Dissertation Proposal

On reactive transport in heterogeneous media Ph. D. Dissertation Proposal Ph. D. Program in Civil Engineering

On reactive transport in heterogeneous media Ph. D. Dissertation Proposal By:

Leonardo David DONADO GARZÓN Ph. D. Student School of Civil Engineering of Barcelona

[email protected] www.h2ogeo.upc.es/ldonado Adviser:

Dr. Xavier SANCHEZ-VILA Associate Professor Department of Geotechnical Engineering and Geo-Sciences

[email protected] Co Advisers:

Dr. Marco DENTZ Researcher Department of Geotechnical Engineering and Geo-Sciences

[email protected] &

Nelson OBREGÓN NEIRA, Ph. D. Associate Professor National University of Colombia

[email protected]

Ph.D. Program in Civil Engineering School of Civil Engineering of Barcelona Technical University of Catalonia UPC © Barcelona July 2005

CONTENTS Pag. ABSTRACT……………………………………………………………………………………vi RESUMEN…………………………………………………………………………………….vii RESUM……………………………………………………………………………………….viii

1.

INTRODUCTION .......................................................................................1

2.

PROBLEM STATEMENT ..........................................................................5

2.1.

PROBLEM APPROACH............................................................................................................................5

2.2.

MOTIVATION........................................................................................................................................6

2.3.

PERTINENCE ......................................................................................................................................... 7

3.

THEORETICAL FRAMEWORK ................................................................9

3.1.

TRANSPORT OF REACTIVE CONSTITUENTS...............................................................................9

3.2. WATER/ROCK INTERACTIONS ................................................................................................... 11 3.2.1. Adsorption / Desorption .......................................................................................................................... 11 3.2.2. Mineral Precipitation/Dissolution ........................................................................................................12 3.2.3. Reaction Rates ...........................................................................................................................................12 3.3.

TRANSPORT IN FRACTURED MEDIA............................................................................................ 13

3.4.

MATRIX DIFFUSION ........................................................................................................................ 13

3.5.

MULTIRATE MASS TRANSFER (MRMT) ...................................................................................... 14

3.6.

DILUTION .................................................................................................................................................. 17

4.

STATE OF THE ART ..............................................................................19

4.1.

REACTIVE TRANSPORT ..................................................................................................................... 20

4.2.

MULTIPLE RATE MASS TRANSFER........................................................................................... 20

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4.3.

DILUTION .............................................................................................................................................22

4.4.

APPLICATIONS .................................................................................................................................. 24

5.

OBJECTIVES AND SCOPE.................................................................... 27

5.1.

GENERAL OBJECTIVE ...........................................................................................................................27

5.2.

SPECIFIC OBJECTIVES .....................................................................................................................27

6.

METHODOLOGY & WORK PLAN ......................................................... 29

6.1.

WORK PLAN.............................................................................................................................................29

6.2.

SCHEDULE............................................................................................................................................30

7.

PRELIMINARY & EXPECTED RESULTS .............................................. 33

7.1. PRELIMINARY RESULTS .....................................................................................................................33 7.1.1. Tracer tests in fractured media ............................................................................................................. 33 7.1.2. Results on reactive transport in heterogeneous media ................................................................... 39

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7.2.

EXPECTED RESULTS ............................................................................................................................ 46

8.

REFERENCES ........................................................................................ 49

9.

MRMT APPROACH APLICATIONS ........................................................IX

LIST OF FIGURES Pag.

FIGURE 1.1 SKETCH OF THE TRANSPORT PROCESSES IN THE ENVIRONMENT (FROM TINDALL ET AL., 1999)............. 2 FIGURE 3.1 SCHEMATIC OF MOBILE-IMMOBILE DOMAIN MODEL (FROM CUSHEY, 1996) ...................................... 10 FIGURE 3.2 CONCEPTUAL MODEL OF MATRIX DIFFUSION (TOP) AND MATRIX BLOCK GEOMETRIES (BOTTOM) (FROM CARRERA ET AL., 1998)...................................................................................................................... 14 FIGURE 3.3 DEUTERIUM BREAKTHROUGH CURVE IN EL BERROCAL (ADAPTED FROM DONADO ET AL., 2005)..... 15 FIGURE 3.4 (A) MASS DISTRIBUTED NORMALLY VERSUS (B) MASS IN TWO SMALL PLUGS (ADAPTED FROM KITANIDIS, 1994).......................................................................................................................................... 18 FIGURE 7.1 CONSTRUCTING THE CONDUCTIVE NETWORK. (A) FROM INDIVIDUAL FRACTURES TO 1-D ELEMENTS. (B) GOING FROM A 3-D DFN TO A MESH OF 1-D ELEMENTS. GRAYSCALE REPRESENTS FAMILIES. (FROM DONADO ET AL., 2005B) ............................................................................................................................... 35 FIGURE 7.2 OBSERVED (DOTS) VS. COMPUTED DRAWDOWN (LINES) AFTER CALIBRATION OF A PUMPING TEST (PUMPING IN S14.2 AND THREE ADDITIONAL OBSERVATION POINTS) IN EL BERROCAL USING A DFN APPROACH. TWO DIFFERENT FITS (AMONGST THE 20 NETWORKS AVAILABLE) ARE PRESENTED; NETWORK A, CORRESPONDING TO ONE OF THE BEST ONES, AND NETWORK B, WHICH PROVIDES AN AVERAGE FITTING. (FROM DONADO ET AL., 2005B).................................................................................................................... 37 FIGURE 7.3 STEPS FOR A SEMIAUTOMATIC CALIBRATION OF A BTC IN A DFN MODEL (FROM DONADO ET AL., 2005B) ......................................................................................................................................................... 38 FIGURE 7.4 OBSERVED (DOTS) VS. COMPUTED CONCENTRATIONS (LINES) AFTER CALIBRATION OF A TRACER TEST IN EL BERROCAL USING A DFN APPROACH. (A) DEUTERIUM: ADVECTION – DISPERSION – MATRIX DIFFUSION MODEL. (B) URANINE: ADVECTION – DISPERSION MODEL (FROM DONADO ET AL., 2005B). ....... 39 FIGURE 7.5 RELATION BETWEEN REACTION RATIO AND TIME WITH DIFFERENT SPACES, EC. 7.32 ....................... 46 FIGURE 7.6 RELATION BETWEEN REACTION RATIO AND SPACE WITH DIFFERENT TIMES, EC. 7.32........................ 46

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LIST OF TABLES Pag.

TABLE 3.1 DENSITY FUNCTIONS FOR PHYSICAL PROCESSES (FROM HAGGERTY ET AL., 2000) ...............................16 TABLE 6.1 SCHEDULE OF THE WORK-PLAN...........................................................................................................30

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AKNOWLEDGEMENTS

I would like to acknowledge to my adviser, Dr. Xavier Sanchez-Vila for his priceless advices. I would like to thank to Dr. Marco Dentz, (Ramon & Cajal Researcher) for his valuable math guidance. I would also like to recognize the support of my ALβAN Co-Tutor, Dr. Nelson Obregón Neira, Associate Professor of the National University of Colombia. Additionally, I would express gratitude to the support of the Programme ALβAN, European Union Programme of High Level Scholarships for Latin America, identification number E03D22383CO in the Research & Education Project “CHARACTERIZATION OF THE EFFICIENCY OF AQUIFER REMEDIATION METHODS IN HETEROGENEOUS MEDIA”. In the end, this dissertation is carried on the framework of the European Project FUNMIG. Fundamental Processes of Radionuclide Migration, 2005 – 2009.

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ABSTRACT An aquifer typically displays a high spatial heterogeneity in its physical and chemical characteristics. Flow and transport processes as well as reaction are dominated by spatial variability and in general very different from the corresponding processes observed in homogeneous media. Detailed knowledge of heterogeneity structure is generally not available and the detailed micro-scale description of the relevant processes is neither efficient nor desirable for field-scale modeling of reactive transport. The objective of the proposed project is to develop analytical and numerical tools that allow for an effective large-scale description of reactive transport processes in heterogeneous formations on the basis of the known micro-scale reaction and transport dynamics. At this point, two main lines of research that will lead us toward this aim are reported on. Flow and transport in highly heterogeneous geological media by inverse modeling are studied. The observed breakthrough curves give clear evidence of anomalous transport patterns (tailing) possibly caused by sorption to the solid matrix and mass transfer (matrix diffusion) between mobile and immobile regions. Multirate mass transfer (MRMT) represents a semi-phenomenological approach to model such anomalous transport features. MRMT quantifies the impact of small-scale spatial heterogeneities on the effective transport behavior in terms of the temporal memory function. This effective transport description and geochemical reactions are coupled. The derived explicit analytical expressions for the reaction rates of sorption processes and precipitation/dissolution reactions allow for a systematic study of the impact of matrix diffusion on such processes. Subsequently, the following procedure is upscaling of groundwater transport parameters, and generalizing large-scale expressions. Finally, reaction rates of geochemical processes in heterogeneous media will be obtained. The generalization of hydrodynamics processes in the aquifer at large-scale will be carried out by means of upscaling mechanisms and quantification of mixing.

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RESUMEN Un acuífero presenta usualmente alta heterogeneidad especial en sus características físicas y químicas. Los procesos de flujo y transporte así como los de reacciones químicas son dominados por la variabilidad espacial y en general con un comportamiento muy diferente a los correspondientes procesos en medios homogéneos. Generalmente, la estructura detallada de la heterogeneidad del medio no es conocida, y la descripción particular de los procesos relevantes en micro escala no es ni eficiente ni la deseada para la modelación del transporte reactivo a escala de campo. El objetivo de la tesis es desarrollar herramientas analíticas y numéricas que permitan una descripción a escala de campo de los procesos de transporte reactivo en medios heterogéneos con base en el conocimiento de las reacciones y la dinámica de transporte a micro escala. Hasta el momento, se cuenta con dos tipos de resultados obtenidos. Los primeros se refieren al estudio del flujo y transporte en medios geológicos altamente heterogéneos, por medio de modelación inversa. Se calibraron curvas de llegada de solutos que presentan una clara evidencia de patrones anómalos de transporte (colas alargadas), posiblemente causados por adsorción de la matriz sólida o transferencia de masa (difusión en la matriz) entre las regiones inmóviles o móviles. La transferencia de masa multitasa (MRMT, por sus siglas en inglés) representa una aproximación semi-empírica para modelar dichas anomalías en el transporte. MRMT cuantifica el impacto de las heterogeneidades espaciales de pequeña escala en el comportamiento efectivo del transporte en términos de una función temporal de memoria. Estas expresiones analíticas de las velocidades de reacción se relacionan con las reacciones geoquímicas. La derivación explicita de expresiones analíticas para la velocidad de reacción de procesos de adsorción y precipitación/disolución permite el estudio sistemático del impacto de la difusión en la matriz de dichos procesos. El siguiente procedimiento a llevar a cabo es el de escalado de los parámetros de transporte y la generalización para expresiones en escala de campo. Finalmente, el proyecto espera determinar la velocidad de reacción en medios heterogéneos y lograr generalizar los procesos hidrodinámicos en acuíferos a escala real, mediante mecanismos de escalado y la cuantificación de procesos de mezcla.

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RESUM Un aqüífer presenta usualment alta heterogeneïtat especial en les seves característiques físiques i químiques. Els processos flux i el transport així com els de reaccions químiques són dominats per la variabilitat espacial i en general molt diferent als processos corresponents a mitjans homogenis. Generalment, l’estructura detallada de l’heterogeneïtat del mitjà no és coneguda, i la descripció detallada dels processos rellevants en micró escala no és ni eficient ni la desitjada per a la modelació del transport reactiu a escala de camp. L’objectiu de la tesi és desenvolupar eines analítiques i numèriques que permetin una descripció a escala de camp dels processos de transport reactiu en mitjans heterogenis amb base en el coneixement de les reaccions i la dinàmica de transport a micró escala. Fins al moment, es conta amb dos tipus de resultats obtinguts. Els primers es refereixen a l’estudi del fluix i transport en mitjans geològics altament heterogenis, per mitjà de modelació inversa. Es van calibrar corbes d’arribada de soluts que presenten una clara evidència de patrons anòmals de transport (cues allargades), possiblement causats per adsorció de la matriu sòlida o transferència de massa (difusió en la matriu) entre les regions immòbils o mòbils. La transferència de massa multi tesa (MRMT, per les seves sigles en anglès) representa una aproximació semi - empírica per a modelar aquestes anomalies en el transport. *MRMT quantifica l’impacta de les heterogeneïtats espacials de petita escala en el comportament efectiu del transport en termes d’una funció temporal de memòria. Aquestes expressions analítiques de les velocitats de reacció es relacionen amb les reaccions geoquímiques. La derivació explicita d’expressions analítiques per a la velocitat de reacció de processos d’adsorció i precipitació/dissolució permet l’estudi sistemàtic de l’impacta de la difusió en la matriu d’aquests processos. El següent procediment a portar a terme és el d’escalat dels paràmetres de transport i la generalització per a expressions en escala de camp. Finalment, el projecte espera determinar la velocitat de reacció en mitjans heterogenis i assolir generalitzar els processos hidrodinàmics en aqüífers a escala real, mitjançant mecanismes d’escalat i la quantificació de processos de barreja.

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Chapter 1 1. INTRODUCTION Groundwater plays an important role in the development of the world’s water-resource potential. As such it has to be protected from the increasing threat of subsurface contamination. The growth of population and industrial and agricultural production, coupled with the resulting increased requirements for energy development, has for the first time in man’s history begun to produce quantities of waste greater than what the environment can easily absorb. The pressure to reduce surface pollution is in part responsible for the use the subsurface for waste disposal (Freeze & Cherry, 1979). The main sources of pollution (Figure 1.1) are municipal and industrial waste disposal, agricultural activities, petroleum spills, mining activities and less frequent, but potentially more harmful, toxic and radioactive waste. Coastal aquifers can suffer salt-water intrusion as a consequence of over-exploitation. In all of these problems the interaction of transport in groundwater and geochemical reaction has to be represented correctly. The problems of groundwater quality degradation are in many ways more difficult to overcome, because of the heterogeneities inherent in subsurface systems and they can be very difficult to detect. Quantitative and qualitative understanding of contaminant transport in groundwater is a prerequisite for the proper prediction of natural attenuation processes, design of waste disposal facilities and remediation of contaminated sites. This understanding is complicated. On one hand, solutes undergo a wide range of (bio)geochemical reactions. On the other hand, spatial heterogeneities in the physical and chemical properties of the medium and temporal fluctuations due to external forces (such as recharge) lead to non-trivial scale dependence of transport and reaction phenomena (Dentz, 2005). While this is widely recognized, current reactive transport models do not take into account the complicated dynamics inherent to heterogeneous reactive transport system.

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L. D. Donado G. – Ph. D. Dissertation Proposal © 2005

Figure 1.1 Sketch of the transport processes in the environment (From Tindall et al., 1999)

Spatial variability and uncertainty in characterizing the flow domain play an important role in the transport of contaminants in porous media: they affect the path lines followed by solute particles, the spread of solute bodies, the shape of breakthrough curves, the spatial variability of the concentration, and the ability to quantify any of theses accurately (Rubin, 2003). Actually it is well known that while the classical advection-dispersion equation holds at the local (short range) scale, it cannot be generally extended to larger scales. Therefore, some semi-empiric approaches to characterize groundwater flow and transport in heterogeneous media at the intermediate to large scale have been developed in the last few years. The most widely used methods are: (1) Continuous Time Random Walk (CTRW), (2) Multirate Mass Transfer (MRMT) and (3) Fractional AdvectionDispersion Equations (FADE) (Dentz, 2005). The MRMT has been used to characterize the effect of heterogeneity on transport and proved to be a proper representation of transport process at local scale for many geochemical or physical reactions that causes the solute to behave in a non-conservative manner. As an example, the distribution of mass transfer rates is influenced by several soil characteristics as the type of minerals as well as the geometry of the porous lattice (Haggerty and Gorelick, 1998). Thus, MRMT represents a framework to integrate the impact of heterogeneity on physical and geochemical processes. This is the approach we will pursue in this Thesis. In summary, the objective of the proposed Thesis is to develop a methodology that explains and correctly models of the transport of solutes that react amongst themselves and with the solid matrix in heterogeneous media using an approach based on the MRMT model. Both analytical and numerical methods will be developed to fulfill this objective. Some preliminary results are presented in this report.

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Chapter 1. Introduction

The outline of the dissertation proposal is as follows. The justification and pertinence of the dissertation are presented in chapter two. Chapter three states the physical and the mathematical framework. Concepts and principles about transport in heterogeneous media, reactive transport, the multirate mass transfer approach, and dilution are defined in this chapter. Chapter four brings forward the state of the art of reactive transport in heterogeneous media. It starts with the actual advances in the multirate mass transfer approach applied to the solution of reactive transport problems in heterogeneous media. The precise objectives (both the main as well as some subsidiary objectives of this research are stated in chapter five. In chapter six the methodology to obtain the objectives is described. Work-plan consists on five stages clearly presented. It starts with the compilation of the state of art, which is actually raised in this document. The following stages comprise: (1) analyzing breakthrough curves of solutes in heterogeneous media, (2) developing analytical expressions for reaction rates for some reactive transport processes as matrix diffusion, precipitation/dissolution and sorption/desorption, (3) properly quantifying the difference between mixing and dispersion and (4) generalizing dynamic process in large scale with the use of upscaling mechanism. Chapter seven shows the preliminary obtained results. Two types of results are presented. The first one refers to the analysis of tracer and hydraulics tests in heterogeneous media. The second one refers to the first analytical expressions for reactive transport. Finally, an appendix with all the detailed process of solution of the preliminary results is presented for completeness. To conclude, the achieved results are separated in two sets. The first one includes the calibration and analysis of breakthrough curves of conservative solutes in fractured geological media. This analysis was presented in June 2005 in ModelCARE 2005 Conference in The Netherlands (Donado et al., 2005a) Using a discrete fracture network as a conceptual model, this part of the research obtained good fits of the advection – dispersion – matrix diffusion model (Carrera et al., 1998), which is based on the concept of a matrix diffusion term can be used to model properly a memory function term. This memory function term incorporates all the fluctuations and variations of the porous media caused either by a real physical process of diffusion into pseudo-stagnant zones, or as an upscaling term incorporating heterogeneity. The second set includes preliminary results of the generation of analytical expressions for the reaction rate of reactive transport processes with a general multi-rate mass transfer term are presented are presented. Firstly, the expressions are for homogeneous media, where all the flow and transport parameters are constants are shown. Then, the problem of transport of solutes undergoing precipitation/dissolution, as well as multirate mass transfer processes to account either for sorption or just heterogeneity is tackled. The main result is obtaining an expression for the precipitation/dissolution rate at every point in space and time, as an extension of the work of De Simoni et al. (2005). The methodology is based on expressing solute transport in terms of conservative components, solving afterwards the speciation problem to obtain the concentration of the actual

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species. The methodology is further applied to transport driven by a step-input function of concentration, solved by Ogata in 1970 for a conservative solute. The final expressions are mathematically very complex. Using a simplification of the memory function for one or more sites, it is possible to obtain simpler formulations that can be analyzed in a more convenient way.

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Chapter 2 2. PROBLEM STATEMENT 2.1. PROBLEM APPROACH During recent years, a large body of research has been carried out about aquifer remediation systems. One of the main objectives has been to provide industry with new techniques to reduce the quantity of solutes present in the subsurface (soils and aquifers) to acceptable limits, so that water can be of some use. Many predictive techniques for the removal of contaminants overestimate the quantity of removed contaminant and underestimate the required time to achieve acceptable levels of solutes in the water. Therefore, in general the existing methods for the prediction of solute removal are neither accurate, nor conservative. Numerical modeling of those processes suggests that a possible reason for the differences is the high degree of heterogeneity of the aquifer media, coupled with a wide range of physical and geochemical reactions. This research project will study the influence of the heterogeneity of the aquifer (by using a different spatially distributed hydraulic conductivity) and chemical reactions on the efficiency of aquifer remediation (Donado and Sanchez-Vila, 2003). The aim of the dissertation is to develop a methodology that explains and correctly models the transport of solutes that react amongst themselves and with the solid matrix in heterogeneous media. Both analytical and numerical methods will be developed towards this objective. To extend the local state results to the field scale is necessary to perform an upscaling process. It will be approached in the framework of a stochastic model for the spatially and temporally fluctuating system characteristics.

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Expected results include the identification and evaluation of effective transport parameters and reaction rates, as well as a systematic study and quantitative characterization of the large scale transport and reaction dynamics, which are, in general, different from the ones found on the local or microscale. These findings can explain the relative failure of current models to study and remediate groundwater contamination and lead to new methods, which are more based on flow controlled mechanisms to enhance the mixing of reactants (Dentz, 2005).

2.2. MOTIVATION There are many factors that complicate the understanding of the transport of solutes in natural formations. They are (1) reaction between pollutants themselves and with the porous media matrix; (2) heterogeneity of the hydraulics properties (singularly hydraulic conductivity); (3) heterogeneity of the parameters controlling mass transfer processes; (4) cross-correlation between both heterogeneous parameters; and (5) temporal fluctuations of external recharge or boundary conditions. All of them produce dramatic changes in the spatial and temporal concentration distributions, with respect to those of transport in homogeneous media. The transport coefficients and the reaction rates delineated in a homogeneous media are not generally pertinent in large scale heterogeneous media. In fact, the length of the pollutant plume obtained with local dispersion is different than the one achieved with the large scale dispersion coefficient. This one varies several orders of magnitude respect to the local dispersion coefficient (e.g. Sudicky et al., 1979; Smith and Schwartz, 1980; Freyberg, 1986). However, the role of the heterogeneity in the mixing of reactive solutes and its impact in the effective reaction and in the dynamics of the transport is not well known yet. It represents an important and active research frame (Islam et al., 2001, van der Lee and De Windt, 2001). The knowledge of reactive and local scale transport processes and their impact in spatial and temporal fluctuations are essential for the understanding of the hydrodynamics of the aquifer, and for the correct design of disposal storage sites or remediation techniques. In summary, this dissertation has as an objective to advance in the understanding of the chemical processes that are relevant in the fate of contaminants in natural media. All of this is useful for evaluating the role of the spatial heterogeneity and temporal fluctuations in the efficient transport of pollutants.

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Chapter 2. Dissertation Description

2.3. PERTINENCE The dissertation is developed in the framework of the integrated project of the European Union, FUNMIG (Fundamental Processes of Radionuclide Migration, 2005 – 2009). Its main objective is the study of chemical reactions and the processes of migration of radionuclide in clay, granite and salts. All of this implies Upscaling and deduction of effective parameters that control the reactive transport of multiple species. Also, this project is done with the support of the Programme ALβAN, European Union Programme of High Level Scholarships for Latin America, identification number E03D22383CO.

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Chapter 3 3. THEORETICAL FRAMEWORK The aim of the thesis is defining new expressions for the reactive transport processes in heterogeneous media. For this reason, a theoretical framework of the principal concepts is shown. First of all, this chapter presents an explanation of the transport of reactive solutes in porous media. Secondly, it is presented a brief description of two types of reactions which will be analyzed in terms of its reaction rate. Furthermore, a special case of diffusion in porous media is shown. The matrix diffusion is on hand in all the porous media, especially in fractured media, and couples to many other processes, leading to highly nonlinear effects. Finally, this chapter presents a brief description of the concept of dilution, and the multiple rate mass transfer approach for the transport governing equation, two issues that are of high relevance in this Thesis. The chapter ends with a summary of applications of this approach in real situations.

3.1. TRANSPORT OF REACTIVE CONSTITUENTS The transport of reactive solutes is studied with the same equation as that of conservative ones, with the effect of reactions included as sink/source terms (Freeze and Cherry, 1979). Changes in concentration can occur because of chemical reactions that take place entirely within the aqueous phase or because of the transfer of the solute to or from other phases such as the solid matrix of the porous media, or the gas phase in the unsaturated zone. The reactions that can alter contaminant concentrations in groundwater flow systems are classified by Freeze and Cherry (1979) in six groups: (1) adsorption/desorption reactions, (2) acid/base

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reactions, (3) solution/precipitation reactions, (4) oxidation/reduction reactions, (5) ion pairing or complexation, and (6) microbial cell synthesis. This work aims at the understanding of water/rock interactions, disregarding microbial effects. The transfer by any chemical process of contaminant mass from the pore water to the solid (immobile) part of the porous media causes the contaminant front to be retarded. As a contaminant plume advances along flow paths, this retardation is the net result of transfer of part of the contaminant mass to the solid phase (De Marsily, 1986). This means that in homogeneous media, the reaction can be represented by a retardation factor. The movement through the porous media is illustrated in the Figure 3.1. Flow occurs only in the mobile water regions where the contaminant is transported by advection and dispersion (Rubin, 2003).

Figure 3.1 Schematic of mobile-immobile domain model (From Cushey, 1996)

In some cases the solute interacts with another mobile species, causing facilitated transport, which eventually can cause the contaminant front to go faster than in a homogeneous media. This issue will not be addressed in the Thesis Dissertation, nor is it in the present report.

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Chapter 3. Theoretical Framework

3.2. WATER/ROCK INTERACTIONS Different processes affect the mobility of the groundwater and its components through the rock matrix, which consist of inorganic (minerals and amorphous compounds) and organic matter. Solids are important to the geochemistry of the system because they are the primary sources and sinks of dissolved constituents. These processes are briefly presented in this chapter. 3.2.1.

Adsorption / Desorption

Adsorption is the removal of a dissolved species from solution by attachment to the surface of a solid. Desorption is the release of the species back into the solution. Adsorption effectively lowers the resident concentration in the mobile phase (the one that can eventually be sampled). Ion exchange is a type of adsorption/desorption phenomenon that primarily impacts the major cations in solution and material with a porous lattice containing fixed charges (Deutsch, 1997). This process is governed by a parameter called Cation Exchange Capacity, which is a function of the amount of inorganic clay minerals and organic humus present in the solid phase. Anion exchange may also occur on clay minerals, but in much lesser extent than cation exchange, because of the dominant fixed negative charge on the clay mineral surface (Deutsch, 1997). The cation exchange is the capacity available to groundwater will be a function of the amount of inorganic clay minerals and organic humus present in the solid phase. Anion exchange may also occur on clay minerals, but much lesser extent than cation exchange because of the dominant fixed negative charge on the clay mineral surface (Deutsch, 1997). In addition to the charge on the anion, the smaller the radius of the ion, the closer it can approach the surface and the greater the affinity of the surface for the ion. However, ions in solution are surrounded by water molecules; therefore their “effective” radius is the radius of the solvated ion and not the bare ion. The larger the bare ion, the smaller the solvated radius; thus for the monovalent ions the affinity of the solid exchanger is as follows (Deutsch, 1997):

Cs + > Rb + > K + > Na + > Li + The equilibrium «constants» (K) for ion exchange reactions are not actually constant and vary with pH because of the pH-dependent charge on the solid surfaces, and it varies with total solute concentrations, when heterovalents ions (such as K+ and Mg+2) are exchanging. Furthermore, K varies with the type of exchanger present in the solid phase. For these reasons, the K values for ion exchange reactions may better be considered exchange coefficients instead of equilibrium constants (Deutsch, 1997). Perhaps the most common impact of ion exchange on water quality is the case of seawater intrusion into freshwater aquifer. Seawater is predominantly a NaCl solution, and the high sodium concentration in the intruding seawater will drive the ion exchange reaction in the direction to adsorb Na+ and release the divalent cations. The ion exchange retards the movement of Na+ relative

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to an ion such Cl- in the seawater that is not as strongly affected by ion exchange or other water/rock interactions (Selker et al., 1999). The impact of cation exchange on a system can be reasonably well predicted from knowledge of the cation exchange capacity and the concentrations of major cations. The pH is important in determining the available anion exchange capacity, which is generally much lower than the cation exchange capacity (CEC). It is also important in the exchange capacity of the materials with organic matter present in a significant portion. 3.2.2.

Mineral Precipitation/Dissolution

A mineral in contact with groundwater represents a geochemical system consisting of a solid phase and a solution phase. If an instantaneous reaction is assumed, at any given time the concentration of the ions is in equilibrium with the mineral, which means that at every point dissolution and precipitation of the solid are in balance and solution concentrations of the mineral components are fixed at the equilibrium values. The solubility of a mineral may be a function of the temperature, pH, Eh, concentration of complexing ligands, and ionic strength of the solution. The equilibrium constants for the minerals are all temperature dependent; for example, the carbonate minerals become less soluble as the temperature increases (Deutsch, 1997), so that, e.g., cooling will lead to enhanced dissolution. If the components of a mineral form complexes with other dissolved species in solution, then the effective dissolved concentrations (activities) of the components are reduced requiring more of the mineral to dissolve to achieve equilibrium concentration levels. The ionic strength of a solution is a measure of the ion shielding that occurs around charged dissolved species. Ion shielding lowers the activity of dissolved species; therefore the higher the ionic strength, the larger the shielding and also the larger the solubility of minerals in contact with solution (Selker et al., 1999). 3.2.3.

Reaction Rates

The use of equilibrium constants and distribution coefficients to calculate solution concentration as a function of mineral solubility, adsorption/desorption, and ion complexation assumes that the system comes to equilibrium with respect to all of its phases (solution, gas and solids). Equilibrium is a good assumption for reactions that take place only in solution. The one notable exception is redox equilibrium, where it has been found that equilibrium does not commonly exist between redox couples and the measured redox potential (Deutsch, 1997). Adsorption of solutes onto the surfaces of solids phases is also generally considered a fast reaction, especially on the time scale of groundwater residence time.

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3.3. TRANSPORT IN FRACTURED MEDIA The behavior of the reactive transport in fractured media is of interest for the design of storage of radioactive wastes. Fractured media are intrinsically heterogeneous, as flow takes place only in a fraction of the full domain (the rock matrix is mostly considered impervious). Fractured media can be represented by four kinds of conceptual models: (1) an equivalent porous media, (2) a discrete fracture porous media, (3) mixed models, including an equivalent porous media with the most relevant water conductive features included explicitly, and (4) a network of channels. The first one assumes a volume segment of the medium that is sufficiently large to be described by hydraulic conductivity and porosity averaged over the bulk mass. The second one is based on the representation of the fractures as conductive elements. It poses a fundamental problem which is the large number of parameters that are involved. The third one combines the advantages of the previous two. The last one has seldom been used in practical applications. One of the particular phenomena regarding transport in porous media is the existence of diffusion of solute toward the immobile rock matrix. This important process is explained in the following section.

3.4. MATRIX DIFFUSION Matrix diffusion is a transport mechanism by which solutes transfer from the water-flowing portions of permeable media to the non-flowing portions (matrix), and vice versa through molecular diffusion (Carrera et al., 1998). Thus, the process is controlled by the exchange between mobile and immobile zones (Figure 3.2). The effects of matrix diffusion are diverse and most important whenever matrix porosity (volume of immobile voids) is large. First of all, an apparent retardation respect to solutes that do not enter the matrix is caused by the intrusion of the solute by diffusion in a large volume of voids, but also sorption sites can be reached by the solute through diffusion causing further retardation. Moreover, under such conditions, sorption is not instantaneous, but delayed. The solute apparently behaves as if affected by kinetic sorption (Carrera et al., 1998). Another effect of matrix diffusion is tailing on breakthrough curves (BTC). Figure 3.3 shows a BTC of El Berrocal Site fitted with an Advection-Dispersion-Matrix Diffusion model. This fit presents the tailing present in BTC because of the sorption processes. This means that solutes take a long time to come out of the matrix because concentration gradients are small. Since heterogeneity also causes tailing on breakthrough curves, it is sometimes difficult to distinguish between these two processes (Carrera et al., 1998). The fact that matrix diffusion causes effects (retardation and tailing) similar to other mechanisms (heterogeneity and kinetic sorption) is a source of difficulties in solute-transport

13


modeling. Mathematically, all three mechanisms can be modeled by means of memory functions that incorporate the effects of nonlinear transport. Indeed, experimental breakthrough curves can be fitted to a number of numerical models that adapt matrix diffusion process in its solution, as in Donado et al. (2005a).

Figure 3.2 Conceptual model of matrix diffusion (top) and matrix block geometries (bottom) (From Carrera et al., 1998)

3.5. MULTIRATE MASS TRANSFER (MRMT) Mass transfer between immobile and mobile zones is a consequence of simultaneous processes that cause small-scale variations in the aquifer properties. These processes are listed in Haggerty and Gorelick (1995). The multirate mass transfer is a nonlinear relationship between two different phases (e.g. mobile and immobile) with a full distribution of rate coefficients. This process is coupled to an advective-dispersive solute transport model. This model allows incorporating small-scale variation of rates and types of mass transfer by using a series of first-order equations to represent the contribution of each of the mass transfer rate (Haggerty and Gorelick, 1995).

14


0.012

-1

Concentration (mg L )

0.010 0.008 Mesured Simulated

0.006 0.004 0.002 0.000 0

5

10

15

20 25 Time (days)

30

35

40

45

Figure 3.3 Deuterium Breakthrough Curve in El Berrocal (Adapted from Donado et al., 2005)

The physical model of mobile-immobile mass transfer may be written in a general form (Haggerty and Gorelick, 1995):

∂cm ∂c + β tot im = L ( cm ) ∂t ∂t

3.1

Where cm [M L-3] is aqueous concentration in the mobile zone, cim [M L-3] is the sorbed concentration in the immobile zone for chemical models or the aqueous concentration in the immobile zone for others models, βtot [dimensionless] is the “capacity ratio” and is the ratio of total contaminant mass in the immobile zone to the mass in the mobile zone at equilibrium defined by:

β=

φim + (1 − f ) ρb K d Rimφim = φm + f ρ b K d Rmφm

3.2

Where ρb [M/L3] is the bulk density of the porous media, f [dimensionless] is the fraction of sorbed phase in sorption equilibrium with the mobile zone, Rim [dimensionless] is the retardation factor associated with the immobile zones and Rm [dimensionless] is the retardation factor associated to mobile zones. In [3.1] L ( cm ) is the advective – dispersive term which is defined by: L ( cm ) = L ( cm )advective + L ( cm )dispersive

φm L ( cm ) = − q∇cm + ∇ ( D∇cm )

3.3

15


Where q [L T-1] is Darcy’s flow and D [L2 T] is the hydrodynamic dispersion tensor. Finally, the expression [3.1] could be represented by (Wang et al. 2005):

∂cm φim + Γ ( t ) = L ( cm ) ∂t φm ∂cm φim + ∂t φm

∞

∫

∂c f (α ) im dα = L ( cm ) ∂t

3.4

0

Where α [dimensionless] is the rate coefficient and f(α) is probability density function of the continuous set of first-order rates. Haggerty et al. (2000) state the density functions various types of mass transfer as given in Table 3.1.

Table 3.1 Density functions for physical processes (From Haggerty et al., 2000)

Model

f (α )

Linear Sorption

βtot δ (α − α i ) ∞

Diffusion: finite layer

∑ j =1

2 ⎡ 2 j − 1) π 2 Da ⎤ ( δ ⎢α − ⎥ 2 4a 2 ⎥⎦ ( 2 j − 1) π 2 ⎢⎣

8β tot

∞

Diffusion: spheres

∑ βπ δ α j =1

6 j2

tot 2

⎛ 2 2 Da ⎞ ⎜ − j π a2 ⎟ ⎝ ⎠

Da: Apparent diffusivity a: Half thickness of the domain

The governing differential equation for the domain is given as:

∂cim 3.5 = α ( cm − cim ) ∂t Where α = α´/Rim, and where α´ is the first-order rate coefficient for a physical model with no sorption.

16


3.6. DILUTION Kitanidis (1994) defined the differences between spreading and dilution, and introduced the concept of the dilution index. Spreading is associated with the stretching and deformation of a contaminant plume, whereas dilution is associated with the increase in volume of the fluid occupied by the solute. In homogeneous media both are characterized by the same parameters (the dispersion coefficients), but in heterogeneous media, the plumes are irregular in shape, and the dispersion coefficients deduced from tracers tests are not reliable measures of the rate at which a tracer plume spreads about its centroid. In conclusion, the rates of dilution and spreading can be quite different in heterogeneous media. The index that quantifies the degree of dilution of a tracer in a system is defined to discrete case by:

⎡ E = ∆V exp ⎢ − ⎣⎢

∑ n

k =1

⎤ Pk ln ( Pk ) ⎥ ⎦⎥

3.6

Where E is the dilution index and p is the concentration normalized by the total mass of solute. The definition can be extended to a continuous case as:

⎡ E ( t ) = exp ⎢ − ⎣

∫

V

⎤ p ( x, t ) ln ( p ( x, t ) ) dV ⎥ ⎦

3.7

The dilution index should satisfy the following criteria (Kitanidis, 1994): 1.

It should correspond to our intuitive notions of what is degree of dilution.

2. It should be proportional to the volume of fluid occupied by the solute. 3. The maximum degree of dilution for a certain setup should correspond to the concentration that formally maximizes the value of the index. Spreading does not necessarily mean dilution. If the distribution of the mass is bell shaped (Figure 3.4a) dilution and spreading are similar concepts. If, otherwise, the plume consists of two small plugs of equal mass, the plume is spread out, but dilution is practically nil. Two cases are characterized by the identical mass, centroid location, and dispersion, but the degree of dilution is certainly lower in Figure 3.4b (Kitanidis, 1994). Intuitively, dilution implies that the mass is distributed over a large volume of water.

17


c

a

c

0.4

4

0.3

3

b

0.2

2

1

0.1

-3

-2

-1

0

1

x/d

2

3 -3

-2

-1

0

1

x/d

2

3

Figure 3.4 (a) Mass distributed normally versus (b) Mass in two small plugs (Adapted from Kitanidis, 1994)

Another important concept is the reactor ratio, M. It is a shape factor, which measures how stretched and deformed the plume is (Thierrin and Kitanidis, 1994). It is defined as the ratio of the actual dilution index to the maximum dilution index that is theoretically possible:

M = E / Emax

3.8

The reactor ratio is a dimensionless number between 0 and 1 that describes how complete is the actual dilution compared to the theoretically maximum dilution that is possible given the setup (Kitanidis, 1994). A value near to 0 indicates incomplete dilution, and a value near 1 indicates a high degree of dilution.

18

Chapter 4 4. STATE OF THE ART The generation of new expressions for reactions effects in the process of transport in any type of media is fundamental for the understanding of the hydrodynamics of the media. In this chapter, a revision of the present state of the knowledge is stated. The knowledge of processes is very well known, but this knowledge is not necessarily true when we consider transport in heterogeneous media. There are also many outstanding issues, such as the behavior of solute transverse spreading. Procedures to upscale flow and transport variables are still under development and there is quantitative understanding only of the influence of density, two phases fluids, and complex reactive properties of solutes (Dagan, 2002). Reactive transport in groundwater is defined by the interaction of two types of processes: dispersive and diffusive transport in groundwater flow through the porous media and chemical reactions. The understanding of these types of processes and the reaction mechanisms is necessary. There is huge quantity of literature on the impact of the spatial heterogeneity in the effective transport of an individual solute. The majority of studies in this topic evaluate the transport in terms of effective macrodispersion coefficients. Because of this, the actual approaches do not calculate dispersion and mixing induced by the heterogeneity for relevant lengths and times (Kitanidis, 1994) and does not take into account that the transversal dispersion plays an important role in reactive transport (Klenk and Grathwohl, 2002). As such, present developed concepts in stochastic hydrogeology are limited to characterizing mixing of sorbing solutes and calculating the effective reaction velocity in heterogeneous media. An immense quantity of studies determines effective transport coefficients as well as dispersion and retardation, for instantaneous linear sorption. They are not able to modify the

19


dynamics of the reactions and the transport in the upscaling process. This circumstance leads to an erroneous representation of solute transport at large scales, as effective coefficients are generally used in an invalid dynamic equation. For this reason, anomalous phenomenon in transport as the tail in the breakthrough curves (e.g. Donado et al., 2005b) and atypical solute distributions in highly heterogeneous media are not well represented by the actual methods (Adams and Gelhar, 1992). In this chapter, the first section reviews the actual methodology to characterize reactive transport, and a first approximation for the analysis of the reactions in homogeneous media, with a typical advection/dispersion model. Then, the state of the art of new types of transport models to correctly represent the reaction processes is presented.

4.1. REACTIVE TRANSPORT Reactive transport in groundwater is defined by two types of processes: (1) dispersiondiffusion transport in the porous media, and (2) chemical reactions as well as acid/base and redox reactions, complex formation, biodegradation, radioactive decay, ion exchange, sorption/desorption and precipitation/dissolution of minerals. A classical classification divides reactions into two types: (1) homogeneous in aqueous phase and (2) heterogeneous reactions with interactions between liquid and solid phases (Rubin, 1983). Saaltink et al. (1998) presented a general mathematical formulation for solving the reactive transport problem in homogeneous media, and Molins et al. (2004) presented a new formulation for decoupling components in reactive transport in homogeneous media. The complexity of the fate of pollutants and geochemical processes has inhibited the development of analytical solutions for multicomponent heterogeneous reactions such as precipitation/dissolution (De Simoni et al., 2005). They presented a procedure to solve groundwater reactive transport in the case of homogeneous and heterogeneous equilibrium reactions induced by mixing different waters.

4.2. MULTIPLE RATE MASS TRANSFER The multiple rate mass transfer approach (MRMT) has been studied in last few years. Haggerty and Gorelick (1995) present a multirate model incorporated into the advective-dispersive equation that allows modeling of small scale variation in rates and types of mass transfer by using a series of first order equations to represent each of the mass transfer processes. They compared the multirate model to the standard first-order and diffusion models of mass transfer, and shown that a

20

Chapter 4. State of the art

combination of these can be represented exactly with the multirate model (Haggerty and Gorelick, 1995) further developed solutions to the multirate equations under conditions of no flow, fast flow and radial flow to a pumping well. Carrera et al. (1998) stated that matrix diffusion is a process that can be defined as a multirate mass transfer process, representing the matrix diffusion term as a convolution incorporating a memory function. Chen and Wagenet (1995) proposed a model for describing solute nonequilibrium transport influenced by sorption-site heterogeneity in porous media and analyzed in terms of its time moments. They concluded that higher moments immediately become highly sensitive to the rate of sorption statistics, rendering a substantial deviation between two-site and a general gamma site. A method for determining temporal moments of concentration for a solute subject to first-order and diffusive mass transfer in steady velocity fields was presented by Harvey and Gorelick (1995). They stated that the moments of all orders have the same form as the steady state non reactive transport and that the form of the moments for different mass transfer models gives close results for reactive transport in heterogeneous media. They conclude that if the velocity field is known, the mass transported past any point or out any boundary, can be calculated without any knowledge of the spatial pattern of mass transfer coefficients and without knowledge of weather mass transfer is occurring. Haggerty et al. (2000) found an expression for the late-time concentration, that present an asymptotic behavior in the tracer test BTCs with rate-limited mass transfer as a function of the advection time, the zeroth moment of the injection impulse and the mean residence time in the immobile domain. They conclude that if rate-limited mass transfer causes a BTC to behave as a power law at late-time, then the underlying density function of rate coefficients must also be a power law. Cunningham and Roberts (1998) examined the effects of nonuniform grain size on the transport of sorbing solute by means of the temporal moments. They concluded that higher-order temporal moments depend on the higher-order moments of the distribution of diffusional timescales and showed that in simulation of aquifer remediation, the appropriate choice for contaminant desorption rate depends upon the remediation goal. A methodology for evaluating the temporal moments of solutes undergoing linear rate mass transfer processes in heterogeneous media based on a Lagrangian approach was presented by Lawrence et al. (2002). The methodology was applied to desorption from an initially contaminated aquifer. Sanchez-Vila and Rubin (2003) presented a methodology for evaluating the unconditional and conditional moments of travel time for sorbing solutes, with heterogeneous sorption properties. The methodology is applicable for any flow configuration and for a wide range of mass transfer ratelimited linear processes. The low-order moments for reactive solutes can be obtained as a function of those for conservative tracers. The unconditional low-order moments can be obtained by means of a stochastic approach. They derived the conditional moments too, to any type (hard or soft) of information.

21


Haggerty and Gorelick (1998) modeled mass transfer process in soil columns. They revised the variability in mass transfer rates and quantified the effects of multiple, simultaneous mass transfer processes. They also reviewed the validity of conventional first-order and diffusion models to represent the mass transfer in subsurface systems. In conclusion, they said that the lognormal distribution of diffusion rate coefficients represent mass transfer much better than a single rate coefficient model. Gwo et al. (1998) evaluated the incorporation of mesoscale structure to fractured media and microscale hydrodynamics into a solute model for predicting solute movement. Their results suggest that mesoscale spreading of tracer in structured porous media might be largely attributed to interregion mass transfer. Maraqa (2001) showed that the mass transfer coefficient is dependent on system parameter including pore water velocity, length scale, retardation coefficient and particle or aggregate size. Also, predictive relationships between mass transfer coefficients to residence time were developed. Zinn and Harvey (2003) studied upscaling of groundwater flow and solute transport in two-dimensional fields with different patterns of conductivity and connectivity. They concluded that the large variations in velocity produce significant mass transfer behavior (e.g. tailings) and the mobile-immobile domain mass transfer may occur for a given amount of conductivity variability. Haggerty et al. (2004) compared the pore-water velocity and residence times, as well as the experimental durations of many experiments. They concluded that the estimated mass transfer time scale (inverse of the mass transfer rate coefficient) is better correlated to residence time and the experimental duration than to velocity. Dentz and Berkowitz (2003) showed that immobile concentrations are given as the convolution in time of the mobile concentrations and a memory function. They used the Laplace transform to develop explicit expressions for the time behavior of the flux concentration, as well as for the center of mass velocity and the (macro) dispersion coefficients of the solute distribution. Parallel work by the same group (Margolin et al., 2003) was oriented to relate the macroscopic transport behaviors of passive and sorbing tracers in heterogeneous media, by comparing the linear MRMT with the Continuous Time Random Walk Approach. They opened new perspectives for modeling transport in heterogeneous media. The last work was presented by Wang et al. (2004). They solved an integraldifferential equation in Laplace space for governing equations describing advective-dispersive transport with multirate mass transfer between mobile and immobile domains. This function is not limited to flow fields with constant velocities.

4.3. DILUTION Since the pioneering work on dilution by Kitanidis (1994), some authors have deepened in the application of the concept to characterize transport in heterogeneous media. An application of this theory is shown in Thierrin and Kitanidis (1994), where an analysis of the rate of dilution of a

22


conservative nonreactive tracer in two well known field experiment is done. In this paper, the dilution index and the reactor ratio (See Chapter 3) are evaluated. The experiments showed that the dilution index increases monotonically and linearly with time. They found that the maximum concentration was more erratic and more susceptible to sampling error. Kapoor and Kitanidis (1996) concluded that the spatial second moment and mean concentrations are insufficient measures of dilution, because concentration fluctuations can be large. Kapoor and Kitanidis (1997) further analyzed the behavior of a nonreactive solute undergoing advection and diffusion in a spatially random divergence-free flow field. They showed that the velocity variability and the smoothing effects of diffusion need to be included in a formulation of the spatial covariance of the concentration to obtain a qualitatively correct description of the concentration fluctuations. Finally, Kapoor and Kitanidis (1998) concluded that the concentration fields become more irregular with time. At early travel times, dilution and spreading can be severely disconnected. However, at large travel times, the volume occupied by the solute approaches that apparent from its spatial second moments. Finally, they concluded that the spatial second moment is increasingly disconnected from dilution, as time progresses. Cirpka and Kitanidis (2000a) concluded that analyzing the temporal moments of a breakthrough curve of a conservative tracer at a point within the domain gives additional information about the dilution of the tracer. From the local first and second central moments an apparent seepage velocity and an apparent dispersivity of mixing can be derived. Cirpka and Kitanidis (2000b) concluded that this information is not enough to distinct between spreading and mixing of the tracer. Further, they transferred results of conservative transport to reactive transport in cases where mixing is governed by local scale dispersion, Their analysis was based on the spatially integrated breakthrough curve as a result of advective-dispersive transport in independent stream tubes with identical Peclet number but different seepage velocity. Cao and Kitanidis (1998) studied flow and transport of a nonsorbing tracer in an idealized pore channel. They developed a methodology for the evaluation of macroscopic parameters based on homogenization and illustrated it with an example of dilution at a small scale to determine dilution. Cirpka and Kitanidis (2001) proposed a method to infer pore-scale transverse dispersivity values in a helical (designed to study radial potential flow) porous media that are difficult to determine with existing methods. They also concluded that spreading of a conservative tracer in potential flow is dominated by radial velocity distribution. Pannone and Kitanidis (1999) presented a tensor that describes the kinetics of dilution of a plume of a nonsorbing solute, demonstrating that the coefficient of variation of concentration at the center of the plume becomes proportionate to 1/t, using a Lagrangian approach. They also derived expressions for the geometric mean of the dilution index and the reactor ratio. Pannone and Kitanidis (2001) also presented a Lagrangian methodology to develop an analytical expression for the spectral covariance of the concentration of a conservative solute applicable to large times after injection.

23


Pannone and Kitanidis (2004) studied the asymptotic behavior of dilution parameters for loghydraulic conductivity functions. They showed that the two-particle covariance coincides with the variance of the centroid location. Investigating flow and solute transport in heterogeneous unsaturated soil under gravity-driven flow conditions, Cirpka and Kitanidis (2002) applied direct simulation Monte Carlo method. They concluded that dilution and macrodispersion rates differed significantly at high infiltration rates.

4.4. APPLICATIONS In 1993, Connaughton et al. presented a study of contaminated soil samples. They did some desorption experiments to adapt models incorporating continuum behavior of the rate coefficients. They used a gamma function to describe the long term desorption experiments. Pedit et al. (1994) treat sorption equilibrium and rate properties as continuously distributed random variables described by log-normal and gamma density functions. A finite difference formulation for contaminant transport including a distribution of contaminant mass transfer rates between water and soils was developed by Culver et al. (1997). They showed that continues gamma distribution of the rates presents better robust results than the two site model. Several applications of the multiple rate mass transfer approach have been published since its definition (Haggerty and Gorelick, 1995). Gooseff et al. (2003) used the solute transport and multirate mass transfer – linear coordinates (STAMMT-L) to interpret slug-tracer injection tests, and Haggerty et al. (2001) investigated MRMT as explanation for the behavior of a single well injection withdrawal test (SWIW) conducted in a fractured media. McKenna et al. (2001) analyzed convergent flow tracer tests with single and multiple rate mass transfer models. All of them observe that the MRMT model is capable of accurately modeling the observed data. Deitsch et al. (2000) quantified rate of sorption and desorption of sorbents using a batch methodology. They used a gamma function to generate the distribution of first-order rate coefficients. They concluded that desorption rate limitations can be interpreted within a context through the soil organic matter, and the distributed-rate model showed to adequately simulate various degrees of mass transfer limitations. Flemming and Haggerty (2001) analyzed experiments to model the presence of multiple matrix diffusivities. Bradford et al. (2000) studied the dissolution behavior of residual tetrachloroethylene (PCE) in chemically heterogeneous soils. They used rate limited dissolution and desorption, and a two-parameter power function expression for the lumped mass transfer coefficient, obtaining reasonable predictions. Sale and McWhorter (2001) used a multiple analytical source superposition

24


technique (MASST) to estimate aqueous mass transfer rates to removal dense nonaqueous phase liquid (DNAPL) from subsurface zones. Schumer et al. (2003) achieved a solution of a fractal mobile/immobile model for solute transport performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase, including the fractal immobile phase with numerical simulations and capturing the behavior of tracer plumes in heterogeneous aquifers.

25

26

Chapter 5 5. OBJECTIVES AND SCOPE 5.1. GENERAL OBJECTIVE The main objective of the dissertation is to develop analytical expressions to represent the reaction rates of geochemical processes of reactive transport as well as adsorption/desorption, precipitation/dissolution, matrix diffusion. Multiple Rate Mass Transfer Approach as well as dilution theory will be applied to achieve these formulations.

5.2. SPECIFIC OBJECTIVES The general objective can be split into a number of specific objectives: ¾ To define specific indicators of mixing for quantitative evaluation of real situations. ¾ To quantify the effect of heterogeneity in hydraulic conductivity as well as in the parameters characterizing reaction processes upon mixing, by means of a multicomponent reactive transport approach. ¾ To generalize the multiple species sorption reactions to large scale aquifers in equilibrium situations by means of upscaling mechanisms, taking into account concepts such as competitive sorption of the different species. ¾ To characterize the behavior of reactive solutes in heterogeneous media by means of analysis and modeling of the breakthrough curves, including multi-component transport.

27

28

Chapter 6 6. METHODOLOGY & WORK PLAN This chapter aims at showing how the planned objectives are expected to be reached. The methodology gives an idea about the logic structure and the scientific strictness of the research process. It initiates with the stating of the fundamental hypothesis and finishes with the way to analyze, interpret and present the results. In this part of the document the procedures, techniques and activities to do the investigation are outlined.

6.1. WORK PLAN The work plan proposed comprises several stages. The first one includes careful revision of the relevant literature on the subject. It is expected that it will be possible to quantify the degree of efficiency of some remediation techniques. Secondly, one way to define the processes to derive the parameters at a given scale is by means of tracer tests. Some tracer tests were carried out in El Berrocal (a site of experimentation in center Spain) project (Rivas et al., 1997). In the framework of HIDROBAP-II project those tracer test were reinterpreted using a different conceptual model (Donado et al., 2005a). In this stage of the dissertation, tracer tests in heterogeneous media (fractured geological media) were interpreted using a calibration process within an inverse problem approach. Representative mean values of the system were used. Some of the results confirm the presence of a sorption process in the transport dynamics. Matrix diffusion is evidently present even in conservative solutes, as tailing is widely present in all breakthrough curves (see Figure 3.3).

29


The actual development in reactive transport permits decoupling the mechanisms of reaction and transport. Nowadays it is possible to state expressions for reaction rates and concentration of reactive species. These formulations can be analytically developed for all homogeneous media and for some heterogeneous media too. Nevertheless, expressions can be numerically evaluated for more complex geometries, although the model will be deterministic. In terms of the stochastic model, only one realization is considered. This project casts the study in a stochastic approach. The third phase in the analysis will concentrate on the formulation of an adequate process for the characterization of the chemical processes that produce reactive transport in aquifer as matrix diffusion evidenced in the preceding stage. First of all, precipitation/dissolution will be analyzed. After it, adsorption/desorption will be incorporated and then the matrix diffusion mechanism. All of these will be done using a multiple rate mass transfer approach in homogeneous and heterogeneous media. In the following stage upscaling will be addressed. Upscaling is the mechanism by means of which parameters derived at some given local (small) scale are exchanged and applied to a larger scale, without changing the underlying governing equation. It will be studied in the following order: (1) Reaction rates, (2) mixing/dispersion and (3) additional geochemical processes. This mechanism integrates the physical heterogeneity with the temporal fluctuations. With the analytical expressions obtained in previous activities, the space-temporal fluctuations on the reaction rates are examined. This mechanism will be applied in equilibrium conditions The extent of mixing and dispersion will be quantified using the dilution index (Kitanidis, 1994), as well as the concepts related to effective and ensemble values of dispersion. Further, the methodology adopted will be based upon the decoupling methodologies of De Simoni et al. (2005) or Molins et al., ( 2004), both of which allow a separated study of transport of conservative species, reactive transport, and mixing/dispersion processes. Most probably some numerical simulations will be required. parameters will be obtained.

Finally, upscaled transport

Traditionally, upscaling has been focused in the parameters. However, the transport processes are characterized by dynamic equations that have different behavior in local and large scales. These equations include general memory terms caused by variations in space and/or temporal fluctuations. For this reason the multiple rate mass transfer approach is going to applied, casting the problem into finding the temporal memory functions, and solving the resulting nonlocal equations. This final stage includes the identification and the quantification of the dynamic processes on large scale with effective equations.

6.2. SCHEDULE The following chart illustrates the work-plan and its activities in the period of execution of the dissertation. Table 6.1 Schedule of the work-plan

30

Chapter 6. Methodology and Planning

31


32

Chapter 7 7. PRELIMINARY RESULTS

&

EXPECTED

7.1. PRELIMINARY RESULTS The results of the dissertation are going to be published in papers in journals. The first one is in preparation to be send to Journal of Contaminant Hydrology. It deals with the modeling of hydraulic and tracer tests in fractured media. Preliminary results of this stage of the research were presented in ModelCARE 2005. Secondly, there are some preliminary results of the stage of analytical solutions for the reaction rates of reactive transport. These results are shown in the following sections.

7.1.1.

Tracer tests in fractured media

Fractured geological media present high complexity to groundwater modeling. The main reasons are geometrical complexity in the location and extent of fractures, plus heterogeneity and anisotropy in hydraulic parameters. Groundwater flow and solute transport in fractured media can be analyzed by four groups of models: equivalent porous media (EPM), discrete fractured network (DFN), porous media with embedded fractures (mixed approach), or channel networks. All of them have been applied widely in forward problems. For inverse problem, though, DFN pose a fundamental problem which is the large number of parameters involved in the calibration process, as

33


due to heterogeneity, each element would have widely different hydraulic parameters associated making inverse modeling a challenging problem. Thus, DFN’s must be cast in a statistical framework, as there is a large amount of uncertainty in the actual location, size and hydraulic parameters of each individual fracture. An alternative calibration method that can be applicable to DFN is proposed. The methodology is based on the idea of reducing the number of values to be calibrated based on the concept that a DFN is a realization of a Spatial Random Function. This method has been applied to the interpretation of hydraulic and tracer tests in the granitic batholith of El Berrocal (Spain), within the project HIDROBAP-II financed by ENRESA and the Nuclear Security Council of Spain. The quality of the fits obtained permits to conclude that DFN models are appropriated to study groundwater flow and solute transport in fractured media since its results are comparable with calibrations obtained using a mixed approach porous media model (Ruiz et al., 2001). ¾

Methodology

The conceptual model consists of five (5) fracture families, defined a priori based on tectonical criteria. Each network is composed by a number of fractures whose location, size (radius) and orientation are geometrically defined by fractal distributions or probability density functions. These distributions are supported by extensive field data (Nita et al., 2004). Each fracture is associated an aperture value from a predefined probability function. Each individual fracture keeps track of the fracture family it belongs (this last point being of outmost importance in our methodology). The next step corresponds to the solution of the PDE’s governing groundwater flow and solute transport in the DFN. The methodology presented here is a modification of the channel model developed by Cacas et al. (1990). It starts by finding the conductive fracture network that is the fractures that are connected among themselves and also connected to the boundaries, therefore capable of conducting water. Actually, despite the fracture is modeled as a disk; actual flow takes place only in a small part of it that is in channels associated to the most conductive features within a given fracture. The critical point is the way these channels are interconnected, forming a 3-D network of 1-D elements. A simplified way to consider channels is represented in Figure 7.1(a) (HIDROBAP, 1998). One of the advantages of this methodology with respect to that of Cacas et al. (1990) is that with this scheme each element still belongs to a given disk, and therefore it is possible to associate each element to a given fracture family. Fig. 1(b) shows the full procedure of going from a DFN to a mesh of 1-D elements while keeping the concept of fracture family. The parameters associated to any given 1-D element are hydraulic conductivity (K) and storage coefficient (S) in the case the groundwater flow equation is solved, and porosity (ф) and longitudinal dispersivity (α) for transport. In each individual element, the actual parameter is equal to the product of two terms: (1) a specific value that comes from a statistical distribution a priori (e.g. a lognormal distribution for K), and (2) a scaling factor (“zone” or “family parameter”) which is an unknown

34

Chapter 7. Preliminary and expected results

value, the same for all elements associated to a given family. The latter value is the one calibrated by means of the inverse problem. As an immediate consequence, only a reduced number of parameters (one or two per family) are to be estimated. The data used for calibration can be steady-state head data, or that coming from a pump (heads) or a tracer test (concentrations).

DISK 3 DISK 2

2 1 DISK 1

3 (a)

(b)

Figure 7.1 Constructing the conductive network. (a) From individual fractures to 1-D elements. (b) Going from a 3-D DFN to a mesh of 1-D elements. Grayscale represents families. (From Donado et al., 2005b)

Regarding boundary conditions (BC), they must be applied to all elements that intersect a given external volume used in the definition of the DFN. This can be done by using a given geometry (e.g. parallelepiped) and associating a boundary condition to every face. In the flow problem BC can be constant head, constant flow or mixed. In transport the BC depend whether in a given node water flows in (with external concentration), or flows out, carrying the actual concentration in the system. Calibration is done using TRANSIN II (Medina et al., 1996). The code requires observed values of heads and or concentrations at a given number of observation points and estimates the parameters that best fit the observed values. Observation points should belong to an element which is part of the conductive network. In the transport problem it is possible to incorporate a number of processes to account for nonconservative species. These processes can be adsorption, first-order radioactive decay and matrix diffusion. Whenever one of the processes is incorporated to the conceptual model, some additional parameters arise. These new parameters are also associated to families, despite any given element could have a different coefficient drawn from an a priori statistical distribution if necessary. In this type of meshes numerical instabilities can cause both the direct or inverse problems not to converge. This might call for reducing the length of the 1-D finite elements, therefore increasing the number of nodes, which leads again to CPU problems.

35


¾

Results Flow Simulations

The methodology is tested by means of the calibration of a combination of hydraulic and tracer test carried out in the framework of the El Berrocal Project (Rivas et al., 1997). Twenty different statistical equiprobabilistic representations of the media were generated and analyzed. The DFN was modeled as defined by 5 fracture families. A pumping test was simulated and the interpretation was performed using the methodology presented in this paper. Initially only the flow parameters were calibrated. The parallelepiped block simulated was 600 m × 600 m × 300 m, with the pumping and observation points located around the mid point in the domain. Boundary conditions applied in the numerical model where no flow in the top and bottom boundaries and zero drawdown elsewhere indicating that the influence radius of the test was assumed to be less than 300 m. The calibrated parameters were a single value for hydraulic conductivity (K) and storage coefficient (S) for each independent fracture family, leading to a total of 10 parameters to estimate. Amongst the 20 initial networks, all of them unconditional to the location of specific fractures, six of them produced excellent results in terms of fitting the transient head data. Figure 7.2 shows one of the best obtained fits, compared to an average one. The match between computed and observed drawdown is similar to the one that was obtained in Rivas et al. (1997) applying a mixed approach, indicating that the methodology presented could compete in terms of data matching. In the observation point S13.1 it was impossible to get a good fitting, either by means of a DFN or a porous equivalent model. During the process of flow calibration the fracture family number 4 was insensible, meaning that the same fits were obtained independently of the zone value used. This means that in the vicinity of the test area very few, if any, of the conducting elements belong to that particular family. This result was repeated in most of the 20 DFN analyzed. Transport Simulations The proposed methodology was extended to transport analysis in the six networks which provided the best fit in the flow simulation. A tracer test with two different tracers (uranine and deuterium) was analyzed. The flow configuration is the result of a dipole, with pumping at the well (0.1 m3 d-1) and simultaneous pumping (0.2 m3 d-1) at the injection point. Boundaries were checked so that no mass was leaving the domain.

36


Time (days) 0

1

2

3

4

Time (days) 5

6

7

8

0

0

1

2

3

4

5

6

7

8

0 S 2.1

-5

-0.5

Drawdown (m)

Drawdown (m)

-10 -15 -20 Netw ork A

-25

S 13.1

-1

-1.5 Netw ork A

-2

Netw ork B

Netw ork B -30

-2.5

Time (days)

Time (days) 0

1

2

3

4

5

6

7

0

8

-0.2

S 13.2

Drawdown (m)

Drawdown (m)

4

5

6

7

8

S 15.2

-0.6

-1.5 -2

-3

3

-0.4

-1

-2.5

2

0

0 -0.5

1

-0.8 -1 -1.2 -1.4

Netw ork A Netw ork B

-1.6

Netw ork A

-1.8

Netw ork B

-2

Figure 7.2 Observed (dots) vs. computed drawdown (lines) after calibration of a pumping test (pumping in S14.2 and three additional observation points) in El Berrocal using a DFN approach. Two different fits (amongst the 20 networks available) are presented; Network A, corresponding to one of the best ones, and Network B, which provides an average fitting. (From Donado et al., 2005b)

A double porosity model was considered. The processes considered were advection, dispersion and matrix diffusion. The total number of estimated zonal parameters was nine (the longitudinal dispersivity, six porosities: one for each family (5) and that of the immobile zone; and two diffusivities corresponding to the mobile and immobile zones). The fitting between observed (HIDROBAP, 1998) and computed concentrations at the extraction point is acceptable and also comparable with the one obtained using an equivalent porous media for conceptualization. The inverse problem technique is not so easy to apply in transport solution. It must be applied after a manual adjust for the parameters to get an approximated shape of the breakthrough curve (BTC) (Figure 7.3). First of all, the molecular diffusion coefficients of the matrix (1 × 10-5 m d-1) and the fracture families (1 × 10-1 m d-1) were fixed to typical values. Second, the immobile porosity was fixed to represent the BTC tail (1), while the mobile porosity (the same value for all the families) was settled to reach the peak position (2). In addition, the dispersion coefficient was increased to shape properly the BTC (3). To start the process, mass tracer and the dispersion coefficient were only calibrated (4), then, the porosities (mobile and immobile) were calibrated automatically (5).

37


0.012

2 Peak Time

-1


0.010 0.008 4

0.006

5

3

0.004

Tail

0.002

1

0.000 0

5

10

15

20 25 Time (days)

30

35

40

45

Figure 7.3 Steps for a semiautomatic calibration of a BTC in a DFN model (From Donado et al., 2005b)

Figure 7.4 shows the fitted BTC for the network number 10 (which provides also the best fit of the flow problem). The Deuterium curve was best fitted using an advection – dispersion – matrix diffusion model (Figure 7.4(a)), while a model without matrix diffusion could fit very well the Uranine BTC (Figure 7.4(b)). This indicates the need for a retention process in the former. Consistency in the parameters comes from the same parameter values for porosities in both curves. The values calibrated for longitudinal dispersivity and diffusivity are different for both tracers.

¾

Conclusions

Twenty different realizations of a DFN were used to simulate pumping and tracer tests in fractured media. Eighty percent (80%) of the DFN analyzed present head objective functions with acceptable values, thirty percent (30%) with objective function values that can be considered excellent from the head fitting standpoint. The methodology adopted allows calibration a relatively short number of parameters, deeming the calibration process possible. The actual fitted parameters (zonal T and S values) vary with each simulation. Families that do not contribute to flow can be detected by this method. The tracer tests interpretation allows estimating additional parameters, such as porosity, longitudinal dispersivity, as well as matrix diffusion parameters (φm, Dm).

38

0.012

3.0

0.010

2.5 Concentration (mg L )

0.008

-1

-1



0.006 0.004 0.002

2.0 1.5 1.0 0.5

0.000 0

5

10

15

20

25

30

35

40

0.0

45

0

Time (days)

5

(a)

10

15

20 25 Time (days)

30

35

40

45

(b)

Figure 7.4 Observed (dots) vs. computed concentrations (lines) after calibration of a tracer test in El Berrocal using a DFN approach. (a) Deuterium: Advection – Dispersion – Matrix Diffusion model. (b) Uranine: Advection – Dispersion model (From Donado et al., 2005b).

In conclusion, the methodology presented allows calibrating the parameters corresponding to a DFN obtaining matches qualitatively and quantitatively as good, or better, as those obtained using a conceptual model of an equivalent porous medium. The definite advantage of the method is that the DFN simulated is not unique, and therefore this methodology is easily and immediately cast in a geostatistical framework, and therefore it is easier to find a physical meaning to the calibrated values.

7.1.2.

Results on reactive transport in heterogeneous media

In this section preliminary results on the analysis of reactive transport in homogeneous or heterogeneous media by means of a multirate mass transfer equation are presented. The starting point is writing a general form of the multirate mass transfer equation:

∂c φm m + φim ∂t

∞

∫

f (α )

∂cim dα = L(cm ) ∂t

7.1

0

For linear mass exchange between the mobile and immobile phases, cim is a linear functional of cm on [7.1] can be written as:

∂c ∂ φm m − φim cim0 g (t ) − φim ∂t ∂t

t

∫ g t −τ c (

)

m

(τ ) dτ = [ −q∇cm + ∇·(D∇cm ) ]

7.2

0

39


Or:

∂c φm m − φim cim0 g (t ) − φim ∂t

t

∫ H t −τ c

) m (τ )dτ + φim cm β = [ −q∇cm + ∇·(D∇cm )]

(

7.3

0

Where H(t) is the derivate of g(t). The memory function g(t) depends on the particular mass exchange mechanism and rate distribution. ¾

Precipitation / dissolution

The general problem analyzed by means of (7.2) is that of two species that are in equilibrium with a mineral phase, so that concentrations are governed by simple precipitation / dissolution in a binary system. The principal assumption is that both species are governed by a mass transfer term with the same parameters. This can represent three quite different conditions: (1) both species experience multirate sorption with the same distribution of sorption rates; (2) the MRMT term accounts for heterogeneity in the physical properties of the medium; or (3) the term represents matrix diffusion of otherwise conservative species, with the same diffusion coefficients. Assuming the ionic strength is small, the mass action law can be written in terms of concentrations, rather than of activities. The two species are therefore related by:

B1 + B2 R

S3s

7.4

cm1 ⋅ cm2 = K id ,

7.5

The transport equations for the two species are:

φm

∂cmi

∂ − φ c g (t ) − φim ∂t ∂t 0 im imi

∫

t

( )

g (t − τ )cmi (τ )dτ = L cmi + r

i=1,2

7.6

0

r corresponding to the reaction (either precipitation or dissolution) rate. Based on De Simoni et al. (2005) it is possible to define a conservative component

um = cm1 − cm2

7.7

The solution for the reaction rate is given by (more details can be looked up in the appendix).

40


⎧ ∂ ⎪⎛ 0 ∂cmi 0 ⎞ − cim r = φim ⎨⎜ uim ⎟ g (t ) − i ∂um ∂t ⎠ ⎪⎩⎝

t

∫ 0

⎡ ∂cm ⎤ ⎫⎪ ∂ 2 cmi ∇umT ( D∇um ) g (t − τ ) ⎢ i um (τ ) − cmi (τ ) ⎥ dτ ⎬ − 2 ⎢⎣ ∂um t ⎥⎦ ⎪ ∂um ⎭

7.8

Using [7.5] and [7.7], and replacing the memory function g(t), it is obtained: ⎧⎛ 0 ⎛ 0 uim ⎪ u r = φim ⎨⎜ im ⎜ −1 + um2 + 4 K id ⎪⎩⎜⎝ 2 ⎜⎝ ... +

∞

2 K id u + 4 K id 2 m

⎞ ⎟ − c0 ⎟ imi ⎠

∫

α f (α )dα −

t 0

(u

∞

⎞ ⎟ α f (α )e −α t dα − ⎟ ⎠0

∫

2 K id

2 m

t

+ 4 K id )

∫ 0

⎫ ⎪

∞

∫

2 K id um2 + 4 K id

α 2 f (α )e−α ( t −τ ) dα dτ ⎬ + ... ⎪⎭

τ 0

7.9

∇umT ( D∇um )

3/ 2

Using the suggested pdf for the multirate mass transfer process in several singularities (Haggerty et al., 2000):

∑ ββ δ (α N

p (α ) =

i

− αi )

7.10

tot

i =1

Then:

∑ N

f (α ) = β tot ⋅ p (α ) = β tot

i =1

βi δ (α − α i ) = βtot

∑ β δ (α N

∑β N

− α i ) with β tot =

i

i =1

fi

7.11

i =1

The reaction rate for this reactive process is: ⎧⎡ 0 ⎛ 0 uim ⎪ u r = φim ⎨ ⎢ im ⎜ −1 + ⎢ 2 ⎜ um2 + 4 K id ⎩⎪ ⎣ ⎝ ... +

2 K id u + 4 K id 2 m

∑ N

t

i =1

⎤ ⎞ 0 ⎥ ⎟ − cim i ⎟ ⎥ ⎠ ⎦

∑β α N

i

i =1

t

∑β α ∫ N

i

e

−α i t

−2

K id

2

i

i =1

i

0

u + 4 K id 2 m

⎫ 2 K id ⎪ β iα i ⎬ − ∇umT ( D∇um ) 3/ 2 2 ⎪⎭ ( um + 4 K id )

e −αi ( t −τ ) dτ + ... τ

7.12

When the reaction is only one-site, the expression can be simplified to:

41


⎧ 0 0 ⎞ ⎞ uim ⎪⎛ u ⎛ 0 ⎟ ⎟ − cim r = φim β tot ⎨⎜ im ⎜ −1 + αβ tot e−α t − α 2 i 2 ⎜ ⎟ ⎜ ⎟ 2 um + 4 K id ⎠ ⎪⎩⎝ ⎝ ⎠ ⎫ 2 K id 2 K id ⎪ ... + ∇umT ( D∇um ) α⎬− 3/ 2 2 2 um + 4 K id t ⎪⎭ ( um + 4 K id )

t

∫ 0

2 K id u + 4 K id 2 m

e −α ( t −τ ) dτ + ... τ

7.13

It can be proved that when the rate of mass transfer tends either to infinity or to zero, the reaction rate is equal to that found by De Simoni et al. (2005): lim r = lim r = −

α →∞

α →0

(u

2 K id

2 m

+ 4 K id )

3/ 2

∇umT ( D∇um )

7.14

And when the reaction rate is variable, the multirate mass transfer term can be observed

¾

Matrix diffusion Recalling the expression for the multirate mass transfer: ∞

Γ (t ) =

∫

∂c 0 f (α ) im dα = −cim g (t ) − ∂t

0

t

∫

H (t − τ )cm (τ )dτ + cm α

7.15

0

With this solution (Haggerty et al., 2000): 0 Γ ( t ) = cm g ( 0 ) − cim g (t ) − cm ∗ g '

7.16

For Finite Layer the pdf and the memory function are, (Haggerty et al., 2000): ∞

f (α ) =

∑ ( 2 j − 1) π 8β tot

2

j =1

∞

g (t ) =

∑ 2β j =1

42

tot

⎡

δ ⎢α − 2 ⎢⎣

2 ( 2 j − 1) π 2 Da ⎤

4a 2

⎡ ( 2 j − 1)2 π 2 Da ⎤ Da exp ⎢ − t⎥ a 4a 2 ⎢⎣ ⎦⎥

⎥ ⎥⎦

7.17

7.18


And the solution for the MRMT term is: Γ ( t ) = 2 β tot

... +

Da a

π 2 Da 4a 2

⎧ ⎪ 0 ⎨cm − cim ⎪⎩

∞

∑ j =1

⎡ ( 2 j − 1)2 π 2 Da ⎤ exp ⎢ − t ⎥ + ... 4a 2 ⎢⎣ ⎥⎦ t

∞

∑( 2 j − 1) ∫ 2

j =1

0

⎫ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎪ exp ⎢ − τ τ τ t c d − ⋅ ⎥ ( ) ( ) ⎬ m 2 4 a ⎢⎣ ⎥⎦ ⎪⎭

7.19

So finally, the reaction rate is: r = 2 β tot

... +

Da a

π 2 Da 4a 2

⎧ ⎪ 0 ⎨cm − cim ⎪⎩

∞

∑ j =1

⎡ ( 2 j − 1)2 π 2 Da ⎤ exp ⎢ − t ⎥ + ... 4a 2 ⎢⎣ ⎥⎦ t

∞

∑( 2 j − 1) ∫ 2

j =1

0

⎫ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎪ T exp ⎢ − τ τ τ t c d − ⋅ ⎥ ( ) ( ) ⎬ − f 3 ( um ) ∇um ( D∇um ) m 2 4 a ⎢⎣ ⎥⎦ ⎪⎭

7.20

And for diffusion in spheres, the pdf and memory functions are respectively (Haggerty et al., 2000):

6 β tot ⎛ D ⎞ δ ⎜ α − j 2π 2 2a ⎟ 2 2 a ⎠ ⎝ j =1 j π ∞ D D ⎞ ⎛ g ( t ) = ∑ 6 β tot 2a exp ⎜ − j 2π 2 2a t ⎟ a a ⎠ ⎝ j =1 ∞

f (α ) = ∑

7.21

7.22

So that, the multirate mass transfer term is: Γ ( t ) = 6 β tot

⎧ Da ⎪ 0 c − cim 2 ⎨ m a ⎪ ⎩

∞

∑ j =1

D ⎞ D ⎛ exp ⎜ − j 2π 2 2a t ⎟ + π 2 2a a ⎠ a ⎝

t

∞

∑ ∫ j

2

j =1

0

⎫ ⎪ ⎡ 2 2 Da ⎤ exp ⎢ − j π 2 ( t − τ ) ⎥ ⋅ cm (τ ) dτ ⎬ a ⎣ ⎦ ⎪⎭

7.23

And the reaction rate is: ∞ ⎧ D ⎞ D ⎪ ⎛ 0 exp ⎜ − j 2π 2 2a t ⎟ + π 2 2a r = 6φim β tot ⎨um − uim a ⎠ a ⎝ j =1 ⎩⎪ − f3 ( um ) ∇umT ( D∇um )

Da a2

∑

t

∞

∑ ∫ j

j =1

2

0

⎫ D ⎪ ⎡ ⎤ exp ⎢ − j 2π 2 2a ( t − τ ) ⎥ ⋅ um (τ ) dτ ⎬ a ⎣ ⎦ ⎪⎭

7.24

43


¾

Application

Next, these solutions are applied for the reaction rate for a particular solution of the conservative solute transport problem. For simplicity we select an example when this solution is given analytically. In particular we study the solution for um ( x, t ) represented by the step-function input with the following initial and boundary conditions

u m ( x, t = 0 ) = 0

x≥0

um ( x = 0, t ) = um0

t≥0

u m ( x = ∞, t ) = 0

t≥0

7.25

The function that represents the function is:

u m ( x, t ) =

⎛ x − vt um0 ⎡ ⎢erfc ⎜ ⎜ 2 ⎢ ⎝ 2 Dx t ⎣

⎞ ⎛ vx ⎟ + exp ⎜ ⎟ ⎝ Dx ⎠

⎛ x + vt ⎞ ⎟ erfc ⎜⎜ ⎠ ⎝ 2 Dx t

⎞⎤ ⎟⎥ ⎟ ⎠ ⎦⎥

7.26

Where erfc is the complementary error function and it is represented by:

erfc ( z ) =

2

π

∞

∫

2

e − p dp

7.27

z

The strong dependence of the first term turns this function into:

u m ( x, t ) =

⎛ x − vt um0 erfc ⎜ ⎜2 Dt 2 x ⎝

And replacing in the reaction rate, it can be written as:

44

⎞ ⎟ ⎟ ⎠

7.28


r ( x, t ) = 2φim β tot

... +

... −

⎛ x − vt Da ⎧⎪ um0 ⎨ erfc ⎜⎜ a ⎪2 ⎝ 2 Dx t ⎩

π 2um0 Da 8a 2

∞

∑ j =1

⎞ 0 ⎟ − uim ⎟ ⎠

∞

∑ j =1

⎡ ( 2 j − 1)2 π 2 Da ⎤ exp ⎢ − t ⎥ + ... 4a 2 ⎢⎣ ⎥⎦

I1

⎫ t ⎪ 2 2 ⎡ ( 2 j − 1) π Da ⎤ ⎛ x − vτ ⎞ ⎪ 2 ⎟ dτ ⎬ − ... ( 2 j − 1) exp ⎢ − ( t − τ )⎥ ⋅ erfc ⎜⎜ 4a 2 2 Dxτ ⎠⎟ ⎪ ⎢ ⎥ ⎝ ⎣ ⎦ 0 ⎪ ⎭

∫

K id ( um0 ) Dx 2

x − vt 4 π ( Dx t ) 2 ⎛ u 0 2 ⎜( m) ⎜ 4 ⎝ 3

⎡ ⎛ x − vt ⎢erfc ⎜ ⎜2 D t ⎢⎣ x ⎝

2 ⎞ ⎞⎤ ⎟ ⎥ + 4 K id ⎟ ⎟⎥ ⎟ ⎠⎦ ⎠

3/ 2

7.29

⎡ 1 ( x − vt )2 ⎤ exp ⎢ − ⎥ ⎢⎣ 4 Dx t ⎥⎦

This expression is mathematically very complicated. Now some simplifications are sought. As the general form of the reaction rate is: ∂c φim c g (t ) + φim u g (t ) 2 + φim ∂u 0

0

t

∫

∂c ∂ ∂u c2 (t − τ ) g ( t ) dτ − φim 2 ∂u ∂t ∂u

0

t

∫

∂ 2c ∂ 2 u (t − τ ) g ( t ) dτ − 22 D ( ∇u ) = r ∂t ∂u

7.30

0

Let’s suppose that g(t) is an exponential where c2 ( t ) − c2 (τ ) = c2 ( t ) − c2 ( t ) +

⎛ ⎝

φim g ( t ) ⎜ c 0 + u 0

∂c2 ∆τ , then ∂t

∂c2 ⎞ ∂ 2 c2 2 ⎟ − 2 D ( ∇u ) = r ∂u ⎠ ∂u

7.31

Yielding: ⎛ ⎛ ⎜ ⎜ ⎜ ⎜ u0 ⎜ −α t ⎜ 0 r ( x, t ) = φimαβ tot e ⎜ c + −1 + 2⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎝

2 ⎞⎞ ⎛ ⎡ 1 ( x − vt )2 ⎤ ⎞ ⎟⎟ x =0 2 ( um ) Kid ⎜⎜ exp ⎢− 4 D t ⎥ ⎟⎟ ⎟⎟ x ⎣⎢ ⎦⎥ ⎠ ⎟ ⎝ ⎟ + 3/ 2 ⎟⎟ 2 2 ⎛ ⎡ x =0 ⎞ ⎡ u x =0 ⎤ ⎤ ⎛ ⎞ ⎛ ⎞ ⎟ ⎟ x vt − u x − vt ⎢ m erfc ⎜ ⎟ ⎥ + 4 K id ⎟ ⎟ 2π t ⎜ ⎢ m erfc ⎜ ⎟ ⎥ + 4 K id ⎟ ⎜ 2 D t ⎟⎥ ⎜ ⎟ ⎜ ⎟ ⎟ 2 ⎟ D t 2 ⎢⎣ 2 ⎢ x ⎠⎦ x ⎠⎥ ⎝ ⎝ ⎦ ⎠⎠ ⎝⎣ ⎠

⎛ x − vt umx = 0 erfc ⎜ ⎜2 D t 2 x ⎝

⎞ ⎟ ⎟ ⎠

7.32

The following figures show the behavior of the reaction rate as a function of time and space, respectively.

45


1E-01 9E-02 8E-02 7E-02

r

6E-02 5E-02 4E-02 3E-02 2E-02 1E-02 0E+00 0

1

2

3

4

5

6

t

7

8

9

10

Figure 7.5 Relation between reaction ratio and time with different spaces, Ec. 7.32

1E-01 9E-02 8E-02 7E-02

r

6E-02 5E-02 4E-02 3E-02 2E-02 1E-02 0E+00 0

1

2

3

x

4

5

6

Figure 7.6 Relation between reaction ratio and space with different times, Ec. 7.32

7.2. EXPECTED RESULTS The expected results of the research will be published in journals. There are three more projected publications. The first one is related to mixing/dispersion quantification. The second one

46


is about upscaling process and the last one to the generalization of dynamics processes in large scale. Moreover, some presentation of the advances will be offered in international (as well as a few national) conferences and congresses.

47

48

8. REFERENCES Abramovitz, M. and I. A. Stegun, (1972) Handbook of Mathematical Functions, Dover, Mineola, New York. Adams, E. E., and L. W. Gelhar, (1992) Field study of dispersion in a heterogeneous aquifer, 2, Spatial moment analysis. Water Resour. Res., 28 (12), 3239 – 3308. Bradford, S. A., T. J. Phelan and L. M. Abriola, (2000) Dissolution of residual tetrachloroethylene infractional wettability porous media: correlation development and application. J. Cont. Hydrol., 45, 35 – 61. Burden, R. L. and J. D. Faires, (2001) Numerical Analysis. Seventh Edition. Brooks/Cole. Thomson Learning. Pacific Groove, CA. Cacas, M. C., E. Ledoux, G. de Marsily and B. Tillie, (1990) Modeling fracture flow with a stochastic discrete fracture network: calibration and validation. 1: The flow model. Water Resour. Res., 26 (3), 479-489. Cao, J., and P. K. Kitanidis, (1998) Pore-scale dilution of conservative solutes: An example. Water Resour. Res., 34 (8), 1941 – 1949. Carrera, J., X. Sanchez-Vila, I. Benet, A. Medina, G. Galarza and J. Guimera, (1998) On matrix diffusion: formulations, solution methods and qualitative effects. Hydrogeol. J., 6, 178 – 190.

49


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54

Appendix. Multirate mass transfer approach applications

Appendix 1 9. MRMT APPROACH APLICATIONS MULTIRATE MASS TRANSFER FOR PRECIPITATION / DISSOLUTION TRANSPORT PROBLEM

Multirate mass transfer equation:

∂c φm m + φim ∂t

∞

∫

f (α )

∂cim dα = L(cm ) ∂t

0

∞

If

Γ (t ) =

∫

f (α )

∂cim dα ∂t

0

ix

n


∂cim = α (cm − cim ) ∂t

Diffusion equation

o

Boundary Condition!!! ∞

Where α [dimensionless] is the mass transfer coefficient and is variable.

f (α ) = β tot ⋅ p (α ) , where β tot =

∫ f (α ) d α

is the total capacity factor

0

∞

(Lawrence et al. 2002) and p (α ) is the pdf to represent α. Therefore,

∫ p(α ) dα = 1

. Moreover, φm [dimensionless] is the mobile porosity, φim

0

3

[dimensionless] is the immobile porosity, cm [M / L ] is the mobile concentration and cim [M / L3] is the immobile concentration.

Solving the diffusion equation:

∂cim = −α cim + α cm . It’s a linear and first order differential equation to solve like this: ∂t

y ' = a ( x ) y + b( x )

y = e∫

The solution is:

→

a ( x ) dx

cim = e

−α t

∫

⎛ − ∫ a ( x ) dx ⎞ b( x) + constant ⎟ ⎜ e ⎝ ⎠

⎛t ⎞ ⎜ eατ α c (τ )dτ + c 0 ⎟ m im ⎜⎜ ⎟⎟ ⎝0 ⎠

∫

t

0 −α t im

cim = c e

+e

−α t

∫

α eατ cm (τ )dτ

0

t

0 −α t im

cim = c e

∫

+ α e −α ( t −τ ) cm (τ )dτ

p

0

x


It is necessary to found the derivate of the solution as a function of mobile concentration. The derivate respect to the time is:

∂cim ∂ 0 −α t = −α cim e − ∂t ∂t

t

∫α

e −α ( t −τ ) cm (τ )dτ q

0

Now, it is necessary to found the second term of the MRMT equation:

∂c ∂ 0 −βt f (α ) im = −α f (α )cim e − ∂t ∂t

Multiplying by f(α)

t

∫

α f (α ) e −α (t −τ ) cm (τ )dτ

r

0

∞

Integrating respect to α

∫ 0

∞

∫ 0

∞

∫

∂c 0 −α t f (α ) im dα = − α f (α )cim e dα − ∂t 0

∞

∫ ∫ 0

∞

∫

∂c ∂ 0 f (α ) im dα = −cim α f (α )e− β t dα − ∂t ∂t 0

t

∂ α f (α )e−α (t −τ ) cm (τ )dτ dα ∂t 0

t ∞

∫∫α

f (α )e−α ( t −τ ) cm (τ )dα dτ

0 0

∞

Let it defines the memory function:

∫

g (t ) = α f (α )e −α t dα

t

0

Now, the previous equation (second term of the MRMT equation) is:

xi

s


∞

∫

∂c ∂ 0 f (α ) im dα = −cim g (t ) − ∂t ∂t

0

t

∫

t

∫

t

g (t − τ )cm (τ )dτ =

0

∫

u

0

∂ ∂t

If the derivate is made applying the Leibniz’s theorem:

∂ ∂t

g (t − τ )cm (τ )dτ

g (t )

∫

g (t )

h ( t − τ ) dτ =

f (t )

∫

f (t )

∂ ⎛∂ ⎞ ⎛∂ ⎞ h ( t − τ ) dτ + ⎜ g ( t ) ⎟ h ( t,g ( t ) ) − ⎜ f ( t ) ⎟ h ( t, f ( t ) ) ∂t ⎝ ∂t ⎠ ⎝ ∂t ⎠

∂ g (t − τ )cm (τ )dτ + g (t − t )cm (t ) − 0 ∂t

0

∞

∫

Applying in the MRMT equation:

∂c 0 f (α ) im dα = −cim g (t ) − ∂t

0

∞

∫

As g (0) = α f (α ) dα = α , then 0

∞

∫ 0

∞

∫

∂c 0 f (α ) im dα = −cim g (t ) − ∂t

0

∂c 0 f (α ) im dα = −cim g (t ) − ∂t

∫ g t −τ c '(

) m (τ )dτ + g (0)cm

0

∞

t

∫ g t −τ c '(

t

∫

) m (τ ) dτ + cm α , and if g ′(t ) = H (t ) = α 2 f (α )e−α t dα then

0

0

t

∫ H t −τ c (

) m (τ )dτ + cm α

v

This equation is the same form (Wang et al., 2005)

0

The transport equation could be written in two ways:

∂c ∂ 0 1. Integral – Differential: φm m − φim cim g (t ) − φim ∂t ∂t

t

∫ g t −τ c (

) m (τ )dτ = L(cm )

0

xii

nm


2. Integral:


t

∫ H t −τ c

) m (τ )dτ + φim cm α = L(cm )

(

nn

0

The advective – dispersive term of the equation is:

1. Integral – Differential:

L(cm ) = −q∇cm + ∇·(D∇cm ) , because (1) and (2) now are:

∂c ∂ φm m − φim cim0 g (t ) − φim ∂t ∂t

t

∫ g t −τ c

) m (τ ) dτ = [ −q∇cm + ∇·(D∇cm ) ]

(

no

0

2. Integral:


t

∫ H t −τ c (

) m (τ )dτ + φim cm β = [ −q∇cm + ∇·(D∇cm )]

0

xiii

np


The problem is simple precipitation / dissolution. The principal assumption is that both species has the same distribution coefficient.

B1 + B2 R

nq

S3 s

cm1 ⋅ cm2 = K id , but K id is variable

nr

um = cm1 − cm2

ns

The transport equations are:

Calculating (16):

φm

∂cmi ∂t

0 − φim cim g (t ) − φim i

It means that cmi = f ( um ,K id ) !!!

∂ ∂t

∫ g t −τ c t

(

)

mi

( )

(τ )dτ = L cmi + r

nt

i=1,2

0

⎧ ∂u ∂ ⎪ φm m = L ( um ) − φim ⎨uim0 g (t ) − ∂t ∂ t ⎪⎩ ⎧ ∂u ∂ ⎪ φm m − L ( um ) = φim ⎨uim0 g (t ) − ∂t ∂t ⎪⎩

t

∫ 0

t

∫ 0

⎫ ⎪ g (t − τ )um (τ )dτ ⎬ ⎭⎪ ⎫ ⎪ g (t − τ )um (τ )dτ ⎬ ⎭⎪

xiv

nu


∂cmi

Applying the Chain Rule,

( )

L cm2

( )

L cm2

∂t

=

∂cmi ∂um ∂cmi ∂K id , and replacing in the expression. From De Simoni et al. (2005), from (15) and (16), it has in (17): + ∂um ∂t ∂K id ∂t

⎧ ∂cm2 ⎫ ∂cm2 ∂ 2 cm2 T ⎡⎣ −q ⋅∇um + ∇ ⋅ ( D∇um ) ⎤⎦ + ⎡⎣ −q ⋅∇K id + ∇ ⋅ ( D∇K id ) ⎤⎦ + ∇ um ( D∇um ) + ⎪ ⎪ 2 ∂K id ∂um ⎪ ∂u2 ⎪ =⎨ ⎬ 2 2 ∂ cm2 T ⎪ ∂ cm2 ⎪ T ⎪2 ∂u ∂K ∇ um ( D∇K id ) + ∂K 2 ∇ K id ( D∇K id ) ⎪ m id id ⎩ ⎭ 2 2 2 ∂cm2 ∂cm2 ∂ cm2 T ∂ cm2 ∂ cm2 T L ( um ) + L ( K id ) + = ∇ um ( D∇um ) + 2 ∇T um ( D∇K id ) + ∇ K id ( D∇K id ) 2 ∂u2 ∂K id ∂um ∂K id ∂um ∂K id 2

From De Simoni et al. (in press), (19) could be simplified (Kid is constant):

( )

L cm2 =

∂cm2 ∂u2

L ( um ) +

∂ 2 cm2 ∂um

2

∇T um ( D∇um )

om

Replacing (20) in (17):

φm

∂cmi

∂ − φ c g (t ) − φim ∂t ∂t 0 im imi

t

∫ 0

⎧⎪ ∂cm ⎫⎪ ∂ 2 cm2 T g (t − τ )cmi (τ )dτ = ⎨ 2 L ( um ) + ∇ um ( D∇um ) ⎬ + r 2 ∂um ⎩⎪ ∂u2 ⎭⎪

on

(21) gives a expression for the reaction:

⎧ ⎧⎪ ∂cm ⎡ ∂cm ⎤ ⎫⎪ ∂ 2 cm2 T ∂ ⎪ 0 i 2 −⎢ ∇ um ( D∇um ) ⎥ ⎬ − φim ⎨cimi g (t ) + r = ⎨φm L ( um ) + 2 ∂t ⎢⎣ ∂u2 ∂um ∂t ⎥⎦ ⎪⎭ ⎪⎩ ⎩⎪

t

∫ 0

⎫ ⎪ g (t − τ )cmi (τ )dτ ⎬ ⎭⎪

xv

oo

nv


Applying the Chain Rule,

⎧ ⎧⎪ ∂cm2 ∂um ∂cm2 ∂K id ∂cm2 ∂ 2 cm2 T ∂ ⎪⎫ ⎪ 0 r = ⎨φm + D φ L ( um ) − u u − ∇ ∇ − ( ) m m ⎬ im ⎨cim2 g (t ) + 2 ∂ u ∂ t ∂ K ∂ t u u t ∂ ∂ ∂ ⎪⎭ 2 m id m ⎪⎩ ⎪⎩ ⎧ ⎫⎪ ⎧ ∂cm ⎛ ∂u ∂ ⎪ 0 ⎞ ∂ cm2 T r = ⎨ 2 ⎜ φm m − L ( u m ) ⎟ − u u D φ ∇ ∇ − m m⎬ im ⎨cimi g (t ) + 2 u t u t ∂ ∂ ∂ ∂ ⎠ m ⎩ m⎝ ⎪⎩ ⎭⎪ 2

t

∫ 0

t

∫ 0

⎫ ⎪ g (t − τ )cm2 (τ ) dτ ⎬ ⎭⎪

⎫ ⎪ g (t − τ )cmi (τ ) dτ ⎬ ⎭⎪

op

Recalling (18) in (21), r is achieved: ∂cm2 ⎡⎢ 0 ∂ r = φim uim g (t ) − ∂um ⎢ ∂t ⎣

⎧ ⎪⎛ 0 ∂cmi 0 r = φim ⎨⎜ uim − cim i u ∂ m ⎪⎩⎝

t

∫ 0

⎤ ∂2c ⎧ ∂ ⎪ 0 m2 T ⎥ g (t − τ )um (τ )dτ − g (t ) + ∇ um D∇um − φim ⎨cim i ⎥ ∂um 2 t ∂ ⎦ ⎩⎪

⎞ ∂ ⎟ g (t ) − t ∂ ⎠

t

∫ 0

t

∫ 0

⎫ ⎪ g (t − τ )cmi (τ )dτ ⎬ ⎭⎪

⎡ ∂cm ⎤ ⎫⎪ ∂ 2 cmi g (t − τ ) ⎢ i um (τ ) − cmi (τ ) ⎥ dτ ⎬ − ∇umT ( D∇um ) 2 u u ∂ ∂ m ⎣⎢ m t ⎦⎥ ⎪⎭

oq ID Form!!!

And solving the derivate, in the Integral Form:

⎛ φim ⎜ uim0 g (t ) + r= ⎜⎜ ∂um ⎝ ∂cmi

t

∫ 0

⎞ ∂ 2c ⎧ ⎪ 0 mi T D φ H (t − τ )um (τ )dτ − um α ⎟ − u u ∇ ∇ − m ( m) im ⎨cimi g (t ) + ⎟⎟ ∂um 2 ⎠ ⎩⎪

xvi

t

∫ 0

⎫ ⎪ H (t − τ )cm (τ )dτ − cm α ⎬ ⎭⎪


Ordering it, the following expression is obtained:

r = φim

∂cmi ∂um

u g (t ) + φim 0 im

∂cmi ∂um

t

∫

H (t − τ )um (τ ) dτ − φim

0

⎛ ⎛ ∂cmi 0 ⎞ 0 ⎜ ∂cmi + r = φim ⎜ uim g (t ) − cim g t ( ) φ ⎟ im i ⎜⎜ ∂um ⎝ ∂um ⎠ ⎝ ⎛ 0 ∂cmi 0 r = φim ⎜ uim − cim i ∂um ⎝

⎛ ⎞ ⎜ g t ( ) φ + ⎟ im ⎜⎜ ⎠ ⎝

t

∫ 0

∂cmi ∂um

t

um α − φ c g (t ) − φim

t

∫

0 im imi

0

t

H (t − τ )um (τ ) dτ −

0

∫

H (t − τ )cm (τ )dτ + φim cm α −

∫ 0

∂ 2 cmi ∂um 2

∇umT ( D∇um )

⎞ 2 ⎛ ∂cmi ⎞ ∂ cmi ∇umT ( D∇um ) H (t − τ )cmi (τ )dτ ⎟ − φim ⎜ um α + cmi α ⎟ − 2 ⎟⎟ u u ∂ ∂ m ⎝ m ⎠ ⎠

⎡ ∂cm ⎤ ⎞ ∂ 2 cmi ⎛ ∂cmi ⎞ H (t − τ ) ⎢ i um (τ ) − cmi (τ ) ⎥ dτ ⎟ − φim ⎜ um − cmi ⎟ α − ∇umT ( D∇um ) or I Form!!! 2 ⎟ ∂um ⎢⎣ ∂um t ⎥⎦ ⎟ ⎝ ∂um ⎠ ⎠

Let’s define r1, as the MRMT term of the reaction rate.

⎛ 0 ∂cmi 0 − cim r1 = φim ⎜ uim i ∂um ⎝

⎛ ⎞ ⎜ + g t ( ) φ ⎟ im ⎜⎜ ⎠ ⎝

t

∫ 0

⎡ ∂cm ⎤ ⎞ ⎛ ∂cmi ⎞ H (t − τ ) ⎢ i um (τ ) − cmi (τ ) ⎥ dτ ⎟ − φim ⎜ um − cmi ⎟ α ⎢⎣ ∂um t ⎥⎦ ⎟⎟ ⎝ ∂um ⎠ ⎠

os

As um = cm1 − cm2 and K id = cm1 ⋅ cm2 , It’s achieved the next relations:

cm2 =

−um + um2 + 4 K id ,ot 2

u ⎛ um um = m ⎜ −1 + 2 2 ⎜ ∂um um + 4 K id ⎝

∂cm2

um 1⎛ = ⎜ −1 + ∂um 2 ⎜ um2 + 4 K id ⎝

∂cm2

⎞ ⎟ ou ⎟ ⎠

⎞ ∂cm2 u ⎛ um ⎟ and um − cm2 = m ⎜ −1 + 2 ⎟ 2 ⎜ ∂um um + 4 K id ⎠ ⎝

xvii

and

∂ 2 cm2 ∂um

2

=

(u

2 K id 2 m

+ 4 K id

)

3/ 2

ov

⎞ −u + u 2 + 4 K −2 K id m id ⎟− m = = f ( um ,K id ) 2 ⎟ 2 u K 4 + m id ⎠

pm


∂cm2 ∂um

0 = uim

0 ⎛ 0 uim uim ⎜ −1 + 2 ⎜ um2 + 4 K id ⎝

⎞ ⎟ ⎟ ⎠

pn

Replacing (27), (28), (29), (30) and (31) in (26), it is achieved: ⎧⎛ 0 ⎛ 0 uim ⎪ u r = φim ⎨⎜ im ⎜ −1 + um2 + 4 K id ⎪⎜ 2 ⎜⎝ ⎩⎝

⎞ 0 ⎟ − cim i ⎟ ⎠

⎫ ⎛t ⎞ ⎞ ∂ 2 cmi ⎟ g (t ) + ⎜ H (t − τ ) [ f (um , K id ) ] dτ ⎟ − [ f (um , K id ) ] α ⎪⎬ − ∇umT ( D∇um ) 2 t τ ⎜ ⎟⎟ u ⎟ ∂ ⎜ m ⎪ ⎠ ⎝o ⎠ ⎭

⎧⎛ 0 ⎛ 0 uim ⎪ u r = φim ⎨⎜ im ⎜ −1 + ⎜ 2 ⎜ um2 + 4 K id ⎝ ⎩⎪⎝

⎞ ⎟ − c0 ⎟ imi ⎠

⎞ ⎟ g (t ) − ⎟ ⎠

∫

t

∫

H (t − τ )

o

2 K id u + 4 K id 2 m

dτ + τ

⎫ ⎪

2 K id u + 4 K id 2 m

α ⎬− t

⎭⎪

(u

2 K id

2 m

+ 4 K id )

3/ 2

∇umT ( D∇um )

po

Replacing g(t), H(t) and α , a expression for reaction rate is achieved: ⎧ 0 0 ⎞ uim ⎪⎛ u ⎛ 0 ⎟ − cim r = φim ⎨⎜ im ⎜ −1 + i 2 ⎜ ⎟ um + 4 K id ⎠ ⎪⎩⎜⎝ 2 ⎝ 2 K id − ∇umT ( D∇um ) 3/ 2 2 ( um + 4 Kid )

∞

⎞ ⎟ α f (α )e −α t dα − ⎟ ⎠0

∫

t

∫ 0

2 K id um2 + 4 K id

∞

∫α f (α )e 2

τ 0

xviii

−α ( t −τ )

⎫ 2 K id ⎪ dα dτ ⎬ + 2 um + 4 K id ⎪⎭

∞

∫α f (α )dα

t 0

pp


Using the pdfs defined by Haggerty et al. (2000) in (33), the following expression is obtained: M2

0 ⎛ u0 ⎛ uim = ⎜ im ⎜ −1 + φim ⎜ 2 ⎜ um2 + 4 K id ⎝ ⎝

∞

⎞ ⎟ α f (α )e −α t dα − ⎟ ⎠0

⎞ 0 ⎟ − cim i ⎟ ⎠

r1

M3

∫

t

∫

2 K id um2 + 4 K id

0

M1

∞

∫α

2

f (α )e

−α ( t −τ )

2 K id

dα dτ +

um2 + 4 K id

τ 0

∞

∫α

f (α )d β

pp b

t 0

The following step is to determine the integrals M. The following is the procedure: 1.

If the analyzed model is a linear multirate limited process for sorption, the model is sorption f (α ) = β tot δ (α − α1 ) (Lawrence et al., 1998).

∑ N

p (α ) =

For several singularities:

i =1

βi δ (α − α i ) , then f (α ) = βtot ⋅ p (α ) = β tot βtot

∑

∞

∫

Using the Dirac Delta’s property:

f ( x ) δ ( x − a ) dx = f ( a )

−∞

∞

M1 =

∫α

f (α )dα =

0

M2 =

∫

∫ α ∑ β δ (α ∞

i

0

∞ −α t

α f (α )e dα =

0

∫α 0

∫

∞

2

f (α )e

−α ( t −τ )

− α i ) dα =

i

αe

∑ β δ (α i

0

∑β α N

− α i ) dα =

i =1

∫α

pq

i

i =1

N

−α t

∞

dα =

∑β α N

i =1

0

∞

M3 =

N

i

e

− α ( t −τ )

∑ β δ (α i

i =1

e −α i t

pr

i =1

N

2

i

∑β α N

− α i ) dα =

i

2 −α i ( t −τ ) i

e

i =1

Replacing (34) (35) and (36) in (33b), it gets:

xix

ps

N

i =1

βi δ (α − α i ) = βtot

∑ β δ (α N

i

i =1

∑β N

− α i ) with β tot =

i =1

fi .


⎧⎛ 0 ⎛ 0 uim ⎪ u r = φim ⎨⎜ im ⎜ −1 + ⎜ 2 ⎜ um2 + 4 K id ⎝ ⎩⎪⎝ −

(u

2 K id

2 m

+ 4 K id )

3/ 2

⎞ ⎟ − c0 ⎟ imi ⎠

t

∑β α e − ∫ N

i

i

i =1

2 −α i ( t −τ )

um2 + 4 K id

0

∑β α e N

2 K id

−α i t

i

τ

i

dτ +

i =1

um2 + 4 K id

∑ N

2 K id t

i =1

⎫ ⎪ β iα i ⎬ ⎪⎭

pt

∇umT ( D∇um )

⎧⎛ 0 ⎛ 0 ⎞ uim ⎪ u 0 ⎟ − cim r = φim ⎨⎜ im ⎜ −1 + i 2 ⎜ ⎟ ⎜ 2 um + 4 K id ⎠ ⎪⎩⎝ ⎝ 2 K id − ∇umT ( D∇um ) 3/ 2 2 ( um + 4 Kid )

1)

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

∑ N

i =1

t

∑ ∫ N

β iα i e

−α i t

βiα i

−2

i =1

K id

2

0

um2 + 4 K id

e

−α i ( t −τ )

dτ +

τ

um2 + 4 K id

∑ N

2 K id t

i =1

⎫ ⎪

β iα i ⎬ ⎭⎪

pu

If N = 1 , For one-site:

⎧ 0 0 uim ⎪⎜⎛ uim ⎜⎛ −1 + r = φim ⎨ ⎜ 2 ⎜ um2 + 4 K id ⎝ ⎩⎪⎝

⎞ 0 ⎟ − cim i ⎟ ⎠

⎞ ⎟ β totα e −α t − 2β totα 2 ⎟ ⎠

⎧⎛ ⎪⎜ 0 ⎛ 0 ⎞ uim ⎪ u 0 ⎟ − cim r = φim β tot ⎨⎜ im ⎜ −1 + i 2 ⎜ 2 ⎜ ⎟ ⎪⎜ m + 4 K id ⎠ ⎝u

⎪⎜ f1 ⎩⎝

t

∫ 0

K id um2 + 4 K id

e

−α ( t −τ )

τ

dτ +

2 K id um2 + 4 K id

⎫ 2 K id ⎪ ∇umT ( D∇um ) β totα ⎬ − 3/ 2 2 ⎪⎭ ( um + 4 K id ) t

⎫ ⎞ t ⎪ ⎟ 2 K id 2 K id 2 K id ⎪ −α ( t −τ ) ⎟ αβ e −α t − α 2 α⎬− ∇umT ( D∇um ) e dτ + 3/ 2 2 2 ⎟ tot 2 um + 4 K id um + 4 K id ⎪ ( um + 4 K id ) ⎟⎟ 0

⎪

f f2 2 f3 ⎠ τ t ⎭

∫

xx

pv


⎧ 0 ⎪⎛ u 0 ⎞ α e −α t − α 2 r = φim β tot ⎨⎜ im f1 ( um ) − cim i ⎟ 2 ⎠ ⎪⎩⎝

t

∫ f (u ) e 2

m

−α ( t −τ )

τ

0

0.7

⎫ ⎪ dτ + f 2 ( um ) t α ⎬ − f3 ( um ) ∇umT ( D∇um ) ⎪⎭

qm

2

0.6 1.5

0.5 0.4

1

0.3 0.2

0.5

0.1 0

5

10 x

15

20

xe− x

0

5

10

x

15

20

25

x 2e− x

2) If α → ∞ then (From (40)) ⎧ 0 ⎪⎛ u 0 lim r = φim β tot ⎨⎜ im f1 ( um ) − cim i α →∞ 2 ⎪⎩⎝

⎧0 If lim α e −α t = δ ( t ) = ⎨ α →∞ ⎩1

(

lim α 2 e −α ( t −τ ) = lim α

α →∞

α →∞

⎞ α e −α t − ⎟ αlim →∞ ⎠

t

∫ 0

⎫ ⎪ f 2 ( um ) τ lim α 2 e −α ( t −τ ) dτ + f 2 ( um ) t lim α ⎬ − f 3 ( um ) ∇umT ( D∇um ) α →∞ α →∞ ⎪⎭

otherwise ⎧0 −α t −τ and lim α e ( ) = δ ( t − τ ) = ⎨ α →∞ if t = τ ⎩1

) ( lim α e α →∞

−α ( t −τ )

otherwise if t = τ

) = δ (t − τ ) ( lim α )

qn

qo

qp

α →∞

xxi


Replacing (42) and (43) in (41): ⎧ 0 ⎪⎛ u 0 lim r = φim β tot ⎨⎜ im f1 ( um ) − cim i α →∞ 2 ⎪⎩⎝

⎞ ⎟δ (t ) − ⎠

⎧ 0 ⎪⎛ u 0 lim r = φim β tot ⎨⎜ im f1 ( um ) − cim i α →∞ 2 ⎪⎩⎝

⎞ α ⎟ δ ( t ) − αlim →∞ ⎠

0 ⎪⎧⎛ u 0 lim r = φim β tot ⎨⎜ im f1 ( um ) − cim i α →∞ 2 ⎩⎪⎝

⎫⎪ ⎞ α f 2 ( um ) τ + f 2 ( um ) t lim α ⎬ − f3 ( um ) ∇umT ( D∇um ) ⎟ δ ( t ) − αlim →∞ α →∞ ⎠ ⎭⎪

⎧⎪⎛ u 0 0 lim r = φim β tot ⎨⎜ im f1 ( um ) − cim i α →∞ ⎪⎩⎝ 2

⎞ α f 2 ( um ) τ + f 2 ( um ) t lim α ⎟ δ ( t ) − αlim →∞ α →∞ ⎠

∫ 0

⎫ ⎪ f 2 ( um ) τ δ ( t − τ ) lim α dτ + f 2 ( um ) t lim α ⎬ − f 3 ( um ) ∇umT ( D∇um ) α →∞ α →∞ ⎪⎭

(

(

lim r = − f3 ( um ) ∇umT ( D∇um ) = −

α →∞

t

t

)∫ 0

(

)

(

)

2 K id

( u + 4 Kid ) 2 m

3/ 2

)

⎫ ⎪ f 2 ( um ) τ δ ( t − τ ) dτ + f 2 ( um ) t lim α ⎬ − f 3 ( um ) ∇umT ( D∇um ) α →∞ ⎪⎭

⎫⎪ T ⎬ − f 3 ( um ) ∇um ( D∇um ) ⎪⎭

∇umT ( D∇um )

qq

The expression (44) is the same for De Simoni et al. (2005). 3) If α → 0 then: ⎧ 0 ⎪⎛ u 0 lim r = φim β tot ⎨⎜ im f1 ( um ) − cim i α →0 ⎪⎩⎝ 2

⎞ α e −α t − ⎟ αlim →0 ⎠

t

∫ 0

⎫ ⎪ f 2 ( um ) τ lim α 2 e −α ( t −τ ) dτ + f 2 ( um ) t lim α ⎬ − f3 ( um ) ∇umT ( D∇um ) α →0 α →0 ⎪⎭

xxii

qr


⎧0 If lim α e −α t = δ ( t ) = ⎨ α →0 ⎩1

(

)(

otherwise if t = τ

and lim α e α →0

)

(

lim α e −α (t −τ ) = lim α lim e −α (t −τ ) = δ ( t − τ ) lim α

α →0

Í

α →0

α →0

−α ( t −τ )

α →0

⎧0 = δ (t − τ ) = ⎨ ⎩1

otherwise if t = τ

)

qt

This limit is undefined as long as L’Hopital Theorem is applied. lim α →0

(

)

(

)

1 1 1 = = → undefined −t αe 0 ×1 0

⎧⎪⎛ u 0 ⎫⎪ T 0 ⎞ − + lim r = φim β tot ⎨⎜ im f1 ( um ) − cim δ lim α lim α t f u f u ( ) ( ) ( ) ⎬ − f3 ( um ) ∇um ( D∇um ) ⎟ m m 2 2 i t α →0 τ 0 → α →0 α ⎪⎩⎝ 2 ⎪⎭ ⎠ ⎧⎪⎛ u 0 0 lim r = φim β tot ⎨⎜ im f1 ( um ) − cim i α →0 2 ⎩⎪⎝

⎞ α f 2 ( um ) τ + f 2 ( um ) t lim α ⎟ δ ( t ) − αlim →0 α →0 ⎠

lim r = − f3 ( um ) ∇umT ( D∇um ) = −

α →0

(u

2 K id

2 m

+ 4 K id )

3/ 2

qs

⎫⎪ T ⎬ − f3 ( um ) ∇um ( D∇um ) ⎭⎪

∇umT ( D∇um )

qu

The expression (48) is the same for De Simoni et al. (2005).

xxiii


MATRIX DIFFUSION Recalling the expression (9)b: ∞

Γ (t ) =

∫

∂c 0 f (α ) im dα = −cim g (t ) − ∂t

0

∫

∫ H (t − τ )c (τ )dτ + c m

m

α

0

∞

Γ (t ) =

t

f (α )

∂cim 0 dα = cm α − cim g (t ) − cm ∗ H ∂t

0

0 Γ ( t ) = cm g ( 0 ) − cim g (t ) − cm ∗ g '

qs

∞

g (0) =

∫β

f ( β )d β

0

xxiv


From Haggerty et al (2000)

xxv


For the case of Diffusion in Finite Layer:

a: half thickness of the immobile domain Da: Apparent diffusivity ∞

2 ⎡ 2 j − 1) π 2 Da ⎤ ( f (α ) = δ ⎢α − ⎥ 2 4a 2 ⎥⎦ ( 2 j − 1) π 2 ⎢⎣

8βtot

Finite Layer:

g (t ) =

∑ j =1

2 βtot

⎡ ( 2 j − 1)2 π 2 Da Da exp ⎢ − a 4a 2 ⎢⎣

⎤ t⎥ ⎥⎦

The following are the moments of the memory function: ∞

g ( 0) =

2βtot

⎡ ( 2 j − 1)2 π 2 Da Da exp ⎢ − a 4a 2 ⎢⎣

2 βtot

⎡ ( 2 j − 1)2 π 2 Da ⎤ Da D exp ⎢ − t ⎥ = 2β tot a 2 a a 4a ⎢⎣ ⎥⎦

∑ j =1

∞

g (t ) =

∑ j =1

∂ g' ( t ) = ∂t g' ( t ) = −

∞

∑

2 βtot

j =1

π 2 Da 2 2a

3

⎡ ( 2 j − 1)2 π 2 Da ⎤ Da exp ⎢ − t⎥ = − a 4a 2 ⎢⎣ ⎥⎦ ∞

β tot

⎤ D 0 ⎥ = 2β tot a a ⎥⎦

∑ j =1

( 2 j − 1)

∑

∑ j =1

j =1

⎡ ( 2 j − 1)2 π 2 Da ⎤ exp ⎢ − t⎥ 4a 2 ⎢⎣ ⎥⎦

⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎡ ( 2 j − 1)2 π 2 Da ⎤ Da exp t⎥ = − 2 β ⎢ ⎥ ⎢− tot a 4a 2 4a 2 ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦

⎡ ( 2 j − 1) π Da ⎤ exp ⎢ − t⎥ 4a 2 ⎢⎣ ⎥⎦ 2

2

∞

∞

2

Replacing them in the MRMT term, the expression (47) is obtained.

xxvi

∞

∑ j =1

⎡ ( 2 j − 1)2 π 2 Da 2 ⎤ ⎡ ( 2 j − 1)2 π 2 Da ⎤ exp t⎥ β ⎢ ⎥ ⎢− tot 2a 3 4a 2 ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦


Γ ( t ) = 2 βtot

Γ ( t ) = 2 βtot

Γ ( t ) = 2 β tot

Γ ( t ) = 2 βtot

Γ ( t ) = 2 β tot

Da D 0 cm − 2cim βtot a a a

Da D 0 cm − 2cim βtot a a a

Da D 0 β tot a cm − 2cim a a

Da D 0 cm − 2cim βtot a a a ⎧ Da ⎪ 0 ⎨cm − cim a ⎪ ⎩

∞

∑ j =1

∞

∑ j =1

∞

⎛ π 2D 2 ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎧⎪ a ⎜− exp ⎢ − t c − ∗ β ⎥ ( ) ⎨ m ⎜ 2a 3 tot 4a 2 ⎢⎣ ⎥⎦ ⎪ ⎝ ⎩ t

∞

∑

⎡ ( 2 j − 1)2 π 2 Da exp ⎢ − 4a 2 ⎢⎣

⎤ t⎥ + ⎥⎦

∞

⎡ ( 2 j − 1)2 π 2 Da exp ⎢ − 4a 2 ⎢⎣

⎤ π 2D 2 a β tot t⎥ + 3 2 a ⎥⎦

j =1

∑ j =1

∞

∑ j =1

∫ 2a

π 2 Da 2 3

⎤ π 2D a t⎥ + 2 ⎥⎦ 4a

j =1

0

⎡ ( 2 j − 1)2 π 2 Da ⎤ π 2 D 2 a t⎥ + βtot exp ⎢ − 2 3 a a 4 2 ⎥⎦ ⎣⎢

⎡ ( 2 j − 1)2 π 2 Da exp ⎢ − 4a 2 ⎣⎢

∑

βtot

t

j =1

∫∑

j =1

t

∞

∑∫ j =1

t

2

∫ 0

0

2

⎡ ( 2 j − 1)2 π 2 Da ⎤ t τ − ⎥ ⋅ cm (τ ) dτ ( 2 j − 1) exp ⎢ − ( ) 4a 2 ⎢⎣ ⎥⎦ 2

( 2 j − 1)

2

j =1

0

2 j − 1)

∑

⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎞ ⎫⎪ t ⎥ ⎟⎬ ( 2 j − 1) exp ⎢ − 4a 2 ⎢⎣ ⎥⎦ ⎟ ⎪ ⎠⎭

∞

∞

∑(

∞

⎡ ( 2 j − 1)2 π 2 Da ⎤ exp ⎢ − ( t − τ )⎥ ⋅ cm (τ ) dτ 2 4a ⎢⎣ ⎥⎦

⎡ ( 2 j − 1)2 π 2 Da ⎤ t τ − ⎥ ⋅ cm (τ ) dτ ( 2 j − 1) exp ⎢ − ( ) 4a 2 ⎣⎢ ⎦⎥ 2

⎫ ⎡ ( 2 j − 1)2 π 2 Da ⎤ exp ⎢ − ( t − τ )⎥ ⋅ cm (τ ) dτ ⎬⎪ 2 4a ⎣⎢ ⎦⎥ ⎭⎪

ALL THE SERIES ARE CONVERGENT!!!! In t=0, NO!!!

xxvii

qt


For the case of diffusion in sphere, the following are the pdf and the memory function (Haggerty et al., 2000) ∞

f (α ) = ∑ j =1

6 β tot ⎛ D ⎞ δ ⎜ α − j 2π 2 2a ⎟ 2 2 jπ ⎝ a ⎠

∞

and

g ( t ) = ∑ 6 β tot j =1

Da D ⎞ ⎛ exp ⎜ − j 2π 2 2a t ⎟ 2 a a ⎠ ⎝

And the moments are: 1

∞

g ( 0 ) = ∑ 6 β tot j =1

g ( t ) = 6 β tot

Da a2

Da D ⎞ D ⎛ exp ⎜ − j 2π 2 2a 0 ⎟ =6 β tot 2a 2 a a ⎠ a ⎝ ⎛

∑ exp ⎜⎝ − j π

g ' ( t ) = −6π β tot 2

∞

2

j =1

Da 2 a4

∞

∑j j =1

2

2

Da a2

⎞ t⎟ ⎠

D ⎞ ⎛ exp ⎜ − j 2π 2 2a t ⎟ a ⎠ ⎝

Replacing the moments en the MRMT term, the following expression is obtained: Γ ( t ) = cm 6 β tot

Γ ( t ) = cm 6 β tot

Γ ( t ) = cm 6 β tot

Da D 0 6 β tot 2a − cim 2 a a

Da D 0 6 β tot 2a − cim 2 a a

Da D 0 − cim 6 β tot 2a 2 a a

∞

∑ j =1

∞

∑ j =1

∞

∑ j =1

⎡ D2 ⎛ 2 2 Da ⎞ exp ⎜ − j π 2 t ⎟ − cm ∗ ⎢ −6π 2 β tot a4 ⎢ a ⎠ a ⎝ ⎣ D ⎞ ⎛ exp ⎜ − j 2π 2 2a t ⎟ − a ⎠ ⎝

t

∫ 0

⎡ 2 ⎢ −6π 2 β tot Da ⎢ a4 ⎣

D ⎞ D2 ⎛ exp ⎜ − j 2π 2 2a t ⎟ + 6π 2 β tot a4 a ⎠ a ⎝

∞

∑ j =1

∞

∑ j =1

∞

t

j =1

0

∑∫

⎤ ⎛ 2 2 Da ⎞ ⎥ j exp ⎜ − j π 2 t ⎟ a ⎠⎥ ⎝ ⎦ 2

⎤ D ⎡ ⎤ j 2 exp ⎢ − j 2π 2 2a ( t − τ ) ⎥ ⎥ ⋅ cm (τ ) dτ a ⎣ ⎦⎥ ⎦

D ⎡ ⎤ j 2 exp ⎢ − j 2π 2 2a ( t − τ ) ⎥ ⋅ cm (τ ) dτ a ⎣ ⎦

xxviii


⎧ Da ⎪ 0 ⎨cm − cim a2 ⎪ ⎩

Γ ( t ) = 6 β tot

∞

∑ j =1

D ⎞ D ⎛ exp ⎜ − j 2π 2 2a t ⎟ + π 2 2a a ⎠ a ⎝

t

∞

∑ ∫ j

2

j =1

0

⎫ ⎡ 2 2 Da ⎤ ⎪ exp ⎢ − j π 2 ( t − τ ) ⎥ ⋅ cm (τ ) dτ ⎬ a ⎣ ⎦ ⎪⎭

qu

ALL THE SERIES ARE CONVERGENT!!!! In t=0, NO!!!

r = φim Γ ( t ) u − f 3 ( um ) ∇umT ( D∇um )

Recalling the rate reaction,

m

From (48): Matrix Diffusion in spheres: Γ ( t )1 = 6 β tot

Da a2

Γ ( t )2 = 6 β tot

⎧ ⎪ 0 ⎨cm1 − cim1 ⎩⎪

Da a2

∞

∑

⎧ ⎪ 0 ⎨cm2 − cim2 ⎪⎩

j =1

∞

∑ j =1

t

∞

∑ ∫

D ⎞ D ⎛ exp ⎜ − j 2π 2 2a t ⎟ + π 2 2a a ⎠ a ⎝

j

2

j =1

D ⎞ D ⎛ exp ⎜ − j 2π 2 2a t ⎟ + π 2 2a a ⎠ a ⎝

0

⎫ D ⎡ ⎤ ⎪ exp ⎢ − j 2π 2 2a ( t − τ ) ⎥ ⋅ cm1 (τ ) dτ ⎬ a ⎣ ⎦ ⎪⎭

qv

⎫ D ⎡ ⎤ ⎪ exp ⎢ − j 2π 2 2a ( t − τ ) ⎥ ⋅ cm2 (τ ) dτ ⎬ a ⎣ ⎦ ⎪⎭

rm

⎫ ⎡ 2 2 Da ⎤ ⎪ exp ⎢ − j π 2 ( t − τ ) ⎥ ⋅ um (τ ) dτ ⎬ a ⎣ ⎦ ⎪⎭

rn

t

∞

∑ ∫ j

2

j =1

0

∞

t

(49) minus (50): Γ ( t )u = 6 β tot m

⎧ Da ⎪ 0 ⎨um − uim a2 ⎪ ⎩

∞

∑ j =1

D ⎞ D ⎛ exp ⎜ − j 2π 2 2a t ⎟ + π 2 2a a ⎠ a ⎝

∑ ∫ j

2

j =1

0

Replacing (51) in the reaction function:

r = 6φim β tot

Da a2

⎧ ⎪ 0 ⎨um − uim ⎪⎩

∞

∑ j =1

D ⎞ D ⎛ exp ⎜ − j 2π 2 2a t ⎟ + π 2 2a a ⎠ a ⎝

t

∞

∑ ∫ j

j =1

2

0

⎫ D ⎡ ⎤ ⎪ exp ⎢ − j 2π 2 2a ( t − τ ) ⎥ ⋅ um (τ ) dτ ⎬ − f 3 ( um ) ∇umT ( D∇um ) ro a ⎣ ⎦ ⎪⎭

xxix


In the same way for infinite layer:

r = 2 β tot

Da a

⎧ ⎪ 0 ⎨um − uim ⎪⎩

∞

∑ j =1

⎡ ( 2 j − 1)2 π 2 Da exp ⎢ − 4a 2 ⎢⎣

⎤ π 2D a t⎥ + 2 4 a ⎥⎦

t

∞

∑

( 2 j − 1)

2

j =1

∫ 0

⎫ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎪ T exp ⎢ − t τ u τ d τ − ⋅ ⎥ ( ) ( ) ⎬ − f 3 ( um ) ∇um ( D∇um ) ro m 2 4 a ⎢⎣ ⎥⎦ ⎪⎭

____ In summary, till the moment, this is the shape of the expression for the reaction rate with MRMT approach.

r = φim Γ ( t ) − f 3 ( um ) ∇umT ( D∇um )

Recalling the reaction rate, 0 ⎛ u0 ⎛ uim r = φim ⎜ im ⎜ −1 + ⎜ 2 ⎜ um2 + 4 K id ⎝ ⎝

−

(u

2 K id

2 m

+ 4 K id )

3/ 2

⎞ 0 ⎟ − cim i ⎟ ⎠

∞

⎞ ⎟ α f (α )e −α t dα − ⎟ ⎠0

∫

t

∫

∞

∫α f (α )e

2 K id u + 4 K id 2 m

0

2

−α ( t −τ )

dα dτ +

τ 0

∇umT ( D∇um )

⎛ u0 0 ⎞ r = φim ⎜ im f1 ( um ) − cim g (t ) − i ⎟ ⎝ 2 ⎠ ⎛ u0 0 r = φim ⎜ im f1 ( um ) − cim i 2 ⎝

⎞ ⎟ g (t ) − ⎠

t

∫ f (u ) g '(t ) dτ + f (u ) g (0) − f (u )∇u 2

m

τ

2

m

t

3

m

T

( D∇um )

T

( D∇um )

m

0

t

∫ f (u ) g '(t ) dτ + f (u ) g (0) − f (u )∇u 2

m

τ

2

m

t

3

m

m

0

0 ⎫⎪ ⎪⎧⎛ u 0 ⎞ r = φim ⎨⎜ im f1 ( um ) − cim g ( t ) − g ' ( t ) * f 2 ( um ) τ + f 2 ( um ) t g ( 0 ) ⎬ − f 3 ( um ) ∇umT ( D∇um ) i ⎟ ⎪⎩⎝ 2 ⎪⎭ ⎠

xxx

2 K id um2 + 4 K id

∞

∫α f (α )dα

t 0


For the case of Linear Sorption: ⎧⎛ ⎪⎜ 0 ⎛ 0 ⎞ uim ⎪ u 0 ⎟ − cim r = φim β tot ⎨⎜ im ⎜ −1 + i 2 ⎜ 2 ⎜ ⎟ ⎪⎜ m + 4 K id ⎠ ⎝u

⎪⎜ f1 ⎩⎝

⎫ ⎞ t ⎪ ⎟ 2 K id 2 K id 2 K id ⎪ ⎟ αβ e −α t − α 2 ∇umT ( D∇um ) α⎬− e −α ( t −τ ) dτ + 3/ 2 2 2 ⎟ tot 2 um + 4 K id um + 4 K id ⎪ ( um + 4 K id ) ⎟⎟ 0

⎪

f2 f2 f3 ⎠ τ t ⎭

∫

The shape of the equation is: ⎧ 0 ⎪⎛ u 0 r = φim β tot ⎨⎜ im f1 ( um ) − cim i 2 ⎪⎩⎝

⎞ −α t 2 ⎟α e − α ⎠

t

∫ 0

⎫ ⎪ f 2 ( um ) τ e −α ( t −τ ) dτ + f 2 ( um ) t α ⎬ − f 3 ( um ) ∇umT ( D∇um ) ⎪⎭

From (47) it is achieved an expression for Matrix Diffusion in infinite layer:

r = 2φim β tot

Da a

⎧ ⎪ 0 ⎨um − uim ⎩⎪

∞

∑ j =1

⎡ ( 2 j − 1)2 π 2 Da ⎤ π 2 D a exp ⎢ − t⎥ + 2 2 4 4 a a ⎣⎢ ⎦⎥

t

∞

∑

( 2 j − 1)

j =1

2

∫ 0

⎫ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎪ T exp ⎢ − − ⋅ t τ u τ d τ ⎥ ( ) ( ) ⎬ − f3 ( um ) ∇um ( D∇um ) m 2 4 a ⎣⎢ ⎦⎥ ⎭⎪

From (48), it is obtained an expression for Matrix Diffusion in spheres:

r = 6φim β tot

Da a2

⎧ ⎪ 0 ⎨um − uim ⎪⎩

∞

∑ j =1

D ⎞ D ⎛ exp ⎜ − j 2π 2 2a t ⎟ + π 2 2a a ⎠ a ⎝

t

∞

∑ ∫ j

j =1

2

0

⎫ D ⎪ ⎡ ⎤ exp ⎢ − j 2π 2 2a ( t − τ ) ⎥ ⋅ um (τ ) dτ ⎬ − f 3 ( um ) ∇umT ( D∇um ) a ⎣ ⎦ ⎪⎭

xxxi


Applying the case of step-function input If um ( x, t ) is represented by the step-function input, from Ogata (1970), it has:

u m ( x, t = 0 ) = 0

x≥0

um ( x = 0, t ) = um0

t≥0

u m ( x = ∞, t ) = 0

t≥0

The function that represents the function is:

u m ( x, t ) =

erfc ( z ) =

⎛ x − vt um0 ⎡ ⎢erfc ⎜ ⎜2 Dt 2 ⎢ x ⎝ ⎣

2

π

∞

∫ z

e

− p2

⎞ ⎛ vx ⎟ + exp ⎜ ⎟ ⎝ Dx ⎠

⎛ x + vt ⎞ ⎟ erfc ⎜⎜ ⎠ ⎝ 2 Dx t

⎛ x − vt dp . Then erfc ⎜ ⎜2 Dt x ⎝

⎞ 2 ⎟= ⎟ π ⎠

⎞⎤ ⎟ ⎥ , where erfc is the complementary error function and it is represented by: ⎟⎥ ⎠⎦

∞

∫

2

e − p dp . The following picture shows the error function.

x − vt 2 Dx t

xxxii


-4

-2

1

2

0.5

1.5

0

2

x

1

4

0.5

-0.5

-1

-4

erf ( z )

-2

erfc ( z )

The strong dependence of the first term turns this function into:

⎛ x − vt um0 u m ( x, t ) = erfc ⎜ ⎜2 Dt 2 x ⎝

⎞ ⎟ ⎟ ⎠

xxxiii

0

2 x

4


Replacing for diffusion equations:

⎧ 0 ⎛ x − vt ⎪ um r ( x, t ) = 2φim β tot ⎨ erfc ⎜⎜ 2 ⎝ 2 Dx t ⎩⎪ 2 K id Dx ∂2 − u 3/ 2 2 m ( u 2 + 4 K ) ∂x

⎞ 0 ⎟ − uim ⎟ ⎠

⎧ ⎛ x − vt Da ⎪ um0 ⎨ erfc ⎜⎜ a ⎪2 ⎝ 2 Dx t ⎩

⎞ ⎟ − u0 ⎟ im ⎠

Da a

m


−

∑

⎤ π 2D a t⎥ + 2 ⎦⎥ 4a

∑

∞

⎡ ( 2 j − 1)2 π 2 Da exp ⎢ − 4a 2 ⎣⎢

⎤ π 2D a t⎥ + 2 a 4 ⎦⎥

∞

j =1

t

∞

⎡ ( 2 j − 1)2 π 2 Da exp ⎢ − 4a 2 ⎣⎢

( 2 j − 1)

2

j =1

∫ 0

⎫ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎥ − ⋅ t u d exp ⎢ − τ τ τ ( ) m ( ) ⎪⎬ 2 4a ⎥⎦ ⎪⎭ ⎣⎢

id

2 K id Dx

⎛⎡ 0 ⎛ ⎜ ⎢ um erfc ⎜ x − vt ⎜⎜ ⎢ 2 ⎜2 D t x ⎝ ⎝⎣


∞

2

⎞⎤ ⎟ ⎥ + 4 K id ⎟⎥ ⎠⎦

⎧ ⎪ ⎛ x − vt Da ⎪ um0 ⎨ erfc ⎜⎜ a ⎪2 ⎝ 2 Dx t ⎪ ⎩

⎞ ⎟ ⎟⎟ ⎠

3/ 2

j =1

0 ⎛ x − vt ∂ 2 ⎡ um ⎢ erfc ⎜ 2 ⎜ ∂x ⎢ 2 ⎝ 2 Dx t ⎣

⎞ 0 ⎟ − uim ⎟ ⎠

K id ( um0 ) Dx 1 − 2 2⎛ 0 ⎡ ⎛ ⎞⎤ ⎜ ⎢ um erfc ⎜ x − vt ⎟ ⎥ + 4 K id ⎜⎜ ⎢ 2 ⎜ 2 D t ⎟⎥ x ⎠⎦ ⎝ ⎣ ⎝

∑

∑

2

⎞ ⎟ ⎟⎟ ⎠

∞

3/ 2

j =1

( 2 j − 1)

j =1

∫ 0

⎡ ( 2 j − 1)2 π 2 Da ⎤ u0 ⎛ x − vτ ⎥ ⋅ m erfc ⎜ − exp ⎢ − τ t ( ) 2 ⎜ 2 4a ⎝ 2 Dxτ ⎣⎢ ⎦⎥

⎞ ⎫⎪ ⎟ dτ ⎟ ⎬ ⎠ ⎪⎭

⎞⎤ ⎟⎥ ⎟ ⎠ ⎥⎦

⎡ ( 2 j − 1)2 π 2 Da exp ⎢ − 4a 2 ⎢⎣

⎛ x − vt ∂2 ⎡ ⎢ erfc ⎜ ⎜2 D t ∂x 2 ⎢ x ⎝ ⎣

∑

t

2

⎤ π 2u0 D m a t⎥ + 2 8 a ⎥⎦

⎞⎤ ⎟⎥ ⎟⎥ ⎠⎦

The last one is the reaction rate for matrix diffusion in infinite layer.

xxxiv

∞

∑ j =1

I1 ⎫

t ⎪ 2 ⎡ ( 2 j − 1) π 2 Da ⎤ ⎛ x − vτ ⎞ ⎪ 2 ⎟ dτ ⎬ ( 2 j − 1) exp ⎢ − ( t − τ )⎥ ⋅ erfc ⎜⎜ 4a 2 2 Dxτ ⎟⎠ ⎪ ⎢⎣ ⎥⎦ ⎝ 0 ⎪ ⎭

∫


It

is

necessary

⎛ x − vt ∂2 ⎡ ⎢ erfc ⎜ 2 ⎜2 D t ∂x ⎢ x ⎝ ⎣

to

calculate

the

derivate

⎛ x − vt ∂ ⎡ ⎢erfc ⎜ ⎜2 D t ∂x ⎢ x ⎝ ⎣

⎡ 1 ( x − vt )2 ⎤ ⎞⎤ 1 exp ⎢ − ⎥ ⎟⎥ = ⎟⎥ π Dx t ⎢⎣ 4 Dx t ⎥⎦ ⎠⎦

and

the

second

derivate

⎡ 1 ( x − vt )2 ⎤ ⎞⎤ x − vt exp ⎢ − ⎥ ⎟⎥ = ⎟⎥ 2 π ( D t )32 ⎢⎣ 4 Dx t ⎦⎥ x ⎠⎦

Replacing in the reaction equation, it is achieved:


−

⎧ ⎪ ⎛ x − vt Da ⎪ um0 ⎨ erfc ⎜⎜ a ⎪2 ⎝ 2 Dx t ⎪ ⎩

⎞ 0 ⎟ − uim ⎟ ⎠

∞

∑ j =1

⎡ ( 2 j − 1)2 π 2 Da exp ⎢ − 4a 2 ⎢⎣

K id ( um0 ) Dx 2

x − vt 4 π ( Dx t ) 2 ⎛ u 0 2 ⎜( m) ⎜⎜ 4 ⎝ 3

⎡ ⎛ x − vt ⎢erfc ⎜ ⎜2 D t ⎢⎣ x ⎝

2

⎞⎤ ⎟ ⎥ + 4 K id ⎟⎥ ⎠⎦

⎞ ⎟ ⎟⎟ ⎠

3/ 2

⎤ π 2u0 D m a t⎥ + 2 8 a ⎥⎦

⎡ 1 ( x − vt )2 ⎤ exp ⎢ − ⎥ ⎢⎣ 4 Dx t ⎥⎦

xxxv

∞

∑ j =1

I1 ⎫

t ⎪ 2 ⎡ ( 2 j − 1) π 2 Da ⎤ ⎛ x − vτ ⎞ ⎪ 2 ⎟ dτ ⎬ ( 2 j − 1) exp ⎢ − ( t − τ )⎥ ⋅ erfc ⎜⎜ 4a 2 2 Dxτ ⎟⎠ ⎪ ⎢⎣ ⎥⎦ ⎝ 0 ⎪ ⎭

∫

is:


Using Maclaurin series (Burden & Faires, 2001): erfc ( z ) = 1 −

( −1) x 2 n+1 ( 2n + 1) n ! π∑ n=0

2

∞

n

Now, the integral I1 is:

t

I1 =

∫ 0

⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎛ x − vτ t τ exp ⎢ − − ⎥ ⋅ erfc ⎜ ( ) 2 ⎜2 Dτ 4a x ⎝ ⎣⎢ ⎦⎥

⎞ ⎟ dτ = ⎟ ⎠

t

∫ 0

⎡ ⎢ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎢ 2 exp ⎢ − ( t − τ )⎥ ⋅ ⎢1 − 2 4a π ⎣⎢ ⎦⎥ ⎢ ⎢ ⎢⎣

∞

∑ n=0

⎛ x − vτ ( −1) ⎜⎜ − ⎝ 2 Dxτ ( 2n + 1) n! n

t

I1 =

∫ 0

⎡ ( 2 j − 1)2 π 2 Da ⎤ 2 exp ⎢ − ( t − τ )⎥ dτ − 2 4a π ⎢⎣ ⎥⎦

t

∫ 0

⎡ ( 2 j − 1)2 π 2 Da ⎤ exp ⎢ − ( t − τ )⎥ ⋅ 2 4a ⎢⎣ ⎥⎦

∞

∑ n=0

⎛ x − vτ ( −1) ⎜⎜ − ⎝ 2 Dxτ ( 2n + 1) n! n

⎞ ⎟ ⎟ ⎠

2 n +1

dτ

⎛ x − vτ ( −1) ⎜⎜ − ⎝ 2 Dxτ ( 2n + 1) n! n

t

⎡ ( 2 j − 1)2 π 2 Da ⎤ 2 exp τ I1 = t − − − ⎢ ⎥ ( ) 2 2 2 4 a π ⎥⎦ 0 ( 2 j − 1) π Da ⎢⎣ 4a 2

t

∫ 0

⎡ ( 2 j − 1)2 π 2 Da ⎤ exp ⎢ − ( t − τ )⎥ ⋅ 2 4a ⎢⎣ ⎥⎦

∞

∑ n=0

⎛ x − vτ ( −1) ⎜⎜ − ⎝ 2 Dxτ ( 2n + 1) n! n

⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎪⎫ 2 ⎪⎧ I1 = t ⎥⎬ − ⎨1 − exp ⎢ − 2 4a 2 π ⎢⎣ ⎥⎦ ⎪⎭ ( 2 j − 1) π 2 Da ⎪⎩ 4a 2

t

∫ 0

⎡ ( 2 j − 1)2 π 2 Da ⎤ exp ⎢ − ( t − τ )⎥ ⋅ 2 4a ⎢⎣ ⎥⎦

xxxvi

∞

∑ n=0

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

2 n +1

dτ

2 n +1

dτ

⎞ ⎟ ⎟ ⎠

2 n +1

⎤ ⎥ ⎥ ⎥ dτ ⎥ ⎥ ⎥⎦


Using Simpson’s method to solve the integral: t

I1 =

∫ 0

⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎛ x − vτ exp ⎢ − ( t − τ ) ⎥ ⋅ erfc ⎜⎜ 2 4a ⎢⎣ ⎥⎦ ⎝ 2 Dxτ

⎞ ⎟ dτ ⎟ ⎠

⎧⎡ ⎫ ⎡ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎛ x − vτ ⎞ ⎤ ⎛ x − vτ ⎞ ⎤ ⎪ ⎢exp ⎢ − ⎪ ⎥ ⎢ ⎥ erfc 3 exp erfc t t − ⋅ + − − ⋅ τ τ ⎥ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ( ) ( ) 2 2 ⎜ 2 D τ ⎟⎥ ⎜ 2 D τ ⎟⎥ 4 a 4 a ⎪⎢ ⎢ ⎢ ⎥ ⎢ ⎥ x x ⎝ ⎠ ⎦τ = 0 ⎝ ⎠ ⎦τ = t ⎪⎪ ⎣ ⎦ ⎣ ⎦ ⎣ 3 t ⎪⎣ 3 = ⎨ ⎬ 2 2 8 3⎪ ⎡ ⎡ ⎡ ( 2 j − 1) π 2 Da ⎤ ⎡ ( 2 j − 1) π 2 Da ⎤ ⎛ x − vτ ⎞ ⎤ ⎛ x − vτ ⎞ ⎤ ⎪ + ⎢ exp ⎢ − ⎟⎥ ⎟⎥ ⎪ ( t − τ )⎥ ⋅ erfc ⎜⎜ ( t − τ )⎥ ⋅ erfc ⎜⎜ ⎪+3 ⎢ exp ⎢ − ⎟ ⎟ 4a 2 4a 2 ⎪ ⎢⎣ ⎝ 2 Dxτ ⎠ ⎥⎦τ = 2t ⎢⎣ ⎝ 2 Dxτ ⎠ ⎥⎦τ =t ⎪ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ 3 ⎩ ⎭

⎧ ⎡ ⎛ t ⎞⎤ +∞ 2 ⎪⎡ 2 − x v ⎢ ⎜ ⎟⎥ ⎤ ⎡ ⎤ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ( 2 j − 1) π Da 2t ⋅ erfc ⎜ ⎛ x⎞ 3 ⎟⎥ ⎪ ⎢ exp ⎢ − ⎢ ⎥ + − ⋅ t erfc 3 exp ⎢ ⎥ ⎥ ⎜ ⎟ ⎪⎢ ⎢ 4a 2 4a 2 3 ⎦⎥ ⎜ t ⎟⎥ ⎝ 0 ⎠ ⎥⎦ ⎢ ⎣ ⎣⎢ ⎦⎥ ⎪⎣ ⎜ 2 Dx ⎟ ⎥ ⎢ 3 ⎠⎦ ⎝ t⎪ ⎣ I1 = ⎨ 8⎪ ⎡ ⎛ 2t ⎞ ⎤ ⎡ 1 ⎜ x−v ⎟⎥ ⎪ ⎢ ⎡ ( 2 j − 1)2 π 2 Da t ⎤ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎛ x − vt ⎢ 3 ⎟ ⎥ + exp ⎢ − 0 ) ⎥ ⋅ erfc ⎜ ⎪+3 ⎢ exp ⎢ − ⎥ ⋅ erfc ⎜ ( 2 2 ⎜2 D t 4a 3 ⎥⎦ 4a ⎜ 2t ⎟ ⎥ ⎢ ⎪ ⎢ ⎢⎣ ⎢⎣ ⎥⎦ x ⎝ 2 D ⎢ ⎣ ⎜ ⎟ ⎢ ⎥ x ⎪ 3 ⎠⎦ ⎝ ⎩ ⎣ ⎧ ⎛ vt 2 2 2 2 ⎪ ⎜ x− ⎡ ⎤ ⎡ ⎤ 2 j − 1) π Da 2 j − 1) π Da ( ( t⎪ 3 I1 = ⎨exp ⎢ − t ⎥ + 3exp ⎢ − t ⎥ ⋅ erfc ⎜ 2 2 8⎪ 4a 6a ⎜ t ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎜ 2 Dx ⎪⎩ 3 ⎝

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎞ ⎤⎥ ⎪ ⎟ ⎪ ⎟⎥ ⎪ ⎠ ⎦⎥ ⎪ ⎭

⎞ ⎛ 2 ⎟ ⎜ x − vt ⎡ ( 2 j − 1)2 π 2 Da ⎤ 3 ⎟ + 3exp ⎢ − t ⎥ ⋅ erfc ⎜ 2 12a ⎟ ⎜ 2t ⎣⎢ ⎦⎥ ⎟ ⎜ 2 Dx 3 ⎠ ⎝

xxxvii

⎞ ⎟ ⎛ ⎟ + erfc ⎜ x − vt ⎜ ⎟ ⎝ 2 Dx t ⎟ ⎠

⎫ ⎞ ⎪⎪ ⎟ ⎟⎬ ⎠⎪ ⎪⎭


Recalling it to the reaction rate: ∞ ⎧u0 ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎛ x − vt ⎞ 0 ⎪ m erfc ⎜ exp ⎢ − t⎥ ⎟−u ⎜ 2 D t ⎟ im 4a 2 ⎪2 ⎢⎣ ⎥⎦ x ⎠ ⎝ j =1 ⎪ ⎪ ⎧ ⎛ vt ⎪ ⎪ ⎜ x− ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎡ ( 2 j − 1)2 π 2 Da ⎤ 3 ⎪ ⎪exp ⎢ − t ⎥ + 3exp ⎢ − t ⎥ ⋅ erfc ⎜ 2 2 Da ⎪ ⎪ ⎜ 4a 6a t ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ ⎨ ⎪ ⎜ 2 Dx a ⎪ π 2u0 D ∞ 3 2 t ⎝ m a ( 2 j − 1) ⎪⎨ ⎪+ 2 8⎪ 8a ⎛ ⎪ 2 ⎞ j =1 2 − x vt ⎟ 2 ⎜ ⎡ ⎤ ⎪ ⎪ ( 2 j − 1) π Da 3 ⎟ + erfc ⎛⎜ x − vt ⎞⎟ ⎜ t erfc ⎥ ⎪ ⋅ ⎪+3exp ⎢ − ⎜2 D t ⎟ ⎜ 12a 2 2t ⎟ ⎢⎣ ⎥⎦ ⎪ ⎪ x ⎠ ⎝ ⎜ 2 Dx ⎟ ⎪ ⎪ 3 ⎝ ⎠ ⎩ ⎩

∑


−

∑

K id ( um0 ) Dx 2

x − vt 4 π ( Dx t ) 2 ⎛ u 0 2 ⎜( m) ⎜⎜ 4 ⎝ 3

⎡ ⎛ x − vt ⎢erfc ⎜ ⎜ ⎝ 2 Dx t ⎣⎢

2

⎞⎤ ⎟ ⎥ + 4 K id ⎟ ⎠ ⎦⎥

⎞ ⎟ ⎟⎟ ⎠

3/ 2

⎫ ⎪ ⎪ ⎪ ⎞ ⎫⎪ ⎟ ⎪⎪ ⎟ ⎪⎪ ⎟ ⎪⎪⎬ ⎟ ⎪⎪ ⎠⎪ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎭⎭

⎡ 1 ( x − vt )2 ⎤ ⎥ exp ⎢ − ⎢⎣ 4 Dx t ⎥⎦

Using Maclaurin series (Burden & Faires, 2001): erfc ( z ) = 1 −

( −1) x 2 n+1 , then: erfc ⎛ x − vt ⎞ = 1 − ⎜ ⎟ ⎜2 D t ⎟ ( 2n + 1) n ! π∑ x ⎠ ⎝ n=0

2

∞

n

The reaction rates can be calculated form the following equation:

xxxviii

⎛ x − vt ⎞ ⎟ ( −1) ⎜⎜ ⎟ ⎝ 2 Dx t ⎠ ( 2n + 1) n! n

2

π

∞

∑ n =0

2 n +1



⎧ ⎡ ⎪ ⎢ ⎪u0 ⎢ ⎪ m ⎢1 − 2 ⎪2 ⎢ π ⎪ ⎢ ⎪ ⎢ ⎪ ⎣ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Da ⎪⎪ ⎨ a ⎪ ⎪ ⎪ π 2 um0 Da ⎪+ 8a 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩

⎫ ⎪ ∞ ∞ ⎪ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎪ ⎥ t exp ⎢ − 2 ⎪ a 4 ⎢ ⎥ ⎣ ⎦ n=0 j =1 ⎪ ⎪ ⎪ 2 n +1 ⎪ ⎧ ⎫ ⎡ ⎤ ⎛ ⎞ vt ⎪ ⎪ ⎪ ⎢ ⎥ ⎜ x− ⎟ ⎪ n ⎪ ⎪ ⎢ ⎥ 3 ⎟ ( −1) ⎜ ⎪ ⎪ ⎪ ⎢ ⎥ ⎜ t ⎟ ⎪ ⎪ ⎪ ⎥ ∞ ⎜ 2 Dx ⎟ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎡ ( 2 j − 1)2 π 2 Da ⎤ ⎢⎢ 2 3⎠ ⎪ ⎪⎪ ⎥ ⎝ t ⎥ + 3exp ⎢ − t⎥ ⎢ 1− ⎪exp ⎢ − ⎪⎪ ⎥ 4a 2 6a 2 ( 2n + 1) n ! π n =0 ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ ⎢ ⎪ ⎪⎪⎪ ⎥ ⎪ ⎪⎬ ⎢ ⎥ ⎪ ⎪⎪ ⎢ ⎥ ⎪ ⎪⎪ ⎢ ⎥ ∞ ⎪ ⎪⎪⎪ ⎢ ⎥ 2 t ⎣ ⎦ ( 2 j − 1) ⎪⎨ ⎬⎪ 2 n +1 8⎪ ⎡ ⎤ ⎪⎪ ⎛ ⎞ j =1 2 ⎢ ⎥ ⎪ ⎪⎪ ⎜ x − vt ⎟ n ⎢ ⎥ 3 2 n +1 ⎪ ⎪ ⎪ ⎜ ⎟ ⎡ ⎤ ( −1) ⎛ x − vt ⎞ ⎢ ⎥ ⎢ n ⎪ ⎜ 2t ⎟ ⎥ ⎪⎪ 1 − ⎜ ⎟ ( ) ⎢ ⎥ ∞ ∞ D 2 2 ⎪ ⎜ ⎟ 2 x ⎜ ⎟ ⎢ ⎥ ⎪⎪ ⎡ ⎤ 2 Dx t ⎠ 2 3 ⎠ ⎥ ⎢ ⎝ ⎝ ⎪+3exp ⎢ − ( 2 j − 1) π Da t ⎥ ⎢1 − 2 ⎥ ⎪⎪ ⎥ + ⎢1 − π ⎪ 2n + 1) n ! 12a 2 ( 2n + 1) n ! ( ⎥ ⎪⎪ π n =0 ⎢⎣ ⎥⎦ ⎢⎢ n=0 ⎥ ⎢ ⎪ ⎥ ⎪⎪ ⎢ ⎥ ⎢ ⎪ ⎥⎦ ⎪⎪ ⎢ ⎥ ⎣ ⎪ ⎪⎪ ⎢ ⎥ ⎪ ⎪⎪ ⎢⎣ ⎥⎦ ⎪⎩ ⎪⎭⎪⎭ ⎛ x − vt ⎞ ⎟ ( −1) ⎜⎜ ⎟ ⎝ 2 Dx t ⎠ ( 2n + 1) n !

2 n +1

n

∑

⎤ ⎥ ⎥ 0 ⎥ − uim ⎥ ⎥ ⎥⎦

∑

∑

∑

−

∑

K id ( um0 ) Dx 2

x − vt 4 π ( Dx t ) 2 ⎧ ⎪ ⎪ u0 2 ⎪( m ) ⎨ ⎪ 4 ⎪ ⎪ ⎩ 3

⎡ ⎢ ⎢ 2 ⎢1 − ⎢ π ⎢ ⎢⎣

∞

∑ n=0

⎛ x − vt ⎞ n ⎟ ( −1) ⎜⎜ ⎟ ⎝ 2 Dx t ⎠ ( 2n + 1) n !

2 n +1

3/ 2

⎡ 1 ( x − vt )2 ⎤ exp ⎢ − ⎥ ⎢⎣ 4 Dx t ⎥⎦

2 ⎫ ⎤ ⎪ ⎥ ⎪ ⎥ ⎪ ⎥ + 4 K id ⎬ ⎥ ⎪ ⎥ ⎪ ⎥⎦ ⎪ ⎭xxxix

∑


∂c ∂ φm 2 − φim cim0 2 g (t ) − φim ∂t ∂t

t

∫

g (t − τ )c2 (τ )dτ = [ −q∇c2 + ∇·(D∇c2 )] + r

0

φm

∂c2 ∂ 0 g (t ) − φim g * c2 = [ −q∇c2 + ∇·(D∇c2 )] + r − φim cim 2 ∂t ∂t

∂c ∂ φm 2 − φim cim0 2 g (t ) − φim ∂t ∂t

t

∫

g (τ )c2 (t − τ )dτ = [ −q∇c2 + ∇·(D∇c2 )] + r

0

Using the following theorem to operate the derivate of an integral (Leibniz)

∂ ∂t

g (t )

∫

f (t )

g (t )

h ( t − τ ) dτ =

∫

f (t )

∂ ⎛∂ ⎞ ⎛∂ ⎞ h ( t − τ ) dτ + ⎜ g ( t ) ⎟ h ( t,g ( t ) ) − ⎜ f ( t ) ⎟ h ( t, f ( t ) ) ∂t ⎝ ∂t ⎠ ⎝ ∂t ⎠

The general expression for the reaction rate is: 1 0 ⎡t ⎤ c20 ∂c2 ∂ ⎛∂ ⎞ ⎛∂ ⎞ 0 ⎢ − φim cim2 g (t ) − φim φm c (t − τ ) g ( t ) dτ + ⎜ t ⎟ c2 ( t − t ) g ( t ) − ⎜ 0 ⎟ c2 ( t − 0 ) g ( 0 ) ⎥ − [ −q∇c2 + ∇·(D∇c2 ) ] = r ⎢ ∂t 2 ⎥ ∂t ⎝ ∂t ⎠ ⎝ ∂t ⎠ ⎣0 ⎦

∫

⎡t ⎤ ∂c2 ∂ 0 c2 (t − τ ) g ( t ) dτ + c20 g ( t ) ⎥ − [ −q∇c2 + ∇·(D∇c2 )] = r φm − φim cim2 g (t ) − φim ⎢ ⎢ ∂t ⎥ ∂t ⎣0 ⎦

∫

xl


⎡ ∂c2 0 − φim cim2 g (t ) − φim ⎢ c20 g ( t ) + φm ⎢ ∂t ⎣

t

∫ 0

∂c φm 2 − φim cim0 2 g (t ) − φim c20 g ( t ) + φim ∂t

⎤ ∂ c2 (t − τ ) g ( t ) dτ ⎥ − [ −q∇c2 + ∇·(D∇c2 )] = r ⎥ ∂t ⎦ t

∫

∂ c2 (t − τ ) g ( t ) dτ − [ −q∇c2 + ∇·(D∇c2 )] = r ∂t

0

∂c φm 2 − φim cim0 2 + c20 g (t ) + φim ∂t

(

)

t

∫

∂ c2 (t − τ ) g ( t ) dτ − [ −q∇c2 + ∇·(D∇c2 )] = r ∂t

0

∂c φm 2 − φim c 0 g (t ) + φim ∂t

t

∫

∂ c2 (t − τ ) g ( t ) dτ − [ −q∇c2 + ∇·(D∇c2 ) ] = r ∂t

(1)

0

Applying the chain rule:

∂c2 ∂c2 ∂u = ∂t ∂u ∂t

An recalling the reaction rate expression:

∂c ∂u φm 2 − φim c 0 g (t ) + φim ∂u ∂t

t

∫

∂ ∂u c2 (t − τ ) g ( t ) dτ − [ −q∇c2 + ∇·(D∇c2 ) ] = r ∂u ∂t

0

xli


∂c ∂u − φim c 0 g (t ) + φim φm 2 ∂u ∂t

t

∫ 0


t

∫ 0

⎡ ∂c ∂c ⎛ ∂c ⎞ ⎤ ∂ ∂u c2 (t − τ ) g ( t ) dτ − ⎢ −q 2 + 2 ⎜ D 2 ⎟ ⎥ = r ∂u ∂t ∂x ∂x ⎝ ∂x ⎠ ⎦ ⎣ ⎡ ∂c ∂ 2c ⎤ ∂ ∂u c2 (t − τ ) g ( t ) dτ − ⎢ −q 2 + D 22 ⎥ = r ∂u ∂t ∂x ∂x ⎦ ⎣

Developing the second derivate: 2

∂ 2 c2 ∂ ∂c2 ∂ ⎛ ∂c2 ∂u ⎞ ∂ ∂c2 ∂u ∂ ∂u ∂c2 ∂ ∂u ∂c2 ∂u ∂ 2u ∂c2 ∂ 2 c2 ⎛ ∂u ⎞ ∂ 2u ∂c2 , in general: = = ⎜ + = + = ⎜ ⎟ + ⎟= ∂x 2 ∂x ∂x ∂x ⎝ ∂u ∂x ⎠ ∂x ∂u ∂x ∂x ∂x ∂u ∂u ∂x ∂u ∂x ∂x 2 ∂u ∂u 2 ⎝ ∂x ⎠ ∂x 2 ∂u

∂ 2 c2 ∂c 2 ∇c2 = 2 ( ∇u ) + 2 ∇ 2u ∂u ∂u An expression for the reaction is:


t

∫ 0

⎡ ∂c ⎤ ∂ 2c ∂c ∂ ∂u 2 c2 (t − τ ) g ( t ) dτ − ⎢ − 2 q∇u + 22 D ( ∇u ) + 2 D∇ 2u ⎥ = r (2) ∂u ∂t ∂u ∂u ⎣ ∂u ⎦

For both species, the equation is the same, and if u = c1 − c2

∂u φm − φim u 0 g (t ) + φim ∂t

t

∫

∂ u (t − τ ) g ( t ) dτ − [ −q∇u + ∇·(D∇u ) ] = 0 ∂t

(3)

0

xlii

, with u 0 = c10 − c20 , then u satisfies the equation for one specie.


(2) – (3) *

∂c2 : ∂u


(2)

t

∫ 0

∂c (3)* 2 ∂u

⎡ ∂c ⎤ ∂ 2c ∂c ∂ ∂u 2 c2 (t − τ ) g ( t ) dτ − ⎢ − 2 q∇u + 22 D ( ∇u ) + 2 D∇ 2u ⎥ = r ∂u ∂t ∂u ∂u ⎣ ∂u ⎦

∂c ∂u ∂c ∂c φm 2 − φim u 0 g (t ) 2 + φim 2 ∂u ∂t ∂u ∂u

t

∫

∂c ∂ u (t − τ ) g ( t ) dτ − 2 [ −q∇u + ∇·(D∇u )] = 0 ∂t ∂u

0

∂c ∂u ∂c ∂c φm 2 − φim u 0 g (t ) 2 + φim 2 ∂u ∂t ∂u ∂u

t

∫ 0

∂c ∂ ⎡ ∂c ⎤ u (t − τ ) g ( t ) dτ − ⎢ − 2 q∇u + 2 D∇ 2u ⎥ = 0 ∂t ∂u ⎣ ∂u ⎦

∂c φim c g (t ) + φim u g (t ) 2 + φim ∂u 0

0

t

∫

∂c ∂ ∂u c2 (t − τ ) g ( t ) dτ − φim 2 ∂u ∂t ∂u

0

⎛ ⎝

2 ∂c2 ⎞ ∂c2 ⎛ ∂u ⎡ ∂c2 ∂u 2 ⎞ ⎤ ∂ c2 φ * g * g D ( ∇u ) = r + − − ⎜ ⎟⎥ ⎟ im ⎢ 2 u t u t u ∂u ⎠ ∂ ∂ ∂ ∂ ∂ ⎝ ⎠⎦ ⎣

⎛ ⎝

2 ⎡ ∂c2 ∂u ∂c2 ⎞ 2 ⎛ ∂u ∂c2 ⎞ ⎤ ∂ c2 + − − φ * g * g D ( ∇u ) = r ⎟ im ⎢ ⎜ ⎟⎥ 2 u t t u u ∂u ⎠ ∂ ∂ ∂ ∂ ∂ ⎝ ⎠⎦ ⎣

φim g (t ) ⎜ c 0 + u 0

∫ 0

Applying convolution theorem to (4):

φim g (t ) ⎜ c 0 + u 0

t

xliii

∂ 2c ∂ 2 u (t − τ ) g ( t ) dτ − 22 D ( ∇u ) = r ∂t ∂u

(4)


t

∂c ⎞ ⎛ φim g (t ) ⎜ c 0 + u 0 2 ⎟ + φim ∂u ⎠ ⎝

∫

⎧ ∂c ⎞ ⎪ ⎛ φim ⎨ g ( t ) ⎜ c 0 + u 0 2 ⎟ + ∂u ⎠ ⎝ ⎪⎩

t

⎧ ∂c ⎞ ⎪ ⎛ φim ⎨ g ( t ) ⎜ c 0 + u 0 2 ⎟ − ∂u ⎠ ⎝ ⎪⎩

0

∫ 0

t

∫ 0

If t ′ = t − τ → τ = t − t ′

⎧ ⎪ φim ⎨ g ( t ) cim0 2 + K id + ⎪⎩

(

)

∂ 2 c2 ∂u ⎡ ∂ ∂ 2 ⎤ − − − c ( t τ ) c ( t ) g t d τ D ( ∇u ) = r ( ) 2 2 2 ⎢ ⎥ ∂t ⎣ ∂u ∂u ∂u ⎦

⎫ 2 ∂u ( t ) ∂ 2 ⎪ ∂c ⎡⎣ c2 ( t − τ ) − c2 ( t ) ⎤⎦ g ( t ) dτ ⎬ − 22 D ( ∇u ) = r ∂t ∂u ⎪⎭ ∂u ⎫ 2 ∂u ( t − τ ) ∂ 2 ⎪ ∂ c2 ⎡⎣c2 ( t ) − c2 ( t − τ ) ⎤⎦ g (τ ) dτ ⎬ − 2 D ( ∇u ) = r ∂τ ∂u ⎪⎭ ∂u then

t

∫ 0

dt ′ = −dτ

⎫ 2 ∂u (τ ) ∂ 2 ⎪ ∂ c2 g (t − τ ) ⎡⎣c2 ( t ) − c2 (τ ) ⎤⎦ dτ ⎬ − 2 D ( ∇u ) = r ∂τ ∂u ⎪⎭ ∂u

If g(t) is an exponential where c2 ( t ) − c2 (τ ) = c2 ( t ) − c2 ( t ) +

⎧ ∂c ⎞ ⎪ ⎛ φim ⎨ g ( t ) ⎜ c 0 + u 0 2 ⎟ + ∂u ⎠ ⎝ ⎪⎩

t

∫ 0

∂c2 ∆τ , then ∂t

⎫ 2 0 ∂u (τ ) ∂ 2 ⎪ ∂c ⎡⎣c2 ( t ) − c2 (τ ) ⎤⎦ dτ ⎬ − 22 D ( ∇u ) = r g (t − τ ) ∂τ ∂u ⎪⎭ ∂u

The new simplified expression for the reaction rate is: ⎛ ⎝

φim g ( t ) ⎜ c 0 + u 0

∂c2 ∂u

2 2 ⎞ ∂ c2 − D ( ∇u ) = r ⎟ 2 ⎠ ∂u

(6)

xliv


1e+20

4

8e+19

3

6e+19 2 4e+19 1

2e+19

0

2

4

1

2

x

6

8

10

3

4

5

0

1

2

x

3

60 50 40 30 20 10 0

⎛ ⎝

φim g ( t ) ⎜ c 0 + u 0

x

g ( t ) = α i β tot e −αit = 3 ⋅ 20e −3t

∂c2 ⎞ ∂ 2 c2 2 Dx ( ∇u ) = r ⎟− ∂u ⎠ ∂u 2

xlv

4

5


Replacing the derivates in the simplified expression, it is achieved that:

um 1⎛ = ⎜ −1 + ∂um 2 ⎜ um2 + 4 K id ⎝

∂cm2

⎛ x − vt u x =0 u ( x, t ) = m erfc ⎜ ⎜2 Dt 2 x ⎝ u x =0 ∂ um = − m ∂x 2

⎞ ∂ 2 cm2 2 K id ⎟ and = 3/ 2 2 ⎟ ∂um ( um2 + 4 Kid ) ⎠ ⎞ ⎡ 1 ( x − vt )2 ⎤ ⎛ x − vt ⎞ ⎤ 1 ∂ ⎡ exp ⎢ − ⎥ , it is achieved that: ⎟⎥ = − ⎟ And it has that: ⎢erfc ⎜⎜ ⎟⎥ ⎟ ∂x ⎢ 4 Dx t ⎦⎥ 2 π D t D t ⎢ x x ⎝ ⎠ ⎣ ⎣ ⎦ ⎠

⎡ 1 ( x − vt )2 ⎤ exp ⎢ − ⎥ π Dx t ⎢⎣ 4 Dx t ⎥⎦ 1

And then replacing the moments of the memory function:

g ( t ) = αβ tot e −α t

The new expression for the reaction rate is: ∂c ⎞ ∂ 2 c 2 ⎛ r ( x, t ) = φim g ( t ) ⎜ c 0 + u 0 2 ⎟ − 22 Dx ( ∇u ) ∂u ⎠ ∂u ⎝

⎛ um 1⎛ r ( x, t ) = φimαβ tot e −α t ⎜ c 0 + u 0 ⎜ −1 + 2 ⎜ 2 ⎜ um + 4 K id ⎝ ⎝

2 ⎞⎞ 2 K id ⎛ ∂um ⎞ ⎟⎟ − D x⎜ ⎟ 3/ 2 2 ⎟⎟ ⎝ ∂x ⎠ ⎠ ⎠ ( um + 4 K id )

xlvi


⎛ ⎛ ⎜ ⎜ ⎜ ⎜ 1 ⎜ r ( x, t ) = φimαβ tot e −α t ⎜ c 0 + u 0 ⎜ −1 + 2⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎝ ⎛ ⎛ ⎜ ⎜ ⎜ ⎜ u0 ⎜ r ( x, t ) = φimαβ tot e −α t ⎜ c 0 + ⎜ −1 + 2⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎝

⎞⎞ ⎟⎟ 2 ⎟⎟ ⎛ ⎡ 1 ( x − vt )2 ⎤ ⎞ umx = 0 2 K id Dx ⎟⎟ − ⎜− exp ⎢ − ⎥⎟ 3/ 2 ⎟⎟ ⎛ 2 2 ⎜ D t 4 π D t 2 ⎢ ⎥⎦ ⎟⎠ ⎞ x = x 0 x ⎡u ⎣ ⎛ x − vt ⎞ ⎤ ⎝ ⎛ x − vt ⎞ ⎤ ⎟ ⎟ ⎜ ⎡ umx = 0 m ⎟ K erfc 4 + ⎢ ⎜ ⎟⎥ erfc ⎜ ⎢ ⎟ ⎥ + 4 K id id ⎟ ⎟ ⎜ 2 D t ⎟⎥ ⎜ 2 D t ⎟⎥ ⎟ 2 ⎟⎟ ⎜ ⎢ 2 x ⎠⎦ x ⎠⎦ ⎝ ⎝ ⎣⎢ ⎠⎠ ⎝ ⎣ ⎠ ⎛ x − vt umx = 0 erfc ⎜ ⎜2 D t 2 x ⎝

⎞ ⎟ ⎟ ⎠

2 ⎞⎞ ⎛ ⎡ 1 ( x − vt )2 ⎤ ⎞ ⎟⎟ x =0 2 ( um ) Kid ⎜⎜ exp ⎢− 4 D t ⎥ ⎟⎟ ⎟⎟ ⎢⎣ ⎥⎦ ⎠ x ⎝ ⎟⎟ + 3/ 2 ⎟⎟ 2 2 ⎛ ⎞ x =0 ⎡ u x =0 ⎤ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎟ ⎟ x − vt u x − vt ⎢ m erfc ⎜ ⎟ ⎥ + 4 K id ⎟ ⎟ 2π t ⎜ ⎢ m erfc ⎜ ⎟ ⎥ + 4 K id ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ 2 2 D t ⎟ 2 D t ⎢⎣ 2 ⎥ ⎢ x ⎠⎦ x ⎠⎥ ⎝ ⎝ ⎦ ⎠⎠ ⎝⎣ ⎠

⎛ x − vt umx = 0 erfc ⎜ ⎜2 D t 2 x ⎝

⎞ ⎟ ⎟ ⎠

Again, the expression is mathematically complex. Solving numerically,

⎛ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 0 u ⎜ −α t ⎜ 0 r ( x, t ) = φimαβ tot e ⎜ c + ⎜ −1 + 2 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎝

⎛ ⎜ x =0 ⎜ um 2 ⎜1 − 2 ⎜ π ⎜ ⎜ ⎝ ⎡ ⎛ ⎢ ⎜ ⎢ u x =0 ⎜ 2 ⎢ m ⎜1 − 2 π ⎢ ⎜ ⎢ ⎜ ⎜ ⎢⎣ ⎝

∞

∑

∑ n=0

⎞⎞ ⎟⎟ ⎟⎟ ∞ ⎟⎟ 2 ⎟⎟ ⎛ ⎡ 1 ( x − vt )2 ⎤ ⎞ ⎟⎟ x =0 2 n=0 ( um ) Kid ⎜⎜ exp ⎢− 4 D t ⎥ ⎟⎟ ⎟⎟ ⎢⎣ ⎥⎦ ⎠ x ⎟⎟ + ⎝ 3/ 2 ⎟ ⎟ 2 2 2 n +1 2 n +1 ⎛⎡ ⎞ ⎤ ⎤ ⎞ ⎛ ⎞ ⎟ ⎟ n ⎛ x − vt ⎞ n ⎛ x − vt ⎞ ⎜⎢ ⎟ ⎟⎥ ⎜ ⎟⎥ ⎟⎟ ⎟ ( −1) ⎜⎜ ⎟ ∞ ( −1) ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟⎥ ⎢ u x =0 ⎜ ⎟⎥ ⎟⎟ 2 ⎝ 2 Dx t ⎠ ⎝ 2 Dx t ⎠ ⎟ ⎥ + 4 K id ⎟ ⎟ 2π t ⎜ ⎢ m ⎜ 1 − ⎟ ⎥ + 4 K id ⎟ ⎜⎢ 2 ⎜ ⎟ ( 2n + 1) n! ⎟ ⎥ ( 2n + 1) n! ⎟ ⎥ π n =0 ⎟⎟ ⎜ ⎟ ⎟⎥ ⎢ ⎜ ⎟⎥ ⎟⎟ ⎟⎥ ⎜ ⎟⎥ ⎜⎢ ⎟ ⎟⎟ ⎠⎦ ⎝ ⎠ ⎣ ⎦ ⎠⎠ ⎝ ⎠ ⎛ x − vt ⎞ ⎟ ( −1) ⎜⎜ ⎟ ⎝ 2 Dx t ⎠ ( 2n + 1) n! n

2 n +1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

∑

xlvii


Using another type of approximation of the error function:

⎛ ⎛ ⎜ ⎜ ⎜ ⎜ 0 u ⎜ −α t ⎜ 0 −1 + r ( x, t ) = φimαβ tot e ⎜ c + 2⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎝

2 ⎞⎞ ⎛ ⎡ 1 ( x − vt )2 ⎤ ⎞ ⎟⎟ x =0 2 ( um ) Kid ⎜⎜ exp ⎢− 4 D t ⎥ ⎟⎟ ⎟⎟ ⎢⎣ ⎥⎦ ⎠ x ⎝ ⎟⎟ + 3/ 2 ⎟ ⎟ 2 2 ⎛ ⎡ x =0 ⎞ ⎡ u x =0 ⎤ ⎤ ⎛ ⎞ ⎛ ⎞ ⎟ ⎟ − x vt u x − vt ⎢ m erfc ⎜ ⎟ ⎥ + 4 K id ⎟ ⎟ 2π t ⎜ ⎢ m erfc ⎜ ⎟ ⎥ + 4 K id ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 2 ⎟ 2 2 D t ⎟ ⎥ ⎢ x ⎠⎦ ⎝ ⎝ 2 Dx t ⎠ ⎥⎦ ⎣⎢ ⎠⎠ ⎝⎣ ⎠

⎛ x − vt umx = 0 erfc ⎜ ⎜ 2 ⎝ 2 Dx t

⎞ ⎟ ⎟ ⎠

(

)

From Abramowitz and Stegun (1964) : erf x = 1 − a1t + a2t 2 + a3t 3 + a4t 4 + a5t 5 e − x + ∈ ( x ) , t = 2

1 , ∈ ( x ) ≤ 1.5 × 10−7 and 1 + px

p = 0.3275911 , a1 = 0.254829592 , a2 = −0.284496736 , a3 = 1.421413741 , a4 = −1.453152027 , a5 = 1.061405429 ,

(

)

(

)

And erfc x = 1 − erf x then: erfc x = 1 − ⎡1 − a1t + a2t 2 + a3t 3 + a4t 4 + a5t 5 e − x + ∈ ( x ) ⎤ = a1t + a2t 2 + a3t 3 + a4t 4 + a5t 5 e − x − ∈ ( x )

⎣

2

⎦

erfc x = ( 0.254829592t − 0.284496736t 2 + 1.421413741t 3 − 1.453152027t 4 + 1.061405429t 5 ) e − x

t=

2

1 1 = 1 + px 1 + 0.3275911x

⎡ 0.254829592 0.284496736 1.421413741 1.453152027 1.061405429 ⎤ − x2 erfc x = ⎢ e − + − + 2 3 4 5⎥ ⎢⎣1 + 0.3275911x (1 + 0.3275911x ) (1 + 0.3275911x ) (1 + 0.3275911x ) (1 + 0.3275911x ) ⎥⎦ And recalling that: If c10 = c20 → u 0 = 0 , then: c20 =

4 K id 2

= K id , c 0 = cim0 + c20 = cim0 + K id 2

2

xlviii

2


Replacing in (5)

⎧ ⎪ φim ⎨ g ( t ) cim0 2 + K id + ⎪⎩

(

)

⎧ ⎪ φim ⎨ g ( t ) cim0 2 + K id + ⎪⎩

(

)

⎧ ⎪ φim ⎨ g ( t ) cim0 2 + K id + ⎪⎩

(

)

⎧ ⎪ φim ⎨ g ( t ) cim0 2 + K id − ⎪⎩

(

)

t

∫ 0

t

∫ 0

t

∫ 0

t

∫ 0

⎫ 2 ∂u ⎡ ∂ ∂ 2 ⎪ ∂ c2 ⎤ − − − c t τ c t g t d τ D ( ∇u ) = r ) ⎬ 2( 2 ( )⎥ ( ) 2 ⎢ ∂t ⎣ ∂u ∂u ⎦ ⎪⎭ ∂u ⎫ 2 ∂u ( t ) ∂ 2 ⎪ ∂c ⎡⎣c2 ( t − τ ) − c2 ( t ) ⎤⎦ g ( t ) dτ ⎬ − 22 D ( ∇u ) = r ∂t ∂u ⎪⎭ ∂u ⎫ 2 ∂u ( t − τ ) ∂ 2 ⎪ ∂c ⎡⎣ c2 ( t ) − c2 ( t − τ ) ⎤⎦ g (τ ) dτ ⎬ − 22 D ( ∇u ) = r ∂t ∂u ⎪⎭ ∂u ⎫ 2 ∂u ( t − τ ) ∂ 2 ⎪ ∂c ⎡⎣ c2 ( t ) − c2 ( t − τ ) ⎤⎦ g (τ ) dτ ⎬ − 22 D ( ∇u ) = r ∂τ ∂u ⎪⎭ ∂u

The following is the reaction rate simplified expression

(

)

φim g ( t ) cim0 + K id − 2

∂ 2 c2 2 D ( ∇u ) = r 2 ∂u

Replacing the memory function:

(6)

(

)

φimαβ tot e −α t cim0 + K id − 2

∂ 2 c2 2 D ( ∇u ) = r 2 ∂u

xlix

(5)


And the limits are the same as De Simoni et al. (2000) ⎡ ∂ 2 c2 ∂ 2 c2 2⎤ 2 0 + − D ∇ = − D ( ∇u ) K u lim r = lim ⎢φimαβ tot e −α t cim ( ) id ⎥ 2 2 2 α →0 α →0 ∂u ∂u ⎣ ⎦

(

)

⎡ ∂ 2 c2 ∂ 2 c2 2⎤ 2 0 K u + − ∇ = − lim r = lim ⎢φimαβ tot e −α t cim D D ( ∇u ) ( ) id ⎥ 2 2 2 α →∞ α →∞ u u ∂ ∂ ⎣ ⎦

(

r = φimαβ tot e

−α t

(c

0 im2

)

)

+ K id −

⎛ x − vt umx =0 erfc ⎜ If u ( x, t ) = ⎜2 Dt 2 x ⎝

(u

2 K id

2

+ 4 K id )

3/ 2

⎛ ∂u ⎞ Dx ⎜ ⎟ ⎝ ∂x ⎠

2

⎞ ⎟ then r = φimαβ tot e−α t cim0 + Kid − ⎟ ⎛ ⎡ x =0 ⎠ ⎜ um

⎛ x − vt ∂ ⎡ And it has that: ⎢erfc ⎜ ⎜2 D t ∂x ⎢ x ⎝ ⎣

(

)

2

2 K id

⎛ x − vt erfc ⎜ ⎢ ⎜2 D t ⎜⎢ 2 x ⎝ ⎝⎣

⎡ 1 ( x − vt )2 ⎤ ⎞⎤ 1 exp ⎢ − ⎥ , it is achieved that: ⎟⎥ = − ⎟⎥ π Dx t ⎢⎣ 4 Dx t ⎥⎦ ⎠⎦

2

(

0 r = φimαβ tot e −α t cim + K id 2

)

⎛ ⎡ 1 ( x − vt )2 ⎤ ⎞ ( u ) Kid ⎜⎜ exp ⎢− 4 D t ⎥ ⎟⎟ x ⎣⎢ ⎦⎥ ⎠ ⎝ + 3/ 2 2 ⎛ ⎡ x =0 ⎞ ⎤ ⎛ ⎞ u x vt − 2π t ⎜ ⎢ m erfc ⎜ ⎟ ⎥ + 4 K id ⎟ ⎜ 2 D t ⎟⎥ ⎜⎢ 2 ⎟ x ⎠⎦ ⎝ ⎝⎣ ⎠ x =0 2 m

l

2 ⎞ ⎞⎤ ⎟ ⎥ + 4 K id ⎟ ⎟⎥ ⎟ ⎠⎦ ⎠

3/ 2

⎛ ∂u ⎞ Dx ⎜ ⎟ ⎝ ∂x ⎠

2


2

If erfc ( z ) = 1 −

r = φimαβ tot e

−α t

π

(c

0 im2

∞

∑ n=0

⎡ ⎢ n 2 n +1 ⎢ ⎛ ⎞ − 1 x ( ) x − vt 2 then erfc ⎜ ⎟ = ⎢1 − ⎜ ⎟ π ( 2n + 1) n ! ⎝ 2 Dx t ⎠ ⎢ ⎢ ⎣⎢

)

+ K id +

(u )

x =0 2 m

⎛⎡ ⎜⎢ ⎜ ⎢ x=0 u 2π t ⎜ ⎢ m ⎜⎢ 2 ⎜⎢ ⎜⎢ ⎝⎣

⎡ ⎢ ⎢ 2 ⎢1 − π ⎢ ⎢ ⎢⎣

⎛ x − vt ⎞ ⎟ ( −1) ⎜⎜ ⎟ ⎝ 2 Dx t ⎠ ( 2n + 1) n!

∞

∑ n=0

⎛ ⎡ 1 ( x − vt )2 ⎤ ⎞ K id ⎜ exp ⎢ − ⎥⎟ ⎜ ⎢⎣ 4 Dx t ⎥⎦ ⎟⎠ ⎝ ∞

∑ n =0

2 n +1

n

x − vt ⎞ n⎛ ⎟ ( −1) ⎜⎜ − ⎟ ⎝ 2 Dx t ⎠ ( 2n + 1) n!

2 n +1

2

2 ⎞ ⎤⎤ ⎟ ⎥⎥ ⎟ ⎥⎥ ⎥ ⎥ + 4 K id ⎟ ⎟ ⎥⎥ ⎟ ⎥⎥ ⎟ ⎥⎦ ⎥⎦ ⎠

li

3/ 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦⎥

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