Coupled wedge waves

Abell, B.; Pyrak‐Nolte, L. J.

doi:10.1121/1.4821987

Cited by 7 publications

(12 citation statements)

References 43 publications

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“…The velocity starts at the Rayleigh wave velocity (Rayleigh, ) and increases to the bulk shear wave velocity. In a similar fashion, the coupled wedge wave (Abell & Pyrak‐Nolte, ) dispersion curve is also captured when one normalized stiffness is high (e.g.,

\frac{ω}{ω_{c 1}} = 1 0^{3}

), and the other is varied (e.g.,

\frac{ω}{ω_{c 2}} = [1 0^{4}, 1 0^{- 2}]

). Thus, the surfaces in Figure fully encapsulate the possible modes supported by any portion of the intersecting fracture plane geometry shown in Figure .…”

Section: Numerical Analysismentioning

confidence: 94%

“…This derivation follows a similar logic to the derivation for a coupled wedge wave (Abell & Pyrak-Nolte, 2013) but applies to four blocks instead of two. Intersection waves are assumed to propagate along the x direction for the intersection shown in Figure 2.…”

Section: Theoretical Derivationmentioning

confidence: 99%

“…Using this notation, the equation of motion can be expressed as

ρ \frac{\partial^{2} u_{α}}{\partial t^{2}} = \sum_{β = 1}^{3} \frac{\partial T_{α, β}}{\partial x_{β}} .

For media with symmetric elastic moduli (i.e.,

C_{αβμν}^{(b)}

) the stress tensor can be simplified to include the elastic modulus tensor and strains (Abell, ; Sokolova et al, ). The displacement amplitudes are then expanded in a linear series of Laguerre functions, which represents the wedge geometry (Moss et al, ), as

Ũ_{α}^{(b)} = \sum_{m, n = 0}^{\infty} a_{m, n}^{(α, b)} φ_{m} (ςky) φ_{n} (εkz),

where φ are the Laguerre functions (Abell & Pyrak‐Nolte, ; Maradudin et al, ; Moss et al, ), ς is (+,−,−,+) and ε is (+,+,−,−) for media (1,2,3,4), respectively. In general, 15 expansion terms are sufficient in the sum of equation .…”

Section: Theorymentioning

confidence: 99%

“…The resulting secular equations for the intersection are

a_{mn}^{(α, 1)} = G_{mnpq}^{(α, β, 1)} ((), - τ_{q}^{(β)} - γ_{p}^{(β)})

a_{mn}^{(α, 2)} = G_{mnpq}^{(α, β, 2)} ((), τ_{q}^{(β)} - σ_{p}^{(β)})

a_{mn}^{(α, 3)} = G_{mnpq}^{(α, β, 3)} ((), H_{q}^{(β)} + σ_{p}^{(β)})

a_{mn}^{(α, 4)} = G_{mnpq}^{(α, β, 4)} ((), - H_{q}^{(β)} + γ_{p}^{(β)})

where the stresses are defined between the media such that ( H , τ , σ , γ ) are between media (4‐3, 1‐2, 2‐3, 1‐4) shown in Figure and G is defined in Appendix B. It should be noted that this definition and set of secular equations differ slightly from the coupled wedge wave and Rayleigh‐Stoneley waves (Abell & Pyrak‐Nolte, ; Sokolova et al, ) because of the geometry and complexity of coupling four media instead of two.…”

Section: Theorymentioning

confidence: 99%

“…The wagging mode is like a dog's tail, which oscillates side to side, and is the most common mode for the wedge wave. The coupled wedge wave can support both wagging (more common) and breathing modes up to retrograde motion, when the fracture plane is fully welded, supporting a Rayleigh wave (Abell, ; Abell & Pyrak‐Nolte, ). Throughout this discussion, the terms wagging and breathing will be used to illustrate these types of motion.…”

Section: Introductionmentioning

confidence: 99%

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Intersection Waves

2017

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Intersections in a fracture network control the connectivity of the flow paths through rock. The long near‐linear geometric nature of fractures makes them difficult to identify and characterize. We present a new type of elastic wave, an intersection wave, which travels along an intersection and is sensitive to the coupling between two orthogonal fractures that define the intersection. Group theory for C2v and C4v point groups predicts sets of propagating elastic waves confined to the fracture intersection. Along with the use of the wave equation and displacement discontinuity boundary conditions, the dispersion relationships for intersection waves are predicted. Experimental ultrasonic measurements on a nonwelded linear intersection between two orthogonal, synthetic fractures in aluminum confirm the existence of multiple modes that travel between the speed of wedge waves (sub‐Rayleigh waves) when the intersection is completely, and bulk shear waves, when the intersection is closed, as predicted by theory. Between these two limits, the intersection behaves as a nonwelded contact and yields these new intersection waves that are dispersive and sensitive to the coupling along the intersection. Intersection waves provide the foundation for new geophysical approaches for characterizing the hydraulic connectivity of fracture networks.

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\frac{ω}{ω_{c 1}} = 1 0^{3}

), and the other is varied (e.g.,

\frac{ω}{ω_{c 2}} = [1 0^{4}, 1 0^{- 2}]

). Thus, the surfaces in Figure fully encapsulate the possible modes supported by any portion of the intersecting fracture plane geometry shown in Figure .…”