Module-oriented modeling of reactive transport with HYTEC

Computers & Geosciences 29 (2003) 265–275

Module-oriented modeling of reactive transport with HYTEC

Jan van der Lee*, Laurent De Windt, Vincent Lagneau, Patrick Goblet

Centre d’Informatique G !eologique, !Ecole des Mines de Paris, 35, rue Saint Honor!e, 77300 Fontainebleau, France

Received 22 January 2002; received in revised form 26 April 2002; accepted 14 June 2002

Abstract

The paper introduces HYTEC, a coupled reactive transport code currently used for groundwater pollution studies,

safety assessment of nuclear waste disposals, geochemical studies and interpretation of laboratory column experiments.

Based on a known permeability field, HYTEC evaluates the groundwater flow paths, and simulates the migration of

mobile matter (ions, organics, colloids) subject to geochemical reactions. The code forms part of a module-oriented

structure which facilitates maintenance and improves coding flexibility. In particular, using the geochemical module

CHESS as a common denominator for several reactive transport models significantly facilitates the development of new

geochemical features which become automatically available to all models. A first example shows how the model can be

used to assess migration of uranium from a sub-surface source under the effect of an oxidation front. The model also

accounts for alteration of hydrodynamic parameters (local porosity, permeability) due to precipitation and dissolution

of mineral phases, which potentially modifies the migration properties in general. The second example illustrates this

feature.

r 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Transport; Geochemistry; Modeling; Numerical methods; Variable porosity; Uranium migration; Radioactive waste

1. Introduction

Reactive transport models combine bio-geochemical

reactions with hydrological processes such as advective

ground-water flow, diffusion and dispersion. They are

increasingly used to understand and to predict the

migration behavior of aqueous or colloidal species in

natural systems (e.g. Lichtner, 1996; Gwo et al., 2001),

to understand the near-field of a waste repository as a

whole (Trotignon et al., 1998; van der Lee and De

Windt, 2001) and to achieve a conceptual and quanti-

tative understanding of individual reaction pathways,

(van Cappellen and Gaillard, 1996).

Among the several numerical methods available for the

integration of bio-geochemical processes in hydrodynamic

transport models, the operator splitting method combined

with the sequential iterative approach has several advan-

tages. For example, the approach allows stand-alone fully

featured codes to be used, thus facilitating development,

testing and maintenance. Also, the operator splitting

method allows the use of a modular-oriented modeling

approach which has major advantages, as outlined next.

Transport and chemistry are solved one after the

other within a single time step. Since both parts are

dependent on each other, an accurate solution is

obtained only after several iterations within the time-

step. The iterative sequential approach is notorious for

failing to converge, even for moderately complex

systems, which is the major disadvantage of the operator

splitting approach. Sometimes, iterative improvement of

the solution is not considered (Pfingsten, 1994; Sch.afer

et al., 1998; Yabusaki et al., 1998) which indeed avoids

numerical instabilities. Different studies clearly demon-

strate, however, the critical importance of iterative

improvement when applying the sequential approach

(Yeh and Tripathi, 1991; Carrayrou et al., 2002).

A second approach is to use a tightly coupled, implicit

approach (e.g. Steefel and Lasaga, 1994; van Cappellen

and Wang, 1996), integrating the reaction equations into

the transport equation and then solving the fully

*Corresponding author. Tel.: +33-164-69-4703; fax: +33-

164-69-4702.

E-mail address: [email protected] (J. van der Lee).

0098-3004/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0098-3004(03)00004-9

coupled algebraic-partial differential problem. Thus, the

problems related to the iterative improvement of the

sequential approach are avoided. Also, larger timesteps

can be used for otherwise stiff problems (Steefel and

MacQuarrie, 1996). The approach involves complex

mathematics, however, and cannot readily use indepen-

dently available modules.

Organic and inorganic colloids are omnipresent in

natural media and especially in subsurface systems. One

of the first colloidal alerts was given two decades ago by

researchers who found that radionuclides such as

plutonium and americium traveled over a significant

distance in colloidal form (Champ et al., 1982; Travis

and Nuttall, 1985). These conclusions have been

confirmed more recently by Kersting et al. (1999) and

colloidal transport has been recognized as an important

mechanism for migration of chemicals in natural

systems in general (e.g., McCarthy and Zachara, 1989;

van der Lee, 1997). Multi-component transport codes

which also account for reactive colloidal species are

nevertheless scarce. Part of the absence of colloidal

migration in current codes is undoubtedly related to the

absence of databases including colloids and reactions

with surfaces in general. The default database of the

geochemical code CHESS is a first attempt to fill this

gap and includes several organic and inorganic colloids.

HYTEC accounts for colloidal transport as illustrated in

one of the examples outlined in this paper.

The operator splitting method allows modular model

development which significantly improves the develop-

ment efficiency and overall performance of the resulting

software. The modular coding approach originally

adopted for HYTEC has been extended and HYTEC

has become part of a broader simulation platform, used

by several institutions in France. The model is con-

stantly subject to testing and verification exercises to

achieve the quality standards needed for performance

assessment of radioactive waste repositories. Generally,

the agreement with comparable codes is satisfactory. As

outlined in more detail later, the discretization method

of the transport equation or the implementation of the

reaction-transport method may nevertheless lead to

discrepancies (De Windt et al., 2003).

2. Global model design

After almost one decade of development, HYTEC has

evolved to a versatile and complex simulation tool and

forms part of a global simulation platform. The global

organization of the platform is schematically illustrated

by Fig. 1. The common denominator is the geochemical

module CHESS, written in object-oriented C++ and

highly optimized for coupling purposes. All function

calls require specific classes, hence calling the library

also requires a C++ code. Therefore, all non-C++

codes linked to CHESS communicate through specific

interface modules. JCHESS is an exception: written in

java, JCHESS provides a graphical user interface to run

CHESS as a stand-alone geochemical code.

Currently, two other reactive transport codes are

based on CHESS. CHEMTRAP, which forms part of a

software system developed by EDF (Electricity of

France), communicates with the wrapper code META-

CHESS via PVM.1 The advantages of PVM are similar

to those of MPI, outlined later. ALLIANCES clusters

several simulation tools developed by the French nuclear

research institute CEA, and ANDRA, responsible for

the final disposal of radioactive waste. The platform

communicates with a wrapper code ALLICHESS via a

scripting language based on Python.

HYTEC, written in C++, is linked with CHESS at

compile time and calls different hydrogeological trans-

port models via MPI.2 Some advantages of this

approach are:

* cost efficiency: a transport model is frequently

developed to run stand-alone. The MPI-based

coupling approach allows the code to be preserved,

since, even under HYTEC, it still runs as an

independent process. No profound restructuring but

only a minimum of specific-purpose coding is

required;* less error-prone: the modules are easily tested as they

exist as independent codes. As long as the commu-

nication interface is respected, the overall model will

run correctly even after major changes in one of the

modules;* more flexibility: using MPI allows the developer’s

preferences and efficiency to be preserved. One

scientist will develop the tools in Fortran, using

finite elements (FE) (METIS is such a code): another

prefers C or C++, using finite volumes (FV) (R2D2,

for example);* parallel computing: the experience with MPI is readily

recycled for parallelization of computationally time-

consuming modules.

Some disadvantages could be mentioned as well:

* HYTEC requires a networked multi-processor com-

puter system;* the approach requires numerous TCP/IP based

system calls, which have a certain cost with respect

to calculation time. The calculation time penalty is

only small, less than 5% for an isolated network (De

Windt and van der Lee, 2000).

1Parallel Virtual Machine, see http://www.epm.ornl.gov/

pvm.2Message Passing Interface, http://www-unix.mcs.anl.

gov/mpi.

J. van der Lee et al. / Computers & Geosciences 29 (2003) 265–275266

http://www.epm.ornl.gov/pvm.

http://www.epm.ornl.gov/pvm.

http://www-unix.mcs.anl.gov/mpi.

http://www-unix.mcs.anl.gov/mpi.

Currently, three transport codes can be selected by the

user at run-time. Supplying different transport codes is

useful since some numerical approaches are more or less

adapted to specific problems, as outlined in following

section.

3. Transport processes

HYTEC searches for an accurate solution to the

multi-component transport problem according to a

sequential iterative approach with a severe, normalized

convergence criterion. In order to simplify notations, let

us introduce a specific-purpose transport operator, L:Accordingly, transport of aqueous, colloidal or gaseous

species, denoted by index j; in a saturated medium is

described by the following equation (e.g., Cederberg

et al., 1985; Yeh and Tripathi, 1989):

@ðoTjÞ@t

¼ Lð *TjÞ; ð1Þ

where o denotes the porosity of the medium, *T the

mobile concentration of the species and Tj is the total

concentration of some species such that Tj ¼ *Tj þ %Tj ; %T

is the immobile concentration. For a multi-dimensional

transport system including advection, diffusion and

dispersion, the transport operator yields:

Lð *TjÞ ¼ r � ðoDjr *Tj � U *TjÞ; ð2Þ

where Dj includes diffusion and dispersion which can be

species-specific (see e.g., de Marsily, 1986) and U is the

filter (or Darcy) flow velocity. Note that the porosity is

explicitly included in the equations since this entity is

subject to change in time and space due to geochemical

processes such as precipitation, dissolution or clogging

of pores by colloidal retention. Eq. (2) only slightly

changes for transport in unsaturated media (e.g. Gwo

et al., 2001). Assuming laminar and isothermal flow in

porous media, the velocity field U can be obtained from

the permeability field k and Darcy’s law (Darcy, 1856,

pp. 581–594; Bear, 1972):

U ¼ �k

mrðp þ rgzÞ: ð3Þ

To solve Eq. (1), we need the relationship between T and*T: This relationship is a function of the bio-geochemical

reactions and notably those involved in immobilizing

part of the species, such as surface complexation, ion

exchange, precipitation and dissolution of mineral

phases but also colloidal retention (clogging, polymer-

ization). Let us introduce a second specific-purpose

operator, R; which integrates the entire geochemical

model (CHESS in our case) and is used to provide a

precise estimate of the mobile fractions, *T as a function

of the total concentrations:

*T ¼ RðTÞ: ð4Þ

Since R depends on T ; iterative improvement is required

to achieve an accurate solution of the system as a whole.

Fig. 2 provides, schematically, a chart of the computa-

tional flow within one time step.

Note the dependency on the porosity o; which also

alters the mineral concentrations—even if they are

chemically inert. This, at first sight, surprising phenom-

enon originates from the fact that, in geochemical

models, mineral concentrations are expressed in molar

or moles per kg of solvent. Since the amount of solvent

(H2O per unit volume of medium) changes, with

porosity, the concentration of matrix-forming minerals

also changes, sometimes with a strong impact on the

overall solution chemistry (Lagneau, 2000). A second-

level iterative procedure is needed to converge to

coherent values of T ; *T and o; all considered as primary

unknowns in HYTEC.

The current version of HYTEC corrects the effective

diffusion coefficient De for porosity changes according

CHEMTRAP ALLIANCESJCHESSHYTEC

RT1

MET

CHESS

JCHESS

MPI link link

++ linkPVM linkM

C++FortranJava

R2D CEA-ANDRA

Fig. 1. Global structure of HYTEC and related simulation tools based on CHESS.

J. van der Lee et al. / Computers & Geosciences 29 (2003) 265–275 267

to a modified version of Archie’s law (Archie, 1942;

Winsauer et al., 1952):

DeðoÞ ¼ De0

o� oc

o0 � oc

� �m

; ð5Þ

where m is an empirical coefficient and oc is a threshold

under which diffusion stops. Allowing a variable

permeability forms part of current studies.

3.1. Transport models and numerical methods

HYTEC farms out the numerical resolution of the

equations involved in ground-water flow to specialized

models. Currently, three codes have been interfaced,

RT1D, METIS and R2D2. All codes handle transport

which can be either advective–dispersive or diffusive.

RT1D is a one-dimensional, finite differences migration

model mainly used for simple column systems. METIS

solves the diffusivity equation and transport of heat and

mass in unsaturated and saturated porous media

(Goblet, 2002). The code uses a 2D/3D FE approach

with concentration defined at the nodes and linear

interpolation between adjacent nodes. R2D2 is written

in C++ and based on FV which is particularly useful

for treatment of the variable porosity problem. This

code uses a flexible grid based on Thiesen polygons and

hence easily adapted to complex geometries. All

boundary conditions normally used by modeling hydro-

geologists are available.

The models can be selected at run-time by the

user, allowing the numerical approach to be adapted

to the specific problem studied. For example, the FE

approach lacks precision in the definition of an initial

mass or concentration confined in a specific geometric

area, due to the interpolation of concentration between

adjacent nodes. This is schematically illustrated by

Fig. 3, which shows the mass-distribution over two

elements in a one-dimensional system (assuming

linear interpolation). The overall FE interpolation

effect can be corrected by a simple geometrical rule

(mass to volume ratio), as illustrated for FE2: Still,

the FE approach tends to smear out the zone

boundaries, which can be amended by grid refinement

only. FD and FV approaches, which use a constant

concentration in each element, are precise with respect

to this problem.

Fig. 2. Diagram of consecutive actions within one time step as implemented by HYTEC. T denotes total concentrations, *T mobile

concentrations, L transport operator (hydrology) and R reaction operator (geochemistry).

c

x

c

x

c

x

FV FE1 FE2

Fig. 3. Illustration of mass distributions for FV and FE approaches. Too much mass is defined for uncorrected FE approach (FE1).


The FV approach also provides a more precise

solution in the case of a variable porosity. Mineral

concentrations (which indirectly define the porosity) are

calculated at exactly the same point where the hydro-

dynamic model uses the porosity, in the center of a

representative volume. With a transport code based on

FE, mineral concentrations are evaluated at the nodal

points but the transport solver uses the elemental

porosities and the values have to be translated.

Translation leads to errors, especially at points of

sudden changes of porosity where an accurate descrip-

tion is mostly needed. Fig. 4 schematically illustrates the

problem, which is all the more critical since mineral

concentrations depend on the porosity. The only way to

reduce these errors is to refine the grid. For multi-

component reactive transport models, however, this will

increase the number of geochemical calculations (one

per grid element). We recall that the geochemical solver

requires much more CPU time than the transport solver.

Another disadvantage of the FE approach is that

velocity fields (and concentration gradients) are not

continuous at element borders. It is therefore impossible

to evaluate fluxes precisely through element segments. In

coupled transport models, the evaluation of fluxes and

accurate mass conservation is critical, especially when

the models are used for performance assessment of

waste repositories with a finite and well-defined source

term.

On the other hand, FE and derived approaches have

the major advantage of dealing with strongly anisotropic

systems since the exact vector of the hydraulic- and

concentration gradients can be calculated. FD and FV

only evaluate the normal component of the gradients

and are therefore not well suited for anisotropic systems.

The mixed hybrid finite elements method (MHFE)

seems to merge the advantages of both FE and FV

approaches. The module-oriented approach of HYTEC

straightforwardly allows coupling to a MHFE-based

transport module. Nevertheless, the MHFE approach is

significantly more complex than FE or FV and increases

the number of primary unknowns of the system, thus

increasing CPU time. So far, the need to include a

transport module based on MHFE has not been brought

forward.

4. Geochemical reactions

While migrating through the pores of the geological

medium, matter contained by the solution is subject to

(micro)biological and geochemical reactions. CHESS,

developed from scratch for coupling purposes, accounts

for the geochemistry of reactive transport. Most

frequently used thermodynamic databases are available

with CHESS. Although not yet operational, biochemical

reactions also form part of the CHESS module since

they strongly depend on the local solution chemistry

(pH, O2 content, Eh, etc.). For a complete conceptual

outline of the geochemical reactions, the reader is

referred to general texts (e.g. Morel and Hering, 1993;

Bethke, 1996; van der Lee, 1998). The following

summarizes the main parts of CHESS.

4.1. Speciation and aqueous complexation

Speciation includes reactions of the species with re-

spect to H2O; inorganic ligands (HCO�3 ;OH�;Cl�;

SO2�4 =S2�;F�;HPO2�

4 ; etc.), organic ligands (EDTA/

DTPA, citrate, lactate, humic and fulvic acids, etc.) or

the redox state of the solution (Eh, pe, Fe(II)/Fe(III)

etc.). At equilibrium, the concentration of each species

Sj can be written as a function of the concentration of a

set of Np basis or primary species, denoted by Ci; and a

thermodynamic formation constant Kj :

Sj ¼Kj

gj

YNp

i¼1

ðgiCiÞaji : ð6Þ

This is a general mass-action law with stoichiometric

coefficients a and activity coefficients g: The total

concentration of basis species Ci is given by

Ti ¼XNs

j¼1

ajiSj ð7Þ

which yields the mole balance. All T are imposed by the

recipe of the solution. Hence the equation for the ith

mole balance has a solution if

Ti �XNs

j¼1

ajiSj ¼ 0: ð8Þ

The problem is to find the roots (the values of Ci) of all

sets of equations simultaneously. Although a variety of

methods are available, the method which has proved to

be the most successful root-finding routine for a

chemical system is the Newton–Raphson method. The

Newton–Raphson method converges quadratically near

the root but tends to fail for trial solutions too far

from the root. As outlined in van der Lee (1998),

CHESS uses the Newton–Raphson method enhanced by

a polishing factor to improve its global convergence

capacity. Several other techniques are also available to

concentrationporosity

concentrationporosity

Finite elements Finite volumes

Fig. 4. Positions of concentration- and porosity–evaluation

points in FE and finite-difference/volumes methods. FE

requires interpolation to achieve approximative solution.


improve the standard Newton–Raphson method (Car-

rayrou, 2001).

4.2. Solution–surface interface reactions

Solution–surface interface reactions allow accounting

for acid–base reactions with solid matter, surface

complexation, ion exchange in clayey soils or in fracture

infill material, surface (co)precipitation, and so on.

Electrostatic effects are included and the user may select

the constant capacitance, the double layer or the triple

layer model. Note that interface reactions take place at

immobile phases but also at mobile, colloidal phases.

Colloids not only tend to increase the apparent

solubility of chemicals but they also tend to act as

efficient transport agents for strongly sorbing metals

(McCarthy and Zachara, 1989).

4.3. Precipitation and dissolution

Solids precipitate only and exactly at the point where

the ion activity product (IAP) equals or exceeds the

solubility constant (which is the inverse of the solid’s

formation constant). Therefore, they have to be treated

as exceptions with respect to other species which exist at

all times. Precipitation and dissolution of minerals and

colloidal matter changes the solution characteristics,

notably the pH. Besides immobile minerals, CHESS also

allows precipitation and/or dissolution of colloidal

(hence mobile) phases. Precipitation does not necessarily

lead to hydrodynamic retention if the neoformed phase

is colloidal. The precipitation or dissolution process

possibly changes the reactivity of the solid surface, i.e. it

creates or destroys functional groups or sorption sites in

general.

CHESS accounts for reaction kinetics, generally

mixed with the thermodynamic equilibrium approach.

The following kinetic reaction for a solid (mineral,

colloidal) species, S is used by the code:

dS

dt¼

ApkpWpðOa � 1Þb if OX1;

�AdkdWd ð1� OcÞf if Oo1;

(ð9Þ

where subscripts p and d refer to precipitation and

dissolution, respectively, k denotes a kinetic constant in

mol=m2=s; A is the volumetric surface area expressed in

m2=m3; a; b; c and f are arbitrary power-constants, used

to fit the law to experimental data, O is the saturation

state (IAP over Ks or IAP times the formation constant,

K). For example, for zinc hydroxide, the variable is

defined as

O ¼ KZnðOHÞ2 ½Znþ½H2O2½Hþ�2: ð10Þ

W is a factor including reaction-catalyzing or -inhibiting

species and may be different for precipitation and

dissolution. Typically, W consists of one or more

activities raised to some power, the latter being a fitting

parameter, for example:

W ¼ ½Hþm½O2ðaqÞn: ð11Þ

As illustrated by Eq. (9), the precipitation law is not

necessarily the same as the dissolution law, even if it

concerns one single solid phase. The time-span used by

CHESS to react kinetically is the timestep used by the

transport model. Internally, however, CHESS may slice

the time-span if needed (i.e. minerals which dissolve

within the time span): this way chemical reaction

kinetics are entirely independent from the transport

model.

5. Applications

5.1. Uranium migration in the subsurface

Oxic or mildly reducing dissolution of uraninite, and

the subsequent migration of uranium, are relevant

processes for accidental radioactive pollution. The

hydrogeological and chemical reactions of such subsur-

face systems are closely coupled, and characterized by

strong redox fronts, kinetics and sorption on minerals

and colloids. This context is used by the following

example to illustrate several features of a reactive

transport model. This case has been presented as a

bench-mark study in a separate paper (De Windt et al.,

2003).

As illustrated schematically by Fig. 5, rain water

infiltrates in a subsurface aquifer and leaches a zone

enriched in uraninite (5% weight content). We assume

that rainwater is in thermodynamic equilibrium with

atmosphere gases (O2; CO2). The main minerals of the

aquifer are calcite and quartz with a minor fraction of

goethite (2.5%). The pH of the aquifer is about 9 and

the carbonate alkalinity is relatively low. The percola-

tion rate yields 2:5 m=yr for a porosity of 40%. The

15 m

2

Rain water

2UOenriched zone

subsurface aquifer

A

A’

Fig. 5. Schematic illustration of uranium migration case.


longitudinal and transverse dispersivities are, respec-

tively, 1 and 1:5 m; the temperature is 25C:We used METIS with a 2D grid and Dirichlet

conditions at the upper parts. The EQ3/6 thermody-

namic dataset of Wolery (1992) was selected, using only

a subset of redox couples and additional experimental

data for uranyl silicates (De Windt et al., 2003). Kinetic

parameters were set for schoepite precipitation ðk ¼10�11 mol=m2=s; Ap ¼ 10�3 m2=m3 (see Eq. (9)). The

intrinsic constants of Gabriel et al. (1998) were used to

model surface complexation of UO2þ2 and UO2OHþ on

both goethite (immobile) and hydrous ferric oxides

(colloidal, mobile).

Fig. 6A illustrates the simulated concentrations of

dissolved oxygen ðO2ðaqÞÞ; an indicator of redox

conditions, after 100 yr: UO2 acts as a reducing buffer

and the oxidative dissolution of UO2; which is assumed

to be in equilibrium in this instance, consumes the

oxygen of the groundwater. The concentration of the

dissolved uranium, U(aq), depends on precipitation of

secondary uranium minerals, sorption on the solid

phase, dispersion and dilution. Fig. 6B illustrates the

case where U(aq) is controlled by kinetic precipitation of

schoepite downstream from the enriched zone. If,

however, pure thermodynamic equilibrium is assumed

for all other uranium phases, uranophane is the most

stable mineral. Downstream migration is retarded due to

sorption on goethite surfaces: as a result, the overall

uranium solubility is reduced by 4 orders of magnitude

(Fig. 6C).

As is expected in most subsurface systems, migration

of uranium species will be hindered by sorption on

mineral surfaces, such as goethite. On the other hand,

colloidal transport may act as a vehicle for uranium and

for heavy or toxic metals in general, which tend to

strongly sorb on these particles. Modeling of uranium

migration including the colloidal effect is illustrated by

Fig. 6D. Here we assumed a flux of 50 mg=l hydrousferric oxides (HFO) colloids migrating downwards from

the upper right corner. All other conditions are equal to

those of Fig. 6C. The colloids only moderately enhance

the solubility but enable a wide-spread dissemination of

(A) (B)

(C) (D)

Fig. 6. Concentration profiles ðmmol=lÞ calculated with HYTEC after 100 yr: (A) dissolved oxygen, (B) mobile uranium controlled by

schoepite kinetic precipitation, (C) mobile uranium controlled by uranophane precipitation and sorption on goethite, (D) same as (C)

with asymmetric transport by colloidal hydrous ferric oxides.


uranium. Colloidal transport in this case is significant

only when uranium is strongly sorbed on HFO—

retention by mineral goethite dominates otherwise.

5.2. Sealing at a clay–concrete interface

The second example application of the code forms

part of on-going research on the long-term stability of

cement and clay barriers of a radioactive waste

repository (Lagneau, 2000). Concrete was simulated by

a typical cement including mainly portlandite, CSH 1.8

(a calcium-silicate hydrate with Ca/Si ratio of 1.8) and

small concentrations of ettringite and calcium mono-

sulfo-aluminate. The initial pore fluid is representative

of a fresh concrete, with a high pH and significant levels

of Na and K (Bateman et al., 1999). The MX-80 clay is a

montmorillonite-rich bentonite from Wyoming (USA),

often proposed as a backfill material. In this study, it has

been simulated by its major components (Lajudie et al.,

1995): mainly montmorillonite-Na and quartz, plus

some K-feldspar, calcite and kaolinite. The initial pore

fluid composition is fixed by the equilibrium with MX-

80 minerals.

An artificial, non-reactive tracer is injected at constant

concentration at the beginning of the cement zone,

in order to follow the transport behavior within

the evolving barrier system. At the beginning of

the simulation, the cement has a porosity of 0.2

and an effective diffusion coefficient of 2:2�10�11 m2=s: For the clay, these values are 0.3 and 5:1�10�11 m2=s; respectively. The system is purely diffusive,

the diffusion coefficient varies with the porosity accord-

ing to Eq. (5), m ¼ 1:5 and oc ¼ 0: The chemistry was

considered at pure thermodynamic equilibrium and ion

exchange was not taken into account. The database

(EQ3/6 thermodynamic dataset) was restricted to a

subset of selected minerals, including scolecite, K-

feldspar, albite and talc, for the degradation of the clay

minerals, and enriched with experimental data for the

cement degradation, notably Ca-depleted CSH (CSH 1.1

and CSH 0.8).

The evolution of the clay–concrete interface is

principally a degradation of the clay minerals due to

the alkaline attack. On the cement side (i.e. the first

50 cm of the system), portlandite is dissolved due to a

lowering pH, after diffusion of the clay–water solution

in the cement. With the dissolution of portlandite, the

pH decreases and CSH 1.8 precipitates. Closer to the

interface, the CSH 1.8 eventually dissolves entirely to be

replaced by the CSH 1.1 (Fig. 7, 1000 yr). A massive

CSH 1.1 precipitation at the interface, however, leads to

a significant reduction in porosity (Fig. 7, bottom

graph). This is expected: sealing of clayey material due

to the intrusion of cementous solutions and precipitation

of secondary phases has been demonstrated experimen-

tally (Adler et al., 2001; Roussel, 2001).

The decrease in porosity also reduces the diffusion-

driven migration of the tracer (Fig. 8). After 50 yr;the tracer profile has reached a quasi-stationary state.

The apparent discontinuity in the profile is due to the

difference between initial diffusion coefficients in the

0

2.5

5

7.5

10

12.5

15

c (m

ol/l

poro

us m

ed.)

portlanditeCSH 1.8CSH 1.1

Kfeldsparcalcite

montmorNaquartz

0

2.5

5

7.5

10

12.5

c (m

ol/l

poro

us m

ed.)

portlanditeCSH 1.8CSH 1.1

K feldsparcalcite

montmor Naquartz

0

0.1

0.2

0.3

0.4

0.5

0.6

30 40 50 60 70

poro

sity

distance (cm)

t=50 yt=500 y

t=1000 yt=2500 y

Fig. 7. Upper two figures: evolution of mineral concentrations

near cement–clay interface at 1000 and 2500 yr (zoom at

interface at 50 cm; concentrations in mol per liter of porous

medium in order to eliminate all porosity effects on mineral

concentration). Lower figure: porosity profiles at different

times.

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200

c (u

mol

/l)

distance (cm)

t=50 yt=500 y

t=1000 yt=2500 y

Fig. 8. Tracer profiles near interface (at 50 cm) for 50, 500,

1000 and 2500 yr (concentration is imposed at left limit of

system).


cement and clay zones. After 500 yr; nearly total sealing

at the interface is obtained, illustrated by a steep

gradient of the tracer concentration. Despite the

gradient, which increases diffusive migration, the

effective diffusion coefficient approaches zero, thus

effectively stopping tracer leakage from the cement zone.

Similarly, geochemical reactive species will be affected

also by the porosity drop as well as the dissolution fronts

in the clay. For example, an interesting side-effect is the

rise of pH in the cement after sealing, thus re-generating

CSH 1.8 from CSH 1.1 (Fig. 7, 2500 yrÞ: The direction

of reaction is inverted from what would be predicted in a

simulation without the sealing effect.

6. Convergence and performance

Operator-split reactive transport models based on the

iterative sequential approach are notorious for failing to

converge, even for moderately complex systems. Sophis-

ticated algorithms for speeding up convergence, e.g.

multi-dimensional minimization methods (downhill sim-

plex method, Powell’s method, the conjugate gradient

method or simulated annealing methods) are not well

suited due to the high degree of interdependence of the

variables. The simple iterative procedure betweenL and

R cannot be amended. However, several recovery

techniques and notably the variable time-step option

significantly increase the probability of success.

Oscillations often occur at the zero total-concentra-

tion level (which is, for reasons of convenience in

geochemical solvers, a small value such as 10�25 molal)

and related to machine-precision and round-off errors.

Therefore, a robust geochemical module which takes

care of these numerical issues significantly helps to

improve the global convergence of the model. As

illustrated by Fig. 2, HYTEC recovers from diverging

or slowly converging situations by reducing the current

timestep. Inversely, the model increases the timestep

when possible. Hence HYTEC adapts the timestep to

the numerical stiffness of the problem.

Assuming that the programs are highly optimized,

computer power still remains a limiting factor in reactive

transport modeling. As always, there is a trade-off

between computing time and memory resources but

reactive transport models are intrinsically computation

intensive. Profiling case studies have been performed in

order to locate the time-consuming parts of the model.

For a simple case study (dissolution front of quartz in a

sand-column system), B67% of the total CPU time is

spent on geochemical calculations (this percentage is

slightly increased to B69%; if dissolution kinetics are

used). For more complex geochemical systems, this

percentage tends to increase to 87% (diffusive transport

with a sequence of precipitation/dissolution fronts), up

to B99% for very complex systems with mixed redox/

precipitation/cation-exchange reactions. Similar values

have been reported in the literature, ranging from 85%

to 95% for relatively simple systems (Yeh and Tripathi,

1991) up to 99.9% for more complex calculations (Liu

and Narashimhan, 1989).

Profiling of HYTEC shows that 75–85% of the CPU

time is used to evaluate logarithmic functions, which

cannot be improved programmatically. Nevertheless,

considerable reduction of the calculation time can be

achieved by using parallel computing techniques.

According to a preliminary study, one can expect a gain

in overall computing time of n—minus the time needed

for communication— for very complex systems, n being

the number of processors available. Parallelization of

HYTEC is currently under development.

7. Summary and conclusion

In order to provide insight into fundamental issues of

mobility and retention processes, a mechanistic hydro-

geological transport model is required which takes into

account the bio-geochemical reactions. Progressively

evolving to completely integrated tools, reactive trans-

port models are also increasingly complex. This paper

presents HYTEC, a reactive transport model which

surmounts part of the complexity by adopting a module-

oriented model development. The central part of the

modeling platform is the geochemical module CHESS,

developed as a library and optimized for coupling

purposes. CHESS is currently used as a geochemical

engine for several coupled transport models.

The module-oriented structure of the code has several

major advantages such as increased efficiency in devel-

opment, maintenance and testing. Different commu-

nication protocols are used: HYTEC uses MPI which

appears to be the most efficient, and also provides the

structure for massively parallel computing. Another

advantage is the possibility to select, at runtime,

different transport models (written according to either

finite elements, finite difference or finite volumes).

HYTEC fulfills many of the modeling-desiderata

currently brought forward as critical for risk assessment

in general and radioactive waste performance assess-

ment in general. It handles hundreds of species

simultaneously in complex heterogeneous systems,

accounts for organic and inorganic colloidal transport,

and deals with steep redox-precipitation- and dissolution

fronts. HYTEC also includes the effects of variable

porosity, due to geochemical (clogging) reactions on the

hydrodynamic regime. The first application illustrates

several of these aspects in the example of uranium

migration in a sub-surface system. The second applica-

tion shows how the model can be used to simulate

phenomena at time scales inaccessible by any direct

experimental approach.


Micro-biologically catalyzing reactions in subsurface

systems are recognized to affect the fate of organic and

inorganic contaminants (Rittmann and van Briesen,

1996). Microbially governed (kinetic) reactions and their

link to the redox state of important species are currently

developed as an extension of the module CHESS.

Because of the high CPU needs to solve sophisticated

systems, HYTEC will be further improved by paralle-

lization of the code, based on MPI.

For critical applications such as performance assess-

ment of radioactive waste storage, a thorough verifica-

tion of the conceptual and numerical parts of the model

becomes a key issue. Development of HYTEC forms

part of an ongoing national research program (PGT,

P #ole G!eochimie-Transport) where confidence building,

verification and bench-mark studies are central topics.

Finally, we continue to put efforts on the vital subject of

thermodynamic database development, notably with

respect to adsorption on inorganic and organic phases.

Acknowledgements

The authors acknowledge the Electricit!e de France

(EDF), the Commissariat "a l’ !Energie Atomique (CEA)

and the Institut de Protection et de S #uret!e Nucl!eaire

(IPSN, now IRSN) who co-sponsored the development

of HYTEC and related scientific simulation tools. We

also thank C. Steefel for his insightful review and

constructive comments.

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Module-oriented modeling of reactive transport with HYTEC

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