Abstract

This paper presents the application of the ensemble Kalman filter (EnKF) method to the integration of well test data into heterogeneous reservoir models generated from geological and geophysical data. EnKF does not require computing the gradient of an objective function, and hence can be applied easily with any reservoir simulator, and more importantly, is far more efficient than a gradient-based history matching procedure when the forward model is represented by a reservoir simulator. In the procedure for integrating pressure transient data considered here, the static geological/geophysical data are assumed to be encapsulated in a multivariate probability density function characterized by a prior mean and covariance for the joint distribution of the porosity and permeability fields. As the prior mean of the property fields obtained from the core and log data can sometimes be erroneous, a partially doubly stochastic model is applied to account for the uncertainty of the prior mean. In the double stochastic model, a correction to the prior mean is adjusted together with the heterogeneous field during history matching. The method is tested with synthetic heterogeneous single-layer and two-layer reservoirs. Excellent data matches are obtained with EnKF in a small fraction of the time that would be required for a gradient-based history-matching process and the observed data fall within the uncertainty bounds of the ensemble data predictions. In the layered reservoir case, the uncertainty in rock property fields is further reduced by integration of production log data (layer flow rates). We show that problems of statistical inconsistency and poor data matches that are sometimes encountered when matching production data with EnKF do not occur when matching pressure data with the EnKF implementation used here. This undoubtedly occurs because the dynamical system (reservoir simulator equations) is much more nearly linear for the single-phase flow problems considered here than for multiphase flow cases.

Introduction

In today's environment, it is common to make critical reservoir development decisions based on models developed from geological and geophysical data interpretation prior to obtained any long-term production data. Although well-testing data are often available, in most cases well test analysis is customarily interpreted based on analytical solutions that assume single-phase flow for a homogeneous reservoir model, or at most, assumes some simple form of heterogeneity, e.g., permeability variation in only the radial direction (Oliver, 1990) or only the linear direction (Levitan and Crawford, 2002), or layered reservoirs with each layer homogeneous (Lefkovits et al., 1961; Larsen, 1981; Kuchuk et al., 1986; Ehlig-Economides and Joseph, 1987; Kuchuk and Habashy, 1996; Larsen, 1999). A very nice summary of layered-reservoir solutions is given in Larsen (1988). Even though this paper is two decades old, it contains most of the analytical layered-reservoir solutions in use today. While the interpretation of pressure transient data using those analytical solutions are of enormous value, the objective of this work is to integrate pressure transient data into a complex highly heterogeneous geological model. As when development decisions are made, the only dynamic data available may be those obtained from an extended well test, it is hope that conditioning a geological model directly to well testing data may reduce the uncertainty in the original geological model and aid early development decisions.

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