The Sea Water Intrusion (SWI2) Package
for Modeling Vertically-Integrated
Variable-Density Groundwater Flow in
Regional Aquifers with the U.S.
Geological Survey Modular Groundwater
Model (MODFLOW-2005)
Mark Bakker, Frans Schaars, Joseph D. Hughes, Christian D. Langevin, and Alyssa M. Dausman
1
Table of Contents
Introduction ......................................................................................................................................4
Purpose and Scope .................................................................................................................................... 4
Acknowledgements ................................................................................................................................... 4
Mathematical Formulation of Vertically-Integrated Variable-Density Groundwater Flow in Coastal
Aquifers .............................................................................................................................................5
Conceptual Model ..................................................................................................................................... 5
Governing Equations for Variable-Density Groundwater Flow ................................................................. 5
Basic Approximations................................................................................................................................ 6
Mathematical Derivation of Vertically-Integrated Variable-Density Groundwater Flow in Aquifers ....... 6
Vertical Leakage Between Aquifers .......................................................................................................... 8
Solution Procedure .................................................................................................................................... 8
Sea-Water Intrusion (SWI) Package Implementation ......................................................................... 10
Finite-difference Solution of the Groundwater Flow Equation ............................................................... 10
Finite-difference Solution for Zeta Surfaces ............................................................................................ 11
Sinks and Sources .................................................................................................................................... 11
Tip and Toe Tracking ............................................................................................................................... 12
Simulation of aquifers and aquitards. ..................................................................................................... 12
Pulling or pushing an interface to another layer .................................................................................... 12
Simulation of the coastal boundary ........................................................................................................ 13
Adaptive Time Stepping .......................................................................................................................... 13
Using the SWI Package ........................................................................................................................... 13
Output Files and Post Processing ............................................................................................................ 13
Tips for Designing MODFLOW-2005 Models Using the SWI Package..................................................... 14
Example Simulations ........................................................................................................................ 16
Example 1: Rotating interface................................................................................................................. 16
Example 2: Rotating brackish zone ......................................................................................................... 17
Example 3: Intrusion in a two-aquifer system......................................................................................... 17
Example 4: Upconing below a pumping well in a two-aquifer system ................................................... 18
Evaluation of Adopted Approximations ............................................................................................ 20
Problem Setup ......................................................................................................................................... 20
Density Inversion ................................................................................................................................ 21
Dispersion ........................................................................................................................................... 21
Vertical Resistance to Flow ................................................................................................................. 21
Conclusion ............................................................................................................................................... 21
References....................................................................................................................................... 22
Appendix 1. SWI2 Data Input Instructions ......................................................................................... 23
Figures............................................................................................................................................. 27
2
3
Introduction
Purpose and Scope
The Sea Water Intrusion (SWI) package is intended for the modeling of regional seawater intrusion with
MODFLOW. The evolution of the three-dimensional density distribution may be simulated through time
while the effects of the density distribution on the flow are taken into account explicitly. The main
advantage of the SWI package is that each aquifer may be modeled with a single layer of cells. An
existing MODFLOW model of a coastal aquifer may be modified to simulate seawater intrusion through
the addition of one input file. The SWI package can simulate interface flow, stratified flow, and
continuously-varying density flow.
The implemented formulation is based on the use of vertically integrated fluxes. Vertically integrated
fluxes were introduced for variable density flow to describe instantaneous flow fields by Weiss (1982)
and Maas and Emke (1988). Strack (1995) used vertically integrated fluxes to develop a potential flow
formulation for variable density flow. Bakker (2003) developed the formulation used in SWI. The original
SWI package was implemented in MODFLOW2000. The current implementation (SWI2) is in
MODFLOW2005 and has a number of additional features over the MODFLOW2000 version.
Acknowledgements
Development of the original SWI package was made possible through grants of the Georgia Coastal
Incentive Grants Program, administered by the Georgia Department of Natural Resources, and through
financial support of the Amsterdam Water Supply (now Waternet) in The Netherlands. The authors thank
Theo Olsthoorn of Waternet and the Delft University of Technology for his suggestions and support.
Vaughan Voller of the University of Minnesota provided valuable suggestions for the tip and toe tracking
algorithm. Debbie Borden of the University of Georgia extensively tested the tip and toe tracking
algorithm.
4
Mathematical Formulation of Vertically-Integrated VariableDensity Groundwater Flow in Coastal Aquifers
Conceptual Model
In coastal aquifers, the density of the groundwater is a function of the salinity. Here, the density in each
aquifer is discretized vertically into a number of zones bounded by curved surfaces. A schematic vertical
cross-section of an aquifer is shown in Fig. 1a; the thick lines represent the surfaces. The elevation of
each surface is a unique function of the horizontal coordinates. The SWI package has two options. For the
first option, called the stratified option, the water has a constant density in each zone and the density is
discontinuous from zone to zone (Fig. 1b). For the second option, called the variable density option, the
surfaces bounding the zones are iso-surfaces of the density and the density varies linearly in the vertical
direction in each zone and is continuous from zone to zone (Fig. 1c).
Governing Equations for Variable-Density Groundwater Flow
A Cartesian x, y, z coordinate system is adopted with the z-axis pointing vertically upward. Darcy's law
for variable density flow may be written as (Post et al, 2007)
qx  k

x
qx  k

x
 

qz  k 
 
 x

(1)
where qx, qy, qz are the components of the three-dimensional specific discharge vector,  is the freshwater
head, k is the freshwater hydraulic conductivity, and  is the dimensionless density ratio defined as
  f
f

(2)
where f is the density of freshwater. The derivation is presented here for an isotropic aquifer, but may be
repeated easily for an isotropic aquifer where the principal components of the hydraulic conductivity
tensor point in the x, y, and z directions.
The freshwater head is defined as

p
z
f g
(3)
where p is the water pressure. The freshwater hydraulic conductivity is defined as
k
 f g
f
(4)
where  is the intrinsic permeability of the aquifer and f is the viscosity of freshwater.
Variations in the viscosity are neglected (the viscosity of freshwater is approximately equal to the
viscosity of seawater). The difference between the hydraulic conductivity of freshwater and the hydraulic
conductivity of seawater is generally only a few percent and is neglected.
5
Basic Approximations
Four main approximations are made in the derivation of the formulations used in the SWI package:
1. The Dupuit approximation is adopted and is interpreted to mean that the resistance to flow in the
vertical direction is neglected (e.g. Strack, 1989, p.36).
2. The mass balance equation is replaced by the continuity of flow equation in the computation of
the flow field (the Boussinesq-Oberbeck approximation; e.g. Holzbecher, 1998, p. 32); density
effects are taken into account through Darcy's law.
3. Effects of dispersion and diffusion are not taken into account.
4. Inversion is allowed between aquifers but not within an aquifer. Inversion means that saltier
(heavier) water is present above fresher (lighter) water, often resulting in the vertical growth of
fingers (e.g., Wooding, 1969). The SWI package is intended for the modeling of regional
seawater intrusion, which is generally on a scale well beyond the size of the fingers.
Mathematical Derivation of Vertically-Integrated Variable-Density
Groundwater Flow in Aquifers
The derivation in this section is based on Bakker (2003). Groundwater in an aquifer is discretized
vertically into N zones separated by curved surfaces. Zones and surfaces are numbered from the top
down; zone n is bounded on top by surface n (Fig. 2). The elevation of surface n is represented by the
function n(x,y). The elevation of the top of the saturated aquifer is called 1; this can be either the
elevation of the (semi) confining layer or the phreatic surface. The elevation of the bottom of the aquifer
is called N+1.
Flow in an aquifer is expressed in terms of the freshwater head h(x,y) at the saturated top of an aquifer
and in the elevations of the surfaces between the zones (2 through N). The vertically integrated
horizontal flow vector below surface p is called U p and is defined as
Up  
p
 N 1
qdz
(5)
where q is the horizontal specific discharge vector q = (qx ,qy ) . The Dupuit approximation is adopted,
which means that the vertical pressure distribution is approximated as hydrostatic and the freshwater head
 at any elevation z may be expressed in terms of the freshwater head h(x,y) at the top of the aquifer as
z
 ( x, y, z )  h    ( x, y, z )dz 
1
(6)
The horizontal specific discharge vector q as a function of the vertical coordinate z in an aquifer is
obtained from substitution of (6) for  in (1)
z
q  k h  k    ( x, y , z )dz 
1
(7)
6
where Ñ is the two-dimensional gradient vector Ñ = (¶/ ¶x,¶/ ¶y) . As surface 1 is the saturated top of
the aquifer, U1 represents the total horizontal flow in an aquifer, so that continuity of flow in an aquifer
may be written as
h
 q z , t  q z ,b  
t
U 1   S
(8)
where S is the storage coefficient of the aquifer, qz,t and qz,b are the vertical specific discharge at the top
and bottom of the aquifer, respectively, and  is a source term. Similarly, continuity of flow below surface
p may be written as
U p  n
 p
 p
t
p  1, 2,
,N
(9)
where n is the effective porosity and p is a source term that may include leakage from underlying or
overlying aquifers. Combination of (5)-(9) gives the following set of N partial differential equations
(T h)  S
 p( p p )  n
h
q z , t  q z ,b    R
t
 p
t
  p  Rp
p  2,
(10)
,N
(11)
where T is the transmissivity of the aquifer, p is defined as the transmissivity below surface p, p is a
measure for the variation of the density between zones, and R and Rp are pseudo-source terms.
For stratified flow, p, R, and Rp are defined as
1   1 ,
 p   p  p 1
p  2,3,
,N
(12)
N
R    n( n n )
(13)
n 1
N
Rp  ( p h)    n( p  n )
p  2,3,
N
(14)
n 1
n p
where p is the dimensionless density of zone p.
For variable density flow, p is defined as the dimensionless density along (iso)surface p, and p is
computed using (12), but with the average dimensionless density, p, of zone p
 p  ( p  p 1 ) / 2
(15)
The terms R and Rp are defined as
N
N * 1
n 1
n  p*
R    n( n n )    n[ n( n   n 1 )]
(16)
7
N
N * 1
n 1
n p
n  p*
Rp  ( ph)    n( p n )    n[ n( n   n 1 )]
p  2,3,
N
(17)
where p is the transmissivity of zone p, p is a measure of the variation of the density over zone p:
 p  ( p 1  p ) / 6
(18)
and p* is the number of the first surface below the top of the aquifer that is larger than or equal to p, and
N* is the number of the last surface above the bottom of the aquifer. Notice that R and Rp reduce to the
equations for stratified flow, (13) and (14), when p = 0 for all p.
Vertical Leakage Between Aquifers
Vertical leakage qz,i-1/2 between layer i-1 and layer i is computed as follows
qz ,i 1/2  Li (ht ,i  hb ,i 1 )
(19)
where Li is the leakance between layer i and layer i-1, hb,i-1 is the freshwater head at the bottom of layer i-1, ht,i
is the head at the top of layer i. The head at the bottom of layer i-1 may be expressed as
(20)
hb,i 1  hi 1  hi 1
where hi-1 is defined as the difference between the freshwater head at the top of layer i-1 and the bottom
of layer i-1, computed for stratified flow as
N
hi 1   n ( n   n 1 )
(21)
n 1
When water leaks upward or downward from one layer to the next, it is possible that lighter water leaks
upward into an overlying aquifer with heavier water at the bottom. In that case, the lighter water is added as a
source term to the zone with the same water. This may be interpreted to mean that fingers grow infinitely fast.
Similarly, it is possible that heavier water leaks downward into an underlying aquifer with lighter water at the
top. Using the same paradigm, the heavier water is added as a source term to the zone with the same heavier
water. In exceptional cases, it is possible that lighter water leaks upward, but only heavier water is present in
the cell above it. In that case, the density of the lighter water is changed to the density of the heavier water,
which is also referred to as the full mixing model. This will cause an error in the volumetric budget of a zone
and is cancelled by the volumetric budget of the zone that the water is added to, so that the overall water
budget remains correct. In a similarly exceptional case it is possible that heavier water leaks downward, but
only lighter fluid is present in the cell below it. In that case, the density of the heavier water is changed to the
density of the lighter water.
Solution Procedure
Initial values must be specified for all dependent variables in each aquifer: the fresh water head h at the
saturated top of each aquifer, and the elevations n of surfaces 2 through N in each aquifer. Boundary
conditions may be specified using standard MODFLOW packages. The boundaries of surfaces 2 through
N may potentially be moving. The boundary conditions for all surfaces are flux-specified during a
timestep. When a surface intersects the bottom of an aquifer, the normal flux is set to zero. When a
8
surface intersects the top of an aquifer, the normal flux is set to the normal component of the total flow in
the aquifer. At the end of each time step, cells along the boundary of each surface are evaluated to
determine whether the boundary should be moved horizontally, as will be discussed in the next chapter.
Given values for the head and elevations of the surfaces at time t, the head and elevations of the surfaces
at time t+t are computed as follows. Differential equation (10) is solved to compute the head at time
t+t, using the elevations of the surfaces at time t to compute the pseudo source term R. In case of
multiaquifer flow the heads in all aquifers are solved simultaneously, as in regular MODFLOW. Next,
equations (11) are solved to compute the elevations of the surfaces at time t+t. This is done for each
aquifer separately, as the fluxes between model layers are known from the head solution. The terms Rp are
computed using the head values at time t+t and the elevations of the surfaces at time t. This procedure is
equivalent to keeping the flow field fixed during a time step.
9
Sea-Water Intrusion (SWI) Package Implementation
SWI is designed such that a MODFLOW model for single-density flow may be converted to a seawater
intrusion model through the addition of one input file to the model, and specification of the SWI input and
output files in the NAME file. No changes are needed in the input files of any of the other packages. The
only additional change that may be required is how the coastal boundary is modeled, as explained later.
The formulation of the SWI package is described in this chapter, as well as guidelines for the construction
of a SWI model.
Finite-difference Solution of the Groundwater Flow Equation
The groundwater flow equation (10) is solved using the standard MODFLOW finite difference approach.
The model domain is discretized in NROW rows, NCOL columns and NLAY layers. Both aquifer and
aquitards are treated in the same manner. The head in row i, column j, layer k is written as hi,j,k. The
discharge out of the block through the right face of the cell is called qi,j+1/2,k. The discharge out of the block
through the front face of the cell is called qi+1/2,j,k. The discharge out of the block through the lower face of the
cell is called qi,j+1/2,k. In a single-density model, the latter discharge may be computed with a central in space
difference scheme as
qi , j ,k 1/2  CVi , j ,k 1/2 (hi , j ,k ,bot  hi , j ,k 1,top )
(22)
where CVi,j,k+1/2 is the is the vertical conductance in row i and column j between layers k and k+1. The head in
layer k represents the head at the bottom of the aquifer and needs to be corrected for density differences (Eq.
(20)). The head in layer k+1 is the head at the top of the aquifer and doesn’t need to be corrected
qi , j ,k 1/2  CVi , j , k 1/2 (hi , j , k  hi , j , k 1  hi , j , k )
(23)
where the term hi,j,k is defined for stratified flow as
N
hi , j ,k   n ( i , j ,k ,n   i , j ,k ,n 1 )
(24)
n 1
The MODFLOW finite difference equivalent of the governing differential equation (11) is given in (5-1)
of Harbaugh (2005), except for the pseudo-source term R and the head correction h, and yields
CRi , j 1/2,k (him, j 1,k  him, j ,k )  CRi, j 1/2,k (him, j 1,k  him, j ,k ) 
CCi 1/2, j ,k (him1, j ,k  him, j ,k )  CCi 1/2, j ,k (him1, j ,k  him, j ,k ) 
CVi , j ,k 1/2 [him, j ,k 1  him, j ,k ]  CVi , j ,k 1/2 (him, j ,k 1  him, j ,k ] 
m
i , j ,k i , j ,k
P
h
 Qi , j ,k  SSi , j ,k ( DELR j DELCi DELVi , j ,k )
him, j ,k  him, j,1k
t m  t m 1

Rim, j,1k  CVi , j ,k 1/2 him, j,1k 1  CVi, j ,k 1/2 him, j,1k
(25)
where CR, CC and CV are the conductance factors along rows, columns and normal to layers,
respectively, P and Q are coefficients for a general external source term (see Eq. 2-21 in Harbaugh,
2005), SS is the specific storage of the cell, DELR, DELC, and DELV are the cell sizes in row, column
and vertical directions, respectively. The superscripts m and m-1 indicate the beginning and end of time
step m (hence, they are not to be interpreted as powers). A finite difference approximation of the pseudosource term R for stratified flow (13) is
N
Rim, j,1k    SRi , j 1/2,k ,n ( im, j11,k ,n   im, j,k1 ,n )  SRi , j 1/2,k ,n ( im, j11,k ,n   im, j,k1 ,n ) 
n 1
SCi 1/2, j ,k ,n ( im1,1j ,k ,n   im, j,k1 ,n )  SCi 1/2, j ,k ,n ( im1,1j ,k ,n   im, j,k1 ,n ) 
(26)
10
The index n indicates the number of the surface. SR and SC with last subscript n are the conductance
factors below surface n along the row and column directions, respectively, which are computed as
SCi , j ,k ,n   n ( i , j ,k ,n  BOTi , j ,k ) / DELVi , j ,k CCi , j ,k
(27)
SRi , j ,k ,n   n ( i , j ,k ,n  BOTi , j ,k ) / DELVi , j ,k CRi , j , k
(28)
where n is given in (12).
Finite-difference Solution for Zeta Surfaces
Once a solution for the head in the aquifer is obtained for time m, the fluxes between layers are computed
with (23). As the fluxes between layers are known, the location of the zeta surface at time m may be
computed for each layer separately. Movement of the zeta surfaces is governed by differential equation
(11), which is again similar to the differential equation solved by standard MODLFOW-2005. The finite
difference approximation of (11) is
SRi , j 1/2,k ,n ( im, j 1,k ,n   im, j ,k ,n )  SRi , j 1/2,k ,n ( im, j 1,k ,n   im, j ,k ,n ) 
SCi 1/2, j ,k ,n ( im1, j ,k ,n   im, j ,k ,n )  SCi 1/2, j ,k ,n ( im1, j ,k ,n   im, j ,k ,n ) 
SNi , j ,k ( DELR j DELCi DELVi , j ,k )
 im, j ,k ,n   im, j,k1 ,n
t m  t m 1
 Gi , j ,k  Rim, j,1k ,n
(29)
where SN is the effective porosity, G is the known source term, and Rim, j,1k , n is the pseudo source term Rn,
which may be computed for stratified flow as
Rim, j,1k ,n  SRi , j 1/2,k ,n (him, j 1,k ,n  him, j ,k ,n )  SRi , j 1/2,k ,n (him, j 1,k ,n  him, j ,k ,n ) 
SCi 1/2, j ,k ,n (him1, j ,k ,n  him, j ,k ,n )  SCi 1/2, j ,k ,n (him1, j ,k ,n  him, j ,k ,n ) 
N
  SR
i , j 1/2, k , p
( im, j11,k , p   im, j,k1 , p )  SRi , j 1/2,k , p ( im, j11,k , p   im, j,k1, p ) 
p 1
pn
SCi 1/2, j ,k , p ( im1,1j ,k , p   im, j,k1 , p )  SCi 1/2, j ,k , p ( im1,1j ,k , p   im, j,k1 , p ) 
(30)
Sinks and Sources
All standard MODFLOW packages may be applied to SWI models. When the package requires the
specification of a head, for example the GHB package, this head must be the equivalent freshwater head
at the top of the layer. The water type of sinks and sources is specified in the SWI input file through the
ISOURCE parameter. Each cell has one value for ISOURCE, hence one model cell can only have sinks
and sources of one type. There are three options for the specification of the water types of sinks and
sources:
11
1. ISOURCE > 0. Sources and sinks are of the same type as water in zone ISOURCE. If such a zone
is not present in the cell, sources and sinks are placed in the zone at the top of the aquifer.
2. ISOURCE = 0. Sources and sinks are of the same type of water as the water at the top of the
aquifer.
3. ISOURCE < 0. Sources are of the same type as water in zone ISOURCE. Sinks are of the same
type of water as the water at the top of the aquifer. This option is useful for the modeling of the
ocean bottom where infiltrating water is salt, yet exfiltrating water is of the same type as the
water at the top of the aquifer.
Tip and Toe Tracking
At the end of every timestep, it must be determined whether the boundaries of the surfaces that separate
the zones move horizontally. Recall that the boundaries of each surface are flux-specified during a
timestep. As such, the surfaces can move up or down along the boundary during a timestep. The boundary
of a surface is either near the bottom of an aquifer (referred to as a toe) or near the top of an aquifer
(referred to as a tip). Here, the algorithm for dealing with a toe is discussed; the algorithm for a tip is
analogous. The algorithm works separately along the rows and columns of a numerical grid.
Consider the surface shown in Fig. 3. At the end of a timestep, the slope i of the surface in boundary cell j
is determined as shown. If this slope is larger than a specified maximum threshold value, imax, the surface
is moved into the adjacent empty cell (Fig. 3). This is accomplished by lowering the elevation of the
surface in the boundary cell by a small amount  and at the same time specifying the elevation of the
surface in the adjacent empty cell to be  above the base of the aquifer. As such, continuity of flow is
ensured (an appropriate adjustment is made if the cell sizes are not equal). Conversely, if the thickness
between the surface in cell j and the bottom of the aquifer is smaller than a minimum threshold value,
min, the surface is moved into the adjacent non-empty cell. This is accomplished by lowering the
elevation in the toe cell to the bottom of the aquifer and increasing the elevation of the surface in the
adjacent non-empty cell by the same amount. The above described algorithm requires the specification of
three parameters: the maximum slope, imax, the minimum thickness min, and the small amount .
Extensive experimentation has shown that transient simulations are rather insensitive to reasonable
choices of these parameters. As a rule of thumb, it is sufficient to specify the maximum slope imax by
taking a representative value of the slope of a surface in an aquifer. Reasonable values for  and min
may then be chosen as
  0.1imax x
 min  0.1
(31)
(32)
where x is the cell size in the direction of seawater intrusion. Separate values are used for the maximum
slope of the toe cells and the maximum slope of the tip cells.
Simulation of aquifers and aquitards.
Both aquifers and aquitards need to be modeled as regular layers, i.e., quasi-three-dimensional layers may
not be used (Harbaugh, 2005 p.2-16).
Pulling or pushing an interface to another layer
During pumping in a model layer where a zeta surface is present in the model layer below it, it is possible
that a zeta surface is pulled upward through the bottom of the model layer. Simlarly, when a freshwater
bell is growing, for example due to artificial recharge, it is possible that a zeta surface is pushed through
12
the bottom of a model layer to the layer below it. This upconing downconing into another layer is handled
automatically by SWI provided cell sizes and time steps are taken small enough. An example of both
upconing an downconing is presented in Example 4.
Simulation of the coastal boundary
The coastal boundary needs to be simulated with multiple cells extending into the ocean. It is
recommended to use GHB cells to represent the ocean bottom. In case the ocean bottom doesn’t extend
into the ocean (when the coastline is formed by a rocky cliff, for example), additional cells with a high
hydraulic conductivity need to be added into the ocean, representing the water in the ocean (similar to
modeling lakes with high-k cells). The reason for extension of the model below the ocean or into the
ocean is that the tip cell of the last surface may not be at the boundary of the model. If this occurs, there is
generally no way for saltwater to enter the model and the amount of saltwater in the model domain will
remain constant, which is not correct. Extension of the model boundary at the ocean is shown in
Examples 3 and 4.
Adaptive Time Stepping –
Will be added by Joe.
Using the SWI Package
The main input of the SWI file is the initial position of the zeta surfaces. A value needs to be specified for
every cell in the model. The zeta surface is placed at the top of a model layer when a value is entered that
is above the top of the model layer and similarly the zeta surface is placed at the bottom of a model layer
when a value is entered that is below the bottom of the model layer. For the case that a surface is present
at only one point in the vertical everywhere, this means that the same grid of zeta values may be entered
for every model layer and SWI will determine in which cells the elevation of the zeta surface falls
between the top and model of each layer (see also Example 3). It is important to start with a reasonably
smooth variation of the zeta surface. In reality it is not possible for a surface to be discontinuous of have
sharp changes in gradient. When an irregular zeta surface is entered, it may take a while in a transient
simulation before a physically realistic variation is reached and it may take very small time steps to get
there.
Head simulations using the SWI package may be steady-state or transient, as specified in the
DISCRETIZATION file. Simulation of the change in the density (salinity) distribution is always
transient. A steady-state head simulation combined with a transient change in the position of the zeta
surfaces is often reasonable, as the zeta surfaces commonly react much slower to a change in the system
than the head. When the head is simulated as steady-state, it means that aquifer storage is neglected and
the heads change instantaneously to a change in the system (for example a well that turns on). An
additional benefit of using the steady-state option is that larger time steps may be used than when the
transient option is used, as the required time step for the head simulation is often smaller than for the zeta
simulation. It is important to realize that the steady-state option does not mean that the steady-state
density distribution is computed. The steady-state density distribution can only be computed by running
the model long enough until steady-state is reached.
Output Files and Post Processing
Output files from a SWI simulation consist of standard MODFLOW output plus the SWI output file. The SWI
output file stores the position of every zeta surface of every model layer for the same time steps for which
heads are recorded. The format of the SWI output file is the same as a standard MODFLOW budget file. Heads
and drawdowns are written in terms of equivalent freshwater heads at the top of each model layer.
There are two main differences in the standard MODFLOW output files that are unique to SWI. First, the
LIST file records the volumetric budget for the entire model, as is recorded for regular MODFLOW models. In
13
addition, it records the volumetric budget for each zone. The volumetric budget for the entire model includes
one new term SWIADDTOCH, which needs to be added to the CONSTANT HEAD term to obtain the correct
value, and may only be non-zero if the constant head cell is located in a cell where the ISOURCE term is not 1.
The volumetric budget of each zone contains three terms. The first term, BOUNDS + STORAGE is the
volume change caused by flow to or from model boundaries plus volume changes due to changes in aquifer
storage. Model boundaries is the cumulative term for all sinks and sources in the model, including but not
limited to recharge, wells and general head boundaries. The second term, CONSTANT HEAD, is the volume
change due to flow to or from constant head boundaries. The third term, ZONE CHANGE, is the volume
change due to a changes in the size of the zone. The inflow caused by ZONE CHANGE represents the
decrease in the volume of the zone while the outflow caused by ZONE CHANGE represents the increase in the
volume of the zone.
Second, the flow terms in the cell-by-cell flow file are expressed as volumetric fluxes rather than mass fluxes
and represent the total flux in the model layer. The fluxes in the cell-by-cell flow file need to be adjusted by
the terms SWIADDTOFLF, SWIADDTOFRF, and SWIADDTOFFF, for the lower face (LF), right face (RF)
and front face (FF), respectively. These values are written to the cell-by-cell flow file or a separate file when
the appropriate flag is set in the SWI input file. Note that separate fluxes need to be computed for each zone to
be able to do particle tracking. Such computations can be carried out using the information in the cell-by-cell
flow file and the position of the zeta surfaces, but they are currently not provided.
Tips for Designing MODFLOW-2005 Models Using the SWI Package
A good approach to start a SWI model is to start with a cross-sectional model for interface flow (one zeta
surface). The cross-sectional model for interface flow may be used to obtain insight in the flow pattern,
and to evaluate appropriate grid resolution, aquifer parameters, hydrologic stresses, computer run times,
and solver limitations. Once the cross-sectional model gives reasonable results, the model can be
extended to two dimensions or additional surfaces may be added, again in a step-wise fashion.
A first estimate of an appropriate grid resolution may be obtained using existing guidelines for singledensity models (e.g., Anderson and Woessner, 1992). For SWI simulations, an additional constraint is
that cell sizes need to be chosen small enough that cells contain zeta surfaces with elevations between the
top and bottom of the aquifer. When the cell size is too large, the zeta surface is at the top of the model
layer in one cell and at the bottom of the model layer in the adjacent cell. This may be seen as an aquifer
where the aquifer is entirely salt in one cell and entirely fresh in the adjacent cell. SWI computes the
vertical movement of each zeta surface only for zeta surface elevations between the top and the bottom of
the aquifer, so the cell size must be chosen small enough to have at least a few zeta surface elevations
between the top and bottom of the aquifer.
The choice of the time step is another important issue. It is recommended to follow an iterative approach
to determine an accurate time step. Start with short simulations with short time steps and increase the
simulation time and time steps when the results indicate that this is realistic. A good first estimate is that
the elevation of a zeta surface in a cell should not change too much in one time step. A good uppper limit
is 20% of the aquifer thickness. An additional consideration in the selection of the time step is that the tip
or toe can move laterally only one cell during a time step. If the time step is too large, this means that the
tip or toe may move too slowly, which results in a very steep zeta surface at the tip or toe. It is generally
recommended to start a simulation with small time steps to smooth-over any coarseness in the initial
variation of the zeta surface specified by the user.
The SWI results that are typically evaluated at the end of a run include the ASCII output listing file and
binary files that contain equivalent freshwater heads at the top of each model layer, position of each zeta
zeta surface in each model layer, and cell-by-cell flows. Users should remember that flow lines will be
perpendicular to contours of equivalent freshwater head only in a horizontal plane. As the computed
heads are the freshwater heads at the top of a model layer, flow will only be normal to contours of this
head when the top of the aquifer is horizontal.
14
Most sinks and sources in a seawater intrusion model are either fresh (e.g., natural recharge) or salt (e.g.,
infiltration at the ocean bottom). There are commonly no brackish sources in a seawater intrusion model
unless a brackish source of water is present. In a physical system, additional brackish water may be
formed through mixing processes such as dispersion and diffusion. In the absence of brackish sources and
since mixing processes are not represented in a SWI model, the amount of brackish water generally goes
down during a simulation.
15
Example Simulations
Example 1: Rotating interface
The first example concerns interface flow in a vertical plane. The plane of flow is 250 m long, 40 m high,
and 1 m wide. The origin of an x,y,z coordinate system is chosen at the upper left-hand corner of the
plane. Flow is confined at all times and the top and bottom of the aquifer are impermeable and horizontal
(Fig. 4a). The hydraulic conductivity is k = 2 m/d and the effective porosity is n = 0.2. There is a constant
flow Qx0 = 1 m2/d from left to right (gradient of 0.0125). The aquifer and water are treated as
incompressible. The interface is, initially, at a 45 degree angle from (x,z) = (80,0) to (x,z) = (120,-40)
(Fig. 4a). The density of the fresh and saltwater are  = 1000 kg/m3 and  = 1025 kg/m3, respectively.
Hence, the dimensionless density difference of the fresh and saltwater are  = 0 and  = 0.025,
respectively (computed with (2)).
The domain is discretized into 50 cells of 1 m wide, and the time step is 2 d. A source of 1 m3/d is
specified in cell 1. MODFLOW requires the specification of one head-specified point (the rotation of the
interface is independent of this choice). In this model, the head is fixed to 0 m in the last cell.
There are two zones (a freshwater zone and a seawater zone) and thus one surface between them
(NPLN=1). Flow is treated as stratified (ISTRAT=1) and the elevation of the interface is written to a file
called swiex1.zta every 50 time steps. The total number of time steps is 200, and thus the position of the
interface is printed at 100, 200, 300, and 400 days. The slope of the interface varies from 1 (initial
position) to about 0.2 (when it reaches the corner of the domain). The maximum slope of the toe and tip is
chosen to be 0.2 (TOESLOPE=TIPSLOPE=0.2), and the  and min parameters are computed
according to (31) and (32) (ZETAMIN=0.01 DELZETA=0.1). Since this is stratified flow with 2 zones, 2
 values must be specified (0 and 0.025). The source/sink terms are specified to be fresh water (zone 1)
everywhere (ISOURCE=1) except in cell 1 where saltwater enters the model and ISOURCE equals 2.
Example 1 is summarized in Table 1.
Table 1. Summary of Example 1
Model size: Lx = 250 m, Ly = 1 m.
Cell size and number of cells: x = 5 m, y = 1 m, Nx = 50, Ny = 1.
Simulation: t = 2 d, ttot = 400 d.
Aquifer: Layer 1: k = 2 m/d, H = 40 m, n = 0.2
Sources: layer 1, row 1, column 1: Q = 1 m3/d.
Fixed head: cell 50: h = 0 m.
SWI: Type: stratified, TIPSLOPE = 0.2, TOESLOPE = 0.2,  = 0.1 m, min = 0.01 m.
Density: 2 zones: 1 = 1000 kg/m3, 2 = 1025 kg/m3.
Results are compared to the exact Dupuit solution of Wilson and Sa Da Costa (1982) and are shown in
Fig. 4b; the dots are the exact solution. The constant flow Qx0 results in an average velocity of vx = 0.125
m/d. The exact solution is a rotating straight interface of which the center moves to the right with this
velocity.
The volumetric budget for each zone is recorded in the LIST file. The budget is discussed here for the end
of the simulation. The total inflow into zone 2 (the salt water) consists of 400 m3 of saltwater inflow from
the boundary (the inflow on the right side of the model) plus a ZONE CHANGE of 3.3245 m3
representing “inflow” caused by the interface moving downward. The total outflow of zone 2 consists of a
ZONE CHANGE of 403.3245 m3 representing “outflow” caused by the interface moving upward. The
total increase in the volume of the saltwater zone is computed by subtracting the inflow ZONE CHANGE
from the outflow ZONE CHANGE, which gives 400 m3, which is indeed equal to the total inflow of
saltwater from the boundary.
16
Example 2: Rotating brackish zone
The second example is a modification of the previous example and includes three zones and no inflow at
the boundary. A 300 m long section of the same aquifer is considered and all boundaries are
impermeable. The groundwater is divided into 3 zones: fresh, brackish, and salt water, with dimensionless
densities of  = 0,  = 0.0125, and  = 0.025, respectively. First, flow is treated as stratified. Initially, at
time t = 0, both interfaces are straight and make a 45 degree angle with the horizontal and run from (x,z) =
(150,0) to (x,z) = (190,-40), and from (x,z) = (110,0) to (x,y) = (150,-40) (Fig. 5a). The head is fixed to 0
at the top of cell 1. The brackish zone will rotate to a horizontal position through time.
The domain is discretized into 60 cells of 5 m wide, and the time step is 2 d. There are three zones and
thus the initial elevations of two surfaces are specified. The shape of the brackish zone is computed at t =
2000 days by taking 1000 time steps of 2 days. The maximum slope of the toe and tip is chosen to be 0.4,
and the  and min parameters are computed according to (31) and (32). Example 2 is summarized in
Table 2.
Table 2. Summary of Example 2
Model size: Lx = 300 m, Ly = 1 m.
Cell size and number of cells: x = 5 m, y = 1 m, Nx = 60, Ny = 1.
Simulation: t = 2 d, ttot = 2000 d.
Aquifer: Layer 1: k = 2 m/d, H = 40 m, n = 0.2
Fixed head: cell 50: h = 0 m.
SWI: Type: both, TIPSLOPE = 0.4, TOESLOPE = 0.4,  = 0.2 m, min = 0.02 m.
Density: 3 zones: 1 = 1000 kg/m3, 2 = 1012.5 kg/m3, 3 = 1025 kg/m3.
Results at t = 2000 days are compared to results obtained with SEAWAT (Langevin et al., 2008) in Fig.
5b. For SEAWAT, the aquifer is discretized into 12,000 cells of 1 by 1 meter and the transport equation is
solved with the TVD option in MT3DMS, without diffusion or dispersion. The grey zone in Fig. 5b is the
zone in SEAWAT corresponding to 5% to 45% salt, and 55% to 95% salt and the solid lines are the
elevations of the interfaces as computed by SWI. The brackish zone rotates slightly faster in SWI than in
SEAWAT. The faster rotation by SWI may be caused by the Dupuit approximation, while the slower
rotation by SEAWAT may be caused by numerical dispersion.
For illustration purposes, the same problem is solved with the variable density option. This requires two
changes in the SWI input file. First, the ISTRAT parameter must be set to 0 to indicate variable density
flow (rather than stratified flow). Second, the NU variable now needs 4 input values: the value along the
top of zone 1, the two surfaces, and the bottom of zone 3, so that the item 4 in the SWI input file
becomes:
0.000000 0.000000 0.025000 0.025000
Notice that this gives an average value of  = 0.0125 in the brackish zone. The position of the surfaces
after 2000 days is not that much different from the stratified simulation (Fig. 5c).
Example 3: Intrusion in a two-aquifer system
The third example concerns interface flow in a two-aquifer system. The plane of flow is 4000 m long and
1 m wide. Both aquifers are 20 m thick and are separated by a leaky layer with a thickness of 1 m. The
origin of the x,y,z coordinate system is chosen at the lower left-hand corner of the plane. The hydraulic
conductivities of the top and bottom aquifer are 2 m/d and 4 m/d, respectively, while the vertical
hydraulic conductivity of the aquitard is 0.01 m/d. The effective porosity of all layers is 0.2. The left 600
m of the model extends below the ocean floor; the freshwater head at the ocean bottom is 50 m. The
leakance for outflow from the aquifer into the ocean is 0.02 d-1 (resistance of 50 days). There is a
comprehensive flow of 0.03 m2/d towards the ocean (distributed according to the transmissivities over
17
both aquifers at the right boundary). Initially the interface between the fresh and seawater is straight and
is at the top of aquifer 1 at x = -100 and has a slope of -0.025 (dashed line in Fig. 6). The dimensionless
density is  = 0.025. After 1000 years, the interface approaches its steady position. Next, the inflow at the
right boundary is halved and the intrusion of the saltwater wedge is simulated.
Each aquifer is discretized into 200 cells of 20 m long. Two stress periods of 1000 years are simulated
with a time step of 2 years. During the first stress period, the last cell in the top aquifer has a source of
0.01 m3/d, and the last cell in the bottom aquifer a source of 0.02 m3/d; in the second stress period these
values are halved. The ocean bottom is modeled with General Head Boundary (GHB) cells. The head in
the GHB cells is 50 m and the COND value is computed as the Vcont (0.02) of the ocean bottom
multiplied by the area of the cell (20). The water in the GHB cells represents the ocean and is salt. The
ISOURCE parameter is set to -2 in the SWI package so that water that infiltrates into the aquifer from the
GHB cells is salt, while water that flows out of the model at the GHB cells is of the same type as the
water at the top of the aquifer. The tip and toe tracking parameters are a TOESLOPE of 0.02 and a
TIPSLOPE of 0.04, a DELZETA of 0.06 and a ZETAMIN of 0.006. Example 3 is summarized in Table
3. The positions of the interface are shown every 200 years for the first stress period in Fig. 6a. The initial
position of the interface is indicated with the dashed line. The steady-state position, computed with the
Ghyben-Herzberg formula is shown in red and is approached after 1000 years. The movement of the
saltwater tongue inland due to the reduction in flow towards the coast is shown in Fig. 6b, where the
interface is shown every 200 years starting from the dashed position.
Table 3. Summary of Example 3. Units: meters, days, and kilograms
Model size: Lx = 4000, Ly = 1, 3 layers
Cell size and number of cells: x = 20, y = 1, Nx = 200, Ny = 1
Simulation:
Stress period 1: t = 2*365, t1 = 1000 years, t2 = 1000 years.
Layer 1: k = 2, H = 20, n = 0.2
Layer 2: kv = 0.01, H = 1, n = 0.2
Layer 3: k = 4, H = 20, n = 0.2
Sources stress period 1, layer 1, row 1, column 200, Q = 0.01; layer 3, row 1, column 20, Q = 0.02
Sources stress period 2, layer 1, row 1, column 200, Q = 0.005; layer 3, row 1, column 20, Q = 0.01
GHB cells: layer 1, row 1, columns 1-30, head = 50, Cond = 40
SWI: TIPSLOPE = 0.04, TOESLOPE = 0.01,  = 0.06, min = 0.006
Density: 1 = 1000, 2 = 1025
Example 4: Upconing below a pumping well in a two-aquifer system
The fourth example concerns flow below a square island with infiltration and a pumping well. The island
is 2050 by 2050 meters and consists of two aquifer layers that extend below the sea bottom. Each layer is
20 m thick and has a hydraulic conductivity of 10 m/d. The recharge is 0.4 mm/d. The model is extended
below the sea bottom for 500 m along all sides. Initially, the interface between freshwater and saltwater is
1 m below the surface on the island and at the sea bottom below the sea; the dimensionless density
difference is 0.025. The growth of the freshwater bell is simulated for 200 years, after which a pumping
well with a discharge of 250 m3/d is started in layer 1 at (x,y) = (0,300) (the origin of the coordinate
system is at the center of the island). For the first simulation, the well pumps for 30 years, after which the
interface almost reaches the top of the top aquifer layer. In the second simulation, an additional well
pumping saltwater with a discharge of 25 m3/d is started in layer 2, 12 years after the freshwater well
starts in layer 1. The saltwater well is intended to prevent the interface from upconing into the top layer.
The model is discretized into 61 by 61 square cells of 50 by 50 m. The wells are located in cell (31,36).
The time step is 0.2 year for the first stress period, and 0.1 year for the other stress periods. The tip and
18
toe tracking parameters are a TOESLOPE and TIPSLOPE of 0.005, a DELZETA of 0.025 and a
ZETAMIN of 0.0025. Example 4 is summarized in Table 4.
Results are shown in a cross-section along the centerline of the island through the wells (Fig. 7). The
position of the interface is shown every 20 years for simulation period 1, during the build-up of the
freshwater bell (Fig. 7a). After 200 years, the interface approaches the steady-state position. The position
of the interface for Simulation 2 is shown every 3 years in Fig. 7b. The interface cones up into the top
layer after approximately 14 years. The position of the interface for Simulation 3 is shown every 3 years
in Fig. 7c. The first 12 years, the interface is the same as for Simulation 2. After 12 years, the saltwater
well is started in the bottom layer, which slows down the upconing. The position of the interface vs. time
in the bottom layer is shown at the location of the wells (x,y) = (0,300) in Fig. 4d. The blue line represents
Simulation 2 and the red line represents Simulation 3; the red line lies on top of the blue line for the first
12 years. The effect of starting the saltwater well after 12 years draws the interface back down, but
eventually it is going up again.
Table 4. Summary of Example 4. Units: meters, days, and kilograms
Model size: Lx = 3050, Ly = 3050
Cell size and number of cells: x = 50, y = 50, Nx = 61, Ny = 61
Simulation:
Stress period 1: t = 200 years, t = 0.2
Simulation 1, Stress period 2, t = 30 years, t = 0.1
Simulation 2, Stress period 2, t = 12 years, t = 0.1, Stress period 3, t = 18 years, t = 0.1
Layer 1: k = 10, H = 20, n = 0.2
Layer 2: k = 10, H = 20, n = 0.2
Recharge on the center 41 by 41 cells: 0.0004
Well stress period 2 and 3, layer 1, row 31, column 61, Q = 250
Well stress period 3(simulation 2), layer 2, row 31, column 61: Q = 25
GHB cells: layer 1, 10 cells along edge of model on al sides: head = 50, Cond = 62.5
SWI: TIPSLOPE = 0.005, TOESLOPE = 0.005,  = 0.025, min = 0.0025
Density: 1 = 1000, 2 = 1025
19
Evaluation of Adopted Approximations
The objective of this section is to evaluate the effect of the three main approximations of the SWI
formulation, which are (see Chapter 2):
1. Dispersive or diffusive mixing between freshwater and seawater are not represented
2. Inversion of seawater overlying freshwater is only allowed between aquifers, not within an
aquifer, and the vertical exchange between these zones is approximated
3. Vertical movement of the surface within one model layer is based solely on continuity where
resistance to vertical flow is neglected.
Several studies investigated the effect of the Dupuit approximation. Seawater intrusion models based on
the Dupuit approximation yield accurate results for many practical problems of interface flow (e.g. Bear
and Dagan, 1964), even if the slope of the interface is relatively steep (up to 45 degrees; Chan Hong and
Van Duijn, 1989). Strack and Bakker (1995) showed that adoption of the Dupuit approximation for
variable density flow gives accurate results for the instantaneous flow field. It also compares well with
fully 3D numerical solutions for a rotating brackish zone (Bakker et. al, 2004).
Here, SWI results are compared to results obtained with SEAWAT (Langevin et al. 2008), which
simulates fully coupled variable-density groundwater flow and solute transport, including mixing,
inversion, and vertical resistance to flow. For SEAWAT to simulate variable-density flow and transport
accurately, however, aquifers have to be discretized more finely, particularly in the vertical direction. The
increased discretization is often required to minimize numerical dispersion near the saltwater-freshwater
interface as well as to represent convective flow patterns. This may result in excessive model run times,
which may be prohibitive for the development of regional-scale saltwater intrusion models.
Results from previous investigations of the two programs indicates that they produce similar results,
provided that the SEAWAT grid was finely discretized and numerical dispersion was minimized (Bakker
et al. 2004). The discussion here evaluates when SWI simulations are appropriate, considering the effects
on saltwater transport of density inversion, dispersion, and vertical anisotropy. (This discussion is based
on the paper by Dausman et al., 2010).
Problem Setup
To evaluate the appropriate usage of SWI, SWI and SEAWAT simulation results are compared for
conditions of (1) density inversion, (2) varying degrees of dispersion, and (3) varying degrees of vertical
resistance to flow. Results from a single SWI simulation are compared against multiple SEAWAT
simulation results. The conceptual model is a 2-dimensional representation of a two-aquifer system (Fig.
8). The aquifer and simulation parameters are listed in Table 5. For the SEAWAT simulation, transport
was solved using the finite-difference method with upstream weighting. The general-head boundary cells
representing the ocean have a salinity value of 35 g/L total-dissolved solids (TDS). For all three tests, the
initial conditions represent a density inversion, as shown by saltwater in aquifer 1 overlying freshwater in
aquifer 2 over a portion of the simulation domain. In the first test of density inversion, the SEAWAT
model has dispersivities set to zero and the ratio of vertical to horizontal hydraulic conductivity set to one.
For the second test, the results from SEAWAT simulations with increasing values of longitudinal
dispersivity (and with transverse dispersivity set to 10% of the longitudinal value) are compared with the
SWI simulation results. For the third test, SEAWAT simulation results using decreasing values of vertical
hydraulic conductivity are compared to the SWI simulation results.
Parameters
Row
Columns
SWI
1
200
SEAWAT
1
800
20
Layers (total)
2
82
# layers in aquifer 1
1
40
# layers in aquifer 2
1
40
# layers for aquitard
N/A
2
Horizontal Hydraulic Cond. Aq. 1
2
2
(m/d)
Horizontal Hydraulic Cond. Aq. 2
4
4
(m/d)
VCONT Aquitard (day-1)
0.01
0.01
Freshwater Inflow aquifer 1 (m2/day)
0.005
0.005
2
Freshwater Inflow aquifer 2 (m /day)
0.01
0.01
Porosity
0.2
0.2
Longitudinal Dispersivity, αL (m)
N/A
0-100
Range
Anisotropy ratio (Kvert/Khor) Range
N/A
0.001-1
Total time period modeled (years)
500
500
Approximate model run time
2 seconds
3 hours
Table 5. Parameters used in evaluation of the adopted approximations
Density Inversion
At 100 years, the models match well except in the area of the density inversion (Fig. 9). With SWI, the
vertically migrating seawater from aquifer 1 is added to the seawater in aquifer 2. The final location of the
interface (Fig. 9) at 500 years shows that the two model results are almost identical. However, the toe of
the interface for the SEAWAT model is slightly seaward of the SWI surface. It should be noted that, as
dispersivity is set to zero in the SEAWAT model, theoretically there should be no transition zone from
saltwater to freshwater. As seen in Fig. 9 there is a thin transition zone as a result of numerical dispersion.
Dispersion
Model results reveal that as dispersivity increases, the location of the toe moves seaward (Fig. 10).
Results indicate that if the width of the transition zone from freshwater to seawater is less than 15% of the
horizontal scale of the problem, SWI would likely produce valid results (Fig. 10). These results indicate if
dispersivity is estimated to be greater than a few meters (Fig. 10), a fully-coupled variable-density flow
and transport model might be necessary for more accurate results. Calibrated values of the dispersivity for
seawater intrusion models in the literature are generally less than a couple of meters (e.g. Oude Essink,
2001); therefore, SWI would likely be appropriate in most saltwater intrusion models. It is also important
to note that the SWI-simulated interface surface is landward of SEAWAT-simulated transition zone.
Vertical Resistance to Flow
As vertical resistance to flow is increased for the SEAWAT simulations, the toe of the interface only
moves seaward when the anisotropy ratio (Kvert/Khor) is smaller than 0.01 (Fig. 4). As with the results
from the dispersion test, the SWI-simulated interface is landward of all SEAWAT-simulated 17.5 mg/L
TDS contours.
Conclusion
21
It is concluded that for the investigated problem, the approximations in SWI produce reasonable estimates
of the location of the saltwater-freshwater interface. SWI results may not be realistic when the dispersion
across the interface is large, when the anisotropy ratio is very small, or in some systems where inversion
occurs and a significant amount of vertical fingering is observed. For simulations where SWI and
SEAWAT results diverge, the SWI-simulated interface tends to be more landward than the SEAWATsimulated transition zone. In terms of evaluating the risk of saltwater intrusion to freshwater resources,
these results could be considered more conservative. Because of the computational savings in SWI with
run times of a few seconds, as compared to SEAWAT with run times of a few hours, it is preferred and
valid to use in many groundwater models if the run times of full variable-density flow and transport
models are prohibitive such as in a regional-scale model.
References
Anderson, M.P., and W.W. Woessner. 1992. Applied groundwater modeling: Simulation of flow and
advective transport: San Diego, Calif., Academic Press, 381 p.
Bakker, M. 2003. A Dupuit formulation for modeling seawater intrusion in regional aquifer systems.
Water Resources Research, 39(5), 1131-1140.
Bakker, M., G.H.P. Oude Essink and C.D. Langevin. 2004. The rotating movement of three immiscible
fluids − a benchmark problem. Journal of Hydrology, 287, 271-279.
Bear, J. and G. Dagan. 1964. Some exact solutions of interface problems by means of the hodograph
method. Journal of Geophysical Research, 69 (8), 1563-1572.
Chan Hong, J.R. and C.J. van Duijn. 1989. The interface between fresh and salt groundwater: A
numerical study. IMA Journal of Applied Mathematics. 42. 209-240.
Harbaugh, A.W. 2005. MODFLOW-2005, the US Geological Survey modular ground-water model -- the
ground-water flow process: U.S. Geol. Surv. Techniques and Methods, vol. 6-A16.
Holzbecher, E. 1998. Modeling density-driven flow in porous media, Principles, numerics, software.
Springer Verlag, Berlin.
Langevin, C.D., D.T. Thorne, Jr., A.M. Dausman, M.C. Sukop, and W. Guo. 2008. SEAWAT Version 4:
A Computer Program for Simulation of Multi-Species Solute and Heat Transport. U.S.
Geological Survey Techniques and Methods Book 6, Chapter A22, 39 p.
Maas, C and M.J. Emke. 1988. Solving varying density groundwater problems with a single density
computer program. Natuurwetenschappelijk Tijdschrift. 70, 143-154.
Oude Essink, G.H.P. 2001. Salt Water Intrusion in a Three-dimensional Groundwater System in The
Netherlands: A Numerical Study. Transport in Porous Media, 43(1), 137-158.
Strack, O.D.L. 1989. Groundwater Mechanics. Prentice Hall, Englewood Cliffs, NJ.
Strack, O.D.L. 1995. A Dupuit-Forchheimer model for three-dimensional flow with variable density.
Water Resources Research, 31(12), 3007--3017.
Strack, O.D.L., and M. Bakker. 1995. A validation of a Dupuit-Forchheimer formulation for flow with
variable density. Water Resources Research, 31(12), 3019-3024.
Weiss, E. 1982 A model for the simulation of flow of variable-density ground water in three dimensions
under steady-state conditions. USGS Open file report 82-352.
Wilson, J.L. and A. Sa da Costa. 1982. Finite element simulation of a saltwater/freshwater interface with
indirect toe tracking. Water Resources Research. 18(4), 1069-1080.
Wooding, R.A. 1969. Growth of fingers at an unstable diffusing interface in a porous medium or HeleShaw cell. Journal of Fluid Mechanics. 39(3), 477-495.
22
Appendix 1. SWI2 Data Input Instructions
The SWI-2005 file contains solver variables, data values for the different zones, algorithm parameters, the
initial positions of the surfaces, the type of sources and sinks, and zeta observation locations. Optional
variables are indicated in [square brackets].
FOR EACH SIMULATION
1.
Data:
NPLN ISTRAT NOBS ISWIZT ISWIBD ISWIOBS
Module: URWORD
2a. Data: NSOLVER IPRSOL MUTSOL
Module: URWORD
IF NSOLVER = 2
2b. Data: MXITER ITER1 NPCOND ZCLOSE RCLOSE RELAX NBPOL DAMP [DAMPT]
Module: URWORD
3.
Data: TOESLOPE TIPSLOPE ZETAMIN DELZETA [NADPTMX] [NADPTMN] [ADPTFCT]
Module: URWORD
4.
Data:
NU(ISTRAT=0: NPLN, ISTRAT=1: NPLN+1)
Module: U1DREL
FOR EACH SURFACE
FOR EACH LAYER
5.
Data:
ZETA(NCOL,NROW)
Module: U2DREL
FOR EACH LAYER
6.
Data:
SSZ(NCOL,NROW)
Module: U2DREL
FOR EACH LAYER
7.
Data:
ISOURCE(NCOL,NROW)
Module: U2DREL
IF NOBS > 0
FOR EACH NOBS
8.
Data:
OBSNAM LAYER ROW COLUMN
Module: URWORD
Explanation of variables read by the SWI2 Package
NPLN—Number of surfaces (planes). This equals the number of zones minus one.
ISTRAT—Flag indicating density distribution.
0 – density varies linear between planes.
1 – density is constant between planes.
NOBS—Number of OBS observation locations. 1
ISWIZT—Flag and a unit number for ZETA output
If ISWIZT> 0, unit number for ZETA output
If ISWIZT≤ 0, ZETA will not be recorded.
ISWIBD—Flag and a unit number for BUDGET output. When this option is selected, corrections to the
cell by cell flows computed by MODFLOW will be written to the same or different file (depending
23
on the unit number). Corrections are called SWIADDTOFLF, SWIADDTOFRF, and
SWIADDTOFFF, for the lower face (LF), right face (RF) and front face (FF), respectively
If ISWIBD> 0, unit number for BUDGET
If ISWIBD≤ 0, BUDGET will not be recorded.
ISWIOBS—Flag and a unit number for OBS output
If ISWIOBS> 0, unit number for ASCII OBS
If ISWIOBS= 0, OBS will not be recorded.
If ISWIOBS< 0, |ISWIOBS| unit number for binary OBS.
NSOLVER—Flag indicating solver used to solve for ZETA surfaces
If NSOLVER= 1, the MODFLOW DE4 solver will be used.
If NSOLVER= 2, the MODFLOW PCG solver will be used.
IPRSOL—is the printout interval for printing convergence information. If IPRSOL is equal to zero, it is
changed to 999. The maximum ZETA change (positive or neg- ative) and residual change are
printed for each iteration of a time step whenever the time step is an even multiple of IPRSOL.
MUTSOL—is a flag that controls printing of convergence information from the solver.
If MUTSOL= 0, tables of maximum head change and residual each iteration will be printed.
If MUTSOL= 1, only the total number of iterations will be printed.
If MUTSOL= 2, no information will be printed.
If MUTSOL= 3, information will only be printed if convergence fails.
MXITER—Maximum number of outer iterations—that is, calls to the solution routine. 2
ITER1—Maximum number of inner iterations—that is, iterations within the solution routine.
NPCOND—flag used to select the matrix conditioning method for the MODFLOW PCG solver
(NSOLVER= 2). NPCOND is not specified if NSOLVER is 3.
If NPCOND= 1, is for Modified Incomplete Cholesky (for use on scalar computers).
If NPCOND= 2, is for Polynomial (for use on vector computers or to conserve
computer memory).
ZCLOSE—is the ZETA change criterion for convergence, in units of length. When the maximum
absolute value of ZETA change from all nodes during an iteration is less than or equal to ZCLOSE,
and the criterion for RCLOSE is also satisfied (see below), iteration stops.
RCLOSE—is the residual criterion for convergence, in units of cubic length per time. The units for length
and time are the same as established for all model data. (See LENUNI and ITMUNI input variables
in the Discretization File.) When the maximum absolute value of the residual at all nodes during an
iteration is less than or equal to RCLOSE, and the criterion for ZCLOSE is also satisfied (see
above), iteration stops.
RELAX—is the relaxation parameter used with NPCOND= 1. Usually, RELAX= 1.0, but for some
problems a value of 0.99, 0.98, or 0.97 will reduce the number of iterations required for
24
convergence. RELAX is only used if NSOLVER is 2 and NPCOND is 1. RELAX is not specified
if NSOLVER is 3.
NBPOL—is only specified when NSOLVER= 2 and used when NPCOND= 2 to indicate whether the
estimate of the upper bound on the maximum eigenvalue is 2.0, or whether the estimate will be
calculated. NBPOL= 2 is used to specify the value is 2.0; for any other value of NBPOL, the
estimate is calculated. Convergence is generally insensitive to this parameter. NBPOL is not used if
NPCOND is not 2.
DAMP—is the damping factor and is only specified when NSOLVER= 2 . It is typically set equal to one,
which indicates no damping. A value less than 1 and greater than 0 causes damping.
If DAMP> 0, applies to both steady-state and transient stress periods.
If DAMP< 0, applies to steady-state periods. The absolute value if used as the dampening factor.
DAMPT—is the damping factor for transient stress periods and is only specified when NSOLVER= 2 .
DAMPT is enclosed in brackets indicating that it is an optional variable that only is read when
DAMP is specified as a negative value. If DAMPT is not read, then the single damping factor,
DAMP, is used for both transient and steady-state stress periods.
TOESLOPE—Maximum slope of toecells.
TOESLOPE—Maximum slope of toecells.
TIPSLOPE—Maximum slope of tipcells.
ZETAMIN—Minimum elevation of a plane before it is removed from a cell for a rule how to compute
this parameter).
DELZETA—Elevation for a plane when it is moved into an adjacent empty cell (see for a rule how to
compute this parameter).
NADPTMX—Maximum number of SWI timesteps per MODFLOW timestep. If NADPTMX is not
specified or is less than 1, it is changed to 1.
NADPTMN—Minimum number of SWI timesteps per MODFLOW timestep. NADPTMN is not
specified if NADPTMX is not specified or is equal to 1. If NADPTMN is not specified, it is
changed to NADPTMX.
ADPTFCT—is the factor used to evaluate tip and toe thicknesses and control the number of SWI time
steps per MODFLOW time step. When the maximum tip or toe thickness exceeds the product of
TOESLOPE or TIPSLOPE the cell size and ADPTFCT, the number of SWI timesteps is increased
to a value less than or equal to NADPT. When the maximum tip or toe thickness is less than the
product of TOESLOPE or TIPSLOPE the cell size and ADPTFCT, the number of SWI time steps
is decreased in the next MODFLOW timestep to a value greater than or equal to 1. ADPTFCT is
not read if NADPTMX is equal to NADPTMN.
NU—Values of the dimensionless density
ISTRAT = 1 – Density of each zone (NPLN+1 values).
ISTRAT = 0 – Density along each surface (NPLN values)
25
ZETA—Initial elevations of the surfaces.
SSZ—Effective porosity
ISOURCE—Source type of any external sources or sinks, specified with any outside package (i.e. WEL
package, RCH package, GHB package). There are three options:
If ISOURCE> 0 – Sources and sinks are of the same type as water in zone ISOURCE. If such a
zone is not present in the cell, sources and sinks are placed in zone at the top of the aquifer.
If ISOURCE= 0 – Sources and sinks are of the same type of water as at the top of the aquifer.
If ISOURCE< 0 – Sources are of the same type as water in zone ISOURCE. Sinks are of the same
type of water as at the top of the aquifer. This option is useful for the modeling of the ocean
bottom where infiltrating water is salt, yet exfiltrating water is of the same type as the water
at the top of the aquifer.
OBSNAM—is a string of 1 to 12 nonblank characters used to identify the observation. The identifier need
not be unique; however, identification of observations in the output files is facilitated if each
observation is given a unique OBSNAM.
LAYER—is the layer index of the cell in which the ZETA observation is located.
ROW—is the row index of the cell in which the ZETA observation is located.
COLUMN—is the column index of the cell in which the ZETA observation is located.
26
Figures
Figure 1. Conceptual models. (a) Vertical cross-section of an aquifer; (b) density distribution along A-A' for stratified flow; (c)
density distribution along A-A' for variable density flow
Figure 2. Schematic vertical cross-section of an aquifer including the numbering of the zones and surfaces
27
Figure 3. The toe-tracking algorithm
28
Figure 4. (a) Setup of example 1, (b) Position of interface at t = 100, 200, 300, and 400 days. Exact solution (line) and SWI
solution (dots) at center of cells
29
Figure 5. Setup of example 2 (top), boundary of brackish zone after 2000 days; shaded area is SEAWAT (middle), and
stratified (solid) versus variable density option after 2000 days (bottom). Need to add SEAWAT results
30
Figure 6. Results for Example 3. Position of the interface every 200 years starting from the dashed position, where the red
line is the steady-state position according to Ghyben-Herzberg (top), and intrusion of the interface every 200 years after the
outflow towards the coast is halved, starting from the dashed position
31
Figure 7. Results of example 4 along centerline of island through wells. (a) Infiltration for 200 years, position of interface
every 20 years. (b) 30 years of pumping freshwater well in layer 1, position of interface every 3 years. (c) 30 years of pumping
freshwater in layer 1 with a saltwater well added in layer 2 after 12 years, position of interface every 3 years. (d) elevation of
interface at location of well in layer 2 vs. time, position without saltwater well (blue) and with saltwater well started after 12
years (red).
32
Figure 8. Problem setup for evaluation of main approximation
Figure 9. The SWI results (squares) and the transition zone simulated with SEAWAT at 5, 17.5 and 30 g/L TDS at
100 and 500 years
Figure 10 SWI results (crosses) and the 50% seawater line (17.5 mg/L TDS) from SEAWAT (black/grey lines) with a range of
dispersivity values. Longitudinal dispsersivity is shwon (transverse dispersivity is 10% of longitudinal dispersivity). The graph
is the longitudinal dispersivity vs. the width of the transition zone (1 gL – 34 g/L TDS) from SEAWAT simulations.
33
Figure 11. SWI results (crosses) and the 50% seawater line (17.5 mg/L TDS) obtained with SEAWAT (black/grey lines) with
different anisotropy ratios (vertical divided by horizontal hydraulic conductivity). The graph shows the anisotropy ratios vs.
the difference between the toe of the SWI interface and the toe (50% seawater line or 17.5 mg/L TDS) from the SEWAT
simulation
34