Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes

A new method of local grid refinement for two-dimensional block-centered finite-difference meshes is presented in the context of steady-state groundwater-flow modeling. The method uses an iteration-based feedback with shared nodes to couple two separate grids. The new method is evaluated by comparison with results using a uniform fine mesh, a variably spaced mesh, and a traditional method of local grid refinement without a feedback.Results indicate: (1) The new method exhibits quadratic convergence for homogenous systems and convergence equivalent to uniform-grid refinement for heterogeneous systems. (2) Coupling the coarse grid with the refined grid in a numerically rigorous way allowed for improvement in the coarse-grid results. (3) For heterogeneous systems, commonly used linear interpolation of heads from the large model onto the boundary of the refined model produced heads that are inconsistent with the physics of the flow field. (4) The traditional method works well in situations where the better resolution of the locally refined grid has little influence on the overall flow-system dynamics, but if this is not true, lack of a feedback mechanism produced errors in head up to 3.6% and errors in cell-to-cell flows up to 25%.     

Fast transport simulation with an adaptive grid refinement

One of the main difficulties in transport modeling and calibration is the extraordinarily long computing times necessary for simulation runs. Improved execution time is a prerequisite for calibration in transport modeling. In this paper we investigate the problem of code acceleration using an adaptive grid refinement, neglecting subdomains, and devising a method by which the Courant condition can be ignored while maintaining accurate solutions. Grid refinement is based on dividing selected cells into regular subcells and including the balance equations of subcells in the equation system. The connection of coarse and refined cells satisfies the mass balance with an interpolation scheme that is implicitly included in the equation system. The refined subdomain can move with the average transport velocity of the subdomain. Very small time steps are required on a fine or a refined grid, because of the combined effect of the Courant and Peclet conditions. Therefore, we have developed a special upwind technique in small grid cells with high velocities (velocity suppression). We have neglected grid subdomains with very small concentration gradients (zero suppression). The resulting software, MODCALIF, is a three-dimensional, modularly constructed FORTRAN code. For convenience, the package names used by the well-known MODFLOW and MT3D computer programs are adopted, and the same input file structure and format is used, but the program presented here is separate and independent. Also, MODCALIF includes algorithms for variable density modeling and model calibration. The method is tested by comparison with an analytical solution, and illustrated by means of a two-dimensional theoretical example and three-dimensional simulations of the variable-density Cape Cod and SALTPOOL experiments. Crossing from fine to coarse grid produces numerical dispersion when the whole subdomain of interest is refined; however, we show that accurate solutions can be obtained using a fraction of the execution time required by uniformly fine-grid solutions.     

Three-dimensional local grid refinement for block-centered finite-difference groundwater models using iteratively coupled shared nodes: a new method of interpolation and analysis of errors

This paper describes work that extends to three dimensions the two-dimensional local-grid refinement method for block-centered finite-difference groundwater models of Mehl and Hill [Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes. Adv Water Resour 2002;25(5):497–511]. In this approach, the (parent) finite-difference grid is discretized more finely within a (child) sub-region. The grid refinement method sequentially solves each grid and uses specified flux (parent) and specified head (child) boundary conditions to couple the grids. Iteration achieves convergence between heads and fluxes of both grids. Of most concern is how to interpolate heads onto the boundary of the child grid such that the physics of the parent-grid flow is retained in three dimensions. We develop a new two-step, “cage-shell” interpolation method based on the solution of the flow equation on the boundary of the child between nodes shared with the parent grid. Error analysis using a test case indicates that the shared-node local grid refinement method with cage-shell boundary head interpolation is accurate and robust, and the resulting code is used to investigate three-dimensional local grid refinement of stream-aquifer interactions. Results reveal that (1) the parent and child grids interact to shift the true head and flux solution to a different solution where the heads and fluxes of both grids are in equilibrium, (2) the locally refined model provided a solution for both heads and fluxes in the region of the refinement that was more accurate than a model without refinement only if iterations are performed so that both heads and fluxes are in equilibrium, and (3) the accuracy of the coupling is limited by the parent-grid size—a coarse parent grid limits correct representation of the hydraulics in the feedback from the child grid.     

A Parallel PCG Solver for MODFLOW

In order to simulate large-scale ground water flow problems more efficiently with MODFLOW, the OpenMP programming paradigm was used to parallelize the preconditioned conjugate-gradient (PCG) solver with in this study. Incremental parallelization, the significant advantage supported by OpenMP on a shared-memory computer, made the solver transit to a parallel program smoothly one block of code at a time. The parallel PCG solver, suitable for both MODFLOW-2000 and MODFLOW-2005, is verified using an 8-processor computer. Both the impact of compilers and different model domain sizes were considered in the numerical experiments. Based on the timing results, execution times using the parallel PCG solver are typically about 1.40 to 5.31 times faster than those using the serial one. In addition, the simulation results are the exact same as the original PCG solver, because the majority of serial codes were not changed. It is worth noting that this parallelizing approach reduces cost in terms of software maintenance because only a single source PCG solver code needs to be maintained in the MODFLOW source tree.     

Spatial Periodic Boundary Condition for MODFLOW

Small‐scale hyporheic zone (HZ) models often use a spatial periodic boundary (SPB) pair to simulate an infinite repetition of bedforms. SPB's are common features of commercially available multiphysics modeling packages. MODFLOW's lack of this boundary type has precluded it from being effectively utilized in this area of HZ research. We present a method to implement the SPB in MODFLOW by development of the appropriate block‐centered finite‐difference expressions. The implementation is analogous to MODFLOW's general head boundary package. The difference is that the terms on the right hand side of the solution equations must be updated with each iteration. Consequently, models that implement the SPB converge best with solvers that perform both inner and outer iterations. The correct functioning of the SPB condition in MODFLOW is verified by two examples. This boundary condition allows users to build HZ‐bedform models in MODFLOW, facilitating further research using related codes such as MT3DMS and PHT3D.     

Comparison of sparse‐grid geometric and random sampling methods in nonlinear inverse solution uncertainty estimation

A new uncertainty estimation method, which we recently introduced in the literature, allows for the comprehensive search of model posterior space while maintaining a high degree of computational efficiency. The method starts with an optimal solution to an inverse problem, performs a parameter reduction step and then searches the resulting feasible model space using prior parameter bounds and sparse‐grid polynomial interpolation methods. After misfit rejection, the resulting model ensemble represents the equivalent model space and can be used to estimate inverse solution uncertainty. While parameter reduction introduces a posterior bias, it also allows for scaling this method to higher dimensional problems. The use of Smolyak sparse‐grid interpolation also dramatically increases sampling efficiency for large stochastic dimensions. Unlike Bayesian inference, which treats the posterior sampling problem as a random process, this geometric sampling method exploits the structure and smoothness in posterior distributions by solving a polynomial interpolation problem and then resampling from the resulting interpolant. The two questions we address in this paper are 1) whether our results are generally compatible with established Bayesian inference methods and 2) how does our method compare in terms of posterior sampling efficiency. We accomplish this by comparing our method for two electromagnetic problems from the literature with two commonly used Bayesian sampling schemes: Gibbs’ and Metropolis‐Hastings. While both the sparse‐grid and Bayesian samplers produce compatible results, in both examples, the sparse‐grid approach has a much higher sampling efficiency, requiring an order of magnitude fewer samples, suggesting that sparse‐grid methods can significantly improve the tractability of inference solutions for problems in high dimensions or with more costly forward physics.     

基于方差分量估计的局部似大地水准面精化拟合模型

局部似大地水准面精化中需要进行GPS水准与重力两种水准面的拟合,其中两种观测量精度的权比确定是关键问题,本文提出基于方差分量估计的方法解决这一问题.根据此思路,文中还利用某给定区域的实际资料,对常用的几种似大地水准面拟合方法——多项式拟合、多面函数拟合以及曲面样条插值的函数模型按不等权拟合似大地水准面进行试验.通过某3...     

Permeameter data verify new turbulence process for MODFLOW

A sample of Key Largo Limestone from southern Florida exhibited turbulent flow behavior along three orthogonal axes as reported in recently published permeameter experiments. The limestone sample was a cube measuring 0.2 m on edge. The published nonlinear relation between hydraulic gradient and discharge was simulated using the turbulent flow approximation applied in the Conduit Flow Process (CFP) for MODFLOW-2005 mode 2, CFPM2. The good agreement between the experimental data and the simulated results verifies the utility of the approach used to simulate the effects of turbulent flow on head distributions and flux in the CFPM2 module of MODFLOW-2005.     

Comparison of scaling methods for waveform inversion

Waveform inversion can lead to faint images for later times due to geometrical spreading. The proper scaling of the steepest-descent direction can enhance faint images in waveform inversion results. We compare the effects of different scaling techniques in waveform inversion algorithms using the steepest-descent method. For the scaling method we use the diagonal of the pseudo-Hessian matrix, which can be applied in two different ways. One is to scale the steepest-descent direction at each frequency independently. The other is to scale the steepest-descent direction summed over the entire frequency band. The first method equalizes the steepest-descent directions at different frequencies and minimizes the effects of the band-limited source spectrum in waveform inversion. In the second method, since the steepest-descent direction summed over the entire frequency band is divided by the diagonal of the pseudo-Hessian matrix summed over the entire frequency band, the band-limited property of the source wavelet spectrum still remains in the scaled steepest-descent directions. The two scaling methods were applied to both standard and logarithmic waveform inversion. For standard waveform inversion, the method that scales the steepest-descent direction at every frequency step gives better results than the second method. On the other hand, logarithmic waveform inversion is not sensitive to the scaling method, because taking the logarithm of wavefields automatically means that results for the steepest-descent direction at each frequency are commensurate with each other. If once the steepest-descent directions are equalized by taking the logarithm of wavefields in logarithmic waveform inversion, the additional equalizing effects by the scaling method are not as great as in conventional waveform inversion.     

FloPy Workflows for Creating Structured and Unstructured MODFLOW Models

FloPy is a Python package for creating, running, and post-processing MODFLOW-based groundwater flow and transport models. FloPy functionality has expanded to support the latest version of MODFLOW (MODFLOW 6) including support for unstructured grids. FloPy can simplify the process required to download MODFLOW-based and other executables for Linux, MacOS, and Windows operating systems. Expanded FloPy capabilities include (1) full support for structured and unstructured spatial discretizations; (2) geoprocessing of spatial features and raster data to develop model input for supported discretization types; (3) the addition of functionality to provide direct access to simulated output data; (4) extension of plotting capabilities to unstructured MODFLOW 6 discretization types; and (5) the ability to export model data to shapefiles, NetCDF, and VTK formats for processing, analysis, and visualization by other software products. Examples of using expanded FloPy capabilities are presented for a hypothetical watershed. An unstructured groundwater flow and transport model, with several advanced stress packages, is presented to demonstrate how FloPy can be used to develop complicated unstructured model datasets from original source data (shapefiles and rasters), post-process model results, and plot simulated results.     

Comparison of two methods for calculating tidal loads

The spherical harmonic expansion up to order n = 1440 of the oceanic tidal heights is derived for the waves of the CSR3.0 model. On the basis of the load Love numbers for the PREM model, the oceanic load effect in gravity is calculated. It is shown that the results provided by the spherical expansion technique and the method utilizing the Green’s function satisfactorily agree even at the order of the expansion n = 720. The obtained model predictions are compared to the oceanic load effect measured by the Global Geodynamics Project (GGP) network of superconducting gravimeters.     

Comparison of six different methods for calculating traveltimes

We study six different methods for the calculation of seismic traveltimes. All methods yield traveltimes at all points of a regular grid.
The methods examined comprise three different variants of finite-difference (FD) eikonal solvers, the graph method, wavefront construction and a combined FD and Runge–Kutta method.
The main points of investigation are computational time, accuracy and memory requirements. We took measures to obtain a high level of both general validity and clear understanding of the results. We used a profiling program to be able to measure the time that the actual core algorithm needs, thus avoiding any overhead of highly system-dependent in-/output operations.
The comparison shows that no single method is the most appropriate but that the choice depends on the task to be fulfilled. The FD eikonal solver that uses expanding squares proves to be best suited for models which are not too complicated because it offers the best compromise between speed and accuracy, whereas wavefront construction should be applied to complex media because of its superior reliability which then justifies the much higher computational times.     

Coupling local discontinuous and continuous galerkin methods for flow problems

In this paper, we discuss the local discontinuous Galerkin (LDG) method applied to elliptic flow problems and give details on its implementation, focusing specifically on the case of piecewise linear approximating functions. The LDG method is one a family of discontinuous Galerkin (DG) methods proposed for diffusion models. These DG methods allow for very general hp finite element meshes, and produce locally conservative fluxes which can be used in coupling flow with transport. The drawback to DG methods, when compared to their continuous counterparts, is the number of degrees of freedom required to compute the solution. This motivates a coupled approach, discussed herein, where the solution is allowed to be continuous or discontinuous on a node-by-node basis. This coupled approximation is locally conservative in regions where the numerical solution is discontinuous. Numerical results for fully discontinuous, continuous and coupled discontinuous/continuous solutions are given, where we compare solution accuracy, matrix condition numbers and mass balance errors for the various approaches.     

Efficient Storage of Large MODFLOW Models

We present a methodology for storing the bulkier portions of a set of MODFLOW input and output files in a compressed binary format using the HDF5 library. This approach results in compression ratios of up to 99% with no significant time penalty. The highly compressed format is particularly beneficial when dealing with large regional models or Monte Carlo simulations. The strategy is focused on the list‐ and array‐based portions of the input files including the cell property and recharge arrays, and is compatible with models containing parameters, including pilot points. The utilities are based on a modified version of the MODFLOW code and are, therefore, compatible with any standard MODFLOW simulation. We present used cases and instructions on how to use the utilities.     

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.     

A MODFLOW Infiltration Device Package for Simulating Storm Water Infiltration

This article describes a MODFLOW Infiltration Device (INFD) Package that can simulate infiltration devices and their two‐way interaction with groundwater. The INFD Package relies on a water balance including inflow of storm water, leakage‐like seepage through the device faces, overflow, and change in storage. The water balance for the device can be simulated in multiple INFD time steps within a single MODFLOW time step, and infiltration from the device can be routed through the unsaturated zone to the groundwater table. A benchmark test shows that the INFD Package's analytical solution for stage computes exact results for transient behavior. To achieve similar accuracy by the numerical solution of the MODFLOW Surface‐Water Routing (SWR1) Process requires many small time steps. Furthermore, the INFD Package includes an improved representation of flow through the INFD sides that results in lower infiltration rates than simulated by SWR1. The INFD Package is also demonstrated in a transient simulation of a hypothetical catchment where two devices interact differently with groundwater. This simulation demonstrates that device and groundwater interaction depends on the thickness of the unsaturated zone because a shallow groundwater table (a likely result from storm water infiltration itself) may occupy retention volume, whereas a thick unsaturated zone may cause a phase shift and a change of amplitude in groundwater table response to a change of infiltration. We thus find that the INFD Package accommodates the simulation of infiltration devices and groundwater in an integrated manner on small as well as large spatial and temporal scales.