Probability Distribution Functions for Geomechanical Properties from Well Log Data

Transcription

1 Final Probability Distribution Functions for Geomechanical Properties from Well Log Data T. de Gast Master of Science Thesis Civil Engineering and Geosciences

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3 Final Probability Distribution Functions for Geomechanical Properties from Well Log Data Master of Science Thesis For the degree of Master of Science in Geo-Engineering at Delft University of Technology T. de Gast November 1, 213 Faculty of Civil Engineering and Geosciences (CEG) Delft University of Technology

4 The work in this thesis was supported by Shell. Their cooperation is hereby gratefully acknowledged. Copyright Civil Engineering and Geosciences All rights reserved.

5 Abstract Reliability-based calculations and the identification of uncertainties in geomechanical calculations are receiving an increasing amount of attention to evaluate risk. This poses multiple challenges, of which one is the variability of input parameters such as geomechanical properties, geological structures and distribution of material properties. In this thesis, the use of reliability geomechanical analysis in oil and gas reservoirs is investigated. The focus lies on obtaining Probability Density Functions from well log data, to investigate the effect of stochastic material properties on geomechanical analysis and to identify important uncertainties other than material properties in geomechanical analyses e.g. geometry and depletion pressure distribution. A method of deriving stochastic material properties from well log data is presented and stochastic material properties are described using Probability Density Functions. A test is introduced to select the most appropriate Probability Density Function in terms of best fit. The most appropriate Probability Density Function can be obtained by comparing a Probability Density Function to the histogram of a set of well log data. In addition to this, it was observed that removing linear depth trends reduce the uncertainty in most observed material properties. Stochastic material properties have been used as input for a series of probabilistic geomechanical analyses. The main geomechanical reliability response studied in this thesis is large scale subsidence caused by pressure changes in a reservoir. Large scale subsidence has also been compared to a small scale Shear Capacity Utilisation response. The effects of stochastic material properties on subsidence are compared to Shear Capacity Utilisation in geomechanical reliability response. For subsidence it was found that, increasing the resolution of uncertainty, i.e. increasing the amount of layers, reduces the variation of results. For Shear Capacity Utilisation geomechanical reliability, increasing the resolution of uncertainty The results increases the variation of results. For the geomechanical analyses, example real reservoir data are used to study uncertainties in geomechanical reliability analyses. Data from a real case were used to illustrate the effects of reservoir geometry and depletion pressure distribution on the calculated subsidence above a depleting reservoir. The different geomechanical analyses demonstrate the need to include Master of Science Thesis Final T. de Gast

6 ii spatial variation of the reservoir, i.e. the variation of the pressure distribution and both vertical and lateral variation in material properties. T. de Gast Final Master of Science Thesis

7 Contents Acknowledgements xi 1 Introduction Background and motivation Workflow Goal Scope Research questions Contents of the thesis Background literature Introduction Geomechanical behaviour Reservoir compaction Geomechanical properties Reservoir copmpaction related geomechanical issues Uncertainties and uncertainty analysis Summary Geomechanical property determination from well log data Introduction Selected literature Geophysical measurements Density logs Sonic logs Porosity logs Geomechanical properties Master of Science Thesis Final T. de Gast

8 iv Contents Properties from well log data Properties from laboratory data (conversion functions) Analysis and discussion Summary and conclusion Deterministic reservoir analysis Introduction Selected literature Case study Fielddata Geomechanical calculation of the reservoir Geomechanical properties Geomechanical response Summary and conclusion Statistical data analysis Introduction Selected literature Example data set Deriving a Probability Density Function (PDF) from data Displaying selected data Probability density functions Area fit test Linear correlation of data Distribution of properties Analysis and discussion Summary and conclusion Stochastic formation properties in geomechanical calculations Introduction Selected literature Static properties of reservoir Stochastic analysis Stochastic geomechanical properties Multiple layers Location of formation with stochastically determined properties Different PDFs Global failure versus local failure Three dimensional geomechanical model using stochastic properties Analysis and discussion Summary and conclusion T. de Gast Final Master of Science Thesis

9 Contents v 7 Discussion Introduction Setup of the problem Geomechanical properties Numerical modelling Summary Conclusion Introduction Research conclusion Updated workflow Recommendations for further development of CORA Geometric uncertainties Pressure uncertainties Random field realisations A Overview Statistics Sheet 91 B Used programs 11 B-1 Subsidence calculator- Fortran B-2 Formation failure calculator- Fortran References 113 Glossary 117 List of Acronyms Master of Science Thesis Final T. de Gast

10 vi Contents T. de Gast Final Master of Science Thesis

11 List of Figures 1-1 Deterministic workflow for variability Subsidence due to reservoir compaction, [Fjar et al., 28] SCU determination from shear stress Well fluid penetration Expected subsidence using the provided dataset Density log Pressure wave velocity log Shearwavevelocitylog Porosity log Geology horizons Boundaries of formation Boundaries of formation Boundaries of formation Boundaries of reservoir Boundaries of reservoir Boundaries of formation Pressure field Pressure field Pressure field Pressure field Pressure field Pressure field Pressure field Well log three: Stiffness Average depletion pressure average geometry Master of Science Thesis Final T. de Gast

12 viii List of Figures 4-22 Local depletion pressure average geometry Average depletion pressure local geometry Local depletion pressure local geometry Processeddata Data to Histogram Uniform probability density function Normal probability density function Lognormal probability density function Truncated normal probability density function Goodnessoffittest Linearization of data Polynomial fit Linear trend removal Linear trend removal Subsurface geomechanical discretisation Images of the layering of reservoirs in the geomechanical analysis The effect of an increasing number of layers on geomechanical response The effect of location of stochastic formation properties on geomechanical response The effect of different PDFs on geomechanical response A comparison between subsidence and SCU Stochastic geomechanical response of the case study Illustrating different responses of three dimensional subsidence Static Young s modulus E s Current Workflow Suggested Workflow T. de Gast Final Master of Science Thesis

13 List of Tables 3-1 Conversion function shapes used in this thesis Average (static) geomechanical properties Four reservoir cases Geomechanical subsidence response figures 4-21 until Level of fit (F fit ) Reduction coefficient in standard deviation Effect of linear trend removal on standard deviation Geomechanical properties and their distributions Stochasticgeomechanicalparameters StochasticgeomechanicalparametersforfivePDFs Master of Science Thesis Final T. de Gast

14 x List of Tables T. de Gast Final Master of Science Thesis

15 Acknowledgements I would like to thank my supervisors for their assistance during the writing of this thesis... By the way, it might make sense to combine the Preface and the Acknowledgements. This is just a matter of taste, of course. Delft, University of Technology November 1, 213 T. de Gast Master of Science Thesis Final T. de Gast

16 xii Acknowledgements T. de Gast Final Master of Science Thesis

17 A model is an appropriate simplification of reality. The skill in modelling is to spot the appropriate level of simplification, to recognise those features which are important and those which are unimportant. David Muir Wood

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19 Chapter 1 Introduction 1-1 Background and motivation As oil and gas demand stays high, the economic benefits of developing reservoirs in more difficult stratigraphy (geological formations) increases. Besides the development of more difficult reservoirs, depleted reservoirs can be used to store or dispose gas. The formations that are influenced by the process of extracting or injecting formation fluids/gasses are heterogeneous and show different properties throughout the formation. The identification of formations and mechanical properties is based on well log interpretation, laboratory tests on core samples and geological studies. Failure of a formation which seals the reservoir can cause loss of containment and can have severe environmental, financial and reputation consequences. To predict, address and prevent containment issues, studies using geomechanical analysis to calculate the effect of pressure changes in the reservoir formations on surrounding formations are carried out. However, because the formation properties change throughout a formation, there is a level of uncertainty and variability in the geomechanical model. The effect of this variability can be analysed using software which is able to compute variability in geomechanical properties e.g. the Shell in house COntainment Reliability Analysis (CORA) framework. In this thesis, the CORA framework is used as the software to analyse the variability in the geomechanical properties. The CORA framework is setup to assess the reliability of different factors that influence the design of a reservoir development and prevent containment related issues. CORA can be used to make decisions based on the uncertainties in the different aspects of the reservoir development. From well log data, geomechanical input parameters can be derived. Well log data are scattered due to formation properties changes and measurement errors. This scatter in the data can lead to a variety of input parameters for a geomechanical analysis. This study focuses on capturing this variability and using statistics to study the effect of variability on the outcome of the geomechanical analysis. Part of the study focuses on the correlation of well log data to formation properties and aims at using less parameters or combined information to get a more accurate geomechanical parameter distribution. This study can be used in reliability analysis using geomechanical statistics in software e.g. the in-house CORA tool. Master of Science Thesis Final T. de Gast

20 2 Introduction 1-2 Workflow Current methods to include variability in a geomechanical calculation are based on deterministic approaches. In practice, three values are determined from the formation properties, a low property estimate (p1), the average property estimate (p5) and a high property estimate (p9). Based on these values, geomechanical analyses are carried out. In the workflow below, instead of a deterministic value, a Probability Density Function (PDF) is computed from the field data to be used as an input in stochastic geomechanical analysis. In order to obtain a PDF, geomechanical properties are derived from well log data. First, geomechanical properties have to be computed based on conversion functions such as porosity based Young s modulus. In addition to the derivation of formation properties, the sub surface model computed by the geologists/reservoir engineers and pressure depletion scenarios from reservoir engineers are combined to build a reservoir model including boundary conditions e.g. initial stress conditions, pressure distribution and formation properties. Once the reservoir model including boundary conditions are entered, a geomechanical calculation can be started. The results of these geomechanical calculations can be presented in the form of deformations (subsidence), failure of formations (Shear Capacity Utilisation (SCU)) or fault activation. These problems can be put into two general failure modes global and local failure. Figure 1-1: Deterministic workflow for variability Each step listed in figure 1-1 has its variability and uncertainty. The CORA program allows specifying of these uncertainties and combining the different uncertainties to obtain a range of geomechanical responses. T. de Gast Final Master of Science Thesis

21 1-3 Goal Goal The goal of a probabilistic project is to understand and improve the assessment of geomechanical response analysis using geomechanical analysis. This is done by analysing the response of geomechanical calculations on different types of variability. The main part of this thesis aims at obtaining stochastic geomechanical properties from well log data and assessing the effect of these properties on the calculated geomechanical response of a reservoir compaction. The goal of this thesis is to combine a transparent and accurate approach to obtain stochastic properties from well log data. In addition this thesis aims to address important factors in dealing with uncertainty and statistical geomechanical analysis in petroleum related predictions. The factors addressed in this thesis are Stiffness (distribution), geometry and depletion pressure distribution. 1-4 Scope The assessment of reliability using stochastic geomechanical analysis will be improved by identifying steps in the workflow where (deterministic) assumptions are made and identifying what effect these assumptions have on a stochastic geomechanical analysis. Constitutive geomechanical model, in each geomechanical calculation, assumptions are made. As a rule of thumb, the more assumptions that are made, the faster the calculation will give a result but each assumption will limit the validity range of the calculation. This thesis will calculate geomechanical response based on a linear elastic constitutive model. Relations of well-log data and geomechanical properties, different geomechanical analyses and constitutive models require different properties. At different locations, relations between distinct material types and their mechanical behaviour can change. Shell has provided relations between well-log data and linear elastic geomechanical properties. This thesis will start out assuming that all the provided relations are exact solutions. However, there are different relations between well-log data to a single geomechanical property. The difference of these relations and their correlations will be part of the thesis. Heterogeneity in formations, soil formations are heterogeneous by nature. This means that before going from geomechanical properties from well log data to geomechanical properties of soil formations the response of the heterogeneous system has to be taken into account. This translation from local properties to formation properties will be part of the thesis. Instrument measurement uncertainties, Shell has done research to measure uncertainties of instruments used in well log identification and their effects on the measurement results and has concluded these uncertainties are minor. This will therefore not be part of this thesis. Master of Science Thesis Final T. de Gast

22 4 Introduction 1-5 Research questions Based on the goal and scope of this thesis one main research question has ben formulated. This main question has led to three sub questions which this thesis aims to answer: What is the effect of the variability of geomechanical properties, their PDF and the effect of the numerical discretization in a geomechanical analysis on reliability predictions of failure mechanisms? How can the variability of material properties be obtained from well log data? What are the effects of different types of uncertainties, unrelated to stochastic geomechanical properties, in a geomechanical analysis? What is the effect of heterogeneity in formation properties, specifically the stiffness properties, on geomechanical analyses and how should this heterogeneity be implemented in reliability based analysis? 1-6 Contents of the thesis First, the considered geomechanical problem is presented and the corresponding geomechanical failure mechanisms that will be addressed in this thesis are mentioned. Measured geomechanical properties have a variability which can be captured using statistics (chapter 5). In order to compute the geomechanical behaviour with the right stochastic geomechanical properties, two methods to determine geomechanical properties are presented and compared (chapter 3). In chapter 4, an example dataset and reservoir are presented and an elaboration on three dimensional effects on global geomechanical failure mechanisms analysed using several deterministic geomechanical models. Several methods to compute a PDF from well log data are presented in chapter 5. The derived PDF can be used in stochastic geomechanical analysis. The use of stochastic property values instead of deterministic property values in a geomechanical analysis can lead to an improved reliability analysis. In the second part of this thesis, the effect of variability in geomechanical reliability studies is explored (chapter 6); in addition, a stochastic analysis on the provided case data is presented. This thesis deals with two types of correlation, (i) the mathematical correlation of a point set and (ii) the correlation (functions) between measured geophysical properties and geomechanical properties. To prevent a misunderstanding in these two types of correlations, the latter type of correlation is called conversion in this thesis. In this thesis, reliability graphs are presented. The reliability (y-axis) is defined as the probability of a value being smaller than a certain value (x-axis). T. de Gast Final Master of Science Thesis

23 Chapter 2 Background literature 2-1 Introduction The subject of this thesis is the reliability prediction of geomechanical response with a focus on geomechanical issues related to reservoir compaction. In order to understand the basics of the geomechanical concepts and geomechanical uncertainties in this thesis, this chapter presents an overview of relevant background literature. The aim of this study is to present a method to define the appropriate stochastic geomechanical properties belonging to a geomechanical reliability analysis. These geomechanical properties are obtained from geophysical well log data and laboratory data on core samples. In addition to obtaining the appropriate geomechanical properties, the research focuses on the effect of modelling uncertainties in geomechanical reliability analyses. In order to keep this chapter concise, further reviews of the literature related other chapters are given at the beginning of each chapter. 2-2 Geomechanical behaviour Soils and sedimentary rocks consist of small particles; the pore space between the particles is filled with water, oil and/or gas. In mechanics, this is modelled as a saturated (or at least partially saturated) porous medium. The deformation of fully saturated porous media depends upon the stiffness of the porous material, but also upon the behaviour of the fluid and/or gas within the pores. The deformations will be considerably hindered by the pore fluid if the permeability of the material is small and the viscosity of the pore fluid is high. A linear relationship between stress and strain is the simplest link that can be proposed in geomechanical analyses. This linear elastic relationship is used to calculate the deformation of a soil body due to an applied stress. This linear relationship does not take plasticity of the material into account, [Fjar et al., 28; Wood, 24]. Master of Science Thesis Final T. de Gast

24 6 Background literature The advantage of linear elastic behaviour is that these calculations are easy and relatively fast to compute. Simplified geometries and uniform pressure decline can be solved using available analytical solutions. For more complex geometries and pressure declines in a geomechanical analysis there are (semi)analytical solutions available, which depending of the size of the geologic model and its complexity it takes a few seconds or minutes to finish. Whereas, more complex stress-strain relations can require several hours finish. Because shorter computing time, an analytical linear elastic material solution will be used instead of a finite element method for the geomechanical reliability analysis of this thesis. For more advanced soil behaviour including plastic deformations, finite element modelling can be used Reservoir compaction The theory linking the pore pressure to the effective pressure was originally developed by Terzaghi and it has been studied extensively since, [Verruijt, 212]. Extraction of oil or gas from the reservoir leads to a decrease of pore pressure and an increase of effective stress. The reduction in reservoir pressure as a function of place and time depends on many factors such as the mobility, solubility, density and compressibility of various pore fluids as well as on the reservoir boundary conditions, [Geertsma, 1973]. An increase in effective stress leads to compaction of the reservoir. Because the pore pressure is linked to the production, the reservoir compaction is linked directly to production. Geertsma derived an analytical solution to compute a compacting reservoir in a homogeneous linear elastic axis-symmetrical half space. Where the decline of fluid pressure in connection with the withdrawal of fluid from an underground reservoir gives rise to a change in volume of both reservoir volumes and reservoir rock, [Geertsma, 1957]. The solution holds for the depletion of reservoirs, provided the lateral dimensions are large compared with their height, which causes reservoirs to deform predominately in the vertical plane, [Geertsma, 1973]. Other analytical solutions have been suggested by various authors. A solution capable of calculating reservoirs in a linear elastic layered system, [Kuvshinov, 27] has been implemented in the Shell in-house software QuickBlocks (QB) which is used in for the geomechanical calculations in this thesis Geomechanical properties The properties to calculate the geomechanical response of a reservoir can be determined in several ways. The two most used in petroleum geomechanical analysis are (i) geophysical surveys, in which seismic wave velocity is used to determine the (dynamic) linear-elastic properties and (ii) laboratory tests which replicate in situ conditions on a drilled rock sample which can be used to determine the (static) linear elastic properties. At very small strains the stiffness of soils can be determined relatively simply and reliably from measurements of wave velocity, [Atkinson, 2]. Originally seismic surveys only measured pressure waves, since a few years the use of shear waves is receiving importance. Even if used alone, shear-wave observations generally allow the determination of subsurface parameters that are of more direct interest to geotechnical engineers than those that can be derived from the observation of pressure waves. The joint observation of pressure and shear waves (if T. de Gast Final Master of Science Thesis

25 2-3 Reservoir copmpaction related geomechanical issues 7 possible, shear waves of two mutually perpendicular directions) has the largest information potential,[verruijt, 26]. The elastic response of a porous material may be significantly affected by the presence of a pore fluid. In poorly consolidated sediments, the pressure wave velocity for a water saturated material can be several times larger than it is when the material is dry. The impact of the pore contents is higher for pressure waves than for shear waves. Tests that have been done in a laboratory are more controlled than tests that are done in the field. This means that information obtained about the material behaviour can be directly derived from laboratory observations while in seismic surveys the material response is measured insitu, which leads to uncertainties in measurements because the measuring conditions are less controlled than laboratory. However, for laboratory tests the material itself is disturbed when it is taken from a formation to the laboratory. And taking a sample to the laboratory is economically less attractive than obtaining material behaviour parameters from seismic surveys. The difference between dynamic and static difference is not straightforward. For example, the stiffness obtained by seismic wave velocities is generally different from the static stiffness. However, the fact that elastic waves are mechanical disturbances means that there has to be a fundamental connection between seismic surveys and rock mechanics. Normally, the dynamic moduli are larger than the corresponding static ones. The difference is largest for weak rocks, and is reduced with increasing confinement pressure, [King, 1969]. The major difference between static and dynamic measurements comes, among other things, from the strain amplitude, not from the strain rate. Static moduli, measured as slopes of stress-strain curves, differ from small strain amplitude dynamic (elastic) moduli because of plasticity or nonlinear effects. To further understand the origin of the static-dynamic discrepancy, it is important to notice that the static and dynamic moduli are equal for a homogeneous, elastic material like steel, [Ledbetter, 1993].Thus the physical origin of this discrepancy is likely to be related to the heterogeneous microstructure of the rock, [Fjar et al., 28]. In petroleum geomechanical analysis, seismic survey data are more available than laboratory data. This has led to a range of empirical relations between rock strength and physical properties, [Chang et al., 26]. 2-3 Reservoir copmpaction related geomechanical issues Changes of pressure in a reservoir lead to deformations. These deformations are not constrained to the reservoir itself but also involve the soil around the reservoir, leading increased stress in formations above and around the reservoir. These increased stresses can lead to fault slip, top seal failure or damage to the extraction or injection well. This can create leakage of fluids or gasses from the reservoir. Leakage from the reservoir can lead to environmental contamination. Geomechanical analyses or reservoirs are aimed at assessing issues relating to the production process of a reservoir e.g. bore hole stability, containment issues, operational safety and environmental safety. The geomechanical analysis for these reservoir issues can be divided Master of Science Thesis Final T. de Gast

26 8 Background literature Figure 2-1: Subsidence due to reservoir compaction, [Fjar et al., 28] into two types of failure mechanisms,(i) strain driven mechanical issues (ii) stress driven mechanical issues. A failure occurs when a set maximum value is exceeded, this maximum value is either derived from the technical limitations of the used equipment, e.g. the ground surface should not subside below a certain value to keep a required clearance for an offshore platform, or this maximum value is obtained from formation properties, e.g. a maximum tensile or shear stress is accepted to prevent faults from activating or formations to crack. Subsidence (illustrated in figure 2-1) is a good example of global failure mechanism, when using satellite positioning and imaging, the easiest failure mechanism to measure. If the subsidence is too large, the clearance for oil-producing platforms can go below the acceptable level. If the clearance is below the acceptable level, the production must be halted. If a reservoir is near shore, civil structures, for instance water retaining embankments, can subside. Subsidence of civil structures can lead to damages to real estate or inundation if a water retaining structure fails because of the subsidence. Subsidence and compaction do not have to occur at the same time a time gap due to consolidation time of clays and other impermeable formations can occur, [van der Knaap and van der Vlis, 1967]. Shear Capacity Utilisation (SCU), equation 2-1 (illustrated in figure 2-2) is a good example of a stress driven failure mechanism. SCU compares the calculated shear stress to the maximum allowable shear stress. One method to obtain the maximum allowable shear stress is by using the Mohr coulomb failure envelope where the maximum shear stress depends on the friction angle and cohesion of the material. SCU = τ τ max (2-1) In this thesis subsidence is used to illustrate the behaviour of global failure mechanisms and SCU is used to illustrate the behaviour of local failure mechanisms. T. de Gast Final Master of Science Thesis

27 2-4 Uncertainties and uncertainty analysis 9 Figure 2-2: SCU determination from shear stress 2-4 Uncertainties and uncertainty analysis From a geotechnical engineering perspective, there are four sources of uncertainty.(i) Natural variability, this can be treated as the major source of uncertainty in geotechnical engineering, which is due to the inherent randomness of natural processes. (ii) Knowledge uncertainty, this is what can be attributed to one s lack of measuring and modelling of the world and hence is mostly subjective. (iii) Operational uncertainties, which are due to human-induced parameters and are not considered in mathematical models of engineering performance. (iv) Decision model uncertainties: these include issues which are considered to be socially-contributed, involving social objectives, [Baecher and Christian, 23]. The limited amount of data and the uncertainties in determining the geomechanical parameters needs improvement on two main aspects: (i) A systematic approach to deal with uncertainties (this thesis). (ii) A clear definition of failure. The definition of failure is based on the fourth uncertainty by giving reliability to the calculated value. This thesis will deal with the first and second uncertainty using stochastic geomechanical analysis. Alternative: apart from a deterministic approach, a Probability Density Function (PDF) is required to describe the uncertainty of a geomechanical parameter. A normal distribution function is often a good initial assumption, but among others, uniform, log-normal and truncated normal distributions are also available to describe uncertainties properties. Data available from field measurements are used to determine the shape of the PDF. Probabilistic geomechanical analysis is becoming more available to geomechanical engineers. The reliability analysis of a subsidence calculation has been done before e.g. a case study on the subsidence near a heavily urbanized Italian coast, in which Geertsma s model to calculate subsidence was used in combination with a Monte Carlo probabilistic risk simulation by Cassiani and Zoccatelli [1998]. In this study, the reservoir shape was modelled using superposition of the different linear elastic solutions. 2-5 Summary This chapter gives a background of the topics related to this thesis. The combination of the increase in computation speed and a three dimensional approach to compute reservoir deformation, can reinforce reliability analysis. The power of a stochastic calculation is that a range of geomechanical responses can be computed based on observed data. Probabilistic Master of Science Thesis Final T. de Gast

28 1 Background literature methods are tools to provide an insight into the geomechanical responses. This makes it possible to make a geomechanical reservoir reliability analysis in which the measured natural uncertainties of the soil are taken into account. The geomechanical response is a function of the geomechanical constitutive model (linear elastic in this thesis) and the measured data. T. de Gast Final Master of Science Thesis

29 Chapter 3 Geomechanical property determination from well log data 3-1 Introduction To predict the geomechanical response of pressure changes in a reservoir, the subsurface structure including the reservoir is modelled in a geomechanical model. The accuracy of the calculated geomechanical response depends on how well the geometry of the reservoir, the stress distribution in and around the reservoir and the constitutive geomechanical model which describes the geomechanical stress and strain behaviour represent the actual reservoir conditions. A constitutive model requires input parameters. The input parameters required depend on the constitutive model which describes the material behaviour. For the linear elastic constitutive model, used in this thesis, the input parameters required are the Young s modulus (E) and Poisson s ratio (ν). These geomechanical properties are obtained by soil investigation. In order to obtain information about the reservoir and surrounding formations, geophysical properties are acquired in a drilled well. These geophysical properties include measurements of electric resistance, gamma radiation, sonic speeds, bulk modulus and porosity. The recording of the different geophysical properties in depth is called a well log. The geophysical data from a well log is used to determine geomechanical properties using conversion functions. In addition to well log data, core samples can be obtained from reservoirs and surrounding formations to be tested in a laboratory. From the laboratory tests geomechanical properties can be obtained through a direct correlation of applied stress and recorded strain. For an investigation into the geomechanical properties for a reservoir analysis, geophysical data are more commonly available than laboratory data. This chapter introduces methods to convert geophysical properties using different conversion functions to geomechanical properties and second, where to apply what type of property. In this chapter there are two types of properties introduced: (i) dynamic properties, denoted with subscript d. These properties are typically obtained from sonic velocity measurements. Master of Science Thesis Final T. de Gast

30 12 Geomechanical property determination from well log data (ii) Static properties, denoted with a subscript s. These properties are typically obtained from laboratory tests. Using these tests, conversion functions between geophysical and geomechanical properties can be obtained. The presented formulas in this chapter will be used throughout this thesis to obtain the geomechanical input parameters for the geomechanical models. 3-2 Selected literature Predicting and measuring geomechanical properties has been and still is a challenging field of research. In this literature overview, a selection is made of research focusing to observe the differences between dynamic properties and static properties. In these papers, dynamic properties can be obtained from sonic velocities and static properties can be obtained from applying stress to a material and observing the strain. An overview of the history of well logging techniques is given by Riboud and Schuster [1971] the techniques presented in this study are still in use for in situ measurements. Today, the measurements and their interpretation techniques have matured since the publication of this overview. Geophysical well logging techniques can be used to get several in situ measurements. For this study, bulk weight, sonic velocities, porosity and gamma ray logs are used to determine geomechanical properties. For non-porous materials like steel, the difference between static and dynamic elasticity has been studied by Ledbetter [1993]. From this study he concluded that an engineered material with no cracks or pore fluids shows a difference between the static and dynamic properties which are within the uncertainty of the static measurements and therefore dynamic moduli can be used compute the material properties for steel. Unification of parameter studies has a long history. A brief comprehensive review of the fundamentals of rock mechanics is written by Cheatham and Gnirk [1967]. The mechanical behaviour of rocks under simulated environmental conditions (laboratory tests) and in situ behaviour (wave propagation) is discussed, but the paper focuses mainly on well-borehole stability, drilling and fracturing. Different failure mechanisms linear elastic, viscoelastic and plastic deformations are presented. A difference in elastic constants of material is shortly discussed but not further explored. The study by King [1969] discusses the anisotropy observed in determining the Young s moduli and tested sandstone in an apparatus using dynamic and static measurements. In laboratory tests on dry samples, it was found that the E d increases as the hydrostatic pressure increased (up to 55 MPa) on the sample is increased. As the sample approaches the maximum stress, the rate of E d drops. For ν d after a sharp increase the value approaches a constant value. It was found that E s is always lower than E d but as the stress increased, the ratio E d /E s becomes less. The behaviour of ν s is similar to ν d however the value of ν s is always lower. It is argued that this behaviour is due to the closing of the (micro) cracks. It is also argued that anisotropic stiffness is caused by the orientation and attenuation of (micro) cracks. Laboratory measurements of ultrasonic wave velocities in rocks obtained from the Campi Flegrei volcanic system and their relation to other field data are described by Zamora et al. [1994]. This study to the dynamic rock properties, velocity dispersion effects in large volumes T. de Gast Final Master of Science Thesis

31 3-3 Geophysical measurements 13 of rocks and small volumes rock in laboratory the effect of micro cracks, effects of temperature and pressure provides elastic moduli calculated from elasticity theory. For saturated samples, ultrasonic velocities measured at room conditions are slightly greater (1-2%) than the available V p sonic velocities measured in situ. This difference can be explained by velocity dispersion as well as the effects of the different pressure-temperature conditions existing at depth. Therefore ultrasonic measurements are valuable for estimates of sonic and seismic velocities. Ultrasonic and sonic velocities increase with depth. Several studies link laboratory data directly to well log data. A study of North Sea shale properties is done by Horsrud [21]. Following Horsrud [21], Chang et al. [26] compared several empirical conversion formulas. A poor fit of the measured data is found and the main critic is that the relations were proposed to fit a subset of data; therefore they do work but not necessarily for all geological formations. Advanced sonic techniques allow for measurement of anisotropy in situ. Sayers et al. [29] used these advanced measurements in combination with literature correlations to make his predictions. Using these data, empirical correlations for prediction of shale mechanical properties have been developed. From these correlations, static mechanical properties can be predicted from various sources such as sonic wire line log and acoustic measurements on cuttings and at various stages in the process of drilling a borehole. The P-wave velocity in the shale is a key parameter in several of the correlations. These correlations can be used as an engineering tool to provide more reliable and more continuous estimates of mechanical properties of shales, keeping in mind that the validity of the correlations should be checked when used in other geological and geographical areas. Other sources of uncertainty also exist (e.g. core-damage effects and temperature effects). A study by Mossop [212] discusses the appropriate use of E s and E d in geomechanical analysis. The pore pressure diffusion time-scale is linked to stiffness. A large time-scale behaves undrained and is therefore stiffer (E d ) than a small diffusion and time-scale "softer" (E s ). This is linked to the depletion of the reservoir, and the suggestion is made to apply static parameters to the depleting reservoir and dynamic parameters to the surrounding formations. Where Mossop [212] links the stiffness difference between dynamic and static measurements to the diffusion of pore pressure King [1969] has shown that even for dry sandstones, there is a difference in static and dynamic stiffness. Difference from dry sandstone are related to the unconsolidated nature of the rock King [1969] used in his laboratory tests. As the pressure increased cracks closed and the static and dynamic stiffness converged. This suggests that the influence of the pore content is significant. This supports the paper by Mossop [212] stating that the proper geomechanical properties for a reservoir are the static properties as the reservoir depletion is caused by mobilising the pore content. In surrounding formations, the mobility of the pore content is restricted requiring dynamic properties. This suggests, to measure the overburden stiffness, only dynamic properties are required. and to measure reservoir stiffness, laboratory tests are require. 3-3 Geophysical measurements In order to obtain information about the subsurface, non-intrusive geophysical measurements are carried out in a drilled well. In general, geophysical measurements are non-intrusive allow- Master of Science Thesis Final T. de Gast

32 14 Geomechanical property determination from well log data ing for fast data acquisition. Logging the measured geophysical properties to the trajectory of the well relates the measured geophysical properties to depth. This is a well log. In this thesis, three types of geophysical measurements are used. Density logs to obtain the bulk weight of a material, sonic logs to obtain the sonic velocities and porosity logs to determine the sandstone properties. These measurements make use electricity, radiation and mechanical vibrations and are applied on different conversions to obtain the geophysical properties. During logging and drilling, the well is filled with a bore fluid. This fluid is used to cool the drill-bit, prevent borehole collapsing inward, prevent flow of formation fluids and bring cuttings to surface. The fluid is kept at a slightly higher pressure than the formation porefluids by control of the fluid density. Because of this pressure difference the fluid infiltrates to certain extend into the surrounding rocks. The penetration depth depends on different factors like pressure difference, permeability of the formation and viscosity of the fluid. Figure 3-1 is a representation of the invasion zones. The reservoir rock fluids are displaced by the well fluid (the flushed zone), a transition zone, where the well fluids did not fully displace the formation fluids and the un-invaded zone. The well fluids influence the geophysical measurements and these should be corrected for the presence of well fluids. Figure 3-1: Penetration depth of the well fluid, [after Serra, 1986]. T. de Gast Final Master of Science Thesis

33 3-3 Geophysical measurements Density logs A measurement of the bulk density of the formation is based on the reduction of gamma ray flux trough a medium. This reduction of gamma flux is due to Compton scattering. Compton scattering occurs when a gamma ray collides with an electron. During this collision, a part of the energy of the gamma ray is transferred to the electron. The higher the density of the material, the higher the amount of electrons in the material will be, and thus the higher the reduction of the gamma ray energy. Because this process is based on electron density, the measured values have to be corrected for bulk density. The bore fluid and the roughness of the well edge have an effect on the measured bulk and electron density. The density logs compensate for the mud cake and roughness of the well edge. A typical setup to compensate is to get two sets of senders and receivers with different spacing. This different spacing leads to a different penetration depth. A closed spaced sender and receiver has a low penetration depth while a larger space has a higher penetration depth. Thus, a comparison between these two measurements can correct for the mud cake and the roughness of the drilled well Sonic logs A measurement of the sonic velocity of a formation is based on the arrival time of a mechanically induced wave. The sonic tool emits a sonic signal which is detected at receivers further down the well. The sonic tool measures the time it takes for sound pulses to travel through the formation (Δt log ). The results of sonic measurements are in general displayed on a log in μs/m against depth. The formation travel time can be interpreted in terms of seismic velocity of the formation. While the sonic signal is travelling down the well, it reflects and refracts against the well wall, which changes the orientation of the seismic velocities. There are two seismic velocities which are important to the determination of geomechanical properties, these are the pressure wave velocity (V p ) and the shear wave velocity (V s ). While the V p wave has the same orientation as its propagation vector, V s is perpendicular oriented on the V p wave and its propagation vector. More recent tools measure V s in two directions. This gives the travel time of the sonic signal in three directions. A three directional sonic signal gives an insight into possible stiffness anisotropy. The sonic signal is influenced by well irregularities, inclination of the well, well fluids, (micro) cracks in the formation rock and the well wall. For these influences, the measured data has to be corrected. The depth of the investigation depends on the frequency of the sonic signal, the energy of the sonic signal and the distance between the transmitter and the receiver. In a well log, the sonic signal is generally measured in the invaded zone and over several decimetres into the well wall Porosity logs The density and neutron tool are both used to determine the porosity; they operate using different geophysical properties, where the density tool measures the bulk density by means of gamma ray and Compton scattering. The neutron tool measures the hydrogen density. Master of Science Thesis Final T. de Gast

34 16 Geomechanical property determination from well log data With a combination of both tools, different formation fluids can be identified. In gas bearing reservoirs, the recorded neutron porosity is lower and the bulk density is reduced. Compared to the response in a similar water/oil bearing formation, these effects can be significant depending on the gas saturation in the invaded zone. 3-4 Geomechanical properties Geomechanical properties can be obtained in different ways. One way is by laboratory testing, and another way is by converting geophysical logs to geomechanical logs using elastic theory. Most laboratory tests represent in situ conditions in a uniaxial compaction test. Generally two types of testing are carried out on selected reservoir rock: (i) Pore-pressure depletion, where the axial stress is kept constant and pore pressure and radial stress are varied (ii) Constant pore-pressure test, where the pore pressure is kept constant and the axial and the radial stresses are varied. These laboratory tests simulate stress conditions of a depleting reservoir under uniaxial conditions. In section 3-4-1, the part of the elasticity theory used to obtain geomechanical properties is presented. Because of the high frequency and small strains, the geomechanical properties obtained with these measurements are within the dynamic domain. To obtain geomechanical properties using elasticity theory, sonic velocity and bulk moduli are required. In this thesis, geomechanical properties obtained using sonic velocity and bulk moduli are called dynamic properties and are denoted with subscript d. In section 3-4-2, several empirical conversion functions have been presented. These conversion functions are obtained from laboratory studies where they have been tested under static boundary conditions. There are several types of conversion functions which make use of sonic velocity and porosity to derive geomechanical properties. In this thesis, geomechanical properties obtained using conversion functions (which have an empirical root) are called static properties and are denoted with subscript s Properties from well log data The theory of elasticity is a well-established and methods to obtain geomechanical properties have been included in many textbooks such as Wood [24]; Verruijt [26]; Fjar et al. [28]. Obtaining geomechanical properties from well log data using the elasticity theory has several advantages e.g. well logs contain a large set of data points. Compared to a laboratory test, obtaining a geomechanical property from the well log is cheap. However, as discussed in the literature review, geomechanical properties obtained from sonic velocities are best suited to describe geomechanical properties in non-reservoir formations. In this thesis, the following formulas using sonic velocities are used to obtain dynamic geomechanical properties: T. de Gast Final Master of Science Thesis

35 3-4 Geomechanical properties 17 E d = ρvs 2 (3Vp 2 4Vs 2 ) Vp 2 Vs 2 (3-1) E d = dynamic Young s modulus [MPa] ρ = bulk weight of the material [kg/m 3 ] V p = pressure wave velocity [m/s] V s = shear wave velocity. [m/s] ν d = V p 2 2Vs 2 2(Vp 2 V 2 s ) (3-2) ν d = dynamic Poisson s ratio [ ] V p = pressure wave velocity [m/s] V s = shear wave velocity. [m/s] Properties from laboratory data (conversion functions) Material samples retrieved from a reservoir can be tested in laboratories. During these tests, either the external pressure on the sample is increased or decreased, or the internal pressure in the sample is decreased or increased. These changes in pressure lead to deformations in the sample. From the combination of the pressure differences and deformations, geomechanical properties are derived depending on the constitutive model used in a geomechanical calculation. A vast amount of studies have tried to link the laboratory geomechanical properties to geophysical measurements resulting in a series of conversion functions. Table 3-1 includes an example of these conversion functions for sandstone and shales. These conversion functions have been used in this thesis to determine static geomechanical properties. Table 3-1: Conversion function shapes used in this thesis Sandstone Shales log(σ UCS )=log(a 1 ) a 2 φ log(σ UCS )= a 3 + a 4 log(a 5 V p a 6 ) log(σ UCS )= b 1 + b 2 log(b 3 V s ) E s = c 1 σ UCS E s = c 2 σ UCS E s = d 1 E d d 2 E s = d 3 E d d 4 ν s = e 1 ν d ν s = e 2 ν d ν s = f 1 + f 2 φ ν s = f 3 σ UCS = Unconfined Compressive Strength [MPa] φ = Porosity [-] V p = Pressure wave velocity [m/s] V s = Shear wave velocity [m/s] E = Young s modulus [MPa] ν = Poisson s ratio [-] a i,..., f i = Shell restricted constants [-] Master of Science Thesis Final T. de Gast

36 18 Geomechanical property determination from well log data 3-5 Analysis and discussion Geophysical properties are used to derive dynamic geomechanical properties. For stiffness, the geophysical properties sonic velocities and bulk modulus are used. However, the geophysical measurements in a well log are influenced by the roughness of the well and the well fluid used to keep the well; if the formation is porous the well fluid can enter the formation displacing the original pore content. The mechanical behaviour of a formation material depends on the solid matrix, the pore content and the mobility of the pore content. If the pore content is immobile, the material behaviour of the formation material is influenced by a combination of the pore content and solid matrix. The pressure wave of a sonic velocity measurement travels through the formation in such speeds that the pore content shows immobile behaviour. These dynamic properties can be used for non-reservoir layers with low permeability and thus low pore content mobility. For reservoir layers and layers with high permeability dynamic properties give (in general) an overestimation of the elastic geomechanical properties. To estimate the geomechanical properties for reservoir formations, static geomechanical properties are required. A static property depends on the solid matrix because the pore content in a reservoir is mobile. Static properties can be derived in laboratory tests. It is generally accepted that the static and dynamic properties are related and therefore conversion functions from dynamic properties to static properties are widely used. However, there is still a large uncertainty in obtaining proper static geomechanical values. And for each reservoir the static properties need to be determined using new laboratory tests on core material. The advantage of the dynamic properties is that there is a large amount of data available over the different formations. This large dataset gives an insight in the fluctuation of the geomechanical properties. The fluctuation of dynamic properties can be used to estimate the fluctuation of static properties. However, in the reservoir formations static properties are required to model the geomechanical behaviour. These properties are obtained from laboratory data. However, fluctuations in a laboratory dataset are not observed since the dataset is too small and often taken from different well logs. 3-6 Summary and conclusion With the use of geophysical measurements, geomechanical properties can be obtained. These geomechanical properties are obtained from small strains and high vibrations and result in undrained (dynamic) material properties. Dynamic material properties depend on the rock and its pore content and can be used to describe the stiffness of undrained material behaviour. Especially for formation outside of the reservoir, for the impermeable shale formations the dynamic properties can be used since the pore contents are immobile. For a reservoir formation, there is drained (static) material behaviour. These static material properties can be obtained from laboratory tests. There is a large amount of conversion functions for static material properties from geophysical measurements. Currently, there is not one available formula which is widely accepted to convert dynamic properties to static properties and thus core material of the reservoir still has to be tested in the laboratory. T. de Gast Final Master of Science Thesis

37 Chapter 4 Deterministic reservoir analysis 4-1 Introduction Before starting the analysis of the geomechanical material properties and their distribution in different formations (chapter 5) and being able to analyse the effect of stochastic geomechanical properties on the geomechanical response of a compacting reservoir (chapter 6), a deterministic analysis of the subsidence caused by a compacting reservoir is carried out in the current chapter. The subsidence of a reservoir depends on several factors. The three main factors in a geomechanical calculation are (i) the geometry of the reservoir (ii) the distribution of the pressure difference caused by fluid extraction from reservoir and (iii) the (constitutive) geomechanical properties of the reservoir. In this chapter, the importance and effect of the geometry of the reservoir and the distribution of the pressure difference in the reservoir and their effect on the subsidence is analysed. To predict the geomechanical response of a compacting reservoir, a reservoir model and field data is needed in order to build a geomechanical model. Shell has provided the following field and reservoir data which are presented in this chapter: Geometry of the reservoir and surrounding formations. Reservoir depletion pressure scenarios. Geophysical well log data. The analyses of the geomechanical response are done with the use of the in-house Shell program QuickBlocks (QB). QB calculates the geomechanical response assuming linear elastic material behaviour in a half space. The program is able to include the field geometry and pressure distribution in the reservoir in a geomechanical analysis. The geomechanical properties E en ν and an initial stress orientation in the formations are used to predict the stress and strain changes in field model. In this chapter, the geomechanical Master of Science Thesis Final T. de Gast

38 2 Deterministic reservoir analysis response used to illustrate the effects of the geometry and pressure distribution in a reservoir the subsidence at sea bed level is chosen. This will provide insight in the effect of different geometries and pressure distributions on a large domain. 4-2 Selected literature The prediction of subsidence caused by reservoir compaction is topic of many studies, the selected literature give an overview of different types of analysis. From linear elastic analytical solution to the computation of subsidence using finite elements able to model geological structures and pressure distributions in a reservoir. In several papers, Geertsma presents an analytical solution for a depleting reservoir in an isotropic linear elastic porous medium (reservoir). Uniaxial-compressibility coefficient of the reservoir rock can be determined from laboratory measurements Geertsma [1957]. The solution proposed by Geertsma is linear elastic. This solution does not take the presence of a hard basement rock and/or an extreme deformable reservoir rock into account Geertsma [1966]. For the compressibility value of the reservoir formation in Geertsma s solution the uniaxial-compressibility is used. This property can be obtained from laboratory tests as well as through V p and ρ well log data Geertsma [1973]. Because the reservoir compaction is calculated from the uniaxial-compressibility coefficient in an analytical axis symmetric analysis, the subsidence can only be calculated for a disk shaped reservoir with a large diameter compared to its height and a uniform depletion pressure. A Study in more complex pressure distributions and geology where the subsidence above oil producing formations was monitored during production and the monitoring data was linked to geomechanical calculations van der Knaap and van der Vlis [1967] showed that fluid extraction causes an increase in effective stress which in turn causes compaction. Subsidence bowls formed at surface above three extraction points followed the liquid production from the reservoir. This indicates that the principle source of energy in compaction is the energy by which the reservoir is producing. Implying that when working with complex pressure depletions e.g. during production in a reservoir, Geertsma s solution is not valid since in general there is no uniform pressure in a producing reservoir. A study in alternative reservoir shapes to Geertsma s disk solution, with inclined and anticlinal reservoirs looking at the three dimensional features (inclination, anticline, variable horizontal thickness) of a reservoir was executed by Suzuki et al. [24]. This study showed that three dimensional features have a significant effect on the subsidence for shallow reservoirs, as the reservoir depth increases the effect of the three dimensional features decreases. In addition to the general accepted linear elastic analytical solution by Geertsma, finite element programs have the advantage calculate more advanced geological structures, advanced geomechanical behaviour and coupled behaviour flow and strain behaviour between the reservoir rock and reservoir fluid leading to complex pressure fields. The drawback of these finite element programs is that they take a long time to calculate. Several studies are done in computing a three dimensional probabilistic reservoir model. The focus in these studies lie on predicting a reservoir model based on the available field data Massonnat [1999] with a combination of field data and regional data one is able to predict the geological heterogeneity Dronamraju et al. [25]. A large part of these studies include T. de Gast Final Master of Science Thesis

39 4-3 Case study 21 inversion techniques. These techniques can be used to constrain the stochastic realisations of reservoirs. To calculate elastic deformations of a complex pressure field in a layered medium, a study by Kuvshinov [27] provides a reflectivity method for geomechanical calculations. The difference with finite element methods is that the solution inside the reservoir layer should match the conditions at the reservoir boundaries and thus coupled problems cannot be taken into account. The main advantage of this method compared to Geertsma s solution and finite element programs is that complex geometries and pressure differences can be computed fast. The disadvantage compared to finite element programs is that there is only a linear elastic material behaviour. This theory has implemented in the in-house Shell program QuickBlocks (QB) which is used for analyses in this thesis. 4-3 Case study One of the issues related to subsidence is the clearance between an production platform and the sea level. A platform requires a certain space between the sea level and the operating deck. Based on a subsidence calculation the location of an operating platform can be determined. In order to create accurate predictions of subsidence caused by production from a reservoir. Different types of data are presented in this chapter. These data are part of a real case, have been provided by Shell and contain: well log data, geological boundaries and pressure depletion scenarios. With the provided data, the expected subsidence can be calculated. In this chapter different assumptions about the geometry and pressure distribution are made. This will show the difference in and importance of using geological models and depletion pressure scenarios. in figure 4-1 the calculated subsidence based on the provided data is shown. For the stiffness properties, the average Young s modulus and average Poisson s ratio are used. Master of Science Thesis Final T. de Gast

40 22 Deterministic reservoir analysis Figure 4-1: Expected subsidence using the provided dataset 4-4 Field data All methods and calculations in this thesis have been tested or illustrated using a data set belonging to an existing reservoir. From the provided data set of the reservoir, five well logs have been provided. These well logs contained density, sonic, porosity and gamma radiation measurements. The well log data have been included in figures 4-2 until 4-5. The well logs have contained several geological formations. The depth of these formations has been provided by a geologist per well log and has been taken into account in the analysis. Systematic noise has been removed from the well log data before this data was provided. T. de Gast Final Master of Science Thesis

41 4-4 Field data 23 5 Well log 1 Well log 2 Well log 3 Well log 4 Well log Depth [m] Density [g/cm3] Figure 4-2: Density log Master of Science Thesis Final T. de Gast

42 24 Deterministic reservoir analysis 5 Well log 1 Well log 2 Well log 3 Well log 4 Well log Depth [m] DTC [us/ft] Figure 4-3: Pressure wave velocity log T. de Gast Final Master of Science Thesis

43 4-4 Field data 25 5 Well log 1 Well log 2 Well log 3 Well log 4 Well log Depth [m] DTS [us/ft] Figure 4-4: Shear wave velocity log Master of Science Thesis Final T. de Gast

44 26 Deterministic reservoir analysis 5 Well log 1 Well log 2 Well log 3 Well log 4 Well log Depth [m] porosity [ ] Figure 4-5: Porosity log T. de Gast Final Master of Science Thesis

45 4-4 Field data 27 Geology and geometry The seabed is situated at 12 m below sea-level. The information about the geology and geometry of the reservoir was provided by geologists. The overburden above the sandstone reservoir are sealing shale formations in these shale formations there are layers of sandstone. The reservoir is divided into two parts, an upper part and a lower part. The sealing layers have been divided into three different formations. In between the reservoir formation there is a thin sealing layer which divides the pressures of the two reservoirs. The reservoir is connected to an aquifer. In figure 4-6 a crossection of the different formations are shown. In figures 4-7 until 4-12 the shapes of the formation horizons are shown these figure it can be read that the reservoir is inclined and has an oval shape. The horizons between the three formations are less curved as a formation is shallower. The seabed floor has some peaks and valleys. 5 1 z direction [m] Seabed Formation boundary 1 Formation boundary 2 Top of reservoir formations Reservoir boundary Bottom of reservoir formations x direction [m] x 1 4 Figure 4-6: For this reservoir, five boundaries have been defined. The first surface is the sea bed floor and top of the model, bottom two surfaces are the horizons above the reservoir Master of Science Thesis Final T. de Gast

46 28 Deterministic reservoir analysis Top formation boundary Depth below sea leve y direction [m] x 1 1 Bottom formation boundary 15 y direction [m] x direction [m] x 1 Figure 4-7: Boundaries of formation 1 Top formation boundary Depth below sea leve 15 y direction [m] x 1 18 Bottom formation boundary y direction [m] x direction [m] x 1 Figure 4-8: Boundaries of formation 2 T. de Gast Final Master of Science Thesis

47 4-4 Field data 29 Top formation boundary Depth below sea leve y direction [m] x 1 Bottom formation boundary y direction [m] x direction [m] x 1 Figure 4-9: Boundaries of formation 3 Top formation boundary Depth below sea leve y direction [m] x 1 Bottom formation boundary y direction [m] x direction [m] x 1 Figure 4-1: Boundaries of reservoir 1 Master of Science Thesis Final T. de Gast

48 3 Deterministic reservoir analysis Top formation boundary Depth below sea leve 15 y direction [m] x 1 29 Bottom formation boundary 15 y direction [m] x direction [m] x 1 Figure 4-11: Boundaries of reservoir 2 Top formation boundary Depth below sea leve 15 y direction [m] x 1 Bottom formation boundary y direction [m] x direction [m] x 1 Figure 4-12: Boundaries of formation 4 T. de Gast Final Master of Science Thesis

49 4-4 Field data 31 Pressure distribution Reservoir modelling, where the flow and pressure distribution of the reservoir fluids was computed was used to calculate the different pressures distribution in the reservoir at different times. These calculations result in several different pressure distributions (pressure fields) within the reservoir. From these different pressure fields one is selected to be analysed in this chapter, the pressure fields shown in 4-13 until In this figure, red is a low depletion pressure and blue is a high depletion pressure. The depletion pressure is given in pressure difference compared to the original pressures before extracting. It can be seen that in the curved (high) end, the pressure difference is the lowest whereas in the middle (near the extraction) the pressure difference is highest. These pressure differences are placed in the corresponding reservoir formation and represent the depleting reservoir. Outside of these depletion pressure fields there is no depletion modelled, leading to high depletion pressures near the edge of the modelled boundary. 15 Pressure distribution first reservoir Pressure [MPa] y direction [m] x Pressure distribution second reservoir 15 y direction [m] x direction [m] x 1 4 Figure 4-13: Pressure field, relisation 1 25 Master of Science Thesis Final T. de Gast

50 32 Deterministic reservoir analysis Pressure distribution first reservoir Pressure [MPa] y direction [m] x 1 Pressure distribution second reservoir y direction [m] x direction [m] x 1 Figure 4-14: Pressure field, relisation 2 Pressure distribution first reservoir Pressure [MPa] y direction [m] x 1 Pressure distribution second reservoir y direction [m] x direction [m] x 1 Figure 4-15: Pressure field, relisation 3 T. de Gast Final Master of Science Thesis

51 4-4 Field data 33 Pressure distribution first reservoir Pressure [MPa] y direction [m] x 1 Pressure distribution second reservoir y direction [m] x direction [m] x 1 Figure 4-16: Pressure field, relisation 4 Pressure distribution first reservoir Pressure [MPa] y direction [m] x 1 Pressure distribution second reservoir y direction [m] x direction [m] x 1 Figure 4-17: Pressure field, relisation 5 Master of Science Thesis Final T. de Gast

52 34 Deterministic reservoir analysis Pressure distribution first reservoir Pressure [MPa] y direction [m] x 1 Pressure distribution second reservoir y direction [m] x direction [m] x 1 Figure 4-18: Pressure field, relisation 6 Pressure distribution first reservoir Pressure [MPa] y direction [m] x 1 Pressure distribution second reservoir y direction [m] x direction [m] x 1 Figure 4-19: Pressure field, relisation 7 T. de Gast Final Master of Science Thesis

53 4-5 Geomechanical calculation of the reservoir Geomechanical calculation of the reservoir The geometry of the reservoir and the pressure field have been presented in figures 4-6 and This data will be used as an input for a geomechanical calculation. This geomechanical calculation calculated the subsidence of the seabed using linear elastic material behaviour. The calculations are done in QB which uses the calculation method proposed by Kuvshinov [27]. QB computes the displacements at specified points; therefore to the vertical sides of the model there are no boundary conditions other than an infinite elastic medium. At the bottom of the QB model there is a fixed boundary. In order to prevent artificial boundary stiffness at the bottom of the model an extra boundary has been added deeper than the known well log data at a depth of 1 m. Artificial boundary stiffness occurs when the stiffness of the bottom of the model interacts with the reservoir behaviour, this can lead to higher displacements at surface. The initial orientation of the stresses in the geomechanical model has an effect on stress related geomechanical behaviour. This effect has not been taken into account in this thesis and therefore the Cartesian directions correspondent with the primary stress directions and no anisotropic stress is taken into account Geomechanical properties The constitutive model used in the geomechanical calculation is linear elastic. Using the static stiffness conversion functions presented in chapter 3 on the geophysical well log data of well log three has led to the in figure 4-2 presented stiffness log. In the stiffness log the different formations, corresponding to the formations presented in figure 4-6 been highlighted using different colours. The seabed is located at 12 m below sea level however there is no data available from the seabed until 7 m below seabed. Therefore the properties obtained for the first formation are assumed to be representative up-to the seabed floor. The difference in the conversion functions of sandstones and shales can be seen in figure 4-2. Where sandstone, the E s based on the porosity, is variable, the shale, the E s based on V s is almost linear and barely varies in depth. The distinctions between sandstone and shales is obtained from geologists. From the data presented in figure 4-2, the geomechanical properties for the analysis are used. The properties derived in this chapter, are the average value for each (reservoir) formation from the well log. The average properties are presented in table 4-1. For the 7 m from seabed until the first formation which is not accounted for in the well log, the properties of formation 1 are used. The distinction between the different material types is neglected. Table 4-1: Average (static) geomechanical properties E ν [MPa] [-] Formation Formation Formation Reservoir Reservoir Formation Master of Science Thesis Final T. de Gast

54 36 Deterministic reservoir analysis Depth [m] Formation 1 28 Formation 2 Formation 3 Reservoir 1 Reservoir 2 Formation Young s modulus [GPa] (a) E s 18 2 Formation 1 Formation 2 Formation 3 Reservoir 1 Reservoir 2 Formation 4 22 Depth [m] Poisson s ratio[ ] (b) ν s Figure 4-2: In this well log the stiffness parameters E s and ν s are presented. Different colours correspond with different formations T. de Gast Final Master of Science Thesis

55 4-5 Geomechanical calculation of the reservoir Geomechanical response Deterministic subsidence of four different compacting reservoirs have been analysed. The calculations were done using the QB program. In order to obtain insight in the subsidence response of a compacting reservoir, the effect of different methods to model a reservoir are calculated. Four different reservoir models have been used. The four models have different reservoir geometry and different pressure distributions. in table 4-2 the different reservoir models are listed. Table 4-2: Four reservoir cases rectangular plane area reservoir actual reservoir shape Uniform pressure Case 1 figure 4-21 Case 3 figure 4-23 Actual pressure Case 2 figure 4-22 Case 4 figure 4-24 For the analysis the total depletion pressure, in the system has been kept in the same order of magnitude for each reservoir. However, despite of having an exact match between the different depletion pressures of the systems, the difference in subsidence response is striking. The response of the different methods and abstraction levels are shown in figures 4-21 until The two properties which give the most information about the subsidence are the maximum subsidence and the volume of subsidence. The volume of subsidence is determined by the compaction of the reservoir and the maximum subsidence is determined by the location where the most compaction is and where this compaction occurs in respect to the sea bed floor. These properties of the calculated subsidence are presented in table 4-3. Table 4-3: Geomechanical subsidence response figures 4-21 until 4-24 Figure Maximum subsidence [m] Volume of subsidence [m 3 ] 3.2e8 3.2e8 3.1e8 3.2e8 Table 4-3 give the geomechanical response parameters of the four different methods. The volumes of subsidence are in the same order of magnitude however the maximum subsidence differs. The volume of subsidence is dictated by the stiffness of the reservoir and the pressure difference in the reservoir. However, the maximum subsidence is influenced by local geometry and depletion pressure differences. A closer look at the different responses shows how the depletion pressures and geometry work together to influence the shape of the subsidence bowl. The rectangular plane area reservoir, with horizontal formation boundaries and uniform depletion pressure, subsidence is shown in figure 4-21 and shows a uniform subsidence bowl. Because there is no local difference in the geomechanical model besides the absence of depletion pressure outside of the reservoir boundaries, the subsidence is uniform and inside the subsidence bowl there are no local variations of subsidence. The solutions of these uniform pressure distributions and simple shaped reservoirs are identical to solutions obtained from analytical solutions. The subsidence bowl shown in figure 4-22 is not uniform but concentrated in the middle of the reservoir. The geometry of the reservoir is still in a rectangular plane with an increased depletion pressure in the middle of the reservoir. This increased depletion pressures in the Master of Science Thesis Final T. de Gast

56 38 Deterministic reservoir analysis middle of the reservoir causes an increased compression in the middle of the reservoir. This increased compression leads to the increased subsidence in the middle of the subsidence bowl. In figure 4-23 the subsidence bowl is concentrated on one side. The geometry of the reservoir is oval and inclined, with a uniform depletion pressure. The inclination of the reservoir causes the compression to occur closer to the seabed floor at the side where the reservoir is closest to the surface. Because the compression is closer to the seabed floor the volume of the compressed reservoir is projected onto a smaller area leading to a larger subsidence near the high end of the reservoir. The combination of the inclined reservoir and the local increase in depletion pressure work together to form the shape of the subsidence bowl. Figure 4-24 shows both the peak of figure 4-22 and the increased subsidence near one side of Effect of the inclination of the reservoir is less visible, this is caused by a decreased depletion pressure,compared to the uniform case in figure 4-21, near the inclination. From figures 4-21 until 4-24 it is seen that the distribution of pressure and modelling of geometry has a large impact on the subsidence bowl. If there is no local pressure change or no local geometry change there is a uniform subsidence bowl. As soon as the geometry changes the subsidence bowl is deeper at the point where the reservoir is closest to the surface. While keeping the geometry horizontal, a local increased pressure difference corresponds with the point where the subsidence is increased. The subsidence calculated from a combination with both complex geometry and pressure has an increased subsidence near the point where the reservoir is closest to the surface and an increased subsidence near the extraction point. This shows that three dimensional effects have a large impact on the geomechanical response. The subsidence predictions show that, in subsidence predictions three dimensional effects should not be neglected. And in a reliability analysis for a global geomechanical mechanism, the lateral variation of material properties, the uncertainty in reservoir boundaries and the reservoir pressure differences are factors which should be implemented. However, from literature Suzuki et al. [24] it is known that as the reservoir lies deeper under the surface, lateral variation and local pressure difference average out. T. de Gast Final Master of Science Thesis

57 4-5 Geomechanical calculation of the reservoir 39 Figure 4-21: Average depletion pressure average geometry Master of Science Thesis Final T. de Gast

58 4 Deterministic reservoir analysis Figure 4-22: Local depletion pressure average geometry T. de Gast Final Master of Science Thesis

59 4-5 Geomechanical calculation of the reservoir 41 Figure 4-23: Average depletion pressure local geometry Master of Science Thesis Final T. de Gast

60 42 Deterministic reservoir analysis Figure 4-24: Local depletion pressure local geometry T. de Gast Final Master of Science Thesis

61 4-6 Summary and conclusion Summary and conclusion In this chapter, the field data of an example reservoir have been presented. The presented example reservoir illustrates the effects of the variability on the geomechanical response of a reservoir and enables test the methods presented in this thesis on real data. In addition to the reservoir data, a subsidence analysis using the QB program has been carried out. In this subsidence calculation the effect of different approaches in modelling a reservoir geometry and pressure difference have been shown. The result of this analysis shows the differences in response caused by differed types of approach in modelling the formation boundaries and pressure distributions. It has been shown that three dimensional reservoir structures and complex pressure fields are important factors to predict the geomechanical (subsidence) response. A further analysis of the variability of the reservoir pressure distribution and reservoir geometry is not part of this thesis but it is clear these aspects of the reservoir simulation should not and cannot be neglected in a probablistic geomechanical analysis. Master of Science Thesis Final T. de Gast

62 44 Deterministic reservoir analysis T. de Gast Final Master of Science Thesis

63 Chapter 5 Statistical data analysis 5-1 Introduction Information gained from the field measurements is intrinsically varied: geological formations and material properties vary as well as systematic errors which occur in the measurement techniques. Various measuring techniques are used to obtain a wide range of material properties. These measurements are often carried out in drilled wells and give therefore a vertical profile of the material properties. Measurement techniques have been discussed in chapter 3. The effect of variable material properties on geomechanical analyses will be further discussed in chapter 6. This chapter presents a method to describe the variability of measured material properties as a type of mathematical function known as a Probability Density Function (PDF). A PDF can be used as input property into a reliability software package e.g. the in-house Shell COntainment Reliability Analysis (CORA) package and describes the uncertainty of a given value. In chapter 4, the example dataset was used to obtain an appropriate PDF for CORA. From this well log, the density and sonic velocities which were measured were converted to the Young s modulus. The area of fit test is introduced to select a proper PDF to represent the uncertainty of a property. After the introduction to represent material properties in a PDF, methods to detect and remove data trends are discussed. Removing trends in data can reduce the uncertainty of a material property. In general, any PDF can be used to represent the data. The methodology presented to test accuracy of the PDF to the data can be used for each chosen PDF. In this thesis, a limited number of PDFs have been tested and analysed for two purposes (i) to demonstrate how different PDFs can be used and investigated (ii) to maintain compatibility with CORA which is used for the probabilistic calculations. The analysis of other PDFs than presented in this chapter is open to further investigation and out of scope of this thesis. Master of Science Thesis Final T. de Gast

64 46 Statistical data analysis 5-2 Selected literature Statistical data analysis is done in several disciplines. Each discipline has their own use for using statistics. For engineering purposes, especially for engineering with natural materials, statistics can be used to describe the natural variability of the encountered materials. A simple and descriptive overview of uncertainty and risk analysis in the domain of Business Dynamics is given by Rodger and Petch [1999]. For Several PDF properties are given including examples where these PDF apply. In addition to presenting the PDFs, the R 2 test which test linear correlation is presented and the outcome of the R 2 tests are discussed. A study to limit the effects of (mainly) systematic noise from in situ measurement has been done by Shier [24]. The goal of this study is to remove systematic inaccuracies and not to remove the genuine lithological anomalies. Removing noise from the measurements is a step that has to be done before addressing geomechanical statistics. In the field of civil engineering, more extensive studies to statistical material properties for soft soils are done. The variability of soil in Phoon et al. [1999] is evaluated as a function of inherent soil variability, measurement error and conversion uncertainty. The contribution of these components to the variability depends on the site conditions, equipment used for measurements, control during testing and the transformation model. For studies where random field generators are used, the inherent material variability and scale of fluctuation are important parameters. Vertical scales of fluctuations are obtained from vertical measurements e.g. Cone Penetration Test (CPT) or well-log data. Lateral scales of fluctuations are harder to obtain. For shallow depths, they can be obtained by placing multiple close spaced CPT tests. For large geomechanical problems carrying out this many CPT tests are economical less promising. One method to obtain a lateral scale of fluctuation could be by means of seismic inversion. However, to the knowledge of the author this has not been researched in subsidence predictions. Overall, the interpretations of geomechanical field data are far from consistent. In a study to the variability of engineering judgement Bond [211], over a hundred professional engineers were asked to make an estimate of a geomechanical parameter based on CPT and SPT data. Bond found little agreement in the responses. This is an argument in favour of using a uniform approach to assess geomechanical parameters. Geostatistics are also applied in the petrophysical industry. In the study of ario Atencio et al. [2] three dimensional seismic data and geostatistical methods were used to select sites where exploration wells were drilled. The geostatistical methods have proven to be a valuable tool in the selection of these sites. 5-3 Example data set In this chapter, the method to obtain a PDF from data and test which PDF should be chosen is explained using example dynamic Young s modulus (E d ) data from well log 3. From the log of E d stochastic material properties are derived for one of the six formations formation. The process to derive a PDF from a data set x i is illustrated. The formation boundaries are based on the geologic boundaries provided with the field data presented in chapter 4. The T. de Gast Final Master of Science Thesis

65 5-4 Deriving a PDF from data 47 well log is presented in figure 5-1. In this figure the Young s modulus is derived from V p, V s and ρ data. Comparing the formations above and below the 27 m boundary shows that the variability of the data above the boundary is a lot lower than the variability below the boundary Formation 1 Formation 2 Formation 3 Reservoir 1 Reservoir 2 Formation 4 22 Depth [m] Young s modulus [GPa] Figure 5-1: The data set x i processed in this chapter is the Young s modulus from the well log data three. In this chapter only formation two is analysed, this data are presented in green Based on the geological structure provided by geologists, the boundaries of the example formation used in this chapter to illustrate are at the depths of 1986 m and 2529 m. The boundaries of the different formations and reservoirs are shown in figure 5-1. The part which is analysed is the second formation, this formation lies above the boundary of 27 m and has a relative low variability compared to the (reservoir) formations below the 27 m boundary. 5-4 Deriving a PDF from data Geological studies, seismic surveys and well log data provide information about the properties of geological formations. To study the reliability of geomechanical responses, the geomechanical variability of each formation can be assessed. To assess the variability of a formation, several steps are taken. (i) The formation and its boundaries are defined. (ii) Data of the formation is collected and an appropriate PDF is selected. (iii) The correlation coefficient of Master of Science Thesis Final T. de Gast

66 48 Statistical data analysis the data is calculated. These steps are detailed in the sections below Displaying selected data The distributed data shown in figure 5-2 can be converted to a histogram. This histogram is then approximated by a PDF and used in paragraph to test the accuracy of a PDF. A histogram is a graphical method that counts the number of data points in a given interval and converts this to a bar. The height of the bar is equal of the counted data normalised by the total number of data points. These intervals are called bins and by normalising the data, the area of all bins combined is equal to one. For small amounts of data, the number of bins may affect the accuracy test of a PDF. Baecher and Christian [23] suggested an analytical value for small amounts of data (equation 5-1). This equation assumes a normal distribution and is in general suitable for distributions up to 2 samples, [Hyndman, 1995]. As an alternative equation 5-2 is proposed, this equation estimates the width of the bin based on the amount of samples and the calculated standard deviation. k =1+3.3log 1 (n) (5-1) h =3.7σn 1/3 (5-2) k is the amount of bins,n is the number of data points, σ is the calculated standard deviation and h is the bin width. The amount of bins should be enough to capture the peaks and the weight of the data mass. Equation 5-2 is the better estimate for larger sample sizes. The binwidth provided by equation 5-2 is used to derive the histogram from the data in the selected formation in figure Formation 2.7 Formation Depth [m] Normalised frequency [ ] Young s modulus [GPa] (a) Data Young s modulus [GPa] (b) Histogram Figure 5-2: The data selected in figure 5-1 is repeated in (a) and then converted to the normalised histogram in (b). A visual interpretation of the histogram in figure 5-2 (b) provides a first estimation of which would PDF fits best to the data. Because the first bin of the histogram in figure 5-2 has a value of.45 and the last few bins have a value near., the expected PDF is either a log-normal distribution or a truncated normal distribution. T. de Gast Final Master of Science Thesis

67 5-4 Deriving a PDF from data Probability density functions Four different PDFs are presented in this chapter, these four PDFs are embedded in CORA. These PDFs are : (i) a uniform distribution (ii) a normal distribution (iii) a log-normal distribution (iv) a truncated normal distribution. To determine the input for each PDF, four properties from the data set x i, with size n, should be derived. These four properties are: the minimum value, equation 5-3; the maximum value, equation 5-4; the average of all values, equation 5-5; the deviation of the mean, equation 5-6; a = min (x i ) (5-3) b = max (x i ) (5-4) μ = 1 n x i (5-5) n i=1 σ 2 = 1 n (x i μ) 2 (5-6) n i=1 These properties are derived from the example formation data set shown in figure 5-2 (a) and have the following values, it should be noted that these values vary for each individual dataset and subset: a = 1.35 GPa b = 9.36 GPa μ = 2.64 GPa σ =.9 GPa The derived values are used to compute the four PDFs shown in figures 5-3 until 5-6. Using the minimum a and maximum b, the uniform probability density function can be obtained, without disregarding any data, the uniform probability density function is given by equation 5-7. f(x) = { 1 b a a x b x<aorx>b (5-7) In figure 5-3 the uniform probability density function is plotted over the histogram. The uniform PDF has a poor fit to the data, the high peak at the left side of the histogram is neglected and the tail at the right side is overestimated. This leads to an overestimation of the material property. For this material property this will mean that the stiffness is overestimated and compaction of the formation is underestimated. Master of Science Thesis Final T. de Gast

68 5 Statistical data analysis.7.6 Formation 2 Uniform distribution Normalised frequency [ ] Young s modulus [GPa] Figure 5-3: The corresponding uniform probability density function Using the average μ and standard deviation σ, the normal probability density function can be obtained. The normal distribution function is given by equation 5-8. f(x) = 1 σ (x μ) 2 2π e 2σ 2 (5-8) In figure 5-4, the normal probability density function is plotted over the histogram. The normal PDF overlaps with the peak of the histogram. However, the tops of the two left histogram bins are not covered by the normal PDF and low values are covered from the values 6. to 1. GPa. This can lead to an unreasonably low estimate of the material property, an underestimation of the Young s modulus leads to an overestimation of the material deformation which leads to an overestimation of subsidence. Using the average μ and standard deviation σ, the log normal probability density function can be obtained. The log normal distribution function is given by equation To correct for the log normal scale, the correction equations 5-9 and 5-1 have been applied to μ and σ. In addition to the corrections, a shift can be introduced to tune the distribution to the histogram. ) σ log = ln (1+ σ2 μ 2 (5-9) μ log = ln(μ) 1 2 σ log (5-1) T. de Gast Final Master of Science Thesis

69 5-4 Deriving a PDF from data Formation 2 Normal distribution Normalised frequency [ ] Young s modulus [GPa] Figure 5-4: The corresponding normal probability density function f(x) = 1 (x ɛ) 2πσ 2 log e (ln(x ɛ) μ log ) 2 2σ log 2 (5-11) In figure 5-5, the log normal probability density function is plotted over the histogram. The log normal PDF covers the tops of the bin at the sides of the histogram. There is a little area of the log normal PDF outside of the minimum and maximum of the histogram and shape of the log normal PDF follows the tops of the bins. Using the minimum a, maximum b, average μ and standard deviation σ, the truncated normal probability density function can be obtained. The truncated normal function is given by equation To correct for truncation and keep the area equal to one, equations 5-14 until 5-17 are applied. ξ = x μ σ α = a μ σ β = b μ σ φ(u) = 1 exp ( 1 ) 2π 2 u2 (5-12) (5-13) (5-14) (5-15) Master of Science Thesis Final T. de Gast

70 52 Statistical data analysis.7.6 Formation 2 Lognormal distribution Normalised frequency [ ] Young s modulus [GPa] Figure 5-5: The lognormal probability density function Φ(v) = erf ( v 2 ) (5-16) erf(w) = 2 π w e t2 dt (5-17) f(x) = { 1 σ(φ(β) Φ(α)) φ(ξ) a x b x<aorx>b (5-18) In figure 5-6, the truncated normal probability density function is plotted over the histogram. The advantage of truncating the normal PDF is that there is no area of the truncated normal PDF. However, like the normal PDF, the bins at the left of the histogram have a poor fit compared to the log normal PDF. T. de Gast Final Master of Science Thesis

71 5-4 Deriving a PDF from data Formation 2 Truncated normal distribution Normalised frequency [ ] Young s modulus [GPa] Figure 5-6: The corresponding truncated normal probability density function Area fit test To test the fit of the PDFs on the data set, the area fit test is used. This test compares how well the area of a PDF matches the area of the histogram. The area fit test is given in equation F fit = 1 n 2 (2 A outsidehistogram A i,(pdf) A i,(histogram) ) (5-19) i=1 In which, A outsidehistogram is the area of the PDF outside of the histogram, i gives the bin number, A i,(pdf) is the area from bin edge to bin edge of the PDF and A i,(histogram) is the area of the bin. The area fit test gives values of fit-coefficient (F fit )fromzerotoone. A value of zero means that the histogram and PDF do not match, a value of one means that the histogram and PDF have a perfect match. Detailed interpretation values are suggested in table 5-1. Figure 5-7 plots the test values for the four PDFs derived from our data set. Table 5-1: Level of fit (F fit ) F fit level of fit. No.65 Low.75 Medium.85 Good.95 Very good 1. Perfect Master of Science Thesis Final T. de Gast

72 54 Statistical data analysis Area fit test [ ] Uniform Normal Lognormal Truncated normal Figure 5-7: Goodness of fit test In section 5-4-1, both the truncated normal and log normal distribution were on visual interpretation suggested to have good fits on the data set. Based on the area fit test, comparing the log normal distribution and the truncated normal the log normal distribution has the better fit to the data. 5-5 Linear correlation of data Geomechanical properties are often linearly correlated for instance the E d and ν d of the example formation as shown in figure 5-8. A linear function can approximate two material properties with first order accuracy. A linear function can be calculated using the least squared estimate method see(equation 5-2). [ ] c = d n n y i i=1 n 1 n y i x i i=1 i=1 n yi 2 n (5-2) xy i i=1 i=1 Where c is the offset for the linear function and d is the linear function, the linear function is provided by equation f(x i )=c + dx i (5-21) In the following example the linear correlation between Young s modulus x i (used before in this chapter) and Poisson s ratio y i is calculated for part of the data and presented in figure 5-8. T. de Gast Final Master of Science Thesis

73 5-5 Linear correlation of data 55.5 Formation 2 linearisation of data.1.15 Poisson ratio [ ] Young s modulus [GPa] Figure 5-8: Linearization of data using least squared estimate. where c =.49 and d =.2 To compute how well the linear function fits the data set, the linear correlation is computed. The linear correlation is computed using the R squared test given by equation To compute the R squared test, the sum of squares (SS tot ) and the residual sum of squares (SS res ) have to be calculated using equations 5-23 and μ y = 1 n y i (5-22) n i=1 n SS tot = (y i μ y ) 2 (5-23) i=1 n SS err = (y i f(x i )) 2 (5-24) i=1 R 2 =1 SS err (5-25) SS tot The range of the results for the R squared test ranges from zero to one. A value zero implies that there is no linear correlation. A value of one implies a perfect linear correlation. In chapter 3, several linear correlations are given. This thesis is not aimed at establishing significance levels to the correlations. However it should be noted that in geomechanical analyses often linear correlation is already assumed for R 2 values from.5 and higher. For the example data set, the Young s modulus and Poisson s ratio have a linear correlation coefficient of.98 [-]. This value can be used as an input for geomechanical reliability calculations. For an reliability analysis, the correlation of these properties should, if possible, be inserd in the stochast. Master of Science Thesis Final T. de Gast

74 56 Statistical data analysis Formation boundaries Before starting geomechanical calculations, the geological structure is derived from geological studies: material retrieved during the drilling of the wells and seismic surveys. The geological structure boundaries for each formation are defined. An appropriate PDF is selected for each formation. It should be noted that, if the data suggest an extra formation boundary, e.g. a sudden shift in the data, this formation boundary should be added to the geomechanical calculation. To investigate if extra boundaries are required, there are two options: (i) a manual reinterpretation of the available data (ii) a numerical reinterpretation of the available data e.g. using the deflection points of a high order least squares polynomial. A detailed study of numerical methods to identify formation boundaries is beyond the scope of this thesis. However, an attempt to automatically identify trends in the data by use of a high order polynomial is made. automatic boundary detection In order to detect multiple trends in the geomechanical properties, a high order polynomial is computed trough the dataset by means of a least squared estimate. This polynomial follows the general trend of the data. Because this polynomial follows the data changes in the data can be identified by identifying changes in the polynomial. In figure 5-9 a 6th order polynomial is plotted trough the data. The inflection points, the points where the polynomial changes direction, are presented. Depth [m] Formation 1 Formation 2 Formation 3 Reservoir 1 Reservoir 2 Formation 4 Polynomial inflection points Young s modulus [MPa] x 1 4 Figure 5-9: Sixth order Polynomial fitted through the E d dataset The changes in trend identified by the inflection points, are similar to the global changes in the data trends. However, the local changes in the data are not identified by this method. Higher order inflection points do not increase the precision of these inflection points. Because the polynomial is fitted to the data, as the order of the polynomial increases the precision of near the top and bottom of the data is decreased. Because of this type of inaccuracy the method is not further developed in this thesis. T. de Gast Final Master of Science Thesis

75 5-6 Distribution of properties 57 One option to improve the accuracy of an automatic boundary detection is by implementing spline interpolation to identify local trends, spline interpolation makes use of the individual data points, However, because the data points are scattered this type of analysis is expected to be to precise. One option which combines the polynomial fit and the spline interpolation is the smooth spline interpolation. In the smooth spline interpolation a weight of influence can be applied to either the polynomial or the spline interpolation. This method could allow identifying the local trends while disregarding the local variation. 5-6 Distribution of properties The method presented in this chapter to select the PDF of is applied to the geophysical material properties measured in the different formations. The results of the analyses are presented in table 5-2. In the data analyses the linear trend has been calculated for the different formations and this trend has been removed. An example of selecting the linear depth trend and removing it from the well log E d has been presented in figure Formation 1 Formation 2 Formation 3 Reservoir 1 Reservoir 2 Formation 4 Linear trend Formation 1 Formation 2 Formation 3 Reservoir 1 Reservoir 2 Formation 4 Depth [m] 24 Depth [m] Young s modulus [MPa] x 1 4 (a) Linear trend explanation Young s modulus [MPa] x 1 4 (b) linear trend removed Figure 5-1: Linear trend removal In table 5-2 the truncated normal distribution and the lognormal distribution are the two most common PDFs to have the highest area of fit. However, the difference between the log normal, truncated normal and normal distribution is not big And in most cases either of three distributions can be chosen to represent the data. Making use of the different PDF attention should be given to the fit coefficient, as mentioned a fit coefficient of.5 is equal to only 5% overlap between the PDF and the histogram. It is advised when selecting a PDF, to only use a fit coefficient of.85 or higher. The data can be manipulated to improve the fit coefficient for the different PDFs or reduce the standard deviation the linear depth trend has been removed from the data. If the trend is removed from the data, by manipulation of data this manipulation should be added back into to the geomechanical discretization. This will reduce uncertainty while keeping the information form the well log. The effect of removing linear trends to reduce uncertainty is shown in table 5-2. The changes in the fit coefficient and reduction coefficient of the standard deviation are shown. It is clear from this analyses that not all removal of the linear depth Master of Science Thesis Final T. de Gast

76 58 Statistical data analysis trend is effective. For instance, in formation 2, the standard deviation of the dynamic Young s modulus and Poisson s ratio does not change however the fit coefficient does increase. Table 5-2: Reduction coefficient in standard deviation ρ V p V s Φ E d ν d E s ν s Original data, fit coefficient Formation 1.95 L.76 T.75 L.96 T.84 L.79 L.37 N.55 U Formation 2.88 L.87 T.85 L.88 T.88 L.85 T.63 T.84 T Formation 3.86 N.86 L.9 T.8 L.86 L.68 T.48 U.66 T Reservoir 1.78 L.75 T.78 L.72 T.76 T.76 L.27 T.77 L Reservoir 2.71 T.68 T.83 T.75 L.68 L.68 T.41 U.62 T Formation 4.84 L.74 L.77 L.83 T.78 T.81 L.86 T.81 L Linear trend removed, fit coefficient Formation 1.96 L.77 T.77 T.94 T.86 T.79 T.68 T.61 T Formation 2.91 L.87 T.83 T.88 L.86 L.85 T.82 T.82 L Formation 3.91 L.89 T.9 T.84 L.79 L.72 T.62 T.77 T Reservoir 1.92 L.66 T.88 T.76 T.84 L.73 L.75 L.73 L Reservoir 2.82 T.7 T.74 T.86 L.69 L.74 T.64 T.72 T Formation 4.89 L.84 L.78 L.87 T.8 T.88 T.9 T.88 T Reduction coefficient of the standard deviation Formation Formation Formation Reservoir Reservoir Formation F fit PDF O >.95 N = Normal distribution O >.85 L = Lognormal distribution O >.75 T = Truncated normal distribution O >.65 U = Uniform distribution O <.65 As mentioned, not all fit coefficients increase after removal of the linear trend. This is among others caused by extreme values which shift as the linearization is removed but are not enough to decrease the width of the histogram. In addition, for some properties, a reversed lognormal distribution would be favourable. The effect of a trend removal can be predicted from the angle of the linear trend and the R 2 value of the property to depth. A high R 2 value and steep angle reduces the standard deviation of the property significantly. Knowing the fit coefficient and the effect of linear depth trend removal can challenge the engineer in choosing a stochastic material property. If a linear depth trend is available this trend should be removed in order to obtain the variability of the material property. However, removing the linear trend can also lead to a reduction in fit coefficient. To choose the right type of stochastic property (linear trend removed or not) two values are suggested: (i) if the reduction of the standard deviation is larger than.1 after linear trend removal and (ii) if the reduction of fit coefficient is not larger than.1. These suggested values are an indication of when to use the linear trend removed stochastic property and when to use the original stochastic property. For further research a study to the removal of linear depth trend and its effect on reliability analyses in combination with the fit coefficient and reduction coefficient is advised. However, while manipulating the data e.g. by linear depth trend removal the T. de Gast Final Master of Science Thesis

77 5-6 Distribution of properties 59 engineer should be careful to blindly follow the highest reduction in variability. Formation properties boundaries based During the processing of the data, trends are observed in formations 2 and 3. By adding three extra boundaries in the well log, these trends split into individual linear trends. The added boundaries are: a boundary at 22 m in formation 2, two boundaries in formation 3 one at 2627 m and one at 2716 m. The trends are clearest observed in the well log containing the V s values. A new analysis is done on the fit of the PDF and again the linear trends are removed. The added boundaries and linear trend removal are presented in figure Formation 2a Formation 2b Formation 3a Formation 3b Formation 3c Linear trend 18 2 Formation 2a Formation 2b Formation 3a Formation 3b Formation 3c Depth [m] 24 Depth [m] Young s modulus [MPa] (a) Linear trend explanation Young s modulus [MPa] (b) linear trend removed Figure 5-11: Linear trend removal The removing of the linear trend from the sub-formation has a large impact on the standard deviation of the material property. In table 5-3 the effect of the splitting of the reservoir in linear trends and removing these trends on the standard deviation is shown. In formation 2 the standard deviation is roughly reduced in half and comparing formation 3 to formation 3b leads to a reduction of about one fifth. implementing these reduced standard deviations and linear trends into a geomechanical analysis will greatly reduce the variation of the compaction, reducing the variability of local and global failure mechanisms. Table 5-3: Effect of linear trend removal on standard deviation σ σ lin σ σ lin Formation Formation 2a 8 47 Formation 2b Formation Formation 3a Formation 3b Formation 3c σ = standard deviation σ lin = standard deviation after trend removal The reduction in standard deviation of E d after removing the linear trend is large. The reduction of the other properties is shown in table 5-4. In this table, it is clear that identifying trends boundaries based on V s significant reduction for all properties except for ρ, and the static stiffness properties E d, ν d, where in some cases the standard deviation is not reduced but increased. Master of Science Thesis Final T. de Gast

78 6 Statistical data analysis Table 5-4: Geomechanical properties and their distributions ρ V p V s Φ E d ν d E s ν s Original data, fit coefficient Formation 2a.81 L.83 T.77 L.81 L.84 L.77 T.7 L.72 L Formation 2b.85 L.82 T.86 T.91 T.84 T.81 L.57 T.83 T Formation 3a.7 L.74 T.88 T.66 T.87 L.87 L.33 T.85 L Formation 3b.84 L.9 L.99 U.69 U.81 L.77 T.33 U.71 T Formation 3c.63 T.87 T.89 T.76 T.92 L.73 L.44 T.75 T Linear trend removed, fit coefficient Formation 2a.86 L.79 T.87 T.81 L.85 L.77 T.7 L.77 L Formation 2b.93 L.91 L.93 T.88 N.88 L.89 T.73 L.77 T Formation 3a.81 L.83 T.85 N.81 L.8 L.79 T.53 L.82 T Formation 3b.91 L.91 T.95 T.83 L.91 L.82 N.83 T.9 N Formation 3c.74 T.88 T.85 T.82 L.9 L.74 T.75 T.73 T Reduction coefficient of the standard deviation Formation 2a Formation 2b Formation 3a Formation 3b Formation 3c F fit PDF O >.95 N = Normal distribution O >.85 L = Lognormal distribution O >.75 T = Truncated normal distribution O >.65 U = Uniform distribution O <.65 If a parameter such as Young s modulus is depends on effective stress, which increases with depth. the derived PDF will be skewed and the uncertainty will have a large uncertainty range. In general there are three types of depth dependency for statistical geomechanical properties: (i) a depth trend where μ changes(ii) a depth trend where σ changes (iii) a scale of fluctuation where the fluctuation changes over depth. Linear trend removal should be applied as long as the fit coefficient after removal is not lowered by a large amount and the standard deviation does reduce. If this does not happen and visual interpretation of the data does suggest an extra boundary, these boundaries should be applied. For stiffness properties, V s is a good property to identify trends on, since this property is less influenced by pore content then V p. When applying extra boundaries in data, these boundaries should be applied to all properties. And a geomechanical software package should be able to apply the removed linear trend on the data. 5-7 Analysis and discussion The method described in this chapter can be used to derive statistical input properties for reliability analyses. The method to derive and test PDFs which can be used to select a proper PDF is presented by analysing the PDF which fits on the Young s modulus derived in formation 2 of well log 3. Different properties have been analysed using the presented method. In this analysis the focus lied on reducing the uncertainty of a material property and thus T. de Gast Final Master of Science Thesis

79 5-8 Summary and conclusion 61 improving the reliability of the calculation. The standard deviations of the properties found in the different geological formations have been compared to the standard deviation of the properties after identifying linear trends and splitting the formation in different parts. The effect of splitting the formations in different parts large, this is best visible when comparing the initial standard deviation of the different formations to the standard deviations after removing the linear trend of the sub-formations. Reducing the standard deviation significantly, this reduces the uncertainty in the reliability analysis. While analysing the different properties and their distribution, unique PDFs to describe properties where not found, what was found is the fit coefficient for the normal, truncated normal and lognormal distribution functions often had a difference no more than.1 [-], in these cases, the shape of the different distributions are similar. If the fit coefficients are similar for different PDFs it is advised to use a non-truncated PDF a non-truncated distribution accounts for values which are not measured but are within the range of expectation. 5-8 Summary and conclusion This chapter describes the process of converting well log data to PDFs as an input for geomechanical reliability analysis. The well log data are first divided into the geological formations that were identified by the geologists. Then for each formation, the measured values are converted to the PDFs to be compared. In this chapter, four PDFs, a uniform distribution, a normal distribution, a log normal distribution and a truncated normal distribution have been compared. The overlap of the distributions with the histogram is computed using an area fit test the PDF with the highest fit coefficient is then selected. This PDF can then be used as an input in the geomechanical calculations. In the analysis of the well log data, it was found that certain properties are not necessarily described by a single PDF. by computing the linear trend and R 2 value, an estimate can be made on how much the removal of the linear depth trend influences the standard deviation of a property. When trends are removed, the geomechanical calculation engine should be able to enter the trend into the analyses. The methods described have been implemented in an excel sheet and can be used when working with CORA. An overview in the use of the excel sheet can be found in appendix A. Master of Science Thesis Final T. de Gast

80 62 Statistical data analysis T. de Gast Final Master of Science Thesis

81 Chapter 6 Stochastic formation properties in geomechanical calculations 6-1 Introduction In the previous chapters, the analysis of the field data and the conversion from geophysical well log data to model input parameters have been introduced. This chapter will look at the effect of the data distributions on geomechanical reliability analysis and how different methods of discretization affect the reliability. The stochastic properties in the geomechanical analyses are computed using a Monte Carlo analysis initialised by the COntainment Reliability Analysis (CORA) software package. CORA computes stochastic parameters based on the input parameters and selected Probability Density Functions (PDFs). These parameters are forwarded to the QuickBlocks (QB) program in which a geological structure and depletion pressure is modelled. To analyse the effect of the stochastic input using CORA in combination with QB, different methods have been analysed to represent the formation property variability: Dividing the reservoir in multiple layers; Varying the stochastic parameters in different parts of the geomechanical model; Using different PDFs to represent material variation; Different dimensions in failure mechanism e.g. the local failure mechanism Shear Capacity Utilisation (SCU) failure versus the global failure mechanism subsidence. The global failure mechanism subsidence is analysed using the formation geometry and depletion pressure of the field data presented in chapter 4. The geomechanical properties and variability have been computed from well log 3 of the field case. In this chapter, reliability graphs are presented. The reliability (y-axis) is defined as the probability of a value being smaller than a certain value (x-axis). Master of Science Thesis Final T. de Gast

82 64 Stochastic formation properties in geomechanical calculations 6-2 Selected literature In this selection of literature, different studies to failure mechanisms are presented these studies include stochastic reservoir subsidence predictions are performed and advanced three dimensional finite element analyses to predict reservoir responses. Subsidence induced shear failure is often a local failure mechanism. Failure oriented around casings was studied by Hamilton et al. [1992] in this study a high likelihood of failure along the weaker shale beds was observed. Using finite element calculations, bedding slip is predicted at an approximate subsidence. In this study it was found, a high stiffness contrast e.g. a shale bed included in sandstone, has a higher chance to slip then a thick shale bed. In a study using a large data set of rock mechanic properties determined by laboratory data, Cassiani and Zoccatelli [1998] used derived C m values to calculate subsidence along the Italian coast. A critical issue in subsidence prediction is the choice of the physical parameters. Subsidence simulations using semi-analytical and finite element simulations showed no risk of subsidence induced damage even for a worst case scenario. A risk based analysis has been performed by applying a Monte-Carlo simulation with the available C m data and a Geertsma simulation. This approach yields to the same conclusion as the finite element method. Inversion techniques, such as described by Fokker [22], are tools which helps to limit the number of probabilistic realisations. An ideal tool to be used with inversion techniques should be able to compute the effect of multiple formation layers and changes in geological horizons. Using an inversion program requires a certain amount of knowledge, of geologic horizons, formation pressures and distribution of these pressures, formation properties and measured responses e.g. pressures and subsidence. In combination with the measured responses, this will result in a set of possible scenarios. These tools can be used once extracting or injecting reservoirs is started to condition the probabilistic analysis. By coupling pressure differences during depletion and geomechanical deformation in a finite element mesh, Lewis et al. [23] has studied geomechanical responses during producing reservoir. Rock deformation is coupled to permeability changes and fluid pressures. This study focusses mainly on the movement of the reservoir content for production purposes. By using a fully coupled stochastic geomechanical model, a study to land subsidence has been carried out by Ferronato et al. [26]. The geomechanical uncertainty has been modelled in an axis-symmetrical coupled FE model varying C m properties. This study shows that the correlation length has a significant impact on the confidence interval. The predicted subsidence proved to be almost insensitive to the aquiver depth. The stress changes around a disk-shaped reservoir intersected by a fault were studied by Orlic and Wassing [212] they observed a large difference in stress near the edges of the reservoir and at a fault slippage. Geomechanical simulations identified the fault areas as most sensitive to fault reactivation. Fault zones along the interface between the reservoir and surrounding formations may already be critically stressed before depletion. 6-3 Static properties of reservoir To compare the results of the different input approaches, the geometry of the reservoir and geomechanical model is kept simple, a rectangular plane area with a fixed volume. The T. de Gast Final Master of Science Thesis

83 6-4 Stochastic analysis 65 differences in the different geomechanical models are that horizontal layers are added dividing the formation into equal parts. The simple reservoir has a plane square area of 5 m by 5 m and a height of 113 m. The seabed floor is 12 m below sea level and the top of the reservoir is placed 2786 m below sea level. There are six geological formations identified of which three geological formations above the reservoir. The reservoir consists of two geological formations and one geological formation below the reservoir until the end of the measurements. The base of the reservoir is chosen at 1, m to prevent boundary conditions to affect the geomechanical response. Figure 6-1: Subsurface geomechanical discretisation, size of the reservoir 5 m x 5 m x 113 m Figure 6-1 shows a static calculation of the square reservoir and subsidence bowl at 12 m depth. The subsidence bowl is calculated using the P5 values of the normal distributed stochastic variables. For normal distributed stochastic parameters μ is equal to the P5. The small plane square area of the reservoirs, result in a circular subsidence with the maximum calculated subsidence is located above the centre of the reservoir. 6-4 Stochastic analysis The main question of a reliability analysis in geomechanical calculations is which stochastic properties have the largest effect on the geomecahnical reliability response. If there are properties, mechanisms or methods that govern the reliability results, it can be wise to remove certain stochastic properties from the reliability analysis to improve the computation time. This can be done if the effect of the removed stochastic property is small compared to the dominating stochastic property. To test if there are any dominating factors in the analysis, different input methods are analysed and are compared in this chapter. these methods are: (i) increasing the amount of stochastic stiffness properties in the formations (adding layers), (ii) changing the type of PDF for the stochastic properties and (iii) comparing a global failure mecahnism (subsidence) to a local failure mechanism SCU. For subsidence, several responses are important: (i) The maximum subsidence (ii) The area of subsidence below a failure value (iii) The volume of subsidence and (iv) The volume of subsidence below a failure value. The maximum allowable subsidence before area/volume of failure occurs is chosen to be.488 m. In this thesis, the subsidence calculations have been done using the QB software in combination with CORA.For each analysis 1 Monte Carlo simulations have been computed. The output of these programs is a point grid with Master of Science Thesis Final T. de Gast

84 66 Stochastic formation properties in geomechanical calculations displacements. The analysis of the CORA subsidence output is done using the Fortran code listed in appendix B-1. This program gives as an output five responses: maximum subsidence, volume of failure, area of failure, volume of subsidence and volume of failure normalised by total subsidence volume. The analysis of SCU output is done using the Fortran code listed in appendix B-2. The depletion pressure throughout the reservoir is kept constant at 25 MPa. The stochastic geomechanical reliability results are presented in a Cumulative Distribution Function (CDF). 6-5 Stochastic geomechanical properties To perform a stochastic analysis, the geomechanical properties have been extracted from well log 3. In chapter 5, the process to extract PDF data from a data set has been introduced. This process has been used to derive the values given in table 6-1. Table 6-1: Stochastic geomechanical parameters E μ E σ F fit ν μ ν σ ν min ν max F fit [MPa] [MPa] [-] [-] [-] [-] Formation Formation Formation Reservoir Reservoir Formation E = static Young s modulus ν = static Poisson s ratio F fit = fit coefficient These values are used as input for the stochastic geomechanical analyses performed in this chapter. If there is no mention of a special PDF, the normal PDF is used for the Young s modulus and a truncated normal PDF is used for the Poisson s ratio. The truncation boundaries for the Poisson s ratio are. and.5. It is clear from the fit coefficient that most properties have a poor fit on the histogram. This is because using static properties, a difference has been made between a shale correlation and a sand correlation. The difference between these two type of material causes a discontinuity in the histogram. This does not work well with a continuous PDF leading to a low fit coefficients Multiple layers In a stochastic analysis using CORA in combination with QB, it is important to keep in mind what the effect is of the geomechanical response to the amount of individual stochastic layers in the system. This chapter shows how increasing number of stochastic layers affects the geomechanical response of subsidence. The reservoirs, drawn in figure 6-1, are split in equal parts (figure 6-2). T. de Gast Final Master of Science Thesis

85 6-5 Stochastic geomechanical properties z=direction [m] y direction [m] x direction [m] 6 8 (a) 1 layer reservoir 3 25 z=direction [m] y direction [m] x direction [m] 6 8 (b) 4 layer reservoir 3 25 z=direction [m] y direction [m] x direction [m] 6 8 (c) 16 layer reservoir Figure 6-2: Images of the layering of reservoirs in the geomechanical analysis Master of Science Thesis Final T. de Gast

86 68 Stochastic formation properties in geomechanical calculations For calculation efficiency purposes, the 2-layered and 8-layered systems are not included in the calculations. In addition to splitting the different reservoirs, the formations around the reservoirs are split as well. It should be noted that the pressure and the size of the reservoir is kept constant in all realisations and only the Young s modulus and Poisson s are varied. The results of the analysis where all the layers were increasingly divided in 1, 4 and 16 layers is presented in figure 6-3. In this figure, it is clear that the amount of layers has a large impact on the reliability results of the stochastic calculation. In each formation the stochastic properties of the divided parts are equal. The stochastic properties of the different layers are uncorrelated. The figure shows that for all results, the widest range of solutions originates in the single-layered system. Reliability [ ] layer per formation 4 layers per formation.1 16 layers per formation P5 marker Maximum allowable subsidence [m] (a) Maximum allowable subsidence Reliability [ ] layer per formation 4 layers per formation.1 16 layers per formation P5 marker.5 1 Volume of failure [m3] x 1 7 (b) Volume of failure Reliability [ ] layer per formation 4 layers per formation.1 16 layers per formation P5 marker Area of failure [m2] x 1 6 (c) Area of failure Reliability [ ] layer per formation 4 layers per formation.1 16 layers per formation P5 marker Calculated Volume [m3] x 1 7 (d) Calculated volume of the grid Figure 6-3: The effect of an increasing number of layers on geomechanical response As the amount of layers per formation increases, the range in the geomechanical response decreases. This decrease in geomechanical response range is a result of the averaging of stiffness. E.g. where a 1-layer per formation system can have a high or a low value, a 2- layer per formation system can have a high and a low value leading to an average subsidence response. The reliability, defined as the probability of a value being smaller than a certain value is read from figures 6-3 (a-d). For the maximum allowable subsidence the P5 value (.488 m) for a one layered system with geomechanical properties which have a normal PDF T. de Gast Final Master of Science Thesis

87 6-5 Stochastic geomechanical properties 69 is equal to a reliability of.5 [-]. For four layers the reliability for a maximum allowable subsidence of.488 m decreases to.45 [-]. For sixteen layers the reliability for a maximum allowable subsidence of.488 m decreases to.43 [-]. When computing a response using a single layer per formation system. The chance of overestimating and under estimating the stiffness of the subsidence is large. This over- or underestimation is caused by using the small variations from the well log and projecting them one-on-one to the large formation properties. To remove this over and under estimation, the amount of layers per formation is increased, leading to a decrease in response range. This however, takes more time to compute the individual realisations. One method to keep computing with one layer per and one PDF per formation is to pre calculate the stiffness response of the small scale variation and convert this to the large scale stiffness variation Location of formation with stochastically determined properties From figure 6-3 it can be derived that the amount of layers has a large impact on the reliability. This raises the question if the location where a stochastic layer is applied is of importance to the geomechanical response. If a formation dominates the geomechanical response, it can mean that less stochastic layers are needed for the stochastic geomechanical analysis. This will improve the calculation speed and the convergence of the statistical solutions in the stochastic geomechanical analysis. Stochastic material properties are applied on selected formations to analyse if there is a main factor that dominates the geomechanical response. To analyse the effect of the location where the stochastic are applied the stochastic formations are turned on or off depending on where a formation is situated in respect to the reservoir. In figure 6-8 the individual stochastic formations are compared in a 4 layers per formation system. In figure 6-3, the geomechanical response is noted as "whole domain" with a specific number that indicates how many layers per formation are used. In order to determine whether there is dominant factor in these calculations, the stochastic geomechanical analysis where 4 layers per formation are used is compared to location depended stochastic geomechanical analysis. This comparison is between the different activations of stochastic properties in the reservoir. In the "above the reservoir" plot, only layers that lie above the reservoir are stochastic layers. The reservoir and the layer below the reservoir are kept constant on the average value. The same goes for "in the reservoir" and "below reservoir". Here, only the mentioned layers are stochastic and the other layers are kept constant at the average value. The results of the analysis clearly show a dominating factor: the results of the solution of the stochastic properties in the reservoir and stochastic properties in the whole domain overlap. From figure 6-8 it can be derived that the formations in which the reservoir is situated, dominates the subsidence response. The subsidence response of the reservoir is nearly identical to the responses of the whole domain solution. Stochastic formation properties above and below the reservoir have very little impact on the subsidence behaviour, because the very low impact of the formations below the reservoir, the reliability is converted to a step function. For the area of failure, the layer above does add a bit to the reservoir response but its influence is of a different order of magnitude compared to the subsidence response of the reservoir. The reason the stochastic properties in the reservoir is dominating the subsidence is, reservoir compaction causes the volume change. If the stiffness properties in the reservoir change, the Master of Science Thesis Final T. de Gast

88 7 Stochastic formation properties in geomechanical calculations Reliability [ ] Stochastics Above.2 Stochastics In Stochastics Under.1 Stochastics Whole P5 marker Maximum allowable subsidence [m] (a) Maximum allowable subsidence Reliability [ ] Stochastics Above.2 Stochastics In Stochastics Under.1 Stochastics Whole P5 marker.5 1 Volume of failure [m3] x 1 7 (b) Volume of failure Reliability [ ] Stochastics Above.2 Stochastics In Stochastics Under.1 Stochastics Whole P5 marker Area of failure [m2] x 1 6 (c) Area of failure Reliability [ ] Stochastics Above.2 Stochastics In Stochastics Under.1 Stochastics Whole P5 marker Calculated Volume [m3] x 1 7 (d) Calculated volume of the grid Figure 6-4: The effect of location of stochastic formation properties on geomechanical response compacting volume of the reservoir changes as well. In the formations outside of the reservoir, the volumetric change of the reservoir is constant and only small influences are being observed in the subsidence Different PDFs In the previous paragraphs, the stochastic formation properties have been normal PDFs. Based on the data provided by Shell, other PDFs can be produced form this data. In chapter 5, the area fit test is introduced to test which PDF the most appropriate fit to the data. This raises the question if a PDF would be different than a normal PDF used to describe a stochastic property for a formation. Does this have an effect on the reliability calculation and is this effect significant? Figure 6-5 shows the effect of different PDFs on the geomechanical response. The different types of PDFs have been computed using the same stochastic input values listed in table 6-2. All calculations are carried out in a four layer per formation system and only the reservoir properties are changed. The presented stochastic input values have been converted to be compatible to the input for the log normal stochastic values. T. de Gast Final Master of Science Thesis

89 6-5 Stochastic geomechanical properties 71 Table 6-2: Stochastic geomechanical parameters for five PDFs E μ E σ E min E max E shift ν μ ν σ ν min ν max ν shift [MPa] [MPa] [MPa] [MPa] [MPa] [-] [-] [-] [-] [-] Reservoir Reservoir E = static Young s modulus ν = static Poisson s ratio Reliability [ ] Uniform PDF Normal PDF.2 Log normal PDF Shifted log normal PDF.1 Truncated normal PDF P5 marker Maximum allowable subsidence [m] (a) Maximum allowable subsidence Reliability [ ] Uniform PDF Normal PDF.2 Log normal PDF Shifted log normal PDF.1 Truncated normal PDF P5 marker.5 1 Volume of failure [m3] x 1 7 (b) Volume of failure Reliability [ ] Uniform PDF Normal PDF.2 Log normal PDF Shifted log normal PDF.1 Truncated normal PDF P5 marker Area of failure [m2] x 1 6 (c) Area of failure Reliability [ ] Uniform PDF Normal PDF.2 Log normal PDF Shifted log normal PDF.1 Truncated normal PDF P5 marker Calculated Volume [m3] x 1 7 (d) Calculated volume of the grid Figure 6-5: The effect of different PDFs on geomechanical response The results of the analysis show a different subsidence response for the different PDFs. The P5 markers for the different PDFs are scatterd illustrating a large difference in subsidence response. This difference is clear when regarding the volume of failure. When using the shifted log normal PDF this results in a high reliability (.95) of no volume of failure, the normal PDF results in a low reliability (.45) of no volume of failure. The different PDFs has an impact on the reliability result. In addition to the volume of failure, an example reliability of a maximum allowable subidence of.488 m has a varying reliability between.45 [-] (normal PDF) and.97 (shifted lognormal PDF). This shows the importance of a proper PDF derived from the input data. Master of Science Thesis Final T. de Gast

90 72 Stochastic formation properties in geomechanical calculations Global failure versus local failure Because not only subsidence, which is a global mechanism, is of importance in petroleum geomechanical issues but also local mechanisms like SCU are important. In this section, behaviour of SCU in a stochastic analysis is compared to the behaviour of subsidence in stochastic analysis. For the analysis of the SCU, stochastic material properties are applied to the formations above the reservoir and in the reservoir. In addition to the location of the stochastic properties, the amount of layers per formation is increased. This is then compared to the maximum subsidence gained from the reservoir. To compare the two different geomechanical behaviours the CDF of both responses are normalised and the standard deviation of two normalised CDFs are taken. This standard deviation is plotted as a normal CDF in figure 6-6. The benefit of normalising the two CDFs is that this approach will tell if there is a difference in geomechanical response for local and global related behaviour. Reliability [ ] Subsidence 1 layer in SCU 1 layer in.2 Subsidence 4 layers in SCU 4 layers in.1 Subsidence 16 layers in SCU 16 layers in Geomechanical response [ ] Reliability [ ] Subsidence 1 layer in SCU 1 layer above.2 Subsidence 4 layers in SCU 4 layers above.1 Subsidence 16 layers in SCU 16 layers above Geomechanical response [ ] (a) Geomechanical response by changes above the reservoir (b) Geomechanical response by changes in the reservoir Figure 6-6: A comparison between subsidence and SCU The results of the analysis drawn in figure 6-6 show the response of two different stochastic approaches. It can be seen that (a) As the number of layers above the reservoir is increased, the subsidence averages out and converges to a single solution for infinite amount of layers. However, as the number of layers increase above the reservoir the value for SCU diverges (b). When applying stochastic properties to the reservoir only, both the subsidence and the SCU converge. From this it can be concluded that the behaviour of SCU depends on the stiffness contrast around the location where the SCU is calculated. The higher the stiffness difference, the higher the SCU value. Because the SCU was only calculated above the reservoir, the largest stiffness differences were obtained as the stochastic formations were applied above the reservoir Three dimensional geomechanical model using stochastic properties In chapter 4, the case reservoir has been presented with non-horizontal boundaries and a pressure field have been presented. In this analysis, the reservoir properties are stochastic while the complex geometry and pressure field stay deterministic. From the results of the T. de Gast Final Master of Science Thesis

91 6-5 Stochastic geomechanical properties 73 analyses it can be concluded that, the shape of the subsidence bowl does not change. However, due to the changes in reservoir volume, the subsidence bowl is stretched (figure 6-8). The results of a probabilistic analysis using normal PDF properties in the reservoir are shown in figure 6-7. In order to present a difference in stochastic geomechanical response in failure behaviour, two additional pressure fields have been analysed, which have been presented in figure 6-7 (c). Reliability [ ] Case perssure field P5 marker Maximum allowable subsidence [m] (a) Maximum allowable subsidence Reliability [ ] Case perssure field P5 marker Calculated Volume [m3] x 1 8 (b) Calculated volume of the grid Reliability [ ] Alternative pressure field 1.1 Alternative pressure field 2 P5 marker Volume of failure [m3] x 1 8 (c) Area of failure, these areas of failure curves are derived from two alternative pressure fields. Figure 6-7: Stochastic geomechanical response of the case study The stochastic analysis in 6-7 shows a similar reliability curve of the maximum subsidence and volume change behaviour in the different simulations as the square mock reservoir. However, if this is combined with a failure mode, the non-uniform subsidence bowl influences the reliability curve of the assessed failure. In this case, there is a higher subsidence near the extraction point. This will fail at first, and if there is a larger volume change, the additional parts will start failing. The subsidence bowls for the minimum, average and maximum subsidence are shown in figure 6-8. If only the material properties are changed in the analysis, and the geometry and pressure field are kept constant, there is no change in the shape of the subsidence bowl. If these local properties like pressure and reservoir geometry are changed as well, and if in addition to the Master of Science Thesis Final T. de Gast

92 74 Stochastic formation properties in geomechanical calculations (a) Minimum calculated subsidence (b) Maximum calculated subsidence (c) P5 subsidence Figure 6-8: Illustrating different responses of three dimensional subsidence stochastic material properties for an entire formation, the material properties of the reservoir are varied laterally within the reservoir, the shape of the subsidence bowl will change due to the different strain distributions within the reservoir. This change of strain distribution is expected to have an impact on the shape of the subsidence bowl. The largest impact for this change will be measured in the behaviour of failure analysis. 6-6 Analysis and discussion The analyses in this chapter show that subsidence response is governed by the properties of the reservoir. The formations surrounding the reservoir have little to no influence on the subsidence response. The main effect on the volume calculations in the subsidence response are the compressibility of the reservoir, the size of the reservoir and the total pressure difference in the system. The maximum subsidence and the distribution of the subsided volume depend on the reservoir geometry, the pressure field and the distribution of stiffness. The analysis of multiple layers in the reservoir showed for subsidence a convergence to a single solution. This convergent behaviour is due to the uncorrelated stochastic properties in the different T. de Gast Final Master of Science Thesis

93 6-7 Summary and conclusion 75 layers: a correlated property distribution by e.g. a random field stochastic approach is able to introduce local property variation in the reservoir model. The additional analysis of SCU, a local mechanism, suggests a high importance to local property distribution and difference. In the calculation of the stress, the main factor that influences stress is the stiffness distribution near the stress points. The largest SCU values are computed at alternating high and low stiffness s near the point of the calculated SCU. An uncorrelated distribution of material properties lead to bigger differences in the material properties than a correlated property distribution. 6-7 Summary and conclusion In this chapter, an analysis of the geomechanical response of reservoir depletion has been performed. The effect of different layers per reservoir formation on the subsidence and SCU has been studied. Different types of formation property PDFs have been applied to the reservoir. It was found that the average stiffness of the reservoir, the reservoir volume and the depletion pressure in the reservoir are the main factors that affect the subsidence volume. However, the distribution of the subsided volume, which dictates the maximum subsidence and are governed by the distribution of pressure in a reservoir, lateral stiffness distribution and reservoir specific geometry. The effects of stochastic material properties on subsidence and Shear Capacity Utilisation in geomechanical reliability response differ. In respect to the subsidence geomechanical reliability, increasing resolution of uncertainty, (increasing amount of layers) reduces the uncertainty. In respect to the Shear Capacity Utilisation geomechanical reliability, increasing the resolution of uncertainty The results increases the range of geomechanical responses. Master of Science Thesis Final T. de Gast

94 76 Stochastic formation properties in geomechanical calculations ïż T. de Gast Final Master of Science Thesis

95 Chapter 7 Discussion 7-1 Introduction In this thesis, several uncertainties in addressing geomechanical responses have been mentioned, such as geological structure, depletion pressure and geomechanical parameter distribution. Along with these, uncertainties suggestions were made to improve future reliability studies. This chapter aims to highlight and discuss important topics raised. The different issues raised are discussed in three sections. In the first section, the setup of the problem is discussed including the required initial information. The second part focuses on obtaining an appropriate Probability Density Function (PDF) for geomechanical parameters and dealing with unknowns in the site characterisation. The last part deals with the setup of the subsurface model and corresponding uncertainties. 7-2 Setup of the problem Global failure vs. local failure Before preforming a geomechanical reliability analysis, it is important to realise the mechanism and scale of the addressed geomechanical problem. There will be a large difference for example in the assessment of local and global failure mechanisms. The difference will be in the scale of the geomechanical model and in the resolution of the geomechanical model. For local failures, a global subsurface model will underestimate the local variability which lead to a less accurate assessment of the geomechanical response. For a global failure, local variations in a subsurface model will be of less influence and result in a longer computing time without additional precision. It is expected that both the required geomechanical properties and the behaviour of the geomechanical mechanism change for different problems. Pressure depletion scenarios The driving force of subsidence is the pressure difference induced by injecting in or extracting from a reservoir. For all mechanisms studied in this thesis the distribution of the pressure Master of Science Thesis Final T. de Gast

96 78 Discussion difference in the reservoir is one of the uncertainties in a reliability analysis. The effect of the uncertainty of different depletion scenarios has not been studied in this thesis. However, it has been shown in this thesis that depletion pressure scenarios significantly affect the shape of the subsidence bowl. The effect the pressure depletion scenarios have on the subsidence bowl leads to the conclusion that, applying a variation in depletion pressure fields will be beneficial for the reliability study. This however depends on how variable the different pressure field scenarios are. Application of variation in pressure depletion scenarios is expected to lead to a larger range of geomechanical responses. Geological model The location, orientation and shape of the reservoir and possible discontinuities such as faults, have a large impact on the pressure distribution and geomechanical response. Variation of discontinuities such as faults can lead to a change in depletion pressure scenarios, which lead to a difference in reservoir compaction. In general a detailed knowledge of reservoir boundaries is only available in discrete locations (wells). The shape of the reservoir is based on the knowledge obtained from these discrete locations and seismic surveys. This means that there is still uncertainty in the geological structure. This means that the location orientation, shape and size of the reservoir can differ from what is assumed. The more information is available the smaller the variation of geological uncertainties will be. Still, application of uncertainties in the geological model is expected to lead to a larger range in geomechanical responses. 7-3 Geomechanical properties Describing variable geomechanical properties using PDFs It is known that the geomechanical properties vary. This variation of geomechanical properties can be described using PDF. The selecting of an appropriate PDF to describe the PDF is an important feat in a geomechanical reliability analysis. In this thesis, the most appropriate PDF has been selected from: Lognormal, normal, truncated normal and uniform distribution. Analysis of the geophysical and geomechanical properties there were no PDFs who were favourable for a type of data. However, the uniform distribution has a very low fit coefficient for most data. The lognormal and (truncated) normal distributions are most common PDFs to be chosen in terms of fit while analysing the data. An advantage of non-truncated PDFsis, unmeasured weaknesses are taken into account in a geomechanical reliability analysis. The advantage of truncated PDFs is that non real property values are not taken into account e.g. the Poisson s ration can be limited between. [-] and.5 [-]. A truncated PDF is useful to exclude non mechanical values. However, care should be taken to use truncation to limit the range of values within the range of possible mechanical values. Coupling laboratory and field data Predicting the stiffness of a reservoir is done using geophysical data from a well log, which has a high information density over depth. This geophysical data can be converted to stiffness by either elasticity theory (based on non-porous media) or empirical conversion functions obtained from laboratory data. These two type of stiffness determinations compute different values for stiffness. The advantages of laboratory data are direct measured responses from core samples. Because the cores are retrieved from the formation trough the well, it is expensive to retrieve a core sample. Because of the economic factor in retrieving a core sample the disadvantages of laboratory data are only a small amount of laboratory data is tested. A T. de Gast Final Master of Science Thesis

97 7-3 Geomechanical properties 79 combination of the well logs and the laboratory datasets are favourable to compute a PDF. The difficulty is to obtain the empirical conversion functions required for the site and material. One method to obtain the empirical conversion function is to collect laboratory data of similar material types, this will become a base dataset for a material type. This base dataset can then be conditioned to laboratory data from the project. If this effort is not made, it leads to empirical conversion functions that are based only on local data. If a empirical conversion function is based on local data only the dataset might be skewed. Such a skew can occur if the core samples taken from the reservoir are taken from weak zones. Conditioning the larger dataset to the smaller dataset could be done by comparing measured V s of the different datasets. The V s is a property which is directly related to stiffness and is less affected by the pore content of the sample than V p. Multiple well logs In this thesis, the data and analysis of the data has only been undertaken on one well log. When there are multiple well logs through a formation, these well logs can be combined to obtain the global distribution properties and global trends in the data. The issue with combining these well logs is that these well logs can cross formations at different depths. Besides corssing the formation at different depths, a formation can have a different thickness in each well log. In order to differentiate between local variations and consistent formation variations in multiple well logs can be compared. A consistent formation variation can be a sudden spike which occurs in different well logs. To analyse the data of multiple well logs it is required to find these common trends in the data. The identification of these common trends is required to link a formation in different well logs. This can be done by removing some variability in the data, for instance by a moving average over the data, this reduces scatter and extreme values while keeping the global variability intact. When common trends have been identified, these can be used as markers to scale the different well logs. This method of scaling will link the consistent formation markers in a formation over multiple well logs. After scaling the well logs, the variability of the material property over the field can be computed using an appropriate PDF. Trend removal It has been shown in this thesis, that the removal of trend can reduce the calculated variation of a parameter. Trend removal applied on a consistent trend, preferably if there is a logical explanation for the trend is preferable. If trends are removed, without consideration of a logical (mechanical) explanation, false trend assumptions can be made. For example in formation 3 in figure 7-1 removing the linear trend increases the F fit from.27 [-] to.75 [-]. However, this increase is not caused by a linear trend in the data, the reason the F fit increases is because the properties of the shale materials overlap the properties of the sandstone materials. Removing the linear trend removes the difference between the shale and sandstone data points. Varying geology and upscaling In the dataset provided the boundaries of the formations were upscaled from the local boundaries. The result of this upscaling is that there are several material types in a formation. This has led to a very poor F fit value for the static stiffness properties. In order to identify the variation of the two properties an extra scale of fluctuation needs to be introduced. The first scale of fluctuation is the variation of the stiffness of a material. The second scale of fluctuation is the variation of two materials e.g. alternating sandstones and shales and will be called scale of persistence. Master of Science Thesis Final T. de Gast

98 8 Discussion identifying the different scale of persistence makes it possible to obtain a PDF for one material and another PDF for the second (or third) material. This leads to higher F fit values for the individual materials. For three dimensional reliability analysis identifying the scale of persistence is as important (if not more important) than identifying the scale of fluctuation Depth [m] Formation 1 Formation 2 Formation 3 28 Reservoir 1 Reservoir 2 Formation Young s modulus [MPa] Figure 7-1: Static Young s modulus E s In figure 7-1, the effect of two types of material and corresponding conversion functions illustrated. The first quarter of formation 2 has a different conversion function from the last quarter of the same formation. It is important to study the effects of horizontal and vertical scales of persistence and variation before upscaling a formation property. If this is not done, unnecessary inaccuracy will be incorporated in a reliability analysis. Obtaining scales of fluctuation The important factor in dealing with computing the effect of different scales of fluctuation and persistence is how to identify the scales of fluctuation. When dealing with variation without taking the different scales of variation into account the obtained PDFs will result in a overestimation or underestimation of the variation in geomechanical responses. The advantage of taking account of different types scales of variation is that a co-variation can be modelled. A co-variation of stiffness leads to a gradual transition of weak and strong zones. For local failures this co-variation will lead to a smaller range of geomechanical responses. In general vertical scales are easier to obtain than horizontal scales. One option to obtain the horizontal scales might be to use seismic surveys to identify weak and strong spots in the subsurface. Another option is to obtain the required horizontal scales from outcrops and geological studies. T. de Gast Final Master of Science Thesis

99 7-4 Numerical modelling Numerical modelling Property estimation In this thesis, the effects of vertical variation of stiffness parameters in probabilistic calculations have been shown. The effects of pressure distribution and geometry on the shape of the subsidence bowl show the possible impact lateral variation may have on the calculated subsidence. To include the lateral variation (and possible linearization) in geomechanical reliability analysis is recommended. Addressing the lateral variation can be done in several ways; one option is to use random fields as a method to deal with the scales of variation and persistence. Stiffness vs. Geometry vs. Pressure The combination of geometry, pressure and stiffness are important factors in calculating a geomechanical response. However, it is discussed to study the impact each individual variation/uncertainty has on the calculated geomechanical response. What is clear from this study and literature is that for subsidence, the local variation of a reservoir such as shape, inclination, pressure distribution and stiffness distribution become more important as the depth to size ratio decreases e.g. a large reservoir which is located deep beneath the surface has a more averaged impact than a large reservoir located at shallow depth which has a more localised impact. The advantage of combining the three different factors in a reliability study is, the resulting calculated geomechanical response is more accurate. The disadvantage is, the computation time is longer because the number of uncertainty dimensions increase for a same convergence level for the reliability result. An option to reduce the amount of computation time and to reduce the amount of realisation can be to preselect realisations based on likelihood of failure. An option is to use an analytical realisation (few seconds) solution to decide to start a numerical realisation (several minutes). Improving reliability during production For a reservoir which is already producing and because of this has a geomechanical response, the range of geomechanical responses can be reduced based on measurements. These measurements help to exclude certain scenarios and to back-calculate new scenarios. This will condition the subsurface model of the field reduce the uncertainties. 7-5 Summary In this chapter important topics in the research have been highlighted and discussed. These highlights include: preselecting uncertainties based on the type of failure which is expected. Including pressure and geometrical uncertainties in reliability analyses. Computing variability in different scales (formation, materialtype and material) and over multiple well logs. removing trends and the danges of removing trends. The importance of lateral and local variation in reliability analysis. And limiting the range of geomechanical responses by coupling monitoring while producing from a reservoir to the reliability analysis. Master of Science Thesis Final T. de Gast

100 82 Discussion T. de Gast Final Master of Science Thesis

101 Chapter 8 Conclusion 8-1 Introduction In this thesis, numerical simulations of geomechanical problems related to reservoir compaction have been addressed. It is found that, the variability and uncertainties in the prediction of reservoir depletion related failure mechanisms are discussed. The main question in this thesis is: What is the effect of the variability of geomechanical properties, their Probability Density Function (PDF) and the effect of the numerical discretisation in a geomechanical analysis on reliability predictions of failure mechanisms? In this thesis, only a limited number of reservoir compaction related failure mechanisms are analysed. These compaction related failure mechanisms are Shear Capacity Utilisation (SCU) and subsidence. The focus of in these analyses was subsidence, while SCU was briefly analysed to illustrate the different behaviour of local and global failure mechanisms. The main question was split in the following sub-questions: How can the variability of material properties be obtained from well log data? This question was addressed in chapter 5. What is the effect of different types of uncertainties unrelated to geomechanical properties, in a geomechanical analysis? This question was addressed in chapter 4. What is the effect of heterogeneity in formation properties, specifically the stiffness properties, on geomechanical analyses and how should this heterogeneity be implemented in reliability based analysis? This question was addressed in chapter 6. The used example field data assured the use of a realistic reservoir shape and depletion scenario, and provided a set of real well log data to test the procedures to derive PDFs from the well logs. Master of Science Thesis Final T. de Gast

102 84 Conclusion In chapter 3 the well log data have been introduced as geophysical measurements in a borehole or well logged in depth. The precision of these measurements depends on a number of factors including the fluid in the well and the roughness of the well sides. From the geophysical properties, geomechanical properties can be obtained by means of conversion functions. Some of these conversion functions are based on elasticity theory and some are empirical. Dynamic geomechanical properties are obtained using high frequency sonic pulses, measuring the velocity of these pulses through the formation. These velocities are converted to stiffness properties using elasticity theory. Static geomechanical properties are obtained under laboratory conditions. Conversion functions are obtained by correlating laboratory data to geophysical properties and by creating a conversion function from this correlation. The material properties in a well log can vary. In chapter 5, a method is present to mathematically describe the variation of material properties by use of a PDF is presented. To obtain a proper PDF from material properties in a well log, the overlap of the histogram and the PDF is calculated to compute a fit coefficient (F fit ). The F fit is used to test how well a PDF represents the data from the well log. The different properties of well log 3 have been analysed on geological scale and by adding boundaries in the well log at depths where a trend break was observed. The best fitting PDFs have been presented and the effect of linear trend removal on the standard deviation of the dataset was shown. Along with presenting the field data, chapter 4 uses deterministic analyses to analyse the effect of the reservoir geometry and the pressure depletion scenario on the calculated subsidence and shape of subsidence bowl. These deterministic calculations showed that local pressures in the reservoir and the geometry of the reservoir have a large effect on the distribution of the subsided volume. Because the material properties in a reservoir vary, a stochastic analysis is performed using variable geomechanical input parameters. Stochastic geomechanical analyses initialised using the Shell programs COntainment Reliability Analysis (CORA) and QuickBlocks (QB) are presented in chapter 6. The effect of the geomechanical properties and a method that models these properties in a geomechanical analysis is tested. In order to analyse the effect of the variable material properties, a reservoir which has a square pan area with a fixed geometry and depletion pressure is used. The geometry of the subsurface is divided in three formation types: the formations above the reservoir, the formations at reservoir height and the formations below the reservoir. In order to study the effect of stochastic properties on subsidence, steps were investigated: 1. The effect of the amount of stochastic layers per formation in the geomechanical model; 2. Computing stochastic layers only in one formation type of the calculation model to study which formation largest influence on the stochastic geomechanical behaviour; 3. Different PDF functions are used to describe the geomechanical properties; 4. The difference between global and local failure behaviour is analysed; 5. The field data shows the effect of three dimensional effects on subsidence and the failure mechanisms. Since subsidence is a global mechanism and some geomechanical response like SCU are local, step one and two have been compared for the reliability behaviour of the maximum subsidence T. de Gast Final Master of Science Thesis

103 8-2 Research conclusion 85 and the maximum SCU value. As a final step is where the reservoir data provided is used in combination of stochastic material properties. 8-2 Research conclusion The findings of this thesis regarding well log data and geomechanical property distributions are related to the sub-questions: Sub question 1 : How can the variability of material properties be obtained from well log data?: In this thesis, the process of extracting data from well log data to input parameters for stochastic geomechanical analysis is done using PDFs. To test if a proper PDF is selected, the area of fit test is used on several PDFs. This test has proven to be useful to select the PDF and using the fit coefficient, the fit of the PDF to the data is analysed. Removing linear (depth) trends from the data is proven to reduce the standard deviation of a material property. An attempt has been made using a polynomial fit and the inflection points of this polynomial to identify changes in trends. However, a manual interpretation of the data was proven to be more effective. Sub question 2a: What is the effect of different types of uncertainties unrelated to geomechanical properties in a geomechanical analysis? : The subsidence bowl resulting from the compaction of a reservoir depends on the stiffness of the formations in the reservoir and the pressure difference in the reservoir. The combination of these factors dictates how much volume is displaced. The subsided volume increases, as the depletion pressure increases or the stiffness of the reservoir decreases. In order to predict the maximum subsidence and subsidence failure modes, it was found that the local distribution of the pressure and the shape of the reservoir geometry govern the shape of the subsidence bowl above the field. With these observations in mind, it is expected that introducing a lateral variability of the formation properties has a significant impact on the shape of the subsidence bowl. Sub question 2b: What is the effect of heterogeneity in formation properties, for these studies the stiffness properties, on geomechanical analyses and how should the heterogeneity be implemented in a reliability based analysis?: It was found that there is a large difference in the geomechanical behaviour of global and local failure mechanisms. The differences lead to the following conclusions: Using stochastic (uncorrelated) formation properties, the subsidence is dominated by the reservoir properties and the response converges to a unique solution as the amount of layers increase. This is because the effects of the different volume reductions are superimposed and lead to a single (average) response. Master of Science Thesis Final T. de Gast

104 86 Conclusion The SCU increases with the local stiffness contrast, the reliability response becomes more uncertain as the SCU as the amount of layers per formation is increased. This is because SCU is influenced by stiffness contrast and an increasing amount of (uncorrelated) layers can lead to a higher contrast. The difference in geomechanical response of subsidence related failure mechanisms with the use of PDFs can be significant. In this thesis, it was shown that for reservoir which has a square plane area that with the use of a lognormal distribution (shifted to improve the fit on the histogram) more than 95% the stochastic realisations did not fail. Based on the same well log, using a uniform distribution only 45% of the stochastic realisations did not fail. 8-3 Updated workflow During this thesis the reliability assessment of the geomechanical response was carried out using CORA in combination with QB. The pressure distributions and reservoir geometry where provided by Shell and where a deterministic set of properties. The material properties of the reservoir formations are based on PDFs obtained from provided well log data. The geomechanical model is then based upon these properties. A study in the behaviour of the stochastic behaviour of the geomechanical response has shown that depending of a stress or strain failure mechanism, the behaviour of the stochastic geomechanical simulation diverges from a single solution or converges to a single solution. This has led to a workflow used in this thesis, shown in figure 8-1 where the pressure distribution and reservoir geometry are not varied and the geomechanical model is based on either a local difference (stress) or a global difference (strain) resulting in a geomechanical response. Figure 8-1: Current Workflow T. de Gast Final Master of Science Thesis

105 8-4 Recommendations for further development of CORA Recommendations for further development of CORA It was found in this study that the geometry and pressure distributions have a large impact on the geomechanical response e.g. for subsidence the distribution of the subsided volume depends on the local pressure and geometry. This conclusion has led to a new recommended workflow in which uncertainties in pressure distribution and reservoir geometry are taken into account. This updated workflow is shown in figure 8-2. In this workflow, the uncertainties of the pressure distribution and the uncertainties are taken into account. These uncertainties should be obtained from reservoir simulations and uncertainties in the reservoir geometry. Figure 8-2: Suggested Workflow In addition to including the uncertainties of the pressure distribution and reservoir geometry, it is recommended to change the input of the material properties from a PDF to a statistical method which takes into account not only the uncertainty of a material property but also the variation rate of a property. E.g. Random field generation. This alternative method will influence the local stiffness which has a large on all studied geomechanical failure mechanisms in this thesis except for the volume of subsidence Geometric uncertainties Geometric properties of the reservoir are based on seismic surveys and interpolation between known points e.g. well-logs. Seismic surveys are used to study the propagation of sonic waves from the survey point, through sealing formations and the reservoir back to the receiver often on the same height as the transmitter. As the sonic signal propagates trough more material, the resolution of the sonic signal decreases. This creates an uncertainty in the reservoir geometry, part of this uncertainty can be corrected by well logs and part of this uncertainty Master of Science Thesis Final T. de Gast

106 88 Conclusion is corrected by statistic processes e.g. kriging. It is recommended to study the effect of these geometric uncertainties stochastic geomechanical calculations Pressure uncertainties The pressure distribution in a reservoir computed based on expected pressure in the reservoir and a calculation of the flow through a reservoir. This flow depends on the reservoir properties such as porosity and permeability. These calculations are based on stochastic realisations of permeability and porosity distributions in the reservoir. The results of the pressure distributions vary. It is recommended to study the effect of different pressure fields in a reservoir and their effect on the geomechanical response Random field realisations Geomechanical properties vary throughout a formation. This variation is both vertically and horizontally. In order to model these variations random field methods to model these variations are developed. These random field methods compute variability on a local scale and generate a spatially correlated property field. Because the local failures mechanisms are generaly triggered by stiffness contrast and these methods reduce the stiffness contrast by computing a stiffness distribution which is correlated to its neighbour. It is therefore recommended to implement random field realisations for stress depended failure mechanisms. Strain related failure mechanisms will benefit less from the local variation. However, a lateral stiffness change can have a high influence on the reliability. In order to use these random field realisations, a vertical scale of fluctuation Θ v should be derived from vertical observations e.g. well log data. The horizontal scale of fluctuation Θ h is harder to determine since horizontal observations are required to obtain this property, an suggestion is to obtained the horizontal scale of fluctuation from seismic data provided the resolution of the seismic signal enough to identify lateral variation. Trend analysis One of the issues determining the scale of fluctuation is to identify a depth trend. These trends may differ from each formation or even within a formation. A general accepted method to obtain these trends is to assume a linear correlation. A different approach would be to search for repeating trends in different well logs and base compute the scale of fluctuation based on these trends. methods to find repeating trends is to apply a least squared estimate polynomial function through the data sets or to apply a spline interpolation on the dataset. A combination of the two is the smooth spline interpolation and where a smoothing parameter is applied on the spline interpolation. The two extreme results the smooth spline can produce are the spline interpolation if no smoothing occurs and the least squared polynomial if there is a high smoothing. T. de Gast Final Master of Science Thesis

107 8-4 Recommendations for further development of CORA 89 Large domain computation Using random field brings its own set of issues. Besides obtaining scales of fluctuation, computing a random field on a small scale in a large domain is expected to be computational expensive, one option to reduce the computing time is to create a set of random field realisations on a smaller scale and use the principle of superposition to project these realisations on a larger domain, this saves computation time and the different sets of random fields can be placed at random in the large domain. The reverse is also possible, to compute stress based failure mechanisms, from the large domain computation the global strain development is obtained. This global strain can be projected on the random field to compute stress related geomechanical behaviour. Master of Science Thesis Final T. de Gast

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