SEISMIC PRINCIPLES

(((((((((((((((((((((((( Chapter 1. SEISMIC PRINCIPLES

-Chapter 1

SEISMIC PRINCIPLES

FIG. 1-1 COMPONENTS OF STRESS

In the above figure, the stress acting upon the faces can be resolved into components. (xy denotes stress parallel to the x- axis and perpendicular to y axis. The stress with the same subscripts is known as normal stress, and with different subscripts are as shearing stress.

Strain:

Strain can be defines as the change occurred in shape and dimension due to stress. There are certain fundamental type of strain, e.g. normal strain and shearing strain.

Dilatation:

The change in volume per unit volume is called dilatation and it is represented

by (.

Hooks Law:

Within elastic limit, stress is proportional to strain. If we define Hooks Law in isotropic media, we get:

(ii = (( + 2( (ii,i = x,y,z

Where ( and ( are known as Lames constant.

Elastic constant:

Youngs modulus,

E = (xx /(xxFor most materials, E is of the order of a mega bar (1012 dyne/cm2).

Poissons ratio,

( = (yy = (zz

(xx (xxUnder hydrostatic pressure, P,

(xx = (yy = (zz = -P

(xy = (yz = (zx = 0

Bulk modulus, K, is the ratio of the pressure to the dilatation.

K = -P/(Substituting these values into the Hooks law, the following relations can be obtained.

E = ((3( + 2();(ii = ( . ; K = 3( + 2( (( + ()

2(( + ()

3

Above relations are strictly within static equilibrium state, where wave equation comes in to remove that restrictions.

Seismic Wave

Wave is a disturbance which travels through the media. When the disturbance (() is in terms of volume change, we denote (=(, and rotational is (=(c along the x-axis.

The above is geometrical aspect of waves, ( is also the function of time (t).

Rock particles deformation motion components using Laplacian transformation:

2/t2 = (2 = 2 /x2 + 2 /y2 + 2 /z2 ..(1)

In one dimension Z in the second law of Newton of motion equation:

1/V22 /t2 = 2 /z2 .(2)Solution:

= ei(wt kz z)

= cos(wt-kzz) + i sin (wt-kzz) (3)

kz : wave number of Z component+ : propagation in +Z direction

- : propagation in Z directionAssumption:

Homogeneous isotropic medium

The simplest form of time variation can be expressed by sine or cosine, such as:

( = A cos 2( k (x Vt);

( = A sin 2(k (1x + my + nz - Vt);

( = B/r cos 2(k (r + Vt)

Where A or B/r means the amplitude of the wave (().

He following terms are frequently used in seismic survey.

Wave length,

( = 1/k,

where k is wave number, k = 2(/(

Period,

T = (/V

Frequency,

( = V/(Type of Seismic Waves:

a) Body wave (P wave and S wave)

b) Surface wave (rayleigh and Love wave)

P wave is known by various name such as dilatational, longitudinal, irrotational,or compressional. It is the first event during an earthquake recording. P wave is defined as,

( = ( (+ 2(

(S wave or shear, transverse, rotational wave is the second event observed in an earthquake records. S wave is defined as,

( ( ((((

( is always larger than (.

If B/d = ( ,

(2 =(2 / (2 = ( x ( . = ( . = - (

(( + 2( (+ 2( 1 - (

As ( tend to zero, value increase up to maximum value, 1/(2

Thus, the S wave ranges from zero up to 70% of P wave velocity.

S wave does not propagate through fluids, as ( = 0.

Surface wave include Rayleigh waves or Love waves known as ground roll, which travel along the free surface of the solid materials.

The amplitude of the Rayleigh waves decreases exponentially with depth. Surface waves are usually found in reflection records with velocities ranging from 100 1,000 m/sec. Approximately.

For the interpretation purpose, the surface waves are essential to eliminate during recording and processing. Propagation of the Love wave is horizontal and transverse with the speed of S wave. It is very seldom to get the Love wave in the course of seismic prospecting because only vertical ground motion is recorded. Love wave is taken up during the earthquake seismology observation.

Seismic Noise

In seismic records, there are two types of information, signal and noise.

Coherent Noise follows across few traces, whereas incoherent noise is dissimilar on all traces. However, closely spaced geophones will give some extent coherent look incoherent noise or random noise. In case of random noise, sum of n random signal is proportional to (n because random noise is out of phase. Whereas sum of n- coherent in phase signal is equal to n. Therefore,

S/N = n / (n = (n

Attenuation of noise is done by adding several random noise, which cancel with each other as they are out of phase. Common depth point method is widely used and is very effective in cancelling several kinds of noise.

Wave Fronts and Ray

FIG. 1-2 WAVE FRONTS AND RAYSWavefronts are the expanding spheres of energy emanating from the source.

Rays are lines that represent the direction of propagation of the wavefronts, and are perpendicular to the wavefronts.

If we begin at the source and connect points on successive wave fronts by perpendicular lines, we have the directional description of the wave propagation. The connecting lines form a ray, which is a simple representation of a three-dimensional phenomenon.

Remember that when we use a ray diagram we are referring to the wave propagation in that particular direction; that is, that wave fronts are perpendicular to the ray at all points.

Spherical Divergence

1/V22 /t2 = 1 /r (r2 - /r) r

Solution is = 1 f (r Vt)

r

A:SPHERICAL WAVE FRONT

B:PLANE WAVE FRONT

ASSUME A B

VALID FOR r LARGE

OR NORMAL INCIDENCE

SPHERICAL DIVERGENCE

CORRECTION MADE IN PROCESSING

FIG. 1-3 SPHERICAL DIVERGENCEElasticity

Seismic exploration naturally depends on the propagation of waves in elastic media. We shall consider the elastic properties of rocks as if they were homogeneous and isotropic, elasticity theory becomes more complicated without these assumptions.

In seismic exploration the continuities in the measured effects serve to indicate any departures from uniform conditions which are interpreted in terms of depth, nature or attitude of geologic units below the surface.

The elastic properties of matter are described by elastic constant formula as follows:

( = k 2/3( = (E .

(5)

(1 + ()(1 - 2()

k = ( + 2/3(= (E .

(6)

(1 + ()(1 - 2()

n = ( = E .

(7)

2(1 - 2()

E = ((3( + 2()

(8)

(( + ()

( = ( . = E - 1

(9)

2(( + () 2(

Where: k - Bulk modulus

n - Shear modulus (Rigidity)

E - Youngs modulus

( - Poissons ratio

( , ( - Lames constants

Elastic wave is composed of several sorts:

Body wave is the wave which is transmitted through the body of material and consisted of two waves:

Longitudinal wave or P-Wave for which the motions of the particles of medium are parallel to the direction of propagation.

Vp = ( k 4/3 n = ( ( + 2(

(

(

( = density

Transverse (Shear) or secondary wave or S-wave for which the motion of particles of the medium are perpendicular to the direction of propagation

Velocity of S-wave is:

Vs = ( n/(= ( E 1 .

( 2(1 + ()

Motion characteristics

Motion characteristics

Compressional or P wave

Rotational or Shear waves

FIG. 1-4. COMPRESSIONAL AND SHEAR WAVES

Surface wave is the wave at boundary, and consisted of two waves:

Rayleigh waves (R) are waves at surface of semi-infinite elastic solid. The motion is a sort of combination of longitudinal and transverse vibrations giving rise to an ellipsoidal motion of the particle; the velocity of R-wave is:

VR = 0.9194 ( (/(

if the special case of ( =

Particle motion is elliptical and retrograde, that is the motion at the top of the ellipse is toward the source. The magnitude of the motion decreases with depth.

FIG. 1-5 RAYLEIGH WAVE (GROUND ROLL)

Love wave (L-wave) are transverse waves propagated in surface, which depend on the wavelength and vary between that of transverse waves in the surface layer and that of transverse waves in the lower medium. The ratio of velocities of longitudinal wave (P-wave) and transverse wave (S-wave) is:

Vp = (k/n + 4/3 = ( ( + 2 = ( (1 - ()

Vs

( (1/2 - ()

If ( = 0.25, the ratio will be Vp/Vs = ( 3 = 1.73

For other values of Poissons ratio, the velocity ratios are:

(00.10.20.30.40.5

Vp/Vs1.411.501.631.872.45(

If ( = , the ratios of the velocities of the 3 types of waves are:

VP : VL : VR = 1 : 0.5773 : 0.5308

Reflection and Transmission Coefficients - Zoeppritz(s Equation

The relative portions of the energy transmitted and reflected are determined by the contrast in the acoustic impedances of the rocks on each side of the interface. It is difficult to precisely relate acoustic impedance to tangible rock properties but, the harder the rock the higher is the acoustic impedance.

The acoustic impedance of a rock is the product of its density and the velocity of longitudinal or compressional seismic wave through it, (V, designated Z.

Consider a P-ray of amplitude A0, normally incident on an interface between two media of differing velocities and densities.

A transmitted of ray of amplitude A2 travels on through the interface in the same direction as the incident ray, and a reflected ray of amplitude A1 returns to the source along the path of the incident ray.

(A) NORMAL INCIDENCE

(B) P AND S WAVES

(A)REFLECTED AND TRANSMITTED WAVES ASSOCIATED WITH A WAVE NORMALLY INCIDENT ON AN INTERFACE OF ACOUSTIC IMPEDANCE CONTRAST

(B)REFLECTED AND REFRACTED P- AND S- WAVES GENERATED BY A P-WAVE OBLIQUELY INCIDENT ON AN INTERFACE OF ACOUSTIC IMPEDANCE CONTRASTFIG. 1-6 ZOEPRITZS EQUATIONS

The transmission coefficient is the ratio of the amplitude to the incident amplitude:

T = A2/A1

When a P-ray strike an interface at an angle, both reflected and transmitted P-rays are generated as in the case of normal incidence. However, some of incident compressional energy is converted to the reflected and transmitted shear rays which are polarized in vertical plane. Zoeppritz(s equation gives the amplitudes of the four components as a function of the angle of incidence. The converted rays contain information that can help identifying fractured zones in the reservoir rocks. In this text, however, we shall discuss compressional waves only.

Snell(s Law

Snell(s Law, originally applied to light and optics, applies equally well to seismic waves and the earth. For a reflected ray, Snell(s Law states that the angle ( between the reflected ray and the normal to reflecting surface is equal to angle between the reflected the incident ray and the normal to reflecting surface. In seismology, of course, the reflecting surface is the boundary between two layers having different acoustic impedances.

(A) SNELLS LAW

(B) CRITICAL ANGLE

(A)Part of an obliquely incident ray is reflected at the angle of incidence. And part is transmitted at an angle that depends on the ratio of the velocities in the two layers.

(b)A head wave is generated in the upper layer by a wave propagating through the lower layer along the boundary.FIG. 1-7 SNELLS LAW AND CRITICAL ANGLE

Critical Angle and Head Waves

When the velocity is higher in the underlying layer there is a particular angle of incidence, known as the critical angle, (c , for which that angle of refraction is 900. This gives rise to critically-refracted ray that travels along the interface at the higher velocity V2 with equation as follows:

(Sin (c/V1 = (Sin 900)/V2

(c = Sin-1(V1/V2)

This wave, known as a head wave, passed up obliquely through the upper layer toward the surface, as shown in figure 1-7 (b).

Reflection Travel Time

The underlying principle of reflection method is as simple as that of calculating the distance of, say a wall by time required for en echo to be reflected back from the wall and the speed of propagation of sound waves.

Lets calculate travel time and thickness of subsurface layer parallel to surface (see figure 1-8):

Travel time= T(ERG) = t = ER + RG

V0 V0

= 2 ( (X2/4 + h20 t = ( ( X2 4h20)

V0

V0

h0 = 1 ( (V20t2 X2)

2

The portion of the incident energy that is not reflected is transmitted ray travels through the second layer. The transmitted ray travels though the second layer with changed direction of propagation, and is referred to as a refracted ray. Snell(s law of refraction states that the ratio of sine of the angle ( to the velocity is a constant. For a refracted P-ray;

(Sin (1)/V1 = (Sin (2)/V2Sin (1 /Sin (2 = V1/V2were the subscripts refer to layer 1 and layer 2, respectively. See figure 1-7 (a).

Refraction

In the refraction method of seismic prospecting, the quantity observed is the time between the initiation of the seismic wave by an explosion and the first disturbance indicates by seismic detector at a measured distance from the shot point.In general, the refractions take place according to Snells law (see figure 2-2):

i,r (or 1, 2)-the angles between the normal to boundary and the rays

V0 ,V1-velocities of different medium

In horizontal plane and discontinuity, refraction will be determined by (see Fig.2):

FIG. 1-9

TE = T(EABG) = f (x, ho, V0 ,V1)

= EA + AB + BG

V0 V1 V0

= 2 ho . + X - 2 h tan (0.1

V0 cos(0.1 V1 V1

= X + 2 ho - 2 ho sin(0.1 sin(0.1

V1 V0 cos(0.1 cos(0.1 V0 TE= X + 2 ho ( 1 - sin2(0.1 )

V1 V0 cos(0.Travel time = TE= X + 2 ho cos(0.1

V1 V0

Thickness ho can be calculated as follows:

Intercept time (I):

I = 2 ho cos(0.1

V0

ho = I V0 = I V0 V1 .

2cos(0.12( V21 V20

ho = 1 I V0 V1 .

2 ( V21 V20

Huygen(s Principle

This principles states that every point on the primary wave front surface is a source of secondary wavelets. The position of the wave front at a later instant then is found by constructing a surface tangent to all secondary wavelets. This concept is a very powerful tool for understanding all types of wave propagation, from electromagnetic waves to seismic waves.

Huygens( principle, illustrated in figure 1-13, regards each point on the advancing subsurface wave as a source that generates a new wavefront, which radiate in all directions. It explains one of the most important mechanism by which a propagating seismic pulse loses energy with depth.

Spreading primary wavefronts

Secondary wavefronts

FIG. 1-13 HUYGENS( PRINCIPLE

Diffraction

When seismic waves strike any irregularity along a surface such as a corner or a point where there is a sudden charge of curvature, the irregular feature acts as a point source for radiating waves in all directions in accordance.

Figure 1-14 illustrates a buried corner at A, from which waves, exited by radiation downward from a source at the surface, spread out in all directions paths which are rectilinear as long as the velocity is constant. A diffracted wave reaches the surface first at a point directly above the edge because the path is the shortest at this point. The amplitude of a diffracted wave falls off rapidly with distance from the nearest point to the source; diffracted events are frequently observed on seismic records but not always recognized.

FIG. 1-14

FIG. 1-8

hO

X

R

G

E

EXPLORATION GEOPHYSICS FOR GEOLOGIST AND ENGINEER1 11Course Instructor: DR. Prihadi S.A.

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