Acoustics of Porous Media

+ cp

tr

t:

(2.72}

and using (2.66), we obtain:

cp div U

..!!... = M

cp) tr t:

- ({J -

(2.73)

It is also known that, under the assumption of small displacements, the defmition of K 11 yields:

(2.74)

- p = K 11 div U

Replacing in Eq. (2. 73) div U and tr e by their values as a function of p [Eqs. (2. 74) and (2.71)], we fmd:

{3-cp

1

cp

-=--+M

· K,

(2.75)

K 1,

Equations (2.70), (2.75) and the equation: K 0 = K 1 - {3 2 M

(2.76)

give us K 1 as a function of K 0 , K 1 , and K,:

1

Kf

1]

1

1

cp [ K,- K 11 + K,- K 0

:0 [~. -:IJ + ~. [ ~. - ~]

(2.77)

This equation is known as Gassmann's equation ( 1951 ). Gassmann obtained it directly by considerations of elementary elasticity, without requiring the use of p and M. The reader can refer to White (1983 b). It should be emphasized. however, that it is implicitly understood that the porosity is uniform throughout the sample. Note that Eqs. (2.70), (2.75) and (2.77) are quite compatible with limit cases. For a solid medium corresponding to P = cp = 0, the expected values K 1 = K. = K 0 and M -+ oo are obtained. For a fluid medium corresponding toP= cp = 1, we obtain K 0 = 0 and KI=Kit=M.

We have stated that a fluid medium corresponds to P= 1 and K 1 = K 11 = M ( = ).1 because J.1. = 0). These equalities can help to understand the significance of the displacements U and u. In order for a medium to be a fluid. the macroscopic stresses given by (2.65) must be identified with - pbii• hence: ().1 - {JM)

tr t: =

M~({J-

II

(2.78)

Since this must be satisfied whatever the local increase in fluid content eand whatever the apparent macroscopic volumetric strain tr t: , the previously mentioned equalities are obtained. P = 1 and i.1 = M. The value of P = 1 introduced in (2.65), jointly with (2.58) and (2.601, where cp = 1 and tr e = div u, then leads to: - p = M di\' U

(2.79)

----------------------

1 2

WAVE PROPAGATION IN SATURATED POROt.:S MEDIA

69

which corresponds to the equation of perfect fluid behavior, for small displacements, where M is the fluid bulk modulus. This helps to understand why U, the average fluid displacement, is not identifted with the macroscopic displacement u, when the medium is totally fluid. In fact, ifU were equal to u, the rise in water content would be~== Oand(2.78) would only yield ).1 = (JM. Thus (J would not ncecessarily be equal to 1 and one could not obtain (2. 79). In fact. u is the macroscopic average on the reference geometric element considered, while U is an average of the displacement oi the fluid contained in this element and of the fluid which has left or has entered it 18 l. Simultaneously, in the general case, u is not the average displacement of the skeleton, but the average, on the reference geometric element, oi the displacement of the fluid part and of the skeleton part.

2.2.2.3

Dissipation pseudo-potential

Dissipations are assumed to result only from the relative fluid/matrix movement. In the neighborhood of equilibrium, it can be stated that the flow vector wand the associated force X are linked by a linear equation such as:

w1 =

:KiJXJ

(2.80)

Onsager's principle, based on microscopic considerations of symmetry, also stipulates that the tensor :K = (:Kii) is symmetrical. Based on (280), a dissipation pseudo-potential [)can be introduced (for example see Germain, 1973),which is a positive defmitequadratic form of representative matrix -!"- 1 , such that: 1r . • []) =- w:K- 1 w

2

Xi=

-

ao ow,

(2.81a) (2.81b)

The term pseudo-potential, due to Eq. (2.81b), recalls that the concept is only justified in the vicinity of thermodynamic equilibrium. In the isotropic case, the tensor :K - 1 is proportional to unity (-!" - 1 = :Kl) and the previous equations become : 1 .2 [) = 2:£ Wi 1 .

--w, X ,:K

(2.82a) (2.82b)

Darcy's classic law can be recognized in Eq. (2.82b) if :K is identified with the hydraulic permeability of the medium and force X with the opposite of the pressure gradient. As shown below, this identification is only possible in steady~state conditions.

(8) This difficulty stems from the fact that the macroscopic des."7iption is a priori Lagrangian, whereas that concerning the fluid is Eulerian. Nevertheless only small displacemnts are considered for the solid and fluid and the two descriptions are equivalent.

70

WAVE PROPAGATION IN SATURATED POROt:S MEDIA

2.2.2.4

2

Kinetic energy

Since the wavelength A is assumed to be large in comparison with the dimensions of the macroscopic elementary volume dQ (i.e. dQ - / 3 and l, .1 ~ 1), one can restrict the expansion of the volumetric density C of kinetic energy to the quadratic terms: 2C = p,ii;U; + 2p.,,..U;Ii·;

+ p,..li';li·;

{.2.83)

The absence of terms such as w;wi is due to the assumed isotropy. The specific case in which no overall movement occurs (i.e. w = 0) serves to identify p., with the average density in Eq. (2.83): p.,

= p = (1- c/J)p, +
{2.84)

where p. and p1 are the matrix and fluid densities respectively. The terms Puw and p,.. will be identif1ed subsequently.

2.2.2.5

Equations of movement

Let l be the volumetric Lagrangian density deftned by: 12.85)

D..=C-\1

Hamilton's principle (for example, see Achenbach, 1973) ~tipulates that, among all the possible paths between any two points in time t 1 and t 2 , the one that prevails will be the one that makes stationary the integral over time and space of the Lagrangian density and of the work ofthe dissipated forces, where the latter are derived from a (pseudo)-potential. Ifl depends on the parameters q;, i]; and qi,i• and the potential of dissipative forces UJ on i]; . . the variational calculus reflects this condition by the classic Euler equations:

a aL a aa.. cL aiD --+----+-=0 at aq_l

axj aqi,j

cq;

oiJ;

12.86)

where q1 = u1 or w; in our case. The application of (2.86), by using the expressions of ID (2.81a), C and V (2.83) and (2.63), leads to the equations of movement: uli.i =

- P.i

pii;

+ p.,..,w;

1 = P.,wul- + p,..w;- + .)('

(.2.87a) .

W;

(.2.87b)

These equations serve to identify the parameters Puw and Pw· If there is no average relative fluid movement with respect to the overall macroscopic movement (i.e. w = 0): the f1rst equation is reduced to the equation of movement in a continuous medium (see Section 2.1.2), while the second must be reduced to the equation of movement in a fluid. thus Puw = p1 . Now if the fluid is at rest (w = - )ii;-; il;

--

-

~-

- - -

-~~~~~~~~~~~~~~~~~-

12.88)

t '~

r I

2

WAVE PROPAGATION IN SATURATED I'()ROUS MEDIA

71

This shows that, if overall acceleration occurs. ~ force must be exerted on the fluid to prevent its average displacement. For its inertial rart, this coupling force is:

j

(p 1 -

p,..)iif

(2.89)

To describe this coupling effect, similar to the qtass effect added in the analysis of the movement of an obstacle in a fluid, it is usual tO introduce a parameter a called the tortuosity parameter 191 such that: a

12.90)

Pw=;pP! I

One must necessarily have: a~

(2.91)

1 I

because the coupling force (1 - a)p1 ii1 must o~viously be opposed to the overall acceleration. This condition (2.91) ensures the n~n-negative character of the quadratic form associated by (2.83) with the kinetic ener*y density C. Like parameter {J, the tortuosity a is related not only to porosity but alsd to the geometry of the medium where the flow occurs. Hence the equation relating a to 4> ~s not biunivocal. Note, however. that a · must tend toward 1 as

sed: a

1[1 i:]

= l ;j; +

(2.92)

As already observed, the equation linking a w1th

F(c/J)c/J,

(2.93)

+ p 1 ii~ + p,.l\·i)

(2.94)

Equation (2.87b) is thus written: wi

=-

f(P.i

As previously noted, it is only in the case of st~ady flow ii = w = 0 that Eq. (~.94) is identif1ed with the classic Darcy's law. In the unstdady case, inertial effects are added and the comparison of (2.94) with (2.80) shows that the force associated with w1 is: Xi= - (P.I + P!iii 1+ p,.l\\)

(2.95)

(9) This tortuosity related to a dynamic process is differenl in principle from that defmed in Secuon 1.4.1, which is only related to the description of the porous mediuiil· In reality, these two quantities are of the same order of magnitude, because the same geometric characteristjc is involved.

72

WAVE PROPAGATION IN SATURATED POROUS MEDIA

2

Hence it should be noted that, even in a case of a zero pressure gradient, a fluid flow can be generated by inertial forces alone. It is customary to derive the classic Darcy's law (in steady-state conditions) from the Navier-Stokes microscopic equations for a fluid of viscosity '7, by making an analogy with Poiseuille's 'law (Mandel. 1950). This gives rise to the expression of the hydraulic permeability or mobility in the form: K

f='1

(2.96)

where "· the absolute permeability, depends only on the geometry of the porous medium. Note however that, as it appears here, Darcy's law only results from considerations of linearity in the vicinity of equilibrium. Hence, as already pointed out (Mat heron, 1967), the linear character of Darcy's law, obtained by the standard procedure mentioned above, actually only results from the linear character of the Navier-Stokes equations for flows at sufficiently low velocity. Only the form (2.96) of f involves the law of linear viscous behavior which these equations imply. To conclude, this law in the form of(2.94), as noted in the introduction, can be obtained by a rigorous homogenization method, once again based on the validity of the NavierStokes equations, but in this case in unsteady-state conditions. Moreo,·er, this method shows that for wavelengths that are large only in comparison with the dimensions of the elementary flow channels, and no longer in comparison with the dimensions of the macroscopic element, as required in the method developed here, the permeability is no longer absolute, but related to the frequency of the wave which generates the flow. Previously this result had been qualitatively found by Biot (1956 b).

2.2.3

Wave propagation

The introduction of the equations of stresses as a function of displacements u and U in Eq. (2.87) yields the equations of motion in the form:

+ 211) grad div u + Y grad div U - 11 curl curl u = P11 ii + P12 0 + b(u- iJ} Ygrad div u + R grad div U = p 12 ii + P:z 2 0- b(u- VJ (i.

(2.97a) (2.97b)

in which we noted :

i. R P12

+ Mcp(cp- 2P) = l 0 + M(P- c/J) 1 • Y= = Mcp 2 , p 11 = p + cpp1(a- 2)

=

i.1

= tPPJ(1- a),

P22

= acppf,

.\lcfJ(P- )

(2.97c)

cp2

b

=f

Note that Eqs. (2.97) can be written for the limit cases of the perfect fluid and the solid. In fact, if the medium is a perfect fluid, the set of parameters to be considered is cp = P=a= 1, i.1 = M and 11 = b = 0 (%-+ + Xi, '1 = 01. Equation (2.97a) disappears, and Eq. (2.97b) gives the dynamic equation of perfect fluids under the assumption of small displacements: M grad div u = p 1 0 (2.98)

2

WAVE PROPAGATION IN SATURATED POROUS MEDIA

73

Note also that Eq. (2.97a) disappears naturally and not by assuming U = u (see Section 2.2.2.1 ). Now, if the medium is solid, letting P = (jJ = 0, a -. + oo, Eq. (2.97b) then gi\·es:

I

,o =- a
(2.99)

~I

while Eq. (2.97a) gives:

~I

Equation (2.100) is clearly the equation of the dynamics of elastic solids, since the second member is cancelled owing to (2.99). Note that this cancellation takes place independently of the manner in which tortuosity tends to infmity and permeability to 0, since the equations are not related to these asymptotic behaviors. Moreover, if decoupling occurs (see Section 2.2.4), P=

11

ij

(A. 1 + 2J.L)graddivu- J.LCurlcurlu- p,.ii

2.2.3.1

=a
1 ()

+ b(u- lJ)

12.100)

Existence of a slow P wave

Let us f1rst examine the case without dissipation (b = 0) and let us consider waves such . that:

(Uu)

= (grad 4>1) = ad cJ) grad 4> 2 gr

12.101)

These waves correspond to dilatational waves (P waves) that are irrotational (curl u =curl U = 0). By introducing (2.101) in (2.97) with b = 0, the following equation is obtained for the potential vector:

R172cJ) - MiD =

0

12.102)

where R and M are the rigidity and mass matrices respectively:

"""

"""

R = """

(A. + 2J.L Y

Y)• M = (Ptt P12) """ P12 P22

~

12.103)

As the matrix R - l M is positive defmite and symmetrical, it has two real positive eigenvalues which we denote V~, and V~ •. In the eigenvector reference system, Eq. 12.102) can then be written :

J72cJ)* -

_I 0) ( v~.

_1_

iD• = o

12.104)

o v2P2

where cJ)* is determined from cJ) by the change in reference system. We can identify here two equations of decoupled waves, corresponding to the two Pwave velocities Vr, and Vp 2 , and to the two characteristic movements defmed by the eigenvectors of R - t M. It can be shown (Biot, 1956a) that one of the characteristic movements corresponds to a movement in which overall and fluid displacements are in phase, and the second to a movement in which the displacements are out of phase. The

74

WAVE PROPAGATION IN SATURATED POROUS MEDIA

2

wave corresponding to the latter case is called the slow wave or the wave of the second kind. This terminology derives from the fact that the associated velocity Vp,, as shown below, is much lower than the velocity Vp, of the in·phase movement waves, called waves of the f1rst kind. These waves correspond to classic P-waves, with which they merge in the absence of fluid. Now let us consider the shear waves (S waves) or isovolumetric waves (div u = div U = 0) such that: u) =(curl 'I' 1 ) = curl 'I' (U curl '1' 2

(2.105)

Equation (2.105), introduced into (2.97), leads to the equations: 2

17 'I' 1

-

2

'I' = _ where velocity

Vs

is given by:

Vs

vz1 'I' 1 = s

0

(P12)'1'1

(2.106)

P22

=( ~ P12)i P11

(2.107)

P22

Since the fluid does not respond to shear forces, it only influences the shear wave through inertial effects. Its movement is evidently in phase with the overall movement [see (2.106) where P121P22 ~ 0]. We have only discussed the isotropic case here. For the non-isotropic case, the results naturally extend as follows: along the principal anisotropy directions if a compressional wave exists in the purely elastic case, two compressional waves, one slow and one · standard, must be considered within the framework ofBiot's theory. From the qualitative standpoint, the shear waves are altered in the same way as in the isotropic case, in accordance \\ith the processes that we shall now discuss.

2.2.3.2

Wave velocities and attenuations

Let us now consider the general case with dissipative effects (b =F 0). Introducing (2.101) into (2.97), we obtain: R 17 2 $ = A cj,

where

~

+ Mii>

(2.108)

is the damping matrix given by:

A=(_:-:)

(2.109)

Let us examine the case of plane harmonic P waves propagating in the x direction, such that: ~~>1 ~~>2

= 4> 10 exp [i(kx- wt)] = 4> 20 exp [i(kx- wt)]

(2.110)

1

1·

I I

T I

2

WAVE PROPAOATION.IN SATURATED POROUS MEDIA

75

where cf> 10 and c1>;z 0 are constants, and·w and k are the angular frequency and wave number respectively. The introduction ofthese equations into (2.108) gives a system of two equations for cf> 1 0 and cf> 20 • The Kramer's determinant of this homogeneous system must be zero in order for cf> 10 and cf> 20 to have non-zero values. This leads to an equation linking k with w. This equation allows two solutions k,., and k,. 2 , which correspond to the flfSt and second kind P waves previously discussed, and which are now complex owing to attenuation effects: k,.,(w) = Re k,.,(w)

+ i Im

k,.,(w)

(2.111)

Introduced into (2.110), these equations show that the imaginary part is the attenuation coefficient a.,., such that the corresponding wave amplitude is proportional to exp (- :x,.,x). while the real part is related to the velocity v,.,: (Re k,.,(w)) V,.,(w)

=w

(2.112)

The same procedure can be followed for the shear wave. Figures 2.7 to 2.12 show the curves relative to these calculations. The P and S wave velocities are normalized respectively by v,. and Vs whose expressions are: 1

V,.=C'I~2J.Ly Vs

=

(~t

(2.113) (2.114)

These expressions correspond to the velocities of the P and S waves if no relative movement occurs between the fluid and the overall movement (closed system). These are reference -velocities corresponding-to the limit velocities of the farst kind P wave and the S wave, wlie~ the frequency tends towards zero. Effectively, within this quasi-static limit, the effects of inertia contrast and dissipative effects become negligible. The frequencies are normalized by a characteristic frequency fc which depends on the ftltering medium :

(2.115)

The attenuation coeffacients are normalized for the compressional and shear waves respectively by :

21t/c

a,.=--

v,.

21t/c

(2.116)

as=--

Vs

The curves in Fags. 2.7 to 2.12 represent the relative variations of velocities and attenuations for the three wave types, with a constant ratio of fluid bulk modulus to solid bulk modulus (i.e. K 11 / K. = constant) equal to 0.06. This value corresponds to a watersaturated siliceous matrix (Wyllie et al., 1963). We varied the ratio ofthe open system bulk modulus to the solid bulk modulus (i.e. K 0 / K. = 0.3, 0.6 and 0.9). The broken, solid and dotted line curves correspond respecti\'ely to the porosities

76

1.0006

WAVE

PROPAG.-\T!O~

r-

Fig. 2.7 Velocity of P1 waves of the fvst kind vs. frequency for different values of K 0 K. (0.3, 0.6 and 0.9) and different porosities (10%. 20% and 30%).

Porosit~'ll.

.·

10 -20

1.()()()4

vi'

H- --I 30

2

I"!' SATL:RATED POROUS !\lEOlA

.6~.··

1

Vp

v 1.0002

v

1.0000 I

se; !1?.7:':':':'

0

.03

• .12

.09

.06

.15

f/fc

l PorosityT ~ % .0010 Li ......

I 10

......

j20

I

·~

- · - I 30

I

K

~- .9 K - .6

I

I

,

,,

5

.9

ap1 •p

I

I

,

•"

,-

a/

.0006

;', .3 , , 2~ .6/77

Fig. 2.8 Attenuation of P1 waves of the f1rst kind vs. frequency for different values of K 0 iK, (0.3, 0.6 and 0.9) and different porosities (10%, 20°1o and 30% ).

... ... ,_. ~'.9~······ -···

.0002

0

.03

.06

.09 t/fc

.12

.15

2

77

WAVE PROPAGA110N IN SATURATED POROUS MEDIA

.20 , . . . - - - - - - - - - - - - - - - - - - - - ,

Fig. 1.9 Velocity of P2 waves of the seeond kind cslow waves) vs. frequency for different values of K 0 ! K, (0.3, 0.6 and 0.9) and different porosities (10%, 20% and 30%).

~ I

-

.15

Vp2

.10

Vp

.05

.00~----~------L------~------~----~

.00

.03

.09

.06

.12

.15

f/fc

.6

.9

I

1 1

.6

.6

.9

1 1 1

.6

.3

I

I

.5

.4

ap2

ap

.3

.2

Fig.l.lO Attenuation of P2 waves of the second kind (slow waves) vs. frequency for different values of K 0 /K, (0.3, 0.6 and 0.9) and different porosities (10%, 20% and 30%).

.1

,

,.......... I -----

20 30

.0

.oo

.03

.06

.09 f/fc

.12

.15

78

I. p~~~s-i~~ .I

1.006 ~

Fig. 2.11 Velocity of shear S waves vs. frequency for different values of K 0 /K, (0.3. 0.6 and 0.9) and different porosities (10%, 20%, 30%).

9'>% 10

20 1.004

2

WAVE PROPAGATION IN SATURATED POROUS MEDIA

~------I

I

30

Vs V 5 1ol

Ko

Forany - Ks

1.002

.,. ,.,. .- "'

_,-'

_,

.,""

""

.--'

... -··

1.000 .00

.03

.06

.09

.15

.12

fife

.006 ~------------------------------------~ Porosity

I

¢%

.005

10 20 30

.004

erg

as

.003

.002

Fig. 2.12 Attenuation of shear S waves vs. frequency for different values of K 0 /K, (0.3, 0.6 and 0.9) and different porosities ( 10%, 20%, and 30%).

For any

_.... ____ .,..._,_-"":""..............,

.001

;'

....

··········· ·············

.000 .00

..

Ko Ks

.03

.06

.09 fife

.12

.15

T I 'Cl

·'

WAVE PROPAGATION IN SATuRATED POROUS MEDIA

79

remaining parameters of the theory are inferred from the above by the equations of Gassmann (2. 70), (2. 75) and (2. 77) for M, fJ and K 1 , and by Berryman's equation (2.92) for tortuosity.

~

().0\

For the three wave types, the velocity can be observed to rise with increasing frequency. This is explained by the fact that the inertial forces increase simultaneously. Indeed, these inertial forces being different for the fluid and the solid part, they generate a differential movement between the fluid and the fluid/solid combination due to permeability effects (Darcy's law). It implies that the overall movement defmed by u involves less fluid, whatever the type of wave considered. Hence as the mass involved in overall movement declines progressively with increasing frequency, the velocity increases with frequency. However, the differential movement thus facilitated by the increase in frequency causes increasing dissipation. This dissipation is proportional to the square of the angular frequency. For the three waves, this means increasing attenuation with frequency. Note also the more highly attenuated character of the slow wave in comparison with the other two (see scales), a characteristic that also limits the increase in velocity V,., corresponding to the high frequencies. This leads us to the following paradoxical situation: the more the phenomenon might tend to occur due to the velocity increasing as the frequency increases, the more this phenomenon would be attenuated.

l"t~

l~

15

2

.I

i

From the general standpoint, note also that permeability only affects the abscissa scale, normalized by frequency f. defmed by Eq. (2.115). More speciftcally, it can be shown numerically that, for commonly used fluids (water, glycerine, kerosene), it is the parameter ,, and hence the viscosity, that is mainly involved, independent of the density or bulk modulus K 11 . As the permeability~ = K/'1 approaches 0 (or towards infmity), the frequency f. tends inversely towards infmity (or towards 0). The rise in the curves with increasing frequency on an absolute scale is accordingly less (or more) pronounced. This agrees with the physicalevidence: the lower (of ~igher) the.~rmeability, the less (or more) are the differential movements (fluid,'matrix) privileged and the less (or more) Biofs effects are pronounced. Once again, however, a paradoxical situation results: the lower the viscosity 'I and hence the higher the permeability ~.the greater the attenuation. This is fairly easy to understand, because the lower the viscosity, the more the differential movement may be pronounced (i.e. less fluid is involved in the overall movement) and hence wis greater and the dissipation given by (2. 72a) (but tempered by the inverse of ~) increases. Actually, therefore, there is no para
80

2

2

WAVE PROPAGATION I]'; SATURATED POROCS MEDIA TABLE 2.3 SoME VALVES OF BlOTS CHARACTERISTIC FREQL"ESCY

~

s

Fontainebleau sandstonem .. Fontainebleau sandstone''' .. Tight sand m . .............. Cordova Cream limcstonel2l . Sintercd glass 131 ••••••••••••

~

Characteristic frequency Porosity IPermeabili t y {mD) (%)

Water

Normal oil

Heavy oil

80 MHz 30kHz 1 GHz 4.5 MHz 42kHz

800 to 4000 MHz 300 to 1500kHz 10 to 50 GHz 45 to 230 MHz 420 to 2100 kHz

8 to 400Hz 3 to 15 MHz 100 to 500 GHz 450 to 2 300 MHz 4.2 to 21 MHz

~

t

'I= I cP) 141 ('1= 10to50cP)(41 ('I= IOOto500cP)( 41

5 20 8 24.5 28.3

to-• 1000 2 10- 2 9 1000

(1) Bourbie and Zinszner (1985). (2) Carmichael (1982). (3) Plona and Johnson (1980). (4) Viscosity 'I is expressed in ccntipoiscs (1 cP = 1 mPa . s).

A glance at the situation depicted in the table tends to show that the very porous Fontainebleau sandstone and sintered glass display the same behavior concerning slow waves. In fact, the slow wave is also sensitive to the pore radius through the skin effect. This fact will be discussed in the experimental veriftcation in the next Section. The curves in Figs. 2. 7 to 2.12 also serve to analyze the influence of the ratio K 0 / K. and of porosity
-----

'

~ ! I .

'

2

WAVE PROPAGATION IN SATURATED POROUS MEDIA

81

a function of porosity. Note also that, as for the shear wave, the tortuosity effects decrease with porosity (less mass added). This merely reinforces the effects discussed above, albeit slightly (see shear wave). A number of considerations can already be drawn from the previous analysis to identify the existence of this slow wave experimentally. It is ftrst necessary for the system to be open and for the permeability to be sufftciently high (low viscosity) for average fluid movements relative to the matrix modeled by this theory to be possible, and for the slow wave to be able to propagate at sufftcient velocities. A critical description of an experiment in which this slow wave was identified (Piona, 1982) is given in Section 2.2.6. A fmal remark concerns behavior at low frequencies. In this range, the term Mi:b, proportional to jl, can be ignored in comparison with the term A$, which is proportional to f. Equation (2.108) is thus reduced to: V2 $ = R - 1 A$

where

R -

-1

~

Knowing that Cl» = (

(R + y

b

-

- (i.

+ 2p)R -

:J

(2.117)

-R- y ) i. + 2p + y

Y - Y- i. - 2p

(2.118)

and substracting the two equations for cP 1 and cP 2 obtained

from (2.117), one obtains : C 0 V 2 (cP 1

-

cPz) =


cPz

(2.119)

where we noted :

+ 2p)R - Y2 -JlM _ .v i. + 2!1 + 2 l' + R

_ (i. C o-

A.0 i. I

+ 2p + 2p

+ + 4f3p

_ .v K 0 4/3p -JlM__;;---:-':-:~ KI

(2.120)

Considering the equations relating pressure p, displacements u and U, and potentials cP 1 and cP 2 , Eq. (2.119) can also be written:

CoVzp = p

(2.121)

Thus at low frequency the slow wave corresponds to a diffusive type of propagation mode, governed by the scalar diffusion equation (2.121) for pressure with a hydraulic diffusivity coefficient CD. This remark is due to Chandler and Johnson (1981), who showed that the analysis of Rice and Clearly ( 1976) was also included in the theory of Biot. This is even more remarkable because Rice and Cleary introduced the fluid bulk modulus separately from the equation of state of the fluid, whereas Biot's theory treats the matrix . and fluid on the same level from the outset.

2.2.4 Biot's theory and Terzaghi's law For the case of total saturation. Terzaghi's law is frequently used in geotechnics. This law breaks down the macroscopic or total stresses qii into so-called effective stresses

at.

I

82

WAVE PROPAGATION IN SATURATED POROUS MEDIA

2

which are the stresses exerted on the solid skeleton, and into hydrostatic stresses - pi>ii• which are the stresses pre\"ailing in the fluid, so that: (2.122) (jjj = pi>jj

at -

Accordingly, one may well ask what are the connections between Biot's theory and Terzaghi's law. If an experiment is performed at zero pressure (p = 0), the total stress tensor is identif1ed with that of the effective stresses. Equation (2.66) then gives: p = 0

tr

at= 3K

0

tr

8

(2.123)

For low porosities (c/> < .20%), by an elementary theory of effective moduli, it is moreover easy to show that: (2.124) p=O tr = 3K,(1 -
at

where K, is the matrix bulk modulus. From (2.124) and (2.123): K 0 = K,(l- c/>)

(2.125)

It is now possible to use Gassmann's equation (2.77) and Eqs. (2.70) and (2.75) to obtain:

f3

= c/>

K1

= K,(1- c/>) + K 11 c/>

(2.126)

K,z

i\1 = c/>

Using (2.122) and (2.1261 and Biofs equation (2.65), the following general identity is obtained: (2.127a) tr afi = 3(1 -
(2.127b)

Thus Terzaghi's law is more restrictive than Biot's theory, since Eqs. (2.126) reveal the presumably additive character of the bulk moduli. In a closed system, the rheological model attached to the saturated porous medium in relation to compressibility effects consists of two springs in parallel, with. constants K,(1- , concerning the respective contributions of the skeleton and the fluid. In an open system, only the skeleton participates (spring with constant K,(l :.___ c/>) = K 0 ), while the share accounted for by the increase in fluid content in the apparent macroscopic deformation is cf> tr 8 ({3 = cf>). This assumption of additivity, which is implicit in Terzaghi's law, leads to a decoupling between the fluid behavior law (2.127b) and that of the skeleton (2.127a). To evaluate the effective stresses, it is therefore assumed that the medium consists exclusively of the solid skeleton, leading to the average pressure K. tr e. The pressure effect is subtracted and the result multiplied by the factor (1 - c/>) representing the effective volumetric proportion of solid skeleton (2.127a). The fluid behavior law (2.127) involves only the volumetric strain of the fluid, div U, which corresponds to the above decoupling. Note here that the strain tensor 8 concerns the average or macroscopic displacement on the solid/fluid combination (e is not related to the solid part alone). Moreo,·er, Biot's theory concerns only small disturbances, and in (2.127) 8 is assumed to be small, as well as U, which is a Lagrangian quantity.

_,

;f'

I

. ·t

2

WAVE PROPAGATION IN SATIJRATED POROUS MEDIA

The introduction of equations (2.126) in (2.97c) gives by (2.1041:

)2 1

v. = ( p,

).0

+ 2Jl

p.(l - t/>)

i;,~

(2.128)

1

Vp, =

"

83

(~;'Y

where the effects of inertial coupling due to tortuosity (a = 1) are ignored, together with any dissipation process. With these assumptions, it is clear that the wave of the ftrst kind involves only the matrix and occurs at zero pressure (open system). while the wave of the second kind concerns only the fluid (~i = 0). This is perfectly understandable, since Terzaghi's law, which is mainly reflected by the assumption fJ = 4> in relation to Biot's theory, assumes a priori that the action of the fluid can be considered as an external action exerted on the matrix or solid skeleton. The equation (2.122) inserted in (2.87a) for steady state conditions therefore leads to:

~at cxi

_ OX; op = o

(2.129)

and the action of the fluid on the solid skeleton is reflected by imaginary volumetric forces of intensity - grad p. It should also be noted that, with the implicit assumption ofTerzaghi's law (fJ = t/>), if the fluid bulk modulus is much lower than that of the solid skeleton (K 11 ~ K 0 ), and if tortuosity effects are taken into account and viscosity effects ignored, the wave velocities are given by: 1

i.0 + 2Jl

V. _ [

p.(l - t/>)

P, -

1

Vp

>

+ tf>p1 (1 -a

=[KJ,J ap

I

Vs =

[

p.(l- t/>)

Jl

+ t/>pf(l- a

·>J Jl

(2.130)

•)

The equations (2.130) can hence lead to an experimental estimation of tortuosity, if a sound choice is made of the saturating fluid (low compressibility and low viscosity). As we have shown, assuming small displacements both in the solid part and in the fluid part, Terzaghi's law applies within a more restricted set of assumptions than Biot's theory. As we know, however, the central value ofTerzaghi's law lies elsewhere and concerns flow problems. In fact, the decoupling assumption helps to analyze substantial fluid movements (flow in an earth dam, for example) independently of small strains of the solid skeleton. To do this, the pressure f1eld in the fluid is determined by Darcy's law (where gravity effects may be accounted for), and the law of conservation of mass (101 for the fluid which, in this type of calculation, is usually assumed to be incompressible (div U = O)uu. 110) Note that the laws of conservation of mass in Biot's theory are automatically satisfted to the nearest second order, since displacements are assumed to be small. I 11) In this case, U becomes a Eulerian quantity (fmite transformation), and (2.1~7b), which is only valid for small displacements, no longer holds and is replaced by Darcy·s law.

84

2

WA\"E PROPAGATION IN SATURATED POROUS MEDIA

By reinserting the pressure fteld thus determined in (2.129) (in which gravity forces are taken into account). we can obtain the equilibrium equations for the solid part alone, with considerable latitude in selecting the constitutive law. Hence, apart from the Eqs. (2.130), which are interesting for the determination of tortuosity, the respecti,·e values ofBiot's theory and Terzaghi's law are not concerned with the same ftelds of application, and these theories are complementary rather than competitive.

2.2.5

Geerstma-Smit equations

As we have shown. the correct solution for the ftrst kind wave velocity within the framework of Biot's theory is relatively easy to obtain, but not very practical for any actual application. Nevertheless, we have pointed out (Table 2.2) the high values of critical frequencies fc for commonly encountered rocks. This makes it possible to obtain an approximate solution for the velocities by developing the second order equations in f !fc (Geerstma and Smit, 1961):

vzp,

v!, + =

v•(!c) f

2

0-

(2.131)

v; + v~(~)

where 1-'0 and V.., are the zero frequency and infmite frequency velocities respectively within Biot's theor'y'"presented here (low frequency hypothesis). By implicitly using Gassmann's equation (2.77), these authors obtain the following expressions for V0 and Vx :

).!..

V0 = (

4 z Kf+-Jl 3 [see Eq. (2.113)] p

Voc=(p.(l-cf>)+~p,-ll-a- )

(2.132)

1

·

4

[ K, + J" +

(

P

K

)(

t/>-a -1 + 1 - Ks - I1 Pt K

)])y

_K(l_2cf>a-

0

' K ¢

(I - K:- ~) K, + K''

1

.

(2.133)

where K 1 is giYen by Gassmann's equation (2.77) and p = (1- cf>)p. + p1 . It should be noted forthwith that, if tortuosity a is infmite, Eq.t2.133) is reduced to Eq. (2.1321. because there is no longer any relative movement of the fluid and solid (infmite

"""

----~--~-------

2 __

,

WAVE PROPAGATION IN SATURATED POROUS MEDIA

85

inertial coupling). and hence no permeability effect. Note also that Eq. (2.133) is reduced to Eqs. (2.130) using the assumptions:

i

'

Ko

1- K =

•

t/J,

K 11 ~ K 0

which have precisely been used to state (2.130).

I

lj 'f i

j

I -~

'"""

2.2.6 2.2.6.1

Experimental verifiCation Qualitative aspects of Biot's model

We have just described a method for modeling wave propagation in a saturated porous medium. The model presented reveals the existence of three propagation waves within such a medium, in other words two compressional waves and one shear wave. The particular feature and main interest of this model reside in the prediction of the second compressional wave, or slow wave. In fact, this slow wave does not exist in a classic solid and isotropic medium. Hence it is essential to prove the validity of Biot's theory to identify the existence of such a slow wave experimentally. We shall ftrst describe the qualitative procedure that enabled Plona (1980) to verify this theory experimentally, and then go on to the results obtained. A saturated porous medium is a medium formed by the interpenetration of two phases. One of them is the solid phase and constitutes the matrix of the material concerned, and the second is the liquid phase, constituting the saturating fluid. This interpenetration can occur in t'to different ways, shown schematically in Fig. 2.13 (Plona, 1982). In one of the two cases (Fig. 2.13a), a discontinuity exists in the liquid phase, consisting of a group of disconnected pores. with continuity of the solid phase. In the second case (Fig. 2.13b), both the liquid and solid phases are continuous. Two types of wave (a compressional wave and a shear wave) can propagate in an isotropic solid, while only a single type of wave, a compressional wa,·e, can propagate in a liquid. Since a saturated porous medium consists of two phases, one solid and the second liquid, it therefore appears logical to be able to observe three waves, namely two compressional waves and one shear wave. For this to be true, it is necessary for the liquid and solid phases to be continuous. In this way the waves can propagate within a given phase. This means that the porous medium must be of the second type described above (Fig. 2.13b). In fact, every natural porous medium possesses both types of porosity (disconnected and connected~ so that the liquid that participates in the motion of the slow wave is merely the fraction of liquid contained in the connected porosity. Finally, it is important to realize that the coarse image by which one considers that, among the two compressional waves, one moves within the liquid and the second in the solid, is false. In fact, the porous medium is a material constituted of a solid and a liquid phase coupled together. A more accurate image can be derived by representing the sample as a system of two springs whose eigen vibrations consist of one vibration in phase and one vibration out of phase. These two types of motion have been modeled and discussed previously in Section 2.2. This remark will enable us to understand more clearly the necessary conditions for the observation of a progressive slow wave. If the liquid is not

86

WAVE PROPAGATION IN SATURATED POROUS MEDIA

2 ~

t.

·~

Fig. 2.13 Schematic diagrams of porous media. a. The fluid phase (white) is discontinuous, the solid phase (cross-hatched) is continuous. b. The fluid and solid phases are continuous. (after Plona. © 1982 IEEE, New York, NY).

viscous, no viscous coupling force occurs at the liquid/solid interface, while, by contrast, if the liquid is very viscous, a substantial coupling exists, preventing differential liquid/solid movement. This clearly shows the importance of the viscosity of the interstitial fluid. Furthermore, as we well know, the intensity of the viscous coupling force depends on the incident wave frequency. At infmite frequency, a viscous fluid acts as if it had no viscosity, whereas at low frequency even low viscosity can give rise to substantial coupling. In fact, considering the intensity of this viscous force, it can be observed that it decreases rapidly "ith increasing distance from the liquid/solid interface (Fig. 2.14). Moreover. this effect can be characterized by a skin depth d3 • This skin depth:

{2i1 • .J-;;;;

d =

(2.134)

is proportional to the square root of the viscosity and inversely proportional to the square root of the frequency and of the fluid density. This clearly shows that the propagation of the slow wave becomes even more observable as the fluid viscosity decreases and the wave frequency increases. The density term in (2.134) is there to remind us of the inertial ·problems of setting any mass in motion. Hence it is very important for the observation of ·Biot mechanisms and for a given sample that this depth d. should be much smaller than the average acces radius of the pores.

"'

2

WAVE PROPAOA110N IN SATURATED POROUS MEDIA

!I

j:

'l 't.

FLUID-

,, I

SOLID-

t
f:

v

!l

0

Wall set in motion

y

f

t>o ,

~

v

Tr-ient motion of the fluid

Fig. 1.14 Fluid velocity proCde due to sudden movement of the walls of the capillary tube.

In sandstones, for example, the average access radius is relatively large, about l to 5 pm for permeabilities ranging from a few mD to 100 mD. This implies that, for an ultrasonic experiment, using water as the saturating fluid, the skin depth d. is about 0.5 to l pm. The ratio of skin depth to access radius is hence too high to be able to observe the slow wave directly, as the viscous effects at the fluid/matrix interface have prevailed over the possibilities of fluid/matrix motion out of phase. This means that the slow wave must be observed indirectly, for instance by analyzing the changes in signature undergone by the different recorded signals. In brief, we have shown that the following properties are required for the observation of a progressive slow wave(l2):

t, ,r

'libli.

th ...

itt;

(a) (b) (c) (d) (e)

l(

"-·.

eet

Continuity of liquid and solid phases, open system. High frequency content of the incident wave. Low saturating fluid viscosity (high hydraulic permeability). High saturating fluid density Oess important). High pore size and pore access radius, high absolute permeability.

For a clear observation of the two movements described above (in phase and out of phase), the difference in velocities between fluid and solid must be high enough to ensure signifacant separation between the two movements. Consequently, it is important to have a fluid that is much more compressible than the solid matrix.

:v

a{\,

l t

lV 6

tiar

l

87

~

~~~~

(12) The term progressive is used here in opposition to diffusive .

....

~~

88

2.2.6.2

WAVE PROPAGATION IN SATURATED POROUS MEDIA

2

Experimental results

*

Having made these preliminary remarks, Plona's experiment appears simple. In effect, to observe the slow wave, it is necessary to allow fluid and solid movements, and thus to transmit and record the waves far from the solid. This means avoiding bonding the transducers to the sample, which would prevent the recording of the slow wave. The experiment is conducted in a water-filled tank (Plona, 1980, 1982) (Fig. 2.15) and the angle of incidence of the signal emitted with respect to the porous medium may vary to observe

Shear wave

Compressional

~

waves in water A

i 0

8

I Receiver I

Fig. :us Diagram showing mode conversions and refractions at the different interfaces: the reflected waves are not indicated (after Plona and Johnson. ~ 1980 IEEE, New York, NY). the angular dependence of the process. The transducers employed have a central frequency of 2.25 MHz. The velocities of the different waves are measured with an accuracy of 3%. Figure 2.16 (Plona, 1982) shows the results obtained with a sample of sintered glass of porosity

(}~· =

.

SID

vw

-1 _ _

Vp,

(2.135)

~,

r ll

2

I

89

WAVE PROPAGATION IN SATURATED POROUS MEDIA

W-'er

Sample

0: 0°

1"'~1 m4t. A

C

D

E

F

I

11 1

A

c

A:OirctP 1 C, E. G: multiples

E

W-'tr

0: Oirct stow wave F: Convertld WIVe

G

Sampl•--+---,r0

(b)

e< ep· c

I

A

--- Slow mode __ Normet mode

1

....

..A o.A 50 mY

F

_;,.I

' ' I

B

A: Dirct P 1 wave 8: Oirct S wave 0: Oirct slow wave 5ps

0

Fig. 2.16 Signals recorded at different angles of incidence for a material made of sintered glass spheres. L 6 = 0". b. 0 < 6 < ~· (after Plona and Johnson. © 1980 IEEE, New York, NY).

·: ~

where V.., = sound velocity in water, Vp, = f1rst kind P1 wave velocity in the sample. Three arrivals are observed, whereas, in the case of a non-porous solid, only two would have been observed. Arrival A is the standard compressional wave, arrival B is the shear wave, and arrival D is the slow wave. Figure 2.17 (Plona, 1980) shows the variation in recordings as a function of angle of incidence. Fig. 2.17a and b resume Fig. 2.16. The recording in Fig. 2.17c was made with an angle of incidence() between the two critical angles related to the P1 and S waves, that is to say: where

O:•

o:· < () < (P,

(2.136)

. - 1•:.. _ 0: =SID Js

(2.137)

is given by (2.135) and:

where V5 is the shear wave velocity in the sample.

lij

90

2

WAVE PROPAGATIO!' IN SATURATED POROUS MEDIA

As observed, there is no longer a transmitted standard compressional wave. However, the S wave and the slow wave continue to be transmitted. If the angle of incidence(} is progressively increased, until it exceeds the critical value 0:. Fig. 2.17d is obtained, in which a single arrival is observed, namely the slow waYe. Finally, it has not been possible to cause the slow wave to disappear: no critical angle exists for the slow wave, since its velocity is lower than that of water [see for example the approximate formula (2.130)]. (a) I

: !

r

L.

,.;

~~

9 =0°

l

!

I

iI

I i I

F

G

I

A

A

ofo
C 0

B

E

I

0

mIMIll B

0

l

d)

9'
I

\ I

I

I

I

i 50mV\

L___,__

I

A IV l

.!_L.-

I

L

i

I

5 .•__

Fig. l.l7 Signals recorded after propagation in the water/porous solid/water system for different angles of incidence (sintered glass).

= 00. b. 0 < 8 < 1980).

a. 8

Of:' c.

~·

< 8<

Of

d. lfi: < 8 < 90" (after Plona,

Plona's results were obtained using various types of porous material. Figure 2.18 (Plona, 1982) offers a glance at the different materials used. The ftrst class of materials consists of different sintered materials I steel, titanium, Inconel). The second class includes ceramics produced by different manufacturers (Coors, 3M, Filtros). These ceramics are normally used as ftltration materials. The third class comprises materials manufactured by Plona in the Schlumberger laboratories at Ridgefteld. They are sintered glasses whose

·--------- -----

i1. l

2

'1:.'

4

Sintered glass beads # 1 beads # 2 beads # 3 3 \f-55 3 \f-40 Coors Porous steel Porous titanium Porous lnconel Filtros # 1

tP

r

v,.,

Ys

v,..

28.3 18.5 10.5 34.5 30.0 41.5 48.0 41.0 36.0 40.0

50 20 10

4.05 4.84 5.15 2.76 2.91 3.95 2.74 2.72 2.12 4.65

2.37 2.93 2.97 1.41 1.62 2.16 1.54 1.79 1.15 2.91

1.04 0.82 0.58 0.91 0.96 0.96 0.92 0.91 0.93 0.94

55 40

55 20 30 90 60

.

...,.

91

WAVE PROPAGATION IN SATURATED POROUS MEDIA

·-

=porosity, = average pore size (IUJI), I j. = fast compressional velocity (krn/s), Is =shear velocity (km/s). lj., =slow compressional velocity (km/s),

ll>

r

Fig. 2.18 List of the different porous materials analyzed (after Plona.© 1982 IEEE, New York, NY).

porosity can be varied. Figure 2.18 shows that all these porous materials display a behavior of the type described by Biot. The slow wave is always slower than the wave propagating in the saturating liquid, which is water in this case. The amplitude of the measured slow wave may be very high. Figure 2.19 provides an example of a seismogram recorded in a sintered steel with a grain size of20 ~tm. But~ we have seen, the amplitude also depends on the ratio of the wavelength ofthe signal transmitted to the pore size, or, in fact. for the materials analyzed, to the grain size. Figure 2.20 highlights this effect. To do this. the frequency - and hence the wavelength - of the transmitted signal is kept constant, while the pore size varies. In Figure 2.20a, the pore size is 15 ~tm. Considering th..: ~kin depth [see Eq. (2.134)], a value of about half a micron is found. already accounting for a significant fraction of the pore size. Thus an incipient viscous effect and a tortuosity effect are observable (see the beginning of Chapter 2: as

il'l

92

WAVE PROPAGATION IN SATURATED POROUS MEDIA

~lmtt!. 8

A

2

Fig. 1.19 Refracted arrivals at non-normal incidence for 20 J.LII1 sintered steel (after Plona and Johnson. © 1980 IEEE, 1\ew York, NY).

0

A:

Normal P1 wave

B:

S wave

0: Slowwave

n~.\tt= A

0

8

I'" I~ A

8

0

lcl

- r\\ !A. ~

50 mv

2 (L•

i

Fig. 2.20 Refracted arrivals at non-normal incidence for 3 \1: samples (after Plona and Johnson, 1980. © 1980 IEEE, 1\ew York, NY). . a. Pore size 15 j.lm, weak slow wave. b. Pore size 55 j.lffi, energetic slow wave. c. Pore size 175 j.lm, scattering: no slow wave.

pedormed with porosities greater than 10%, but a theoretical extension to the value 0 was obtained on the curve. Since the slow wave is due to the presence of a liquid inside a porous medium, it is perfectly logical for its velocity to tend towards 0 with decreasing porosity (a tends towards infmity as 4>· tends towards 0). The experiments described here concern artiftcial materials other than rocks. As we have alrea4y emphasized, owing to the pore size of a usual rock, it is very difficult, if not impossible, with usual fluids (such as water) to have a slow wave propagating within a porous medium, and the slow wave is generally attenuated immediately. Experiments are currently being conducted by Plona and Johnson and their colleagues to develop a technique to resolve this problem. The general principle consists in using liquid helium II, whose viscosity is zero, as the saturating fluid. The preliminary results obtained in these conditions have revealed the existence of a slow wave in a rock. The use of liquid helium as the saturating fluid is extremely interesting for two main reasons. Johnson (1980) showed that the slow wave mechanism was actually the

~'f:

2

~

WAVE PROPAGATION IN SATURATED POROUS MEDIA

93

generalization of the fourth acoustic wave in superfluid helium II. Moreover, with this material (i.e. negligible viscosity and fluid compressibility ~ matrix compressibility), Biot's expressions for the velocities are simplifted and are given by Eq. (2.130), assuming the applicability of Terzaghi's law (see Johnson and Plona, 1982). Hence it is possible to measure tortuosity a by using superfluid helium. for which the above equations are valid to the nearest 0.01% (Johnson et al., 1982). Johnson eta/. f 1982) also showed that these equations are valid to within 10% for water-saturated sintered glass. Incidentally, this parameter can be related to other types of measurements, such as the refractive index of the fourth wave of a superfluid He II (Johnson and Sen, 19811 or the electrical conductivity or the formation factor (Johnson et al.. 1982). 6

Fused glaa beads

s

~

::!!. ~

l!

;

2 200m\D

_,,1..---'"

•

•

•• • Slow wave

Ol,,~

0

10

20

30 Porosity (%I

40

50

Fig. 2.21 Measured velocities of·P, Sand slow waves vs. porosity for samples of sintered glass, The P ~ave and S wave velocities in solid glass are 5.69 and 3.46 km/s respectively (after Plona. © 1982 IEEE, New York, NY1.

In conclusion, Plona demonstrated the existence of a slow wave very close to the one predicted by Biot's theory. He showed that this wave could only exist as a propagation wa,·e if the following conditions were satisfted: (a) Continuity of the solid and liquid phases (i.e. possibility of differential fluid and liquid motion). (b) Sufftciently high incident wave frequency and sufl'lciently low fluid viscosity (i.e. weak wave attenuation mechanism). (c) Incident wavelength sufl'lciently large in comparison with pore size to avoid scattering, while the pore size must be adequate to avoid viscous effects at the wall (skin depth effect). (d) Very different fluid and solid bulk moduli in order to separate clearly the two compressional waves. Moreover, Plona and Johnson (Eq. (2.130)] revealed the possibility of employing approximate equations to determine the different velocities in the saturated porous medium. The equations make use of tortuosity a characterizing the geometry of a porous medium.

...

94

WAVE PROPAGATION IN SATURATED POROUS MEDIA

2

2.3 CONCLUSION We have considered a theory modeling wave propagation in saturated porous media. This theory, whose experimental conftrmation has been discussed above. identifted a number of mechanisms related to the presence of fluids and the permeability of the medium. It has been shown that these mechanisms occur to a signiftcant extent in the case of high mobilities and frequencies. The usual permeabilities and viscosities are generally inadequate to cause them to appear. The frequencies at which the mechanisms described are invoked tend to be too high (see Table 2.3) to apply the theory, since it is restricted to wavelengths that are large in comparison with the dimensions of the flow channels where ftltration occurs. For a wave propagating in an infmite two-phase medium (i.e. one which does not ··encounter" boundary conditions). wave attenuation should be looked for in the absolute local motion of the fluid, and no longer in the motion ofthe fluid with respect to the matrix, although certain experimental results (for example Morlier and Sarda, 1971) appear to demonstrate a relationship between permeability and attenuation. Note for the time being the theoretical work of Datta (1975), Mavko and Nur (1979) and Walsh (1976) who deal with these problems, which we shall discuss in detail in the subsequent Chapters. Howe,·er, it must be observed that, in these processes, the entire fluid part is involved, both the fluid found in the disconnected porosity and the fluid in the connected porosity. In addition, contrary to the framework ofBiot's theory, these mechanisms occur throughout the whole saturation range. One may well question the practical interest of Biot's theory for the study of wave propagation in an infmite medium. However, for the problems in which boundary conditions are imposed on the flows, pressures and displacements, the phenomena identifted by Biot's theory, which are mainly related to pressure gradients, must play a major role. These include the important problems of seismic reflection at interfaces, and the problems related to surface waves in sonic logging. Geerstma and Smit (1961) made a theoretical analysis of the problem ofreflection, and showed that a slow wave was always generated at the interface. This slow wave may account for a non-negligible portion of the total energy, although, due to its attenuation, it is not generally observable. Rosenbaum (1974) pointed out the importance of the role of permeability on the attenuation of Stoneley waves at the interface of a saturated porous medium and a fluid, in the plane case and in the case of guided waves in wells. These studies will be discussed in Chapter 6, which will demonstrate the necessity of a careful analysis of wave propagation for seismic applications. Biot's theory also justifies certain general considerations. The assumptions of this theory include the notion of scale related to wavelength and hence to frequency. The seismic profiles correspond to low frequencies ( ~ 50 Hz), while sonic logging corresponds to higher frequencies (~ 10 to 20kHz). In all cases, it is considered that these wa,·es propagate in homogeneous and generally isotropic and stratifted media. In nature, the media that the wave travels through are porous, inhomogeneous, and anisotropic to varying degrees. The use of exact laws of propagation should allow a better understanding of the complexity of these media, so that data can be available on their petrophysical properties and their fluid content. Strictly speaking, these exact laws do not exist. From a

r ;'l ~'

.j,:. '-

'

a

2

WAVE PROPAGATION IN SATURATED POROUS MEDIA

95

practical point of view, the laws used must always be chosen according to the specif1c problem investigated (boundary conditions, geometries) to the tool employed (frequencies, wavelengths, energy), and to the parameters to be quantif1ed (water content. permeability, porosity). ' One may be tempted to complicate a model such as Biot's model (for example, by introducing the viscoelasticity of the matrix) to clarify a number of measurements. In our opinion, this complication is extremely liable to be prolonged indefmitely, without any hope of success. In fact, the complexity of the porous medium (see Chapter 1) is such that it is totally unrealistic to try to construct a general model for porous media. On the other hand, it is necessary to account for in situ conditions and for the frequency range to be investigated to adapt the theory to the problem at hand. To each problem corresponds a different theory, which relies on a specific physical parameter to yield an observable effect. In the laboratory, it is essential to press forward with a qualitative phenomenological analysis of the problems (which is the preponderant effect in a given experiment?). an adjustment of simple models to the experiment, and a suff1cient number of measurements to acquire statistical knowledge that is virtually "rock type., by "rock type ... As we have already stressed on several occasions, for the time being, the complexity of the natural media under consideration dashes any hopes of a unique theory based on continuum mechanics. However, in the beginning of this Chapter, we pointed out that, for a given physical phenomenon, a porous medium could be replaced by an equivalent homogeneous medium with very specif1c properties. In other words, the porous medium is no longer · considered with its complexity, and only the result of its interaction with a given physical phenomenon is important. This sets the stage for a new trend in investigating porous media, which may fmally lead to a clear understanding of certain observed mechanisms. For problems of acoustic propagation, and for certain types of process, the rock can be replaced by a homogeneous and linear viscoelastic medium. The study of this type of medium will be discussed in the next Chapter.

· Appendix 2.1

WAVE PROPAGATION IN A NON-ISOTROPIC ELASTIC MEDIUM The introduction of(2.7) into the constitutive law (2.25) of a linear elastic medium, and the result of this introduction in the equilibrium equations (2.24) gives: cijkluj.kl

= pii;

(2A.ll

For isotropic media satisfying (2.28), Eq. (2A.l) is merely Eq. (2.39) set in another form. Let us now consider a wave front of unit normal vector n and normal velocity V at geometric point M belonging to this front. If f(xt, t) is the equation of the wave front, note that V is defmed by :

96

WAVE PROPAGATION IN SATURATED POROUS MEDIA

-j

V=

2

(2A.2)

1

[~ u:">2Tand thus represents the velocity of a geometric point moving on the normal n between time t, when it coincides with M, and a later time t + dt. Let us now consider acceleration waves, characterized by a jump A of the acceleration vector when crossing the geometric surface representing the wave front. We shall set: [ii~c]

= Ale

(2A.3)

where(·] represents the ..jump" operator and consists in determining the difference in the quantity· on either side of the wave front. If the second partial derivatives with respect to time satisfy (2A.3), we know that the second partial derivatives with respect to a space variable and to another space variable or to time must satisfy the equations of kinematic compatibility stated by Hadamard (1949). These equations are: n"n'

[ U·~cl]=-A·. 2 1

}.

V

•

.

U·~c= }.

nt

--A. v 1

(2A.4)

If the second derivatives of a quantity satisfy an equation, their discontinuities or jumps must satisfy the same equation when crossing the wave front. By applying the jump operator [·] to Eq. (2A.l ), and taking account of (2A.3) and (2A.4), we obtain: AljAi-pV 2 A;=0

(2A.5)

Alj = C;"i'n"n'

(2A:6)

where (Alj) is defmed by :

This represents the acoustic tensor relative to direction n considered. This acoustic tensor is symmetrical and the associated quadratic form is positive defmite. Consequently, the endomorphism associated with the acoustic tensor has three mutually orthogonal eigenvectors associated with the positive eigenvalues A1 = p Vj (J = 1, 2, 3) satisfying: det (Alj - A 1 <5;) = 0

(2A.7)

Hence, at any point of a wave front, three mutually perpendicular polarization directions exist, and each is associated with a velocity V,. These directions display a complex dependence on the direction n concerned, through the associated acoustic tensor defmed by (2A.6). In the isotropic case, Eq. (2.28) leads to:

i.<5;i<5"' + J1(<5;"<5i, + <5u<5j,)

(2A.8)

The introduction of (2A.8) into (2A.6) and (2A.5) yields: (A. + J1){D • A)n + (/1 - p V 2)A = 0

(2A.9)

cii"'

=

This equation is satisf1ed either ifn and A are colinear, or if their respective coefficients are zero. The former case corresponds to longitudinal waves. In fact, by setting A = An, which corresponds to a longitudinal polarization, we have: P

v; = i. + 211

(2A.l O)

..,....-

----------

2

------------·--·----

WAVE PROPAGATION IN SATURATED POROUS MEDIA

.

97

The second case corresponds to transverse waves. The transverse polarization n . A = 0 thus leads to : pV~ = Jl

(2A.ll)

The results found for harmonic plane waves, concerning polarizations in an isotropic elastic medium (see Section 2.1.5), are hence valid for any shape of wave front. In the most general anisotropic case, in which the tensor Ciitl displays no particular symmetry apart from the natural symmetries (2.26) and (2.27), the characteristic polarization directions are normally random and depend continuously on the vector n. It is only if the material exhibits certain additional particular symmetries that certain polarization directions are preferential. Let us thus consider a case of standard anisotropy which is the orthotropy'of revolution (or transverse anisotropy). Let us assume for the purpose that the orthotropy axis is parallel to the physical direction OxJ. This means that all the directions are equivalent in plane (Ox 1 , Ox 2). In this case, it can be shown that, ofthe 21 independent components for tensor cij/d in the general case (see Section 2.1.3), only five independent components remain, which are C 1111 = C 2222 , CJJJJ C 1122 , C 22 JJ = CuJJ• C 1313 = C 2J 2J· The remaining components that are not inferred from natural symmetries (2.26) and (2.27) are zero, except C 1212 given by: 1 (2A.l2) Cu12 = 2 (Cuu- Cuu) Considering (2A.6) in the case of the orthotropy of revolution, it is easy to see that two types oflongitudinal wave and two types of transverse wave can propagate. To begin with, when n = nJ is oriented along the orthotropic axis OxJ, a longitudinal wave (A= LlnJ) with velocity .jC 3333 /p and two transverse waves (A. nJ = 0) with velocity jC1313 /p = JC 2J 23 /p are seen. Subsequently, if n belongs to the plane normal to the ortbotropy axis(n. nJ = O),alongitudinalwave(A == Lln)ofvelocityjC 1111 /p == JC 2222 /pandtwo transverse waves, one polarized along nJ(A =An) with velocity jC 1313 /p = jC2323 /p identical to the f1rst type of transverse wave, and the second polarized in the plane normal to the orthotropy axis (A= Lin x n3 ) with velocity 12 ufp. Finally, the four independent velocities are:

JC

• P waves:

Vr==/¥

n= n3 A= Lin

I

n . n3 = 0

Vp=/¥=FF

(2A.l3)

• S waves:

A. n =0

n = n3 n . n3 = 0, n . n3

= 0.

rc;;;;

A • n = 0 } Vs == [C;;;; == A=LinJ ~p ~p

A = Lin

X

OJ

J's = ~

T '

I

wave propagation and vibration effects in viscoelastic med1a (unidimensional)

3

INTRODUCTION We have shown in the previous Chapter that the average motion of the fluid with respect to an elastic matrix led to invoking a dissipative mechanism and hence to wave attenuation. However, we showed that, apart from the case in which boundary conditions are involved, this mechanism could be considered as negligible in most cases. The cause of dissipation is to be found in the absolute local motion of the particles, both solid and liquid. This Chapter is devoted to the mechanical modeling of this phenomenon by the introduction of viscoelastic models. It is not intended to provide orders of magnitude of the parameters that we shall introduce (which can be found in Chapter 5), nor to discuss the relevance of these models concerning the behavior of rocks at the passage of a wave. We again consider the medium to be homogeneous. In other words, the models developed take account, from a macroscopic standpoint (that of the measurement), of the average dissipation on a representative volume of fluid and solid, without drawing a distinction between the share accounted for by the fluid and that relating to the solid. The fme analysis (microscopic) will be dealt with from a qualitative standpoint in Chapter 5. The motion considered here is hence the average macroscopic displacement u of the fluid/matrix combination introduced in the previous Chapter. In Chapter 2, for the matrix itself, we considered a rigorously elastic constitutive equation; in other words the material has no strain memory. This is the simplest case. Real materials, and rocks in particular, macroscopically display irreversibilities of behavior: they dissipate energy when subjected to deformation. This dissipation, apart from the Biot type of mechanism (Chapter 2), stems from many sources. The most probable mechanisms involved for rocks are discussed in Chapter 5. They include capillary forces, thermal effects, intergranular friction, and local fluid movements. These microscopic, irreversible effects are therefore numerous and complex,

100

WAVE PROPAGATION IN VISCOELASTIC MEDIA

3

and any macroscopic model (in the sense intended here) that would attempt to describe them fully, even if available, would be of extreme complexity and of very limited scope. Consequently, we shall only consider the simplest models here, namely linear viscoelastic models. In fact, as we shall show in Chapter 5, these models are well suited for the description of a broad class of dissipative processes, resulting from rapid, smallamplitude variations in strain due to waves that propagate in rocks. These models require the knowledge not only of the present stress and strain values, but also of past values, and are therefore called memory models. To investigate and quantify these effects in actual materials, it is essential to understand what is generally called their delayed behavior. In this way, the time effects, particularly those associated with strain and stress velocities, are introduced quite naturally. The following study is deliberately restricted to uniaxial or unidimensional behavior of the materials that we will examine. The reader interested in an exposition of the general case can refer to several works (Fliigge, 1975, Christensen, 1982, Salen.;on, 1983).

3.1

DELAYED BEHAVIOR OF MATERIALS

The experimental study of the behavior of materials as a function of time can be undertaken by two main types of test: creep tests and relaxation tests .

.

3.1.1

Creep tests

In this type of test, the sample, previously at rest, is subjected to a constant load a 0 from time t 0 • An analysis is made of the variation with time of the strain e(t), or its equivalent, the creep function f(t) defmed by: e(t) = f(t, t 0 ; a 0 ) ao

(3.1)

In the following analysis, we shall consider only materials whose properties do not vary with time (non-aging materials). For the wave propagation experiments dealt with here, this assumption can be accepted without restriction in view of the measurement time scale (i.e. wave passage time). Under this assumption, the function f depends only on the interval t - t 0 and (3.1) becomes: e(t)

-

ao

= f(t - t 0 ;

a0)

(3.2)

Figure 3.1 shows a standard creep experiment. The loading time sequence consists in the application of a given stress a 0 which is kept constant for the time interval (t0 , t 1 ) during which no additional stress is applied. The strain response is shown schematically in Fig. 3.lb. At time t 0 , an instantaneous strain e0 is usually observed such that:

T

I I

3

WAVE PROPAGATIOl' IN VISCOELASTIC MEDIA

ao

101

(3.3)

eo=Mo

where M 0 is the instantaneous elastic modulus (Young's, shear, bulk, depending on the experiment). After t 0 , the strain increases with imposed load, giving rise to creep. This is the phase that determines the function f defmed by (3.1). At time t 1 , when the stress is e

0

Creep

p -----1, =-~~---~-~ -----_1:::

ao ~---

--

0

to

0

t 1

to

tl

(a) Stress

(b) Strain

Fig. 3.1

Standard creep test.

removed, an instantaneous unloading occurs such that, if e(tl) is the strain just before unloading. and e(t just after, we have:

n

Lie

= e(tn -

e(tl)

=-

e0

(3.4)

Equation (3.4) is only true because the material is assumed to be non-aging (if not, Lie :;C - e0 ). After time ti, the strain continues to decrease, leading to recovery. This recovery may be partial, in which case the strain tends to residual non-zero strain e~ (see for example Maxwell's model, Fig. 3.10a), or may be total, in which case e00 = 0 (see for example Voigt's model, Fig. 3.10b).

3.1.2 Relaxation tests In this second type of test, the experiment consists in imposing a constant strain e0 on a sample initially at rest, from time t 0 • The stress a(t), required to produce this strain, is then analyzed as a function of time. This is equivalent to analyzing the so-called relaxation function r(t) which, assuming the materials are non-aging, is defmed by: a(t)

-

Bo

= r(t -

t 0 ; e0 )

(3.5)

Figure 3.2 shows a standard relaxation test. To produce a strain e0 at time t 0 instantaneously, a load a0 must be applied instantaneously. After t 0 , to maintain the same strain e0 , it can be seen that the required stress decreases, because stress relaxation occurs. If zero strain is wanted after time t 1 , it is

102

WAVE PROPAGATION IN \-ISCOELASTIC MEDIA

Relaxation

E

~

oo Eo

3

1--0

0 to

(tl -,

rOto--~

a.,

tl

1----

O(ttl

(al Strain

(bl Stress

Fig. 3.2 Standard relaxation test.

necessary to apply an instantaneous jump Au in the stress to be imposed, which, for a nonaging material, is .Ju = - u 0 . Subsequently, the stress to be imposed becomes increasingly low, and tends to a non-zero stress value (partial stress cancellation) or tend to zero (total stress cancellation).

3.2

LINEAR VISCOELASTIC BEHAVIOR

Equations (3.1) and (3.5) only assume the non-aging character of the viscoelastic constitutive law. In addition, linearity consists in assuming that the creep function f and relaxation function rare independent of u 0 and e0 , namely:

= u 0 f(t- t 0 ) u(t) = e0 r(t - t 0 )

e(t)

(3.6a) (3.6b)

Equations (3.6a) and (3.6b) help to obtain the law oflinear viscoelastic behavior for any loading history. In fact, linearity implies Boltzmann's superposition "principle" which states that the effects can be added. This gives: e(t) =

'f 'f

du

f(t - r) -d dr r

- :o

u(t) =

-x:

r(t- rl

de

d

r

dr

+ ~ f(t- r;)(Au);

(3.7a)

1

+~

r(t- rJ(Ae);

(3.7b)

I

Equations (3.7a) and (3.7b) state that the history of stress u (or strain e) can be considered as the superposition ofinfmitesimal steps du(r)[or de(r)] and fmite steps (Au); [or (Ae)J at time r;. The term step means that the quantity involved is imposed and then

~ 3

WAVE PROPAGATIOS IN VISCOELASTIC MEDIA

103

kept constant from time t considered. One can then formally write the Eqs. (3.7) more briefly in the form : e(t) =

'I 'I

da f(t - r) -d dr r

-:r:

a(t) =

r(t - r)

-::c

de

d

r

dr

(3.8a)

(3.8b)

where the integrals and derivatives are implicitly understood in the sense of distributionsu 1• This convention for derivatives and integrals is always presumed in the following discussion. Moreover, in accordance with geophysics convention, we shall use Eq. (3.8b) and not Eq. (3.8a), although these equations are obviously equivalent. Causality implies that stress a(t) will not be influenced by the future of the strain (i.e. by e(r) for r > t). Equation (3.8b) can then be rewritten in the form : a(t) =

I

_,

d r(t - r) de dr

-x

(3.9)

't

where it is presumed that, because of causality: r(T) = 0

for r < 0

(3.10)

Deducing (3.9) from (3.7b) may appear somewhat artificial. However, the introduction of the formalism of distribution and the constraint of causality is extremely useful in dealing with discontinuities and for using the Fourier transform instead of the Laplace transform in applications to dynamics, as we shall show subsequently. Integrating by parts (in the sense of distributions, both on rand e) and assuming that e(r) = 0 fou < - t 0 [i.e. e(- oo) = 0), (3.9) and (3.10) yield the equation: +oo

u(t)

I

= _x

m(t - r)e(T) dT

(3.11)

where dr dT

m=-

(3.12)

Equation (3.11) shows that a is written in the form of a convolution product (again in the sense of distributions) that we shall denote more briefly:

a= m•e

(1) A discontinuity .1/ at timer; for the function f corresponds to a Dirac mass Jfo(rdf/dr, if the latter is considered as a distribution.

(3.13)

tJ for the derivative

104

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

3.3 DYNAMICS OF UNIDIMENSIONAL LINEAR VISCOELASTIC MEDIA, FIRST CONCEPTS OF THE QUALITY FACTOR Q 3.3.1

Complex modulus

Let us consider the Fourier transform F(w) of a function f(t) defmed by 121 : F(ro) =

f

Conversely: f(t) =

+:c

21 1t

(3.14a)

f(t) exp (- iwt) dt

_ 00

f

+X>

(3.l4b)

F(w) exp (iwt) dro

-:10

The use of the Fourier transform in viscoelasticity problems is quite convenient because we know that the Fourier transform of a convolution product is the product of the Fourier transforms. Hence, by applying the Fourier transform to Eq. (3.13), we obtain: (3.15)

I(w) = M(w)E(w)

where M(w), the Fourier transform of m(t), is called the complex modulus. M(w) can be separated into its real and imaginary parts: M(ro) = MR(w)

+ iM1(w)

(3.16)

For a linear elastic medium. Eq. (3.13) is nothing other than Hooke's law, which is written: (3.17) u= M0 e This shows that a linear elastic medium satisfies : m(t)

= M 0 t5(t), MR = M 0 , M, = 0

(3.18)

where t5(t) is the Dirac function. Equations (3.16) and (3.18) suggest that the imaginary part M 1(ro) characterizes the dissipation of viscoelastic models, while the real part MR(w) is associated with the instantaneous response (at angular frequency w). The following Chapters therefore develop the study of this complex modulus. Before dealing with the unidimensional problem, it should be noted that the three-dimensional (isotropic) generalization of (3.9) is: +::c [ d de; .. ] (3.19) uii(t) = _"' J.(t- t) dr (tr e) t5ij + 2f.,l(t - t) d;J dt

will

I

giving rise to the defmition of complex Lame coefficients. (2) A capital letter denotes the Fourier transform ofthe function considered: F for f, M form, :E for u, E for e etc. This transform is understood in the sense of distributions. Hence the Fourier transform of a harmonic distribution: exp (iw 0 t) is 2m5(w- w 0 1(see Eq. (3.14b)]. However, this transform does not exist in the sense of ordinary functions.

r I I

3

105

WAVE PROPAGATION IN VISCOELASTIC MEDIA

3.3.2 Harmonic problems Let us assume that a strain e(t) is imposed on a sample such that( 31 : e(t) =em exp (iw 0 t)

(3.20a)

Inserting (3.20a) in (3.13) and taking the Fourier transform of the equation obtained yields: I"(w)

= emM(wo)27tb(w- Wo)

(3.20b)

where the Fourier transform of exp (iw0 t) is 2nb(w- w0 )(see footnote in Section 3.3.1). Equation (3.20b) can alternatively be written in the form: I"(w)

= 2ne'" IMI exp [iq>(w0 )]b(w- w0 )

(3.21)

where the phase (/) and the modulus IMI satisfy: .\f.(wo) tan q>(w0 ) = \1 ( ); MR = IMI cos •

q>;

. M 1 = IMI sm

q>

(3.22)

R Wo

To return to the time domain, it is necessary to take the inverse Fourier transform of Eq. (3.20b). The real stress a(t) is the real part of the expression obtained. The inverse transform (3.14b) then gives rise to:

.

(3.23a)

a(t) = Re [e'"M(w 0 ) exp (iw0 r)] 1

a(t)

= Re { e'"[Mi(w0 ) + Mf(w0 )]2 exp 1

a(t) = e..,[Mi.(w 0 )

[i(w 0 t

+ Mf(w0 )]2 cos (w0 t + q>)

+ q>)]}

(3.23b) (3.23c)

In the harmonic problem. stress and strain are hence out of phase by a quantity q> directly related to the viscosity of the medium. This viscosity therefore causes a lag between response and excitation in the steady-state problem defmed by (3.20a). As the angular frequency w 0 tends towards 0, the influence of viscosity becomes lesser and the static limit(41 is obtained by (3.22) and (3.23):

a= e..,MR(O)

(3.24)

Equation (3.24) should be compared with (3.17). Hence it is natural to associate. the viscoelastic behavior examined with the elastic behavior defmed by modulus MR(O) = M 0 , called the relaxed modulus. However, it is very important to stress that this

(3) The complex notation is e'ident: the physical quantity is the real pan of the complex quantity e(t) = £, cos w 0 t. This will always be understood in the following discussion, if not mentioned. (41 Note here that, since the inenia terms are ignored in principle (sample of negligible mass). the term static means that the time is suiTiciently long (pulsation w;;; 1/r very small) for relaxation at imposed strain e., to be complete, and MR(O) is merely the relaxed modulus (Section 3.4.1).

106

WAVE PROPAGATION IN VISCOELASTIC MEDIA

3

in no way prejudges the elastic energy (reversible) that can be stored by the viscoelastic material analyzed for a motion defmed by (3.20a) and (3.23 ). Thus it is not because: a ( t,

= 2mr roo ;

n

1, ... )

= 0, e(t,)

= e.,.MR(w 0 )

(3.25)

=o

that it can be stated that the elastic energy at times t .. is equal to e!MR/2. The evaluation of this energy can only be undertaken with additional data about the '1 iscoelastic behavior analyzed. This could, for example, be the case of the rheological models that we shall examine in Section 3.4.1. Let us now consider the energy dissipated over a period. This energy is written: LIW

=

!: ae dt

f

(3.26)

0

since the elastic energy is reversible over a period T = 2n 'w 0 . Considering that the quantities concerned in (3.26) are real, one obtains from (3.20a) and (3.23): Ll W = ne!M 1(w 0 )

(3.27)

The energy dissipated per period is hence directly proportional to the imaginary part of the complex modulus. Note that defmition (3.26) shows that Ll W can be measured by the area of the closed curve described in the plane [e(t), a(t)] during a period. Eliminating the time between (3.20a) and (3.23), it can be shown that this curve is an ellipse (Fig. 3.3). In the plane related to dimensionless coordinates (e(t)/e ... , a(t)/em IMI ), the main axes of the ellipse are plotted by the major and minor bisectors. The ratio of the half-axes is 1/tan 2 ({J/2, the major axis being plotted by the major bisector and equivalent to 2 cos 2 qJ/2. CJ (t)

Fig. 3.3 Stress-strain cycle in the plane (e(t)/em. u(t)/tmiMI).

3

WAVE PROPAGATION IN VISCOELAmC MEDIA

In the elastic case, the ellipse is reduced to the major axis because, in this case, lf' IMI = MR. The intersection of the ellipse with the major axis is given by:

e = ± e,.. cos lf'/2,

107

= 0 and

u = ± IMI e,. cos lf'/2

The fact that A W depends on the frequency is characteristic of viscous behavior, because this implies that the power dissipated depends on the strain velocities which are proportional to w. For instance, in the case of dry friction, which depends only on the limiting stress, this dependence disappears. In conciusion, we must stress that, for harmonic problems, the expressions of phase shift qJ (3.22) and of dissipated power (3.27) depend only on the data of the complex modulus.

3.3.3 Wave propagation 3.3.3.1

Wave propagation and attenuation

In dynamics, the unidimensional equilibrium equation is written (Section 2.1.5.2): (3.28)

u.x = pii

The introduction of the behavior Eq. (3.13) into (3.28) gives the equation of unidimensional waves in linear viscoelasticity by replacing e by oujox. This gives: 13.29)

m•u.xx = pii

Let us now consider a displacement of the form : u(x, t) = u0 exp [i(w 0 r- k*x)]

(3.30a)

In this expression k* is a priori a complex quantity and Eq. (3.30a) can also be written: u(x, t)

= u0 exp [- :x(w0 )x] exp

[i(w 0 t - kx)]

(3.30b)

with k* = k- icx, where k and ex are real quantities. A solution such as (3.30b) thus corresponds to a traveling wave with attenuation :~(w0) at angular frequency w 0 • We fmally obtain the equation: k 0 V0

= k(w 0 ) V(w 0 ) =

w0

13.31)

where k and V are the wave number and phase velocity respectively at angular frequency Wo.

The introduction of Eq. (3.30b) into Eq. (3.29) and the derivation of the Fourier transform of the result obtained gives: - (MR

+ iM 1)(k- icx)2 + pw'f, = 0

13.32)

By separating Eq. (3.32) into the real and imaginary parts, the following two equalities are obtained: k

= Wo

/P(IMI + MR) 21MI2

(3.33a)

108

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

oc

p(IMI- M.J

= Wo

(3.33b)

21MI 2

1

where IMI =(Mi. + Mf}2. Assuming weak viscoelastic effects, i.e. M 1 ~ MR, Eqs. (3.31) and (3.33) lead us to: V(w 0 ) =

JMR~wo)

2oc

(3.33c)

M1

(3.33d)

-;:= MR

Thus the wave's phase velocity at angular frequency w 0 is given (to the nearest second order) by the same equation as in elasticity, if care has been taken to replace M 0 [refer to (3.17)] by MR(w 0 ). Moreover, the ratio M.JMR is again observed to appear in the expression of attenuation deriving from the dissipati,·e character of viscoelasticity.

3.3.3.2

Variation in time of a free wave packet: group velocity and phase velocity

Let us begin by analyzing in the elastic case the qualitative behavior of u(x, t) for a very simple specific case: u(x, t) corresponds to the superposition of three waves of amplitudes Ak Ak u(k 0 ), lf2u(k 0 ), 1 '2u(k 0 ), and of respective wave numbers k 0 , k 0 - 2 and k 0 + 2

(angular frequencies. w u(x, t)

= u(k0 )

{

0,

w0

-

Aw T

and

Wo

+

Aw) : T

exp [- i(k 0 x- w 0 t)]

+~exp[ -{(k

0 -

~k)x-(w0 -A;)t]]

+~ exp [- {(k, + ~nx

= u(k 0 ) [ 1 +cos ( x Ak 2

Aw)] exp [- .

- tT

1(k 0 x- w 0 t)]

-('"•

+ ~;)r]J} (3.34)

Let us firSt consider the initial timet = 0. In this case, the modulus lu(x, 0)1 is maximum at x = 0 because, for this value, the three waves are in phase and interfere constructively (Fig. 3.4). As we deviate from the value x = 0, the waves mutually go out of phase, causing a modulation oflu(x, 0)1 which decreases the amplitude away from x = 0. The interference becomes completely destructive when the phase difference between exp (ik 0 x) and exp [i(k 0 Ak/2)x] is equal to ± nand lu(x, 0)1 = 0 at x = ± Ax/2 where Ax is given by:

+

AxAk

= 4n

(3.35)

This equation indicates that the width Ax of the function lu(x, 0)1 (distance between two successive zeros) increases with decreasing Ak. In a more general manner, for a signal

T

3

109

WAVE PROPAGATION IN VISCOELASTIC MEDIA

I !

I

II

ko+~ 2 ko

~

ko-T

llx

-T

Fig. 3.4 Example of the composition of three waves. The result is given on the lower curve [see Eq. (3.34)] (after Cohen-Tannoudji et al., Mecanique quantique.© 1977, Hermann, Paris).

whose spectrum is different from 0 over a band Ak or .1w, this means that the narrower the band, the more spread out the wave "packet". Let us now consider a later time. Equation (3.34) shows that the maximum of lu(x, t)l, which was found at x = 0 at time t = 0, is now at the point :

=

XM

Aw

(3.36)

Ak t

and not at point x = (w 0 fk 0 )t = V0 t. The physical origin of this result appears in Fig. 3.5. Part (a) of this ftgure represents the position at time t = 0 of three adjacent maxima (1), (2)

Ak ko+T ko Ak ko--· 2

1,,) 1(1) 1(1)

1(2) 1(2) 1(2)

1(1)

1(3)

1(1)

1(3)

1(1)

1(3)

1(2) 1(2) 1(2)

1(3) 1(3) 1(3)

X

(a)

X

0

0

t

(b)

xM(O)

t

xJt(t)

Fig. 3..5 Constructive interference effects: group velocity concept (after Cohen-Tannoudji et al., Mecanique quantique. © 1977, Hermann. Paris).

IIO

WAVE PROPAGATION IN VISCOELASTIC MEDIA

3

and (3) for each of the real parts ofthe three waves. The maxima identified by subscript (2) coincide at x = 0, and constructively interfere and therefore correspond to the peak ju(x, 0)!. In the case of Fig. 3.5, where the velocity increases with k (the most frequent case), the maxima (3) of each wave merge. After a certain time interval, the situation shown schematically in Fig. 3.5b is thus obtained: the maxima with subscript (3) coincide and give the position of the maximum xM(t) oflu(x, t)l. One can therefore clearly see in Fig. 3.5 that xM(t) is not equal to V0(t) but is given by Eq. (3.36). In the more general case in which the signal corresponds to a Fourier spectrum of amplitude U (k), varying slowly with k and being non-zero over a narrow band centered at k 0 • the center of the corresponding wave packet can be obtained in a similar way by the socalled stationary phase method. Without going into mathematical details, which are unnecessary here, the displacement in the time domain is actually given by the transform [Eq. (3.14b)]: u(x, t) =

;1t J_+oooo ["{k) exp [i[kx -

J

w(k)t] dk

(3.37)

Since U(k) is assumed to vary slightly. the most significant contribution to the signal corresponds to constructive interferences in time t at position x. or, for a properly stationary phase:

c

(3.38)

ck [kx - w(k)t]iko = 0

Hence for: XM

= V,(ko)t;

V,(ko)

= dw dk

I

(3.39)

ko

where V, is the group velocity of the wave packet centered at k 0 and corresponds to the velocity at the peak value. One may well question the relevance ofthe foregoing remarks for the viscoelastic state. The quantity U(k) is transformed into U(k) exp [- cx(k)x]. To apply the method of the stationary phase, it is necessary for this quantity, U(k) exp [- cx(k)x]. to vary slightly in the frequency band considered. This condition is satisfied if the value of cx~k) is small, an assumption that has been experimentally justified for most rocks (see Section 5.1.2). In this case, the foregoing argument leading to 13.39) is valid. If not, no conclusions can be drawn without additional data. Note however that it is the interference effects that led to the concept of group velocity. The group velocity V,(k 0 ) is different from the phase velocity V0 when the phase velocity is frequency-dependent. The medium is then stated to be dispersive. In elastic media, the interferences leading to a dispersive character are of geometric origin and are generally produced by multiple reflection effects (wave guides). Hence, in an infmite homogeneous elastic medium, w 0 is always equal to k 0 V0 , the group velocity coincides with the phase velocity, and the frequency content of a given signal does not change. For an infmite homogeneous viscoelastic medium, this does not apply, because of dispersion due to intrinsic dissipation. The group velocity does not coincide with the phase velocity, and,

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

111

when a wave is no longer monochromatic, but centered around an angular frequency w 0 , the signal peak propagates at the group velocity, hence:

V, = dw dk

I=

V(ko)

lo

+ ko dV dk

I

(3.40a)

ko

or

_!_ __ 1 __ ~dvl

V,-

V(w 0 1

V 2 (w 0 ) dw

(3.40b) wo

As a rule. in an elastic medium in which dispersion is due to interferences of geometric origin (wave guides, Rayleigh waves), the group velocity decreases with frequency (its minimum is called the Airy phase velocity). When the group velocity decreases with frequency. dispersion is said to be normal. If not, it is said to be inverse. In a viscoelastic medium in which dispersion is of intrinsic origin, the phase velocity usually increases with frequency. and dispersion is therefore inverse( 51 • Note however that the concept of group velocity, obtained by the stationary phase method, imposes a minimum propagation distance for the wave to be grouped in packets. Before this minimum distance is obtained, this concept is irrelevant. In a viscoelastic medium, owing to the preferential attenuation of the high frequencies, for relatively high attenuations this grouping in packets may not necessarily occur, and the notion of group velocity no longer makes sense. Hence the velocity given by the arrival time does not correspond to an easily identified velocity. This problem will be examined in the next Chapter. The foregoing defmition of group velocity is based on a kinematic reasoning. For a wave propagating in a homogeneous linear elastic medium, it can be shown, by applying the Hamilton principle used in the previous Chapter (see for example Eringen and Suhubi. 1975, Achenbach, 1973), that the group velocity is also the energy propagation velocity. i.e.: V

= (P)

9

(IHI)

=

>.

(P (V+C)

(3.41)

where P is the power per unit area, and IHI the total energy density (sum of the kinetic energy C and elastic energy V), while the notation ( ·) indicates that the average is taken over a period

T((·)

=

~

tr·

dt). In the unidimensional case, (3.41) would be written:

( uu '> vg = (lf2pu2)

~,(1 '2u~)

(3.421

On the other hand, since the stationarity principle of the Lagrangian C - \ no longer applies to viscoelastic media due to dissipation, the energy propagation velocity of a narrow-band centered signal is neither the group velocity nor the phase velocity. In the two- or three-dimensional case, for a pure monochromatic wave (i.e. no interference effects) that is not homogeneous (planes of equal phase not parallel to planes of equal amplitude. see Chapter 6). the energy propagation velocity once again is neither the group

(5) An example is the case of Biot's media of the previous Chapter.

112

WAVE PROPAGATIO!'< IN VISCOEL...STJ(' MEDIA

3

velocity [i.e. defmed by (3.40)] nor the phase velocity. Only in infmite homogeneous linear viscoelastic media do homogeneous monochromatic waves (as always satisfied in unidimensional conditions) have an energy propagation ,·elocity equal to the phase velocity (Borcherdt, 1973). From the experimental standpoint, with respect to rocks in an infmite medium where no interference effects occur (as dispersion of geometric origin is obYiously possible in viscoelastic media), one must consider the relative importance of the concept of group velocity. Since attenuations are often such that the parameter "1. is small (low or medium attenuations), dispersion of intrinsic origin is also low, and the group and phase velocities may be merged as a ftrst approximation. Moreover, ifthe ray path for a given experiment is short, signiftcant dispersion, in effect, has no time to occur. Using Eq. (3.30b), the attenuation of a monochromatic wave can be obtained by: 1 dA d oc = - - - = - - - (In A) 16 ' A dxM dx.\1

(3.43a)

or a:= _ _ 1 _ I n A(x.\1,) x.\1 2

-

xM,

for x.\1, < x.v,

A(x.\1,)

(3.43b)

where A is the wave amplitude, and x.v• xM, and x.v, different observation positions. For a wave packet, the general concept of attenuation cannot be extracted from the dispersion effects discussed above. For moderate attenuations. however, Eqs. (3.43a) and (3.43b) can be appplied, taking the signal peak value equal to A. The attenuation determined can then be considered as the attenuation at the central frequency of the signal.

3.3.4 Quality factor, prelill)inary notions The importance of the breakdown of the complex modulus into real and imaginary parts has already been discussed. It is now possible to defme a quality factor Q which quantitatively characterizes the dissipation of the medium by: Q(CJ) = MR(w) M,(CJ)

(3.44)

Note that this defmition involves only the complex modulus. It also applies to any medium modeled by linear viscoelasticity, although it does not draw on any particular viscoelastic model like those which we shall examine in the next Section (Section 3.4). The quality factor is dimensionless. Since the imaginary part of the modulus is zero for a nondissipative elastic medium, the factor Q is infmite. By contrast. a zero quality factor implies an infmitely attenuating medium, without any transmission quality.

(6) The unit of attenuation is the neper per unit length. :x can also be expressed in dB unit lengtl:. and one can write: %dB ur:it len,C:

= 8.686

2:~p

unit 1entth

~

I

3

WAVE PROPAGATION II' VISCOELASTIC MEDIA

II3

Note that deftnition (3.44) is intrinsic and makes no reference to experiments which lead to its measurement, experiments which will be examined subsequently. For the most general linear viscoelastic medium, the developments in the foregoing sections enable Q to be related to other apparent parameters. The factor Q is fust related to the phase shift cp(w) between stress and strain measured on a sample subjected to a harmonic excitation [see Eq. (3.221]: Q(w) = tan q>(w) l Q(w) = cp(w)

Q~ l

(3.45a) (3.45b)

By (3.33a) and (3.33b), it is also shown that attenuation at angular frequency w 0 is related to Q by:

~-1 a=l;,J2 Jl+ 1 ~

Wo

(3.46at

Q2

or alternatively

Q = w0

[t _cx ~~] 2

2aV0

w0

(3.46bl

For Q ~ 1, these equations are written: Q ~ Wo = rcfo 2cxV0 aV0

(3.46cl

B = 2nC""'"

(3.471

Let us consider the value: LIC""'"

where C....," and LIC""'" are respectively the maximum kinetic energy and the decrease in this energy over a wavelength A. C sing (3.30b), it can easily be shown that: B = 2r.[l - exp (- 2aA)] - l

(3.48al

B = 2rc{ 1- exp (- 4n(ji+Q2- Q)]} -1

(3.48bl

which can also be written :

and for small cx(Q

~

1), we therefore have: B~Q

(3.491

114

3.3.5

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

Example of application of viscoelastic modeling to porous media

Porous media and rocks will be examined in the light of the viscoelastic models and discussed in the following Chapters. However, we should like to present here an example in which the macroscopic origin of viscoelastic modeling can be explained microscopically. To do this, let us consider a medium in which the porosity consists of spherical cavities of radius r, isolated from each other (Fig. 3.6). DiffriCted wave

z

,"::/ --

""'

I

··o, ,_,, 1

...

' _,

_..... I

.,- .....

.....

I

- ..... 0\I

... I

I ' .....

-' lo' I

'

'

I

I

..... I

,-

l(

-,/

'o, '.... _........

(p) . . -.. , I,0 __ \

p

Source

...

2r

I I

0\I

' _....

I

....

""'

Receiver

/

Fig. 3.6 Schematic porous medium consisting of spherical cavities in a homogeneous matrix.

Let us now assume that an incident P wave (see Chapter 2) propagates in this medium. The P and S wave velocities of the medium corresponding to the supposedly elastic matrix· are denoted Vr and J--S. The incident wave is expressed by: u

= u 0 exp

[i(kx - rot)]

(3.50)

where kVp = w. The incident power per unit area is written: P0

= Uxxu

(3.51)

Stress u xx is determined from (3.50) by: q xx =

(A

+ 2J-t)U,x

(3.52)

where A. and J-1 are the Lame constants (see Chapter 2). The average incident power over a period, defmed by: 2"

POv =

w

27t

Jol"' P

0

dt

(3.53)

can then be written from (3.50), (3.51) and (3.52):

POv =

-

~ (A. + 2J-t)wku~

(3.54)

3

liS

WAVE PROPAGATION IN VISCOELASTIC MEDIA

When the incident wave meets a spherical cavity, an elementary scattering process occurs. The scattered wave is the superposition of two waves, an S wave and a P wave. This scattered wave radiates energy in all directions, and this energy is subtracted from the incident wave energy. The latter is accordingly attenuated. To quantify the elementary scattered power, it is customary to introduce the notion of scattering cross-section SE which is the ratio of the average power scattered over a period to the average incident power per unit area. Hence it has the dimensions of an area. Many authors have investigated scattering effects by inclusions. Ying and Truell ( 1956) in particular determined the expression of the scattering cross-section for a spherical cavity. If the incident wavelength is large in comparison with the cavity radius r, they found: SE = gk•r6 m (3.55) If the spherical cavities are sufficiently distant from each other, i.e. if the number of spherical cavities n per unit volume is small (in practice, if the porosity is lower than 20%), it can be considered that no interactions occur between the scattered waves (no interference, and no multiscattering effects). The scattering cross-sections can then be added (see for example Waterman and Truell, 1961). This additivity of scattering crosssections allows the approximation to the f1rst order of multiple scattering to be made (see for example Ishimaru, 1978). In this approximation. it is considered that the direct wave is no longer the incident wave defmed by (3.50), but a wave attenuated by the elementary scatterers on the path already traveled. More specifically, consider a direct wave of average power P"v per unit area. In the volume dx d}' dz, the number of spherical cavities is ii dx dy dz. By defmition of the scattering cross-sections and from the principle of their additivity, the average power lost - dP"r by the direct wave on a path dx satisfies: - dP"v dy dz = iiSEP"c dx dy dz (3.56) from which, by integration: P"v

= POv exp (- 2cxx),

u

= u0 exp (- cxx) exp [i(kx -

rot)]

(3.57)

where attenuation oc is: 1 -s oc=2nE

(3.58)

Equation (3.33d) and the definition of the quality factor Q (3.44) then give rise to:

Q-t = iiSE k

(3.59)

(7) The precise value of the coeiT!cient g is:

41t

g=9

~

[ 3+ 40

(V.)s

2+32 Vs

L

nfVrYl

2

J

-2(vJ +3(vJ +16C,J 3 Vr

2

2 Vr

9

Vr

4]

If the wavelength is no longer large compared with the cavity radius, the expression of SE differs from (355t and the frequency-dependence to the fourth power disappears.

116

WAVE PROPAGATION IN VISCOELASTIC MEDIA

3

Hence, when the elementary microscopic process corresponds to scattering effects, Eq. (3.57) shows that a viscoelastic effect occurs macroscopically. However, the limits of the analysis should be noted forthwith, apart from the limitation on porosity mentioned above. Equation (3.55) corresponds to the so-called Rayleigh approximation or long-wave approximation. It assumes that the wavelength A, defmed by: 2n Vp A=-=-

f

k

(3.60)

is large compared with the inclusion radius, or more precisely:

kr < 0.1

(3.61)

An example of a radius of 1 J1. and a P wave velocity of 5000 m/s lead to a frequency limit fc of about 4 MHz, a limit that is generally an order of magnitude or more above applications of classic seismics (fteld or laboratory). Moreover, ncan be approximated by the expression :

cP

n = r3

(3.62)

where ¢ is the porosity. This consists in isolating the cavities in the tangent spheres of radius R = r/¢ 1 ' 3 • Combining (3.55) and (3.59) this gives:

Q- 1

=

c/Jgk3r3

(3.63)

Equation (3.63) shows that Q- 1 is proportional to the cube of the frequency, and also to the cube of the cavity radius. Hence, at the seismic to ultrasonic frequencies, scattering processes are negligible. They only become important at the very high frequencies. Emphasis must be placed on the fact that it would be misleading (or useless) to attempt to reverse the law of viscoelastic behavior defmed by (3.63) in the time domain. Apart from the high frequency limit fc of the analysis, other dissipative mechanisms with far more importaJ.l,t effects occur in the low frequency ranges (i.e. f ~ fc), where scattering processes are negligible. Thus, if the cavities are ftlled with a fluid, it is the viscosity of the fluid that plays the most important role in dissipation. Walsh (1969) pointed out that a fluid inclusion in a homogeneous matrix satisfactorily obeys the standard model developed in Section 3.4.1. The central angular frequency c.o, [see (3.76)] is then equal to ep.f'f, where e is the aspect ratio< 81 ofthe fluid-ftlled cavity, '7 the viscosity of the fluid, and J1. the shear modulus of the matrix. Other authors investigated the problem of wave propagation for a concentration of cavities ftlled with viscous fluids (in particular Datta, 1975). While these investigations are in themselves interesting in attempting to relate macroscopic viscous models to microscopic effects whose physical origin is clearly understood, they are nevertheless limited, because the experimental conftrmations are not conclusive. This is due to the fact that many other dissipation processes occur. They are analyzed in Chapter 5.

(8) The aspect ratio is the ratio of the two extreme dimensions of the inclusion: for a sphere this is l, for an ellipsoid with a major axis a and minor axis c. this is cfa and e ~ l.

.,-

:I

i

3

WAVE PROPAGATION IN VISCOELASTlC MEDIA

117

3.4 IMPORTANT VISCOELASTIC MODELS 3.4.1

Rheological models, new defmitions of the quality factor

Rheological models are often used for theoretical support in describing uniaxial behavior which displays dissipative effects of viscous origin. We shall show their general interest later. These rheological (viscous) models consist of networks of two elementary models, the spring and the dash-pot shown in Fig. 3.7. 11

E

~

IINIIIIN

•

(a)

a=Ee

(b)

Fig. 3.7 Elementary linear viscoelastic models. a. Spring b. Dash-pot.

If the force acting on the element is denoted by a and t denotes its extension, the equations of beha\ior are:

• For the spring: a= Et

(3.64)

,t

(3.65)

• For the dash-pot: t1

=

In its simplest form, this equation represents a linear and viscous behavior which linearly relates the force (or stress) t1 exerted on the element to the extension rate (or strain rate) e to which the element is subjected. We shall now examine the case of a commonly used model, the so-called u standard" or Zener model, shown in Fig. 3.8.1f t 0 and t 1 are the extensions (strains) of the two springs, using Eqs. (3.64) and (3.65) one obtains:

e = t 0 + e1 a= Eoto a= Ettt +'let

(3.66)

Eo

11

Fig. 3.8 Standard or Zener model.

118

WAVE PROPAGATION IN VISCOELASTIC MEDIA

3

where£ is the total strain of the model. It is then easy to deduce from (3.66) the equation:

a+ r'X)a

= E 0 r 00

e+ E

00

(3.67)

£

where we have set: 1: cc

=

I

'I

1

1

(3.68)

-=-++ E 1 ·Eoo E0 E 1

E0

The parameter r is called the characteristic relaxation time as it is representative of an experiment in which the strain is imposed [ d~:fdt = 0 in (3.67)] and in which the relaxation of the stress (to be imposed) to maintain this strain is observed (see Section 3.1.2). The parameter Eoo is the delayed elastic modulus; after an infmite time interval, it linearly relates the stress and strain (a= t = 0). On the other hand, £ 0 is the instantaneous modulus. It can then be shown that the solution of (3.67) can be written: X>

I

a(t) =

0£ r(t - r) -d dr

t

-oo

(3.69)

1:

where the expression of r(t) is given by: r(t)

= E + (E0

r(t)

=0

00

E aol exp ( - r

-

~)

t

~0

(3.70)

t < 0

The function r(t) is hence the relaxation function of the standard model, which had been defmed in general by Eqs. (3.5) and (3.6b). Proceeding as in Section 3.2, one can determine from (3.69): a(t)

I

+oo

= _

with

m(t - r)~:(r) dr

(3.71)

00

. dr m=dt

(3.72)

(3.70) and (3.72) can then be used to obtain the complex modulus relative to the Zener model: M(w)

= M0

1+r 00 r 0 w 2 •

+

t

2 00

W

2

•

+ zM 0 w

r 0 -r, • · 1 + ! 200 W

(3.73)

where we have set:

Eo _ !!_ M 0 = E-r_

to = r :c E

-

E1'

(3.74)

00

The complex modulus .\f(w) tends towards M 0 = £ 00 as w tends towards 0. M 0 is then called the relaxed modulus. Motion has become infmitely slow, and the dash-pot does not play a role (t = 0); the Zener model (Fig. 3.8) behaves like two springs in series E 0 and E 1 , or a single spring with a constant E'XJ defmed by (3.68). If the angular frequency tends towards infmity, the complex modulus tends towards M oc = E0 • As motion becomes very fast (t = oo ), the dash-pot blocks the movement ofthe springE 1 and only spring E0 reacts.

tr d

~~

3

119

WAVE PROPAGATIOS IN VISCOELASTIC MEDIA

The quality factor is determined from the defmition (3.44) and Eq. (3.73) by:

Q=

1 + t:/0 t 0 ci 0(t 0 -

t

(3.75)

,,J

The maximum 1/Q representing maximum dissipativity is obtained for the central angular frequency rom: w

m

0.76)

= .ytot:r: ~

and is therefore 1/Q(ro,.,)=

to- tao ~

2-y t 0 t

(3.77)

00

Figure 3.9 gives the variations of Q and MR(ro) as a function of angular frequency ro.

.------..----r-----.----.----....-----.

1.1

0.16

,.0_,._

2-;r;;:;:..

~MRP(w)

1.0

0.12

0.9

0.08

a-1

0.04

0.8 1

""'m ==

-;:r;;;: I

0.7

-=r=

1

-3

-2

I -1

I I

0

I

7't:

I

2

3

0

Angular frequency (log IClllt)

Fig. 3.9 Three-parameter solid (Standard) model.

The Zener model represents the network of three elements. Two other simpler models are defmed by the limit cases. One of them is the Kelvin-Voigt model, obtained for E0 ~ oo. This is the prototype of solid rheological models, because the delayed elastic modulus Eoo is non-zero. The second is the Maxwell model, obtained for £ 1 ~ 0. This is the prototype of fluid models. because the delayed elastic modulus tends towards 0; in other words the delayed strain is infmite and the material "flows" indefmitely (see Fig. 3.10).

i~

120

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

~~ ;I I

~

'I

Eo

1-f

If-IN

•

a

I lal

Time

.E~

'I

::!:J---

I!

en

a

El

~=I I

I Time

Fig. 3.10

lbl

Maxwell (a) and Kelvin-Voigt (b) models and their creep functions.

The Maxwell and Kelvin-Voigt models are very important because it can be shown (Mandel, 1966) that any linear viscoelastic solid can be represented by a series network of Kelvin-Voigt models and a spring (constant M 00 ), representing the instantaneous elasticity (Fig. 3.11 ), or in an equivalent manner by a parallel network of Maxwell models and a spring (constant M 0 ) representing the delayed elasticity. If the representation by Kelvin-Voigt models is adopted, in a similar manner to (3.66), this gives the equations: II

e = e0

u

=M

00

e0

+

L

i= 1

(3.78a)

e1

= E 1e1 + rue1

(3.78b)

For elements in series, the inverse of the resulting complex modulus is the sum of the inverses of the complex moduli of the elements. For the Kelvin-Voigt model, (3.78a) leads to: MkJJ) = E1 + i'71w

(3.79)

Hence the complex modulus satisftes: 1

1

1

= - + : LM-(w) M(w) M"" 1

(3.80)

1

For harmonic problems, it has been shown (Section 3.3.2) that, for a harmonic strain e = e,. cos wt, the stress is written :

u=

Bm

lMl cos (wt + qJ)

(3.811

I

I

ir 3

121

WAVE PROPAGATION IN VISCOELASTIC MEDIA

llj

llj-1

111

-1

M_

E1

eo

----+ Ej-1

a

Ei

Mj

e1

Ej

ei-1

Fig. 3.11 Representation of a linear viscoelastic solid using Kelvin-Voigt models.

The introduction of (3.81) in (3.78b) gives: IMI

+ cp- cp1)

e1 =e.., IMJI cos (c.ot IMI

e0 = e.., -

Moo

(3.82)

+ cp)

cos (c.ot

where we have set:

+ iM1;

M = IMI exp (icp) = MR

(3.83)

M 1 = IM11 exp (icp1)

The elastic energy at time t is written:

_1 1 \1-2 M:ceo

2 + "1 4- 2 E1e1

(3.84)

J

Hence from (3.82): 2

_ 1 2 2 [cos (c.ot + cp) \1 - 2 e.., IMI Mao

£

"

1 + 7' IMl

2

cos (c.ot

+ cp -

J

(3.85)

cp1)

From (3.80) and (3.83) it can be inferred that:

1

. .

IMI (cos cp -Ism cp) =

l

"

£

."

'IJc.o

1 MCC) + 7'1Ml-' 7'1Ml

(3.86)

Equations (3. 79), (3.85) and (3.86) then lead successively to:

\1

=~

2

2 [IMI 4 e... M QO

2

"

+ L...j

IMI £ 1 cos 2(c.ot + cp) IMI 2 IMJ·1 2 + M QO 2

IMI~ £ cos 2(c.ot + ~ IM 2 1

1

~ e2[cos 2c.ot Re ( Mao+ Ei MJ V =- e!Mo + 4 • M

2

1

4

1 e!MR \1 = 4

M

2

[(M2 M2) Moo + MJ £1

J

cp1)

(3.87a)

(M2 + EJ M2)] Mf

)

· 20Jt 1m M

-SID

+ 41 e! Re

+ cp -

exp (2ic.ot)

•

J

(3.87b) (3.87c)

122

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

From (3.80) we obtain: 1 eM

1 oMj

- =· IM~-ow M2 cw J

(3.88)

J

Hence with (3.79): oM OW

M2

~! 2

w - =I - 2 j Mj

i'1jW

=Ij -Mj2 (Mj- Ej)

(3.891

Equations (3.89) and (3.80) then lead to: 2 M2 M2 M2 E-M oM 1 M=-+ I-=-+ I-2-+wM oo i Mi Moo ow 1 Mi

(3.90)

Finally this equation introduced into (3.87c) gives the following expression for elastic energy:

W=

~ e!MR + ~ e! Re [ (M- w ~~) exp (2iwt)J

{3.91)

This expression had already been obtained by Bland (1960) and by O'Connell and Budiansky (1978), but in a less direct manner. Equation (3.91) gives the maximum elastic energy W""'" in the form :

wmax = 41 e! {

M R+

[r\ M R- w eM CWR)

oM ) -p:} J 2

2

+ ( M, -

w aw'

(3.92)

Thus, as stressed in Section 3.3.2, the maximum elastic energy, contrary to a classic error which consists in treating it as 1/2e!MR, depends in the general case on the real and imaginary parts of the modulus and their derivatives. Based on (3.91), however, the average elastic energy Wau is written very simply in the form: Wa., =

f T 1

2Jt

T=c;-

\1 dt =

1

4 e!MR

(3.93)

0

This is valid in the general case of viscoelastic models assuming a discrete breakdown as in Fig. 3.11. Since the energy dissipated per cycle is (see Section 3.3.2): JW

= ne!M1

(3.94)

Eqs. (3.93) and (3.94) immediately lead to a new defmition of Q:

Q = 4nWa., AW

(3.95)

In the general case of a rheological model, Q can therefore only be defmed from the average elastic energy and not from the maximum elastic energy. Nevertheless, this is possible for low and medium attenuations (Q > 10). In fact, the quantities MR and M 1, the real and imaginary parts of the complex modulus, are related, through their very defmition, by the Kramers-Kronig integrals (Nowick and Berry, 1972). These relationships are linear, and it can be shown that, if M 1 = 0, then oMRfow = 0 (see Appendix 3.1). This result can be understood by noting that the existence of an attenuation

wr-

•1

il

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

123

in a material (M1 :f:- 0) imposes the existence of a velocity dispersion [V = V(w) or what is equivalent MR = M~,(w)]. The functions M,(w) and c.\I.Rfow therefore display equivalent asymptotic behavior when attenuation becomes low: Q > lO ou M,

~

MR

<=>

oMR) ( M,. ow ~ MR

(3.96)

Accordingly we have:

Q > lO

1

2

\/,.." ~ 2 e,.MR

(3.97)

which is valid for only low dissipations, contrary to (3.95). Hence in the case of low attenuations, an approximate expression of the quality factor is:

2n\/,.." Q 1 Q ~ LtW ; ~

(3.98)

However, the demonstrations that led to (3.95) and (3.98) concern viscoelastic materials for which the representation by Kelvin-Voigt models corresponds to the discrete breakdown of Fig. 3.11. The term discrete does not imply that the elements are fmite in number, but that their characteristics E1 and 'IJ are isolated values on the positive real axis. The creep spectrum is said to be discrete and the material is said to have a short memory, because only the present stress and strain time derivatives are involved in the constitutive equation [see for example Eq. (3.67)]. If this is not the case, the representation by rheological modeling loses its value, since in particular, Eqs. (3.95) and (3.98) can no longer be demonstrated (although often accepted) because the discrete summation (3.84) of the energies by means of hidden parameters such as the strains e0 and e1 is no longer possible. The material is said to have a long memory, because all the past stress and strain values are involved in the constitutive equation which is available only in the form of a convolution product. To overcome this problem and to account for experimental observations that will be discussed, we can therefore, as in the following Section, defme the model directly using the qu.ality factor, rather than infer it a priori from a rheological model. The value of these models (for Constant or Nearly Gonstant Q) is obvious at the operational level, but it must be kept in mind that. in this new defmition, we lose the advantage of an explanation of apparent macroscopic behavior by hidden parameters, an explanation that leads to the development of rheological models.

3.4.2 Models defmed from the quality factor 3.4.2.1 NCQ model (Nearly Constant Q) Certain experimental results (for example, Murphy, 1982, Spencer, 1981) have shown that the quality factor remains constant over wide frequency ranges. The idea of Liu eta/., (1976) was hence to construct a mathematical model for which Q would be nearly constant, by direct superposition of Zener models, all having the same relaxed modulus M 0 but with different central angular frequencies [see (3.76)]. This yields the frequency dependence for 1/Q and velocity as shown in Fig. 3.12.

..·~I 124

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

~I

I

10' 8

r

0.01

1/0

10 4

10'4

0.005

0 4.8

v (km/sl

4.7

Phase velocity

4.6

4.5 10' 4

10'.

1

10 4 Frequency (Hz)

Fig. 3.12 Typical dispersion relations for the inverse of the quality factor and velocity for the NCQ model (after Liu et al., 1976).

The same result is obtained if the Zener model superposition is no longer discrete but continuous (Liu et al., 1976). The NCQ modc;l's dispersion equation is then: V(wl) ~ 1 + _1_ In col V(co2) 1tQ w2

(3.99)

Other authors have reached this conclusion mathematically by assuming Q is independent offrequency over a wide band (Azimi et al., 1968, Strick, 1970). Lomnitz ( 1957) also arrived at the same result by using an experimental creep law of the form: 0 t
\

where q, d and M 0 are experimental constants. 3.4.2.2

CQ model (Constant Q)

The naturally following mathematical step consists of building a model with Q strictly independent of w. This model was derived by Kjartansson (1979). The practical advantage of this CQ model in comparison with the NCQ models is clear. For the NCQ models, the dispersion relations and the frequency dependence of the quality factor are merely approximate relations, valid only for quality factors over 30. Moreover, the NCQ models all imply the introduction of a parameter related in some way or other to the frequency band over which Q is constant. The width and cutoff frequency of this band appear to be perfectly arbitrary, and the physical implications of this cutoff frequency vary according to the models (Lomnitz, 1957, Futterman, 1962, Strick, 1967, Liu et al., 1976). The Constant Q model is very simple mathematically, and is completely specif1ed by two parameters,

.;;r3

125

WAVE PROPAGATION IN VISCOELASTIC MEDIA

namely the wave phase velocity at a reference frequency and the value of Q. The creep function used by Kjartansson ( 1979) has the following form :

0 f(t) = CQ

{

t~O

1 M_rtl

t

-1-

?v\ Co)

27

(3.101)

t~O

where r is the classic gamma function (see, for example, Abramovitz and Stegun, 1972) and M 0 is the modulus of the complex modulus at the reference angular frequency w 0 = 2n/t0 • This creep function has already been analyzed by Bland ( 1960). It implies that the complex modulus is: 2

iw ) M(w) = M 0 ( Wo

2 Y=

w M 0 I Wo

1 Y exp

[i1t}' sgn (w)]

9 1 )

(3.102)

The quality factor Q is then given by: 1 - = tan (1tf')

(3.103)

Q

Figure 3.13 shows the behavior ofthe creep function vs. time for the Constant Q model. It is interesting to note the absence of instantaneous elastic strain for this model, unlike the previous Lomnitz model (1957). Let us examine the form and properties of the impulse response of this model. To analyze the impulse response amounts to examining how a Dirac delta function c5(t) is propagated: (3.104)

u = u0 c5(t)

This pulse is emitted at a reference abscissa x 0 in a medium satisfying the wave equation (3.29). Since the plane wave solution is exp (-ax) exp [iw(r- xfV)], the Fourier transform of the impulse response (after a travel distance x) is: H(w) = u0 exp (- tXX) exp (- iw

~)

(3.105)

For the Constant Q model, it is shown using (3.33), (3.102) and (3.103) that: ex = tan

~ sgn (w) ; ' ~ = Vo (~oy Vo =

I: r

(3.106)

....!...:.---'--

cos

1t}'

2

hence the function H(w).

(9) The sgn (·) function is the function which yields the sign of the quantity inside the parentheses.

126

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

0= 10

0=1 1.2 3

-!

-8 1.1 -~

-~ ~ 2

c.

..

i

1

c

c

-~

"£

u;

1

II)

0

.9 .8

0

2

6

4

8

0

10 Time

2

1.002

"B

1.01

1.001

~

'a.

i

10

8

0= 1000

0=100

c ·;;

6

Time

1.02

i

4

c.

i

1

1

c

!

~

.999 .998

.98 0

2

6

4

10

8

0

2

4

6

10

8

Time

Time

Fig. 3.13 Creep function oftheConstant Q model (after Kjartansson. © 1979 AGU).

The impulse response h(t) is then the inverse Fourier transform of H(w). This response is shown in Fig. 3.14for a unit pulse(u 0 = !)emitted at x 0 t 0 = QV0 as a function of time tft 0 [see (3.101)]. An important property of the impulse response is i

(X )t"=""Y

1 TocLfoc-oc -

A

V0

(3.107)

where T is the wave travel time for distance x, A the signal width, and A its maximum amplitude. This relation can be written: T A= b(Q) Q

(3.108)

where the function b(Q) depends on Q. It can be shown (K.jartansson, 1979) that b(Q) is virtually constant for Q > 20, which allows the development of a measuring technique for Q called the measurement of the signal width or measurement of rise time (see Chapter 4). To compare the Constant Q model with the NCQ models, the creep function (3.101) can be written in the form :

{~) = Mor(! + ly) exp [ 2y InC:)J

~-----

-------~~--~--

(3.109)

~

3

127

WAVE PROPAGATION IN VISCOELASTIC MEDIA

0.6r-----;:----------, 0=1

i' "

0.4

II~

0.2

0

5

o'

I

10 Time

5

~

5

..i

0.4

i

!c

0.2 01....---'-...L.--===~---L..J

105

100

______ ,

10

,,

15

20

0.6r--------------,

0=100

95

1:

Time

0.6 r - - - - - - - - - - - - ,

I

Oc 10

i !G 0.2~'

0.41

i

0.2~ 0·

110 Time

0= 1000

~

995

II::=-±-

1000

1005

II

1 010 Time

Fig. 3.14 Impulse response for the Constant Q model (after Kjartansson. (C) 1979 AGU).

Assuming that expansion:

y

is small (i.e. Q > 10), /(t) can be approximated by its ftrst order CQ

~~) ~ Mor(~ + 2y) [ 1 + 2y ln r:]

(3.110a)

1(1 + 2y) = 1 - 2yf3 to the fust order

(3.110b)

Moreover: where

p is the Euler constant: f3 ;;;;: 0.57721 (Abramovitz and Stegun. 1972)

(3.110c)

~~) ~ ~0 [1 + 2{1n (:J + P]] = ~ 0 [1 + 2y In rrJ

(3.111)

and hence

In the Lomnitz equation (3.100), the quantity d. inverse of a time, is very large in comparison with the time resolution of the experiment. Therefore 1 + td ~ td, which when introduced into (3.1001, implies that: 1 /(t) ~ -M (1 NCQ

0

which is clearly of the same form as /(t). CQ

+ q In td)

(3.112)

128

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

The same thing may be observed by starting with the dispersion equation (3.106):

v=

Vo

I:J' ~ vo(l ;J +yIn

(3.113)

in which Eq. (3.99) can also be recognized. The Constant Q model is hence the same as the NCQ model for quality factors greater than 10. This is not surprising. The Constant Q model is a limit of the NCQ model constructed from an inftnite set of Zener models. The Constant Q model offers the advantage of being easy to use in various situations, due to the frequency independence of Q. Physically, however, its validity is not generally greater than that of any other models.

3.5 VIBRATIONS IN VISCOELASTIC MEDIA 3.5.1

Traveling waves and vibrations

Let us consider a cylindrical rod of an attenuating material, of length l, initially at rest and ftee at both ends. For a short instant, let us now impose a force at the end x = 0. for example by an impact, and receive the signal at the other end x = l. The recorded signal is shown in Fig. 3.15. A series of pulses at intervals Tis fust observed. These different arrivals repeat the shape of the ftrst pulse, but their amplitude decreases and their duration lengthens with increasing distance from the original instant. A damped sinusoid of pseudo-period T gradually appears. This signifies the transition from unsteady state conditions to damped vibrations or pseudo-steady state conditions. The equations of the problem are the unidimensional wave equation (Section 2.1.5.2), for an elastic medium, which is recalled here :

u.xx

1 ..

(3.114)

=Vi u

where VE is the extension wave velocity and the boundary conditions: u.x(O, t) = f(t); f(t) = 0 u.x(l, t) = 0

t<0

(3.115)

where f(t) is the force exerted at extremity x = 0, normalized by the product ES (E Young's modulus, S rod cross-section). Since the derivative u,x also satisftes the wave equation, the analysis will be based on this quantity. By making a change in variables: X

Zt = t -

VE

X

z2

= t + VE

(3.116)

~ 3

129

WAVE PROPAGATION IN VISCOELASTIC MEDIA

0.4ms

.........

I I

I I

I

I

I

I

T

I

I : • T .:. T --+--'--~

1 ms

........... Fig. 3.15 Experimental signal recorded at x = I. The source is a short pulse at the origin. The sample is a Plexiglas rod. The top recording is a magniftcation of the frrst milliseconds of the bottom one.

the· wave equation then becomes: (3.117)

u,x%1%2

This equation implies that the general form of the solution of the wave equation is:

u,.,(x, t)

= t/l(t-

;J

+ q>(t +

;J

(3.118)

The function t/1 corresponds to a wave propagating in the direction of increasing x, while the function q> corresponds to a wave propagating in the direction of decreasing x. These functions thus correspond to the successive reflections occurring at both ends of the bar (at x = 0 for t/1, and x = I for q>). More precisely, it can be shown (Courant and Hilbert, 1962, for example) that the solution to (3.114) and (3.115) is: 00

X ) u,.,(x,t)=f ( t - -

vE

+L

•• 1

[

-f

(

2nl) ( v£

2nl)] v£

X - - - +f t -X- - t+

v£

vE

(3.119)

The f1rst term represents the propagation of the signal imposed at x = 0, while the second and t~rd terms correspond to the n•h reflections at the ends x = I and x = 0. The beginning of the signal in Fig. 3.15 thus corresponds to the successive reflections of the original pulse. In relation to (3.119), however, established for an elastic case, the dissipative processes deform the signal in two ways: on the one hand, energy is lost, and the peak

130

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

amplitude decreases, and on the other, the signal spreads due to the dispersive character of the waves (i.e. the velocity depends on the frequency). This spreading corresponds to the one observed for a Dirac delta function and a CQ model in paragraph 3.4.2.2. The question arises how to interpret the later appearance of the damped sinusoid. Letus particularize the function f by a brief excitation modeled by a Dirac 15(t). Equation (3.119) then gives:

( VEX) + L

u.x(x,t)=b t - -


[

X -2n/) -b ( t + - +b ( t -X- -2n!'J] (3.120)

VE

n;!

VE

VE

VE

Taking the Fourier transform [Eq. (3.14)] of this equation, we obtain: GO

U,x(x, k*) = exp (- ik*x)

+

L

(exp [- ik*(x + 2nl)]- exp [ik*(x- 2nl)])

(3.121)

n;l

where k*, the wave number, is given by: k* VE = w. The equation can be expressed alternatively in the form : :X:

U,..,(x, k*) = 2i exp (- ik*l) sin k*(l - x)

L

exp (- 2ik*lp)

(3.122)

p;O

which, taking the sum of the geometric progression, leads to: U ' x(x, k*) =

sin k*(l- x) . , ., [1 - exp (- 2ik*lp)], p Stn .

-+

+ oc

The integration of this equation gives: u (x, k*) = ucolt + uincolt

(3.123)

(3.124)

where ucolt =cos k*(l- x)

k* sin k*l • lncolt _ _

U

-

exp (- 2ik*lp) cos k*(l- x) for P -+ k* sin k*l

+ 00

(3.125)

The term ucoll is the coherent part of the signal representing the contribution of constructive interferences of the reflections to the general signal. The incoherent part uincolt, however, has no limit. Its phase is random, except precisely at frequencies such that: k* _ mn I

(3.126)

m-~

At these frequencies, ucolt tends towards infmity. This fmally gives the schematic spectrum in Fig. 3.16, whose irregularity results from the contribution of uincolt which varies at each reflection. · Equation (3.123) corresponds to the non-dissipative elastic case. If one considers a viscoelastic attenuating medium, the wave number becomes complex (Section 3.3.3.1): k*

= k- ia

a> 0

(3.127)

r

3

WAVE PROPAGATION IN VISCOELASTie MEDIA

131

1-

-!

.~

..

Q.

E

~ §

~ i5

),

.J."-

w,

____, ·'-"'3

"'2

· Angular frequency

Fig. 3.16 Schematic diagram of the spectrum in the elastic case for any point x.

When introduced in (3.123), this expression shows that the incoherent part now tends towards 0, and we obtain:

u = ucoh = cos k(l -

x)

k sin kl

(3.128)

Hence, in the viscoelastic case, U has a limit whatever the frequency, with local maxima at frequencies close to:

w,

m1t

k, = VE(w,) = -,-

m = 1, 2...

(3.129)

For sufficiently low attenuations, the velocity Vt:(w) can be considered as independent of the frequency, and the Eqs. (3.129) give the local maxima to the nearest second order. After a sufficient number of reflections, the signal is mainly composed of damped sinusoids of angular frequencies w,.. ·As a rule, attenuation increases with frequency, and the contributions of the sinusoids to displacement disappear sooner for larger values of m. The results are shown schematically by the spectra in Fig. 3.17. Note that the period T in Fig. 3.15 is simply 21t/w 1 = 21/Vt:, a time which clearly corresponds to a round-trip of the transient wave, and explains the spacing Tofthe pulses of the ftrst part of the signal. One can thus qualitatively explain the experimental result in Fig. 3.15: the original signal propagates, is reflected successively at the ends, and, due to dissipative effects, decreases in amplitude. Only the frequencies corresponding to W 111 contribute significantly to the signal, because they correspond to constructive (in phase) interferences of the different reflections, and thus give a signal with a sufftciently high amplitude to be able to propagate without being attenuated too rapidly. To directly attempt to derive the pseudosteady state asymptotic solution requires making a modal analysis ofthe system, in other words ignoring the transient conditions and developing the solution on the normal modes

132

WAVE PROPAGATION IN VISCOELASTIC MEDIA

3

defmed by the characteristic angular frequencies w,. This modal analysis will be examined in the next Section.

~

a =a ~ E ~

§

"§. i5

"'t

"',

"'2

Angular frequency

Fig. 3.17 Schematic diagram of successive spectra (p .1') in the viscoelastic case.

3.5.2 Modal analysis 3.5.2.1

Normal modes

Let us consider an elastic rod and determine the standing waves of angular frequency w which may be established therein. The corresponding displacement can be taken in the form: u(x, t) = Re [u(x) exp (iwt)] (3.130) The introduction of (3.130) in (3.114) gives:

u,xx

+ k2 u =

0

k =

(I)

VE

(3.131)

The solution is straightforward: u(x) = A sin kx

+ B cos

kx

To obtain the asymptotic solution for a bar free at both ends, it is necessary to satisfy the boundary conditions: or

u,..,(x = 0 or x = l, t) = 0

(3.132)

A=O, sinkl=O

(3.133)

leading to the characteristic angular frequencies (or eigenvalues) and to the normal modes: . mn mn mnx k, = - -, w, = - - VE, u,(x) =cos - -, m = 1, 2... (3.134) 1 1 1

r

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

133

l

I

which are precisely the angular frequencies ro, discussed in the previous Section which correspond to constructive interferences. The term .. normal modes" arises from the fact that these modes are mutually orthogonal, in the sense of the integral scalar product: (2/l) J:u,u" dx = {

~

m :F n m=n

(3.135)

This orthogonality means that the inertial forces in modem, proportional to w!u,, do not contribute to the displacements u" of mode n :F m. Let us now consider a viscoelastic bar subjected to linear external excitation p(x, r). The equation of motion is (Section 3.3.3.1):

a2 u

m(r) * ax2 (x, r) = pii -

s1 p(x, t)

(3.136)

where Sis the cross-section of the bar. By applying the Fourier transform we obtain: d2U

M(ro) dx 2 (x, ro)

pw 2 U-

=-

1

S P(x, ro)

(3.137)

Let us now consider the associated elastic bar defmed by the modulus M(w) = M(O) = M 0 1

and the velocity VE = (.\10 / p)2. Let us break down the solution of the viscoelastic problem to the normal modes of this associated elastic bar : U=

L a,(w)u,(x)

(3.138)

"' where u..,(x) is defmed by (3.134). We multiply (3.138) by u" and integrate. The orthogonality of the normal modes (3.135) gives the following as the expression for the generalized coordinate a,.(ro):

ff

U(x, w)u, dx

a.,.(ro) =

(3.139)

u! dx

By similarly multiplying (3.137) by u., and integrating, we obtain: M(wl

2

f

2 ddxU 2 u., dx = - pw

f

1 Uu.., d~ - S

f

Pu., dx

(3.140a)

Integrating by parts and taking account of boundary conditions (3.132), we obtain: M(w)

f

2

u dx U ddx;'

+ pw2

f

1 Uu., dx = - S

f

Pu., dx

(3.140b)

The normal modes satisfy: d2u ~-"' dx 2

2 + w., v2

E

u,. = 0

(3.141)

134

3

W.WE PROPAGATION IN VISCOELASTIC MEDIA

Equations (3.138) and (3.139) introduced into (3.140) thus give: 1

a,.(w)

f

P(x, w)u,.(x) dx

(3.142)

= R,. [M(w)/M0 ]w!- w 2

where R,. = pS

f u! dx

(3.143)

For the extension modes examined here, R,. = p Sl/2. However, we shall leave R,. aside, because the previous developments remain valid for other types of vibration (bending, torsion, etc.) which do not necessarily satisfy R,. = constant. These other types of vibration will be analyzed experimentally in the next Chapter.

3.5.2.2

Forced vibrations and free vibrations

Let us ftrst consider a harmonic excitation of angular frequency ro 0 exerted at point x 0 P(x, w)

= 27tP0 b(x- x 0 )b(w- w0 )

:

(3.144)

By (3.138) and (3.142), the response is: r· _ '- -

~

L..

.. = 1

27tP0

u,.(x 0 )u,.(x)

2

2

~( _ ) u W Wo

2

R,.w, M(ro)/M 0 -ro fw,.

(3.145)

By introducing M(w) = .\fR(w) + i.\11(w) into (3.145) and inverting the expression, we obtain the time domain expression. where we have formally replaced w 0 by w: U (X,

t)

=

~ 1

.,-;;: 1

u,.(x 0 )u,.(x) cos (wt - ({),.)

P0

-2

R,.w,. [(MR/M0

-

w 2 /w!) 2

1

(3.146)

+ (MJIM0 ) 2]2

where the phase difference ({),. is given by : MJIM 0 tan({),.= MR/Mo- w 2 /w!

(3.147)

Let us now consider free vibrations, obtained by an impact. The excitation, assumed to be exerted at x = 0, is thus:

= P0 b(x)b(t) P(x, w) = P0 b(x)

p(x, t)

(3.148)

Equation (3.148) introduced into (3.142) with (3.138) leads to:

U=

L _!l_

u,.(O)u,.(x) .

,. R,.w! M(ro)/M 0

-

w 2 fw!

(3.149)

Without any additional assumption about M(w), it is impossible to proceed further. Let us therefore consider the case in which damping is slight. In this case, in the vicinity of an angular frequency w,., one can write: (f) ::: (1)'" '

M(w)

~

MR(ro,.)[1

+ 2ie,.J

(3.150)

r I

3

I~

WAVE PROPAGATION

135

VISCOELASTIC MEDIA

where ~ ... is the reduced damping defmed by:

2e .. ~ .\.f,(w.,.)

(3.151)

.\.f~~.(w.,.)

•

Due to the slight damping, the maxima of(3.149) are clearly separated, and it is possible to invert (3.149) in the time domain:

u(x, r) =

I

m:l

RPo

2 mWm

u..(O)u.,.(x) exp (-

e. . w.,.t) sin co,..t

(3.152)

This equation corresponds to the qualitative result of the experiment in Section 3.5.1.

3.5.3 Defmitions of the quality factor using vibrations When dealing with vibration methods, the quality factor can be defmed using Eqs. (3.146) and (3.152). Let us fust consider free vibrations in the case ofs1ight damping (Q ~ 1). By filtering, each term of the series in Eq. (3.146) can be isolated, and each of term then corresponds to a damped sinusoid (Fig. 3.18). The logarithmic decrement b., is then defmed as the logarithm of the ratio of two successive maxima, or as the logarithm of the ratio of two displacements, for a time interval equal to a pseudo-period: f> ... =In

u(t) u(t

)

+ 2nfw...

(3.153)

by (3.151) and (3.152): 1 Q(w...) ~ 2e ...

·1 •

-!

£

l

Tm=211/W

m

.-1

1[

= b,.

Q~1

u ltol•- ~mwm(t-tol •

Llto+T I /

~mf

Fig. 3.18 _ Damped sinusoid corresponding to the mth characteristic frequency [Eq. (3.152)].

(3.154)

136

WAVE PROPAGATION IN VISCOELASTIC MEDIA

3

For slight damping, the quality factor is thus related simply to reduced damping and to the logarithmic decrement. The quality factor is obtained in the vicinity of the characteristic· angular frequency Wm· In the case of the impact experiment previously described, the first characteristic frequency is the ohly one to make a significant contribution [factor 1/w! in (3.152)]. Hence only Q(wd can be determined. To obtain the quality factor at higher frequencies by the logarithmic decrement, it is preferable fJCSt to subject the bar to forced excitations near the desired angular frequency wm, and then interrupt the excitation. The free vibrations then occur at the desired angular frequency given the adequate initial conditions. Let us now consider the case offorced vibrations. Each term of the series (3.146) can be placed in the form: Po

u..,(x, t) = - R 2 cos (wt - rpm)A..,u,.(x)u,.(x 0 )

,.w,.

(3.155)

where A.., is the amplitude term: 1

A,.=

1

[(Ma/M0

-

(3.156)

w 2 /w!} 2 + (M.fM 0 ) 2 ]2

Assuming slight damping (Q ~ 1), we know from the Kramers-Kronig relations that Ma ~ M 0 (see Section 3.4.1 and Appendix 3.1) and the amplitude term is reduced to: 1

A,.~

1

[(1 -

(3.157)

w2 /w;.) 2 + (MJ!Ma) 2]2

Assuming slight damping (Q ~ 1), the contributions corresponding to each characteristic angular frequency are clearly separated, which means that if the exciting angular frequency is close to an angular frequency w..,, only the term A,. has a significant contribution in the series (3.146). Figure 3.19 thus shows the influence of modes 2 to 5 on the response with excitation close to the flfst characteristic frequency in the case of a slightly attenuating material Q =50 and strongly attenuating material Q = 3. For the case of Q = 50, it is apparent that the higher harmonics can be totally ignored, but this becomes less valid for higher frequency or lower Q (Fig. 3.20). Assuming Q ~ 1, one can then proceed with a classic measurement of the quality factor, called the frequency sweep. Let us determine two angular frequencies w 1 and wu (see Fig. 3.21) such that:

A!(Wr) A!(Wu) = w.., w,. =

2

A (1)

2

(3.158)

We can then show that:

Q~

w,. wu - w 1

= w.., = _1_

A..,w

2e,

Q~ 1

(3.159)

where A,w = wu- w 1 quantifies the spectrum width and rises with increased damping.

----- ----------------------------

T

3

137

WAVE PROPAG.... TION II' VISCOELASTIC MEDIA

I

I

E

1

0.4

~

0.2

ol

I 1000

. "'

:r

1500

.

=:e'

I

1

2000

J

2500 Frequency (Hz)

Fig. 3.19 Contribution of harmonics 2 to 5 to the shape ofthc f1rst harmonic. The model used is the Constant Q model. The modulus M 0 is 5 GPa, the reference angular frequency is 1000 s- 1 • The density of the material is 2.5 gjcm 3 , and the sample length is ..W em.

0

F---;----=w-· I 3000

'

1 3500

'

J 4000

r

:

1

~

F.-quency (Hzl

Fig. 3.20 Contribution of harmonics 1, 3, 4 and 5 to the shape of the second harmonic. The parameters of the model are the same as in Fig. 3.19.

The measurement of the phase difference between excitation and displacement offers another means to determine ru1 and wu. In fact, for MR;;;; M 0 , the expression of tan cp.,. (3.147) immediately shows that: tan cp.,(w 11 ) = -tan cp,..(ru1) = 1

(3.160)

The excitation and the response show a phase difference of ± :: 14. In the case of rheological models. we have shown in Section 3.4.1 how the factor Q could be defmed from the energy dissipated over a cycle and the average or maximum elastic energy involved during a cycle. These defmitions apply to an experiment in which the excitation is the stress and the strain is the response (i.e. no inertia terms, sample dimensions negligible). One may well ask whether this applies to a rod in which the dimensions are no longer negligible.

I I

l

t

138

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA A({l)

1

a= 011 _n

AM(O}

1

VV2AM(0}

w

n, Fig. 3.21

n=-

"'m

nu

Principle of the measurement of Q by frequency sweep.

To answer this question we consider the assumption by which damping is sufficiently slight for the response to be given by (3.155), when excitation occurs near the angular frequency w,. Owing to the conservation of energy, the dissipated energy A W is the energy furnished to the system during a cycle, because the elastic energy (reversible) is totally restored at the end of the cycle. The dissipated energy, for the case of excitation giYen by (3.144), can be written:

-; f 2"

AW =

du

(3.161)

P0 dt"' (x 0 , t) dt

0

The introduction of Eq. (3.155) for u,(x0 , t) into the foregoing equation gives: Ll W

=

1tP~

R

u!(x0 ) sin cp,.

2

,.w,. [(MR/M0 -

1

(3.162)

ru 2 /ru!) 2 + (M1/M 0 ) 2 ]2

For rheological models, the average elastic energy V,., in the case of slight damping, for which MR ~ M 0 , can be written from (3.93) in the form:

V,.,=4M 1 0

f'

s!dx

(3.163)

0

where s, is the maximum deformation at point x, namely: s,. = max ou,(x, t)

•

t

OX

Introducing the expression of s,., taken from (3.155), in (3.163), and substituting the expression ofsin cp, deduced from (3.141), we obtain, after a few calculations: 47tV,..,

Q ::::-LIW

Q~

1

(3.164)

-

~------

T

3

--

------~-

WAVE PROPAGATION IN VISCOELASTIC MEDIA

139

I

I I

I

This shows that the local expression (3.95) can be generally applied to the resonant bar with the difference that Eq. (3.164) for the resonant bar is valid only for dampings such that Q ~ 1, whereas (3.95) was valid assuming rheological models alone. Simultaneously, it can be shown that: 27tV""'.x

Q ~ AW

Q~ 1

(3.165)

where \1 ""'x is the maximum energy.

3.6 CONCLUSIONS CONCERNING THE QUALITY FACTOR AND FINAL REMARKS Table 3.1 shows all the ways to deftne or approximate the expression of the quality factor Q, whose reference defmition is M .. 1M 1, and notes which experiments underlie these defmitions. Each of the experimental techniques (resonance or propagation mechanisms using standing or traveling waves) allows the direct or indirect measurement of a speciftc quality factor Q. In fact, every value of the quality factor Q = (M .. I M 1) is associated with the type of excitation analyzed, and hence with the elastic modulus concerned. A bar in extension allows the measurement of QE = E~./E., where E =E.. + iE1 is the complex Young's modulus. A measurement performed on shear waves allows the measurement of Q5 = JLRIJL~o where JL = Jl1. + ip1 is the complex shear modulus. We can also defme Q.c = Ka/ K 1, where K = K .. + iK1 is the complex bulk modulus even if a direct experimental measurement seems ditTtcult to realize. Winkler and Nur (1979) showed that, for slight attenuations (i.e. by ignoring the products ofthe imaginary parts of the elastic moduli) and by using the correspondence principle (Fung, 1965) (i.e. by replacing the relations between real terms of Table 2.2 by the same relations but in complex terms), it is possible to obtain simple relations between Qp, Q5, QE and Q.c: (1 - v)(l - 2v) 1 + v 2v(2 - v) Qp =~Qs 3 1 - 2v 2(v + 1) -=--+---:-

QE

QK

1+ v

3(1- v) ~= Qp

Qs

(3.166)

2(1 - 2v) Qs

The Poisson's ratio vis obtained from the velocities by the equation: v=

v~ - 2 v~ 2 2 = 2(Vp- V 5 )

Vi -

2 v~ 2 2Vs

(3.167)

TABLE SUMMARY OF QUALITY FA<-TOR

Parameters concerned

Ct,

w,

v

tiC.,.,., C...,... A, et

Propagation

Propagation Harmonic excitation on sample

Vav• tiW Harmonic excitation on resonant bar

\!max• tiW

Harmonic excitation on sample

DEFINITIONS

Viscoelastic models

Sections

All

3.3.4

Q ~ 2etV

All

3.3.4

Q ~ 2nC..,.. LICmax

All

3.3.4

Q ~ 4nVa• LIW

Rheological

AnyQ

Q~l

I Q=tan cp

Q~-

Q =-~(I- C(2~) 2etV w2

Possible experiments Harmonic excitation on sample

cp

3.1 (Q = M,jM1)

c_..

--=(1-exp [-2etA])-

Lie_..

Q

4nV•• =---

tiW

I

cp

(I)

1

3.4.1

3.5.3

2nV max -- tiW

Q~-

3.4.1 Rheological

Harmonic excitation on resonant bar Ll.,w

15.,

'"'

Harmonic excitation on resonant bar Free vibrations on resonant bar Free or harmonic vibrations on resonant bar

-----------~- ---~~---·

35.3 Q~ w,.,

Ll.,w 1t

Q~-

15.,

l

Q~-

2~ ..

All

3.5.3

All

3.5.3

All

3.5.3

3

WAVE PROPAGATION IN VISCOELASTIC MEDIA

141

where the different velocities are related by 2

V,. = V2

_

v:(4 Vi 3Vi-

Vi)

Vi

V~(3V~- 4V~) v~-

E-

Vi

(3.168)

2 2 4 2 VK=V,.-3Vs

The ratio Q,./Qs can also be written as a function of (V,./V5 ) 2 :

Q,.= Q5

"'' K

4

I+ Jill

v~

-2

Vs

(3.169)

Finally, it can be shown that one of the following relations always occurs:

QJt > Q,. > QE > Qs Q" < Q,. < QE < Q5 Q"

for high V,./Vs ratios (e.g. in dry or totally saturated rocks), for low V,./Vs ratios (e.g. for partially saturated rocks),

= Q,. = QE = Qs.

3. 7 CONCLUSIONS In this Chapter we have considered the problem of wave propagation in viscoelastic media mainly from the phenomenological standpoint. More speciftcally, experiments (creep, relaxation) ftrst led us to the laws of linear viscoelastic behavior defmed by the convolution product (3.9). In some cases, it was possible to obtain a representation of these laws by the rheological models developed in Section 3.4.1. In most cases, however, the creep and relaxation functions are only implicitly known through the quality factor Q, and only over limited frequency bands. Nevertheless, the physical origins of the validity of these different models are not generally established in an indisputable manner. Note that the use of a linear \iscoelastic model enabled us to defme the quality factor Q in the form MR/M 1, and to relate the different ways to defme various quality factors which, a priori cannot be deduCed from each other by the equations in Table 3.1. Hence, for the linear viscoelastic model. the quality factor concept is quite independent of the type of experiment concerned, and is an intrinsic parameter of the medium. For linear non-viscoelastic models, the defmitions of Table 3.1 normally allow a characterization of various damping effects rather than one damping effect. Thus Biot's theory clearly leads to the notion of attenuation ex in propagation, and hence to a quality factor (c.o/2cxV), and yet this quality factor is not equivalent to the one that could be determined by an experiment with a resonant bar (White, 1983). From the practical standpoint, the "quality factor" parameters obtained by experiment can subsequently be compared with each other in the light of the assumption of linear

142

3

WAVE PROP:\GATION IN VISCOELASTIC MEDIA

viscoelastic behavior. It is clear that this is merely a model that may not necessarily represent reality. Laboratory experiments nevertheless show that, as a ftrst approximation, this model is satisfactory and can serve to analyze field problems, for example. The many causes of dissipation, and their relative preponderance according to the frequency range analyzed, normally preclude any attempts to make a too sophisticated defmition of a complex modulus. and hence a factor Q (and even more the creep and relaxation functions), by refming the models developed in this Chapter. The procedure required can be identif1ed clearly. It proceeds from a dual standpoint. experimental and theoretical. From the experimental standpoint, it consists of analyzing the variations of the quality factor for each type of rock, which appears to be the most interesting quantity to measure, apart from velocities, as a function ofvarious parameters (frequency, porosity, water content, etc.), for which care has been taken, as much as possible, to isolate these effects in an ad hoc experiment. From the theoretical standpoint, it consists of a careful study to understand the major physical effects responsible for the observed attenuation at the frequency studied. This makes it possible to model the processes, or rather represent them by viscoelastic models such as those developed in this Chapter. These models thus appear more as working tools rather than as an end in themselves. Adapted to the rocks analyzed, they can therefore provide guidance for a mathematical modeling of the processes involved, not only at the level of the laboratory sample, but at the level of f1eld seismics. We shall discuss these points in their various aspects in the subsequent Chapters.

Appendix 3.1

THE KRAMERS-KRONIG RELATIONS It has been shown in Section 3.2 that: u(t)

f

+JC

= _cc

m(t - r)e(r) dr

(3A.l)

where m is the derivative in the sense of distributions of the relaxation function r: dr m = dt

with r(r) = 0, r < 0, r(O+) = r0

(3A.2)

Since the complex modulus is the Fourier transform of min the sense of distributions: +or:: dr M(w) = MR(w) + i.\11(w) = d exp (- iwt) dt (3A.3)

f

-0::

t

the expressions (3A.2) give: MR(w) = r 0

M1(w)

=-

+[

f -

+:x: Xl

o::dr -d cos wt dt 0::

d

t

..!... sin CJt dt dt

(3A.4)

--3

WAVE PROP:\GATION IN VISCOELASTIC MEDIA

143

since the discontinuity of the relaxation function rat t = 0 is a Dirac delta function r 0 c5(t) for its derivative, considered in the sense of distributions. In (3A.4), unlike (3A.3), we deal with integrals of ordinary functions. Similarly, one can introduce (see Section 3.3.2) a function such that : ~>+ao

e(t) =

I

df

j(t - t)O'(t) dr

.. -:o

By introducing the Fourier transform J(w) = la(w) Eqs. (3A.4), we obtain: Ja(w) == io

f f

+

+ao

+ iJ1(w)

simultaneously in the

df

-d cos wt dt t

-ao

+:o

J 1(w) = -

(3A.5)

j=dt

(3A.6)

df

-d sin wt dt t

-X)

Note that Eqs. (3A.l) and (3A.5) require that: J(w)M(w) = 1 Hence in particular : J(oc).\f(oc) = J 00 M 00 =jor0 = 1 J(O).\f(O) = 1 0 Mo =j 00 r 00 = 1

,_

(3A.7)

In fact, we know (see, for example, Sections 3.2 and 3.4.1) that the instantaneous or unrelaxed modulus r 0 = r(t -+ o+) and the delayed or relaxed modulus r x = r(t -+ oo) correspond respectively to the moduli Mao (inf10ite angular frequency) and M 0 (zero angular frequency) in the Fourier domain. By eliminating the relaxation and creep functions between (3A.4) and (3A.6), we obtain (see, for example, Nowick and Berry, 1972) the so-called Kramers-Kronig relations: 1

2w M aW ( ) = Mo+-

2 M 1(w) =n:

la(w)

J 1(w)

i

0

n:

ao

1t

00

o

M 1(«) d« -----

«

(Ma(«) - Mo)

f

1 +2 . = v.YJao 1t 0

'w = :_

i

i"' (

00

J•(«)

-

Mao

d«

«2 - ( l )

« d«

1 ) la(«)--

0

(l)

2 OC

«

wl -

.-

-

(l)

(3A.9)

- 2d« --

W

(3A.8)

-«

•

where the integrals are taken as principal values. Thus with Eq. (3A.8), one can show that, if M 1 = 0, then Ma(w) = M 0 is a constant.

---

·~

.....1 1.

<

~

•

<':1

;;,1~

1.

.").-,~f~H·

~-..,.·

~,J

.,;,

4 experimental techniques for measuring velocties and attenuations '-

INTRODUCTION

, __ "--•

.....

·

'-

The prnioul Chapter attempted to show that the concepts of velocity and attenuation were complex, even at the lewl of their dcf'mitiou. We have stressod that eac;h of these definitions stemmed from a different experimental technique, and that. mathematically, the deftnitions could be coqsidered to be equivalent for low or medium attenuations. This statement is true from the mathematical standpoint: in other words, for each experiment, each measuriag metftOCI· was idealized formodeliil;.ln fact. the many interference effects, or, more simply. experimental dift'tculties encountered, mean that the measured quantity is not exactly the one anticipated or that it contains a substantial error. For instance, the l'DC81UleiDCIIIt of aueauatioa :M quilc d6cult. ovea ita the laboratory. It is nec:enary to extract tho.._ we WMl ftoiD . a·Witolt.,..:of illlelfenuas-~ ~ multiple nleclioM. ........ etc.)wlaoletl'ccts MOIRORI or less foraeeable and imply the aeed lor ~ This llmlals tlae impoetaace. of a souad knowledp of expcriJnoaaalteeluaiquaforiiWUUriaavelocityandaueauaboaforaaoocJ undentancliDJ oldie reliability- ~ oftbe .... aad to he able to GOmpue various sots of uta found in tlac literature. Three maia caM&oria o(.meuuremata can. be dist~: ·. (a) Measurements us~ travolioa waVes. (b) Measurements usin& vibratiaa systems. (c) Measumnents usia& stress/strain curves which are distinguished from the second catqory, althoup it uses an cquivttent ~~tation system. For each~ we shall examine tbt·eom:ctions to be applied to the raw data;

~

~

146

4

TECHNIQUES FOR MEASURING VELOCITIES AND ATTENUATIOJI;S

4.1 MEASUREMENTS 'USING WAVE PROPAGATION 4.1.1 · Difficulties Measuring methods using travdiag. waves are especialLy interesting because, by defmition, the mechanisms involved are propagation processes.~imilar to those of seismic exploration. Naturally, for laboratory measurements, the frequency range is totally different (around I MHz) as compared with the frequencies used in the fteld (50 Hz in seismics and 10kHz in logging) (see Ftg. 4.1). The wavelengths are hence quite different, and the mechanisms responsible for the deformation of ultrasonic and seismic signals are not necessarily the same. Furthermore, the velocity dispersion processes associated with the presence of attenuations become significant. Exploration geophysicists are thoroughly familiar with the problems encoun~.in tryW, to ~late a seismic section to a sonic log (Goetz er al., 1979). The extrapolation of the results obtained in the laboratory to the fteld is therefore a difficult problem. e.mq.,•• I

10'1

•

100 SEISMIC

~it- 4.l

lchoiiiiUI.... ·, .....,..

EJqltoq1ioft

I 10

,'o; ACOuSTIC

. ·;I

tal ..

I

1o-

l.illiloreiDry

I 101

I

~,

10"

ULTRASOtiiC

.FUQUCDCf.~ of ~.~ur-.cn&L,

~

· NeverthelesstJaemetllodsthat\Vemddcscrib'eareWOtrllall)ta,..blc·tof'llllet__.. aswellaslaboratorylifnals.lnthe'fieli.·a.--Gfeft'ectstlfstwebotltdle.,.....tioft and shape of tksipal itself. ·For scillllic prGpaptkm, tile naaitt di«tcuuciee . from the ignorance of the distance traveled by dlle wave. 11lil waw.ce.uinlfdtlft·not follow the straipt ray path generally 8JIUM, 'but rather a eutVeCl path~ to''thc prOifeSSive variation in the acoustic pMpertiesoftlte formationsaw:ouatered. Moreover, even if straight ray paths are assumed, the depth of the different interfaeet is unknoWb. At best, if the multiple refte«iOM have been elinlllrated cort11atly. one may determine the travel time to the different interfaces. The int~~l vel~y ~t~~n two i~terfaces is then determined by the maximum likelihood enctgy metftod. Fin-~:Y, iq ~g,J seismics, the existence of a poorly-known weatb~ :z:one (WZ), w~icb, .is iDJwnioaen~us and . attenuating, further adds to the djfftCul~ in ~Cnninins the vel()Qty. In brief, it is assumed that we can distinguish variatiOnS of 1'Yo to 2o/o on seismic velocities, but that the absolute value of these velocities .is only known widaill 100.4. ·As lot acoustic loging, the formations encountered by the propaptins wave are better known, but mechanisms such as wave scattering at the fractures may give rise to errors in the velocity (e.s. cycle skipping). In the case of very high attenuations, the signal received may be very weak and the velocity measurement very disturbed. We shall be discussing a laboratory example below.

.ra

...

•

"'·

j

'

----T -..__..-

·-/

•'~'

~·

I - }t"·I I

I -

~

;

4

l'l!dfNtQul!s POl( ~·~AMD1A'I"n!NUATroNs

147

We have jUst descrit'Jed some- of· the dilfle'bltiet eac6Untered in the fteld in the · · measurement of velocity; Hence itisdear that the~t ofattftuationwiHbe ~ motedift1cvltlto adlieft.lnfact, tile attenuation ofbtlerest to us is the tnrrlPISic attenuation

of the medium, namely that relafe(Ho·flle interadion between the wave and the porous medium and its saturating flui~ u opposed to the extrinsic attenuation. which depends on the geometry of the beds and on the source (scattering. internal multiples. geometric divergence, etc.). In the fteld, in fact,•the simple propagation of the wave in the rock formation is modified by,pttering. and by internal multiples that constitute energy losses unrelated to the lois dUe 'to the iM1uticity c:Jf tfN,media tl"''fetsecl. It is~ difficult to correct these drects (Schoenberger and Levin. 1974~ ~is why reliable attenuation measurements in the freld are taken either in well-known aDdllomoseneohs regions, using well-to-weU propap~a. for cupaple. or by .~JeCOI'diaas obtained ia.t.he wella and by ~rial siinaJa that have. travclo4 . . . .._t.patbs. A second major source of dilf~eulty is tbe ~of~ ItisacocraUY NIUDlCd that the initial wave.ftoJ&til~whereK.U.:r.it~~tnJir~ Tbismattcris welllaaown to sipal pr~l .ptb~ w~ to ~ tlaeir nKX)rdiags for seomctric diver~, apply a U.. A»m!dioo·io r' (~here /l. f!l Oaad ~the t ~n«Pectcd for a spboric:al wave. The ~- tbc qJIJDCrical value of the factor , remains CJDpirical..AAotllef ~of problems io ~a. .ualio~ais4be biah·noise level in.

di-..or

scisaait~ TM~of-.iad~~oot~lyrclatcdtotbe desired~ For~plc.iathe~,..U~ica,~wavcs(pstudo-RayJeiab or

Stoneley) could constitute .. noise" if one were interested in the refracted S arrival. This rapid description of the processes involved in the deformation of the signal points out the extreme difr)tUky obtainina meaain,.ul resUlts in the raeld.

or

·--

·,_

With rupect to laboratoryJDeUurements, some of the problems mentioned above c:an be eliminated. To the ..-lium analyzed is well(!) known, the interfaces deftned, and the ray paths relat~ely clc8r, pftll'llly yieldina accurate Velocity measurements ("itbin J'!. ~ Yet a 4iftlculty arilea ila ,..._.,accurately the dme or the.Jirst arrival in an atteJtuatina ~ Weshall.,....i~~~~t ttu.l.~ in detaiHt~ the next Sec:tion. As for attenuatioa meuumpenta, multiple reflections-.cl iatcrference arrivals can be eliminated 1 MHz (for ccntimctric · (e.a. by usia& samples of *'*tuate size). For~ saqaples), however, the lfODletric ~vcrpnce effect. ~Y ~important, and, as a rule, operations are conducted at discanoes &om the ICtUfCC JUch that the emitted wave is vi11UAUy a plaac wave. the raaJe Qt. this wOrk\fta iet~l de.pends on the site ·of the transducer, tho waveleaath emitted, and the distance between the emission and obsavatioapeints. In this zone, sinoetlle wave front is virtuaUy planar it is assumed that the amplitude decay of the displaeement due to the anclasticity of the medium is exponential.

beam "'tb,

leas....._

'-._.../

If the wavelenath becomes comparable to the transducer diameter, diffraction may occur. The correction required can nonnaUy only be made empirically (Trudl et al., 1969). Another problem, typical of laboratory propaption measurements, concerns coupling problems betweeaube·ttaasctucer AJMlthe sample(ltt6 a the reprodUcibility oflossts due to'coupliaJ). Mete ....-. tbe frelllim 1b8UW tolwd'empirically (Truell n al., 1969). when -~ tllvdw.l ift ·ultrasollic measurements, the Finally, O'WiOJ· to samPles must haw :pr«!11efy parallt1 Skies. This l'loblem ft5 ilr¥estiaatcd experimentally by Truell atld Oates (1963), wllo' shM.red th-.'\he ~of imperfect paraiWism

th..,

,_

~

•

148

TECH:"lQl:ES FOR ME!,SUIUN(i

4

VELOClT~ AND-~TIENUATIOl"S

depended on the inverse of the quali.Ut ~r ofthe ~rial used. and on the frequency employed. A lars,c; part of~ diff¥:ul~ listed abQvct can be .solved experimentally. Ne,·ertheless, attenuation measuremc:a.ts by wav• pr9DQ&tiOO remain a ~ult matter, albeit easy to design and implemeat. The reliability of!~ methods and the acx:uracy of the measurements of the quality factor are at bc;lt .10%.

4.1.2 1\-leasuremeat prbaciples _. 4.1.2.1

~xpe~

tecltaiques

,Ai

Velocity m_easantaeats

As shown in the prtvious Chapter, ~ion 3,.3.3.2, tfwrfne8SUI'enlent of velocity in an attenuating medium by means of tile Y ·a X/t (where j and· r are the travel distance and propagation tinie respectively) jives usi hmriOgelleous qdantity for velocity, but one that has no particular phyliead'signifrcance for t~signal analyt.ed. Contrary to the measurement of the distance x, Wbidl is nQnllally acetrrate anct easy to make, the measurement of time r is ditrteult becau_, a l*tje number'offac:tors add to the inaccuracy, such as length. shape and repett'tion rate of the etecttk:at pulse, 1he break of the emitted wave (directly related to the rise time 11etifted in Fig. 4: 12), the cbarteteristics of the · rertiver system. and \-he·rransducet aditl'fttstenill'l ~ A nUmber oft~ problems can be etiminated by using a reference *ftlple witll welMmoWil acoustic pioperties.

eetuatidtl

,•

.1~

~.·

-1

t"·'

l

t

t 0 .........

. ."'

.

,. '" .

?'lf'

-1

T-. ..).

0

0.5· -!

1

.......,.._.(MHZ)

Fig. 4.2 Signal in time (left) and in the frequency domain (right) used for the model. ., i~. i

J-

;

'

Nevertheless. the p:oblem.. ~~ ·with wa~ riso .time JDoasurcmenu r"mains uQSQlved. Let us consider two ~pies.
~

;;,--

'-'

"-----' "-../

..

·~Mix.M_.,ItiYI~IitlMfli'WlMSAf.f*"'*tllliS

Note also that, for the Constant

'-

Q 1\,lQde~. d1e phase velocity is pven ))y:

v-

'-

Vo(;;y

l

l

i' =- tan- 1

'----

'-'--

._.,-

'"-../

'-

,_

-

~

"--'"

'-

(4.2)

-

Q

7t

'-

(4.1)

with

'-~

'-

149

The greatest di«crence in phase velocity for the two samples is therefore obtained at the maximum sipikaftt bquency. Frequencies lbove t»0 /2n propapte faster in tile more attenuatina . mediUm (Q • 10) than in the less «tenuatina medium (Q • 100). The ()ppOJite occurs rot frequencies below m0 /2x. The lipal, which bu traveled throUJh dMUDOCiiUDl orquality factor Q • 10 hence anivesbefore the ODe that bu traveled tbrouah the medium or quality factor Q ... 100. and will be wider. This is lhowu in Fig. 4.3. To~~~~UU~Wdae~illpropapU. tiiDcbotwecn tbc twosipall.and hence the dilenmce in propaptioo velocity, it is necessary to set a level bdCMV which it is impossible to distinpish the signal from the noise. H this level is set at 1°/o, the relative differeDce ill velocitic$ is 4.2%. whereas if the level is set at 4.%. tbis difference is .-educed to 1.7%. l'bis~ is obsetft'Me on actual slPIJs,lll
amvaiS

(FiJ. 4.4).

or

.

.

even

.

.

·~--~~--~----~~----~----~_.----~~

Ju

·~

~

oI

I

0

1.1 h2>r

I

'-

''-

-G.I

~'-

'~

-1.' 0

'

I

I

5

I

I

•g

1

"'

I

,

tO .

•

..

I

I

I

I

I

15

I

I

I

20 TiiM (Ia)

'-

life. 43

Sipali recGrdedafter p~ itta,COMtat Q thateial. their maximal amplitude being nonnalu.d ten.· ·

'---

L

j

' 150

TiCMNIQUES FOR MtiASURJNG Va,OCITIES AND .4 TIENUATIONS

4

!

l

0.5

--

----

Initial signll

"

0•100

(l_·~"-

~

~,

'

0

0.6

0

Ffc. itA

F~CMtfal

Spectrum of sipals recorded ·after propaptioll in a constallt Q

material.

Hence ia the~ of Q == 100. ~Jlc farst ~v~ cop-esJX>nds to an averaae frequency ro 1 higher than the a~ frequency Wa l,)f tJie second aiTival in the case of Q == 10 : The contri.,utipn of the frequ.cncy co, ha$ ~smaller than the,averase nqtse. This makes it very difticult to C
"

quality factor has varied (for example, by changins the saturatiot\). For average attenuations (Q ~ 10 to 20~ an .. error" of the foregoing type of about l% can be t!xpected due to the technique employed. Nevertheless, the determination of propaption time remains the. CISSential problem. Figure 4.5 shows an exnmee~ple. TJte actual signal emitted has propagated through

II

" o.s 0

., - -----------

I I I

.....

-O.S ..

-1

s

~

~

~

a

I•

•I

51'S

~

•

_,

~

G

~

Time ~Psi

Actual 'lilaal tta.t lw ;W.voled throUJh 40 DlJP of FontaineiNeau sandstone with a measured Q ;; 2.

Fil- 4.5

"

.-,

"

'_,

,--

~--~--~------------------------------------------------------------------------------------------------------_J

'-

..

'~ "--'

~~~...-.·-~~~

1$1

a sample of Fontainebleau sandstone with a quality factor Q ~ 2. The uncertainty on the starting location of the ftrst arrival is about 1 p.s. while tbe travel time is aboqt 22 ps, makin& a relative error of So/o. . Hcacc it can be considered t"-j&Jae ~aventiolaal ~~ bcrc to measure velocities yields velocity values with an uncertainty of about l to 2% according to the attenUation·~ the material in~tipted: : · Yet 1llCUIJ'femcftts ttrat are~ more pllysic:ally·nteanla&fUI (phase \'elocity) and more accUrate can be perfOrmed by develepins a -spedftcsy*'in \'dt)'cities (see, for example,

for

Trueftdal.,I969);Wewillbritflyd~prepolidbyRoaezaadBader(t984)

i

for the measutement of phase telodtieS in l . .(sie'flla. '4.6). A moaochromatic wave is emitted in two different liquids, a non-attenuatiq reference liquid wit1r lmoWII wlocity and the test sample. As a ftrst approximation, the velocity cliftierencc A Y is related to the difference in propaption time At. This difference in propaption time is meesurcd by a calibrated Pbuc analyzcrusins the tM- ....... Scattering and tempeqlWR6etJ ~rtm.lly aqlipblc sinee-.... ~.-..- llellfc>r the sipal piopaptins in the refcftace liquid and ~n the_tcst ,..pic. Fiaally, this technique requires the emission of- lipals having prcc:Uc frequCnc:y Control to eliminate attenuation etrects. The velocities are measured with accuracy of about l.S crn/s.

cere..........

-~

an

~-

Acot.dc

'-

lftllliftlltlniiA

..-.

ChlnMIA

'-'

,._

)

cl'nlctor

........

'--

, .. ,,

'-._/

"---

'-

'-

4.l.a.2 AWrlfir.,... ...- •' ••••• Two main techniques are used to analyze wave propaption in an attenuatina medium: pu)IMcho methods and tra~ssion m~tlloda. n.Jmere~ principles are the same, but tbat application reveals •differences in det,Ut . , -•

hJu-«luJ metlunJs In pulse-echo methods, the sipal emittec;l by a''pie:~XN~Iectrie-ayltal bonded to the 1 sample underaocs muffipte refleCtiotls at. t~ ~ ~·plane the attenuation is determined from the ~ 1Un&ifkude o( two · sutcessive multiple retlec:tions : .. . L

ittt41'face.

'-

•.-.l..tn(~) .· 2L r

'-'

• \_

~~

;,

'

wa•

(4.3)

__________

' lS2

4

TECKNfQUESFOR twt!AtiURDrGV!Lotmi!S' AN'D·ATI'ENUATIOMS

where L = sample length, A 1 = spectral amplitude of a multiple reflection, A 2 = spectral amplitUde of the next trmltiple refleCtiOn.

· ·,. ' ,·

Note that, as we showed in Section 3.3.3.2, thisequan~~ valid only for tl:te QaSC of an atteo.uation «that isJtQ~ t091alp"'and ~a m~odlr.o~tiC siB~.~ PlCthod 8$SWDCS negligible losses. at the, d~ intc!f{ac;cs. e~i~y the. tr~/bOod/sample interfac:es. ~ver, iUS4UI.QOS~ ~t l<.ls~ a conditio~t r_.y rcalizcd for e~riJneats without any ~~ ~ FF1o 4:7 S:ehematically s~ws the experimcatal setup. ~· -,

•

SLm:::I=~~I

.>•

f. l

-~ .'

~

l J

·'·

'

Fla. 4.7 Diagram of a pulse-echo ex~ with q'*tz tr~uc:cr. ' .:~.- :\ "~ '.'c:. :::-.. :.. . / ' -·\,, ..,->~:.- ; ·-. . ~- " This method raises an McJi!fOpa( ~- .,t ia......... id ri.ake Q1casurcments on hiply attenuating samples, be<:ause all tile energy, •.•bsc)rbed in t"' paths inside the ~pi~, 1'tle solution is to reduce the loo.llb of the sample. It is also . . . .., ~e lure

·1i. .

'tted .. by theFinally transducer. propaga .. tioo dit~'nam . ely thatthickness thew.aveol the th' sample. one ID\IIIt ·e ·~that:tbe rcftec:ti.S on the edges of the sample res.W..,~.&MCMCU.Vff\;'1 it · . ·· ,·:·" ftoalbe.toua:e arri"-,arv.* direct signal, 01' have negligible energy'm Com it. Tliis problem is in fact fairly difticult to solve, and involves the theory of bounaecl ~- The interested reader can refer _to HostCD and ~mP,fJ (l~f4~ . ; , •. ~v. ,.d . • . . ;r ,1.. , . W10kler and Plona (1982) used a Jllodirtcation of'dlis technique; Tbaii' ·~pd consisted in comparing the reflections obtained at the two interfaces formed by a fnt bUffer and the top of the sample, and a second buffer and the bottom ofthe sample (Fig. 4.8~ The overall system could then be placed in a hip-pressure ce1f. for dffs setup, attenuation is Jiwn by.: enu.··

I

I

.

.

I.

. F ·t·t~.Yt J4(f») ' 2' . oc = 2L ln IR 12 (w)l :A'(ro) tt - R, 1{~)lt ·

(4.4)

where L • == Jength of ~e,~ .. , ... A(w).. A'(co) oo;,~~t~ of tpe tw~ ~1ons tQ be a>m~d. ~1 2 (w) = retlcct4oa ~t 1'1 ~rftce 1/2. R23 (w) == reflection codttcicnt at interface 2/3.

It should be noted in Eq. (4.4) tha~ tbC ~tion coefficients depend on freq~y. This is due to the anelastic character of' tlie materials concerned (see Section 6.2). Various --..

-,

•

'\

.......

¥[. :

.:

ns

~1~·· '9L61 "sRWpwdwd .S) ~ ~ liDW ~,...'*!P 3!-11~ (:) pue q ._6.t "I!.:J) as1l:) J(W UJ •laAta»J pn JOU!1D511~ ,qtuilii~ . ....., lUO»JP.P OIAl lOJ J)Op.JOOQJ SfHI!I oql lupedQIO:) ~rp UOf,lRQll1t ·amttOQl. Ol a{qiJIOd l! Ia~ INJ. .lpOJCl

af.J1f , , 1IUOJ a1lp ~~r. .... ~~ ;.trr~J ut.~*-"~~·~··,.,., IMU~lct~t! ..-ftlt~Rt~WtJPftMJ~'IJCM1·11f·~

lft,Pl'«) pn ·~ • • ~~saoo~~ ui1lllb fPfi ~N 'M6t 'D!WfU?ft) poJ 1'3!JPU!fA3 8 .8f· ~ OJ BJ~ ~.a. p»JtDUal DA81A IK(l pua •. _ . " ' Q! 'lw.A ~ llf.dar~ ~ 'M·t "1!.:1 uJ ~q!lftlla p:llappao3 :Kl 08:) atefuan :M{l JO sap!5 ~ql UIOlJ SUO!P'IJN pUlr ·~ QJ U1lql lsaf IJ' l(d1llal 01(1)0·~ Mil 86•t "~!.:1 UJ ·(n.\p:».t pn JOUJUII'Wl) UO!'~ Ja)t\p5U11Jl pue

Pm"'.

sr.-...

ms:.rcfiiawforttlg.~~·-·~---A·*••*-*.tlJO-'tw• aqi·ao ,.nq·~ub,sdlaoii.Mt1«'.fq ~~IJ·,..._,,.....WM·M.f·~ fJUO*·~ (Mip·,q.,,...,., ~),.,.., ...,.,....,.l 'q .

.·

•

d

·uOflvnuaue ~ :M{l JO tuauwnsam JO .(:)e.Jn:X)y ·~a~dum__.,..., .........._,UWJ•~'fqfnod aa suo~~ ·saovpaJUJ JU~:,)JP.patp Jlt su~oo lu!Jdno:) puw iUJl:,)ll1tOS ft1pi\I~OO~ lsnm suo~

lOJ o/e01 tnoqe pue %1 Jnoqe · SJ D!JJ."''(aA

e

~)l:lel'

--+k-...---tt I I

I

~.1--~.....-

I I

I

: l ..._ ,

-.-.~-~~~~~~~ .... AIICidJ--ov-·

~~·~----~====r

TECH~"IQt;'t!S l'<»t 'MEASURTNG VELOCITtES :"ND .4 TT£NUATIONS

154

4

Tarif, 1986). In fact. for all transmission methods, one of the most reliable techniques for obtaining the atten~tion from the recorded signals consists in comparing the spectral ~lUnplitudes at dift'~ren.t frequencies. Because attenuation implie5 a preferential loss of the h(gft" f~quencies. a change in the total spectrum will therefo~ occur. The spectral amplitude of a wave can be written (Ward and Toksoi, 19711:

(4.5)

.4(/. Xo) Ill= Gx0 A,(/) exp [- 2(/lxo]

.,

where G

= coefficient including effects of ~ometri(: divergence,

x0

= distance traveled,

transmission and reflection, rx(f)

A,(f)

= attenuation coeft'tciettt. = receiYer response.

The ratio of spec:;tral amptit~
In It is also

A2

(f

'x2

) = rx(f)[x 2

x 1]

-

G1

(4.6)

+ ln -G ;

,.

tnowa tbac.,at the intorclel' in a:-ol: .

1tf' rx(f) = Q(f) V

-...

(4.7)

of

For a given source, the width of tM! 'frequency speCtn0n the transmitted signal is rela~ivelytitnited~ so1b·t~ft em lSt ~!Mto be iftd:epend'eM el'fteqBeftCY. WC shall pro\ide an expet'bnetrtal jdifie:atift· Ofttfisiamtmpt'ioe itt Chaplet S; ·'Bquation (4.6) is theref'ote' writtea·:' · ', "· ' ·· ·

.... -+•U x,> • (x A2(f, :cJ), ,Qf ;1C ·

.,,

The term

In

GV~

is

1 .... :

~~v + 118 ~

.

G:

-'

(4.8)

in~ ~ f~· for a s-plierical ~- ActuaRy,

experitnentat.UU~ts co~ tilt validity of'this bjpottiesis m'liiaJtl111tteftuating samples. The quality faCtor Q'~is therefon~dbtaiMd~· me8suriDg tht Slope of the curve.:

.

. g(ft-~ tf;(/, ~!)

·:r.

. ,·

'

.4:2(/....~:2)

The same method. stiahtly modified, was applied successfully by Tok50z et al. (1979), (Fig. 4.10). They comp8ted the spectra obtained for a reference sample and the studied sample, both samples being of the same length and geometry. These spectral comparison ~niqueJ are routinel~· employed.in, the laboratory and in the fteld. They require a signal as uncontaminated as possible by otkr arrivals. Here also, if the studied material is highly attenuating, the signal recorded after propagation widens, the frequency content decreases, and the attenuation measurement becomes less accurate. In Fig. 4.11, the signal width is 3 J.lS for a Q of 200 and 9 J.lS for a Q of 3.5 after propagation through a Co~t QIJ}e
.,

/''\

..-..,

---.

:;;,----~ '-.../,

~'t'dlt-ill.iSiJ1IIMI~~--

4

! c

''-

I

~

l

~

.I i

'-·

.5

\

I

I

\

-

' .... ..... __

i

I

'-

)

'---'

.,__ ~

lJ

1St'

s-w-

1.0t II

" ::®

!

o.n: v\

.5

c

I I 'I

"--

I I

I I

~

ot

~

0

I I

...

s-w-

I

1.0

0.5

1.5

,_.Mf!CY CMHzl

Fr~CMHtl

'---

'-·-

1.0

Fit. •Ut Example of attenuation measurcmenta by spectral ratio in Navajo sandstone (after Tok.Oz ettd., 1979).

"---"----' '--

'--

-

'-'-~

•___, ''--'

betomes annoyina. For sliJhdy attcnuatina materials, the spectral ratio method is inaccurate, since the llope of the rclfC$Sion JUte if _toot~. the main diftic:ulties or these measurements lD the laboratory aacltb the faelchrites &OIIil eouplina problems. In the field. another difficulty may arise· ftOaa CIODtamiaatioD by iaterferina tipals in borehola:,H...-ver,the major ~ty,~_ .,_$PDCIJalJ&tio tectmique..arisea with pometMdiYfflllalll .-,~--J\on-planar wave fronts. Corrections arc therefore Dledecl for ablolutc 0 measun:meots. These corrections are lbeoretically siJDple, bat involw: tedious c:aJculationa. Papadakis (1976) computed these types of correction for bomopoeousliquid media and his mults do not seem to tit the data for solid materials (Tarif. 1986). Hence. the abtolute accurac:y of Q measurements Jiven in Table 4.1 must'- iaterpreted cautiously, especiaUy for hip Q values. It is also important~ mention d)e tilJC • -~Uf: bued on an empirical equation of Gladwin and Stacey (19741: ·

o.e.or

pr..._ ..........................

'-

'--

- f -

'--• ·~

where Q can vary in· SJ*le but

tnne

to +

r:

c {r l

d~Jl41K . . . . on

• wave rl5c '(5ee FiJ. 4.12), • travel time, t 0 , C • two constafttl de.,..._t ea tbo ...._

t

T '---

'--

--.· '--

ck -

--

frequency and

.(4.9)

156

TECN~E,S Fell MEASUIUNG

4

VELOCITIES AND ATTENUAl;IONS

0•2IDO

•--~ fi i · I I I\ I

I

<

I

-~.5b

_,I

...

0

I 5

I

I 10

I

'

I

I

w

I

I 15

I

I

I

I 20

I

Tlmo Clal

O•U

~

I Or----__} -OJ ·.·

.:.u

I

I

0

I

·~i'

!,,1

e',~i:.,"f'•·•·c~-

l

I"

•-'-'

Tlm!l""'

:.

·~. · • ·· in of dift'emlt attenuation and' for. tlle>illiM'flli&ilt siltill ~ ·Q IIIIOdllt;

.fla. .f.tl Synthetic Jipak •rec&ded afWi

m ·a · teria · ls.

"" ··?

l'uu4.t 0RDEil OF MAG~1Tt:'DE OF ACCURACY OS THE MEASUllEMENT OF Q BY SPECTilAL RATIOS

-

....

.fQ.'Q (%) Remark

s

3

Q

>

}()00.0

SOOAI

Signal prac- Weak sipal tically unobservable because hillhl - y attenuated

so 10 to 20%

Preferential

operf.tint range

100

;;; SO%

1

>SO%

The slope of the spectral ratio on 'flllicb Q dcpl'llds il Vf6Y low, and itsI determination is : inaccurate

I

very inaccurate ~

.--.

"" '"

'-------,---------------------------------~-·-~--~---

.,

~

"---~

;4

:II!CDIN,...,,_..

.....,'""!!I~llttp_.~~~

IS?

-'-----'

"--'

I Tm. ~

Fie- 4.11 Possible ddtnition of the rise time of-a signal. '---"

·~-

Kjartansson (1979), showed that a similar theoretical equation could be derived for the Constant (2 model (i.e: a quality 'factor independont off~) and for a Dirac pulse, namely: T' Q =C(4.10) T

'---....-"---

where T' = pseudo-period of the wave, C ;:; constant (for a Jivcn source). As a aeneral nlle for BllJ, source, Eq. (4.9) remains empirical. In general, the timeintegrated form of the cquatieo is used : . T ,T ""' To + C Q (4.11)

·---'----'

-----

Blair and Spathis (1982). showed thaf...guantity To is a flUlCtiOft of the source, but cannot be relate4 simply to its rise time,t nil: ~o-constant C is a sliptly variable function of Q (f~ Q > 20) and alsO depends ·on,_. source (Blair and Spatbis, 1982). Stewart, 1984, theoretically calculated Eq. (4.11) for a llarae number of artificial sources (window, Dirac, and intermediate sources). and showed that. the constant C could vary from 2 for a Heaviside funcmon to 0.~ for a Dirac delta rui.ction. Hence, assuming a relationship ofthc.type~.11) between rise time and quality factor, an attenuation measurement c:an be obtained from the rise time, if the values of the constantl --r 0 and Care lalowllbya-tlpetimatoaa rd'erelltl sample. Uafortunatcly, the relationship betWeen- me time and- quality -Mfer it· not as simple as Eq. (4.11) implies. Figures 4.13 and 4.14 show a synthetic example (Tarif and Bourbie, 1986). The relationship between r and Q is clearly a one-to-one correspondc~ and approximately ·ill die slope C aM ill the ordinate intetolpl (t 0 ) with linear, but a chaaae dccrcasit'lg Q : T Q < IS t ~ o.G7l + 8.295 Q

oecun

Q > ~s r

"-..-

=o.tS6 + oJs7 QT

158

4

TECHNIQL"ES FOR ME.,Sl"Rf:SG VELOCITIES' AND -' lTENL" ATIONS

f =i 0.5 .1

j

l

~

c)l

I

'rd

0

I

l

5

1

----.,

10

F~IMHzl

TimtU&sl

Fi&- .C.U A theomical reference .,aiJOill used -in FiJ. 4.14. The central frequency is 1.35 MHz. Time domain signal (left I. Frequency domain (right).

2

""'

11.8 ~

1J1.2

T

= 0.114 +0.311 T

0

a:

,

••

T

, ,.-o~+o:~1e 0

"'

.. 0

:r

3 " · ,.,.., tlmlfQ .,., ' '

~-

;s

8

Fit- 41.C Rise time t~t as a f..... of T/IJ. The diffenma rise times computed using the Constant Q~ aD4 the signal of fig. 4.13.

~

In practice-; this makes the applieatioa of th4: method di~uk. It becomes necessary to know the entire cur1·e t = t(Q) to be really able to obtain an absolute value oti Q in all cases. Naturally it is always possible to compare the quality factor Q for two measurements : the greater the rise time, the greater the attenuation. Tarifand Bourbie(l986) point out a possible solution to this problem. They show that the curvet= t(Q) can be simulated point by point using a Constant Q model. and that ~

"'

~~-------------------------------------------

., -"' '

-:;rr--

_D ci_\-..~......:v-....,

...___~

I '--../

'--../

'-.....-

'--.__/

,_/ ''-

'-'

'~

,__ ~'

4

TECH~JQ~FOR ~~~l.ko V£i&!tiES ~ti ATTENtJAilONs c:=:::;;

159

this simulati9n is el~y comparable to ':the true curve, the essential point bcins tha't the initial souni in the simUlation m~t bC as close as possible to the soul'
to

or

desree

_/

--

1.8 1.4

'-.._/

~~

1

11.2

.! a:

1

'--'

.8 ~

.8

'-

...

2

0

'-

'-

""'

..

5

8

·,

~

Comparison between spectral ratio and ri$e ,tbk methods to c;alc:ulate (2. Measurements of Q by the spcc:tra) ratio technique are taken from Bout. . and Zinszner, 198~. ,

'~

3

Fie- 415

'~

'-

1

Trwet*-/Q . .

'---'

I

It is oblervable that both ;mtt~ are more or less .valent. tn fact, the abscissa (value of Q) of the pob•tJ in 4.t'$1S o~taineC;l ~y lilc;aiuriDJ tbupec:tral ratio, whereas the ordinate (t) is aleasured on the reCorded sipaJ. A one-t~ ~pondcnce is obtained between t aDd T /Q which can be approximated by the straiaht line:

r...

t-

0.451

T

+ 0.208 Q

""-

Note that for high values of Q (Q > SO) both methods are equally inaccurate. This is because, at a Jive:n sampliaa rate, the rise time is too coanely sampled to measure slight attenuation. Similarly, the slope of the spectral ratio is very low, and it is therefore diflicult to make an accurate measurement, as we have already noted. For values of Q < SO. the accuraey of both measurements appears to be more or less identical. However for large attenuations, the rise time method, which utilizes only the ftrst '-'

'-' ~'

'--'

'---

,_

160

4

TECHNIQL'ES FOil MEASURING VE!pCITIES AND ATTENUATIONS

quarter period, is much less subject to the pro~ of jnterfering signals that may influence the spectral ratio method.. I~ must be. kept iq ..Und that a quarter period may not be representative of, the frequency content of si,aluUs several periods in leilgth. Hence, the rise time analysis before and after propaption ~)' not accurately measure the signal as a whole. One physically unrealistic example is :sh'en in Fia. 4.16, in which the rise time decreases after propaption throuah an ~tte~uating m~um. As a rule; however, the situation is much more favorable. Let us recall that, as for the spectral ratio technique, aeometric diffraction (irregular wave fronts ~ue tp the fQ1ite size of transducers and samples) remains a concemfor rise time measurements (Tarit 1986). Finally, in the fi~ld, it appears that the application of the rise tune ptcthOd. yields better results than spectral ratios (Arditty et al., 198.2). ·

---,

------

'

I

·~

,, .cJH l

.----.

~

j·:·

A 1\ vo

I

>:,

"

~

~

''-',-.i_{+...

,,,.

,_

"

Fia- (.16 Schematic example of a sipa1 before and after propagation in an

at-.atiaa
"

.

~

. ---......_,

-._

·~

·""'\

'

-...;p-

..

-~~

'J

'-.._/

4

ft!alf·--~~

l61

4.2 MEASUilEMENTS VSJNG;\1BitATING SYSI'BMS. (STANDING'W·AVES) '-

'-"

1'hese meth()ds have- OR1y becD used ia the laboratory.

4.2.1 ·~

·~

-··

'--'

'

:

Diffacaldes

As for propaption methods, we shaH fint dalya the dift'tculties of application of these measurements from the aeneral standpoint. The main handicap of these methods results from the fact that, by their very clefmition. they involve standing waves. whereas the only waves anaJtaed in the ftetd ..-........ •aws. It may prove dift'teuk te apply the results obtained for standing waves to traveling waves. espcfcially for attenuation measurements. Nevmhllea,tlte J*Yious.a.pter lhowedaat it •asf!OIIible torelaw two. t)!pCS of meatOfelllent eM a if &be attenua&ioll was aot too great. In practice, many ..iftt~M'fem~Ge" ll&tltl illaybee.,citidby ,_...... •vet. For-example. in a poNMtatwated • ..._....._. Wbta1i1ataayaiw rile tor..-a,.W~DCDtsofthe pen~ fluid ilt reletion to dae skeletal lluae or liot typo. aovemeats (W!Ute, 1983) w:~ influence the intrinsiC' at11eoalioa.••••Htt nt·(IIIM Section-4.2.3.2). However, it is customary to operate at frequencia of about 1 to 10 kHz, which happen to be near the frequency of acoustic louinJ. It is .V.~ to operate at much lower frequencies near SO liz which will be of interest in . . . . stta4ies.

t._

bid•••ioul• ...,_,__.rod

·~

·--· "----'

'--

"-·~'-....-

.i"'

4.2.2,. ~ ....~•.. Two methods of measurement are discuAed, one \ISing forced vibl'atiou and the other Ulia1 free vibrations. For free oscillationl, the sample il vibrated at one of its NSOnant frequencies for a short period of time. ~s the ~t of the decay in displacement amplitHe for,~succeuin )I!ICUdo-perioda, as ~ showed in Chapter 3 (Fig. 3.18), aives the value of attenuation u a function of the logarithmic decrement or the quality futor Q: ,}c .. ,

1C .

A2

'IC

·~:·In-•;;

",

where A 1 and A 2 are the signal amplitudes for two consecutive oscillations. Assuming that the displacement amplitude decay isexp (-Ill), which is reasonable for attenuations that are not larp (see Section 3.$.3), one call also write:

Q-::,·: -: '-....-

...__,.

~

'-

(4.12)

J

(4.13)

162

T~tQUES

4

FOR MEASURING YELOCm£5 AHD ATTE!!Ilh\TIONS

As we have stressed above, the use of free oscillations to measure.attenuation is only feasible if the sample. Is oxcdled at o~ of its~. freq~ In general, if the excitation frequency is dift'erecat from. ~. . ., :the ~ance frequencies, the signal amplitude is too small to obtain a reliable measurement. Vibration at the resonant frequency allows for a measurement of velocity [see Eq. (4.14)]. The second experimental technique involves fomcd O$CilatiQOs. TIM tat sample is subjected to a continuous sinusoidal force of predetermined frequency. The sample is in the form of a rod with a diameter much smaller (if possible) than its length. The frequencies at which the bar resonates depends on the elastic properties of the test material and on the length L of the rod. The relationship between these parameters is pven by the equation: '

,;.,

'·'

2L/

Y=Af-....;.;...... . n

.-

(

-

(4.14)

. (.

-~)

- IP\

ror exteasionat modes v. - v. - ..;; · and me.r ~ _v- ~ -, 'VI'f •

-

In this equation,

•

,·

J.,.

'

•

•

I

<

f is the ~-- f'reqacnc.y, A is die -viwatioft, waveliRt'h. E the

Y~smod'ulus,ptheshearlbOd..,,.aa,lhttbe~Theterm~·exteaeional

mode" means that the rod is subjected to a longitudinal extension/compression force. The term .,.shear JBOde" mea~~~ that the rocl·it subjec&ed te a ·shear Ioree. For the ftfturahnode-(the 1'0d issetJ.jeded m •lMDdi.q.foroe). the ~-vo no lORBer equidilcMt.l'hc-tnPII8t. . :wlbcity is tllM of..lbe extouional w~ aJ14l,.&he frequeneyfvelecity relationship is .Wen t., SdniiDer « al. (1~.7~: : .

. ,. , ·_; -~<-. -~ Y£

27t1Jf

:;;. ,'·, -{J =-. -"fPFr

1111 .......;2'•

J;,

'<,

~-

::.

-·

(4.15)

where

R, =-= radius of gyration,

f

= resonance frequency,

.

m - a constant depending on the order of the rcsonabc::C ~and, r•ttt~e.:rn:e boundaty condiUt>Qs. C
,. ..,.,.

To obtain the attenuation value with forced oscillations, whatever the mode anafyzed, the frequency spectrum is scanned in or4tr to d~be the resonance peak entirely (see Fig. 3.21), and the value of Q is thea given by one of the equations of Section 3.5, for example; . (t)

Q=~

(4.16)

where co is the resonance angular frequency and Aw the width of the resonance peak of the displacement at 1/.j2 times the maximum height of this peak.

..--..

..._.-....._

/

....

'

~~·T .. '------' "-...--"

~frbjl~~,--~~

lli3

:

4.2.3 Experimeatal setups The experimental setups for harmonic: cxqtatbl of a rock sample can be divided into two main categories. pendulums and resonant krs.

'"-../

'-

4.2.3.1

Pendulums

The farst category has been used for many years tor rocks (e.g. Pesclnick and Outerbridge. 1961 ). The experiment consists or subjleting a vertically suspended rock bar to a harmonic excitation, with a hip ..,... ofillatia mass, possibly attached to the lower end to increase the resonance period of the system (the sample{mertia-mass combination) (tee FiJ. 4.17). ~To avoid plal'inB tile 8l1llple uacler tellsion, some experimenters balance the sample(mcrtia-mass ~bination by a co1111ler'WeiJht acting through a pulley by \\'bich alterations of the vibration of the system are avoided. Two vibration modes are possible, the torsional mode and tbe bcndina mode. In the torsional mode, tbe fundamental resonance frequency oftbe ovenllsystcm is in~ly proportional to the square root of the system's moment of inertia:

1-ft "-../

"----'

where I is tbe moment of polar inertia, and~ is a torsioa constant. The moment of inertia and torsion constant are related to tbe cbaracteristicdimcuioas oftbe sample (length and diameter). The veloclty is measured by means of the fi&i.4itY modulus which, for a cylindrical rod clamped atone end and free at the otbcr one ~b Land diameter d, is:

'~

3.

'-

The

r

Sl21L 2

,~

"-'

(4.17)

p=

'~

.

(4.18)

measured~· v··= .jiiiP is the phase ~It the ...... frequency f.

A$ we previously pointed out, the resoftiMI'. .. . , cap be deaeucd bJ incccuina - inertia of the system. This system penDfts'it'Nit• ....... at ~ low frequencies

( < 10- 2 Hz), and is rlbt normally used for frequllltildl. . lhaft tO Hz. Theexprasion

for the resonancefrequencyoftbebendinl . . . . . . . . . . . . .~~oftbe

"'--'

system is aaueb more corpplicatcd and theafeN will 1lOt t.e tiveft. The mtensted reader can refer to the work'ofSehreibcr et al. (1973~ In cOildusion, the use of pendulums is relatively cas)', and ODe oftJae oftlJ dift'aculties

consists ofminimilina all tile encrJY losses other tbaa._.OCI.WIIiiaam the rock. It is also possible to use small salftples and to make meuurements at ~ low frequencies. However, the current trend leans more to the use of the resonant bar technique, which we shall examine in detail below.

'---' '-._,

,

__ -...'---'

'-~

"--"

4.2.3.2

Resonant bar

As we have just emphasized, thiS it aarretldy tbi Molt' widely used experimental setup today. The experiment consists in subjectiq a rock bar,ltept horizontal either by supports or by suspensions (Fig. 4.18), to a harmonic excitation.

164

TECH:-.
fO~. r,I~Sl:RING

4

VELOCITlES AND.-\TT£Nl:A TIOSS

'

I

" R«kiii!IPfe

L•gemoment of inertia

,

•. ?J

Simplified

L

dilaru1

;

(by

sz:ti pc~n

tTtm$111ission and 4ZUIU,UJI4""by

La..nns from

Sf:lsmit

waves: r(lt/imiM,

WJUtc. C 1965.McGraw HilL SY).

----..

0 I I

----..

f .

~/~

~

I I I

fi

1

2~. 4, •• 7

•

R. :•Rockllllllllt A W1 W2 : We9!tund Inertia erm c ::Air- coli '· J :: l'ivoc tnd IIIPIIbire

ROCI
~-c.n. ...............

. SWe~encll*l.....-.

S

M

Soriile clamp

G

.(b)

: Light source

:!Minor

:Scale

(c)

b. and c. Actual diagram (after Peselnick and Outerbridge.

Fia- 4.17

© 1961 AGU).

T~on ~ndulwns.

-~

' '-------,-----------------------------------------·--~------

...

---

•(.' ®

·.· .·

.'

.

.··

.............. ·piii.(IJu..........., It! JMIW -.met UOPwtPP n R OSIIIIW:) ~uq ~dW!S. 'sUO!...WUlJ IAOf.puw 1110!11lf1PIO INJ JO ;no ttp 10.:1 i6J't '1!.:1 all) IUOpWffFISO poo.IOJ 'PfA\ J0 (~"J ...S ~· J Act J*!W!I I! fwul!s a~pu -n) SUO!PJIIPSO aq 11)J4l ~ lflt{f110 -u -,a o-.ocJc1o a.p Ol ~Uoq )(1!1' p:llS lfiWS V .
'*" aq

*'· _,

AI~

5! UQPWl~Xa 3p't~ 'q?Ol J0.:1 "UOJtlnl-.lflltls.(JM qp lfW!JaPRU

,JO sprpw JOJ ~'lSD ~(U.Q AIJBJaUaJ ~· SlaOJn,ps ~ JAOf Na~&~Jnq ~ 3fl8l'OJJ»P JO (S~6J 'lUVH,O ~~Ul '.\quqn0J. ~~l.a!'i .(q palp~ aq 8'1:) IUOfl11.J'l!A ~'1.1

Q w:;;..===~-;~~-=----_-7,1 Ii

~

1~

---

166

TECHNIQUES FOR MEAS\HUNG. VEL<)CffiES AND ATTESUATIONS

4

As we have shown, the fundamental resonance frequency is given by:·

v

J=2L

(4.19)

where V = wave phase \'elocity.

L

= bar length.

.---.,

For a given sample, it is possible to alter the resonance frequen.:y by changing the length ofthe bar. As we have s,bown. three~ ~n be excited: the lolljitudinal mode (the rod is subjected to a t.ensioa}c:olnprcssion ~~.~ torsio~;tatmoc,te ud the bendina mode. The loagitudinaland headina modes cmable the tnC!MUrement of Y ouaa's modultll E thrOUJh velocity"£=

fp.an,dthe~rrespoudin,a'l\la"lityfactor.~(seeSection 3.6~ The torsion

mode enables the measuremat of the shear modulus (p) lhroup velocity Vs, and quality factor Qs. The values v,. and Q,. are calculated from the measured vatue1 using the equations in SectiQIJ 3.6. A. aumbor 9( "'""'ectiOJIJ must be applied to· the rcsoaaace frequency values to obtain absolute volecity ~t$. By usia& the uniclimcasional wave equation, we have ignored the inertial effects in transverse directions, and we must correct for lateral extension. Strictly speaking, it is necessary to analyze the bar in three dimensions. These calculations show th•t a ratio of wavelength to bar diameter determines the mapitude of the eo~TCCtion to be made to a.e·frequency and modulus values. Equations and charts oftbese corrections can be foun4 Uitpinner and Tefft ( 1961) and in Schreiber er ol. ( 1973). A .. slenderness" ratio of 10 (length diameter) issuflkient for these corrections to be negUgible for harmonicS or a -..atively to~.- order. The samples can also be jacketed for tests under ,pressuwe aMt "M a $"'en saturatiod, a jacketing which requires further e«>.~ti~~ (Ga~r: ~! oi••J 964 and Winkkr. 1979). In the laboratory, it is difl'acult to analyze samples ~ tbiJJJ nt .lo~ and consequently to 10 down to frequencies lower than SOO Hz. Trttmq~JJ97'1) l'CCOIIUnended the use of the addition of end masses &o efl'cdively lower tbe frequency by chaftliQg the momeet of iaedia and only slightly alter the fiaidity of the S)'Jtem. This apin requires specihc corrcc:tioas (Birch and Bancroft, 1938). To study variation :::=nifal PfOperties as a function ofexcitation frequency, it is necessary to analya b . . of a hilh order. However, these harmonics interfere with each other and tbe inftuence o( one harinonic on its neighbors can be observed to increase as its order rises and (2 decreascis. He~. the resonance peak, which was previously symmetrical, becomes· asymnietrlcal (FiJ, 4.2C)t. It may be observed that the effect is m~able (as)'Dlltl~ri4:&1 peak) iftbc quality factor is less than or equal to 10. Other corrections must be m•de when working with rock samples To begin with, differential movements may exist between the;fluid and solid in a saturated sample (see Biot's theory, Chapter 2). Ia a resonan,t bar, f&teral exteasion processes are capable of generating these movements (W..ite, lft3). f.ig.are 4.21 illustrates a test to identify this phenomenon. The attenuation and veloctlyirl a homogeneous bar of Vosges sandstone of porosity 22% and permeability close to 100 mD were measured accurately as a function of saturation (solid curve). This bar was then cut lonjatUdinally into four plates. These plates were then bonded together and a very thin aluminum foil inserted between them to

'

'

~

.-~

~.

_,

.---.,

'"'' ·~

OOl ,

•

...

~

•

..... 01.

. .. t"L

••

·-t-..............,...

I'L

................ 00&

•

01

01.

"'

I

I

•

l I

01

OS 0 OOL OOO &

•

OSL ~

JO

-.n

,.,.l}lallatll:) lOJ~ ~qlftu tttt .toJ ·.-.r OOUtiUOIU , . ac~etts _... 'lt4 lZMt ..,.....,~

~lllml JMIOI*

c

P:~::r:~:..,.....r~....,..~~...,....,.;.::;:.-.,.

L91

:·

168

TECHNIQUES FOR MEA.WJJNO Vf&LOCITlaAND ATTENUATIONS

4

guarantee total impermeability from one plate to another. The attenuation and velocity were then measured on this foliated bat (dotted curve). The lensth and diameter of both bars (homogeneous and foliated) were the same. It may he observed that the attenuation at very high saturations(> 85%) has~ by 35°/o, whereas for saturations less than 80%, the decrease is only 17%. Certain problems were added to the measurement, such as rigidiftcation (and hence an increase in velocity) of the bar due to~the introduction of the . aluminum foil Nevertheless, this result appears to ascribe some importance to the Biot process in this type of experiment, with high water saturation (i.e. high relative permeability). One must also note that a rock is not a truly homopneous ()bject, and the velocity and attenuation measurCd using the ~~91= peak may not necesSarily be indicative of the average velocity and attenuation Of tllC'sample. Figure 4.22 shows the manrement ·taba 011 • · •sample oonsistiq of a piece of aluminum and a piece of plexipas oftbetame''fenath bonded tOfetber. The velocity and attenuation measured on the ftrst harmonic are very close to the velocity and attenuation of plexiglas alone. It is therefore essential to have samplef·as homogeneous as possible, a criterion of homogeneity being tlae regularity of the frequency distance bet~een resonance peaks. (

I

3045

I 10.5 ~

R

12040

;I

I Jl o~.L ..., 3GIIIt

I I ~ I!

c.·:.

1020

- .. ·':A

, _ _ (Hal

,.., ,..,

181100

Fig. 4.ll Experimental recording of resonance peaks vs. frequency for a twoelement bar (aluminumfplexipas).

Another problem related to measurements on rock samples isthe·non-linearity of the Qstrain relationship. If ~ deformation caused by excitation is excessive, the Q-strain relationship becomes non-liflear and ~g: iatriuically different from fteld studies far from the source (see next_ Chapter)..Fisure 4.~3 sh()WS one example of the change in a resonance peak at high amplitude of deformation. 'Qais ch8Jlle shows the dependence of velocity and quality factor on t)ac."tr,ain &JQPlitude (Fig. 4.24). The resonance bar method, as well as the pendulum method (see paragr. 4.2.3.1~ is reliable and matively easy to it'rlJ'le.-t. Aceuraoy oliM meaatare111ientd ~ iS between 5 and 10% for Q leSs tllan 1~. P'orlmver •uenuations(fl > tOO;forft.amPle), the accuracy declines. In the extreme case, for metal~ (or Which Q is nearly infinite ( > 10,000), the

......,

.........

........ lOU 1! pmpita 'Jwq laftOIOJ ~l pn '=trn<*U • ~ Ol ~!S$0dd. Apl\lJ!A~i! l!'((' > ()) ......WIH• .tfltl!q 1! 1lltlp1ll MpJ! •uA~ •suo!ltm~l· n II""' n la!l~ so:~·ot > JOj 'At {lftOqV pta ot < ~O.J tnoq•

D .;.,·o

Z'

: lUOJP3D 1!~ S!'P UJ top!OOJM ~~'(ON& ~(I'A A1pop\ MpJO hwlft038 ~tt.L

-.sGf tm)!lftd1 llqlO

·

. ptJW UO!}V!PWJ JO S~ ~ Ol Ullm)WA »pUll ~npuoo ~

ptnoqs lDW!JadX:J

•(6L61

'nwu..A\ nt.JW) apt\)!fdunt U!'J.QI ... IJO!I.IOI U! UO!fii'Gallll . . .
t

I

{ZHt kluenbai:J

~

..----.,

~

i.

'

'""" '"""'-

-'

--.

I

I

..

l

'"""

"

,..-.., -~'""'

170

TECHNIQUES

FO~

MEASURING VELOCITIES AND ATTENUATIONS

4

""'

4.3 MlTHODS USING STRESS/STRAIN CURVES These methods ~e qtJite silnilar to lhe foregoing. since the types of excitation are the same. As we have already stated above(see Fig. 3.3), the ener&flosS per oscillation cycle can be measured directly from the hysteresis of the stress/strain carves for different loading and unloading cycles far from the reSonance of the system (Gordon and Davis, 1968. McKavanagh and Stacey, 1974). The U$C ofthis method by these authors has been limited to torsion experiments and uniaxial stresses. The major diffu:ulty is that the adaptation of this method to small strains requires very high measu~t aa:Uracy. To obtain a reliable numerical value, it is necessary to achieve excollent repeatability of the excitation stress function. The value of the quality factor Q is then pven by the equation: 4nV.,•.

Q == L1W

"' -, ~.

~-

(4.20)

where V.., is the average energy stored during a loading cycle and L1 W is the energy dissipated per loading cycle. Spem:er ( 1981) (Fig. 4.25) directly measured ·the phase difference between stress and strain at even higher frequencies and lower strains~ For the phase difference measurement, the quality"fact()r is given by: I Q•tanqJ

"' ',

"' .......,

(4.21)

The velocity is obtained by measurilll the elastif; modulus relating stress and strain. The ratio between stress all4l stnllti lives tlie uttplitude of Young's modulus (dilatation waves)

I

from which the velocity is determined (Spencet, 1981):

I

V=!J

with

lui = lEI tel

(4.22)

"'

i

I ,, 1,

In this equation, p is the tkosity, u the st~ and e the strain. This velocity Y is, slisll_.Y lower than the phase velocity:

f

21EI 2

I

""'

i

V.== ,

~[lEI+

£a]

[see Eqs. (3.31) and (3.33a)]

(4.23)

Here also, an accurate representation of the sinusoidal excitation funetion is aecessary. These rnethods are relatiVely diff'JC'illt to implement, ·given the accuracy required for the small strains involved (s <' to-•). lltetrequency range for seismic applicatiOn spans from a few Ht to a few dozen ·Hz(and eVeri a few hundred Hz for the Spencer apparatus). The accuracy of measurements reported by Spencer is le5s than s•;o for values of Q'of about 20. Accuracies of 1 to 2% can be expected for velocity measurements though error increases in the case of rigid materials. In this type of experiment, it is essential to operate far from the resonance frequency.

I

'-

_!

' ----..

~,,

-, """""''"' --.,

'---'-,_~

4

TECHNIQUES FOR MEASURING VELOCI'f1t!S ANn ATtENUATIONS

171

"-

.,

SECTION THROUGH THE LABORATORY APPARATUS A.ftoclc$1111111e B.,__......._,

-,_~

C. m.lace:ment ltlnlduclr

D. Elaltromtldl...-. Jbaker E. Shaker uble F. Tllble extender

G. Tlble -.-nsfon H. S1llel billet . I. 8aetam _ . .

J.

T.. tndplate .

K. Cleinping ri. . L. Filllurefortlla~

'-~

........_..tubbar,...

M. Flelcllle

'-

~....,_,

\..../

F ... 4.15 Diaaram Of a pendulum t"or measumnents Of phase differ· ence between strefi .aed strain (ifter Spencer. C 1981 AGU).

'-..._/

'~

-~

~

'---·

4.4·' CONCLUSIONS

~

'-.../

~

"--' ~

'--

"~

____.

The many ttclmiques for measuring attenuatioa are dift'acult to ~pltmcnt, both in the fteld and jn tile laboratory, and the accuracy or.-.ul11ment is atncrallY not very high. Moreover, Che freqt;ICilcy range uied in these methods varies, as well as the type of wave involved (ltalldina or travelins waves). Velocities, however, can be measured quite accurately (a f%1 Neyerthelcss,.tlteJypc of V41locity IMUure4 (phase velocity. group velocity) -.aries from one technique to another, and the analysis of velocity dispersion is only feasible for atandina-wave experiments in which the ·phase velocity is measured directly. For propaption experimejts, any accurate mcasvmnent of velocity require$ its own experimentat setlp, since systelnS for the sim\lltaneous measurement of velocity and attenuation arc not suffiCiently acanate for both parameters. The. main results are. summariied in Table 4.2. ·

"-·

"-

'-· ~

...._)

r TABLE

4.2

MeTHODS FoR MP.ASURtNu veuJCtTII'.S AND ATTI!NUATtoNs. SuM~ARY

Field of application.

Type or method

Method

r.eid

Laboratory

Standinj wavel ·. Vibration ..

Ease or implementution

Pendulum

LF, a rew Hz

No

•esonant bar

a''tHz • 10kHz ..

No

DkHz to S MHz

No('l)

~::.

..

Puhc...M:1.0

Fairly easy

thue.wcity

Fairly easy

Pbue~y

Easy

Oill'erent from follow- Rcnection amplitude ratio or spectral ratio ing velocities • phase

\

~~

...

...,,<

... ~

-,,,

Tfll vCiina wavea

&rOUP .•.,.,

Trasll!li111illut:

S)(l kHz to

5 MHz SeismM:at

well-to- Very easy well In wells tt· a"h kHz

"

'··

..

tT-£

.. dift'elilnoe

Di~

cycle

A rew Hz to a few No dozen Hz

l"luhc: ·

) ' ) )

)

)

m•umncnt Up to 400Hz ofphuc

)

---

)

OlciJI•ionl I • forCed •. •.. • free

"'

prttpqa·

Dilerent &em follow- Spectral ratio. Rise ingvcl~

)

No

'

..

time

.phue

.poup:' >"-.J

·.:~11oft

-Stl'fl!l/ftrain

Oscillations • foR:ed • free

lion

....

"Pr~tien

Measuring tcchftiquc

Measured velocity

propaga·

'

Dift'R:Uk

- phase velocity

Di5sipated 'energy

Difftcult

- phase velocity

Phase difference t1 -I:

)'

i J

)

(

(

(

(

I

(

(

(

(

(

---------------- -----

{

(

(

(

(

(continued)

Reliability

Stras/strain Plwe difl'erence

Suwr.tARY

Reliability and acx:uracy oaQ

and accuraq

Method

oilY

Corrections

Remarks

Yes 0.5% ifQ > 20 I %ifQ<20

Yes

Yes 0.5% ifQ> 20 l%ifQ<20 1 ifQ< 3

Yes Yes but tabulated ... 5%ifQ < IOOanciQ > 3

PuJie.eeho

2 .. 3% depeadiq 08 Q. ReliabiB if Q aot too low. ltelatiw mees•rementJ more HOCUratc

Coupliaa problema''' imply uareliatlility. Accuracy > 10 %. Relati¥e ntea!Mtremenl5 more ~KX-'Ufllte

Many and empirical = difFICUlt

With buffers, possibility of study under pressure and better reliability

Traft~~millsion

I to 2 ~. depeftdina on Q. ltdiable if Q > 10

Unreliable 111 > 1o-1•. Accuracy depends on Q

Many and empirical = dill'lcull

In the f~eld, ri5C time appc:11rs to he better

,,_,.cycle

YCII lto 2

s to 10 •;.

YCII

Sliafll

YCII

Sliaht if accurate

Standing wave~~ Vibration lt.esollant bar

Travelifta waves Propaption

(((\

~

Menioos FOtt MEASURING VELOCmES AND ATTl!NUATJONs.

Pendulum

((,

--------

TABLE 4.2

Type of method

(

(

~.

Direct meuurement Yes lto2% ofplwe

II) The aa:ui'IICy values

Yes but tabulated

Torsion. bendina Low frequency measure-

- S •;. iCQ < lOOancl Q > 3

ments

5%forQ~20

ai\lefl for YtltltlltJflio: prOfl"plion MCthoola refer mtJf'C to

¥Blues

if

Torsioa, bending Low frequency measurements

system

very Fur from resona~nce Torsion and uniaxial stras

system

very Far from resonance Extensional s
accurate

ul' rcprodlleibility, sitK:c it ill very dill'oc:ult lu ublain 1he nh•nlull: value nl Q.

~

~

/i,}.:.;.!>"':!.:f. .. y:f~i' {!~I· ~:·!_J·J:jr'[_"f~"'

'---"

5

'-

'--

''---'

WcNe

propagation in porous media

resu~s

and mechanisms

''---

'

'---'

~J·

'-

" ·~

INTROOOCI'ION We have 10 far ill&roduced a number oftlleorctieai(Cbapters 2 ud 3) and experimental (Cbapter 4) toelseaabliaa usto raodelaad •raNK dal lllllduaaioal propcrtios of porous media. We also dlowed iD Chapler 1 that to .,..U of~ JMdia in the absolute IOMC wu relatively ........... A timple conaept·sucrh•-tltat of porosity must be split up ialo . a large number of categories. In this Chapter, we have decided to begin by .,..,..... a mambcr of result$ of acoustic measurements in the laboratory by hiahliahtina the influence of a number of important parameters. Usina this qualitative knowledp, we shall then try to understand the mechanisms responsible for the process~~~.-.-.. deal with 'tile inverse problem, that is to say in situ measurements for which the unknowns are the properties of the propaptiaa media. We shall show how certain lderatuq aauks cu be used and provide some of the most widely employed CRJpirk:al ~~ns. '

j"

"

•

'-

'-'--.-'

'-

"-

----,___. ·___. -"~

'-,...·

'--

5.J RESULts ANn 1\li'.CIU.NISMS . IN THE LABORATORY The earliest syltematic meuurements or variatioM In the acotlltic properties of rocks appeaftld in die 1950(twos. At the·time, muy f1JIIrl1llis,illduditla Wyllie, GfiJory and

Gantner, inYeltiptedthe variations in the wlocitiet

or...... waves in porous media as

a funcrioll of such pltl'lltftetert a porosity, ·aturat.ion aild prasuro. 'f'hey also cleak with the problems raised by the measurement of intrinsic attenuation. At the same time, Nafe and Drake hepn anatp.ina ~tics. in 09IU RQOI' lledimcnts. with a partic;ular focus on the preblem of colllf)IClion. This type or ......... ~ down ia dae 19(i()s. but similar ia~iJations were revived in the early t91Gs·by_ re~ inchtdina Domenico and Nur, whose studies of variatiOns of velOCity as i fUnction of prdSure, saturation and

176

RESULTS AND MECHAI'ISMS

s

interstitial fluid viscosity are well-known. At the present time. many areas draw on the knowledge of velocities and, though more rarely, of attenuations. These include many subjects in civil engineering and geophysics. Concerns and motivations are different in each of these areas: they involve the analysis of thermal mechanisms for geothermal energy, high pressure, high temperature and time measurements for nuclear energy, zero pressure measurements in soil mechanics, and quality control of concrete for construction. Yet the primary area for research on velocities and attenuations remains petroleum geophysics. The considerable growth and the pioneering role played by the petroleum laboratories arc nat rally grounded on the fundamental importance of seistnic reflection surveys for petroleu exploration, and on acoustic logging in wells. This also explains s, and especially sandstones (whose importance as reservoirs is well why sedimentary r known), offered the ost popular area for experimental research. In the following pages, we shall try to avoid estricting ourselves to experimental curves available for sandstones. pointed out that few investigations dealing with carbonate rocks, However, it must clays and shales exis in the literature. One must realize that the results presented concern clearly defmed samp es, and that these~rnalts.can only be extrapolated to other samples with all due precaut ons. We shall fust examine the results obtained by velocity investigations, which will enable us to highlight a number of trellds. We shall thea demonstrate the difftculty of deflfting a siftlle "WJocitY'•parameter •oharactcri:re the bdha,ior of a giWil sample. We shall then go on to ~ts of afteiR'Ja!Wn. The interpretation of theBe resUlts by means of physical -meciNMtsms will follOw. FiftaDy ,..., ,shall take the .tewpoint el the investigator and develop some empirical laws for velocities and attenuatioDI and give their limitations, for various. in~. ·

'""'

-~.

'""'

•.v«

~

S.l.t 5.1.1.1

Velocities_. Jlllllt es •

Rocks in the earth's crust are srib,;ect't6 high stresses. To understand the processes observed, it is therefore essential to understand the role of pressure on acoustic parameters. To begin with, we shall examine the effects of different types of pressure on velocity, namely confming pressure Pc (pressure to which the sample is subjected), pore pressure Pp (pressure of tJte. flp1.·4. i.,nsia.F the por~~ and the differ~ntial or ••effective pressure" Pefl ( = p • ._ pP)fH. We Wilt~ ~bow the e1fes'Ct of a uniaxial stress.

a. Effective pressure Figures 5.1 and 5.2 aivo a ~"r of c~amplcs of V.Jriati
.

--..

...__

-.

but this increase-dcpends.svb!;~ly ~n·~ type of rock co~oed. PractialUy no (1) The eft'e<:tive Strc:IS is normaRy ~ U the stress to which the $01id skeleton is SUbjected, i.e. the difl'eftllee betWeen the lithostatiC ·streu-...ct tk l)'drostetic prtSsure. This is derivWII from nrza,m.._ law diKUSSed in Scetion 2.2.4. At • rule, tlais ~vc ~trcacaa be trtatca u tile differealial pressure, Wbic:h is~ diffefCJl<:e between confmins pressure. and the. ~re press!'re (see ~ ur and B~ :rice. 1971).

~---

·-"

""

......

- T

I

\.....'-'

...............

-

7

1

I

5

8

v,

-;

l4

::!

js

~

l:

8

Vs

2

Vs

3

0

100 2110 300 EffectiM ,....,. CMhl

408

2a.--....a......---'--......1'--...J 0

(a) 7

Webnlck 'dolomite

i::! '-c~

'-

fs ... >

'--'

4

108 200 300 Efhctivt ..,_.. CMhl 1iI'Oytrlrutt .

v, 8

Vp

J.

--

~ 3

.,..

400

(I)

I

"

...

I

~

I>

I -----'-·

S.t.

4

Sat. llld dry

vs ~,

100 200 EffMtlve ,.._.. fMPal 5

\_

300 (a)

I

!3

Sat. lftd dry

l&

Vp

•

~

'~

......_,,_..,.. .,

I

4

I>

.

.,, .,....:..,__ .:.,

3"-----.,a.----'----..J

2

'4

s.t.ll'id dry

3

0 '-

0

10 20 30 Effectlwe.,....,. CMPtl

'

• 40

<')

vs ___.__

2 0

•

I

100

200

Effective,_,,. {Mhl Fit.~~ J~UCDQ; of poreaape o• ~14ft«tivc pr•ure ~hips (after Nur and Murphy, edited by lrutin and Hsieh, 1981 and Nur, personal correspondence). a. Microcracked rocks (ultrasonic measurements). Ia. Massillon sandstone (resonant bar), Solcnhofen- limestone (ultrasonic measurements). '-

-.___

4 Pierre shtle

Sit.

3

i

~'...

~

.~

!

........ "'

.,.,.

..

",..P Dry

_......-o--.......... - .. _-o Dry

2

~s.t.

Vs

110

100

150

&ftec:tlve pressunt (Mh)

Ch..k 3.5

sw

•=306%

~~~~··-==;-1~

::::;

-

i

10

seo

3 }-,

,"1,<

fu >

$w

s

0

2

10 . 40 20

80

~to

100_ _ _..___ _ _ 1.5 .__ 0

10

~..-

_ _ _;r..;,;.;_ _ _L---.....J

20 30 Effective ,..._,. IMflll

40

Fia. 5.2 Velocity/etTec:tive pressure relationships for Pierre shale (after Tosaya. 1982) and chalk (after Greaory. 1976} (ultrasonic measureinents).

"

i".'.

,...~u~,·

~:.f.i

h'L~.

·~~-{11,

J;·;~ ;<:t..,n;,

h~: :Y~:r~en~

J

._.~

'----'

~

t

·:·

'-

~.

'-

179

increase occurs in Solenhofcn limestone, whereas cbc P wave vcloacy rises from 4 to 6 km/s for dry Westerly granite. 1'he iftereue in I' waw wlocily vs. e«ectiYe 9fC8IUie is much smaller for a saturated sample than tot a dry ....,te.. and tbis applies to all the samples in Fia. 5.1. On the other hand, the shear waw veloCity is vinually unaffected by the presence of water in the medium. Also observable is the existence of a maximum pressure above which the Velocity reriWns constant Howewr in Piem shale (FiJ. 5.2), no plateau can be obSetwd at maximum pressure applied: The samples shown in Fij. 5.1 are 6lirfy weU4cnowtt porous media (see Chapter 1). Granites, dolomite and Bedford limestone are well known for their essentially microctack porosity. Solenhofen limestone and MusiiJon S8ftdttone, however; exhibit a pore apace consistina of pores with aspect ra1irGI dOle to t··.mc~~ we call ~·spherical" porosity, aad cracts are uncommon. The samples in Fit. 5.2 display c:oniJ)Iel pen apaoea, eitber because of the presence of clays or due to the pbysicc:M:hemical sensitivity of the c:alcite making up the chalk. Til~ Fig. S.l shows-that the effect ofeonftaiq ~is direCtly related to the number of cracks (two-dimensional pores whid'r tend to desteuily) existinJ in the sample. Denis et al. (1979) measured the ultrasonic velodties of dry samples at atmospheric pressure and succeeded in calculatiag a fraeturina index (eontinuity illdex). Fiptt 5.3 illustrates the

'-

'-

~A'MI)~

!

'--~

'--

''-,_...-

0.&

'-'

o.•

~

~

"---

~ Yo

~

0.1

~·

0.2 '--'

'--'

.

0.1

1

2

S

•

1

I

Crack~ tX tO~ol · • "--"

F'IJ. 5.3

Influence of microcrack porosity on tbe velocity/effective pressure relationship (compressional wave) (ultrasoaic: measurcmettts) Iafter Nur and Murphy, edited by Brulin and Hsieh, 1981)..·

--'----' ''-.--

Vo = velocity at atmospberic: pressure, V1 == velocity at 1 OPa,

&V- V1

-

V0 •

0, Oak. aJl ~tone; FO,, duqitt; Jl~.·. 9!~.ap; TO, Tro~· sranitc; WD, Webatuclt. dolomtte; WG, Welterly JfU'ife: "'-'• "Stone Jranitt: CO, Casc:o Jl.

,_,....-..../

gnmfte.

'-.-

'-··~ '----'

J

·

180

~LTS.

AND MECHANISMS

5

extent. of fracture porosity by the inc~·ia velocity ,with ~~·The .ordinate in the diagram represents the normalizl0d4i&rcnce bctw,eeothe compressional velocity without pnssure and that uader a prcuurc of 1 .PPa for dry samples: the ,greater the number of crackS; dto more the velocity. v.aric$ with pressure. As already noted, the P~rrc ~pies used in FiJ. 5.l display a different velocity function dllan Fis. 5.1. The~ RlateJu as a fuiKltion ofp~ure, which represents the closure of its .. last" crack for the ~pic ;~nc::emod. is not oluervcd for this rock at 120 MPa. In fact, Jones and Waoa (l981) .obMfvcd this phe1lome.non continuing up to 0.4 GPL It is possible. &Nt tbc .~ in velocity as a function of pressure shows. in addition to thccontinuousdosurcofthc ~a.:ks,.an aJipment of the day crystals in the miDimwa shear streaJtb plueflosaya, 1982)-;aa.Jk. tlae$CCOnd examplein Fia. 5.2. is very c:omplicated ~ the stnacture 8f ~ material ~ ac:cordiq to the pressure applied (creep). We shall .net 10 .into detail for Qjthcr Pierre shale or chalk. assuming that the variation in velocity versus pressure are only important if the strw:ture of the material iavestiptedis aot GMqcct itq:venibly: by the ~L For clays as well as c:balk, tbore is goect reason. to s~ :At~ in the CUfVe of velocity vs. cft"~ive

-.le.

'

furtlaer

pr.essurc. .

We have mown that the increase in velocity with prasure resultS from tbc closure of the cracks, and that this closure is reflected by a greater rigidity of the material under pressure (i.e. an increase in the correspondins elastic modulus~ In fact, it must be remembered that any velocity V can be expressed in the form:

v=

Jo/;

(5.1) ~.

where M is the elastic modulus, aad p the density, and that, consequently, at constant p, an increase in elastic modulus implies in velocity. The behavior of cracks and pores under confmina pressure was modeled by Walsh (1965, 1969) and Wu (1966) for a pore or crack included in a matrix. The equations ~ by K, the bulk modulus. and p., the shear modulus of the rock for the dry sample are- the follo\\illl:

arise

(t + !.)e , _!_ (1 + !.) ,

_!_ == _1 K K1

A

.!_

B

JJ

=

~

(S.2) (5.3)

e

JJJ

where K 1 and p 1 are tbe solid moduli, e the aspect ratio oftbe pore or crack (e = 1 for a sphere, e ~ 1 for a crack), tj) the porosity, and A and B are constants depending on the characteristics of the medium an4 close to 1. For the saturated sample: 1 1 . (5.4) K ~ K; (l + A'tj))

.!_ JJ

~ _!_ P1

(1 + !.),e B'

(5.5)

where .A.' llDd I!" arc consta~ts deP.eqd•rw on K 1 and Jlt ($D4 ft,U.id bulk .modulus for A') and close to l.lt can be seen that, if e = ·t, the eff'ect of the pore on the moduli is negliJible

-.,

'-'-

-

r ·f. ; ~

' !

-- .. ,. 'lti!Siiii$ ~ M£Ct4ANiiMS '

5

f8l

to-

2, the effect on for low porosities (about the order ofmasnitude of • ). However, i(e == the bulk modulus of the dry sample isconsiderable. On the other hand, it is nepgible for a saturater.i sample. For a livcn sample, it i$ necessa11 to integrate the effects of type (5.2) to (S.S) for all the pores and cracks. The more pressure is applied. the more cracks are closed and the less the moduli are altered. EquatiOns (5.:zrn,~s.S,.als9 bclp to understand the di8'ercn
"'

o•ine

0.35.,_,.

;.·.

0

.... lllldllone

A Dry DW....,pp•O • W..,pc•toMPa

20

...,

to'

- ..

10

£ffwtlw ......... ......, '

lnftuenc:e of saturatina fluid_.~ pressure on Poilsn'• nttio of Berea sandstone Iafter Nur and Murphy, edited by Brulin and Hsicll, 1981).

Ftc- 5.4

... hN,....,. . . etlll/itlilll~ The resulti hi Fijs. S:,, S.~ .~ 5. 7 ~w clearly that the important parameter controlling velocity is the. .ivc pressure. tletail, it is observed that, at a pven effective pressure, the higher the pore pressure (i.e. the higher the confaaing pressure), the higher the

In

4.0

4:0

MassiHon undstone

'·'~· ••

~ 3» ~

t

Vp

>

.1::

8

' ....

•

Pc•80MI'I· _,

~· Vs

1~.:,:...__ _

1.5

0

·a

:

• -

•

..

13.0 f

-

~ 2.5

2.oL,.

...

•

v,

u

2.5

"s

2.0

1.5 '--_ _......_ _ __...._

20

40 60 lttectM.,.... ....,

80

20

0

_,.._~-------''-'

..........

40

~

60

60

5.0

3.8

4.5

i

v,

3.4

• .,p...,

4.0

~

13.2

St ,._, SlndttoM

•

Ys

13.0

13.5

•

>

>

2.8

3.0

Vs

2.5

0

2.8

.J.l

L 20

40

60

..,~o

80

20

0

Effective ..,._. MPa

40

60

110

Effective .,.._. MPa

FJa. 5.5 y.locity/efl'ective pressure relaUQRSh,ips for cases of zero pore pressure (p., ... ()) ana '(l()Ditant confaning pressure (p, = 80 MPa) (ultrasonic measurements) (after Jonea; 1983).

,;,

...... ....._ 4

~Pett=Pc

13.5 ~

PeffilinMJita Peff=21 MPe

J

Peft•14r.'IPI . Peff=7 MPa

-

3 Peff=O

.

. 0

20

40

80

Confining pressure (MPal

~;

fitJ 5.6 "••prn•ionat ,..w velocities in

aetea ..-ollC VJ. ~fipjq~te and eltective p-cssure (ultr&sonic measurements) (after Wjftie er lll., 19$8). ·

,_.,-"

r

•

J 4

3.8 . - - - - - - - - - - - - - - ,

...

-a

13.4

,, .. ,

3.5

t

Pp""41 MP•

.-,=o

~

i
m·

--·-~

v,

Drv

""_,.,. t

I.

I

--4

~

1.1L

Pp = 0

p,•1MP•

"

100

• ,,•41 ...

lu

O.IL-.._ _,_____...__ _..__

Pp =41 MP•

• ,,-,1 w.

-a Ju fJA,

"p,=41MP•

50

e

1:. ,.

2.5

12

32

..

, , .. 1 . ..

__..~

21

150

Cclnflnlnt,...,,. ......""

50 7t •• IH.ctlw.....-MP8

l'frl. 5.7 lducace of port .,-JPlHmd ceafiiUoa pressure OJI ~ io Piette .... (ultrascmic inebu~ (aflfl' Tosaya, 198~. ". _ ,·

'~

velocii1(FJ8. 5.S). Y« the d~betweeo Ute vdocity~ ~ut"ewrves for the cases of zero pore pnsaure aDd constant confanina pn:aa.e is very sliaht, and in many cases lies within thcacwracy ol the ox~t. Fi~Urcs $.6_11l4 S.7 show the increase in velocity II a fuilction of codaina 'fhiSUN for dilrereat..- presiUl'CS. The general conclusions are the _ ..... ofelfective pressure. It is also ~ that, at a pven con(apina ~ure. the vclec:it)' is.~ for flow pore pressure ia cc:qpariiOft witla a IU~ .,.,. ~ · ~-~ tbe increase in velOcity wida dKtive prcuure. " "" From a 4I'JIIIitaeiw tiewpeint, ads JK.a.llble dlat t~~e~cliPt iDctcale in velocity correspondiaJ to hiP pore pRSSures with a pven effecti\ie .....re can Mtelated to the correspondiftl increase in confming pressure, which slipdy increases the riJidity of the matrix and hence the velocity of the medium.

......_,.....,_,_.fllaalion 1'bi:

c.

~

. , _ , - . . . . . . . .,_..~

The uriatioasofvelocity as a fuactic.aof"Miexial'ltn:ls_..doscly 4opeadeQt OD two parameten:tlle rau.G'thisltals tofhefaih,....,...ofdle...aplcaadthedirec:tioaof application ol&he atfeU in relldiDD.*the-lWOPI. . . . cliaetiaa 1oftbe waws analpled.ln

fact,dependialoo th&mapliUickof*cllHII~.._ theesistiqcr~ks aredosod (low stress) or new cracks arc created (high stress).

184

5

RESUI..'R A!'fD M6CHANISMS 0

VN 1~

------------------

'

Fig. SJia 11\ftuonce of a uniuialatress on P and S velocities ill a sample of Westerly granite (after Lockner~ al. C 1977 AGU). Left: schematic view Gl' ~xperimJntal setup. Right: normalized velocities as a function of failure strength.

a!WII

i

uf 4.4

25 20 ,,

·i

4.2

tO

-

~

4.8

' sv 3:1F-._

p

SH

3.t

\

30

:::4.015 ~.8 L

o <;.

3.1

-~

L-.a..

o•

..

:::· &

,...•. .,., ;·····-·...................,............... j_

••

~·d

~~ 2.7~:: :?;-;;tee;:

-,,c

·td '.to!

••

to•

F.. UJ .. V~I~ty ~&Jopyill4~ oll.•~Jranitesampk bl a uniaxial sttesr. n.e stm~tsapPHed mtt.ra~·D • ft'. The SH wave ts polarized

perpendicular to the stress direction for any 8. The SY •'&ve is polarilled in . tile pine coaeainiaa•tbe:*- direetion<(aftcr Nur an4 Murphy, edited by 8Nlia aDd IUiola. 1"1).

At low stresses, the cracks perpendicular to the stress are closed preferentially, inducing a clear anisotropy in the sample. In Fig. 5.8a, at stresses lower than half of the failure strength of the Westerly granite investigated, the Wteeity inereases (closure of the crack-s). It then drops sharply at high stressel. . . .mt ef craeks). The velocities are lllClUUrCid in a direction perpendicular to the strea·la As. Ub, byeon.tnst, the wlocities are measured in all the direeticms afid.at IDlJICh ..._. lttelacS (pndudias any aution of cradcs). The anisotropy induced by the uniaxiAl sUms· is dearly shown by the diffe~WJCC in velocity measurements in each direction.

----Stereoscopic setup PAV, qJ= 6%

sw

100%

sw

100%

() 1 Atm.Press : 5 MPa

v

(m/s}

1Cl0qo

5400

I 5435 20

32 500/).m

z 220,

qJ_= 3.8%

(5"

1

Atm.Press. 5 MPa I

(m~sl 4358

1001o

172

14800 84 I

I

Fig. 5.9 SEM photographs of two pore casts of low-porosity Fontainebleau sandstones obtained by the technique described in Chapter I (after Bourbie and Zinszner, 19851.

186

. t-

~

s

RESULTS AND MECHANISMS

..V~ONfflii-Dry

GN£1SS-0ry

..,. ...............

-~· .\:~_ _

~6

f)t 14

(J

'

·~~-·

o i~ ~·tl Unudalt

~

.L•'AtiliaU.

Oil ~iatioft; l L........ '

,

;;'a ,

'

1

,

1

,

tl· · ,

....,..... ~~" .40 .: . •

'

'

,

'.:,'.~~:....:w .,,

1

' .,

Cal

...........

1110

~

er-------~------~----~~

.-,·.~~

Sw•100% 3 Vp4SO

.-v,4fSo

5

~

• _v33

l::':!. ~

i>

v22 = Vn v21

4

L:.'-: •

2

• v,3 =V31

't

2 0I

I

"~

..

'

)!rf.J .. J'il ·.. lioJ

,

....

Aaother~~-.pp&~~~ un~~~tC,.,of~~pvcn in S.9. N-~--

DI•n twoF~iaiD-~~able

po~ a·~·&fti:Jilatk:al chediiltry (99.9%q~;w.r.. ......... by F"

ultrasonic methods at a central frequency of 500 kHz with and without uniaxial stress of SO MPa. The velocities and atteauations behaved differently under the effect of the uniaxial stress, and the velocities ·are quite different for the two samples. Figure 5.9 allows a qualitative explanation. The pore casts (see Chapter l) are also quite different. In addition to a "three-dimensional" porosity, one of the two samples (Z220) shows abundant grain boundaries that act like cracks. At zero stress, these cracks make

~

r'

,,~-

'- f·~·t·

·'.

" "--

atsOl'tt'AMY' r.M.if..\t...MiP'

5

18?

the skeleton compressible, hence the low \·elocity (and high attenuation). The application ·
"-.;!!"

of a uniaxial stnsspartly dotes the cracks (those perpendicular to the stress direction) and the velocity riles substntially. These induced anisotropies arc naturally quite di«erent froiJl the intrinsic velocity anisotropies frequently observed ill rocks. three examples of which are given in Fig. 5.1 0.

'-' '-

5.1.1.2 Velocities aad saturatiells

"--

Natural porous media are always saturated with fluids and the influence of these fluids on acoustic propertia is essential. We will prescnt farst the effect of water/air saturation, and then ao on to the other saturating fluids, cmphasizittg the role of the viscosity of these fluids.

'-

'-

"-'- '-·

~

"--· "-·

I

s.t.ndi011 As already shown (fi&- S, 1 for ~x~ple). velogty measurements arc quite different for ••dry"< 21 or 100% lablrated samples. Fipre 5.11 gives aa eumple of velocity measurements venus pressure for dry, partially. and fully saturated Massillon sandstone. For P waves at partial saturation, the velocity is lower than the .. dry" velocity, which is lower than the velocity at l
L,

5

'-

.

Masillon undrtone

•

'-.._-

4

Dry .

4

...

Fullyut.

Dry

;:!a

''--'

I I I I I

' -~;··n' '._/

''-

"--

0 0

10

20

30

Efflctlve pressure IMPel

•1

l'fl. Ul Veloeity/W*f .......UOD relatioubip ia M...won u.adrltonc 'VI. coalaainc,...... (~ ~·~\at et Ill.. 1910). waves uad. f9f sp~ wr~ t~ onq.eflc:Qt d~ t~ in.trod~o~ of water jnto the dry sample is a density inc,rease lq~DJ··to a~~ in velocity Eq. (5.1)]. By eontrast, when COIIJI),tele sat\aratio,a is ~ me '~. become more' difftcult to compreis,

t'he

t•

(2) Ccmlictirallie Ulbltuib· ~aallle lCim ..dry .. Pfe\-ails amoq the cfilerent tUdlots. 'For sOme. tlae t.n. ... , .ellfllidl·to.....,yeliMditMMIIJ(I.e. ,;m tbe,.....4f......_ifa dae rock.lldlol1lld ia _,...

ao••••bcric

we,..un

..,._IICIC:!ItfCiiq oo~. For~ "Po~ ~,UIIdv VICIIUIL As llaow further on in this Section, the water adsorbed plays a fundamental rote on the velocities (and attenuations). This ambipity or te!'JftinolOI)' may explain certain apparent c:ontradictions in the variOus experimental results.

._.,

-~,

•188

5

RESULTS Al'll) MECHANISMS

4

,.......-...........

1-

~

·-'"'

l

11.5 -1311Hz

~

~

8 3 ~

40

20

0

......,_._ ··:Ia

eo

.............. .,, "''

2~--~~--~----

100

80

.......,.....__ .-:a.

2 ~ ~.,

1.8 '111.8

l~

VE 15911-1187 Hzl

~o-o--o--o-o •

11

tto ......1

1.4

-~

Ju ::>

v5 1315 . eu Hd

•

w-..aw.tionl"l

~

v 5 13115-385 Hzl

~

~

4

Ve 1571 ·847Hz)

L...

•

------

0.8

....~

to

...

~·~

z

40

~eo

ro

80

~

~

',

W.... ldlrlltion I,. I

till .

Fla. !.ll Velocity/water saturatiob relatic.-hip fot different materials (resonant bar). Curve (IJ) after Bourbie and Zinszner, 1985. Curves (a) and (e) after Murphy, 1982.

increasing the velecity..F:or S waves, it ·may be ummed that the liquid bas M effect on velocities and that the eft'eet observed is exd~rvely a density effect; For granite, the experiments ofNur and ~urphy (FiJ~ 5.12) show that the "dry" velocity is lower than the velocity at partial saturatfop, which iS lo\11er than 't~ velocity 100"/o saturation. In this case, the porosity is too low for the density effect t
at

"' ~

'-'~

,-

~ -~

'-- !

s

'119

·~

Nr 111UmiGn (S) 100

50

0

Pc!l35MPa

'-'--

'-'-....·

'-

I·

'---'

10.
I

·'---' '-'-·~

'---'

0

'-

'-'

.............. "', !10

100

··~" behavior of l'lltlocitY .a a Juacqon of wa~r · saturation for c:ontOiidated lediments 8ncl a-~ ~.of 35 MPa · (alter Greaory. 1976). .

Ffa. U3

'-' '--·

''-...-

With rcsptet to puita. ~ty JDa'lllll with ••utur.U.n for satuutions above 20o/.. For all dterects..._., tho.,.,.,wa~;~ it virtQily~t of water satutatioa tlftor the &au. •tNds 2%. ~_Nutilio,n ~aadAoao, ~ ia very porous. a sliabt demucl in velocitil,a iJ observed d¥C to, &be densilJ effect~ JSq. (5.1)]. Ia dba,orous saacilloae.lor r~a& bar~ Umilarbebavioris observed to daat clel4.:ribe4 ·by .Gfesory (1976).1or ~-waves (Fi&- S.l3) :even if the loqitudinal v~ia Fil- S.l2 are•tonsional veloeiQaaad aot P wave vcloQtic$(~ QMptcr4). ltilntativelycuyto,apltliatbtbelaavlorof~ia . . . .iUoa-.saadstoneforwater saturations .t 2 to 100~•. From the qualitative standpoint, in fact, the clastic modulus is iadependent of saturatioD for saturation froua 2 to 95%, 'ililereas the density (p = (1 - fl)p. + t;p1 S. p. is the dtrtsity of the solid, p1 the density of the liquid, S.., water saturation)ibcreases with saturation, leadin.s to a slipt decrease in velocity. At total ' ~aturatioa. dae water ~ tends to . . _ the material (for lonlitudinal wavet).aad thewtocity·risa.:For Sftftlnoluidc«ectooeun, asw U.C altudystated.

«"

......

,

(3) 1k dality of aid• aelfiafHe ill .oaapariloll.with dlole of the hid and solid. '-

~

190

5

&ESVLTS AND MECHANISMS

Gassmann's equation [see Section 2.2.2.2, Eq. (2.77)] serves to quantify this type of process. In the case of total saturation, this equation can obviously be adapted by replacing the fluid bulk modulus by: 1 1 l /(=(1-S,..)K+S ... K (5.6) /It

/1

H


/H) and Silanoi ions

0

..

,

flz

where K 11 , and K 111 are the bulk moduli of the two fluids, and S,.. is the saturation with fluid 2. This presumes that one of the two fluids (fluid 1) is totally included in the second (fluid 2). The use of Eqs. (2.77) and (5.6) jives a behavior similaT to that observed for Massillon sandstone at water saturations of 2 to ·100%. In fact, these equations constitute the zerofrequency approximation of the eqvationsdeveloped by Biot (sec Chapter 2, Eq. (2132)]. For measurements taken at ultrasonic frequencies (sec Fig. 5.13 for example), the application of Eq. (2.132) does not ,yield a very good result (sec Fig. 5.14a~ In Fig. 5.14b, Domenico (1976), uses the Geerstma-Smit .. high frequency" equation [see Section 2.2.5, Eq. (2.133)] in which be varies the tortuosity parameter from 1 to inftnity. The result is not better than itt the case of Fig. 5.14a. Various reasons can be postulated for the disagreement. To begin with, the equation used does not account for permeability effects, and, at a jiven tonuosity, the permeability effect is negli@ible only if f/1. is greater than 5 (Gecrstma, 1961). Possibly even more important is the fact that this equation presumes a por~ity uniformly distributed throughout the sample, and that any gas is totally included in the fluid. In fact, partial saturation of the sample is not uniform (see for example Bourbie and Zinszner,1984). Gas may~ occur in a partly continuous form, in which case the sample is truly three-phase and requires a more sophisticated Biot theory (Brutsaert, 1964). Even so, the non-uniformity makes analysis difftcult. The Geerstma-SIIlit equations are interestJna to describe overilt1 behavior, but are defmitely not intended for systematic use., · · Note that neither Gassmann's nor the Geerstma-Smit equations help to explain the behavior of velocities at low saturation and at frequencies in the neishborhood of 1 kHz (Fig. 5.12). To explain tbe~ble ftrilltion in velOCity at very lew sataration, Oark er at. (1980)invokedehydrati0ft ndstifreftins Of day miaerals that are in~tact with the grains. This explanation is ~. tieeause a velocity variation df'!M same type was observed 1ft porous gl•ss (\tycOt) and in sand (Hardin et al., 1963). materials from which clay mitlenils are afMieiftt. Two mechanisms may explain the behavior of velocities at very low saturation. To 1»e8in with, the surface ora quartz snriD possesses a suffteient negative ~to ioni!e the tltinftlm ofwater covering ittParts, 1984). HelrlCe it appears that the pore surface of all sands and sandstones is hydroxylated. A doUble, charged layer is fOrmed betWeen the wateniht Ole silicate SUffiCe (Fig. 5'. ts and Murphy, 1982b) with. molecules of water

'

6 'si/

'si/

-~

~

.The .

hydropn bonds between the surflce b)!droxyls ...amoleeulcs of water are broken which leads toaclissipatioD.·ofentrl)'(teeSeeticm S.U.lb; Tittl8anll«dll.,19B0,5peaccr.1981), and the free surface energy, namely the work required to produce an increase in free surface area, is decreased by the breakaae of these bonds. The. adsorbed water also exerts a signiftcant effect in reducina this free energy(Murphy, 1982 b). The twofold deereate in free

' ~

' ~

~fi'J.jfj; ~~

5

t9t'

2

ju

~j

'-· s*.. ~-

·.tij

••1.1 , ....

1-

2

I

.~

,,_:,,.

8 ~

I

t

0

20

~;

. ..

JO

100

I

.....

I

I

'

......,...Vp I

4.

0

-.~'

fu~

I

'

',·'

f Coltlpulllll

:

-

.,_"'*

i

i

v,

.I

'

I

waw UQ1r11!on I"I

''

I

181

I

,, '---

-·-

Fie- 5.14 Com.Puison of velocities · - ur...... calatl&ted· Ia wandlt~ ··~ computatioa . . . a........ ..,..., [100 Eq. (2.132)). J.iaht, computation usiils Riot's theory [see Eq. (2.133)]. In this latter case a denotes the tortuosity (after Domenico, 1976). .

~

10

so

·- t

•• s ·- tt'

10

·--~

eo w.... _.....,",

JO

.,

too

tbl .• ~_;

or the material and hence a 16.-er wlocit;'. A 1eeOftd catqOry of fofcft 1My explaitl tOck beltavior at low satutation; aamely ca'pilatyforees. lk-rele ofcapi~Wyfot'Ch itmati'Vely well blown for uDCODSOlidated tedimentl. A schematic aad qualitative examplcjs .provided by a pile of dry sand to which wawis ~ually ..... Tt.-elttllllplc is only a qualitatiw4escription and cetramly cannot be usld as a proof, sitiCC muay parameters are not taken into enerJY(adsor~ 'iltld elc!Ctrochemieal) implies •'JI'e*fcomptiance

'-

'-

.,__._.,.

,,,.

.......

.,._........

PiJ. 5.15 ~~orWUitadtorbedonsramsotiilica(lfter Murphy, 1982b).

·~·

192

R~UL~

5

ANQ MECQANISMS

consideration, such as the adhesive forces between the grains. Initially, the sand is dry and the solid skeleton non-rigid: velocity is therefore.low. As soon as a small amount of water is added, capillary forces act between the grain., the solid skeleton exhibits greater rigidity. By adding water further, these forces disappear and the apparent rigidity (and hence velocity) decreases and then becomes virtually ~t until high saturation. At high saturation, the incompressibility of the flui4 itself plays a role and the velocity again rises. The qualitative variation in velocity in theund pile is hence dosely comparable to the variation observed in laboratory experimonts on rock samples (see for example Fig. 5.12) except for "infmitesimal" saturationsf'very dry.. to .. dry"). In lalldstones, a capillary mechanism of the type d~~ abowe is therefore plausible, as the capillary preesure at low saturation guarantees better cohesion of the material. FiaaUy, it should be noted that ttae ~ ~-~ PJ:OPQIICCI (cbemiRI boPds and capillary forces) are both related to the speCifiC surface atea of the porous medium, and we shaD show that this remark raises a; problem for attenualiott at low saturation. Ia conclusion, for dry rocks (saturation less than 2%); substantial variatiort in velocity is observed, which can be explained by a decrease in the sui:f'a6e energy, resulting from the combination of a variation in capillary pressure and an adsorption mechanism. In all cases, we have seen that the·~ ofvelocitl on saturation,-is v~y Jliabt, both for P waves and for S waves. aa4b ·water saturatiOIIS bonreea >10 ed 90%. Couoq._.tly, velocities are not aood indicators of the quantity ef ... pniiMlDt in .... pons.

aoo

tree

b.

Ty~ ofm~~rati•g fluid

The type of saturating fluid influences the acoustic wave velOcity. if only because of the compressibility of the fluid. An example of this variation in velocity is given in tippee. tbcftuidv~y.(andhcnce Fig. 5.16. ~urthermorc,u is*-t.IJ ~-... its temperature) intl'4Cqa:s tbl: v~ cyeo _,o thafl !be ~ type of aaturatina fluid. This is wby ~~Y}tcuapcf.'ture relat~ are diseuiiCMl in the aext Sef;tion.

in.,.

i

........ ... .--,..

·---

•

......

~._~--._~~~~~--._--~-----.

.,

·.

if~

~................

~

3

-·

~

I

i: 0

,, .. 3 ...

Pp•3MI'•

Yp

o

aoc

0

~.12PC

• tSOOC 0

10

20

,_,-3 ...

'1, '·

30

40

50 0

2S0e

0 -~

~ '~ o tSOOC

10

20

250C 1250C

-, ISOOC 30

40

50 0

10

•

30

40

50

DiffemltW ,...,. IMh)

Fia- 5.16 lnfluenee of temperature on P wave velocitJ -in Venezuelan sands Jor dim:rc~saturJlliD& flWds (ultruonK: ~relDClltJ) (~ Tosayt. et Ill., 1985).

~

;:--

---s

':.·

"

JIIUIIfl AN»~

193

'-

5.1.1.3 Veloddes and te.,enawes The burial of rocks in sedimentary basins implies that tl\eY are not only subject to pressure e&cts, but also to temperature effccts;' Tile 'averap temperature Jl'lldient in the crust is around 1-c per lO m of ICdiments. Weshall tirst discuss the ~effect in the broad sense, and then go on to the proble!Jas associated with the chanp in viscosity, and then ftnaUy deal with problems of phase chanp under the efrect of temperature. '----"

'-"--·._..-

a.

Tempa'tlt,.e

Velocity varies slightly with temperature. For example, the velocity in distilled water increases by 1•1o as the temperature rises from 10 to IOOOC (Kaye and Laby, 1973) and the velocity in quartz decreues by 0.4% as the temperatures varies by IOO"C (Carmicbacl, Vol. Ill, 1984). This shows that temperature exerts a very slight intrinsic eft'ect on velocity. In fact, a number ofextrinsic factors cause the velocity to vary with temperature, and these indude di&renees in compressibility between the clitferen.t matrix dameatl aad banc:e di«erences in pressure, chanps in the viscotity of tlie'tauttatHra fluid, and phase cbangcs.

.......

'---'

~'~

........

I

1... Ju~ 1-

•.o ... '--

'~

'-

'--

0

f1l. 5.17.

'-

'-'-../

'---"

_......

?iF

..........

_.

. . ___.:

,::po-_..,. ' ~p '• ~-~-!_ Wet

1

1Clo

-

--- '

••

-

I

•

c:.tlnlnt......,. CW.)

.

'

!

I

400

1100

Compressio. Col • aaJ ~tics vs. temperature in Water.ly puitc.

Compariloa hetwccD dry aDd satwated cues for different confaaiaa pnuurea (ultatoDic _,utemeota) (after Nur, .1980).

For lowporosityroeks such as pauite and pbbro (FiS. 5;17) the decrease in velocity with temperature is nearly alwaytieu than 5% for a tcmpanture inaeasc of about lOO'C (Carmicblcl. Vol. II, 1912). F . , . 5.18 and S.l9 o&r a nuaber of•...,... of velocity variations with temperatUI'f'for Jaip porOiity sandstones. It may be observed that, for dry and water-saturated ..-tones, vciOClity ~with risina temperature. The~ in velocity observed in Fig. S.19d 'at partial~,:._. from aa irrewnible structund chanFin the r9'k, ~ aJ¥lwn VfltY clearly by the hytt~ f.ucpt in this specific the deeroale ia 'ftlleaity is .-er•y fairly slipt. For a tempcratu(e rise of lOO"C, the velocity varies by a maxiuium ota fC'fl perwnt anoon as the confuainapn~~SUre is~ . obecnred in sandltoncs ._, ~ be explained by the dilremn Velocity thermal expansion ~ts betwcea the~ COilstQ,nts of the solid, wbich include clays and .,artz. ·

case.

vanalioris

~-~

3CJCIIDc

0

• 4000C

3.5

_Fit- 5.18 P and S wave ·velocities vs.

Vp

temperature in brine-saturated Boise sandstone (u_ltrasonic measurements) (after TiMUi, 1917). .. . .

J...- cr--o_o_O.....C.....a....,.

....... ~-•.s•a'

13.2& ... ~

I! .2

il

.........~~

r

t

·c

;;:

s.

'---...o.

13

•

~1.75-

,. •

I

Vs

(.)

•

I

0

&

eo

40

20

US·

10 100 l20 140-

Fia. 5.19 Variations in velocities in Berea sandstone vs. temperature. a. Dry Berea sandatone (ultrasonic measurements) (alter Mobarek, 1971). b. c. Fully saturated Berea sandstone (resonant bar) (.rter Jones, 1983). d. Partially water saturated Berea sandstone (resonant bar) (after Jones, 1983).

Temper.cu,. C0c1

.

""'-;,:.

........._

4

)1,. 3.8

i

~

3.6

J3.4 I I

w..rlllVt'atMI

~

.

1.80

··J.,.

~

•

'

it . ·'

'ii ... J

p,=4.5MPa

...............

~'c-10Mh

.:l

0

3.2

20

3

-_..

I

I

4&

(al

~~

rflna.

..... •

10

~

1.2

>

r-..

•

0

220C 0.95 cp

•

&aC'C

jl.~.I:C'

e . IMat 0.34cp 6

_Pp•O.SM,_

5

lei

120

I I

1.8

1.4

100

2.2t'

:I

J

2.0

Ju J

10

T....,.cur.t'cl . Pp"'Pc •O.S·MPa

...... ..,...

20

eo

40

(b)

A.

Ve

10

itooc· o.aq, 15

Confining prwan (M.-.1

20

20 (dl

40

60

10

100

T.....,...,. lOCI

140

~.

"-'~

'---' '---" ~

'-'-../

~

·~

'--

"·~ ·~

---

J

(~'

5

RI!SUL1i

AMI)

"-··-

""'-

~-••'··.

195

ViSCfl6ity ofs,..-, /11M lnvestiptions on velocity variations as a function of saturating fluid viscosity are conducted by varying the fluid viscosity by heatina the sample. Velocity variations in solids as a function of temperature~ ftellilible, and. as the variation in velocity of the fluid as a function oftemperatu,re is small in comparison with that orthe viscosity, raising the temperature is equivalent to observiq velocity variations with viscosity. For glycerol, for example, from - 77 to + 100"C, the viscosity decreases by ten orders of magnitude, while its compressibility only varies by a factor of 3 (Nur and Simmons. 1969~ Figures 5.20 and 5.21 provide a number ofexamples. In Barre granite or very low porosity Bedford limestone, and in the sandstone samples, a rise in velocity is observed with increasins vi$cosity for saturatins fluids sudl as pycerol and oil. In the case of water (Fig. 5.21) the velocity is inclepenc:lcet of temperature to within measurement accuracy, and the viscosity of water is virtually invariable in the interval concerned (in comparison with that of oil). The biJher the saturation of the medium with viscous liquid, the Jllore important the vitcolity Clffoct (Fip. 5.20 and 5.21), 1be slisht variation of voloc.ity in the fluid, as opposed to the high variation of velocity of saturated rock with tanperature (20 to 60%) shows that the etrect observed is clearly that of viscosity. From a tbeoretical staadpoiat, Walsh (1969) de-veloped£qs. (5.4) and (5.5), assDmins that the inclusion .CODSickred was 'riiCDIII. To .do this, he replaced the ricliaity ot the fluid phue by korf,_ wJlere «»is the anplar frequency and 'I the viscosity. He fouad tbat the c«ectivo shear mocJulus I' is liven by:

b.

1 + (CO'f/1Jlo) 2 p • l'o l'oll't + ((J)'f/tl'o)'-

·~

(5.1)

'-../

·~·

'~

"-"

while the bulk modulus K varies sliptly.ln this equation Jlo is the relaxed shear modulus (at c» :1:: 0) equal to the solid modutus, 1'1 the relaxed modulus oftbe composite, and e the aspect ratio of the iaclusion. This shows that an inclusion of viscous ftuid in this model behaves approximately like a standard (or bner) model (tee Ctlaptcl' 3). Qualitatively, increasins the temperature is equivaltnl to loweriq the fluid viscosity, and thus. reduces the rigidity of the material and its velocity. · VISCosity etrects are still noduli)' W1dentood, and recent investiptions. especially those of Nur et al. (1984) may laelp te clarify thee results. Hence, we will leave tbe subject for future discussion. c.

~

'-...-

'--

~~

'-·~·

"---'

'-" '---"

r~MM,cJi.,ge

The ftnal effect on the viscosity variation fl a phue chanae, either by vaporization or by solidiftcation. Timur{1968) observed an increue in velocity in water-saturated rock when the water fr'*(tee Fia- .5.22). Spetzler; and Andenon(l.)eblierved a wide variation in P and S wave velocities when partial meltins bepn. Experiments due to DeVilbiss (1980) and Tosaya et al. (1985) on different types of rock show a signifant increase in velocity at the water-steam transition staae (Fia. 5.22). _ · This effeet lw beea oblervtd to be stroapr in lou porous and less permeable materials (e.g. Jranite), than ill porous·end permeable materials (e.J. -sandstone). It is possible that the specifac behavior ~compact materials result from a non-uniform distribution of the of the pru. cha.nac.ln low ~rmeability media and for relatively pore pressure at the •

Bedford •irnest-

1.0

t.O

Vp

~

o.• -

t

0.7

J!.u

-· J

0.8.

Vs

c

0.8

I

' j

I

I

0

-2

2

4

•

6

Log v~ lpoiMI

(a)

10

2.5 ~---------------.., 1Qil" GivaN~

~0

I

I

G.7

I

P-6

- - - IO"Giyowol

-··- o.v-...

2.25

i

•

I

v,

>

1.5

-··--........... :: -·. a -~

-20

0

20

4

•

II

,._,.,_. Varia~ in velocity vs. ,;scosity in JIJarol amples. L Normalized. P addS wave vel.x:ities in 1ledfenl liDI!IstOtle (ultrasonic :nasun:-

••rated

··-

~~Nur,l~.

Ia. Nonuatized shear modulus in Barre J,ranite and Bedford limestone (ultra-

some measufements) (after Nur. 1980}. c. S wave ·velddty in Boile saadstoae for

1.25

(e)

2

Lot~--·

f')

,,

-80 -60

0

-2

-4

2 1.75

~

~

80

80

100

various glycerol saturations

TemperatureiOcl

~~esoaant

'"-

bar} (after Nur et al., 1984). ~

.

1<.-Riw .... 3.6

' V..,..,....oil ....

p.•JOW..

'·•10.W. u ----.

-

l

3.2

1• 3

•

II

2.8

100Sell

1

.....

f J ~J

-

•.

•

-.....:: •

...101* lr 50% oi

10

1 1CIO flO ~... (fig

-bl'

................. ell 2.4

. . ·-- ,..,.. ..

2 0

50

100

150

T......-tocl

-

2

f.5

.o

lOG

hence

Fig. 5.21 Influence of tempei-ature. (and of saturating ftuid viscosity) M P wave velocity in water- and oil-saturated sandstone (ultrasonic measurements) (after Tosaya et al., 198S).

~

-

,_

,_.-

r _y

y I

y y

'

I

r :

I

.

• 4.1

,''.:/?U.·~

I

t_P

'I

........,

i

~

!'4.8

l

6Temp./

t

.. 4.4

4.2

1

ICI

WATIR

4 ~

34

20 tl

11

I

II

0.

'

~- -1-12-18-20-24-.21

....... t"Q

,.,

1.1

WATER

SftAM

i !u 1

............... '0......, ......

I

&SIIftt ............

. ,M

0.7

Jl'

~u

0

0.2

0.4

...

..... ..,..,.. ,

0.1

0.8

1

1.2

1.4

1.8

.. lbl

f1a. 5.22 ltdhaencc ofpbase'ehaqe ohaturatinl fluid on wloclty (ultritonic measurements).

a. P wave velocity in Berea sandstone (ultrasonic measurements) (after '----' '--->

"---

Timur, 1968).

Ia. P and S wave velocities in StPeter and Berea sandstones and Westerly aranite (ultrasonic measurements) (after DeVilbiss, 1980).

198

RESULTS AND

5

MECHANIS~S

'

high-frequency waves, the assumption that the pore pressure is not uniform is reasonable. Finally, the compressional vei9Cities display virtually no variation at the time of the water/steam transition in samples which are only partially water-saturated (Tosaya eta/., 1985).

5.1.1.4 Velocities ud frequency The foregoing Chapters have shdWn that, in attenuating media, velocity is slightly dependent-on frequency: the media are said to be dispersive. We have compared a large number or results, without going into details about the frequency used. The frequencies used for the measurements do vary widely, and, for laboratory experiments, range approximately from 1 kHz to l MHz.ln Fig. 5.23, we show both ultrasonic and resonant bar measurements in similar samples. Ia this ftgure, it is not possible to compare directly the two compressional velocities v,. . d VE. These; velocities correspond to different boundary conditions but, knowina the Poisson's ratio, are related by Eq. (3.168). For an average Poisson's ratio (0.2-0.3), J'£ is smaller than v,. of 5 to 15%. in low to moderately attenyatin& samples (Sierra White aranitt an4 Fort \ltticMt ~), variations of l 0 to 20% betwcon ultrasonic and resonant bar measurements are oblerved. These variations can be explained partly using a Constan&Q model (see Fig. 5.24 for granite) and partly because ultrasonic: measurements have been made under a small uniaxial stress. For the highly attenuating Massillon sandstone, the difference between ultrasonic and resonant bar measurements is important and much bigger than the one we would get· from a Constant Q model (see Fig. 5.24)even if 10 to 20%4flhis difference is taken into account by the effect of u•xial stress. A Zener model as discussed previously may reconcile ultrasonic and resonaJl! bar ~Cifs. This reconciliation problem is an experimental example of bow diffiCUlt it is to measure a reliable velocity dispersion on a . wide frequency ranae. ... Finally, if a high frequency is used, effects~& to scatterin& are observed (Fig. 5.25, Winkler, 1983). Velocity then decreases with rising frequeac)C. and this is an extrinsic: effect. The higher frequencies .~ p~ and llence travel a longer path. The velocity calculated on the basis of a dircw;t p,tdljs thus artiftcially decreased. By contrast, the lowest frequencies are only slightly scattered, if at all, and their apparent velocity is the true one. In conclusion, velocities vary rather slightly with frequency ifattenuation is not high. We shall show, on the other hand, that variations of attenuation as a function or frequency may be significantly greater.

I I

-

I

I I

Ii

5.1.1.5 Velocities and strains The resulting strain from· a passillg \\'ave f. dependent on ttae amplitude of the wave. Excessive strain may involve sevetalmechanisms liable to alter the elastic properties of the materials analyzed. Figure S.26 shows the effeCt of,Jttain ampti4Kie on velocity. When this amplitude exceeds a valueofapprQximatelylO"'•, tb,e-velocitydecreases, but only slightly. Moreover, dry material is obsen'~ to display a higher threshold than partially saturated material, and the dependence ofvelocity on deformation above the threshold is areater in partially saturated materials. Pressure also has a similar effect. The higher the pressure, the higher the ,threshold. and the lesser this strain amplitude dependence. Mavko (1979)

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--, Fig. 5.14 Influence of frequency on velocities as a function of saturation. Baaed on the low-frequency velocity ....ta (resonant ber), the ultrasonic veiQc:iaies are computed with a Constant Q DiCrdel Note that Sierra White granite appears to display Constant Q behavior contrary to Berea sandstone. In Berea sandstone, water ~turation is not computed, and only drying time is given (after De Vtlbiss. 1910). · ~.

_.......

4 . . . .11011....._

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1000

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Fit- 5.15

lntl~

of frequency on oomprcuional velocity in dry and

sabN'ated Massillon sandstone (after Winkler. C 1983 AGU).

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.--.... ~

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202

llESULTS AND MECHANISMS

5

showed theoretically that the 8{ain-to-sfain friction proc;ess was a threshold mechanism involving a non-linear relationship between velocity and straia nu- will be discussed further in Section 5.1.2.5. To conclude, Winkler's calculations (personal communication) showed that the strain amplitude of standard' seismic sources was less .than to-• in far f~eld for any source employed (artiftcial or natur~and hence for the range of uses in geophysics, velocity can be considered to be independent of strain.

II

~

I I I

5.1.1.6 Slmunary We have shown the following properties concernin& variations in velocity as a function of physical paramet~rs: • Velocity increases with coafiDia& ·1'ft'SSU£e aftd differential pressure. The pressure parameter is the ditTerential or efl'ective pressure, which is equal to the difference between the conftning pi'CS$ure and the pore pl§SIIre. Grain contacts and cracks play an essential role in this increase; hence, for crack-free rocks, velocity is virtually independent of pressure. • Velocity depends sliahtly on water saturation for two-phase water/air mixtures and water saturations that are neither too IQW ( < 10%)nor too·biab (> 90%). For dry and 100% saturated samples, velocity increases signiftcantly. • The viscosity of the saturatina fhttd is a parameter that strongly influences velocity. Velocity increases with viscosity. • The role of temperature is slight. An increase of lOO"C in temperature causes a decrease in velocity of only a few per cent. • Phase changes of the fluid in a porous medium cause the velocity to vary substantially. • Velocity depends on frequency ils.JD~liis meditiln; and thus increases slightly with it. • For the low strain usplitudes used in seismic experiment$, velocity is independent of strain.

5.1.2 Attenuations: results We have deliberately decided to present the resultS of attenuation measurements here without any explanation oftbemecbaains involved, other than descriptive comments. In fact, if the behavior observed for velocities could be explained, either using models or qualitatively, it is because the velocities have been,studied and measured for a very long time. By contrast, attenuations are more ditTtcult to measure. They are far more variable and the few mechanisms involved ar~ mostly slibject to consid~ble debate. We shall therefore present the te5ults first, and then~ in Sectipn $.1.3, )Ve 'ShaH try to understand what causes tbis attenuation by interp.-etation _of the data;

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I

l f

I I I

i'

I j I

I

I I I I

~

'-../

~r---------------------------~

.'----'

Gqnfte

.,;_ ·pv

t'~·

1- ·':

-~r---------------------------~ i~"'t2

..............

'-· 20

~

Q

10

40

.30 \.....-.

20

.., .

Confitllne .....,., CMPel .

:fr·;; f •

w~------~------_.------~--------J 0 tiG wo ttiG 200 Conflllllle ,._. .......

fbi

~

•• ~1

Colorldo oil ......

-~

0

1... :' !::::-

15

~~

tO

I-~

I

I " J..

5 0~~----~------~------~~-----J 50 .. 110 aoo Confinilll~twa)

lei

f'il- 5.f7 Influence of confmins pr~ure ~n quality factor. \ .. Pbue dift'erei!OI ..........~ qat, ,..am~ (after .GOrcloo aa4 .Davis.

C 1968 AGU). .. Berea sandstone and c. Colorado' shale (ultrasoaic ..-uremcnts). A clistinetioll is tnade .lle:NIIiin wa• plltdel alld ~ to- tile baldiJt8 (der .Jota&oa adlolcako e,lM ACM.ij.

---..s

lldfonll.__

80 110

a,.. '---

!10

tOO

200

Olfferenliat .,._.IMI'al.

FiJ. 5.28

Influence of pressure on quality factor for Bedford limestone (ultrasonic measurements). Note the sipiftcant hysteresis, probably due to the creation of microcracks (after Johnston and Tolts<>z. C 1980 AGU).

-~

204

5

RESULTS AND MECHANISMS

5.1.2.1

Attenuations and

pre~&Ures

As in the case of velocities. the essential parameter controlling attenuation is the eft'ective pressure. A decrease in attenuation is normally observed with increasing pressure for P and S waves (Fig. 5.27). One exception is that of the Colorado shales, in which attenuation is virtually independent of pressure. Considerable anisotropy is nevertheless observed. Also, as for velocity, the be!havior of attenuation as a function of pressure depends greatly on the presence of microcracks. Figure 5.28 iUustrates this problem. In this experiment, the Bedford limestone sample undergoes deterioration of its pore network by the creation of new microcracks. Tile hysteresis curve shows that attenuation is greater in the presence of microcracks, and when they are able to open. This very important role played by microcracks is demonstrated in Fig. 5.29, in which the two Fontainebleau 150 120

~ p

~ PAV= 6%

5001&HI

3001-

~

00'1 =0

90

1p ·1110

60

30

l

..

~··

..

.,~1'1-=l 40

60

80

100

W1ter satur1tion (%)

Fit- 5.29

p

120

.

,~.

~=3.8%

cr 1= 0 cr = 5 MPI 1

0

o -o-, -io

.,.J .....~.. -.....

60

0

20

0...~ ~.4>

240

ecr1 = 5MPa

z 220

X

Qi

500kHz

o L--'--~--'--~ 20

40

60

80

100

W1ter satul'ltlon (%I

Influence or microcticts on attenuation in Fontainebleau

sandstone (same samples as m: Fit- S.9Hafter Bourbie ~nd Ziftii'Zilef, 198S).

sandstones of Fi&- 5~.are aulyzocl feN- atteauaU. hy tlw ultrasoaic •thod. As for the velocities, it can be observfd that the lllDChtollewitlaout pain contalctsil unaffected by the uniaxial stress, and that its attenuation is very slight By contrast the sandstone with grain contacts and very high attenuation is very sensitive to the uniaxial stress.

I'

I I

5.1.2.2 Attenuations aad saturaftoat a.

Wflter/.U sat,.atiOII

Figure 5.30 gives two examples of variations in attenuation as a function of saturation. The attenuation of P and S waves is slight fordryrdeka:~efootnote( 2 1 in Section 5.1.1.2a), but high for partially saturated rocks. Attenuation in sat~ted rocks is intermediate, and the quality factor Q5 always appear to be lower dlan Q,.-' 100% saturation. Figure 5.31 shows various types ofvariations in attcnuatil'taas a function of saturation. These measurements were taken around 1 kHz by the resonant bar technique. For granite, it may be observed that S wave attenuation increases steadily with water saturation.

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1979 AGU).

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lff20

30

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..

70r-------~--------------------~

eo ,..,., sandnDM

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Dry Q

25

•

76

40

tOO

Confilliftl , _ . . (MPal

30

.. Navajo sandstone (ultrasoll\£

Nllilliolllldnone W..UIIntld

measurements) (after Johastoti ·

aad TobOz.

C 1980 AGU).

so

100 150 Dltt.rantial , _ . .....

·--"------"

Fie- 5.30 pressure.

InRuenc:e of saturation on attenuation as a function of effective

20

206

5

RESULTS AND MECHANISMS

~ r-----------------------------~ • - 18.1"

Fontainebleau sandstone (after JouriMe and Ziasmer. 1985).

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320

tt:'-

240

180

r

(1.5 ·3kHz)

~1•

~·-·

_•..._ •....__.••./ •J FontliMbleeu~

80

0.

f I

I

I

.,

«»

0

I

• I

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e

eo

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watar uturation C1' I

100 Sierra Wl'ltte granite

.,

._ Sierra White granite (after Murphy, 1982).

60'

1¥ 40

20

0 0

20

40

80

80

100

·.Watar......,..~)

10

r---------------------~----------, Vyoor porous gl•

e. Vycor porous glass (after Winkler and Nur, 1982).

lfle 0 0

Fit- 5.31 Influence measurements.

of

saturation

20

on

40 60 Water taturation I" I

attenuation.

Resonant

80

100

bar

-------,--------------------------------------------------------------------

----

'-·

5

:!OT

RI!SUL'TS AND M!CHAMISMS

.........._

'-.

20

o....... ....._, 0 c:-.lno ..,._:

10

~

15

I

~ 10 0

I

·-----., 2 3 .--.... volatl..

4

,,.,.. 5.32 loRuence of slight traces of liquid on 1.5

w.w --...c,.ttto..., •........, Cal

b.

attenuation (re10out ball.

a. (after Clark et al. C 1980 AGU). k_ (after Tittmann et al., published by AGU, 19~).

J'ny low stltlll'lllitM: "tlry" rMics (.,.,. ,.,.._ory collllltioiU) Attenuation is very ~itive to.tbe.presence of~aces ofliquh~ ia the porous medium.

Figure 5.32 provides an iHustration ·ofthis. AttenUation rises with the addition of liquid. and increase$ accordina to the type 0( sample. Fiaure 5.32b shows t~ this increase depellds on the types of fluid present in the porous medium, Fluids consisting of polar moleeules inc:rease attenuation more substantially than other fluids.

c.

s.tw.n. tee~

Several techniques are available to obtain a Jiven saturation. For a water lair mixture, for t~xample, the saturated sample can be dried, eeetrifuaed, or depi'CSiurized to introduce the liquid. All these techniques yield a different distribution of the two ftuids in the porous medium (sec Fig. S.33bJ. Attenuation measui'CIDWltS depend heavily on the saturation tecbniques (see FiJ. 5.lla), Ud the attenuation peat shifts or disappears dependin1on the saturatien technique employed~ · 5.1.2.3 Atteauadoas aad temperatures •

a. ,,...,_.e Fipres 5.34 and· S.35 show a decreaic..il} attCJWation with risinl temperature. This decrease is much pnter than that of velt'Jtify,. sift<:e attenuatioo varies by a fKtor of S between 20 aftd 1OOOC (Fig. 5.36}. Tbe eti'Kt Of temperature appears to be less significant at . partial saturations. b.

J'isctnity

Chanaes in viscosity of the saturatina ftuid are pnerally obtained by chanses in temperature. Fiaures 5.37 to 5.39 summarize the results obtained. althoush the dependence of attenuation on viscosity may be hiJhly complicated in detail (Fig. S.~9a);

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CENTRIFUGE

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DEP.RESSURI%ATION

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~

200

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300

Zl

..:

lt. Redrawn SEM photographs. Fluid distributions in the pore space are displayed by epoxy injection. Air bubbles are shown in black.

· a. Ultrasonic measurements.

F'IJ. 5.33 Inftuence or saturation technique on atte~uationjsaturation relationship (after Bourbie · and Zinszner, 1984).

v=:

90

CENTRIFUGE

............... "',

70

DRYING

10

~

ll!

J>\i J...

Water-ion"'l

70

DEPRESSURIZATION

I!!

90

100

'\.....~

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'~ '-

40

50

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·i 40

Pc"'10MPII

"

•

•

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t-

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40

6 E e S

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lf!l30

~ Ttmf*ature /

E t

s 'r..,..,._'

20

'-

10

10

100

120

t40

.t......... tocl

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FiR- 5.36 lnftuem:e of temperature cydcl on attenuation (nsonant partially saturatlkf 8era samlstcme (after Jones. 1983).

AI

...... ,,.,

confming pressures, p, • 4.5 MPa and p~ - 10 MPa (alter Jones, 1983).

50

.,

I

100

lnftucnc:e of temperature on attenuation (resouant bar) in a water-saturated Berea sandstone (constant pore and

'---

40

I

Fia- 5.35.

of temperature on attenuation (raonant bar) in a water-saturated Berea sandstone. Pore pressure is O.S MPa (after Jones, 1983).

20

I

_,.in

a

I

120

~.

--..

11

0.5

I~ o -4

-2

4

2

0

6

8

s

l

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j

p

I 10

• ...;4

2

0

-2

Viscosity

Vilc:osity (Poise ·log scale)

4

8

6

TO

!Poise- Log 1C81el

Fig. 5.37 Influence ofsaturating Ruid visclosity on attenuation (Barre granite), The attenuations have been normalized to the atteauadon of the dry sample (ultrasonic measurements) (after Nur and Simmons, 1969).

$

'

Fi&- 5.38 Influence

of saturating fluid viscosity on attenuation. Bedford limestone (ultrasonic measurements) (after Nur. C 1971 AGU).

100

--.

'

1000

-a 10

-4

-2

•

0 4 2 8 Viscosity !Poise • IOf*:elel

10.

140 ~==30%

120

'

2-4kHz Glycwot 11turation 80%

.-100

ipso Fig. 5.39 Influence of saturating fluid viscosity (temperature) on attenuation in a partially glycerol-saturated Boise sandstone (JCIOnant bar) (after Nur et al., 1984).

'In 40

.,

20

. o' , , -80

,

, -20

1

,

,

20

,

60

,

, 100

Temperlture (°CI

.,

"

'

~-----,--------------------------------~--------------------------~----------·--------

~T I mut~-~,,~~

~

I

J2

I

p

P

-"-v..ltlon

1-

11I' z

2tl

V..-.laoilund

0.11-

1.

1

.............,......

4

~I

...._s

j

13

1.

l

!1 J

Pc•10MPe Temp.110·c

2

..r I . . • ........ ..........

J

ASelnt ......

lrine awreM

I

llu

0.4

J,fl I

I

I 0

0.2

0.4

0.8

0.1

1

1.2

1.4

1.8

Pore..-..IMI'al

ol I' 0

1•1

, __

I

I

I

1

2

3

4

Pen,_. IMPel

5

8

lbl

Fit- !AO lnOuepce of phase ~hange of a saturating fhrid (waterfsteam) (ultrasonic measurements). L SandstOPCS and granite (after De Vilbiss. 1980). ._ Venezuela 011 ~d (after Tosoya et til., l98S).

'--

GO

attaauation

The existellCe of one or more attenuation peaks appears to bedeilonstrated clearly by the results given. These peaks are intrinsic to the liquid itself. c.

Pluu~ clttllw~

As already discussed. temperature can cause phase changes within the porous medium. As in the case of velocities, changes in attenuation as a function. of phase ~F of the

saturating ftuici are sigaif~nL Fipre SAO ofers two examples. A wicle. variation in attenuation is observed at tbe vapor/liquid transition. These wide variations are only observed on samples that are totally saturated with liquid. For partially water-saturated samples, these \-ariations are not observed (Tosaya et al., 1985).

5.1.1.4 AtteautltldM _. ........,. We have shown that velocities depended sliptly on frequency over a limited frequency range. On the other hand, attenuation may vary considerably (see Fip. 5.41 to 5.45). Measurements taken by Gordon and Davis (1968) reveal a slight variation of attenuation in granites and quartzites at frequencies ranging from 0.6 to 50 mHz. Quality factor peaks

J

212

5

ll£St;lTS Al\10 MECIJAI'ISMS

•

Gr.,ite +·GIY<*ine

~

30>oG•.,;N 6

Quartzite

~20

~a

0

•

10

0

~

0

e-

..,.

QD

tb

COil

,_

10

0.1

100

Frequency (mHzl

Fia. 5.41 Frequency depeJKtence of attenuation. Granite and quartzite (after Gordon and Davis.
I

9 Dry a.r.. sandstone

[

8~

7t-

E

.. -- 4~~1 s

50

I

Wet Berea sandstone

~

""' -

6

30

o-: ~ 3J

5

tp· '

:

;J"O%

4

-o., s

~ 2ti'

E 20

3

~·-

lPc' · =0..5.MPa · aS Pc., 10 MPa

2 ..,. •• E S

I 0 . 0

oE I

I

. . I

2

I

I

4

I

I

6

I

I

s

I

10

=t • '

5

0

Pp tip ..• ·a5

J .·

•

tip ··o

.

Gh

0

Frequency (kHzl

Fi&~

1tc • 10 Wa

5.41 Frequency dependence of a~n. sandstone (after Winkler and Nur, 1982).

2

4

8

8

Frequency (kHz)

OfJ; and saturated Berea

"

r ~

125

·.;.

100

.oe

t..

"-') )\

Mlllltlon •ldltone

• E

• s

75

I

1¥ 50

'---

':t:~

25

1--:: 0 O.Q2

1

0.1 0.2

2

20 ~~---'---'---'-=,__ 0

10

f'*IUifiCY {kHz) L

11

tp

Massillon sandstone with different laturatioas.

I

-

-- -·

12

8 4

,.20 r[-~w--.;_100-,.----_,.;...,J b. Porous glass (Vycor) with different saturations.

11

/-...., s

12

//~ F /• E

8

FiJ. 5.43 Frequency dependence of attenuation (after Murphy, 1982).

'-

-~

4

0

I

t

3

~·

Vyoor,poraul .... I

f

5

7

I

'

fr~Q~MMY

I

I

I

11

13

15

CkHzl

'--

50 WinptiiWldiUIM lletlltiUd

40

•-7kHz • -200kHz

10

~20

'--·

10 0 0

10

20 30 40 50 . 5tlllc:IM ...........

eo

10

Y .. 5..u Frequency dependence of attenuation as a function of e8'ectivc pressure in a saturated Wingate sandstone (after Tittlnann et al. C 1981 AGU).

...

214

RESULTS AND

"l

---

70

5

MEC'HA~"ISMS

I •

~

I

50

,.,.,.,.//\! .--~: '""m~• ...•..• ...... ..• ... .0.... T::-· l ......

~]

_~

n

3D

I

20

-

~·

• •, - . .

I

I

I

eo

40

10

I

•

10

100

20

eo

40

w- uturMien I")

10

100

w..., _.,., 1%1

F'JI. 5.45 Frequency dependence of attenuation as a function of saturation. Massillon sandstone (after Murphy, 1982).

125 MaliliO'I undstone

1()0

---Drv _:

75

~ 50

/

.,

..,"

25

,., ,/""

""

0

0

0.5

1 1.5 F,..-y lltHI)

2

2.5

Fit- 5.46 Frequency depeD4caae ofatteauation. Scattering effect, Massillon, sandstone (after Winkler. C 1983 AGU). ----,

r".

....______

-,-

----~-----~~~~-~-~--~~~~-~---~~-~~~~~-----~~---

r '--'-

---

~

~

"----

1 $

'I )1

~ fl I I

....:::::::.

215

arc ob&crvcd for pal1ially and fully aturatcd ~ at t'Nqucndcs in tbe

'---

·~

...a.u AMD MICIIANIDIS

5

neilkborboocl of S to 10 kHz (appiOXialately the frcqucacies employed ill acoustic Vyoer porous Jlass also cliapUn--beha¥icrr ·timiJat to s*dstones, twnely atteauation is inllepeftdent of freqUIIM)' for the a., sample. and biPIY dependent on · frequency at around 10 kHz with increasing sample saturation. Figure 5.45 shows that the position of the attenuation peak as a function of saturation varies with frequency. Figure 5.44 shows that the difference between _ attenuations at two given frequencies louin~).

decreases with rising effective press~ At sufficiently high frequencies (in comparison with grain sizeL scattering effects are observed. as in the case of velocities (Jee Fla. 5.46), and scatterint induces an increase in attenuation with rising frequency at frequendes ......- daan 1 MHz. The behavior observed corresponds fairly closely to a scattering-type mechanism (~ion 3.3.3.5, Sayers, 1981, Winkler, 1983). This mechanism OCICUrs only if the transmitted wavelenith is comparable to the arain siD (Devaney et al.. 1982). Tbc _<:entral frequency of the scattering proc:ess is given by: -L

~

-~

fD

3(M)! P.

=R

1

(5.8)

where

R • p-ain radius, P. = density of the material, M = elastic modulus of the wave, and, in the case of sqdstoneS. the fttquencyf.o is around 3 MHz. 5.1~5

AttelmatioM _. ltnias

As we ba"' seen, ~ocities arc independent o(sttam for the strain range observed iR ppbysics ( < 10- 6 ). Attenuation bebaves silaUrly (Fip. 5.47 to 5.49): "---'---

(a) For deformations lower than a threshold '• of about 10- 6 • Q does not vary with strain. (b) The threshold e. increucs with risiaa prcssu~e and decreases with inacuina water saturation. (c) For e peater than e., attenuation decreases with risiag pressure and increucs with incrcasinJ water saturation. The dependence of attenuation on strain resultia,a from adtreshold.e«eot, namely gftinto-&rain friction (Mavko, 1979), appears to be Jlelligible at seismic strains.

'---

Variations in attenuation as a function of the ,various above can be llimmarized as f'oii
'--

'-._.-'

.'--

Physical paraaeters considered

• AtteauatiOa decreases db risina efteetive pressure, sinc:e the effects of pore pressure and conf'taina pressure are not inetependen( The p n ·oontacts and cracks play an important role, and their presence causes su\Jitu~ attenuation and considerable dependence of attenuation on pressure.

~

100

10

't::=:: :~·: ·. =:. a-~

~ ~::: 1

:

-~

:

:

:

~~

1

f .

FiJ. 5.47 lnflueru:e of strain amptitude on attenuation of different rocks and crystals (lonpmdinal ..CUation at 90 kHz) (lfter Oordoa and Davis. C 1961 AGU).

.

' ~\

"

j

0.1 ~

~

Quartz single crySUf

0.01

to-&

to-7

to-"

,~

...............·,.-s

10-8

..,-3

.----._,

10 • - llndstone

Pc•1MPa

8

~ Fig. 5.48 Influence of straia am• plitude on attenuation. Berea sandstone under different confining and pore pressures (resonant bar) (after Winkler et al. Cl 1979 Macmillan Journals Ltd~

8

0

-o

..... 0.

_ _ _ _ _...,;,.,...,..-:.------• Pc • 5 MPa

4

...,. Strain amp! itl,lde

50

t e M...mon Sw • II S- 70Hz 6 M-illon Sw = 0 S - 1 kHz

40

~o

OttfWII

'~w

•

o" lltolt tem

1~30

Fie. 5.49 Influence of strain am20 10

r.

-·

o~

~~ i

•

~ <,.-oo"''

__________._________~._--------L---------~ -7

-8

·-s

plitude on attenuation for different saturatiQlls (t~nion pendulwn~ Massillon sandstone (after Nur and Murphy, edited by J,ru.lixl and Hsieb. 1981) and Ottawa sand (alter Stott, 1979).

__

~,

-4

Strain .mplltude (tOil tettel ~

~

~

"'

~"-'

rj I:;

5

"

• - - dqlalds coasidorably
I - '

J

'~

·~

'--

...,.,.,~ND'~Nars

217

_,oil

or of P*r liquid· ia 1tie pomus medium atteouatioa Spifaatly, and the atteauetioa of *Y rooks is low. The quality factor Or tcema atwa,s than Qs at l 00% saturation. Attenaatioa also depends on the distribution of air bubbles (or of aonwettins Ruici~ within the porous medium. • Rising temperature detrcascs attenuation signiftcantly. • Attenuation is hilblY sensitive· 'to the viSCQSity of the saturatins fluid. One or more attenuation peaks are observed. • Phase cbanps of the fluid cause substantial cbanses in attenuation. • The quality factor depends stronJly on frequency at frequencies dole to tO kHz, and at· f~ ptator tbu l MHz. 1'hc i,avene of the ~uality factor 4isplaJS a peak arowad lO kHz. • Ia lliaic aptdmeuts, the quality factor is illdependeat of strain amplitude. • Figure S.SO SUlbnlari7.a some l:.l these resUlts for a ••tjpical* sandstone, Massillon

.-ter

sandstone.

-

f

'---'

'--

'--

..... 5;11 . . . .tic reprt~eotation or aaturatiob (after Murp~ty, 1982).

actcmuation vs. frequency and · ·

5.1.3 A.__:_.lllisms We aucaeclod in aeasuria& the attenuation)IIII'WIDOtet' eheracterizinathe loss of energy a ,.._ wave duriJtl ita. propqlltioa du'oulla a &iven materiaL We thus obsened that this pa,....r, jult like velocity, varied: cxmsiderably and depoDded strODIIy Oft lecll ~ tetaplntUe ·udnother physical dfccts. It remains to undentaJHlho9i thiS loss ofeaeraya:cars. How does tMinteraction between the wave and underpM by

·~

218

5

RISVLTS AND MECHANISMS

the porous medium dilsipate enersy tlarough a· Joule dfect? What is alae souroe of the intrinsicaUenuatioa?Which mechaliismsarc N8pOnSiblc under the coaditions that are of interest to teismic exploration? UDJite velocity, attenuation il still.poorty uaderstood. and many mochanisms that we dcscftbc in attempting 10 answer diose qwatioos are merely hypotheses. This. area ofrodc physics is: jn a staa of constant cbansc. We will ftrst examine the role ofintergranularf'rietion. which was tbefocusofimportant work in the 1970s (Walsh, 19(4 Jobnston et al., 1979). We lhall then exlnsidcr ~everal linear mechanisms, distinguishing those that apply to .. dry" rocks (in laboratory conditions) and those applicable to partially or fUlly saturated rocks.

5.1.3.1 lnterpuutar f~ Recent ·experimental measurements (Wiftkler et Jil., 1979, Wialcler and Nur, 1982, Stewart et al., 1983, Murphy, 1982 b, Stoll, 1979) were made at strains ranghlJfrom to-• to to-•, and revealed the iadoponcloace of velomy aiJd ~tioa .............. from strain amplitude, if the ta~r is less than 10- 6 (fjas 5.26 agd 5.47 to 5.49). Moreover, we have mentioned that Mavko (1979) shows that intergranular friction is a .. non-linear mechanism with a threshold effect. This mechanism is triggered only if the deformation amplitude is higher than a given value. Finally, the Winkler and Nur calculations (1982) also served to show that the strain amplitude in seismic experiments was always less than 10- 6 • Hence all these .facts suggest that intergranular friction is not an important phenomenon in the attenuation of seismic waves. Yet Figs. 5.26and 5.47 to 5.49 bave-lbotbown.thejmportancc=ofthe presence of water, even in very small-proportions, in altering both the threshol$1 and the amplitude of variation in mechanical properties as a fUnction of strain. this implies a sort of lubrication achieved by the ftrst layers of water 'between the dift'ercnt ·arams.

5.1.3.2 Att----'••••mws

f ~dry" ... very d&fatly tU!h ..,. rocks mq..-fN

The term ••dry" rock li..a..mat dndear. The fOGies not totally free of moisture. In fact, they ate dry~ under laboratory ~ns (atmospheric pressure and a given humidity) and not rMJP~en f~~~ oven. The degree of water saturation of a ••dry" rock hence UCS:~efil-"and a maximum of 0.5%. Typical experimental results obtained on a Massilli!llandstone with 0 to 10% saturation are given in Fig. 5.51. The very low saturation valUQS are o~~ by ~lo.Wing equilibrium to ~ur between the sample and an atmosphere with a given relative humiditf.. ~ mc:Uurements of velocity (Fig. 5.12) and attenuation were taken by Murphy (1982) using a resonant bar. The measurement frequencies are the resonance frequencies of the bar and are therefore nV/2L(n- orderofresonance,L- bar length, V""' volqe,it,at~Jiequcncy);Tbe variations in measurement frequency shown in Fig. 5.51 (namely 599/997 Hz and 385/653 Hz) result from the wide variation in wlocity from 0 to 1 "• sablratiott (see Fig. S.12). By exlntrast, the resonanc:e,fnlqueQcy for water satimttio111 of I to 10'/o is virtually constant; For saturations of 1.5 to 10%; attenuation is constant, whereas for saturations less thaa 1.5% the variation in attenua:Pon is sipificant. Attenuation rises from 60to 70% for saturation varyinl'from6%to sligbtly-leDthaa 1%. A peat appears at

~"

/"

'-..---'

'---'

~I

-...

5

~

2-19

lOr----------------------------------,

''-'

...

-......--.;.--

10

-,-; ~

''-

50

'-'

E (ti88 • 917 Hz)

lf940 3D

t

'-

20

'toL---_.~--~~--~~--~~--~

0

2

4

•

8

,10

.... llllntioft ~) '-

n.. 5.51

Extensional aad sbeat attetauatioos (1000/Q) iD Musillon sandstont vs. saturation (0 to tG-!.)(after Murphy, 1982). '

'-'

-\_./

1%. 1bis peat pollibly results from the ,,...._..,. depeDdeace of 1000/Q. ·If ibe were .taken at coastaat lro.qucac:,, tllc _pe¥ would probably be absent. However, tbia explaaation requires ~ ~af~PUtioa. We shall now cumioc a munber of pouible mcd1anisms offering aa qualitative explaDatioD of attaluatiop behavior at low ~n. ~

.

DUJoeiJtitM ill parlt Mason n crt (1910) SlltJest that~ oftbe titieatc surface are set in motion by tbeacoastitwaw. This is a rather debatable~ sbicetbe leftldl ofdislocatiobs in quartz and the forc::es rcsistina the dislocatiou involved are such that tbe corresponding resoaancc &equencics are mudl too high to be excited. L

'--

'-

....._,

'---

'-' '-

'--

c,.,._ ,_.

lt. lluMIIP sf lu shown above for velockjes, the porous surface all suds and sandstones is bydroxylated. Tittmann et al. (1980)and SpeDCOf{l981) augcstcd that part of the energy is diuipated by the bnakagc of hydrolia ~.-.eca tbe . awfagc'h)'droxyls and the water moleculc:s, thus subJtantially doc:rouiag the fRJC surface encrJY. This attenuation model is directly depeadent on the specific surface aQa of the porous medium: the greater the specific area, the Val~ of,apeciftc area and attenuation in Vycor (Fig. S.43) and in Massillon sandstone are as~ follows:

or

Ji*•* . ......,

"-----'

~ate&

Attenuation

(tr/Jf

"--~

'"-----' ·~

'-../

~

Vyoor ............................ . Mauilloa~ ........-....... .

a :ZOO •

10

1000/Q ~

2

;; 10 to 25

1,. 220

5

RI!SULD ANI) MECHANISMS

~.

Their incompatibility is easily ob$ervable assuming an attcnuatio~tmecbanism exclusively at the surface. Hence, to explain Che results iJl Fig. 5.51 exclusively by means of the breakage of hydrogen bonds is inad~uate. We shall discuss in the next Section one possibility of explaining the incompatibility observed between speciftc surface area and attenuation.

C11piOtuy forc~s We have shown for velocities that in the case of low saturations capillary forces may play an important role. For attenuation at very low water saturations, part of the energy dissipated may also result from capillary mechanisms. The mec:banism involved is again associated with the specifK: lPlfaa: area of the porous medium, and the Vycor/sandstone problem (see previous paragr.) again appears to arise. In fact, the v•lue of attenuation due to capillary forces depends considerably on the microstructue of the porous medium (surface roughness, aspect ratio and size of capillary tubes) and the specif1c area per se plays a minor role. . Attenuation is hence the combination ofmicrocapillary hysteresis (viscous dissipation on rough surfaces combined with the breakaac of ch~ bonds) and of mechanisms linked to the motion of Ouids (-' saturations pater than 1 to 2%).

c.

~.

--"

~.

~

5.1.3.3 . Atteaaatlea ~~ ._ partially 01' f.,Uy .._.... roc:b Figure 5.52 shows the variation in attenuatiOn at water saturations ranainJ from 10 to 100%, and concerning the same MassiHoti sail'istone'as in F~ S.St.ne variation -in extensional measurement frequency results here from a sharp rise in velodty(see Fig. 5.12) at saturations approaching 100%. The lfteasurement f'requeftcy is hence mually Constant for all saturations in shear mode, and for saturations ranging from 10 to 95% in extensional mode. Fipre 5.52 re~ dlat at~. is coD$idcrabJy affectod bJ water saturation. ExtcDiional atteauation ·readies a . . . .\Uil around ~% Aturation, wbile abear

~

-----..

70

Miitalllon sandstone

•·2a

80

~.

60

!.!2• "

30 20 10 20

40

60

80

100

Wdlttltiii'Mion~l

Fla. 5.Sl Exten•ional and sbtar attenuations (1000/Q) in Maalilloo sandstone va. saturation (10 to 100%) (after Murphy,1982).

"

., "

....,

5

'-../

----,:1

~

'--

_,

:t

ld!IVlti'

"* tid!a~A. . .

121

attenuation shows a maximum at 100% saturatiqn. Moreover, extensional attenuation

(Qi 1 ) is always greater than shear attenuation (Qi 1 ) except at very JUab saturation ( > 97% ~ Also observable (Fia. s;43) is a wide variation of Qi 1 and Qi 1 as a function of frequency, with a peak at around 5. kHz. Finally, as we have shown (Fig. 5.4S), the attenuation peak as a function of saturation varies in position and amplitude with frequency. These properties (peat at 5 kHz, satur~on, depebdence) arc approximately independent of the type of rock analyied, or rather, the dcpendetl4:C on the type of rock is very slight in compari~n with cQnaes in porosity and permeability (see Table 5.1). We shall review tbct.111eehanisms that arc m01fcommonly used today to explain these experimental L

res~tlts.

DUI«tditHH ;, ,....,, As discussed above in Section 5.l.3.2a,: tbis mechanism appears to be inadequate.

i!t

b. llrHk"6e of elwlltkld 6ollb . At high saturations, this medllnism, rolateclto the; spccif'tc surface area of the porous

medium, is undoubtedly nealiJiblc. Moreover, tlle frequency ~ce of attenuation at around 1 kHz appean to be cHfBcUtt to ezplaip by an ioctelse in the number of broken chemic::al bonds, as these bonds raoJtate only at Vt:r'J biab {Nquenc:y. The attenuation peak at hiJh saturation also CiUillOt be ftplaioed ·by this mechanism. The brcaka1e of chemical bonds is therefore unacceptable as an attenuation mechanism at high saturation.

·--

c. Cqilltlry forces . At low saturations, the iPfluaiC:e 'O(t;:apillary:forccs was due to a sort of microcapillary

hysteresis. At high saturatio• tlia mechanism is heace <*taialy less active, and cannot· possibly explain 'the attenuation pt8k observed (see FiB- S.S2). '

d.

Tlln11t0-rela..,

An acoustic wave propaptins in a pven me4ium acts by ~poling a sudden change in stress on the medium coDCel'Did.. Bccause it is a rapid pt~ wave propaption may be considered a process that is controlled macr~y by t~ adiabatic properties of the material. Microsc:opically~the ~ is veryheteroaaneeu.. siDce each grain and each pore with its Ouid respollds adiabatically in accordanco with its own thermomechanical , properties. The rapid strain variations due. to tbc wave front impose temperature variations throup thermomecbanical coupliq. These temperature variations are heteroaeneous due to the heteroaeneity of the microscopic constituants of the porous medium. Macroscopically, tcnaperatW'! equilibrium' is obtained through thermal conduction. A tllmnal relaxaticlll.dtus takes'))lace an4 iavolves a phase shift between macroscopic stress and . strain. &foreovcr, if two phales of the same component (waterjsteam)~prelent,masatiaasks(vapoj!izatio8/conclensation)takeplacefromone phase to the other to auaraaU~Q tlermodyuamic equilibrium. This mechanism was analyzed theoNtjcally by kjart.,.,.... (1919 b); He prtdkted a dcpendctl4:C of attenuation on saturation and on freQuency. The cenn.t i.laxatfon frequeacy is pven by:

''~

'--"'

hl

h-~

'~-

..__._

....

~~

fg

TAIIU:

5.1

FRF.QIJF.N('Y Df.I'P.NUf.N('E 01' ATI'IINIJATION

Per meP()tosit' ' ability (%}

Type

of sample

(md)

Maximum attenuation

....

lnvcstljatcd

rrequency

freq~ncy

"

Bere8 saadsto.ie .......

20

617

2kHz

Berea ....dstone .· .. ' ...

20

tooo

· 2 k,Jfz

Navajo sandstone .....

ll

;;l:O.S kHz

·u

;;. O.$.kHz '().017 kHz ;;.0.5 kHz

Expc;ri.-entat

conditions

'"

0.4 kHz-3 kHz

4>.4

Reference

kHz~3.5-:.tHz

Jones and N ur, Warer · sat IH'ated (1983) and Pc aad p, up to 20 MPa . Jones and Nur, As &oo-ve· (1983)

Nav¥» sandstone . ·• ..• Spor~ liniestone .•..• OWa *!a aruite .....

14

t

3 Hz-500 Jk

.. -. 3Hz-~ HZ 3 Hz-'OOHJ 3Hz• Hz

...

23

737

4to.kUz

JO Hz-tOk~

Perous Blass. (Vycor) ...

28

0.01

7109kHz

1 kHz-f2 'kHz

Barca sandstone .:·......

20

500

3 to 6kHz

1kHz-8kHz

Ma.'lllilkm

~~andatonc

From JOnes (1983}.

Spencer, (1981) Saturated (various Ouidsl Spencer (1981) Low saturation Spencer (1981) Water .satura&ecl Spencer (1981) W•r saturated Murphy (1982) Vau:illble · satUration Murphy (1982) Variabtc saturation \Vater saturated Winklcr~d Nur (1 2) (10 MPa}

!i

I

-

1111

• ) )

)

)

)

~r .._..

-;,

~i

·:, $

~

~

~·

h - pore half-width, D • &bermal dift'usivity of the composite material.

In the case of Massillon sandstone, fr is approximately 10kHz. which is close to the resonance frequency observed (Fig. 5.43). Figure 5.53 Jives the frequency dependence of attenuation obtained by thermo-elastic effect ill relation to that of the standard model previously analyzed It may be observed that the thermo-elastic peak is much broader than the peak of the standard model. This iS normal. because the thermal relaxation process is a diffusive one. The width of the thermal relaxation peak is incompatible with the narrowness of the peaks observed on Massillon sandstone (Fig. 5.43). Figure 5.54 shows tl\e variation of attcauation witb •saturation. The ~um is obtained around 99% saturation. which docs not correspond to tbc experimental results. Figure 5.55 shows that attenuation increases with rising temperature. However some recent data reported by Jones (1983) show the opposite behavior. Finally, the thermal relaxation process fails to explain the caergy ~ obJet •ed in shear. Nevertheless, the foregoing comparisons between theory and experiment must be adjusted appropriately. In fact, the theory is developed for traveling waves, and the experiments employed standing waves (resonant bar method). In the present state of our knowledge. it is diffiCUlt to 8SICSS the scope of this difference. Yet it appears that thermoelastic processes are negligible, at least at temperatures below lOO"C. CluurKe ;, chellfictll ~llilibriltm Acoustic plane waves propagating in a liquid generate local variations in temperature and pressure in the range of 2 • l 0- 3oK and 3 ·• l 0 3 Pa ccspectively (ltalsiag, 1975). Any chemical system in equilibrium subject to sucb· a dist\U'bance will fnld its equilibrium changed in tbe form of a modification of onc.or more of its thermodynatftic constants. This modiftcation is possible by the passaae Qf individual dlllllleculcs from~ state to another, aad hence a dissipation of ~IY· Thi~ dissipation clqJends on the frequency of the disturbance transmitted. At high freq~ the chaJliiS are too rapid for the system to dissipate energy, and at low frequencies the reaction has the time to adapt so as to be nearly reversible. Thus an attenuation peak will be· observed for an intermediate frequency. The type of chemical reaction propolcd by Schmidt et al. (1986) is the proton exchange reaction between free water and bound water, and whose resonance frequency is about a few kHz. It is worthwhile noting that Schmidt also succeeded in modeling the ditsipation of .tectromapotic waves due to the e~mical reactions. He showed that the fnqueaty atteuation peak occurred at the same frequeacy as for the acoustic waves if changes caused by conductivity are ipored. Fipre · 5.S6 shows that similar behavior was observed for clcctromagaetic and aoouatic waves. This model raises two problems: it is a r~ of the speciflc surface area (see foreaoing Sections) and it fails to explain variatioas as a t\mction of saturation. Nevenbeless, it helps to explain the behavior of granite-lit-e rocks as a function of saturation (Fig. S.31b). In this case, attenuation increascs.with risin1 water saturation. Qualitatively, the porous medium of a aranitic matcritl coasists essentially of microcracks. The additipn of water to the porous medium, as satwatioa iacreases, gradually saturates a growing numhcr of crac:ks. The attenuation, which results from proton exchanges between free water and boWld water, can only increase as the number of saturated cracks increases, in other words with an increase in

e.

"--'

~

'-

'--

'..._..

2l3

where

~·~ '--

amfl.n iA~-~~

5

I'' o.sr---~r--------~------~.--------r--------,----,

Fig. 5.53 Comparison of theoretical attenuations of standard model aDd thermoelastic dil'uaive model (after Kjartans-

aA

son. 1979b).

01

:=:

.d'::

, ---

,

~---

1

,

71== :=;=--

L

'

J't ~:

' t

Ftwquiftey Cwrl

!fJ! Fit 5.54 Tlleoretical thermo-elastic ..uenuation vs. saturation and for dilrerent cfepdls (alter KlartaassoD, tmb~ .

.

....

8.1

~ flwlian ... "''

Fie- .5.58

Theotetical

t~

attenuation vs. temperature in three watersaturated These three samples have aa identical matrix and dil'enmt porosities. The dry velocity is the same (1 kmfs) and ttac satu-

samples.

lp

rated velocities are 2, 0

200

300

3 and 4 km/s (after Kjartaasson, 1979 b).

Tempemure C"CI

~

------~~~--~~-~-~~~~---~---------------'

;;:-

~

"----'

5

,_,..,.._~

l2S

''---"

•r--r--~-r--~-r--,-~--~~~-

o

..,..tGHI

• tlllli'Os , .. - 2.1 kHz

'-'

50 0

----

0

40 "----

I·

"--'

'--


'--"

20 '-

'-

10 '-'

.,

0..___,__....___._ _,__.__....._...._....,.,___.___, 0

-~

'-

'--

'---

,____.'---" ·~

~

100

Comparison of attenuations of e~omaptic and a<:oustic waves vs. saturation. Massillon sandstoae (after Sclmtidt et til., 19&6).

saturatioa.lil tile case fl FOillaiadlleau saaclstooe(fi&. 5.3la) we fmct one example of the same ptOCIIII. 'l1!le additionofthltftnUte tOOAI ofwaterftlls the ~fain contacts of the pore networt.leadilla to a wry apid rile iD attftuatkm. In an interpretative sense, Fis. S.3lb for aranite is analoaous to the very low saturation behavior of Fontainebleau sandstoJle. At very low saturations, as we have shown. other attenuation mechanisms are also invoked (hydroxyl bonds. capillary forces~

Ft.UI/IOU IMrtilll c~ This mechanism, associated with Biot's theory (Chapter 21. accounts for relative movements of the ftuid and solid. It is a bulk mechanism •bic:h, u we haw streaed, requires a hiahly permeable sample. This theory predicts _a substantial frequcacy depende~ of attenuation in the ranse 10 to 100kHz for materials such as Massillon sandstone. At the seismic fiequenclcs and at common permeabilities, the losses predicted by this ~octet are insisftiftc:ant. Moreover, MOchizuki (1982), recently showed that this mechanism implied attenuation values that were too low in comparison with the fortaoina data. The equations of Biot ( 1956) and Mochizuki (1982) show tbat the c:titical frequeaey ia proportioaa1 to tbe fluid ~· aacl that attenuation clecreaset with tisina viscosity. Thil result ia in eontradictic,.urith,tbc experimental results of Jones and Nur (19U) for oumple (FiJ. 5.57). t

"-

....,_.....•

40

Fla- 5.56

\,__..,-"

"----

20

226

5

RESULTS AND MECHANISMS

ll

(GPIII

~--·

615

40

g.

I

~30r 20 10

0.2

Pc= 10MPa pp=4.5 MPa

·' 0.4 0.6

"

1

2

fNqiMncy X

Dyn~~nic

vilclllty (kHr • d')

Fia- 551 Attenuation of S waves arid dyl\lmic modulut as a function of the product of frequency by dynamic viscosity (resonant bar) (after Jones and Nur. C 1983 AGU).

~

~

'

Finally, this model predicts a dependence of the c:cntral frequency on the inverse of permeability, a dependenc:c that has never been actually observed (see Table 5.1).

~-

re#utltiolt of tM "*''ltiiW p.i4 If the fluid phase is sufl-.ciently viscous. viscous stresses opposed to the motion are added to pressure. A .unplc cakulatioq (Nur, 1971, \,\'tWa. 1N9} -~ lhows that the riequcaCies at a~ oporatioaal arc \"et)' biJb for fluids such as water, and for liaht oils. In fact, ~- ~trallrequency of the mechanism is g.

1

JrUCOIII

! ,

which

Jiven by:

eva

_mecba.., --ep

f/

Q)e- -

(5.1())

where

e - aspect ratio of the pores, p .... shcaE modulus of the matrix. '1 - viscosity of the intentitial fluid. -

For water, cue is about 109 to 1012 Hz in satura1ed -sandstones. However., for rocks in the state of partial melt, this mechanism may~ important at the seismic frequencies.

._ Rep.djlu This is the fluid flux between the peaks aad trdups ofa strain wave, or the flux from a high stress •• regioll" to a low s&ntss ••repon ". 'Fhi$mochanism concems a group or several pores. It depends on the permeability of the rock and the wave amplitude. The transmitted

',

~

----

-

=T ""'~.....

'

..........

-....----

5

227

wave period lllust be or the l8llle order ofmapaudc utho PRBSurcrcluation time for the mechanism to be siplifacant. MaximutD aatenuation is obtaiaed when the period alld relaxation time are identical. or for a frequency given by:

y2

I= 4n2Co

(5.11)

where Y is the seismic velocity and C 0 the hydraulic dift'usivity. We have for the hydraulic diffusivity«41 : CD= KKt,

"

(5.12)

where " is the pcrmeab,ility~ ~-the porQSity, K 11..the bulk moclulus of the fluid and 'I its viscosity. -' For the usual values o( the fGftgoia)a parameters: ._, : ;

'

I a too ~Hz '--

This frequCncy is hiper ~l'l t~'one fouo4 in,·~ abQve Jne4$Urements. Furthermore, it depends linearly on the inverse permeability, Wbidl is not experimentally observed (see Table 5.1). Fipally~ the depeadelieeof'frequcncton\tiscosityiS theinverseotth&t observed by Jones and Nur (1983){Yaa. SS1). Hence~-~ does not appear appropriate.

or

'-·

i. "Stpiln" or "MJf'b/1 "jlllw " This is allOCher rnedla1dsi'D of~ relauticm, but tile exciting mechanism is external, the ''squirt"or ''sqvilh"ofthelui4iatllcpotbutcavity. The .. squirt low .. or the "squish flow" ._...... . . . mvolw:~lieleal1'110tioa d the -fluid -ia the porou cavity with a hiP Reynolds - • · Tbis mecllbi• it i.Utialiad bJ tbe ,....,.. of the wave. In fact, the ~of the . . . • tate capila., tubea , . _ .... a1ocat pore pre~~ure which iltl the fluid in -...loa __... lllle ,.,._, 1",_ • tllitoMical ~ it 'has boell deAtonlttated that Ibis t)tpe of metlllanilm'tOUld by a standard Yitcoelastic model (a.,ter 3) (MaW. aad Mur, -1979; Pa1altr ldld Tra.totia, 1981, and O'ConneU aad BlldiMiky, 19'71). The reaent.aperimeatal data efJoaes and Nur(l983)(Fig. 5.57)or tlloseofSponcler(t98l)a'PPe&Uooodtma;belraWor'oftbeZenertype.ln this model, the liquid low can Oftly Ufke place iftbc ~is tower tba1l a limit frequellC)' (Murphy, l982b):

be..,......

'---

Ji .!

f&,

b"-;;

~5.13)

where b • dime~'ofthe fluid dr()P, K [I - bulk modtd"s or the ftuid,

p1

-

ftQid density.

(4) c, coeft'tc:ient is actually the one cWmed by Eq. (2.120) if TerzaJhi's hypothesis is made [M ""' K fl{•• Eq. (2.126)] and 10 - i.1 •

'-

228

s

USULTS AND M£CHANIINS

This limit frequency is about l GHz ill sandstollCI and sands. The central relaxation frequency is given by (Palmer aad Trawma. 1981): s /,= K e (5.14)

.-,

where

e - aspect ratio, K. =bulk modulus of the skeleton, 'I = fluid viscosity.

For high porosity sandstone, this frequency is about 1 to 10kHz, and the attenuation amplitude is approximately the S&lDe as that in Massillon sandstone at 500 Hz. This mechanism thus qualitatively explains tlie frequency dependehce obsei-ved in rock samples. We shall now examine the clefendence of attenual;ion on saturation. At total saturation, in compression, the intentitial liq~id offers resistance due to its low compressibility, and low pressure gradients are aenerated inside the porous medium. The resulting attenuation is therefore sli&ht By contrast, in s~. the pr~ure gradients generated are ~ter and the R$ult~ llttelluation is hiaJlef. Henee the result that Qi 1 is greater than Qi 1 at total saturatio~ (Fi,P. 5.31 aild ·s.52). , At partial saturation, the problem is totaUy dift'ereilt. The wat«jr,as mixture in the pores is highly compressible, and the extensional attenuation (Qi 1 ) dominates (Q£ 1 > Qi 1 ). Also the existence of an attenuation peak as a function of saturation is observed. This peak occurs in the Massillon sandstone investigated by Murphy,1982, at;a-water satwation of about 80o/e.lt can al.sobo._.Wi&. S.
......

of'_.._

Hence the ··squirt flow" process appears to partially and qualitatively explain the attenuation behavior observed. However, no theory i~ ye~ ,.dequate to provide a more quantitative 'iew of the process. It should therefore come as no surjrlse that, given the complexity of the porous medium, no single global or local model based on arains and spherical cavities can be adapted in detail to the fme description of mechanical behavior. The question remains whether it is necessary to complicate the models used ad infinitum.

j ~

--..

~~T \_/

·s

ialttif.AWJ~

229

5..1 RESULTS AND'MlCHANisMs CONCERNiNG IN SITU MEASUREMENTS 5.2.1

~

Intrecluction

The measurement results that we presented in s.ction S.l Wet"Cobtaincd by &aboratory experiments. The value of this type of measurement is obvious both for an understanding of the mechanisms oblerved and for the applicatioll of the results in the fteld. Nevertheless.

"---'

it is essential to be able to measure velOc:ities and atteauations in situ for many pnctical reasons. For example, the reconsttuction Of tile subsuiface acometry requires the blowlecfae of velocities in situ. We areiliterested here in the r¢1ationship between acoustic measuremelltl ad petrophJiical dwactotistic:t Of in Jitu materials. In fact, considering the rite in oil prices UftOaJ other factors, the need 'bas arisen in petroleum popb)'lic:s to determine not only the 'pometry of the beds, l>ut also their lithology, fluid content, porosity and permeability. Sonic weD louin& ori&fnally intended as a .. simple" conftrmation of depth for seismic sections, bas been 0.0 to approach those problems due to its finer depth resolution. It is clear that the in situ mciasurement of acoustic properties is an inverse problem widt ftiiiPICl to laboratory JRea~UN~DCDts. In the laboratory~ the petrophysical proportiel(porosity aadpenatabitity.ofthesampleare weD known and She meuuremeat is·intonded to oblerw the wriatioa in acoustic: propertie$ as a fuaction of physical parameters (twessule. temperatare). Ia the fteld, by contrast, the· s;hysical conditions (pressure, temperature) are relatively well known, and the· geopbytic:ilt is illterestcd in determining the petrophysical cbaraeteristics of the formations from the

·.._

'-

'-./.

acoustic properties measured.

The comparison ~ field meuuremera.tl aad laboratory measurements is not an easy one. It is often tacad witll_problelas such ..- dia~ induced by the temperature and stress reoond.itioniq of t.bc samplei 1a fac:l. AOt uusual to ftnd that laboratory measurements display· a lt)'ltereais iadic:itiaa -~·. inevcmbiiity of a structural c:banp durin& a measurement cycle. Tbia mca01 tlaat tlae results obtained cannot be representative of the behavlot of rocb buried \lDdel' kBOmcten of iediments for millions

'~.

"..i•

of years.

~

'-

~

ddcb:••••

.._. ~

i

i

'-

But the most important poiD& is &bat the frequencies employed in the laboratory are often quite different from 'dlele employed in situ, namely about SO Hz for seismic proapectina and 10 tlb for Well -lolling. The ebaraetoristie waveleftgth in seismic prospecting is thus quite dilrcrbt from the ~ waYCJenath in the laboratory, aDd, since the resolution of the measurement is proportional to this wavc:!ength, in situ meuuremeats tlwt itftelrate fonutioa that are tarely wUform. Fmally, the c:onstderlable frequeftcy ·depeDCieMe of the pt«111ill (espeCially attenuation, aee for example Fit- 5.4Sr~tt ~ direCt uttapdtalten of labOratory ·rei\llts to the fteld; Howewr, laboratory ~ ~- the existence of Jimit vahtea for velockiel ad a~ uader.-(Ap. S.l 'ad J.27foreumple). These limit values are usually ealklid ·tetmibal wloch:les . - tltmtdal atte~ntatkms (Wyllie tt al., 1951) and occur at efrectiw pNIIUI'II of abOUt ·lOO'M~ ~n1 to sediment thickaeDes in kilometen. ·

I

j -·

~-___,

230

5

RESULTS AND MECHANiiMS

In practice, and particularly for velocities, users have developed a number of empirical laws, enabling them to deal with thcr inverse prot)~ they face. We shall examine the results obtained for velocities, amt theft 'for attenuations, and draw conclusions on the usefulness of these empirical taws. -~'

5.2.2 Velocities ::.,;..-...,4

5.2.2.1

Ia situ velodty measunlllelltl 0

""·

In seismic prospecting vdocities are determined by intervals, usina the reflections obtained on a single mirror point. The relative error in the determination ·or these velocities is about 1 to 2%, whe~ absolute accuracy is about 10%.1'hese measurements ~ taken with P and· S waves . ~ to dle t}rpe of source employed. The reader

--.._

wisbina to familiarize ~If with _seismic re~on measurina techniques can refer for example to the work of CQrdier (1~83). In wdlloain& the determination of velocity is standard, as the distance traveled is ac:cura~ly known. The absolute accuracy is very hiah,

about 1%.

-----~

5.2.2.2 Veloclties aad porosity measure~Detlts

of Wyllie et GL. (1956,. .1958, 1962) reveal a simple relationship between velocity and porosity (Fip. 5.58, 5.59 and 5.60) for saturated samples under suft'ldent stress (terminal velocity). aacl of similar mincraiOJical oamposition, naJDCly sandstones and ICdimentary rocks in this case. Tile equation for P waves is called the Wyllie equation:

The

l • 1-t; -==-+-v v, v,.

-,~

(S.lS)

where 4> is the porosity, ~·the \'etocity ofJhc qturated rock, ~i the velocity in the fluid, and V,. the velocity in the rock matrix. This meaDS ~bat, if the type of rock, saturating ftuid and velocity are kno1m, the porosity can be catculated. The matrix velocities are given below for three major families of 'fOCks (from Schtmilberaer Co., 1971).

v. (mfs)

Sandstones .......... ; •...•.•... Limestone& ................... .

Dolomites ................ , · · · ·

S488to S950 6!tOO to 1000 7000.to 1925

This extremc.l.y ~pie equat.ioD noaed1eless requires many precautions for its use. The vcloQties are :quito different at shallow cicptbdrom those aivcn by Eq. (5.15). Fiaurc 5.59 shows that, at biah.poroliUu. thee~~ poiJUadeYiatefrom theaverase.curve (it ia not possible to apply a suftlcieot stress to~ uoconsolidatod.samplc,s, and the concept of terminal velocity boc:omes mea~). Fisuro 5.61 dearly shows that, at atmospheric pressure, the velocities of Fontaiacbleau ~roaes differ substan&ially from the average velocity (up to 40%). The apP.ication of a uaiuial stress of S MPa reduces tllis deviation from the average velocity to a maximum of 15 to 20%.

i'

I I',

I

I

I

~,

~

~

'-'-

I

'i.··''"'· t,: ·~ (

Velocity Cllmltl

3.5

3

4

4.6

'~<
:·~

Ytllollty lkmltl

5

35 110

•• •

30

80

25

~

_60 ~

20

~so

~

·g :

15

40

............

10

e Tripolhl

5

·W-......S 90

.,

Trwel

tm. Clts/ftl

0

120

110

100

eo

70

QI

5()

I

I

I

I

I

110

140

120

100

10

F~&-

,

1.1'

30

25

!:zo ~

J ·~

11 10 6

0 ''

f

1).2

I

II

o.a

Poillon's ratio

·~.

FJa. 5.60

lS" I 10 40

5.59 Compressional velocity/~rosity relatioasbip for siliceous rocb UDder . uniaxial 1tre1s (ultrasonic measurements) (after WyUie et al, 1958).

'

35

I

,.,

Trft time Cltslftl

Fig. SSI Compressional velocity/porosity relationship. Comparison between laboratory measurements and Wyllie's equatioll (ultrasonic measurements) (after Gre,ory.

1977).

I

200

,

110

I

160

I

I

140 , .

J . . ' .. •

t

'

I

..

'

I 40

.

Travel tin. Cltslftl

Experimental ret.tionship. Dear waft veloc:ity and Poisson's ratio as a function of porosity (ultrasonic measurements) (after Wyllie et Ill., 1962).

232

5

RESULTS AND !otECHANISMS

•• ••

5.5

5

•• 1·· • •

?:

j

.: .....

•

Sw=100%

4.5

~

• •

I

4

500kHZ

I

•••• •

3.5 3

•

•

s

0

20 •.:·

15

10

• 30

25

PorCIIity "')

,,. .:1

..

!5

8?: ...5 l

•

...t

b

... • I

•

4 3.5

•

Sw•100%

.... ••

500kHz <..',

• •

3

u 0

5

15 20 Porosity I'K. I

10

25

30

Fit- 5.61 Co~~ velocity. porosity relatioubip in Fontainebleau Sandstone (ulpsonic Dle&surementa) (after Bourbie and Zinszner, 1983). L Atmospheric ple8SIItC. ft. S MPa uaiuia1 sttess.

The only intrinsic velocity being teqnina} velocity, it is the only 'one that should be introduced into an equation such as that of Wyllie. Thu&~ndercompacted materials fail to satisfy Wyllie's equation. Moreover, it has been pointed Qut that, at very high and very low porosities, Eq. (5.15) is 'inadequate. Jd,Oy authors have proposed a modified Wyllie equation, such as Nafe and Drake-(1963) arid more recently Raymer et a/.(1980). The latter propose the following equatipns :._, · Consolidated rocks:

.<35%

V =(I -

t/>) 2 V,+ tj>V1

:

(5.16)

Unconsolidated ocean floor sediments:

t/> > 45%

1

t/>

l-t/>

~::~·*""'"?VI" pv• p1 v 1 p.Vr

(5.17)

.,

>"

~

-....-

1 I

·s

~tt~tliD~~

133

The comparison with Wyllie's equation is given in Fig. 5.62. Figure 5.63 gives a porosity /velocity Idationship for uncoasolidat61 ocean floor scdimcats, and Fig. 5.64 shows a correlation with experimental data and average regression lines for sandstones, limestones and dolomites. To a certain degree, it is therefore possible to relate velocities and porosity in a biunivocal manner. The extrapolation of porosity from velocity will have to be performed with precaution, as Eqs. (5.1 5), (5.16) and (5.17) are empirical experimental relationships and not physical laws.

S.l.l.3 Veloddes aad dentlty The densities of the materials encountered in ~ry basins vary relatively sliptly in comparison with velocities. Nate and Drah(t963)nd Gardner er al.(t974) proposed a relationship between the termiDat P wave velocity for saturated samples and the density. Figures 5.65 and 5.66 show this experimental relationship. The curve ~ be approximated by the follo\\ina equa~on: p- 0.31 V0· 25

(5.18)

where p ==density in aJcm 3 , V = velocity in m/s. Moreover, we know that the density of a given totally saturated material is related to the densities of the matrix p., of the saturatiq Ruid ·p1 , and the porosity t/J:

p .. p,(l -

~

+ tflp,

(5.19)

Equation (5.18) .:an therefore be transformed by means of £4. (5.19) into a Wyllie type of equation: (5.20)

V- V(•• p., Pt)

Wyltie'sequatiea c:u be-expralltdiChcmatically in thefortll V = V(t/J, V,, V1 ) where V, and Y1 ate tbeiftatrisud ftuictWiotities respectively. Hence Eq. (5.18) involves only the densities and not the elastic moduH, unlike Wyllic"s equation. It is therefore a less accurate equation than Wyllie's, and its application is mainly limited to the determination of an averaae density from the measurement of a terminal velocity.

5.1.2.4 Velodties ud day eonteat Itcccnt measuten~oents(De Martini et al., 1976, Tosaya and Nur, 1982. and Kowallis et al., 1983) show that -at.ionsbips exjst for water-saturated samples between compfessional or sbcar.velociticS,and p<»rQSity or clay content. The experimental results arc given iB Fip. 5.67 an4 ~.68. 11ae relationshipi between Vs, porosity • and volumetric clay content (c.) are(To.-Ya and Nur, 1982):

v,.,

V,.{km/a)• - 2Aca - 8.()(p J's(km/s) ~,..::.. ~~Jc.- 6.3q, for a differential pressure of 40 MPa.

j

+ 5.8 + 3.7

(5.21) (5.22)

234

R!:SULTS AND

5

M~CHANISMS

100'

80

~60

·i2

j_40

20

O''"

I

80

60

.., I

I

I

120

140

,., I

I

I

180

200

T.....t time (l.cs/ft)

Fig. 5.62

Modifacations ofWyiJie•s equation proposed by Raymer et al., 1980.

Velocity (kmlsl

100

•.•

•.•

·-

\

Fft. $.0

Experimental comparison of travel

time .-d porosity in ocean floor sediments

\

(after Raymer et al., 1980).

\

\

4!

~

Rt- 1M

I

Porosity/travel time Nlationship

Cor clil"oreat typeJ oC rock (after Raymer et

C\f» oo/

al.. 1980).

c:4,•

Vlloclty lkmls) ·

% /O oolacD

7.5 8.5

•

8

," I

3.5

(

3

2.5

,,

... -----. --

4o

e~

i3G ~

40~----~----._----~----~ 180 110 200

210

Travel time (la/ft)

4.5

so~~~~~T-~---,-----r------~----~

0/0 0

' o

u s

J

l~

~

... ,

.,..

1

~ le1Urftld

. , ...

cv,, • 100 mlsl

10 0 40

50

80

70

80 90 100 Travel tm. fj.q(ft)

110

120

130

140

....,7

--- ,, --·

-"~- ~-

f1a. 5.65

~ Compreuional velocity/ daiJity rdatioaabip iD various sedimcau (after Na(c and Drake.

8

I

C 1963 Wiley ~Dterscieacle).

' I

5

i:!!. -~

J

4

I 3

2

"'~----

I

1~--------------_.--------------~ 1 2 3

I

llulkchNity 7

• I

~

14

f.

.... '

3

2.5

2

..... " ' Compreuional veiotJM.y/ density reJatioDibip for clift'ilpat types rock (after Gardner et Ill;

1.5

or

1974).

1.1

2

u

2.4

2.1

lulk IMnlltv (log ....l

'-._..

2.8

3

6r-------------------~

3.5

Vs

Vp

P-eff • 40 MPI

6.5

Pett=40MPa

s.-too"

s.-=100"

i

i

.>/.

i

~

4.5

"jj ~ 2 3.5

31

0

I

Jh.

20

40

~

'

80

1.5 1

·'

•

0

80

20

40

"\

60

CIIY content ('\ vol.)

Cliy content '" vol.l

(a)

(b)-

i

'

80

Fia. 5.67 Velocity clay content relationship as a funetion cl porosity for saturated samples. L Compressional velocity (after Tosaya and Nur. C 1982 AGU). b. Shear velocity (after Tosaya., 1982).

5

I

v,. Pc•tOMh

0" 4 .............

5

.f

I

I

3

2

10

11

'20 POI'OIIty

Fft. 5.68

"'J

25

30

Compressional velocity 'clay content relationship as a function of porosityfor dry samples (after Kowallis et al. C 1983 AGU).

'-'-

5

USULTS

AND~·

237

These eqations allow a better estanation of porosities if the velocities aaa tbe ~verage day C01ltent • kaowa. They olwioully depend on the differeatial. preuure applied. Finally, h1ce the fonlaoiDi eqpations, ·. tltey empirically enable the eVIIIaation of one parameter if the mnaininJ two are bown.

5.2.2.5 Velocities .... CIIIDpAction The burial of rocks in sedimeritary basins, for example, generates compaction processes due to the lithostatic pressure. Compaction is the~ in porosity due to the effect of overburden pressure. Different geological bodies obey compaction laws that are more or less woO kaown but aenerallY distinct. The knOwledge of these compaction laws and the meuuremeat of velocity variaboaa with depth can thus help to provide an idea of the geological characteristics of a layer or a JI'OUp of layers investigated. Faust (1951) developed an empirical law from some 500 velocity surveys (Fig. 5.69). He showed that, in shale and sand sections, the relationship between compressional velocity and depth was: 1

V,.==B(~n

(5.23)

where ~ • depth in meters, B = constant equal to 46.6 for Faust's samples, v,. - compressional velocity in m/s, ~ == age of the sediment in years. In carbonate sections, a general equation of type I5.23) is difftcQit to determine. This is because tu role of compaclioa is-leas uniform titan in shale and sahd sections. J•owsky (1970) nevertheless proposed a qepession of the'ame type (Fia- S.70). or sands, Gardner et al. (1974) and Domenico (1977) proposed the ,-elocity/deptll relationships shoym in Fis- S.7L.:Siilhl~amobservable in comparison with faust"s rclatioe that result from specific pressure aacl coatpacd6n elwacteristics.ltiliatefestinj to note that a power

f.

I

'--

I

dependance on depth. ~.was also found by G~ (1,51) for the P wave propagation velocity in a helagonal packing of spheres (COrdiet, 1983). The usc of Faust's relation or an intCJf&ted law: s t==B'~

where tit the vertical propaption time, offers a auess at the type of lediments analyzed. If the law applies, it is pneratly a normally compacted shale and sand section. If not, it may consist of carbonates or evaporites. or even of hi&hlY tectonized rocks, or series that have undergoae erosion after b\lrial. In this latter case velocities are higher than the ones given by Faust's relation. If velocities are lower, the formation may consist ofundercompacted shales or very biJb porosity series. Finally, as we pointed out for~ previous relations deanna with in situ velocities this is only an empirical law to qproximaiHhe depee of compaction of deep sediments. Several variations of Faust's relation exist, and these are neither more or less accurate, but merely indicative of slightly differetrt sediments.

5

Fig. 5.69 Compressional velocity/depth relationship for different ~ series {after Faust, 1951).

-;;4

l

~

f

l !

3

D .luralic·Tri.aic

.

oea.-us

A~OE-

•

2

"-••~

0'-'lln

0

o Tertlety

......._,

·3

2

4

Qlpth ClPnt

Fig. 5.70 Compressional velocity/depth relationship for differeat .lithoJoaics (after . Jankowsky, 1970).

0.5

.,,

Shaly 1u...-,..._;on liM

e ~

1u 2

...('

...-.onn.. ,

•

''

2.5 '---'---''----'----'U...--.....;L..-_ _ _ _.....J 800 500 400 300 200 100 TliiMl tllne .,.,..,

e ~

J ._.,,... lrlne-...ct

Fig. 5.71 Compressional velocity/depth relationship for different types of sand (after Domenico, 1977).

2

• 2

2.5

3

, _ OM!ocity (kllllsl

• 3.5

'-"~

·~

s 5.2.1.6 ,.

~

I

dSULW'Atlt

~--'

239

V,.f Vs and Poisson's ratio

The ratio ( v,. /V5 ) of lonsitudinal velOcities ( V,.) to transverse velocities ( V1 ) corresponds one-to-one with Poisson's ratio v: Jj.""

1$

J2(l -

Y)

(5.24a)

1 - 2v

or

,. = O.S(V,./Vsf -

l

(S.24b)

(V,./I's)l.- 1

The c:alculation of Poi110n's ratio._ requiftlll, tbe limultaacous measurement of the velocities 1',. and J's. The compilatioa made by LawrtM{pcnoul c:ommuaic:ation) helps to show that the different types of roCk display. rather different Poisson's ratios (see Fig. 5.72). ·-'-

·~

............

~

.we•• •

Dry••~_. . . . . . . . . . .

'-

. 0

0.1

l'-

•

0.2

•

.~~

~.3

-

•

OA

o.a

......... ,.tlo

J11e. 5.71

Averqe Poiuon's ratios for di«erent lithoJoaiea (after Lavcqne,

penonal correaponcleDee~

._/

'-

Hip values of Poisson's ratio ( > 0.35) (or high ratios Y,/l's) correspond to UllCOIUOlidatcd rocks. Compact rocks dispJe;y P~'arati9!1~ 02 aaci0.3S.Yt'bile ps sands have a very low Poisson's ratio, about O.l.ln water-saturated sands, by contrast, Poislea's natio ia about 0.4. A awnbcr of average values of P aod S wave velocities are pvea in Table 5.2. 1be for:qoiaa c:orrelalions bctweea ~ type of rock,. saturation and Poissen's ratio (Fig. 5.72) are merely indicative values .Uowiaa the ppbysicist to focus on the polosical cbarac:tcristics of.tbc rOICks inVC!It.i8ted.

'-

5.U7 Sa-mary • ia :sial Wllodty ......,...... ''-

r=

3.E

.__/

.__/

All the empirical equations that we have pointed out show that, if v,. is known, or preferably v,. and J's, a pas can be made u to the Bthol()ay arid the porosity (but virtually nothins about the saturation) of the material stUdied. This is not illtended to determine

precise values but to set limits between which dW: true nsponse. should lie. It is also important to point out that, strictly spcakins, tbe foreaoina equations are strictly applicable only to the samples investigated. since variations exist from one type of sample to another.

240

s

ResuLTS AND MECHANISMS TABLE

5.2

AVERAGE PROPAGATION VELOCITIES OF p AND AND AVEitAOI! ROCK DENSmi!S

Type of formation Scree, vegetal soil ............ Dry sands .................. Wet sands .................. Saturated shales and clays ..... Marls ..•......... , ......... Saturated shale and sand sections Porous and saturated sandstones Litnestones .•..........•..... Chalk ...................... Salt ........................ Anhydrite ..............•.... Dolomite ................... Granite ..................... Basalt ..... , ...............•. Gneiss ...................... Coal ....................... Water .... :: .....•.......... Ice ......................... Oil ............. ·.............

S

WAVES

Pwave velocity (m/s)

S wave velocity (m/s)

Density (g/cm 3 )

300-700 400-1200 1500-2000 11 ()()..2500

100-300

1.7-2.4 1.5-1.7 1.9-2.1 2.0-2.4 21-2.6 2.1-2.4 2.1-2.4 2.4-27 1.8-2.3 2.1-2.3 29-3 2.5-2.9 2.5-2.7 2.7-3.1 2.5-2.7 1.3-1.8 1 0.9 0.6-0.9

~~

J

100-SOO 400-600 200-800 750-1500

500-150 800-1800 2080-3500 3500-6000 2000-3300 2300-2600 1100-1300 4500-5500 2500-3100 4000-SSOO 2200-3100 3500-6500 1900-3600 4!00-6000 .2500-3300 28410-3400 ~ 4400-5:200 21c».3200 2200-2700 1000-1400 1450-1500 3400-3800 17004900 1200-125<> ...

'·

Density of constituent crvstal (g/cml).

2.65 quartz 2.65 quartz

2.65 271 2.71 21

-

quartz calcite calcite halite

··~

-

(Ca, Mg)C0 3 1.8-2.99

'

------

After Laverpe, personal correspondence.

·~

!.2.3 Attenuations ~.1.3.1'

Ia situ .......................

Measuring in situ attenuation is a delicate problem, and fteld measuretneftts found in· the literature are oftetl debatable. More6ver, the attenuation measured by various authors is generally the total attenliatiotl, in other words the sum of the intrinsic attenuation directly related to the porous medium, and the extrinsic attenuation resulting from the geometry of the subsurface and of the source, scattering, etc. The most reliable field measurements are taken either by means of weiJ.loging tools such as the EVA (SI tool at frequencies around 10 kHz, or by means of VSP (vertical seismic profiles) at the seismic frequencies. Table S.3 Jives some ...rences aQd results of field measurements. We shall _ discuss these techniques ip the tuwCbaptcr. .In this Section, we. have decided to present so~e results and ho~s conq:J-nina t~ use of ahen1,1ation m~urements.

~'

'~

(S) The EVA tool (Elf Aquitaine trade-mark) has S transmitters and 12 receivers.

\

j

r

~

I

(((

(

..._

(((((

(

(

(

,'.),'.

~~~--

...~

--

V.U.IIF.S Of

Type of rock

Location

Limon (Colonulo)

Depth 1m)

Measurement frequency (Hz)

Q,. apparent

50-450

32

SO-«ll

2 181

Sandy day Clay/saM

()..3 3-30 30-150

IS0-300

50-400

Clay/saad ·•

117D-1770

Saadlaad . . . .

Same .,_ _,. saMr

1'710-2070

" 125

So.at.....·T8aJ

Sandbaab.Jftll tiltyahale

Southeast Texat

Mostly . . Sand (23%} aad clay Sand (2C)-4l afld

2010-2850 9$0-1560 JSf0-1800 t8Ci).2100

Oult Coast (30 km south or

.. Ofl'shore lovisiana. (~ .

.

/1':

Southeast Teaas Hcauf~t

Loam/saitdfcl&y

Sands and..._

Houstottt.

--

clal

l.ime!etotte an4 clta Sand(4S%, qct day

Sand(~· aifd'~)'

Sea (('anada)

otfshore

SiliccoU. chaJit

6GO-IS60

From Carmichael (1984) and Goldberg fl98S).

50-400 50-400

"125

"us " 80 " 80 c;80 c;80

15 136 67 > 273 28 52 >'273 ., 30· 41 > 273

>1020

._ RO 15-40 4()..70

!i4CJ- 1193 945-1311

12S 425

278-442

5000-1 5,000

442-5H2

on avcraJlC 5000-1 5,000 2H

1590-1755 ~1320

Baltimore

Silieco111 Chalk with porcellanitejoina.·

~

--

'_:

___

TAILI! 5.3 Q MF.A.~IRF.Il IN SF.Il1MI1NT!I

0-225

Pierre sllale

-

28 55

References

corrected

McDonald (1958)

..

Tudos (1969)

et

and

Reid "5 ·,.:

·( ,,;

:;·

Haup(I9Bt) ;:

46

> 273 34 94 43 67

Oanlcy and Kanasewich (19110)

68

on average

at.

'

67 > 273 31 •·109 > 273 37

Goldberg (1985)

( l, ~

242

5

RESULTS A"N"D MECHANISMS

5.2.3.2

Results

The pessimism expressed above must be moderated by the results obtained in the laboratory. In fact, attenuation is a difftcult parameter to measure. but provides considerable infonnation concerning the medium investigated. especially with respect to. saturation. Figure S.73 shows that knowledge of attcmuations and velocities provides a better defmition of saturation than that allowed by knowledge of velocities alone.

--.---4 ~

,r

0 0

Vp

0

I

I 0 01>0 oO

I

° ~

0

0

r...itlon Andst-

•Drv

6 . .,

2 1-

6

• P-'illly a1Ur~ Fully a1Ureted Sierr• White gr.nite

, t::

•

0 Plrt;.Hy a1Ur~

o

•Drv

I

2.0

I

1.5

-

• Drv

l I

I

I

I

::j

1.5

2

2.5

3

~to

0,

0~

0.0

-

• Pattilllly . . .reted 0 fully~1MI

Sirrll White gr•nite

6• '(6 I

1.5

~

•Orv 0 f'ltrt;.Hy •1Ur111MI

-

.... 0 0

0

I

9_

2

0

I

2.5

Vp/Vs

Vp/Vs

C•l

(bl

3

"""te

FiJ. 5.73 Massillonsaa4ltoac and Sierra graaite. The arrow indicates increasing effective pteflures (resonant bar) (after Winkler, 19791. L Relationship V, as a function of V,.fV5 • . b. Relationship Q; 1/Qi 1 as a function of V,./V,.

Although. oa the averap.~tenua~ rises with porosity, Figures S.i4 aad S.1S show that attenuation/porosity. relationships· appear to be less simple than velocity/porosity relationships. Hence, with. the attenuat;iort parameter,lhe possibility exists of obtaining comple11lentar)' data ab"Gut the porous medium. Nur n Gl. (1980) pointed out that S wave attenuation and permeal>ility couldtJe correlated; Wesball discuss a possible explanation of this in the next Chapter. . . .

S.2.4 Conclusions on in situ measurements Field measurements lead us to derive a number of empirical laws for seismic velocities. These empirical laws enable the geoP-hysiciSt to pinpoint the petrophysical properties of tbe materials analyzed (porosity, debsity, -~Y content, compaction). It has not been possible to establish empirical laws fot attenuations, because in situ measurements are still too few in number and inaccurate. Ultimately, it is likely that reliable in situ measurements of attenuation will be obtained. These measurements will then allow an evaluation of other petrophysical properties, such as permeability and fluid content.

~-

.4

,__,~

5

' j

-

243

USULTS AND MECHANISWS

1000r---------------------------------

o,

·~

I

0

L-1.0

tocl

I

\00

t

.

' ...

....

lgneoulll'ld rocks

*Me_...e.lc

101-

t

o

..

Ll"*'-

....._

....... _ol 10

1

0,1

Porolity

FiJ. 5.74

(~I

Attenuation vs. porosity in a number of rocks (after Johnston et al.,

1979~

'-~

~

500 L 400

1-

1~:t 100 L

•

Op (500 kHzl

•

a1 =0 sw = 100%

•

• • •• •• • •• • •

.,..

10

.

•• •

'

15

•

Op (&00 kHzl

200

= 5MPa Sw= 100~

a,

•

!..!!!9 150 Op

•

260

,.

!10

•'

0

20

25

30

0

...

5

•• 10

Porality(~)

•

•'••• •

• ,• •

100

•

• 15

20

25

PorOiity (~1.

J'ia. 5.75 Compressional wave attenuation/porosity relationship in saturated Fontainebleau sandstone (after BourbiC and Zinsmer, 198S).

...

30

•

v~r ~ ~

. i

i

'f.

:':l

',,;,

6

.. ;

~~

i waves and intertaces

'--.../

A

INTRODUCI'ION

'-.-/

'-.."-_

"'---'

·-·

_/

"---'

In Chapters 2 and 3, we have studied wa¥e propaption in i'ffinite poroelastic and viscoelastic media. In the subsurface. acoustic waves are reflected and transmitted at interfaces which are characterized either by contrasts in elastic properties (acoustic impedance contrasts) or in anelastit properties(attenuation or permeability contrasts). To interpret field data, it is esscali8l .to UIMlentaaclllow theloioterfaces modify the reflection and transmission of waves. Moteover the existence of interfaces allows the aencration of interface waves (Raylciab or Stoneley wa•) w~b may also be used for the detcrmiDation oflitllolopcal -~ petrophysical propcriies of the subsurfaee. In this Chapter concemia& iataface p:~~ ~.,_..,we will study firSt •UI'ated porous media, and secondly viscoelastic media. Finally, we will show that the two models are complementary, and must be used jointly mcertain cases to correctly interpret fteld

....

-..__/ ~

"-../

6.1 WAVE PROPAGATION

IN SATIJRAT£1> POROUS MEDIA DISCONTINUITY- EPFBCTS

---~

'--' ....,

____

_/

~

we--... in the«JJldusionaaelaapter 2 th&&.c.t'~ porous mediamodelod by Biot's theory, permeability plays a role essentially in the presence of discontilluities. More precisely, the discontinuities, by the bo\lndary conditions that they imply, induce pRSSUre aractients which modify the ftow. Heace they jive rise to attenuation. The poasible existence of a second kind P wave also implies sipifac:ant chances in correspondina clutod)'UIIIie ~. . In the flnt Sec;&ion of dais~. wethaU ~the boandary con4itioDs tMt must be satisfied at the interface of two saturated porous media. The IOCXIIld Seotion will deal with wave reftections at this interface. The fmal Section will show what happens to the

~,

246

WAVES AND INTERFACES

c~todynamic

6

interface waves. We will also examine a typical problem in borehole

seiamics in which a source is immersed in a fluid and acts near a permeable interface.

6.1.1

Boundary conditions

Let us consider an interface S separating two saturated porous media denoted by superscripts (1) and (2) respectively. The unit normal attached to this interface and oriented from medium 1 towards medium 2 is denoted n, with components "r The direct normal vector ton is denoted t with components t1 (see Fig. 6.1 ). The notations are those of Chapter 2: displacements and macroscopic stresses ui'1 and ai7, pressures p
-,

lSI "

M.dlum1

•n

Medlum2

Fil- 6-t Notations at a Jiven mterrace. At the interface, in additiOn to the fteld equations valid in each medium and developed in Chapter 2, ne1v coaditib..S 'ftltlit 'be satisfied. The first deals with the continuity of macroseopic displacements. 11Us is tile tmemadc cbndffion:

'' 1411 .,., ui 21

(6.1) '

The second condition is the CODICI'Vation of fluid mass and hence the continuity of' the flow through the interface. This amounts to writins the continuity of the filtration velocity component in the direction n, which we shall denote w,.:

w-

11-

.:.ell,. = W!lln "'I I I I

"

(6.2)

The fmal conditions to be expressed are the so-called natural conditions, derived from Hamilton's principle and ~with the~Mcroetopk:ltresseso17 and pressures P''' at the interface. These conditions were obtaieed by Deietiowia aad Sk.alak (1963). We shall adopt a presentation here ,tbae is close te their oWn, but in agreement with that of Chapter 2. Just as in Section 2.2.2.3, where we defaned a volumetric pseudo-potential ofdissipation, we shall also define here a peeudo-pomuial of dissipation per unit a~ea o. relati¥e to the interface, defined by :

D,=-1-~

2rc, "

(6.3)

where "• has the dimension of a hydraulic permeability per unit length. The parameter "• characterizes the permeability of the interface,, and hence the interCOftftCCtion of the two porous media (iee Fig. 6.2).

J

'""

'--'

-

r l

i

.,.,..~ 't111'1t1VMB

8

2i47

Fipre6.2acorrespoadstoacueinwbidlaUdacbannelaaamllllicate(thl'illterfaceis said to be opcB) aDd "• is theNfore illfmite (K, • co, ao dissipation). Fipre 6.2c n:p~aents the cue of aeoaled iMerfacoaad K, 0, 10•tbat, u shown lator on, O(no filtration). Figure 6~ (0 < K, < co) com.ponds to an iatermecliate cue. Tbaa ill a borehole. the parameter K, can model the effect of a ''mudcake";-orof a tlooded zoae. if the latter is sufkitntly thin to be treated u a resion without thickness.

w.-

=

Open lnwflce

hrtillly open ..........

................ Medium 1

lllldium2 --._,

Cal

rzz::J sOlid

(bl ph-.

(cJ

c:JUquldpt._

Fi&. 6.l Simp~ cJil&rap ohn intorfacc botwee&. two porous media (after Deresiewic!:i and'Stllal; 1963). ·· ' · ,. ' ' , The variational formulation of Chapter 2 can then be resumed, addint the surface potential ·t() the velumettio 4illip.don poteDdaa dclft1led by &q. (4S.3~ · Hamilton's principle al'wAy& leads to-thnalile lieN eqMdont reiMed to tk medium concerned [i.e. uu- t111 Mld p • ~. w,.·"f'', ., - flf, I • l·or 2· m (2.87}]. FurthermoN. under conditio•(6.1)and·(6.l). dletoll...._ift1e8ral iiiMMIId IIIICUGelled:

'-

J.
'-

d.t

d.t - 0

(6.4)

~vc*the v~tiou W..-4Jw•. ~f;lwe.vanatioAaare~' ud the

portioG of dac, 4atc.rf8cc S is IUtJittarJ, iq. (6.4) daw:_. ...,....t41J ~ the nullity ol the two intcaraJa. Thus from tbe lint oae.: ~Jln1 •

o1J•n1

(6.S)

Condition (6.5) represents the 'continuity of the macroacopic stress vector at the interface crossing. This is the equivalent of:

~~-~ on a surface where the surface force ~ is. imposed. The W11 a - K,(pCll- p(ll)

-~

~~ ~

condition gives: (6.7)

Condition(6.7), in which iJp - ,C2l - p(l) represents the difference in pressure prevailing on either side of the interfac:c (or of a mudcake), is nothiq other than Darcy's law governing the fluid flow across the interface. It may therefore be noted that, for

'-;'

.

2118

6

WAVES AND INTERFACES

(open interfKC~·the fmitcness ef:tbe flow gives rise to a zero pressure difference (~p- pl 21 - rP 1 - 0). For "•- 0 (sealed interface). no flow is obtained (w. - 0). It is dear that "• depends notoaly on the eeotaet geometry between lbe two media; but must also depend oa the frequency (iloanbaum. 1974).. if the latter cornsponds to a wavdeagththat is not large in comparison with the dimeosioDJ of the elementary volume, but only in comparison with those of the. channels in wbicb the Oow takes place. In this case. permeability is no longer absolute but relative to the frequency, on which it depends (BioL 1956 b). The experimental determination of "•• for application to .actual cases (mudcalce for example), is nonetheless a delicate matter. Hence aU the theoretical investigations consider oaly the limiting cases "• "" 0 and "• = (Geerstma and Smit, 1961. Rosenbaum. 1974, Feq aiad Johnsoo, 1983-). . Equations (6.1), (6.2), (6•.S)and(6.7) are knee the so-called nat1,U'al boundary conditions that must be satislied at .tbo interface ·or two saturated porous media. It must be emphasized that, contrary to the inconect preaontation adopted by Deresiewicz and Slcalalc (1963). for a Hamiltonian fQnnuJation of the problem, o~ Cannot, independent of this formulation, introduc:x: the constitliti\'ic equation (6.7) govetning tbe ftow at the interface. In fact, this equation is a consequence of Hamilton's principle. To apply the principle, it is necessary to account for all dissipations, adding the potential of dissipation per unit area (6.3) to the volumetric poten&ials of dissipation. Giveri their clear physical signiftcance, one could obviously hav~ introduced (6..5) and postulated (6.7). However, a pan from the fact that this presentation woUld not bcin~t with the Hamiltonian presentation of the theory developed in Chapter 2, the procedure would also have obscured the so-called natural character of (6.5) and (6. 7), by conferring on them a more ·heuristic value. Two specifiC eases "iH be~ below. F~ eoaaiGor the case or a {fee surface. Equations (6.1) and (6.2) then . . . . . .r. Equatioa(66) iueplaced .by Eq. (4.6). where we ICIC j, == Ia t6.:7), P' 21 • 0 and bene~ p .. p'- 11 • 0, ~\IICtolal interc()niiiOetion exists (open interface, '"• •· cc ). Ia the cue :Of a. frce. •udae.. ;mC i*lCiitio• arc. tilus:

" • - JO

o.

tln=O Free surface { . 11 1 p==O

(6.8)

The second cue tonsidered ii that Of the ·reftection. at·tlle iBtetfaee of a pOrous 'Bledium (medium l)ilndofaftuidmedium{tliedhlm 2).1nthisc:aseEq. (6.1)cfisappean.Moreo~. since the fluid corresponds to unit porosity, the ftltration vector seen from thetluid side is reduced to \\1 21 = Ul 21 - fl1 1 ' where iq 2J is the average fluid velocity. Furthermore, the stresses in the fluid are reduced to a hydrostatic pressure, whereas (6.7) remains unchanged. Finally we obtain: ·

w = W. 11 n- = (UP1-

l

II

Interface Porous mediUm l~fluid2

l

I

o1] 1n1n1 = o1]"n1t1 == 0

w. =

l

p<21

......

il.ll)n. I I

(6.9)

- "•(p(l) - pill)

.

' ., ........

'=-'-'r ·~·

,

tl49

WA¥81 A.-'IMIIli'MIIS

...

~·

·~

'--·~

j

'---'

6

~ -~,

''--'

•v--"

·-.'---'

'---'

6.1.2 Wave reflectloas at the interface of two saturated porous ...U. In this Section it is assumed that the problems dealt with are two-dimensional. the interface is planar, the waves_. p~anecu • harmonic wa• aad that the incident wave travels from medium 1 to medium 2. Ia dMsic elutodyumic:s. an iDc:ideat P or Sl' wave reflected anbe interface of two clastic tolids paerally lives rise to four typea of wa'Ve, two transmitted and two nlected. P aDdS I' (for example. Ewiot a al., 1951). Tbc SH wa~ is of less interest because of the at.nce of convenioa. and will not be conaidered further. In poroelutic:ity, tbe famewcn of the study, the possibility exists of pneratina six types of wave, tbla. traaaitted aad three r•c:ctl
6.1.11 Case of .,..lladdeaee

"---'

Let us farst introduce the ratio Z of acoustic impedances correspondinl to the incident

~

wave: "--'

z _!.e.!!!

.__.

In (6.10). Vis the wave velocity for a closed meclium (i.e. no relative motion between the ftuid and the overall movement). With the notations from Chapter 2, it may be recalled

(6.10)

(pY).

-

~=

C''; l#J)' l

Incid.ont P1 wave: V • V,.., ~'--"

l

Incident $ wave:

V•

Ys • (~Y

(6.11)

~

(1) It is always possible to approaima&e ,vave frpD~ ~ ~y low curvature loc:aUy by tlMir taaaent planes. For any wave front. a couJ)Iitla OCICUfs ~'thf lriSftt pometry and the iaterface, and it is Uceual')' to employ 6Der theories such u til<* oh&)'S (Cerveny ~r Iii., 1917).

---~

'---'

·-·

...

250

6

WAVES AND INTERFACES

i

Let us also introduce a reference angular frequency (Chapter 2t defmed by:

al

.

41

= 2nfc =-;;;:

(6.12)

PpJf

where~

is the hydraulic p'Cnneability.lt ma;v be recalled (see Chapter 2) that we is usually very lar,e, thus allowing expensiODS with respect to •tw<. One can now determine the rdlection C()Cft'tcients R and traasmission coefficients T. These coefficients are dehned as the ratio of the amplitUde of the displacement corresponding to the wave a1Udyzed (reflected or transmitted) to that of the displacement correspondiq to the incident wave. Note that, coatnry to the dastodynamic; case, these coefficiems are complex because of aUIID\Iation processes dar to the two-phue character of the materials involved. tbis at:tataation ·leadiBa to differeaces of phase. The computations corresponding to the different typc5 of reflection were conducted by Geerstma and Smit (1961) and Deresiewicz and Rice ( 1964). We shall only comment on their results here. L

CIIU of- ltu:i*W S

""'

WCN

For an S wave at normal incidence, only the S waves are evidently transmitted and reflected. lbe coefftcients are then given by:

zz (. ~w ~w )z 2[ z2. .( w·. w )2 ·T··• -:--1 + Z I +.4(1+ zl' 7t al,_ Q7 R-

1- z [ T+Z 1 + (1 + Zf

y1

-

y2

exp (i tan

where }' = Pt

1

.-..,

8,.)

(6.13)

,.

2

.

1

.

for media l and 2

p

9.1 d

1

1

exp (i tan"" 8y)]

i2

'

_ ]

:.Zzi (¥• ;; -"yz ~)

(6.14)

(J))

(J) OT = -z- ( ¥2-1'1 --;-·

1+

z

~

(1)1

These equations correspond to an expansion to the nearest second order in wfw< [i.e. up to (w/wc) 2 ]. Note, to begin with, that, if w - 0, the classic reflection (1 - Z)/(1 + Z) and transmission 2/(1 + Z) cocfticimts are obtained: Moreover. they depend on w2 , thus differing from the same problem ·in viscoelasticity (see Section 6.2.3.2). The flow conditions at the interface, characterized by the parameter Ks. have no effect on reflection and transmission of the S waves. This is nonaal. ~¥cause shear waves induce no pressure and hence no flow at the interlace. The influence of Biot's mechanisms is only felt by the dissipation induced by the competitive effects of permeability and inertial forces, thus the only forces responsible for flow in the media. but not across the interface. Thus transmission is complete (T = 1) in the ftrst ord~r term of w d if Z = 1. In the second order term of wfw<, the order in which the two-phase character prevails for the case of S

~

""' .-..,,

--.., .-..,

·--.., ~

""

-.

~-,­

'- ,i "---"

:..

2Sl

ft"'WI•·AD~

'

waves, it is also necessary that the quantity ,Pp1 /pof = :Kp}lp be continuous [substitution of(6.12) and(6.14) into (6.13)]. In fact, this quantity quantifies the foregoing competitive effects. These d'ects ·must .then act similarly in both media in order for the phase dift'crcncc between Ruid motion and macroscopic motion to be the same on either side of the interface, and for transmission to be complete.

b. CIIIC of a iltciMIII P1 w••c The tr&Mmisaion and rctlcction codlieieata corrcspOildina to aa W:idont P1 wavoand a transmitted or rcftected ~wave arc denoted yw and ~. Tbc sublaiptj is 1 fot waves of the r.m kind and 2 for dlc wave of the seconcl kind or slow wave. For rcftected and transmitted P w~ves, we !)btain : " .. . . .. ., . . 1

R'u •

.

fUl -

1-Z 1

+ z (1 + 60 ) exp (i tan - 1 80 )

,m 1- z) exp [- i tan-• (1~ - z8 )] +2Z•·(~1-8 1 0

0

(6.1S)

.

where (6.16) with

'[(1 - pm)rrryJ fJ l

. . m== fJM ·g= .1.1 + 21',

.

(6.17)

for media l and 2 and where fJ and M arc the coefftcients of 1Siot"$ theory (see Chapter 2. Section 2.2.2.1 ).:;· , ~• For reflected ud tralll-.itte&t 1\ waYCS: ·

.· R'1• • 91(1-ft ;D,)exp{i ~- 1 ·(1-lB:a- ~s>] (6.18)

TC 21 •

-

/

~ f 1(1 -92 + 8,-IJ•J eltp [t taft- (82 ·+ 84 )] 1

"'•

where

81 =

Jj(t- ~ Z

m1

m1

-

m2

80

Bz == (~ :-. 1 . ~~·2 ) 1 - z2 Bo 1 + Z rc.p 1 V,, m1 - m2 ~· . .

~ ~dico/aft -
.1 fla ~-:flit .. ·----.,zZ (lz oft

8

-

,~

.

(6.19)

2S2

6

WAVES AND INTERFACES

with

4-, t/131fPt p(p,fp- m)

for media. 1 and 2

(6.20}

In (6.20). q is a shape factor that depends on the geometry ottbe pore network. It from a more elaborate theory than the one developed in Chapter 2 (Biot, 1956 average value is Among other factors, it is related to tortuosity" by ah, acoefticient eharacceriziog thecapitlary CI'OSS•aectioa (Ia ~ 8 fot' ein:uJar JeCbOIII, h for infmite penny-shaped cracks).

r-

JS.

The foregoing equations correspOnd to an expansion ·with respect to (J)/(.1)~. expressions thus show that the influence of the two-pbUe dtaracter of the 1

concerned is felt to the first order in (Q)/0>1i for tb.e ftlflection of a P1 wave, as opposed to the case of the reflection of an SV wave, for which the influence is only felt at tbe frrst order in (.1)/0Je. This inftuence is mainly exerted through the contrast, of tbe quantity m defmed by (6.17). Ifthis quantity]J continuous (i.e. m1 - ,.2 ), Eq. (6.16) gives 80 = 0 and Eqs. (6.17) then show that the reflection coeft"tcicnts AC 1 ' and transmission coefticients TC1 1 are reduced to the classic cocfticicnts of elastodynamic:s (1 - Z)/(1 + Z) and 2/(1 + Z). This is understandable considerina tJle ~QCC Let us recall the defmition of the macroscopic stress (see Chapter 2):

or ....

t~,1 =

If we set

t~.,

A.1 tr e ~iJ + 2p eu- PMe~,1

(6.21)

= 0 in (6.21), oy being the direction ~ to the interface, we obtain:

tr ~ ==

mt

(6.22)

siru:c tr • is,. in this case. equal to ~· The quantity 1/m thus quantifiCS the share accounted for by the inaeue io water content in the apparent volumetric strain tr .... zerO 4W!JQ~ . . . .M:opie--' stuss (u., == 0). Ifm 1 = m2 , this share is the same on both sides of the interface. relative to the respective voiUJJleitric ttrain: Thus, for m1 • m2 , dae traJ1Jinissioa and Rlloction of the P1 wave of the first kind can take place with the same elastodynamic coeft"'cients, because of the absence of au atUitioltalsta;cu at tbc iater(ace with respect to the elastodynamic case. Simultaneously, note that in the absence ofan impedance contrast (i.e. Z = 1). the contrast of the quantity m leads to the generation of a reflected P1 wave dependent on .;;;;!Oi. Mechanisms of this type were identified in the analysis of the effect of attenuation in viscoelastic media (Bourbie, 1982, Bourbie ami Nar, 1984). This will be discussed in the second part of this Chapter.

e

We have just shown that the contrast Am= m 1 - m2 uerted a considerable influence on the reflection and tran~on coeftic:icntuelative to the P1 wave corresponding to the classic compressional wave in elastodynamics. Thus total transmission occurs for .1m = 0. However, even when m1 = m2 • slow waves er MCODd kind P2 waves are always generated. This is understandable because it is tho ~tual movement of the interface (and not the additional macroscopic stress, which is 'zero for m1 == m2 ) which generates slow waves in return. For m 1 = m2 , this effect nevertheless depends on w/O>c to the first order (factor 8182) whereas it depends on (w/w1 112 in the general

case. .

~-r;_ '---' '

~".;.

'-' ~

'

253'

WAVII . . DSI~·

It shoulcl ftnaUy be noted that the permeability .of tbe interface .rc, bu ao eft'ect (to the tirst onlcr in CIJ/of} ncept t1uuaJb the factor f 1 82 [aee (6.18)]. aad hence only for slow waYeL This is UDCientaa4able becaute it is always· the mac:t'OICOI'ic moW~DCtDt of tbe interface that can generate a fluid flow MI'OII:dlc iDtelface ill the ·ctirectioa Gppt)Sitc to the propaption direction of the iacident waws. Equations (<6.12) and (6.1S) to (6.19} pvc Dllults correspoadiag to a number of limit . cases. Let us therel'orc auume thatoeeofdle two media is impermeable. Tbis correspoads to Ulumina for this medium aa iBflnite ~ anautar frequency [.1'" -= 0 in (6.12)]. Now let u assume that the iDc:ideat waw canes from tlae permeable medium (i.e. crfz ..,. + GO). This does not aeacnto U}' aipifiamt~YC ChaJlF for tbe problem of the iacideat S wave.(aft- w.~ .... +eo (6..UO•. Fer th8 problem of an im:idcnt P waw from. a pcrmcabk naedium,. _Ecp. (6.1S) ta {6.1sa) show that, to tbe aearat second order in ...;;;;r;;i:

a

H«ll==l-Z

1 + z•

y
p; R«ll _ ~

; ..,...

(0 ( .d ...-..·up l+Z of

Jt)2

(6.23)

Is is clear that no P2 wa:ve is ~ in the impermeable medium (T' 21 - 0). The signiftcant cft'ect is that the two-pbue daancter is only exerted on the reflected P2 wave to the second order in and no lonsef to the f1rst order, as in the case of two permeable media. This is understandal\le in the tllht oltbe forqoina explanations, since, with respect to the elastic case, a macroscopic:.,..._. efthe iatetflleecuealy have an influence on the P2 wave aencrated i11 tile ~ '1iledium. The lesser sipifteance of the result is shown by a transfer to • ~ order -"'· ~· Now, if the wave CO~DCJ from the impermeable medium(~ • Of, oft .... -+ GO), the efl'ec:t is obviouSly reversed (R( 21 - 0 and

...;;;;r;;i

ym ~ on (J)/iff). Another interestinlspeciftecase is tbe one in which one of the two mec:f'1a is a'ftuid.ln the limitina cue ol a. fluid, the parameters introctuced into Biot'a theory (Chapter 2) are

·~

reduced to:

-·

ll ==0

no c:ouplina force. unit tortuosity unit porolity infuute permeability, no dissipation no shear

;.1 == M

fl-1

a •l

~ ... +ex> ..,-

These conditions have an evident physical sipifteance which was discussed in Chapter 2. In ~. an infinite hydraulic permeability correspoo4s to an inftnite absolute permeability rc (see Chapter 2. :II" • "'"· where If is the viscosity). The boundary conditions (6.9) correspond to a perfect OWd, IUdl that the viscosity is zero. ·We observe a certain paradox. becaue the porous medium is saturated by the same fluid, aad its viscosityistbesoaroeofnon-nealiliblediaipativee«ects(flnitepermeabitityoftbeporous medium). But in fact, this paradqx -~ - ""rWtly exist. The diSsipative elfects are actually due to the fad that fluid. Dow occu'n within a complicated pore network which

'-

,_

(6.~4)

•

'~ 254.

6

WAV£5 AND INTERFACES

~·

slows down its proBR*ion by giriag rise to shear stresses of viscous orilin at the liquid solid interi'ace. In tbecue of a luid medium. this interface does not exist, and the assumptiOA of a perfect luid can be ablined. because disaipatiOA within the Ruid is nqligible far from the walls lor I~ viscoJity Ruids. Substituting (6.24) into (6.1 5) to (6.19) to determine the case of the Ruid/porous medium interface is in fact irrelevant, becaual, accordins to (6.24), w" -+ 0 and the limited developments in w/aJ~ for tludluicl pbue are ao loftFr valid. The entire problem lllUst therefore be reconsidered, butitasedoaboulldary eonditions(6.9), to develop a function of wlaf whtn of is the characteristic aagutar frequency of the other pcnneabte medium. Geerstma and Smit (19tH) pafonaed the Corresponding computation. The qualitative results agree with the tJC110l'81· reaul&s, .ia other words the pneration of reflected and transmitted P1 waves with a firit order eotrective term in j(i{ai, the generation of a slow P2 wave in the permeable medium with a ftrst order coefftcient in j(i{ai.

Approximllte 1fU18IIitlllks oftlll ,_,.,.,._ We have used the experimental data of Wyllie et al. (1962), concerning three types of sandstone with widely differing pcnneabilities:

c.

porpsity t; = 29.7% ·permeability K = 1900 ml>

Teapot sandstone Berea

~ndstorie

porosity

f

=·

pel'JDel.~ty

i 9o/o K

= 200 mD

FoxAills saa4t~ ..·' porolity • ... 1.4% ~y K- 32.5 mD The saturating fluid is water. The~-- data a~ stun~ in the following table: wa~ <~turadaa ~) . . VISCOSity ...................•...•. ·...................... . Density ........................ ' ....................... . Sound velocity .....................•.....................

1 cP to* kl/arl lSOO m/s

~

GraiDS (silica) Density ...............................•.•.•••..•••....... Bulk modulus ............•.......•... ~ .........•......... Matrix Berea sandstone P wave velocity S wave velocity T~pot sandstone P wave velocity s wave velocit:r F oxhills sandstone P wave: velocity

2650 kg/m 3 3.19 x 1010 kg/m/s 2

................................ · · .. · · · · ....................................... .

3670 mjs 2170 mis

.. ' ....... ; ........ • .... · ... · · · · · · · · · · · ·

3048 m/s 186S m/s

. . • . . . • . . . • • . • . • • . . . • • . . . . ••.•••.. • .. · · ·

44SOm/s 2S1S m/s

...........,...•••....• ' ................. .

S wave velocity •.•••.. ~ ••••..•••••.•, ••..•••.••••.••••.•

Tortuosity is assumed .to be constant and ~ual to a == 3. . . ' '~

...

:;;r·~-

"---' '-....-

·~~~yft)'~OW..

~

255

We have considered two types of ititerfage~ (a) The interface between two porous media (two of the above tbree Ulldstones). (b) The interface between water and ono of the above .lbree sandstoDCI. ·

In the ftrst case. we consider an incident P wave (Fie. 6.3) and an incidentS wave (Fig. 6.4) coming from one of the two porous media. P1l

lA(1)

Par-medium1

tRC2l

JOl

'--·

yC11

~

rnecllurft 2

6r--------------------------, '-

4

1-<2)1

3

-'5

Prop.t•f2

I-J2

~-

,, 0

--~--'!.~!~_:..

···········-·········"" ..................................... -~

rl' . . . .

•••···•••••••

.

0

·~

F......,....CIIHzt

28

30

ofthe~lilileclioecoetftcleM

R'-dtbe 1\ wave ....._.toModulwa the elutk rel'ectioa ...-..(ot tbt P, wawt • a fuDctieD (cue an wave). Tbe curves identilted as

... U ''--

\0.

..,_lilftdatJFOldtills Uflllltt.

or frequency or incident P1 Me follows: porous medium (I)/porous II'IO!Itium (., The wave approaches from medium (1). Note that. for the reftec:tion and traDsmission <:Oeffic:ients 11< 11, ym and 2l, the quantities A.. A. ., , and are lower

T<

R
I

IT(2)1 T_.

than 10%. The symbol el stands for elastic: and meus that the reflection or tnnvri,;Oil coeffiCieats are~ at-. ia.-.betwelm two dutic media or that daae ooctT&<;icnts are QQIDPIUid.at ZeFO frequency.

'-

ln-tbe IICODCI caK, we consider an iacililent I' wave comiat from the Suid medium (water) (Fig. 6.S). To identify the differences betweea poro-clulielllobavior aa4 purely elastic behavior in terms of wave roflectioa and tru•i!lsioa. Fip. 6.3_to 6.S show. as a functioe, ofliequency (up to ~ut 30 kHz. i.e. frcqueQel used in aco~ wdl-loqiq expcrhDcnts): ' (a) The modulus of the mative .deviations of poro-'elastic Rfltction R", and

~

256

6

WAVE$ 4ND lii.IERFACES

transmissiQn T 1il coeft'tcients ofthe P1 wave(i = l)and the Swave(i = S)in relation to the elastic reflection (R.,) and transmission (T.,) coefticients (i.e. at zero frequency hence zero Biot effect) (Fags. 6.4 abd 6.S). (b) The modulus of the reflection R', and transmission "fC3tcoell'lcionts of the P2 wave related to the elastic reflection and transmission coafticients respectively of the P1 wave (Figs. 6.3 and 6.5).

s RfSI ...._medium1 ...._medium2

4r---------------------------~~ jR(S) _

RJ

3

laJ "2 ·0· 1

0

Fie· 6.4

~

- .1 'f ...... ----~~fioldlit ..," ............ :r.;-;r.~-

• 30

20 10 fN~a~lncvlkHz)

Modulus of the relative deYlation

IRfSlR~ l

$...,_

A.,

or the poro-elastic

reflection coeffacient in relation to the elastic reftection c:oe8'1Cient as a fuJM:tionoffreq~ofaa..._a

Tbe~ideQcif'IOd:u

follows: porous aaediual (1)/porout ....._ (2J.l"he waw apprcwfaea (rom Jlllldium (1). Note tJiat: jl"'t - T.dfiT..! < 1% We shall emphasize the following points: • Effects on P1 and S wawi are anater inldection thaD in transmiaion. In the case of an incident S wave (Fig. 6.4), the n6lctiot) deviations, although sli!)lt ( < SOlo). are always greater than the transmission deviations ( ronounted for the ·s wa"" than for the P1 wave. Thus the relath·e deviatien of the poro-elastlc' teftection coelftcients in relation to the elastic coeffiCients. although slight in the cased an incident S.wave (Fia- 6.4) ( ~ 4o/o at 30 kHz),

~.

~

•/ :uu ' •• );•' i•

?::,·:_~.if

J'J ,

...,....,........,..

yl21

·~----------------~~--~~~ jattl_ ...~"20

~

"-·''"

_____ ... _... .....1......_~=----.,__.. ..... ...,,............... :;..,...-"................-..

..

-·--

0

---:········~-·--···············-

10

20

•

Fnquency lkHit

(a)

•r---------------------------~ 10

trt''- yJ,. jTJ

\

Pro.,.'t '" 5 -~

-~"::-·--·~~

___ :. .. ,

,. , _ .... - - ---.:.,,,~-~-~............... ·····-···-····-·············--··········-··· 10

<•)

10 f~·Hzt .•

•

40

.--~-------------------,

!;.

~"(e)

,..,,•12 ~-~~-...... -------... ----~..-n. ........... --·-····~··-·--······-······-··················

to

io Frequency lkHzl

F1a-

6.! Modulus of the ftllative deviatiens of porCHiaStic mlection coeft'tcient IR'1l - R.d/IR..I (4) . . and transmiuion coeft'tcient OT111 - T.,) /IT.d) {It) or the P~ wave in relation to the elutic reftection and transmission coeft'tcieniS ~ively, and modulus or the por04laatic transmission coelftcient Tl 21 (e) or the p 2 wave related to the elastic transmisaion. ~ . . . fuactioA of rrequeacy (cale or an incident p 1 wave).

"--'

•

258

WAVES AND JSTERFACES

6

is greater than the same deviation in the case of an incident P1 wave (Fig. 6.3) ( ~ 1%) which is not shown. • As predicted. the effects are much greater if one of the two media is fluid (see scales in Fig. 6.5) than if both media are porous (see scale in Figs. 6.3 and 6.5). • The reflection Rf2 , and transmission T'·n coeff'tcients of the P2 wave are not at all negligible (Fig. 6.5c and also Fia- 6.3) compared with. the reflection and transmission coefficients of the P1 wave. Thus, in thecase of Teapot sandstone (fig. 6.5c), the ratio of the transmission coefficient T 121 of the P2 wave and the elastic transmission coefficient ofthe P1 wave reaches 30% at 30kHz in very permeable Teapot sandstone (K = 1900 mD). • Tho deviations are proportional to the square-Qf the frequency (Fig. 6.4) in the case of an incident S wave, aad. to the square root of the frequency (Figs. 6.3 and 6.5) in the case of an incident P wave. Moreover, the elfocts are stron,er with more permeable media. • It must be emphasized that dle orders of magnitude given above for the transmission and reflection coefficients concern an open interface. The effects are slight for a sealed interface (one order higher in w/af).

In the light of these results, it appears that if both media are porous and permeable, or if one is fluid, the P 2 wave reflection and ta,lnsmission coefficients are not at all negligible. The generation of reflected or transmitted P 2 waves participates in the total energy balance and contributes in a non-neJligible manner t~ alter the P 1 wave reflection coefficients. However, owing to the high attenuation of P 2 waVes in comparisoo with P 1 waves (see Chapter 2), direct observation oftb~former appears to be impossible. Hence the non-negligible signature of the P2 waves must be found on other types of wave: In conclusion, it should be noted that the main influence is exerted on reftection and not on transmission. and that it must themore appear .Jti the recordings. Th.e two media considered must nevertheless both be permeable and the interface open for a P2 wave to be generated with a noticeable effect:

6.1.2.2 Aaalysis of reflecdoa CNI die free surface of ll seml-inf•nite saturated porous medium We have selected this specific case to IUgblight another important property of reflection

processes in saturated porous media, namely the existence of inhomogeneous waves. The computations raise no theoretical difficulties, and it suffices to satisfy the boundary conditions (6.8) by superJ,esing tbe incident wave and the reflected waves necessary to carry them out. However, the computations remain cumbersome and we shall only stress the qualitative character of the results obtained by Deresiewicz and Rice (1962). A wave is defmed by its potential II such that :

II

= 110 exp i(wt -

k* • r)

(6.25)

where k* is the wave vector and r the position vector of the point considered. Due to attcnuatiQn JllCCbanisms,,k• is a ~plex v~r: k* • k - iA

(6.26)

• • 110 up (-A. r) up [i(an- k. rt]

(6.27)

which gives for (6.25):

-,

~

~-

-T

I

·~

I

6

2S9

~AV!SANDI~

The norms of vector A and k arc written:

I

tAl== A

(6.28)

fkl- k

where A, inverse of a Jenath. is an attenuation, whereas k is the wave number such that: kV = ro (6.29) where V is the wave velocity. A wave is said to be inhomogeneous if the \"ectors k and A arc not colincar. In other words, the planes of equal amplitude (A • r = constant) arc not parallel to the planes of equal phase (k • r - constant). These inhomogeneous waves arc encountered in another coatext '**ly comparable to the one examined here, that of the reflection at the interface of two viiCOelastic media (Bourbie, 1982, BourbiC and GonzalezSerrano, 1983). This problem wiU be dealt with thoroughly in the next part of this Chapter, which can be referred to for a more detailed analysis of the inhomopncous character of the waves. The most important qualitative results for the problem examined here are the following. In the case of an incident P1 or SY wave, the charactcristM:s efrcftected waves of the same type as the incident P1 and SY wave dift'er relativfiy IJiPtly from the same waves rcftected in elastodynamics. They are homoaencous, sliahtlY dispersive and dissipative, and this dissipation depends on w/of to the lint order. By contrast, the reflected P2 wave is highly 1

dispersive and depends on (ro/of}¥io the fu'st order. ltsattenlMtion is proportional to rofw• l .

.

in a dircctioR panllel to tbe free surface, and to (w/of)-l' in the normal direction. Deresiewicz aftd 6.'e (lf62) auaamcaD)' analyzed the expcrimtwltal da1a ofFatt (1959) correspondina to a kerosene-saturated ~."tortuosity was assumed to be 1.01. We Jive here the results~ to an iDddent 1'1 wave 4:w wbidl.thc Biot effects arc particularly important, especially at low anatcs. of illCidcace (Glose to DOtlD&l incidence). This is due to the polarization of the wave involvina fluid phase movements pcrpendic~ to the free surf.,.,. For an incident P1 wave, the variations of phase velocity normalized by velocity Vr as a function offrcqucncy are in the neiabborhood of 10-" and 10- 2 respectively for reflected P1 and SV waves. For the retlected P2 wave, these.variations arc about to-• (see Fia. 6.6). Apin note the sliaht difereDces between the dispersive eharactcristic:s of the inhomoaeaeous waves (see F"tJ. 6.6, solid curves) and the dispersi:re characteristics of the homogentous 'WtWeS (dotted curves in the same ftgarc), analyzed in Chapter 2, and propaptina in the same porous medium (kerosene-saturated Sandstone), albeit infmite, namely in the absence or diJcontinuities. The attenuations arc much hi8her in the direction perpendicular to the free surface thart in the parallel direction. This was forelceablc, sinc:e. durina these .reflections, the Biot effects occur, especially for flows normal to the discontinuities. It seems clear that, during reflection, the free surface condition will considerably favor the differential movement of the liquid phase in relation to the solid phase, essentially in a normal direction to thC surface. The difl'erential m~ in tbilntiNc:tionwill theft be the main souroc of energy dissipation. For the rcftectcd P1 wave, attenuation characterbed by the ratio A. k is too low to be signiftcant, this ratio being about to-•. For the reflected SVwave, this ratio is about 10- 2•

'---

•• 1.()()04

.3 Yp __:1..2

v,

-.::..1 Yp UXI02

lnhofncla-neaus-

Vp

'

0

1.0000

I

s«:

I

t

0

r

3

Vt/fc

3

2

vm;

2

0

I

I

.I

:::.. Yp .49

.48

0

1

vm;

2

3

· Fia- 6.6 P . ve1ocily of il'l~ r~ P1 wave (Y,..~ P2

wa~ (V,.2 ) and S V wave (J'.)(solid lines) compared with the phase veloCities of the saae types of. wava(P1 , P2 ancl·SY)ill the eae olullo........- IDII)de.of propaption .ia aa iaMR. I&'IJ8dium (dotted ·~ 'l'boee ~ are

~ .~· tJac vcloatf of '• ~vca (DO dissipatiOil). ~.abtcissa is the square root ol the bquenc:y.~ to the~ hquency fc (cue of aft incident 1'1 wate). 111e iatilo VJ.!Yr is ta'bll eqlilld to 0.483 (noadiMipati-ve c:ue) (ilftei: DeaMwicz aad .Rice, 1~ .

~ :t \ 2L

l 0

0

I ""'

r,-o~ 1

I

=

;

•..~.

t__.. ...

~

2

~

Fit- 6.7 Normal, ceaaponeot of thc atten..UOn Vfllilt"' for the ..aow wave normalized to the norm of the propagation vector as a func:tion of ,1flfc (incidence P1. wave) (after ~ and Rice. 1962).

.----..,

~

-~...___

-

~-·

6

261

WAVES AND IHTUPACES

30

r I -

r3 p1 p2

I

I

0

30

10

r,

sv

10

·~--------------------------~ flo.U

20

\fiif:.,: ·4t

r2

'

·.

' .. ' . .

-

Vffc· .1

~or~c,or <•

thAI Pa wave and{3 of

SV wave

vs. anateof~ { 1 ohbe P1

1Rve 1962).

Deresiewicz and Rice, .

ViTfc. 0

8

30

10

to

r,

The intcrestiq qualitative. aspec:t is that, at the

.alnc frequenq, it rises with ana1c of

iDc:icJeDce (defmed in relation to - normal to cfie free surface). This results from the peiMilatiea of perticle 1DotioD ·J*pCDdicular -~ tbc propaption direction. For the rcflccted P2 wave, the normal component of attenuation is very hip (see Fia. 6.7) b1U is pradically independent of ansJ.e of incidence.·

or

or

Fipn 6.8 shows the variations anales reflection { 2 and { 3 of reflected P2 and SV waves as a function of anpc of iaddence C1 of the P1 wave. It is clear that the angle of reflcction ef the P1 wave is tbe .... as tUt of the iac;idcBt waw. Moreover, DQte that the variation of ar&ale of nflec&ioa ', .. a ru.:tien of antic of incidence differs from the clanical SaeiJ.Descartes law of er.t~: This eift'ect 1Mcomcs more pronounced with increasing deviation from· aiOimaJ incidence. The anJies of reflection { 2 and increuc with the ratio . the ratio with which the effects due to the two-phase character of the studied- media increase. This can be explained by the dispersion of the reflected waves which, conversely, impoee an increase in the anpe of reftection as a function of frequency at faxed analc of incidence, to satisfy the Snell-Descartes laws.

'1

.;c;r;;,

c,

1.0

-l

I

182 0 (I)

I

IR1>/ 0.2

...

/

.........

"'

L

I

I

e,

I 180

178 90

f1

30

0

80

30

0

0 90

80

r1 0.06

0.04

IR2'!

r

I

fife• 1.11

.~

0.021

.........

I

.:t------90

A .1

of

. a:t<:: 30

0

., 80

r,

o.e

0.4

~

m,:m

~

'

-

··~-_::I o____ ~

---

30

0 .·!·

"'··

80

90

r1

'""'"~·.-.

180

jacslj

113

171

l---.

tl

I

Z1.a-=-

..

30

80-

I

I I

'. I

I 90

r1 30

80

r1

Fla. 6.9 Moduli fRt"' and phde 11of reflec:tiot1 ~ts (in potential) of P1 (i • 1), P1 .(i• 2) and SV (i • l ar S) waves u a function of anJle of incidence ' ' (~ 1'1 wave). ~~Yel ~e in4exod by the nonnaliad frequency f [. (after Deresiewicz. qd Rice, 1962).

.,

-----

"---'~

r I

-~f '

I

I

'WA\U'Aktr~Ad!s

6

Figure 6.9 shows the

,-ariation of moduH and phases of refteC:tion coetftcients RU1 U= l(P1). 2(P1), 3(SY)) vs. the anJle of incidence { 1 • Here the cOefticicnt All' is ctef'tned from tbe potentials related to the wave studied by:

T

-. ....

-I

-,

I

::r ~

..-

'-

...,-

_1

:1

.J

263

j

= l, 2, 3

Ru1 =

i; = IR
(6.30)

1)

where ()0 is the incident wave amplitude. Note that the reflection coefficients corresponding to the P1 and SV waves differ slightly from those calculated in elastodynamics (i.e. w/ol = 0 in the curves). The only signiftcant difference is a wider phase d~rence whidl is independent of the angle of incidence. as compared with the elastic cast for the SV wave. This increase (in absolute value) results essentially from dissipative effects which slow -down particle motion. Moreover, as above, for a ftxed angle of incidence { 1 , the more the conditions for twophase effects are favored (rising w/Cif), the larger amplitude P2 wave is generated at the free surface, which was foreseeable. In fact, coetTICient IJIC 2 ~ increases with w/ol' for a ftxed angle of incidence and by contrast, the reflection coetTtcicnt IJIC 1 ~ corresponding to the P1 wave decreases to satisfy the enersy balance. The SV wave reflection coeft"tcient IA«511 displays slightly more complex l;)ehavio~, which we can S11DUD&rize as follows: At high angles of inc:idence (about > 650), and due to its polarization, the SV wave sets partides in motion in the solid phase. esscntialy in a direction perpendicular to the free surface, while the liquid phase rerbains practiallty at rest, since it does not transmit shear forces. This has the eft"cct of generating difrerent'-1 movements between the solid phase and the fluid, which induce two-phase effects, fuoring the generation of P2 waves and consequently the attenuation of SV waves. As ptedicted, the effect is intensified with rising frequency. At angles of incidence close to no~l, the mechanism described above is no longer effective for the s1· wave, but for the I\ wave. To summarize, an incident P (or SY) wave at the-interface between two porous media gives rise to three types of reflected wave (P1 , ]\,and SY) and to three types of transmi_tted wave (P1 , P2 and SY). 1bt wavtlitt the iatcrfacle: ..........., ildaolnOI'ftCO\JS (planes of constant phase diltmct from planes iJr CODitUt'amJiitude) except c:asc of the refJccted wave, which is of the same type as the incident wave. Moreover, the Biot eft'ccts on reflections and transtnissions are more pronounced at aqles of incidence approaching normal, and hence ~ no tcmlt; be ipored. -

in-.

6.1.3 laterfaee probltllll ltetwee11 porous ua.atetl metlia Applicadoa to aeOUitic ......... 6.1.3.1

Jaterface ·wues

•• ll•ylft6h ··~~·

In classic elastodyrWnics at the f~ surface of a _._infmite .,lid, a so-callocl surface wave can propagate, namely the Rayleijh wave (see for example Lord Rayleigh, 1885 or Viktorov, 1967). This is a non-dispersiye inllemopneous wave whose amplitude decays exponentially with depth. Particle motion O<:f\-1" flong two transverse components with a phase difference or 7r./2 and contained in the Ulittal plane (plane deftned by the wave

'--

264

6

W.\VU.AND JNTERFACES

vector and the nonn,alto the$~). The ~~t~tmnity ofthe polari,zation vcctor.describcs an ellipse in the direct.ioa retfoar• near the~ aad PfO&rade in 4cpth. The horizontal component is cancelled at depth Q.2 118 , where Aa is the associated wavelcagth (sec Fig. 6.10). MJJJJJJJJJJ JJJJJJJJJJ

JJ)JJW)j)))))))

~

JC

Prclpegition

~~

nnn JJJj ns;nn;j;;ilnn ;;;;; ;;;;;; Jh;sJ; --... x

l

ftrOIMIJition

'

z

AW2

lbl

'

f'll, 6;.. DistriHtioo.otveklfity fleW (a) ud 4i~ <')of particles ol Ill isotropic scmHafmi!e medium at the passage of a llayleiah WJVe (after A\lld, 1973).

.

..

We shall now examine what happens to this Raylciab wave if the propaptioo medium in Fig. 6.10, for the same geometry, is a saturated porous medium described by Biot's theory. Let us consider potentials of the form:

== t;1 ~ (- r1z) exp [i(kJx- a>t)] j == 1, 2, 3 (6.31) Each potential defmes a P1 U == 1), P2 U a; 2)aad S{i • 3)waveasindic:atedin Chapter 2. ~J

To identify a surface wave of the Rayleigh type consists of determining whether the boundary conditions (6.8) can be satisfted by a superposition of waves dtfmed by potentials of type (6.31). If this is possible, the real part Re(kl) and the imaginary part Im (kl) in (6.31) respectively defme a Rayleigh wave phase velocity V8 and a corresponding attenuation ~. in the ox directioa ~ (()

~e (kl) ""' v.(w) 1m 1kl) == «a(a>)

(6.32)

''

WAWiAiM~Aa!s

6

285

The general eafculatiolll oorrespondins ~ dais procedure were performed by Deresiewicz(1962). Based on these c:alculatiolll, oneoan clemoBstrate that an expansion in w/of yields the equations which must be satisfied by Va and «.a in the form: 1

2

1

1

../i ,.2q, v. (•- v!)1(w)1 _4 (•-~)2( v~)l- (2 _ v~) 1 _ Y,. g V,. V of Ys . V 5

5

(

6.33>

vl t - 2 V2) - .fi.«a {(3 vivi V2vi- 2(V2vi.,. . 1)} ==

_(w)

l

2

of

~'-<

'----'

"---

''-.--"

'---'

·~·

~ ,.~
g

V5

Y,.

V,.

V5

(6.34)

Rayleigh's classic equatioll(wbere ro/af • O)can be RJCOpized in the second member of Eq. (6.33). The ftrst member, where m and (J ~ de(ined in (6.17), corresponds to the twophase effect. This ftrst member depends on theJrequcncy. and heDce, contrary to classic elastodynamics, Rayleigh waves in saturated porous media are dispersive. Figure 6.11 shows the Rayleigh velocity as a function of frequency, and also attenuation, or, more precisely, 1000/Qa where Qa is the quality factor derived from (6.34). The ratio of the bulk modulus K 11 of the fluid (water) to the bulk modulus K. of the solid constituting . the matrix (silica) is kept constant and equal to O.OS9(- K1 ,JKJ:. This value is the same as for a water-saturated Berea saadstoac(Wyllie «&.1962). Asia Claapter 2, the ratio of the bulk modulus of the open system to the bulk modulus of the solid has also been varied (i.e. K 0 /K.- 0.2, 0.4 and 0.6). The broken line, solid line aad dotted line curves correspond respectively to the pGft)lilies t; ... 30, 20 and. 10%. The other parameters involved are derived from Gassmann's equation •d Berrynia11's equation for tortuosity [sec Eq. (2.92) in Chapter 2]. It may be observed ia Fig. 6.11a that the velocity fint decreases with rising frequency, and that, at a pvea frequency, it increases witb the ratio K 0 /K, . .This is easy to understand: liace the surface is free~ the pressure p ia ..., fluid is :lAII'O on this surface and a depth gradient op/o: is created, c:ausina differential m0¥011leftt between .the ftuid and solid. The movement correspondina te the·~ wave iS obtaiaed by the superposition of three potetltials(6.31),eacbc.:oi'reSpondina to a P1 waw,an Swave and a slow P2 wave. As sboWD ia Chapter 2, however, the P1 and S waves have veloc:itiel which rise very slowly with frequency, whereas tbo P2 wave has a vetodty wbicla inCreases rapidly (from zero at :lAII'O frequency). Hence as the frequency rises, the contribution of the P2 wave dominates. Since the velocity of this wave is much lower than Va(O) (the velocity of Rayleigh waves at zero frequeacy), the velec:ity oldie Jlay,lei&h WM1J4ooroa.scs-' the frequency rises. This initial dec:nue is mdeady leis pronounced if K.ofK.. approacba 1, because this causes a less pronounced difFerential movement of tho ·ftukl. At ·hilber frequencies, the Rayleigh, wave velocity increues as a function of frequency. This result agrees with the ftndinp of Dercsicwicz, 1962). This is explained by the fact that, at the higher frequencies, the P2 wavei are too attenuated to be sipifJCant. In this case, the increasing effects of the P1 and S wave velocities prevail. The curves corresponding to 1000/Q11 (Fig. 6.1lb) vary in the oppotite way to the curw:s correspondia; to the velocities (i.e. inereasing at low frequencies, decleasina at hilb &equencies for frequencies 110t plotted in Fig. 6.11 but studied by Deresiewi<2, 1962). This is ~rtnal. becaUte the more significant the P2 wave amplitude the jreater its attenuati~ and "'" wrsa.

266

6

WAVES AND INTERFACES

,.,......, 1.012

.. _,' .

~,

'"''

/

10

,.,.....

20 30

f..·~

"'"" .......·:;/1 ... ··

/

- - ...... -:...: ... ··:·-:;;:..·,""

VR

Vo

7

.4

~

1.000

I

········

.2 .. ""

,t;"

-- , ...

.1198

.4

.2

......... ....... ' _,______ . --..... -··· ..~.2

~

Cal

~--···-~ '

~

.oa

0

.08-

.•

.12

'Ks

l

.15

fife

I i

20

~ity

...... 15

-

""'' JO

~.

~---~~-L . . ~...

!...9!!!

/

OR

/

10

,./ -~ ,

(

.2/ ., ,.,

, ..· "' .. ··

'

.... ······

.6

(

········.4 ·······:·····

( I

5 (b)

0 .0

.03

.08 fife

••

.12"

l

.15

F'ia· 6.11 Phase velocity 'fa) aad ittvreild ctuality factor<') of the Rayleigh

wave. Tbe phase Velocity is~ to the phase velocity in the absence of dissipation (i.e. at zero n.~). the frequency is. normalized to· the

characteristic frequency fc. The curves arc drawnfor varyina ratio Ko!K•.

b. Stouky

WIIHS

In classic elastodynamM:s, at the iate~ of two. elastic half-spaces, a aeneralizcd Rayleigh wave can propaaate, wh~ amplitude c,iecreases ~xpooentially with distance from the interface. This wave. identified by Stooeley. does not-always exist. Its existence was investigated by several authors, and particularly by Cagniard (1962). The reader can ~

'""'

'-'-·

'-

t/Jfhl$ ANfi~Aeil

6

261

senerat

'-

'-

refer to MikloWitz (1978) for a diiCUuion. For the ease ofa solid/fluid interface, the Stoneley wa~ always eXists. . Now let us consider the case in'wbich one of the two mtdia is a perfect fluid and the second a saturated porous medium. To identify the existence of an interf8ice wave of the Stoneley type consists in determining whether the bOUndary conditions (6.9) can be satisfled by a superposition ()f waves defined by potentiak similar to (6.31) [thn~e potentials (P1 , P2 and S waves) for the porous medium, and one potential (P wave) for the fluid medium]. "fhesecalculations raise no difl'reulties but are tedious, and the reader can refer to Rasolofosaon and Coussy (1985). Figure 6.12 shows the variation of velocity V5, of the Stoneley waves and the ratio 1000/Q51 where Qs, is the ctuality factor, as a function of frequency for different values of

K 0 /K•.

1.CIOit'' . YSt

~ Open

1.ooi0

Vo

.til (a) 0

.01

AMI

.12

.()8

fife :q~:,

'-

.a

_....,_ ::ru~l .

'!Sa.

Ost

. .3•KofK.

---. . --------L-?tttz=--=--····--··· .....

_.... -~-

..

..

.oe

fife

Fie- 6.ll Phase

.

a;;-:;.t J (It)

.12

velocity (a) and invene,cpility factor (lt).of the. StoadcY .· wave. The phase velodt¥ is •onu.lizecl tp the pbase velocity in the a~ of ~ip&tio!l.
'

268

6

WAVES AN~ INTERFACES

The PQrosity is assumed here to.~ 19o/.. and tortuosity is 3 (see Berryman's formula. Eq. (2.92) in Chapter 2). The ratio of the bulk modulus K I«. of t)lc Ouid (water) to the bulk modulus K. of the solid constituting the matrix (silic.~t.) ~ kept constant and equal to K 11 /K, = 0.059. The solid curves comspolld to ~e case of tbc open Ouid/solid interface (i.e. JC1 = ao) and the brokeo.liQCS to the case of the scaled or the closed interface (i.e. "• - 0). Like tbc R,aylciah wave in the case of porous media. the Stoneley wave is now dispersive. For a better unders&andjng of the curves obtained. Fig. 6.13 shows schematically the geometric representation of the increase in the ratio K 0 / K, at constant porosity. Poroua medium

fluid

Poruulllllllium

Fluid

Fluid-=· Fluid__.

Fluid

Fluid

KJKs "\.1

lowKJK,

Fi&. 6.13 Schematic diagram of a Ouiclfconstant porosity porous medium interface. Note the influence of K 0 /K. Oft 40 pore sh'e". ·

-,,

To presume increastDJ K 0 K •• at constant t/J and hence at co~tant saturated connected pore volume, corresponds in r.ct to tic ~x.i:stence of illCldsinglyJme pore channels (see Chapter 2). Let us ftrst consider the case efthc scaled-mterface. f"br this sealed interface, no fluid exchange occurs between the porOU& medium and the fluid medium. Hence there is no contribution from the P2 wave in the porous medium, because of the absence of any possibility of movement out of phase between the overall system and the fluid part. Accordingly, the fwo-phase effect will only be exerted througb 1'1 and S waves. Since the velocity of these waves increases with frequency (see Chapter 2). the velocity of the Stoneley waves will'also increase as a functioll of ftequency. These dift'erential inertial effects, increasing with frequenq, will incrnse IB~dilference between the motion of the fluid and the overall system. This differential movement is respollsible for attenuation phenomena and therefore implies that l088LQ61 increases as a fUnction of frequency. At a given frequency, these effects Will obvioutly be less pronounced for a rising ratio K 0 /K•• since, in this case, dift'erential movement will occut tt.ith lesser facility. Let us now consider the case of the open interface. tn this case, fluid exchange occurs between the porous medium and the fluid medium. In the porous medium, this fluid exchange allows for a contribution from tile P2 wave, for which the Ouid movement and the overall movement are out of phase. Thus the case of the open interface for the Stoneley wave leads to the same phenomena as for the Rayleigh wave, which explains the variation in velocities: an initial decrease, followed by an increase as a function of frequency. Signif1caot difFerences rtevertheless exist. In fact, the pressure gradient prevailina in the ftuicJ at the interfa~ is l(,)wer for smaller ratios K0 /K •. For a suft'1ciently SQ1all KoiK•• the channels arel10t so. thin (riC- 6.13a) and

•

..-....,

'

---~ '---' "-./

'-

''-' '----'

'-"-./

'-

'---

"-

"-----~

'----

-~

"---'

'----

WA'WIAHB~

'

:!69

the reduction in cross-section throush which the fiUidftiJws toWards the porous medium is leis drastic. Tbe- pteiiUI'e 8f'8dield is tbenlf'ore llal'fOWW, Iivia~ rile to a sanaDer Ouid cxchaftae between the two mediL The contribution of the-~\ wave, as well as the dilsipation d• to Row across the iaterface, is ......... ICCOfCiiatly. 1'llis aplaiDt why, in contrast to Rayleip waves, the velocity eurveaconapondinsto K 0 /K, • 0.3 lie above those corrapondins to K 0 K, - 0.6 and 0.9, and why, at low frequencies. attenuation (1000/QSr) is less for K0 /K, 0.3 than for hisher Ke/K, ratios. At hip fn:queocies, however, with increasins K 0 / K,, attenutiOn decNases because the attenuation due to P1 and S waves dominates in this cue. The velocity curve correspondins to K 0 / K, • 0.91ies above that correspondin& to Ko/K, • 0.6. because the channels become exceaively thin, and the pressure sradient prevailins at the interface is inadequate to auarantee sipiftcant flow as K 0 /K, approaches l.

=

c. Slulutulrl As a general conclusion concernina interface waves, we can state that the two-phase cbaracter of saturated porous aaoclia ~Mba 1M RaJleilh and Stoneley waves dispersi\'e. However, the effects can be iporecl fer the ltayleiala wve, both for Yeloeities and for attenuation (see scales in Fia. 6.11). The same applies to the velocities for Stoneley waves. On the other band, the attenuation ofStonelcy waves due to the tv.·o-pbase character is no longer neJlilible in the case of an open interrace. At low frequencies, in fact, an open interface leads to Ouid excbanacs IJetween the media, and hence a sipiftcant contribution of the Pz wave, correspondint to out-of-phase fluid/matrix movements. Dissipation and attenuation then reach a maximulllat a liven -frequellcy for intermediate values of K 0 / K, (K 0 /K, • 0.6). allowing large pressure ~tadiea:ts anCf SGbstantial fluid exchanges. Inertia effects become domiaant at IBP fN\quencies, and ateteetl on P, and S waves. 'fbeseefl'ects are also aecentuated by the open dtaracter of the. ietotface. For applk:ations to wen loainr. it is iftttrestial-to aulya the claaqe cauled by the two-phase character of the medium in comparison with the elastodynamic framework, wbea a seismic source is plaeed mtt.e·Yicitdty of a'I\IM/penlable rneclit.tm interface. We shall examine this problem in the next Section. ' -

6.1.11 .............. wJcilllty ., 11 fl:dtlfOIMI - · - .Wterface We have just examined the aeneraJ properties of sutface waves (ltayJeiab and Stoneley) propaptina at the pfaae surface or a semi-inf'mite saturated porous medium, and at tbC plue interface betweea a fluid and a saturated porous medium. These waves were a~ iadependently, ill other Words 1rithout poltUiatiDa any hypothesis on the source generatina them. We shaD now examine problems correspoJl'dinS to sources immetsed in a Ouid, near a petmeable interface. We shaD examine the thanacs aenerated by the permeable porous media in comparison with the classic elastic media. We shall see that the two-phase eft'ects alter the waves propaptinJ at the iatedace. irrespective of the cylindrical or plane geometry. The study wiD be carried out with the aim of fmdint a suitable method for determinint the permeability of porous media, tbroup its sipature in the ,enerated waves (i.e. modification of the recorded signal). We shaD f1nt describe an analytical method of resolution of the problem, and then analyze the ~results of sewral numerical simulations.

270

'

WAVI;SANP lNTEI\FACES

a. Alflllytical resolutitM qJ tlw ,...,_,

~

The problem is defmed by the pollletry of the propqating media and by the source characteristics. The geometries aaalyr.ed corrapond. for example. to the follo•ing problems: a plaae interfacx betwcea a 8uid and a saturated porous medium t Fig. 6.14._ a layer of fluid of constant thickness lyina between .two identical saturated porous media (Fig. 6.1 5), or a circular borehole filled with Ouid in a surrounding formation consisting of a saturated porous medium (Fig. 6.16). The sources are defmed by their aeometrical characteristics (point source. line source or cylindrical source) and the time sipal omitted (Dirac delta function, Ricker wavelet). All these problems can be dealt with in the same way. We propose below an analytical resolution algorithm. ~

Choice of a system of coordiDteL For example, cylindrical coordinates in Figs. 6.14 and 6.16 and rectangular (2 D) coordinates in Fig. 6.15 for the case of a lint source.

Defulitioll ef potelltial fuc:tioM. Tlaese potential functions serve to completely describe the vibration state in each of the propapting media: (a) In the fluid, assumed to be perfect and incompressible, and where the source is immersed two scalar potentials must be defined, one corresponding to the solution obtained without any soura; and the secoi)d coticsponding to the superposition of a source. (b) In the porous GJ.edWm. it iua~~ry to dell~ three potentials, two scalar ( 41 1 and 4Jz) and one vectorial 'I' ~5ec Ch~ 2~ Due to the spccif1c symmetries. the potential 'I' actually has OAly one aon-zero eoptpOnent, the orthoradial component in the case of cylindrical,aymmctry (FiJ. 6.1~ aad the normal component to the plane of the figures in the bidimcJWonal case (Fip. 6.14 and 6.15). ~doaofdlecrlltidoala...;....,._...,_• • . _

and the porous medium (Section 6.1.1 ),

"

The complete resolution of the problem hence amounts to the determination of all these potentialfuactions.lt is very coammietlttouse mecbods based on Fourier-type transforms (Bracewell,1978, for example), in die tilDe and spac:edomainin the two-dimensional case, or of the Hankel type (Oitkinc and Proudnikov, 1978. and Bracewell. 1978, for example) in the space dom~n in thC three-c:timeJlSional case \\
,...

..,t -

v12

~

'il't

u d). 4S(r) • == fi(t,.,(z iJ 11

1

where /(t) VI

= source fu~tion. = wave velocity in t~ fluid,

nr

for i

= 1,2

(6.35)

~"

-~

~

__

,

"\

z

-. P!nt:l

Wft!I*JI-.IOof

IJII/J/Il/1111//J!IIJII//f«U

..... --1

------

:

s

z

P!ftl:l

p

z 9

272

IS(z- d)

WAVES AND lNTERFA(;ES

6

;_r> =Dirac disrbution product and, ' {1 = Kronecke s delta 0

6 11

ifi=l if i :#= 1.

Equation (6.35) ex that the potential fl 1 destftbes the vibratory state in the infmite fluid in the presen of a so...rce (second member of (6.35) non-zero) and the potential fl 2 corresponds t the ··reflected" waves (second member= 0). After the time Fourier tra sform (t -+ w) and then the zeroth order Hankel transform (r -+ k) of Eq. (6.35), ~ utions of this equation are eafily put into the form: •

k 1

fl2

where

h} = k2

-

'

) _ F(w) exp (- h1 1z- dl) hI

z, w - 2w2

k, z, w) = fl2 0 (k, w) exp (- h1z)

(6.36)

1· ··

w2 V2 with Re(h1 > 0,

~notes

fl1(k, z, w) the ouble transform (Fourier and Hankel) of the potential z, t), 11>20 (/c, w) is an arbitrary runction ot k and w which are the conjugate variables of r and t in the two foregoing transformations.

fl1(r,

The necessary uniform behavior oflthe poklnlitl1s at infmil)' (radiation condition) justifies the condition Re(h1 ) > 0 and the rejection of the solution fl 2 in exp ( + h1 :). The complete resolution of the problem amounts to the determination of the unknown function fl 20 (k, w) which allows the determination of potential 11>2 related to the fluid, and the determination of potentials •., •~ aad 'I' related to the porous medium. These unknown functions are computed t.y bbfe.Vansforming the equations and determining the boundary COnctitfou to be;atisfied at the m.terfaces. These conditions are expressed in the following matrix fow: [M](X) '"" [}f)

(6.37)

_where • [X] is the unknown column matri'lt' whose elements are unknown transformed functions [for example fl 20 (k, w)] to be dttermined. • [Y] is a column matrix of the same dimension as X, whose elements depend exclusively on the source spectrum (freq\acncy and wave number). • [M] is a square matrix depending exclusively on the propagating media (geometrical and mechanical characteristics) and totaUy independent of the characteristics of the source.

"

In the absence of a source, the matrix [Y:J is zero and syatem (6.37) is then reduced to: [M][X) = [0] (6.38) where [0] is tbe column matrix composed of demnts that are all zero.

"

I

i

_j

...._,

"

, - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ----------·- ---

T .._,''---'

,_ :~

'---'

m

ftJmfftiij~'

'

In thisQSC, to deviate from the trivial sOlution {X] - O,correspoildiDJ to all potentials equal to 0. or to a si*e at rat, it is llecellary tocancdthecletermiuilt ofsystem (6.31)or: dct [AI) - 0 (6.39) Equation (6.39) actually constitutes the velocity dispersion relation ohhe system, which

serves to dctcnnine the characteristic propaption modes of w,vcs corresponding to the pomctry analyzed. Equations of this type were discussed in Section 6.1.3.1, enabling us to

~

describe the dispersion and attenuation functions of the Rayleigh and Stoneley waves. These characteristic modes are obviously independent of the source. In the presence of sources, the sotutions to system (6.35) are given by: '---'

X

det (WO] , - dct [M]

(6.40)

where

is the transform of the fla unknown potential. • [~ is a square matrix obtained ltrnplacina the ,.. column by the column vector (Y].

• X1

'----

Since the unknown transformed functions are all perfectly determined by Eq. (6.40), the vibration of the space is determiDed by takina tbeir inverse transforms. Thus, for example, ia the cue C'll F'll- 6.14, tbe.,..... dillribUtioll iac6eeuid is determiDed .. fDUows: By clebition: Pr(r, z, t) • - Pr[11 (r; z, r) + 4S~(r, z,' t)] '

'-

(6.-41)

where f'tckll Mel the fluid dcDiitJ (AcbaaiNichi lJU, for-ple). HCJl4:0,., Fouriw and zerotb o.- Hankel uaasrona cl (6M). we .obtaia!. ,

pf{lc, Z, fD) • p1aPf.t(lc, t-, Gf) + .2(~ %, m)J

\.._-

(6.42)

where •• and • 2 were previously detcrmiDed by Bq. (6.40). The pressure distribution in the ftuid is then obtained by inverse trauaform of Eq. (6.42), ~: ,

~

, A,r, z, t) - ;.

'-

f J

_+..,.., _+..,.., p/co•r•. (i, ~~ tiJ) ....2(t, z, co)]Jo(kr)t dk dco (6.43)

.. .....,.. of,.,.,...., ..........

'---'

The simatations that we pnwat were performed to.y lleeeabaum (1974). based on the expcrinaeatal data of Wyllie n Gl. ( 1962) (aee Section 6.1.2.lc~ Only two spedficconditions

at the iDtelface were ualyakt: (a) Free interface (b) S.W. irlterfaGe

·-..

t~'-

·

'

"• • oo.

"• -

o.

To aDalyze the eft'ect ofpermeabDityJllore pndlery'ia each simulation, the results for each type of sandstone were rcplacecl by die ·rMitl obbtibed for a fictitious equivalent material Qf the same dlarKt...... but widllowcr pmacabilitY.:

,_

(a) S mD instead of 1900 m'D for Tupot salldltone. (b) s·mn instead of 200 mD foi'lerea taridstofte. · (c) 0.1 mD instead· of 3l.S, 10 or' I iaD fOr Poxhilts Andstone.

~'---

_... '~

'

274

WAV~

AND

INT~RfACES

6

The source employoci is a pressure so~ who~ tfequency spectrum is such that its phase is zero and its moclulqs.c:oostaat bet~ frequencies / 11 and .f•• and zero.on either side of the frequencies f 1 ( < f 11) and f 1 ( > f.). Between f 1 and fs and between / 2 and / 8 the spectrum amplitude rolloff is a cosine taper (see Fig. 6.17).

I

l

'a

.

,,_..,.. 'H

'2

Fig. 6.17 Frequency spectrum of source function.

Two JeOIDetries wcte iiM:dipted by ROICilbaum: plaae (fia. 6.14) and cylindrical geometry (Fig. 6.16). However, it should be recalled that the Biot's effects occur essentially at discontinuities which, by the bo.undar)' conditions that they imply. induce pressure gradients which alter the flows. This shows that, on one hand. the geometry of propagating media can only play a secondary 1'0~ ·and why, on the other, pcrmeabi1ity and ftow conditions at the interface have the ~same .qualitatift iRflueac:e for both aeometries. We shall therefore mainly discuS$ the l'esults concernins a plane interface, but compare them to the ones obtained for cylindrical syinmetry to emphasize the foregoing remarks.

Different wa•e type& Several type$ of waves boundary:

are

senerated by . . immened sourc:e ncar a fluid-solid ~

• Body waves: (a) A conical P wave rcfr~ P oa .the interf:ac:e (dcoot~ P below) constitutina the f~rst arrival. (b) A conical P wave refracted Son the interface (denoted S) constitutina the second arrival: this wave only exists if the S wave velocity in the solid is greater than the P wave velocity in the fluid. (c) A direct P wave in the fluid (denoted D), which does not .. meet- the interface: this wave is often muted by tbc. StQilcley wave whose velocity is (see Section 6,1.3.1) close to the wave velocity in~ fluid. • In the case of the cylindrical ~ry (F~g. 6.16) and of the equivalent plane geometry (Fig. 6.15), ''multi-reflected n piQed. wa,ves which display an i.Qfinity of modes, all very dispersive and attenuated, e~ in elastodJ1W11ia (Paillet and White, 1982): the zeroth mode of these guided waW~. slightly dispersive, is also called the Stoneloy mode

~

'

"~T

~ Ji

'

- ~I

'-;

--'-" '--

'"-

,, 1

----· '-'-.,..-

'-.,..-

~

-·-

1:~

·- ·;4-

WAVES AND~

6

21S

because, in elastodynamks, the velocity correspoading to the high frequency linut JJ tbe velocity of the Stoaeley wave (the Wllvelength beiDa very small. the. cwvature Gf· the interface doll not have any eft'ect). Its low frequency limit is the familiar tube wave in borehole seismology. • In the case of a plane interface. a Stoneley wave (denoted St) previously described in Section 6.1.3.1, which is non-dispersive and non-attenuated in elastodynamics. In the simulations and actual recordings (acoustic logs) these multi-refiected modes (except the 2CI'oth mode) are very often masked by other arrivals due to their small amplitude. Hence we shall only analyze the conical P wave, the conical S wave and the Stoneley wave, whose amplitudes are significant.

Results of 101M . . . . . . . . Different theoretical sipals are shown in Fip. 6.18 and 6.19 to simulate signals recorded in Berea sandstone. As a ..-. the anivals correspond to rather low amplitudes for the P wave, moderate for the S wave, and very energetic for the Stoneley wave. Variations of the amplitude ratio obser'Wid fortb.e thtee types ofnvc(P. Sand Stoneley) in the case of a Jiven porous material, and of those observed in the case ofthe equivalent material having 11M sune cba.racteristics but very low permeability, are shown in Figs. 6.20 and 6.21. The sipal is recorded in lbctfluid., the receiver beiDt at 3m from the source and at the same depth. Infl~nce

of the source frequency spectrum.

~

For Teapot and Berea sandstones with high and medium permeability respectively, high frequency and low frequeacy sources w~ tested. The phenomena are more pronounced (see Fig. 6.20) at the hiper frequencies, which is expected because high frequencies favor two--phase eft'cCts.

....... '-"" '-..-~

-

lrt}lvence of interfae. flow cmuliJioM. (open or 8flled interface). The effect of an opea or sealedioteffacc is prKtlCitly neali&ible Oft the P wno and the S wave (sec F11- 6.20~ However it iuadical on tbe stoneley wave (compare Figs. 6.19a and

,__,_

6.20 to Fia. 6.19c:~

·-.__.

Infl~nce

'--'

'--

of permetJbility.

Rosenbaum's results are summarized in Table 6.1, which gives the calculated attenuations per unit lonath obMrved for each type of wave. .....__._

The effect on the P wave is plllactioally nepaibJe. Therefore for example. in a very permeable sandstoae (J'apot) atltlnlation is lea than 1 dB/m. The effect on the J wave is sipificant and relativoly independent of interface Oow conditions. This is a particularly intmlltina result for the inverse problem of measuring the mobility of the ftuid directly from fteld data. The effect of the Stolldey wave is considerable. bul de~nds heavily on the interface ftow conditions. For the hiJh permeability Teapot sandstone, open interface attenuation is 20 dB/m and sealed interface attenuation is 1 dB/m. Note that &a results pwa fol: low~ Foxbilli sandstone (JCC Table 6.1) are only refOI'ellCIO iafoanadoa, aad aN aot .at all OCJIDIW~ to the other raults (Berea and Teapot saDdttones). 'I'M fnlquency specm.. of the source is much richer in high

,___ '--~

"--' "-~

~

--~

"---'

--'-.,..-

"--'

41

,

•I

I

,

O+St

•I

I

I

D+St

I s.lecl ....._

I~

..t J •I 1110

IC

J

:=_

~i: .I

I

J~

·~

0

-

...

1100

"*limO

- .

10110

-

...

1000

100

Ti!lllelft ...........

1 ~

l~ • ~ j~ ~

1100

~0

~

l,

Fi&- 6.18 Tbcoretical sci.smopuDI for a plue ~ted~ between a saturated

~~ere& sandstone aad water. The tranamitter and receiver are near the interface. and the transmitter/receiver distance is 3 m. The source frequency spectrum is aiven in fia. 6.17 (/1 • 0, f•- 10kHz, 1... lO kHz, / 2

J

.J.

• 30kHz) (after Roaeabum, 1914).

J

J,_

,

•& •••

I s

,

O+lt

·-

.....,_

s-led

-.

'

I tr-

I

&

---~AA.._____L,

.----AAA-

~,,.---~-----, ~v~

-40«11

- -_, .. -,__ ....... r

I

I

1

r. + -- - -I

I

····mO

I

-li

='l

11::-

~i

--t: -1

•

!

I

J_,

....

1-:J

J 1400

Theoretical seismOaraJDs for a plane interfaee between a saturated Berea lalldltone an4 water. The U'aMiDithr aacl reoeiwr aN aear the interface, and the traasrDitter/reatiYW distance-is 3 IQ. The IOUia (~ spectrum is pven in Fis. 6.17 (/1 • 15 kHz, f. "" 25 kHz, In = 35 kHz, / 2 ... 45 kHz) (after Rosenbaum, 1974).

Ji1c. 6.19

~

~:

·=t·~ limO

I

~

l·~

• •

~---.

TliMinmer-lds

~

......, ....,._

d

c.

'

l

.I

.t-··UIIfll---

-=. 2001110

...("""

O+lh

I

1100

_f

-l

~ --.

........,

-----.. '\

~~

i

(

(

(

(

(

('

( (\

(

(

(

(

\

(

(

0

(

.~

II

"'r (

(

._.....

----------------· BEREA SANDSTONl (IC • 20D mOl

,......,_... 1.0

II t·' I!,=,

r

:.

":·

.01 ~

.01

11

-

__ ,

1.0

.........,. 75

dlst-c-t

Openlnt.t-

.___,

0.5

' : :::·· li r

·.

~s

1.0

~~ ,.Q1

~

1.0

lu I

._..........,

.

......

Open ......

TEAI'OT SANDSTONE I " : I 900 mOl

31110

._

I!,=

J

.1

J

.1

I .. f1

I

St

. ) t.o

.01

,01 ______ L 11

31110

...............

dlltMiatlcml

.5 .1

cllltllllltcml

•

.01

i

r ~~ 1.0

li

r-

-P.St.S

r .01

=-

Il ·' ,.~ J

75

300

Sou___._

~-

,

.01

.01

~:

.1

J•

.01

!-

1.0

1.0

j

.5

i-:r l

. 75

..

.......... .......... 300

1 I! a!

"!:::

.1

___ ,

1.0

I

.5

I1

.1

's

j.OI

.01

I

-e

,

1

I

z

.Ot ~-"-'-'-"""--75 300 Source-r-w

\

,011

1

1

0

0

75 31110 Soun:e-realwr

~-1

Fig. 6.10 P ~tnd S rerracted wave and Stonelcy wave (St) amplitude variation!! Rll a (unction or transmitter/i'eceiver distance. These amplitudes are normalized to the amplitudes computed for an equivalent low permeability (S mD) medium. Case of plane geometry in Berea and Teapot sandttones (after l.oseabaum, 1974~ L

,L.-.

,.

._lcml

II

(

( (\ ..( ....,..,..... ____ (

I

•

',. ~

FOXHILLS SANDSTONE

St

1.0 ,.... 0

E

"' ~ H I(

1·5

t::t::!:::

-

•

1.0 ~

~:

1

l .5 rJ

) I

.1

20

40

eo

"'

::!!

"@0

...

-:z

_j_

.1

2040

so

~Neeiverd..._

>

"'!

St

l

I z

i

Open int..tKe

SNied inwt.ce

10

~

eo

~""

Soun:e-,_w., disu.ce (em}

(em}

0 __:~

..,

1.0 ....

I

.I

I

i

..

2 I(

• .5

.

...

~:::::!

:· St

1.0

-!t"'

I~~ ~: I:"

Q

"'~ ~ ;;l l,j

St

)

J

..

1 I

I

.1

•j!:

.

r-

1

Ill

'

.1 20

Carnl

-=-~ . ~

.<·· :>-

2040eo80 ~iverd..._

_J "' Ill

40

~~

10 80

S o u - - - d - . - (em}

-.0

-~ z

..'_...l

<

i ~

I

I

i

u

I(

1.0

L"

• • •

I

r P,St,S

I

•

1.0

.I 1-

• .5 ~

.1

I

J

!

~~

:';

I

I

20

40

Sou~vrr

I

I

eo80

d.._ (CI'II}

•

~

I ::::::

-Ill

P,S St

£

-·"' ··.~ s "'

~z

~ ~< ;;l

.1

I

20 40

_j_

eo

80

Sour~~lcml

P ~ Stefrac:ted "!ave and Stoneley wave (St) amplitude variations . as a function or transautter/receiver distance. These amplitudes are normaliad to the amplitudes computed for an equivaleat low permeability (0.1 mD) medium. Case o!plaacpometry iD Foxbilksandatoae.. The.sourcc frequency spectrum is given in Fig. 6.17 (/1 =60kHz, / 2 = 160kHz) (after Rosenbaum. 1974).

z

-E<

F~~o 6.21

-------r----------------------------------------------------------------~

' ..

~

(

~

(

(

(

I

(

,..

(

(

(

(

(

'

(

(

\

(

(

(

(

(

(

(

(

(\

TAKLF. ATTENUATIONS OF INTEIFACI!

P,

6.1

SAND STONI!LF.V WAVF.S C"ALC"ULATF.D I'OR THF. C"ASF. 01' PLANF. (if.OMF.lRY

I'Oit VAIUOUS TYPF.'I Ot' SANI>!."TONF. WITH lllt'I'F.RF.Nl" pt;Utf.I\IIIUTIF.S

.

Type

orroct

Teapot •ddstone

1900

BerCallliMktone

200

frequency

spectrum

(mD)

Attenuation

Cetttt-al

F~equency

Permeability

ofiOUrc:e

~

IS_, 45kHz

1.2

Pwave

Swave

Stoneley

Open

0.4

3.5

20.s·

Sealed

8.9

4.0

0.9

Open

8.9

4.1

11.6

Sealed

0.9

3.5

0.9

Open

ts

5.9

17.1

Sealed

1.S

6.2

~o

Open

~o

3.2

10

Sealed

~e

3.2

~o

Open

~o

~o

J.5

Scaled

~o

~o

~tO

frequcney

IS to 45kHz

0.2

·32.5

(dB/In)

Interface

3

60to 1601Hz ,.

Poxhilts. samistcine

'"7.

3

60 to 160kHz

3

60 '*' 160

tO

I

T,uu 6.2 CoNI'A&IION Nl

Sowce

spectrum

BI!TWII!N A'l'fafUATtoNS CALCULATED IN I'LANI! AND CVLJNOiliCAL OBOMITitY

Bal,4 SANlJSTONI! 011 ~

Pwave I Swave 0.9

0.9

Stonefey

3.5

Source spectrum

11.6

4.1

IS to 45 tHz Sealed

20(). mD AJIID POaOSITY • •

Attctauatlon ~B/mJ

Interface Open

I( •

I

19 %

AND

P0a //f.

• 0.2 Attenuation (dB/m)

Interrace P wave

Swave

Stoneley

I

4

3.4

0.7

2.7

()pea IS to 45kHz

0.9

Sealed

-~---~

Plane geometry

Borehole (24 em diameter)

(FIJ. 6.14)

(fiJ. 6.!6)

0.6 ....

-------

I

,

'---~-~

----

(

280

6

WAVES AND INTERFACES

freqacacies (central spectrum frequency/cha!1lcteristic frequency of Biot's theory ;; 3), since it is also necessary to use very bish frequencies to observe sipif1cant effects at low permeabilities (see Table 6.1 and Fia. 6.21). Furthermore. for comparison with the attenuations previously observed for plane geometry, we have summarized the same results for the attenuations computed in cylindrical geometry (see Table 6.2) in Berea sandstone. As we previously pointed out, the effect of the source spectrum, permeability, and intctrface conditions of the different waves is similar. However, the attenuation of the Stoneley mode is smaller for the cylindrical case than the attenuation of the Stoneley wave for the planar case (see Table 6.2). The interpretation of these observations is given in Fig. 6.22. In the case of the refracted P wave at the interface the movement of the ftuidis more or less in phase with the overall movement (tee Chapter 2). This means that the differential movement between the two phases (the source of attenuation in Biot's theory) is relatively small Moreover, particle motion, due to its polarization, iS parallel to die iatert'ace, so that the interface conditions are unin:lportant because the filtration velocity vector is also parallel to the interface.

p._.,. li'ropegetlon

fluid

..

s-w-

-~-

fllulcl

Solid

Solid

,.. ........llltllliGft

...... lloc'.......

........... .................... ......... Dllfhllllllt.......... to .

................ _........., ......,..._ .......... •..

.. .................. lkrt

.-,

StoneltyPrQPIIIIdan

Proplgltlon

..................... •

·r.

'

.~

~~~- ~ r o

Dlplh

"''

\

.....,

tafluence or relative movement or fluid phase in relation to solid phue aad interface conditions on refracted I', S and Stoneley waves.

Fta. 6.21

Phyiical iaterpretation.

In the case of a refracted S wave at the interface, since the liquid phase does not respond to shear forces, no pressure aradient occurs and hence there is no fluid exchanse across the interface. Thus the open or sealed cbal1lcter ofthe interface has a nealigible effect on the S wave. However, since the interface is set in motion, flow occurs in the porous medium by inertial coupling (Darcy's law and inertiaUerms), and not by the pressure gradient etTect, which is zero. Thus differential movement occurs between the fluid and solid skeleton of the permeable porous medium affectina the S wave attenuation. Finally, permeability and interface conditions exert a considerable influence on Stoneley waves, as discussed above (Section 6.1.3.1). However, the effects in the

'--, ~

~

"'

.._.-

• 6

·~

. . . .ANI)ftim!lt.ti'.«:!!S

neighborhood of the interfa<:e arc incrcasinpy pronounced for the Stoncley.wave and at. higher tiequency (lower penetration depth). since Stoae1cy, particle movement is elliptically polarized and hence ditfcrential movement ocx:urs between the fluid and the porous medium. I tifluence of distance

·~

_/

'~

'--'

'-

·-..__._ '-'

,_ '.._

of transmitters and receil'ef"S from the interface.

All the simulations described above apply to transmitters and receivers very close to the interfaces. In Fig. 6.23c. the sourc:c is lcK:atoclat 10 em from the interface. and the porous medium is a highly permeable Teapot sandstone. The effect ofdittance between the sourc:c and interface is clearly undetectable for the P and S waves(c:o.aaparefigs. 6.23a and 6.23b with Fig. 6.23c). wlaercaa the same distance (10 em) implies that the Stonelcy wave amplitude is attenuated by about 30 dB. Stoneley attenuation is approximately the same as that due to the difference between a sealed interface and an open interface (;;; 20 dB, Table 6.1). This is expected, because Stoncley amplitude dccreucs exponentially from the interface. whereas P and S amplitudes decrease as the invcnc of the distance from the interface.

6.1.3.3

.....

211

Coadullloas

To conclude tbitaalylil, aucl from a practical studpoint; if research is guided in the direction of invcrtiilg acouatic lhcasurem•u for mobilitJ ofdadluid, independcndy of the interface Dow conditions, it is of primiry importance to Ute bish frequency sources (that is to say frequency dose to the chaaeteristic frequency). However at these high frequencies, the Stoneley wave is strongly attenuated and hence diffiCUlt to observe. In practical applicaqons, it will be necessary to compromise, in terms of tiequency, if the Stoneley wave is desired. AI it has been thown, the Stoneley wave is most influenced by permeability, in the case of an open inttrface. It is important to note that the fmt artivals (P wavea) are •• inlensitive" to permeability for any interface conditions. F"mally ·4- S wave is especially interesting because its attenuation is sensitiw to the :variation iD pcrmcability and i~t of interface flow conditions whicb arw,. . .t for in tiWIIliU1JI'elllCRt. . The transmitters and rec::civers must be placed relatively'*- to the interfaces for these c6c:ta to be observed. It is nevertheless difttcult to evalute permeabilitics leu tbaQ to .aD'(see Fig. 6.2le and Fia. 6.2U) cspccially if the interface is -.Jed. Laboratory experimcnu on borehele models of non-porous epoxy and sand (Schoenbera et al., 1911) or porous concrete (Cben..l9U) materials have revealed wide gaps betwoea the RtuiU olailild by ~ aDd tlaose predicted by theory. The possibility of sample hctcrepoeity has been suaested as an explanation. For in situ acoustic loging experiments, the problem of the inversion of acoustic measurements for permeability still remaiaa :poorly .....~ For the. aiAae bling. it .appears that, in many ca1e1, ue is oftina maa-of a ftUJDiaer of e&Dpirica1 folmu&as, dtlvoid of any solid physical foundations, which work only in rare favorable eonlipratiOM. However, the importance of this inverse problem for hydrocarbon recovery and reservoir evolution justiftcs a major effort in this area to understand the physical phenomena and develop an experimental proaram. Some references on field applications of permeability effects on interface waves are given in Chapter 7, Section 7.2.

. •(tL6t 'Umwefua80lf £ 5! ~S!P »A!**J'»U!UJIUQ ~)--- pu1J NOl. . . . ~~ puaua U;»MlaQ acnJj»J9} aQid 11 10j ~~oaq.J. .£r9 'lt.!i

---owtps

Jltp}{W

c-. tlllt.L

i" i"' 'i'

I

I

0

l

01

woo,....-.

-.-ut flll.UI04

'---"'ln---...;..-~o

,__.....

·.~

t

lS>Q

I

t

s

d

01

....._,_,.,

•.-.u!"'lctl

• t

t

..,.,..UI pejft$

q

lStQ

IIWIOIIIOS

01

'

s

d

01 -...ul"'lOl

.........

-p-.-,.

'-

0

-.-ut

liMO

•

·~

t

lStO

9

t s

01 (JH'Iot=HJ:~,.IJ) ZH'I 9t>-O allll\4adl801f10$

t d

S3.JV.:fll3l.NI QNV S3AV.'A

QIIIOQ8L•,.

1' L'&t• • , 3NO~NVS .LOdV:i.l.

-

~

.,

·*AVIS AN&~IWAd!s

283

6.2 REFLECI10N AND TRANSMISSION IN VISCOELASTICITY

·~

We have dilcussod the intemum in Biot type media ia tbe tint pan of this Chapt«. In this second part. we shaD examine the chanJCS made by the introdactioa of iat~ in a viscoelastic medium. We shall develop the arguments in a thne-climeasioul space, and then. for quality-factor-related problems, we shall limit ourselves to the two-dimensional case. We will then calculate reflection and transmission coeft'Jcients and examine in detail the ditterences with respect to,the elastic case. Tbe problem ~ ielerface waves will be discussed very rapidly.

-~

6.2.1

Wave equation in viscoelastic media

We showed in Chapter 3 that the unidimensional c:dl!lstitutiveequation between stress a and strain e of viscoelastic materials is written in one dimension :

a=m•e

(6.44)

The extension to three dimensions yields:

a.1(t) = C1.;t1(t) • £111 (t)

(6.4S)

In this equation, Einstein's aotation is implied,. with IUIIIIDadon OD the repeated subscripts. In the isotropic case, Eq. (6.45) is reduced to: ·~

qlj(t) =

·-

'•J( K(t)- 32Jl(C)) ·~~(t) + 2p(t)Hu(t)

wbcrc 61} - K.roooc*cr 4clta;

·{ 8,1 -o ~I}- 1 ~

(6.46)

iflt'J if i • j

K(t)- bulk modulus, p(t) = shear modulus. The equilibrium equations are:

a,1J"" pu,.

(6.47)

where u1 is the f'A component of the displacement Equation (6.47) caa be ICWI'ittlln by iDtroducina the constitutive Eq. (6.46) in Eq. {6.47):

•

pii1 - 61{ K(t)- ~t)] u•..t&' + p{l) •ui.JI + p(t) • "JJJ "--·

(6.48)

284

6

WAVES AND .U'ITEJt.f'A.CES

In this equation, displacement u is a function of the space [r = (x1)] and time (t) · variables. Takina tbe Fourier~ 1Jf (6.4&), we obtain: - pw 2 U

= ( K(co) +It(~))

an•

8

+ p(co)

V2 U

(6.49)

where K(co) and p(~are tlte FoariettnnaformsofK(t)andp(t)R"JSpectively, and 6 is the ~olumetric 1train - div U. U - U(r, 0)) is the Fourier transform of •(t, t). Let us fmd U in the form: U = arad 4> + curl .P (6.50) with the condition ctiv 'I' • a By combining Eq. (6.50) with (6.49) we obtain:

172 4>

V2 ,

..

+ kJ 2 4> = 0 + kJ 2 '1' = 0

(6.51a) (6.5lb)

with

14

pw2

2

K(co)

•

(6.52)

+ 4/3Jl(co)

kJ2 = pcol p(co}

(6.53)

or

1

1

kJ == ~~ V,.(QJ) • (K(co) + 4/3p(co))'l • ().(co) + 2p(co))2 Y,.(co}

.

p

co kf • Ys(a>)" V,(m)

.

=

(

Jl~)

r !

P

(6.54)

(6.55)

The quantities K(co), p(co), V,.{co), V,(co), kJ and kJ are complex quantities. The chosen branch for the square roots in Eqs. (6.54) and (6.55) correspond topositive real parts. This choice will be understood in the ~g discussion. The general plane wave solution in the time domain of Eqs. (6.51} is:

4> == •o exp [i(cot - k• • r)]

(6.56)

In this equation, k• is a complex vector which we can separate into its rCa1 and imqinary parts: k• = k- iA

(6.57)

where k is the propagation vector and A is the attenuation vector. The solution (6.56) to Eq. (6.51) is then written:

4> = 4>0 exp (- A • r) exp [i(cot - k • r)]

(6.58)

As a rule, the vectors k and A are not panllel, and in this case the wave is inhomogeneous. If the angle between k and A is zero, the wave is homogeneous.

'

-,

-;;or ,_.~

,.,

-~

-6 '"-'

· . . . . . . MIDINUIIPAC!ES

PhyiM:ally, the wave ampliludc must no& Daile in dac propaption diRiction.:implying that the anale 7 between dae wctoa k aacl A must satisfy:

-~

·~

O~y<

2•

1be existence of a non-zero anale y expNISCS that the planes with constant phase and planes of constant amplitude are not parallcl. Fiaure 6.24 shows schematically the behavior of a homogeneous wave (y - 0) at the interface between two attenuating liquid&. 1be transmitted wave is automatically inhomogeneous.

·~

·~

---

...

~

.... 6.JA ......... ud .IIIW"watt fll &.-pilule medium (Schw•tjc a..,..~

'-'

WMe Uta vilc:ociMtic

'-

,_

Equations (6.Sl), (6.56) and (6.S7) live us: '-...-'

k• • k• - .., ..... fAI 1 -- '211AIIkl COI1

(6.S9a)

~ /X»1 k•. k• • - • iiiif [Ma - iMJ M IMI

(6.S9b)

and '-

\-

where M • M(QJ) is the ooaplex modulus of the wave, namely:

""'-

'-

4 M(QJ) • K(QJ) + j p(w) • l((l)) + 2~t(w) for a P wave, and

'-' ~~

\_

·~

M(fO) • for an S wave.

~t((l))

286

6

WAYI!i!h\ND JNTERFACES

In two dimensions (x,

z coordinates) Eq. (6.58) is written:

tfJ = tfJ0 exp [- (Axx

+A:z)] exp [i(rot- (k"x + k=z)]]

(6.60)

and Eq. (6.59) can be expressed in the form: 2

(kx - iAs-)2

+ (k: - iAJ2 =

p:;

(6.61)

or

k!2 where V is the complex velocity

+ k: 2 = pro

'

2 _ w2

(6.62)

M- y2

~med

-,,

by:

V=l=;_ Equation (6.59) enables us to calculate

lkl IAI =

= { ~ [ Re kt {

lkl and IAI as a function of kt- and 1

+ (lYJ 2

2

~ [ - Re kt

+ (I::J~J

2

2

+ (
2 2 kt )

r

~.

y:

1

1

YJ

(6.63)

y 1

(6.64)

The phase velocity of the plane waves considered is equal to:

v

=

' (0

k

(6.65)

lkl 2

By means or Eqa. (6.63) and t6-64), it c:aa be 'Sbelwn that this volocity is· higher for a homogeneous wave than for an inhomogeneous wa'Ve (Borcherdt, 1973). Finally, it is interesting to note that, in the case or homogeneous waves, the foregoing equations are simplifaed. If 6 is the angle propagation measured from the vertical (Fig. 6.25) we obtain:

k! =

~- iA,. •

(Jkl- iiAI) sia 8 == ro

j{;

sin 6

(6.66)

II

X

~r"'k z

---...

A';

~

II '

-k -,

z

Fig. 6.25 Attenuation and propagation vectors for homogeneous Oeft) and inhomogeneous (right) waves.

-----.. -~,

"'

.,........

WAV!f'MftntftdFAees

6

28?

The aDJic tJ is given, whether· the wave is homogeneous or iohOJno~ by:·

""

(6.67)

lkl

sin 6 •

which is written, in the case of the homogeneous wave:

. 8 = Vi + Yf -k"

v..

SID

(6.68)

(I)

where Va and V. are respectively the real and imaginary parts of the complex velocity V = foJP and equal to: (6.69)

v. 6.1.2

-('.vt;M·t

(6.70)

Energy balance and quaHty factor

6.1.1.1

Energy balance

We have shown that the displacement uWi8flcd Bq. (6.48), or that its Fourier transform was the solution to Eq. (6.49~ for a plane wave, it is always possible to write Eq. (6.49) formally as a function of u (see for example Bourbie and Gonzalez-Semno, 1983):

pi= (K(w)

+ p~td)] arM 8 + p(w)V2u

(6.71)

with 8- div u.

__

The actual displacement vector is the real part Ua of the displacement new equation verified by Ua: ·

11.

By deriving Eq.

(6.71), we obtain a

,

pl. - ( Ka +

~) .... 8~ + l'tV2. . +

![

(K, +

~) .... Ba + Jlt vzu.]

(6.72)

The .sublcripts R and I indicate the real and imaginary parts respectively of the quantities considered. · The~ ~ua~ is olQined \lY detenniaina the scalar product of Eq. (6. 72) by lia (Linds&y, ~~~After tranSrormatioq (llorctierdt, 1973), we obtain:

Pa) 8a2 + 21lallaa.)llaJ•J 1 ] ot0 [p2 ~2 + 21 ( Ka +. T

1 [(~

+(I)

I

+

Jl•) 8a. l + llilla,..JUaJ . ] 3

. [(Ka+3Pa).Baila+ •1 ( Ka+3Pa) 8atia · )

-dtv

+ p i [ ,..(U. • div 11a +

!

Pt(litt • div

(6.73)

lia)]

288

6

WAVES AND INTEitFA(;ES

The intcaration of (6.73) on. a vol&UU. D ofsurfacc SJi~ us the equation:

:r L

E dO+

fn 0 dQ = -

II•.

dS

(6.74)

where E == sum of kinetic and potential .energy densities, 0 = dissipated energy per unit volume, I =energy flux per unit time.

6.2.2.2

Quality factor

The knowlcdae of these quantities now enables us to calculate the quality factor defmed in a manner compatible with one of the 1D defmitions. We have decided to defme the inverse of the quality factor in the same way as Borcherdt (1977), namely:

Q_1

= _1 loss of eneraY denlity per forced oscillation cycle2x

peak energy density stored during a cycle

( ) 6 75 ·

Using the results of Borcherdt (1973), the values of the different quality factors are obtained: (a) For P and SV waves:

Q_ 1

M1 + ll•Mf/IMI 1 tan1 y - Ma + ~taMl/21Ml1 aaa1 y _

'·

(6.76)

where for P waves:

4 M == M(
(b) For SH waves, Q- is given by: 1

1

Q _1 _

lla[1

6.2.2.3

+ Jll./114 1 tan1 YaP 1 + (1 + Ill 11Jll 1 tan 1 YsHJiJ

2p.[t

(6.77)

Coostaat Q model in two dimeosioas

We showed in Chapter 3 that KJartansspn (1979) bad developed a riaor~ thcotetical model for which the one-dimenSional quality factor was ripously ilickpcrldent of frequency. The quality factor considered by Kjartansson was the one defined by:

Q""l == : :

(6.78)

Equations (6. 76) and (6. 77) are the same as Eq. (6.78) in the case of homoaencous waves (y == 0). For inhomogeneous waves, these equations are different, as the angle y between the planes of constant phase and oonstant amplitude atfects the value of the quality factor.

'--

(

\

(

(

/

(

\

I

(

(

(

(

(

(

(

(

(

I

(

\

(

i \

(

(

(

(

(

(

(

(

'

(

{

(

(

(

(

(

( I

~~

CottsTANT

TAIU! 6.3 Q llroDBU IN ONI! AND TWO DIMI!NSIONS

Constant Q mqdelln two dimensions

SH waves

p(al} = p(CIJo)

QiJ-

SV waves

,.(01} _

(w)z'

p(m) •

0).

tan.,

OiJ = 2 tan JCI

p(wo)(':.Y'

+ lin2 Kl tan 2 .,,.lr I + (l + sift 2 . , tan2 "'1111,. ,t (l

p(Ofo<:r'

lw)"' p(Q)) - p(We)(C»t

tan--

Qi.,' • tan .p l + 1/2 smi •P tan2 "/sv

Qi.J-

1 + ltin*"' tan

(t)

M{GJ) •

M(oto{:)~

2

"/sv

.... lictuid medium

Q;' •lan trar

P waves

JI(Q))- Jl(t»o<:r·

(2)

M(w) • M(roo)

Q;·---

(iw)a. Wo

p(w) • p(wo)

and

1 + l'(a~o)fM("'o)

,-a I'

Al(w)

(4) Trivial

CllliC

= M(•o)(;iw)l
a;'- tan •«

M• .. o Ma

2

•)2'

~· It{«») • p(~Gi;

J

Wo

2 •

sm2 •ar tan2 y,.

"' -tan" 1 + p(aJo)/2M(Q)o) sift (3)

(fw)

Q;'-o

XIX

tan 2 "/r

, tan u • 2 tan

•P

290

W.-\ VES A:>;D

I~"TERF ACES

6

This is hence no longer an intrinsic property of the material, but a combined property of the medium and the type of wave analyzed. It can be shown (Bourbie. 1982) that, in the general case. a rigorous Constant Q model does not exist in two dimensions for an inhomogeneous P wave. In very specific cases, where the material analyzed reacts in a non-independent manner to longitudinal and shear waves, a Constant Q model exists. The results are summarized in Table 6.3. Nevertheless, it is important to realize that the variation. as a function of angles i'r· i'sv. y58 , of quality factors Q-p1 , Q.S/ and Q5J. defmed by Eqs. (6.76) and (6.77) is slight. Figures 6.26, 6.27 and 6.28 provide an example of these variations for a particularly unfavorable case in which attenuations are extremely high. It may be observed that Q -;.1 , Qiv1 and Qi,} are independent of the angles }' if these anaies are smaller than 70 to 75 degrees. The hilh values of y, as we will shoW in the next Section. are only obtained in the case of an incident homogeneous wave for angles close to the critical angle. Hence we can consider that the quality factor is virtual~· independent of the inhomoaeneity angle y, which amounts to stating that the quality factor can be assumed to be equal to the defmition (6.78) even for the two-dimensional problem. The Constant Q model of Kjartansson can be considered as a two-dimensional Constant Q model.

6.2.3 ReOections aad transmisllons in two dimensions 6.2.3.1

Theory

The details of certain calculations, particularly for the case of homogeneous waves, can be found in Cooper (1967), Lockett (1962), Borcherdt (1977) and Bourbie (1982). Let us

consider an interface between two attenuating media The variables associated with the medium in· which the incident.wave propagates will be denoted with a subscript lor 2 according to whether an incident of reftected ~·ariable is involved. The '-ariables associated with the medium in which the waves are transmitted will be denoted by a prime('). The computations will be carried out in potentials, since their use simpliftes the wave equation. Finally, time dependence will be understood and equal to exp (iaJt). The incident wave in the most general case is the sum of a P wave and an SV wave. If x is the horizontal (i.e. along the interface) coordinate and z the vertical coordinate, the system of equations to be resolved is the foJiowing :

• = A 1 exp (- i(kx + dz)] + A 2 exp (- i(kx- dzl]

(6.79)

+ B 2 exp (- i(kx- hz)]

(6.80)

'I'- B 1 exp (- i(kx

+

hz)]

., ==A" exp [- i(k'x

+ d'z)]

(6.81)

'I''= B' exp [- i(k'."C

+ h'z)]

(6.82)

R.e td) ;il: 0

(6.83a)

k2

IV•ll

+ d 2 = -.,_-_ I.+ 2p

k2

+ h2 =

pwl

-

J.l

Re (h) ~ 0

~,

(6.83b)

,,

'--

'-· -·

', __

I

.30

.-.·;

I

,.

~

·-~'j:;r

.· '-:'t·:,-.

Vp=2

hp = 0) =5

v$ = 1.2

0s lls =ot""s

:.._~~"· ~.~

f'il. 6.26 Inverse of quality factor 1JQ1 vs. angle between attenuation vector and propqation vector (after Bourbie, 1982).

•18 1

Osv

.t21::----------.. . --.06

~~_.--~--~~--~--._~~-L~

0

w

~

~

~

•

~

~

•

~

Propegation . •tt8nUitlon ..... (dig.)

~r---------------------------------~ Vp =: Op hp "'01 = 5 v 5 = ;,2

.24

.18

cl; .12

FJe. 63!1 . lnvene ot qalifJ facft,t

1/Qsr va. aqle.,...... attimua- ; o

tion vector and propagation vector (after BourbiC. 1982).

..oe OI 0

I

w

I

~

I

~

Propeption.

I

~

I

I

I

~

~

~

,....rian...........

I

~

I

~

30r---------------------------------~ v5 •1,2 Ostts .,., ·•

.24t .18[

..;.. .12

..t

'

"'""'

.oe o~--~~--_.--~--~~~~--~--~ 0 w ~ ~ ~ ~ • ~ ~ • Propegation • Anenuetlan engle Idee-I

Fi&- 6.l8 Inverse of quality factor l/Q111 vs. anp between attenuation vector and propaption vector (after ~urbie. 1982).

292

WAVES_ANP ,

y + d'l ...

2

•.P_(I)_ • ,

k'l

6

I~RFACES

+ h'l = p w

Re (d') ;;11: 0

(6.83c)

2

Re (h') ;,:: 0

p'

(6.83d)

In these equations, the quantities k. d, h, k', d' arul h' and the amplitudes are complex quantities, and are related to the propagation k and attenuation A vectors. Displacements and stresses are given as a function of potentials by: (1) Displacements:

a.

o'P

(6.84a)

u% -ax - - iJz -

u

:r

a• o'P

==-+az ox

(6.84b)

(2) Stresses :

... a.. (1 u

. a2• A.

_

02• 2

oz

+ 2P ax cz

a2•

0 2 .,

a2 ")

iJxl + (l + 2")

a2"

(6.85a)

= p ( 2 ~~ az - az'-' + ax 2

(6.85b) -~,

The reflection and transmission coefftc:ients are obtained by appling the boundary conditions at the interface. Three alternatives are avMlable. namely a liquid/solid interface, a solid/liquid interface, and a solid/solid interface. We shalupidly examine the results of the computations in the three cases.

a.

LitiiiiiiNIItl -~

Since the incident wave propagates in a liquid medium, we have p Continuity of displacements and stresses implies: U:r

= 0 and B 1 = B 2 = 0.

•a(

= a;x a.,.- u;z

(6.86)

0 We obtain k • k' and:

A2 p'd[(h'a- k2 ) 2 + 4k 2 d'h']- pd'(h' + P)2 Al == R == p' d[(h~:t - k2)2 + 4k2tl' It'] + ptl'{lt'-» + k2)2 2

A'

2pd(h'• - k') + 4k2 d'h'] + pd'(h':

+ P)2

<6·88)

4p4tl'k(#l'a + k2 ) = Ts = p'd[(h',t- k')2 + 4Jc2d'h'] + pd'(h''

+ k2)2

(6.89)

As • T r -.p'd[(h' B Al

(6.87)

1

-

k2)2

·

I

·w.wM-MellftiiU'AaS

6

293

b. Soll4//i411Ul hltajla Sinc:c the transmitted wave propaptcs in a liquid. we have p' • 0, B' continuity equations arc written:

= 0.

The

"• =,; flu= t1.,.

=

0

(6.90)

a;.

Here again, this implies that k = k'. Two alternatives arc available for the incident wave, namely an incident P wave or an incident S wave: (1) Incident P wave,.. Bt Az

= Rrr =

= 0: p' d(hz

+ k2)2 + 41c;2dd'hp- pd'(hz- kz)z h~

(6.91)

Bz _ 4kptl4'(h 2 - k2) At • R,.s = p'i.(Jil + ~ti)f'+ 4k2d4'hp + pd'(hi _ k 2 )1

(6.92)

At

A' A1

.t

Jn.'i .. • 'h'i .

H

4 •

2pd(h4

111.

•

•u•'i

1

k")

-

= Trr = p' d(hi + ti)z + 4kiUi;p + pd'(hl _

k'2)z

(6.93)

(2) Incident S wave • At • 0:

!!! s.

R,.

_,k:(U'hp :- pd'(h2- A:lt- p'd(hl + kl)l _,

JIL4

•

..1.1 .

s. A'

Bt

'-

------~-

• ••• . ... 4 ••,.

- 41ul'hp(h 2

Az '~

..,. 4

R,- p' d(h2

-

k2 )

+ P)l + ,4'(hi - k2)2 + 4i(!M' lap _ 4/ulllp(h 2

(6.94)

(6.95)

+ k2 )

= Tsr- p'd(h2 + fc'1)2 .;·pd'(lt2- k2)1 + 4k2dd'hp

(6.96)

e. So#U/HIU later/"" The continuity equations are writteft:

u., = ~~~ "• = u; a•.,= a'" ••• - a'., Here aaain we find the c:ondition k = k'.

~ ..

__

(6.97)

294

'

WAVES AND INTERFACES

In the case of an incident P wave (B 1 = 0), we deftne the quaat.itics: A2 Rp p = A,

B2

Rrs=-

A,

(6.98)

A'

T,,=A, B'

Trs=-

A,

solutions of the linear system: dfftp = 11

(6.99)

where

k

h k.

-d d=

- k -d' - lk4' p'

-2k4p ~ (h2 - k2 )/l ( (h2 _ P)p 2 - 2khp - p'(h'

~

~,=

(

R,.,.) Rrs

,g I'Jfl

. T, T rs

In the case of an incidat S

wave (A~ •

=

-

k2 )

J

h' - k (h' 2 - k)p' - 2kh'p'

( - dk J.·.. -

- 2kdp - p(h2 - k 2 )

0). we dCODc the quantities:

A2 Rsr = -

B,

Ba

Ass='B,

(6.100)

A'

Ts,.=Bt

.

B'

Tss=-

B,

solutions of the linear system : dfft5

= f!J

(6.101)

where

Jls =

RsrJ (Tss Rss Tsr

{!} =

- hk J (p(h2k2)

- 2kla,4

~

.,__.--

----

'1

,,

:.t:·;

WAVasANDI!\iEaf:ACES ·-· .}

29S

It is interestin& to note that, in aU cases, the condition k == k' must be satisfied. This implies a generalized Snell-Descartes law. In fact, for a single type of wave (P or S), we have: -~

ka = ka => lk1 1sin 81 = lk~l sin 82 k1

= kl

=> IAtl sin (8, - ;·,)

-~-

= lk'l sin 8'

(6.102)

= IA2I sin (lJ2- 'Yl) = IA'I sin (8' -y')

(6.103)

From which. by means of Eqs. (6.63) and 16.64) we obtain:

82

•d,

(6.1048)

l'l

= i"t

(6.104b)

An inhomogeneous P (or S) wave is therefore reflected as a P (or S) wave with the same inhomogeneity angle and with an angle of reflection equal to the angle of incidence. The values of 8' and i are more complicated, but can be iaferred similarly from the foregoing equations. Figures 6.29 and 6.30 show two examples of transmission angles (8', i) in viscoelastic media, for a solid/liquid interface and a homogeneous iQcident wave. The angle 8' for the attenuatilla case is very close to the angle 8' for the el&Jtic cue except near the critical angle of incidence. which in efcct does not exist for the attenuatina case. The angle "'( increases with angle of incidence up to the .. critical" angle for which y' is approximately

90". -~-

'v

'-

The foregoing calculations show that the different reflectioa and transmission coeftcients are complex numbers irrespective of the angle of inc:ideooc. In Fig. 6.31 we have compared the retlection coeft'ac:icnt for a non-attenuating liquid/solid interface with the one obtained for the S&me interface, but with an attenuating solid with Constant Q behavior. The incident wave is assumed to be homogeneous, and the reflection coefac:icnt for the intcrt'ue betwceaatteDU&tial~ is cak:ulatld for a~ aqular frequency (co0 - 1 Hz) in the Constut Q . . - . . VutuaUy-u c:Uenmae is obeenedbctween the two reficction coelftcients for angles less than the critic:al angle, thoup a larger difference occurs at anaJes greater than the critical angle. The interested reader can fmd other examples in Bourbii and Gonzalez-Serrano 11'983) for solid/solid interfaces. -

6.2.3.2 laterfaee effect of attenuatioa As we have shown, attenuation modiftes the modulus and phase of tho. reflection and transmission coefticients. BourbiC (1986) showed tbat. for any linear vitcoelastk model. the reflection coeftcient at normal incidence for a plane wave could be expressed in the ftrst order (i.e. for moderate attenuations) in the form: R

~ Pt V,o- P2 v20 +..!.. (-•-- _1_) In t~l + ~ L! Pt V,o

+ P2~'2o

2lt Qot

Qo2

CUo

4 ~t

- _1_, Qo2

(6105) ·

The-subscript 0 indicates-that the velocities .v, and atteauatioDI (ljQ,) arc taken at a · . reference anplar fr~ w0 • This gtfteral formula clearly explains the effect observed in Fig. 6.31 for a Constant Q lllOdel, ia other words the modikation of tbe modulus and pbue. However; it also shows

110

""

Vp •1.2 V'p•4 v·s=1

80 .., 70

l -eo fso 5

:[ I

~leo t SOL J I

,..,..,.

):

.I

Cp•10 O'p=50 o·5 =20

I

I

'

,_

40

30,

,__

v,= 1.2 V'p=4 V's = 1

I

'

....

o

ro

20

80

so

lnclc*>ce .ngte

Idea-l

30

40

ro

80

o

110

ro

20

30

40

80

ro

eo

80

110

lncidel)ce ..... 14111-1

f1l. 6.29

Transmission angle vs. angle of incidcacc for a mud/solid interface (after Bourbie, 1982). Left: elastic case. Right: viscoelastic case.

r

110

lt ..~ 70.

I ~~

Vp•1,2

0,.•10

V'p=4

O'p=80

V's•1

O'sa20

I

I

.._

' "

J

J .ro

ro

so • ...................

20

30

40

•

110

f1a; 63t· Aap~ attenUation tector anlpropapdoinec:tor vs.eJie · of iJieidence lor • mud/Solid iatert'ace (after 8oudK6. 1982); 0

I

-

•

i ...

.a

L~

i

I

. 1

I

'5

!

j

0-135

J 01

o

I

I

I

I

I

to

20

30

40

80

C)<-f

80

lnclc*>ce engle ldeg.)

70

I

I

80

110

-1101 o

I

10

!

20 .

I

.1

·I

I

30 40 10 eo lftcldenoe .,.ie ldll-)

-r---r=--' 10

80

'

110

Fit- 6.31

Modulus and phase of reOection coeffiCient vs. aqle of ioeideD:cc for a liqWd/IOiid interface. The solid curve is for the viscoelastic case and the dotted curve for the elastic one. The velocities are v,, ... 1.5 km/s, v,. = 2.5 km/s and Vs. = 1.2 km/s. The attenuations for the viscoelastic case are liven by Q,, = oo, Q,. - 14 and O.ra • 10 (after Bourbie, 1982).

~

-----------------------------------------~---~

~

F j:

6

wAVIS'·.\111)~·

297

that the clastic effects (a
Flaure 6.32 summariles tbe results .obtainecf m aniplit\tde. Ro li

'

I

I

No .........

'UnaoiiiOiidltld ....

lnterl- efftc:t

I

I

A-.~··&~= a:::::a--

I

elatic

pnltM I I"' Ti~ ... uncls

1

reflection _.,.,._ O.ot

ol 0

~;-........ ·~

'-lr!Ft 111Gb ...a.....JL....-....__.__

,

~

40

•

,.

10»10400

Q

,.. U2

AOU).

Jntcnaee .

effect .•

or a,tteauatiotl (._ lloutW· and Nur. C '

·"'

I

6.1.4 ltlterlaa .wA'fes ill

r

I I

.o..-• '-"

·

.· J:

y

r

1984

Left: sbftP66ed·dlalts li'htJ the . . . . . . . . . . ....UCucl vilcoelutic releetion ~.. (%). . . Ript: some "cbatacteriltic" VU. olila .,..._..._

I

·

i)

~

..._

s.-.

We havt lhowa in Chapktt 3 aa4 in 6.2. J &bat die CQQ&Doas of elasticity are easy to write mathematically when tbl .....aptioa of olaaticity is replaced by linear vitoooluticit)'.lndeed, tbe co,........_. bo,applied to replace the real clastic moduli by complex fr~-4ep • deat oaas ia aU e&utodyaami~ equations. The modiftcations of the equations for the propaption pf Rayleigh and Stoneley waves, assumina viscoelasticity, are then easy to derive. No other fundamental phenomena other than propaption of these waves in viscoelastic media need to be taken into account This was not the case in poroclastic media where fluid flow at the interface has to be taken into ac:a>unt. More theoretical details on in~ waws in vilcodastic: media can be found in Borcherdt (1971, 1973).

,..,..._,uld

. ~--.

298

WAVES AND INTERFACES

6

6.3 GENERAL.CONCLUSION Two types of constitutive laws for rocks have. been cJ,iscuss,ed in this boQk. First we have considered rocks as a combination of a solid skeleton and pore fluids. In this case, displacements both in the. fluid and in the solW skeleton were considered (see Chapter 2 and Chapter 6. Section 6.1 ). But the extreme complexity of a porous medium makes this approach too simplistic in most cases. The macro~c behavior of. rocks could not be obtained from microscopic laws. Attemativ~ly, we can consider an equivalent homogeneous material which WCJuld react to acoustic waws like the rock itseU: Linear viscoelasticity is an appealing model since it ~nts the e«ect of fluid inlide the rock and it is relevant to the small strain amplitudes encountered in seismic exploration (see Chapter 3 and Section 6.2). For wave propagation in~~- and for the Creque~~ ranae relevant to f~eld measurements, macroscopic fluid motion with respect to the matrix has a negligible Cft'ect (see Chapter 2). Rock behavior can be modeled effectively by viscoelasticity. The anelastic part is then quantified by an intrinsic quality factor Q.,e· In the presence of interfaces such as a borehole wall, macroscopic fluid flow is no longer negligible. At the mud/borehole interface, the free surface conditions ~e important, and fluid flow effects modify the wave propagation. Viscoelastic phenomena also add to macroscopic fluid flow effects. Therefore, to interpret acoustic data in a borehole, it is necessary to separate the two ~ts. To the r1rst order, they can be considered independent:

Qf..' .• flM:, (2) + Q;. t

~.

~\

(6.106)

This formula is simila,r:to the one obtained by Goldberg et al. (1984) on simulated data for P wave refraction. In their example, the total apparent attenuation is approximately the sum of a borehole iluid. at.tc;nua~p ao4 &Q .attenuation ,ip the viscoelastic medium. This observation implies that, even t()f p waves, there is Sonic tluid interface effect. The detenaination of~ aUQW4 a meaaure .of intrinsic ~~ty and mud cake effects. The determination of Q;;, 1 , which is an intrinsic; parameter,· aliQws a measure of porosity, porous structure and saturatillg fhdda·of the rock. If both terms can be obtained independently from the data. then several properties of the rock can be understood. .It is, then, essential for the interp~ of~ acoustic·loss to model the porous media as a viscoelastic Biot medium. Unpublished experimental results (Rasolofosaon, personal correspondence, 1985) support dw1.hypothesis (6.1 06) and show that Q;. 1 sheuld not be neglected in the interpretadoft of the data. In situ, it may be diffiCult to oBtain ill4ependefttmeuom of Q; 1 and Q;;;,. The double effects of attenuation may then become an *tade practical ,apptieationa. :,.

m

~,

Q.,,.

(2) is delmed from attenuatiotus in Yi~asticity. This def'mition is·soalewba1 abusive: this quality factor cannot be related to the other deftnitions, as was possible in viscoelasticity (see Chapter 3}.

.......

'-~

''----' "----'

7

some applications in petrdeum geophysics

INTRODUCI'ION (

In the previous Chapters, we lhoYt'!DCl the extreme compleuty of the relationship

I

between acoustic waves and porous media. Both velocities and attenuations vary substantially with porosity, ftuid contont, temperature, and pressm, and it is not always possible to der1ne the boundaries between the inftuence of the different parameters. Hence

I

.._

!I

~

' 'I I

'

~

~

"-...--

.,__ I

I

~

I

II

I ~ : I

.._

..._

'-~

it is easy to realize the dift'JCUity of the inverse problem, which consists in characterizing the porous mcdiulll through the chanps occurring in che w.ave traveling through it. Yet this very diffiCUlty is rdated to the amount of information that we can anticipate from solving this problem. The followilla table (fable 7.1) summarizes ~ salient features of the ~elations developed ia tht ~ Chapters between ~· tbnlc major families of parameters, namely ex.terDal CODditions (fresat~e, tempctatuu); the characteristics of the porous medium, and its acoustic ~ (veloi:iity, atteaUation). Of these three poups of parameters, oaly two are iadepeadent In fact. pressu~ and temperature conditions inftuence the~ oftbe porounl'liediUID without truly altering the mechanical propertie!lol'the eoDIIi-.tive ~ Tlaedrec:t of pressure and temperature can therefore be replaced by an eft'ect on the fluid daatacteristics or O$ the pore structure. Faced with the abundanc:e of ~elationships which to discourage any detailed analysis, it is necesaary toproOeed by successive eliminations of variables. Depending on the major application imolved, this elimination assumes various forms. The Simplest case is that of JDCaSurenaeats aimed at the nondestructive testiag of materials. In this case, the external eonditions can be accurately controlled. Knowing the ftllid content, one can then focus on one cbaractcriiUc of the porous medium. For those analyses, the measurement of acoustic properties is rdatively a«uraae and easy, provided the sample aaalyzcd is in direct coatact with the measuring iastruments (as opposed to remote measurement in f~eld . studies). Oae can ~ deaip a laqe number of experimeats to explore a specif1c relationship between particular parameters, keeping the other parameters constant. Nondestructive testing of materials has been developed, mainly based on velocity measurements. The additional quantitative consideration of the attenuation factor can

soem

~

·--

-~--

.

.....___._:.___

)

)

)

,,

•••

-

-

·-

~-

7.1

-~~

--'--~~

~lftellum

Characteristics or

~

••

..

•••

•••

-

-

•

•(?)

•••

,,

Attenuation

•• ••

••

Velocity

averap, ••• strong.

-

------.-.. -------

neaHJiblfl. • sliJht, "

~

Gat/liquid aaturation. Geometric distribution or phases ....

01P (plwc diaiaM ......

Type and physical state of

fluid content

Peftneab_iiity . ~ ............

Porosity •.... • •.•.........

Teatw!e: pn contacts, c:racb ...•.....•.........

-

Comptessional

-

-

• -

•

••

Velocity

Body waves

)

••

•••

•

*(?)

••

-

Attenuation

Shear

Acoustic properties

8tJMMAlY Of' NAIN POROUS MEDIUM/ACOUSTIC PROPERTY INTERACTIONS

TYpe oholid pbua •....•..

- - --------.,,---,-·

IDI1uedce: ("l) indeterminate, -

'

..

-

-

--

-

-

-

Temperature

•••

-

- -

prcaare

Effective

External conditions

~

TUU!

?

••

••

?

-

-

Surface waves

.,

'---'

10MB~ . . . . . . . . .

-

MIIIIJIICII

''----"

only help to improve future testins. We give below two examples of possible applications for petroleum-~ &abo..,...:

• We showed (in Chapter 5) that microcracb. under low effective stress, substantialy

affect the quality factor Q of a porous medium. Thus attonuation measurements on core samples from a well can serve to determine the orientation of the micrcxneb" anisotropy, and consequently furnish data on the state offractur'iq of a reservoir or a description of in

--~

situ streslel. • We also pointed out how the velocity and espec:ially the-attenuation of ultrasonic waves could vary as a function of ps-saturation and the distribution of this ps for a given saturation. This. ofrers a much needeed analytical method to study twopbue flows iD porous media. ·

or

aeometric

In this Chapter, we have decided to focus our presentation on applications iD petroleum aoopbysica. Firat .- all, the taraota ill ........... IIOPhYtics as aftoa far from the ll.lC8IUiiJII aad rer:ordia& IJiteiQ. It il .~ - M Y for &be acoustic waves tr&Dimitted lO adeq-.Jy .,...U.te tM IU~ 'l1lil ,...uatioa depth illimite4 by the IDCfiJ lOIIIIIUitained by tho wa• as ia tile~ (eaer&J lolaes which we have called the iDtriasic ud utri1tir .........uo-). For deep penetration. attauatioa lll\llt be smallaad the IOUNC ~tow. Oadle Giber baed, for louin& IDMIUia~•• which thc ••riaatlltcm. tile aco.UC frequcaciet ar:e hiaJaer. Tile follewial table(Tai*1.l) liltl._i_..,.of,.......ofthc wavclcaltha used in · the dift'ereot IOiamic tedmiques.

it,....,....,. taqet.,.._ •.,_ •• .. T.uu 7.2

5uMMAilY OF

.........,_

:~~e..

-~

.I ''----"

'j

lI I

,.,.., ..., ....

DIFFiaENr. ~ ~·,,··t.IIIO IN ioaAcrtcAL AI'I'UCA~ ·r. '

•x JOl

•

•

14~\~';

10'

ID

, .... ·~IMIIII:.

,,

~· .....

..

........... plelpictlii ,,

.....

•• ..~a

.,.

w

; . .....-c ..... .

4

xto·•

.. ,...... ....... 106

t.ow••....Y ...... i,jla,_·

Two major aroups emcrp, separated by neatly two orders of mapitude in frequency:

(1) Low-frequency seitmie prospectia& (arouad 50 Hz) il intended to detect iDterestina aeoloaie&l horizons under sedi~ ........ea ap to ...,.a •tilol8etcrs. This technique, whose main application ia the aeometric description of the subsurface, is ~to . . for tMMIOIIIiC ............ oftbiaW. widl respect to

..a,-,

the~~

(l) Hi.....,..1HIIlCY-.. proepe;ti•a (a w Hz)u..., fOf .....ICIDIIlta ill well$, .-...a at die puUcular WI ef ........., MCl it very siiDilar in tcduaiquc to. eleelria8lloaiDJ. 1111oWIII-. . . .tiwlJ . .y to . . and aM resolatiGJl is much bctecr tbata low-freftun.cy Pl'GIIMIDtiu. However the rook vohlmes·teated are lialited arouacl tbc wellL

''---'

302

SOME AM.ICAT1i0NS'IN PETROLEUM GeoPHYSICS

7.1 7.1.1

7

LOW FREQUENCY SEISMIC PROSPECfiNG

Geaeral

To analyze the changes undergone by an acoustic signal traveling through a bed. it is necessary to be able to distin,&uish between the reflectiom at the top and bott~ of the bed. In other words layers ofminimum thickness clo~Jo the wavelength (- 20-30 m) must be distinguishable (Widess, 1973). Therefore limits are imposed for interpretation of largescale exploration seismic prospecting (conventional seismic prospecling> in which the large wavelengths employed are not sensitive to the reservoir beds most frequently encountered. Hence any increase in the maximum UDble frequency in the recording enhances the knowtedse of the subsurfaa. nus is adlN\Ikt in high-resolution seismic prospecting. in which frequeli<:ies in the 1'11DJC of lOOHz'llfe 'still ob8ervftle after propaption in the subsUrface. It is ob\lious that tile illvestiptiOn depth Of hi@h-reselation seismic prospectint is less than that of~· seismic prospecting, but hip resolution is essential f~r an accurate iderttikatien fJf the JitholoJy: of a thin bed or· of a sha11ow reservoir. The fleqUeacy eentelltd' tlt6 teeerclochignat Clln also be mcr.sect tJy limiting the distance traveled in the subsurface. By recording in a borehole, vertical•a«d multiple offset seismic proftles·permit shOJ!ter'trav.d·patblfo neorders inside the borehole. Hence the technique of reservoir seismics emerged, very dose to. conventional seismic prospecting in its methods of measurement and analysis, and whose initial purpose was the calibration of seismic profiles. Its bet~er def1niti~ makes it easier to acquire high ·· resolution data about the' acoustic properties orreservoirs. These problems of ticaleean be partly limited in certain cases by using the-knowledJC of the interfaces obtained from Chapter 6. Whatever its frequency, a wave i$'icll:cted ®m discontinuities of elastic or anelastic properties, for instance when travetm, from a gassaturated bed. to a liquid-~~ bed: ,_.. rMion can be detect~ even ·it' the waveleaath bed thickness ratio is less than favorable (Widell.. 1973). This variation in the reflection, ·as a function of vti'hltions in • or~ .for exalftJ'Ie,"elln provide valuable data about the interface. w~, abaU provide an ~xample for bright spot characterization in· the followifti·Sectioa: ·

7~1.2

The··expNSiiott c.»tmtntioaatMilmic prG~p~Cting applies to tdeetion seilmics used· in petroleum exploration. As a f1rst approximation, the soarcaand receivers an located on the ground ,or at sea leveL Itt ~ seismic piOipeeting, ()Wing to the tarp number of shots and geopbOMS,· several amvals ant ftJCOI'ded for tbc sanle ·rellectioa point correspondillg to daerent transMitterlftleliver distallces (ol&ets). The Ule of. the first arrival times helps to 4otermine the 'Velocity aad depth of each bed (see, for example, Waters, 1978 or Grau, 1985). As a rule, seismic prof1les are obtained with compressional

"' ~

' 1

L_~~------------~------------------~--~~----

'-

s6Mi 'AHIU&1isM$<'tif'~ !~<;

7

·-

wave sources. However, a number of sources exist (Marthorw, S wave Vibrators) that enable the recordiq ·of shear wave seismic sections.

7.1.2.1

'-

303''

Ca. .laed .aeof P .... SWIMS

The combiacd usc of P and S waves allows the measurement of the P and S wave velQCitics for a given ~ and hcn4::e the detcnnination of the Poisson's ratio. As we showed previously (Section S.2.2.6), this ratio provides information about the type of formation traversed by the wave and helps to limit the problem. Nevertheless, to obtain a more thoroush knowledge of the litholoQ, it is necessary ,to employ more elaborate techniques, which no longer Use only the information given by propaptioa time.

7.1.2.2 Sipal .....ysis

·-

''-

The mea~Un~~Deat of.Mwh velecitiea ia reOectioa aacl refraction prospectiaJ requires oaly tJae kaowJedJe of ·the arrival tiJDel ol tbe ~ lipb. Aa we have shown (Chapcer 4), a sipal wbicb ptopaptes in an .uenuatins 'JJOilOUI medium undcriJOCS. cl1aaaa in spectral COBteaC, 10 that the fGna of tbis dwale ia tbe lipatare of the io.-actioa between the wave- and a poroua medium ol given :properticl.(porosity, permeability, saturation). In face, a wave ~ ia the subsutfloo'uadclrsocs other typos of tantformatioa, cveo. ill an elutic 111111-iilfmite lllldiUJL . For a wave witlaanoa-plaMwavetn.t(aplawi&W or c:yliadriallf'otaa•ple}, the wave amplitude decreases with propagation distallCC. The total cnerJY, proportional to the square of the displa<:emcnt amplitude at the souro:, is distributed on the wave front whose area increases during propagation, and hence the displacetWDt ampitu4,e decreases. . Thus, for a spherical wave front (!of i~~ an explosive soun;e), the. total area of the wave froat is 4d'1r1 at a given time t, for a ~ of the medium V, ~ beli<:e the wave amplitude per uait area is proportlonal to 1/t and thus ~ :Withdme.'t:tiis etrect is known as ~di~.ltsipUWlytkenall t~frequeQCies. tt~wa~ no loa,Fr

propaptes in a sinsle. ~•.b\Jt iit·.~·.series fl·a~· btcl$, tll,e.~ _transmitted undergoes other changes.~ CbanFI ~teto traasmission and1'eftection i.Mchanisms at the different interfaces. Many iavcstiptioas on fteld seismocrams have revealed that these processes affect not only the amplitude, but also the frequcaey. c:oatent of dte tr&QS!Dit&cd •ve (O'~y a.od AMtc)', 1971, -Sdloqaberpr and ~ .~974). Other more com~ pr~sucb as ac::atteriq. . . . dcct the s.,ctral abape or,~ sipal received. · Two types of attenuatioa caJ1 be disti~ The tint, which we sbaU ~ intrinsic atteauatipn, has 10 far bcp.our maiawbjfa diacuMion.lt clw~ the aaelasticity of the material the wave' has thro\lp._ t1lc second. v.:ltidlwe Shall cafl extrinsic attenuation, isclw'ac~ ofthe-pomctr)' of'~ su~ue and Qfthe soUQ:C.lt has two etrects oa the seismic sipal; an apparatt ndueliOn in amplitude (the missinctner&Y is not lost but has beea delayed ~ale to.. tbe . . _ , peth.s traveled by the wave). and a frequency fllterin& dl'ec:t for mul&iple rc~Ject.ioas.. siuai~N. to .~ dJect of iatrinaic a~ten~ti9n (Schoenberpr aad LeviQ, 1974. M.odol et Gl..lt$2).1~ fact, it is therefore vc[y dift'tcult to separate intrinsic: and extrinlic aUCB~~ b ~pte Spe0ccr et al, 1?82).

aonc

or

<-

-·-

(tl ·~

b&istered lfllde..mark of lrrstitur F~ts tlu Mrilte.

304

SO,_E

APPUCATIO~

IN

PET&O~UM

Gl!OJ'HYSICS

7

a. Seismk #rfllicr.,¥ Seismic stratigraphy uses the fact that a Jivca ~ ~Cquence (altematina shale/sand, delta deposits, seabed slumps) corresponds to a given sipature on the seismic signal, making it possible to obtain qualitative li&hol~·te. 1lae IUin iatfation is to use the extrinsic attenuation of the signal. In fact, a cyclic ~ namely one consisting of a regular success..ion of beds (fo.r instance sha.le and sand~'ons),. possesses a clearly defined spectral signature. The low frequencies, those whose w gths are tona in comparison with ~ thicknesses, travel as if they were passing thr up one homoaeneous medium, whereas the hip frequencies undergo multiple ret1 •ons within the sequence. The resulting frequency content is characteristic of the so· and the sequence analyzed (see for example Morlet et al., 1982). The morphological analysis of seismic sections is iCOJlductcd by means of various techniques of amplitude restitution and signal processjng, such as the aJlalytical signal (Bracewdl; 1971~ Various examples are available in (Tanaer « Ill., 1979, Sheriff and Tanaer, 1979, Lacaze « Ill., :1978, et til., 1979). Note that this tecbuiquc is oaly applicable to clearly defined p · teq\IOftCIS, aDd that it requires priorknowJodpeftbepolouiDtMrepoaiiWCICip · siaceiO\'eralsequenoeseaahave the same sipature. Where it does apply, however, tbia ~ terVCS to determine lhe contours of1be nsenroic and of deposita. Finally. altiMJUih it mainly involves a sublluface geometric effect (and hence extriasic·to the porouuaodiuat), die euminetion not ODly of arrival ~ "- alsolipal shafes is UICiful for the remote lithOlotic determination of the subsurface.

*litemule aadiuey

b. .AIUI/y$1$ Dj /iriglt 6]1DfS In a seismic section, the amplit\ICles ot Cfle retlectioas observed are prally fairly constant, with m~or co~oiu for elects of ae'ometric diveraence. In certain ~ · however, bilher ~ncrgy· refteetioD$ are observed in the form·oflocal amplitude anomalies (see Fi&- 7.,1). 'ThClle ~ c8Ued ~ .,o~. ~ JeOPlayslcists have observed for a lona time-t~~· bN,bt spoti ocCasiOaally identified the pretOilOC or ps at the interrace concerned. ·· · · VelocityCODCI .... The investigations of Domenico (l~4) showocl that tbe refteetiVity of a Shale/Oil sand interface was quid: different from that of a shale/gas' sand interface. The seismic velocities vary when a very small ;pnount of gas (S% or even less) is introduced into the system (see Fig. 7.2~ The reflection amplitude rises liipiftcaatly'hithk·cae. Tbis amplitude is not a linear functioll of the amount of gas preiCilt in the reservoir. Unt'Ortunatcdy, no biunivocal relationship·exisu between bright spots and the preseilcC ota ps reservoir. Some wells drilled on bright spots proved to be dry. The local amplitude auoJnaly results in this cae either from taterat facies variations, or from constructive (or destructive) interference between near tefld:tors. Hence it is essential' to be able to distinpnh betwee~~ bright spots with gas and without gas. To do this. all the traCes·~ for tbe same reflec:tion point (corresponding to different transmitterfftlCeiver distances allCl heD<:e to dilferent angles of incidence) must be used. Recent investiptiOils (Ostrander, 1982 and Backus, 1982) show the importance of a fmc analysis of amplitude variations with angle of incidence. Ostrander (1982), for example, pointed out that the gas sands have low Poisson's ratios

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APPLICATIONS IN PETROLEUM GEOPHYSICS

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305

306

SOME APPLICATIO!'IIS IN PETROLEUM GEOPHYSJCS

7

(see Fig. 5. 72 in Chapter 5) (Poisson's ratio;: 0.1) with respect to shale and sand section (Poisson's ratio;: 0.3 to 0.4) and causes the P wave reflection amplitude to rise with increuina anp of ;ncideDcc. He also provides an example. In the case of a lateral facies variation. no uaiform ¥aria1ioB ill ttfllcUoa amplitude witb an,te of incidence should be

observecl. since tbe Poiuon~s raa of tbe different;becb are similar. Another approach. lot ~tin,a,.s eft'ectt in briabt Spots.~., comparing the P and S wave recor(Jinp correspondiafto the saJDe profde. Since S waves ate less sensitive to the presence ofps tho Pwaves,a,.. bript spot visible on the:Pwaveteetion will not appear on that of the S wave. Naturan,Y. inthe·c:ase of a lithologic variation, the bright spot will exist on both P aactS wave -'iOns (Ensley, 1985).

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The invcsti,ations so far d~ wore only concerned'. with the analysis of the properties of seismic veloc:ilies.Yet wbow (see SectioQ $.1.1.2) that the P and S wave v~are relam-elyml...._..ltr.&bt~ ofpsforuturations~from 5 to 95%, and thereforeCODSlitute. . . .v.IJ.fODrsaturat~ i.,._(Ors. By~ acoustic attenuations are biJhf1 --~ to ~ differen~ saturations (see Sectioa 5.1.2.2), and attenuation readies a peak for a ps ·sa.-atkm from 20 to ~4. Bourbie aod Nur ( 1984); showed that ~ waves exertea:cwo au.uarion e1rects. •·t:ft'ect on propaption and an e«ect at tbc iaterface (Section 6.1.'.2).: The p~ of an attenuation contrast at an interface ~ the roftection....... which woukl haft been aecordcd in the absence of attenuation coatrast 'l"hh ..pitude of dris elfec:t, depends 011 the absolute value of tbe .. elastic.. Rflection~ ~~ atteDuation).(scc Fig.. 6.9).1'bit attenuation contrast is only ob~rvable on the amplitude at angles of incidence smaller than 20 to 30". Moreover. attenuation corttrasts act on the phase of the reflected signal, altering its behavior as a function of anate of incidence. In the purely crlastic case, a reflection changes polarity abruptly after ~ing through zero, whereas, in the attenuating case, this change in polarity occurs as a cqnge of phase (Bourbie, 1982). Figure 7.3 gives an example of this behavior. A detailed analysis· of ttte behavior of relections as a function of angle of incidence is also necessary. tot in situ atten~ation contrasts, the measurement of the absolute value of intrinsic attemultion is no longer relevent since the relative variation of attenuation at the interfac:e suftidently describes the effect. This can then be implemented easily in synthetic seismoarams (see for example Jones, 1983).

Conclusion on bright spots For the analysis of bright-spats by velocity and attenuation measurements. we have shown the need to study the variations of reflections as a (pnction of offset. To do this. it is essential to have a high sipal/noise ratio and to work on unprocessed data. Few analyses ofthis type have yet been conducted to determine whether it is truly (statistically I possible to differentiate between bript spots_with and without gas. Nevertheless, the few results obtained so far are promising.

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7.1.2.3 Three-component sa.dle$ Conventional seismic prospecting generally records a single component of particle displacement at each rec:eiver. Recent investigations (i.e. Prunier, ·1981) have demonstrated the value for onshore seismic prospecting of a simultaneous recording of the three

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of the velocity vector. However the application of three-component reoordiQp to two- and three-dimensional Seismic prospecting requires an understanding of the propaaation m~hamsms iq the weattlefed zone (WZ). This undercompacted and under~nsolidated zoiie displays hip attenuations(~ ~ 10 to 20) and any interpretation of three-component particle l'ilodon must takt thc$e attenuations and th~r anisotropy into account. Recent examples (Dubesset nd tliet, 1984) reveal differential behavior

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SOME APPLICATIOSS II' PETROLEUM GEOPHYSICS

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between the different components of velocity, which is not yet fully understood but appears to result from differential attenuation. Therefore the future of experimental threecomponent analyses should provide a better idea of the anisotropy of the propagating media.

7.1.3 Rese"olr seismies

* and hence higher This expression is applied to a hisher frequency('> 100-.lSO Hz) resolution kind of conventioaal seismic pros~ J»caetration is shallow, and the dcimition ia reservoirs. application of this method mainly adlio\ra bed« The frequency iDcrcase c:an be obtained by wttJcaJ SeisDaic profiliaa (VSP). For these profiles, the SOUI'Ce remains located ()a tfte .JI'Ound, but the receiver is pia~ in a borehole right below the source. This ~- ope travel path throop the hi,lhly attenuating weathered zone (WZ). However, Iince tbese .travel paths ate subvertical, tbe zone analyzed is limited to the formation doSe te .~ well. An ofsct VSP tcdmicp.re was developed to overcome this drawback. in Which the s~nirte is no ~aer near tbe borehole but offset by up to a few hundred meters~ In this. c;asc. the zone analyzed iJ cxtend.fd to formations that are further from the borehole (see Fit. 7.4~. ·'

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Diagram of an offset VSP.

In VSPs. velocities and attenuations •re easier to measure than in standard seismic prospectina. Measurements can ~ QlJde iteratively relativ~ to. depth locations. Since the subsurface JCOII\etry is well defined, it is possible to determine the extrinsic •ttenuation from total attenuation. This m~es it much easier to obtain. infqrmation about the formations and their fluid content from signal analyiis. ' .

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Variation in fluid phase or pore

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a. Ste11111 jrtHII tUt«:titHH TltemeasURme~~ts presented in Chapter S showed that the liquid/ps phuecbanp was marked by a sipiticant chansc in velocities anchttteauations. Fiprc 7.5 siva an example ofa tiekl recording obtained by McEvilly er .,.(1978~ Thueianicdata COilCCrninga single event were obtained at diferent stations. The station located immediately nat to the caldera (station 3) shows a 10\\• ratio of P to S wave amplitudes. This is similar to the experimental results by De Vilbiss (1980) and c:orresponds to the minimum of the water/steam transition (sec Fig. 5.40). This is not an example in a reservoir, because there the procc11 oec:urs at a much laqer ICile, but it IUJIIIls UICful. results that ·can be aaticipated for steam front mOnitoriq.

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Nur0912)SUUOSfOdtileucofaetwocbofroc:eivendurinasteamOoodinaoranyothcr enhaJK.Od racovcry operatioa. inorclcr to4Cler~Pinc ~ ud attenuations within the reservoir. The monitorin& of ,-ariations in velocities· and attenuations would otter a knowlcdp of the gas phase distribution within the raervoir. A procedure of this type, adapted to an undc!JrOUnd gas storasc problem, is discussed by Blondin and Mari (1984).

1». A.IHtonncl reunoir JWe.,., We have shown that velocities and attenuations depend on the effective pressure, namely the dift'erence · bet\¥een CODAllins ~ (litbostatk) an.4 J10re pressure {hydrostatic). In the cMe of Jrilh-pressUte raervoin, the bi8h pore pressUre couatcrbalaaces the confiDills prts~t~te yie14iaa atmMIUIIIow I' and wave velocities and abnormally high attenuations. The determination of the acoustic parameters or their relative variations at the resen·oir jaterf~. &:Ould thus help to identify. hisb·pressure

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310

SOME APPLICATIONS IN PETROLEUM GEOPHYSICS

7 .1.3.2

7

Analysis of fractured soaes

a. Nlllurtll fr«turi~~g We discussed in Sections S.l.l.l and S.l.2.1 the int~nce offrac:turina on velocities and attenuations. In a saturated rock. fracturina reduees the velocities and increases the attenuations. This is one reason why fractured zones are acoustically distinc:t from the surrounding zones. Other processes are also involved, such as wave scatterina by the fractured zone, if the transmitted wavelength is about the same as that of tbe fractured unit block size. Well·to-well seismic prospecting is the ideal tool for analyzing these shallow fractured zones. Fractured zoaes arc more eeoustically vitiblc with lower effective pr~un:. in other words at a given depth with increasing pore pressure. This may also cxplaia die hiah amplitude reflections sometimes observed in deep seismic prospecting in crustal tectonic zones. If the permeability of the rock surrounding a fractured zone is sutftciently low, the fractures may preserve overpressures over long time IC81es (see for example tbc Wind River Range overthrust (nearly 30 km_deep), Jones, 1983). b.

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The technique of hydraulic fracturina to improve hydrocarbon recovery in a given reservoir involves injecting a pressurizeG Ouid into a well to fracture the reservoir rock. These pressurj2ed fractures exert a 'tiplacaat effect on the transmitted waves. and also generate reflected waves due to the contrasts in acoustic properties. ReOections are generally greater with ~ waves ~ with compressional waves across a hydraulic fracture, since the aheat moduli aa. ......Y tow; The ideal seismic cocbaique for .....,... ~ hydra~ fr~urina process is well-towell seismic prospectiaa (see Glf~,1914 In ibiS ·metJaod. acoustic waves are transmitted and ft'Jeeived iR two ..ens.. a:h huladrad meten •part. This elimiaates all problems associated with the weathertd'zone, so that various horizons can be observed with relative a uracy. The maximum frequencies received are on the order of a few hundred Hz fo well-t&-well distances 'of 200 to 800 m. By means of wlocity and ·attenuation me urements, it is p0$si~ to delimit the horiZons and to determine the length and strike of the fractures induced during injection. Unfortunately, the well-to-well technique has so far been used very little, due partly to the possible damage caused in the wells by seismi shots. Recently, due t& efrset VSP dewlopments, this technique is reappearing in hysies and sbould. prove its drectivcncss in the years to come.

7.2 I FULL WAVEFORM ACOUSTIC LOGGING ~~

seismic prospectins in boreholes is a broad term for sonic logging with analyses. The $ODic sonde, up to 15 m long (like the EVA tll tool) has ers and receiveTL B9th tr~milllion and reception take place in the

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borehole at frcquenacs of 10 to 20kHz. After propagation through the formation and mud, the full waveform is recorded. Two types of waves can be distinguished. namely the two waves refracted along the borehole wall (P and S wave). and the surface or guided waves due to the presence offluid and to the cylindrical geometry (see Fig. 7.6~ In perfectly elastic media, P and S waves are non-dispersive, whereas guided pseudo-Rayleigh and Stoneley waves are dispersive. An example of a recording obtained with the EVA tool is shown in Fig. 7.7. which reveals t~ coinplexity of a recording of this type. In some cases. neither a refractefl~ ~vc aor a_ Stoncley wave is obeened. while. in other zones. a refracted wa...a ~ ... aoStoaeley'WPe; Muy theOretical studies have been conducted to illaodet ........ ~ ~don iD .~ media (see for example Rosenbaum. 1974 or . . . . 6.·ll. ~fl• dloiU&:~ for Qan1ple Cheng and Tok.OZ, 1981 or~,a~.;l-~ ..... ~also ~used to test the validity of the . . . (!"i~*d:E~ ltJ3.adChen. 1984).

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Experimental .-esults ud in situ show that it is Q/wi/lys possible to obtain the compressional velocities v,. and $bear velocities Vs of the formation, by using either a standard source (as in the EVA tool) or by using a dipole or quadrupole source such as in the Exxon tool (Winbow and Rice. 1984). For the time being, measurements of attenuation Q,. and Qs are not routine, although interestins results have been obtained in some speciftc cases (Goldberc et al., 1984b, Huang and Hunter, 1984, Mathieu and .

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Tok50z, 1984). These authors clearly point out the relationship existing between the amplitude ratio of P and S arrivals (Goldbcq et al., 1984) and the Stoneley wave amplitude (Huang and Hunter 1984, Mathieu and ToksOz, 1984). and the fracturinc and hence the local fracture permeability. These results should not surprise us. In faa, they agree completely with the conclusions of Chapter 6, showing the importance of attenuations (and hence amplitudes) of refracted S and Stoneley waves for the measurement of a permeability that may not be due to fracture permeability only but to matrix permeability as well.

The relatively short wavelencths (a feW dozen em). the knowledce of the P and S wave velocities, and hence of the Poisson's ratio, and a certain approach to permeability, make these techniques the most promising of all those that have yet examined. It is in this type of approach that a sound knowledac of acoustic problems in porous media is essential to allow maximum use of recordinp of the type shown in Fig. 7.7.

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bibliography REFERE~CE

BOOKSUI

Ac:hmbach. J. D .• WD~ propogtllioll in «Mtic solids, Notth Holland. Amsterdam, 1973. Blud, D. IL, 1M tlwory of lbwr riscotltutlt:lty, Perpmon Pless, New York, 1960. Ccrveay, V., Molodtov, I. A., PSCDCilt 1., Ray IMthod ill uismology, Univ. Karlova. Prape, Czechoslovakia, 1917. Dullieo, F. A. L., Pof"fnu nwd/11: jiJiid trfiMIIOI't tllld por~ stf11Ctlll'e, AcademK: Press. New York, 1979. FunJ, Y. C., FOIIIfdDtiOIU of solitJ ~McluMics, PmltKe.HaU. New York, 196S. Miltlowitz. J., 1M tlwory of ~lastlc WfWU lllld K"tlrr p;Ms. Nonh Holland, Amsterdam. 1978. Nowick, A. s.. Berry. 8. S.. An tltlstic r~hvcDtion in crysuii/IM solids. AcademK: Press, New York, 1972. Schreiber. E., Anderson, 0. L.. Sop, N., Eltutic constants llltd tlwir IMliSIN'.,..,.,, McGraw Hill, New York,

1973. /nJil6~ QIIQ/ysis lltfllm~~themtltiCtll m~~rplrology, AcadanK: Press. New York. 1982. TnaeD, IL. Eltlaum, C~ Chick, B. B., llltrwMMic,.dwJd,r in MIJ sltll~ plrpics. AAdemic: "-.New Yorlt, 1969. Vilttorov. I. A., Rll.t·/rigll tiiUi 1.-rb -..ts, Plenum Preis, Nev.· York, 1967. Waters. K. H., Reflection uismology: a too/for energy resource exploration. Whitey lntC!fiC. Pub.• New York, 1978. White, J. E., Ulftlngrotllfll6(111111/: lfillictuion of uismic K"D\YS. Elsevier, Amsterdam, 1983.

Serra, J.,

REFERENCES Abramovitz, M., S~.l. A., HtiiWiboolc ofmt~tlwmtllicfllfi/llctiotts, US Govemmeat Printin1 OfCK:c, 1972. Achenbach. J. D., lfa~ [lf'OINI6tllloll ill «•tic IOiids. Norda Hollaad. Amst4ltda1ll, 1973. Alldrieux. S., "Un JDOCiilc de matiriau ~: applicatioa aux rocha et aux bCioal", Thais, ENPC, Paris, 1983. Anpheister, G., PhysicQ/ propertin ofroc/u, subvolume b. Sprinaer-Verlaa, Berlin, 1912. An:hie, G. G., .. Elecuic:alllllislivi&)' uanaid iDeo~ analysis iDtapmation", TrlfiiS AIME. I"'· S4-63, 1942. Alditty, P. C., AJwns. G., Staron, Pia., "State oftbe art,.Qil EVA data pnx:essiq: an improwrment iD subsurf.ce imaamg•·. presented at the 52nd AlmwJ/Int. SEG Jleetillg. Dallas, 1982.

~~-

Auld. B. A., ACOIIStic /abllffll-NS in solids, VoL-~. Wile)"lnlcl¥., New York. 1973. Azimi, S. :\., Kalinin. A. V., Kalinin, V. V., Pivonrov, B. L .. ··Impulse and transient characteristics of media with linear an4 quadratic absorptioa laws". Physic. Solid. L;uth (Eiljlish Ed.). 88-93, 196&. Bldtus, M. M., ''The reflection seismoJram in a solid layered earth", PaperS 16.S, presented at SEG -~ring, Dallas. 1982. Bec:quey, ~-. LaverJDe, M .• Willm, C., "Acoustic impedance lop computed from seismic: traces··. G«Jpptys.,
'~

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BIBLIOGRAPHY

Bensoussan. A .• Lions. J. l., Papanicolaou, S .. As.mlptotic Amsterdam. 1978.

ana~rsis

for periodic structures, Nonh Holland.

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-~

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321

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Hommap au Prof. J. Coulomb, Ed. CNU. Ptril, 1 • Nur, A., "Seismic imatillliaeahllaced oil ftiOIMI)'", SH;I)t)£ M•tllw, Tulsa, hperllo.10610. ApdJ 1982. Nur, A., Byerl.ee, J. N., ··All exact effective stratlawforelasticdetOrmadoa ofrocb willa ftuicl .., J. GHpll. bs., 16, 6414-6411, 1971. • Nur, A., Murphy, W., "Wa,·e velocities and atteauationinporousmedia with fluids", Proc.ofl#lh Int. Conf. on synars, S~ )I 1-32:7, 1911. · Contlnlutr lftOtll& of Nur, A., Simmons, G .• "The effect ofviscolity oh Ouid phase on velocity in low porosity rocb", Earth a1111 PloMt Sc. Lnt.• 1, 99-108, 1969. . . Nur, A., Tosaya, C., Vo-Thanh, D., "Seismic DlODitorln& of thermal enhanced oil recovery processes'\ Paper .RS6, 54th SEG M«tin6. Atlanta, 1984. Nur, A., Walls, J. D., Winkler, K.., De V'tlbils, J., .. E«ecu of fluid saturation on waws in porous rock and relalioDt to liydraulic permeability", SPE }tltlnlltl• .tSD-4!8. 1910i. Nur, A., W'mkler, K .... Thenrlulfficltiea.*laict. . in rock&" ( .U.I), ~. 45, S91·S9~ 1980. O'Connell, lt. J., Budiansk.y, B.. "Seismic wlecitica ill~- sa&!Dted crackocholida", J. (hop/1. /l.u., 1t, S412-S426, 1974. .

dist:r'"

ill..-........._

O'Connell, It J., Budiansk.y, 8., ''Vill:od··~~~~ IOiicls",J. ~.IW., 76, 2022-2034, 1977. . . . . . .. . ·. .

O'Connell. ll. J., Budiansky, B., "Measures ot~ in ~illedia", O.Oplry. Ra. Lftt., 5, s-8, 1971.

J

O'Doherty, R.. F., Anstey, N. A., "Rettections on amp&~". GetJplt. P.rosp•• 19, 430-4S8, 1971. O'Hara,$, G., "l.nu-ce of~~ ud ~JhiiciOP tJao frequmc:y-dqptDIIIeat attcuuation of elastic waves 'n Berea sandstone", Php/l:ill rnfew ..l, 32, 413-488, 198!. · Ostrander, J., '6Piane wa-Ye rerlectiOD coelfldenta ror'ps sands at non normal anates ortncidence'\ , . , No. SJ6.4 p~ted at 52nd SEG Meetilfl, DaJiu, 1982. Paillet, F. L., White, J. E., ··Acoustic model of propaption in tile borehole and their relationship to rock properties",~~-· 47, 1215-1228, 1982. ' . Palmer, I. D., Traviolia, M. L., "AtteauatiOD by ~quirt flow in under-saturated ps sancla ", Geophys., 45, 17801192, 1981. ! . Pallet, M., et IIJI., La mkaniqw des rochu qplilplh IIIIX 011vr1Jfft de Gblie Civil, ENPC, Paris, 1976.

w.

Papadakis, E.i P., "Ultrasonic difl'rac:tion frOm apert~ willa applications to pulse measurement& and crystal physics", ill: Physical ..lCOtUtic, XI, Ed. Muon Tlwnton, Ac.-Preu, Chap. 3, lSl-211, 1976. Parts, G. A., "Surface and intcdacial free . . . . . of' ........ J. G.o,/t .. ICes., 8, 3997-4008, 1984•.

liilalc

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.PadDick. L., ,Outcrkidp. yt. F.. "Ill . ...a lUI4 sbcu . . .u.o(Soleabofcn JiiDestoae over a frequmcy ranp of 10 cyclci ... ....t",/. ~,.,... "- 581-SII,.t96t.

WJ.J .. ~ B. f .• ~ W.J.. "~ofair-~udoil-aircapillarypreuurcdatain the study of pore struc:ture and ftnid cfistrmutioJa , $1'. JOfiT1IIll, $5-61, 1966. . · ·

etutic

~

h.

Picltett, G., "Equations for computinJ coMtants flquraJ ,.cl19rsioaal RIOUilt &cquciacies of vibration of prisms and cvlindcrs", Am. Soc. Ten. A#(IIB. Proc., 45. 846-860, 1945. Pjumann, E., Dusclultko:R.. "Usc ofpQrc casts S'!M io pore pometry", Sell. Petrology. «l.

115!-1151, 1910.

study

ad

Jm'u.

. ' . .. ' Plona, T., "()bscrvation of a JeCOncl bulk~-~ an a poMUS medium at ultrasoaic frequencies",

..lppl. Phys. Lett., ._ 259-26t, t980. · Plona, T. J., "Acoustics of Ruid-saturated pi)I'Obt inedia.... Vlfraonlci-Sprrtp. IEEE, 2, 1044-1048; 1912. Ptona, T. J ., Johnsoil, D. L., .. Eipeiimelital Mudy' Of the two bulle COibprcSSional modes in water-satunted porous ltnlc:tures .., Vltrt1Sottk2 S1f'IP. 16EE, ...,!12. ..-,_ · Pnalicr, A., "RecMrcbe en tnlitaacat -.....•r1adidtdea_..SlNIIIDll troiseodlpolaDtcs, Application sur un exemple reel", Thesis, Univ. de Purcell, "Capill8ry ,_..,.... 'l1lllir ••• IMI'CUIY ....the caii:Watioa of penlleltbility thercf'nrm", J. of Petroleum Tedurology, 39-47, 1949. Qnimby,S.L.,"OatJaeexperimeat.tdetenlliDatiollefdll:viNosityofvibratinatolidi".P•,,.••.• zs.ssa..s73. 192S.

SUa.,.., ••· t

-~·

....

322

IIIIUOORAPHY

Rasolofosaon.. P.. Coussy. 0., "Piopaption des ondes ~urs dans les milieux poreux satures - Effets d'intenace. ~ panie··. Rttr. lnst. Franr. du Ntr*, 40, 78S-t02. l98S. Rassing. J .• "l:ltraSonic maxation specbomefry'' ;,: Clwrniad tlllll biologietll IUJI«U II/ rBaxlllion Sf'«trotnetr_t·. Ed. by Wyn:Jons. Reidal. nouan.l !~7S. Lord Rayleigh. ··On waves propapted along the plane surface of an elastic solid''. Prot:. Loru~Df~. Matlt. Soc .. 17. 4-11. 1SS5. Ra~mer. L. L.. Hunt. E. R .. Gardner. J. S., "An improved sonic! transit time-to-porositytransform",SPWLA. :Jst .4nn. L.ogg. Symp .. July 1980. Rice. J. R .. Cleary. M. P .. ··Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents··. Re'·· Geoplr. and S~ce Phys.• 14. 227-:41, 1976. Rogez. D .• Bader. M .. "Tltrasonic ,·elocity dispersion in liquids betwea .3.3 and 330 MHz llling a high ~lution phase measurement techPique", J. AcOJMI. Soc. Am•• 16. }67-172. 1984. Rosenbaum. J. H .• "Synthetic mic:rosei~: loUidl ia.pon*S foRDaticla", Geo,ltys•• 14-32, 1974. Sa~. J.. YUcoilturtctte. PNaa de t•ac. NatioUIIt dll Poatut Cbauslies. Paris, 1983. Sayers. C. M .• "lltrasoninelocity dispersion in porous materials". J. Phys. D. : Appl. Phys.• 14, 413-420, 1981. Schlum~ Co.• Log intn'pretatitllt ~. Ed. by !dtiU11lb!!tJer Limited 1977. Schmidt. E. J.• \'o-Tbanb. D .• Nur, A.. Auld, .B., "AA:oustic and electromapetic energy absorption in clay bearing rocks at low fluid saturation ... to be published', (kop/iys.. 1986.

~·~

».

Schoenberg. M.. Manetta. T .. Aron, J., Porter, R., "Spa<:e-t.ime .lependance of acoustic waves in a borehole ... J ..·koust. Soc. Am., 70. 1496-1$07, 1981. · Schoenber,er. M .• Le'in. F. K., "Apparent attenuation due. to intrabed multiples", Geophys.• 39,278-291, 1974. . . .· . Schone. P. A.. ·• Chalk diagenesis &Qd' itS rdaliori tO 'pettOiell!Ji exploration". _.APG. S, 982-1009, 1977. ~hreiber, E~. Anderson. 0. L., Sop, N., Eitutic ~tants and their tneiUIITemetrt, Mc(Jraw Hill, New York, 1973. . . . . . . . . . .

..

Serra. J.• /mage analysis and mathematiCDI !1f0rphology. ~ fress, New York. 1982. Sheriff. R. E .. Taner, M. T .. ··Seismic attrlbute me;.$~reshelp defnacreservoifs". Oi/andGtuJourrull. 112-186. 1979. . Spencer. J. W .. "Stress relaxation at low frequencies i~ Ouid saturated rocks: attenuation and modulus dispersion". J. Geoph. Res .. 1303-l812. 1981. Spencer. T ..-\ .. Sonnad. J. R., Butler, T. M., "Seismic Q. Stratiy.lphy or dissipation", Gef!phys., 47, 16-24, > • .• •·• . • . . ' 1982. . ' ..

~tzler._.H.. :'M~· D. L.. "nC elf~ o( ~ture ~partial mellinJ Oil velocity and at$eouation in a · sunple bm&r) system . J. Geoph. J&s .. 73, 6051~. tMII. Spitmtt. s.. Teft, W. E., .. MethdclfOr~lllfidlmi<:lfn:sorwtc:drequeoc:ieallblltorcalc:11latingelastic moduli from these frequencies", AM.. Soc. Tm. Mam. Proe.• 61, l:!!l•tm. 1961. ·

rise

s«.. "·

Stewart. tt. G .• ··Q uct the uct a~ ptil~e", aeoPb,, J. JtD.r. Astr. 193-805, 1984. Stewan. R. R., TolcsOz, M. N., Timur, A., "Strain dependent attenuation: observations and a proposed medumism... J. Geoph. Rn.. •. S46-SS4, 1913. · StoU. R. D .• "Experimeptal s t . of~ttenuation in sediment$", J. Acoust. Soc. Am.• 66, 1152-1160, 1979. Strick. E.. "Tiie determination of' Q, viseosity arid creep ·cun·es from waw propaption measurements", Geoph. J. Roy. Astr. Soc., 13, 197-218! 1967. . · Strick. E.• " A predicted pedestal effect for pulse propagation in eonstant Q IOlids ". Geoplv•s.• 35, 387-403, 1970. Suquet. P.•. "Piasticite ct homopgeisation". ~ U~versi¢ P. et M. Curie. Paris 1982. Swanson, B. F .... Visualizing pores 8JlCi nonwettina phase .in por:ous rock", J. Pet. Tech.. Jan, 10-18,. 1979. Tauer. M. T .. Koehler. F .. Sheriff, R. E., "Complex trac1ranalysis". GftJPhJ'I.• 44, 1041-1063, 1979. Tarif, P., Bourbii. T., •• Experu.atal ~ botwten apeetral ratio and rise time teclmifrues for the measurement of attenuation··. to be published, G• . Prosp., 1986. Tarif, P.. -M~de I'attenuation par.prop~~ptioa d'ultruoa: application aI'etude de l'aailotropie ",Thesis, Unhrersite Paris \11, 1986. Tsmur. A.• "Velocity ofcompreuional waves ill pomusmedia at permalrosttempcraturee", Geop6ys .• 33, S84595. 1968.

ran or

,

..

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~~

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3~3

BIBLIOGRAPHY

Timur.A.. MTcmpaaturcdcpcndcnccof~onalandslaqrwaveveloc:itielinfocb",~ys.• 4l.9S09S6, 1977. Tittmann. B. R .. "Internal friction measurements and their implications in seismic Q-struc:turc models of the crust''. AGU ~oplt.t·s. MOllO,., Ja, The earth's crust, 197-213. 1977. Tittmann. B. R .• Oark, V. A., Richardson. J. M .• speitcer. T. \\' .. "Possible mechanism for seismic attenuation in rocks containing a small amount of ,·olatiles". J. Geoplr. Rts.. IS. 5199-5208, 1980. Tittmann. B. R .• Noadler, H .• aark, V. .-\ .. Ahlberg. L.A.. Spmcer, T. W .. "Frequencydependenccofseismic attenuation in saturated rocks", Geoph. Res. Lm .. 8, 36-38. 1981. Tok50z. M. N .. Johnston. D. H., Timur. A .. "Attenuation of seismic waves in dry and saturated rocks: I. Laboratory measurements", Geophys.. 44, 681-690. 1979. Tosaya. C., "Acoustical propenies of clay-bearing rocks". PhD. Thesis. Stanford Univ., Calif., USA. 1982. Tosaya.C.. Nur. A., ••EffectsofdiaJtneSisandclaysoncompressional velocities in rocks", Geoph. hs. Lett•• 9, 1982. Tosaya. C., Nur•.A .• Vo-Thanh, D .• Da Prat. G .. "Laboratory seismic methods for remote monitorins of thermal EOR". -~ Cnll. Europ. ·"" l'am;/fnrarinn de Ia rkrtf'<'rarinn tht phrnle. Rome. Arril 19RS. Truell. R.. Oates. W .... Effect of lack of parallelism in sample fa.:es on measurement of ultrasonic attenuation .., J. ACOIISI. Soc. •.fm., 35, 1382·1385, 1963. Truett. R.. Elbaum. C., Chick, B. B., Ultrasonic merlrods in solid sttJie physics. Academic Press. New York, 1969. Tullos, F. N., Reid. A. C., "Seismic attenuation of Gulf coast sediments", ~ophys., 34, 516-S28, 1969. Viktorov. I. A.. Rayleigh tmd IAmb "·a1·es, Plenum Press, New York. 1967. Walsh, J. B~ "The effect of cracks on the compressibility of rocks". J. ~oph. Res.. 10, 381-385, 196S. Walsh, J. B., "Seismic wave attenuation in rock due to friction", J. Geoplr. Res.• 71. 2591-2599, 1966. Walsh, J. B., "New analysis of attenuation of panially melted rock", J. ~- Res., 14, 4333-4337, 1969. Ward. R. W., t ok50z. M. N., "Causes of rqional variation of magnitude". Bull. Seism. Soc. Am., 61, 649-670, 1971. Wardlaw. N.C.. Taylor, R. P., "Mercury capillary pressure curves and the interpretation of pore structure", Bull. Ctm. Petrol. Geol., 14. 225-262. 1976. Waterman, P. C.. Truell, R., "Multiple scattering of waves". J. of Math. Physics. 1, 4, 512-538, 1961. Waters, K. H., hjlection seismology: a toolfor energy resource t.Tploration. Wiley lntersciencc Pub., New York, 1978. White, J. E., &ismic wans: rtJdiotion. trtDUmission and atlenUIJiion, McGraw Hill. New York, 1965. White, J. E., "Q of fluid saturated rods", PtJper S 4UJ6. AGll Meeting. May 1983. White, J. E., c:rulergrotUUI SOillld: wlictJiion of seismic t\'Ql'es. Elsevier. Amsterdam, 1983 b. Widen, M. B.. "How thin is a thin bed?", Geophys .. 31, 1176-1180, 1973. Winbow, G. A .. Rice, J. A., "Theoretical performance ofmultipole sonic loainl tools", PtJper BHG 2.7, 54th SEG Jleetittg, Atlanta. Dec:. 1984. Wmlder, K., "The effects of pore fluids aacl frictional slidina on seismic attenuation", PhD. Thesis, Stanford University, Calif., USA, 1979. Wmlder, K., "Frequency dependent ultrasonic properties of high porosity sandstone ... J. ~ph. Res.• a, 94939499, 1983. Willkler, K., Nur, A., "Pore fluids and seismic attenuation in rocks", Geoph. Res. Lett .• 6, 1-4, 1979. Willkler. K., Nur, A., "Seismic attenuation: effects of pore fluids and frictional slidina", Geophys.• 41, l-IS, 1982. Wia1tler. K., Nur, A., Gladwin, M., "Friction and seismic attenuation in rocks", .\'ature,l77. 528-531, 1979. Winkler, K., Plona. T., "Technique for measuring ultrasonic velocity and attenuation spectra in rocks under pressure", J. Geoph. hs .• 17, 10776-10780, 198:!. Wu, T. T., "The effect of inclusion shape on the elastic moduli of a two phase material", Intern. J. Solids Structures, l, 1-10, 1966. Wyllie, M. R. J .• Gregory, A. R., Gardner, L. W ., .. Elastic wa\·e velocities in heterogeneous and porous media", Geoplrys .• l, 41-70. 1956. Wyllie, M. R. J.. Gregory, A. R., Gardner, G. H. F .... An experimental investiption of factors affectins elastk wave velocities in porous media", Geophys.. 13. -'59-493, 1958.

s-s.

·~

..___

'·~·

---~

•

324

BIBLIOGRAPHY

Wyllie, M. R. J., Gardner. G. H. F., Gregory, A. R., "Studies of elastic wave attenuation in porous media··, ~•• 17, ~S89, 1962, lllrd d&t:ussi011: G#opltya., 21, 1074, 1963. Yin& C. F., Truell, R., "Scattering of a plane longitudinal wave in an isotropically elastic solid .., J. of Appl. Pltp. 7:1, 1086-1097, 1956. Zinsmer, B., Meynot, Ch .• ··Visualisation des proprietes capillaires
~-----

•

~

~

"---"

author index '-..__

"'--

'-

Brutsaert, w.. 190 Budiansky, B., 58, 65, 122 Butler, T. M., 304 Bycrlee, J. D., 176, 184

Abramovitz, M., 125, 127 Achenbach, J. D., 49, 70, Ill. 273 Ahlberg, L. A., 213 ADdenon, D. L., 123, 124, 195 Andenoa, 0. L., 162, 163, 166 Andrieux, 63 Aqenheistcr, G., 57 Anaona, F. A., 241 Anstey, N. A., 303 Archie, G. G., 71 Arditty, P. C., 160 Arens, G., 160, 304 Aron, J., 281 Asce, A.M., 190 Asce, F., 190 Auld, B. A., 223, 225, 264 Azillli, A., 124

s..

"'--·

.._

.

Capiarcl, L., 266 Caqil1, G. S., 13, 16 Camwm, P. C., 35 Carmichael, R. S., 80, 193, 241 Cutqna, J. P., 298, 311, 313 Caye, ll., 13 Cerveny, v .• 249 Claandler, N., 81 Chang, s. IC.., 311 Chell, S. T., 281, 311 Chalg, C. H., 311 Cbkk, B. B., 147, 151 Qoquette, P. W., 14, 18 Ouistensen. R. M., 100 Cart. V. A., 190, 207, 213, 21P Cleary, M. P., 81 aiel, C., 307 Coben-Tannoudji, C., 109 Cooper, M. F. Jr., 290 Cordier, J.P., 230, 237 CouraDt, ll., 129 Coussy, 0., 266

s.

Backus, M. M., 304 ·~

Bader, M., 151

BaDcrof't, D., 166. "'--

'---'

~

'---' '-

~

'-.-c

Beard, D. C., 233 Becquey, M., 304 Belllouuan, A., 64 Berry, B. S., 122, 143 Berryman, J. G., 71 Babers, D. N., 219 Biot, M. A., 64, 65, 67, 72. 7J, 22S, lin:ll, F., 166 Blair, D. ll., 157 Blan4, D. R., 122, 125 .Bioadin, L., 309

Borcherdt, R. D., 112, 286. 287, 288, 290. 291 Bourbie, T., 80, 157, 158, ISS, Ill, 190, 204, 206, 208, 232, 243, 252, 259. 287.290, 291, 295, 296, 297' 306, 307 Bousquie, P., 29 Bowler, J .• 146 Brace, W. F., 34 Bracewell, R.N., 270, 304

--

248. 252 Danbura, J. s.. 233 Da Prat, G., 192, 196, 197. 207, 211 Datta, s. IC.., 94 Davis, H. T., 17 Davis, L.A., 170, 203, 2ll. 212, 216 DelfiDCI', P., 13, 20 De Martini, D. C., 233 Denis, A., 179 Demiewic:z. H., 246, 247. 248, 250, 258,260.261, 262,264 Desclwnps, M., 152

'._/

_/

·._/

~

326

AUTHOR INDEX

Devaney, A. J .. 215

Grau, G., 302

DeVilbiss, J. W., 187, 195, 197, 200, 211, 242, 309 Dhawan, G. K., 29 Ditkine, V., 270 Diu. B., 109

Gregory, A. R., 75,177,181,189,130,231.133,235. 237, 254, 265. 273 Grittl, R. E., 12 Guillot, D., 14

Domenico. S. N., 190. 191, 237, 238, 304, 305 Drake, C. L.. :!32, 233. 235 Droschak, D. M .• 166 Dubesset, M .• 307 Dullien, F. A. L., II. 23, 29, 30 Dupa, L. L., 146 Duschatko, R., 13

Elbaum, C., 147. 151 Ensley, R. A., 306 Eringen A. C .. Ill Eshelby, J. D .. 58 Etienne, J ., 29 Everhart, A. H .. 3lt Ewing, W. M., 49, 249

Fatt, 1., 259 Faust, L. Y., 237. 238 Feng, D. L., 248 Flugge, W., 100 Fourgeau, E., 304

Fung, Y. C., 53

Hadamard, J., 96 Hardin, B. 0., 190 Hauge, P. S., 241

Hickman, W. B.• 31 Hilbert, D., 129 Hosten, B., 152 Huang, C. F., 311, 313 Hunt, E. R., 232, 234 Hunter, J. A., 311, 313

~~

~

~~

lshimaru, A., liS

Jacquin, Ch., 35, 44 Jankowsky, W., 237, 238 Jardetzky, W. S., 49, 249 Johnson D. L., 71, 80, 81, 88, 89. 91, 92, 93, 248 Johnston, D. H., 154, ISS, 203, 205, 218, 243 Jones, L. E. A., 180, 233, 236

Jones, T.,l82, 186,194,209,222,223,226,227,306,

~

310

Futterman, W. L., 124

Gai'Perin, E. 1.. 310 Ganley, D. C., 241 Gant, W. T., 311 Gardner, G. H. F., 75, 166, 182,230, 231,233,235, 237, 254, 265, 273 Gardne~J. 8.,232,234 Gardner, L. W .. 230, 231, 233, 237 Gassmann, F., 68, 237 Geerstma, J., 84. 94, 190, 248,250, 2S4 Germain, P .. 49. 66, 69 Giard, D., 304 Gladwin, M. T .. ISS, 201, 216, 218 Goetz, J. F., 146

Kalinin, A. V., 124 KaliniD, V. V., 124

Kan, T. K., 298, 313 Kanamori, H., 123, 124 Kanasewich, E. R., 241 Kaye, G. W. C., 193 Kjartansson, E., 125, 126, 127, 157, 221, 224, 218, 305,307 Koehlet, F., 304 Kojima, H., 93 Kowallil, B.J., 233, 2.36 Kuo, J. T., 219 Kuster, G. T., 58

Goldbug. D., 241, 298, 311, 313

Laby, T. H., 193

Gonzalez-Serrano, A., 259, 287, 29S Gordon, R. B.• 170, 203, 211, 212, 216

Lalc)e, F., 109

/-

..-......

~-----

Lacue, J., 304 ---.._

-----~

_...,

A~'iWolix

Nowick. A. S., 122. 143 Nur, A .. 62, 94. 139. 176, 177. 179, 181, 184, 187, 188,192,193.195,196,197.201,205,206.210, 211,212,216.218,222,223, 226,227,233, 236. 242, 252. 297, 306. 309

Landau. L., 49 Lanon. R. G., IS Lawrpc, M., 239, 240, 304 Le Fournier, J., 29 Levin. F. K .• 147, 303 Levine. H .• 215 Levy. Th., 64 Lifehitz, E., 49 Lindsay, R. B., 287 Lions. J. L., 64 Liu, H. P., 123, 124 Lockett, F. J., 290 Lockner, D. A., ISS 124 Lomnitz, Louis. P., 304

m.

Oates. w .. 147 O'Connell, R. J .. 58. 6S, 122 O'Doherty, R. F., 303 O'Hara, S. G., 165 Ostrander, W. J., 304 Outerbridge, W. F., 163, 164

c ..

Paillet. F. L., 274

Mahood .• G. A., 309 Majer. E. L., 309 Mandel, J., 33, 49, 53, 64. 66, 72, 120 Mari. J. L., 309 Marigo, J. J., 58 Marzetta, T., 281 Mason, W. P., 219 Matheron, G., 18, 72 Mathieu, F., 3tl, 313 Mavko, G. M., 58, 94, 198; liS, 2'18, '~

Me Skimin, H. J., 153 Meynot, a., 29 Miklowitz, J., 266 Mills. R. L., 241 Mobarek, S. A. M., 194 Mochizuki, s., 225 Molotkov, I. A., 249 Monic:ant, R., t I Mor1et, J., 304 Mortier, P., 94

,_

~~'

"---

'~

22'7

Me Donald, F. J., 241 Me Evilly, T. V., 309 Me Kavanash, B., 170

'~

l

Morrow, N. IL, 23 Murphy, W. F. III, 123, 177, 179, 181. 184, 187, 188, 190,191,199,201,206,213,214.216,218,220, 222,227,229

Nafe. H. E., 2~2, 233, l!S Noadler, H., 213

'327

Palmer. I. D., 227, 229 Panet, M., 57, 177 Papadakis, E. P., 153. ISS Papanicolaou, S., 64 Parks, G. A., 192 Puierb, F., 93 ~ L., 163, 164 Pickell, J. J., 29 Pickett. G., 162 Pierrot. R., 13 Pittmann, E., 13 Pivovarov, B. L., 124 Plona, T., 80, 81, 85. 86, 88, 89. 98. 91, 92, 93, 152, 153, 213 Porter, R., 281 Pray, L. C., 14, 18 Preis, F., 49, 249 Prevosteau, J. M., 13 Proudnikov, A., 270 Prunier, A., 306 l'leacik. I., 249 Purtell, w. R., 26

Quimby, s. L., 165 Rqot, J. P., 13 RuolofolllOD, P., 266, 298 Raising, J., 223 Lord Raylcip, 263 bymer, L. L., 232. 234 Reid, A. C., 241

-~j

328

Al:THOR INDEX

Reynaud, R .• 304 Rice, J. A., 311 Rice, J. R., 81

Rice, J. T., 2SO, 258, 2~. 26l, 262 Richardson, J. M., 207, 219 Richart, F. E., 190 Robinson, J. H., 233 Rogez, D., 151 Rosenbaum, J. H., 94,248,273.274,276. 277.278. 282.311 Salen~n. J., 100 Sanchez-Palencia, E., 64 Sarda, J. P., 94 Sayers, C. M., 215 Scala, c., 93 Schechter, B.S., 309 Scblumberger, Co., 230 Schmidt, E. J., 223, 225 Schoenberg, M., 281 Schoenberger, M., 147, 303 Schone, P. A., 34.

Schreiber, E., 162, 163, 166 Scriven, L. E., 17 Sen, P. N., 71, 93

Sensbusb. R. 1., 241 Serra, J., 18 Sherift, R. E.. 304 SieJ(Jed, R. W., 311 Simmons, G., 195, 210 Skalak, R., 93, 94, 246, 247, 248 Smit, D. C., 84, 94, 248, 250, 254 Soga, N., 162, 163, 166 Sonnad, J. R., 304 Spathis, A. P., 157 Spencer, J. W., 123, 171, 222, 227 Spencer, T. A., 304 Spencer, T. W., 190, 207, 213, 219 Spetzler, H., 195 Spinner, S., 166 Stacey, F. D., 155, 170

Staron, Ph., I~ Stegun, I. A., 125, 127 Stewart, R. G., 157 Stewart, R. R., 218 Stoll, R. D., 218 Strick, E., 124

Suhubi, S., 111 Suquet, P., 63, 64 Swanson, B. F., 31 Tanner. M. T., 304 Tarif. P., 154, 155, 157, 158. 160 Taylor. R. P., 26 Teft. W. E., 166 Timur. A .• 154, ISS, 194. 195, 197. 218. 243 Tittmann, B. R.• 166, 190. 207, 213, 219 Tok50z, M. N., 58,.154, 155,203,205,218,243,311,

-~

313

Tourenq, C., 177 Tosaya, C., 178, 180, 183, 186, 192, 195, 196, 197, 210, 211, 233, 236

Tra,ioUa, M. L., 227, 228 Truell, R., 115, 147, 151 Truesdel, A. H., 309 Tullos. F. N .• 241

...--...~

Van Nostrand, R. G., 241 Viktorov, I. A., 263 Vo-Thanh, D., 192, 195, 196. 197,210. ~II, 223,225 Walls. J. D., 187, 242 Walsh, J. B., 58, 94, 116, 180, 184, 195. 218, 2l6 Wang, lt.·F., t80, 233,236 Ward, R. W., 154 Wardlaw, N. C., 26 Waterman, P. C., 115 Waters, K. H., 302 White, J. E., 68, 141, 153, 161, 164, l~. 241,1171 Widess, M. 8., 302 Willis, D. G., 65, 67 Willis, M. E., 311 Willm, C., 304 Winbow, G. A., 311 Winkler, K., 58, 139, 152, 153, 165, 166. 169, IB7, 197,200,201,205,206.212,214,215,216,218, 222, 242 Wu, T. T., 58, 180

~-""""'

~~

Wyllie,M. R.J., 75,166,182.230,231,254,265,273

Ying, C. F., liS Zinsmer, B., 29, 80, IS9, 185, 188, 190. 204, 206, 208. 232, 243

~

~

...--...

-----:'1

....... ''--' '

.

'---' '----'

subject irdex '·.,____

'---'

'-' '''-"

------

Act.orbed water, 219, 225 Ac:ces radius (porous medita), ~6. 29, 87 Atlnl (material), I 00 Air)' ..... Ill A8qiiAollte, 216 Allltydrite, 235, 240 Allilotropy iaduced, 183, 184 intrinsic, 186 ~pasation,61, 74,96 transverse, 97 Aaale (eritical), 88 Aaortllollte, 220 Ane.atloa capillary forces, 220, 221 contrast, 306 deftnition, 15, 107, 1 t: extrinsic, 147, 303 frequency, 211-214, 2::2 in situ measurements. 240, 243 interface, 279-281 intrinsic, 147, 305 mechanisms, '63, 217, 229 measurements, 15 I ~.156

permeability, 275 "---" "--'

'---'

'-----' "--· '-'

'-"-'

attenuation contrast, 306 velocity contrast, 304

Caldera, 309 Capmary, Calilarity, 22 desorbtion, 24 equilibrium, 23 force, 191, 220, 221

pressure, 24 visualization, 29, 32 ear.a.Kozeay fonmda, 35,47 Casco (see Gruite) C...uty prillcifle, 103 Claalt (see u.e.toae)

a ...ca~._.

hydrosen, 219 Pray dellllfacatloll 15, 16 Oay (see also, Sllale), ll, 12, 34, 235, 236--240

a...-ue· ...

phase change, 211 viscosity, 207, 210 saturation, 2()4..207 methods, 207, 208 scatterin& 214, 215 strain, 215 temperature, 207, 209

Colli, 240 COCGidao (see Saadst011e) Celoralle(seeSIIIIes) eo.,.ctloa, 237 CotDctlfllle,23 COIIdaulty bldex, 179 ConloYa ereaaa (see I' eltDae) C....... l47, 153 inertial, 71, 83, 225

Barre (see Graaite) BMalt,216,240 Wfonl (see IU.IIt1M1e) Berea (see s..tst..) IJerrymaa's foranlla, 79. 265 Bimodal perGiimelric SJiftb ... 27,

8riPt ...... 304-308

breakage, 219, 221

pressure, 203, 204 saturatina fluid

-~

--------

Blot's dleory, 63-84. 141, 161, 166, 190. 225, 250, 259, 263 (experimental veriftcation of), 85 Bolle (see Saadltoae) ~.. prladple, 102 .............. 152

tbermomechaDica 65

viscous, 87

rT

COYMiuce, 18, 40 Creep fuctloa, 100, 124, 12S

'-'

_.. ..::£:;-

330

SUBJECT INDEX

Cracks, 179. 180, 186, 204 growth, 57 porous medium, 21 closure, 57 Cross-sectioa (scatterilll), 115

Damping (re4uced), 136

Dare:,·

Ia~·. 31, 69. 281 unit, 31 Decay (lopritluaic), 135, 161 Delayed behamr (coastlbltlve law), 54, 56, 100 Desorbtioa (capillary), 24 Diatomite, 231 Diffusion, 81 Diffushity (hydraulic), 81, 227 Dilatation (vollllllie), 52, 55 Diopsiclite, 216 Dileoatiauides aad wafti;24S..l49 Dislocation, 219, 221 Dispersion, 128 inverse, Ill normal, Ill Rayleigh waves, 265 Stonely waves, 267 Dissipation, 269 Diveraeace(spreadllll), 147, 153, t'5s, t60, 303 Dolomite, 16, 17, 32,230,234,235,240 Webatuck, 177, 179 Ih1Uaa,e,23, 29,32 Dry rock (defulitioa), 187 Duaite, 179, 216

Echo method, 151, 172, 173 ~. 132,273 llgeafrequency, 131, 136 Elastic constaats, 62 Elasticity (linear), 54 Eblltodynamfcs, 49 Eaeray average, 122, 138 balance, 287 elastic, 106, 121, 138 kinetic, 70 maximum, 122, 139

EVA tool, 159, 240, 310-312

Exteasional IROde, 162 Fadel variation, 304 Faust's fennula, 237 Feldspar, 17 Filtratioa velocity, 246 Flexural..-, 162 Flow (1Rultiphase), 38 Foataiaellleu (see s..lstoae) For~Ratiolt faetor, 71 Fort Ualoa (see S.Ddstoae)

~

-,,

Fourier.......,_, 104 Foxbills (see s..lstoae) Fraehuila ' (hydraulic), 310 (natural), 310 Free 'ribratloas, 134, 161

,,

~

F~

band, 146, 301 central, 228 ch~stic,

75,80 (eigen), 130, 131, 136 resonance, 162 sweep, 136, 162 thermal relaxation, 221 Frldloa. 216

c .........

fOI'IIIIIIa, 68, 82, 84, 190, 265 Gcentma .... Smlt ........ 84, 190 Geopltysia (petrale-), 301 Gneiss, 186, 240 Granite, 212, 216, 239, 240 Barre, 184,195,210 Casco, 179 Oklahoma, 222 Sierra White, 182, 188, 199, 200, 206. 242 Stone, 179 Troy, 177, 179 Westerly, 177, 179, 184, 193, 197, 211 Grain contact,9 size, 19,44

~-

-~~

~

---, ' ~

Halite, 235, 240 Hamilton's prilldple, 70, 111, 246

~

' ~

~

'"'""

-~~ ~. 16, 17,2?;28,32, 37

Haakel . . . . , _ , 270

attenuation, 243 Bedford attenuation, 203, 210 ve~ty, 171, 179, 195. 196 chalk, 27, 28, 39 attenuation, 207, 241 velocity, 178, 179,240 Indiana, 207 Oak Hall, 179 porosity/permeability relation., 34 relative permeability, 39 Solenhofen, 177, 179 Speraen, 222 ~ty, 179, 230, 234, 235, 231-ol40 ~ 146,247,255,2&1

Hannoaic:, 137, 167 lleteropaelty, 167 -~

''-

~

'--

'---

~ (minimum volume), 40, 4i, 42 Biot's theory, 63 Hooke's law, S4 HydrauUc fraetwiDI, 310 Hydrogea bollll, 190, 209 Hydroxyla, 190, 219, 225 ,:

- · ...., . , 18, 19 llllbiWdoa, 23, 29, 32

lncelapr! Dllitr ... '--'

I

I

S6

M....,. (see Sa~Mtstwe)

Impulse ........ 125 ladiua (SH I I !ROlle) bljecdoD (epoxy), 21, 29 laerdal coupling. 71, 83, 225 effects, 81 lntercryltallia (ponul ......), 18

Madlemadc:al ~. 18 MaxweiiiBCMiel, 119, 120 Mlcroeraeb, 9, 21, 179, 204, 301

MoWIIty, 72 MCNiel

Interface eff~(attenuation),29S

open or sealed, 247, 268-270

laterfereace, 108, 109, 133

IJiter&ruulM' (ponul . . . . .),

13, 16, f8, 35

~(paroal---), 16.'t« lrrotadoltal ...,_ellt's. 59

'-

331

Jn's.......,26

~

Kelrut-Veip ...... 119,"120 Kera RiYer (see SaillllsteM) Kr~~~~en-Kroal& .........., 122. 142

'

Constant Q, 124-128, ISO, 200, 287 Kelvin-Voigt, 119, 120 Maxwell, 119, 120 Nearly Constant Q, 123 standard, 117, 123 20,288 Zener, 117, 123

~

complex, 104, 112, 142. 285 delayed, 118, 143 effective, $1 instantaneous, 118, 143 reflection and transmission c:oefticieat, 263 relaxed, lOS static, 58 MoNk (porous me6Jiia), 16, 18 M .. cake, 247, 248 Myloaite, 186

(

---------~

-~

l..qrutiu, 70, Ill

Lame's ea•t•..., 54, 62, 61, 104 Laplace's ,......., 22

Lateral extealioll eofta:tiDM. 166

Na•ajo (SH Saadstone) N.--Stokes equadoa, 72 Neller (deflaldoa), 112 Nolle, 147, ISS, 161 NGIIIinearity, 168

-,_

_-,J

332

SUIUECt INDEX

Oak Hall (.rn I..ilaelltoae) Offset VSP, 308 Oalager's priadple, 69 Orthotropy' 97 Ottawa (see Sand)

Pressure confming, 176, 181 differential, 176 effective, 176, 181 hydrostatic, 56, 67 pore, 176, 181 Pyroxeaite, 216

"'·

'

\

Packing (spheres), 13, 14

.....Utude.

129, 130 Pendulum, 163-169, 172, 173 Peak

Q (see Quality faeter) Quadratic form, 66 Quartzite, 179, 212, 216 QaaUtyfaetor(defmitioa), 112, 117, 125, 140, 170, 288 attenuation, 113 kinetic energy, 113 2 0 equation. 286 in situ measurement. 241 Q•• 139 QK,, 139 Q_,, 298 Q,., 139 Q•• 265 Qs. 139,289 QSr, 267 Que• 298

Percolation theory, 15, 36 PenaeaWIIty absolute, 31, 7l bulk, 31 global, 31 hydraulic, 69, 72 interface wave, 247, 275 . ! matrix, 31 measurement. 33 monophasic, 31 porosity, 34-36 relative, 38, 39 Permeameter, 33 Phase chana~ (flllicl), 207, .211 Phase shift, 134, 261 stress-strain, 113, 170,.173

Pierre (see Shale) Polarity cbaage, 307

Racliolarite, 231 RQP,.20, 40

Polseuille's law, 33

Raylftala

Poissoa's ratio,. 54, SS, 62, 139, 181, 231, 239, 304. 306

..

Radius ac:ces·(poroas Blediam), 26, 29, 87 Reflectioa coefficient, 152, 250, 255-263, 29().:296 modulus, 263, 295 phase, 263, 295 interface, 249-258 multiple, 147, 303 Relaxation, 101, 11&, 142 time, 118 thermal, 221, 224 viscous, 226 Resouace peak, 162, 166

Poteatial strain, 56, 66 dissipation, 69 surface dissipation, 246

;

-t•

Resoaant bar, 137, 163. 169, Restam'ed .._. mediad. 24

I

,

approximation, I t6 (pseudo), 311 wave, 263

Pore cast, 12, 17, 21, 37, 45, 187

Porosimetry (mercury), 26 '· Porosity, 10 clay, II connected, II fiSSure, 21 measurement, 10 occluded, 11, 36 total, II, 46 trapped, II, 25

~,

~

P-.elllllddty, lS4 Po~BMtrics~,27,28,44

~.

··'

---.....-..,

17~ 173

Rheological lllOdel, 117

Rise time, 1~6, 148, 155-160

"' ~

~.

\.____

'-

'--

I

,

I

"-

!

'--

I

'-

I

'--

I

'-

I

~

'--~

I

'-

I

'~

I

\.____

I

'--

'-'-....-

'--'

I

I I

~

'- ,,__,

I.

1..~

I

'------

I

'~

I

3l3

Saiat Peter (.sw SaMIIeae)

~11,89,91

Salt. 235, l40

Skill, effect, 86 depth, &7

s.-......... fonladoal.238,240, 241,304 '-._,

;;

~IMJD;I

s..l. s..t.to.e, 234, 235,238.241, 243 ~.254,273,279

attenuation, 203, 209, 211, 212. 216.222 velocity, 181, 182, 194, 197, 200, 201 Boise, 194, 196, 207, 210 Coconino, 207 Fontainebleau, 43-47, 185 attenuation, 159, 204, 206, 208, 243 velocity, ISO, 186, 188, 232 Fort Union, 199 Foxhills, 2S4, 279 Kern River, 196 Massillon attenuation, 169, lOS, 212, 214, 216, :us, 219, 222, 22S, 242 velocity,l77,179,182,187, 18&,199,200, 201 Navajo, 205, 222 Ottawa, 216, 231 porosity/permeability relation., )4, 44 relative permeability, 39 Saint Peter, 182, 197, 211 shaly, 27, 28, 37 Teapot,2S4,273,279 Venezuela, 192, 196, 211 Vosges, 166-168 Wingate, 207, 213 Satwation

irreductib1e, 24, 39 methods, 207, 208 residual, 25, 39 Seale, IC8hl ......... 40, 94, 302 Scatteriac. 91, 115, 146, 147, I», 197, 211, 214,303 cross-section, II S Seia1ic p~upediag borehole, 308 full waveform ~. 3l0 , low frequency, 302 '' ·, reflection, 302 reservoir, 302, 308 statigraphic, 304 three-component, 306 well-to-well, 310 Sbale Colorado, 203 Pierre, 178, 180, 182, 186, 241 Shape factor, 116, 180 Sllear ..... 162 Sierra White (see Grufte)

Soleala.r. (see U.llhlle) Source (.a-lc:). 269 SpecifiC area, 22t gravity (rocks). 240 Spectral ratio, 1S3-l 56 s,erp. (.r« Umestoae)

s.at--.227 Sf(alllt flow, 227 Staq4*,128 ~ ............ 110

Stall rroat, 309 Stelle (see Graaite)

Stoaeley mode, 274 wave, 266, 311 Stylolidle, 16

StniD principal directions, S l small, Sl, S3 teasor, 49 ~eyde,10S

s......

compressional, SS effective, 81

'I

iI

shear," telliOr, S2 traction,SS Sweep (fnlpleaey). 136, 162

SJDdledc rrf

i

J ~\

......., 276; 282

I

Syttaa open, 75,79 closed, 75, 79

Teapot (see Saadstoae) Terupi'alaw, 81-84, 93, 176 Textwe ...aysls, 18, 19

l'henlapa••en\....,., 12 ~221,224

nta .etioll (rock). 12, 14, 32 'J'IIree.colllpoMDt reccrii..,

306

• >.-/ (

'~

_:J;I

Tor.Honal~, 166 Tortuosity, 35, 71, 80, 81, 83, 93, 191, 252; 254 Traasmission coefficient, 250, 255-263, 290-296 modulus, 263, phase, 263, 295 method, 153, 172, 173 Tripolite, 231 Troy (see Granite) Tube wave, 275

m

o t \.\,~as- C\' ~). Unsteady-state, 128 \..c:..\ Y"CE.J<.-4 Ch~"do "C

Vector

attenuation, 260, 284 propagation, 284 Venezuela (see Saad) Velocity average value rocks, 230, 240 clay content, 179, 233, 2l_6 compaction, 237-239 complex, 286 dispersion, 197 energy, Ill extensional, 128, 141, 166 frequency, 197 group, 108-112 in situ measurement, 23o;.240 interval, 146 measurement, 148 phase, 108,149,260,286 porosity, 23() pressure, 177-186 Rayleigh, 265 saturating fluid phase, 196 viscosity, 192-196 saturation, 187-192 specific gravity, 233-235 Stoneley, 267 strain, 198 temperature, 193 uniaxial stress, 182 Vibrations, 128 . forced, 134, 161 free, 134, 161 · _,quality factor, 135

Viscoelasticln· (liaear), 107-ll2 interface ~aves, 297 reflection and transmission, 283-297 v-...coupllaa,87 ~. 194. 204, 210 V~tlon

capillary properties. 29, 30 porous medium, 12 VSP (vertical seismic profdiag), 240, 308 Vugy (Jiorous IDellium), 15, 16 Vycor, 191, 206. 213, 219, 222, 228

'

.,

'

~·'

\

·----..

Water content, 66, 252 Waftl dilatational. 59 equation, 61 extensional. 63, 128, 132 homogeneous, 284 inhomogeneous, 258, 284 interface. 263-269 longitudinal. 60, 63 primary. 58. 73 propagative. 128 Rayleigh, 263 refracted, 274 secondary, 60 shear, 70, 74 slow (second kind), 73-82, 83, 88., 252 standing, 161-169 Stoneley, 266, 311 transverse, 60 tube, 275 wave guidt~ 110 WeiMtuel (SH Debaite) Well loallll (acoustic), 146, 247, 255, 281 Westerly (see Graaite) Wettallility, 23 Wettillc/IIOil •"ddlli .... 22 Wi~~&&te (see Saadstone) WOCMI's metal, 29 WyDie's formula, 230-233

.,

~-,

.,. .,

-,

·,

...__

-.

' -.

---...

·-..

' ----.,

~,

YoiiRI'S modulus, 54, 57, 170

'·

Zener model, 117' 123

......

"

' \

~,

J