MATHEMATICAL MODELS OF SEISMICS IN COMPOSITE MEDIA: ELASTIC AND PORO-ELASTIC COMPONENTS

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1 Electronic Journal of Differential Equation, Vol ), No. 184, pp ISSN: URL: or MATHEMATICAL MODELS OF SEISMICS IN COMPOSITE MEDIA: ELASTIC AND PORO-ELASTIC COMPONENTS ANVARBEK MEIRMANOV, MARAT NURTAS Abtract. In the preent paper we conider elatic and poroelatic media having a common interface. We derive the macrocopic mathematical model for eimic wave propagation through thee two different media a a homogenization of the exact mathematical model at the microcopic level. They conit of eimic equation for each component and boundary condition at the common interface, which eparate different media. To do thi we ue the two-cale expanion method in the correponding integral identitie, defining the weak olution. We illutrate our reult with the numerical implementation of the invere problem for the implet model. 1. Introduction Thi article i devoted to a decription of eimic wave propagation in compoite media Q R 3, coniting of the elatic medium Ω 0), poroelatic medium Ω, which i perforated by a periodic ytem of pore filled with a fluid, and common interface S 0) between Ω 0) and Ω ee Figure 1, 2). That i, Q = Ω S 0) Ω 0) and Ω = Ω f Γ Ω, where Ω i a olid keleton, Ω f i a pore pace liquid domain), and Γ i a common boundary olid keleton-liquid domain. The tructure of the heterogeneou medium Q i too complicated and make hard a numerical imulation of eimic wave propagation in multicale media. The main difficulty here i a preence of both component olid and liquid) in each ufficiently mall ubdomain of Q. It require to change the governing equation from Lame equation to the Stoke equation) at the cale of ome ten micron. There are two baic method to decribe phyical procee in uch media: the phenomenological method and the aymptotical one which i baed on the upcaling approache. The phenomenological approach for wave propagation through a poroelatic medium [4, 5] lead, in particular, to Biot model [1]-[3]. It baed on the ytem of axiom relation between the parameter of the medium), which define the given phyical proce. But, there can be another ytem of axiom defining the ame proce. Thu, it i neceary chooe the correct authenticity criterion of the mathematical decription of the proce. It can be, for example, the phyical experiment. A a rule, each phenomenological model contain ome 2010 Mathematic Subject Claification. 35B27, 46E35, 76R99. Key word and phrae. Acoutic; two-cale expanion method; full wave field inverion; numerical imulation. c 2016 Texa State Univerity. Submitted May 28, Publihed July 12,

2 2 A. MEIRMANOV, M. NURTAS EJDE-2016/184 et of phenomenological contant. Therefore, one can achieve agreement between the uggeted theory and elected erie of experiment changing thee parameter. Figure 1. Domain in conideration The econd method, uggeted by Burridge and Keller [6] and Sanchez-Palencia [7], baed on the homogenization. It conit of: 1) an exact decription of the proce at the microcopic level baed on the fundamental law of continuum mechanic,and 2) the rigorou homogenization of the obtained mathematical model. To explain the method we conider a characteritic function χ 0 x) of the pore pace Ω f. Let L i the characteritic ize of the phyical domain in conideration, τ i the characteritic time of the phyical proce, ρ 0 i the mean denity of water, and g i acceleration due gravity. In dimenionle variable x x L, w α w τ L, t t τ, F F g, ρ ρ ρ 0, the dynamic ytem for the diplacement w and preure p of the medium take the form [6, 7, 8]: ϱ 2 w = P + ϱf, 2 1.1) P = χ 0 α µ Dx, w ) + 1 χ 0)α λ Dx, w) + χ 0 α ν w ) p) I, 1.2) p + α p w = ) Equation 1.1)-1.3) are undertood in the ene of ditribution a correponding integral identitie. They are equivalent to the Stoke equation for the velocity v = w v ϱ f = P p f + ϱ f F, + α p,f v = 0, 1.4) P f = α µ Dx, v) + α ν v) p ) I 1.5) and preure p in the pore pace Ω f and the Lame equation ϱ 2 w 2 = P + ϱ F, p + α p, w = 0, 1.6) P = α λ Dx, w) pi 1.7) for the olid diplacement w and preure p in Ω. At the common boundary Γ velocitie and normal tenion are continuou: w = v, P n = P f n. 1.8) Here n i a unit normal to Γ.

3 EJDE-2016/184 MATHEMATICAL MODELS OF SEISMICS 3 In 1.1)-1.8), Dx, u) = 1 2 u + u ) i the ymmetric part of u, I i a unit tenor, F i a given vector of ditributed ma force, α p = α p,f χ 0 + α p, 1 χ 0 ), ϱ = ϱ f χ 0 + ϱ 1 χ 0 ), α τ = α ν = L gτ 2, α µ = 2ν α τ τlg ρ 0, 2µ α τ τlg ρ 0, α λ = α p,f = ϱ f c 2 f α τ Lg, 2λ α τ Lg ρ 0, α p, = ϱ c 2 α τ Lg, where µ i the dynamic vicoity, ν i the bulk vicoity, λ i the elatic contant, ϱ f and ϱ are the repective mean dimenionle denitie of the liquid in pore and the olid keleton, correlated with the mean denity of water ρ 0, and c f and c are the peed of compreion ound wave in the pore liquid and in the olid keleton repectively. Figure 2. The pore tructure The mathematical model 1.1) 1.3) can not be ueful for practical need, ince the function χ 0 change it value from 0 to 1 on the cale of a few micron. Fortunately, the ytem poee a natural mall parameter ε = l L, where l i the average ize of pore. Thu, the mot uitable way to get a practically ignificant mathematical model, which approximate 1.1) 1.3), i a homogenization or upcaling. That i, we uppoe the ε-periodicity of the olid keleton, let ε to be variable, and look for the limit in 1.1) 1.3) a ε 0. There are different homogenized limiting) ytem, depending on of α µ, α λ,... Some of thee number might be mall and ome might be large. We may repreent them a a power of ε, or a function depending on ε. Let µ 0 = lim ε 0 α µ ε), c 2 f,0 = lim ε 0 α p,f 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 2,0 = lim ε 0 α p, ε), λ α λ 1 = lim ε 0 ε 2. It i clear that the choice of thee limit depend on our willing. For example, for ε = 10 2 and α = we may tate that α = 2 ε 1 2, or α = 0.02 ε 0. It i

4 4 A. MEIRMANOV, M. NURTAS EJDE-2016/184 uual procedure when we neglect ome term in differential equation with mall coefficient and get more imple equation, till decribing the phyical proce. The detailed analye of all poible limiting regime ha been done in [8, 9]. To decribe the eimic in two different media elatic and poroelatic), having a common interface we mut choe one of the two method dicued above. The firt method ugget only ome guee, while the econd method ha a clear algorithm for the derivation of the boundary condition. That i why we chooe here the econd method. We derive new eimic equation in each component elatic and poroelatic) and the boundary condition on the common boundary. For thee boundary condition the very little i known and only for the liquid filtration ee for example [10]). For three different et of µ 0, λ 0,... for each component we derive three different mathematical model, which decribe the proce with different degree of approximation. We tart with the integral identitie, defining the weak olution w ε and p ε, and ue the two-cale expanion method [11, 12], when we look for the olution in the form w ε x, t) = wx, t) + W 0 x, t, x ε ) + ε W 1x, t, x ε ) + oε), p ε x, t) = px, t) + P 0 x, t, x ε ) + ε P 1x, t, x ε ) + oε) with 1-periodic in the variable y function W i x, t, y), P i x, t, y), i = 0, 1,... Thi method i rather heuritic and may lead to the wrong anwer. But our guee are baed upon the trong theory, uggeted by G. Ngueteng [13, 14]. For the rigorou derivation of eimic equation in poroelatic media, which dictate the correct two-cale expanion, ee [8]. Finally, to calculate limit a ε 0 in correponding integral identitie, we apply the well-known reult lim F x, x ε 0 Ω ε, t) dx dt = F x, y, t)dy ) dx dt 1.9) Ω Y for any mooth 1-periodic in the variable y Y function F x, y, t). 2. Statement of the problem For the ake of implicity we uppoe 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,f =, α p, = c 2. Let Y be a unit cube in R 3, Y = Y f γ Y. We aume that pore pace Ω ε f i a periodic repetition in Ω of the elementary cell εy f, the olid keleton Ω ε i a periodic repetition in Ω of the elementary cell εy, and the boundary Γ ε between a pore pace and a olid keleton i a periodic repetition in Ω of the boundary εγ. Detailed decription of the et Y f and Y i done in [8]. From thee uppoition, χ 0 x) = χ ε x) = χ x ε ), where χy) i a 1-periodic function uch that χy) = 1 for y Y f and χy) = 0 for y Y.

5 EJDE-2016/184 MATHEMATICAL MODELS OF SEISMICS 5 For a fixed ε > 0 the diplacement vector w ε and preure p ε atify Lame ytem ϱ 0) 2 w ε 2 = P 0), p ε + c 2,0 w ε = 0, 2.1) P 0) = α 0) λ Dx, wε ) p ε I 2.2) in the domain Ω 0) for t > 0, and the ytem 1.1)-1.3) with χ 0 = χ ε x), ϱ = ϱ ε = ϱ f χ ε + ϱ 1 χ ε ), and α p = α ε p = α p,f χ ε + α p, 1 χ ε ) in the domain Ω for t > 0. On the common boundary S 0) = {x = x 1, x 2, x 3 ) R 3 : x 3 = H} the diplacement vector and normal tenion are continuou: lim w ε x, t) = lim w ε x, t), x 0 S 0), 2.3) x Ω 0) x Ω lim P 0) x, t) e 3 = lim Px, t) e 3,, x 0 S 0), 2.4) x Ω 0) x Ω where e 3 = 0, 0, 1). The problem i complemented with the boundary condition P 0) e 3 = p 0 x, t)e 3, x = x 1, x 2 ) 2.5) on the outer boundary S = {x = x 1, x 2, x 3 ) R 3 : x 3 = 0} for t > 0 and homogeneou initial condition w ε x, 0) = wε x, 0) = ) Let ςx) be the characteritic function of the domain Ω and ϱ ε = 1 ς)ϱ 0) + ςϱ ε, α ε p = 1 ς) c 2,0 + ς α ε p. Then the above formulated problem take the form ϱ ε 2 w ε 2 = 1 ς)p 0) + ςp ), 2.7) p ε + α p ε w ε = 0, 2.8) P = χ ε α µ Dx, wε ) + 1 χε )α λ Dx, w ε ) χ ε α ν p ε + pε) I, 2.9) where in 2.9) we have ued the conequence of 2.8) in the form ς χ ε α ν wε ) = ς α χε ν p ε. Equation 2.7) i undertood in the ene of ditribution. That i, for any mooth function ϕ with a compact upport in Q the following integral identity ϱ ε 2 w ε 2 ϕ + 1 ς)p 0) + ςp ) ) : Dx, ϕ) + p 0 ϕ) dx dt = ) hold. We call uch olution the weak olution. In 2.10) = Q 0, T ) and the convolution A : B of two tenor A = A ij ) and B = B ij ) i defined a A : B = tra B) = 3 i,j=1 A ijb ji. Uing tandard method one can prove that for any poitive ε > 0 and given mooth function p 0 there exit a unique weak olution of the problem 2.7)-2.9)

6 6 A. MEIRMANOV, M. NURTAS EJDE-2016/184 which make ene to the integral identity 2.10). We look for the limit of the weak olution for the following cae: I) µ 0 = λ 0 = λ 0) 0 = 0, µ 1 = λ 1 =, 0 ν 0 <, λ 0) 0 = lim ε 0 α 0) λ ; II) µ 0 = λ 0 = λ 0) 0 = µ 1 = ν 0 = 0, λ 1 = ; III) µ 0 = ν 0 = 0, 0 < λ 0, λ 0) 0, µ 1 <. 3. Homogenization: cae I) According to [9], the two-cale expanion for the weak olution of the problem 2.7)-2.9) under condition I) ha the form w ε x, t) = wx, t) + oε), p ε x, t) = px, t) + oε), 3.1) where lim ε 0 oε) = 0. The ubtitution 3.1) into 2.10) reult in the integral identity χ x ε )ϱ f + 1 χ x ε )) ) 2 w ϱ 2 ϕ χx ε )α ν p ) + p) ϕ dx dt + p 0 ϕ) dx dt + Ω 0) T Ω 0) T ϱ 0) 2 w 2 ϕ p ϕ)) dx dt = oε). 3.2) Now we ue 1.9) and after the limit in 3.2) a ε 0 arrive at the integral identity ˆϱ 2 w 2 ϕ mν 0 p + p) ϕ) dx dt + p 0 ϕ) dx dt + ϱ 0) 2 3.3) w 2 ϕ p ϕ)) dx dt = 0, where ˆϱ = mϱ f + 1 m)ϱ and m = Y χy)dy. Next we rewrite 2.8) a 1 ς) c 2,0 + ς χ x ε ) + ς 1 χ x ε )) ) p ε c 2 + w ε = ) We multiply the reult by a mooth function ψx, t) with a compact upport in Q, and integrate by part over domain : ψ 1 ς) c 2 + ς χ x ε ),0 + ς 1 χ x ε )) ) p ε c 2 ψ w ε) dx dt = ) A above, we ubtitute 3.1) into 3.5) and pa to the limit a ε 0: ψ 1 ς) c 2 + ς m ς1 m) ) ),0 c 2 + p ψ w f c 2 dx dt = ) Integral identitie 3.3) and 3.6), complemented with initial condition wx, 0) = w x, 0) = 0, 3.7) form mathematical model I) of eimic in compoite media. In fact, thee identitie contain the differential equation in Ω and Ω 0) and the boundary condition on S and S 0).

7 EJDE-2016/184 MATHEMATICAL MODELS OF SEISMICS 7 Let ϕ be a mooth function with a compact upport in Ω 0). Rewriting 3.3) a ϱ 0) 2 w + p) ϕ dx dt = 0 2 Ω 0) T and uing the arbitrary choice of ϕ we conclude that ϱ 0) 2 w = p 3.8) 2 in the domain Ω 0) for t > 0. For function ϕ with a compact upport in Ω, 3.3) implie ˆϱ 2 w 2 = p + mν 0 p ), ˆϱ = mϱ f + 1 m)ϱ 3.9) in the domain Ω for t > 0. Now, if we chooe ϕ = ϕ 1, ϕ 2, ϕ 3 ) with a compact upport in Q and ϕ 3 x, t) 0 for x S 0), then the integration by part in 3.3) together with 3.8) and 3.9) reult in where Therefore, S 0) T p x 1, x 2, t) = px 1, x 2, H 0, t), lim px, t) = lim x Ω 0) x Ω p p + + m ν 0 p + )) ϕ 3 dx dt = 0, p + x 1, x 2, t) = px 1, x 2, H + 0, t). ν 0 p px, t) + m x, t)), x 0 S 0). 3.10) Finally, for function ϕ with a compact upport in Ω 0) and ϕ 3 x, t) 0 for x S the integration by part in 3.3) together with 3.8) reult in S T p p 0 )ϕ 3 dx dt = 0, or px, t) = p 0 x, t), x S. 3.11) In the ame way a above, it can be hown that 3.6) implie continuity equation and m + 1 c 2,0 p + w = ) 1 m) ) p + w = ) c 2 in the domain Ω 0) and Ω repectively, and the boundary condition lim e 3 wx, t) = lim e 3 wx, t), x 0 S 0) 3.14) x Ω 0) x Ω on the common boundary S 0). Differential equation 3.8), 3.9), 3.12), and 3.13), boundary condition 3.10), 3.11), and 3.14), and initial condition 3.7) contitute the mathematical model I) in it differential form.

8 8 A. MEIRMANOV, M. NURTAS EJDE-2016/184 For thi cae we put 4. Homogenization: cae II) w ε x, t) = 1 ς)wx, t) + ςχ x ε )wf,ε) x, t) + ς 1 χ x ε )) w x, t) + oε), p ε x, t) = px, t) + oε), where w f,ε) x, t) = W f) x, t, x ε ), 4.1) and W f) x, t, y) i a 1-periodic in the variable y function. The ubtitution 4.1) into 2.10) reult in the integral identity p 0 ϕ) dx dt + ϱ 0) 2 w Ω 0) 2 ϕ p ϕ)) dx dt T ϱf + χ x W f) ε ) 2 2 x, t, x ε ) + ϱ x 1 χ ε )) 2 w ) ) ϕ p ϕ) 2 dx dt xε wf,ε) = α µ χ )Dx, ) : Dx, ϕ) dx dt + oε), ΩT which hold for any mooth function ϕx, t). Let w f) x, t) = ς lim ε 0 χ x ε )Wf) x, t, x ε ) = ς Y f W f) x, t, y)dy 4.2) be the weak limit of the equence {w ε }. Then after the limit a ε 0 we arrive at the integral identity p 0 ϕ) dx dt + ϱf 2 w f) = 2 Ω 0) T ϱ 0) 2 w 2 ϕ p ϕ)) dx dt + ϱ 1 m) 2 w 2 ) ϕ p ϕ) ) dx dt = ) Note that the term α µ Dx, wf,ε) ) in the right-hand ide of 4.2) converge to zero α due to the uppoition lim ε 0 α µ = lim µ ε 0 ε = 0: α µ D x, wf,ε) x, t) ) = α µ D x, Wf) x, t, x ε )) + α µ ε D y, Wf) x, t, x ε )). The ubtitution of 4.1) into the continuity equation 2.8) lead to the integral identity ψ 1 ς) c 2 + ς χε,0 + ς1 χε )) p c 2 ψ 1 ς)w + ςχ ε W f) x, t, x ε ) + ς1 χε )w ) ) dx dt = oε). 4.4) The limit here a ε 0 reult in ψ 1 ς) c 2 + ς m,0 c 2 + f ς1 m) ) p+ c 2 ψ 1 ς)w + ς w f) + ς1 m)w ) ) dx dt = )

9 EJDE-2016/184 MATHEMATICAL MODELS OF SEISMICS 9 A in previou ection we conclude that integral identitie 4.3) and 4.5) imply differential equation in Ω 0) and Ω and boundary condition on the boundarie S 0) and S. Namely, in the domain Ω 0) the diplacement vector w and preure p of the olid component atify the eimic ytem ϱ 0) 1 c 2,0 2 w = p, 4.6) 2 p + w = ) In the domain Ω the diplacement vector w of the olid component, diplacement vector w f) of the liquid component, and preure p of the medium atify the eimic ytem 2 w f) ϱ f 2 + ϱ 1 m) 2 w 2 = p, 4.8) m 1 m) ) + p + w f) ) c m)w = ) On the common boundary S 0) the diplacement vector w, w, and w f) and preure p atify continuity condition lim px, t) = lim px, t), 4.10) x Ω 0) x Ω lim e 3 wx, t) = lim e 3 w f) x, t) + 1 m)w x, t) ). 4.11) x Ω 0) x Ω Finally, on the outer boundary S, A above, we have to add the initial condition: px, t) = p 0 x, t). 4.12) wx, 0) = w x, 0) = w x, 0) = w x, 0) 4.13) = w f) x, 0) = wf) x, 0) = 0. The obtained ytem of differential equation and boundary and initial condition i till incomplete. We have no differential equation for the liquid diplacement w f). To find the miing equation we pa to the limit ε 0 in 4.2) with tet function ϕ ε in the form ϕ ε x, t) = hx, t)ϕ 0 x ε ), where h i a mooth function with a compact upport in Ω and ϕ 0 y) i a mooth function with a compact upport in Y f that i ϕ ε vanihe outide of the pore pace Ω f ). For an arbitrary function ϕ 0 y) the term p ϕ ε become unbounded a ε 0: ϕ ε = x hx, t) ) ϕ 0 x ε ) + 1 ε hx, t) y ϕ 0 x ε )). Therefore, we require that condition y ϕ 0 y) = 0, y Y f, 4.14)

10 10 A. MEIRMANOV, M. NURTAS EJDE-2016/184 ϕ 0 y) = 0, y γ 4.15) hold. The term α µ Dx, wf,ε) ) : Dx, ϕ ε ) in the right-hand ide of 4.2) converge to α zero becaue of the aumption lim ε 0 α µ = lim µ ε 0 ε = lim ε 0 αµ ε = 0: 2 α µ D x, wf,ε) x, t) ) : Dx, ϕ ε ) = α µ D x, Wf) x, t, x ε )) + 1 ε D y, Wf) x, t, x ε ))) : 1 h) ϕ0 + ϕ 0 h) ) + h 2 ε D y, ϕ 0 x ε ))) = oε). Here a matrix a b i defined a a b) c = ab c), for any vector a, b, and c. Thu, the limit a ε 0 in 4.2) reult in the integral identity Yf 2 W f) ϱf 2 h p h) ) ) ϕ 0 dy dx dt Yf 2 W f) = hx, t) ϱf 2 + p ) ) ϕ 0 dy dx dt = 0, 4.16) which hold for any mooth function hx, t) with a compact upport in Ω and for any mooth olenoidal function ϕ 0 y) with a compact upport in Y f. By arbitrary choice of hx, t), 4.16) implie Yf ϱf 2 W f) 2 + p ) ϕ 0 dy = ) Thi identity mean that the function ϱ 2 W f) f + p ) i orthogonal to any olenoidal 2 function. Therefore there exit ome 1-periodic in the variable y function Πx, t, y) uch that 2 W f) ϱ f 2 + p = y Π 4.18) in the domain Y f for any parameter x, t). There i one equation 4.18) for two unknown function W f) and Π. To derive the econd equation we put in 4.4) ψ = ε hx, t) ψ 0 x ε ) with arbitrary mooth function hx, t) and arbitrary mooth 1-periodic function ψ 0 y) and pa to the limit a ε 0: ) hx, t) χy) ψ 0 y) W f) x, t, y)dy dx = 0. Y After reintegration we obtain the deired microcopic continuity equation W f) = 0, y Y f. 4.19) A rigorou theory ee [13, 8, 9]) upplie the ytem 4.18), 4.19) with the boundary condition W f) x, t, y) w x, t) ) ny) = )

11 EJDE-2016/184 MATHEMATICAL MODELS OF SEISMICS 11 on the boundary γ with the unit normal ny), and the homogeneou initial condition Problem 4.18)-4.21) ha been olved in [9]: W f) x, 0, y) = Wf) x, 0, y) = ) ϱ f 2 W f) 2 = ϱ f 2 2 I i=1 y Π i e i ) p + ϱf 2 2 ), 4.22) where Π i y), i = 1, 2, 3 are olution to the periodic boundary value problem Thu, y Π i = 0, y Y f, y Π i e i ) ny) = 0, y γ. where ϱ f 2 w f) 2 = mϱ f 2 2 B f) 2 = mi i=1 B f) 2 p + ϱ f 2 2 ), 4.23) Y f y Π f) i dy e i. 4.24) Differential equation 4.6)-4.9), 4.23), boundary condition 4.10)-4.12), and initial condition 4.13) form the mathematical model II) of eimic in compoite media. 5. Homogenization: cae III) According to [8] the et of criteria III) dictate the form of the two-cale expanion: w ε x, t) = 1 ς)wx, t) + ς χ x ε )wf,ε) x, t) + ς 1 χ x ε )) w x, t) + εw ε x, t) ) + oε), p ε x, t) = 1 ς)px, t) + ς χ x ε )p f x, t) + ς 1 χ x ε )) p ε x, t) + oε), 5.1) where w f,ε) x, t) = W f) x, t, x ε ), wε x, t) = W x, t, x ε ), pε x, t) = P x, t, x ε ), and W f) x, t, y), W x, t, y), P x, t, y) are 1-periodic in the variable y function. Next we expre the preure p ε in the olid component in Ω uing the continuity equation 2.8) and two-cale expanion 5.1): p ε x, t) = c 2 w x, t) + y W x, t, x ε )) + oε). 5.2)

12 12 A. MEIRMANOV, M. NURTAS EJDE-2016/184 The ubtitution of 5.1) and 5.2) into 2.10) reult in the integral equality p 0 ϕ) dx dt + ϱ 0) 2 w Ω 0) 2 ϕ + λ 0) 0 Dx, w) pi) : Dx, ϕ) ) dx dt T ϱf + χ x W f) ε ) 2 2 x, t, x ε ) + ϱ x 1 χ ε )) 2 w ) ) ϕ 2 dx dt x + λ 0 1 χ ε )) N 0) : Dx, w ) + D y, W x, t, x ε )))) : Dx, ϕ) dx dt = χ x ε ) αµ ε D y, Wf) x, t, x ) ε )) p f I : Dx, ϕ) dx dt + oε), 5.3) which hold for any mooth function ϕx, t). In 5.3) N 0) = J ij J ij + c2 I I, J ij = 1 ) ei e j + e j e i, λ 0 2 i,j=1 {e 1, e 2, e 3 } i a tandard Carteian bai, and the fourth-rank tenor A B i the tenor direct) product of the econd-rank tenor A and B: A B) : C = AB : C) for any econd-rank tenor C. After the limit in 5.3) a ε 0 we arrive at the integral identity where Ω 0) T ϱ 0) 2 w 2 ϕ + λ 0) 0 Dx, w) pi) : Dx, ϕ) ) dx dt ϱf 2 w f) + p 0 ϕ) dx dt ϱ 1 m) 2 w ) ) ϕ 2 dx dt + λ 0 N 0) : ) ) 1 m)dx, w ) + Dy, W ) Y : Dx, ϕ) dx dt Ω T = mpf I ) : Dx, ϕ) dx dt, w f) = W f) Yf, F A = A F y)dy, A Y. 5.4) To pa to the limit a ε 0 in the continuity equation 2.8) we rewrite it a an integral identity and ue the repreentation 5.1): 1 ς) ψ c 2 p + ς χ x,0 ε )p f c 2 + ς 1 χ x ) ) f ε )pε c 2 dx dt ψ 1 ς)w + ς χ x ε )W f + ς 1 χ x 5.5) ε )) ) w dx dt = oε). In the limit a ε 0 reult in the integral equality ψ 1 ς) c 2 p + ς m,0 c 2 p f + ς c ) f 2 P Y ) dx dt ψ 1 ς)w + ς w f) ) + ς1 m)w dx dt = 0, 5.6)

13 EJDE-2016/184 MATHEMATICAL MODELS OF SEISMICS 13 which hold for any mooth function ψ vanihing at S. Finally, we rewrite the continuity equation in the pore pace Ω f a the correponding integral identity ψ ΩT χ ε p ε + χ ε w ε) dx dt = ψ ΩT χ ε p ε + w ε 1 χ ε ) w ε) dx dt χ = ψ ΩT ε p ε ψ) w ε ψ1 χ ε ) w ε) dx dt, and apply the two-cale expanion 5.1): ψ c 2 χ x f ε )p f ψ) χ x ε )Wε f + 1 χ x ε )) ) w ψ 1 χ x ε )) w + y W ) ) dx dt = oε). 5.7) In the limit a ε 0 reult in the deired integral equality m ψ c 2 p f ψ w f) ) ψ y W Y dx dt = ) f The localization of 5.4), 5.6), and 5.8) give a the Lame ytem ϱ 0) 2 w 2 = P0), P 0) = λ 0) 0 Dx, w) pi, 5.9) p + c 2,0 w = ) in the domain Ω 0) for t > 0, the macrocopic dynamic equation 2 w f) ϱ f 2 + ϱ 1 m) 2 w 2 = P, 5.11) P = λ 0 N 0) : ) 1 m)dx, w ) + Dy, W ) Y mpf I 5.12) for the olid component and the macrocopic continuity equation m p f + w f) = y W Y 5.13) for the liquid component in the domain Ω for t > 0. The ame localization of 5.4) and 5.6) alo provide the boundary condition P 0) e 3 = p 0 e ) on the outer boundary S with the unit normal e 3, and the continuity condition lim P 0) x, t) e 3 = lim Px, t) e 3, 5.15) x Ω 0) x Ω lim wx, t) e 3 = lim w f) x, t) + 1 m)w x, t) ) e ) x Ω 0) x Ω on the common boundary S 0) x 0 with the unit normal e 3. More detailed mathematical analyi how that lim wx, t) = lim 1 m)w x, t) 5.17) x Ω 0) x Ω

14 14 A. MEIRMANOV, M. NURTAS EJDE-2016/184 for x 0 S 0). Unfortunately we have no poibility to prove the tatement due technical reaon. Differential equation and boundary condition are upplemented with initial condition wx, 0) = w x, 0) = w x, 0) = w x, 0) 5.18) = w f) x, 0) = wf) x, 0) = 0. However, the obtained ytem 5.9)-5.18) i till incomplete. We need two more differential equation for W and W f). More preciely, we have to expre the term Dy, W ) Y and y W Y by mean of function Dx, w ) and p f and rewrite 5.12) and 5.13) a m P = λ 0 N : Dx, w ) p f C, 5.19) p f + w f) = C 0 : Dx, ) + c 0 λ 0 p f. 5.20) To find the miing equation for the function W let u conider the integral identity 5.3). A in previou ection, we chooe tet function in the form ϕ ε x, t) = ε hx, t) ϕ 0 x ε ), where h i an arbitrary mooth function with a compact upport in Ω vanihing on S, and ϕ 0 y) i an arbitrary 1-periodic mooth function with a compact upport in Y. The limit in 5.3) a ε 0 with choen tet function reult in ) λ 0 1 χy) N 0) : Dx, w ) h Y + Dy, W ) ) ) χy) mp f I ) : Dy, ϕ 0 )dy dx dt = 0 After a localization we obtain the differential equation ) y λ 0 1 χy) N 0) : Dx, w ) + Dy, W ) ) ) mp f χy) = ) in the domain Y, which i undertood in the ene of ditribution. That i, a a uual differential equation y N 0) : Dx, w ) + Dy, W ) )) = ) in the domain Y. In the ame way uing tet function with a compact upport localize at γ we derive the boundary condition λ 0 N 0) : Dx, w ) + Dy, W ) )) n = mp f n 5.23) on the boundary γ. Here n i a unit normal to γ. The problem 5.18), 5.19) i completed with the periodicity condition on the remaining part Y \γ of the boundary Y. Let U ij) 2 y) and U 0) 2 y) be olution of periodic problem y 1 χ) N 0) : J ij) + Dy, U ij) 2 ) ))) = 0, 5.24) y 1 χ) N 0) : Dy, U 0) 2 ) + I)) = )

15 EJDE-2016/184 MATHEMATICAL MODELS OF SEISMICS 15 in Y. Then where Thu where D ij = 1 2 W x, t, y) = i,j=1 U ij) 2 y)d ij x, t) + m λ 0 p f x, t) U 0) 2 y), ui + u ) j, = u 1, u 2, u 3 ), Dx, w ) = x j x i Dy, W ) Y = Dy, U ij) 2 ) Y D ij + m p f Dy, U 0) 2 λ ) Y 0 i,j=1 D ij J ij). i,j=1 = Dy, U ij) 2 ) Y J ij)) : Dx, ) + m p f Dy, U 0) 2 λ ) Y, 0 i,j=1 y W Y = N = N 0) : λ 0 N 0) : 1 m)dx, ) + Dy, W ) Y ) mpf I = λ 0 N : Dx, ) p f C, y U ij) 2 Y D ij + m p f y U 0) λ 0 i,j=1 2 Y = y U ij) 2 Y J ij) : Dx, ) + m y U 0) 2 λ ) Y pf, 0 i,j=1 1 m) C 0 = J ij J ij + i,j=1 Dy, U ij) 2 ) Y J ij)), 5.26) i,j=1 C = mi Dy, U 0) 2 ) Y, 5.27) y U ij) 2 Y J ij, c 0 = y U 0) 2 Y. 5.28) i,j=1 The derivation of the equation for W f) repeat in it main feature the argument of the previou ection. We chooe the tet function ϕ ε in 5.3) a ϕ ε x, t) = hx, t) ϕ 0 x ε ), where h i a mooth function with a compact upport in Ω and ϕ 0 y) i a mooth 1-periodic olenoidal function with a compact upport in Y f. After the limit a ε 0 and localiation we arrive at the differential equation 2 W f) ϱ f 2 = µ 1 D y, Wf) ) y Π f) p f 5.29) in the domain Y f for t > 0. Here, a in previou ection, we alo mut define a 1- periodic in y function Π f) x, t, y), which appear due to the choice of tet function.

16 16 A. MEIRMANOV, M. NURTAS EJDE-2016/184 The miing equation i derived from the continuity equation in it integral form 5.5) in the ame way a in the previou ection and coincide with 4.19). According to [8] and [9] the ytem 5.29), 4.19) upplie with the boundary condition W f) x, t, y) = w x, t) 5.30) on the boundary γ, and the homogeneou initial condition W f) x, 0, y) = Wf) x, 0, y) = ) Problem 4.19), 5.29)-5.31) ha been olved in [9]: W f) = x, t) + = x, t) + Π f) x, t, y) = t i=1 0 t i=1 0 i=1 W f) i y, t τ) p f 2 w,i x, τ) + ϱ f x i τ 2 x, τ)) dτ f) ) 2 w W i y, t τ) e i pf x, τ) + ϱ f τ 2 x, τ)) dτ, t 0 Π f) i y, t τ) p f 2 w,i x, τ) + ϱ f x i τ 2 x, τ)) dτ, where = w,1, w,2, w,3 ) and {W f) i, Π f) i }, i = 1, 2, 3, are olution to the following periodic initial boundary value problem 2 W f) i ϱ f 2 = µ 1 D y, Wf) i ) y Π f) i, y, t) Y f 0, T ), 5.32) y W f) i y, t) = 0, y, t) Y f 0, T ), 5.33) W f) i Thu, w f) x, t) = W f) x, t, y)dy Y f where = m x, t) + W f) i y, 0) = 0, ϱ f y, 0) = e i, y Y f, 5.34) W f) i y, t) = 0, y, t) γ 0, T ). 5.35) t 0 B f) 3 t) = B f) 3 t τ) px, τ) + ϱ f 2 w τ 2 x, τ)) dτ, i=1 5.36) Y f W f) i y, t)dy e i. 5.37) Differential equation 5.9)-5.11), 5.20), and 5.36), boundary condition 5.14)- 5.17), initial condition 5.18) and tate equation 5.19) and 5.26)-5.28) contitute the mathematical model III) of eimic in compoite media. 6. One dimenional model for the cae I): numerical implementation Direct problem. For the ake of implicity we conider the pace, which conit of the following ubdomain: Ω 1 = {x R : 0 < x < H 1 }, Ω 2 = {x R : H 1 < x <

17 EJDE-2016/184 MATHEMATICAL MODELS OF SEISMICS 17 H 2 }, and Ω 3 = {x R : x > H 2 }. Differential equation 3.8), 3.9), 3.12), and 3.13) reult in 1 2 p ĉ 2 x) 2 = div 1 ˆρx) p + mν 0 p )) where 1 ĉ 2 = m + 1 m) c 2 and ˆρ = mρ f + 1 m)ρ are repectively average wave propagation velocity and average denity of the medium. Applying now the Fourier tranformation we arrive at d 2 ˆP dx 2 + ˆρω 2 1 mν0 iω)ĉ ˆP 2 = 0 6.1) where ˆP x, ω)-the preure obtained after Fourier tranform. c 2 f Figure 3. Scheme of arrangement of layer maratainur.jpg Depending on the exact phyical propertie, the edimentary rock zone i divided into three ubdomain. The value of the geometry of pore, vicoity of fluid, denity of rock, and velocity of eimic wave conidered in each layer to be different. In the experiment in order to get numerical olution, it aumed that the firt medium i a hale, the econd medium i oil aturated andtone, and the third medium i a limetone ee Fig.3). Let u uppoe that there i a plane wave which propagate from. Then the general olution of equation 6.1) for < X H 1 in the cae ν 0 = 0 i written down a: ˆP 1 = exp { iω ˆρ1 ĉ 1 } x + A 2 exp { iω ˆρ1 ĉ 1 } x. 6.2) The general olution of equation 6.1) for H 1 x < H 2 in the cae ν 0 > 0 i repreented a: { iω ˆρ { 2 ˆP 2 = B 1 exp ĉ 2 1 mν 0 iω x} iω ˆρ } + B 2 exp 2 ĉ c mν 0 iω x. f c 2 f 6.3) Finally the general olution for x H 2 in the cae ν 0 = 0 will be the following: { } ˆρ3 ˆP 3 = D 1 exp iω x. 6.4) ĉ 3

18 18 A. MEIRMANOV, M. NURTAS EJDE-2016/184 Continuity condition in contact media will be: ĉ 2 1 ĉ 2 2 [ ˆP 1 iω mν 0 c 2 f d dx [ ˆP 1 iω mν 0 c 2 f [ ˆP 2 iω mν 0 c 2 f d dx [ ˆP 2 iω mν 0 c 2 f ˆP 1 ] H1 0 = [ ˆP 2 iω mν 0 ˆP c 2 2 ] H1+0 f 6.5) ˆP 1 ] H1 0 = ĉ 2 d 2 dx [ ˆP 2 iω mν 0 ˆP 2 ] H ) c 2 f ˆP 2 ] H2 0 = [ ˆP 3 iω mν 0 ˆP c 2 3 ] H2+0 f 6.7) ˆP 2 ] H2 0 = ĉ 2 d 3 dx [ ˆP 3 iω mν 0 ˆP 3 ] H ) Thee relation are nothing ele but the ytem of linear algebraic equation for the coefficient A 2, B 1, B 2, D 1 which can be eaily reolved by any direct method. Thee coefficient are ued in order to contruct the olution in time frequency domain and after invere Fourier tranform in time the olution in the time domain can be eaily recovered ee Fig.4). c 2 f Figure 4. Propagation of eimic wave in different layer Invere problem. In invere problem [15] except ˆP x, ω) the value H 1, H 2, ĉ 2, ν 0, m are unknown a well. To determine thee value one need ome additional information about olution of the direct problem - data of invere problem. Uually they are given a function P ω) at X = 0. The mot widepread way i to earch for thee value by minimization of the data mifit functional being L 2 norm of the difference of given function and computed for ome current value of unknown parameter: F i H i 1, H i 2) = F i H i 1, ĉ i 2) = ωn ω 1 ˆP i ω, H i 1, H i 2) P ω, H 1, H 2 ) 2 dω 0 6.9) ωn ω 1 ˆP i ω, H i 1, ĉ i 2) P ω, H 1, ĉ 2 ) 2 dω ) ωn F i H2, i ĉ i 2) = ˆP i ω, H2, i ĉ i 2) P ω, H 2, ĉ 2 ) 2 dω ) ω 1 Here P ω,...,...) i the given wave field at X = 0, while ˆP ω,...,...) are wave field computed for ome current value of the deired parameter. In our numerical experiment the minimum i earched by the Nelder-Mead technique [17], Fig.5).

19 EJDE-2016/184 MATHEMATICAL MODELS OF SEISMICS 19 Figure 5. Simple cheme of Nelder-Mead for two variable regular implex Recovery of H 1 and H 2. Behavior of the data mifit functional for thi tatement i repreented in Figure 6 and 7. A one can ee thi functional i convex and ha the unique minimum point. Therefore thi invere problem i well reolved. Figure 6. Minimization of the functional F H 1, H 2 ). urfwveteng.jpg Figure 7. Level line of the functional F H 1, H 2 ). contour200eng.jpg Recovery of H 1 and c 2. Now we come to the non convex functional and therefore invere problem may have few olution ee Figure 8 and 9). Recovery of H 2 and c 2. Thi tatement alo generate non convex functional, but now it ha excellent reolution with repect to H 2 ee Figure 10 and 11).

20 20 A. MEIRMANOV, M. NURTAS EJDE-2016/184 Figure 8. Minimization of the functional F H 1, ĉ 2 ). Figure 9. Level line functional F H 1, ĉ 2 ). Figure 10. Minimization of the functional F H 2, ĉ 2 ). Concluion. In thi publication we have hown how to derive mathematical model for compoite media uing it microtructure. A a rule, there i ome et of model depending on given criteria µ 0, λ 0,... of the phyical proce in conideration. For a fixed et of criteria the correponding model decribe ome of the main feature of the proce. In the paper the implet invere problem wa dealt with - recovery of elatic parameter of the layer by Nelder-Mead algorithm. In the future we are planning to etablih connection upcaling procedure and cattered wave and apply on thi

21 EJDE-2016/184 MATHEMATICAL MODELS OF SEISMICS 21 Figure 11. Level line functional F H 2, ĉ 2 ). bae recent development of true-amplitude imaging on the bae of Gauian beam for both reflected and cattered wave [18, 19]. Reference [1] M. Biot; Theory of propagation of elatic wave in a fluid-aturated porou olid. I. Lowfrequency range. Journal of the Acoutical Society of America. 28, ). [2] M. Biot; Theory of propagation of elatic wave in a fluid-aturated porou olid. II. Higher frequency range. Journal of the Acoutical Society of America. 28, ). [3] M. Biot; Generalized theory of eimic propagation in porou diipative media. J. Acout. Soc. Am. 34, ). [4] J. G. Berryman; Seimic wave attenuation in fluid-aturated porou media, J. Pure Appl. Geophy. PAGEOPH) 128, ). [5] J. G. Berryman; Effective medium theorie for multicomponent poroelatic compoite, ASCE Journal of Engineering Mechanic 132 5), ). [6] R. Burridge, J. B. Keller; Poroelaticity equation derived from microtructure, Journal of Acoutic Society of America 70, No ), [7] E. Sanchez-Palencia; Non-Homogeneou Media and Vibration Theory, Lecture Note in Phyic, Vol. 129, Springer-Verlag, [8] A. Meirmanov; Mathematical model for poroelatic flow, Atlanti Pre, Pari, [9] A. Meirmanov; A decription of eimic eimic wave propagation in porou media via homogenization. SIAM J. Math. Anal. 40, N3, ). [10] A. Mikelić; On the Efective Boundary Condition Between Diferent Flow Region. Proceeding of the 1. Conference on Applied Mathematic and Computation Dubrovnik, Croatia, September ), [11] A. Benouan, J. L. Lion, G. Papanicolau; Aymptotic Analyi for Periodic Structure, North-Holland, Amterdam, [12] N. Bakhvalov, G. Panaenko; Homogenization: averaging procee in periodic media, Math. Appl., Vol. 36, Kluwer Academic Publiher, Dordrecht, [13] G. Ngueteng; A general convergence reult for a functional related to the theory of homogenization. SIAM J. Math. Anal ), [14] G. Ngueteng; Aymptotic analyi for a tiff variational problem ariing in mechanic, SIAM J. Math. Anal ), [15] A. Fichtner; Full Seimic Waveform Modelling and Inverion, Springer- Verlag Berlin Heidelberg. 2011) pp [16] M. I. Protaov, G. V. Rehetova, V. A. Tcheverda; Fracture detection by Gauian beam imaging of eimic data and image pectrum analyi, Geophyical Propecting, in-pre 2015). [17] J. H. Mathew, K. D. Fink; Numerical method uing Matlab. 4th edition, 2004, Prentice-Hall Inc., chapter 8, pp

22 22 A. MEIRMANOV, M. NURTAS EJDE-2016/184 [18] M. I. Protaov, V. A. Tcheverda; True amplitude imaging by invere generalized Radon tranform baed on Gauian beam decompoition of the acoutic Green function, Geophyical Propecting, 592), ). [19] M. I. Protaov, V. A. Tcheverda; True-amplitude elatic Gauian beam imaging of multicomponent walk-away VSP data, Geophyical Propecting, 606), ). Addendum poted on November 28, 2016 The editor from Zentralblatt informed u that a big portion of thi article coincide with the article Seimic in compoite media: elatic and poroelatic component by Anvarbek Meirmanov; Saltanbek Talapedenovich Mukhambetzhanov and Marat Nurta Sib. Elekron. Mat. Izv. 13, 75-88) 2016) Zbl ). The Electron. J. Differential Equation requeted an explanation from the author. They replied that two co-author ubmitted the manucript to two different journal, and each eventually publihed it without conulting the other. They write, It my fault that I did not control the proce. There i not any other explanation. Now I do not know what I hould do. Maybe the bet way here i to remove the paper from the ite, if it i poible. I apologize once again, your incerely, Anvarbek Meirmanov. Since the article i already publihed, the EJDE editor poted thi explanation. We recommend that co-author inform each other about their ubmiion. End of addendum. Anvarbek Meirmanov School of Mathematical Science and Information Technology, Yachay Tech, Ibarra, Ecuador addre: ameirmanov@yachaytech.edu.ec Marat Nurta School of mathematic and cibernetic, Kazakh-Britih Technical Univerity, Almaty, Kazakhtan addre: marat nurta@mail.ru