Simultaneous characterization of geological shapes and permeability distributions in reservoirs using the level set method

Abstract

Abstract An important technique in modern reservoir management is the use of production data for the characterization of permeability distributions with more than one lithology from production data. A major difficulty which arises in these situations is that sharp discontinuities of permeabilities between these lithofacies need to be reconstructed at the same time with more smoothly varying profiles inside of them. So far, techniques have been developed which recover either the interfaces or a smoothly varying profile, but not both of them. Our group has developed recently a novel method for modelling and reconstructing geological shapes in reservoirs from production data. This method is based on a level set representation of the shapes. In our previous work[1,2] we have assumed that the permeability distribution in the different regions of the reservoirs do not vary significantly such that they can be approximated by a constant value which was approximately known from prior information. In this paper we will present an extension of this new method which is able to recover at the same time the shapes of the different geological regions and variations of the permeability distribution in each of these regions. In each step of our new reconstruction technique the production data are used in order to calculate an update for the shape of the two lithofacies and at the same time a pixel-based correction of the parameter profiles inside of these regions. Only one simulation of a two-phase flow simulator and a corresponding adjoint simulator are necessary for calculating both updates. In our paper we will present numerical results in 2D for realistic situations of a water flooding process. We show that our method is able to reconstruct two lithofacies of complicated shape and the corresponding permeability distributions simultaneously from these production data. Introduction The history matching problem is an important component of a reservoir simulation. In order to optimize this process, the reservoir engineer needs to know the rock properties inside the reservoir. Due to its importance in field development plans, the history matching problem has been widely studied, and many solution techniques have been developed to date. For example, given the measured data, the physical parameter distributions (e.g., permeabilities) inside the reservoir can be sought using a pixel based inversion technique.[3–10] One important difficulty in this approach is that the corresponding inverse problem is severely ill-posed, so that strong regularization needs to be applied in order to stabilize the inversion process and to obtain a well-defined solution. Moreover, in many situations there exist more than one type of rock in the reservoir in different regions, which makes the problem even worse. In these cases, the pixel based inversion techniques typically don't allow for representing sharp discontinuities which exist between these regions. However, if there is more than one type of rock in the reservoir, it is necessary to know the location of these region interfaces in addition to the physical properties inside each of the regions in order to obtain a reliable model. Therefore, most of the automatic history matching tools used today are not well-suited for being applied to reservoirs with unknown facies boundaries. In the literature, there exist several works focusing on alternative automatic history matching techniques based on geological shape definitions. One example is the use of shape triangularization[11,12] which, however, requires to have prior information of the facies location available. Other methods use geostatistical approaches,[13,14] which on the other hand require to generate a considerable number of realizations. Alternative level set based approaches have been proposed very recently in the literature.[1,2,15,16] Our scheme presented in the work Villegas et al.[1,2] differs in several aspects from the approach in Lien et al.[15,16] We apply a so-called ‘adjoint scheme’ to calculate the sensitivities during the reconstruction, and as regularization tool we use an adapted filtering operator to be applied in each step of the inversion problem. This combination of the adjoint scheme and the novel regularization scheme allows us to characterize reservoirs with production history on a regular PC in a relatively short time.