Integrating PVT Properties for the
Description of Well Responses in
Gas Condensate Reservoirs
A Thesis
Submitted to the Faculty of Graduate Studies and Research
in Partial Fulfillment of the Requirements
for the Degree of
Master of Applied Science in Petroleum Systems
Engineering
University of Regina
by
Jiawei Li
Regina, Saskatchewan
June 2015
Copyright 2015: Jiawei Li
UNIVERSITY OF REGINA
FACULTY OF GRADUATE STUDIES AND RESEARCH
SUPERVISORY AND EXAMINING COMMITTEE
Jiawei Li, candidate for the degree of Master of Applied Science in Petroleum Systems Engineering, has presented a thesis titled, Integrating PVT Properties for the Description of Well Responses in Gas Condensate Reservoirs, in an oral examination held on May 1, 2015. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: Dr. Chun-Hua Guo, Department of Mathematics & Statistics
Supervisor: Dr. Gang Zhao, Petroleum Systems Engineering
Committee Member: Dr. Daoyong Yang, Petroleum Systems Engineering
Committee Member: *Dr. Yee-Chung Jin, Environmental Systems Engineering
Chair of Defense: Dr. Paul Laforge, Electronic Systems Engineering *Not present at defense
I
ABSTRACT
A gas condensate reservoir exhibits complex behaviors when the bottomhole pressure
falls below the dew point pressure at a given reservoir temperature. When the condensate
oil begins to drop out from the gas, a two-phase fluid system develops and a bank of
condensate oil builds up, inducing severe productivity losses. While the production rate is
constant, different mobility zones are formed around the wellbore corresponding
respectively to the original-gas-in-place (OGIP) away from the well, the condensate bank
with only gas flow, and two-phase gas and oil flow near the wellbore. Thus, the behaviors
of gas condensate systems are complex and difficult to interpret.
In this thesis, a single well model is built to evaluate the dynamic performance of an
infinite and homogeneous gas condensate reservoir. Firstly, apparent compressibility is
defined by integrating PVT properties. The application of modified pseudo-pressure and
pseudo-time linearizes the partial differential equations with the non-linearity caused by
gas properties. Secondly, a three-region method accounts for the composition changes in
the reservoir. Fluid flow towards the well during depletion can be divided into three
concentric main flow regions, from the wellbore to the reservoir. An analytical model
could have been built directly from the three-region method. Thirdly, on the basis of the
three-region method, the semi-analytical model is developed by dividing the whole
reservoir into multiple sub-radial regions. In the modeling process, the discretized sub-
radial regions are hydraulically coupled with nearby sub-radial regions so that an ultimate
linearized system is generated to obtain bottomhole pressure responses. Finally, a moving
boundary is also taken into consideration to investigate the difference between a
II
consistent boundary model and a moving boundary model. All models have been
validated and can be successfully used to analyze pressure and production data of gas
condensate production wells.
This thesis has contributed to production from gas condensate reservoirs with detailed
studies on the inherent PVT properties, condensate banks, and the interference of
adjacent regions. The modeling results provide reliable perspectives of transient pressure
analysis in gas condensate reservoirs and help characterize and estimate the drainage
areas of the three regions mentioned above, which is critical in gas condensate reservoir
development. Furthermore, this model builds a consolidated foundation for further
investigation of reservoir heterogeneity in the development of unconventional reservoirs.
III
ACKNOWLEDGEMENTS
I would like to acknowledge the opportunity offered by Dr. Gang (Gary) Zhao for
me to do my graduate studies at the University of Regina. This work, which summarizes
two years of research, is conducted under the direction of Dr. Gary Zhao. I am deeply
appreciative of Dr. Zhao’s advice, guidance and encouragement throughout our friendly
collaboration.
I would also like to give thanks to my group members: Mr. Lei Xiao, Mr. Chang Su,
Mrs. Jianli Li, Mr. Shuai Chen, Mr. Wanju Yuan, Ms. Yue Zhu, and Mr. Ning Ju. I am
also grateful to my friends here: Mr. Deyue Zhou, Ms. Xiaoyan Meng, Mr. Yu Shi, Mr.
Sixu Zheng, Mr. Zhongwei Du, Mr. Yanbin Gong, Mr. Longyu Han, Mr. Hao Yang, Mr.
Hongyang Wang, Mr. Tuo Huang, Mr. Yulong Zhao, Mr. Jinkai Liu, Mr. Zhaoqi Fan,
Mrs. Xiaoli Li, Mrs. Yu Xie. Thank you very much for your care and friendship to me
during my stay in Regina.
IV
DEDICATION
I wish to show my respect to my parents, Mr. Ronghua Li and Mrs. Yuxin Si. Many
thanks and many appreciations are extended to them for their love, support,
encouragement and understanding throughout my life.
This thesis is dedicated to my grandfather who left me and my family in 2013.
V
TABLE OF CONTENTS
ABSTRACT ................................................................................................................ I
ACKNOWLEDGEMENTS ........................................................................................... III
DEDICATION ............................................................................................................. IV
TABLE OF CONTENTS ................................................................................................. V
LIST OF TABLES ....................................................................................................... VIII
LIST OF FIGURES ........................................................................................................ IX
NOMENCLATURE ..................................................................................................... XIII
CHAPTER 1 INTRODUCTION ............................................................................... 1
1.1 Gas Condensate Reservoir ................................................................................ 1
1.2 Pressure Transient Analysis .............................................................................. 1
1.3 Objectives of This Thesis ................................................................................... 2
1.4 Outline of the Thesis .......................................................................................... 3
CHAPTER 2 LITERATURE REVIEW ................................................................... 4
2.1 Gas Condensate Flow Behavior ........................................................................ 4
2.1.1 Gas condensate characterization ................................................................... 4
2.1.2 Flow behavior ............................................................................................... 7
2.2 Pseudo-pressure................................................................................................ 10
2.2.1 Single-phase ................................................................................................ 10
2.2.2 Steady-state ................................................................................................. 12
2.2.3 Three-Region .............................................................................................. 17
VI
2.3 Pseudo-time ....................................................................................................... 23
2.4 Mathematical Formulation of Well Test Analysis ........................................ 25
2.5 PVT Measurement ........................................................................................... 26
2.5.1 Constant composition expansion (CCE) experiment .................................. 26
2.5.2 Constant volume depletion (CVD) experiment .......................................... 28
2.6 Field Cases ........................................................................................................ 30
CHAPTER 3 PVT INTEGRAL ............................................................................... 32
3.1 PVT Properties ................................................................................................. 32
3.2 Total Compressibility....................................................................................... 43
CHAPTER 4 THREE-REGION MODEL ............................................................. 48
4.1 Model Description ............................................................................................ 49
4.2 Model Demonstration ...................................................................................... 51
4.3 Relative Permeability Model ........................................................................... 54
4.4 Analytical Solution ........................................................................................... 57
CHAPTER 5 SEMI-ANALYTICAL MODELING ............................................... 60
5.1 Mathematical Formulations of Semi-analytical Model ................................ 60
5.2 Results and Discussion ..................................................................................... 64
5.2.1 Model validation ......................................................................................... 65
5.2.2 Effect of boundary ...................................................................................... 70
5.2.3 Effect of transmissibility ratio .................................................................... 76
5.2.4 Effect of storability ratio ............................................................................. 83
5.2.5 Effect of sub-segment number .................................................................... 91
5.2.6 Flow rate profile .......................................................................................... 97
CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS ........................ 100
VII
6.1 Conclusions ..................................................................................................... 100
6.2 Recommendations .......................................................................................... 101
REFERENCES ........................................................................................................... 102
VIII
LIST OF TABLES
Table 2.1 Typical Composition and Characteristics of Three Fluid Types (Wall,
1982) .................................................................................................................... 6
Table 3.1 Well stream: Measured and Calculated (Suwono et al., 2012) ........... 33
Table 3.2 Heptane properties: Measured and Calculated (Suwono et al., 2012) 33
Table 3.3 Four Different Compositions for Gas Condensate Reservoirs ............ 34
IX
LIST OF FIGURES
Figure 2.1 Ternary Visualization of Hydrocarbon Classification (Whitson and
Brule, 2000) ....................................................................................... 5
Figure 2.2 Phase diagram of a gas-condensate system (Li et al., 2005) .............. 9
Figure 2.3 Single-phase pseudo-pressure as a function of pressure (Al-Hussainy
et al., 1965) ...................................................................................... 11
Figure 2.4 Pressure-saturation relationship estimated by the steady-state model
(Al-Ismail, 2010) ............................................................................... 14
Figure 2.5 Two-phase steady-state pseudo-pressure as a function of real
pressure (Boe et al., 1989) .............................................................. 16
Figure 2.6 Gas-condensate three-flow regions (Penuela and Civan, 2000) ....... 18
Figure 2.7 Pressure-saturation relationships estimated by the three-region model
(Al-Ismail, 2010) ............................................................................... 21
Figure 2.8 Three-region pseudo-pressure as a function of pressure (Fevang and
Whitson, 1996) ................................................................................. 22
Figure 2.9 The relationship between real time and pseudo-time (Agarwal, 1979)
......................................................................................................... 24
Figure 2.10 Schematic of a CCE experiment for an oil and a gas condensate
(Whitson and Brule, 2000) ............................................................... 27
Figure 2.11 Schematic of CVD Experiment (Whitson and Brule, 2000) .............. 29
Figure 3.1 Liquid volume as a function of pressure for Composition 1 ............... 38
Figure 3.2 Liquid volume for different compositions ........................................... 39
X
Figure 3.3 Gas compressibility factor as a function of pressure for different
compositions .................................................................................... 41
Figure 3.4 Gas viscosity as a function of pressure for different compositions .... 42
Figure 3.5 Total compressibility as a function of pressure for Composition 1 ..... 46
Figure 3.6 Total compressibility for different compositions ................................. 47
Figure 4.1 Schematic of radial three-region composite model (Gringarten et al.,
2000) ............................................................................................... 50
Figure 4.2 Gas & Oil relative permeability curves ( ) ............................... 55
Figure 5.1 Schematic of a radial semi-analytical model ...................................... 63
Figure 5.2 Comparison of dimensionless pseudo-pressure and dimensionless
pseudo-pressure derivative from semi-analytical model and Kappa 67
Figure 5.3 Comparison of dimensionless pseudo-pressure and dimensionless
pseudo-pressure derivative from semi-analytical model and Kappa 68
Figure 5.4 Comparison of pressure from pseudo-pressure between semi-
analytical model and Kappa ............................................................. 69
Figure 5.5 Dimensionless pseudo-pressure curves for different radii of boundary
between Regions 1 and 2 ................................................................ 71
Figure 5.6 Dimensionless pseudo-pressure derivative curves for different radii of
boundary between Regions 1 and 2 ................................................ 72
Figure 5.7 Dimensionless pseudo-pressure curves for different radii of boundary
between Regions 2 and 3 ................................................................ 74
Figure 5.8 Dimensionless pseudo-pressure derivative curves for different radii of
boundary between Regions 2 and 3 ................................................ 75
XI
Figure 5.9 Dimensionless pseudo-pressure responses for different
transmissibility ratios ........................................................................ 78
Figure 5.10 Dimensionless pseudo-pressure derivative responses for different
transmissibility ratios ........................................................................ 79
Figure 5.11 Dimensionless pseudo-pressure responses for different
transmissibility ratios ........................................................................ 81
Figure 5.12 Dimensionless pseudo-pressure derivative responses for different
transmissibility ratios ........................................................................ 82
Figure 5.13 Dimensionless pseudo-pressure responses for different storability
ratios ................................................................................................ 85
Figure 5.14 Dimensionless pseudo-pressure derivative responses for different
storability ratios ................................................................................ 86
Figure 5.15 Dimensionless pseudo-pressure responses for different storability
ratios ................................................................................................ 89
Figure 5.16 Dimensionless pseudo-pressure derivative responses for different rD2
for the same value of storability ratios ............................................. 90
Figure 5.17 Dimensionless pseudo-pressure responses for different sub-segment
numbers ........................................................................................... 93
Figure 5.18 Dimensionless pseudo-pressure derivatives responses for different
sub-segment numbers ..................................................................... 94
Figure 5.19 Dimensionless pseudo-pressure responses for different sub-segment
numbers ........................................................................................... 95
XII
Figure 5.20 Dimensionless pseudo-pressure derivatives responses for different
sub-segment numbers ..................................................................... 96
Figure 5.21 Dimensionless flow rate profile for the drawdown test at different
dimensionless times ........................................................................ 98
Figure 5.22 Dimensionless flow rate profile for different mobility ratios .............. 99
XIII
NOMENCLATURE
Notations
a = attraction parameter
= gas formation volume factor, RB/scf
= oil formation volume factor, RB/STB
b = van der Waals covolume
= gas compressibility, 1/psi
= formation compressibility, 1/psi
= oil compressibility, 1/psi
= total system compressibility, 1/psi
= water compressibility, 1/psi
CR = diffusivity ratio
CS = storability ratio
CT = transmissibility ratio
= non-Darcy factor
= immiscibility factor
= thickness of reservoir, ft
= modified Bessel function of zero order
= modified Bessel function of zero order
XIV
= gas relative permeability, mD
= maximum relative permeability for gas, mD
= immiscible gas relative permeability, mD
= miscible gas relative permeability, mD
= oil relative permeability, mD
= maximum relative permeability for oil, mD
= permeability, mD
L = molar fraction of liquid
= the value of pseudo-pressure
= reference pseudo-pressure
= pseudo-pressure
= dimensionless pseudo-pressure
= number of sub-segments
= exponent ranging from 1 to 6
= exponent ranging from 1 to 6
= dew point pressure, psi
= reservoir external boundary pressure, psi
= initial reservoir pressure, psi
= boundary pressure between Region 1 and Region 2, psi
= wellbore flowing pressure, psi
XV
= dimensionless flow rate in region i
= standard gas flow rate, mscf/d
= producing gas/oil ratio, scf/STB
= solution gas/oil ratio, scf/STB
= radius, ft
= dimensionless radius
= solution oil/gas ratio, STB/scf
= gas saturation
= gas critical saturation
= oil saturation
= oil residual saturation
= initial water saturation
= water critical saturation
t = time, hr
= pseudo-time
= dimensionless pseudo-time
= velocity of gas phase, ft/s
V = molar fraction of vapor
= molar volume
z = compressibility factor
XVI
Greek Letters
= Forchheimer constant
= effective Forchheimer constant
= density of gas at standard conditions,
= density of gas in gas phase,
= gas density,
= density of solution gas in oil phase,
= oil viscosity, cP
= gas viscosity, cP
= porosity, fraction
Subscripts
D = dimensionless variable
g = gas phase
i = initial condition, or, sub-segment
o = oil phase
1,2,3 = Region 1,2,3
1
CHAPTER 1 INTRODUCTION
1.1 Gas Condensate Reservoir
With further exploration of oil and gas resources, gas condensate reservoirs are
becoming more and more important and are scattered all over the world. Large gas
condensate reservoirs are distributed in the Shearwater field in the North Sea,
Shtokmanovskoye Russia, the Arun and Senoro fields in Indonesia, the Dina field in
China, the Karachaganak field in Kazakhstan, the offshore North field in Qatar, the South
Pars field in Iran and the Cupiagua field in Colombia (Li et al., 2005).
Gas condensate reservoirs are different from conventional oil and gas reservoirs in that
complex behaviors will be exhibited when wells are produced below the dew point
pressure. When the bottom-hole pressure falls below the dew point pressure, condensate
oil begins to drop out and accumulates. As a result, a condensate bank is created, which
has a serious effect on production performance.
1.2 Pressure Transient Analysis
Due to the complex behaviors and the two-phase flow, it is difficult to evaluate well
performance of production wells in gas condensate reservoirs. Many efforts have been
made in order to solve such problems. In 1949, Muskat found that a condensate bank
builds up around the producing well once the bottomhole pressure falls below the dew
point pressure. Kniazeff et al. (1965) identified that two more regions other than the
condensate bank exist in the reservoir from the numerical simulations. The radial model
2
that considers the flow of individual components and accounts for component mass
transfer between phases was used to predict the performance of a producing well in a
reservoir containing a rich gas condensate reservoir (Roebuck et al., 1969). Fussell (1973)
modified the radial model developed by Roebuck et al. (1969) to study long-term single
well performance in three condensate reservoirs. O’Dell and Miller (1965) presented a
simple method based on steady state flow concepts that can be used to estimate quickly
the well deliverability. A unique relationship between pressure and saturation was
developed by Boe et al. (1989). Jones and Raghavan (1988) used a fully implicit model
to simulate the well responses in a gas condensate system by modifying the steady-state
theory. Thompson et al. (1993) presented an analytical solution for well testing in gas
condensate reservoirs. Fevang and Whitson (1996) proposed the three-region model to
model the well deliverability in a gas condensate reservoir. Whitson et al. (1999) showed
that the relative permeability in gas condensate systems should include three parts by
considering the capillary number effect and the non-Darcy flow effect. Gringarten et al.
(2000) showed that three regions exist with different liquid saturations when pressure
falls below the dew point pressure. A method to characterize condensate bank was
proposed by Bozorgzadeh and Gringarten (2006).
1.3 Objectives of This Thesis
To measure the PVT properties of gas condensate systems and use the PVT
properties for the calculation of pseudo-variables;
To propose a new definition of total compressibility by integrating PVT
properties;
3
To provide proper definitions of pseudo-pressure and pseudo-time in order to
linearize the diffusivity equations in the mathematical model; and
To develop mathematical models (analytical and semi-analytical) to investigate
the performance of an integrating gas condensate reservoir system.
1.4 Outline of the Thesis
The thesis is composed of six chapters. Chapter 1 briefly introduces the gas
condensate reservoirs and major relevant research objectives. Chapter 2 provides a
literature review that includes the flow behavior of gas condensate reservoirs,
development of pseudo-pressure and pseudo-time and mathematical foundations for well
testing. Chapter 3 presents the PVT measurement for gas condensate flow: Constant
composition expansion and constant volume depletion; PVT properties simulation for a
gas condensate system and the definition of total compressibility by integrating PVT
properties. Chapter 4 shows the development of a three-region model and relative
permeability for gas condensate reservoirs. Chapter 5 proposes a semi-analytical model
on the basis of the three-region model. Chapter 6 summarizes major conclusions of this
thesis and provides recommendations for further study.
4
CHAPTER 2 LITERATURE REVIEW
2.1 Gas Condensate Flow Behavior
2.1.1 Gas condensate characterization
For gas condensate reservoirs, single-phase gas exists in the reservoir at the beginning
of production, but yields small amounts of oil on the ground. The gas condensate system
has a composition consisting largely of methane and small fractions of intermediate and
heavy ends (typically, approximately : 87%, : 9% and : 4%) (Kamath, 2007).
Different hydrocarbons systems are classified by their colors, densities and gas-oil ratios
(Gravier et al., 1986). The composition of a gas condensate system compared with other
hydrocarbon systems is shown in Figure 2.1. Table 2.1 gives typical compositions and
characteristics of different hydrocarbon systems.
Most known gas condensate reservoirs are discovered in the ranges of 5000 ft to
10000 ft deep, 3000 psi to 8000 psi and 200 to 400 (Roussennac, 2001). At the
temperature between critical temperature and cricondentherm temperature, condensation
drops out from gas when the pressure falls below the dew point pressure. The reservoir
fluids will separate into: gas phase and oil phase. With pressure continuing to decrease,
more condensate oil drops out and will reach a maximum volume. Gas condensate fluids
can be divided into lean, medium-rich, or rich, depending on the range of their
condensate to gas ratio (Kgogo and Gringarten, 2010). A lean system may yield
approximately 10 STB/MMscf (2% maximum condensate), and a rich system could yield
as much as 20% condensate, i.e. 300 STB/MMscf (Kamath, 2007).
5
Figure 2.1 Ternary Visualization of Hydrocarbon Classification (Whitson and Brule,
2000)
6
Table 2.1 Typical Composition and Characteristics of Three Fluid Types (Wall,
1982)
Component Black Oil Volatile Oil Condensate Gas
Methane 48.83 64.36 87.07 95.85
Ethane 2.75 7.52 4.39 2.67
Propane 1.93 4.74 2.29 0.34
Butane 1.60 4.12 1.74 0.52
Pentane 1.15 2.97 0.83 0.08
Hexanes 1.59 1.38 0.60 0.12
C7+ 42.15 14.91 3.80 0.42
Molecular wt C7+ 225 181 112 157
Gas-Oil Ratio
SCF/Bbl
625 2000 182000 105000
Liquid-Gas Ratio
Bbl/MMSCF
1600 500 55 9.5
Tank Oil Gravity API 34.3 50.1 60.8 54.7
Color Green/Black Pale
Red/Brown
Straw White
7
2.1.2 Flow behavior
The flow behaviors of a gas condensate reservoir are characterized by the phase
envelope of the fluids and the condition of the reservoir. Figure 2.2 shows a typical
envelope: the pressure – temperature (P – T) diagram. The phase envelope is affected by
the composition and condition of the reservoir. A bubble point line and a dew point line
meeting at the critical point are shown in this phase envelope. In the process of
isothermal expansion, the first bubble of gas will vaporize from the liquid phase on the
bubble point line. In contrast, the first droplet of liquid will condense from the gas phase
on the dew point line. When pressure is above the cricondenbar or temperature is above
the cricondentherm, only one phase (liquid or gas) can exist. At the critical point, the
liquid and gas phase cannot be distinguished because the composition and all other
intensive properties of the two phases become identical (Al-Ismail, 2010). Gas reservoirs
and gas condensate reservoirs are determined by their initial reservoir conditions.
For gas reservoirs, if the reservoir temperature is above the cricondentherm, path –
is an example for the reservoir during the isothermal expansion and will not enter the
two-phase region. Therefore, gas flow remains consistent in the reservoir and the
reservoir fluid composition remains during the depletion process.
For gas condensate reservoirs, if the reservoir temperature is between the critical
temperature and cricondentherm, path – is an example for the reservoir during the
isothermal expansion. During the isothermal expansion, when path – reaches at the
dew point line, retrograde condensation will occur in the reservoir, leading to the changes
of the reservoir fluid compositions. At the beginning, the condensate saturation is low
and the mobility of condensate is almost zero; only gas flows in the reservoir and the
8
condensate accumulates due to heavy components of gas dropping out from the gas flow.
After the condensate saturation reaches the critical condensate saturation, the condensate
will start to move in the reservoir and as a result, gas and oil both flow. Condensate will
redissolve in the gas phase and the condensate volume will decrease if the depletion is
further continued.
9
Figure 2.2 Phase diagram of a gas-condensate system (Li et al., 2005)
10
2.2 Pseudo-pressure
2.2.1 Single-phase
In gas well test analysis, the non-linear partial differential equations describing gas
flows are transformed into a linear form by the use of pseudo-pressure, which is similar
to the liquid flow equations. Therefore, the analytical solutions for liquid flow equations
can be used for gas flow equations. Al-Hussainy et al. (1965) and Al-Hussainy and
Ramey (1966) proposed the definition of pseudo-pressure as:
(2-1)
where is the reference pressure, is the viscosity and z is the compressibility factor.
The following assumptions are made in the derivation of the equations:
1) The medium is homogenous.
2) The flow is laminar and isothermal.
3) The flowing gas has a constant composition.
The single-phase pseudo-pressure is applied to the dry gas. For the gas condensate
system, the assumption is that the liquids (condensate and water) are immobile and the
variation in the relative permeability for the gas phase is negligible, so the single-phase
pseudo-pressure can be used (Raghavan et al., 1995).
In a gas condensate reservoir, the gas compressibility factor and gas viscosity can be
considered as a function of pressure, so the integral can be calculated. The single-phase
pseudo-pressure as a function of pressure is plotted in Figure 2.3.
11
Figure 2.3 Single-phase pseudo-pressure as a function of pressure (Al-Hussainy et
al., 1965)
0.00E+00
2.00E+08
4.00E+08
6.00E+08
8.00E+08
1.00E+09
1.20E+09
1.40E+09
1.60E+09
0 1000 2000 3000 4000 5000 6000 7000
m(p
), p
si2 /
cp
Pressure, psi
12
2.2.2 Steady-state
In gas condensate reservoirs, two phases will exist when the bottomhole pressure
drops below the dew point pressure. The two-phase steady-state theory to predict the
performance of a single well in a gas condensate reservoir was first proposed by O’Dell
and Miller (1967). However, the productivity predicted by the O’Dell-Miller theory is not
accurate on the basis of examination by Fussell (1973) because the O’Dell-Miller theory
cannot provide an accurate prediction for the saturation profile in the two-phase region.
Boe et al. (1989) suggested techniques to determine sandface saturation with a
relationship between pressure and saturation by applying the Boltzmann transformation.
However, the Boe et al. theory (1989) cannot be applied under certain circumstances
because the saturation cannot always be an expression of a single value of Boltzmann
variables. The steady-state theory was modified by Jones and Raghavan (1988).
Two flow regions are assumed for the steady-state model:
Region 1: An inner region below the dew point pressure where gas and condensate are
present and mobile.
Region 2: An outer region above the dew point pressure where only gas exists and
flows.
The two-phase pseudo-pressure was introduced to provide a better description of the
condensate reservoir integral:
(2-2)
13
where is the reference pressure, is oil viscosity, is gas viscosity, is oil
relative permeability, is gas relative permeability, is the oil formation volume
factor, and is the gas formation volume factor.
Region 1:
In order to solve the Equation (2-3), the correlation between relative permeability and
pressure should be known. In a gas condensate reservoir, the oil/gas relative permeability
ratio can be expressed as:
(2-3)
where L and V refer to the equilibrium molar fraction of liquid and vapor derived from
flash calculations, respectively. The left hand side is a function of saturation and the right
hand side is a function of pressure. This assumption implies that the overall composition
of the flowing mixture at any location in the reservoir is the composition of the original
reservoir fluid and a region where the composition of the flowing mixture is changing
does not exist (Al-Ismail, 2010).
Condensate saturation could be evaluated from the condensate/gas relative
permeability ratio by using the relative permeability curves (Figure 2.4).
14
Figure 2.4 Pressure-saturation relationship estimated by the steady-state model (Al-
Ismail, 2010)
15
The gas and oil relative permeabilities as functions of saturation are expressed as the
following forms:
(2-4)
(2-5)
The gas compressibility factor and gas viscosity can be derived as a function of
pressure from the flash calculation.
Region 2:
For region 2, the method to calculate the integral is the same as that for the single-
phase pseudo-pressure. The gas compressibility factor and gas viscosity can be also
derived as a function of pressure from the flash calculation.
The results of the two-phase steady-state pseudo-pressure calculation are shown in
Figure 2.5.
16
Figure 2.5 Two-phase steady-state pseudo-pressure as a function of real pressure
(Boe et al., 1989)
0.00E+00
1.00E+07
2.00E+07
3.00E+07
4.00E+07
5.00E+07
6.00E+07
7.00E+07
8.00E+07
0 2000 4000 6000 8000
m(p
), p
si2/c
p
Pressure, psi
17
2.2.3 Three-Region
In gas condensate reservoirs, the condensate will drop out from the gas when the wells
are producing under the dew point pressure, leading to compositions changes in the
reservoir. On the basis of many observations of the three flow regions for many gas
condensate systems, Fevang and Whitson (1996) developed a simplified method to
calculate the pseudo-pressure integral. This method has been generally applied in many
cases of gas condensate systems study. Figure 2.6 shows a schematic of three flow
regions in a gas condensate reservoir.
Region 1: Region 1 is the region around the wellbore where both gas and oil flow
simultaneously. The flowing composition (GOR) in Region 1 is considered to be constant,
which means that the gas entering Region 1 has the same composition as the produced
well-stream. Region 1 is the main source of productivity losses in a gas condensate
reservoir.
Region 2: Region 2 is the region where condensate drops out from gas and builds up,
only gas flows and oil is immobile. Condensate saturations in Region 2 are closely
approximated by a liquid-dropout curve from a constant-volume-depletion (CVD)
experiment. The liquid-dropout curve from the CVD experiment is used to approximate
condensate saturations in Region 2. The size of Region 2 is the largest at early times just
after the reservoir pressure drops below the dew point pressure, which decreases with
time as Region 1 expands (Fevang and Whitson, 1996).
Region 3: Region 3 is the region where the pressure is above the dew point pressure
for the gas condensate system. Only gas exists in Region 3. When the pressure drops
below the dew point pressure, Region 2 will appear.
18
Figure 2.6 Gas-condensate three-flow regions (Penuela and Civan, 2000)
19
On the basis of three flow regions for gas condensate systems, a method to calculate
the pseudo-pressure integral was developed by Fevang and Whitson (1996):
Total
Region 1
Region 2
Region 3
(2-6)
where is the total pseudo-pressure, is the wellbore flowing pressure, is the
boundary pressure between Regions 1 and 2, is the dew point pressure, is the
reservoir boundary pressure, is the initial water saturation, is the oil viscosity,
is the gas viscosity, is the oil relative permeability, is the gas relative permeability,
is the oil formation volume factor, is the gas formation volume factor, and is the
solution gas/oil ratio.
Region 1: The modified Evinger-Muskat approach is used to solve the Region 1
pseudo-pressure (Fevang and Whitson, 1996). At pressure , the PVT properties
can be found directly. For , it is defined where in the PVT table (Fevang
and Whitson, 1996). The equation defining producing GOR is proposed:
(2-7)
and is used to calculate the gas/oil relative permeability ratio as a function of pressure:
20
(2-8)
where is the solution oil/gas ratio, and is the producing gas/oil ratio.
In Equation (2-8), PVT properties are known as functions of pressure. Equation (2-8)
can be considered to be equivalent to the following equation:
(2-9)
For Region 1, similar to the steady-state method, Equation (2-9) is used to estimate the
condensate-gas relative ratio as a function of pressure. The condensate saturation can be
evaluated from the condensate/gas relative permeability ratio by using the relative
permeability curves.
Region 2:
The Region 2 integral is evaluated by use of , where condensate saturation is
estimated as a function of pressure from CVD relative volume (Fevang and Whitson,
1996). Then, the condensate saturation as a function of pressure is illustrated in Figure
2.7. The gas relative permeability in Region 2 is shown as follows:
(2-10)
Region 3:
Only gas PVT properties are found in the Region 3 integral, where the single-phase
pseudo-pressure function can be found.
The three-region pseudo-pressure as a function of pressure is plotted in Figure 2.8.
21
Figure 2.7 Pressure-saturation relationships estimated by the three-region model
(Al-Ismail, 2010)
22
Figure 2.8 Three-region pseudo-pressure as a function of pressure (Fevang and
Whitson, 1996)
0.00E+00
1.00E+07
2.00E+07
3.00E+07
4.00E+07
5.00E+07
6.00E+07
7.00E+07
8.00E+07
0 1000 2000 3000 4000 5000 6000 7000 8000
m(p
), p
si2
/cp
Pressure, psi
23
2.3 Pseudo-time
Al-Hussainy (1966) developed pseudo-pressure to apply analytical solutions for gas
flow. However, in the governing diffusivity equation, the viscosity and compressibility
corresponds to pressure for gas flows. In addition, the porosity for the formation can also
be considered to change with pressure. As a result, the diffusivity equation for gas flow is
non-linear when expressed in terms of pressure, time variables. In order to linearize the
non-linear equations, Agarwal (1979) defined pseudo-time which integrates the variations
of gas viscosity and compressibility as a function of pressure:
(2-11)
where is viscosity, is total system compressibility.
The first definition of the total system compressibility in multi-phase conditions was
made by Martin (1959), neglecting the rock compressibility. On the basis of Martin’s
work, Ramey (1964) included the formation compressibility by defining the total system
compressibility, as:
(2-12)
where is formation compressibility, is oil compressibility, is water
compressibility, and is gas compressibility.
The definition of proposed by Ramey (1964) has been generally used for computing
pseudo-time. The relationship between real time and pseudo-time is shown in Figure 2.9.
24
Figure 2.9 The relationship between real time and pseudo-time (Agarwal, 1979)
0.00E+00
1.00E+07
2.00E+07
3.00E+07
4.00E+07
5.00E+07
6.00E+07
0 20 40 60 80 100 120
Pse
ud
o-t
ime
, psi
/cp
Real Time, hour
25
In gas condensate reservoirs, the definition of pseudo-time also takes the two phases
into consideration. Heidedari and Shahab (2011) introduced one form of two phase
pseudo-time as:
(2-13)
where is the reference pseudo-pressure, m is is pseudo-pressure, is oil saturation,
is gas saturation, is the oil formation volume factor, is the gas formation volume
factor, and is the solution gas/oil ratio.
Application of pseudo-pressure and pseudo-time makes the analytical solutions
available to these non-linear equations, which are generally used to analyze and model
well tests.
2.4 Mathematical Formulation of Well Test Analysis
The diffusion equation for the pressure transient of a well test is written as:
(2-14)
There are assumptions for the derivation of the diffusion equation, which are listed as
follows (Horne, 1995):
1) Darcy’s Law applies;
2) Single phase flow;
3) Porosity, permeability, viscosity and compressibility are constant;
4) The pressure gradient in the reservoir is small;
26
5) Fluid compressibility is small; and
6) Gravity and thermal effects are negligible.
2.5 PVT Measurement
In the laboratory, constant composition expansion (CCE) and constant volume
depletion (CVD) experiments can be performed to analyze phase behavior of the gas
condensate fluids.
2.5.1 Constant composition expansion (CCE) experiment
The CCE experiment could be performed for a gas condensate mixture, also for oil
mixtures. Figure 2.10 illustrates the procedure for the CCE experiment. A known mass
of reservoir fluid fills in a closed cell. Temperature is kept constant, often at the reservoir
temperature. The reservoir fluid is brought to the condition above the saturation pressure,
ensuring that the fluid is in single phase.
The experiment starts at a pressure higher than the initial reservoir pressure that is
always considered as the saturation pressure. For a gas condensate mixture, this means
the experiment starts at a pressure above the dew point pressure. The initial volume is
measured and the volume increases step by step. The volume and pressure at every step
should be measured and recorded. In addition, the saturation point should be recognized.
The CCE experiment provides detailed data about the saturation pressure at the
reservoir temperature and about the relative volume of gas and oil during the experiment.
27
Figure 2.10 Schematic of a CCE experiment for an oil and a gas condensate
(Whitson and Brule, 2000)
28
2.5.2 Constant volume depletion (CVD) experiment
The CVD experiment is also an experiment performed for gas condensate and volatile
oil reservoirs producing by pressure depletion. The stepwise procedure of a CVD
experiment is shown schematically in Figure 2.11. Similar to the CCE experiment, a
certain amount of reservoir fluids is filled into a cell that is kept at a constant temperature,
usually at the reservoir temperature. The cell is constructed in the same way as for the
CCE experiment, but something should be done in order to allow the depletion of gas
during the experiment, such as equipping the cell with a valve to control the volume.
The CVD experiment starts at the dew point pressure for a gas condensate mixture. In
the experiment, the pressure and relative volume should be measured and recorded at
every step, which is similar to the CCE experiment. The valve can be used to keep the
total volume constant during pressure depletion.
The CVD experiment provides data that can be used directly, including: 1) a reservoir
material balance that gives average reservoir pressure vs. recovery of total well-stream,
sales gas, condensate, and natural gas liquids; 2) produced well-stream composition and
surface products vs. reservoir pressure; and 3) average oil saturation in the reservoir
(liquid dropout and revaporization) that occurs during pressure depletion (Whitson and
Brule, 2000).
29
Figure 2.11 Schematic of CVD Experiment (Whitson and Brule, 2000)
30
2.6 Field Cases
Many field investigations have been conducted over the last decades to understand the
flow behaviors in gas condensate reservoirs. Behrenbruch and Kozma (1984) interpreted
results from well testing gas condensate reservoirs by comparing theory and field cases.
In the case of high potential gas-condensate wells in good permeability reservoirs, the
following requirements are needed for accurate deliverability forecasting: good
estimation of fluid conditions; estimations of formation properties; and detailed
knowledge of active flow (Initial well deliverability may depend on the depth of the test
zone in the reservoir) (Behrenbruch and Kozma, 1984).
For a very thick gas condensate reservoir, the vertical variation in compositions can be
estimated by considering: detailed results from recombination fluid samples; and
measurement of gas gradient with a high accuracy pressure gauge (Behrenbruch and
Kozama, 1984). In the very high temperature KAL-5 gas condensate well in the
Moslavacka Gora formation in Yugoslavia, a successful hydraulic simulation was
performed (Economides et al., 1989). In 1991, S gnesand discussed the effect of
retrograde condensate blockage on the long-term well performance of vertically fractured
gas condensate wells and presented a method to correct the effect of condensate blockage
by using the concept of the time-dependent skin factors.
Raghavan et al. (1995) considered practical factors in analysis of gas condensate wells
and summarized two conclusions: It is possible to relate the relative permeability values
to pressure and use the resulting analogue to evaluate pressure-buildup tests in a
quantitative manner; and the saturation profile at shut in governs the shape of the pressure
buildup trace and the success of the two-phase analogue is dependent on the ability to
31
estimate this profile. Diamond et al. (1996) developed a method to estimate probabilistic
well deliverability in the Britannia Gas Condensate Field based on log and core data.
Marhaendrajana and Kaczorowski (1999) proposed a rigorous and coherent approach for
the analysis of well test data from a multi-well reservoir system; all of the available well
test data are collected from the giant Anrun Gas Field (Sumatra, Indonesia). Kool et al.
(2001) outlined the metrology and procedure to obtain a representative formation fluid
sample that may be used for compositional and PVT analysis.
The modified black-oil model was tested against the fully compositional model and
performances of both models were compared by using various production and injection
scenarios for a rich gas condensate reservoir (Izgec et al., 2005). A novel approach was
introduced in the use of two-phase pseudo-pressure for the interpretation of gas
condensate well test data in naturally fractured reservoirs (Mazloom et al., 2005). The
Fetkovich method was chosen to evaluate reservoir productivity and well future
production performance in conjunction with well test analysis based on test draw-down
data (Zheng and Marius, 2006).
32
CHAPTER 3 PVT INTEGRAL
3.1 PVT Properties
Fluid behavior for gas condensate reservoirs is not only a function of pressure but also
dependent on compositions. The first step of PVT simulation for a gas condensate
reservoir is to determine the compositions comprising the well stream. The compositions
used are a sample from the Senoro field, a major gas condensate field in East Indonesia.
The mole fraction of each component is shown in Table 3.1, in which the lab
experimentally measured data and calculated data give an acceptable match with less than
2% of deviation (Suwono et al., 2012). The mole fraction is generally characterized
by the application of the gamma distribution model (Whitson, 1983). The results are
tabulated in Table 3.2.
In Table 3.3, Composition 1 represents the composition data from the Senoro field,
which is used as the standard. Three simplified compositions are derived from the
composition data from the Senoro field. Different compositions form different gas
condensate systems. The differences of Compositions 2, 3 and 4 are the mole fraction of
and because and are the main sources of condensate liquid during
the production process. The total mole fraction of and is assumed to be
constant in Compositions 2, 3, and 4. In Composition 2, occupies most of the total
mole fractions of and . In Compositions 2 and 3, the most part is and
respectively.
33
Table 3.1 Well stream: Measured and Calculated (Suwono et al., 2012)
Table 3.2 Heptane properties: Measured and Calculated (Suwono et al., 2012)
34
Table 3.3 Four Different Compositions for Gas Condensate Reservoirs
Composition
1 (Suwono et
al., 2012)
mol %
Composition
2 mol %
Composition
3 mol %
Composition
4 mol %
1.0808 1.5000 1.5000 1.5000
0.9093 1.5000 1.5000 1.5000
84.8293 80.0000 80.0000 80.0000
5.1132 5.0000 5.0000 5.0000
2.9694 3.0000 3.0000 3.0000
0.9332 0.5000 0.5000 0.5000
1.1031 0.5000 0.5000 0.5000
0.5853 2.5000 0.5000 0.5000
0.4767 2.5000 0.5000 0.5000
0.5773 2.0000 6.0000 2.0000
1.4224 1.0000 1.0000 5.0000
35
The key part of PVT simulation is to use the Peng-Robinson Equation of State (PR
EOS). Because compositions, volumes, and temperatures are known at all times, in
single-phase liquid or vapor regions, the resulting pressure is calculated directly by the
equation of state. In two-phase regions, a flash calculation is required. In a typical EOS
flash calculation, pressure is known and the equation of state calculates phase volumes. A
commercial simulator (CMG WinProp 2011) is used to perform a two-phase calculation
on the basis of the PR EOS. The PR EOS is written as follows (Peng and Robinson,
1976):
(3-1)
where P is pressure, a is the attraction parameter, b is the van der Waals co-volume, R is
the universal gas constant, T is absolute temperature and is the molar volume.
Real gas properties are computed as a function of pressure under isothermal
conditions through the PR EOS in the model. A gas condensate reservoir is divided into
three regions (Figure 2.6). The governing partial differential equations for gas and oil in
Region 1 are written as (Boe et al., 1989):
(3-2)
(3-3)
where is oil saturation, is gas saturation, is oil viscosity, is gas viscosity,
is oil relative permeability, is gas relative permeability, is the oil formation
volume factor, is the gas formation volume factor, is the solution gas/oil ratio, is
36
the solution oil/gas ratio, t is time, is the initial reservoir pressure, is the porosity, k
is the absolute permeability, and r is the radius.
Though both gas and oil exist in the gas condensate reservoir, the effects of the oil
phase are ignored due to its small amount and poor flow ability, and only the gas phase is
considered in this thesis.
Equation (3-2) is nonlinear due to the high compressibility property of gas. In order to
linearize the non-linear diffusivity equation, pseudo-time and pseudo-pressure are applied:
(3-4)
(3-5)
where is pseudo-pressure for Region 1, is pseudo-time for Region 1, and is the
apparent compressibility for Region 1.
With the application of pseudo-pressure and pseudo-time, the Equation (3-2) can be
transformed into:
(3-6)
Equation (3-6) is similar to the diffusivity equation for fluids, so the solutions for
fluids with constant compressibility, viscosity and porosity can be used.
During the process of integral of pseudo-pressure and pseudo-time, the gas flow
properties can be calculated through the two-phase flash calculation and from empirical
correlations. Liquid volume, the gas compressibility factor and gas viscosity are used for
37
demonstration by applying the compositions of a gas condensate system (Table 3.1 and
Table 3.2).
Figure 3.1 illustrates the liquid (condensate) volume changes as a function of pressure
in a gas condensate system for Composition 1. When pressure falls below the dew point
pressure, the condensate will drop out. In a certain range of pressure, there will be more
and more condensate accumulating. After the accumulation of condensate reaches the
maximum, the volume of condensate decreases due to the fact that some compositions
cannot be kept in liquid phase under that pressure.
In Figure 3.2, the accumulative liquid volume shows obvious differences due to the
change of compositions. More hydrocarbons that have heavier molecular weight mean
more liquid volume dropping out from the gas flow in a gas condensate reservoir. In
addition, the dew point pressure also changes significantly corresponding to the
compositions. There is a tendency that a small change of heavier hydrocarbons can affect
the liquid volume curve much more significantly. With the increase of heavier
hydrocarbons, the dew point pressure of each gas condensate system increases.
38
Figure 3.1 Liquid volume as a function of pressure for Composition 1
0
0.05
0.1
0.15
0.2
0.25
800 1000 1200 1400 1600
Liq
uid
Vo
lum
e, %
Pressure, psi
39
Figure 3.2 Liquid volume for different compositions
0
1
2
3
4
5
6
7
8
0 500 1000 1500 2000 2500 3000 3500 4000
Liq
uid
Vo
lum
e, %
Pressure, psi
Composition 1
Composition 2
Composition 3
Composition 4
40
As shown in Figure 3.3, the different compositions have direct effects on the curves
of compressibility factor Z because this factor can be calculated directly through the PR
EOS. In addition, the gas compressibility curves also depend on the intermolecular forces
of gases. Starting at a low pressure, there is enough space for attraction forces to be
dominant in gas molecules with the increase of pressure, leading to a smaller gas
compressibility factor. When the space in the gas molecules reduces to a critical value,
the repulsive factor will be dominant. Then, the gas compressibility factor increases
gradually with the increase of pressure.
Figure 3.4 describes the viscosity as a function of pressure. The gas viscosity shows a
small difference because the main compositions of gas flow are similar for each gas
condensate system. In CMG Winprop, the Pederson correlation is expected to give better
liquid viscosity prediction for light and medium oils than the JST model (Suwono et al.,
2012). Therefore, the modified Pederson method (Pederson and Fredenslund, 1987) is
expected to have better estimation for viscosity modeling.
41
Figure 3.3 Gas compressibility factor as a function of pressure for different
compositions
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0 1000 2000 3000 4000 5000
Co
mp
ress
ibili
ty F
acto
r Z
Pressure, psi
Composition1
Composition 2
Composition 3
Composition 4
42
Figure 3.4 Gas viscosity as a function of pressure for different compositions
0
0.005
0.01
0.015
0.02
0.025
0.03
0 500 1000 1500 2000 2500 3000 3500 4000
Vis
cosi
ty, c
p
Pressure, psi
Composition 1
Composition 2
Composition 3
Composition 4
43
3.2 Total Compressibility
As shown in Equation (3-4), the definitions of pseudo-time contain the total
compressibility. In general, a reservoir consists of rock and pore space occupied by oil,
gas, and water. The total compressibility is defined as follows (Ramey, 1964):
(3-7)
The computation of pseudo-time using the total compressibility defined in Equation
(3-8) is referred to as conventional pseudo-time (Rahman et al., 2006). Pseudo-time is
intended to unify and take the effects of the following variables with pressure into
consideration: time, gas viscosity, gas compressibility, porosity, and fluid saturation.
Based on the above consideration, a new definition of the total compressibility for gas
flow is given (Rahman et al., 2006). A form of total compressibility for a gas condensate
system is proposed by taking oil saturation changes into consideration (Bozorgzadeh and
Gringarten, 2006). Xiao and Zhao (2013) also provide a definition of the total
compressibility for the foamy oil flow in the reservoir. The following equations illustrate
the derivation of the total compressibility in a gas condensate reservoir. The derivation
includes two phases, gas and oil. If water exists in the reservoir, it is assumed to be
immobile and then water saturation is considered to be consistent, as initial water
saturation.
The partial differential equation (3-2) in Region 1 is used for demonstration. The
right-hand side of Equation (3-2) is written as follows:
(3-8)
44
On the right-hand side of Equation (3-8), applying the pressure dependent variables,
the following equation is obtained:
(3-9)
The variation of porosity with pressure can be expressed using the formation
compressibility as:
(3-10)
Submitting Equation (3-10) into Equation (3-8) yields:
(3-11)
As shown in Equation (3-11), the definition of is expressed as follows:
(3-12)
Mathematical developments for the total compressibility factor are strictly valid with
the definitions of pseudo-pressure and pseudo-time. The total compressibility factor is
also a function of pressure. Applying the compositions of a gas condensate reservoir
mentioned in Tables 3.1 and 3.2, Figure 3.5 shows the the total compressibility as a
function of pressure for Composition 1. The total compressibility increases with the
45
pressure in a very small range, showing an approximate linear relationship. Since the
pressure ranges for condensate liquid dropping out are variant for different compositions,
so the relative pressure range is set here in order to compare the trends of total
compressibility. In Figure 3.6, the total compressibility for Compositions 1 and 2 almost
remains the same with pressure. For Composition 3, it increases gradually with pressure.
The total compressibility increases very rapidly corresponding to pressure for
compositions. Figure 3.6 reflects the great effect of compositions for a gas condensate
system on the total compressibility.
46
Figure 3.5 Total compressibility as a function of pressure for Composition 1
47
Figure 3.6 Total compressibility for different compositions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 100 200 300 400 500 600
Ct,
1/p
si
∆p , psi
Composition 1
Composition 2
Composition 3
Composition 4
48
CHAPTER 4 THREE-REGION MODEL
In general, reservoirs with different physical properties have been studied by using
analytical or numerical composite reservoir models. The pressure behavior of composite
reservoirs has been analyzed in many cases. All of these cases can be classified in three
large groups: two-region composite reservoir models (Ambastha and Ramey, 1989a;
Ambastha and Ramey, 1989a; Ambastha, 1988; Chu et al., 1993; Xiao and Zhao, 2013),
three-region composite reservoir models (Ambastha and Ramey, 1992; Issaka and
Ambastha., 1997) and multi-region composite reservoir models (Acosta and Ambastha,
1994; Zeng and Zhao, 2008; Zhu et al., 2012).
In 1967, O’Dell and Miller proposed the pseudo-steady theory to account for two-
phase flow in the gas condensate reservoirs after a certain time of production. In pseudo-
steady theory, a pseudo-steady state flow occurs for different mobility or storativity ratios
between two regions. Then a gas condensate reservoir can be divided into two regions: 1)
Two phases (oil and gas) both flow; and 2) Only gas exists and flows. Based on many
observations of gas condensate systems, Fevang and Whitson (1996) proposed an
accurate simple model of a gas condensate reservoir to account for flow property
differences by dividing the whole reservoir into three regions: 1) an inner near-wellbore
region where both gas and oil flow simultaneously; 2) a region of condensate buildup
where only gas is flowing; and 3) a region containing only single-phase reservoir gas.
Therefore, analytical solutions for a composite model which describes three flow regions
depending on different flow properties are applicable and useful to characterize pressure
responses and production responses for wells in gas condensate reservoirs.
49
4.1 Model Description
A radial single-porosity reservoir whose outer boundary is infinite, and where top and
bottom are considered, is used. A single production well is located in the center of the
reservoir. Figure 4.1 shows a schematic of a production single well in a radial composite
reservoir. The bottomhole pressure is below the dew point pressure. All three regions are
assumed to develop in the reservoir. The region boundary will be considered to be
constant during the short testing time. The pressure responses caused by gas flow are
evaluated because main compositions comprising a gas condensate reservoir are gas
phase. The gas flow includes the gas phase flow and gas components in the oil phase
flow.
50
Figure 4.1 Schematic of radial three-region composite model (Gringarten et al., 2000)
51
4.2 Model Demonstration
Based on Darcy’s Law and mass conservation, the governing partial differential
equation for the flow of the gas component in a gas condensate reservoir is yielded. There
are some assumptions to derive the diffusivity equations, which are listed as follows:
Thickness of the reservoir is constant;
Gravity is ignored;
Darcy’s Law is applicable;
Temperature is constant; and
Capillary pressure is ignored.
On the basis of previous model (Boe et al., 1989; Fevang and Whitson, 1996;
Heidedari and Shahab, 2011), the governing partial differential equations are written as:
(4-1)
(4-2)
(4-3)
where
(4-4)
52
(4-5)
Submitting Equations (4-4) and (4-5) into Equations (4-1), (4-2) and (4-3) to eliminate
from both sides:
(4-6)
(4-7)
(4-8)
The initial and boundary conditions are:
(4-9)
(4-10)
(4-11)
where is oil saturation, is gas saturation, is oil viscosity, is gas viscosity,
is oil relative permeability, is gas relative permeability, is gas relative permeability,
is the oil formation volume factor, is the gas formation volume factor, is the
solution gas/oil ratio, is the molar density of gas at standard conditions, is the
molar density of gas in the gas phase, is molar density of solution gas in the oil phase,
is the wellbore radius, is the radius of the inner region, is the radius of the
condensate bank, t is time, is the initial reservoir pressure, is the porosity, k is
absolute permeability, is the standard gas flow rate, and h is the thickness of the
reservoir.
53
As shown in Equations (4-6), (4-7) and (4-8), the equations are non-linear due to the
properties of gas. In order to linearize the equations, pseudo-pressure and pseudo-time are
applied here.
(4-12)
(4-13)
(4-14)
(4-15)
(4-16)
(4-17)
where is pseudo-pressure, is pseudo-time, is the total compressibility.
Applying the definitions of pseudo-pressure and pseudo-time to Equations (4-6), (4-7),
and (4-8), the partial differential equations can be written as follows:
(4-18)
(4-19)
(4-20)
54
After transformations of pseudo-time and pseudo-pressure, the partial differential
equations for three flow regions should be considered as a whole in order to obtain the
solutions of the mathematical model. A uniform expression of pseudo-time needs to be
adopted instead of three different pseudo-time expressions. Xiao and Zhao (2013)
demonstrates the detailed process of using a pseudo-time form to represent another one in
a two-region radial composite reservoir model. Acosta and Ambastha (1994) provides
more information about uniform expression for a multi-region reservoir model. Based on
their work, the following forms under the conditions in terms of two-phase pseudo-
variables are obtained:
(4-21)
(4-22)
(4-23)
4.3 Relative Permeability Model
The oil/gas relative permeability , are functions of pressure (Section 2.2.3). The
gas/oil relative permeability data are calculated by the Corey power-law relationship
(Corey, 1954; Brooks and Corey, 1966; Ali et al., 1997). The relative permeability curves
that only contain gas and oil phases are shown in Figure 4.2.
(4-24)
(4-25)
55
Figure 4.2 Gas & Oil relative permeability curves ( )
56
where and
are maximum relative permeability for oil and gas, respectively, is
oil residual saturation, is gas critical saturation, is water critical saturation, and
are exponents ranging from 1 to 6.
In near-wellbore regions, there is a phenomenon that oil saturation decreases and gas
relative permeability increases due to low interfacial tensions at high gas flow rates
(Gondouin et al., 1967). This is caused by the high capillary number, and is also called
‘positive coupling’ (Boom et al., 1995; Henderson et al., 2000). The definition of
capillary number that is the ratio of viscous to capillary force is given by (Moore and
Slobod, 1955):
(4-26)
The method of which gas relative permeability depends on capillary number is
proposed as (Whitson et al., 1999):
(4-27)
(4-28)
(4-29)
The inertial high velocity gas flow in gas condensate reservoirs is one source of
additional pressure drop. On the basis of the Forchheimer equation, the non-Darcy factor
is shown (Forchheimer, 1901; Whitson et al., 1999):
(4-30)
57
(4-31)
where is the immiscibility factor, is immiscible gas relative permeability, is
miscible gas relative permeability, is the non-Darcy factor, is the Forchheimer
constant, is the effective Forchheimer constant, is gas mass density, and is the
Darcy velocity of the gas phase.
4.4 Analytical Solution
Through the application of pseudo-pressure and pseudo-time, the non-linear partial
differential equations are transformed into linear ones. Region 1 is used as the reference
region here. The dimensionless equations can be written as:
(4-32)
(4-33)
(4-34)
where is dimensionless pseudo-pressure, is dimensionless pseudo-time, is the
dimensionless radius, is the diffusivity ratio (Zhao and Thompson, 2002).
(4-35)
(4-36)
The diffusivity ratio can be derived from the transmissibility ratio and storability
ratio .
58
(4-37)
(4-38)
The expression of the transmissibility ratio and storability ratio are shown as
follows:
(4-39)
(4-40)
(4-41)
(4-42)
After the Laplace transformation, the general solutions for each region are given by:
(4-43)
(4-44)
(4-45)
where is the Laplace variable.
For calculating the transmissibility and storability ratios, it is reasonable to find a
pressure that presents a region.
For example, assuming the initial reservoir pressure is 4000 psi and the dew point
pressure is 3500 psi, the area where the pressure is above the dew point pressure in the
reservoir is Region 3. The pressure range of integration for pseudo-pressure is 3500 -
4000. It is easy to find a pressure to represent the whole pressure range. In another
59
method, the integration can be calculated under the same reasonable pressure range for
each region (100 psi).
Based on the flowing equations and boundary conditions, the analytical solution for
the three-region model can be obtained:
(4-46)
When , the bottomhole pressure can be calculated.
60
CHAPTER 5 SEMI-ANALYTICAL MODELING
5.1 Mathematical Formulations of Semi-analytical Model
Zeng and Zhao (2008) have developed a semi-analytical model to quantify the
transient pressure behavior of vertical wells with non-Darcy flow in the reservoir. This
semi-analytical model is also applied to evaluate the early-period SAGD by interpreting
the temperature falloff data (Zhu et al., 2012). A semi-analytical model is built on the
basis of the three-region model to evaluate the performance of gas condensate reservoirs.
The whole reservoir is divided into many sub-segments (Figure 5.1). The
mathematical model for sub-segment , , can be written as (Zeng and Zhao,
2008):
(5-1)
where is the Laplace variable in sub-segment , and are inner and outer
boundary radii for sub-segment , is the dimensionless pseudo-pressure for sub-
segment and is the dimensionless flow rate in region i.
The analytical solution in the Laplace domain for segment is:
(5-2)
Applying the boundary conditions (5-1) helps generate the coefficients, and .
Then, the dimensionless pseudo-pressure in sub-segment can be written as a linear
61
equation in terms of flow rates on the boundaries of every sub-segment (Zeng and Zhao,
2008):
(5-3)
where and are combinations of the Bessel functions of the local Laplace variable.
Combining all sub-segments generates a linear tri-diagonal system.
(5-4)
A
nnnn
nnnnnn
AA
AAA
AAA
AA
,1,
,11,12,1
3,22,21,2
2,11,1
(5-5)
B
n
n
B
B
B
B
,1
1,1
2,1
1,1
(5-6)
and
62
C
n
n
C
C
C
C
1
2
1
(5-7)
are functions of time and radius, represent flow rates and is the residual.
63
Figure 5.1 Schematic of a radial semi-analytical model
64
5.2 Results and Discussion
The semi-analytical model for a single well has been programmed. The basic
procedure is to calculate variables in the Laplace Domain, then apply the Stehfest
inversion algorithm to transform the variables into real domains (Stehfest, 1970). The
type curves generated by the semi-analytical model are in dimensionless form responding
to the pseudo-pressure and pseudo-time. The transmissibility and storability ratios, which
are defined in Chapter 4, are used to characterize the differences among the three regions.
In order to match the type curves generated by this model, testing data from the field
need to be transformed into dimensionless pseudo-pressure and dimensionless pseudo-
time. The following analysis aims to examine the effects of the inner boundary between
Regions 1 and 2, the outer boundary between Regions 2 and 3, permeability, total
compressibility, numbers of sub-segments and flow rate distribution in the reservoir.
65
5.2.1 Model validation
No direct validation of the semi-analytical model is available due to the lack of field
data. Here the commercial software Kappa Ecrin 4.12 is used for model validation
because it can provide an accurate analytical solution for a homogeneous model and a
two-region radial composite model. The number of the semi-analytical sub-segments is
80 and the dimensionless radius (rD) of every sub-segment is 10 (This can guarantee the
appearance of radial flow in the known dimensionless time). When ordering the
transmissibility ratio , the diffusivity ratios ( and ) are assumed
to be equivalent to the mobility ratio while the stability ratio . Then the
three-region model will become a homogeneous model. The dimensionless pseudo-
pressure and dimensionless pseudo-pressure derivative generated from the semi-
analytical model are compared with those generated from Kappa.
Figure 5.2 shows that both the pressure and pressure derivative curves give a good
match. When , , , , and the diffusivity ratios
and are both equal to 0.5, the pressure and pressure derivative curves are also
identical (Figure 5.3). Figures 5.2 and 5.3 illustrate that the semi-analytical model has
accurate well performance compared with Kappa. For Figure 5.4, the dimensionless
pseudo-pressure and dimensionless pseudo-time are transformed into the values of
pseudo-pressure and pseudo-time. Then, the real pressure and real time are calculated
from the pseudo-pressure and pseudo-time directly. Figure 5.4 shows that the
comparison of real pressure is almost the same, proving that the pseudo-pressure and
pseudo-time in the semi-analytical model are accurate and reasonable. The identical
66
results have validated this semi-analytical model built on the basis of the three-region
model.
67
Figure 5.2 Comparison of dimensionless pseudo-pressure and dimensionless pseudo-
pressure derivative from semi-analytical model and Kappa
0.1
1
10
0.1 1 10 100 1000 10000 100000 1000000
Dim
en
sio
nle
ss P
seu
do
-pre
ssu
re a
nd
De
irva
tive
Dimensionless Time
Semi-analytical mD
Semi-analytical dmD
Kappa mD
Kappa dmD
CT21=1, CT31=1 CS21=1, CS31=1 rD1=100, rD2=400
68
Figure 5.3 Comparison of dimensionless pseudo-pressure and dimensionless pseudo-
pressure derivative from semi-analytical model and Kappa
0.1
1
10
0.1 1 10 100 1000 10000 100000 1000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re a
nd
de
riva
tive
Dimensionless time
Semi-analytical mD
Semi-analytical dmD
Kappa mD
Kappa dmD
CT21=0.5, CT31=0.5 CS21=1, CS31=1
69
Figure 5.4 Comparison of pressure from pseudo-pressure between semi-analytical
model and Kappa
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
0.000001 0.0001 0.01 1 100
Pre
ssu
re, p
si
Time, hour
Semi-analytical
Kappa
70
5.2.2 Effect of boundary
In a gas condensate reservoir, the radius of each region determines the drainage and
has a direct effect on the transient pressure responses of well testing. Figure 5.5 and 5.6
show the dimensionless pseudo-pressures and dimensionless pseudo-pressure derivatives
of different radii of the boundary between Regions 1 and 2 (rD1) while the boundary
between Regions 2 and 3 (rD2) is constant. The number of semi-analytical sub-segments
is 80 and the dimensionless radius (rD) of every sub-segment is 10 (This can guarantee
the appearance of radial flow in the known dimensionless time).The parameters of each
region are assigned to be: , , , , which causes
different deliverability of each region. In Figure 5.5, the dimensionless pseudo-pressure
curves are different from each other due to the change of rD2. Smaller values of rD2 result
in higher dimensionless pseudo-pressure, which means that more pressure drops in the
reservoir. This is because the transmissibility in Region 1 is better than those of Regions
2 and 3, which means fluids flow less easily in Regions 2 and 3. Figure 5.6 demonstrates
that the effects of different values of rD1 on three dimensionless pseudo-pressure
derivative curves are obvious. Significant variances appear after the pressure disturbance
reaches the Region 2 on the dimensionless pseudo-pressure curves and three
dimensionless pseudo-pressure derivative curves become the same at late time because
the value of rD2 is constant. The value of the boundary between Regions 1 and 2 can
affect the pressure responses.
71
Figure 5.5 Dimensionless pseudo-pressure curves for different radii of boundary
between Regions 1 and 2
0.1
1
10
100
0.1 1 10 100 1000 10000 100000 1000000 10000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re
Dimensionless time
rD1=100
rD1=300
rD1=500
CT21=0.5, CT31=0.25 CS21=1, CS31=1 rD2=700
72
Figure 5.6 Dimensionless pseudo-pressure derivative curves for different radii of
boundary between Regions 1 and 2
0.1
1
10
0.1 1 10 100 1000 10000 100000 1000000 10000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re d
eri
vati
ve
Dimensionless time
rD1=100
rD1=300
rD1=500
CT21=0.5, CT31=0.25 CS21=1, CS31=1 rD2=700
73
Figures 5.7 and 5.8 show the dimensionless pseudo-pressures and dimensionless
pseudo-pressure derivatives of different radii of the boundary between Regions 2 and 3
(rD2) while the boundary between Regions 1 and 2 (rD1) is constant. The number of semi-
analytical sub-segments is 80 and the dimensionless radius (rD) of every sub-segment is
10 (This can guarantee the appearance of radial flow in the known dimensionless time).
The parameters of each region are assigned as , , ,
, which causes different deliverability of each region. In Figure 5.7, smaller
values of rD1 result in higher dimensionless pseudo-pressure, which means that more
pressure falls in the reservoir. This is because the transmissibility in Region 1 is better
than those of Regions 2 and 3, indicating fluids flow less easily in Regions 2 and 3.
Figure 5.8 demonstrates that the effects of different values of rD2 on three dimensionless
pseudo-pressure derivative curves are obvious. Significant variances appear after the
pressure disturbance reaches Region 2 on the dimensionless pseudo-pressure curves,
while three dimensionless pseudo-pressure derivative curves become the same at later
time because the value of rD1 is constant.
On the basis of Figure 5.4 - 5.8, the boundary between Regions 1 and 2 (rD1) and the
boundary between Regions 2 and 3 (rD2) have significant effects on pressure responses.
74
Figure 5.7 Dimensionless pseudo-pressure curves for different radii of boundary
between Regions 2 and 3
0.1
1
10
100
0.1 1 10 100 1000 10000 100000 1000000 10000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re
Dimensionless time
rD2=300
rD2=500
rD2=700
CT21=0.5, CT31=0.25 CS21=1, CS31=1 rD1=100
75
Figure 5.8 Dimensionless pseudo-pressure derivative curves for different radii of
boundary between Regions 2 and 3
0.1
1
10
0.1 1 10 100 1000 10000 100000 1000000 10000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re d
eri
vati
ve
Dimensionless time
rD2=300
rD2=500
rD2=700
CT21=0.5, CT31=0.25 CS21=1, CS31=1 rD1=100
76
5.2.3 Effect of transmissibility ratio
The differences between each region in a gas condensate reservoir are in terms of
transmissibility ratios and storability ratios. The effects of transmissibility are discussed
here. The transmissibility ratio includes three parameters: permeability ratio, viscosity
ratio and reservoir thickness. Generally, the thickness of a reservoir is considered to be
constant. The permeability ratio is positively correlated to the transmissibility ratio and
the viscosity is negatively correlated to the transmissibility ratio on the basis of the
definitions in Chapter 4.
The number of semi-analytical sub-segments is 80 and the dimensionless radius (rD) of
every sub-segment is 10 (This can guarantee the appearance of radial flow in the known
dimensionless time). The storability ratios are equal to 1 ( , ). The
values of the boundaries between Regions 1 and 2 (rD1) and between Regions 2 and 3 (rD2)
are constant.
In Figure 5.9, the dimensionless pseudo-pressure curves vary from each other due to
different values of transmissibility between Regions 1 and 2 (CT21). The larger value of
CT21 leads to a larger pressure drop. This is because the transmissibility in Region 1 is
better than that of Region 2, which means fluids flow less easily in Region 2. Finally, the
slopes of the dimensionless pseudo-pressure curves are the same due to the fact the fluid
flow ability in Region 3 is the same as the others. Figure 5.10 shows total differences on
pseudo-pressure derivative curves in the area of Region 2, which are caused by the
transmissibility ratios. A larger distinction between Regions 1 and 2 will result in a larger
variance. When pressure disturbance reaches the boundary between Regions 2 and 3, the
pseudo-pressure derivative curves gradually become the same line with the value of 0.5,
77
which means they reach radial flow. This is because the transmissibility ratios between
Regions 1 and 3 are the same for these cases.
78
Figure 5.9 Dimensionless pseudo-pressure responses for different transmissibility
ratios
0.1
1
10
100
0.1 1 10 100 1000 10000 100000 1000000 10000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re
Dimesionless time
CT21=5
CT21=2
CT21=1
CT21=0.5
CT21=0.2
CT31=1,CS21=1, CS31=1 rD1=100, rD2=400
79
Figure 5.10 Dimensionless pseudo-pressure derivative responses for different
transmissibility ratios
0.1
1
10
0.1 1 10 100 1000 10000 100000 1000000 10000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re d
eri
vati
ve
Dimensionless time
CT21=5
CT21=2
CT21=1
CT21=0.5
CT21=0.2
CT31=1, CS21=1, CS31=1 rD1=100 ,rD2=400
80
For Figures 5.11 and 5.12, the transmissibility between Regions 1 and 2 is kept
constant (CT21=0.5) for three different values of the transmissibility between Regions 1
and 3. Another case is the homogenous reservoir (CT21=1, CT31=1), which is used as the
standard. In Figure 5.11, three different regions can be identified clearly from the
dimensionless pseudo-pressure curves. The first distinction on the dimensionless pseudo-
pressure curves in Figure 5.11 can be used to identify the existence of Region 2. The
second distinction among the three dimensionless pseudo-pressure curves (CT21=0.5) in
Figure 5.11 is caused by the different transmissibility ratios between Regions 1 and 3. As
in Figure 5.11, the larger value of transmissibility between Regions 1 and 3 (CT31) leads
to a larger pressure drop. Figure 5.12 also reflects the differences of each region. When
pressure disturbance reaches the boundary between Regions 1 and 2 (rD1), the standard
case (CT21=1, CT31=1) can be classified from three other cases due to the different
transmissibility ratios. Then, the distinction appears when the pressure disturbance
reaches the boundary between Regions 2 and 3 (rD2) due to different transmissibility
ratios (CT31) in Figure 5.12. Finally, radial flow appears for every pseudo-pressure
derivative curve.
81
Figure 5.11 Dimensionless pseudo-pressure responses for different transmissibility
ratios
0.1
1
10
100
0.1 1 10 100 1000 10000 100000 1000000 10000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re
Dimensionless time
CT21=0.5, CT31=1
CT21=0.5, CT31=0.5
CT21=0.5, CT31=0.25
CT21=1, CT31=1
CS21=1, CS31=1 rD1=100, rD2=400
82
Figure 5.12 Dimensionless pseudo-pressure derivative responses for different
transmissibility ratios
0.1
1
10
0.1 1 10 100 1000 10000 100000 1000000 10000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re
Dimensionless time
CT21=0.5, CT31=1
CT21=0.5, CT31=0.5
CT21=0.5, CT31=0.25
CT21=1, CT31=1
CS21=1, CS31=1 rD1=100, rD2=400
83
5.2.4 Effect of storability ratio
The storability ratio is the other part that forms the diffusivity ratio. Based on the
definitions in Chapter 4, the storability ratio includes the initial porosity ratio, the total
compressibility ratio, and the reservoir thickness ratio (Equation 4.41 and 4.42). The
reservoir thickness ratios are generally considered to be constant, which means equal to 1.
The initial porosities in Regions 1, 2 and 3 are different due to the condensate dropping
out from the gas. And the total compressibility of every region is different from the others.
As a result, the storability ratio is positively correlated to the total compressibility ratio
and the initial porosity ratio. As shown in Figure 3.8, different compositions for the gas
condensate system lead to different values of total compressibility corresponding to
pressure. Due to the fact that the storability ratio is negatively correlated to the diffusivity
ratio, the total compressibility has a direct effect on the diffusivity ratio.
For Figures 5.13 and 5.14, the number of semi-analytical sub-segments is 80 and the
dimensionless radius (rD) of every sub-segment is 25 (This can guarantee the appearance
of radial flow in the known dimensionless time). The transmissibility ratios are assumed
to be constant (CT21=1, CT31=1). In addition, the value of storability between Regions 1
and 3 (CS21) is also equal to 1.
In Figure 5.13, the dimensionless pseudo-pressure curves show a small difference in
the area of Region 2, which is caused by the different values of storability ratios for
Region 2. Differences on the dimensionless pseudo-pressure curves exist only when the
pressure reaches the boundaries between Regions 1 and 2 (rD1) and between Regions 2
and 3 (rD2). Figure 5.14 shows that larger values of storability ratio between Regions 1
and 2 lead to higher humps when the pressure turbulence reaches Regions 2 and 3. Three
84
dimensionless pseudo-pressure derivatives will finally reach the radial flow after certain
humps. This is because the transmissibility ratios between each region are equal to 1. For
Region 2, the radial flow does not appear because the length of Region 2 is short.
85
Figure 5.13 Dimensionless pseudo-pressure responses for different storability ratios
0.1
1
10
0.1 1 10 100 1000 10000 100000 1000000 10000000 100000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re
Dimensionless time
Cs21=0.1
Cs21=0.2
Cs21=0.4
CT21=1, CT31=1, CS31=1 rD1=250 ,rD2=1000
86
Figure 5.14 Dimensionless pseudo-pressure derivative responses for different
storability ratios
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1 10 1000 100000 10000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re d
eri
vati
ve
Dimensionless time
Cs21=0.1
Cs21=0.2
Cs21=0.4
CT21=1, CT31=1, CS31=1 rD1=250, rD2=1000
87
The Figure 5.15 will show the existence of radial flow in Region 2. For Figure 5.15,
the number of semi-analytical sub-segments is 200 and the dimensionless radius (rD) of
every sub-segment is 25 (This can guarantee the appearance of radial flow in the known
dimensionless time). The transmissibility ratios are assumed to be constant (CT21=1,
CT31=1). Figure 5.15 is used to prove that storability ratios have a direct effect on the
humps on the dimensionless pseudo-pressure curves. The area of Region 2 should be long
enough to ensure the appearance of radial flow. For the case (CS21=0.8, CS31=1), when
pressure reaches the boundary between Regions 1 and 2 (rD1), a hump appears due to the
difference from the storability ratio (CS21) and then the derivative curve reaches to a value
of 0.5 during the period of the radial flow area due to the same transmissibility (CT21=1).
When the pressure disturbance reaches the boundary between Regions 2 and 3 (rD2),
another hump appears which is opposite to the first hump. This is also caused by the
difference from the storability ratio (CS21=0.8, CS31=1). Finally, the radial flow period is
reached. The other two cases (CS21=1, CS31=1.25 and CS21=0.8, CS31=0.8) can be taken as
the application of two region composite models and be used to validate the results from
the other case (CS21=0.8, CS31=1).
For Figure 5.16, the number of semi-analytical sub-segments is 80 and the
dimensionless radius (rD) of every sub-segment is 10 (This can guarantee the appearance
of radial flow in the known dimensionless time). The transmissibility ratios are assumed
as constant (CT21=1, CT31=1). The storability ratios are also given the same value
(CS21=0.1, CS31=0.1). Figure 5.16 shows that the dimensionless pseudo-pressure
derivative curves are the same for different values of the boundary between Regions 2
and 3 (rD2). There is a hump existing when the pressure disturbance reaches the boundary
88
between Regions 1 and 2 (rD1). The increase of the boundary between Regions 2 and 3
has no effect and no hump appears at the late period. This is because the storability ratios
for Regions 2 and 3 are the same (CS21=0.1, CS31=0.1). This can be used to validate the
semi-analytical model because the transmissibility ratio and storability ratio between
Regions 2 and 3 are totally the same, not affected by the boundary.
89
Figure 5.15 Dimensionless pseudo-pressure responses for different storability ratios
0.1
1
1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08 1.00E+09
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re d
eri
vati
ve
Dimensionless time
Cs21=1, Cs31=1.25
Cs21=0.8, Cs31=0.8
Cs21=0.8, Cs31=1
CT21=1, CT31=1 rD1=125, rD2=4250
90
Figure 5.16 Dimensionless pseudo-pressure derivative responses for different rD2 for
the same value of storability ratios
0.1
1
0.1 1 10 100 1000 10000 100000 1000000 10000000 100000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re d
eri
vati
ve
Dimensionless time
rD2=200
rD2=400
rD2=600
CT21=1, CT31=1 CS21=0.1,CS31=0.1 rD1=100
91
5.2.5 Effect of sub-segment number
The semi-analytical model is based on dividing the whole reservoir into many sub-
segments. This section discusses the effect of the number of sub-segments (N). The
radius of every sub-segment (rD) is corresponding to the sub-segment number. The
chosen number of sub-segments and the radius of every sub-segment can guarantee the
appearance of radial flow in the known dimensionless time. The transmissibility and
storability ratios are assumed to be constant (CS21=1, CS31=1). And another case is
simulated here to be used as the reference.
For Figure 5.17 and Figure 5.18:
Case 1: N=40, rD=20, CT21=0.5, CT31=0.25.
Case 2: N=80, rD=10, CT21=0.5, CT31=0.25.
Case 3: N=200, rD=4, CT21=0.5, CT31=0.25.
Case 4: N=80, rD=10, CT21=1, CT31=0.25.
In Figure 5.17, the dimensionless pseudo-pressure curves for Cases 1, 2 and 3 are
totally the same. However, they are different from Case 4, which is used as the standard.
The values of the dimensionless pseudo-pressure curves for Cases 1, 2 and 3 are larger
than that of Case 4 due to different transmissibilities. As shown in Figure 5.18, the
dimensionless pseudo-pressure derivative curves for Cases 1, 2 and 3 are also the same.
Case 4 is used to identify the existence of Region 2 here because the transmissibility
ratios between Regions 1 and 3 are the same for all cases (CT31=0.25). Based on the
above statements, the numbers of sub-segments have no effect on the pressure responses
calculated from the semi-analytical model.
92
For Figure 5.19 and Figure 5.20:
Case 1: N=6, rD=10, CT21=0.5, CT31=0.25.
Case 2: N=60, rD=1, CT21=0.5, CT31=0.25.
Case 3: N=6, rD=1, CT21=1, CT31=1.
Case 4: N=6, rD=1, CT21=0.5, CT31=0.5.
Figures 5.19 and 5.20 are used to examine the effects of the sub-segment number by
application of a smaller sub-segment number and a smaller radius of every sub-segment.
Similar to Figure 5.17, the dimensionless pseudo-pressure curves for Cases 1 and 2 are
the same in Figure 5.19. Cases 3 and 4 show an obvious distinction from Cases 1 and 2
due to different transmissibility ratios. In Figure 5.20, Case 3 reaches the period of
radial flow first and then does Case 4. Cases 1 and 2 are last to reach the radial flow.
Cases 3 and 4 are used to show the existence of Regions 1 and 2 here. On the basis of the
above statements, smaller values for the sub-segment number and corresponding radius
still have no effect on the pressure responses calculated from the semi-analytical model.
On the basis of Figures 5.17 - 5.20, the numbers of sub-segments have no effect on
the pressure responses.
93
Figure 5.17 Dimensionless pseudo-pressure responses for different sub-segment
numbers
0.1
1
10
100
0.1 1 10 100 1000 10000 100000 1000000 10000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re
Dimensionless time
N=40, rD=20, CT21=0.5, CT31=0.25
N=80, rD=10, CT21=0.5, CT31=0.25
N=200, rD=4, CT21=0.5, CT31=0.25
N=80, rD=10, CT21=1, CT31=0.25
CS21=1, CS31=1 rD1=100, rD2=400
94
Figure 5.18 Dimensionless pseudo-pressure derivatives responses for different sub-
segment numbers
0.1
1
10
0.1 1 10 100 1000 10000 100000 1000000 10000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re d
eri
vati
ve
Dimensionless time
N=40, rD=20, CT21=0.5, CT31=0.25
N=80, rD=10, CT21=0.5, CT31=0.25
N=200, rD=4, CT21=0.5, CT31=0.25
N=80, rD=10, CT21=1, CT31=0.25
CS21=1, CS31=1 rD1=100, rD2=400
95
Figure 5.19 Dimensionless pseudo-pressure responses for different sub-segment
numbers
0.1
1
10
100
0.1 1 10 100 1000 10000 100000 1000000 10000000 100000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re
Dimensionless Time
N=6,rD=10,CT21=0.5,CT31=0.25
N=60,rD=1,CT21=0.5,CT31=0.25
N=6,rD=10,CT21=1,CT31=1
N=6,rD=10,CT21=0.5,CT31=0.5
CS21=1, CS31=1 rD1=10, rD2=30
96
Figure 5.20 Dimensionless pseudo-pressure derivatives responses for different sub-
segment numbers
0.1
1
10
0.1 1 10 100 1000 10000 100000 1000000 10000000 100000000
Dim
en
sio
nle
ss p
seu
do
-pre
ssu
re d
eri
vati
ve
Dimensionless time
N=6, rD=10, CT21=0.5, CT31=0.25
N=60, rD=1, CT21=0.5, CT31=0.25
N=6, rD=10, CT21=1, CT31=1
N=6, rD=10, CT21=0.5, CT31=0.5
CS21=1, CS31=1 rD1=10, rD2=30
97
5.2.6 Flow rate profile
Different points located in the reservoir have different pressure drops. For a single
well model, the pressure is decreased at different radii that are not the same as in the area
of pressure disturbance.
For Figure 5.21, the number of the semi-analytical sub-segments is 80 and the
dimensionless radius (rD) of every sub-segment is changing in order to calculate the
dimensionless flow rate profile at different dimensionless times. The transmissibility and
storability ratios are set as: CT21=1, CT31=1, CS21=1, CS31=1. Figure 5.21 shows the
dimensionless flow rate profile for the drawdown test at different dimensionless times.
When the dimensionless time is longer, the area where the flow forms is larger, which
represents the area of pressure disturbance.
For Figure 5.22, the number of semi-analytical sub-segments is 80 and the
dimensionless radius (rD) of every sub-segment is also changing in order to calculate the
dimensionless flow rate profiles for different transmissibility ratios. Figure 5.22 shows
the flow rate profile for different mobility ratios at the same time, illustrating that the
mobility ratio has a great effect on the pressure disturbance spread.
98
Figure 5.21 Dimensionless flow rate profile for the drawdown test at different
dimensionless times
0.00
0.20
0.40
0.60
0.80
1.00
1 10 100 1000 10000
Dim
en
sio
nle
ss F
low
rat
e
Dimensionless Radius
tD=10
tD=100
tD=1000
tD=10000
tD=100000
CT21=1, CT31=1 CS21=1, CS31=1
99
Figure 5.22 Dimensionless flow rate profile for different mobility ratios
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1 10 100 1000 10000
Dim
en
sio
nle
ss F
low
Rat
e
Dimensionless Radius
M12=M23=0.2
M12=M23=1
M12=M23=5
CT21=5, CT31=25 CT21=1, CT31=1 CT21=0.2, CT31=0.04
CS21=1, CS31=1 tD=1000
100
CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS
6.1 Conclusions
The compositions of gas condensate reservoirs have direct effects on PVT
properties. Liquid volume, compressibility factor and viscosity are affected by
different compositions and heavier components have more significant effects.
A modified definition of total compressibility is proposed for the first time and
total compressibility for four different compositions of gas condensate systems
are performed , which reflect that a small increase of heavier components still
have obvious effects.
A three region radial composite model has been developed semi-analytically to
investigate dynamic performances of an infinite homogeneous retrograde gas
condensate reservoir based on multiple radial composite regions around the
testing well. Pseudo-variables that integrate the pressure dependent properties
have been applied with corresponding to relevant properties of gas.
The total compressibility is an important factor that affects the well testing type
curves for a gas condensate reservoir.
Transmissibility and storability ratios between different regions are the most
important factors to characterize effects of fluid properties on transient pressure
responses.
101
Different values of the boundaries between three regions determine the well
testing type curves directly with respect to constant values of transmissibility and
storability ratios.
The range of the region should be large enough to prove the effects of storability
ratios between three regions.
The numbers of sub-segments don’t affect the transient pressure responses when
other parameters are the same.
With known PVT properties, this model can provide reliable perspectives of
transient pressure analysis in retrograde condensate reservoirs and helps
characterize and estimate the drainage areas in a gas condensate reservoir.
6.2 Recommendations
1) It is necessary to analyze field testing data by the application of the analytical
model and semi-analytical model proposed in the thesis.
2) More laboratory experiments should be conducted to characterize the PVT
properties and measure the permeability of a gas condensate reservoir in details.
3) The semi-analytical model can be extended to describe liquids rich shale reservoirs
by combining source function.
4) Non-Darcy effect should be examined on more field cases.
102
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