Effective-Stress Coefficients of Porous Rocks Involving Shocks and Loading/Unloading Hysteresis

Abstract

SummaryA critical review, examination, and clarification of the various issues and problems concerning the definition and dependence of the effective-stress coefficients of porous-rock formations is presented. The effective-stress coefficients have different values for different rock properties because the physical mechanisms of rock deformation can affect the various rock properties differently. The alteration of petrophysical properties occurs by the onset of various rock-deformation/damaging processes, including pore collapsing and grain crushing, and affects the values of the effective-stress coefficients controlling the different petrophysical properties of rock formations. The slope discontinuity observed in the effective-stress coefficients of naturally or induced fractured-rock formations during loading/unloading, referred to as a shock effect, is essentially related to deformation of fractures at less than the critical effective stress and deformation of matrix at greater than the critical effective stress. The hysteresis observed in the effective-stress coefficients of heterogeneous porous rocks during loading/unloading is attributed to elastic deformation under the fully elastic predamage conditions, and/or irreversible pore-structure-alteration/deformation processes.A proper correlation of the Biot-Willis coefficient controlling the bulk volumetric strain is developed using the data available from various sources in a manner to meet the required endpoint-limit conditions of the Biot-Willis coefficient, ranging from zero to unity. The modified power-law equation presented in this paper yields a physically meaningful correlation because it successfully satisfies the low-end- and high-end-limit values of the Biot-Willis coefficient and also provides a better quality match of the available experimental data than the semilogarithmic equation and the popular basic power-law equation. It is shown that the semilogarithmic correlation cannot predict the values of the Biot coefficient beyond the range of the data because it generates unrealistic values approaching the negative infinity for the Biot coefficient for the low-permeability/porosity ratio and unrealistically high values approaching the positive infinity for the high-permeability/porosity ratio. The basic power-law equation is not adequate either because it can only satisfy the low-end value but cannot satisfy the high-end value of the Biot coefficient. The correlation developed in this paper from the modified power-law equation is effectively applicable over the full range of the Biot-Willis coefficient, extending from zero to unity. To the best of the author's knowledge, this paper is the first to present an effective theory and formulation of the convenient correlation of the Biot-Willis poroelastic coefficient that not only satisfies both of the two endpoint-limit values of the Biot-Willis coefficient but also produces the best match of the available experimental data.