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Advances in Water Resources 86 (2015) 32–45

Contents lists available at ScienceDirect

Advances in Water Resources

journal homepage: www.elsevier.com/locate/advwatres

Salinization in a stratiﬁed aquifer induced by heat transfer

from well casings

Jan H. van Lopika,b,∗, Niels Hartoga,b, Willem Jan Zaadnoordijkb,c, D. Gijsbert Cirkelb,

Amir Raoofa

aUtrecht University, Department of Earth Sciences, Budapestlaan 4, 3584 CD Utrecht, The Netherlands

bKWR Watercycle Research Institute, Groningenhaven 7, 3433 PE Nieuwegein, The Netherlands

cDelft University of Technology, Water Resources Section, Faculty of Civil Engineering and Geosciences, Stevinweg 1, 2628 CN Delft, The Netherlands

a r t i c l e i n f o

Article history:

Received 21 January 2015

Revised 24 September 2015

Accepted 25 September 2015

Available online 9 October 2015

Keywords:

Thermo-haline convection

Wellbore heat transmission

Salinization

Non-linear density equation

SEAWAT

a b s t r a c t

The temperature inside wells used for gas, oil and geothermal energy production, as well as steam injection,

is in general signiﬁcantly higher than the groundwater temperature at shallower depths. While heat loss

from these hot wells is known to occur, the extent to which this heat loss may result in density-driven ﬂow

and in mixing of surrounding groundwater has not been assessed so far. However, based on the heat and

solute effects on density of this arrangement, the induced temperature contrasts in the aquifer due to heat

transfer are expected to destabilize the system and result in convection, while existing salt concentration

contrasts in an aquifer would act to stabilize the system. To evaluate the degree of impact that may occur

under ﬁeld conditions, free convection in a 50-m-thick aquifer driven by the heat loss from penetrating hot

wells was simulated using a 2D axisymmetric SEAWAT model. In particular, the salinization potential of fresh

groundwater due to the upward movement of brackish or saline water in a stratiﬁed aquifer is studied. To

account for a large variety of well applications and conﬁgurations, as well as different penetrated aquifer

systems, a wide range of well temperatures, from 40 to 100 °C, together with a range of salt concentration (1–

35 kg/m3) contrasts were considered. This large temperature difference with the native groundwater (15 °C)

required implementation of a non-linear density equation of state in SEAWAT. We show that density-driven

groundwater ﬂow results in a considerable salt mass transport (up to 166,000 kg) to the top of the aquifer in

the vicinity of the well (radial distance up to 91 m) over a period of 30 years. Sensitivity analysis showed that

density-driven groundwater ﬂow and the upward salt transport was particularly enhanced by the increased

heat transport from the well into the aquifer by thermal conduction due to increased well casing temperature,

thermal conductivity of the soil, as well as decreased porosity values. Enhanced groundwater ﬂow and salt

transport was also observed for increased hydraulic conductivity of the aquifer. While advective salt transport

was dominant for lower salt concentration contrasts, under higher salt concentration contrasts transport was

controlled by dispersive mixing at the fresh-salt water interface between the two separate convection cells

in the fresh and salt water layers. The results of this study indicate heat loss from hot well casings can induce

density-driven transport and mixing processes in surrounding groundwater. This process should therefore be

considered when monitoring for long-term groundwater quality changes near wells through which hot ﬂuids

or gases are transported.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Oil and gas deposits, as well as exploitable geothermal energy,

are typically found in reservoirs well below the depths of exploitable

fresh groundwater supplies. Therefore, wells for conventional oil

∗Corresponding author at: Utrecht University, Department of Earth Sciences, Envi-

ronmental Hydrogeology Group, Budapestlaan 4, 3584 CD Utrecht, The Netherlands.

Tel: +31 631697963.

E-mail address: j.h.vanlopik@gmail.com,j.h.vanlopik@uu.nl (J.H. van Lopik).

and gas, shale gas and geothermal energy production fully penetrate

shallow fresh water aquifers. The temperatures of wellbores during

oil production [6], gas production [11], geothermal energy produc-

tion [13] and hot water or steam injection [50], can be signiﬁcantly

higher (e.g., T>40 °C) than the typical temperatures of the shallow

aquifers in moderate climates (10–20 °C). The temperature difference

causes heat transfer to the surrounding formations and cooling of

the ﬂuid or gas that is ﬂowing inside the well [35]. The resulting

temperature at the wellhead is important for operational drilling

and injection/production aspects, such as determining viscosity and

http://dx.doi.org/10.1016/j.advwatres.2015.09.025

0309-1708/© 2015 Elsevier Ltd. All rights reserved.

J.H. van Lopik et al. / Advances in Water Resources 86 (2015) 32–45 33

Fig. 1. Thermally induced density-driven ﬂow due to heat transfer from a well to the aquifer and its effect on fresh-salt stratiﬁed groundwater (parameter values belong to the

reference scenario used in this study).

thereby ﬂowing pressure changes in a heavy-oil well, or estimating

steam quality during steam injection. However, to our knowledge,

the thermal effects of wellbore heat losses for the aquifers that

receive the heat have not been addressed.

Many studies have investigated density-driven ﬂow in porous

media under the inﬂuence of temperature contrasts in groundwater

systems (e.g. [4,15]). This can occur both at large scale, like geother-

mal convection in geological basins [4], or at small scale such as

the upward density-driven ﬂow of injected hot water during high

temperature aquifer thermal energy storage (e.g. [2,27,42,44]). In

addition to temperature, density-driven groundwater ﬂow is affected

by salt concentration contrasts. The effect of salt concentration and

temperature gradients on convective ﬂow patterns in thermo-haline

systems were studied by [4,17,29,36,39]. A few studies have focused

on thermo-haline convection on ﬁeld scale; e.g. the effect of hyper-

saline cooling canals on aquifer salinization [16] and the transport of

hot, brine water plumes [30].

Thermal convection under the inﬂuence of a vertical heat source

like a vertical ﬂat plate [3] or long vertical thin blades [32] in a porous

medium has been investigated numerically. These geophysical ex-

amples suggest that hot wellbores penetrating cooler aquifers could

thermally induce density-driven groundwater ﬂow.

In many aquifers, fresh groundwater overlies denser, saline water

and salinization by mobilization of the underlying saline water is con-

sidered a major threat to fresh groundwater resources and drinking

water production [5,34,48]. Local thermally induced density-driven

ﬂow in the vicinity of hot well casings could therefore result in mixing

and deterioration of the groundwater quality (see Fig. 1). To explore

this possibility, we simulated transient temperature and salinity de-

pendent density-driven groundwater ﬂow along a hot wellbore. We

used SEAWATv4, and further include a non-linear density equation of

state, to apply to various thermal conditions, salt concentration con-

trasts and aquifer properties.

2. Theory and methodology

2.1. Wellbore heat transmission

Fluid or gas ﬂowing in the wellbore loses heat to its surroundings

by thermal conduction due to the difference between wellbore ﬂuid

and surrounding aquifer temperature during injection or production

operations. The heat transfer from the wellbore is proportional to

the thermal resistance of the well system, including the tubing wall,

annulus, casing wall and cement sheets. Ramey [35] introduced

an approximate, analytical solution for wellbore heat transmission

to estimate wellhead temperature as a function of wellbore depth

and the operational time. He has developed solutions for ﬂuids and

perfect gasses, assuming steady-state ﬂuid ﬂow in the wellbore and

transient heat conduction into the formation. An overall heat transfer

coeﬃcient was introduced to account for the total thermal resistance

of the well system. Other studies introduced methods to account for

multiple formation layers with different physical properties [50], for

real gas production [11], or for two-phase ﬂow in the wellbore [13].

Wellhead temperature distributions during oil, gas and geothermal

energy production, as well as temperature distributions of steam and

hot water injection applications, show that the difference between

wellbore ﬂuid temperature and surrounding formation at shallow

depths can be signiﬁcant with temperature differences larger than

30 °C[6,11,13,35,50]. However, the effective temperature of the outer

well casing may differ from the wellbore ﬂuid temperature, depend-

ing on the total thermal resistance of the well system. In general,

thermal resistance of steel casings and tubings can be neglected,

while insulating materials like cement sheets and annuli ﬁlled with

liquid or gas have a high thermal resistance. According to Ramey

[35], heat transfer through the different thermal resistance elements

of the wellbore is considerably faster than heat transfer in the sur-

rounding formation and, therefore, may be assumed as a steady-state

solution.

2.2. SEAWAT

We have used SEAWATv4 [10,24] to model density-driven

groundwater ﬂow induced by heat transfer from a hot well cas-

ing. SEAWATv4 is a coupled version of the simulation programs

for groundwater ﬂow, MODFLOW2000 [12] and for multi-species

mass transport, MT3DMS [51], together with a variable density and

viscosity package. This enables the simulation of variable-density

groundwater ﬂow combined with heat and multi-species solute

transport. The differential equation for solute transport takes into

34 J.H. van Lopik et al. /Advances in Water Resources 86 (2015) 32–45

account advection, dispersion and molecular diffusion:

³1+ρbKd

θ´δ(θC)

δt=∇hθ³Dm+αq

θ´(∇C)i−∇(qC)−q′

sCs(1)

where Cis the concentration of the solute (kg/m3), tis the time (d),

qis the speciﬁc discharge, ρbis the bulk density (kg/m3), Kdis the

distribution coeﬃcient (m3/kg), Dmis the molecular diffusion coeﬃ-

cient (m2/d), θis the porosity, α(m) is the dispersivity coeﬃcient, qs

is the source or sink (m/d) and Csis the source or sink concentration

(kg/m3). We have followed Langevin [24] and employed solute

transport equations to simulate heat transport in the aquifer:

µ1+1−θ

θ

ρscps

ρcp f ¶δ(θT)

δt

=∇·θµλfθ+(1−θ)λs

θρcp f

+αq

θ¶(∇T)¸−∇(qT)−q′

sTs(2)

The molecular diffusion coeﬃcient, as input parameter, can be used

to describe conductive heat transport by deﬁning a bulk thermal

diffusivity, DT(m2/d),as:

DT=λfθ+(1−θ)λs

θρcp f

(3)

where λfand λs(W/m °C) are the thermal conductivity of water and

solid phase, respectively, cpf is the heat capacity of water (J/kg °C)and

ρis the water density (kg/m3). Moreover, thermal equilibrium be-

tween water and the solid phase during heat transport is represented

by a thermal distribution factor, KdT (m3/kg):

KdT =cps

ρcp f

(4)

where cps (J/kg °C) is the heat capacity of solid phase. Therefore,

thermal retardation can be described in a similar manner as solute

retardation:

RT=1+ρb

θKdT where ρb=ρs(1−θ)(5)

where ρs(kg/m3) is the solid phase density.

SEAWATv4 uses the Oberbeck–Boussinesq approximation (e.g.

[4]) to assume a constant ﬂuid density in the heat transport equation,

while ﬂuid density is a function of salt concentration and tempera-

ture in the buoyancy term [40]. Assuming a constant ﬂuid density

(ρ) and the heat capacity of the ﬂuid (cpf) simpliﬁes the heat trans-

port equation (Eq. (2)) and enables the use of a constant bulk ther-

mal diffusivity (DT) and thermal distribution factor (KdT). We have

used a ﬂuid density (ρ) of 1000 kg/m3to calculate these parameters.

At high temperature and salt concentration contrasts the Oberbeck–

Boussinesq approximation becomes insuﬃcient [4,21]. In our sim-

ulated scenarios large temperature contrasts between well casing

(80 °C) and background groundwater temperatures are used, as well

as large salt concentration contrasts (0–35 kg/m3). Maximum ﬂuid

density is that of seawater (Cs=35 kg/m3) at a temperature of 15 °C

with a value of 1027 kg/m3. Close to the wellbore, the water is heated

up to 71.5 °C, resulting in a ﬂuid density of 977 kg/m3. Considering

these densities, the likely change in the bulk thermal diffusivity term

(DT) and the thermal distribution factor (KdT) can be calculated with

Eqs. (3) and (4). The relative error for DTand KdT is ±2.5% while using

the ﬂuid density of 1000 kg/m3as a reference. This means that the re-

sulting minimum and maximum DTvalues are 0.124 and 0.129 m2/d,

respectively. For KdT, these values are 1.86Å10−4and 1.96Å10−4m3/kg

and the error in heat transport due to the Oberbeck-Boussinesq ap-

proximation will be negligible [40].

The Boussinesq approximation in SEAWAT also simpliﬁes the con-

tinuity equation such that the volume of water is not inﬂuenced

by temperature and salinity, but only by head [21]. This means the

source terms nρbT∂T/∂tand nρbC∂C/∂tare neglected with respect to

ρSs∂h/∂t, where bTand bCare the thermal and solute coeﬃcient of

water expansivity, respectively. Our study has ranges of temperature

from 20 to 60 °C, salinity from 0 to 35 kg/m3, and head variation of

0.5 m. Using values of bT=4.3Å10−4°C−1and bC=7.3Å10−4m3/kg,

this means this assumption is not warranted. However, the volume

changes do not have a large inﬂuence on the ﬂow in our study.

The inﬂuences on the volume are strongest near the well, uniformly

over the thickness of the aquifer. The related pressure changes dissi-

pate laterally so that the upward convection along the well hardly is

affected.

2.3. Equation of state

Equations of state describe the temperature and salinity depen-

dence of properties of water, such as density, viscosity, heat capacity,

and thermal conductivity. SEAWATv4 allows for equations of state for

ﬂuid density and viscosity. The following viscosity equation [46] is

used in SEAWAT:

µ(C,T)=2.394 ·10−5·³10 248.37

T+133.15 ´+1.92 ·10−6(C−C0)(6)

where µis the dynamic viscosity (kg/m s), Cis the salt concentration

(kg/m3) and Tis the temperature of the ﬂuid.

The current standard SEAWATv4 computer code allows only for a

linearized form of the density equation of state:

ρ=ρf+dρ

dC (C−C0)+dρ

dT (T−T0)(7)

This linear form of the density equation of state is used in many

groundwater ﬂow and heat transport codes, like SUTRA [46] and

HST3D [20]. However, the density–temperature relationship is in-

deed non-linear (e.g. [26,37,38,41,49]). It can only be linearized with a

δρ/δTerror of less than 5% for temperature ranges in the order of 2 °C

for low water temperatures (∼20 °C) and for temperature ranges in

the order of 10 °C for higher water temperatures (∼60 °C) (see Fig. 2).

For a wide range of temperature and salinity, Sharqawy et al. [38]

derived an empirical non-linear density relationship based on exper-

imentally derived datasets for both salinity and temperature at 1 atm

pressure from Isdale and Morris [18] and Millero and Poisson [26]:

ρ(T,S)=(999.9+2.034 ·10−2T−6.162 ·10−3T2

+2.261 ·10−5T3−4.657 ·10−8T4)

+µ802.0S

1000 −2.001 S

1000 T+1.677 ·10−2S

1000 T2

−3.060 ·10−5S

1000 T3−1.613 ·10−5³S

1000 ´2

T2¶(8)

where Sis the salinity (g/kg) and Tis the temperature (°C). This den-

sity relationship (Fig. 3) is valid for a temperature range of 0–180 °C

and a salinity range of 0–160 g/kg with an accuracy of ±0.1%.

Previous simulation studies on density-driven groundwater ﬂow

have used different density equations of state (Table 1). Thorne et al.

[40] and Langevin et al. [25] veriﬁed SEAWATv4 by modeling the

Henry–Hilleke problem with the temperature and salt concentration

ranges shown in Fig. 3. Their linear approximations for the density–

temperature relationship are shown in Fig. 2. Holzbecher [15] and

Tsang et al. [42] have used a non-linear density–temperature rela-

tion (Table 1) based on empirical relationships derived by Tilton and

Taylor [41] and Wooding [49], respectively.

The conditions for density-driven groundwater ﬂow in our study

span large temperature contrasts between a well casing (up to 100 °C)

and background groundwater temperatures of 15 °C, as well as large

salt concentration contrasts (0–35 kg/m3). First, we have investigated

the effect of using a non-linear density equation in the SEAWATv4

code for the reference scenario (Case 1) and compared these re-

sults with a scenario run with the conventional SEAWATv4 code us-

ing the linear density equation of state (Case 1.1). A linear density–

temperature approximation for the conditions considered in this

J.H. van Lopik et al. / Advances in Water Resources 86 (2015) 32–45 35

Fig. 2. Dependence of density–temperature gradient on temperature based on the density equation of state as listed for each study in Table 1, while considering fresh water

(S=0 g/kg). The gray line shows the linear approximation over the temperature range 15–80 °C at a salinity value of 17.5 g/kg based on Eq. (8)[38]. These dρ/dT gradient is used

for our modeling study (Case 1.1).

Fig. 3. Water density [kg/m3] at different temperature and salinity values calculated

with Eq. (8) [38], where the used salinity and temperature ranges are shown for the

studies listed in Table 1 and the present study.

study was derived from Eq. (8) by linear interpolation of the mini-

mum and maximum density values. For the linear approximation of

the reference scenario (Case 1.1), the temperature range of 15–80 °C

and the average salinity value of the aquifer system (17.5 g/kg) were

used. This results in a dρ/dT gradient of −0.434 (Fig. 2, Case 1.1). The

linear density approximation over this temperature range yields a de-

viation from the non-linear relationship (Eq. (8), with S=17.5 g/kg) at

both low and high ends, respectively with an overestimation and un-

derestimation of the density–temperature gradient by approximately

a factor 2. To determine the effect of deviations caused by the lin-

earization of the density relationship on modeling density-driven

ﬂow, we have also implemented the empirical non-linear equation

(Eq. (8)), developed by Sharqawy et al. [38], in the SEAWATv4 code.

An iterative algorithm was implemented to calculate the ﬂuid den-

sity from the salt concentration (Cs) and temperature (see Fig. 4). Clo-

sure of the iterative process was set at a relative density difference of

1Å10−3. The source code implementation is given in Appendix A.

2.4. Model setup

We considered a homogeneous conﬁned sandy aquifer with a

horizontal interface between fresh and saline groundwater at 40 m

depth (10 m above the aquifer bottom). The groundwater ﬂow was

simulated axi-symmetrically, following the approach of Langevin

[23] which has been validated for transport of solutes [47] and

heat [45].

We assumed a constant effective temperature of the outer well

casing, to simulate the heat loss from the well. Analytical and ﬁeld

Table 1

Density equation of state in previous variable-density groundwater modeling studies.

Study T-range [°C] Cs-range [kg/m3] Density equation of state Code

Langevin et al. [25] 6.1–73.9 0–35 ρ=1000 +0.78C−0.392(T−14)SEAWAT

Thorne et al. [40] 5–50 0–35.7 ρ=1000 +0.7C−0.375TSEAWAT

Vandenbohede et al. [44] 7.85–48 0 ρ=1000 −0.375(T−10)SEAWAT

Tsang et al. [42] 20-55.4 0 T>25 °C: ρ=996.9[1 −3.17·10−4(T−25)

−2.56 ·10−6(T−25)2]T<25 °C:

ρ=996.9[1 −1.87 ·10−4(T−25)]

CCC

Holzbecher [15] 0-40 0 ρ=(1−(T−3.9863)2

508929.2·T+288.9414

T+68.12963 )·1000 FAST_C(2D)

36 J.H. van Lopik et al. /Advances in Water Resources 86 (2015) 32–45

Fig. 4. Flowchart for the non-linear density calculation in the SEAWAT code.

studies have shown that in many cases the bore-face temperature at

a given depth can be considered constant [22,35]. The heat transport

through the well casing elements can be considered rapid compared

to heat transport into the surrounding aquifer and can be considered

steady-state [35]. In addition, we assumed that the effective well cas-

ing temperature (Tc) is constant over the entire aquifer thickness of

only 50 m. Using Ramey’s solution [35], this is a reasonable assump-

tion for this limited depth range.

In the axi-symmetric transient model, a radius of 500 m and

aquifer thickness of 50 m was used. The grid resolution in the model

domain is 1r=0.25 m by 1z=0.5 m for [0.5 <r<23], 1r=0.5 m

by 1z=0.5 m for [23 <r<180] and 1r=1 m by 1z=0.5 m for

[180 <r<500]. The well system is 0.499 m thick, where the ﬁrst

column, representing the wellbore, is deﬁned as a no-ﬂow boundary.

The second column is only 0.001 m thick and represents the outer

well casing with a constant effective well casing temperature (Tc). The

initial aquifer temperature is 15 °C and the initial salt concentration

is 0 kg/m3for [0 <z<40] and Cs[40 <z<50] (see Table 2). The

top and bottom are no-ﬂow and no-heat ﬂux boundaries. The outer

boundary has a constant temperature, salt concentration and head

that equals the initial conditions. The Preconditioned Conjugate Gra-

dient 2 (PCG2) was used to solve for groundwater ﬂow. The method

of characteristics (MOC) was applied to solve the advection part of

transport, with a Courant number of 0.1. The convergence criterion of

relative temperature was set to 10−10 °C to simulate heat conduction

with a high bulk thermal diffusivity accurately [45].

2.4.1. Model parameters and sensitivity

First, the standard SEAWATv4 with a linear density relation was

applied and results for the reference scenario (Table 2) were com-

pared with the results from a simulation with the newly imple-

mented non-linear relation. Subsequently, the non-linear relation

was used for the subsequent simulations to investigate the sensitivity

of modeling results on the main input variables of well casing temper-

ature (Tc), salt concentration contrasts (Cs), as well as aquifer proper-

ties (Table 2). A non-linear density–temperature relation without the

density effects of salinity (see Eq. (8),S=0 kg/m3) was used for Case

3.3. The same conditions were assumed as for Case 3.1 with a tracer

concentration (Ctr) of 1.0 kg/m3to investigate the effect of small salt

concentration gradients on density-driven ﬂow and salt mass trans-

port. The values for aquifer properties were varied within the range

expected for sandy aquifers, using Bear [1] and Fitts [8] for hydraulic

conductivity and porosity, as well as Molz et al. [27] and the Engineer-

ing Toolbox [7] for thermal conductivity and heat capacity. Heteroge-

neous sandy aquifers with small clay layers could have elevated heat

capacities, since average clay heat capacity is 920 J/kg°C[7], which is

higher than the value of 800 J/kg°C used for sand. Therefore, a sce-

nario with a higher heat capacity of the aquifer than used for the

reference scenario was also considered (Case 6.2). For all scenarios,

the viscosity dependence on temperature is taken in account using

Eq. (6) in SEAWATv4.

Other hydraulic properties and thermal properties (Table 3) were

kept constant between the various simulations. A longitudinal dis-

Table 2

Summary of the input parameters analyzed in the sensitivity analysis. Underlined values indicate a variation on the reference scenario (Case 1).

Case Variation Tc[°C] Cs[kg/m3]kh[m/d] kv[m/d] λs[W/m °C] cps [J/kg °C] θ[dimensionless]

1 (Ref.) 80 35 15 1.5 3 800 0.35

2.1 Tc=0.5Tref

c40 35 15 1.5 3 800 0.35

2.2 Tc=0.67Tref

c60 35 15 1.5 3 800 0.35

2.3 Cs=1.33Cref

s100 35 15 1.5 3 800 0.35

3.1 Cs=1/35Cref

s80 1 15 1.5 3 800 0.35

3.2 Cs=10/35Cref

s80 10 15 1.5 3 800 0.35

3.3aCtr =1/35Cref

s80 1 15 1.5 3 800 0.35

4.1 kh=0.33kref

h80 35 5 0.5 3 800 0.35

4.2 kh=3.0kref

h80 35 45 4.5 3 800 0.35

4.3 kv=5kre f

v80 35 15 7.5 3 800 0.35

5.1 λs=0.67λref

s80 35 15 1.5 2 800 0.35

5.2 λs=1.33λref

s80 35 15 1.5 4 800 0.35

6.1 cps =0.9cref

ps 80 35 15 1.5 3 720 0.35

6.2 cps =1.1cref

ps 80 35 15 1.5 3 880 0.35

7.1 θ=0.6θref 80 35 15 1.5 3 800 0.21

7.2 θ=1.2aθref 80 35 15 1.5 3 800 0.42

aFor Case 3.3, a non-linear density–temperature relation was used without taking in account the density effects of the tracer concentration (Ctr).

J.H. van Lopik et al. / Advances in Water Resources 86 (2015) 32–45 37

Table 3

Hydraulic and thermal aquifer properties used for all simulations.

Properties Parameter value

Speciﬁc storage Ss=1Å10−4m−1

Solid phase density ρs=2650 kg/m3

Heat capacity of the ﬂuid cpf =4186 J/kg °C

Thermal conductivity of the ﬂuid λf=0.580 W/m °C

Molecular diffusion Dm=8.64Å10−5m2/d

Longitudinal dispersivity αl=0.5 m

Transversal dispersivity αT=0.05 m

persivity value (αl) of 0.5 m and a transverse dispersivity value (αT)

of 0.05 m were chosen, resulting in a grid Peclet number of 1 near the

wellbore. SEAWAT uses a single set of dispersivity values which are

applied to both heat and solute transport.

2.5. Metrics to quantify overall salinization and compare

scenario results

The distribution of salt in the top of the aquifer (upper 25 m) was

used to quantify the salinization and to facilitate comparison of the

scenarios. Several metrics were deﬁned for this purpose.

2.5.1. Cumulative salt mass and its associated center of mass

The transport of salt was quantiﬁed using the cumulative amount

of salt mass in the upper 25 m of the aquifer and the associated cen-

ter of mass. The cumulative salt mass is calculated using a summa-

tion over the salt mass in each grid cell in the model domain (725

columns, 100 layers):

MT=θ

725

X

j=0

100

X

k=1

C(r,z)¡r2

j+1πzk−r2

jπzk¢(9)

where MTis the cumulative salt mass (kg), rjis the radial distance at

column j(m), and zkis the depth at layer k(m). The following equa-

tions give the coordinates rcM and zcM (m) of the center of mass of the

salt mass MTfrom Eq. (9):

rcM =Ãθ

725

X

j=0

100

X

k=1

C(r,z)¡r2

j+1πzk−r2

jπzk¢rj+1!/MT

(10)

zcM =Ãθ

725

X

j=0

100

X

k=1

C(r,z)¡r2

j+1πzk−r2

jπzk¢zk!/MT

2.5.2. Maximum radius of salinization

We used the maximal radial extent of salt concentrations above

0.1 kg/m3(100 ppm) close to the top of the aquifer (Rmax at z=1 m) to

quantify the maximum spreading of salt mass in the lateral direction.

2.5.3. Dimensionless analysis of thermo-haline convection

A dimensionless analysis is applied on the cases in the sensitivity

analysis (Table 2) in order to characterize the buoyancy forces in our

system. Both the temperature and the salt concentration affect the

buoyancy term in the ﬂow equation [4].

To determine the ratio between the buoyancy tendency due to the

salt concentration contrast and the buoyancy tendency due to the

temperature contrast, the Turner, or buoyancy, number is used [4]:

B=1ρs

1ρT

(11)

where 1ρTis the maximum change in ﬂuid density due to tempera-

ture (kg/m3) and 1ρsis the maximum change in ﬂuid density due to

solute concentration (kg/m3). The dimensionless sensitivity analysis

for all scenarios is listed in Table 4.

Table 4

The maximum change in ﬂuid density due to temperature contrasts (1ρT), the maxi-

mum change in ﬂuid density due to salt concentration contrasts (1ρs) and the associ-

ated buoyancy number for each scenario.

Case Parameter 1ρT[kg/m3]1ρs[kg/m3]B[dimensionless]

1Ref 27.13 27.15 1.00

2.1 Tc=40 °C 6.71 27.15 4.05

2.2 Tc=60 °C 15.67 27.15 1.73

2.3 Tc=100 °C 40.63 27.15 0.67

3.1 Cs=1 kg/m327.13 7.76 0.29

3.2 Cs=10 kg/m327.13 0.785 0.03

4.1 kh=5 m/d 27.13 27.15 1.00

4.2 kh=45 m/d 27.13 27.15 1.00

4.3 kv=7.5 m/d 27.13 27.15 1.00

5.1 λs=2 W/m °C 27.13 27.15 1.00

5.2 λs=4 W/m °C 27.13 27.15 1.00

6.1 cps =720 J/kg °C 27.13 27.15 1.0 0

6.2 cps =880 J/kg °C 27.13 27.15 1.00

7.1 θ=0.21 27.13 27.15 1.00

7.2 θ=0.42 27.13 27.15 1.00

3. Results

First, the results for the reference scenario simulated with both

the standard linear density equation of state and the newly imple-

mented non-linear one are presented. Subsequently, the results of the

sensitivity analysis (Table 2) are presented.

3.1. Reference scenario (Case 1)

In the reference scenario (Case 1), the effect of heat transfer from

a hot well casing with a constant effective temperature of 80 °C for

40 years is simulated for a condition of fresh (0 kg/m3) groundwater

overlying groundwater with a seawater salt concentration (35 kg/m3,

Table 2).

3.1.1. Predicted salinization with linear and non-linear density

equation of state

Case 1 is the simulation of the reference scenario with the non-

linear density equation (Eq. (8)). The results show that density-

driven upward groundwater ﬂow along the wellbore induced by heat

transfer from the well casing led to considerable salinization of the

initially fresh upper part of the aquifer after 30 years (Fig. 5). Two

distinct thermal convection cells developed: one in the dense, salt

water at the bottom of the aquifer and another one in the fresh water.

The radial extent of these convection cells increases over time due

to the continuous heat transfer from the well casing. After hot saline

groundwater reaches the top of the aquifer, it is transported laterally.

It then gradually cools down and dilutes by mixing with the cooler,

fresh, groundwater.

The total salt mass in the top 25 m appears to increase exponen-

tially with time until it reaches a linear trend after 30 years, with

a rate of ∼4000 kg/year (Fig. 6A). The distance of the center of salt

mass and the maximum lateral spreading of salinization (Rmax) show

a rapid initial increase after which the increase tapers down (Fig. 6B

and C).

Modeling of the reference scenario (Case 1) was also done at dif-

ferent grid size (dr =0.33 m by dz =0.33 m) for [0.5 <r<154] and

dr =1 m by dz =0.33 m for [154 <r<500] to investigate the im-

pact of grid resolution on the model results. The effects on the main

metrics to quantify overall salinization (Section 2.5) are small, result-

ing in error percentages of total cumulative salt mass and associated

center of mass in the range of 0.18–1.95% and 0.33–1.02% respectively.

Therefore, no further grid reﬁnement was done and the grid conﬁgu-

ration described in Section 2.4 and was used for all simulations.

Case 1.1 is the reference scenario simulated with a linear den-

sity equation. A δρ/δCgradient of 0.73 was used for the linear salt

38 J.H. van Lopik et al. /Advances in Water Resources 86 (2015) 32–45

Fig. 5. (A) Salt concentrations [kg/m3] and (B) temperature [°C] at t=30 years for the non-linear reference scenario (Case 1) with the Darcy velocity vectors. The dot shows the

center of mass of cumulative stored salt in the model domain above the line [0 <z<25 m] over time and the red line shows Rmax. (For interpretation of the references to color in

this ﬁgure legend, the reader is referred to the web version of this article.)

concentration–density relationship (Eq. (7)) and a dρ/dT gradient

of −0.434 for the linear density–temperature relationship (line 1.1,

Fig. 2). The results of this simulation reveal trends similar to those

of the non-linear equation (Fig. 6). However, the linear density equa-

tion results in signiﬁcant overestimation of the cumulative salt mass:

140,437 kg after 30 years instead of 70,867 kg (Fig. 6A). The relative

error percentage for the cumulative salt mass increases from 81%

after 5 years up to 103% after 40 years, respectively. Also the lat-

eral spreading in the top of the aquifer is signiﬁcantly overestimated

while using the linear density equation. For instance, the r-coordinate

of the center of mass and the maximum lateral spreading of saliniza-

tion after 30 years is respectively 66.1 m and 103.5 m in the linear

density case instead of 53.6 m and 78 m in the non-linear density case

(Fig. 6B and C). The average relative error percentages for r-coordinate

of the center of mass and Rmax are respectively 24% and 32%.

The observed overestimation can be explained by the differ-

ences in the developed density gradients between the two cases. At

the lower end of the groundwater temperature range, the density–

temperature gradient for the non-linear density function is signif-

icantly lower than for the linear density approximation (line 1.1,

Fig. 2). Consequently, the upward advective salt mass transport at

low temperature ranges are signiﬁcantly overestimated by the linear

case (Case 1.1). Therefore, the associated mobilization of salt water

occurs over a consistently larger radial distance from the wellbore. In

addition to differences in upward advective salt mass transport, dis-

crepancies in salt mass transport between the cases with linear and

non-linear density equation of state are affected by differences in salt

water mixing by dispersion at the fresh-salt water interface. Differ-

ent lateral transport lengths and ﬂow velocities along the fresh-salt

water interface lead to variable extents of dispersive salt mixing.

It should be noted that convection is determined by the overall

density distribution that is affected by both temperature and salt con-

centration contrasts. Thermal retardation plays an important role in

the difference between advective salt transport and heat transport

during thermo-haline convection [30]. Salt mass is approximately

transported at the pore velocity (q/θ), while heat is retarded due to

thermal equilibration between the water and the solid grains (see

Eq. (4)). Note that the salt water front at Rmax has moved along both

borehole length and the top of the aquifer, while the heat front has

only moved along the top of the aquifer. Consequently, the thermal

front moves much slower laterally away from the wellbore in the top

of the aquifer than the salt front (as illustrated by Fig. 5). This means

that at larger radial distance the vertical stabilizing temperature gra-

dient on density decreases, and therefore the destabilizing gradient

of salt concentration on density increases. If this destabilizing gradi-

ent of salt concentration is high enough and the horizontal ﬂow ve-

locity is low enough, downward transport of salt mass occurs at Rmax

(Fig. 5A). The lower density-temperature gradient in the temperature

range of 15–20 °C for the non-linear density equation (Case 1) com-

pared to the linear density approximation (Case 1.1) (line 1.1, Fig. 2),

results in a faster destabilizing effect of the salt concentration gradi-

ent on water density at closer radial distances from the well for Case

1. Therefore, lateral transport of salt mass in the top of the aquifer

(Rmax) is overestimated for the linear density equation.

3.1.2. Reference scenario: breakthrough curves

Two observation points were added to the reference scenario

to illustrate how the salinization and heating of the groundwater

progress over time. Obs-1 is placed at a radial distance of 0.75 m from

the wellbore, at a depth of 25 m and Obs-2 at a radial distance of

40 m from the wellbore in the upper 1 m of the aquifer (see Fig. 5).

The upward transport of saline groundwater results in a progressively

increasing but slightly ﬂuctuating salt concentration in the vicinity

of the wellbore (Fig. 7A). The salt concentration over time in Obs-1

increases from 4.7 kg/m3(4700 ppm) after 1.5 years to 16.5 kg/m3

(16,500 ppm) after 40 years. The groundwater temperature in Obs-1

J.H. van Lopik et al. / Advances in Water Resources 86 (2015) 32–45 39

Fig. 6. (A) Cumulative salt mass, (B) associated center of mass and (C) the maximum

spreading of salt mass in the lateral direction, Rmax, in the upper half of the aquifer

[0 <r<500 m], [0 <z<25 m] for the non-linear density scenario (Case 1) and the

linear density scenario (Case 1.1). The relative error percentage between Case 1 and

Case 1.1 is used to quantify the overestimation of these parameters while using the

linear density equation.

reaches steady state in 1 year and remains constant at a temperature

of 70.5 °C (Fig. 7A). Consequently, the upward Darcy velocity in Obs-1

after 1.5 years is 0.059 m/day and decreases slightly due to the in-

creasing salt concentration and density of the groundwater (Fig. 8).

At a radial distance of 4.75 m, no elevated salt concentrations break-

through occurs and the upward Darcy velocity increases up to a value

of 0.011 m/day after 10 years in the fresh water layer. The strenght of

the lower salt water thermal convection cell is signiﬁcantly lower, re-

sulting in upward Darcy velocities which are a factor 0.52 lower than

observed in the fresh water layer at a radial distance of 4.75 m.

In the shallow observation point Obs-2 (Fig. 7B) elevated temper-

ature breakthrough occurs after 3.7 years, while elevated salt concen-

trations (at 0.1 kg/m3) breakthrough occurs after 6.7 years. In the ob-

servation point Obs-2, the salt concentrations vary strongly over time

between values of 2.00 (2000 ppm) and 2.45 kg/m3(2450 ppm) af-

ter 30 years. Moreover, the variability in salt concentrations increases

over time (Fig. 7B).

3.2. Parameter sensitivity analysis

Sensitivity analysis was conducted to test how the salinization is

affected by various factors. A set of 6 parameters was varied with re-

spect to the reference scenario which is shown in Table 2. Three dif-

ferent temperatures at the well casing (Case 2) were simulated, as

well as three different salt concentration contrasts with the overlying

fresh water (Case 3), two hydraulic conductivities (Case 4), two ther-

mal conductivities (Case 5), two solid phase heat capacities (Case 6),

and two porosities (Case 7).

3.2.1. Effect of well casing temperature (Case 2)

The well temperature at shallow depth can vary over a wide range,

depending on the technical application of the well as well as its pro-

duction depth and insulation. In the simulations, the rate of salt mass

transport and lateral spreading in the top of the aquifer were clearly

affected by different effective well casing temperatures (Fig. 9). The

highest casing temperatures resulted in the strongest increase in ver-

tical transport and lateral spreading of salt mass in the upper part

of the aquifer (Fig. 9). For a casing temperature of 100 °C (Case 2.3),

the temperature is high enough to overcome the density difference

between saline and fresh water (B=0.67, Table 4). Consequently, ad-

vective transport of salt mass was observed along the whole well-

bore length. This was not observed for Cases 1, 2.1 and 2.2, where the

buoyancy number B<1 (Table 4). The accumulation of transported

salt mass into the top of the aquifer is disproportionally large with

respect to the increases in effective temperature of the well casing.

For example, the increase of temperature from 40 to 60 °C yields a

6.2–9 fold increase in total salt mass transport over the whole time

range. In contrast, the lateral salt spreading increases progressively

less at higher well casing temperatures. Low well casing tempera-

tures (Tc=40 °C) already result in distinct lateral spreading of salt

mass in the top of the aquifer (Case 2.1) (Fig. 10).

3.2.2. Effect of salt concentration contrast (Case 3)

The reference scenario (Case 1) has fresh groundwater

(Cs=0 kg/m3) overlying water with a salt concentration of 35 kg/m3,

which is representative of intruding seawater in coastal areas (e.g.

[48]). However, smaller salt concentration contrasts may be present,

e.g. due to long-term progressive salinization [33]. Cases 3.1 and 3.2

have salt concentration contrasts of 1 and 10 kg/m3, respectively.

The rate of salt mass transport for Case 3.1 is only a factor 2.1 lower

than for Case 1, while for Case 3.2 it is approximately a factor 1.2

higher (Fig. 11A). Although salt mass transport is highest for a salt

concentration contrast of 10 kg/m3, the lateral spreading, as reﬂected

by the radial center of mass and Rmax, is slightly higher for the case

with a lower (1 kg/m3) salt concentration contrast (Fig. 11B and C).

Two convection cells developed in the reference scenario (Case

1). In contrast, upward convective ﬂow is not restricted at the fresh-

salt interface for Cases 3.1 and 3.2. The buoyancy numbers B for both

Cases 3.1 and 3.2 are lower than 1 (Table 4). Hence, the well casing

temperature of 80 °C is suﬃciently high to decrease the density of

the water with a salt concentration of 1 or 10 kg/m3below that of

the initially fresh water density at 15 °C in the vicinity of the well-

bore. Therefore, the saline groundwater moves upward over the en-

tire thickness of the aquifer (Fig. 12A).

The stabilizing effect of salt concentration on the density contrasts

is not taken into account for Case 3.3. The effect on the cumulative

salt mass transport is the same as for Case 3.1 (Fig. 11A). However,

lateral spreading of salt mass in the top of the aquifer was a factor

1.1 higher for Case 3.3 than for Case 3.1 (Figs. 11C and 12). Hence,

the destabelizing effect of the salt concentration on density contrasts

during lateral transport of salt mass in Case 3.1 results in a smaller

radial extent of the convection cell over time compared to Case 3.3.

3.2.3. Effect of hydraulic conductivity (Case 4)

The hydraulic conductivity of the aquifer inﬂuences groundwater

ﬂow velocities. Decreasing khand kvvalues by a factor of 3 (Case 4.1),

the cumulative salt mass transported into the top of aquifer decreases

40 J.H. van Lopik et al. /Advances in Water Resources 86 (2015) 32–45

Fig. 7. Breakthrough curves at observation points (A) Obs-1 and (B) Obs-2 for respectively salt concentration and temperature for the reference scenario (Case 1).

Fig. 8. Upward Darcy velocity over time at different radial distances in the vicinity of

the wellbore 25 m depth.

by a factor of 3.0 compared to the reference case (Fig. 13). The con-

trary is observed for Case 4.2, where khand kvare increased by a fac-

tor of 3. This results in increased cumulative salt mass in the top of

aquifer by a factor of 2.5. Case 4.3 has a higher anisotropy ratio, where

kvis raised to 7.5 m/day. This results in the largest salt mass trans-

ported into the top of the aquifer after 15 years of all considered sce-

narios. The downward transport of salt mass at the outer boundary

of the convection cell is also increased for both Cases 4.2 and 4.3 and

salt water is already transported to a depth below 25 m after 24 and

17 years, respectively. Consequently, continuous salt mass loss out of

the upper aquifer domain [0 <z<25 m] results in a slower increase

of cumulative salt mass for these cases (Fig. 13A and B).

3.2.4. Different thermal properties of the aquifer (Cases 5 and 6)

The thermal conductivity and heat capacity of the solid phase in-

ﬂuence heat transport via the bulk thermal diffusivity term (Eq. (3))

and the thermal retardation factor (Eq. (5)), respectively. A higher

thermal conductivity of the solid phase increases the heat conduction

from the well casing to the aquifer and the density-driven groundwa-

ter ﬂow. Varying the thermal conductivity for the solid phase within

Fig. 9. (A) Cumulative salt mass, (B) associated center of mass and (C) Rmax in the upper

half of the aquifer [0 <r<500 m], [0 <z<25 m] for Case 2.

J.H. van Lopik et al. / Advances in Water Resources 86 (2015) 32–45 41

Fig. 10. (A) Salt concentrations [kg/m3] and (B) temperature [°C] at t=30 years for the scenario with a well casing temperature of 40 °C (Case 2.1) with the Darcy velocity vectors.

The dot shows the center of mass of cumulative stored salt in the model domain above the line [0 <z<25 m] over time and the red line shows Rmax. (For interpretation of the

references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

a realistic range showed a moderate impact on the convection. The

salt mass transported into the top of the aquifer increased by a factor

of 1.2 for the upper bound (Case 5.2) and decreased by a factor of 1.2

for the lower bound (Case 5.1) after 30 years (Fig. 13).

Besides thermal conduction, heat transport is determined by ther-

mal retardation which is described as a function of the heat capacity

of the solid (Eq. (5)). The effect of thermal retardation is small (Case

6). A higher heat capacity of the solid (Case 6.2) resulted in more

absorption of heat into the solid matrix during heat transport and

thereby a higher thermal retardation factor. Consequently, the salt

front moves faster laterally away from the well than the thermal front

and cooling of the salt front occurs at closer radial distances from the

well. This results in a faster destabilizing effect of the salt concentra-

tion gradient on water density during lateral salt transport and hence

a reduced maximal lateral spreading (Rmax). The reverse is observed

for Case 6.1, since the adsorption of heat into the solid matrix is lower

than the reference scenario.

3.2.5. Effect of aquifer porosity (Case 7)

Porosity affects heat transport by bulk thermal diffusivity (Eq. (3))

and thermal retardation (Eq. (5)). SEAWATv4 allows only modeling

with effective porosity, while heat conduction is determined by the

overall, bulk porosity. Hence, thermal conduction in the aquifer will

be slightly overestimated when considering effective porosity instead

of bulk porosity, since the bulk thermal diffusivity term (DT) will be

lower in the simulations. The total cumulative salt mass in the top

of the aquifer for Case 7 indicates that the parameter sensitivity for

varying porosity is high (Fig. 13). The lowering of the porosity by a

factor of 0.6 (Case 7.1) increased the salt mass transport by a factor

of 1.14. An increased porosity by a factor 1.2 results in decreased salt

mass transport by a factor of 1.04. The decrease and increase in salt

mass transport are mainly due to the difference in advective salt mass

transport. For Case 7.1, the pore velocity (q/θ) at a given Darcy velocity

is increased by a factor 1.7, while for Case 7.2 the pore velocity is de-

creased by a factor 1.2. In reality such porosity changes would likely

be accompanied by permeability changes that mitigate the impact of

porosity.

4. Discussion

Simulations show that the heat loss from well casings can induce

upward convection of salty groundwater, leading to salinization of

overlying fresh groundwater. To the best of our knowledge, this is the

ﬁrst study that considers the impact of hot well casings on groundwa-

ter ﬂow and salt concentration gradients in aquifers. Here, the prac-

tical implications, such as monitoring requirements and the impli-

cations of modeling density-driven ﬂow under the conditions pre-

sented in this study will be discussed.

4.1. Implementation of non-linear density equation of state

We have simulated density-driven ﬂow over large temperature

and salt concentration ranges induced by the heat transfer from a hot

well casing. The resulting salt and heat transport in the aquifer was

shown to occur over relatively short distances (<100 m). The imple-

mentation of a non-linear density equation of state [38] in SEAWAT

was required for accurate results, as the linear approximation causes

relative errors up to 103% for the cumulative salt mass transport.

Other numerical simulations of convection induced by temperature

gradients use a non-linear relation between density and temperature

only (e.g. [15,42]). This study uses a non-linear density dependence

on both salt concentration and temperature.

Generally, heat transport mechanisms are classiﬁed as natural

(free) convection if ﬂow is solely induced by ﬂuid density differences

and forced convection if ﬂow is forced by external sources, like a

hydraulic gradient. Previous studies have shown that heat transport

42 J.H. van Lopik et al. /Advances in Water Resources 86 (2015) 32–45

Fig. 11. (A) Cumulative salt mass, (B) associated center of mass and (C) Rmax in the

upper half of the aquifer [0 <r<500 m], [0 <z<25 m] for Case 3.

by forced convection over relative small time ranges can be simu-

lated with linear density approximations in the SEAWAT code (e.g.

[25,44]). However, in the present study, heat transport is solely due

to free convection under the inﬂuence of continuous heat conduction

from the well casing over a large time range (up to 40 years). The

radial temperature distribution in the model domain varies spatially

and temporally. No sharp temperature front between the hot water

mass and colder ambient groundwater is observed (shown in Fig. 5B).

Both temperature and salt concentration distributions determine the

extent of the convection cells and the associated convective transport

over time. This is reﬂected in the difference between the simulations

with linear and non-linear density relations (Fig. 6).

A non-linear density relation in SEAWAT enables numerical sim-

ulations of problems where thermally induced free convection in

aquifers occurs over large temperature and time ranges. A disadvan-

tage would be the increased run times associated with the use of a

non-linear density relation. The higher accuracy of the implemented

non-linear density equation comes at the price of a larger compu-

tational runtime: 50 h compared to 20 for Case 1.1 with a Core i7-

4770@3.40 GHz processor.

4.2. Well casing temperatures of exploration wells

The range of well casing temperatures used in this study is repre-

sentative for temperatures encountered for conventional oil and gas

wells, geothermal energy production, as well as hot water or steam

injection [6,11,13,35,50]. However, little information is available on

the actual temperature at the outside of the well casing. The actual

temperature is an important parameter changing for different practi-

cal applications. This means that for a speciﬁc production or injection

operation detailed information about the wellhead temperature over

depth, the thermal resistance of the well system and the actual tem-

perature gradient over time in the well system is required to simulate

heat transfer into the aquifer as accurately as possible.

Decline in production rates results in decreasing well casing tem-

peratures. Therefore, the effects of thermally induced density-driven

groundwater ﬂow on upward salt transport are expected to be lower

when the duration of heat transfer through the wellbore decreases to

a few years. For example, typical initial shale gas production rates of

major U.S. shale gas ﬁelds are around 30 million m3per year, while

production rates drops to values around 4 million m3per year af-

ter 5 years of operation [43]. The analytical function by Ramey [35]

predicts that such low production rates will result in small tempera-

ture differences (<5°C) between the wellbore ﬂuid and the shallow

aquifer.

4.3. Fresh-salt interfaces and salt concentration contrasts

Fresh-salt interfaces are common in coastal areas (e.g. [48]). In

general, these interfaces have relative small dispersed transition

zones between fresh and saline waters compared to other stratiﬁed

fresh-salt aquifer systems. Usually, coastal regions are densely popu-

lated and fresh water reserves are scarce. Salinization poses problems

for drinking water, agriculture, and ecological habitats [34,48]. This

study can be used to provide insight into possible thermal impacts

on groundwater ﬂow and salinization risks near deep wells penetrat-

ing fresh water lenses.

Fresh-salt groundwater stratiﬁcation also occurs in regions with

brackish paleo-groundwater, or halite-rich formations [5,33]. For

these scenarios, the lower salt concentration contrasts of 1 and

10 kg/m3(Cases 3.1 and 3.2) are realistic. However, a less sharp dis-

persed fresh-salt transition should be considered under these condi-

tions [33]. Salt mass transport from the upper part of the diffusive

transition zone could then occur at lower temperature contrasts. For

example, for Case 3.2, the salt concentrations in the upper part of a

diffusive transition interface will be lower compared to a case with a

sharp salt concentration contrast of 10 kg/m3. Consequently, the re-

quired temperature rise to overcome the buoyancy difference with

cold fresh groundwater will be lower. Therefore, it is likely that salt

transport occurs over a larger radial distance along the wellbore if a

wide diffusive fresh-salt transition zone is considered.

Moreover, direct upward salt transport over the entire aquifer

thickness occurs for the lower salt concentration contrasts (1 and

10 kg/m3) of Cases 3.1 and 3.2 (Section 3.2.2). The density difference

for the reference scenario is too large to allow direct groundwater

ﬂow along whole borehole length at the simulated effective well cas-

ing temperatures (Cases 1 and 2.1–2.3). Consequently, the effect of

salt mixing by dispersion between the salt and fresh water layer along

the fresh-salt interface becomes more important for this case with

two separate convection cells.

4.4. Uncertainties in the simulation results

Dispersion along the fresh-salt interface controls the salt trans-

port to the top of the aquifer for large salt concentration contrasts

when two convection cells develop (e.g. Case 1). Such enhanced salt

water mixing along a fresh-salt water interface due to the formation

of thermal convection cells was observed in the lab experiment of

Henry and Hilleke [14] and numerical simulations of this experiment

[25]. In this experiment, warm fresh water was injected in a large

sand-ﬁlled tank and ﬂowed on top of water with a salt concentration

of 35 kg/m3. Heat pads were located at the lower and right part of the

J.H. van Lopik et al. / Advances in Water Resources 86 (2015) 32–45 43

Fig. 12. (A) Salt concentrations [kg/m3] for Case 3.1 and (B) salt concentrations [kg/m3] for Case 3.3 at t=30 years with the Darcy velocity vectors. The dots show the center of

mass of cumulative stored salt in the model domain above the line [0 <z<25 m] over time and the red line shows Rmax. (For interpretation of the references to color in this ﬁgure

legend, the reader is referred to the web version of this article.)

Fig. 13. Sensitivity of the (A) cumulative salt mass, (B) associated center of mass, and

(C) Rmax in the upper half of the aquifer [0 <r<500 m], [0 <z<25 m] after 15 and 30

years. ∗Note that for Cases 4.2 and 4.3 salt has been transported downward, below the

25 m reference line at the outside of the convection cell.

sand-ﬁlled tank, resulting in thermal convection cells in the lower left

part of the tank in both the fresh and the salt water layer. The mix-

ing by these convection cells widened the fresh-salt interface. For a

simulation scenario without taking into account the temperature de-

pendency of the density, less salt mass transport was observed due to

the absence of thermal convection cells.

This illustrates that the mixing along a fresh-salt interface is

largely controlled by advection and transverse dispersion [4,31]. The

presented simulations used dispersivity values of αl=0.5 m and

αT=0.05 m. Enhanced solute mixing by dispersion will result in in-

creased salt mass transport to the top of the aquifer. For example, a

scenario with dispersivity values of αl=1.0 m and αT=0.1 m re-

sulted in an increase in salt mass transport by a factor of 1.14. There-

fore, the dispersivity value should be carefully considered for simula-

tions of speciﬁc ﬁeld cases with fresh-salt interfaces (e.g. [9]).

Small ﬂuctuations of salt concentrations with time are observed

in the vicinity of the wellbore, when the salt concentration contrast

of the interface is large (such as in Case 1, Fig. 7A). This can be ex-

plained by the difference in diffusivity between heat and salt. The

initial situation is stable due to the salt concentration gradient. The

heat transfer from the wellbore causes destabilization of the system.

This is called the diffusive regime of Double Diffusive Convection and

instabilities can arise during this phenomenon [4]. At the top, hot salt

water moves laterally over colder fresh water. This is called the ﬁn-

gering regime of Double Diffusive Convection [4,17,39]. If the desta-

bilizing salt concentration gradient in the top of the aquifer is large

(e.g. Case 1 and Case 3.2), it results in instabilities and ﬂuctuating

salt concentrations over time (Fig. 7B). The salt concentration ﬂuctu-

ations due to Double Diffusive Convection decreases with enhanced

salt mixing by dispersion.

Grid resolution, dispersivity values, and the use of a 2D axi-

symmetric domain could have an effect on the simulation of unsta-

ble upward ﬂuid ﬂow along the wellbore and local salt concentration

ﬂuctuations (e.g. [4,19]). Therefore, the salt concentrations should be

44 J.H. van Lopik et al. /Advances in Water Resources 86 (2015) 32–45

measured frequently in the ﬁeld (e.g. using automatic electrical con-

ductivity (EC) data loggers) to obtain insight in the mixing and the

average salt concentrations at the top of the aquifer.

This study did not consider heat loss from the aquifer into the

conﬁning layers. Horizontal heat transport from hot well casings

to surrounding aquitards will be conduction dominated, while heat

transport in the aquifer occurs by both conduction and convection.

As illustrated in Fig. 5B, the upward convective heat transport results

in heat accumulation and hence a larger radius of elevated aquifer

temperature at the top of the aquifer. The radius of elevated aquifer

temperature at the bottom of the aquifer is reduced by the convec-

tive inﬂow of cooler native groundwater. Heat conduction into and

from the aquifer can be expected for the underlying and overlying

aquitards, respectively. However, the thermal conductivity is gener-

ally high for clay layers compared to the thermal conductivity of

sandy aquifers [1,7]. Hence, heat transfer from the well casing into

the aquitard by thermal conduction is expected to be higher than for

the aquifer. Therefore, heat conduction into the aquitard is expected

to have a limited impact on the lateral transport of salt in the top of

the aquifer.

4.5. Practical considerations

The present study shows the possibility of signiﬁcant upward

transport of salt around hot well casings. Field evidence of this phe-

nomenon does not exist to the best of our knowledge. This would re-

quire spatially detailed and frequent long-term monitoring to obtain

quantitative insight on such thermo-haline convection in the vicinity

of hot wells. The simulations of the present study were carried out for

an axi-symmetric homogeneous sandy conﬁned aquifer domain. This

section addresses the implications of different ﬁeld conditions.

Axi-symmetric ﬂow condition strongly reduces simulation run-

ning times, but does not allow simulation of regional groundwater

ﬂow. However, regional ﬂow will increase the net heat transfer from

the hot well casing to ambient groundwater and cause a different

temperature distribution around the well. This will change the ther-

mal convection cells in both salt and fresh water layer.

The aquifer was assumed to be homogeneous. However, variations

in hydraulic and thermal properties of the aquifer near the wellbore

would impact the density-driven groundwater ﬂow and salt mass

transport. Aquifer heterogeneity could have a large impact on con-

vective ﬂow patterns in thermo-haline systems [28]. For example,

horizontal clay lenses restrict density-driven ﬂow and may lead to

separate convection cells above and below a clay lens. Consequently,

salinization of the upper part of the aquifer might be limited or even

prevented.

Unconﬁned aquifers were not tested in this study. For an uncon-

ﬁned aquifer, the temperature at the water table is controlled by the

atmospheric temperature. Continuous heat loss at the top results in

less intensive convection. Consequently, salt mass transport and lat-

eral spreading of salt in the top of the phreatic zone are expected to

be lower as for conﬁned aquifers. In addition, unconﬁned aquifers are

recharged with fresh water by precipitation which will dilute trans-

ported salt water in the top of the phreatic zone.

5. Conclusions

To the best of our knowledge, this is the ﬁrst study that shows that

heat loss from well casings may cause mixing due to density-driven

ﬂow and mobilize fresh-salt water interfaces, resulting in salinization

of shallow fresh groundwater. A non-linear density relationship as a

function of both salt concentration and temperature had to be imple-

mented in the SEAWATv4 code to simulate this phenomenon accu-

rately. The total transported salt mass to the top of the aquifer ranged

between values of 3150 and 166,000 kg after 30 years for the simu-

lated scenarios. Lateral spreading of salt mass in the top of the aquifer

occurred over radial distances up to 91 m. During lateral transport of

salt water along the top of the aquifer, ﬂuctuating salt concentration

breakthrough over time occurs due to destabilizing salt concentration

gradients.

For large salt concentration contrasts, such as a fresh-salt (seawa-

ter) interface, groundwater heating results in the formation of sepa-

rate convection cells in the fresh water layer and the salt water layer.

At smaller salt concentration contrasts, the heated salt water layer

becomes less dense than the cooler native groundwater. Hence, up-

ward advective transport of salt along the wellbore occurs over the

entire thickness of the aquifer.

The convection and salt transport is stronger for scenarios with

increased temperatures of the well casing, increased horizontal and

vertical hydraulic conductivities, decreased porosity and increased

thermal conductivity of the aquifer. Therefore, these aspects should

be considered when assessing the potential impact of hot well casings

for site-speciﬁc conditions. The results of this study indicate heat loss

from hot well casings can induce density-driven groundwater ﬂow

and salinization of shallow fresh groundwater. This process should

therefore be considered when monitoring for long-term groundwa-

ter quality changes near wells through which hot ﬂuids or gases are

transported.

Acknowledgments

The authors thank four anonymous reviewers for their con-

structive feedback, which allowed us to improve the manuscript

signiﬁcantly.

Appendix A. Modiﬁed density equation in SEAWAT

The calculation steps for the density calculation (Fig. 4) are out-

lined in this appendix. The non-linear density equation (Eq. (8)) has

been implemented in the VDF package of SEAWAT. The code for the

function CALCDENS in the VFD1.f ﬁle has been adapted. This function

calculates ﬂuid density for each grid cell from the concentrations of

each MT3DMS species, which is in our model salt concentration (Cs)

and temperature (T).

First the salt concentration (in kg/m3) and temperature (in °C) are

read. Subsequently, the iterative algorithm is performed by solving

Eqs. (A.1)–(A.3):

S=Cs/ρn−1(A.1)

ρn=(999.9+2.034 ·10−2T−6.162 ·10−3T2

+2.261 ·10−5T3−4.657 ·10−8T4)

+µ802.0S

1000 −2.001 S

1000 T+1.677 ·10−2S

1000 T2

−3.060 ·10−5S

1000 T3−1.613 ·10−5S

1000

2

T2¶(A.2)

rρ=|ρn−ρn−1|(A.3)

Closure of the iterative scheme is set to residual rρ<1−10−3.

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