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

Abstract

Reservoir simulation to predict production performance requirestwo 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 insuffcient 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.

Ensemble-Based Relative Permeability Estimation Using B-Spline Model

“…The non-parametric model is far more general and flexible as there is no assumption regarding the shape of the relative permeability curves. For example, the B-spline model has been used in various studies to represent the relative permeability curves Yang and Watson 1991;Kulkarni and Datta-Gupta 2000;Okano et al 2005Okano et al , 2006Chen et al 2008;Eydinov et al 2009). …”

“…Reynolds et al (2004) used the modified Levenberg-Marquardt algorithm to estimate the absolute and relative permeability of a three-phase flow system. Okano et al (2005Okano et al ( , 2006 proposed a method to estimate relative permeability at the coarse scale, using the neighborhood approximation algorithm in order to encapsulate the sub-grid heterogeneity in the model. The relative permeability range was restricted using physically based prior information.…”

Relative Permeability Estimation from Displacement Experiments Using EnKF Method

“…The non-parametric model is far more general and flexible as there is no assumption regarding the shape of the relative permeability curves. For example, the B-spline model has been used in various studies to represent the relative permeability curves (Watson et al, 1988;Yang and Watson, 1991;Kulkarni and Datta-Gupta, 2000;Okano et al, 2005;Chen et al, 2008;Eydinov et al, 2009).…”

Simultaneous Estimation of Relative Permeability and Porosity/Permeability Fields by History Matching Production Data

“…In this paper, we also quantify the uncertainty in relative permeabilities together with the porosity and permeability fields and the uncertainty in the future reservoir performance prediction. Okano et al (5) also considered quantifying uncertainty in relative permeability using B-spline representation in a coarse-scale simulation. However, their work focused on the estimation of relative permeability to account for the loss of heterogeneity caused by upscaling, and the rock property fields (porosity and permeability) were not adjusted during history matching.…”