Simulation of Waves in Poro-Viscoelastic Rocks Saturated by Immiscible Fluids: Numerical Evidence of a Second Slow Wave

Abstract

We present an iterative algorithm formulated in the space-frequency domain to simulate the propagation of waves in a bounded poro-viscoelastic rock saturated by a two-phase fluid. The Biot-type model takes into account capillary forces and viscous and mass coupling coefficients between the fluid phases under variable saturation and pore fluid pressure conditions. The model predicts the existence of three compressional waves or Type-I, Type-II and Type-III waves and one shear or S-wave. The Type-III mode is a new mode not present in the classical Biot theory for single-phase fluids. Our differential and numerical models are stated in the space-frequency domain instead of the classical integrodifferential formulation in the space-time domain. For each temporal frequency, this formulation leads to a Helmholtz-type boundary value problem which is then solved independently of the other frequency problems, and the time-domain solution is obtained by an approximate inverse Fourier transform. The numerical procedure, which is first-order correct in the spatial discretization, is an iterative nonoverlapping domain decomposition method that employs an absorbing boundary condition in order to minimize spurious reflections from the artificial boundaries. The numerical experiments showing the propagation of waves in a sample of Nivelsteiner sandstone indicate that under certain conditions the Type-III wave can be observed at ultrasonic frequencies.

Numerical simulation of ultrasonic waves in reservoir rocks with patchy saturation and fractal petrophysical properties

“…This algorithm is applied to perform numerical experiments in a real sandstone saturated by two fluids (gas-water). In previous works we dealt with homogeneous models [10,41,42], in which we verified numerically the propagation of fast and slow waves in this kind of media.…”

“…with ber(θ l ) and bei(θ l ) being the Kelvin functions of the first kind and zero order and θ l = a l p √ ωρ l /µ l , a l p = 2 KK rl A 0 /φ, where A 0 denotes the Kozeny-Carman constant [29,42].…”

“…The space-time solution is obtained by solving (3.5), (3.6) for a finite number of frequencies and an approximate inverse Fourier transform [18]. The definition of the iteration (3.5), (3.6) can be extended to the case of larger subdomains j , see [42].…”

Acoustic waves in unsaturated soils

“…Since fluctuating capillary pressure allows the pore fluids to experience distinct time-dependent pressure changes, a dilatational mode in addition to the two predicted by the Biot [1962] model is expected. On the basis of numerical simulations, Santos et al [2004] proposed that this third mode involves the nonwetting fluid moving out-of-phase with the wetting fluid while it moves in-phase with the solid framework. This proposal was elucidated by Lo et al [2010], who demonstrated numerically that excitation of the third dilatational mode actually does not affect the solid framework significantly, but instead causes the two pore fluids to move out-of-phase, which is consistent with this mode arising from capillary pressure fluctuations.…”

Acoustic wave propagation simulation in a poroelastic medium saturated by two immiscible fluids using a staggered finite-difference with a time partition method

“…Employing Santos's model, Ravazzoli et al [28] investigated the acoustic and mechanical properties of reservoir sandstone by two immiscible hydrocarbon fluids, under difference saturation and pressure conditions. Santos et al [29] considered the effects of capillary pressure and the reference pressures of the immiscible fluids on the stress-strain relations, as well as the friction effects between the oil and water phases. They conducted numerical modeling using a finite-element method in the space-frequency domain.…”