A Rigorous Mathematical Approach for Petrophysical Properties Estimation

Problem statement: Over the last few decades, the oil industry has shown a growing interest in the new risk analysis methodologies aimed at the evaluation of the uncertainties associated with reservoir exploitation. In particular, the effort is made to take all possible sources of uncertainties into account so that not only the strengths but also the potential weaknesses of each possible technical and economic exploitation strategy are highlighted. Approach: The main parameters used to calculate strategic information, such as the Hydrocarbon Originally In Place (HOIP) and to define proper field development plans, are porosity, fluid saturations and Net To Gross (NTG). These quantities are typically obtained through the log interpretation process, which is an inverse problem where the main petrophysical characteristics are calculated as the acceptable minimum of a cost function. The cost function describes the discrepancy between measured and simulated logs with the latter being reproduced on the basis of an assumed formation model. Results: The results of the calculation process can be affected by several uncertainties related to the physics and calibration of the measuring tools, the identification of the proper formation model and the quantification of the formation model coefficients. An effective and robust methodology able to provide a reliable evaluation of petrophysical properties and the assessment of the associated uncertainties is presented and discussed in this study. The log interpretation process was approached as a linearly constrained optimization problem, solved by coupling a Lagrange-Newton method with a primal active set algorithm. Conclusion: The evaluation of the uncertainties was obtained by coupling the optimization algorithm with the Monte Carlo approach. The results obtained by the application of the methodology to a real case, where the interpretation was complicated by a poor characterization of the reservoir fluids, are also presented the study.