Numerical Simulation for Flow in 3D Highly Heterogeneous Fractured Media

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Numerical Simulation for Flow in 3D Highly Heterogeneous

Fractured Media

H. MustaphaJ. Erhel

J.R. De Dreuzy

H. Mustapha INRIA, SIAM Juin 2005

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Outline

Fractured media geometrical model flow fluid model

Mesh requirements for Finite Element Methods numerical method mesh generation difficulties

New approach for computing flow based on a projection method main idea and some examples quality of the mesh and precision of the computed solution

Conclusions and future work

H. Mustapha INRIA, SIAM Juin 2005

3

Outline

Fractured media geometrical model flow fluid model

Mesh requirements for Finite Element Methods numerical method mesh generation difficulties

New approach for computing flow based on a projection method main idea and some examples quality of the mesh and precision of the computed solution

Conclusions and future work

H. Mustapha INRIA, SIAM Juin 2005

4

Geometrical model

• Discrete fracture network. • Impervious matrix.

• Only the fractures are considered.

• Network = Set of fractures.

H. Mustapha INRIA, SIAM Juin 2005

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Equations Q = - K*grad (h) div (Q) = 0

Boundary conditions

Flow fluid model

Fixed head

(Dirichlet)

Q.n = 0(Neuman

n)

H. Mustapha INRIA, SIAM Juin 2005

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Outline

Fractured media geometrical model flow fluid model

Mesh requirements for Finite Element Methods numerical method mesh generation difficulties

New approach for computing flow based on a projection method main idea and some examples quality of the mesh and precision of the computed solution

Conclusions and future works

H. Mustapha INRIA, SIAM Juin 2005

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Numerical method

Mixed Hybrid Finite Element Method complete 3D mesh of the network mesh of each fracture with identical intersections (conforming mesh)

Global linear system Assembled by all corresponding fractures linear systems Direct linear solver

Conforming mesh

H. Mustapha INRIA, SIAM Juin 2005

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Geometrical complexity: fractures and network

Origin of the complexity

• Important number of intersections.

• Existence of small intersections.

• Existence of zones containing small angles => need mesh refinement to improve the quality.

Our approachTo modify the complex

configurations• What is the simplifications criteria ?

• What is the loss in precision ?

• What are the CPU time and memory capacity improvements ?

• The blue counter is the fracture border.

• The red lines are the intersections with the cube borders.

• The black lines are the intersections with the other fractures of the network.

H. Mustapha INRIA, SIAM Juin 2005

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Network properties• size : 18• number of fractures: 285

Generation and quality of the mesh for the fracture networks

Mesh After refinement

0 20 40 60 80 1001201401601800

4

8

12

16

% N

um

ber

of

ang

les

Angle in degree

Distribution of angles

H. Mustapha INRIA, SIAM Juin 2005

zoom before refinement

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Outline

Fractured media geometrical model flow fluid model

Problem presentation geometrical complexity in two scales: fractures and

network classical mesh generation for fracture networks

New approach for computing flow based on a projection method main idea and some examples quality of the mesh and precision of the computed solution

Conclusions and future works

H. Mustapha INRIA, SIAM Juin 2005

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Main ideaStep (1)

A

B

X

Y

C

D

2D Projection

3D projection

Generalization

H. Mustapha INRIA, SIAM Juin 2005

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Main ideaStep (2)

2D projection

2D projection

H. Mustapha INRIA, SIAM Juin 2005

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Results of our approachExample 1: projection method

H. Mustapha INRIA, SIAM Juin 2005

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Results of our approachExample 2: projection and generation mesh

H. Mustapha INRIA, SIAM Juin 2005

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Generation and mesh quality using the new approach

0 20 40 60 80 100 120 140 160 1800

4

8

12

16

% N

um

be

r o

f an

gle

s

A ngle in degree

Distribution of angles

0 20 40 60 80 100 120 140 160 1800

4

8

12

16

% N

um

be

r o

f an

gle

s

A ngle in degree

Distribution of angles M esh with projection M esh without projection

Mesh with projection

Network properties• size : 18 number of fractures: 285

H. Mustapha INRIA, SIAM Juin 2005

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Our new approach:

• Use a projection method as a simple criteria

• Lead to a reduced configuration

• Allows to mesh complex fracture networks

Questions to address:

•What is the loss in precision ?

•What are the CPU time and memory capacity improvements ?

Summary

H. Mustapha INRIA, SIAM Juin 2005

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Precision of computed solution

L

Q

h1

h2

Math formulas:

Global equivalent permeability:

Average flow across the fractures: , N: number of fractures

Average flow across the intersections: , M: number of intersections

Errors:EN =

EF =

EI =

MRI

Rh2))-(h1*Q/(LK

NRF

1ref1refapp ||F||/||F-F||

1ref1refapp ||I||/||I-I||

|K| / |K - K| refrefapp

H. Mustapha INRIA, SIAM Juin 2005

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0 20 40 60 80 100 120 140 160 1800

4

8

12

16

20

% N

um

be

r o

f an

gle

s

A ngle in degree

Mesh with projection Mesh without projection

Very small mesh step

Kref 45.9017

Kapp 45.9015

EN 4×10-6

EF 1.8×10-5

EI 2×10-5

• Solution obtained with high precision

• Approximate solution can be used as a reference solution

Precision of computed solution

With MHFE method

Network properties• size : 18•number of fractures: 285

H. Mustapha INRIA, SIAM Juin 2005

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Examples of mesh and computed solution

3 fractures

Head450 fractures

Head

Head

1800 fractures

h1

h2

H. Mustapha INRIA, SIAM Juin 2005

Network mesh: 285 fractures

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Precision and numerical error

•Loss in precision: 5%•Good quality of the mesh

0 20 40 60 80 100 120 140 160 1800

4

8

12

16

20

% N

um

be

r o

f an

gle

s

A ngle in degree

Mesh step=0.08 Mesh step=0.11 Mesh step=0.15 Mesh step=0.19

0,06 0,09 0,12 0,15 0,18 0,210,0

1,5

3,0

4,5

M esh step

EN×100 EF ×100 E I ×100

Reference solutionMesh step : 0.08

H. Mustapha INRIA, SIAM Juin 2005

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mesh steplinear system

sizeCPU time

0.08 313 776 27.65

0.09 277 169 19.21

0.11 185 611 12.12

0.13 131 029 7.65

0.15 98 025 5.18

0.17 76 798 3.98

0.19 62 460 3

0.21 50 588 2.36

•Memory usage decrease: 80% •CPU time decrease: 90%

H. Mustapha INRIA, SIAM Juin 2005

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Outline

Fractured media geometrical model flow fluid model

Problem presentation geometrical complexity in two scales: fractures and network classical mesh generation for fracture networks

New approach for computing flow based on a projection method main idea and some examples quality of the mesh and precision of the computed solution

Conclusions and future works

H. Mustapha INRIA, SIAM Juin 2005

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Conclusions

• Mesh generation of complex fracture networks.

• Approximate geometry by projection.

• The loss in precision of computed solution is very small.

• The improvement in CPU time and memory capacity is very important.

Future work

• Parallel computation.

• Stochastic experiments.

H. Mustapha INRIA, SIAM Juin 2005