Level-set techniques for facies identification in reservoir modeling

Abstract

In this paper we investigate the application of level-set techniques for facies identification in reservoir models. The identification of facies is a geometrical inverse ill-posed problem that we formulate in terms of shape optimization. The goal is to find a region (a geologic facies) that minimizes the misfit between predicted and measured data from an oil–water reservoir. In order to address the shape optimization problem, we present a novel application of the level-set iterative framework developed by Burger in (2002 Interfaces Free Bound. 5 301–29; 2004 Inverse Problems 20 259–82) for inverse obstacle problems. The optimization is constrained by (the reservoir model) a nonlinear large-scale system of PDEs that describes the reservoir dynamics. We reformulate this reservoir model in a weak (integral) form whose shape derivative can be formally computed from standard results of shape calculus. At each iteration of the scheme, the current estimate of the shape derivative is utilized to define a velocity in the level-set equation. The proper selection of this velocity ensures that the new shape decreases the cost functional. We present results of facies identification where the velocity is computed with the gradient-based (GB) approach of Burger (2002) and the Levenberg–Marquardt (LM) technique of Burger (2004). While an adjoint formulation allows the straightforward application of the GB approach, the LM technique requires the computation of the large-scale Karush–Kuhn–Tucker system that arises at each iteration of the scheme. We efficiently solve this system by means of the representer method. We present some synthetic experiments to show and compare the capabilities and limitations of the proposed implementations of level-set techniques for the identification of geologic facies.