leak-off process in oil-based drilling fluids-overveldt
TRANSCRIPT
AES/PE/11-14 A CT scan aided core-flood study of the leak-off process in oil-based drilling fluids 08-07-2011 Andrea Simone van Overveldt
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Title : A CT scan aided core-flood study of the leak-off process in oil-based drilling fluids Author : Andrea Simone van Overveldt Date : July 2011 Professor : Prof. dr. Pacelli L.J. Zitha Supervisors : dr. Hua Guo, Ing. Gerard de Blok Graduation Committee : Prof. dr. P.L.J. Zitha dr. Hua Guo dr. R. Dams dr.ir. E.S.J. Rudolph dr. P. van Hemert TA Report number : AES/PE/11-14 Postal Address : Section Petroleum Engineering Section Faculty of Civil Engineering and Geosciences Department of Geotechnology Delft University of Technology P.O. Box 5028 The Netherlands Telephone : (31) 15 2781328 (secretary) Telefax : (31) 15 2781189 Copyright ©2011 Section for Petroleum Engineering Section All rights reserved. No parts of this publication may be reproduced, Stored in a retrieval system, or transmitted, In any form or by any means, electronic, Mechanical, photocopying, recording, or otherwise, Without the prior written permission of the Section for Petroleum Engineering
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Acknowledgements
I truly enjoyed the time I spend at the TU Delft. I would like to express my gratitude to many people who
helped me during the last months I worked towards finishing my Masters degree for Petroleum
Engineering.
First of all I would like to express my appreciation for the guidance by my supervisors; Pacelli Zitha, for his
thoroughness, shared knowledge and supervision throughout the whole thesis. Hua Guo, for introducing
me too the drilling fluid experiments and her contribution too my work. Gerard de Blok for his advice
concerning the use of fluids in the field and the talks we had about my wish to pursue a career in drilling.
Rudy Dams, Susanne Rudolph and Patrick van Hemert for agreeing to be part of my committee next to
Pacelli Zitha and Hua Guo.
For the assistance in the laboratory I would like to thank Henny van der Meulen, Jolanda van Haagen, and
Dirk Delforterie. Joost van der Meel for assisted me with optical measurements and Arjan Thijssen assisted
in performing the SEM measurements. Thanks to Wim Verwaal for help using the Micro-CT. Frans
Korndorffer from Shell, Rudy van Campenhout and Tom Opstal from 3M for their help with the drilling
fluid components. Special thanks go out to Mark Friebel and Ellen Meijvogel-de Koning for their
thoroughness, setting up the experiments and assisting with using the CT scanner.
I thank Siavash Kahrobaei for his help processing the CT scan data. Of course thanks to Rahul, Raymond,
Guido, Nanne, Matthijs, Bouwe, and Machiel for the serious and less serious talks we had over the last
year, drinking liters of coffee. Dorien Frequin for being such a good friend in this world filled with men,
these last six years were amazing.
Most of all I would like to thank my parents in so many ways. Their unconditional love and support have
always guided me through. Even when my plans caused them sleepless nights, their support was endless.
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Abstract
An experimental study on the leak-off of oil based drilling fluid sandstone cores is reported. First we revised the
theoretical models for the rheology of the drilling fluid, the flow behavior of drilling fluids in the drill pipe and
annulus, and filtration mechanisms. Then systematic static leak-off experiments were carried out using an
innovative method where CT scans taken at time intervals were used to visualize and accurately quantify
infiltration of fluids in a sandstone core. Different compositions of oil based drilling fluids were investigated, to
examine the influence of various particles on the external filter cake and internal filtration. Scanning electron
microscopy was used to characterize the external filter cake and internal filtration. The results give accurate
measurements of the filtration volume of the drilling fluids. Depending on the composition of the drilling fluid,
the formation of external filter cake could be visualized on CT images. The core flow experiments are
matched to the theory for linear static filtration. The results lead to new insights concerning the build of
external filter cake and internal filtration. The experiments use real sandstone cores giving more realistic data
than using an API press test and filter paper. This work creates a basis for future improvement of oil based
drilling fluid, by providing a better understanding of mechanisms involved in leak-off control.
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Index 1. Introduction......................................................................................................................................................... 6
2. Oil Based Drilling Fluids ....................................................................................................................................... 7
3. Theoretical Model ............................................................................................................................................... 8
3.1 Rheological model ......................................................................................................................................... 8 3.2 Flow in the Drill Pipe...................................................................................................................................... 9 3.3 Flow in Annulus ........................................................................................................................................... 10 3.4 Linear filtration ............................................................................................................................................ 11 3.5 Radial Filtration............................................................................................................................................ 15 3.6 Application of theory to well case............................................................................................................... 19
4. Experimental Method........................................................................................................................................ 19
4.1 Experimental Setup ..................................................................................................................................... 19 4.2 Preparation of Oil Based Drilling Fluids ....................................................................................................... 20 4.3 Core flow procedure.................................................................................................................................... 21
5. Results and Discussion....................................................................................................................................... 22
5.1 Base drilling fluid ......................................................................................................................................... 22 5.2 Base drilling fluid with barite....................................................................................................................... 30 5.3 Base drilling fluid with Gilsonite .................................................................................................................. 34 5.4 Base drilling fluid with barite and Gilsonite ................................................................................................ 38 5.5 Summary of the filtration volumes.............................................................................................................. 43
6. Conclusions and Recommendations.................................................................................................................. 44
6.1. Conclusions................................................................................................................................................. 44 6.2. Recommendations...................................................................................................................................... 45
7. Nomenclature.................................................................................................................................................... 46
8. References ......................................................................................................................................................... 47
Appendix I: Theoretical Model .............................................................................................................................. 50
Flow in drilling pipe and annulus with no movement of drill string.................................................................. 50 Rheological Models ........................................................................................................................................... 51 Flow in Drill Pipe................................................................................................................................................ 53 Flow in Annulus ................................................................................................................................................. 57 Linear filtration.................................................................................................................................................. 60 Radial Filtration ................................................................................................................................................. 67 Model Validity ................................................................................................................................................... 73
Appendix II: Handling Fluid Losses during Drilling................................................................................................. 74
Appendix III: Additional Images Core Flow Experiments ...................................................................................... 76
Base case drilling fluid ....................................................................................................................................... 77 Base case drilling fluid with barite..................................................................................................................... 78 Base case drilling fluid with gilsonite ................................................................................................................ 80 Base case drilling fluid with barite and Gilsonite .............................................................................................. 82 Base Oil .............................................................................................................................................................. 84
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1. Introduction
The operation of drilling oil and gas wells safely and efficiently is directly tied to using the appropriate drilling
fluids. The main functions of drilling fluids include providing sufficient pressure to prevent influx of reservoir
liquids, transporting the cuttings to the surface, providing wellbore stability, cooling and lubricating drill string
and drill bit, sealing permeable formations. The formulation of drilling fluids is specially designed to accomplish
these functions. Their composition is rather complex and the physical properties as well as the rheological
behavior of drilling fluids have been subjects of research for decades.
Exploration and production of oil is moving to depths in excess of 10.000 meters (Jellison et al. 2008). There is
an increasing demand of stable drilling fluids that can withstand high pressure and temperature (HPHT)
prevailing at those frontier environments. Oil-Based Drilling Fluids are commonly recognized for their stability
at high temperature. Other advantages of Oil-Based Drilling Fluids include superior lubricating characteristics
and being effective against various types of corrosion (Mihalik et al. 2002; Bland et al. 2002; Bourgoyne et al.
1991).
Permeable zones or fractures in the formations being drilled can lead to leak-off of the drilling fluid into the
formation (Al Ubaidan et al. 2000; Romero et al. 2006). The focus of this study is on fluid loss into the matrix
and more specifically on ways to minimize the loss of oil based drilling fluids. Formulation of drilling fluids is
such that when the leak-off occurs, external filter cake is formed on the face of the formation and thus leak-off
is minimized. There is always invasion of the matrix because filter cake formation takes a certain time and
external filter cake is not completely impermeable. Understanding the leak-off processes and preventing it
requires knowledge of both filter cake build-up (external filtration) on the formation surface and matrix
invasion (internal filtration). There is widely consensus that the external filter cake permeability should be as
low as possible. Effects of circulation are not taken into account in this study. However circulation of the drilling
fluid causes erosion of the external filter cake and therefore influences the build up of the external filter
cake.(Liu et al. 1996).
Internal filtration concerns also the produced water re-injection (PWRI) and water production from artesian
wells. Extensive work was performed on a better understanding of internal filtration (alias deep bed filtration)
in the context of PWRI, proposing a variety of models of the deposition inside a core (Iwasaki 1937; Herzig et al.
1970; Bedrikovetsky 2001; 2002; 2003; 2004; Obeta et al. 2010; Rousseau et al., 2008; Pang and Sharma, 1997).
Various publications have reported on experiments where a suspension containing hematite particles is flown
across sandstone investigating internal filtration. The experiments are carried out in a CT scanner (Al-Abduwani
2005; Ali et al. 2005, Saraf et al 2008; Obeta et al. 2010). Other techniques have been used to investigate
internal filtration such as scanning electron microscopy, x-ray and nuclear magnetic resonance (Bailey et al.
2002).
For the problem at hand we expect external filter cake build-up (external filtration) and deep bed filtration
(internal filtration) to be both present or even to occur simultaneous. There are various types of particles and
with different particle size distributions. Particles that are smaller than pores will flow through the filter cake
and be retained in the matrix. Pang et al. (1997) propose the transition time (t*) concept: the transition time
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marks the time where no particles invade the formation any longer as the formation is plugged and the initial
layer of external filter cake starts to form. Al-Abduwani et al. (2005) suggests that internal filtration continues
after the formation of an external filter cake.
The objective of this study was to investigate the filter-cake build-up and internal filtration and to determine
the role of the various components of the drilling fluid could play in controlling leak-off. To provide a solid basis
for the experiments, the flow of the drilling fluid in the drill pipe and annulus coupled with leak-off was first
analyzed. The objective is to gain understanding of internal and external filtration mechanism and match
theory to experimental data. Attention will be paid to mechanisms involved with internal filtration and external
filter cake build-up. The build-up of the external filter cake was considered in detail. Testing of drilling fluid
formulations and leak-off is traditionally by the use of the API press test. In this research a new method is
proposed for testing drilling fluids. The experiments consisted of controlled core flow tests through sandstone
core samples. CT scans will be used to visualize and quantify the infiltration of fluids in the cores. The
microstructure of the filter cake and penetration of particles in to the core will be studied using Scanning
Electron Microscopy (SEM).
2. Oil Based Drilling Fluids
In addition to oil and water, oil based drilling fluids contain various other components as shown in Table 3. An
emulsifier is necessary to disperse the water phase. A wettability control agent prevents the water phase to
agglomerate with the mostly water wet formations. Similar to the emulsifier the wettability control agent is a
surfactant. Solids are added to the drilling fluid to increase its viscosity. For viscosity control bentonite is often
used. Barite or calcium carbonate can be used for density control. Drilling into the reservoir formation, calcium
carbonate is often preferred, over barite. Calcium carbonate is easier to clean up by matrix acidizing than barite.
To maintain a certain alkalinity, lime (Ca(OH)2) is added to the drilling fluid. Finally, fluid loss control agents
decrease filtration into the formation (Bourgoyne et al. 1991).
To prevent leak-off, the drilling fluid should be formulated so that an external filter cake is formed rapidly and
the cake permeability is minimized. Several additives that prevent the drilling fluid flowing into the formation
and thus reduce substantially the leak-off have been used in the past and were extensively investigated
(Longeron et al. 1998; Mihalik et al., 2002; Hua et al., 2011). Examples of additives for fluid loss control are
polymers, lignite, asphalt and manganese oxide. The prevailing concept is that additives favor the formation of
an external filter cake as fast as possible to diminish the leak-off of drilling fluid into the formation. In the
1940’s fluid loss was controlled using bentonite. Moderate fluid loss control was achievable; the viscosity of
this mixture is high and can therefore help controlling leak-off (Nelson 1990). By the late 1950’s, the
introduction of carboxymethyl-hydroxyethylcellulose (CMHEC) hailed the beginning of use of polymer as fluid
loss agents (Mueller, 1992). Several polymers and copolymer additives were used as fluid loss control agents. In
the last decades, additives such as Gilsonite which are soluble in oil soluble or partially-soluble fluids were
widely used in drilling fluids (Aston, 2002).
The influence of the several components and the concentration of these components have been investigated
by Mihalik et al. (2002) using API filter paper. They state that a wetting agent has a positive effect on fluid loss
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control as it disperses the clay particles in the drilling fluid. They also suggest that platey particles are
preferable over granular particles for fluid loss control. In their studies they found that the larger granular
particles barite and calcium carbonate are ineffective for fluid loss control. Hua et al. (2011) suggest that
emulsified water droplets are there to fill the gaps between particles in the external filter cake. However, the
stability of emulsified water droplets and interaction between the emulsified water droplets and particles are
important factors to determine the role of emulsion, but these factors could not be examined clearly.
It is desirable that the control agent has a deformable structure to fill up the void space in the external filter
cake between emulsified water droplets and solid particles. Gilsonite is a naturally-occurring, solid organic
material which is classified as an asphaltene, and has the characteristic that it is deformable. It is a
relatively pure hydrocarbon without significant mineral impurities. Gilsonite has a softening point around
188 °C (Davis, 1988). The use of Gilsonite will be investigated in this research. Another example for a fluid loss
control agent can be the use of polymers. This could be interesting as polymer forms a gel-like continuous
phase with the water.
3. Theoretical Model
For the purpose of connecting the leak-off of drilling fluids and the circulation in the drilling systems, a
theoretical model is made for the flow in the drill pipe and the annulus. This is followed by a theoretical
analysis of the leak-off combining external and internal filtration. Below only the main formulas of the model
are represented. Appendix I gives a more complete overview of the equations and their detailed derivations.
3.1 Rheological model
As we have already mentioned, drilling fluids are complex from both physical-chemical and rheological
viewpoints (Bourgoyne, 1991, Bird et al., 1987, Macosko, 1994). They are non-Newtonian, i.e. their viscosity
depends on the shear rate or equivalently their shear stress is not proportional to the shear rate. Several
models describe non-Newtonian behavior. The Herschel-Bulkley model is a recognized model to approximate
the rheological behavior of drilling fluids (Kelessidis et al. 2006, Wang et al. 1999, Bourgoyne 1991, Bird et al.
1987, Macosko 1994). The Herschel-Bulkley model is characterized by three parameters namely, the
consistency index K, the flow index n and the yield stressτ 0 .
0 0,
ndu
K fordr
τ τ= + ≥τ ττ ττ ττ τ (1)
Where τ is the shear stress, du
dr is the shear strain rate.
The Herschel-Bulkley model can be reduced to Power law, Bingham and the Newtonian model. When Setting
0 0τ = and 0α = the flowrate for the Power law model is obtained, in addition that n=1, the flowrate for the
Newtonian Model can be determined. If n=1 but 0 0τ ≠ then the flowrate for the Bingham model is obtained
The Herschel-Bulkley can be seen as being a more general non-Newtonian fluid model. The Herschel Bulkley
model is preferred above the Power law and Bingham Plastic model for drilling fluids. The reason for this
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preference is that Herschel Bulkley is more accurate in predicting the behavior of drilling fluids. A certain
reluctance to use the model lies in the fact that the derivation for the three parameter model is complex.
3.2 Flow in the Drill Pipe
The three parameter Herschel-Bulkley model is used to describe flow in cylindrical drill pipe. An analytical
solution is provided for the three parameter model (Kelessidis et al., 2006, Wang et al. 1999, Bourgoyne et al.,
1991, Bird et al., 1987, Makosco, 1994). There is a region around central core of the fluid that has a shear stress
less than the yield stress. This region is delimited by the plug radius Rp. In the plug area 0 pr R≤ ≤ ,the velocity
is constant and fluid moves as a rigid plug. The radius at which there is an unsheared portion of the fluid, Rp, is
given by:
2pR
τ=∆
Where ∆ stands for dp
dz.
The following assumptions are made: 1) the drill string is not being rotated, 2) sections of open hole are circular
in shape and of known diameter, 2) the drilling fluid is incompressible 3) the flow is isothermal and 4) there are
no gravity effects.
In Figure 1 the velocity profile in a drill pipe is sketched. Figure 1 also gives an impression of the shear stress
profile within the drill pipe.
r r
τu
τ0
τ0
Rp
R
Figure 1: Laminar Herschel Buckley Flow in a Cylindrical Pipe
The flow rate in the drill pipe is given by equation 2.
( )π ατ τ α ατ
+ + + + = − + − + + + + +
1/ 13 2 2
0
2 1 1 1 4 5 1(1 )
1/ 1 2 2(3 1) 2(2 1)(3 1)
n
w
w
K R n n nQ
n K n n n (2)
where 0 0
wR
τ τ ατ
= =
10
By using n=1 but 0 0τ ≠ the flowrate for the Bingham model is obtained (equation 3), this equation is also
known as Reiner-Buckingham equation.
π α α− = − +
440 4 1
18 3 3
LP PrQ
K L (3)
3.3 Flow in Annulus
Establishing a velocity and flowrate profile for a Herschel-Bulkley model is too complicated. The rheology
model used to approximate the flow in the annulus is the Bingham Plastic Model. For laminar flow of
Newtonian and non-Newtonian fluid it is not possible to use a mean hydraulic radius, which early literature
suggests can be used for turbulent flow of non circular sections (Binder 1943 and Moody 1944). Laird (1957)
defined the annular velocity for Bingham Plastic fluids in an annular geometry. An overview of the flow profile
in the Annulus is given in Figure 2.
For engineering applications Laird proposes a function of the flowrate in an annulus in a simplified version. The
simplification that is made is similar to Bingham’s (1922) simplification for pipe flow. The simplification
assumes that for reasonable high pressure the dimensions of the plug can be neglected, thus rn=rp.
R2
R1
Flow
rp
rn
Figure 2: Flow in Annulus
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Under these conditions the flow rate is given by equation 4.
2 2 24 4 3 3 3 22 12 1 0 2 1 0 0 2 1
2
1
( ) 8( ) ( ) 4 3 ( )
8 3lnA
p R RQ R R R R r r R R
Rl
R
π τµ
∆ −
= − − − − + + + (4)
In equation 4 r0is given by equation 5.
2 22 2 1
02
1
2ln
R Rr
R
R
−=
(5)
3.4 Linear filtration
3.4.1. External filter cake
Hua et al. (2011) proposed a physical model to describe the static filtration process. Here a summary of this
model is presented. Figure 3 illustrates the drilling fluid containing particles filtering through a core. It is
assumed that fluid and particles are incompressible in the filtration process. Here it is considered that that only
solid particles in the drilling fluid, contribute to the formation of the external filter cake. The role of emulsified
water droplets in the drilling fluid might also play an important role in the formation of external filter cake, but
is not taken into account in this model. The flow is laminar isothermal at the given pressure and flow rate. The
volumetric flow through a filter cake and filter paper is described by Darcy’s law. In the filtration process, the
number of particles in the drilling fluid that has been filtered is equal to the number of particles deposited in
the filter cake at any time t.
k,φ
lc,kc,φc
S
Figure 3: Overview Filtration Process
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The behavior of filtration will be described by:
( ) ( )0 0
dV tQ = F V t + 1
dt
Where:
( )( )f f p
0 0
f c c p f
k k cΔPQ = S ; F =
η l k 1- 1- c l Sφ
In the above equations kc and kf are respectively the permeability of filter cake and filter paper, Δp Is the total
pressure drop, c f
l ,l are respectively thickness of the filter cake and filter paper, η Is the viscosity of mud and S
is the cross-sectional area of filter paper. c
φ Is porosity of filter cake. V(t) stands for the volume of the filtrate.
Finally cp is the concentration of particles in the drilling fluid.
The general solution for the filtration volume is given by equation 6.
( ) ( ) ( )0 s
2
0 s 0 0 s
s
0
Q t, t < t
V t = F V + 1 + 2F Q t - t - 1, t > t
F
(6)
Here ts and Vs are spurt time and spurt loss volume of filtrate respectively.
When s sV = 0, t = 0 ;
( ) 0 0
0
1+ 2F Q t - 1V t =
F (7)
In the condition0 0
2F Q 1≫ , equation 7 can be approximated as:
0
0
2QV(t) = t
F (8)
Equation 8 shows that the filtrate volume is proportional to the square root of the time after the spurt loss
time.
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3.4.2. Deep-bed filtration
Now the model for the internal filtration is presented. Figure 3 already gave a schematic overview for the
filtration through a core. The idea behind the filtration of particles into the core is that particles larger than the
pore size will not enter the core and particles smaller than the pore do enter the core. For the model proposed,
it is assumed that the pore size distribution is very narrow and the size of the particles determines whether
particles penetrate the core or not. This situation is represented in Figure 4. The yellow area represents the
fraction of particle c1 whose radius is larger than the pore radius. These particles cannot penetrate the core and
will therefore form the external filter cake. The grey area represents the fraction of particles c2 that are smaller
than the pore size. These particles penetrate the core.
rparticle>rporerparticle<rpore
Figure 4: Distribution of particle size
The classical deep filtration model can be used to describe the deposition inside the core. It consists of the
mass conservation equation and the kinetic equation (Iwasaki 1937; Herzig et al. 1970; Sharma and Yortsos
1987). It is assumed that the porosity is constant and the particle diffusion and dispersion coefficient is
negligible (D=0).This work will illustrate the simplest case, in which the deposition coefficient is constant:
0λ λ= .
A summary of the governing equations:
(1 )t x tc u cφ φ σ∂ + ∂ = − ∂ (9)
0t ucσ λ∂ = (10)
Here c is the concentration of suspended particles, σ is concentration of deposited particles, φ the effective
porosity of the sandstone core and u is the velocity. The equations are to be solved with the following.
The initial conditions are:
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Position (x)
σ(x,
t)
t1t2t..t..t..t..t..t..tn
Position (x)
C(x
,t)
t1 t.. tnt.. t.. t.. t.. t.. t.. t..
( , 0), 0c x t x= > , ( , 0) 0, 0x t xσ = = >
And the boundary conditions are:
2( 0, ) , 0c x t c t= = > , ( 0, ) 0, 0x t tσ = = >
The solution for c(x,t) and σ (x,t) are given in equation 11 and 12. These equations are valid for x < vt and
0 for x > vt , u
vφ
= .
2( , ) exp( )c x t c Kx= − , where 0(1 )K φ λ= − (11)
( ) ( ) ( )0 2σ x,t = λ c vt - x exp -Kxφ (12)
In Figures 5 and 6, c(x,t) and ( , )x tσ are given as function of the position x. In Figure 5 the different lines for t
represent the moving front of suspended particles. For ( , )x tσ at times t1,..,tn the different graphs for
concentration of deposited particles are shown.
Figure 5 : Suspended particles Figure 6 : Deposited particles
The total concentration of particles in the porous medium is the sum of the suspended particles and deposited
particles and can be expressed as
( , ) ( , ) (1 ) ( , )C x t c x t x tφ φ σ= + − (13)
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Figure 7 gives the graph for the total concentration of particles where at a certain time t, the concentration in
the drilling fluid and external filter cake in constant for every position. The concentration of the particles in the
porous medium decreases with position x.
C(x,t)
x0
Drilling Fluid
Filter Cake
Porous Medium
Top Core
Figure 7: Graph for total concentration
3.5 Radial Filtration
3.5.1. External filter cake
A derivation of static filtration equations was made for linear flow in equations 6-8. Now these equations are
derived for a radial geometry. Figure 8 gives schematically the situation in the well.
R2
R1
rf
rc
Pin Pout
Figure 8: Wellbore with external filter cake
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The volume of the particles for a radial geometry is given by equation 14. Where cp depicts the concentration
of particles, V(t) the filtration volume.
2 2
2
( )(1 ) ( )
1
p
p c C p
p
c V tV R r h c V
cφ π= − − = =
− (14)
The pressure difference over the external filter cake and small portion of formation is as follows:
fcPP
in out in mid mid outP P P P P P P
∆∆
∆ = − = − + −����� �����
(15)
Assuming that 2 ;c c c cR r l l r= + ≪ and 2 ;f f f fr R l l r= + ≪ , the pressure difference over the filter and a small
portion of the formation is depicted in equation 16.
2 2
12 2
f f f fc c
f c w f c f
k l l kl lQ QP
k h k R l k h R k l
µ µπ π
∆ = + +
≃ (16)
The leak-off volume V(t) obeys the following ordinary differential equation
[ ]0 1
( )( ) 1
dV tQ F V t
dt= + (17)
In which F1 is given by:
1
f
c f
kF
k l
α= (18)
22(1 )(1 )
p
p c c
c
c R l hα
φ π=
− − (19)
The general solution of for the filtration volume is given in equation 20:
( ) ( ) ( )0 s
2
1 s 1 0 s
s
1
Q t, t < t
V t = F V + 1 + 2F Q t - t - 1, t > t
F
(20)
17
Using that the spurt volume and spurt time (Vs and ts) are equal to zero equation 20 becomes:
( ) 1 0
1
1+ 2F Q t - 1V t =
F (21)
Further simplifying by the condition1 0
2F Q 1≫ , gives:
0
0
2QV(t) = t
F (22)
3.5.2 Deep-bed filtration
A linear model has been proposed for linear deep bed filtration. Now a 1D radial deep bed filtration model is
presented (Pang et al. 1997; Bedrikovetsky et al. 2010). The radial cross section is given in Figure 9.
ksk
rs
R2
ksk
rs
R2
Figure 9: Radial section
The problem requires now knowing the velocity u(r,t) and σ(r,t). The relevant conservation equations read:
( ) ( ) 0c + u c = 1- σ; ru =t r t rφ φ∂ ∂ ∂ ∂ (23)
From the second conservation equation:
( )ru = A t (24)
If injection is done a constant flow rate q is known, A(t) is time dependent coefficient such that:
ru = q 2π
18
The solution for the equation for the concentration is:
( ) ( )( )
2 2 f
f
c exp[-K r - R ], r < r tc(r,t) =
0, r > r t
(25)
By integration of the kinetic equation we obtain also:
( ) ( ) ( )( )
22 20 2f f
f
exp[-K r - R ]λ cr t - r , r < r t
σ(r,t) = 2 r
0, r > r t
(26)
With, ( ) ( )22f Rr t = + qt πhφ
The profile of σ(r,t) and c(r,t) for deep-bed radial filtration is sketched in Figure 10.
r
c(r,t)
σσσσ(r,t)
R2 rs
Figure 10: Profile for σσσσ(r,t) and c(r,t)
19
3.6 Application of theory to well case.
In appendix II an overview is given of the procedures involved with fluids during drilling. If in drilling operations
losses are severe (above 3-5m3/h) drilling operations are put on hold and Loss Control Material (LCM) is added
to the drilling fluid. Normally the drill string is pulled up a few stands before the drilling fluid with LCM is
pumped and spotted in the wellbore. If the depths of the formation, which causes these losses, are known, the
drilling fluid with LCM can be pumped in the annulus above the permeable zone, after which circulation will be
paused for approx. 1-2 hours. Not circulating the drilling fluid gives LCM time to form an external filter cake
across the loss zone, without eroding the external filter cake.
The linear model for the external filter cake and internal filtration, can be related to our experiments in chapter
5 . The models for radial filtration build give a better approximation of the situation in a well. The models for
flow in a drill pipe and annulus are not directly connected to the experimental results. They give a better
understanding of the flow in a well, and can possibly be used to monitor the flow in a well and observe changes
in the flow possibly due to occurrence of leak-off.
4. Experimental Method
4.1 Experimental Setup
The experimental setup used in this study is shown schematically in Figure 11. A photo of the setup is also
shown in Figure 12. The core holder is made of PEEK which is transparent to X-rays. On the top of the core
holder there are two tubings, one which connects to the airstream and another to the syringe used for
injection of the drilling fluid. The core holder is vertically placed in the CT scanner, such that scans are parallel
to axis of the core. The experimental setup has an outlet valve, to ensure that the drilling fluid does not
penetrate into the core during injection.
The core used is a Bentheimer sandstone core: details about the core can be found in Table 1. The cores are
caste in Araldite self-hardening glue. After hardening of the glue, the core is machined such that the length of
cores is one third of the core holder. The cores are dried in an oven for 24-48hours. The glue is estimated to
penetrate 2mm into the core which is negligible. The CT scans are obtained using a third generation SAMATOM
Plus4 Volume Zoom Quad slice scanner (Table 2). More details about the CT scanner can be found elsewhere
(Nguyen, 2003).
20
Figure 12: Experimental setup place in CT scanner
Table 1:Characteristics Sandstone Cores Table 2: Settings CT- Scanner
4.2 Preparation of Oil Based Drilling Fluids
For the preparation of the oil based drilling fluid the primary emulsifier OmniMul was dissolved in the base oil,
while using an IKA overhead stirrer. The rotation speed of stirrer was increased gradually until 5000 rpm, and
stirring was continued for five minutes. Lime Ca(OH)2 was dissolved in water in a separate beaker and intensely
mixed using a magnetically stirrer. The brine was added while mixing the oil phase: stirring continued for five
more minutes. The bentonite, secondary emulsifier OmniChem and the fluid loss control additive were added
consecutively while the emulsion phase was being stirred. Finally, barite was added and the drilling fluid was
stirred with IKA stirrer at a rate of 3000 rpm for a length of 10 minutes. Table 3 gives the characteristics of the
components used for the drilling fluid mixtures.
Porosity (%) 20±1
Material Bentheim
Permeability (D) 1.3±0.2
Diameter (cm) 3.8 ± 0.1
Length (cm) 5.8 ± 0.1
Pore volume( cm3) 13± 0.5
Pore Size Average (µµµµm) 17±5
Description Conditions
Voltage 140 kV
Current 250 mA
Slice thickness 1 mm
Voxel size 0.3 mm
Scan Mode Sequential
Figure 11: Experimental Setup
21
Table 3: Characteristics drilling fluid components
Component Specific Gravity Weight%
Base oil= Sipdrill 2/0 0.81 29.4
Water 1.00 9.1
Lime= Ca(OH)2 2.24 0.9
Viscosity control= Bentonite 2.40 1.3
Density control=Barite 4.20 56.9
Fluid loss control=Gilsonite 1.00 0.5
Wettability control=OmniChem 1.00 0.4
Emulsifier=OmniMul 0.95 1.5
4.3 Core flow procedure
The experiments were carried out at the ambient temperature. The cores (Table 1) used dry at the start of the
experiment. The core is placed in the core holder and before any drilling fluid was injected, a CT scan image
was made. All the CT images were shot in the center of the core. This dry core CT scan image was obtained to
determine precisely the position of the core holder in the scanner but also to have dry core reference data.
Next the drilling fluid was carefully injected during one minute using a syringe. As already mentioned, during
injection of the drilling fluid the outlet valve is closed. A CT scan was made before pressure was applied. Then a
pressure of 7 Bar was applied by turning the pressure regulator on, simultaneously the outlet valve was opened
to start the core flow experiment. After the start of the core flow experiment, CT scan images were made at
time intervals over a period of 30 minutes.
A range of core flow experiments was performed varying the composition of the drilling fluid systematically,
starting with a base case drilling fluid (Table 4) checking the effect of each component individually and then
adding the components one by one. The different experiments are listed in Table 5. Each core flow experiment
was repeated to check the reproducibility of the measured leak-off volumes. The characterization of internal
filtration and the filter cake is only carried out for the first experiment of each set. For each set the same
drilling fluid recipe is prepared twice. Although the same formulation and procedure were used, slight
difference in the two mixtures of each set can exist. For characterization of the external filter cake and internal
filtration a Philips XL30 ESEM scanning electron microscope is used.
22
Table 4: Components in Group1 Table 5: Composition drilling fluids experiment
5. Results and Discussion
In this chapter the results of the leak-off experiments are discussed. Only the most representative experiments
are discussed. Additional Figures and images can be found in appendix III. For the first experiment it is
explained how the results are processed, which is similar to the other experiments.
5.1 Base drilling fluid
5.1.1. CT images and attenuation profiles
Figure 13 shows the intensity profiles for the CT scans from the leak-off experiment (Exp. 1A-1B in Table 5). The
inset is the CT-image shot 7 seconds after the start of the experiment. The horizontal axis represents the
position of the core: zero mm corresponds to the top of the inset image. The vertical axis gives the attenuation
coefficient measured by the CT scanner. The attenuation coefficient for air was -1000 HU, in the data
processing the value of the air is set to zero. The wiggly horizontal lines are respectively the air, the drilling fluid
and the sandstone core. These lines are not straight as the different values represent the different media
attenuating the X-rays. In the sandstone core there will be an extra effect caused by flow of the drilling fluid
into the core sample. The vertical lines are the interfaces, respectively due to the attenuation contrast between
air and drilling fluid and between drilling fluid and the core.
The inset image displays the following features: the yellow part at the bottom is the Bentheimer sandstone
core. The light blue box on top of the core is the drilling fluid. Above the drilling fluid, the dark blue part
indicates air. The red dots on the corners of the bottom of the core are O-rings for the sealing function.
Number Exp. Components
1A Base Drilling Fluid
1B Base Drilling Fluid
2A Base Drilling Fluid + Barite
2B Base Drilling Fluid + Barite
3A Base Drilling Fluid + Gilsonite
3B Base Drilling Fluid + Gilsonite
4A Base Drilling Fluid +Gilsonite+ Barite
4B Base Drilling Fluid +Gilsonite+ Barite
Components Base Drilling Fluid
Sipdrill 2/0 ( Base Oil)
Water
Emulsifier 1 (OmniMul)
Viscosifier (Bentonite)
Emulsifier 2 (OmniChem)
Lime
23
0 20 40 60 80 100 120 140 1600
500
1000
1500
2000
2500
3000
Position Core [mm]
Att
enua
tion
coe
ffic
ient
0 sec.7 sec.19 sec.33 sec.47 sec.71 sec.142 sec.171 sec.231 sec.291 sec.351 sec.470 sec.589 sec.890 sec.1188 sec.1486 sec.1784 sec.
0
500
1000
1500
2000
2500
3000
3500
Figure 13: Intensity profile for experiment 1A, with an inserted CT Image corresponding
At every time step 4 parallel images are shot with a spacing of 0.5 mm from each other. The images are
averaged (Figure 14) and used for determination of the intensity profiles. The filtrate volume over time is
determined from the shifting interface between air and drilling fluid in the measured intensity profiles. This
method proves to be rather accurate with an error of about 0.01 mm, comparable to that of the HPHT tests
(Hua et al. 2011). The red rectangle in Figure 13 indicates the area zoomed into in Figure 15. In this Figure, the
shift of the interfaces between air and drilling fluid is more is more clearly visible. This shift is converted into
the filtration volume.
Images shot at distance 0.5mm Averaged Image
Figure 14: Images after 7 seconds start of experiment, thickness 1 mm and spacing 0.5mm
24
56 57 58 59 60 61 62 63 64 65 66
0
200
400
600
800
1000
Position Core [mm]
Att
enua
tion
coe
ffic
ient
0 sec.7 sec.19 sec.33 sec.47 sec.71 sec.142 sec.171 sec.231 sec.291 sec.351 sec.470 sec.589 sec.890 sec.1188 sec.1486 sec.1784 sec.
Figure 15: Interfaces between air and drilling fluid for experiment 1A
Figure 16 shows four of the CT images obtained from experiment 1A at different times. In total 17 CT scan
images where shot over the duration of the test (30 minutes) but these 4 images were selected to visualize the
leak-off process. On the last image a very small light blue layer is present, which possibly indicates the external
filter cake. To better visualize the filtration volume and the external filter cake the CT images are subtracted
from the start image. In Figure 17 the top blue layer indicates the volume that penetrated into the core the
second blue layer on top of the formation surface is the external filter cake. It is clearly visible that at the spurt
time the external filter cake already started to form. Efforts to measure the thickness of the external filtercake
throughout the core flow experiment should be made, as thickness of the external filter cake is too small for
the settings of the CT scanner. The filter cake is roughly estimated to have a maximum thickness of 1 mm .
Table 6: Components Base Drilling Fluid
Start 33 Seconds ± 15 Minutes ± 30 Minutes 0
500
1000
1500
2000
2500
3000
3500
Figure 16: CT Images for Experiment 1A
Components Weight%
Sipdrill 2/0 69
Viscosifier (Bentonite) 3
Water 21
Emulsifier 1 (OmniMul) 4
Emulsifier 2 (OmniChem) 1
Lime 2
25
0
500
1000
1500
2000
2500
3000
3500
7 Seconds ± 15 Minutes ± 30 Minutes
Figure 17: Subtracted CT Images
5.1.2. Filtration volume
The filtration volume plotted in Figure 18 increases linearly as a function of the square root of time after the
spurt time. The linear relationship is determined using the least-squares method. The linear dependent V v.s.
t1/2
behavior agrees with the model that proposed in the theory section in equation 6-8. The spurt loss volumes
for experiment 1A and 1B are respectively 4.00±0.0.5 and 3.08±0.0.5 ml. Our model for the static filtration
contains the parameter for the flow rate during the spurt loss Q0. The spurt loss volumes are determined by
extrapolation and using the intersection of the linear fit and zero time. This method is common practice
(Bourgoyne et al. 1991) however the spurt time cannot be determined exactly. The measured spurt time
reported is actually the time it took after applying the pressure and taking the first CT image. Thus Q0 can only
be estimated roughly from the results. For all the experiments carried out the slope for the spurt loss is
determined by the connecting the point for the start with the point for the first image after the start of the
experiment. This suggests linear behavior. However the linear model for the spurt loss states: 0( )V t Q t= for
t<ts, thus the slope for the spurt plotted as function of the square root of time should be quadratic. Would it be
possible to make more measurements during the spurt, it is expected that this slope would show quadratic
behavior.
26
0 1 2 3 4 5 60
1
2
3
4
5
6
t1/2 [min]
Filt
ratio
n V
olum
e (m
l)
Experiment5Linear Fit Exp 5 Experiment 16Linear Fit Exp16
V(t)=0.27 t1/2+4.00R2=0.95
V(t)=0.30t1/2 +3.08R2=0.97
Figure 18: Filtration volume as function of the square root of time for experiment 1A and 1B
The experimental procedure was adjusted during the performance of the experiment sets 1-4. For part of the
experiments it was possible to control the pressure using an outlet valve outside the CT scanner room instead
of inside the CT room. The time to take the first CT images after applying and opening the outlet valve could be
reduced form 11 seconds experiment to 3-4 seconds. It is possible that the spurt time is even smaller than 3-4
seconds. This indicates that process of internal filtration and build up of filter cake is very fast, because the
largest amount of leak-off occurs during the spurt. The uncertainty in the spurt time occurs in every
experiment.
5.1.3 Characterization of the external and internal filter cakes
The external filter cake and the internal pore space of the core are examined using scanning electron
microscopy (SEM). The top part of the core (Figure 19) is used for characterization in the SEM (Figure 20). The
contrast in the images is determined by the atomic number of the elements. In addition an element analysis is
carried out. The particles are smaller than the size of the beam, therefore the spectrum shows the collective
compositions at the interested area.
The thickness of the external filter cake was found to be 0.60+0.05 mm. The dark and grey particles represent
bentonite and lime respectively. Traces of white barite particles are also present from other experiments, these
amounts are negligible. Throughout the external filter cake large and small pores are visible. The tiny pores are
probably due to the emulsified water droplets that evaporate upon drying the filter cake.
27
Glue
Sandstone
External Filtercake
Figure 19: Top core after drying
Figure 21 gives an overview of the elements present in the internal pore space and elements in the external
filter cake. Table 7 indicates which elements correspond to a specific particle type. From the elemental analysis
it is possible to get an impression of the contribution of the different particles to the external filter cake and to
the internal filtration. The elemental analysis reveals that the external filter cake is a combination of bentonite
and lime particles. It is clear from the elemental analysis and SEM images that both bentonite and lime
particles are present in the internal pore space as well. Figure 21 shows that Al and Si the markers of Bentonite
are present in both the external and the internal filter cake. The ratio Al/Si for the external cake is totally
different than that for the internal cake. This is because Si is the main component of sandstone rock. For a
quantitative analysis of the internal filtration of bentonite in all experiment sets an element analysis of a pure
sandstone core is needed.
In the theoretical model it was assumed that particles will pass through the external filter cake and penetrate
into the core if they are smaller than the pore size and that particles larger than the pore size will not enter the
core. A more complex picture emerges from the above results. Seemingly, small particles not only penetrate
the porous medium, but also they are part of the external filter cake, filling up gaps between larger particles.
Possibly the large particles initiate the formation of the external filter cake and small particles plug the pore
space in the cake, causing leak-off to slow down.
28
External Cake Thickness:599µm External Filtration
Internal Filtration Internal Filtration
A
B
Figure 20: SEM Images experiment 1A. The dark grey particles A indicate lime and the light grey particles B indicate
bentonite How about the very dense white particles?
29
Table 7: Chemical formulas particles in the drilling fluid
Element Chemical Formula/Content Elements that mark particles
Bentonite Al2O34SiO2H2O Si, Al
Lime Ca(OH)2 Ca
O Na Mg Al Si S K Ca Fe0
5
10
15
20
25
30
35
40
45
Elements
Wt%
External CakeInternal Filtration
Figure 21: Analysis for experiment 1A. Displayed weight percentage of elements in the external filter cake and internal
filtration
30
5.2 Base drilling fluid with barite
5.2.1. CT images and attenuation profiles
In the second experiment barite is added to the drilling fluid (see Exp 2A-2B in Table 5). In Figure 22 the drilling
fluid is shown as the bright red phase, due to the strong attenuation of the barite. The images contain a lot of
artifacts. The CT attenuation coefficient induced by barite dominates in the drilling fluid phase, which hinders
their detailed analysis. To get an idea of the filtration of this drilling fluid into the core a different approach is
used. The images are subtracted from the start image. In Figure 23 it is visualized how much drilling fluid
disappears from the top of the drilling fluid (represented as the thin red layer) and thus penetrates into the
core. By subtracting the first image form the other images (Figure 24) it is visualized how barite penetrates into
the core (the light blue phase). It is not possible to visualize the external filter cake for this experiment due to
the overwhelming attenuation of the barite in the drilling fluid.
Table 8: Components base drilling
fluid with Barite
Start 11 Seconds ±14.5 Minutes ± 27 Minutes0
500
1000
1500
2000
2500
3000
3500
Figure 22: CT Images for Experiment 2A
11 Seconds ±14.5 Minutes ± 27 Minutes0
500
1000
1500
2000
2500
3000
3500
11 Seconds ±14.5 Minutes ± 27 Minutes0
500
1000
1500
2000
2500
3000
3500
Figure 23: Ct images subtracted from the start CT Image Figure 24: Start Images subtracted from other CT Images
Component Weight%
Sipdrill 2/0 29.5
Viscosifier (Bentonite) 1.3
Water 9.10
Emulsifier 1 (OmniMul) 1.6
Emulsifier 2 (OmniChem) 0.4
Lime 0.9
Barite 57.2
31
5.2.2. Filtration volume
The measured spurt loss volumes for experiment 2A and 2B are respectively 0.87±0.05 ml and 1.21±0.05 ml.
The linear dependence of the leak-off volume on the square root of time after spurt time is also observed in
this case. The slopes have values of 0.13 ml and 0.14 ml. The presence of barite improvises the leak-off control
compared to the experiment with the base case drilling fluid. It should be noted that barite is added mainly for
weight control, thus making the particle density of the drilling fluid significantly higher than that of the drilling
fluid used in the experiments with the base case drilling fluid. Here we show that barite also contributes to
fluid loss control.
0 1 2 3 4 5 60
1
2
3
4
5
6
t1/2 [min]
Filt
ratio
n V
olum
e [m
l]
Experiment 2ALinear Fit Exp 2AExperiment 2BLinear Fit Exp 2BV(t)=0.13 t1/2+0.89
R2=0.99
V(t)=0.14 t1/2+1.21R2=0.96
Figure 25: Filtration volume as function of the square root of time for experiment 2A and 2B
5.2.3 Characterization external filter cake and internal filtration
The thickness of the external filter cake measured using SEM images (Figure 28) was found to be 2.00 ± 0.05
mm. The pore space near the surface of the core is plugged with mainly barite particles. The external filter cake
is a combination of barite, bentonite and lime particles. The bright white barite particles are clearly visible on
the SEM images. Several larger and tiny pores are visible in the external filter cake.
The external filter cake show a more densely packed combination of larger and smaller particles than the
previous experiment. This again confirms that the actual case for internal filtration is more complex than what
was proposed in the theory section for linear deep bed filtration. It is suspected that internal filtration and
external filter cake build up takes place simultaneously.
Barite particles seem to form agglomerates in the pore space, filling up the pores. From both the elemental
analysis and SEM images it can be argued that barite is the main component in the pore space as well as in the
external filter cake. Adding barite to the drilling fluid favors both the formation of an external filter cake and
32
the internal plugging of the pore space. This might be due to the fact that barite has a wide range of particle
sizes, which can contribute to a less permeable filter cake than previous case where barite is not present. In
this experiment the spurt loss volumes of 0.87 mland 1.21 ml are significantly smaller than the spurt loss
volume of 4.00 ml and 3.08ml for the previous experiment. The external filter cake is significantly thicker (2mm)
than that of the previous experiment (0.6mm). The role of barite in our experiments seem to be efficient in
contrast to the suggestion of Mihalik et al. 2002 that Barite does not seem to play a role in leak-off control.
The theory for linear deep bed filtration a model for the deposited particles was proposed in chapter 3, where
the deposited particles decrease exponential with depth. The experiments suggest that the largest portion of
deposition into the core takes place in the first few seconds and occurs at shallow depth. Images from the
micro-CT support this idea of a shallow penetration of barite. As can be clearly seen on micro-CT images and
these particles are present at a depth of 1.0±0.1 mm into the core. Figure 26 shows a cross section of the top of
the core displaying white barite particles. The top layer is the external filter cake packed with barite particles.
In the core it can be seen that some particles penetrated deeper into the core (up to 0.5 mm) and that plugging
of the pore space is even shallower.
Distance = 0.36 mmDistance: 0.36 mm
Figure 26: Cross section of top of the core. in the micro-CT scanner images grey particles represent the barite
External Filtercake
Sandstone
Glue
Figure 27: Top core after drying
33
B A
C
External Cake Thickness:2000µm External Filtration
Internal Filtration Internal Filtration Figure 28 : SEM Images experiment 1A. The dark grey particles A indicate lime, the light grey particles B indicate
bentonite and the white particles C is barite
Table 9: Chemical formulas particles in the drilling fluid
Element Chemical Formula/Content Elements that mark particles
Bentonite Al2O34SiO2H2O Si, Al
Lime Ca(OH)2 Ca
Barite BaSO4 Ba , S
34
Start 2 Seconds ± 14 Minutes ± 30Minutes0
500
1000
1500
2000
2500
3000
3500
C O Na Al Si S Ca Ba Fe0
5
10
15
20
25
30
35
40
45
50
Elements
Wt%
External CakeInternal Filtration
Figure 29: Analysis for experiment 2A. Displayed is the weight percentage of elements in the external filter cake and
internal filtration
5.3 Base drilling fluid with Gilsonite
5.3.1. CT images and attenuation profiles
In this experiment (see Table5, Exp 3A-3B) Gilsonite is added (Table 10), and barite is left out of the drilling fluid.
The drilling fluid is again displayed as a light blue phase. The last CT image after thirty minutes shows a tiny
light blue layer on top of the core surface. This layer might be indicating the external filter cake.
Table 10: Components Base drilling fluid
with Gilsonite
Figure 30: CT Images for Experiment 3A
Components Weight%
Sipdrill 2/0 68.2
Gilsonite 1.2
Water 21.0
Emulsifier 1 (OmniMul) 3.7
Viscosifier (Bentonite) 2.9
Emulsifier 2 (OmniChem) 0.8
Lime 2.2
35
Experiment 3A and 3B out show large difference in the spurt loss and linear square root behavior (Figure 33).
To investigate this difference also the subtracted CT images for experiment 3B are plotted in Figures 31 and 32.
Experiment 3A has a lower spurt loss than experiment 3B. The images in Figure 30 corresponding to
experiment 3A suggest that the external filter cake is thinner than that of experiment 3B. This might be due to
that particles have already settled on the formation surface between the time the drilling fluid was injected
and the core flow experiment was started.
2 Seconds ± 14 Minutes ± 30Minutes0
500
1000
1500
2000
2500
3000
3500
3 Seconds ± 15 Minutes ± 30 Minutes0
500
1000
1500
2000
2500
3000
3500
Figure 31: Subtracted CT Images for Experiment 3A Figure 32: Subtracted CT Images for Experiment 3B
5.3.2. Filtration volume
The spurt loss volumes for experiment 3A and 3B are respectively 0.95 ±0.05 ml and 1.87 ±0.05 ml. The spurt
loss and total filtration volume are smaller than those off or the base drilling fluid (see section 5.1.2), but larger
than the drilling fluid with added barite. In the field Gilsonite is used as leak-off control agent and added to the
drilling fluid when losses occur in the well. This experiment confirms that Gilsonite is effective in preventing
leak-off.
The difference in spurt loss and filtration volume for the repeated experiments 3A and 3B is much higher than
in the other experiments (Figure 18 ,25, 40). The spurt loss in the experiment 3B is twice that of experiment 3A.
The slope of V(t) v.s. t1/2
in experiment 3A is approximately 30% smaller than in experiment 3B. Other
experiments, in contrast, show spurt loss variations in the order of 20-30% and much smaller differences in the
slope of V(t) v.s. t1/2
. With use of the CT images already a possible explanation was given for this difference
other explanations could be:
1) The CT image belonging to the start of experiment 3A (Figure 30), shows some leakage of fluid in the right
top corner of the core. This might indicate that some of the fluid has already leaked-off into the core before
applying pressure.
2)The preparation of the drilling fluid was not performed exactly like in other experiments for one or both of
the experiment within this set.
36
0 1 2 3 4 5 60
1
2
3
4
5
6
t1/2 [min]
Filt
ratio
n V
olum
e [m
l]
Experiment 3A Linear Exp 3AExperiment 3B Linear Fit Exp 3B
V(t)=0.16 t1/2+1.14R2=0.94
V(t)=0.24 t1/2+2.16R2=0.98
Figure 33: Filtration volume as function of the square root of time for experiment 3A and 3B
5.3.3 Characterization external filter cake and internal filtration
The measured thickness of the external filter cake with SEM is approximately 0.84 ±0.05 mm. The external filter
cake contains lime, bentonite and Gilsonite particles. Bentonite and Gilsonite have also penetrated the pore
space. The external filter cake shows large holes, similar to the external filter cake in the experiment for the
base case drilling fluid.
External Filtercake
Sandstone
Glue
Figure 34: Top of core experiment 3A, after drying. Scale in cm
37
Internal FiltrationInternal Filtration
External Cake Thickness:836µm External and Internal Filtration
Figure 35: Images of external filter cake and internal filtration for experiment 3A, using SEM
Table 11: Chemical formulas particles in the drilling fluid
Element Chemical Formula/Content Markers Element Analysis
Bentonite Al2O34SiO2H2O Si, Al
Lime Ca(OH)2 Ca
Gilsonite C, H, N, S, O N, S
38
C N O Na Mg Al Si S Cl K Ca Ba Fe0
5
10
15
20
25
30
35
40
Elements
Wt%
External CakeInternal Filtration
Figure 36: Element analysis for experiment 3A, displayed is the weight percentage of elements in external filter cake
and internal filtration
5.4 Base drilling fluid with barite and Gilsonite
5.4.1. CT images and attenuation profiles
In this experiment (see Table5, Exp 4A-4B) Gilsonite and barite are added (Table 12).Again barite particles
cause strong radiation of the drilling fluid phase, limiting the information that can be obtained from the CT
images. Figure 38 and 39 give a better expression of the filtration of the drilling fluid. It is clearly visible that
barite penetrates again into the core.
Table 12: Components Base drilling
fluid with barite and gilsonite
Start 3 Seconds ±16 Minutes ± 30 Minutes0
500
1000
1500
2000
2500
3000
3500
Figure 37: CT Images for Experiment 2A
Components Weight%
Sipdrill 2/0 29.4
Barite 56.9
Water 9.1
Emulsifier 1 (OmniMul) 1.5
Viscosifier (Bentonite) 1.3
Emulsifier 2 (OmniChem) 0.4
Lime 0.9
Gilsonite 0.5
39
3 Seconds ±16 Minutes ± 30 Minutes0
500
1000
1500
2000
2500
3000
3500
3 Seconds ±16 Minutes ± 30 Minutes0
500
1000
1500
2000
2500
3000
3500
Figure 38: Ct images subtracted form the start CT Image Figure 39: Start Images subtracted from other CT images
5.4.2. Filtration volume
The spurt loss volumes for experiment 4A and 4B are respectively 0.58±0.05 ml and 0.79±0.05 ml. The spurt
loss and total filtration volume have the lowest values of all the experiment reported. This suggests that for the
experiments performed a combination of particles in the form of bentonite, lime, barite and Gilsonite is most
efficient in leak-off control. The combination of a high particle density induced by the addition of barite and
Gilsonite that serves as a leak-off control agent establishes a very efficient leak-off control. The result of the
two experiments are similar with no larger difference in the spurt loss, thus this indicates that the formulation
for this drilling fluid is reproducible.
0 1 2 3 4 5 60
1
2
3
4
5
6
t1/2 [min]
Filt
ratio
n V
olum
e [m
l]
Experiment 4ALinear Fit Exp 4AExperiment 4BLinear Fit Exp 4B
V(t)=0.18 t1/2 +0.58R2=0.79
V(t)=0.17 t1/2+0.79R2=0.69
Figure 40: Filtration volume as function of the square root of time for experiment 4A and 4B
40
5.4.3 Characterization external filter cake and internal filtration
The measured thickness of the external filter cake with SEM (Figure 42) is approximately 2.76 ±0.05 mm. This
drilling fluid mixture seems to have wide range of particle sizes, which causes build-up of a dense external filter
cake. The particles deposited in the external filter cake are mostly barite, but lime, bentonite and Gilsonite are
also present. Compared to the experiment where barite was added to the base case drilling fluid, the pore
space contains less barite particles, but contains a mix of different components. The pore space is also not filled
up densely by particles in comparison to the experiment where barite was added and Gilsonite was not present.
This could indicate that the presence of Gilsonite induces the formation of a densely packed external filter cake,
preventing the penetration of individual particles into the core.
A densely packed filter cake blocking drilling fluid to penetrate into the core is desirable. From the experiments
without barite it was possible to visualize the external filter cake on the CT images. The external filter cake
seems to already take substantial form during the spurt. It is expected that the initial build-up of the external
filter cake is due to particles larger than the pores not penetrating into the core; simultaneously internal
filtration takes places of particles smaller than the pore penetrating into the core. However these smaller
particles are also necessary to build-up an impermeable filter cake. Instantaneous formation of an
impermeable external filter is not possible but internal filtration should be limited to an absolute minimum. A
possible option to obtain faster an impermeable filter cake is to have a continuous gel like phase in the drilling
fluid which quickly forms an impermeable layer on the core surface.
In the introduction the concept of transition time was mentioned (Pang and Sharma 1997). The transition time
marks the time where no particles invade the formation any longer as the formation is plugged and the initial
layer of external filter cake starts to form. The last experiment shows a pore space that is not completely
plugged with particles. However this experiment has the most efficient leak off control and the thickest
external filter cake. This indicates that the pore space does not have to be plugged internally before build up of
the external filter cake can take place.
Sandstone
External Filtercake
Glue
Figure 41: Top core after drying
41
External Cake Thickness:2761µm External and Internal Filtration
Internal FiltrationInternal Filtration
Figure 42: Images of external filter cake and internal filtration for experiment 3A, using SEM
42
Table 13: Chemical formulas particles in the drilling fluid
Element Chemical Formula/Content Markers Element Analysis
Bentonite Al2O34SiO2H2O Si, Al
Lime Ca(OH)2 Ca
Barite BaSO4 Ba
Gilsonite C, H, N, S, O N
C N O Na Mg Al Si S Cl K Ca Ba Fe0
5
10
15
20
25
30
35
40
45
50
Element
Wt%
External CakeInternal Filtration
Figure 43: Element analysis for experiment 4A, displayed is the weight percentage of elements in external filter cake
and internal filtration
43
5.5 Summary of the filtration volumes
The leak-off volumes measured for each set of experiments were averaged and then plotted together in Figure
44. The reproducibility of the experiments will now be examined further. The experiment with added Gilsonite
and barite (experiment 4A and 4B) seem to be rather reproducible. In the experiment with Gilsonite added
(experiment 3A and 3B) to the drilling fluid, the difference in filtration volume between the two experiments is
large. It is possible that the high particle density induced by the barite contributes to a more reproducible
result. To further investigate the reproducibility, the experiments with similar formulations of drilling fluid
should be carried out in series of more than two experiments.
Some explanations for the differences in the results could be: 1) Small differences in composition can lead to
differences in particle density in the mixtures used. 2) The pressure applied is set to be 7 Bar, but this value can
divert as it is dependent on the use of the airstream in other places in the laboratory. 3) Depending on the time
between injection and applying the pressure it is possible that some of the heavy particles have sank to the
core surface in the core holder, causing premature formation of the external filter cake. 4) There can be
difference in the flatness of the core surface, penetration of the Araldite self-hardening glue and pore sizes on
the core surface.
Again it is pointed out that 0( )V t Q t= for t<ts . Q0 is estimated form the experimental data, and is used the plot
the slope for the spurt in figure 44.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
3
4
5
6
t1/2[min]
Filt
ratio
n V
olum
e [m
l]
Experiment 1Linear Fit Exp 1Experiment 2 Linear Fit Exp 2Experiment 3Linear Fit Exp 3Experiment 4Linear Fit Exp 4
V(t)=0.29 t1/2 + 3.54
V(t)=0.20 t1/2 + 1.61
V(t)=0.13 t1/2 + 1.04
V(t)=0.18 t1/2 +0.69
Figure 44: Plot of experiment 1-4. In which the result for each set of experiments are averaged and plotted. The slope
for the spurt loss is determined by estimating Q0 from the experimental data.
44
6. Conclusions and Recommendations
6.1. Conclusions
• The combination of the CT scanner core flow experiments and the characterization of the external
filter cake and internal filtration proved to be an efficient tool to quantify filtration and explain the
mechanisms involved.
• An external filter cake was visible on the CT images in experiments where no barite is present in
the drilling fluid. This enabled visualization of the external filter cake in time. During spurt the
external filter cake had already formed substantially. When barite was present in the drilling the
noise of barite was too strong for a proper visualization of the formation of an external filter cake.
The buildup of filter cake is very fast and the thickness is too small to be accurately measured with
the current settings of the CT scanner.
• The spurt loss seems to take place in a time frame between 0 and 3 seconds. The spurt loss
volume is the determining factor in the total amount of filtration volume. Future research into
controlling leak-off should focus on lowering the spurt loss volume.
• The theory for the linear square root behavior seems to match with our experimental results in all the
experiments. The experiments show that the mechanisms involved with deep filtration are more
complicated than the theory proposed for linear deep bed filtration in the theory section. Small
particles not only penetrate into the core but also contribute to build-up of an impermeable external
filter cake.
• Adding Gilsonite improves leak-off using the base case drilling fluid as reference, addition of barite
is however more efficient than addition of Gilsonite. Addition of barite particles to the base case
drilling fluid, results in a pore space filled up with barite agglomerates. The addition of both barite
and Gilsonite to the base case drilling fluid is most efficient in controlling fluid leak-off, and builds
up the most densely packed filter cake. However, in addition of barite, the density of particles
dramatically increases in the mud; the gravity might play a role in the filtration process. If Gilsonite
is present in the drilling fluid, the internal pore space is not completely filled up with particles. The
role of Gilsonite is not yet fully understood.
• Build up of external filter cake take place simultaneously with internal filtration. The pore space
does not have to be completely filled with particles to establish leak-off control
45
6.2. Recommendations
This study forms a basis for future improvement of oil based drilling fluid, by providing a better understanding
of mechanisms involved leak-off control. Various new insights were obtained on filtration behaviour using
advanced core flow experiments carried out in a CT scanner. Future work into drilling fluids using a similar
experimental procedure is now suggested:
• The reproducibility of the experiments has to be investigated more. Future experiments should be
performed in series of more than two experiments to rule out uncertainties.
• This study focuses on static filtration. To verify the role of erosion of the external filter cake and
contribution of circulated particles to build up of the external filter cake, dynamic filtration
experiments should be carried out.
• Further investigation has to focus on the mitigation of the spurt loss by investigating the use of
novel additives to favor the formation of a fast and efficient build of an external filter cake.
• The use of barite caused problems with the radiation from this specific component. For future
experiments it should be considered to use CaCO3 instead of barite. It would then also be possible
to visualize the external filter cake on the CT images.
46
7. Nomenclature K =consistency index
n = flow index
τ0 =yield stress
τ = shear stress
Rp = plug radius
R =radius drill pipe
P0 =pressure start drill pipe
PL =pressure bottom drill pipe
l =length of well
V(t) =filtration volume
ts =spurt time
kc, kf = permeability of filter cake and filter paper
Δp = total pressure drop
η = viscosity of mud
S = area of filter paper
cφ = porosity of filter cake
V(t) = volume of filtrate
cp = concentration of particles in the mud
D = dispersion coefficient
x = distance along the core in the direction of flow
c = concentration of suspended particles
C = total concentration of particles
c1 = Concentration particles in external filter cake
c2 = Concentration particles internal filtrate
σ =concentration of deposited particles
φ = effective porosity of the sandstone core
u = the velocity
λ = the deposition coefficient,
lc =thickness filter cake
lf = thickness internal filtration
rc = radius center well to filter cake
rf = radius center well to penetration depth internal filtrate
47
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50
Appendix I: Theoretical Model
Flow in drilling pipe and annulus with no movement of drill string
Governing Equations
In order to describe the motion of drilling fluids, we recall the basic governing equations. The motion of any
fluid is described by the equations of change expressing the conservations of mass, momentum and energy
(Bird et al. 1987). Table A.1 introduces the general form for the conservation of mass, momentum and energy.
Table A.1: Equations of Change
Equations of Change
Mass: ρ ρ∂ = −∇ ⋅
∂( )
tu
Momentum: ( ) ( )ρ ρ π ρ∂ = − ∇ ⋅ − ∇ ⋅ +∂t
u uu
Energy: ( ) ( ) ( )ρ ρ π∂ = − ∇ ⋅ − ∇ ⋅ − ∇∂⌢ ⌢
:U U qt
u u
The momentum and energy equations, both contain an additional momentum flux. In addition to the
momentum transport by flow, there is also momentum transferred by virtue of the molecular motions and
interactions within the fluid. This additional momentum is represented by the second order tensor p= +π δ τπ δ τπ δ τπ δ τ .
The assumption of isothermal flow makes the energy balance equation negligible.
In this study we are concerned with the flow of fluid in a pipe and in the annulus between the pipe and the
formation. The natural system of coordinates or this problem is the cylindrical polar coordinates ( )θ, ,r z . For
the present problem the velocity is along the axis θ and invariant by any rotationθ , i.e. ( ) ˆ= u r zu . Assuming
flow to be fully developed and non-inertia, equation 1 gives the momentum equation of laminar flow in a pipe:
= − ( )dp d rr
dz dr
ττττ (1)
Introducing the Shear stress ττττ , the force required to maintain the shear rate. Where the Shear
rate;γ =ɺ /dv dr is the velocity change dv, with displacement dr. The Shear Rate is the relative velocity with
which one layer moves with respect to an adjacent layer.
51
Rheological Models
Drilling fluids are too complex to be characterized by a single viscosity value. For most drilling fluids the
viscosity depends on shear rate. Fluids that do not follow a direct proportional relationship between shear
stress and shear rate are classified as non-Newtonian fluids. Drillings fluids are mostly psuedoplastic of nature,
meaning that apparent viscosity decreases with increasing shear. Besides that, drilling fluids are also generally
thixoptric, the apparent viscosity decreases with time after the shear rate is increased to a new constant value.
Thixotropic fluids have the characteristic to be shear and time dependant (Bourgoyne et al. 1991; Bird et al.
1987, Macosko 1994). Fluids with more clay and drilled solids are thixotropic. Figure A.1 represents the
different rheological models, where shear stress is plotted as a function of shear rate.
Figure A.1: Various Rheological models. (Schlumberger)
The flow index determines if the model is shear thickening/yield-dilatant (n >1) or shear thinning/yield-pseudo
plastic (n<1). Paint is an example of a shear thinning fluid. Drilling fluids are commonly shear thinning.
Power Law Model
The model is defined by:
γ= =ɺn
n dK K
dr
uττττ (2)
Where τ represents the shear stress,γɺ is the shear rate, K represents the consistency index of the fluid and n
is called the flow-behavior index. The power law model can be used to represent a psuedoplastic fluid (n<1), a
Newtonian Fluid (n=1), or a dilatant fluid (n>1). Equation 2 is only valid for laminar flow.
The power law model approximates the typical drilling profile for the lower range for the shear and, can thus
be used for that region.
52
The Bingham Plastic Model
A Bingham plastic will not flow until the applied shear stress τ exceeds a minimum value 0τ . The yield stress
0τ is the shear stress that should be overcome to start fluid flow. These equations are only valid for laminar
flow. Hence the Bingham model for 0τ>ττττ can be expressed by:
0pµ γ τ= +ɺττττ (3)
The Bingham Plastic model is a good approximation for the viscosity profile of a typical drilling fluid in the
medium shear rate ranges. The predominant factor affecting this part of the viscosity profile is the
concentration of inert solids. The model deviates significantly from measured data in the lower shear rate
range corresponding to the annular region of the well.
The Herschel Buckley Model
The Herschel-Bulkley model is a recognized model to approximate the rheological behavior of drilling fluids.
The Herschel-Bulkley model can be reduced to Power law, Bingham and the Newtonian model. When Setting
0 0τ = and 0α = the flowrate for the Power law model is obtained, in addition that n=1, the flowrate for the
Newtonian Model can be determined. If n=1 but 0 0τ ≠ then the flowrate for the Bingham model is obtained
The Herschel-Bulkley can be seen as being a more general non-Newtonian fluid model. The Herschel Bulkley
fluid needs similar to the Bingham model a certain yield stress, to initiate flow. The model is characterized by
three parameters namely, the consistency index K, the flow index n and the yield stressτ0 .
τ τ= + ≥0 0,
nd
K fordr
uτ ττ ττ ττ τ (4)
In which τ is the shear stress, d
dr
u
is the shear strain rate.
The Herschel Bulkley model is preferred above the Power law model and Bingham Plastic model for drilling
fluids. The reason for this preference is that Herschel Bulkley is more accurate in predicting the behavior of
drilling fluids. The hesitance to use the model lies in the fact that the derivation for the three parameter model
is complex. Below the velocity and flow rate for Herschel-Bulkley flow in a pipe and annulus are derived. For
the derivations the articles of Kelessidis et al. (2006), Wang et al. (1999), Bourgoyne et al. (1991), Bird et al.
(1987), and Makosco (1994) are used.
53
Flow in Drill Pipe
Assumptions:
i. the drill string is placed concentrically in the casing or the hole
ii. the drill string is not being rotated
iii. sections of open hole are circular in shape and of known diameter,
iv. the drilling fluid is incompressible,
v. the flow is isothermal.
The shear stress is presented in equation 5:
= − + 11
2
dp Cr
dz rττττ (5)
The shear stress should be finite at r = 0, thus C1 = 0. Hence
= − 1
2
dpr
dzττττ (6)
The drill pipe is considered to be a cylindrical a sketch of the flow profile with the parameters of interest is
given in Figure A.2. There is a region around central core of the fluid that has a shear stress less than the yield
stress. This region delimited by the plug radius Rp. In the plug area 0 pr R≤ ≤ , the velocity is constant and fluid
moves as a rigid plug. The radius at which there is an unsheared portion of the fluid is given by equation:
2,pR
τ=∆
Where ∆ stands for dp
dz (7)
Plug Region
Rp
R
Figure A.2: Laminar Flow in a Drilling Pipe, with plug region.
54
Now the wall shear stress is introduced in eq. 8 and eq. 9.
−= − = 0( )1
2 2
Lw
P P RdpR
dz Lττττ (8)
τ= w
r
Rττττ (9)
1
0 2
ndpd
K rdzdr
τ + = −u, because 0
du
dr≤ ,
du
dr is defined for in eq. 10.
1
0
1
2
ndur
dr Kτ∆ = − − −
(10)
u=0 at r=R. The fluid velocity u is given by:
( )τ τ τ ττ
+ + = − − − − +
1 1
0 0
1 1
1
n n
n n
w w
w
n KR ru
n K R K (11)
For the interval of the plug area ≤ ≤0 pr R , the fluid velocity is given by:
τ ττ ττ
+ + − = − − − +
1 1
0 0
1
n n
n npw w
p
w
Rn Ru
n K R K K K (12)
In Figure A.3 the velocity profile in a drill pipe is sketched. Figure A.3 also gives an impression of the shear
stress profiles within the drill pipe.
r r
τu
τ0
τ0
Rp
R
FigureA.3: Laminar Herschel Buckley Flow in a Cylindrical Pipe
55
For τ =0 0 the power law is recovered.
( )+ − = − +
1 1
01
1 2
nn n
LP P RnR ru
n L R (13)
If in addition if n=1 we obtain the expression for the velocity of a Newtonian fluid in a pipe:
( ) − − = − = −
2 220 01 1
2 2 4
LP P R P PR r R ru
L R L R (14)
If n=1 butτ ≠0 0 , then the velocity for the Bingham model is obtained:
ττ
− = − − − −
22
0 0( )1 1
4
L
w
P PR ru
L R For τ≥ 0ττττ (15)
ττ
−= −
2
0 0( )1
4
L
w
P PRu
L For τ τ≤ 0 (16)
The flow rate Q can be derived as follows:
π π= =∫ ∫ ɶ1
2
0 0
2 ( ) 2 ( )
R
Q u r rdr R u x xdx (17)
Where = rx
R, and 0 0
wR
τ τ ατ
= =
α
α
π
= + ∫ ∫
1
2 1 2
0
2Q R u xdx u xdx (18)
Where 1
u is the velocity for τ τ≤ 0 and 2
u the velocity forτ τ≥ 0 .
( )π ατ τ α ατ
+ + + + = − + − + + + + +
1/ 13 2 2
0
2 1 1 1 4 5 1(1 )
1/ 1 2 2(3 1) 2(2 1)(3 1)
n
w
w
K R n n nQ
n K n n n (19)
56
For the Power lawτ =0 0 , α = 0 the flowrate becomes:
π τ π − = = + +
113 3
0( )
3 1 3 1 2
nnLw P P RnR nR
Qn K n LK
(20)
In addition that n=1, the flowrate for Newtonian fluid becomes:
π τ π π− − = = =
113 3 4
0 0( ) ( )
4 4 2 8
L Lw P P R P PR R RQ
K LK KL (21)
If n=1 butτ ≠0 0 , then the flowrate for the Bingham model is obtained:
440 4 1
18 3 3
LP PRQ
K L
π α α− = − + (22)
This is the Reiner-Buckingham equation.
57
Flow in Annulus
Assumptions
In addition to the assumptions made for flow in a drill pipe, it is assumed that the drill pipe is placed concentric
in the annulus and flow in the annulus is in opposite direction of flow in the drill pipe. Figure A.4 gives a sketch
of the annulus and the flow profile. Establishing a velocity and flowrate profile for a Herschel-Bulkley model is
too complicated, the rheology model used to approximate the flow in the annulus is the Bingham Plastic Model.
R2
R1
Flow
rp
rn
Figure A.4: Flow in Annulus
Looking at the flow profile of a Bingham plastic there is an unsheared plug with boundaries rn and rp, in the
middle of the concentric annular flow. For laminar flow of Newtonian and non-Newtonian fluid it is not
possible to use a mean hydraulic radius, which early literature suggest can be used for turbulent flow of non
circular sections (Binder, 1943 and Moody, 1944). Laird (1957) defined the annular velocity for Bingham Plastic
fluids in an annular geometry. The definition of a Bingham fluid as given in equation 3 will be used in terms of a
hypothetical cylindrical body in which a drill string is concentrically placed. The force and velocity of the fluid is
acting the direction the flow. Laird uses the following boundary conditions:
1. There is no slip at the walls of the annulus, = =2) ( ) 01u(R u R .
2. The shear stress resisting ∆p is insufficient to overcome the yield stress, = = 0( ) ( )n pu r u r u .
3. From the reasoning behind it is also concluded that the viscosity must reduce to zero at the boundary
and inside the plug, = =( )( )
0pn
du rdu r
dr dr.
In the following the equations from Laird will be used to give the velocity and flow rate profile for Bingham
Plastic flow in the annulus.
58
The velocity profile for < <1 nR r r is given in equation 23.
τµ ∆= − + − +
2
1 0
1( ) ln
4
pru r A r r B
l (23)
The velocity profile for < <2 pR r r is given in equation 24.
τµ ∆= − + + +
2
2 0
1( ) ln
2
pru r A r r C
l (24)
A1, A2 ,B and C are constants of integration. There is no slip at the annulus walls, this gives the boundary
conditions: = =1 2( ) ( ) 0u R u R . These boundary conditions can be used to find B and C in terms of A1 and A1,
τ∆= − +2
11 1 0 1ln
4
pRB A R R
l (25)
τ∆= − −2
22 2 0 2ln
4
pRC A R R
l (26)
By equating the sum of the shear forces at the annular walls to the pressure drop times annular area in
equation.
2 2 1 22 1 0 1 2 1 2
( ) ( )( ) 2 ( )
du R du Rp R R R R R R
dr drπ π τ µ µ ∆ − = + + −
2 2 1 1 2 22 1 0 1 2 0 1 0 2
1 2
( ) ( )2 2 2
p pR A pR AR R R R R R
l l R l Rτ τ τ
∆ ∆ ∆− = + − − + + − +
Using equation 23 and 24 1( )du R
dr and 2( )du R
dr can be determined and r=R1 and R2, then A1 is found to be equal
to A2 so A1=A2=A. To find A the following boundary condition is used:
= = 0( ) ( )n Pu r u r u , where u0 is the velocity of the annular plug.
τ τ∆ ∆− − − − + + −=
2 2 2 2
2 1 0 2 1 0
2
1
( ) ( ) (( ) ( )4 4
ln
p n p n
n
p
p pR R r r R R r r
l lAR r
R r
(27)
59
Now the annular flowrate is determined by first using:
2
1
02 ( ) 2 ( ) 2
pn
p n
rr R
A
R r r
Q ru r dr ru r dr u rdrπ π π= + +∫ ∫ ∫ (28)
Equating the shear and pressure forces on the boundary of the annular plug gives (equation 29) a boundary
condition as given in equation 30.
π π τ∆ − = +2 2
0( ) 2 ( )p n p np r r l r r (29)
τ− =∆
02( )p a
lr r
p (30)
Using the boundary condition from equation 30, the flowrate in the annulus is given in equation 31.
2 2 2 2 2 2 2
2 1 2 14 4 4 4
2 12 2 2 2 2 2 2 20 0
2 1 1 21
2 3 2 33 3
20
( ) ( )( )2 2 16 8
( ) ( )16 16
ln ( ) ( )( ) ( )(( )16 4 4
2 (
2 2 6 2 3
p n
A n p
p n n p p n
n p p n p pn
p pR R r r R R
p p l lQ R R r r
R pl lr r r r R R R R r r
R l
r r r r r rr R R
π πτ τµ µ
π τµ
∆ ∆ − − + − − − ∆ ∆ = − + − + ∆ − − + − + + −
++ + + − − −3
1 )
6
(31)
For engineering applications Laird proposes a simplified version of equation 31. The simplification that is made
is similar to Bingham’s (1922) simplification for pipe flow. The simplification assumes that for reasonable high
pressure the dimensions of the plug can be neglected, thus rn=rp. Under these conditions the flow rate is given
by equation 32.
2 2 24 4 3 3 3 22 12 1 0 2 1 0 0 2 1
2
1
( ) 8( ) ( ) 4 3 ( )
8 3lnA
p R RQ R R R R r r R R
Rl
R
π τµ
∆ −
= − − − − + + +
(32)
In equation 32 r0 is given by equation 33.
2 22 2 1
02
1
2ln
R Rr
R
R
−= (33)
60
Linear filtration
External filter cake
Hua et al. (2011) proposes a physical model to describe the static filtration process. Here a summary of this
model is presented. Figure A.5 illustrates the drilling fluid containing particles filtering through a core.
Figure A.5: Overview Filtration Process
It is assumed that fluid and particles are incompressible in the filtration process. Here it is considered that only
solid particle in the drilling fluid contribute to the formation of the external filter cake. The role of emulsified
water droplets in the drilling fluid might also play an important role in the formation of external filter cake, but
is not taken into account in this model. The flow is laminar isothermal at the given pressure and flow rate. The
volumetric flow through a filter cake and filter paper is described by Darcy’s law. In the filtration process, the
number of particles in the mud that has been filtered is equal to the number of particles deposited in the filter
cake at any time t.
The leak-off volume V(t) obeys the following ordinary differential equation.
( ) ( )0 0
dV tQ = F V t + 1
dt
Where
( )( )f f p
0 0
f c c p f
k k cΔPQ = S ; F =
η l k 1- φ 1- c l S
61
In the above equations kc and kf are respectively the permeability of filter cake and filter paper. Δp Is the total
pressure drop .c f
l ,l Are respectively thickness of the filter cake and filter paper. η Is the viscosity of the drilling
fluid and S is used to describe the area of filter paper. c
φ Is the porosity of the filter cake. V(t) stands for the
volume of the filtrate. Finally cp is the concentration of particles in the drilling fluid. The filtrate volume is given
by:
( ) ( ) ( )0 s
2
0 s 0 0 s
s
0
Q t, t < t
V t = F V + 1 + 2F Q t - t - 1, t > t
F
(34)
Here ts and Vs are spurt time and spurt loss volume of filtrate respectively.
When s sV = 0, t = 0 ;
( ) 0 0
0
1+ 2F Q t - 1V t =
F (35)
In the condition0 0
2F Q 1≫ , equation 35 can be approximated as:
0
0
2QV(t) = t
F (36)
Equation 36 shows that the filtrate volume is proportional to the square root of the time after the spurt loss
time.
62
rparticle>rporerparticle<rpore
Deep-bed filtration
A model is presented for the internal filtration of drilling fluid. Figure A.5 gave a schematic overview for the
filtration through a core. The idea behind the filtration of particles into the core is that particles larger than the
pore size will not enter the core and particles smaller than the pore do enter the core. The drilling fluid
contains particles in a range of sizes. Particles in the drilling fluid will penetrate into the porous medium, build
up the filter cake, or stay in suspension the drilling fluid. The porous medium has a certain pore size range. The
particles will also have a certain size distribution. Particles that are too large to enter the porous medium will
be retained in the external filter cake. Particles that are smaller than the pores can filtrate into the core. The
particle size distribution and pore size distribution overlap. Figure A.6 sketches this situation. The light blue
space represents the area where particles are too large to enter, and the pore spaces are too small for these
particles to pass. For the model proposed, it is assumed that the pore size distribution is constant and the size
of the particles determines if particles penetrate. This situation is represented in Figure A.7. The yellow area
represents the fraction of particle c1 whose radius is larger than the pore radius. These particles cannot
penetrate the core and will therefore form the external filter cake. The grey area represents the fraction of
particles c2 that are smaller than the pore size. These particles penetrate the core.
F(r)
r
Particle Size Pore Size
Figure A.6: Distribution of particles size and pore size Figure A.7: Distribution of particle size
(1) (2)( )p pV V V t V= + + (37)
In equation 37, (2)
pV is the volume of the particles penetrated into the core, ( )V t is the volume of the filtrated
fluid, (1)
pV is the volume of the particles in the filtercake. (1)
pV Can also be given as:
(1)
1pV c V= �
(1)
1
pVV
c= �
(1)
(1) (2)
1
( )p
p p
VV V t V
c= + + �
(1) (1) (2)
1 1 ( )p p pV V c c V t V = + + .
Finally (1)
pV is rewritten in the form that is presented in equation 38:
63
(1) (2)1
1
( )1
p p
cV V t V
c = + −
(38)
For (1)
pV = ( )1 c cSlφ− equation 38 becomes:
(2)1
1
(1 )(1 )( )c c
p
c SlV t V
c
φ− − = + (39)
(2)( ) pV t V+ = Flux
The classical deep bed filtration model is a system that exists out of the mass conservation equation and the
kinetic equation. The classical deep filtration model can be used to describe the deposition inside the core
(Iwasaki 1937; Herzig et al. 1970; Sharma and Yortsos 1987). Assuming that the porosity is constant, the
dispersion coefficient is negligible (D=0), and the particle diffusion is negligible as well. This work will illustrate
the simplest case, in which the deposition coefficient is constant : 0λ λ=
The simplified mass conservation is given by equation 40
(1 )t x tc u cφ φ σ∂ + ∂ = − − ∂ (40)
Where c the concentration of suspended particles, σ is concentration of deposited particles, φ effective
porosity of the sandstone core and u is the velocity.
The following functions and derivations are used to obtain an expression for c(x,t).
( , )t f cσ σ∂ = Where 0( , )f c ucσ λ= (41)
The initial conditions are:
( , 0), 0c x t x= > , ( , 0) 0, 0x t xσ = = >
The boundary conditions are:
2( 0, ) , 0c x t c t= = > , ( 0, ) 0, 0x t tσ = = >
64
The solution is as follows:
( )ˆ( , ) ( )c x t c x H vt x= − , this form of functions is called Anzats, H represents a heavy side function, and u
vφ
=
The derivative of the heavy side function is given by a delta function. c is partially derived to t and x.
ˆ( , ) ( ) ( )t
xc x t c x t
v∂ = ∂ − (42)
1ˆ ˆ( , ) '( ) ( )x
x xc x t c x H t c x t
v v v
∂ = − − ∂ −
(43)
0
1ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )
x x x xc x t v c x H t c x t vc x H t
v v v v vλ = ∂ − + − − ∂ − = − −
The derivative of c is given by the following
0ˆ'( ) ( )c x c xλ= − � 0
ˆln ( )c x x constantλ= − + � 0ˆ( ) exp( )c x constant xλ= −i (44)
Using the boundary condition: ˆ( 0, ) (0)c x t c= = � 2ˆ(0)c c=
2constant c= � 2 0ˆ( ) exp( )c x c xλ= − . Using the last equation, the solution for c(x,t) is given in equation 45.
2 0( , ) exp( )x
c c x t c x H tv
λ = = − −
(45)
Subsequently a relationship for σ is determined.
0t ucσ λ∂ = (46)
Equation 46 can also be written using a heavy side function:
0ˆ( ) ( )t
xuc x H t
vσ λ∂ = − (47)
65
Position (x)
σ(x,
t)
t1t2t..t..t..t..t..t..tn
Position (x)
C(x
,t)
t1 t.. tnt.. t.. t.. t.. t.. t.. t..
Use following conditions
xt
v< � 0tσ∂ = ,
xt
v> � 0
ˆ( )t uc xσ λ∂ =
0 2ˆ( , ) ( ) ( )x t uc x t f xσ λ= + (48)
For t=0 , σ as function of x and t is : 1 2( ,0) ( ) ( )x f x f xσ = =
Equation 49 displays the function of σ .
0( , ) ( , )x t utc x tσ λ= (49)
( ) ( ) ( )0 2σ x,t = λ c vt - x exp -Kxφ
In Figure A.8 and A.9, c(x,t) and ( , )x tσ are given as function of the position x. In Figure A.8 the different lines
for t represent the moving front of suspended particles. For ( , )x tσ at times t1,..,tn the different graphs for
concentration of deposited particles are shown.
Figure A.8: Suspended particles Figure A.9 : Deposited particles
66
The total concentration of particles is the sum of the suspended particles and deposited particles is given in
equation 50
( , ) ( , ) (1 ) ( , )C x t c x t x tφ φ σ= + − (50)
Figure A.10 gives the graph for the total concentration of particles where for a certain time t, the concentration
in the drilling fluid and external filter cake are approximated to be constant for every position. The particles
concentration in the porous medium decreases with an increase for position x.
C(x,t)
x0
Drilling Fluid
Filter Cake
Porous Medium
Top Core
Figure A.10: Graph for total concentration
67
Radial Filtration
External filter cake
Equation 34-36 gave a set of equations for the filtration volume V(t) in a static situation. A similar set of
equations is used to describe filtration for radial flow. During mud circulation through the drill pipe and the
annulus, filter cake will be formed on the formation surface. Figure A.11 gives an overview of the cross section
of the situation, if filter cake is formed. lc Is the thickness of the filter cake.
R2
R1lc
rc rf
Figure A.11: Cross Section of the well, where lc is the thickness of the external filter cake.
To give an approximation of the pressure in the well, the effects of the formation of external filter cake are
taken into account.
R2
R1
rf
rc
Pin Pout
Figure A.12: Wellbore with Filter Cake
Now these equations are derived for a radial geometry as this is the simplistic situation in the well, as given in
Figure A.12. The volume of the particles for a radial geometry is given by equation 51. Where C depicts the
concentration of particles, V(t) the filtration volume. It is assumed that fluid and particles are incompressible in
68
the filtration process. For now it is only considered solely particles in the mud, contribute to build up of the
filter cake. The flow is laminar isothermal at the given pressure and flow rate.
2 2
2
( )(1 ) ( )
1
p
p c C p
p
c V tV R r h c V
cφ π= − − = =
− (51)
In which h is the length of well/
Equation 51 can be rewritten as; 2 2
2( ) (1 )(1 ) ( )p p c Cc V t c R r hφ π= − − − .
The pressure difference over the filter cake and small portion of formation is as follows:
fcPP
in out in mid mid outP P P P P P P
∆∆
∆ = − = − + −����� �����
Using equation 52 and 53, the pressure difference over the filter and a small portion of the formation are
depicted in equation 54.
2ln2
Lc
c c
Q RP
k h r
µπ
∆ = (52)
2
ln2
fLf
f
rQP
k h R
µπ
∆ = (53)
2 2
2 2
ln ln ln ln2 2 2
f f fL L L
c c f f c c
r k rQ R Q Q RP
k h r k h R k h k r R
µ µ µπ π π
∆ = + = +
(54)
Assuming that; 2 ;c c c cR r l l r= + ≪
The following approximation can be made;
2ln ln ln 1c c c
c c c
r l lR
r r r
+= = +
� 2
2
ln c c
c c
l lR
r r R= ≈
Similar to the above it can be assumed that: 2 ;f f f fr R l l r= + ≪
This gives the following approximation:
2 2 2 2
ln ln ln 1f f f f fr r l l l
R R R R
+ ≈ = + ≈
69
Using the approximations equation 54 becomes:
2 2
12 2
f f f fc c
f c w f c f
k l l kl lQ QP
k h k R l k h R k l
µ µπ π
∆ = + +
≃ (55)
Rewriting equation 55:
21
f f cw
f c f
k h k lrP Q
l k l
πµ
∆ = +
(56)
By using the following:
2
2 2
2 2 2
2
( )( )
c
c c c
l R
R r R r R r
≈ ⋅
− = − +� �� �
. Equation 51 can be used to find an equation for lc as given in
equation 57.
22(1 )(1 )( )
p c c
p
c R l hV t
c
φ π− −= � ( )cl V tβ= (57)
In which β is given by equation 58.
22(1 )(1 )
p
c c
c
C R l hβ
φ π=
− − (58)
Substitute equation 57 in equation 56 gives:
22 ( )
1f f
L
f c f
k h kR V tP Q
l k l
π βµ
∆ = +
(59)
Rewrite as:
[ ]0 1
( )( ) 1
dV tQ F V t
dt= + (60)
In which, 1
f
c f
kF
k l
α= .
70
To obtain a solution for V(t), equation 60 is integrated
[ ]( )
0 1' ( ) 1
s s
V tt
t V
Q dt F V t dV= +∫ ∫
The general solution of equation 60 for the whole filtration process is:
( ) ( ) ( )0 s
2
1 s 1 0 s
s
1
Q t, t < t
V t = F V + 1 + 2F Q t - t - 1, t > t
F
(61)
Using that the spurt volume and spurt time (Vs and ts) are equal to zero equation 61 becomes:
( ) 1 0
1
1+ 2F Q t - 1V t =
F (62)
Further simplifying by the condition1 0
2F Q 1≫ , gives:
0
1
2QV(t) = t
F (63)
Equations 61-63, are similar to the equation set 34-36. The difference however is that now radial filtration is
presented. Although the core flow experiments carried out for this research are static, radial filtration gives a
more realistic situation for filtration in the wellbore.
71
Deep-bed filtration A linear model has been proposed for linear deep bed filtration. Now a 1D radial deep bed filtration model is
presented (Pang et al. 1997; Bedrikovetsky et al. 2010). The radial cross section is given in Figure A.13
Figure A.13: Radial section
The problem is requires now knowing the velocity u(r,t) and s(r,t). The relevant conservation equations read:
( ) ( ) 0c+u c = 1- σ; ru =r rt tφ φ∂ ∂ ∂ ∂
σ(r,t) obeys:
( )tσ = λ σ uc∂
From the second conservation equation:
( )ru = A t
If injection is done a constant flow rate q the constant is known A(t) is constant such that:
ru = q 2π
The particle conservation equation becomes thus:
-1 -1t r1 1 1
qc + r r c = -Kr r c; q = ;K = 1-
2πkhφφ∂ ∂
ksk
rs
R2
ksk
rs
R2
72
The Darcy’s equation reads:
( ) -1ru = -k σ μ P∂
The solution for the equation for the concentration is:
( ) ( )( )
2 2 f
f
c exp[-K r - R ], r < r tc(r,t) =
0, r > r t
By integration of the kinetic equation we obtain also:
( ) ( ) ( )( )
22 20 2f f
f
exp[-K r - R ]λ cr t - r , r < r t
σ(r,t) = 2 r
0, r > r t
With ( ) ( )22f Rr t = + qt πhφ
The profile of σ(r,t) and c(r,t) for deep-bed radial filtration is sketched in Figure A.14
r
c(r,t)
σσσσ(r,t)
R2 rs
Figure A.14: Profile for σσσσ(r,t) and c(r,t)
73
Model Validity
For the model described for Herschel Bulkley flow through the drill pipe and annular Bingham plastic flow
through the annulus assumptions are made, which in the real case will not hold up. Now some of these
assumptions (concentric flow and no rotation of drill string) will be shortly discussed and references will be
made to articles which explain these cases in detail.
Eccentric Annular Flow
In the real situation the drill pipe will not be situated concentrically in the borehole. In practice the borehole
will be eccentric. Especially when deviated and horizontal well paths are considered an eccentric geometry is a
more realistic situation. Luo et al. (1999) offers an analysis of eccentric annular flow. In their analysis they use
the concept that an eccentric annulus is replaced by an infinite number of concentric annuli. They present for
both the power-law and Bingham Plastic fluids analytical solutions for the shear stress, shear rate, velocity and
volumetric flow rate/pressure gradient.
Rotation of Drill Pipe -
Would the drill pipe be rotated than both the translation motion in the z-direction and the rotational flow have
to be taken into account. Would there be rotation in the drill string two directions of movement can be
considered the equation for the velocity in cylindrical polar coordinates is presented for this situation in
equation 64.
ˆ ˆ( , ) ( , )u r z u r z zθ= +u (64)
In practice a rotating drill string is the real situation during drilling operations. Due to the complexity of
considering rotation of the drill string this is not taken into account. There are however various studies
preformed on the subject of a rotating inner cylinder. These studies used various rheological models,
concentric and eccentric annuli. Escudier et al., 2002 have studied different rheological models (Power-Law,
Herschel–Bulkley, Carreau and Cross) combining the effects of an eccentric annulus and with rotation of the
inner cylinder. Modeling the flow between two cylinders with rotation of the inner-cylinder is also used in
rheometers approximated as Couette flow (Bird et al., 1987, Bourgoyne et al., 1991 and Macosko,1994).
Marken et al., 1992 suggest that the principle of Couette flow is useful for a slow circulation rate of the drilling
fluid. In case of higher flow rates it gets a lot more complicated.
74
Appendix II: Handling Fluid Losses during Drilling This overview is meant to illustrate how in the field losses are treated during drilling. If in drilling operations
losses are severe (above 3-5m3/h) drilling operations are put on hold and Loss Control Material (LCM) is added
to the drilling fluid. Losses can be classified in the following 3 categories:
a) Seepage losses when 90-99% of the drilling fluid returns.
b) Partial losses with a 35-89% return of the drilling fluid
c) Severe losses with only 0-35% returns.
Although the above classifications are generally recognized to be correct, one has to bear in mind that a return
rate expressed in percentage is very subjective, because the effect of the rate of circulation as well as the effect
of annular circulation friction (known as ECD or Equivalent Circulating Density) are not taken into account.
Occasionally there are partial losses during circulation and no losses when circulation is stopped, thus clearly
showing the difference between dynamic losses and static losses as a result of ECD and possible erosional
effect of the external filter cake.
There is a wide variety of LCM on the market, ranging from natural fibers, such as nutshells to rock particles,
such as calcium carbonate as well as different types of polymers. Normally the drill string is pulled up a few
stands before the drilling fluid with LCM is pumped and spotted in the wellbore. If the depths of the formation,
which causes these losses, are known, the drilling fluid with LCM can be pumped in the annulus above the
permeable zone, after which circulation will be paused for approx. 1-2 hours. Not circulating the drilling fluid
gives LCM time to form an external filter cake across the loss zone, without eroding the external filter cake.
Figure A.15 gives a schematic overview of the drilling fluid pumped into the wellbore in relation to the depth of
the permeable zone.
75
Height up to where Mud with LC is pumped
Permeable Zone
Figure A.15: Schematic overview of well with Permeable zone and Drilling fluid containing LCM
The bit is pulled such that it is at least opposite the bottom of the loss zone. Then the LCM is spotted across the
loss zone and above (about 50+ m). The drilling fluid with LCM can then plug the loss zone. In some instances
immediate success can be seen, i.e. no static losses and drilling fluid level remains at surface, whereas other
instances we notice that the level drops until it finds its equilibrium, a measurable distance from surface. At
that time, it is unknown if the LCM will stop the dynamic losses. It is necessary to wait until going back to
bottom and start circulation once more.
The choice of components and LCM in the drilling fluid is very dependent on the formation that is drilled, i.e. if
the pore sizes of the formation are known and if the formation is part of the reservoir. If the pore sizes of the
permeable zone are well known the LCM to use is easier to determine. Otherwise the choice is often made to
use LCM in which the particles vary in size (small, medium and large sizes). Oil and/or gas will produce from the
reservoir once the well is completed. It is important that the zones from which production will take place later
are not impaired by invasion of drilling fluid. The choice of component and LCM in the drilling fluid should take
this possible formation damage of the reservoir zone into account. Drilling the reservoir with CaCO3 is
preferred over the use of barite.
The choice of LCM, concentration of LCM, composition of the overall drilling fluid and pumping strategies are
made with the help of detailed guidelines for fluid losses and the experience we have gained over the many
years using these products.
76
Appendix III: Additional Images Core Flow Experiments
The results and discussion section reported on eight experiments that were carried out in four sets. Besides
these four sets eleven other experiments were carried out. From these experiments five experiments failed
due to incorrect performance of the experimental procedure or failure of the CT scanner. Four experiments
failed due to the fact that no outlet valve was used. In this appendix the results of an additional experiment will
be reported. The fluid injected in this additional experiment was base oil. The experimental procedure slightly
deferred, as the outlet valve remained closed during the whole experiment. One other additional experiment
was carried out using polymer plus a crosslinker as leak-off control agent. The polymer and crosslinker are
expected to form a continuous phase with water emulsified water droplets. However this experiment did not
show any improvement of leak-off control compared to the base case scenario. It was concluded that further
investigation has too take place into controlling the size of the emulsified water droplets, which is outside the
scope of this thesis. Table A.2 gives an overview of the experiments from which experiments 1-4 are reported
in the thesis and experiment 5 will be discussed in this appendix. For the experiments 1-4 the remaining CT
images and intensity profiles will be given which were not shown in the Results chapter.
Table A.2: Overview experiments
Nr. Components
1A Base Drilling Fluid
1B Base Drilling Fluid
2A Base Drilling Fluid+ Barite
2B Base Drilling Fluid+ Barite
3A Base Drilling Fluid+ Gilsonite
3B Base Drilling Fluid+ Gilsonite
4A Base Drilling Fluid+Barite+Gilsonite
4B Base Drilling Fluid+Barite+Gilsonite
5 Base Oil
Table A.3: Components Group 1
Components Group 1
Sipdrill 2/0
Viscosifier (Bentonite)
Water
Emulsifier 1 (OmniMul)
Emulsifier 2 (OmniChem)
Lime
77
Base case drilling fluid
The results for this experiment are discussed in the results chapter in the thesis. In this appendix the CT images
and the intensity profile for experiment 1B is given as well. Both the CT images and the intensity for
experiment 1A and 1B show similar behavior.
Start 4 Seconds ± 15 Minutes ± 30 Minutes0
500
1000
1500
2000
2500
3000
3500
Figure A.16: CT Images Experiment 1B
0 20 40 60 80 100 120 140 1600
500
1000
1500
2000
2500
3000
Position Core [mm]
Att
enua
tion
coe
ffic
ient
0 sec.410 sec.18 sec.24 sec.30 sec.41 sec.82 sec.141 sec.240 sec.300 sec.421 sec.601 sec.840 sec.1141 sec.1388 sec.1741
Figure A.17, Intensity profile for experiment 1B
78
Base case drilling fluid with barite
The results for this experiment are discussed in the results chapter in the thesis. The CT images for experiment
2A and 2B are influenced strongly but the radiation of barite, which hinders information. Both the CT images
and the intensity profiles for experiment 2A and 2B show similar behavior, but differ much in comparison to
the experiment where no barite is present. This reason for this difference is that the drilling fluid for the
experiment with barite in the drilling fluid has a higher attenuation coefficient than the Bentheimer sandstone
core. If barite is not present in the drilling fluid, the attenuation coefficient op the Bentheimer sandstone core
is higher than that of the drilling fluid.
Start 4 Seconds ± 13 Minutes ± 30 Minutes0
500
1000
1500
2000
2500
3000
3500
Figure A.18: CT Images Experiment 2B
79
0 20 40 60 80 100 120 140 1600
500
1000
1500
2000
2500
3000
3500
4000
4500
Position Core [mm]
Att
enua
tion
coe
ffic
ient
0 sec.11 sec.28 sec.41 sec.56 sec.73 sec.89 sec.115 sec.145 sec.170 sec.269 sec.329 sec.340 sec.416 sec.491 sec.666 sec.797 sec.921 sec.1045 sec.1344 sec.1574 sec.
Figure A.19: Experiment 2A
0 20 40 60 80 100 120 140 1600
500
1000
1500
2000
2500
3000
3500
4000
4500
Position Core [mm]
Att
enua
tion
coe
ffic
ient
0 sec.4 sec.10 sec.17 sec.23 sec.30 sec.39 sec.61 sec.99 sec.131 sec.192 sec.253 sec.374 sec.495 sec.607 sec.784 sec.1021 sec.1191 sec.1487 sec.1773 sec.
Figure A.20: Experiment 2B
80
Base case drilling fluid with gilsonite
Start 3 Seconds ± 10 Minutes ± 30 Minutes0
500
1000
1500
2000
2500
3000
3500
Figure A.21: CT images Experiment 3B
0 20 40 60 80 100 120 140 1600
500
1000
1500
2000
2500
3000
Position Core [mm]
Att
enua
tion
coe
ffic
ient
0 sec.11 sec.18 sec.24 sec.34sec.44 sec.78 sec.138 sec.198 sec.300 sec.489 sec.840 sec.1200 sec.1794 sec.
Figure A.22: Experiment 3A
81
0 20 40 60 80 100 120 140 1600
500
1000
1500
2000
2500
3000
Position Core [mm]
Att
enua
tion
coe
ffic
ient
0 sec.3 sec.9 sec.15 sec.22 sec.29 sec.39 sec.72 sec.120 sec.180 sec.239 sec.300 sec.420 sec.600sec.900 sec.1202 sec.1500 sec.1799 sec.
Figure A.23: Experiment 3B
82
Base case drilling fluid with barite and Gilsonite
Again barite is present in the drilling fluid hindering detailed information from the CT images. The results for
this experiment are discussed in chapter 5.
Start 3 Seconds ± 16 Minutes ± 30 Minutes0
500
1000
1500
2000
2500
3000
3500
Figure A.24: CT Images Experiment 4B
83
0 20 40 60 80 100 120 140 1600
500
1000
1500
2000
2500
3000
3500
4000
4500
Position Core [mm]
Att
enua
tion
coe
ffic
ient
0 sec.3 sec.9 sec.15 sec.22 sec.28 sec.39 sec.60 sec.120 sec.179 sec.240 sec.300 sec.420 sec.540 sec.953 sec.1200 sec.1500 sec.1814 sec.
Figure A.25: Experiment 4A
0 20 40 60 80 100 120 140 1600
500
1000
1500
2000
2500
3000
3500
4000
4500
Position Core [mm]
Att
enua
tion
coe
ffic
ient
0 sec.3 sec.9 sec.16 sec.22 sec.29 sec.39 sec.59 sec.89 sec.120 sec.178 sec.239 sec.299 sec.419 sec.599 sec.778 sec.959 sec.1200 sec.1499 sec.1799 sec.
Figure A.26: Experiment 4B
84
Base Oil
The outlet valve remained closed throughout the whole experiment. Thus leak off the base oil into the core is
limited by the compression of the air .The pressure applied is 7 Bar. It is assumed that before the pressure is
applied the pressure in the core holder is atmospheric (1 Bar). The assumption is made that the ideal gas law is
valid:
1 1 2 2PV P V=
Where P1=1 Bar and V1= the volume before pressure is applied of the air. P2= 7 Bar and V2= the volume after
the pressure applied. Gives that 2 1
1
7V V=
This predicts that 1/7 of the injected base oil will penetrate into the core. The base oil injected is approximately
80 ml. After, 33 minutes 14.0 ml of base oil penetrated into the core. The filtration volume is a bit more than
1/6 of the injected base oil. The reasons for the filtration volume not being exactly 1/7 of the injected volume
can be: 1) The injected volume is not exactly 80 ml. 2) The atmospheric pressure and applied pressure are not
exactly 1 and 7 bar.
The spurt loss for the base oil is 13.8 ml. The large spurt loss is due the absence of particles and no filter cake
will be formed. On top of that the permeabilty of the Bentheimer sandstone being 1 Darcy is very good.
Start 6 Seconds ± 10 Minutes ± 33 Minutes0
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1000
1500
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3500
Figure A.27: CT images Experiment 5
85
0 20 40 60 80 100 120 140 1600
500
1000
1500
2000
2500
3000
Position Core [mm]
Att
enua
tion
coe
ffic
ient
0 sec.6 sec.27 sec.50 sec.73 sec.96 sec.173 sec.368 sec.613 sec.1987 sec.
Figure A.28: Intensity profiles experiment 5
0 1 2 3 4 50
2
4
6
8
10
12
14
t1/2 [min]
Filt
ratio
n V
olum
e [m
l]
Experiment 5Linear fit Exp 5V(t)=0.03 t1/2 +13.8
R2=0.77
Figure A.29: Filtration volume as function of the square root of time for Experiment 5