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Motivation Modelling 2-Scale Model Local PDEs Simulation Mathematical and Computational Modelling of Saturated Poroplastic Materials J.Alexei Lu zar Obreg on, SEN/UFF M. Murad, LNCC/MCT VIII Workshop in Partial Differential

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SLIDE 1

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Mathematical and Computational Modelling of Saturated Poroplastic Materials

J.Alexei Lu´ ızar Obreg´

  • n,

SEN/UFF

  • M. Murad,

LNCC/MCT

VIII Workshop in Partial Differential Equations Rio de Janeiro

25-28/08/2009

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 2

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Mathematical and Computational Modelling of Saturated Poroplastic Materials

J.Alexei Lu´ ızar Obreg´

  • n,

SEN/UFF

  • M. Murad,

LNCC/MCT

VIII Workshop in Partial Differential Equations Rio de Janeiro

25-28/08/2009

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 3

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Outline of presentation

1

Motivation

2

Pore–Scale Modelling

3

Two-scale Poroplastic Model

4

Local PDE’s Effective Parameters Discrete Local PDE’s Numerical Results

5

Reservoir Simulation

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 4

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Motivation

Geomaterials: multiscale porous media

Large-scale (reservoir). Darcy-scale (laboratory). Pore-scale.

  • Solid

Large-scale Pore-scale Darcy-scale Phase Fluid Phase

Irreversible behavior Compaction is usually caused by changes at the microscopic level within the rock mass.

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 5

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Classical Macroscopic Modelling

div

  • σeff

s

− αp0 = 0 σeff

s

= C

  • Ex
  • u0

− Eeff

p

  • α : Ex

∂u0 ∂t

  • − div
  • K∇p0

= 1 β ∂p0 ∂t Fsat

  • σ0

T, p0

≡ Fdry

  • σeff

s

  • ?

dEeff

p

dt = γ ∂Fdry ∂σeff

s

  • σeff

s

  • Fdry
  • σeff

s

  • ≤ 0,

γ ≥ 0, γFdry

  • σeff

s

  • = 0,

γ dFdry dt

  • σeff

s

  • = 0.

Theory, Coussy (1995), Mechanics of Porous Media Experimental Results!, Pietruszczak and Pande (1995), Int. J.

  • Num. Anal. Meth. Geomech., v.19

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 6

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Pore–Scale Modelling

Asymptotic expansion technique of Homogenization

Auriault & Sanchez-Palencia (1977) Terada (1998) (Linear Coupling) Suquet (1985) (Non–linear one phase)

Separation of scales: ǫ = l/L ≪ 1. Constitutive laws defined at the pore-scale. Up–scale Effective Medium

Homogenized Domain Stokesian Fluid l L Elastoplastic Solid l L < < Continuum Mechanics Laws Darcy-scale Pore-scale

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 7

Motivation Modelling 2-Scale Model Local PDE’s Simulation

PDE’s at Microscopic Model

Stokesian Fluid: divv = 0 divσf = 0 σf = −pI + 2µE (v)    in Ωf

Ω Γ s f f Ωs

Elastoplastic Solid: divσs = 0; σs = aEe (u) Ee (u) = E (u) − Ep; ∂Ep ∂t = γ ∂f ∂σ (σs) γf (σs) = 0 and f (σs) ≤ 0 γ ≥ 0 γ ∂f ∂t (σs) = 0              in Ωs Boundary conditions v = ∂u ∂t and σsn = σf n

  • n Γsf .

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 8

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Homogenization Technique

Family of domains Ωǫ composed of periods Y ǫ. An ǫ-model in Ωǫ consists scaled PDE’s on copies Y ǫ ≃ Y . Associate to each {uǫ, σǫ

s, γǫ, Eǫ p, σǫ f , vǫ, pǫ} a two-length

scale dependence. Determine the effective macroscopic description equivalent to the asymptotic behavior: ǫ → 0.

= y x ε ε ε ε ε

=0,333 =0,2 =0,1

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 9

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Asymptotic expansion technique of Homogenization

Each Φǫ(x) = {uǫ, σǫ

s, γǫ, Eǫ p, σǫ f , vǫ, pǫ} expand in terms of ǫ

Φǫ(x) = Φ0(x, y) + ǫΦ1(x, y) + ǫ2Φ2(x, y) + · · · Fluid: divyv0 = 0; divxv0 + divyv1 = 0; σ0

f = −p0I;

σ1

f = −p1I + 2µEy

  • v0

; divyσ0

f = 0;

divxσ0

f + divyσ1 f = 0

Elasto-plastic solid: σ0

s = a

  • Ex
  • u0

+ Ey

  • u1

− E0

p

  • ;

divy

  • aEy
  • u0

= 0; divx

  • aEy
  • u0

+ divyσ0

s = 0;

divxσ0

s + divyσ1 s = 0;

∂E0

p

∂t = γ0 ∂f ∂σ0

s

  • σ0

s

  • ;

γ0f (σ0

s) = 0;

γ0 ∂f ∂t

  • σ0

s

  • = 0 .

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 10

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Two-scale poroplastic model derived from Homogenization

  • divx
  • σeff − αp0

= 0 σeff = C

  • Ex
  • u0

− Eeff

p

  • −divx
  • K∇xp0

+ α∗

p : Ex

∂u0 ∂t

  • = 1

βp ∂p0 ∂t        in Ω Eeff

p

≡ C−1 a

  • E0

p − Ey

  • unl

α∗

p ≡ α −

∂G ∂Ex (u0) βp ≡ 1 β + ∂G ∂p0 −1 G ≡ divyunl = G

  • Ex
  • u0

, p0

  • divy
  • a
  • Ey
  • unl

− E0

p

  • = 0

∂E0

p

∂t = γ0 ∂f ∂σ0

s

  • σ0

s

  • σ0

s = a

  • (II + Ey (ξ)) Ex
  • u0

+ Ey (η) p0 +Ey

  • unl

− E0

p

  • γ0f (σ0

s) = 0

γ0 ≥ 0 f (σ0

s) ≤ 0

γ0 ∂f ∂t

  • σ0

s

  • = 0

                     in Ys a

  • Ey
  • unl

− E0

p

  • n = 0
  • n

∂Ysf

s p

α* βp εp

eff

y x

Y

Ω

C

α K

(u 0)

εX

p

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 11

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Local Elastoplastic Problem

α∗

p = α −

∂G ∂Ex (u0) βp = 1 β + ∂G ∂p0 −1 G ≡ divyunl Given Ex

  • u0 (x, t)
  • and p0 (x, t), find {u1, E0

p}

divyσ0

s = 0

σ0

s = a

  • Ex
  • u0

+ Ey

  • u1

− E0

p

  • ∂E0

p

∂t = γ0 ∂f ∂σ0

s

  • σ0

s

  • γ0f (σ0

s) = 0

γ0 ≥ 0 f (σ0

s) ≤ 0

γ0 ∂f ∂t

  • σ0

s

  • = 0

                   in Ys a

  • Ex
  • u0

+ Ey

  • u1

− E0

p

  • n = −p0n on ∂Ysf .

u 0)

εx

(u 0)

εx

( ( u

0)

εx

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 12

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Effective yield surface

Overall strength domain D: admissible macroscopic

  • σeff

s , p0

admissibility ≡

  • σeff

s , p0

equilibrium with local stresses ∂D ≡ Feff

extr

  • σeff

s , p0

extremal surface [Suquet (1985)]

eff

F

=F

e

D

e x t r e f f init Outer boundary

D

Initial Surface =0 Closure of the strictly elastic states Extremal Surface =0

  • f the macroscopic

strength domain

Initial surface Feff

init

  • σeff

s , p0

= 0 Determination: strain-paths or stress paths [Michel et al. (1999)].

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 13

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Fully discrete formulation of the local problem

Point-projection algorithm Prescribed Ex

  • u0,n

and p0,n.

  • Ξk −Υk ⊗ ΥT

k

  • Ey
  • ∆u1,n

h,k

  • , Ey
  • w1

h

  • 1 = −F1

k

  • u1,n

h,k

  • where

F1 u1,n

h

  • =
  • a
  • Ex
  • u0,n

+ En

y

  • u1,n

h

  • − E0,n

p,h

  • , En

y

  • w1,n

h

  • 1 +
  • p0,n, w1

h

  • ∂Ysf

Ξk =   a−1 + ¯ γn

k

∂2f n

k

∂

  • σ0,n

h,k

2   

−1

Υk = Ξk ∂f n

k

∂σ0,n

h,k

  • ∂f n

k

∂σ0,n

h,k

T Ξk

  • ∂f n

k

∂σ0,n

h,k

  • Rede Siger. SEN/UFF– LNCC/MCT

VIII WORKSHOP PDE /Rio-2009

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SLIDE 14

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Elastoplastic Periodic Cell

Elastoplastic solid obeying Mohr–Coulomb criterion Cell geometries

X Y

I

X Y

II

X Y

III

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 15

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Mesh and Boundary Conditions

I II III

l l l l

  • x

y

l

  • x

y

  • x

y

ry ry ry x r Ysf

e

Ysf

e

Ysf

e

po po po

x

l

r x r

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 16

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Computational Results for Effective Coefficients

Non-linear poromechanical parameters α∗

p and βp

Strain-path procedure Geometry I, porosity 0.126

X Y

3.50 3.75 4.00 4.25 4.50 4.75 5.00 1.00 1.50 2.00 2.50 3.00

tr(εx(u0))

|βp|/E

p0=00 p0=20 p0=40 p0=60 p0=80

x10-3 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 1.00 1.50 2.00 2.50 3.00 3.50

tr(εx(u0)) tr(α*

p) p0=00 p0=20 p0=40 p0=60 p0=80

x10-3

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 17

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Stress Constitutive Laws

For a compresion experiment

100 200 300 400 500 MPa 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

tr(σeff

s )

tr(εx(u0))

σ0

T+αp0

p = 80 p = 60 p = 40 p = 20 p = 00

x10-2

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 18

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Level curves for the initial yield surfaces

Geometry I, porosity 0.4

X Y

−125 −100 −75 −50 −25 25 50 MPa −125 −100 −75 −50 −25 25 50MPa

Σy Σx

σ0

T+αp0

p0=0 p0=50 p0=100

−125 −100 −75 −50 −25 25 50 MPa −125 −100 −75 −50 −25 25 50MPa

Σy Σx

σ0

T+p0I

p0=0 p0=50 p0=100 Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 19

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Level curves for the initial yield surfaces

Geometry II, porosity 0.091

X Y

−225 −200 −175 −150 −125 −100 −75 −50 −25 25 50 MPa −350 −300 −250 −200 −150 −100 −50 50 MPa

Σy Σx

σ0

T+αp0

p0=0 p0=30 p0=60 p0=90 p0=100

−175 −150 −125 −100 −75 −50 −25 25 50 75 MPa −300 −250 −200 −150 −100 −50 50 100MPa

Σy Σx

σ0

T+p0I

p0=0 p0=30 p0=60 p0=90 p0=100 Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 20

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Level curves for the initial yield surfaces

Geometry III, porosity 0.40

X Y

−175 −150 −125 −100 −75 −50 −25 25 50 MPa −175 −150 −125 −100 −75 −50 −25 25 50 MPa

Σy Σx

σ0

T+αp0

p0=0 p0=50 p0=100

−125 −100 −75 −50 −25 25 50 75 MPa −125 −100 −75 −50 −25 25 50 75 MPa

Σy Σx

σ0

T+p0I

p0=0 p0=50 p0=100 Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 21

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Numerical reconstruction of initial surface

Geometry III, porosity 0.4

X Y

σ0

T+αp0 −140 −120 −100 −80 −60 −40 −20 20 40 60

Σ1(MPa)

−160−140−120−100 −80 −60 −40 −20 20 40 60

Σ2(MPa)

20 40 60 80 100

p0(MPa)

σ0

T+p0I −120 −100 −80 −60 −40 −20 20 40 60

Σ1(MPa)

−140−120−100 −80 −60 −40 −20 20 40 60

Σ2(MPa)

20 40 60 80 100

p0(MPa)

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 22

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Linear aproximation of the initial surface

Compression range (σn, τmax)

20 40 60 80 100 MPa 20 40 60 80 100 120 140 MPa

τmax σn

σ0

T+αp0

p=000 p=050 p=100

20 40 60 80 100 MPa 20 40 60 80 100 120 140MPa

τmax σn

σ0

T+p0I

p=000 p=050 p=100

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 23

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Effective Parameters of Linear aproximation

The linear relation ≡ Mohr Coulomb function with effective cohesion dependent on pore-pressure

50 60 70 80 90 100 110 120 MPa 20 40 60 80 100 MPa

Ceff Effective Cohesion Pore−Pressure Ceff=60.437+0.27 p0 Ceff=60.1738+0.525 p0

σ0

T+p0I

σ0

T+αp0

20.0° 25.0° 30.0° 35.0° 40.0° 45.0° 20 40 60 80 100 MPa

φeff Effective Friction Angle Pore−Pressure φeff=31.32+0.0190 p0 φeff=31.57+0.0471 p0

σ0

T+p0I

σ0

T+αp0

ceff

sat ≡ ceff dry + p0 tan φ

Feff

init,sat

  • σeff

s , p0, φ, ceff sat

  • ≡ Feff ,

init,dry

  • σeff

s , φ, ceff dry + p0 tan φ

  • Rede Siger. SEN/UFF– LNCC/MCT

VIII WORKSHOP PDE /Rio-2009

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SLIDE 24

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Modified model with effective yield surface

Neglecting the hardening effects from initial to extremal surfaces and assuming that plastic strain obeys principle of maximum plastic dissipation divx

  • C
  • Ex
  • u0

− Eeff

p

  • − αp0

= 0 α∗

p : Ex

∂u0 ∂t

  • − divx
  • K∇xp0

= 1 βp ∂p0 ∂t dEeff

p

dt = γeff ∂Feff

init

∂σeff

s

  • σeff

s , p0

Feff

init

  • σeff

s , p0

≤ 0, γeff ≥ 0, γeff Feff

init

  • σeff

s , p0

= 0, γeff dFeff

init

dt

  • σeff

s , p0

= 0                        boundary and initial conditions

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 25

Motivation Modelling 2-Scale Model Local PDE’s Simulation Effective Parameters Discrete Local PDE’s Numerical Results

Fully discrete formulation of macro-problem

  • Ξ0

k −Υ0 k ⊗ Υ0,T k

  • Ex
  • ∆u0,n

H,k

  • , Ex
  • w0

H

  • 0−
  • ∆p0,n

H,k, divxw0 H

  • 0=−F1
  • u0,n

H,k, p0,n H,k

  • div
  • ∆u0,n

H,k

  • , q0

H

  • + ∆tK
  • ∇x
  • ∆p0,n

H,k

  • , ∇xq0

H

  • = −F2
  • u0,n

H,k, p0,n H,k

  • with Ξ0

k and Υ0 k

Ξ0

k =

  C−1 + ¯ γ0,n

k

∂2Feff

init,k

∂

  • σ0,n

H,k

2   

−1

Υ0

k =

Ξ0

k

∂Feff

init,k

∂σ0,n

H,k

  • ∂Feff

init,k

∂σ0,n

H,k

T Ξ0

k

  • ∂Feff

init,k

∂σ0,n

H,k

  • Rede Siger. SEN/UFF– LNCC/MCT

VIII WORKSHOP PDE /Rio-2009

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SLIDE 26

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Reservoir Compactation and Surface Subsidence

Heterogeneous reservoir: only variability in permeability Boundary conditions

impervious O

impervious well LOAD LOAD a H H

y x y free drainage

Sandstone Formation

x

Incremental Load applied

Load (t) t*

  • Fractal realization for permeability

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 27

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Plastified Regions

Feff

ini

  • σeff

s , p0

(Modified) Feff

ini

  • σeff

s

  • (Classic)

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 28

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Well discharge

Evolution of discharge for classical and effective poroplastic model with ceff = ceff (p0)

10 20 30 40 50 60 70 80 90 [m3/s] 0.0 0.2 0.5 0.8 1.0 1.2 1.5 1.8 2.0 Q Time[days]

Discharge

Classic Poroplastic Model Modified Poroplastic Model Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009

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SLIDE 29

Motivation Modelling 2-Scale Model Local PDE’s Simulation

Conclusions

In the framework of the homogenization theory

Non-validity of Terzaghi’s principle for Poroplasticity. Derivation and computation of the non-linear coupling poromechanical parameters appearing in the overall mass balance. Computation of the initial surfaces and the hardening effects induced by pore-pressure. Impacts of the effective model on reservoir compaction due to fluid withdrawal.

Now Working: Large–Scale Heterogeneus Reservoir

Stochastic Plastic Parameters: Pore Collapse Pressure, Stochastic Permeability. Unsaturated Porous Media (oil/water).

Rede Siger. SEN/UFF– LNCC/MCT VIII WORKSHOP PDE /Rio-2009