A Lagrange multiplier method for flow in fractured poroelastic … · 2019. 6. 27. · 0.000358...
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A Lagrange multiplier method for flow in fractured poroelastic media Eldar Khattatov † , Ivan Yotov † , Ilona Ambartsumyan † , Paolo Zunino ‡ † Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania, USA; ‡ Department of Mechanical Engineering & Materials Science, University of Pittsburgh, Pittsburgh, Pennsylvania, USA; 1. Model Problem: Coupled Stokes-Biot System • Groundwater flow and flow in fractured porous media • Cardiovascular flow Figure 1. Applications Figure 2. Stokes-Biot model Biot system of poroelasticity in Ω p : u p = Darcy velocity, p p = pressure, η = displacement -∇ · σ p (η )= f p ∂ ∂t (s 0 p p + α∇· η )+ ∇· u p = q p , K -1 u p = -∇p p , where σ p (η )= λ p (∇· η )I + 2μ p D(η ) - αp p I is a poroelastic stress tensor, α is Biot-Willis constant and K is a permeability tensor. Stokes flow in Ω f : u f = Stokes velocity, p f = fluid pressure ρ f ∂ u f ∂t -∇· σ f = f f , ∇· u f = q f , where σ f = -p f I + 2μ f D(u f ) is a Cauchy stress tensor. Interface conditions on Γ fp : Mass conservation: u f · n =( ∂ η ∂t + u p ) · n Balance of normal fluid stress: -(σ f n) · n = p p Continuity of momentum: σ f n = σ p n No slip or Beavers-Joseph-Saffman condition: • u f · τ = ∂ η ∂t · τ •-(σ f n) · τ = c BJS (u f - ∂ η ∂t ) · τ 2. Discretization & numerical analysis The discretization • is derived on simplicial grids in 2D and 3D; • uses standard conforming finite elements for Stokes and elasticity equations and mixed finite elements for Darcy; • allows the fluid and poroelastic region grids to be non-matching through the interface; • utilizes a Lagrange multiplier λ on the interface to weakly impose the continuity of flux; • uses the space for λ being the normal trace of Darcy velocity space. Figure 3. Example of lowest order FE spaces: (P 2 ×P 1 ) × (RT 0 ×P 0 ) ×P 1 ×P 0 Discrete weak fomulation Find (u f,h ,p f,h , u p,h ,p p,h , η p,h ,λ h ) in V f,h × W f,h × V p,h × W p,h × X p,h × Λ h such that (ρ f ∂ t u f,h , v f,h )+ a f (u f,h , v f,h )+ a d p (u p,h , v p,h )+ a e p,h (η p,h , ξ p,h ) +a BJS (u f,h , η p,h ; v f,h , ξ p,h ) + b f (v f,h ,p f,h )+ b p (v p,h ,p p,h ) +αb p (ξ p,h ,p p,h )+ b Γ (v f,h , v p,h , ξ p,h ; λ h ) = f (v f ,h , ξ p,h ), (∂ t s 0 p p,h ,w p,h ) - αb p (∂ t η p,h ,w p,h ) - b p (u p,h ,w p,h ) -b f (u f,h ,w f,h )= q (w f,h ,w p,h ), b Γ (u f,h , u p,h ,∂ t η p,h ; μ h )=0, With Backward Euler used for discrete time derivative d τ u n := τ -1 (u n - u n-1 ) Convergence analysis of the discrete formulation With k f ,s f ,k p ,s p ,k s denoting the order of polynomials used in the discretization of Stokes, Darcy and elasticity equations respectively, there holds ku f - u f,h k L ∞ (L 2 ) + ku f - u f,h k L 2 (H 1 ) + ku p - u p,h k L 2 (L 2 ) +kp p - p p,h k L ∞ (L 2 ) + kη p - η p,h k L ∞ (H 1 ) + d-1 X j =1 νc BJS kK -1/4 j ( (u f - u f,h ) - ∂ t (η p - η p,h ) ) · τ f,j k L 2 (L 2 (Γ fp )) ≤ C h k f ku f k L 2 (H k f +1 ) + h k f ku f k L ∞ (H k f +1 ) + h k f ∂ t u f L 2 (H k f +1 ) +h s f +1 kp f k L 2 (H s f +1 ) + h k p +1 ku p k L 2 (H k p +1 ) + h k s ∂ t η p L 2 (H k s +1 ) +h k p +1 kλk L 2 (H k p +1 (Γ fp )) + h k s kη p k L ∞ (H k s +1 ) + h k p +1 kλk L ∞ (H k p +1 (Γ fp )) +h s p +1 kp p k L ∞ (H s p +1 ) + h k p +1 k∂ t λk L 2 (H k p +1 (Γ fp )) + h s p +1 kp p k L 2 (H s p +1 ) 3. Convergence, BJS condition and continuity of flux Convergence study on a reference domain Figure 4. Reference grid Parameter Units Value Young’s modulus (Pa) 10 10 Fluid density (kg/m 3 ) 1.0 Dynamic viscosity (Pa s) 1.0 Lame coefficient (Pa) 5/12 · 10 10 Lame coefficient (Pa) 5/18 · 10 10 Hydraulic conductivity (m 2 /Pa s) Id Mass storativity (Pa -1 ) 1.0 Biot-Willis constant 1.0 BJS coefficient 1.0 Total time (s) 1.0 u f · n = 10, u f · τ =0 on Γ inflow u p · n =0, on Γ left p p =0, on Γ top ∪ Γ right ∪ Γ bottom η =0, on Γ top ∪ Γ right ∪ Γ bottom Table 1. Convergence study using the lowest order elements h ku f - u f,h k l 2 (H 1 ) rate ku f - u f,h k l ∞ (L 2 ) rate 1/20 1.33e+00 0.00 9.58e-03 0.00 1/40 5.11e-01 1.38 1.87e-03 2.36 1/80 2.31e-01 1.15 3.88e-04 2.27 1/160 9.24e-02 1.32 7.07e-05 2.46 h ku p - u p,h k l 2 (L 2 ) rate kp p - p p,h k l ∞ (L 2 ) rate 1/20 5.46E-02 — 4.37E-02 — 1/40 3.26E-02 0.74 2.04E-02 1.10 1/80 1.63E-02 1.00 8.74E-03 1.22 1/160 8.26E-03 0.98 2.91E-03 1.59 h kη - η h k l ∞ (H 1 ) rate 1/20 1.07E-01 — 1/40 5.95E-02 0.85 1/80 2.92E-02 1.03 1/160 1.05E-02 1.48 Effect of BJS condition Figure 5. Effect of BJS condition 0.586 Pressure 0 5.33e-11 Displacement 0 Free slip c BJS =0 0.71 Pressure 0 2.24e-09 Displacement 0 Slip with friction c BJS =1 Continuity of flux Table 2. Jump in fluxes across the interface, R Γ fp = R Γ fp u f · n f + ( ∂ η p ∂t + u p ) · n p Lagrange multiplier Nitsche h R Γ 1 R Γ 2 R Γ 1 R Γ 2 1/20 4.44E-12 3.86E-12 2.75E-01 2.75E-01 1/40 1.97E-12 1.97E-12 4.87E-03 4.87E-03 1/80 4.23E-13 4.24E-13 1.54E-03 1.54E-03 1/160 1.07E-13 1.06E-13 3.85E-04 3.85E-04 4. Applications to reservoir simulation Injection-production example Figure 7. Physical grid Figure 8. Phases Parameter Units Value Young’s modulus (KPa) 10 7 Fluid density (kg/m 3 ) 897.0 Dynamic viscosity (KPa s) 10 -6 Lame coefficient (KPa) 5/12 · 10 7 Lame coefficient (KPa) 5/18 · 10 7 Hydraulic conductivity (m 2 /KPa s) (200, 50) · 10 -6 Mass storativity (KPa -1 ) 6.89 × 10 -2 Biot-Willis constant 1.0 BJS coefficient 1.0 Total time (s) 300 u f · n = 10, u f · τ =0 on Γ inflow u p · n =0, on Γ left p p = 1000, on Γ top ∪ Γ right ∪ Γ bottom η =0, on Γ top ∪ Γ right ∪ Γ bottom Figure 9. Last step of injection (top) and production (bottom) phases Pressure 7.139e-5 0.00014 0.00021 0.000e+00 2.856e-04 Displacement 1651.8 2304.8 9.989e+02 2.958e+03 Pressure 1.942e-5 3.884e-5 5.826e-5 0.000e+00 7.767e-05 Displacement 852.74 7.158e+02 1.000e+03 Velocity 1651.8 2304.8 9.989e+02 2.958e+03 Pressure 2.1522 4.3045 6.4567 0.000e+00 8.609e+00 Velocity 852.74 7.158e+02 1.000e+03 Pressure 0.38149 0.76298 1.1445 8.487e-06 1.526e+00 Velocity 8.4738 2.215e-01 1.152e+01 2.8246 5.6492 Velocity 1.467 6.760e-02 2.024e+00 0.489 0.978 Heterogeneous permeability example Young’s modulus: E = E 0 (1 - φ 0.5 ) 2.1 Figure 10. Permeability, porosity and Young’s modulus, SPE data. 2e+04 Permeability 0.00371 0.4 Porosity 0 1e+07 Young’s mod. 3.42e+06 Figure 11. Last step of injection phase 0.000358 Displacement 0 10.2 Pressure -203 2.72e+04 Velocity 1.6e-11 15 Velocity 0 5. References [1]I. Ambartsumyan, E. Khattatov, I. Yotov and P. Zunino. A Lagrange multiplier method for flow in fractured poroelastic media. Preprint. [2]M. Bukac, I. Yotov, R. Zakerzadeh and P. Zunino. Partitioning strategies for the interaction of a fluid with a poroelastic. Comput. Methods Appl. Mech. Engrg. 292 (2015) 138170.
Transcript of A Lagrange multiplier method for flow in fractured poroelastic … · 2019. 6. 27. · 0.000358...
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A Lagrange multiplier method for flow in fractured poroelastic mediaEldar Khattatov†, Ivan Yotov†, Ilona Ambartsumyan†, Paolo Zunino‡
† Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania, USA;‡ Department of Mechanical Engineering & Materials Science, University of Pittsburgh, Pittsburgh, Pennsylvania, USA;
1. Model Problem: Coupled Stokes-Biot System
•Groundwater flow and flow in fractured porous media
•Cardiovascular flow
Figure 1. Applications
Figure 2. Stokes-Biot model
Biot system of poroelasticity in Ωp:up = Darcy velocity, pp = pressure, η = displacement
−∇ · σp(η) = fp∂
∂t(s0pp + α∇ · η) +∇ · up = qp,
K−1up = −∇pp,
where σp(η) = λp(∇ · η)I + 2µpD(η) − αppI is a poroelastic stresstensor, α is Biot-Willis constant and K is a permeability tensor.
Stokes flow in Ωf :uf = Stokes velocity, pf = fluid pressure
ρf∂uf∂t−∇ · σf = ff ,∇ · uf = qf ,
where σf = −pfI + 2µfD(uf) is a Cauchy stress tensor.
Interface conditions on Γfp:
Mass conservation:uf · n = (
∂η∂t + up) · n
Balance of normal fluid stress:−(σfn) · n = pp
Continuity of momentum:σfn = σpn
No slip or Beavers-Joseph-Saffman condition:
• uf · τ =∂η∂t · τ
• −(σfn) · τ = cBJS(uf −∂η∂t ) · τ
2. Discretization & numerical analysis
The discretization
• is derived on simplicial grids in 2D and 3D;
• uses standard conforming finite elements for Stokes and elasticityequations and mixed finite elements for Darcy;
• allows the fluid and poroelastic region grids to be non-matchingthrough the interface;
• utilizes a Lagrange multiplier λ on the interface to weakly imposethe continuity of flux;
• uses the space for λ being the normal trace of Darcy velocityspace.
Figure 3. Example of lowest order FE spaces: (P2 × P1)× (RT 0 × P0)× P1 × P0
Discrete weak fomulation
Find (uf,h, pf,h,up,h, pp,h,ηp,h, λh) in Vf,h × Wf,h × Vp,h × Wp,h ×Xp,h × Λh such that
(ρf∂tuf,h,vf,h) + af (uf,h,vf,h) + adp(up,h,vp,h) + a
ep,h(ηp,h, ξp,h)
+aBJS(uf,h,ηp,h; vf,h, ξp,h) + bf (vf,h, pf,h) + bp(vp,h, pp,h)
+αbp(ξp,h, pp,h) + bΓ(vf,h,vp,h, ξp,h;λh) = f(vf ,h, ξp,h),
(∂ts0pp,h, wp,h)− αbp(∂tηp,h, wp,h)− bp(up,h, wp,h)−bf (uf,h, wf,h) = q(wf,h, wp,h),
bΓ(uf,h,up,h, ∂tηp,h;µh) = 0,
With Backward Euler used for discrete time derivative
dτun := τ−1(un − un−1)
Convergence analysis of the discrete formulationWith kf , sf , kp, sp, ks denoting the order of polynomials used in thediscretization of Stokes, Darcy and elasticity equations respectively,there holds
‖uf − uf,h‖L∞(L2) + ‖uf − uf,h‖L2(H1) + ‖up − up,h‖L2(L2)+‖pp − pp,h‖L∞(L2) + ‖ηp − ηp,h‖L∞(H1)
+
d−1∑j=1
νcBJS‖K−1/4j
((uf − uf,h)− ∂t(ηp − ηp,h)
)· τ f,j‖L2(L2(Γfp))
≤ C(hkf‖uf‖L2(Hkf+1) + h
kf‖uf‖L∞(Hkf+1) + hkf∥∥∂tuf∥∥L2(Hkf+1)
+hsf+1‖pf‖L2(Hsf+1) + hkp+1‖up‖L2(Hkp+1) + h
ks∥∥∂tηp∥∥L2(Hks+1)
+hkp+1‖λ‖L2(Hkp+1(Γfp)) + hks‖ηp‖L∞(Hks+1) + h
kp+1‖λ‖L∞(Hkp+1(Γfp))
+hsp+1‖pp‖L∞(Hsp+1) + hkp+1 ‖∂tλ‖L2(Hkp+1(Γfp)) + h
sp+1‖pp‖L2(Hsp+1)
)
3. Convergence, BJS condition and continuity of flux
Convergence study on a reference domain
Figure 4. Reference grid
Parameter Units ValueYoung’s modulus (Pa) 1010
Fluid density (kg/m3) 1.0Dynamic viscosity (Pa s) 1.0Lame coefficient (Pa) 5/12 · 1010Lame coefficient (Pa) 5/18 · 1010Hydraulic conductivity (m2/Pa s) IdMass storativity (Pa−1) 1.0Biot-Willis constant 1.0BJS coefficient 1.0Total time (s) 1.0
uf · n = 10, uf · τ = 0 on Γinflowup · n = 0, on Γleft
pp = 0, on Γtop ∪ Γright ∪ Γbottomη = 0, on Γtop ∪ Γright ∪ Γbottom
Table 1. Convergence study using the lowest order elements
h ‖uf − uf,h‖l2(H1) rate ‖uf − uf,h‖l∞(L2) rate1/20 1.33e+00 0.00 9.58e-03 0.001/40 5.11e-01 1.38 1.87e-03 2.361/80 2.31e-01 1.15 3.88e-04 2.27
1/160 9.24e-02 1.32 7.07e-05 2.46h ‖up − up,h‖l2(L2) rate ‖pp − pp,h‖l∞(L2) rate
1/20 5.46E-02 — 4.37E-02 —1/40 3.26E-02 0.74 2.04E-02 1.101/80 1.63E-02 1.00 8.74E-03 1.22
1/160 8.26E-03 0.98 2.91E-03 1.59h ‖η − ηh‖l∞(H1) rate
1/20 1.07E-01 —1/40 5.95E-02 0.851/80 2.92E-02 1.03
1/160 1.05E-02 1.48
Effect of BJS conditionFigure 5. Effect of BJS condition
0.586
Pressure
0
5.33e-11Displacement
0
Free slip cBJS = 0
0.71
Pressure
0
2.24e-09Displacement
0
Slip with friction cBJS = 1Continuity of fluxTable 2. Jump in fluxes across the interface, RΓfp =
∫Γfp
(uf · nf +
(∂ηp∂t + up) · np
)Lagrange multiplier Nitsche
h RΓ1 RΓ2 RΓ1 RΓ21/20 4.44E-12 3.86E-12 2.75E-01 2.75E-011/40 1.97E-12 1.97E-12 4.87E-03 4.87E-031/80 4.23E-13 4.24E-13 1.54E-03 1.54E-03
1/160 1.07E-13 1.06E-13 3.85E-04 3.85E-04
4. Applications to reservoir simulation
Injection-production example
Figure 7. Physical grid
Figure 8. Phases
Parameter Units ValueYoung’s modulus (KPa) 107
Fluid density (kg/m3) 897.0Dynamic viscosity (KPa s) 10−6
Lame coefficient (KPa) 5/12 · 107Lame coefficient (KPa) 5/18 · 107Hydraulic conductivity (m2/KPa s) (200, 50) · 10−6Mass storativity (KPa−1) 6.89× 10−2Biot-Willis constant 1.0BJS coefficient 1.0Total time (s) 300
uf · n = 10, uf · τ = 0 on Γinflowup · n = 0, on Γleft
pp = 1000, on Γtop ∪ Γright ∪ Γbottomη = 0, on Γtop ∪ Γright ∪ Γbottom
Figure 9. Last step of injection (top) and production (bottom) phasesPressure
7.139e-5
0.00014
0.00021
0.000e+00
2.856e-04
Displacement
1651.8
2304.8
9.989e+02
2.958e+03
Pressure
1.942e-5
3.884e-5
5.826e-5
0.000e+00
7.767e-05
Displacement
852.74
7.158e+02
1.000e+03
Velocity
1651.8
2304.8
9.989e+02
2.958e+03
Pressure
2.1522
4.3045
6.4567
0.000e+00
8.609e+00
Velocity
852.74
7.158e+02
1.000e+03
Pressure
0.38149
0.76298
1.1445
8.487e-06
1.526e+00
Velocity
8.4738
2.215e-01
1.152e+01
2.8246
5.6492
Velocity
1.467
6.760e-02
2.024e+00
0.489
0.978
Heterogeneous permeability exampleYoung’s modulus: E = E0(1− φ0.5)
2.1
Figure 10. Permeability, porosity and Young’s modulus, SPE data.
2e+04Permeability
0.00371
0.4Porosity
0
1e+07Young’s mod.
3.42e+06
Figure 11. Last step of injection phase
0.000358Displacement
0
10.2
Pressure
-203
2.72e+04
Velocity
1.6e-11
15Velocity
0
5. References
[1] I. Ambartsumyan, E. Khattatov, I. Yotov and P. Zunino. A Lagrange multipliermethod for flow in fractured poroelastic media. Preprint.
[2] M. Bukac, I. Yotov, R. Zakerzadeh and P. Zunino. Partitioning strategies for theinteraction of a fluid with a poroelastic. Comput. Methods Appl. Mech. Engrg.292 (2015) 138170.
