Mechanistic 3D finite-difference simulation of borehole spontaneous potential measurements

Abstract

The quantitative interpretation of borehole spontaneous potential (SP) measurements via Nernst’s equation often relies on limiting assumptions such as shallow mud-filtrate invasion, negligible streaming potentials, and uniform borehole symmetry. To overcome these limitations while honoring the governing physics of coupled mass transport associated with SP phenomena, we have developed a 3D finite-difference algorithm to simulate borehole SP measurements acquired across water-bearing rocks that incorporates electrochemical, membrane, and electrokinetic SP. The algorithm is based on a mechanistic description of nonequilibrium thermodynamics that enables its coupling with a fluid flow simulator to quantify the effects of continuously varying properties within permeable formations due to mud-filtrate invasion. Numerical modeling of SP measurements acquired under complex petrophysical and geometric conditions enables uncertainty quantification in the estimation of formation-water resitivity, location of bed boundaries, or detection of permeable beds while accounting for shoulder-bed effects, borehole deviation, and borehole eccentricity. Our results indicate that for well trajectories with a relative dip of less than 30°, the assumption of perpendicular beds does not entail significant errors in SP-related calculations, thereby reducing CPU time by a factor of at least 1.76. In vertical wells, SP provides the best resolution possible because deviated wells or dipping beds result in more extended and pronounced shoulder-bed effects. Furthermore, electrokinetic effects can be neglected for commonly used pressure overbalance ranges. In cases in which electrokinetic contributions are not negligible, we conclude that they are more significant when the rock permeability is in the two-figure millidarcy range. Finally, the simulation algorithm enables hypothesis testing to determine the origin and conditions under which SP shale-baseline shifts may occur. The latter shifts can signal vertical variations in salt concentration, which are crucial in the estimation of water saturation and detection of aquifer compartments.