CH-1 Rheological Models

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Semester January 2015 By: ASIF ZAMIR


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The concepts of shear rate and shear stress apply to all fluid flow, and can be

describe in term of two fluid layers (A and B) moving past each other when a force

(F) has been applied.

Viscosity

Fluid Properties and Types

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When a fluid is flowing, a force exists in the fluid that opposes the flow. This force

is known as theshear stress. It can be thought of as a frictional force that arises

when one layer of fluid slides by another. Since it is easier for shear to occur

between layers of fluid than between the outer most layer of fluid and the wall of

a pipe, the fluid in contact with the wall does not flow. The rate at which one

layer is moving past the next layer is theshear rate. The shear rate is therefore a

velocity gradient. The formula for the shear rate is

Drilling Fluid Properties

Viscosity

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In the most general sense, viscosity describes a substances resistance to flow.

Hence a high-viscosity drilling mud may be characterized as "thick," while a low-

viscosity mud may be described as "thin."

Viscosity (m), by definition, is the ratio ofshear stress(t) toshear rate (g):

Unit: PaS, NS/m2

, kg/ms, cp, dyneS/cm2

, lbfS/100ft2

Drilling Fluid Properties

Viscosity

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The simplest class of fluids is called Newtonian. The base fluids (freshwater,

seawater, diesel oil, mineral oils and synthetics) of most drilling fluids are

Newtonian. In these fluids, the shear stress is directly proportional to the shear

rate. The points lie on a straight line passing through the origin (0,0) of the graph

on rectangular coordinates. The viscosity of a Newtonian fluid is the slope of this

shear stress/shear rate line. The yield stress (stress required to initiate flow) of a

Newtonian fluid will always be zero. When the shear rate is doubled, the shear

stress is also doubled. When the circulation rate for this fluid is doubled, the

pressure required to pump the fluid will be squared (e.g. 2 times the circulation

rate requires 4 times the pressure).

Fluid Types

Newtonian Fluids

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The shear stress at various shear rates must be measured in order to characterize

the flow properties of a fluid. Only one measurement is necessary since the shear

stress is directly proportional to the shear rate for a Newtonian fluid. From this

measurement the shear stress at any other shear rate can be calculated from the

equation:

Fluid Types

Newtonian Fluids

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When a fluid contains clays or colloidal particles, these particles tend to bump

into one another, increasing the shear stress or force necessary to maintain a

given flow rate. If these particles are long compared to their thickness, the

particle interference will be large when they are randomly oriented in the flow

stream. However, as the shear rate is increased, the particles will line up in the

flow stream and the effect of particle interaction is decreased. This causes the

velocity profile in a pipe to be different from that of water. In the center of the

pipe, where the shear rate is low, the particle interference is high and the fluid

tends to flow more like a solid mass. The velocity profile is flattened. This

flattening of the velocity profile increases the sweep efficiency of a fluid in

displacing another fluid and also increases the ability of a fluid to carry larger

particles.

Fluid Types

Non-Newtonian Fluids

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Arheological model is a description of the relationship between the shear stress

and shear rate. Newtonslaw of viscosity is the rheological model describing the

flow behavior of Newtonian fluids. It is also called the Newtonian model.

However, since most drilling fluids are non-Newtonian fluids, this model does not

describe their flow behavior. In fact, since no single rheological model can

precisely describe the flow characteristics of all drilling fluids, many models have

been developed to describe the flow behavior of non-Newtonian fluids. Bingham

Plastic and Power Law are discussed. The use of these models requires

measurements of shear stress at two or more shear rates. From these

measurements, the shear stress at any other shear rate can be calculated.

Fluid Types

Non-Newtonian Fluids

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Fluid Types

Rotational Viscometer

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The Bingham Plastic model has been used most often to describe the flow

characteristics of drilling fluids. It is one of the older rheological models currently

in use. This model describes a fluid in which a finite force is required to initiate

flow (yield point) and which then exhibits a constant viscosity with increasing

shear rate (plastic viscosity).

Fluid Types

Bingham Plastic Fluids

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The two-speed viscometer was designed to measure the Bingham Plastic

rheological values for yield point and plastic viscosity. A flow curve for a typical

drilling fluid taken on the two-speed Fann VG meter is illustrated in Figure below.

The slope of the straight line portion of this consistency curve is plastic viscosity.

Fluid Types


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Most drilling fluids are not true Bingham Plastic fluids. For the typical mud, if a

consistency curve for a drilling fluid is made with rotational viscometer data, a

non-linear curve is formed that does not pass through the origin, as shown in Flow

diagram of Newtonian and typical mud. The development of gel strengths causes

the y-intercept to occur at a point above the origin due to the minimum force

required to break gels and start flow. Plug flow, a condition wherein a gelled fluid

flows as a plug with a flat viscosity profile, is initiated as this force is

increased. As the shear rate increases, there is a transition from plug to viscous

flow. In the viscous flow region, equal increments of shear rate will produce equal

increments of shear stress, and the system assumes the flow pattern of a

Newtonian fluid.

Fluid Types


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Fluid Types


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The Power Law model attempts to solve the shortcomings of the Bingham Plastic

model at low shear rates. The Power Law model is more complicated than the

Bingham Plastic model in that it does not assume a linear relationship between

shear stress and shear rate. However, like Newtonian fluids, the plots of shear

stress vs. shear rate for Power Law fluids go through the origin.

Fluid Types

Power Law Model

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This model describes a fluid in which the shear stress increases as a function of

the shear rate mathematically raised to some power. Mathematically, the Power

Law model is expressed as

t =Kgn

Where:

t = Shear stress

K= Consistency index

g = Shear rate

n = Power Law index

Fluid Types

Power Law Model

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Plotted on a log-log graph, a Power Law fluid shear stress/shear rate relationship

forms a straight line in the log-log plot. The slope of this line is n and K is

the intercept of this line. The Power Law index n indicates a fluids degree of

non-Newtonian behavior over a given shear rate range.

Fluid Types

Power Law Model

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n = Power Law index or exponent

K = Power Law consistency index or fluid index (dyne secn/cm2)

q1 = Mud viscometer reading at lower shear rate

q2 = Mud viscometer reading at higher shear rate

w1 = Mud viscometer RPM at lower shear rate

w2 = Mud viscometer RPM at higher shear rate

Fluid Types

Power Law Model

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A rotational viscometer containing a non-Newtonaian fluid gives a dial reading of

12 at a rotor speed of 300 rpm and a dial reading of 20 at a rotor speed of 600

rpm. Determine the rheological model of this fluid in two cases: Bingham model

and Power Law model:

Bingham model:

Power Law model:

Fluid Types

Example

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Newtonian Fluid Flow Model

Non-Newtonian Fluid Flow Models

Bingham Plastic Model

Power Law Model

RHEOLOGICAL MODELS

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Application of Rheological Model

They are used to identify the type of fluid flow through pipe and annulus

Pressure Losses

The circulation system consists of: pump, surface connections

(stand pipe, hose, swivel and Kelly), drill pipe, drill collars, bit,

annulus between drill collars and hole, annulus between drill

pipe and hole, mud return lines, and mud tanks.

Friction of fluid through these parts causes pressure losses

The calculation of this pressure losses depend on four parts

Surface connection losses

Pipe losses

Annular losses

Losses across the bit

Losses depend on the type of fluid used and the type of flow

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RHEOLOGICAL MODELS

Pipe losses takes place inside drill pipe and drill collars

They are designated as P2 and P3

Annular losses takes place around drill collars and drill pipe

They are designated as P4 and P5

The magnitude of these pressures depends on

Dimension of pipe

Mud rheological properties, weight, plastic viscosity, and

yield point

Type of flow, laminar or turbulent

Several models exist to measure pressure losses

Only two are models will be used: the Bingham plastic and the

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RHEOLOGICAL MODELS


The following are the equations used for calculations

Pipe Flow

Determine average velocity

Determine critical velocity

If Va > Vc, flow is turbulent

If Va < Vc, flow is laminarInternal




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RHEOLOGICAL MODELS

Power Law Model


Pipe Flow


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RHEOLOGICAL MODELS

Power Law Model

Pipe Flow

If Va > Vc, flow is turbulent If Va < Vc, flow is laminar

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RHEOLOGICAL MODELS



Annular Flow


Determine critical velocity

If Va > Vc, flow is turbulent

If Va < Vc, flow is laminar

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RHEOLOGICAL MODELS

Power Law Model


Annular Flow

If Va > Vc, flow is turbulent If Va < Vc, flow is laminar

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RHEOLOGICAL MODELS

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CH-1 Rheological Models

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