Numerical Simulation of Finite Difference Method for Wave Fields in Biphase Media

In computational geophysics and seismology, numerical simulation of wave fields in biphase media represents a significant research area. Biphase media typically refer to porous materials composed of solid matrices and pore fluids, such as water-saturated sandstones and hydrocarbon reservoirs. The wave propagation characteristics in these media are considerably more complex than in single-phase media.

The Finite Difference Method (FDM) serves as an effective numerical approach for simulating such wave propagation. The core methodology involves discretizing continuous space and time into grid points, replacing differential operations with finite difference approximations. For biphase media, this typically requires solving the Biot equations or their modified versions, which describe the coupled motion between solid matrices and pore fluids. In code implementation, this translates to solving systems of partial differential equations using discrete difference operators.

Key implementation considerations include: Spatial Discretization: Employing staggered grid techniques to handle different physical quantities (displacements, stresses, etc.) - in practice, this means assigning velocity and stress components to different grid positions to improve numerical accuracy Temporal Discretization: Using explicit time integration schemes like central difference methods, where the implementation typically follows leapfrog time-stepping algorithms Boundary Condition Handling: Implementing absorbing boundary conditions (such as PML - Perfectly Matched Layers) to minimize artificial reflections at computational domain edges Stability Analysis: Applying Courant-Friedrichs-Lewy (CFL) conditions to determine maximum allowable time steps for numerical stability Physical Parameter Configuration: Setting material properties including solid density, fluid density, porosity, permeability, and other Biot parameters through structured input arrays

This simulation methodology finds extensive applications in hydrocarbon exploration, groundwater resource assessment, and seismic hazard prediction, providing numerical experimental foundations for understanding wave field characteristics in complex geological formations. By adjusting medium parameters through parameterized input files, researchers can investigate wave propagation patterns under various geological conditions, enabling systematic sensitivity analysis and model validation.