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
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 978-82-326-1102-7 (print) ISBN 978-82-326-1103-4 (digital) ISSN 1503-8181 Doctoral theses at NTNU, 2015:223 Printed by NTNU Grafisk senter
Q l = A h Q l A h
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
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
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
ϕ C t + U C (ϕ C) q m =0 C = D e
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
σ = D I ε e I σ ε e I D I I = 1 ν ν ν 0 0 0 E ν 1 ν ν 0 0 0 ν ν 1 ν 0 0 0 1+ν 0 0 0 0.5 ν 0 0 0 0 0 0 0.5 ν 0 0 0 0 0 0 0.5 ν E ν ε d = J σ σ ε d J k 1 n 0 0 J = 0 ks 1 0 0 0 kt 1 k n k s k t
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 )
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) = 3 3 2 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
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
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 2 2 + 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
Δε d = ed Δσ ed Δε d Δε d irr = γ d Qd σ F d Q d
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 )
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)
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
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
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
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
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
σ t p σ t = σ α p α α
Δf = T ΔPdV V
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
Δf el = T e ΔσdV V e Δσ Δσ = σ t σ f Δf el Δf el
Appendix A
Δσ = I Δε e
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
Δε d = J Δσ Δσ Δε d J k 1 n 0 0 J = 0 ks 1 0 0 0 kt 1 k n k s k t
[ (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
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
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
E K n / ε = t σ
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
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 )
120 85 50 15-20 -55-90 -125-160 -195-230 0-0.2-0.4-0.6-0.8-1 -1.2-1.4-1.6 [mm]
0.0024 0.0015 0.001 0.0005 0
23.0 12.7 7.62 3.88 2.14 1.18 0.65 0.36 0.2
1 ν ν ν 0 0 0 I = E ν 1 ν ν 0 0 0 ν ν 1 ν 0 0 0 1+ν 0 0 0 0.5 ν 0 0 0 0 0 0 0.5 ν 0 0 0 0 0 0 0.5 ν E ν Δε p = γ p Qp σ
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
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
Appendix B
accepted for publication in International Journal of Rock Mechanics and Mining Sciences, DOI: 10.1016/ j.ijrmms.2015.09.011
θ θ X Y Z Z X Y
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
Δσ = ep Δε Δσ Δε Δε =Δε e +Δε p ep ep I E ν ep c Φ Φ Ψ Ψ Φ
x y z Δε d = ed Δσ Δσ Δε d ed ed F J k 1 n 0 0 J = 0 ks 1 0 0 0 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 ] 1 2 + σ xx c f ) 2 ] 1 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
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
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
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
X Z Y Z XZ θ θ.....
Overburden pressure Out Flow
.
0.003 0.0027 0.0024 0.0021 0.0018 0.0015 0.0012 0.0009 0.0006 0.0003 0 10 3
Appendix C
= 1 P μ P P q (ϕ) U = q ϕ = k k ij
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
β α c ρ R k 10 10 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
Δε t = t Δσ t ε t σ t t t = I + T J I J I = 1 I 1 ν ν ν 0 0 0 I = E ν 1 ν ν 0 0 0 ν ν 1 ν 0 0 0 1+ν 0 0 0 0.5 ν 0 0 0 0 0 0 0.5 ν 0 0 0 0 0 0 0.5 ν E ν k 1 n 0 0 J = 0 ks 1 0 0 0 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
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
xz z x y x y z...
yz % 40.5 40.4 40.3 40.15 40.0 yz
yz J yz
J yz
Bibliography