Seismic Wave Equations in Tight Oil/Gas Sandstone Media

TL;DR

This paper derives simplified seismic wave equations for tight oil/gas sandstone media using volume averaging, facilitating better parameter inversion and gas detection, especially when considering compressional waves in gas.

Contribution

It introduces a new macro-scale motion equation system derived from volume averaging tailored for tight sandstone media, simplifying Biot equations.

Findings

Derived simpler motion equations suitable for parameter inversion.

Established explicit relationships between wave equation coefficients and medium parameters.

Simplified equations applicable to compressional wave analysis in gas detection.

Abstract

The paper is devoted to the derivation of a combined system of motion equations for solid and fluid in isotropic tight oil/gas sandstone media through volume averaging theorems (VAT). Based on the features of the media, four physical assumptions are proposed as the foundation for our derivation. More precisely, volume averaging theorems are applied to the micro-scale motion equations for both the solid and the fluid as well as to the stress-strain relations, resulting in a combined system of macro-scale equations for the tight oil/gas sandstone media. It is worth noting that the four assumptions may not be satisfied in the whole region. Nevertheless, since the characteristic diameter for applying VAT ranges between $10^{-6}$ meters and dozens of meters, we may split the entire domain into several sub-domains such that the four physical assumptions are satisfied in each sub-domain. By choosing a proper characteristic diameter of an averaging volume, we derive a formula for the fluid average pressure in terms of the divergence of the average displacement from the continuity equation of the fluid. As a result, the motion equations derived in this paper are simpler than the Biot equations, and are more suitable for inversion of porous medium parameters. When the fluid is gas and the compressional wave is considered, the derived motion equations can be simplified to the diffusive-viscous wave equation. Moreover, the explicit relationship between the coefficients in this equation and medium parameters is very important for gas detection in tight gas sandstone.