Stiffness Formula and Conditioning Analysis for General Brown–Korringa Fluid Substitution

Source abstract

Subsurface rocks commonly contain fluids and exhibit elastic anisotropy. Fluid substitution in anisotropic media is therefore important in geophysical applications. The general fluid-substitution relation given by Brown and Korringa is expressed in terms of elastic compliance, whereas seismic wave propagation is formulated in terms of stiffness. In this study, a direct stiffness-domain formula for anisotropic fluid substitution is derived using the Sherman–Morrison formula. The undrained stiffness matrix is written as a rank-one update of the frame stiffness matrix. The derived formula is analytically equivalent to the conventional compliance-inversion form, and the singular condition is identified through determinant analysis. Numerical calculations for an anisotropic Berea sandstone model demonstrate that variations in the pore-fluid bulk modulus cause nonlinear and component-dependent stiffness changes. As the singular limit is approached, the undrained compliance matrix becomes increasingly ill-conditioned. The compliance inversion error reaches O (10−6), whereas the error of the direct rank-one expression remains on the order of 10−16 over the tested range. The proposed formula provides a direct stiffness-domain implementation of Brown–Korringa fluid substitution. It improves numerical accuracy over conventional compliance inversion near the singular limit.