HYDROLOGICAL PROCESSES
Hydrol. Process. 29, 2052–2064 (2015)
Published online 1 October 2014 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/hyp.10354
Using MODFLOW code to approach transient hydraulic head
with a sharp-interface solution
Carlos Llopis-Albert1* and David Pulido-Velazquez2,3
Instituto Geológico y Minero de España (IGME), C/ Cirilo Amorós, 42 – Entreplanta, 46004 Valencia, Spain
Instituto Geológico y Minero de España (IGME), Urb. Alcázar del Genil, 4-Edif. Zulema, Bajo, 18006 Granada, Spain
Departamento de Ciencias Politécnicas, Escuela Universitaria Politécnica, UCAM Universidad Católica San Antonio de Murcia, Murcia, Spain
1
2
3
Abstract:
Saltwater intrusion problems have been usually tackled through analytical models because of its simplicity, easy implementation
and low computational cost. Most of these models are based on the sharp-interface approximation and the Ghyben–Herzberg
relation, which neglects mixing of fresh water and seawater and implicitly assumes that salt water remains static. This paper
provides insight into the validity of a sharp-interface approximation defined from a steady state solution when applied to transient
seawater intrusion problems. The validation tests have been performed on a 3D unconfined synthetic aquifer, which include
spatial and temporal distribution of recharge and pumping wells. Using a change of variable, the governing equation of the steady
state sharp-interface problem can be written with the same structure of the steady confined groundwater flow equation as a
function of a single potential variable (ϕ). We propose to approach also the transient problem solving a single potential equation
(using also the ϕ variable) with the same structure of the confined groundwater flow equation. It will allow solving the problem
by using the classical MODFLOW code. We have used the parameter estimation model PEST to calibrate the parameters of the
transient sharp-interface equation. We show how after the calibration process, the sharp-interface approach may provide accurate
enough results when applied to transient problems and improve the steady state results, thus avoiding the need of implementing a
density-dependent model and reducing the computational cost. This has been proved by comparing results with those obtained
using the finite difference numerical code SEAWAT for solving the coupled partial differential equations of flow and densitydependent transport. The comparison was performed in terms of piezometric heads, seawater penetration, transition zone width
and critical pumping rates. Copyright © 2014 John Wiley & Sons, Ltd.
KEY WORDS
seawater intrusion; analytical solutions; sharp interface; coastal aquifers; critical pumping rates; Ghyben–Herzberg;
calibration
Received 5 March 2014; Accepted 2 September 2014
INTRODUCTION
Salt water intrusion (SWI) occurs in coastal freshwater
aquifers when the different densities of both the saltwater
and freshwater allow the salt water to intrude into the
freshwater aquifer. These aquifers are usually supporting
large populations, irrigated agriculture and industrial
areas, so that the demand of groundwater withdrawal
may exceed the recharge rate, thus favouring the SWI.
The encroaching seawater will encounter an area known
as the transition zone, where the freshwater and saltwater
mix and form an interface. The unknown concentration of
the fluid within the transition zone makes the problem
complex (Bear, 1999). In addition, the interface moves
back and forth through time naturally because of
*Correspondence to: Carlos Llopis-Albert, Instituto Geológico y Minero
de España (IGME), C/ Cirilo Amorós, 42 – Entreplanta, 46004 Valencia,
Spain.
E-mail: cllopisa@gmail.com
Copyright © 2014 John Wiley & Sons, Ltd.
fluctuations in the recharge and pumping rates. The
seawater intrusion processes, recent advances and future
challenges were investigated in Werner et al. (2013),
while the effect of climate change on sea water intrusion
was analysed by Sherif and Singh (1999).
The scarcity of reliable estimates of parameters and
variables and the computational burden in most real cases
prevents the use of variable-density models (Sreekanth
and Datta, 2010). Furthermore, the computational burden
implies a major disadvantage in coupled management
simulation–optimization models for control and remediation of SWI due to multiple calls of the simulation model
by the optimization algorithm (Dhar and Datta, 2009).
Then SWI problems are usually tackled by common
simplifications such as the sharp-interface approach and
the Ghyben–Herzberg relation because of its simplicity in
terms of required parameters and computational burden.
On the one hand, the sharp-interface approach entails that
there is no mixing between fresh and salt water, which is
one of the most distinctive features of SWI and its
USING MODFLOW TO APPROACH TRANSIENT HEAD WITH A SHARP INTERFACE
associated dynamics (Henry, 1964; Bear, 1979). On the
other hand, the Ghyben–Herzberg relation can be applied
when the interface is practically stabilized, which means
steady state flow conditions (Essaid, 1999). Through such
approximations, several authors derived or applied
analytical solutions to deal with SWI problems in coastal
aquifers (Strack, 1976; Mahesha, 1996; Iribar et al., 1997;
Dagan and Zeitoun, 1998; Cheng and Ouazar, 1999; Naji
et al., 1999; Sakr, 1999; Cheng et al., 2000; Mantoglou,
2003; Mantoglou et al., 2004; Park and Aral, 2004;
Masciopinto, 2006; Paster et al., 2006; Mantoglou and
Papantoniou, 2008; Pool and Carrera, 2011; Shi et al.,
2011; Mas-Pla et al., 2013).
The sharp-interface approach for steady state flow
conditions can be solved using single-density groundwater codes using a change of variable (e.g., Mantoglou,
2003; Bakker and Schaars, 2013). In this sense, the wellknown MODFLOW code can be used (McDonald and
Harbough, 1988). The changes of variable are common
technique to solve groundwater flow problems with
numerical and analytical solutions (Pulido-Velazquez
et al., 2006, 2007a, b; Capilla et al., 2009).
Although the MODFLOW code is not the most
appropriate to modelling coastal aquifer, it is still very
employed to asses some management alternatives in these
aquifers (e.g., van Camp et al., 2013).
Finally, a sensitivity analysis of the validity of the sharpinterface approach for steady state flow conditions and with
regards to groundwater flow and mass transport parameters
was developed by Llopis-Albert and Pulido-Velazquez
(2013). Different scenarios of spatially distributed recharge
values and spatial wells placement were analysed in this work.
This paper goes a step further by discussing the validity
of a sharp-interface approach, based on the Ghyben–
Herzberg relation, when extended to deal with transient
flow conditions. Furthermore, the sharp-interface equation is solved using MODFLOW, thus easing the
implementation of the model. A few works have dealt
with transient SWI problems using the sharp-interface
approach, although the equation is not solved using
directly MODFLOW (Eichert et al., 1982; Essaid, 1999;
Cartwright et al., 2004; Bakker et al., 2013). Then, Essaid
(1999) presented the SHARP model, which is a quasi
three-dimensional, numerical model that solves by a finite
difference scheme the equations for coupled freshwater
and saltwater flow separated by a sharp interface in
layered coastal aquifer systems. Each aquifer is represented by a layer in which flow is assumed to be
horizontal. Cartwright et al. (2004) developed a sharpinterface model to examine and quantify the response of
the salt–freshwater interface in a coastal aquifer to a
wave-induced groundwater pulse. Bakker et al. (2013)
developed the seawater intrusion (SWI2) package for
MODFLOW. This package is based on the Dupuit
Copyright © 2014 John Wiley & Sons, Ltd.
2053
approximation. It does not account for diffusion and
dispersion and has no need for vertical discretization of the
aquifer since the water is discretized into zones of different
densities. A comparison between SWI package and
SEAWAT was also performed by Dausman et al. (2010).
The comparison led to successful results, and it was
determined that when the diffusion and dispersion effects
are important the use of the SWI package is discouraged.
The main difference of the present work regarding
previous models dealing with transient sharp-interface
solutions is that we propose to solve a governing equation
with the same structure of the steady confined groundwater
flow equation as a function of a single potential variable (ϕ).
It would allow a direct use of MODFLOW (without
requiring any additional package integrated into it). Then
this approach allows that already developed groundwater
flow models can be directly used to tackle SWI problems by
means of its well-known graphical interfaces with only a
simple change of variables in some parameters of the
groundwater flow equation. This will ease the implementation of SWI problems for a large number of modellers who
are only familiar with the use of MODFLOW as presented in
its graphical interfaces. It will also allow a direct integration
of SWI problems into already developed groundwater
management models based on MODFLOW simulation
(e.g., Pulido-Velazquez et al., 2008; Llopis-Albert et al.,
2014 using the code GWM (Ahlfeld et al., 2005).
The Eigenvalue Method techniques could be also
applied to obtain an efficient conceptual solution (PulidoVelazquez et al., 2007b) of the confined governing
equation. Therefore, it could be directly employed to
solve the proposed approximation to integrate coastal
aquifers in complex river basin management models (with
many elements and aspects to be analysed) (PulidoVelazquez et al., 2007b, 2011a, b). Furthermore, this
approach also allows the direct integration of SWI
problems into already developed models for dealing with
optimal contamination remediation, conjunctive use and
groundwater management strategies and designs (e.g.,
GWM (Ahlfeld et al., 2005), SOMOS (Peralta , 2004) or
MGO (Zheng and Wang, 2003).
In addition, different transient tests regarding spatial
and temporal distribution of recharge and pumping wells
has been simulated. The validation tests have been
performed on a 3D unconfined synthetic aquifer, thus
allowing taking control of all variables and parameters.
The hydrological parameters of the potential steady
equation can be obtained after a transformation of their
counterparts parameters in the flow equation, which
provides the fresh piezometric head (h). But the additional
parameters introduced to approach the variability of the
solution in the time have to be calibrated.
We will show that the proposed sharp-interface
approach may provide accurate enough results when
Hydrol. Process. 29, 2052–2064 (2015)
2054
C. LLOPIS-ALBERT AND D. PULIDO-VELAZQUEZ
applied to transient problems. This has been proved by
comparing the results of the sharp-interface approach with
those obtained using the finite difference numerical code
SEAWAT for solving the coupled partial differential
equations of flow and density-dependent transport. The
comparison was performed in terms of piezometric heads,
seawater penetration, transition zone width and critical
pumping rates. Furthermore, this work also shows how
results are improved under steady state conditions; that is,
piezometric heads are closer to those provided by densitydependent transport models after the calibration process.
MATERIALS AND METHODS
As abovementioned, we compare a sharp-interface solution
based on the Ghyben–Herzberg approach with a numerical
three-dimensional variable-density flow simulations. The
governing equation of the sharp-interface approach
(a potential equation deduced from a change of variable
with the same structure of the confined groundwater flow
equation) is solved using the numerical flow model
MODFLOW (McDonald and Harbough, 1988; Harbaugh
et al., 2000), whereas the variable-density flow simulations
are solved using SEAWAT (Guo and Langevin, 2002).
The hydrological parameters of the steady equation of the
sharp-interface solution are obtained after a transformation
of their counterparts parameters in the flow equation, which
provides the fresh piezometric head (h). Note that the
hydraulic conductivity (K) of the sharp-interface equation
(Equation (2)) is the hydraulic conductivity (K′) of the
groundwater flow equation divided by d (the distance
between the aquifer base and the sea level; see Figure 1)
(Mantoglou, 2003).
But the additional parameters introduced to approach
the variability of the solution in the time need to be
calibrated. Then the specific storage of the porous medium
(Ss) of the sharp-interface equation (Equation (2)) is not the
storage of the aquifer (Ss′) used to calibrate flow models, and
no equation is available to state their relationship.
Finally, in order to perform a sensitivity analysis and
taking into account that K can vary spatially by several
orders of magnitude (e.g., Llopis-Albert and Capilla,
2009, 2010a, b), we will also compare these results with
those obtained calibrating also K.
The conditional data are the piezometric heads (h)
obtained from solving the flow equation, which after a
transformation (Equation (1)) provides the potential flow
(ϕ) of the sharp-interface equation to be calibrated. The
calibration of the parameters of the sharp-interface
governing equations is carried out by means of the
inverse model PEST (Doherty, 2004). This paper is
intended to show how the proposed sharp-interface
approximation may provide accurate enough results when
applied to transient simulations. The validity of the sharpinterface approach in transient simulations is defined in
terms of piezometric heads, seawater penetration, transition zone width and critical pumping rates. In addition,
the use of the numerical model MODFLOW ease the
implementation of the sharp-interface approach in
coupled management simulation–optimization models
for control and remediation of SWI, thus reducing the
computational burden that prevents the use of variabledensity models. The governing equations of such
approaches are following presented:
Governing equations based on a single potential formulation
The single potential formulation implies some simplifications of the SWI problem in coastal aquifers. The sharpinterface assumption neglects mixing of fresh water and
seawater (Henry, 1964; Bear, 1979). In addition, the
interface is considered to be practically stabilized, so that
steady state flow conditions are required to be applied
(Essaid, 1999). The Dupuit hydraulic assumption is used to
obtain two-dimensional equations, by averaging the flow
equation in vertical direction. The depth of the interface
below the sea level (ξ) is deduced from the Ghyben–
Herzberg relation (Figure 1), whereas the freshwater depth
from the free surface to the interface is represented by b. By
means of such assumptions, Strack (1976) derived a singlepotential theory across the saltwater and freshwater zones.
Many other authors have used this formulation more
recently (e.g., Cheng and Ouazar, 1999; Cheng et al.,
2000; Mantoglou, 2003; Mantoglou et al., 2004; Park and
Aral, 2004). The flow potential ϕ for an unconfined aquifer
is defined as follows:
Zone 1 :
Figure 1. Cross section of an unconfined coastal aquifer and parameters of
the sharp-interface approach
Copyright © 2014 John Wiley & Sons, Ltd.
Zone 2 :
i
1h 2
hf ð1 þ δÞ d 2
2
2
ð 1 þ δÞ ϕ¼
hf d
2δ
ϕ¼
(1)
Hydrol. Process. 29, 2052–2064 (2015)
2055
USING MODFLOW TO APPROACH TRANSIENT HEAD WITH A SHARP INTERFACE
ρ ρ
where δ ¼ sρ f ¼ Δρ
ρf ≈ 0:025 being ρs and the density of
f
salt water and freshwater, respectively; hf is the freshwater
piezometric head with reference to the impermeable bottom
of the aquifer ρf; and d is the height of the aquifer from
the horizontal aquifer base to the sea level. In this paper,
we propose to approach also the transient problem
solving a single potential equation (using also the ϕ
variable) with the same structure of the confined
groundwater flow equation.
Eventually, the differential equation to be solved either
by analytical or numerical approximations is
∂
∂ϕ
∂
∂ϕ
∂ϕ
K
þ
K
þ N Q ¼ Ss
∂x
∂x
∂y
∂y
∂t
(2)
This equation is not deduced mathematically in a
formal way. It is only based on the fact that the potential
equation deduced by Strack (1976), which is analogous to
the confined groundwater flow equation, provides good
approximations for steady state SWI problems. So, we
propose to add a term to approach the variability on the
time, maintaining the structure of the potential confined
equation.
Therefore, we pursue to provide insight about the validity
of this equation by comparing results with a densitydependent model for a set of different transient simulations,
which include spatial and temporal distribution of recharge
and pumping wells.
The equation is solved as a one-zone problem through
the use of proper boundary conditions. The parameters of
this equation are the specific storage of the sharp
approximation (different to the real porous medium)
(Ss); the hydraulic conductivity of the sharp approximation (that can be deduced from the hydraulic conductivity
and the distance d, as was mentioned previously) (K), the
recharge distributed over the surface of the aquifer (N)
and the pumping rate from the aquifer wells (Q). We have
used the finite difference groundwater flow model
MODFLOW (McDonald and Harbough, 1988) to solve
Equation (2). The interface location ξ is calculated as
follows:
Zone 1 :
ξ ¼ d;
z ¼ 0;
Zone 2 :
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ϕ
ξ¼
;
ð1 þ δÞδ
when
z ¼ d ξ;
ð1 þ δÞδ 2
d ≤ϕ
2
when 0 ≤ ϕ ≤
ð1 þ δÞδ 2
d
2
(3)
Copyright © 2014 John Wiley & Sons, Ltd.
Accordingly, the piezometric hf is determined from
Zone 1 :
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
hf ¼ 2ϕ þ ð1 þ δÞd 2 ;
when
Zone 2 :
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2δϕ
hf ¼
þ d;
ð 1 þ δÞ
when
ð1 þ δÞδ 2
d ≤ϕ
2
0≤ϕ≤
ð1 þ δÞδ 2
d
2
(4)
The toe of saltwater wedge (τ) is located at ξ = d
(Figure 1). From Equation (4), this means that the toe is
located where ϕ takes the following values:
ϕ ðxτ ; yτ Þ ¼
ð1 þ δÞδ 2
d
2
(5)
Governing equations for flow and transport in SEAWAT
The coupled partial differential equations for variabledensity flow and transport are solved by means of the
three-dimensional finite difference numerical code
SEAWAT (Guo and Langevin, 2002). It considers
multispecies solute and heat transport, together with the
effect that fluid viscosity variations have on resistance to
groundwater flow. The SEAWAT model comprises
MODFLOW (McDonald and Harbough, 1988) and
MT3DMS (Zheng and Wang, 1999) into a single program
that conserves fluid mass, rather than fluid volume, while
considers equivalent freshwater head as the primary
dependent variable. Additionally, there are well-known
preprocessors and postprocessors of such programs to
ease the process to create SEAWAT datasets and
visualize results. SEAWAT iteratively solves the coupled
partial differential equations on account of groundwater
flow causes the redistribution of solute concentration,
which leads to alter the density field, and therefore, the
groundwater movement. On the one hand, the groundwater flow is calculated in MODFLOW from freshwater
head gradients and relative density–difference terms. On
the other hand, the resulting groundwater flow field is
used for the solute transport model MT3DMS. It takes
into account the processes of advection, molecular
diffusion and mechanical dispersion. Eventually, an
updated density field is calculated from the new solute
concentrations and incorporated back into MODFLOW as
relative density–difference terms. In this way, the
coupling between flow and transport is carried out
through a synchronous time stepping approach that cycles
between solutions of flow and transport models.
The formulation of the partial differential equation for
variable-density groundwater flow in porous media is
Hydrol. Process. 29, 2052–2064 (2015)
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C. LLOPIS-ALBERT AND D. PULIDO-VELAZQUEZ
subsequently presented (Langevin et al., 2007):
μ0
ρ ρ0
∇z
∇· ρ K 0 ∇h0 þ
μ
ρ
¼ ρSs;0
(6)
∂h0
∂ρ ∂C
þθ
ρs q′s
∂t
∂C ∂t
with
ρ0
μ
K0
h0
Ss,0
t
θ
C
q′s
the fluid density [ML3] at the reference concentration and reference temperature
dynamic viscosity [ML1 T1]
hydraulic conductivity tensor of material saturated
with the reference fluid [LT1]
hydraulic head [l] measured in terms of the
reference fluid (commonly freshwater) of a specified concentration and temperature.
specific storage [l1], defined as the volume of
water released from storage per unit volume per
unit decline of h0
time [T]
porosity [–]
salt concentration [ML3]
source or sink [T1] of fluid with density ρs
The transport of solute mass in SEAWAT is formulated
by the following partial differential equation (Langevin
et al., 2007):
ρb K kd ∂ θC k
1þ
¼ ∇· θD·∇C k ∇· qC k
θ
∂t
q′s qCks
(7)
with
ρb
K kd
Ck
D
q
C ks
bulk density (mass of the solids divided by the total
volume) [ML3]
distribution coefficient of species k [L3 M1]
concentration of species k [ML3]
hydrodynamic dispersion coefficient tensor [L2T1]
specific discharge [LT1]
source or sink concentration [ML3] of species k
Parameter estimation process using PEST
The hydrological parameters of the steady equation of
the sharp-interface solution are obtained after a transformation of their counterparts parameters in the flow
equation. Furthermore, when extending the solution to
transient problems, there is no equation to state the
relationship between the specific storage of the porous
medium (Ss) of the sharp-interface equation (Equation (2))
and the storage of the aquifer (Ss′) of the groundwater flow
Copyright © 2014 John Wiley & Sons, Ltd.
equation. In addition, in order to perform a sensitivity we
will also compare results with those obtained calibrating
also K.
To spatially calibrate these hydrogeological parameters
of the sharp-interface equation, we have used the
parameter estimation model PEST (Doherty, 2004) (i.e.,
the hydraulic conductivity and the storage coefficient of
Equation (2)). The aim is to report the same transient
piezometric field (h) as that given by the flow equation
after reversing the transformation once the sharp-interface
equation is solved.
PEST is a nonlinear parameter estimation procedure
that is based on calculating the mismatch, in the least
squares sense, between the simulated values (given by the
approximation using the finite difference groundwater
flow equation or its equivalent single potential formulation) and the observed values, and then evaluating how
best to correct that mismatch. Subsequently, after solving
the groundwater flow equation, the fresh piezometric head
(hm) is transformed into the flow potential (ϕ) using
Equation (1). Consequently, these values are used as
conditional data of the sharp-interface model to be
calibrated. Finally, the PEST model adjusts model input
data and runs the model again until a defined maximum
number of optimization iterations are reached or a
convergence criterion is achieved. With the calibrated
parameters, we will prove that a better approximation
between the results of the sharp-interface model and the
SEAWAT model are achieved. This comparison is
performed in terms of seawater penetration, transition
zone width and critical pumping rates. The final goal is to
show that after the calibration process the sharp-interface
approach may be extended to transient conditions while
providing accurate enough results.
Because of the implementation of the parameter
estimation algorithm (Gauss–Marquardt–Levenberg algorithm), the calibration process is performed with high
computational efficiency. It is based on an iterative
process implying a linearization of the relationship
between model parameters and model-generated observations using a Taylor expansion about the currently best
parameter set. This entails to obtain the derivatives of all
observations with respect to all parameters. After solving
the problem, a better parameter set is obtained, and the
new parameters tested by running the model again.
Therefore, the problem is to minimize the following
objective function (ϕ):
ϕ ¼ ðc co Jðb bo ÞÞt Qðc co Jðb bo ÞÞ (8)
where
c
co
is the experimental observation vector
is the model-calculated observation vector
Hydrol. Process. 29, 2052–2064 (2015)
USING MODFLOW TO APPROACH TRANSIENT HEAD WITH A SHARP INTERFACE
J
Q
b
bo
u = (b bo)
is the Jacobian matrix. Jij is the derivative
of the i’th observation with respect to the
j’th parameter
is the observation weights matrix (square
and diagonal) whose i’th diagonal element
qii is the square of the weight wi attached to
the ith observation
is the new parameter vector
is the current parameter vector
is the parameter upgrade vector
Applying the Taylor’s theorem,
c ¼ co þ Jðb bo Þ
(9)
and the upgrade vector u becomes
1
u ¼ ðJt QJÞ Jt Qðc co Þ
(10)
Finally, the parameter covariance matrix is
C ðbÞ ¼ σ 2 ðJt QJÞ
1
(11)
where σ 2 represents the variance of each of the elements
of c.
NUMERICAL SIMULATIONS
The numerical simulations have been carried out in a 3D
unconfined synthetic aquifer for different scenarios of
recharge and pumping. This situation represents many of
the SWI problems in coastal aquifers (Gimenez and
Morell, 1997; Gómez Gómez et al., 2003). The aim is to
compare the transient solutions of the sharp-interface
approach (based on the Ghyben–Herzberg relation)
solved using the numerical model MODFLOW with
those obtained solving the coupled partial differential
equations of flow and density-dependent transport using
the code SEAWAT. The comparison is performed in
terms of piezometric heads, seawater toe penetration,
transition zone width and critical pumping rates. Furthermore, we also compare the transient piezometric heads of
the flow model with those obtained with the sharpinterface approach. We are intended to show how after the
calibration process we can use the sharp-interface
approach under transient situations, solved in
MODFLOW, without the need of implementing a
density-dependent model. This would make it appropriate
to be integrated into groundwater and river basin
management models (e.g., Pulido-Velazquez et al.,
2008; Peña-Haro et al., 2010, 2011; Molina et al.,
2013a,2013b; Llopis-Albert et al., 2014) while reporting
the same piezometric field as the flow model and
providing good results enough in terms of seawater toe
Copyright © 2014 John Wiley & Sons, Ltd.
2057
penetration, transition zone width and critical pumping
rates when compared to the density-dependent model.
Additionally, this work is also intended to show how
results are improved under steady state conditions after
the calibration process.
In order to define more precisely the conceptual model
and to have a complete control of all parameters, a
synthetic aquifer has been used. Then, the discrepancies
caused by other sources different than the use of the two
approaches are reduced.
On the one hand, the coastal aquifer has been modelled
with SEAWAT using a three-dimensional block-centred
finite difference grid. A schematic cross section of the
aquifer is depicted in Figure 2. All frontiers are defined
with boundary conditions of no-flow are except the right
one, which represents the sea in the coastal aquifer. The
sea boundary has been defined with a prescribed head of
0 m. Transport boundaries of prescribed concentrations
have also been defined. The left contour, which represents
the freshwater boundary, has been defined with concentrations of 0 mg/l. The right contour, which represents the
coastal boundary, has been defined with concentrations of
35 mg/l. The finite difference scheme of the model is
made up of 61 columns and 60 rows in the horizontal
plane, with a cell size of 25 × 25 m. Because the numerical
experiment has been designed presenting symmetry with
respect to the X axis only half of the domain is simulated.
Hence, the domain extends over a size of 1500 m × 1500 m.
SEAWAT requires transient simulations so that in order
to compare its results with those obtained under steady state
conditions in the sharp-interface approach, a large enough
period of time has been simulated (25 years), which allows
reaching flow steady state conditions at the aquifer.
In order to ensure a good accuracy in the model results,
some requirements are needed (Guo and Langevin, 2002).
First, for models with an unconfined layer, the centre
elevation for each cell in the upper layer should be
slightly higher than the highest expected elevation of the
water table. Second, the grid resolution in the vertical
Figure 2. Cross section of the numerical test aquifer, boundary conditions
and pumping and observation wells. The spatial location of the wells is as
follows: W1 = row 11, column 17; W2 = row 20, column 17
Hydrol. Process. 29, 2052–2064 (2015)
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C. LLOPIS-ALBERT AND D. PULIDO-VELAZQUEZ
direction requires a much greater level of detail to represent
the complex flow patterns near areas of high concentration
gradients. A vertical discretization of 10 layers has been
used. The layers are defined as confined/unconfined layers,
where the transmissivity varies, with a thickness of 10 m for
the first layer and 0.5 m for the rest of them. Third, a spatially
uniform cell volumes are defined, which are less prone to
numerical instabilities than models designed with variable cell
volumes. Therefore, the elevation of the bottom of the first
layer is at 0 m, while it takes the value of 5 m for the layer
number 10. Eventually, the aquifer has been defined with a
large extend compared to its depth as required by the sharpinterface approach (Mantoglou and Papantoniou, 2008).
A density of freshwater and seawater of 1000 kg/m3
and 1025 kg/m3 is defined, respectively. The slope of the
linear equation of state that relates fluid density to solute
concentration is defined as 0.7143. The hydraulic
conductivity, specific storage, specific yield and effective
porosity are defined accordingly to the calibrated values
as explained below. The third-order TVD scheme was
used to solve the advection, with a Courant number of
0.75. Dispersion values of 0.1 and 0.01 have been defined
for the relations between the horizontal and vertical
transverse dispersivity and the longitudinal dispersivity,
respectively. The effective molecular diffusion has been
considered as zero.
On the other hand, the sharp-interface approach has
been solved numerically using MODFLOW. A different
vertical discretization with only one layer has been used,
as required by this approach, of 5 m. The Preconditioned
Conjugate Gradient with the modified Incomplete
Cholesky method has been used to solve the flow
problem. Table I presents the different cases analysed
and also indicates parameters and stress periods where the
recharge and pumping stresses are applied within the
temporal discretization of the transient simulations.
Parameters and recharge and pumping rates have been
chosen to ensure that the toe of saltwater wedge falls
inside the domain, thus allowing a better comparison
between the two approaches. Four fully penetrating wells
are considered, which are active or inactive depending on
the case analysed. The spatial location of the pumping
wells in terms of the number of row and column is,
respectively, 11 and 17 for W1 and 20 and 17 for W2
(Figure 2). In addition, we have defined 36 observation
boreholes, which information is used as conditional data
(Figure 2). Therefore, for steady state simulations, 36
piezometric head values are used as conditional data in
the parameter estimation process, whereas for transient
simulations, two measures taken at each observation
borehole and at different times are used, i.e., a total of 72
conditional data. These transient measurements were
selected for covering the whole temporal horizon.
The different cases cover a set of groundwater flow and
mass transport parameters, and different scenarios of
spatially distributed recharge values and spatial wells
placement. The parameters of the sharp-interface equation
(Equation (2)) have been jointly calibrated. Both
homogeneous and heterogeneous estimated parameter
fields have been obtained using PEST. For homogeneous
cases only, one value of the hydraulic conductivity and
the storage coefficient are jointly calibrated. For heterogeneous cases, the parameters to be calibrated are defined
as strips parallel to the coast. Six strips or zones are
considered to provide accurate enough results. Therefore,
six calibrated values are obtained for steady state cases 1
and 3 (six values of hydraulic conductivity), while 12
calibrated values are obtained for transient cases 2 and 4
(six for the hydraulic conductivity and six for the storage
coefficient).
The sharp-interface approach due to its limitations
does not consider longitudinal dispersion and vertical
conductivity.
The temporal discretization for transient simulations of
both approaches is 25 years, where the simulation time
unit are days. The initial heads (h) or flow potential (ϕ) in
transient simulations has been defined as the flow
conditions of the steady state cases.
The critical pumping rates in coastal aquifers are
defined as the maximum permissible discharge without
Table I. Cases and parameters analysed
Cases/Parameters
1
2
3
Recharge (mm/year)
20
80
Wells (m3/day)
NO
40 ON–OFF each
6 months
NO
Layer thickness (m)
Hydraulic conductivity
(m/day)
Specific storage (1/m)
Specific yield
Steady/Unsteady flow
5
10
–
–
Steady
Copyright © 2014 John Wiley & Sons, Ltd.
5
10
0.001
0.05
Unsteady
62.5 W1 and W2
always activated
5
10
–
–
Steady
4
5
80
80
62.5 W1 and W2
62.5 W1 and W2
ON–OFF each 6 months ON–OFF each 6 months
5
variable
10
10
0.001
0.05
Unsteady
0.001
0.05
Unsteady
Hydrol. Process. 29, 2052–2064 (2015)
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USING MODFLOW TO APPROACH TRANSIENT HEAD WITH A SHARP INTERFACE
salinizing the well. On the one hand, for the densitydependent transport model, a 0.1% of mixing with
seawater at the well is adopted as the salinization
threshold in the density-dependent transport model (Pool
and Carrera, 2011). Therefore, the toes of the interface for
the sharp-interface approach were compared with the
0.1% isoconcentration lines of the density-dependent
transport model. On the other hand, the critical pumping
rates for the sharp-interface approach were defined as
those where the X coordinate of the toe penetration match
up with the stagnation point (xs; see Figure 1). The
stagnation point (xs) is the point in the streamline that
separates water flowing into the well from that flowing
into the sea. When increasing the pumping rate at the
well, the interface migrates landward and the stagnation
point migrates seaward. Eventually they cross, seawater
enters into the well capture zone and the well becomes
salinized. Hence, the critical pumping rate Qc is reached
when ξ = c and x = xs (Figure 1).
This formulation leads critical pumping rates and toe
penetrations to be calculated based on ‘potential constrains’. It protects the wells from intrusion by keeping a
potential at the wells larger than the toe potential, as
Þδ 2
shown by Equation (4). That is, ϕ i > ð1þδ
2 d ; i = 1, …,
m, where m represents the number of wells. In contrary,
the ‘toe constraint’ formulation protects the wells from
saltwater intrusion by avoiding the toe of the interface
to reach the wells, i.e., xτi < xi. Mantoglou (2003)
showed that ‘potential constraint’ formulations provide
simpler solutions and smaller and safer pumping rates
since the wells are not salinized if there was a small
increase of pumping rates or a small decrease of aquifer
recharging rates.
RESULTS AND DISCUSSION
The criterion to evaluate simulation results after the
calibration process is based on performance measures of
Equation (12) (ωi). It compares the piezometric head (hs)
obtained from solving in MODFLOW the sharp-interface
equation (i.e., solving Equation (2) and transforming ϕ
to h) with those obtained solving the flow equation (hm):
ηh ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
2
ωi hsi hm
i
i¼1;…;mh
Therefore, the performance measure ηh is defined as the
square root of a weighted
mean of the square departures
of piezometric head hsi from the measured heads hm
i ,
where are the weights assigned to measurements hm
i (i = 1,
…, mh), which are defined to add up to the unity and with
the same value.
The K is calibrated for steady state simulations,
whereas K and Sy are jointly calibrated for unsteady
cases. In addition, those parameters are shown as a
homogeneous value for the whole domain or as different
values corresponding to strips parallel to the coast
(heterogeneous values). Table II presents the performance
measure ηh, which reports a good agreement for all for the
different cases and assumptions. However, because of the
greater smoothness of the hydraulic gradient, steady state
cases and scenarios entailing only recharge (without
pumping wells) lead to lower ηh values. The performance
measurement is even lower, as expected, when different
parameters are calibrated for the whole domain (heterogeneous cases). Note that the performance depends on the
condition data selection (number of data, location, times
when were taken) in transient simulations, so that a better
results could be achieved if more conditional data had
been used. We have only conditioned to 72 data spatially
and temporal distributed to reproduce the scarcity of data
in real case studies. Additionally, results also show that
before the calibration procedure (i.e., with the initial set of
parameters), the agreement can be considered good for
most real cases. This is an important conclusion since we
could use directly a transient sharp-interface model to
deal with seawater intrusion problems without performing
a calibration process and implementing a densitydependent model. Figures 3 and 4 reinforce this idea.
Figure 3 shows the agreement with regard to h for the
initial set of parameters and between the sharp-interface
equation, the flow equation (both solved using
MODFLOW) and the SEAWAT model. Figure 4 presents
the same results for the heterogeneous set of parameters
and all cases after the calibration procedure. Again, the
Table II. Performance measurements (h) after the parameter calibration process for the different cases and scenarios
Initial
parameters
Calibrated parameters
(homogeneous)
Cases
ɳh
0.05380
0.01694
Cases
ɳh
0.05233
0.00999
Calibrated parameters
(heterogeneous)
Initial
parameters
Calibrated parameters
(homogeneous)
0.00365
0.06611
0.02259
0.00302
0.08697
0.02457
1
2
3
Copyright © 2014 John Wiley & Sons, Ltd.
Calibrated parameters
(heterogeneous)
0.01250
4
0.02040
Hydrol. Process. 29, 2052–2064 (2015)
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C. LLOPIS-ALBERT AND D. PULIDO-VELAZQUEZ
Figure 3. Comparison of the piezometric heads at observation well #10, for all cases and using the initial set of groundwater parameters, between the
sharp-interface approach, the flow model (both solved using MODFLOW) and the SEAWAT model
Figure 4. Comparison of the piezometric heads at observation well #10, for all cases and using the calibrated groundwater parameters, between the
sharp-interface approach, the flow model (both solved using MODFLOW) and the SEAWAT model
good agreement with regard to piezometric heads
between the three approaches in all cases is clear, even
for transient simulations. Moreover, the steady cases are
improved since results get closer. This good agreement
has been observed for all observation wells and cases, as
shown for instance in Figure 5. The agreement of the rest
of the boreholes and cases is not presented for the sake of
conciseness. Taking into account that a good approximation has been obtained with the initial parameters for all
Copyright © 2014 John Wiley & Sons, Ltd.
the analysed cases, we expect that the initial parameter
can be employed in many cases. Nevertheless, we have
been working with synthetic aquifers where all parameters and operating conditions are perfectly known. The
approximation in some complex real aquifer could be
worse, and the calibration process would be helpful to
correct the initial mismatch.
Case 5 has been designed to show the good behaviour
in terms of piezometric head of the sharp-interface model
Hydrol. Process. 29, 2052–2064 (2015)
USING MODFLOW TO APPROACH TRANSIENT HEAD WITH A SHARP INTERFACE
Figure 5. Comparison of the piezometric heads at observation well #19,
for transient cases and using the calibrated groundwater parameters,
between the sharp-interface approach, the flow model (both solved using
MODFLOW) and the SEAWAT model
under unsteady simulations and with a spatially variable
height of the aquifer from the aquifer base to the sea level
(d). This is because the transformation between both
approaches (i.e., between the flow potential, ϕ, and the
piezometric head, h) makes use of such parameter, as
shown in previous sections. Figure 6 depicts the good
agreement of both approaches under these conditions for
two different observation wells.
Furthermore, unsteady cases could be only calibrated
regarding the storage coefficient due to the good agreement
observed for the initial parameters. In this sense, the
performance measure ηh for the calibrated parameters is
0.05864 for case 2, 0.04278 for case 4 and 0.13606 for case 5.
However, the extension of the sharp-interface approach
to deal with transient seawater problems should not be
defined only in terms of piezometric heads but also in
terms of seawater penetration, transition zone width and
critical pumping rates. Llopis-Albert and PulidoVelazquez (2013) showed that these SEAWAT results
for steady simulations strongly depend on groundwater
transport parameters and aquifer recharge and pumping
scenarios, which may lead the discrepancies between both
approaches (i.e., sharp-interface approach and densitydependent transport model) to be not suitable in many
cases. Furthermore, the validity of the sharp-interface
approach depends on whether the analysis focuses on the
Copyright © 2014 John Wiley & Sons, Ltd.
2061
Figure 6. Comparison of the piezometric heads at observation well #10,
for case 5 and using the calibrated groundwater parameters, between the
sharp-interface approach and the flow model
results of the transition zone width or on the toe
penetration discrepancies for a given critical pumping
rates. This means that for a specific set of hydrogeological
parameters, a better approximation regarding the transition zone width but a worse approximation in terms of the
relative distance between the toe penetration of both
approaches could be achieved. In this sense, results of
transient simulations reaffirm the conclusions that a better
approximation between both approaches, in terms of the
toe penetration, is attained for higher values of longitudinal dispersion and molecular diffusion coefficients. For
instance, for case 2, at the simulation time equal to
4470 days, the toe penetration for the sharp-interface
approach takes place at a distance of 978.1 m from the
coast, whereas for the SEAWAT model 143.3 m when the
longitudinal dispersion coefficient is equal to 30 m
and the molecular diffusion is not considered. However,
when the longitudinal dispersion coefficient is defined
as 220 m and the molecular diffusion as 1.62 m2/d, the
isoconcentration line corresponding to the 0.1% of
the salinity level in seawater is 975.6 m far away from
the coast. Then, a value much closer to the sharp-interface
approach is obtained. These have been proved to be true
for all transient cases and times. These lower discrepancies can be explained by the fact that the sharp-interface
approach overestimates the penetration of the salt wedge,
and when higher values of those coefficients and longer
Hydrol. Process. 29, 2052–2064 (2015)
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C. LLOPIS-ALBERT AND D. PULIDO-VELAZQUEZ
simulation times are considered, the salt-wedge intrusion
becomes larger in the density-dependent transport model.
In contrary, with regard to the transition zone width, the
validity of the sharp-interface approach for transient
simulations is only valid at early times, because with
longer times, the transition zone width becomes larger
and the salinity contours extend farther inland. The
previous results are in line with those obtained for critical
pumping rates. We have used the term critical pumping
rate because, although we are dealing with transient
simulations, the pumping rates are kept constant during
the stress periods in which they are activated. In this
sense, the wells are first salinized at time equal to
9306 days. We have somehow ‘calibrated’ the parameters
(longitudinal dispersion and molecular diffusion coefficients) until a good agreement, in terms of critical
pumping rate, between the sharp-interface approach and
SEAWAT is attained. For this situation, the critical
pumping rate for both approaches is the same and equal to
116.5 m3/d at each pumping well. The ‘calibrated’
groundwater transport parameters are: longitudinal dispersion coefficient equal to 220 m and molecular
diffusion coefficient equal to 1.62 m2/d. On the one hand,
Figure 7 shows how the 0.1% isoconcentration lines of
the density-dependent transport model reaches the
pumping wells at t=9306 days, which has been adopted
as the salinization threshold to obtain the critical pumping
rate. On the other hand, Figure 7 also depicts for sharpinterface approach and the same time, the curve where the
X coordinate of the toe penetration match up with the
stagnation point, which has been considered the criteria to
obtain the critical pumping rate in this approach. Again, we
observe how high values of groundwater transport
coefficients lead to better agreement regarding the toe
penetration and a worse agreement regarding the transition
zone width (Figure 7). This is because the sharp-interface
approach neglects mixing of fresh water and seawater,
whereas for density-dependent model, high values of
groundwater transport parameters and longer times lead to
Figure 7. Criteria for computing the critical pumping rates for the densitydependent transport model (0.1% isoconcentration lines reaches the
pumping wells) and the sharp-interface approach (curve where the X
coordinate of the toe penetration match up with the stagnation point)
Copyright © 2014 John Wiley & Sons, Ltd.
a larger transition zone width at time that the salinity
contours extend farther inland.
As a final remark, the discrepancies between both
approaches may vary, for a particular set of hydrogeological
parameters, according to future scenarios of climate change,
water demand, water availability, and socioeconomic,
environmental and political issues (e.g., Pulido-Velazquez
et al., 2011b, 2014).
CONCLUSIONS
This paper provides insight into the validity of a sharpinterface approach to deal with seawater intrusion in coastal
aquifers under transient flow conditions. This is achieved by
comparing results between three different approaches under
both steady and unsteady flow conditions and for a set of
heterogeneous groundwater flow and mass transport
parameters, and different scenarios of spatially distributed
recharge values and pumping well rates. On the one hand,
the comparison is performed between the sharp-interface
approach, based on the Ghyben–Herzberg approach, and the
groundwater flow model, both solved using MODFLOW.
On the other hand, the same results are obtained using the
finite difference numerical code SEAWAT, which solves
the coupled partial differential equations of flow and
density-dependent transport. The comparison is performed
in terms of piezometric heads, toe penetration, transition
zone width and critical pumping rates. The numerical
experiment has been carried out in a 3D unconfined
synthetic aquifer, where we show how the sharp-interface
approach provides a good agreement in terms of piezometrics heads regarding the density-dependent transport model.
This is true for the initial set of groundwater parameters,
while after the calibration process of the parameters of the
sharp-interface equation, results become even closer to
those obtained with SEAWAT. This leads to the conclusion
that a transient sharp-interface model could be used to deal
with SWI problems without performing a calibration
process and implementing a density-dependent model.
The simplicity and the low computational cost of this
approach would make it suitable to be integrated into
groundwater management models at a river basin scale.
However, with regard to the toe penetration, transition zone
width and critical pumping rates the agreement between
both approaches is strongly influenced by the aquifer
properties and future management and climate scenarios.
We have shown that for this synthetic case study is possible
to somehow calibrate the groundwater transport parameters
(longitudinal dispersion and molecular diffusion coefficient)
to obtain the same toe penetration and critical pumping rates
for both approaches, while the transition zone width
becomes worse with time. Furthermore, results have shown
that discrepancies between both approaches may not be
Hydrol. Process. 29, 2052–2064 (2015)
USING MODFLOW TO APPROACH TRANSIENT HEAD WITH A SHARP INTERFACE
suitable in many cases. Given the difficulty of properly
identifying the groundwater parameters and providing
general rules applicable to all aquifers and situations, these
results can be used as a guideline for modellers, helping
them to determine if the sharp-interface approach is
appropriate for a particular case study. The decision will
be based on the particular hydrogeological parameters of the
aquifer as well as on current and future recharge and
pumping scenarios. This is because these future scenarios
also depend on the global climate change, socioeconomic
developments and needs of water in the region, environmental issues and policy decisions. Therefore, all these
issues should be taken into account when deciding if the
sharp-interface approach is suitable or not. It should be
noted that the validity of the sharp-interface approach under
transient flow conditions is maybe suitable with the current
situation on a particular aquifer but not under future
management and climate scenarios. Hence, if the sharpinterface approach makes use of future projections for
solving coupled simulation–optimization models for control
and remediation of SWI problems, the validity of the sharpinterface approach should require a deeper analysis. Finally,
the developed methodology represents an advantage
regarding previous models dealing with transient sharpinterface solutions, since groundwater flow models developed in MODFLOW can be directly used to tackle SWI
problems by means of its well-known graphical interfaces.
In addition, the use of MODFLOW also allows the full
integration of the SWI problem into already developed
groundwater management models at a river basin scale or
models for dealing with optimal contamination remediation,
conjunctive use and groundwater management strategies
and designs.
ACKNOWLEDGEMENTS
This research has been partially supported by the
GESHYDRO project (CGL2009-13238-C02-01) of the
Spanish Ministry of Science and Innovation (Plan
Nacional I+D+I 2008–2011) and the GESINHIMPADAPT project (CGL2013-48424-C2-2-R) of the
Spanish Ministry of Economy and Competitiveness (Plan
Estatal I+C+T+I 2013–2016) and from the
subprogramme Juan de la Cierva (2011) of the Spanish
Ministry of Economy and Competitiveness.
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