Simulation of Porosity Field Using Wavelet Bayesian Inversion of Crosswell GPR and Log Data

In this paper, we present a novel approach to simulate porosity fields constrained by borehole radar tomography images. The cornerstone of the method is the bayesian analysis of the approximation wavelet coefficients of a petro-physical analogue. The method is tested with a two-dimensional porosity field generated from a digital picture of a real sand deposit. The porosity field is translated into electrical properties and a cross-hole tomography synthetic survey is modeled using a finite-difference modeling algorithm. In parallel, an analogue deposit is created based on the geological knowledge of the area under study. The analogue porosity field is converted into electrical property fields using the same equations as previously. A synthetic GPR tomography is also computed from the latter. Wavelet decomposition of both measured and analogue tomograms and porosity analogue fields is then calculated. Based on the assumption that geophysical data carry only the large-scale information about the geological model, statistical analysis of the approximation coefficients of each variable is carried out. From the measured tomogram approximation coefficients and the cross statistics evaluated on the analogues, the approximation of the real porosity field is inferred using bayesian inference. Finally, based on the geostatistical relationships between wavelet coefficients across the different scales, all the porosity wavelet detail coefficients are simulated using a standard geostatistical simulation algorithm. The wavelet coefficients are then back transformed in the porosity space. The final simulated porosity fields contain the large wavelengths of the measured radar tomogram and the texture of the analogue porosity field.