The stability of poro-elastic wave equations in saturated porous media
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RESEARCH ARTICLE - APPLIED GEOPHYSICS
The stability of poro‑elastic wave equations in saturated porous media Fansheng Xiong1 · Weitao Sun1,2 · Jiawei Liu3 Received: 2 April 2020 / Accepted: 4 November 2020 © Institute of Geophysics, Polish Academy of Sciences & Polish Academy of Sciences 2020
Abstract 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. Keywords Wave equations · Porous media · Volume averaging · Stability analysis
Introduction The formulation of the poro-elastic wave equations is an important topic of petroleum engineering and geophysics for a long time (Biot 1941; Burridge and Keller 1981; De Communicated by Michal Malinowski (CO-EDITOR-INCHIEF)/Liang Xiao (ASSOCIATE EDITOR). Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11600-020-00508-y) contains supplementary material, which is available to authorized users. * Weitao Sun [email protected] Fansheng Xiong [email protected] Jiawei Liu jw‑[email protected] 1
Zhou Pei‑Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China
2
School of Aerospace Engineering, Tsinghua University, Beijing 100084, China
3
School of Geoscience and Info‑Physics, Central South University, Changsha 410083, China
la Cruz and Spanos 1985; Pride et al. 1992; Dvorkin and Nur 1993; Chapman et al. 2002; Carcione et al. 2004). Biot (1956, 1962) proposed wave equations in porous elastic media saturated with a single fluid based on Lagrange’s equations. There is a vast extensive works about wave propagation models in porous medium based on or similar with Biot’s theory (White 1975; Dutta and Ode 1979; Mavko and Nur 1979; Dvorkin et al. 1995; Johnson 2001; Sun et al. 2016, 2018). It is worth noting that Biot theory assu
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