Borehole Geophysics Acoustic logging measurements - Review of ...

Borehole Geophysics Acoustic logging measurements - Review of basic physics background - Concept of P- and S-wave measurements and logging tools - Tube waves - Seismic imaging - Synthetic seismograms - Field application examples (oil&gas detection, gas hydrates) - Cross-hole seismic imaging (Guest-Lecture by Gerhard Pratt, Queens) - Vertical Seismic Profiling (Guest-Lecture by Gilles Bellefleur, NRCan)

EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

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Borehole Geophysics

Acoustic logging exploits the fact that acoustic velocity is strongly dependent on the sediment type, pore-fluid filling, and temperature/pressure conditions. The mathematical derivation of the acoustic/elastic wave equation is rather complex and will be dealt with in another class (EPSC-330, Earthquakes and Earth structure). However, for the interested mind, the derivation of the wave equation and stress-strain relationships are given in the textbook by Hearst et al., 2000: “Well logging for physical properties”, pp 256 - 261. For the purpose of this course, we will deal mainly with the more basic concepts and focus on the application side and go through the various dependencies of acoustic velocity on local sediment type and pore-fluid filling as well as bore-hole specific phenomena (guided waves).


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Borehole Geophysics

Waves in an elastic medium An elastic medium can be described as one that recovers completely after being deformed. Most of the phenomena in acoustic logging can be described by the theory of elastic waves in elastic media. In case of non-elastic behavior, new terms (attenuation, Q-factor) are introduced to describe the deviation from the elastic realm. For the theory of elastic waves, it is important to understand the concept of stress and strain. Stress (σ) is defined as force per unit area. If the force is perpendicular to the surface on which it is exerted, it is called a normal stress, if it is tangential it is called shear stress. When an elastic body is under stress changes in shape and size occur, called strains (ε). If an elastic body is under stress, the change in shape is described by displacement vectors. Consider the rectangle below. If all corners received the same displacement (indicated by the arrow), then there is no change in shape or volume, i.e. there is no strain! If, however, only on corner is displaced, then there is an effective strain applied.


Slide S2-3

Borehole Geophysics

Displacements are described as vectors, and their three components to describe the differential movement into the x, y, and z direction are called u, v, and w. In the most general theory, those three vectors are again functions of the three Cartesian coordinates, i.e. u = u(x,y,z) etc. Similar to normal stresses, one defines normal strain: ε xx =

∂u ∂v ∂w ; ε yy = ; ε zz = ∂x ∂y ∂z

(EQ 2-1)

The sum of all the normal strains is the effective change in volume of the elastic body, also denoted as dilatation (Δ).

Hooke’s law In an elastic medium, the strain is directly proportional to the stress that produced it. The total strain is the sum of all strains produced by the individual stresses.; therefore each strain is a linear function of all the stresses (and vice versa): σii = λΔ + 2μεii, i=x,y,z

(EQ 2-2)

σij = μεij, i,j = x,y,z; i≠j

(EQ 2-3)


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Borehole Geophysics

Young’s-modulus; linear relation between stress and strain Longitudinal stress (ΔF / A) / Longitudinal strain (ΔL / L)

Bulk-modulus, describing compressibility of rock Volume stress (ΔP) / Volume strain (ΔV / V)

Shear-modulus, describes ability of rock to be sheared

shear stress (τ) / shear strain (ε); ε = tan (Θ) EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

Slide S2-5

Borehole Geophysics

In the above equations (2-2) and (2-3) the quantities λ and μ are known as Lame’s constants. Since μ is a measure of resistance to shear, it is also call the shear modulus. Using Hooke’s law, the standard relations among the elastic moduli are defined as: Young’s modulus E: Bulk modulus K: Poisson’s ratio ν:

E=

μ (3λ + 2μ ) λ+μ

(EQ 2-4)

3λ + 2μ 3

(EQ 2-5)

K=

ν=

λ

2(λ + μ )

(EQ 2-6)

Fluids cannot sustain shear stresses. Therefore μ for a fluid is always zero and K = λ.


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Borehole Geophysics

We distinguish two types of body waves: P-waves

(also referred to as primary wave)

S-Waves (also referred to as secondary wave) P-wave: An elastic body wave or sound wave in which particles oscillate in the direction the wave propagates. S-wave: An elastic body wave or sound wave in which particles oscillate perpendicular to the direction the wave propagates. Two types of S-waves are distinguished: SH (horizontal) and SV (vertical).

Without further derivation, we can define the P-wave speed (VP) and S-wave speed (VS) as a function of the elastic parameters and density.


VP =

λ + 2μ ρ

(EQ 2-7)

VS =

μ ρ

(EQ 2-8) Slide S2-7

Borehole Geophysics Surface waves.

There are two principle kinds of surface waves, named after the originators of the theories describing them: (1) Rayleigh waves:

SV waves, with a coupled P-wave component.

Particle motion is vertical and perpendicular to wave propagation.

(2) Love Waves:

SH waves

particle motion horizontally and perpendicular to wave propagation

Source: http://www.allshookup.org/quakes/wavetype.htm EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

Slide S2-8

Borehole Geophysics In borehole geophysical applications there is a special type of surface/interface wave that is often the dominant wave-form: Tube waves or Stoneley-waves

In addition to the interface (or tube-) wave, there is a special phenomenon in boreholes, as they act as wave-guide resonators. This is similar to the occurrence of soundchannels in the ocean, where sound is trapped inside a channel that is defined by water masses of different densities. The effect of the wave-guide is that the original spectrum of the acoustic source of the logging tool is greatly altered and (similar to what one experiences with musical instruments) is dominated by modes (overtones). The resonance effect (depending on the diameter of the borehole) creates preferential propagation of certain wave modes and types. The individual modes are also dispersive, creating group- and phase velocity. This dispersion creates an “artificial” amplitude decay that is different from the intrinsic (rockproperty) attenuation (Q, α).


Slide S2-9

Borehole Geophysics A quick reminder about group and phase velocity …

Surface waves are highly dispersive, that means that the waveform changes and spreads out over time (or equivalently with distance from the earthquake). To understand dispersion it is essential to know whether it is the phase (v) or group speed (u) of the wave that is observed. Phase speed (v) is the speed of which a particular phase or element of the wave packet travels. Group speed (u) is the speed of the envelope of the wave packet. From Stacey, 1992 (page 217)


Slide S2-10

Borehole Geophysics A quick reminder about group and phase velocity …

Each frequency of the wave packet will propagate with its appropriate group velocity, and this group velocity is constant for a given frequency, but different for different frequencies. Thus, the location of a given frequency in the wave packet from successive receiving stations will plot on a straight line passing through the origin. The slope of this line is the group velocity. EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

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Borehole Geophysics

In logging we want to exploit the head-waves for P- and S-motion. In plane wave analysis, a wave front is a plane and normal to such a plane is the direction of propagation, or the ray. Refraction/Reflection of plane waves occurs at an interface between two media with a difference in wave speeds c1 and c2 or densities.

Wavefronts are generated using Huygen’s principle.


Slide S2-12

Borehole Geophysics

t C → B = t A→ D

CB = AB sin α 1 AD = AB sin α 2

CB AD = V1 V2

Snell’s law:

sin α 1 sin α 2 = ≡p V1 V2

⇒

sin α 1 V1 = sin α 2 V2

(EQ 2-9)

with the constant p as ray-parameter.

V1 V2 α1 is called the critical angle of incidence. If a ray impinges the interface at that critical angle, the refracted (or head) wave will propagate in half-space II parallel to the interface at the speed of V2. In case of an open borehole, V1 is the velocity of the drilling-fluid (mud or water), V2 is the formation velocity. If V2 > V1, then sin α2 > sin α1; and in the limit of sin α2 = 1, we get:


sin α 1 =

Slide S2-13

Borehole Geophysics

Illustration of headwaves (a) and their representative raypaths (b) for a single transmitter (T) and receiver (R) in a logging tool.

Source: “Well logging for physical properties”, Hearst et al.,2000, page 289 EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

Slide S2-14

Borehole Geophysics Back to the wave-guide …


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Borehole Geophysics Basic acoustic logging tools In most common acoustic transducers the piezoelectric effect is used to convert mechanical wave energy into electrical signals. Many kinds of crystals (e.g. quartz) have a lattice structure such that, when stress is applied to a slab of the crystal in the proper direction with respect to the crystal axes, the centers of positive and negative charge in the crystal are separated and a surface charge is induced. Transducers can be used in both ways – receiver and transmitter. Single-transmitter tools: Commonly, the transducer is mounted along the axis on the outside of a logging tool. It is usually coupled to the drilling fluid either by being in direct contact to the fluid or immersed in a liquid-filled bag that in turn is in contact to the drilling fluid. Care must be taken to isolate the transmitter from the receiver to avoid acoustic signal transmission through the tool itself. The choice of frequency is critically important (see above examples) for the generation of useful acoustic logs.


Slide S2-16

Borehole Geophysics

In surface seismic data acquisition, the broad-band source is chosen under consideration of expected target depth, required resolution, velocity and attenuation. For logging tools, the criteria are different and are mainly determined by the size of the borehole. Optimum logging tool response occurs when the source excites a single pair of P- and S-wave peaks in the borehole response. Frequencies of commercial logging tools vary, but are normally within the kHz domain (1050 kHz). A single pair of transducer/receiver is often insufficient and 2 (or more) pairs are combined, which is called a borehole compensated system.


Slide S2-17

Borehole Geophysics

Example of the influence of source frequency (f), borehole radius (R) and the result in source/borehole response spectrum.


Slide S2-18

Borehole Geophysics

The tilt of the tool in the right image is exaggerated and in reality the tilt-angle (Θ) is very small relative to the length of the tool (L > 5 m) and thus can be neglected in the averaging of the travel time and wavepaths. 0.5 *(t21-t22+t12-t11) = 1/V * L* cos(Θ) t21 = travel-time from transmitter 1 to receiver 2


Slide S2-19

Borehole Geophysics Display of acoustic log-data

Display of full waveform as a wiggle-plot


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Borehole Geophysics

Display of full waveform as a variable density plot (also in color).

Source: “Well logging for physical properties”, Hearst et al.,2000, page 293


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Borehole Geophysics

It is often of better use to display the coherence of the waveform, which in the example in the right shows the powerful nature of separating the interface wave phase (here the constant lowvelocity event) from the actual formation velocity of interest.

Source: IODP Expedition 311, Logging Summary EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

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Borehole Geophysics

But what we really want is a single velocity log! How does one process and interpret the acoustic waveform data?

If identification of the P- and S-wave is easy, first-arrival picking yields travel-times (Δt), which can be converted to velocity if the distance between source (transmitter) and receiver is known. Commercial sonic logs are often displayed in terms of slowness, which is the inverse of velocity, and has the unit μs/ft or μs/m. The picking process is tedious and thus is often automated (guided picking through search for the maximum in waveform coherence). Manual quality controls are always required to ensure optimum data quality. Sometimes special processing techniques are required to remove tube waves and/or separate modes in the measured spectrum. Some processing steps will be explained as part of the Vertical seismic profiling chapter and on X-hole imaging. EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

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Borehole Geophysics

Next Tuesday Vp and Vs as function of rock properties Field examples Next Thursday VSP X-hole


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