Unsaturated poroelastic responses of an inclined borehole in a porous medium saturated with two immiscible fluids incorporating the concept of the equivalent pore pressure

Conventional drilling design inaccurately predicts the mud density required for borehole stability by assuming the porous rock is fully saturated with a single fluid while in reality it may be saturated with multiple fluids. However, the unsaturated poroelastic responses arising from the coupling of multiphase flow and mechanical deformation in a porous rock around a borehole have rarely been studied either numerically or analytically. For an inclined borehole in a formation saturated with two immiscible fluids, this paper presents a semi-analytical Laplace-type unsaturated poroelastic solution that takes into account the effect of interfacial energy by incorporating the equivalent pore pressure concept for an unsaturated (under-saturated) porous medium. The solution can be used not only to analyze the borehole stability during drilling but also to evaluate the poroelastic responses of a deeply excavated circular unlined tunnel in a formation saturated with water and gas. Based on the abovementioned fundamental Laplace-type solution, one further firstly proposes analytical unsaturated poroelastic solutions under four types of time-domain conditions including the instantaneous, modified instantaneous, short-time, and long-time ones. These different time-domain solutions can well guide the drilling engineer to design a more appropriate mud weight that minimizes the borehole collapse (breakout) from a conservative perspective. It is considered where the formation is saturated with water and oil, and the main results show that the phase-pressure differences that are determined by the two concepts of the equivalent and average pore pressure increase with increasing interfacial energy. The commonly used assumption that the formation is fully water-saturated overestimates the safe mud pressure window for borehole stability at early times but underestimates it at later times. Contrary to the classical elastic solution, the poroelastic solutions including the unsaturated and saturated ones show that the location of the borehole breakout does not always occur at the borehole wall. It first occurs at the borehole wall, then in the formation, and finally at the borehole wall as time progress. This is the result of variations in pore pressure and effective stresses with time. It suggests that engineers introduce the time-dependence and location-dependence into a mud-weight design to ensure borehole stability.