Summary
This paper applies the ensemble Kalman filter (EnKF) to history match a North Sea field model. This is, as far as we know, one of the first published studies in which the EnKF is applied in a realistic setting using real production data. The reservoir-simulation model has approximately 45,000 active grid cells, and 5 years of production data are assimilated. The estimated parameters consist of the permeability and porosity fields, and the results are compared with a model previously established using a manual history-matching procedure. It was found that the EnKF estimate improved the match to the production data. This study, therefore, supported previous findings when using synthetic models that the EnKF may provide a useful tool for history matching reservoir parameters such as the permeability and porosity fields.
Introduction
The EnKF developed by Evensen (1994, 2003, 2007) is a statistical method suitable for data assimilation in large-scale nonlinear models. It is a Monte Carlo method, where model uncertainty is represented by an ensemble of realizations. The prediction of the estimate and uncertainty is performed by ensemble integration using the reservoir-simulation model. The method provides error estimates at any time based on information from the ensemble. When production data are available, a variance-minimizing scheme is used to update the realizations. The EnKF provides a general and model-independent formulation and can be used to improve the estimates of both the parameters and variables in the model. The method has previously been applied in a number of applications [e.g., in dynamical ocean models (Haugen and Evensen 2002), in model systems describing the ocean ecosystems (Natvik and Evensen 2003a, 2003b), and in applications within meteorology (Houtekamer et al. 2005)]. This shows that the EnKF is capable of handling different types of complex- and nonlinear-model systems.
The method was first introduced into the petroleum industry in studies related to well-flow modeling (Lorentzen et al. 2001, 2003). Nævdal et al. (2002) used the EnKF in a reservoir application to estimate model permeability focusing on a near-well reservoir model. They showed that there could be a great benefit from using the EnKF to improve the model through parameter estimation, and that this could lead to improved predictions. Nævdal et al. (2005) showed promising results estimating the permeability as a continuous field variable in a 2D field-like example. Gu and Oliver (2005) examined the EnKF for combined parameter and state estimation in a standardized reservoir test case. Gao et al. (2006) compared the EnKF with the randomized-maximum-likelihood method and pointed out several similarities between the methods. Liu and Oliver (2005a, 2005b) examined the EnKF for facies estimation in a reservoir-simulation model. This is a highly nonlinear problem where the probability-density function for the petrophysical properties becomes multimodal, and it is not clear how the EnKF can best handle this. A method was proposed in which the facies distribution for each ensemble member is represented by two normal distributed Gaussian fields using a method called truncated pluri-Gaussian simulation (Lantuéjoul 2002). Wen and Chen (2006) provided another discussion on the EnKF for estimation of the permeability field in a 2D reservoir-simulation model and examined the effect of the ensemble size. Lorentzen et al. (2005) focused on the sensitivity of the results with respect to the choice of initial ensemble using the PUNQ-S3. Skjervheim et al. (2007) used the EnKF to assimilate seismic 4D data. It was shown that the EnKF can handle these large data sets and that a positive impact could be found despite the high noise level in the data.
The EnKF has some important advantages when compared to traditional assisted history-matching methods; the result is an ensemble of history-matched models that are all possible model realizations. The data are processed sequentially in time, meaning that new data are easily accounted for when they arrive. The method allows for simultaneous estimation of a huge number of poorly known parameters such as fields of properties defined in each grid cell.
By analyzing the EnKF update equations, it is seen that the actual degrees of freedom in the estimation problem are limited equal to the ensemble size. One is still able to update the most important features of large-scale models. A limitation of the EnKF is the fact that its computations are based on first- and second-order moments, and there are problems that are difficult to handle, particularly when the probability distributions are multimodal (e.g., when representing a bimodal channel facies distribution).
This paper considers the use of the EnKF for estimating dynamic and static parameters, focusing on permeability and porosity, in a field model of a StatoilHydro-operated field in the North Sea. The largest uncertainty in the model is expected to be related to the permeability values, especially in the upper part of the reservoir where the uncertainty may be as large as 30%.