Coupled Fluid Flow and Elastoplastic Damage Analysis of Acid. Stimulated Chalk Reservoirs

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Ēᾍιδης Παπάγος
6 χρόνια πριν
Προβολές:

1 Nazanin Jahani Coupled Fluid Flow and Elastoplastic Damage Analysis of Acid Stimulated Chalk Reservoirs Thesis for the degree of Philosophiae Doctor Trondheim, October 2015 Norwegian University of Science and Technology Faculty of Engineering Science and Technology Department of Engineering Design and Materials

2 NTNU Norwegian University of Science and Technology Thesis for the degree of Philosophiae Doctor Faculty of Engineering Science and Technology Department of Engineering Design and Materials Nazanin Jahani ISBN (print) ISBN (digital) ISSN Doctoral theses at NTNU, 2015:223 Printed by NTNU Grafisk senter

17 Q l = A h Q l A h

18 h = p ρg = ρg μ μ ρ g A q = p = τ μ p U q (ϕ) U = q ϕ U = w2 f 12μ τ p w f τ p τ p τ n ˆn τ p = τ τ n τ p = τ (τ ˆn)ˆn

19 U = w2 f (τ (τ ˆn)ˆn) 12μ U = w2 f 12μ (δ ij n i n j )τ = ϕ w2 f 12μ (δ ij n i n j ) i j = k [ ] k = ϕ w2 f 12μ [ ] [ ] =(δ ij n i n j ) i j δ ij n i n j ˆn i j x y z 2w f H = 2ϕH 3μ w3 f (δ ij n i n j ) i j

20 z d x dip y d y d x y z x x y z z x y ρϕ + U = q s t q s ϕ U =0

21 ϕ C t + U C (ϕ C) q m =0 C = D e

24 ϕ C a + U C a = (ϕ C a ) α c (C a ) t α c ϕ t = βα cc a ρ R ρ R β

26 σ = D I ε e I σ ε e I D I I = 1 ν ν ν E ν 1 ν ν ν ν 1 ν ν ν ν ν E ν ε d = J σ σ ε d J k 1 n 0 0 J = 0 ks kt 1 k n k s k t

27 Δσ = ep Δε Δε =Δε e +Δε p σ η F F (σ, η) < 0 F (σ, η) =0 F (σ, η) > 0 p m p m = 1 3 (σ xx + σ yy + σ xx )

28 J J = 3J 2 J 2 1 J 2 = 2 [S2 x + Sy 2 + Sz 2 ] + σxy 2 + σyz 2 + σzx 2 S x = σ xx p m S x = σ xx p m S x = σ xx p m J 2 θ θ (3θ) = J 3 J 2 J2 J 3 J 3 = S x S y S z +2σ xy σ yz σ zx S x σ 2 yz S y σ 2 zx S z σ 2 xy Δε p = γ p Qp σ Δε p Q p γ p

29 J, P m F p = { } (φ) (θ) J 2 (θ) ζ [2 (2θ) 1] + 3 [ p m (φ)+c (φ ) ] Q p = { } (ψ) (θ) J 2 (θ) ζ [2 (2θ) 1] + 3 [ p m (ψ)+c (ψ ) ] ζ φ c Q p φ ψ ψ φ φ ini φ peak φ = φ ini + (φ peak φ ini ) 2 ε pl ε peak pl ε peak pl (ε peak pl ) 2 ε pl = ε peak pl φ res φ =(φ peak φ res ) e Υ(ε pl ε peak ) + φ res

30 c c ini c res c = c ini η (ε pl ε peak pl ) Υ η ε pl ε pl = 2(ε xx,pl ε v,pl 3 )2 +(ε yy,pl ε v,pl 3 )2 +(ε zz,pl ε v,pl 3 )2 + ε 2 xy,pl + ε2 yz,pl + ε2 zx,pl ε v,pl J p m F p =3 J M 2 (p 2 m p m p cc ) M p cc p c ε v,pl p cc = p c ( ) b ε 0 ε 0 b p c p c

31 Δε d = ed Δσ ed Δε d Δε d irr = γ d Qd σ F d Q d

32 x y z [ (σ F d ) 2 ( ) ] 1 xy σ 2 2 = + xz + σ xx c f μ y μ y μ z [ (σ Q d ) 2 ( ) ] 1 xy σ 2 2 = + xz + σ xx (α) μ z σ xx σ xy σ xz c f μ y μ z μ y = (Φ r + α y ) μ z = (Φ r + α z ) Φ r α y α z x z α α y α z α y = α y0 ( m Δε d y ) α z = α z0 ( m Δε d z )

33 x Z Y Y X Z xyz x y z Δε t =Δε e +Δε p +Δε d x,y,z x, y, z x z x x y =ΛT y z z l x = (x,x) m x = (x,y) n x = (x,z) Λ= l y = (y,x) l z = (z,x) m y = (y,y) m y = (z,y) n y = (y,z) n y = (z,z)

34 ε ε u x u x l x Λ m x Λ n x Λ = l y Λ m y Λ n y Λ l z Λ m z Λ n z Λ ε d = ε d σ = T σ T lx 2 m 2 x n 2 x 2 l x m x 2 m x n x 2 l x n x ly 2 m 2 y n 2 y 2 l y m y 2 m y n y 2 l y n y T = lz 2 m 2 z n 2 z 2 l z m z 2 m z n z 2 l z n z l x l y m x m y n x n y m x l y + l x m y n x m y + m x n y n x l y + l x n y l y l z m y m z n y n z m y l z + l y m z n y m z + m y n z n y l z + l y n z l x l z m x m z n x n z m x l z + l x m z n x m z + m x n z n x l z + l x n z xx yy zz xy yz xz ε d = F J σ ε d = F J T σ T = 1 ε d = T J σ

35 Δε e =( I + T J )Δσ t t = I + T J D t = [( ep ) 1 +( ed ) 1] 1 σ = t ε t ep (ε t ε d )= ed ε d

36 f x 0 f(x 0 +Δx) f(x 0 )+ f x x 0 Δx σ tr 1 = σ 0 + t 1 Δε t 1 Δε p σ f p 1 = σ tr 0 I Δε p

37 Δε p = γ p Qp σ γ p σ f d 1 = σ tr 0 1 J Δεd Δε d irr = γ d Qd σ γ d Δσ f = σ f p σ f d =0 ΔF p (σ p, η p )= Fp σ p Δσp + Fp η p Δηp ΔF d (σ d, η d )= Fd σ d Δσd + Fd η d Δηd Δσ p = I Δε p = I γ p Qp σ Δη p = γ p Qp ε Δσ d = 1 J Δεd = 1 Qd J γd σ

38 Δη d = γ d Qd ε γ p γ p = ΔF p F p T Q p σ I σ Fp η p η p ε p γ d γ d = ΔF d F d T σ 1 Q d J σ Fd η d η d ε d

41 σ t p σ t = σ α p α α

43 Δf = T ΔPdV V

44 p f g Δf = g Δu g n g = T t dv = T e t e dv V e=1 V e t ep V Δε = Δu σ tr = σ 0 +Δσ e = σ 0 + t eδε t e

45 Δf el = T e ΔσdV V e Δσ Δσ = σ t σ f Δf el Δf el

53 Appendix A

57 Δσ = I Δε e

58 I Δε e F p Q p F p Q p F p = { } (φ) (θ) J 2 (θ) ζ [2 (2θ) 1] + 3 [ p m (φ)+c (φ ) ] Q p = { } (ψ) (θ) J 2 (θ) ζ [2 (2θ) 1] + 3 [ p m (ψ)+c (ψ ) ] J 2 p m θ ζ φ c Q p φ ψ ψ φ Δε p ep J 2 p m

59 Δε d = J Δσ Δσ Δε d J k 1 n 0 0 J = 0 ks kt 1 k n k s k t

60 [ (σ F d ) 2 ( ) ] 1 xy σ 2 2 = + xz + σ xx c f μ x μ x μ z [ (σ Q d ) 2 ( ) ] 1 xy σ 2 2 = + xz + σ xx (α) c f μ x μ z μ z μ x = (Φ r + α x ) μ z = (Φ r + α z ) Φ r α x α z x z α α x α z α x = α x0 ( m ε d x ) α z = α z0 ( m ε d z ) ed Δε d

61 Δε t =Δε e +Δε p +Δε d Δε t =( I + T J )Δσ t I J t t = I + T J 6 3 { σ xx σ xy σ xz } T { } T = σxx σ yy σ zz σ xy σ yz σ xz T { ε xx ε xy ε xz} T = { εxx ε yy ε zz ε xy ε yz ε xz } T

62 D t = [( ep ) 1 +( ed ) 1] 1 t Δ =Δ Δ Δ Δε m σ km n = σ m 1 + t nδε t n m m n m k σ k n = t n ε t n = ep n (ε t n ε d n)= ed n ε d n

63 E K n / ε = t σ

64 σ k n ε d n = T J σ k n σ k n = σ k n σ k n ep n+1 σ k n+1 σ k n ed n+1 σ k n+1 ε n+1 = J σ n+1 ε d n+1 = T ε Δε d n+1 = ε d n ε d n+1 Δε d n+1 =0 (m +1) σ k+1 n+1 = ep n+1 (εt n+1 ε d n+1) =σ k n+1 + ep n+1 Δεd n+1

65 E ν Φ Ψ c k n k s k t c f α x α z Φ r..... / / /. σ zz ( 1 ΔU z = σ zz + 1 H k n E H z x y x y z )

66 [mm]

70 1 ν ν ν I = E ν 1 ν ν ν ν 1 ν ν ν ν ν E ν Δε p = γ p Qp σ

71 γ p γ p γ p = F σ FT σ I Δε T Q I σ F κ Δσ = ep Δε κ ε p ep Δε Δε =Δε e +Δε p [D ep ] I γ p =0 ep = I I Q FT I σ σ FT Q I σ σ F κ κ ε p γ p > 0 1 ed J γ d =0 = 1 J 1 Q d F d T J σ σ 1 J γ d > 0 F d σ 1 Q d J σ Fd μ μ ε p γ d Δε d = γ d Qd σ lx 2 m 2 x n 2 x 2 l x m x 2 m x n x 2 l x n x = l x l y m x m y n x n y m x l y + l x m y n x m y + m x n y n x l y + l x n y l x l z m x m z n x n z m x l z + l x m z n x m z + m x n z n x l z + l x n z

72 l x = (d) (a) m x = (d) (a) n x = (d) l y = (a) m y = (a) n y =0 l z = (d)cos(a) m z = (d) (a) n z = (d) d a

73 Appendix B

75 accepted for publication in International Journal of Rock Mechanics and Mining Sciences, DOI: / j.ijrmms

78 θ θ X Y Z Z X Y

79 q = μ P P μ = k [ ] k k = w2 f 12 w f [ ] [ ] =δ ij n i n j δ ij n i n j ˆn U q (ϕ) U = q ϕ

80 Δσ = ep Δε Δσ Δε Δε =Δε e +Δε p ep ep I E ν ep c Φ Φ Ψ Ψ Φ

81 x y z Δε d = ed Δσ Δσ Δε d ed ed F J k 1 n 0 0 J = 0 ks kt 1 k n k s k t [ (σ F d ) 2 ( xy σ = + xz μ x μ z [ (σ Q d ) 2 ( xy σ = + xz μ x μ z ) 2 ] σ xx c f ) 2 ] σ xx (α) c f μ x μ z μ x = (Φ r + α x ) μ z = (Φ r + α z ) Φ r α x α z x z α α x α z

82 α x = α x0 ( m ε d x, ) α z = α z0 ( m ε d z, ) α α = α x + α z 2 Δε e =( 1 I + T J )Δσ X Y Z x y z t t = 1 I + T J t D t = [( ep ) 1 +( ed ) 1] 1

83 ϕ β0 k. α w f = w f0 (1 (α)) w f0 w f w f0 =. ϕ β = ϕ β0 + ε v ϕ β ϕ β0 ε v

84 E ν Φ Ψ c k n k s k t c f α x α z Φ r..... / / /. σ p σ = σ α p α α

85 X Z Y Z XZ θ θ.....

86 Overburden pressure Out Flow

87 .

91 Appendix C

95 = 1 P μ P P q (ϕ) U = q ϕ = k k ij

96 y z x d dip d k ij k k ij k ij = δ ij n i n j n i n j δ ij a d x k k k = 2 3 fw3 f f w f ρϕ + t =0 ϕ =0

97 β α c ρ R k m 2 ϕ ϕ C a + t C a = (ϕ C a ) α c (C a ) D e α c ϕ t = βα cc a ρ R ρ R β

98 Δε t = t Δσ t ε t σ t t t = I + T J I J I = 1 I 1 ν ν ν I = E ν 1 ν ν ν ν 1 ν ν ν ν ν E ν k 1 n 0 0 J = 0 ks kt 1 k n k s k t x,y,z x x, y z 6 3 { σ xx σ xy σ xz } T { } T = σxx σ yy σ zz σ xy σ yz σ xz lx 2 m 2 x n 2 x 2 l x m x 2 m x n x 2 l x n x = l x l y m x m y n x n y m x l y + l x m y n x m y + m x n y n x l y + l x n y l x l z m x m z n x n z m x l z + l x m z n x m z + m x n z n x l z + l x n z

99 E ν k n k s k t α / / / l x = (x,x) m x = (x,y) n x = (x,z) l y = (y,x) l z = (z,x) m y = (y,y) m y = (z,y) n y = (y,z) n y = (z,z) p σ t σ t σ t = σ t α p α xy

100 xz z x y x y z...

101 yz % yz

102 yz J yz

103 J yz

104

105 Bibliography

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