A closer look at non-uniqueness during dynamic data integration

Abstract

Characterizing heterogeneous permeable media using dynamic data such as transient pressure, tracer or multiphase production history typically requires the solution of an inverse problem. These inverse problems apart from being computationally intensive are ill-posed by essence. The solutions of such inverse problems present two undesirable characteristics, instability and non-uniqueness. In order to overcome these difficulties, the addition of a regularization term is required in the inversion procedure. These regularization techniques can be grouped, depending on the approach selected, as being stochastic or deterministic. Both methods have been described and used with success in related sciences like geophysics and groundwater resources. Our objective is to show analytically and graphically the interrelation between these two techniques as well as their usefulness for obtaining better inverse solutions. In this thesis we review several concepts regarding regularization techniques for inverse problems. We determine an analytical relationship between the two existing regularization approaches and show graphically similarities between them. Next, several realizations of permeability fields are obtained by the inversion of production data without any type of regularization criteria as well as applying each of the regularization techniques. We then perform a comparative analysis of the results obtained. These models are qualitatively compared with the true solution. Finally, recognizing the importance of the relative weighting of the regularization term and the data misfit, its optimal value is investigated. The regularization techniques studied here are distribution is reconstructed using water-cut history from producing wells. This synthetic example addresses some of the key issues regarding the behavior of the solutions of the inverse problem. As a result of this study, a practical set of guidelines regarding several factors which affect the uniqueness of the inverse problem is presented.