Wave Propagation in Solid-Fluid Layered System Analysis
wave propagation in a solid-fluid layered system, this study dives into frequency-dependent slowness surfaces, caustics, dispersion analysis, and more. Detailed diagrams and matrices illustrate the dynamics of waves in different layers.
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Waves in solid/fluid layered system Alexey Stovas, NTNU Yuriy Roganov, Tesseral ROSE 2017
Outline Wave propagation in solid-fluid layers Frequency dependent slowness surface From phase to group domain Caustics Dispersion Analysis Conclusions
Solid-fluid layers 1 1 1 = = = 2 2 2 : , , Slowness surfaces in individual layers q p q p q p f 2 2 2 f = v Ea : Solid ( ) T ( ) T = = d P d S u P u S a v , , , , , , , A A A A u u 1 3 13 33 p q p q q q p p = E ( ) ( ) 1 2 1 2 2 2 2 2 2 2 2 2 pq p pq p ( ) ( ) 1 2 1 2 2 2 3 2 2 3 2 2 p pq p pq = w E b : Fluid f ( ) T ( ) T = = d S u S b w , , , B B u 3 33 q q f f f f = E f f f f f
Solid-fluid layers Let us consider the diagonal matrices and propagator matrices ( ) i q H 1 exp 0 0 0 2 ( ) i q H 1 0 exp 0 0 2 = , ( ) i q H 1 0 0 exp 0 2 ( ) i q H 1 0 0 0 exp 2 i q H f exp 0 2 = f i q H f 0 exp = = 1 P Q E E E E ( ) solid 2 1 ( ) fluid f f f
Periodic fluid-solid system Fluid z1 z2 ( ) ( ) ( ) z w ( ) ( ) ( ) z = = = w Qw z z 2 1 v Pv z z Solid z3 4 2 Qw z4 z5 5 4 Fluid z
Monodromy matrix : At the boundaries between liquid and solid layers and are continuous and Fv Gw ( w v ) ( ) z ( ) z = = 0 w Mw 3 = = 33 13 5 1 , , M = QFPGQ where 0 0 1 0 0 0 0 1 2 12 21 r r i = F 1 11 12 r r 21 22 r r + 11 12 r r r r 21 22 r r r r f M = P P P P 11 22 r r 32 31 34 31 11 12 21 22 2 1 i f 1 0 0 0 0 1 11 12 r r 21 22 r r = G ( ) i qH is an eigenvalue of matrix M exp Similar approach is proposed by Schoenberg, 1983; 1984
Frequency dependent slowness surface = 11 12 r r r r Hq 2 tan 2 21 22 2 H 11 12 r r r r ( ) = , arctan q p 21 22
f = 100 Hz Slowness surface : Horizontal velocities A F E r C H = , : 0 12 r = = = : : 0 0 0 22 11 r r E : =0.001 f = 10 Hz 21 H : Model = = 4.0 2.2 1.5 = km s km s km s Slow S*-wave D B f = 1000 Hz G Fast S*-wave f = 3 2.7 1.0 = = g cm g cm P-wave 3 f A C F 0.001 H km ( ) ,2011 Korneev
Group velocity surface: fluid effect at f=10Hz = 0.1 = 0.5 = 0.001
Group domain G : Caustics + B 21 22 11 r r 12 r r r r r r D = + D 11 12 21 22 G A C E H F D G
Horizontal velocity dispersion H=1m : Horizontal velocities P wave A r Fast S wave V E r F V min ( ) = , 0 is weakly dispesive S V 12 max S V min ( ( ) = F , 0 is weakly dispesive max 22 ) = F , 0 V C r is dispesive H=10m min 11 Slow S wave V F V min ( ( ) ) = S , 0 F r is weakly dispesive max 12 S V = S , 0 V H r is dispesive max min 21 S V min
P wave vertical velocity ( ) ( ) 1 1 fH fH fH fH + + tan tan tan tan 1 1 fH f f = arctan ( ) f ( ) ( ) v 1 1 fH fH fH fH 0 P tan tan 1 tan tan f f = f f f Backus 0 1 1 ( ) ( ) = + + 1 ( ) 0 f 2 P 2 2 f v 0
Analysis (large p and large frequency*) ( ) 1 1 1 max , , If p f ( ) ( ( ) x i tan tanh ix i ) ( ) ix = lim tan x Sq q Sq q iS q iS q f f = = = = , , , 11 r 12 r r r 21 22 q q ( ) 2 f = + 1 2 + 2 4 2 2 4 S p q q p f : Solutions q i S = ( ) = non Scholte wave propagating wave 0 ( )
Analysis (zero frequency) 0 f ( ) p ( ) p 2 f 2 q q 1 p v ( ) ( ) p ( ) ( ) = + + 2 1 1 q ( ) f 2 2 PL 1 f = 0.1 2 ( ) = 2 1 = v Lamb wave velocity 0.01 PL 2 = 0.001 Generally, it is different from Backus, however, gives similar results for small .
Analysis: zero limit (inherited slip) ( ( 1 2 ) ) ( ( ) ) ( ( ) ) 2 ( ( ) ) 1 2 + 2 2 2 4 tan 4 tan p fHq p q q fHq 1 fH ( ) ( ) = lim tan tan q fHq fHq 2 0 + 2 2 2 4 tan 4 tan p fHq p q q fHq ( ) a homogeneous anisotropic layer with the slip at both interfaces + + + + + 2 2 2 2 ( ) = lim 30 q f Hz 0 2 2 2 1 1 p ( ) ij c = = lim , q 0 2 0 0 1 4 2 2 1 p f 2 0 0 lim q A homogeneous anisotropic medium with the slip smeared within the whole medium 0 0 f . 2 2 = = : 0 2 1 Anisotropic parameters and 2 2 ( ) sameresultcanbeobtained fromBackus
Stop bands for vertically propagated P wave ( ) + 1 fH 1 1 R R fH f f = tan tan 0 = 0.1 0 f = 0.2 = 0.3 Pass Pass Stop Stop
Fluid substitution (Utsira sand, f=30Hz) CO2 Water 1. Vp0&eps are weakly dependent on frequency; 2. Anisotropic parameter is strong negative and frequency dependent. d=1cm CO2 Water d=5cm
Conclusions The frequency-dependent slowness surface is defined for waves propagating in a solid-fluid system (periodical). It consists of three sheets corresponding to P- and two S-waves. Both S*-waves have triplications. The low- and high-frequency limits are defined. The frequency dispersion is computed for horizontally propagating S-waves. The stop-bands for P-wave are defined. When the fluid layer thickness tends to zero, we obtain the medium with inhereted slip. It is shown that S-waves are sensitive to fluid substitution.
