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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
Bingham Plastic Fluids
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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
Bingham Plastic Fluids
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Fluid Types
Bingham Plastic Fluids
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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
Internal
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RHEOLOGICAL MODELS
Bingham Plastic Model
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
The following are the equations used for calculations
Pipe Flow
Determine average velocity
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RHEOLOGICAL MODELS
Power Law Model
Pipe Flow
If Va > Vc, flow is turbulent If Va < Vc, flow is laminar
Internal
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RHEOLOGICAL MODELS
Bingham Plastic Model
The following are the equations used for calculations
Annular Flow
Determine average velocity
Determine critical velocity
If Va > Vc, flow is turbulent
If Va < Vc, flow is laminar
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RHEOLOGICAL MODELS
Power Law Model
The following are the equations used for calculations
Annular Flow
If Va > Vc, flow is turbulent If Va < Vc, flow is laminar
Internal
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RHEOLOGICAL MODELS
Internal
Semester January 2015 By: ASIF ZAMIR
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