UNIVERSITY OF OKLAHOMA
GRADUATE COLLEGE
EVALUATION OF EQUIVALENT CIRCULATING DENSITY OF DRILLING
FLUIDS UNDER HIGH PRESSURE-HIGH TEMPERATURE CONDITIONS
A THESIS
SUBMITTED TO THE GRADUATE FACULTY
in partial fulfillment of the requirement for the
degree of
MASTER OF SCIENCE
(Petroleum Engineering)
By
Oluseyi Harris
Norman, Oklahoma
2004
EVALUATION OF EQUIVALENT CIRCULATING DENSITY OF
DRILLING FLUIDS UNDER HIGH PRESSURE-HIGH TEMPERATURE
CONDITIONS
A THESIS APPROVED FOR THE
MEWBOURNE SCHOOL OF PETROLEUM AND GEOLOGICAL
ENGINEERING
BY
Chair: Dr. Samuel Osisanya
Dr. Subhash Shah
Member:
Member:
Dr. Djebbar Tiab
©Copyright by Oluseyi Harris 2004 All Rights Reserved.
ACKNOWLEDGEMENTS
The author wishes to express his profound gratitude and appreciation
for Dr. Samuel Osisanya. His guidance, moral and financial support, and
encouragement were invaluable. The author would like to thank the members
of the thesis committee, Dr Samuel Osisanya, Dr Subhash Shah, and Dr.
Djebbar Tiab for their helpful comments and suggestions. Heartfelt thanks go
to Dr. Subhash Shah for his assistance in allowing use of WCTC facilities in
performing research for this thesis. The author wishes to extend special
thanks to colleagues whose assistance and encouragement was invaluable
during the course of this research work- Ricardo Michel-Villazon, Aristotelis
Pagoulatos, Kayode Aremu, Kola Ayeni.
The author wishes to thank his other half, Lola for always being there.
The author would also like to express immeasurable gratitude towards his
parents for their constant and unwavering support and faith. Last and most
importantly, thanks and praise are extended to God almighty who alone
makes all things possible.
Oluseyi Harris
Norman, Oklahoma July, 2004
iv
TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS ..............................................................................iv
LIST OF TABLES ........................................................................................viii
LIST OF FIGURES.........................................................................................ix
ABSTRACT....................................................................................................xi CHAPTER PAGE
1. FORMULATION OF THE PROBLEM .........................................................1 1.1. Introduction ..........................................................................................1
1.2. Literature Review .................................................................................3
1.3. Objectives and Scope of Work...........................................................13
1.4. Study Organization ...........................................................................14
2.FUNDAMENTAL CONCEPTS FOR ESTIMATION OF EQUIVALENT STATIC AND CIRCULATING DENSITY ....................................................15 2.1 Equivalent Static density ......................................................................15
2.2 Estimating Equivalent Static Density....................................................18
2.2.1 Compositional Models ...................................................................18
2.2.1.1 Volumetric Models for Mud Constituents ..................21
2.2.2 Empirical Models ...........................................................................23
2.3 Equivalent Circulating density ..............................................................23
2.4 Frictional Pressure Loss.......................................................................24
2.5 Fluid Rheology .....................................................................................26
2.5.1 Bingham Plastic Model ..................................................................27
2.5.2 Power Law Model ..........................................................................28
2.5.3 Herschel-Bulkley Model.................................................................30
2.5.4 Casson Model ...............................................................................31
v
2.5.5 Ellis Model .....................................................................................31
2.5.6 Carreau Model...............................................................................32
2.6 Temperature and Pressure Dependent Rheological Parameters.......33
2.6.1 Temperature/Pressure Dependent Plastic Viscosity....................33
2.6.2 Temperature Dependent Yield point..............................................35
2.7 Bingham Plastic Pressure Loss Equations......................................36
3.DRILLING FLUID TEMPERATURE PROFILE ESTIMATION ...................40 3.1 Heat Transfer in the Wellbore ..............................................................41
3.2 Analytical Method.................................................................................43
3.2.1 Assumptions of Analytical Model ..................................................43
3.2.2 Heat Balance in the DrillPipe........................................................44
3.2.3 Heat Balance in the Annulus ........................................................45
3.2.4 Heat Flow in the Formation and System Heat Balance ................46
3.3 Numerical Method................................................................................50
3.3.1 Equations Governing Heat transfer in the Wellbore and Formation
...............................................................................................................50
3.3.2 Discretizing Heat Flow Equations for Finite difference Analysis ....53
3.4 Summary.........................................................................................68
4.DEVELOPMENT AND VALIDATION OF THE DYNAMIC DENSITY SIMULATOR AND MODELLING OF DYNAMIC DENSITY .......................69 4.1 Program Lay-Out.............................................................................70
4.2 DDS Program Execution .................................................................71
4.2.1 General Well Parameters Form ...................................................71
4.2.2 Mud Properties Form ...................................................................77
4.2.3 Formation Properties Form ..........................................................77
4.2.4 Heat Transfer Coefficients Form..................................................77
4.2.5 Results and Results Form ...........................................................80
4.3 Equations used in DDSimulator Program........................................82
4.3.1 Fluid Properties ...........................................................................82
4.3.2 Temperature Profile Estimation ...................................................83
vi
4.3.3 Equivalent Hydrostatic Head and ECD ........................................84
4.4 Model Validation..............................................................................84
4.5 Dynamic Density Estimation............................................................91
Summary..................................................................................................107
5.SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS...................108 5.1 Summary.......................................................................................108
5.2 Conclusions...................................................................................109
5.3 Recommendations ........................................................................110
NOMENCLATURE ......................................................................................112
REFERENCES............................................................................................115
APPENDIX ..................................................................................................119 Code for DDSimulator Program ...............................................................119
vii
LIST OF TABLES TABLE PAGE
4.1 : Well and mud circulating properties for a gulf coast well…………….85
4.2 : Simulated Well Conditions………………………………………………92
4.3 : Results of Well Simulation………………………………………………92
4.4 : Well simulation results for parameters detailed in Table 4.2 with
gG = 0.015 oF/ft……………………………….……………..……….…..96
4.5 : Well simulation results for parameters detailed in Table 4.2 with
gG = 0.025 oF/ft………………………………………………….….……96
4.6 : Well simulation results for parameters detailed in Table 4.2 with
inlet fluid temperature = 80 oF……………..……………….….….…….97
4.7 : Well simulation results for parameters detailed in Table 4.2 with
circulation rate = 210 gal/min…………………………………..……...102
4.8 : Well Simulation Results for Parameters Detailed in Table 4.2
with Circulation Rate = 300 bbl/hr…………………………………….105
viii
LIST OF FIGURES FIGURE PAGE
1.1 : Schematic Diagram of Fluid in the Well bore
at the Start of Circulation………………………………………………..9
2.1 : Volumetric behavior of various liquids under varying conditions of
temperature and pressure.....………………………………………….17
2.2 : Shear-thinning in a typical non-Newtonian Fluid….....……………...29
2.3 : Flow curves for time-independent fluids……..……………………….29
3.1 : Schematic of Heat Balance for Fluid Circulating in a Wellbore…....42
3.2a : Solution grid for Finite Difference Analysis…..……………………....51
3.2b : Heat Flow at Formation Annulus Boundary…..……………………...51
3.3 : Finite Difference Grid…...………………………………………………54
3.4 : Heat Balance at Bottom-Hole…..……………………………………..60
4.1 : DDSimulator Program Flow Chart…...………………………………..72
4.2 : Title Form…..……………………………………………………………73
4.3 :DDSimulator Launch Command Button…..…………………………..74
4.4 : Well Parameters Form………..………………………………………..75
4.5 : Mud Properties Form………………………..………………………….76
4.6 : Formation Properties Form…………………………………………....78
4.7 : Heat Transfer Coefficients Form……..…………………..…..……….79
4.8 : Results Form………………………………………………..…………..80
4.9 : A Sample Temperature Profile Using Excel Graph Feature…..…...81
4.10 : Plot of Temperature Profile For Gulf Coast Well…………………….86
ix
4.11 : Well Temperature Profile While Circulating Field Salt Water………89
4.12 : Temperature Profile For Gulf Coast Well…………………………….90
4.13 : Temperature Profile in 17200ft well after 5hrs……………………….93
4.14 : Annular Pressure Profile in 17200ft well after 5hrs………………….94
4.15 : Equivalent Circulating Density in 17200ft well after 5hrs…………...94
4.16 : Temperature Profile in 17200ft well after 5hrs……………………….97
4.17 : Annular Pressure Profile in 17200ft well after 5hrs………………….98
4.18 : Equivalent Circulating Density in 17200ft well after 5hrs…………...98
4.19 : Temperature Profile in 17200ft well after 5hrs……………………….99
4.20 : Annular Pressure Profile in 17200ft well after 5hrs………………….99
4.21 : Equivalent Circulating Density in 17200ft well after 5hrs………….100
4.22 : Temperature Profile in 17200ft well after 5hrs………………………100
4.23 : Annular Pressure Profile in 17200ft well after 5hrs…………………101
4.24 : Equivalent Circulating Density in 17200ft well after 5hrs…………..101
4.25 : Temperature Profile in 17200ft well after 5hrs………………………103
4.26 : Annular Pressure Profile in 17200ft well after 5hrs…………………103
4.27 : Equivalent Circulating Density in 17200ft well after 5hrs………..…104
4.28 : Temperature Profile in 17200ft well after 5hrs………………..…..…105
4.29 : Annular Pressure Profile in 17200ft well after 5hrs…………..…..…106
4.30 : Equivalent Circulating Density in 17200ft well after 5hrs………..…106
x
ABSTRACT
The effects of the temperature and pressure conditions prevalent in
high temperature/high pressure wells on the equivalent circulating density of
drilling fluids in a circulating wellbore as well as the bottom-hole pressure are
studied. High temperature conditions cause the fluid in the wellbore to
expand, while high pressure conditions in deep wells cause fluid
compression. Failure to take these two opposing effects into account can lead
to errors in the estimation of bottom-hole pressure on the magnitude of
hundreds of psi. The rheological behavior of drilling fluids is also affected by
the temperature and pressure conditions.
A Bingham plastic model was used to simulate the temperature and
pressure dependent rheological behavior of the drilling fluids studied, with the
rheological parameters expressed as functions of temperature and pressure.
Analytical and numerical methods for estimating the temperature profile in a
circulating well-bore were studied. A simulator called DDSimulator was
developed using visual basic to simulate the wellbore during circulation. This
simulator can develop the temperature and pressure profiles of a wellbore
during circulation, and compute the bottom-hole pressure and equivalent
circulating density taking into account the temperature and pressure
conditions in the wellbore. The Crank-Nicolson numerical discretizing scheme
was employed in the DDSimulator for the evaluation of the temperature
profile.
xi
From the results of the DDSimulator, it was found that the geothermal
gradient has a great effect on the bottom-hole temperature and pressure, and
the equivalent circulating density that will occur in a circulating well-bore. It
was also found that the inlet pipe temperature did not have a significant effect
on the bottom-hole temperature and pressure. This is even more the case in
deep wells, and in areas with high geothermal gradient. The circulation rate
plays an important role in the bottom-hole temperature and pressure that will
occur in circulating well.
The major technical contribution of this work is the development of the
DDSimulator. The density and rheological properties of the drilling fluid in the
wellbore can be estimated in order to adequately evaluate the bottom-hole
pressure during fluid circulation. DDSimulator allows the evaluation of the
bottom-hole pressure taking into account the variation in the volumetric and
rheological properties of the drilling fluid under high temperature and high
pressure conditions in the wellbore. The effects of variation in the inlet fluid
temperature, circulation rate, and geothermal gradient are explored and
discussed in this work.
xii
Chapter 1
FORMULATION OF THE PROBLEM
1.1. Introduction
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
1
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 deepwater wells1, the seabed
temperature can be as low as 30 oF 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
be very pressure and temperature dependent 2,3. Rheological parameters
such as viscosity and yield stress affect frictional pressure losses during the
2
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.
1.2. Literature Review
Numerous publications4-10 have dealt with the behavior of equivalent
static density of drilling fluids in response to variations in pressure-
temperature conditions. Various models4-10 have been proposed in order to
characterize this relationship, with some models being empirical in nature,
and some being compositional. The compositional model4-5 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
3
solid phase is inert. Hoberock et al4 proposed the following compositional
model for equivalent static density of drilling fluids.
( )⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+
+++=
111,
2
1
2
1
1122
w
wvw
o
ovo
vccvssvwwvoo
ff
ffffTP
ρρ
ρρ
ρρρρρ (1.1)
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 al5 applied the Hoberock et al4 compositional model
successfully to model volumetric behavior of diesel-based and mineral oil-
4
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
oF 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% over the range of temperature
and pressure examined.
Sorelle et al6 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. Kutasov8 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 al7 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 oF. 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 al4. 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.
5
Kutasov8 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
6
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 al11
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 al3 on shear stress/shear rate data, the
7
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 al11. 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
8
changes in temperature. Research on this effect is still at a very early stage
and will not be taken into account during this study.
Cooler fluid from drill pipe cooling down the annulus
Warmer fluid from bottom-hole warming up the upper annulus
Drill Pipe
Figure 1.1- Schematic Diagram of Fluid in the Well bore at the Start of Circulation
9
Alderman et al3 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 al3 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 al3 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
10
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. Ramey13 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 Swift14 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.
11
Arnold15,16 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 al17
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 al18 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 Touboul19 created a numerical simulator for designing and
evaluating down-hole circulating temperatures during drilling and cementing
operations in deep-water wells. Zhongming and Novotny20 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.
12
1.3. Objectives and Scope of Work
The main objective of this work is to evaluate changes in drilling fluid
density with variations in temperature and pressure conditions and
characterize how these changes in equivalent circulating density are affected
by the composition of the drilling fluid. Specifically, the objectives of this work
are;
1. Evaluate changes in static density of drilling fluids with changes in the
temperature-pressure conditions
2. Evaluate changes in the rheological behavior of drilling fluids with
changes in the temperature-pressure conditions and ascertain the
degree of the resultant effect on the dynamic ECD.
3. Evaluate different methods of estimating the circulating fluid
temperature gradient in the well bore and the effects on frictional
pressure loss and hence on ECD.
The objectives of this work are accomplished with the development of a
Dynamic Density Simulator. This simulator was developed in the visual basic
language and will allow the estimation of the equivalent circulating density
under high pressure and high temperature conditions.
13
1.4. Study Organization
The fundamental concepts involved with hydrostatic pressure and
frictional pressure loss are discussed in Chapter Two. The most commonly
used rheological models for characterizing drilling fluid flow in conjunction
with frictional pressure loss calculation methods are also discussed. The
equations that express viscosity as a function of temperature and pressure
will be discussed here. Chapter Three discusses the heat transfer equations
in the well bore and the analytical and numerical methods for estimating the
temperature profile in a circulating well. Chapter Four covers the development
of the ECD simulator for high-pressure/high-temperature wells. The chapter
covers the modeling procedure, model validation, and discusses the results.
Chapter five covers the summary of the results, conclusions, and
recommendations.
14
Chapter 2
FUNDAMENTAL CONCEPTS FOR ESTIMATION OF EQUIVALENT STATIC AND CIRCULATING DENSITY
In today’s drilling industry, deeper and hotter wells are increasingly
being drilled. In order to maintain proper well control, prevent lost circulation,
and accurately analyze fracture gradient data, it is of paramount importance
to accurately predict the density of the drilling fluids used in drilling these
wells, under high temperature-high pressure conditions. Drilling fluids in
general become compressed under high pressure, and expand with
temperature. Hence, their down-hole densities are often quite different from
their surface densities, which are usually measured during drilling operations.
The fundamental concepts of equivalent static density and equivalent
circulating density are reviewed in this chapter.
2.1 Equivalent Static density
The equivalent static density of a drilling fluid is an expression of the
hydrostatic pressure exerted by the fluid. Hydrostatic pressure in turn can be
defined as the pressure exerted at any point by a vertical column of fluid. The
hydrostatic pressure is a function of the density of the fluid, and the height of
the fluid column. Hydrostatic pressure is expressed in field units as follows.
15
P = 0.052ρh (2.1)
Where,
P = pressure, psi
ρ = fluid density, lbm/gal (ppg)
h = height of fluid column, ft
This simple equation assumes the fluid in question to be incompressible. If
the temperature and pressure in the mud is low, the use of constant surface
mud density in conjunction with the above equation will yield a reasonable
approximation of the bottom-hole density.
Equivalent static density however, must take into account the effects of
temperature and pressure conditions present in the well. Excluding these
factors in the estimation of bottom-hole pressure in the case of deep, hot
wells, can yield figures that are in error by hundreds of psi. Figure 2.1 shows
the effects of temperature and pressure on the density of various base liquids
that can be used in drilling fluids. As expected, these figures show that
density increases with increasing pressure, but decreases as temperature
increases. However, as depth increases, temperature effects tend to
dominate pressure effects, so that the net result is decreasing mud density
with increasing depth.
16
oF
Figure 2.1- Volumetric Behavior of Various liquids
Under varying conditions of Temperature and Pressure4
17
2.2 Estimating Equivalent Static Density
There are two main methods of characterizing the variation in
equivalent static density of drilling fluids in response to changes in
temperature and pressure conditions; empirically based models, and
compositional models. The empirical model provides explicit empirically
derived equations for estimating mud density at various temperature-pressure
conditions. The compositional model however takes into account the
volumetric behavior of each of the individual mud constituents in response to
variations in temperature and pressure.
2.2.1 Compositional Models
The compositional model proposed by Hoberock et al4 is derived as
follows. In order to account for the variation in compressibility across the
different constituents present in a drilling fluid, it is necessary to perform a
material balance on the drilling fluid as a whole. In the model, it is assumed
that all solids present in the drilling fluid are incompressible. Consider a
drilling fluid that consists of oil and water phases, solid weighting material,
and chemical additives. The volume and weight of the drilling fluid at some
reference temperature (p1, T1) would be expressed as follows
18
V1 = Vo + Vw + Vs (2.2)
W = ρo1Vo + ρw1Vw + ρsVs (2.3)
Where,
Vo, Vw,Vs = volume of oil, water, and solids
W = weight
ρo1, ρw1 = density of oil and water phases at reference
conditions (p1, T1)
ρs = density of solid content
Ideal mixing is assumed in Eqs. 2.2 and 2.3. Once the drilling fluid is
subjected to a new set of temperature-pressure conditions (p2, T2), the
volume of the fluid will change due to the compressibility of the liquid phases.
The new drilling fluid volume is thus expressed as
V2 = Vo + Vw + Vs + ∆Vo + ∆Vw (2.4)
From the law of conservation of mass, the change in volume of the liquid
phases can be expressed as follows.
oo
ooo VVV −⎟⎟
⎠
⎞⎜⎜⎝
⎛=∆
2
1
ρρ (2.5)
ww
www VVV −⎟⎟
⎠
⎞⎜⎜⎝
⎛=∆
2
1
ρρ (2.6)
From Eqs. 2.3 and 2.4, the new mud density at T2 and p2 will be as follows.
( )wocswo
sswwoom VVVVV
VVVTp∆+∆+++
++=
ρρρρ 1122, (2.7)
19
where the subscript m refers to the drilling mud. Substituting Eqs. 2.5 and 2.6
into Eq. 2.7 and dividing by the original total volume at pressure p1 and
temperature T1, the following equation is obtained.
( )
1112
1
12
1
1111
11
22 ,
VV
VV
VV
VV
VV
VV
VV
VV
Tpcsw
w
wo
o
o
cc
ss
ww
oo
m
+++
+++=
ρρ
ρρ
ρρρρρ (2.8)
Consider the volume fraction fx of each component to be
VVf x
x = (2.9)
where Vx = volume of component x
fx = volume fraction of component x
V = total volume
Taking into account that
fo + fw + fs = 1 (2.10)
then Eq. 2.8 can be expressed as
( )⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+
++=
111,
2
1
2
1
1122
w
ww
o
oo
sswwoom
ff
fffTp
ρρ
ρρ
ρρρρ (2.11)
( )⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+
=111
,
2
1
2
1
122
w
ww
o
oo
mm
ffTp
ρρ
ρρ
ρρ (2.12)
where ρm1 is the mud density at temperature T1 and pressure p1. From the
above equation, it is evident that the mud density at elevated temperatures
and pressures can be predicted based on knowledge of the mud constituents
and the volumetric behavior of each constituent relative to variations in
20
temperature and pressure. Various authors have proposed equations
expressing the volumetric behavior of water, and oil phases that may be
present in drilling fluids.
2.2.1.1 Volumetric Models for Mud Constituents
Sorelle et al6 proposed the following empirical expressions for the
volumetric behavior of the water phase, and diesel oil No. 2.
ρo = Ao + A1(T) + A2(p-po) (2.13)
ρw = Bo + B1(T) + B2(p-po) (2.14)
where
Ao = 7.24032 Bo = 8.63186
A1 = -2.84383 * 10-3 B1 = -3.31977 * 10-3
A2 = 2.75660 * 10-5 B2 = 2.37170 * 10-5
The equation for the water phase was obtained by curve fitting data from
tables of physical constants, while that of the diesel oil No. 2 was obtained by
curve fitting data from experiments.
Kutasov 21 analyzed the pressure-density-temperature behavior of
water, and proposed the following similar empirical equation.
(2.15) ( ) ( ) ( )[ ]20
70
40
6 10*0123.510*2139.210*0997.33619.8 TTTTppw e −−−−− −−−
=ρ
where po and To represent standard temperature (59 oF) and pressure (14.7
psia).
21
Isambourg et al7 proposed a nine parameter model to express the
volumetric behavior of liquids in response to variations in temperature and
pressure. The model is expressed as follows.
Vr(p, T) = k00 + k01T + k10P + k02T2 + k20P2+k11pT
+ k12pT2 + k21p2T + k22p2T2 (2.16)
where Vr(p, T) is the volumetric ratio at pressure p, and temperature T.
( )( )oo TpVolume
TpVolumeratioVolumetric,,
= (2.17)
Eq. 2.16 is valid in the range of 14.5 to 20,000 psia, and 68 to 392 oF and can
be used to estimate fluid density at elevated temperatures and pressures in
the following manner.
( ) ( ) ( )( )TpV
TpVTpTpr
oorooff ,
,,, ρρ = (2.18)
The above equation is a simplification of the compositional model proposed
by Hoberock et al4. Isambourg et al7 also proposed the following equation to
express variations in the density of solid weighting material with changes in
temperature-pressure conditions.
( ) ( )( )[ ] ([ )]osos
ooss ppbTTa
TpTp−+−+
=1*1,, ρρ (2.19)
where
as = 0.8*10-4 oC-1, thermal expansivity of barite
bs = -1.0*10-5 bar-1, compressibility of barite
22
2.2.2 Empirical Models
Kutasov8 proposed the following empirical three-parameter equation of
state to describe the volumetric behavior of drilling fluids.
(2.20) ( ) ( ) ( )[ ]2ooo TTTTpp
mom e −±−−−= γβαρρ
where,
ρmo = mud density at standard pressure and temperature
(14.5 psia, 59oF)
α, β, γ = empirical constants
Kutasov’s model applies to both water-based and oil-based drilling fluids, and
treats the particular drilling fluid as a continuous phase. Hence, knowledge of
the volumetric behavior of each of the constituents of the drilling fluid is not
required.
2.3 Equivalent Circulating density
The equivalent circulating density of a drilling fluid can be defined as
the sum of the hydrostatic head of the fluid column, and the pressure loss in
the annulus due to fluid flow. It is expressed as density term at the point of
interest.
23
( frictionchydrostatiecd PPh
∆+∆=052.01ρ ) (2.21)
where,
ρecd = equivalent circulating density (lb/gal)
∆Phydrostatic = Hydrostatic head of fluid column (psi)
∆Pfriction = Pressure drop due to friction in the annulus (psi)
As stated before, the hydrostatic pressure of the drilling fluid is affected by the
temperature-pressure conditions present in the well-bore, and the depth of
the well-bore. The frictional pressure loss term in the above equation however
is affected by the well-bore and drill string geometry, fluid rheology, and the
pump rate or fluid flow rate.
2.4 Frictional Pressure Loss
The frictional pressure loss is the loss in pressure during fluid flow due
to contact between the fluid and the walls of the flow conduit. When fluid
moves past the solid interface, a boundary layer is formed adjacent to the wall
of the flow conduit. The viscous property of the fluid creates a variation in the
flow velocity normal to the solid interface, ranging from zero at the pipe wall
with a no-slip assumption and maximum velocity at the edge of the boundary
layer. This variation in fluid velocity represents a loss in momentum and a
resistance to flow. The associated pressure loss is directly proportional to the
24
length of the flow conduit, the fluid density, the square of the fluid velocity,
and inversely related to the conduit diameter.
LD
vfp ∆=∆22 ρ (2.22)
where
∆p = frictional pressure loss
f = Fanning friction factor
ρ = fluid density
∆L = conduit length
V = fluid velocity
D = pipe diameter
In the case of non-circular flow conduits, the diameter parameter is replaced
by the equivalent diameter.
w
fe P
AD 4= (2.23)
where
De = equivalent diameter
Af = cross-sectional area
Pw = wetted perimeter
The variable “f” in equation 2.22 is known as the Fanning friction factor. The
friction factor can be defined as the ratio between the force exerted on the
walls of a flow conduit as a result of fluid movement, and the product of the
characteristic area of the flow conduit and the kinetic energy per unit volume
of the fluid.
25
2.5 Fluid Rheology
Rheology can be defined as the science and study of the deformation
and flow of matter, in this case drilling fluids. It is also the characteristic of the
particular fluid in reference to its flow behavior. Rheological models seek to
characterize this flow behavior by developing relationships between applied
shear stress, and the shear rate of the fluid. Based on the nature of this
relationship, fluids in general can be classified as Newtonian, non-Newtonian,
and visco-elastic fluids.
Newtonian Fluids- Newtonian fluids are fluids in which the ratio between
applied shear stress, and the rate of shear is constant with respect to time
and shear history. The relationship characterizing Newtonian fluids is
expressed mathematically as follows:
γµτ &= (2.24)
where,
τ = shear stress
γ& = shear rate
µ = viscosity
Examples of Newtonian fluids are water, light hydrocarbons, and all gases.
Non-Newtonian fluids- Non-Newtonian fluids are fluids whose viscosity varies
with time and shear history. This class of fluids can be further subdivided into
time-dependent and time-independent fluids. Time-dependent fluids are
26
fluids, in which the viscosity varies with time at a constant shear rate, while
time-independent fluids are fluids whose viscosity is constant over time at a
constant shear rate. Most drilling fluids are non-Newtonian fluids.
Visco-elastic Fluids- These are fluids which exhibit both viscous and elastic
behavior. When subjected to stress, they deform and flow like true fluids, but
once the stress is removed, they regain some of their original state like solids.
Examples of visco-elastic fluids include flour dough, and polymer melts.
The following are the rheological models that characterize the various
types of non-Newtonian fluids.
2.5.1 Bingham Plastic Model
The Bingham plastic model is a two-parameter time-independent
rheological model that accounts for the stress required to initiate fluid flow in
viscous fluids. This initial stress is referred to as the yield stress. Once the
yield stress is overcome, the fluid is represented as a Newtonian fluid, which
is shown by the linear relationship between the applied stress and the rate of
shear. The constitutive equation for the Bingham plastic model is given as
follows.
γµττ &po += (2.25) oττ >
where
τo = yield stress µp = plastic viscosity
27
Although the Bingham plastic model does account for the yield stress, it can
be inadequate for characterizing some types of drilling fluids, as it does not
account for their shear thinning property.
2.5.2 Power Law Model
The power law model is also a time-independent two parameter
rheological model like the Bingham plastic model. However, where the
Bingham plastic model expresses a linear relationship between shear stress
and shear strain, the power law model uses a non-linear relationship which
can better characterize the shear-thinning characteristics of most common
drilling fluids. The following is the constitutive equation for the power law
model.
(2.26) nkγτ &=
where
k = consistency index
n = flow behavior index
k and n are constants characteristic of a particular fluid. k is a measure of the
consistency of the fluid, the higher the value of k the more viscous the fluid; n
is a measure of the degree of non-Newtonian behavior of the fluid. In cases
where the flow behavior index is equal to 1, the power law model describes
the behavior of a Newtonian fluid. In situations where the flow behavior index
is between 0 and 1, the fluid is referred to as pseudoplastic and shear-
28
thinning. Shear-thinning refers to the reduction in viscosity with the shear rate.
The limiting viscosity is known as the viscosity at infinite shear, ∞µ (Fig. 2.2).
∞µ
µο
µ
γ
Figure 2.2- Shear-thinning in a typical non-Newtonian Fluid
Shear Rate
Dilatant
Newtonian
Pseudo-plastic
Bingham Plastic
Yield Pseudo-plastic
Shear Stress
Figure 2.3- Flow curves for time-independent fluids
29
When the flow behavior index is greater than 1 the fluid is referred to
as dilatant and shear thickening. This is shown in Fig. 2.3. The dimensions of
k are dependent on the value of n; hence k is not a material property but an
empirical constant. In general, the shear-thinning behavior is more desirable
in drilling fluids; hence drilling fluids tend to be pseudo-plastic fluids. Shear-
thinning behavior is desirable in drilling fluids because it allows the fluid to
carry cuttings even while it is at rest due to the fluid thickness, and at the
same time lowers pumping costs because the fluid becomes thinner and
easier to pump as it is sheared.
2.5.3 Herschel-Bulkley Model
The Herschel-Bulkley model is a time-independent three parameter
rheological model that accounts for both the yield stress, and the non-linear
relationship between shear stress and shear rate exhibited by most drilling
fluids. The constitutive equation is given below.
no kγττ &+= oττ > (2.27)
where
k = consistency index
n = flow behavior index
τo = yield stress
The Herschel-Bulkley model is also used widely in the oil industry to
characterize drilling fluids as well as fracturing fluids.
30
The above three rheological models are the most commonly used in
the oil industry for the characterization of drilling fluids. There are, however,
various other rheological models that can and have been used. The following
are a few of these models.
2.5.4 Casson Model
The Casson model is a two-parameter model originally developed in
order to characterize the rheology of pigment-oil suspensions. The
constitutive equation is given as follows.
γττ &ko += oττ ≥ (2.28)
where
τo = yield stress
k = Casson model constant, similar to the consistency index for
Power-law model
The Casson model is also commonly used to characterize the rheology of
blood.
2.5.5 Ellis Model
The Ellis model is a three-parameter model which accounts for a
Newtonian region at low shear rates, while still expressing a power law type
dependence at higher shear rates. These are the initial flat plateau and
31
successive straight line segments of Fig. 2.2. The constitutive equation is
given as follows.
γ
ττ
µγτµ α
&&
⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
+
== −1
21
1
oa (2.29)
where
µa = apparent viscosity µo = low shear rate viscosity
τ1/2 = shear stress @ µa = µ0/2 α = Ellis model parameter
From eq. 2.29, a plot of ⎟⎟⎠
⎞⎜⎜⎝
⎛−1ln
a
o
µµ versus
21
lnττ will yield a slope of (α-1).
Hence, if µo and τ1/2 are known, α can be estimated. µo refers to the viscosity
at very low shear rates, i.e. as τ tends to zero.
2.5.6 Carreau Model
The Carreau model is a four-parameter model developed in order to
account for the entire flow curve shown in Fig. 2.2, i.e. the two Newtonian-like
flow regions at very high shear rates and very low shear rates characterized
by flat plateaus and the power-law region in the middle. The constitutive
equation for the Carreau model is as follows.
32
( )[ ]( )2
121
−
∞
∞ +=−− n
o
γλµµµµ
& (2.30)
where
oµ = low shear rate viscosity
∞µ = viscosity at infinite shear
λ = time constant
n = exponential constant
The λ parameter is a time constant calculated from the point on the flow curve
where the flow behavior transitions from the lower Newtonian region to the
power law region.
2.6 Temperature and Pressure Dependent Rheological
Parameters
In order to estimate the flow behavior of drilling fluids under high
temperature-high pressure conditions, the following variation on the Bingham
plastic model proposed by Politte22 will be applied.
2.6.1 Temperature/Pressure Dependent Plastic Viscosity
Politte22 analyzed rheological data for diesel based drilling fluid and
found the plastic viscosity tracked the behavior of the base oil. Hence, the
plastic viscosity of the oil-based drilling fluid is normalized with the viscosity of
the base oil. The plastic viscosity will be normalized with the viscosity of the
33
base fluid at reference conditions. The steps of this method are detailed as
follows:
1. Measure the plastic viscosity of the drilling fluid at reference conditions
(PV0).
2. Calculate the base oil viscosity at the reference conditions (µo) and at
the temperature and pressure conditions of interest (µT,P).
3. Calculate the plastic viscosity at the conditions of interest using the
following equation.
o
PToPT PVPV
µµ ,
, = (2.31)
Politte22 concluded that this procedure will be valid regardless of the type of
base oil used. He obtained the following equations from the analysis of diesel
viscosity data.
( ) ⎟⎠⎞⎜
⎝⎛ +++++
= ρρµ
111111
110GFPETPDTBACTPP (2.32)
1000 ≤ P ≤ 15000
75 ≤ T ≤ 300
(2.33) 222
22222 TFTEPDPCPTBA +++++=ρ
A1 = -23.1888 A2 = 0.8807
B1 = -0.00148 B2 = 1.5235*10-9
C1 = -0.9501 C2 = 1.2806*10-6
D1 = -1.9776*10-8 D2 = 1.0719*10-10
E1 = 3.3416*10-5 E2 = -0.00036
F1 = 14.6767 F2 = -5.1670*10-8
34
G1 = 10.9973
Where
µ = viscosity (cp)
ρ = density (lb/gal)
T = temperature (oF)
P = pressure (psi)
Further analysis with other oils by Politte led to the conclusion that Eqs. 2.32
and 2.33 are applicable for estimating the downhole plastic viscosity
regardless of the type of base oil used.
2.6.2 Temperature Dependent Yield point
Politte22 concluded from his analysis of rheological data for emulsions
that the yield point is not a strong function of pressure, and becomes
progressively less as temperature increases. The effects of temperature on
the yield point are, however, hard to predict, as there are chemical as well as
particle effects that have to be considered. Politte22 advises that in situations
where it may be important to know the precise behavior of the drilling fluid,
the yield point should be measured on a viscometer capable of such
measurements. If the equipment is not available, he provides the following
steps based on an empirical equation obtained from the analysis of diesel oil
based drilling fluid.
35
1. Measure the yield value at the reference conditions (YVTo).
2. Calculate the yield value of the drilling fluid at the temperature of
interest (YVT) using the following equation.
23
133
23
133
−−
−−
++++
=oo
yoy TCTBATCTBAττ (2.34)
90 T ≤ ≤ 300
A3 = -0.186
B3 = 145.054
C3 = -3410.322
Where
τy = yield point (lbf/100ft2)
T = temperature (oF)
Since yield value is dependent on the chemical attractions between the
particles present in the drilling fluid, Eq. 2.34 cannot be used to estimate the
yield value of drilling fluids that have base fluids of significantly different
chemistry from No. 2 diesel oil.
2.7 Bingham Plastic Pressure Loss Equations
In order to evaluate frictional pressure loss, it is first necessary to determine if
the flow is laminar or turbulent. The apparent viscosity of the fluid is first
calculated using the following equations.
36
Apparent Newtonian viscosity in pipes-
v
dypa
τµµ
66.6+= (2.35)
Apparent Newtonian viscosity in the annulus-
vdey
pa
τµµ
5+= (2.36)
where
µa = apparent viscosity (cp) d = pipe Diameter (in)
µp = plastic viscosity (cp) de = equivalent annular
τy = yield point (lbf/100ft2) diameter (in)
v = average fluid velocity
The apparent viscosity is then used in place of the Newtonian viscosity in
order to calculate the Reynolds number according to the following equation.
a
dvNµρ928
Re = (2.37)
where
ρ = fluid density (lb/gal)
d = pipe diameter or equivalent annular diameter (in)
v = average fluid velocity (ft/s)
aµ = apparent viscosity (cp)
A Reynolds number greater than 2100 indicates turbulent flow. Depending on
the flow regime, the frictional pressure drop can be calculated using the
following equations.
37
Laminar Flow
Pipe
Ldd
vP yp
f ∆⎟⎟⎠
⎞⎜⎜⎝
⎛+=∆
2251500 2
τµ (2.38)
Annulus
( ) ( ) Ldddd
vP yp
f ∆⎟⎟⎠
⎞⎜⎜⎝
⎛
−+
−=∆
122
12 2001000τµ
(2.39)
Turbulent Flow
( ) 395.0log41Re10 −= fN
f (Colebrook Equation) (2.40)
where p
dvNµρ928
Re = (2.41)
∆PF = Frictional pressure drop (psi)
∆L = Flow conduit length (ft)
ρ = fluid density (lb/gal)
d = pipe diameter or equivalent annular diameter (in)
v = average fluid velocity (ft/s)
pµ = plastic viscosity (cp)
d1 = inner annular wall (in)
d2 = outer annular wall (in)
If the flow regime is turbulent, once the friction factor has been obtained, the
frictional pressure drop can be found with Eq. 2.22.
38
2.8 Summary
For the purposes of this study, the compositional model for
characterizing the volumetric behavior of drilling fluids as expressed in Eq.
2.12 will be applied in conjunction with Eqs. 2.14 and 2.33 to express the
behavior of the major fluid constituents. In order to characterize the flow
behavior of the drilling fluid under high temperature-high pressure conditions,
the Bingham plastic model with temperature/pressure dependent model
parameters will be applied. The Bingham plastic model was chosen because
it is the most commonly used rheological model on the oil field and models
the behavior of a wide variety of fluids.
39
Chapter 3
DRILLING FLUID TEMPERATURE PROFILE ESTIMATION
As fluid flows in the wellbore, it absorbs heat from the formation,
causing a rise in its temperature. This rise in temperature in turn can lead to
changes in the fluid’s volumetric and rheological behavior, and thus the
frictional pressure drop. This effect is more pronounced in deep high
temperature wells and fluids with temperature sensitive rheological
properties11. Estimation of fluid temperature in the drill pipe and the annulus is
thus necessary in order to calculate the frictional pressure drop for a number
of well construction operations.
Fluid temperature within the wellbore will vary with depth and time with
this variation being especially pronounced at early times when the
temperature within the wellbore has not stabilized appreciably. The
temperature profile within the wellbore can be estimated analytically, or
numerically. This chapter gives a description of the heat transfer processes
that take place in the wellbore and the methods for estimating the
temperature profile.
40
3.1 Heat Transfer in the Wellbore
Figure 3.1 shows a schematic of drilling fluid circulating in the wellbore
and the associated heat transfer process over a differential element of length
∆z. The figure shows heat flow from the formation into the annular section
through convection (qfa). This rate of heat flow by convection into the annulus
is much greater than the rate of heat conduction in the formation. This is due
to the relatively low heat conductivity of the formation. This fact will be
important when modeling the heat transfer process in the wellbore. The fluid
within the drillpipe receives heat from the annulus via convection on the pipe
surface on the inside and outside of the drill pipe, and conduction through the
drillpipe itself (qap). There is heat flow in and out of the differential elements
within the drillpipe and annulus due to the bulk flow of fluid (qp(z), qp(z+Dz),
qa(z+Dz), qa(z) respectively).
Two methods have emerged for estimating the temperature profile in
the wellbore during circulation. They are the analytical method and the
numerical method. The analytical method entails solving the equations
governing heat transfer in the wellbore analytically, that is, assuming constant
fluid and formation properties. This method is best applied to systems of
simple geometry as in the case of a single casing string and inner drill pipe.
The numerical method uses a finite difference scheme to represent the
wellbore/formation system. Systems of great complexity can be better
handled using this method, and it has the added advantage of allowing
variable fluid and formation properties.
41
z
z +
qfa
qap
qa(z+∆
Annulus
Formation
qa(z)
qa(z+∆
qp(z)
qa(z+∆
Ta
Tp
qa(z)
Drill Pipe
Figure 3.1- Schematic of Heat Balance for Fluid Circulating in a Wellbore
42
3.2 Analytical Method
The temperature of the fluid within the drillpipe and the annulus is
described by two coupled ordinary differential equations. The temperature in
the formation is determined by the geothermal gradient coupled with the
transient formation heat conduction function, f(tD)13. The function accounts for
the un-steady state heat conduction in the formation. In order to solve these
equations, boundary conditions are required. The boundary conditions
applied are as follows.
Boundary Conditions
• The inlet fluid temperature coming into the drillpipe at the surface.
• The fluid temperature in the drillpipe and the annulus are equal at the
bottom-hole.
3.2.1 Assumptions of Analytical Model
The assumptions used in deriving and solving the equations governing
heat transfer within the wellbore are stated below16.
• The analytical method assumes constant fluid properties.
• Heat generated by viscous forces, friction, and changes in potential
energy are negligible.
• The formation is radially symmetric and infinite with respect to heat flow.
• Heat flow within the wellbore is rapid compared to heat flow within the
formation. Hence, heat flow within and across the wellbore conduits is
43
assumed to be steady-state, and heat flow within the formation is
assumed to be transient.
3.2.2 Heat Balance in the DrillPipe
Heat enters the differential element in the drillpipe from two sources;
bulk fluid flow qp(z), and from convection and conduction through the drillpipe
wall, qap. Heat leaves the differential element through bulk fluid flow qp(z+∆z).
The heat balance of the differential element in the drillpipe yields the following
equation
qp(z) + qap = qp(z+∆z) (3.1)
where,
qp(z) =zpflTmc (3.2)
qap = ( ) ( )( )dzzTzTUr papp −π2 (3.3)
qp(z+z)= zzpflTmc
∆+ (3.4)
to yield
( ) ( )( )zzpflpappzpfl TmcdzzTzTUrTmc
∆+=−+ π2 (3.5)
rearranging,
( ) ( )( 02 =−− zTzTUrz
Tmc papp
pfl π )
δδ
(3.6)
where,
m = mass flow rate of drilling fluid (lb/hr)
cfl = fluid Heat Capacity (Btu/lb-oF)
44
Tp = temperature of drillpipe fluid as a function of depth (oF)
Ta = temperature of annular fluid as a function of depth (oF)
rp = radius of drillpipe (ft)
Up = equivalent heat transfer coefficient across pipe wall
(Btu/hr-ft2-oF)
z = depth (ft)
3.2.3 Heat Balance in the Annulus
Heat enters the differential element in the annulus from the formation
by convection (qfa), and through bulk fluid flow (qa(z+∆z)). Heat leaves the
differential element through convection and conduction through the pipe wall
(q(ap)) and through bulk fluid flow (qa(z)). This process yields the following
equation.
qfa + qa(z+Dz) = qap + qa(z) (3.7)
where
qfa = ( ) ( )( dzzTzTUr aiaa )−π2 (3.8)
qa(z+Dz) = zzaflTmc
∆+ (3.9)
qap = ( ) ( )( )dzzTzTUr papp +π2 (3.10)
qa(z) = zaflTmc (3.11)
to yield
( ) ( )( )zzaflaiaa TmcdzzTzTUr
∆++−π2
( ) ( )( )zaflpapp TmcdzzTzTUr ++= π2 (3.12)
45
rearranging,
( ) ( )( ) ( ) ( )( ) 022 =++−−zTmcdzzTzTUrdzzTzTUr a
flpappaiaa δδππ (3.13)
where
ra = radius of annulus (ft)
Ua = equivalent heat transfer coefficient across formation/annulus
interface (Btu/hr-ft2-oF)
Ti = temperature at interface between Formation and Annulus (oF)
3.2.4 Heat Flow in the Formation and System Heat Balance
The heat flow from the formation is given by the following equation.
( ) ( ) ( )( dzzTzTtfkq iFD
Ff −= )π2 (3.14)
where
kF = formation thermal conductivity (Btu/ft-oF-hour)
TF = temperature of Formation according to the undisturbed
geothermal gradient (oF)
f(tD) = dimensionless time function
The dimensionless time function16 is given as
( )( )
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−
=
tr
trEi
tfa
a
D
α
α
4exp
421
2
2
(3.15)
FF
F
ck
ρα = (3.16)
46
where
ρF = formation density (lb/gal)
cF = formation heat capacity (Btu/lb-oF)
Ei = exponential Integral function
It can be observed that the heat flow from the formation should be equal to
the heat flow into the annulus by convection. Thus, qf = qfa. The temperature
of the interface between the formation and the annulus can thus be eliminated
as follows.
( ) ( ) ( )( ) ( ) ( )( )dzzTzTUrdzzTzTtfk
aiaaiFD
F −=− ππ 22
rearranging,
( ) ( )
( ) ( )aaD
F
aaFD
F
i
Urtfk
TaUrTtfk
Tππ
ππ
22
22
+⎟⎠⎞⎜
⎝⎛
+⎟⎠⎞⎜
⎝⎛
= (3.17)
From Eq. 3.2,
pp
a TdzdT
T += β (3.18)
dzdT
dzTd
dzdT ppa += 2
2
β (3.19)
where,
pp
fl
Urmcπ
β2
= (3.20)
Inserting Eqs. 3.17, 3.18, and 3.19 into Eq. 3.13 will yield an equation that is
in terms of Tp alone. The equation is given below.
47
02
2
=+−− Fppp TT
dzdT
dzTd
βσβ (3.21)
where,
( )⎟⎟⎠
⎞⎜⎜⎝
⎛ +=
Faa
DaaFfl kUr
tfUrkmcπ
σ2
(3.22)
The formation temperature is based on the geothermal gradient. Therefore,
TF = TFs + gGz (3.23)
Where,
TFs = surface formation temperature (oF)
gG = geothermal gradient (oF/ft)
Eq. 3.21 then becomes,
zgTTdzdT
dzTd
GFsppp −−=−− βσβ 2
2
(3.24)
The solution of the above ordinary differential equation is given below.
(3.25) ( ) GFsGzz
p gTzgeCeCzT βγγ −+++= 2121
where
αβ
αβββγ
242
1++
= (3.26)
αβ
αβββγ
242
2+−
= (3.27)
From Eq. 3.25,
( )
Gzzp geCeC
dzzdT
++= 212211
γγ γγ (3.28)
48
By inserting Eqs. 3.25 and 3.28 into Eq. 3.18, we obtain the following
equation for the temperature in the annulus.
( ) ( ) ( ) FsGzz
a TzgeCeCzT +++++= 2211 11 21 βγβγ γγ (3.29)
In order to obtain the constants C1 and C2, the following boundary conditions
are applied16.
@ z = 0 Tp(z) = Tps
@ z = L Tp(z) = Ta(z)
where,
Tps = fluid temperature at drillpipe inlet or at the surface (oF)
L = the total vertical depth of the well (ft)
By applying the boundary conditions, the following expressions are obtained
for the constants C1, and C2.
12
21 12
2
γγγ
γγ
γ
LLdiff
LG
eeTeg
C−
−= (3.30)
12
12 12
1
γγγγγ
γ
LLdiff
LG
eeTeg
C−
+−= (3.31)
where,
Tdiff = (TFs – Tps – βgG) (3.32)
49
3.3 Numerical Method
The numerical method involves solving the equations governing heat
flow in the wellbore and formation, using finite difference technique. Heat
transfer in the wellbore is assumed to be steady-state, while heat transfer
between the formation and annulus is treated as unsteady-state heat flow.
The solution grid used is shown in Fig. 3.2.
3.3.1 Equations Governing Heat transfer in the Wellbore and Formation
The equation of conservation of energy for a control volume inside the
drill pipe20 is given as
( ) ( ) ( )[ ] ( ) ( )t
tzTcr
ztzT
mctzTtzTtzUr pflp
pflpapp ∂
∂+
∂
∂=−
,,,,,2 2ρππ (3.33)
where
rp = radius of pipe (ft)
Up = heat Transfer coefficient across boundary layer on outer pipe
surface, pipe wall, and boundary layer on inner pipe surface.
(Btu/hr-ft2-oF)
Ta = temperature in the annulus (oF)
Tp = temperature in the drillpipe (oF)
m = mass Flow rate of drilling fluid (lb/hr)
cfl = heat capacity of drilling fluid (Btu/lb-oF)
50
r
z
Annuulus DrillPipe
Formation
Figure 3.2a- Solution grid for Finite Difference Analysis
q1 q2
Heat Flow by Convection
Heat Flow by Conduction
Control Volume in
the Annulus
Control Volume in
the
Figure 3.2b- Heat Flow at Formation Annulus Boundary
51
The equation for conservation of energy for a control volume inside the
annulus20 is given as follows:
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]tzTtzTtzUrtzTtzrTtzUr pappaafaa ,,,2,,,,2 −−− ππ
( ) ( ) ( )z
tzTcmt
tzTcrr afl
oa
flpa ∂∂
−∂
∂−=
,,22ρπ (3.34)
where
ra = annular radius (ft)
Ua = heat transfer coefficient across annulus/formation interface
(Btu/hr-ft2-oF)
TF = formation Temperature (oF)
The temperature in the formation is given by the following equation20.
( ) ( )t
trzTr
trzTrrr
FF
∂∂
=⎟⎠⎞
⎜⎝⎛
∂∂
∂∂ ,,1,,1
α (3.35)
where
α = formation transmissivity (kF/ρcF)
kF = formation conductivity
ρ = formation density
cF = formation heat transfer coeeficient
Note that heat flow in the formation is assumed to occur radially only. At the
boundary of the formation, heat exchange between the formation and annulus
is governed by Eq. 3.36. The equation is derived by performing a heat
balance about a sufficiently small control volume within the formation adjacent
to the annulus.
52
@ r = ra
( ) ( ) ( ) ( )⎥⎦⎤
⎢⎣⎡
∂∂
+−−r
trzTkrtzTtrzTtzUr aFFaaaFaa
,,2,,,,2 ππ
( )t
trzTcrr aFFa ∂
∂∆=
,,2 ρπ (3.36)
The first term on the left hand side of Eq. 3.36 represents the rate at which
heat is leaving the formation boundary by convection (q1 in Fig. 3.2b). The
second term on the left hand side of the equation represents the rate at which
heat enters the control volume by conduction (q2 in Fig. 3.2b). The right-hand
side of Eq. 3.36 represents the rate at which heat accumulates in or is lost
from the control volume at the formation boundary, leading to changes in
temperature.
3.3.2 Discretizing Heat Flow Equations for Finite difference Analysis
The solution of the equations governing heat transfer in the wellbore
and formation will be propagated across the formation and well bore using the
finite difference grid shown in Fig. 3.3. The solution will be advanced starting
at the outer boundary of the formation in the r-direction until the temperature
field is mapped for the entire formation cross-section at a particular time-step.
The temperature in the wellbore and formation are expressed as follows.
Tp(z, t) = Tp(i∆z, n∆t) = ( )nipT
Ta(z, t) = Ta(i∆z, n∆t) = ( )niaT
TF(z, r, t) = Ta(i∆z, j∆r, n∆t) = ( )njiFT ,
53
r
j = 0 j = 1 j = 2 i = 0
i = 1
i = 2
i = 3
z
FormatioWellbore
Figure 3.3- Finite Difference Grid
54
where,
i = depth coordinate j = radial coordinate n = time coordinate
Two discretization schemes were considered for the equations describing
heat flow in the wellbore (3.33 & 3.34). They were first discretized using
explicit finite differences as follows.
Eq. 3.33 can be expressed as
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )t
TTcr
zTT
cmTTTT
Urnip
nip
flp
nip
nip
fl
onip
nip
nia
nian
ipp ∆
++
∆
−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−
++
−−−
12111
222 ρππ
for i = 1,2,3,…,I-1,I n = 1,2,3,…N-1,N (3.37)
Parameters bearing coordinate n are known while parameters bearing
coordinate n+1, i.e. at the next time step, are not known. Equation 3.37 is
rearranged with the known parameters on right-hand side and the unknown
parameters on the left-hand side as follows.
( ) ( ) ( ) ( )nip
flp
ppfl
o
nia
flp
pnia
flp
pnip T
crz
UrzcmtT
crtU
Tcr
tUT
1211
−−+
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∆
⎟⎠⎞
⎜⎝⎛ ∆−∆
+∆
+∆
=ρπ
π
ρρ
( ) nip
flp
ppfl
o
Tcrz
Urzcmt
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
∆
⎟⎟⎠
⎞⎜⎜⎝
⎛∆+∆
−+2
1ρπ
π
(3.38)
Equation 3.38 in essence, expresses the temperature at the particular
coordinate of interest as a weighted average of temperatures at spatial
coordinates located nearby in the pipe and annulus, at the previous time-step.
55
Stability Criterion- The coefficients of the known pipe and annulus
temperatures (i.e. in the present time-step) in Eq. 3.38 must be positive in
order to arrive at a stable solution of the equation. A negative coefficient
would not make physical sense, because it would be saying that the hotter
the temperature is at a coordinate near or at the particular coordinate of
interest for the present time-step, the colder the temperature will be at the
particular coordinate of interest at the next or future time-step. Therefore, the
coefficients must all be greater than or equal to zero. In fact, it is usually
desirable to avoid zero coefficients. Application of this rule to Eq. 3.38, results
in the following constraints.
⎟⎠⎞
⎜⎝⎛ ∆+
∆≤∆
ppfl
oflp
Urzcm
crzt
π
ρπ 2
(3.39)
(3.40) ppfl
oUrzcm π∆≥
The equation describing heat flow in the annulus (Eq. 3.34) is discretized as
follows.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−
+−
⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−
+ −−−−
222
222 1111
nip
nip
nia
nian
ipp
nia
nia
niF
niFn
iaa
TTTTUr
TTTTUr ππ
( ) ( ) ( ) ( ) ( )
tTT
crrzTT
cmnia
nia
flpa
nia
nia
fl
o
∆+
−+∆−
−=+
−1
221 ρπ
for i = 1,2,3,…,I-1,I n = 1,2,3,…N-1,N (3.41)
Equation 3.41 is rearranged, with known parameters on the right hand side of
the equation and unknown parameters on the left hand side as follows.
56
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )nip
flpa
ppnip
flpa
ppniF
flpa
aaniF
flpa
aania T
crrtUr
Tcrr
tUrT
crrtUrT
crrtUrT 2212222122
11 −
∆+
−
∆+
−∆
+−
∆=
−−+− ρρρρ
( ) ( )nia
flpa
ppaafl
o
Tcrrz
UrzUrzcmt1221 −
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−∆
⎟⎠⎞
⎜⎝⎛ ∆+∆+∆
−+ρπ
ππ
( ) ( )nia
flpa
ppaafl
o
Tcrrz
UrzUrzcmt
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−∆
⎟⎠⎞
⎜⎝⎛ ∆−∆−∆
+ 22ρπ
ππ (3.42)
Stability Criterion- Application of the rule of positive coefficients to Eq. 3.42
results in the following constraints.
( )
⎟⎠⎞
⎜⎝⎛ ∆+∆+
−∆≤∆
ppaafl
oflpa
UrzUrzcm
crrzt
ππ
ρπ 22
(3.43)
(3.44) ppaafl
oUrzUrzcm ππ ∆+∆≥
Boundary conditions in the Wellbore- The following boundary conditions are
applied to the wellbore.
@ i = 0 (pipe inlet) ( ) psnip TT = for all n.
@ i = imax (bottom-hole) ( ) ( )nia
nip TT = for all n.
where
Tps = temperature of fluid at pipe inlet
The second boundary condition states that at the bottom of the well, the pipe
and annulus fluid temperatures are equal. Note that Eq. 3.38 is solved from
57
the surface to the bottom in the direction of fluid flow, while Eq. 3.42 is solved
from bottom-hole to surface, also in the direction of fluid flow.
The explicit method detailed above was tested and found to be too
slow, that is, it took a longer time to converge on an answer. This was largely
due to the constraints placed on the time step. An alternative discretizing
scheme, the Crank-Nicolson method26, was then used. This scheme yields an
efficient, easy to use finite difference scheme which gives more accurate
solutions without constraints on the time step used.
Using the Crank-Nicolson scheme, Eq. 3.33 is discretized as follows.
( ) ( ) ( ) ( ) ( ) ( ) ( ) nip
nia
nipp
nip
nia
nipp TTUrTTUr −−+− +++ θπθπ 122 111
( ) ( )( )
( ) ( ) ( ) ( )t
TTcr
zTT
zTT
cmnip
nip
flp
nip
nip
nip
nip
fl
o
∆
−+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∆
−−+
∆
−=
+
−+
+
−
+
+
1211
11
11
21
2ρπθθ (3.45)
Eq. 3.45 is rearranged as
( ) ( ) ( ) ( ) 11
112
11 2
22
+
+
+++
− ∆+⎟
⎟⎠
⎞⎜⎜⎝
⎛+
∆+
∆− n
ipfl
o
nip
nipp
flpnip
fl
o
Tz
cmTUr
tcr
Tz
cmθθπ
ρπθ
( )( ) ( ) ( ) ( ) ( )( )nip
fl
o
nip
nipp
flpnip
fl
o
Tz
cmTUr
tcr
Tz
cm1
2
11
2121
2 +−−
∆−⎟
⎟⎠
⎞⎜⎜⎝
⎛−+
∆+−
∆= θθπ
ρπθ
( ) ( )( ) ( ) ( ) 11212 +++−+ nia
nipp
nia
nipp TUrTUr θπθπ (3.46)
Note that in Eq. 3.46, all the terms on the left hand side are unknowns while
all the terms on the right-hand side are known except the last one which is the
temperature in the annulus at the depth of interest during the current time
step. This problem is solved by taking an initial guess of the temperature
profile in the annulus. This guess is the temperature profile at the previous
58
time-step. Hence, the last term on the right-hand side of Eq. 3.46 will be a
known term.
The equation describing heat flow in the annulus (Eq. 3.34) is
discretized using the Crank-Nicolson method as follows.
( ) ( ) ( ) ( ) ( ) ( ) ( ) nia
niF
niaa
nia
niF
niaa TTUrTTUr −−+− +++
0,11
0,1 122 θπθπ (3.47)
( ) ( ) ( ) ( ) ( ) ( ) ( ) nip
nia
nipp
nip
nia
nipp TTUrTTUr −−−−− +++ θπθπ 122 111
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∆−
−+∆−
−∆
−−= −+
+−
++
+
zTT
zTT
cmt
TTcrr
nia
nia
nia
nia
fl
onia
nia
flpa 21
211
11
11
122 θθρπ
Equation 3.47 is rearranged as
( ) ( ) ( ) ( ) ( ) ( ) 11
11122
11 2
222
++
++++− ∆
−⎟⎟⎠
⎞⎜⎜⎝
⎛++
∆
−+
∆nia
fl
o
nia
niaa
nipp
flpania
fl
o
Tz
cmTUrUr
tcrr
Tz
cmθθπθπ
ρπθ
( )( ) ( ) ( ) ( ) ( ) ( ) ( nia
niaa
nipp
flpania
fl
o
TUrUrt
crrT
zcm
⎟⎟⎠
⎞⎜⎜⎝
⎛−−−−
∆
−+−
∆−= − θπθπ
ρπθ 12121
21
22
1 )
( )( ) ( ) ( )( ) ( ) ( )( )niF
niaa
nip
nipp
nia
fl
o
TUrTUrTz
cm0,
11 12121
2θπθπθ −−−+−
∆− +
( ) ( ) ( ) ( ) 10,
111 22 ++++ ++ niF
niaa
nip
nipp TUrTUr θπθπ (3.48)
Note that all the terms on the left hand side of Eq. 3.48 are unknowns while
all the terms on the right hand side are known with the exception of the last
term. The sixth term is already known because the temperature profile in the
drill pipe at any given time-step is evaluated before the temperature profile in
the annulus. The problem of the last term is solved by making an initial guess
59
of the temperature profile in the immediate adjacent formation. The initial
guess is taken to be the temperature profile at the previous time step.
Boundary Conditions in the Wellbore - A heat balance is performed at the
bottom of the wellbore taking into account the bottom-hole boundary
condition. Figure 3.4 shows a diagram of the heat balance.
( )1max−ipT
( ) maxiT
imax
`
Figure 3.4- Heat Balance at Bottom-Hole
60
Note,
Timax = ( )maxipT = ( ) maxiaT
Since @ i = imax (Bottom-hole), ( )maxipT = ( ) maxiaT
Performing a heat balance about the bottom-hole volume element in the
wellbore yields the following equation.
( ) ( ) ( ) ( ) tTczrTTzUrTcmTcm flaiiFaaifl
o
ipfl
o
∂∂∆
=−∆
+−− 22
2 2maxmaxmax1max
ρππ (3.49)
Where T = Tp, Eq. 3.49 is discretized as follows.
( ) ( ) ( ) ( ) ( ) ( ) ( ) 1max
10max,max1max
1max
11max
1 ++
−
++
−−∆+−−+− n
ipniFaa
nip
nipfl
onip
nipfl
oTTzUrTTcmTTcm θπθθ
( ) ( ) ( ) ( ) ( )tTT
czrTTzUrnip
nip
flanip
niFaa ∆
−∆=−−∆+
+
max1
max2max0max, 2
1 ρπθπ (3.50)
Eq. 3.50 is rearranged as
( ) ( ) 1max
21
1max 2++
− ⎟⎟⎠
⎞⎜⎜⎝
⎛
∆
∆−−+ n
ipfla
fl
onipfl
oT
tzcr
cmTcmρπ
θθ
( )( ) ( ) ( ) ( ) 10max,max
2
1max 211 +
−∆−⎟
⎟⎠
⎞⎜⎜⎝
⎛
∆
∆−−−+−= n
iFaanip
flafl
onipfl
oTzUrT
tzcr
cmTcm θπρπ
θθ
( )( ) ( ) ( )( )niaaa
niaaa
niFaa TzUrTzUrTzUr max
1max0max, 11 θπθπθπ −∆−+∆+−∆− + (3.51)
Equation 3.46 in conjunction with Eq. 3.51, thus form a tridiagonal system of
equations. The system of equations is solved easily using the Thomas
algorithm26.
Where T = Ta, Eq. 3.49 is discretized as follows.
61
( ) ( ) ( ) ( ) ( ) ( ) ( ) 1max
10max,max1max
1max
11max
1 ++
−
++
−−∆+−−+− n
ianiFaa
nia
nipfl
onia
nipfl
oTTzUrTTcmTTcm θπθθ
( ) ( ) ( ) ( ) ( )t
TTczrTTzUr
nia
nia
flania
niFaa ∆
−∆=−−∆+
+max
1max2
max0max, 21 ρπθπ (3.52)
Eq. 3.52 is rearranged as follows
( ) ( ) ( )( )nipfl
onipfl
oniafl
o
aafla TcmTcmTcmzUr
tzcr
1max1
1max1
max
2
12 −
+
−
+ −+=⎟⎟⎠
⎞⎜⎜⎝
⎛+∆+
∆
∆θθθθπ
ρπ
( ) ( )( )niFaa
niFaa TzUrTzUr 0max,
10max, 1 θπθπ −∆+∆+ +
( ) ( ) ( )nia
flaaafl
oT
tzcr
zUrcm max
2
211 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
∆
∆+−∆−−−+
ρπθπθ (3.53)
Equation 3.47 is solved in conjunction with Eq. 3.53 to yield the temperature
profile in the annulus.
Heat Flow Equations in the Formation
Equation 3.35 governs heat flow in the formation and is discretized
using the Taylor series approximation of derivatives in conjunction with the
Crank-Nicolson method of finite differences. The Crank-Nicolson method
results in an implicit set of linear algebraic equations which must be solved
simultaneously. However, this method has the advantage of being
unconditionally stable. There is no constraint set on the size of the time-step
to be used. Equation 3.35 is thus discretized as follows.
62
( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( )( ) ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛
∆
+−+
∆
+−=
∆
− +−+
+++
−+
21,,
11,
2
11,
1,
11,,
1, 22
2 r
TTT
r
TTTt
TT njiF
njiF
njiF
njiF
njiF
njiF
njiF
njiF α
( ) ( ) ( ) ( )
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∆
−
∆+
∆
−
∆+ −+
+−
++
rTT
rjrTT
rj
njiF
njiF
njiF
njiF
21
21
21,1,
11,
11,α (3.54)
Rearranging Eq. 3.54 so that unknowns are on the left-hand side,
( ) ( )( ) ( ) nji
njiF
rrnjiFr
nijiF
rr Taj
aTaTaj
a,
11,
1,
1, 24124 φααααα =⎟
⎠⎞⎜
⎝⎛ −−+++⎟
⎠⎞⎜
⎝⎛ − +
+++
− (3.55)
where,
( ) ( )( ) ( )njiF
rrnjiFr
nijiF
rrnji Ta
jaTaTa
ja
1,,,, 24124 +− ⎟⎠⎞⎜
⎝⎛ ++−+⎟
⎠⎞⎜
⎝⎛ +−= αααααφ
( )2rtar ∆
∆=
Formation Boundary Conditions- At the formation and annulus interface, the
boundary condition can be expressed as follows.
aaFaF
F TUTUr
Tk =+∂
∂− (3.56)
Equation 3.47 can be expressed as follows.
( ) ( ) ( ) ( )n
iaaniFa
niF
niF
F TUTUrTT
k =+∆
−− −
0,1,1,
2 (3.57)
Rearranging Eq. 3.57 for time-steps n and n+1, we obtain the following.
( ) ( ) ( ) ( )niF
F
niF
niaa
niF T
krTTUT 1,0,1,
2+
∆−=− (3.58)
( ) ( ) ( ) ( ) 11,
10,
111,
2 ++++− +
∆−= n
iFF
niF
niaa
niF T
krTTUT (3.59)
63
Note that there in Eqs. 3.49 and 3.50, there is a node j = -1. This is an
imaginary node located outside of the formation in the annulus. Near the
boundary, the equation describing internal heat flow in the formation (Eq.
3.35) is replaced with the following equation26.
2
2
2rT
tT FF
∂∂
=∂
∂ α (3.60)
This is done to avoid the apparent singularity which occurs at the node j = 0.
Equation 3.60 can be expressed as
( ) ( )t
TT njiF
njiF
∆
−+,
1,
( ) ( ) ( )( )
( ) ( ) ( )( ) ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛
∆
+−+
∆
+−= +−
++
++−
21,,
11,
2
11,
1,
11, 22
22
rTTT
rTTT n
jiFn
jiFn
jiFn
jiFn
jiFn
jiFα (3.61)
rearranging at node j = 0, we obtain
( ) ( )( ) ( ) ni
niFr
niFr
niFr TaTaTa 0,
11,
10,
11, 21 φααα =−++− +++
− (3.62)
where
( ) ( )( ) ( )niFr
niFr
niiFr
ni TaTaTa 1,0,,0, 21 ααφ +−+= − (3.63)
The temperatures at the imaginary node j = -1 are eliminated from Eqs. 3.62
and 3.63 by inserting Eqs. 3.58 and 3.59 to obtain
( ) ( ) ni
niFr
niFa
Frr TaTU
kraa 0,
11,
10, 2221 φααα =−⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∆++ ++ (3.64)
64
where
( ) ( )niaa
Fr
niaa
Fr
ni TU
kraTU
kra ∆
+∆
= + 22 10, ααφ
( ) ( )niFr
niFa
Frr TaTU
kraa 1,0, 2221 ααα +⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∆+−+ (3.65)
Equations 3.64 and 3.65 express the temperature in the formation at the
formation-annulus interface. Note that in Eq. 3.65, the first term on the right
hand side contains the annulus temperature for the future time step. This
does not create a problem because at the time when the temperature in the
formation is evaluated, the temperature in the wellbore for the future time-step
has already been evaluated. Hence, the temperature in the annulus for the
future time step is already known. At the outer boundary of the formation, a
sufficient radius of consideration is chosen at which the temperature
disturbance caused by the flow of fluid in the wellbore is no longer felt in the
formation. This length is generally taken to be 10 ft from the wellbore14. At this
outer boundary, the temperature conforms to the undisturbed geothermal
gradient.
@ j = J for all n (3.66) ( ) GGFSn
jiF TzigTT =∆+= **,
Equations 3.55, 3.64, and 3.66 thus form a complete set of linear algebraic
equations which describe the temperature at every node in the formation.
They can be expressed in matrix form as follows.
(3.67) [ ] Ψ=Ω T
This is done at each depth and at each time-step n as follows:
65
[ ]
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=Ω
100000
00000001
KK
MM
MM
K
CBA
CBACBA
( )( )( )
( ) ⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
=
+
+
+
+
1,
12,
11,
10,
nJiF
niF
niF
niF
T
TTT
T
M
M
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
=Ψ
−
G
nJi
ni
ni
ni
T1,
2,
1,
0,
φ
φφφ
M
where
⎟⎠⎞⎜
⎝⎛ −= 24
rr aj
aA αα ( )raB α+= 1 ⎟⎠⎞⎜
⎝⎛ −−= 24
rr aj
aC αα
Equation 3.56 is a tridiagonal system of linear algebraic equations, and can
thus be solved using the simple but efficient Thomas Algorithm.
Numerical Procedure-
The following is a summary of the steps taken in the numerical solution.
1. The initial conditions of the system are specified (time t = 0). The initial
temperature conditions in the formation conform to the geothermal
gradient. The initial temperature conditions in the wellbore also
66
conform to the formation geothermal gradient. This condition is chosen
because it is found that after sufficient time during a trip, the
temperature of the fluid in the wellbore is equal to temperature of the
formation.
2. The temperature profile in the drill pipe is evaluated first using Eqs.
3.46 and 3.51. It is first necessary to guess the temperature profile in
the annulus at the current time step in order to evaluate the drill-pipe.
The initial guess is taken to be the temperature profile in the annulus at
the previous time step.
3. Based on the newly evaluated drill-pipe temperature, the annular
temperature profile is evaluated using Eqs. 3.48 and 3.53. Note that it
is necessary to guess the temperature profile in the immediate
adjacent formation at the current time-step. The guess chosen is the
temperature profile in the previous time step.
4. The temperature profile in the formation is then evaluated at the
current time step based on the newly evaluated annulus profile. The
results of the procedure are then compared with the initial guesses. If
the error is insignificant, the next time step is evaluated. If there is
significant error, the whole procedure is repeated with the current
temperature profiles in the annulus and formation being used as the
guesses. This procedure is repeated until the calculations are
completed for the total circulation time.
67
3.4 Summary
The analytical and numerical methods for estimating the temperature
profile within the wellbore and formation have been described. It is assumed
that heat flow in the wellbore occurs rapidly in comparison to heat flow in the
formation. Heat flow in the wellbore occurs by bulk fluid flow, convection
across surface films on the pipe conduit and on the outer wall of the annulus,
and conduction through the pipe wall. Heat flow in the formation occurs by
conduction in the radial direction only. For the numerical method, the
equations governing heat flow in the wellbore and formation are solved using
the Crank-Nicolson method which results in a system of linear algebraic
equations. The numerical method is used in the development of the Dynamic
density Simulator.
68
Chapter 4
DEVELOPMENT AND VALIDATION OF THE DYNAMIC DENSITY SIMULATOR AND MODELLING OF DYNAMIC DENSITY Drilling operations are being conducted to deeper depths as the need
to supplement dwindling hydrocarbon resources forces exploration into more
unconventional environments. Deeper wells mean the drilling fluid will
encounter higher temperatures and pressures. As discussed in previous
chapters, it is not expected that the volumetric and rheological properties of
the fluid remain constant under these conditions. It is thus necessary to
predict these downhole conditions and their impact on the drilling fluid
behavior. This allows for precise drilling fluid selection and preparation, and
accurate estimation of the maximum allowable pump pressure.
The following chapter contains a description, development, and
validation of the Dynamic Density Simulator (DDS) and analysis of the results
of equivalent circulating density estimation under high-temperature/high-
pressure conditions. The DDS program is a predictive tool that will allow the
drilling engineer to predict the down-hole temperature/pressure conditions
that will be encountered and the resultant change in drilling fluid rheological
behavior. The simulator was written using Visual Basic for applications
automated through Microsoft Excel. The user interface is integrated with
Excel and initiated with a command button that is integrated into the main
69
worksheet menu. This format was chosen because it allows ease of use and
accessibility. It also allows manipulation of generated results with Excel
utilities such as worksheets and graphs. The following is a detailed
description and explanation of the program layout.
4.1 Program Lay-Out
The program interface is executed with a series of user forms, which
will accept data pertaining to the well bore, drilling fluid, and formation
parameters and return the temperature profiles in the wellbore and formation,
pressure losses in the wellbore and the ECD of the circulating fluid. The user
can navigate between forms and input data at leisure using the “back” and
“next” buttons. Once all the parameter values have been entered into the
program, the results are displayed on a “results” form. The following is the
sequence of forms used in the program.
1. “frmStart”- This is the starting form and the form that is displayed
when the program is initiated. On this form, the option is given to initiate a
new well profile.
2. “frmWellProps”- This form allows input of the well bore parameters. These
include the total vertical depth of the well, drill string dimensions such as
inner and outer drill pipe diameters, drill bit dimensions, and information
such as the inlet pipe temperature and the circulation rate.
3. “frmMudProps”- This form accepts the mud parameters such as
rheological data, volumetric and constituents data, and thermal data.
70
4. “frmFormationProps”- This form allows input of the thermal properties of
the surrounding formation.
5. “frmHeatTransfer”- This form allows input of the heat transfer coefficients.
6. “frmResults” - This form displays the frictional pressure losses and ECD
results.
Figure 4.1 shows a step-by-step sequence of data entry, computation and
results display.
4.2 DDS Program Execution
The first form that is displayed once the program is executed is the
program title page. From this form, a new well profile can be initiated. Figure
4.2 shows a screen capture of the title form. The form is initiated by clicking
on a command button that is integrated into Excel’s set of main menu
commands as shown in the screen capture in Fig. 4.3. The following
sequence of forms will be described as follows.
4.2.1 General Well Parameters Form
On this form, the dimensions and configuration of the drill-string are
specified. These include the dimensions of the drill-pipe, heavy-weight drill-
pipe and drill collars. The bit size, circulation rate, inlet pipe temperature and
total vertical depth are also specified. A screen capture of this form is shown
in Fig. 4.4.
71
START (Initiate Well
Profile)
INPUT WELL PARAMETERS
INPUT MUD PARAMETERS
INPUT FORMATION PARAMETERS
Drill String Configuration/Geometry
Bit Configuration
TVD, Circulation Rate, Inlet Pipe Temperature
EVALUATE WELLBORE TEMPERATURE PROFILE
CALCULATE BOTTOM-HOLE PRESSURE & ECD
DISPLAY
INPUT HEAT TRANSFER COEFFICIENTS
STOP
Reference Conditions
Thermal Properties
Density and Constituents
Rheological Properties
Figure 4.1- DDSimulator Program Flow
Chart
72
Figure 4.2- Title Form
73
DDSimulator Command
Button
Figure 4.3- DDSimulator Launch Command Button
74
Figure 4.4- Well Parameters Form
75
Figure 4.5- Mud Properties Form
76
4.2.2 Mud Properties Form
The properties of the drilling fluid are entered on this form. These
include the rheological properties such as plastic viscosity and yield strength,
the density and constituents, and the thermal properties such as thermal
conductivity and heat capacity. The density and rheological parameters are
obtained at certain reference temperature and pressure. These reference
conditions are also entered. Figure 4.5 shows a screen capture of the form.
4.2.3 Formation Properties Form
The properties of the formation are entered on this form. These
properties include the density, geothermal gradient, surface temperature, heat
conductivity, specific heat capacity, and maximum radius of interest. The
maximum radius of interest refers to the minimum radius at which the
formation no longer sees the temperature disturbance as a result of
introducing the drilling fluid into the well. This radius is usually about 10-ft
from the well-bore. Figure 4.6 shows a screen capture of the formation
properties form.
4.2.4 Heat Transfer Coefficients Form
The overall heat transfer coefficients for heat transfer across the
annulus-formation interface, and across the drill-pipe wall are entered on this
form. Figure 4.7 shows a screen capture of the heat transfer coefficients form.
77
Figure 4.6- Formation Properties Form
78
Figure 4.7- Heat Transfer Coefficients Form
79
4.2.5 Results and Results Form
Once all the system parameters have been specified, the simulator can
then evaluate the temperature profile inside the drill-pipe and annulus and the
resultant variation in density and rheological parameters. The frictional
pressure drop in the annulus and drill-pipe, the bottom-hole pressure and the
ECD are then determined. The results are presented on the result form and
temperature profile is presented graphically using the Excel graph feature.
The result form is shown in Fig. 4.8 and a sample temperature profile is
shown in Fig. 4.9.
Figure 4.8- Results Form
80
Figure 4.9- A Sample Temperature Profile Using Excel Graph Feature
81
4.3 Equations used in DDSimulator Program
The DDS program uses the numerical method to evaluate the
temperature profile in the wellbore and near formation environment. Taking
the temperature profile into account, the simulator computes equivalent
hydrostatic head and frictional pressure loss in the wellbore during circulation.
The equations used in the simulator are as follows:
4.3.1 Fluid Properties
The density of the fluid in the wellbore is computed using the
compositional method according to the following equation.
( )⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+
=111
,
2
1
2
1
122
w
ww
o
oo
mm
ffTp
ρρ
ρρ
ρρ (2.12)
The volumetric behavior of the oil component is calculated as follows:
(2.33) 222
22222 TFTEPDPCPTBAo +++++=ρ
where
A2 = 0.8807 B2 = 1.5235*10-9
C2 = 1.2806*10-6 D2 = 1.0719*10-10
E2 = -0.00036 F2 = -5.1670*10-8
The volumetric behavior of the water component is calculated as follows:
ρw = Bo + B1(T) + B2(p-po) (2.14)
where
Bo = 8.63186 B1 = -3.31977 * 10-3 B2 = 2.37170 * 10-5
82
The density of the fluid in the annulus is computed for 200 discrete lengths of
the wellbore, that is, the total vertical depth is divided into 200 discrete
lengths. The temperature dependent plastic viscosity of the oil component of
the fluid in the well bore is calculated according to the following equation.
( ) ⎟⎠⎞⎜
⎝⎛ +++++
= ρρµ
111111
110GFPETPDTBACTPP (2.32)
1000 ≤ P ≤ 15000
75 ≤ T ≤ 300
where
A1 = -23.1888 B1 = -0.00148 C1 = -0.9501
D1 = -1.9776*10-8 E1 = 3.3416*10-5 F1 = 14.6767
G1 = 10.9973
The steps used in the DDSimulator to calculate the plastic viscosity and yield
value of the drilling fluid are detailed in Sections 2.6.1 and 2.6.2 of Chapter 2.
The apparent viscosity of the drilling fluid is calculated according to Eqs. 2.36
and 2.36. The apparent viscosity is calculated for 200 discrete lengths of the
drill pipe.
4.3.2 Temperature Profile Estimation
The temperature profiles in the drill pipe and annulus are evaluated
implicitly using the Crank-Nicolson method. This discretizing scheme was
chosen because it is an efficient, easy to use scheme that allows for accurate
solutions without constraints on the time step used. The equations and
solutions steps used are detailed in Chapter 3.
83
4.3.3 Equivalent Hydrostatic Head and ECD
The hydrostatic head for each of the discrete sections of the wellbore
for which the density is known is computed according to Eq. 2.1. In order to
compute the frictional pressure loss, the flow regime must first be known.
Thus, the Reynolds number is first computed according to Eq. 2.37. If the flow
regime is laminar, the frictional pressure drop is computed using Eq. 2.38 or
2.39. If the flow regime is turbulent, the friction factor is computed according
to Eq. 2.40 and 2.41. The frictional pressure loss is then computed according
to Eq. 2.22. The equivalent circulating density is then calculated according to
Eq. 2.21.
4.4 Model Validation
As stated previously, the numerical method is applied in the
DDSimulator for temperature profile estimation in the wellbore. This method is
chosen because it can model more complex geometries than the analytical
method. In order to validate the temperature estimation capability of the
simulator, fluid circulation in a Gulf-Coast well was modeled. The well
parameters are as detailed in Table 4.1. The numerical results from the
DDSimulator were compared with the results obtained using the analytical
method. Fig. 4.10 shows a plot of the temperature profile in the pipe and
annulus obtained with the numerical and analytical methods.
84
Well Geometry Well Depth, ft 15000 Drill Stem OD, in. 6-5/8 Drill-Bit Size, in. 8-3/8 Circulation Rate, bbl/hour 300 Mud Properties Inlet Temperature, oF 75 Plastic Viscosity, cp 20.9 Yield Strength, lbf/100 ft2 35.3 Thermal conductivity, Btu/ft-oF-hour 1 Specific Heat, Btu/lb-oF 0.4 Density, lb/gal 10 Formation Properties Thermal conductivity, Btu/ft-oF-hour 1.3 Specific Heat, Btu/lb-oF 0.2 Density, lb/cu ft 165 Surface Earth Temperature, oF 59 Geothermal Gradient, oF/ft 0.0127
Table 4.1 WELL AND MUD CIRCULATING PROPERTIES FOR A GULF COAST WELL14
85
0
2000
4000
6000
8000
10000
12000
14000
16000
0 50 100 150 200 250 300
Temperature (oF)
Dep
th (f
t)
Pipe Numerical Annulus NumericalGeothermal Gradient Pipe AnalyticalAnnulus Analytical
Figure 4.10- Temperature Profile For
Gulf Coast Well
86
Figure 4.10 shows the good agreement for the temperature profile
between the numerical model and the analytical method. The maximum
deviation between the two methods was less than 2%. The predicted flowing
bottom-hole temperature in the well matched the observed flowing bottom-
hole temperature of 186 oF. Figure 4.10 also shows that the maximum
temperature in the well-bore may not occur at the bottom of the hole. As seen
in the figure, the maximum temperature in the well-bore for this particular
case occurs in the annulus several feet above the total vertical depth (TVD).
This agrees with observations made by several authors13-17.
The shape of the temperature profile occurs as a result of the heat flow
equilibrium attained by the fluid as it flows down the drill-pipe and up the
annulus. As the fluid flows down the drill-pipe, it gains heat from the annular
fluid thereby increasing in temperature until it reaches the bottom of the hole.
Once the fluid enters the annulus, it starts to lose heat to the relatively cooler
drill pipe. However, for a certain length in the annulus, the formation is still
hotter and some heat is lost to the annular fluid. The annular fluid thus
increases in temperature, until the heat lost to the drill-pipe is greater than the
heat gained from the formation or the annular fluid temperature is actually
higher than the formation temperature, whichever occurs first. This process
results in the highest well-bore temperature occurring in the annulus some
length above the TVD, and the unique shape of the temperature profile.
The numerical model was also used to simulate reported field data.
The following data was obtained from a well in Matagorda County, Texas23.
87
The well had been drilled and cased with 5-1/2-in., 17 lbf/ft casing, and 2-1/2-
in. tubing was set without a packer at 8650-ft. Tests were conducted on the
well measuring bottom-hole circulating temperature while circulating field salt
water at 84 and 252 gal/min. The initial undisturbed static bottom-hole
temperature was 250 oF, with a temperature gradient of 2.03 oF/100 ft. The
well was circulated at 84 gal/min for 2 hours and 40 minutes. The bottom-hole
temperature of the fluid dropped to 213 oF. The well was then logged and
circulated again at 252 gal/min. The bottom-hole temperature at the beginning
of the second circulation period was 224 oF. The well was circulated for 56
minutes, at the end of which the bottom-hole temperature dropped further to
196 oF. These conditions were simulated using the numerical method, and
the results are shown in Figs. 4.11 and 4.12.
The bottom-hole temperature at the end of first circulation period was
estimated to be 230 oF. This value has an 8% difference relative to the actual
measured value of 213 oF. The estimated bottom-hole temperature at the end
of the second circulation period was 199 oF. This is a 1.5 % difference from
the actual measured value of 196 oF.
88
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0 50 100 150 200 250 300
Depth (ft)
Tem
pera
ture
(o F)
Pipe Numerical Annulus Numerical Geothermal Gradient
Circulation Rate- 84 gal/min Circulation Time- 2 hrs 40 min Fluid- Field Salt Water
Temperature (oF)
Figure 4.11- Well Temperature Profile While Circulating Field Salt Water
89
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0 50 100 150 200 250 300
Depth (ft)
Tem
pera
ture
(o F)
Pipe Numerical Annulus Numerical Geothermal Gradient
Circulation Rate- 252 gal/min Circulation Time- 56 min Fluid- Field Salt Water
Temperature
Figure 4.12- Temperature Profile For Gulf Coast Well
90
4.5 Dynamic Density Estimation
The effects of temperature and pressure on the equivalent circulating
density in a high temperature-high pressure well were simulated using the
DDSimulator. The properties of the first well that was simulated are detailed
in Table 4.2. The temperature profile for the well is shown in Fig. 4.13. The
temperature profile indicates that the temperature in the well-bore is higher
than the formation temperature for a large portion of the hole. As fluid moves
from the bottom of the hole upwards in the annulus, it looses heat to the pipe,
and for about 2000 ft up the annulus. It also gains heat from the formation.
Beyond this point, the annular temperature is higher than the formation
temperature. Thus, heat moves from the annulus into the formation as well as
into the drill-pipe. The rate of heat transfer across the pipe wall is very high
due to the high heat conductivity of steel and the high flow rate. Hence, the
temperature profiles in the annulus and drill pipe are very close.
The results of the ECD calculations are detailed in Table 4.3. The ECD
based on constant fluid properties (i.e. independent of the
temperature/pressure conditions) was evaluated for comparison. The bottom-
hole pressure taking into account the temperature-pressure dependence of
the fluid properties was 218 psi lower than the bottom-hole pressure obtained
using constant fluid properties. This is due to the volumetric behavior of the
drilling fluid. The decrease in the density of the fluid due to temperature is
more pronounced than the increase in density due to the pressure. Hence,
91
the bottom-hole pressure is less than one would expect if the density of the
drilling fluid remained constant.
General well Properties Well Depth (L) 17200 ft Outer Drill Pipe Radius (rp) 0.208333 ft Annulus Radius (ra) 0.354167 ft Circulation Rate 400 bbl/hr Circulation Time (hr) 5 hr
Inlet Mud Temperature (Tps) 120 oF Mud Properties Viscosity (µfl) (@ reference conditions) 50.82 lb/(ft-hour) Yield Value (@ reference conditions) 10 lbf/100ft2
Thermal Conductivity (kfl) 1 Btu/(ft-oF-hour) Specific Heat (cfl) 0.4 Btu/(lb-oF) Density (ρfl) (@ reference conditions) 16.8 lb/gal Oil Fraction 0.594 Water Fraction 0.066 Formation Properties
Thermal Conductivity (kF) 0.3 Btu/(ft-oF-hour) Specific Heat (cF) 0.21 Btu/(lb-oF) Density (ρF) 165 lb/ft3 Surface Earth Temperature (TFs) 70 oF
Geothermal Gradient (gG) 0.020 oF/ft
Temperature/Pressure
Dependent Constant Property Difference
Bottom Hole Pressure (psi) 22020 22238 -218
ECD (ppg) 24.6 24.9 -0.3
Table 4.3 Results of Well Simulation
Table 4.2 Simulated Well Conditions
92
Figure 4.14 shows a plot of the bottom-hole pressure versus depth. It
can be observed that there is a steady increase in pressure as the depth
increases with the final bottom-hole pressure obtained with constant fluid
density being higher than that obtained with a temperature/pressure
dependent fluid density. Figure 4.15 shows the temperature/pressure
dependent behavior of the fluid density. As depth increases the equivalent
circulating density continues to decrease as a result of the greater effect of
fluid expansion due to temperature, as opposed to compression due to the
increased down-hole pressure.
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 50 100 150 200 250 300 350 400 450
Temperature (oF)
Dep
th (f
t)
Pipe Numerical Annulus Numerical Geothermal Gradient
Circulation Rate- 400 gal/minCirculation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ft
Figure 4.13- Temperature Profile in 17200-ft well after 5 hrs
93
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 5000 10000 15000 20000 25000
Dep
th (f
t)
Annular Pressure (psi)
Variable Fluid Properties Constant Fluid Properties
Circulation Rate- 400 gal/minCirculation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ft
Annular Pressure (psi)
Figure 4.14- Annular Pressure Profile in 17200-ft well after 5 hrs
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5
Dep
th (f
t)
Equivalent Circulating Density (ppg)
Temperature/Pressure DependentConstant Fluid Properties
Circulation Rate- 400 gal/min Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ft
Equivalent Circulating Density (ppg)
Figure 4.15- Equivalent Circulating Density in 17200-ft well after 5 hrs
94
The well parameters detailed in Table 4.2 were simulated using water
based drilling fluid. The results are shown in Table 4.4, and in Figs. 4.16 to
4.18. The results showed a similar trend to that obtained with oil-based drilling
fluid with the ECD obtained taking pressure and temperature conditions into
account being lower than the ECD calculated assuming constant fluid
properties.
The effects of the temperature gradient of the formation were also
studied. The same well properties as detailed in Table 4.2 were used. The
results are displayed in Tables 4.5 and 4.6, and in Figs. 4.19 to 4.24. As seen
in Tables 4.5 and 4.6, increase in the geothermal gradient results in a larger
difference between, the bottom-hole pressure estimated taking into account
the temperature/pressure conditions, and the bottom-hole pressure estimated
assuming constant fluid density and viscosity with the constant property
bottom-hole pressure being higher. A geothermal gradient of 1.5 oF/100-ft
results in a difference of 110 psi while a geothermal gradient of 2.5 oF/100-ft
results in a difference of 325 psi. These results show the higher fluid
expansion that occurs with a higher geothermal gradient. This trend is also
displayed in Figs. 4.21 and 4.24. Failure to take this effect into account during
drill operations could lead to the occurrence of a kick and possibly a blow-out.
95
Table 4.4 Well Simulation Results for Parameters Detailed in Table 4.2 with Water-Based mud
Temperature/Pressure
Dependent Constant Property Difference
Bottom Hole Pressure (psi) 18585 18792 -207
ECD (ppg) 20.8 21.0 -0.2
Temperature/Pressure
Dependent Constant Property Difference
Bottom Hole Pressure (psi) 22128 22238 -110
ECD (ppg) 24.7 24.9 -0.2
Table 4.5 Well Simulation Results for Parameters Detailed in Table 4.2 with gG = 0.015 oF/ft
96
Temperature/Pressure
Dependent Constant Property Difference
Bottom Hole Pressure (psi) 21913 22238 -325
ECD (ppg) 24.5 24.9 -0.4
Table 4.6 Well Simulation Results for Parameters Detailed in Table 4.2 with gG = 0.025 oF/ft
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 50 100 150 200 250 300 350 400 450
Temperature (oF)
Dep
th (f
t)
Pipe Numerical Annulus Numerical Geothermal Gradient
Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- WBM Geothermal Grad- 0.02 oF/ft
Figure 4.16- Temperature Profile in 17200-ft well
after 5 hrs
97
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
20.5 21 21.5 22 22.5 23 23.5 24 24.5D
epth
(ft)
Equivalent Circulating Density (ppg)
Temperature/Pressure DependentConstant Fluid Properties
Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- WBM Geothermal Grad- 0.02 oF/ft
Equivalent Circulating Density (ppg)
Figure 4.17- Equivalent Circulating Density in 17200-ft well after 5 hrs
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 2000 4000 6000 4000 16000 18000 20000
Dep
th (f
t)
8000 10000 12000 1
Annular Pressure (psi)
Variable Fluid Properties Constant Fluid Properties
Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- WBM Geothermal Grad- 0.02 oF/ft
Annular Pressure (psi)
Figure 4.18- Annular Pressure Profile in 17200-ft well after 5 hrs
98
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 50 100 150 200 250 300 350
Temperature (oF)
Dep
th (f
t)
Pipe Numerical Annulus Numerical Geothermal Gradient
Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.015 oF/ft
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 5000 10000 15000 20000 25000
Dep
th (f
t)
Annular Pressure (psi)
Variable Fluid Properties Constant Fluid Properties
Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.015
Annular Pressure (psi)
Figure 4.19- Temperature Profile in 17200-ft well after 5 hrs
Figure 4.20- Annular Pressure Profile in 17200-ft well after 5 hrs
99
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
24.5 25 25.5 26 26.5 27 27.5 28 28.5D
epth
(ft)
Equivalent Circulating Density (ppg)
Temperature/Pressure DependentConstant Fluid Properties
Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.015
Equivalent Circulating Density (ppg)
Figure 4.21- Equivalent Circulating Density in 17200-ft well after 5 hrs
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 100 200 300 400 500 600
Temperature (oF)
Dep
th (f
t)
Pipe Numerical Annulus Numerical Geothermal Gradient
Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.025 oF/ft
Figure 4.22- Temperature Profile in 17200-ft well after 5 hrs
100
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 5000 10000 15000 20000 25000
Dep
th (f
t)
Annular Pressure (psi)
Variable Fluid Properties Constant Fluid Properties
Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.025 oF/ft
Annular Pressure (psi)
Figure 4.23- Annular Pressure Profile in 17200-ft well after 5 hrs
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5
Equivalent Circulating Density (ppg)
Dep
th (f
t)
Temperature/Pressure DependentConstant Fluid Properties
Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.025 oF/ft
Equivalent Circulating Density (ppg)
Figure 4.24- Equivalent Circulating Density in 17200-ft well after 5 hrs
101
The effect of varying the inlet temperature of the drilling fluid was also
studied. The well parameters detailed in Table 4.2 were simulated with an
inlet temperature into the drill pipe of 80 oF. Although the return temperature
coming up the annulus was reduced to 87 oF, the bottom-hole temperature
and pressure after 5 hours of circulation did not change significantly from the
values obtained with an inlet temperature of 120 oF. The results are shown in
Table 4.7 and in Figs. 4.25 to 4.27. The ECD profile in the well during
circulation also did not change appreciably from the case of 120 oF pipe inlet
temperature. This trend indicates that the inlet temperature over a certain
range does not play an important role in the overall wellbore heat transfer
mechanism compared to the geothermal gradient.
Temperature/Pressure
Dependent Constant Property Difference
Bottom Hole Pressure (psi) 22029 22238 -209
ECD (ppg) 24.6 24.9 -0. 3
Table 4.7 Well Simulation Results for Parameters Detailed in Table 4.2 with Inlet Fluid Temperature = 80 oF
102
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 50 100 150 200 250 300 350 400 450
Temperature (oF)
Dep
th (f
t)
Pipe Numerical Annulus Numerical Geothermal Gradient
Circulation Rate- 400 bbl/hrCirculation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ftInlet Temp = 80 oF
Figure 4.25- Temperature Profile in 17200-ft well
after 5 hrs
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 5000 20000 25000
Dep
th (f
t)
10000 15000
Annular Pressure (psi)
Variable Fluid Properties Constant Fluid Properties
Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ft Inlet Temp = 80 oF
Annular Pressure (psi)
Figure 4.26- Annular Pressure Profile in 17200-ft
well after 5 hrs
103
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
24 24.5 25 25.5 26 26.5 27 27.5
Equivalent Circulating Density (ppg)28 28.5
Dep
th (f
t)
Temperature/Pressure DependentConstant Fluid Properties
Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ft Inlet Temp = 80 oF
Equivalent Circulating Density (ppg)
Figure 4.27- Equivalent Circulating Density in 17200-ft well after 5 hrs
The effects of the circulation rate on the bottom-hole pressure were
also studied. The well parameters in Table 4.2 were simulated with a
circulation rate of 300 bbl/hr. The results are shown in Table 4.8 and Figs.
4.28 to 4.30. The difference in the bottom-hole pressure estimated with
constant fluid properties and temperature-pressure dependent properties rose
to 297 psi. This could be due to the fact that the bottom-hole temperature
increases slightly at the lower rate and will thus result in increased expansion
of the drilling fluid and a greater reduction in the fluid density. The circulation
rate thus plays a great role in the temperature and pressure profiles that will
occur in a circulating well.
104
Table 4.8 Well Simulation Results for Parameters Detailed in Table 4.2 with Circulation Rate = 300 bbl/hr
Temperature/Pressure
Dependent Constant Property Difference
Bottom Hole Pressure (psi) 18791 19089 -298
ECD (ppg) 21.0 21.3 -0.3
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 50 100 150 200 250 300 350 400 450
Temperature (oF)
Dep
th (f
t)
Pipe Numerical Annulus Numerical Geothermal Gradient
Circulation Rate- 300 bbl/hrCirculation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ftInlet Temp = 120 oF
Figure 4.28- Temperature Profile in 17200-ft well after 5 hrs
105
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 5000 10000 15000 20000 25000
Dep
th (f
t)
Annular Pressure (psi)
Variable Fluid Properties Constant Fluid Properties
Circulation Rate- 300 bbl/hr Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ft Inlet Temp = 120 oF
Annular Pressure (psi)
Figure 4.29- Annular Pressure Profile in 17200-ft well after 5 hrs
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
20.5 21 24 24.5 25
Dep
th (f
t)
21.5 22 22.5 23 23.5
Equivalent Circulating Density (ppg)
Temperature/Pressure DependentConstant Fluid Properties
Circulation Rate- 300 bbl/hr Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ft Inlet Temp = 120 oF
Equivalent Circulating Density (ppg)
Figure 4.30- Equivalent Circulating Density in 17200-ft well after 5 hrs
106
Summary
A drilling hydraulic simulator called DDSimulator was developed using
Microsoft Visual Basic for Applications automated through Excel. This format
was chosen for ease of use and accessibility and access to Excel’s powerful
dynamic graphing capability. The simulator allows estimation of the
temperature profile in the wellbore during circulation and estimation of the
frictional pressure drop under high-temperature/high-pressure conditions. The
complete code for the DDS program is documented in the Appendix A.
High-temperature/high-pressure well conditions were simulated. It was
found that the bottom-pressure in the well is lower for the oil based mud that
was simulated when the temperature-pressure conditions prevalent in the
well-bore during circulation are taken into account. This indicates that the
effect of the fluid expansion due to temperature was more pronounced than
the effect of compression as a result of the increased pressure down-hole.
Temperature thus plays a more pronounced role in this particular case. This
is further confirmed by the further drop in bottom-hole pressure with
increasing geothermal gradient. The inlet temperature of the drilling fluid into
the drill-pipe was not found to have a significant effect on the bottom-hole
pressure even though it had an effect on the return temperature out of the
annulus. The circulation rate was found to play an important role in the
temperature profile that develops in a well during circulation.
107
Chapter 5
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS
5.1 Summary
The objective of this study was to explore the effects of the
temperature and pressure conditions prevalent in high temperature/high
pressure wells on the equivalent circulating density of the drilling fluid and on
the bottom-hole pressure. The high temperature conditions cause expansion
of the drilling fluid while the high pressure conditions at deeper depths cause
compression. In the industry, these two counter effects were thought to
cancel each other resulting in constant surface fluid density throughout the
length of the well-bore during circulation. However, industry experience has
shown that predicted bottom-hole pressures assuming constant surface fluid
properties are often in error by hundreds of psi.
The above objective was achieved with the development of a simulator
called DDSimulator. This is a hydraulic simulator that computes the bottom-
hole pressure and ECD throughout the length of a circulating well-bore taking
into account the temperature and pressure conditions in the well-bore. The
program can compute the temperature and pressure profile in a circulating
well. The effects of temperature and pressure on the density and viscosity of
drilling fluids was studied in chapter 2, along with frictional pressure loss
108
estimation. Analytical and numerical methods for estimating the temperature
profile in a circulating well-bore were studied in chapter 3. The Crank-
Nicolson numerical discretizing scheme was employed in the DDSimulator for
the evaluation of the temperature profile in a circulating well. Taking the
temperature profile into account the program then estimates the frictional
pressure loss, equivalent circulating density, and bottom-hole pressure using
methods detailed in Chapter 2.
In the case of the oil based drilling fluid that was simulated, it was
found that the bottom-hole pressure estimated taking into account
temperature and pressure conditions, is lower than if the fluid properties are
taken to be independent of temperature and pressure. This indicates that the
temperature effect of fluid expansion is more pronounced than the
compression effect due to pressure. Thus, a reduction in the fluid density
occurs. It is also important to note that the increased temperature results in
lower fluid viscosity and thus lower frictional pressure drop.
5.2 Conclusions
Based on the simulations that were performed, the following
conclusions were drawn.
1. Temperature and pressure effects play an important role in the bottom-
hole pressure that will occur in deep hot wells.
2. Higher geothermal gradients lead to lower bottom-hole pressure.
109
3. The inlet pipe temperature does not have a significant effect on the
bottom-hole pressure.
4. Higher circulation rates result in lower bottom-hole temperature and
higher bottom-hole pressure.
5. The objectives of the study were achieved by developing the Dynamic
Density Simulator. The simulator allows evaluation of the bottom-hole
pressure and equivalent circulating density taking into account the
temperature and pressure conditions in the well-bore.
5.3 Recommendations
It is recommended that a similar set of circulating wellbore simulations as
detailed in this study be carried out before the commencement of drilling
operations where it is known that high pressure and temperature conditions
may be encountered. This will minize the occurrence of common drilling
problems such as
• Premature intake of formation fluid into the wellbore (kick).
• Formation damage
• Unnecessary trips
thereby reducing the total drilling cost.
The following areas have been identified for further improvement.
• Simulations should be carried out with more varieties of drilling fluids,
including synthetic oil based drilling fluids, and drilling fluids with
110
chemical additives such as surfactants, flocculants, and fluid loss
reducers. Some experimental work may be required, as data of the
volumetric behavior as well as rheological behavior of drilling fluid
components with respect to temperature and pressure are not
abundant in the literature.
• Simulations should be carried out under deep water conditions where
cold temperatures and multiple temperature gradients are
encountered.
111
NOMENCLATURE
Ao, A1, A2 = Empirically determined parameter in Sorrelle et al6 model
Af = Cross-sectional area
as = 0.8*10-4 oC-1, thermal expansivity of barite
Bo, B1, B2 = Empirically determined parameter in Sorrelle et al6 model
bs = -1.0*10-5 bar-1, compressibility of barite
D = Pipe diameter
De = Equivalent diameter
f = Friction factor
fx = volume fraction of component x
f vo, fvw, fvs, fvc = Fractional volume of oil, water, solid weighting material, and
chemical additives, respectively
h = height of fluid column, ft
k = consistency index
L = Conduit length
n = flow behavior index
P = pressure, psi
Pw = Wetted perimeter
P1, P2 = Pressure at reference and condition “2”
T 1, T2 = Temperature at reference and condition “2”
V = Total volume
Vo, Vw,Vs,Vc = Volume of oil, water, solids and chemical additives
112
Vx = volume of component x
W = Weight
YV = Yield value (lbf/100ft2)
∆Phydrostatic = Hydrostatic head of fluid column (psi)
∆Pfriction = Pressure drop due to friction in the drill string and annulus
(psi)
∆p = Frictional pressure loss
α = Ellis model parameter
γ& = shear rate
λ = time constant
µa = apparent viscosity
µ = viscosity
µo = low shear rate viscosity
µp = plastic viscosity
∞µ = viscosity at infinite shear
ρ = fluid density, lbm/gal (ppg)
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
ρo1, ρw1 = Density of oil and water phases at reference conditions (p1,
T1)
113
ρs, ρc = Density of solids content and chemical additives
ρecd = equivalent circulating density (lb/gal)
τ = shear stress
τo = yield stress
τ1/2 = shear stress @ µa = µ0/2
A1 to G1 = Empirical parameters in temperature/pressure dependent
diesel
viscosity equation
A2 to F2 = Empirical parameters in temperature/pressure dependent
diesel
density equation
A3,B3, C3 = Empirical parameters in temperature/pressure yield value
equation for oil based drilling fluid
114
REFERENCES
1. Davison, J.M., Clary, S., Saasen, A., Allouche, M., Bodin, D., Nguyen,
V.A.: “ Rheology of Various Drilling Fluid Systems Under Deepwater
Drilling Conditions and the Importance of Accurate Predictions of
Downhole Fluid Hydraulics”, SPE 56632, Houston, Oct 3-6, 1999.
2. Houwen, O.H., Geehan, T.: “ Rheology of Oil-Base Muds”, SPE 15416,
New Orleans, LA, 5-8 Oct, 1986.
3. Alderman, N.J., Gavignet, A., Guillot, D., Maitland, G.C.: “High-
Temperature, High-Pressure Rheology of Water-Base Muds”, SPE
18035, Houston, TX, 2-5 Oct, 1988.
4. Hoberock, L.L., Thomas, D.C., Nickens, H.V.: “Here’s How
Compressibility and Temperature Affect Bottom-Hole Mud Pressure”,
OGJ, Mar 22, 1982, p. 159.
5. Peters, E.J., Chenevert, M.E. and Zhang, C.: “A Model for Predicting
the Density of Oil-Based Muds at High Pressures and Temperatures”,
SPEDE (June 1990) 141-148; Trans., AIME, 289.
6. Sorelle, R.R., Jardiolin, R.A., Buckley, P., Barios, J.R.: “Mathematical
Field Model Predicts Downhole Density Changes in Static Drilling
Fluids”, SPE 11118, New Orleans, Sept 26-29, 1982.
7. Isambourg, P., Anfinsen, B.T., Marken, C.: “Volumetric Behavior of
Drilling Muds at High Pressure and High Temperature”, SPE 36830,
Milan, Italy, Oct 22-24, 1996.
115
8. Kutasov, I., and Sweetman, M.: “Method Predicts Equivalent Mud
Density”, OGJ, Sept 24, 2001, p. 57.
9. Babu D. R.: “Effects of P-ρ-T Behavior of Muds on Static Pressure
During Deep Well Drilling”, SPE 27419, SPEDC, June 1996, pp. 91-97.
10. McMordie Jr., W.C., Bland, R.G. and Hauser, J.M.: “Effect of
Temperature and Pressure on the Density of Drilling Fluids”, SPE
11114, New Orleans, Sept. 26-29, 1982.
11. Rommetveit, R., Bjorkevoll, K.S.: “Temperature and Pressure Effects
on Drilling Fluid Rheology and ECD in Very Deep Wells”, SPE 39282,
Bahrain, 23-25 Nov, 1997.
12. Baranthol, C., Alfenore, J., Cotterill, M.D., Poux-Guillaume, G.:
“Determination of Hydrostatic Pressure and Dynamic ECD by
Computer Models and Field Measurements on the Directional HPHT
Well 22130C-13”, SPE 29430, Amsterdam, 28 Feb-2 Mar, 1995.
13. Ramey, H.J., Jr: “Wellbore Heat Transimission,” JPT(April 1962) 427-
35
14. Holmes, C.S., Swift, S.C.: “Calculation of Circulating Mud
Temperatures,” JPT(May 1970) 670-74
15. Arnold, F.C.: “Temperature Profile During Heated Liquid Injection,” Int.
Comm. Heat Mass Transfer, Vol. 16, pp. 763-72.
16. Arnold, F.C.: “Temperature Variation in a Circulating Wellbore Fluid,”
Journal of Energy Resources, Vol. 112, pp. 79-83.
116
17. Kabir, C.S., Hasan, A.R., Kouba, G.E., Ameen, M.M.: “Determining
Circulating Fluid Temperature in Drilling, Workover, and Well-Control
Operations,” SPE 24581, Washington, DC, Oct 4-7, 1992.
18. Marshal, T.R., Lie, O.H.: “A Thermal Transient Model of Circulating
Wells: 1. Model Development,” SPE 24290, Stavanger, Norway, May
25-227, 1992.
19. Romero, J. and Touboul, E.: “Temperature Prediction for Deepwater
Wells: A Field Validated Methodology,” SPE 49056, New Orleans,
Sept. 27-30, 1998.
20. Chen, Z., Novotny, J.: “Accurate Prediction Wellbore Transient
Temperature Profile Under Mulitple Temperature Gradients: Finite
Difference Approach and Case History,” SPE 84583, Denver, Oct 5-8,
2003.
21. Kutasov, I.M.: “Water FV Factors at Higher Pressure and
Temperatures,” Oil & Gas J. (Mar, 20, 1989) 102-104.
22. Politte, M.D.: “Invert Oil Mud Rheology as a Function of Temperature,”
SPE13458, New Orleans, Mar 6-8, 1985.
23. Tragasser, A.F., Crawford, P.B., Horace, R.: “A Method for Calculating
Circulating Temperatures,” Journal of Petroleum Technology, Vol. 19,
pp. 1507-1512, 1967.
24. Raymond, L.R.: “Temperature Distribution in a Circulating Drilling
Fluid,” Journal of Petroleum Technology, Vol. 21, pp. 333-341, 1969.
117
25. Muneer, T., Kubie, J., and Grassie, T.:Heat Transfer-A Problem
Solving Approach, Taylor & Francis Group, New York and London,
2003; pg 231.
26. Ozisik, M.N.:Finite Difference Methods in Heat Transfer, CRC Press,
Boca Raton, AnnArbor, London, and Tokyo, 1994; pg 99-137.
27. Kraus, A.D., Aziz, A., and Welty, J.:Extended Surface Heat Transfer,
John Wiley & Sons, 2001; pg 165-172, 181-190.
28. Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids, Clarendon
Press, Oxford, 1986; pg 220-224.
29. Welty, J.R., Wicks, C.E., Wilson, R.E., Rorrer, G.: Fundamentals of
Momentum Heat and Mass Transfer, John Wiley & Sons, 2001.
30. Dusinbere, G.M.:Heat Transfer Calculations by Finite Differences,
International Textbook Company, Scranton, Pennsylvania, 1961; pg 8-
23, 103-106.
118
APPENDIX Code for DDSimulator Program
119
Option Explicit 'This is a temperature profile object in which the methods to compute the pipe and annular 'temperature profiles are contained Dim Tax As WellProfile 'The following list conatins the depth coordinates at which temperature will be computed Dim Depth(200) As Double Private Sub ConstPropPressureDrop_Click() 'This sub computes the frictional pressure drop in the pipe and annulus with constant 'fluid properties Dim PlasticP As Double Dim PlasticA As Double Dim YieldP As Double Dim YieldA As Double Dim vP As Double Dim vA As Double Dim muP As Double Dim muA As Double Dim ReP As Double Dim ReA As Double Dim DPP As Double Dim DPA As Double Dim DppTotal As Double Dim DpaTotal As Double Dim i As Integer Dim imax As Integer imax = Tax.iTotal DppTotal = 0 DpaTotal = 0 'Compute Frictional Pressure Drop For i = 0 To imax If i > 0 Then 'compute plastic viscosity in cp PlasticP = Tax.RefMudPlasticViscosity PlasticA = Tax.RefMudPlasticViscosity 'Compute yield point in lbf/100ft^2 YieldP = Tax.RefMudYieldValue
120
YieldA = Tax.RefMudYieldValue 'Compute velocity in pipe and annulus vP = 4 * Tax.mRate / (7.48 * 3.142 * ((2 * Tax.rpi) ^ 2) * Tax.RefMudDensity) vA = 4 * Tax.mRate / (7.48 * 3.142 * (Tax.de ^ 2) * Tax.RefMudDensity) 'Compute apparent viscosity in pipe and annulus muP = PlasticP + (6.66 * YieldP * 2 * Tax.rp / vP) muA = PlasticA + (5 * YieldA * Tax.de / vA) 'Compute Reynold's # in pipe and annulus ReP = Tax.ReynoldsNum(Tax.RefMudDensity, vP, (2 * Tax.rp), muP) ReA = Tax.ReynoldsNum(Tax.RefMudDensity, vA, Tax.de, muA) 'Compute frictional pressure drop DPP = Tax.PressureDrop(ReP, (2 * Tax.rp), 0.01, Tax.DeltaZ, Tax.RefMudDensity, vP) DPA = Tax.PressureDrop(ReA, Tax.de, 0.03, Tax.DeltaZ, Tax.RefMudDensity, vA) DppTotal = DppTotal + DPP DpaTotal = DpaTotal + DPA End If Worksheets("AnnulusTemperature").Cells((i + 2), 12).Value = DppTotal Worksheets("AnnulusTemperature").Cells((i + 2), 13).Value = DpaTotal Next i End Sub Public Sub LoadProperties_Click() Set Tax = New WellProfile Tax.AnalyticalConstants End Sub Private Sub ComputeTempProf_Click() Dim i As Integer 'The "Interval" refers to the distance between depths at which temperature will be computed
121
Dim Interval As Double Interval = Tax.TVD / 200 For i = 0 To 200 Depth(i) = i * Interval Tax.AnalyticalComputeTemp Depth(i) Cells(i + 16, 8).Value = Tax.TPipe Cells(i + 16, 9).Value = Tax.TAnnulus If i < 300 Then End If Next i End Sub Private Sub NumericalTemperature_Click() 'Note that terms bearing an "N" at the end signify data at the time step that 'is currently being evaluated. Dim FormTemperature() As Double Dim PipeTemperature() As Double Dim AnnTemperature() As Double Dim FormTemperatureN() As Double Dim PipeTemperatureN() As Double Dim AnnTemperatureN() As Double 'These matrices store the pressure profile with the pipe and annulus Dim PipePressure() As Double Dim AnnPressure() As Double Dim PipePressureN() As Double Dim AnnPressureN() As Double 'Dim Uaaa() As Double 'Dim Uppp() As Double 'mud density in the pipe and annulus Dim rhoPipe() As Double Dim rhoAnnulus() As Double 'i - (depth), j - (radius), n - (time)
122
Dim i As Integer Dim j As Integer Dim n As Integer Dim imax As Integer Dim jMax As Integer Dim nMax As Integer 'This array is used to store the old pipe temperature values Dim store1() As Double Dim store2() As Double 'heat transfer coefficient for inner pipe surface Dim hi As Double 'error between guess and solution Dim err As Double Dim check1 As Double Dim check2 As Double Dim check3 As Double 'Terms used to evaluate the frictional pressure loss Dim PlasticP As Double Dim PlasticA As Double Dim YieldP As Double Dim YieldA As Double Dim vP As Double Dim vA As Double Dim muP As Double Dim muA As Double Dim ReP As Double Dim ReA As Double Dim DPP As Double Dim DPA As Double Dim DppTotal As Double Dim DpaTotal As Double imax = Tax.iTotal jMax = Tax.jTotal nMax = Tax.nTotal ReDim FormTemperature(imax, jMax) ReDim PipeTemperature(imax) ReDim AnnTemperature(imax) ReDim PipePressure(imax)
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ReDim AnnPressure(imax) ReDim FormTemperatureN(imax, jMax) ReDim PipeTemperatureN(imax) ReDim AnnTemperatureN(imax) ReDim PipePressureN(imax) ReDim AnnPressureN(imax) ReDim rhoPipe(imax) ReDim rhoAnnulus(imax) 'ReDim Uaaa(iMax) 'ReDim Uppp(iMax) ReDim store1(imax) ReDim store2(imax) 'Worksheets.Add.Name = "PipeTemperature" 'Worksheets.Add.Name = "AnnulusTemperature" 'Worksheets.Add.Name = "FormationTemperature" 'Set initial conditions in the formation and annulus Tax.InitializeGrid FormTemperature(), PipeTemperature(), AnnTemperature(), PipePressure(), AnnPressure(), rhoPipe(), rhoAnnulus(), imax, jMax 'The following code computes temperature in the formation and wellbore For n = 1 To nMax 'Set initial guess for Ta(i,n+1), Ua(i,n+1), Tp(i,n+1), Up(i,n+1), Pp(i,n+1), and Pa(i,n+1) For i = 0 To imax AnnTemperatureN(i) = AnnTemperature(i) FormTemperatureN(i, 0) = FormTemperature(i, 0) 'PipePressureN(i) = PipePressure(i) AnnPressureN(i) = AnnPressure(i) 'rhoPipeN(i) = rhoPipe(i) 'rhoAnnulusN(i) = rhoAnnulus(i) Next i Do 'Evaluate the pipe Tax.EvaluatePipe FormTemperature(), FormTemperatureN(), PipeTemperature() _ , PipeTemperatureN(), PipePressure(), PipePressureN(), AnnTemperature() _
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, AnnTemperatureN(), AnnPressure(), AnnPressureN(), imax 'Re-evaluate Pp(i,n+1) with the newly obtained Tp(i,n+1) For i = 0 To imax If i = 0 Then PipePressureN(i) = 14.7 Else rhoPipe(i - 1) = Tax.MudDensity(PipeTemperatureN(i - 1), PipePressureN(i - 1)) PipePressureN(i) = PipePressureN(i - 1) + 0.052 * rhoPipe(i - 1) * Tax.DeltaZ End If Next i 'Evaluate the Annulus Tax.EvaluateAnnulus FormTemperature(), FormTemperatureN(), PipeTemperature() _ , PipeTemperatureN(), AnnTemperature(), AnnTemperatureN(), PipePressure() _ , PipePressureN(), AnnPressure(), AnnPressureN(), imax, store1() 'Re-evaluate Pa(i,n+1) with the newly obtained Ta(i,n+1) For i = 0 To imax If i = 0 Then AnnPressureN(i) = 14.7 Else rhoAnnulus(i - 1) = Tax.MudDensity(AnnTemperatureN(i - 1), AnnPressureN(i - 1)) AnnPressureN(i) = AnnPressureN(i - 1) + 0.052 * rhoAnnulus(i - 1) * Tax.DeltaZ End If Next i 'Evaluate the formation Tax.EvaluateFormation FormTemperature(), FormTemperatureN(), AnnTemperature() _ , AnnTemperatureN(), AnnPressure(), AnnPressureN(), imax, jMax, store2() err = 0 'check for convergence
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For i = 0 To imax check1 = ((AnnTemperatureN(i) - store1(i)) ^ 2) ^ (1 / 2) check2 = ((FormTemperatureN(i, 0) - store2(i)) ^ 2) ^ (1 / 2) check3 = ((PipeTemperatureN(imax) - AnnTemperatureN(imax)) ^ 2) ^ (1 / 2) If check1 > err Then err = check1 End If If check2 > err Then err = check2 End If If check3 > err Then 'err = check3 End If Next i Loop Until err <= 0.05 'update the temperature in the formation and wellbore For i = 0 To imax For j = 0 To jMax FormTemperature(i, j) = FormTemperatureN(i, j) Next j AnnTemperature(i) = AnnTemperatureN(i) PipeTemperature(i) = PipeTemperatureN(i) PipePressure(i) = PipePressureN(i) AnnPressure(i) = AnnPressureN(i) Next i 'Display results in formation For i = 0 To imax For j = 0 To jMax Worksheets("FormationTemperature").Cells((i + 2), (j + 2)).Value = FormTemperatureN(i, j) Next j Next i Next n 'Display results in wellbore For i = 0 To imax Worksheets("AnnulusTemperature").Cells((i + 2), 3).Value = AnnTemperatureN(i) Worksheets("AnnulusTemperature").Cells((i + 2), 2).Value = PipeTemperatureN(i)
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Next i DppTotal = 0 DpaTotal = 0 'Compute Frictional Pressure Drop For i = 0 To imax If i > 0 Then 'compute plastic viscosity in cp PlasticP = Tax.MudPlasticViscosity(PipeTemperatureN(i), PipePressureN(i)) PlasticA = Tax.MudPlasticViscosity(AnnTemperatureN(i), AnnPressureN(i)) 'Compute yield point in lbf/100ft^2 YieldP = Tax.MudYieldValue(PipeTemperatureN(i), PipePressureN(i)) YieldA = Tax.MudYieldValue(AnnTemperatureN(i), AnnPressureN(i)) 'Compute velocity in pipe and annulus vP = 4 * Tax.mRate / (7.48 * 3.142 * ((2 * Tax.rpi) ^ 2) * rhoPipe(i - 1)) vA = 4 * Tax.mRate / (7.48 * 3.142 * (Tax.de ^ 2) * rhoAnnulus(i - 1)) 'Compute apparent viscosity in pipe and annulus muP = PlasticP + (6.66 * YieldP * 2 * Tax.rp / vP) muA = PlasticA + (5 * YieldA * Tax.de / vA) 'Compute Reynold's # in pipe and annulus ReP = Tax.ReynoldsNum(rhoPipe(i - 1), vP, (2 * Tax.rp), muP) ReA = Tax.ReynoldsNum(rhoAnnulus(i - 1), vA, Tax.de, muA) 'Compute frictional pressure drop DPP = Tax.PressureDrop(ReP, (2 * Tax.rp), 0.01, Tax.DeltaZ, rhoPipe(i - 1), vP) DPA = Tax.PressureDrop(ReA, Tax.de, 0.03, Tax.DeltaZ, rhoAnnulus(i - 1), vA) DppTotal = DppTotal + DPP DpaTotal = DpaTotal + DPA End If Worksheets("AnnulusTemperature").Cells((i + 2), 4).Value = rhoPipe(i) Worksheets("AnnulusTemperature").Cells((i + 2), 5).Value = rhoAnnulus(i) Worksheets("AnnulusTemperature").Cells((i + 2), 6).Value = PipePressureN(i)
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Worksheets("AnnulusTemperature").Cells((i + 2), 7).Value = AnnPressureN(i) Worksheets("AnnulusTemperature").Cells((i + 2), 8).Value = DppTotal Worksheets("AnnulusTemperature").Cells((i + 2), 9).Value = DpaTotal Next i End Sub The following Code details the methods and characteristics of a well-bore profile object. Option Explicit 'Formation Properties Public kF As Double Public cF As Double Public FormationDensity As Double Public alpha As Double 'The maximum formation radius that will be considered (ft) 'a.k.a r-infinity Public rMax As Double 'Undisturbed formation temperature at the maximum depth considered Public Tmax As Double 'Mud Properties (densities in lb/gal) Public RefMudDensity As Double Public RefMudPlasticViscosity As Double Public RefMudYieldValue As Double Public OilFraction As Double Public WaterFraction As Double Public mRate As Double Public cfl As Double Public kfl As Double 'Reference mud conditions Public RefTemp As Double Public RefPress As Double 'annular radius Public ra As Double
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'outer pipe radius Public rp As Double 'inner pipe radius Public rpi As Double 'heat conduction coefficient of pipewall Public kp As Double 'heat conduction coefficient of cement Public kcement As Double 'equivalent diameter of the annulus Public de As Double 'Heat Transfer coefficients across pipewall and across annulus/formation interface 'in Btu/(hour-ft2-oF) Public Ua As Double Public Up As Double Public time As Double Public beta As Double Public gG As Double Public TVD As Double Public Tdiff As Double Public TFs As Double Public Tps As Double 'Analytical parameters Public DTime As Double Public DTimeFunc As Double Public sigma As Double Public gammaOne As Double Public gammaTwo As Double Public COne As Double Public CTwo As Double 'Temperature in the annulus and pipe (oF) terms used in analytical analysis Public TAnnulus As Double Public TPipe As Double 'time, depth and radius intervals Public DeltaT As Double Public DeltaZ As Double Public DeltaR As Double 'terms used in numerical analysis- see class initialize for definitions Public ar As Double
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Public az As Double 'Total number of depth, radial, and time steps Public iTotal As Double Public jTotal As Double Public nTotal As Double Private Sub Class_Initialize() mRate = Range("B20").Value cfl = Range("B11").Value kfl = Range("B10").Value kF = Range("B14").Value cF = Range("B15").Value FormationDensity = Range("B16").Value ra = Range("B5").Value rp = Range("B4").Value rpi = Range("B34").Value kp = Range("B25").Value Ua = Range("B24").Value Up = Range("B23").Value rMax = Range("B22").Value alpha = Range("B21").Value time = Val(Application.Worksheets("Sheet1").CirculationTime.Value) gG = Range("B18").Value TVD = Range("B3").Value TFs = Range("B17").Value Tps = Range("B7").Value kcement = 0.025 RefTemp = Range("B27").Value RefPress = Range("B28").Value RefMudDensity = Range("B29").Value RefMudPlasticViscosity = Range("B30").Value RefMudYieldValue = Range("B31").Value OilFraction = Range("B32").Value WaterFraction = Range("B33").Value de = 2 * (ra - rp) 'The following constants are declared for convenience beta = mRate * cfl / (2 * Pie * rpi * Up) Tdiff = TFs - Tps - beta * gG
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'Tdiff = Tps - TFs + beta * gG Tmax = TFs + gG * TVD 'numerical intervals DeltaT = 0.05 DeltaZ = TVD / 200 DeltaR = rMax / 100 'terms used in numerical analysis ar = DeltaT / (DeltaR ^ 2) az = DeltaT / (DeltaZ ^ 2) 'compute iTotal If ((TVD / DeltaZ) - Int(TVD / DeltaZ)) < 0.5 Then iTotal = Int(TVD / DeltaZ) Else nTotal = Int(TVD / DeltaZ) + 1 End If 'compute jTotal If ((rMax / DeltaR) - Int(rMax / DeltaR)) < 0.5 Then jTotal = Int(rMax / DeltaR) Else jTotal = Int(rMax / DeltaR) + 1 End If 'compute nTotal If ((time / DeltaT) - Int(time / DeltaT)) < 0.5 Then nTotal = Int(time / DeltaT) Else nTotal = Int(time / DeltaT) + 1 End If End Sub Private Function ExpInt(X As Double) As Double Rem C Rem DOUBLE PRECISION FUNCTION EXPIN(X) Rem C Rem C Rem C EXPIN is the function which computes Rem C the exponential integral. Rem C Rem C E1(x)=SIGMA(x---->infinity)(exp(-t)/t)dt Rem C Rem C
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Rem IMPLICIT REAL*8(A-H,O-Z) Rem COMMON AS,BS,S,ISTEP Rem C IF (X.LE.0) WRITE(10,10) X Dim Sign As Double Dim XX As Double Dim A1 As Double Dim A As Double Dim B As Double If (X <= 0#) Then Sign = -1# Else Sign = 1# Rem GoTo 2 End If XX = Abs(X) Rem C 10 FORMAT(//,2X,'X must be a positive number X= ',D19.8) If ((XX >= 0#) And (XX <= 1#)) Then GoTo 1 End If Rem C Rem C 1<=X<infinity. 5.1.56 Abramowitz & Stegun. Rem C A1 = XX ^ 4 A = A1 + 8.5733287401 * XX ^ 3 A = A + 18.059016973 * XX ^ 2 + 8.6347608925 * XX A = A + 0.2677737343 B = A1 + 9.5733223454 * XX ^ 3 B = B + 25.6329561486 * XX ^ 2 + 21.0996530827 * XX B = B + 3.9584969228 ExpInt = A / B / XX / Exp(XX) GoTo 2 Rem Return Rem C Rem C 0<=X<=1 5.1.53 Abramowitz & Stegun. Rem C 1 A = -0.57721566 + 0.99999193 * XX - 0.24991055 * XX ^ 2 A = A + 0.05519968 * XX ^ 3 - 0.00976004 * XX ^ 4 A = A + 0.00107857 * XX ^ 5
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ExpInt = A - Log(XX) / Log(Exp(1#)) Rem Return Rem End Rem C 2 ExpInt = Sign * ExpInt End Function Public Function Pie() As Double 'This function computes Pi Pie = Application.WorksheetFunction.Pi End Function Public Function Absolute(X As Double) As Double 'This function returns the absolute value Absolute = Application.WorksheetFunction.Abs(X) End Function Public Function NatLg(X As Double) As Double NatLg = Application.WorksheetFunction.Ln(X) End Function '---------------------------TRIDIAGONAL ALGORITHM Public Sub ThomasAlgorithm(A() As Double, B() As Double, C() As Double, D() As Double, X() As Double, n As Integer) Dim i As Integer For i = 1 To n B(i) = B(i) - A(i) * C(i - 1) / B(i - 1) D(i) = D(i) - A(i) * D(i - 1) / B(i - 1) Next i ' Back Substitution X(n) = D(n) / B(n) For i = n - 1 To 0 Step -1 X(i) = (D(i) - C(i) * X(i + 1)) / B(i) Next i End Sub
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Public Sub AnalyticalConstants() 'The following code will calculate the constants "sigma", "gammaOne", '"gammaTwo", "COne", and "CTwo" 'mRate = mass flow rate, cfl = fluid heat capacity, kF = formation 'conductivity, ra = annular radius, Ua = annular heat transfer coeff, 'aplha = k/(rho*cF), time = length of time of fluid circulation, 'beta = m*cfl/(2*Pi*rp*Up), gG = formatin temp grad. , TVD = Total 'vertical depth, Tdiff = (Tfs - Tps - beta*gG) 'Calculate dimensionless time "tD" 'DTime = (ra ^ 2) / (4 * alpha * time) DTime = (-1 * ra ^ 2) / (4 * alpha * time) 'Calculate dimensionless time function "f(tD)" DTimeFunc = 0.5 * ExpInt(DTime) / Exp(DTime) sigma = mRate * cfl * ((kF + ra * Ua * DTimeFunc) / (2 * Pie * ra * Ua * kF)) 'Compute "gammaOne" and "gammaTwo" gammaOne = (beta + ((beta ^ 2) + 4 * sigma * beta) ^ (1 / 2)) / (2 * sigma * beta) gammaTwo = (beta - ((beta ^ 2) + 4 * sigma * beta) ^ (1 / 2)) / (2 * sigma * beta) 'Compute "COne" and "CTwo" COne = (gG - (Exp(gammaTwo * TVD) * gammaTwo * Tdiff)) / (gammaTwo * Exp(gammaTwo * TVD) - gammaOne * Exp(gammaOne * TVD)) CTwo = (-1 * gG + (Exp(gammaOne * TVD) * gammaOne * Tdiff)) / (gammaTwo * Exp(gammaTwo * TVD) - gammaOne * Exp(gammaOne * TVD)) 'COne = (gG + (Exp(gammaTwo * TVD) * gammaTwo * Tdiff)) / (gammaTwo * Exp(gammaTwo * TVD) - gammaOne * Exp(gammaOne * TVD)) 'CTwo = (gG + (Exp(gammaOne * TVD) * gammaOne * Tdiff)) / (-gammaTwo * Exp(gammaTwo * TVD) + gammaOne * Exp(gammaOne * TVD)) End Sub Public Sub AnalyticalComputeTemp(z As Double) 'Computes temperature in the drill pipe and annulus at a particular depth TAnnulus = AnalyticalAnnularTemp(z) TPipe = AnalyticalPipeTemp(z)
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End Sub Private Function AnalyticalAnnularTemp(Depth As Double) As Double 'Computes the temperature in the annulus at a particular depth AnalyticalAnnularTemp = (1 + beta * gammaOne) * COne * Exp(gammaOne * Depth) + (1 + beta * gammaTwo) * CTwo * Exp(gammaTwo * Depth) + gG * Depth + TFs End Function Private Function AnalyticalPipeTemp(Depth As Double) As Double 'Computes the temperature inside the drill pipe at a particular depth AnalyticalPipeTemp = COne * Exp(gammaOne * Depth) + CTwo * Exp(gammaTwo * Depth) + gG * Depth + TFs - beta * gG End Function Public Function ReynoldsNum(rho As Double, v As Double, D As Double, mu As Double) As Double 'Computes the Reynold's number 'density (rho) is in lb/gal, velocity (v) is in ft/hr, equivalent diameter (D) is in ft 'apparent viscosity (mu) is in lb/ft-hr ReynoldsNum = (7.48 * rho * v * D) / mu End Function Private Function NusseltNum(Re As Double, Pr As Double, D As Double, L As Double) As Double 'This function computes the Nusselt number 'Re & Pr - dimensionless 'D - ft 'L - ft If L = 0 Then L = DeltaZ / 2 End If If Re < 2300 And L / D < (8 / (Re / Pr)) Then 'Seider and Tate(1936) correlation - Laminar NusseltNum = 1.86 * (Re * Pr * (D / L)) ^ (1 / 3) ElseIf Re < 2300 And L / D > (8 / (Re / Pr)) Then
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NusseltNum = 6.49 ElseIf ((2300 <= Re) And (Re <= 10000)) Then 'Hausen(1943) correlation - Transition NusseltNum = 0.116 * ((Re ^ (2 / 3)) - 125) * (Pr ^ (1 / 3)) * (1 + (D / L) ^ (2 / 3)) ElseIf Re > 10000 Then 'Seider & Tate(1936) correlation - Turbulent NusseltNum = 0.027 * (Re ^ 0.8) * (Pr ^ (1 / 3)) End If End Function Public Function FrictionFactor(Re, Roughness, Diameter) As Double 'The friction factor is computed using the Swamme and Jain correlation If Re < 2300 Then FrictionFactor = 64 / Re Else FrictionFactor = 1.325 / (NatLg((Roughness / Diameter) / 3.7) + (5.74 / (Re ^ 0.9))) ^ 2 End If End Function Public Function PressureDrop(Re As Double, Diameter As Double, Roughness As Double _ , L As Double, rho As Double, v As Double) As Double 'Re- Reynold's # 'D - diameter in question (ft) 'e - Pipe roughness (ft) Dim F As Double If Re < 2100 Then F = 64 / Re Else F = 1.325 / (NatLg((Roughness / Diameter) / 3.7) + (5.74 / (Re ^ 0.9))) ^ 2 'f = (0.79 * NatLg(Re) - 1.64) ^ (-2) End If PressureDrop = F * rho * (L / Diameter) * (v ^ 2) * 7.48 / (2 * 32.174 * (3600 ^ 2) * (12 ^ 2))
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End Function Private Function WaterDensity(T As Double, P As Double) As Double 'This function computes the water density at the given temperature and pressure 'The correlation used in this function was obtained from Sorelle (1982) 'This function gives water density in lb/gal 'P - psi, T - oF WaterDensity = 8.63186 + (-3.31977 * 10 ^ -3) * T + (2.3717 * 10 ^ -5) * P End Function Private Function OilDensity(T As Double, P As Double) As Double 'This function computes the oil density at the given temperature and pressure 'The correlation used in this function was obtained from Politte (1985) 'This function gives density in (g/cm^3) 'P - psi, T - oF OilDensity = 0.8807 + 1.5235 * (10 ^ -9) * P * T + 1.2806 * (10 ^ -6) * P _ + 1.0719 * (10 ^ -10) * (P ^ 2) + (-0.00036) * T _ + (-5.167 * 10 ^ -8) * T ^ 2 End Function Private Function OilViscosity(T As Double, P As Double) As Double 'This function computes the oil phase viscosity at the given temperature and pressure in "cp" 'P - psi, T - oF Dim A As Double Dim B As Double Dim C As Double Dim D As Double Dim E As Double Dim F As Double Dim G As Double A = -23.1888 B = -0.00148 C = -0.9501 D = -1.9776 * 10 ^ -8 E = 3.3416 * 10 ^ -5 F = 14.6767 G = 10.9973
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OilViscosity = P * ((T * P) ^ C) * 10 ^ (A + B * T + D * T * P + E * P + F * OilDensity(T, P) + G / OilDensity(T, P)) End Function Public Function MudDensity(T As Double, P As Double) As Double 'This function Computes the mud density in lb/gal at the given temperature 'and pressure using the compositional model 'P - psi, T - oF MudDensity = RefMudDensity / (1 + OilFraction * ((OilDensity(RefTemp, RefPress) / (OilDensity(T, P))) - 1) + WaterFraction * ((WaterDensity(RefTemp, RefPress) / WaterDensity(T, P)) - 1)) End Function Public Function MudPlasticViscosity(T As Double, P As Double) As Double 'This function computes the plastic viscosity of the mud at the given temperature and pressure 'in centipoise (cp) 'P - psi, T - oF MudPlasticViscosity = RefMudPlasticViscosity * OilViscosity(T, P) / OilViscosity(RefTemp, RefPress) End Function Public Function MudYieldValue(T As Double, P As Double) As Double 'This function computes the yield value at the given temperature and pressure 'in lbf/100ft^2 'P - psi, T - oF Dim A As Double Dim B As Double Dim C As Double A = -0.186 B = 145.054 C = -3410.322 If T >= 90 Then MudYieldValue = RefMudYieldValue * (A + B * (T ^ -1) + C * (T ^ -2)) / (A + B * (RefTemp ^ -1) + C * (RefTemp ^ -2)) Else MudYieldValue = RefMudYieldValue
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End If End Function Public Function ConvectiveHeatTransCoeff(T As Double, P As Double, L As Double _ , D1 As Double, D2 As Double) As Double 'This function computes the heat transfer coefficient 'T - oF, P - psi, L - ft, D - ft Dim rho As Double Dim mu As Double Dim velocity As Double Dim Re As Double Dim Pr As Double Dim Nu As Double 'Compute fluid density in lb/gal 'rho = MudDensity(T, P) rho = RefMudDensity 'Compute fluid viscosity in lb/ft-hr 'mu = 2.42 * MudPlasticViscosity(T, P) mu = 2.42 * RefMudPlasticViscosity 'Compute fluid velocity in ft/hr velocity = 4 * mRate / (7.48 * 3.142 * (D1 ^ 2) * rho) 'Compute Reynold's # Re = ReynoldsNum(rho, velocity, D1, mu) 'Compute Prandtl # Pr = mu * cfl / kfl 'Compute Nusselt # Nu = NusseltNum(Re, Pr, D1, L) 'Compute heat transfer coefficient ConvectiveHeatTransCoeff = Nu * kfl / D2 End Function Public Sub InitializeGrid(FormTemp() As Double, PipeTemp() As Double, AnnTemp() As Double, PipePress() As Double, Annpress() As Double, rhoP() As Double, rhoA() As Double, imax As Integer, jMax As Integer)
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Dim i As Integer Dim j As Integer Dim hi As Double 'Set initial conditions in the formation and annulus i.e. n = 0 PipePress(0) = 14.7 Annpress(0) = 14.7 For i = 0 To imax For j = 0 To jMax FormTemp(i, j) = TFs + gG * i * DeltaZ Next j PipeTemp(i) = TFs + gG * i * DeltaZ AnnTemp(i) = TFs + gG * i * DeltaZ 'Specific well initial conditions 'PipeTemp(i) = 134.7 + 0.0047 * i * DeltaZ 'AnnTemp(i) = 134.7 + 0.0047 * i * DeltaZ 'Set the initial hydrostatic pressure profile in the wellbore If i > 0 Then rhoP(i - 1) = MudDensity(PipeTemp(i - 1), PipePress(i - 1)) rhoA(i - 1) = MudDensity(AnnTemp(i - 1), Annpress(i - 1)) PipePress(i) = PipePress(i - 1) + 0.052 * rhoP(i - 1) * DeltaZ Annpress(i) = Annpress(i - 1) + 0.052 * rhoA(i - 1) * DeltaZ End If Next i End Sub Public Sub EvaluateFormation(FormTemp() As Double, FormTempN() As Double _ , AnnTemp() As Double, AnnTempN() As Double, Annpress() As Double, AnnpressN() As Double _ , imax As Integer, jMax As Integer, store() As Double) 'This sub evaluates temperature in the formation and at the formation boundary Dim A() As Double Dim B() As Double Dim C() As Double
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Dim D() As Double Dim X() As Double Dim E() As Double Dim F As Double Dim G() As Double Dim H As Double Dim K() As Double Dim L As Double Dim Q As Double Dim Uaa As Double Dim i As Integer Dim j As Integer ReDim A(jMax) ReDim B(jMax) ReDim C(jMax) ReDim D(jMax) ReDim X(jMax) ReDim E(jMax) ReDim G(jMax) ReDim K(jMax) 'Store the old guess for Tf(i,0,n+1) For i = 0 To imax store(i) = FormTempN(i, 0) Next i F = 1 + alpha * ar L = 1 - alpha * ar H = 1 + 2 * alpha * ar Q = 1 - 2 * alpha * ar For i = 0 To imax Uaa = ConvectiveHeatTransCoeff(AnnTempN(i), AnnpressN(i), ((100 - i) * DeltaZ), de _ , de) 'This is for a specific cased hole with cement Uaa = ((1 / Uaa) _ + (Pie * (4.892 ^ 2) * NatLg(5.5 / 4.892) / (2 * Pie * kp * DeltaZ)) _
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+ (Pie * (4.892 ^ 2) * NatLg(6.108 / 5.5) / (2 * Pie * kcement * DeltaZ))) ^ (-1) 'Uaa = Ua 'construct matrix equation for formation at particular depth coordinate i For j = 0 To jMax If j = 0 Then A(j) = 0 B(j) = H + alpha * ar * 2 * DeltaR * Uaa / kF C(j) = -2 * alpha * ar D(j) = (alpha * ar * 2 * DeltaR * Uaa / kF) * AnnTemp(i) _ + (Q - alpha * ar * 2 * DeltaR * Uaa / kF) * FormTemp(i, j) _ + (2 * alpha * ar) * FormTemp(i, (j + 1)) _ + (alpha * ar * 2 * DeltaR * Uaa / kF) * AnnTempN(i) ElseIf j = jMax Then 'A(j) = 0 'B(j) = 1 'C(j) = 0 'D(j) = TFs + (gG * i * DeltaZ) E(j) = ar * alpha * ((1 / (4 * j)) - 1 / 2) G(j) = ar * alpha * (-(1 / (4 * j)) - 1 / 2) K(j) = ar * alpha * (-(1 / (4 * j)) + 1 / 2) A(j) = E(j) B(j) = F C(j) = 0 D(j) = K(j) * FormTemp(i, (j - 1)) + L * FormTemp(i, j) _ + (-G(j)) * (TFs + (gG * i * DeltaZ)) - G(j) * (TFs + (gG * i * DeltaZ)) Else E(j) = ar * alpha * ((1 / (4 * j)) - 1 / 2) G(j) = ar * alpha * (-(1 / (4 * j)) - 1 / 2) K(j) = ar * alpha * (-(1 / (4 * j)) + 1 / 2) A(j) = E(j) B(j) = F C(j) = G(j) D(j) = K(j) * FormTemp(i, (j - 1)) + L * FormTemp(i, j) _ + (-G(j)) * FormTemp(i, (j + 1)) End If
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Next j ThomasAlgorithm A, B, C, D, X, jMax 'update the temperature in the formation grid For j = 0 To jMax FormTempN(i, j) = X(j) Next j Next i End Sub Public Sub EvaluateAnnulus(FormTemp() As Double, FormTempN() As Double _ , PipeTemp() As Double, PipeTempN() As Double, AnnTemp() As Double, AnnTempN() As Double _ , PipePress() As Double, PipePressN() As Double, Annpress() As Double _ , AnnpressN() As Double, imax As Integer, store() As Double) Dim A() As Double Dim B() As Double Dim C() As Double Dim D() As Double Dim X() As Double Dim E As Double Dim F As Double Dim G As Double Dim H As Double Dim L As Double Dim M As Double Dim i As Integer Dim UaaN As Double Dim UppN As Double Dim hoN As Double Dim hiN As Double Dim Uaa As Double Dim Upp As Double Dim ho As Double Dim hi As Double 'These parameters will be used to compute the bottom-hole
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'annular temperature Dim Re As Double Dim mi As Double Dim fa As Double Dim so As Double Dim theta As Double ReDim A(imax) ReDim B(imax) ReDim C(imax) ReDim D(imax) ReDim X(imax) 'Store the old guess for Ta(i,n+1) For i = 0 To imax store(i) = AnnTempN(i) Next i E = mRate * cfl / (2 * DeltaZ) 'The following computes the bottom-hole annular temperature theta = 2 / 3 UaaN = ConvectiveHeatTransCoeff(AnnTempN(imax), AnnpressN(imax), ((200 - imax) * DeltaZ) _ , de, de) 'This is for a specific cased hole UaaN = ((1 / UaaN) _ + (Pie * (4.892 ^ 2) * NatLg(5.5 / 4.892) / (2 * Pie * kp * DeltaZ)) _ + (Pie * (4.892 ^ 2) * NatLg(6.108 / 5.5) / (2 * Pie * kcement * DeltaZ))) ^ (-1) 'UaaN = Ua Re = mRate * cfl mi = Pie * ra * UaaN * DeltaZ fa = 7.48 * MudDensity(AnnTemp(imax), Annpress(imax)) * Pie * (ra ^ 2) * DeltaZ * cfl _
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/ (2 * DeltaT) 'fa = 7.48 * RefMudDensity * Pie * (ra ^ 2) * DeltaZ * cfl / (2 * DeltaT) so = Re * theta * PipeTempN(imax - 1) + Re * (1 - theta) * PipeTemp(imax - 1) _ + mi * theta * FormTempN(imax, 0) + mi * (1 - theta) * FormTemp(imax, 0) _ + (-Re * (1 - theta) - mi * (1 - theta) + fa) * AnnTemp(imax) AnnTempN(imax) = so / (Re * theta + mi * theta + fa) For i = 0 To (imax - 1) 'These are the heat transfer coefficients at the current time-step UaaN = ConvectiveHeatTransCoeff(AnnTempN(i), AnnpressN(i), ((200 - i) * DeltaZ) _ , de, de) 'This is for a specific cased hole UaaN = ((1 / UaaN) _ + (Pie * (4.892 ^ 2) * NatLg(5.5 / 4.892) / (2 * Pie * kp * DeltaZ)) _ + (Pie * (4.892 ^ 2) * NatLg(6.108 / 5.5) / (2 * Pie * kcement * DeltaZ))) ^ (-1) hoN = ConvectiveHeatTransCoeff(AnnTempN(i), AnnpressN(i), ((100 - i) * DeltaZ) _ , de, de) hiN = ConvectiveHeatTransCoeff(PipeTempN(i), PipePressN(i), (i * DeltaZ) _ , (2 * rpi), (2 * rpi)) UppN = ((1 / hiN) + (Pie * (rpi ^ 2) / (hoN * Pie * rp ^ 2)) _ + Pie * (rpi ^ 2) * NatLg(rp / rpi) / (2 * Pie * kp * DeltaZ)) ^ (-1) 'These are the heat transfer coefficients at the previous time-step Uaa = ConvectiveHeatTransCoeff(AnnTemp(i), Annpress(i) _ , ((200 - i) * DeltaZ), de, 2 * ra) 'This is for a specific cased hole Uaa = ((1 / Uaa) _ + (Pie * (4.892 ^ 2) * NatLg(5.5 / 4.892) / (2 * Pie * kp * DeltaZ)) _ + (Pie * (4.892 ^ 2) * NatLg(6.108 / 5.5) / (2 * Pie * kcement * DeltaZ))) ^ (-1)
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ho = ConvectiveHeatTransCoeff(AnnTemp(i), Annpress(i) _ , ((100 - i) * DeltaZ), de, 2 * rp) hi = ConvectiveHeatTransCoeff(PipeTemp(i), PipePress(i), (i * DeltaZ), 2 * rpi _ , 2 * rpi) Upp = ((1 / hi) + (Pie * (rpi ^ 2) / (ho * Pie * rp ^ 2)) _ + Pie * (rpi ^ 2) * NatLg(rp / rpi) / (2 * Pie * kp * DeltaZ)) ^ (-1) 'Uaa = Ua 'UaaN = Ua 'Upp = Up 'UppN = Up F = 2 * Pie * ra * UaaN G = 2 * Pie * rp * UppN H = 7.48 * MudDensity(AnnTempN(i), AnnpressN(i)) * Pie _ * ((ra ^ 2) - (rp ^ 2)) * cfl / (DeltaT) 'H = 7.48 * RefMudDensity * Pie * ((ra ^ 2) - (rp ^ 2)) * cfl / (DeltaT) L = 2 * Pie * ra * Uaa M = 2 * Pie * rp * Upp Select Case i Case 0 A(i) = 0 B(i) = F * theta + G * theta + H C(i) = -E * theta D(i) = -E * (1 - theta) * (AnnTemp(i) - 2) _ + (H - L * (1 - theta) - M * (1 - theta)) * AnnTemp(i) _ + E * (1 - theta) * AnnTemp(i + 1) + L * (1 - theta) * FormTemp(i, 0) _ + M * (1 - theta) * PipeTemp(i) + F * theta * FormTempN(i, 0) _ + G * theta * PipeTempN(i) - E * theta * (AnnTempN(i) - 1)
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Case (imax - 1) A(i) = E * theta B(i) = F * theta + G * theta + H C(i) = 0 D(i) = -E * (1 - theta) * AnnTemp(i - 1) _ + (H - L * (1 - theta) - M * (1 - theta)) * AnnTemp(i) _ + E * (1 - theta) * AnnTemp(i + 1) + L * (1 - theta) * FormTemp(i, 0) _ + M * (1 - theta) * PipeTemp(i) + F * theta * FormTempN(i, 0) _ + G * theta * PipeTempN(i) + E * theta * AnnTempN(imax) Case Else A(i) = E * theta B(i) = F * theta + G * theta + H C(i) = -E * theta D(i) = -E * (1 - theta) * AnnTemp(i - 1) _ + (H - L * (1 - theta) - M * (1 - theta)) * AnnTemp(i) _ + E * (1 - theta) * AnnTemp(i + 1) + L * (1 - theta) * FormTemp(i, 0) _ + M * (1 - theta) * PipeTemp(i) + F * theta * FormTempN(i, 0) _ + G * theta * PipeTempN(i) End Select Next i ThomasAlgorithm A, B, C, D, X, (imax - 1) 'Update annulus temperature in finite difference grid For i = 0 To (imax - 1) AnnTempN(i) = X(i) Next i End Sub Public Sub EvaluatePipe(FormTemp() As Double, FormTempN() As Double, PipeTemp() As Double _ , PipeTempN() As Double, PipePress() As Double, PipePressN() As Double _ , AnnTemp() As Double, AnnTempN() As Double, Annpress() As Double, AnnpressN() As Double _ , imax As Integer) Dim A() As Double Dim B() As Double Dim C() As Double Dim D() As Double Dim X() As Double
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Dim E As Double Dim F As Double Dim G As Double Dim H As Double Dim L As Double Dim M As Double Dim mi As Double 'This parameter is used to weight the numerical solution ' at the bottom of the hole. A value of half gives the 'Crank-Nicholoson scheme Dim theta As Double Dim i As Integer Dim Uaa As Double Dim Upp As Double Dim ho As Double Dim hi As Double ReDim A(imax) ReDim B(imax) ReDim C(imax) ReDim D(imax) ReDim X(imax) theta = 2 / 3 E = mRate * cfl / (2 * DeltaZ) 'Store the old guess for Tp(i,n+1) 'For i = 0 To iMax ' store(i) = PipeTemp(i, n) 'Next i 'This sets the inflow pipe temperature PipeTempN(0) = Tps For i = 1 To imax ho = ConvectiveHeatTransCoeff(AnnTemp(i), Annpress(i), (i * DeltaZ), de, de)
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hi = ConvectiveHeatTransCoeff(PipeTemp(i), PipePress(i), (i * DeltaZ), 2 * rpi _ , 2 * rpi) Upp = ((1 / hi) + (Pie * (rpi ^ 2) / (ho * Pie * rp ^ 2)) _ + Pie * (rpi ^ 2) * NatLg(rp / rpi) / (2 * Pie * kp * i * DeltaZ)) ^ (-1) 'Upp = Up F = 7.48 * MudDensity(PipeTempN(i), PipePressN(i)) * Pie * (rp ^ 2) * cfl _ / (DeltaT) 'F = 7.48 * RefMudDensity * Pie * (rp ^ 2) * cfl / (DeltaT) G = 2 * Pie * rp * Upp H = 2 * Pie * rp * Upp Select Case i Case 1 A(i - 1) = 0 B(i - 1) = F + G * theta C(i - 1) = E * theta D(i - 1) = E * (1 - theta) * PipeTemp(i - 1) + (F - H * (1 - theta)) * PipeTemp(i) _ - E * (1 - theta) * PipeTemp(i + 1) + H * (1 - theta) * AnnTemp(i) _ + G * theta * AnnTempN(i) + E * theta * PipeTempN(i - 1) Case imax Uaa = ConvectiveHeatTransCoeff(AnnTemp(i), Annpress(i) _ , ((200 - i) * DeltaZ), de, 2 * ra) 'This is for a specific cased hole Uaa = ((1 / Uaa) _ + (Pie * (4.892 ^ 2) * NatLg(5.5 / 4.892) / (2 * Pie * kp * DeltaZ)) _ + (Pie * (4.892 ^ 2) * NatLg(6.108 / 5.5) / (2 * Pie * kcement * DeltaZ))) ^ (-1) 'Uaa = Ua L = mRate * cfl
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M = 7.48 * MudDensity(PipeTempN(i), PipePressN(i)) * Pie * (ra ^ 2) * DeltaZ _ * cfl / 2 * DeltaT 'M = 7.48 * RefMudDensity * Pie * (ra ^ 2) * DeltaZ * cfl / 2 * DeltaT mi = Pie * ra * Uaa * DeltaZ A(i - 1) = theta * L B(i - 1) = -theta * L - M C(i - 1) = 0 D(i - 1) = -(1 - theta) * L * PipeTemp(i - 1) + ((1 - theta) * L - M) _ * PipeTemp(i) - mi * theta * FormTempN(i, 0) _ - mi * (1 - theta) * FormTemp(i, 0) + mi * theta * AnnTempN(i) _ + mi * (1 - theta) * AnnTemp(i) Case Else A(i - 1) = -E * theta B(i - 1) = F + G * theta C(i - 1) = E * theta D(i - 1) = E * (1 - theta) * PipeTemp(i - 1) _ + (F - H * (1 - theta)) * PipeTemp(i) _ - E * (1 - theta) * PipeTemp(i + 1) _ + H * (1 - theta) * AnnTemp(i) + G * theta * AnnTempN(i) End Select Next i ThomasAlgorithm A, B, C, D, X, (imax - 1) 'Update pipe temperature in finite difference grid For i = 0 To (imax - 1) PipeTempN(i + 1) = X(i) Next i End Sub
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