numerical simulation for flow in 3d highly heterogeneous fractured media

1

Numerical Simulation for Flow in 3D Highly Heterogeneous

Fractured Media

H. MustaphaJ. Erhel

J.R. De Dreuzy

H. Mustapha INRIA, SIAM Juin 2005

2

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


3

Outline




Conclusions and future work


4

Geometrical model

• Discrete fracture network. • Impervious matrix.

• Only the fractures are considered.

• Network = Set of fractures.


5

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

Boundary conditions

Flow fluid model

Fixed head

(Dirichlet)

Q.n = 0(Neuman

n)


6

Outline




Conclusions and future works


7

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


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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.


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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


zoom before refinement

10

Outline


Problem presentation geometrical complexity in two scales: fractures and

network classical mesh generation for fracture networks




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Main ideaStep (1)

A

B

X

Y

C

D

2D Projection

3D projection

Generalization


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Main ideaStep (2)

2D projection

2D projection


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Results of our approachExample 1: projection method


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Results of our approachExample 2: projection and generation mesh


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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


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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


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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


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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


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Examples of mesh and computed solution

3 fractures

Head450 fractures

Head

Head

1800 fractures

h1

h2


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


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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%


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Outline


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




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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.


numerical simulation for flow in 3d highly heterogeneous fractured media

Documents

siam juin

mustapha inria

mesh quality

projection methodmain

computing flow

need mesh refinement

computed solutionconclusions

projection methodh