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

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.