Adaptive Ensemble-Based Optimisation for Petrophysical Inversion

Abstract

This article introduces a new optimisation algorithm designed for sampling the solution space of non-linear, non-convex quadratic problems. This method has been specifically designed for inversion problems where multiple distinct scenarios must be explored, which would not be achievable with a standard ensemble-based optimisation method (EnOpt). A prior ensemble is used to sample the uncertainty of the parameters before updating each member of the ensemble independently in order to find the minimum of an objective function. This update is given by a Gauss–Newton-like approach, where the first-derivative matrix is adaptively estimated from sub-ensembles of parameters and their corresponding forward model responses. As the first-derivative matrix is statistically computed from an ensemble of realisations, the forward model does not need to be known and the algorithm is independent of it. The final ensemble provides an estimation of the uncertainty after the inversion process. The efficiency of this adaptive ensemble optimisation (A-EnOpt) method is first tested on simple two-dimensional problems where known mathematical functions are used as forward models. The results show that the method minimises the objective functions and samples the final uncertainties of the problems to a better degree than a standard EnOpt. The A-EnOpt method is then applied to a real heavy-oil field petrophysical inversion. Porosity, shale fraction and fluid saturations are inverted under continuity and Lagrange constraints, conditioned by P-impedance models from stochastic seismic inversion. The updated properties are incorporated in a fluid flow model of the field. The simulation results produce a better match to historical production data at one well than equivalent flow simulations using the properties before inversion. This shows that conditioning by seismic data improves the quality of the geological models.