Poroelastic modeling - low-frequency case

The classic poroelastic theory of Biot, developed in 1950's, describes the propagation of elastic waves through a porous media containing a fluid. This theory has been extensively used in various fields dealing with porous media: seismic exploration, oil/gas reservoir characterization, environmental geophysics, earthquake seismology, etc. In this work we use the Ursin formalism to derive explicit formulas for the analysis of propagation of elastic waves through a stratified 3D porous media, where the parameters of the media are characterized by piece-wise constant functions of only one spatial variable, depth. There is considered the lowfrequency limit of the Biot equations. Introduction Poroelastic models are used in geophysics and petroleum engineering, where porous media filled with fluid and/or gas is of great interest. The best-known poroelastic theory was developed by Maurice Biot, see Biot (1956a) and Biot (1956b). There are many works devoted to the development and application of analytical/semi-analytical methods for wave propagation analysis in stratified elastic media, see, for instance, Thomson (1950), Haskell (1953), Brekhovskih (1960), Kunetz and d’Erceville (1962), Ursin (1983), and Molotkov (1984). The development of similar methods in the case of stratified porous media is very important too, see Allard et al. (1989), Baird et al. (1999), Molotkov (2002), and Carcione (2007). The Ursin formalism gives a unified treatment of electromagnetic waves, acoustic waves, and the isotropic elastic waves in plane layered media. Recently, this formalism was applied to the Pride equations for simulation of the electrokinetic phenomena in layered media, see White and Zhou (2006). In this work we apply Ursin’s method for solving the Biot system in the case of the 3D poroelastic plane layered media. In the exposition of results, we follow basically to the White and Zhou work. Although the results obtained by White and Zhou allow, under certain conditions, to split Pride’s equations and select only the poroelastic part, we examine the case of a more complete poroelastic system, characterized by presence in the Darcy law of an inertial force connected with the effective density of pore fluid. Method 1. Problem. We shall consider wave propagation in a porous half-space R = Rk k=1 k=N ∪ , composed with stratified layers Rk = x = (x1,x2 ,x3 ≡ z)∈R 3 : zk < z < zk+1 { } , where 0 = z0 < z1

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