History Matching Channelized Reservoirs Using the Ensemble Kalman Filter
Authors: Lorentzen Rolf Johan , Flornes Kristin Margrethe , Naevdal Geir 
Published: March 2015

Abstract
Even though the ensemble Kalman filter (EnKF) is widely used, history matching reservoirs with facies description has proven to be a major challenge. A preferred technique for estimating large-scale facies fields within the petroleum industry is still missing. In this paper we present a new approach to this problem. Instead of applying the EnKF directly on facies realizations, the approach applies a transformation of facies fields to a specific level-set function, representing distances between facies types. This ensures better agreement with the EnKF Gaussianity assumptions, and the method always returns facies realizations with geological authenticity. The method also offers large flexibility in generating the initial ensemble, which can be done using any geostatistical tool. Further, no modifications of the standard EnKF equations are needed.
The methodology is evaluated on two synthetic examples with increasing complexity. In both examples we consider reservoirs with channel structure. The results presented show that the updated models give large improvements in matching the measurements, and the uncertainty of the models is decreased. Further, recovery of the true petrophysical parameters is highly dependent on sufficient information in the measurements, but in one of the examples considered we are able to completely recover the true channel structure. Additional improvements in the quality of the updated facies fields are obtained by proper handling of the distances close to the reservoir boundaries, and conditioning on specific statistical measures to better preserve prior information about channel properties.

Introduction
The ensemble Kalman filter (EnKF) (Evensen [1], Aanonsen et al. [2]) is one of the most promising tools for assisted history matching of reservoir models. The filter has in particular proven to work well on cases where petrophysical properties are modeled using Gaussian random fields with variogram models. However, many operative oil and gas reservoirs have challenging complex geological structures - channelized reservoirs being one example. These structures typically have rapid spatial variations in the petrophysical properties (e.g., permeability and porosity). The effect of applying the EnKF directly on such reservoirs is that the characteristics of the fields are lost, i.e., sharp gradients are smeared out (dissipated).
Several approaches for using EnKF on complex reservoirs have been proposed. The goal is to ensure that the updated fields are facies realizations, or at least approximations of such, and that good history match is obtained. The methods involve use of truncated pluri-Gaussian methods (Liu and Oliver [3, 4]), Gaussian mixture models (Dovera and Della Rossa [5], Sun et al. [6]), discrete cosine transformations (Jafarpour and McLaughlin [7]), kernel methods (Sarma and Chen [8]), and level-set methods (Moreno et al. [9]). Many of the above mentioned methods give promising results, but there are still drawbacks related to e.g., flexibility in how the initial ensemble may be generated, or the method involves complex and large modifications of the standard EnKF implementation.
Our goal is to develop a methodology which is both simple, flexible and able to preserve facies structure for the updated fields. To achieve this, we propose a method motivated by the work of Moreno et al. [9], in the sense that level-set functions are used to represent the facies fields. However, in Moreno et al. [9], only perturbations of a fixed "base case" are updated, and it is questionable whether the real uncertainty is properly described by the prior ensemble. The method we propose is not limited to updating perturbations, and the initial ensemble can be computed using any geostatistical tool. This means that using our methodology, the initial ensemble can be conditioned on the correct facies types at the well locations.
The method relies on a transformation of the facies field to a specific level-set function - a signed distance function (representing distances between opposite facies types). The distances are then updated using the EnKF, and converted to petrophysical properties when the reservoir simulator is run to the next assimilation time. The advantage with this transformation is that the ensemble of parameters to be estimated is closer to a Gaussian distribution, which is in better agreement with the EnKF Gaussianity assumptions. In addition, we are guaranteed that the updated fields are facies realizations.