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The reactive transport benchmark proposed by GdR MoMaS. Presentation and first results

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The reactive transport benchmark proposed by GdR MoMaS: Presentation and first results

• J. Carrayrou*,1 and V. Lagneau2

* corresponding autor 1Institut de Mécanique des Fluides et des Solides, UMR ULP-CNRS 7507

2 rue Boussingault – 67000 Strasbourg, France. carrayro@imfs.u-strasbg.fr tel: (+33) 03 90 242 916 – fax: (+33) 03 88 614 300

2Centre de géosciences Ecole des Mines de Paris

35 rue Saint Honoré – 77305 Fontainebleau cedex, France Abstract: We present here the actual context of reactive transport modelling and the major numerical challenges. GdR MoMaS proposes a benchmark on reactive transport. We present this benchmark and some results obtained on it by two reactive transport codes HYTEC and SPECY. Keywords: Reactive transport, benchmark, numerical model

The activities of the research group MoMaS (Mathematical Modelling and Numerical Simulation for Nuclear Waste Management Problems - 2002-2007) are centred around scientific computing, design of new numerical schemes and mathematical modelling (upscaling, homogenization, sensitivity studies, inverse problems, ...). This GdR proposes to the scientific community a benchmark on reactive transport modelling for year 2007. We will present the subject of this benchmark and its specificities; the expected results and some results obtained by two separate organisations. The objectives of this benchmark are to help the development or the implementation of new numerical methods for reactive transport specific problems, give some answers about strategic questions on this subject and propose an intercomparison tool for reactive transport modellers. The strategic questions deal mainly with operator splitting (OS) option. Since 1989 (Yeh and Tripathi, 1989), OS are the most widely used approaches for reactive transport modelling (van der Lee and De Windt, 2001). The choice of the most adapted OS method (iterative or not) is still open. Moreover, even if OS methods are easier to program and faster to compute, they introduce splitting errors (Barry et al., 1996;Carrayrou et al., 2004;Valocchi and Malmstead, 1992). For this reason, some recent works deal with global approach. This benchmark is intended to help compare and understand the advantages of each approach. The proposed exercises consist on a three level problem, ranked by increasing chemical difficulties: “easy” deals with instantaneous equilibrium; “medium” with equilibrium and kinetic; and “hard” with equilibrium with precipitation and

• kinetic. Hydrodynamic is proposed for 1D and/or 2D flow in a heterogeneous media.

We will first present a short overview of the methods used for reactive transport modelling and of the numerical difficulties associated with it. In a second part, we will describe briefly the benchmark proposed and we will underline the numerical difficulties expected. We will then present some of the results obtained by two independent research groups using different resolution methods. Numerical methods and difficulties. Description The reactive transport equation for porous media is written, under the instantaneous equilibrium assumption (Rubin, 1983;Steefel and McQuarrie, 1996):

( ) ( ) (

j j j

M F M

T T uT D T t ω ω ∂ + = −∇ + ∇ ⋅ ∇ ∂

)

j

M

(1) where is the total mobile concentration for each component and is the total immobile concentration; ω is the porosity of the media the velocity of the flow and D the dispersion.

j

M

T

j

F

T u We consider a set of chemical reactions among several species. After relabelling, we assume that each reaction may be written so that each product derives from a unique set of components. We also distinguish between mobile (in solution) and immobile (on the solid matrix) species. Reactions among mobile species are written as

SLIDE 3

, 1

1,

NxM i j j i j

a X C i NcM

=

=

∑

• (2)

where the , are the (mobile) components, and the are the secondary species. are stoechiometric coefficients. Reactions between mobile and immobile species are written as (we assume a single immobile component S , with stoechiometric coefficients ):

j

X 1, , j Nx =

• M

i i

s CS

=

= ⋅

∑

1, , j NxM =

• s

i i

,

i

C

, i j

a

, i j

as

, , 1

1,

NxM i j j i s i j

as X as S CS i NcS

=

+ =

∑

• (3)

Each chemical reaction gives rise to a mass action law, and we have a conservation law for each component. Conservation laws used for transport equations are:

, 1

j

NcM M j i j i

T X a C

=

= + ∑ and T a (4)

, 1

j

NcS F i j i

S

M

T = and (5)

, 1

S

NcS F i i

T S as CS

=

= + ⋅

∑

For each component, conservation laws are:

, , 1 1 NcM NcS j j i j i i j i i

T X a C as CS

= =

= + ⋅ + ⋅

∑ ∑

and (6)

, 1 NcS i s i i

TS S as CS

=

= + ⋅

∑

For each aqueous and fixed species, the mass action laws are:

i

C

i

CS

,

1

i j

NxM a i i j j

C K X

=

= ⋅ ∏ and (7)

,

1

i j i s

NxM as as i i j j

CS Ks X S

=

= ⋅ ⋅

∏

Overview of the methods Combining the transport equations and the chemical laws leads to a non linear differential algebraic system. There are two families of methods (Yeh and Tripathi, 1989) for solving this system. The global approach (Miller and Benson, 1983;Shen and Nikolaidis, 1997) requires the discretisation and the resolution of the entire system. The operator-splitting (OS) one requires a separated resolution of both transport and chemistry operator (Appelo et al., 1998;Carrayrou, 2001;van der Lee et al., 2003). After the funding paper of (Yeh and Tripathi, 1989), OS methods have been widely preferred. Once the approach is chosen, one should select a way to discretise the resulting equation(s) in time and space; to linearise the algebraic system(s) coming from chemistry; and to solve the linearised systems. Several schemes for time discretisation have been tested: explicit (Cederberg et al., 1985) implicit or Cranck-Nicholson (van der Lee et al., 2003); adaptive time step based on heuristic (van der Lee et al., 2003) or predictor-corrector (Belfort et al., 2007) approach. Various numerical schemes for space discretisation haven been adapted to reactive transport: finite volume(van der Lee et al., 2003), finite element, discontinuous and mixed hybrid finite element (Carrayrou et al., 2003), particle tracking (Ginn, 2001), ELLAM (Younes et al., 2006), multilevel wavelets (Cruz et al., 2002). The most used method for linearization of the chemical algebraic system is the Newton-Raphson one (for global approach and for OS). Nevertheless, it has been reported and shown (Brassard and Bodurtha, 2000;Carrayrou et al., 2002;van der Lee, 1997) that this method is subject to non convergence problem. Many methods are proposed to overpass this problem (Carrayrou et al., 2002). The methods used for solving the linearized system produced by reactive transport and their influence have not been studied in the literature (to the best of our knowledge). Numerical difficulties Regarding the reported specificity of reactive transport modelling, we can list a set of important numerical key-points:

• Spatial variations.
• Time variation.
• Numerical diffusion.
• Non physical oscillation.
• Convergence problems.
• Association kinetic-equilibrium

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Spatial variations: reactive transport phenomena induce often the apparition of some sharp moving fronts separated by very large quasi homogeneous zones. This phenomenon has been widely experimentally reported (Lefevre et al., 1993). An accurate description of the sharp fronts requires a fine space discretisation whereas an efficient description of homogeneous zones needs a large mesh size. Some numerical solutions haven been proposed to this problem, by using ELLAM methods (Younes et al., 2006), wavelets methods (Cruz et al., 2002) or by adapting the OS scheme (Carrayrou et al., 2003). Other numerical methods can be used to get a solution to this problem. As an example, the moving mesh techniques are actually poorly used but we think that they can give an efficient solution. By refining the mesh around the sharp front and following them during their displacement; by de-refining the mesh after the passage of the front; a moving mesh method can combine a precise description of the sharp fronts and a low computing cost. Multilevel wavelets (Cruz et al., 2002) presents another solution to this problem by using a high level of wavelet around the front and the basis wavelet for homogeneous zones. Time variation: reactive transport phenomena are often characterized by the succession of some very fast evolutions and long steady-state periods. One needs very small time steps to get a nice description of the fast evolution but (very) large time steps are useful to reduce the computing effort during the simulation of the long steady state periods. Some efficient solutions are proposed by authors using adaptive time steps (Belfort et al., 2007;van der Lee et al., 2003). The numerical diffusion problem is well known for non reactive transport. Due to some specific reaction such as precipitation and dissolution, this problem is amplified. Indeed, the reaction of precipitation occurs if the product of the reactant concentrations is greater than the solubility product. Numerical diffusion induces a diminution of the

• concentrations. A consequence is that some precipitation reactions never occur if numerical diffusion is too important.

Some ways to reduce the numerical diffusion are proposed in Carrayrou et al. by using discontinuous finite elements, by Younes et al. (2006) using ELLAM or by Cruz et al. (2002) using multi level wavelets. It is well known that some non physical oscillation can occur during the simulation of non reactive transport (Hoteit et al., 2002) and the methods to avoid them are well known: respect of the Courant-Friedlisch-Levy (CFL) criterion for

• example. Unfortunately, the interactions between transport and chemistry and the non linearities introduced by chemistry

leads to some oscillation even if these kind of criterion are respected (Carrayrou, 2001). The problem of non convergence of the Newton Raphson method for solving the non linear chemical operator has been reported and some solution have been given by van der Lee (1997), Brassard and Bodurtha (2000) or Carrayrou et al. (2002). One of the most used OS scheme is the standard iterative one (van der Lee et al., 2003;Yeh and Tripathi, 1991). This scheme requires Picard iteration between the transport and chemistry operators. It is well known that these iterations do not converge if the time step is too large, that is if the initial guess is too far away for the final solution. Actually, the selected solution is to reduce the time step if the Picard iterations do not converge after a prescribed number of iterations. To our knowledge, none of the authors presenting a global approach report some convergence problem but we think that these problems occurs in the case of global approach too. But in the case of global approach, the chemical non linearities are associated in the same matrix to the linear transport operator. In this case we cannot predict if the addition of the transport will reduce the non convergence problem or if it will increase it. Chemical systems can be described using the instantaneous equilibrium assumption or a kinetic description. Historically (Cederberg et al., 1985;Rubin and James, 1973;Yeh and Tripathi, 1989), reactive transport in porous media has been described using instantaneous equilibrium. The first reason of this choice is a scientific one, the second is material. Because the flow is porous media is usually slow (1 meter per day) and the chemical reactions in water are very fast (10-9 s for acid – base reactions), the local equilibrium assumption is valid (Valocchi, 1985). If the reactions become slower, this assumption is not valid any more and a kinetic description should be used. A review of the classification and the condition for sufficiently slow or fast reaction can be found in (Rubin, 1983). Practically, data bases giving equilibrium constant for each reaction are very detailed whereas it is very hard to obtain the kinetic rate constants. The recent developments of reactive transport modelling (van der Lee and De Windt, 2001) make the use of kinetic chemistry more frequent. Many reactive transport codes propose a description of the chemical system using a mixed kinetic and equilibrium description (Appelo et al., 1998;Carrayrou, 2001;Gerard et al., 1998;van der Lee et al., 2003). Equilibrium description is used for fast reaction and kinetic description is used for fast reaction. Nevertheless, the numerical part of this problem has not been extensively studded whereas it is well known that kinetic rate law may by stiff (Sandu et al., 1997) and generate very specific difficulties. The 1D easy level benchmark and the expected difficulties By reading this synthesis of the numerical difficulties and the wide variety of proposed solution, one can easily understand that a global comparison of all the possibilities cannot be done by a single research group. The reactive

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transport benchmark of GdR MoMaS (http://www.gdrmomas.org/ex_qualifications.html#geochimie) has been developed to gather in one quite simple reactive transport problem all the numerical difficulties reported previously. By this way, we hope to motivate many researchers to test, develop and adapt numerous methods for reactive transport modelling and to share their results on the same intercomparison exercise. The flow domain is heterogeneous with hydrological and chemical heterogeneities. This is shown in Figure 1. The medium A (majority of the domain) has a low porosity and a low reactivity (concentration of immobile component). The medium B (small barrier in the middle) has a high porosity and reactivity. This will induce some sharp reaction fronts as

• pposed to very large homogeneous zones.

Figure 1: Scheme of the 1D problem The chemistry for easy test case is pure equilibrium with a few numbers of components (5) and secondary species (7) as shown in Table 1. Chemical non linearities are important because stoechiometric coefficients varies from -4 to 4 and equilibrium constants from 10-12 to 1035. Table 1 : Equilibrium for easy test case

1

X

2

X

3

X

4

X S K

1

C

• 1

1.00E-12

2

C 1 1 1

3

C

• 1

1 1

4

C

• 4

1 3 0.1

5

C 4 3 1 1.00E+35

1

CS 3 1 1 1.00E+6

2

CS

• 3

1 2 1.00E-01 Total (m.L-3)

1

T

2

T

3

T

4

T TS Initial for medium A

• 2

2 1 Initial for medium B

• 2

2 10 Injection [ ] 0,5000 t ∈ Imposed total concentration at inflow boundary Inflow for 1D 0.3 0.3 0.3 Zone 1 for 2D 0.3 0.3 0.3 Zone 2 for 2D 0.3 0.3 0.3 Leaching [ ] 5000,... t ∈ Imposed total concentration at inflow boundary Inflow for 1D

• 2

2 Zone 1 for 2D

• 2

2 Zone 2 for 2D

• 2

2 The modelling time is quite long, 5 000 time units, to impose the modelling of long steady state periods.

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An estimation of the computing time is proposed for this benchmark: the time unit is defined as time computing time needed to run the product of two (1000 x 1000) real matrix. Numerical model tested HYTEC First results are given by HYTEC (V Lagneau) on its standard version. HYTEC is a distributed numerical simulation code for transport of chemically reactive elements, heat and solute transport in porous media (van der Lee et al., 2003). It is based on the geochemical speciation code CHESS and the flow and transport model R2D2. The chemical equations are solved by CHES, a versatile geochemical speciation code, and consequently allows for the simulation of solute complexation, solids precipitation/dissolution, several models of sorption (surface complexation, ion exchange), impact of temperature, isotopic fractionation and radioactive decay. All reactions can be described using the local equilibrium hypothesis, kinetic control or mixed equilibrium/kinetics. No specific limitations are fixed on the number

• f species, which derive automatically from the choice of the thermodynamic database.

The hydrodynamic module R2D2 (Lagneau, 2003) simulates flow in saturated or unsaturated media as well as heat and multicomponent solute transport. The code uses a finite volume scheme based on a Voronoi (nearest-neighbour) spatial discretisation in 1, 2 or 3D. The discretisation scheme can be chosen between centred (default for the dispersion) and upstream (default for the advection). The time discretisation is one step and can be set from fully explicit to implicit, with default Crank-Nicholson. The transport-chemistry coupling is based on the sequential iterative approach (Yeh and Tripathi, 1991). The initial guess at each new timestep is carefully devised, on a two-step basis, in an attempt to minimise the number of coupling

• iterations. Convergence is tested after each chemical speciation resolution, with a criterion on the evolution between to

iterations of the fixed fractions for each component. The iterative loop includes the porosity update, which allows for a proper handling of the feedback of chemistry on transport. The flow and heat equations are not included in the iterative loop (Figure 2). Figure 2: schematic organisation of the coupled reactive transport code HYTEC. Transport Chemical speciation Porosity update Reactant transport Mineral volume Diffusion coeff. Velocity field Flow End of timestep Timestep optimisation Permeability update HYTEC includes an automatic optimisation of the time step: the time step increases (decreases) if the number of coupling iteration is less (greater) than a user defined value (default 20 iterations), within certain limits (total duration and sampling, Courant-Friedlisch-Levy criterion if applicable). This enables to increase the time steps, while limiting the stiffness of the coupling. SPECY Second results are given by SPECY (J. Carrayrou). This code uses a non iterative OS scheme. In this OS scheme, the transport of the mobile total concentration of each component is first solved.

j

n M

T

SLIDE 7

( ) ( ) (

j j

M M

T uT D T t ω ω ∂ = −∇ + ∇ ⋅ ∇ ∂

)

j

M

(8) By adding the fixed total concentration in the cell at previous time step , we get the total concentration of each component in each cell of the mesh.

1

j

n F

T

− 1

j j

n n n j M F

T T T

−

= + (9) The chemistry is then solved for each cell independently. It gives the concentration of each species and the new repartition between mobile and fixed total concentration. By eliminating the iterations between transport and chemistry, we increase the OS errors but the computing effort needed by one time step is lower. This allows smaller mesh size or time steps and avoids non-convergence problems. Space discretisation is discontinuous finite element time explicit for advection and mixed hybrid finite element, time implicit for dispersion (Carrayrou et al., 2003). Discontinuous finite elements are very efficient for modelling sharp fronts but needs the respect of CLF criterion. Small time steps should be used. Mixed hybrid finite element gives a good solution

• f diffusion problems without non physical oscillations.

The chemistry is solved separately, using a combined algorithm (Carrayrou et al., 2002). To reduce the risk of non convergence, a Chemically Allowed Interval (CAI) is defined. The research of the solution will be conducted into this

• CAI. Because the Newton-Raphson method is not efficient far from the solution, the Positive Continuous Fractions

method is used as pre conditioner until a relative error on the mass balance of 95 %. Once the research is sufficiently close to the solution, we use the Newton-Raphson method. Comparison and analysis Results shown here are obtained by HYTEC with 420 nodes and a CPU time of 294.24 on a 2-processor computer (transport and chemistry are solves sequentially on separate processors, without parallelisation gain); and by SPECY with 220 nodes and a CPU time of 402.23 on a one processor computer. Figure 3 shows the concentration profiles at time 10 for a reactive aqueous species C4. A small peak is detected by both codes around coordinate 0.1; but HYTEC provide a higher peak than SPECY. A mesh refinement analysis proves that the higher peak is the most accurate solution. We think that the better result given by HYTEC comes here from the iterative OS scheme.

0,0 0,5 1,0 1,5 1E-23 1E-22 1E-21 1E-20 1E-19 1E-18 1E-17 1E-16 1E-15 1E-14 1E-13 1E-12 1E-11

Concentration Space C4 - HYTEC C4 - SPECY

1000 2000 3000 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08

concentraion Time C5 HYTEC C5 SPECY

Figure 3 : Concentration profile à time 10 for species C4 given by HYTEC and SPECY Figure 4: Elution curves for species C5 at z = 2.1 given by HYTEC and SPECY Figure 4 presents the elution curves for species C5 given by both codes. The oscillations presented by these elution curves are very important whereas both codes use numerical methods to avoid non physical oscillations: HYTEC is full implicit and SPECY uses discontinuous and mixed hybrid finite elements. In this case, it seems that these oscillations come from chemical non linearities. As it can be seen in Table 1, species C5 is one of the most non linear species. Theses

SLIDE 8

• scillations are not today explicated. We don’t know if they have a physical signification or if they are related to a

numerical problem. Often are these kind of oscillations related to the mesh size but the comparison between SPECY (220 nodes) and HYTEC (420) nodes does not confirm this explanation. This question is still open. Figure 5 shows the elution curves for two components. X1 is a tracer and both codes provide the same solution. X2 is

• reactive. In this case both codes provide similar results. Whereas it uses a smaller mesh size, HYTEC needs less

computing time than SPECY to get the same result. By using an adaptive time step strategy, HYTEC reduces the number

• f time steps and is then faster. Because discontinuous finite element are time explicit and must respect the CFL criteria,

SPECY cannot use an adaptive time step.

50 100 150 200 250 5000 5050 5100 5150 5200 5250 5300 1 2 3 4 5 6

Concentration Time X1 (tracer) - HYTEC X2 - HYTEC X1 (tracer) - SPECY X2 - SPECY

Figure 5: Elution curves for components X1 and X2 at z = 2.1 given by HYTEC and SPECY Conclusion The numerical problems associated to reactive transport modelling are various. The optimal solution is not unique. We show here that discontinuous finite element reduces the non physical oscillations but are not compatible with time step adaptation. These first results clearly show that this benchmark is discriminative versus the methods used and can give helpful answers for the development of future reactive transport codes. Acknowledgements We greatly thank GdR MoMaS for supporting this work. References Appelo, C. A. J., E. Verweij and H. Schafer, A hydrogeochemical transport model for an oxidation experiment with pyrite/calcite/exchangers/organic matter containing sand, App. Geochem., 13(2), 257-268, 1998. Barry, D. A., C. T. Miller and P. J. Culligan-Hensley, Temporal discretisation errors in non-iterative split-operator approaches to solving chemical reaction/groundwater transport models, J. Contam. Hydrol., 22(1-2), 1-17, 1996. Belfort, B., J. Carrayrou and F. Lehmann, Implementation of Richardson extrapolation in an efficient adaptive time stepping method: applications to reactive transport and unsaturated flow in porous media, Transport in Porous Media, in press, -, 2007. Brassard, P. and P. Bodurtha, A feasible set for chemical speciation problems, Computers & Geosciences, 26(3), 277-291, 2000. Carrayrou, J., Modélisation du transport de solutés réactifs en milieu poreux saturé, Ph.D. Université Louis Pasteur Strasbourg I, 2001. Carrayrou, J., R. Mosé and P. Behra, New efficient algorithm for solving thermodynamic chemistry, AIChE J., 48(4), 894-904, 2002. Carrayrou, J., R. Mosé and P. Behra, Modelling reactive transport in porous media: iterative scheme and combination of discontinuous and mixed-hybrid finite elements, C. R. Acad. Sci. , Ser. II Univers, 331(3), 211-216, 2003.

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van der Lee, J., Modélisation du comportement géochimique et du transport des radionucléides en présence de colloïdes., Ph.D. Ecole Nationale Supérieure des Mines de Paris, 1997. van der Lee, J. and L. De Windt, Present state and future directions of modeling of geochemistry in hydrogeological systems, J. Contam. Hydrol., 47(24), 265-282, 2001. van der Lee, J., L. De Windt, V. Lagneau and P. Goblet, Module-oriented modeling of reactive transport with HYTEC, Computers & Geosciences, 29(3), 265-275, 2003. Yeh, G. T. and V. S. Tripathi, A model for simulating transport of reactive multispecies components: model development and demonstration, Water Resour. Res., 27(12), 3075-3094, 1991. Yeh, G. T. and V. S. Tripathi, A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components., Water Resour. Res., 25, 93-108, 1989. Younes, A., P. Ackerer and F. Lehmann, A new efficient Eulerian-Lagrangian localized adjoint method for solving the advection-dispersion equation on unstructured meshes, Adv. Water Res., 29(7), 1056-1074, 2006.

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