dividual workings at considerable distances from the edge sections of the co,non worked-out space. LITERATURE CITED i. 2.

G . I . Barenblatt and S. A. Khristianovich, "Roof cavin 8 during mining," Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, No. ii (1955). S . A . Khristianovich and S. V. Kuznetsov, "Stressed state of a rock mass in carrying out winning operations," in: Mine Pressure [in Russian], No. 59, VNIMI, Leningrad

(1965). 3.

4. 5.

S . V . Kuznetsov and N. S. Khapilova, "Determination of the instant of roof caving," Fiz.-Takh. Probl. Razrab. Polezn. Iskop., No. 5 (1969). N . I . Muskhelishvili, Some Basic Problems of Mathem-tical Elasticity Theory [in Russian], Moscow (1966). G . P . Cherepanov, Mechanics of Brittle ~ailure [in Russian], Nauka, Moscow (1974).

NONLINEAR WAVES IN SOILS AND FISSURED ROCK V. N. Nikolaevskii

i. Dynamic force processes that result in the failure of earth materials are normally recognized to be nonlinear waves. Advances in the field of nonlinear-wave experiment and theory make it possible to perform computations with confidence at the present time and to predict the dampin 8 of strong shock waves near the points of their excitation, as well as the effect of these waves on ore masses and structures [1-3]. Principal to this is that failure sets in at the moment that the static ultimate strength of the medium is reached, while movement of the failed mass occurs under conditions of limiting Coulomb flow with attendent dilatancy-shear volume changes. The procedure of viscous damping, which, moreover, facilitates mathematical analysis of the dynamics of the medium in the translimiting state, has been introduced additionally for specific calculations. The following are the most significant among the recent achievements in mine dynamics. Nikolaevskii [3] has shown convincingly that not only the dynamic strength, but also the critical stress-concentration factor for the tip of a crack are a functional problem in dynamics [4], and not a material characteristic of the medium. It is observed that transverse waves are split in earthquake-prone zones b y virtue of dilatation anisotropy, depending on their polarization. These waves, the displac--~nts in which are orthogonal to coplanar cracks, correspond to the lower stiffness and arrive at the observation point later [5]. Note that familiar effects of strain localization [2] also appear in the dynamics [6], and the developing slip surfaces possess strong-rupture properties [7]. In addition, mechanical processes that determine variations in seismic waves at remote distances have not been well understood until now. It is reliably established that the arrivals of waves at observation stations correspond to calculations performed in accordance with linear dynamic elasticity; however, the law governing their orderliness (the proportionality of first-degree damping of the vibration frequency [8]) is outside the framework of linear viscoelastic models. -Experimental investigation, for example, of interaction between waves and the vibrator has led to the conclusion concerning the existence of nonlinear dynamic processes in a loose sedimentary stratum [9]. Vil'chinskii's experiments [9] expose the dominant frequencies (-25 Hz) of beach sands, which correspond not to the stratification of the rock, but to its microstructure. This has been confirmed by the correspondence between the plastic-emission spectra of sand (in the stage of sublimiting plasticity) and weak stress waves that had passed through the mass. It was pointed out that the dominant frequency is 40 Hz for clays, i00 Hz for weathered granites, and i0 Hz for O. Yu. Shmidt Institute of Earth Physics, Moscow. Translated from Fiziko-Tekhnicheskie Problemy Raz~abotki Polezykh Iskopaemykh, No. 6, pp. 31-38, November-December, 1988. Original article submitted May 23, 1988. 0038-5581/88/2406-0513512.50

9

1989 Plenum Publishing Corporation

513

if

G

Et p/

6

Fig. i

Fig. 2

dense fissured soils. The possibility of converting wave energy to lower frequencies also assumes major significance for the problem of identifying blasts of moderate intensity [101.

The International Symposium on Nonlinear Seismology ~ was convened in Suzdal' in 1987, where experimental investigations and heuristic search for equations of the evolution of slightly nonlinear waves were discussed. Evolution wave equations motivated by mmchanistic notions assume special significance. Since nonlinearity is assumed to be minor, the mathematical model constructed should correspond, in a first approximation, to linear elastic dynamics, while the evolution equation should represent the possibility of an increase (instability) in oscillations from the narrow interval of the dominant frequencies. In connection with the problem of improving oil recovery from strata, moreover, it is of interest to develop notions concerning transfer in a porous medium subjected to vibration. We can state that "sonic-wind" mechanisms and internal convection durin E ultrasonic activity from the mouth of the well onto the stratum have been discussed earlier [11]. The causes of the development of dominant frequencies of a rock and the possible mechanism of the vibration effect on a watered-flooded oil reservoir will be addressed below. 2. The equations of the uniform dynamics of fragmented rock include mass and impulse balances

~.f

+ . ~op, - -- 0,

.~tv+ -'~" ap,~,= "~', Oa

where p is the density, v is the displacement rate, and o is the stress. supplemented by determination of the strain e through field v De

Oe

a~

0~

b7---6~ + ~ - ~ -- ~ -

(2.1)

They should be

(2.2)

Fragmented rock is represented as a set of oscillating blocks (grains), with contacts being modeled by elastic and viscous .elements (Fig. i). Since contacts "through a liquid" end direct contacts are possible, two types of each of the elements should be introduced, and the "equilibrium" and "frozen" scales of the oscillating blocks should also differ. The phenomenological relation used is presented in Fig. 2. Determining relationships should be compiled separately for spherical and deviator components of the stress and strain tensors for nonuniform movements. For uniform waves, they reduce to the relationships

for the following values of the coefficients:

Mt

b, -- 0~--z , ao = Ezz, al =" (Ex + Ezz) O + (Etz + E,) 0,,

(2.4)

at--Eli ( ~m'+ ~~ n+ e o , ) +(EI+E,)0O,, ~Major papers presented at the symposium are published in the journal "Physics of the Earth and Planetary Interiors," Vol. 50, No. 1 (1988). 514

a3 - , (M] + Mn) O + { M~ + 3It~ e ~ , z+~ E. ]) 0.. .,. MIMII a4 ~ + OO,MI, fl,~ == 3.

MIMII

~I

/at O ~=~-'i.

0.,

(2.4)

~,

O. :=, E'-'?"

Let us introduce the running coordinates

=- 'q~Cz - ct), with variation of the length and t i m e form

t

(Il

0,

c) _

+ ~ . p (u - - c) Il -- a--T'

[~ r ae

Tn-

{

a

(3.1)

q << i, Eqs. (2.1)-(2.3) will then assume the

R-+a-~-p --

-~- n~ - ~ 1

scale:

~-aaP

T,l

't -- (1/2) qot

ae

Ov

-~-+ ( u - - : ) ~ = a'7"'

bo + p-, b, q.,

m

[L 2• n"-"

qml

c

"

ol/

+

N

(3.2) ~-

-

We can then substitute in Eq. (3.2) the expansions in terms of the small parameter q for the dynamic variables

P -- P~ + riP' + v12P2+ . . . .

....

v--qvt+Vl~V2+

We can assmne s~multaneously that 8 = 1 + a and ~ >> i. of n), Eqs. (3.2) will then yield

Ol --' --pOCYh

(3.3)

In first approximation (in terms

~l "=--Ill~C,

p, -- Oo(Ii,lc), @ -- gt,/Oo.

(3.4)

In a second approximation, we have

-'~'~ (Poll, - - P,r

t 0p1 - i " ' ~ " + o ~ P'u*'

-

l

av l

(a= + pocIl,) - ---f- Po'~"

E,

' -

a (ce=+Ils)-- .-~-i ~a% + II,.~--a,, F, 0-[ boa: -- aoe: + TlIll ~,.. - - ca~

--

b~

am i l Vl

,if=

7

+

(3.5) *l "--~ +

-- c

-- T.

System of equations (3.5) has a determinant equal to zero, since Eqs. (3.4) are valid. The condition of con~patibillty of system (3.5) leads to the equation

0(

+)

a-[ EnF--cF~ + ca..i[ --0

(3.6)

in terms of the amplitude of the displacement rate in a first approximation v x - v. Let us reduce Eq. (3.6) to the ordinary form of the evolution equation for a wave packet, noting that q = V/c is the Mach number, v - c, [~] = v[T] = c[x], and, finally, av

v

au

I

ov

a--.[ :~l>q -~- -~,~. ,-,-, ,] -~- ~ .

(3.7)

515

In Eq. (3.6), consequently, let us retain the terms containing the smaller exponents of before the oldest derivatives, and neglect the derivatives of the same order, but with smaller coefficients. Then, I

,~v

Z = T p o - ~,

~(~0u

F--

;

0v)

~-+v-~T,

( o/' ~_~

(3.8) 8

~

mp

lh'1

rc,)- i, 4.

z > o;

re,)- o,

: < o.

The resultant equation assumes the form (n - 5, m - 3)

- c

~

rc~_,)bp-- re._,)

v.

(4. i)

p--l If we now eliminate the distortions of the length and time scales, i.e., use the coordinates X = x - ct, T = t/2 and the amplitude of the velocity V ffi Qv, Eq. (3.6) will assume the form n

aV OV a'Y + V ~--~-- p--I y. .~

~+~V OXp$i

(4.2)

with variable-sign coefficients

A~+ l

--

c ( - - C)P r(m-~)bp - - r(._p) p0c2 ,

(4.3)

where all bp end ap values are determine~. Note that when n - 1 and A 2 > 0, Eq. (4.2) converts to Burgers' equation; when n = 2, A s = 0, and A 3 < 0, it converts to Cortevega-de Writs' equation. Variants have been encountered in the theory of waves on the surface of liquid films, in the theory of combustion and chemical concentration waves, and in the theory of waves in a fluid with gas bubbles. Cases of n = 3 when A2, A 3 , end A~ < 0, and even n - 4 when A2, A3, A~, and As < 0 have been proposed to describe coherent structures in turbulent flows, i.e., the concept of negative turbulent viscosity has actually been used. It can be demonstrated that Eq. (4.2), which is sixth-order, may correspond to the dominant frequencies in a narrow interval of the vibration spectrum. Let us find the displacement rate in the form of V - V 0 exp i(,,T - kx), where ~ is the frequency, end k is the wave number, linearizin E (4.2) with respect to the level V,. The dispersion equation

- k v , - - I-4., i e + 1.4,1 ~ + i ~ ( l ~ l - - I A ,

I ~ + I A, I k')

(4.4)

indicates that the quantity Im,-, which defines damping, has a minimum If this minimum number Tun,.,- 0, terms dropped on the basis of est~m-te (3.7) should be retained to calculate

the r e a l ~:~ping.

F i n a l l y , i f t h e c o n d i t i o n }A=! - IA~lk ~ + IAs}k ~ < 0 i s f u l f i l l e d ,

vibra-.

tions with a wave number from the region k: ~ ~ k= ~ k~=, where

(4.5) are ~ s t a b l e , and the corresponding frequencies are termed dominant [12]. The exponential increase in the amplitudes of these vibrations is significantly large when compared to the extremely small displacement rates in ordinary seismic waves. It is restricted by nonlinear effects, including those due to the coupling of oscillating masses. As a matter of fact, we have

A= --

516

Po

'

P-~'

A~

*) - - o f EI+E'Se, + xh , xh-

\

P0

[ x" E.8, + Ex8

.

Mix P0 '

] (4.6)

.

. ~

H

in conformity with determinations (2.4). Let us estimate the dominant frequency w d, which can be used with the conditions E I ~ F~ << EII, ~ >> ~I, xI, ~II" Then,

r 1/']~ ,,.,r

1/'~-..,__r 1I~".

Ar

where dc is the dispersion of the velocity (which is small in the case of geomaterials). The wavelength k and the extent to which the vibration amplitude increases Im~ are evaluated from the equations ---~Nxn

E, -- xn~7",

The condition of the existence of roots ( 4 . 5 ) I ~ I I > 2 1 ~ n l ; that is a s ~ = e d to be f u l f n l e d ,

I m s - - En p x| I

requires t h a t

(4.8)

IA~l = > 4JA=]JA, I, i . e . ,

Let us perform approximate evaluations. In moist beach sands, ~d - 25 Hz, c ~ i00 m/ sec, and ~II " 4 mm [13]. According to (4.7), dominant frequencies are possible, if EII/E~ ~ 106 , and the wavelength k - 4 m. For large-scale seismic vibrations, ~d " 1 Hz, c ~ 6 km/ sec, ~c - i00 m/sec, xii - i00 m, and k - 6 km. Equation (4.8) indicates that an increase in the scale (union) of the oscillating blocks (grains) reduces the increase in the amplitudes of the dominant vibrations. A conglomerate of I0 sand particles [13], each of which was 0.4 mm in size, may therefore have been an elementary oscillator in the beach sand. 5. In the case of the saturation of a porous body with a dual-phase fluid, the flows of the phases are controlled by the generalized Darcy law [2] ~

)

where ~(=) is the viscosity and p(=) is the pressure of the = phase, and k is the permeability of the body. The phase permeabilities f[~) for an "oil + water" system are presented in FiE. 3 (curve 1 - without external compression; 2 -- compression at 35 MPa). It is apparent that the phase permeability fi~ returns to zero when the value of the bulk degree of oil saturation s is lower than the threshold saturation s(~ (-30% in Fi E. 3), end this is caused physically by the extent to which the drops of oil are dispersed in the water mass (Fig. 4). It is precisely the recovery of oil from a watered stratrum, therefore, that has a limit of 70%. There is also a threshold mobility for the water (~40% in FiE. 3), if considerable oil (gas) is contained in the stratum.

7

2

2O $w 0

20

, 60

Fig. 3

.S

Fig. 4

517

Q,. m31day 0~I w a t e r

,,r ot 8 I-5o~ ------2

o

6ZO -t~O 810 "II0

/ 0.00~

800 " f O 0

~ ? O~Q .-

~ooooO0 ~oooooo

n ~0~

,

I L ~

?gO " n 15

I

rl

I

I

I

J

II

I

!

I

Ig Z! ZJ Z$ Z7 Z9 31 Z

,

,

,

~

5

I

8

I

!

t tO

10 tZ Ig76

Fig. 5 The phase permeabilities may become momentarily other than zero, even when s < s(O), if vibrational agitation creates clusters [14], i.e., continuous channels through which the flow of oil is restored (Fig. 4). In his own experiments, even Leverett [15] noted that an increase in drop size will lead to an increase in phase permeabillties. Cluster formation under vibration (for gas bubbles penetrating through a network) is described in [16]. To confirm this viewpoint, let us turn to field data on variations in the extent to which oils were water-flooded after the Chernogorsk earthquake (July 28, 1976). Characteristic curves (Fig. 5) have been taken from Osika [17] : case I - Western Gudermes Yormatlon, a distance of 50 km from the epicenter, well No. 183; case II - the NovoEroznensk Formation, a distance of 80 kin, total separation; case I I I - the Shamkal-Bulak Formation, a distance of 130 kin, well No. 12; and, case IV - the Makhachkalinsk Formation, 170 kin, well No. 220; curves 1 and 2 denote oil and water, respectively. The daily yield and the date of the shocks and measurements are plotted as ordinate and abscissa, respectively; K is the class (energy) of the shock, and the difference per unit corresponds to a tenfold energy increase. It is apparent that in cases I, II, and IV, distinct, although time-restrlcted, increases in the amall fraction of oil (If, IV), or water (I) occur in well production. Totally similar data on water-flooded wells of the Gash oil field after the earthquake in Dagestan (May 14, 1970) have been published by Voitov, Kissin et el. [18], while Osika [17] cites data on an increase in the gas condensate in the yield of water-flooded well No. 222 in the Makhachkalinsk field after the Buinak earthq,,-~e (January 10, 1975). To explain the increase in the large proportion of oil with a constant degree of waterflooding (case III), we must proceed with consideration of another possibility. It is known that the redistribution of tectonic stresses, which lead, as a rule, to a reduction in well production immediately prior to the shook, occurs during earthquakes [18]. After the shock, the total yields may increase, for example, due to additional compression of the seam and an increase in rock pressure. For case III, It is critical that the relative permeability of the oil increases with compression precisely for average degrees of saturation (see Fig. 3), although the mobility threshold does not change, i.e., does not affect final oil recovery. Let us understand that vibrations intensify viscous deformations of strata.

518

According to Kissin [19], the degree of water-flooding in oil wells increases prior to an earthquake, and, conversely, the percentage of oil in water-flooded wells in increased. It is apparent from comparison of [17] and [19] that the entire matter rests in foreshock vibrations. Natural seismicity accelerates the migration of oil, and, consequently, even its accumulation in zones of seismic activity. Moderate increases (by 50%) in the production of wells situated from the center of a powerful blast by a distance known to exceed the zone of explosive crushing of rock (0.4-2.0 m/kg I/3 of explosive) [3, 20] should also be explained by seismic activity. Like Nikolaevskii's estimate [3] of the energies of earthquates for their magnitudes, the experience gained in these studies suggests a disproportionately large energy required for a commercially meaningful vibration-caused increase in oil recovery. The effect of dominant vibrations of block ore masses for which reconstruction of the microstructure of the pore spaces is possible is therefore extremely important [13, 21]. LITERATURE CITED i. 2. 3.

M . A . Sadvoskii (ed.), Mechanical Effect of an Underground Blast [in Russian], Nauka, Moscow (1971). V. S. Nikiforovskii and E. I. Shemyakin, Dynamic Failure of Solids [in Russian], Nauka, Novosibirsk (1979). V. N. Nikolaevskii, Mechanics of Porous and Fissured Media [in Russian], Nedra, Moscow

(1984). 4. 5. 6. 7.

8. 9. i0.

W. G. Knauss and K. Ravi-Chandar, "Some basic problems in stress wave dominated fracture," Int. J. Fracture, 27 (1985). S. Crampin, "The basis for earthquake prediction," Geophys. J. R. Soc., 91 (1987). M. A. Meyers and L. E. Murr, Shock Waves and Rapld-Deformation Phenomena in Metals [Russian translation], Metallurgiya, Moscow (1984). V. N. Nikolaevskii, "Mechanics of geomaterials. Complex models," Results of Science and Engineering, Series on Mechanical Deformations of Solids [in Russian], Vol. 19, Vsesoyuznyi Instltut Nauchnoi i Tekhnicheskoi Informatsli, Moscow (1987). K. Kasahara, Earthquake Mechanics [Russian translation], Mir, Moscow (1985). Problems of Nonlinear Selsmics [in Russian], A. V. Nikolaev and I. N. Galkin (eds.), Nauka, Moscow (1987). V. N. Nikolaevskii (ed.), Nonlinear Wave Processes [Russian translation], Mir, Moscow

(1987). ii. 12. 13. 14. 15. 16.

17.

M. L. Surguchev, O. L. Kuznetsov, and E. M. Simkin, Hydrodynamic, Acoustic, and ThermalCycling Effects on Oil-Bearing Strata [in Russian], Nedra, Moscow (1975). M. A. Blot, Mechanics of Incremental Deformations, Wiley, New York-~ondon-Sydney (1965). N. A. Vil'chinskaya and V. N. Nikolaevskil, "Acoustic emission and seismic-signal spectra," Izv. Akad. Nauk SSSR, Fiz. Zemli, No. 5 (1984). J. Za~n~n, Disorder Models [Russian translation], Mir, Moscow (1982). M. S. Leverett, "Movement of water-~il mixtures in loose sands," Methods for Improving Oil Recoveryln Strate [Russian translation], Gostoptekhizdat, Moscow--Leningrad (1948). N. M. Belyaev, V. A. Degtyarev, and S. V. Chekhuta, "Experimental investigation of the effect of vibrations on the retentivity of capillary reticular phase separators," Numerical Modeling of Hydrogasdynamic Flows [in Russian], Dnepropetrovsk (1987). D. G. Oslka, Fluid Regimes of Seismically Active Regions [in Russian], Nauka, Moscow

(1981). 18. 19. 20. 21.

A. V. Nikolaev and I. G. Kissin (eds.), Hydrogeodynamic Earthquake Precursors [in Russian], Nauka, Moscow (1984). I. G. Kissin, "On anomalous variations in the degree of watering in oil wells prior to an earthquake," Dokl. Akad. Nauk SSSR, 270, No. 3 (1983). A. A. Bakirov and E. A. Bakirov, Use of Underground Nuclear Explosions in the OilRecovery Industry [in Russian], Nedra, Moscow (1981). L. Gomes and L. Graves, "Stabilization of beach sand by vibration," Highway Research Board Bulletin, No. 325, Washington, D.C. (1962).

519