Quantification of Uncertainty in Relative Permeability for Coarse-Scale Reservoir Simulation

Reservoir simulation to predict production performance requires two steps: one is history-matching, and the other is uncertainty quantification in forecasting. In the process of history-matching, rock relative permeability curves are often altered to reproduce production data. However, guidelines for changing the shape of the curves have not been clearly established. The aim of this paper is to clarify the possible influence of relative permeabilities on reservoir simulation using the uncertainty envelope. We propose a method for adjusting the shape of relative permeability curves during history-matching at the coarse scale, using the Neighbourhood Approximation algorithm and B-spline parameterisation. After generating multiple history-matched models, we quantify the uncertainty envelope in a Bayesian framework. Our approach aims at encapsulating sub-grid heterogeneity in multi-phase functions directly in the coarse-scale model, and predicting uncertainty. In this sense, the framework differs from conventional procedures which perturb fine-scale features, upscale the models and evaluate each performance. In addition, B-spline parameterisation is flexible allowing the capture of local features in the relative permeability curves. The results of synthetic cases showed that the lack of knowledge of the subgrid permeability and the insufficient production data provoked a substantial amount of uncertainty in reservoir performance forecasting. Introduction Reservoir simulation is routinely employed in the prediction of reservoir performance under different depletion and operating scenarios. This practical use of reservoir simulation requires two steps: one is history-matching, and the other is uncertainty quantification in forecasting. In the traditional approach, a single history-matched model, conditioned to production data, is obtained, and is used to forecast future production profiles. Since the history-matching is non-unique, the forecast production profiles are uncertain. Recently, in order to take account of the non-uniqueness of the inverse problem, a new methodology for uncertainty quantification has been introduced to the petroleum industry. The Markov Chain Monte Carlo method has been adopted by [1, 2, 3, 4], along with the Neighbourhood Approximation [5, 6], in order to investigate parameter space. Here, the requirement for reservoir modelling is to generate multiple history-matched models which encapsulate the effect of the detailed features in a reservoir. In general, as the cell size of a model gets smaller, the accuracy for capturing the details improves. In reservoir modelling studies, the geostatistical approach has been employed to generate multiple realisations at the fine scale. Then, because simulation of the fine-scale model is usually too time consuming, the number of cells is reduced by upscaling to conduct history-matching. However, upscaling techniques have some problematic aspects in real situations. For example, given complete details of the fine-scale features, two-phase upscaling could be performed to calculate pseudo relative permeabilities for every coarse-grid cell in each direction. The pseudos then require grouping into a limited number of tabular functions for coarse-scale simulation, [7, 8]. Unfortunately, this procedure is not feasible, as it is time consuming and results may not be robust. On top of those issues on upscaling, the task of evaluating multiple realisations of geostatistical models still remains a research issue. For example, if you try to adjust a correlation length in geostatistical simulations to history-match the model, you need to take into account the variance of the multiple realisations as Copyright 2005, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Europec/EAGE Annual Con in Madrid, Spain, 13-16 June 2005. This paper was selected for presentation by an SPE Program Committee followi ntained in an abstract submitted by the author(s).