Travel Time Distributions Derived from Particle Tracking in Models Containing Weak Sinks

A travel time distribution based on a particle-tracking analysis in a ground water model containing weak sinks is often uncertain because whether a particle is discharged or allowed to pass through a weak sink is unresolved by particle-tracking theory. We present a probability-based method to derive an objective travel time distribution in models containing weak sinks. The method discharges a fraction of the particle at the weak sink and allows the remaining fraction to pass through the weak sink. The weight of the discharged fraction depends on the ratio of the sink flux to the influx into the weak sink cell. We tested this approach on a coarse (100 × 100 m) and a fine (25 × 25 m) horizontal resolution regional scale ground water model (34.5 × 24 km). We compared the travel time distributions in a small subcatchment derived from particle-tracking analysis with one derived from a transport model. We found that the particle-tracking analysis with the coarse model underestimated the travel time distribution of the catchment compared to the transport solution or a particle-tracking analysis with the fine model. The underestimation of travel times with the coarse model was a result of a large area covered by sink cells in this model and the more accurate flow patterns simulated by the fine model. The probability-based method presented here compares favorably with a solute transport solution and provides an accurate travel time distribution when used with a fine-resolution ground water model.     

On Modeling Weak Sinks in MODPATH

Regional groundwater flow systems often contain both strong sinks and weak sinks. A strong sink extracts water from the entire aquifer depth, while a weak sink lets some water pass underneath or over the actual sink. The numerical groundwater flow model MODFLOW may allow a sink cell to act as a strong or weak sink, hence extracting all water that enters the cell or allowing some of that water to pass. A physical strong sink can be modeled by either a strong sink cell or a weak sink cell, with the latter generally occurring in low‐resolution models. Likewise, a physical weak sink may also be represented by either type of sink cell. The representation of weak sinks in the particle tracing code MODPATH is more equivocal than in MODFLOW. With the appropriate parameterization of MODPATH, particle traces and their associated travel times to weak sink streams can be modeled with adequate accuracy, even in single layer models. Weak sink well cells, on the other hand, require special measures as proposed in the literature to generate correct particle traces and individual travel times and hence capture zones. We found that the transit time distributions for well water generally do not require special measures provided aquifer properties are locally homogeneous and the well draws water from the entire aquifer depth, an important observation for determining the response of a well to non‐point contaminant inputs.     

How processing digital elevation models can affect simulated water budgets

For regional models, the shallow water table surface is often used as a source/sink boundary condition, as model grid scale precludes simulation of the water table aquifer. This approach is appropriate when the water table surface is relatively stationary. Since water table surface maps are not readily available, the elevation of the water table used in model cells is estimated via a two-step process. First, a regression equation is developed using existing land and water table elevations from wells in the area. This equation is then used to predict the water table surface for each model cell using land surface elevation available from digital elevation models (DEM). Two methods of processing DEM for estimating the land surface for each cell are commonly used (value nearest the cell centroid or mean value in the cell). This article demonstrates how these two methods of DEM processing can affect the simulated water budget. For the example presented, approximately 20% more total flow through the aquifer system is simulated if the centroid value rather than the mean value is used. This is due to the one-third greater average ground water gradients associated with the centroid value than the mean value. The results will vary depending on the particular model area topography and cell size. The use of the mean DEM value in each model cell will result in a more conservative water budget and is more appropriate because the model cell water table value should be representative of the entire cell area, not the centroid of the model cell.     

Nested circulation modelling of inter-tidal zones: details of a nesting approach incorporating moving boundaries

Nested circulation models developed to date either exclude the flooding and drying process or prohibit flooding and drying of nested boundaries; they are therefore ill-suited to the accurate modelling of inter-tidal areas. The authors have developed a nested model with moving boundaries which permits flooding and drying of both the interior domain and the nested boundaries. The model uses a novel approach to boundary formulation; ghost cells are incorporated adjacent to the nested boundary cells so that the nested boundaries operate as internal boundaries. When combined with a tailored adaptive interpolation technique, the approach facilitates a dynamic internal boundary. Details of model development are presented with particular emphasis on the treatment of the nested boundary. Results are presented for Cork Harbour, a natural coastal system with an extensive inter-tidal zone and a complex flow regime which provided a rigorous test of model performance. The nested model was found to achieve the accuracy of a high resolution single grid model for a much lower computational cost.     

LE TRITIUM DANS LES PRÉCIPITATIONS DE L'ASIE DU SUD-EST ET LE PHÉNOMÉNE DE LA MOUSSON EN 1966 ET EN 1967

Abstract

The problem of non-steady flow of water in a soil-plant system can be described by adding a sink term to the continuity equation for soil water flow. In this paper the sink term is defined in two different ways. Firstly it is considered to be dependent on the hydraulic conductivity of the soil, on the difference in pressure head between the soil and the root-soil interface and some root effectiveness function. Secondly the sink is taken to be a prescribed function of the soil water content. The partial differential equation applying to the first problem is solved by both a finite difference (FD 1) and a finite element (FE 1) technique, that applying to the second problem by a finite difference approach (FD 2). The purpose of this paper is to verify the numerical models against field measurements, to compare the results obtained by the three numerical methods and to show how the finite element method can be applied to complex but realistic two-dimensional flow situations. Two examples are given. The first concerns one-dimensional flow and it compares numerical results with those obtained experimentally in the field from water balance studies on red cabbage (Brassica oleracea L. ‘Rode Herfst’) grown on a clay soil in the presence of a water table. The second example describes two-dimensional flow in a complex field situation in the Netherlands where flow takes place under cropped field conditions through five anisotropic layers. Water is supplied to the system by infiltration from two unlined ditches and is withdrawn from the system by evapotranspiration and by leakage to an underlying pumped aquifer.     

An improved approach for assigning pumping rates to heterogeneous aquifer models

A common assumption of ground water models formulated using a block-centered finite-difference method is that a well is located at the center of a cell regardless of its actual location. Due to this assumption, errors are introduced in the spatial distribution of simulated heads. This paper presents an alternative approach for assigning the pumping rates of wells that are located off cell centers. This approach consists of assigning the pumping rate not only to the cell in which the well is located but also to adjacent cells, taking into account the length of the well screen, the hydraulic conductivity, and the distance from the well to the center of its cell. The advantage of this alternative approach over the conventional one is illustrated with a test problem of a synthetic aquifer. Statistical measures of error indicate a much better model fit when pumping rates of wells are distributed over several cells.     

Beitrag zur analytischen Beschreibung von Transportprozessen im Grundwasserleiter bei diffusem Stoffeintrag

A model is described for the relation between water quantity and water quality in the aquifer with the diffuse substance input by the gravitation water being taken into account. The model is based on a partial differential equation of the second order for a flow pipe. For constant transport parameters, a sink of the kinetics of the first order and a time-dependent source term an exact analytical solution is presented and explained by a model. At a diffuse substance input, the dispersion has only a slight influence on the transport of substances contained in water. Potential applications of these solutions to different problems are mentioned.     

A Stable and Efficient Numerical Algorithm for Unconfined Aquifer Analysis

The nonlinearity of equations governing flow in unconfined aquifers poses challenges for numerical models, particularly in field-scale applications. Existing methods are often unstable, do not converge, or require extremely fine grids and small time steps. Standard modeling procedures such as automated model calibration and Monte Carlo uncertainty analysis typically require thousands of model runs. Stable and efficient model performance is essential to these analyses. We propose a new method that offers improvements in stability and efficiency and is relatively tolerant of coarse grids. It applies a strategy similar to that in the MODFLOW code to the solution of Richard's equation with a grid-dependent pressure/saturation relationship. The method imposes a contrast between horizontal and vertical permeability in gridblocks containing the water table, does not require "dry" cells to convert to inactive cells, and allows recharge to flow through relatively dry cells to the water table. We establish the accuracy of the method by comparison to an analytical solution for radial flow to a well in an unconfined aquifer with delayed yield. Using a suite of test problems, we demonstrate the efficiencies gained in speed and accuracy over two-phase simulations, and improved stability when compared to MODFLOW. The advantages for applications to transient unconfined aquifer analysis are clearly demonstrated by our examples. We also demonstrate applicability to mixed vadose zone/saturated zone applications, including transport, and find that the method shows great promise for these types of problem as well.     

Improved water table dynamics in MODFLOW

The standard formulation of a block-centered finite-difference model, such as MODFLOW, uses the center of the cell as the location of a cell node. Simulations of a dynamic water table can be improved if the node of a cell containing the water table is located at the water table rather than at the center of the cell. The LPF package of MOD-FLOW-2000 was changed to position a cell's node at the water table in convertible cells with a water table. Improved accuracy in the upper regions of an unconfined aquifer is demonstrated for pumping from a partially penetrating well. The change introduces a nonlinearity into the solution of the flow equations that results in slightly slower convergence of the flow solution, 7% slower in the presented demonstration. Accuracy of simulations is improved where vertical flow is dominated by a moving water table, but not when a large water table gradient dominates over the water table movement.     

Predicting Collector Well Yields with MODFLOW

Groundwater flow models are commonly used to design new wells and wellfields. As the spatial scale of the problem is large and much local‐scale detail is not needed, modelers often utilize two‐dimensional (2D) or quasi three‐dimensional models based on the Dupuit‐Forchheimer assumption. Dupuit models offer a robust set of tools for simulating regional groundwater flow including interactions with surface waters, the potential for well interference, and varying aquifer properties and recharge rates. However, given an assumed operating water level or drawdown at a well screen, Dupuit models systematically overpredict well yields. For design purposes, this discrepancy is unacceptable, and a method for predicting accurate well yields is needed. While published methods exist for vertical wells, little guidance is available for predicting yields in horizontal screens or collector wells. In plan view, a horizontal screen has a linear geometry, and will likely extend over several neighboring cells that may not align with rows or columns in a numerical model. Furthermore, the model must account for the effects of converging three‐dimensional (3D) flow to the well screens and hydraulic interference among the well screens; these all depend on the design of a specific well. This paper presents a new method for simulating the yield of angled or horizontal well screens in numerical groundwater flow models, specifically using the USGS code MODFLOW. The new method is compared to a detailed, 3D analytic element model of a collector well in a field of uniform flow.     

Numerical simulation of an unsaturated flow equation

A numerical model for an unsaturated flow problem by using the finite element method is established in order to simulate liquid moisture flow In an unsaturated zone with homogeneous soil and deep subsurface water, and with different initial and boundary conditions. For infiltration or evaporation problems, a traditional method usually yields oscillatory non-physics profiles. However, nonoscillatory solutions are obtained and non-physics solutions for these problems are evaded by using the mass-lumped finite element method. Moreover, the kind of boundary condition is handled very well. Project supported by the National Key Project of Fundamental Research ”Climate Dynamics and Climate Prediction Theory“ and China Postdoctoral Science Foundation.     

Use of upscaled elevation and surface roughness data in two-dimensional surface water models

In this paper, we present an approach that uses a combination of cell-block- and cell-face-averaging of high-resolution cell elevation and roughness data to upscale hydraulic parameters and accurately simulate surface water flow in relatively low-resolution numerical models. The method developed allows channelized features that preferentially connect large-scale grid cells at cell interfaces to be represented in models where these features are significantly smaller than the selected grid size. The developed upscaling approach has been implemented in a two-dimensional finite difference model that solves a diffusive wave approximation of the depth-integrated shallow surface water equations using preconditioned Newton-Krylov methods. Computational results are presented to show the effectiveness of the mixed cell-block and cell-face averaging upscaling approach in maintaining model accuracy, reducing model run-times, and how decreased grid resolution affects errors. Application examples demonstrate that sub-grid roughness coefficient variations have a larger effect on simulated error than sub-grid elevation variations.     

Two‐dimensional physically based finite element runoff model for small agricultural watersheds: I. Model development

Soil erosion by water is the root cause of ecological degradation in the Shiwalik foothills of Northern India. Simulation of runoff and its component processes is a pre‐requisite to develop the management strategies to tackle the problem, successfully. A two‐dimensional physically based distributed numerical model, ROMO2D has been developed to simulate runoff from small agricultural watersheds on an event basis. The model employs the 2‐D Richards equation with sink term to simulate infiltration and soil moisture dynamics in the vadoze zone under variable rainfall conditions, and 2‐D Saint‐Venant equations under the kinematic wave approximation along with Manning's equation as the stage‐discharge equation for runoff routing. The various flow‐governing equations have been solved numerically by employing a Galerkin finite element method for spatial discretization using quadrilateral elements and finite difference techniques for temporal solutions. The ROMO2D computer program has been developed as a class‐based program, coded in C + + in such a way that with minor modifications, the model can be used to simulate runoff on a continuous basis. The model writes output for a runoff hydrograph of each storm. Model development is described in this paper and the results of model testing and field application are to be presented in a subsequent paper. Copyright © 2008 John Wiley & Sons, Ltd.     

Recharge process and aquifer models of a small watershed

Abstract

Kanchanapally watershed covering an area of about 11 km² in Nalgonda district, Andhra Pradesh, India is located in granitic terrain. Groundwater recharge has been estimated from a water balance model using hydrometeorological data from 1978–1994. The monthly recharge estimates obtained from the water balance model formed input for the groundwater flow model during transient model testing. The groundwater flow model has been prepared to simulate steady state groundwater conditions of 1977 using the nested squares finite difference method. The transient groundwater flow model has been tested during 1977–1994 using the estimated recharge values. The present study helped verify the usefulness of monthly recharge estimates for accounting dynamic variations in recharge as reflected in water level fluctuations in hydrographs.     

APPLICATION OF A QUASI-2D HYDRO-MORPHOLOGICAL MODEL TO THE ARGENTINEAN PARAN(A) RIVER

Herein a simplified quasi-two dimensional horizontal hydro-morphological mathematical model is presented. The governing equations for the quasi-2D horizontal time-depending flow field are represented by the well-known approach of interconnected cells. New discharge laws between cells are incorporated. The model is capable of predicting temporal changes in water depth, velocity distribution,sediment transport, bed elevation, as well as water and suspended sediment exchanges between main stream and flood plains. An application of the model to the middle reach of the Argentinean Parana River is presented. Satisfactory results were obtained during model calibration, validation and application.     

Modeling ground water flow in alluvial mountainous catchments on a watershed scale

In large mountainous catchments, shallow unconfined alluvial aquifers play an important role in conveying subsurface runoff to the foreland. Their relatively small extent poses a serious problem for ground water flow models on the river basin scale. River basin scale models describing the entire water cycle are necessary in integrated water resources management and to study the impact of global climate change on ground water resources. Integrated regional-scale models must use a coarse, fixed discretization to keep computational demands low and to facilitate model coupling. This can lead to discrepancies between model discretization and the geometrical properties of natural systems. Here, an approach to overcome this discrepancy is discussed using the example of the German-Austrian Upper Danube catchment, where a coarse ground water flow model was developed using MODFLOW. The method developed uses a modified concept from a hydrological catchment drainage analysis in order to adapt the aquifer geometry such that it respects the numerical requirements of the chosen discretization, that is, the width and the thickness of cells as well as gradients and connectivity of the catchment. In order to show the efficiency of the developed method, it was tested and compared to a finely discretized ground water model of the Ammer subcatchment. The results of the analysis prove the applicability of the new approach and contribute to the idea of using physically based ground water models in large catchments.     

Coupled and splitting bedload sediment transport models based on a modified flux-wave approach

Numerical modeling of free-surface flow over a mobile bed with predominantly bedload sediment transport can be done by solving the shallow water and Exner equations using coupled and splitting approaches.The coupled method uses a coupling of the governing equations at the same time step leading to a non-conservative solution.The splitting method solves the Exner and the shallow water equations in a separate manner,and is only capable of modeling weak free-surface and bedload interactions.In the current study,an extended version of a Godunov-type wave propagation algorithm is presented for modeling of morphodynamic systems using both coupled and splitting approaches.In the introduced coupled method the entire morphodynamic system is solved in the form of a conservation law.For the splitting technique,a new wave Riemann decomposition is defined which enables the scheme to be utilized for mild and strong interactions.To consider the bedload sediment discharge within the Exner equation,the Smart and Meyer-Peter&Müller formulae are used.It was found that the coupled solution gives accurate predictions for all investigated flow regimes including propagation over a dry-state using a Courant-Friedrichs-Lewy(CFL)number equal to 0.6.Furthermore,the splitting method was able to model all flow regimes with a lower CFL number of 0.3.     

MODFLOW/MT3DMS-based simulation of variable-density ground water flow and transport

This paper presents an approach for coupling MODFLOW and MT3DMS for the simulation of variable-density ground water flow. MODFLOW routines were modified to solve a variable-density form of the ground water flow equation in which the density terms are calculated using an equation of state and the simulated MT3DMS solute concentrations. Changes to the MODFLOW and MT3DMS input files were kept to a minimum, and thus existing data files and data files created with most pre- and postprocessors can be used directly with the SEAWAT code. The approach was tested by simulating the Henry problem and two of the saltpool laboratory experiments (low- and high-density cases). For the Henry problem, the simulated results compared well with the steady-state semianalytic solution and also the transient isochlor movement as simulated by a finite-element model. For the saltpool problem, the simulated breakthrough curves compared better with the laboratory measurements for the low-density case than for the high-density case but showed good agreement with the measured salinity isosurfaces for both cases. Results from the test cases presented here indicate that the MODFLOW/MT3DMS approach provides accurate solutions for problems involving variable-density ground water flow and solute transport.     

A multiscale method for subsurface inverse modeling: Single-phase transient flow

High-resolution geologic models that incorporate observed state data are expected to effectively enhance the reliability of reservoir performance prediction. One of the major challenges faced is how to solve the large-scale inverse modeling problem, i.e., to infer high-resolution models from the given observations of state variables that are related to the model parameters according to some known physical rules, e.g., the flow and transport partial differential equations. There are typically two difficulties, one is the high-dimensional problem and the other is the inverse problem. A multiscale inverse method is presented in this work to attack these problems with the aid of a gradient-based optimization algorithm. In this method, the model responses (i.e., the simulated state data) can be efficiently computed from the high-resolution model using the multiscale finite-volume method. The mismatch between the observations and the multiscale solutions is then used to define a proper objective function, and the fine-scale sensitivity coefficients (i.e., the derivatives of the objective function with respect to each node’s attribute) are computed by a multiscale adjoint method for subsequent optimization. The difficult high-dimensional optimization problem is reduced to a one-dimensional one using the gradient-based gradual deformation method. A synthetic single-phase transient flow example problem is employed to illustrate the proposed method. Results demonstrate that the multiscale framework presented is not only computationally efficient but also can generate geologically consistent models. By preserving spatial structure for inverse modeling, the method presented overcomes the artifacts introduced by the multiscale simulation and may enhance the prediction ability of the inverse-conditional realizations generated.     

Modeling surface and ground water mixing in the hyporheic zone using MODFLOW and MT3D

We used a three-dimensional MODFLOW model, paired with MT3D, to simulate hyporheic zones around debris dams and meanders along a semi-arid stream. MT3D simulates both advective transport and sink/source mixing of solutes, in contrast to particle tracking (e.g. MODPATH), which only considers advection. We delineated the hydrochemically active hyporheic zone based on a new definition, specifically as near-stream subsurface zones receiving a minimum of 10% surface water within a 10-day travel time. Modeling results indicate that movement of surface water into the hyporheic zone is predominantly an advective process. We show that debris dams are a key driver of surface water into the subsurface along the experimental reach, causing the largest flux rates of water across the streambed and creating hyporheic zones with up to twice the cross-sectional area of other hyporheic zones. Hyporheic exchange was also found in highly sinuous segments of the experimental reach, but flux rates are lower and the cross-sectional areas of these zones are generally smaller. Our modeling approach simulated surface and ground water mixing in the hyporheic zone, and thus provides numerical approximations that are more comparable to field-based observations of surface–groundwater exchange than standard particle-tracking simulations.