Hierarchic Modeling and History Matching of Multi-Scale Flow Barriers ...

HIERARCHIC MODELING AND HISTORY MATCHING OF MULTI-SCALE FLOW BARRIERS IN CHANNELIZED RESERVOIRS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Hongmei Li August 2008

c Copyright by Hongmei Li 2008

All Rights Reserved

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I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

(Dr. Jef Caers) Principal Adviser


(Dr. Hamdi Tchelepi)


(Dr. Stephan Graham)

Approved for the University Committee on Graduate Studies.

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Abstract Deep-water channelized reservoirs often consist of multi-scale architectures containing large-scale channel belts, middle-scale individual channels and small-scale channel infill components. These architectural elements are defined between bounding surfaces. Along these hierarchical bounding surfaces, thin shale drapes may be present as flow barriers that compartmentalize the reservoirs. Because the distribution of shale drapes is dominated by these bounding surfaces, they themselves form multi-scale reservoir heterogeneity. The key problem addressed in this thesis arises in the situation where multi-scale shale drapes are present along channel, channel belt and/or valley bounding surfaces, but the channel locations are uncertain or unknown. In order to reduce this uncertainty, first a realistic representation of the channel distribution should be obtained and constrained to hard data; then the channel and drape locations should be calibrated to the production data. We propose a coupled geologic modeling and history matching method where the reservoir architecture composed of channels are simulated with pre-defined stacking patterns, the shale drapes are then simulated along the bounding surfaces using multiple-point statistics techniques; channel and hole locations are gradually perturbed until the corresponding flow responses match the field production data. The perturbation during the history matching honors the individual channel geometry and the conceptual channel stacking patterns. In other words, the perturbation is geologically consistent. The reservoir architecture modeling in this work involves defining channel deposition fairways (valleys) based on seismic data, modeling long sinuous channels and placing them into defined fairways such that all data are matched. This work adopts v

a stratigraphic-based modeling approach. In this stratigraphic-based approach, individual channels are simulated using the YACS method (Alapetite et al., 2005). This method is fast and conditions to well data under the assumption that channel sand can be identified in the well data. To stack multiple channels reproducing the conceptual stacking pattern model, the migration ratio and overlap ratio are used as input parameters in the simulation process. To match the production data, channel location is perturbed using the gradual deformation method, the continuity of shale drapes is perturbed using the probability perturbation method. Tests on a reservoir analog of deepwater confined-channel systems demonstrated the feasibility of proposed modeling and history matching workflow. The sensitivity study showed that the hole proportion along individual channels and channel location are the most important parameters to flow. The history matching results also revealed that performing one step perturbation of channel location and hole proportion is enough to obtain desired history-matched geologic models. The prediction results have shown that the proposed workflow can be used to generate geologic models with better prediction power than randomly selected geologic models. Application of the proposed workflow to a realistic deepwater confined-channel reservoir showed that, in case of high NTG (07-08) reservoirs, both the scheme of perturbing channel and hole location and the scheme of perturbing hole location along fixed channels can achieve history matching. However, the former perturbation scheme is more efficient since the perturbation is in full 3D and consistent to geological conceptual model. Case study also demonstrated that including prior information of variability of shale drape leads to more realistic geologic models.

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Acknowledgements First and foremost, I wish to express my deepest gratitude to my advisor, Prof. Jef Caers, who has guided me over the years. Jef is one of the best teachers I have ever had. He has been a consistent source of knowledge, advice and encouragement. I have benefitted greatly from his thoughtful guidance and many insightful suggestions. For his careful reading of many drafts of this dissertation and other SCRF papers, for his unwavering enthusiasm and support of this work, for his many contributions to my education at Stanford, I owe him many thanks. I would like to thank Prof. Andre Journel for his wisdom and for imparting some of his knowledge to me through our many meetings and classes. It has been an honor and a privilege to have learned Geostatistics from him. This work would not have been possible without the contributions of Frans van der Vlugt, Mark Barton, Carlos Pirmez and Omer Alpak from Shell. Shell provided invaluable model dataset and constructive suggestions regarding channel stacking pattern simulation and history matching. I would like to thank Prof. Hamdi Tchelepi and Prof. Steve Graham for reading this dissertation and providing valuable comments and suggestions. I am also thankful to Prof. Roland Horne for being my committee member. Prof. Mark Zoback kindly chaired my Ph.D. oral defense and is gratefully acknowledged. I would like to express my gratitude to the SCRF research team for the valuable advice and help provided and for making research a lot of fun. Dr. Scarlet Castro shared with me her PPM framework written in python. Her help is greatly appreciated. My thanks also go to Lisa Stright for providing me with details on shale drape modeling and perturbation, and to Alex Bourcher and Ting Li for their help and guidance with vii

SGEMS. I owe my greatest gratitude to the Department of Energy Resources Engineering, all the faculty, staff and my fellow students. In particular, I would like to thank Prof. Margot Gerritsen who is not only providing excellent engineering courses, but also hosting fun women’s dinner. I am also thankful to Ginni Savalli and Thuy Nguyen. They are always there to help me through all kinds of paper work. The SCAPE friends have made my stay at Stanford truly pleasant. Their help and friendship are very much appreciated. Before I came to US, I was a geologist. My master thesis advisor Prof. Chris White at Louisiana State University help me make a smooth transition from a pure geologist to a reservoir engineer and led me into the door of geostatistic reservoir modeling. Without his guidance and recommendation, I could not have had the chance to study at Stanford. I feel lucky to have had Prof. Chris White as my master thesis adviser at the very beginning of my engineer adventure, and I am deeply grateful to him. Special thanks goes to my dear friends Wenjuan Lin and Junpeng Yue for their friendships. Wenjuan shared with me the challenging experience of the first year. I thank her for always being there to support. Junpeng always encourage me and give me confidence to go through many tough situations. Each one of you is an amazing friend and I have been fortunate to have met you. Last but not least, I would like to thank my family. My husband Deqiang Wang has been always there for me during the ups and downs. I thank him for his love, support and encouragement. I would like to express my gratitude to my parents for their unconditional love and confidence in me, and my brother Shicheng for taking part of my responsibility to take care of our parents.

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Contents Abstract

v

Acknowledgements

vii

1 Introduction

1

1.1

Reservoir architecture modeling . . . . . . . . . . . . . . . . . . . . .

3

1.2

Flow barriers modeling . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.3

Geologically consistent history matching . . . . . . . . . . . . . . . .

11

1.4

Proposed modeling approach . . . . . . . . . . . . . . . . . . . . . . .

15

1.5

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2 Proposed modeling workflow 2.1

2.2

19

Reservoir architecture modeling . . . . . . . . . . . . . . . . . . . . .

20

2.1.1

Individual channel modeling . . . . . . . . . . . . . . . . . . .

21

2.1.2

Channel simulation parameters . . . . . . . . . . . . . . . . .

25

2.1.3

Channel stacking pattern modeling . . . . . . . . . . . . . . .

25

2.1.4

Well data conditioning . . . . . . . . . . . . . . . . . . . . . .

33

2.1.5

Shale drapes modeling . . . . . . . . . . . . . . . . . . . . . .

40

2.1.6

Summary of the architecture modeling . . . . . . . . . . . . .

43

Geologically consistent history matching . . . . . . . . . . . . . . . .

45

2.2.1

Channel stacking pattern perturbation . . . . . . . . . . . . .

45

2.2.2

Behavior of a chain of stacking pattern realizations . . . . . .

56

2.2.3

Shale drape perturbation . . . . . . . . . . . . . . . . . . . . .

60

2.2.4

Perturbation procedure for history matching . . . . . . . . . .

62

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3 Workflow testing on a synthetic data set

67

3.1

Introduction to the synthetic dataset . . . . . . . . . . . . . . . . . .

68

3.2

Sensitivity study of shale drapes parameters . . . . . . . . . . . . . .

69

3.3

Modeling parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

3.4

History matching results . . . . . . . . . . . . . . . . . . . . . . . . .

85

3.5

Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

4 Applications to a realistic turbidite reservoir

97

4.1

Information available for history matching . . . . . . . . . . . . . . .

4.2

Setting up “true” cases . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.3

Perturbing shale drapes in high NTG reservoir . . . . . . . . . . . . . 107

4.4

Perturbing shale drapes by regions . . . . . . . . . . . . . . . . . . . 109

4.5

98

4.4.1

Region sensitivity study . . . . . . . . . . . . . . . . . . . . . 113

4.4.2

History matching results . . . . . . . . . . . . . . . . . . . . . 115

Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5 Conclusions and future work

122

5.1

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.2

Recommendations for future work . . . . . . . . . . . . . . . . . . . . 125

Bibliography

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List of Tables 3.1

Seven factors and level set up . . . . . . . . . . . . . . . . . . . . . .

80

3.2

Plackett-Burman design with 7 factors . . . . . . . . . . . . . . . . .

80

3.3

Water cut(%) response @ 900 days for PB experiment runs and analysis 81

3.4

Water breakthrough time (days) for PB experiment runs and analysis

3.5

Field oil recovery efficiency(%) @900 days for PB experiment runs and

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analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

3.6

Global NTG for different channel belts of reservoir analog

85

4.1

NTG for different channel belts . . . . . . . . . . . . . . . . . . . . . 100

4.2

Variograms used for channel porosity and permeability simulation . . 101

4.3

Description of simulation model for “true” production simulation . . 105

4.4

Four factors and level set up . . . . . . . . . . . . . . . . . . . . . . . 115

4.5

The two level 24−1 design . . . . . . . . . . . . . . . . . . . . . . . . . 115

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. . . . . .

List of Figures 1.1

Hierarchy for a deepwater channelized reservoir. Shale drapes could be deposited on the channel belt margins, on channel margins within individual channel belts and on cross-laminations within individual channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

Schematic diagram showing the influence of shale drapes on reservoir connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

3

Example result from MPS. Where two channels cross, MPS can not delineate the fine scale flow barriers that separate these two channels.

1.4

2

7

A synthetic 2D fracture training image (left, 400x300 grid) with 13% fracture (blue lines) and one realization using snesim (right, 200x200 grid). The fracture patterns shown in training image are not reproduced in the simulated realization.

. . . . . . . . . . . . . . . . . . .

8

1.5

A hierarchic workflow to simulate multi-scale flow barriers. . . . . . .

17

1.6

Coupled geologic modeling and history matching of multi-scale flow barriers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.1

Definition of parameters used to describe the channel geometry . . . .

21

2.2

The workflow for individual channel modeling . . . . . . . . . . . . .

24

2.3

The example showing curvilinear channel belt case . . . . . . . . . . .

24

2.4

The original potential map in Figure 2.3 showing the channel geometry

2.5

before adding noise map . . . . . . . . . . . . . . . . . . . . . . . . .

24

Channel potential maps with different noise histogram variances . . .

26

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2.6

Channel potential maps with different noise variogram ranges: top row is for ranges along channel orientation direction, bottom row is for ranges perpendicular to this direction . . . . . . . . . . . . . . . . . .

2.7

26

The potential maps (bottom) and their corresponding original potential maps (top) generated with different potential gradients . . . . . .

27

2.8

Two parameters used for channel stacking pattern modeling . . . . .

29

2.9

Schematic graph showing channel stacking patterns with different pattern parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.10 A channel belt body (left) and its bounding surface (right). Channels will be filled into the space within the channel belt limit . . . . . . .

29

2.11 The channel stacking pattern modeling process showing how channel is filled into belt (continue to next page) . . . . . . . . . . . . . . . .

30

2.12 The channel stacking pattern modeling process showing how channel is filled into belt. Left column is the architecture model, middle column are the realizations for migration ratio, and right column are the realizations for overlap ratio . . . . . . . . . . . . . . . . . . . . . . .

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2.13 Three channel belts case. Individual channels are filled into each belt until its net-to-gross reached. All the channels are confined by channel belt limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.14 Channel bounding surfaces extracted from Figure2.11 . . . . . . . . .

32

2.15 Schematic graph showing the well facies data and their interpreted channel sections

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.16 Synthetic uniform distributions of two pattern parameters . . . . . .

35

2.17 Synthetic example demonstrating the stochastic interpretation of well facies data. The same well data can result in multiple channel section stacking pattern realizations . . . . . . . . . . . . . . . . . . . . . . .

36

2.18 Schematic example showing the compatible/incompatible interpretation when multiple wells present . . . . . . . . . . . . . . . . . . . . .

37

2.19 Well channel section data (left) and its original potential map (right). The channel section geometry-maximum width and thickness- should be the same as the defined channel cross section geometry . . . . . . xiii

38

2.20 Well conditioning for interpreted channel sections. Well 1,2,3 could be connected with one channel (left); However, well 3,4,5 can be connected within another channel (right). Hence two interpretations are possible.

39

2.21 Three conditional realizations (bottom) of channel potential conditioned to 10 wells (top) . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.22 Three conditional realizations (bottom) of channel potential conditioned to 20 wells (top) . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.23 One realization (right column) of a channel complex conditioned to interpreted channel stacking pattern (left column top) at well location

42

2.24 One example of shale drape training image. Red color represents holes, and blue color indicates shales . . . . . . . . . . . . . . . . . . . . . .

44

2.25 The workflow of shale drape modeling . . . . . . . . . . . . . . . . . .

44

2.26 Different noise maps (middle row) added to the same original potential map (top row) result in different channel geometries (0-potential line in bottom row) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

2.27 Same noise maps (top row) added to different original potential map (middle row) result in different channel geometries (0-potential line in bottom row) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

2.28 Channel pattern parameters sampled from their distribution function

49

2.29 A uniform distribution is transferred into Gaussian distribution . . .

50

2.30 Two Gaussian realizations are perturbed to generate one new realization 51 2.31 Transfer the perturbed realization back to uniform distribution . . . .

51

2.32 A chain of channel realizations with different perturbations applied to the pair of channel stacking pattern parameters . . . . . . . . . . . .

52

2.33 Four realizations of individual channel potential maps with different perturbation parameters applied to their noise maps. Note the channel centerlines are roughly same. . . . . . . . . . . . . . . . . . . . . . . .

54

2.34 Channel complex realizations with different perturbation parameters applied to the individual noise maps. Note the channel locations are roughly the same (left column) but the channel cross-section changes with different perturbations (right column, y=20). . . . . . . . . . . . xiv

55

2.35 Channel realization (a) and the shale drapes (b) along channel boundaries (blue color represents shale drapes and red color is for scour holes); wells are located in the reservoir thickness map(c); (d) is the plot of water cut for two producers. . . . . . . . . . . . . . . . . . . . . . . .

57

2.36 Channel realization (a) independently generated from the one shown in Figure 2.35 and the shale drapes (b) along channel boundaries (blue color represents shale drapes and red color is for scour holes); (c) is the reservoir thickness map, wells are located at the same positions as shown in Figure 2.25c; (d) is the plot of water cut for two producers. 2.37 Realizations with different perturbation parameters . . . . . . . . . .

58 59

2.38 Water cut curves for a chain of 10 realizations. The initial realization is for the model in Figure2.35a . . . . . . . . . . . . . . . . . . . . . .

59

2.39 Objective function of a chain of 10 realizations . . . . . . . . . . . . .

60

2.40 A synthetic reservoir water saturation model with 20 channels and 3 producers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

2.41 The region defining procedure . . . . . . . . . . . . . . . . . . . . . .

64

3.1

A deep-water channelized reservoir analog. The brown and yellow colors are for channel fill facies, green color is for shale drapes and blue color is for scour holes. The model is constructed using surface-based grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

69

Analog model (upper) is converted into Cartesian grid (middle) and upscaled to a coarse scale model (bottom). The color reprsents different channel objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

3.3

Coarse channel belt geometry identified from reference analog model.

71

3.4

The channel belt geometry is restored into original shape and their

3.5

boundaries are traced for later channel simulation . . . . . . . . . . .

72

Two level of heterogeneities: channel belts and channels . . . . . . . .

73

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3.6

Shale drape training images (top 2 rows) and two reservoir models. In training images, the red color represents scour holes and blue background the shale drapes. Round vs. ellipsoid is for hole geometry, small vs. large is for hole size, and 0.1 vs. 0.5 is for hole proportion. .

78

3.7

Relative permeability for flow simulation model. . . . . . . . . . . . .

79

3.8

Effect charts for flow responses. . . . . . . . . . . . . . . . . . . . . .

79

3.9

Channel geometry parameter derived from analog . . . . . . . . . . .

86

3.10 Channel pattern parameter distribution: the experimental CDF (blue dots) and the analytic CDF function obtained by regression (solid lines). 86 3.11 The 3D hole distribution models (middle row) and the cross sections (bottom row) for belts and channels (middle row) using the corresponding hole training images (top row). The dark red indicates holes. The training image grid dimension is 100 by 100 and the hole models are 50 × 50 × 50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

3.12 Observed production responses from the synthetic reservoir. . . . . .

89

3.13 Optimization performance for two step perturbations. . . . . . . . . .

91

3.14 History match results for two step perturbation. . . . . . . . . . . . .

92

3.15 Reference model vs. history match geologic model. . . . . . . . . . .

92

3.16 Well data vs. simulated facies at well locations. Different color represents different channel section. . . . . . . . . . . . . . . . . . . . . . .

93

3.17 Well water cut and field oil recovery predictions using the history matched models and randomly picked models. . . . . . . . . . . . . .

94

3.18 Well bottom hole pressure predictions using the history matched models and randomly picked models. . . . . . . . . . . . . . . . . . . . . .

95

3.19 A channel hole training image (right) with proportion different with the reference (left). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

3.20 The optimization performance for the case that both hole location and proportion are perturbed. The hole proportion converges to the reference with the mismatch decreases. . . . . . . . . . . . . . . . . .

96

3.21 Flow response Predictions using the history matched models and randomly picked models. . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

96

4.1

Stratigraphic interpretation of WC reservoir. Gray color is for valley region. Other colors are for different channel belts. . . . . . . . . . .

4.2

99

Scour hole training images for valley, belt and channel hole simulation. The red color objects are scour holes, and blue background is shale drapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.3

Scour hole realizations for valley, belt and channel using the training images shown in Figure 4.3. . . . . . . . . . . . . . . . . . . . . . . . 102

4.4

Oil and water relative permeability curves. . . . . . . . . . . . . . . . 103

4.5

Geologic properties used for flow simulation. . . . . . . . . . . . . . . 104

4.6

Reservoir model showing reservoir structure, the geologic regions, and well configuration.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.7

Water cut and bottom hole pressure profiles for “true” case 1 and 2. . 107

4.8

Oil saturation distribution after 5 year production. . . . . . . . . . . 108

4.9

Ten Water cut and bottom hole pressure profiles for both non-history matched models and history matched models performed using pseudo 3D perturbation scheme. Blue line is for non-history matched model, red line is for history matched model, and black line is for “true” data. 110

4.10 Ten Water cut and bottom hole pressure profiles for both non-history matched models and history matched models performed using true 3D perturbation scheme. Blue line is for non-history matched model, red line is for history matched model, and black line is for “true” production data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.11 Optimization performances corresponding to 10 history matching results shown in Figure 4.9 and Figure 4.10. Blue color is for pseudo-3D perturbation scheme, purple color is for full 3D scheme. . . . . . . . . 112 4.12 The effect chart of examined factors listed in Table 4.4 on flow responses. Red lines are decision limits. If the effect bar exceed red, this means that factor is significant to the corresponding flow response with 95% confidence level. . . . . . . . . . . . . . . . . . . . . . . . . 116 4.13 Streamlines showing the water flow path. . . . . . . . . . . . . . . . . 117 4.14 History matching results assuming two regions in the reservoir. . . . . 118 xvii

4.15 Optimization performance corresponding to the history matched results shown in Figure 4.14. . . . . . . . . . . . . . . . . . . . . . . . . 119 4.16 History matching results assuming the reservoir as one region. . . . . 119 4.17 Optimization performance corresponding to the history matched results shown in Figure 4.16. . . . . . . . . . . . . . . . . . . . . . . . . 120 4.18 Reference and history-matched geologic models. . . . . . . . . . . . . 120 4.19 Oil and water saturation distribution after 5 years production. . . . . 121

xviii

Chapter 1 Introduction One of the primary goals of 3D geological modeling is to provide a geologically reasonable, numerical representation of the geology for input into flow simulators, in order to predict the behavior of the subsurface reservoir fluid flow under various hydrocarbon recovery scenarios. In most flow simulations, the permeability spatial distribution is the most influential input because it conditions the flow path (Weber, 1986; Hewett, 1986). It has long been recognized that flow in heterogeneous porous media is mainly controlled by the degree of continuity and connectivity of permeability - flow barriers, e.g. shales, or high permeability channels or faults (Giordano, 1985; Lake, 1986). Randomly disconnected small shale bodies may not affect fluid flow, whereas a small proportion of connected shale drapes may control flow and thus sweep efficiency and recovery. Therefore, a realistic representation of the distribution of high-flow channels or flow barriers is often essential to obtain reliable production prediction. Shale drapes have been increasingly reported and studied as flow barriers for channelized reservoirs, including fluvial channels (MacDonald, 1993) and turbidite channels (Weimer et al., 2000). Channelized reservoirs often show multi-scale architectures such as large-scale channel belts, middle-scale single channels and small-scale channel infill facies (Figure 1.1). These architectural elements are defined between bounding surfaces (Clark and Pickering, 1996b). Along these hierarchical bounding surfaces, thin shale drapes may be present as permeability barriers that compartmentalize the reservoirs. Studies have shown that shale drapes have stronger impact 1

2

CHAPTER 1. INTRODUCTION

Figure 1.1: Hierarchy for a deepwater channelized reservoir. Shale drapes could be deposited on the channel belt margins, on channel margins within individual channel belts and on cross-laminations within individual channels.

on the flow behavior than the shale facies distributed inside channels (Weimer et al., 2000). Figure 1.2 is a schematic diagram showing the effect of shale drapes on reservoir connectivity. Due to the drape presence as barriers (green curvilinear curve), oil in the disconnected channels (brown-colored) is left unswept even when these channels are stacked and appear connected to producers (yellow-colored). Since the geometry and distribution of shale drapes are governed by the erosional surfaces of channel belts and the channel-body margins which are at different scales, they represent multi-scale heterogeneities. Characterizing the distribution of these multi-scale flow barriers and representing them properly in a flow simulation model are critical to reservoir behavior prediction and management. The multi-scale nature of flow barriers calls for a hierarchical modeling approach in which the large-scale reservoir architecture is modeled first, the shale drapes are then simulated within this architecture framework.

1.1. RESERVOIR ARCHITECTURE MODELING

3

Figure 1.2: Schematic diagram showing the influence of shale drapes on reservoir connectivity.

1.1

Reservoir architecture modeling

A channelized reservoir consists of deposits formed by multiple channels. Depending on the variation of interaction between lateral and vertical amalgamation during the growth of channel systems, channel deposits demonstrate different stacking patterns(Clark and Pickering, 1996a, b). These three-dimensional mosaics of permeable (reservoir) and impermeable (non-reservoir) rock bodies of various sizes, shapes and arrangements are termed the sedimentary architecture. The individual rock bodies such as channels are architectural elements( Mutti and Normark, 1987). In the case that shale drapes are deposited as basal lags along the individual channel bounding surfaces, the proper representation of the spatial distribution of architectural elements is important to account for the connectivity of reservoir sand bodies. Reservoir architecture modeling in this work refers to modeling the hierarchic stacking patterns of multi-scale architectural elements. The architecture hierarchy of channelized reservoir ranges from seismic-scale (canyons, belts) to sub-seismic scale (channels and internal facies). In order to attach shale drapes to the architectural element boundaries, the simulated architectural elements should have clearly defined

4


geometry. Hence, the desirable architecture modeling technique should be such that it is capable to simulate channel stacking patterns delineated by bounding surfaces. At the same time it should be flexible for data conditioning. For seismic-scale architecture elements such as channel belts, it is easy to include them deterministically into reservoir models. This is because we can deterministically identify and interpret these large-scale object geometry from seismic data. Therefore we can input the interpreted channel belt geometry into reservoir models as a container for finer scale channels. For sub-seismic channels, a stochastic modeling approach is required to model uncertainty and integrate various types of data. Channels have been an important target for stochastic modeling because they control the fluid flow in the reservoir as conduits. Various channel simulation methods have been developed to capture the geologically realistic channel geometry. This section will review important stochastic simulation approaches used for channel simulation and their limitations will be addressed correspondingly. Object-based methods are also referred to as the Boolean methods. Object-based techniques were pioneered by Haldorsen and Lake (1984), Haldorsen and Chang (1986) and Stoyan et al. (1987). In object-based modeling, the geological heterogeneities such as sand and shale are defined as a set of objects. For example, in the case of channel-type reservoirs (fluvial or submarine), the sand facies can be represented by sinuous channels and the shale facies can be represented by elliptical objects. The dimensions of each object are defined using a limited set of parameters, usually in terms of a distribution of thickness, width, sinuosity, etc. After these objects are defined, their mutual spatial relationship such as object erosion rule, object attraction/repulsion rule will be established. Then, the following stochastic simulation simply consists of placing these objects into reservoir model with specified spatial relationship, and moving these objects around to match local data (wells and seismic). Compared to pixel-based methods, object-based methods provide more realistic shapes such as curvilinear channels. However, with this approach it is harder, if not impossible, to honor exactly geobodies interpreted from seismic. Another longstanding disadvantage with object-based reservoir modeling methods lies in the conditioning to well data, particularly in the presence of many wells. To overcome this


5

problem, Shmaryan and Deutsch developed a rule-based approach (Shmaryan and Deutsch, 1999) for fast conditioning to well data. In the rule-based approach, the speed is achieved through direct conditioning of the undulation channel centerline to the sand intervals by means of 1-D conditional sequential Gaussian simulation. The disadvantage is that this approach is unable to create large channel complexes in the same manner as it creates individual channels within a channel complex. Surface-based modeling approach has been introduced recently (Deutsch and Xie, 2001; Pyrcz and Deutsch, 2005) to better simulate internal and external geometries and stacking patterns of architectural elements associated with turbidites. In a surface-based approach, a conformable grid is used to explicitly model the location and the thickness of the channel and channel drapes. With this modeling approach, the thin shale layer could be represented with surfaces, and the cells contained between the two surfaces created as flow barriers could be represented either with transmissibility modifier or zero vertical permeability values during flow simulation (Begg and King 1985; Jackson and Muggeridge, 2000; Willis and White, 2000; Weimer et al., 2000). However, three limitations exist. First, adequate horizontal discretization is required to preserve a smooth surface shape, which will result in a large number of cells and make the flow simulation time-consuming; Secondly, as the lever of conditioning increases or channel size increases, it becomes more difficult to match well data. As a result, greater computational effort is needed to generate models; Thirdly, because both the channel location and shale drapes location are uncertain, they need to be perturbed during history matching of the reservoir model, which would mean changing the conformable grid automatically. Currently, such gridding is not yet at an adequate level of robustness to be made automatic. Alternatively, one could look at process-based methods for generating channels and shale drapes (Bridge and Leeder, 1979; Lopez, 2003). Process-based methods attempt to simulate fundamental geological processes to produce a numerical representation of the reservoir geology. These approaches include the rigor of the physics of sedimentation and depositional processes thus have the advantage to generate realistic geological heterogeneity. However, enormous difficulties arise when it comes to conditioning process-based models to well data.

6


More recently, pixel-based multiple-point statistics (MPS) techniques have been developed by Strebelle (2002) to simulate geologically realistic models. This approach has been applied to capture the curvilinear features in the deepwater settings (Strebelle et al., 2003). It allows simulations of 3D facies geometries and distribution using multiple-point statistics and training images. Multiple-point statistics can capture correlations that are geologically complex. Training images are non-conditional and purely conceptual depictions of the geological patterns deemed relevant for a particular reservoir. It can be derived from outcrop observation, expert knowledge or geophysics (see an example in Caers et al. 2003). The MPS algorithms extract geological patterns from the training image and reproduce them in simulated models at the same time being constrained by seismic and well data. MPS methods make it possible to simulate complex geometries such as channels, meanders and preserve the relations between facies, while honoring well information as well as any other secondary constraints such as a seismic-derived probability map of facies presence. The problems with this method are: (1) the facies patterns are reproduced as a whole, hence in the final model one has no knowledge of an individual object (eg., an individual channel). As a result, we can not identify single channels from MPS realizations due to channels cross-cutting each other. For example, Figure 1.3a shows one MPS simulated realization with channels (red) coded as 1 and non-channel (gray background) as 0. Figure 1.3b is a part of this realization where two channels cross-out each other. We can notice the cross-cutting in the horizontal direction, however in the vertical direction, it is impossible to identify the channel boundary because two channels have same numerical code (color). One possible solution is to code different channels with different numbers thus boundaries can be identified (Figure 1.3c). To achieve that, we need to build a large and pattern-rich training image to guarantee enough repetition of different cross-cutting patterns present in the training image, however the algorithm then requires several GBs of memory (Strebelle, 2002). (2) fine-scale low-proportion features such as shale drapes are difficult to model with MPS. This is because low-proportion patterns do not repeat frequently enough in the training image. Therefore, the simulated geometry and distribution of low-proportion shale drapes may not follow the patterns shown in the training image (Figure 1.4).


7

In 2005, Alapetite et al. developed Yet Another Channel Simulation(YACS) ap-

Figure 1.3: Example result from MPS. Where two channels cross, MPS can not delineate the fine scale flow barriers that separate these two channels. proach to generate geologically realistic fluvial channels while allowing for relatively easy conditioning to multiple well data. The core of this approach lies in the association of a fairway with the channels to be simulated. In this approach, a potential field is first defined within the fairway, then this potential map is mapped into thickness resulting in generating a channel inside the fairway. To add the necessary sinuosity to the channel, a correlated noise is stochastically simulated and added to the potential field. Conditioning to well data is obtained by transforming well observations into channel thicknesses which will constrain the simulation of noise. This method is fast and can condition to well data under the assumption that individual channel sand bodies can be identified in the well data. But as with other Boolean approaches, it cannot necessarily reproduce accurately the channel stacking patterns observed from outcrop analog or from training image. As stated above, these modeling methods can either simulate geologically realistic individual channels or channel patterns as a whole. None of them is capable of

8


Figure 1.4: A synthetic 2D fracture training image (left, 400x300 grid) with 13% fracture (blue lines) and one realization using snesim (right, 200x200 grid). The fracture patterns shown in training image are not reproduced in the simulated realization. simulating channel stacking patterns delineated by bounding surfaces.

1.2

Flow barriers modeling

Flow barriers are impermeable deposits or very low permeability deposits. They develop in many depositional environments, such as fluvial, deltaic, shoreface or shallow/deep marine. These impermeable barriers can form by depositional processes such as mud draping, or by digenetic processes such as calcite cementation. In this work, we focus on thin shale drapes formed along erosionally confined channel bases. Field and outcrop study have shown that repeated episodes of erosional cutting filling with coarse clastics - draping by mud during periods of quiescence - partial excavation of mud drapes are a feature of some erosionally confined channels (Gardner and Borer, 2000; Mayall et al., 2006). In contrast, some other channels systems, or segments of channels, lack drapes, depending on a variety of factors (e.g., Lowe, 2004; Hubbard et al., 2008a, b). In this thesis, we deal with the channel end-member characterized by impermeable shale drapes. The basal shale drapes within erosionally confined channels have been increasingly described from a number of outcrops (e.g., Gardner and Borer, 2000; Gardner et al., 2003; Eschard et al., 2003). These drapes generally are very thin (cm to m) and below seismic resolution, but they could form

1.2. FLOW BARRIERS MODELING

9

permeability barriers. Therefore, presence of channel base drape can have a significant impact on oil recovery because they represent one of the main uncertainties in the development of a turbidite channel reservoir. Outcrop data and production data from real field have provided ample evidence of the presence of “holes” in shale drapes. Due to the presence of holes, shale drapes are discontinuously distributed along the channel bounding surfaces. These discontinuous thin shales often are not correlated between wells. Even the “hole” distribution trend may be predictable if they related to the locus of strong current flow within a channel, their exact location within channels and the individual channel location are uncertain. Hence their location will be modeled using stochastic techniques. Stochastic simulating the spatial distribution of shale drapes is difficult. The first difficulty is related to data resolution. Sparse well data can not provide sufficient information on where these shales are located or on their complex geometry as well as continuity. The vertical resolution of seismic surveys (above 10 m) is too coarse to image these thin shale drapes. As a result, the coverage and geometry of shales are difficult to infer clearly from well or seismic data alone. The second difficulty is related to gridding these features in a high resolution geostatistical model or in a coarsened flow simulation model: even if we can gain sufficient knowledge about shale drapes distribution and gridding, it is difficult to capture them with traditional modeling techniques. The thickness of shale drapes could be on the order of centimeters whereas a simulation model grid is usually 10s or 100s of meters in resolution. The details of the channel geometry and fine scale flow barriers can potentially be captured with a very high-resolution highly unstructured grid (as shown in Figure 1.1), however the ultimate goal of reservoir modeling is flow prediction hence highresolution models need to be coarsened to make flow simulation feasible. In such coarsened model, even with the use of unstructured grids, the continuity of shale barrier may be destroyed. Beside the modeling difficulties, there are multiple sources of uncertainties that need to be taken into account. A first source of uncertainty is the shale drape proportion/location. The proportion and location govern the amount of communication between each of the separate channels. As previous stated, the location, geometry

10


and continuity of shale drapes are difficult to infer from well data and seismic data alone. Although outcrop data may provide a conceptual idea of shale drape properties, such information is often 2D. Another uncertainty source is the location of the individual channel body and channel belt itself. Quite often channel belts can be identified from seismic data, but individual channel bodies within these belts are below seismic resolution ( Slatt and Weimer, 1999; Abreu, 2003). Outcrop studies may provide knowledge about the type of channel stacking, channel width, depth and sinuosity. However, the actual location of channels in the reservoir is case-dependent thus requires one to build a high-resolution geological model constrained to actual reservoir data. Because the geometry of shale drapes is governed by the geometry of channel boundaries along which they were deposited, the location of the channel body will have strong impact on the reservoir performance, even in high net-to-gross system in which channels have a strong degree of lateral migration due to the absence of mud-rich levees (Clark and Pickering, 1996a). Previous studies on discontinuous shale drapes have used either simple 2D generic models, in which shales are located randomly within a homogeneous background (Richardson, 1978; Martin, 1984; Begg et al., 1985; Jackson and Muggeridge, 2000) or detailed 3D models of specific depositional environments derived directly from outcrop data or modern analog (White and Barton, 1997; Willis and White, 2000; Li and White, 2003; Pranter, 2007). All these studies concentrated on planar or inclined bedding surfaces. Stright et al. (2005) used a “semi-stochastic” technique to simulate shale drapes along the curvilinear surfaces of channels. The model first specified the geometries of channel bounding surfaces, then stochastically placed the ellipsoid holes (with background as shale drapes) on each surfaces using conditional multiple-point geostatistical simulation. All of these studies focus on a single hierarchy. And the governing surfaces of architectural elements are fixed or deterministically included into the models. Other approaches for modeling thin shales rely on more ad hoc “engineering” approaches. For example, thin shales are implicitly modeled by altering relative permeability, kv/kh ratios or pore volume and transmissibility multipliers. In reality,

1.3. GEOLOGICALLY CONSISTENT HISTORY MATCHING

11

engineers often perturb these properties by trial and error to match production history. The success or failure of the history matching endeavor largely depends on the experience, fortitude and sometimes good fortune of the engineer. The main limitation of such an approach is that geological information is often ignored or worse, the single deterministic model generated becomes geologically inconsistent. Given these difficulties, there is a need for a coupled geologic modeling and history matching engineering methodology which integrates both static and dynamic data simultaneously. We propose an approach where thin shale barriers referring to different scales are modeled hierarchically in a manner that is directly linked to a reservoir flow simulator transmissibility, rather than attempting to explicitly represent them by gridding. This approach will prove to be appealing in the context of matching the production history since channel location, shale proportion and shale location need to be perturbed until the historical production data is matched.

1.3

Geologically consistent history matching

The lack of geologic information to appropriately reproduce the reservoir heterogeneity is one of the most significant sources of uncertainty. To reduce this uncertainty, production data are often integrated into geologic models. Integration of production data or history matching is an optimization process which involves the definition and minimization of an objective function. This objective function measures the difference between the historical production data and the equivalent simulated responses. It is minimized through iteratively adjusting the geological model. Once a model has been history matched, it can be used to simulate future reservoir behavior with a higher degree of confidence, especially if the adjustments are consistent with geological information. In order to obtain a history matched geological model, most methods rely on modifying the initial geostatistical realization, proceeding by trial and error to obtain a history match. As we know, the history matching process is essentially an inversion of reservoir parameters from the historical production behavior. Because

12


this inversion problem is ill-posed, the history matched geological models are not always consistent with the data, other than dynamic, or do not always preserve the geological concept believed to exist. To overcome this problem, geologically consistent history-matching methods have been introduced. During geologically consistent history-matching process, the objective function is minimized through systematical perturbation of the local conditional distributions with a perturbation parameter, rD that is calibrated using the available production data. Based on the way the reservoir realization is perturbed, there are two perturbation methods: one is the gradual deformation method (GDM) (Hu, 2000), the other is the probability perturbation method (PPM), developed by Caers (2003).

GDM The objective of the gradual deformation is to create continuous perturbation of an initial realization, such that the perturbed realization matches better the production data. All the perturbed realizations are sampled from the same prior model as the initial realization. The perturbation is achieved by linearly combining two independent standard Gaussian random functions Y1 and Y2 with identical covariance: Y (rD ) = Y1 cos(rD ) + Y2 cos(rD )

(1.1)

Whatever the rD perturbation parameter, Y has the same mean, variance and covariance as Y1 and Y2 . Furthermore, Y is also a Gaussian random function because it is a sum of two Gaussian random functions (Hu, 2000). Two independent realizations y1 and y2 of Y1 and Y2 can form a continuous chain of realizations y(rD ), which depends on the parameter rD . This feature can be used to calibrate realizations to production data. The basic idea is to optimize rD based on some mismatch between field response and the one simulated on y(rD ). In geostatistical practice, sequential simulation is often used to generate conditional realizations. This takes two steps: first, one builds a probability distribution model conditioned to the hard data and pre-simulated values; secondly, one draws a


13

value from this probability model using the random number with a uniform distribution in [0, 1]. In this sequential simulation framework, GDM modifies the realizations by perturbing the random numbers used to draw from the conditional probability distribution model. The idea is to gradually perturb a realization of a uniform vector. This vector contains the random numbers that are used for drawing. The perturbation is achieved through Eq.1.1. Suppose u is a vector of random numbers with uniform distribution in [0, 1]. The vector u is first transformed into Gaussian domain as y. Next, y is perturbed using Eq.1.1, generating a perturbed Gaussian vector y(rD ). Then y(rD ) is back-transformed into a uniform vector of perturbed random numbers u(rD ). This u(rD ) is a perturbation of u, so when u(rD ) is used in sequential simulation, a perturbation is achieved. GDM was initially presented as a tool to deform realizations of Gaussian-related stochastic models. By deforming gradually Gaussian realizations, the objective function varies smoothly with a gradual change in the perturbed realizations. As the resulting variations are continuous, this method is of interest for gradient-based optimizations. Later, the gradual deformation method was extended to Boolean or object-based simulations. Hu (2000, 2003) suggested applying the gradual deformation method to modify the object locations and the number of objects in a simulation. However, the sudden appearance and disappearance of objects induces strong discontinuity in the objective function. Such behavior is not desirable for gradient-based optimizations. To alleviate this undesirable feature, Le Ravalec et al. (2004) reformulated the gradual deformation method for Poisson probabilities. The reformulated GDM allows to now smoothly add or remove objects from a Boolean simulation, thus reduces the discontinuity of the objective function, but does not fully eliminate them. PPM Unlike GDM that perturbs the random number used to draw from probability distribution, PPM perturbs the probability model itself that is used to generate conditional simulation. This method is developed by Caers (2003) and Caers and Hoffmann (2006). It relies on the development of multiple-point statistics (MPS). MPS is a new geostatistical technique that allows generating conditional stochastic simulations

14


reproducing the training image-based geological heterogeneity (Strebelle, 2002). The PPM offers solutions by extending the data integration framework of MPS to production data: accounting for production data yet being constrained by prior geological information. The idea is to convert both static data B and dynamic data D into probabilities P (A|B) and P (A|D) respectively, where A is the property to be simulated (such as channel occurs at specific location), then combine these individual probabilities into a single probability model P (A|B, D); finally, the reservoir models are generated by sequentially drawing from P (A|B, D). To combine individual probabilities into one model, Jounel’s tau model is used, and this method is expressed as follows: x = a

τ1 b c τ2 c a

(1.2)

where x=

1 − P (A|B) 1 − P (A|D) 1 − P (A) 1 − P (A|B, D) ,b = ,c = , and a = P (A|B, D) P (A|B) P (A|D) P (A)

P (A) is the global proportion of A occurring; The parameters τ1 and τ2 account for redundancy between the data B and D, but generally, they are chosen as τ1 =τ2 =1, which amounts to a form of standardized conditional independence. The combined probability model, P (A|B, D), for unit τ values is equal to the following expression: P (A|B, C, D) =

1 a = 1+x a + bc

(1.3)

While P (A|B) can be directly inferred by scanning the training image for equivalent data events A|B, P (A|D) is defined as follows considering a binary spatial variable case: P (A|D) = (1 − rD )i0 (u) + rD P (A) ∈ [0, 1]

(1.4)

Where rD is a parameter between [0,1] that controls how much the model is perturbed and i0 (u) is the initial realization of I(u) with

1.4. PROPOSED MODELING APPROACH

( I(u) =

15

1 if a given facies occurs at u 0 else

Similar to gradual deformation method, PPM parameterizes the perturbation using a single parameter rD that can be optimized. The optimization uses the Brent method but it could be any one-dimensional optimization routine. The advantage of Brent is that it does not require derivatives, and it is efficient and robust (Hoffman, 2005). Although PPM was initially based on MPS, the method is not limited to multiplepoint statistics or even sequential simulation. Any technique that uses conditional probabilities allows one to use the probability perturbation method.

1.4

Proposed modeling approach

To model a multi-scale reservoir architectural geometry, a hierarchical workflow (Figure 1.5) is adopted which is routinely used in reservoir modeling. Within this workflow, large-scale architectural elements (for example channel belt) are modeled first, their corresponding bounding surfaces are then identified and the shale drapes are simulated on these surfaces; next, the second level hierarchic architecture elementsindividual channels within channel belts are simulated, the corresponding bounding surfaces are captured, and shale drapes are modeled; then, lithofacies are simulated within each channel, finally continuous petrophysical properties such as porosity and permeability are assigned on a by-facies basis. Within the hierarchical workflow, available or known data must be honored at all times to obtain a reliable geological model, that is, each level of hierarchy model must be consistent with all available data with corresponding scales. In this work, for the purpose of studying shale drapes, the lithofacies within channel will only take into account sand which means channels are fully filled with sand facies. Properties such as porosity and permeability are modeled using traditional

16


sequential Gaussian simulation. The shale drapes along the bounding surfaces will be simulated using MPS and treated as edge property without volume. For reservoir architectural modeling, this involves defining channel deposition fairways (belts) based on seismic data, modeling long sinuous channels and placing them into defined fairways such that all data are matched. Because we assume that channel belts can often be identified from seismic data (with some associated uncertainty), the second level hierarchic architecture modeling- individual channel geometry and stacking pattern modeling are the more challenging part in this workflow. To integrate production data into the reservoir modeling workflow, this research will adopt the probabilistic perturbation methods (PPM and GDM) because these geologicaly consistent perturbation methods are capable of accounting for geological continuity consisting of strongly connected, curvi-linear geological objects such as channels. PPM/GDM couples the geological modeling and history matching in an automatic workflow. The stochastic nature of PPM/GDM makes assessment of the uncertainty of reservoir connectivity possible. The main challenge in the history matching part is how to modify channel locations to match production history while honoring the observed or believed stacking patterns. Figure 1.6 shows the proposed coupled modeling and history matching workflow based on the previous discussion. The probabilistic perturbation is performed in a hierarchical framework which is consistent with previously stated hierarchic workflow, that is, it starts modeling shale drapes associated with channel belts, then individual channels are simulated within channel belts and the associated shale drapes are simulated for each channel. To match the production data, the shale drape locations along both belt and channel boundaries as well as the channel locations are perturbed stochastically in an order defined by sensitivity study.

1.5

Thesis outline

This dissertation consists of five chapters. The proposed hierarchic modeling and history matching workflow is presented in Chapter 2. This chapter also explains the

1.5. THESIS OUTLINE

17

Figure 1.5: A hierarchic workflow to simulate multi-scale flow barriers. well data conditioning process and probabilistic perturbation process. The synthetic examples are provided to illustrate each of them. Chapter 3 tests the proposed workflow on a confined-channel reservoir analog. The sensitivity study is first performed to determine the most sensitive parameters such as shale drape geometry, proportion and location. Then, the proposed workflow is applied to check its feasibility. Chapter 4 presents a realistic case study based on an offshore West Africa turbidite reservoir, where the modeling and history matching workflow has been applied with the purpose of determining the location of channels and shale drapes in the reservoir. Chapter 5 discusses the major findings and contributions of this dissertation, as well as suggestions for future work.

18


Figure 1.6: Coupled geologic modeling and history matching of multi-scale flow barriers.

Chapter 2 Proposed modeling workflow The proposed modeling workflow in Figure 1.6 is composed of two parts: (1) static geologic modeling by integrating various static data such as well logging, seismic, outcrop analog; (2) dynamic data integration using a probabilistic perturbation scheme that ensures consistency between reservoir models developed from one stage to the next. The final goal is to provide multiple geologic models that can be used to predict the reservoir performance with a higher degree of confidence . The major contribution in this proposed modeling workflow consists of a new channel stacking pattern modeling technique as well as perturbation method that modifies channel location consistently with the conceptual stacking pattern model. For channelized reservoirs, the first level reservoir architecture - channel belt/complex is generally observable directly from seismic data, while the individual channels are below seismic resolution. Therefore, the channel stacking pattern modeling or the second level reservior architecture modeling is the most challenging one. The proposed modeling method makes use of the distribution function of pattern parameters to simulate and perturb stacking patterns. This chapter will first introduce the static geologic modeling process which includes individual channel modeling, channel stacking pattern modeling, shale drape modeling and well data conditioning. Following that, the geologic consistent perturbation process will be presented and a 3D synthetic example is provided to test the feasibility of proposed perturbation scheme. 19

20

CHAPTER 2. PROPOSED MODELING WORKFLOW

2.1

Reservoir architecture modeling

The static geologic modeling in this dissertation focuses mainly on reservoir architecture modeling. This is because fine-scale reservoir architecture at sub-seismic scale is challenging to model. In this work, we assume the first level hierarchic reservoir architecture-channel belt/complex is above seismic resolution and can be integrated into reservoir model deterministically. On the other hand, individual channels are below seismic resolution and require a stochastic modeling approach. Hence the reservoir architecture modeling here refers to the second level hierarchic reservoir architecture modeling or channel stacking pattern modeling. The second level hierarchic reservoir architecture modeling (channel stacking pattern modeling) workflow consists of two components: individual channel modeling and channel stacking pattern modeling. The criteria for choosing a specific modeling approach include the ability to match hard data, to define object (channel) boundaries (for the purpose of attaching shale drapes), to reproduce a given stacking pattern and to be CPU efficient. The present work adopts a stratigraphic-based modeling approach. In this stratigraphic-based approach, individual channels are simulated using the YACS method (Alapetite et al., 2005) because this method allows the simulated channels to be continuous throughout the modeling area, to have defined boundaries and to stay within an observed fairway (valley). The method is stochastic, rather than genetic or process-based technique; it only attempts to mimic the latter. But as will be shown, this method is fast and conditions to well data under the assumption that individual channel sand bodies can be identified form well data. To put multiple channels together reproducing desired stacking pattern, stacking pattern parameters such as overlap ratio and migration ratio are simulated stochastically.


2.1.1

21

Individual channel modeling

To perform the YACS approach (Alapetite, et al., 2005), the following parameters are required to specify channel geometry (Figure2.1):

Figure 2.1: Definition of parameters used to describe the channel geometry

• Channel orientation (α): the direction a channel extends; • Channel wavelength (λ): the distance between adjacent peaks or troughs; • Channel amplitude (A): the distance between adjacent trough and peak; • Channel width (W): the maximum width of the channel; it is located at the center of the channel; • Channel thickness (H): the maximum thickness of the channel; it is located at the center of channel;

22


• Channel cross-section geometry is defined by a parabola h(x) = H

1−

2(x − x0 ) W

2 !

where x0 is the location of channel center point and x is the position on the cross-section to be computed. For simplicity, in this dissertation we assume all the channels have symmetric cross section. We also assume all the channels have regular sinesoid geometry described by a set of geometry parameters listed above and don’t consider the unanticipated change in sinuosity.

The core of the YACS method lies in the association of a

fairway with the channels to be simulated. In this method, a single channel simulation starts by selecting a particular depositional surface. On this surface, a channel belt is defined and a so-called “potential field” is computed within this belt. In order to create sinuosity and realistic channel geometry, a stochastic perturbation is applied to this original potential field. The variations then control channel geometry such as sinuosity, amplitude. The perturbed potential field is then mapped to channel thickness using a transform function. Placing the thickness underneath the selected depositional surface results in generating a channel inside the channel belt. The main steps for single channel modeling are: 1. Select a deposition surface and define the channel belt boundaries on this surface (Figure 2.2a); 2. Compute the original potential values between two boundaries by interpolation (Figure 2.2b). One boundary is set to negative and the other one is positive. The absolute number of these two extremes corresponds to the bottom of the channel belt at which the channel is located; 3. A correlated noise is simulated using Sequential Gaussian simulation (sgsim) (Figure 2.2c); The simulated noise is normal distributed with mean 0. Its standard deviation is linearly related to the amplitude of simulated channel.


23

The principle direction of the variogram is the orientation of the channel belt. The range in that direction is linearly related to channel wavelength; 4. A perturbed potential map (Figure 2.2d) is obtained by adding the simulated noise to the original potential map; Keep the positive potential values in Figure 2.2d and flip the negative potential values into positive, the 0-isopotential line is the channel centerline (Figure 2.2e); 5. Once the channel centerline is located, the channel region is defined (Figure 2.2f) since we know the channel width; 6. Apply a transfer function h(d) = H

1−

2d W

2 !

on the channel region to obtain a channel thickness map (Figure 2.2g); At the 0-potential point on Figure 2.2f, d is 0, and d increases towards the channel boundary along the potential gradient direction. 7. Paint the thickness underneath the depositional surface; a channel in 3D space forms (Figure 2.2h). In the case that the channel belt is curvilinear ( Figure 2.3a), the original potential map is interpolated between the boundaries of channel belts (Figure 2.3b), then a correlated noise field is simulated (Figure 2.3b). The principal axes of the noise variogram are defined by the potential gradient direction. The principal direction of the variogram is the orientation of the channel belt. Performing Step 4 to add the simulated noise to the original potential map, the 0-potential values within the perturbed potential map delineates the desirable channel centerline (dark blue) (Figure 2.3e). Comparing Figure 2.3e with the channel geometry showing in the original potential map (before adding the noise map) (Figure 2.4), the perturbed channel geometry is more realistic. Then we can perform Step 5 - 7 to generate a 3D channel body within this curvilinear channel belt.

24


Figure 2.2: The workflow for individual channel modeling

Figure 2.3: The example showing curvilinear channel belt case

Figure 2.4: The original potential map in Figure 2.3 showing the channel geometry before adding noise map


2.1.2

25

Channel simulation parameters

In this modeling method, the channel geometry is controlled by the simulated noise (Step 3) which is generated using Sequential Gaussian Simulation. To obtain a desirable channel geometry, the noise statistics such as the variogram and histogram of the noise can be obtained to generate desirable channel geometry parameters such as channel wavelength, amplitude and sinuosity. Figure 2.5 shows the perturbed channel potential maps with different noise histogram variances. As the noise variance increases, channel amplitude and sinuosity increases. The channel wavelength is related to the noise variogram range along the channel orientation direction. From Figure 2.6 top row we observe that the channel wavelength increases with noise variogram ranges along the channel orientation direction. Figure 2.6 bottom row shows channel geometries with different noise variogram ranges perpendicular to the channel orientation direction. The channel amplitude decreases with an increase in the variogram range perpendicular to the channel orientation. Besides geostatistical parameter related to noise simulation, the potential gradient in the original potential map also affects the final channel geometry (Figure 2.7). Figure 2.7 top row are the original potential maps generated using different potential gradients. The 0-isopotential line is located at the same position on these maps. The bottom row shows the perturbed potential maps depicting channel centerline geometry after adding the same noise. It shows that with potential gradient increase, channel amplitude decreases.

2.1.3

Channel stacking pattern modeling

Reservoir architectural geometry is reflected in the channel stacking pattern. This dissertation uses two parameters to define the stacking pattern, i.e. the overlap ratio and the migration ratio. The reason for choosing these two parameters is that the channel stacking architecture is controlled by the interaction between a lateral and vertical amalgamation process (Clark & Pickering, 1996). The migration ratio (Figure 2.8 left) is the ratio of horizontal distance (x) between two adjacent channel center points and the channel amplitude. It constrains the lateral distance between

26


Figure 2.5: Channel potential maps with different noise histogram variances

Figure 2.6: Channel potential maps with different noise variogram ranges: top row is for ranges along channel orientation direction, bottom row is for ranges perpendicular to this direction


27

Figure 2.7: The potential maps (bottom) and their corresponding original potential maps (top) generated with different potential gradients

two adjacent channels. The overlap ratio (Figure 2.8 right) is the ratio of vertical overlap thickness (h) between two channels and the channel maximum thickness (H). It constrains the vertical distance between two adjacent channels. Figure 2.9 depicts a cartoon showing channel stacking patterns with different pattern parameter combinations. In this work, these two parameters are combined to determine the location of the channel object centerline relative to an adjacent channel. We assume that the probability distribution functions of these two parameters can be obtained either from outcrop study or process-based models. For demonstration purposes, a uniform distribution for migration ratio and overlap ratio is used in the following synthetic cases. We also assume the net-to-gross ratio for each channel belt can be obtained. Based on the superposition, the younger channel (deposited on upper surface) erodes the older one (deposited on lower surface) if they are in contact with each other. Given the predefined pattern parameter distributions and net-to-gross ratio as well as the erosion rules, the architecture modeling is performed as follows:

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1. In a channel belt (Figure 2.10), if there are no well data to be conditioned to, first simulate a single channel at the channel belt top center using the simulation method stated in section 2.1.1 (Figure 2.11a); if there are wells passing through this belt, then first generate channels fitting all the interpreted well channel sections; 2. Draw a value for the migration ratio and overlap ratio from their corresponding distribution functions, and use these ratios to obtain the location relative to the previously simulated channel; simulate a new channel centered at this location (Figure 2.11b). 3. If the simulated channel does not fully stay within the channel belt, then it is rejected and step 2 is repeated until the new channel is completely within channel belt; 4. Repeat step 2-3 to generate a new channel within the channel belt until the given net-to-gross ratio is approximately reached (Figure 2.12c - h); 5. Repeat step 1-4 for each channel belt in the reservoir (Figure 2.13); The architecture modeling is performed from top to bottom which appears to contradict the sequence of deposition. There is no doubt that the channels can be generated from base to top following the deposition rule. However, the proposed modeling sequence is more favorable when a vertical proportion curve needs to be taken into account. The vertical proportion curve obtained through well and seismic data specifies the proportion of all sand as a function of vertical elevation (or depth). This information will provide a constraint on the number of channel to be simulated for each layer. Because channel thickness simulated in the upper layer of the grid will contribute to the sand proportion for the current layer of the grid, it is reasonable to generate channels from top to bottom in order to honor the vertical proportion curve. We should notice that all the individual channels are continuous throughout the simulation domain if they are not eroded. These channels stay within the predefined belts as expected. Once these channel complexes are generated, their bounding surfaces can be traced for shale drapes modeling (Figure 2.14).


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Figure 2.8: Two parameters used for channel stacking pattern modeling

Figure 2.9: Schematic graph showing channel stacking patterns with different pattern parameters

Figure 2.10: A channel belt body (left) and its bounding surface (right). Channels will be filled into the space within the channel belt limit

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Figure 2.11: The channel stacking pattern modeling process showing how channel is filled into belt (continue to next page)


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Figure 2.12: The channel stacking pattern modeling process showing how channel is filled into belt. Left column is the architecture model, middle column are the realizations for migration ratio, and right column are the realizations for overlap ratio

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Figure 2.13: Three channel belts case. Individual channels are filled into each belt until its net-to-gross reached. All the channels are confined by channel belt limits

Figure 2.14: Channel bounding surfaces extracted from Figure2.11


2.1.4

33

Well data conditioning

Well data conditioning is performed in the individual channel modeling process. The individual channel modeling approach adopted in this dissertation is flexible to condition to well data because the only stochastic engine is pixel-based Sequential Gaussian Simulation. The basic idea of conditioning well data is that the simulated noise at the well locations must be such that the resulting potential, consisting of a noise component and the original potential (which maps back into channel thickness) is close to zero (i.e., close to channel centerline). Based on this idea, the interpreted channel thicknesses at the well locations are first converted into potential values, then transformed to noise values. These noise values are used to condition Sequential Gaussian Simulation (Alapetite, 2005). Unlike the Boolean simulation “move-until-fit” process, this approach directly places the generated object to the locations corresponding to the well data. In other words, the channels in this approach are generated directly, not through iterative type perturbation, to match the well data; this makes the conditioning fast. However, the method of conditioning is not as general as traditional sequential simulation. Certain assumptions need to be made. Most importantly, the well data need to be interpreted in terms of architectural elements. Such interpretation is subject to uncertainty (not considered in this dissertation), as further detailed in the next section. Well data in this case are facies values known along the well path. The first step in the well data interpretation is to separate channel and non-channel facies from well data. Next, the channel facies are assigned to different channel sections (Figure 2.15). This assignment is essentially a process of geological interpretation that takes into account all the geological information available about the stacking patterns (this dissertation assumes channel has determined and constant dimensions). Incorporating the channel stacking pattern information into well data interpretation is necessary because the stacking architecture of channel sand bodies has a strong control on the interconnectivity between channels. Similar as Viseur’s method (Viseur et al., 1998), this dissertation uses two parameters to define the stacking pattern, i.e. the overlap ratio and the migration ratio. As a result, the interpretation of the well data is stochastic. For each interpretation, we draw pattern parameters from their distribution

34


functions and then convert them into a channel section center position relative to the adjacent channel section. In other words, for the same well data set, we obtain multiple channel sections with different stacking patterns. All of their pattern parameters follow the same given distributions. Next, we will use a synthetic example to explain this interpretation process. Suppose we know the channel stacking pattern parameter probability distribution functions for the reservoir under study (Figure 2.16) and we also know individual channel geometry parameters: wavelength λ = 25m, amplitude A=12m, orientation α =0, thickness H=9m and width W=15m. All the channels have constant geometry parameters. For the well column shown in Figure 2.17, the sand thickness is 20 feet which

Figure 2.15: Schematic graph showing the well facies data and their interpreted channel sections is much larger than an individual channel thickness H (9 feet in this case). This indicates that the well passes through multiple channels. Therefore an interpretation is required to assign channel sand section in well to channel bodies. A pair of stacking pattern parameter values is drawn using their respective probability distribution functions shown in Figure 2.16. This pair of values is used to obtain the location of a channel section centerline relative to its adjacent one. Once we have simulated


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Figure 2.16: Synthetic uniform distributions of two pattern parameters

channel section centerline position, a channel section is added into the well sand facies with this simulated centerline position. We repeat the random drawing of pattern parameter ratios and the channel section generation process until the sand facies are fully filled by channel sections at the well location. The final result is one stochastic interpretation (eg. Figure 2.17, Case1). If we repeat the above process several times, we can obtain multiple interpretation results (Figure 2.17). Note that in this case we assume no knowledge of the channel boundaries in the well data. If such information were available, the uncertainty in the interpretation could be further reduced. For a case with multiple wells, the individual well facies data interpretation follows the previously stated interpretation process. In addition, the interpretation should consider well correlation data and channel geometry parameters such as sinuosity and amplitude. This will ensure that the interpreted multiple well data are compatible. In other words, if two wells are close to each other (distance less than the channel amplitude) and have sand facies at the same interval, the interpreted channel sections from these two wells should not conflict with the channel sinuosity information if they are on the same depositional surface. For example, in Figure 2.18, the blue dash line represents a predefined channel geometry in map view. Well A and B are within the channel amplitude region in the x− direction and very close to each other in the y− direction. If these two wells have interpreted channel sections on the same depositional surface (such as purple channel section for well A and the green section for well B), then these two channel sections are candidates to belong to the

36


Figure 2.17: Synthetic example demonstrating the stochastic interpretation of well facies data. The same well data can result in multiple channel section stacking pattern realizations


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same channel. However, in the left interpretation the channel section centerline locations are not consistent with the channel sinuosity information thus is incompatible data, while the right one is compatible. In practice, it may not be trivial to generate “compatible data”, especially when many wells are present. In this work we assume the interpreted channel sections are compatible when multiple wells are present. Having the interpreted channel sections along the well path, the tops and center

Figure 2.18: Schematic example showing the compatible/incompatible interpretation when multiple wells present points as well as channel thicknesses at the well locations for these channel sections are recorded. Next, the hard noise data will be computed based on these interpreted results. The computed hard noise data will be honored during noise simulation using sgsim in order to fit a channel object to the interpreted channel section. The main steps are explained using an example in Figure 2.19: 1. Calculate the distance (d) between the well location (point A in Figure 2.19) and the channel centerline (point C in Figure 2.19) using W d= 2

r

h 20 1 − , so for data in Figure 2.19, d = H 2

r 1−

8 = 6.8 15

2. Generate the original potential map with predefined potential gradient dP (=0.75). The 0-isopotential line (original channel centerline) passes through

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the interpreted channel center point (point C). Read the original potential value Porg (=10.5) at the well location A; 3. Calculate the potential value at the well location A: Pw = dP ×d = 0.75×6.8 = 5.1 4. Calculate the noise at the well location: Nw = Pw − Porg = 5.1 − 10.5 = −5.5 The result in Step 4 is hard noise data for noise simulation. In this example, the simulated noise is normal distributed with mean 0 and variance 25. When a shale interval occurs at well location, simply assign a large Pw to avoid the shales falling close to the 0-potential line (the channel centerline) on the perturbed potential map. For the multiple-well case one first identifies all channel sections whose top fall

Figure 2.19: Well channel section data (left) and its original potential map (right). The channel section geometry-maximum width and thickness- should be the same as the defined channel cross section geometry within the same depositional surface (or layer). These channel sections potentially belong to the same channel object. To check if these channel sections belong to the same channel, one calculates the centerline distances between channel sections in the direction perpendicular to the channel orientation direction. If channel sections are within a channel amplitude range, they belong to the same channel (Figure 2.20 left). However, the same channel section (such as the section of well 3 in Figure 2.20) could belong to a different channel (Figure 2.20 right). In this specific case we need to make


39

a decision to deterministically set the channel section to a channel. This means that two interpretations are possible leading to two different interpreted hard data sets (hard data uncertainty). Once we know the conditioning data points for one channel, the actual channel

Figure 2.20: Well conditioning for interpreted channel sections. Well 1,2,3 could be connected with one channel (left); However, well 3,4,5 can be connected within another channel (right). Hence two interpretations are possible. centerline is equated to the average value of channel sections’ center point locations. Then the previous single channel simulation approach is applied to generate a channel passing through these channel sections by means of conditional sequential Gaussian simulation of the noise. Figure 2.21 shows the case where 10 channel sections interpreted from 10 wells belong to one channel (on the same depositional surface). In the top row figure, 10 points indicate well locations and their color represent channel section thicknesses at the well locations. The bottom figures are three realizations conditioned to the top well thickness data. We observe that the larger thickness the point has, the closer it is located to the 0-potential line which represents the channel centerline. Figure 2.22 shows the 20-well case, the well data are perfectly honored and multiple realizations can be obtained. Figure 2.23 shows the case where one well vertically passes through different layers, and there are three interpreted channel sections overlapping one another. Based on the interpreted thickness information, three channels are simulated and the interpreted channel stacking patterns at the well location is reproduced.

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Figure 2.21: Three conditional realizations (bottom) of channel potential conditioned to 10 wells (top)

2.1.5

Shale drapes modeling

As previously stated, shale drapes are associated with erosional bounding surfaces. Although shale drapes may exhibit a varying degree of coverage, they are often very thin (cm to m) compared to sand bodies. Therefore, ignoring their volume effects, shale drapes will be simulated on their associated bounding surfaces in 2D space. The simulated 2D shale drapes can be easily converted into 3D transmissibility multipliers in flow simulation model for shale drapes effects study (Stright, 2005). In this work, multiple-point statistics (MPS) program snesim will be used for shale drapes modeling. To perform MPS, a shale drape training image representing the reservoir conceptual shale drape distribution model is required. Figure 2.24 is one example of


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Figure 2.22: Three conditional realizations (bottom) of channel potential conditioned to 20 wells (top)

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Figure 2.23: One realization (right column) of a channel complex conditioned to interpreted channel stacking pattern (left column top) at well location


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the shale drape training image. The red color indicates scour holes and the background represents shale drapes covered on the bounding surface. In fact, if shale drapes only cover a small portion of the bounding surfaces (less than 40% areally), the reservoir connectivity is not affected by these shale drapes (Li and White, 2003). On the contrary, if bounding surfaces are fully draped by shales, and only a small portion of scour holes are present among the shale drapes, the location and proportion of scour holes will have significant impact on reservoir connectivity (Li, 2003; Stright et. al., 2006). This is because the holes may connect two channels if they are in contact with each other. Therefore, instead of simulating shale drapes, we will simulate holes distribution on the bounding surfaces. The modeling process is as follows (Figure 2.25):

1. Extract the bounding surfaces of the architecture elements (such as belts and channels) from the simulated architectural model 2. Flatten these bounding surfaces into multiple 2D surfaces 3. Apply snesim on these 2D surfaces to generate holes 4. Fold back 2D surfaces with simulated holes into their original 3D space This process is applied for each hierarchy. Finally a multi-scale shale drapes model can be generated.

2.1.6

Summary of the architecture modeling

This section presented static geologic modeling techniques to simulate channel stacking patterns and shale drapes distribution along multi-scale bounding surfaces in a channelized reservoir. The complete static geologic modeling workflow is shown in Figure 1.5. In this workflow, the large-scale reservoir architecture is modeled first, the shale drapes are then simulated within this architecture framework. The architecture modeling approach adopted in this dissertation is essentially a stratigraphic-based modeling approach. It is fast and can condition to well data, at the same time the

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Figure 2.24: One example of shale drape training image. Red color represents holes, and blue color indicates shales

Figure 2.25: The workflow of shale drape modeling


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channel stacking patterns can be reproduced. However, as we mentioned before, this modeling process works under some assumptions. As a consequence, the quality of results will be dependent on the agreement between the stated geological modeling hypothesis and the actual observed data. The first assumption is that the probability distribution functions of pattern parameters are available or can be derived from some sources. We also assume that the net-to-gross ratio of each channel belt is known. As a result, the number of channels within channel belts is determined based on this net-to-gross ratio input. In practice, obtaining the pattern parameter distribution functions that reflect the geological pattern features is not an easy job; and determining the net-to-gross ratio for each channel belt may be challenging.

2.2

Geologically consistent history matching

In the previous section, the reservoir architecture modeling technique and the shale drapes modeling technique are introduced to integrate various geologic information from different sources. However, building geologic models is just the first step. The next important step is to perturb these geologic models to match production data. The perturbation should be consistent with the geologic concept and the static data. Once a model has been history matched, it can be used to predict reservoir performance or control reservoir performance with higher reliability. From the proposed workflow in Figure 1.6 we can see that there are two types of perturbation: channel stacking pattern perturbation and shale drapes perturbation. The next section will describe the perturbation techniques for these two properties and the history matching procedure based on these geologically consistent perturbation mechanisms.

2.2.1

Channel stacking pattern perturbation

Before developing our geologically consistent perturbation technique, we need to state clearly what is meant by “geological consistency”. For stacking channels, geological

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consistency mainly means preserving the channel stacking pattern and channel geometry. In this work, individual channels are continuous over the entire simulation domain if they are not eroded by other channels. If a portion of channel is moved from one location to another, other portions of the same channel should be moved in order to preserve user-defined channel geometry parameters such as wavelength and amplitude. Furthermore, because deep-water confined channel reservoirs often exhibit a wide variety of stacking patterns (Clark and Pickering, 1996a, b), the perturbation of the channel locations should not destroy the interpreted or observed stacking pattern if such pattern is deemed correct. In other words, when we modify individual channels, the modified channel should keep the same geometry parameters, and the perturbed channel complex should preserve the stacking pattern believed to exist. From the previous individual channel simulation process described in Section 2.1.1 we can observe that there are two components that potentially affect channel locations: (1) the noise map which has impacts on individual channel geometry; (2) the channel centerline (0-isopotential line) location in the initial potential map which directly controls where the channel is located. Figure 2.26 shows different channel potential maps (bottom row) obtained by adding different noises (middle row) to the same original potential map (top row). The 0-isopotential lines in the bottom row potential maps of Figure 2.26 are the channel centerlines, and they can be used to represent channel shapes. Because the same original potential map is used, the channel centerlines in the perturbed potential maps have similar location. But they have different geometry due to the different noise realizations used. Figure 2.27 shows the same noise map (top row) being added to the different original potential maps (middle row). The final curvilinear channels exhibit different geometry and are located at different positions. In the proposed channel stacking pattern simulation workflow, the channel centerline locations are simulated by random drawing of stacking pattern parameters. More precisely, a pair of migration ratio and overlap ratio is first simulated from their corresponding distribution models, then converted to a channel centerline whose location is placed relative to the previously generated channel. Therefore each channel is associated with a pair of simulated stacking pattern parameters. These pairs of pattern parameter realizations can be


47

perturbed in order to change channel locations during history matching, at the same time preserving the stacking pattern. The stacking pattern is fully defined by the distribution of migration ratio and overlap ratio. As observed from these two examples, modifying the channel centerline location will result in a significant change of the channel position, whereas deforming the noise map will cause slight variations of channel geometry, but the channel centerline location remains the same. As previously stated, modifying a pair of pattern parameter realizations will

Figure 2.26: Different noise maps (middle row) added to the same original potential map (top row) result in different channel geometries (0-potential line in bottom row)

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Figure 2.27: Same noise maps (top row) added to different original potential map (middle row) result in different channel geometries (0-potential line in bottom row)


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change channel locations. This modification results in an entire channel object being modified because a pair of pattern parameter realizations determines one channel centerline location. Obviously PPM is not suitable for moving channel centerline locations because PPM modifications are essentially pixel-based not object-based. Hence the gradual deformation method will be applied to modify channel location. To perturb channel geometry by modifying the noise map, this work will adopt the PPM approach. These two perturbations will share the same optimization processes, i.e. they will share the same perturbation parameter. The optimization process adopts the Brent method because it does not require derivatives, hence the discontinuity in the objective function behavior in GDM is not an issue here. Gradual deformation of channel locations The key idea is to perturb channel locations, but maintain the interpreted or observed channel stacking patterns. GDM is applied to deform two channel stacking pattern parameters which are then used to determine the deformed locations relative to adjacent channels. This perturbation process is explained using a synthetic example with two uniformly distributed stacking pattern parameters (see Figure 2.12).

O 1. Record the migration ratio (uM i ) and overlap ratio (ui ) for each channel i

(here assuming these parameters are uniform distributed) and store them into O M O O M a parameter vector u=(uM 1 ,u2 ...un ,u1 ,u2 ...un ) (Figure 2.28)

Figure 2.28: Channel pattern parameters sampled from their distribution function

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2. Transfer the recorded channel pattern parameters from uniform distribution into Gaussian distribution (Figure 2.29) ;

Figure 2.29: A uniform distribution is transferred into Gaussian distribution

3. Generate another set of Gaussian realization for channel pattern parameters and combine them with the ones in Step 2 (Figure 2.30) using equation: y(rD ) = y1 cos(rD ×

π π ) + y2 sin(rD × ) 2 2

(2.1)

4. Transfer the deformed realization y in Step 3 back to uniform distribution (Figure 2.31) 5. Use these deformed ratios to obtain the locations relative to the adjacent simulated channel; simulate new channels centered at this location using the same noise maps as the initial patterns. Figure 2.32 shows a synthetic example of how the initial channel positions are deformed when different perturbation parameter values are applied. We notice that with an increase in rD , the magnitude of perturbation of the channel position increases, but the stacking pattern is maintained during perturbation. This is because the deformed pattern parameters always have the same distribution functions as the initial ones


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Figure 2.30: Two Gaussian realizations are perturbed to generate one new realization

Figure 2.31: Transfer the perturbed realization back to uniform distribution

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Figure 2.32: A chain of channel realizations with different perturbations applied to the pair of channel stacking pattern parameters


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Probability perturbation of channel geometry As we mentioned before, the noise map affects the channel geometry. Probability perturbation can be applied to modify the noise map. This modification will change channel geometry but keep the channel centerline location fixed. The noise is a continuous property, while traditional PPM is often applied to the categorical case, where a parameterized soft data P (A|D) is used to create the perturbation. Hence, instead of calculating P (A|D) and combining it with prior data using the tau model, collocated sequential Gaussian simulation (cosgsim) is performed using a correlation coefficient,ρ with as the soft data the previous ‘best” realization. Similar to the perturbation parameter, the correlation coefficient quantifies the amount by which this current best realization is perturbed. The correlation coefficient and the perturbation parameter are related through ρ = 1 − rD . When ρ = 1 (rD = 0), the soft data is honored exactly, and the current best realization is reproduced. Conversely when ρ = 0 (rD = 1), the soft data provides no information, and the realization is maximally perturbed into a new equiprobable realization. Figure 2.33 shows how the initial individual channel geometry changes with different perturbation parameter values (rD values). The channel centerline locations in different perturbed maps (0-isopotential line in the left column figures) have roughly the same location, but the channel geometry varies. As rD increases, the channel geometry variation becomes larger. Figure 2.34 shows multiple channel geometry deformations with different perturbation parameters. The initial channel complex realization is the same as shown in Figure 2.32. By perturbing the noise maps of each individual channel, the channel geometries change. Although channel centerline locations remain almost constant as demonstrated in Figure 2.33, the channel geometry variations will cause the channel cross-section to move from one position to another. This results in the channel crosssection changes (right column).

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Figure 2.33: Four realizations of individual channel potential maps with different perturbation parameters applied to their noise maps. Note the channel centerlines are roughly same.


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Figure 2.34: Channel complex realizations with different perturbation parameters applied to the individual noise maps. Note the channel locations are roughly the same (left column) but the channel cross-section changes with different perturbations (right column, y=20).

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2.2.2

Behavior of a chain of stacking pattern realizations

Two independent realizations (Figure 2.35 & 2.36) are provided to illustrate the previous methodology to perturb channel locations. The objective of the study is to generate a chain of realizations between these two reference realizations with different perturbation parameters and check the objective function variation. Figure 2.35a and 2.36a show channel complex realizations with 7 channels continuously distributed within a valley. The reservoir net-to-gross is 0.35. The dimension of the reservoir grid is 70 × 100 × 60 with grid scale 25 × 25 × 1 m. Channels are fluid conduits with permeability 1000m and porosity 25%. The permeability and porosity of the background are 5md and 5%, respectively. Shale drapes are simulated and added onto the channel bounding surfaces to increase reservoir heterogeneity (Figure 2.35b & 2.36b). There are two producers and one injector (Figure2.35c). Due to reservoir heterogeneity, these two producers have different water cut curves (Figure 2.35d and 2.36d). Next the proposed perturbation method is applied to generate perturbed realizations with rD ranging from 0.1 to 0.9 (Figure2.37). In more detail, the vectors u1 and u2 of pattern parameters used for reservoir models in Figure 2.35a and 2.36a respectively are transformed into Gaussian domain as y1 and y2 . Then a series of perturbations are applied to these two vectors using Eq.2.1 with rD varying between 0.1 and 0.9. The perturbed pattern parameters are then back-transformed into their original distribution domain and used to determine channel locations. This results in a chain of reservoir models with varying amount of perturbations. The shale drapes along the channel bounding surfaces within each perturbed realization are simulated using the same random seed as the reference models. Figure 2.38 shows the water cut curves of two wells for these 10 cases. The objective function is defined as the sum of the square differences between the values of water cut at each time step of the initial realization (Figure 2.35a) and those of each perturbed realization. Figure 2.39 shows the evolution of the objective function vs. the rD values. The variation of the objective function is reasonably smooth which means that the perturbation can be effectively applied to history match the model. In summary, A method is presented in this section for history matching of the


57

Figure 2.35: Channel realization (a) and the shale drapes (b) along channel boundaries (blue color represents shale drapes and red color is for scour holes); wells are located in the reservoir thickness map(c); (d) is the plot of water cut for two producers.

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Figure 2.36: Channel realization (a) independently generated from the one shown in Figure 2.35 and the shale drapes (b) along channel boundaries (blue color represents shale drapes and red color is for scour holes); (c) is the reservoir thickness map, wells are located at the same positions as shown in Figure 2.25c; (d) is the plot of water cut for two producers.


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Figure 2.37: Realizations with different perturbation parameters

Figure 2.38: Water cut curves for a chain of 10 realizations. The initial realization is for the model in Figure2.35a

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Figure 2.39: Objective function of a chain of 10 realizations channel stacking patterns in channelized reservoirs. This method allows constraining channel stacking patterns to production data using gradual deformation method and/or probability perturbation method. By adjusting certain aspects of geostatistical channel reservoir model such as channel stacking pattern parameters and noise map while history matching, the method makes it possible to reduce the uncertainty in geostatistical models. Although gradual deformation method is not suitable for discontinuous objective function, the optimization algorithm - 1D Brent methodadopted in probability perturbation method will make it work for non-differentiable objective functions.

2.2.3

Shale drape perturbation

Shale drapes along channel belts and channels are simulated using MPS, where shale drapes vs scour holes are modeled as a binary indicator variable ( Section 2.1.5). Since PPM is a suitable perturbation method for MPS facies simulation (Caers, 2002; Hoffman, 2005), this work will use PPM to perturb shale drapes for both channel belts and individual channels. Stright (2006) demonstrated that shale drapes can be history matched using the Single-Region PPM method. In her work the whole reservoir was treated as one single region and a single perturbation parameter rD is used for entire reservoir. As a result, the whole reservoir is perturbed by a similar amount


61

(Hoffman, 2005). The same approach will be adopted in this dissertation when the field has few wells. In the case that a field has many wells, production data from one well (area) may match history whereas production data from another well (area) may not. Hence perturbing the reservoir model globally may not be efficient. In fact, if many wells are present, the single region PPM may not be efficient at all. To overcome this problem, a Multi-Region PPM was proposed (Hoffman, 2005), which allows different amounts of perturbation to be applied to various areas of the reservoir. The workflow of Multiple-Region PPM is discussed fully in Hoffman’s dissertation (Hoffman, 2005). The idea of Multi-Region PPM is to perturb the properties that influence production data for one region differently than the properties that affect production for another region by optimizing perturbation parameter for each region. The workflow for Multi-Region PPM is similar to the Single-Region PPM, that is, it also consists of an inner loop and an outer loop. The inner loop finds the optimum realizations between the initial realization and an equiprobable realization. The outer loop consists of replacing the initial realization with the previous optimum realization and changing the random seed. However, there are two differences in optimization procedure: the first difference is that the objective function is calculated differently since multiple perturbation parameters are defined instead of one parameter; consequently, the second difference is that Multi-Region PPM optimizes multiple parameters jointly while Single-Region PPM optimize one parameter. In order to perform Multiple-Region PPM, the region geometry should be defined first. One option is to use a streamline method to define regions. For a synthetic reservoir model shown in Figure 2.40, it takes three steps (Figure 2.41): 1. The streamline simulation is run using the simulation model described previously. This will provide us the streamlines defining flow paths to production wells . 2. Having the streamline distribution, we can identify a set of streamlines that go to the same production well. All gridblocks hit by this set of streamlines are assigned to that well. These gridblocks form a region that principally affect

62


this production well. For example, if we have three production wells, this correspondes to three sets of streamlines and three regions are assigned to these wells. 3. After we have identified regions in 3D model, the next step is to assign regions to channel boundaries along which the shale drapes are present. This is done by simply taking the channel boundaries in 3D channel region model. After the regions are defined, the holes distribution can be perturbed on these bounding surfaces using a Multi-Region PPM. One should notice that, becuase the hole locations will change per realization, this region defining procedure should be done at every iteration. In some cases, different channel belts are formed in very different depositional settings, such as different erosion energy, different bypass period. This could result in different shale drape coverage for each channel belt and geological regions of reservoir model. In this case, the regions are fixed for whatever channel and shale drapes are distributed. In order to perform history maching efficiently, we may need to assign different rD to different geological regions or channel belts. This could be done by performing sensitivity study to check which production well is most sensitive to which geologic region, then assign rD to that region based on the objective function calculcted from that well(s).

2.2.4

Perturbation procedure for history matching

The calibration of the perturbation parameter rD and the subsequent geologic model updating requires the implementation of an interface between the geologic modeling algorithm and the flow simulator. In this dissertation, the geologic modeling algorithm is the main program that includes all the tasks in the probabilistic perturbation approaches for dynamic data integration. The main program is implemented within SGEMS framework using the Python script. The flow simulator (@ Eclipse) is also executed within the main program. Because of the hierarchic nature of the shale drapes in channelized reservoirs, the


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Figure 2.40: A synthetic reservoir water saturation model with 20 channels and 3 producers.

perturbation will be performed hierarchically. For example, the shale drapes along channel belt and the distribution of individual channels within channel belts are perturbed simultanously until a predefined threshold A is reached, then the shale drapes along the individual channels are perturbed to reach the final threshold B which is smaller than A. In this case, the perturbation has two stages: perturbation of larger scale geologic features and perturbation of the small scale ones. Both stages share the same probabilistic perturbation procedure with two-loop Markov chain (Hoffman, 2005). The inner loop finds the optimum realizations between the initial realization and another equiprobable realization. The outer loop consists of replacing the initial realization with the previous optimum realization and changing the random seed. Within the inner loop, the perturbation is performed through the optimization of a 1-D “free” parameter rD , that is between [0,1] and controls how much the initial model is perturbed. Once we have a rD value, the shale drapes distribution will be updated using PPM (Eqn.1.3) and the channel distribution will be updated using

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Figure 2.41: The region defining procedure


65

GDM (Eqn.1.1). Then a new multiscale shale drape realization is generated that is a perturbation from the original realization. There may exist a value of rD , such that the corresponding realization matches the production data better than the initial realization. Hence finding an optimum realization is equivalent to finding the optimum rD value. The optimal rD value is found by minimizing the objective function,

rDopt = min O (rD ) = DS (rD ) − D

(2.2)

Where DS (rD ) is the simulated data and D the observed field data. O(rD ), the objective function, measures the mismatch between the simulated production data and observed field data. For N production variables over T time steps, the objective function is : v u T N uX X (Simi,t − Histi,t ) 2 O(rD ) = t Histi,end 1 1

(2.3)

At the end of each inner loop, the optimum realization will be used as the starting realization for the next outer loop, and the random seed will be changed to introduce new realizations. The iterative loop is repeated until a tolerance in the ojective function (mismatch) is met or a fix number of outer iteration is reached. The implemented code for history matching can be described in the following steps: 1. Perform proposed geologic modeling algorithim to stochastically generate an initial realization; 2. A Markov chan iterative updating process is started with the initial realization. The Markov chan forms the outer loop of updating. Every outer loop includes following procedures. • Change random seed • Making an initial guess of rD ; • Update probability model for shale drapes (PPM) and the channel pattern

66


parameters (GDM) for channels using the same rD . Generate a updated realization; • Perform a 1-D optimization process to get the rD that result in best match to the production data. This is a calibration process of rD with the production data and it is called the inner loop. The step of inner iteration is controlled by a fix number. • Update the stochastic realization using the best model (with the minimum objective function). 3. Repeat Step 2 (outer loop) until a tolerance in the objective function has been reached or a fix number of outer iteration is met.

Chapter 3 Workflow testing on a synthetic data set The proposed modeling workflow in Figure 1.6 is an iterative process of geologic modeling and perturbation for history matching. The results of the workflow depend on the input modeling parameters such as the distribution functions of pattern parameters, individual channel object geometry parameters or shale drape proportion since these parameters will determine whether the geologic model built is realistic. It is also dependent on the perturbation scheme for establishing the optimal rD because different perturbation scheme may results in different rate of convergence. The sensitivity of geologic parameters to the objective function also affects the convergence rate, hence the final result of the workflow. Synthetic reservoirs are useful for such an integrated modeling and flow simulation study because the shale drape distributions are known. Using synthetic reservoirs, we can check if the proposed geologic modeling technique reproduces channel stacking patterns given the geologic reasonable input parameters. We can perform sensitivity study to find the most sensitive geologic parameters. Based on sensitivity study, we can also evaluate the impact of different perturbation schemes on the convergence rate.

67

68

CHAPTER 3. WORKFLOW TESTING ON A SYNTHETIC DATA SET

3.1

Introduction to the synthetic dataset

Figure 3.1 is an analog of a deep-water incised canyon fill reservoir from offshore West Africa, where well log and core data exhibit significant vertical and lateral facies variation and channel-base shale drapes presence. This analog has 5 channel belts and 20 channels. The belt 1 to 5 is numbered from young (top in the model) to old (bottom in the model). Repeated episodes of erosion, bypass, and deposition result in a complex heterolithic fill and multiple significant bypass surfaces that increase potential for baffles and barriers to flow within the canyon fill complex. In this synthetic reservoir, shale drapes deposited during bypassing periods are discontinuously distributed along individual channel boundaries and channel belt boundaries. These thin (cm to m), multi-scale shale drapes are potential flow barriers that could compartmentalize the reservoirs. Therefore this analog is an excellent tool to study modeling of multi-scale shale drape distribution. The size of this analog model is 3000m × 2000m × 64m. The grid dimension is 100 × 100 × 209. As shown in Figure 3.1, this analog is created using surface-based gridding. The shale drapes are treated as thin layers and the individual channels are thicker layers. Such flexible gridding allows explicit capture of the shale drapes. However, this fine scale gridding results in a large number of cells and will make the flow simulation very time-consuming, and since both the channel location and shale drapes location are uncertain, they need to be perturbed during history matching of the reservoir model, which would mean changing the conformable grid automatically. Currently, such gridding is not yet at an adequate level of robustness to be made automatic. As a result, this surface-based fine-scale geologic model is first converted into a Cartesian grid, and then upscaled to a coarse grid with 50 × 50 × 50 cells. The coarse grid is chosen based on the rule that the geometry and continuity of individual channels are deformed as little as possible (Stright, 2005). Figure 3.2 is the upscaled geologic model with Cartesian grid. This model contains 20 channels. In order to study the multi-scale shale drape distribution, the geometry of individual channel belts is first traced explicitly from the analog model (Figure 3.3) and then their original shape is restored if they are eroded by the younger belts (Figure 3.4).

3.2. SENSITIVITY STUDY OF SHALE DRAPES PARAMETERS

69

The assumption here is that the channel belt geometry is known, for example, they are identified from seismic data. Once the channel belt model is created, we have two levels of heterogeneities caused by shale drapes: channel belts and individual channels (Figure 3.5). Then we can simulate shale drapes along belt and channel boundaries, and treat this model as a reference model. Next, we can test the proposed workflow on this synthetic case: simulate channel distribution within channel belt regions, simulate shale drapes along belt and channel boundaries, perturb channel location and drape location to match the production data generated from our reference model.

Figure 3.1: A deep-water channelized reservoir analog. The brown and yellow colors are for channel fill facies, green color is for shale drapes and blue color is for scour holes. The model is constructed using surface-based grid.

3.2

Sensitivity study of shale drapes parameters

Shape drape size, geometry, proportion and location for both channel belts and channels have the potential to affect flow responses. In reality, all of these parameters are uncertain with varying degree. Understanding their impact on fluid flow behavior will improve the reliability of geologic modeling and the efficiency of history matching. Once the sensitivity of these parameters are identified, this information can be

70


Figure 3.2: Analog model (upper) is converted into Cartesian grid (middle) and upscaled to a coarse scale model (bottom). The color reprsents different channel objects.


71

Figure 3.3: Coarse channel belt geometry identified from reference analog model.

72


Figure 3.4: The channel belt geometry is restored into original shape and their boundaries are traced for later channel simulation


73

Figure 3.5: Two level of heterogeneities: channel belts and channels

used to guide the data acquisition or collection to obtain reliable geologic modeling parameters; it also can be used to help design the perturbation scheme by focusing perturbation on the most sensitive parameters, resulting in an efficient history matching. Experimental design methodology has been used in reservoir engineering applications such as uncertainty modeling (Damsleth et al., 1992; Egeland, et al., 1992; Bu and Damsleth, 1996), sensitivity studies (Jones et al., 1995; Willis and White, 2000; White et al., 2001, 2003), performance prediction (Chu, 1990), upscaling (Narayanan, 1999), history matching (Eide et al., 1994) and development optimization (Dejean and Blanc, 1999). Experimental design is a strategy in which the factors are varied simultaneously in a series of experimental runs (3D geologic modeling) according to a predefined design matrix to obtain the experimental response. Therefore, a design is a set of factor-value combinations for which responses are measured (Myers and Montgomery, 1995; Box and Draper, 1987). The input parameters that are varied are called factors. The responses are obtained by measurement or modeling. Once

74


the designed simulation is completed, the response surface modeling can be used to test the relative importance of the input factor to the output response statistically. In this way, the experimental design methods ensure that accurate conclusions can be drawn about the entire experiment with just a few experimental runs. The first step in a designed simulation study is to identify the factors that may influence responses. Factor ranges should include all feasible factor values. Factor values are usually scaled and coded as factor levels (Montgomery, 2000) such as [-1,1] for two-level factorial design. Then these coded factors are combined with specified design scheme. 3D geologic models are generated based on all the possible factor combinations. Next, the responses are measured or modeled using these geologic models. Factor-responses often have linear relationship. Therefore, this technique works well for non-spatial factors because the response often behaves linearly. For spatial factor such as facies type, experimental design is not suitable due to the discrete nature of some of the input geostatistical parameters and the possible non-linear variation of the flow response. Because the primary goal in this work is to select or screen out the few important main effects from the many less important ones, assuming their interaction an order of magnitude less important, the screening design that is intended to find the few significant factors from a list of many potential ones (Barrentine, 1999; Montgomery, 2001) is a suited choice. The routinely used screening design is the Plackett-Burman (PB) design (Friedmann et al, 2001, 2003). The PB design can be used to study n − 1 factors in n runs in which n is dividable by four. That is, 7 factors can be studied in 8 runs, 11 factors in 12 runs, etc. Therefore, such design can practically handle a large number of factors. It is a two-level design which allows efficient estimation of main effects of all factors being studied, ignoring the factor interactions. In this two-level factorial design, each factor is assigned to its maximum or minimum value (±1) in all possible combinations with other factors. Factors considered Seven non-spatial geologic parameters were examined (Table 3.1). Channel belt scour hole size (BS), channel scour hole size (CS): the area of individual scour holes for both belts and channels are varied from small to large (Figure 3.6). The small hole size is coded as lower level -1, and the large hole size is


75

coded as higher level +1. Hole shape for belts (BG) and channels (CG): round shape and ellipsoid shape are considered for both belts and channels. The round shape is arbitrarily coded as level -1 and the ellipsoid as level +1. Hole proportion for belt (BP) and channels (CP): the areal coverage of shale drapes along bounding surfaces is varied from 0.1 to 0.5. The lower limit 0.1 is coded as level -1 and the upper limit 0.5 is coded as level +1. Channel location (CH): except the analog model, another channelized reservoir model is simulated which also has 20 channels but channels have different distribution than the analog (Figure 3.6). The reference analog model is assigned as lower level -1, and the new generated one as higher level +1. Flow responses examined Three flow responses were checked: water breakthrough time (WT) which is the days when the water cut reaches 0.1; water cut (WC) after 900 days production and field oil recovery (FOE) after 900 days production. Experiment design An eight-run (factor combinations) Plackett-Burman was selected for the experimental design study, and 6 realizations are generated for each run. Table 3.2 shows the PB design for these 7 factors. The symbol ‘+’ indicates higher level (+1), and ‘-’ refers to lower level (-1). Flow simulation model The reservoir simulation models have the same dimension as the coarse analog model described in Section 3.1. In order to perform stochastic simulation of shale drapes along bounding surfaces, shale drape training images are first generated according to the combination shown in Table 3.2 (Figure 3.6). The MPS program snesim was performed to simulated shale drape distribution along channel belt boundaries and channel boundaries. The shale drapes property is explicitly included into simulation model. If channel boundary and belt boundary overlaps (share the same grid) and both boundaries have holes, the transmissibility multiplier for this grid will be 1, otherwise 0. But if channel boundary does not tie to belt boundary, when holes are present on belt boundaries or channel boundaries, the grid multiplier will be 1. One injector injects water into one side along the channel direction (end of north side), one producer produces oil from the other side of the model. The channel reservoir has constant porosity (20%) and permeability (1000 mD). The model is a two-phase oil-water system. The viscosities of oil and water are 1.0 centipoise (cp).

76


The density of the oil is 40lb/f t3 and the water density is 62.24lb/f t3 . The oil-water relative permeability curves are shown in Figure 3.7. To observe the holes effect, the oil and water mobility is set to 1 and there is no gravity and capillary pressure are present. Analysis The first step in analyzing this experiment is to convert the data table into a spreadsheet format. Table 3.3 ~ Table 3.5 are the responses for different factor combination runs and the corresponding analysis results. The first left column number ‘1, 2...8’ is for 8 runs or factor combinations. Each factor combination is recorded in the following column as ‘+’ and ‘-’. For each run or factor combination, 6 statistically generated geological models are flow simulated, the corresponding responses are recorded afterwards as ‘Obervations 1, 2...6’. Based on these recorded responses, the sensitivity analysis are performed as follows: Step1 : Calculate the mean and variance of the 6 sample data point for each run. The means are recorded in the row of W C in Table 3.3, W T in Table 3.4 and F OE in Table 3.5. The variances are recorded in the row of S 2 in the three tables. Step2 : Sum up the mean values that correspond to higher level(’+’) for each facP P P tor, and record them in the row of W C+ , W T+ and F OE+ in Table 3.3 ~ Table 3.5 correspondingly; Sum up the mean values that correspond to lower level(’-’) P P P for each factor, and put them in the row of W C− , W T− and F OE− . Step3 : Calculate the mean of summed means for level low and high for each factor, and put them into the row of W C + and W C − in Table 3.3, W T + and W T − in Table 3.4, F OE + and F OE − in Table 3.5. Step4 : Calculate main effect from analysis tables, which is the difference between the mean of summed means for higher level (‘+’) and lower level (‘-’). For example, the main effect of belt hole size (BS) for water cut (WC) in Table 3.3 is: 0.730-0.745 =-0.015. Step5 : Create the Pareto chart of effects (Figure 3.8). Step6 : Calculte the estimated standard deviation Se :


v ! u k u X Se = t Si2 /k

77

(3.1)

i=1

where k is the number of runs, in this case k=8. Step7 : Calculate the effective standard deviation Sef f :

Sef f = Se

p 4/N

(3.2)

where N is the total number of sampled data. It is the product of the number of runs (k) and the replicates for each run. In this case N = 8 × 6 = 48. Step8 : Determine degrees(df ) of freedom and t-statistic.

df =(# of replicates per run -1)(# of runs) In this case, df =(5)(8)=40 and tdf =40;α=0.05 =2.021 Step9 : Compute decision limit based on t and Sef f . DL = tSef f

(3.3)

In this work, the decision limits are calculated at the 95% level of confidence (α=0.05). Any effects shown absolute value greater than the decision limit(DL) was considered statistically significant on the responses. As shown in Table 3.3 ~ Table 3.5, for water cut, hole shape for channel belt(BG), hole size for channels(CS), hole proportion for channels (CP) and channel locations (CL) are significant factors that affect individual well water cut profile at the 95% level of confidence. Among these four factors, the hole proportion for channels (CP) and channel locations (CH) are the most significant factors ( Fig 3.8). For the water breakthough time, except hole size for channels (CG), other six factors are significant at the 95% level of confidence, and the hole proportion for channels(CP) is the most significant factor, the channel locations (CH) the second significant one (Fig 3.8). For field oil recovery rate, all these seven

78


factors are significant at the 95% level of confidence, and the most significant factor is hole proportion for channels (CP), the scond significant one is channel location (CH) (Fig 3.8). In summary, the hole proportion for individual channels and the individual channel location are the most sensitive factors that affect reservoir fluid flow behavior in this case. This information will help the following history matching process performed in a efficient manner: only perturb the most sensitive geologic factors, that is, hole proportion for channel boundaries and channel location.

Figure 3.6: Shale drape training images (top 2 rows) and two reservoir models. In training images, the red color represents scour holes and blue background the shale drapes. Round vs. ellipsoid is for hole geometry, small vs. large is for hole size, and 0.1 vs. 0.5 is for hole proportion.


Figure 3.7: Relative permeability for flow simulation model.

Figure 3.8: Effect charts for flow responses.

79

80


Table 3.1: Seven factors and level set up Factor Hole size of belt(BS) Hole size of channel(CS) Hole shape of belt(BG) Hole shape of channel(CG) Hole proprtion of belt(BP) Hole proportion of channel(CP) Channel location(CH)

Factor level -1 1 Small Large Small Large Round Ellipsoid Round Ellipsoid 0.1 0.5 0.1 0.5 Analog model Modified analog model

Table 3.2: Plackett-Burman design with 7 factors

Run 1 2 3 4 5 6 7 8

BS + + + + -

BG + + + + -

BP CS + + + + + + + + -

CG + + + + + -

CP + + + + -

CH + + + + -

+ + + + -

2.918

W C+

W C−

1 2 3 4 5 6 7 8

P

P

0.730

0.745

-0.015 0.045

0.013

40

0.05

2.021

0.026

W C+

W C−

Effect Se

Sef f

df

α

t

DL

2.979

BS

Run

+ + + + -

BP

0.028 -0.008

0.723 0.741

0.751 0.733

2.893 2.965

3.005 2.933

+ + + + -

BG

+ + + + + -

CG

+ + + + -

CP

0.027 0.000 -0.112

0.724 0.737 0.793

0.751 0.737 0.681

2.894 2.949 3.173

3.004 2.949 2.725

+ + + + -

CS

-0.104

0.789

0.685

3.157

2.741

+ + + + -

CH

2

3

4

5

6

WC

S2

0.719 0.897 0.888 0.884 0.767 0.819 0.829 0.0054

0.598 0.604 0.606 0.630 0.609 0.584 0.605 0.0002

0.797 0.790 0.758 0.735 0.747 0.809 0.773 0.0009

0.688 0.898 0.882 0.860 0.867 0.804 0.833 0.0061

0.747 0.779 0.791 0.793 0.783 0.742 0.773 0.0005

0.724 0.745 0.698 0.695 0.718 0.752 0.722 0.0006

0.706 0.692 0.739 0.806 0.713 0.771 0.738 0.0019

0.627 0.633 0.644 0.599 0.600 0.648 0.625 0.0005

1

Observations

Table 3.3: Water cut(%) response @ 900 days for PB experiment runs and analysis

3.2. SENSITIVITY STUDY OF SHALE DRAPES PARAMETERS 81

19 -57 29.6 8.5 40 0.05 2.02 17.3

Effect Se

Sef f

df

α

t

DL

504

466

W T−

447

485

W T+ 30

461

491 -20

486

466 -17

484

467 235

358

594 183

384

567

1865 2018 1843 1944 1937 1433 1537

+ + + + -

W T−

+ + + + -

1941 1789 1964 1863 1870 2374 2270

+ + + + + -

CH

W T+

+ + + + -

CP

P

+ + + + -

CG

P

+ + + + -

CS

+ + + + -

BP

1 2 3 4 5 6 7 8

BG

BS

Run 708 418 516 410 335 464 740 276

430 526 475 300 455 723 353

2

712

1

265

724

417

316

458

528

441

688

3

307

697

417

338

404

516

409

706

4

Observations

272

714

448

208

377

498

437

704

5

261

728

380

311

427

516

368

719

6

289

721

430

301

425

517

417

706

1247.6

209.6

988.6

2299.9

1312.6

113.1

722.2

107.4

W T S2

Table 3.4: Water breakthrough time (days) for PB experiment runs and analysis

82 CHAPTER 3. WORKFLOW TESTING ON A SYNTHETIC DATA SET

+ + + + 1.997

F OE+

F OE−

1 2 3 4 5 6 7 8

P

P

+ + + + -

BP

+ + + + -

CS

0.021 0.005 0.023 -0.014 0.0009 0.0003 40 0.05 2.02 0.0005

Effect Se

Sef f

df

α

t

DL

0.486 0.477 0.496

0.478

F OE−

0.737 0.500 0.482

0.499

1.944 1.909 1.983

1.966 2.001 1.927

+ + + + -

BG

F OE+

1.913

BS

Run

+ + + + -

CP

+ + + + -

CH

0.001 0.139 0.086

0.488 0.419 0.458

0.489 0.558 0.544

1.953 1.677 1.834

1.957 2.233 2.177

+ + + + + -

CG

2

3

4

5

6

F OE S 2

0.431 0.473 0.421 0.459 0.0007

0.595 0.0003

0.453 0.357 0.337 0.339 0.381 0.358 0.371 0.0019

0.591 0.589 0.587 0.581 0.589 0.63

0.510 0.510 0.514 0.500 0.515 0.463 0.502 0.0004

0.455 0.409 0.397 0.433 0.289 0.426 0.402 0.0034

0.479 0.436 0.454 0.417 0.427 0.462 0.446 0.0005

0.558 0.556 0.575 0.566 0.563 0.538 0.559 0.0002

0.472 0.485 0.47

0.576 0.573 0.577 0.578 0.578 0.581 0.577 0.000

1

Observations

Table 3.5: Field oil recovery efficiency(%) @900 days for PB experiment runs and analysis

3.2. SENSITIVITY STUDY OF SHALE DRAPES PARAMETERS 83

84


3.3

Modeling parameters

As described in Chapter 2, in order to perform the proposed geologic modeling method, the input parameters such as channel geometry parameters (wavelength, amplitude etc.) and channel stacking pattern parameters (migration ratio, overlap ratio) need to be specified first. Generally, it is not easy to obtain information on these parameters from seismic data because the objects are often below seismic resolution. Outcrop study can provide some data but outcrop data is often limited to 2D. For this synthetic study, we are in the ideal situation that parameters required for geologic modeling are available. In order to obtain the information about the channel geometry, the individual channels are extracted from the 3D synthetic model. For each channel, five geometry parameters including amplitude (A), wavelength (λ), orientation (θ), maximum width (W ) and thickness (H) are measured. In this synthetic model, there are 20 channels. Hence, there are 5 sets of 20 sample measurements for these five geometry parameter. The averaged values of each set samples for these geometry parameters are shown in Figure 3.9. For stacking pattern parameters, each individual channel centerline is identified, their locations in x - direction (channel orientation is in y-direction) are recorded. Then, the horizontal distance (x) between two adjacent channels is calculated which is the difference between their horizontal location values. And the migration ratios are derived by dividing the channel amplitude (A) to horizontal distance data (x). To obtain the overlap ratio data, every channel top surface z is recorded. Then, the vertical distance h0 between two adjacent channels is calculated which is the difference between their surface z values. The vertical overlap ratio is calculated by dividing maximum thickness (H) to (H-h0 ). For 20 channels within 5 belts, there are 6 sample values for both overlap ratio and migration ratio. The cumulative distribution functions of these two parameters are calculated using each 6 data points (Figure 3.10 black dots). A regression is applied to obtain analytic cdf functions for these two parameters (Figure 3.10 solid line). Therefore, the cdf

3.4. HISTORY MATCHING RESULTS

85

function for migration ratio (M R) used in this example is as follows: p = 0.7113ln(M R) + 1.9486, M R ∈ [0.06, 0.27] The cdf functions for overlap ratio (OR) is: p = 4.9693(OR) − 3.9576, OR ∈ [0.79, 1] In addition of channel geometry and pattern parameters, the net-to-gross (NTG) ratio is required for each channel belt. For simplicity, channel is treated as “sand” facies (NTG=1) and background shale is “non-sand” (NTG=0). After restoring the eroded parts of the channels, the corresponding net-to-gross is calculated for channel belt 1 ~ 5 respectively (Table 3.6). Table 3.6: Global NTG for different channel belts of reservoir analog Channel belt Global NTG Belt 1 0.68 Belt 2 0.65 Belt 3 0.65 Belt 4 0.68 Belt 5 0.60

3.4

History matching results

The proposed modeling and history matching procedure is applied to perturb multiscale shale drape/hole locations to match the historical production data for this synthetic reservoir. The reference channel distribution model is shown at the bottom in Figure 3.2. The shale drape models for channel belts and channels are generated using the shale drape modeling procedure presented in Chapter 2 (Figure 3.11). With the same flow simulation parameters as sensitivity study, the reference flow model is set up with two injectors and three producers. The injectors were controlled by

86


Figure 3.9: Channel geometry parameter derived from analog

Figure 3.10: Channel pattern parameter distribution: the experimental CDF (blue dots) and the analytic CDF function obtained by regression (solid lines).


87

surface injection rate, and the producers were controlled by total liquid rate. Figure 3.12 shows the field oil saturation model (top) and three producers flow responses (bottom) for 1000 days production. These production profiles are treated as the observed data. The first 600 days production profiles will be history matched, and the last 400 days production profile will be used to test the prediction reliability of the history matched geologic models. The objective function for the history match was defined using the sum of difference in observed and simulated water cut and bottom hole pressure at the three producers. During the perturbation process, the geologic model is generated using the modeling parameters derived in previous section. The geologic modeling will be constrained to hard data from 5 wells (Figure 3.16). The perturbation is performed in two steps: first the channel location and hole location along belt boundaries are modified simultaneously (using same rD ). Note that the hole location along individual channels is not perturbed, but since each outer loop changes the random seed for hole simulation, their location changes randomly due to a change of random seed input into the MPS algorithms that generate the hole; Secondly, the hole location along channel boundaries is perturbed while channel location and hole distribution along belts remain fixed. During the perturbation of hole location along channel boundaries, the hole models are simulated using the hole training images shown in the top row of Figure 3.11. This means the perturbation only modifies hole location, the hole proportions is taken as the same one of the reference. Both steps have 10 outer iterations. During the whole perturbation process, the hole proportion for channels and belts are the same as the reference. In other words, the hole proportion is not modified during history matching. Figure 3.13 shows the optimization performance for these two steps. In the first step (purple dots), the mismatch drops down from 4.2 to 0.87, while in the second step (green dots), the mismatch slightly decreases from 0.87 to 0.85. The reason why the first step objective function drops considerably more than the second one is because the channel location mainly controls the hole location along the channel boundaries. In the first step perturbation, when the channel location is modified to optimally match production data, as a consequence, the hole location is optimized with the channel location. As a result, the mismatch decreases rapidly. The perturbation in the second

88


Figure 3.11: The 3D hole distribution models (middle row) and the cross sections (bottom row) for belts and channels (middle row) using the corresponding hole training images (top row). The dark red indicates holes. The training image grid dimension is 100 by 100 and the hole models are 50 × 50 × 50.


Figure 3.12: Observed production responses from the synthetic reservoir.

89

90


step is to fine tune the hole location based on the first perturbation result to further match the production data. This fine tuning process often requires a large portion of the CPU time to reduce a small amount of mismatch. Therefore, it is suggested that in most case the first step perturbation may be enough to get a satisfactory history match. Figure 3.14 shows the history matched results corresponding to the optimization performance shown in Figure 3.13. Figure 3.15 compares the history matched channel distribution model with the reference. These two models have very similar stacking patterns. The reference has 20 channels while the history matched one has 21 channels. Figure 3.16 checks the well data conditioning in the history matched model. It shows the simulated facies at five well locations are consistent with the well data. Note that this is true because the well hard data interpretation is exact (not true in reality). Once the history matched geologic models are obtained, prediction is assessed for the next 400 days. Figure 3.17 and Figure 3.18 are prediction plots of well water cuts, bottom hole pressure and field oil recovery for multiple history matched cases (left column) and non-history matched cases (right column). Comparing with the non-history matched cases, the history matched models provide narrow ranges of the future water cuts and field oil recovery around the reference curves. This clearly demonstrates that the history matched models have the capability to provide a more reliable prediction. In the above example, we assume the hole proportions are known and only the hole location is uncertain. In practice, it would be difficult to obtain an accurate target hole proportion from data. Moreover, as shown in previous sensitivity study, the proportion of holes along individual channels is one of the most sensitive factors for flow responses. Therefore, one will often need to perturb both hole location and proportion to match history. In the next history matching example, with the same production data and well configurations as the example above, a hole training image (Figure 3.19) with target proportion different from the reference is used. Hence, for channels both hole location and proportion need to be perturbed, while for belts only the hole location is perturbed since it is less sensitive to the flow response. The


91

history matching procedure now consists only of the first perturbation step (with 10 outer iterations), that is, perturbs the hole location for belts, the channel location and the hole proportion for channel boundaries. Figure 3.20 shows the optimization performance. The blue curve reflects the inner loop mismatch, the purple curve the outer loop mismatch and the green curve the channel hole proportion evolution during perturbation. The mismatch between the simulated flow responses and the historical data decreases considerably. Meanwhile, the channel hole proportion converges from its original 20% to the reference 10%. The initial guess and final history matched results, as well as the corresponding predictions are shown in Figure 3.21. Again, the history matched model has prediction results closer to the reference compared with the randomly picked models.

Figure 3.13: Optimization performance for two step perturbations.

92


Figure 3.14: History match results for two step perturbation.

Figure 3.15: Reference model vs. history match geologic model.

3.5. CHAPTER SUMMARY

93

Figure 3.16: Well data vs. simulated facies at well locations. Different color represents different channel section.

3.5

Chapter summary

The synthetic example studied in this chapter is a very realistic representation of reservoir architecture for deep-water confined channel system from offshore West Africa. Hence, it can be used to test the feasibility of proposed modeling and history matching workflow. We can also derive the channel stacking pattern information from this analog for future real case. The sensitivity study will help set up efficient perturbation scheme in the real history matching process. The sensitivity study shows that the hole proportion along individual channels and channel location are the most sensitive parameters for flow responses, though other parameters are also significant with 95% level of confidence. This means when we perturb the geologic model for history matching, the efficient perturbation parameters are channel location and channel hole proportion. The history matching results also show that performing one step perturbation of channel location and hole proportion is enough to obtain desired history-matched geologic models. The second step perturbation of channel hole location has very slow convergence rate because the channel location which is associated with hole location is already optimized in the first step perturbation. The history matching and prediction results have demonstrated that the proposed workflow can be used for modeling and history matching of multi-scale shale drapes. The final history matched geologic models have better prediction power than randomly selected geologic models.

94


Figure 3.17: Well water cut and field oil recovery predictions using the history matched models and randomly picked models.


95

Figure 3.18: Well bottom hole pressure predictions using the history matched models and randomly picked models.

Figure 3.19: A channel hole training image (right) with proportion different with the reference (left).

96


Figure 3.20: The optimization performance for the case that both hole location and proportion are perturbed. The hole proportion converges to the reference with the mismatch decreases.

Figure 3.21: Flow response Predictions using the history matched models and randomly picked models.

Chapter 4 Applications to a realistic turbidite reservoir The dominant reservoir types in offshore West Africa (WA) are large, erosionally confined deepwater channel complexes developed within slope valley (Abreu, et al., 2003). High resolution seismic data and extensive outcrop studies have improved our understanding of these types of deepwater channels. Numerous and comprehensive studies have focused on the specific aspects of channel morphology, deposition process and stacking patterns. These large erosinally confined channels are typically 50-100m thick, several kilometers wide and long. They are characterized by a repeated cutting and filling process. This process has very important implication for reservoir connectivity (Mayall and O’Byrne, 2002). The facies at the base of each channel is critical to the connectivity across larger erosional belts. If these facies are composed of shale drapes they may result in barriers within reservoir. Another feature for deep-water channelized reservoir is that they tend to be sand-rich. They often have very high (0.6-0.8) net-to-gross ratio (NTG). This chapter will study a reservoir model based on a real turbidite reservoir in offshore West Africa. The valley and channel belt geometries are interpreted from 3D seismic data. However, the depositional facies filling the belts cannot be easily inferred from the seismic data due to the limited resolution. The net-to-gross ratios for channel belts are obtained from the company database of analog reservoirs. The 97

98 CHAPTER 4. APPLICATIONS TO A REALISTIC TURBIDITE RESERVOIR

production data and core data from the real reservoir provide ample evidence of shale drape presence. These drapes act as barriers to compartmentalize the reservoir. But the proportion and spatial distribution of shale drapes in the reservoir are uncertain. The objective of this chapter is to apply the proposed geologic modeling and history matching workflow to perform the history matching study. First, the information available for this study is described. Next, we demonstrate that, in case of high NTG reservoir, perturbing shale drapes in the true 3D domain while remaining consistent with its geologic conceptual description leads to more efficient and favorable history matching results as compared to perturbing in pseudo-3D domain. Finally, a process of perturbing the geologic model by regions with unknown shale drape proportion in each region is conducted and the results are presented. To assign different perturbation parameters rD to different regions, a sensitivity study is performed to relate the producers to different geologic regions.

4.1

Information available for history matching

In this WA reservoir, the seismic data quality is of sufficient quality to identify stratigraphic architecture including valley and belts framework (Figure 4.1). Figure 4.1 (a) shows the surface-based model of valley and belt geometry interpreted from seismic data. The grid dimension is 100 × 200 × 34. The reservoir dimension is 4200 × 7200 × 100m. Figure 4.1 (b) shows the upscaled Cartesian grid model with grid dimension as 50 × 50 × 50. This Cartesian grid will be used for geologic modeling and history matching. The reservoir volume and individual channel belt volumes are deemed known and retain for this study. In other words, these channel belts are taken as containers for the following individual channel filling with pre-defined conceptual staking patterns. Based on geologic studies of well logs, this reservoir is classified into two regions. Region 1 is the within valley region. This region has higher NTG (0.8) and less shale drapes along belt and channel edges. Region 2 is formed by the parts outside of the valley. This region has slightly less NTG (0.7) and more shale drapes. In order to fill the individual channels into channel belt containers, the channel

4.1. INFORMATION AVAILABLE FOR HISTORY MATCHING

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Figure 4.1: Stratigraphic interpretation of WC reservoir. Gray color is for valley region. Other colors are for different channel belts.

100CHAPTER 4. APPLICATIONS TO A REALISTIC TURBIDITE RESERVOIR

belt’s NTG need to be known. Based on the company database of analog reservoirs as well as well logs interpretation, the NTG for each belt are obtained and listed in Table 4.1. These NTG values are assumed certain. This study will make use of the channel geometry information (Figure 3.9) and channel stacking pattern information (Figure 3.10) derived from reservoir analog in Chapter 3. The only changes are channel dimensions. To adapt the channel dimension to this realistic reservoir dimension, the channel maximum thickness is increased from 15m to 18m, the maximum width is increased from 300m to 500m. Well-log interpretation from the real reservoir provides the range of porosity Table 4.1: NTG for different channel belts Channel belt NTG Belt 1 0.8 Belt 2 0.8 Belt 3 0.8 0.75 Belt 4 Belt 5 0.75 Belt 6 0.75 Belt 7 0.75 0.7 Belt 8

and permeability for channel facies. Based on these information, the histograms of porosity and permeability are generated. The porosity for channels follows normal distribution with mean 0.28 and standard deviation 0.01. The permeabilities are also normally distributed with mean 2200md and standard deviation 200md. The background shales have very low permeability (mean 25md) and porosity (mean 0.03). In this study porosity and permeability are simulated using “Sequential Gaussian Simulation” (sgsim) by facies (channel/non-channel). The variograms listed in Table 4.2 are used to simulate both porosity and permeability distribution, for both channel and non-channel facies. The final porosity and permeability models are obtained after “cookie-cutting” the porosity and permeability realizations using channel distribution realization.

4.2. SETTING UP “TRUE” CASES

101

Table 4.2: Variograms used for channel porosity and permeability simulation

Type Nugget Ranges Angle

Channel

Background shale

Exponential 0 2000m/800m/20m 0/0/0

Exponential 0 1000m/1000m/10m 0/0/0

Shale drapes distribution along valley, belt and channel boundaries have large uncertainty since there is few quantitative information available from the real field. Based on well-log data, it was interpreted that the average areal proportions of shale drapes along valley/belt/channel are 0.55/0.65/0.65. A shale drape parameter sensitivity study (Chapter 3) has shown that the most important parameter is the shale drape (or scour hole) proportion. The scour hole geometry and shape play a lesser role for reservoir connectivity. Therefore, this study will use three different scour hole training images for valley, belt and channel (Figure 4.2). Figure 4.3 shows the scour hole realizations for valley, belt and channel using these training images. The production data from real reservoir is not available. Instead, the PVT table, the oil-water relative permeability curves (Figure 4.3) and fluid and rock properties were provided. These information will be used to build “true” production profiles for our study. The next section explains the process of obtaining “true” production data.

4.2

Setting up “true” cases

The production data from the actual reservoir were not made available by the operating companies. In order to perform history matching study, synthetic production data are generated using the PVT information from the real reservoir. The geologic model characterizing channel distribution is first simulated using the proposed geologic modeling approach (Figure 4.5a). Next, the shale drapes are generated along


Figure 4.2: Scour hole training images for valley, belt and channel hole simulation. The red color objects are scour holes, and blue background is shale drapes.

Figure 4.3: Scour hole realizations for valley, belt and channel using the training images shown in Figure 4.3.


103

Figure 4.4: Oil and water relative permeability curves.

valley, belt and channel edges. Then the porosities and permeabilities are simulated based on facies model (Figure 4.5 b and c). The shale drapes properties are included into the simulation model as multipliers. If channel boundary, belt boundary and valley boundary overlap (share the same grid) and three overlapped boundaries have holes, the transmissibility multiplier for this grid will be 1, otherwise 0. Same for the situation that two of these three boundaries overlap. But if these three boundaries do not overlap, then when holes are present on channel boundaries, belt boundaries or valley boundaries, the multiplier will be 1. Figure 4.5d is an example of the transmissibility distribution in x− direction accounting for the shale drapes presence. Figure 4.5d shows that even in a reservoir with high poro/perm properties, reservoir connectivity may be reduced due to the presence of drapes. The simulation grid is dipping 4 degree to the west (along x− direction) and 6 degree to the south (along y− direction) (Figure 4.6). There is one producer and one injector in Region 1 (purple colored region in Figure 4.6), one producer and one injector in Region 2 (gray-colored region in Figure 4.6). The model is a two-phase oilwater system. Table 4.3 summarizes the simulation model description for generating production data. The same water saturation, well configuration, fluid properties and relative permeability curves will be used for each waterflooding simulation. Therefore the significant variables are reservoir architecture and shale drape distribution. In this work, there are two “true” cases created, i.e. One-region case and Two-


Figure 4.5: Geologic properties used for flow simulation.

Figure 4.6: Reservoir model showing reservoir structure, the geologic regions, and well configuration.


105

Table 4.3: Description of simulation model for “true” production simulation

Simulation property and description Simulation model Simulation period, years Grid (Cartesian) Active grid block Grid block dimensions, m3 Permeability Reference depth, m Initial pressure at 3000m, psi Water-Oil contact, m Oil gravity (AP I) Water injectors Injection-control rate, Stb/day Injection-BHP upper limit, psi Oil producers Production- Liquid rate control, Stb/day Production- BHP lower limit, psi

Value Black oil 5 50 × 50 × 50 28364 84 × 144 × 2 Kx = Ky , Kz /Kx = 0.1 3000 4500 4000 30 2 200000 6000 2 175000 1800


regions case. These two cases share the same channel facies model, porosity and permeability models shown in Figure 4.5 a, b and c, but have different shale drape distribution along belt and channel boundaries. Since there are 4 wells drilled (2 producers and 2 injectors), the geologic models are constrained to channel facies data from these 4 wells. One-region true case: The two geologic regions contain similar hole proportions for both channels and belts. More specifically, scour hole proportions are 0.45/0.4/0.3 for valley/belt/channel. In this case we will compare two perturbation schemes: (1) full 3D perturbation: both channel location and hole location are perturbed with known target hole proportions; (2) psuedo-3D perturbation: channel locations are fixed, only scour hole location is perturbed with known target hole proportions. The question we will address concern is: In high NTG reservoir, is perturbing shale drapes enough for history matching? Two-region true case: Two geologic regions have different scour hole proportions for both belts and channels. Region 1 has hole proportions as 0.4 /0.35 for belts and channels respectively; while Region 2 has a lesser proportion, 0.2/0.15 for belts and channels respectively. This case is used to test the importance of including prior (region) information for history matching. Two perturbation schemes are performed: (1) perturbation by region: channel location and hole location as well as proportion are perturbed assuming two regions; (2) perturbation without region: channel location and hole location as well as proportion are perturbed assuming one region. Both perturbation schemes assuming channel hole proportion is uncertain or unknown. This means the target hole proportion along channels will also be perturbed. The flow simulations are run for 5 years. Figure 4.7 shows “true” production data- water cut and bottom hole pressure profiles for both cases. The oil saturation distributions after 5 years of production and water injection are shown in Figure 4.8. Compared with the no-shale-drape case, the presence of shale drapes renders some oil in the reservoir unswept. Since the two-region case is more heterogeneous than the one-region case, there is more oil left in the reservoir in former case. Next sections will perform the proposed geologic modeling and history matching workflow to match these synthetic production data using various perturbation schemes.

4.3. PERTURBING SHALE DRAPES IN HIGH NTG RESERVOIR

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Figure 4.7: Water cut and bottom hole pressure profiles for “true” case 1 and 2.

4.3

Perturbing shale drapes in high NTG reservoir

If one believes that, in high NTG reservoirs, almost every channel connects with one another, then only shale drape location matters to fluid flow, hence, perturbing shale drape distribution is sufficient for history matching. However, the stratigraphic architecture could play an important role since channel location controls the location of shale drapes. In order to check the impact of facies architecture (with which shale drape are associated) on flow responses, this section conducts two perturbation schemes to match the “true” production data for One-region case: pseudo-3D perturbation vs. full 3D perturbation. Shale drape target proportions are assumed known. As mentioned in the previous section, the same well configuration, fluid properties and relative permeability curves are used for each waterflooding simulation. This


Figure 4.8: Oil saturation distribution after 5 year production.

ensures that the significant variables are reservoir architecture and shale drape distribution. Two perturbation schemes start with the same initial geologic model including channel facies distribution, porosity/permeability distribution and shale drape distribution. To make these two perturbation schemes comparable, each follows the same random path when performing stochastic optimization (PPM). Ten initial geologic models are provided, and 10 history matching runs are performed for both perturbation schemes, each has 15 outer iterations. Figure 4.9 shows the production profiles simulated from 10 initial geologic models (blue-colored lines) and the history matched models (purple-colored lines) for pseudo3D perturbation scheme. Figure 4.10 is for the full 3D perturbation scheme. These two figures show the same production profiles for the initial geologic models (blue lines) since they start from the exact same initial guess. We can observe that some initial models match the production data pretty well without any history matching process. This is because in this case we assume the shale drape proportion for

4.4. PERTURBING SHALE DRAPES BY REGIONS

109

both belts and channels are certain. In fact, from the sensitivity study performed in Chapter 3, we retained that the shale drape proportion is the most important factor for flow. By assuming the shale drape proportions known, the uncertainty of geologic models is reduced in a great degree. Although the uncertainty is greatly reduced, comparing the history matched results (purple lines) with non-history matched ones (blue lines), the former still provides smaller uncertainty for both perturbation schemes. Comparing the history matching results of these two perturbation schemes, full 3D perturbation provides more effective results given the same number of outer iterations. Figure 4.11 displays the optimization performances for these two perturbation schemes. Even though the mismatch of the results obtained by the pseudo-3D perturbation scheme drops faster in the first couple iterations, the final mismatches are slightly less optimal than the ones obtained from full 3D perturbation scheme. This demonstrates full 3D perturbation is more efficient. In the following section, all the history matching processes are performed using a full 3D perturbation scheme.

4.4

Perturbing shale drapes by regions

As introduced in Section 4.1, this realistic reservoir has two geologic regions (Figure 4.6). These two regions have different amount of scour holes (or shale drape) for both belts and channels. Each region has one producer and one injector. With few wells available, there is a high degree of uncertainty on shale drape proportions for valley, belts and channels. This is often the case in actual reservoir applications. To improve prediction power of the history matched models, the perturbation should consider uncertainty in hole proportions. In other words, the perturbation should not only modify the hole location, but also the hole proportions. Furthermore, different regions may require different amount of perturbation to efficiently converge to different hole proportions present in each region. This section is to conduct the history matching process to match the production data of the two-regions true case generated in Section 4.2. Two perturbation schemes are performed: multi-region perturbation vs. single-region perturbation. Both schemes automatically modify the hole proportion during history matching


Figure 4.9: Ten Water cut and bottom hole pressure profiles for both non-history matched models and history matched models performed using pseudo 3D perturbation scheme. Blue line is for non-history matched model, red line is for history matched model, and black line is for “true” data.


111

Figure 4.10: Ten Water cut and bottom hole pressure profiles for both non-history matched models and history matched models performed using true 3D perturbation scheme. Blue line is for non-history matched model, red line is for history matched model, and black line is for “true” production data.


Figure 4.11: Optimization performances corresponding to 10 history matching results shown in Figure 4.9 and Figure 4.10. Blue color is for pseudo-3D perturbation scheme, purple color is for full 3D scheme.


113

process. The purpose of this study is to check the importance of including prior information on the presence of region for history matching. In order to perform multi-region perturbation, producers need to be assigned to different regions for the purpose of objective function calculation (Hoffman, 2005). In the multiple region history matching approach, the number of objective functions should be consistent with the number of regions such that the optimization process perturbs optimally for each regions. Meanwhile, we need to know which factor(s) is most sensitive to which flow response(s), and which flow response can be used as an indicator for hole proportion adjustment (increase or decrease). Therefore, an experimental design is performed.

4.4.1

Region sensitivity study

Table 4.4 lists four factors examined. A one-half fraction design is performed and Table 4.5 shows the factor combinations for 8 runs. Each run has 5 realizations. Hence a total of 40 flow simulations are performed. Flow responses in both producer 1 (in Region 1) and producer 2 (in Region 2) are checked, including water breakthrough time (WT), water cut (WC) at the end of 5 years production, well bottom hole pressure (BHP) at the end of 5 year production. The sensitivity analysis is conducted in the same manner as the one described in Section 3.2. Figure 4.12 displays the effects of four factors on these three flow responses. From this sensitivity analysis result, we can obtain the following information:

• Hole proportion along channels in Region 2 is the most sensitive factor. This is because the number of channels in Region 1 is less than the one in Region 2 (12 vs.20), same as the number of channel belts. This means Region 2 is more heterogeneous than Region 1. Therefore, hole proportions in Region 1 have less impacts on water movement as compared to Region 2. As shown in Figure 4.13, water injected by Injector 2 flows towards Producer 1 (Region 1) or Producer 2 (Region 2), while almost all the amount of water injected by Injector 1 flows


towards Producer 1 (Region 1). • BHP can be used as evidence to assign the producers to different regions. As shown in Figure 4.12, the channel hole proportion in Region 1 has the strongest impact on BHP for Producer 1 in Region 1, and channel hole proportion in Region 2 for Producer 2 in Region 2. • BHP is used as an indicator for channel hole proportion perturbation. This is because BHP is most sensitive to channel hole proportion. In this study, the channel hole proportion is modified by the following equation: P ropHoleU pdate = P ropHoleCurrent + ll × rD /10 Where rD is the PPM perturbation parameter, ll is an indicator (1 or -1). The sign of ll is determined using simulated BHP information. If the simulated BHP is higher than reference BHP data, this means the simulated reservoir model has more energy than reference reservoir. The reason of having higher energy is because the simulated reservoir model has less shale drapes (or higher hole proportion) such that the injected water front moves more homogeneous (hence slower) than the one in reference reservoir, and pushes more oil towards the producer, resulting in higher bottom hole pressure for that producer. Hence the channel hole proportion for the region that this producer located is decreased. If the simulated BHP is lower than reference BHP data, this means the injected water front moves faster towards the producer due to higher shale drape proportion (lower hole proportion), hence we need to increase hole proportion in the region that this producer located. In this study we choose threshold -10 psi and 10 psi as match criteria. If the difference between reference BHP and simulated BHP is smaller than -10 psi, ll =-1; if the difference is larger than 10 psi, ll=1; otherwise ll=0.

• BHP and WC are used to calculate objective function since both are sensitive flow responses. The objective function is calculated as:


115

Table 4.4: Four factors and level set up Factor level -1 1 proportion of belt in Region1 (HPBR1) 0.2 0.4 proportion of channel in Region1 (HPCR1) 0.15 0.35 proprtion of belt in Region2 (HPBR2) 0.2 0.4 proportion of channel in Region2 (HPCR2) 0.15 0.35

Factor Hole Hole Hole Hole

Table 4.5: The two level 24−1 design

Run 1 2 3 4 5 6 7 8

HPBR1 HPCR1 + + + + + + + +

HPBR2 + + + +

HPCR2 + + + +

v v u T u T 2 uX (W C S − W C H ) 2 uX (BHPtS − BHPtH ) t t t t O(rD ) = + H H W Cend BHPend 1 1

4.4.2

History matching results

After the sensitivity study, history matching processes are performed. Figure 4.14 shows the history matched production profiles using multi-region perturbation. Figure 4.15 is the optimization performance for this perturbation scheme. With the


Figure 4.12: The effect chart of examined factors listed in Table 4.4 on flow responses. Red lines are decision limits. If the effect bar exceed red, this means that factor is significant to the corresponding flow response with 95% confidence level.


117

Figure 4.13: Streamlines showing the water flow path.

optimization process proceed, the objective function converges to the tolerance (1.0) (Figure 4.15a) and the hole proportions converge from the initial guesses (0.5 for Region 1 and 0.3 for Region 2) to the references (0.35 for Region 1 and 0.15 for Region 2) respectively (Figure 4.15b). If we assume the whole reservoir as one region, and apply single-region perturbation, the historical data can also be matched. Figure 4.16 shows the history matched production profile using one region perturbation scheme. Figure 4.17 shows the corresponding optimization performance for this perturbation. The channel hole proportion in the reservoir converges to an average value (0.25) of the channel proportion in “true” reservoir, while the objective function decreases towards the tolerance. Hence both perturbation schemes can provide us satisfactory history matching. Figure 4.18 are reference and history-matched channel distribution models. Both perturbation schemes provide very similar history-matched geologic models in terms of the number of channels and the channel stacking patterns. If we compare the water and oil saturation at the end of 5 years production from both history matched models with the one from “true” model (Figure 4.19), the history matched model obtained using multi-region perturbation scheme (Figure 4.19 b) has similar water and oil distribution as the “true” model (Figure 4.19a), while the fluid distribution within history matched model using single-region perturbation (Figure


4.19c) is quite different than the “true” model, though there are many oil left in the history matched reservoir model. The results demonstrate that including the prior information of variability of holes by regions improves the prediction power of the history matched models.

Figure 4.14: History matching results assuming two regions in the reservoir.

4.5

Chapter Summary

In this chapter, we applied proposed geologic modeling method to built channel facies models based on the available static information of a real offshore West Africa reservoir. In order to obtain “true” production data, two “true” shale distribution cases, including One-region case and Two-region case, are generated and the corresponding production data are simulated. Then the proposed modeling and history matching method was applied to simulate and perturb the geologic models until their flow responses match “true” production data. Different perturbation schemes have been conducted to address various issues. The learning from this case study is summarized as follows:


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Figure 4.15: Optimization performance corresponding to the history matched results shown in Figure 4.14.

Figure 4.16: History matching results assuming the reservoir as one region.


Figure 4.17: Optimization performance corresponding to the history matched results shown in Figure 4.16.

Figure 4.18: Reference and history-matched geologic models.


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Figure 4.19: Oil and water saturation distribution after 5 years production. • In very high NTG (07-08) reservoirs, the presence of shale drapes has significant impact on reservoir connectivity; • For very high NTG (07-08) reservoirs, both the scheme of perturbing channel and hole location and the scheme of perturbing hole location along fixed channels can achieve satisfactory history matching. However, the former perturbation scheme is slightly more efficient since the perturbation is in true 3D and consistent to geological conceptual model; • In the case that the shale drape proportion is different in different region of reservoir, it is suggested to apply multi-region perturbation to obtain more realistic geologic models. Even single-region perturbation can obtain satisfactory history matching, the history matched models may not have reliable prediction power due to ignorance of the important prior information.

Chapter 5 Conclusions and future work 5.1

Conclusions

A methodology for modeling and history matching of multi-scale flow barriers in channelized reservoirs has been presented. With this methodology, reservoir models containing multi-scale facies architecture and associated flow barriers are constructed that match production data and consistent to geologic data, such as well logs and conceptual channel stacking patterns. Within the geologic modeling process, individual channels are simulated using an object-based approach. Unlike traditional object-based method in which a 3D object geometry is directly created based on given geometry parameters, the object-based modeling method adopted in this work first generates a 2D pixel-based thickness map that reflects the channel geometry parameters, then paints the thickness in 3D forming an individual channel. Compared to traditional pixel-based approach, the adopted modeling method can provide explicitly the channel geometry needed for attaching shale drapes. Compared to traditional object-based approach, the introduction of a pixel-based property into object simulation makes the hard data conditioning more flexible. Furthermore, this idea makes the perturbation of individual channel geometry possible. By perturbing the probability model used to generate a channel thickness map, the channel geometry in 3D is modified. However, the underlying Boolean stochastic model is preserved. 122

5.1. CONCLUSIONS

123

One of the most novel aspects of this methodology is the channel stacking pattern modeling technique, as well as perturbation method that modifies channel location consistently with the conceptual stacking pattern model. A channel stacking pattern in this work is generated using two pattern parameters: migration ratio (MR) and overlap ratio (OR). By simulating a set of pairs of realizations of MR and OR from their corresponding probability distribution functions, one channel stacking pattern is constructed, and the number of channels is the same as the number of pair of MR and OR. With this modeling approach, the channel location can be modified by changing the pattern parameter realizations using the gradual deformation method. The perturbed channels still follow the same conceptual stacking pattern model. Using the presented channel stacking pattern modeling and perturbation technique, discontinuous shale drapes associated with curvilinear channels can be perturbed in true 3D space. As shown in Chapter 2, shale drapes are simulated on multiple 2D surfaces, each surface corresponds to one channel erosional boundary. Because shale drapes are associated with channel boundaries, perturbing shale drape distribution by moving channel location provides more freedom to the perturbation, at the same time the perturbation remains consistent with the geologic concept. The realistic case study in Chapter 4 has demonstrated that even in high NTG reservoirs where almost all the channels touch with each other, true 3D perturbation scheme is still more efficient and robust as compared to a pseudo-3D perturbation (perturbing shale drape distribution with fixed channel location). For medium-low NTG reservoirs where channel location is the first-order controlling factor for reservoir connectivity, perturbing shale drape distribution with movable channel location is crucial for efficient history matching. The success of the presented modeling and history matching approach largely relies on the availability and quality of input information, that is, conceptual channel stacking pattern model, channel geometry parameters and net-to-gross ratio (NTG) for individual channel belt. In this dissertation, we assume NTG for each belt is available from geologic study. Tests on a synthetic case in Chapter 3 has demonstrated how channel geometry parameters and probability distribution functions of channel stacking pattern parameters can be obtained from reservoir analog. This information

124

CHAPTER 5. CONCLUSIONS AND FUTURE WORK

is used as input for a realistic reservoir study in Chapter 4. To match production data for reservoirs with multi-scale heterogeneity, a sensitivity study is often required to determine which scales most significantly impact flow, and what combination of flow responses should be used for the objective function. In the case that the reservoir has different geologic regions, a sensitivity study is a useful tool to assign producers to different regions for objective function calculation purposes. The synthetic belt-channel system reservoir study in Chapter 3 has showed the hole proportion along individual channels and channel location are the most sensitive parameters for flow responses. The realistic valley-belt-channel reservoir study in Chapter 4 used a similar sensitivity study to assign producers to different geologic regions. The analysis revealed that the hole proportion along channels is the most significant factor that impact production data, hence reservoir connectivity. The proposed modeling and history matching workflow in this dissertation is initially developed for simulating shale drapes in deep-water confined channel reservoirs. These thin shale drapes are discontinously distributed along hierarchically erosional surfaces. However, not all turbidite channel reservoirs have shale draped along the channel boundaries. Only if the available data from field, such as cores, well-logs, production logging test (PLT), pressure information and 4D seismic data infer the presence of shale drapes or other type of barriers that are associated with reservoir architecture, the presented workflow can be applied to generate flow barrier distribution models with reduced uncertainty. From this point of view, the proposed modeling and history matching workflow is general and can be applied to many other depositional settings such as fluvial channel system and deltaic distributary channel system, or different type of barriers such as fracture systems and various cements, that can be associated stratigraphically with reservoir architectures. In the case that we are uncertain about the shale draped channel models, scenario-based geological modleing should be performed, that is, shale draped channel scenario and non-shale draped channel scenario. For the shale draped channel scenario, the proposed workflow can be appropriately applied to simulate shale distribution models.

5.2. RECOMMENDATIONS FOR FUTURE WORK

5.2

125

Recommendations for future work

Data integration is a fundamental concept in reservoir modeling. The objective is to explicitly account for all of the available data including geologic data, geophysical data and production data. The final goal is to provide multiple realistic geologic models with prediction power. A large part of the ongoing research in geostatistic modeling is to devise techniques that can accommodate a great variety of data. This dissertation has been devoted to developing and testing proposed modeling technique to integrate well data, conceptual stacking patterns from reservoir analog and production data. Suggestions for future work to improve and broaden this modeling workflow are discussed below. • Further data conditioning Other data can be integrated into reservoir models, including vertical sand proportion curve, areal sand proportion map, well-testing interpreted shale drapes or channel width, trend map of shale drapes distribution, and seismic-derived probability cube. In practice, some or all of these information are often available through extensive studies on well, seismic and production data. Developing conditioning techniques to integrate these data will make the proposed modeling workflow more practical. A vertical proportion curve specifies the proportion of all channel sand as a function of vertical elevation z or stratigraphic time, and an areal proportion map specifies the sand proportion as a function of areal location (x,y). The basic idea of constraining these two types of proportion data is as follows: in the channel complex simulation process, each simulated channel is rasterized on a grid, and the current vertical proportion curve and areal proportion map are updated. Because the channels passing through wells are simulated first, at some point there will be no channels needed in the current depositional layer for well data. Then extra channels outside of well data can be added into the belt until the vertical proportion at the current layer is met. Next, the current areal proportion map is updated and the simulation goes to next layer. This process is repeated until the vertical proportion curve and areal proportion map

126


are honored. This idea however requires some modification of the channel complex modeling method described in this dissertation, for example, certain rules should be added to ensure the simulated channels follow conceptual stacking pattern, at the same time the given proportion curve and areal map are honored. The interpreted channel thickness, width or shale drape geometry from welltesting data can be constrained directly using the presented modeling technique. The thickness and width can be constrained in the same way as well data conditioning described in Chapter 2. Channel thickness is first converted to noise hard data, and then this noise hard data is conditioned during correlated noise simulation. Channel width can be constrained in Step 5 of individual channel simulation (Chapter 2 Section 2.1.1) by using the interpreted channel width to define channel regions instead of using a predefined width. Shale drapes geometry or trend map can be easily conditioned when performing pixel-based MPS simulation. The most challenging part of data conditioning is for seismic-derived sand probability cube. The first challenge is how to correlate pixel-based probabilities to objects, the second challenge is how to honor probability cube and conceptual channel stacking pattern simultaneously. As described in Chapter 2, essentially channels are simulated using an object-based method. More specifically, a 2D thickness map is first simulated, and then painted into 3D forming channel body. The location of the channel is determined by stacking pattern parameter probability distribution functions. One possible avenue of integrating probability cube is as follows:

– In an explicitly defined channel belt container, start from the top surface of this belt region, check the probability cube around this location and find a point (x,z) that has maximum probability, then treat this point as the first channel initial centerline location and generate one channel. Channel geometry is simulated in a way that is consistent with the probability map


127

underlying the surface z. This could be done by (1) sum up the probability below surface z down to surface z − H (H is channel maximum thickness) and normalize it, then we get a 2D probability map. (2) integrate this 2D probability map when simulating correlated noise map for channel thickness. – To determine the next channel location, first the probability map of migration ratio P robM R and probability map of overlap ratio P robOR are calculated around point (x,z) using their representative probability distributions, then a cross section of seismic-derived probability cube P robseis passing point (x,z) is extracted, next we combine these three 2D probability maps and find the most likely point (x0 ,z 0 ) (with maximum probability value) on the combined probability map. This point is the location that the next channel initial centerline passes through. Once we have the initial centerline, we can simulate another channel. The next channel location will be based on point (x0 ,z 0 ). • Testing shale effects in other flow regimes In this dissertation, shale effects are considered in 3D, that is, if shale drape is present, the transmissibility multiplier in x−, y− and z− directions could be 0 depending on the specific channel configuration. In order to test shale drape impacts on reservoir connectivity, only immiscible flow with favorable mobility ratio (M ≤ 1) is considered in flow simulation model. There is no gravity, no capillary pressure included. As we know, the effect of a given geologic heterogeneity on reservoir performance during immiscible flow depends not only on the nature of that heterogeneity, but also upon the fluid properties and the flow regime. For example, the geometry and evolution of the waterfront during water flooding depends on the relative mobility of the oil and water and the relative importance of viscous and gravity force. Future work on shale drape effects can extend to (1) unfavorable case (M > 1); (2) gravity dominated case; (3) including capillary pressure. • Gridding and upscaling

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Using surface-based gridding, each sedimentary body is defined between two stratigraphically consecutive sharp boundaries. Hence, each layer represents one object. The advantages of using a surface-based grid to characterize multiscale flow barriers are: (1) thin shale drape can be explicitly modeled as one layer in surface-based reservoir model, instead of implicitly treated as surface property in structured coarse Cartesian grid; (2) because the vertical grid cell size varies with object (channel or drape) thickness, surface-based gridding has potential to preserve a smooth channel surface shape with affordable number of cells for flow simulation; (3) surface-based gridding can be directly linked to the concept of stratigraphic sequence. For example, hierarchic sequence boundaries interpreted from seismic data and well data can be used to construct multiscale surfaces in reservoir model. This creates reservoir model grids reflecting geologic architecture. However, as described in Chapter 1 Section 1.1, there are some limitations in terms of history matching. One major concern is that during history matching, this conformable grid need to be automatically updated with the reservoir model perturbed. But such gridding is not yet robust enough to be made automatic. Hence, future research work on robust surface-based gridding is a promising direction in building geologically realistic reservoir with affordable computation cost for flow simulation. Instead of using a uniform Cartesian grid with uniform upscaling ratio, another alternative is to use upscaling adaptive gridding approach in a Cartesian grid. Within this approach, no upscaling is performed in the areas of fine-scale reservoir model with high degree of permeability variation, such as channel boundaries attached by drapes. If permeability variation is small, such as within a single channel region, upscaling is performed in this area. This adaptive upscaling approach can provide a coarse-scale model that contains fewer grid cells than would be required when using uniform upscaling in Cartesian grid to achieve the same accuracy during simulation.

• Extend to other systems One immediate extension is to add multi-scale fractures into reservoirs with


129

multi-scale shale drapes. With two types of flow barriers, each has multi-scale nature, the geologic modeling, gridding and upscaling is getting much more complicated and challenging. However, studying the effects of these flow barriers will provide important guidance for similar reservoir prediction and management. Another interesting and challenging deposition system to be studied is deepwater distributary lobes. Some authors (e.g., Sprague et al., 2002) argue that lobes show a similar stratigraphic hierarchy as channel deposits. If scouring (erosion) is significant in the upper lobe environment, fine-grained deposits could sometimes be removed, such that good quality sand bodies within lobes will be connected. Taken in this sense, stacking pattern of lobes will control reservoir connectivity. Therefore, knowing individual lobe geometry will help locating flow barriers and good-quality sand bodies. For this reason, the presented channel stacking pattern modeling technique can be extended to simulate lobe distribution. However, unlike the confined channel system studied in this dissertation, distributary lobes exhibit non-stationarity, that is, lobe complex is diverging from proximal to distal, and individual lobe also shows diverging geometry. This non-stationary nature makes the geologic modeling more challenging compared to channel system. For example, how to define a transfer function to convert simulated potential map into lobe thickness while condition to well data? How to characterize erosion vs. deposition within individual lobes? How to integrate progradation and retrogradation trend into geologic models? All of these are very interesting and practical questions to be addressed..

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