Plane waves in porous or cracked media with internal friction

Theoretical and Applied Fracture Mechanics 29 (1998) 161±167

Plane waves in porous or cracked media with internal friction B.P. Sibiriakov

*

Institute of Geophysics, Siberian Branch of Russian Academy of Sciences, 3 Universitetsky prospect, 630090 Novosibirsk, Russian Federation

Abstract Asymmetry of compression and shear in porous media is considered. Since the shear causes in®nite stresses across the contacts, the models of partial creep in shear are used to exclude the in®nite stresses. The inelastic behavior in the shear nonlinear equations of motion corresponding to the Korteweg±de Vries (KdV) equations in one-dimension lead to purely imaginary nonlinear term. This causes absorption and dispersion of P and especially S waves. Ó 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction Frequency di€erence of P and S wave is a wellknown fact in seismology and multi-wave seismic prospecting. A great deal of work has been done to explain this phenomenon [1]. There are two independent quantities in the elasticity theory, namely, density and stress or density and velocity. A speci®c time (and frequency) may occur when geometric dimensions are present such as ®nite source dimension, or mean thickness of some layered structure. Finite source dimensions actually form the lower frequency of S waves rather than the P-waves. The physical reason for this is that it is impossible to create a spherically symmetric ®eld of shear waves. Flow of the shear deformations from the higher stress areas to the low stress areas may not be suciently fast though faster than the shear wave velocity. Layered structures cause dispersion of both P and S waves which can degenerate in limited cases. *

Fax: +7 3832 35 01 32; e-mail: [email protected].

0167-8442/98/$19.00 Ó 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 9 8 ) 0 0 0 2 7 - 5

If the structures are periodic and the period is small relative to the wave length, the dispersion degenerates into hexagonal anisotropy. For relatively thick layers, all waves are divided into physically interpretable sums in accordance with the ray method. Shallow layers with period thickness of 0.1±0.2 wave length produce shear waves with lower frequency than the analogous compression waves. The aforementioned phenomenon is not completely understood. First the frequencies of both P and S waves are one or two orders lower than those generated from the real source geometry. For example, steel tubes of 20 cm in diameter dug into the ground have been used. The tube was ®lled with porous absorber. It provided asymmetric e€ect of the charge of one detonator or several inside the tube. Good quality seismograms and phase inversion of SH waves were achieved by use of steel cover so as to exclude inelastic impact. Similar methods of exclusion of strong impacts were used in cases of striking a blow into the end-wall of a massive horizontal stand (railway sleeper). The observations were carried out as a

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B.P. Sibiriakov / Theoretical and Applied Fracture Mechanics 29 (1998) 161±167

rule on head waves of SSS and PPP types. The ratio of S and P waves frequencies was found to be approximately equal to the ratio of their velocities [2]. The frequencies of both P and S waves were ten times lower than those of the wave velocity to the source dimension. It was clear that both wave frequencies evolved but the rates di€ered. In any case, the approximately equal lengths of P and S waves were found in the various frequency ranges. Study of wave absorption showed that in the ultrasonic frequency range, the absorption amplitude depended only on deformations of 10ÿ6 ± 10ÿ7 order of magnitude. For the lower wave amplitude values the measurements also showed the enrichment of spectra with low frequencies that was interpreted as a cause of absorption [3]. The frequency behavior of the compression shear and converted waves were considered to be caused mainly by inelastic action and the microstructure of real rocks. The microstructure may also give rise to highly localized stress concentration points invalidating the Hooke's law although elastic behavior may be assumed on the average. Considered in this work are the ®niteness of microstructure dimensions and the occurrence of stress and deformation concentrating points and their in¯uence on the process of wave propagation in the porous and cracked media. 2. Equation of motion 2.1. Porous medium Porous and cracked media may be described by the porosity f and a speci®c surface r0 with the inverse length dimension. The typical mean microstructure dimension `0 is related to r0 and f by the relation r0 `0 ˆ 4…1 ÿ f †

…1†

that is known in integral geometry [4]. The discrete nature of the structure re¯ected by the ®nite dimension `0 results in nonequivalence of the di€erence and di€erential operators. Equivalence takes place only in case of `0 ! 0. Derivatives of the order higher than the second thus appear in the differential equations.

The translation operator for the function u…x  h† is known to be p…u…x†† ˆ u…x† exp…hDx †;

…2†

where Dx ˆ @=@x. The di€erence operator is determined as ÿ
ÿ
u x ‡ h2 ÿ u x ÿ h2 D1 ˆ  h     1 h ÿh ˆ u…x† : …3† exp Dx ÿ exp Dx h 2 2 The second di€erence corresponding to @ 2 =@x2 takes the form     2 1 h ÿh : …4† Dx ÿ exp Dx u…x† 2 exp h 2 2 2.2. Nonlocal e€ects Classical continuum mechanics assumes that the cause and e€ect are localized. This, however, may not apply to micro-nonhomogeneous media, where nonlocal e€ects may lead to di€erences of many orders. For porous media containing liquids and gases, the stress ®eld varies strongly on the surface of each grain from the point of contact between the grains to the point of contact with liquid or gas. Hence, continuous mechanics formulation would require the concept of real ®eld averaging and relating the ®eld to any structure point, e.g. the grain center of gravity or the center of a cube with the dimensions of `0 determined by speci®c surface of r0 (for cracked media). One of the simplest versions of averaging is the translation p…u…x†† ˆ n…x† exp…hDx †

…5†

operator for the u…x ‡ h† function. The arithmetic mean of the ®eld is P ‰u…x†Š ˆ 16 u…x†‰ exp…hDx † ‡ exp…ÿhDx † ‡ exp …hDy † ‡ exp …ÿhDy † ‡ exp …hDz † ‡ exp …ÿhDz †Š;

…6†

where P is the averaging operator. The operator may take a probabilistic meaning as X p…1; x; y†u…y†: …7† P ‰u…x†Š ˆ Mx u…x; 1† ˆ

B.P. Sibiriakov / Theoretical and Applied Fracture Mechanics 29 (1998) 161±167

Note that all sums of u…x† are positive and add up to 1. The operation p(1, x, y) is the probability of transfer from the neighboring point y to point x in one step if 1 ; …8† 2` where ` is space dimension and k runs from 1; 2; . . .  `; hÿk ˆ ÿhk . In Eq. (7), Mx is mathematical expectation. This averaging concept justi®es the derivatives approximation by central di€erences. The linear operator A ˆ P ÿ E (where E is identify operator) has long been noticed to be a discrete analog of the Laplace operator h2 =2D [5]. For the elasticity theory the highest derivatives of the fourth order with small factor h2 appear in the form of   h2 @ l…D Dli † ‡ …k ‡ l† D div l …9† 2 @xi

p…1; x; y ‡ hk † ˆ

and in one-dimension case, it is just a summand …k ‡ 2l†h2 =2uxxxx that is familiar in one-dimensional chain theory [6]. 2.3. Dissipation energy It was found in contact mechanics [7] that for elastic particles with ®nite area of contacts the forces normal to the contact surface do not cause in®nite stress at any contact point. However, the translations in the direction of the contact plane create in®nite stress near the edge of the contact surface no matter how small is the disturbance. To exclude these in®nite stresses, a model of partial creeping of particles was created in contact mechanics [7]. A similar situation takes place at the crack tips. For discrete solids, an asymmetry appears in the behavior of normal and shear of microstructure which is not present in continuous media. A known solution of the shear problem may be used to obtain an expression for the dissipation energy, say DW . For the shear of two grains, it has been found that [7] DW ˆ

1 1 2ÿm 3 s: 18ap P0 l

…10†

163

In Eq. (10), DW is the energy lost for a cycle of loading and unloading; a the contact radius; p the friction coecient; m the Poisson's ratio; l the grain material shear modulus; s the tangent stress; and P0 the normal contract stress. Thus, the hysteresis loop may be approximated by an ellipse with the major axis proportional to s (or deformation c) and the minor axis to s2 (or c2 ). The equation for an ellipse in complex form is given by s ˆ x ‡ iy ˆ c cos u  ic2 sin u;

…11†

where ÿ1 6 cos u 6 1. The tendency s2 ) 0 determines the single-valued relation between stresses and deformation; this corresponds to zero hysteresis loop area and dissipative energy. The generalization of Eq. (10) for the case of arbitrarily placed contacts and arbitrary direction of action forces may bepobtained by using a tangential  stress of intensity Dr2 ; where Dr2 is the second (square) invariant of deviator of stress tensor. In this case, Dr2 ˆ …rik nk †2 ÿ …rik ni nk †2 ;

…12†

where the ®rst term is the square of the absolute value of the load vector P12 ; and the second term is the square of the load acting over the contact normal, i.e., 2

Pn2 ˆ …rik ni hk † :

…13†

Dissipative force is the Euler variational derivative of the dissipation energy with respect to the generalized coordinate rik and then with respect to the coordinate xk in accordance with the least action principle, i.e., f1 ˆ i

@ @DW : @xk @rik

The Hook's law is thus modi®ed as ÿ p
rik ˆ kHdik ‡ 2leik 1 ‡ is De2 ;

…14†

…15†

where De2 is the second invariant of the deviator deformation tensor. Generalized form of Eq. (15) is based on the fact that it coincides with the known equation [7] for one-dimensional case. Moreover, the voluminal force created by inner stresses @rik =@xk is invariant with respect to the coordinate system rotation. It follows that

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B.P. Sibiriakov / Theoretical and Applied Fracture Mechanics 29 (1998) 161±167

p @ De2 1 @De2 1 @De2 2 eik ˆ p eik ˆ p a eim : @xk 2 De2 @xk 2 De2 @xm km …16†

Note that

To be noted also is that p p X @ De2 @ De2 a2km ˆ 1; eik ˆ eim : @xk @xm k

where e0 is the amplitude value of deformation. The + and ± correspond to the loading and unloading processes. In nondimensional coordinates,

…17†

Using the least action principle, a purely imaginary dissipative force is obtained. p @rik ; …18† fi ˆ isl De2 @xk where 2 ÿ m r0 l s ˆ 2p g…1 ÿ g†n; 3 a pP0

r0 r0 : gˆ 3

ss 1 1 3=2 lD r0 z ; pP0 e2 ce …2c2 †2

x ; k1

c20 ˆ

k ‡ 2l l or ; q q

c0 ` ` ˆ ; k1

…23†

…24†

where k1 is the mean wave length. The equation of motion becomes utt ˆ uxx ‡ 2buxxxx  2ieux
uxx ;

…25†

`20 =ik21 .

…19†

…20†

where M is the constant depending on the creep line con®guration; z the mean length of a crack rectilinear segment; ce the elasticity limit of deformation; r the ratio VS =VP , and ss ˆ lcc . In this case, the factor 1=ce accounts for the nonlinear term. An in®nite friction coecient p ! 1 results in the absence of creep and elastic deformation of structure. The equations of motion become    h2 @ div u l Du1 ‡ D Dui ‡ …k ‡ l† 2 @xi  h2 @eik p De2 ‡ D div u ‡ 2lis 2 @xk 1 @De2 ˆ q ui : …21† ‡ 2lis p 2 De2 @xk For the one-dimensional case 1 utt ˆ uxx ‡ `20 uxxxx  ieux
uxx : c20

x ˆ

1 ÿ m=2 lnc0 ; a pP0

where b ˆ

Here, r0 is the speci®c surface and n the mean number of contacts. The dissipation force containing terms quadratic in the deformations cannot be ignored because of the factors l=pP0 and r0 =a. For the cracked media, an analogous problem was solved in Ref. [8] for the rigid-plastic case. The dissipative function was found in the form DW ˆ iM

e ˆ e0

…22†

3. Plane waves In discrete medium P and S wave are taken on the average such that the mean intensity p of the tangential stress of the S wave is 3=2…1=c† times higher than the compressional wave, i.e., r 2 …26† ceS : eP ˆ 3 The shear wave velocity is 1=c times lower than that of the compressional wave. bS ˆ

1 b : c2 P

…27†

The fourth derivative is the result of discretization of the medium. For sedimentary rocks, e and b are of the order of 10ÿ2 ±10ÿ4 . With accuracy of e and b squared, Eq. (9) may take the form @m @m @3m  iem ÿ b 3 ˆ 0; @g @n @n

…28†

where m ˆ ux and g ˆ t ÿ x;

nˆtÿx

…29†

are the characteristic arguments. Such a reduction has been carried out before [6], and it is related to the fact that di€erentiation with respect to g coordinate leads to small values of e or b. An ideal wave process would not depend on the g coordinate or the values of the second order and may

B.P. Sibiriakov / Theoretical and Applied Fracture Mechanics 29 (1998) 161±167

be neglected. Eq. (28) is a Korteweg±de Vries (KdV) equation with a purely imaginary nonlinear term. In the case of b ˆ 0; it has an exact solution m ˆ F …n  iegm†:

…30†

In general, it has an approximate solution with the accuracy of e and b squared: m…n; g† ˆ F ‰n  iegm ÿ k 2 …n†gŠ ˆ F …f†;

F 000 …0; t† ‡ k 2 …t†F 0 …0; t† ÿ 3F 00 …0; t†

@2f …0; t† ˆ 0: @n2 …32†

In Eq. (32), the factor of the second derivative is of the ®rst order of triviality relative to e and b. Hence, for the zero approximation, Eq. (32) may be simpli®ed to F 000 …0; t† ‡ k 2 …t†F 0 …0; t† ˆ 0:

…33†

It is applied to locations where the ®rst derivative F 0 …0; t† does not vanish. It is not dicult to see that the arbitrary function of the arguments f ˆ n  ieg…m ‡ C† and f ˆ n  ieg…m ÿ C†

correct for comparatively short distances x from the source (up to ®ve wave lengths). The result is  ÿ
2 F …f† ˆ ÿIm n ÿ 2iexV …n† ÿ 2bk 2 x  exp ‰ ÿ ix1 …n ÿ 2iexV …n† ÿ 2bk 2 x†Š :  exp ‰a…n ÿ 2iexV …n† ÿ 2bk 2 x†Š …37†

…31†

where F is the arbitrary function while k 2 …n† satis®es the equation

…34†

also satisfy Eq. (28). Here, C is an arbitrary constant. A nondecreasing function V …m† can be constructed for m to increase monotonously. This leads to a solution that decreases due to friction for both on loading and unloading. An approximate solution in the form of an arbitrary function of the f argument allows the evolution of a plane P or S wave over an interval. At the boundary x ˆ 0; there results   t ; …35† F …t† ˆ P0 1 ‡ ieV …t† ÿ bk 2 …n†

165

4. Discussion of results and conclusions The solution in Eq. (36) describes the evolution with the Berlage impulse distance which is set by the equation P …0; t† ˆ t2 eÿat sin …x1 t† with x ˆ 0. Unlike the classical KdV equation, the imaginary nonlinear term causes absorption of the wave only but not the increase of the front steeples of the waves. The dispersion term expands the impulse as in the classical KdV equation. The nonlinear term causes both p absorption and expansion of  the impulse; it is 3=2…1=c† times higher for S waves. The dispersion term leaves the absorption una€ected and casts the low frequencies in the forefront leaving the higher frequencies behind. That may mistakenly be considered as absorption. Large values of both dispersion and absorption for shear waves are observed such that S wave frequencies become lower than that of P waves. Both waves should have the same lengths in the limit [2].

where P0 ; …0; t† is the set impulse at the boundary x ˆ 0. For an arbitrary argument f; an implicit solution is obtained:   f : …36† F …f† ˆ P0 1 ÿ ieV …f† ÿ bk 2 …n† An explicit form of Eq. (36) may be found by making the approximation V …f†  V …n† and the fact that k 2 …n† is close to x2t ; where xt is the visible impulse frequency. This approximate solution is

Fig. 1. Spectra of P and S waves at di€erent distances from the source.

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B.P. Sibiriakov / Theoretical and Applied Fracture Mechanics 29 (1998) 161±167

Fig. 2. Direct P and S waves registered by one vertical and four horizontal devices.

In Fig. 1, the impulse changes of plane P and S waves are shown for di€erent distances from the source. Curve A corresponds to the same P and S wave spectrum while Curve B and C correspond to P wave and S wave spectrum, respectively. As an example, the Berlage impulse was set with the index n ˆ 2. It is evident that S waves have higher absorption and expansion than P waves. Eq. (28) shows that in the di€erent frequency ranges, there are actually di€erent equations of motion, since for long and short waves the in¯uence of dispersion and absorption di€ers signi®cantly. The observation of direct waves using di€erent orientation of borehole devices shows a clear dif-

ference between P and S waves frequencies in sand and clay sediments in West Kazakhstan. In Fig. 2, the record of direct waves excited by a small amount of detonator near the borehole ori®ce is shown. The container of horizontal receivers orientated in di€erent azimuths holds one vertical device registering compression and then shear waves. The depth of the device immersing was 14 m. The horizontal devices hardly register compression waves. It is clear from Fig. 2 that visible frequencies of S waves are half as high as the ones of P waves that were emitted with the same source. The ratio VS =VP is about 0.5. In Fig. 3, a three-component record is shown for PPP and SSS head waves in the same region excited by BB load in a borehole ®lled with water. On the vertical Z-component, a PPP wave is clearly seen, and on the horizontal X -component ± an SSS wave. The ratio of visible frequencies is approximately equal to the velocities ratio of P and S waves.

References

Fig. 3. Head PPP and SSS waves excited by explosion in a borehole.

[1] B.P. Sibiriakov, Excitation of Spheric SH Waves in Elasticoplastic Media Shear and Converted Waves in Seismic Prospecting (in Russian), Nedra, Moscow, 1985, p. 111. [2] N.N. Puzyrev, A.V. Trigubov, L. Yu Brodov et al., Seismic Prospecting Using the Shear Converted Waves Technique (in Russian), Nedra, Moscow, 1985, pp. 133±207.

B.P. Sibiriakov / Theoretical and Applied Fracture Mechanics 29 (1998) 161±167 [3] J.E. White, Underground Sound Application Of Seismic Waves, Elsevier, Amsterdam, 1983, p. 136. [4] F.A. Usmanov, Foundations of Mathematical Analysis of Geological Structures (in Russian), FAS UzSSR, Tashkent, 1977, pp. 120±124. [5] B.B. Dynkin, A.A. Yushkevich, Theorems and Problems on Markov's Processes (in Russian), Nauka, Moscow, 1967, p. 232.

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[6] A.I. Kunin, Theory of Microstructured Elastic Media (in Russian), Nauka, Moscow, 1975, p. 510. [7] K.I. Johnson, Contact Mechanics, Cambridge Univ. Press, Cambridge, 1985, pp. 250±260. [8] B.P. Sibiriakov, Microplasticity of grain media and its e€ect on seismic waves, in: Russian Geology and Geophysics, vol. 34, Alerton Press, New York, 1993, pp. 88±98.