FLOW IN FRACTURES COUPLED WITH DEFORMATION EFFECTS ... · Marie Curie Workshop on Flow and...

______________________________________________________________________________________________________________________________________________________________________Marie Curie Workshop on Flow and Transport in Industrial Porous Marie Curie Workshop on Flow and Transport in Industrial Porous Media Media –– Utrecht Utrecht –– 1212--16/11/0716/11/07

FLOW IN FRACTURES COUPLED WITH DEFORMATION EFFECTS:

APPLICATION TO SEALING PROBLEMS

D. LASSEUX, C. VALLET

Other contributors C. MARIE, E. PERROY, G. CARILLONCollaboration: LaMCoS, LTDS (Lyon - France)

TREFLE – Université Bordeaux 1Esplanade des Arts et Métiers - 33405 TALENCE Cedex – France

Acknowledgements: CNRS, EDF, SNECMA, CNES


ContextContext

Flow through the contact between rough deformable surfaces

Static seal under severe thermodynamic conditionsDirect metal/metal tight contactApplications: spatial and nuclear industry, etc.

General problem:General problem:


DifficultiesDifficulties

Many parameters controlling the leak(some not controlled by the designer!):

- type of surfaces (texture, dimensions, … local defaults)- contrast of scales (from material grain size to contact scale)- surface materials- tightening (level, uniform, localized,

micro & macro deformations…)- type of fluid (gas, liquid, SC)- thermodynamic conditions (T, P, phase change)- operating conditions (tightening relaxation, vibrations,cycling etc.) - …


Literature is relatively poor:

* Gas flow in the rarefied regime(Armand et al., 1964; Lejay, 1967; Polycarpou and Etsion, 2000)

* Experiments in particular situations(Matsuzaki and Kazamaki, 1988; Yanagisawa et al., 1990, 1991)

* Contact models(Greenwood and Williamson, 1966; Nayak, 1973; Bush et al., 1975;McCool, 1985; Bushan, 1988; Kogut and Etsion, 2003)

Literature surveyLiterature survey

To be completed:Gas versus liquid? (Amesz, 1966)General approach (comparisons predictions-experiments)Analysis in terms of scales of surface defaults


GoalsGoals

… in the short term:- perform careful experiments under "well controlled" conditions- try to derive "complete" models- perform direct comparison between predictions and experiments- identify relevant parameters - how they can be controlled

… in the long term:- diagnose the expected leak on an existing configuration- improve the design for more efficient contact seals


- 1 rough surface, 1 "perfectly" flat(surface-sum, O'Callaghan and Probert, 1987)

- Deterministic and random textures(turned, sand-blasted and lapped surfaces)

RestrictionsRestrictions

- Fluid: liquid- T and P constant- Uniform load

Sand-blasted LappedTurned


OutlineOutline

1. ExperimentExperimental set-upResults

2. Physical models (transport)Micro-scaleMacro-scale

3. ComparisonsImportance of macro-scale defaults

4. Some more analysisImpact of some large scale defaults


ExperimentExperiment

Temperatureregulated water N2

Gas Phase Chromatography(GPC)

Sapphire

Load cell

P

Leaking fluid underpressure(Solute)

Solvent loop

Sample(metallic surface)

Hydraulic pump

Hydraulicjack

Reservoir

F

Peristaltic pump

Pressure transducer

Viscous conditionViscous condition

Temperatureregulated water

Gas Phase Chromatography(GPC)

Sapphire

Load cell

Solvent loop

Sample(metallic surface)

Hydraulic pump

HydraulicjackF

Peristaltic pump

Diffusive conditionDiffusive condition


Experiment (cont'd)Experiment (cont'd)

Patent CNES (France : N°02.01930 – Europe : N° 03.290358.5)


Experiment (cont'd)Experiment (cont'd)

Ra =0.5; 1 µm

20

1.0

φ 50

φ 40 0.3

Ra =0.5; 1 µm

20

1.0

φ 50

φ 40 0.3

Ra =0.5; 1 µm

20

1.0

φ 50

φ 40 0.3

1.0

φ 40 0.3

1.0

φ 40 0.3


Experiment: Results (viscous condition)Experiment: Results (viscous condition)

Eprouvette A

y = 0.50x + 2.49R2 = 0.999

y = 1.18x + 8.83R2 = 0.999

y = 1.86x + 13.63R2 = 0.998

0

50

100

150

200

250

0 20 40 60 80 100 120 140 160

t (min)

mas

s (m

g)

P=9bar P=9bar P=9bar P=20bar P=20bar P=30bar

Δ

Δ

Δ

ΔΔ

Δ

Turned 316L surface - Contact pressure: Pca = 265 MPaApparent contact area: 38 mm2 - Loadoad: 10 000 N: 10 000 N

Mass of solute versus time (GPC)


Contact pressure:Pca = 700 MPa

m = 1.8 10- 4 t + 9.46R2 = 0.5

9

9.5

10

10.5

0 500 1000 1500

t (min)

mas

s (m

g)ΔP=30bar

m = 5.6 10 - 5 t+ 9.89R2 = 0.10

9

9.5

10

10.5

11

0 500 1000 1500 2000 2500 3000t (min)

mas

s (m

g)

ΔP=10bar m = 1.2 10 - 4 t + 9.71R2 = 0.18

9

9.5

10

10.5

0 500 1000 1500t (min)

mas

s (m

g)

ΔP=20bar

Mass of solute versus time (GPC)Experiment: Results (viscous condition)Experiment: Results (viscous condition)


Mass flow-rate versus pressure drop

y = 0.3059x + 0.0119R2 = 0.9893

y = 0.1104x - 0.0021R2 = 0.9999

y = 0.0516x - 0.0025R2 = 0.9995

y = 0.0397x - 0.0113R2 = 0.9942

y = 0.0108x + 0.0012R2 = 0.9984

y = 0.004x + 0.0005R2 = 0.9992

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15ΔP (bar)

Q (m

g/m

in)

50Mpa75Mpa100Mpa125Mpa150Mpa300Mpa



Experiment: Results (viscous condition)Experiment: Results (viscous condition)Q is linear versus ΔP at a given load

( ) ieieii rrµ/rrrµrπ −≈=

ΔPKlnΔPK

2Q ρρ K: transmissivity (m3)

1.0E-091.0E-081.0E-071.0E-061.0E-051.0E-041.0E-031.0E-021.0E-01

0 200 400 600 800Apparent contact pressure Pca (MPa)

K (µ

m3 )

Sample B1Sample B2


1.0E-09

1.0E-07

1.0E-05

1.0E-03

1.0E-010 200 400 600 800

Pca (M Pa)

K (µ

m3 )

Sample ASample B1Sample B2

B A


Unload

Ra=0.5µmRa=1.1µm



1.0E-091.0E-081.0E-071.0E-061.0E-051.0E-041.0E-031.0E-021.0E-01

0 200 400 600Pca (MPa)

K (µ

m3 )

Turned B1Turned B2Sand-BlastedLapped

Ra=1.1µm

Ra=1µm

Ra=0.4µm


Experiment: Results (diffusive condition)Experiment: Results (diffusive condition)

ie

ei

iei

ei

i rrcc

rrrcc

r −−

≈−

=)(D

ln)(D

2πQ effeff Deff: effective "diffusivity" (m3/s)

1.E-12

1.E-10

1.E-08

1.E-060 100 200 300 400 500

Pca (MPa)

Def

f (m

3 /s) Turned B1

Turned B2


Experiment: error estimationExperiment: error estimation

* K estimated fromQ(ΔP) estimations

0

0.05

0.1

0.15

0.2

0.25

0 10 20 30ΔP (bar)

Q (m

g/m

in)

* Q estimated fromm(t) measurement

y = 9.0789x + 413.33R2 = 0.9938

0500

100015002000

0 50 100 150t (min)

m (m

g)


m(t) is determined form the GPC calibration curve

0

5

10

15

0 5 10mass ratio

Pic

area

ratio

Estimation of Q, K, calibration coefficient etc.and their associated uncertainties

Gauss Markov parameter estimation method



Illustration on the flow-rate Q ctM jbj += Q

⎟⎠⎞

⎜⎝⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Q

1

1 11 c

t

t

M

MM

X

mbm

b

321

MMM

Optimal estimation with minimal variance

( ) MXPXXPc tt 111

Q−−−=⎟

⎠

⎞⎜⎝

⎛ with ( )

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

2

22

21

0

0

cov

m

MP

σ

σσ

O

andif mass measurement errorsare uncorrelated

( ) 11

Qcov

−−=⎟⎠

⎞⎜⎝

⎛ XXPc t



⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛ +−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

∑ ∑

∑∑

∑ ∑∑= =

−−

=

−

=

−

= =

−

=

−−m

j

m

jjbjMM

m

jbjM

m

jjM

m

j

m

jjM

m

jjMM

tM

Mt

tt bjbj

bjbj

bjbjbj 1 1

22

1

2

1

2

1

2

1

2

1

222

1Qσσ

σσ

σσσ

1

1

2

1

2

1

222

1

22

−

= =

−

=

−−

=

−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∑ ∑∑∑

m

j

m

jjM

m

jjMM

m

jMQ tt

bjbjbjbjσσσσσand

Same approach to estimate m(t) from calibrationand to estimate K (and Deff) from Q and ΔP

Result: error on K is less than 10%error on Pca can reach 10%


Complete determination of optimal parametersand associated errors


Physical models: microPhysical models: micro--scalescale1. Viscous flowLow Re, stationary, isothermal flow; rigid surfaces.

Stokes model:

0. =∇ vv20 ∇+−= μp∇

0=v at solid walls

v, p

xyz

α

h 0

l 0

B.C.

Hyp:slope is small everywhereh0/l0=ε <<1

u, v ~ u0 x, y ~ l0

w ~ w0 z ~ h0

v(u,v,w)


Physical models: microPhysical models: micro--scalescale0*

**

*** =++ zyx wvuMass: with 00 uw ε=

Momentum:

withε

μ0

00

hup =

( ) ***

***

***

2**0 zzyyxxx uuup +++−= ε

( ) ***

***

***

2**0 zzyyxxy vvvp +++−= ε

( ) ***

***

***

2**

20 zzyyxxz wwwp +++−= − εε

( )*** , yxp only integrate x and y momentum eqs over z*

( )**2***

* 21 zhzpu x −=

( )**2***

* 21 zhzpv y −=

0*** =⋅+⋅+⋅ zy wvu enenen x ε at the wallsB.C.


Physical models: microPhysical models: micro--scalescale

xphdzu

dydQq

hx

x ∂∂

−=== ∫ μ12

3

0

Integrate over z: dimensional form

yphdzv

dxdQ

qh

yy ∂

∂−=== ∫ μ12

3

0

Reynolds or lubricationapproximation(Zimmerman and Bodvarsson,1996)

Integration of the mass balance equation:

0)(00 =+∂

∂

+∂

∂ ∫∫zw

y

dzv

x

dzuzz

i.e. for z=h 0=∂

∂+

∂∂

yq

xq yx



2D viscous flow problem: q(qx,qy)

ph∇−=

μ12

3q

0=⋅nq

0=⋅∇ q

in β

in β

on Cβσ


Physical models: microPhysical models: micro--scalescale1. Diffusive flowFickian, stationary, isothermal; rigid surfaces

0. =∇ jcD∇−=j

0=⋅nj at solid walls

c

xyz

α

h 0

l 0

Hyp: slope is small everywhere h0/l0=ε <<1

jx, jy ~ j0 x, y ~ l0

jz ~ j1 z ~ h0

B.C.


Physical models: microPhysical models: micro--scalescale0*

***

** =++ zyx jjjMass: with 01 jj ε=

Fick's law along ez *

***

2zcjz ∂∂

−=ε

),( yxcc = only Integrate over z to get (dimensional form)

xcDhdzjq

h

xx ∂∂

−== ∫0 y

cDhdzjqh

yy ∂∂

−== ∫0

0=∂

∂+

∂∂

yq

xq yx

0=B.C. ⋅+⋅ yyx qq enen x at solid walls

* x and y Fick's law

* mass


2D diffusive problem: q(qx,qy)


cDh ∇−=q

0=⋅nq

0=⋅∇ q

in β

in β

on Cβσ


Physical models: microPhysical models: micro--scalescaleSame problem for diffusion and viscous flow

ω∇−= kq

0=⋅nq

0=⋅∇ q

in β

in β

on Cβσ

Diffusion: and Dhk = c=ω

Viscous flow: and μ12

3hk = p=ω


Physical models: macroPhysical models: macro--scalescaleUpscaling of the micro-scale model

Method: surface averaging(Whitaker, 1999)

∫=β

ϕϕS

dSS1

∫=β

ϕϕβ

β

SdS

S1

* Averaging theorem:

∫+∇=∇βσ

ϕϕϕC

dCS1

* 2 averages:


Physical models: macroPhysical models: macro--scalescaleResult: model at the RES scale

βω∇⋅−= Hq0=⋅∇ q

( )bIH ∇+= k

b is solution of the closure problem (micro-scale):

( ) kk ~−∇=∇⋅∇ b in βnbn =∇⋅− on Cβσ

0=bb is periodic and

kkk −=~

Diffusion:

Viscous flow:

βcD ∇⋅−= Dq ( )bID ∇+= h

β

μp∇⋅−=

Kq ( )bIK ∇+=12

3h


Physical models: macroPhysical models: macro--scalescale

Solution of the closureAperture field h(x,y) is known on a regular discrete gridh is constant on each cell

Cell i

Interface ij

Cell j

b is periodic

ijn

0=b

02 =∇ ib( ) ( )IbnIbn +∇⋅=+∇⋅− jjii kk

ji bb =

02 =∇ jb

Numerical scheme: finite volume O(Δx2,Δy2)


Comparisons: experiment Comparisons: experiment vsvs predictionpredictionMicro-scale data: surface roughness measurement usingwhite light interferometry (LTDS, Lyon)Resolution : x-y: 0.1 to 1µm z: 2 nm

Turned surface (B1):Computational flow-chart

z(x,y)

Deformation

Percolation

Closure resolution

h(x,y)

h(x,y)h3(x,y)/12

K D

(Flat rigidplane)


Comparisons: experiment Comparisons: experiment vsvs predictionpredictionDeformation: elasto-plastic or purely plastic (erosion) model

Result on turned surface B1

Transmissivity at Pca=200 MPaKyy (µm(µm33))

ElastoElasto plastic plastic ((LaMCoSLaMCoS)) 0.94 0.94

ElastoElasto plasticplastic(LTDS)(LTDS)

1.121.12

PurelyPurely plastic plastic (Erosion)(Erosion) 1.031.03

x

y


Comparisons: experiment Comparisons: experiment vsvs predictionpredictionOn surface B1: no radial flow: leak is along the spiral groove

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+010 200 400 600 800

Pca (MPa)

Q (m

g/m

in/b

ar) Computed

Experiment B2Experiment B1

Topological description is uncomplete


Comparisons: experiment Comparisons: experiment vsvs predictionpredictionIdea: larger scale defaults must be taken into account

Default order1 Form default2 Waviness

3 and 4 Roughness

Flatness measurement by fingering (Talyrond) on the rough surface

(Default on sapphire surface < 56nm)

1 div.=0.1µmPeriod ~30° - Amplitude ~1.2µm


Comparisons: experiment Comparisons: experiment vsvs predictionpredictionReconstruction of the complete annular contact

-0.8-0.6-0.4-0.2

00.20.40.60.8

0 20 40 60 80 100 120 140

Circonférence (mm)

Déf

auts

de

plan

éité

s (µ

m)

+

-3-2-10123

0 50 100 150 200 250 300(µm)

(µm

)

Circumference (mm)Wav

ines

s (µm

)

µm

µm


0

100

200

300

4000

25

50

75

100

125

150

175

-2.50-1.250.001.252.50

-2.50 -1.25 0.00 1.25 2.50

(µm)

125 mm

0.3 mm

0

100

200

300

4000

25

50

75

100

125

150

175

-2.50-1.250.001.252.50

-2.50 -1.25 0.00 1.25 2.50

(µm)

125 mm

0.3 mm

Comparisons: experiment Comparisons: experiment vsvs predictionpredictionΔr

Q

L

X

Δr

x

QY

Δr

Q

L

X

Δr

x

QY

Q

L

X

Δr

x

QY

YX

C.L.* Y periodic* X: Dirichlet

(on p or c)


Comparisons: experiment Comparisons: experiment vsvs predictionpredictionLocal velocity map

0 25 50 75 100 125 150 175

0

25

50

75

100

125

0 20000 40000 60000 80000 100000

(µm/s)

Pca=200 Mpa – Local radial flow


0 25 50 75 100 125 150 175

0

25

50

75

100

125

0 100 200 300 400 500

(µm/s)

Comparisons: experiment Comparisons: experiment vsvs predictionpredictionLocal velocity map

Pca =600 Mpa – Circumferential flow in the spiral groove


Comparisons: experiment Comparisons: experiment vsvs predictionpredictionB1 and B2 surfaces – Viscous flow

1.0E-09

1.0E-07

1.0E-05

1.0E-03

1.0E-010 100 200 300 400 500 600 700 800

Pca (MPa)

K (µ

m3 )

Experiment B1Experiment B2Computed


Comparisons: experiment Comparisons: experiment vsvs predictionpredictionB1 and B2 surfaces – Diffusion (D=10-9m2/s)

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01100 200 300 400 500

Pca (MPa)D

=Def

f/D

(m)

Experiment B1Experiment B2Computed


Some more analysis: turned surfacesSome more analysis: turned surfacesLarge effect of 2nd order (waviness) defaults

Analyze 3 types of defaults (viscous flow)

Pure waviness Conical defaultSkew default

Origin: - LF spindle vibrations,- excessive tightening during machining, - remaining stress within the material,- spindle and cutting tool movement axes not perpendicular,- non-uniform chip.

Roughness amplitude (sine): 5µm; contact width: 300µm;contact length: 100mm; default amplitude: 1µm


Pure waviness

0.000001

0.00001

0.0001

0.001

0.01

0.1

1-2.5 -2 -1.5 -1 -0.5 0 0.5 1

Enfoncement

K

10 périodes5 périodes

0 50 100 150 200 250

0

20

40

60

80

X

Y

0 50 100 150 200 250

0

20

40

60

80

X

Y

0 50 100 150 200 250

0

20

40

60

80

X

Y

0 50 100 150 200 250

0

20

40

60

80

X

Y

Effective contact area contours

Deformation

10 periods5 periods

Some more analysis: turned surfacesSome more analysis: turned surfaces


Pure waviness

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

0 500 1000 1500

Pca (Mpa)

K (µ

m3 )

Surface plane Ondulation (5)Ondulation (10)Ondulation (15)

Flat surface Waviness (5)Waviness (10)Waviness (15)



0.000001

0.00001

0.0001

0.001

0.01

0.1

1-2.5 -2 -1.5 -1 -0.5 0 0.5 1

Enfoncement

K

0 50 100 150 200 250

0

20

40

60

80

X

Y

0 50 100 150 200 250

0

20

40

60

80

X

Y

0 50 100 150 200 250

0

20

40

60

80

X

Y

0 50 100 150 200 250

0

20

40

60

80

X

Y


Deformation

Conical defaultSome more analysis: turned surfacesSome more analysis: turned surfaces


Conical default

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

0 500 1000 1500

Pca (Mpa)

K (µ

m3 )

Surf. PlaneConiqueFlat surfaceConical



0.000001

0.00001

0.0001

0.001

0.01

0.1

1-2.5 -2 -1.5 -1 -0.5 0

Enfoncement

K

10 sommets

20 sommets

0 50 100 150 200 250

0

25

50

75

X

Y

enfoncement = -0.9

0 50 100 150 200 250

0

20

40

60

80

X

Y

enfoncement = -1

0 50 100 150 200 250

0

25

50

75

X

Y

enfoncement = -1.3

0 50 100 150 200 250

0

25

50

75

X

Y

enfoncement = -2


Skew default

10 peaks20 peaks

Deformation



1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

0 500 1000 1500

Pca (Mpa)

K (µ

m3 )

Surf. PlaneGauche (5)Gauche (10)Gauche (15)

Skew default

Flat surface Skew (5)Skew (10)Skew (15)



1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 100 200 300 400 500

Pression de Contact (en Mpa)

K (µ

m)

Surf. PlaneConiqueGauche (5)Ondulée (5)

Flat surface ConicalSkew (5)Wavy (5)

Pca (MPa)



Some conclusionsSome conclusions- Static seal performance of direct metal/metal contactstrongly depends on many parameters.- Texture of both machined surfaces is one the most important.- Many different scales of defaults are usually involved.- The complete structure of the contact might be representedby a simplified two-scale approach.- For turned surfaces, some macro-scale defaults (conical)are preferable.

- Data obtained from very careful experiments are a key supportto assess validity of models that include a complex combinationof mechanisms.-Viscous flow (h3) and diffusion (h) provide two separateddiscriminating tests of aperture description and evolution.


Flat non-deformable plane

h

h

- Aperture fields h(x,y) are identical

- Elasticity of the material is acombination of the two original ones

- Hardness of the material is thatof the softer one

SurfaceSurface--sum conceptsum concept

2

22

1

21

*111

EEEνν −

+−

=


ErosionErosion

Aperture field under load after erosion

Ai

Position cutting plane

Compute effective contact area

Test:

∑∑ ==i

ii

ii AHApW

Cutting plane

W applied load, H material hardness

FLOW IN FRACTURES COUPLED WITH DEFORMATION EFFECTS ... · Marie Curie Workshop on Flow and...

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Transcript of FLOW IN FRACTURES COUPLED WITH DEFORMATION EFFECTS ... · Marie Curie Workshop on Flow and...