Search

SUMMARY The stability of partial differential equations determines the properties of their solutions. This study focuses on the stability analysis of the equations describing wave propagation in fluids-saturated porous media. We briefly introduce the stability analysis method for the wave propagation equations and discuss the adverse effects on the solutions. In this way, the first part of this paper is mainly devoted to the analysis of the Tuncay and Corapcioglu's (TC) model, which describes the dynamic behaviour of porous media saturated with two immiscible fluids. It is pointed out that the TC model allows spatially bounded but time-exponentially exploding solutions and may yield unstable numerical results. Based on the deduced unstable factors, we construct a stable equivalent fluid (SEF) model. We rigorously analyse the stability of the SEF model using the energy method. For predicting the influence of saturation on wave velocity, the robustness of this model is preserved due to its consistency with the original TC model. Furthermore, the numerical simulations of the wavefields show that the results of the TC model exponentially increase with time after the initial effective wave signal, which does not occur in the SEF model curves. This indicates the necessity of considering the stability from a mathematical point of view during the construction of physical model. It could be useful to merge the mathematical stability theory with the geophysical wave propagation modelling theory.

Poro-elastic wave equations are one of the fundamental problems in seismic wave exploration and applied mathematics. In the past few decades, elastic wave theory and numerical method of porous media have developed rapidly. However, the mathematical stability of such wave equations have not been fully studied, which may lead to numerical divergence in the wave propagation simulation in complex porous media. In this paper, we focus on the stability of the wave equation derived from Tuncay’s model and volume averaging method. By analyzing the stability of the first-order hyperbolic relaxation system, the mathematical stability of the wave equation is proved for the first time. Compared with existing poro-elastic wave equations (such as Biot’s theory), the advantage of newly derived equations is that it is not necessary to assume uniform distribution of pores. Such wave equations can spontaneously incorporate complex microscale pore/fracture structures into large-scale media, which is critical for unconventional oil and gas exploration. The process of proof and numerical examples shows that the wave equations are mathematically stable. These results can be applied to numerical simulation of wave field in reservoirs with pore/fracture networks, which is of great significance for unconventional oil and gas exploration.

Simulating and predicting wave propagation in porous media saturated with two fluids is an important issue in geophysical exploration studies. In this work, wave propagation in porous media with specified structures saturated with two immiscible fluids was studied, and the main objective was to establish a wave equation system with a relatively simple structure. The wave equations derived by Tuncay and Corapcioglu were analyzed first. It was found that the coefficient matrix of the equations tends to be singular due to the inclusion of a small parameter that characterizes the effect of capillary stiffening. Therefore, the previously established model consisting of three governing equations may be unstable under natural conditions. An improved model based on Tuncay and Corapcioglu’s work was proposed to ensure the nonsingularity of the coefficient matrix. By introducing an assumption in which one fluid was completely wrapped by the other, the governing equation of the wrapped fluid was degenerated. In this way, the coefficient matrix of wave equations became nonsingular. The dispersion and attenuation prediction resulting from the new model was compared with that of the original model. Numerical examples show that although the improved model consists of only two governing equations, it can obtain a result similar to that of the original model for the case of a porous medium containing gas and water, which simplifies the complexity of the calculations. However, in a porous medium with oil and water, the predictions of dispersion and attenuation produced by the original model obviously deviate from the normal trend. In contrast, the results of the improved model exhibit the correct trend with a smooth curve. This phenomenon shows the stability of the improved model and it could be used to describe wave propagation dispersions and attenuations of porous media containing two immiscible fluids in practical cases.

Reservoir rocks are heterogeneous porous media saturated with multiphase fluids, in which strong wave dissipation and velocity dispersion are closely associated with fabric heterogeneities and patchy saturation at different scales. The irregular solid inclusions and fluid patches are ubiquitous in nature, whereas the impact of geometry on wave dissipation is still not well-understood. We have investigated the dependence of wave attenuation and velocity on patch geometry. The governing equations for wave propagation in a porous medium, containing fluid/solid heterogeneities of ellipsoidal triple-layer patches, are derived from the Lagrange equations on the basis of the potential and kinetic energies. Harmonic functions describe the wave-induced local fluid flow of an ellipsoidal patch. The effects of the aspect ratio on wave velocity are illustrated with numerical examples and comparisons with laboratory measurements. The results indicate that the P-wave velocity dispersion and attenuation depend on the aspect ratio of the ellipsoidal heterogeneities, especially in the intermediate frequency range. In the case of Fort Union sandstone, the P-wave velocity increases toward an upper bound as the aspect ratio decreases. The example of a North Sea sandstone clearly indicates that introducing ellipsoidal heterogeneities gives a better description of laboratory data than that based on spherical patches. The unexpected high-velocity values previously reported and ascribed to sample heterogeneities are explained by varying the aspect ratio of the inclusions (or patches).