Rio de Janeiro

May 2019

CHARACTERIZATION OF CARBONATE AND SANDSTONE SAMPLES

THROUGH TRACER TESTS AND COMPUTER SIMULATIONS

Gabriel de Belli Correia

Projeto de Graduação apresentado ao

Curso de Engenharia de Petróleo da Escola

Politécnica, Universidade Federal do Rio de

Janeiro, como parte dos requisitos necessários à

obtenção do título de Engenheiro.

Orientadora: Thaís M. G. Silveira, M.Sc.

Coorientador: Paulo Couto, D.Eng.

iii

Correia, Gabriel de Belli

Characterization of Carbonate and Sandstone Samples

through Tracer Tests and Computer Simulations / Gabriel

de Belli Correia – Rio de Janeiro: UFRJ / Escola

Politécnica, 2019

XI, 50 p.: il.; 29,7 cm.

Orientadora: Thaís M. G. Silveira

Projeto de Graduação – UFRJ / Escola Politécnica /

Curso de Engenharia de Petróleo, 2019.

Referências Bibliográficas: p. 57-61.

1. Traçador Químico. 2. STANMOD. 3. Coreflood. I.

Silveira, Thaís M. G.. II. Universidade Federal do Rio de

Janeiro, UFRJ, Engenharia de Petróleo. III. Título.

iv

I dedicate this work to:

My father who never stopped

believing in me, Silnei;

My mother who gave me every

support that I needed, Ana;

My brother who was always by my

side in everything, Davi.

v

ACKNOWLEDGMENTS

I would like to thank all of the team that worked with me at LRAP, for the hard work,

the fun times and where I could learn a lot. Especially Alex for his patience and fellowship

during my period at the lab.

I would like to thank Prof. Paulo Couto for his guidance throughout my entire course,

for the opportunity to work at LRAP, and for introducing me to my advisor Thaís.

I especially thank my advisor Thaís, who taught me everything at the lab, embraced

my work and guided me through it. She always gave her best and offered me every opportunity

to grow along the way.

I also would like to thank all my friends and teachers along this course, especially Prof.

Santiago for accepting my invitation to be member of my defense evaluation committee and for

inspiring me through his classes.

I would like to thank all members of the evaluation committee for the opportunity to

contribute to this work and their participation.

And I also would like to add that this research was carried out in association with the

ongoing R&D project registered as ANP n 20163-2, “Análise Experimental da Recuperação

de Petróleo para os Carbonatos do Pré-sal do Brasil através de Injeção Alternada de CO2 e

Água", sponsored by Shell Brasil under the ANP R&D levy as “Compromisso de Investimentos

com Pesquisa e Desenvolvimento”.

vi

Resumo do projeto de Graduação apresentado à Escola Politécnica/UFRJ como parte dos

requisitos necessários para a obtenção do grau de Engenheiro de Petróleo.

CHARACTERIZATION OF CARBONATE AND SANDSTONE SAMPLES THROUGH

TRACER TESTS AND COMPUTER SIMULATIONS

Gabriel de Belli Correia

Maio/2019

Orientadora: Thaís M. G. Silveira

Coorientador: Paulo Couto

Curso: Engenharia do Petróleo

Um cenário atual de exploração limitada e campos maduros demandam novas técnicas

de recuperação de petróleo. No Brasil, a descoberta dos campos do Pré-sal, ou seja, de

reservatórios carbonáticos heterogêneos, fomentou a implementação de métodos de

recuperação avançada de petróleo (EOR) desde o início dos projetos de exploração. Porém, os

métodos de recuperação avançada apresentam alguns riscos e podem gerar diferentes problemas

operacionais, tais como: breakthrough precoce em poços de produção, redução de injetividade,

corrosão, incrustações, precipitação de asfaltenos e formação de hidratos. Entretanto, a

mitigação desses riscos pode ser realizada se houver uma caracterização do meio poroso de

forma adequada. Nesse sentido, o presente trabalho tem como objetivo contribuir para a

caracterização do meio poroso de amostras carbonáticas através da corroboração de ensaios

experimentais com traçador químico e simulação computacional. Para cumprir o objetivo

proposto, um aparato de escoamento em meio poroso foi construído para realização dos testes

experimentais. Duas amostras de rocha carbonática foram utilizadas, além de uma terceira

amostra de arenito, para fins comparativos. As três amostras foram submetidas a testes com

traçador químico não reativo de iodeto de sódio e posteriormente simuladas

computacionalmente com o auxílio do programa STANMOD, a fim de obter os parâmetros que

vii

governam o fluxo no meio poroso. Os resultados obtidos demonstraram ótima correlação entre

os dados experimentais e de simulação, evidenciando que ambos os testes podem contribuir

para a correta caracterização do meio poroso de rochas carbonáticas.

Palavras-chave: Traçador químico, Carbonatos, Coreflooding, STANMOD.

viii

Abstract of Undergraduate Project presented to Escola Politécnica/ UFRJ as a partial fulfillment

of the requirements for the degree of Engineer.

CHARACTERIZATION OF CARBONATE AND SANDSTONE SAMPLES THROUGH

TRACER TESTS AND COMPUTER SIMULATIONS

Gabriel de Belli Correia

May/2019

Advisor: Thaís M. G. Silveira

Co-advisor: Paulo Couto

Course: Petroleum Engineering

The present scenario of limited exploration and mature fields demands new techniques

of oil recovery. In Brazil, the discovery of the pre-salt fields, heterogeneous carbonate

reservoirs, promoted the implementation of enhanced oil recovery (EOR) methods from the

beginning of exploration projects. Still, advanced recovery methods present some risks and can

generate different operational problems, such as: early breakthrough in production wells,

reduction of injectivity, corrosion, scale deposition, asphaltene precipitation and hydrates

formation. However, mitigation of these risks can be handled if there is an adequate

characterization of the porous medium. As such, the present work aims to contribute to the

characterization of carbonate samples both experimentally using chemical tracer tests and by

means of computational simulations. In order to achieve the proposed objectives, a core flood

apparatus was built to perform the experimental tests. Two carbonate samples were used, in

addition to a third sandstone sample for comparative purposes. The three samples were

submitted to chemical tracer tests with non-reactive sodium iodide, which were later analyzed

numerically using the STANMOD software for the purpose of obtaining the parameters that

govern fluid transport through the porous medium. The results obtained showed excellent

ix

correlation between the experimental and simulation data, thus demonstrating that both tests

can contribute to the correct characterization of the porous medium of carbonate samples.

Keywords: Chemical Tracer, Carbonates, Coreflooding, STANMOD.

x

LIST OF TABLES

Table 1. Ionic composition, total dissolved solids (TDS), pH, CO2 solubility and density (ρ) of

the brines used in this work. ..................................................................................................... 36

Table 2. Initial values of the coefficients used for the BO, EB and IL_AT simulations. ......... 40

Table 3. Initial values of the coefficients used for the IL simulation. ...................................... 40

Table 4. Conventional core analysis data ................................................................................. 41

Table 5. Calibration curve of sodium iodide concentration versus solution density. ............... 41

Table 6. Results from the tracer test on the Edward Brown limestone. ................................... 42

Table 7. Results from the tracer test on the Boise sandstone. .................................................. 43

Table 8. Results from tracer test experiment on the Indiana limestone. .................................. 44

Table 9. Results from tracer test on the Indiana limestone after the destructive carbonated

seawater injection test. .............................................................................................................. 45

Table 10. CFITIM mode summary of parameters results. ....................................................... 53

Table 14. BO non-linear least squares analysis, final results. .................................................. 53

Table 15. EB non-linear least squares analysis, final results.................................................... 53

Table 16. IL non-linear least squares analysis, final results. .................................................... 53

Table 17. IL_AT non-linear least squares analysis, final results. ............................................. 54

Table 15. BO CFITIM mode parameters iterations. ................................................................. 62

Table 16. EB CFITIM mode parameters iterations. ................................................................. 62

Table 17. IL CFITIM mode parameters iterations. .................................................................. 63

Table 18. IL_AT CFITIM mode parameters iterations. ........................................................... 63

Table 19. BO concentrations observed and fitted with its respective residuals’ outputs. ........ 64

Table 20. EB concentrations observed and fitted with its respective residuals’ outputs.......... 65

Table 21. IL concentrations observed and fitted with its respective residuals’ outputs. .......... 67

Table 22. IL_AT concentrations observed and fitted with its respective residuals’ outputs.... 68

xi

LIST OF FIGURES

Figure 1. Dead-end pore schematic. ......................................................................................... 19

Figure 2. Concentration Profile (on Arithmetic-Probability Paper) Resulting from Diffusion of

a Tracer into Water (D0 = 1 x 10-5 cm²/sec). (PERKINS, JOHNSON, 1963). ......................... 26

Figure 3. Typical Tracer Effluent Concentration Profiles (KANTZAS, BRYAN, TAHERI,

2018) ......................................................................................................................................... 27

Figure 4. Tracer profiles of different pore classes. (SKAUGE et al., 2006) ............................ 30

Figure 5. CFITIM window of the STANMOD software.......................................................... 33

Figure 6. CFITIM window of the STANMOD software to input the initial transport and

reaction parameter estimates. ................................................................................................... 33

Figure 7. Edward Brown limestone core sample (20cm length and 3.79cm diameter). .......... 34

Figure 8. Boise sandstone core sample (20cm length and 3.79cm diameter). ......................... 35

Figure 9. Indiana limestone core sample (20cm length and 3.79 diameter). ............................ 35

Figure 10. Indiana limestone core sample (20cm length and 3.79cm diameter) after

destructive tests with carbonated seawater injection. ............................................................... 35

Figure 11. Schematic of tracer test instrumental setup. ............................................................ 37

Figure 12. Core wrapping procedure. ....................................................................................... 38

Figure 13. Calibration Curve of Sodium Iodide concentration versus Density of the Tracer

Fluid. ......................................................................................................................................... 42

Figure 14. Effluent tracer dimensionless concentration versus dimensionless injected pore

volume of samples IL_AT and IL. ........................................................................................... 46

Figure 15. Effluent tracer dimensionless concentration versus dimensionless injected pore

volume of samples BO and EB. ............................................................................................... 46

Figure 16. Effluent tracer dimensionless concentration versus dimensionless injected pore

volume of samples BO, EB, IL and IL_AT. ............................................................................. 47

Figure 17. Dimensionless fluid injected ( U = (I-1)/sqrt(I)) versus the effluent tracer

dimensionless concentrations (C/Co) from the IL tracer test. .................................................. 48

Figure 18. Dimensionless fluid injected ( U = (I-1)/sqrt(I)) versus the effluent tracer

dimensionless concentrations (C/Co) from the IL_AT tracer test. ........................................... 49

Figure 19. Dimensionless fluid injected ( U = (I-1)/sqrt(I)) versus the effluent tracer

dimensionless concentrations (C/Co) from the EB tracer test. ................................................. 49

Figure 20. Dimensionless fluid injected ( U = (I-1)/sqrt(I)) versus the effluent tracer

dimensionless concentrations (C/Co) from the BO tracer test. ................................................ 50

xii

Figure 21. BO STANMOD graphical output comparing experimental data with the fitted

curve. ........................................................................................................................................ 51

Figure 22. EB STANMOD graphical output comparing experimental data with the fitted

curve. ........................................................................................................................................ 51

Figure 23. IL_AT STANMOD graphical output comparing experimental data with the fitted

curve. ........................................................................................................................................ 52

Figure 24. IL STANMOD graphical output comparing experimental data with the fitted curve.

.................................................................................................................................................. 52

xiii

LIST OF CONTENTS

LIST OF TABLES ................................................................................................................... X

LIST OF FIGURES ............................................................................................................... XI

1. INTRODUCTION ........................................................................................................... 15

Motivation ........................................................................................................... 16

Objectives ............................................................................................................ 16

Organization of the Text ..................................................................................... 16

2. LITERATURE REVIEW ............................................................................................... 18

Petrophysical Concepts ....................................................................................... 18

2.1.1. Porosity .................................................................................................... 18

2.1.2. Permeability ............................................................................................. 20

Carbonates ........................................................................................................... 21

The Equation of Continuity ................................................................................. 22

2.3.1. Dispersion Coefficient ............................................................................. 24

2.3.2. The Capacitance Model ........................................................................... 26

Tracer Tests to Characterize Reservoirs Samples ............................................... 29

STANMOD Software .......................................................................................... 30

3. EXPERIMENTAL METHODOLOGY ........................................................................ 34

Materials .............................................................................................................. 34

3.1.1. Core Samples ........................................................................................... 34

3.1.2. Fluids ....................................................................................................... 36

3.1.3. Experimental Setup ................................................................................. 36

Experimental Procedures ..................................................................................... 37

3.2.1. Core Preparation ...................................................................................... 37

3.2.2. Basic Petrophysical Analysis .................................................................. 38

3.2.3. Tracer Tests ............................................................................................. 39

Simulation Setup ................................................................................................. 39

4. RESULTS AND DISCUSSION ...................................................................................... 41

Experimental Results ........................................................................................... 41

Simulation Results ............................................................................................... 50

xiv

5. CONCLUSIONS .............................................................................................................. 56

6. REFERENCES ................................................................................................................ 57

7. APPENDIX ...................................................................................................................... 62

15

1. INTRODUCTION

Despite the growth of renewable energy sources, petroleum represents 33% of the

global primary energy consumption and, in Brazil, up to 46%, according to a recent BP

statistical review of world energy (BP, 2018). However, limited exploration and mature fields

demands a more effective recovery rate. Therefore, according to Gurpinar (2018), petroleum

industries have becoming increasingly interested in applying enhanced oil recovery (EOR)

techniques to field operations.

Thus, in order to meet the rising energy demand, Brazil has moved exploration of oil

reserves further away from the continent and into the sea, where new discoveries have been

made in an area known as Pre-salt. Commercial exploitation of the pre-salt began with the Lula

field (formerly called the Tupi field) in 2006. But, due to many reservoir uncertainties,

Petrobras, along with its partners, have decided to emphasize their development strategy based

on intensive information gathering, extended well tests (EWTs), multi-well production pilots

and the implementation of EOR methods (COSTA FRAGA et al., 2012).

PIZARRO e BRANCO (2012) concluded in their study that “Lula field reservoir

studies indicate that implementing an EOR miscible method, CO2 and/or gas injection

alternated with water (WAG), can be beneficial to the ultimate recovery.”. As such, it is

noteworthy that the most common operational problems with WAG are: early breakthrough in

production wells, reduced injectivity, corrosion, scale deposition, asphaltene precipitation and

hydrate formation (CHRISTENSEN; STENBY; SKAUGE, 2001).

Looking more closely to early breakthrough problems, they are usually caused by

channeling or override, which can be prevented with a better reservoir characterization

(CHRISTENSEN; STENBY; SKAUGE, 2001). Even though studies of CO2 flows through

carbonate rocks date back to the 1960’s, there are only a few studies in the literature related to

Pre-salt conditions. So, a need exists for better understanding WAG flow through carbonate

reservoir at Pre-salt scenarios.

It is within this context that the Laboratory of Advanced Oil Recovery (LRAP) aims

to study techniques that increase oil production, especially in Brazilian Pre-salt fields. This

study was developed within the WAG Experimental (WAGEx) Project at LRAP, where its main

objective is to carry out multidisciplinary experimental and theoretical investigations on the

alternating injection of water and carbon dioxide (WAG) in carbonate reservoirs thereby

quantifying the mechanisms and parameters related to hydrocarbon flow in the porous medium.

16

Performing tracer tests combined with computational simulations can add more

information on the characterization of core samples, which in turn can provide a better

understanding of the complex mechanisms and parameters involving the core flooding

experiments at LRAP.

MOTIVATION

Pre-salt reservoirs in Brazil provide a new and unique scenario that have not yet been

studied much, while EOR methods has become an industry reality, as mentioned on the

introduction above. For that reason, the motivation of the present work was the application of

tracer tests to carbonate rocks, together with computational simulations, so the results may

contribute to the development of ongoing projects at this university aligned with the Brazilian

focus on national reservoir engineering problems.

OBJECTIVES

The objectives of this study are to: 1) perform coreflooding experiments using a

chemical tracer, known as tracer tests, on two limestones and one sandstone; 2) estimate the

parameters that govern the tracer fluid flow through the porous media (dispersion, flow fraction,

and mass transfer) using analytical solutions of the convection-dispersion equation with the aid

of simulation software; and 3) evaluate and compare the results obtained from the experimental

tests and the simulations.

Note that the experimental tests and simulations using the sandstone are not the focus

of this study. The sole purpose is hence a comparison using the carbonate samples.

ORGANIZATION OF THE TEXT

This study followed a sequence of 7 chapters, starting with the introduction, then

literature review, experimental methodology, results and discussion, conclusions, references

and finishing with the appendix.

The next chapter presents an overview of basic petrophysical concepts of porosity and

permeability, followed by the characteristics of carbonate formations. Then we present the

different developed models in the literature to simulate the tracer transport through porous

medium, a sequence of studies about the use of tracer tests to characterize carbonate pore

structure and the STANMOD software used to simulate tracer tests.

17

Chapter 3 presents the experimental methodology, divided into two main sections:

materials and procedures. The materials section shows the core samples used, the fluids

prepared and the experimental apparatus built in order to perform the tracer tests. While the

procedures section explains the basic petrophysical analysis to measure porosity and

permeability, the steps to perform the tracer tests and the simulation setup required by the

STANMOD software.

The experimental and simulation results are presented over chapter 4. The effluents’

tracer concentration and the pore volumes injected are plotted in a graph, then the obtained

profiles are discussed and compared with the estimated parameters from the computer

simulations.

Chapter 5 summarizes the conclusions of this study and the author’s recommendations

for a future work.

References and appendix can be found in chapters 6 and 7, respectively.

18

2. LITERATURE REVIEW

Following the objectives presented previously, this chapter introduces the physical

concepts which supports studies of solute transport in porous media, a literature review

concerning tracer tests and the concepts behind the simulation software used in the analysis of

the experimental data.

PETROPHYSICAL CONCEPTS

Since it is of interest in reservoir engineering the fluids within a rock reservoir, it is

important to study the rock properties related with the volume of fluid inside it and their ability

to flow through the porous media. Rosa (2006) pointed out that: “Information concerning rock

properties, such as the properties of fluids, are decisive factors for the study of the behavior of

petroleum reservoirs and, therefore, their collection and interpretation must receive special

attention through an exhaustive and meticulous work process.”

The next two sections will focus on the physical concepts of porosity and permeability,

which are important in terms of conceptualizing more complex equations that will follow.

2.1.1. POROSITY

Rocks do not form a homogeneous formation; they consist of mineral grains of all

shapes and sizes compressed together with pores in between them. Thus, since pores structures

are complex, their volume can differ from one rock to another. For reservoir engineering, pore

structures and their connections are important topics since they determine whether the

formation fluids can be stored or be transported.

The term given to the amount of space available for storage of hydrocarbons is

porosity. Quantitatively, this property is the ratio between the volume of pores and the total

volume (bulk volume).

During rock formation, grains are compressed together and pores are formed in

between them. But not all pores are connected, and some of them become isolated. This leads

to two distinct types of porosity, namely:

• Absolute porosity

• Effective porosity

While absolute porosity is defined as the ratio of the total pore space in the rock to that

of the bulk volume, effective porosity considers only the interconnected pores from a standpoint

19

of flow through the porous medium. Therefore, a rock may have considerable absolute porosity

and yet have little conductivity to fluid for lack of pore interconnections, leading to a low

effective porosity. There are different ways to determine the effective porosity, from visual

methods to laboratory measurements. According to Lucia (2012), “the most accurate method

of measuring porosity is the helium expansion method”, where pore volume is a result of a

pressure difference. Note that this study will adopt the term porosity many times to describe the

effective porosity, because it represents the interconnected pore space that contains the

recoverable hydrocarbon fluids.

During the cementation process in consolidated rocks as the pore space is being filled

with cementing material, significant reduction in porosity may take place. For intergranular

materials that are poorly to moderately well cemented, the total porosity is approximately equal

to effective porosity. For more cemented materials and some carbonates, significant differences

in absolute and effective porosity values may occur, which can lead to dead-end pores or

stagnant pockets (Figure 1). These pores belong to the class of interconnected pores but

contribute very little to fluid flow processes. Having only a constricted opening to the flow

path, they allow fluid in them to be practically stagnant. Even though fluids inside these dead-

end pores are stagnant, they may be important to other mechanisms of flow such as diffusion

and dispersion.

Figure 1. Dead-end pore schematic.

Porosity is also commonly classified according to its genesis, known as primary and

secondary porosities. The porosity is that developed originally in the process of deposition

20

forming the rock is called primary, while secondary porosity is a result of subsequent geological

processes like the creation of fractures or solution cavities. Rocks having primary porosity are

more uniform in their characteristics than those rocks in which a large part of the porosity is of

the secondary type. Materials which have suffered from secondary processes, like fracturing

and dissolution, are more common with carbonates, leading to a complex core configuration.

In fact, Rosa (2006) states that two or more systems of pore openings may occur in such rocks

as a result of leaching or fracturing of the primary rock material.

Reservoir rocks may generally show large variations in porosity vertically but less so

parallel to the bedding planes. Limestones present a more complex porosity variance along its

parallel and vertical axes, since their porosity usually are of a secondary type. As a result of

secondary porosity mechanisms, carbonate formations can be very heterogenous, even at the

scale of meters to decimeters (CORBETT, BORGHI, 2013).

2.1.2. PERMEABILITY

While porosity is a property that relates to the volume of fluids (e.g. hydrocarbons)

that can be stored, permeability relates to the rate at which fluids can be recovered. It means

that permeability is related to the flow capacity of a medium. When commonly compared with

electrical conductors, it represents the inverse of resistance which the porous medium offers to

flow.

In 1856, French engineer Henry Darcy developed an equation, known as Darcy’s Law,

to express the absolute permeability of a single-phase fluid flowing through a porous medium.

Darcy’s law has been used by petroleum engineering ever since:

𝑄 =𝑘𝐴∆𝑝

𝜇𝐿 (2.1)

where Q is rate of flow, k is permeability, A is the cross-section area of the sample, ΔP is the

differential pressure, µ is fluid viscosity and L is the length of the sample.

Note that Eq. 2.1, takes in consideration the following conditions as summarized by

Rosa (2006):

• Isothermal, permanent and laminar flow;

• Incompressible, homogeneous fluid with isobaric viscosity;

21

• Homogenous porous medium.

It´s also important to highlight that permeability is a vector quantity in that the

permeability vary in different directions, with the vertical permeability being commonly less

variable than the horizontal permeability. Therefore, permeability is often represented as a

vector in the x, y, and z directions. However, for core analysis using cylindrical core samples,

single axial flow is usually assumed, therefore resulting in a single permeability value.

According to Lucia (2012) reservoir permeability values range considerably from less

than 0.01 millidarcy (mD) to well over 1 Darcy, with “Darcy” being a practical unit of

permeability (in honor of Henry Darcy). A porous material has a permeability equal to 1 Darcy

if a pressure difference of 1 atm will produce a flow rate of 1 cm³/sec of a fluid with 1 cP

viscosity through a cube of 1 cm in side length.

CARBONATES

According to Schlumberger (2019), “it is estimated that more than 60% of the world's

oil and 40% of the world's gas reserves are held in carbonate reservoirs”. Distinctive and unique

aspects of carbonate rocks are their predominantly intrabasinal origin, their primary dependence

on organic activities for their constituents and their susceptibility to modification by post-

depositional mechanisms. These three features are significant to distinguish the productivity of

carbonate rocks from other sedimentary rocks including sandstone and shale.

Carbonates are formed in special environments in that they are biochemical in origin

(LUCIA, 2002). Organisms play an important role and have a direct role in determining the

reservoir quality. Processes like compaction, lithification and other diagenetic events result in

large variations in the reservoir quality of carbonates. Carbonates are particularly sensitive to

post-depositional diagenesis, including dissolution, cementation, recrystallization,

dolomitization, and replacement by other minerals. Abundant unstable aragonite (in bioclasts

and cements) converts to more stable low-magnesium (or high-magnesium) calcite. Calcite can

be dolomitized readily, thereby sometimes increasing porosity. Complete leaching of grains by

meteoric pore fluids can lead to textural inversion, which may enhance reservoir quality through

dissolution, or occlude reservoir quality through cementation. Burial compaction fracturing and

stylolithification are common diagenetic effects in carbonates, thus creating high-permeability

zones and permeability barriers or baffles, respectively (ROSA, 2006).

22

Carbonates are characterized by different types of porosity involving unimodal,

bimodal and other complex pore size distributions, which result in wide permeability variations

for the same total porosity, after leading to complex porous media therefore, whose

producibility are generally difficult to predict.

In Brazil, the largest and most important carbonate reservoirs are in the Pre-Salt fields.

Located along the southeast shore covering an area of approximately 800 km long and 200 km

wide, they encompass the basins of Campos, Santos and Espírito Santo. According to ANP data

(2016), the proven reserves of the Pre-Salt are approximately 7.25 Bboe, corresponding to more

than half of Brazil’s total proven reserves of 12.6 Bboe. Within a few years, those large oil

reserves have been shown their potential by breaking records in many aspects of national oil

production. As reported by the ANP bulletin of January 2018, the total Pre-Salt production is

1,723 Mboe/d, of which 1,065 Mboe/d are from a single field (Lula).

In addition to the large production and all of the known problems associated with this,

the Pre-Salt carbonate reservoirs are located in a deep offshore environment; their locations

may reach up to 300 km from the coast, with depth ranging from 5,000 and 6,000 meters below

the sea level, and with salt layers thicknesses of up to 2,000 meters as pointed out by Costa

Fraga et al. (2014). As noted by Moczydlower et al. (2012), the carbonate lithology of those

reservoirs usually shows a higher degree of heterogeneity than sandstones, therefore making

Pre-Salt a unique formation, being very heterogenous and with many production challenges.

Few known analogs exist, with none being in ultra-deep water (COSTA FRAGA et al., 2014).

THE EQUATION OF CONTINUITY

As presented by Bird, Stewart and Lightfoot (1960), applying the law of conservation

of mass of species “i” in a multi-component mixture to an arbitrary control volume of fluid, we

obtain the equation of continuity as follows:

𝜕𝑐𝑖

𝜕𝑡+ ∇. 𝑛�̅� = 𝑟𝑖

(2.2)

Here 𝑐𝑖 is the concentration, 𝑛�̅� is the mass flux vector, 𝑟𝑖 is the source/sink term, t is time and

the i the species involved. To define the mass flux of species i in time and space, a constitutive

23

equation provided by Fick’s first law, Eq. (2.3), is used as demonstrated by Bird, Stewart and

Lightfoot (1960).

𝑛�̅� = 𝑐𝑖�̅� − 𝜌𝐷𝑜∇𝑚𝑖 (2.3)

Substituting Eq. (2.3) into Eq. (2.2) gives:

𝜕𝑐𝑖

𝜕𝑡+ �̅�

∂𝑐𝑖

∂x= 𝐷𝑜

𝜕2𝑐𝑖

𝜕𝑥2 + 𝑟𝑖 (2.4)

where �̅� is the mass average velocity, 𝜌 is the fluid mass density, 𝐷𝑜 is the molecular diffusion

coefficient, 𝑚𝑖 is the mass fraction of species i and x is distance in the flow direction.

Treating the control volume as a multiphase porous medium leads to a very

complicated situation since the pore and flow geometry are too complex to be modeled.

Looking for a solution to this problem, Bear (1972) used a continuum approach on a coarser

level to describe the fluid flow and solute transport in porous media. He defined this fictitious

continuum medium as a “macroscopic control volume”, in which the representation of a porous

medium must be much larger than an individual pore or grain, but much smaller than the entire

flow domain. The porosity of the macroscopic control volume must then be representative of

the porous medium as a whole. Another aspect of this continuum approach is that the properties

of the control volume are treated as averages.

Following this approach, the solute transport equation during one-dimensional flow is

defined as (Bear, 1972):

𝜕𝑐𝑖

𝜕𝑡+ �̅�

∂𝑐𝑖

∂x= 𝐾𝑙

𝜕2𝑐𝑖

𝜕𝑥2 + 𝑟𝑖 (2.5)

where the molecular dispersion 𝐷𝑜 and the velocity �̅� on Eq. (2.4) are replaced by the dispersion

coefficient 𝐾𝑙 and the apparent linear velocity �̅�, respectively. It is important to point out that

the dispersion coefficient results from molecular diffusion in the direction of flow, coupled with

transverse molecular diffusion due to the presence of velocity profiles (as in capillary tubes

defined by Nunge and Gill, in 1970) and additional mechanical mixing arising from velocity

variations due to the complex nature of the pore structure.

24

For a single solute and no sink/source, Eq. (2.5) reduces to:

𝜕𝑐

𝜕𝑡+ 𝑣

∂c

∂x= 𝐾𝑙

𝜕2𝑐

𝜕𝑥2 (2.6)

with the implicit assumptions:

a. Homogenous porous medium of constant cross-section;

b. Bulk flow in the axial direction at a constant interstitial velocity;

c. Constant fluid density;

d. Constant dispersion coefficient;

e. Incompressible porous medium;

f. Uniform concentration distribution perpendicular to the flow direction (time is

“long” enough for the convection-dispersion model to hold, see Perkins and

Johnston, 1963; Nunge and Gill, 1970);

g. No solute source/sink.

Several solutions to Eq. (2.6) can be found in the literature (MANNHARDT, NASR-

EL-DIN, 1994). In general, they are various combinations of the error function, differing

according to the boundary conditions imposed.

2.3.1. DISPERSION COEFFICIENT

As shown by Nunge and Gill (1969), dispersion in a capillary tube is a result of the

mechanisms of molecular diffusion and varying velocity profiles across the flow channels.

However, those are not the only two mechanisms affecting the dispersion in a porous medium.

Dullien (1979) pointed out that porous media can be compared with a set of non-uniform

capillary tubes entangled, each one with variable cross-sections and different conductivities.

Thus, the flow runs at various angles from the axial direction, mixing and re-splitting at the

pore junctions. Such a random network structure induces a mechanical dispersion process of

the fluids depending on the medium characteristics (BEAR, 1972), and has been analyzed by

various investigators.

Bear (1972) states that in general the dispersion coefficient in porous media is a

second-order tensor that depends on local variations of the velocity and various porous media

characteristics. Even though the dispersion coefficient is anisotropic, it is commonly reduced

25

to a longitudinal component parallel to the flow directions, and a transversal component

perpendicular to the flow. However, experiments have shown that the longitudinal component

is larger than to the transversal component often by a factor of about 10 (BLACKWELL, 1962;

FRIED, COMBARNOUS, 1971).

To estimate the dispersion coefficient, Brigham et al. (1961) described a simple

procedure based on the effluent of a typical laboratory core flood experiment with a tracer. Such

an experiment generally involves a porous medium that is fully saturated with a fluid, to which

then a miscible fluid or tracer, such as an ionic solution, or a dye or radioactive material, is

injected at a known, constant flow rate. Effluent samples are subsequently collected at the core

exit and analyzed for tracer concentrations.

The variable t time can be transformed into a dimensionless time, often referred to as

por volume (BRIGHAM et al., 1961; BRIGHAM, 1974):

𝑡 =𝑉

𝑉𝑝

𝐿

𝑣 (2.7)

where 𝑉 is the volume injected, 𝑉𝑝 is the pore volume and 𝐿 is the core length. The solution for

Eq. (2.6) is therefore given by Brigham et al. (1961) and 𝑃𝑒 is Peclet number:

𝑐

𝑐𝑜=

1

2[erfc (

2√𝐾𝑙/𝑣𝐿)] =

1

2[erfc (

2√1/𝑃𝑒)] (2.8)

Where:

= 𝑉/𝑉𝑝−1

√𝑉/𝑉𝑝 (2.9)

And 𝑃𝑒 the Peclet number given by:

𝑃𝑒 = 𝑣𝐿

𝐾𝑙 (2.10)

26

When λ is plotted against the percentage mass concentrations of displacing fluid in the

effluent on arithmetic probability coordinates (Figure 2), and provided the convection-

dispersion model holds, a straight line is expected which the longitudinal dispersion coefficient

can be obtained using the following equation (BRIGHAM et al., 1961; PERKINS,

JOHNSTON, 1963):

𝐾𝑙 = 𝑣𝐿 [90−10

3.625]

2

(2.11)

Figure 2. Concentration Profile (on Arithmetic-Probability Paper) Resulting from Diffusion of a Tracer into

Water (D0 = 1 x 10-5 cm²/sec). (PERKINS, JOHNSON, 1963).

Alternatively, the dispersion coefficient can be determined by plotting experimental

effluent concentration profiles against volume injected and adjusting 𝐾𝑙 until Eq. (2.8) fits the

experimental data.

2.3.2. THE CAPACITANCE MODEL

The literature contains numerous analytical solutions to Eq. (2.6) with different

boundary conditions. Some of these works have been compiled by Mannhardt and Nasr-El-Din

(1994), and include Danckwerts (1953), Gershon and Nir (1969), Bear (1972), Brighham (1974)

and Coats and Smith (1964).

27

The main difference between the boundary conditions is the characterization of the

medium as a finite system or an infinite system with a step change in input concentration.

Gershon and Nir (1969) and van Genuchten and Alves (1982) compared those solutions of the

convection-dispersion model, Eq. (2.6), with different boundary conditions. They have

concluded that for Peclet numbers high enough, larger than 30 (VAN GENUCHTEN, ALVES,

1982) the solutions converge and the definition of the boundary conditions becomes less critical

Tracer test in porous media are often characterized by effluent profiles with

symmetrical S-shape given by the solutions of Eq. (2.6) at high Peclet numbers (Figure 2),

normally larger than approximately values of 30 to 50 can be considered high in this context

(BRENNER, 1962).

(a)

(b) (c)

Figure 3. Typical Tracer Effluent Concentration Profiles (KANTZAS, BRYAN, TAHERI, 2018)

However, due to heterogeneities in porous media, such as the presence of dead-end

pores and preferential flow paths, profiles often deviate from the sigmoidal S-shape. As a result,

a probability plot of these tracer concentrations is concave upward at the high concentration

end (Figure 3.c).

28

In order to adjust Eq. (2.6) to account for diffusion or mass transfer into the stagnant

volume, Carberry and Bretton (1958) introduced the concept of capacitance in a porous

medium, which implies the presence of non-flowing fluid in the medium. Later, various

capacitance-models were created. Among them the work of Coats and Smith (1964) can be

highlighted, which was based on Deans (1963) capacitance-model, where the main difference

is the introduction of two new parameters in the model: a stagnant volume fraction 𝑓, and a

mass-transfer factor 𝑀.

The equations of the model by Coats and Smith in differential form are given by Eq.

(2.12) through Eq. (2.16):

𝐾𝑙𝜕2𝑐

𝜕𝑥2 − �̅�∂c

∂x= 𝑓

𝜕𝑐

𝜕𝑡+ (1 − 𝑓)

𝜕𝑐∗

𝜕𝑡 (2.12)

(1 − 𝑓)𝜕𝑐∗

𝜕𝑡= 𝑀(𝑐 − 𝑐∗) (2.13)

subjected to the initial condition:

𝑐(𝑥, 0) = 0 for 𝑥 ≥ 0 (2.14)

and the boundary conditions:

𝑥 = 0, 𝑣𝑐0 = 𝑣𝑐 − 𝐾𝑙𝜕𝑐

𝜕𝑥 (2.15)

𝑥 → ∞, 𝑐(∞, 𝑡) = 0 (2.16)

Some analytical solutions to the capacitance model and discussions of boundary

conditions can be found in papers by Coats and Smith (1964), Brigham (1974), de Smedt and

Wierenga (1979a,b), and Patel and Greaves (1987). However, this work will focus on the

analytical solution presented by Coats and Smith (1964).

29

TRACER TESTS TO CHARACTERIZE RESERVOIRS SAMPLES

A tracer is defined as a trackable substance added to a fluid in order to characterize its

flow through a porous media. As pointed by Zemel (1995), there are two main types of tracers,

chemical tracers and radioactive tracers. The present study is based on the use of chemical

tracers, which are those that can be identified and measured in the effluents by general analytical

methods; while radioactive tracers are those detected by their emitted radiation.

Tracer tests have been used in the petroleum industry for many years in numerous

works to describe heterogeneity of a sample, both experimentally and theoretically

(MOCTEZUMA-BERTHIER, FLEURY, 2000). Basically, a tracer test consists of the

displacement of an initial fluid, which saturates the rock sample, by another fluid containing a

tracer. As the tracer step injection enters the sample, its concentration will gradually increase

due to the convection and dispersion mechanisms previously discussed (see section 2.3.1 and

2.3.2), while the measured concentration curves can provide valuable information about the

rock heterogeneity.

Bretz et al. (1984) concluded a spatial correlation between the pore size detected on

the scale of thin sections and preferential flow paths of tracer fluid through porous media,

leading to an early-breakthrough and tailing. Their findings indicated that “wide pore size

distribution and preferential flow paths are characterized in the Coats Smith model by high

dispersion coefficients and low flowing fractions”.

Dauba et al. (1999) performed tracer tests to determine the existence of longitudinal

heterogeneity such as preferential paths, fractures or double-porosity porous media. Their goal

was to identify the samples according to their heterogeneity as an initial guidance of perform

coreflooding tests to determine relative permeability curves. They found that the long tail of the

tracer profile indicated to high permeability heterogeneity.

Moctezuma-Berthier and Fleury (2000) used tracer tests at two flow rates and different

viscosity ratios to obtain both a dynamic and static characterization of their sample. They

reproduced their experiments with a numerical simulation and concluded that their sample was

best described by a model designed for layered systems with large correlation length.

In 2004, Hidajat et al. performed tracer tests in six vuggy carbonate samples to

improve the cores-analyses. Their objective was to improve permeability estimates from NMR

responses for carbonates by including vug connectivity. As a result of their tracer tests, they

30

pointed to a high permeability flow path which is representative of tracer profiles with very

early breakthrough and long tails behavior.

Skauge et al. (2006) studied the application of tracer test with the aid of the capacitance

model developed by Coats-Smith, to differentiate carbonates samples and their pore classes.

Distinct tracer responses were observed associated with different carbonates pore classes,

thereby allowing easy classification in groups having a common profile. For instance, Figure

4.b shows results for Chalky-Micro Pore type samples which present a well-defined profile that

indicates a negligible amount of inaccessible pores, and a low to insignificant fraction of dead-

end pores. On the other hand, Figure 4.a shows Intercrystalline Patchy-Meso Pore type samples

which indicating a low fraction of inaccessible pores and more tailing due to a higher fraction

of dead-end pores.

(a) (b)

Figure 4. Tracer profiles of different pore classes. (SKAUGE et al., 2006)

STANMOD SOFTWARE

STANMOD which stands for STudio of ANAlytical MODels, is a windows-based

computer software package for evaluating solute transport in porous media using analytical

solutions of the convection-dispersion equation. The software is in the public domain and

available at www.pc-progress.com/en/Default.aspx?stanmod (accessed 27 december 2018).

As presented by its developers, the software integrates seven separate codes that have

been popularly used over the years for a broad range of one-dimensional and multi-dimensional

solute transport applications. This study simulated using the CFITM model which applies for

one-dimensional transport. All of the models can be run for direct (forward) problems, and

31

several, among them the CFITIM, can also be run for inverse problems (VAN GENUCHTEN

et al., 2012).

For the inverse analyses performed in this study, the software uses a Marquardt-

Levenberg type weighted nonlinear least squares optimization approach (MARQUART, 1963)

to obtain estimates of the parameters. The approach requires an initial estimate of the

parameters to be obtained. Depending on the problem being considered, factors such as the

magnitude of the measurement errors and the number of parameters being optimized, its

convergence is very sensitive to the initial values chosen. When the optimization of the

parameters lacks a well-defined global minimum, or may have several local minima in

parameter space, the authors suggest “repeating the minimization problem with different initial

estimates of the optimized parameters, and then selecting those parameter values among the

different runs that provide the lowest value of the objective function O(b).” (VAN

GENUCHTEN et al., 2012).

𝑂(𝑏) = ∑ 𝑤𝑖[𝑐𝑖∗(𝑥, 𝑡) − 𝑐𝑖(𝑥, 𝑡; 𝑏)]2𝑛

𝑖=1 (2.16)

where n is the number of concentration measurements; 𝑐𝑖∗(𝑥, 𝑡) are observed concentrations at

time t and location x (in one, two, or three dimensions); 𝑐𝑖(𝑥, 𝑡; 𝑏) represent corresponding

model predictions for the vector b of unknown transport parameters; and 𝑤𝑖 are weights

associated with a particular concentration data point.

Among the codes built in the software, CFITIM as detailed by van Genuchten (1981),

presents analytical solutions for physical nonequilibrium transport. They used the phrase

“physical nonequilibrium” to refer to a porous medium with a mobile and immobile region. The

governing equations are based on those presented by Coats and Smith (1964), previously shown

in section 2.3.2 by Eq. (2.12) and Eq. (2.13):

𝛽𝑅𝜕𝐶𝑚

𝜕𝑇=

1

𝑃𝑒𝑚

𝜕2𝐶𝑚

∂𝑋2 −𝜕𝐶𝑚

𝜕𝑋− 𝜔(𝐶𝑚 − 𝐶𝑖𝑚) (2.18)

(1 − 𝛽)𝑅𝜕𝐶𝑚

𝜕𝑇= 𝜔(𝐶𝑚 − 𝐶𝑖𝑚) (2.19)

32

where R is the retardation factor, β accounts for fraction of the flux that is located in the mobile

region, ω refers to the mass transfer coefficient for exchange between the mobile and immobile

regions, Pe is the Peclet number, T is the dimensionless time, X is the dimensionless position

and the subscripts m and im refer to the mobile and immobile regions of the soil, defined as:

𝑃𝑒𝑚 =𝑣𝑚𝐿

𝐷 (2.20)

𝑅 = 1 +𝜌𝑘

𝜃 (2.21)

𝛽 =𝜃𝑚+𝑓𝜌𝑘

𝜃𝑅 (2.22)

𝜔 =𝛼𝐿

𝑞 (2.23)

where ρ is the dry soil bulk density, and k is a linear partitioning or distribution coefficient of

the solute between the liquid and solid phases, f is the fraction of sorption sites located in the

mobile region, α is a first-order mass transfer coefficient, 𝑣𝑚 is the pore-water velocity for the

mobile phase (q/θm) and θ is the water content.

The CFITIM code can be used with two inlet boundary conditions (x = 0). One is a

“first-type”, with a constant concentration boundary condition, and the other one is the “third-

type”, with a constant flux boundary condition:

𝐶(0, 𝑡) = 𝐶𝑜 (2.24)

(−𝐷𝜕𝐶

𝜕𝑥+ 𝑣𝐶)|

𝑥=0= 𝑣𝐶𝑜 (2.25)

Figure 5 shows the CFITIM window where users select the type of transport model

“Type of Model” and the imposed boundary conditions, the option of running as an inverse or

direct problem defined as “Type of Problem” in the lower left corner of Figure 5, the maximum

number of permitted iterations in the inverse problem (20 in this example) and the number of

data points that are fitted (27 in this example).

33

Figure 5. CFITIM window of the STANMOD software.

The inverse problem allows the analytical solution of Eq. (2.17) and Eq. (2.18) to be

fitted to the experimental data leading to estimates of up to five parameters at the same time.

Those parameters are: the Peclet number (Pem), the retardation factor (R), the dimensionless

nonequilibrium (β) having values between 0 (all nonequilibrium) and 1 (all equilibrium), the

dimensionless mass transfer coefficient (ω), and the amount of solute mass entering the column,

described as dimensionless pulse time (To). Figure 6 shows the window where the initial

transport and reaction parameters are entered.

Figure 6. CFITIM window of the STANMOD software to input the initial transport and reaction parameter

estimates.

34

3. EXPERIMENTAL METHODOLOGY

This chapter covers the complete experimental methodology. First, it will present the

materials (samples) that were used, followed by the procedures that were performed. The

experimental results will be presented over chapter 4.

MATERIALS

To achieve the proposed objectives, a bench apparatus setup was built in order to

perform the coreflooding tests. Each core was submitted to basic petrophysical analyses to

obtain their pore volume and absolute permeability.

This section details the core samples selected, the fluids prepared, the instrumental

setup and the core wrapping.

3.1.1. CORE SAMPLES

Four samples obtained from Koçurek Industries Inc. (Caldwell, Texas) were used in

this study: an heterogenous Edward Brown limestone (EB) (Figure 7), a heterogenous Boise

sandstone (BO) (Figure 8), and one homogenous Indiana limestone (IL) (Figure 9). The latter

sample, Indiana limestone, was also submitted to a destructive coreflooding experiment using

carbonated seawater injection. Hence, the sample named IL_AT (Figure 10) refers to the

Indiana limestone after its destructive test.

Figure 7. Edward Brown limestone core sample (20cm length and 3.79cm diameter).

35

Figure 8. Boise sandstone core sample (20cm length and 3.79cm diameter).

Figure 9. Indiana limestone core sample (20cm length and 3.79 diameter).

Figure 10. Indiana limestone core sample (20cm length and 3.79cm diameter) after destructive tests with

carbonated seawater injection.

At this point, it is important to detail that sample IL_AT (Figure 10) had a wormhole

due to a special core analysis (SCAL). Although the destructive test was not part of this work,

it is important to mention that the presence of CO2 can change the porous medium by dissolution

of the carbonates, and therefore may affect the tracer test results. The SCAL was subjected to

coreflooding under reservoir conditions, with the aim to analyze the effects of different flooding

waters in secondary and tertiary modes of injection. In the secondary mode, the goal was to

evaluate the oil recovery after DSW flooding. But, in tertiary mode, the objective was to

evaluate whether the production of oil would restart by injecting carbonated seawater (DSW

saturated with CO2) as an EOR method. After the SCAL experiments, a conventional core

analysis and tracer test was performed to evaluate petrophysical properties and contribute to

36

rock characterization. A complete summary of all displacements performed on that core is

available elsewhere (DREXLER et al., 2019).

3.1.2. FLUIDS

Normal brine was prepared by dissolving 0.2 wt% of sodium chloride (NaCl) and 0.8

wt% of calcium chloride (CaCl₂) in deionized water. Sodium iodide (NaI reagent grade, Sigma-

Aldrich) was used to prepare the brine used for the single-phase tracer test. Sodium iodide was

chosen due to its non-adsorbing properties, availability and economical value. In order to

prepare the tracer solution, sodium iodide was added in a normal brine solution up to the

concentration of 10 wt%. The ions concentrations, pH and density of the brines used are in

Table 1.

Table 1. Ionic composition, total dissolved solids (TDS), pH, CO2 solubility and density (ρ) of the brines used in

this work.

Ion Normal Brine (ppm) Normal Brine + NaI (ppm)

Na+ 527 2061

Ca2+ 1464 1464

Cl- 3399 3399

I- - 8466

ρ (g/ml) at 25 °C

and 14.7 psi 1.00 1.06

3.1.3. EXPERIMENTAL SETUP

A scheme of the coreflood experiments setup used for the tracer tests is shown in

Figure 11. The setup includes: a core holder, an hydraulic pump, two accumulators, two

pressure transducers and an automatic sampler.

37

Figure 11. Schematic of tracer test instrumental setup.

The core holder was made of a resistant corrosion material, capable of withstanding a

working pressure of 10.000 psi and a temperature of 302 °F. They came with regular inlet and

outlet tip ports of 1/4” and a distribution pattern to contact the total face area of the core. Piston

cells and all lines were made of Hastelloy ® C276 of 1/8”. A syringe pump, Quizix QX-10K,

was used, which has a working pressure capability of 10.000 psi and a flow rate up to 1800

cm³/h. The pressure transducers, produced by Quartzdyne company, were used to calculate the

inlet and outlet pressures of the core.

EXPERIMENTAL PROCEDURES

In this section the experimental procedures are divided in two parts: first the basic

petrophysical analysis, which measured pore volume and permeability, and then the core

flooding involving the chemical tracer.

3.2.1. CORE PREPARATION

Before placing the cores samples inside the core holder, each one went through a

standardized wrapping procedure to isolate them (Figures 12). First, all of the lateral area of the

core was wrapped with PTFE tape, followed by a layer of aluminum foil and a final layer of

aluminum tape to seal the rock and to allow the fluids to flow only in the axial direction. Inside

the cylindrical core holder, a Viton® rubber sleeve confined the core with an overburden

pressure of 500 psi above the inlet pressure, to prevent any fluid from bypassing the core.

38

Figure 12. Core wrapping procedure.

3.2.2. BASIC PETROPHYSICAL ANALYSIS

To measure the pore volume of each sample, tests were performed in the laboratory.

Using a (Helium) pore volume apparatus. The apparatus consisted of one cylinder with known

volumes filled with helium, connected to the core holder also with a known volume containing

the rock sample. With the system all closed, the initial pressure was measured, after which the

inlet valve was opened to allow the gas to expand into the core holder. Once equilibrium was

established in the system, a new pressure was imposed, which corresponded to the final volume,

from which the pore volume could be calculated.

39

To determine the absolute permeability, core flood tests were performed using

nitrogen. The cores were submitted to a constant flow, with two transducers placed on the core

holder inlet and outlet in order to measure the differential pressure. The absolute pressure was

calculated using Darcy’s law. Since a gas flow was used, the Klinkenberg effect (ROSA;

CARVALHO; XAVIER, 2006) was taken into account, but found to be considered negligible.

3.2.3. TRACER TESTS

Prior to commencing the tests, the core samples were saturated with a normal brine

(0.2 wt% NaCl and 0.8 wt% CaCl2). A step-input tracer test was performed by injecting 10

wt% NaI added to the normal brine. Slugs of 2 pore volumes of tracer were injected, followed

by 2 to 3 pore volumes of brine at constant flow rates of 50 cm³/h. The analytical method to

identify sodium iodide in normal brine was gravimetry, which determines the amount of an

analyte (the ion being analyzed) through the measurement of mass. Effluents for this purpose

were collected and analyzed using a density meter (Mettler Toledo DM40). A calibration curve

was used to correlate sodium iodide concentrations to the solution density. Finally, tracer

concentrations versus injected pore volumes were plotted. The results of the tracer tests,

including adsorption and desorption processes, and the obtained calibration curves are shown

in the results section.

SIMULATION SETUP

The STANMOD software discussed on section 2.5, was used for estimating transport

parameters through an inverse problem by fitting results generated with one of the built-in

analytical solutions to the experimental data. Since the experimental results from the probability

plots suggested the use of an advective-dispersive model, the CFITIM model, discussed in

section 2.5, was chosen. The boundary condition chosen was the “third-type”. Maximum

number of iterations chosen was 30.

40

Table 2. Initial values of the coefficients used for the BO, EB and IL_AT simulations.

No Name Initial Value

1 Peclet 100.000

2 RetFac 1.000

3 Beta 0.950

4 Omega 0.500

5 Pulse 2.000

Table 3. Initial values of the coefficients used for the IL simulation.

No Name Initial Value

1 Peclet 100.000

2 RetFac 1.000

3 Beta 0.950

4 Omega 0.300

5 Pulse 2.000

Several Peclet and Beta combinations were tested within a range of reasonable values.

However, all simulations converged to the same final estimates as obtained with the values

shown in Tables 2 and 3. Table 3 shows that for the Indiana sample, the initial estimate of ω

was 0.3 instead of 0.5, to better convergence. We note that even though the initial estimates

were changed, the program still converged to similar solutions, which reinforced the choice of

the CFITIM model as proper. However, depending of the initial estimates, the number of

iterations required to reach convergence fluctuated, which translated to more or less computer

time, but still very fast.

Note that the retardation factor is given by Eq. (2.20), where k is a linear partitioning

or distribution coefficient of the solute between the liquid and solid phases. When k is zero, the

retardation factor (RetFac) becomes equal to 1.

The Pulse Time (Pulse), To index in Eq. (2.20), represents the amount of solute mass

entering the column, or likewise, the total pore volume of tracer solution injected in each test.

Therefore, the initial estimate value was exactly 2 pore volumes as mentioned as a part of the

experimental procedures (section 3.2.3).

41

4. RESULTS AND DISCUSSION

This section presents the experimental results, obtained in the laboratory, and the

simulation results, obtained with the STANMOD software; using the methodologies discussed

in Chapter 3.

EXPERIMENTAL RESULTS

Prior to the core flood tests using a non-reactive tracer, conventional core analysis

were performed, as described on section 3.2.1, to determine dry weight, length, diameter, total

pore volume and the absolute permeability of each sample. Results can be found in Table 4.

Table 4. Conventional core analysis data

Sample id Dry

weight (g)

Length

(cm)

Diameter

(cm)

Pore volume

(cm³)

Porosity

(%)

Abs.

permeability

(mD)

IL 510.18 20 3.79 34.04 15.06 5.40

IL_AT 509.11 20 3.79 31.46 13.91 3.98

EB 362.53 20 3.79 73.00 33.62 265

BO 385.90 20 3.79 65.26 28.62 2904

As discussed in section 3.2.2, the results obtained from the tracer test experiments are

shown in Tables 6, 7, 8 and 9, while the calibration curve is shown in Table 5 and Figure 13.

Table 5. Calibration curve of sodium iodide concentration versus solution density.

Tracer

Concentration Density (g/cm³)

0% 1.0033

20% 1.0156

20% 1.0157

40% 1.0273

40% 1.0274

60% 1.0395

60% 1.0397

80% 1.0495

80% 1.0516

80% 1.0503

100% 1.0635

42

Table 6. Results from the tracer test on the Edward Brown limestone.

PV NaI

density

Tracer

Concentration PV

NaI

density

Tracer

Concentration

0.03 1.0032 0.0000 1.94 1.0584 0.9253

0.07 1.0035 0.0000 2.01 1.0589 0.9338

0.12 1.0035 0.0000 2.07 1.0595 0.9439

0.16 1.0035 0.0000 2.14 1.0596 0.9456

0.20 1.0035 0.0000 2.21 1.0604 0.9591

0.24 1.0033 0.0000 2.35 1.0611 0.9709

0.28 1.0035 0,0000 2.42 1.0614 0.9759

0.32 1.0035 0,0000 2.49 1.0616 0.9793

0.36 0.9938 0,0000 2.55 1.0619 0.9844

0.40 1.0035 0,0000 2.62 1.0617 0.9810

0.44 1.0028 0,0000 2.69 1.0600 0.9523

0.49 1.0035 0,0000 2.76 1.0561 0.8865

0.53 1.0004 0,0000 2.83 1.0504 0.7903

0.57 1.0038 0,0041 2.90 1.0444 0.6891

0.61 1.0043 0,0125 2.97 1.0386 0.5913

0.65 1.0060 0,0412 3.03 1.0343 0.5187

0.69 1.0084 0,0817 3.10 1.0310 0.4630

0.73 1.0114 0.1323 3.17 1.0288 0.4259

0.77 1.0152 0.1965 3.24 1.0259 0.3770

0.81 1.0190 0.2606 3.31 1.0249 0.3601

0.86 1.0226 0.3213 3.38 1.0232 0.3314

Figure 13. Calibration Curve of Sodium Iodide concentration versus

Density of the Tracer Fluid.

43

0.90 1.0259 0.3770 3.44 1.0217 0.3061

0.94 1.0292 0.4327 3.51 1.0202 0.2808

0.98 1.0315 0.4715 3.58 1.0189 0.2589

1.02 1.0344 0.5204 3.79 1.0154 0.1998

1.06 1.0365 0.5558 3.86 1.0146 0.1863

1.18 1.0398 0.6115 3.92 1.0137 0.1711

1.25 1.0425 0.6571 3.99 1.0129 0.1576

1.32 1.0465 0.7245 4.06 1.0121 0.1442

1.39 1.0490 0.7667 4.13 1.0114 0.1323

1.46 1.0509 0.7988 4.20 1.0107 0.1205

1.53 1.0522 0.8207 4.27 1.0102 0.1121

1.60 1.0532 0.8376 4.33 1.0097 0.1037

1.66 1.0548 0.8646 4.47 1.0086 0.0851

1.73 1.0552 0.8713 4.54 1.0083 0.0800

1.80 1.0562 0.8882 4.61 1.0079 0.0733

1.87 1.0576 0.9118 4.75 1.0071 0.0598

Table 7. Results from the tracer test on the Boise sandstone.

PV NaI density Tracer

Concentration PV NaI density

Tracer

Concentration

0.04 1.0024 0.0000 2.24 1.0625 0.9945

0.08 1.0006 0.0000 2.40 1.0624 0.9928

0.13 1.0020 0.0000 2.47 1.0589 0.9338

0.18 0.9856 0.0000 2.55 1.0528 0.8308

0.22 1.0024 0.0000 2.63 1.0445 0.6908

0.31 0.9995 0.0000 2.70 1.0357 0.5423

0.36 1.0023 0.0000 2.78 1.0286 0.4225

0.41 1.0021 0.0000 2.86 1.0234 0.3348

0.50 1.0023 0.0000 2.93 1.0195 0.2690

0.54 1.0030 0.0000 3.09 1.0153 0.1981

0.59 1.0053 0.0294 3.16 1.0134 0.1661

0.64 1.0096 0.1020 3.24 1.0125 0.1509

0.68 1.0131 0.1610 3.32 1.0104 0.1155

0.73 1.0203 0.2825 3.39 1.0100 0.1087

0.77 1.0252 0.3652 3.47 1.0091 0.0935

0.82 1.0309 0.4613 3.62 1.0070 0.0581

0.86 1.0340 0.5136 3.70 1.0064 0.0480

0.91 1.0396 0.6081 3.78 1.0058 0.0379

0.96 1.0428 0.6621 3.85 1.0058 0.0379

1.00 1.0464 0.7229 4.01 1.0045 0.0159

1.05 1.0500 0.7836 4.24 1.0042 0.0109

44

1.09 1.0522 0.8207 4.31 1.0021 0.0000

1.14 1.0545 0.8595 4.39 1.0036 0.0000

1.19 1.0559 0.8831 4.47 1.0036 0.0000

1.25 1.0575 0.9101 4.54 1.0035 0.0000

1.32 1.0584 0.9253 4.70 1.0028 0.0000

1.40 1.0589 0.9338 4.77 1.0031 0.0000

1.48 1.0594 0.9422 4.85 1.0032 0.0000

1.55 1.0596 0.9456 4.93 1.0028 0.0000

1.63 1.0609 0.9675 5.00 1.0029 0.0000

1.71 1.0613 0.9743 5.08 1.0030 0.0000

1.78 1.0617 0.9810 5.16 1.0028 0.0000

1.86 1.0618 0.9827 5.23 1.0031 0.0000

2.01 1.0624 0.9928 5.31 1.0030 0.0000

2.17 1.0624 0.9928 5.39 1.0029 0.0000

Table 8. Results from tracer test experiment on the Indiana limestone.

PV NaI density Tracer

Concentration PV NaI density

Tracer

Concentration

0.01 1.0064 0.0000 2.79 1.0656 0.9964

0.11 1.0064 0.0000 2.89 1.0648 0.9829

0.21 1.0064 0.0000 2.98 1.0625 0.9442

0.31 1.0064 0.0000 3.08 1.0562 0.8381

0.41 1.0064 0.0000 3.18 1.0459 0.6647

0.51 1.0064 0.0000 3.28 1.0344 0.4711

0.61 1.0069 0.0082 3.38 1.0240 0.2960

0.71 1.0093 0.0486 3.48 1.0165 0.1698

0.81 1.0153 0.1496 3.58 1.0118 0.0907

0.91 1.0254 0.3196 3.68 1.0092 0.0469

1.00 1.0369 0.5132 3.78 1.0079 0.0250

1.10 1.0475 0.6917 3.87 1.0072 0.0132

1.20 1.0550 0.8179 3.97 1.0069 0.0082

1.30 1.0594 0.8920 4.07 1.0068 0.0065

1.40 1.0622 0.9391 4.17 1.0067 0.0048

1.50 1.0637 0.9644 4.27 1.0067 0.0048

1.60 1.0645 0.9779 4.37 1.0067 0.0048

1.70 1.0649 0.9846 4.47 1.0067 0.0048

1.80 1.0652 0.9896 4.57 1.0067 0.0048

1.90 1.0653 0.9913 4.67 1.0065 0.0014

2.09 1.0652 0.9896 4.77 1.0065 0.0014

2.19 1.0654 0.9930 4.86 1.0065 0.0014

2.29 1.0656 0.9964 4.96 1.0064 0.0000

45

2.39 1.0654 0.9930 5.06 1.0064 0.0000

2.49 1.0656 0.9964 5.16 1.0064 0.0000

2.59 1.0657 0.9981 5.26 1.0064 0.0000

2.69 1.0658 0.9997

Table 9. Results from tracer test on the Indiana limestone after the destructive carbonated seawater injection test.

PV NaI density Tracer

Concentration PV NaI density

Tracer

Concentration

0.03 1.0028 0.0000 2.02 1.0591 0.9371

0.06 1.0027 0.0000 2.12 1.0546 0.8612

0.14 1.0029 0.0000 2.32 1.0442 0.6857

0.25 1.0030 0.0000 2.42 1.0372 0.5676

0.32 1.0029 0.0000 2.52 1.0290 0.4293

0.39 1.0034 0.0000 2.63 1.0231 0.3297

0.49 1.0048 0.0210 2.73 1.0187 0.2555

0.60 1.0088 0.0885 2.83 1.0151 0.1948

0.70 1.0166 0.2201 3.03 1.0114 0.1323

0.80 1.0258 0.3753 3.13 1.0099 0.1070

0.90 1.0360 0.5474 3.23 1.0087 0.0868

1.00 1.0441 0.6841 3.34 1.0079 0.0733

1.10 1.0499 0.7819 3.44 1.0078 0.0716

1.20 1.0539 0.8494 3.64 1.0072 0.0615

1.41 1.0576 0.9118 3.74 1.0068 0.0547

1.51 1.0586 0.9287 3.84 1.0061 0.0429

1.61 1.0595 0.9439 4.05 1.0059 0.0395

1.71 1.0594 0.9422 4.15 1.0058 0.0379

1.81 1.0601 0.9540 4.25 1.0057 0.0362

1.91 1.0604 0.9591 4.35 1.0055 0.0328

46

Observed tracer concentration profiles are shown in Figures 14 to 16 by plotting the data in

Tables 6 to 9.

Figure 14. Effluent tracer dimensionless concentration versus dimensionless injected pore volume of samples

IL_AT and IL.

Figure 15. Effluent tracer dimensionless concentration versus dimensionless injected pore volume of samples

BO and EB.

47

Figure 16. Effluent tracer dimensionless concentration versus dimensionless injected pore volume of samples

BO, EB, IL and IL_AT.

As shown in Figure 14, the IL sample presented a relatively symmetrical profile close

to 50% C/Co at 1 PV injected, as expected, because of its homogenous nature. However, the IL

sample after the destructive test (IL_AT) showed an asymmetrical profile. Note also that the

breakthrough for the IL curve occurred at 0.61 PV injected, while at the IL_AT curve, it

occurred at 0.41 PV injected. As discussed in chapter 2, the appearance of preferential flow

paths and inaccessible pores can be related to the breakthrough curve shifted to earlier times.

The dissolution caused by the carbonated seawater dissolved parts of the porous medium, and

plugged small paths, thus explaining this behavior. Therefore, the creation of new inaccessible

pores and preferential flow paths accelerated tracer flow through the sample. The right side of

Figure 14, shows a small increment of the tailing. As presented in section 2.4, tailing effects

are related with the presence of dead-end pores, most likely created due to porous media

dissolution.

The BO sandstone and the EB limestone on Figure 15 show asymmetrical

breakthrough adsorption curve, which was expected since both samples had heterogenous pore

distributions. Another aspect of a heterogenous medium is the breakthrough moving to earlier

pore volumes, in contrast with a homogenous medium that would have a breakthrough

occurring at or close to 1 PV. This is because in a homogenous sample all tracer fluid would

travel as uniform flow, while heterogenous samples present preferential flow paths. BO and EB

showed a breakthrough occurrence at 0.59 and 0.57 PV injected.

48

The Boise sample presented a faster tracer response when compared with EB and IL,

since it reached a higher tracer concentration for the same PV injected. However, the Edward

Brown showed a higher tailing effect and asymmetry than the other samples. Thus, indicating

that this sample was the most heterogenous with many dead-end pores (section 2.4).

Before moving to the software simulation results, a short graphical analysis was made,

as discussed in section 2.3.1 and 2.3.2, about the recommendation made by Brigham (1974, p.

96): “Generally one should plot the data in this form first [probability plot], for if they follow a

straight line it is immediately clear that the diffusion equation should be used rather than the

dead-end pore equation.”

Figure 17. Dimensionless fluid injected ( U = (I-1)/sqrt(I)) versus the effluent tracer dimensionless

concentrations (C/Co) from the IL tracer test.

49

Figure 18. Dimensionless fluid injected ( U = (I-1)/sqrt(I)) versus the effluent tracer dimensionless

concentrations (C/Co) from the IL_AT tracer test.

Figure 19. Dimensionless fluid injected ( U = (I-1)/sqrt(I)) versus the effluent tracer dimensionless

concentrations (C/Co) from the EB tracer test.

50

Figure 20. Dimensionless fluid injected ( U = (I-1)/sqrt(I)) versus the effluent tracer dimensionless

concentrations (C/Co) from the BO tracer test.

Notice that the data in Figures 17 to 20 are concave upward at the higher

concentrations. This is a typical profile for core samples containing dead-end pore volumes

(BRIGHAM, 1974), which suggests the use of the capacitive model, like the one proposed by

Coats and Smith (1964), discussed in section 3.2.2., and further used with STANMOD (next

section).

SIMULATION RESULTS

The results from the STANDMOD software consist of a graphical and a text output.

The graphical output, Figures 21 to 24, shows the effluent concentrations versus pore volume,

where the experimental data are represented by red circles, and the fitted data by a blue line.

The text output contains the initial values of coefficients, the observed and predicted data, the

number of iterations until convergence of all parameters, the correlation matrix between the

parameters, a non-linear least squares analysis of the parameters results, and a comparison

between the experimental and fitted data.

51

Figure 21. BO STANMOD graphical output comparing experimental data with the fitted curve.

Figure 22. EB STANMOD graphical output comparing experimental data with the fitted curve.

52

Figure 23. IL_AT STANMOD graphical output comparing experimental data with the fitted curve.

Figure 24. IL STANMOD graphical output comparing experimental data with the fitted curve.

As shown in Figures 21 to 24, all simulations visually presented a good fit of the

experimental data, since experimental and simulated values deviated only slightly from each

other, with a maximum deviation of 6.3%. All of the calculated curves in figures 20 to 23 were

obtained with the physical non-equilibrium (dead-end) transport model.

53

Table 10. CFITIM mode summary of parameters results. BO EB IL IL_AT

Peclet 53.0 88.7 48.1 24.9

RetFac 0.92 1.19 1.02 0.97

Beta 0.85 0.66 0.98 0.85

Omega 0.41 1.06 0.01 0.19

Pulse 1.89 2.09 2.27 1.62

Iterations 7 11 18 15

The iterative least-squares technique could, for all cases, converge within the

maximum range of 18 iterations. Also note that for all cases: the retardation factor was around

1 and all simulations converged to a pulse time close to 2, except for the IL_AT simulation,

which converged to a value 40% lower. That may well be due to leakage we observed from the

inlet tube caused by a loose connection. During the IL_AT tracer test, a loose connection

allowed a small amount of fluid to leak out of the piston cell. Still, its earlier detection didn’t

compromise the test results and a visual estimate of 10 ml of fluid loss was taking into account

through adjustment of the number of injected tracer pore volumes.

Table 11. BO non-linear least squares analysis, final results.

95% Confidence Limits

VAR NAME VALUE S.E.COEFF. T-VALUE LOWER UPPER

1 Peclet 52.97928 8.7661 6.04 35.2998 70.6588

2 RetFac 0.91859 0.0081 113.32 0.9022 0.9349

3 Beta 0.84980 0.0179 47.41 0.8137 0.8860

4 Omega 0.41437 0.1178 3.52 0.1768 0.6519

5 Pulse 1.89395 0.0076 247.87 1.8785 1.9094

Table 12. EB non-linear least squares analysis, final results.

95% Confidence Limits

VAR NAME VALUE S.E.COEFF. T-VALUE LOWER UPPER

1 Peclet 88.74028 25.4021 3.49 37.8707 139.6098

2 RetFac 1.19429 0.0111 108.02 1.1721 1.2164

3 Beta 0.65882 0.0172 38.40 0.6245 0.6932

4 Omega 1.05724 0.1207 8.76 0.8154 1.2990

5 Pulse 2.08794 0.0108 193.68 2.0664 2.1095

Table 13. IL non-linear least squares analysis, final results.

95% Confidence Limits

54

VAR NAME VALUE S.E.COEFF. T-VALUE LOWER UPPER

1 Peclet 48.07467 0.6392 75.21 46.7727 49.3767

2 RetFac 1.01609 0.0042 240.19 1.0075 1.0247

3 Beta 0.97758 0.0038 260.38 0.9699 0.9852

4 Omega 0.01461 0.0029 5.06 0.0087 0.0205

5 Pulse 2.26857 0.0016 1416.71 2.2653 2.2718

Table 14. IL_AT non-linear least squares analysis, final results.

95% Confidence Limits

VAR NAME VALUE S.E.COEFF. T-VALUE LOWER UPPER

1 Peclet 24.85832 3.9996 6.22 16.6896 33.0271

2 RetFac 0.96635 0.0214 45.06 0.9225 1.0101

3 Beta 0.85427 0.0192 44.41 0.8150 0.8936

4 Omega 0.19266 0.0760 2.53 0.0374 0.3479

5 Pulse 1.61812 0.0146 111.01 1.5883 1.6479

The standard error coefficient (S.E. COEF.) of the regression provides the absolute

measure of the typical distance that the data points fall from the regression line. The S.E.

COEFF. is in the units of the dependent variable. While T-VALUE is the value of the parameter

divided by the S.E. COEFF.. All simulations presented reasonable fit measures with S.E.

COEFF. values not higher than 28.6% of the parameter value, therefore resulting in 95%

confidence windows within similar orders of magnitude.

Note that in Table 16, the IL sample flowing fraction parameter (Beta) converged to

0.97758, nearly 1, which would reduce the capacitance model to the classical convection-

dispersion equation, discussed in sections 2.3.1 and 2.3.2. In that case, it is expected that

approximately 50% of the normalized concentration occurs at the 1 PV injected. Section 4.1

presented the IL tracer test result (Figure 14 and Table 6), where at 1 PV the injected normalized

concentration was 51.3%. And since the convection-dispersion equation only depends on a

single parameter, the Peclet number, it is also expected a low mass transfer parameter (Omega),

around 0, or a very high value indicating very rapid exchange, thus making the immobile (dead-

end) region mostly part of the mobile region. In Table 16, the IL sample presented a low Omega

value of 0.01461.

For IL_AT sample, the occurrence of porous media dissolution is expected as

discussed in section 4.1. Still, Table 17 shows an increase in the Omega value from 0.01 to

0.19, and a decrease in the Beta value from 0.98 to 0.85. That is expected, since dissolution

55

occurred in the sample to create dead-end pores, thus increasing the importance of mass transfer

between the mobile and immobile region and decreasing the mobile flowing fraction.

Comparing the profiles in Figure 14, the IL sample showed a wider plateau than the

IL_AT. As Eq. (2.6) is evaluated, a lower IL Peclet number is expected since this parameter is

responsible for narrowing the peak of the capacitance model analytical solution. Thus, the

simulation results confirm the prediction with a 24.9 Peclet value for the IL_AT sample against

a 48.1 Peclet value for the IL sample. The same happens for the BO and EB samples, as shown

in Figure 14 for the BO curve, which presented a wider plateau than the EB curve with Peclet

values of 53.0 and 88.7, respectively.

In terms of the Beta and Omega parameters, the sample with the highest Omega and

lowest Beta was the heterogenous EB sample, which confirms the presence of dead-end pores

and the visual tailing effects observed. At the same time, the sample with the lowest Omega,

highest Beta and the most symmetrical profile is the homogenous IL sample.

56

5. CONCLUSIONS

The objective of this study is to gather information on carbonates porous media by

performing tracer tests and computer simulations. Note that the computational simulations,

which supported the experimental tracer tests, were possible thanks to the availability of an

open source software STANMOD (STudio of ANAlytical MODels), along with the

experimental infrastructure of UFRJ. Some conclusions can be drawn from the coreflooding

experiments performed in the laboratory and the computational simulations:

1. The heterogeneity of rock samples at the pore scale could be analyzed, showing

a significant effect on the tracer tests and the software simulation results.

2. Examination of the tracer tests results and software parameters showed that

carbonated water injection into the Indiana limestone created a preferential path, but still had

little influence on forming dead-end pores.

3. Heterogenous samples presented more complex parameter and visual results

than the homogenous sample. Thus, the efficiency of tracer tests and the software simulations

to detect dissolution can be diminished since results may not be so evident. This issue may

deserve more judicious evaluation in future studies.

4. STANMOD software successfully simulated all tracer test results, without any

problems on estimating the transport parameters. This was true also for the homogenous IL

sample, where the Beta parameter converged to 1, thereby reducing the capacitance model to a

convection-dispersion formulation.

5. Tracer tests and computational simulations proved to be a feasible practice and

can become a routine part of core analysis for rock characterization. We note that tracer effluent

concentrations are simple and easy to measure, that an open source software with a robust

coding is available for the simulations, and tests are relatively quick providing valuable

information on the transport parameters.

Regarding future work, tracer tests and computational simulations should be applied

to carbonates samples for characterization, before and after destructive SCAL experiments.

Since heterogeneities at the core scale play a significant role in laboratory core flood

experiments. Therefore, they may be helpful on SCAL studies focused on EOR methods, for

instance, by guiding flow rate and even water-gas ratio choices of WAG and SWAG methods.

At present, such prior inputs at SCAL studies investigating EOR methods are hard to determine

and often adjusted by trial and error.

57

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62

7. APPENDIX

Table 15. BO CFITIM mode parameters iterations.

Iteration SSQ Peclet RetFac Beta Omega Pulse

0 1.653017 100.00000 1.00000 0.95000 0.50000 2.00000

1 0.891444 74.16696 0.98697 0.93100 0.15554 1.98380

2 0.290441 30.32618 0.96806 0.91775 0.09207 1.96327

3 0.041279 28.29534 0.88738 0.91521 0.18809 1.90220

4 0.034972 46.16384 0.91807 0.82993 0.50148 1.89281

5 0.025384 53.24964 0.91976 0.85005 0.40602 1.89447

6 0.025347 52.95826 0.91859 0.84981 0.41447 1.89394

7 0.025347 52.97928 0.91859 0.84980 0.41437 1.89395

Table 16. EB CFITIM mode parameters iterations.

Iteration SSQ Peclet RetFac Beta Omega Pulse

0 1.328607 100.00000 1.00000 0.95000 0.50000 2.00000

1 0.953936 70.23881 1.00776 0.93408 0.20902 2.00942

2 0.487603 32.69506 1.02482 0.91775 0.20275 2.02870

3 0.161684 15.35221 1.09626 0.86621 0.16716 2.07362

4 0.097475 17.41819 1.10569 0.86301 0.29469 2.08496

5 0.069900 24.33231 1.15726 0.75766 0.57787 2.09806

6 0.057432 38.08569 1.18123 0.69929 0.82566 2.09385

7 0.054338 66.72517 1.19692 0.64437 1.11683 2.08868

8 0.045481 85.49010 1.19516 0.65955 1.04200 2.09033

9 0.045392 88.67148 1.19423 0.65861 1.05952 2.08794

10 0.045391 88.76445 1.19429 0.65882 1.05714 2.08798

11 0.045391 88.74028 1.19429 0.65882 1.05724 2.08794

63

Table 17. IL CFITIM mode parameters iterations.

Iteration SSQ Peclet RetFac Beta Omega Pulse

0 1.099844 100.00000 1.00000 0.95000 0.30000 2.00000

1 0.142373 60.50272 1.04603 0.80653 1.21551 2.18872

2 0.008492 54.43877 1.00093 0.88754 1.37733 2.26485

3 0.001222 61.11548 1.00534 0.91069 1.29466 2.26553

4 0.001150 62.01469 1.00552 0.91939 1.02050 2.26579

5 0.001119 60.72628 1.00554 0.92887 0.83235 2.26580

6 0.001084 59.07300 1.00532 0.93864 0.65704 2.26580

7 0.001039 57.29536 1.00504 0.94878 0.49016 2.26580

8 0.000978 55.39247 1.00476 0.95927 0.33314 2.26581

9 0.000888 53.31376 1.00451 0.96998 0.19238 2.26582

10 0.000744 51.04816 1.00445 0.98009 0.08159 2.26586

11 0.000603 48.93226 1.00526 0.98667 0.02116 2.26600

12 0.000581 48.13073 1.00902 0.98462 0.01138 2.26721

13 0.000461 48.00623 1.01358 0.98023 0.01309 2.26820

14 0.000444 48.05141 1.01540 0.97831 0.01425 2.26849

15 0.000442 48.06952 1.01591 0.97777 0.01454 2.26855

16 0.000442 48.07373 1.01604 0.97763 0.01460 2.26856

17 0.000442 48.07454 1.01608 0.97760 0.01461 2.26857

18 0.000442 48.07467 1.01609 0.97758 0.01461 2.26857

Table 18. IL_AT CFITIM mode parameters iterations.

Iteration SSQ Peclet RetFac Beta Omega Pulse

0 2.408621 100.00000 1.00000 0.95000 0.50000 2.00000

1 1.785396 67.70265 0.98687 0.92855 0.16131 1.97413

2 0.831282 23.97714 0.96803 0.89728 0.01985 1.91764

3 0.175470 14.23020 0.94601 0.87199 0.11258 1.77658

4 0.023583 20.28265 0.94349 0.86810 0.20980 1.62102

5 0.021221 22.89361 0.96880 0.86273 0.13329 1.62134

6 0.020537 25.37586 0.94867 0.86784 0.20344 1.61356

7 0.020442 23.44564 0.96865 0.86020 0.14833 1.62027

8 0.019940 25.38124 0.95526 0.86164 0.20518 1.61473

9 0.019749 24.17440 0.96658 0.85750 0.17353 1.61885

10 0.019680 25.45592 0.96178 0.85518 0.20795 1.61621

11 0.019660 24.53637 0.96748 0.85504 0.18281 1.61873

12 0.019634 25.25191 0.96418 0.85421 0.20239 1.61711

13 0.019622 24.86850 0.96625 0.85440 0.19216 1.61809

14 0.019621 24.85805 0.96634 0.85427 0.19265 1.61812

15 0.019621 24.85832 0.96635 0.85427 0.19266 1.61812

64

Table 19. BO concentrations observed and fitted with its respective residuals’ outputs.

No Pore

Volume

Concentration Residual

Obs. Fitted

1 0.540 0.000 0.021 -0.021

2 0.590 0.029 0.056 -0.026

3 0.640 0.102 0.115 -0.013

4 0.680 0.161 0.180 -0.019

5 0.730 0.282 0.277 0.006

6 0.770 0.365 0.359 0.006

7 0.820 0.461 0.460 0.002

8 0.860 0.514 0.533 -0.020

9 0.910 0.608 0.613 -0.005

10 0.960 0.662 0.678 -0.016

11 1.000 0.723 0.721 0.002

12 1.050 0.784 0.764 0.019

13 1.090 0.821 0.792 0.028

14 1.140 0.860 0.821 0.038

15 1.190 0.883 0.844 0.039

16 1.250 0.910 0.867 0.043

17 1.320 0.925 0.889 0.036

18 1.400 0.934 0.909 0.025

19 1.480 0.942 0.925 0.017

20 1.550 0.946 0.937 0.009

21 1.630 0.968 0.948 0.020

22 1.710 0.974 0.957 0.017

23 1.780 0.981 0.964 0.017

24 1.860 0.983 0.970 0.013

25 2.010 0.993 0.979 0.013

26 2.170 0.993 0.986 0.007

27 2.240 0.995 0.988 0.006

28 2.400 0.993 0.983 0.010

29 2.470 0.934 0.950 -0.016

30 2.550 0.831 0.855 -0.024

31 2.630 0.691 0.706 -0.016

32 2.700 0.542 0.564 -0.022

33 2.780 0.422 0.420 0.002

34 2.860 0.335 0.312 0.023

35 2.930 0.269 0.244 0.025

36 3.090 0.198 0.152 0.047

37 3.160 0.166 0.126 0.040

38 3.240 0.151 0.103 0.048

39 3.320 0.116 0.085 0.031

65

40 3.390 0.109 0.072 0.037

41 3.470 0.094 0.059 0.035

42 3.620 0.058 0.041 0.017

43 3.700 0.048 0.034 0.014

44 3.780 0.038 0.028 0.010

45 3.850 0.038 0.023 0.015

46 4.010 0.016 0.016 0.000

47 4.240 0.011 0.009 0.002

48 4.310 0.000 0.008 -0.008

Table 20. EB concentrations observed and fitted with its respective residuals’ outputs.

No Pore

Volume

Concentration Residual

Obs. Fitted

1 0.530 0.000 0.002 -0.002

2 0.570 0.004 0.008 -0.004

3 0.610 0.013 0.021 -0.009

4 0.650 0.041 0.048 -0.007

5 0.690 0.082 0.089 -0.007

6 0.730 0.132 0.143 -0.011

7 0.770 0.197 0.205 -0.009

8 0.810 0.261 0.269 -0.008

9 0.860 0.321 0.344 -0.022

10 0.900 0.377 0.396 -0.019

11 0.940 0.433 0.442 -0.009

12 0.980 0.471 0.482 -0.010

13 1.020 0.520 0.516 0.004

14 1.060 0.556 0.547 0.009

15 1.180 0.612 0.624 -0.013

16 1.250 0.657 0.663 -0.006

17 1320 0.725 0.697 0.027

18 1390 0.767 0.729 0.038

19 1460 0.799 0.757 0.042

20 1530 0.821 0.782 0.038

21 1600 0.838 0.806 0.032

22 1660 0.865 0.823 0.041

23 1.730 0.871 0.843 0.029

24 1.800 0.888 0.860 0.029

25 1.870 0.912 0.875 0.037

26 1.940 0.925 0.889 0.037

27 2.010 0.934 0.901 0.033

28 2.070 0.944 0.910 0.033

66

29 2.140 0.946 0.920 0.025

30 2.210 0.959 0.929 0.030

31 2.350 0.971 0.945 0.026

32 2.420 0.976 0.951 0.025

33 2490 0.979 0.956 0.023

34 2550 0.984 0.961 0.024

35 2620 0.981 0.963 0.018

36 2690 0.952 0.952 0.001

37 2760 0.886 0.904 -0.018

38 2830 0.790 0.815 -0.024

39 2.900 0.689 0.707 -0.018

40 2.970 0.591 0.608 -0.017

41 3.030 0.519 0.539 -0.020

42 3.100 0.463 0.476 -0.013

43 3.170 0.426 0.424 0.001

44 3.240 0.377 0.381 -0.004

45 3.310 0.360 0.342 0.018

46 3.380 0.331 0.307 0.024

47 3.440 0.306 0.280 0.026

48 3.510 0.281 0.251 0.030

49 3580 0.259 0.225 0.034

50 3790 0.200 0.161 0.039

51 3860 0.186 0.143 0.043

52 3920 0.171 0.130 0.041

53 3990 0.158 0.116 0.042

54 4060 0.144 0.103 0.041

55 4.130 0.132 0.092 0.041

56 4.200 0.121 0.081 0.039

57 4.270 0.112 0.072 0.040

58 4.330 0.104 0.065 0.038

59 4.470 0.085 0.051 0.034

60 4.540 0.080 0.045 0.035

61 4.610 0.073 0.040 0.033

62 4.750 0.060 0.032 0.028

67

Table 21. IL concentrations observed and fitted with its respective residuals’ outputs.

No Pore

Volume

Concentration Residual

Obs. Fitted

1 0.510 0.000 0.000 0.000

2 0.610 0.008 0.007 0.001

3 0.710 0.049 0.047 0.002

4 0.810 0.150 0.154 -0.004

5 0.910 0.320 0.328 -0.008

6 1.000 0.513 0.507 0.007

7 1.100 0.692 0.684 0.007

8 1.200 0.818 0.815 0.003

9 1.300 0.892 0.898 -0.006

10 1.400 0.939 0.945 -0.006

11 1.500 0.964 0.969 -0.005

12 1.600 0.978 0.981 -0.003

13 1.700 0.985 0.987 -0.002

14 1.800 0.990 0.989 0.000

15 1.900 0.991 0.991 0.001

16 2.090 0.990 0.992 -0.003

17 2.190 0.993 0.993 0.000

18 2.290 0.996 0.993 0.003

19 2.390 0.993 0.994 -0.001

20 2.490 0.996 0.994 0.002

21 2.590 0.998 0.994 0.004

22 2.690 10.000 0.995 0.005

23 2.790 0.996 0.994 0.002

24 2.890 0.983 0.986 -0.003

25 2.980 0.944 0.948 -0.004

26 3.080 0.838 0.840 -0.002

27 3.180 0.665 0.666 -0.001

28 3.280 0.471 0.468 0.003

29 3.380 0.296 0.295 0.001

30 3.480 0.170 0.170 0.000

31 3.580 0.091 0.092 -0.001

32 3.680 0.047 0.048 -0.002

33 3.780 0.025 0.026 -0.001

34 3.870 0.013 0.016 -0.003

35 3.970 0.008 0.011 -0.003

36 4.070 0.006 0.008 -0.002

37 4.170 0.005 0.007 -0.002

68

Table 22. IL_AT concentrations observed and fitted with its respective residuals’ outputs.

No Pore

Volume

Concentration Residual

Obs. Fitted

1 0.390 0.000 0.003 -0.003

2 0.490 0.021 0.026 -0.005

3 0.600 0.088 0.111 -0.022

4 0.700 0.220 0.243 -0.023

5 0.800 0.375 0.398 -0.023

6 0.900 0.547 0.543 0.004

7 1.000 0.684 0.661 0.023

8 1.100 0.782 0.747 0.034

9 1.200 0.849 0.808 0.041

10 1.410 0.912 0.880 0.032

11 1.510 0.929 0.900 0.029

12 1.610 0.944 0.915 0.029

13 1.710 0.942 0.926 0.016

14 1.810 0.954 0.935 0.019

15 1.910 0.959 0.943 0.016

16 2.020 0.937 0.947 -0.010

17 2.120 0.861 0.925 -0.063

18 2.320 0.686 0.720 -0.034

19 2.420 0.568 0.569 -0.001

20 2.520 0.429 0.427 0.002

21 2.630 0.330 0.304 0.026

22 2.730 0.256 0.223 0.032

23 2.830 0.195 0.168 0.027

24 3.030 0.132 0.105 0.027

25 3.130 0.107 0.087 0.020

26 3.230 0.087 0.074 0.013

27 3.340 0.073 0.063 0.010

28 3.440 0.072 0.055 0.017

29 3.640 0.061 0.043 0.019

30 3.740 0.055 0.038 0.017

31 3.840 0.043 0.033 0.010

32 4.050 0.040 0.026 0.014

33 4.150 0.038 0.023 0.015

34 4.250 0.036 0.020 0.016

35 4.350 0.033 0.018 0.015