Adaptive Ensemble-Based Optimisation for Petrophysical Inversion
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Adaptive Ensemble-Based Optimisation for Petrophysical Inversion Rémi Moyen1 · Théophile Gentilhomme2
Received: 29 January 2020 / Accepted: 1 October 2020 © International Association for Mathematical Geosciences 2020
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
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Rémi Moyen [email protected]
1
CGG, 27, avenue Carnot, 91341 Massy, France
2
Idiap Research Insitute, Martigny, Switzerland
123
Math Geosci
than equivalent flow simulations using the properties before inversion. This shows that conditioning by seismic data improves the quality of the geological models. Keywords Inversion · Petrophysics · Ensemble · Seismic · Porosity
1 Introduction This article presents a new adaptive ensemble-based optimisation (A-EnOpt) method for sampling the solution space of non-linear inversion problems in order to obtain an estimation of the uncertainties with different plausible solutions. This method is in particular applied to petrophysical inversion to illustrate its application to geophysical problems where several realistic parameter configurations can reproduce the observed data. Non-linear inverse problems are usually solved using global or local optimisation methods, and must include a stopping criterion which is often related to the data uncertainty (Tarantola 2005). Global optimisation methods aim to directly find a glo
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