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Editorial Reviews
From the Publisher
'A very comprehensive and complete book spanning all the aspects of stress within the accessible Earth. It is particularly useful in the fields of oil industry, geothermics and seismic hazard.' Tectonophysics'… a strong and authoritative treatment … Professor Zoback's book will be a valued guide and reference to geoscientists and engineers.' International Journal of Rock Mechanics and Mining Sciences
'A major advantage of the book is … that it provides an excellent crossover between aspects of structural geology and reservoir engineering  a link that is all too often overlooked. … It is customary to have an 'ah but …', bit towards the end, but I genuinely have very few quibbles with this book. … I would recommend it to any geologist working in a situation where faults or fractures interact with the presentday stress field.' Tim Needham, Needham Geoscience, Illkley
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Cambridge University Press
9780521770699  Reservoir Geomechanics  by Mark D. Zoback
Excerpt
1 The tectonic stress field
My goals in writing this book are to establish basic principles, introduce practical experimental techniques and present illustrative examples of how the development of a comprehensive geomechanical model of a reservoir (and overlaying formations) provides a basis for addressing a wide range of problems that are encountered during the lifecycle of a hydrocarbon reservoir. These include questions that arise (i) during the exploration and assessment phase of reservoir development such as the prediction of pore pressure, hydrocarbon column heights and fault seal (or leakage) potential; (ⅱ) during the development phase where engineers seek to optimize wellbore stability through determination of optimal well trajectories, casing set points and mud weights and geologists attempt to predict permeability anisotropy in fractured reservoirs; (iii) throughout the production phase of the reservoir that requires selection of optimal completion methodologies, the prediction of changes in reservoir performance during depletion and assessment of techniques, such as repeated hydraulic fracturing, to optimize total recovery; and (ⅳ) during the secondary and tertiary recovery phases of reservoir development by optimizing processes such as water flooding and steam injection. Chapters 1– 5 address basic principles related to the components of acomprehensive geomechanical model: the state of stress and pore pressure at depth, the constitutive laws that commonly describe rock deformation and fractures and faults in the formations of interest. Chapters 6– 9 address wellbore failure and techniques for using observations of failure to constrain stress orientation and magnitude in wells of any orientation. Chapters 10– 12 address case studies that apply the principles of the previous chapters to problems of wellbore stability, flow associated with fractures and faults and the effects of depletion on a reservoir and the surrounding formations.
Why stress is important
The key component of a comprehensive geomechanical model is knowledge of the current state of stress. Wellbore failure occurs because the stress concentrated around the circumference of a well exceeds the strength of a rock (Chapters 6 and 10). A fault will slip when the ratio of shear to effective normal stress resolved on the fault exceeds its frictional strength (Chapters ^{4}, 11 and 12). Depletion causes changes in the stress state of the reservoir that can be beneficial, or detrimental, to production in a number of ways (Chapter 12). As emphasized throughout this book, determination of the state of stress at depth in oil and gas fields is a tractable problem that can be addressed with data that are routinely obtained (or are straightforwardly obtainable) when wells are drilled.
In this chapter, I start with the basic definition of a stress tensor and the physical meaning of principal stresses. These concepts are important to establish a common vocabulary among readers with diverse backgrounds and are essential for understanding how stress fields change around wellbores (Chapters 6 and 8) and in the vicinity of complex structures such as salt domes (as discussed at the end of the chapter). I also introduce a number of fundamental principles about the tectonic stress field at a regional scale in this chapter. These principles are revisited at scales ranging from individual wellbores to lithospheric plates in Chapter 9. While many of these principles were established with data from regions not associated with oil and gas development, they have proven to have broad relevance to problems encountered in the petroleum industry. For example, issues related to global and regional stress patterns are quite useful when working in areas with little preexisting well control or when attempting to extrapolate knowledge of stress orientation and relative stress magnitudes from one area to another.
Stress in the earth’s crust
Compressive stress exists everywhere at depth in the earth. Stress magnitudes depend on depth, pore pressure and active geologic processes that act at a variety of different spatial and temporal scales. There are three fundamental characteristics about the stress field that are of firstorder importance throughout this book:
Knowledge of stress at depth is of fundamental importance for addressing a wide range of practical problems in geomechanics within oil, gas and geothermal reservoirs and in the overlaying formations.
In short, the in situ stress field in practice is determinable, comprehensible and needed to address a wide range of problems in reservoir geomechanics. The in situ stress field at depth is remarkably coherent over a variety of scales. These scales become selfevident as data from various sources are analyzed and synthesized.
It is relatively straightforward to measure, estimate or constrain stress magnitudes at depth using techniques that are practical to implement in oil, gas and geothermal reservoirs. Hence, the state of stress is directly determinable using techniques that will be discussed in the chapters that follow.
In this chapter I review a number of key points about the state of stress in the upper part of the earth’s crust. First, we establish the mathematical terminology that will be used throughout this book and some of the fundamental physical concepts and definitions that make it possible to address many practical problems in subsequent chapters. While there are many excellent texts on elasticity and continuum mechanics that discuss stress at great length, it is useful to set forth a few basics and establish a consistent nomenclature for use throughout this book. Next, the relative magnitudes of in situ stresses are discussed in terms of E. M. Anderson’s simple, but powerful, classification scheme (Anderson 1951) based on the style of faulting that would be induced by a given stress state. This scheme leads naturally to some general constraints on stress magnitudes as a function of depth and pore pressure. These constraints will be revisited and refined, first in Chapter 4 where we will discuss constraints on stress magnitudes in terms of the strength of the crust and further refined when we incorporate information about the presence (or absence) of wellbore failures (Chapters 7 and 8).
In the next section of this chapter I briefly review some of the stress indicators that will be discussed at length in subsequent chapters. I do so in order to review synoptically some general principles about the state of stress in the crust that can be derived from compilations of stress information at a variety of scales. The overall coherence of the stress field, even in areas of active tectonic deformation and geologic complexity is now a demonstrable fact, based on thousands of observations from sites around the world (in a wide range of geologic settings). We next briefly review several mechanisms that control crustal stress at regional scale. Finally, we consider the localized rotation of stress in the presence of near frictionless interfaces, such as salt bodies in sedimentary basins such as the Gulf of Mexico.
Basic definitions
In simplest terms, stress is defined as a force acting over a given area. To conform with common practice in the oil and gas industry around the world I utilize throughout the book calculations and field examples using both English units (psi) and SI units (megapascals (MPa), where 1 MPa = 145 psi).
To be more precise, stress is a tensor which describes the density of forces acting on all surfaces passing through a given point. In terms of continuum mechanics, the stresses acting on a homogeneous, isotropic body at depth are describable as a secondrank tensor, with nine components (Figure 1.1 left).
so that the order of the subscripts is unimportant. In general, to fully describe the state of stress at depth, one must define six stress magnitudes or three stress magnitudes and the three angles that define the orientation of the stress coordinate system with respect to a reference coordinate system (such as geographic coordinates, wellbore coordinates, etc.).
Figure 1.1 Definition of stress tensor in an arbitrary cartesian coordinate system (Engelder and Leftwich 1997), rotation of stress coordinate systems through tensor transformation (center) and principal stresses as defined in a coordinate system in which shear stresses vanish (right).
In keeping with the majority of workers in rock mechanics, tectonophysics and structural geology, I utilize the convention that compressive stress is positive because in situ stresses at depths greater than a few tens of meters in the earth are always compressive. Tensile stresses do not exist at depth in the earth for two fundamental reasons. First, because the tensile strength of rock is generally quite low (see Chapter 4), significant tensile stress cannot be supported in the earth. Second, because there is always a fluid phase saturating the pore space of rock at depth (except at depths shallower than the water table), the pore pressure resulting from this fluid phase would cause the rock to hydraulically fracture should the least compressive stress reach a value close to the value of the pore pressure (Chapter 4).
Once a stress tensor is known in one coordinate system, it is possible to evaluate stresses in any other coordinate system via tensor transformation. To accomplish this transformation, we need to specify the direction cosines (, as illustrated in Figure 1.1) that describe the rotation of the coordinate axes between the old and new coordinate systems. Mathematically, the equation which accomplishes this is
There are two reasons why the ability to transform coordinate systems is of fundamental importance here. First, once we know an in situ stress field in some coordinate system, we can compute stresses in any other. For example, if we know the stress state in a geographic coordinate system, we will show how it is possible to derive the stress field surrounding a wellbore of arbitrary orientation (Chapter 8) to address problems of stability (Chapter 10), or along a fault plane (Chapter 5) to gauge its proximity to frictional failure and slip (Chapter 11). Another reason why tensor transformation is important is because we can choose to describe the state of stress at depth in terms of the principal stresses (i.e. those acting in the principal coordinate system), making the issue of describing the stress state in situ appreciably easier. The principal coordinate system is the one in which shear stresses vanish and three principal stresses, _ fully describe the stress field (as illustrated in the right side of Figure 1.1). In the principal coordinate system we have diagonalized the stress tensor such that the principal stresses correspond to the eigenvalues of the stress tensor and the principal stress directions correspond to its eigenvectors:
Relative stress magnitudes and E. M. Anderson’s classification scheme
In applying these concepts to the earth’s crust, it is helpful to consider the magnitudes of the greatest, intermediate, and least principal stress at depth (S_{1}, S_{2}, and S_{3}) in terms of S_{v}, S_{Hmax} and S_{hmin} in the manner originally proposed by E. M. Anderson and alluded to above. As illustrated in Figure 1.1 and Table 1.1, the Anderson scheme classifies an area as being characterized by normal, strikeslip or reverse faulting depending on whether (i) the crust is extending and steeply dipping normal faults accommodate movement of the hanging wall (the block of rock above the fault) downward with respect to the footwall (the block below the fault), (ⅱ) blocks of crust are sliding horizontally past one another along nearly vertical strikeslip faults or (iii) the crust is in compression and relatively shallowdipping reverse faults are associated with the hanging wall block moving upward with respect to the footwall block. The Anderson classification scheme also defines the horizontal principal stress magnitudes with respect to the vertical stress. The vertical stress, S_{v}, is the maximum principal stress (S_{1}) in normal faulting regimes, the intermediate principal stress (S_{2}) in strikeslip regimes and the least principal stress (S_{3}) in reverse faulting regimes. The dip and strike of expected normal, strikeslip and reverse faults with respect to the principal stress are discussed in Chapter 4.
Relative stress magnitudes and faulting regimes
Figure1.2 E. M. Anderson’s classification scheme for relative stress magnitudes in normal, strikeslip and reverse faulting regions. Earthquake focal mechanisms, the beach balls on the right, are explained in Chapter .
The magnitude of ^{5}S_{v} is equivalent to integration of rock densities from the surface to the depth of interest, z. In other words,
where ρ(z) is the density as a function of depth, g is gravitational acceleration and is the mean overburden density (Jaeger and Cook 1971). In offshore areas, we correct for water depth
here ρ_{w} is the density of water and z_{w} is the water depth. As ρ_{w} ∼ 1 g/cm^{3} (1.0 SG), water pressure (hydrostatic pressure) increases at a rate of 10 MPa/km (0.44 psi/ft). Most clastic sedimentary rock has an average density of about 2.3 g/cm^{3} which corresponds to a porosity of about 15%. This results in a vertical principal stress that increases with depth at a rate of 23 MPa/km (or conveniently, ∼1 psi/ft). Correspondingly, the magnitudes of the two horizontal principal stresses increase with depth. Some of the practical problems associated with the computation of S_{v} using equations () and () relate to the facts that density logs frequently measure anomalously low density when the well is rugose and density is often not measured all the way up to the seafloor when drilling offshore. This is illustrated by the density log in Figure 1.3. The density log (top figure) is somewhat noisy and no data are available between the seafloor (1000 ft below the platform) and 3600 ft. This makes it necessary to extrapolate densities to the seafloor where the density is quite low. Integration of the density log using equation () yields the overburden stress as a function of depth (middle figure). The rate at which the overburden stress gradient increases with depth is shown in the lower figure. Note that because of the water depth and low densities immediately below the seafloor (or mud line), the overburden stress gradient is only 0.9 psi/ft at a depth of 14,000 ft, even though density exceeds 2.3 g/cm^{3} below 8000 ft.
According to the Anderson classification scheme, the horizontal principal stresses may be less than, or greater than, the vertical stress, depending on the geological setting. The relative magnitudes of the principal stresses are simply related to the faulting style currently active in a region. As illustrated in Figure 1.2, the vertical stress dominates in normal faulting regions (S_{1} = S_{v}), and fault slip occurs when the least horizontal principal stress (S_{hmin}) reaches a sufficiently low value at any given depth depending on S_{v} and pore pressure (Chapter 4). Conversely, when both horizontal stresses exceed the vertical stress (S_{3} = S_{v}) crustal shortening is accommodated through reverse faulting when the maximum horizontal principal stress (S_{Hmax}) is sufficiently larger than the vertical stress. Strikeslip faulting represents an intermediate stress state (S_{2} = S_{v}), where the maximum horizontal stress is greater than the vertical stress and the minimum horizontal stress is less (S_{Hmax} ≥ S_{v} ≥ S_{hmin}). In this case, faulting occurs when the difference between S_{Hmax} and S_{hmin} is sufficiently large. The angle between the principal stress directions and the strike and dip of active faults is discussed in Chapter 5.
Third, an implicit aspect of Andersonian faulting theory is that the magnitudes of the three principal stresses at any depth are limited by the strength of the crust at depth. An obvious upper limit for stress magnitudes might be the compressive strength of rock. In fact, a more realistic upper limit for the magnitudes of principal stresses in situ is the frictional strength of previously faulted rock, as essentially all rocks at depth contain preexisting fractures and faults (Chapter 4).
Of critical interest in this book is the current state of stress (or perhaps that which existed at the onset of reservoir exploitation) because that is the stress state applicable in the problems of reservoir geomechanics considered in this book. Hence, a point about Figure 1.2 worth emphasizing is that the figure shows the relationship between states of stress and the style of faulting consistent with that stress state. In some parts of the world
Figure1.3 Illustration of how integration of density logs (upper figure) can be used to estimate overburden stress at depth (center figure) or overburden stress gradient (lower figure). Variability of the density logs, as well as the fact that they are often not obtained over the total depth of interest, leads to uncertainty in the calculated overburden (see text).
there is a close correspondence between the current stress field and largescale active faults in the region. Western California (discussed below) is such a region. However, in other regions, the current stress state is not consistent with largescale geologic structures because those structures evolved during previous tectonic regimes, in some cases, regimes that have not been active for tens, or even hundreds, of millions of years. In fact, in some parts of the world there is a marked disagreement between currently active tectonic stresses and the largescale geologic structures defining oil and gas. One example of this is the Tampen Spur area of the northern North Sea (mentioned below and discussed in detail in Chapter 9) where earthquake focal mechanisms and direct stress measurements indicate that there is currently a compressional (strikeslip and reverse faulting) state of stress in much of the area, but the principal geologic structures are those associated with extension and basin formation (normal faulting and subsidence) at the time of opening of the North Atlantic in Cretaceous time, more than 70 million years ago. As discussed in Chapter 9, the compressional stresses in this area appear to arise from lithospheric flexure associated with deglaciation and uplift of Fennoscandia in only the past 20,000 years. In some places in the northern North Sea, after tens of millions of years of fault dormancy, some of the normal faults in the region are being reactivated today as strikeslip and reverse faults in the highly compressional stress field (Wiprut and Zoback 2000). The opposite is true of the eastern foothills of the Andes in Colombia and the Monagas basin of eastern Venezuela. Although extremely high horizontal compression and reverse faulting were responsible for formation of the largescale reverse faults of the region, the current stress regime is much less compressive (strikeslip to normal faulting) (Colmenares and Zoback 2003).© Cambridge University Press
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