Nonlinear flow and well test analysis in porous media

192 pages

English

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Nonlinear flow and well test analysis in porous media , livre ebook

EDP Sciences - Zhao Lun , Fengjun HAO , Chen Li , Jincai WANG , Libing FU

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192 pages

English

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A propos
Informations
Extrait

Description

Based on the analysis of factors influencing fluid infiltration in porous media, this book systematically summarizes the characteristics and expressions of low-velocity nonlinear flow and high-velocity nonlinear flow infiltration in porous media and provides a set of evaluation methods. Using the exponential formula, the starting pressure gradient formula, and the binomial equation of motion, the authors present a detailed comparison and analysis of the production, pressure, and dimensionless background pressure of the nonlinear flow and Darcy linear flow for steady and unsteady flow. In addition, based on the equation of motion of the starting pressure gradient, a mathematical model of the one-, two-, and three-medium nonlinear seepage flow is established, and approximate analytical solutions are given while the graph of the corresponding well test curve is drawn. Finally, based on the mathematical model of the well test established from the exponential equation of the high-velocity nonlinear flow motion, the atypical well test curve and the relational surface of the time- and space-varying infiltration index are obtained. The authors also discuss the relationship between reservoir and fluid properties and the nonlinear flow test curve. This book is intended to serve as a reference for technical personnel, researchers, teachers, and students involved in oil and gas development. Its research contents provide a theoretical basis for the identification of water flow dominant channels in the long-term water injection development of high-water-cut oilfields, profile control and water shut-off, productivity evaluation of carbonate reservoirs and formation parameters.

Preface..................................................... III

CHAPTER 1

Mechanism of Nonlinear Flow and Analysis of Characteristics............ 1

1.1 Analysis of Factors Affecting Fluid Flow in Porous Media .......... 1

1.2 Mechanism of Nonlinear Flowin Low Permeability Reservoirs....... 3

1.2.1 Characteristics of Nonlinear Flow in Low Permeability

Reservoirs........................................ 4

1.2.2 Model of Nonlinear Flow in Low Permeability Reservoirs ..... 6

1.3 Mechanisms of Stress Sensitivity in Low-Permeability Reservoirs..... 9

1.3.1 Stress Sensitivity Characteristics of Low-Permeability

Reservoirs........................................ 9

1.3.2 Stress Sensitivity Modelfor Low Permeability Reservoirs ..... 10

1.4 Characteristics of High-Velocity Nonlinear Flow ................. 13

1.4.1 Introduction....................................... 13

1.4.2 Characteristics of High-Velocity Nonlinear Flow ............ 15

1.4.3 Judgment Method for High-Velocity Nonlinear Flow ........ 17

1.4.4 Equation for Description of High-Velocity Nonlinear Flow .... 19

CHAPTER 2

Model of Low-Velocity Nonlinear Flow in Single Media ................. 25

2.1 Threshold Pressure Gradient Model of Low-Velocity Nonlinear Flow . . 25

2.1.1 Stable Flow Model.................................. 26

2.1.2 Unstable Flow Model................................ 26

2.1.3 Dimensionless Well Test Model Under Unstable Flow ........ 28

2.2 Exponential Model of Low-Velocity Nonlinear Flow .............. 34

2.2.1 Stable Flow Model.................................. 35

2.2.2 Unstable Flow Model................................ 37

2.2.3 Dimensionless Well Test Model of Unstable Flow ........... 41

CHAPTER 3

Theory of Well Test for Low-Velocity Nonlinear Flow in Multiple Media .... 47

3.1 Mathematical Model of Low-Velocity Nonlinear Flow in Double

Media................................................. 47

3.1.1 Motion Equation................................... 48

3.1.2 Channeling Equation................................ 48

3.1.3 State Equation..................................... 49

3.1.4 Continuity Equation................................ 50

3.1.5 Simplified Models of Matrix Permeability and Fracture

Porosity.......................................... 50

3.1.6 Simplified Model of Matrix Permeability ................. 52

3.2 Theory of Well Test of Low-Velocity Nonlinear Flow in Double Media 53

3.2.1 Mathematical Model of Well Test in Dual-Porosity Reservoir

and Its Solution.................................... 53

3.2.2 Mathematical Model Considering Wellbore Storage Effect

and Skin Factor.................................... 56

3.2.3 Characteristics of Well Test Curve of Low-Velocity Nonlinear

Flow in Double Media............................... 56

3.2.4 Analysis of Factors Affecting the Well Test Curves

of Low-Velocity Nonlinear Flow in Double Media Reservoirs . . 59

3.3 Mathematical Model of Low-Velocity Nonlinear Flow in Triple

Media................................................. 67

3.3.1 Ideal Model of Triple Media........................... 68

3.3.2 Mathematical Model and Its Solution for Triple Media

Reservoirs........................................ 69

3.3.3 Well Test Curve Characteristics of Low-Velocity Nonlinear

Flow in Triple Media Reservoirs........................ 73

3.3.4 Analysis of Factors Affecting the Well Test Curves

of Low-Velocity Nonlinear Flow in Triple Media ............ 76

CHAPTER 4

Well Test of Nonlinear Flow Considering Stress Sensitivity in Low

Permeability Reservoirs........................................ 89

4.1 Factors Affecting Stress Sensitivity and Mathematical Models ....... 89

4.1.1 Change Characteristics of Stress Sensitivity ............... 89

4.1.2 Factors Affecting Stress Sensitivity ...................... 92

4.1.3 Mathematical Model of Stress Sensitivity ................. 94

4.2 Nonlinear Stable and Pseudo-Stable Flow Model Considering Stress

Sensitivity.............................................. 97

4.2.1 Productivity and Pressure Models of Stable Flow Considering

Stress Sensitivity................................... 97

4.2.2 Productivity and Pressure Models of Pseudo-Stable Flow

Considering Stress Sensitivity.......................... 97

4.3 Nonlinear and Unstable Well Test Considering Stress Sensitivity ..... 99

4.3.1 Assumptions and the Model........................... 99

4.3.2 Solution and Charts for Well Test Model Considering Stress

Sensitivity in Low Permeability Reservoirs ................ 101

4.4 Well Test in Fracture-Pore Double Media Considering Stress

Sensitivity.............................................. 108

4.4.1 Sensitivity of Fracture Permeability ..................... 108

4.4.2 Well-Test Model of Double Media Considering Stress

Sensitivity and Typical Curves......................... 111

CHAPTER 5

High-Velocity Nonlinear Stable Flow Model ......................... 119

5.1 Darcy Linear Radial Flow.................................. 119

5.2 Binomial Nonlinear Radial Flow ............................. 120

5.3 Exponential Nonlinear RadialFlow ........................... 122

5.4 Comparison of the Calculation Results for Darcy’s Linear Flow

and Nonlinear Flow...................................... 123

5.4.1 Producing in a Constant Production .................... 124

5.4.2 Producing at a Constant Bottom-Hole Pressure ............ 129

CHAPTER 6

Transient Well Test Under High-Velocity Nonlinear Flow ............... 139

6.1 Exponential High-Velocity Nonlinear Flow Model ................ 139

6.1.1 Exponential High-Velocity Nonlinear Flow Model ........... 139

6.1.2 Analytical Solution to the Exponential High-Velocity

Nonlinear Flow Model............................... 141

6.1.3 Numerical Solution for the Exponential High-Velocity

Nonlinear Flow Model............................... 144

6.1.4 Results and Discussion............................... 148

6.2 Binomial High-Velocity Nonlinear Flow Model .................. 152

6.2.1 Establishment of Flow Model.......................... 152

6.2.2 Solution of the Model................................ 153

6.2.3 Results and Discussion............................... 155

CHAPTER 7

Transient Well Test of Moving Boundary in High-Velocity Nonlinear Flow . . 159

7.1 Assumptions............................................ 159

7.2 Establishment and Solution ofthe Mathematical Model ........... 160

7.3 Results and Discussion.................................... 161

Nomenclatures............................................... 167

Appendix: Virtual ArgumentInteger Bessel Function .................. 173

References.................................................. 177

Sujets

Fluids

Sciences et techniques

Mechanics

Science

Ingénierie

Informations

Publié par	EDP Sciences
Date de parution	14 décembre 2023
Nombre de lectures	2
EAN13	9782759831067
Langue	English
Poids de l'ouvrage	17 Mo

Informations légales : prix de location à la page 0,9150€. Cette information est donnée uniquement à titre indicatif conformément à la législation en vigueur.

Extrait

Current Natural Sciences

F L U I D M E C H A N I C S

Libing FU, Jincai WANG, Li CHEN, Fengjun HAO and Lun ZHAO

Nonlinear Flow and Well Test

Analysis in Porous Media

F L U I D M E C H A N I C S

ISBN : 978-2-7598-3105-0

9 782759 831050

Current Natural Sciences

Nonlinear Flow and Well Test Analysis in Porous Media

Libing FU, Jincai WANG, Li CHEN, Fengjun HAO and Lun ZHAO

Based on the analysis of factors influencing fluid infiltration in porous media, this book systematically summarizes the characteristics and expressions of lowvelocity nonlinear flow and highvelocity nonlinear flow infiltration in porous media and provides a set of evaluation methods. Using the exponential formula, the starting pressure gradient formula, and the binomial equation of motion, the authors present a detailed comparison and analysis of the production, pressure, and dimensionless background pressure of the nonlinear flow and Darcy linear flow for steady and unsteady flow.

In addition, based on the equation of motion of the starting pressure gradient, a mathematical model of the one, two, and threemedium nonlinear seepage flow is established, and approximate analytical solutions are given while the graph of the corresponding well test curve is drawn.

Finally, based on the mathematical model of the well test established from the exponential equation of the highvelocity nonlinear flow motion, the atypical well test curve and the relational surface of the time and spacevarying infiltration index are obtained. The authors also discuss the relationship between reservoir and fluid properties and the nonlinear flow test curve.

This book is intended to serve as a reference for technical personnel, researchers, teachers, and students involved in oil and gas development. Its research contents provide a theoretical basis for the identification of water flow dominant channels in the longterm water injection development of highwatercut oilfields, profile control and water shutoff, productivity evaluation of carbonate reservoirs and formation parameters.

www.edpsciences.org

Current Natural Sciences

Libing FU, Jincai WANG, Li CHEN, Fengjun HAO and Lun ZHAO

Nonlinear Flow and Well Test Analysis in Porous Media

Nonlinear Flow and Well Test Analysis in Porous Mediafirst edition was originally published in Chinese in 2022, by Science Press, ISBN 9787030696779. This translation is published by arrangement with Science Press.

Printed in France

EDP Sciences–ISBN(print): 9782759831050–ISBN(ebook): 9782759831067 DOI: 10.1051/9782759831050

All rights relative to translation, adaptation and reproduction by any means whatsoever are reserved, worldwide. In accordance with the terms of paragraphs 2 and 3 of Article 41 of the French Act dated March 11, 1957,“copies or reproductions reserved strictly for private use and not intended for collective use”and, on the other hand, analyses and short quotations for example or illustrative purposes, are allowed. Otherwise,“any representation or reproduction–whether in full or in part–without the consent of the author or of his successors or assigns, is unlawful”(Article 40, paragraph 1). Any representation or reproduction, by any means whatsoever, will therefore be deemed an infringement of copyright punishable under Articles 425 and following of the French Penal Code.

The printed edition is not for sale in Chinese mainland.

Science Press, EDP Sciences, 2023

Preface

Our world is diversified and complicated. The development and change of anything in our world are not affected by only a single factor, but should be the result of the comprehensive influence of various factors. The effects of various factors on things are not constant but show the characteristics of temporal and spatial variation. In most cases, the joint effect of multiple factors is a nonlinear problem. The velocity of an object falling in the air is nonlinear in time and space. The trajectory of objects thrown into the air varies nonlinearly over time. Nonlinearity occurs widely in natural and social development. Fluid flow in the porous media is a nonlinear problem. When fluid flows in the porous media, the flow velocity and patterns change spatially over time, and the influence factors and degree also change sig nificantly. With significant progress and development of technologies, especially those in modeling method and simulation, several complicated problems of fluid flow and well test in the porous media have been solved. Nevertheless, in previous publications, the nonlinear flow problems are still illustrated with the traditional theories, and this fails to meet the needs of research and field application. The goal of this publication is to introduce the mechanisms and laws of nonlinear flow in porous media and the well test models. The basic principles and methods of well test interpretation and some definite solutions are not illustrated here. Moreover, we give insights into the nonlinear phenomena in academic study or daily life, and the solutions to the nonlinear problems. A lot of achievements have been made in study on nonlinear flow in porous media, especially lowvelocity nonlinear flow. There is few systematic study on nonlinear flow in porous media. Based on analysis of factors affecting fluid flow in porous media, we summarize the flow characteristics of lowvelocity nonlinear flow and highvelocity nonlinear flow in porous media and their expressions, and establish a set of methods for judging the flow patterns, which is the first highlight in this publication. To describe steady and transient flow, we carry out comparison of production, pressure, pressure derivative and their variation in nonlinear flow and

DOI: 10.1051/9782759831050.c901 Science Press, EDP Sciences, 2023

Preface

Darcy linear flow based on the exponential, threshold pressure gradient and bino mial motion equations. Moreover, we plot the pressure and pressure derivative curves, which is the second highlight of this publication. The third highlight lies in establishment of mathematical models of nonlinear flow in single, double, and triple media based on the threshold pressure gradient motion equation. We provide the approximate analytical solutions and well test curves, which are references for engineering application. There are multiple flow patterns in the formation, and highvelocity nonlinear flow usually occurs around the nearwellbore zone. When the distance from the bottom hole increases, the flow pattern gradually transits from nonlinear flow to linear flow. It is estimated that the motion equation follows the exponential expression, and the critical radii of nonlinear flow and Darcy flow change spatially over time. The flow pattern in the nonlinear flow zone also changes spatially over time. The well test model of highvelocity nonlinear flow is established by incor porating initial and boundary conditions. The typical well test curves of dimen sionless bottomhole pressure and pressure derivative and flow index are calculated with the differential method. The correlation surface of flow index with time and space is obtained. In addition, the effects of reservoir and fluid properties on non linear flow are discussed. This publication can be only completed with support from colleagues and friends, to whom the most sincere gratitude is delivered, and their valuable sug gestions and help are highly appreciated. Moreover, our most special thanks are for experts and scholars who have been cited in this publication. The authors make their best to illustrate expressions clearly, thus to deliver their thoughts and understandings, especially the modeling process and solution calcu lation. Nevertheless, there are still defects, and any suggestion and advice to further improve the publication are welcomed.

Contents

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 1 Mechanism of Nonlinear Flow and Analysis of Characteristics. . . . . . . . . . . . 1.1 Analysis of Factors Affecting Fluid Flow in Porous Media. . . . . . . . . . 1.2 Mechanism of Nonlinear Flow in Low Permeability Reservoirs. . . . . . . 1.2.1 Characteristics of Nonlinear Flow in Low Permeability Reservoirs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Model of Nonlinear Flow in Low Permeability Reservoirs. . . . . 1.3 Mechanisms of Stress Sensitivity in LowPermeability Reservoirs. . . . . 1.3.1 Stress Sensitivity Characteristics of LowPermeability Reservoirs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Stress Sensitivity Model for Low Permeability Reservoirs. . . . . 1.4 Characteristics of HighVelocity Nonlinear Flow. . . . . . . . . . . . . . . . . 1.4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Characteristics of HighVelocity Nonlinear Flow. . . . . . . . . . . . 1.4.3 Judgment Method for HighVelocity Nonlinear Flow. . . . . . . . 1.4.4 Equation for Description of HighVelocity Nonlinear Flow. . . .

CHAPTER 2 Model of LowVelocity Nonlinear Flow in Single Media. . . . . . . . . . . . . . . . . 2.1 Threshold Pressure Gradient Model of LowVelocity Nonlinear Flow. . 2.1.1 Stable Flow Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Unstable Flow Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Dimensionless Well Test Model Under Unstable Flow. . . . . . . . 2.2 Exponential Model of LowVelocity Nonlinear Flow. . . . . . . . . . . . . . 2.2.1 Stable Flow Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Unstable Flow Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Dimensionless Well Test Model of Unstable Flow. . . . . . . . . . .

CHAPTER 3 Theory of Well Test for LowVelocity Nonlinear Flow in Multiple Media. . . . 3.1 Mathematical Model of LowVelocity Nonlinear Flow in Double Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III

1 1 3

4 6 9

9 10 13 13 15 17 19

25 25 26 26 28 34 35 37 41

3.2

3.3

Contents

3.1.1 Motion Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Channeling Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 State Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Continuity Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Simplified Models of Matrix Permeability and Fracture Porosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 3.1.6 Simplified Model of Matrix Permeability. . . . . . . . . . . . . . . . . Theory of Well Test of LowVelocity Nonlinear Flow in Double Media 3.2.1 Mathematical Model of Well Test in DualPorosity Reservoir and Its Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Mathematical Model Considering Wellbore Storage Effect and Skin Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Characteristics of Well Test Curve of LowVelocity Nonlinear Flow in Double Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Analysis of Factors Affecting the Well Test Curves of LowVelocity Nonlinear Flow in Double Media Reservoirs. . Mathematical Model of LowVelocity Nonlinear Flow in Triple Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Ideal Model of Triple Media. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Mathematical Model and Its Solution for Triple Media Reservoirs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Well Test Curve Characteristics of LowVelocity Nonlinear Flow in Triple Media Reservoirs. . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Analysis of Factors Affecting the Well Test Curves of LowVelocity Nonlinear Flow in Triple Media. . . . . . . . . . . .

CHAPTER 4 Well Test of Nonlinear Flow Considering Stress Sensitivity in Low Permeability Reservoirs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Factors Affecting Stress Sensitivity and Mathematical Models. . . . . . . 4.1.1 Change Characteristics of Stress Sensitivity. . . . . . . . . . . . . . . 4.1.2 Factors Affecting Stress Sensitivity. . . . . . . . . . . . . . . . . . . . . . 4.1.3 Mathematical Model of Stress Sensitivity. . . . . . . . . . . . . . . . . 4.2 Nonlinear Stable and PseudoStable Flow Model Considering Stress Sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Productivity and Pressure Models of Stable Flow Considering Stress Sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Productivity and Pressure Models of PseudoStable Flow Considering Stress Sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Nonlinear and Unstable Well Test Considering Stress Sensitivity. . . . . 4.3.1 Assumptions and the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Solution and Charts for Well Test Model Considering Stress Sensitivity in Low Permeability Reservoirs. . . . . . . . . . . . . . . . 4.4 Well Test in FracturePore Double Media Considering Stress Sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48 48 49 50

50 52 53

67 68

89 89 89 92 94

97 99 99

101

108

Contents

4.4.1 4.4.2

Sensitivity of Fracture Permeability. . . . . . . . . . . . . . . . . . . . . WellTest Model of Double Media Considering Stress Sensitivity and Typical Curves. . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 5 HighVelocity Nonlinear Stable Flow Model. . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Darcy Linear Radial Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Binomial Nonlinear Radial Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Exponential Nonlinear Radial Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Comparison of the Calculation Results for Darcy’s Linear Flow and Nonlinear Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Producing in a Constant Production. . . . . . . . . . . . . . . . . . . . 5.4.2 Producing at a Constant BottomHole Pressure. . . . . . . . . . . .

CHAPTER 6 Transient Well Test Under HighVelocity Nonlinear Flow. . . . . . . . . . . . . . . 6.1 Exponential HighVelocity Nonlinear Flow Model. . . . . . . . . . . . . . . . 6.1.1 Exponential HighVelocity Nonlinear Flow Model. . . . . . . . . . . 6.1.2 Analytical Solution to the Exponential HighVelocity Nonlinear Flow Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Numerical Solution for the Exponential HighVelocity Nonlinear Flow Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Binomial HighVelocity Nonlinear Flow Model. . . . . . . . . . . . . . . . . . 6.2.1 Establishment of Flow Model. . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Solution of the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 7 Transient Well Test of Moving Boundary in HighVelocity Nonlinear Flow. . 7.1 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Establishment and Solution of the Mathematical Model. . . . . . . . . . . 7.3 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Nomenclatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Virtual Argument Integer Bessel Function. . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VII

108

111

119 119 120 122

123 124 129

139 139 139

141

144 148 152 152 153 155

159 159 160 161

167 173 177