Reducing Uncertainties in Production Forecasts by Constraining Geological Modeling to Dynamic Data

Building reservoir geological models that are consistent with all available information is necessary to reduce uncertainties in production forecasts. The stochastic/geostatistical approach is the only feasible way of integrating all kinds of data ranging from the geological knowledge to the production history. A stochastic model is chosen that accounts for geological knowledge, and geostatistical simulation provides realizations of the stochastic model. These realizations can be conditioned to static data at well locations to reduce the model uncertainty in the area where data are available. All along the different stages of a field life more and more dynamic data are available. This paper addresses the issue of generating model realizations that account for available dynamic data to further reduce uncertainty in the underlying geological model. A general tool has been developed to build geostatistical realizations that match both the available geological and dynamic data. The core of this tool is the Gradual Deformation Method that allows consistent modification of a realization in a continuous way. This method can be combined with a nonlinear optimization method for constraining realizations of a stochastic model to dynamic data. This approach is operational for Gaussian or nonGaussian pixel based models. Model realizations can be deformed globally or locally while preserving the geostatistical characteristics. Several examples are provided to illustrate different features of the gradual deformation approach. Introduction Geostatistics, originally developed for mining industry, has been increasingly gaining success in the petroleum industry for the last twenty years. This is mainly because geostatistics provides the potentiality of consistently integrating various sources of information at different scales and also the possibility of efficiently generating multiple realizations for uncertainty evaluation. Several methods and algorithms have been developed for building reservoir models in a wide range of geological environments. Although geostatistical models help geologists doing better 3D interpretation of geological phenomena, reservoir engineers still not commonly use them for performing history matching and production forecasting. The main obstacles for the use of geostatistics by reservoir engineers are the huge number of parameters in a geostatistical model and the lack of an efficient tool for consistent modification of geostatistical realizations. This led us to develop the gradual deformation method. In this paper, we first present the principle of the gradual deformation of geostatistical models. Then, we review the most widely used geostatistical algorithms for generating reservoir geological models. We focus on the applicability of gradual deformation method to these simulation algorithms. Finally, we describe the inverse approach that combines an optimization method and the gradual deformation method for constraining or updating geostatistical reservoir models. Gradual deformation methods Geostatistical simulation can provide independent realizations of a stochastic reservoir model, which are consistent with geological and geophysical information. However, in most cases, these realizations do not reflect the dynamic behavior observed on the real field. Reservoir engineers need to modify these realizations for matching well-test data and production history. Traditional practice of history matching, by zonation, multiplication, adding permeability barriers etc., destroys the consistency and the spatial continuity of the initial stochastic model. This often leads to reservoir models with low predictability. Consequently, this prevents geostatistics from SPE 56703 Reducing Uncertainties in Production Forecasts by Constraining Geological Modeling to Dynamic Data L.Y. Hu, M. Le Ravalec, G. Blanc, F. Roggero, B. Noetinger, Institut Français du Pétrole, A. Haas, B. Corre, Elf Exploration & Production 2 L.Y. HU ET AL. SPE 56703 being routinely used by reservoir engineers. It has been urgent to build an efficient tool for modifying a realization while keeping it consistent with the geological information. The gradual deformation method was born and developed in such a background. Gradual global deformation. The idea for building deformation tool is to consider the stochastic reservoir model as a spatio-temporal process. At any time, the state of this process corresponds to a realization of the reservoir model. In the Gaussian framework, a convenient way of building a spatio-temporal Gaussian random function consists in combining two independent standard Gaussian random functions Y1 and Y2 with identical covariance: Y t Y t Y t ( ) cos sin = + 1 2 It can be shown that, for any t , Y t ( ) is a standard Gaussian random function which shares the covariance of Y1 and Y2 . Given two independent realizations of Y1 and Y2 , we get a continuous chain of realizations of Y t ( ) . The basic idea of the gradual deformation method is to modify realizations by changing parameter t . Figure 1 shows an example of a continuous train of realizations. By gradually deforming a Gaussian realization, we gradually deform all Gaussian-related models (e.g. lognormal model, truncated Gaussian model etc.). Moreover, instead of using two independent realizations, it is possible to create a multidimensional process of realizations by using several independent realizations. This provides more flexibility for deforming realizations 14. Gradual structural deformation. In many cases, there is not enough data to infer accurately the structural parameters of a stochastic model (e.g. its mean, variance, covariance function etc.). These structural parameters are often given in terms of intervals or a priori distributions. Therefore they must also be considered as fitting parameters. This involves the deformation of a stochastic model with respect to its structural parameters. Most of the stochastic simulation algorithms require at least the specification of the covariance before generating realizations. In order to deform a realization while modifying simultaneously its structural parameters, it is necessary to use a stochastic simulation algorithm that separates the generation of random numbers from the imposition of a structure 15. Denote by X a standard Gaussian white noise and L the covariance operator, then