Multi-level parameter structure identification for two-phase porous-media flow problems using flexible representations Inga Berre a, * , Martha Lien b , Trond Mannseth a,b a Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N-5008 Bergen, Norway b CIPR – Centre for Integrated Petroleum Research, University of Bergen, Realfagbygget, Allégaten 41, N-5007 Bergen, Norway article info Article history: Received 6 May 2009 Received in revised form 29 September 2009 Accepted 2 October 2009 Available online 12 October 2009 Keywords: Parameter identification Two-phase porous-media flow Reparametrization abstract We consider identification of absolute permeability (hydraulic conductivity) based on time series of pres- sure data in sparsely distributed wells for two-phase porous-media flow. For this problem, it is impossi- ble to recover all details of the parameter function. On the other hand, a coarser, approximate recovery may be sufficient for many applications. We propose a novel solution approach, based on reparametriza- tion, for such approximate identification of the parameter function. We use a nonlinear, composite rep- resentation, which is detached from the computational grid, allowing for a flexible representation of the parameter function at many resolution levels. This is utilized in a sequential multi-level estimation of the parameter function, starting at a coarse resolution, which is then gradually refined. The composite rep- resentation is designed to allow for smooth as well as sharp transitions between regions of nearly con- stant parameter value. Moreover, it facilitates the estimation also of the structure and smoothness of the parameter function itself. As a limiting case, the chosen representation is reduced to a zonation with implicit representation of the interior boundaries that is equivalent to a level-set representation. A moti- vation for the selected representation and the multi-level estimation is presented in terms of an analysis of sensitivity and nonlinearity. Numerical examples demonstrate identification of coarse-scale features of reference permeability distributions with varying degree of smoothness. Comparisons show how the multi-level strategy stabilize the identification and avoid local minima of the objective function com- pared to a single-level strategy. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction The need to characterize the absolute permeability (hydraulic conductivity) of porous media is rooted in important applications within groundwater flow, petroleum-reservoir flow, hydrothermal and enhanced geothermal systems, and flow in the context of car- bon dioxide sequestration and storage of nuclear waste. The per- meability may vary on different length scales, and, typically, permeability values span several orders of magnitude. Dynamic data from wells provide important information in per- meability identification. Utilization of this data results in an in- verse problem, but for any detailed permeability identification, the problem is severely ill-posed due to the spatially sparse distri- bution of wells. In addition to the problems stemming from the ill- posedness, high nonlinearity of the mapping from permeability to data causes difficulties in the parameter estimation. To gain a better posed problem, a variety of regularization tech- niques has been developed [6,7,26,27,29,31,41]. One approach for regularization is to apply a reduced representation of the unknown parameter function. Traditionally, a solution candidate to the in- verse problem is found by minimizing an objective function with respect to a set of unknown coefficients in a predefined represen- tation. Because the resolution power of the data, as well as the structure of the permeability field, is unknown, it is difficult to find a sufficiently good representation prior to the estimation. One way to address this problem is to consider the extended inverse prob- lem, where one seeks to determine both a representation war- ranted by the available information, and the associated coefficients in the representation [32,37]. Even though a detailed permeability identification is not war- ranted from dynamic data alone, such data may be sufficient for estimation of large-scale structures, like lithofacies, in the sense that their location and permeability values may be approximately identified. Prior knowledge about the shapes or even the existence of such large-scale structures is often very limited, and therefore, difficult to utilize for regularization. To achieve regularization based on a reduced representation, the permeability is often represented by a zonation [21,22]. Tradition- ally, the reservoir is partitioned into a predefined set of zones that each are assumed to be of unknown, but constant, permeability. 0309-1708/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2009.10.002 * Corresponding author. Tel.: +47 5558 2856; fax: +47 555 89672. E-mail addresses: inga.berre@math.uib.no (I. Berre), martha.lien@cipr.uib.no (M. Lien), trond.mannseth@cipr.uib.no (T. Mannseth). Advances in Water Resources 32 (2009) 1777–1788 Contents lists available at ScienceDirect Advances in Water Resources journal homepage: www.elsevier.com/locate/advwatres