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

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

1"

2"

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

3"

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

4"

!  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

5"

!  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.

6"

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

7"

!  Analytical model for Berea sandstone"

1, 2, 3: dilatational waves t: shear wave

8"

!  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"

9"

( )

( )

( ) ( )( )

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)"

10"

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 ??

11"

!  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))

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

13"

!  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

14"

!  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

15"

( )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

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

17"

!  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

Thank you for your attention

18"