Modeling Elastic Waves with Lattice Spring Model

Xia, Muming (China University of Petroleum) | Zhou, Hui (China University of Petroleum) | Zhang, Qingchen (China University of Petroleum) | Chen, Hanming (China University of Petroleum) | Dou, Yuzhao (China University of Petroleum)

OnePetro 

Summary

In this paper, the Lattice Spring Model (LSM) is adopted in forward modeling of elastic waves propagation in solid medium by combination with the Verlet Algorithm. Different from the traditional methods, such as Finite Difference Method (FDM), Finite Element Method (FEM) etc., LSM is a new method which is not based on the wave equations, but on the microcosmic mechanism that causes wave propagation. Firstly, the origin and history of LSM is introduced. Secondly, the theoretical framework of LSM is elaborated and a stability condition for the evolution of this system is deduced. Then, some numerical results of LSM are demonstrated and they are compared with the wave fields obtained by FDM. Finally, a brief conclusion is drawn based on the previous discussions.

Introduction

First devised by Grest and Webman in 1984, Lattice Spring Model (LSM) is a collection of linear springs connected at nodes distributing on a cubic lattice used for describing solid medium (Grest and Webman, 1984; Hassold and Srolovitz, 1989). In order to model materials of different Poisson’s ratios, angular springs are added to the original linear spring system (Wang, 1989). Ladd and Kinney (1997) developed this model by taking the idea of elastic element to improve its calculation precision. Such a simple model is sufficient to simulate heterogeneous elastic medium, and its application can be seen in modeling deformation and failure (Ladd and Kinney, 1997; Buxton et al., 2001; Zhao et al., 2011).

As is known to all, extensive research has been performed to solve the dynamic problems involving waves, and FDM is the most frequently used numerical method, which solves the wave equation by finite difference approximation of its partial derivative (Toomey and Bean, 2000). Yim and Sohn (2000) adopted a model similar to LSM for visualization of ultrasonic waves, but the evolution of wave fields are calculated by FDM. Pazdniakou and Adler (2012) made a further introduction of LSM and laid the foundation for its potential application in wave propagation in porous media in the low frequency band. Xia et al. (2014) modeled P waves from low frequencies (seismic frequency) to high frequencies (sonic log frequency) by importing a stability conditional for LSM dynamics.