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A method for characterizing reservoir heterogeneity using continuous measurement of tracer data

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A method for characterizing reservoir heterogeneity using continuous measurement of tracer data

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Content i

A Method for Characterizing Reservoir
Heterogeneity using Continuous Measurement of
Tracer Data

By NOHA N. NAJEM
A Dissertation submitted to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In partial fulfillment for the degree of
DOCTOR OF PHILOSOPHY
PETROLEUM ENGINEERING
December 2019

Copyright©2019 Noha N. Najem

ii

Acknowledgements

I dedicate my work to my Mother Hanaa A. Gougou (Former Director of Legal Affairs, Ministry Of
Communications, Kuwait) and my Father Dr. Nabeel M. Najem (Consultant Dermatologist, Head
of Dermatology Department, Al-Adan Hospital, Kuwait) who have been the inspiration in my life,
and to my family Wafaa Gougou, Tammam Najem, ElSaid GouGou, Yahia Gougou , Hisham Najem,
Hussam Najem, Rana Najem, Faisal Najem ,and Hana Najem.
To my Grandfather Mohammed Mahmoud Najem and Grandfather Ahmed GouGou who paved the
way of success for the generations of each of their families.
I would also like to thank those who have made this journey possible from Kuwait Petroleum
Corporation (KPC), Kuwait Oil Company (KOC), and the University of Southern California (USC):
Dr. Adel Al-Abbasi (Former DMD, KPC)
Mr. Jamal Jaafar (Former CEO, KOC)
Mr. Hashem Hashem, CEO KPC
Mr. Emad Sultan, CEO KOC
Mr. Bader E. Al-Attar, DCEO Planning & Finance, KOC
Mr. Khaled Al-Otaibi, DCEO MP &TS, KOC
Mr. Khaled Al-Sumaiti (Former DCEO, KOC)
The late Dr. Mohammed Osman, Specialist KOC
Dr. Iraj Ershaghi, USC
Dr. Fred Aminzadeh, USC
Dr. Roger Ghanem, USC
Dr. Christian Jessen, USC
Dr. Birendra Jha, USC

iii

Abstract

Properties that control displacement processes, particularly in (IOR) and enhanced oil recovery
(EOR) operations, are controlled by heterogeneity in the subterranean reservoirs. In this study,
we focus on the merits of using continuous monitoring of injected tracers to more accurately
investigate reservoir direction of fluid transport, flow performance, and reservoir connectivity.
Tracer technology has been sparsely used in the past as an essential method in determining
reservoir heterogeneities and flow directions. In this study, we introduce a novel concept of
tracer analysis with the assumption of continuous monitoring of tracer effluent concentration.
Combined with water cut data and by performing inverse modeling using a new method we
realize significant improvements in the reservoir characterization process.
The modeling is based on the assumption of real-time measurement of the effluent concentration
of tracer data using downhole detection units; these devices that can accurately quantify injected
tracers in the reservoirs are gradually getting into the market. As such, this study focuses on the
powerful impact of the proposed methodology.
To test the concepts, we generated synthetic continuous real-time concentration measurement.
In this report, we present the workflow of the proposed research method. Initially, we prepared
a derivation of the analytical model. We then mathematically derived the transport velocity using
the analytical model by coupling the diffusivity and concentration equations. This allows relating
transport velocity to indicate flow direction and in-situ permeability.
We also conducted some core-flooding experiments using tracers such as saline solution in Berea
Sandstone (USA), and performed other experimental work including the use of Fluorobenzoic
acid and its derivatives (FBA, 2-FBA, and 2,4 FBA), as well as HTO on actual field cores from
Kuwait oilfields to determine the model parameters and to refine a more realistic synthetic
model.
Finally, a dynamic simulation multipoint model is developed to examine and assess the results,
thus creating a dynamic model based on the real-time tracer concentration data. A genetic
algorithm technique (Differential Evolution) is used to validate the results.

iv

2 TABLE OF CONTENTS

ABSTRACT III
1 CHAPTER 1: INTRODUCTION 1
1.1 MOTIVATION 1
1.2 RESEARCH OBJECTIVE 1
1.3 LITERATURE REVIEW 2
1.3.1 AN OVERVIEW ON TRACERS AND TRACER APPLICATION 2
1.3.2 TRACER TYPES 2
1.3.3 LITERATURE REVIEW ON DISPERSION AND TRACERS IN POROUS MEDIA 4
1.3.4 DISPERSION IN POROUS MEDIA - TAYLOR DISPERSION 7
1.3.5 ADVECTION – DISPERSION EQUATION 10
1.3.6 LONGITUDINAL DISPERSION AND TRANSVERSE DISPERSION IN POROUS MEDIA 17
1.3.7 DISPERSIVITY 23
1.3.8 SUMMARY 30
2 CHAPTER 2: MATHEMATICAL MODELING COUPLING OF THE DIFFUSIVITY EQUATION
AND THE CONCENTRATION EQUATION 31
2.1 INTRODUCTION 31
2.2 RELEVANCE OF THE TRANSPORT EQUATION AND TRACER CONCENTRATION 31
2.2.1 OVERVIEW 31
2.2.2 THE ADVECTIVE FLUX 34
2.2.3 THE DISPERSIVE FLUX 36
2.3 AN OVERVIEW OF ADVECTION-DISPERSION AND DISPERSION 38
3 CHAPTER 3: CORE FLOODING EXPERIMENT TO DETERMINE LONGITUDINAL
DISPERSION 41
3.1 SCOPE OF WORK 41
3.2 EXPERIMENT 1: MEASURING CONCENTRATION USING CORE FLOODING IN BREA CORE 41
3.2.1 SCOPE OF WORK 41
3.2.2 EXPERIMENT DESIGN 41
3.2.3 RESULTS AND CONCLUSION 42
3.3 EXPERIMENT 2: DETERMINING LONGITUDINAL DISPERSION COEFFICIENT USING CORE FLOODING IN
KUWAIT CORE SAMPLES 42
3.3.1 FICKIAN MODEL 42
3.3.2 CAPACITANCE MODEL 43
3.3.3 CALCULATING POROSITY OF ROCK 44
3.3.4 SCOPE OF WORK 45
3.3.5 TECHNICAL REQUIREMENTS & GIVEN PARAMETERS 46
3.3.6 EXPERIMENT PROCEDURE AND DESIGN 47
3.3.7 RESULTS 53
3.3.8 INTERPRETATION OF RESULTS 64
v

3.3.9 CONCLUSION 70
3.4 A STUDY OF UPSCALING TECHNIQUES OF LABORATORY-SCALE DISPERSION TO FIELD SCALE 70
4 CHAPTER 4: NUMERICAL SIMULATION OF FORWARD MODELS (VELOCITY, TRACER
CONCENTRATION PROFILES) 72
4.1 ONE-DIMENSIONAL TRACER SIMULATION MODEL 72
4.1.1 RESULTS OF CASE NO.1 ONE-DIM PERMEABILITY XYZ= 1D WITH TRACER (CASE) 73
4.1.2 RESULTS OF CASE NO 2 ONE-DIM PERMEABILITY XYZ= 10D Z=1D WITH TRACER (TRACER 2) 74
4.2 3-DIMENSIONAL TRACER SIMULATION MODEL 75
4.3 SYNTHETIC SIMULATION MODELS 78
4.3.1 EFFECT OF DISPERSION COEFFICIENT 78
4.3.2 SYNTHETIC MODELS (VELOCITY PROFILE) 79
5 CHAPTER 5: AN OVERVIEW OF EVOLUTIONARY OPTIMIZATION ALGORITHMS 82
5.1 EVOLUTIONARY COMPUTING AND GENETIC ALGORITHMS 82
5.2 DIFFERENTIAL EVOLUTION 83
5.3 COMMONLY USED GENETIC OPTIMIZATION ALGORITHMS 83
5.3.1 PARTICLE SWARM OPTIMIZATION 83
6 CHAPTER 6: INVERSE MODELLING, OPTIMIZATION, AND VALIDATION OF DYNAMIC
MODEL TRACER CONCENTRATION RESULTS USING DIFFERENTIAL EVOLUTION (SINGLE
PARAMETER) 85
6.1 WHY USE DIFFERENTIAL EVOLUTION 85
6.2 OPTIMIZATION AND VALIDATION OF DYNAMIC MODELS USING DIFFERENTIAL EVOLUTION 85
7 CHAPTER 7: A COMPARATIVE STUDY AND RESULTS OF EVOLUTIONARY ALGORITHMS
USING NOVEL PARAMETERS TO IDENTIFY HIGH PERMEABILITY CHANNELS (DUAL
PARAMETER) 103
7.1 INTRODUCTION 103
7.2 CHALLENGES WITH TRADITIONAL INVERSE MODELLING 104
7.3 DIFFERENTIAL EVOLUTION 106
7.3.1 INVERSE MODELLING METHOD USING SYNTHETIC TRACER AND WATER CUT DATA 108
7.4 FORWARD MODELS AND DATA SETS 110
7.4 FURTHER STUDY: A COMPARISON OF FOUR EVOLUTIONARY ALGORITHM METHODS FOR INVERSE
SIMULATION 117
7.5 INTERPRETATION OF RESULTS 126
8 CHAPTER 8: RESULTS OF INVERSE SIMULATION OF DAILY WATER CUT AND TRACER
SYNTHETIC DATA 129
9 CHAPTER 9: CONCLUSION 134
10 CHAPTER 10: FUTURE WORK 136
10.1 COMBINING SEISMIC INTERPRETATION AND CONTINUOUS TRACER DATA 136
10.2 EXPANDING THE RESEARCH METHOD TO SIMULATE MULTI-LAYER RESERVOIR CASE STUDIES 137
10.2.1 CHALLENGES OF SIMULATING MULTI-LAYER RESERVOIR 138
vi

10.2.2 EXPANDING THE RESEARCH METHOD CONTINUOUS TRACER DETECTION FOR MULTI-LAYER
RESERVOIR 139
11 REFERENCES 142

vii

Index of Tables

Table 2.1: FBA Isomers Used. _______________________________________________________________________________________ 4
Table 2.1: Summary of the Definitions for Transport Equation. ________________________________________________ 32
Table 3.1: The mean residence time of tracers in the systems. __________________________________________________ 44
Table 3.2: The calculated porosity of core samples. _____________________________________________________________ 45
Table 3.3: Parameters of Core Samples from Kuwait. __________________________________________________________ 46
Table 3.4: Chemical Composition of Brine for the Experiments. ________________________________________________ 48
Table 3.5: Equipment used in the Experiments. __________________________________________________________________ 50
Table 3.6: Summary of Experiment Parameters and Procedure. _______________________________________________ 52
Table 3.7: 2-FBA Concentration Data. ___________________________________________________________________________ 54
Table 3.8: Core I Experiment B HTO Concentration Data. ______________________________________________________ 56
Table 3.9. 2, 4-DFBA Concentration Data _______________________________________________________________________ 59
Table 3.10: Core I Experiment C HTO Concentration Data. _____________________________________________________ 60
Table 3.11: Core I Experiment C 4-FBA Concentration Data. ___________________________________________________ 62
Table 3.12: The mean residence time of tracers in the systems. ________________________________________________ 64
Table 3.13: The calculated porosity of core samples. ____________________________________________________________ 64
Table 3.14: The fitting parameter of tracer response curves of Core I. _________________________________________ 65
Table 3.15 Dispersivity Results. ___________________________________________________________________________________ 70
Table 4.1: One-Dimensional Simulation Models. _________________________________________________________________ 72
Table 4.2: Case No.1 Model Properties. ___________________________________________________________________________ 73
Table 4.3: Concentration vs Time Results (Tracer2). ____________________________________________________________ 74
Table 7.1: Parameterization of S-Shaped Synthetic Model. ____________________________________________________ 111
Table 7.2: Development Strategy for the S-Shaped Synthetic Model. __________________________________________ 112
Table 7.3: Parameterization of S-Shaped Synthetic Model. ____________________________________________________ 113
Table 7.4: Development Strategy for S-Shaped Synthetic Model. ______________________________________________ 114
Table 7.5: Parameterization of S-Shaped Synthetic Model. ____________________________________________________ 115
Table 7.6: Development Strategy for S-Shaped Synthetic Model. ______________________________________________ 116

viii

Index of Figures

Figure 2.1: Dispersion in Capillary Tube (Bear, 1972). ............................................................................................................. 7
Figure 2.2: Non-uniformity of the microscope flow velocity. .................................................................................................. 9
Figure 2.3: Tracer Concentration Vs Time Relation of Column Outflow. ......................................................................... 11
Figure 2.4: Schematic view of the fracture-matrix system. ................................................................................................... 15
Figure 2.5: Longitudinal Dispersion Coefficient for a random network of capillaries. ............................................... 17
Figure 2.6: Effect of rate and viscosity level on dispersion in 40-200 mesh sand on transverse dispersion
coefficient. ................................................................................................................................................................................................. 18
Figure 2.7: Typical longitudinal and transverse dispersion profile.................................................................................... 18
Figure 2.8: Comparison of the network model results for longitudinal (upper solid line) and transverse
(lower solid line) dispersion coefficients with experimental data. ..................................................................................... 20
Figure 2.9: Definition of two-layer porous media; and (b) schematic illustration of Taylor's dispersion in
media. .......................................................................................................................................................................................................... 21
Figure 2.10: Dispersion results in Berea Core for both single and two-phase flow at various flow rates (Kz is
the longitudinal dispersion,D is the molecular diffusion coefficient, Uz is the interstitial velocities, r is the
particle radius)........................................................................................................................................................................................ 23
Figure 2.11: Field and laboratory dispersivity data. ................................................................................................................ 25
Figure 2.12: (a) Longitudinal dispersitivity versus the scale of observation identified by the type of
observation and type of aquifer; (b) Ratio of longitudinal dispersivity and transverse dispersivity. ................... 27
Figure 2.13: (a) Relationship between effective porosity/porosity and test scale; (b) Relationship between
longitudinal dispersivity and test scale. ........................................................................................................................................ 28
Figure 2.14: Relationship of longitudinal dispersivity to scale of measurement for consolidated media........... 29
Figure 2.15: Relationship of longitudinal dispersivity to scale of measurement for unconsolidated sediments.
....................................................................................................................................................................................................................... 29
Figure 2.1: Mass In and Out of Control Volume. ......................................................................................................................... 32
Figure 2.2: Control Volume Element. .............................................................................................................................................. 34
Figure 2.3: Representative Element. ............................................................................................................................................... 36
Figure 2.4: Dispersive Mass Transfers. ........................................................................................................................................... 36
Figure 3.1: Tracer concentration vs Time. ................................................................................................................................... 42
Figure 3.2: Dynamic Tracer Test System. ..................................................................................................................................... 49
Figure 3.3: 2-FBA-response curve of Core I at a flow rate of 0.02 mL/min. .................................................................... 54
Figure 3.4: Core IB HTO Curve. ......................................................................................................................................................... 56
Figure 3.5: 2,4-DFBA response curve of Core I at a flow rate of 0.04 mL/min. .............................................................. 58
Figure 3.6: HTO response curve of Core I at a flow rate of 0.07 mL/min. ....................................................................... 60
Figure 3.7: Core IC – 4-FBA................................................................................................................................................................. 61
Figure 3.8: FBA tracer response curves vs. time after injection of Core I at different flow rates (0.02 mL/min,
0.04 mL/min and 0.07 mL/min). ...................................................................................................................................................... 63
Figure 3.9: HTO tracer response curves vs. time after injection of Core I at different flow rates (0.04 mL/min
and 0.07 mL/min). ................................................................................................................................................................................. 63
Figure 3.10: 2FBA tracer curve of Core I at a flow rate of 0.02 mL/min with the capacitance model. ................ 67
Figure 3.11: HTO tracer curve of Core I at a flow rate of 0.04 mL/min with the capacitance model. ................. 67
Figure 3.12: 24DFBA tracer curve of Core I at a flow rate of 0.04 mL/min with the capacitance model. .......... 68
Figure 3.13: HTO tracer curve of Core I at a flow rate of 0.07 mL/min with the capacitance model. ................. 68
Figure 3.14: 4FBA tracer curve of Core I at a flow rate of 0.07 mL/min with the capacitance model. ................ 69
Figure 3.15: Correlation between longitudinal dispersion coefficient and mean flow velocity. ............................. 69
Figure 4.1. 1-D Deterministic Model (applies to all 1-D Cases). ........................................................................................... 73
ix

Figure 4.2: Concentration vs Time Results (Case). .................................................................................................................... 74
Figure 4.3: Concentration vs Time Results (Tracer2). ............................................................................................................. 75
Figure 4.4: Permeability 1 mD. ......................................................................................................................................................... 75
Figure 4.5: Permeability 5 mD. ......................................................................................................................................................... 76
Figure 4.6: Permeability Narrow Channel Base 1mD. ............................................................................................................. 76
Figure 4.7: Permeability Wide Channel Base 1mD. .................................................................................................................. 76
Figure 4.8: Permeability Curved Channel Base 1mD. ............................................................................................................... 77
Figure 4.9: Results of Simulation...................................................................................................................................................... 77
Figure 4.10: 15 X 15 Quarter 5 spot pattern results. ................................................................................................................ 78
Figure 4.11: 15 X 15 Quarter 5 spot pattern simulation results. ......................................................................................... 79
Figure 4.12: Velocity Profile for the homogeneous case (5mD). .......................................................................................... 80
Figure 4.13: Velocity Profile for Narrow channel. ..................................................................................................................... 80
Figure 4.14: Velocity Profile for S Curve channel. ..................................................................................................................... 81
Figure 6.1: 1 Dimensional Model 2000 x 10 x10. ....................................................................................................................... 85
Figure 6.2: 1 Dimensional Model 2000 ft. x 10ft x 10ft (Forward Results)...................................................................... 86
Figure 6.3: 1 Dimensional Model 2000 ft. x 10ft x 10ft (Match Results). ......................................................................... 87
Figure 6.4: Region definition and ranges. ..................................................................................................................................... 87
Figure 6.5: Simulation results for all cases. ................................................................................................................................. 88
Figure 6.6: Simulation results for all cases. ................................................................................................................................. 89
Figure 6.7: Simulation results for all cases. ................................................................................................................................. 90
Figure 6.8: Best Match Case. .............................................................................................................................................................. 91
Figure 6.9: Narrow Channel Case. ................................................................................................................................................... 92
Figure 6.10: Forward Model results for P1 & P2/ P3 &P4. .................................................................................................... 93
Figure 6.11: Initial Model for Initialization. ................................................................................................................................ 93
Figure 6.12: Forward Simulation Initial Model for Initialization. ...................................................................................... 94
Figure 6.13: Initial Model for Initialization and objective function. .................................................................................. 95
Figure 6.14: Permeability Ranges for Differential Evolution Algorithm. ......................................................................... 96
Figure 6.15: Permeability Ranges for Differential Evolution Algorithm. ......................................................................... 96
Figure 6.16: Matching for Well No 2............................................................................................................................................... 97
Figure 6.17: S-Curve Synthetic Model............................................................................................................................................. 98
Figure 6.18: S-Curve Synthetic Model Forward Concentration Curve Results. .............................................................. 99
Figure 6.19-Initial Model for S Curve. ......................................................................................................................................... 100
Figure 6.20: Forward Concentration Results for Initial Model. ........................................................................................ 100
Figure 6.21: Forward Concentration Results for Initial Model Vs. Objective Function. .......................................... 101
Figure 6.22: Permeability ranges for S-Curve Model. ........................................................................................................... 101
Figure 6.23: Match cases for S Curve. .......................................................................................................................................... 102
Figure 7.1 Differential Evolution Method. ................................................................................................................................. 106
Figure 7.2: Synthetic Model 5 Spot Pattern with Narrow Channel Permeability (5mD). ....................................... 111
Figure 7.3: S-Shaped Synthetic Model. ........................................................................................................................................ 113
Figure 7.4: 5 Spot Patter with Narrow Channel Multiple Permeability. ....................................................................... 115
Figure 7.5: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm techniques
(Producer 1) Water Cut Data (objective function). ............................................................................................................... 119
Figure 7.6: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm techniques
(Producer 1) Tracer Data (objective function). ...................................................................................................................... 120
Figure 7.7: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm techniques
(Producer 1) Tracer Data (objective function). ...................................................................................................................... 121
Figure 7.8: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm techniques
(Producer 2) Tracer Data (objective function). ...................................................................................................................... 122
Figure 7.9: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm techniques
(Producer 3) Water Cut Data (objective function). ............................................................................................................... 123
x

Figure 7.10: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm techniques
(Producer 3) Tracer Data (objective function). ...................................................................................................................... 124
Figure 7.11: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm techniques
(Producer 4) Tracer Data (objective function). ...................................................................................................................... 125
Figure 7.12: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm techniques
(Producer 4) Water Data (objective function). ....................................................................................................................... 126
Table 7.13: Root Mean Square Error of Predicted Permeability vs. Actual Permeability. ..................................... 127

1

1 Chapter 1: Introduction

1.1 Motivation

World demand for increased oil production has led to a focus on the issues of declining primary
oil production. High-efficiency production methods are required. However, Improved Oil
Recovery (IOR) techniques, which include secondary and enhanced oil recovery (EOR,) when
applied, can only recover a small fraction of the oil reserves. This is due to reservoir uncertainties
and poor well placements. This application of improved oil recovery has led to a need to optimize
reservoir characterization. Although extensive efforts have been made in research in terms of
seismic imaging, pressure transient using interference and pulse testing, and mathematical
analytics, limited discussions in literature have focused on continuous measurement of tracer
concentration through permanent downhole detectors.
Tracers are increasingly being used for effective reservoir monitoring in the Oil & Gas industry.
In the past, tracers applied in the petroleum industry were usually radioactive, but with the
development of chemical tracer technology in the last several decades, chemical tracers have
largely replaced radioactive tracers. Today, different approaches are being studied to apply
environmentally friendly non-chemical tracers in the reservoir. Ecologically friendly non-
chemical tracers should satisfy fundamental conditions in the sense that they should be
chemically and biologically stable under different reservoir conditions, and do not exhibit
sorption on rock surfaces.
We need such tracers to investigate reservoir direction of flow and flow performance, reservoir
connectivity, and reservoir properties that control displacement processes, particularly (IOR)
and (EOR) operations.
Limited studies have been published on dynamic models using continuous tracer measurement.
The use of continuous tracer measurement requires a focus on two areas: techniques that
improve reservoir characterization by further providing a more accurate model of heterogeneity
and by utilizing such characterization to assess the well placement of injectors and producers in
IOR and EOR operations.
1.2 Research Objective

The objective of this research is to show a new modeling technique to dynamically model
heterogeneity, assuming we have access to data from permanent downhole detectors, which
continuously measure tracer concentration.
To accomplish this, we investigated the effect of longitudinal dispersion and velocity
mathematically and examined the fundamental equations, which serve as the basis. We then
2

coupled the equations to determine the relationship of tracer concentration to longitudinal
dispersion, and velocity and investigated the various work of several authors who have published
in this area.
We also conducted several laboratory core flooding experiments by using a passive miscible
tracer in water using several cores including Berea Sandstone and carbonate samples from
different producing fields /reservoirs from Kuwait to obtain and analyze the velocity and
longitudinal dispersion values.
In the absence of continuously measured tracer concentration data, we used several synthetic
simulation models for our analysis. We used extensive numerical simulation on several synthetic
models to study the behavior of reservoir response under various permeability models. Finally,
we validated the results through assisted history matching and inverse modeling using a new
adaptation of the differential evolution technique.
1.3 Literature Review

1.3.1 An Overview on Tracers and Tracer Application

The selection of a suitable tracer is fundamental in any tracer injection operation and differs from
laboratory scale to field scale. One must consider both the safety and environmental factors, and
potentially the mixing issues with groundwater. Careful selection of tracers is a critical issue in
any field operation in terms of the use of passive or active tracers in the laboratory or the oilfield.
In principle, a passive tracer is being carried by the fluid phase. Active tracers interact or may
react with existing fluids in the system or with the rock in the subsurface. Selection of tracers
should fulfill the following criteria:
-it can be detected with a low detection limit,
-it does not change and be stable under reservoir conditions,
-it follows the fluid phase that is being marked and have a minimal partitioning into other fluid
phases (unless the use of the tracer is to partition into different phases),
-ideally, it should have no adsorption on the reservoir rock and
-it should have minimal or no impact on the environment.
When obtaining the results after the implementation of tracers, a proper interpretation of the
tracer concentration curves will reflect the subsurface conditions. The interpretation can provide
information about the fluid saturation and rock petrophysical properties. The more accurate the
information, the more we would be able to minimize the cost associated with the use of expensive
fluids such as surfactants, micellar fluids, and polymers.

1.3.2 Tracer Types

3

Over the years, different tracer types have been deployed in various injection operations.
Chemical tracers such as halides and thiocyanate were commonly used in the 1980s. In general,
the best tracers to use are the radioactive tracers with a reasonably long half-life; however,
contamination of the reservoir with radioactive tracers can negatively impact the future
application of similar tracers. These tracers are, however, are more effective because of the high
sensitivity of detection equipment.
There have been growing use of non-radioactive tracers, fluorobenzoic acid and its forms (FBA,
2-FBA, 2, 4 FBA, and 4-FBA), and these tracers are still being used in several tracer injection
operations. In 2015, new high-temperature gas and water tracers, as well as specific tracers such
as tracers for heavy oil EOR and functional nano-tracers, were introduced.
The tracers used in this work are FBA isomers (organic acids COOH) and have been widely used
as oilfield tracers in the industry. Isomers have the same chemical compound but different
physical structures. A variety of derivates of benzoic acids (fluorobenzoates) are used as tracers:
(Malcome et al. 1980, Stetzenbach, Jensen and Thompson, 1982; Bowman, 1984a; Bowman and
Gibbens, 1992; McCarthy, Howard, and McKay, 2000) Christian Leibundgut, Piotr Maloszewski).
There are 16 FBA isomers or derivatives that exhibit similar physiochemical properties and
environmental behavior due to the number and the position of the fluorine atom in the benzene
ring (Hu and Moran, 2005). According to Juhler and Mortensen (2002), all of these can serve as
tracers. FBAs typically chosen for hydrological studies are Difluoro-Benzoic Acids (DFBAs).
Fluorobenzoic acids do not occur naturally and are, therefore, suitable for use as hydrological
tracers. Their pKa values (acid dissociation constant) are relatively low. Therefore, FBA tracers
are predominantly negatively charged under most environmental conditions. As anions, they are
readily soluble in water and nonvolatile. FBAs are commonly described as conservative tracers
with low levels of sorption and degradation under laboratory and field conditions. However,
some studies show that there are significant degradation and retardation in substrates rich in
clay or organic carbon matter (Bowman and Gibbens, 1992; Haynes 1994). Seaman (1998)
observed adsorption of FBAs to hydrous Fe-oxides (Hu and Moran 2005). McCarthy et al. (2000)
indicate that FBAs can be useful as nonreactive tracers as long as the pH is approximately two pH
units above FBAs specific pKa. FBAs can be analyzed using ion chromatography (Pearson,
Comfort and Inkeep, 1992, Bownman 1984b, and Caldiga and Greibrokk, 1998).

4

The solubility of FBA ranges from 1.2 g/L of H20 (4-FBA) – 7.2 g/L of H2O (2-FBA). This is shown
in Table 1.1 below:
Table 1.1: FBA Isomers Used.
Sr.
No
Name Chemical Compound Solubility in H2O Melting Point
1 2-Fluorobenzoic
Acid (2-FBA)
FC6H4CO2H

Soluble 121-124 °C
2 2,4-
Difluorobenxoic
Acid
F2C6H3CO2H Soluble 188-190 °C
3 4-Fluorobenzoic
Acid (4-FBA)
FC6H4CO2H

Very slightly
soluble (cold
water) – freely
soluble (hot
water)
182-184 °C
Source: www.sigmaaldrich.com

1.3.3 Literature Review on Dispersion and Tracers in Porous Media

Since the 1950s dispersion phenomena have become the primary interest in many different fields
such as physics, chemical engineering, soil mechanic, geology, petroleum engineering,
environmental engineering, geothermal, and hydrogeology. Dispersion plays an essential role in
solute transport processes in porous media; examples include, pollutant transport in
groundwater, and miscible displacement during enhanced oil recovery processes.
Dispersion refers to the spreading of solute in a continuous phase of another material. In porous
media, dispersion is also called hydrodynamic dispersion, which strongly influences the mixing
of two miscible fluids. The hydrodynamic dispersion consists of two components: transverse
dispersion, which occurs perpendicular to the main flow direction and longitudinal dispersion,
which occurs parallel to the main flow direction. Experimental results show that transverse
dispersion is smaller than longitudinal dispersion (Blackwell, 1962; Perkins and Johnston, 1963;
Sahimi, 1982).
Two mechanisms contribute to the hydrodynamic dispersion: molecular diffusion and
mechanical dispersion. Molecular diffusion is characterized by the random motion of molecules
due to the gradient of concentration, i.e., from an area of high concentration to an area of low
concentration. The molecular diffusion in porous media is smaller than the molecular diffusion in
solution. The solid matrix of a porous medium, of course, acts locally as a separator of streamlines
and thus as a barrier to diffusion (Sahimi, 1982). Mechanical dispersion results from variations
5

in local velocities caused by microscopic heterogeneities. These velocity variations arise because
the fluid in the center of a pore space travels faster than the fluid near the wall and because the
diversion of flow paths around individual grains of porous material causes variations in average
velocity among different pore spaces (Anderson, 1979).
Moreover, dispersion phenomena are affected by both microscopic and macroscopic effects. The
macroscopic dispersion or field-scale dispersion, which results solely from variations in the
permeability of the porous system is related to the scale of the heterogeneity as well as the
distribution function of the permeabilities. Previous studies showed that the field and laboratory-
scale dispersion could differ by a factor of 10 (Warren, 1964).
Mathematically, the magnitude of dispersion is expressed in equation (Harleman et al., 1963;
Brigham, 1961; Blackwell, 1962):

where,
m
D is the molecular diffusion of solute in porous media,
α is the dispersivity of porous media,
v is the mean flow velocity, and
β is a parameter depending on porous media.

The longitudinal dispersion can be determined from the following relationship:
where
𝐷 𝑜 𝐹𝜙
is the molecular diffusion of solute in porous media (D 0: the diffusion coefficient in solution,
F: the formation electrical resistivity factor, ϕ: the porosity),
αL is the longitudinal dispersivity, and
v is the mean flow velocity.
In multiphase flow, dispersion becomes more complicated. Dispersion in each of the phases
depends on the pore space occupied by that phase. Sahimi (1982) showed that longitudinal and
transverse dispersion in a given phase depends on the phase saturation. Longitudinal dispersion
increases with phase saturation, and transverse dispersion also increases, but more slowly.
Dispersivity is a measure of dispersion that is usually estimated by matching the effluent history
of an injected tracer to the solution of the convection-dispersion equation (Lake, Adepoju, Johns,
2015). Both summaries of field observations (e.g., Gelhar et al., 1985, 1992) and theoretical

β
m
αv D D + =

(Eq. 1)

𝐷 𝐿 =
𝐷 𝑜 𝐹𝜙
+𝛂 𝐿 𝑣 1.2

(Eq. 2)
6

evaluation (e.g., Gelhar and Axness, 1983; Gelhar, 1987, 1993) have shown that dispersivity is a
function of the heterogeneity of a geologic formation. The dispersivity can be measured in a
laboratory experiment. Megascopic dispersivity is different from field-scale dispersivity.
Megascopic dispersivity is usually on the order of 0.01 cm to 1 cm compared with field
dispersivities ranging from 0.1 m to 100 m (Anderson 1984; L. W. Gelhar et al. 1992; H. Li 1995;
Dirk Schulze-Makuch 2003). In an isotropic porous medium; the transversal dispersivity is
approximately 15 times smaller than the longitudinal dispersivity for such a system (Menzie,
1988).
In enhanced oil recovery using solvent flooding, miscible displacement depends on the dispersion
processes taking place in the heterogeneous porous media. Dutta (1984) showed that the
dispersivity plays a vital role in hydrocarbon recovery at very low recovery rates. Moreover, data
on dispersion effect can be used to improve the design of solvent slug size for miscible flood
processes and to interpret the miscible displacement experiments in the laboratory (Greeves,
Patel 1982). Estimation of dispersivity is, without a doubt, one of the important parameters that
can enhance reservoir simulation.
Tracer tests (H. Li 1995; C. Welty & L. W. Gelhar 1994; Jui-Sheng Chen et al. 2005) are the
simplified approaches to the evaluation of dispersivity in both field-scale and laboratory-scale
experiments. The dispersivity is calculated using the three methods proposed by Poulin (1985)
that include graphical solution (Brigham’s method), the analytical solution of the dispersion
model, and finite difference solution of the dispersion model.
Based on previous studies, this review refers to dispersion phenomena from the literature,
including dispersion coefficient and dispersivity, which provides some academic backgrounds for
subsequent research.
7

1.3.4 Dispersion in porous media - Taylor dispersion

Over the years, many theoretical models for describing hydrodynamic dispersion in porous media
have been proposed.
Taylor (1953) and Aris (1956) conducted early experiments to analyze dispersion through solute
transport in capillary tubes. Taylor (1953) considered the movement of a conservative tracer in
laminar Newtonian flow in long capillary tubes. Tracer spreads in the flow direction in a capillary
tube of radius R. Due to the parabolic velocity distribution across the tube, tracer molecules
initially on a plane perpendicular to the tube’s axis, will lie at a later time on a paraboloidal surface
as shown in Figure 1.1 (Bear 1972).

Figure 1.1: Dispersion in Capillary Tube (Bear, 1972).
Taylor (1953) showed that the solute concentration in the capillary tube could be described by:

If molecular diffusion is neglected, Taylor gave the expression of dispersion :

Where d is the capillary’s diameter, D 0 is the molecular diffusivity in solution, and v is the average
velocity. In the later work, Aris (1956) showed that if molecular diffusion is not left out, the
dispersion coefficient is as follows:

It is noted that only longitudinal dispersion and axial molecular diffusion were considered and
transverse dispersion, of course, was limited on the wall of the capillary tube. However, the

2
2
x
C
D
t
C
L


=



(Eq. 1)

0
2 2
48D
v d
D
L
=

(Eq. 2)

0
2 2
0
48D
v d
D D
L
+ =

(Eq. 3)
8

analysis of a single capillary tube can be adapted to study dispersion in porous media if the
medium is a “network” of capillaries, randomly orientated, connected by pore throats (M.
Gutierrez Neri, 1978).
Dispersion phenomena related to the mixing of two miscible fluids in porous media is controlled
by two primary mechanisms, which include molecular diffusion and mechanical dispersion.
Molecular diffusion is the result of random motions of the fluid particles (Brownian motion) due
to the presence of a concentration gradient (Brown, 1828), i.e., the movement of particles follow
a random walk from an area of high concentration to an area of low concentration. The
relationship between the net flux of a solute due to dispersion processes and the concentration
of a solute was first considered by Graham (1850). Graham concluded that the flux caused by
diffusion is proportional to the concentration difference of solute in a solution. Based on
Grahams’s conclusion, Fick (1855b) developed the 1st law of diffusion, where diffusion of a solute
at a point x in time t in the aqueous solution can be represented by:
where J is the solute net flux [M/L
2
], and C is solute concentration [M/L
3
]. Einstein
(1905a) supposed that if a spherical solute molecule in a solvent is made up of much smaller
molecules, then the molecular diffusivity D0 of a particle can be modeled by:
where k is the Boltzman constant, T is the absolute temperature, µ is the viscosity of the solvent,
and R is the radius of the solute (M. Gutierrez Neri, 1978).
In porous media, molecular diffusion is called effective molecular diffusion and is smaller than
the molecular diffusion in solution due to the presence of a solid matrix. The effective molecular
diffusion can be defined as,
where τ is the tortuosity factor (Bear, 1972), F is the formation of electrical resistivity factor,


is the porosity. A ratio of 1/τ equal to 2/3 has been observed in experiments in unconsolidated
media (Fried & Combarnous, 1971; Bear, 1972).
Mechanical dispersion results from variations in local velocities caused by microscopic
heterogeneities. The spreading of solute due to mechanical dispersion relates to two basic
mechanisms: velocity differences within pores in microscopic scale and path differences due to
the tortuosity in porous media. As a result, the dispersion in porous media is a function of the
fluid velocity. This is shown in Figure 1.2 (Ne-Zheng Sun et al., 1996).

x
C
D J


− =
0

(Eq. 4)

6 ππμ
kT
D
0
=

(Eq. 5)


= =
F
D D
D
e
0 0


(Eq. 6)
9

Figure 1.2: Non-uniformity of the microscope flow velocity.
Scheidegger (1957) used a statistical approach to analyze dispersion. He showed there are two
possible forms of the dispersion coefficient: the first directly proportional to velocity where
represents the complete mixing inflow channel,
and the second is proportional to the velocity squared where describes the mixing is not
complete:
Some classical experimental works on single-phase flow dispersion are those of Harleman et al.
(1892), Brigham et al. (1961) and Blackwell (1962). Experimental data have been correlated
with:
The expression of longitudinal dispersion (D L) and transverse dispersion (D T) are respectively:

where α L, α T is the longitudinal dispersivity and transverse dispersivity respectively. The terms
β L and β T are the exponential parameters of the velocity. The terms β L and β T depend on the nature
of the porous medium structure and the flow condition. Legaski et al. (1967) determined the
largest and smallest values of β L, which are 1.28 and 1.13, respectively. β L equaling to 1.2 is the
result obtained from experiment investigation of hydrodynamic dispersion in linear miscible
displacements carried out by Brigham et al. (1961). Sahimi (1984) studied dispersion in two- and
three-dimensional pore networks using Monte Carlo simulation and showed nonlinear
dependence of the longitudinal dispersion coefficient on the mean velocity of flow where β L equal

D ~ v

(Eq. 7)

D ~ v
2

(Eq. 8)

β 0
αv
F
D
D +

=

(Eq. 9)

L
β
L
0
L
v α
F
D
D +

=

(Eq. 10)

T
β
T
0
T
v α
F
D
D +

=

(Eq. 11)
10

to 1.27. Harleman et al. (1963) reported the smallest value β T 0.7. Hassinger et al. (1968) claimed
the largest value β T, equals unity. Blackwell (1962) reported β T to be about 0.97. The experiments
show that β has a value from 1 to 2.

1.3.5 Advection – Dispersion Equation

The solute transport equation in porous media is derived from the continuous equation, where
the change of the solute flux into and out of a fixed elemental volume within the flow domain is
equal. The solute flux in the porous medium is influenced by two basic physical processes:
hydrodynamic dispersion, which describes the movement of solute according to the flow
direction with mean flow velocity in porous media and advection, which occurs as a result of
mechanical mixing and molecular diffusion. The mathematical description of solute transport in
porous media is well-known as the advection-dispersion equation.
Analytical solution of the advection-dispersion equation can be used to interpret the miscible
displacement in laboratory and field tests to understanding the mechanisms of dispersion and
identifying dispersion parameters. This review refers to three mathematical models of solute
transport in porous media: the Fickian model, Capacitance model, and the Single-fracture matrix
model.
The classical, Fickian advection-dispersion transport equation for conservative solute (with no
sink/source) is
where
C is the solute concentration [M/L
3
],
Φ is the porosity [fraction],
Dij is the dispersion coefficient tensor [L
2
/T], can be represented by the matrix (i, j = 1, 2,
3 is an indicator of dimension):
q i is Darcy flux (the ratio of the flow rate and the cross-sectional area), [L/T].
For a bulk flow in the x-direction and anisotropic porous medium, the dispersion tensor is

( ) C q
x x
C
D
x t
C
i
i j
ij
i


−












=


 

(Eq. 12)











=
33 32 31
23 22 21
13 12 11
D D D
D D D
D D D
D
ij

(Eq. 13)
11

where D L and D T are the longitudinal (or axial) and transverse (or radial) dispersion coefficients
[L
2
/T], respectively.
One-dimensional Fickian advection-dispersion transport equation for the conservative solute in
saturated, homogeneous, isotropic porous media under steady-state, uniform flow can be written
as:

where ,
v x is the mean pore velocity (is the ratio of the Darcy flux and the porosity) [L]/[T], and x is the
distance between the tracer injection position and observation position [L], t is time [T].
If a nonreactive tracer at concentration C 0 is continuously injected into the homogeneous
granular medium under steady-state, uniform flow, the mechanical dispersion and molecular
diffusion occur and the breakthrough curve (i.e., the tracer concentration versus time at the
outflow) is characterized by an S-shape as shown in Figure 1.3 (Freeze and Cherry, 1979).

Figure 1.3: Tracer Concentration Vs Time Relation of Column Outflow.
If the initial concentration in the model is zero, the boundary conditions are described
mathematically as:
Ogata (1970) provides the following solution:











=
T
T
L
D
D
D
D
0 0
0 0
0 0

(Eq. 14)

x
C
v
x
C
D
t
C
x L


−


=


2
2

(Eq. 15)

( ) 0 0 , = x C

0  x

( )
0
, 0 C t C =

0  t

( ) 0 , =  t C

0  t

(Eq. 16)
12

Analytical solutions under other boundary conditions are described by Rifai et al. (1956), Ebach
and White (1958), Ogata and Banks (1961), and others.

The Fickian advection-dispersion transport equation first introduced by Taylor (1953) and Bear
(1961), among others, is inadequate for representing field-scale solute transport in highly
heterogeneous porous media. Laboratory-scale miscible displacement experiments shows that
these heterogeneities could cause enhanced dispersion, early breakthrough, and long tailing,
which are difficult to model with the Fickian advection-dispersion equation (H. Greenhorn, 1991;
B. Zachara, 1993; C. Zheng & Fred J. Molz, 2000].
An alternative to the single-porosity advection-dispersion model is the dual-porosity or dual-
domain mass transfer model. The porous media contains the dead-end pore zone where solute
can migrate by molecular diffusion processes. The solute exchange between the mobile and
immobile zone and represented with the first-order mass transfer equation depends on the
concentration difference between the two phases.
The concept of dual-porosity was first introduced by Coats & Smiths (1964). The general mass
balance equations for the solute transport (S.Brouyere & A. Dassargues, 1999; C. E. Feehley & C.
Zheng, 2000; W. J. Bound & P. J. Wierenga, 1990) are:
ϕm is the porosity of the mobile zone [fraction],

where,
f: the flow fraction
ϕ im is the porosity of the immobile zone [fraction],

















+








+








−
=
t D
t v x
erfc
D
xv
t D
t v x
erfc
C
C
L
x
L
x
L
x
2
exp
2
2
1
0

(Eq. 17)
𝜑 𝑚 𝜕 𝐶 𝑚 𝜕𝑡
+ 𝜑 𝑖𝑚
𝜕 𝐶 𝑖𝑚
𝜕𝑡
=
𝜕 𝜕 𝑥 𝑖 (𝜑 𝑚 𝐷 𝑖𝑗
𝜕 𝐶 𝑚 𝜕 𝑥 𝑗 ) −
𝜕 𝜕 𝑥 𝑖 ( 𝑞 𝑖 𝐶 𝑚 ) (Eq. 18)

𝜑 𝑖𝑚
𝜕 𝐶 𝑖𝑚
𝜕𝑡
= 𝐾 ( 𝐶 𝑚 − 𝐶 𝑖𝑚
)
(Eq. 19)

  f
V
V
V
V
V
V
pore
pore
m m
m
= = =

(Eq. 20)
13

f: the flow fraction
Φ is the total porosity (equal to ϕ m + ϕ im) [fraction],
K is the mass transfer rate coefficient [1/T],
Cm is the solute concentration in the zone of mobile water [M/L
3
],
Cim is the solute concentration in the zone of immobile water [M/L
3
].
The factors affecting mass transfer include sorption, measurement scale, and fluid velocity
(Griffioenet et al., 1998). The larger the rate constant, the faster transport occurs between the
mobile and immobile zones for a given concentration difference.
As  → K , the connection is instantaneous, and the system acts like a single-domain model with
the porosity approaching total porosity. As 0 → K the system behaves as a single-domain model
with total porosity equal to ϕ m.
One-dimensional dual-porosity advection-dispersion transport equation for conservation of
tracers in homogeneous porous media can be written as:

where,
β is the ratio of the dead-end pore volume and the flow volume (dead-zone parameter)
v x
*
is the mean flow velocity
v x is the mean pore velocity (is the ratio of the Darcy flux and the porosity)
D L is the longitudinal dispersion coefficient [L2/T]
K
*
: the equivalent degradation coefficient,

( )   f
V
V
V
V
V
V
pore
pore
im m
m
− = = = 1

(Eq. 21)

𝜕 𝐶 𝑚 𝜕𝑡
+ 𝛽 𝜕 𝐶 𝑖𝑚
𝜕𝑡
= 𝐷 𝐿 𝜕 2
𝐶 𝑚 𝜕 𝑥 2
− 𝑣 𝑥 ∗
𝜕 𝐶 𝑚 𝜕𝑥

(Eq. 22)

𝛽 𝜕 𝐶 𝑖𝑚
𝜕𝑡
= 𝐾 ∗
( 𝐶 𝑚 − 𝐶 𝑖𝑚
)
(Eq. 23)

f
f −
=
1


(Eq. 24)

f
q
f
v
v
x x
x

= =
*

(Eq. 25)
14

With conditions:

At x = 0:

At x
 →
:

The solution of the above equation is given by Coats and Smiths (1964) has the form:
where, J, γ,  ,p, a 1, a 2, Z, w are a function of the ratio of the dead-end pore volume and the flow
volume, mean flow velocity, dispersion coefficient. The solution was numerically evaluated by
Simpson’s rule of numerical integration.
The applicability of analytical solutions of the Fickian model and the Capacitance model was
proven through miscible displacement experiments, such as the one reported by Brigham (1974),
and Donaldson (1976).
Brigham (1974) investigated mixing in short cores where the mixed zone is large compared with
the core length. The modified solution of the Fickian model and the Capacitance model were used
to match the effluent concentration curve. A good agreement between model and experiment
result was achieved.
A procedure for the determination of the macroscopic dispersion coefficient for Bandera, Berea,
Cottage Grove, Noxie, and Torpedo sandstone using tracer technique was presented by
Donaldson (1976). Tritium and sodium chloride was used as a conservative tracer. Both
analytical and finite difference solution of the Fickian model and Capacitance model were applied
to analyze the tracer response curves. Experimental results showed that the Berea sandstone
exhibited the sharpest displacement front and the least amount of tailing, making it an optimum
selection for miscible displacement experiments. The tracer concentration profiles of the other
cores exhibited a high degree of asymmetry, which is reflected in the large varies of mass transfer
coefficient for matching. It is observed that the effluent concentration profiles of non-adsorbing
tracers do not show the tailing effect and can be fitted by the Fickian model.

 f
K
K =
*

(Eq. 26)
( ) ( )
( )
x
t x C
D t x vC t x vC
m
im m


− =
,
, ,
0

(Eq. 27)

( ) 0 , → t x C
m

(Eq. 28)

( ) ( )  dZ w ZJ a w ZJ a
a a
e e
C C
p
J
m m
− + −
+
=








−
sin cos
2
2 1
0
2
2
2
1
2
cos 1
2
0
 


(Eq. 29)
15

Diffusion and advection processes control the solute transport in fractured - porous matrix media.
Fractures present much higher permeability and as such, flow through fractures often dominates
the overall flow behavior, on the other hand, matrix typically dominates the total pore volume of
the system. As a result, solute velocity in fracture - porous matrix media is much higher than in
unfractured rock or unconsolidated soils (Berkowitz, 2002).
Previous studies pointed out solute transport in fracture-matrix porous media plays an essential
role in many processes such as geological disposal of nuclear waste, contaminant transport
during groundwater contamination, enhanced oil recovery in naturally fractured reservoirs, and
greenhouse gas sequestration in fractured geological formations (Trivedi 2009).
The analytical model of advection-dispersion transport in a fracture with a diffusive exchange
with a porous matrix was first proposed by Tang et al. (1981). The system under study consisted
of a single fracture represented by two smooth parallel plates with a constant aperture 2b,
embedded in a homogeneous porous rock. This is shown in Figure 1.4 (Tang et al. 1981).

Figure 1.4: Schematic view of the fracture-matrix system.
The transport processes in the single fracture - matrix system can be described by two coupled,
one-dimensional equations, along with the initial and boundary conditions. The equations of
continuity in the fracture and porous matrix are as follows:

with the initial and boundary conditions given by:
( ) 0 0 , = x C
f

( ) ( ) t C t C
in f
= , 0

( ) 0 , =  t C
f

and the diffusion equation in the porous matrix is:

f f
b z
p
p p
f
f
f f
f
C R
z
C
b
D
x
C
D
x
C
u
t
C
R   −


+


+


− =


=
1
2
2

(Eq. 30)
16

with the initial and boundary conditions provided by:
( ) 0 0 , , = z x C
p

( ) ( ) t x C t b x C
f p
, , , =

( ) 0 , , =  t x C
p

where the subscripts f and p, refer to the fracture and pore space of the rock matrix, respectively;
x and z are the coordinates along and perpendicular to the fracture plane, respectively; t denotes
the time; c represents the concentration of the solute; R the retardation coefficient; D is the
coefficient of longitudinal dispersion in the fracture or the coefficient of pore diffusion in the
homogeneous rock matrix; u is the groundwater velocity in the fracture; is the porosity; λ is
the decay constant; and C in(t) describes the concentration of the solute at the origin of the fracture
as a function of time.
The solution to C f in the Laplace domain can be written as:

where t w is the mean water residence time; The diffusive mass-transfer parameter G is defined as
(Mahmoudzadeh et al. 2013, 2016)
Pe is the Peclet number
and
with s being the Laplace variable.


p p
p
p
p
p
C R
z
C
D
t
C
R  −


=


2
2

(Eq. 31)

( )






+ + − 





= S R S G t
Pe
Pe Pe
C C
f w in f
4
1
2
exp
2
exp

(Eq. 32)
Pe = ux/Df, (Eq. 33)
S = s + λ (Eq. 34)

B
D R
G
P P P

=

(Eq. 35)
17

1.3.6 Longitudinal dispersion and Transverse dispersion in porous media

Josselin De Jong (1958) and Saffman (1959) studied longitudinal dispersion in porous media
based on the medium being regarded as a “network” of capillaries. The model consists of a
random, statistically isotropic, network of straight, circular capillaries, several capillaries starting
and finishing at each junction. The case is considered where the Reynolds number of the flow
through the capillaries is small so that Poiseuille flow exists in each capillary or pore.
Saffman (1959) showed that the longitudinal dispersion is a logarithmic function of the Peclet
number𝑃𝑒 =
𝑣𝐿
𝐷 0
, where, 𝑣 is the average velocity in the capillary, L is the length of the capillary,
𝐷 0
is molecular diffusion). Saffman’s results can be sketched by Perkin and Johnston (1963), as
shown in Figure 1.5 (Saffman 1959, modified by Perkins and Jounston 1963).

Figure 1.5: Longitudinal Dispersion Coefficient for a random network of capillaries.
In porous media, the longitudinal and transverse dispersion coefficient can be calculated using
tracer method by injecting a tracer into a media which is saturated with another fluid and
measuring the tracer concentration at the exit end of the model as a function of time.
Brigham (1961) has shown a method for determining the longitudinal dispersion coefficient in
packs of granular material using the plot of
p
p
V V
V V
/
1 / −
=  versus the percent of tracer
concentration, where V is the volume injection and V p is the pore volume of the model. Brigham
investigated the effect of displacing rate, bead size, length of travel, the diameter of bead pack,
and viscosity ratio on miscible displacement in porous media. He used oil and kerosene as
miscible fluids. The porous media were packs of relatively uniform spherical glass beads and
consolidated sandstone cores. The experiments showed that the more significant mixing found in
the small-diameter pack. At a low flow rate, the ratio of D/D 0 = 1/F  approached to 0.7.
Blackwell et al. (1959) and Carman (1956) found a similar result for spherical glass bead packs.
The term 1/F  will commonly vary between 0.15 and 0.7, depending on the lithology of porous
media. At the high flow rate, the dispersion coefficient is characterized by the following relation
for sandstone and glass bead packs, D/D 0 = a.1.2P e, where a is a function of the viscosity and the
inhomogeneity of the porous medium. As a result, the mixing zone increases with velocity.
Blackwell (1962) investigated the microscopic mixing process. In the mixing process, one fluid is
displaced from a porous medium by a second fluid, which is miscible with the first. Longitudinal

18

and transverse dispersion coefficients were measured for a range of rates, sand sizes, fluid
viscosities, and column lengths. The primary fluids system used was distilled water and a 0.05-
normal aqueous solution of potassium chloride. They obtained fluids of higher viscosity by adding
glycerin to the primary fluids. The results for investigation of the effect of rate and viscosity on
the transverse dispersion coefficient showed that the transverse dispersion is dominated by
molecular diffusion for the value of Pe less than 0.4. For Pe greater than 4, D T/D 0 increases with
Pe. Convective processes dominate the transverse dispersion. On the other hand, the longitudinal
dispersion coefficient was about 24 times the transverse dispersion coefficient for 20-30 mesh
Ottawa sand and 40-200 mesh silica sand at a high flow rate. The results for the investigation of
the effect of sand size and column length on the transverse dispersion coefficient are shown in
Figure 1.6. In the region where the convective phenomena is dominant for all sands (Pe > 6), the
transverse dispersion coefficient increases as the particle radius decrease.

Figure 1.6: Effect of rate and viscosity level on dispersion in 40-200 mesh sand
on transverse dispersion coefficient.
According to Blackwell (1962), transverse dispersion is affected by two mechanisms: the lateral
motion of a particle along these streamlines, combined with the exchange of particles between
streamlines by molecular diffusion both within the individual flow channels and at junctures.
On the other hand, the profile of tracer concentration along a line perpendicular to the direction
of fluid movement will be observed as the typical S-shaped. Similarly, the transverse dispersion
coefficient can be determined by the plot of the percent tracer concentration versus the distance
from 50 percent tracer concentration X (Perkin and Johnston 1963). Typical longitudinal and
transverse dispersion profile is shown in Figure 1.7.

Figure 1.7: Typical longitudinal and transverse dispersion profile.
19

The longitudinal and transverse dispersion coefficient can then be calculated with
2
10 90
625 . 3





 −
=
X X
L
v
D
T
(Eq. 37)
By the dimensional analysis, in general, the coefficient of dispersion is a function of the Peclet
number, P e = v dp/D 0, where d p is the mean grain size (Bear, 1972). Perkins and Johnston (1963),
Fried and Comparnous (1971) and Bear (1972) identified flow regimes based on the dependence
of longitudinal dispersion and transverse dispersion in porous media on Peclet number.
Region I. (P e < 1). D L/D 0 = 1/  . In this region, molecular diffusion dominates the dispersion
process. The variable

is also known as the tortuosity factor. Its value has been found for packs
of unconsolidated material to be about 1.5. This means that the ratio D L/D 0 approaches about 0.67
instead of unity (Bear, 1972). For consolidated porous media, e.g., sandstones, 1/

has been
found to be much higher than 1.5 (Brigham et al., 1961).
Region II. (P e ~ 0.4 - 5). D L/D 0 is not reported. In this region, the effects of mechanical dispersion
already appear and are considered of the same order of magnitude as molecular diffusion.
Region III. (P e ~ 5 - 300). D L/D 0 ~ 0.5Pe b(where 1 < b < 1.2). In this power-law regime,
mechanical dispersion is the predominant mechanism, but molecular diffusion effects are still
visible.
Region IV. (P e ~ 300 - 105). D L/D 0 ~ 1.8Pe. Here, pure convective or mechanical dispersion
dominates the dispersion mechanisms, and the dispersion coefficient is linearly related to Pe.
Region V. (P e > 105). D L/D 0 is not reported. In this region, turbulence effects appear on the
dispersion process. The corresponding fluid velocities are rather large and fall outside the validity
of Darcy’s Law.
D L and D T are approximately equal to P e < 1, i.e., molecular diffusion dominates the dispersion
process. As P e increases through the range 1 < P e < 1000, both D L and D T increase with P e, although
D L increases slightly faster (Hackert, 1996).
Transverse mixing, although smaller than longitudinal mixing, is still very important in
contaminant transport, as it acts to smooth solute concentration fluctuations and dilute the plume
(Kapoor and Gelhar, 1994; Kittanidis, 1994). More recent studies have focused on the importance
of transverse dispersion in the mass transfer of contaminants across the capillary fringe (Klenk
and Grathwohl, 2002) and in reactive transport during natural attenuation of continuously
emitting contaminant sources (Cirpka et al., 2006) [B. Bijeljic, 2006].
B. Bijeljic (2006) provided a fundamental understanding of how pore structure controls
dispersion using a measure of the transit times that particles spend when moving between
2
10 90
625 . 3





 −
=
 
vL D
L

(Eq. 36)
20

neighboring pores. The pore network model representing Berea sandstone was used to describe
flow and diffusion at the pore scale (um) to compute the longitudinal dispersion coefficient at a
larger scale (cm to m). The solute transport can be viewed as a series of steps as the solute moves
between pores. The probability that a particle just arrived at a pore will subsequently first reach
the nearest pore space in a time increment t to t + dt is

where t1 is the mean advective transit time, t 2 is a late-time cut-off for t 2>t>t 1. Approximately,
( )
( ) 

+ − t
t t ~
where β is a parameter characterizing the porous medium heterogeneity. The
results of comparison of both longitudinal and transverse asymptotic dispersion coefficients
obtained from the pore-scale network model with experimental results in previous studies
(Pfannkuch 1963; Kandhai 2002; Khrapitchev 2003; Seymour 1997; Stöhr 2003; Han 1985;
Harleman & Rumor 1963, Hassinger & von Rosenberg 1968; Gunn, D. J. and Pryce 1969; Seymour
1997; Khrapitchev 2003) is shown as Figure 8. The predictions are in good agreement with the
large body of experimental data. As a result, the macroscopic behavior of dispersion in porous
media as a function of Pe is explained.
For longitudinal dispersion, molecular diffusion is the only mechanism of fluid mixing at the low
Pe number. The ratio D L/D 0 being smaller than unity because of the reducing mean free path
related to the porous medium matrix. At Pe = 0.1, the transition regime starts, where both
diffusion and advection have an impact on mixing. At Pe=10 advection starts to dominate in the
dispersion process. The particles are flowing fast through the throats but still have enough time
to sample some of the low-velocity regions near the walls. The best fit of model results in the
regime 10 < Pe < 400 for DL ~ Pe1.2. For Pe > 400, the dependence of D L with Pe is linear due to
purely advection dominated, in which particles have not sufficient time to move significantly by
molecular diffusion. Comparison of the network model results for longitudinal and transverse
dispersion coefficients with experimental data is shown in Figure 1.8.

Figure 1.8: Comparison of the network model results for longitudinal (upper solid line) and transverse
(lower solid line) dispersion coefficients with experimental data.
( ) ( )
( )
2
/ 1
1
/
t t
e t t r A t
− + −
+ =



(Eq. 38)
21

Many authors have agreed that dispersion coefficients are scale-dependent, in which, field-scale
dispersion (megascopic dispersion) is much larger than laboratory-scale dispersion
(macroscopic dispersion). The term "field-scale" refers to vertical dimensions of tens to hundreds
of meters and horizontal dimensions of tens of meters to kilometers. In the laboratory, the scale
is usually up to one meter (Gelhar 1985).
Warren (1964) investigated the effects of permeability variations such as the scale of
heterogeneity and distribution function of permeabilities on macroscopic dispersion. The
influence of macroscopic dispersion on laboratory experiments was also studied. A miscible
displacement process in a three-dimensional, heterogeneous porous medium can be evaluated
by means of experimental computation based on the Monte-Carlo model. The permeabilities are
assumed to be log-normal distribution, while the porosities are considered to be a normal
distribution. The results shows that the effect of porosity variations on macroscopic distribution
is second-order with respect to the effect of permeability variations. Laboratory experiments do
not yield a valid measure of macroscopic dispersion due to the scale of heterogeneity.
Lake and Hirasaki (1981) explained the large field-scale dispersion observed in stratified porous
media based on the Taylor dispersion theory. The porous medium was assumed to consist of two
homogeneous layers of contrasting permeability (k), porosity (  ), and thickness (h) as shown in
Figure 1.9.

Figure 1.9: Definition of two-layer porous media; and (b) schematic illustration
of Taylor's dispersion in media.
Transverse dispersion governs miscible displacement in two-layer media. If no transverse
dispersion, the tracer response curve manifests the heterogeneous character with longitudinal
dispersion superimposed on each layer's breakthrough. When transverse dispersion is
moderate, some of the solute molecules move through the layer boundary, causing a distortion of
the tracer response curve. For large transverse dispersion, the effluent history no longer
manifests heterogeneous character, and the medium behaves as a single-layer medium with
increased dispersion - effective longitudinal dispersion. This effective dispersion coefficient is
larger than the corresponding homogeneous (intralayer) dispersion coefficient. The stratification
tends to increase the effective dispersion for two layers because the difference in the velocity of
the two-layer causes a solute to travel with different velocities in each layer.
Transverse dispersion between layers may be characterized by transverse dispersion number,
NTD, is defined as:
22

Where, L is the system length, H is the system transverse dimension, α T is the transverse
dispersivity,

F is the porosity contrast (the ratio of porosity of two layers), and F k is the
permeability contrast (the ratio of permeability of two layers). When NTD <0.2, the system
behaves as a two-layer system with no transverse dispersion. When NTD >5, the system behaves
as if it were a single layer with a breakthrough curve is given by

where ( )
1
/
−
Pe
N is the effective inverse Peclet number, x D is the dimensionless longitudinal
coordinates, and Q D is the dimensionless cumulative injection.
Moreover, Lake and Hirasaki (1981) proposed a multilayer combination method to obtain
layered reservoir models from core data and estimate of field-scale dispersion.
Most of the enhanced oil recovery (EOR) processes used today were first proposed in the early
1970s. Enhanced oil recovery is a tertiary phase in oil production, which is displacement
processes involves injecting a fluid (such as gases, polymer, surfactant, etc.) into an oil reservoir
to increase oil recovery. Miscible displacement depends strongly on dispersion processes taking
place in the oil reservoir. The dispersion coefficient is an important parameter used to improve
the design of solvent or surfactant slug size for a miscible flood process.
Kinzel (1989) investigated the influence of an oil phase saturation on dispersion in Berea
sandstone core using chloride ions as a tracer to provide quantitative information useful to the
prediction of the motion of reservoir fluids in enhanced oil recovery. The analytical solution of
the advection-dispersion equation presented by Ebach and White (1958) was used to determine
the dispersion coefficient. Dispersion results in Berea Core for both single and two-phase flow at
various flow rates are shown in Figure1.10. The results showed that low oil saturation has no
effect on the dispersion in the water phase, but high oil saturation (~55%) results in a dispersion
coefficient of 2.2 times higher than in water alone. It is found that the presence of an immiscible
phase has a pronounced effect on the dispersion coefficient in porous media.

k
T
TD
F
F
H H
L
N


=

(Eq. 39)

( )
















−
−
− 
1 2
1
2
1
/
Pe D
D D
N Q
Q x
erf C

(Eq. 40)
23

Figure 1.10: Dispersion results in Berea Core for both single and two-phase flow at various flow rates (Kz is
the longitudinal dispersion,D is the molecular diffusion coefficient,
Uz is the interstitial velocities, r is the particle radius).
M. Greaves et al. (1982) described a series of experiments conducted in unconsolidated porous
media to study the effects of velocity, viscosity ratio, surfactant concentration, salinity gradient,
alcohol concentration, permeability, phase-type, phase behavior, phase saturation and gravity on
the value of the dispersion coefficient that concerned with improving the efficiency of enhanced
oil recovery technique. The results showed that the dispersion coefficient increases as the
permeability decreases. The rate of dispersion is independent of the viscosity ratio in the range
of 1.0 to 0.9. The dispersion coefficient is proportional to the 1.2th power over the range of
velocities studied. The results of dispersion increase as the saturation of immobile oil phase
decreases. We observe no detectable effect on dispersion due to surfactant concentration or the
salinity gradient.

1.3.7 Dispersivity

Theis (1962, 1963) realized that the variations in dispersivity values according to scale
experiments could be due to the wide distribution of permeabilities and flow velocity through
aquifers.
Fried et al. (1972) presented a few dispersivity values equal to 0.1 – 0.6 m for local scale (for each
layer of the aquifer), 5 – 11 m for the global scale (total aquifer thickness). Fried found no scale
effect on the transverse dispersion coefficient and suggested that its value could be obtained from
laboratory results (Moazed 2009; C. Rajanayaka et al. 2003).
Oakes and Edworthy (1977) conducted two-well pulse and radial injection experiments in a
sandstone aquifer and found that the obtained dispersivity values for total depth to be 2 to 4 times
that in the values for discrete layers.
24

Klotz et al. (1980) from both laboratory and field experiments in loose soil found that the
longitudinal dispersion coefficient increases linearly with longitudinal dispersivity and mean
flow velocity with the exponent of the velocity in the range of 1.07 to 1.1. The exponent of the
velocity for transverse dispersion is the same as longitudinal dispersion. It is showed that
longitudinal dispersivity is dependent on the scale of experiments, and the ratio of longitudinal
dispersivity and transverse dispersivity is approximately equal to a value of 20.
Wood (1981) proposed a method for determining values of dispersivity for large-scale regional
aquifer system using the concept of hydrochemical facies. Concentration contours in the space of
conservative solutes or isotopes were obtained. Curves relating concentration contour and pore
volume of fluids are compared to the one-dimension analytical solution of the dispersion
equation developed by Ogata and Banks (1961) to determine dispersivity:
In which, C is the output concentration, C 0 is the input concentration, x is distance, t is time, v is
the velocity, and D L is the longitudinal dispersion coefficient.
Watson and Jones (1981) determined the value of dispersivity for Bungendore fine sand by
matching numerically obtained profiles with those obtained from an adsorption experiment
under constant concentration conditions. A value of 0.0001 m was found to be satisfactory
(Reddy, 2015).
A series of laboratory column and field tracer tests to investigate the longitudinal dispersivity in
a sandy stratified aquifer were conducted by Pickens and Grisak (1981). The average longitudinal
dispersivity of 0.035 cm was obtained for three laboratory tracer tests with a packed column with
a length of 30 cm. The analysis of the concentration history of Single - well tracer test showed
longitudinal dispersivity of 3 cm and 9 cm for flow length of 3.13 and 4.99 m, respectively. The
result of Interwell tracer test showed that dispersivity equals 50 cm with the distance between
wells is 8 m. It was observed that the scale dependency of longitudinal dispersivity for the study
site has a relationship of αL = 0.1 L, L is the mean travel distance.
Arya et al. (1988) investigated the behavior of dispersivity in field-scale miscible displacement
using numerical simulation in two dimensional, randomly heterogeneous medium . The paper
reported the log-log least-squares fits of the longitudinal dispersivity for field data
α L = 0.229L0.755
and for all the experiment and field data
α L = 0.044L1.13
which based on data from numerous miscible flood experiments (L is the measured length scale).
It is observed that dispersivity is a scale-dependent property as shown Figure 1.11. This paper
also showed that the magnitude of megascopic dispersivity depends on heterogeneity, aspect
ratio, and weakly on diffusion.

( ) ( )


















+








+










−
=
t D
vt x
Erfc
D
xv
Exp
t D
vt x
Erfc
C
C
L
L
L
2
1
2
1
2
1
0

(Eq. 41)
25

Figure 1.11: Field and laboratory dispersivity data.
Kelkar (1988) studied the effect of small scale (or core scale) heterogeneities on the effective
dispersivity value in field-scale using Amoco's finite element simulator. It is assumed that the
permeability varies in a log-normal distribution. The effective dispersivity was then calculated by
matching the concentration profiles to the solution of the one-dimensional convection-dispersion
equation in the Laplace space under the identical reservoir and flow conditions. It is observed
that the effective dispersivity is affected by the degree of heterogeneity, the average length of
heterogeneity, the length of the system, and the manner in which the permeability values are
spatially distributed. The effect of heterogeneity becomes significant if the coefficient of variance
is greater than 0.4. The impact of dimensionless scale length on effective dispersivity is
insignificant for dimensionless scale values less than 0.01. The effective dispersivity increases
almost linearly with the length of the system for constant dimensionless scale length. Kelkar
(1988) also proposed a simple equation used to qualitatively estimate the effects of small-scale
heterogeneities on the effective dispersivity αe on porous media:

In which, α is rock dispersivity, V is the Dykstra-Parsons' coefficient, SL is the dimensionless
characteristic length, Pe is the Peclet number.
Menzie (1989) presented a series of displacement experiments for measuring dispersion and
dispersivity on Berea sandstone, Saint Peter sandstone, unconsolidated sand packs and plug
samples from oil field cores. During the miscible displacements, the effluent concentrations of the
displacing fluid were determined. The longitudinal dispersion coefficient, DL, was then calculated
from the equation proposed by Perkins and Johnston (1963). Longitudinal dispersivity, αL was
then derived from:

2 . 1 3 . 2 6 . 2
9 . 7 1 Pe SL V
e
= 





−



(Eq. 42)
26

The experimental results showed that for the Berea sandstone cores, both the dispersion
coefficient and the dispersivity increased with the permeability of the cores under some specific
flow. A generalized equation governing this relationship can be written as:
The empirical correlation of dispersion coefficients and viscosity ratio is obtained:
where
D s is the standard dispersion coefficient,
D L is the dispersion coefficient at any given viscosity ratio,
µ r is the viscosity of the displaced fluid,
µ d is the viscosity of the displacing fluid,
R v = (µ r/µ d) is the dynamic viscosity ratio, n is the empirical constant exponent. Menzie (1989)
concluded that the standard dispersion coefficient could be used to predict the slug size
requirement to recover a certain amount of solute in a solvent flooding process.
Wierenga and Van Genuchten (1989) conducted solute transport experiments on several small
and one large column packed with the same sandy soil material. Chloride and tritium were used
as tracers for water movement. Tracer response curves were then compared with solutions of the
advection-dispersion equation. The obtained dispersivity values ranged from 1 cm for the small
columns to about 5 cm for the 6m long column.
Young (1990) reported the feasibility of using physical dispersion relationships to
account for the fluid mixing associated with viscous fingering. The dispersivity was
dependent upon viscosity gradient by this form:
where α* is the dispersivity for constant viscosity displacement, β is a constant, µ is the dynamic
viscosity, and x is the direction of flow. Dispersivity calculated from the above equation gives a
reasonable match with the experimental data. In addition, a numerical method for simulating
dispersion effects for systems with phase changes was developed.
Neuman (1990) excluded the data with an apparent length scale more significant than 3500 m
for theoretical reasons. Although the data is widely scattered, the regression analysis has the best
fit line with a narrow 95% confidence bands. The equation of the best fit line is:
v D
L L
= 

(Eq. 43)
( ) c k aLog D
L
+ =

(Eq. 44)
( )
n
d r s
−
=   / D D
L

(Eq. 45)

x
Ln
e


=


 
*

(Eq. 46)
27

where α is the dispersivity, and L s is the field scale (Reddy, 2015).
A review of dispersivity observations from fifty- nine different field sites conducted by Gelhar
(1992) pointed out that longitudinal dispersivity increases linearly with the observation scale as
shown in Figure 1.12. The longitudinal dispersivities ranged from 10
-2
to 10
4
m for scales ranging
from 10
-1
to 10
5
m.

Figure 1.12: (a) Longitudinal dispersitivity versus the scale of observation identified by the type of
observation and type of aquifer; (b) Ratio of longitudinal dispersivity and transverse dispersivity.
In all of the cases, the transverse dispersivity is about one-third of longitudinal. Gelhar (1992)
concluded that the variations in dispersivity reflect the influence of differing degrees of aquifer
heterogeneity at different sites.
H. Li (1995) investigated the influence of the test scale on effective porosity and longitudinal
dispersivity by using field tracer tests and laboratory tracer tests. Br- and Cl- were selected as
tracers in these experiments. The results are shown in Figure 1.13.
For the field tracer test, the velocity distribution of the flow field and layer permeability were
calculated by using the finite element method (FEM),the advection-dispersion equation was then
used to analyze, numerically, the tracer test results for the determination of dispersivity. For the
laboratory tracer test, the dispersivity of rock cores was calculated by matching the tracer
breakthrough curve to the solution of the one-dimensional convection-dispersion equation given
by Ogata and Banks (1961). As the test scale increases from laboratory scale (several
centimeters) to field scale (several thousand meters), effective porosity decreases and
longitudinal dispersivity increases due to the heterogeneity of porous media.

46 . 1
0175 . 0
s
L = 

(Eq. 47)
28

Figure 1.13: (a) Relationship between effective porosity/porosity and test scale; (b) Relationship between
longitudinal dispersivity and test scale.
Mahadevan (2002) investigated three types of dispersion in permeable media to the estimation
of true dispersivity in field-scale permeability. There are (1) echo dispersivity – estimated from
Single-well tracer tests, (2) transmission dispersivity - estimated from Interwell-tracer tests and
(3) local dispersivity – observed at a point as tracer flows past it. A collection of Single-well tracer
test data from Majoros and Dean (1980) was analyzed to estimate echo dispersivity using the
UTCHEM simulator. The estimated echo dispersivities agreed with field-measured dispersivity.
UTCHEM simulator was also used to generate a layered permeable medium and a stochastic
permeability field. Both Interwell-tracer tests and Single-well tracer tests were conducted on
these models. As a result, transmission dispersivity should be greater than echo dispersivity.
Moreover, numerical simulation results showed that the local dispersivity is always less than the
transmission dispersivity and higher than the echo dispersivity depending on the amount of
heterogeneity, the autocorrelation structure of the permeability field, and vertical permeability.
Makuch (2005) reviewed longitudinal dispersivity obtained from laboratory experiments,
aquifer tests, and modeling results conducted by 109 different authors for different types of
geological media. The results are shown in Figure 1.14 and Figure 1.15. Dispersivity values for
consolidated media were subdivided as basalts, granites, sandstones, and carbonate rocks, while
unconsolidated sediments were divided into three reliability classes. The relationship of
longitudinal dispersivity with measurement scale for specific types of geological media was
described by the empirical power law:
where a is longitudinal dispersivity [L], c a parameter characteristic for a geological medium [L1-
m], L the flow distance [L], and m the scaling exponent.
The results showed that scaling exponent for consolidated and unconsolidated geological media
varied between 0.40 - 0.92, and 0.44 - 0.94, respectively.

m
L
L c. = 

(Eq. 48)
29

Figure 1.14: Relationship of longitudinal dispersivity to scale of measurement for consolidated media.

Figure 1.15: Relationship of longitudinal dispersivity to scale of measurement for unconsolidated sediments.
Jui-Sheng Chen (2006) presented a two-dimensional mathematical model to investigate the effect
of scale-dependent dispersion on non-axis symmetrical solute transport in a radially convergent
flow field. The Laplace transform finite difference technique is applied to solve the proposed
equation in cylindrical coordinates. The comparison between the breakthrough curve obtained
from the scale-dependent dispersivity model and from the constant dispersivity model can
produce the same curves of averaging concentration only at observation distance near the
extraction.
Moazed (2009) studied the dependence of dispersivity on the thickness of aquifer on a physical
model using a rectangular Plexiglas tank. Sodium chloride with an electrical conductivity of 14
dS/m was selected as a conservative tracer. The dispersivity of aquifers was determined. The
results showed that the dispersivity of sandy porous media is independent of particle size. In
homogeneous sandy aquifers with coarse and medium particle size, the dispersivity is
independent of aquifer thickness.
Baise (2013) calculated the longitudinal dispersivities using the self-potential signals. They
proposed several approaches to deduce the longitudinal Dispersivity in a sand column during
experiments of tracer tests, by measuring the output fluid concentration and self-potential signals
in the electrodes inserted into sands. Values of longitudinal Dispersivity calculated at the top of
the column by means of the analytical solution is applied to the breakthrough obtained by a
30

conductivity meter, coupled and uncoupled hydrogeological inversion approach is applied on the
self-potential signals (Reddy 2015).

1.3.8 Summary

This review has presented a brief evaluation of hydrodynamic dispersion phenomena and
dispersivity in porous media. Hydrodynamic dispersion governs the spreading of miscible fluids
in porous media. There are two mechanisms that contribute to the hydrodynamic dispersion:
molecular diffusion - characterized by the random motion of molecules due to the gradient of
concentration and mechanical dispersion – derived from the velocity gradient present in each
pore channel and tortuosity and pore size difference along a given flow path. At a low flow rate,
molecular diffusion dominates. At high flow rate, mechanical dispersion is the dominant process
controlling dispersion phenomena. The dispersion coefficient is proportional to mean flow
velocity and dispersivity. As a result, dispersivity is a measure of dispersion. Dispersivity is
usually estimated in both field-scale and laboratory-scale by matching the effluent history of a
conservative tracer to the solution of the convection-dispersion equation. The past studies
indicated that the calculated dispersivity values from laboratory do not coincide with the values
obtained in the field. The reason for that difference may include the effects of heterogeneities in
reservoir properties and reservoir stratification.

31

2 Chapter 2: Mathematical Modeling Coupling of the Diffusivity Equation
and the Concentration Equation

2.1 Introduction

To investigate uncertainty in reservoir characterization using tracers, it is important to
understand the fundamental basics of mass transport, advection, and diffusion and
dispersion equations as well as the diffusivity equations. These equations will be the
underlying basis for a look at coupling the relationship diffusivity equation and the dispersion
equation, which is the mathematical basis of this research.

The law of the conservation of mass or principle of mass conservation states that for any
closed system to transfer of matter and energy, (i.e., systems that do not allow mass transfer
in or out of the system), the mass of the system over time will remain constant. Therefore, the
quantity of mass is conserved over time.
It is worth noting that because mass can neither be created nor destroyed, it may be
rearranged in space or entities where it may be changed in a form such in the case of chemical
reactions.

2.2 Relevance of the Transport Equation and Tracer Concentration

2.2.1 Overview

The mass transport equation consists of three components concentration C, mass flow rate
M and flux J. These components are defined in terms of mass [m], dimension or length [L] and
time [t].
Concentration is defined as the relative amount of a given substance or solute contained
within a solution or in a particular volume of space; it is the quantity or amount of solute per
unit volume of solution. Hence, concentration C is mass per unit volume [m/L
3
].
Mass flow rate M is defined as the mass per unit time [m/t]. The steady and continuous flow
of the mass into a particular defined area is flux J. Flux is defined as Mass flow rate through
the unit area [m/L
2
t]

32

Table 2.1 shows the summary of definitions, symbols and units for the transport equation.
These definitions are particularly useful in the case of tracers, conservative or passive tracers
remain constant in fluid, whereas reactive tracers (such as compounds undergoing a mutual
chemical reaction) grow or decay with time.

Table 2.1: Summary of the Definitions for Transport Equation.
Term Symbol Definition Units
Concentration C Mass per unit
volume
m/L3
Mass flow rate M Mass per unit time m/t
Flux J The mass flow rate
through unit area
m/t L2

To derive the transport equation, we will look at the mass balance for a control volume where
the transport occurs only in the x-direction, with the assumption that the transport equation
applies to a conservative/ passive tracer, the control volume is constant with time, and the
flux J can be fluid, dispersion, etc.).
If we assume the diagram in Figure 2.1 shows the control volume, then the mass entering the
control volume and leaving the control volume will be depicted as follows:

Figure 2.1: Mass In and Out of Control Volume.
Hence,
Because V is the volume,

33

dividing by V:

The change in flux is changing in the x-direction as a gradient of
𝜕𝐽
𝜕𝑥
, so

Substituting Jout gives

𝜕𝐶
𝜕𝑡
=
𝐴 𝑉 ( 𝐽𝑖𝑛 − ( 𝐽𝑖𝑛 +
𝜕𝐽
𝜕𝑥
Δx)) (Eq. 53)

Because Δx= V/A
Then A/V is 1/ Δx
Substituting this in the equation yields:

𝑉
𝜕𝐶
𝜕𝑡
= 𝐴 ∙ 𝐽𝑖𝑛 − 𝐴 ∙ 𝐽𝑜𝑢𝑡
(Eq. 49)

𝜕𝐶
𝜕𝑡
= 𝐴 /𝑉 ∙ 𝐽𝑖𝑛 − 𝐴 /𝑉 ∙ 𝐽𝑜𝑢𝑡
(Eq. 50)

𝜕𝐶
𝜕𝑡
=
𝐴 𝑉 ( 𝐽𝑖𝑛 − 𝐽𝑜𝑢𝑡 )
(Eq. 51)

𝐽𝑜𝑢𝑡 = 𝐽𝑖𝑛 +
𝜕𝐽
𝜕𝑥
Δx

(Eq. 52)

𝜕𝐶
𝜕𝑡
=
1
Δx
( 𝐽𝑖𝑛 − 𝐽𝑖𝑛 −
𝜕𝐽
𝜕𝑥
Δx)
(Eq. 54)

34

Expressing the above in 3- dimensional space

2.2.2 The Advective Flux

Advection is defined as the transfer of matter by the flow of a fluid, usually when referring to
advection; it is referred to as horizontal transfer of matter by the fluid flow.
The model in Figure 2.2 depicts advective flux; advection occurs only towards one direction
in a time interval.

Figure 2.2: Control Volume Element.
We will assume that particles move only in the positive direction of x, and Δx is the distance
a particle can pass in a time interval of Δt.
In this case, the number of particles (i.e., mass) moving from control volume I to control
volume II in the time interval Δt can be calculated using:
N is defined as the number of particles, i.e., mass passing from volume I to volume II in time
interval Δt , C is the concentration of any material dissolved in water in control volume I

𝜕𝐶
𝜕𝑡
= −
𝜕𝐽
𝜕𝑥

(Eq. 55)

𝜕𝐶
𝜕𝑡
= − (
𝜕 𝐽 𝑥 𝜕𝑥
+
𝜕 𝐽 𝑦 𝜕𝑦
+
𝜕 𝐽 𝑧 𝜕𝑧
) (Eq. 56)

35

(M/L
3
) , Δx is the distance [L], and A is the cross-section area between the control volumes
(L
2
).

Dividing by Δt will give the number of particles passing from I to II in unit time:

dividing by cross-sectional area (A):

where the advective flux 𝐽 𝐴𝐷𝑉 is the number of particles passing from I to II in unit time per
unit area.
Hence, the Advective Flux:

𝑁 = 𝐶 ∙△ 𝑥 ∙ 𝐴 (Eq. 57)
𝑁 /Δt = ( 𝐶 ∙△ 𝑥 ∙ 𝐴 ) /Δt (Eq. 58)
𝐽 𝐴𝐷𝑉 = 𝑁 /𝐴 Δt = ( 𝐶 ∙△ 𝑥 ) /Δt (Eq. 59)

𝐽 𝐴𝐷𝑉 = lim
△𝑡 →0
( △ 𝑥 /Δt ∙ C ) = C
∂x
∂t

(Eq. 60)

𝐽 𝐴𝐷𝑉 = C
∂x
∂t

(Eq. 61)

36

2.2.3 The Dispersive Flux

The Dispersive flux can be analyzed with the simple conceptional model as described in
Figure 2.3; in this case, dispersion occurs towards both directions in a time interval.

Figure 2.3: Representative Element.
Δx is the distance, which a particle can pass in a time interval of Δt. The assumption is that
the particles move in the positive and negative directions.
In addition, a particle doesn’t change its direction during the time interval of Δt and that the
probability of moving to positive and negative x-direction is equal for all particles. This means
that the probability to move to positive and negative x directions is 50% for all particles.
Looking at both components of the dispersive mass transfer in Figure 2.4, one from control
volume I(q1) to control volume II, and the second from control volume II to control volume I
(q2).

Figure 2.4: Dispersive Mass Transfers.

𝑛 1 = 0.5 ∙ 𝐶 1
△ 𝑥 ∙ 𝐴

(Eq. 62)

37

If we divide both sides by time

and defining the relationship between Concentrations C as

Substituting 𝐶 2
we get:

The Number of particles passing from I to II in unit time is

If we divide both sides by Area A to get Flux which is the number of particles passing from I
to II in unit time per unit area:

𝑛 2 = 0.5 ∙ 𝐶 1
△ 𝑥 ∙ 𝐴

(Eq. 63)

𝑁 = 𝑛 1
− 𝑛 2

(Eq. 64)

𝑁 = 0.5 ∙△ 𝑥 ∙ 𝐴 ( 𝐶 1
− 𝐶 2
)

(Eq. 65)

𝑁 /△ 𝑡 = ( 0.5 ∙△ 𝑥 ∙ 𝐴 ( 𝐶 1
− 𝐶 2
) ) /△ 𝑡

(Eq. 66)

𝐶 2
= 𝐶 1
+
𝜕𝐶
𝜕𝑥
△ 𝑥

(Eq. 67)
𝑁 /△ 𝑡 = ( 0.5 ∙△ 𝑥 ∙ 𝐴 ( 𝐶 1
− (𝐶 1
+
𝜕𝐶
𝜕𝑥
△ 𝑥 )) ) /△ 𝑡 (Eq. 68)

𝑁 /△ 𝑡 = ( 0.5 ∙△ 𝑥 ∙ 𝐴 ( 𝐶 1
− 𝐶 1
−
𝜕𝐶
𝜕𝑥
△ 𝑥 ) ) /△ 𝑡

(Eq. 69)

𝑁 /△ 𝑡 = ( −0.5 ∙△ 𝑥 ∙ 𝐴 𝜕𝐶
𝜕 𝑥 △ 𝑥 ) /△ 𝑡

(Eq. 70)

38

Hence the equation for dispersive flux is :

2.3 An overview of Advection-Dispersion and Dispersion

The general transport equation, advective flux, and dispersive flux were derived in the
previous sections.
Combining them yields:

General transport equation:

The advective flux:

The Dispersive Flux

𝐽 𝐷𝐼𝑆𝑃 = 𝑁 /𝐴 △ 𝑡 = −0.5 ∙△ 𝑥 ∙
𝜕𝐶
𝜕𝑥
△ 𝑥 /△ 𝑡
(Eq. 71)

𝐽 𝐷𝐼𝑆𝑃 = 𝑁 /𝐴 △ 𝑡 = −0.5 ∙ ( △ 𝑥 )
2
∙
𝜕𝐶
𝜕𝑥
/△ 𝑡

(Eq. 72)

𝐽 𝐷𝐼𝑆𝑃 = −( 0.5 ∙ ( △ 𝑥 )
2
/△ 𝑡 )∙
𝜕𝐶
𝜕𝑥

(Eq. 73)

𝐽 𝐷𝐼𝑆𝑃 = −𝐷 ∙
𝜕 𝐶 𝜕𝑥

(Eq. 74)

𝜕𝐶
𝜕𝑡
= −
𝜕𝐽
𝜕𝑥

(Eq. 75)

𝐽 𝐴𝐷𝑉 = C
∂x
∂t

(Eq. 76)

39

Defining J as the sum of the advection and dispersion flux:

Substituting J in the general transport equation:

substituting:

We know that the term
∂x
∂t
is velocity u in the x-direction hence,

where:

𝐽 𝐷𝐼𝑆𝑃 = −𝐷 ∙
𝜕𝐶
𝜕𝑥

(Eq. 77)

𝐽 = 𝐽 𝐴𝐷𝑉 + 𝐽 𝐷𝐼𝑆𝑃

(Eq. 78)

𝜕𝐶
𝜕𝑡
= −
𝜕 𝜕𝑥
( 𝐽 𝐴𝐷𝑉 + 𝐽 𝐷𝐼𝑆𝑃 )

(Eq. 79)

𝐽 𝐴 𝐷𝑉
= C
∂x
∂t

(Eq. 80)

𝐽 𝐷𝐼𝑆𝑃 = −𝐷 ∙
𝜕𝐶
𝜕𝑥

(Eq. 81)

𝜕𝐶
𝜕𝑡
= −
𝜕 𝜕𝑥
(
∂x
∂t
∙ C ) −
𝜕 𝜕𝑥
(−𝐷 ∙
𝜕𝐶
𝜕𝑥
)

(Eq. 82)

𝜕𝐶
𝜕𝑡
= − 𝑢 𝜕𝐶
𝜕𝑥
+ 𝐷 ∙
𝜕 2
𝐶 𝜕 𝑥 2

(Eq. 83)

40

𝜕𝐶
𝜕𝑡
is concentration over time with units [M/L
3
T]
−𝑢 𝜕𝐶
𝜕𝑥
velocity time concentration over space [L/T ]*[M/L
3
*L] =[M/(L
3
*T)]
𝐷 ∙
𝜕 2
𝐶 𝜕 𝑥 2

[M/L
3
T]
The Advection- Dispersion Equation for non-conservative materials

The advection-dispersion equation for non-conservative or reactive materials can be added
to account for reaction kinetics can be determined by adding the term ∑ 𝑘 ∙ 𝐶 [M/L
3
T]

In addition, external loads and sinks can be added including external loads; other sources and
sinks which must be given in units [M/L
3
T

𝜕𝐶
𝜕𝑡
= − 𝑢 𝜕𝐶
𝜕𝑥
+ 𝐷𝑥 ∙
𝜕 2
𝐶 𝜕 𝑥 2
− 𝑣 𝜕𝐶
𝜕𝑦
+ 𝐷𝑦 ∙
𝜕 2
𝐶 𝜕 𝑦 2
− 𝑤 𝜕𝐶
𝜕𝑧
+ 𝐷𝑧 ∙
𝜕 2
𝐶 𝜕 𝑧 2

(Eq. 84)

𝜕𝐶
𝜕𝑡
= − 𝑢 𝜕 𝐶 𝜕𝑥
+ 𝐷𝑥 ∙
𝜕 2
𝐶 𝜕 𝑥 2
− 𝑣 𝜕𝐶
𝜕𝑦
+ 𝐷𝑦 ∙
𝜕 2
𝐶 𝜕 𝑦 2
− 𝑤 𝜕𝐶
𝜕𝑧
+ 𝐷𝑧 ∙
𝜕 2
𝐶 𝜕 𝑧 2
+∑ 𝑘 ∙ 𝐶

(Eq. 85)
41

3 Chapter 3: Core Flooding Experiment to Determine Longitudinal
Dispersion

3.1 Scope of Work

The objective of the experimentation is to study and investigate core flooding using different
tracer types. Several experiments were conducted using different parameters. The design of
the first experiment was a basic core flooding experiment to determine the concentration in
a Brea core.
3.2 Experiment 1: Measuring Concentration using Core Flooding in Brea Core

3.2.1 Scope of Work

The objective of this experiment is to investigate and study measuring effluent concentration
in a carbonate core using available materials in the lab; in this experiment, a dry core is
saturated with NaCl solution (2%) and displaced by NaCl solution at a higher concentration
(4%).
3.2.2 Experiment Design

Materials:
4% NACL solution
2% NACL solution
Core Length=6" saturated Brea core (12.25 cm)
Core Volume =30 cc
Flow Rate /Pumping Rate =0.2 cc/min

42

3.2.3 Results and Conclusion

The concentration of the NaCl solution vs time is shown in Figure 3.1.

Figure 3.1: Tracer concentration vs Time.

It was observed that the concentration of the NaCl solution was detected at a time expected
later than the calculated. This resulted in a revisit of the experimental design to improve the
design and selection of the tracer.
3.3 Experiment 2: Determining Longitudinal Dispersion Coefficient using Core
Flooding in Kuwait Core Samples

This section presents a series of tracer tests conducted with Kuwait core samples in pulse
mode. The tracer concentration profiles are then matched with the analytical solution of the
convection-dispersion equation to obtain the dispersivity of the cores.
3.3.1 Fickian model

The classical, Fickian advection-dispersion transport equation for conservative tracers (with
no sink/source) is (C. Erin Feehleyand Chunmiao Zheng & Fred J. Molz, 2000)

where,
0
0.5
1
1.5
0 20 40 60 80 100 120
C/Co
Time (min)
C/Co vs Time

𝜙 𝜕𝐶
𝜕𝑡
=
𝜕 𝜕 𝑥 𝑖 (𝜙 𝐷 𝑖𝑗
𝜕𝐶
𝜕 𝑥 𝑗 ) −
𝜕 𝜕 𝑥 𝑖 ( 𝑞 𝑖 𝐶 )

(Eq. 86)

43

C: Solute concentration, [M]/[L
3
]
ϕ: Porosity, [fraction]
Dij: Dispersion coefficient tensor, [L
2
]/[T]
qi: Darcy flux (is the ratio of the flow rate and the cross-sectional area), [L]/[T].
One-dimensional Fickian advection-dispersion transport equation for conservative tracers in
homogeneous porous media (ϕ = const):

D L: The longitudinal dispersion coefficient, [L2]/[T]
v x: The mean pore velocity (is the ratio of the Darcy flux and the porosity) [L]/[T].
If the tracer is injected as pulse mode to the infinite one-dimensional system, the obtained
concentration over space and time is

where
M: the mass of tracer, [M]
A yz: cross-section area of porous media, [L2]
x: the distance between the tracer injection position and observation position [L]
t: time [T].

3.3.2 Capacitance Model

The Fickian advection-dispersion transport is first introduced by Taylor (1953) and Bear
(1961), among others, has been shown to be inadequate to represent field-scale solute
transport in very heterogeneous porous media. Laboratory-scale miscible displacement
experiments showed that these heterogeneities could cause enhanced dispersion, early
breakthrough, and long tailing, which are difficult to model with the Fickian advection-
dispersion equation (Haselowand Greenkorn,1991; Brusseauand Zachara, 1993).

𝜕𝐶
𝜕𝑡
= 𝐷 𝐿 𝜕 2
𝐶 𝜕 𝑥 2
− 𝑣 𝑥 𝜕𝐶
𝜕𝑥

(Eq. 87)

𝐶 ( 𝑥 , 𝑡 ) =
𝑀 𝐴 𝑦𝑧
𝑒 −( 𝑥 −𝑣𝑡 )
2
4𝐷𝐿𝑡 √4𝜋 𝐷 𝐿 𝑡
(Eq. 88)

44

An alternative to the single-porosity advection-dispersion model is the dual-porosity or dual-
domain mass transfer model. The porous media contains the dead-end pore zone in which
solute can migrate by molecular diffusion processes. The solute exchange between the mobile
and immobile zone is represented with the first-order mass transfer equation depends on the
concentration difference between the two phases.
The concept of dual-porosity was first introduced by Coats & Smiths (1964).
The factors affecting mass transfer include sorption, measurement scale, and fluid velocity
(Griffioenet et al. 1998). The larger the rate constant, the faster transport occurs between the
mobile and immobile zones for a given concentration difference.

3.3.3 Calculating Porosity of Rock

Among the three core samples, only Core I has the porosity value given (pore
volume/porosity is 3 mL/0.04). The porosity of two remaining cores was estimated by HTO
concentration profiles obtained from the core flood experiment.
The pore volume (V pore) of the core can be determined by the following equation:

where,
t HTO: mean residence time of HTO in the core
Q: flow rate

where, C is the HTO concentration, and t is the time since tracer injection.
The porosity is the ratio of pore volume and the bulk volume of the core. Results of mean
residence time calculation and pore volume, the porosity of the core samples are given in
Table 3.1 and Table 3.2.
Table 3.1: The mean residence time of tracers in the systems.

𝑉 𝑝𝑜𝑟𝑒 = 𝔨 𝐻𝑇𝑂 𝑄

(Eq. 89)
𝔨 =
∫ 𝐶 𝑡 𝑑𝑡
∞
0
∫ 𝐶 𝑑 𝑡 ∞
0
=
∑ 𝐶 𝑡 ∆𝑡 𝑖 ∑ 𝐶 ∆𝑡 (Eq. 90)

45

Experiment

Tracers
The Mean Residence time in system (min)
Core I (SA-454) Core II (RA-
258)
Core III (SA-
129)
HTO FBA HTO FBA HTO FBA
A HTO+2FBA 350.2 609.8 544.5
B HTO+2,4DFBA 133.5 137.9 317.7 332.5 818.0 735.8
C

HTO+4FBA 73.3 76.4 205.6 214.8

Table 3.2: The calculated porosity of core samples.

Parameters
Core I
(SA-454)
Core II
(RA-258)
Core –III
(SA-129)
EXP B EXP C EXP B EXP C EXP A EXP B
Pore Volume
(mL)

3.3

3.6

17.5

17.0

5.8

7.0
Average
Vpore
3.5 17.3 6.4
Porosity 0.05 0.05 0.3 0.3 0.08 0.09
Average
porosity
0.05 0.3 0.09

3.3.4 Scope of Work

The scope of work is to conduct core flood experiments on three (3) field core samples from
Kuwait. The investigation was conducted using Fluorinated Benzoic Acids as tracers to obtain
tracer response curve C(t) in terms of tracer concentration (mg/L) vs. time for further
calculation.

46

The Concentration vs. time curves was used to derive the longitudinal dispersion values,
which is the objective of this experiment.
Three (3) core samples sizing approximately 4 inches in length and 1.5 inches in diameter. In
order to confirm detectability, Tritiated Water (HTO) was added to FBAs as the ideal water
tracer for further reference.
3.3.5 Technical Requirements & Given Parameters

In these experiments, we used values equivalent to field values and downscaled them:

- Flooding fluids: brine (35,000 mg/L)
- Temperature (175 F ~ 79.4oC)
- Injection flow rate (5000 bbl – 10000 bbl/day) ~ 596 – 1192 m3/day down to lab
experimental scale 0.01 to 1.0 mL/min
- Tracers: chemical tracer such as FBAs + Tritiated water HTO
- Core samples: 3 core samples provided with parameters as shown in Table 3.3.

Table 3.3: Parameters of Core Samples from Kuwait.
Seq No. Core No. Well No Depth, ft Size
1 Core I SA-129

6865.6

D=3.9 cm
L=6.5 cm
2 Core II RA-258

7501.90

D=3.8 cm L=4.9cm

3 Core III

SA-454

7064.67

D=1.5”(3.8cm)
L= 6.15 cm

47

3.3.6 Experiment Procedure and Design

The schedule of tasks to fulfill the experimental design is as follows:
Schedule of Tasks

1) Setting up the experiment, as shown in Figure 3.2 below.
2) Core sample preparation
3) Core packaging
4) Core flush for cleaning
5) Water saturation
6) Water core flooding to stable
7) Tracer injection as a pulse
8) Collection of samples
9) Tracer analysis
10) Conclusions

Materials and equipment
Materials
- A core sample of Well No. SA-454 at a depth of 7064.67 ft.
- A core sample of Well No. RA-258 at a depth of 7501.90 ft.
- A core sample of Well No. RA-129 at a depth of 6865.6 ft.
- Brine: synthesized of distilled water and MERCK chemicals at a concentration of 35000
mg/L as required. The chemical compositions were referred to Middle Marrat injection brine,
a reference reservoir in Kuwait as shown in Table 3.4.

48

Table 3.4: Chemical Composition of Brine for the Experiments.
Compositions Concentration/parameters
Salinity 35,000 mg/L
EC 54.3 uS/cm
TDS 37.4 g/L
pH 6.38
Density 1.025 g/mL
Na+ 10,996 mg/L
Ca2+ 2,168 mg/L
Mg2+ 230 mg/L
K+ 450 mg/L
Cl- 21,405 mg

Tracers:
Three FBA compounds (Sigma Aldrich, 98%):
2-FBA (2-Fluorinated Benzoic Acid),
2,4-DFBA (2,4 Di-fluorinated Benzoic Acid)
4-FBA (4-Fluorinated Benzoic Acid)
Tritiated Water (HTO) – RCA (USA)
Other:
- Distilled Water (Lab grade)
- Toluene (Lab grade)
- Isopropanol (Lab grade)
- Methanol (Lab grade)
- CO2 gas (Lab grade)
- Vacuum oil

49

- Epoxy Resin
- Shrinkable Teflon tube, Alumina foil, Viton sleeve
Equipment
- Dynamic tracer test system as shown in Figure 3.2.
- The equipment used in the experiments is listed in Table 3.5.

Figure 3.2: Dynamic Tracer Test System.

50

Table 3.5: Equipment used in the Experiments.
Sr.No Equipment Name Parameter
1 LC-10AD pumps (Shimadzu)

Flow rate: 0.001~5 mL/min
(10 ~ 400 kgf/cm2)

2 Pressure gauges (U.S.GAUSE)

0-5,000 psi

3 Pressure regulator

10,000 psi

4 Hydraulic hand pump

10,000 psi

5 Sampling fractional collector (Model 1220 –
ISCO)

6 Gas Chromatograph Analyzer GC/MSD Agilent
7890A/5975B

For Detection of FBA
7 Ion Chromatograph Analyzer Dionex ICS-5000+

For Detection of FBA/HTO
8 Liquid Scintillation Counter Packard 3100

For Detection of HTO
9 Tubing, valve, fitting (SS316L-Swagelok)

1/16 inches

10 Sample glass vials

2-4 ml

51

Experimental Procedure

Step 1. Preparation of core samples
- Polishing the core sample
- Wrapping the core sample by three layers: shrinkable Teflon tube, aluminum foil, and Viton
sleeve
Step 2. The core sample is vertically placed into the holder and installed into the flow lines.
Step 3. Pumping hydraulic oil into the holder to wrap core.
Step 4. Set up the system to be stable at a temperature of 80 °C.
Step 5. For cleaning the core, distilled water is pumped through the core at a determined flow
rate for 2 hours. Check the stability of pumping pressure.
Step 6. Pumping solvent through the core in sequence for hours: toluene, isopropanol,
methanol to clean residual oil in the core sample.
Step 7. CO2 is pumped through the core at low pressure (~ 20 atm) for about 20 minutes.
Step 8. Prepared brine is pumped through the core for about 48 hours. At the same time,
water is taken to determine the exact flow rate.
Step 9. Injection of tracers into the system in pulse mode.
Step 10. Collection of the sample with an estimated frequency.
Step 11. Collected samples are then processed to analyze the tracer concentration in samples.
FBA compound is analyzed by GC/MS procedure.
Step 12. When the sampling is completed, the system is kept flowing for 15 hours to ensure
that no residual tracer still remains in the system.
Summary of experiment parameters and procedure are shown in Table 3.6.

52

Table 3.6: Summary of Experiment Parameters and Procedure.
Data Core I (SA-454) Core II (RA-258) Core III (SA-129)
Core length 6.15 cm 4.9 cm 6.5 cm
Core Diameter 3.8 cm 3.8 cm 3.9 cm
Pore volume of core 3 mL

17.5 mL

7mL
Pumping Rate
EXP A

0.02 mL/min

0.012 mL/min

EXP B

0.04 mL/min

0.06 mL/min

0.01 mL/min

EXP C

0.07 mL/min

0.09 mL/min

Pulse Tracer Volume 0.3 mL

0.3 mL

0.3 mL

Tracer FBA/HTO
EXP A 2-FBA: 4660 mg/L
HTO: 70565 Bq/mL
2-FBA: 4660 mg/L
HTO: 70565 Bq/mL
2-FBA: 4660 mg/L
HTO: 70565 Bq/mL
EXP B 2,4-DFBA: 3894
mg/L
HTO: 72395 Bq/mL
2,4-DFBA: 3894 mg/L
HTO: 72395 Bq/mL
2,4-DFBA: 3894
mg/L
HTO: 72395 Bq/mL
EXP C 4-FBA: 1200 mg/L
HTO: 70093 Bq/mL
4-FBA: 1200 mg/L
HTO: 70093 Bq/mL
4-FBA: 1200 mg/L
HTO: 70093 Bq/mL
Pumping Pressure/
Wrapping Pressure

EXP A

12 atm/ 42 atm

65 atm/68 atm

53

EXP B

30 atm/ 46 atm

4 atm/ 20 atm

34 atm/41 atm

EXP C

50 atm / 65 atm

4 atm/ 24 atm

Differential pressure
between inlet-outlet
of core test

EXP A

11 atm

64 atm

EXP B

29 atm

3 atm

33 atm

EXP C

49 atm

3 atm

Dead Volume
(Valves, tubing...)
1.5 mL 1.5 mL 1.5 mL
Temperature
80°C 80°C 80°C

3.3.7 Results

Tracer experiment is conducted with a flow rate of 0.02 mL/min corresponding to using 2-
FBA and HTO as tracers. However, due to leak during the injection of HTO, only 2-FBA tracer
was injected correctly. The 2-FBA-response curve and 2-FBA concentration data are shown
in Figure 3.3 and Table 3.7, respectively.

54

Figure 3.3: 2-FBA-response curve of Core I at a flow rate of 0.02 mL/min.

Table 3.7: 2-FBA Concentration Data.
Time (min) Sampled effluent volume
(mL)

Pore Volume Fraction
Vpore = (Sampled Effluent
Volume/Pore Volume)

C (mg/L)

33.0 0.55 0.2

0.0

66.0 1.10

0.4

0.0

99.0 1.65

0.6

0.0

132.0 2.20

0.7

3.4

165.0 2.76

0.9

48.5

55

Time (min) Sampled effluent volume
(mL)

Pore Volume Fraction
Vpore = (Sampled Effluent
Volume/Pore Volume)

C (mg/L)

198.0 3.31

1.1

292.5

264.0

4.41

1.5

166.3

297.0

4.96

1.7

113.8

330.0

5.51

1.8

72.7

462.0

7.72

2.6

42.4

561.0

9.37

3.1

22.3

660.0

11.02

3.7

14.4

759.0

12.68

4.2

8.6

924.0

15.43

5.1

3.3

1386.0

23.15

7.7

1.1

1485.0 24.80 8.3 1.0

1584.0 26.45 8.8 1.1

56

Time (min) Sampled effluent volume
(mL)

Pore Volume Fraction
Vpore = (Sampled Effluent
Volume/Pore Volume)

C (mg/L)

CORE I- Experiment B Results
Tracer experiment is conducted with a flow rate of 0.04 mL/min corresponding to using 2, 4-
DFBA and HTO as tracers. The HTO-response curve and HTO concentration data are shown
in Figure 3.4 and Table 3.8, respectively.

Figure 3.4: Core IB HTO Curve.

Table 3.8: Core I Experiment B HTO Concentration Data.
Time (min)

Sampled effluent
volume (mL)

Vpore
Pore Volume Fraction
Vpore = (Sampled Effluent
Volume/Pore Volume)
C (Bq/mL)

20.5

0.75

0.3

0.0

57

Time (min)

Sampled effluent
volume (mL)

Vpore
Pore Volume Fraction
Vpore = (Sampled Effluent
Volume/Pore Volume)
C (Bq/mL)

41.0

1.50

0.5

0.0

61.5

2.25

0.8

0.0

82.0

3.00

1.0

1116.3

102.5

3.75

1.3

4279.0

123.0 4.50 1.5 3075.5
143.5 5.25

1.8

1800.0

164.0

6.00

2.0

1010.7

184.5

6.75

2.3

592.8

205.0

7.50

2.5

402.7

225.5

8.25

2.8

289.2

266.5 9.75 3.3

148.7

307.5

11.25

3.8

73.8

58

Time (min)

Sampled effluent
volume (mL)

Vpore
Pore Volume Fraction
Vpore = (Sampled Effluent
Volume/Pore Volume)
C (Bq/mL)

348.5

12.75

4.3

36.2

389.5

14.25

4.8

13.3

430.5

15.75

5.3

0.0

The 2, 4-DFBA-response curve and 2, 4-FBA concentration data are presented in Figure 3.5
and Table 3.9, respectively.

Figure 3.5: 2,4-DFBA response curve of Core I at a flow rate of 0.04 mL/min.

59

Table 3.9. 2, 4-DFBA Concentration Data
Time (min) Sampled effluent
volume (mL)

Vpore
Pore Volume
Fraction
Vpore = (Sampled
Effluent
Volume/Pore
Volume)
C (mg/L)
41.0 1.50 0.5 0.0
61.5 2.25 0.8 0.1
82.0 3.00 1.0 157.0
102.5 3.75 1.3 258.8
123.0 4.50 1.5 150.8
143.5 5.25 1.8 63.2
164.0 6.00 2.0 43.1
184.5 6.75 2.3 30.6
225.5 8.25 2.8 18.0
266.5 9.75 3.3 8.7
307.5 11.25 3.8 4.6
348.5 12.76 4.3 3.8
471.5 17.26 5.8 2.8
533.0 19.51 6.5 2.6

CORE I- Experiment C Results
Tracer experiment is conducted with a flow rate of 0.07 mL/min corresponding to using 4-
DFBA and HTO as tracers. The HTO-response curve and HTO concentration data are shown
in Figure 3.6 and Table 3.10, respectively.

60

Figure 3.6: HTO response curve of Core I at a flow rate of 0.07 mL/min.

Table 3.10: Core I Experiment C HTO Concentration Data.
Time (min) Sampled effluent
volume (mL)
Vpore
Pore Volume
Fraction
Vpore = (Sampled
Effluent
Volume/Pore
Volume)
C (Bq/mL)
7.8 0.55 0.2 0.0
15.6 1.10 0.4 0.8
23.4 1.65 0.6 0.0
39.0 2.75 0.9 54.3
46.8 3.30 1.1 3001.2
54.6 3.85 1.3 6118.7
62.4 4.40 1.5 5532.0
78.0 5.50 1.8 2776.8

61

101.4 7.15 2.4 1022.2
132.6 9.35 3.1 397.7
148.2 10.45 3.5 315.0
195.0 13.75 4.6 2.6
210.6 14.85 5.0 1.7
226.2 15.95 5.3 1.0
241.8 17.05 5.7 0.8
265.2 18.70 6.2 0.5
288.6 20.35 6.8 0.3

The 2,4-DFBA-response curve and 2,4-FBA concentration data are presented in Figure 3.7
and Table 3.11, respectively.

Figure 3.7: Core IC – 4-FBA.

62

Table 3.11: Core I Experiment C 4-FBA Concentration Data.
Time (min) Sampled effluent
volume (mL)
Pore Volume Fraction
Vpore = (Sampled Effluent
Volume/Pore Volume)
C (mg/L)
7.8 0.55 0.2 0.2
15.6 1.10 0.4 0.2
23.4 1.65 0.6 0.2
31.2 2.20 0.7 2.5
39.0 2.75 0.9 71.7
46.8 3.30 1.1 187.8
54.6 3.85 1.3 189.9
70.2 4.95 1.7 99.7
93.6 6.60 2.2 21.1
109.2 7.70 2.6 10.4
124.8 8.80 2.9 9.4
140.4 9.90 3.3 4.9
156.0 11.00 3.7 4.6
171.6 12.10 4.0 3.8
202.8 14.30 4.8 3.7
226.2 15.95 5.3 3.9
234.0 16.50 5.5 3.8
257.4 18.15 6.1 3.1
280.8 19.80 6.6 2.7
304.2 21.45 7.2 2.2

63

The expression of tracer response curves in the form of concentration vs. time since tracer
injection C(t) and vs. injected volume C(Vpore) is showed in Figure 3.8 and Figure 3.9.

Figure 3.8: FBA tracer response curves vs. time after injection of Core I at different flow rates (0.02
mL/min, 0.04 mL/min and 0.07 mL/min).

Figure 3.9: HTO tracer response curves vs. time after injection of Core I at different flow rates (0.04
mL/min and 0.07 mL/min).

64

3.3.8 Interpretation of Results

The porosity is the ratio of pore volume and a bulk volume of the core. Results of mean
residence time calculation and pore volume, the porosity of the core samples are given in
Table 3.12 and Table 3.13.
Table 3.12: The mean residence time of tracers in the systems.

Experiment

Tracers
The Mean Residence time in system (min)
Core I (SA-454) Core II (RA-
258)
Core III (SA-
129)
HTO FBA HTO FBA HTO FBA
A HTO+2FBA 350.2 609.8 544.5
B HTO+2,4DFBA 133.5 137.9 317.7 332.5 818.0 735.8
C

HTO+4FBA 73.3 76.4 205.6 214.8

Table 3.13: The calculated porosity of core samples.

Parameters
Core I
(SA-454)
Core II
(RA-258)
Core –III
(SA-129)
EXP B EXP C EXP B EXP C EXP A EXP B
Pore Volume
(mL)

3.3

3.6

17.5

17.0

5.8

7.0
Average
Vpore
3.5 17.3 6.4
Porosity 0.05 0.05 0.3 0.3 0.08 0.09
Average
porosity
0.05 0.3 0.09

65

The tracer response curves of Core I are characterized by enhanced dispersion and long
tailing effect, which are difficult to model with the Fickian advection-dispersion equation.
Many authors (Coats & Smiths 1964; S.Brouyere & A. Dassargues 1999; C. E. Feehley & C.
Zheng 2000; W. J. Bound & P. J. Wierenga 1990) postulated that such results must be due to
capacitance effects.
The porous media contains t dead-end pore zone in which solute can migrate by molecular
diffusion processes. The solute exchange between the mobile and immobile zone is
represented with the first-order mass transfer equation depends on the concentration
difference between the two phases.
In this work, the experiments using HTO and FBAs as tracers were carried out at different
flow rates (0.02 mL/min, 0.04 mL/min, and 0.07 mL/min).
Longitudinal dispersion coefficient (DL), flow fraction (f) and mass transfer coefficient (K)
were determined by fitting below to the tracer concentration profiles (Figure 3.10, Figure
3.11, Figure 3.12, Figure 3.13, Figure 3.14).

It was concluded that for each flow rate, tracer curves have the same longitudinal dispersion
coefficient. The same flow fraction must be applied to all tracer curves.
The fitting parameters of tracer response curves of Core I are shown in Table 3.14, where the
actual travel time of tracers is corrected with Tshift - the travel time in the dead volume
(valves, fittings). As a result, the flow fraction of rock is 0.63, i.e., the porosity of the immobile
zone equals 0.015.
𝐶 𝑚 ( 𝑥 , 𝑡 ) = 𝐿 −1
[𝐶𝑚 ( 𝑥 , 𝑠 ) ] = 𝐿 −1
[
𝑀 𝐴𝑦𝑧 1
√Δ
𝑒 (
𝑣 −√Δ
2𝐷𝐿
)𝑥 ]
Δ = ( 𝑣 𝑥 )
2
+ 4𝐷 𝐿 (𝑠 +
𝐾𝛽𝑠 𝛽𝑠 + 𝐾 ∗
)
Table 3.14: The fitting parameter of tracer response curves of Core I.
Parameter EXP A EXP B EXP C
HTO 2FBA HTO 2,4DFBA HTO 4FBA
Tshift
(min)
68 42 42 23 23
M/Ayz
[M]/cm2
1900 31000 1500 50000

1300

66

qx
(cm/min)
0.0012 0.0028 0.0028 0.0056

0.0056
x (cm) 6.15 6.15 6.15 6.15

6.15

Ф 0.04 0.04 0.04 0.04

0.04

DL
(cm2/min)
0.005 0.013 0.013 0.03

0.03

K (min-1) 0.00024 0.00064 0.0004 0.0016

0.001
F 0.63 0.63 0.63 0.63

0.63
NRMSD

0.34 0.16 0.49 0.35

0.50

67

Figure 3.10: 2FBA tracer curve of Core I at a flow rate of 0.02 mL/min with the capacitance model.

Figure 3.11: HTO tracer curve of Core I at a flow rate of 0.04 mL/min with the capacitance model.

68

Figure 3.12: 24DFBA tracer curve of Core I at a flow rate of 0.04 mL/min with the capacitance model.

Figure 3.13: HTO tracer curve of Core I at a flow rate of 0.07 mL/min with the capacitance model.

69

Figure 3.14: 4FBA tracer curve of Core I at a flow rate of 0.07 mL/min with the capacitance model.
By using the relationship between longitudinal dispersion and mean flow velocity as shown
in Figure 3.15 the longitudinal dispersivity of rock can be calculated. As a result, the
longitudinal dispersivity αL equals 0.18 cm.

Figure 3.15: Correlation between longitudinal dispersion coefficient and mean flow velocity.

70

3.3.9 Conclusion

The experiments of tracer injection through 3 cores were implemented using FBAs and HTO
as tracers. Seven injections at different flow rates were conducted to obtain the tracer
response curves. The tracer data can be used for interpretation of the dispersivity of the
cores. The results are shown in Table 3.15.
The 1D analytical solutions of single porosity model (Fickian model) and double porosity
model (Capacitance model) were applied to fit the tracer concentration profiles.
Core I and Core III were matched with a double porosity model, which gave the dispersivity
0.18 cm and 0.17 cm, respectively.
Core II was curve matched with the single porosity model but decomposed into 3 component
channels that have the dispersivity values 0.29. 0.88 and 0.35 cm.

Table 3.15 Dispersivity Results.

All determined values of dispersivity of the cores are in the usual range from 0.01 cm to 1.0
cm.

3.4 A Study of Upscaling Techniques of Laboratory-Scale Dispersion to Field Scale

The closest study found to field upscaling to this research pertaining to the upscaling of is the
work done by John, Bryant, Lake, Jennings et al (SPE113429-MS). Although they don’t
address laboratory upscaling, they look at it from a simulation and field-scale perspective.
We understand from their study that dispersity data compiled over many lengths show that
values at typical inter-well distances are about two to four factors order of magnitude higher
than those measured on cores.
This gave rise to a necessity for investigation on upscaling techniques from laboratory level
to field scale. Lake et al. (2008) use flow reversal by non-numerically simulating the tracking
of particles on 3D high-resolution models at the field scale. The flow reversal tests
Core sample Longitudinal
dispersivity (cm)
Porosity Flow fraction Analytical
solution
Core I SA-454 0.18 0.04 0.63 Capacitance
model
Core II RA-
129
0.29 (Channel 1)
0.88 (Channel 2)
0.35 (Channel 3)
0.30 1.00 Fickian model
with multi-
channel
Core III SA-
258
0.17 0.09 0.58 Capacitance
model

71

differentiate between dispersion due to spreading and mixing. Flow reversal as defined by
Lake et al. is in the absence of temperature effects and turbulence effects local velocities
reverse themselves on reversing the boundary conditions (Flekkoy, 1997). Because
heterogeneity increases the amount of mixing, particularly in the transverse direction, it is
worth looking at.
Lake et al. (2008) studied in their paper mentioned above the different aspects of dispersion
and flow reversal, including its effects on mixing and spreading, correlation structure
(reservoir size and correlation in permeability field), local mixing, permeability variance,
permeability anisotropy, travel time. They compared their field-scale results with the
simulated results to observe their work but did not directly address the laboratory upscaling,
but rather the comparison between non-numerical simulated model and field work.
72

4 Chapter 4: Numerical Simulation of Forward Models (Velocity, Tracer
Concentration Profiles)

4.1 One-Dimensional Tracer Simulation Model

A summary of all the models are given in Appendix B. For the one-dimensional synthetic
model has been created to simulate different model scenarios as summarized in Table 4.1.

Table 4.1: One-Dimensional Simulation Models.
Case
No
Name PermX
(Darcy)
PermY
(Darcy)
PermZ
(Darcy)
Por
o
NTG
Ratio
Strategy Tracer
Included
Special
Feature
1 (CASE) 1 1 1 0.3 1 01-01-
2016 to
01-03-
2016
Yes None
2 (Tracer
2)
10 10 1 0.1
5
1 01-01-
2016 to
01-03-
2016
Yes Anisotrop
y
3 Case_1_
Disp
1 1 1 0.3 1 01-01-
2016 to
01-03-
2016
Yes Include
Dispersio
n
4 1D
Adsorpt
ion
1 1 1 0.3 1 01-01-
2016 to
01-03-
2016
Yes Include
Adsorptio
n
5 Case 1 1 1 0.3 1 01-01-
2016 to
01-03-
2016
Yes Velocity
Profile

73

4.1.1 Results of Case No.1 One-Dim Permeability xyz= 1D with Tracer (Case)

1-D Deterministic model and model properties are presented in Figure 4.1 and Table 4.2
respectively. The result of concentration vs time is shown in Figure 4.2.

Figure 4.1. 1-D Deterministic Model (applies to all 1-D Cases).

Table 4.2: Case No.1 Model Properties.
Model Properties (Case) Value Unit
Permeability X 1 Darcy
Permeability Y 1 Darcy
Permeability Z 1 Darcy
Porosity 0.3
Prediction Strategy 01-01-2016 – 01-03-2016 (3
months)
Date Range

74

Figure 4.2: Concentration vs Time Results (Case).

4.1.2 Results of Case No 2 One-Dim Permeability xyz= 10D z=1D with Tracer (Tracer 2)

1-D permeability model properties are presented in Table 4.3. The result of concentration vs
time is shown in Figure 4.3.
Table 4.3: Concentration vs Time Results (Tracer2).
Model Properties (Case) Value Unit
Permeability X 10 Darcy
Permeability Y 10 Darcy
Permeability Z 1 Darcy
Porosity 0.15
Prediction Strategy 01-01-2016 – 01-03-2016 (3
months)
Date Range

75

Figure 4.3: Concentration vs Time Results (Tracer2).

4.2 3-Dimensional Tracer Simulation Model

A three-dimensional synthetic model has been created to simulate different model scenarios
as summarized in Figure 4.4-4.8. The result of concentration vs time is shown in Figure 4.9.

Figure 4.4: Permeability 1 mD.

76

Figure 4.5: Permeability 5 mD.

Figure 4.6: Permeability Narrow Channel Base 1mD.

Figure 4.7: Permeability Wide Channel Base 1mD.

77

Figure 4.8: Permeability Curved Channel Base 1mD.

Figure 4.9: Results of Simulation.

78

4.3 Synthetic Simulation Models

4.3.1 Effect of Dispersion Coefficient

To carry out this research, we have assumed several synthetic models. Our first model is a
quarter five-spot pattern.
The model (15x15 grid) is a homogenous model with a permeability of 1md with an injector
and producer placed at the corner of the grid, as shown in Figure 4.10.

Figure 4.10: 15 X 15 Quarter 5 spot pattern results.
The purpose of the design of this model is to run the forward model simulation of the
concentration profile to determine if the inclusion of the dispersion coefficient value has an
effect on the concentration profile.
The simulation strategy assumed constant rate injection at the injector well. The results are
depicted in Figure 4.11.

79

Figure 4.11: 15 X 15 Quarter 5 spot pattern simulation results.

The results show the attenuation of the concentration profile due to the inclusion of the
dispersion coefficient. This indicates that the incorporation of dispersion coefficient values
in simulation does as shown by the results have an effect on the simulated concentration
profile.
The green line depicts the constant injection rate as an example (6500 STB/d).
The red line depicts the results of the concentration profile without the inclusion of
dispersion. The blue line represents the concentration profile with the dispersion coefficient
included in the simulation. It is clear that there is a difference in the simulation of the
concentration profile due to the dispersion coefficient.
4.3.2 Synthetic Models (Velocity Profile)

Synthetic Models were created to observe the effects of simulating velocity. The velocity of is
an essential parameter as we had mentioned in the analytical section dispersion coefficient
is a function of diffusion and a quotient of parameters which include velocity.
This research shows that the equivalent permeability is a function of velocity and dispersion
coefficient, hence the velocity profile is simulated for various synthetic models as shown in
Figure 4.12-4.14 which will later be used.

80

Case 1: Velocity Profile (Homogeneous Case) (5md)

Figure 4.12: Velocity Profile for the homogeneous case (5mD).

Case 2: Velocity Profile (Narrow Channel Case)

Figure 4.13: Velocity Profile for Narrow channel.

81

Figure 4.14: Velocity Profile for S Curve channel.

82

5 Chapter 5: An Overview of Evolutionary Optimization Algorithms

5.1 Evolutionary Computing and Genetic Algorithms

In 1962, Holland introduced a form of computing that mimics or replicated the Darwinian
natural evolution process. This form of computing mimics evolution in the sense of the
reproduction of a population through element mutation. This general form of computing falls
under the general term “Evolutionary Computing”. Evolutionary Computing, a broad term,
encompasses Genetic Algorithms (GA), Evolutionary Programming (EP), Evolutionary
Strategies (ES), Classifier Systems (CS), Genetic Programming (GR) and Cellular Automata
(CA) are some of the evolutionary methods a referenced by [Aminzadeh and de Groot, 2006].
Genetic Algorithms are a stochastic optimization algorithm and have gained popularity in
recent years due to their robust techniques. Genetic Algorithms have been widely used in
several industries including the Oil and Gas industry as an optimization technique used in
conjunction with inverse modelling or history matching. Genetic algorithms are robust and
efficient as opposed to purely analytical optimization methods which are restricted by
differentiability and the need for derivatives.
As referenced by [Aminzadeh and de Groot, 2006], some of the applications of genetic
algorithms are in the geosciences, oil exploration and production. In reservoir engineering,
genetic algorithms have been used by Guerreiro et al. (1998) in application of genetic
algorithms for a synthetic quarter five-spot model and Wong (2002) and Nikravesh et al.
(2003) for reservoir engineering applications. Mohaghegh (2003) and Johnson and Rogers
(2003) used genetic algorithms for production optimization.
Genetic Algorithms are stochastic algorithms composed of these processes: initialization of a
population, mutation, and selection. These processes are used to reach an optimum value for
any particular objective function. These stochastic algorithms initialize a population for
optimization in a global search space. The population is enhanced through mutation and
selection to obtain the optimized objective function. The advantage of using genetic
algorithms is their ease of use and robustness.
Some common genetic algorithm methods are Differential Evolution, Particle Swarm
Optimization; other methods include Response Surface (a.k.a Proxy Model) and the Simplex
(Nelder Mead) method in their paper on global optimization strategies. We have the above
methods to find optimum values of permeability to match tracer concentration data in our
research.

83

5.2 Differential Evolution

Differential Evolution (DE) was developed by Storn and Price in 1996. [Al-Nemer et al., 2015]
define differential evolution as “ a stochastic search that works by creating a new potential
agent-position by combining the position of randomly chosen agents from its population, and
updating the agent’s current position in case of an improvement to its fitness (Kennedy and
Eberhart, 1995). It is proven to be efficient in searching for global optimization over
continuous spaces. DE uses a population of agents, called chromosome, to search for the
optimum solution. Better chromosomes are updated, by a process called mutation, after each
generation while the weak chromosomes are removed. DE algorithm is influenced by three
operators, mutation, crossover, and selection, in addition to the three parameters which are
population size Np, scale factor F and Crossover Probability CR (Youyunao et al.2009). In its
simplest form DE algorithm is used to optimize an objective function of D-dimension in the
following steps:
Initialization: an initial population is randomly created. They are assumed to be distributed
uniformly. Each chromosome is a candidate solution. Chromosomes are vectors of the input
parameters of the objective. The objective function is evaluated at each chromosome, and this
population is considered the current.
Selecting parents: for each chromosome xi, three different chromosomes are randomly
selected xr1, xr2 and xr3 such that xi≠ xr1 ≠ xr2 ≠ xr3.
Mutation: a random number Rnd is selected such that Rnd is a member of [1,…D]. one can
then create a new trial population X .
The fitness of the trial population is compared with the current population. For each
chromosome xi in the trial population, if the fitness is better or equal to the corresponding
chromosome in the current population xci, then xi=xci. The chromosome with the best fitness
among the population is also selected to be the Pob(best).
Steps 2 to 4 are repeated until the algorithm converges (Vitaliy Feoktistov, 2006).
Some studies have suggested the following values of the behavioral parameters: Np ≈ 40,
F≈0.6 and CR ≈ 0.9 ( Magnus Erik Hvass Pedersen, 2011). “

5.3 Commonly Used Genetic Optimization Algorithms

5.3.1 Particle Swarm Optimization

[Al-Nemer et al., 2015] describe some of these methods in detail amongst others.

84

Particle Swarm Optimization (PSO) was introduced by Kennedy and Eberhart (1995). This
model mimics the social model of the swarm movement of birds or a school of fish learning
from its food search/predator experiences by optimizing behavior. In this algorithm,
particles also known as agents are randomly selected in the search space. Each particle
maintains the parameters of the objective function, the history of the best-achieved result
and the velocity along each dimension upon completion of the evaluation of the objective
function at each particle. We calculate velocity to determine the direction of movement using
formula which is a function of the particle’s best-discovered position, and the swarm’s best-
discovered position (Magnus Erik Hvass Pedersen, 2011).
The PSO algorithm steps are as follows:
-Initialize the population at random positions and velocities. Select the best particle fitness
and best global fitness to be an infinite positive number.
-Evaluate the objective function for each particle.
-For each particle, if current fitness < best particle fitness then best particle[i]= current
particle and best particle fitness = current fitness.
-For each particle, if current fitness< best global fitness, then best particle= current particle
and best fitness= current fitness.
-Change the velocity of each particle according to the above equation.
-Go to step 2 until the best global fitness is <= tolerance value

85

6 Chapter 6: Inverse Modelling, Optimization, and Validation of Dynamic
Model Tracer Concentration Results using Differential Evolution (Single
Parameter)

6.1 Why use Differential Evolution
Differential Evolution are widely used in the various industries. In the Oil & Gas sector , it is
commonly used in some application. We introduce a novel use for Differential Evolution
given the parameters assuming real-time tracer and water cut data in the following
sections.
6.2 Optimization and Validation of Dynamic Models using Differential Evolution

The differential evolution technique was run on a 1D model (2000x10x10) as shown in Figure
6.1 to determine permeability.

Figure 6.1: 1 Dimensional Model 2000 x 10 x10.
The synthetic model with one injector and one producer placed on the horizontal axis is a 1D
model with the permeability of different values at five intervals. Interval 1 (1md), Interval 2
(200md), Interval 3(1000md), Interval 4 (600md) and interval 5 (50md).
To run the evolutionary algorithm (differential evolution), we started with a target model as
a homogenous model with permeability set to 400md.

86

The forward simulation was run to determine the tracer concentrations results for the
homogeneous case. The tracer concentration was defined as the objective function in order
to run the evolutionary algorithm to determine permeability (target variable).
Figure 6.2 is the results of the tracer curves vs. time for the homogenous case.

Figure 6.2: 1 Dimensional Model 2000 ft. x 10ft x 10ft (Forward Results).
Next, the forward simulation for the heterogeneous (objective case) is run, a comparison
chart is shown in Figure 6.3.

87

Figure 6.3: 1 Dimensional Model 2000 ft. x 10ft x 10ft (Match Results).

It is shown that the heterogeneous case shows a higher peak (Concentration =0.45) and
earlier arrival time by one month for the 1D case.
The Differential Evolution algorithm inputs include:
Initial Permeability Value for Model = 400md
The model was divided into five regions equivalent to the synthetic model (heterogeneous
case).
The ranges for the minimum permeability was set to 0.1 multipliers (factor), and the
maximum permeability multiplier was set to 10. Region definition and ranges are described
in Figure 6.4.

Figure 6.4: Region definition and ranges.

Each region used the same minimum and maximum multiplier (factor). The differential
evolution algorithm generated 109 cases as shown in Figure 6.5.

88

Figure 6.5: Simulation results for all cases.

The cases with the minimum objective function error (root mean square) were case number
65.

89

Figure 6.6: Simulation results for all cases.

Case No. 65 provided the closest match with a peak value of Concentration of 0.45.
This was chosen from the best 5 cases which matched the objective function (heterogeneous
case)

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Figure 6.7: Simulation results for all cases.

As expected, the best five non-unique solutions were selected with the lowest root mean
square error of the objective function (concentration). Because this is a stochastic method, it
will generate several non-unique solutions in comparison to deterministic methods.
We found that region 1 and region 5 were outlier values, this is because the regions had no
influence on the concentration profile because the wells were located in region 2 (injector)
and region 4 (producer), hence flow will be affected by these regions and the regions in the
range of influence in between (regions 3, 4).

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Figure 6.8: Best Match Case.

The differential evolution technique was run on a 3D model (300ft x 300ft x10ft) to determine
permeability. Although the base of this model is homogeneous (1md), complexity arises from
the high permeability pattern of the channel.

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The Narrow Channel synthetic model is shown below:

Figure 6.9: Narrow Channel Case.

The model constitutes two wells (P2 & P4) lying outside the channel and two well (P1 & P3)
inside the high permeability channel.

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Figure 6.10: Forward Model results for P1 & P2/ P3 &P4.
(P1 & P3) show identical results and an earlier arrival time than wells (P2 &P4). Wells (P2 &
P4) shows matching concentration curves in the simulation model.
To run the inverse model using the differential evolution technique, an initial value of
permeability of 1md is assumed.
The initial model is below:

Figure 6.11: Initial Model for Initialization.

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We will run the forward simulation model to obtain the initial tracer concentration profiles
of the producer wells. The results of the simulation run shown in Figure 6.12 show an exact
match of all wells; this is typically the case as the model is heterogeneous and the fluid is
equally distributed at a uniform rate.

Figure 6.12: Forward Simulation Initial Model for Initialization.

We initialize the differential evolution technique by setting the target of the heterogeneous
model (N Channel results) and running the initial cases for each well.

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The differential evolution technique was run on a 3D model (300 x 300 x10) to determine
permeability. Although the base of this model is homogeneous (1md), complexity arises from
the irregular pattern of the S-Curve.

Figure 6.13: Initial Model for Initialization and objective function.

The Model is divided into 2 permeability regions; we begin by assuming each region has the
same permeability.
The model starts with the same perm value in all regions = 1 mD. We define the minimum
permeability to be tested to be 1multiplied by a factor of 0.1.
The maximum permeability will be determined as1 multiplied by a factor of 10*1 = 10 mD.
Hence the factor range is from 1 to 10.
Each region will be tested separately, and the new cases inputs will be defined by Differential
Evolution Algorithm. The Objective function is (to be matched results): Tracer Production
Concentration resulted from the Narrow Channel case.

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Figure 6.14: Permeability Ranges for Differential Evolution Algorithm.
The results of the differential evolution algorithm are as Figure 6.15 for wells P1 * P3:

Figure 6.15: Permeability Ranges for Differential Evolution Algorithm.

The results from running the Differential Evolution is Channel Perm Obtained = 5.4 mD in
comparison to the actual permeability of the channel, which is Perm = 5 mD.

For the wells located outside the channel P2 & P4, we begin by comparing the initial case:

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Figure 6.16: Matching for Well No 2.

The results show that after running Differential Evolution Channel, Perm Obtained = 0.1 mD
in comparison to the actual permeability 1md.
Hence the observation is that the differential Evolution algorithm succeeded to produce the
objective tracer production concentration only inside the channel (high perm).

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Figure 6.17: S-Curve Synthetic Model.

Figure 6.18 is the forward simulation run for the production concentration of the S-Curve
channel. The concentration profiles of Production Wells P1, P2, P2, and P4 are displayed
versus time.

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Figure 6.18: S-Curve Synthetic Model Forward Concentration Curve Results.

It is observed that the wells in the high permeability curve display early arrival time.
To determine the permeability of the model, we assume the permeability is unknown and
define the tracer curves above as the objective function.

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Figure 6.19-Initial Model for S Curve.
The concentration profile of the homogeneous case is shown in Figure 6.20, as expected, all
wells display identical arrival times:

Figure 6.20: Forward Concentration Results for Initial Model.

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For the individual wells, we compare the concentration profiles to use the inverse modeling
evolutionary method to sensitize on permeability distribution until the objective function is
optimized.

Figure 6.21: Forward Concentration Results for Initial Model Vs. Objective Function.
We divided the Model into two regions in order to run the simulation. Assuming each region
having the same initial permeability value (homogenous value of 1md).
We then set the range multipliers to be (0.1- 10); hence, the minimum permeability (1md)
and the maximum permeability is 10md.
Each region will be tested separately, and the new cases inputs will be defined by Differential
Evolution Algorithm
The objective function is (to be matched) is the Tracer Production Concentration resulted
from the S-Curve case. The settings are set in the simulator as Figure 6.22:

Figure 6.22: Permeability ranges for S-Curve Model.

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The results were as shown in Figure 6.23, the best curve vs. the actual permeability using
differential evolution:
(Differential Evolution): Results
Channel Perm Obtained = 6.2 mD Actual Perm = 5 mD
Outside Channel Perm Obtained = 0.9 mD Actual Perm = 1 mD

Figure 6.23: Match cases for S Cu
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7 Chapter 7: A Comparative Study and Results of Evolutionary Algorithms
using Novel Parameters to Identify High Permeability Channels (Dual
Parameter)

7.1 Introduction

Reservoir uncertainty is among the highest challenges in realizing an increase in the recovery
factor. To understand the sweep efficiency, one needs to characterize the reservoir
heterogeneity. In particular, characterizing high permeability channels is essential.
In most reservoirs, an estimated 15-20% of oil in place is naturally produced. Under
waterfloods, in circumstances when water cut increases, production is compromised because
water is moving in high permeability channels. As such, identifying and understanding the
direction of the flow of water is imperative, which leads to improved sweep efficiency.
Ultimately the enhanced sweep efficiency means higher recovery.
In many cases, inverse modeling is used to reduce uncertainty and optimize simulation.
Common inverse modeling parameters included production data. In this research, we used
differential evolution as a method to optimize inverse modeling. Differential Evolution has
been used in some disciplines in petroleum engineering such as drilling (Awotunde and
Mutaisiem, 2014), rock mechanics (Gutierrez and Zia, 2011) and well testing (Moussa and
Issaka, 2017). Other applications include reservoir characterization (Hajizadeh and
Demyanov, 2010), (Hajizadeh and Demyanov ,2009) and Decker and Mauldon, 2006) using
production rates. Through our research, it has been noticed that unlike oil or production
rates, the use of tracer data is not common as an input to inverse modeling. Hence, we
developed a hypothesis to deploy the assumption of the use of continuous tracer data for
inverse modeling and computer-aided history matching using differential evolution.
This proposed method uses differential evolution with input parameters as continuous tracer
data in conjunction with production data, in particular, water cut data, as an input to the
inverse model. The model runs a statistical evolutionary model, differential evolution, to
determine the permeability ranges. A commercial simulator uses the input from the
differential evolution model results to calculate permeability using an inverse method. A
comparison of the objective function (historical tracer data and water cut data) is made to
determine the case with the lowest error.
We selected one stochastic method to run the inverse to estimate the permeability channel
and later compared it to other evolutionary algorithms. The differential evolution method
uses vector differences for a perturbing vector population to minimize errors and is

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developed to optimize the estimation of real parameter and real-valued functions. The
method defines an objective function, which is the input parameter used for history matching.
In this research, we introduce the novelty of defining the objective function as the tracer
concentration in conjunction with production data (water cut) to identify high permeability
channels. The algorithm then performs hundreds of combinations of stochastic runs based
on parameter ranges of permeability which are then used to run the reservoir simulator in
order to refine the values to optimize the objective function.
The results have shown that the use of tracer data in conjunction with water cut data and
evolutionary algorithms and in particular, differential evolution prove to be a useful
technique in identifying high permeability channels. Validation is done by uses similar class
algorithms and comparing them to differential evolution.
7.2 Challenges with Traditional Inverse Modelling

In this section we discuss the challenges of inverse modeling using only production data or
using tracer data (from intermittent samples) without any additional data parameters and
some limitation of the traditional I inverse modeling methods.
Oilfield inter-well tracer tests are an efficient tool for understanding reservoir heterogeneity,
well-to-well communication, and fluid dynamics (Zemel 1995, Dugstad 2007). Tracers have
also been performed before enhanced oil recovery (EOR) operations to optimize flooding
efficiency (Cheng et al.2012, Sanni et al.,2015, Sanni et al. 2017a, Serres Piole, et al. 2012).
Tracer data has been incorporated in the reservoir history matching processes (Allison, Pope,
and Sepehrnoori 1991, Illiassov and Datta-Gupta 2002, Valestrand et al. 2010, Huseby et al.
2010). There are still many advantages and uses incorporating tracer data to characterize
reservoirs and optimize the recovery factor by prudent well placement.
Many authors have used tracer data as an input to the ensemble-based approach (ENKF) to
maximize the different objective functions (Chen, Oliver, and Zhang 2009, Chaudhri et al.
2009, Chen and Oliver 2010b, Su and Oliver 2010, Jafroodi and Zhang 2011, Do and Reynolds
2013, Fonseca, Leeuwenburgh, et al.2014), [Chen and Poitzch,2018]. Tracers are arguably the
only direct reservoir interrogation method that provides immediate and unambiguous
information on the injector/producer communication once the tracer breakthroughs from
producers are observed. Despite their appealing features, tracers are largely underused in
the industry for various reasons including the cost factors.
They show that history match, including tracer data, resulted in a better match of the field
production data with smaller standard deviation. They demonstrated that including tracer
data resulted in more distinct geological features when observing the history matched
geological maps and production optimization with higher NPV (objective function) produced.
It was demonstrated in [Chen and Potizch,2018] +4.3% increase of final NPV was observed
with the use of history matching with tracer data.

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Other authors such as [Sing , Maucec and Knabe 2014] use a stochastic history matching
algorithm ( Monte Carlo-based) with efficient sampling from the given distribution of
properties. The authors conducted sensitivity analysis on physical properties such as
porosity, permeability, relative permeability, net to gross (NTG), etc. to match production
data ( oil/water/gas flow rate) and bottom hole pressure [BHP]. When applied to a multi-
million grid cell model in the Middle East (Maucec et al. 2013), it was realized that one would
require faster-converging algorithms. This is the case in large simulation models with many
wells having significant mismatch before history matching and with model realization
missing connectivity. Examples include streamlining simulation where connectivity exists
between producer and injector or connectivity from aquifers to match water cut or gas cap
to match GOR, or static models with high uncertainty in terms of history matching
parameters. The authors did not test their assisted history matching (AHM) method within a
large-scale reservoir to test the efficiency of their approach.
[Cacelliere, Verga and Viberti, 2011] discuss the benefits and limitations of assisted history
matching (inverse modeling) techniques. They state that the selection of the adequate
optimization algorithm is not trivial, and the number of independent variables involved in
complex reservoir simulation does not make the solution of the optimization problem a
standard procedure hence, an optimization methodology for a wide variety of reservoirs is
almost impossible. The calibration of a reservoir model suffers from non-uniqueness. As such,
inverse problems are based on insufficient constraints and data [Schaaf et al. 2008]; thus,
several combinations of parameters might exist capable of satisfactorily matching the past
dynamic behavior of the system. Further, there is a challenge when calibrating parameters at
the same time because the behavior of the reservoir models and the interdependencies is
exceptionally complex. Hence, the complexity of the problem can be addressed with a large
number of parameters that are dealt with at the same time using statistic and stochastic
algorithms (as opposed to the deterministic methods).
In the past only performance data has been primarily used for inverse modeling related to
reservoir characterization. It is apparent that using only production data for inverse
modeling is insufficient. Further, when incorporating tracers, only single pulses or samples
of tracers have been used. In other published literature, multiple injections at different times
have also been used. In our research, we hypothesize that it is more comprehensive to have
a measure of continuous tracer data to ensure a full picture with no missing data points.
Missing the first breakthrough points may lead to considerable uncertainties.
Because most inverse modeling optimization methods are somewhat capable of
incorporating or assimilating a vast range of data types ( with some limitation) into the
objective function, the data types may include traditional data ,i.e., fluid rates and pressures,
or more specialized and innovative data from 4D seismic surveys or results from tracer tests.
Hence, it is essential to reduce uncertainty in inverse modeling techniques as the best
reference simulation case is selected among several reservoir characterizations for forecasts
by incorporating such data.

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7.3 Differential Evolution

Differential evolution is one of many genetic algorithms. It is a global algorithm for
solving problems and was discovered by Storn and Price in 1995 as an attempt to solve a
polynomial fitting problem. A workflow of the differential evolution algorithm is summarized
in Figure 7.1.

Figure 7.1 Differential Evolution Method.
Differential evolution has proven to be very efficient in different computer science and
engineering fields (Price et al., 2005) (Chakraborty, 2008). Differential Evolution has been
used in some disciplines in petroleum engineering such as drilling (Awotunde and Mutaisiem,
2014), rock mechanics (Gutierrez and Zia, 2011) and well testing (Moussa and Issaka, 2017).
Other applications include reservoir characterization (Hajizadeh and Demyanov ,2010),
(Hajizadeh and Demyanov ,2009) and Decker and Mauldon, 2006).
Differential Evolution is a population-based stochastic search algorithm that uses Np D-
dimensional parameter vectors and evolves this population by simple arithmetic operations
on the vectors.
As described in (with Storn and Price, 1995) the generic form of the differential evolution
algorithm is as follows:
Initialization:
Select a random first population (starting population)
Like other stochastic methods, in the differential evolution, the original population is
randomly scattered in the search space.
Each individual member in the population is a vector of real numbers.
All vectors are uniquely indexed from 1 to Np for bookkeeping.
After the initialization of the algorithm and obtaining objective function values, in the second
step, two vectors are randomly selected among the current population.

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Then, the difference vector between two selected members is computed.
In the next step, the difference vector is multiplied by a real number called scaling factor (F
is a member of [0, 2]).
The scaling parameter controls the amount of perturbation introduced the difference vector.
Depending on the selected value for F, the difference vector may become smaller or larger
than its original size.

Mutation:
We then select another member in the population, and we add the scaled difference vector to
this new individual.

Cross Over:
Another vector is selected and is added to the scaled difference to form the trial vector.
This forms the trial vector that can be written as:
𝒗 𝑮 +𝟏 = 𝒓 𝟏 ,𝑮 + 𝑭 ( 𝒓 𝟐 ,𝑮 − 𝒓 𝟑 ,𝑮 )
𝒓 𝟏 is the base vector which we add the scaled difference vector and 𝒓 𝟐 and 𝒓 𝟑 are two other
vectors chosen from the population?
Selection:
-In differential evolution, each trial vector competes against the population vector of the same
index.
-If the trial vector has a lower objective function value comparing to the first individual
number, it will be selected to proceed to the next generation.
-The above procedure is repeated for each individual in the Np population to form the next
generation of solutions.

The differential evolution algorithm is used because this method accommodates practical
nonlinear problems. Such problems are difficult to solve analytically, differential evolution is
used to find approximate solutions to such issues and has shown to be effective on a broad
range of optimizations.
Different studies also show that the differential evolution algorithm achieves produces better
results in comparison with other optimization methods such as simulated annealing (Storn

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and Price, 1995) and separate implementation of genetic algorithms (Cruz et al., 2003)
(Biesbroek, 2006).
Hence, in this study, we introduce a new application of differential evolution by using dual
parameters (tracer data and water cut data) to identify high permeability channels using the
differential evolution technique. We further extend our study by comparing the results of the
inverse modeling to other genetic algorithms and conclude with the results.

7.3.1 Inverse Modelling Method using Synthetic Tracer and Water Cut Data

In the absence of a real data set, we will use a synthetic model to test our hypothesis. Initially,
we will define several reservoir models with defined permeability channel patterns. We will
then run the forward simulation models and obtain the results. The results of the forward
simulation will be production rates (water cut), pressure, and tracer concentration profiles.
The results are interpreted and verified. Following this, the results are used as an input data
set to the inverse model simulation runs to determine heterogeneity by matching tracer
concentration and water cut to determine the permeability value using differential evolution.
This method introduces a novel technique to indirectly determine heterogeneity by using the
differential evolution method to match tracer concentration and water cut to determine the
permeability value.
. This method uses synthetic data in the absence of continuously measured tracer data to
create several model variations to support the hypothesis. This method represents an inverse
method to indirectly to model heterogeneity using new model parameters and the
assumption of continuous measurement. The inverse modeling technique used is differential
evolution.
In our simulation, the differential evolution method that was applied was as follows:
Initially, the synthetic models with known heterogeneity (permeability distributions) were
created using a commercial simulator and the results of the forward model simulation for
water cut and tracer concentrations output.
The water cut and tracer concentrations results were then defined as the parameters for the
objective function.
The 5-Spot patterns with known permeability and heterogeneity distribution are shown in
synthetic model figures below are run separately. These will formulate the independent cases
of forwarding simulation. The concentration profile and the water cut profile will be used as
objective functions in the inverse model.
Now, to prove our hypothesis, we run the inverse model using the differential evolution
algorithm.

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We use differential evolution to minimize the objective function (matching tracer
concentration (c) and water cut (WC))
The equations are taken at the n-th time step:
𝑀 2 = ∑
( 𝑐 𝑠𝑖𝑚 − 𝑐 ℎ𝑖𝑠𝑡𝑜𝑟𝑖𝑐𝑎𝑙 )
0.5 ∗ 𝑐 ℎ𝑖𝑠𝑡𝑜𝑟𝑖𝑐𝑎𝑙 𝑁 𝑛 =1
+
( 𝑤 𝑐 𝑠𝑖𝑚 − 𝑤𝑐
ℎ𝑖𝑠𝑡𝑜𝑟𝑖𝑐𝑎𝑙 )
0.5 ∗ 𝑤𝑐
ℎ𝑖𝑠𝑡𝑜𝑟𝑖𝑐𝑎𝑙
where
N is the number of observations
c is the concentration (unit less)
WC is the water cut

We use a python code to define water cut and tracer data as an objective function in our
commercial simulator.
Because the commercial simulator does not have this function as default, the python code is
customized as follows and used as an input to the commercial simulator:

diff=WUTRPROD-WTRPROD2
export (diff, name=’diff’)
diff2=WUTRPROD + WUTRPROD2
export (diff2, name=’diff2’)
diff4=if_then_else (WUTRPROD>WUTRPROD2,WUTRPROD-WUTRPROD2, WUTRPROD2-
WUTRPROD)
export (diff4, name=’diff4’)
where
WUTRPROD: Tracer Production Concentration for each well vs. time (current case)
WUTRPROD2: Tracer Production Concentration for each well versus time (Synthetic Case)
diff: Positive difference for each well versus time
diff2: negative difference for each well versus time
diff 4: Absolute difference (objective to reach sigma= 0 for each well vs. time)

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In our method, the starting population is a previously found set of permeability within a
realistic range (in this case permeability =1md for our synthetic model) supplied as the
starting population. In this case, we use a homogeneous grid of a single-valued permeability
= 1md.
The commercial simulator then created vectors with permeability values. A sample vector is
shown below for an n x m grid where all p are randomly selected permeability values, in our
case all equal to 1md initially:
[
𝑝 1 𝑝 2 𝑝 3 … … . . 𝑝𝑛
𝑝 2 𝑝 3 𝑝 4 … … … …
𝑝𝑚 … … … … … … . .
]
The permeability values are now input to the forward simulator.
The water cut and tracer concentration are calculated with the assumption of the initial
population.
The differential evolution algorithm now compares the results of the simulator with the
synthetic output objective function (water cut and tracer data from the forward model of the
synthetic model).
The root mean square (RMS) is calculated after several hundred runs the lowest difference is
the optimum

7.4 Forward Models and Data Sets

Several models were used to test this hypothesis; in this study, we show the simulation
results of three synthetic models with varying heterogeneities (i.e., permeability
distributions).
The model parameters and description are shown below:

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Model 1- Five Spot Pattern – Narrow Channel Synthetic Model

Figure 7.2: Synthetic Model 5 Spot Pattern with
Narrow Channel Permeability (5mD).
Table 7.1: Parameterization of S-Shaped Synthetic Model.
Parameters Units Prior Range
Permeability mD Narrow Channel
(5 mD)
Grid (1mD)

Model Size

Grid 100 ft x100 ft
x10 ft
50 I * 50 J * 5 K
Cell: 2ft x 2 ft x 2
ft
Porosity - 0.3
PVT Data - Dead Oil Case
Pb = 400 psi
API = 30
o

Initialization
Initial Pressure =
2000 psi

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Development Strategy:
Table 7.2: Development Strategy for the S-Shaped
Synthetic Model.
Simulation Period 3 years
Wells 1 The injector in the
middle
4 Producers at edges

Injector Control Max BHP of 2500 psi
Tracer Injection
Process
Tracer Name: “00T”
Tracer injection Started at
t =0 day (1 Jan. 2016)

Tracer injection
concentration: 0.1
(bbl/bbl)/day

Tracer Injection stopped
at t = (6 Jan. 2016)

Tracer injection
concentration: 0.0
(bbl/bbl)/day

Producer
Control:

Liquid Rate: 1000 BLPD
Min BHP = 400 psi (to
ensure no gas formed at
reservoir)

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Model 2- Five Spot Pattern – S-Shaped Channel Synthetic Model

Figure 7.3: S-Shaped Synthetic Model.
Table 7.3: Parameterization of S-Shaped Synthetic Model.
Parameters Units Prior Range
Permeability mD S-Shape (5 mD)
Grid (1mD)

Model Size

Grid 100 ft x100 ft
x10 ft
50 I * 50 J * 5 K
Cell: 2ft x 2 ft x 2
ft
Porosity - 0.3
PVT Data - Dead Oil Case
Pb = 400 psi
API = 30
o

Initialization
Initial Pressure =
2000 psi

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Development Strategy:
Table 7.4: Development Strategy for S-Shaped Synthetic
Model.
Simulation Period 3 years
Wells 1 The injector in the middle
4 Producers at edges

Injector Control Max BHP of 2500 psi
Tracer Injection
Process
Tracer Name: “00T”
Tracer injection Started at t
=0 day (1 Jan. 2016)
Tracer injection
concentration: 0.1
(bbl/bbl)/day

Tracer Injection stopped at
t = (6 Jan. 2016)

Tracer injection
concentration: 0.0
(bbl/bbl)/day

Producer
Control:

Liquid Rate: 1000 BLPD
Min BHP = 400 psi (to
ensure no gas formed at
reservoir)

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Model 3- Five Spot Pattern- Narrow Multi Permeability Channel

Figure 7.4: 5 Spot Patter with Narrow Channel
Multiple Permeability.
Table 7.5: Parameterization of S-Shaped Synthetic Model.
Parameters Units Prior Range
Permeability mD S-Shape (5 MD)
Grid (1mD)

Model Size

Grid 100 ft x100 ft x10 ft
50 I * 50 J * 5 K
Cell: 2ft x 2 ft x 2 ft
Porosity - 0.3
PVT Data - Dead Oil Case
Pb = 400 psi
API = 30
o

Initialization
Initial Pressure =
2000 psi

116

Development Strategy:
Table 7.6: Development Strategy for S-Shaped Synthetic
Model.
Simulation Period 3 years
Wells 1 The injector in the
middle
4 Producers at edges

Injector Control Max BHP of 2500 psi
Tracer Injection
Process
Tracer Name: “00T”
Tracer injection Started at
t =0 day (1 Jan. 2016)

Tracer injection
concentration: 0.1
(bbl/bbl)/day

Tracer Injection stopped
at t = (6 Jan. 2016)

Tracer injection
concentration: 0.0
(bbl/bbl)/day

Producer
Control:

Liquid Rate: 1000 BLPD
Min BHP = 400 psi (to
ensure no gas formed at
reservoir)

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Forward Model Simulation Results (Water Cut vs. Time and Tracer Concentration Vs Time)
The 5 spot patterns contain five wells; four producers denoted as P1, P2, P3, and P4 and an
injector well i-1 located in the middle of the grid with the various variations as discussed in
the study.

Results of forward Simulation:
We have run several models to prove the hypothesis that using continuous tracer data in
conjunction with production data can identify high permeability channels. The differential
evolutionary method is used in conjunction with a commercial reservoir simulator to
determine permeability values with the lowest possible uncertainty.
The results showed reasonably good accuracy in comparison to the original synthetic model
permeability values.
Our next challenge was to answer the inquiry, was our selection of the differential evolution
algorithm wise, how does compare to results from other evolutionary algorithms?

7.4 Further Study: A Comparison of Four Evolutionary Algorithm Methods for
Inverse Simulation

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To further our study of our hypothesis for the selection of differential evolution method, in
particular, we ran the same models using three additional conventional evolutionary
methods (Particle Swarm (PS), Response Surface (RS) and Simplex Model (SM)) and
compared the results as shown in the results section below.
In what follows, we show the results of the inverse model runs for Figure 4 Synthetic Model
with the different permeabilities. Four different procedures (differential evolution (DE),
Particle Swarm (PS), Response Surface (RS) and Simplex Model (SM) of Inverse Model
Algorithms were run for the 4 wells. The following represents the inverse model matching
results in comparison to the historical data results (objective function) for water cut data and
tracer data.
1-Producer 1 (P1)
The inverse modeling results using the differential algorithm shows the closest match to
the historical values of the production data. The water cut data was matched using four
different evolutionary algorithms as shown in Figure 7.5. The same range for water cut
amounts to run the algorithm 0.1- 1 was used for all the evolutionary algorithms. The
differential evolution algorithm showed the closest match with the lowest root mean
square (RMS) at 20% showing a permeability in the region with permeability value 5 MD.
The simplex method showed the least matching results were the simplex method.

119

Figure 7.5: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm
techniques (Producer 1) Water Cut Data (objective function).

The inverse modeling results using the differential algorithm shows the closest match to
the historical values of the production data . The water cut data was matched using four
different evolutionary algorithms as shown in Figure 7.6. The same range for water cut
amounts to run the algorithm 0.1- 1 was used for all the evolutionary algorithms. The
differential evolution algorithm showed the closest match with the lowest root mean
square (RMS) at 20% showing a permeability in the region with permeability value 5 MD.
The simplex method showed the least matching results .

120

Figure 7.6: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm
techniques (Producer 1) Tracer Data (objective function).

2-Producer 2 (P2)
The inverse modeling results using the differential algorithm shows the closest match
to the historical values of the production data . The water cut data was matched using
four different evolutionary algorithms as shown in Figure 7.7. The same range for
water cut amounts to run the algorithm 0.1- 1 was used for all the evolutionary
algorithms. The differential evolution algorithm showed the closest match with the
lowest root mean square (RMS) at 20% showing a permeability in the region with
permeability value 5 MD.
The simplex method showed the least matching results.

121

Figure 7.7: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm
techniques (Producer 1) Tracer Data (objective function).

The inverse modeling results using the differential algorithm shows the closest match
to the historical values of the production data. The water cut data was matched using
four different evolutionary algorithms as shown in Figure 7.8. The same range for
water cut amounts to run the algorithm 0.1- 1 was used for all the evolutionary
algorithms. The differential evolution algorithm showed the closest match with the
lowest root mean square (RMS) at 20% showing a permeability in the region with
permeability value 5 MD. The simplex method showed the least matching results.

122

Figure 7.8: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm
techniques (Producer 2) Tracer Data (objective function).

3- Producer 3 (P3)
The inverse modeling results using the differential algorithm shows the closest match to
the historical values of the production data . The water cut data was matched using four
different evolutionary algorithms as shown in Figure 7.9. The same range for water cut
amounts to run the algorithm 0.1- 1 was used for all the evolutionary algorithms. The
differential evolution algorithm showed the closest match with the lowest root mean
square (RMS) at 20% showing a permeability in the region with permeability value 5 MD.
The simplex method showed the least matching results.

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Figure 7.9: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm
techniques (Producer 3) Water Cut Data (objective function).

The inverse modeling results using the differential algorithm shows the closest match to
the historical values of the production data . The water cut data was matched using four
different evolutionary algorithms as shown in Figure 7.10. The same range for water cut
amounts to run the algorithm 0.1- 1 was used for all the evolutionary algorithms. The
differential evolution algorithm showed the closest match with the lowest root mean
square (RMS) at 20% showing a permeability in the region with permeability value 5 MD.
The simplex method showed the least matching results .

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Figure 7.10: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm
techniques (Producer 3) Tracer Data (objective function).

4-Producer 4 (P4)
The inverse modeling results using the differential algorithm shows the closest match to
the historical values of the production data . The water cut data was matched using four
different evolutionary algorithms as shown in Figure 7.11. The same range for water cut
amounts to run the algorithm 0.1- 1 was used for all the evolutionary algorithms. The
differential evolution algorithm showed the closest match with the lowest root mean
square (RMS) at 20% showing a permeability in the region with permeability value 5 MD.
The simplex method showed the least matching results .

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Figure 7.11: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm
techniques (Producer 4) Tracer Data (objective function).

The inverse modeling results using the differential algorithm shows the closest match to
the historical values of the production data . The water cut data was matched using four
different evolutionary algorithms as shown in Figure 7.12. The same range for water cut
amounts to run the algorithm 0.1- 1 was used for all the evolutionary algorithms. The
differential evolution algorithm showed the closest match with the lowest root (RMS) at
20% showing a permeability in the region with permeability value 5 MD. The simplex
method showed the least matching results .

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Figure 7.12: Inverse Model Results from Synthetic Model 3 using four evolutionary algorithm
techniques (Producer 4) Water Data (objective function).
7.5 Interpretation of Results

In the first model (Narrow Channel), the results show an exact match for P1, P2, P3, and P4
following 132 runs. We compare the four methods : (Differential Evolution (DE), Particle
Swarm (PS), Response Surface (RS) and Simplex Model (SM)). Comparison of the best inverse
model match results for water cut and tracer data for this particular case showed very close
results similar to the actual. The differential evolution method showed the nearest results,
and the simplex method showed the least match. A summary of the root means square (RMS)
error is shown in Table 7.13.

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Table 7.13: Root Mean Square Error of Predicted Permeability vs. Actual Permeability.
Model
(Best
Match)
Inverse
Match
Water Cut &
Tracer Data
Perm at
region
P1
Perm at
region P2
Perm at
region P3
Perm at region
P4
Original
Value
(Perm)
0 1 5 2 3
Diff Evol
(DE)
Predicted
Value
0.0009 0.9729 4.9004 1.8301 2.8763
Diff Evol
(DE)
RMS error
4.3% 2.7% 2.0% 8.5% 4.1%
Simplex
Method
(SM)
Predicted
Value
0.0535 1.1765 9.2765 0.9725 7.9499
Simplex
Method
(SM)
RMS error
79.9% 17.7% 85.5% 51.4% 165.0%
Particle
Swarm
(PS)
Predicted
Value
0.00174836
9
0.956280
467
4.27360731
4
2.136474667 2.547087124
Particle
Swarm
(PS) RMS
error
10.2% 4.4% 14.5% 6.8% 15.1%

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The Differential Evolution Model run shows an exact match in the Permeability =1 region
with a 2.7 % deviation; the results showed a value of 0.972. For the area with Permeability=
5, the results showed a 2.0% deviation with a value of 4.900. For the area with
Permeability=2, the results showed an 8.5 % deviation with a value of 1.830. For the area
with Permeability = 3, the results showed a variation of 4.1% with a value of 2.876.
In comparison, the Particle Swarm Model run shows a close match in the Permeability=1
region with a 4.4 % deviation and a value of 0.956. For the area with Permeability=5, there
was an acceptable deviation of 14.5% with a value of 4.273. For the area with Permeability
=2, there was an acceptable deviation of 6.8% with a value of 2.136. Finally, for the area with
Permeability= 3, the variation of 15.1% deviation with a value of 2.547.
The comparative results show that both the Differential Evolution and Particle Swarm
method show relatively close results to the objective synthetic data.
In comparison, the Simplex Model run shows a close match in the Permeability=1 region with
a 17.7 % deviation and a value of 1.176. For the area with Permeability=5, there was a high
deviation of 85.5% with a value of 9.276. For the area with Permeability =2, there was a high
deviation of 51.4% with a value of 0.972. Finally, for the area with Permeability= 3, the
variation was out of range with a 165.0% deviation with a value of 7.949. The least matching
result was shown by the Response Surface Model Run, which shows a variance for the region
of Permeability = 1 of 6.6% with a value of 0.933. For the area with Permeability = 5, there
was a deviation of 70.4% with a value of 1.479. For the area with the Permeability=2 was a
very high deviation of 258.3 % with a value of 7.166. For the area of Permeability =3 a
difference of 109.8 % and a value of 6.294.

Response
Surface
(RS)
Predicted
Value
0.36872613
3
0.933936
207
1.47919363
1
7.166473243 6.294974799
Response
Surface
(RS)
RMS error
111.3% 6.6% 70.4% 258.3% 109.8%

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8 Chapter 8: Results of Inverse Simulation of Daily Water Cut and Tracer
Synthetic Data

We demonstrate the results of running the Perm (S Shape Channel). Figure 8.1 shows
applying the daily tracer production.

Figure 8.1: Daily Tracer Production Forward Simulation Results.
We see that the concentration shows daily proxy measurement for concentration
data. We now demonstrate the production data results as shown in Figure 8.2.

Figure 8.2: Daily Proxy Water Cut Forward Simulation Results.

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Now, Assuming the actual perm distribution is unknow and the concentration curve is
known we apply the research method technique as shown in Figure 8.3 below:
Starting from assumed homogeneous model assuming that we do not know the shape of the
permeability channel, and using the output known tracer concentration from the forward
simulation and then applying the research method using inverse modeling we obtain the
results.

Figure 8.3: Daily Proxy Water Cut Forward Simulation Results.

Figure 8.4 shows the results of the initial assumption, a homogeneous model with known
permeability:

Figure 8.4: Results of the Tracer Daily of Homogenous Proxy Model (Initial Model).

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Figure 8.5: Results of the Water Cute Daily of Homogenous Proxy Model (Initial Model).
The results used to run the differential evolution technique as shown in Figure 8.6 (a) and
Figure 8.6 (b) for tracer concentration and water cut.

Figure 8.6: (a): Inverse Modelling of Tracer Concentration Daily Data.

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Figure 8.6 (b): Inverse Modelling Water Cut Concentration Daily Data.
The results are shown in Figure 8.7 (a) and Figure 8.7 (b).

Figure 8.7: (a) Tracer Concentration Daily Data Match Results.

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Figure 8.7: (b) Water Cut Daily Data Match Results.
The interpretation of the results show that that including daily tracer concentration data and
water cut data give more accurate results. The channel permeability obtained from the
technique is 4.9 md in comparison to the actual value of 5 md.
The outside channel permeability inverse simulation obtained is 1 md showing and exact
match compared to the actual value of 1 md. Hence concluding that more frequent data in this
case a proxy model with the assumption of daily recorded measurement of water cut and
tracer data will give more accurate results.
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9 Chapter 9: CONCLUSION

Tracers are an essential source of information particularly prior/post to waterflood
secondary and tertiary EOR operations; however they have not been used to their fullest
potential in the Oil & Gas industry. Undetected high permeability channels in the reservoir
may disrupt costly EOR operations, which lead to low recovery efficiency.
Identifying high permeability channels will allow better operational design and optimal well
locations, which will lead to a higher recovery efficiency and the prevention of the loss of
millions of dollars in failed operations.
We aimed to investigate that the use of continuous measurement of tracer concentration data
in conjunction with production history data and accuracy in input for the simulation can
unravel some uncertainty about the heterogeneity of the reservoir.
Our research shows that more accurate results on determining heterogeneity in terms of
identifying high permeability channels are obtained when using inverse modeling techniques
with tracer and production data as inputs.
We conducted experiments to first understand the phenomena behind the physical factors
when injecting tracers into a porous media.
We chose to run our laboratory experiments on carbonate core samples of high producing
reservoirs in Kuwait since the majority of reservoirs are carbonates.
We injected commonly used non-invasive, relatively save FBA (Fluorobenzuate Acid)
alongside radioactive HTO as tracers to measure the effluent concentration to estimate and
permeability from the experiments.
We conducted literary research and concluded that the dispersion (mixing) phenomena
should not be ignored for more accurate results in simulation.
We concluded our experimentation with laboratory range values of the dispersion coefficient
and used these values in our forward simulation and obtained more accurate results on the
tracer effluent concentration measurement.
To investigate the implementation of the use of continuous tracer data to identify high
permeability channels, we designed a workflow for forward and inverse simulation modeling
and created ideal synthetic models with known permeabilities to compare our results.
The measurement of continuous data requires permanent downhole gauges that provide
uninterrupted, accurate measurement of the tracer concentration. In the absence of the
existence of such certified and commercial gauges and hence, continuous data from the actual

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oilfields, we created ideal synthetic models with known permeabilities to compare our
simulated results.
In our simulation, we designed one-dimensional (1D) and two-dimensional synthetic models
(2D) with known permeability parameters. We concluded from the forward simulation that
when running the simulation models with the dispersion coefficient, there were more
accurate results of the tracer concentration that matched our ideal model.
To investigate the use of tracer concentration for different variations of complex and non-
complex heterogeneity (homogeneous and heterogeneous models), we used synthetic data
with the assumption of continuous measurement and noncontinuous measurement. We
concluded that continuous measurement gave more accurate simulation results that matched
our ideal model.
We conducted several literature reviews on the use of tracer data; we did not come across
any published papers in the sourced references (see references section of our dissertation)
that assumed continuous measurement of data in their simulation.
To verify our results, we conducted a literature review to determine the best technique to use
to verify our conclusions. We verified our results using inverse modeling techniques.
We used a modern approach for inverse modeling by using genetic algorithms.
Our investigation showed that no apparent commercial simulator present today contains
modules that use continuous tracer data as input to computer-aided history matching genetic
algorithm techniques.
Hence, we considered differential evolution as the best approach and developed python code
to allow the simulator to read continuous tracer data as input
We concluded from our comparison of differential evolution using tracer data on our
synthetic model produced more accurate results compared to other common genetic
algorithm inverses modeling techniques such as (Simplex Algorithm, Particle Swarm
Algorithm and Response Surface Algorithm.
Although the use of continuous tracer data to identify permeability proved highly accurate
results matching the synthetic model, we aimed to conclude whether the use of tracer data
alone without the use of production history data would be sufficient.
We ran a further investigation on the synthetic models of various heterogeneity user tracer
data in conjunction with production data (typically used alone or with pressure data in past
literature in history matching). We concluded that in all cases, results are more accurate by
using tracer data and production history data (historical water cut data).
Hence, we conclude in our research that the use of continuous tracer data in conjunction with
production data (historical water cut) as an input to Differential Evolution, a modern inverse
modeling genetic algorithm gives accurate results in identifying heterogeneity of reservoirs
and in particular high permeability channels.

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10 Chapter 10: FUTURE WORK

Our research focus was on several combinations of heterogenous models with variations in
channel permeability. This is because our objective was to detect high permeability channels.
The models used in this research are single layer models. We believe that further variations
of proxy or synthetic models will allow further insight in adopting real time continuous
measurement in conjunction with water cut data using inverse modeling of differential
evolution.
These future expansions and the challenges are explained in detail in the next sections:
10.1 Combining Seismic Interpretation and Continuous Tracer Data

While continuous tracer measurement (CTM) provides the opportunity to identify
heterogeneity in the reservoir along the available wells, the lateral heterogeneity information
is hard to assess. In addition, although. CTM can help monitor dynamic in changes reservoir
properties (permeability, fluid saturation, etc.,) either directly or indirectly, at the well
locations, it is difficult to extrapolate such information away from the wells. Seismic data can
be helpful in filling in the large gaps within the xyz coordinates in the static terms (using 3D
seismic) and changes in time (x, y, z and t), in dynamic terms (using 4D seismic or at
continuous micro-seismic measurements).
As it is discussed in Aminzadeh et al (2013), the goal of reservoir characterization is to use
all the available date to create a model for the reservoir with as accurate estimates of the
reservoir properties as possible. The key word here is “all the available data”. Thus, as we
produce from the reservoirs, new data becomes available. This includes the production data,
updated decline curves, and possibly CTM data as well as new seismic data. Creating an
updated reservoir model or “dynamic model” is an important step to better understand any
important changes in the reservoir characteristic. This information is crucial to do a better
job in reservoir management and optimize production. It is also important when we need to
make certain interventions such as enhanced oil recovery (to increase permeability) or
artificial lift (to increase pressure), or do in-fill drilling in an optimum manner.
There are many challenges associated with linking seismic data with CTM, production data,
petrophysical data to create to dynamic reservoir models. The main one, referred to as SURE
challenge in Aminzadeh (2009) and further highlighted in Aminzadeh and Dasgupta (2013)
is recolonizing the fundamental differences in Scale, Uncertainty, Resolution and
Environment in different data types. While we are not claiming to have the answer to all
those challenges including how best to address CTM and 4D seismic data integration. Table
10.1, adopted from Aminzadeh and Dasgupta (2013) is a starting point to further investigate
the problem. In general, integration of geoscience (seismic, petrophysics, core and
microseismic data) and reservoir engineering data (production data, reservoir simulation,

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water cut, CTM) may be facilitated by machine learning and data analytic methods to address
the challenge.

Table 10.1: Some of the Challenges of Integration of 4D- Seismic and Reservoir Engineering Data.
Link 4D seismic to reservoir properties Integrate 4D seismic with reservoir
engineering
Repeatability - acquisition, reprocessing,
cross-equalization.
Integration -- huge amounts of data,
disparate data sets at different Scales,
Uncertainty, Resolution and Environment
(SURE Challenge).
Interpretation -- rock physics, quantitative. Accelerate integration loop to increase
benefit of data and use of machine learning
to supplement human intelligence
Lack of calibration data--validation of
different methods.
Parameterization. Use of well log data and
Continuous Tracer Measurement (CTM)
Decoupling of reservoir properties --
pressure and saturation.
Non-uniqueness can be reduced through
use of CTM. And Machine learning
approaches
Definition of 4D Seismic attributes Automated history matching -- misfit
function, optimization (e.g. genetic
algorithm,) stopping criteria, Genetic
algorithm
10.2 Expanding the Research Method to Simulate Multi-layer Reservoir Case
Studies

Our research presents the novel inverse modelling technique based on genetic optimization
algorithm to interpret continuous tracer data and historical water cut data in an oil reservoir.
The technique tested on single-layer synthetic numerical models gives accurate results in
identifying heterogeneity of reservoirs.
The research focus is combinations of heterogeneous models with variations in channel
permeability, in which, the initial step is testing on single-layer models. The expansion of
application the proposed technique in case of multilayer reservoir models will be an exciting
research direction in the future. However, this will encounter not simple challenges.
Multi-layer reservoirs or stratified reservoirs are one kind of heterogeneous reservoir consist
of two or more layers, in which petrophysical properties of each layer such as porosity,

138

permeability, layer thickness, total compressibility, and initial skin factor are different. These
properties strongly affect the behaviour of injected fluid (Mohammadreza Jalaji et al. 2016).
In a layered reservoir with communicating layers, the injected fluid spreads from higher
pressure layers to lower pressure layers because of a different pressure gradient, which is
well-known as cross-flow phenomena. Natural cross-flow is referred in case of the pressure
equilibrium condition between layers, whereas forced cross-flow is referred in case pressure
equilibrium is not achieved due to production activities (Mohammadreza Jalaji et al., 2016).
In a layered reservoir with non-communicating layers, cross-flow isn't introduced. The
problem becomes more complicated with each layer as an individual flow system with
different conditions.
The interpretation of permeability in a multi-layer reservoir model based on tracer data and
historical water cut data using inverse modelling technique is not an easy mission because of
some reasons. First, the complexity of interlayer flow within a reservoir, in which each layer
has different petrophysical properties that depend on layer depth and deposition
environment. Second, the production rate at the wellhead and historical of the whole
reservoir are controlled by porosity, permeability, the layer thickness of each layer, the
interaction of layers through a wellbore and external boundary and interaction between
layers characterized by cross-flow effect (Eskandari Niya et al. 2012, Faruk Civan et al. 2013).

10.2.1 Challenges of simulating multi-layer reservoir

Reservoir modelling has become a powerful tool since the 1950s in the oil and gas
exploitation revolution helps to interpret reservoir parameters as well as to predict problems
related to extraction efficiency (Jørg E. Aarmes et al. 2007). Modelling multi-layered reservoir
problem using numerical methods was first proposed in the 1960s. Jacquard and Jain (1965)
used a number of homogeneous cells of constant reservoir properties to describe a single-
layered reservoir. Regression analysis techniques were then applied to determine the
permeabilities of the cells (Soleimani et al. 2017).
Simulating of multi-layer heterogeneous reservoirs depends on many factors such as detailed
sedimentary and stratigraphic data of reservoir, the appropriate simulator solution methods
and options and good interpretation of measured data (Jørg E. Aarnes et al. 2017, A.Y.
Assadoor et al. 1989).
Basic data requirements for general modelling that can be listed are porosity, permeability,
organic compositions, compressibility, stress, boundary, fluid PVT properties, wettability,
flow rate, historical of production (A. Satter et al. 2015). Until nowadays, available
information of the reservoir is quite discrete due to the limit in the measurement methods
and costs. Some introduced methods include seismic, well testing, core testing with their own
advantages and disadvantages. Seismic survey is reported that can give sort X-ray image of a
reservoir, but require high cost with a normal resolution of ten meters. This, of course, makes
challenges in the case of multi-layer reservoirs. Well testing gives the representative
information of the vicinity well. The more detailed information is regarded from core tests

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using CT scan, X-ray technique, tracer technique, however, testing on a small scale of
centimeters is one of the most limitations of this method. Moreover, the detailed data
obtained from well testing and core testing are not representative of the whole reservoir and
extrapolations applied in the model always have a certain degree of uncertainty (Jørg E.
Aarnes et al. 2007).
It can be said that fluid mechanics in oil reservoirs is controlled by a variety of physical
processes which are described using complex non-linear partial differential equations (P.G.
O. Ossai et al. 2018). By applying the finite difference approximations, the numerical method
allows modelling the reservoir in the form of a set of grid cells with a certain degree of
accuracy. It can be said that fluid mechanics in oil reservoirs is controlled by a variety of
physical processes which are described using complex non-linear partial differential
equations. By applying the finite difference approximations, the numerical method allows
modelling the reservoir in the form of a set of grid cells with a certain degree of accuracy.
Each grid cell has specific factors including dimension - depends on size and type of reservoir,
the scope of simulation, rock and fluid properties - in which, the most important factors are
porosity and permeability. The larger and the more heterogeneous the model such as multi-
layer reservoir requires a larger number of grid cells (E. Aarnes et al. 2007, A. Satter et al.
2015).
The previous studies pointed out that most of the challenge problems in simulating multi-
layer model are related to (1) the high level of detail required in the input data including areal
and vertical grids, fluid/layer contacts, saturation, and PVT functions, (2) the initialization
model problems - setting up exactly gravity-capillary equilibrium conditions for layers, (3)
the historical matching problems may be influenced by cross-flow effect (E. Aarnes et al.
2007, A.Y. Assadoor et al. 1989). This demands the improvement in computational methods
that ensures the consistency of a large number of data and properly describes special fluid
flows in the model.

10.2.2 Expanding the Research Method Continuous Tracer Detection for Multi-layer reservoir

Tracer technique is known as a helpful method provides important information about
reservoir heterogeneities serving evaluation of sweep efficiency. In the case of multi-layer
reservoir, the tracer response curve is the total response curve contributed by individual
layer responses. The inversion problem here is the determination of the number of the layers
and its properties based on the tracer breakthrough curve representing the multi-layered
system. The shape and peaks of the tracer response curve give information about the level of
stratification. The tracer response curve of individual layers can be obtained by the
decomposition of the total response curve using the optimization technique. The
breakthrough time, as well as the height and width of the peaks, reveal the permeability of
each layer (Maghsood Abbaszadeh- Dehghani 1984, S. Mishra et al. 1990).
The first semi-analytic model for interpretation of tracer response curve in the study on
evaluation heterogeneity of a five-spot pattern was introduced by Brigham and Smith in 1965

140

(Brigham and Smith, 1965). Dehghani (1984) presented an effective optimization technique
is presented to deconvolute individual layer responses. The technique was tested to analyze
a tracer profile from a field five-spot system (Maghsood Abbaszadeh- Dehghani 1984). Tong
Shen et al. (2017) presented a new approach to evaluate limited cross-flow between layers
of a stratified reservoir using tracer. The proposed method produces satisfactory results
when testing on both numerical simulation and field examples (Tong Shen et al., 2017).
For years, chemical and radioactive tracer have been applied in oil reservoir to investigate
well connectivity, oil saturation and reservoir heterogeneous. Although the technical benefits
of traditional tracers have been well demonstrated and recognized, they still have limitations
about background noise, low detection sensitivity due to dilution effects, high costs of
materials and analysis, time-consuming due to laboratory procedures, potential
environmental contamination, health requirements. Along with the outstanding
development of nanotechnology in many areas in recent years (e.g. medicine, paint, optics,
electronics, and the environment), nanoparticles are expected to become potential tracers to
provide an effective solution to address these challenges (Agenet et al. 2012, Ellis et al. 2016,
Xiang-Zhao Kong et al., 2018).
Optical nanoparticles-based tracers such as carbon-based fluorescent nanoparticles, silica-
based fluorescent nanoparticles, and rare-earth ion-doped nanoparticles are well-known in
applications related to the investigation of the petroleum reservoir. In which, the continuous
concentration data of tracer can be detected in real-time using modular optical spectrometer
is one of the most outstanding advantages (Ellis et al. 2016, Ellis et al. 2017, Agenet et al.
2018, H. Ow et al. 2018, M.O. Metidji et al. 2019).
Basic sensing techniques for measurement of fluorescent nanoparticles-based tracer meet
limited due to the natural fluorescence of crude oil in the sample. Y-J. Chuang et al. (2016)
introduced a new detection method used for LiGa5O8:Cr3+ near-infrared persistent
luminescent nanoparticles as a tracer nano-agent to suppress the background fluorescence.
The initial results showed the tracer detection limit of about 200 ppm in synthetic seawater
containing 5 wt.% crude oil.
N. Agenet et al. (2018) described the design and characterization of fluorescent silica nano-
objects using core flooding test at room temperature and in the absence of crude oil. The test
showed the recovery result of 98% upon injection of 0.5 ppm solution of nanoparticles.
Ellis et al. (2016, 2017) introduced a type of carbon-based nanoparticles (ADOTS) which
synthesized through a hydrothermal treatment process. A sensing system detecting
fluorescent nanoparticles in real-time was developed with the analysis time of 5 minutes per
sample and the detection limit at the ppb level. The sensing system employs three distinct
processes: oil-water separation, filtration of the separated production water and detection of
the fluorescent nanoparticles. The report also gave the future design of sensing system in the
form of "analyser box".
In recent years, the validation of tracer measurement in real-time has become a revolution,
however, still faces many challenges include controlling the purification process after

141

producing the nanoparticles and the formation of a stable dispersion, as well as, the accuracy
of analytical methods (M. O. Metidji et al. 2018).
As a result, the challenges of applying continuous tracer measurement in multi-layer
reservoir consists of type tracer and detection method, as well as, tracer interpretation using
an optimization technique.

142

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Abstract (if available)

Abstract Properties that control displacement processes, particularly in (IOR) and enhanced oil recovery (EOR) operations, are controlled by heterogeneity in the subterranean reservoirs. In this study, we focus on the merits of using continuous monitoring of injected tracers to more accurately investigate reservoir direction of fluid transport, flow performance, and reservoir connectivity. ❧ Tracer technology has been sparsely used in the past as an essential method in determining reservoir heterogeneities and flow directions. In this study, we introduce a novel concept of tracer analysis with the assumption of continuous monitoring of tracer effluent concentration. Combined with water cut data and by performing inverse modeling using a new method we realize significant improvements in the reservoir characterization process. ❧ The modeling is based on the assumption of real-time measurement of the effluent concentration of tracer data using downhole detection units

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Asset Metadata

Creator Najem, Noha N. (author)

Core Title A method for characterizing reservoir heterogeneity using continuous measurement of tracer data

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Petroleum Engineering

Publication Date 12/11/2019

Defense Date 08/27/2019

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag continuous measurement,OAI-PMH Harvest,reservoir heterogeneity,tracer,tracer data

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Aminzadeh, Fred (committee chair), Ershaghi, Iraj (committee member), Ghanem, Roger (committee member)

Creator Email najem@usc.edu,noha_najem@hotmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-252332

Unique identifier UC11674959

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Legacy Identifier etd-NajemNohaN-8048.pdf

Dmrecord 252332

Document Type Dissertation

Rights Najem, Noha N.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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