reflection and transmission by a double porosity layer ... · 4"! hypothesis of double...
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Reflection and transmission by a double porosity layer obeying Berryman-Wang theory
F. KACHKOUCH, H. FRANKLIN, A. TINEL, A. ALEM, H. WANG
Université*Le*Havre*Normandie,*Laboratoire*Ondes*et*Milieux*Complexes*LOMC**
UMR*CNRS*6294*A*LabEx*EMC³*
Symposium on the Acoustics of Poro-Elastic Materials
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PLAN
Dispersion)curves)for)Berea)sandstone))
Transmission)characteristics)for)a)layer)of)Berea) Sandstone)
Double)porosity)model
Main)objectives
Evaluation)of)the)clogging)by)ultrasonics)testing
Conclusion)&)prospects
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o Characterization of double porosity layers by acoustic methods - Accessibility to double porosity parameters (porosities, permeabilities,
bulk modulus…)
- Evaluation of the clogging/pollution phenomenon due to the deposition of fine suspended particles resulting from a water flow caused by rainwater infiltration.
- Estimation of erosion.
Main)objectives
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ν(1) ϕ(1)
Microporosity (Matrix porosity)
ν(2) ϕ(2)
Macroporosity (fractures porosity)
h&ps://www.researchgate.net/figure/
234848514_fig6_FIG>6>A>
semilogarithmical>plot>of>the>FID>for>
the>fully>water>saturated>Berea>
sandstone"
! Robu © ! Berea Sandstone (confirmed)
Porous"Glass"Beads"Borosilicateglas"
3.3"ROBU"
"(ф"="4mm)""
Views"on"Electronic"scanning"microscope
k22 > k11
ν(1)ϕ(1) > ν(2)ϕ(2)
Distribution profile of the matrix porosity in a grain (porosimeter)
Double)porosity)model
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! Hypothesis of double porosity medium (Berryman and Wang (Int. J. Rock Min. (2000))
- Extension of Biot’s theory to account for both pores and fractures (phenomenological)
- Continuous medium (low frequency waves considered)
- Isotropy (randomly oriented fractures, no preferred axis for fluid flow)
- Fluid in matrix and fractures is the same but the two fluid regions may be in different states of average stress (distinguished by their superscripts)
Consequence of this model : the increased number of independent coefficients describing the medium (inertial, drag and stress-strain) with respect to Biot's initial theory.
Alternative theory : C. Boutin and P. Royer Geophys. J. Int. (2015) Match with Berryman and Wang theory at low frequency but more complicated to implement
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! Constitutive equations for isotropic double porosity poroelastic medium
( ) ( ) ( ) ( )( )1 1 1 1ν φ= −w U u
( ) ( ) ( ) ( )( )2 2 2 2ν φ= −w U u
( ) ( ) ( )1 1 212 22 23P C e C Cξ ξ= − + +
( ) ( ) ( )2 1 213 23 33P C e C Cξ ξ= − + +
( ) ( ) ( ) ( ) ( )1 1 2 22 2 +αβ δδ αβ αβ αβσ µ ε δ µε ξ ξ δ' (= − + − * +H C C
- Stress tensor and fluid pressures
- Relative fluid-solid displacements
Cij : related to the material properties and to the generalized poroelastic expansion and storage coefficients
H, C(1), C(2) : moduli of the dPP medium depending on Cij
( ) ( )1 2,u, U U : the displacement vectors of the solid frame, the micropore fluid and the macropore fluid.
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Parameters Symbols and units Berea sandstone (from BW Int. J. Rock
Min. 2000)
Water
Density of grains ρs (kg/m3) 2600
Unjacketed bulk modulus for the whole Ks (GPa) 39
Unjacketed bulk modulus for the matrix Ks(1) (GPa) 38.306
Total porosity ϕ 0.1926
Matrix porosity ϕ(1) 0.178 Fracture porosity ϕ(2) 1
Matrix permeability k(11) (m2) 10-16 Fracture permeability k(22) (m2) 10-12
Volume fraction occupied by matrix ν(1) 0.9822 Volume fraction occupied by fractures ν(2) 0.0178
Biot-Willis parameter for the whole α 0.8462 Biot-Willis parameter for the matrix α(1) 0.7389
Biot-Willis parameter for the fractures α(2) 1 Overall tortuosity a 2.9460 Matrix tortuosity a(1) 3.3090
Fracture tortuosity a(2) 1 Bulk modulus of undrained porous frame Ku (GPa) 15.2
Jacketed bulk modulus of matrix K1 (GPa) 10 Jacketed bulk modulus of fractures K2 (GPa) 0.108
Jacketed bulk modulus of porous frame K (GPa) 6 Shear modulus for drained medium µ (GPa) 5.478
Density ρf (kg/m3) 1000 Pore fluid bulk modulus Kf (GPa) 2.3
Viscosity η (Pa s) 10-3
! Material data table
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! Analytical model for Berea sandstone"
1, 2, 3: dilatational waves t: shear wave
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! Basic equations of Berryman-Wang theory
" Boundary conditions (filtration occurs across the interfaces at z = ± d/2)
( ) ( )1 20z z z zu u w w= + +
( )10 =P P
( )20 =P P
0 zzP σ− =
0 xzσ=
Continuity of fluid volume
Continuity of fluid pressure for phase 1
Continuity of fluid pressure for phase 2
Continuity of normal stress
Disappearance of tangential stress
( )
( )
( )
( )1 1 23 1 1 23
1 1 23 1 1 23
12
τ τ
τ τ
+ −
− +
+# $+ −
# $ % &=% & % &− +' ( % &
' (−
∑ ∑
∑ ∑
S S A Acyc cyc
S S A Acyc cyc
RT
C i C i
C i C i
X X
X X
( ) ( )
( ) ( )
1 1
2 2
Fm m Fn nmn
Fm m Fn n
C i C i
C i C i
τ τ
τ τ• •±
•
• •
± ±=
± ±X
" Reflection and transmission coefficients
( ) ( ) ( ) ( )1 1 23 1 1 23 2 2 31 3 3 12τ τ τ τ+ + + ++ = + + + + +∑ S S S S S S S ScycC i C i C i C iX X X X
•"="S"or"A""""
(1),"(2)":"fluid"phases"
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( )
( )
( ) ( )( )
2 cot 2tan 2
jjfmSFm mz
tjmztAFm
C K dL
K dC
ρ
ρ
" # " #=$ % $ %$ % & '& '
l
( )( )
( )( )
cot 2 cot 2tan 2 tan 2
Sm mz tzm m
Am mz tz
C K d K dLN M
C K d K d! " ! "! "
= +# $ # $# $% & % & % &
( ) ( )1 2m m m mτ τ τ τ= + +%
They account for the squirting out and the suction of the saturating fluid under the effect of the c o m p r e s s i o n a l a n d s h e a r deformations of the elastic frame."
They account for the deformations of the elastic frame and express the coupling of each of the dilatational wave with the shear wave."
j=1, 2 (fluid phases), m=1, 2, 3 (wave numbers)
It depends on several parameters constituting the medium as the density, and also on the acoustical properties as the wave numbers (perturbation of the Cs and Ca)"
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Complex « usual » modes obtained with real angles
Complex « unusual » modes obtained with imaginary angles
K.#E.#Graff###Wave#Mo.on#in#Elas.c#Solids#1975#
"
Absence of A0-like mode ???
Partial branch only for S0
V1max = 3270 m/s
V2max = 550 m/s
V3max = 35 m/s
Vtmax = 1552 m/s
Transmission)characteristics)for)a)layer)of)Berea))Sandstone)
M e a s u r a b l e experimentally ??
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! Obtained with the transition operator
in relation with symmetrical modes
" ( 1) / 2= + −ST R T i
Ang
le (d
egre
e)
0
10
20
30
40
50
60
70
80
90
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (kHz)0 20 40 60 80 100
10
20
30
40
50
60
70
80
90
1
SDP2
SDP 3
SDP 4
SDP
0
SDP
( )12jk jf k c d= + tk tf k c d=
k = positive integer j = 1, 2, 3
Cut-off frequencies for symmetrical modes
Enhancement of the vertical mode visualisation (related to the second dilatational wave)
10 11 1216351, 49053, 81756,...f f f= = =
0 1 2 315520, 31040, 46560, 62080,...= = = =t t t tf f f f
θ = 72°
θ = 26°
Dispersion)curves)for)Berea)sandstone))
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DPS2 DPS
5 DPS
8
12"
Frequency (kHz)
Ang
le (d
egre
e)
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
SDP2
SDP 3
SDP 4
SDP
0
SDP
Frequency resonances at θ = 0 °
Angular resonances at f = 30 kHz
A b s e n c e o f s p i k e s corresponding to the shear wave because of their small width of resonance
First dilatational wave
DPS1
DPS2
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! Effect of the total porosity variation on the transmission coefficient
ϕt = 10% ϕt = 19%
Frequency (kHz)
Ang
le (d
egre
e)
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
- The spikes shift toward the low frequecy - The resonance width is more larger for a small porosity
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! Effect of permeabilities variation on the transmission coefficient
k22 = 10-12 m², k11 = 10-16 m² k22 = 10-10 m², k11 = 10-14 m²
Frequency (kHz)
Ang
le (d
egre
e)
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
- The spikes shift toward the low frequecy - The increase of the transmission amplitude (above the second critical angle)
For prospects : study of parameters affecting the propagation of Lamb modes
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( )212 1
11g p
g gg p
T RR R
R R= +
−
12
11g p
gg dpp
T TT
R R=
−
z
x O
d Double porosity layer
Fluid
Fluid
2
1
3 t
Reflected Wave
Incident wave
Transmitted wave
θ
θ
Aluminum plate
Aluminum plate
Rg2
Tg2
" Fabry-Perot method :
Fluid – Aluminium plate – fluid Fluid – DPP – fluid (previous study) Fluid – Aluminium plate – fluid
Rg1, Tg1
Rp, Tp
! Model for handling a layer of granular medium (Robu)
D = 0 The model presented"
Characterization of the clogging by injecting fines particles of clay with a water flow
Evaluation)of)the)clogging)by)ultrasonics)testing
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coefficients
exp0
SCS
=
0num
TCT
=
16"
FFT of the temporel signal transmitted through the model filled with DPP
FFT of the temporel signal transmitted through the model filled with water
Sensitivity of the acoustic measures to the clogging phenomenon
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! Particularities observed for the fundamental modes in the dispersion curves of a dpp layer compared to an elastic layer.
! The third dilatational wave plays no role for the characterisation of the Lamb waves in this frequency range.
! This analytical study can be also of interest as a first approach for
understanding the acoustic behaviour of cancellous bones (media with multiple porosities).
! The study of clogging by ultrasonic methods can lead subsequently to the development of non destructive methods of predicting phenomena affecting hydraulic structures.
Conclusions
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Thank you for your attention
18"