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Advances in Water Resources Vol. 22, No. 6, pp 567±575, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter

A grid generating algorithm for simulating a
¯uctuating water table boundary in heterogeneous
uncon®ned aquifers
A.S. Crowea,*, S.G. Shikazea & F.W. Schwartzb
a

National Water Research Institute,Canada Centre for Inland Waters, P.O. Box 5050, Burlington, Ont., Canada L7R 4A6
b
Department of Geological Sciences,The Ohio State University,125 South Oval Mall,Columbus, OH 43210, USA
(Received 14 December 1997; revised 6 October 1998; accepted 12 October 1998)

An algorithm is presented for generating ®nite element grids that can be used to
calculate the position of a ¯uctuating water table and the formation of seepage
faces within a heterogeneous uncon®ned aquifer. Our approach overcomes limitations with existing techniques by allowing the water table to rise or decline

through hydrostratigraphic boundaries yet maintains numerical and conceptual
accuracy with respect to hydrostratigraphic geometry. The algorithm involves (1)
limited stretching or shrinking of elements along the water table if the change in
the position of the water table is small with respect to the vertical grid spacing,
and (2) the addition or removal of nodes and elements in the ®nite element mesh
along the water table as the change becomes large with respect to the vertical grid
spacing. This technique is applicable to any 2-D or 3-D ®nite element code that
contains an automatic ®nite-element grid generator. Ó 1999 Elsevier Science
Limited. All rights reserved
Key words: ®nite element method, free surface, grid generation, groundwater,
hydrogeology, modelling, uncon®ned aquifer, water table.

1 INTRODUCTION

For example, in some schemes the elevation of the nodes
along the water table is ®xed (i.e., the grid does not
deform) throughout the simulation but the calculated
value of heads along the water table can change2,14,17,18.
With the geometry of the cells remaining the same
throughout the simulation, errors can result because the

hydraulic head along the water table is not equal to the
elevation of the water table.
Other schemes match water-table elevations with
hydraulic heads by allowing the mesh to deform through
time1,4±12,14,16. The elements that are deformed might
include only the top row of elements or all rows in the
domain. In most of these cases, the elements are stretched vertically, but a few of these algorithms allow the
grid to deform both vertically and horizontally. Although this approach more accurately approximates the
position of the water table in a homogeneous domain,
problems can arise in heterogeneous domains. If the
mesh expands or contracts through layer boundaries,
the initial hydrostratigraphy is not preserved (Fig. 1b),

The objective of this paper is to present a method for
generating ®nite element grids that can change in response to a water table that rises or falls through
hydrostratigraphic boundaries, and still maintains numerical and conceptual accuracy with respect to
hydrostratigraphic geometry. Speci®cally, our algorithm
calculates the position of a ¯uctuating water table, the
formation of seepage faces and the distribution of hydraulic head within a heterogeneous uncon®ned aquifer.
Although we make use of a two-dimensional domain

discretized with triangular ®nite elements, the method
proposed here can be incorporated into any 2-D or 3-D
®nite element code that contains an automatic ®nite element grid generator.
This work provides an important step forward over
other approaches that are limited in their capabilities.
*

Corresponding author.
567

568

A. S. Crowe et al.
the ®nite element grid will always conform to the
hydrostratigraphic geometry.

2 METHOD
In a free-surface problem, both the hydraulic head distribution and the water table con®guration are unknown. The main criterion for accurately simulating the
position of the water table is that the elevation of a node
i along the water table, d(x,t), is equal to the hydraulic

head at the water table node, h(x,t), at all times. Also, if
the water table is at ground surface, a seepage face will
form and the value of hydraulic head will be equal to the
elevation of the ground surface.
Our approach considers only the saturated part of the
¯ow system (below the water table). Positive and negative ¯uxes are used as boundary conditions along the top
of the domain, causing the water table to rise or fall.
Fig. 2 shows a typical cross section and boundary conditions that are described below. The governing equation for transient groundwater ¯ow in the saturated
zone (S in Fig. 2) is:


o
oh
oh
ˆ Ss ;
…1†
Kij
oxi
oxj
ot

where Kij is the hydraulic conductivity tensor [L/T], h is
hydraulic head [L], Ss is the speci®c storage coecient
[Lÿ1 ], t is time [T], xi is the coordinate vector [L] and i,
j ˆ 1,2. The initial conditions are:
h…x; z; 0† ˆ ho …x; z†;

…2†

d…x; z; 0† ˆ do …x; z†;

…3†

where d is the elevation of the free surface (F in Fig. 2)
above a datum [L], d0 is the initial elevation of the free
surface [L], h0 is the initial hydraulic head [L]. The
boundary conditions for eqn (1) are:
h…x; z; t† ˆ H …x; z; t†;
Fig. 1. Illustration of the geometry of the hydrostratigraphic
units modelled by the reforming of a ®nite element grid as the
water table rises; (a) initial grid and hydrostratigraphic unit

geometry, (b) grid after several time steps generated with a
method which stretches elements, (c) grid after several time
steps generated our algorithm which adds nodes and elements.

resulting in inaccurate calculations of both the position
of the water table and the hydraulic head distribution
within the saturated domain. Finally, with elements able
to deform in this manner, numerical inaccuracies can
exist causing problems with numerical instabilities and
the aspect ratio should the elements stretch or compress
excessively.
Our algorithm involves the addition or removal of
nodes and elements along the water table, as the water
table rises or falls, respectively (Fig. 1c). By adding and
removing nodes and elements, the vertical grid spacing
throughout the saturated zone remains constant. Hence,

Kij

on bÿc (Fig. 1)

@h
ni ˆ ÿQ…x; z; t†;
@xi

d…x; z; t† ˆ h…x; z; d; t†;

…4†
…5†

on F …Fig: 2†;

…6†

Fig. 2. Schematic cross section showing the computation domain (S), the free surface (F), seepage face a±b, and constant
head boundary b±c.

Simulating a ¯uctuating water table boundary
Kij



@h
@d
ni ˆ R ÿ Sy
n3 ;
@xi
@t

h…x; z; t† ˆ z;

on F …Fig: 2†

on aÿb …Fig: 1†;

…7†
…8†

where H is the hydraulic head on a constant head
boundary [L], R is the rate of vertical recharge along the

free surface [L/T], ni is the unit outward normal vector,
Sy is the speci®c yield and Q is the ¯ux along a speci®ed¯ux boundary [L/T].
Equations (2) and (3) are initial conditions that
specify the hydraulic head values and the elevations of
the water table at the start of the simulation. Equation
(4) represents a ®rst-type or Dirichlet boundary condition where speci®ed values of hydraulic head are assigned along the boundary. The value of the speci®ed
head at these boundaries can change in time, and constant head nodes can be turned o€ and/or on during a
simulation. Equation (5) represents a second-type, or a
Neumann boundary condition where a speci®ed ¯ux
across a boundary is assigned. Equations (6) and (7)
represent the boundary conditions along the free surface, depending on whether or not recharge ¯uxes are
present. Equation (8) represents a seepage-face boundary where the hydraulic head is equal to elevation of
the ground surface. The boundary value problem de®ned by eqns (1)±(8) is described in further detail by
Neuman and Witherspoon15.
The two-dimensional form of eqn (1), subject to initial and boundary conditions, is solved in a vertical cross
section using a standard ®nite-element technique. Although, we use a triangular ®nite-element mesh, the
procedure would apply with most element types (e.g.,
quadrilateral ®nite elements, 3-D domains). The ®niteelement equations are formulated using the Galerkin
method16. Our algorithm for generating the ®nite-element grid satis®es the following conditions:
· the position of the water table can rise or fall over

time as a result of boundary conditions that can
change in time,
· all nodes along the water table are located at
d(x,t) ˆ h(x,t),
· a single element does not cross over the interface
between two hydrostratigraphic units (i.e., nodes
located at the interface),
· seepage faces form where nodes are located at
z(x,t) ˆ h(x,t) ˆ zg (x,t) (where zg is the elevation
of the ground surface).
Our method involves a combination of a limited
stretching of elements and/or the addition or removal of
nodes and elements along the water table. If the change
in the position of the water table is small with respect to
the vertical grid spacing, the elements along the water
table are stretched or compressed. If the change in position is large with respect to the vertical grid spacing,
new elements and nodes are added or removed.
The ®rst step is to discretize the computational domain into triangular ®nite elements using an automatic

569

mesh generator. This initial mesh depends on the geometry of the domain, the boundary conditions and the
initial elevation of the water table. The grid spacing (Dx,
Dz) is small relative to the scale of the problem in order
to represent the hydrostratigraphic units and to position
nodes along the interface between these units. The grid
generator assigns an elevation to each node along the
uppermost row of the mesh that is equal to the assigned
value of hydraulic head of the water table at that node.
The algorithm uses an iterative solution to determine
the elevation of the water table and values of hydraulic
head. Fig. 3 illustrates the adjustment of the ®nite-element mesh. At the beginning of a time step, the algorithm ®xes the elevation of the nodes along the water
table and calculates the hydraulic heads within the ¯ow
domain (Fig. 3(a),(g)). Next, nodes along the water table are moved to a position where the elevation of each
node equals its value of hydraulic head (Fig. 3(b),(h)).
Because only the nodes along the water table move, only
the top row of elements stretch or compress. Changing
the vertical dimension of an element produces a new
vertical spacing of Df. All remaining elements below the
uppermost row of elements remain at a constant vertical
spacing of Dz (Fig. 3(b),(h)). At the end of each iteration, numerical convergence is tested by calculating a
residual de®ned as the sum of the absolute di€erence
between the head at the nodes along the water table
(h(x,t)) and the elevation (d(x,t)) of this node:
residual ˆ Rjd…x; t† ÿ h…x; t†j

…9†

The solution has converged when the residual is less
than a user-de®ned tolerance or closure criterion (we
have chosen a tolerance of 0.001 which allows convergence within about 5 iterations). Moving to the next
time step, the algorithm repeats this process with the
opportunity to change the mesh again (Fig. 3
(c),(e),(i),(k)).
The procedure outlined above is used with most ®nite-element codes that allow the grid to deform as the
shape of the ¯ow domain changes. However, in our
method, at the beginning of each new time step, if an
element is stretched more than 1/4Dz beyond a regular
grid spacing (Df > 5/4Dz) we form a new node and a
new element. The new node is inserted at the regular Dz
spacing, and the new element is inserted along the water
table with a vertical element spacing of Dfnew ˆ
Dfold ) Dz (Fig. 3(e)). If an element stretches less than
1/4Dz beyond the regular grid spacing (Df < 5/4Dz),
only the top two elements are stretched, and a new node
is not inserted (Fig. 3(b)). These two stretched elements
are formed from the regular Dz spacing to the present
position of the water table where Dfnew ˆ Dz + Dfold
(Fig. 3(f)). Similarly, if a node at the water table declines
by more than 3/4Dz of the regular grid spacing (Df <
1/4Dz), the node immediately below this water table
node is removed (Fig. 3(k)). If the decline of a water
table node is less than 3/4Dz, the z position of this node

570

A. S. Crowe et al.

Fig. 3. The procedure through which a ®nite element stretches/compresses and is added/removed as the water table rises or falls,
respectively.

is simply lowered to the current value, thereby compressing the ®nite element, with no removal of nodes or
elements (Fig. 3(h)).
Because all elements, except those at the water table,
are maintained at the original vertical grid spacing of
Dz, unit boundaries remain unchanged. The extent to
which our ®nite element grid maintains the geometry of
the layers versus a grid generated by a method which
only stretches elements is illustrated by Fig. 1. The only
instance where the mesh may not coincide with the unit
boundaries occurs when the water table passes into a
new geologic unit. Initially, the changes in the water
table elevation can result in water-table elements
stretching less than 1/4Dz from Df ˆ Dz (i.e., Df <
5/4Dz). If such a stretched element exists at the interface
between two units, the stratigraphy will not be preserved
and the boundary will actually be between 0 and 1/4Dz
units higher than the actual position. With time, the

water table will continue to rise to a point where a new
element will form, at which time the stratigraphy will be
once again be accurately represented and placed at the
proper Dz spacing. The error resulting from this slight
misrepresentation is very small with respect to the actual
position of the water table at that particular time step
and at that element location. Hence, as the water table
continues to rise and the correct unit boundaries are
represented, this error will diminish. However, there
may be a case in this situation where the scheme can
have diculty converging to a stable solution. This
problem will be addressed in more detail below.
Our algorithm, like many codes that only stretch elements, also allows seepage faces to form when the
water table intersects the ground surface. Nodes and
elements along the seepage face are inserted as the
seepage face expands or removed as the seepage face
contracts according to the above criterion. Nodes are

Simulating a ¯uctuating water table boundary
not allowed to be repositioned above the ground surface. At the ground surface, nodes are redesignated as
constant-head nodes along the seepage face. If the water
table falls below the ground surface, the constant-head
nodes along the seepage face revert to regular nodes.
2.1 Limitations
Convergence problems can develop when the water table
moves through a boundary between geologic units that
have a large contrast in hydraulic conductivity. For
example, if the water table rises from a unit of low to
one of high hydraulic conductivity, the elements in the
lower K unit will stretch until the hydraulic heads increase beyond 1/4Dz (Fig. 3(d)). After this, new watertable elements will form, but these new elements will
have the higher K value assigned to them, and the low K
elements will shrink to the regular element size
(Fig. 3(e)). Because these new elements have a higher K,
the hydraulic head along the water table can decrease
resulting in a drop in the water table. If these high K
elements shrink to below 3/4Dz, they will be removed
and the low K elements will be stretched upwards
(Fig. 4(c)). This can result in an increase in hydraulic
head and result in the formation of new high K elements
yet again (Fig. 4(b)). This entire sequence can repeat
itself with the water table at this node oscillating
(Fig. 4(a)) and convergence might never be achieved.
We have provided two solutions to rectify this
problem. An algorithm is included within the code that

571

identi®es these oscillations. With the ®rst solution, the
criterion for forming a new element is decreased from
1/4Dz to 1/10Dz, and the criterion for removing an element is increased from 3/4Dz to 9/10Dz for the current
time step. After the water table rises across the boundary between elements with a low (lower element) and
high (upper element) hydraulic conductivity, it will form
a new element. However, when the water table subsequently falls due to the new element with a high hydraulic conductivity, it will not remove this element
because the water table is above 1/4Dz. This ®x provides
convergence in most cases.
In some instances where the contrast in the hydraulic
conductivity of the two hydrostratigraphic units is very
large, the above solution may still not lead to convergence. In this case, it is recommended that the user
change the hydraulic conductivity of the cell containing
the element where the oscillation occurs. The computer
code can identify the node causing the problem for the
user. We recommend that the user start with a value of
hydraulic conductivity that is midway between those of
the two units. This second solution will always results in
convergence.

3 APPLICATION
This section compares simulations using our adaptive
griding algorithm with a scheme that allows various
numbers of rows of elements to deform. Speci®cally, we
allow the upper-most row or all of the rows of elements
to deform. These latter cases are the techniques incorporated in other codes. The comparison involves three
di€erent simulation problems. The ®rst is the simple case
of a homogeneous, isotropic aquifer. The second case
provides a layer with a contrasting hydraulic conductivity through which the water table rises. The third case
represents a more realistic domain containing numerous
layers and lenses of contrasting hydraulic conductivity.
Fig. 5 presents a schematic of the domain and the
boundary conditions for the ®rst two cases. The initial
water table elevation is ®ve metres, which coincides with
the initial top boundary of the domain. With time, the
water table will rise due to recharge and the size of the
domain in the vertical direction will increase as new elements are formed. Recharge is applied uniformly during each time step and across the top boundary
(Table 1). A constant head node is speci®ed at the top
right corner of the domain with a hydraulic head value
of ®ve metres. All other boundaries in the domain are
no-¯ow.
3.1 Case 1 ± Homogeneous aquifer

Fig. 4. Example of a condition where convergence cannot be
attained causing (a) oscillations in a portion of the water table,
due to the successive (b) addition of an element, and (c) the
removal of an element.

The homogeneous case is used to compare our algorithm to others, including an analytical solution, to
check the accuracy of our method. The physical and

572

A. S. Crowe et al.

Fig. 6. Comparison of three current numerical methods that
involve only element stretching with our algorithm; Case 1 ±
homogeneous aquifer.
Fig. 5. Conceptual model and boundary conditions for the
sensitivity analysis for Case 1 and Case 2.

numerical parameters for Case 1 are tabulated in Table 1. Fig. 6 shows results from the three di€erent solution methods at a time equal to 100 days and at
steady-state. The hydraulic head along the water table is
plotted versus the horizontal (x) distance. At steady
state (approximately 3460 days), the maximum elevation
of the water table is approximately 2 m above the
starting water-table elevation. For this simple case, all
three solution schemes produce essentially identical results. This outcome is expected because although elements can deform signi®cantly, hydraulic parameters in
all elements remain the same and vertical stretching of
the elements is minimal which precludes problems related to the aspect ratio.
Fig. 6 also compares results obtained from the numerical methods to an analytical solution developed by
Marino13. Although the match among the solutions is
very good at early times (i.e., 100 days), the results of the
analytical solution deviate slightly at later times as the
water table rises. This deviation comes from limiting
assumptions in the approach (Bear3, page 365). Because
the numerical solutions are 2-D, they account for ver-

tical ¯ow. However, the analytical solution presented by
Marino13 is based on the Dupuit-Forcheimer assumption, and hence does not consider vertical ¯ow. The
accuracy of the analytical solution also decreases as the
height of the water table increases.
3.2 Case 2 ± Layered aquifer
Case 2 di€ers from Case 1 in that it includes a layer of
lower hydraulic conductivity at an elevation of 4±6 m
(Fig. 5). This example is a severe test designed to contrast the performance of our algorithm with others that
stretch elements and generally do not attempt to preserve the pattern of layering. As Case 1 showed, if the
water-table ¯uctuations does not encompass di€erent
units, solutions will be comparable. Because the initial
domain is only ®ve metres thick, the simulation techniques that involve only stretching of elements will not
preserve the location and thickness of this low conductivity layer. Our scheme, however, preserves the geometry of the low-K layer. Fig. 7 shows the hydraulic head

Table 1. Hydrogeological and numerical parameters used in the
simulations

Dx
Dz (initial size)
Dt; deltin
Dtmax
K1
K2
n
Ss
Sy
Recharge

Case 1

Case 2

4.0 m
0.5 m
0.5 days, 1.02
5 days
10ÿ5 m/s
10ÿ5 m/s
0.3
0.0005
0.2
18 cm/yr

4.0 m
0.5 m
0.5 days, 1.02
5 days
10ÿ5 m/s
2 ´ 10ÿ6 m/s
0.3
0.0005
0.2
51 cm/yr

NOTE: deltin is the factor by which to increase each successive
time step.

Fig. 7. Comparison of three current numerical methods that
involve only element stretching with our algorithm; Case 2 ±
layered aquifer.

Simulating a ¯uctuating water table boundary
at the water table versus horizontal distance for the
three di€erent schemes. As the simulations proceed, inaccuracies related to the treatment of the low conductivity layer become apparent for schemes that only allow
the elements to stretch. If one row of elements is stretched, the steady-state water table (6000 days) exceeds
that from our method by approximately 2 m. The excess
head develops because the low-K unit is enlarged unrealistically by element stretching. The low hydraulic
conductivity layer is stretched from its original thickness
of 2 m to a maximum thickness of 8.5 m. If all elements
are stretched, the water table at steady state (4100

573

days) is lower by approximately 0.1 m, as compared to
our method. The match between our algorithm with that
in which all rows of elements are stretched is more favorable because the thickness of the low conductivity
layer remains close to the actual thickness as represented
by our algorithm. However, the position of the low
conductivity layer is distorted. Here the thicknesses of
the lower most unit (K ˆ 10ÿ5 m/s) and the middle layer
(K ˆ 2 ´ 10ÿ6 m/s) are stretched from 4 and 2 m, respectively to a maximum of 8 and 3 m, respectively.
This case was designed to compare techniques that
preserve or do not preserve patterns of layering. While

Fig. 8. Cross sections for the heterogeneous case showing (a) hydrostratigraphy and initial position of the water table. The
hydrostratigraphy and ®nite element grid at steady state as generated by (b) our algorithm, and (c) an element stretching algorithm.

574

A. S. Crowe et al.

the code cross-veri®cation points to signi®cant di€erences between our algorithm and those that simply
stretch elements, it is not possible to compare the accuracy of the methods without an appropriate analytical
solution. Solution methods that stretch elements over
predict the elevation of the water table because they
exaggerate the thickness of the low permeability layer.
3.3 Case 3 ± Heterogeneous aquifer
The ®nal example includes various zones of contrasting
hydraulic conductivity, ranging from 1 ´ 10ÿ4 to
1 ´ 10ÿ7 m/s. Fig. 8(a) shows the domain, and the initial
water table (located at an elevation of 5 m). Note that
the region above the initial water table also contains
various lenses of di€erent hydraulic conductivity. These
lenses above the water table will not be considered in the
methods that involve only element stretching. Fig. 8(b)
shows the steady state water table and the computational domain using our method. The low and high
hydraulic conductivity lenses above the initial water
table are accurately represented by our algorithm.
Fig. 8(c) shows the domain and water table at steady
state from the method that involves stretching all rows
of elements, and clearly shows that the hydrostratigraphy is not accurately maintained. Fig. 9 shows the
steady state water tables generated by our algorithm, in
comparison with the techniques that stretch one or all of
the rows of elements. The methods that involve element
stretching over-estimate the water table by as much as
one metre.
The solution methods that simply stretch the elements produce results that can be dramatically di€erent
from those produced by our method. Although there is
no analytical solution with which to compare the results,
this example clearly shows that our algorithm not only
maintains the hydrostratigraphy better than the other
methods, but also considers units which the stretching
algorithms miss as the water table rises. Because our
algorithm maintains the hydrostratigraphic geometry,

whereas the other algorithms do not, it is apparent that
our solution is more representative.
4 CONCLUSIONS
Several models are capable of simulating both the hydraulic head distribution and water-table con®guration
within a uncon®ned aquifer. The simplest scheme involves calculating the position of the water table with a
®xed grid. The calculated heads along the water table
can change but the position of the nodes along the water
table remain ®xed. Other models calculate the position
of the water table by having nodes and cells/elements
stretch or compress in order to match the elevation of
the water table nodes to the respective values of hydraulic head. However, these models are accurate only
in cases where the water table ¯uctuates within a single
homogeneous unit. They are not appropriate for systems in which a rising or falling water table passes
through hydrostratigraphic unit boundaries. In such
cases the stretching of elements distorts the geometry
and changes the hydraulic parameters of the domain
through which the water table ¯uctuates.
The algorithm presented here calculates the position
of a ¯uctuating water table as it rises or falls through
hydrostratigraphic unit boundaries, and still maintains
the geometry of these units. More importantly, it
maintains the distribution of hydraulic parameters by
careful regeneration of the grid as appropriate. Our algorithm involves the addition or removal of nodes and
elements along the water table, as the water table rises or
falls, respectively. By adding and removing nodes and
elements, the vertical grid spacing throughout the saturated zone remains constant. Hence, the ®nite element
grid will always conform to the geometry of the units.
An iterative technique is required to match the elevation
and the hydraulic head of each node along the water
table. Convergence is generally attained within ®ve iterations. This technique is applicable to any 2-D or 3-D
®nite element code that contains an automatic ®nite-element grid generator.
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Fig. 9. Comparison of three current numerical methods that
involve only element stretching with our algorithm; Case 3 ±
heterogeneous aquifer.

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