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Comparison of local grid refinement methods for MODFLOW

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Many ground water modeling efforts use a finite-difference method to solve the ground water flow equation, and many of these models require a relatively fine-grid discretization to accurately represent the selected process in limited areas of interest. Use of a fine grid over the entire domain can be computationally prohibitive; using a variably spaced grid can lead to cells with a large aspect ratio and refinement in areas where detail is not needed. One solution is to use local-grid refinement (LGR) whereby the grid is only refined in the area of interest. This work reviews some LGR methods and identifies advantages and drawbacks in test cases using MODFLOW-2000. The first test case is two dimensional and heterogeneous; the second is three dimensional and includes interaction with a meandering river. Results include simulations using a uniform fine grid, a variably spaced grid, a traditional method of LGR without feedback, and a new shared node method with feedback. Discrepancies from the solution obtained with the uniform fine grid are investigated. For the models tested, the traditional one-way coupled approaches produced discrepancies in head up to 6.8% and discrepancies in cell-to-cell fluxes up to 7.1%, while the new method has head and cell-to-cell flux discrepancies of 0.089% and 0.14%, respectively. Additional results highlight the accuracy, flexibility, and CPU time trade-off of these methods and demonstrate how the new method can be successfully implemented to model surface water-ground water interactions.
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Comparison of Local Grid Refinement Methods
for MODFLOW
by Steffen Mehl
1
, Mary C. Hill
2
, and Stanley A. Leake
3
Abstract
Many ground water modeling efforts use a finite-difference method to solve the ground water flow equation,
and many of these models require a relatively fine-grid discretization to accurately represent the selected process
in limited areas of interest. Use of a fine grid over the entire domain can be computationally prohibitive; using a
variably spaced grid can lead to cells with a large aspect ratio and refinement in areas where detail is not needed.
One solution is to use local-grid refinement (LGR) whereby the grid is only refined in the area of interest. This
work reviews some LGR methods and identifies advantages and drawbacks in test cases using MODFLOW-2000.
The first test case is two dimensional and heterogeneous; the second is three dimensional and includes interaction
with a meandering river. Results include simulations using a uniform fine grid, a variably spaced grid, a traditional
method of LGR without feedback, and a new shared node method with feedback. Discrepancies from the solution
obtained with the uniform fine grid are investigated. For the models tested, the traditional one-way coupled
approaches produced discrepancies in head up to 6.8% and discrepancies in cell-to-cell fluxes up to 7.1%, while
the new method has head and cell-to-cell flux discrepancies of 0.089% and 0.14%, respectively. Additional results
highlight the accuracy, flexibility, and CPU time trade-off of these methods and demonstrate how the new method
can be successfully implemented to model surface water–ground water interactions.
Introduction
Many numerical models of ground water flow use
finite-difference methods to discretize and solve the gov-
erning partial differential equation. These models often
require a highly refined finite-difference grid such that
the solution can be accurately simulated in areas of inter-
est, for example, (1) regions of large variations in hy-
draulic gradient occurring over relatively small spatial
scales caused by, for example, pumping/injection wells
(water supply wells and artificial aquifer recharge), riv-
ers, and drains, (2) areas of site-scale contaminant migra-
tion within a regional model, and (3) heterogeneity
structures such as thin lenses, confining units, faults,
pinch-outs, and fractures.
Using a fine grid over the entire domain (referred to here
as global refinement) can be computationally intensive—
both in terms of CPU time and memory requirements—
and intractable in some cases; other alternatives are
needed. This work investigates three methods to address
this problem. First, the methods and their characteristics
are discussed. Two cases are then used to compare the
methods in terms of accuracy, CPU time, and memory
requirements.
Description of the Methods
Variable Spacing
A common approach to provide more refinement in
an area of interest is to use a finite-difference grid with
variable spacing such that the grid spacing is small where
needed and gradually increases away from this area.
While this approach reduces the computational time com-
pared to globally refining the grid, it generally results in
peripheral refinement because conventional finite-difference
1
Corresponding author: U.S. Geological Survey, 3215 Marine
Street, Boulder, CO 80303; (303) 541-3078; fax (303) 447-2505;
swmehl@usgs.gov
2
U.S. Geological Survey, 3215 Marine Street, Boulder, CO
80303; mchill@usgs.gov
3
U.S. Geological Survey, Tucson, AZ 85719; saleake@usgs.gov
Received May 2004, accepted September 2005.
Copyright ª2006 The Author(s)
Journal compilation ª2006 National Ground Water Association
No claim to original US government works.
doi: 10.1111/j.1745-6584.2005.00192.x
792 Vol. 44, No. 6—GROUND WATER—November–December 2006 (pages 792–796)
grids require that the grid lines in each direction extend
out to the model boundaries. This requirement has two
important implications: (1) if refinement is needed in
more than one area of the domain, using a variably
spaced grid often results in a relatively fine grid over the
entire domain, thus losing much of its computational
advantage over global refinement and (2) in addition to
introducing extra nodes and thus more computations, this
approach can produce finite-difference cells with large
aspect ratios, which can lead to numerical errors (de
Marsily 1986, 351).
This approach demonstrates the shortcomings of a
traditional finite-difference method vs. a finite-element
method in that the grid is not flexible. The variably
spaced finite-difference grid really is a brute-force solu-
tion to the problem of local-grid refinement (LGR). De-
spite these drawbacks, this method of refinement remains
an accurate and viable alternative in some circumstances.
Nevertheless, working with these grids (construction,
data input, and postprocessing) is more arduous than with
uniformly spaced grids.
Telescopic Mesh Refinement with No Feedback
The drawbacks of the variably spaced grid were sig-
nificant enough that other methods of LGR were sought.
One common alternative is often called telescopic mesh
refinement (TMR). This technique combines two or more
different-sized finite-difference grids—usually a coarse
grid, which incorporates regional boundary conditions, and
a locally refined grid, which focuses on the area of inter-
est. In ground water modeling, the link between the
coarse and local grids is most commonly accomplished
by first simulating the coarse grid and using its results to
interpolate heads and fluxes, or a combination of both,
onto the boundaries of the local grid (for example, Ward
et al. 1987; Leake et al. 1998; Davison and Lerner 2000;
Hunt et al. 2001).
This approach is fairly straightforward, flexible, and
relatively easy to implement but has some serious limita-
tions. Because the coupling communication between the
two grids occurs in one direction, from the coarse grid to
the local grid, there is no feedback from the local grid to
the coarse grid. Thus, after running the coarse- and local-
grid models, significant discrepancy can occur in either
the boundary heads or fluxes, whichever one was not used
to couple the grids, leaving the modeler unsure of whether
the coupling is adequate or not. Naturally, these discrep-
ancies along the boundary are propagated into the interior
of the model where high accuracy is desired, but the
extent of this propagation is unknown and can vary. The
heads and the fluxes across the interfacing boundary
can be checked for both models, but if they do not
match, there is no formal mechanism for achieving better
agreement.
While this approach is attractive for its simplicity
and computational efficiency, it can be inaccurate. The
problem is that in some cases this method works well,
while in other cases it does not, and thus the burden is
placed on the modeler to check the consistency between
the two grids to determine how well this approach works
for their particular case (Leake and Claar 1999, 5–7). It is
the authors’ opinion that these checks are seldom done in
practice, and therefore use of this approach without rec-
ognizing and quantifying the potential pitfalls can pro-
duce misleading results.
Methods with Feedback
The shortcomings of the traditional TMR approach
were recognized, and alternative methods were developed
that have a rigorous numerical coupling while incurring a
minimum of computational cost. Much of the work in-
volving the coupling of two different grids was pioneered
by the petroleum industry. Edwards (1996) concisely re-
views many of these methods, including their strengths
and weaknesses. Recent alternatives within the ground
water modeling field involve directly coupled (Haefner
and Boy 2003; Schaars and Kamps 2001) and iteratively
coupled methods (Sze
´kely 1998; Mehl and Hill 2002a,
2004; Mehl 2003). Both approaches incorporate feed-
backs from the local grid to the coarse grid, ensuring that
heads and fluxes are consistent between both grids.
Figure 1a shows a locally refined grid embedded
within coarser grid. The directly coupled method modifies
the finite-difference equations to account for the irregular
geometry along the interface of the two grids. Unfortu-
nately, this produces matrices that are different from the
diagonally dominant banded matrices that result from a
conventional finite-difference discretization. Therefore,
solvers that can accommodate these more difficult matri-
ces are required. In contrast, the iteratively coupled method
uses both flux and head boundary conditions, which are
updated iteratively, to couple both grids. In this case, the
coarse-grid solution is used to interpolate heads onto the
boundary nodes of the locally refined model. After simu-
lating the local model, the fluxes across the adjoining
interface are calculated and applied as a specified-flux
boundary on the coarse model for the subsequent iteration
(Figure 1b). This process is repeated until the head and
a
b
Figure 1. (a) Two-dimensional schematic of the locally
refined grid. The interface area denoted by the dashed line is
shown in greater detail in (b), and illustrates flux balance
across the local refinement interface. Darker shading is
material represented by the local grid, lighter shading is
material represented by the coarse grid, and no shading is
material at the interface. dnode of the coarse grid only;
snode of the coarse grid that is shared with a boundary
node of the local grid; node of the local grid only. The
coarse grid is inactivated here after the initial coarse-grid
simulation, so the coarse grid has a hole in it; specified-
head boundary node of the local grid determined by interpo-
lation from the coarse grid.
S. Mehl et al. GROUND WATER 44, no. 6: 792–796 793
flux change along the interfacing boundary is deemed neg-
ligible. Because the coupling occurs through the bound-
ary conditions, and these appear on the right-hand side of
the matrix equations, the finite-difference stencil is not
modified and the matrices maintain their regular structure.
Therefore, standard solvers can be used without modifi-
cation. This method is being developed for MODFLOW
and is publicly available with MODFLOW-2005 (Mehl
and Hill 2005).
In either case, the coupling is more rigorous than the
traditional TMR approaches, but the price for this rigor-
ous coupling comes in the form of increased computa-
tional costs. It will be shown later that the accuracy of
this rigorous coupling is necessary to produce good re-
sults for applications involving river-aquifer interactions.
Thus, it is the authors’ opinion that these approaches pro-
vide a good trade-off between the computational effi-
ciency and flexibility of TMR methods and the accuracy
of variably spaced grids and therefore are a good solution
to the problem of LGR.
Comparisons
Two synthetic test cases were designed to contain
features that emulate those important in many ground
water models. The first test case exhibits contrasts in
transmissivity and sharp changes in gradient caused by
a pumping well, while the second test case examines the
representation of river-aquifer interactions in three dimen-
sions. Using the uniform fine grid as the reference solution,
comparisons of the locally refined solutions are based on
the head and flux solution within the locally refined area,
CPU time, and memory requirements, and are used to
draw conclusions about advantages and drawbacks of the
LGR methods considered.
Test Case 1-Two-Dimensional, Heterogeneous
with Pumping
The first synthetic test case simulates steady-state
flow with a pumping well in a heterogeneous transmissiv-
ity field, as shown in Figure 2. The block-like trans-
missivity structure can be represented explicitly on all the
grids without changing values across the interface between
grids. A model with a uniform fine grid over the entire
domain provided a reference solution for comparison with
results from three methods of LGR. The fine grid has 450
rows and 972 columns, with cell dimensions of 1.028 and
1.0 m in the east-west and north-south directions, respec-
tively. For the LGR methods, the embedded grid has 100
rows and 154 columns with cell dimensions the same as
the fine grid. The area of the locally refined model is
shown enclosed in a dashed line in Figure 2, which in-
cludes the pumping well that extracts 5.5 m
3
/s. The
coarser outer grid has 50 rows and 108 columns, with
cell dimensions of 9.25 and 9.0 m in the east-west and
north-south directions, respectively. The ratio of refinement
is 9:1 (nine local cells span the width of one coarse
grid cell).
The variably spaced grid is implemented in MOD-
FLOW-2000 and has grid spacing equivalent to the local
grid in the area surrounding the well, with increasing grid
spacing away from this area. Two TMR methods are
considered in this example. They are implemented using
MODTMR (Leake and Claar 1999) and represent the
traditional TMR methods using one-way coupling with
either heads or fluxes (labeled TMR-Head and TMR-
Flux, respectively). For the TMR-Flux simulations, a sin-
gle head along the boundary was fixed so that a unique
solution could be obtained (Leake and Claar, 1999, 7).
The iteratively coupled method used here is described
by Mehl and Hill (2002a) and was implemented in
MODFLOW-2000.
For all methods, the heads and fluxes within the inte-
rior 36% of the local model domain are compared to the
uniform fine-grid results, and the discrepancies are used
to judge the accuracy of the local grid solution methods.
The model grids were designed such that the grid spacing
and node locations within the interior of the refined
region are identical so that interpolation errors would not
affect the calculation of the discrepancies. At each loca-
tion, the percent head discrepancies are calculated as the
difference between the head from the fine-grid model and
the head from one of the other models normalized by
dividing by the head from the fine grid and multiplying
by 100 to obtain a percent. The absolute values were
averaged to provide an overall measure of discrepancies.
This same procedure was used to investigate discrepan-
cies in the cell-to-cell fluxes in the same interior 36% of
the local model domain. These results are shown in
Table 1. CPU times are also compared.
The results shown in Table 1 indicate that there is
a clear trade-off between accuracy of the variably spaced
grid and CPU time of TMR methods. The iteratively cou-
pled method provides a compromise in this trade-off.
Another interesting result is that coupling using fluxes
(TMR-Flux) provides more accurate flux results along the
boundary, but less accurate flux results in the interior
compared to coupling with heads (TMR-Head), as pointed
out by Mehl and Hill (2002b). This result is counterintui-
tive and demonstrates one of the pitfalls of the TMR
methods—lack of consistency on the boundary can propa-
gate both head and flux discrepancies into the interior of
the refined region, diminishing the accuracy where it is
needed most. The errors are propagated from the
Table 1
Comparison of Head and Flux Discrepancies from
the Fine-Grid Solution and CPU Times for Several
Grid Refinement Schemes for the Interior 36% of
the Local Model Domain
Gridding
Average %
Head
Discrepancy
Average %
Cell-to-Cell Flux
Discrepancy
CPU
Time (s)
Fine grid (‘‘truth’’) 0.000 0.000 716
Variably spaced 0.023 0.034 57
TMR-Head 0.393 2.140 3
TMR-Flux 6.801 7.074 4
Iteratively coupled 0.089 0.140 28
794 S. Mehl et al. GROUND WATER 44, no. 6: 792–796
boundary into the interior via a diffusion process, as dis-
cussed by Mehl and Hill (2004). Even though the itera-
tively coupled method has a feedback, discrepancies are
still introduced along the boundary interface from the
abrupt change in grid spacing and resolution, and there-
fore it is not as accurate as the variably spaced grid.
Test Case 2—Three-Dimensional, Homogenous
Model of River-Aquifer Interactions
A second synthetic test case was created to test the abil-
ity of the LGR methods to represent small-scale features of
river-aquifer interactions. Figure 3 shows the plan view of
a meandering river in contact with a homogenous, uncon-
fined aquifer and the planar area of local refinement. The
refinement extends vertically through half the thickness
of the aquifer. The river has a linear drop in stage from
inlet to outlet and is represented using MODFLOW’s
River Package. Constant-head boundaries are placed on
the east and west sides to provide a background gradient;
no-flow boundaries span the north and south edges of the
domain. The hydraulic conductivity of the riverbed is
equal to that of the aquifer to maximize the interaction
between the two. The riverbed conductance is assigned
according to the area of the river that intersects each cell.
For the locally refined grids, all the properties (riverbed
conductance, vertical conductance, etc.) are treated the
same as they are in the globally refined grid.
For this case, no comparisons are made using a varia-
bly spaced grid, which is particularly awkward to work
with because of the meandering river. The river leakages
in the locally refined grids are compared to the river leak-
ages obtained using a globally refined model with the
equivalent local-grid spacing throughout the entire
domain. The ratio of refinement is 3:1 in all three grid di-
mensions; thus, 27 local-grid cells occupy the volume of
a single coarse-grid three-dimensional cell. Table 2 shows
the average of the absolute value of the percent discrep-
ancies of the river leakage. Essentially, the numbers
indicate how much improvement in memory and CPU
time is lost to accuracy by using local instead of global
refinement.
The results in Table 2 show that the decrease in
RAM and CPU time can be significant for the locally
refined grids vs. the globally refined equivalents. The
globally refined grid requires 1,476,225 nodes and 107
MB of RAM, while the locally refined model requires
154,850 nodes and 13.7 MB of RAM. The result also sug-
gests that little accuracy is lost by refining the grid locally
instead of globally, if a feedback is included (iteratively
coupled). However, if a feedback is not included (TMR-
Head), discrepancies in river leakage relative to the fine-
grid solution are 11.6% for this problem.
Conclusions
The three methods of LGR considered here—(1) a
variably spaced grid; (2) a traditional TMR method; and
(3) a rigorously coupled LGR method—all have advan-
tages and drawbacks. The variably spaced grid can be
viewed as a brute-force solution that is accurate but can
be computationally intensive and lacks flexibility and ele-
gance. The TMR methods are conceptually simple, easy
to implement, and computationally very efficient but can
be inaccurate in ways that often are not obvious to the
4.250x100
Transmissivity
m2/s
Pumping well
Constant head =1.0 m
No-flow boundary
No-flow boundary
999 m
450 m
1.200x103
4.306x102
1.611x102
1.350x101
XX
X X X X
X
X X X
XX
X
X X X
X
X
Constant head =10.0 m
N
Figure 2. Synthetic test case that is two dimensional and
heterogeneous. Area of local refinement is indicated by the
dashed rectangle.
1029.4 m
1544.1 m
No Flow
Area of local refinement
No Flow
N
Constant head = 49.75 m
Constant head = 44.75 m
Figure 3. Meandering river and planar area of refinement.
The ratio of refinement is 3:1 and extends in all three
dimensions.
Table 2
Comparison of Memory, CPU Time, and Mean River Leakage Discrepancies in the Locally
Refined Grid vs. a Globally Refined Grid. Values in Parentheses Represent
the Percent Reduction vs. the Globally Refined Grid
Discretization Grid Size RAM Usage (MB) CPU Time (s) % River Leakage Discrepancy
Globally refined 405 3405 39 107 1913 0
TMR-Head 163 3190 35 13.7 76 11.6
Iteratively coupled 163 3190 35 13.7 (87%) 811 (58%) 1.72
S. Mehl et al. GROUND WATER 44, no. 6: 792–796 795
user. Use of these methods without carefully considering
the potential errors can produce misleading model results.
Local grid refinement methods that include a feedback to
rigorously couple the grids produce results that are con-
sistent between both grids and thus have better accuracy
than the traditional TMR methods. These methods are
more flexible and computationally more efficient than
variably spaced grids. These trade-offs make this a good
approach to use for models requiring LGR.
Acknowledgments
This article benefited from reviews by C. Neville,
D. Feinstein, and J. Ward, and suggestions from C. Zheng.
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The objective of the present paper is to develop a methodology that could allow the representation of the analytical hyporheic flux equation model (AHF) in a numerical model done in MODFLOW. Therefore, the scope of the research is to show the viability of the methodology suggested in a real case (Biebrza river, Poland, Europe). Considering that the model requires extensive manipulation in the creation of the packages, a test phase through the seepage package of MODFLOW is carried out with the aim of representing the river package of MODFLOW. FloPy is the tool chosen to develop this implementation due to the versatility of manipulating the packages available in MODFLOW through coding. The obtained results showed a correct implementation of the AHF model using the example of the Biebrza River. The results obtained will enable a better understanding regarding the modelling of the interaction between the river and the aquifer, considering streams with specific geometries where the depth is dimensionally higher than the width.
... Traditionally, LGR (Local Grid Refinement) methods are used to subdivide grid cells in corner-point grids, see e.g. Mehl et al. (2006). Corner-point grids are discussed in Section 3. ...
Thesis
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PhD thesis in Computational Geoscience at the Department of Energy Resources, Faculty of Science and Technology, University of Stavanger, Norway. Abstract: Development of numerical methods for effective updates of an earth model, with a particular focus on real-time decision support while drilling (geosteering). The methods include a) local updates of the geological structure in an earth model grid, b) multi-resolution management of both the geological structure and the grid enabling local control of their resolutions, and c) local-scale uncertainty management of the interpretation of the geological structure in the grid. The aim is to pave the way for an always updated multi-realization 3D model at optimal resolution while drilling in complex formations, suitable for real-time decision support under uncertainty. The methods are general and could be improved to enable effective generation of earth model grids at fit-for-purpose resolution for any subsurface application.
... 等的模拟。此外,为了提高模拟精度,还可以使用 MODFLOW-LGR 对地下水水 分运动模型的网格进行局部细化 [18] 。在非饱和地下水模拟方面,Šimunek 等人开 发的 HYDRUS 模型则得到了较为广泛的应用,它可以用来模拟一维到三维非饱 和地下水中水分、热量和溶质的运移转化 [19] ,并且可以与 MODFLOW 耦合来研 究包气带和饱和地下水带之间的交互 [20] 。为了方便上述模型的构建和结果展示, ...
... Early TMR methods initiated with the work of Ward et al. (1987) and Duffield et al. (1987) in the late 1980s. Mehl et al. (2006) compares LGR methods in regards to computing requirements and percent discrepancy in a groundwater flow model. The study concluded that iteratively coupled LGR performed with higher accuracy than traditional TMR methods and was considered more efficient than the computationally intensive, yet highly accurate, global refinement and variably spaced grid methods. ...
Thesis
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Recent studies have concluded that stream reaches are not simply gaining or losing to groundwater but are best described as a mosaic of exchanges that contrast between flow paths of varying lengths and directions which inherently influence solute residence times. These residence times directly affect chemical speciation of solutes such as salts, nitrate, selenium, and uranium and have the opportunity to undergo microbial dissimilatory reduction in the shallow riparian zone and the deeper sub-surface. To improve water quality and the overall health of these natural systems requires engineering intervention supported by reliable data and calibrated models. A threedimensional numerical flow model (MODFLOW-UZF2) is used to simulate unsaturated and saturated groundwater flow, with linkage to a streamflow routing model (SFR2), for a 5-km reach of the Arkansas River near Rocky Ford, Colorado. The reach-scale model provides increased discretization of previous regional-scale models developed for the Arkansas River Basin, using 50 x 50 m grid cells and dividing the Quaternary alluvium that represents the unconfined aquifer into 10 layers. This discretization facilitates an enhanced view of groundwater pathways near the river which is essential for future solute transport evaluation and for consideration of alternative best management practices. Model calibration is performed on hydraulic conductivity (K) in the upper three layers, K in the lower seven layers, and specific yield (Sy) of the entire aquifer by applying an Ensemble Kalman Filter (EnKF) using observed groundwater hydraulic head and stream stage data. The EnKF method accounts for uncertainty derived from field measurements and spatial heterogeneity in parameters calibrated using a Monte-Carlo based process to produce 200 realizations in comparison to error-prone measurements of hydraulic groundwater hydraulic head and stream stage as calibration targets. The calibrated transient model produced Nash-Sutcliffe Efficiency (NSE) values of 0.86 and 0.99, respectively, for the calibration and evaluation periods for calibration targets using the ensemble mean of realizations. Realizations of calibrated parameters produced by the EnKF exhibit the equally-likely spatial distributions of aquifer flow and storage characteristics possible in the area, while MODPATH simulations display the associated groundwater flow paths possible under such conditions. The mean residence time of a streamdestined fluid particle within the riparian zone was estimated as 1.8 years. Simulated flow paths to the stream were highly variable given different geologic conditions produced by EnKF, with flow paths to some stream reaches being traced to different groundwater sources and transit times differing sometimes by decades. The simulated average annual groundwater return flow to the stream was 70 m3 d-1. Simulated average annual return flow was highly variable along the study reach and ranged from -250 to a little over 250 m3 d-1 with a CV of 1.4. The mean percentage of shallow (within the top three model layers) groundwater return flow to flow in aquifer layers beneath the stream was 27% with a CV of 0.58. Simulated groundwater flow paths were superimposed upon a map of shallow shale units residing in the study region, demonstrating how groundwater flow paths may interact with or contact regional seleniferous shale layers. Results hold major implications for biogeochemical processes occurring in the sub-surface of the riparian area and the hyporheic zone that have an important influence on solute concentrations. Results may be used to aid decision makers in the implementation of best management practices and to further understand contaminant sources and fate.
Article
Multi-scale modeling of the localized groundwater flow problems in a large-scale aquifer has been extensively investigated under the context of cost-benefit controversy. An alternative is to couple the parent and child models with different spatial and temporal scales, which may result in non-trivial sub-model errors in the local areas of interest. Basically, such errors in the child models originate from the deficiency in the coupling methods, as well as from the inadequacy in the spatial and temporal discretizations of the parent and child models. In this study, we investigate the sub-model errors within a generalized one-way coupling scheme given its numerical stability and efficiency, which enables more flexibility in choosing sub-models. To couple the models at different scales, the head solution at parent scale is delivered downward onto the child boundary nodes by means of the spatial and temporal head interpolation approaches. The efficiency of the coupling model is improved either by refining the grid or time step size in the parent and child models, or by carefully locating the sub-model boundary nodes. The temporal truncation errors in the sub-models can be significantly reduced by the adaptive local time-stepping scheme. The generalized one-way coupling scheme is promising to handle the multi-scale groundwater flow problems with complex stresses and heterogeneity.
Technical Report
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Ground-water models are commonly used to evaluate flow systems in areas that are small relative to entire aquifer systems. In many of these analyses, simulation of the entire flow system is not desirable or will not allow sufficient detail in the area of interest. The procedure of telescopic mesh refinement allows use of a small, detailed model in the area of interest by taking boundary conditions from a larger model that encompasses the model in the area of interest. Some previous studies have used telescopic mesh refinement; however, better procedures are needed in carrying out telescopic mesh refinement using the U.S. Geological Survey ground-water flow model, referred to as MODFLOW. This report presents general procedures and three computer programs for use in telescopic mesh refinement with MODFLOW. The first computer program, MODTMR, constructs MODFLOW data sets for a local or embedded model using MODFLOW data sets and simulation results from a regional or encompassing model. The second computer program, TMRDIFF, provides a means of comparing head or drawdown in the local model with head or drawdown in the corresponding area of the regional model. The third program, RIVGRID, provides a means of constructing data sets for the River Package, Drain Package, General-Head Boundary Package, and Stream Package for regional and local models using grid-independent data specifying locations of these features. RIVGRID may be needed in some applications of telescopic mesh refinement because regional-model data sets do not contain enough information on locations of head-dependent flow features to properly locate the features in local models. The program is a general utility program that can be used in constructing data sets for head-dependent flow packages for any MODFLOW model under construction.
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The effluent from a former coal-carbonisation plant in Nottinghamshire has contaminated the underlying aquifer with ammonium and organic compounds. In terms of flow and contamination, the site has experienced a complex history. MODFLOW has been used to unravel the past flow directions and to provide a basis for solute transport and biodegradation modelling. The modelling has shown that a fine grid is required to represent local flows to prevent incorrect site interpretations. A telescopic mesh technique was essential for this study to enable the site features to be adequately represented while including the regional hydrogeological influences. The influence of grid size to numerical dispersion was investigated for the MT3D computer program. The 'method of characteristics' and 'hybrid method of characteristics' modules of MT3D were found to be relatively free from numerical dispersion for all the grid sizes investigated. However, the 'modified method of characteristics' suffered extensively, and a linear relationship between grid size and numerical dispersion was demonstrated for this complex model.
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The available data base and hydrogeologic information for the Chem-Dyne hazardous waste site, located in southwestern Ohio, are typical of many Superfund sites. Data are localized because investigations conducted at these sites are for the purpose of defining contaminant plumes. Little or no data are available to characterize the regional flow systems which impart a controlling influence on the rate and direction of contaminant migration. To simulate groundwater flow and contaminant transport in this setting, a telescopic mesh refinement (TMR) approach is appropriate. This approach provides the means of accurately incorporating regional controlling factors into smaller model domains and also increased grid resolution in areas of critical importance. At the Chem-Dyne site a finite-difference model was applied at three scales: regional, local, and site. This application of TMR integrates the regional and local flow system characteristics to analyze the effectiveness of a proposed remedial action at the site scale. The TMR modeling approach permits effective flow and transport model construction and calibration to provide quantitative analysis of system response to a groundwater extraction-injection well clean-up system. Calibration of the flow models demonstrates the regional importance of induced river infiltration to groundwater pumping centers and the control of surface water features on the direction of plume migration. The site-scale transport model reveals that a significant portion of the volatile organic contaminants are not captured by the preliminary proposed system.
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Classical cell centered discretization of the reservoir simulation pressure equation on anh-adaptive grid results in anO(1/h) leading truncation error at the grid interface. A new flux continuous finite volume correction is presented together with an improved version of a previously proposed correction. While both corrections eliminate the leading error, the new correction exhibits the best convergence rates and has the following properties: the resulting matrix is in general symmetric positive definite, diagonally dominant for locally isotropic spatially varying coefficients, convergence toO(h) is demonstrated, and support of the standard approximation is retained, ensuring a fully implicit implementation. Results computed by the uncorrected classical scheme and both correction schemes are compared for two phase flow simulations with multilevel dynamic local grid refinement.
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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.
Book
This book combines two separate themes: a description of one of the links in the chain of the water cycle inside the earth's crust, i.e., the subsurface flow; and the quantification of the various types of this flow, obtained by applying the principles of fluid mechanics in porous media. The first part deals with the concept of water resources. The second part is necessary in order to quantify ground water resources. It points the way to other applications, such as solutions to civil engineering problems including drainage and compaction; and transport problems in porous media, including aquifer pollution by miscible fluids, multiphase flow of immiscible fluids, and heat transfer in porous media, i.e.e, geothermal problems.
Article
The effluent from a former coal-carbonisation plant in Nottinghamshire has contaminated the underlying aquifer with ammonium and organic compounds. In terms of flow and contamination, the site has experienced a complex history. MODFLOW has been used to unravel the past flow directions and to provide a basis for solute transport and biodegradation modelling. The modelling has shown that a fine grid is required to represent local flows to prevent incorrect site interpretations. A telescopic mesh technique was essential for this study to enable the site features to be adequately represented while including the regional hydrogeological influences. The influence of grid size to numerical dispersion was investigated for the MT3D computer program. The ‘method of characteristics’and ‘hybrid method of characteristics’modules of MT3D were found to be relatively free from numerical dispersion for all the grid sizes investigated. However, the ‘modified method of characteristics’suffered extensively, and a linear relationship between grid size and numerical dispersion was demonstrated for this complex model.
Article
A new method of local grid refinement for two-dimensional block-centered finite-difference meshes that uses an iteration-based feedback to couple two separate grids has been developed. Its convergence properties have been evaluated and comparisons with alternative methods have been completed (Mehl and Hill, in review a). This work further investigates a difficulty encountered with the traditional telescopic mesh refinement (TMR) methods that lack a feedback.Results indicate: (1) Coupling the coarse grid with the refined grid in a numerically rigorous way that allows for a feedback can improve the coarse grid results; this improvement is not possible using the TMR methods because there is no feedback. (2) The TMR methods work 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 6.8% and errors in cell-to-cell fluxes up to 7.1% for the case presented. (3) For the TMR methods, coupling using flux boundary conditions produces significant inconsistencies in the head distribution at the boundary interface. TMR inaccuracies can substantially effect parameter estimation (Mehl and Hill, in review b).
Article
Many small-scale ground water models are too small to incorporate distant aquifer boundaries. If a larger-scale model exists for the area of interest, flow and head values can be specified for boundaries in the smaller-scale model using values from the larger-scale model. Flow components along rows and columns of a large-scale block-centered finite-difference model can be interpolated to compute horizontal flow across any segment of a perimeter of a small-scale model. Head at cell centers of the larger-scale model can be interpolated to compute head at points on a model perimeter. Simple linear interpolation is proposed for horizontal interpolation of horizontal-flow components. Bilinear interpolation is proposed for horizontal interpolation of head values. The methods of interpolation provided satisfactory boundary conditions in tests using models of hypothetical aquifers.
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A numerical technique for spatial (lateral and vertical) zooming in finite-difference multiaquifer ground water flow models with a point-centered finite-difference scheme is presented. A composite, rectangular finite-difference mesh is used, allowing for different mesh resolutions and/or layering in hierarchically associated windows of the flow domain. An iterative procedure, called mesh interface simulator (MIS), is developed to link the parent and child meshes along their boundaries, referred to as interfaces. MIS equates the piezometric head along and the lateral flux across the interface. A numerical example of four interbedded meshes in a two-aquifer system with spatial zooming is evaluated. The results of numerical simulation are compared to an analytical solution to assess the overall approximation error of the numerical finite-difference and MIS procedures.