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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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