Wave propagation in heterogeneous, porous media: A velocity‐stress, finite‐difference method

Abstract

A particle velocity‐stress, finite‐difference method is developed for the simulation of wave propagation in 2-D heterogeneous poroelastic media. Instead of the prevailing second‐order differential equations, we consider a first‐order hyperbolic system that is equivalent to Biot’s equations. The vector of unknowns in this system consists of the solid and fluid particle velocity components, the solid stress components, and the fluid pressure. A MacCormack finite‐difference scheme that is fourth‐order accurate in space and second‐order accurate in time forms the basis of the numerical solutions for Biot’s hyperbolic system. An original analytic solution for a P‐wave line source in a uniform poroelastic medium is derived for the purposes of source implementation and algorithm testing. In simulations with a two‐layer model, additional “slow” compressional incident, transmitted, and reflected phases are recorded when the damping coefficient is small. This “slow” compressional wave is highly attenuated in porous media saturated by a viscous fluid. From the simulation we also verified that the attenuation mechanism introduced in Biot’s theory is of secondary importance for “fast” compressional and rotational waves. The existence of seismically observable differences caused by the presence of pores has been examined through synthetic experiments that indicate that amplitude variation with offset may be observed on receivers and could be diagnostic of the matrix and fluid parameters. This method was applied in simulating seismic wave propagation over an expanded steam‐heated zone in Cold Lake, Alberta in an area of enhanced oil recovery (EOR) processing. The results indicate that a seismic surface survey can be used to monitor thermal fronts.

Acoustic detection of buried objects in 3-D fluid saturated porous media: numerical modeling

“…An analytical solution for the particle velocity field in a homogeneous, fluid-saturated poroelastic medium subject to a point source in 3-D space or a line source in 2-D space can be obtained in a closed form via potential functions [9]. In a poroelastic medium with an ideal nonviscous fluid, a purely dilatational source will only excite P waves given by scalar potentials where distance; source time function; and ratios between the solid and fluid motion for the fast P-wave and the slow P-wave, and the coefficients; and determined by the regularity conditions; and velocities of the fast P-wave and the slow P-wave, respectively.…”

“…The FDTD method is then used to solve the Biot equations [8]. Similar to the velocity-stress finite-domain (FD) method [9], a velocity-strain, FD method is developed in a staggered grid for heterogeneous poroelastic media. In this method, Biot equations [8] are reformulated into first-order equations to arrive at a leap-frog system in a staggered grid both in time and space domains.…”

Using a Poroelastic Theory to Reconstruct Subsurface Properties: Numerical Investigation

“…This method was also used to solve the coupling between seismic and electromagnetic waves (Garambois & Dietrich, 2002;Haartsen & Pride, 1997). The poroelastic equations have been solved in 2D and 3D cases, mainly using finite difference schemes (Carcione, 1998;Dai et al, 1995;Masson & Pride, 2010;O'Brien, 2010) in the time-space domain. For discretisation issues, the equations (1) should be decomposed in propagative and diffusive parts, which have to be solved independently (Carcione, 1998).…”

Numerical modeling of 2D seismic wave propagation in fluid saturated porous media using graphics processing unit (GPU): Study case of realistic simple structural hydrocarbon trap

“…On the basis of Biot's theory of poroelasticity, analytical solutions for propagation velocities and reflection coefficients are available for simple [5]. The finite-difference (FD) method for Biot's equations has been formulated in several ways, central difference FD method in displacement [6], velocity-stress predictor-corrector FD method [7]. To simulate the wave propagation in unbounded media, the perfectly matched layer (PML) method is implemented as an absorbing boundary condition (ABC) [8].…”