Drilling fluids are in general
complex heterogeneous mixtures of
various types of base fluids and
chemical additives that must remain stable
over a range of
temperature and pressure conditions. The properties of these
complex
mixtures, such as equivalent static density (ESD) and the
rheological
properties of the fluid mixture determine pressure
losses in the system while
drilling. It is often assumed that these
properties and thus the equivalent
circulating density (ECD) are
constant throughout the duration of drilling
activities. This
assumption can prove to be quite wrong in cases where there

is large variation in the
pressure/temperature conditions, such as in high
pressure-high
temperature (HPHT) wells, and deep-water drilling, where low
temperature conditions are encountered very close to the sea bed.

In HPHT wells, as the total
vertical depth increases, there is an
increase in the bottom-hole
temperature, as well as the hydrostatic head of
the mud column.
These two factors have opposing effects on equivalent
circulating
density. The increased hydrostatic head causes increase in the

equivalent circulating density due to
compression. The increase in
temperature on the other hand, causes a
decrease in the equivalent
circulating density due to thermal
expansion. It is most often assumed that
these two effects cancel
each other out. This is not always the case,
especially in high
temperature, high pressure wells.

Large variations in equivalent
circulating density can also occur when
drilling in deep water
environments where relatively cold temperatures are
encountered in
the riser, near the ocean bed. In deep-water wells, the seabed
temperature can be as low as 30 F and hydrostatic pressures at the
bottom

of the riser will be 2700 psi, with a
mud density of 8.6 lb/gal and a water depth
6000-ft. The low
temperature conditions can cause severe gelling of the
drilling
fluid, especially in oil-base muds (OBM). Failure to take this effect
into
account will result in underestimation of the equivalent
circulating density of
the drilling fluid.

Errors in the estimation of
equivalent circulating density have an
especially disastrous effect
when drilling through formations with a small gap
between pore
pressure, and the pressure at which the formation will fracture.
In
such cases, the margin for error is very small and thus, the
equivalent

circulating density must be estimated
precisely. Disregarding pressure and
temperature effects in this
case can lead to greater probability for the
occurrence of kicks,
and blow-outs due to under-balanced pressure or fluid
loss to the
formation (lost circulation and formation damage) due to

overbalance pressure.
Various
experimental studies have also shown drilling fluid rheology to
. Rheological parameters
be very pressure and
temperature dependent
such as viscosity and yield stress affect
frictional pressure losses during the

flow of drilling fluids. Failure to
take into account the dependence of these
parameters on
temperature-pressure conditions can result in obtaining
erroneous
values for equivalent circulating density, which takes into account
the hydrostatic head of the drilling fluid as well as the pressure
loss it

experiences during flow.

The focus of this research is to
study the effect of pressure and
temperature on equivalent static
density as well as equivalent circulating
density of drilling
fluids.

Numerous publications have
dealt with the behavior of equivalent
static density of drilling
fluids in response to variations in pressure temperature conditions.
Various models have been proposed in order to
characterize this
relationship, with some models being empirical in nature,

and some being compositional. The
compositional model characterizes the
volumetric behavior of
drilling fluids based on the behavior of the individual
constituents
of the drilling fluid. Hence, prior knowledge of the composition of
the drilling fluid is required for application of the compositional
model.

In the compositional model, the
density of any solids content in the
drilling fluid is taken to be
independent of temperature and pressure. It is
assumed that any
change in density is due to density changes in the liquid
phases. It
is also assumed that there are no physical and/or chemical
interactions between the solid and liquid phases in the drilling
fluid, or that the
solid phase is inert. Hoberock et al proposed the
following compositional
model for equivalent static density of
drilling fluids.

where,

ρο1, ρw1 = density of oil
and water at temperature T1 and
pressure P1, respectively

ρο2, ρw2 = density of oil
and water at temperature T2 and
pressure P2, respectively

f vo, fvw, fvs, fvc =
fractional volume of oil, water, solid weighting
material, and
chemical additives, respectively
.

P1, P2 = pressure at
reference and condition “2”

T 1, T2 = temperature at
reference and condition “2”

Application of the compositional
model requires some knowledge of
how the densities of each liquid
phase in the mud, usually water and some
type of hydrocarbon, change
with changes in temperature and pressure. The
static mud density at
elevated pressure and temperature can be predicted

from knowledge of mud composition,
density of constituents at ambient or
standard temperature and
pressure, and density of liquid constituents at
elevated temperature
and pressure.

Peters et al applied the
Hoberock et al compositional model
successfully to model volumetric
behavior of diesel-based and mineral oil-
based drilling fluids. In
their study, they measured the density of the individual
liquid
components of each drilling fluid at temperatures varying from
78-350
F and pressures varying from 0-15,000 psi. Using this data in
conjunction
with Hoberock et al’s compositional model, they were
able to predict the
density of the drilling fluids at the elevated
temperature-pressure conditions.

The model predictions yielded an error
of <1 and="and" examined.="examined." of="of" over="over" p="p" pressure="pressure" range="range" temperature="temperature" the="the">

Sorelle et al proposed
equations expressing the relationship between
water and hydrocarbon
(diesel oil No. 2) densities, and temperature/pressure
for use with
the compositional model with some success. Kutasov analyzed
pressure-density-temperature behavior of water and proposed a
similar

equation, which was reported to yield
very low error in predicting water
densities in the HTHP region.

Isambourg et al proposed a
nine-parameter polynomial model to
describe the volumetric behavior
of the liquid phases in drilling fluids, which is
applicable in the
range of 14.5-20,000 psi and 60-400
F. This model
characterizes
the volumetric behavior of the liquid phases in the drilling fluid
with respect to temperature and pressure, and is applied in a
similar
compositional model to that proposed by Hoberock et al. The
model also
assumes that all volumetric changes in the drilling fluid
is due to the liquid
phase, and application of the model requires a
very accurate measurement of
the reference mud density at surface
conditions.

Kutasov proposed an empirical
equation of state (EOS) model for
drilling fluids to express the
pressure-density-temperature dependent
relationship. As is the case
for the compositional model, mud density using
Kutasov’s empirical
equation of state is evaluated relative to its density at
standard
conditions (p= 14.7 psi, T = 60 oF). He applied the equation of
state
with a temperature-depth relationship in order evaluate
hydrostatic pressure
and equivalent static density as a function of
depth.

Babu9 compared the accuracy of
the two compositional models
proposed by Sorelle et al4 and Kutasov8
respectively, and the empirical model
proposed by Kutasov8 in
predicting the mud weights for 12 different mud
systems. The test
samples consisted of 3 water based muds (WBM), 5

OBM’s formulated using diesel oil No.
2, and 4 OBM’s formulated using
mineral oil. Babu9 found that the
empirical model yielded more accurate
estimates for the
pressure-density-temperature behavior of a majority of the
muds over
the range of measured data more accurately than the
compositional
model. He also concluded that the empirical model has more
practical
application because unlike compositional models, it is not hindered
by the need to know the contents of the drilling fluid in question.

Drilling fluids contain complex
mixtures of additives, which can vary
widely with the location of
the well, and sometimes with different stages in the
same well. This
was especially apparent in the behavior of the drilling fluids
prepared with diesel oil No. 2. Different oils available under the
category of
diesel oil No. 2 that were used in the preparation of
OBM’s can exhibit
different compressibility and thermal expansion
characteristics, which were
reflected in the
pressure-density-temperature dependent behavior of the fluids
prepared with them.

Research has also been reported
on characterizing drilling fluid
rheology at high temperature/high
pressure conditions. Rommetveit et al approached their analysis of
shear stress/shear rate data at high temperature
and pressure by
multiplying shear stress by a factor which depends on
pressure,
temperature and shear rate. Coefficients of this multiplying factor
are fitted to shear stress/shear rate data directly without
extracting rheological
parameters such as yield stress first. This
eliminates the need to characterize
the behavior of each rheological
parameter relative to pressure and
temperature changes. In essence,
they obtain an empirical model in which
the effects of variation in
all rheological parameters that describe fluid flow
behavior are
lumped together.

Another approach to the
analysis of temperature and pressure effects
on drilling fluid
rheology is to consider the effect of temperature and pressure
changes on each rheological parameter that describes the behavior of
the
fluid. The two most common models3 considered for such an
analysis are the

Herschel-Bulkey/Power law model and the
Casson model which is an
acceptable description of oil based mud
rheology. Of these two models, the
Herschel-Bulkley model is the
most robust, as it is a three parameter model
as opposed to the
Casson model which is a two parameter model. In the

analysis performed by Alderman et al on
shear stress/shear rate data, the
Herschel-Bulkley/Power and Casson
models were considered. The behavior
of each rheological parameter
in these models with respect to changes in
temperature and pressure
was investigated. They studied a range of fluids
covering
un-weighted and weighted bentonite water-based drilling fluids with

and without deflocculant additives.

In order to estimate equivalent
circulating density, it is important to take
into account the
effects of temperature and pressure on fluid rheology. Two
methods
are proposed to accomplish this by Rommetveit et al. They
propose a
stationary or static method and a dynamic method. In both
methods,
the contributions of hydrostatic and frictional pressure losses in
high
pressure/high temperature wells to the equivalent circulating
density were
considered. The variation in temperature vertically
along the well bore is
taken into account for both models, and
drilling fluid properties are allowed to

vary relative to temperature.

The dynamic method however,
also takes into account transient
changes in temperature i.e. change
in temperature over time. This effect is
especially important in the
case where circulation has been stopped for a
significant amount of
time. The drilling fluid temperature will begin to
approach the
temperature of the formation. Once circulation commences
again as
shown in Fig. 1.1, the lower part of the annulus will be cooled by
cold fluid from the drill string and the upper part of the annulus
will be warmed
by hotter fluid coming from the bottom-hole. During
this transient period, fluid

density and rheological characteristics
can change rapidly due to rapid
changes in temperature. Research on
this effect is still at a very early stage
and will not be taken
into account during this study.

Figure 1.1- Schematic
Diagram of Fluid in
the Well bore at the Start of Circulation

Alderman et al performed
rheological experiments on water based
drilling fluids over a range
of temperatures up to 260 oF and pressures up to
14,500 psi, using
both weighted and unweighted drilling fluids. Rheograms
were
obtained for the water based drilling fluids, holding temperature
constant

and varying pressure, and vice versa.
It was found that the Herschel-Bulkley
model yielded the best fit to
the experimental data. Other models that were
investigated are the
Bingham plastic model, and the Casson model which
some authors argue
is the best model for characterizing oil-based drilling fluid

rheology.

For the Herschel-Bulkley model,
it was found that the fluid viscosity at
high shear rates increased
with pressure to an extent, which increases with
the fluid density,
and decreases with temperature in a similar manner to pure
water.
Alderman et al found the yield stress to vary little with
pressure-
temperature conditions. The yield stress remained
essentially constant with
respect to temperature until a
characteristic threshold temperature is attained.

This threshold temperature was found to
depend on mud composition. Once
this threshold is reached, the yield
stress increases exponentially with 1/T.
Alderman et al also found
that the power law exponent increased with
temperature, and
decreased with pressure. This lead them to conclude that

the Casson model will become
increasingly inaccurate at these two extremes,
that is, at high
temperature and low pressure.

The estimation of ECD under
high temperature conditions requires
knowledge of the temperatures
to which the drilling fluid will be subjected to
downhole. As the
fluid is circulated in the wellbore, heat from the formation
flows
into the wellbore causing the wellbore fluid temperature to rise.
This

process is more pronounced in deep, hot
wells where the temperature
difference between the formation and the
well-bore fluid is greater. The
process is very dynamic at early
times, that is, at the commencement of
circulation, with great
changes in fluid temperature occurring over small

intervals of time.

There are two major methods for
estimating the down-hole
temperature of drilling fluid. The first is
the analytical method. This method
assumes constant fluid
properties. Ramey solved the equations governing
heat transfer in a
well bore for the case of hot-fluid injection for enhanced oil
recovery. His solution permits the estimation of the fluid, tubing
and casing
temperature as a function of depth. He assumed that heat
transfer in the well
bore is steady state, while heat transfer in
the formation is unsteady radial
conduction.

Holmes and Swift solved the heat
transfer equations analytically for
the case of flow in the
drillpipe and annulus. They assumed the heat transfer
in the
wellbore to be steady state. However, they used a steady-state
approximation to the transient heat transfer in the formation. They
justified

this assumption by asserting that the
heat transfer from the formation is
negligible in comparison to the
heat transfer between the drill pipe and
annular sections due to the
low thermal conductivity of the formation.

Arnold also solved the heat
transfer equations analytically for both
the hot-fluid injection
case and the fluid circulation case. However, in
circulation case,
he did not assume steady state heat transfer in the
formation. He
represented the transient nature of heat flow from the formation
with a dimensionless time function that is independent of depth16.
Kabir et al also solved a similar set of equations, but for the case
of flow down the
annulus and up the drill pipe. They also assumed
transient heat flow in the

formation, and evaluated a number of
dimensionless time functions.

The second method of estimating
fluid temperature during circulation
involves allowing the fluid
properties such as heat capacity, viscosity, and
density to vary
with the temperature conditions. This method involves solving
the
governing heat transfer equations numerically using a finite
difference

scheme. Marshal et al created a model
to estimate the transient and steady-state temperatures in a well
bore during drilling, production and shut-in using
a finite
difference approach.

Romero and Touboul created a
numerical simulator for designing and
evaluating down-hole
circulating temperatures during drilling and cementing
operations in
deep-water wells. Zhongming and Novotny developed a finite
difference model to predict the well bore and formation transient
temperature
behavior during drilling fluid circulation for wells
with multiple temperature
gradients and well bore deviations.

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