PROOST: Object oriented approach to multiphase reactive transport. modeling in porous media.

Transcription

1 1 2 3 PROOST: Objet oriented approah to multiphase reative transport modeling in porous media. P. Gamazo a, L.J. Slooten b,e, J. Carrera b,e, M.W. Saaltink,e, S. Bea d and J. Soler a a Departamento del Agua, CENUR LINO, Universidad de la Repúblia, Gral. Rivera 135, 5 Salto, Uruguay, gamazo@unorte.edu.uy, jsoler@unorte.edu.uy b Institute of Environmental Assessment and Water Researh (IDAEA), CSIC, /Lluis Solè Sabarìs, s/n, 828 Barelona, Spain, luitjan.slooten@up.edu, jesus.arrera@idaea.si.es Dept. Geotehnial Engineering and Geosienes, Universitat Politenia de Catalunya, UPC, /Jordi Girona 1 3, 834 Barelona, Spain, maarten.saaltink@up.edu d CONICET IHLLA Repúblia de Italia 78, 73, Azul, Buenos Aires, Argentina, sabea@faa.unien.edu.ar e Assoiated Unit: Hydrogeology Group (UPC CSIC) Abstrat Reative transport modelling involves solving several nonlinear oupled phenomena, among them, the flow of fluid phases, the transport of hemial speies and energy, and hemial reations.. There are different ways to aount this oupling that might be more or less suitable depending on the nature of the problem to be solved. In this paper we aknowledge the importane of flexibility on reative transport odes and how objet oriented programming an failitate this feature. We present PROOST, an objet oriented ode that allows solving reative transport problems onsidering different oupling approahes. The ode main lasses and their interations are presented. PROOST performane is illustrated by the resolution of a multiphase reative transport problem where geohemistry affets hydrodynami proesses Introdution Reative transport models are tools that help to understand the hydrauli and hemial behavior of natural and artifiial porous media. It has been used to solve a broad range of 1

2 problems like groundwater remediation (Loomer et al. 21), nulear waste disposal (MaQuarrie and Mayer 25) and CO2 sequestration (Zhang et al. 212) among others, from miro sale (Trebotih et al. 214) to field sale problems (Sassen et al. 212). Modeling reative transport in porous media involves simulating several oupled phenomena: phase flow, solute transport, and reations. It may also involve multiphase flow, heat transport and porous media deformation (Steefel et al. 214,). These phenomena may be omplex to model individually, and modeling together brings on new diffiulties assoiated with oupled effets (Lihtner 1996). Whih oupled effets have to be onsidered and the optimal solution strategy for the oupled equations depend on the nature of the problem to be solved and may vary signifiantly from ase to ase (Zhang et al. 212). The ideal reative transport ode would have to use an aurate, robust and effiient numerial approah. However, it is diffiult to obtain these goals with a single numerial approah. Therefore, onessions have to be made and different oupling alternatives have to be hosen at different levels. Numerial auray is generally preferred on other issues when solving modeling researh appliations. On the other hand, when solving field sale problems, effiieny and robustness have priority while auray remains within the bounds of the unertainty assoiated with model parameters (Yeh et al. 212). Two big family of methods were addressed to aount for the oupling between solute transport and hemial reation proesses: (1) the Operator Splitting (or Sequential Iterative (or NON iterative) Approah, and (2) the Global Impliit or Diret Substitution Approah (Saaltink et al. 21). As regards the first one (i.e., the sequential methods), whether iterative (SIA) or not (SNIA) adopt operator splitting tehniques that effetively deouple omponent transport equations. As regards the last one, diret substitution approahes (DSA) solve both transport and hemial reations simultaneously. A number of authors have studied the numerial performane of these methods (Steefel and MaQuarrie, 1996), and they onlude that in spite of the fat that the DSA is more aurate and robust, there are ases where the 2

3 SIA is more onvenient from an effiieny auray point of view. In addition, SNIA may be appropriated for senario with Courant number smaller than 1 (Xu et al. 212). Some reative transport odes are able to work with both of these approahes (CRUNCHFLOW, Steefel 29; DUMUX, Flemish et al. 211; HYDROGEOCHEM, Yeh et al, 21; PFLOTRAN, Lihtner et al. 213; RETRASOCODEBRIGHT et al. Saaltink et al. 24.), while others use the fully impliit approah (NUFT, Hao et al. 212; MIN3P, Mayer et al. 212), or different variants of operator splitting tehniques (CORE, Samper et al 29; HYDRUS PHREEQC (HP1), Jaques et al. 211; HYTEC, Lagneau and Van Der Lee 21; IPARS, Wheeler et al. 212; OPENGEOSYS, Li et al. 214; ORCHESTRA, Meeussen 23; PHAST, Parkhurst et al. 21; PHREEQC, Parkhurst and Appelo 213; PHT3D, Prommer and Post 21; RT3D, Johnson and Truex 26; STOMP, White and MGrail 25; TOUGHREACT, Xu et al. 211). On a more omplex level is the oupling between phase onservation and reative solute transport. Most reative transport odes deouple phase onservation (i.e. flow equation) from reative transport alulations (RT3D, MIN3P, PFLOTRAN, PHAST, RETRASOCODEBRIGHT, HYTEC, TOUGHREACT). This approah is onvenient in most ases, but a numerially oupled solution will generally be more suitable when the phenomena involved are highly physially oupled. One example of this ould be found in problems related with the CO 2 sequestration in brine aquifers, whih has prompted the development of odes that solve oupled multiphase flow and reative transport (Fan et al. 212) and even mehanial deformation (Zhang et al. 212). Likewise, Wissmeier and Barry (28) showed that the onsumption of water due to hydrated mineral preipitation an have impats on flow and solute transport for unsaturated flow problems. These impats an be even more important if gas transport is also onsidered beause water ativity, whih ontrols vapor pressure, is affeted by apillary and osmoti effets. Moreover, ertain mineral paragenesis an fix water ativity (produing an invariant point), ausing the geohemistry to ontrol vapor pressure, whih is the key variable for vapor flow (Risaher and 3

4 Clement, 21). In suh ases, deoupling is not appropriated. Formulations that are able to represent these effets are omplex to implement sine they should onsider all oupled phenomenon and a variable number of omponents in spae and time. While most reative transport odes onsider a single tehni for the resolution of the partial differential equation some odes an adopt more than one. In Table 1 the supported disretization method and oupling strategies for different reative transport odes are detailed. Table 1 Supported disretization method and oupling strategies for different reative transport odes Reative transport modeling in fratured media might also require flexibility regarding the way the medium is onsidered. Important hanges in fluid pressures and solute onentrations will propagate rapidly through the frature system, while exhanges with the matrix bloks will 4

5 our slowly. To aount this, some reative transport odes have inluded Multiple Interating Continua modeling (TOUGHREACT, PFLOTRAN). In short, for reative transport modeling the adopted oupling tehniques, the partial differential equation disretization method and the way as the domain is onsidered, may be problem dependent. Therefore, a reative transport ode should inlude several solution approahes to be used in a broad range of problems. Moreover, in order to ensure its use for present and future problems, it must have an extensible design. A number of authors have pointed out that objet oriented (OO) programming failitates the implementation of these features (Commend and Zimmermann 21, Filho and Devloo 1991) The sientifi ommunity has been adopting OO tehniques for problem solving sine the end of the last entury (Forde et al. 199, Slooten et al. 21, Wang and Kolditz 27). But only in the last deade have OO odes been developed for reative transport modeling. Meysman et al. (23) developed an OO reative transport ode for a single fluid phase. Gandy and Younger (27) developed an OO multiphase reative transport ode for pyrite oxidation and pollutant transport in tailing ponds. Shao et al. (29) inlude reative transport alulations into a Thermo Hydro Mehani OO framework adopting a sequential non iterative approah (SNIA). Bea et al. (29) developed an OO module apable of solving reative transport for a single phase onsidering the SNIA, SIA or DSA approah. However, all of these odes, and most of the proedural reative transport odes, have a predefined strategy for dealing with oupling effets. Partiularly, they do not allow for hanging number and definitions of hemial omponents when solving flow and reative transport in a oupled way. The objetive of this paper is to present an OO struture for reative transport that an aommodate different level of physial and hemial proesses oupling. The struture presented here is apable to model from single phase SIA problems to fully oupled multiphase reative transport problems. In addition, the best of our knowledge, it is the first OO tool apable to aount the ourrene of invariant points (e.g., for referene see Risaher 5

6 and Clement) in a reative transport problem. This is an extreme ase where geohemial proesses signifiantly affet fluid flow and the number and definitions of hemial omponents may vary signifiantly in spae and time. This struture has been implemented in PROOST whih was programmed in FORTRAN 95 following the OO paradigm, and until now ould solve single phase reative transport by the SIA method and a fully oupled multi phase reative transport by the DSA method Equations to solve Reative transport modeling implies establishing several onservation priniples, like mass or energy onservation, expressed as partial differential equations (PDE), and several onstitutive and thermodynami laws (suh as retention urve or mass ations laws) expressed as algebrai equations (AE). Dary s law is used to represent momentum onservation. In this setion we present a generi onservation equation to represent onservation priniples in reative transport problems. We onsider in detail the speies and omponent onservation and we briefly present the onstitutive and thermodynami laws General onservation equation 133 Conservation of a physial entity an be expressed as 134 A F j, (1) t Where A is the amount of per unit volume of medium, j, is the flux of due to the driving fore (e.g. advetion or diffusion), and F is a sink soure term. Sine time and spatial derivatives are involved, onservation equations usually take the form of a partial differential equation (PDE) Speies and omponent onservation equation 6

7 The onservation of a speies i belonging to phase, whih is a partiular ase of equation (1) has the following expression: Where Ne Nk i, L i, Sej, irej Skj, irkj fi (2) t j1 j1 is the volumetri ontent of phase, i, is the speies i onentration in phase, Se is the stoihiometri oeffiient of the equilibrium reation j for the speie i, j, i re is the reation rate of the equilibrium reation j, and Ne is the number of equilibrium j reations. Sk, j, i rk and Nk are analogous to j Se, j, i re and Ne but for kineti reations. f j i is an external sink soure term, and involving advetive and diffusive dispersive proesses: Mobile phase fluxes L is the linear transport operator for the mobile phase i, i, D, i L q are alulated aording to Dary s law: q j (3) q K p g (4) 152 where K, p and are the ondutivity tensor, pressure and density of the phase respetively. Diffusive dispersive fluxes j D, i are alulated aording to Fik s law: where diff D and the tortuosity. diff disp j D D D (5) D, i i, i, disp D are the diffusion and dispersion tensor for phase respetively and is Note that the general sink soure term of equation (1) F involves several different terms in equation (2): 159 Ne F Se re Sk rk f Nk (6) j, i j j, i j i j1 j1 7

8 There is no expliit expression for the equilibrium reation rates that the orresponding mass ation law is satisfied. Therefore, re, their value has to be suh j re values an be written as a j funtion of both transport and hemial proesses (De Simoni et al. 25). A ommon approah to avoid dealing with these terms is to formulate the onservation of omponents as a linear ombination of speies that remain unaffeted by equilibrium reations. As suh, equilibrium reative rates disappear from the onservation equations of omponents (Steefel and MaQuarrie 1996). However, omponents may involve speies belonging to different phases, therefore onservation equation for omponents have to be written: t t ui, ui, L ui, ku f (7) i ui Where ui, and ui, are the i omponent onentration in mobile phases and immobile phases respetively, and k u i is a linear ombination of the kineti terms that affet the speies omposing the omponent. We onsider as immobile phases minerals and fluid solid interfae, despite the fat an interphase is not a phase from a thermodynami point of view. Note that the omponent onservation equation (7) has the same struture as equation (2). The main differene is that a omponent ui may be present in more than one phase, while a speies i belongs to a single phase. There are several ways of defining omponents and therefore some freedom in the hoie of omponents. This has led to formulations that try defining omponents that do not affet eah other, suh as those proposed by Molins et al. (24), Kräutle and Knabner (25) and Hoffmann et al. (21). Saaltink et al. (1998) introdued a definition that eliminates speies whose ativities are known and onstant. That is the ase of minerals, that are onsidered as pure phases, so that their ativity equals unity. Also, the ativity of water an be assumed unity for the ase of diluted solutions. Minerals, often onsidered as onstant ativity speies, might appear or disappear from portions of the domain due to preipitation dissolution proesses. Therefore, under equilibrium assumption, 8

9 the dimension of the omponent vetor, the number of omponents, may be different at eah disrete point in spae and vary in time. This inreases the diffiulty of solving equation (7) sine the matrix system to be solved has a dynami size, whih signifiantly affets the ode. One all omponent onservation and geohemial equations have been solved, all speies onentrations are known. Equilibrium reation rates re are then alulated form speies j onservation equation (2). If onstant ativity speies have been eliminated from the omponent definition, their onentration must also to be alulated from equation (2) Constitutive and thermodynami laws The literature provides several models for density, visosity and diffusion oeffiients of mobile phases. These parameters are usually expressed as an expliit funtion of phase omposition, pressure and temperature. Several models express saturation and relative permeability as an expliit funtion of apillary pressure and surfae tension. All these relations lead to a loal system of equations, whih is valid at every point of the domain. Thermodynami relations also form part of this loal system of equations. The most important of these are the hemial equilibrium reations, whih may be expressed by means of mass ation laws, as often done in reative transport. Also required are models for the alulation of ativity, suh as Debye Hükel (1923), or Pitzer (1973) and expressions for kineti rate laws (suh as Monod or Lasaga, Mayer et al. 22). Minor hanges on the solid matrix, like porosity hanges due to mineral dissolutionpreipitation or logging, may also be expressed as algebrai equations (Soleimani et al. 29). More omplex mehanial proesses, like deformation or onsolidation, involve momentum onservation equation and have to be solved as a PDE (Villar et al. 28). Constitutive and thermodynami relationships define a set of algebrai equations (AE) that have to be solved together with the onservation equations (PDE). 29 9

10 Numerial solution of the equations Methods suh as finite element or finite differenes, among others, are normally used to approximate time or spae derivative terms in PDEs. Appliation of suh methods leads to a set of equations that represent the onservation priniple for disrete portions of the domain (representing nodes or ells). The urrent version of PROOST supports two methods: the Finite Elements and the Mixed Finite Elements. Contrary to onstitutive or thermodynami laws, these equations are not loal, that is, equations at a disrete point are funtion of variables at other disrete points. As onstitutive and thermodynami models (AEs) involve variables that appear in the PDE, both AEs and PDEs may have to be solved simultaneously. Generally, the resulting set of equations is non linear, whih makes their solution more diffiult. As mentioned in the introdution different approahes an be adopted for solving these oupled sets of equation: independently, sequentially, iteratively or oupled OO analysis of reative transport modeling and PROOST lass organization Aording to the OO philosophy, the numerial solution of reative transport an be represented by a group of interating objets. These objets belong to lasses whih define ommon types of data and funtionality. Aording to Filho and Devloo (1991), defining suitable lasses is the first and perhaps the most important step in software design under OO Our analysis was based on the following abstration: reative transport modeling is onsidered as a set of equations (PDEs and AEs), representing the onservation of hemial speies, that need to be solved in a ertain domain. These equations involve several variables or fields (suh as onentrations, density or porosity) whih are also defined over portions of the same domain. The domain is disretized and fields are defined over the disretized spae (nodes or ells). Using disretization tehniques (suh as finite element or finite differenes methods) PDEs are onverted into a set of algebrai equations whih represent a disrete version of the 1

11 PDE. For eah disretized time interval, this set of equations an simultaneously be solved with the AE or using an operator splitting tehnique The above desription points to a natural lass struture for our problems. The PDEs share attributes suh as terms in the equation, state variables or domain definitions, and also share funtionalities suh as omputing the balane or the matries for the disretized PDE. Therefore, we find it natural to define a lass, termed Phenomenon, to identify PDEs. In the same fashion, we define Proess as the lass whose instanes will be speifi terms in the PDE (e.g. advetion, dispersion, et.). The lass Meshfields defines objets representing various properties defined over spae (and time). To deal the geohemial proesses we use the lass CHEPROO (CHEmial PRoesses Objet Oriented, Bea et al. 29). All these objets produe the terms for the (non linear) disretized PDEs, whih are solved with the funtions of the lass Solver. The lass organization desribed above is shown in Figure 1 and its detailed desription is given below. 11

12 Figure 1 Organization of main lasses in PROOST. Eah box represents a lass with its attributes and methods. A paradigm is show below eah lass Phenomenon lass PDEs are a entral ingredient of reative transport modeling. All PDEs represent a onservation priniple. All of them onsist of different terms, like storage, flux divergene or soure terms and are subjet to initial and boundary onditions. Therefore, we define a lass for representing PDEs. We term this lass Phenomenon. Note that a number of authors have also defined similar lasses in their analysis (Boivin and Ollivier Gooh 24, Kolditz and Bauer 24, Meysman et al. 23). But the main differene here is that in our ase, the Phenomenon objet is omposed by several objets of the lass Proess whih represent the different terms 12

13 of the PDE. This is a key aspet that failitates ode reuse, as will be shown below in the Proess lass desription. Beside the Proesses that define the PDE, the initial and boundary onditions are also the main attributes of the Phenomenon lass. Methods inlude the omputations of balanes or the ontribution to matries omprising the disrete version of the PDEs. The values of the solution variables or unknowns will be obtained from the solution of this matrix system. Initial onditions and Dirihlet boundary onditions are defined as a Meshfields and are handled by the Phenomenon Class. The rest of boundary onditions, as an be expressed as different terms of the PDE, are represented by instanes of the Proess lass. The Dirihlet boundary onditions has the partiularity of imposing the state variable value over different parts of the domain. For this reason there are handled diretly by the Phenomenon. A Phenomenon objet an be used to represent a single onservation priniple (suh as speies mass or energy) or several onservation priniples with similar equations, like omponents onentrations. For this latter ase, the Phenomenon lass makes use of the fat that the same onservation equation applies to all omponents, and therefore only one PDE has to be defined whih applies to all omponents Proess lass The terms that ompose the PDE (e.g., storage or advetion) and the boundary onditions that onstrain it (e.g. leakage) are represented by the Proess lass. The atual nature of this term is defined via inheritane by speialization lasses (Figure 1) The main attributes of this lass are the time and spae where the Proess is applied (for example the loation of a pumping well for a sink soure Proess) and the fields it involves (the pumping rate in this example). Methods inlude the omputation of the proess ontribution to the system matrix or to the global balane. All these are performed by using methods of the lass Mesh, where all disretization integration information and methods are enapsulated. 13

14 The Proesses objets are the terms that onstitute the onservation equations. A Phenomenon an be formulated by ombining different Proesses. This lass failitates the extensibility of the ode beause only the new terms (new speialization of the lass Proess) have to be programmed to extend the set of equations that an be solved. It also allows reusing ode, sine the same type of Proesses an be used for different onservation equations. For example, a diffusive proess for a mass onservation equation and an energy onservation equation are a different instane of the same lass. Another example is the extension of the omponent onservation equation from single phase to multiphase (equation (7)). For this ase, all proesses have to be repliate for eah mobile phase. This an be easily done by onsidering new instanes of the same Proess objets There are ertain limitations regarding the kind of Proesses that an be added to a Phenomenon, and are related to the numerial method hosen for solving it. The nature of the onsidered Proess has to be supported by the numerial method. For example, in its urrent implementation, advetive terms annot be onsidered when solving a PDE with the Mixed Finite Element Method Most boundary onditions are represented with objets of the lass Proess. Imposed fluxes and variable dependent fluxes are onsidered through a Sink Soure objets, whih are speialization of the lass Proess. As mentioned before Dirihlet boundary onditions are handled by the Phenomenon lass Mesh lass There are different tehniques to solve PDEs numerially. All these tehniques share an approah for disretizing the spatial domain (suh as nodes, elements or ells) and methods to integrate (or differentiate) the terms (Proess) of a PDE (Phenomenon) to produe a matrix system from whih the disrete solution of the PDE an be obtained. 14

15 Thus, all the data and funtionality regarding spatial disretization and the disretizationintegration methods for solving PDE (suh as finite element or finite differenes) define a lass that we term Mesh. A number of authors have defined similar lasses in their analysis. However, most of them separate the domain disretization from the integration methods in different lasses (Commend and Zimmermann 21, Wang and Kolditz 27). This integration was made beause, despite of the fat that both methods an share a mesh (elements and nodes), the mesh topologial data required might be different. For example, Mixed Finite Elements and Finite Elements an both use the same mesh, but Mixed Finite Elements needs extra information about edges for 2D problems or sides for 3D. Another differene between these two methods is that while Finite Elements gives a ontinuum salar field for the solution over the mesh, Mixed Finite Elements gives a vetor field. Therefore, some aspets of the spatial disretization are related to the integration method, and that is why both are onsider in a single objet in PROOST. The main attributes of the Mesh lass are the domain disretization information (suh as nodes or ell oordinates and onnetivity between these disrete elements). Methods inlude yielding information of spae disretization (suh as the number of disrete elements and their geometrial information), integrating the different terms of the onservation equation (Proesses) over the domain, and evaluating spatial properties of variables suh as gradients. The Mesh lass allows inorporating new disretization integration numerial methods by adding new speializations of the lass. Two speializations of the lass Mesh are urrently implemented in PROOST: the Finite Elements and the Mixed Finite Elements Meshfield lass Another important element of reative transport modeling is the AEs that represent onstitutive and thermodynami laws. Constitutive laws express one field as a funtion of others. Thus a lass termed Meshfield is defined to represent the projetion of different salar, 15

16 vetor or tensor fields (suh as pressure, flux or ondutivity) in the disrete domain. The main attributes of this lass are the values and derivatives of a field for the disrete entities (nodes, elements or ells) and the parameters of the funtion or onstitutive laws they represent. The main methods of the lass are to alulate its values and derivatives, and to interpolate its values over any point of the domain. Among others, Meshfield is used to represent retention urves, relative permeability urves and dispersion oeffiients. For example a flux Meshfield objet defined as q T h, an alulate its values and its derivatives to transmissivity T and head h fields. When a Meshfield represents one of the solution variables of the problems, like head in the previous example, its values are set by the solver lass. The Meshfield lass failitates ode extension sine new onstitutive laws an be easily added to the ode by reating new speializations CHEPROO lass Geohemial alulations for the omponent onentrations and kineti rate laws of equation (4) are in fat onstitutive laws. Hene, we treat them as a speialization of a Meshfield, whih we term Chemial Meshfield. Many geohemial variables affet the evolution of the system but do not appear expliitly in any PDE (e.g. the ativity of aqueous speies). For this reason and also beause of the omplexity of some geohemial alulations, all geohemial models and omputations are enapsulated into a single objet of a lass termed CHEPROO. Only the hemial variables that appear in PDE (suh as omponent onentration or density) are stored in a Chemial Meshfield. The CHEPROO lass uses a module with the same name, with an internal lass hierarhy inluding lasses like speies, phase and reation (Bea et al.29). CHEPROO attributes inlude the geohemial models, suh as those for ativity oeffiients, density or kineti rates laws, and the hemial data assoiated to eah disrete point of the problem, suh as onentrations 16

17 or omponents definition. CHEPROO inludes methods for alulating the values and derivatives of hemial variables (like omponent onentration) with respet to the solution variables of the PDE, and to dump them into Chemial Meshfield objets. CHEPROO also ontrols the number of hemial omponents. For some formulations, like the one of Saaltink et al. (1998), the number of omponents may hange in time and spae. Thus, CHEPROO has to provide information about the omponents in order to establish the dimension of the final matrix system to be solved Solver lass A oupling strategy (oupled or deoupled) needs to be hosen when solving several PDEs. A solution tehnique for non linear systems (Newton Raphson or Piard) is also needed. An objet of the Solver lass will be in harge of solving a number of PDEs with a hosen solution strategy: Independently, there are no rossed influenes between Phenomena (for example hanges on porosity due to hemial hanges are not onsidered when solving fluid phase onservation) Sequentially, influenes between Phenomena are onsidered lagged in time (for the porosity example, hanges due to hemistry in time t are onsidered for flow in time t +dt) Iteratively, all Phenomena are alternately solved until no signifiant hanges on linking variables ours (for the porosity example, flow and transport are solved alternately until no signifiant hanges in porosity ours) Coupled, all Phenomena are solved at one. Solver attributes inlude the set of Phenomena, the oupling strategy, the time disretization parameters or the onvergene riteria. Methods are required for assembling and solving the disretized PDE system, for time integration. To address these, Solver uses other lasses. For instanes, matrix systems are handled by a lass termed Matrix that enapsulates matrix data and solution tehniques for linear systems. 17

18 Solver is the lass that ontributes most to the flexibility of the ode sine it an be used to solve several onservation equations following different strategies. For example, it might be used to solve first a steady state phase onservation equation (for phase flow alulation) and then a transient omponent onservation. Or it an be used to solve simultaneously the omponent and energy onservation Component onservation Phenomenon for the SIA and DSA approah Despite the fat that the SIA and DSA are two approahes for solving the same Phenomenon, (the omponent onservation equation), the way this Phenomenon is formulated in PROOST depends on the hosen approah When solving omponent onservation equations with the DSA approah the input Phenomenon for PROOST should be the same as in equation (7). However, for the SIA approah immobile speies storage and kineti reations are treated as a sink soure term : 397 f u k (8) SIA i i, ui t Thus the omponent onservation equation is written only in terms of mobile omponent onservation: t ui, L ui, fsia f i u (9) i The Proost lass organization allowed implementing the SIA method without many modifiations. The SIA sink soure term was represented with the preexisting sink soure Proess lass. This proess evaluates the values of the sink soure term, whih are given by a Meshifield, and alulates its ontribution to the disretized PDE system. By doing this, all the 18

19 45 46 omplexity of this term is enapsulated in the lass Cheproo, whih sets the values of the SIA soure term in a Chemial Meshfield Solution proedure sheme for a time step The interation between PROOST objets an be illustrated by the solution of a time interval for a reative transport problem onsidering the DSA method. The flow diagram is shown in Figure 2, from whih 15 relevant points have been identified. 19

20 Figure 2 Flow diagram of a time interval resolution for a reative transport problem in PROOST 2

21 Solver establishes the size of the matrix system to be solved. This size depends on the number of oupled phenomena and the dimension of eah state variable. Reall that omponent onservation dimension an be different for eah disrete point and may hange among the iterative proess. 2. Solver assembles the matrix system to be solved. To this end, Solver requests eah Phenomenon for its ontribution. 3. Phenomenon requests for the ontribution of all its Proesses. 4. Eah Proeses request the values of all the Meshfields to whih it is related. 5. Meshfield omputes its values. 6. CHEPROO alulates Chemial Meshfield values. 7. Mesh omputes the ontribution of the Proess to the matrix system. 8. Matrix solves the matrix system. 9. Solver updates the alulated solution variables (onentrations, temperature or pressures) in CHEPROO. 1. CHEPROO alulates the onentration of all speies from these values (speiation). If there are signifiant hanges on hemial omposition, like omplete dissolution of minerals in equilibrium or preipitation of new ones, geohemial alulation might not onverge. If that is the ase, the length of time interval is redued and the resolution proedure is restarted. The user sets the ideal time step, but if the resolution of the matrix system (whih goes from step 2 to 11) exeeds a ertain number of iterations, also set by the user, the time step is redued. 11. Solver ontrols the onvergene of the PDEs linearization and resolution proess. When onvergene is reahed all variables involved in the phenomena are known, exept equilibrium reations rates that were eliminated when solving omponent onservation (equation (7)). These rates an be alulated from the speies onservation equations (equation (2)). In order to avoid the formality of formulating both omponents and speies 21

22 Phenomenon, this is done by onsidering an alternative omponent definition; eah mobile speies is onsidered a omponent. Therefore, the result of the balane of the new omponent onservation will be the produt of the stoihiometri oeffiient and the equilibrium reation rates. These aspets are illustrated with an example in the next setion. 12. CHEPROO hanges omponent definition (eah mobile speies is onsidered a omponent). This step allows solving speies onservation equations with the same struture used for omponent onservation equations. This is one of the advantages of Proost lass organization. More details on this partiular aspet will be given in setion Phenomenon omputes balane (similar to step 3). 14. CHEPROO alulates equilibrium reation rates from Phenomenon balane. Some reative transport formulations, like the one of Saaltink et al. (1998), eliminate onstant ativity speies, like minerals, from omponent omposition. These speies onentration an be alulated one the equilibrium reation rates are known. 15. If the formulation onsidered eliminates onstant ativity speies, the number of omponent is affeted by the disappearane or appearane of minerals. Therefore, omponent definition has to be ontrolled after the eliminated speies were alulated. If omponent definition hanges the resolution proedure has to be started for the new definition, if not the resolution proedure for the time step is finished Code implementation The ode presented results from merging and expanding two existing odes: PROOST and CHEPROO. The original design of PROOST was already apable of solving different phenomenona, in a oupled or deoupled way, by onsidering different tehniques for the resolution of non linear system (suh as Newton Raphson or Piard). However, suh a design only allowed solving Phenomenon objets that had one salar field as unknown. Also 22

23 Phenomenon Proesses had to be written expliitly as a funtion of the unknown variable. These featured lashed with the resolution of omponent onservation, espeially when the DSA approah is onsidered. The solution of omponent onservation equations involves onsidering the onservation equation of several omponents. As the number of omponents and its definition might hange in time and spae (beause of omplete dissolution or appearane of new mineral speies), the number of Phenomenon onsidered would also have to vary. In order to avoid this diffiulty, and as the same Proesses affet all omponent onentrations, only one Phenomenon is onsidered whih applies to a vetor variable: the omponent onentration vetor. Therefore, Phenomenon and Proess lasses were expanded to handle a vetor variable whose size may hange in time and spae. Proesses where originally designed to represent terms of PDEs that diretly involve the unknowns of the problem (i.e. main state variables of the phenomenon: pressure for flow equation, onentration for transport equation). For example, all Proesses in a onservative transport problem involve the solute onentration variable, whih is also the unknown of this problem. When solving reative transport by the DSA method Proesses are formulated in terms of omponent onentrations, but the unknowns of the problem are the primary speies onentrations. Therefore, Phenomenon and Proess lasses were expanded so they an be formulated in terms of any variable and not neessarily the unknown. Originally, CHEPROO was apable of solving single phase reative transport problems. CHEPROO uses a matrix system alulated by another onservative transport ode to formulate and solve the reative transport problem (Bea et al. 28). In order to take advantage of PROOST s flexibility we hoose to formulate and solve the multiphase reative transport equations in PROOST instead of CHEPROO. Therefore, CHEPROO was added to the PROOST struture with the only purpose of performing the hemial alulations (speiation) and provide geohemial variables values and derivatives. 23

24 Besides adding new servies to make hemial variables available outside its module, several improvements were made in CHEPROO. Phase properties like density, visosity and enthalpy, and apillary effet on water ativity were added. The PROOST lass organization allowed representing all this hemial variables in the lass Chemial meshfield. By doing this all the work related to the evaluation, update and dependeny of these fields to others (like pressure or temperature) is done by pre existing methods. Also a new speiation algorithm that uses the Newton Raphson method had to be programmed in CHEPROO due to the high nonlinearity of onentrated solutions. CHEPROO and PROOST were programmed in FORTRAN 95 following the OO paradigm. This language was hosen for its high popularity among hydrogeologists and its exellent performane reputation. Even though FORTRAN is not a full objet oriented language it an diretly support many of the important onepts of OO programming. Details about OOP onepts in FORTRAN an be found in Akin (1999), Carr (1999), Deyk et al. (1998), Gorelik (24), Maley et al. (1996) and Norton et al. (1998) Appliation: 6.1. Appliation Desription In order to illustrate the lasses introdued before, some aspets of the solution proedure sheme for a time step (generially desribed in setion 4) are shown for a onrete appliation. We present the modeling of a 24 m olumn of porous gypsum subjeted to a onstant soure of heat, in whih a signifiant evaporation ours. We will fous on the omponent onservation equation. This syntheti example was designed for illustrating the interation between hydrodynami and geohemial proesses and it is desribed in detail by Gamazo et al. (212). Due to this interation a ompositional formulation was adopted and therefore no phase onservation equations are expliitly solved. The finite element method was used for the spatial disretization. One of the most interesting aspet of the appliation is how the equilibrium reation rates were alulated. This implies solving a different onservation equation, speies onservation instead of omponents. The PROOST struture allowed to alulate the equilibrium reation rates by using preexisting methods. 24

25 This appliation example inludes gypsum, liquid water and vapor, dissolved and gaseous, alium and sulfate (main omponents of gypsum besides water) and 2 onservative speies potassium and hloride (see Table 2). It also onsiders the ourrene of anhydrite, whih may preipitate as a result of gypsum dehydration. Note that the oexistene in equilibrium of anhydrite and gypsum an fix water ativity and therefore produe invariant points (Risaher and Clement 21). As was exposed above, to our knowledge, PROOST is the first multiphase reative transport apable of modeling this senario. 526 Table 2 Chemial speies and reations onsidered h2 o, h2 o,,, Ca, So, K, Cl, gypsum, anhydrite l g l g 4 H O H O 2 ( l) 2 ( g) K Pa Kgh o x1 exp T mol 11 2 vapour ( l) ( g) K 8 Pa Kgh2o 2.9x1 mol Ca SO anhydrite K anhydrite Ca SO H O gypsum ( l) K gypsum Initial omponent definition Pore solution was initially onsidered in equilibrium with gypsum with a mineral volumetri ontent of.6. The inoming energy heats the olumn, whih inreases evaporation and redues saturation degree at the top. This indues an asending non statured flow of liquid water. At ertain point a desending evaporation front appears followed by an also desending gypsum dehydration front in whih anhydrite preipitates (see Figure 3). Note that this seond front has a signifiant effet on vapor flow. 25

26 Figure 3 Liquid saturation, mineral mass, evaporation rate and vapor flux for the upper 8 m of the olumn. Note that besides the typial evaporation front assoiated to the drying front there is a seond evaporation front assoiated to hydrated mineral dissolution. This seond front has a signifiant effet on vapor flow. 26

27 When the simulation starts the whole domain has the same mineral omposition and therefore the omponent onservation equations for all nodes are the same (see Table 3 for omponent definition): 54 K K K l g l l g g f g aq Ca So 4 g aq Ca So 4 aq Ca t t q D 2 So4 qg Dg Cl Cl Cl h2o 2 2 h2o 2 2 g 2 l So h o l So h2o 2 h o 4 g h2o f g h2o l So g 4 (1) This implies that the number of omponents is the same for the entire domain. This aspet is ontrolled by a single objet of the Cheproo lass, and affets almost all lasses: from the Solver, in harge of alulating the dimension of the system to be solved, to the Meshfiled, in harge of storing field values and their derivatives to state variables Table 3 Component definition for different mineral ombinations (from up to down: only gypsum, gypsum and anhydrite, only anhydrite) and the one omponent per mobile speies omponent definition U 1. U U gypsum anhgyp u u u u u K CaSO4 Cl ho 2 2SO4 u u u u K CaSO4 Cl 2 2 H2O Ca SO4 K Cl H2O anh gyp l l g g H2O Ca SO4 K Cl H2O anh gyp l l g g

28 U anhydrite u u u u u u u u U1 u u u u u K CaSO4 Cl ho 2 ho 2 l l Ca SO4 K Cl ho 2 g l 2 2 H2O Ca SO4 K Cl H2O anh gyp l l g g H2O Ca SO4 K Cl H2O anh gyp l l g g Despite of having several omponents, eah with its own onservation equation or phenomenon, PROOST treats omponents as entities pertaining to one phenomenon. This simplifies the ode s internal operability and problem definition, sine it allows taking benefit of the fat that several proesses affet speies in the same way. For example the storage, advetion and diffusion dispersion proesses in equation (1) affet all speies from a phase in the same way. For these proesses the ontribution to the system matrix are alulated for all omponents together. Enapsulation allows onfining to the Proess lass all the omplexity assoiated to the fat that proesses an be part of one or a set of partial differential equations. Currently the only proess that ats differently over eah speies is the sink/soure proess The Cheproo objet also defines whih speies and variables will be onsidered as state variables for Newton Raphson system. When only gypsum is present in the system, the states 558 variables assoiated to omponent onservation equations are: K 2 l Ca Cl p l. The rest 559 of speies (, g h 2 o, 2 g So ) are seondary and its values are alulated by Cheproo by 4 28

29 onsidering mass ations laws. Reation rates and non mobile speies onentrations are alulated in a subsequent step In order to understand the physial meaning of omponent onservation equations, it is helpful to assoiate state variables to speifi omponents. For example, eah of the speies 564 hosen as state variables (, K, 2 and l Ca Cl ) an be onsidered as the onstituents of the four first omponents of the onservation equations (1). The assoiation of the liquid pressure ( p l ) to a speifi omponent is not straight forward. Liquid pressure is related to liquid saturation whih affets all omponents. Sine the last omponent in equation (1) ontains all water speies and only involves seondary speies, liquid pressure an then be assoiated to its main variable. However, variables like ativity oeffiients, density, visosity, gas pressure and liquid saturation, depend on all state variables and make the system fully oupled. Nevertheless, the exerise of defining a main variable for every omponent provides a more profound knowledge about variables dependeny, whih may be relevant for some ases as will be shown in setion 6.3. For that ase water speies is eliminated from the omponent equation and both alium and sulfate are defined as seondary variables As an be seen in Figure 2, matrix system assembling is the ore of a time interval resolution. It involves all the lasses shown in Figure One the system is solved there are still unknown variables to be alulated: the eliminated speies onentration and the equilibrium reation rates These variables an be alulated by onsidering the speies onservation equation. In order to avoid formulating a different phenomenon the PROOST lass organization allows using the same struture as used for alulating omponent onservation for speies onservation. This is one of the advantages of the Proost lass organization. The same phenomenon is onsidered 29

30 and only the omponent definition is hanged. The new omponent definition onsiders every mobile speies as a omponent (see Table 3): 585 K K K l l l 1 g g f g g 1 r 2 2 Ca 2 Ca Ca 1 aq g aq t q Daq g g 2 2 So t 4 So 2 4 q D 2 So 4 1 rh o Cl r gyp Cl Cl h2o l h2o l h2o 1 2 l h2o g 2 2 h o g h o f g h2o 1 g (11) Note that all the proesses in equation (11) are analogous to equation (1), exept the last one. This is the only term in equation (11) that has unknown variables ( r, r 2 h o, r gypsum ), the other terms involve known variables. In order to alulate these unknown variables the U 1 omponent definition is onsidered and a general method of the proess lass, balane, is used to alulate all terms at the right hand side of equation (9): K K K r l l l r g g f g g r 2 2 gyp Ca 2 Ca aq g Ca aq r q Daq g 2 gyp t So t q Dg So 4 So4 Cl Cl Cl rh2 o2r gyp h2o l h2o l h2o l r h2o h2o g 2 h o g h2o f g h2o g (12) The result is used by Cheproo to alulate the reation rates (evaporation, volatilization of dissolved and gypsum preipitation). Note that the number of equations exeeds the number of unknowns (eight and three, respetively). In theory, solution of all equations should give the same reations rates. For simpliity we used the least square method for the solution of equation (9). One the reation rates are alulated, the mole variations of mineral speies an be omputed (gypsum for this ase). If a mineral is ompletely depleted or if the solution has beome saturated for a new mineral, omponents should be redefined and alulations for the time step realulated. 3

31 Anhydrite preipitation As the system evolves over time, water ativity dereases at the top of the olumn due to osmoti and apillary effet, and anhydrite starts to preipitate. When anhydrite and gypsum oexists a singularity, known as invariant point, ours and water ativity remains onstant (Risaher and Clement 21 and Gamazo et al. 211). Combining the mass ations law for anhydrite and gypsum (equation (13)) the fixed water ativity value an be obtained: 67 a a K K 2 2 Ca SO 4 anhydrite gypsum ah2 ( l ) 2 2 Ca SO h 22 4 ( l ) gypsum anhydrite a a a K K (13) Under this senario gypsum dissolves and anhydrite preipitates at a rate that ensures this fixed water ativity: gypsum anhydrite 2H2O (14) This implies a singular omponent definition (see Table 3) whih results in the following omponent onservation equation: 613 K K K l l l g g aq g aq aq g t 2 2 t q 2 D 2 2 q g f g 2 Ca So Dg 4 Ca So4 Ca So4 Cl Cl Cl (15) Note that all forms of water have been eliminated from the omponents onservation equation. For all nodes where anhydrite and gypsum oexists these new omponents have to be onsidered, and the states variables assoiated to onservation equation will be: Cl, p l. K, l, This onservation equation, and the orresponding states variables, may make it diffiult to assoiate state variables to speifi omponents. As in the previous system, the state variables K, l, Cl an be assoiated to the first, seond and fourth omponent respetively. Hene, the remaining variable, liquid pressure, must be assoiated to the third omponent of equation (15). Although this third omponent only ontains Ca 2+ and SO 4 2 and no H 2 O, it still 31