The Phononic Lattice Solid With Fluids For Modelling Non-Linear

Geophys. J. Inf . (1994) 117, 529-538

The phononic lattice solid with fluids for modelling non-linear solid- fluid interactions

Lian-Jie Huang and Peter Mora Dkpartement de Sismologie, Insfifut de Physique du Globe de Paris, 4 Place Jussieu, F-75252 Paris Cedex 05, France

Accepted 1993 November 8. Received 1993 July 12

SUMMARY The phononic lattice solid has been developed recently as a possible approach for modelling compressional waves in complex solids at the microscopic scale. Rather than directly modelling the wave equation, the microdynamics of quasi-particles is simulated on a discrete lattice. It is comparable with the lattice gas approach to model idealized gas particles but differs fundamentally in that lattice solid particles carry pressure rather than mass and propagate through a heterogeneous medium. Their speed may be space and direction dependent while the speed of lattice gas particles is constant. Furthermore, they may be scattered by medium heterogeneities. Lattice sites in the phononic lattice solid approach are considered to be fixed in space for all time. Lattice site movements (i.e. deformations) induced by the passage of a macroscopic wave are particularly important for a fluid-filled porous medium considering that non-linear solid-fluid interactions are thought to play a role in attenuation mechanisms. We take lattice site movements into account in the phononic lattice solid and name the approach ‘the phononic lattice solid with fluids (PLSF)’ because it could lead to an improved understanding of the effect of solid-fluid interactions in wave propagation problems. The macroscopic limit of the Boltzmann equation for the PLSF yields the acoustic wave equation for heterogeneous media modified by shear and bulk viscosity terms as well as the second-order term in macroscopic velocity (for the PLS) and additional non-linear terms due to the lattice site movements. It is hoped that PLSF numerical simulation studies of waves through digitized rock matrices may lead to an improved understanding of attenuation mechanisms of waves in porous rocks.

Key words: attenuation, Boltzmann equation, lattice gas, lattice site movements, phononic lattice solid, solid-fluid interactions.

1 INTRODUCTION

The lattice gas (LG) (Hardy, Pomeau & de Pazzis 1973; Hardy, de Pazzis & Pomeau 1976; Frisch, Hasslacher & Pomeau 1986), which connects with theoretical physics, computational physics and dedicated machine technology (e.g. parallel computers), has developed rapidly as an approach to simulate complex physical phenomena at the microscopic scale. It has successfully simulated fluid flow (d’Humi&res & Lallemand 1987; Rivet et al. 1988), flow in porous media (Rothman 1988a; Chen et al. 1991b), constant speed sound waves (Rothman 1987), two-phase flow (Rothman 1988b), magnetohydrodynamics (Chen & Mat- thaeus 1987; Chen et al. 1991a). The lattice Boltzmann equation method (LBE) is an alternative approach to the LG for the study of hydrodynamic problems (Higuera 1988;

McNamara & Zanetti 1988; Higuera, Succi & Benzi 1989). It uses real numbers instead of bits to represent particle distributions. An important property of the LBE is that simulations for flow in simple and complex geometries have the same speed and efficiency, while all other methods, including the spectral method, are unable to model complicated geometries efficiently (Chen et d. 1992).

The phononic lattice solid (PLS) simulates the microscopic physics of quasi-particles carrying pressure in order to model compressional waves in heterogeneous media at the macroscopic scale (Mora 1992; Huang & Mora 1993). Mora (1992) presented the PLS based on a finite-difference scheme to solve the transport equation for quasi-particles. This method required a small lattice spacing in order to converge to the macroscopic result (typically 60 grid points per wavelength in wave propagation problems) and smooth

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530 L.-J. Huang and P. Mora

velocity models (an interface was not sharp but implicitly had the width of the grid spacing). Huang & Mora (1993) developed the PLS by interpolation which has high precision for sharp interfaces and can therefore handle sharp solid-fluid interfaces in porous rocks. The PLS can be used for modelling compressional wave phenomena in highly complex media which are not easily handled by classical finite-difference methods because of the geometrical complexity of interfaces and the presence of pore fluids leading to complicated scattering effects as well as non-linear interactions.

Lattice sites, representing a medium, move as a macroscopic wave propagates through the medium. This results in non-linear interactions between heterogeneities such as the solid matrix and pore fluids. The behaviour of fluids such as oil, gas or water is non-linear as well as viscous and hence theoretical analyses are difficult due to the nature of the pore fluids coupled with the geometrical complexity of rock matrices.

We develop a method named 'the phononic luttice solid with fluids (PLSF)', which takes lattice site movements into account in the PLS, and hence provides a tool to study these non-linear effects on wave propagation in heterogeneous acoustic media.

We derive the Boltzmann equation for the PLSF. Then we use the Chapman-Enskog expansion for the particle number density to second order in macroscopic velocity to derive the macroscopic wave equation including non-linear terms which, coupled with viscoclastic and scattering effects, may be partially responsible for attenuation mechanisms. Numerical examples demonstrate the overall wave attenuation due to the combination of these effects which appears to be 'constant Q' in some regimes when frequencies are low and possibly tends to viscous behaviour a t higher frequencies.

Ultimately, an elastic PLS is required for the solid matrix (i.e. one supporting S as well as P waves) in order to apply the method t o the study of real rocks. While an elastic PLS is under development, none has yet been implemented (Maillot & Mora 1993), so we restrict the present study to heterogeneous acoustic media supporting only P waves.

2 LATTICE SITE MOVEMENTS

A solid matrix such as that of a fluid-filled porous medium shown in Fig. l (a) , move with time as a wave propagates through the medium. The window indicated by the small rectangle in the figure is enlarged in Fig. l (b) in order to highlight the triangular lattice of the 2-D phononic lattice solid model. In order to capture additional non-linear effects due to heterogeneity and finite deformation on wave propagation, we take into account the lattice site movements as shown in Fig. l(b). In Fig. l(b), we have only shown the lattice links for the lattice sites A and B with heavy Iines. The arrow represents the induced fluid flow.

Wave phenomena including the effects of lattice movements will be simulated using the phononic lattice solid by interpolation which is valid for any velocity and impedance contrasts (Huang & Mora 1993). The formulation of the Boltzmann equation is first presented for the PLSF starting from the Boltzmann equation for the PLS.

Figure 1. (a) The schematic illustration of fluid motions in a fluid-filled porous medium induced by thc passage of a macroscopic wave. The brighter regions represent the fluid and the darker regions represent the solid matrix. (b) Blow-up of the window shown in (a) illustrates the distortion of the triangular lattice. The fine lines are the original undistorted lattice and the heavy lines depict how the lattice sites A" and B" move in response to the passage of a macroscopic wave to positions A and B.

3 T H E BOLTZMANN E Q U A T I O N FOR T H E PLSF

Consider a medium without intrinsic anisotropy. The aim is to derive the Boltzmann equation for the PLSF, namely, to take the lattice site movements into account in the Boltzmann equation for the PLS. The first-order differential version of the Boltzmann equation for the PLS without lattice movements is (see Mora 1992; Huang & Mora 1993)

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Modelling non -linear solid-fluid interactions 53 1

where

(4)

In these equations, N, is the number density of particles moving in direction a = (1,2, . . . , b) where b is the total number of directions for a lattice site (i.e. 4 for a square lattice and 6 for a triangular lattice in the 2-D case), x denotes the position of a lattice site, Af is the time step, Ax: is the vector between lattice sites in the a-direction, Axo= (AxtI = const and c(x) is the particle speed with AN: representing the collision term for the Bosons (Higuera 1988) and AN: the scattering term. The Boolean variables S, and Sk define input and output states with a (collision) transition probability from input state S to output state S‘ of A(S-+S’) . The reflection coefficient at x for a particle moving in the a-direction is denoted R,(x) .

In the classical PLS, the lattice sites were fixed in space for all time. However, the lattice sites are associated with a given (space-dependent) medium property and, therefore, should move as the medium distorts. Consider two points x and x - Ax, in the a-direction with an infinitely small separation of Ax in space (in a Lagrangian coordinate system), where Ax = lAx,\. Quasi-particles of pressure propagate from x - Axm to x with a velocity of E ( x , t ) + v,(x, t ) rather than c(x), where E ( x , t ) is the time-dependent medium velocity determined by the density and the bulk modulus of the medium, and u, is the component of the medium particle velocity in the a-direction. Therefore, the PLSF quasi-particle transport equation is

and the direction and time-dependent dimensionless effective speed is

(7)

where E(x, t) is given by (see Mora 1992; Huang & Mora 1993)

where x’ is the point at which the medium particle moves to x (see Fig. 2 for the 2-D case), D is the number of space dimensions, K is the bulk modulus and 7, is the line density of the link given by

m Ax P ( x , f) = - , (9)

where m is the mass of the medium within the region between x - Ax, and x. The density of the medium within

Figure 2. Illustration of medium particle movements: the medium in the infinitely small region from x - Axe to x comes from the medium in the infinitely small region from x’ - Ax; to x’.

the region between x’ - Ax; and x’ is

where Ax; is the distance between the points at x‘ - Ax; and x’ (see Fig. 2).

We restrict our formulation in the following to the 2-D triangular lattice case.

From Fig. 2, we have

= {[AX - u,(~, t ) - $u,+l(x, t ) + &,+Z(X, f)] - [ -U,(X - AX,, t ) - ; U , + ~ ( X - AX,, t)

+ &,+Ax - Ax,, f l I2 + ac-r.,+,cx, t ) + u,+z(x, 111

+ [u,+i(x - Ax,, t ) + u,+z(x - Ax,, t ) ] ) ’ , (11) where u, is the component of displacement in the a-direction and u,+,, u , + ~ are those in the ( a + 1)- direction and (a + 2)-direction, respectively. To first order in the derivative of displacement, it follows from eq. (11) that

A x k - A X [ I - u,(x, t ) - U,(X - AX,, t ) Ax

_ - 1 U,+I(X, t ) - u,+,(x - Ax,, t )

1 u,+*(x, t ) - U,+Z(X - Ax,, t )

2 Ax

2 Ax I. (12) +- From eqs (9), (10) and (12) and again to first order in the derivative of displacements, we obtain

- _ 1 u,+,(x, t ) - U,+I(X - Ax,, t ) 2 Ax

(13) 1 u,+z(x, t ) - u,+z(x - AX,, t ) +- 2 Ax

Fig. 3 illustrates the axes along a fixed 2-D triangular lattice at the point x at which the medium particle comes from the point at x‘. In the figure, the xL-axis represents the perpendicular axis of x,-axis. To first order, the Taylor’s expansion of the quasi-particle speed c(x‘) is



X'

Figure 3. Illustration of the axes along a fixed 2-D triangular lattice at the point x at which the medium particle comes from x'.

where u i is the component of displacement along the xi-axis given by

v3 u;(x , t ) = - [U,+I(X, t ) + u,+2(x, 41. 2

On the other hand, we have

ac 1 ac d3 ac

ax,,, 2ax, 2 3x; dc 1 ac d3 dc

ax,+* 2ax, 2 ax:

- +---,

+---- -=---

It follows from eqs (16) and (17) that

Substituting eq. (18) into eq. (14) and making use of relation (15) yields

a C c(x ' ) = c(x) - - u,

3%

Substituting eqs (13) and (19) into eq. (8) yields

q x , t ) = c(x) - - ac(x) u,(x, t ) ax,

Notice that, in the above derivation, we have dropped terms involving higher than first-order derivatives considering that those terms lead to terms involving higher than second-order derivatives in the following derivation for the macroscopic equation.

The differential version of the reflection coefficient R, is (see Mora 1992)

Ax dZ 2 2 axj -/'

R , - + - - - e .

where Z(x) = p ( x ) c ( x ) / l b is the impedance, c ( x ) / l b the macroscopic wave speed of a medium and emj is a component of e, which is given by

for the case of a 2-D triangular lattice. Using eq. (21), substituting eq. (20) into eq. (7) and then

substituting it into eq. ( 5 ) , dividing both of its sides with At, taking the limits of At-0 and A x + O and dropping terms involving higher than second-order derivatives yields.

where c z is the effective speed of quasi-particles given by

and d N t I d t denotes the rate of change of particle number density due to scattering processes given by

Eq. (23) is the Boltzmann equation for the PLSF. We see that it has the same form as the Boltzmann equation for the PLS (see Mora 1992; Huang & Mora 1993) but the velocity of quasi-particles in the PLSF is space-, time- and direction-dependent even for a medium without intrinsic anisotropy.

4 T H E MACROSCOPIC LIMIT

The macroscopic pressure P and velocity v i are, respectively, defined by

and

The component of the macroscopic velocity in the a-direction is therefore

2 0 v, = - vieai.

b


Modelling non-linear solid-fluid interactions 533

This can be verified by substituting eq. (28) into the relation where

Eijkl enienjeakeal a

vi = C v,emi

and using the identity

ff

We also obtain

From eq. (28), it follows that

and

OI D bPG(P) C N,emie,, = ___

Dc2 The energy and quasi-momentum conservation of

particles during their interaction (collision) processes implies that -- bPF (-+-'--k6,) dv, dv. dv

2D dxi ax, ax, b ( D -2)PFdvk - 6, -

20' axk and

Summing eq. (23) over direction a and making use of the definitions (26) and (27) and the relations (31), (32) , (37) and (42) yields

(33)

The Chapman-Enskog expansion of N , to second-order in macroscopic velocity is (see Wolfram 1986)

D + 2 1 N , = P + pcv,eni + ~ PQ,,[ G(P)-pviv, -

2

where G ( P ) is a dimensionless function of P which is determined by the conservation of momentum,* F is an undetermined coefficient, and Qmii is given by

The corresponding equation for the PLS (see Mora 1992) only involves the first two terms on the left-hand side of eq. (43). All other terms in eq. (43) result from the lattice site movements. The last term involves the second-order term in macroscopic velocity, an effective pressure term as well as the shear and bulk viscosity terms as discussed later.

On the other hand, multiplying eq. (23) with [De,;l(bc)], summing it over a, using the definition (27) and the relations (30), (31) , (33), (41) and (42) yields

1 Q,, = eaieaj - - D 6. . t/. (35)

By using eqs (25), (31) and (34), noting that

1 - 3 vkvk] 6, (37)

and (44)

where P is given by eq. (51) in the next section. Within the curly brackets in eq. (a), the first term is the second-order term in macroscopic velocity (of the Navier-Stokes equation) with an effective density of p, given by

P p, = - G(P) .

C 2 (45)

The second term corresponds to the effective pressure P, in the medium given by * It corresponds to C ( p ) in the lattice gas automaton that leads to

non-Galilean invariance. Galilean invariance can be recovered by the particular choice of the equilibrium values for the average occupation numbers N, (Qian 1990).



The third and the fourth terms are the shear viscosity term and the bulk viscosity term, respectively. The shear viscosity is therefore

PF v = - p

and the bulk viscosity is

( D - 2)PF '= 2 0 '

(47)

Eq. (44) without its last three terms is the result for the PLS but taking into account the non-linear term in macroscopic velocity as well as bulk and shear viscosity terms. The last three terms in eq. (44) result from lattice site movements.

Therefore, the macroscopic behaviour of the PLSF model is similar to a real fluid to second order in macroscopic velocity but allows for a heterogeneous bulk modulus. We see that the bulk viscosity is zero for 2-D cases, which is the same as was found for the lattice gas (HCnon 1987).

5 FINITE S T R A I N TERM

The classical strain versus displacement with finite strains is

1 " 2 ( ax, ax, ax, ax, 1 au, au, au,au,

E =- -+-+--.

The pressure is therefore

(49)

where the second-order term in derivatives of displacement is the finite strain term.

On the other hand, making use of eqs (38) and (40) in eq. (43) and then integrating it over time t yields.

where P,, = P ( x , f = 0) and the constant H is defined by

3 0 2 b ( D + 2) '

H E

and P, is given by

Eq. (51) involves the terms of second order in the derivatives of displacement. Comparing eq. (51) with eq. (50), we see that the first term of second order in the derivatives of displacement in eq. (51) is the finite strain term appearing in eq. (SO). Therefore, the PLSF has equivalently taken the effects of finite strain into account but

with constant H different from 1 due to the manner of the model discretization (cf. the G ( P ) term).

6 NUMERICAL E X A M P L E S

6.1 Attenuation without the effect of lattice site movements

We use an acoustic model with 2048 x 32 grid points in a triangular lattice to study the attenuation of waves propagating in a heterogeneous medium. The model is shown schematically in Fig. 4 where the white regions represent the solid matrix (sands grains) with a velocity of 2.5 km s-' and a density of 2.2 g cmp3 and the black regions represent a viscous fluid (water) with a velocity of 1.5 km s-' and a density of 1.0gcm-'. The 'solid-fluid interfaces' in the model are situated at lattice sites. The absolute value of the normal incident reflection coefficient at solid-fluid interfaces is 0.571 4286. Note that the model we used is a long narrow rectangle with circular boundaries in z rather than a box such as in Fig. 4. Hence, the model is periodic in z which should have only a minimal effect on our results relative to a box model considering we propagate a plane wave along the rectangle in the following experiments. We use normalized particle speeds of 0.3 of the solid and of 0.18 of the fluid in the PLS calculations. The mean particle speed is, therefore, 0.226 6807 calculated by averaging slowness and the mean density is 1.618892. The centres of the grains are separated from each other by a distance of 8 f 1 grid spacings in the horizontal and vertical directions. The diameters of the grains range randomly from 5 to 7 grid spacings and hence their average diameter is about 6 grid spacings. A pressure line source is set up at the left border of the model and the receivers are placed in a horizontal line

Figure 4. The heterogeneous model used for the attenuation study. The white regions represent the solid matrix (sand grains) with a velocity of 2.5 km s- ' and a density of 2.2 g cm-3 and the black regions represent the viscous fluid (water) with a velocity of 1.5kms- ' and a density of l . O g ~ m - ~ . Note that the maximum strain induced by the plane wave propagation ranges approximately from 0.14 to 0.39 during the simulation experiment.


Modelling non-linear solid-fluid interactions 535

0 2000 4000 6000 8000 Time Step

(c) Spect ra 10 01 I

Frequency

" I I

I . , , , . , . , . . . 0 2000 4000 6000 8000

Time S t r p

I . , ,

20 40 60 Frequency

Figure 5. The windowed and tapered (50 points at both ends) seismograms calculated by the PLS at the positions of: (a) 100 and (b) MM) grid points from the left border of the model with their amplitude spectra (c) and the natural logarithm of the spectral ratios versus frequency (d). Note that the weaker events trailing the first arrival in (a) and (b) are due to scattering effects which also contribute to overall attenuation. In (c) the solid line corresponds to the seismogram in (a) and the dashed line to the seismogram in (b). In (d) the solid linc is the natural logarithm of the spectral ratios of thc spectra (the solid one to the dashed one) shown in (c) and the dashed line is a linear fit to the spectral ratios up to the frequency of 42.

through the centre of the model. The line source consists of 32 point sources each of which is the first derivative of a Gaussian time history with an amplitude of 0.005 and a wavelength of about 60 grid points for the average particle speed, 80 grid points for the particle speed of the solid and 48 grid points for the particle speed of the fluid.

We assume that the solid has essentially no viscosity (i.e. is perfectly elastic) but that the fluid has a significant viscosity. Therefore, we choose the mean value of the particle number density for the solid regions as 0.450 for which the viscosity is approximately zero, and that for the fluid regions as 0.300 for which the viscosity coefficient is less than about 0.008 (See Mora & Maillot 1991). We used the PLS by interpolation to calculate the pressure wave propagation for 8192 time steps. Then we selected the seismograms at (100,16) and (600,16) in windows of 500-5000 time steps and 3500-8000 time steps, respectively,

as shown in Figs 5(a) and (b), to compute the corresponding Fourier spectra using a Fast Fourier Transform (FFT) algorithm. The corresponding Spectral amplitudes are shown in Fig. 5. The natural logarithm of the spectral ratios of the spectra in Fig, 5(c) is shown in Fig. 5(d) as a solid line. The dashed line in Fig. 5(d) is a linear fit to the solid line up to the frequency of 42 units which corresponds to a wavelength of about 1/14 of the average diameter of the grains in the model. This indicates that the result supports the constant Q (the quality factor) model for low frequencies (see Toksoz, Johnston & Timur 1979) because the quality factor and the attenuation coefficient a are related linearly as follows:

(54)


536 L.-J. Huang and P . Mora

where f is the frequency. Beyond the frequency of 42 units, scattering effects and viscosity seem to dominate and the attenuation coefficient grows increasingly rapidly with frequency.

6.2 Attenuation including the effect of lattice site movements

In the numerical implementation for the PLSF, we calculate number densities of quasi-particles at moving lattice sites. Transmission and reflection coefficients as well as collision rules of quasi-particles in a moving lattice were taken to be the same as those for the case without lattice site movements. Therefore, the following numerical study has only taken into account the effects of lattice site movements on the changes of the quasi-particle speed and lengths of lattice links. The effects of lattice distortion on transmission and reflection coefficients and collision processes of quasi-particles will be studied in a future work.

tii 0 il I . , . I . , . I , . ' " ' ' I '

0 2000 4000 6000 6000 Time Step

(4 Spectra In 01

20 4 0 60 80 Frequency

We use the same model as above to calculate the pressure wavefield for 8192 time steps using the PLSF and study the attenuation including the effect of lattice site movements. The results corresponding to Fig. 5 are plotted in Fig. 6. In order to compare the attenuation for both cases, we combine Figs 5(d) and Fig. 6(d) together and show them in Fig. 7 where the solid lines are the curve or line in Fig. 6(d) and the dashed lines those in Fig. 5(d). This figure illustrates that there is an additional attenuation due to the lattice site movements.

6.3 The effect of lattice site movements

In order to study the effect of lattice site movements, we plotted the seismograms in Fig. 5(b) (for the PLS) and Fig. 6(b) (for the PLSF) together and showed them in Fig. 8(a) where the solid line is the seismogram in Fig. 5(b) and the dotted line is that in Fig. 6(b). We see that there is some difference between both seismograms, especially between

(b) Seismogram N 0 0

a(

41 I . ~ . I . ~ . 1 ' ~ ' I ' ' ' 1

0 2000 4000 6000 8000 Time Slep

(d) In( t h e sDectral ratios f

I . , , I

F requency 20 40 60

Figure 6. The windowed and tapered (50 points at both ends) seismograms calculated by the PLSF at the positions o f (a) 100 and (b) 6Oo grid points from the left border of the model with their amplitude spectra (c) and the natural logarithm of the spectral ratios versus frequency (d). In (c) the solid line corresponds to the seismogram in (a) and the dashed line the one in (b). In (d) the solid line is the natural logarithm of the spectral ratios of the spectra (the solid one to the dashed one) shown in (c) and the dashed line is the linear fit to the spectral ratios up to the frequency of 42 units.


Modelling non -linear solid-fluid interactions 537

In( the spectral ra t ios ) I

20 40 60 Frequency

@re 7. Comparison of Fig. S(d) (dashed lines) and Fig. 6(d) (solid lines).

i " ' 1 " ' I . . ' ' I " ' I '

0 2000 4000 6000 8000 Time S t e p

(b) Seismogram

n I

1 ' ' ' l . . . r . . . , . ' ' / ~

1 0 2000 4000 6000 8000

Time S t e p

Figure 8. (a) The windowed and tapered (50 points at both ends) seismograms calculated by the PLS (the solid one) and the PLSF (the dotted one) at the positions of 600 grid points from the left border of the model. (b) The section of the seismograms (without tapering) in (a) from 4450 to 8ooO time steps with a larger scale.

20 40 G O Frequency

(b) In( the spectral ratios ) u? 0

1 , . , I

Frequency 20 40 60

Figure 9. (a) The amplitude spectra of the seismograms in Fig. 8(a); (b) the natural logarithm of the spectral ratios versus frequency. In (a) the solid line corresponds to the seismogram for the PLS and the dashed line the one for the PLSF in Fig. 8(a).

both later arrivals. The later arrivals (from 4450 to 8OOO time steps) in the seismograms in Fig. 8(a) are shown in Fig. 8(b) with a larger scale in order to see their difference more clearly. In Fig. 9, (a) is the amplitude spectra of the seismograms in Fig. 8(a) and (b) is the natural logarithm of the spectral ratios (the solid one to the dashed one) of the spectra shown in (a). The solid line in Fig. 9(a) corresponds to the seismogram indicated by the solid line in Fig. 8(a) and the dashed line the one indicated by the dotted line. Fig. 9(b) tells us that the lattice movements lead to an increase in the spectral amplitudes at lower frequencies and a decrease at higher frequencies.

7 CONCLUSIONS

We have developed a phononic lattice solid model (PLSF) that takes into account the lattice site movements induced by wave motions. We denote the approach the phononic


538 L.-J. Huang and P . Mora

lattice solid with fluids (PLSF) because it could lead to an improved understanding of the effect of solid-fluid interactions in wave propagation problems. Comparing the PLSF Boltzmann equation with that of the PLS, we see that they have the same form but the speed of quasi-particles is space-, time- and direction-dependent for the PLSF d u e t o the lattice site movements (i.e. finite strains) even for a medium without intrinsic anisotropy. By making use of the Chapman-Enskog expansion t o second order in macroscopic velocity, we have obtained the corresponding macroscopic equation (momentum conservation equation) for heterogeneous media, which contains shear, bulk viscosity terms, the second-order term in macroscopic velocity-these are the results for the P L S a s well as additional non-linear terms due t o lattice site movements. The non-linear terms introduce an additional attenuation as demonstrated by numerical experimentation. It is hoped that the method outlined will allow studies t o be conducted of attenuation due t o non-linear solid-fluid interactions induced as a wave propagates through a porous rock. This would ultimately require an elastic PLSF to handle elastic waves in the solid matrix and the resolution of outstanding issues such as how t o control independently the PLSF viscosity coefficients and remove non-physical discretization constants.

A C K N O W L E D G M E N T S

L.-J. Huang would like to thank Professor Albert Tarantola for his encouragement. This research has been funded in part by the French National Ministry of Education (MEN) and the National Center for Scientific Research (CNRS). It could not have been done without the support of t he Sponsors of the Seismic Simulation Project (Amoco, CEA (French Atomic Energy Authority), Delany Enterprises Inc., Elf , G O C A D , Inverse Theory & Applications, Norsk Hydro, Shell, Thinking Machines Co. and Total-CFP). T h e Connection Machine was funded by the M E N .

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