A New Package in MODFLOW to Simulate Unconfined Groundwater Flow in Sloping Aquifers

The nonhorizontal‐model‐layer (NHML) grid system is more accurate than the horizontal‐model‐layer grid system to describe groundwater flow in an unconfined sloping aquifer on the basis of MODFLOW‐2000. However, the finite‐difference scheme of NHML was based on the Dupuit‐Forchheimer assumption that the streamlines were horizontal, which was acceptable for slope less than 0.10. In this study, we presented a new finite‐difference scheme of NHML based on the Boussinesq assumption and developed a new package SLOPE which was incorporated into MODFLOW‐2000 to become the MODFLOW‐SP model. The accuracy of MODFLOW‐SP was tested against solution of Mac Cormack (1969). The differences between the solutions of MODFLOW‐2000 and MODFLOW‐SP were nearly negligible when the slope was less than 0.27, and they were noticeable during the transient flow stage and vanished in steady state when the slope increased above 0.27. We established a model considering the vertical flow using COMSOL Multiphysics to test the robustness of constrains used in MODFLOW‐SP. The results showed that streamlines quickly became parallel with the aquifer base except in the narrow regions near the boundaries when the initial flow was not parallel to the aquifer base. MODFLOW‐SP can be used to predict the hydraulic head of an unconfined aquifer along the profile perpendicular to the aquifer base when the slope was smaller than 0.50. The errors associated with constrains used in MODFLOW‐SP were small but noticeable when the slope increased to 0.75, and became significant for the slope of 1.0.     

Nonlinear Flow in Karst Formations

The variation of effective hydraulic conductivity as a function of specific discharge in several 0.2-m and 0.3-m cubes of Key Largo Limestone was investigated. The experimental results closely match the Forchheimer equation. Defining the pore-size length scale in terms of Forchheimer parameters, it is demonstrated that significant deviations from Darcian flow will occur when the Reynolds number exceeds 0.11. A particular threshold model previously proposed for use in karstic formations does not show strong agreement with the data near the onset of nonlinear flow.     

Implementation of Solute Transport in the Vadose Zone into the “HYDRUS Package for MODFLOW”

The “HYDRUS package for MODFLOW” is an existing MODFLOW package that allows MODFLOW to simultaneously evaluate transient water flow in both unsaturated and saturated zones. The package is based on incorporating parts of the HYDRUS-1D model (to simulate unsaturated water flow in the vadose zone) into MODFLOW (to simulate saturated groundwater flow). The coupled model is effective in addressing spatially variable saturated-unsaturated hydrological processes at the regional scale. However, one of the major limitations of this coupled model is that it does not have the capability to simulate solute transport along with water flow and therefore, the model cannot be employed for evaluating groundwater contamination. In this work, a modified unsaturated flow and transport package (modified HYDRUS package for MODFLOW and MT3DMS) has been developed and linked to the three-dimensional (3D) groundwater flow model MODFLOW and the 3D groundwater solute transport model MT3DMS. The new package can simulate, in addition to water flow in the vadose zone, also solute transport involving many biogeochemical processes and reactions, including first-order degradation, volatilization, linear or nonlinear sorption, one-site kinetic sorption, two-site sorption, and two-kinetic sites sorption. Due to complex interactions at the groundwater table, certain modifications of the pressure head (compared to the original coupling) and solute concentration profiles were incorporated into the modified HYDRUS package. The performance of the newly developed model is evaluated using HYDRUS (2D/3D), and the results indicate that the new model is effective in simulating the movement of water and contaminants in the saturated-unsaturated flow domains.     

Simulation of Lake-Groundwater Interaction under Steady-State Flow

MODFLOW is one of the most popular groundwater simulation tools available; however, the development of lake modules that can be coupled with MODFLOW is lacking apart from the LAK3 package. This study proposes a new approach for simulating lake - groundwater interaction under steady-state flow, referred to as the sloping lakebed method (SLM). In this new approach, discretization of the lakebed in the vertical direction is independent of the spatial discretization of the aquifer system, which can potentially solve the problem that the lake and groundwater are usually simulated at different scales. The lakebed is generalized by a slant at the bottom of each lake grid cell, which can be classified as fully submerged, dry, and partly submerged. The SLM method accounts for all lake sources and sinks, establishing a governing equation that can be solved using Newton's method. A benchmarking case study was conducted using a modified model setup in the LAK3 user manual. It was found that when there is a sufficient number of layers at the top of the groundwater model, SLM simulates an almost identical groundwater head as the LAK3-based model; when the number of layers decreases, SLM is unaffected while LAK3 may be at a risk of giving unrealistic results. Additionally, the SLM can reflect the relationship between the simulated lake surface area and lake water depth more accurately. Therefore, the SLM method is a promising alternative to the LAK3 package when simulating lake - groundwater interaction.     

Simulating ground water-lake interactions: approaches and insights

Approaches for modeling lake-ground water interactions have evolved significantly from early simulations that used fixed lake stages specified as constant head to sophisticated LAK packages for MODFLOW. Although model input can be complex, the LAK package capabilities and output are superior to methods that rely on a fixed lake stage and compare well to other simple methods where lake stage can be calculated. Regardless of the approach, guidelines presented here for model grid size, location of three-dimensional flow, and extent of vertical capture can facilitate the construction of appropriately detailed models that simulate important lake-ground water interactions without adding unnecessary complexity. In addition to MODFLOW approaches, lake simulation has been formulated in terms of analytic elements. The analytic element lake package had acceptable agreement with a published LAKI problem, even though there were differences in the total lake conductance and number of layers used in the two models. The grid size used in the original LAKI problem, however, violated a grid size guideline presented in this paper. Grid sensitivity analyses demonstrated that an appreciable discrepancy in the distribution of stream and lake flux was related to the large grid size used in the original LAKI problem. This artifact is expected regardless of MODFLOW LAK package used. When the grid size was reduced, a finite-difference formulation approached the analytic element results. These insights and guidelines can help ensure that the proper lake simulation tool is being selected and applied.     

Approaches to the simulation of unconfined flow and perched groundwater flow in MODFLOW

Various approaches have been proposed to manage the nonlinearities associated with the unconfined flow equation and to simulate perched groundwater conditions using the MODFLOW family of codes. The approaches comprise a variety of numerical techniques to prevent dry cells from becoming inactive and to achieve a stable solution focused on formulations of the unconfined, partially-saturated, groundwater flow equation. Keeping dry cells active avoids a discontinuous head solution which in turn improves the effectiveness of parameter estimation software that relies on continuous derivatives. Most approaches implement an upstream weighting of intercell conductance and Newton-Raphson linearization to obtain robust convergence. In this study, several published approaches were implemented in a stepwise manner into MODFLOW for comparative analysis. First, a comparative analysis of the methods is presented using synthetic examples that create convergence issues or difficulty in handling perched conditions with the more common dry-cell simulation capabilities of MODFLOW. Next, a field-scale three-dimensional simulation is presented to examine the stability and performance of the discussed approaches in larger, practical, simulation settings.     

A Generic Closed-Loop Contaminant Treatment System Package for MODFLOW 6

We present a contaminant treatment system (CTS) package for MODFLOW 6 that facilitates the simulation of pump-and-treat systems for groundwater remediation. Using the “nonintrusive” MODFLOW 6 application programming interface (API) capability, the CTS package can balance flows between extraction and injection wells within the outer flow solution loop and applies blended concentration/mass treatment efficiency within the outer transport solution loop. The former can be important when the requested extraction rate cannot be satisfied by the current simulated groundwater system conditions, while the latter can be important for simulating incomplete/imperfect treatment schemes. Furthermore, the CTS package allows users to temporally vary all aspects of a simulated CTS system, including the configuration and location of injection and extraction wells, and the CTS efficiency. This flexibility combined with the API-based implementation provide a generic and general CTS package that can be applied across the wide range of MODFLOW 6 simulation options and that evolves in step with MODFLOW 6 code modifications and advancements without needing to update the CTS package itself.     

Simulating Piecewise-Linear Surface Water and Ground Water Interactions with MODFLOW

The standard MODFLOW packages offer limited capabilities to model piecewise-linear boundary conditions to describe ground water–surface water interaction. Specifically, MODFLOW is incapable of representing a Cauchy-type boundary with different resistances for discharge or recharge conditions. Such a more sophisticated Cauchy boundary condition is needed to properly represent surface waters alternatively losing water through the bottom (high resistance) or gaining water mostly near the water surface (low resistance). One solution would be to create a new package for MODFLOW to accomplish this. However, it is also possible to combine multiple instances of standard packages in a single cell to the same effect. In this specific example, the general head boundary package is combined with the drain package to arrive at the desired piecewise-linear behavior. In doing so, the standard USGS MODFLOW version can be used without any modifications at the expense of a minor increase in preprocessing and postprocessing and computational effort. The extra preprocessing for creating the input and extra postprocessing to determine the water balance in terms of the physical entities from the MODFLOW cell fluxes per package can be taken care of by a user interface.     

Multiple Well Systems with Non‐Darcy Flow

Optimization of groundwater and other subsurface resources requires analysis of multiple‐well systems. The usual modeling approach is to apply a linear flow equation (e.g., Darcy's law in confined aquifers). In such conditions, the composite response of a system of wells can be determined by summating responses of the individual wells (the principle of superposition). However, if the flow velocity increases, the nonlinear losses become important in the near‐well region and the principle of superposition is no longer valid. This article presents an alternative method for applying analytical solutions of non‐Darcy flow for a single‐ to multiple‐well systems. The method focuses on the response of the central injection well located in an array of equally spaced wells, as it is the well that exhibits the highest pressure change within the system. This critical well can be represented as a single well situated in the center of a closed square domain, the width of which is equal to the well spacing. It is hypothesized that a single well situated in a circular region of the equivalent plan area adequately represents such a system. A test case is presented and compared with a finite‐difference solution for the original problem, assuming that the flow is governed by the nonlinear Forchheimer equation.     

SEEPAGE, a new MODFLOW DRAIN package

The prediction of the location of ground water discharge areas is a key aspect for the protection and (re)development of ground water-dependent wetlands. Ground water discharge areas can be simulated with MODFLOW using the DRAIN package by setting the drain level equal to the topography, while the conductance is mostly set to an arbitrary high value. However, conceptual and practical problems arise in the calculation of the ground water discharge by the DRAIN package as calculated water tables above the land surface, difficult parameterization of the conductance, and large water balance errors. To overcome these problems, a new SEEPAGE package for MODFLOW is proposed. The basic idea of this package is an adaptable constant head cell. It has a variable head, unless the ground water rises above the seepage level, in which case it has a constant head cell. The estimation of the ground water discharge location along a homogeneous, isotropic, linear sloping profile is used to verify the model and to compare it to the DRAIN solution. In an application to three basins in Belgium, it is shown that the SEEPAGE package can be used in combination with the DRAIN package in situations where an upper boundary for a free water table and additional resistance for drainage is required. It is clearly demonstrated that the identification and delineation of regional ground water discharge areas is more accurate using the SEEPAGE package.     

MODFLOW‐Based Coupled Surface Water Routing and Groundwater‐Flow Simulation

In this paper, we present a flexible approach for simulating one‐ and two‐dimensional routing of surface water using a numerical surface water routing (SWR) code implicitly coupled to the groundwater‐flow process in MODFLOW. Surface water routing in SWR can be simulated using a diffusive‐wave approximation of the Saint‐Venant equations and/or a simplified level‐pool approach. SWR can account for surface water flow controlled by backwater conditions caused by small water‐surface gradients or surface water control structures. A number of typical surface water control structures, such as culverts, weirs, and gates, can be represented, and it is possible to implement operational rules to manage surface water stages and streamflow. The nonlinear system of surface water flow equations formulated in SWR is solved by using Newton methods and direct or iterative solvers. SWR was tested by simulating the (1) Lal axisymmetric overland flow, (2) V‐catchment, and (3) modified Pinder‐Sauer problems. Simulated results for these problems compare well with other published results and indicate that SWR provides accurate results for surface water‐only and coupled surface water/groundwater problems. Results for an application of SWR and MODFLOW to the Snapper Creek area of Miami‐Dade County, Florida, USA are also presented and demonstrate the value of coupled surface water and groundwater simulation in managed, low‐relief coastal settings.     

Hydrogeologic unit flow characterization using transition probability geostatistics

This paper describes a technique for applying the transition probability geostatistics method for stochastic simulation to a MODFLOW model. Transition probability geostatistics has some advantages over traditional indicator kriging methods including a simpler and more intuitive framework for interpreting geologic relationships and the ability to simulate juxtapositional tendencies such as fining upward sequences. The indicator arrays generated by the transition probability simulation are converted to layer elevation and thickness arrays for use with the new Hydrogeologic Unit Flow package in MODFLOW 2000. This makes it possible to preserve complex heterogeneity while using reasonably sized grids and/or grids with nonuniform cell thicknesses.     

Nonlinear model reduction of unconfined groundwater flow using POD and DEIM

Nonlinear groundwater flow models have the propensity to be overly complex leading to burdensome computational demands. Reduced modeling techniques are used to develop an approximation of the original model that has smaller dimensionality and faster run times. The reduced model proposed is a combination of proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). Solutions of the full model (snapshots) are collected to represent the physical dynamics of the system and Galerkin projection allows the formulation of a reduced model that lies in a subspace of the full model. Interpolation points are added through DEIM to eliminate the reduced model's dependence on the dimension of the full model. POD is shown to effectively reduce the dimension of the full model and DEIM is shown to speed up the solution by further reducing the dimension of the nonlinear calculations. To show the concept can work for unconfined groundwater flow model, with added nonlinear forcings, one-dimensional and two-dimensional test cases are constructed in MODFLOW-OWHM. POD and DEIM are added to MODFLOW as a modular package. Comparing the POD and the POD-DEIM reduced models, the experimental results indicate similar reduction in dimension size with additional computation speed up for the added interpolation. The hyper-reduction method presented is effective for models that have fine discretization in space and/or time as well as nonlinearities with respect to the state variable. The dual reduction approach ensures that, once constructed, the reduced model can be solved in an equation system that depends only on reduced dimensions.     

Grid cell distortion and MODFLOW's integrated finite-difference numerical solution

The ground water flow model MODFLOW inherently implements a nongeneralized integrated finite-difference (IFD) numerical scheme. The IFD numerical scheme allows for construction of finite-difference model grids with curvilinear (piecewise linear) rows. The resulting grid comprises model cells in the shape of trapezoids and is distorted in comparison to a traditional MODFLOW finite-difference grid. A version of MODFLOW-88 (herein referred to as MODFLOW IFD) with the code adapted to make the one-dimensional DELR and DELC arrays two dimensional, so that equivalent conductance between distorted grid cells can be calculated, is described. MODFLOW IFD is used to inspect the sensitivity of the numerical head and velocity solutions to the level of distortion in trapezoidal grid cells within a converging radial flow domain. A test problem designed for the analysis implements a grid oriented such that flow is parallel to columns with converging widths. The sensitivity analysis demonstrates MODFLOW IFD's capacity to numerically derive a head solution and resulting intercell volumetric flow when the internal calculation of equivalent conductance accounts for the distortion of the grid cells. The sensitivity of the velocity solution to grid cell distortion indicates criteria for distorted grid design. In the radial flow test problem described, the numerical head solution is not sensitive to grid cell distortion. The accuracy of the velocity solution is sensitive to cell distortion with error <1% if the angle between the nonparallel sides of trapezoidal cells is <12.5 degrees. The error of the velocity solution is related to the degree to which the spatial discretization of a curve is approximated with piecewise linear segments. Curvilinear finite-difference grid construction adds versatility to spatial discretization of the flow domain. MODFLOW-88's inherent IFD numerical scheme and the test problem results imply that more recent versions of MODFLOW 2000, with minor modifications, have the potential to make use of a curvilinear grid.     

Evaluation of Bias Associated with Capture Maps Derived from Nonlinear Groundwater Flow Models

The impact of groundwater withdrawal on surface water is a concern of water users and water managers, particularly in the arid western United States. Capture maps are useful tools to spatially assess the impact of groundwater pumping on water sources (e.g., streamflow depletion) and are being used more frequently for conjunctive management of surface water and groundwater. Capture maps have been derived using linear groundwater flow models and rely on the principle of superposition to demonstrate the effects of pumping in various locations on resources of interest. However, nonlinear models are often necessary to simulate head‐dependent boundary conditions and unconfined aquifers. Capture maps developed using nonlinear models with the principle of superposition may over‐ or underestimate capture magnitude and spatial extent. This paper presents new methods for generating capture difference maps, which assess spatial effects of model nonlinearity on capture fraction sensitivity to pumping rate, and for calculating the bias associated with capture maps. The sensitivity of capture map bias to selected parameters related to model design and conceptualization for the arid western United States is explored. This study finds that the simulation of stream continuity, pumping rates, stream incision, well proximity to capture sources, aquifer hydraulic conductivity, and groundwater evapotranspiration extinction depth substantially affect capture map bias. Capture difference maps demonstrate that regions with large capture fraction differences are indicative of greater potential capture map bias. Understanding both spatial and temporal bias in capture maps derived from nonlinear groundwater flow models improves their utility and defensibility as conjunctive‐use management tools.     

Accelerating Groundwater Flow Simulation in MODFLOW Using JASMIN‐Based Parallel Computing

To accelerate the groundwater flow simulation process, this paper reports our work on developing an efficient parallel simulator through rebuilding the well‐known software MODFLOW on JASMIN (J Adaptive Structured Meshes applications Infrastructure). The rebuilding process is achieved by designing patch‐based data structure and parallel algorithms as well as adding slight modifications to the compute flow and subroutines in MODFLOW. Both the memory requirements and computing efforts are distributed among all processors; and to reduce communication cost, data transfers are batched and conveniently handled by adding ghost nodes to each patch. To further improve performance, constant‐head/inactive cells are tagged and neglected during the linear solving process and an efficient load balancing strategy is presented. The accuracy and efficiency are demonstrated through modeling three scenarios: The first application is a field flow problem located at Yanming Lake in China to help design reasonable quantity of groundwater exploitation. Desirable numerical accuracy and significant performance enhancement are obtained. Typically, the tagged program with load balancing strategy running on 40 cores is six times faster than the fastest MICCG‐based MODFLOW program. The second test is simulating flow in a highly heterogeneous aquifer. The AMG‐based JASMIN program running on 40 cores is nine times faster than the GMG‐based MODFLOW program. The third test is a simplified transient flow problem with the order of tens of millions of cells to examine the scalability. Compared to 32 cores, parallel efficiency of 77 and 68% are obtained on 512 and 1024 cores, respectively, which indicates impressive scalability.     

Flow‐Through Stream Modeling with MODFLOW and MT3D: Certainties and Limitations

This paper aims to assess MODFLOW and MT3D capabilities for simulating the spread of contaminants from a river exhibiting an unusual relationship with an alluvial aquifer, with the groundwater head higher than the river head on one side and lower on the other (flow‐through stream). A series of simulation tests is conducted using a simple hypothetical model so as to characterize and quantify these limitations. Simulation results show that the expected contaminant spread could be achieved with a specific configuration composed of two sets of parameters: (1) modeled object parameters (hydraulic groundwater gradient, hydraulic conductivity values of aquifer and streambed), and (2) modeling parameters (vertical discretization of aquifer, horizontal refinement of stream modeled with River [RIV] package). The influence of these various parameters on simulation results is investigated, and potential complications and errors are identified. Contaminant spread from stream to aquifer is not always reproduced by MT3D due to the RIV package's inability to simulate lateral exchange fluxes between stream and aquifer. This paper identifies the need for a MODFLOW streamflow package allowing lateral stream‐aquifer interactions and streamflow routine calculations. Such developments could be of particular interest for modeling contaminated flow‐through streams.     

Representing Pump-Capacity Relations in Groundwater Simulation Models

The yield (or discharge) of constant-speed pumps varies with the total dynamic head (or lift) against which the pump is discharging. The variation in yield over the operating range of the pump may be substantial. In groundwater simulations that are used for management evaluations or other purposes, where predictive accuracy depends on the reliability of future discharge estimates, model reliability may be enhanced by including the effects of head-capacity (or pump-capacity) relations on the discharge from the well. A relatively simple algorithm has been incorporated into the widely used MODFLOW groundwater flow model that allows a model user to specify head-capacity curves. The algorithm causes the model to automatically adjust the pumping rate each time step to account for the effect of drawdown in the cell and changing lift, and will shut the pump off if lift exceeds a critical value. The algorithm is available as part of a new multinode well package (MNW2) for MODFLOW.     

Adjoint simulation of stream depletion due to aquifer pumping

If an aquifer is hydraulically connected to an adjacent stream, a pumping well operating in the aquifer will draw some water from aquifer storage and some water from the stream, causing stream depletion. Several analytical, semi-analytical, and numerical approaches have been developed to estimate stream depletion due to pumping. These approaches are effective if the well location is known. If a new well is to be installed, it may be desirable to install the well at a location where stream depletion is minimal. If several possible locations are considered for the location of a new well, stream depletion would have to be estimated for all possible well locations, which can be computationally inefficient. The adjoint approach for estimating stream depletion is a more efficient alternative because with one simulation of the adjoint model, stream depletion can be estimated for pumping at a well at any location. We derive the adjoint equations for a coupled system with a confined aquifer, an overlying unconfined aquifer, and a river that is hydraulically connected to the unconfined aquifer. We assume that the stage in the river is known, and is independent of the stream depletion, consistent with the assumptions of the MODFLOW river package. We describe how the adjoint equations can be solved using MODFLOW. In an illustrative example, we show that for this scenario, the adjoint approach is as accurate as standard forward numerical simulation methods, and requires substantially less computational effort.     

Ground water flow in a desert basin: challenges of simulating transport of dissolved chromium

A large chromium plume that evolved from chromium releases in a valley near the Mojave River was studied to understand the processes controlling fate and migration of chromium in ground water and used as a tracer to study the dynamics of a basin and range ground water system. The valley that was studied is naturally arid with high evapotranspiration such that essentially no precipitation infiltrates to the water table. The dominant natural hydrogeologic processes are recharge to the ground water system from the Mojave River during the infrequent episodes when there is flow in the river, and ground water flow toward a playa lake where the ground water evaporates. Agricultural pumping in the valley from the mid-1930s to the 1970s significantly altered ground water flow conditions by decreasing water levels in the valley by more than 20 m. This pumping declined significantly as a result of dewatering of the aquifer, and water levels have since recovered modestly. The ground water system was modeled using MODFLOW, and chromium transport was simulated using MT3D. Several innovative modifications were made to these modeling programs to simulate important processes in this ground water system. Modifications to MODFLOW include developing a new well package that estimates pumping rates from irrigation wells at each time step based on available drawdown. MT3D was modified to account for mass trapped above the water table when the water table declines beneath nonirrigated areas and to redistribute mass to the system when water levels rise.