An Earth Modelling Perspective: Seismic Reservoir Characterization

Seismic Reservoir Characterization An Eart E arthh Modelli M odelling ng Perspective Pe rspective EDUCATI TION ON TOUR SERI SERIES ES

EAGE EUROPEAN ASSOCI ASSO CIAT ATIO ION N OF OF GEOSCIENTISTS & ENGINEERS

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Table of contents

Chapter 1 Introduction to Geostatistics and Earth Modelling from Seismic Data

5

1 . 1 Course Overview 1.2 Earth Modelling Workflow with Seismic Data 1.3 Introduction to Geostatistics

5 7 21

Chapter 2 3-D Geostatistical Interpolation & Filtering

43

2.1 Overview 2.2 Simple Kriging 2.3 Ordinary Kriging 2.4 Kriging with Locally Variable Mean 2.5 External Drift Kriging 2.6 Cokriging 2.7 Factorial Kriging 2.8 Factorial Cokriging

43 43 51 52 55 58 67 74

Chapter 3 Stochastic Simulation with Seismic Constraints

79 79 79

3.1 Overview 3.2 Sequential Gaussian Simulation 3.3 Sequential Simulation with LVM or Collocated Cokriging 3.4 SGS with Non-linear Relationships 3.5 Simulation with Downscaling 3.6 FFT-Moving Average Simulation 3.7 G radual Deformation

Chapter 4 Facies Modelling from Seismic Data 4A Overview 4.2 Bayesian Classification 4.3 Sequential Indicator Simulation with S eismic Constraints 4.4 Truncated Gaussian Simulation 4.5 Boolean Simulation Methods 4.6 Multi-Point Statistics Simulation

Chapter 5 Stochastic Inversion 5.1 Overview 5.2 Geostatistical Inversion using SGS Si B ayesian Stochastic Inversion 5.4 Exploiting Geostatistical Inversion Results 5.5 Joint Stochastic Inversion of Elastic and Petrophysical Properties

83 87 89 101 103

105 105 105 120 128 133 133

139 139 139 142 156 163























































Tabl Ta ble e of of con conte tent ntss

Chapter 6 Statistical Rock Physics

165

6.1 Introdu Introduction ction 6.2 Monte Carlo Simulation of Rock Physics Temp Templates lates 6.3 Combining Rock Physics and Spatial Uncertainty 6.4 Uncertainty Propagation using Linearized Analysis 6.5 Rock Physics Inversion using MCS and Bayesian Method Methodss 6.6 Pseudo Well Generation 6.7 Direct Petrophysical Inversion

165 165 170 179 185 190 197

Chapter 7 4-D Earth Modelling

211

7.1 Overview 7.2 Initial Geo Geomo model del Construction 7.3 Downscaling of Dynamic Properties 7.4 PEM-Based Calculation of Elastic Properties 7.5 Computation of 4-D Synthetics 7.6 4-D Interpretation 7.7 4-D Seismic History Matching

211 213 213 214 220 222 225

Future Directions

229

References

233

Acknowlegements

247

List of Main Mathematical Symbols

249

List of Main Abbreviations Abbreviations

251

Index

253























































Chapter 1 Introduction to Geostatistics and Earth Modelling from Seismic Data

1.1 Course Overview Three-dimensional numerical earth models play an increasingly central role in the E&P industry. They are routinely used to plan new wells, calculate hydrocarbon reserves and, when coupled to a flow simulator, predict production profiles. Due to sparse well coverage, earth models are often poorly constrained away from well locations. A key challenge for reservoir geoscientists is therefore the quantitative integration of 3-D seismic data to obtain a more accurate representation of reservoir properties between the wells. Using a combination of theory and case study illustrations, the aim of the course is to review current best practices, emerging techniques and challenges for constraining earth models with seismic information. Geostatistics plays an important role in the construction of earth models. Initially developed in connection with the mining industry by Matheron, geostatistical techniques have been widely adopted in the petroleum industry following the pioneering work of Prof. A. Journel at Stanford University and Prof. H. Omre in Norway, among others. Techniques such as kriging and conditional simulation are now well-established to interpolate reservoir properties between wells and create realistic 3-D heterogeneity models. Commercial implementations of these techniques in geomodelling software packages are widespread, based in part on pioneering public domain software developed in universities (e.g., GSLIB, Deutsch and Journel, 1992). One attractive feature of geostatistics is the availability of powerful techniques such as cokriging and co-simulation for constraining 3-D earth models with seismic information. Later in this chapter, we introduce basic geostatistical concepts and present a typical workflow for constructing 3-D earth models by combining seismic and well data.

In Chapter 2, we review the most popular geostatistical estimation techniques for seismic data integration. Our goal is not to give a complete catalogue of existing techniques but to introduce the most robust techniques and explain their limitations. In Chapter 2, we also discuss the use of kriging-based techniques to design spatial filters and show their application for conditioning seismic data prior to reservoir modelling Stochastic heterogeneity modelling is one of the hallmarks of geostatistics. Following their development in the late 80s and 90s, techniques such as sequential indicator simulation, sequential Gaussian simulation, truncated Gaussian simulation and Boolean simulation are now mature and routinely applied in earth modelling workflows. In Chapter 3, we review the concept of conditional simulation for continuous variables such as porosity, focusing mainly on techniques suitable for incorporating seismic information. In Chapter 4, we introduce stochastic techniques for seismic lithology modelling and explore the link between Bayesian classification and sequential indicator simulation. We also briefly cover the promising new development in Multi Point Statistics simulation, which can be used to simulate more realistic geological features than variogram-based techniques and can also be condi-























































Chapter 1

used to guide the areal interpolation of zone average well data. Today, 2-D mapping has been largely replaced by 3-D earth modelling and the focus is on integrating seismic-derived information to constrain fine-scale 3-D property models. The limited vertical resolution of inverted seismic data has been a major obstacle to the widespread use of seismic information in 3-D property modelling. A number of promising seismic downscaling techniques have been developed recently and will be reviewed in Chapter 3. Since its introduction by Haas and Dubrule in 1994, stochastic inversion remains to this day a very active area of research. This topic is closely linked to geomodelling because stochastic inversion can be constrained directly with fine-scale log data and delivers high frequency impedance volumes that can be integrated in 3-D earth models without downscaling. While deterministic inversion remains the preferred choice in many companies, the application of stochastic inversion has been growing in recent years. In parallel with deterministic inversion, stochastic inversion has evolved from post-stack acoustic impedance inversion to pre-stack implementations in which multiple partial angle stacks are inverted simultaneously to estimate several elastic parameters. The link between deterministic and stochastic inversion is also better understood, following in particular the work of Buland et al. (2003a) in a Bayesian setting. In Chapter 5 we discuss recent developments in stochastic inversion and downstream workflows that exploit multiple realizations of inverted elastic properties for lithology prediction and uncertainty quantification.

Earth models are becoming increasingly complex. They are used to represent properties from several domains: reservoir properties (e.g., facies, porosity and permeability), seismic properties (e.g., acoustic impedance), mechanical properties (e.g., effective elastic moduli) and dynamic properties (e.g., saturation, and pore pressure). Data integration between the different domains poses several challenges, including how to establish relationships between data from different domains and how to merge the data in a common model framework. Geomechanical and 4-D applications also require a full earth model covering not just the reservoir interval but also the overburden. Rock physics provides the necessary link for relating data from different domains (for example a petro-elastic model is used to link seismic velocities to lithology, porosity, saturation and pressure) while geostatistics provides the tools tools to propagate propagate uncertainties in the rock physics transforms. In Chapter 6, we discuss statistical rock physics, which combines together rock physics and geostatistics. We give a number of examples of rock physics-based inversion and stochastic simulation workflows for seismic pore pressure prediction in a 3-D geomechanical earth model and for seismic porosity and lithology estimation in 3-D reservoir models. This chapter also compares sequential inversion workflows, where elastic inversion is followed by a rock physics inversion, with direct petrophysical petrophysical inversion, inversion, where seismic data are inverted inverted directly ffor or rock properties properties such as porosporosity and lithology. The direct petrophysical inversion workflows are becoming more popular because they guarantee coherency between the earth model and seismic amplitude data, which is not the case for traditional cascaded workflows. In Chapter 7, we concentrate on 4-D earth modelling and the integration of dynamic flow simulation results with time-lapse seismic data in the earth model framework.

























































Introduction to to Geostatistics and Earth Modelling from Seismic Data Data

issues of saturation and pressure prediction from time-lapse seismic data, joint inversion of seismic and production data and 4-D geomechanics because these topics have been covered in other recent courses (Calvert, 2005; MacBeth, 2007). Furthermore coverage of these extra topics would require an extended course period. In the course, we use the terms earth model, reservoir model and geomodel interchangeably to mean a 3-D numerical representation of subsurface properties that is obtained by integrating multiple types of data and can be used to perform physical calculations such a flow simulation and that serves as a base for making reservoir management decisions. A reservoir model may be seen as a part of an earth model in that an earth model may represent not just the reservoir but also formations above and below the producing interval.

1.2 Earth Modelling Workflow with Seismic Data Earth modelling workflows are typically organized in three main stages (e.g., Dubrule et al., 1997). 1 . Construction of geometric framework from horizons and faults interpreted from seismic data. 2. Building of 3-D stratigraphic grids that provide a cellular framework for property interpolation. 3. Generation of 3-D property models by well log interpolation. In this section, we review this workflow and focus on the extensions required to incorporate seismic data as a constraint in the construction of 3-D property models. The workflow presented here is not universally applicable. Instead, it highlights some of the key concepts and practical issues that are faced when integrating seismic information.

Main Inputs for Constructing C onstructing Seismic-Constrained Seismic-Const rained Reservoir Model Depth-converted cube of acoustic impedance

Porosity log data at vertical and horizontal wells

Depth-converted horizons defining different reservoir intervals























































C hapter 1

Simultaneous Elastic Inversion € Example Example Workflow Structural Framework

Synthetic Angle Stacks

Initial Elastic Model

Real Angle Stacks

Forward modelling with Zoeppritz equation

Difference Simulated annealing model perturbation Vp, V,, p

<8?

Final model

Fig. 1.2.2

Correction of Residual Errors in Depth Conversion Al

Kft/sec x gm/cc

Map of Interpolat Interpolated ed of depthing corrections

to























































Introduction to Geostatistics and Earth Modelling from Seismic Seismic Data Data

Input data The main input data for the construction of a seismic-constrained earth model are typically the following:

typically specified as regular 2-D 1. Horizon data and fault picks. A number of horizon surfaces, typically grids are used to define the top and base of the different reservoir intervals. Horizons are usually obtained from 3-D seismic interpretation and are depth-converted before integration with the geomodel. Extra stratigraphic surfaces may also be defined between key markers identified from the seismic data. Fault surfaces or fault polygons or poly-lines picked from seismic data are also a key input for the geomodel construction. 2. Well log l og data and well tops. top s. Typically, one or multiple log data tables will be stored for each well containing measured depth, true vertical depth, porosity, lithology, saturation and other properties, properties, for each sample point along the well trajectories. trajectories. When available core data may also be incorporated to calibrate log-derived porosity estimates or calculate a porosity-permeability transform. Data tables, containing well picks for the top and base of the different zones of interest, are also required. Seismic attribute attr ibute volumes. volumes. Cubes of depth-converted seismic attributes are obviously a key input. 3. Seismic Depth conversion of the seismic volumes may be performed by resampling the data between key horizons defined in both time and depth domains. A seismic velocity cube may also be used to constrain the depth conversion process. Figure 1.2.1 1.2. 1 depicts the main inputs in a structurally simple carbonate reservoir reservoir from from the Middle Middle East. In this case, log data at several horizontal wells and a cube of inverted acoustic impedance are two key inputs for the modelling of the reservoir properties. Seismic Seismic Inversion Which seismic attributes should be used as input for reservoir modelling? Experience shows that best results are often obtained when using cubes of inverted attributes such as acoustic impedance. There are two main benefits of using inverted attributes compared to using amplitudes, AVO attributes or other attributes resulting from simple transforms of amplitude data. First, inversion partially removes wavelet effects, increases the resolution and provides impedance values that can be directly correlated with the absolute values of the rock properties stored in our 3-D models. This is not the case of amplitude or AVO attributes which are related to relative changes in reservoir properties across elastic interfaces. Second, inverted impedances can be directly calibrated at well locations with log-derived























































Chapter 1

Multi-Lay Multi-Layered ered Stratigrap Stratigraphic hic Grid Grid Fram Framewor ework k

Aver Averag aged ed logs logs in in grid grid cells cells intersecting the wells Grid block size varies from one rval to the next

Fig. 1.2.4

Stratigraphic Stratigraphic Grids with Different Layering Styles

Proportional

Onlap / Baselap























































Introduction to Geostatisti Geostatistics cs and Earth Modelling Modelling from Seismic Data Data

Furthermore, some inversion tools deliver the inverted elastic properties in stratigraphic grids that can be used directly for reservoir modelling. In the illustrative workflow described below and in subsequent chapters, we will focus mainly on the use of inverted attributes to constrain reservoir models. However, in Chapter 6, we will come back to the issue of constraining earth models directly from seismic amplitude data.

Depth Conversion QC Before embarking on the construction of the earth model, basic data QC steps are required. For example, we need to check that depth-converted horizons match the tops defined at the wells. This will be the case if tops have been used to constrain the depth conversion process. If this is not the case, the horizons need to be flexed locally near the well locations to ensure that input horizons are consistent with the well tops. Another important data QC is to check the quality of the inversion at the well locations after time-depth conversion, as shown in Figure 1.2.3. This is done by sampling the depth-converted inverted cubes along the well trajectories and displaying inverted impedance values side-by-side with log impedance. As shown in the example, inverted results (dotted curve) may not align properly with the log values due to residual errors in the depth conversion. This problem must be rectified, as it will adversely affect the the petro-seismic petro-seismic calibration at later stages of the workflow. Different strategies can be applied to correct the residual depthing errors, such as flexing horizons near the wells or applying residual velocity corrections. This process is often iterative and can be QC by displaying cross plots of log versus inverted impedance before and after depthing corrections (Figure 1.2.3). At this stage of the workflow, a decision is made as to whether or not the seismic data can be used to constrain the reservoir model, based on the quality of the seismic-well tie. Depending on the vertical resolution of the seismic data compared to the thickness of the reservoir zones, a preliminary decision may also be made about the type of property modelling technique that will applied: for example, with high resolution inversion results, we may use a 3-D cokriging procedure while in a thin reservoir zone we may decide to constrain the model using an areal trend map derived from seismic impedance. Structural Modelling and 3-D Gridding Having assembled and QC the data, the first step is to build the structural and stratigraphic framework for the earth model. A 3-D stratigraphic grid is a structured grid composed of hexahedral cells aligned along layers in order to conform to reservoir stratigraphy. When no faults are present,

























































Chapter 1

Comparison omparison of Regular and Stratigraphic Grids Regular Cartesian Grid

Stratigraphic Grid

I IE M111•11111116:11

I I IPP-7AELZ:7111111 1 do 1 1 1 1 1 1 1 1 1 1 0 ! A I M

1 1 / 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1M I N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1E

Stratigraphic coordinates are approximated approximated by (i,j,k) ( i,j,k) indices in the stratigraphic grid

Adapt Adapted ed from R. Moyen Moyen

Fig. 1.2.6























































Introduction to Geostatistics and Earth Modelling from Seismic Data

Vertical ertical Section trough trough Faulted Faulted Stratigraph Stratigraphic ic Grid

Ada Adap pted ted fro from R. Mo Moyen yen

Fig. 1.2.8























































Chapter 1

Log Averaging — Horizontal Wells

Log data are averaged in grid cells intersecting the horizontal well trajectories

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Fig. 1.2.9
























































Grid Flexing before Log Averaging and Well QC `DNS

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Comparison of I original logs and grid p cell averages (white)

3-D Grid Flexing Flexing

Log Averaging into Grid Cells before

1 after

From Cole et al., 2003

Fig. 1.2.10























































Chapter 1

Impact of Log Averaging on Porosity € Impedance Relation o

L o g data: p=-0.82

0

Grid block block data: p=-0 .86

10,000

15,000

20,000 Sonic Impedance (m/sec x gm/cm 3 )

From Doyen et al., 1997

Fig. 1.2.11


























































Potential Problems in Property Transfer between Seismic and Reservoir Grid Stratigraphic grid does not conform to interpreted seismic horizons and faults.

























































Chapter 1

Improv Improved ed Strat Strategy egy for Property Property Transfer Transfer from Seismic Seismic to Reser Reservo voir ir Grid Grid

ta)


























































Porosity - Seismic Impedance Correlation

H = high (1)

L = low (1)

























































C hap ter 1

Variogram Calculation From Stratigraphic Distances

) 2 % 4

0 1 ( m a r g 5 o i r a

Horizontal Horizontal correlati correlation on range 800m


























































between seismic impedance and porosity (or other reservoir properties). This analysis is used to define the statistical correlation parameters for geostatistical interpolation techniques such as cokriging (see Chapter 2).

Having the seismic data available in a stratigraphic grid is also useful for fast exploration data analysis. Figure 1.2.16 shows an example of dynamic graphics analysis. A seismic impedance histogram is constructed after transferring the seismic attribute in a stratigraphic grid. The data range (low impedances) highlighted in the histogram is cross-referenced to the 3-D display where only grid cells in that range are visible. As expected, low impedance anomalies tend to coincide with the position of high porosity horizontal wells. Spatial Continuity Continui ty Analysis The next step in the reservoir modelling workflow is to conduct a variogram analysis to characterize the spatial continuity in the variation of reservoir properties. Variogram calculations are performed



























































Chapter 1

Gaussian Probability Distribution Function




























































Gaussian Cumulative Probability Distribution Function



























































Chapter 1

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Bi-variate Gaussian Distribution Distrib ution f (z) N

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Chapter 1

Gaussian Conditional Distribution and Conditional Expectation















































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































An Earth Modelling Perspective: Seismic Reservoir Characterization

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