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Images - Chapter 2 Snapshots of wave propagation. The jet colorscale displayed here will be used to displayed snapshots throughout this chapter.

Let us drop a stone into a tank containing water. A disturbance of a very short duration occurs at the point of impact. While the deformed area returns to its initial equilibrium, the disturbance gradually expands away from the point of impact. This figure shows three snapshots of this phenomenon, known as wave propagation. The disturbance varies with time and position in space. The speed at which the wave moves from one point to another depends on the physical properties of the medium. Energy decay from similar points also depends on the physical properties of the medium as well as the type of wave (body wave, surface wave, etc.) propagating in the medium. These properties and others that we will introduce later allow us to use wave propagation to probe media, even those as complex as the subsurface.

Unfortunately, we cannot see or directly analyze wave propagation in the subsurface because we are dealing with a dark and compact medium. However, we can put sensors at certain locations at the surface and/or inside the earth, through boreholes, to record the evolution of a disturbance with time.

This narrow view of a disturbance is similar to observing the New York marathon from a specific roadside location instead of from an aerial view in a plane. Although we cannot view all the athletes everywhere at the same time, we will have the opportunity to see and judge the speed of each of them. If more people are positioned at different locations on the route, an even more accurate picture of the race can be formed.


Images - Chapter 2 Snapshots with receiver's positions. Only half of the snapshots are shown because they are symmetric. The rectangular dots indicate the receiver's positions. (b) Particle motion of a fluid model.

This figure shows examples of the narrow view of the wavefield from specific points in the medium, to which sensors are attached. Wavefields recorded at these specific locations as a function of time are called seismograms. Based on the seismograms in this figure, we can see that elements of the medium are displaced from their positions of equilibrium at $t=0$, then restored to these positions after time. We can also notice that the movement of each element of the material is coupled to its adjacent elements. An initial displacement of the first element imparts on the second, the second imparts on the third, etc. The net result of this series of interactions is the wave propagation of the initial pulse. Therefore, if enough sensors are available and properly deployed, seismograms can capture the characteristics of wave propagation needed to probe the subsurface.

To summarize: in petroleum seismology we do not have direct access to snapshots of wave propagation in the earth; our seismic
data are limited to seismograms recorded from sensors deployed either on or just below the surface of the earth, or in boreholes. Therefore, problems of prime importance in petroleum seismology include making sure that sensors effectively measure the desired physical quantities and that they are adequately distributed to capture the main characteristics of wave propagation. To properly address these problems, we need to develop an understanding of the wave propagation theory.

A photomicrograph of a sandstone

Images - Chapter 2 A photomicrograph of a sandstone showing very fine laminations of dark-brown humic organic matter.

Let us begin our description of the continuous-medium assumption by recalling the definition of continuous material. A material is considered continuous if it completely fills the space it occupies, leaving no pores or empty spaces, and if its properties can be described by continuous functions.

Like all substances, rock formations are made of atoms. They contain gaps or empty spaces. This feature is especially true for sedimentary rocks, which comprise most petroleum reservoirs. We will disregard the atom scale (microscopic scale) of rock formations and envision them without gaps or empty spaces. Furthermore, we will assume that mathematical functions (force, stress, displacement, and strain) which enter into wave propagation theory are continuous as well as the derivatives of these functions, if they enter. There is one exception to the continuous medium assumption: the physical properties of rock formations. The mathematical functions describing these properties can contain a finite number of surfaces separating regions of continuity. In other words, rock formations can be said to consist of piecewise-continuous regions, separated by interfaces, where the medium parameters are discontinuous.

The assumption of a continuous medium permits us not only to define stress, displacement, and strain at the particle scale (macroscopic scale) instead of at the microscopic scale but also to use the laws of continuous mechanics to study seismic wave propagation and seismic data.

Two additional assumptions often made about the nature of rock formations are that they are linearly elastic and isotropic. Elastic rock formations can return to their initial equilibrium after deformation and, for linearly elastic media, the force-displacement relationship at any point is linear. These assumptions are valid when forces, resulting displacements, and gradients of displacements are small. The word {\it isotropic} means that the physical properties of rock formations are identical in all directions. It should be clearly understood that the isotropic assumption is completely independent of homogeneous and heterogeneous assumption.

Cartesian coordinates

Images - Chapter 2 Configuration of the rectangular Cartesian coordinates.

The history of a given piecewise-continuous elastic medium will be described by the position of each of its particles as a function of time. We will label the particles by the positions they occupy in space at the fixed time t = 0.

To properly define these positions, let us consider the configuration in this figure, where the position is specified by the coordinates { x, y, z } with respect to a fixed orthonormal Cartesian reference frame with the origin O and three mutually
perpendicular base vectors { i1,i2, i3 } in which each vector has unit length. i3 points vertically downward. Sometimes the coordinates of a point { x, y, z } will be called { x1, x2, x3 }. Thus we will use the vector x={ x, y, z }={ x1, x2, x3 } to label a particle throughout its entire history (x is its position at t=0).

1D, 2D, and 3D media

Images - Chapter 2 Illustration of homogeneous, 1D, 2D, and 3D media.

Under the continuous-medium assumption, a rock formation can be characterized as either homogeneous or heterogeneous. A rock formation is homogeneous if its physical properties are invariant with space and time. Otherwise it is heterogeneous.

Four particular cases of heterogeneous media are commonly cited in petroleum seismology studies:

  • The 1D (one-dimensional) case, in which the physical properties are invariant along the x- and y-axis and with time; i.e., the physical properties vary only along the z-axis,

  • The 2D (two-dimensional) case, in which thevphysical properties are invariant along the y-axis and with time; i.e., the physical properties vary along the x- and z-axis,

  • The 3D (three-dimensional) case, in which the physical properties are invariant only with time; i.e., the physical properties vary along the x-, y-, and z-axis,

  • The 4D (four-dimensional) case, in which the physical properties vary with time as well as with position.

A 4D model describing the Gulfaks field

Images - Chapter 2 An illustration of a 4D model describing the Gulfaks field. This model varies in space as well as in time, with time being the fourth dimension. In this figure the time variable takes only two values: 1985 and 1999. The 1999 survey (bottom) clearly shows the effect of production when compared with the baseline survey of 1985 (top). The change in the seismic reflection strength of the top of the reservoir is related not only to the saturation change but also to the original oil-column height. When water replaces oil, the acoustic impedance in the reservoir increases, causing a dimming effect on what used to be a strong response from the top of the reservoir. The strong seismic response from the oil-water contact (OWC) in 1985 has also been dimmed due to production. Red and yellow colors represent a decrease in acoustic impedance, while blue colors represent an increase in acoustic impedance. Structure and fluid content are shown in the cross-sectional models on the right. Interpretation shows that the smaller oil accumulation (to the left of the fault) has been drained by 1999, while much oil is still to be recovered from the main accumulation (to the right of the fault). (Courtesy Statoil).

Traction forces

Images - Chapter 2 Traction forces acting on a particle of a rock formation.

Elastic waves are associated with the local motions of the particles of a solid medium. Particles of a given medium are displaced, then restored to their initial positions after time. Contrary to external forces (processes external to the medium), the forces responsible for restoring a particle to its initial unstressed position of equilibrium are internal to the medium. For that reason, we will call them internal forces or stresses.

Traction forces

Images - Chapter 2 Traction forces acting on a particle of a rock formation. This case illustrates normal traction forces.

Traction forces

Images - Chapter 2 Traction forces acting on a uniformly compressed bar.

Traction forces

Images - Chapter 2 Traction forces applied to a uniformly sheared block.

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