STOCHASTIC OPTIMIZATION FOR GLOBAL MINIMIZATION AND GEOSTATISTICAL CALIBRATION

Abstract

This study proposes a stochastic optimization technique that uses a gradient-based method as the primary optimization method, as well as a geostatistical conditional simulation to perturb and calibrate parameters at every local minimum. If the optimization process is trapped at a local minimum due to the limitations of the gradient-based method, it generates equi-probable parameter fields using a geostatistical conditional simulation. Among the generated fields, the optimization process selects one that enables the objective function to be reduced below the value of that at the local minimum, and then reactivates the gradient-based optimization. In generating equi-probable parameter fields, a constrained number of points (noted as releasing points) are randomly selected, and spatially correlated values are generated at the releasing points, conditioned to optimum parameters at the local minimum.By applying the stochastic optimization to synthetic permeability fields, it is observed that an inversed permeability field reproduces not only global distribution but also local spatial variability of the reference fields. In addition, the pressure distributions of the inversed and the reference field were much alike. To investigate dynamic properties of the inversed field and the reference field, streamline simulation was performed on both fields. Streamlines of the inversed field showed similar trajectories to those of the reference field, and time of flight (TOF) distribution of the inversed field was analogous to that of the reference field.The stochastic optimization technique proposed in this paper enables an inverse process to converge to a global minimum while preserving geostatistical properties such as mean, standard deviation, and variogram of an original field. Therefore, the stochastic optimization will be efficient in predicting future performance of a field from constrained number of permeability and pressure observation data.