numerical modeling of coupled variably saturated fluid flow and reactive transport with fast and...

Ž .Journal of Contaminant Hydrology 47 2001 379–390www.elsevier.comrlocaterjconhyd

Numerical modeling of coupled variably saturatedfluid flow and reactive transport with fast and slow

chemical reactions

Ž . a,) b,1 b,1Gour-Tsyh George Yeh , Malcolm D. Siegel , Ming-Hsu Lia Department of CiÕil and EnÕironmental Engineering, The PennsylÕania State UniÕersity,

212 Sackett Building, UniÕersity Park, PA 16802 USAb Sandia National Laboratories, Albuquerque, NM 87185 USA

Received 18 August 1999; received in revised form 22 February 2000; accepted 3 April 2000

Abstract

The couplings among chemical reaction rates, advective and diffusive transport in fracturedmedia or soils, and changes in hydraulic properties due to precipitation and dissolution withinfractures and in rock matrix are important for both nuclear waste disposal and remediation ofcontaminated sites. This paper describes the development and application of LEHGC2.0, amechanistically based numerical model for simulation of coupled fluid flow and reactive chemicaltransport, including both fast and slow reactions in variably saturated media. Theoretical bases andnumerical implementations are summarized, and two example problems are demonstrated. Thefirst example deals with the effect of precipitationrdissolution on fluid flow and matrix diffusionin a two-dimensional fractured media. Because of the precipitation and decreased diffusion ofsolute from the fracture into the matrix, retardation in the fractured medium is not as large as thecase wherein interactions between chemical reactions and transport are not considered. The secondexample focuses on a complicated but realistic advective–dispersive–reactive transport problem.This example exemplifies the need for innovative numerical algorithms to solve problemsinvolving stiff geochemical reactions. q 2001 Elsevier Science B.V. All rights reserved.

Keywords: Reactive transport; Numerical models; Geochemistry; Groundwater; Remediation

) Corresponding author. Tel.: q1-814-863-7304; fax: q1-814-863-7304.Ž . Ž .E-mail addresses: [email protected], [email protected] G.-T. Yeh , [email protected] M.D. Siegel ,

Ž [email protected] M.-H. Li .1 Fax: q1-505-284-4023.

0169-7722r01r$ - see front matter q2001 Elsevier Science B.V. All rights reserved.Ž .PII: S0169-7722 00 00164-9

( )G.-T. Yeh et al.rJournal of Contaminant Hydrology 47 2001 379–390380

1. Introduction and background

More than 10,500 individual contaminated sites have been identified at the Depart-Ž .ment of Energy DOE facilities with pollutants including mixtures of hazardous

organics, metals, and radionuclides. Terminal disposal of these wastes requires develop-ment of techniques for waste characterization, performance assessment, and long-termmonitoring technologies that are cost-effective and result in acceptable risks to humanhealth and the environment. Accurate tools for reliable prediction of contaminantmigration and transformation are necessary to support these tasks. Consideration ofequilibrium chemistry, kinetic chemistry, and hydrologic transport and the interactionbetween fluid flow and reactive transport is necessary to be able to reflect thecomplexity of many real systems. For example, solid phase reactions including precipita-tion and dissolution can potentially plug pores or open fractures reducing matrixdiffusion and promoting rapid flow through fractures.

The development of mechanistically based reactive chemical transport models hasŽ .exploded in the last decade Yeh and Tripathi, 1989; Steefel and Van Cappellen, 1998 .

Some models couple transport with equilibrium chemistry, others couple transport withkinetic geochemistry for specific geochemical processes like precipitationrdissolution oradsorption. This paper describes the development and application of LEHGC2.0, amechanistically based numerical model for simulation of coupled fluid flow and reactivechemical transport in variably saturated media. LEHGC2.0 can simulate a full suite ofkinetic and equilibrium geochemical processes simultaneously, while its previous ver-sions dealt with only equilibrium geochemistry. The coupling of this capability withsimultaneous modeling of fluid flows and their interactions with reactive transport givesLEHGC2.0 a robust ability to simulate the geochemistry of the subsurface environment.

2. Governing equations

The governing equations of LEHGC2.0 include three modules: a flow equation, a setof hydrologic transport equations, and a set of geochemical equations. A modifiedRichard equation describes density-dependent fluid flow in variably saturated media. It

Žcan be derived based on continuity of fluid, continuity of solid, motion of fluid Darcy’s. Ž .law , consolidation of the media, and compressibility of water Cheng et al., 1998 . The

governing equations for a reactive system are mass balance equations for aqueouscomponents, adsorbent components, equivalent-balance equations for ion exchange sites,mass action equations for equilibrium species, and reaction rate expressions for kineticspecies. For simplification, we will only present the governing equations for the aqueouscomponents, complexed species, and precipitated species. Neither sourcersink nordecay is considered in the following presentation; detailed descriptions of these equa-

Ž .tions can be found elsewhere Yeh et al., 1998, 1999 .

2.1. Hydrologic transport equations for chemical components

ET Eu Euju q P sL C y C , jgN , 1Ž .Ž .j j j

Et Et Et

( )G.-T. Yeh et al.rJournal of Contaminant Hydrology 47 2001 379–390 381

where u is the moisture content, T is the total analytical concentration of the jthjŽchemical component, P is the total immobile including precipitated, adsorbed, andj

.ion-exchanged concentration of the jth component, C is the total dissolved concentra-j

tion of the jth component, N is the number of chemical components, and L is anadvection–diffusionrdispersion operator. It should be noted that for immobile compo-nents, C s0 and P sT .j j j

2.2. GoÕerning equations for complexed species

Nxx ai kx sa c , ig M yK , 2Ž . Ž .Łi i k x x

ks1

NN kE u xŽ . xi bx fx a x xi ksL x yk x qa c q n ym R , igK . 3Ž . Ž .Ž .Ł Ýi i i i k i k i k k xEt ks1 ks1

2.3. GoÕerning equations for adsorbed species

Nyy ai ky sa c , ig M yK , 4Ž .Ž .Łi i k y y

ks1

NN kE u yŽ . yi by fy a y yi ksyk y qa c q n ym R , igK . 5Ž .Ž .Ł Ýi i i k i k i k k yEt ks1 ks1

2.4. GoÕerning equations for precipitated species

Napp ai k1sa c , ig M yK , 6Ž .Ž .Łi k p p

ks1

NN kE u pŽ . pi bp fp a p pi ksA ya qa c q n ym R , igK , 7Ž .Ž .Ł Ýi i i k i k i k k pž /Et ks1 ks1

where c , x , y and p are the concentrations of the k th aqueous component species,k i i i

the ith complexed species, ith adsorbed species, and the ith precipitated species,respectively; M is the number of chemical species; K is the number of kinetic species;a is the stoichiometric coefficient of the jth component in the ith reactions; a is thei j i

apparent stability constant for the ith equilibrium species; k is the backward ratei

constant of the ith basic kinetic reaction; a b and a f are the modified backward andi i

forward rate constants of the ith basic kinetic reactions, respectively; n and m are thei k i k

stoichiometric coefficients of the ith species in the k th parallel kinetic reactions for theproducts and reactants, respectively; N is the number of parallel kinetic reactions; andk

R is the reaction rate of the k th parallel reaction. The AxB, AyB, and ApB in thek

superscript or subscript position of the notations in the above equations are used to


describe the association of these notations with complexation, adsorbed, and precipita-tionrdissolution species, respectively. A s0 if the ith basic kinetic precipita-i

tionrdissolution reaction is not to occur or A s1 if the reaction is to occur.i

2.5. Coupling between fluid flows and reactiÕe transport

The density-dependence of fluid flows is described via the following constitutiverelationships:

N Na a

rsr q C M , msm q a C M 8Ž .Ý Ý0 j j 0 j j jjs1 js1

where r and m are fluid density and fluid dynamic viscosity with dissolved chemicalconcentrations, r and m are fluid density and fluid dynamic viscosity in pure water,0 0

M is the molecular weight and total dissolved concentration of the jth aqueousj

component, respectively, and a is a weighting parameter for the jth component.j

The interactions between reactive transport and fluid flows are described by theŽ .following dependencies of moisture content u , saturated hydraulic conductivity tensor

Ž . Ž .K , and dispersion coefficient tensor D on concentrations of minerals:so

nSu 1so

us and K sK 9Ž .so soo ž /1qSf 1qSfp p

mmy1DsD u 1yf 10Ž .Ž .0 p

where u is the effective saturated moisture content without the effect of solid species;so

S is the degree of saturation of water, which is a function of the pressure head;w 3f sÝ p V , where p is the precipitated concentration of the ith mineral molrdm ofp i i i

x w 3 xwater and V is the molar volume of the ith mineral dm of solidrmol ; K is thei sooŽreference saturated hydraulic conductivity tensor ignoring the effect of density and solid

.concentration ; n is the fractal for estimating hydraulic conductivity based on particlesize and packing structure; D is the dispersion coefficient tensor computed without the0

Ž .effect of solid species; and m is the Acementation exponentB Dullien, 1979 .

3. Numerical solutions

In LEHGC2.0, the flow equations are discretized with the Galerkin finite elementmethod. The transport equations are discretized with either conventional finite elementmethods or a hybrid Lagrangian–Eulerian finite element method. Nonlinearities in thefluid flow, reactive transport, and their interaction are solved with using Picardlinearization, and any nonlinearity in geochemistry are solved with Newton–Raphsonlinearization.

The two subsystems of equations describing coupled hydrologic transport and mixedgeochemical equilibriumrkinetic reaction problems are solved iteratively. Three differ-ent approaches may be used in LEHGC2.0 to reach a convergent solution. The first


approach is complete iteration between two subsystems, the second approach is the useof operator splitting, and the third approach is the employment of the predictor–correc-tor method. For each time step, a sequential iteration approach is also applied to solvecoupled fluid flows and reactive chemical transport equations until the pressure head,total analytical concentrations of chemical components, and concentrations of kineticallycontrolled complex species meet prescribed convergent criteria.

4. Examples

Nineteen example problems have been developed to demonstrate the ability ofLEHGC2.0 to solve a variety of mixed subsurface flow, solute transport, and chemical

Ž .reaction problems Yeh et al., 1999 . However, due to space limitation, only twonumerical experiments with mostly equilibrium geochemistry are described in this paper.Demonstrative example problems with mixed kinetic and equilibrium reactions and

Ž .verification examples can be found in the LEHGC2.0 document Yeh et al., 1999 .

4.1. Example 1: coupled fracture flow, matrix diffusiÕe, and reactiÕe transport

The first example deals with the effect of precipitation on fluid flow and matrixdiffusion in a two-dimensional fractured media. The domain of interest was based on alongitudinal cross-section of a 0.76-m-high column with a 120-mm-wide fracture in thecenter, running the length of the column. Due to symmetry, one-half of the cross-sectioncan be used as shown in Fig. 1 with a 6=10y4 dm wide fracture on the left-hand side.

Fig. 1. Problem definition for Example 1.


The fractured media is simulated by the three left-hand columns of elements with amoisture content, u , of 1.0 when precipitated chemical species are not present. The topand bottom surfaces are held at a Dirichlet head of 7.615 and 7.6 dm, respectively. Theupper boundary of the domain is assumed to maintain a constant concentration, Dirichletboundary. Neumann boundaries are used for the other three boundaries with ECrExs0for the two vertical boundaries and ECrEzs0 for the bottom boundary. The referencesaturated hydraulic conductivity K in the z-direction for the matrix is K s2.16=soo z

10y6 dmrh; in the fracture, K s237.0 dmrh for material 2, K s222.0 dmrh forz z

material 3, and K s90.0 dmrh for material 4. The matrix elements have a moisturezŽ .content without the presence of precipitated species , tortuosity and bulk density of

0.35, 0.1, and 1.7 kgrdm3, respectively. The molecular diffusion coefficient is 3.6=y4 2 Ž . Ž .10 dm rh. Longitudinal dispersivity a and lateral dispersivity a for allL T

elements is set to 7.6 and 0.0 dm, respectively. A detailed description of the finiteelement grid is found in Yeh et al., 1999.

The chemistry of the system is relatively simple: the chemical components of thesystems are Ca2q, CO2y, Hq, and a conservative tracer. Calcite is present in the matrix3Ž y5 .8.642=10 M in equilibrium with the pore water and fracture water, which contain

y3 2q 2y Ž .10 M Ca and CO total concentrations . During the simulation, a solution 10=3

richer in both total calcium and total carbonate is injected in the fracture. The complexedy q q y Žspecies include OH , CaCO , CaHCO , CaOH , HCO , and H CO . Calcite CaCO3 3 3 2 3 3

Ž ..s is included as solid species subject to kinetic precipitation and dissolution. Thereaction stoichiometries, equilibrium constants and rate constants can be found in Yeh etal., 1999.

Two cases are simulated in this example; in the first case, the flow and reactivetransport are coupled through the effect of precipitation on hydraulic conductivity,porosity, and hydrodynamic dispersion. To reflect the effect of mineral precipitation on

Ž .flow, the fractal exponent in Eq. 9 is set equal to 2.5. The cementation exponent inŽArchie’s law is set equal to 2.0 to reflect the dependence of dispersion coefficient Eq.

Ž .. Ž .10 on precipitationrdissolution reactions. Molar volumes V of 0.03 and 0.037i3 Ž . Ž . Ž .dm rmol are assumed for portlandite, Ca OH s , and calcite, CaCO s , respectively.2 3

In the second case, although the effects of precipitationrdissolution on flow, conductiv-ity and dispersion are not considered, solute transport and chemical reaction still need to

Ž .be coupled because heterogeneous reactions precipitationrdissolution reactions areŽ .involved in the system. A time step size of 0.24 h was used; a total of 6 days 144 h is

simulated using 600 time steps.The results of the simulation are summarized by the profiles of tracer concentration

Ž . Ž .shown in Fig. 2a precipitation–flow coupling and Fig. 2b no coupling . In the firstcase, diffusion into the matrix is diminished due to calcite precipitation in the matrix.The concentrations of tracer in the matrix zone in this case are lower than those withoutthe precipitation effect. In addition, since less tracer can diffuse into the matrix, a largeramount of tracer remains in the fracture zone. As seen in the figures, the concentrationsof the conservative tracer in the fracture zone are much higher when the precipitation

Ž .effect is included. Examination of the simulated flow field not shown shows that thechange of flow velocity is very small because the calcite precipitation in the fracturezone fills only a small portion of the fracture zone.


Ž . Ž .Fig. 2. Concentration profiles of conservative tracer: a with the precipitation effect and b without theprecipitation effect for Example 1 at ts144 h.

4.2. Example 2: adÕectiÕe–dispersiÕe reactiÕe transport to demonstrate species switch-ing.

The second example involves the transient simulation of advection–dispersion andequilibrium complexationrprecipitation reactions to demonstrate the successful applica-tion of species switching in LEHGC2.0. This technique is required when the calculated


Fig. 3. Problem domain, discretization, and initial total concentrations for Example 2.


Table 1List of reactions and thermodynamic data for Example 2

q yH qOH |H O Log K s14.0023q y 2qŽ .Fe qOH |Fe OH Log K sy2.193q y qŽ .Fe q2OH |Fe OH Log K sy5.6723q y Ž .Fe q3OH |Fe OH Log K sy12.5633q y yŽ .Fe q4OH |Fe OH Log K sy21.6043q y 2qŽ .Al qOH |Al OH Log K sy5.003q y qŽ .Al q2OH |Al OH Log K sy10.2023q y Ž .Al q3OH |Al OH Log K sy17.2033q y yŽ .Al q4OH |Al OH Log K sy23.004

q 2y qH qCO |HCO Log K s10.333 3q 2y2H qCO ||H CO Log K s16.683 2 32q 2yCa qCO ||CaCO Log K s3.223 32q q 2yCa qH qCO ||CaHCO Log K s11.443 3

q 2y qH qSO ||HSO Log K s1.994 42q 2yCa qSO ||CaSO Log K s2.304 43q 2y qAl qSO ||AlSO Log K s3.504 43q 2y qŽ .Al q2SO ||Al SO Log K s5.004 4 23q 2y qFe qSO ||FeSO Log K s4.044 43q 2y qŽ .Fe q2SO ||Fe SO Log K s5.424 4 2q 2y qNa qSO ||NaSO Log K s0.074 42q 2y Ž .Ca qCO ||CaCO s Log K s8.483 33q y Ž . Ž .Al q3OH |Al OH s Log K sy9.1133q y Ž . Ž .Fe q3OH |Fe OH s Log K sy4.8932q 2y Ž .Ca qSO ||CaSO s Log K s4.584 4

concentration of an aqueous component species becomes very low and causes a massbalance error. In this example, the system domain is a 10=5000-dm2 porous medium,with 200 equal size elements, a moisture content of 0.3, a constant flow field of 155.5

Ž .dmrday, and a longitudinal dispersivity of 150 dm Fig. 3 . The chemical systemincludes seven aqueous components, 20 equilibrium complexed species, and fourequilibrium precipitated species; thermodynamic data for the chemical reactions aregiven in Table 1. Fig. 3 shows the distributions of the initial total concentrations of eachcomponent in the domain. Three compositional zones are indicated by differences insolution composition and precipitated solids content: Zone 1 contains goethite; Zone 2

Table 2Ž .Incoming total dissolved concentrations M for variable boundary conditions in Example 2

Unit: M ts0 day ts2.5 day ts2.6 day ts120 day2q y3 y3 y3 y3w xCa 7.4854=10 7.4854=10 3.94=10 3.94=102y y4 y4 y3 y3w xCO 1.0=10 1.0=10 2.5=10 2.5=103

3q y2 y2 y9 y9w xAl 4.0=10 4.0=10 9.3077=10 9.3077=102y y1 y1 y3 y3w xSO 2.0=10 2.0=10 4.4424=10 4.4424=104

q y3 y3 y3 y3w xH 1.7071=10 1.7071=10 2.7628=10 2.7628=103q y3 y3 y7 y7w xFe 2.093=10 2.093=10 3.0788=10 3.0788=10q y1 y1 y2 y2w xNa 1.4=10 1.4=10 2.3518=10 2.3518=10


Ž . Ž . Ž .Fig. 4. Concentration profiles of a pH, b total dissolved aluminum, and c precipitation of gibbsite atdifferent times for Example 2.


contains goethite, gypsum and gibbsite, and Zone 3 contains calcite with minor amountsof gypsum and gibbsite. Variable boundary conditions with the incoming total dissolved

Ž .concentrations as listed in Table 2 are assigned on the top nodes 401 and 402 andŽ .bottom nodes 1 and 2 sides. On the right and left sides, a zero flux condition is

imposed.A constant time step size of 0.025 day was used for 400 time steps resulting in a total

real simulation time of 10 days. Without species switching, the simulation stops duringthe first precipitation cycle of the first time step at node 299 while checking the mass

3q Žbalance of the Al component at the first nonlinear iteration between solving solute.transport and chemical reaction . When the species switching option was used to resolve

Ž .ythe error in the mass balance, the aqueous complexed species Al OH was selected by4

LEHGC2.0 to replace Al3q as the new component at several nodes. Subsequently,successful simulations for all 400-time steps were carried out.

Calculated concentration profiles for each of the components are given in Yeh et al.,1999. Due to space limitation, only the results for pH and aluminum are given in Fig. 4,

Žwhere the transport of contaminants is shown with respect to the z-coordinate any two.nodes having the same z-coordinate are identical with flow going from right to left. Fig.

4a shows the pH values computed in this system at different times. When tF2.5 days,the region between zs4000 dm and zs4550 dm becomes more acidic with time. This

Ž .is accompanied by increases in total dissolved aluminum Fig. 4b , sulfate, and iron andŽ .dissolution of gibbsite Fig 4c . When t)2.5 days, the system becomes more alkaline

with time because of the significant increase of the incoming total carbonate and themajor decrease of incoming total sulfate. This also causes the reprecipitation of gibbsiteŽ . Ž . Ž .Al OH s Fig. 4c in the region between zs4000 dm and zs5000 dm.3

5. Summary and conclusion

This paper communicates the development of a mechanistically based coupled fluidflow and reactive chemical transport model including both fast and slow reactions invariably saturated media. The theoretical bases and numerical techniques of the codewere summarized and two example problems were presented. The first problem exam-ined the effects of precipitation on fluid flow and matrix diffusion in fractured porousmedia. Although the flow field was not significantly affected by the precipitation, thediffusion of a conservative tracer was significantly reduced and transport of the tracerthrough the fracture was enhanced by this effect. The example demonstrated the abilityof LEHGC2.0 to show that chemical reactions between major solutes and rocks alongtransport pathways will affect contaminant retardation by matrix diffusion. The secondexample demonstrated the use of species switching by LEHGC2.0 to solve an advec-tive–dispersive–reactive transport problem in which large changes in the concentrationof an aqueous component species caused numerical instability. This example exempli-fied the need for innovative numerical algorithms to solve problems involving stiffgeochemical reactions. Robust mechanistically based numerical models like LEHGC2.0provide tools to more reliably predict the migration and transformation of chemicals inthe subsurface environment. Such capabilities could enhance the ability of environmen-


tal scientists, engineers and decision-makers to analyze the impact of subsurfacecontamination and to evaluate the efficacy of alternative remediation techniques prior toincurring expense in the field.

Acknowledgements

The LEHGC2.0 model was originally developed by the senior author with funding inpart from Sandia National Laboratories under Contract No. AX-9023 and in part fromNational Science Foundation under Grant No. EAR-9708494. Preparation of thismanuscript by Drs. Siegel and Li was supported by the internal laboratory research anddevelopment program of Sandia. Sandia is a multiprogram laboratory operated bySandia, a Lockeed Martin company, for the United States Department of Energy underContract DE-ACO4-94AL85000. Example 2 was provided by Dr. Chen Zhu at Univer-sity of Pittsburgh.

References

Cheng, J.R., Strobl, R.O., Yeh, G.T., Lin, H.C., Choi, W., 1998. Modeling of two-dimensional density-depen-dent flow and transport through saturated-unsaturated porous media. J. Hydrol. Eng., J. Hydrol. Eng.,

Ž .ASCE 3 4 , 248–257.Dullien, F.A.L., 1979. Porous Media. Academic Press, 574 pp.

Ž .Steefel, C.I., Van Cappellen, P. Eds. , Special Issue: Reactive Transport Modeling of Natural Systems. J.Ž . Ž .Hydrol. vol. 209 1–4 , 1–388 Guest Editors .

Yeh, G.T., Tripathi, V.S., 1989. A critical evaluation of recent developments in hydrogeochemical transportŽ .models of reactive multichemical components. Water Resour. Res. 25 1 , 93–108.

Yeh, G.T., Iskra, G., Zachara, J.M., Szecsody, J.E., 1998. Development and verification of a mixed chemicalŽ .kinetic and equilibrium model. Adv. Environ. Res. 2 1 , 24–56.

Yeh, G.T., Li, M.H., Siegel, M.D., 1999. User’s manual for LEHGC: A Lagrangian–Eulerian Finite-ElementModel of Coupled Fluid Flows and HydroGeoChemical Transport through Saturated–Unsaturated Media-Version 2.0. Department of Civil and Environmental Engineering, The Pennsylvania State University,University Park, PA 16802.

numerical modeling of coupled variably saturated fluid flow and reactive transport with fast and...

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