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FEM study of the faults activation

Technische Universität München

Joint Advanced Student School (JASS)

St. Petersburg Polytechnical University

Author: Ulanov Alexander

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Problem significance Geomechanics application:

- Subsidence of rocks

- Sliding of bed near oil well

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Аrea of study Faults activation in deforming saturated porous medium.

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Particularity

Examples of the elastic bodies (3D case and 2D case) with the possible surfaces of slipping.

Interface (contact) element concept

Parameters of the media may discontinue

Nonlinear problem

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Goals and objectives Simulation of joint transient process of diffusion porous pressure and stress state calculation in saturated porous medium.

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Our estimates

Еstimates:

Saturation porous medium - combination of pore space, deformable skeleton and moving fluid.

- Darcy's law for fluid.

- Fluid is compressible.

- Porous medium is isotropic and linear.

- Small deflection.

Examples:sandstone,clay.

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Continuity equation:

udivt

bpk

t

pM

Equilibrium equation:

pbdivG uu

p - pore pressure

u - displacement vector

k - coefficient of permeability

μ - viscosity of the pore fluid

Coupled solution for saturated one-phase flow in a deforming porous medium

G - shear modulus

Biot 1955

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Variational formulation (part 1)

312 )(q, WU

Ω – domain in 2D(3D) space; S - boundary; n – external normal

Variational formulation of equilibrium equation

0: S

TT dsdivGdpbdivdivG qnuIuqququ

pbdivG uu

dsn

ddS

T qu

ququ :

dsfdfdivdfS

nqqq

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)(, 12 Wqp

Variational formulation of continuity equation

Variational formulation (part 2)

udivt

bpk

t

pM

0)(

S

qdspk

dqt

divbqp

kq

tp

M nu

10

cSSSS \21

Sc – surface of contact

0:

cS

nn

S

nT dsdsdbpdivdivdivG qFqFqFqququ

Interface model

0)(

cS

nn

S

n dsqQqQqdsQdqt

divbqp

kq

tp

Mu

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Characteristics of interface layer elements: - infinitesimal thickness - permeability D - stiffness C

0)()(:

cSS

nT dsdsdbpdivdivdivG qquuCqFqququ

)(1 uuCnFFn)(2

uuCnFFn

)( ppDQQ nn

Interface element concept

Goodman 1968

0))(()(

cSS

n dsqqppDqdsQdqt

divbqp

kq

tp

Mu

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Slip computation

Slipping condition (Mohr-Coulomb) :

HnS CK ||

σn - normal stress

σs - shear stress

K - friction coefficient

СH - cohesion stress

Iterative process:

n

s

s

C

C

C

00

00

00

C

Stiffness С:

If contact element is sliding Cs = 0 1 Calculation of strain state.

2 Slip conditional test.

3 Calculation of strain with new stiffnesses form.

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Program structure

Geometry and Grid generation

- Ansys ICEM

- Gambit

Solution of problem- FEM solver

- Optimization of data ( sparce-matrix )

- Iteration lib (ITL MTL)

Processing and result аnalysis

- GID

- Tecplot10

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Mesh generation in Gambit (format .CDB )

GID output

Domain example (1)

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Domain example (2)

Mesh quality adaptation

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Results

No interface (slip) zone Modelling of sliding

- Diffusion effect -Influence of cohesion Coh

-Influence of permeability k

-Influence of nonuniform permeability D

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No interface (slip) zone. Diffusion effect

- Pressure on lateral side is fixed- Zero-initial condition for pressure- No fluid flux in normal direction

Establishment of linear pressure distribution

0nQ

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Influence of cohesion (Coh)

Cohesion Count of elements

2.9 2

2.7 4

2.6 8

- Fixed pressure- Fixed permeability k- External load- Slipping condition

Relative displacement of interface layer

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Influence of permeability k

- Fixed value of cohesion

- Diffusion effect ( P = constant )

- Different value of permeability k permeability

k Сoh Number of

elements

0.1 2.7 22

0.2 2.7 8

0.3 2.7 4

1 2.7 4

- External load

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Effect of nonuniform permeabilityDestruction of rock in contact layer

slslsl BAS , – Slip zone in contact layer

dqqppDdqqppDdxqQqQslcc

xyxysnnn 41

slipisc kkk

kis – isotropic permeability ( no slip case )

kslip – additional component ( appear in slip case )

0)(

cS

nn

S

n dsqQqQqdsQdqt

divbqp

kq

tp

Mu

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- Different value of Ds

- Establishment of linear pressure distribution

- Zero-initial condition for pressure

- Zero displacement

- Sliding on all contact layer

Influence of nonuniform permeability ( part 1 )

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Influence of nonuniform permeability ( part 1 )

Ds=1

Ds=10 Ds=100

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Influence of nonuniform permeability ( part 2 ) - Establishment of linear pressure distribution- Zero-initial condition for pressure

- External load - Ds=100

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Conclusions

The model of coupled solution for saturated one-phase flow in a deforming porous medium is considered.

Influence of various parameters on sliding is investigated .

Goodman interface element concept is used.

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Thank you for your attention !