Journal of Contaminant Hydrology 145 (2013) 90–104
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Journal of Contaminant Hydrology
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Reactive dispersive contaminant transport in coastal aquifers: Numericalsimulation of a reactive Henry problem
H.M. Nick a,⁎, A. Raoof a, F. Centler b, M. Thullner a,b, P. Regnier a,c
a Department of Earth Sciences, Utrecht University, The Netherlandsb Department of Environmental Microbiology - UFZ, Helmholtz Centre for Environmental Research, Leipzig, Germanyc Department of Earth and Environmental Sciences, Université Libre de Bruxelles, Brussels, Belgium
Article history:Received 28 July 2012Received in revised form 10 December 2012Accepted 16 December 2012Available online 23 December 2012
The reactivemixing between seawater and terrestrial water in coastal aquifers influences thewaterquality of submarine groundwater discharge.While thesewaters come into contact at the seawatergroundwater interface by density driven flow, their chemical components dilute and react throughdispersion. A larger interface and wider mixing zone may provide favorable conditions for thenatural attenuation of contaminant plumes. It has been claimed that the extent of this mixing iscontrolled by both, porous media properties and flow conditions. In this study, the interplaybetween dispersion and reactive processes in coastal aquifers is investigated bymeans of numericalexperiments. Particularly, the impact of dispersion coefficients, the velocity field induced by densitydriven flow and chemical component reactivities on reactive transport in such aquifers is studied.To do this, a hybrid finite-element finite-volumemethod and a reactive simulator are coupled, andmodel accuracy and applicability are assessed. A simple redox reaction is considered to describe thedegradation of a contaminant which requires mixing of the contaminated groundwater and theseawater containing the terminal electron acceptor. The resulting degradation is observed fordifferent scenarios considering different magnitudes of dispersion and chemical reactivity. Threereactive transport regimes are found: reaction controlled, reaction–dispersion controlled anddispersion controlled. Computational results suggest that the chemical components' reactivity aswell as dispersion coefficients play a significant role on controlling reactivemixing zones and extentof contaminant removal in coastal aquifers. Further, our results confirm that the dilution index is abetter alternative to the second central spatial moment of a plume to describe the mixing ofreactive solutes in coastal aquifers.
Keywords:Density driven flowReactive transportDispersionSubmarine groundwater dischargeNode-centered finite-volume–finite-element methodDilution index
1. Introduction
Coastal aquifers are dynamic systems that are vulnerable toanthropogenic perturbations such as climate change, risingdemand for freshwater, disruption of natural hydrologicalconditions, and groundwater contamination. Density drivenflow and biogeochemical processes in such aquifers continue toreceive attention as these aquifers are sensitive to salinisationof fresh groundwater by seawater intrusion, the disposal ofwaste and sewage, and the leaching of contaminants (Post,
ll rights reserved.
2005; Werner et al., 2012). These processes are importantcontrolling factors for the groundwater and coastal surfacewater quality (Moore, 1999). In particular, submarine ground-water discharge (SGD) provides a pathway for the transfer ofnutrients from land to the coastal ocean (Burnett et al., 2003)and the reactions among the intruded seawater, fresh ground-water and porous media influence their concentration in SGD(Slomp and Van Cappellen, 2004). The processes at freshwater–seawater interface are thus important for determining subsur-face pathways and fluxes of land-derived chemicals to themarine environment.
Many factors control the dynamics of the freshwater–saltwater interactions, including: density variations induced
91H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
by seawater penetrating into the freshwater (e.g. Abd-Elhamidand Javadi, 2011; Reillya and Goodmanb, 1985); porous mediaheterogeneity (Diersch and Kolditz, 2002; Kerrou and Renard,2010; Li et al., 2009; Mulligan et al., 2007); the width of themixing zone (e.g. Abarca et al., 2007); seasonal fluctuations ininflux of seawater and groundwater (Michael et al., 2005;Taniguchi et al., 2006); tidal effects (Attaie-Ashtiani et al.,2001; Boufadel et al., 2011; Li et al., 2008; Lu et al., 2009;Robinson et al., 2007); dense contaminants leaking to theaquifer (e.g. Simmons, 2005); geothermal convection (Moore,2010; Wilson, 2005); and the chemistry of the fluid andmedium present in the system (e.g. Boluda-Botella et al., 2008;Rezaei et al., 2005; Santoro, 2010).
The mixing between freshwater and seawater in coastalaquifers is a key process controlling the chemistry of SGD. Formixing of non-reactive species in coastal aquifers, it has beenclaimed that the dispersive region along the freshwater–seawater interface is controlled mainly by the flow field andporous material properties such as dispersion coefficients(e.g. Kerrou and Renard, 2010). Pool et al. (2011) showed thesignificant contribution of transverse dispersion in creatingmixing in the interface. Lu and Luo (2010) related the enhancedmixing to kinetic mass transfer between mobile and immobilephases. The detailed study of Abarca et al. (2007) revealed thecontrolling effects of longitudinal and transverse dispersion onthe extent of the unreactive mixing zone. Held et al. (2005) alsopointed out that dispersivemass transport from the seawater tothe freshwater and themedium permeability govern the exten-sion of the intruded seawater in the aquifer.
The knowledge on reactive mixing in contaminated aquifers,under steady state conditions, is also fairly advanced for density-independent problems. It has been suggested that the transversedispersion controls the degradation of contaminant plumes(Cirpka and Valocchi, 2007; Shao et al., 2009), as well as thelength of the plumes (Ham et al., 2004). Werth et al. (2006)showed that flow focusing in high permeability zones leads to asignificant enhancement of transverse mixing and, hence,biodegradation capacity. Transverse dispersion at the fringe ofcontaminant plumes also limits the reactive mixing (Chiogna etal., 2011; Rolle et al., 2009) as do bioclogging processes (Thullneret al., 2004). The micromodel experiments on biomass growthconducted by Zhang et al. (2010) revealed that when theDamköhler number (Da, the ratio of the chemical reaction rate to
Table 1Examples of density driven reactive transportmodels for flow and transport in porousmFVM (finite volumemethod), FEM (finite element method), CVFEM (control volume fimethod), are utilized. Three different commonly used coupling approaches for solving(sequential iterative approach) and DSA (direct substitution approach). Chemical thermGEM (Gibbs energy minimization).
Flow & transportcode
Discretizationmethod
Reaction solver(and calculation method)
Coumet
CORE2D FEM Internal (LMA) SIACrunchFlow IFD Internal (LMA) SNIACSMP++ Hybrid FEFVM BRNS (LMA) SNIAHydroGeoSphere FEM, CVFEM Internal (LMA) SIAHYTEC FEM, FVM CHESS (LMA) SIAOpenGeoSys FEM BRNS (LMA), GEMS3K (GEM) SNIAPHWAT FEM PHREEQC-2 (LMA) SNIARETRASO FEM Internal (LMA) DSASTOMP IFD Internal (LMA) SNIATOUGHREACT IFD Internal (LMA) SIA
themass transfer rate) decreases, the reactive transverse mixingzone becomes wider. These results suggest that the reaction rateconstant together with transverse mixing controls the totalreaction rates in groundwater aquifers.
In contrast to density-independent groundwater, there islimited number of studies on reactive mixing for seawaterintrusion problems. In coastal aquifers, flow is not only affectedby dispersion but also by density variations. In other words, thedispersive density driven flow controls flow field as well aschemical mixing. For instance, a couple of field and numericalstudies have shown that dispersion and density variation areimportant factors affecting nutrient fluxes into the sea (Andersenet al., 2005; Kroeger and Charette, 2008; Spiteri et al., 2008b).However, the mechanisms controlling reactive mixing in costalaquifers remain poorly known. Their dynamic interplay shouldalso be elucidated further as this will ultimately determine thequalitative significance of fluxes of land derived chemicals to themarine environment.
Over the last decade, several flow and transport codes havebeen coupled with multi-component reaction solvers to studyreactive processes for seawater intrusion problems (Table 1).Thesemodelsmake use of various discretizationmethods aswellas different coupling approaches for solving reactive transportproblems (Table 1). The transport code based on structured gridsdoes not lend themselves easily to model complex geometriesand accurate velocity fields as they cannot easily representinclined or curved geological features, such as fractures (Matthäiet al., 2009). Hence, the models based on unstructured meshpose a more rigorous alternative to models based on structuredgrids. Flow and transport models utilizing a first-order schemefail to preserve sharp concentration fronts, since the transportequation has a strongly hyperbolic character (e.g. Gudonov,1959). This makes it desirable to extend the scheme to obtainhigher-order accuracy, although higher order schememay sufferfrom spurious oscillations (Matthäi et al., 2009). The previousnumerical studies on coupled density driven flow and reactivetransport problems are limited (Graf and Therrien, 2007; Post,2005), and many of the most commonly used software focuseither on transport phenomena or reactions (Wissmeier andBarry, 2011).
In this article, themain objective is to advance our qualitativeunderstanding of reactive processes in coastal aquifers whereseawater intrudes by density driven flow. For this purpose, two
edia. Different discretizationmethods: IFD (integral finite difference approach),nite element method) and FEFVM (finite-element node-centered finite-volumereactive transport problems are: SNIA (sequential non-iterative approach), SIAodynamicmodeling exists in twomain variants: LMA (law ofmass action) and
plinghod
References
Samper and Zhang (2006), Yang et al. (2008), SIA Steefel (2008)
Matthäi et al. (2007, 2009), Regnier et al. (2002), current studyBrunner and Simmons (2012), Graf and Therrien (2008)Lagneau and Van Der Lee (2010), Van Der Lee et al. (2003)Centler et al. (2010), Kolditz et al. (2012), Kulik et al. (2012)Mao et al. (2006), Parkhurst (1995)Saaltink et al. (2004)White and McGrail (2005)Xu et al. (2010)
92 H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
flexiblemodels are combined: a) the Complex SystemModellingPlatform (CSMP++)which can simulate both single- andmulti-phase flow in fractured and porous media (Matthäi et al., 2009),and b) the Biogeochemical Reaction Network Simulator (BRNS)capable of solving for kinetically and thermodynamically con-strained biogeochemical reactions (Aguilera et al., 2005; Regnieret al., 2002). In this study, the combined model is employed tostudy the combined effect of density driven flow, dispersion andchemical reactions on transport and reactive processes in coastalaquifers.
This paper is structured as follows. First, the governingequations as well as the combined CSMP++ and BRNS modelare presented. Next themodel is applied to the degradation of adissolved organic carbon (DOC) type contaminant along thefreshwater–seawater interface of a coastal aquifer. Verificationof the developed coupled model by means of an analyticalsolution and an accurate finite element approach are thenincluded. The combined effect of dispersion and reactiondynamics on the degradation of DOC is quantified undertransient and steady state flow conditions for a wide range ofdispersivity and chemical reactivity. Furthermore, the dilutionindex is calculated as an indicator for reactive mixing in coastalaquifers. Finally, the factors controlling the extent of thereactive mixing region are identified.
2. Reactive transport modeling
2.1. Governing equations
Considering the effects of density and viscosity, Darcy'slaw reads,
u ¼ −kμ
∇P−ρgð Þ; ð1Þ
where u is the velocity [LT−1], k is the intrinsic permeabilitytensor [L2], μ is the dynamic viscosity [ML−1T−1], ∇P is thepressure gradient [ML−2T−2], and ρ is the fluid density [ML−3].Parameter g is the gravity vector [LT−2]. Assuming a slightlycompressible fluid and a non-deformable porous medium,conservation of mass is ensured by the continuity equation,
ϕρβw∂P∂t −∇⋅ ρ
kμ
∇P−ρgð Þ� �
¼ ρq; ð2Þ
where βw=∂ρ/ρ∂P, denotes the compressibility of fluid phase[M−1LT2]. t represents time [T], q [T−1] stands for external fluidsources and sinks, and ϕ is porosity[−]. The mass balance forcomponent i (i=1,.., Nt, where Nt is the total number ofcomponents) in a non-deformable porous medium is given by,
ϕ∂ci∂t þ∇⋅ uci−D∇cið Þ þ Ri ¼ c�i q; ð3Þ
where ci denotes the concentration [ML−3] of component i, Ri[ML−3T−1] is the reaction source/sink term and ci
∗ refers to theconcentration of component i at sources. The hydrodynamicdispersion tensor, D (Scheidegger, 1961) is
D ¼ ϕDeff þ αT uj j� �
I þ αL−αTð Þuuuj j ; ð4Þ
where Deff is the effective molecular diffusion with Deff=Dm/τ,Dm denoting the molecular diffusion coefficient [L2T−1] and τ
referring to the tortuosity of the porousmedia. The parametersαL and αT, are the longitudinal and transversal dispersivities/dispersion lengths [L] respectively, I is the unit tensor, and |u| isthe magnitude of the velocity vector.
Eqs. (2) and (3) are coupled by the equations of state whichpresent the fluid density and viscosity as functions of massfraction, temperature and pressure (Diersch and Kolditz,2002). Often the fluid density equation is expressed as a linearfunction of the mass fraction (e.g. Abarca et al., 2007; Younes,2003).
2.2. Numerical approach
2.2.1. BRNSBRNS is capable of handling a comprehensive suite of
multi-component complexation (aqueous and surface) andmineral precipitation and dissolution reactions, and is also ableto treat reaction networks characterized by partial redox dis-equilibrium andmultiple kinetic pathways (Centler et al., 2010;Dale et al., 2009b; Thullner et al., 2005). BRNS has been appliedfor reactive transport problems in porous media such as fate ofphosphorus in saturated aquifers (Spiteri et al., 2007), anaerobicoxidation of methane in marine sediments (Dale et al., 2008),organic matter mineralization pathways in marine sediments(Thullner et al., 2009), influence of bioavailability limitations onbiodegradation at pore scale (Gharasoo et al., 2011), and deg-radation of wellbore cements under CO2 storage conditions(Raoof et al., 2012). BRNS benefits from using a Maple interfaceas a symbolic programming language. The Maple preprocessorgenerates all the necessary Fortran files needed for solving thereaction network. The method is explained in detail in Regnieret al. (2002, 2003) and Aguilera et al. (2005). A Fortran libraryencapsulating the chemical solver of BRNS and the user definedreaction network can be built and linked with other transportcodes (Centler et al., 2010).
Concentrations of reactive species, as well as stoichiometry,rate parameters and equilibrium constants of the reactiveprocesses are used to define a specific reaction network. Thereaction term in Eq. (3) representing the sum of all reactionsaffecting the concentration of a given species i can be writtenas:
Ri ¼XNreaction
m¼1
ami ri; ð5Þ
where Nreaction is the number of reactions, aim is the stoichio-metric coefficient of component i in the reaction m, and rirepresents the rate of the reaction m. If some of the reactionsconsidered are assumed to be at equilibrium, they can bereplaced with algebraic expressions based on mass actionexpressions. By replacing one or more of the ordinary differ-ential equations (ODEs) associated with reaction(s) withalgebraic relations based on a mass action expression in thelocal equilibrium case, the set of ODEs is transformed into a setof differential-algebraic equations (DAEs) (see Regnier et al.,2002, for further details). Examples of applications include thestudy by Spiteri et al. (2008c) for full equilibrium simulationsand the studies by Dale et al. (2009a) and Krumins et al. (2012)for mixed kinetic-equilibrium cases. The form of rate expres-sion, ri, is arbitrary, even nonlinear, and can be a function ofseveral concentrationswithin the system. The compiled library
93H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
contains the kinetic and equilibrium reactions. The generatedFortran files and the solver engine for solving a nonlinear set ofequations ensued from kinetic of multi components areembedded in BRNS. The reactive solver uses a Newton–Raphsonmethod to solve the set of kinetic and equilibrium reactionequations.
2.2.2. CSMP++The computer code CSMP++ can simulate both single- and
multi-phase 2D/3D flow and the resulting transport of dissolvedspecies (Matthäi et al., 2004). It features an object-orientedapplication programmer interface (API), designed for the simu-lation of complex geological processes and their interactions(Coumou et al., 2008; Geiger et al., 2010; Latham et al., 2012;Matthäi et al., 2007; Nick et al., 2011; Paluszny and Zimmerman,2011). This geometrically flexible and stable transport algorithmcan resolve complex geological structures and many orders ofmagnitude of permeability variations which are ubiquitous ingeological formations (Geiger and Matthäi, 2012). CSMP++relies on a non-oscillatory higher-order accurate finite-elementnode-centered finite-volume (FEFV) scheme for solving time-dependent advection–dispersion problems (Matthäi et al.,2009). An algebraic multigrid method for system of equations,SAMG, (Stüben, 1999) is employed as the solver for Eqs. (2) and(3).
Eqs. (2) and (3) are solved implicitly using an operatorsplitting method (Matthäi et al., 2009). First, we obtain thepressure field by solving Eq. (2) on a linear finite-elementdiscretization. The pressure field can be solved in the weakform of Eq. (2) with the standard linear finite-elementBubnov–Galerkin method. Using Euler's method to discretizetime, the space–time integration of Eq. (2) over the domainΩ⊂Rd (d=2,3) for x∈Ω yields the following finite element(FE) integrals,
∫ΩNTϕρβwNdx þ Δt∫Ω∇NTρ
kμ∇Ndx
� �PtþΔt ¼
∫ΩNTϕρβwNdx
� �Pt þ Δt ∫ΩN
Tρ2 kμg∇Ndx þ ∫ΩN
TρqNdx� �
;
ð6Þwhere t and Δt refer to time and time increment, respectively.The superscript T refers to the transposed element interpola-tion function vector or matrix of spatial derivatives, and N and∇N represent its interpolation function vector and matrix ofspatial derivatives obtained from each element, respectively.
Then, the transport equation is solved using a conservativefinite-volume method (FVM) discretized on a virtual finite-volume mesh constructed around the nodes of the finiteelement mesh. In the FEFV method (FEFVM), each finite-element node also is the center of a finite volume created byconnecting element barycentres with midpoints of associatededges (see Fig. 1 of Nick and Matthäi, 2011b). This partitions anelement into as many finite-volume sectors as it has nodes, andtheir combined contribution to the set of linear algebraicequations describing the global transport problem is referredto as a finite-element finite-volume stencil.
The transport equation is solved using a fractional stepmethod in which the diffusion term of the advection diffusionequation is discretized using an implicit finite element method(FEM) and the advection term of Eq. (3) is discretized using animplicit FVM. By analogy with the pressure equation, finite-
element integration of the diffusion term in Eq. (3) over thedomain Ω yields,
∫ΩNTϕNdx−Δt∫Ω∇NTD∇Ndx
� �ctþΔti ¼
∫ΩNTϕNdx
� �cti ;
ð7Þ
where ct and ct+Δt denote the distributed dependent variable, c,at time t and t+Δt, respectively. For the advection term inEq. (3), using piecewise constant FV interpolation functions, Mj,for each finite-volume j and a first-order upwind scheme,integration over volume V gives,
ϕ∫V MctþΔti dV þ Δt∮V n⋅uð ÞctþΔt
i dS ¼ϕ∫V MctidV þ Δt∫V c
�i qdV ;
ð8Þ
where, n represents the normal vector to the outward facingsurface element. To make the transport scheme second-orderaccurate in space, we calculate estimates of the gradient of thetransported variable for each control volume facet. Then, weapply theminmod slope limiter to suppress spurious oscillationsthat occur when the gradient of the transported variable isoverestimated (Pain et al., 2003). This guarantees that thetransport scheme becomes total variation diminishing (Matthäiet al., 2009).
CSMP++ hybrid element stencils allow for the assemblyof finite volumes from arbitrary combinations of tetrahedral,prism, pyramid and hexahedral elements which permit tosimulate complex geological formations (Paluszny et al.,2007).
2.2.3. Numerical procedureCSMP++and BRNS are coupled to simulate reactive multi-
component transport in heterogeneous porous media that arediscretizedwith spatially variably refined unstructured grids toallow a realistic representation of the flow geometry. Wecompute transient flow, advection–dispersion and reaction in asequential manner (e.g. Steefel and MacQuarrie, 1996).
Eqs. (2) and (3) are nonlinear as the fluid density affectingthe velocity field is a function of the salt concentration.Therefore, a Picard iterative method is employed to linearizethe equations (Putti and Paniconi, 1995). Within the iterativestep, Eq. (2) is solved to obtain flow velocity field via Eq. (1).Next, the multi-component advection–diffusion Eq. (3), de-scribing solute transport is solved employing a hybrid FEFVMfor the components that may affect the flow field. This isbecause the flow field needs to be recalculated if the porousmedia properties, such as porosity and permeability, and fluidproperties, such as fluid density and viscosity, change as aconsequence of transport or reactions (e.g. Nick et al., 2009).After the iterative procedure has converged, the Eq. (3) is solvedfor the components which do not affect the flow equation.Applying sequential non-iterative operator splitting, in eachtime step first the transport solution is obtained and then thereaction part is solved using the reactive solver of BRNS. In orderto eliminate possible operator splitting errors the Courantnumber condition, which is defined as uΔt/Δxb1, is imposedduring the simulations (Xu et al., 1999). Since the concentra-tions are piece-wise constant within each finite-volume, theconcentrations of the components at every node are assembledfor BRNS to solve for one reaction time step. BRNS updates the
Start
End
Build geometry
ConfigurationAssign material properties
Assign boundary and initial conditionsSetup computational algorithm
Solve flow
Solve transportOnly for the components affecting the flow
Update Fluid and material properties
Convergence
Solve transportFor the components not affecting the flow
Output results
Solve reactionBRNS
Time loop
Picard iteration loop
Yes
No
Next time step
Fig. 1. Flow chart of the numerical procedure for reactive density driven flow and transport.
94 H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
concentration of reactive components for the next transporttime step (Fig. 1). In principal, similar to Mao et al. (2006), thefluid density and viscosity as a function of species concentra-tion can be evaluated. However, in this work for the sake ofsimplicity we utilize a linear function, as equation of state forfluid properties, only to consider the effect of salt concentrationon fluid density as, ρ=ρ0+ω(ρmax−ρ0), whereω denotes saltmass fraction, and ρmax and ρ0 are the maximum (ω=1) andinitial fluid densities (ω=0), respectively.
2.3. Numerical experiments
Four 2D models of the reactive Henry problem introducedhere (Table 2) are used to examine mechanisms controllingthe fate of nutrient transport in coastal aquifers. Thesemodels are utilized for the verification of the simulationmethod as well as for studying the reactive transport at theseawater freshwater mixing zone in coastal aquifers. Note
that the Henry problem is yet a simplified model, however, inorder to present a more suitable description of the saltwaterintrusion problem in coastal aquifers both velocity depen-dent dispersion and reaction are included.
2.3.1. Verification, conservative steady state problemA vertical cross section through a homogeneous confined
aquifer (Model 1), 2 m×1 m, is meshed with rectangularelements (element size of 5 cm, Fig. 2). Hydrostatic fieldpressure affected by density gradients leads to seawaterintrusion into the model domain through the right boundary(Dirichlet boundary condition), while a constant flux ofgroundwater (3.3×10−5m2s−1) is assigned along the leftboundary. Like Simpson and Clement (2004), flow and porousmedium properties are the same as for the modified Henryproblem. Permeability, porosity, and molecular diffusion are1×10−9m2, 0.35 and 1.886×10−5m2s−1, respectively. A max-imum density ratio of 1.025 between seawater and freshwater,
Table 2Model configurations of the 2×1 m test cases with porosity of 0.35 and intrinsicpermeability of 1×10−9m2. The abbreviations SSt and Tt stand for steady statetransport and transient transport, respectively.
95H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
and a freshwater density of 1000 kg m−3 is used for all exam-ples in this work. Note that the seaward boundary condition forconcentrations switches between Dirichlet and Neumann con-ditions depending on the flow direction at the boundary (Volkerand Rushton, 1982).
2.3.2. Verification, reactive transient problemTo verify the combined CSMP-BRNS model (Model 2), a
redox reaction:
DOC þ O2→CO2 aqð Þ þ H2O; ð9Þ
is solved by CSMP-BRNS employing the FEFVM as well asCOMSOLMultiphysics (COMSOL, 2005;Hayek et al., 2012) usinga FEM. In the simulations seawater contains O2 of 0.2 mM.Anoxic groundwater contains dissolved organic carbon, and aconservative tracer, both of 1 mM. This scenario is chosen byanalogywith the case three of Slomp and Van Cappellen (2004)where anoxic groundwater meets oxic seawater, representing acommon end-member case in such aquifers (Charette et al.,2005; Spiteri et al., 2006; Uchiyama et al., 2000).
The rate law is defined as (Hunter et al., 1998):
r ¼KDOC DOC½ � if O2½ � > KmO2
;
KDOC DOC½ � O2½ �KmO2
h i else
8><>:
where KDOC is the rate constant for decomposition of DOC, andKmO2
is the limiting concentration ofO2.KmO2=3.0×10−2 mM,
and KDOC 1.0×10−3s−1, are used in this example.
L=2
d=1
m
Freshwater discharge
w
x
Y
Fig. 2. Domain and the variables used for analysis of the Henry saltwater intrusiodiscretization used in Model 1.
2.3.3. Transient reactive transport — dispersion sensitivityAnthony (2004) and Abarca et al. (2007) have addressed
the effect of longitudinal and transversal dispersivities on non-reactive seawater intrusion. Here, in order to evaluate this effectfor reactive transport, different combinations of dispersioncoefficients are used to investigate the controlling behavior ofthese coefficients on reactive mixing between seawater andfresh groundwater. Note that some of these values used in thisstudy are selected for demonstration purposes and sensitivityanalysis. As in the previousmodel, aerobic degradation Eq. (9) issimulated; but now for a transient injection of DOC (Model 3).At steady state conditions for the flow, DOC is injected for 1 hthrough the left boundary at a concentration of 1 mM. Thesesimulations are conducted for porous media with differentcombinations of high (h) and low (l) dispersivity values: L1T1(αL=1 mm and αT=1 mm), LhT1 (αL=10 cm and αT=1 mm), L1Th (αL=1 mm and αT=10 cm), and LhTh (αL=10 cm and αT=10 cm). Here, the abbreviations L and T standfor longitudinal and transversal dispersivities, respectively. Notethat a constant flux of fresh groundwater 6.0×10−5m2s−1 andthe molecular diffusion value of 1×10−9 (m2s−1) are used forall these following simulations. The model mesh is also refinedfurther and an element size of 2.5 cm is used.
2.3.4. Steady state reactive transport — reaction rate sensitivityThe effect of dispersivity and aerobic degradation rate, KDOC,
is analyzed by comparing the results obtained for simulationsthat combine longitudinal dispersivities of 1 mm or 10 cmwithtransversal dispersivities of 1 mm, 1 cm or 10 cm for differentKDOC values ranging from 1.0×10−3 to 1.0×10−9 s−1. HereDOC is injected continuously and simulations are carried outuntil steady state is achieved for all components (Model 4).
2.4. Analysis strategy
Following the study of Abarca et al. (2007) three differentcharacteristic parameters are calculated: 1) Dimensionless toepenetration of saltwater, LD=Ltoe/d, which is the penetration ofthe seawater intrusion wedge, Ltoe, measured as the distancebetween the seaward boundary and the point where the 50%mixing isoline intersects the aquifer bottom, normalized by themodel width, d. 2) Normalized averaged width of the mixing
75%
25%
50%
m
Ltoe
Seawaterdischarge
SGD
0.2 Ltoe
0.8 Ltoe
n problem. Shown mesh of rectangular elements corresponds to the spatial
96 H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
zone WD as the average of the vertical distance W betweeniso-concentration lines of 25% and 75% concentration between0.2 LD and 0.8 LD divided byd. 3) Saltwater/freshwater dischargeratio, RD (Fig. 2).
2.4.1. Dilution indexKitanidis (1994) introduced the dilution factor to measure
the true mixing which is defined as:
E ¼ exp −∫Ω Pln Pð Þdx� �
; ð10Þ
where P=c(x,y)/∫Ωc(x,y)dx is the normalized concentrationfield by the integrated mass.
The index E is a measure for both, the area over which themass of a tracer is distributed, and its uniformity. This index islarger for enhanced uniform dilution. It is referred to as “truemixing” as the macro dispersion model employing macrodispersivities based on second central spatial moments of aplume result in an overprediction of mixing-controlled reac-tion rates (Cirpka et al., 1999).
3. Results and discussion
3.1. Verification
First, seawater intrusion into a homogeneous domain isverified against semi-analytical solution of the modified Henryproblem (e.g. Kolditz et al., 1998). This, further, is comparedwith the solution obtained using COMSOL Multiphysics toverify CSMP-BRNS against a standard finite element approach.
3.1.1. Steady state problemThe modified Henry problem (Simmons, 2005) is solved
employing both CSMP-BRNS and COMSOL (Holzbecher, 2005).Seawater intrudes from the seaward boundary until a steadystate between the denser intruded fluid and the lighter re-charging fluid is reached. The iso-concentration contours areshown in Fig. 3. Both CSMP-BRNS and COMSOL model resultsagree well with the semi-analytical solution of the modified
Fig. 3. Steady state iso-concentration contours calculated with FEFVM (CSMP-BRdimensional model is discretized by rectangular elements (Model 1). Note that, in orDirichlet boundary condition is applied for the salt concentration at the seaward bo
Henry problem presented by Simmons (2005). This verificationstep shows that the CSMP-BRNS can solve accurately 2D stablevariable-density problems.
3.1.2. Nonconservative transient problemThis test case compares the transient distribution of non
conservative species achieved with CSMP-BRNS against a welltested numerical approach, COMSOL. The objective is to com-pare the CSMP-BRNS using sequential non-iterative approach(e.g. Walter et al., 1994) against COMSOL using fully coupledalgorithm, in which chemical equations are directly substitutedinto the transport equations so that they are solved simulta-neously, for both transport and reaction processes (e.g. Steefeland Lasaga, 1994). Here, the transient simulation is conductedfor density flow transport and reaction. Aerobic degradationresults from the injection of DOC on the landward boundary,which is injected for 1 h at a concentration of 1 mM, and O2 onthe seaward boundary. As a result of the reaction dissolvedinorganic carbon (DIC) is produced (Eq. (9)). Fig. 4 revealsthat the DOC, O2 and DIC concentration fields after 2 h forboth models are in good agreement. This is confirmed by L2
0.0431, and 0.0079 for DOC, O2 and CO2, respectively. Whilethe fully coupled approach is restricted to simple problems dueto large memory requirements (Bell and Binning, 2004),CSMP-BRNS provides the required flexibility to model multi-component reactive flowproblems in geologicalmedia. Geiger etal. (2009) is an example of utilizing CSMP for multi-componentmultiphase flow problems in fractured porous media. In addi-tion, CSMP-BRNS has been implemented in an object-orientedsimulator, whose modular design allows for further enhance-ments and extensions (Matthäi et al., 2007).
3.2. Dispersivity effects on density driven flow, transport andreaction
Here, we identify the role of longitudinal and transversaldispersivity on transport and, then, reaction for a density
NS), FEM (COMSOL) and the semi-analytical solution are shown. The twoder to be consistent with the semi-analytical solution of the Henry problem, aundary.
Fig. 4. a) DOC, b) O2, and c) DIC iso-concentration contours calculated withthe FEFV and FE methods after 7200 s (Model 2).
Table 3The dimensionless parameters calculated for different combinations of high(h) and low (l) dispersivity values: L1T1 (αL=1 mm and αT=1 mm), LhT1(αL=10 cm and αT=1 mm), L1Th (αL=1 mm and αT=10 cm), and LhTh(αL=10 cm and αT=10 cm) (Model 3).
97H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
driven flow setup. Several transient simulations are con-ducted for porous media with different combinations of highand low values for the dispersivities. Results show that thedimensionless toe penetration of saltwater (LD) decreaseswith increasing dispersion coefficients (Table 3). As ex-pected, the normalized averaged width of the mixing zone(WD) and the saltwater/freshwater flux ratios (RD) becomelarger for the cases with higher dispersion. The results alsoshow that the LD reduction due to an increase of thetransversal dispersivity is more significant when the longi-tudinal dispersivity value is high. While the LD and RD aresensitive to both dispersivity values, the WD is mainlycontrolled by the transversal dispersivity only. The smalleffect of longitudinal dispersivity is limited to high salinityzones. This is in agreement with the results of Abarca et al.(2007). Velocity field and salt concentration distributions arestrongly affecting each other. Considering the dispersive fluxterm in Eq. (3) together with the Scheidegger formulation forcalculating the dispersion Eq. (4), reveals that transverse
dispersion is dominant where the velocity field is perpendic-ular to the salt concentration gradient. In contrast, longitu-dinal dispersion is affecting the flow and transport where thesalt concentration gradient is parallel to the velocity vectors.Thus, as shown in Fig. 5, the saltwater intrusion toe area, andthe zone below the interface at the vicinity of the counter lineof 90% concentration are two regions of the domain wherelongitudinal dispersion should be dominant. However, thefact that the velocity vectors are about two orders of mag-nitude smaller here than the velocity vectors in the freshwaterzone, explainswhy the influence of longitudinal dispersivity onconcentration distribution is limited.
Fig. 6 shows the effect of dispersion coefficients on the DOCdegradation rate (Model 3). The transient results clearly showthat larger mixing coefficients enhance the efficiency of thedegradation process. In particular, increasing transversal dis-persivitywidens the reaction zonewhere the velocity vectors areparallel to the interface between saltwater and freshwater, whileincreasing longitudinal dispersivity causes a broader reactionzone at the toe of the penetrated seawater where the velocityvectors are more perpendicular to the interface (Fig. 5). This isfurther substantiated in Fig. 7 which shows breakthroughcurves ofDOC and tracer concentrations for the four dispersivityscenarios. These breakthrough curves are obtained at thefreshwater outflow along the right boundary of the model.Increasing the longitudinal dispersivity values causes anoma-lous transport, longer tailing, for all components. On the otherhand, increasing dispersivities lowers the peak in the break-through curves and widens the mixing zone, resulting inenhanced degradation of DOC. In Fig. 5 it can also be seen thatthe velocity vectors, which are normal to the concentrationgradient, are larger in the freshwater zone near the interface.This explains why the reactive mixing zone is more developedinto the freshwater zone, leading to an asymmetric concentra-tion distribution in the mixing zone (Fig. 6).
3.3. KDOC and dispersion effects on degradation rate
In Fig. 8, the scaled aerobic degradation rate (R/KDOC)profiles at x=0.5 LD are illustrated for combinations of differentdispersion coefficients and reaction rate parameters. ComparingFig. 8a and b reveals that increasing longitudinal dispersivityexpands slightly the width of the mixing zone towards theseawater zone. This effect ismore pronouncedwhenKDOC is lowand, thus, longitudinal dispersivity is less influential on reactivemixing for fast reactions. An increase in transversal dispersivitywidens the reactivemixing zone significantly and this spreadingis larger when KDOC is low. While increasing longitudinaldispersivity has negligible impact on the maximum value ofthe scaled aerobic degradation rate, increasing transversal
Fig. 5. Velocity distribution at the seawater/freshwater interface is shown for the subdomain (0.5bxb1.0 m, 0byb0.5 m) of Model 3. Contour lines of 10% and 90%concentration are depicted in red (αL=1 mm and αT=1 mm).
98 H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
dispersivity enhances it. This effect is most substantial for fastreactions.
The volume-averaged degradation rate, bR>, increases withincreasing dispersivities because enhanced mixing forces thedecomposition process. As expected, transversal dispersivity hasmore influence on the averaged rate than longitudinal dis-persivity (Fig. 9). To study the effect of KDOC on averaged reactionrate, theDamköhler number, Da=KDOCLtoe/Fsw is used,where Fswis the seawater-flux through the seaward boundary. For thelowest value of the Da number, the dispersivity values controlthe reactive mixing, while the reactive rate, KDOC, is thecontrolling parameter for the higher value of the Da number.The regime shift occurs at Da>4 and Dab0.2 for the reaction
a
c
2.×10-7
Fig. 6. Aerobic degradation rates for the four scenarios with different dispersion coeffi
controlled and dispersion controlled regimes, respectively.Intermediate Da numbers correspond to a reactive mixingcontrolled by both KDOC and the dispersion. The linear propor-tionality of bR> to KDOC, for slow reactions, confirms that thedispersive reactive mixing is not limited by kinetics. This isdifferent for fast reactions as the dispersive reactive mixing islimited by kinetics.
Fig. 8 provides further insight into the reactive mixingdynamics. For KDOC values within the range of 1.0×10−3−1.0×10−5 s−1, a decrease in the rate constant increases thewidth of the reactive mixing zone. This explains why the scaledaveraged rate decreases when KDOC increases. However, lower-ing further the KDOC has no effect on the scaled averaged rate
b
d
6.×10-74.×10-7
cients are illustrated after 2.5 h; a) L1T1, b) LhT1, c) L1Th and d) LhTh (Model 3)
Fig. 7. Flux weighted breakthrough curves at the right (seaward) modelboundary for a) tracer and b) DOC for the four scenarios with differentdispersion coefficients are illustrated (Model 3); L1T1 (black), LhT1 (green), L1Th(red) and LhTh (blue). Note that the concentration values are scaled by theinitial concentrations.
99H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
because themixing zonewidth stays unchanged (Fig 8), and thescaled averaged rate is then only sensitive to the dispersivityvalues. Fig. 8a also shows an asymmetric reaction zone whichmoves towards the groundwater zone as KDOC values decrease.Thus, while the asymmetric shape of the reaction zone isgoverned by inhomogeneous dispersion induced by the flowfield along the freshwater–seawater interface, the reactivitycontrols the location of the reaction zone.
The behavior of the reactive Henry problem is furtherillustrated in Fig. 10. As already shown in Fig. 8, faster reactionsconfine the reactive mixing zone to a narrower zone (Fig. 10a).The reaction zone also moves towards the fresh groundwaterregion when KDOC decreases, as the dispersion in this region islarger than the dispersion in the seawater zone, showing thatits position is sensitive to the rate constant (Fig. 10b). Anincrease in longitudinal dispersivity advances the mixing zoneinto the seawater intruded area and leads to a receding toepenetration (Fig. 10c). The transversal dispersivity has a morepronounced effect on the mixing zone (see also, Abarca et al.,2007; Held et al., 2005; Werth et al., 2006). Although it alsolimits the seawater penetration, themixing zone advances intothe fresh groundwater zone (Fig. 10d). In general, a higherdispersion widens the mixing zone overlapping betweenreactants, but limits penetration of seawater into the aquifer.
3.3.1. True mixingThe dilution index, E, Eq. (10), as a measure of the degree
of true solute mixing, is calculated based on the simulationsfor different dispersivity and KDOC values. The dilution factorcalculated from the tracer results for each combinations ofdispersivity coefficients is used to normalize the E values forthe O2 concentration in Fig. 11a. This normalized valuedescribes how complete is the dilution compared to thedilution of the tracer concentration. The mixing decreases byincreasing the reaction rate constant. It is however lessaffected for constant reaction rates lower than 10−6 s−1,illustrating a well mixed regime where the effect of chemicalreactivity on true mixing is less pronounced. The significantreduction of the dilution index for the incompletely mixedregime and its sensitivity to both chemical reactivity anddispersivity values imply that both the chemical and porousmedia properties are of utmost importance in controlling thetrue mixing.
Similar to the dilution index, the second central spatialmoment values are calculated. No meaningful relationship isobtained between the second moment values and the totaldegradation rates (not shown). A strong relation between thenormalized dilution index and total degradation rate scaled byKDOC values is found. This can be seen in Fig. 11b where thedilution indices are plotted against the scaled total degradationrates. The scaled total rate is linearly proportional to the scaleddilution index. Altering transversal dispersivity affects thedegree of this proportionality. Kitanidis (1994) suggested thatthe second central spatial moment may fail to explain themixing of reactive solutes in porous media. Our result that thetotal degradation rate is proportional to the dilution index butnot to the second spatial moment, also confirms this. However,a broader study is needed to explore the relationship betweenthe dilution index and reactive flow parameters.
3.3.2. Future workThe simulations are limited to the study of a single kinetic
process coupled to a density-driven flow and transport problem.Although reactive processes are undoubtedly more complex innatural settings, this description is nevertheless sufficient andrealistic to elucidate the behavior of reactive mixing in coastalaquifers (Dai et al., 2012; Spiteri et al., 2008a). In the future,additional processes such as precipitation, dissolution, andformation of biofilms could be included as they remain to bestudied in the context of reactive seawater intrusion problems.In addition, the effects of larger scale heterogeneity such as,fractures and impermeable lenses, could be addressed withCSMP-BRNS as this model is specifically geared to includediscontinuities at the material interfaces (Nick and Matthäi,2011a).
In this study, the homogeneous Henry problem model isused, to minimize confounding variables. Future researchshould expand to field scale models. Kerrou and Renard(2010) also pointed out that the results of dispersive seawaterintrusion in two-dimension (2D) can be extrapolated tothree-dimensions (3D) provided that the permeability field isrescaled and dispersivity values are modified accordingly. Therelevance of our 2D results for the reactive mixing case willnevertheless require further assessment in the context of 3Dapplications.
00.250.50.751
0
0.5
1
Salt concentration
Ele
vati
on
(m
)
a
KDOC = 10-3
KDOC = 10-4
KDOC = 10-5
KDOC = 10-6
KDOC = 10-9
Normalized seawater concentration
00.250.50.751
0.1 10 1,000
0
0.5
1
Salt concentration
Ele
vati
on
(m
)
b
00.250.50.751
0
0.5
1
Salt concentration
R/KDOC (μμM) R/KDOC (μM)
R/KDOC (μM) R/KDOC (μM)
Ele
vati
on
(m
)
c
00.250.50.751
0
0.5
1
Salt concentration
Ele
vati
on
(m
)
d
0.1 10 1,000
0.1 10 1,0000.1 10 1,000
Fig. 8. Steady state scaled aerobic degradation rate profiles at x=0.5 LD for different dispersion coefficient combinations (Model 4). Normalized concentrationprofiles are also plotted. a) L1T1, b) LhT1 c) L1Th and d) LhTh.
100 H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
0.1
1
10
100
1,000
0.001
0.01
0.1
1 10 100
<R>
/KD
OC (
μM)
Da
LlTl
LhTl
LlTh
LhTh
LhTm
ReactionControlled
Reaction-Dispersion Controlled
Dispersion Controlled
Fig. 9. Averaged aerobic degradation rates scaled by KDOC versus Da numberat steady state (Model 4) for different scenarios with different combinationsof high (h), medium (m), and low (l) dispersivity values: L1T1 (αL=1 mmand αT=1 mm), LhT1 (αL=10 cm and αT=1 mm), L1Th (αL=1 mm andαT=10 cm), LhTh (αL=10 cm and αT=10 cm) and LhTm (αL=10 cm andαT=1 cm).
c
αL=0.001 m αL=0.1 m
αT=0.001 m
KDOC<0.000001 1/s
αT=0.01 m
αT=0.001 m
dαL=0.1 m
KDOC=0.001 1/s
KDOC=0.001 1/sKDOC=0.000001 1/s
b
αT=0.01 m
αL=0.1 m
KDOC=0.001 1/sKDOC<0.000001 1/s
a
αT=0.001 m
αL=0.001 m
Fig. 10. Response of the reactive transport to dispersivity and reaction ratevariations. Shaded areas depict the mixing zones between contour lines of 10%and 90% salt concentration. The iso-degradation-rate contours, marked by ⇕,represent 10% of maximum degradation rate of the DOC. The extension ofreactive mixing zone due to dispersivity and reaction rate variations is markedby a single (red) arrow. a,b) Decreasing KDOC while αL and αT are fixed, c)increasing αL, and d) increasing αT.
101H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
We present the accuracy and potential of CSMP-BRNS for asimplified example. The combined model presented here,however, is applicable to a wide range of flow and multi-component transport problems in geological media since itmaintains the full flexibility of the original models. Yet, per-forming potential of CSMP-BRNS formore complex applicationsin coastal aquifers needs further demonstration.
4. Conclusions
The CSMP-BRNS model developed in this study is asimulation tool applicable to multi-component reaction-transport density-dependent flow in porous media. Using thismodel, we investigate the interaction between density drivenflow, dispersion and reactive transport.
Similar to reactive-transport in density-independent ground-water systems:
• The Damköhler number as well as dispersivity parameters areimportant parameters governing the biogeochemical dynam-ics in the reaction zone and release of land-derived chemicalsinto coastal waters. Our findings suggest three dispersivemixing reactive regimes: 1) Dispersion controlled regime;2) Reaction–dispersion controlled regime; and 3) Reactioncontrolled regime.
• The dilution index appears to be a good predictor for reactivemixing in the seawater intrusion problem.
In contrast, in seawater intrusion systems:
• An increase in longitudinal and transverse dispersion widensthe mixing zone but shortens the length of the interfacebetween seawater and freshwater, due to rather complexdensity driven flow. The effect of transverse dispersion ismore significant.
• The reactivity of the chemical compounds also plays asignificant role on the position and shape of the reactivemixing zone and, ultimately, on the averaged rates and exportfluxes in reaction controlled regime. Decreasing chemicalcomponent reactivity considerably moves the position of thereaction zone towards freshwater.
Hence, accounting explicitly for combined effect of disper-sion and reaction in upscaled modeling of SGD using modelswith sharp interfaces or box models is essential.
Acknowledgments
This work was generously supported by the King AbdullahUniversity of Science and Technology (KAUST), as part of theSOWACOR project, by the Government of the Brussels-CapitalRegion (Brains Back to Brussels award to P. Regnier), and by theHelmholtz Association via grant VH-NG-338 (GReaTMoDE).
0.0
0.2
0.4
0.6
0.8
1.0
0.000001 0.00001 0.0001 0.001 0.01 0.1 1
E/E
trac
er
K DOC 103 (1/s)
LlTl
LhTl
LlTh
LhTh
LhTm
Complete mixing Incomplete mixing
a
<R> / K DOC = 0.2E/E tracer + 0.0077R = 0.97
<R> / K DOC = 0.1E/E tracer - 0.0493R = 0.97
0.0
0.1
0.2
0 0.2 0.4 0.6 0.8 1
<R>/
KD
OC (
mM
)
E/Etracer
LhTh
LlTh
LlTl
LhTl
b
Fig. 11. Normalized dilution index calculated for different dispersivity combinations versus a) KDOC values, and b) integrated scaled aerobic degradation rate (Model 4).
102 H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
References
Abarca, E., Carrera, J., Sanchez-Vila, X., Dentz, M., 2007. Anisotropic dispersivehenry problem. Advances in Water Resources 30 (4), 913–926.
Abd-Elhamid, H., Javadi, A., 2011. A density-dependant finite element modelfor analysis of saltwater intrusion in coastal aquifers. Journal ofHydrology 401 (3–4), 259–271.
Aguilera, D., Jourabchi, P., Spiteri, C., Regnier, P., 2005. A knowledge-basedreactive transport approach for the simulation of biogeochemical dynamicsin earth systems. Geochemistry, Geophysics, Geosystems 6.
Andersen,M., Nyvang, V., Jakobsen, R., Postma, D., 2005. Geochemical processesand solute transport at the seawater/freshwater interface of a sandyaquifer. Geochimica et Cosmochimica Acta 69 (16), 3979–3994.
Anthony, A., 2004. Mixed convection and density-dependent seawatercirculation in coastal aquifers. Water Resources Research 40, 1–16.
Attaie-Ashtiani, B., Hassanizadeh, S., Oostrom, M., Celia, M., White, M.,2001. Numerical simulation and homogenization of two-phase flow inheterogeneous porous media. Journal of Contaminant Hydrology 49,87–109.
Bell, L., Binning, P., 2004. A split operator approach to reactive transport withthe forward particle tracking Eulerian Lagrangian localized adjointmethod. Advances in Water Resources 27 (4), 323–334.
Boluda-Botella, N., Gomis-Yages, V., Ruiz-Bevi, F., 2008. Influence of transportparameters and chemical properties of the sediment in experiments tomeasure reactive transport in seawater intrusion. Journal of Hydrology 357(1–2), 29–41.
Boufadel, M., Xia, Y., Li, H., 2011. Modeling solute transport and transientseepage in a laboratory beach under tidal influence. EnvironmentalModelling & Software 26 (7), 899–912.
Brunner, P., Simmons, C., 2012. Hydrogeosphere: a fully integrated,physically based hydrological model. Ground Water 50 (2), 170–176.
Burnett, W., Bokuniewicz, H., Huettel, M., Moore, W., Taniguchi, M., 2003.Groundwater and pore water inputs to the coastal zone. Biogeochem-istry 66, 3–33.
Centler, F., Shao, H., Biase, C., Park, C., Regnier, P., Kolditz, O., Thullner, M., 2010.Geosysbrns—a flexible multidimensional reactive transport model forsimulating biogeochemical subsurface processes. Computers & Geosciences36 (3), 397–405.
Charette, M., Sholkovitz, E., Hansel, C., 2005. Trace element cycling in asubterranean estuary: Part 1. geochemistry of the permeable sediments.Geochimica et Cosmochimica Acta 69 (8), 2095–2109.
Chiogna, G., Cirpka, O., Grathwohl, P., Rolle, M., 2011. Transverse mixing ofconservative and reactive tracers in porous media: quantification throughthe concepts of flux-related and critical dilution indices. Water ResourcesResearch 47, W02505.
Cirpka, O., Valocchi, A., 2007. Two-dimensional concentration distributionfor mixing-controlled bioreactive transport in steady state. Advances inWater Resources 30 (6–7), 1668–1679.
Cirpka, O., Frind, E., Helmig, R., 1999. Numerical simulation of biodegradationcontrolled by transverse mixing. Journal of Contaminant Hydrology 40 (2),159–182.
Coumou, D., Driesner, T., Heinrich, C., 2008. The structure and dynamics ofmid-ocean ridge hydrothermal systems. Science 321 (5897), 1825–1828.
Dai, M., Yin, Z., Meng, F., Liu, Q., Cai, W., 2012. Spatial distribution of riverineDOC inputs to the ocean: an updated global synthesis. Current Opinionin Environmental Sustainability 4 (2), 170–178.
103H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
Dale, A., Regnier, P., Knab, N., Jørgensen, B., Van Cappellen, P., 2008. Anaerobicoxidation of methane (AOM) in marine sediments from the skagerrak(Denmark): II. Reaction-transport modeling. Geochimica et CosmochimicaActa 72 (12), 2880–2894.
Dale, A., Brüchert, V., Alperin, M., Regnier, P., 2009a. An integrated sulfurisotopemodel for Namibian shelf sediments. Geochimica et CosmochimicaActa 73 (7), 1924–1944.
Dale, A., Regnier, P., Van Cappellen, P., Fossing, H., Jensen, J., Jørgensen, B.,2009b. Remote quantification ofmethane fluxes in gassymarine sedimentsthrough seismic survey. Geology 37 (3), 235–238.
Diersch, H., Kolditz, O., 2002. Variable-density flowand transport inporousmedia:approaches and challenges. Advances in Water Resources 25, 899–944.
Geiger, S., Matthäi, S., 2012.What canwe learn fromhigh-resolution numericalsimulations of single-and multi-phase fluid flow in fractured outcropanalogues? Geological Society, London, Special Publications 374.
Geiger, S., Cortis, A., Birkholzer, J., 2010. Upscaling solute transport in naturallyfractured porous media with the continuous time random walk method.Water Resources Research 46. http://dx.doi.org/10.1029/2010WR009133.
Gharasoo, M., Centler, F., Regnier, P., Harms, H., Thullner, M., 2011. A reactivetransport modeling approach to simulate biogeochemical processes inpore structures with pore-scale heterogeneities. Environmental Model-ling & Software 30, 102–114.
Graf, T., Therrien, R., 2007. Coupled thermohaline groundwater flow and single-species reactive solute transport in fractured porous media. Advances inWater Resources 30 (4), 742–771.
Graf, T., Therrien, R., 2008. A test case for the simulation of three-dimensionalvariable-density flow and solute transport in discretely-fractured porousmedia. Advances in Water Resources 31 (10), 1352–1363.
Gudonov, S., 1959. The finite-difference method for the computation ofdiscontinuous solutions of the equations of fluid dynamics. MatematicheskiiSbornik 47, 357393.
Ham, P., Schotting, R., Prommer, H., Davis, G., 2004. Effects of hydrodynamicdispersion on plume lengths for instantaneous bimolecular reactions.Advances in Water Resources 27 (8), 803–813.
Hayek, M., Kosakowski, G., Jakob, A., Churakov, S., 2012. A class of analyticalsolutions for multidimensional multispecies diffusive transport coupledwith precipitation–dissolution reactions and porosity changes. WaterResources Research 48 (3), W03525.
Held, R., Attinger, S., Kinzelbach, W., 2005. Homogenization and effectiveparameters for the henry problem in heterogeneous formations. WaterResources Research 41, W11420.
Holzbecher, E., 2005. FEMLAB performance on 2D porous media variabledensity benchmarks. COMSOL Multiphysics User's Conference 2005Proceedings, Frankfurt, Germany, p. 16.
Hunter, K., Wang, Y., Van Cappellen, P., 1998. Kinetic modeling of microbially-driven redox chemistry of subsurface environments: coupling transport,microbial metabolism and geochemistry. Journal of Hydrology 209 (1–4),53–80.
Kerrou, J., Renard, P., 2010. A numerical analysis of dimensionality andheterogeneity effects on advective dispersive seawater intrusion processes.Journal of Hydrology 18 (1), 55–72.
Kitanidis, P., 1994. The concept of the dilution index. Water ResourcesResearch 30 (7), 20112026.
Kolditz, O., Ratke, R., Diersch, H., Zielke, W., 1998. Coupled groundwater flowand transport: 1. verification of variable density flow and transportmodels. Advances in Water Resources 21 (1), 27–46.
Kolditz, O., Bauer, S., Bilke, L., Böttcher, N., Delfs, J., Fischer, T., Görke, U.,Kalbacher, T., Kosakowski, G., McDermott, C., et al., 2012. Opengeosys:an open-source initiative for numerical simulation of thermo-hydro-mechanical/chemical (THM/C) processes in porous media. Environmen-tal Earth Sciences 1–11.
Kroeger, K., Charette, M., 2008. Nitrogen biogeochemistry of submarinegroundwater discharge. Limnology and Oceanography 1025–1039.
Krumins, V., Gehlen, M., Arndt, S., Van Cappellen, P., Regnier, P., 2012. Dissolvedinorganic carbon and alkalinity fluxes from coastal marine sediments:model estimates for different shelf environments and sensitivity to globalchange. Biogeosciences 9, 8475–8539.
Kulik, D., Wagner, T., Dmytrieva, S., Kosakowski, G., Hingerl, F., Chudnenko,K., Berner, U., 2012. Gem-selektor geochemical modeling package:revised algorithm and gems3k numerical kernel for coupled simulationcodes. Computational Geosciences 1–24.
Lagneau, V., Van Der Lee, J., 2010. Hytec results of the momas reactivetransport benchmark. Computational Geosciences 14 (3), 435–449.
Latham, J., Xiang, J., Belayneh, M., Nick, H., Tsang, C., Blunt, M., 2012. Modellingstress-dependent permeability in fractured rock including effects ofpropagating and bending fractures. International Journal of Rock Mechan-ics and Mining Sciences 57, 100–112.
Li, H., Boufadel, M., Weaver, J., 2008. Tide-induced seawater–groundwatercirculation in shallow beach aquifers. Journal of Hydrology 352 (1–2),211–224.
Li, X., Hu, B., Burnett, W., Santos, I., Chanton, J., 2009. Submarine groundwater discharge driven by tidal pumping in a heterogeneous aquifer.Ground Water 47 (4), 558–568.
Lu, C., Luo, J., 2010. Dynamics of freshwater–seawater mixing zone develop-ment in dual-domain formations. Water Resources Research 46, W11601.
Lu, C., Kitanidis, P., Luo, J., 2009. Effects of kinetic mass transfer and transientflow conditions on widening mixing zones in coastal aquifers. WaterResources Research 45, 1–17.
Mao, X., Prommer, H., Barry, D., Langevin, C., Panteleit, B., Li, L., 2006. Three-dimensional model for multi-component reactive transport withvariable density groundwater flow. Environmental Modelling & Soft-ware 21 (5), 615–628.
Matthäi, S., Geiger, S., Roberts, S.G., 2004. The Complex Systems PlatformCSP5.0: User's Guide . (ETH).
Matthäi, S., Geiger, S., Roberts, S.G., Paluszny, A., Belayneh, M., Burri, A.,Mezentsev, A., Lu, H., Coumou, D., Driesner, T., Heinrich, C.A., 2007.Numerical simulation of multi-phase fluid flow in structurally complexreservoirs. In: Jolley, S.J., Barr, D., Walsh, J.J., Knipe, R.J. (Eds.), StructurallyComplex ReservoirsGeological Society of London Special Publication 292,405–429.
Matthäi, S., Nick, H., Pain, C., Neuweiler, I., 2009. Simulation of solutetransport through fractured rock: a higher-order accurate finite-elementfinite-volume method permitting large time steps. Transport in PorousMedia. http://dx.doi.org/10.1007/s11242-009-9440-z.
Michael, H.A., Mulligan, A.E., Harvey, C., 2005. Seasonal oscillations in waterexchange between aquifers and the coastal ocean. Nature 436 (7054),1145–1148.
Moore, W., 1999. The subterranean estuary: a reaction zone of ground waterand sea water. Marine Chemistry 65 (1–2), 111–125.
Moore, W., 2010. The effect of submarine groundwater discharge on theocean. Annual Review of Marine Science 2 (1), 59–88.
Mulligan, A., Evans, R., Lizarralde, D., 2007. The role of paleochannels ingroundwater/seawater exchange. Journal of Hydrology 335 (3–4), 313–329.
Nick, H., Matthäi, S., 2011a. Comparison of three FE–FV numerical schemesfor single- and two-phase flow simulation of fractured porous media.Transport in Porous Media 90, 421–444.
Nick, H., Matthäi, S., 2011b. A hybrid finite-element finite-volume methodwith embedded discontinuities for solute transport in heterogeneousmedia. Vadose Zone Journal 10 (1), 299–312.
Nick, H., Schotting, R., Gutierrez-Neri, M., Johannsen, K., 2009. Modelingtransverse dispersion and variable density flow in porous media.Transport in Porous Media 78 (1), 11–35.
Nick, H., Paluszny, A., Blunt, M., Matthäi, S., 2011. Role of geomechanicallygrown fractures on dispersive transport in heterogeneous geologicalformations. Physical Review E 84, 056301.
Pain, C.C., Eaton, M.D., Bowsher, J., Smedley-Stevenson, R.P., Umpleby, A.P.,de Oliveira, C.R.E., Goddard, A.J.H., 2003. Finite element based Riemannsolvers for time-dependent and steady-state radiation transport.Transport Theory and Statistical Physics 32 (5–7), 699–712.
Paluszny, A., Zimmerman, R., 2011. Numerical simulation of multiple 3Dfracture propagation using arbitrary meshes. Computer Methods inApplied Mechanics and Engineering 200 (9–12), 953–966.
Paluszny, A., Matthäi, S., Hohmeyer, M., 2007. Hybrid finite element-finitevolume discretization of complex geologic structures and a new simulationworkflow demonstrated on fractured rocks. Geofluids 7, 186–208.
Parkhurst, D., 1995. User's Guide to PHREEQC: a Computer Program forSpeciation, Reaction-path, Advective-transport, and Inverse Geochemi-cal Calculations. US Department of the Interior, US Geological Survey.
Pool, M., Carrera, J., Dentz, M., Hidalgo, J., Abarca, E., 2011. Vertical average formodeling seawater intrusion. Water Resources Research 47, W11506.
Post, V., 2005. Fresh and saline groundwater interaction in coastal aquifers:is our technology ready for the problems ahead? Hydrogeology Journal13 (1), 120–123.
Putti, M., Paniconi, C., 1995. Picard and Newton linearization for the coupledmodel for saltwater intrusion in aquifers. Advances in Water Resources 18(3), 159–170.
Raoof, A., Nick, H., Wolterbeek, T., Spiers, C., 2012. Pore-scale modeling ofreactive transport in wellbore cement under CO2 storage conditions.International Journal of Greenhouse Gas Control 11, S67–S77.
Regnier, P., O'Kane, J., Steefel, C., Vanderborght, J., 2002. Modeling complexmulti-component reactive-transport systems: towards a simulationenvironment based on the concept of a knowledge base. AppliedMathematical Modelling 26 (9), 913–927.
Regnier, P., Jourabchi, P., Slomp, C., 2003. Modeling complex multi-componentreactive-transport systems: towards a simulation environment based on theconcept of a knowledge base. Netherlands Journal of Geosciences 1 (82),5–18.
104 H.M. Nick et al. / Journal of Contaminant Hydrology 145 (2013) 90–104
Reillya, T., Goodmanb, A., 1985. Quantitative analysis of saltwater–freshwaterrelationships in groundwater systems Ña historical perspective. Journal ofHydrology 80 (1–2), 125–160.
Rezaei, M., Sanz, E., Raeisi, E., Ayora, C., Vazquez-Sune, E., Carrera, J., 2005.Reactive transport modeling of calcite dissolution in the fresh–salt watermixing zone. Journal of Hydrology 311 (1-4), 282298.
Robinson, C., Li, L., Barry, D., 2007. Effect of tidal forcing on a subterraneanestuary. Advances in Water Resources 30 (4), 851–865.
Rolle, M., Eberhardt, C., Chiogna, G., Cirpka, O., Grathwohl, P., 2009.Enhancement of dilution and transverse reactive mixing in porousmedia: experiments and model-based interpretation. Journal of Con-taminant Hydrology 110 (3–4), 130–142.
Saaltink, M., Pifarré, F., Ayora, C., Carrera, J., Pastallé, S., 2004. Retraso, a codefor modeling reactive transport in saturated and unsaturated porousmedia. Geologica Acta 2 (3), 235.
Samper, J., Zhang, G., 2006. Coupled microbial and geochemical reactivetransport models in porous media: formulation and application tosynthetic and in situ experiments. Journal of Iberian Geology 32 (2),215–231.
Santoro, A., 2010. Microbial nitrogen cycling at the saltwater–freshwaterinterface. Hydrogeology Journal 18, 187–202.
Scheidegger, A., 1961. General theory of dispersion in porous media. Journalof Geophysical Research 66, 3273–3278.
Shao, H., Centler, F., De Biase, C., Thullner, M., Kolditz, O., 2009. Comments onÒtwo-dimensional concentration distribution for mixing-controlledbioreactive transport in steady-stateÓ by oacirpka and ajvalocchi.Advances in Water Resources 32 (2), 293–297.
Simmons, C., 2005. Variable density groundwater flow: from currentchallenges to future possibilities. Journal of Hydrology 13 (1), 116–119.
Simpson, M., Clement, T., 2004. Improving the worthiness of the Henryproblem as a benchmark for density-dependent groundwater flowmodels. Water Resources Research 40 (1).
Slomp, C., Van Cappellen, P., 2004. Nutrient inputs to the coastal oceanthrough submarine groundwater discharge: controls and potentialimpact. Journal of Hydrology 295 (1-4), 6486.
Spiteri, C., Regnier, P., Slomp, C., Charette, M., 2006. ph-dependent iron oxideprecipitation in a subterranean estuary. Journal of Geochemical Explora-tion 88 (1), 399–403.
Spiteri, C., Slomp, C., Regnier, P., Meile, C., Van Cappellen, P., 2007. Modellingthe geochemical fate and transport of wastewater-derived phosphorusin contrasting groundwater systems. Journal of Contaminant Hydrology92 (1), 87–108.
Spiteri, C., Slomp, C., Charette, M., Tuncay, K., Meile, C., 2008a. Flow andnutrient dynamics in a subterranean estuary (Waquoit Bay, MA, USA):field data and reactive transport modeling. Geochimica et CosmochimicaActa 72 (14), 3398–3412.
Spiteri, C., Slomp, C., Tuncay, K., Meile, C., 2008b. Modeling biogeochemicalprocesses in subterranean estuaries: the effect of flow dynamics and redoxconditions on submarine groundwater discharge. Water Resources Research44 (2) (18 pp.).
Spiteri, C., Van Cappellen, P., Regnier, P., 2008c. Surface complexation effectson phosphate adsorption to ferric iron oxyhydroxides along ph andsalinity gradients in estuaries and coastal aquifers. Geochimica etCosmochimica Acta 72 (14), 3431–3445.
Steefel, C., 2008. CrunchFlow, Software for Modeling MulticomponentReactive Flow and Transport. Users Manual. Lawrence Berkeley NationalLaboratory (LBNL).
Steefel, C., Lasaga, A., 1994. A coupled model for transport of multiplechemical species and kinetic precipitation/dissolution reactions withapplication to reactive flow in single phase hydrothermal systems.American Journal of Science 294 (5), 529–592.
Steefel, C., MacQuarrie, K., 1996. Approaches to modeling of reactivetransport in porous media. Reviews in Mineralogy and Geochemistry34 (1), 85–129.
Stüben, K., 1999. Algebraic Multigrid (AMG): An Introduction with Applica-tions. ForschungszentrumInformationstechnik GmbH, Sankt Augustin,Germany.
Taniguchi, M., Ishitobi, T., Shimada, J., 2006. Dynamics of submarinegroundwater discharge and freshwater–seawater interface. Journal ofGeophysical Research 111, 1–9.
Thullner, M., Schroth, M., Zeyer, J., Kinzelbach, W., 2004. Modeling of amicrobial growth experiment with bioclogging in a two-dimensionalsaturated porous media flow field. Journal of Contaminant Hydrology 70(1), 37–62.
Thullner, M., Van Cappellen, P., Regnier, P., 2005. Modeling the impact ofmicrobial activity on redox dynamics in porous media. Geochimica etCosmochimica Acta 69 (21), 5005–5019.
Thullner, M., Dale, A., Regnier, P., 2009. Global-scale quantification of mineraliza-tion pathways inmarine sediments: a reaction-transportmodeling approach.Geochemistry, Geophysics, Geosystems 10 (10), Q10012.
Uchiyama, Y., Nadaoka, K., Rolke, P., Adachi, K., Yagi, H., 2000. Submarinegroundwater discharge into the sea and associated nutrient transport ina sandy beach. Water Resources Research 36 (6), 1467–1479.
Van Der Lee, J., De Windt, L., Lagneau, V., Goblet, P., 2003. Module-orientedmodeling of reactive transport with hytec. Computers & Geosciences 29(3), 265–275.
Volker, R., Rushton, K., 1982. An assessment of the importance of someparameters for seawater intrusion in aquifers and a comparison ofdispersive and sharp-interfacemodelling approaches. Journal of Hydrology56 (3–4), 239–250.
Walter, A., Frind, E., Blowes, D., Ptacek, C., Molson, J., 1994. Modeling ofmulticomponent reactive transport in groundwater: 1. model develop-ment and evaluation. Water Resources Research 30 (11), 3137–3148.
Werner, A., Bakker, M., Post, V., Vandenbohede, A., Lu, C., Ataie-Ashtiani, B.,Simmons, C., Barry, D., 2012. Seawater intrusion processes, investigationand management: recent advances and future challenges. Advancesin Water Resources. http://www.sciencedirect.com/science/article/pii/S030917081200053X.
Werth, C.J., Cirpka, O., Grathwohl, P., 2006. Enhanced mixing and reactionthrough flow focusing in heterogeneous porous media. Water ResourcesResearch 42, W12414.
White, M., McGrail, B., 2005. Stomp (subsurface transport over multiple phase)version 1.0 addendum. Eckechem Equilibrium-conservation kinetic EquationChemistry and Reactive Transport. Pacific Northwest National Laboratory,PNNL-15482, Richland, Washington.
Wilson, A., 2005. Fresh and saline groundwater discharge to the ocean: aregional perspective. Water Resources Research 41 (1), 1–11.
Wissmeier, L., Barry, D., 2011. Simulation tool for variably saturated flow withcomprehensive geochemical reactions in two- and three-dimensionaldomains. Environmental Modelling & Software 26 (2), 210–218.
Xu, T., Samper, J., Ayora, C., Manzano, M., Custodio, E., 1999. Modeling ofnon-isothermal multi-component reactive transport in field scaleporous media flow systems. Journal of Hydrology 214 (1), 144–164.
Xu, T., Spycher, N., Sonnenthal, E., Zhang, G., Zheng, L., Pruess, K., 2010.Toughreact version 2.0: a simulator for subsurface reactive transport undernon-isothermal multiphase flow conditions. Computers & Geosciences 37(6), 763–774.
Yang, C., Samper, J., Montenegro, L., 2008. A coupled non-isothermal reactivetransport model for long-term geochemical evolution of ahlw repositoryin clay. Environmental Geology 53 (8), 1627–1638.
Younes, A., 2003. On modelling the multidimensional coupled fluid flow andheat or mass transport in porous media. International Journal of Heatand Mass Transfer 46 (2), 367–379.
