Supporting Information
Appendix S1: Electronic supplement for ‘Saltwater upconing due to cyclic pumping by
horizontal wells in freshwater lenses’, by Pieter S. Pauw, Anton Leijnse, Sjoerd. E.A.T.M.
van der Zee, and Gualbert H.P. Oude Essink.
Pieter S. Pauw.
Corresponding author: Department of Soil and Groundwater, Deltares, P.O. Box 85467,
3508 AL Utrecht, The Netherlands // Department of Soil Physics and Land Management,
Wageningen University, P.O. Box 47, 6700 AA Wageningen, The Netherlands.
+31 623786887.
pieter.pauw@deltares.nl
Sjoerd E.A.T.M. van der Zee.
Department of Soil Physics and Land Management, Wageningen University, Wageningen,
The Netherlands.
sjoerd.vanderzee@wur.nl
Anton Leijnse.
Department of Soil Physics and Land Management, Wageningen University, Wageningen,
The Netherlands.
toon.leijnse@wur.nl
Gualbert H.P. Oude Essink.
Department of Soil and Groundwater, Deltares, Utrecht, The Netherlands // Department
of Physical Geography, University of Utrecht, Utrecht, The Netherlands.
gualbert.oudeessink@deltares.nl
Abstract (main text)
This paper deals with the quantification of saltwater upconing below horizontal wells in
freshwater lenses using analytical solutions, as a computationally fast alternative to
numerical simulations. Comparisons between analytical calculations and numerical
simulations are presented regarding three aspects: 1) cyclic pumping, 2) dispersion,
and 3) finite horizontal wells in a finite domain (a freshwater lens). Various
hydrogeological conditions and pumping regimes within a dry half year are considered.
The results show that the influence of elastic and phreatic storage (which are not taken
into account in the analytical solutions) on the upconing of the interface is minor.
Furthermore, the analytical calculations based on the interface approach compare well
with numerical simulations as long as the dimensionless interface upconing is below
1/3, which is in line with previous studies on steady pumping. Superimposing an
analytical solution for mixing by dispersion below the well over an analytical solution
based on the interface approach is appropriate in case the vertical flow velocity around
the interface is nearly constant, but should not be used for estimating the salinity of the
pumped groundwater. The analytical calculations of interface upconing below a finite
horizontal well compare well with the numerical simulations in case the distance
between the horizontal well and the initial interface does not vary significantly along
the well, and in case the natural fluctuation of the freshwater lens is small. For
maintaining a low salinity in the well during a dry half year, the dimensionless
analytically calculated interface upconing should stay below 0.25.
Table of contents
1. Introduction
2. Analytical solutions
2.1 Derivation of interface Equations 3 and 10 of the main text
2.2 Derivation of dispersion Equations 6 and 7 in the main text
2.3 Illustration of the analytical calculation of interface upconing due to cyclic
pumping
2.4 Illustration of the analytical calculation of interface upconing below a finite
horizontal well
2.5 Perturbation theory and implications for the applicability of Equations 3 and
10 in the main text
2.6 ‘Critical rise’
3. Parameter combinations used for the comparison between the numerical
simulations and analytical calculations using the 2D model
4. Brief description of the 3D numerical model
5. Numerical simulation results of the reference model
6. List of symbols
7. References
1
Introduction
In this Supporting Information, additional information on the analytical equations and
set-up of the numerical models is presented. Furthermore, some results that were for
brevity omitted in the main text are presented and discussed. Finally, an overview of all
symbols used in the main text and the Supporting Information (except for the ones used
in references/previous studies) is given.
2
Analytical solutions
2.1 Derivation of interface Equations 3 and 10 of the main text
Equation 3 and 10 in the main text are based Equations 43 and 69 of Dagan and Bear
(1968), respectively. Below it is shown how Equation 43 of Dagan and Bear (1968) was
transformed into Equation 3.
Equation 43 of Dagan and Bear is given by:
  x, t  



1Q  1 cosh[ (a  d )] 
 t
1  exp 
  cos( x) d  . (S1)

 0  sinh( a) 
 1 coth( a)   2 coth(b)  

Similar as in the main text, λ is the variable of integration. Furthermore, the variables x
(horizontal coordinate) [L], t (time) [T], a (initial thickness of the fresh groundwater
zone) [L], b (initial thickness of the saline groundwater zone) [L], d (vertical distance
between the interface at t = 0 and the well) [L], and n (porosity), are defined as in the
main text. ζ and Q correspond with ζan (analytically calculated interface depth) [L] and
Qd (the pumping rate of the well per unit length of the well) [L2 T-1]in the main text,
respectively. The subscript 1 refers to fresh groundwater, 2 refers to saline
groundwater.
The integral form Equations 3 and 10 of the main text were solved numerically using
the Python function ‘quad’ (see docs.scipy.org). They can also be transformed into
‘closed-form’ (i.e., not in integral form) solutions, in case it is assumed that the
parameters a and b (initial thickness of the zones of fresh groundwater and saline
groundwater, respectively) are infinite. Dagan and Bear (1968) presented the closedform solution of Equation 43 (Equation 49 in their work), whereas the closed-form
solution of Equation 69 of Dagan and Bear (1968) was presented by Schmorak and
Mercado (1969). An interesting difference between the 2D and 3D equations (both in
integral form and closed form) is that for the 3D equation a steady-state end-point of
the interface can be reached (t  ∞) in case of continuous pumping, whereas for the 2D
equation there is not an end-point, so there is a continuous rise of the interface through
time.
Returning to Equation (S1): using the subscript ‘f’ for subscript 1 and ‘s’ for 2, α1 and α2
are defined as:
f
s
1 
, 2 
.
(S2)
 
 
μf and μf are the dynamic viscosity [M L-1 T-1] of fresh and saline groundwater,
respectively. κ is the isotropic permeability [L2]. Δγ is equal to g(ρs - ρf), where g
(gravitational acceleration) [L2 T-1], ρs (density of saline groundwater) [M L-3], and ρf
(density of fresh groundwater) [M L-3] are defined as in the main text. β1 is equal to nα1
and β2 is equal to nα2. Using the definition of δ (the dimensionless density difference):

s   f
,
f
(S3)
(ρs - ρf) is equal to δρf, so that Δγ is equal to gδρf. Inserting this into Equation (S2) yields:
f
s
.
(S4)
1 
, 2 
 g f 
 g f 
It was assumed that the dynamic viscosity was based on fresh groundwater, such that μs
= μf, α1 = α2, and β1 = β2. Furthermore, the isotropic hydraulic conductivity K [L T-1] is
also based on fresh groundwater, and defined as:
 f g
.
f
(S5)
1
n
, 1   2 
.
K
K
(S6)
K
Therefore:
1 
Using the definitions above, Equation (S1) can be written as:








Q
1 cosh[ ( a  d )] 
 t
 an  d 
cos( x) d  . (S7)
1  exp  n


n
 K 0  sinh( a) 

coth( a) 
coth(b)  
K
 K




Bear and Dagan (1965) derived relationships to transfer an isotropic problem into an
anisotropic one. Following these relationships for this 2D case, the parameters Qd, K,
and x are changed into:
Qd  Qd
Kz
Kz
,
, K  Kz , x  x
Kx
Kx Kz
(S8)
where Kx [L T-1] is the horizontal hydraulic conductivity and Kz [L T-1] is the vertical
hydraulic conductivity, both based on fresh groundwater. Using Equation (S8) in
combination with the reference level at the top of the aquifer results in the following
equation, which is equal to Equation 3 in the main text:



Kz
1 cosh[ ( a  d )] 
 t

 cos( x
 an ( x, t ) 
1

exp
) d   a.

n
n
K
 K x K z 0  sinh( a) 

x
coth(a  ) 
coth(b  )


 Kz
 Kz
Qd

(S9)
In a similar way, Equation 69 of Dagan and Bear (1968) was changed into Equation 10
of the main text.
2.2 Derivation of dispersion Equations 6 and 7 in the main text
Equations 6 and 7 in the main text are based on Schmorak and Mercado (1969). In
applying equations 6 and 7, it is assumed that the flow right below the well due to cyclic
pumping is vertical and that the concentration gradient right below the well is parallel
to the flow lines. Furthermore, transverse dispersion and molecular diffusion are
neglected, such that mixing is assumed to be caused solely by longitudinal dispersion
(Eeman et al, 2011). It is also assumed that the pore water velocity in the vertical does
not vary spatially (i.e., that it is equal to the velocity of the interface), and that the
vertical concentration distribution around interface (i.e., 0.5 times the maximum
concentration) is normally distributed.
The approach of Schmorak and Mercado (1969) is based on a classical solution for onedimensional steady and uniform groundwater flow and advective and dispersive solute
transport in an infinite domain (Ogata and Banks, 1961). In the solution of Ogata and
Banks (1961), initially the solute concentration of the groundwater (C [M L-3]) is 0 kg m3. At time t = 0 and distance xi = 0, the concentration of the groundwater changes to Cs
[M L-3] kg m-3. The concentration for a given distance xi, t, and pore water velocity v [L T1] is given as:
vxi

C
 x  vt 
 x  vt  
D
 0.5  erfc  i

e
erfc  i

 ,
Cs
2
Dt
2
Dt





(S10)
where D is the dispersion coefficient [L2 T-1], accounting for longitudinal dispersion and
molecular diffusion. Ogata and Banks (1961) described that in most applications, the
second term in equation (S10) can be neglected. If the dispersive transport is of interest
relative to the location of the (moving) average concentration (0.5Cs) the product vt can
be changed to 0.5Cs and xi can be changed into xμ [L] (the distance from 0.5Cs) leading to:
 x  0.5Cs 
C
 0.5erfc  
.
Cs
 2 Dt 
(S11)
Note the similarity of this equation with equation 11 in Schmorak and Mercado (1969),
which is written in terms of the error function (erf) rather than the complementary
error function (erfc).
When it is assumed that molecular diffusion can be neglected compared to longitudinal
dispersion and when it is assumed that the mixing is independent on the direction of
flow (Eeman et al., 2012), equation (S11) can be written as:
 x  0.5Cs
C
 0.5erfc  
 2  |s|
Cs
L


,


(S12)
where αL is the longitudinal dispersivity [L] and │s│ is the total travelled distance of
the average concentration 0.5Cs [L], independent on the direction. In terms of the
symbols used in the main text, the depth of the analytically calculated interface ζan is
taken as 0.5Cs and the distance from ζan (i.e., the analytically calculated depth of a
concentration C) xμ is taken as zCan. Using these terms and rewriting (S12) yields:
 C
zCan ( x, t , C )   an ( x, t )  2  L (| s |)erfc1  2  .
 Cs 
(S13)
If the initial concentration distribution is not sharp, the initial thickness of the mixing
zone can be implemented in Equation (S13) by assuming that the concentration in the
mixing zone is normally distributed. In the main text, the initial width of the mixing
zone M [L] was defined as twice the distance between the interface (0.5Cs) and the
depth at which the concentration equals 0.024Cs. Hence,:
0.5M  zCan ( x, t , 0.024Cs )   an ( x, t )  2  L (| s |)erfc 1  0.048  .
(S14)
By inserting the initial width of the mixing zone and longitudinal dispersivity α in
Equation (S14), ∣s∣ can be calculated, which in this case represents the width of the
mixing zone in the form of an equivalent travelled distance of the interface. Using the
symbol η [L] for this equivalent distance, η can be defined as:
2


0.5M


1
2erfc  0.048  

,
 (M , L ) 
L
(S15)
which is equal to Equation 7 in the main text. η can then be added to ∣s∣ in (S13),
which yields:
 C
(S16)
zCan ( x, t , C )   an ( x, t )  2  L (  | s |)erfc1  2  .
C
 s
Equation Error! Reference source not found. is equal to Equation 6 of the main text.
2.3 Illustration of the analytical calculation of interface upconing due to cyclic
pumping
The superposition principle was used for the analytical calculation of cyclic pumping.
This procedure is illustrated using Figure S1, with Ton (time of pumping during a
pumping cycle) [T] = 5 days, Toff (time of no pumping during a pumping cycle) [T] = 10
days, and ncyc (number of pumping cycles) = 3. The three blue lines in Figure S1 each
indicate interface upconing due to pumping with extraction rate Qd. They start at t =
i(Ton+Toff), with i = (0,1,2). The red lines indicate the downward movement of the
interface due to well injection with rate -Qd. These lines start at t = i(Ton+Toff) + Ton, with
i = (0,1,2). ζan is obtained by summing the contribution of all lines throughout the total
pumping period Tp [T](black line in Figure S1).
Figure S1: Illustration of calculating the upward and downward movement of the
interface ζan due to well extraction (q on) and well shutdown (q off), using the
superposition principle.
2.4 Illustration of the analytical calculation of interface upconing below a finite
horizontal well
The superposition principle was also used for the analytical calculation of a horizontal
well with a finite extent. This procedure is illustrated using Figure S2. The following
parameters are used: a = 12 m, b = 18 m, n = 0.3, δ = 0.025, Kx = Kz = 10 m d-1, t = 180 d
(steady pumping), and Q (pumping rate of the well) [L3 T-1] = 12 m3 d-1. The length of
the well Lwell [L] is 100 m. First, the length of the well is first split into n-seg segments.
For every segment, the interface upconing is computed using Equation 10 of the main
text using a discharge Q/n-seg, for a large number of values of r (radial distance from
the well [L], for which x is used in this case) for such a distance from the well that the
interface upconing is negligble. Subsequently, the contributions (i.e., interface upconing)
of each segment are summed. The total contribution equals the interface upconing
below the horizontal well.
For this procedure, it is important that Lwell is split into enough n-seg segments. This is
shown in Figure S2. The centre of the horizontal well is at x = 0 m. In case n-seg = 1
(which is equivalent to a point sink) or n-seg = 2, the interface upconing is erroneously
calculated. In case n-seg = 10, only a small error is made, whereas n-seg = 100 results in
an appropriate calculation of the interface upconing below the horizontal well. The
difference in interface upconing between n-seg = 1 and n-seg = 100 in Figure S2 also
shows the advantage of using a horizontal well over a vertical well (point sink).
Figure S2: Illustration of analytical calculation of the depth of the interface ζan (blue line)
due to pumping by a horizontal well with finite extent using the superposition principle.
The well is located between x = -50 m and x = 50 m.
2.5 Perturbation theory and implications for the applicability of Equations 3 and
10 in the main text
Equations 3 and 10 in the main text are based on equations 43 and 69 of Dagan and
Bear (1968), respectively, which were derived using perturbation theory. Perturbation
theory is a mathematical theory that can be used to derive an approximate solution for a
mathematical problem in case an exact solution does not exist. In perturbation theory,
the mathematical problem is split into an ‘initial’ part, which can be solved directly, and
‘perturbation’ parts (power series), which describe the deviation of the initial part from
total mathematical problem. In hydrological studies that employ perturbation theory it
is commonly assumed that the first few perturbation parts are sufficient to approximate
the total mathematical problem. This assumption is only appropriate if the total
mathematical problem does not differ too much from the initial part in the perturbation
approximation, which is what Dagan and Bear (1968) assumed in their derivation of the
interface solutions.
The most important implication of the assumptions of Dagan and Bear (1968) regarding
the perturbation theory is that their solutions are appropriate only up to a certain
degree of interface upconing (i.e., not deviating too much from the initial situation).
Based on comparison of the interface solutions with laboratory (Hele Shaw)
experiments of steady pumping regimes, Dagan and Bear (1968) found that the
analytically calculated dimensionless interface upconing dζ represented the 0.5Cs
concentration contour as long as dζ was approximately less than 1/3. In case dζ is larger,
the analytically calculated interface underestimates the depth of the 0.5Cs concentration
contour.
2.6 ‘Critical rise’
The term ‘critical rise’ was used by Schmorak and Mercado (1969) (p. 1293), partly for
the limited applicability of the equations of Dagan and Bear (1968) as a result of the
perturbation approximation. Below the critical rise, the equations are appropriate,
whereas above the critical rise errors in the estimated interface upconing can be
expected, as was explained in the previous section. Dagan and Bear (1968) and
Schmorak and Mercado (1969) advised the critical rise as a practical guideline for
maintaining a low salinity in the well, based on their comparisons between the
analytical solutions and laboratory experiments and field data.
Besides that critical rise refers to the limited applicability due to the perturbation
approximation, critical rise also has a physical meaning. Bear and Dagan (1964) and
Dagan and Bear (1968) showed that above the critical rise, the rate of interface rise
accelerates. This was also described Muskat (1946), for the case of water coning below
a well extracting oil. Muskat (1946) furthermore explained that for a stable position of
the interface, the vertical pressure distribution in the saline groundwater should be
hydrostatic, and that at the interface, the vertical pressure gradient in the fresh
groundwater should be equal to the vertical pressure gradient of the saline
groundwater. Close to a pumping well, the flow is strongly convergent and the velocities
(and pressure gradients) increase fast towards the well. Consequently, in case the the
interface is close to the well, the pressure gradient of the fresh groundwater will always
be higher than the pressure gradient in the saline groundwater. This effect causes the
saline water to be accelerated towards the well and ultimately end up there.
In his analysis, Muskat (1946) made two assumptions to tackle the complex
mathematical problem of interface upconing. In terms of fresh and saline groundwater,
the first assumption is that the saline groundwater is at rest, such that the vertical
pressure distribution in the lower liquid is hydrostatic. The second assumption is that
the upconing does not influence the flow in the upper fluid. Bear (1972) explained that
these assumptions are the physical representations of the assumptions of the
perturbation approximation of Dagan and Bear (1964). Hence, above the critical rise it
cannot be assumed that the vertical pressure distribution in the lower fluid is
hydrostatic, and that the flow pattern in the upper flow is not influenced by the upconed
interface.
Critical rise has also been used in the studies of Bower et al. (1999) and Garabedian
(2013), which were mentioned in the introduction of the main text. In both studies, the
critical rise was not used as an applicability condition related to the perturbation theory,
as these steady-state analytical solutions were not based on perturbation theory. Rather,
the goal of these studies was to find the critical pumping rate and corresponding critical
rise in terms of aquifer properties and well characteristics. Essentially, Bower et al.
(1999) and Garabedian (2013) adopted the two assumptions of Muskat (1946).
3
Parameter combinations used for the comparison between the numerical
simulations and analytical calculations using the 2D model
As certain parameters of the 2D model are related, a complete overview of all parameter
combination was, for brevity, not given in Table 1 in the main text. In Table S1, a
complete overview of the parameter combinations is given. Per parameter (bold face),
the reference value is indicated between parentheses and the variation of that
parameter is shown on the same line in the next column. Below that line, the actual
combination of parameters is given. For example, for the parameter variation Qd = 20.0
m3m-1d-1, Ton (period of pumping during one pumping cycle) was set to 0.01 d and Toff
(period of no pumping during one pumping cycle) to 1.99 d. This yields, together with
the reference value for the number of pumping cycles (ncyc = 90), the reference value for
the total pumping period (Tp = 180 d), the total amount of water extracted (Qtot = 18
m3m-1), i.e., 90 (ncyc) * 0.01 d (Ton) * 20 m3m-1d-1 (Qd) = 18 m3m-1. In total, 32 different
parameter combinations were considered.
Table S1: a complete overview of the parameter combinations of the 2D model.
parameter
variations and parameter combinations
units
Qtot (18)
36, 54, 72, 80, 98
m3m-1
Qd
0.4, 0.6, 0.8, 0.88, 1.08
m3m-1d-1
ncyc (90)
Ton & Toff
180, 45, 30, 10, 1
0.5, 2, 3, 9, 90
d
Qd (0.2)
Ton
Toff
20.0, 4.0, 2.0, 0.4, 0.16, 0.13, 0.11
0.01, 0.05, 0.1, 0.5, 1.2, 1.5, 1.8
1.99, 1.95, 1.9, 1.5, 0.8, 0.5, 0.2
m3m-1d-1
d
d
K (10)
1, 5, 50, 100
m d-1
Kx/Kz (1)
Kz
2.0, 5.0
5.0, 2.0
m d-1
Cs (35)
17.5, 8.25
kg m3
αL (0.1)
0.01, 1.0
m
S (0.15 & 1E-5)
Qd
Ton
Toff
0 (Sy), 0 (Ss)
20.0, 4.0, 2.0, 0.4, 0.16, 0.13, 0.11
0.01, 0.05, 0.1, 0.5, 1.2, 1.5, 1.8
1.99, 1.95, 1.9, 1.5, 0.8, 0.5, 0.2
- and m-1
m3m-1d-1
d
d
M (0)
1.4, 2.8
m
4
Brief description of the 3D numerical model
Only the most important aspects of this numerical model are described here, as many
aspects are similar to the 2D numerical model that is described in the main text. In the
3D numerical model an irregular grid was used in view of computational effort. Near the
well, the horizontal cell size is 0.1 m. Away from the well the cell size gradually
increases towards the boundaries. Between the well and the initial mixing zone, 0.1 m
thick model layers were used. Above and below, the layer thickness gradually increases.
The bottom and the vertical planes at x = 0 and y = 0 are no-flow boundaries. Along the
vertical planes at x = 0.5 Lx and y = 0.5 Ly a constant hydraulic (saltwater) head (0 m)
and concentration boundary condition with concentration Cs is applied. The hydraulic
head along these vertical sides therefore follows a hydrostatic pressure distribution.
These flow and concentration boundary conditions remain constant throughout the
simulation.
The initial condition (i.e., the initialization period) was simulated in the numerical
model starting from an initial saline model domain, and simulating the development of a
freshwater lens using subsequent periods Trch (length of the recharge period) and Tdry
(length of the dry period). During Trch, groundwater recharge is simulated over the total
model domain with rate N = 0.002 m d-1 and with concentration C = 0 kg m-3. During Tdry
no recharge was simulated. The initialization period was ended when the total salt mass
in the domain over successive Tdry and Trch periods reached a virtually stable value.
Pumping using the finite horizontal well during Tp (length of the total pumping period,
180 days) was simulated by superposition of point sinks. The extraction rate for every
cell that represented the horizontal well was determined by dividing the total extraction
rate of the well by the amount of columns that were used to simulate the well.
5
Numerical simulation results of the reference model
In Figure S3 streamlines and the groundwater salinity distribution of the numerical
simulation with the reference parameter values at the end of the last Ton (time of
pumping during a pumping cycle) and Toff (time of no pumping during a pumping cycle)
periods are shown. At the end of Ton, the streamlines are approximately horizontal at a
far distance (150 m) from the well and a radial flow pattern is present close to the well.
During Toff a rotational flow pattern is observed near the well, where the saline
groundwater below the well flows down and induces the saline groundwater somewhat
further away from the well to flow upwards.
In Figure S4 the results of the simulation with the reference parameters but without
storage are shown. Comparison with Figure S3 indicates that storage has a large
influence on the flow patterns during cyclic pumping, but that its influence on saltwater
upconing is relatively minor.
Figure S3: Groundwater salinity distribution (kg m-3, shown in colour) and streamlines (in
white) of the numerical simulation with the reference values at the end of the last periods
Ton and Toff. The well is located at z = - 7.0 m.
Figure S4: Groundwater salinity distribution (kg m-3, shown in colour) and streamlines (in
white) of the numerical simulation with the reference values at the end of the last periods
Ton and Toff, except that no storage is simulated. The well is located at z = - 7.0 m.
Mixing in the numerical simulation with the reference values was investigated by
computing the relative contributions of the longitudinal (fαL) dispersion, transversal
dispersion (fαT) and molecular diffusion (fDm) on total hydrodynamic dispersion during
Ton and Toff, using:
f L 
 L v2
Dm | v |  L v2  T v||2
T v||2
f T 
Dm | v |  L v2  T v||2
fDm 
Dm | v |
Dm | v |  L v2  T v||2
where │v│ is the magnitude of the velocity vector [L T-1], v⊥ is the magnitude of the
velocity vector component in the direction of the concentration gradient (i.e.,
perpendicular to the orientation of the mixing zone) [L T-1] v∣∣ is the magnitude of the
velocity vector component perpendicular to the concentration gradient (i.e., parallel to
the orientation of the mixing zone) [L T-1], αT is the transverse dispersivity [L], and Dm is
the molecular diffusion coefficient [L2 T-1]. For further details, the reader is referred to
Eeman et al. (2011).
fαL, fαT, and fDm for the end of the last Ton period are shown in Figure S5. Close to the
well, the mixing is dominated by longitudinal dispersion as the concentration gradient
and the velocity vector are (nearly) parallel here. Further away from the well, the
velocity vector is aligned more perpendicular to the concentration gradient, which
explains the increasing values of fαT. Even further from the well, the flow velocities
decrease such that fαT decreases and fDm increases.
In Figure S6 fαL, fαT, and fDm at the end of the last period Toff are shown. Again, mixing
near the well is dominated by longitudinal dispersion. The results of the other output
time steps, which are not given here, show that longitudinal dispersion is the most
important mixing process below the well not only at the end of Ton and Toff, but also
throughout these periods.
Figure S5: Relative contributions (scaled, from 0-1) of fαL, fαT, and fDm on the total
hydrodynamic dispersion at the end of the last Ton period. In the white region, fαL, fαT, and
fDm are undefined as there is no concentration gradient (no spatial change in
concentration). The location of the well is indicated with a black dot. The depth of the
numerically calculated interface ζnum (i.e., C = 0.5Cs ) is also indicated.
Figure S6: Relative contributions fαL, fαT, and fDm on the total hydrodynamic dispersion at
the end of the last Toff period.
6
List of symbols
Symbol
a
Dimension
[L]
αL
αT
b
[L]
[L]
[L]
C
Cs
C1σ
[M L-3]
[M L-3]
[M L-3]
C2σ
[M L-3]
C1σ-an
C1σ-num
C2σ-an
C2σ-num
Cwellnum
d
D
Dm
dζ
δ
εζ
[M L-3]
[M L-3]
[M L-3]
[M L-3]
[M L-3]
[L]
[L]
[L2 T-1]
[-]
[-]
[L]
ε1σ
[L]
ε2σ
[L]
fαL
-
fαT
-
fDm
-
g
h
λ
κx
κz
Kx
Kz
Lx
Ly
Lwell
M
[L T-2]
[L]
[L]
[L2]
[L2]
[L T-1]
[L T-1]
[L]
[L]
[L]
[L]
n
ncyc
N
η
ρf
ρs
Q
[L T-1]
[L]
[M L-3]
[M L-3]
[L3 T-1]
Explanation
initial thickness of the fresh groundwater zone; the distance
between the initial interface depth and the top of the aquifer
longitudinal dispersivity
transverse dispersivity
initial thickness of the saline groundwater zone; the distance
between the initial interface depth and the bottom of the aquifer
salt concentration of the groundwater
salt concentration of the groundwater with density ρs
concentration at one standard deviation from the average
concentration
concentration at two standard deviations from the average
concentration
analytically calculated value of C1σ
numerically simulated value of C1σ
analytically calculated value of C2σ
numerically simulated value of C2σ
numerically simulated concentration in the well
vertical distance between the interface and the well at t = 0
thickness of the domain
molecular diffusion coefficient
analytically calculated dimensionless interface upconing (ζan/d)
dimensionless density difference (see Equation 4)
difference between the analytically calculated and
numerically simulated interface depth (see Equation 5)
difference in the analytically calculated and numerically
simulated depths of C1σ
difference in the analytically calculated and numerically
simulated depths of C2σ
the relative contribution of the longitudinal dispersion on the total
hydrodynamic dispersion
relative contribution of the transverse dispersion on the total
hydrodynamic dispersion
relative contribution of the molecular diffusion on the total
hydrodynamic dispersion
gravitational acceleration
hydraulic head
variable of integration
horizontal permeability
vertical permeability
horizontal hydraulic conductivity based on fresh groundwater
vertical hydraulic conductivity based on fresh groundwater
horizontal extent of the 3D domain in the x direction
horizontal extent of the 3D domain in the y direction
length of the horizontal well
initial thickness of the mixing zone; twice the distance between
the interface and the depth at which the concentration C equals
0.024Cs
porosity
number of pumping cycles (see Equation 1)
recharge rate
equivalent travelled distance of the interface (Equation 7)
density of fresh groundwater
density of saline groundwater
pumping rate of the well
Qd
Qtot
r
Ss
Sy
sζ
│s│
t
Tp
Ton
Toff
Trch
Tdry
│v│
v⊥
[L2 T-1]
[L3]
[L]
[L-1]
[L]
[L]
[T]
[T]
[T]
[T]
[T]
[T]
[L T-1]
[L T-1]
v∣∣
[L T-1]
μ
μf
x
y
z
zC1σ-an
zC1σ-num
zC2σ-an
zC2σ-num
zCan
ζ
ζan
ζnum
[M L-1 T-1]
[M L-1 T-1]
[L]
[L]
[L]
[L]
[L]
[L]
[L]
[L]
[L]
[L]
[L]
pumping rate of the well per unit length of the well
total amount of extracted fresh groundwater during Tp
radial distance from the well (see Equation 10)
specific storage
specific yield
vertical distance from ζan (see Equation 6)
total travelled distance of ζan/average concentration 0.5Cs
time
pumping period (dry half year, 180 d)
time of pumping during a pumping cycle
time of no pumping during a pumping cycle
length of the recharge period
length of the dry period
magnitude of the velocity vector
magnitude of the velocity vector component in the direction of the
concentration gradient
magnitude of the velocity vector component perpendicular to the
concentration gradient
dynamic viscosity
dynamic viscosity based on freshwater
horizontal coordinate
horizontal coordinate
vertical coordinate
depth of the analytically calculated value of C1σ
depth of the numerically simulated value of C1σ
depth of the analytically calculated value of C2σ
depth of the numerically simulated value of C2σ
analytically calculated depth of a concentration C
depth of the interface between fresh and saline groundwater
analytically calculated interface depth
numerically simulated interface depth
7
References
Bear, J. 1972. Dynamics of Fluids in Porous Media. New York: Dover Publications.
Bear, J. and G. Dagan. 1965. The relationship between solutions of flow problems in isotropic and
anisotropic soils. Journal of Hydrology 3: 88–96.
Bower, J. W., L. H. Motz and D. W. Durden. 1999. Analytical solution for determining the critical condition
of saltwater upconing in a leaky artesian aquifer. Journal of Hydrology 221: 43–54.
Dagan, G. and J. Bear. 1968. Solving the problem of local interface upconing in a coastal aquifer by the
method of small perturbations. Journal of Hydraulic Research 6: 15–44.
Eeman, S., A. Leijnse, A., P.A.C. Raats, and S.E.A.T.M. van der Zee. 2011. Analysis of the thickness of a fresh
water lens and of the transition zone between this lens and upwelling saline water. Advances in Water
Resources, 34, no. 2: 291–302.
Eeman, S., S.E.A.T.M. van der Zee, A. Leijnse, P.G.B. de Louw, and C. Maas. 2012. Response to recharge
variation of thin rainwater lenses and their mixing zone with underlying saline groundwater. Hydrology
and Earth System Sciences and Earth System Sciences 16, no. 10: 3535–3549. doi:10.5194/hess-16-35352012
Garabedian, S. P. 2013. Estimation of salt water upconing using a steady-state solution for partial
completion of a pumped well. Groundwater 51, no. 6: 927–34.
Muskat, M. 1946. The flow of homogeneous fluids through porous media. J. W. Edwards, Inc.
Ogata, A. and R.B. Banks. 1961. A solution of the differential equation of longitudinal dispersion in porous
media. Fluid movement in earth materials, geological survey professional paper 411-A.
Schmorak, S., and A. Mercado. 1969. Upconing of fresh water-sea water interface below pumping wells,
field study. Water Resources Research 5, no. 6: 1290-1311.