. Legend: A amalgamated braided channel-fill with brackish water, B channel-fill point bar sand with brackish water, C stacked/amalgamated channel-fill with very salty water, D floodplain, and E natural levee (Kerr and Jirik 1990; Miall 2010). SN short normal resistivity (dashed line); LN long normal resistivity (solid line). 1,000 ft = 304.8 m

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Well log interpretation is inevitably subjected to an individual’s experience and the purpose of the work. This study does not intend to discuss the uncertainty of computational grids due to different log interpretations. Moreover, it is possible to use the established relationships between log responses and fluvial facies from Galloway (1977), Kerr and Jirik (1990), and Miall (2010) to infer different fluvial facies—for example, Fig. 2 shows some identified fluvial facies based on the established relationships. This study does not intend to identify specific fluvial facies, but focuses on aquifer system construction using identified sand and clay facies from well logs.

Indicator kriging for hydrostratigraphy construction

To construct hydrostratigraphy based on the aforementioned well log interpretation of sand-clay sequences, well logs are first transformed into binary indicator values. The indicator value for sand facies is 1 and for clay facies is 0. The indicator kriging (Johnson and Dreiss 1989) is suitable for handling bimodal heterogeneity. By using a regional geological dip to correlate well logs as shown in Fig. 3a, indicator kriging is performed on inclined surfaces where indicator data are obtained at the intersections with well logs. To make it easier for operating indicator kriging, all well logs are translated vertically to a non-dipping domain. To do so, the vertical translation distance depends on the dip angle and the distance from well log location to a strike line that serves as a pivot as illustrated in Fig. 3b. Then, indicator kriging is performed on horizontal surfaces given horizontal discretization, which can be done by any methods available in the literature. The same horizontal discretization is applied to all horizontal surfaces at different depths. A detailed 3D hydrostratigraphic architecture can be achieved by assembling a large number of horizontal surfaces with fine intervals. This study conducts indicator kriging at horizontal surfaces with one-foot intervals. Finally, the 3D hydrostratigraphic architecture is transformed to a dipped architecture that follows the dip direction.

Fig. 3
figure 3

Translation of well log positions from a a dipping surface to b a non-dipping surface. c An illustration of bed boundary projection for the vertical column (i, j) of a grid. Bed boundaries of neighboring vertical columns are projected to the vertical column (i, j)

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It is noted that the grid generation technique in this study is not limited to indicator kriging. Any geostatistical method can be used to estimate hydrofacies for a surface. The resulting indicator data from horizontal surfaces are used to compute experimental variograms. Then, a variogram model can be derived by fitting to the experimental variograms.

The expected value of the indicator at an unobserved location is obtained by

$$ v\left({\mathbf{x}}_0\right)={\displaystyle \sum_{i=1}^S}{\lambda}_i\kern0.1em I\left({\mathbf{x}}_i\right) $$
(1)

where v(x 0) is the expected value at unobserved location x 0, S is the number of well logs for a horizontal surface, and λ i are the indicator kriging weights. Indicator kriging has been well documented in the literature. Readers are referred to Olea (1999) for more information.

The expected value of indicator function represents the probability that facies at a location x 0 fall into sand facies or clay facies. By giving a cutoff α as follows, distributed sand and clay facies on a horizontal surface can be achieved:

$$ I\left({\mathbf{x}}_0\right)=\left\{\begin{array}{c}\hfill 1\ :\kern0.75em \mathrm{sand}\ if\kern0.5em v\left({\mathbf{x}}_0\right)\ge\ \alpha \hfill \\ {}\hfill 0\ :\ clay\ \mathrm{if}\kern0.5em v\left({\mathbf{x}}_0\right) < \alpha \hfill \end{array}\right. $$
(2)

Determination of a defensible cutoff value is challenging. A value of 0.5 is commonly used for a neutral selection. However, a better cutoff can be determined in a calibration process where facies estimates are subject to additional information, e.g., driller’s logs, total volume of sand or clay facies from electrical logs, etc. (Elshall et al. 2013).

Upscale hydrostratigraphic architecture to a model grid

The MODFLOW model (Harbaugh 2005) uses a structured grid where cells are rectangular in the two-dimensional plane. The 3D model domain is discretized into rows, columns, and layers, ordered in a Cartesian coordinate system. Each grid cell (except for those at model boundaries) is connected to six surrounding cells. Unlike the unstructured grid version of MODFLOW-USG (Panday et al. 2013) that has more flexibility for domain discretization, the challenge for conventional MODFLOW models is that all computational layers in the structured grid must be continuous throughout the model domain. Once a hydrostratigraphic architecture with very fine vertical discretization is generated by indicator kriging, the following two steps are developed to upscale the hydrostratigraphic architecture to a MODFLOW structured grid by merging the same hydrofacies in the vertical direction to reduce the number of layers.

  1. Step 1.

    Project neighboring bed boundaries. To account for the continuity of MODFLOW layers throughout the model domain, Fig. 3c illustrates bed boundary projection for each vertical column of a grid. A vertical column (i, j) of a grid in Fig. 3c originally has four bed boundaries and has direct connections to its neighboring vertical columns which have different bed boundary positions. The bed boundaries of the neighboring vertical columns are projected to the vertical column (i, j) and increase bed boundaries of the vertical column (i, j) to 12. Bed boundary projection is applied to all vertical columns of the grid. By doing so, each vertical column of the grid possesses information of bed boundary positions of its neighboring vertical columns. Bed boundary projection is an important step in order to preserve the continuity of flow pathways, especially through geological faults, pinch-out areas, or narrow connections.

  2. Step 2.

    Determine model layers. Given a desired number of model layers, this study introduces a “ruler” algorithm to assign MODFLOW layer indices to each vertical column. Again, the layer boundaries are required to match the bed boundaries. As shown in Fig. 4a, the start and end of the ruler match the top and bottom boundaries of a vertical column, respectively. The number of major ticks in the ruler represents the number of MODFLOW layers. The number of layers up to a bed boundary for a vertical column is obtained by comparing its bed boundary location to the ruler—for example, a bed boundary located between 0 and 1.5 in the ruler indicates one layer up to the bed boundary, between 1.5 and 2.5 indicates two layers up to the bed boundary, between 2.5 and 3.5 indicates three layers up to the bed boundary, and so forth. When the thickness between consecutive bed boundaries is small, the ruler algorithm is likely to assign two or more bed boundaries with the same number of layers up to their bed boundaries as shown in Fig. 4b. In this case, the ruler algorithm will adjust the numbers to make sure that each bed boundary has a distinct layer index. In the last step, equal thickness of layers is given to segments that need to be divided into two or more layers based on the final assignment of the layer indices to the bed boundaries.

Fig. 4
figure 4

Illustration of the ruler algorithm: a a vertical column of a grid with a distinct number of layers up to each bed boundary, and b a vertical column of a grid with two bed boundaries having the same number of layers. A refers to the number of layers up to a bed boundary and B model layer indices

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Model calibration using parallel computing

Model calibration is the process of adjusting model parameter values until a satisfactory fit between model outputs and field measurements (e.g., heads, concentrations) is achieved. Traditionally, model calibration is performed manually based on trial-and-error methods. This approach is easy to apply, but is laborious and time-consuming; furthermore, trial-and-error methods may not guarantee finding the best solutions because different user’s manipulations may produce dissimilar solutions.

Automatic model calibration using optimization methods is efficient due to their ability to handle a high number of model parameters and the accuracy of solutions. Optimization methods can be classified as derivative-based and non-derivative-based search methods (global search methods). Derivative-based methods converge quickly, but solutions may be trapped to local optima. Global search methods have potential to find near global solutions as well as handling non-differentiable and discontinuous functions. Popular global search methods applied to groundwater model calibration include genetic algorithms (El Harrouni et al. 1996; Wang 1997; Karpouzos et al. 2001), simulated annealing and tabu search (Zheng and Wang 1996), ant colony optimization (Abbaspour et al. 2001), particle swarm (Gill et al. 2006; Krauße and Cullmann 2012), and shuffled complex evolution (SCE) (Vrugt et al. 2003). The common disadvantage of global search methods is that a large number of model runs and iterations are needed in order to reach a near global solution. For a computationally expensive simulation model, this method may become impractical; this problem, however, can be solved efficiently by parallel computing. Reviews and comparisons of methods for model calibration can be found in many books and articles (Cooley 1985; Sun 1994; Hunt et al. 2007; Hill and Tiedeman 2007; Vrugt et al. 2008; Hendricks Franssen et al. 2009; Fienen et al. 2009; Doherty 2015; Yeh 2015). Popular software for automatic groundwater model parameter estimation include PEST (Doherty et al. 1994), UCODE (Poeter and Hill 1999), or MGO (Zheng and Wang 2003).

This study adopts CMA-ES (Hansen and Ostermeier 2001; Hansen et al. 2003) as a global search method to calibrate a groundwater model and estimate model parameters for three key reasons. First, CMA-ES has the capability of obtaining a near global solution and avoiding entrapment in local optima. Second, CMA-ES provides a full covariance matrix of estimated parameters, which can be used to assess parameter estimation and prediction uncertainty. Third, CMA-ES can be implemented in a high-performance computing system to overcome the prohibitive computational cost (Elshall et al. 2015).

Construction of Baton Rouge aquifer system, southeastern Louisiana

The Baton Rouge aquifer system shown in Fig. 5 is part of the Southern Hills regional aquifer system, southeastern Louisiana, USA. The Baton Rouge aquifer system consists of a succession of south-dipping siliciclastic sandy units and mudstones of Miocene through Holocene age, extended to a depth of 3,000 ft (914 m; Meyer and Turcan 1955), and is highly complex as a result of fluvial deposition (Chamberlain et al. 2013). Eleven freshwater aquifers underneath Baton Rouge are the Mississippi River Alluvial Aquifer (MRAA), the “400-foot” sand, the “600-foot” sand, the “800-foot” sand, the “1,000-foot” sand, the “1,200-foot” sand, the “1,500-foot” sand, the “1,700-foot” sand the “2,000-foot” sand, the “2,400-foot” sand, and the “2,800- foot” sand. These aquifers were named by their approximate depth below ground level in the Baton Rouge Industrial District (Meyer and Turcan 1955). The depth of the MRAA in the model domain is similar to the “400-foot” sand. The sand deposition is non-uniform due to spatial and temporal variations in fluvial processes as well as a large amount of missing sand possibly due to erosional unconformity. From the region-scale study of Griffith (2003), the Baton Rouge aquifer system dips south towards the Gulf of Mexico.

Fig. 5
figure 5

Map of the study area, boreholes of well logs (circles), and US Geological Survey groundwater observation wells (triangles). The coordinate system is UTM (meters), Zone 15, NAD83

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The Baton Rouge aquifer system consists of the east–west trending Baton Rouge fault and the Denham Springs-Scotlandville fault. The faults crosscut the aquifer/aquitard sequence in the study area (McCulloh and Heinrich 2013). The Denham Springs-Scotlandville fault is generally thought to have no significant effect on groundwater flow. The historical groundwater data suggests the Baton Rouge fault to be a horizontal flow barrier that separates the freshwater to the north and the saline water to the south. The source of saline water is likely from the expulsion of over-pressured brine fluids, extending vertically upward above the top of Gabriel salt dome all the way to the water table (Anderson et al. 2013). Bense and Person (2006) suggested the Baton Rouge fault to be a conduit-barrier fault. Recent study (Elshall et al. 2015) suggested the Baton Rouge fault and the Denham Springs-Scotlandville fault to be low-permeability leaky faults.

Groundwater naturally flows southward; however, excessive groundwater withdrawals in the area between two faults have caused significant declines of groundwater levels north of the Baton Rouge fault. The two largest pumping areas that cause the most significant drawdown are the Industrial District and the Lula pump station. The pumping wells at the Industrial District were screened from the “400-foot” sand to the “2,000-foot” sand. The “2,000-foot” sand is the most heavily pumped by the industrial wells. The Lula pump station heavily pumps the “1,500-foot” sand for public supply. The declines of groundwater level caused land subsidence (Whiteman 1980) and reversed the flow direction in the vicinity of the Baton Rouge fault to cause saltwater intrusion (Morgan and Winner 1964; Whiteman 1979; Tomaszewski 1996). Observed chloride concentrations are generally much less than a few thousand mg/L (Lovelace 2007). To better understand the impacts of the geological faults and the groundwater withdrawals on groundwater level decline, this study applied the proposed methodology to develop a Baton Rouge groundwater model, which covers the 11 sands.

Saltwater intrusion modeling and subsidence modeling were not conducted in this study. To the best of authors’ knowledge, the extent of subsidence does not affect groundwater flow in the Baton Rouge area. However, potential simulation errors in groundwater levels may occur near or in the south of the Baton Rouge fault where high chloride concentrations are present. Since this study used the developed groundwater model to investigate groundwater budget and flow patterns, the errors due to the density effect would not be significant. Nevertheless, saltwater intrusion modeling and subsidence modeling are important to the Baton Rouge area and will be the focus of future studies.

Results and discussion

Well log interpretation and hydrostratigraphic architecture

Using the method in sections ‘Well log interpretation’ and ‘Indicator kriging for hydrostratigraphy construction’, the study analyzed wireline well logs from 583 boreholes (Fig. 5). The result of binary well log interpretation is shown in Fig. 6a. The number of sand and clay segments in well logs ranges from 3 to 59. It is impractical to manually build correlations between boreholes. The model domain in the planar direction is discretized into 93 rows and 137 columns with a cell size 200 × 200 m, resulting in 12,741 grid cells. Indicator kriging was adopted to construct a hydrostratigraphic architecture with a dip angle 0.29° and a cutoff value 0.40 from a prior study (Elshall et al. 2013). After translating all well logs vertically to a non-dipping domain, indicator kriging was conducted on horizontal surfaces of 1 ft (0.304 m) intervals from land surface to –2,890 ft (–881 m) below NGVD29 that covers the 11 sands. This study adopted an isotropic exponential variogram model (nugget = 0.084, range = 8,600 m, and sill = 0.223) for the indicator kriging (see Fig. 6b). The modeling area was divided into three regions by two fault lines. Hydrostratigraphic architecture for each region was constructed by the indicator kriging; then, the three hydrostratigraphic architectures were put together as shown in Fig. 6b. The presence of a large number of small facies thickness indicates a fluvial deposition environment, involving a wide mixture of grain sizes (e.g., from gravel to clay), a broad range of grain-size sorting (e.g., from poorly sorted to nearly homogeneous facies), and a wide-ranging interconnectivity between lithostratigraphic facies (e.g., at scales ranging from tens of meters to centimeters; Bowling et al. 2005). Figure 6b reveals the complexity of the Baton Rouge aquifer system, such as unconformity of sand units, interbedded clays, isolated sands, coalescences, and pinch-outs.

Fig. 6
figure 6

Construction of hydrostratigraphic architecture from well logs: a distribution of sand and clay segments in boreholes as the result of well log interpretation, b exponential indicator variogram model. Black dots are experimental indicator variograms, and c hydrostratigraphic architecture. The vertical exaggeration is 10

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Moreover, facies displacement at the faults and hydraulic connection across the faults can also be identified. It was found for deeper sands that the vertical offset of the Baton Rouge fault is about 80 m at the “1,200-foot” sand and about 100 m at the “2,000-foot” sand. The vertical offset of the Denham Springs-Scotlandville fault at the “1,200-foot” sand is about 35 m and at the “2,000-foot” sand is about 70 m (Elshall et al. 2013). The offsets increase with depth. Figure 7 presents the architectures of the Baton Rouge fault and the Denham Springs-Scotlandville fault. White areas show potential hydraulic connections formed by juxtaposition of sand units at the faults. The figure indicates complex sand deposition and erosion through the fluvial process and the faulting process.

Fig. 7
figure 7

The architecture of a the Denham Springs-Scotlandville fault and b the Baton Rouge fault. Red and cyan areas indicate clay facies at the north and the south of each fault, respectively. White areas are potential hydraulic connections. The faults lines in the model domain are shown in Fig. 5

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Model grid

It is obvious that a highly complex 3D MODFLOW grid is necessary in order to represent the Baton Rouge aquifer system. Using the proposed method in section 2.3, a grid of 162 model layers was constructed by upscaling the hydrostratigraphy. There are 808,078 active computational cells shown in Fig. 8. The model grid accurately matches the hydrostratigraphic architecture (Fig. 6a) and preserves layer continuity. Each cell is 200 × 200 m with the cell thickness ranging from 3.05 to 30.5 m. The average thickness of the layers is 5.2 m. With this grid generation technique, the pumping wells are correctly positioned in their corresponding sands. Updating the model grid is straightforward when new well logs become available.

Fig. 8
figure 8

Conceptual groundwater model for the Baton Rouge aquifer system, including 11 aquifers, two geological faults, and pumping and injection wells

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Groundwater model conceptualization

Using the generated MODFLOW grid, this section develops a groundwater model for the Baton Rouge area. The conceptual model structure is based on Fig. 8. The simulation period is from January 1, 1975 to December 31, 2014. The historical groundwater head data in the simulation period were collected from 42 US Geological Survey observation wells (Fig. 5). The historical groundwater head data were used to derive a groundwater head distribution for January 1, 1975 as the initial condition for the groundwater model. The historical groundwater head data were also used to derive groundwater head values at the model boundaries as the time-varied specified head boundary condition for the groundwater model. The monthly stress period and monthly time step were adopted owing to available monthly pumpage records (1975–2014) for the study area; therefore, there are 480 stress periods. Surficial recharge in the model domain was neglected due to a confining unit on the top of the MRAA and the “400-foot” sand.

The MODFLOW well package (MNW2; Konikow et al. 2009) was used to model pumping wells screened in a single sand or multiple sands. The time-variant specified head (CHD) package was used for the boundary condition, which is assigned to all boundary active cells. The Baton Rouge fault and the Denham Springs-Scotlandville fault are considered as horizontal flow barriers and their permeability is characterized by the hydraulic characteristic (HC; Hsieh and Freckleton 1993). The two faults were simulated using the horizontal flow barrier (HFB) package. The PCGN solver (Naff and Banta 2008) was used. The model parameters to be estimated are hydraulic conductivity, specific storage, and fault hydraulic characteristic.

Model calibration results and estimated model parameters

The parallel CMA-ES (Elshall et al. 2015) was implemented to minimize the root mean square error (RMSE) between the calculated and observed groundwater heads. Parallel computation was carried using a supercomputer at Louisiana State University, which has 382 compute nodes. Each compute node is equipped with two 10-core 2.8GHz Intel Ivy Bridge-EP processors. Hansen and Ostermeier (2001) and Hansen et al. (2003) recommended a good population size for CMA-ES to be tenfold of unknown variables. Since the number of model parameters to be estimated is 38, the study used a population size of 380. To maximize the efficiency of parallel computing, the number of processors is equal to the population size; therefore, 19 compute nodes were used to conduct model calibration using the parallel CMA-ES.

Given that the algorithm parallelization time is less than 1 s per iteration, the speedup of the parallel CMA-ES is roughly equal to population size. Running the Baton Rouge groundwater model takes about 18 h and using 380 processors for model calibration takes 75 days with 100 iterations. It would need 78 years without parallel computing. The parallel CMA-ES significantly reduced calibration time.

The groundwater model was calibrated using 10,393 transient groundwater heads from 42 US Geological Survey observation wells (Fig. 5) from January 1, 1975 to December 31, 2014. Model calibration found a good match between observed and calculated groundwater heads shown in Fig. 9. The RMSE is 3.76 m. The estimated parameter values of hydraulic conductivity, specific storage and fault hydraulic characteristic are given in Table 1. The calibration result indicates that the Holocene aquifer (MRAA) to the Pliocene aquifer (“1,700-foot” sand) have relatively lower hydraulic conductivity than that of Miocene aquifers (“2,000-foot” sand to “2,800-foot” sand). Specific storage is between 10−4 and 10−5 m−1 for all sands. Hydraulic characteristic for the faults varies across seven orders of magnitude. The calibration result indicates very low permeability for the Denham Springs-Scotlandville fault at the “2,000-foot” sand, which implies very limited freshwater flow across the fault. The calibration result also indicates very low hydraulic characteristic for the Baton Rouge fault at the “600-foot” sand, which limits groundwater flowing from south of the fault. In general, the calibration result supports the previous study (Elshall et al. 2015) that the Baton Rouge fault and the Denham Springs-Scotlandville fault are low-permeability leaky faults that restrict horizontal flow.

Fig. 9
figure 9

Scatter plot for observed groundwater level vs. calculated groundwater level

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Table 1 Estimated model parameters for the Baton Rouge aquifer system. K hydraulic conductivity, S S specific storage, DSS Denham Springs-Scotlandville, BR Baton Rouge; HC hydraulic characteristic
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Groundwater flow simulation and budget analyses

Simulated groundwater levels in December 31, 2014 are presented in Fig. 10 using the estimated parameters in Table 1. The low groundwater level in the aquifers between the faults is caused by the heavy pumping. Groundwater levels decrease in depth indicating more groundwater withdrawal from deep sands. The “2,000-foot” sand has the lowest groundwater level due to heavy pumping in the Industrial District. The downward head difference between the shallow sands and the deep sands warrants the purpose of the connector well EB-1293 (Fig. 8) that connects the “800-foot” sand and the “1,500-foot” sand in order to raise groundwater level in the “1,500-foot” sand (Dial and Cardwell 1999).

Fig. 10
figure 10

Simulated groundwater level in December 31, 2014 in 11 aquifers. Vertical lines are pumping wells. EB-150, EB-151, EB-733, and EB-1253 are screened in both the “2,000-foot” sand and the “2,400-foot” sand

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Moreover, Fig. 10 indicates potential flow from the “2,400-foot” sand to the “2,000-foot” sand due to upward head difference. The simulation result indicates that the “2,400-foot” sand may recharge the “2,000-foot” sand through four pumping wells (EB-150, EB-151, EB-733, and EB-1253) (Fig. 10), which were screened in both sands, when pumps are not running.

The distinct head differences across the faults shown in Fig. 10 are due to low permeability of the faults that restrict horizontal flow. Nevertheless, regarding the concern of saltwater intrusion (Morgan and Winner 1964; Whiteman 1979; Tomaszewski 1996; Lovelace 2007), the modeling result indicates the Baton Rouge fault to be a leaky fault that permits a certain amount of salty groundwater flow northward through the fault.

Analyses of 1975–2014 water budget shown in Fig. 11a–d estimate about 580,000 m3/day of groundwater flow annually into the aquifers between the two faults. About 70% of the total inflow comes from the east and west boundaries of the model domain, about 17% of the total inflow comes through the Denham Springs-Scotlandville fault, and about 13% of the total inflow comes through the Baton Rouge fault. About 581,300 m3/day of annual groundwater outflow is estimated, which is greater than the annual total inflow. The majority of the total outflow is groundwater pumping, accounting for about 59% of the total outflow. Groundwater outflow through the east and west boundaries of the model domain is estimated at about 39% of the total outflow. Groundwater outflow through the faults is very limited. In average, about 100 m3/day of groundwater annually flows southward through the Baton Rouge fault, and about 9,800 m3/day of groundwater annually flows northward through the Denham Springs-Scotlandville fault.

Fig. 11
figure 11

Simulated groundwater budget for the aquifers between the Baton Rouge fault and the Denham Springs-Scotlandville fault. Inflow and outflow (m3/day) through a east boundary of model domain, b west boundary of model domain, c Denham Springs-Scotlandville (DSS) fault, and d Baton Rouge (BR) fault. e Annual pumping rate (m3/day) and annual groundwater storage change (million m3) with respect to the beginning groundwater storage of 1975

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Figure 11e shows the estimated annual storage change with respect to the beginning storage of 1975. The storage change is strongly corresponding to groundwater pumping. Groundwater storage was generally increased before 1995 due to groundwater pumping decrease. Groundwater storage started to fall below the beginning storage of 1975 in response to groundwater pumping increase. The study estimates about 18.4 million m3 of groundwater storage loss in 2014.

Particle tracking

MODPATH (Pollock 2012) was used to track groundwater flow towards pumping wells and through the Baton Rouge fault. Four imaginary particles were placed at each pumping well and at each cell immediately south of the Baton Rouge fault. By using forward tracking and backward tracking, Fig. 12 shows the horizontal-plane projections of 40-year 3D flow paths from January 1, 1975 to December 31, 2014. The longer paths in the “2,000-foot” sand and the “2,400-foot” sand owing to heavy pumping and high hydraulic conductivity. The long outward flow paths from the pumping wells EB-150, EB-151, and EB-733 in Fig. 12d indicate strong recharge from the “2,400-foot” sand to the “2,000-foot” sand. While the model did show recharge from the “2,400-foot” sand to the “2,000-foot” sand in pumping well EB-1253, the outward flow path from EB-1253 is shortened by the strong pumping of its nearby wells.The flow paths across the Baton Rouge fault indicate fault leaky areas. The simulation result shows the possibility that groundwater south of the Baton Rouge fault can reach some pumping wells near the fault within 40 years.

Fig. 12
figure 12

Particle tracking simulation from January 1, 1975 to December 31, 2014 for a MRAA “400–600–800-foot” sands, b “1,000–1,200-foot” sands, c “1,500–1,700-foot” sands, d “2,000-foot” sand, and e “2,400-foot” sand. Squares are pumping and injection wells. Red lines and blue lines represent backward tracking and forward tracking of particle traces, respectively

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Conclusions

This study presents a general framework to develop groundwater models for fluvial-in-origin aquifer systems with a focus on grid generation and model calibration, which are two crucial modeling steps. The developed grid generation technique can handle a large number of well logs and preserve facies geometries of complex hydrostratigraphic architecture by using fine vertical discretization. This includes coalescences, pinch-outs, and narrow hydraulic connections through faults. It improves the shortcomings of the solid method and the pre-defined grid method and reduces model structure error for groundwater models.

The model development framework is successfully applied to the Baton Rouge aquifer system. The well log information and the constructed hydrostratigraphy confirm the complexity of the fluvial-in-origin aquifer system. Potential leaky areas in the geological faults are identified. To meet the current structural complexity, a large number of model layers is needed in order to model the 11 aquifers underneath Baton Rouge. The developed groundwater model is very time-consuming. Using parallel computing is necessary for automatic model calibration.

The calibration result indicates Miocene aquifers have higher hydraulic conductivity than Pliocene-Holocene aquifers. The Baton Rouge fault and the Denham Springs-Scotlandville fault are identified as low-permeability leaky faults. The estimated hydraulic characteristic of the faults varies over seven orders of magnitude. Specifically, the calibration result indicates very limit freshwater flow across the Denham Springs-Scotlandville fault for the “2,000-foot” sand and very limit freshwater flow across the Baton Rouge fault for the “600-foot” sand.

The model result indicates groundwater level decreasing with depth. The “2,000-foot” sand shows the lowest groundwater level owning to heavy industrial pumping. Potential groundwater flow from the “2,400-foot” sand to the “2,000-foot” sand is indicated by the upward head gradient between these two sands.

The water budget analyses for the sands between the geological faults indicate that 70% of the total inflow comes from the east and west boundaries of the model domain and about 30% of total inflow comes through the two faults. It is estimated that 13% of the total inflow comes from the Baton Rouge fault. Groundwater pumping is estimated about 59% of total outflow. The water budget analyses also indicate that groundwater storage is significantly depleted since 1995 due to excessive groundwater pumping. The particle tracking analysis reveals the location of the leaky areas of the Baton Rouge fault. The result also indicates that the groundwater south of the Baton Rouge fault can reach some pumping wells near the fault within 40 years.

The developed grid generation technique is highly flexible to include new well logs and to re-generate structured MODFLOW grids. Future additions in this study may include new wireline well logs and good-quality driller’s logs to update the constructed hydrostratigraphy. Including more geological and geophysical data may lead to a more complex groundwater model. A future study may develop a grid generation technique for unstructured MODFLOW grids (MODFLOW-USG) to handle complex hydrostratigraphy and reduce MODFLOW computation time.