Download Numerical Simulation in Applied Geophysics by Juan Enrique Santos, Patricia Mercedes Gauzellino PDF
By Juan Enrique Santos, Patricia Mercedes Gauzellino
This e-book offers the idea of waves propagation in a fluid-saturated porous medium (a Biot medium) and its program in utilized Geophysics. particularly, a derivation of soaking up boundary stipulations in viscoelastic and poroelastic media is gifted, which later is hired within the applications.
The partial differential equations describing the propagation of waves in Biot media are solved utilizing the Finite point approach (FEM).
Waves propagating in a Biot medium endure attenuation and dispersion results. specifically the quick compressional and shear waves are switched over to sluggish diffusion-type waves at mesoscopic-scale heterogeneities (on the order of centimeters), influence frequently happening within the seismic variety of frequencies.
In a few circumstances, a Biot medium provides a dense set of fractures orientated in choice instructions. while the common distance among fractures is way smaller than the wavelengths of the vacationing speedy compressional and shear waves, the medium behaves as an efficient viscoelastic and anisotropic medium on the macroscale.
The e-book offers a approach make sure the coefficients of the powerful medium utilizing a suite of time-harmonic compressibility and shear experiments, within the context of Numerical Rock Physics. each one test is linked to a boundary worth challenge, that's solved utilizing the FEM.
This procedure bargains an alternative choice to laboratory observations with the benefits that they're low-cost, repeatable and basically unfastened from experimental error.
The varied themes are via illustrative examples of program in Geophysical Exploration. specifically, the results attributable to mesoscopic-scale heterogeneities or the presence of aligned fractures are bearing in mind within the seismic wave propagation types on the macroscale.
The numerical simulations of wave propagation are offered with adequate element as to be simply carried out assuming the information of clinical programming techniques.
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1987]: F(ω ) = 1 − i 4S2 κ xΛ 2 φ and Λ can be calculated from ; where x = ηφ κ −1 , ωρ 8Sκ = 1. 95) The coefficient Λ has the dimensions of length and is a geometrical parameter of the porous medium. 97) = −φ ∇Δ p f , f Δ σi j (u (ω ), u (ω )) = [A e (ω ) + P θ (ω )]δi j + 2μεi j (us (ω )), s s f −φ Δ p f (u (ω ), u ) = P e (ω ) + R θ (ω ). 101) where f(1) and f(2) are external forces in the bulk material and the fluid per unit bulk volume and g(ω ) = Sρ f FI (ω ) −1 ηκ , + φ ω b(ω ) = ηκ −1 FR (ω ).
1990b]. Thus, if Δ Vqc denotes the part of the total change in volume Δ Vq = Vq − V q due to changes Δ pq = pq − pq in the corresponding fluid pressures, since in equilibrium ∇Sq = 0, we see that Sq ξ q = Δ Vq − Δ Vqc Vb =φ Δ Vq − Δ Vqc Vf , q = n, w. 2) Let Kn = Cn−1 and Kw = Cw−1 denote the bulk moduli of the non-wetting and wetting fluids, respectively, Cn and Cw being the corresponding compressibilities. Then, by definition Δ Vqc Δ pq =− , q = n, w. 2) that Δ Sq Δ V f Δ Vqc + − Vf Vq Sq , ξ ∗ = Sn ξ n + Sw ξ w , q = n, w.
Attenuation values are very high at low frequencies, showing that they are diffusion-type waves. After 100 Hz, all curves have a decreasing behaviour, with P2 waves suffering the highest attenuation for the oil case, the lower attenuation for the gas case, and the water case having an intermediate behaviour. After 1 MHz (ultrasonic range), P2 attenuation is negligible and P2 waves become truly propagating waves. 8 Application to a real sandstone Fig. 3 Phase velocity of P1 waves as function of frequency for a sample of Nivelsteiner sandstone saturated by water, oil and gas.