agip fluid dynamics

Eni Agip E&P DivisionCorporate Technical Services

Technological Area: Drilling & CompletionActivity: Mud & Cement Engineering

The Dynamics

ofDrillingFluids

Quaderno Tecnico no. 7

THE DYNAMICS OFDRILLING FLUIDS

INDEX

Introduction i

1 - The Rheology of drilling muds1.1 - Introduction 11.2 - Fluid deformation 21.3 - Flow regimes 3

Plug flow 3 Laminar flow 3

Transitional flow 4 Turbulent flow 41.4 - Determination of the flow regime 5

The Reynolds number 51.5 - Viscosities for Newtonian and non-Newtonian fluids 81.6 - Thixotropy, rheopecticity and gel strength 121.7 - Rheological classification of fluids during flow 14

Plastic fluids 15 Pseudoplastic and dilatant fluids 16 Yield pseudoplastic and dilatant fluids 18

1.8 - Rheological models 19 Models with one constant parameter 19 Models with two constant parameters 20 Models with three constant parameters 20 Models with four constant parameters 21 Models with five constant parameters 21

1.9 - Selecting the rheological model 221.10 - Determining rheological parameters by means of 25

rotational viscosimeter readings Bingham model 27

Ostwald & de Waele model 28 Herschel & Bulkley model 29

1.11 - The effect of pressure and temperature on rheological 31 parameters Plastic fluids 33 Pseudoplastic fluids 37

Yield pseudoplastic fluids 381.12 - Bibliography 40

2 - The hydraulics of drilling muds2.1 - Introduction 432.2 - Laminar flow in circular section 432.3 - Turbulent flow in circular section 522.4 - Critical flow conditions: transition from laminar to 56

turbulent flow2.5 - Laminar flow in annular section 572.6 - Turbulent flow in annular section 652.7 - Critical flow conditions: transition from laminar to 69

turbulent flow2.8 - Bibliography 70

3 - Secondary flows in the hydraulic circuit3.1 - Introduction 723.2 - Surface circuit 723.3 - The bit 773.4 - Determination of the discharge pressure at the mud pumps

(SPP)81

3.5 - Determination of the equivalent circulating density 813.6 - Power required by the mud pump 823.7 - Bibliography 83

Appendix - A brief outline of drilling oil wellsA1.1 - Introduction 84A1.2 - Drilling techniques 84A1.3 - The rig site 87A1.4 - Directional drilling 89A1.5 - The hydraulic drilling circuit 90A1.6 - Bibliography 92

0 17 February

1998

First issue Roberto Maglione Angelo Calderoni Giuseppe Libri

Rev. Date Description Prepared Checked Approved

Circulation of this document is limited to Eni and affiliate Companies

Rig G-125Rig G-125 WellWellRagusa 53 Dir-ARagusa 53 Dir-A

THE DYNAMICS OF DRILLING FLUIDS

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The Dynamics of Drilling Fluids i

INTRODUCTION

This technical manual briefly summarises the theoretical aspects of rheology (chapter 1) and thehydraulics of drilling fluids (chapter II) which are required for calculating flow parameters, pressureprofiles and well head pressure.A part of the theory discussed in these two chapters was taken from existing publications, whileother aspects were developed for the first time, on the basis of Agips experience.Chapter III gives some of the most important examples of flows in the hydraulic circuit, which aredefined as secondary in order to distinguish them from the dominant axial flows inside the drill stringand the annular section. The latter are in fact the flow in the surface circuit (of the drilling rig) andthe flow which can be seen at the bit.Some calculation examples are also given for a simpler and quicker understanding of the practicaluse of the various theories explained.Lastly, the drilling rigsite and well drilling techniques are briefly described in the appendix.

San Donato Milanese, 17th February 1998

The Dynamics of Drilling Fluids 1

1. THE RHEOLOGY OF DRILLING MUDS

1.1 - INTRODUCTION

The inability of a fluid to withstand a tangential force is a fundamental characteristic of liquids. Infact, if we apply a tangential force to a given volume of fluid, the fluid will lose its shape andtends to become deformed. If this deformation is continuous, it is known as flow.The flow of a fluid must always take place within a conduit. Often the conduit does not have acircular section (e.g. pipe), but may have the most varied of shapes, for example a fluid flowingdown an inclined tabletop is enclosed by the table surface underneath and by the atmospheri onthe sides and top.

Circular sections are the most frequent in the hydraulic drilling circuit and represent the typicalform of the inside of the drill string pipes, whether they are drill pipes, heavy weight or drillcollars. The nozzles of the bit, which mud passes through from the drill string to the annulus,belong to this category; these nozzles have a circular section, despite their slightly conical internalshape which promotes mud flow.A second type of section is annular, formed by the gap existing between the drill string and thewell of the hole or casing.

In the analytical procedures of this manual, the two sections, circular and annular, have beenconsidered as concentric in order to simplify the development of both the flow theories andresulting calculations. In reality, the string is hardly ever in line with the hole. In fact duringdrilling, the drill string is nearly always eccentric in relation to the hole axis, leading to a gapwhich is often variable, not associable to well defined geometries and often difficult to analyticallyevaluate. Moreover, while drilling horizontal wells, the drill string nearly always rests on thelower part of the hole, annulling all annular type geometric conditions.The type of conduit significantly affects the behaviour of fluid flow. In fact, all analyses of flowmust consider the geometric form of the conduit, as this greatly affects the complexity of theanalytical development for determining the formulas of the flow parameters.

In general, at low flow rates, fluid flow is the result of parallel fluid layers (laminae) sliding pasteach other at different velocities. The fluid layer adjacent to the wall of the conduit adheres to thesurface (and so its velocity is practically zero), while each subsequent layer slides past itsneighbour with increasing velocity. This type of flow is called laminar flow.At higher flow rates, the laminae lose their orderly movement and randomly crash into oneanother, while an orderly flow only remains very close to the wall of the conduit. This type offlow is called turbulent flow.The change from laminar flow to turbulent flow conditions is called the transitional zone.


1.2 - FLUID DEFORMATION

Figure 1.1. shows the characteristic deformation of an element of fluid, when subject to atangential force. We shall consider two parallel layers placed at a distance of h, both moving at avelocity of v. If we apply a force, F, which is tangential to the upper layer of area A, the velocityof the upper layer will increase about of v; likewise the sliding of a fluid lamina past the otherlamina will increase. The resistance to this sliding movement or frictional drag, is called shearstress.

Fig. 1.1 - Characteristic Deformation of an Element of Fluid Subject to Tangential Force

y

v + v

v

vx (y) (y) (y)..

A

F

h

x

dy

The shear stress, , is commonly defined as the applied force, F, divided by the area over which itacts, A. So the shear stress is:

= FA

(1.1)

The shear stress is commonly interpreted as acting in a direction which is opposite to the directionof the applied force, F. The velocities of two fluid layers, placed at a distance of dy, are given byv and v+dv respectively and are constant. Constant velocity indicates that no net force is actingupon the fluid layers to cause a change in the velocities or an acceleration. As the net force isequal to zero, the applied force, F, is equal in magnitude and opposite in direction to the frictionalforces. Accordingly, the shear stress may be directly defined in terms of the applied force, F.

When a force is applied to initially static fluid, the fluid accelerates from zero velocity to anaverage constant velocity. During this lapse of time, the applied force, F, is greater than that ofthe frictional forces and so the resulting force tends to accelerate the fluid. The state ofequilibrium which is reached may be considered as the state of a flowing system after a very longor even infinite time of flow. Generally speaking, all fluid flow analyses consider the equilibriumsystem as constant in time.


The stress existing between two adjacent fluid layers is correlated with the value .

, known as theshear rate or velocity gradient. The shear rate is defined as the difference in the velocitiesbetween two levels divided by the distance which separates them; in differential terms, it may beexpressed as:

.

.

= dvdy

(1.2)

The relationship between shear stress, , and the shear rate, .

, defines the fluids rheologicalbehaviour to flow. For some fluids, the relationship is linear so, if the shear stress is doubled, theshear rate will also double. These fluids are known as Newtonian fluids. Most of the fluids usedin drilling, however, are non-Newtonian and are instead defined by a more complex relationshipbetween shear stress and shear rate.

1.3 - FLOW REGIMES

During the flow of drilling fluid in a conduit, four different types of flow regime may be present,irrespective of the pipes geometry: plug flow, laminar flow, transitional flow and turbulent flow.

PLUG FLOWIn plug flow, the fluid basically moves as a single, undisturbed solid body or plug. The movementis made possible by the slippage of a thin layer of fluid along the conduit surface. This type offlow usually occurs at low flow rates and above all when the fluid has very marked plasticcharacteristics, with high yield stress values. The Reynolds number of this flow is generally < 100- 300. A flow regime of this type hardly ever occurs during drilling.

LAMINAR FLOWThe laminar flow of a non-Newtonian fluid in a pipe is characterised by the flow of laminae,which are similar to concentric cylindrical shells, which slide past one another like the sections ofa telescope. The velocity of the fluid shell at the pipe wall is zero, while the velocity of the fluidshell at the centre of the section is maximum.

Figure 1.2 shows the bi-dimensional velocity profile. As can be seen, the shear rate, .

, defined inthis case for a pipe with a circular section having radius r, as dv/dr, is simply the inclination of thecurve at any point of the velocity profile. The shear rate is maximum at the wall and zero at thecentre of the pipe.

As the shear stress is directly proportional to the shear rate, for a Newtonian fluid, the shear stressis also maximum at the wall of the pipe and zero at the centre of the pipe.Accordingly, both the shear stress and shear rate are related to the radial distance inside the pipe.The concept of variations in the shear rate and shear stress may cause some confusion.To explain this concept better, we shall consider another, characteristic parameter of the fluids:viscosity.Generally speaking, viscosity indicates the fluids thickness.


In quantitative terms, it isexpressed as the ratio betweenthe shear stress and shear rate,based on the followingexpression:

=

. (1.3)

For a Newtonian fluid, the ratiois constant and the viscosity isthe same at every pointthroughout the fluid, while theviscosity varies with the shearrate for a non-Newtonian fluid.Accordingly, the viscosity variesfrom one point to another insidethe pipe, for a non-Newtonianfluid.

Fig. 1.2 - Velocity Profile of a Fluid with Laminar Flow

dv/dr = 0

rr

flowvelocity

radius

0

Vmax

In general terms, regardless of the type of fluid studied, viscosity is considered as a fixedhomogeneous characteristic for a given substance. This notion is not however very accurate if theflow concerns a non-Newtonian fluid.

TRANSITIONAL FLOWThis flow is marked by a gradual transition from laminar flow to turbulent flow characteristics.Initially, when the flow is still orderly, a random destabilisation of the laminae or concentriccylinders sliding past one another, like telescope sections, may be noted due to the presence ofrandom vortical and chaotic movements.

As the flow rate, and hence the velocity of the flow, gradually increase, the vortical and chaoticmovements in the fluid also increase until they become predominant and annul nearly entirely thelaminar characteristics of the flow. In this case, the flow becomes turbulent.

It is difficult to represent the trend of the flow rate profile in this type of regime, due to therandomness of the vortices, for which there are no analytical expressions that accurately definethis type of fluid.

TURBULENT FLOWTurbulent flow is caused by high flow rates and low fluid viscosity values and is characterised bythe chaotic and disorderly movement of fluid particles. This random movement consists of twovelocity components: one, transversal to the flow and the other longitudinal to the flow. Thelongitudinal velocity tries to move the fluid particles in a parallel direction to the pipe axis, whilethe transversal velocity tries to move the fluid in a direction which is normal to the pipe axis.

Despite this random movement of the fluid particles, the final flow rate profile tends to be uniformalong nearly all the transversal section of the flow, with a part, close to the pipe wall, where thevelocity decreases very rapidly.


Fig. 1.3 - Velocity Profile of a non-Newtonian Fluid withTurbulent Flow

rr

flowvelocity

0

Vmax

radius

Figure 1.3 shows the velocityprofile of a non-Newtonian fluidwith turbulent flow. Due to thechaotic and random movement,the fluid essentially moves as if itwere a plug, the only differencebeing that a thin layer with laminarflow characteristics is present veryclose to the wall. Accordingly theshear rate velocity is high close tothe walls of the pipe, while it islow in other pipe zones.During drilling, an effort is madeto prevent turbulent flow in theannular section, as the turbulencemay cause major hole erosionproblems. However, duringcement jobs, turbulent rather thanlaminar flow is preferred, as itfacilitates removal of the mud cakefrom the side wall, enabling thecement to adhere to the formationsurface and providing betterresults.

1.4 - DETERMINATION OF THE FLOW REGIME

As stated, fluid flow may be plug, laminar or turbulent. A flow region also exists, the transitionalzone, where the flow is neither laminar nor turbulent.Publications report on various methodologies to determine the flow regime existing in a givencondition. The most well known and widely applied uses the Reynolds number, conceived forwater flow and subsequently applied to the flow of fluids with different behaviours, such asplastic, pseudoplastic, yield pseudoplastic or dilatant.

THE REYNOLDS NUMBERThe type of flow is determined by calculating the Reynolds number, while the field of existence ofa flow regime may be determined either by calculating the critical Reynolds number (named afterthe person who discovered it, Sir Osborne Reynolds, who defined this method for determining thecritical velocity of water in pipes in 1883), or the critical velocity. The Reynolds number isdirectly proportional to the pipe diameter, the average velocity of the flow and the fluid density,while it is inversely proportional to the fluid viscosity.It may be considered as the ratio between the inertial forces and the frictional forces of theflowing fluid.

The Reynolds number equation for circular section pipes, using International Systemmeasurement units, is given by the following:


Re =

V di (1.4)

likewise, for the annular section:

Re( )=

V D de (1.4a)

where:

di = diameter of the circular section, mde = internal diameter of the annular section, mD = external diameter of the annular section, mV = average flow velocity, m/s = flow density, kgm/m3

= fluid viscosity, Pas

using British system measurement units, this becomes the following, for the circular section:

Re .= 927 58

Vdi (1.5)

and for the annular section:

Re .( )

=

927 58

V D de (1.5a)

where:

di = diameter of the circular section, inde = internal diameter of the annular section, inD = external diameter of the annular section, inV = average flow velocity, ft/s = flow density, lbm/gal = fluid viscosity, cP (centiPoise)

Sometimes, the following formula, with mixed units (field units) is applied. For the circularsection:

Re .=

2 54 104

V di (1.6)

for the annular section:

Re .( )

=

2 54 104

V D de (1.6a)


where:

di = diameter of the circular section, inde = internal diameter of the annular section, inD = external diameter of the annular section, inV = average flow velocity, m/s = flow density, kgm/dm3

= fluid viscosity, cP (centiPoise)

By performing studies on water flow in a circular section pipe, Reynolds discovered thatturbulence began at an approximate value of 2,100. So, in the case of water, the critical velocitywhich determines the transition from laminar to turbulent flow, may be calculated by simplyapplying the following expression:

VdC i

= 2100

(1.7)

where = 1000 kgm/m3 e = 110-3 Pas (1 centiPoise).

In any case, for every type of non-Newtonian fluid other than water, and for every flow regimepassage, a critical Reynolds number exists which defines the end of a given behaviour and thestart of a new behaviour which is characteristic of the subsequent regime.

As a result, the critical velocity for each passage of regime must be calculated case by case; it isnot a set value, not even for the same family of fluids (for example all fluids with yieldpseudoplastic behaviour) and depends on the relative critical Reynolds number. The complexprocedures for determining this value, which are required to evaluate the passage from one regimeto another, shall be given in the next chapter.

EXAMPLE 1.1

We shall consider a flowing drilling mud with a density of = 1200 kgm/m3, at a flow rate of Q =2450 lt/min (0.0408 m3/s), inside a drill string with a 4.27 diameter (0.1084 m) flowing up theannular section with diameters of 8.535 (0.217 m) and 5 (0.127 m). We shall determine theReynolds number for the two sections, knowing that the mud viscosity inside the pipes is c =62.5 cP (0.0625 Pas) and a = 69.0 cP (0.069 Pas) inside the annular section.


The velocity inside the pipes is given by:

VQ

Am sc = = =

0 0408

4 010844 42

2

.

.. /

while the velocity of the annular section is given by:

VQ

Am sa = =

=0 0408

4 0 217 0127168

2 2

.

( . . ). /

So the Reynolds number inside the pipes is the following:

Re. .

.cc i

c

V d= =

=

1200 4 42 01084

62 5 1091993

while in the annular section it is as follows:

Re( ) . ( . . )

aa e

a

V D d=

=

=

1200 168 0 217 0127

69 1026293

1.5 - VISCOSITIES FOR NEWTONIAN AND NON-NEWTONIAN FLUIDS

For ideal or Newtonian fluids, the property which determines the braking action or resistance toshear, which develops when a fluid layer slides above another, is called viscosity. The viscosity ofa Newtonian fluid, already discussed in paragraph 1.3, is a parameter which correlates therelationship of shear stress with the shear rate, according to the following equation, formulated byNewton (English physicist) in 1687:

= .

(1.8)

In the equation (1.8), is the viscosity and is often called dynamic to differentiate it fromkinematic viscosity. It is constant at a given temperature and pressure.However, real solids and liquids frequently do not have these characteristics, but intermediateones between these two extremes, with equations which relate stress and deformation in a highlycomplex manner. The fluids used for drilling are, for example, non-Newtonian, thixotropic andhave viscoelastic properties.

As already mentioned, fluids tend to resist, to a greater or lesser extent, to continuousdeformation and flow, depending on their viscosity.

The equation (1.8), known as Newtons Law, applies to Newtonian fluids and correlates the shear

stress , the viscosity and the shear rate .

.


Fig. 1.4 - The Action of Velocity as a Result of Sliding

wall pipe

h

F

flowing fluid

L

moving plate, A

L

Figure 1.4 shows the action ofvelocity as a result of sliding, due toforce F applied to the upper layer ofarea A, and to the braking action ofthe liquid.The shear stress, , can be defined asfollows:

= FA

(1.9)

where:

= shear stress, (ML-1T-2)F = applied force, (MLT-2)A = area, (L2)

The measurement unit of in thetechnical system is 1 at (technicalatmosphere) in SI units 1 Pa (Pascal)and in the fps British system - 1 psi(pound per square inches).

Often, and especially when the shear stress is determined in the lab instruments, the measurementunit lb/100ft2 is used. At times, though less frequently, g/100cm2 is used, above all when filling indaily drilling reports.

The conversions from one system to another are as follows:

1 lb/100ft2 = 6.9410-5 psi1 g/100cm2 = 0.9807 Pa

The dynamic viscosity, determined by the equation (1.8) is measured in centiPoise (or Poise) inthe Technical System, in Pas in the International System and in centiPoise (or Poise) in the fpsBritish system.

However, dynamic viscosity is often measured in the British system in lbs/ft2 or in lbm/fts. Theconversions from one system to another are as follows:

1 Poise = 1 g/cms = 1 dyns/m2

1 centiPoise = 10-2 Poise 1 centiPoise = 10-3 Pas 1 centiPoise = 2.088610-5 lbs/ft2 = 6.71910-4 lbm/fts 1 lbs/ft2 = 47880 centiPoise

In practice, the parameter kinematic viscosity, , is often used and is defined as the ratio betweenthe (dynamic) viscosity and density of the considered fluid. The following is therefore obtained:


= (1.10)

where, using Technical System units:

= kinematic viscosity, m2/s = dynamic viscosity, kgps/m2

= fluid density, kgp/m3

in the International System (SI):

= kinematic viscosity, m2/s = dynamic viscosity, Pas = fluid density, Kgm/m3

and in the fps British system:

= kinematic viscosity, ft2/s = dynamic viscosity, lbs/ft2

= fluid density, slug/ft3 (lbs2/ft4)

At times, the kinematic viscosity is also measured in St (Stokes) which is the measurement unit ofthe CGS System. In this case:

1 St = 1 cm2/s

As stated previously, the equation (1.8) is only applicable when the fluid is classified asNewtonian. At given pressure and temperature values, these types of fluids are characterised by aconstant ratio between shear stress and shear rate. All fluids which have no direct proportionalrelationship between the shear stress and shear rate, at constant pressure and temperature, areclassified as non-Newtonian.

Suspensions of solids in liquids are generally non-Newtonian, in the same way as drilling fluidsand cements. Moreover the (dynamic) viscosity of these fluids varies, not only in relation to thechange in the shear rate, but also to the duration of the shearing stress.

As drilling fluids, have a non-Newtonian flow behaviour, they are characterised by viscosityvalues, sometimes called apparent or effective which distinguish them from Newtonian fluids;these are related to the sliding velocity, the shear rate range considered, the chemical-physicalcharacteristics of the fluid and various operative parameters, such as temperature, pressure,setting time and shear history.

Figure 1.5, below, shows the trend of shear stress in relation to shear rate of a non-Newtonian

fluid (yield pseudoplastic) in the diagram (,.

).


Fig. 1.5 - Trend of Shear Stress vs Shear Rate for a non-Newtonian Fluid (YieldPseudoplastic)

shear rate

shea

r st

ress

2

1

1

2.

1.

2

The sliding shear rate greatly affects the velocity of non-Newtonian fluids, which may either

decrease or increase as .

increases. In drilling muds, the apparent viscosities generally decrease

as .

increases (shear thinning). Figure 1.6 shows the trend of apparent viscosity in relation to theshear rate for a fluid with pseudoplastic yield behaviour.

Fig. 1.6 - Trend of Apparent Viscosity vs Shear Rate for a Fluid with Yield PseudoplasticBehaviour

shear rate

1

2

app

aren

t vi

sco

sity

2.

1.


At low .

values, the apparent viscosities are higher as the interactions between the shale and/orpolymer particles in the mud maintain their gel-like structure (developed in static conditions)

which has a strong braking or shear resistance action. As the .

increases, the apparent viscositiesdecrease as the gel-like structure breaks up and the fluid dynamic units trend in the direction ofthe flow, with a reduced braking action.The interactions between particles and the formation of a gel-like structure are determiningfactors in the complexity of the rheological response to different operative factors. Moreover,these interactions are also the cause of the muds thixotropy and viscoelastic properties.

1.6 - THIXOTROPY, RHEOPECTICITY AND GEL STRENGTH

Thixotropy is the ability of a fluid to develop a gel strength over time, i.e. to create a rigidstructure when it is in static or slow moving conditions; this structure may change back to a fluidstate after mechanical shaking or a return of the flow.By applying a sequence of increasing shear rates, a thixotropic material will break in relation toboth time and the maximum shear value applied.Rheopectic fluids instead behave inversely. In fact by applying a deformation velocity, an internalstructure will gradually form in rheopectic fluids.This behaviour is observed at moderate flow values, which are lower than a threshold valuebeyond which the formed structure is destroyed. This rheology is characteristic of different fluidtypes, including aqueous dispersions of bentonite.

Fig. 1.7 - Trend of Weak and Strong Gels in Relationto Time

weak gel

time

gel

str

eng

th

strong gel

Drilling fluids generally have athixotropic behaviour. If the initialshear stress measures the attractiveforces of a fluid in flow conditions,and is a constant parameter for agiven dispersion, the gel strengthmeasures the attractive forces instatic fluid conditions. In the case ofthixotropic systems, such as drillingfluids, the higher the static time is,the greater the increase of the gelstrength.Two different types of gel exist: weakgel (fragile gel), when the gelstrength increases slowly over time,and strong gel (progressive gel) whenthe strength increases more quickly.The latter increase is due to theincrease in the concentration of shaleparticles in the dispersion.Figure 1.7 shows the trend of weakand strong gels in relation to time.

The gel strength is measured using a Fann rotational viscometer, setting the viscometers rotationrate at 3 revolutions per minute. These measurements are recorded for a static mud sample, for 10seconds and 10 minutes respectively; the results are then compared to determine the formationvelocity and characteristics of the gel.


In general terms, strong gels should not be used as they may lead to problems such as excessivepressures when circulation begins (due to the gels breaking up) with potential risks to the wellshydraulic circuit. In fact, for this very reason, the well head pressure may be far higher thannecessary when circulation begins in order to maintain the required flow rate.If the gel strength value is known, the pressure gradient needed for breaking the gel may becalculated and well circulation can begin at the desidered flow rate. As the shear stress is greaterat the walls of the pipes - because the shear rate value is greatest - initial fluid movement willoccur at this point. By relating the shear stress at the walls with to the gel strength, the followingis obtained for the pipes:

p

L dig=

4 (1.11)

while for the annulus:

( )

p

L D deg=

4 (1.11a)

where:

p/L = pressure gradient, Pa/mg = gel strength, Padi = internal diameter of the pipes, mde = external diameter of the pipes, mD = internal diameter of the casing or hole diameter, m

in sas system units, the following is obtained:

p

L dg

i

=

300

(1.12)

( )

p

L D dg

e

=

300 (1.12a)

where:

p/L = pressure gradient, psi/ftg = gel strength, lb/100ft2

di = internal diameter of the pipes, inde = external diameter of the pipes, inD = internal diameter of the casing or hole diameter, in

In very thin annular sections, such as slim holes, with muds that have strong gels, it may benecessary to have high pressures in order to break the gel and begin circulation. This pressure maysometimes exceed the formation fracturing pressure. In these cases, in order to reduce thepressure needed to begin circulation, the drill string can be rotated before to start the pumps so


that they partially break the gel strength and reduce the initial pressure value for starting wellcirculation. Moreover, the pump speed can be increased very slowly, while the drill string is beingrotated, in order to minimise annular pressure losses caused by the mud flow.

EXAMPLE 1.2

We shall consider a well being drilled, with a 5 casing shoe at 4,000 ft and a well with a mudcirculation of 9.5 lbm/gal. Viscometer readings defined a gel strength at 10 minutes of 42lbf/100ft2. The internal diameters of the casing and external diameters of the drill pipes are 4 1/4and 3.7 respectively. In this example we want to find out the existing pressure and equivalentdensity at the casing shoe when circulation starts up again.

The pressure gradient needed to break the gel strength at the pipe walls is given by:

( ) ( )

p

L D dpsi ftg

e

=

=

=

300

42

300 4 25 3 70 254

. .. /

The shoe pressure when the gel beings to break is given by:

p Lp

LL psi= + = + = + =0 052 0 052 9 5 4000 0 254 4000 1976 1016 2992. . . .

while the equivalent density at the shoe is given by:

e mp

Llb gal=

=

=

0 052

2992

0 052 400014 38

. .. /

As can be noted from these calculations, the gel strength creates a major overpressure when fluidcirculation in the well begins. In the example, the increase in pressure is equal to 51% of thestatic pressure. This could cause fracturing problems under the casing shoe if the formationfracturing gradient is exceeded. This problem is, however, not so evident for standard geometrywells, where values well below those for slim geometry wells exist.

1.7 - RHEOLOGICAL CLASSIFICATION OF FLUIDS DURING FLOW

A rheological model represents a fluids behaviour to flow, using a mathematical relationship tocorrelate shear rate and shear stress.Published material contains many equations to define the rheological behaviour of a fluid, yetnone of these universally defines the phenomenon. Generally, drilling muds and cements aretypically non-Newtonian and may be plastic, pseudoplastic, yield pseudoplastic and also dilatant attimes, even though the latter is not very common.Three rheological models are given below, for non-Newtonian fluids: the Bingham model forplastic fluids; the Ostwald & de Waele model for pseudoplastic and dilatant fluids and theHerschel & Bulkley model for yield pseudoplastic and dilatant fluids.

PLASTIC FLUIDS


The ratio between the shear stress, , and shear rate, .

, of these fluids, also called Binghamfluids, is linear and very similar to that of Newtonian fluids.With reference to the flow curve in figure 1.8 (a), the equation which defines this type ofrheological behaviour is as follows:

= + o p.

(1.13)

where:

= shear stress = yield pointp = plastic viscosity

.

= shear rate

The yield point, 0, also termed yield stress and often indicated by the symbol YP, is the positiveintercept on the axis of the shear stress values (for zero shear rates) and p, which is sometimesindicated by the symbol PV as well, is proportional to the inclination of the curve.Unlike Newtonian fluids, this type of fluid does not flow until the applied shear stress, , exceedsa given value 0. After this point, equal increases in the shear stress lead to equal increases in theshear rate which are proportional to p.The apparent viscosity is defined as the relationship between the shear stress and the shear rateand is given by the dip of the line which joins the origin with a general point of co-ordinates (,

.

).

Fig. 1.8 (a) e (b) - Trend of the Shear Stress (a) and Apparent Viscosity (b) vs Shear Ratefor a Plastic Fluid

shear rate

app

aren

t vi

sco

sity

p

(b)

shear rate

shea

r st

ress

11

= = tg 11

11..

lim .=

=

p.

(a)

p

22 = = tg22

22

22

1111


In figure 1.8 (a) the apparent viscosity is given, for different shear rate values (.

1 ,.

2 ) by the thin

continual lines; as can be noted, the viscosity decreases as the shear rate increases until it reachesthe plastic viscosity value of the equation (1.13), as the shear rate tends towards infinite. Thisphenomenon is known as shear thinning. Figure 1.8 (b) shows the trend of apparent viscosity, ,

in relation to .

, which tends to p for high shear rate values.

The Bingham model has been and is still widely applied in the oil industry as it is easy to use andrepresents the behaviour of some drilling fluids commonly employed up until a few years ago,fairly well (such as bentonite muds).

This model was so widely used in the past that one of the most important measurementinstruments for determining rheological parameters, the Fann rotational viscometer, wasexclusively designed and calibrated for directly determining the parameters of the Bingham model.

In any case, this model does not reflect real conditions as it does not accurately represent thetrend of most drilling fluids currently used, especially at low shear rate values, or even, at lowflow rate values. In fact, other rheological models which take account of these trends are used atpresent.

PSEUDOPLASTIC AND DILATANT FLUIDSThese types of fluids have no yield point and the apparent viscosity is a non linear function of theshear rate. In a pseudoplastic fluid, the apparent viscosity decreases as the shear rate valuesincrease; the dip of the flow curve decreases continually and very often tends to a constant valueat high shear velocity rate values.

The rheological behaviour of a dilatant fluid is the opposite of that of a pseudoplastic fluid,because its apparent viscosity increases as the shear rate increases.

The rheological model which best represents the behaviour of these fluids was conceived byOstwald & de Waele, also known as the power law, at two constant parameters. This model is asfollows:

= kn.

(1.14)

where:

= shear stress

.

= shear ratek = consistency indexn = flow behaviour index

Generally speaking, the consistency factor, k, indicates the degree of fluid viscosity and is theanalogous as the apparent viscosity. By increasing the k value, the fluid becomes more viscous.

The exponent, n, called the flow behaviour index, is a quantitative index which may be used toevaluate the behaviour of a non-Newtonian fluid.


The greater the difference of n from 1, in both directions, the more marked the non-Newtoniancharacteristics of a fluid are.In fact the following distinction can be made:

0 < n < 1 pseudoplastic fluidn = 1 Newtonian fluid n > 1 dilatant fluid

Figure 1.9 (a) shows the trend of the flow curves as the value of the flow behaviour indexchanges. Figure 1.9 (b) shows the rheological trend of the relative apparent viscosity for varioustypes of behaviour.

Fig. 1.9 (a) and (b) - Shear Stress trend (a) and Apparent Viscosity (b) vs Shear Rate for aPseudoplastic Fluid

shear rate

app

aren

t vi

sco

sity

shear rate

shea

r st

ress

22

11

11..

22..

11

pseu

dopla

stico

, n

1

22

(a) (b)

n=1

n>1

n


The Bingham model includes the yield point value but does not account for the variation in shearstress at low shear rate values. The power law model describes the flow at low shear rates fairlywell, but does not include a yield point value in its equation. So the best rheological model andthe most suitable for simulating the behaviour of drilling fluids must take account of all theseparameters.

YIELD PSEUDOPLASTIC AND DILATANT FLUIDSYield pseudoplastic fluids have a yield point and apparent viscosity which have no linearrelationship with the shear rate, as already noted for pseudoplastic fluids.In these fluids, the apparent viscosity decreases as the shear rate values increase; the inclination ofthe flow curve, instead, continually decreases and very often tends to a constant value at highshear rate values.The rheological behaviour of a dilatant fluid, with yield point, is opposite to that of apseudoplastic yield fluid, because its apparent velocity increases as the shear rate increases.The rheological model which best represents the behaviour of these fluids was conceived byHerschel & Bulkley, at the beginning of this century, to simulate the behaviour of rubber andbenzene solutions, with three constant parameters. This model is as follows:

= + on

k.

(1.15)

where:

= shear stress = yield point or yield stress

.

= shear ratek = fluid consistency indexn = flow behaviour index

Generally, the consistency index, k, indicates the degree of fluid viscosity and at times isanalogous as the apparent viscosity. By increasing the k value, the fluid becomes more viscous.The exponent n, known as the flow behaviour index, is a quantitative index which may be used toevaluate the behaviour of a non-Newtonian fluid.The greater the difference of n from 1, in both directions, the more marked the non-Newtoniancharacteristics of a fluid are.In fact:

0 < n < 1 pseudoplastic yield fluidn = 1 Bingham fluid n > 1 Dilatant fluid

Figure 1.10 (a) shows the trend of these rheological behaviours, while figure 1.10 (b) shows therelative trend of apparent viscosity in relation to the shear rate.


Fig. 1.10 (a) e (b) - Shear Stress Trend (a) and Apparent Viscosity (b) vs Shear Rate for aPseudoplastic Yield Fluid

shear rate

app

aren

t vi

sco

sity

p

shear rate

shea

r st

ress 22

11

11..

22..

yield p

seudo

plastic

o, n1

2211

(a) (b)

n=1

n>1

n


where 0 is the yield stress. This formula is above all of historical interest, as it is hardly ever usednowadays.

MODELS WITH TWO CONSTANT PARAMETERSThere are many models with two constant parameters, devised to simulate the behaviour ofplastic, pseudoplastic and yield pseudoplastic fluids. One of the most important is:Casson Model, with the constant parameters o and p,

= + o p.

(1.17)

This is widely used in the field of medicine, for studying the rheological behaviour of blood, aswell as in the food industry.

MODELS WITH THREE CONSTANT PARAMETERSThere are various models with three constant parameters, devised to simulate the behaviour ofpseudoplastic and yield pseudoplastic fluids. Some of these are described below:Collins & Graves Model, with the constant parameters A, B and C:

= + ( )( ).

( ).

A B e C1 (1.18)

This model was purposely devised to simulate the behaviour of drilling fluids, with the aim ofhaving a model which could accurately follow both the trend of data with low shear rates andpseudoplastic characteristics and data with high shear rates and plastic characteristics. Moreover,this model does not have points of singularity so it is easier to use - than the Bingham model -with numerical simulators. This model has not been widely adopted and is above all, historicallyimportant.Parzonka and Vocadlo Model, with the constant parameters o, k and n,

1

0

1 1n n nk= +

.

(1.19)

This model is very similar to the Robertson & Stiff model, and the two are sometimes mixed up.It is used to simulate the behaviour of pseudoplastic yield fluids.Gucuyener Model, with the constant parameters o, k and n,

1

0

1n n k= +

.

(1.20)

this model was devised in the mid seventies to simulate the flow of drilling fluids. It is applicableto yield pseudoplastic fluids. However, there has been little development in its application in thefield of fluid rheology.Casson & Shulman model, with the parameters o, p and m,

1

0

11

m mp

m

= +

.

(1.21)


This is an intermediate model between the Casson model (with two parameters) and the Shulmanmodel (with four parameters) valid for yield pseudoplastic fluids, and at times attributed toSaunders. This model is not used often in practice.Robertson & Stiff Model, with the parameters o , k and n:

= +k on( )

.

(1.22)

This model was purposely devised in the mid seventies to simulate the rheological behaviour ofdrilling fluids. The inventors proposed this model, stating that it provided an adequate descriptionof the behaviour of these fluids, comparing the results to the models of Bingham and Ostwald &de Waele (power law). Even though it approximates the flow at low and high shear rate valuesfairly well and has a yield point, the models results are slightly less accurate compared to thoseobtained using the Herschel & Bulkley model, which is preferred for simulating the rheologicalbehaviour of drilling fluids. The former is used above all in North Sea countries, replacing theHerschel & Bulkley model.Papanastasiou model, with the parameters o, and n,

= +

.

( ).

one1 (1.23)

the basic structure of this model is very similar to that of the Collins & Graves model. As thePapanastasiou model incorporates an exponential term in its analytical structure, it is moresuitable, than other models, for simulating the trend of a fluid in a wider behavioural range. It isnot currently used in the oil industry.

MODELS WITH FOUR CONSTANT PARAMETERSFew models have been developed with four constant parameters, due mainly to the difficulty ofdirectly determining the values of the models characteristic parameters. The most important ofthese models is:Shulman Model, with the constant parameters o, k, n and m,

1

0

11

n nm

k= +

.

(1.24)

this was devised in the late seventies. It is probably one of the best models for simulating drillingfluids, but has not been widely used because of application difficulties inherent to the complexitiesof the analytical calculations. It is used above all in Russia and Eastern European countries.

MODELS WITH FIVE CONSTANT PARAMETERSPublished materials report on very few models with five constant parameters. These types arecertainly the most accurate, but the parameters must be solved using sophisticated numericalcalculation procedures, so practical application is difficult. The most important five constantparameter models are:Maglione, Ferrario, Rrokaj & Calderoni (MFRC) Model, with the constant parameters a, b, c, nand m,


1 11

11

/ /. / .

n nm

ma b c= +

+

+

(1.25)

This model is applicable for non-Newtonian plastic and pseudoplastic fluids as well as yieldpseudoplastic and dilatant fluids.The five constant parameters which are characteristic of the fluid are correlated with the fluidproperties as follows:

a yield pointb and c correlated with the fluid viscosity m and n correlated with the flow behaviour

The equation (1.25) summarises many rheological models whereby, with a variation in the abovementioned constants, the expression coincides and adapts to describe the behaviour of a differentfluid.At present, this expression is complex to apply, because of the sophisticated numericalcalculations that have to be performed to solve the differential equation of the model; as a result,its use is limited to predicting the rheological behaviour of drilling fluid from lab data.

1.9 - SELECTING THE RHEOLOGICAL MODEL

As mentioned previously, the Herschel & Bulkley rheological model is the most commonly usedto describe the flow behaviour of a drilling fluid. Most of the other models are either particularcases of the previous model, or have different mathematical equations, approximating the trend oflab data in a more or less accurate way.It is sometimes possible to use other rheological models, when these approximate the actual trendof the drilling fluid to a better extent, compared to the Herschel & Bulkley model, for examplewhenever the model with three constant parameters fails to approximate the rheological behaviourfor the entire field of shear rates studied.This may happen, for example, when the fluid has trends which can no longer be predicted byapplying a parabola equation.One (published) method for determining the capacity of a model to approximate lab data is basedon calculating the coefficient of non linear regression of the interpolator curve, also known as theBest Index Value. This value is defined as the ratio between the sum of the squares of thedeviation of the value calculated from the mean value and the sum of the squares of the deviationof the value measured from the mean value. This is given by the following formula:

( )( )

BIVc i

i

m ii

=

,

,

2

2 (1.26)

where:


= m i

i

n

n

,

(1.26a)

The closer the BIV value is to 1, the better the approximate capacity is, of the lab data whichpredicts the configuration of the rheological model.

A second method for determining the quality of the approximation of a curve to experimental datais to calculate the square of the correlation coefficient, r. In fact, the statistics programmes whichare used to determine the equations for interpolator curves of lab or experimental data, are basedon the square value of r.

This is defined as follows:

( )r

n

n n

i c i i c i

i i c i c i

2

2

22

22

=

.

,

.

,

. .

, ,

(1.27)

where:

n = number of readings at the viscometerc,i = calculated i-th shear stress

i.

= the i-th shear rate

In this case too, the closer the r2 value is to 1, the better the approximation is of the interpolatorcurve to experimental data.

EXAMPLE 1.3

We shall consider the rheological measurements, taken using a Fann VG 35 rotational viscometer,on a polymeric mud, which are shown in table 1.1 below:


RPM [rpm]

shear rate [1/s]

readings[degree]

shear stress[Pa]

3 5.1 3.0 1.536 10.2 4.0 2.04

10 17.0 4.7 2.4030 51.1 7.0 3.5760 102.5 10.0 5.10100 170.3 12.0 6.13200 340.6 17.0 8.68300 510.9 21.0 10.72400 681.2 25.0 12.76500 851.5 28.0 14.30600 1,021.8 31.0 15.83

Tab. 1.1

Fig. 1.11 - Rheological Behaviour in Co-ordinates (, .

) ofthe Polymeric Mud of Table 1.1.

0 200 400 600 800 1 000 1 2000

2

4

6

8

10

12

14

16

18

shear rate [1/s]

shea

r st

ress

[P

a]

Figure 1.11 represents therheological behaviour in co-

ordinates (, .

). As can benoted, the figure shows thetypical parabolic trend of adrilling mud.In this case we want todetermine the degree ofapproximation of rheologicalmodels in terms of lab data.By using the equation (1.27)and a statistical programme,the square value of thecorrelation coefficient, r2, wasdetermined for the mudspecified in table 1.1,adopting the rheologicalmodels most widely describedin publications.The values are shown in table1.2 below.

Rheological Model r2

MFRC 0.99984Casson & Shulman 0.99983Shulman 0.99983Herschel & Bulkley 0.99976Robertson & Stiff 0.99960Parzonka & Vocadlo 0.99960Gucuyener 0.99960Ostwald & de Waele 0.99837Casson 0.99710Papanastasiou 0.99609Collins & Graves 0.99608Bingham 0.98424

Tab. 1.2


As can be seen from the examples, the Herschel & Bulkley model, used the most at present forsimulating pseudoplastic yield fluids, is the most suitable compared to the Robertson & Stiff,Ostwald & de Waele, Casson and Bingham models, for showing the trend of lab data for theentire range of investigated shear rates. Other models, such as MFRC, Shulman and Casson &Shulman, are far better at representing rheological behaviour than the previous models. The latterare however very problematic for practical application due to the complexity of their analyticalformulas, so they are limited to interpreting lab data. Lastly, models such as Parzonka &Vocadlo, Gucuyener, Papanastasiou and Collins & Graves are less accurate in simulatingrheological behaviour, compared to conventional models. To sum up, the Herschel & Bulkleymodel is best for approximations and for its easy and simple practical application.

1.10 - DETERMINING RHEOLOGICAL PARAMETERS BY MEANS OFROTATIONAL VISCOMETER READINGS

The rheological behaviour of a fluid is studied in the lab on samples, by means of flow tests usinga rotational viscometer. One of the most widely used viscometers in the oil industry is the FannVG. 35, shown in figure 1.12 (a). Figure 1.12 (b) shows the principle of operation. A mudsample is put into a cylinder which rotates at different speeds. This cylinder contains a second,coaxial, fixed cylinder; by means of a built-in spring this measures the stress the fluid layer closeto the rotating cylindrical surface is subject to, on the actual surface. The fluid flow in the FannVG. 35 type coaxial cylinder device (figure 1.12 a), is comparable to two cylindrical surfacesmoving in relation to one another.

Fig. 1.12 (a) and (b) - Fann VG 35 Viscometer (a) and Principle of Operation (b)

(a)

liquid

motor

measuring device

di

de

Vr

Vmax

r

N

(b)


If the flow in the annulus of the viscometer is laminar and steady, for a given fluid model, thesystem may be described in mathematical terms; accordingly the instrument makes it possible toobtain the characteristic parameters of the flow and the rheological parameters which determinethe fluids behaviour vis--vis the flow. This is no longer possible when the fluid becomesunstable and turbulent: in this case investigation of the characteristics and flow parameters is nolonger valid and other types of viscometer (for example capillary) must be used.

International standards define the following limits for the ratio between the external and internalradius of the cylinder:

1,00 1,10

Clearly, it is not possible to use a ratio of = 1,00, while, with a ratio of = 1,10 errors are madein calculating the velocity gradient due to the laminar flow value moving into the unstable flowzone range.

This ratio is defined as follows for the Fann viscometer:

= =18415

1 72451 067846

,

,,

cm

cm

Figure 1.12 (b) shows the profile of fluid velocities through the concentric surfaces in the annulusof the viscometer.

The parameters of a rheological model can be determined, starting from a set of measurementstaken with the rotational viscometer, using two different procedures:

Numerical calculation. In this case, the parameters of rheological models are determined byapplying numerical calculation procedures based on minimum squares to the lab data set. Thismethod is generally used to determine the characteristic parameters of highly complex models,where the explicit analytical determination of parameters is extremely difficult. This procedureis very often applied to calculating the parameters of rheological models which are far lesscomplex, as well, where the analytical equation is sometimes very simple. The reason for thisis, when other conditions are equal, the formulas of the models with parameters determinedusing a numerical calculation approximate lab data far better than those models where theparameters are determined analytically.

The quality of a given equation to approximate lab data is quantified by calculating thecoefficient of non linear regression of the formula determined using the equation (1.26) or thesquare of the correlation coefficient given by the formula (1.27). The closer these values are to1, the better the formula used approximates the trend of the lab data studied.

Analytical calculation. This approach is applied above all for the simple and quick

determination of the parameters of a given rheological model. This procedure is now beingused far less and is limited to cases where it is difficult to acquire suitable numerical calculationtools and computers. This method is still used widely at rigsites and all the flow parametersare determined by applying equations based on analytical calculation.

All the formulas for analytically determining the characteristic parameters of some of the mostcommonly used rheological models, Bingham, Ostwald & de Waele (power law) and Herschel &Bulkley, are given below.


BINGHAM MODELThe parameters of this model are determined from two measurements, N2 = 600 rpm and N1 = 300rpm. We therefore have:

p =( )

.600 300

478 9 (1.28)

where:

300 = viscometer reading at 300 rpm, deflection degree600 = viscometer reading at 600 rpm, deflection degreep = plastic viscosity, lbs/100ft2

or, by expressing the plastic viscosity in centiPoise, we have the following:

p = 600 - 300 [centiPoise] (1.29)

While the value of the yield point, directly expressed in lb/100ft2 is given by:

o = 2300 - 600 [lb/100ft2] (1.30)

The formula (1.30) shows that the difference between the doubled value of the reading at 300 rpmwith the reading at 600 rpm gives the initial shear stress (yield point), based on the Binghammodel, directly expressed in lb/100ft2.

EXAMPLE 1.4

We shall consider a water base mud (WBM) with the rheological measurements taken using aFann VG 35 rotational viscometer and shown in table 1.3.

RPM [rpm]

shear rate [1/s]

readings[degree]

shear stress[Pa]

3 5.1 7.0 3.586 10.2 11.0 5.62

100 170.3 26.0 13.28200 340.6 33.0 16.86300 510.9 38.0 19.41600 1021.8 52.0 26.57

Tab. 1.3

In this example we shall determine the rheological parameters, plastic velocity and yield point,using both an analytical and numerical procedure.

By using the expressions (1.29) for plastic viscosity and (1.30) for the yield point, we obtain:


p = 52 - 38 = 14 centiPoise = 0.014 Pas

o = 238 - 52 = 24 lb/100ft2 = 11.49 Pa

the value of the square of the correlation coefficient, r2, using the expression (1.27) will be givenby:

r2 = 0.9679

Calculating the parameters p and o using the numerical calculation procedure, the following areobtained:

p = 0.015 Pas

o = 11.26 Pa

with a value of r2 equal to:

r2 = 0.9702

As mentioned previously, the parameters determined using the numerical calculation procedureapproximate the trend of the studied lab data well, as the r2 value in the second case is closer tothe unit, compared to the value obtained by analytically calculating the parameters.

OSTWALD & DE WAELE MODELThe parameters of this model are obtained from readings taken at the velocities of N1 = 300 rpmand N2 = 600 rpm. The flow behaviour index is given by:

( )n =

log

log

600

300

2 (1.31)

while the consistency index is given by:

kn

lb s ftnn

n=

10664

62 831 0877 100300

1 2..

( . ) [ / ] (1.32)

EXAMPLE 1.5

We shall consider the WBM of the previous example, determining the parameters n and kaccording to the power law model.

Using equations (1.31) and (1.32) the following is obtained:


( )n =

=log

log.

52

382

0 4525

k lb s ft Pa sn n=

= = 10664 38

0 4525

62 831 0877 2 3289 100 11153

10 4525

0 4525

2..

.( . ) . / ..

.

The r2 value, using the expression (1.27) will be given by:

r2 = 0.9970

Calculating the parameters n and k using the numerical calculation procedure, the following areobtained:

n = 0.3289

k = 2.5518 Pasn

with a r2 value equal to:

r2 = 0.9987

Like the previous example, the parameters determined using the numerical calculationapproximate the trend of the studied lab data better.

HERSCHEL & BULKLEY MODELThe Herschel & Bulkley model includes both the Bingham model (valid when fluids have a certainflow threshold) and the power law (applied to pseudoplastic fluids when the flow threshold iszero), in its mathematical formula.As a result, the entire ranges relative to the Bingham model and the power law are covered, andthe actual flow curves of drilling fluids are accurately followed.

The formulae for obtaining the indexes of the model are based on four readings taken with a FannVG 35 rotational viscometer at 100, 200, 300 and 600 rev per minute, and are the following:

( )( )

oPa=

+ 05107

600 100 200 300

600 100 200 300

. [ ] (1.33)

n =

log

log ( )

600 200

300 100

2 (1.34)

( )k

zPa so

z

z

n

n=

05107

20

1

600. [ ]

(1.35)

with:


z =

2 2

600

300

log ( )

log

where:

= 1.06784

EXAMPLE 1.6

We shall consider the WBM of the previous examples, determining the parameters n, k and oaccording to the Herschel & Bulkley model.

By applying the formulas (1.33), (1.34) and (1.35) for high shear rate values, the following isobtained:

( )( ) o Pa=

+

=0 510752 26 33 38

52 26 33 387151. .

n =

=log

log ( ).

52 33

38 26

20 6629

( )k Pa sn=

= 0 5107 52 7151

20 4 42 10678

10678 1

01874 42

4 42

0 6629

. .

. .

.

..

.

.

with:

z =

=2 2

52

38

4 42log ( )

log

.

The r2 value will be given by:

r2 = 0.99342

Calculating the parameters k o using the numerical calculation procedure, the following areobtained:

o = 1.702 Pa

n = 0.4352


k = 1.206 Pasn

with a r2 value given by:

r2 = 0.99965

In this case as well, the parameters determined using the numerical calculation approximate thetrend of the studied lab data better.

Table 1.4 summarises all the values of the square of the correlation coefficient calculated for thethree models, in relation to parameters determined using both the analytical and the numericalcalculation procedure.

Model square of the correlation coefficient, r2

analytical procedure numerical procedureBingham 0.96790 0.97020Ostwald & de Waele 0.99700 0.99870Herschel & Bulkley 0.99342 0.99965

Tab. 1.4

The table shows that, for any given model, the quality of approximation of the curve is alwaysbetter when parameters are calculated using the numerical method. Moreover, the Herschel &Bulkley model best approximates the trend of lab data.

1.11 - THE EFFECT OF PRESSURE AND TEMPERATURE ON RHEOLOGICALPARAMETERS

The rheological properties of drilling fluids under well pressure and temperature conditions mayoften be very different from those measured in ambient pressure and temperature conditions. Infact, at high depths, the pressure applied by the mud column may even exceed 1,400 bar, whilethe temperature is a function of the geothermal gradient and can exceed 260C at the bottom holein the case of very deep wells. In these conditions, the viscosity of muds at depth may be higheror lower than that measured at the surface; in fact an additive which reduces viscosity at surfacepressure and temperature may increase values in bottom hole conditions.

At present, the rheological properties of a drilling fluid are considered independently of wellpressure and temperature in hydraulic calculations. In many cases, this assumption leads to goodapproximations. Indeed, in shallow wells, the changes in temperature are not very great and sothe rheological variations in relation to the temperature are very slight. Moreover, in many wellsthere is a major difference between the pore pressure and fracturing pressure, so any errors madein estimating the circulation pressure do not affect the wells stability or the probability of a kickoccurring.

In very deep wells, with small differences between the pore pressure and fracturing pressure, it ishowever necessary to carefully analyse in detail the effects of temperature and pressure on wellhydraulics and on the probability of a kick occurring. Even recently reported cases of kicks havebeen the result of an approximate and inaccurate evaluation of mud rheology and well hydraulics.


In general, high temperature and pressures can significantly affect the rheological properties ofdrilling fluids in the following ways:

Physically. An increase in temperature decreases the viscosity of the liquid phase, while anincrease in pressure increases the density of the liquid phase and therefore the viscosity.

Chemically. All hydroxides above a temperature of 94 C react with clays. With alkaline basemuds, such as those treated with lignosulfonate, the effect on rheological properties is notsignificant, while muds with a high alkaline content may have major effects, depending on thetemperature and the type of metal ion of the hydroxide. In the case of calcium hydroxide Ca(OH)2the metal ion is calcium. For instance, in the case of muds with a high content of solids treatedwith lime, hydrate aluminosilicates may form, with a consistency which becomes very similar tocement above a temperature of 300 F (149 C).

Electro-chemically. An increase in temperature causes an increase in the ionic activity of any typeof electrolyte and in the solubility of any type of partially soluble salt which may be present in themud. The resulting changes in the ionic equilibrium alter the balance between the repulsive andattractive forces of the particles, and thus change the degree of dispersion and flocculation. Themagnitude and direction of these changes, and their effects on the rheology of muds, variesaccording to the electrochemical properties of the mud.

Due to the many variables present, the behaviour of a drilling fluid at high temperatures, andespecially a water base mud, is unpredictable and still not entirely clear. In fact, even smallchanges to the composition of mud may lead to major changes in rheological characteristics.

Mud rheology at high temperatures and pressures is studied by using various types of viscometer.One of these is the consistometer: it measures the transit time of a bob, magnetically controlled,through a mud sample. It is a useful tool for comparing a wide number of variables, but gives noindication as to the shear rate value during flow, so the data obtained have an empirical value.

To determine the characteristic parameters of plastic (Bingham), pseudoplastic (Ostwald & deWaele) and yield pseudoplastic (Herschel & Bulkley) fluids, required for hydraulic calculations,capillary or rotational viscometers modified for use at high pressures and temperatures have to beused.

The most widely adopted rotational viscometer is the HPHT Huxley & Bertram. This instrumentcan work at pressures of up to 500 bar and temperatures of up to 175C. The 750 ml mudsample, is put inside an autoclave which is electrically heated by the elements welded in a analuminium sleeve. The system is air cooled for temperatures above 120C and water cooled fortemperatures below this value.


Fig. 1.13

The temperature of the sample ismeasured by a thermocouple andcontrolled by a digital PID controllerin cascade, up to a temperature of300C. The sample pressure ismeasured instead by a transducer andcontrolled by a digital PID controller.This value may vary from 0 to 1,375bar. The device can be both manuallyand automatically operated with ashear rate range of 1-1500 s-1 and ashear stress of 0-200 Pa.One of the major problems of theseviscometers is the size which meansthey are normally used in the lab andrarely at the rigsite; another problemis the time needed for the tests: 4hours for performing tests on just onemud sample. Figure 1.13 shows anHPHT Huxley Bertram viscometer.

PLASTIC FLUIDSMany studies have been carried out to assess the effects of pressure and temperature on therheological parameters of a plastic fluid, which comes under the Bingham law, i.e. plastic viscosityand yield point.

Fig. 1.14 - Trend of Plastic Viscosity vsTemperature for a Bentonite Suspension

visc

osi

ty,

cp

temperature, F

50 100 150 200 250

0

20

30

40

50

60

70

80

10

18 ppg bentonite suspension

100 1/s shear rate

plastic viscosity

normalized viscosity of water

Annis and Hiller studied the rheology ofwater base muds at high temperatures,coming to the conclusion that if asuspension is entirely deflocculated, i.e.the reactive clay debris does notcontaminate the mud nor alter itsrheological properties, then plasticviscosity and yield point decrease as thetemperature increases, up to 350 F(177 C), whereas if the mud isflocculated, only the plastic viscositydecreases while the yield point sharplyincreases at higher temperatures up tothe water boiling point.The plastic viscosity of a shalesuspension decreases as the temperatureincreases at high shear rates, as a resultof the decrease in water viscosity.Figure 1.14 shows the trend of plasticviscosity of a bentonite suspension, inrelation to temperature; this trendcoincides nearly exactly with that ofnormalised water viscosity.


The actual viscosity of the same suspension at low shear rates increases as the temperatureincreases. The reason is that high temperatures cause an increase in the attractive forces of theparticles, as shown by the increase in gel strength in figure 1.1.5, and the actual viscosity isaffected by the forces between the particles at low shear rates, but not at high shear rates.Moreover, when the mud isdynamically aged, the degree ofdispersion increasesconsiderably. The actualviscosity of a bentonitesuspension increases at both highand low shear rates, after thesuspension has been dynamicallyaged at a high temperature. Theincrease in viscosity at high shearrates is mainly due to an increasein the degree of dispersion. Thebehaviour of calcium claysuspensions at high temperaturesdiffers from that recorded forsodium clay suspensions and isfar more complex.

Fig. 1.15

50 100 150 200 250 300

temperature, F

20

60

40

80

100

120

140

160

gel

str

eng

th,

lb/1

00ft2

30 minute gel

initial gel

18 ppg bentonite suspension

The repulsive forces between the calcium clay particles are far weaker than those of sodium clay;as a result the effect of high temperatures on the degree of deflocculation is far higher and soplastic viscosity increases.Behaviour at high temperatures varies a great deal depending on the type of mud. For example,salt water base muds are fairly stable because the high electrolyte content prevents the dispersionof clay. The behaviour of gypsum muds is similar to that of calcium montmorillonite. Lime mudsdevelop powerful gel strengths due to the reaction between the hydroxides and the clays, whilemuds with calcium surfactants remain fairly stable at temperatures above 350 F.

Fig. 1.16 - Trend of Viscosity in Relation to Pressure forOBM

1 10 100 1000 10000 100000

1

10

100

1000

visc

osi

ty,

cp

shear rate, 1/s

Temp = 150 F

Mud Weight = 12 ppg

20000 psi

10000 psi

1500 psi

The studies conducted by Annisand Hiller do, however,demonstrate that an accurateevaluation of the rheologicalparameters of water base muds athigh temperatures can only beperformed by taking directmeasurements with a rotationalviscometer at relevanttemperatures.Oil base muds deteriorate to alesser extent compared to waterbase muds and can withstandhigher temperatures. Unlike waterbase muds, the viscosity of thesemuds is mainly affected bypressure, as shown in figure 1.16.

The effects of pressure and temperature on oleofinic inverse emulsion muds is nearly entirelyphysical and the changes in the properties are mainly due to the effect of pressure and temperatureon the viscosity of the continual phase which is usually diesel oil. Combs and Whitmire measured


the actual viscosities using a capillary viscometer at different temperatures and pressures anddiscovered that when the values were normalised with those of diesel oil values, at the sametemperature and pressure, the points all fell on the same curve for each temperature, as shown infigure 1.17.

Fig. 1.17

shear rate, 1/s

1 10 102 103 104 1051

10

102

103

visc

osi

ty m

ud

/ vi

sco

sity

die

sel o

il

150 F - 1500/10000/20000 psi

300 F - 10000/20000 psi

mud weight = 12 ppg

These results demonstrate that viscosity in the well with these types of oil base muds may bedetermined, starting from the viscosity measured at ambient temperature, by means of correctionfactors based on the viscosity of diesel oil at the studied pressure and temperature, seeing as themud basically remains stable.

Fig. 1.18 - Trend of the Yield Point for an AsphaltSuspended in Diesel Oil

0 2 4 6 8 10 12 14 16

pressure, 1000 psig

yiel

d p

oin

t, l

b/1

00 s

q f

t

0

10

20

30

40

100 F

130 F

150 F

170 F

200 F

suspension of asphaltin diesel oil

Oil base muds in asphaltcolloidal suspensionssometimes have a very complexbehaviour. Figures 1.18 and1.19 show the trend of the yieldpoint and plastic viscosityrespectively, measured forasphalt suspended in diesel oil,using the rotational viscometer.As can be noted, the increase inthe yield point and plasticviscosity as the pressurechanges, is greater at highertemperatures than lowertemperatures.


Fig. 1.19 - Trend of Plastic Viscosity for an Asphalt Suspension in Diesel Oil

pressure, 1000 psig

(suspension of asphaltin diesel oil)

pla

stic

vis

cosi

ty,

cp

0 2 4 6 8 10 12 14 160

10

20

30

40

50

60

70

80

100

F

130 F

150 F

170 F

200 F

Sun developed some expressions for determining the rheological parameters of inverse emulsionsynthetic base muds, with plastic behaviour in relation to pressure and temperature.

The variation in plastic viscosity for these muds is given by the following expression:

( )( )

PV P eP

T= + +

9172 0 000368199 2 0 008473

. .. .

(1.36)

whereas the variation in the yield point is given by the following expression:

YP e T= 10 2115 4

..

(1.37)

where:

PV = plastic viscosity, cPYP = yield point, lb/100ft2

T = temperature, FP = pressure, psi

The yield point expression does not include pressure, as the results obtained from laboratoryexperiments show that this parameter, unlike temperature, does not affect the YP variation. Theconclusion is very similar to the conclusions drawn by several authors on the importance ofpressure in the YP variation of diesel based mud.


EXAMPLE 1.7

With an SBM type drilling fluid, having plastic behaviour, we shall determine the yield point andplastic viscosity values at a temperature of 150 F and at a pressure of 10,000 psi.

By using the expressions (1.36) and (1.37) the following are obtained for the plastic viscosity andthe yield point, respectively:

( )( )

PV e cP= + =+

9172 0 000368 10000 85 3199 2 0 008473 10000

150. . .. .

YP e lb ft= = 10 2 22 01 100115 4

150 2. . /.

PSEUDOPLASTIC FLUIDSPublished materials contain little information about the behaviour of pseudoplastic fluids whichadhere to the Ostwald & de Waele model (power law). McMordie et al demonstrated that thebehaviour of oil base muds at high temperatures and pressures can be described using thefollowing expression:

ln ln ln /.

= + + +k n A p B T (1.38)

where:

= shear stress

.

= shear ratek = consistency index of the fluidn = flow behaviour indexA = constant referred to the pressureB = constant referred to the temperature

Fig. 1.20

shear rate, 1/s

10 20 50 100 200 500 100010

20

50

100

200

500

100

shea

r st

ress

, d

ynes

/cm

2

calculated

actual

250 F

- 10000

psig

350 F

- 14000

psig

150 F

- 2000 p

sig

The values A and Bmust however bedetermined separatelyfor every type of mud.Figure 1.20 shows thecorrelation betweenshear stress and shearrate values determinedexperimentally andusing the previousexpression.


YIELD PSEUDOPLASTIC FLUIDSAccurate studies and research on the rheological behaviour of yield pseudoplastic both on waterand oil base fluids have been conducted in the last few years. Rommetveit et al performed studieson OBM and WBM muds using the HPHT viscometer. They noted that the shear stress in thewell could be identified, starting from the known value in standard pressure and temperatureconditions and using a corrective factor in relation to the pressure, temperature and shear rate.This corrective factor is expressed by the following formula:

f p T e eg p T g p T( , , ).

( , ).

( , ) = +1 2 (1.39)

the functions g1(p, T) and g2(p, T) are represented by the sum of squared, bi-linear and linearterms with constant terms. The coefficients of the constant terms are determined from the best fitof the data obtained experimentally.

For oil base muds (OBM), the functions g1(p, T) and g2(p, T) in equation (1.39) can be written asfollows:

( )g p T pT

pp

T17 23 696 0 0008787

58 922 039 10

0 02381, . .

..

.= + + +

(1.40)

( )g p T p2 1882 0 000543, . .= + (1.41)

whereas for water base muds (WBM) the functions g1(p, T) and g2(p, T) in equation (1.39) arethe following:

( ) ( )g p T p T1 5594 0 0001595446 4

130 2, . .

.

.= + +

+ (1.42)

( )g p T T2 1263 0 003913, . .= (1.43)

where, in (1.40), (1.41), (1.42) e (1.43):

= shear stress, Pa

.

= shear rate, s-1

p0 = pressure in standard conditions, 1 barp = pressure, barT0 = temperature in standard conditions, 20 CT = temperature, C

The trend of shear stress in well conditions, at pressure, p, temperature T and the shear rate

.

, can be represented as follows:


p T p T

f p T

f p To o

o o

, , , ,, ,

, ,

. .

.

.

=

(1.44)

while in standard conditions, the shear stress has the following trend:

( ) ( )( )

p T p T k p To o o o o o on p To o

, ,.

, ,. ,

= + (1.45)

finally the function:

f p T e eog p T g p To( , , )

.( , )

.( , )

01 0 2 0 0 = + (1.46)

is the same for general p and T conditions, apart from the fact that it is determined at standardpressure and temperature values.

EXAMPLE 1.8

We shall determine the shear stress value of an oil base mud (OBM) at a pressure of 1,050 bar

and a temperature of 182 C, and a shear rate of .

= 1021.8 s-1, knowing that the rheological

parameters in standard conditions are o = 8.3 Pa, k = 0.135 Pasn and n = 0.65, respectively.

By applying the formulas (1.40) and (1.41), the following are obtained:

( )g p T1 7 23 696 0 0008787 105058 92

1822 039 10 1050

0 02381 1050

1822 5371, . .

..

..= + + + =

( )g p T p2 1882 0 000543 2, . . .4521= + =

while the function given by the expression (1.39) will be:

f p T e e s( , , ) . .43.

. . = + = 2 5371 2 4521 110218 92

from the expression (1.45), the following shear stress value in standard conditions is obtained:

p T Pao o, ,.

. . . .. = + =8 3 0135 10218 20 5

0 65

the expression (1.46) is calculated applying the formulas (1.40) and (1.41) in standard pressureand temperature conditions:

f p T e e so( , , ) . ..

. .0

0 7479 1 8825 110218 490 25 = + =

where:


( )g p T1 7 23 696 0 0008787 158 92

202 039 10 1

0 02381 1

200 7479, . .

..

..= + + + =

( )g p T2 1882 0 000543 1 18825, . . .= + =

So, the shear stress value, in the assumed pressure and temperature conditions, is given by thefollowing expression (1.44):

p T p T

f p T

f p TPao o

o o

, , , ,, ,

, ,.

.

..

. .

.

.

=

= =20592 43

490 25386

1.12 - BIBLIOGRAPHY

1 API BUL 13 D: The Rheology of Oil Well Drilling Fluids

2 Bachvalov N. S.: Metodi Numerici, Editori Riuniti, 1981

3 Bingham E. C.: Fluidity and Plasticity, McGraw Hill, New York, 1922

4 Bird R. B., Stewart W.E. and Lightfoot E. N.: Transport Phenomena, John Wiley & Sons,New York, 1960

5 Bourgoyne A. T. Jr., Chenevert M. E., Millheim K. K. and Young F. S. Jr: Applied DrillingEngineering, SPE, Richardson, TX, 1986

6 Bukhman Y. A., Lipatov V. I., Litvinov A. I. and Mitelman B. I.: Rheodynamics of Non-linearViscoplastic Media, Journal of Non-Newtonian Fluid Mechanics, October 1982

7 Casson N.: Flow Equation for Pigment Oil Suspensions of the Printing Ink Type, Rheology ofDispersed System, Pergamon Press, Oxford, 1959

8 Collins R. E., Graves W. G.: A New Rheological Model for non Newtonian Fluids, paper SPE7654, 1978

9 Darley H. C. H. and Gray G.: Composition and Properties of Drilling and Completion Fluids,Gulf Publishing Co, Houston, 1988

10 Draper N., Smith H.: Applied Regression Analysis, Wiley, New York, 1981

11 Ghiringhelli L.: Reologia dei Fluidi, Agip, S. Donato M.se, 1981

12 Gucuyener I. H.: A Rheological Model for Drilling Fluids and Cement Slurries, paper SPE11487 presentato a Middle East Oil Technical Conference of SPE, Manama, Bahrain, 14-17March 1983


13 Hemphill T., Campos W. and Pilehvari A.: Yield-Power Law Model More Accurately PredictsMud Rheology, Oil & Gas Journal, 23 August 1993

14 Herschel W. H. and Bulkley R.: Measurement of Consistency as Applied to Rubber-BenzeneSolutions, Proc. ASTM (1926), no 26

15 Maglione R.: New Method Determines Flow Regime and Pressure Losses during Drilling andCementing, Oil & Gas Journal, 4 September 1995

16 Maglione R., Ferrario G.: Equations Determine Flow States for Yield Pseudoplastic DrillingFluids, Oil & Gas Journal, 5 February 1996

17 Makosko C. W.: Rheology, Principles, Measurements and Applications, VCH Publishers,New York, 1994

18 Merlo A., Maglione R., Piatti C.: An Innovative Model for Drilling Fluid Hydraulics, paperSPE 29259, presentato a Asia Pacific Oil & Gas Conference, Kuala Lumpur (Malaysia), 20-22March 1995

19 Munson B. R., Joung D. F., Okiishi F.: Fundamentals of Fluid Mechanics, Wiley and Sons,New York, 1994

20 Nelson E. B.: Well Cementing, Elsevier, Amsterdam, 1990

21 Okafor M. N.: Experimental Comparison of Rheology Models for Drilling Fluids, paper SPE24086, 1992

22 Ostwald W. & Auerbach R.: Kolloid Z. , 38, 1926

23 Papanastasiou T. C.: J. Rheol., 1987

24 Parzonka W. & Vocadlo J.: Methode de la Caracteristique du Comportement Rheologique desSubstances Viscoplastiques dapres les Mesures au Viscometre de Couette (Model NouveauTrois Parametres), Rheologica Acta, 4, 1968

25 Rabia H.: Rig Hydraulics, Entrac Software, Newcastle, 1989

26 Robertson R. E. & Stiff H. A.: An Improved Mathematical Model for Relating Shear Stress toShear Rate in Drilling Fluids and Cement Slurries, Soc. of Petr. Eng. Journal, February 1976

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