Porosity inference and classification of siliciclastic rocks from multiple data sets

We develop a Bayesian formulation for joint inference of porosity and clay volume, incorporating multiple data sets, prior information, and rock physics models. The derivation is carried out considering the full uncertainty involved in calculations from unknown hyperparameters required by either rock physics equations (model coefficients) or statistical models (data variances). Eventually, data variances are marginalized in closed form, and the model coefficients are fixed using a calibration procedure. To avoid working with a high-dimension probability density function in the parameter space, our formulation is derived and implemented using a moving window along the data domain. In thisway, we compute a collection of 2D posterior distributions forinterval porosity and clay volume, corresponding to each positionalong the window’s path. We test the methodology on both synthetic and real well logs consisting of gamma-ray, neutron, compressional and shear sonic velocity, and density. Tests demonstrate that integrating the relevant pieces of information about porosity and clay volume reduces the uncertainties associated with the estimates. Error analysis of a synthetic data example shows that neutron and density logs provide more information about porosity, whereas gamma-ray logs and velocities provide more information about clay volume. Additionally, we investigate a change in fluid saturation as a source of systematic error in porosity prediction. A real data example, incorporating porosity measurements on core samples, further demonstrates the consistency of our methodology in reducing the uncertainties associated with our final estimates.