ELEMENTS OF ACOUSTIC WAVES IN POROUS MEDIA

Transcription

1 ELEMENTS OF ACOSTIC WAVES IN POROS MEDIA By George F FREIHA niversiy of Balamand (Spervised y Dr Elie HNEIN) niversié de Valenciennes e d Haina Camrésis (Spervised y Dr Berrand NONGAILLARD & Dr George NASSAR) Sppored y: CNRS (Comié Naional de Recherche Scienifiqe de France)

2 Plan of he Presenaion Theory of homogeneiy & heerogeneiy Inrodcion o poros media and is properies 3 Inrodcion o acosic wave propagaion 4 Lagrangian formalism of he Bio Theory 5 Solion of he eqaion of celeriy and coefficien of aenaion of he differen acosic waves in he Bio Theory 6 Graph presenaion and discssion of he ehavior of he celeriy and aenaion coefficien in fncion of differen parameers

3 HOMOGENEITY & HETEROGENEITY - Homogeneos Medim: Physical Properies Independen of Posiion - Heerogeneos Medim: Change in Properies when Passing from Poin o Poin in he Medim W W According o W, he medim is homogeneos, according o W, i is heerogeneos Two ypes of heerogeneos medim: -Coninos heerogeneos medim -Discree heerogeneos medim

4 INTRODCTION TO POROS MEDIM ( I ) In general, a poros medim has wo phases: The solid phase which can e elasic or rigid The flid phase which can e a viscos and compressile liqid Fier form Grain form Cone form

5 INTRODCTION TO POROS MEDIM ( II ) A microscopic mahemaical descripion of sch medim is impossile ecase of he mechanical and hermal ineracion eween he flid and solid, dring heir displacemen In he macroscopic approach, he size of he elemenary volme ms follow wo essenial condiions If: L he characerisic size of he medim R he size of he elemenary volme ha we are working on d he characerisic dimension of he pores we ms have d << R << L

6 INTRODCTION TO POROS MEDIM ( III ) Characerisics of a poros medim ( I ) Porosiy: β v v flide oal v v f v s v f v f Types of porosiy: -Non conneced porosiy: oflid medim conaining solid inclsions osolid medim conaining flid inclsions -Conneced porosiy: pores are conneced ogeher -Semi-conneced porosiy: pores are conneced o ohers y a very small common srface

7 INTRODCTION TO POROS MEDIM ( IV ) Characerisics of a poros medim ( II ) In mos of he poros medims, he semi-conneced and non-conneced pores are negligile while doing measremens Or sdy will concenrae on he conneced porosiy ype where acosic waves can ravel and where he flid circlaion is possile

8 INTRODCTION TO POROS MEDIM ( V ) Characerisics of a poros medim ( III ) Permeailiy: a facor ha inflences he energeic dissipaion eween solid and flid dring heir displacemen Torosiy: a facor ha gives an idea ao ways of he flid inside he solid medim and i is considered as a facor of he size and geomery of he pores The eqaion of porosiy is given y α v f v f v vdv dv

9 INTRODCTION TO ACOSTIC WAVE ( I ) Acosic waves are resls of mechanical perraion There are wo kinds of waves: - Longidinal waves where he mechanical perraion is done in a direcion parallel o he propagaion - Transversal waves where he mechanical perraion is done in a direcion perpendiclar o he propagaion The medim is considered isoropic so he characerisics of propagaion are independen of he direcion of propagaion

10 INTRODCTION TO ACOSTIC WAVE ( II ) x3 Displacemen vecors are r AA' A' A AA' and B he deformaion of he solid will e he ensor E where he elemens E are defined as BB' A B r BB' B' x E i x j x i j Where i is he direcion of vecor r x in x i direcion e E represens he dilaion of he solid E E33

11 INTRODCTION TO ACOSTIC WAVE ( III ) Force eqaions: a force df j is applied on a elemenary ni of srface ds j This force can e projeced giving s he sress componens τ ds lim j df ds Hooke s law: Hooke s law gives s he relaion eween sress and j srain in he volme τ c kl E kl xj For isoropic solid τ λ eδ μe where λ and μ are he Lame coefficiens τ r ik df jk e is he dilaion of he solid k r r j i r r τ r τ jk xi E is he ensor of srain xk

12 INTRODCTION TO ACOSTIC WAVE ( IV ) Eqaion of propagaion of a harmonic plane wave( I ) The fndamenal eqaion of dynamics gives s he firs relaionship eween force and he local displacemen in he medim ρ i Having τ τ x in isoropic medim j c where ρ is he mass per ni volme of he medim kl E kl ρ i c kl x jxk ( ) ( ) [ ( )] λ μ grad div μro ro ρ Helmholz heorem ( ) gradφ ro ψ r φ r Ψ Δφ and ΔΨ v v L T wo propagaion eqaions

13 Lagrange fncion : A Λ() d Λ Ω L r, ( r ) dv Assming ha he exciaion done on he medim is A, we will have The mos general Lagrangian densiy fncion ms e r & r, & are he emporal derivaives i, j, k, LAGRANGIAN FORMALISM OF BIOT THEORY( I ) Inrodcion o Lagrangian formalism: x i are he spaial derivaives of he field, k j i j r & r & r r (,,,, &,, i ) L, j, k, x x

14 r & r r r r r L (,,,, i, j ) L(,,,, i, j ) r LAGRANGIAN FORMALISM OF BIOT THEORY( II ) Condiions of invariance of he Lagrangian fncion L Since we are dealing wih a homogenos and isoropic medim he fncion L ms e invarian according o roaion and ranslaion r r & ( A B, A, A ) L(, ) L i, j, i, j The Lagrangian is only fncion of he speed and he gradiens of he deformaions r r & r & r e i, j

15 BIOT THEORY ( I ) Condiions of he heory: Small srain for he solid and flid We ms deal wih conneced ype of porosiy The wave lengh ms e o imes greaer hen he dimensions of pores A insan he Lagrangian fncion of he fields (srain of he solid) and r r, ( r ) cominaion of he wo fields r r, (srain of he liqid) is a linear ( r ) r r, ( r ) r r, ( r ), and heir derivaives

16 [ ( ) ( ), L jj ii jj ii jj ii γ ω ω γ ε ε γ ε ε γ ε ε γ ε ε γ ε ε γ ε ε γ ρ ρ ρ ( )( ) ( )( ) ( )( ) ( )( ) a a a a a a a a a a a a jj a ii A o A e,,,, ε ε ω ω ε ε General Lagrangian fncion: BIOT THEORY ( II )

17 ( ) [ ( ) ( ) ( )( ) ( ) ( )]}, A L T α α α α α α α ρ ρ ρ a a a T A A A A, 6, 5, 4 γ γ γ BIOT THEORY ( III ) Wih α γ γ 4 ; α γ γ 5 ; α 3 γ 3 γ 6 ; α 4 γ 4 γ 7 ; α 5 γ 5 - γ 7 ; α 6 γ 6 γ 7 ; α 8 γ 8

18 [ ] ( ) ω d L r d A Ω Ω,, ;, a A a i δ δ a i 3 j a i j j a i a i L x L L BIOT THEORY ( IV ) For a small variaion on any of he fields he exciaion done on he medim ms e saionary Eler Lagrange eqaions for each of he srain fields : exciaion:

19 ( ) ( ) ( ) ( ) ( ), α α α α α ρ ρ ( ) ( ) ( ) ( ) ( ) α α α α α ρ ρ ( ) ( ) ( ) 4 α α α ρ ρ ( ) ( ) 3 α α ρ ρ BIOT THEORY ( V ) and and

20 , M ( ) ( ) ( ) Ρ Q μ ρ ρ Dissipaion can e he variaion of emperare from poin o poin which leads o an irreversile hermo condcion process The dissipaion sress noed M are a linear eqaions of he relaive velociy eween he wo phases ( ) ( ) Q R ρ ρ and BIOT THEORY ( VI )

21 BIOT THEORY ( VII ) The componens of he sress ensors can e fond sing he following eqaion which leads s o a relaionship eween sress and srain in or medim L a τ, a i, j τ s Ρ ( ) ( ) ( ) Q N ( ) ( ) ( ) f τ i p f hδ R Q δ R Q P,Q,R,N are considered as phenomenological parameers in he BIOT heory and are know as coefficien of BIOT They can e fond y doing 3 special physical experimens known as BIOT and WILLIS experimens

22 f s s s s K K h K K h hk K K h Q, f s s s K K h K K h K h R, ( ) N K K h K K h K K K h K K K h h P f s s f s s s 3 4, BIOT THEORY ( VIII )

23 BIOT THEORY ( IX ) Copling in BIOT heory ( I ) The ineracions eween he solid and flid phases are deermined in he eqaions y cerain copling consans The mos imporan wo ypes of copling consan are he mass copling and viscos copling omass copling : sing Lagrangian expression we can define he kineic energy of he medim E c ρ ρ ρ,

24 BIOT THEORY ( X ) Copling in BIOT heory ( II ) Viscos copling: I represens he relaive moion eween he wo phases: F ( ) ( κ) κt κ 4 T iκ ( κ) T ( ) κ er ' ( κ ) i ei '( κ) ( κ ) i ei( κ) Where: κ is a facor of he geomery and dimensions of he pores er and ei are he real and imaginary pars of he Kelvin fncion κ and ( ) ( ) ( 3 ) er κ i ei κ J i κ J is he Bessel fncion of order zero Ber and ei are he real and imaginary pars of he derivaive of he Kelvin fncion ha can e calclaed as following: er ( 3 ) κ ( κ ) ( κ) J i x x dx i er ' i ei '

25 ( ) ( ) ( ) Λ Λ N P Q ρ ρ ( ) ( ) Q R ρ ρ Λ w φ ΛW ψ We already fond he eqaion of wave propagaion in he medim SOLTION OF EQATIONS OF WAVE PROPAGATION IN BIOT THEORY ( I )

26 SOLTION OF EQATIONS OF WAVE PROPAGATION IN BIOT THEORY ( II ) Longidinal waves eqaions ( I ) We sar working on he scalar par alone firs y replacing φ y heir vales:, ψ and φ ρ ψ P ( ) ( ) ψ φ φ Q ψ ρ, ψ ρ ρ ω ~ ρ ω ~ ρ φ ω ~ ρ φ R Harmonic plan waves Wih ψ ω ~ ρ ~ ρ ψ φ ψ φ ( ) ( ) ψ Q φ f r r e ( ) ( ) jω P R, φ Q ψ Q ρ j ; ~ ρ ρ j ; ~ ρ ρ ω ω f ψ φ j ω

27 ~ ~ ~ ~ ρ ρ ρ ρ ω A Ψ Ψ A M Longidinal waves eqaions ( II ) A is he complex densiy marix: R Q Q P M M is he BIOT coefficien marix: Ψ ψ φ Ψ is he poenial vecor : Mliplying he eqaion y M - we will ge: φ φ φ φ δ δ, SOLTION OF EQATIONS OF WAVE PROPAGATION IN BIOT THEORY ( III ) Ψ Ψ M A Marix form

28 SOLTION OF EQATIONS OF WAVE PROPAGATION IN BIOT THEORY ( IV ) Longidinal waves eqaions ( III ) Wih δ and δ he eigen vales of he eigen vecors represening he consan of propagaion of he waves Φ Φ and δ δ ω [ ~ ~ ~ Δ ] ( ) ( Pρ ) Rρ Qρ PR Q ω [ ~ ~ ~ Δ ] ( ) ( Pρ ) Rρ Qρ PR Q Wih Δ ( P ~ ρ R ~ ρ Q ~ ρ ) 4( PR Q )( ~ ρ ~ ρ ~ ρ )

29 SOLTION OF EQATIONS OF WAVE PROPAGATION IN BIOT THEORY ( V ) Longidinal waves eqaions ( IV ) We will ge modes of propagaion associaed o wo longidinal P ~ ρ R ~ ρ ~ Qρ waves one rapid wih phase velociy : v ~ ~ ~ ρ ρ ρ And a slow longidinal wave wih phase velociy: v P ~ ρ R ~ ρ Q ~ ρ ~ ( ~ ρ ~ ρ ρ ) Δ ( ) Δ Normalizaion: P R Q,, H H H σ σ σ H P R Q ρ ρ ρ γ, γ, γ ρ ρ ρ and z V v c L, wih, wih ( ) ρρ ρ ρ ρ ρ βρ β ρ f s

30 SOLTION OF EQATIONS OF WAVE PROPAGATION IN BIOT THEORY ( VI ) Longidinal waves eqaions ( V ) Afer sdying differen geomery of pores, BIOT inrodced a characerisic freqency δ f c πρ πρ γ γ ( ) 6 varying eween and 3 f, and noiced ha κ δ, wih f So he phase velociies of he waves will e:, ~ V s r Re The aenaion coefficiens per cycle of he waves are: ( z ) Ac s r c Im π Re ( z ) ( z ) The aenaion coefficiens per disance of he waves are: Ad s r ( ) π Im z

31 SOLTION OF EQATIONS OF WAVE PROPAGATION IN BIOT THEORY ( VII ) Transversal wave eqaions Now we will work on he vecorial par alone and y heir vales: Λω Λ Ω r r Δω ρ ωρ Ω κ ωω r r r N r r r ( ) F( ) ( ) r r ( ) ( ) ( ) F ρ ωρ Ω κ ωω

32 SOLTION OF EQATIONS OF WAVE PROPAGATION IN BIOT THEORY ( VIII ) Transversal wave eqaions (II) N Having V R we can calclae he velociy of ransversal wave sing: ρ f c ( ) i V γ γ γ ( γ γ ) R f T v f c γ i ( γ γ ) f So he phase velociies of he waves will e:, ~ V Re The aenaion coefficiens per cycle of he waves are: ( T ) Ac Im π Re ( T ) ( T ) The aenaion coefficiens per disance of he waves are: Ad π Im( T )

33 Graphs of he Variaion of he Velociy and Coefficien of Aenaion for he Transversal and Longidinal (Slow & Rapid) Wave

34 Resls in he asence of copling (I) Inflence of he srcral facor δ δ σ σ σ γ γ γ 6/3 8

35 Phase Velociy of he wave in fncion of δ Slow Longidinal Wave Rapid Longidinal Wave Transversal Wave

36 Aenaion per Cycle of he wave in fncion of δ Slow Longidinal Wave Rapid Longidinal Wave Transversal Wave

37 Aenaion per Disance ni of he wave in fncion of δ Slow Longidinal Wave Rapid Longidinal Wave Transversal Wave

38 Resls in he asence of copling (II) Inflence of he raio σ / σ δ σ σ σ γ γ γ σσ 8,,9, 8 8,9, 9

39 Phase Velociy of he wave in fncion of σ σ Slow Longidinal Wave Rapid Longidinal Wave Transversal Wave

40 Aenaion per Cycle of he wave in fncion of σ σ Slow Longidinal Wave Rapid Longidinal Wave Transversal Wave

41 Aenaion per Disance ni of he wave in fncion of σ σ Slow Longidinal Wave Rapid Longidinal Wave Transversal Wave

42 Resls in he asence of copling (III) Inflence of he raio γ / γ δ σ σ σ γ γ γ γγ 8,,9, 8 8,9, 9

43 Phase Velociy of he wave in fncion of γ γ Slow Longidinal Wave Rapid Longidinal Wave Transversal Wave

44 Aenaion per Cycle of he wave in fncion of γ γ Slow Longidinal Wave Rapid Longidinal Wave Transversal Wave

45 Aenaion per Disance ni of he wave in fncion of γ γ Slow Longidinal Wave Rapid Longidinal Wave Transversal Wave

46 Resls wih copling effec Inflence of he coefficien of elasic copling σ δ σ σ σ γ γ γ 8 8,3, 8,,

47 Phase Velociy of he wave in fncion of σ Slow Longidinal Wave Rapid Longidinal Wave Transversal Wave

48 Aenaion per Cycle of he wave in fncion of σ Slow Longidinal Wave Rapid Longidinal Wave Transversal Wave

49 Aenaion per Disance of he wave in fncion of σ Slow Longidinal Wave Rapid Longidinal Wave Transversal Wave

50 THANK YO Qesions???