# A Damped Dynamic Finite Difference Approach for Modeling Static Stress–Strain Fields

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## Abstract

Modeling dynamic and static responses of an elastic medium often employs different numerical schemes. By introducing damping into the system, we show how the widely used time-marching staggered finite difference (FD) approach in solving elastodynamic wave equation can be used to model time-independent elastostatic problems. The damped FD method can compute elastostatic stress and strain fields of a model subject to the influence of an external field via prescribed boundary conditions. We also show how to obtain an optimized value for the damping factor. We verified the damped FD approach by comparing results against the analytical solutions for a borehole model and a laminated model. We also validated our approach numerically for an inclusion model by comparing the results computed by a finite element method. The damped FD showed excellent agreement with both the analytical results and the finite element results.

## Keywords

Finite difference static stress–strain fields numerical modeling damped wave equation effective medium## Notes

### Acknowledgements

We thank Professor Leon Thomsen for discussions about the physical meaning of the damping parameter. We thank Dr. Nishank Saxena for pointing to the finite element code. We appreciate the financial support from UH and SUSTC. We appreciate the UH Xfrac group for providing clusters for computations. We also appreciate the comments from Dr. Hao Hu and Dr. Chen Qi. R. Lin and Y. Zheng are partially supported by NSF EAR-1833058. X. Fang is supported by National Natural Science Foundation of China (grant 41704112) and National Key R&D Program of China (2018YFC0310105).

## References

- Aki, K., & Richards, P. (1980).
*Quantitative seismology: Theory and methods*. Sausalito: University Science Books.Google Scholar - Backus, G. E. (1962). Long-Wave elastic anisotropy produced by horizontal layering.
*Journal of Geophysical Research,**67*(11), 4427–4440.CrossRefGoogle Scholar - Blackstock, D. (2000).
*Fundamentals of physical acoustics*. Oxford: Wiley.Google Scholar - der Levan, A. (1988). Fourth-order finite difference P-SV seismograms:
*eophysics,**53,*1425–1436.Google Scholar - Fang, X., Fehler, M. C., & Cheng, A. (2014). Simulation of the effect of stress-induced anisotropy on borehole compressional wave propagation.
*Geophysics,**79*(4), D205–D216.CrossRefGoogle Scholar - Fjar, E. (2008).
*Petroleum related rock mechanics*. Amsterdam: Elsevier.Google Scholar - Garboczi, E. J. (1998).
*Finite element and finite difference programs for computing the linear electric and elastic properties of digital image of random materials*(p. 6269). Rep: National Institute of Standards and Technology.CrossRefGoogle Scholar - Gregory, R. (2006).
*Douglas*. Classical Mechanics: Cambridge University Press.Google Scholar - Kelly, K., Ward, S., Treitel, S., & Alford, M. (1976). Synthetic seismograms: A finite difference approach.
*Geophysics,**41,*2–27.CrossRefGoogle Scholar - Mavko, G. T., Mukerji, T., & Dvorkin, J. (2009).
*The rock physics handbook*(2nd ed.). Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Meng, C. (2017). Benchmarking Defmod, an open source FEM code for modeling episodic fault rupture.
*Computers & Geosciences,**100,*10–26.CrossRefGoogle Scholar - Meng, C., & Wang, H. (2018). A finite element and finite difference mixed approach for modeling fault rupture and ground motion.
*Computers & Geosciences,**113,*54–69.CrossRefGoogle Scholar - Morse, P., & Feshback, W. (1953).
*Methods of theoretical physics*(Vol. 2). New York: McGraw-Hill Book Co.Google Scholar - Raymer, L. L., Hunt, E. R., & Gardner, J. S. (1980). An improved sonic transit time to porosity transform.
*21st Annual Logging Symposium, Trans. Soc. Prof. Well Log Analysts*, 8–11 July, 1–12.Google Scholar - Reddy, J. N. (2006).
*An introduction to the finite element method*. New York: McGraw-Hill.Google Scholar - Saenger, E. H., Gold, N., & Shapiro, S. A. (2000). Modeling the propagation of elastic waves using a modified finite difference grid.
*Wave Motion,**31*(1), 77–92.CrossRefGoogle Scholar - Schuster, G. (2017).
*Seismic inversion*. Tulsa: Society of Exploration Geophysicists. https://doi.org/10.1190/1.9781560803423.ch8 CrossRefGoogle Scholar - Udias, A., & Buforn, E. (2018).
*Principles of seismology*. Cambridge: Cambridge University Press.Google Scholar - Virieus, J. (1986). P-SV wave propagation in heterogeneous media: Velocity-stress finite difference method.
*Geophysics,**51,*889–901.CrossRefGoogle Scholar - Virieux, J. (1984). SH-wave propagation in heterogeneous media: Velocity-stress finite difference method.
*Geophysics,**49,*1933–1942.CrossRefGoogle Scholar