СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ

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1 ... S e MR ISSN СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ Siberian Electronic Mathematical Reports Том 13, стр (2016) УДК ; DOI /semi MSC 35L52, 35Q35, 35Q74 SEISMIC IN COMPOSITE MEDIA: ELASTIC AND POROELASTIC COMPONENTS A.MEIRMANOV, S.MUKHAMBETHZANOV, M.NURTAS Abstract. In the present paper we consider elastic and poroelastic media having a common interace. We derive the macroscopic mathematical models or seismic wave propagation through these two dierent media as a homogenization o the exact mathematical model at the microscopic level. They consist o seismic equations or the each component and boundary conditions at the common interace, which separates dierent media. To do this we use the two-scale expansion method in the corresponding integral identities, deining the weak solution. Our results we illustrate with the numerical implementations o the inverse problem or the simplest model. Keywords: seismic, two-scale expansion method, ull waveield inversion, numerical simulation. 1. Introduction The present paper is devoted to a correct description o seismic wave propagation in composite media Q R 3, consisting o the elastic medium Ω (0), poroelastic medium Ω, which is perorated by a periodic system o pores illed with a luid, and common interace S (0) between Ω (0) and Ω (see Fig.1). That is, Q = Ω S (0) Ω (0) and Ω = Ω Γ Ω s, where Ω s is a solid skeleton, Ω is a pore space (liquid domain), and Γ is a common boundary "solid skeleton-liquid domain". Meirmanov, A.M., Mukhambetzhanov, S.T., Nurtas, M., Seismic in composite media: elastic and poroelastic components. c 2016 Meirmanov A.M., Mukhambetzhanov S.T., Nurtas M. This research was supported by the Ministry o Education and Science o the Republic o Kazakhstan, agreement 80 rom (grant 2021/GF4). Received November, 29, 2015, published February, 16,

2 76 A.MEIRMANOV, S.MUKHAMBETHZANOV, M.NURTAS The structure o the heterogeneous medium Q is too complicated and makes hard a numerical simulation o seismic waves propagation in multiscale media. The main diiculty here is a presence o both components (solid and liquid) in each suiciently small subdomain o Q. It requires to change the governing equations (rom Lame s equations to the Stokes equations) at the scale o some tens microns. There are two basic methods to describe physical processes in such media: the phenomenological method and the asymptotical one which is based on the upscaling approaches. The phenomenological approach or waves propagation through a poroelastic medium ([4], [5]) leads, in particular, to Biot model ([1]-[3]). It based on the system o axioms (relations between the parameters o the medium), which deine the given physical process. But, there can be another system o axioms deining the same process. Thus, it is necessary choose the correct authenticity criterion o the mathematical description o the process. It can be, or example, the physical experiment. As a rule, each phenomenological model contains some set o phenomenological constants. Thereore, one can achieve agreement between the suggested theory and selected series o experiments changing these parameters. The second method, suggested by R. Burridge and J. Keller [6] and E. Sanchez- Palencia [7], based on the homogenization. It consists o: 1) an exact description o the process at the microscopic level based on the undamental laws o continuum mechanics, and 2) the rigorous homogenization o the obtained mathematical model. To explain the method we consider a characteristic unction χ 0 (x) o the pore space Ω. Let L is the characteristic size o the physical domain in consideration, τ is the characteristic time o the physical process, ρ 0 is the mean density o water, and g is acceleration due gravity In dimensionless variables x x L, w α w τ L, t t τ, F F g, ρ ρ ρ 0, the dynamic system or the displacements w and pressure p o the medium takes the orm [6, 7, 8]: (1) ϱ 2 w t 2 = P + ϱf, (2) P = χ 0 α µ D(x, w t ) + (1 χ 0)α λ D(x, w) + ( χ 0 α ν ( w t ) p) I, (3) p + α p w = 0. Equations (1)-(3) are understood in the sense o distributions as corresponding integral identities. They are equivalent to the Stokes equations (4) ϱ v t = P + ϱ F, p t + α p, v = 0, (5) P = α µ D(x, v) + ( α ν ( v) p ) I

3 SEISMIC IN COMPOSITE MEDIA: ELASTIC AND POROELASTIC COMPONENTS or the velocity v = (6) 77 w and pressure p in the pore space Ω and the Lame equations t 2w %s 2 = Ps + %s F, p + αp,s w = 0, t Ps = αλ D(x, w) p I (7) or the solid displacements w and pressure p in Ωs. At the common boundary Γ velocities and normal tensions are continuous: w = v, Ps n = P n. t Here n is a unit normal to Γ. 1 In (1)-(8) D(x, u) = ( u + u ) is the symmetric part o u, I is a unit 2 tensor, F is a given vector o distributed mass orces, (8) αp = αp, χ0 + αp,s (1 χ0 ), ατ = L, gτ 2 αµ = % = % χ0 + %s (1 χ0 ), 2µ, ατ τ Lg ρ 0 αλ = 2λ, ατ Lg ρ 0 % c2 2ν %s cs2, αp, =, αp,s =, 0 ατ τ Lg ρ ατ Lg ατ Lg µ is the dynamic viscosity, ν is the bulk viscosity, λ is the elastic constant, % and %s are the respective mean dimensionless densities o the liquid in pores and the solid skeleton, correlated with the mean density o water ρ 0, and c and cs are the speed o compression sound waves in the pore liquid and in the solid skeleton respectively. The mathematical model (1) (3) can not be useul or practical needs, since the αν = Figure 1. The pore structure unction χ0 changes its value rom 0 to 1 on the scale o a ew microns. Fortunately, l the system possesses a natural small parameter ε =, where l is the average size L o pores. Thus, the most suitable way to get a practically signiicant mathematical model, which approximate (1) (3), is a homogenization or upscaling. That is, we suppose the ε-periodicity o the solid skeleton, let ε to be variable, and look or the limit in (1) (3) as ε 0.

4 78 A.MEIRMANOV, S.MUKHAMBETHZANOV, M.NURTAS There are dierent homogenized (limiting) systems, depending on o α µ, α λ,... Some o these numbers might be small and some might be large. We may represent them as a power o ε, or as unctions depending on ε. Let µ 0 = lim ε 0 α µ (ε),,0 = lim ε 0 α p, (ε), α µ µ 1 = lim ε 0 ε 2, ν 0 = lim ε 0 α ν (ε), λ 0 = lim ε 0 α λ (ε), c 2 s,0 = lim ε 0 α p,s (ε), λ α λ 1 = lim ε 0 ε 2. It is clear that the choice o these limits depend on our willing. For example, or ε = 10 2 and α = we may state that α = 2 ε 1 2, or α = 0.02 ε 0. It is usual procedure when we neglect some terms in dierential equations with small coeicients and get more simple equations, still describing the physical process. The detailed analyses o all possible limiting regimes has been done in [8, 9]. In order to describe the seismic in two dierent media (elastic and poroelastic), having a common interace we must chose one o the two methods discussed above. The irst method suggests only some guesses, while the second method has a clear algorithm or the derivation o the boundary conditions. That is why we choose here the second method. We derive new seismic equations in each component (elastic and poroelastic) and the boundary conditions on the common boundary. For these boundary conditions the very little is known and only or the liquid iltration (see or example [10]). For three dierent sets o µ 0, λ 0,... or each component we derive three dierent mathematical models, which describe the process with dierent degrees o approximation. We start with the integral identities, deining the weak solution w ε and p ε, and use the two-scale expansion method [11, 12], when we look or the solution in the orm w ε (x, t) = w(x, t) + W 0 (x, t, x ε ) + ε W 1(x, t, x ε ) + o(ε), p ε (x, t) = p(x, t) + P 0 (x, t, x ε ) + ε P 1(x, t, x ε ) + o(ε) with 1-periodic in the variable y unctions W i (x, t, y), P i (x, t, y), i = 0, 1,... This method is rather heuristic and may lead to the wrong answer. But our guesses are based upon the strong theory, suggested by G. Nguetseng [13, 14]. For the rigorous derivation o seismic equations in poroelastic media, which dictate the correct two-scale expansion, see [8]. Finally, to calculate limits as ε 0 in corresponding integral identities, we apply the well-known result (9) lim F (x, ε 0 Ω x ε, t)dxdt = ( F (x, y, t)dy ) dxdt Ω Y or any smooth 1-periodic in the variable y Y unction F (x, y, t). 2. The statement o the problem For the sake o simplicity we suppose that Q = {x = (x 1, x 2, x 3 ) R 3 : x 3 > 0}, Ω (0) = {x = (x 1, x 2, x 3 ) R 3 : 0 < x 3 < H}, Ω = {x = (x 1, x 2, x 3 ) R 3 : x 3 > H}, F = 0, and α p, =, α p,s = c 2 s.

5 SEISMIC IN COMPOSITE MEDIA: ELASTIC AND POROELASTIC COMPONENTS 79 Let Y be a unit cube in R 3, Y = Y γ Ys. We assume that pore space Ω ε is a periodic repetition in Ω o the elementary cell εy, the solid skeleton Ω ε s is a periodic repetition in Ω o the elementary cell εy s, and the boundary Γ ε between a pore space and a solid skeleton is a periodic repetition in Ω o the boundary εγ. The detailed description o the sets Y and Y s is done in [8]. Due to these suppositions χ 0 (x) = χ ε (x) = χ( x ε ), where χ(y) is a 1-periodic unction such that χ(y) = 1 or y Y and χ(y) = 0 or y Y s. For a ixed ε > 0 the displacement vector w ε and pressure p ε satisy Lame s system (10) ϱ (0) 2 w ε s t 2 = P (0) s, p ε + c s,0 2 w ε = 0, (11) P (0) s = α (0) D(x, wε ) p ε I λ in the domain Ω (0) or t > 0, and the system (1)-(3) with χ 0 = χ ε (x), ϱ = ϱ ε = ϱ χ ε + ϱ s (1 χ ε ), and α p = α ε p = α p, χ ε + α p,s (1 χ ε ) in the domain Ω or t > 0. On the common boundary S (0) = {x = (x 1, x 2, x 3 ) R 3 : x 3 = H} the displacement vector and normal tensions are continuous: ε (12) lim x Ω (0) x Ω x x 0 w (x, t) = lim x x 0 (0) (13) lim x Ω (0) x Ω x x 0 P (x, t) e 3 = lim x x 0 w ε (x, t), x 0 S (0), P(x, t) e 3, x 0 S (0), where e 3 = (0, 0, 1). The problem is completed with the boundary condition (14) P (0) s e 3 = p 0 (x, t)e 3, x = (x 1, x 2 ) on the outer boundary S = {x = (x 1, x 2, x 3 ) R 3 : x 3 = 0} or t > 0 and homogeneous initial conditions (15) w ε (x, 0) = wε (x, 0) = 0. t Let ς(x) be the characteristic unction o the domain Ω and ϱ ε = (1 ς)ϱ (0) s + ς ϱ ε, α ε p = (1 ς) c 2 s,0 + ς α ε p. Then the above ormulated problem takes the orm (16) ϱ ε 2 w ε t 2 = ((1 ς)p (0) s + ςp ), (17) p ε + α ε p w ε = 0, (18) P = χ ε α µ D(x, w ε t ) + (1 χ ε )α λ D(x, w ε ) ( χ ε α ν p ε t + p ε) I,

6 80 A.MEIRMANOV, S.MUKHAMBETHZANOV, M.NURTAS where in (18) we have used the consequence o (17) in the orm ς χ ε α ν ( w ε t ) = ς χ ε α ν p ε t. Equation (16) is understood in the sense o distributions. That is, or any smooth unctions ϕ with a compact support in Q the ollowing integral identity (19) ( ϱ ε 2 w ε t 2 ϕ + ( (1 ς)p (0) s + ςp ) ) : D(x, ϕ) + (p 0 ϕ) dxdt = 0 Q T holds true. We call such solution the weak solution. In (19) Q T = Q (0, T ) and the convolution A : B o two tensors A = (A ij ) and 3 B = (B ij ) is deined as A : B = tr(a B) = A ij B ji. i,j=1 Using standard methods one can prove that or any positive ε > 0 and given smooth unction p 0 there exists a unique weak solution o the problem (16)-(18) which makes sense to the integral identity (19). We look or the limit o the weak solutions or the ollowing case: (I) µ 0 = λ 0 = λ (0) 0 = 0, µ 1 = λ 1 =, 0 ν 0 <, λ (0) 0 = lim α (0) ε 0 λ. 3. Homogenization: case (I) According to [9], the two-scale expansion or the weak solution o the problem (16)-(18) under conditions (I) has the orm (20) w ε (x, t) = w(x, t) + o(ε), p ε (x, t) = p(x, t) + o(ε), where lim ε 0 o(ε) = 0. The substitution (20) into (19) results in the integral identity (21) Ω T ( (χ( x ε )ϱ + ( 1 χ( x ε )) ) 2 w ϱ s t 2 + (p 0 ϕ)dxdt + Q T Ω (0) T ϕ (χ(x ε )α ν p ) t + p) ϕ dxdt ( ϱ (0) 2 w s t 2 ϕ p ( ϕ)) dxdt = o(ε). Now we use (9) and ater the limit in (21) as ε 0 arrive at the ollowing integral identity (22) Ω T (ˆϱ 2 w t 2 ϕ (m ν 0 where ˆϱ = mϱ + (1 m)ϱ s and m = (23) Next we rewrite (17) as ((1 ς) c 2 s,0 + ς χ( x ε ) p t + p) ϕ) dxdt + (p 0 ϕ)dxdt Q T + Y Ω (0) T ( ϱ (0) 2 w s t 2 ϕ p ( ϕ)) dxdt = 0, χ(y)dy. + ς ( 1 χ( x ε )) ) p ε c s 2 + w ε = 0,

7 SEISMIC IN COMPOSITE MEDIA: ELASTIC AND POROELASTIC COMPONENTS 81 multiply the result by a smooth unction ψ(x, t) with a compact support in Q, and integrate by part over domain Q T : ( (24) ψ ( (1 ς) Q T c s,0 2 + ς χ( x ε ) c 2 + ς ( 1 χ( x ε )) ) p ε c s 2 ψ w ε) dxdt = 0. As above, we substitute (20) into (24) and pass to the limit as ε 0: ( (25) ψ ( (1 ς) Q T c s,0 2 + ς m ς (1 m) ) ) c 2 + p ψ w c s 2 dxdt = 0. Integral identities (22) and (25), completed with initial conditions (26) w(x, 0) = w (x, 0) = 0, t orm mathematical model (I) o seismic in composite media. In act, these identities contain the dierential equations in Ω and Ω (0) and the boundary conditions on S and S (0). Let ϕ be a smooth unction with a compact support in Ω (0). Rewriting (22) as (ϱ (0) 2 w s + p) ϕdxdt = 0 t2 Ω (0) T and using the arbitrary choice o ϕ we conclude that (27) ϱ (0) 2 w s t 2 = p in the domain Ω (0) or t > 0. For unctions ϕ with a compact support in Ω (22) implies (28) ˆϱ 2 w t 2 = (p + m ν 0 p c 2 t ), ˆϱ = mϱ + (1 m)ϱ s in the domain Ω or t > 0. Now, i we choose ϕ = (ϕ 1, ϕ 2, ϕ 3 ) with a compact support in Q and ϕ 3 (x, t) 0 or x S (0), then the integration by parts in (22) together with (27) and (28) result in where Thereore S (0) T ( p (p + + m ν 0 p + c 2 t )) ϕ 3 dxdt = 0, p (x 1, x 2, t) = p(x 1, x 2, H 0, t), p + (x 1, x 2, t) = p(x 1, x 2, H + 0, t). (29) lim x Ω (0) x Ω x x 0 p(x, t) = lim x x 0 ( ν 0 p p(x, t) + m c 2 t (x, t)), x 0 S (0). Finally, or unctions ϕ with a compact support in Ω (0) and ϕ 3 (x, t) 0 or x S the integration by parts in (22) together with (27) result in S T (p p 0 )ϕ 3 dxdt = 0, or (30) p(x, t) = p 0 (x, t), x S.

8 82 A.MEIRMANOV, S.MUKHAMBETHZANOV, M.NURTAS In the same way as above, it can be shown that (25) implies continuity equations 1 (31) 2 p+ w =0 c s,0 and m (1 m) + p+ w =0 2 c c s2 (32) in the domains Ω(0) and Ω respectively, and the boundary condition (33) lim x x0 x Ω(0) e3 w(x, t) = lim x x0 x Ω e3 w(x, t), x0 S (0) on the common boundary S (0). Dierential equations (27), (28), (31), and (32), boundary conditions (29), (30), and (33), and initial conditions (26) constitute the mathematical model (I) in its dierential orm. 4. One dimensional model or the case (I): numerical implementations 4.1. Direct problem. For the sake o simplicity we consider the space, which consists o the ollowing subdomains: Ω1 = {x R : 0 < x < H1 }, Ω2 = {x R : H1 < x < H2 }, and Ω3 = {x R : x > H2 }. Dierential equations (27), (28), (31), and (32) result in 1 ν0 p 1 2p = div (p + m 2 ) c 2 (x) t2 ρ (x) c t m (1 m) 1 and ρ = mρ + (1 m)ρs are respectively average wave = 2+ c 2 c c s2 propagation velocity and average density o the medium. Applying now the Fourier transormation we arrive at where (34) ρ ω 2 d2 P + P = 0 0 dx 2 (1 mν iω)c 2 c2 where P (x, ω)-the pressure obtained ater Fourier transorm. Figure 2. Scheme o arrangement o layers

9 SEISMIC IN COMPOSITE MEDIA: ELASTIC AND POROELASTIC COMPONENTS 83 Depending on the exact physical properties, the sedimentary rock zone is divided into three subdomains. The value o the geometry o pores, viscosity o luid, density o rock, and velocity o seismic wave considered in each layers to be dierent. In the experiment in order to get numerical solution, it s assumed that the irst medium is a shale, the second medium is oil saturated sandstone, and the third medium is a limestone (see Fig.2). Parameter medium Density (kg/m 3 ) Elastic wave velocity (m/s) Oil Shale Sandstone Limestone Table 1. The average values o density and sound velocity. Let us suppose that there is a plane wave which propagates rom. Then the general solution o equation (34) or < X H 1 in the case ν 0 = 0 is written down as: { } { } iω ˆρ1 iω ˆρ1 (35) ˆP1 = exp x + A 2 exp x. ĉ 1 ĉ 1 The general solution o equation (34) or H 1 x < H 2 in the case ν 0 > 0 is represented as: iω ˆρ 2 iω ˆρ (36) ˆP2 = B 1 exp ĉ 2 1 mν x + B 2 exp 2 0 c 2 iω ĉ 2 1 mν x. 0 c 2 iω Finally the general solution or x H 2 in the case ν 0 = 0 will be the ollowing: { } ˆρ3 (37) ˆP3 = D 1 exp iω x. Continuity condition in contact media will be: (38) [ ˆP 1 iω mν 0 ĉ 3 ˆP 1 ] H1 0 = [ ˆP 2 iω mν 0 ˆP 2 ] H1+0 (39) ĉ 2 d 1 dx [ ˆP 1 iω mν 0 ˆP 1 ] H1 0 = ĉ 2 d 2 dx [ ˆP 2 iω mν 0 ˆP 2 ] H1+0 (40) [ ˆP 2 iω mν 0 ˆP 2 ] H2 0 = [ ˆP 3 iω mν 0 ˆP 3 ] H2+0 (41) ĉ 2 d 2 dx [ ˆP 2 iω mν 0 ˆP 2 ] H2 0 = ĉ 2 d 3 dx [ ˆP 3 iω mν 0 ˆP 3 ] H2+0 These relations are nothing else but the system o linear algebraic equations or the coeicients A 2, B 1, B 2, D 1 which can be easily resolved by any direct method. These coeicients are used in order to construct the solution in time requency domain

10 84 A.MEIRMANOV, S.MUKHAMBETHZANOV, M.NURTAS and ater inverse Fourier transorm in time the solution in the time domain can be easily recovered (see Fig.3). Figure 3. Propagation o seismic waves in dierent layers 4.2. Inverse problem. In inverse problem [15] except ˆP (x, ω) the values H 1, H 2, ĉ 2, ν 0, m are unknown as well. To determine these values one needs some additional inormation about solution o the direct problem - data o inverse problem. Usually they are given as unction P (ω) at X = 0. The most widespread way is to search or these values by minimization o the data misit unctional being L 2 norm o the dierence o given unctions and computed or some current values o unknown parameters. During the numerical simulations, we use the ollowing list o inormation. For these data, the characteristic dimension o depth L 3000m, characteristic time τ 0 3sec, α τ 33, the porosity o the sandstone m 0.007, average size o pores l 30micron, then ε 10 10, µ P a.s, ν 0 3.8cCt = m 2 /s, λ P a. Thereore Coeicients corresponding values(dimensionless) approximate values(by ε) The irst layer is shale α p,s 3.63 ε 1/10 The second layer is sandstone(oil containing area) α p,s ε 2/10 α p, 1.52 ε 1/10 α ν ε 6/5 α µ ε 6/5 µ ε 4/5 α λ 0.22 ε 1/5 λ ε 2 The third layer is limestone α p,s ε 2/10 Table 2. The values o the coeicients used in practice.

11 SEISMIC IN COMPOSITE MEDIA: ELASTIC AND POROELASTIC COMPONENTS 85 (42) F i (H i 1, H i 2) = (43) F i (H i 1, ĉ i 2) = (44) F i (H i 2, ĉ i 2) = ωn ω 1 ˆP i (ω, H i 1, H i 2) P (ω, H 1, H 2 ) 2 dω 0 ωn ω 1 ˆP i (ω, H i 1, ĉ i 2) P (ω, H 1, ĉ 2 ) 2 dω 0 ωn ω 1 ˆP i (ω, H i 2, ĉ i 2) P (ω, H 2, ĉ 2 ) 2 dω 0 Here P (ω,...,...) is the given wave ields at X = 0, while ˆP (ω,...,...) are wave ields computed or some current values o the desired parameters. In our numerical experiments the minimum is searched by the Nelder-Mead technique [17] Recovery o H 1 and H 2. Behavior o the data misit unctional or this statement is represented in Fig.4 and Fig.5. As one can see this unctional is convex and has the unique minimum point. Thereore this inverse problem is well resolved. Figure 4. Minimization o the unctional F (H 1, H 2 ). Figure 5. Level line o the unctional F (H 1, H 2 ) Recovery o H 1 and c 2. Now we come to the non convex unctional and thereore inverse problem may have ew solutions (see Fig.6, Fig.7).

12 86 A.MEIRMANOV, S.MUKHAMBETHZANOV, M.NURTAS Figure 6. Minimization o the unctional F (H 1, ĉ 2 ). Figure 7. Level line unctional F (H 1, ĉ 2 ) Recovery o H 2 and c 2. This statement also generates non convex unctional, but now it has excellent resolution with respect to H 2 (see Fig.8, Fig.9). Figure 8. Minimization o the unctional F (H 2, ĉ 2 ).

13 SEISMIC IN COMPOSITE MEDIA: ELASTIC AND POROELASTIC COMPONENTS 87 Figure 9. Level line unctional F (H 2, ĉ 2 ). Conclusions. In this publication we have shown how to derive mathematical models or composite media using its microstructure. As a rule, there is some set o models depending on given criteria µ 0, λ 0,... o the physical process in consideration. For a ixed set o criteria the corresponding model describes some o the main eatures o the process. In the paper the simplest inverse problem was dealt with - recovery o elastic parameters o the layer by Nelder-Mead algorithm. In the uture we are planning to establish connection upscaling procedure and scattered waves and apply on this base recent developments o true-amplitude imaging on the base o Gaussian beams or both relected and scattered waves ([18], [19]. Conlict o Interests. The authors declare that there is no conlict o interests regarding the publication o this manuscript. Reerences [1] M. Biot, Theory o propagation o elastic waves in a luid-saturated porous solid. I. Lowrequency range, Journal o the Acoustical Society o America,28 (1955), [2] M. Biot, Theory o propagation o elastic waves in a luid-saturated porous solid. II. Higher requency range, Journal o the Acoustical Society o America, 28 (1955), [3] M. Biot, Generalized theory o seismic propagation in porous dissipative media, J. Acoust. Soc. Am., 34 (1962), [4] J. G. Berryman, Seismic wave attenuation in luid-saturated porous media, J. Pure Appl. Geophys. (PAGEOPH) 128 (1988), [5] J. G. Berryman, Eective medium theories or multicomponent poroelastic composites, ASCE Journal o Engineering Mechanics, 132:5 (2006), [6] R. Burridge and J. B. Keller, Poroelasticity equations derived rom microstructure, Journal o Acoustic Society o America, 70:4 (1981), Zbl [7] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, 129, Springer-Verlag, (1980). Zbl [8] A. Meirmanov, Mathematical models or poroelastic lows, Atlantis Press, Paris, [9] A. Meirmanov, A description o seismic seismic wave propagation in porous media via homogenization, SIAM J. Math. Anal. 40:3 (2008), Zbl [10] A. Mikelić, On the Eective Boundary Conditions Between Dierent Flow Regions. Proceedings o the 1. Conerence on Applied Mathematics and Computation Dubrovnik, Croatia, September (1999),

14 88 A.MEIRMANOV, S.MUKHAMBETHZANOV, M.NURTAS [11] A. Bensoussan, J. L. Lions, and G. Papanicolau, Asymptotic Analysis or Periodic Structures, North-Holland, Amsterdam, (1978). Zbl [12] N. Bakhvalov and G. Panasenko, Homogenization: averaging processes in periodic media, Math. Appl., 36, Kluwer Academic Publishers, Dordrecht, (1990). [13] G. Nguetseng, A general convergence result or a unctional related to the theory o homogenization, SIAM J. Math. Anal. 20 (1989), Zbl [14] G. Nguetseng, Asymptotic analysis or a sti variational problem arising in mechanics, SIAM J. Math. Anal., 21 (1990), Zbl [15] A. Fichtner, Full Seismic Waveorm Modelling and Inversion, Springer- Verlag Berlin Heidelberg, (2011), [16] Protasov M.I., Reshetova G.V., Tcheverda V.A., Fracture detection by Gaussian beam imaging o seismic data and image spectrum analysis, Geophysical Prospecting, in-press (2015). [17] J.H.Mathews, K.D.Fink, Numerical methods using Matlab, 4th edition, Prentice-Hall Inc., chapter 8, (2004), , [18] Protasov M.I., Tcheverda V.A., True amplitude imaging by inverse generalized Radon transorm based on Gaussian beam decomposition o the acoustic Green s unction, Geophysical Prospecting, 59:2 (2011), [19] Protasov M.I., Tcheverda V.A. True-amplitude elastic Gaussian beam imaging o multicomponent walk-away VSP data, Geophysical Prospecting, 60:6 (2012), Anvarbek Mukatovich Meirmanov Al-Farabi Kazakh National University, Ave. Al-Farabi, 71, , Almaty, Kazakhstan address: anvarbek@list.ru Saltanbek Talapedenovich Mukhambetzhanov Al-Farabi Kazakh National University, Ave. Al-Farabi, 71, , Almaty, Kazakhstan address: Marat Nurtas Kazakh-British Technical University, Str. Tole bi, 59, , Almaty, Kazakhstan address: maratnurtas@gmail.com