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A knowledge-based reactive transport approach for the simulation of biogeochemical dynamics in Earth systems

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A knowledge-based reactive transport approach for the simulation of biogeochemical dynamics in Earth systems
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   A knowledge-based reactive transport approach for thesimulation of biogeochemical dynamics in Earth systems D. R. Aguilera, P. Jourabchi, C. Spiteri, and P. Regnier  Department of Earth Sciences—Geochemistry, Faculty of Geosciences, University of Utrecht, P.O. Box 80.021, NL-3508 TA Utrecht, Netherlands (aguilera@geo.uu.nl) [ 1 ]  A Knowledge-Based Reactive Transport Model (KB-RTM) for simulation of coupled transport and biogeochemical transformations in surface and subsurface flow environments is presented (http:// www.geo.uu.nl/   kbrtm). The scalable Web-distributed Knowledge Base (KB), which combinesInformation Technology (IT), an automatic code generator, and database management, facilitates theautomated construction of complex reaction networks from comprehensive information stored at the levelof biogeochemical processes. The reaction-centric approach of the KB-RTM system offers full flexibilityin the choice of model components and biogeochemical reactions. The procedure coupling the reactionnetworks to a generalized transport module into RTMs is also presented. The workings of our KB-RTMsimulation environment are illustrated by means of two examples of redox and acid-base chemistry in atypical shelf sediment and an aquifer contaminated by landfill plumes. Components:  9832 words, 9 figures, 5 tables . Keywords:  biogeochemistry; information technology; Knowledge Base; numerical methods; reactive transport. Index Terms:  1009 Geochemistry: Geochemical modeling (3610, 8410). Received  16 December 2004;  Revised  10 March 2005;  Accepted  4 May 2005;  Published  27 July 2005.Aguilera, D. R., P. Jourabchi, C. Spiteri, and P. Regnier (2005), A knowledge-based reactive transport approach for thesimulation of biogeochemical dynamics in Earth systems  , 6  , Q07012, doi:10.1029/2004GC000899. 1. Introduction [ 2 ] Reactive transport models (RTMs) are power-ful tools for capturing the dynamic interplay be-tween fluid flow, constituent transport, and biogeochemical transformations [ Steefel and VanCappellen , 1998]. They have been used to simu-late, among others, rock weathering and soil for-mation [e.g.,  Ayora et al. , 1998;  Soler and Lasaga ,1998;  Steefel and Lichtner  , 1998a, 1998b;  Thyne et al. , 2001;  Soler  , 2003;  De Windt et al. , 2004],nutrient dynamics in river drainage basins andestuaries [e.g.,  Soetaert and Herman , 1995;  Billenet al. , 1994;  Regnier et al. , 1997;  Regnier and Steefel  , 1999;  Vanderborght et al. , 2002], reactivetransport in groundwater, like contamination of aquifers [e.g.,  Engesgaard and Traberg  , 1996;  Brown et al. , 1998;  Hunter et al. , 1998;  Xu et al. ,1999;  Murphy and Ginn , 2000;  Barry et al. , 2002;  Brun and Engesgaard  , 2002;  Thullner et al. , 2004; van Breukelen et al. , 2004], early diagenetic trans-formations in sediments [e.g.,  Soetaert et al. , 1996;  Boudreau , 1996;  Van Cappellen and Wang  ,1996;  Dhakar and Burdige , 1996;  Berg et al. ,2003;  Jourabchi et al. , 2005], benthic-pelagic cou- pling in ocean systems [e.g.,  Soetaert et al. , 2000;  Archer et al. , 2002;  Lee et al. , 2002] and hydrocar- bon migration and maturation in sedimentary basins[e.g.,  Person and Garven , 1994]. By integratingexperimental, observational and theoretical knowl-edge about geochemical, biological and transport  processes into mathematical formulations, RTMs provide the grounds for prognosis, while diagnosticcomparison between model simulations and mea-surements can help identify gaps in the conceptualunderstanding of a specific system or uncertainties G 3 G 3 GeochemistryGeophysicsGeosystems Published by AGU and the Geochemical SocietyAN ELECTRONIC JOURNAL OF THE EARTH SCIENCES GeochemistryGeophysicsGeosystems  Article  Volume 6 , Number 7 27 July 2005Q07012, doi:10.1029/2004GC000899ISSN: 1525-2027Copyright 2005 by the American Geophysical Union 1 of 18  in proper parameterization of biogeochemical pro-cesses [  Berg et al. , 2003;  Jourabchi et al. , 2005].[ 3 ] RTMs have traditionally been developed andused to investigate the fate and transport of aselected set of chemical constituents within agiven compartment of the Earth system (e.g., theearly diagenetic models by  Soetaert et al.  [1996],  Boudreau  [1996],  Van Cappellen and Wang   [1996],  Dhakar and Burdige  [1996], and references citedabove). As a result, they have tended to be envi-ronment and application specific with regards tothe flow regime and the biogeochemical reactionnetwork.[ 4 ] Although the first attempts to develop interac-tive software systems for automatic solution of models based on ordinary and partial differentialequations date back to the creation of digital com- puters [e.g.,  Young and Juncosa , 1959;  Lawrenceand Groner  , 1973;  Mikhailov and Aladjem , 1981,and references therein], such developments have sofar received little attention in the field of reactive-transport modeling. Literature review shows that RTM codes allowing for more flexible definition of state variables and processes without requiring in-depth knowledge of programming or numericalsolution techniques have been developed over thelast decade [e.g.,  Reichert  , 1994;  Chilakapati ,1995;  Regnier et al. , 1997;  Chilakapati et al. ,2000;  MacQuarrie et al. , 2001;  Meysman et al. , 2003a, 2003b;  Van der Lee et al. , 2003].Database tools, such as that developed by  Katsev et al.  [2004], have also been presented recently tothe reactive transport community. Model flexibilityis a critical feature since a major challenge in thefield of reactive transport modeling is the realisticrepresentation of the highly complex reaction net-works (RN) that characterize the biogeochemicaldynamics of natural environments [e.g.,  Mayer et al. , 2002;  Berg et al. , 2003;  Quezada et al. , 2004].At the same time, many field- and laboratory-basedexperiments are also being conducted to identifynovel reaction pathways, quantify reaction ratesand microbial activity levels, describe ecologicalcommunity structures, and elucidate interactions between biotic and abiotic processes. This rapidlygrowing knowledge about biogeochemical trans-formation processes creates a need for efficient means of transferring new experimental findingsinto RTMs.[ 5 ] Here, a unified modeling approach for imple-menting complex reaction networks in RTMs is presented. Our simulation environment is based ona modular approach to facilitate incorporation of new theoretical and experimental information onthe rates and pathways of biogeochemical reac-tions. The key novel feature of the modelingenvironment is a Web-distributed Knowledge Base(KB) of biogeochemical processes, which acts asthe evolving repository of up-to-date informationgained in the field of geochemistry. The implemen-tation of such a library within a simulation envi-ronment is a major step toward the model’sflexibility, because it is at the level of an easilyaccessible open resource, the KB, that process- based theoretical and experimental advances areincorporated in the modeling process. Model gen-eration is conducted via a graphical user interface(GUI) on a Web-based ‘‘runtime’’ server, whichallows for the development of biogeochemicalreaction network modules. That is, informationstored at the level of individual biogeochemical processes can be combined into mathematicalexpressions defining completely the (bio)chemicaldynamics of the system. Since the reaction network is assembled from information stored at the process level, almost any conceivable combina-tion of mixed kinetic and equilibrium reactionscan be implemented in our model architecture.The selected RN can easily be merged withexisting transport models, hence creating a flex-ible framework in which to assemble RTMs. The proposed approach allows the RTM communityto test and compare, in close collaboration withexperimentalists, alternative mathematical descrip-tions of coupled biogeochemical reaction networks.For example, increasingly detailed representationsof biogeochemical processes can be incorporated inthe Knowledge Base. Reaction network modules of increasing complexity may then be assembled andcoupled to surface or subsurface flow models, inorder to determine which level of biogeochemicalcomplexity is adequate to simulate chemical systemdynamics at variable spatial and temporal resolu-tions. By taking a ‘‘reaction-centric’’ approachwhich utilizes the unifying conceptual and math-ematical principles underlying all RTMs, one-dimensional (1D) transport descriptions relevant to many compartments of the Earth system (rivers,estuaries, groundwater or sediments) can be in-corporated in our simulation environment. The proposed approach should thus help overcometraditional disciplinary barriers between the differ-ent subfields of RTMs.[ 6 ] The paper is structured as follows: First, themass conservation equation describing 1D coupledtransport and reaction is briefly presented. A gen-eralized continuum representation is proposed, GeochemistryGeophysicsGeosystems G 3 G 3  aguilera et al.: earth system dynamics  10.1029/2004GC000899 2 of 18  which allows for the simulation of reactive-trans- port problems characterized by different flowregimes and dispersion intensities. A brief descrip-tion on how an existing Automatic Code Generator (ACG) based on symbolic programming [  Regnier et al. , 2002] can be used to create the modelspecific source code necessary to the numericalsolution of the governing equations is then given.We demonstrate how our Web-distributed Knowl-edge Base concept, which combines InformationTechnology with symbolic computing techniques,directs the mathematical formulation of the bio-geochemical reaction network and leads to a mod-eling environment offering full flexibility. Finally,the workings of our KB-RTM are illustrated withtwo contrasting examples of complex redox andacid-base geochemistry in an aquatic sediment andan aquifer, respectively. 2. Mathematical Representation of Reactive-Transport Equations [ 7 ] A one-dimensional continuum representationof coupled mass transport and chemical reactionsin Earth systems can be described mathematically by a set of partial differential equations (PDEs) intime and space of the form x @  C   j  @  t   ¼  @ @   x D   x   @  C   j  @   x      @ @   x v    x   C   j      j   |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}  T  þ X  N  r  k  ¼ 1 l k  ;  j    s k   |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}   R ;  j   ¼  1 ;  . . .  ; m ;  ð 1 Þ where  t   is time and  x  denotes the position alongthe 1D spatial domain. Particular solutions of equation (1) require specification of initial and boundary conditions. Further discussion about thegeneralized continuum representation of theadvection-dispersion-reaction (ADR) equation isgiven by  Regnier et al.  [2002] and  Meile  [2003].The first two terms on the right hand side, insquare brackets, compose the transport operator ( T  ); the last one (  R ) represents the sum of transformation processes (e.g., reactions) affectinga species  j   of concentration  C    j  . Table 1 shows that  x ,  D  and  v   are generic variables which takedifferent meanings depending on the environment considered.  s k   represents, for kinetic reactions, therate of the  k  - th  reaction and  l k  ,   j   is the stoichio-metric coefficient of species  j   in that reaction.Currently, it is assumed that the reaction processes(  R ) have no effect on the physical or transport ( T  ) properties of the system (i.e.,  x ,  D  and  v   areunaffected by reactions). The rate  s k   is of arbitraryform, even nonlinear, and can be a function of several concentrations of the system. Through thiscoupling by the reaction terms, most multicompo-nent problems result in a set of coupled nonlinear  partial differential equations (PDEs) of size  m ,number of species of the reaction network. In theevent that some of the reactions considered areassumed to be at equilibrium, algebraic expressions based on mass action laws are introduced into thesystem of equations to be solved [  Regnier et al. ,2002]. By replacing one or more of the  m differential equations associated with reactions withalgebraic relations based on a mass action expres-sion in the local equilibrium case, the set of ODEs istransformed into a set of differential-algebraicequations (DAEs) [ Chilakapati , 1995;  Hindmarshand Petzold  , 1995a, 1995b;  Brenan et al. , 1989].Thistransformationleadstoasystemof  m equationsto solve, including  m k   equations associated withkinetic reactions, and  m e  algebraic equations basedon mass action expressions.[ 8 ] The numerical solution of the set of discretizedPDEs and DAEs ( @  t   !  D t  ,  @   x  !  D  x ) commonlyrequires the use of implicit methods in order to becomputationally efficient [ Steefel and MacQuarrie ,1996]. We currently make use of the time splittingtechnique, which consists of solving first the trans- port and then the reaction terms in sequence for asingle time step. This method is referred to as the Table 1.  Meaning of the Generalized Variables  x ,  D ,and  v   for Different Environments a  Surface FlowAquaticSediment Groundwater Flow PathSolutes or Suspended Solids Solids Solutes Solids Solutes x  A  1    f f  1    f f v V    flow  w w  +  v    flow  0  V    flow  D K  turb  D b  D b  +  D  sed   0  D disp a From  Meile  [2003]. In porous media flow, distinction between anaqueous and a solid phase must be considered, and  C   is either aconcentration of solute or solid.  A [L 2 ], cross-section area of the surfaceflow channel;  f  [  ], porosity;  V    flow [LT  1 ], externally imposed flowvelocity;  w [LT  1 ], burial velocity defined with respect to the sediment-water interface (SWI);  v    flow [LT  1 ], flow velocity acting only on solutesexternally imposed or from porosity change; usually defined withrespect to the SWI [e.g.,  Boudreau , 1997];  K  turb [L 2 T  1 ], longitudinalturbulent dispersion coefficient;  D b [L 2 T  1 ], bioturbation coefficient;  D  sed  [L 2 T  1 ] =  D mol  /(1    ln( f 2 )), tortuosity corrected molecular diffusion coefficient for solutes at in situ temperature and salinity[  Boudreau , 1997];  D disp [L 2 T  1 ] =  a  L   j V    flow j , longitudinal dispersion,where  a  L  [L] is the longitudinal dispersivity [  Freeze and Cherry ,1979]. GeochemistryGeophysicsGeosystems G 3 G 3  aguilera et al.: earth system dynamics  10.1029/2004GC000899 aguilera et al.: earth system dynamics  10.1029/2004GC000899 3 of 18  sequential noniterative approach (SNIA). Whensolving the reaction part an iterative method isrequired to numerically find the roots of thefunction residuals,  f     j  , which correspond to mass balance equations and, if equilibrium reactions areincluded, mass action equations. This is becausethe reaction terms can be nonlinear functions of the species concentrations. By far the most com-mon approach for finding the root of nonlinear sets of equations is the Newton-Raphson method[  Press et al. , 1992]. This method involves the useof a first degree Taylor series expansion tolinearize the problem for every single iterationstep. The function residuals,  f     j  , representing thereaction network (RN), and the Jacobian matrix,which contains the partial derivatives of thefunction residuals with respect to the unknownconcentrations, are the most important pieces of information required to implement the Newton-Raphson algorithm [e.g.,  Dennis and Schnabel  ,1996]. Once linearized, the resulting problem issolved using linear algebra methods, such as theLU decomposition [e.g.,  Strang  , 1988]. A precisedefinition of the function residuals and the Jaco- bian matrix is given by  Regnier et al.  [2002]. 3. Reactive Transport Modeling Withthe Knowledge Base [ 9 ] Inspection of the transport and reaction oper-ators  T   and  R  in equation (1) shows that thefollowing information is required to define a spe-cific reaction transport application:[ 10 ] 1. Domain definition: spatiotemporal size andresolution (  x tot  , D  x ,  t  tot  , D t  ), where  x tot   and  t  tot   standfor the total domain length and simulation time,respectively, and  D  x  and  D t   are the space and timestep used for numerical integration, respectively.[ 11 ] 2. Transport coefficients  x ,  v   and  D  (Table 1).[ 12 ] 3. Reaction Network (RN): processes; rate parameters and equilibrium constants; species in-volved and stoichiometry.[ 13 ] 4. Boundary (BC) and initial (IC) conditionsfor every species of the RN.[ 14 ] In a modeling environment offering full flex-ibility, this information is specific to each RTMapplication and needs to be automatically translatedinto source code. This task is relatively easy for the physical domain definition, transport parame-ters, BC and IC. However, if flexibility in choos-ing process formulations is also important, thenthe stiff system of differential equations using alinearization method (such as the above-cited Newton-Raphson algorithm) necessitate automaticdifferentiation schemes for the calculation of theterms in the Jacobian matrix. Automatic symbolicdifferentiation offers the advantage of producingderivatives of potentially complicated functionswhich are accurate up to the precision of the programming language used (e.g., FORTRAN), plus the convenience of updating the derivativeseasily if the srcinal functions are changed [ Steefel and MacQuarrie , 1996]. Automated differentiationfor stiff sets of differential equations is one of thekey features of our Automatic Code Generation(ACG) procedure [  Regnier et al. , 2002].[ 15 ] The simultaneous implementation of a libraryof biogeochemical processes into a KnowledgeBase (KB) is an additional crucial component of the proposed simulation environment. The KBmakes it possible to take full advantage of theACG. The integration of these two componentswithin a Web system shows how the combinationof Information Technology with advanced symbolic programming allows the use of the Internet as asoftware provider in the area of Reactive-Transport modeling (Figure 1). 3.1. The Internet as a Software Provider  [ 16 ] An Internet system developed in PHP lan-guage (http://www.php.net) provides the adaptiveJavaScript and HTML code for the Graphical User Interface (GUI), as well as the Web-based ‘‘run-time’’ server to our modeling environment. It isaccessible at http://www.geo.uu.nl/   kbrtm. TheGUI is of evolutionary nature, that is, it dynami-cally adapts to changes in structure and content of the KB system as well as to the selections made bythe user.[ 17 ] Initially, the user accesses an interface for theKB system in the style of a Web form for selection of desired biogeochemical processes(Figure 1, step 1; further detailed in Figure 2).Species-independent physical parameters (defini-tion of spatiotemporal domain and most transport coefficients in Table 1) are also defined at this stage.[ 18 ] Figure 2 gives a detailed description of themodel design procedure. The KB system consistsof a set of biogeochemical processes, containingdefault formulations for reactions, which are avail-able to all users of the system (common KB) andwhich cannot be modified. However, these com-mon processes can be edited if desired, or new GeochemistryGeophysicsGeosystems G 3 G 3  aguilera et al.: earth system dynamics  10.1029/2004GC000899 4 of 18   processes created using a standard template, andstored in a ‘‘private’’ KB library which is inacces-sible to other users. To facilitate model develop-ment, processes can be grouped in different subsets, for instance, in terms of reaction types(see below).[ 19 ] The selected processes specify the user-definedreaction network. The latter is then analyzed todetermine the list of chemical species involved inthe RN. A second Web submission form is subse-quently created to specify all species-dependent  parameters (such as molecular diffusion coeffi- Figure 1.  Structure of the Knowledge-Based (KB) Reactive Transport Model (RTM) environment which uses theWeb as a software provider: 1, form submission; 2, parsing of the ASCII files into MAPLE; 3, MAPLE/ MACROFORT symbolic computing; 4, translation of MAPLE results into FORTRAN; 5, linking and compilation of the FORTRAN code; 6, transfer of the executable file to the user via e-mail. Figure 2.  KB-Internet model design process (detailed description of step 1 in Figure 1): (1) Process edition or creation. Templates can be used to speed up the implementation of new processes. Modified or new process files arestored in a ‘‘private’’ KB which is accessible only to a single user. (2) Process selection from the common and privateKB process pools, and specification of the physical support. (3) Analysis of the selection and creation of adynamically adaptive Web form to input reaction network dependent parameters. (4) Submission of complete modelinformation (processes selected, physical parameters, and species-dependent parameters) to the ACG via our Webserver. GeochemistryGeophysicsGeosystems G 3 G 3  aguilera et al.: earth system dynamics  10.1029/2004GC000899 5 of 18
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