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.