Mechanistic numerical simulation and interpretation of borehole measurements of spontaneous electrical potential acquired in complex petrophysical environments

Borehole measurements of spontaneous electrical potential (SP) are routinely acquired in wells drilled with water-based mud. However, to this day, the interpretation of borehole SP measurements is chiefly limited to imprecise calculations of formation water resistivity and qualitative assessments of volumetric concentration of shale and permeability. This dissertation develops new methods to numerically simulate borehole SP measurements and improve their quantitative interpretation. Interpretation products are water saturation, water resistivity, and radius of invasion of mud-filtrate invasion in permeable rocks, together with their uncertainty. The calculation of formation water resistivity from borehole SP measurements is commonly performed via Nernst’s equation under the assumptions of shallow mud-filtrate invasion, negligible streaming potentials, and water as the only rock-saturating fluid. To circumvent these limitations while honoring the governing physics of coupled mass transport associated with SP phenomena, a three-dimensional finite-difference algorithm is developed to incorporate electrochemical, membrane, and electrokinetic SP phenomena in the simulation of borehole SP measurements. The algorithm implements a mechanistic description of non-equilibrium thermodynamics, which is coupled to a fluid-flow simulator to quantify the effects of time-varying conditions within permeable formations due to mud-filtrate invasion. Simulations indicate that the best spatial resolution of rock properties possible with SP borehole measurements occurs when rock beds are perpendicular to the well; deviated wells or dipping beds give rise to extended and pronounced shoulder-bed effects on SP measurements. It is also found that the simplifying assumption of perpendicular beds relative to the borehole does not cause significant errors in the numerical simulation of borehole SP measurements acquired in well trajectories with a relative dip less than 30°, thereby reducing CPU time by a factor of at least 1.76. Furthermore, electrokinetic effects on SP measurements become negligible for commonly used pressure overbalance ranges. For the interpretation of borehole SP measurements acquired in hydrocarbon-bearing rocks, this dissertation explores whether the difference between borehole SP measurements and Nernst-equation predictions enables the estimation of in situ hydrocarbon saturation of porous rocks. A new petrophysical model is advanced and successfully verified to establish the limits of detectability of hydrocarbon saturation solely from borehole SP measurements. It is found that optimal conditions for the quantification of hydrocarbon saturation from borehole SP measurements take place when (1) capillary forces dominate the process of mud-filtrate invasion, (2) the matrix-pore interface region, known as the electrical double layer, has a relevant impact on the diffusion of counter-ions, and (3) the electrolyte concentration of drilling mud is greater than that of formation water. Three blind tests show that the developed petrophysical model and the mechanistic SP simulation algorithm enable the estimation of hydrocarbon saturation from SP borehole measurements without the need of electrical resistivity measurements or porosity calculations. The estimation is reliable when (a) the volumetric concentration of shale is negligible, (b) the pore network structure is constant throughout the reservoir, and (c) radial invasion profiles are similar to those observed in calibration key wells used to adjust the parameters of the new petrophysical model. Finally, this dissertation develops a new inversion-based method for the interpretation of borehole SP measurements, which concomitantly mitigates shoulder-bed and mud-filtrate invasion effects on SP logs via fast numerical simulations based on Green’s functions. The method delivers layer-by-layer estimates of (a) equivalent NaCl concentration, (b) radius of mud-filtrate invasion, and (c) sodium macroscopic transport number, together with their uncertainty, by progressively matching borehole SP measurements with their numerical simulations. Successful examples of implementation include noisy borehole SP measurements acquired in aquifers with various degrees of petrophysical complexity. Results confirm the possibility of accurately and reliably estimating the electrical resistivity of formation water resistivity solely from borehole SP measurements, i.e., without the need of porosity calculations or fitting parameters from independent core measurements (as is the case with borehole resistivity measurements). Inversion-based interpretation results (a) compare well to those obtained from resistivity and nuclear porosity logs, (b) provide estimates of uncertainty, and (c) can assimilate a priori knowledge of aquifer petrophysical properties in the estimation.