The effect of residual oil on deep bed filtration and formation damage

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The Effect of Residual Oil on Deep Bed Filtration

and Formation Damage

Proefschrift

ter verkrjging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.dr. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 16 april 2007om 12:30 uur door

Mohammad Ali Jumaa

Mohammad Ali Jumaa ALI

ALI

Master of Petroleum Engineering, Colorado School of Mines, USA Bachelors of Petroleum Engineering

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Dit proefschrift is goedgekeurd door de promoter: Prof. dr. P.K. Currie

Samenstelling promotiecommissie:

Rector Magnificus, Technische Universiteit Delft, voorzitter Prof. dr. P.K. Currie, Technische Universiteit Delft, promotor Prof. dr. S.M. Hassanizadeh, Universiteit Utrecht

Prof. dr. D Marchesin, Instituto de Matematica Pura e Applicada, Brasil Prof. dr. ir. G.Ooms, Technische Universiteit Delft,

Prof. dr. W.R. Rossen, Technische Universiteit Delft,

Dr. Mohammad Jassim Salman, Kuwait Institute for Scientific Research Dr. P.L.J.Zitha Technische Universiteit Delft,

reservelid

Prof. dr. ir J.D. Jansen, Technische Universiteit Delft,

Financial Support

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3 Dedication

Dedication

In the name of Allah most gracious most merciful

To my dear mother

for her love, support and understanding

throughout this study

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Conference Proceedings Conference Proceedings Conference Proceedings Conference Proceedings

Ali, M.A.J., P.K. Currie and M. J. Salman (2005). "Measurement of the particle deposition profile in deep-bed filtration during produced water re-injection." SPE-93056. Presented at the 14th SPE Middle East Oil & Gas Show and Conference held in Bahrain International Exhibition Centre, Bahrain, 12–15 March 2005.

Ali, M.A.J., P.K. Currie and M. J. Salman (2005). "Effect of Residual Oil on the Particle Deposition in Deep-Bed Filtration During Produced Water Reinjection." SPE-94483. Presented at the European Formation Damage Conference held in Scheveningen, The Netherlands, 25-27 May 2005.

Ali, M.A.J., P.K. Currie and M. J. Salman (2007). “Permeability Damage due to Water Injection Containing Oil Droplets and Solid Particles at Residual Oil Saturation”, SPE-104608. Presented at the 15th SPE Middle East Oil & Gas Show and Conference held in Bahrain International Exhibition Centre, Bahrain, 11–14 March 2007.

Ali, M.A.J., P.K. Currie and M. J. Salman (2007). “The Effect of Residual Oil on Deep-Bed Filtration of Particles in Injection Water”, SPE-107619. Will be presented at the European Formation Damage Conference held in Scheveningen, The Netherlands, 30 May–1 June 2007.

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Samenvatting

Een on-lijn lineair Röntgen apparaat werd gebruikt voor het onderzoeken van diepe filtratie tijdens water injectie in Bentheimer zandsteenkernen. Een suspensie in water van hematiet deeltjes werd met verschillende stroomsnelheiden en concentraties geinjecteerd, om zo het afzettingsprofiel te bestuderen. De Röntgen attenuatie is sterk gerelateerd aan de dichtheid, en dus is het mogelijk op iedere positie en op ieder tijdstip de hoeveelheid hematiet die afgezet wordt te meten. De overeenkomst van de Röntgen metingen werd bewezen bij verschillende spanningsniveaus en voor monsters met verschillende hematiet concentraties en vloeistofsaturaties. Om het Röntgen signaal te calibreren, werden monsters met een bekende hoeveelheid hematiet gemaakt, en de absorptie coefficient in Lambert’s vergelijking werd bepaald voor iedere component (gesteente, vloeistof in de poriën en hematiet). Herhalings-testen voor de druk en Röntgen metingen bewezen de betrouwbaarheid van de metingen, en een chemische analyse gaf een goede overeenkomst met de Röntgen signalen. De Röntgen signalen tonen veel ruis.

De Röntgen data permitteren een duidelijke separatie tussen de effecten van interne afzetting en externe afzetting (de vorming van een filtercake). Externe filtercake begint bijna direct te groeien na de injectie van een paar porieën-volumes. Dus interne afzetting vindt plaats terwijl de externe filtercake gevormd wordt. Een significant deel van dit proefschrift is gericht op het evalueren van de juistheid van Deep Bed Filtration Theory (een theorie die de afzetting van deeltjes in een poreus medium beschrijft). Hierbij wordt het effect van afzetting op de vorm van de filtratie en permeabiliteit functies bepaald, en een analyse gemaakt van de invloed van stroomsnelheid, concentratie en aanwezigheid van residuale olie.

Het effect van residuele olie saturatie op het afzettingsprofiel werd met monsters met honderd procent water saturatie vergeleken. De aanwezigheid van olie zorgt voor een grotere reductie in permeabiliteit. Er werd ondekt dat er een diepere invasie is met residuale olie, en dat de externe filtercake dikker is. Mogelijke verklaringen voor deze resultaten worden besproken, gebaseerd op SEM analyses en de eigenschappen van de geinjecteerde suspensies.

Het was niet mogelijk om de afzetting van hematiet in de externe filtercake rechtstreeks te meten. Het was ook niet mogelijk om de uiteindelijke totale afzetting te meten, omdat tijdens de verwijdering van het monster een significant deel van de externe filtercake verloren ging. Om een schatting te maken van de variatie van de filtercake permeabiliteit, werd een membraan test gebruikt. Gebaseerd op de resultaten van de membraan test, wordt geconcludeerd dat de drukval over de externe filtercake insignificant is in vergelijking met de drukval over de zandsteen kern.

Methoden om de afzettingsmetingen te gebruiken om de vorm van de filtratie coefficient

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(3) konstante coefficient gebaseerd op een gobale massa analyse en (4) niet-konstante filtratie coefficient gebaseerd op een locale massa analyse. De laatste twee methoden zijn nieuw. Iedere methode gebruikt de Röntgen data, maar op verschillende manieren. De voordelen en nadelen van de vier methoden zijn ook besproken. De afzettingsdata impliceren dat de filtratie coefficient

λ

groter is voor lagere stroomsnelheden dan voor hogere snelheden, en groter voor S condities dan voorw S . or

De conclusies van de Röntgen analyse werden bevestigd door de gemeten reductie in permeabiliteit en de chemische analyse van de afzetting van hematiet. Modellen zijn voorgesteld voor de correlatie tussen de druk-toename en de afzetting in of bij vernauwingen van porieën. Voor de S tests, bevestigen de permeabiliteitsmetingen de w

resultaten van de afzettings analyse. Er is een duidelijk verschil tussen de lage en hoge snelheidsmonsters. Geconcludeerd wordt dat voor hoge snelheid, afzetting in iedere vernauwing minder is, maar dat de penetratie dieper is, resulterend in relatief meer permeabiliteitsreductie. Voor de S monsters is de conclusie minder duidelijk. Van de or

afzettingsdata is de penetratie veel groter dan voor de S monsters. De permeabiliteits w

reductie toont een verschil tussen hoge en lage snelheiden, maar minder duidelijk dan voor de S monsters. w

Ook werd een minder omvattende studie gemaakt van de invloed van olie druppels in het ge-injecteerde water op de interne and externe afzetting en permeabiliteits reductie. De druk werd gemeten, terwijl de uitgang concentraties van beide olie druppels en hematiet deeltjes werden geregistreerd. Van observaties van verschillende concentraties van hematiet deeltjes en olie in water, is het duidelijk dat de diameter van de hematiet deeltjes niet met de concentratie verandert, terwijl de diameter van olie druppels toeneemt met toenemende olie concentratie. Bij toenemende olie concentratie hebben olie druppels de neiging om te coalesceren. Als zowel olie als hematiet aanwezig zijn, dan zijn de kleine en grote olie druppels stabiel in de suspensie, maar de middelgrote druppels hebben aangehechte hematiet deeltjes. Correlatie wordt gemaakt tussen de uitgangs concentratie en de toename van de druk, en de invloed van deeltjes afzetting op de verbeterde residuele olie mobiliteit wordt onderzocht.

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vii vii vii vii

SUMMARY

An on-line linear X-ray apparatus has been used to examine deep bed filtration during water injection in Bentheimer sandstone core samples. Hematite particles suspended in brine were injected at different flow-rates and concentrations to investigate the deposition profile. The X-ray attenuation is strongly related to the density; and therefore the amount of deposited material can quantitatively be measured at any time and distance along the core sample. The consistency of the X-ray measurements at different voltage settings was confirmed for samples with different hematite concentrations and fluid saturations. To calibrate the X-ray signal, samples were prepared with a known amount of hematite, and the absorption coefficient in Lambert's equation was measured for each material (rock, fluids in pores, and hematite). Repeatability tests on pressure measurement and X-ray data confirm the reliability of the measurement, and chemical analysis gave good agreement with the X-ray analysis. The X-ray signals showing the attenuation due to hematite deposition are noisy.

The X-ray data allows a clear separation between the effects of internal deposition and external filter-cake formation. External filter cake starts to form almost immediately after injecting a few pore-volumes. Therefore, internal deposition continues while the external filter cake is being formed. A significant part of this thesis is concerned with evaluating the adequacy of the deep bed filtration theory. This includes determining the form of the filtration and permeability damage functions, and analyzing their dependence on flow-rate and concentration and on the presence of residual oil.

The effect of residual oil saturation on the deposition profile of suspended solids in the injected water was compared with fully brine saturated cores. The presence of oil caused greater apparent damage (reduction of permeability). It was also observed that there is deeper invasion at residual oil saturation than at full brine saturation, and it was inferred from the X-ray measurements that the external filter cake is thicker. Possible mechanisms for this difference in deposition are discussed based on SEM analysis and characteristics of the injected suspensions.

The deposition of hematite in the external filter-cake on the inlet face could not be measured directly during the test. Nor could the final deposition be measured because a large amount was lost during removal of the sample. The membrane test has been used to give an estimation of external filter cake permeability variation with thickness. Based on this membrane test, it is concluded that the pressure drop across the external filter-cake is insignificant compared with the pressure drop in the sandstone core due to internal deposition.

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different ways. The advantages and disadvantages between the four models are also discussed in this thesis. The invasion data implies that the filtration coefficient

λ

is larger for lower flow-rates than for higher flow-rates. The filtration coefficient is larger under pure water conditions (denoted by S conditions) than under water with residual oil w

conditions (denoted as S conditions). or

Permeability decline as determined by the pressure change and chemical analysis of hematite concentration were used to confirm the X-ray analysis. Models are proposed to correlate pressure buildup due to formation damage caused by particle deposition at or near the pore throats. In the case of the S samples, the permeability damage w

measurements confirm the findings of the deposition analysis. There is a clear difference between the low and high flow-rate samples. It is inferred that at high flow-rate, deposition at each pore throat is lower, but the penetration is deeper, leading to relatively more permeability damage. For the S samples, the picture is less clear. From or

the deposition data, the penetration depth is much larger than for the S samples. The w

permeability damage data shows a difference between high and low flow-rates, but not as clearly as for the S samples. w

A less extensive study has also been made of the effect of the presence on oil droplets in the injected water on internal and external deposition and impairment. Differential pressure was measured while monitoring the effluent particle concentration of both solid and oil droplets. Different concentrations of solid in water, oil in water, and solid-oil in water showed that solid particle diameter does not change with increasing particle concentration, whereas oil droplet size increases with oil concentration. Oil particles have higher tendency to coalesce with increasing oil concentration. When both oil and solid are present, then the very small and very large oil droplets are stable in suspension but the medium size oil droplets coalesce and have attached solid particles, confirming the oil wettability preference of hematite. Correlation between the effluent concentration and pressure buildup is made, and the effect of solid deposition in improving residual oil mobility is investigated.

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λ

FROMLABORATORYTESTS... 21

3.6.1 Determining

λ

_avgfrom measured inlet and outlet concentration... 22

3.6.2 Analysis based on global mass conservation... 22

3.6.3 Analysis based on local mass distribution ... 24

3.7 MODELSFORPERMEABILITYIMPAIRMENT ... 25

3.7.1 Linear permeability reduction model ... 26

3.8 SUMMARYANDCONCLUSIONS ... 28

4. 4.4. 4. EXPERIMENTAL SETUP AEXPERIMENTAL SETUP AND PROCEDUREEXPERIMENTAL SETUP AEXPERIMENTAL SETUP AND PROCEDUREND PROCEDUREND PROCEDURE... 29292929 4.1 INTRODUCTION ... 29

4.2 MATERIALSUSED... 29

4.3 X-RAYCALIBRATION ... 29

4.3.1 Measurement of absorption coefficients... 33

4.3.2 Determination of the Hematite concentration from the X-ray Signal... 35

4.4 COREANDFLUIDSPREPARATION... 37

4.4.1 Core preparation ... 37

4.4.2 Brine and hematite preparation... 37

4.4.3 Emulsion preparation ... 38

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4.5 HIGHSHEARSTRESSMIXER...38

4.6 CORESAMPLESATURATION...39

4.7 INJECTIONOFWATERCONTAININGSUSPENDEDSOLIDSANDOILPARTICLES ...40

4.7.1 Flow rate measurement ...42

4.7.2 Pressure measurement ...42

4.8 EXTERNALFILTER-CAKEMEASUREMENT ...43

4.9 PARTICLESIZEANALYZER,VIPA ...46

4.10 HEMATITEWETTABILITY ...50

4.11 SEMANALYSIS...51

4.12 CHEMICALANALYSIS...51

4.13 ZETAPOTENTIALMEASUREMENT...53

4.14 SUMMARYANDCONCLUSIONS ...55

5. 5. 5. 5. EXPERIMENTAL RESULTSEXPERIMENTAL RESULTSEXPERIMENTAL RESULTSEXPERIMENTAL RESULTS ...57575757 5.1 INTRODUCTION ...57

5.2 DEPOSITIONPROFILEFROMX-RAY...58

5.3 FILTERCAKE ...60

5.4 PRESSUREDATA ...64

5.4.1 Injection of brine with hematite...64

5.5 QUALITYCONTROL...65

5.6 CHEMICALANALYSIS...67

5.7 SCANNINGELECTRONMICROSCOPE(SEM) ...69

5.8 ZETAPOTENTIALMEASUREMENT...70

5.9 INJECTIONOFOILY-WATERINTOBRINE-SATURATEDCORES...72

5.10 INJECTIONOFOILYWATERWITHSUSPENDEDHEMATITEINTO RESIDUAL-OIL-SATURATEDCORES ...73

6. 6. 6. 6. DISCUSSION OF RESULTDISCUSSION OF RESULTDISCUSSION OF RESULTDISCUSSION OF RESULTSSSS...76...767676 6.1 INTRODUCTION ...76

6.2 X-RAYDEPOSITIONMEASUREMENTS ...78

6.2.1 Effect of flowrate on external cake formation...79

6.2.2 The effect of concentration on external cake ...79

6.2.3 The effect of velocity on deep bed filtration profiles...80

6.2.4 Invasion depth...82

6.3 DETERMINATIONOFTHEFILTRATIONCOEFFICIENT...85

6.3.1 Constant filtration coefficient determined by fit of the inlet concentration ...85

6.3.2 Linear filtration coefficient model ...87

6.3.3 Filtration coefficient factor from global mass analysis ...90

6.3.4 Filtration coefficient from local mass distribution analysis...94

6.3.5 Comparison of the results for the filtration function ...96

6.3.6 The effect of residual oil on the filtration coefficient ...97

6.4 PERMEABILITYDAMAGEANDPARTICLEDEPOSITION...98

6.4.1 Permeability damage modeling ...98

6.4.2 Discussion of the permeability damage results ...102

6.5 INJECTIONOFOILY-WATER...103

6.6 INJECTIONOFOIL-WATERCONTAININGSOLID...105

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xi xi xi xi 7. 7.7.

7. IMPLICATIONS FOR MATIMPLICATIONS FOR MATRIX REIMPLICATIONS FOR MATIMPLICATIONS FOR MATRIX RERIX RERIX RE----INJECTION OF INJECTION OF PRODUCED WATERINJECTION OF INJECTION OF PRODUCED WATERPRODUCED WATERPRODUCED WATER... 109... 109109109

7.1 INTRODUCTION ... 109

7.2 MODELSFORTHEFILTRATIONCOEFFICIENT... 110

7.3 CONSTANTFILTRATIONCOEFFICIENTMODEL

λ

=

λ

0... 111

7.4 DEPOSITIONSTEP-CHANGEFILTRATIONFUNCTION ... 111

7.5 FLOWRATESTEP-CHANGEFILTRATIONFUNCTION... 115

7.6 SUMMARYANDCONCLUSIONS ... 118

8. 8.8. 8. CONCLUSIONS AND RECOCONCLUSIONS AND RECOMMENDATIONSCONCLUSIONS AND RECOCONCLUSIONS AND RECOMMENDATIONSMMENDATIONSMMENDATIONS ... 120120120120 8.1 MAINACHIEVEMENTS ... 120

8.2 PRINCIPALCONCLUSIONSOFTHESIS ... 121

8.3 RECOMMENDATIONFORFUTUREWORK ... 123

NOMENCLATURES NOMENCLATURESNOMENCLATURES NOMENCLATURES ... 124124124124 REFERENCES REFERENCESREFERENCES REFERENCES... 126126126126 APPENDIX A. DEPOSI

APPENDIX A. DEPOSIAPPENDIX A. DEPOSI

APPENDIX A. DEPOSITION PROFILE FROM XTION PROFILE FROM XTION PROFILE FROM X-TION PROFILE FROM X---RAY SYSTEMRAY SYSTEMRAY SYSTEMRAY SYSTEM ... 137...137137137 APPENDIX B. PHOTOS

APPENDIX B. PHOTOSAPPENDIX B. PHOTOS

APPENDIX B. PHOTOS OF CORE SAMPLES OF CORE SAMPLES OF CORE SAMPLES OF CORE SAMPLES ... 142...142142142 APPENDIX C. PHOTO I

APPENDIX C. PHOTO IAPPENDIX C. PHOTO I

APPENDIX C. PHOTO IMAGES OF EQUIPMENTMAGES OF EQUIPMENTMAGES OF EQUIPMENT...MAGES OF EQUIPMENT... 147...147147147 APPENDIX D. DEPOSI

APPENDIX D. DEPOSIAPPENDIX D. DEPOSI

APPENDIX D. DEPOSITION PROBABILITY ANDTION PROBABILITY ANDTION PROBABILITY ANDTION PROBABILITY AND PECLET NUMBER PECLET NUMBER PECLET NUMBER PECLET NUMBER... 149149149149 APPENDIX E.

APPENDIX E.APPENDIX E.

APPENDIX E. ACCURACY OF DATA ACCURACY OF DATA ACCURACY OF DATA ACCURACY OF DATA ... 154...154154154 ACKNOWLEDGMENT

ACKNOWLEDGMENTACKNOWLEDGMENT

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1.

1. INTRODUCTION

INTRODUCTION

Re-injection of produced water is of increasing importance in the oil industry as water production continues to increase worldwide. Currently, it is estimated that three barrels of water are produced for every barrel of oil produced (Kevin et al. 2003). Re-injection provides an environmentally acceptable solution to the disposal of produced water, and contributes to pressure maintenance when injection takes place into the producing reservoir itself. A rich source of information on this topic summarizing 30 years of water injection, can be found in Hofsaess and Kleintz (2003). Injection can take place under matrix injection or fracturing conditions. In both cases, the performance of the injection well and the distribution of the injected water are strongly influenced by the build-up of formation impairment around the wellbore or at the fracture face.

This impairment is caused by solid particles and small oil droplets in the injected water, which can block the reservoir pores and cause rapid and severe permeability decline depending on the water and reservoir characteristics. Total removal of solids and oil may not be economically feasible or practically possible. Therefore, to minimize or slow the rate of impairment, it is important to understand the relationship between the impairment mechanism and water quality parameters such as the concentration of solids, particle size, oil droplet concentration and injection rate, (Al-Hamadah 1995, Al-Humadhi 1988).

In the experimental study reported in this thesis, different concentrations of oil droplets and solid particles suspended in water were injected into sandstone core samples and the formation damage measured. The challenge with this sort of experiment is that two types of formation damage are taking place simultaneously. On the front surface of the core, particles are deposited to form an external filter cake. Once inside the core, particles are transported through the pores and may be captured and deposited. This process is known as deep bed filtration. Separating these two types of formation damage, and studying them individually is impossible in most experiments, since measurements are normally made only of the inflow and effluent properties of the flowing suspension, together with pressure gradient measurements. Real-time measurement of the capture and deposition process is not possible.

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The equations governing the simple deep bed filtration theory consist of a volumetric balance of conservation of mass and a kinetic equation governing the rate of particle deposition. This deposition rate is governed by the empirical filtration coefficient (Rege and Fogler 1988). The pressure is determined by Darcy’s law, with the permeability dependent on the local deposition to account for the effects of impairment. The X-ray apparatus monitors the internal deposition at any given time and at any distance inside the core and thus provides a direct check of the models. We note again that this check is unaffected by any external deposition taking place on the entrance face of the core, thus overcoming the limitations of many earlier experiments.

However, when measuring the pressure drop, we have not been able to separate directly the internal and from the external effects. The total pressure buildup due to particle deposition was measured, including both internal and external deposition. A membrane test was used to mimic the pressure buildup due to external deposition alone, allowing estimation of the pressure buildup caused by internal deposition alone. This data can then be coupled with the X-ray deposition data to give insight into the dynamic growth of the filter cake caused by external deposition and the permeability impairment caused by internal deposition.

So a significant part of this thesis is concerned with evaluating the adequacy of the deep bed filtration theory. This includes determining the form of the filtration and permeability damage functions, and analyzing their dependence on flow-rate and concentration and on the presence of residual oil.

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2.

2. PRODUCED WATER RE

PRODUCED WATER RE

PRODUCED WATER RE-

-

-INJECTION

-

INJECTION

2.1 2.12.1

2.1 INTRODUCTIONINTRODUCTION INTRODUCTIONINTRODUCTION

Production of hydrocarbons is usually accompanied by the production of water. This produced water consists of formation water and/or water that has previously been injected into the formation. As more oil is produced, the amount of produced water increases. Over the life span of the field, the volume of the produced water will exceed the volume of oil produced, by a factor of 3-6 times (Kevin et al. 2003). Unfortunately, the produced water is not a saleable product, hence, an operator must find ways to handle relatively large amount of water in an environmentally-acceptable manner at the lowest cost. One way of managing this water is to re-inject it for disposal, pressure maintenance or enhanced oil recovery. An important and difficult task in the re-injection process is the prediction of the impact of water quality on well injectivity. This is mainly due to the poor understanding of the deposition mechanisms by which suspended solids and oil droplets present in the produced water are retained by the formation.

Injection of water can occur under either fracturing or matrix conditions. If the bottomhole pressure is below the fracturing pressure, matrix injection will occur. Otherwise, a hydraulic fracture will be created. It is believed that many water injection wells are fractured, because the fracturing pressure of the formation is lowered by the injection of cool water. Field experience shows that several volume of solids can be injected without significant loss of injectivity (Paige and Murray 1994, 1995, Martins et al. 1995). Fracturing may introduce problems as it may cause poor vertical sweep due to fracturing through barriers and early breakthrough (Garon et al. 1988). Generally, produced water may be re-injected with minimal treatment when injected under fracturing conditions. Water quality requirements for matrix reservoirs are more stringent. If the injected water contains solids or oil droplets that are large enough to plug the formation, injectivity will decline. To avoid this, the produced water must be filtered prior to injection, but filtration is costly and it should be balanced against the cost of other alternatives. In some cases it is more profitable to inject unfiltered water under fracturing conditions (Doscher and Weber 1957, Farley and Redline 1968).

In matrix injection projects, thousands of pore volumes typically flow through the near wellbore region where most particles deposit. Understanding and predicting the resulting decline in injectivity requires the knowledge of several parameters involving the water quality, formation characteristics, and the rate of deposition. In the case of injection under fracturing conditions, the same parameters influence the leak-off of particles and fluid through the fracture face and the growth of the fracture with time.

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Kharrat 1995) and perforation plugging (Al-Taq et al. 2005). Chetri et al. (2005) discuss wellbore and perforation clean-up. These effects are not discussed here

2.2 2.2 2.2

2.2 WATER QUALITYWATER QUALITYWATER QUALITYWATER QUALITY

Assessment of water quality is important. The types of solids, fluid droplets and bacteria present in produced water determine the injectivity decline. Chemical composition, size and concentration, salinity, temperature, pH, and reaction gases are all factors influencing injectivity (Kreuger 1986). The main agents causing injectivity decline are suspended solids (specified by the factor Total Suspended Solids, TSS), oil in water (OIW) and bacteria. It is well understood that TSS is by far the most damaging (Hsi et al. 1994) and that losses in injectivity can be expected if water with a high TSS is injected. The TSS measured at the surface is not in fact that experienced at the bottom of the well. Solid content increases with the addition of biocide, corrosion inhibitors and scale inhibitors (Nasr-El-Din and Al-Ghamdi 1996a, 1996b). These contaminants form solid particles (some referred to as “schmoo”) during flow through the pipeline and tubing into the formation. The type of solids may change during the field life cycle of the injection well. Fines an order of magnitude smaller than the formation pore size can cause considerable damage. Fines can come from either the producing or injection formation (Civan 1990, 1992, Ohen and Civan 1991, 1993, 1996) or from completion fluids (Gruesbeck and Collins 1982, Wojtanawicz and Krilov 1987, Bennion et al. 1994, 1995a). For example, iron sulfide particles are almost certainly present in most reservoirs (Martins et al. 1995) and are generally very small but when they are oil-wet they can coagulate and block pore throats.

Loss of injectivity has been shown in some fields to be reversible, which suggests that an improvement of injection water quality leads to a direct enhancement in injection well performance (Van den Hoek et al. 1996, Kevin et al. 2003). This implies that, in some cases, waterflood/disposal performance can be improved, without resorting to a workover, by upgrading the water treatment or by cycling wells between produced water injection and injection of another type of compatible water.

2.3 2.3 2.3

2.3 RETENTION MECHANISMSRETENTION MECHANISMSRETENTION MECHANISMSRETENTION MECHANISMS

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• _{surface sites: the particle is retained on the surface of grain,}

• constriction sites: the particle cannot enter into a pore smaller than its size, • _{cavern sites: the particle is retained in an area formed by several grains.} The following forces influence this retention:

• hydrodynamic forces,

• friction forces: for example, a particle wedged in a pore may have been slightly deformed and may remain in place by friction,

• surface forces: attractive forces such as Van der Waals, and electrical forces which are either attractive or repulsive according to the physicochemical conditions of the suspension,

• chemical forces: when particles are <1 m

µ

some chemical bonding can occur. Given these retention processes, then a number of scenarios are possible for particle capture

(1)-Inertial forces: Particles come close to the surface of the grain and are captured because of the surface forces.

(2)-Straining or screening: Two or more particles arrive at a pore throat at the same time. They plug the entrance to that pore if their combined diameter is larger than the pore entrance itself. Hence, this will result in straining of progressively smaller particles. (3)-Sedimentation: When there is a difference in density between the fluid and particles, the particles are affected by gravity and their velocities are no longer the same as that of the fluid. This will increase the likelihood of particles to contact a grain. Usually the deposition is highly irregular and very few particles deposit in the pore throat. The kinetics that control the deposition of particles on the grain surface are very anisotropic and depend on physical and chemical factors such as pore-scale hydrodynamics, electrostatic charges, pore surface texture, and particle composition.

(4)- Interception: If the suspended particles and the fluid have the same densities, the particles due to their inertia will not be able to follow the smallest tortuosities of the fluid streamline, and hence they will collide with the walls of the pores. a special case of interception is Bridging: a particle is retained by a previously deposited particle (Blauch et al. 1989, Veerapen et al. 2001).

(5) -Diffusion: Small particles will be subjected to random Brownian motion that increases the number of deposition between particles and grains.

Modeling of these deposition mechanisms have been investigated by several researchers as summarized in Herzig et al. (1990) and discussed in Appendix D. Based on our experimental conditions, the deposition probability is highest by interception. Diffusion and sedimentation are found to decrease with increasing velocity, whereas straining increases with velocity. The decrease of interception and inertial forces with increasing velocity is found to be insignificant. We will discuss in Chapter 6 the effect of velocity on pore wall and pore throat deposition.

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reduction is not related to the total amount of particles retained, but to the fraction of deposit in the pore throat area (Al-Abduwani et al. 2001).

Fig.2.1: a) Hydrodynamic Mobilization. b) Internal and external filter cake formation. c) Pore throat plugging. d) Surface Deposition. e) Colloid Expulsion. f) External filter cake formation by large particle. g) Internal filter cake by small particle deposition. From Civan (1999)

2.4 2.4 2.4

2.4 INTERNAL AND EXTERNAL FILTERCAKEINTERNAL AND EXTERNAL FILTERCAKEINTERNAL AND EXTERNAL FILTERCAKEINTERNAL AND EXTERNAL FILTERCAKE

The above considerations apply to the retention of individual particles. However, as the number of retained particles increases, the concepts of internal and external filter-cake become relevant.

Internal Filter Cake: At some density of particle deposition inside the formation, no more particles can flow; instead, they are all retained. This is referred to as internal filter cake and it is usually reached when about 60% to 70% of the pore throat is filled or plugged by particles (Roque et al. 1995, Chauveteau et al. 1998). Barkman and Davidson (1972) suggested that internal filter cake initiates at the injection face of a well, and progresses away from the wellbore. However, internal filter cake initiated by pore bridging is often observed at low velocity but rarely in the high velocity range. This suggests that an internal filter-cake may form deeper in the reservoir (where the velocity is low) and grow toward the well (Veerapen et al. 2001). Eylander (1988) modified Barkman & Davidson’s model and introduced the filter cake porosity parameter. Van Oort et al. (1993) proposed a model to predict permeability decline due to the internal filter-cake, built up by the particles which are smaller than one-third and larger than one-seventh of the pore diameter. He observed that the loss of permeability due to the internal filter cake decreases with increasing flow velocity and those particles that are smaller than one-fourteenth of the pore diameter and have low concentration have a very small effect on permeability damage.

a b

c d

e f

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7777

External Filter Cake: If straining is the most dominant deposition mechanism, then most of the pore throats on the inlet surface of the formation will be plugged in a relatively short time and an external filter cake will be formed by accumulation of subsequent particles outside the formation (Eylander 1988). Similarly, if the concentration of particles is high, then many particles will try to pass through the pore throat causing a log-jam or multi-particle hydrodynamic retardation and formation of an external filter-cake (Pang and Sharma 1997). Donaldson et al. (1977) found that large particles initiate external cake formation. In core tests, the prior treatment of the core can influence the rate at which an external cake is formed, whether the cores are broken or sawed (Smart et al. 1991, Zhang et al. 1994, Todd et al. 1984, and Hsi et al. 1994) and whether they are treated with ultrasound to dislodge loose fines on the surface (Saraf 2005).

Transition Time: Many authors believe that an external filter-cake will be formed after the internal filter-cake has reached a certain critical density or after a certain transition time. Vitthal & Sharma (1992) and Pang & Sharma (1994) calculated the transition time after which the first layer of external filter cake is completely formed. Barkman and Davidson (1972), King (1993) and Wennberg and Sharma (1997) developed an equation for transition time which express the half life cycle of injection well as a function of injection rate, filter cake properties, and formation characteristics. The half life is defined as the time required for the injection rate to decrease to 50% of its initial value. The short coming of the model is that it depends on filter cake measured from membrane test. One of the first techniques of measuring external filter cake properties was developed by Cerini et al. (1946), then further by Kumar (1991), and Hsi et al. (1994). The method consists of passing the injection water through a sintered glass disk at constant pressure and then plotting filtration rate versus cumulative filtration volume. Later development was to replace the sintered glass with membrane filters (Patton 1990). Membrane filters come with different pore opening sizes to mimic the effect of an actual core sample. Because the membrane prevents most of the particles from passing through, the pressure measurements overestimate the pressure differential buildup in an external cake formed on a core sample (in other words, it gives the worst scenario). The membrane test has proven to be not correlating well with actual core sample due to the difference in pore structure and flow behavior (Ershagi et al. 1986).

Generally it is assumed that once an external filter-cake has been formed, no further particles enter the formation. However, experimental observation indicates that internal invasion continues to proceed even with the establishment of external cake (Khatib 1994), and that not only solid size and concentration determine pore plugging, but also solid type and compressibility. The study of external filter cakes in drilling and other operations is extensive (Williams 1940, Jiao and Sharma 1994, Altman and Ripperger 1997, Abu-Sayed et al. 2005). Models are available for predicting external filter-cake buildup in the presence of cross-flow (Stamatakis and Tien 1993, Belfort et al. 1994, Song and Elimelech 1995, and Al-Abduwani et al. 2003,2004,2005a). Most give a prediction of the cake thickness from a force balance assuming a constant average flow.

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8888

embedding/thin section, and post mortem analysis. Non-destructive techniques proposed by Tiller (1995), Zwaag et al. (1997), and Pignon et al. (2000) include ultrasonic methods, optical techniques, static light, impedance measurement, CT-scan, X-ray, SEM, and XRD. In our experimental analysis we use an online non-destructive method, using a linear X-ray apparatus described in Chapter 3.

2.5 2.5 2.5

2.5 DEEP BED FILTRATION THEORYDEEP BED FILTRATION THEORYDEEP BED FILTRATION THEORYDEEP BED FILTRATION THEORY

Deep bed filtration theory models the retention of solid particles suspended in a fluid flowing inside a porous medium. The theory is explained in detail in Chapter 3. The deep bed filtration equations are based on conservation of mass and a simple kinetic equation governing the rate of retention. The kinetic equation assumes that the rate of particle retention is directly proportional to the number of particles available to be captured by the porous medium. The number of particles available is equal to the product of the local suspension concentration and the interstitial velocity. The proportionality factor

λ

, called the filtration function, has unit m-1. This simple filtration function is assumed to describe the complex retention processes described above in Section 2.4. The retention of particles in the formation results in loss of permeability. This loss of permeability can be described by a permeability reduction function

β

. Often

β

is assumed to be linear in the amount of retention.

The deep bed filtration theory was first proposed by Iwasaki (1937), with the assumption that the filtration function

λ

is constant. Ives (1963, 1965) modified the Iwasaki equation and considered that the deposition of particle is highest at the beginning of injection where the probability of particle capture is high. Many authors have looked at ways of determining the filtration function. Most use the assumption that

λ

is constant, in this case

λ

can be determined, in principle, from the effluent concentration in core tests (Wennberg and Sharma 1997). Pang and Sharma (1997) used the classic deep bed filtration for constant

λ

but included a critical value of porosity at which deposition mechanisms change and an external filter-cake starts to grow. Bedrikovetsky et al. (2001, 2002) developed a method for the simultaneous determination of constant

λ

and

β

, using three pressure measurements at the inlet, outlet, and an intermediate point of a core and the effluent concentration. Al-Abduwani et al. (2003, 2004, 2005a) developed a chemical technique determine the final deposition profile along the core length. Later, a CT-scanner was used (Al-Abduwani et al. 2005b). It was concluded that

λ

monotonically decreases with deposition.

2.6 2.6 2.6

2.6 EFFECT OF FLOWRATE ON PARTICLE DEPOSITIONEFFECT OF FLOWRATE ON PARTICLE DEPOSITIONEFFECT OF FLOWRATE ON PARTICLE DEPOSITIONEFFECT OF FLOWRATE ON PARTICLE DEPOSITION

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9999

this model and observed that

λ

is higher is the shallower section of the lower velocity experiment than in the deeper section of the higher velocity experiment. Gruesbeck and Collins (1982) compared deep bed filtration theory with constant

λ

against experimental data. They observed that reversing the flow restored the permeability, but only temporarily if the flow-rate is above a critical velocity, above which presumably retained particles become free. Rege and Fogler (1987) developed a model to predict

λ

accounting for geometric size, fluid velocity, and deposition morphology. It predicts that low velocity leads to high

λ

, a prediction that agrees with experimental results by Soo and Radke (1984) and Donaldson et al. (1977). Van Velzen et al. (1992) developed a model describing injectivity under radial flow conditions and concluded that

λ

and

β

decrease with increasing flow velocity. Kwok et al. (1995) presents a well literature review of experimental and theoretical results for dispersion process accruing in radial flow geometry. We will briefly discuss deep bed filtration with radial flow geometry in Chapter 7.

In our tests, the maximum velocity is 90cc/min, maximum particle diameter 10

µ

m and assuming that the suspension has the same properties of water since it is dilute; the maximum value of the Reynolds number R_e =

ρ

ud_p

µ

is 185 (see Appendix-D). We showed in Appendix D that the Peclet number switches from one linear to another linear at some critical velocity. We believe the critical velocity at which the deposition mechanism changes from pore throat to pore wall is the intercept of the two lines. This will be discussed later in Chapter 6. A particle suspended in a low-Reynolds-number flow experiences two hydrodynamic forces owing to the fluid motion: one is the applied shear force which acts to move the particle in the direction of the pressure gradient, and the other is the drag force opposing relative motion between the particle and the fluid. The balance between these two forces determines the particle velocity. However, if there are non-hydrodynamic interaction forces between the flowing particle and another particle or the porous medium, these may become more important. Some researchers argue that high velocity increases the hydrodynamic forces relative to the non-hydrodynamic forces, leading to rapid pore bridging and blocking of pore throats (Grolimund et al. 1998, Ramachandran et al. 2000, and Ruben et al. 1997). Other researchers claim the opposite, and that high velocity leads to increased surface deposition and lower probability of particle deposition in the pore throats (Roque et al. 1995, and Chauveteau et al. 1998)

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10 1010 10

1 (see Appendix D). These two mechanisms (hydrodynamic shadowing and surface entrainment) suggest that the formation of internal damage that is initiated by pore bridging will be observed at low velocity but rarely in the high velocity regime.

2.7 2.7 2.7

2.7 INJECTION OF WATER CONTAIINJECTION OF WATER CONTAIINJECTION OF WATER CONTAIINJECTION OF WATER CONTAINING OIL DROPLETSNING OIL DROPLETSNING OIL DROPLETSNING OIL DROPLETS

The presence of oil droplets in injected water, without any suspended particles, can also cause limited injectivity decline. Even a small amount of oil can reduce the permeability of the external filter cake to water due to wettability alteration. Larger quantities of suspended oil may impair injectivity by reducing the permeability of the formation to water, especially in injection zones that initially have less than the mobile oil saturation. Oil tends to foul deep-bed filtration and enormous efforts are made to remove the majority of the oil. In one waste water project (Nasr-El-Din and Al-Taq 1996b), the presence of oil in the injection water damaged the well drastically; as the oil saturation increased around the wellbore, the brine permeability decreased, causing wettability alteration and hence injectivity decline. Heavy hydrocarbons and asphaltene can also precipitate and block some of the pore throats. Therefore produced water containing oil in the range of 500mg/l to 5,000mg/l or higher is usually treated before reinjection (Thro and Arnold 1994, Van der Zande et al. 2000, and Janssen 2000). It is theorized that slug of oil can cause as much, if not more, damage to produced water injection wells as continuous injection of suspended oil droplets (Coleman and McLelland 1994).

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11 11 11 11

and plug the formation whereas large oil droplets can deform to smaller oil droplets. Oil droplets with small diameter have low capture probability.

In addition to internal damage, the injection of oily water can form an external filter cake if the pores have the same size as the oil droplets (Ochi et al. 1999). At low pressure the oil droplets behave like solid particles, forming a cake through which no more particles can pass. However, at high pressure, oil droplets deform and continue to flow through the cake into the formation. Oily cakes tend to be compressible and have very low permeability.

2.8

2.82.8

2.8 INJECTION OF WATER CONTAINING SUSPENDED SOLIDS AND OIL DROPLETSINJECTION OF WATER CONTAINING SUSPENDED SOLIDS AND OIL DROPLETS INJECTION OF WATER CONTAINING SUSPENDED SOLIDS AND OIL DROPLETSINJECTION OF WATER CONTAINING SUSPENDED SOLIDS AND OIL DROPLETS As discussed in the previous section, generally damage inside the formation caused by oil droplets is less severe than with solid particles alone. But damage is more severe when oil droplets and solid particles are both present (Mehdizadeh et al. 1992, Zhang et al. 1993). The presence of oil causes solid particles to coalesce, acting as an adherence agent, and increasing the impairment. More oil content in combination with solids leads to increased damage. The complex nature of the solid-oil-water attachment can change with concentration, wettability, surface charge, and other parameters that we may not be aware of (Aderangi and Wasan 1995). In the field, a solid-oil-water emulsion can be formed because of high pressure, high temperature, and high flow-rate, with the addition of scale and corrosion inhibitors. Emulsion droplet diameters are usually in the

0.05–100-m

µ

range (Wasan et al. 1978). Some factors that contribute towards emulsion stability include interfacial tension, ionic surfactants, and fine solids (Deverex 1974, Soo and Radke 1984,1986, and Van den Broek et al. 1999).

Soo and Radke (1986) carried out experiments with oil droplets and solid particles. They found that invasion depth of the oil droplets alone was considerable, but does not contribute to a dramatic decrease in permeability. Permeability impairment was drastic in the presence oil droplets and suspended solids. Zhang (1994) was among the first to use broken face core samples to study permeability damage due to solid and oil droplet invasion. His observations were that if the oil droplets can deform to pass through the pore spaces, then they can travel deep into the formation. His observation agrees with Hsi et al. (1994), who stated that if a large number of oil droplets is present in the produced water, they will be filtered out in the first part of the core resulting in increased local oil saturation. However, due to relative permeability effects, this small increase in oil saturation can significantly reduce effective brine permeability. Very fine solid particles can help in stabilizing emulsion and hence make it difficult to separate oil from solids prior to injection and therefore the emulsion can be carried all the way into the injection stream to cause injectivity decline

Bedrikovetsky et al. (2002) and Var-Jr et al. (2006a,b) considered the application of deep bed filtration theory when both oil droplets and solid particles are present. They concluded that the filtration coefficient

λ

for a suspension of solid particles is higher than

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12 1212 12 2.9 2.9 2.9

2.9 THE EFFECT OF WETTABILITYTHE EFFECT OF WETTABILITYTHE EFFECT OF WETTABILITYTHE EFFECT OF WETTABILITY

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13 13 13 13

3.

3. FILTRATION MODELS

FILTRATION MODELS

3.1 3.13.1

3.1 INTRODUCTIONINTRODUCTION INTRODUCTIONINTRODUCTION

In this chapter, theoretical models are introduced for describing the filtration phenomena which occur during the flow of a suspension of small particles into and within a porous medium. The emphasis will be on Deep Bed Filtration theory, which describes particle transport and deposition within a porous medium. However particle deposition also occurs at the external face of the porous medium, and this combination of external and internal deposition makes the interpretation of conventional experiments difficult. Use of X-ray data enables a distinction to be made between external and internal deposition processes, and makes it possible to examine the relative amounts of particles deposited by each process.

Section 3.2 discusses the mass conservation equation governing deep bed filtration. Section 3.3 gives a short review of the equation describing the process of deposition, known as the kinetic equation, which is needed to complete the description of the model. Section 3.4 gives the general solution for the deposition profile and discusses various models proposed for the internal filtration coefficient introduced in the kinetic equation. Section 3.5 discusses external deposition and the formation of an external filter-cake. 3.6 discusses ways to estimate the filtration coefficient from experimental data. And finally section 3.7 reviews the effect of deposition on permeability and introduces the formation damage factor.

3.2 3.23.2

3.2 BASIC MASS CONSERVATION EQUATIONBASIC MASS CONSERVATION EQUATION BASIC MASS CONSERVATION EQUATIONBASIC MASS CONSERVATION EQUATION

The deep bed filtration equation governs the deposition of particles inside a porous medium. It is concerned only with internal deposition, and does not consider deposition on the face of the porous medium. A volumetric balance on particles on suspension, using conservation of mass, yields the classic one-dimensional deep bed filtration equation (Bedrikovetsky et al. 2001a,b, Barkman and Davidson 1972, Iwasaki 1937)

( c (1 ) ) (uc D c) 0

t

φ

φ σ

x x

∂ ∂ ∂

+ − + − =

∂ ∂ ∂ (3.1)

Here

σ

is the specific deposit (volume of deposited solid per volume of sand), c is volume of suspended solid per volume of fluid,

φ

is the porosity, x is the axial coordinate, u v=

φ

is the approach velocity, where v is the interstitial velocity,

t

is the time, and Dis the dispersion coefficient.

Equation (3.1) can be further simplified by introducing two approximations. The first is by assuming incompressible flow which implies that uis constant. The second assumption, justified by Herzig et al. (1970), is that diffusion is negligible for particles larger than

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14 1414 14

above assumptions, if particle deposition is the only mechanism changing porosity, we also have

φ

=

φ σ

( ) (3.2) However, if the suspended particle concentration c<<1, the variation of

φ

with

σ

can be neglected to yield (1 ) 0 c _u c t t x

σ

φ

∂ + −

φ

∂ + ∂ = ∂ ∂ ∂ (3.3)

In order to completely describe the filtration phenomena in the porous rock, the deposition function

t

σ

∂

∂ must be specified through the kinetic equation described in

section 3.3, and a detailed information can be found in Elimelech et al. (1995). Also a momentum equation must be selected, and relationship between permeability and specific deposition is required. In this work, we will use Darcy’s law as the momentum equation. The permeability/deposition relationship is discussed in Section 3.7.

Experimental data of deep-bed filtration of hematite particles in a sandstone core, obtained using X-ray methods, will be analyzed. For these experiments, the boundary and initial conditions for (3.3) are

c =0,

σ

=0 for x≥0,t≤0

o

c c= for x=0,t >0 (3.4)

where c_o is some function of time, specifying the inflow concentration into the volume of the core x ≥0

3.3 3.3 3.3

3.3 KINETIC EQUATIONKINETIC EQUATIONKINETIC EQUATIONKINETIC EQUATION

The kinetic equation describes the dynamic interaction between the suspended particles (oil droplets or solids), the deposited particles, and the porous medium. Herzig et al. (1970) presented an extensive literature review of the parameters affecting the form of the kinetic equation. They stated that the kinetic equation is affected by the type of deposition and the dominant attraction and repulsive forces. The equation describing this process cannot be obtained strictly theoretically, but must be inferred experimentally. The probability of deposition of particles is affected by several parameters such as:

- the carrier fluid, characterized by viscosity and density

- the suspended particles or droplets, characterized by concentration and density

- the porous media, characterized by permeability, porosity, mean pore throat diameter

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15 15 15 15

These and other factors all play a role in determining the physico-chemical, gravitational and inertial forces governing deposition, influencing effects such as sedimentation with respect to the bulk flow, Brownian motion, interception, etc.

Iwasaki (1937) was the first to define a filtration coefficient

λ

which is a function of the above parameters and has a unit of m-1. He defined

λ

through the Kinetic Equation (3.5)

uc t

σ

λ

∂ = ∂ (3.5)

The filtration coefficient for dilute suspensions is assumed to be independent of c and

u, making the deposition rate linear in suspended mass flux and the filtration coefficient just a function of the deposited concentration

σ

.

( )

σ

λ

= (3.6)

In this approach, the filtration coefficient is not determined from the physical or chemical properties of the suspensions or filter bed; rather, it can only be determined from laboratory experiment with the given suspended particles and filter bed. In general,

λ

changes as particles accumulate within the filter bed, this was also confirmed by Saripalli et al. (2000).

It may be a valid point to question the reliability of kinetic equation (3.5), and especially the simplification (3.6), because of the many assumptions made to derive it. This point will be returned to in the discussion of the experimental results in Chapter 6.

3.4 3.43.4

3.4 SOLUTIONS OF THE DEEP BED FILTRATION EQUATIONSSOLUTIONS OF THE DEEP BED FILTRATION EQUATIONS SOLUTIONS OF THE DEEP BED FILTRATION EQUATIONSSOLUTIONS OF THE DEEP BED FILTRATION EQUATIONS

It is generally accepted to use equations (3.3) and (3.5) for deep bed filtration during water injection. We will assume for the present time that

λ

is a function only of

σ

. Various authors introduced specific assumptions for the form of the filtration coefficient, as discussed below.

3.4.1 3.4.13.4.1

3.4.1 General General General ssssolutionGeneral olutionolution olution

The general form of the solution of the deep bed filtration equation for arbitrary filtration coefficient

λ σ

( )is given by Bedrikovetsky et al. (2001a,b). The basic mass balance equation (3.3) and the kinetic equation (3.5), together with the boundary conditions in (3.4), yield the following

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16 1616 16

where

σ

₀ is the deposited concentration at the inflow boundary, determined from the inflow concentration c t_o( )as follows

0( ) 0 0 ( ) ( ) s s o dy _{u c y dy} y σ

λ

=

∫

(3.9)

Since

λ

>0 it follows from (3.9) that 0(t x v/ ) ( , )x t

σ

− ≥

σ

(3.10)

The bulk concentration c is given by

0

( , ) ( , ) (_o / ) / ( / )

c x t =

σ

x t c t x v−

σ

t x v− (3.11)

In the X-Ray experiments, the measured quantity of hematite includes the deposited and suspended particles. Therefore the overall particle concentration C with respect to the total sample volume is introduced, where

( , ) (1 ) ( , )

C =

φ

c x t + −

φ σ

x t (3.12)

It follows from (3.11) and (3.12) that

0 0 0 0 0 ( , ) ( , ) ( / ) / ( / ) ( ) ( ) (1 ) ( ) C x t x t C t x v t x v C t c t t

σ

φ

φ σ

= − − = + − (3.13)

Also from (3.13) it follows that

0

( , ) ( , ) (_o / ) / ( / )

c x t =C x t c t x v C t x v− − (3.14) The change in porosity due to particle deposition, neglecting the suspended particles, is given by (1 ) (1 ) d d

φ

φ σ

φ

φ σ

= − − ∆ = − = − − (3.15)

where

φ

_dis the damaged porosity, and

φ

is the initial porosity

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17 17 17 17

We now make two approximations, which we will assume in evaluating our experimental data. These assumptions are not valid at very early times in an experiment. The first assumption is that, for a finite core of length X of a few centimeters, (t x v− / )≈tfor all samples, except at very early time; hence the term ( / )x v is neglected in the above equations. This is equivalent to neglecting the term ∂

σ

/ t∂ in (3.3).

Secondly, we note that the maximum concentration used in the laboratory c=200ppm (by mass) and

φ

=0.225, so that

φ

c x t( , )<0.00044 which can be neglected except initially compared with typical values of

(

1−

φ σ

)

=0.004 and higher. Thus we assume

σ

>>c, and that Ccan be approximated by

(

1−

φ σ

)

.

We now define M as the mass in grams of hematite contained inside the pores of the rock in the core volume between

x

iand X (the end of the core)

( , ) ( , ) (1 ) ( , ) i i X X i h h x x M x t =

ρ

A C x t dx

∫

≈

ρ

A −

φ

∫

σ

x t dx (3.16)

where

ρ

_his the density of hematite andAis the cross-sectional area of the core. By conservation of mass, it follows that the rate at which M changes with time is determined by the hematite fluxes into and out of the volume

(

,

)

(

,

)

h i M _{Au c x t} _{c X t} t

ρ

φ

∂ _ _ = _ − _ ∂ (3.17)

where c X t( , ) is the effluent flow concentration c_eff

3.4.2 3.4.23.4.2

3.4.2 Constant filtration coefficientConstant filtration coefficientConstant filtration coefficient Constant filtration coefficient

In the special case where

λ

has a constant value (

λ

₀), the integral of (3.8) gives

0 0 ( , ) ln[ ] (1 ) ( / ) x t _x t x v

σ

φ λ

σ

− = − − (3.18)

But from (3.11) it follows

0 ( , ) ( , ) ( / ) ( / ) o c x t x t c t x v t x v

σ

= − − (3.19)

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18 1818 18 0 0 (1 ) (1 ) 0 0 0 0 ( , ) ( ) , ( , ) 0, 0 ( , ) ( ) , ( , ) 0, 0 ( ) ( ) x o x t o c x t c t e c x t t x t t e x t t t u c s ds λ φ λ φ

σ

λ

− − − −

=

≤

=

≤

=

∫

(3.20)

This model predicts that if the inflow concentration c0is constant, then the effluent concentration c_effwill be also constant in time, and that

(

0

)

0

(

)

ln c c_eff =

λ

X 1−

φ

(3.21) Therefore, using equations (3.12) and (3.20), with the assumption of a constant inlet concentration and constant

λ

, gives the following expression:

0( ) (0, ) o [1 (1 ) 0 ]

C t

=

C t

=

c

φ

+

−

φ λ

vt (3.22) This linear relationship between

vt

and C t₀

( )

only holds if

λ

₀is constant, assuming the inflow concentration

c

0is constant . Therefore if we plot the total concentration values at

x =0 versus

t

, then from the slope we can determine the constant value of

λ

₀. If the relationship is not linear, then the slope of the data gives an approximation of

λ

₀ .

3.4. 3.4. 3.4.

3.4.3333 Linear filtration coefficiLinear filtration coefficiLinear filtration coefficiLinear filtration coefficientententent

In practice

λ

is not constant, but varies with

σ

. A simple form of

λ

is given as a linear function of

σ

, i.e.,

λ σ

( )

=

λ

₀

(

1 b

+

σ

)

(3.23) where b can be positive or negative, but in the case of a negative value of b, the model is valid only if b

σ

<1. Herzig et al. (1970) gave analytical solution for constant inflow concentration

c

0 from (3.6), (3.8) and (3.11),with the assumption(t x v− / )≈t . It is given also by Tien (1989)

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19 19 19 19

From equation (3.24), it is clear to see that if the inflow concentration is constant, a plot of ln[ eff ]

o eff

c

c −c against time

t

yields a straight line, with intercept of

0 (1 )

ln[

e

λx −φ

1]

− − and

a slope of −u c b

λ

_{0 o} . Accordingly, equation (3.24), can be applied, in conjunction with effluent concentration history, to obtain the values of b and

λ

₀. Farajzadah (2005) used this approach and observed the non-linearity at low values of deposition, but he also approximated the relationship with simple line. The draw back of this method is that

λ

₀

becomes zero at some point of

σ

, and that when

σ

is greater than this value

λ

becomes negative. 3.4.4

3.4.43.4.4

3.4.4 Other specific forms of filtration coefficient in the literatureOther specific forms of filtration coefficient in the literatureOther specific forms of filtration coefficient in the literature Other specific forms of filtration coefficient in the literature

Several authors correlated their measurements from laboratory experiments to deduce the form for

λ σ

( ). Their work is summarized in Herzig et al. (1970). Many authors use the constant value for

λ

and correlate the value of

λ

₀ with their experimental parameters. Sakthivadivel (1966) used the linear form (3.23) for

λ σ

( ) with b<0. Ives

(1963, 1967), Ives and Shoulji (1965), Bai and Tien (2000) proposed nonlinear forms for

λ σ

( ) as follows m o      − =

φ

βσ

λ

0 1 (Ives, 1963) 1 ' 2 0 1 0 1 b b

σ

λ

σ

φ

σ

    = + −  ₋   

(Ives and Shoulji, 1965)

1 2 0 0 0 '' 1 1 1 MAX b

σ

ξ

βσ

ξ

σ

λ

φ

σ

      = _ +  _{ } −  _ _ − _       (Ives, 1967) 2 3 0(1 b1 b2 b3 ...)

λ

=

λ

+

σ

+

σ

+

σ

+ (Bai and Tien, 2000)

3.4. 3.4.3.4.

3.4.5555 StepStepStep-Step---chchchchange modelange modelange modelange model

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20 2020 20

( )

1 2 , 0 , c c

λ σ

λ

σ

λ σ

σ

= ≤ ≤ = < (3.25) 3.5 3.5 3.5

3.5 EXTERNAL FILTER CAKEEXTERNAL FILTER CAKEEXTERNAL FILTER CAKEEXTERNAL FILTER CAKE

As discussed in section 3.1, particle deposition occurs not only inside the porous medium but also on the external face of the porous medium. Sections 3.2-3.5 have described the deep bed filtration theory for deposition inside the porous medium. This section is concerned with the deposition in the external filter cake forming on the surface of the porous medium.

As will be discussed in Chapter 4, there are major difficulties in laboratory measurements in decoupling the effect of internal and external deposition. It is even difficult to investigate which mechanism is more dominant and contributes to higher loss in injectivity. For example, it is impossible to measure the pressure buildup at the face of the core due to external deposition. Correlating the deposited mass to pressure buildup will always include the effect of external filter cake. Therefore, it is very important to distinguish between the two deposition mechanisms in order to give a good understanding of particle deposition. In this section we demonstrate that X-ray measurements of internal deposition enable an estimate to be made of the size of the external filter-cake. From mass balance, the total mass of particles entering the core must be equal to the total mass of particles deposited or still suspended in the pore fluid plus the mass of particles in the effluent water. Although the prepared particle concentration in the water tank c_tankentering the flow cell is known, the particle concentration c_oentering the core is unknown and it may not be constant, since some particles are deposited at the core inlet, forming the external filter cake.

The mass that has entered the flow cell up to time

t

_{is given by} inj tank

M

=

c Aut

(3.26)

since

Aut

is the total volume of fluid injected up to time

t

. A plot ofM_inj as a function of time gives a straight line, as shown in Figure 3.3, indicating the mass outflow at the tank. The mass of hematite inside the core at any given time is

M

( )

0,

t

, where

M

( )

x

,

t

is defined

by Equation (3.16). In addition, the effluent concentration

c

eff is known as a function of

time. It follows that the mass of the filter-cake

M

cakeis given by the equation

( )

0

0, t

cake tank eff

The effect of residual oil on deep bed filtration and formation damage