Frequency-domain seismic wave modelling in heterogeneous porous media using the mixed-grid finite-difference method

Xu Liu, Stewart Greenhalgh, Bing Zhou, Mark Greenhalgh

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


To simulate seismic wave propagation in natural porous media, the second-order Biot wave equations in the frequency domain are numerically solved with the (2-D) mixed-staggered grid nine-point stencil finite-difference method using both homogeneous and heterogeneous media approaches. A comparison of numerical results on three synthetic model examples shows that a significant discrepancy may be caused with the homogeneous formulation that ignores the spatial gradients of the medium properties. Such an approach, whilst previously applied, is not suitable to use for reflected/refracted wave investigations. A new and easier method is introduced to obtain the spatial derivative terms in the governing partial differential equation through a rotated coordinate system without using the derivative chain rule. This method is to directly rotate the vectors of the wave equations in the rotated coordinate system to the vectors in the classic coordinate system to constitute the wave equation in the mixed grid system. By using this new method, the exact same results are obtained as those with the classic method using the derivative chain rule. The calculated wavefields of solid particle velocity and fluid flux from the second-order Biot wave equation are used to calculate the pore (fluid) pressure through Biot's first-order constitutive equation. The fluid pressure wavefields are almost entirely devoid of all identified shear waves. Attention is also drawn to the fact that the current finite-difference, frequency-domain modelling methods, designed for the highest frequency and the slowest shear wave velocity in the model, could result in undersampling of the Biot slow P wave.

Original languageBritish English
Pages (from-to)34-54
Number of pages21
JournalGeophysical Journal International
Issue number1
StatePublished - 1 Jan 2019


  • Computational seismology
  • Numerical solutions
  • Wave propagation


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