Stochastic Bayesian inversion of borehole self-potential measurements

Abstract

We propose a mechanistic model to compute and to invert self-potential log data in sedimentary basins and for near-surface geophysical applications. The framework of our analysis is founded in a unified electrical conductivity and self-potential petrophysical model. This model is based on an explicit dependence of these properties on porosity, water saturation, temperature, brine salinity, cementation and saturation (Archie) exponents and the volumetric charge density per unit pore volume associated with the clay fraction. This model is consistent with empirical laws widely used to interpret self-potential logs according to the two limiting cases corresponding to a clean sand and a pure shale. We present a finite element calculation of the self-potential signal produced by sand reservoirs interstratified with shale layers. For layered strata normal to the well, we demonstrate that the 3-D Poisson equation governing the occurrence of self-potentials in a borehole can be simplified to a 2-D axisymmetric partial differential equation solved at each depth providing a common self-potential reference can be defined between these different depths. This simplification is very accurate as long as the vertical salinity gradients are not too strong over distances corresponding to the borehole diameter. The inversion of borehole data (self-potential, resistivity and density well logs, incorporating information derived from neutron porosity and gamma-ray log data) is performed with the Adaptive Metropolis Algorithm (AMA). We start by formulating an approximate analytical solution for the six model parameters (water saturation, porosity, the two Archie's exponents, the pore water conductivity and the volumetric charge density of the diffuse layer). This solution is used for the AMA algorithm to converge in less than 60 iterations at each depth for the real case study. The posterior probability distributions are computed using 50–60 additional realizations. Our approach is applied to a case study concerning a small sedimentary sequence in the Piceance Basin, Colorado, in a series of tight gas reservoirs.