fluid dispersion effects on density-driven thermohaline flow and transport in porous media
TRANSCRIPT
Advances in Water Resources 61 (2013) 12–28
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Advances in Water Resources
journal homepage: www.elsevier .com/ locate/advwatres
Fluid dispersion effects on density-driven thermohaline flowand transport in porous media
0309-1708/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.advwatres.2013.08.006
⇑ Corresponding author. Tel.: +1 225 578 4246; fax: +1 225 578 4945.E-mail addresses: [email protected] (Z. Jamshidzadeh), [email protected]
(F.T.-C. Tsai), [email protected] (S.A. Mirbagheri), [email protected] (H.Ghasemzadeh).
1 Tel.: +98 913 162 5146; fax: +98 21 880 35 516.
Zahra Jamshidzadeh a,1, Frank T.-C. Tsai b,⇑, Seyed Ahmad Mirbagheri a,1, Hasan Ghasemzadeh a,1
a Department of Civil Engineering, K.N. Toosi University of Technology, Tehran, Iranb Department of Civil and Environmental Engineering, Louisiana State University, 3418G Patrick F. Taylor Hall, Baton Rouge, LA, USA
a r t i c l e i n f o a b s t r a c t
Article history:Received 12 December 2012Received in revised form 11 August 2013Accepted 12 August 2013Available online 21 August 2013
Keywords:Porous mediaDensity-driven flowThermohalineFluid dispersionHenry problemElder problem
This study introduces the dispersive fluid flux of total fluid mass to the density-driven flow equation toimprove thermohaline modeling of salt and heat transports in porous media. The dispersive fluid flux inthe flow equation is derived to account for an additional fluid flux driven by the density gradient andmechanical dispersion. The coupled flow, salt transport and heat transport governing equations arenumerically solved by a fully implicit finite difference method to investigate solution changes due tothe dispersive fluid flux. The numerical solutions are verified by the Henry problem and the thermal Elderproblem under a moderate density effect and by the brine Elder problem under a strong density effect. Itis found that increment of the maximum ratio of the dispersive fluid flux to the advective fluid fluxresults in increasing dispersivity for the Henry problem and the brine Elder problem. The effects of thedispersive fluid flux on salt and heat transports under high density differences and high dispersivitiesare more noticeable than under low density differences and low dispersivities. Values of quantitativeindicators such as the Nusselt number, mass flux, salt mass stored and maximum penetration depth inthe brine Elder problem show noticeable changes by the dispersive fluid flux. In the thermohaline Elderproblem, the dispersive fluid flux shows a considerable effect on the shape and the number of developedfingers and makes either an upwelling or a downwelling flow in the center of the domain. In conclusion,for the general case that involves strong density-driven flow and transport modeling in porous media, thedispersive fluid flux should be considered in the flow equation.
� 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Density-driven thermohaline flow and transport modeling inporous media is of interest in many real-world problems, includingsaltwater intrusion into freshwater aquifers, brine flow near saltdomes, and possible spread of nuclear waste into deep saline aqui-fers. In such flow and transport problems, the flow field stronglydepends on spatial–temporal changes of fluid density and viscositythat vary with salt concentration and temperature. As discussed inBear and Bachmat [1] and Bear and Cheng [2], the density-drivenflow and transport problems may also be influenced by the disper-sive fluid flux of total fluid mass, an additional fluid flux in the flowequation, caused by the density gradient and mechanical disper-sion. However, discussions of the dispersive fluid flux in the flowequation and its effects on salt and heat transports are very limitedin the literature.
For highly saline water, such as brine, the concentration of saltcan reach to 300 g/l, which corresponds to water density of about1200 kg m�3 [3]. Large density gradients may cause significantfluid dispersion to take place. To account for the effect of high con-centration gradient and high density gradients on fluid flow andsalt transport in porous media, Hassanizadeh [4] developed anon-linear theory of solute dispersion and used a nonlinear formof Darcy’s law and Fick’s law to account for the dispersive flux oftotal solute mass in brine transport problems. Hassanizadeh andLeijnse [5] experimentally showed that the Fickian description ofsolute dispersion in a porous medium is valid only if small concen-tration gradients are presented in the system. They pointed outthat under high concentration gradients, large amounts of solutemass tend to be dispersed, and the nonlinear dispersive flux mustbe considered in the strong density-driven flow and solute trans-port system. Their experiment showed that the calculated break-through curves for high concentration gradients, based on thelinear form of Darcy’s law and Fick’s law, are significantly differentfrom the measured breakthrough curves. They concluded that thenonlinear theory provides satisfactory results to match the exper-imental data over a wide range of concentration gradients. vanDuijn et al. [3] analyzed brine transport in porous media using a
Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28 13
modified form of the Darcy velocity suggested by Hassanizadeh [4]and obtained a semi-analytical solution for coupled flow and masstransport equations. Their study showed that Fick’s law, based onthe assumption of a linear relation between solute dispersive massflux and concentration gradient, is not valid when high concentra-tion gradients are encountered. Bear and Bachmat [1] derived ageneral macroscopic flow equation which involves both advectiveand dispersive fluid fluxes. However, they did not discuss the ef-fects of the dispersive fluid flux by suggesting that the advectivefluid flux is much larger than the dispersive fluid flux in commoncases of weak density-driven flow. Recently, Bear and Cheng [2]obtained a density-driven flow equation that considers both advec-tive and dispersive fluid fluxes for salt transport. The dispersivefluid flux, based on the linear form of Fick’s law, is proportionalto density gradients. The proportionality is the mechanical disper-sion. Since fluid density is affected by salt concentration as well asheat, this study follows Bear and Cheng [2] derivations and extendsto investigate the effects of the dispersive fluid flux on thermoha-line flow and transport problems based on the benchmarks of theHenry problem [6] and the Elder problems [7,8].
Regardless of the dispersive fluid flux in the flow equation, var-ious numerical methods and models were developed to obtainsolutions of either coupled flow and salt transport equations orcoupled flow and heat transport equations using the finite ele-ment method [9–12], the finite difference methods [13], the inte-grated finite difference method [14,15], the method ofcharacteristics [16], and the lattice Boltzmann method [17,18].Only a few studies obtained solutions from complete coupledflow, salt transport, and heat transport equations. Voss and Pro-vost [19] developed a 3D computer model, SUTRA, to solve thefluid mass-balance equation and unified energy- and solute- bal-ance equation for variable-density, single-phase, saturated–unsat-urated flow and single-species transport. Hughes and Sanford [20]introduced a new version of SUTRA that is capable of simulatingvariable density flow and transport of heat and multiple dissolvedspecies through variably to fully saturated porous media. Theyused the Henry and Hilleke [21] problem to test the variable-den-sity modeling code HST3D [22], which includes temperature andconcentration effects on variable-density flow. They gave a moredetailed discussion of the effects of mesh resolution on thenumerical simulation. Thorne et al. [23] developed a new versionof SEAWAT [13] to simulate simultaneous transport of heat andsolute by allowing fluid density and viscosity as a function of tem-perature and/or solute concentration in the equation of state andused the Henry and Hilleke [21] problem to test their code. Grafand Boufadel [24] used the HydroGeoSphere model to investigatethe effects of viscosity, capillarity and grid spacing on thermal var-iable-density flow under saturated and unsaturated flow condi-tions. They showed that the thermal convective flow is highlysensitive to spatial discretization and grid size. Graf [25] incorpo-rated heat transport within the saturated-zone flow regime intoHydroGeoSphere together with temperature-dependent fluidproperties, such as viscosity and density. This model can be usedfor the case of thermohaline flow and transport in saturated andunsaturated porous media. Ranganatham and Hanor [26] studiedthe brine-density flow around salt domes using SUTRA [19] tosimulate flow pattern due to gradients in either solute concentra-tion or temperature. They considered two cases: (1) Solution ofcoupled density-driven fluid flow and salt transport equations toevaluate salt dissolution around a salt dome, neglecting the effectsof temperature. (2) Solution of coupled density-driven flow andheat transport equations to evaluate flow field pattern, neglectingthe effects of salt concentration. They showed that the thermalgradients have a significant effect on groundwater flow patternsas well as salt concentration gradients. But, the magnitudes of
flow resulting from thermal convection are less than brine-densityflow for similar geometries and sediment properties.
To verify numerical codes for solving density-driven fluid flowand salt/heat transport equations without dispersive fluid flux inthe flow equation, the Henry problem, the thermal Elder problem,and the brine Elder problem are the benchmarks. The Henry prob-lem [6] provides a well-defined semi-analytical solution of seawa-ter intrusion into a confined aquifer. The original boundaryconditions used by Henry [6] have been modified by several stud-ies to make it close to realistic conditions [27–30]. In this study, thenumerical code was verified based on the traditional sea boundarycondition in the Henry problem, and the updated form of seaboundary was not applied here. The thermal Elder problem [7] isa free thermal convection problem that fluid flow is driven bythe fluid density differences due to the variation of temperature.Many studies solved the Elder problem and found different shapeof plumes and flow directions at the center of the domain [31–34]. It is understood that the density difference at the outerextremities of the lower heat boundary creates an unstable flowcondition, causing the upwelling of warm water and forming ther-mal fingering [34,35]. Many numerical factors such as discretiza-tion schemes, grid size, and simplifications in the governingequations that can cause non-unique stationary solutions havebeen reported for the Elder problem [15,24,34,35]. Diersch [36]and Voss and Souza [37] converted the thermal Elder problem intoa brine transport problem. Many researchers have studied thisbenchmark and found different results. They discussed the exis-tence of an upwelling or downwelling flow in the central part ofthe domain at 20-year simulation with different numerical approx-imation methods, discretization schemes, grid size, and density dif-ferences [13,38–40]. Also, any change in the governing equationsand the equation of state may lead to different solutions [40,41].
Verification of a code based on the Henry problem is not chal-lenging because this problem has a unique solution while the Elderproblem has multiple solutions which are associated with thenumber of fingers and the direction of flow in the central part ofthe domain. Diersch and Kolditz [42] pointed out that the solutionsof the Elder problem obtained by various simulation codes dependson grid discretization. Park and Aral [34] evaluated the sensitivitiesof the numerical solution of the Elder problem to the density,velocity and numerical perturbations. They concluded that highdensity differences (greater than 20%) cause physical instabilitythat is inherent in this problem, and for low density differences(less than 20%) the solution is stable and is independent of griddensity. Van Reeuwijk et al. [31] reported that the differences inthe governing equations, grid resolution and employed discretiza-tion techniques can cause different size and shape of fingers in theElder problem. They showed that for low Rayleigh number(Ra < 76), the Elder problem has a unique solution.
This study develops a fully implicit finite difference code to sim-ulate coupled density-driven thermohaline flow and transport inporous media. The dispersive fluid flux [2] is added into the flowequation to investigate the influence of this term on salinity andtemperature solutions. The Henry problem [6], the thermal Elderproblem [7], and the brine Elder problem [8] are adopted to verifythe numerical solution and to investigate the effects of the disper-sive fluid flux on the solutions of these benchmarks given differentdensity differences and dispersivities. Quantitative indicators [43]and grid convergence are evaluated to ensure numerical accuracyand stability. Finally, the numerical code is applied to a thermoha-line problem in anisotropic and non-homogeneous porous mediato investigate the influence of the dispersive fluid flux on the solu-tions. For simplicity, the tortuosity and the thermal conductivityfor the solid and the fluid phases are assumed isotropic in theanisotropic and non-homogeneous numerical example.
14 Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28
2. Mathematical model
In order to simulate coupled groundwater flow, salt transport,and heat transport in a porous medium, the following governingequations, based on the mass and energy conservation laws,Darcy’s law, and Fick’s law, are employed.
2.1. Governing equations
2.1.1. Flow equation with dispersive fluid fluxBased on the mass conservation law, the density-driven flow
equation with the dispersive fluid flux of total fluid mass is [2, p.624]
@ðhqÞ@tþr � ðhqv þ hJqÞ ¼ 0 ð1Þ
where q is the water density [ML�3], v is the seepage velocity[LT�1], h is the porosity, t is time [T], qv is the advective fluid fluxof total fluid mass [ML�2T�1], and Jq is the dispersive fluid flux oftotal fluid mass [ML�2T�1]. The Darcy velocity q = hv for density-driven flow in terms of the freshwater head is [13,44] is
q ¼ �kqf gl
rhf þq� qf
qf
!rz
" #ð2Þ
where k is the intrinsic permeability tensor [L2], l is the dynamicviscosity [ML�1T�1], qf is the reference freshwater density [ML�3],hf is the freshwater head [L], g is the gravitational acceleration[LT�2], and z is the elevation [L]. Hassanizadeh [4] formulated a gen-eral form for the dispersive fluid flux, which is related to the gradi-ents of water density, salt concentration, and temperature. In thelater formulation, the water density is considered as a function ofsalt concentration and temperature. Using the chain rule on thewater density, following the expression in Bear and Cheng [2, p.624], and adding the temperature effect, one can obtain an expres-sion for hJq as follows
hJq ¼ �hDrq ¼ �hD@q@CrC þ @q
@TrT
� �ð3Þ
where C is the salt concentration [ML�3], T is the temperature [H],and D, a second rank symmetric tensor, is the mechanical disper-sion tensor [L2T�1]. Eq. (3) can be expressed as Fickian-type lawfor density dispersion, i.e., �hDrq. It is noted that diffusive fluxof the total mass is identically zero [2, p. 166]. The mechanical dis-persion tensor is expressed by Scheidegger’s formula as follows[2,44]:
D ¼ aT jvjdij þ ðaL � aTÞvivj
jvj ð4Þ
where aL and aT are the longitudinal dispersivity and transverse dis-persivity [L], respectively, |v| is the magnitude of the seepage veloc-ity, and dij is the Kronecker delta.
Substituting Eq. (3) into Eq. (1), the density-driven thermoha-line flow equation is
@ðhqÞ@tþr � ðqqÞ � r � hD
@q@CrC þ hD
@q@TrT
� �¼ 0 ð5Þ
The third term in Eq. (5) shows the effect of the salt concentrationgradient and the temperature gradient on the dispersive fluid flux.It is noted that the difference between density-dependent and den-sity-independent problems is the existence of oq/oC and oq/oT. Theeffect of the dispersive fluid flux depends on the order of magnitudeof the third term in Eq. (5). If O(�hDrq) << 1, the fluid dispersioneffect may be negligible, but the flow is still density-dependent.Moreover, the effect of dispersive flow can also be evaluated bythe Peclet number, the ratio of advective fluid flux to dispersive
fluid flux. The dispersive fluid flux is negligible if it is much smallerthan the advective one [1].
In general, fluid density is a nonlinear function of pressure, tem-perature and salt concentration. This study assumes that the pres-sure effect on the fluid density is much smaller than temperatureand salt concentration effects for the range of pressures consid-ered. Using the linearized approximation, the equation of statefor fluid density is [2]
qðC; TÞ ¼ qf ½1þ aðT � T0Þ þ bC� ð6Þ
where T0 is the reference temperature [H], qf is freshwater densitygiven T0 and zero salinity, a is the thermal coefficient [T�1], and b isthe salt concentration coefficient [M�1L3]. The thermal coefficientcan be approximated by
a ¼ 1qf
@q@T� 1
qf
qTmax� qf
Tmax � T0
� �ð7Þ
where qTmax[ML�3] is the fluid density at the maximum tempera-
ture Tmax [H]. The salt concentration coefficient can be approxi-mated by
b ¼ 1qf
@q@C� 1
qf
qmax � qf
Cmax
� �ð8Þ
where qmax [ML�3] is the maximum water density, and Cmax [ML�3]is the maximum salt concentration.
In spite of the assumption of neglecting the pressure effect inEq. (6), this study does take into account the effect of pressure inthe expression of specific storage later obtained for the mass bal-ance equation (1).
The dynamic viscosity is a nonlinear function of salt concentra-tion and density, which can be expressed as [2, p. 623]
lðC; TÞ ¼ lf 0ðTÞ 1þ 1:85Cq
� �� 4:1
Cq
� �2
þ 44:5Cq
� �3" #
ð9Þ
where lf0 is the freshwater dynamic viscosity, corresponding toC = 0 and temperature T. The temperature-dependent freshwaterviscosity can be expressed as [45]
lf 0ðTÞ ¼ 2:394� 10�5 � 10248:37
Tþ133:15ð Þ ð10Þ
where lf0(T0) is the freshwater viscosity at the reference tempera-ture T0. Therefore, the equation of state for the dynamic viscosity is
lðC; TÞ ¼ 2:394� 10�5
� 10248:37
Tþ133:15ð Þ 1þ 1:85Cq
� �� 4:1
Cq
� �2
þ 44:5Cq
� �3" #
ð11Þ
2.1.2. Salt transport equationIn the absence of source/sink term, the advection–dispersion
equation for salt transport is [30,44]
@ðhCÞ@t�r � ðhDhrCÞ þ r � ðhvCÞ ¼ 0 ð12Þ
where Dh = D + sDm is the hydrodynamic dispersion tensor, whichincludes the mechanical dispersion D and the effective diffusionsDm. Dm is the molecular diffusion coefficient [L2T�1], and s, a sec-ond rank symmetric tensor, is the tortuosity of the porous medium.For isotropic saturated porous media, we have s = sdij and the scalartortuosity s equals s = h1/3 [21]. The effective diffusion sDm dependson both molecular diffusion of the salt in water and porous mediumtortuosity.
Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28 15
2.1.3. Heat transport equationApplying the energy conservation law, the heat transfer equa-
tion in saturated porous media is written in terms of temperatureT as [12]:
@
@tfðð1� hÞqscs þ hqcf ÞTg �r � ðkerTÞ þ r � ðhqcf vTÞ ¼ 0 ð13Þ
where T is temperature [H], qs is the density of the solid phase[ML�3], cs is the specific heat capacity of the solid phase [L2T�2H�1],cf is the specific heat capacity of the fluid [L2T�2H�1], and ke is theequivalent thermal conductivity [tensor] of the fluid and heat con-duction in both solid and fluid phases [MLT�3H�1]. For an isotropicporous media, ke is given below [12]:
ke ¼ ½ð1� hÞks þ hkf �dij þ hqcf D ð14Þ
where ks and kf are thermal conductivity [MLT�3H�1] for the solidand the fluid phases, respectively.
In Eq. (14), the following assumptions are made. (1) It is as-sumed that the temperature of the solid phase and its containedfluid is the same. (2) Thermal energy transferred by radiation is ig-nored. (3) Solid density does not change with time. (4) The externalheat sinks and sources due to chemical reactions (dissolution/pre-cipitation) are negligible. And (5) the thermal conductivities (ks
and kf) are assumed to be isotropic.
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Langevin & Guo [30]+ + + + + + +
this study
Simpson & Clement [28]
Servan-Camas & Tsai [17]
∂C/∂z=0
∂C/∂z=0
C=0
qx=6.6×10-5
qz=0
(a)
(b)
1.024 1.02
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
m/s
Fig. 1. Solution of the Henry problem [6]: (a) boundary conditions and
2.2. Numerical scheme for density-driven thermohaline flow andtransport
To solve the benchmarks of the Henry problem and the Elderproblems, we simulate density-driven thermohaline flow andtransport in two-dimensional cross sections. According to Eq. (5),the extended form of the density-driven saturated flow equationin terms of the freshwater head hf is
qqf
Sf@hf
@tþ ha
@T@tþ hb
@C@t
¼ @
@xkxxqg
l@hf
@x
� �� �þ @
@zkzzqg
l@hf
@zþ
q� qf
qf
!" #( )
þ @
@xhDxxa
@T@xþ hDxxb
@C@xþ hDxza
@T@zþ hDxzb
@C@z
� �
þ @
@zhDzza
@T@zþ hDzzb
@C@zþ hDzxa
@T@xþ hDzxb
@C@x
� �
ð15Þ
where Sf is the freshwater specific storage [L�1]. It is assumed thatthe principal directions of anisotropic intrinsic permeability arealigned with the coordinates in x and z directions.
The extended form of the non-reactive salt transport equation(Eq. (12)), is
1.2 1.4 1.6 1.8 2
75%
25%50%
10%
90%
Cmax
=35 g
/L
qz=0
1.022
1.016
1.01
1.2 1.4 1.6 1.8 2
hf =1.025 m
hf =1 m
isochlor curves and (b) freshwater head distribution and flow field.
Table 1Model parameters for the Henry problem (modified after Servan-Camas and Tsai[17]).
Parameter Symbol Value Unit
Hydraulic conductivity K 1.0 � 10�2 m s�1
Molecular diffusion coefficient Dm 1.886 � 10�5 m2 s�1
Upstream inflow flux Q 6.6 � 10�5 m2 s�1
Specific storage Sf 0 m�1
Porosity h 0.35 –Tortuosity s 1 –Longitudinal dispersivity aL 0 mTransverse dispersivity aT 0 mFreshwater density qf 1000 kg m�3
Seawater density qmax 1025 kg m�3
Seawater concentration Cmax 35 kg m�3
Salt concentration coefficient b 7.14 � 10�4 –
αL (m)
00.05
0.10.15
0.20.25
0.30.35
0.40.45
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
{}
max
/ρ
θρ
Jq
=0.38 αL+0.035
{}
max/ρ
θρ
Jq
Fig. 2. The maximum ratio of dispersive fluid flux (|hJq|) to advective fluid flux(|qq|) versus longitudinal dispersivity for the Henry problem.
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
5
10
15
5
10
15
5
10
15
this studySEAWAT [46]
T =12 qz=0
T =12 qz =0
T =20 qz =0
T =12 qz =0
qx =0 ∂T/∂x=0
qx =0 ∂T/∂x=0
A B
0.2 0.6
0.2
0.6
(a) 2 y
(b) 10
(c) 20
0.2
0.6
Fig. 3. Comparison of 0.2 and 0.6 isothermal curves of the thermal Elder problem with SEand (c) 20 years.
16 Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28
@ðhCÞ@t¼ @
@xhDhxx
@C@xþ hDhxz
@C@z
� �þ @
@zhDhzz
@C@zþ hDhzx
@C@x
� �
þ @
@xkxqf g
lC@hf
@x
� �@
@zkzqf g
lC
@hf
@zþ
q� qf
qf
!( )ð16Þ
The heat transport equation is
½ð1� hÞqscs þ hqcf �@T@tþ cf hT
@q@t
¼ @
@xkexx
@T@xþ kexz
@T@z
� �þ @
@zkezz
@T@zþ kexz
@T@x
� �
þ cf@
@xq
kqf gl
@hf
@x
� �T
� �þ cf
@
@zq
kqf gl
@hf
@zþ
q� qf
qf
!" #T
( )
ð17Þ
The study employs a fully implicit finite difference method tosolve Eqs. (15)–(17). By using the block-centered finite differencediscretization scheme in the flow equation and the Picard iterationscheme for the nonlinear terms, the fully implicit approximate ofthe two-dimensional density-driven flow equation is
A1hnþ1;mþ1fi�1;j þ B1hnþ1;mþ1
fi;j þ C1hnþ1;mþ1fiþ1;j þ D1hnþ1;mþ1
fi;j�1 þ E1hnþ1;mþ1fi;jþ1
¼ F1hnfi;j þ RHSðC; TÞnþ1;m ð18Þ
where A1 to F1 are the numerical coefficients as a function of fluidproperties and solid permeability between two nodes which havebeen calculated based on harmonic mean, n is the time step, andm is the iteration stage in one time step. RHS(C,T) is the coefficient
0 100 200 300 400 500 6000
0
0
0
0 100 200 300 400 500 6000
0
0
0
0 100 200 300 400 500 6000
0
0
0
ears
years
years
AWAT [46] (left) and corresponding flow field (right) at time (a) 2 years, (b) 10 years
Table 2Model parameters for the thermal Elder problem (modified after Elder [7]).
Parameter Symbol Value Unit
Permeability k 1.95 � 10�10 m2
Molecular diffusion coefficient Dm 3.565 � 10�6 m2 s�1
Specific storage Sf 0 m�1
Porosity h 0.1 –Tortuosity s 1 –Longitudinal dispersivity aL 0 mTransverse dispersivity aT 0 mReference density qf 998.2 kg m�3
Minimum density qmin 997.7 kg m�3
Thermal conductivity of fluid kf 0.65 kg m s�3 �C�1
Thermal conductivity of solid ks 1.591 kg m s�3 �C�1
Freshwater dynamic viscosity lf 10�3 kg m�1 s�1
Thermal coefficient a �6.26 � 10�5 �C�1
Maximum temperature Tmax 20 �CReference temperature T0 12 �CSolid density qs 2650 kg m�3
Specific heat capacity of solid cs 0 m2 s�2 �C�1
Specific heat capacity of fluid cf 4.18 � 103 m2 s�2 �C�1
0
2
4
6
8
10
12
14
16
18
20
0 2 4 6 8 10 12 14 16 18 20Time (Year)
SEAWAT [46]Elder [7]this study
Nu
Fig. 4. Comparison of the Nusselt number of the thermal Elder solution with Elder[7] and SEAWAT [46].
Table 3Model parameters for the brine Elder’s problem (modified after Voss and Souza [37]).
Parameter Symbol Value Unit
Permeability k 4.845 � 10�13 m2
Molecular diffusion coefficient Dm 3.565 � 10�6 m2 s�1
Specific storage Sf 0 m�1
Porosity h 0.25 –Tortuosity s 1 –Longitudinal dispersivity aL 0 mTransverse dispersivity aT 0 mFreshwater density qf 1000 kg m�3
Maximum density qmax 1200 kg m�3
Maximum salt concentration Cmax 1 –Freshwater dynamic viscosity lf 10�3 kg m�1 s�1
Salt concentration coefficient b 0.2 –
Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28 17
that depends on salt concentration and temperature in previousiteration stage. The RHS term is related to Jq. The mechanical disper-sion coefficients between two nodes are calculated based on
qz =0
5
10
15
0 50 100 150 200 250 3000
50
100
150
+
++
5
10
15
0 50 100 150 200 250 3000
50
100
150+ + + ++
+
+
+
+
+
+
0.2
0.6
0.60.2
+ + + + + + + + + + Prasad and Simoons [43]
this study (without J )ρ
C=1 C=0
qz =0 C=0
∂C/∂x=0
qx =0 qx =0
∂C/∂x=0
qz =0 A
(a) 1 year
(c) 10 years
Fig. 5. Comparison of 0.2 and 0.6 isochlor curves of the brine Elder problem with Prasad4 years, (c) 10 years, and (d) 20 years.
harmonic mean. Due to limited space, the details in the numericalcoefficients and RHS(C,T) are not shown.
The fully implicit approximate of the two-dimensional salttransport equation is
A2Cnþ1;mþ1i�1;j þB2Cnþ1;mþ1
i;j þC2Cnþ1;mþ1iþ1;j þD2Cnþ1;mþ1
i;j�1 þE2Cnþ1;mþ1i;jþ1
þF2Cnþ1;mþ1iþ1;jþ1 þG2Cnþ1;mþ1
i�1;jþ1 þH2Cnþ1;mþ1iþ1;j�1 þ I2Cnþ1;mþ1
i�1;j�1 ¼ J2Cni;j ð19Þ
where A2 to J2 are the numerical coefficients.Using the same numerical scheme, the fully implicit finite-dif-
ference approximate of the two-dimensional heat transport equa-tion is
A3Tnþ1;mþ1i�1;j þ B3Tnþ1;mþ1
i;j þ C3Tnþ1;mþ1iþ1;j þ D3Tnþ1;mþ1
i;j�1 þ E3Tnþ1;mþ1i;jþ1
þ F3Tnþ1;mþ1iþ1;jþ1 þ G3Tnþ1;mþ1
i�1;jþ1 þ H3Tnþ1;mþ1iþ1;j�1 þ I3Tnþ1;mþ1
i�1;j�1 ¼ J3Tni;j ð20Þ
where A3 to J3 are the numerical coefficients.The linearized system of Eqs. (18)–(20) are solved using the
Gaussian elimination method. At the end of each iteration stage,the fluid density, viscosity, and Darcy velocities are updated bythe following equations:
qnþ1;mþ1i;j ¼ qf ½1þ aðTnþ1;mþ1
i;j � T0Þ þ bðCnþ1;mþ1i;j Þ� ð21Þ
0 50 100 150 200 250 3000
0
0
0
0 50 100 150 200 250 3000
0
0
0
0.6
0.2
0.6
0.2
(b) 4 years
(d) 20 years
and Simmons [43] for mesh ‘teld’ (Dx = 5 m and Dz = 3.75 m) at time (a) 1 year, (b)
18 Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28
lnþ1;mþ1i;j ¼ 2:394� 10�5 � 10
248:37Tnþ1;mþ1
i;jþ133:15
� �1þ 1:85
Cnþ1;mþ1i;j
qnþ1;mþ1i;j
!"
� 4:1Cnþ1;mþ1
i;j
qnþ1;mþ1i;j
!2
þ 44:5Cnþ1;mþ1
i;j
qnþ1;mþ1i;j
!335 ð22Þ
qnþ1;mþ1xi;j
¼ �kxqf g
lnþ1;mþ1i;j
hnþ1;mþ1fiþ1;j � hnþ1;mþ1
fi�1;j
2Dx
!ð23Þ
qnþ1;mþ1zi;j
¼ �kzqf g
lnþ1;mþ1i;j
hnþ1;mþ1fi;jþ1 � hnþ1;mþ1
fi;j�1
2Dzþ
qnþ1;mþ1i;j � qf
qf
!ð24Þ
where Dx and Dz are discretization interval in x and z direction,respectively.
0 50 100 150 200 250 3000
50
100
150
0 50 100 150 200 250 3000
50
100
150
5
10
15
0 50 100 150 200 250 3000
50
100
150
1
1
0 50 100 150 200 250 3000
50
100
150
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
1
1
1
1
(a) mesh
(b) mesh
(c) mesh
(d) mesh
Fig. 6. Isochlor curves for 0.2, 0.4, 0.6 and 0.8 and flow fields in the half-domain for diffe(Dx = 6 m and Dz = 3.75 m), (c) meld (Dx = 5 m and Dz = 5 m) and (d) teld (Dx = 5 m and
The Picard iteration method is used to obtain the numericalsolutions of the nonlinearly coupled equations. The solution proce-dure is as follows. At the first time step, the freshwater head is cal-culated by Eq. (18), where the numerical coefficients and RHS(C,T)are calculated based on the initial condition. Then, the Darcy veloc-ities are calculated by the initial fluid density and viscosity andnew freshwater head. The salt concentration and temperature arecalculated by Eqs. (19) and (20), respectively, where numericalcoefficients are calculated with the initial fluid density and viscos-ity and new Darcy’s velocities. The calculated salt concentrationand temperature are used to update fluid density and viscosity.The freshwater head, salt concentration, temperature arere-calculated by solving Eqs. (18)–(20) using the updated fluiddensity and viscosity. The iteration procedure is repeated untilthe following convergence criteria are met:
jhnþ1;mþ1f � hnþ1;m
f j 6 10�2 ð25Þ
0 50 100 150 200 250 3000
0
0
0
0 50 100 150 200 250 3000
50
00
50
4.02.0 0.6
0.8
0.2 0.4 0.6
0.8
0 50 100 150 200 250 3000
50
00
50
0.20.4 0.6
0.8
0 50 100 150 200 250 3000
50
00
50 0.8
0.20.4
0.6
“held”
“reld”
“meld”
“teld”
rent meshes of the brine Elder problem: (a) held (Dx = 10 m and Dz = 5 m), (b) reldDz = 3.75 m) at 4 years (left) and 20 years (right).
log(
εh m
ax)
log (Δx)(a)
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
αL=αT=0 m, Δz=3.75 m, O (Δx1.83)
αL=αT=4 m, Δz=3.75 m, O (Δx1.40) with Jρ
αL=αT=4 m, Δz=3.75 m, O (Δx1.16) without Jρ
log(
εC
max
)
log (Δx)(b)
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5−1.1
−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
αL=αT=0 m, Δz=3.75 m, O (Δx1.38)
αL=αT=4 m, Δz=3.75 m, O (Δx0.88) with Jρ
αL=αT=4 m, Δz=3.75 m, O (Δx0.91) without Jρ
Fig. 7. Convergence rate for maximum errors ehmax and eCmax at 20 years, the brine Elder problem.
6
8
10
12
14
16
18
20
1 3 5 7 9 11 13 15 17 19 21
Nu
Time (Year)(a) Mesh “held”
Time (Year)(b) Mesh “teld”
6
8
10
12
14
16
18
20
1 3 5 7 9 11 13 15 17 19 21
Nu
Prasad and Simmons [43]this study
Fig. 8. Comparison of the Nusselt number of the brine Elder solution with Prasadand Simmons [43] for (a) mesh ‘‘held’’ and (b) mesh ‘‘teld’’.
Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28 19
jCnþ1;mþ1 � Cnþ1;mj 6 10�2 ð26Þ
jTnþ1;mþ1 � Tnþ1;mj 6 10�2 ð27Þ
Then, the calculations move to the next time step.
3. Model verification
In this section, the numerical code is verified with the Henryproblem, the thermal Elder problem, and the brine Elder problem.
3.1. The Henry problem
The Henry problem has been discussed extensively in the liter-ature as a seawater intrusion benchmark for moderate density-dri-ven groundwater flow in an isotropic and homogenous confinedaquifer. The Henry problem has an exaggerated molecular diffu-sion coefficient and neglects mechanical dispersion (i.e., aL = aT = 0)[6]. The spatial dimension and boundary conditions of the Henryproblem are shown in Fig. 1(a). The top and bottom boundariesare impervious. The flow domain initially is full of freshwater.The initial hydraulic head is set to 1 m everywhere in the domain.The numerical code is tested for the traditional sea boundary con-dition (Dirichlet boundary condition of C and hf). Table 1 lists themodel parameters for the Henry problem. In spite of the enormousmolecular diffusion, the dispersive fluid flux (Jq) is not considereddue to zero dispersivity in the original Henry problem. A grid size
20 Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28
of 21 rows and 41 columns is used to discretize the domain, whichis corresponding to a uniform discretization with Dx = Dz = 0.05 m.The time step is 6 s. The steady-state solution is obtained after300 min. Simulation after 300 min shows no significant changesin the solution. The steady-state solution shown in Fig. 1(a) is com-pared well to the semi-analytical solution [28] and the previouspublished numerical solutions [17,30]. Fig. 1(b) shows the fresh-water head distribution and the flow field. For better clarity, onlya half of the velocity vectors are shown in the Fig. 1(b).
To investigate the Jq effect in the Henry problem, longitudinaldispersivity is systematically increased from 0 to 1. Fig. 2 showsthe maximum ratio of the dispersive fluid flux to the advectivefluid flux in the domain, which increases with increasing longi-tudinal dispersivity for the range given aL/aT = 10, density differ-ence 20% and 300-min simulation time. A linear regressionequation is obtained: max{|hJq|/|qq|} = 0.38aL + 0.035 for Fig. 2.It is clear that the dispersive fluid flux becomes influential asdispersivity increases and cannot be ignored in the flowequation.
0 50 100 150 200 250 3000
50
100
150
0 50 100 150 200 250 3000
50
100
150
0 50 100 150 200 250 3000
50
100
150
0 50 100 150 200 250 3000
50
100
150
5
10
15
0
50
100
150
0
50
100
150
00
50
100
150
(a) Density diffe
With J ρ
Without J ρ
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.20.4
0.6
0.8
(b) Density diffe
(c) Density diff
(d) Density dif
Fig. 9. Isochlor curves in the half-domain of mesh ‘‘teld’’ for aL = aT = 4 m given density
More investigations on the contribution of density differencesand mechanical dispersion to Jq are given to the Elder problems.
3.2. The thermal Elder problem
The thermal Elder problem [7] is used here to test the ability ofthe numerical code to simulate free thermal convection problem.Fig. 3(a) shows the boundary conditions for the thermal Elderproblem. The model parameters are listed in Table 2. The intrinsicpermeability is assumed 1.95 � 10�10 m2 to keep the same Ray-leigh number of 400 in the original thermal Elder problem. The ref-erence density qf is the initial water density. The whole domain isdiscretized into 61 rows and 41 columns, which is correspondingto Dx = 10 m and Dz = 3.75 m. Constant head of 150 m is assignedto point A at the bottom left corner and point B at the bottom rightcorner in Fig. 3(a). Time step of two months is used. The total sim-ulation time is 20 years. Since the dispersivity is zero in the originalthermal Elder problem, the effect of Jq on the numerical solutionsdoes not exist. The purpose here is to compare to existing solutions
0 50 100 150 200 250 3000
0
0
0
0 50 100 150 200 250 300
0 50 100 150 200 250 300
50 100 150 200 250 300rence=10%
0.2
0.4
0.6
0.8
0.20.4
0.6
0.8
0.2
0.40.6
0.8
rence=20%
erence=25%
ference=30%
0.20.4
0.6
0.8
upw
ellin
g
upw
ellin
g
upw
ellin
g
upw
ellin
gup
wel
ling
upw
ellin
gup
wel
ling
dow
nwel
ling
difference (a) 10%, (b) 20%, (c) 25% and (d) 30% at 4 years (left) and 10 years (right).
(b) αL=αT=2 m
(a) αL=αT=1 m0 50 100 150 200 250 300
0
50
100
150
0 50 100 150 200 250 3000
50
100
150
0 50 100 150 200 250 3000
50
100
150
0 50 100 150 200 250 3000
50
100
150
0 50 100 150 200 250 3000
50
100
150
0 50 100 150 200 250 3000
50
100
150
0.20.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.20.4
0.6
0.8
0.20.4 0.6
0.8
0.2 0.4 0.6
0.8
ρ
ρ
upw
ellin
g
upw
ellin
gup
wel
ling
upw
ellin
gup
wel
ling
upw
ellin
g
Without JWith J
(c) αL=αT=3 m
Fig. 10. Isochlor curves in the half-domain of mesh ‘‘teld’’ for 30% density difference given dispersivity (a) aL = aT = 1 m, (b) aL = aT = 2 m and (c) aL = aT = 3 m at 4 years (left)and 10 years (right).
Time (Year)
Nu
(a)
0 2 4 6 8 10 12 14 16 18 204
5
6
7
8
9
10
Time (Year)
Mas
s flu
x (k
g/da
y)
0 2 4 6 8 10 12 14 16 18 200
0.20.40.60.8
11.21.41.61.8
Time (Year)
Salt
mas
s sto
red
(kg)
x 104(b)
(c)
0 2 4 6 8 10 12 14 16 18 200
20
40
60
80
100
120
140
Time (Year)
PD (m
)
(d)
468
1012141618202224
0 2 4 6 8 10 12 14 16 18
Without Jρ
With Jρ
(αL=αT=4 m)
(αL=αT=4 m)
20
Fig. 11. The effect of Jq on the quantitative indicators for the brine Elder problem assuming density difference 30% and aL = aT = 4 m, (a) Nusselt number, (b) mass flux (kg/day), (c) salt mass stored (kg) and (d) maximum penetration depth (PD) of 0.6 isochlor.
Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28 21
that does not consider dispersive fluid flux in order to verify thecode. Fig. 3 shows the numerical solution for 0.2 and 0.6 isothermalcurves and the corresponding flow fields at different times. The
temperature distributions are satisfactory although noticeable dif-ferences in comparing to SEAWAT [46] are expected. The maxi-mum velocity in x direction is vx = 3.83 � 10�6 m s�1 and in z
αL (m)
0
0.1
0.2
0.3
0.4
0.5
0.6
0 2 4 6 8 10 12 14 16 18 20
{}
max
/ρ
θρ
Jq
=0.028 α L+0.014
{}
max/ρ
θρ
Jq
Fig. 12. The maximum ratio of dispersive fluid flux (|hJq|) to advective fluid flux(|qq|) versus longitudinal dispersivity for the brine Elder problem.
Table 4Model parameters for the thermohaline Elder problem (modified after Diersch andKolditz [49]).
Parameter Symbol Value Unit
Permeability k 4.85 � 10�13 m2
Molecular diffusion coefficient Dm 3.565 � 10�6 m2 s�1
Specific storage Sf 0 m�1
Porosity h 0.1 –Tortuosity s 1 –Longitudinal dispersivity aL 0 mTransverse dispersivity aT 0 mReference density qf 998.2 kg m�3
Minimum density qmin 997.7 kg m�3
Maximum density qmax 1200 kg m�3
Maximum salt concentration Cmax 1 –Freshwater dynamic viscosity lf 10�3 kg m�1 s�1
Salt concentration coefficient b 0.2 –Thermal conductivity of fluid kf 0.65 kg m s�3 �C�1
Thermal conductivity of solid ks 1.59 kg m s�3 �C�1
Maximum dynamic viscosity lmax 1.9 � 10�3 kg m�1 s�1
Thermal coefficient a �5 � 10�4 �C�1
Maximum temperature Tmax 1 �CReference temperature T0 0 �CSolid density qs 2650 kg m�3
Specific heat capacity of solid cs 0 m2 s�2 �C�1
Specific heat capacity of fluid cf 4.18 � 103 m2 s�2 �C�1
22 Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28
direction is vz = 2.82 � 10�6 m s�1. The mean velocity calculatedfrom non-zero velocities in the entire domain at 20 years isvx = 0.59 � 10�6 m s�1 and vz = 0.33 � 10�6 m s�1.
The Nusselt number (Nu) is a quantitative indicator which canbe used to evaluate the results of the numerical code. This dimen-sionless number enables the comparison of the actual heat transferacross the bottom boundary of the model to the heat transfer oc-curred solely through conduction. Elder [7] and Dausman et al.[46] calculated the Nusselt number as a function of time. The sim-ilar temporal variations of Nu shown in Fig. 4 for Elder [7], SEAWAT[46] and this study verify our code for solving the coupled flow andheat transport equations.
In the following brine Elder problem, more detailed investiga-tions on grid convergence and effects of density differences anddispersivities on Jq are provided.
3.3. The brine Elder problem
The brine Elder problem is a benchmark for strong density-dri-ven flow and salt transport [8,37]. The boundary conditions forthe brine Elder problem are shown in Fig. 5(a) for a half domain.The model parameters are given in Table 3. Constant head of150 m at point A at the upper left corner is assigned. The simula-tion time is 20 years and is divided into 240 time steps, each of
(a) C at 10 years 0 100 200 300 400 500 6000
50
100
150
5
10
15C=1 0=T0=T0=T qz =0 A B
0.2
0.4
0.6
0.8
∂C/∂x=0∂T/∂x=0qx =0
∂C/∂x=0
∂T/∂x=0qx =0
C=0 C=0
T=1 C=0 qz =0
mesh 61×31mesh 41×21
0 100 200 300 400 500 6000
50
100
150
(c) C at 20 years
0.2 0.4
0.6
0.8
5
10
15
Fig. 13. The thermohaline Elder problem [49]. (a) Isochlor curves and (b) isothermal cDx = 10 m and Dz = 5 m.
which is corresponding to one month. The Rayleigh number is400.
The dispersive fluid flux Jq is not considered in the original brineElder problem due to zero dispersivity. Therefore, the numericalsolution is compared to Prasad and Simmons [43,47]. Fig. 5 showsthe comparison of the numerical solutions for 0.2 and 0.6 isochlorcurves for the fine mesh, ‘‘teld’’ (Dx = 5 m, Dz = 3.75 m) at differenttimes. Diffusion causes the migration of the salt solute into thefreshwater zone, and then the heavier saline water sinks fasterthan the surrounding water, creating a convective pattern. Thenumerical solution in Fig. 5 is in satisfactory agreement with Pra-sad and Simmons [43].
To investigate grid convergence for the code, four spatial dis-cretizations are used: (1) held (Dx = 10 m and Dz = 5 m), (2) reld(Dx = 6 m and Dz = 3.75 m), (3) meld (Dx = 5 m and Dz = 5 m),and (4) teld (Dx = 5 m, Dz = 3.75 m) [43]. Fig. 6 shows the position
(b) T at 10 years 0 100 200 300 400 500 6000
0
0
0
0.8
0.6
0.4
0.2
0 100 200 300 400 500 6000
0
0
0
0.8
0.60.4
0.2
(d) T at 20 years
urves at time 10 years and 20 years for mesh Dx = 15 m and Dz = 7.5 m and mesh
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
without Jwith Jρ
ρ
(a) density difference=15% (b) density difference=20%
(c) density difference=25% (d) density difference=30%
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.20.4
0.6
0.8
0.2 0.4
0.6
0.8
upw
ellin
gup
wel
ling
upw
ellin
gup
wel
ling
upw
ellin
gup
wel
ling
dow
nwel
ling
dow
nwel
ling
Fig. 14. Isochlor curves at 10 years given mesh Dx = 15 m and Dz = 7.5 m for density differences in the thermohaline Elder problem: (a) 15%, (b) 20%, (c) 25% and (d) 30%. Solidlines are solutions with Jq. Dashed lines are solutions without Jq. The dispersivity coefficients are aL = aT = 2 m.
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
ρ
ρ
(a) αL=αT=1 m (b) αL=αT=2 m
(c) αL=αT=3 m (d) αL=αT=4 m
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Without J
With J
Fig. 15. Isochlor curves for mesh Dx = 15 m and Dz = 7.5 m at 5 years for dispersivity (a) aL = aT = 1 m (b) aL = aT = 2 m, (c) aL = aT = 3 m and (d) aL = aT = 4 m given a 30%density difference.
Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28 23
of 0.2, 0.4, 0.6 and 0.8 isochlor curves and the flow fields for thefour discretizations at time 4 years and 20 years. The visual differ-ences in the isochlor curves are due to grid resolution. All solutionsshow consistent direction of flow rotation and display an upwell-ing flow at the center of the full domain (at x = 300 m). Accordingto Kolditz et al. [8], the upwelling flow indicates correct solutions.Moreover, the number of developed fingers is the same for allmeshes. Therefore, the code produces correct solutions for the fourspatial discretizations.
To investigate grid convergence rate for the code, numericalsolutions are obtained for Dx = 5, 10, 15, 25 and 30 m givenDz = 3.75 m. Since the analytical solution is not available, the max-imum errors of fresh groundwater head and salt concentration arecomputed with respect to the finest mesh ‘‘teld’’ at 20 years, i.e.,ehmax ¼maxðjhteld
ij � hijjÞ and eCmax ¼maxðjCteldij � CijjÞ. Fig. 7(a)
shows the convergence rate for groundwater head. For zero disper-sivity aL = aT = 0, it shows almost the second-order convergencerate for groundwater head. For aL = aT = 4 m, convergence rate forincluding Jq is 1.40. The convergence rate without the Jq term isaround the first order. Fig. 7(b) shows the convergence rate forconcentration. Convergence rate for zero dispersivity is 1.38. GivenaL = aT = 4 m, convergence rate with or without the Jq term isaround the first order.
To examine numerical stability and numerical dispersion in salttransport simulation, we check the Peclet number (Pe) and theCourant number (Cu). Using the maximum velocity and moleculardiffusion coefficient, the Peclet number in x direction is Pex = vxDx/Dm = 7.09 for mesh ‘‘held’’ and is reduced to 4.38 for mesh ‘‘teld’’.The Peclet number in z direction is Pez = vzDz/Dm = 2.87 for mesh‘‘held’’ and is reduced to 2.15 for mesh ‘‘teld’’. So to keep Pe < 4
A B
600 m
300 m
Kx=2×10-12 , Kz=2×10-13
Kx=8×10-12 , Kz=8×10-13
Kx=2×10-12 , Kz=2×10-1360 m
40 m
50 m θ=0.45
θ=0.3
θ=0.45
Case (1)
A B
600 m
Kx=8×10-12 , Kz=8×10-13
Kx=2×10-12 , Kz=2×10-13
Kx=8×10-12 , Kz=8×10-1360 m
40 m
50 m θ=0.3
θ=0.45
θ=0.3
Case (2)
A B
600 m
Kx=2×10-13 m2
Kz=2×10-12 m2Kx=2×10-13 m2
Kz=2×10-12 m2
θ=0.45 θ=0.45
Kx=8×10-13 m2
Kz=8×10-12 m2
θ=0.3
A B
600 m
230 m 140 m 230 m
Kx=8×10-13 m2
Kz=8×10-12 m2
Kx=8×10-13 m2
Kz=8×10-12 m2
θ=0.3 θ=0.3
Kx=2×10-13 m2
Kz=2×10-12 m2
θ=0.45
Case(3) Case (4)
300 m
300 m 300 m
150 m
150 m150 m
150 m
230 m 140 m 230 m
Fig. 16. Cases of non-homogenous and anisotropic porous media.
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0.8
0.6
0.4
0.2
0.8
0.6
0.40.2
0.8
0.6
0.40.2
0.8
0.6
0.40.2
0.8
0.6
0.4
0.2
0.8
0.6
0.4
0.2
0.8
0.6
0.4
0.2
(a) Case 1
(b) Case 2
(c) Case 3
(d) Case 4
without Jwith Jρ
ρ0.80.6
0.4
0.2
Fig. 17. 20-year simulation for 0.2, 0.4, 0.6 and 0.8 isochlor curves (left) and 0.2, 0.4, 0.6 and 0.8 isothermal curves (right) for (a) Case 1, (b) Case 2, (c) Case 3 and (d) Case 4 formesh Dx = 10 m and vz = 5 m.
24 Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28
(a) Case 1 (b) Case 2
(c) Case 3 (d) Case 4
0
20
40
60
80
100
120
140
PD (m
)
Time (Year)
0
20
40
60
80
100
120
140
0 2 4 6 8 10 12 14 16 18 20
0 2 4 6 8 10 12 14 16 18 20
0 2 4 6 8 10 12 14 16 18 20
PD (m
)
Time (Year)
20
40
60
80
100
120
140
PD (m
)
Time (Year)
00 2 4 6 8 10 12 14 16 18 20
0
20
40
60
80
100
120
140
PD (m
)
Time (Year)
ρWithout J
With Jρ
Fig. 18. Maximum penetration depth (PD) of 0.6 isochlor after 20 years simulation for (a) Case 1, (b) Case 2, (c) Case 3 and (d) Case 4.
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0 100 200 300 400 500 6000
50
100
150
0.20.4
0.6
0.8
without Jwith J
ρ
ρ
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
6.0 4.0
0.8
0.20.4
0.6
0.8
0.2
0.4
0.60.8
PD (with J ) = 119 mρ
PD (without J ) = 120 mρ
PD (with J ) = 45 mρ
PD (without J ) = 46 mρ
PD (with J ) = 81 mρ
PD (without J ) = 85 mρ
PD (with J ) = 119 mρ
PD (without J ) = 129 mρ
PD (with J ) = 131 mρ
PD (without J ) = 110 mρ
PD (with J ) = 86 mρ
PD (without J ) = 95 mρ
(a) salt distribution for density difference=10%
(b) salt distribution for density difference=20%
(c) salt distribution for density difference=25%
Fig. 19. Isochlor curves at 5 years (left) and 20 years (right) for density difference (a) 10%, (b) 20% and (c) 25%.
Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28 25
as required for stability, Dx < 4.6 m and Dz < 4.6 m. The Courantnumber is 0.11 for mesh ‘‘held’’ given the mean velocity0.42 � 10�6 m s�1 and is increased to 0.27 for mesh ‘‘teld’’ giventhe mean velocity 0.52 � 10�6 m s�1 and one-month time step(2.592 � 106 s). Using the mean velocity and one-month time step,
the numerical dispersion coefficient according to Woods et al. [48]is 0.35 � 10�6 m2 s�1, which is much smaller than the moleculardiffusion coefficient. Therefore, the numerical solutions are stableand reliable since the Peclet number is less than 4 [19], the Courantnumber is less than one, and the numerical dispersion is small.
26 Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28
The Nusselt number (Nu) in the brine Elder problem enablesthe comparison of actual salt transfer across the top boundaryof the model to salt transfer solely through diffusion. Fig. 8 showsthe comparison of the Nusselt number (Nu) with Prasad and Sim-mons [43] for mesh ‘‘held’’ and mesh ‘‘teld’’. Although the differ-ences are noticeable due to using different numerical techniques,the results show similar behavior over time.
To investigate the Jq term due to density differences, Fig. 9shows the isochlor curves for 10%, 20%, 25% and 30% density differ-ences given aL = aT = 4 m. It is clear that the effect of Jq on salt dis-tribution for the density difference higher than 20% becomes moreevident. For the density difference below 25%, an upwelling flow inthe center of the domain is developed regardless of the Jq term.However, the flow pattern is quite different given a 30% densitydifference. In Fig. 9(d), the solution with Jq still shows an upwellingflow while the solution without Jq shows a downwelling flow inthe center of the domain.
To investigate the Jq term due to dispersivities, Fig. 10 showsthe isochlor curves for aL = aT = 1 m, 2 m, and 3 m given a 30% den-sity difference. For aL = 2 and 3 m, the impact of the dispersive fluidflux on solutions is evident. An upwelling flow in the center of thedomain is observed for aL = 3 m or less with or without Jq. The ef-fects of Jq on the quantitative indicators with time, suggested byPrasad and Simmons [43], for a 30% density difference and aL = -aT = 4 m are shown in Fig. 11. The values of the four indicators(the Nusselt number (Nu), mass flux, salt mass stored, and maxi-mum penetration depth (PD) of 0.6 isochlor) show noticeablechanges by considering Jq in the flow equation.
Fig. 12 shows the maximum ratio of the dispersive fluid flux tothe advective fluid flux in the entire domain, which increases withincreasing the dispersivity for the range from 0 to 20 m after 20-year simulation and based on a 20% density difference and aT/aL = 1. A linear regression equation is obtained: max{|hJq|/|qq|} = 0.028aL + 0.014 for Fig. 12. It is clear that for high dispersiv-ity, the advective fluid flux is not dominant. The dispersive fluidflux has to be incorporated in the flow equation.
In conclusion, the numerical solutions from the code are veri-fied to be reliable for solving the brine Elder problem. The disper-sive fluid flux does have an important impact on flow pattern andsalt transport under high density differences and high dispersivity.
4. Model applications
In this section, we investigate the importance of the dispersivefluid flux in the thermohaline flow and transport problem [49]. Theeffect of the dispersive fluid flux on flow and transport in aniso-tropic and non-homogenous porous media are also evaluated.
4.1. Thermohaline Elder problem
Diersch and Kolditz [49] modified the brine Elder problem to athermohaline convection problem. The entire domain and bound-ary conditions are shown in Fig. 13(a). The domain is heated fromthe bottom and high salinity is at the top. The model parametersare listed in Table 4. The reference density qf is the initial waterdensity. The dispersive fluid flux is zero because of consideringzero dispersivity in the original problem [49]. To test grid conver-gence, the following spatial discretizations on the whole domainare used: (1) Dx = 15 m and Dz = 7.5 m and (2) Dx = 10 m andDz = 5 m. Constant head of 150 m is assigned to point A (upper leftcorner) and point B (upper right corner). A time step of 30 days isused. Fig. 13 shows the position of isochlor and isothermal curvesat time 10 years and 20 years. Both coarse mesh and fine mesh pro-duce the same upwelling flow in the center of the domain and thesame number of the developed fingers. The maximum velocity in x
direction for the coarse mesh (41 � 21) is vx = 5.92 � 10�7 m s�1
corresponding to the Courant number Cux = 0.05, and in z directionis vz = 5.98 � 10�7 m s�1 corresponding to the Courant numberCuz = 0.1. Therefore, the solution is stable.
To investigate the dispersive fluid flux due to density differ-ences, Fig. 14 shows the isochlor curves at 10 years for density dif-ferences 15%, 20%, 25% and 30% given aL = aT = 2 m. Similar to theprevious observation, the effect of the Jq term on the salt transportbecomes more evident for density differences more than 20%.Moreover, the density gradient has a considerable effect on the flowpattern in the center of the domain and the shape and the numberof developed fingers in isochlor curves. For a 15% density difference,a downwelling flow in the center of the domain is resulted whiledensity differences 20% and more produce an upwelling flow.
To investigate the dispersive fluid flux due to mechanical dis-persion, Fig. 15 shows the isochlor curves at 5 years for aL = aT = 1,2, 3, and 4 m given a 30% density difference. Increasing dispersivitydisperses more salt in the domain, while remaining the same dis-tribution pattern (two fingers and an upwelling flow in the centerof the domain).
4.2. Non-homogenous and anisotropic thermohaline flow andtransport
This numerical experiment is to study the influence of Jq on thedensity-driven thermohaline flow and transport in non-homoge-neous and anisotropic saturated porous media. Four different casesare shown in Fig. 16 to be studied. They are (a) a three-layered por-ous medium with low permeable layers at the top and the bottomand a high permeable layer in the middle; (b) a three-layered por-ous medium with high permeable layers at the top and the bottomand a low permeable layer in the middle; (c) a three-block porousmedium with low permeable blocks at two sides and a high perme-able block in the center; and (d) a three-block porous medium withhigh permeable blocks at two sides and a low permeable block inthe center. The boundary conditions are the same as in Fig. 13(a).A time step of 30 days is used for simulation. The dispersivitiesaL = aT = 2 m are used for all cases. Other model parameters arethe same as in Table 4. For simplicity, it is assumed that (1) the dis-persivities remain constant in space; and (2) the tortuosity and thethermal conductivity for the solid and the fluid phases are isotro-pic. Therefore, the scalar tortuosity s = h1/2 is used in the hydrody-namic dispersion equation and the equivalent thermalconductivity in Eq. (14) is used for the heat transfer equation.
Fig. 17 shows the comparisons of the salt concentration (left)and temperature (right) distributions with and without Jq at20 years for the 4 cases. It is observed that the effect of Jq on saltand temperature distributions in Case 4 is more evident than otherthree cases. The evolution of the maximum penetration depth (PD)of 0.6 isochlor [43] over time in Fig. 18 also shows the noticeableeffect of Jq in Case 4. In Case 1 and Case 4, Jq causes decrease inPD in comparison with the PD without Jq. In Case 2 and Case 3, aslight difference in PD due to Jq is observed. Fig. 19 shows the ef-fect of Jq in Case 4 given 10%, 20% and 25% density differences.Again, it is observed that Jq has a noticeable effect on salt distribu-tion and PD under high density differences.
5. Conclusions
Dispersive density-driven thermohaline flow and transportmodeling is suggested by including the dispersive fluid flux tothe flow equation. To illustrate the influence of Jq, a fully implicitfinite difference method is developed to solve the coupled flow,salt transport and heat transport governing equations in saturatedporous media. The numerical solutions are verified by the
Z. Jamshidzadeh et al. / Advances in Water Resources 61 (2013) 12–28 27
benchmarks of the Henry problem, the thermal Elder problem, andthe brine Elder problem and are compared well to the publishedsolutions. The calculated Nusselt number for the brine Elder andthe thermal Elder are in a good agreement with that of othernumerical codes. The grid convergence study to the brine Elderproblem shows that the code for both coarse and fine meshes pro-duce the same upwelling flow in the center of the domain and thesame number of developed fingers. Spatial discretization does notcause any considerable changes in the brine Elder solution.
Given a density difference, the increasing maximum ratio of thedispersive fluid flux to the advective fluid flux due to increasingdispersivities indicates the important effect of the dispersive fluidflux in the flow equation. Moreover, high density differences makethe dispersive fluid flux an important role on the shape and thenumber of developed fingers and lead to different isochlor shapesin the brine Elder problem. Also, high density differences make thedispersive fluid flux significant to the quantitative indicators suchas the Nusselt number, mass flux, salt mass stored, and, maximumpenetration depth in the brine Elder problem.
The results of testing different density differences anddispersivities on the dispersive fluid flux in the thermohaline Elderproblem show that both density gradient and mechanicaldispersion are important to produce dispersive fluid flux. For givena density difference, an increase in the mechanical dispersionresults in wider isochlor curves, but the salt transport pattern doesnot change.
The dispersive fluid flux has considerable effects on the thermo-haline flow and transport modeling in non-homogeneous andanisotropic porous media. The arrangement of anisotropic, zonalpermeability and density difference affect the flow pattern, saltand temperature distribution and the maximum penetrationdepth. Therefore, in the general case that involves strongdensity-driven flow and transport modeling in porous media, thedispersive fluid flux should be considered in the flow equation.
Acknowledgments
Zahra Jamshidzadeh was supported by the Iran Ministry of Sci-ence, Research and Technology to conduct research at LouisianaState University. Frank Tsai was supported in part by Grant/Coop-erative Agreement Number G10AP00136 from the United StatesGeological Survey to conduct saltwater intrusion simulation. Thecontents of the study are solely the responsibility of the authorsand do not necessarily represent the official views of the USGS.The authors acknowledge constructive comments from fourreviewers to improve the manuscript.
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