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The thesis of Pichit Vardcharragosad was reviewed and approved* by the following:
Luis F. AyalaAssociate Professor of Petroleum and Natural Gas EngineeringThesis Advisor
R. Larry GraysonProfessor of Energy and Mineral EngineeringGraduate Program Officer of Energy and Mineral Engineering
Li LiAssistant Professor of Energy and Mineral Engineering
Yaw D. YeboahProfessor of Energy and Mineral EngineeringHead of the Department of Energy and Mineral Engineering
*Signatures are on file in the Graduate School
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iv
condensate tank model for the case volatile oil reservoirs are addressed. Further recommended
studies on negative value of decline exponent variable and expanding current capability of the
proposed model are also presented.
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v
TABLE OF CONTENTS
List of Figures .......................................................................................................................... viiList of Tables ........................................................................................................................... ixNomenclature ........................................................................................................................... xAcknowledgements .................................................................................................................. xvChapter 1 Introduction ............................................................................................................. 1Chapter 2 Background ............................................................................................................. 3
2.1 Gas Condensate Hydrocarbon Fluid .......................................................................... 32.2 Modified Black-Oil Model ......................................................................................... 52.3 Zero-Dimensional Reservoir Model .......................................................................... 72.4 Field Performance Prediction ..................................................................................... 82.5 Visual Basic for Applications (VBA) ........................................................................ 12
Chapter 3 Problem Statement .................................................................................................. 13Chapter 4 Model Description ................................................................................................... 15
4.1 Phase Behavior Model (PBM) ................................................................................... 164.1.1 Compressibility Factor .................................................................................... 174.1.2 Vapor-Liquid Equilibrium ............................................................................... 214.1.3 Fluid Property Prediction ................................................................................ 274.1.4 Phase Stability Analysis .................................................................................. 37
4.2 Standard PVT Properties ............................................................................................ 424.2.1 Definitions, Mathematic Relationships, and Characteristics ........................... 434.2.2 Obtaining Standard PVT Properties from Laboratory PVT Reports ............... 514.2.3 Obtaining Standard PVT Properties from a Phase Behavior Model ............... 60
4.3 Zero-Dimensional Reservoir Model .......................................................................... 704.3.1 Generalized Material Balance Equation .......................................................... 714.3.2 Material Balance Equation for a Gas Condensate Fluid ................................. 754.3.3 Phase Saturation Calculations ......................................................................... 804.3.4 Volumetric OGIP/OOIP Calculations ............................................................. 82
4.4 Flow Rates and Flowing Pressures Calculation ......................................................... 844.4.1 Inflow Performance Relationship (IPR) .......................................................... 854.4.2 Tubing Performance Relationships ................................................................. 914.4.3 Nodal Analysis ................................................................................................ 96
4.5 Field Performance Prediction ..................................................................................... 994.5.1 Performance during Plateau Period ................................................................. 1004.5.2 Performance during Decline Period ................................................................ 1034.5.3 Annual Production Calculation ....................................................................... 106
4.6 Economic Analysis and Field Optimization............................................................... 1094.6.1 Simplified Economic Model ........................................................................... 1104.6.2 Field Optimization ........................................................................................... 116
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LIST OF FIGURES
Figure 2-1: Phase Diagram of Typical Gas Condensate Reservoir .......................................... 3Figure 2-2: Distributions of Pseudo Components among Phases in Modified Black-Oil
Model ............................................................................................................................... 5Figure 2-3: Graphical Representation of Zero-Dimensional Reservoir Model ........................ 7Figure 2-4: Typical Field Performance of Gas Condensate Gas and Oil Flow Rates vs.
Time ................................................................................................................................. 9Figure 2-5: Typical Field Performance of Gas CondensateReservoir Pressure,
Bottomhole Flowing Pressure and Wellhead Pressure vs. Time ..................................... 10Figure 2-6: Typical Field Performance of Gas Condensate Cumulative Gas and Oil
Production vs. Time ......................................................................................................... 11Figure 4-1: Graphical Representation of Standard PVT Properties ......................................... 44Figure 4-2: Typical Characteristic of Gas Formation Volume Factor () and Volatilized
Oil-Gas Ratio () for Gas Condensate .......................................................................... 48Figure 4-3: Typical Characteristic of Oil Formation Volume Factor () and Solution
Gas-Oil Ratio () for Gas Condensate ........................................................................... 48Figure 4-4: Graphical Representation of CVD Data used in Walsh-Towler Algorithm .......... 53Figure 4-5: Graphical Representation of Nodal Analysis ........................................................ 96
Figure 4-6: Graphical Representation of Field Optimization .................................................. 116Figure 5-1: Simulated Gas Formation Volume Factor and Volatilized Oil-Gas Ratio of
Gas Condensate ................................................................................................................ 118Figure 5-2: Simulated Oil Formation Volume Factor and Solution Gas-Oil Ratio of Gas
Condensate ....................................................................................................................... 119Figure 5-3: Simulated Specific Gravity of Reservoir Gas ....................................................... 120Figure 5-4: Volumes of Surface Gas Pseudo Component in Reservoir Gas Reservoir Oil,
and Cumulative Gas Production ....................................................................................... 121Figure 5-5: Volumes of Stock-Tank Oil Pseudo Component in Reservoir Gas, Reservoir
Oil, and Cumulative Oil Production................................................................................. 122Figure 5-6: Densities of Surface Gas and Stock-Tank Oil Pseudo Components at First
Stage Separator, Second Stage Separator and Stock Tank Condition .............................. 124
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Figure 5-7: Volumes of Stock-Tank Oil Pseudo Component in Reservoir Gas, ReservoirOil, and Cumulative Oil Production in term of Gas-Equivalent ...................................... 125
Figure 5-8: Total Volumes of Stock-Tank Oil Pseudo Component and Surface Gas
Pseudo Component in term of Gas-Equivalent ................................................................ 126Figure 5-9: Simulated Production Results of Gas Condensate using Simplified Gas
Condensate Tank Model .................................................................................................. 130Figure 5-10: Phase Envelope and Reservoir Depletion Paths at Two Different Reservoir
Temperatures .................................................................................................................... 133Figure 5-11: Simulated Gas Formation Volume Factor and Volatilized Oil-Gas Ratio of
Volatile Oil using Gas Condensate PVT Model .............................................................. 134Figure 5-12: Simulated Oil Formation Volume Factor and Solution Gas-Oil Ratio of
Volatile Oil using Gas Condensate PVT Model .............................................................. 134Figure 5-13: Simulated Production Results of Volatile Oil using Simplified Gas
Condensate Tank Model .................................................................................................. 135Figure 5-14: Mole Fraction Behavior of Vapor Phase Molar Fraction ( ) for Gas
Condensates and Volatile Oils ......................................................................................... 136Figure 5-15: Cumulative Gas and Oil Production vs. Time..................................................... 138Figure 5-16: Total Gas and Oil Flow Rates vs. Time .............................................................. 139Figure 5-17: Reservoir Pressure, Bottomhole Flowing Pressure, and Wellhead Pressure
vs. Time ............................................................................................................................ 140Figure 5-18: Gas Saturation and Specific Gravity of Reservoir Gas vs. Time ........................ 142Figure 5-19: Total Gas Flow Rate () vs. Cumulative Gas Production during Decline
Period .................................................................................................. 143Figure 5-20: Decline Rate () vs. Cumulative Gas Production during Decline Period
() ........................................................................................................... 145Figure 5-21: Annual Expenditure, Annual Total Revenue, and Cumulative Discounted
Net Cash Flow vs. Production Time ................................................................................ 147Figure 5-22: Net Present Value vs. Interest Rate ..................................................................... 148Figure 5-23: Field Optimization Results .................................................................................. 149
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ix
LIST OF TABLES
Table 4-1: Volume-Translation Coefficients for Pure Components (Whitson and Brule,2000) ................................................................................................................................ 31
Table A-1: Pressures and Temperatures for Standard PVT Properties CalculationSubroutine ........................................................................................................................ 163
Table A-2: Physical Properties of Pure Components ............................................................... 163Table A-3: Binary Interaction Coefficients of Pure Components ............................................ 164Table A-4: Volume Translation Coefficient of Pure Components .......................................... 164Table A-5: Reservoir Input Data .............................................................................................. 165Table A-6: Relative Permeability Input Data .......................................................................... 165Table A-7: Standard PVT Properties ....................................................................................... 166Table A-7: Standard PVT Properties (Cont.) ........................................................................... 167Table A-8: Tubing Input Data .................................................................................................. 168Table A-9: Economic Input Data ............................................................................................. 168Table A-9: Economic Input Data (Cont.)................................................................................. 169Table A-10: Field Performance Prediction Input ..................................................................... 169Table A-11: Field Performance Optimization Input ................................................................ 169
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NOMENCLATURE
Normal Symbol Definition
Reservoir drainage area Hyperbolic decline exponent Co-volume parameter of i-th component Formation volume factor Gas formation volume factor Oil formation volume factor Two-phase gas formation volume factor Two-phase oil formation volume factor Overall molar faction of i-th component Formation (rock) compressibility
Deitz shape factor
Non-Darcy coefficient / Tubing Diameter Decline rate Expansivity of formation (rock) Expansivity of reservoir gas Expansivity of reservoir oil Expansivity of reservoir water Efficiency factor of tubing Fannings friction factor Fugacity of i-th component in vapor phase Fugacity of i-th component in liquid phase
Moodys friction factor
Molar fraction of vapor phase
Molar fraction of liquid phase at reservoir condition Molar fraction of liquid phase Molar fraction of liquid phase at first-stage separator Molar fraction of liquid phase at first-stage separator produced fromreservoir gas Molar fraction of liquid phase at first-stage separator produced fromreservoir oil Molar fraction of liquid phase at second-stage separator Molar fraction of liquid phase at second-stage separator produced fromreservoir gas
Molar fraction of liquid phase at second-stage separator produced fromreservoir oil Molar fraction of liquid phase at stock-tank condition Molar fraction of liquid phase at stock-tank condition produced fromreservoir gas Molar fraction of liquid phase at stock-tank condition produced fromreservoir oil Fugacity of i-th component in original fluid
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Fugacity of i-th component in liquid-like phase Fugacity of i-th component in vapor-like phase
Amount of surface gas pseudo component / Gas in place
Amount of gas-equivalent pseudo component Amount of surface gas pseudo component in reservoir gas phase Amount of surface gas pseudo component in reservoir oil phase Amount of cumulative gas injection Amount of cumulative gas production Cumulative gas production at year Cumulative gas production at abandonment condition Cumulative gas production at end of plateau Cumulative gas recovery Incremental of cumulative gas production
Annual gas production at year
Incremental of gas recovery
Reservoir thickness Elevation of upstream node Elevation of downstream node Difference in elevation of downstream and upstream node Absolute permeability of reservoir Effective permeability of reservoir gas Relative permeability of reservoir gas Relative permeability of reservoir oil Volatility ratio of i-th component Tubing length
Temperature dependency coefficient of i-th component
Molecular weight of vapor phase Molecular weight of gas at reservoir condition Molecular weight of i-th component Molecular weight of oil at reservoir condition Molecular weight of oil at stock-tank condition Molecular weight of oil at stock-tank condition produced from reservoirgas Molecular weight of oil at stock-tank condition produced from reservoiroil Molecular weight of liquid phase
Molecular weight of remaining fluid inside PVT cell
Number of component in multi-component hydrocarbon
Mole fraction of excess gas removed from PVT cell Mole fraction of remaining gas inside PVT cell Mole fraction of remaining gas inside PVT cell plus excess gas Mole fraction of remaining oil inside PVT cell Mole fraction of remaining fluid inside PVT cell Amount of stock-tank oil pseudo component / Oil in place Amount of stock-tank oil pseudo component in reservoir gas phase Amount of stock-tank oil pseudo component in reservoir oil phase
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Amount of cumulative oil production Cumulative oil production at year
Cumulative oil production at abandonment condition
Cumulative oil recovery
Reynolds number Incremental of cumulative oil production Annual oil production at year Incremental of oil recovery Original gas in place Original oil in place Pressure Upstream pressure Downstream pressure Average pressure between upstream and downstream
Critical pressure of i-th component
Drawdown pressure inside the reservoir
Pseudocritical pressure Reservoir pressure Reservoir pressure at abandonment condition Reservoir pressure at end of plateau Reduced pressure of i-th component Pressure at standard condition Bottomhole flowing pressure Bottomhole flowing pressure at end of plateau Wellhead pressure Minimum allowable wellhead pressure
Pressure drop from initial reservoir pressure
Total gas flow rate of the field Total gas flow rate of the field at abandonment condition Total gas flow rate of the field during plateau period Total oil flow rate of the field Total oil flow rate of the field at abandonment condition Gas flow rate per well Gas flow rate per well during plateau period Annual average gas flow rate of the field Annual average oil flow rate of the field
Reservoir radius
Wellbore radius Universal gas constant Gas-oil equivalent factor Fugacity ratio of i-th component Solution gas-oil ratio Solution gas-oil ratio at bubble point pressure Volatilized oil-gas ratio Volatilized oil-gas ratio at dew point pressure Target recovery factor at end of plateau
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Fugacity ratio of i-th component in liquid-like phase Fugacity ratio of i-th component in vapor-like phase
Total skin factor
Volume-translate coefficient of i-th component
Mechanical skin factor Average reservoir gas saturation Minimum gas saturation Sum of the mole number of liquid-like phase Average reservoir oil saturation Sum of the mole number of vapor-like phase Average reservoir water saturation Connate water saturation Specific gravity of gas Production time Production time at abandonment condition Production time at end of plateau Temperature Temperature of upstream node Temperature of downstream node Pipe section average temperature Critical temperature of fluid Critical temperature of i-th component Pseudocritical temperature of Reduced temperature of i-th component Temperature at standard condition Fluid velocity
Retrograde liquid volume fraction
Amount of excess gas at reservoir condition Amount of remaining gas phase at reservoir condition Amount of remaining gas phase plus excess gas at reservoir condition Amount of remaining oil phase at reservoir condition Pore volume of reservoir Original volume of PVT cell Molar volume of phase a calculated from EOS Critical molar volume of i-th component Molar volume of vapor phase Molar volume of vapor phase calculated from EOS
Molar volume of liquid phase
Molar volume of liquid phase calculated from EOS Pseudocritical molar volume Amount of water pseudo component in reservoir water Amount of water influx Amount of cumulative water injection Amount of cumulative water production Molar fraction of surface gas pseudo component in reservoir oil Molar fraction of i-th component in liquid phase Molar fraction of stock-tank oil pseudo component in reservoir oil
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Molar fraction of surface gas pseudo component in reservoir gas Molar fraction of i-th component in vapor phase Molar fraction of stock-tank oil pseudo component in reservoir gas
Molar fraction of i-th component in liquid-like phase
Molar fraction of i-th component in vapor-like phase Molar number of i-th component in liquid-like phase Molar number of i-th component in vapor-like phase Compressibility factor of fluid Two-phase compressibility factor Compressibility factor of phase a Average compressibility factorGreek Symbol Definition
Coefficient to adjust relative permeability of reservoir gas
Turbulence parameter
Specific gravity of gas Binary interaction coefficient between i-th and j-th component Tubing roughness Fluid viscosity Viscosity of vapor phase / Viscosity of reservoir gas Viscosity of i-th component at low pressure Viscosity of liquid phase Viscosity of liquid phase at low pressure Viscosity of reservoir oil Fluid density
Density of vapor phase
Density of gas phase at reservoir condition Density of liquid phase Density of oil phase at reservoir condition Density of oil phase at stock-tank condition Density of oil phase at stock-tank condition produced from reservoir gas Density of oil phase at stock-tank condition produced from reservoir oil Pseudo reduced density of liquid phase Density of remaining fluid inside PVT cell Molar density of reservoir gas Molar density of surface gas pseudo component
Molar density of reservoir oil
Molar density of stock-tank oil pseudo component Annual production time Average reservoir porosity Fugacity coefficient of i-th component in vapor phase Fugacity coefficient of i-th component Fugacity coefficient of i-th component in liquid phase Pitzers acentric factor of i-th component
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xv
ACKNOWLEDGEMENTS
First and foremost I would like to thanks my advisor, Dr. Luis Ayala, for his continuous
guidance, support and friendship throughout my graduate study. Without his encouragement and
invaluable advice, this research would not have been completed. Additional thanks are extended
to Dr. Larry Grayson, and Dr. Li Li for their interest and time in serving as my thesis committee.
I would like to express my sincere appreciation to Dr. Turgay Ertekin and Dr. Russel
Johns, and Dr. Zuleima Karpyn for the fundamental knowledge they have taught. I am also very
grateful for educational environment that the faculty and staff of the Department of Energy and
Mineral Engineering have created. I highly thank my sponsor, PTT Exploration and Production
Company, for every support they have given.
Many friends and colleagues have been very supportive. I would like to express my
gratitude to Pipat Likanapaisal, Nithiwat Siripatrachai and Kanin Bodipat who always are good
friends throughout my student life at Pennsylvania State University. I also thank all of my
colleagues for making me have meaningful time and experience.
Finally, but most deeply, I am forever in dept to my family, my father Phiraphong
Vardcharragosad, my mother Pikun Tanarungreung, my sister Pungjai Keandoungchun, and
sisters family, for their support, encouragement, and most importantly their tolerance.
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Chapter 1Introduction
Natural gas is a natural occurring gas which consisting of methane primarily. It plays a
significant role in global economic as one of the main sources of energy. In 2009, world natural
gas reserves equaled 6.29 Trillion Standard Cubic Feet (TCF) while world production reached
106 BCF for the year (EIA, 2011). Conventional reservoirs consist of five different fluid types:
dry gas, wet gas, retrograde gas, volatile oil, and black oils (McCain, 1990). They are
distinguished from each other based on the present of fluid phases inside the reservoir and at
surface production facilities.
Field development and investment decisions in petroleum and natural gas require an
integration of expertise from various areas including geology, reservoir, drilling, completion,
process, and economic. Location and size of reservoirs, production rates and time, total
recoverable volumes, number of wells and platforms, drilling and completion techniques,
processing facilities scheme, cost and revenue, etc. are examples of information required for
adequate field development decisions. Field performance indicators consist of information
regarding flow rates, pressures, and production time is very important for field development. If
field performance indicators are satisfactorily predicted, the hydrocarbon field could be
developed using the best possible exploitation strategy while optimizing its economic
performance. If not, the field might end up with too many wells, processing facilities that are too
large, or wrong equipment sizing which can jeopardize profits or even lead to significant losses of
investors capital.
In modern age, computer simulation is used to simulate various types of mathematical
models which can couple geological, fluid property, reservoir, production network, processing
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facilities, and economic information. Field performance could be predicted by integrating these
models together. However, the required type of mathematical model needs to be carefully
selected to be able to perform the calculation most effectively. For reservoir characterization, for
example, the modeler might utilize either afully dimensional numerical model- which can aptly
capture all reservoir heterogeneities and geometry by discretizing it into many small grids -, or a
zero-dimensional model - which assumes average reservoir and fluid properties across the
domain. For fluid behavior characterization, the modeler might select either a fully compositional
modelbased on the use of an equation of state and detailed fluid composition data, or a black-oil
model - which uses the pseudo-component concept and relies on PVT laboratory results.
Selection of those models generally depends on availability of input data, time constraint, and
required accuracy of simulation results. In this study, a zero dimensional model coupled with a
black-oil PVT fluid description is implemented for the study of field development optimization
strategies in retrograde natural gas reservoirs.
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Chapter 2Background
2.1 Gas Condensate Hydrocarbon Fluid
A gas condensate, retrograde gas condensate, or retrograde gas, is one of the five
reservoir fluid types (McCain, 1990). The typical phase envelope of gas condensate reservoirs is
shown in Figure 2-1. Gas condensates contain more intermediate and heavy hydrocarbon
components more than dry gases or wet gases. As shown in Figure 2-1, their reservoir
temperature is located in between the fluids critical temperature and their cricondentherm. The
reservoir depletion path of a gas condensate fluid typically crosses the dew point line and a liquid
phase appears at reservoir pressures lower than that of the dew point. The presence of liquid
phase in the reservoir significantly increases the system complexity, even if this liquid phase does
not flow and is very unlikely to be produced under normal production conditions.
Figure 2-1: Phase Diagram of Typical Gas Condensate Reservoir
Re
servoirPressure
Reservoir Temperature
Reservoir
Depletion
Path
Surface
Depletion
Path
Critical Point
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The general characteristics of gas condensate reservoir fluid can be summarized as
follows (Walsh and Lake, 2003):
Initial Fluid Molecular Weight: 2340 lb/lbmol Stock-Tank Oil Color: Clear to Orange Stock Tank Oil Gravity: 4560 API C7-plus Mole Fraction: 0.010.12 Typical Reservoir Temperature: 150300 F Typical Reservoir Pressure: 15009000 psia Volatilized Oil-Gas Ratio: 50300 STB/MMSCF Primary Recovery of Original Gas In Place: 70%85% Primary Recovery of Original Oil In Place: 30% - 60%
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2.2 Modified Black-Oil Model
A black-oil fluid model is a fluid characterization formulation which represents multi-
component hydrocarbon mixture in terms of two hydrocarbon pseudo components, namely the
surface gas and stock-tank oil pseudo components. In a traditional black-oil model, the
solubility of the surface gas pseudo component in the reservoir oil fluid phase is taken into
account while the solubility of stock-tank oil pseudo component in reservoir gas phase is
neglected. The modified black-oil model which also called two-phase two-pseudo component
model does not neglect the stock tank oil solubility in the gaseous reservoir phase, thus
including both solubility variables into the formulation. Figure 2-2 shows the distribution of
surface gas and stock-tank oil pseudo components among reservoir gas and reservoir oil phases.
Figure 2-2: Distributions of Pseudo Components among Phasesin Modified Black-Oil Model
The assumptions behind the modified black-oil PVT model can be summarized as
follows (Walsh and Lake, 2003 and Whitson and Brule, 2000):
There are two pseudo components which are surface gas and stock-tank oil. There are two fluid phases which are reservoir gas (vapor) and reservoir oil
(liquid) phases.
Surface GasStock-Tank
Oil
Surface Gas Stock-Tank Oil
Reservoir Gas
Phase
Reservoir Oil
Phase
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Surface gas pseudo component is reservoir fluid which remains in gas phase atstandard condition.
Stock-tank oil pseudo component is reservoir fluid which remains in oil phase atstandard condition.
The reservoir gas phase, which is reservoir fluid remains in vapor phase atreservoir condition, consists of surface gas and stock-tank oil pseudo
components.
The reservoir oil phase, which is reservoir fluid remains in liquid phase atreservoir condition, consists of surface gas and stock-tank oil pseudo
components.
Properties of surface gas and stock-tank oil pseudo components remain the samethroughout the reservoir depletion.
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2.3 Zero-Dimensional Reservoir Model
The Material Balance Equation (MBE) is a specialized type of mass balance equation that
combines mass balance equations of all pseudo components present in the reservoir into single
equation. The MBE is also called zero-dimensional reservoir modelor tank model because it
assumes that a reservoir behaves like a homogeneous tank with average rock and fluid properties
across the domain. Pressure, temperature, and compositional gradients are thus neglected. MBEs
can be derived from integrating diffusivity equations over space and time.
Figure 2-3: Graphical Representation of Zero-Dimensional Reservoir Model
(Source: http://www.joe.co.jp/english/menu2-5.html)
The following assumptions are implemented in traditional in zero-dimensional reservoir
models:
Reservoir is isothermal Reservoir is under thermodynamic equilibrium condition There are no chemical and biological reaction in reservoir Capillary pressures of reservoir fluids are negligible Gravitational gradients in reservoir are negligible Pressure gradients in reservoir are negligible
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Figure 2-5: Typical Field Performance of Gas CondensateReservoir Pressure, Bottomhole Flowing Pressure
and Wellhead Pressure vs. Time
Figure 2-5 demonstrates that, during field development calculations, reservoir pressure
() decreases as production time increases because more oil and gas are being removed from thereservoir. Wellhead pressure () is also continuously decreased in time in order to maintain thegas flow rate () per well during the build-up and plateau periods. After that, once wellheadpressure () reaches the minimum allowable wellhead pressure at surface conditions, theplateau gas flow rate () cannot be maintained any longer and the decline period starts.Bottomhole flowing pressure (
) changes along with changes in reservoir pressure (
) and
wellhead pressure () in order to provide the required pressure drop within the reservoir andproduction tubing.
Pressure
Production Time
pr
pwf
pwh
Build-up Plateau Decline
Minimum Allowable Wellhead Pressure
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2.5 Visual Basic for Applications (VBA)
Visual Basic for Applications (VBA) is a programming language from Microsoft. The
program is built into most MS-Office applications i.e. MS-Word, MS-Excel, MS-Access. Users
can use VBA to create calculation subroutine and control user interface features such as menus,
toolbars, worksheets, charts, etc (Walkenbach, 2007). VBA can only run within the host
application, and not as a standalone application. VBA is functionally rich, and flexible. Because it
is built into MS-Office applications, VBA subroutines will be able to execute so long as those
applications are available on computer machines. MS-Excel with built in VBA is a very favorable
platform for developing simulations. The main reasons are that most of engineers are familiar
with MS-Excel application and MS-Excel itself is user-friendly software with many useful built-
in features. Excels worksheets could be used as table to store input data. Simulation results could
be easily stored in the tabular form and displayed on various types of built-in chart.
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black oil PVT formulation for fluid properties calculation. These models are relatively simple, but
fast, reliable, and robust. Results show that the proposed model is able to predict field
performance while faithfully capturing the most salient characteristics of gas condensate
reservoirs. In addition, optimization on targeted variables can be accomplished without difficulty.
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Chapter 4Model Description
The proposed field performance predictor has been developed using Microsoft Excel with built-in
Visual Basic for Applications (VBA) subroutines. Workflow begins with the simulation of
standard black-oil PVT properties, which could be done either based on standard PVT laboratory
results (such as the Constant Volume Expansion or CVD) or via a phase behavior model based on
cubic equations of state. Next, field performance data is calculated by integrating a zero-
dimensional reservoir model, standard PVT fluid properties, well performance models for flow
rates and pressure calculation, and production constraints. Based on this, an economic analysis
can be performed based on simplified economic model. Finally, optimization on target variables
can be carried out by evaluating field performance and net present value repeatedly for different
and plausible production scenarios. The proposed simulation tool has been designed to simulate a
single gas condensate reservoir based on the continuous drilling of identical wells placed at
different locations of the reservoir area. Wellhead pressure is used to control gas flow rate.
Reservoir pressure is used as the abandonment criteria. Optimization variables are target recovery
factor at end of plateau and total number of wells. Those variables could be re-selected by simple
modification in the VBA code. However, optimization variables can be made independent for a
real field operation.
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4.1 Phase Behavior Model (PBM)
A phase behavior calculation (or a flash calculation) is used to predict the phase behavior
of a reservoir fluid at an equilibrium condition. A standard phase behavior model consists of four
main calculation modules; namely, compressibility factor calculations, vapor-liquid equilibrium
calculations, fluid properties predictions, and phase stability analysis, which must be fully
integrated to perform the flash calculation. The calculation starts with the determination of
number of co-existent phases or phase stability analysis. If fluid is found in a single phase (stable)
condition, fluid properties are calculated based on the available information on overall fluid
composition. If fluid is found in a two-phase (unstable) condition, composition and molar fraction
of each phase are determined using vapor-liquid equilibrium calculations. Then properties of each
co-existent phase are calculated based on fluid composition of that phase (Ayala, 2009b).
Input data consists of pressure, temperature, overall composition, physical properties,
binary interaction coefficients, and volume translation coefficient of each pure component. Peng-
Robinson Equation-of-State (PR EOS) is used to calculate Pressure-Volume-Temperature (PVT)
relationship of the reservoir fluid (Peng and Robinson, 1976). Vapor-liquid equilibrium is
assumed and an overall species material balance for a two-phase system is enforced. The output
from a PBM subroutine consists of number of phases, molar fraction, composition, molecular
weight, compressibility factor, density, adjusted density and viscosity of each fluid phase.
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4.1.2 Vapor-Liquid Equilibrium
Two main components are considered in order to predict properties of multi-component
hydrocarbon in Vapor-Liquid Equilibrium (VLE) condition: material balance considerations and
thermodynamic considerations. Iterative procedure is applied until the solution that satisfies both
criteria can be determined.
Mater ial Balance Considerations
Rachford and Rice objective function, which is derived from enforcing an overall species
mass balance in a two-phase multi-component system, is utilized to calculate molar fraction of
each phase (Rachford and Rice, 1952):
Equation 4-5
where:
= molar faction of i-th component = volatility ratio of i-th component =
= molar fraction of i-th component in vapor phase
= molar fraction of i-th component in liquid phase = molar fraction of vapor phase
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After solving for from the objective function, molar fraction of liquid phase iscalculated from Equation 4-6, composition of vapor phase is calculated from Equation 4-7, and
composition of liquid phase is calculated from Equation 4-8.
Equation 4-6
Equation 4-7
Equation 4-8
Thermodynamic Considerations
According to the second law of thermodynamics, any system in equilibrium, such as a
VLE condition, must have the maximum possible entropic state under the prevailing conditions.
For such condition to be established, thermodynamics shows that net transfer of heat, momentum,
and mass between both phases must be zero. Thus, temperature, pressure, and every species
chemical potential in both phases must be equal to each other.
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Chemical potential cannot be measured directly. However, equality of chemical potential
can be represented by equality of fugacity between both phases. Fugacity is the pressure
multiplier to correct non-ideality and to make ideal gas equation work for real gas during Gibbs
energy calculations. In a VLE condition, fugacity of liquid phase must be equal to fugacity of
vapor phase. Equation 4-9 is used to calculate fugacity for vapor phase while Equation 4-10 is
used for liquid phase.
Equation 4-9
Equation 4-10
where:
= fugacity of i-th component in vapor phase
= fugacity of i-th component in liquid phase = fugacity coefficient of i-th component in vapor phase = fugacity coefficient of i-th component in liquid phase= molar fraction of i-th component in vapor phase= molar fraction of i-th component in liquid phase = pressure {psia}
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For the generalized formula of cubic EOSs discussed above, fugacity coefficients can be
calculated using Equation 4-11 (Coats, 1985) below. Definitions of parameters are the same as
definitions used in Equation 4-1. It should be noted that is equal to for calculating fugacitycoefficient of a liquid phase and is equal to for calculating fugacity coefficient of a vaporphase.
Equation 4-11
Volatility ratio () is equal to ratio between the gas composition and the liquidcomposition during an equilibrium condition. For a system with a VLE condition, is equalto
. By substituting Equation 4-9 and Equation 4-10 into definition of volatility ratio, volatility
ratio can be expressed in terms of fugacity coefficients as follows.
Equation 4-12
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The Successive Substi tut ion Method
From material balance consideration, molar fraction of vapor phase and composition of
each phase are functions of volatility ratios and overall composition. Volatility ratios themselves
are also function of composition of each phase. Thus, an iterative procedure is needed in order to
perform VLE prediction and honor the fugacity equality constraint. The following procedure is
used to perform two-phase flash calculation (Whitson and Brule, 2000, p.52-55).
First, initial guesses of volatility ratios are calculated using Equation 4-13 as proposed by
Wilson (Wilson, 1968). Rachford and Rice objective function (Equation 4-5) is then solved using
a standard Newton-Raphson iterative method. Then, the compositions of each phase are
calculated using Equation 4-7 and Equation 4-8.
Equation 4-13
Next, the fugacity values of each component in both liquid and vapor phases are
calculated using Equation 4-9 through Equation 4-11. Successive Substitution Method (SSM) is
utilized to update volatility ratios (Equation 4-14) for a next iteration as shown below
Equation 4-14
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where:
= volatility ratio of i-th component at iteration level n = fugacity of i-th component in liquid phase at iteration level n = fugacity of i-th component in vapor phase at iteration level n
Once volatility ratios are updated, convergence criteria presented in Equation 4-15 must
be checked. If the criteria are not satisfied, the procedure is repeated by solving Rachford and
Rice objective function and recalculating phase compositions and resulting fugacities until
convergence is attained.
Equation 4-15
The SSM algorithm is expected to have slow convergence rate near the critical point. To
avoid this problem, accelerated SSM algorithm has been proposed. The algorithm proposed by
Michelsen (Michelsen, 1982b) or the algorithm proposed by Merah et al(Merah et al, 1983) are
examples of well-known ASSM algorithms.
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4.1.3 Fluid Property Prediction
Molecular Weight
Molecular weight of vapor and liquid phases are weighted average of molecular weight
of all pure components, as shown below
Equation 4-16
Equation 4-17
where: = molecular weight of vapor phase {lb/lbmol} = molecular weight of liquid phase {lb/lbmol} = molecular weight of i-th component {lb/lbmol} = mole fraction of i-th component in vapor phase
= mole fraction of i-th component in liquid phase
= number of components in the multi-component hydrocarbon
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Density
Density of each phase is calculated from Equation 4-18 and Equation 4-19.
Equation 4-18
Equation 4-19
where:
= density of vapor phase {lbm/ft3} = density of liquid phase {lbm/ft3}
= molecular weight of vapor phase {lbm/lbmol}
= molecular weight of liquid phase {lbm/lbmol} = molar volume of vapor phase {ft3/lbmol} = molar volume of liquid phase {ft3/lbmol }
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Molar volume of each phase is calculated from real gas law (Equation 4-20), then,
adjusted by using volume-translation technique.
Equation 4-20
where
= calculated molar volume of phase a from EOS {ft3/lbmol}= compressibility factor of phase a = universal gas constant {10.732 psi-ft3/R-lbmol} = temperature {R} = pressure {psia}
As discussed by Whitson and Brule (p.51) and Danesh (p.141-143), calculated molar
volume from real gas law can be adjusted by implementing volume-translation or volume-shift
technique (Whitson and Brule, 2000 and Danesh, 1998). This technique improves volumetric
calculation of liquid phase, which is the main problem of two-constant EOSs, without altering
VLE prediction results. The volume translation technique, originally introduced by Martin and
further developed by Penelous et aland Jhaveri and Youngren, can be summarized as follows
(Martin, 1979, Penelus et al, 1982, and Jhaveri and Youngren, 1988):
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31 Equation 4-23
where:
= component-dependent volume-shift parameter = co-volume parameter of i-th component= volume-translate coefficient of i-th component
Table 4-1: Volume-Translation Coefficients for Pure Components (Whitson and Brule, 2000)
Component PR EOS SRK EOS
N2 -0.1927 -0.0079
CO2 -0.0817 0.0833
H2S -0.1288 0.0466
C1 -0.1595 0.0234
C2
-0.1134 0.0605
C3 -0.0863 0.0825
i-C4 -0.0844 0.0830n-C4 -0.0675 0.0975
i-C5 -0.0608 0.1022
n-C5 -0.0390 0.1209
n-C6 -0.0080 0.1467
n-C7 0.0033 0.1554
n-C8 0.0314 0.1794
n-C9 0.0408 0.1868
n-C10 0.0655 0.2080
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Viscosity
Viscosity of vapor phase is calculated from the correlation proposed by Lee et alin 1966
(Equation 4-24 through Equation 4-27).
Equation 4-24
Equation 4-25
Equation 4-26
Equation 4-27
where:
= viscosity of vapor phase {cp} = density of vapor phase {lbm/ft
3
} = molecular weight of vapor phase {lbm/lbmol} = temperature {R}
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Equation 4-30
Equation 4-31
where:
= viscosity of liquid phase at low pressure {cp} = molar fraction of i-th component in liquid phase = viscosity of i-th component at low pressure {cp} = viscosity parameter of i-th component {cp -1} = reduce temperature of i-th component ( )
= temperature {R}
= critical temperature of i-th component {R} = critical pressure of i-th component {psia} = molecular weight of i-th component {lbm/lbmol} = number of componentsViscosity parameter of liquid phase is calculated from Equation 4-32 to Equation 4-35.
Equation 4-32
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Pseudo reduced density of the liquid phase is calculated from Equation 4-36 and
Equation 4-37 shown below.
Equation 4-36
Equation 4-37
where:
= pseudo reduced density of liquid phase = density of liquid phase {lbm/ft3}
= molecular weight of liquid phase {lbm/lbmol}
= pseudocritical molar volume of liquid phase {ft3/lbmol} = critical molar volume of i-th component {ft3/lbmol} = molar fraction of i-th component in liquid phase = number of components
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Step 6: Calculate the fugacity of vapor-like and liquid-like phases based on the calculated
mole fraction from Step 5. Equation 4-9 , Equation 4-10 and Equation 4-11 are utilized.
Step 7: Calculate the fugacity ratio corrections for successive substitution update of the values.
Equation 4-44
Equation 4-45
where:
= fugacity ratio calculation of i-th component in vapor-like phase = fugacity ratio calculation of i-th component in liquid-like phase = fugacity of i-th component in original fluid = fugacity of i-th component in vapor-like phase = fugacity of i-th component in liquid-like phase
= Sum the mole numbers of vapor-like phase
= Sum the mole numbers of liquid-like phase
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Step 8: Check whether convergence criteria is achieved
Equation 4-46
Step 9: If convergence is not obtained, update values
Equation 4-47Step10: Apply criterion to check whether a trivial solution has been obtained
Equation 4-48
Step 11: If a trivial solution is not indicated, go to Step 3 for the next iteration.
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4.2 Standard PVT Properties
The standard PVT properties used to describe a two-phase, two-pseudo component fluid
model (black oil model) relies on the definition and calculation of four basic properties,
namely: gas formation volume factor (), oil formation volume factor (), volatilized oil-gasratio (), and solution gas-oil ratio (). These PVT properties are required inputs for a zero-dimensional reservoir model. In this study, these required PVT properties can be obtained from
either a laboratory fluid analysis, typically a Constant Volume Depletion (CVD) test, or from a
phase behavior model (PBM) calculation. If the PVT/CVD laboratory report is available, the
resulting PVT properties are calculated using Walsh-Towler algorithm (Walsh and Lake, 2003).
A template has been prepared using MS-Excel worksheet for this purpose. In the absence of a
PVT lab report, a PBM calculation is implemented which combines Walsh-Tolwer method with
the work of Thararoop in 2007 (Thararoop, 2007). This PBM subroutine does not only extend the
flexibility of the main simulator significantly, but also provide very useful information about fluid
properties which could help in thoroughly analyzing the depletion characteristics of the given gas
condensate fluid.
The specific gravity of reservoir gas is required for flow rate and flowing pressure
calculations, as it will be discussed below. The specific gravity of a reservoir gas can be obtained
from either the laboratory fluid analysis or from molecular weight calculations derived from
PBM. If the lab analysis is available, compositions of the produced wellstreams reported in the
experimental depletion study based on the Constant Volume Depletion (CVD) test are used to
calculate molecular weight of reservoir gas. If the lab report is unavailable, the molecular weight
of the reservoir gas is obtained directly from flash/PBM calculation results. Specific gravity of
reservoir gas is equal to molecular weight of reservoir gas divided by molecular weight of air.
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4.2.1 Definitions, Mathematic Relationships, and Characteristics
A clear understanding of the definitions of standard PVT black oil properties that are
used to characterize two-phase, two-pseudo component fluid models is crucial for their
meaningful calculation and prediction. These definitions, mathematic relationships, and their
most significant features have been summarized below (Walsh and Lake, 2003; Whitson and
Brule, 2000).
Definitions
Figure 4-1shows the graphical representation of the definitions of the standard PVT
properties used in the formulation of two-phase, two-pseudo component fluid model (or modified
black-oil model). In this figure, the gas phase at reservoir condition () results from the mixingof certain amounts of surface gas () and stock-tank oil () pseudo components. The oilphase at reservoir condition (
) results from the mixing of certain amounts of surface gas (
)
and stock-tank oil () pseudo components. The produced gas phase at surface condition ()(not shown in the figure) would consists of the combination of surface gas pseudo component
produced from gas phase at reservoir condition () and surface gas pseudo component liberatedfrom oil phase at reservoir condition (). By the same token, the produced oil phase at surfacecondition () (not shown in the figure) consists of stock-tank oil pseudo component producedfrom oil phase at reservoir condition (
) and stock-tank oil pseudo component condensed from
gas phase at reservoir condition ().
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component in oil phase at reservoir condition is thus defined as . Clearly, + = 1. Theirformulas are summarized in Equation 4-53 through Equation 4-56.
Equation 4-53
Equation 4-54
Equation 4-55
Equation 4-56
Mathematic Relati onships
If only one mole of reservoir fluid is considered, volumes at reservoir condition, and
, can be represented by molar density at reservoir condition,
and
, respectively.
Similarly, volumes at surface condition, , , , and , can be represented by molarfraction of pseudo component in reservoir fluid and molar density at surface condition, , , , and , respectively. If we substitute these definitions into equations for
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Figure 4-2: Typical Characteristic of Gas Formation Volume Factor ()and Volatilized Oil-Gas Ratio () for Gas Condensate
Figure 4-3: Typical Characteristic of Oil Formation Volume Factor ()and Solution Gas-Oil Ratio () for Gas Condensate
Rv-VolatilizedOil-GasRatio
Bg-GasFormationVolumeFactor
Reservoir Pressure
Dew Point Pressure
Rv
Bg
Rs-SolutionGas-OilRatio
Bo-OilFormationVolumeFactor
Reservoir Pressure
Dew Point Pressure
Bo
Rs
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As shown in Figure 4-2, gas formation volume factors () are expected to increase withdecreasing reservoir pressure () because the denominator, , in Equation 4-57 approaches zero.Volatilized oil-gas ratio () will remain constant because all parameters in Equation 4-59 remainthe same. Constant values of, , and result from the constant composition of gas phase inthe reservoir. Once dew point conditions are reached, Figure 4-2 also shows that the volatilized
oil-gas ratio () is expected to decrease with decreasing reservoir pressure, mainly because ofdecreasing and increasing values in Equation 4-59. Driven by the condensate drop out thatdevelops in the reservoir below dew point conditions, the reservoir gas will start to contain less
heavy hydrocarbon molecules that can be produced as condensate at surface condition. As a
result, the fraction of stock-tank oil () in the reservoir gas decreases while fraction of surfacegas () increases ( + = 1). As pressure depletion progresses, and if it gets low enough, thevolatilized oil-gas ratio () trend would be reversed.
Figure 4-3 illustrates that at reservoir pressure above the dew point there is no liquid
phase at reservoir condition and therefore no calculations of
and
can be directly performed
from their definitions. Once dew point conditions are crossed, oil formation volume factor () isexpected to decrease with decreasing reservoir pressure mainly because of increasing and values in Equation 4-58. As pressure decreases, more surface gas pseudo component will be
liberated from the oil phase. As a result, the molar fraction of stock-tank oil pseudo component in
oil phase () becomes higher and the density of oil phase at reservoir condition () alsoincreases. Similarly, the solution gas-oil ration (
) will be expected to decrease with decreasing
reservoir pressure because of the increased molar fraction of stock-tank oil pseudo component in
oil phase () and decreasing molar fraction of surface gas pseudo component in oil phase () inEquation 4-60. Even though oil formation volume factors () and solution gas-oil ratios ()cannot be calculated directly because of the lack of an actual liquid phase at reservoir from their
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definitions, Walsh and Lake suggest employing the following relationships for oil formation
volume factor () and solution gas-oil ratio () as place-holder values above the dew point:
Equation 4-61
Equation 4-62
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4.2.2 Obtaining Standard PVT Properties from Laboratory PVT Reports
In a laboratory PVT test, a representative sample of the reservoir fluid is subjected to a
series of depletion steps that try to closely mimic or reproduce the expected pressure depletion
path followed by the fluid during reservoir production. Temperature of the test is maintained
constant and equal to prevailing reservoir temperature. Resulting volumes of each phase (liquid
and vapor) are recorded along with the pressure at which the record is made. Fluid composition
and physical properties of the produced fluids are also analyzed. The typical standardized PVT
tests conducted for gas condensate fluids are the Constant Composition Expansion (CCE) and
Constant Volume Depletion (CVD) tests. Details of these PVT tests can be found in many
petroleum engineering textbooks (McCain, 1990; Denesh, 1998; Whitson and Brule, 2000, Walsh
and Lake, 2003); thus, they will be discussed very briefly in this manuscript.
In a CCE test, the reservoir fluid sample is placed inside a PVT cell and is pressurized to
a pressure equal to initial reservoir pressure, while maintaining a constant temperature inside the
PVT cell equal to reservoir temperature. Pressure inside the cell is then decreased to a next lower
pressure level by isothermal expansion. The new volume of each phase is recorded. This process
continues until abandonment pressure conditions are reached. In the CCE testing process, no fluid
is taken out the cell and therefore the overall composition of reservoir fluid inside the PVT cell
remains constant while the volumes and densities of each the co-existing phases below dew point
conditions do change with cell pressure.
In a CVD test, a reservoir fluid sample will be placed inside the PVT cell and pressurized
to the dew point pressure, while the temperature of the PVT cell is kept constant at reservoir
temperature. Then, pressure of the cell will be lowered to the next pressure level by isothermal
expansion. After that, a portion of gas phase inside the cell is produced (i.e., removed out of the
cell) so that the cells volume is restored back to the original cell volume at dew point conditions.
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The volume that the liquid phase occupies inside the PVT cell is recorded and the excess
(produced) gas analyzed. Depletion study which provides the resulting cumulative production
data at every pressure level is recorded and is used during the calculation of the standard PVT
properties from laboratory PVT fluid test report.
In this study, a calculation template is prepared in MS-Excel worksheet. The Walsh-
Towler algorithm is implemented to convert the results from the CVD experiments into the
standard table of PVT properties for a gas condensate fluid. Walsh-Towler algorithm is
summarized below.
Walsh-Towler Algori thm
Walsh-Towler algorithm is one of the methods used to calculate standard PVT properties
for gas condensate based on CVD testing results (Walsh and Towler, 1995; Walsh and Lake,
2003). This algorithm is relatively simple because it based on enforcing material balance
constraints around the PVT cell at every pressure level during the PVT lab test. The algorithm
was originally proposed by Walsh and Towler in 1995 and was later modified by Walsh and Lake
in 2003. By directly using data from a CVD report, this algorithm is implicitly assuming that
actual field separator conditions of the surface production system is the same as those surface
condition used during the CVD PVT test. It also assumes that only the gas phase at reservoir
condition can be recovered and that any condensate drops out inside the reservoir will remain
immobile during reservoir life.
One of the constraints of using this method is the availability of cumulative production
data at surface conditions because such data is not always performed or reported for every CVD
experiment. If such cumulative production data at surface conditions is not available in the CVD
report, it is customarily recommended to implement surface flash calculations using Standings
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K-values to reproduce them (Walsh and Lake, 2003). The algorithm also requires a high accuracy
and reliability of the CVD report in order to obtain a healthy and physically meaningful set of
derived standard PVT properties. It can be demonstrated that small error in the data reported by a
CVD test can result in PVT property values which are physically impossible (e.g., negative
values). And even when the data reported by the CVD report is highly reliable, the Walsh and
Towler algorithm can still lead to unphysical values for standard PVT properties. This limitation
results from combining the two-phase two-pseudo component (black oil) model with material
balance calculation around the PVT cell. This limitation will be discussed in detail in Chapter 5.
Walsh-Towler algorithm consists of six sequential steps which must be fully completed at
every given pressure level before moving to the next pressure. One pre-calculation is also needed
before starting the algorithm. The variables and their nomenclature employed in the sequence of
calculations are graphically illustrated in Figure 4-4.
Figure 4-4: Graphical Representation of CVD Data used in Walsh-Towler Algorithm
Vg,j
Vo,j
Reservoir Condition Surface Condition
Gfg,j
Nfg,j
Gfo,j
Nfo,j
Reservoir Gas
Reservoir Oil
Surface Gas
Stock-Tank Oil
Vg,j
Gfg,j
Bg = ----------- Rv = -----------
Nfg,j
Gfg,j
Bo = ----------- Rs = -----------
Vo,j
Nfo,j
Gfo,j
Nfo,j
VT
VEG,j
PR PDew PR < PDew Gpj
Npj
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Pre-calculation: In this step, the total cumulative volumes of surface gas () and stock-tank oil () pseudo components produced from the reservoir fluid, and the resulting volume ofPVT cell () are calculated for the dew point condition. The volume of surface gas pseudocomponent () is calculated from the summation of cumulative gas recovery from 1 st stageseparator, 2nd stage separator, and stock tank for all available pressures - from dew point
conditions to the last reported (abandonment) pressure. The volume of stock-tank oil pseudo
components () is equal to cumulative oil recovery from stock tank for all available and reportedpressures (dew point to abandonment). These data are obtained from the calculated cumulative
recovery reported in the depletion table.
PVT cells volume is calculated from the definition of gas formation volume factor
(Equation 4-63). The gas formation volume factor () is calculated from Equation 4-57.Compressibility factor of gas phase () can be obtained from the CVD report. Mole fraction ofsurface gas pseudo component in the reservoir gas () is equal to divided by the volume of gasequivalent at the dew point (
) which is usually taken as 1000 MSCF.
Equation 4-63
Volatilized oil-gas ratio at dew point () is calculated from Equation 4-64, while oilformation volume factor (
) and solution gas-oil ratio (
) are calculated from Equation 4-61
and Equation 4-62, respectively.
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Equation 4-64
Step 1: Find and : Starting at the dew point, the volume of surface gas pseudocomponent released from the excess gas () at each pressure is calculated from the summationof cumulative gas recovery from 1st stage separator, 2nd stage separator, and stock tank. Volume
of and stock-tank oil pseudo component released from the same excess gas () at each pressureis equal to cumulative oil recovery from stock tank. These data are obtained from the calculated
cumulative recovery reported in the depletion table. Incremental of and from pressure levelj-1 to pressure level j are calculated from Equation 4-65 and Equation 4-66. Please note that
pressure level j begins from zero at the dew point (j=0). , , , and are alsoequal to zero.
Equation 4-65
Equation 4-66
Step 2: Find
and
: Total volume of surface gas (
) and stock-tank oil (
) pseudo
components released from both reservoir gas and reservoir oil at pressure level j are calculated
from Equation 4-67 and Equation 4-68. It should be noted that pressure level j begins from zero at
the dew point (j=0), and and are equal to and , respectively.
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56 Equation 4-67
Equation 4-68
Step 3: Find and : Volume of oil phase at reservoir condition at pressure level j() is calculated from Equation 4-69. Retrograde liquid volume fraction at pressure level j(), can be obtained from CVD report. Volume of gas phase after excess gas removal atreservoir condition at pressure level j () is calculated from Equation 4-70. Note that pressurelevel j begins at zero at dew point conditions (j=0)
Equation 4-69
Equation 4-70
Step 4: Find , , and : Molar fraction of reservoir fluid which remains in thePVT cell at pressure level j (
) is calculated from Equation 4-71. For this calculation, two-
phase compressibility factor () data can be obtained from the CVD report. Molar fraction ofexcess gas which is removed from PVT cell at pressure level j () is calculated from Equation4-72. Molar fraction of gas phase which remain in PVT cell at pressure level j () is calculatedfrom Equation 4-73. Compressibility factor of gas () is also obtained from the CVD report.
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Please note that pressure level j begins from zero (j=0) at the dew point. and at dewpoint are equal to 1.0 while at dew point is equal to zero.
Equation 4-71
Equation 4-72
Equation 4-73
Step 5: Find and : Volume of surface gas pseudo component produced fromreservoir gas at pressure level j () is calculated from Equation 4-74. Volume of stock-tankpseudo component produced from reservoir gas at pressure level j () is calculated fromEquation 4-75. It is important to note that pressure level j begins from zero at the dew point (j=0).
and
at dew point pressure are equal to
and
, respectively.
Equation 4-74
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Equation 4-75
Step 6: Find and : Volume of surface gas pseudo component produced fromreservoir oil at pressure level j () is calculated from Equation 4-76. Volume of stock-tank oilpseudo component produced from reservoir oil at pressure level j () is calculated fromEquation 4-77.
Equation 4-76
Equation 4-77
After completing all six steps outline above for the given pressure level, Equation 4-49
through Equation 4-52 are now directly used to calculate the standard PVT properties. All
applicable unit conversion factors must be checked and adjusted properly. The calculation
process is systematically repeated for all pressure levels until all reported data in the CVD report
have been considered and abandonment conditions have been reached.
Standard PVT properties at pressures higher than the dew point are calculated based on
the properties at dew point pressure. Gas formation volume factor () is the product of gasformation volume factor at dew point pressure and relative volume obtained directly from CCE
testing results. The relative volume is the ratio between total volume of hydrocarbon at reservoir
conditions and the volume at saturated conditions. For under-saturated gas condensate system,
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relative volume is equal to the ratio between at specified pressure and at dew point pressure.Volatized oil-gas ratio () is equal to volatilized oil gas ratio at dew point pressure. Oilformation volume factor () and solution gas-oil ratio () are calculated from Equation 4-61and Equation 4-62, respectively.
Finally, it is very important to mention that, in Walsh-Towler algorithm, volumes of
pseudo components produced from the reservoir oil (step 6) do not actually come from direct
surface measurement. In a CVD test, the oil inside the cell is never produced (is assumed
immobile) so surface data for produced oil is not available.. Instead, these values are indirectly
calculated based on the enforcement of mass balance constraints around the PVT cell. Therefore,
actual oil formation volume factor () and solution gas-oil ratio () calculated from actualsurface flashes of the reservoir fluid might be significantly different from the ones estimated
using these indirectly calculated surface volumes. If the calculated and resulting from theapplication of this algorithm do not agree with the physically acceptable trends or values, the
results should be disregarded and the laboratory results have to be adjusted.
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direct calculation of saturation pressure at the prevailing reservoir temperature could be also
alternatively employed (Whitson and Brule, 2000).
The initial amount of mole of the reservoir fluid sample inside PVT cell () iscalculated from Equation 4-78. Standard condition is set to be 14.7 psia and 520 R. Volume ofinitial reservoir fluid in term of gas equivalent () is assumed to be 1.0 MMSCF which is usedas the basis for the calculation.
Equation 4-78
The associated volume of PVT cell () is calculated from Equation 4-79. Molecularweight () and density () are obtained by performing flash calculation on initial reservoirfluid composition at the dew point condition.
Equation 4-79
The molar fractions of surface gas (
) and stock-tank oil (
) pseudo components in
reservoir fluid are calculated from Equation 4-80 and Equation 4-81. Molar fraction of liquid
phase at first-stage separator () is obtained by performing flash calculation on initialreservoir fluid composition at first-stage separator condition. Molar fraction of liquid phase at
second-stage separator () is obtained by performing flash calculation on liquid composition
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from first-stage separator at second-stage separator condition. Molar fraction of liquid phase at
stock-tank condition () is obtained by performing flash calculation on liquid compositionfrom second-stage separator at stock-tank condition.
Equation 4-80
Equation 4-81
Total volume of surface gas () and stock-tank oil () pseudo components initiallypresent in the reservoir fluid are calculated from Equation 4-82 and Equation 4-83. Value of
379.56 is molar volume of gases at standard condition which is constant. Molecular weight
() and density () of oil at stock-tank condition are obtained from flash calculationresults at stock-tank condition. Please note that these values (G and N) are not being obtained by
cumulative adding cumulative production values at every pressure level, as done in the original
Walsh and Tower algorithm. Chapter 5 will present a discussion on this regard and justification.
Equation 4-82
Equation 4-83
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Step 2: Find and : The volume that the gas phase occupies at reservoircondition before the removal of the excess gas at every pressure level j () is calculatedfrom Equation 4-87. The volume of reservoir oil phase present at pressure level j () iscalculated from Equation 4-88. Molecular weight and density of gas and oil phases at reservoir
condition at pressure level j (, , , ) are obtained by performing flashcalculation on overall composition from pressure level j-1, at pressure level j. Note that pressure
level j begins from zero at the dew point (j=0). is equal to and is equal to zero.
Equation 4-87
Equation 4-88
Step 3: Find and : The volume of reservoir gas phase after excess gas removal atpressure level j () is calculated from Equation 4-89. Volume of excess gas at reservoircondition at pressure level j () is then calculated from Equation 4-90.
Equation 4-89
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separator at second-stage separator condition. The fraction of liquid phase at stock-tank condition
recovered from reservoir gas at pressure level j ( ) is obtained by performing flashcalculation on liquid composition from second-stage separator at stock-tank condition.
Equation 4-93
Equation 4-94
Step 6: Find and : Volume of surface gas () and stock-tank oil ()pseudo components in reservoir gas at pressure level j are calculated from Equation 4-95 and
Equation 4-96. The value of 379.56 is molar volume of gases at standard condition which is a
constant for ideal gases. Molecular weight (
) and density (
) of oil at stock-tank
condition recovered from reservoir gas at pressure level j are obtained from flash calculation
results at stock-tank condition in Step 5.
Equation 4-95
Equation 4-96
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Step 7: Find and : The molar fractions of surface gas () and stock-tank oil() pseudo components in the reservoir oil at every pressure level j are calculated fromEquation 4-97 and Equation 4-98. The fraction of liquid phase at first-stage separator recoveredfrom reservoir oil at pressure level j ( ) is obtained by performing flash calculation oncomposition of reservoir oil at pressure level j, at first-stage separator condition. The fraction of
liquid phase at second-stage separator recovered from reservoir oil at pressure level j ( ) isobtained by performing flash calculation on liquid composition from first-stage separator at
second-stage separator condition. The fraction of liquid phase at stock-tank condition recovered
from reservoir oil at pressure level j ( ) is obtained by performing flash calculation on liquidcomposition from second-stage separator at stock-tank condition.
Equation 4-97
Equation 4-98
Step 8: Find and : The volume of surface gas () and stock-tank oil ()pseudo components in reservoir oil at pressure level j are calculated from Equation 4-99 and
Equation 4-100. The value of 379.56 is molar volume of gas at standard condition which is
constant. Molecular weight () and density ( ) of oil at stock-tank condition recoveredfrom reservoir oil at pressure level j are obtained from flash calculation results at stock-tank
condition in Step 7.
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Equation 4-99
Equation 4-100
Step 9: Find and : Remaining moles of reservoir fluid inside PVT cell at pressurelevel j ( ) is calculated from Equation 4-101. Overall composition of i-th component insiderPVT cell at pressure level j () after gas removal is updated by implementing Equation 4-102.Note that pressure level j begins from zero (j=0) at the dew point. is equal to . Liquidcomposition () and vapor composition () of i-th component at pressure level j are obtainedby performing flash calculation on overall composition from pressure level j-1, at pressure level j.
Equation 4-101
Equation 4-102
After completing all nine steps outlined above at every given pressure level, Equation
4-49 through Equation 4-52 will be used to directly calculate standard PVT properties. All unit
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conversion factors must be checked and properly adjusted. This calculation process must be
continuously repeated for the every pressure level until abandonment pressure is reached.
Standard PVT properties at pressures higher than the dew point are calculated based on
available properties at dew point pressure. Gas formation volume factor () is calculated fromgas formation volume factor at dew point pressure using Equation 4-103. The ratio between
( ) at dew point pressure and ( ) at specified pressures above the dew point is equivalent toratio between volume of reservoir gas () at specified pressures above the dew point and volumeof reservoir gas (
) at dew point pressure. Volatized oil-gas ratio (
) is equal to volatilized oil-
gas ratio at dew point pressure (). Oil formation volume factor () and solution gas-oil ratio() are calculated from Equation 4-61 and Equation 4-62, respectively.
Equation 4-103
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4.3 Zero-Dimensional Reservoir Model
The Material Balance Equation (MBE) (also known as zero-dimensional reservoir model
or tank model) is a mass balance statement that combines mass balance equations of all pseudo
components present in the reservoir fluid. The assumptions behind a tank model have been
already addressed in Section 2.3. Walsh and Lake (2003) have presented a generalized form of
material balance equation that could be used for the analysis of depletion performance for all five
types from reservoir fluids, based on the work originally published by Walsh (1995). They also
developed the MBE specialized for gas condensate fluids by simplifying the generalized MBE for
the conditions particular to these kind of fluids. Section 4.3.1 discusses and presents the GMBE
proposed by Walsh as implemented in this study.
In zero-dimensional reservoir model, cumulative productions of pseudo components and
saturations of reservoir fluids are calculated as functions of reservoir pressure, standard PVT
properties, and initial reservoir condition. This model treats a reservoir as a homogeneous tank;
thus only average reservoir pressure and average PVT properties are required as the model inputs.
In this study, a VBA subroutine has been developed to simulate cumulative oil and gas
productions as well as their saturations as a function of reservoir pressure, by implementing the
MBE specialized for gas condensate fluids. Most of the time, the MBE is used to simulate the
results explicitly as a function of time and depletion. However, if some target outputs are
specified, such as cumulative recovery at end of plateau, an iterative procedure would need to be
implemented in order to honor the additional constraint.
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Equation 4-104
where:
= volume of surface gas pseudo component in reservoir gas at initialcondition {SCF}
= volume of stock-tank oil pseudo component in reservoir oil at
initial condition {STB}
= volume of water component in reservoir water at initial condition{STB}
= pore volume at initial condition {RB} = volume of water influx {RB}
= cumulative gas production {SCF}
= cumulative gas injection {SCF} = cumulative oil production {STB} = cumulative water production {STB} = cumulative water injection {SCF} = expansivity of reservoir gas {RB/SCF} = expansivity of reservoir oil {RB/STB} = expansivity of reservoir water {RB/STB} = expansivity of formation (rock) {Dimensionless} = gas formation volume factor {RB/SCF} = oil formation volume factor {RB/STB}
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Expansivity of reservoir fluid is defined as the total expansion of a unit mass of reservoir
fluid between two reservoir pressures at the same reservoir temperature. Expansivities of
reservoir gas, reservoir oil, and reservoir water are calculated from Equation 4-105, Equation
4-106, and Equation 4-107, respectively. Expansivity of formation (rock) is defined in a different
form from fluid expansivity and is calculated in terms of formation (rock) compressibility as
indicated by Equation 4-108.
Equation 4-105
Equation 4-106
Equation 4-107
Equation 4-108
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where:
= two-phase gas formation volume factor {RB/SCF} = two-phase oil formation volume factor {RB/SCF} = formation (rock) compressibility {psi-1} = pressure drop from initial reservoir pressure {psi}
The two-phase formation volume factor implemented above is defined as the ratio
between total volume of reservoir fluid (gas and oil phases) and total volume of the pseudo
component. Two-phase formation volume factor of gas () and oil () are calculated fromEquation 4-109 and Equation 4-110, respectively. If reservoir is a single phase gas reservoir, two-
phase gas formation volume factor () will be equal to gas formation volume factor () whiletwo-phase oil formation will remain undefined. Similarly, if reservoir is single phase oil reservoir,
two-phase oil formation volume factor () will be equal to oil formation volume factor ()while the two-phase gas formation volume factor will remain undefined.
Equation 4-109
Equation 4-110
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4.3.2 Material Balance Equation for a Gas Condensate Fluid
GMBE can be simplified significantly when condensate drop out, developed below dew
point saturation conditions in the reservoir, is considered immobile. The immobile condensate
assumption is a fairly reasonable one for gas condensates; however, it cannot be applied for other
types of reservoir fluid (Walsh and Lake, 2003). The Simplified Gas Condensate Tank model,
SGCT, is derived from Generalized Material Balance Equation with the following additional
assumptions:
Reservoir is under-saturated at initial reservoir pressure Expansivities of water and formation are negligible There is no water influx, water production, and water injection There is no gas injection Condensate drop out in the reservoir is immobile
Gas Condensate Performance Below Dew Poin t
At initial undersaturated conditions, the volume of surface gas pseudo component in
reservoir gas at initial condition is equal to the Original Gas In Place (OGIP or G) while and the
volume of stock-tank oil pseudo component in reservoir oil at initial condition is equal to zero.
Equation 4-111 is the SGCT model after applying all these additional assumptions:
Equation 4-111
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This SGCT model can be further manipulated in order to obtain a more useful form by
dividing it through by and substituting by . After that, finite differenceapproximation is applied, resulting in expressions for the calculation of incremental oil and gas
production. As a result, Equation 4-112 through Equation 4-118 are a set of equations that can be
used to calculate reservoir performance from SGCT model.
Equation 4-112
Equation 4-113
Equation 4-114
Equation 4-115
Equation 4-116
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Equation 4-121
Equation 4-122
Equation 4-123
It should be noted that this set of equation (Equation 4-120 to Equation 4-123) only
applies for pressures above dew point. The pressure level j begins from zero at initial reservoir
and end at the last pressure above the dew point. , , , and are equal tozero at the initial reservoir pressure. The calculation has to be completed at one pressure level
before moving to the next pressure level.
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4.3.3 Phase Saturation Calculations
One of the methods to derive a phase saturation equation is to combine the mass balance
equation for the stock-tank oil pseudo-component with the saturation equation constraint in order
to eliminate gas saturation parameter ( ) and then combine the resultingequation and volumetric OGIP calculation equation so that the pore volume variable is
eliminated. The resulting saturation equation for the oil phase is shown in Equation 4-124.
Equation 4-124
where:
= average reservoir oil saturation = average reservoir gas saturation = average reservoir water saturation = cumulative oil production {STB} = original oil in place {STB} = average reservoir porosity = gas formation volume factor {RB/SCF} = oil formation volume factor {RB/STB} = volatilized oil-gas ratio {STB/SCF} = subscript for initial co