This paper describes the application of the iterative ensemble smoother to the history matching of the Norne field in the North Sea, which has a moderately large number of wells, a variety of data types, and a relatively long production history. Particular attention is focused on the problems of identification of important variables, generation of an initial ensemble, plausibility of results, and efficiency of minimization.
Introduction
The Norne field produces from an oil and gas reservoir discovered in 1991 offshore Norway. The full-field model consists of four main fault blocks that are in partial communication and many internal faults with uncertain connectivity in each fault block. There have been 22 producers and nine injectors in the field. Water-alternating-gas injection is used as the depletion strategy. Production rates of oil, gas, and water of the 22 producers from 1997 to 2006 and repeat formation tester (RFT) pressure from 14 wells are available for model calibration. The full-field simulation model has 22 layers, each with dimension of 46×112 cells. The total number of active cell is approximately 45,000.
The Levenberg-Marquardt form of the iterative ensemble smoother [-Levenberg-Marquardt ensemble randomized maximum likelihood (LM-EnRML)] is used for history matching the Norne full-field model using production data and RFT pressure. The model parameters that are updated include permeability, porosity, and net-to-gross ratio at each gridblock; vertical transmissibility at each gridblock for six layers; transmissibility multipliers of 53 faults; endpoint water and gas relative permeability of four reservoir zones; depth of water/oil contacts; and transmissibility multipliers between a few main fault blocks. The total number of model parameters is approximately 150,000. Distance-based localization is used to regularize the updates from the LM-EnRML. The LM-EnRML is able to achieve improved data match compared with the manually history matched model after three iterations. Updates from the LM-EnRML do not introduce artifacts in the property fields as are produced in the manually history matched model. The automated workflow is also much less labor intensive than manual history matching.
Ensemble-based methods use information from an ensemble of reservoir models to compute directions of change to the model. The most popular of these methods, the ensemble Kalman filter (EnKF), has been successfully applied on synthetic test cases in which well-by-well data match required many degrees of freedom.
With one exception, applications of ensemble-based history matching to real field data have used the EnKF, in which data are assimilated sequentially in time, and both model variables (e.g., permeability, porosity, fault transmissibility multipliers) and state variables (e.g., pressure, saturations, concentrations, temperature) are updated at each data assimilation time. Despite the good performance of the EnKF, practical limitations on the need for restarting of the reservoir simulator with updated state variables (e.g., phase saturations and pressure) will dictate that an ensemble smoother is the preferred methodology for data assimilation. The ensemble smoother is similar to traditional automatic history matching in that only the model variables, such as permeability, porosity, and fault transmissibility multipliers, are updated during an iteration, and data from all times are included in a single objective function. State variables, such as pressure and water saturation, are computed using the reservoir simulator.
The Norne Field and the Simulation Model
The Norne field is an oil and gas field in the Norwegian sector of the North Sea. The original oil in place is approximately 160 million std m3. Production began in November 1997 and peaked in 2001 when approximately 11 million std m3 of oil were produced. By 2012, oil production had decreased to 600,000 std m3. The field has been produced from a total of 22 producers through the end of 2006 and had injected water and gas in nine injectors. Production data has been made available for history matching by the operator. RFT pressures were acquired in 14 wells at times ranging from 0.3 years after start of production to 7.4 years after start of production. The RFT data have not been released in digital form but can be obtained from figures in reports released as part of a benchmark study.
A reference simulation model, which had been manually history matched, was released as part of the benchmark study. The dimensions of the reference model are 46×112×22, with 44,927 active cells.
Parameterization
The selection of history-matching variables for ensemble-based methods is quite different from many other types of history-matching methods. Here, it is important to approximate the prior uncertainty in the model by sampling all variables that are uncertain and that have an effect on the mismatch of prediction with data. Because most reservoir properties are uncertain, this can result in a very large number of variables to be updated in the data assimilation. Normally, the identification of uncertain variables would be done by an integrated team of geoscientists and engineers familiar with the field and the technologies of modeling and history matching. In the case of the Norne field, the authors had to rely on available reports and experience in history matching. In the end, the properties that were modeled as uncertain are the following: gridblock porosity (44,927 parameters); gridblock permeability (44,927 parameters), which is assumed to be correlated with porosity; gridblock net-to-gross ratio (44,927 parameters); gridblock vertical transmissibility multiplier for six layers (13,309 parameters); fault transmissibility multiplier (53 parameters); endpoint water and gas relative permeability of four zones (eight parameters); and transmissibility multiplier between a few fault blocks (three parameters). The total number of model parameters used in the ensemble smoother, therefore, is approximately 150,000.
Gridblock values of porosity, permeability, and net-to-gross ratio can almost always be assumed to be uncertain. For the Norne field, the prior estimates for these quantities had been obtained by kriging values that had been estimated from well logs along well trajectories. It was not clear how uncertain the log-interpreted values should be or how great the spatial variability might be. The authors chose to model the uncertainty of the gridblock-based properties as a Gaussian random field with an isotropic exponential covariance (in horizontal directions) and practical range of 26 gridblocks (approximately 2500 m). The mean of the gridblock-based properties is the kriged maps from well-log data. The standard deviations of log-transformed permeability, porosity, and net-to-gross ratio are 1.0, 0.05, and 0.1, respectively. The initial covariance was assumed to be stationary, but the means were not.
Flow in the vertical direction can often be characterized by vertical permeability, but the Norne field has a number of cemented calcareous layers that appear to be barriers to vertical fluid flow. The thickness of these layers are typically a small fraction of the thickness of the reservoir layers, so they are better described through the use of vertical transmissibility multipliers than by the use of vertical permeability.
Fault transmissibility multipliers are drawn from Gaussian distribution. The means were from the released reference model.
Uncertainty in the ensemble smoother is represented by an ensemble of models. In most runs, the ensemble size is 100 realizations, but one run was made with 200 realizations for evaluation of the effect of ensemble size.
Production and RFT Data
Monthly production and injection rates are available for all wells from November 1997 to December 2006. The producers are under reservoir volume constraint and the injectors are under water or gas injection rate constraint (both rates are provided by the operator). The oil, water, and gas production rates were then used as data to be matched. In order to properly account for the effect of the choice of units and the possibility of differences in measurement accuracy for the various observations, it is necessary to choose the weight to be applied to each observation. In a Bayesian formulation with no model error, the weight is given by the variance in the measurement error. In fact, it is not possible to remove the model error, so the effective measurement error is typically increased to account for the model error. This process is rather ad hoc and is often based on an examination of the plots of data to see how much variability is apparent in the data. That is what has been done here. The noise in data is modeled as Gaussian random variables with zero mean. The standard deviations of the noise are 100 m3/d for oil production rate, 200 m3/d for water production rate, and 20 000 m3/d for gas production rate. These refer to rates at standard conditions.
Localization of Updates
In the iterative ensemble smoother approach used in this study (LM-EnRML), each observation is associated with one column of the coefficient matrix. Please see the complete paper for the equations associated with the LM-EnRML.
Because the elements of the coefficient matrix are computed from a relatively small ensemble, it is necessary to apply some type of localization to prevent ensemble collapse and to reduce the effect of spurious correlations on the updates. Please see the complete paper for the updating equation that was used.
Updated Reservoir Models
While the iterative ensemble smoother and all EnKF-like methods are designed to preserve the spatial variability and make reasonable changes to models, the reality is that unreasonable changes can occur if the size of the ensemble is small or if the initial uncertainty estimate is inconsistent with the actual observations. In particular, if uncertainty in some important parameters is neglected, the result may be that large changes must be made to other variables. Therefore, it is important to review the updated models for plausibility.
Fault transmissibility multipliers are important for compartmentalization and for controlling the direction of fluid flow in the Norne field.
In ensemble-based methods, realizations of the model are generally considered to be of fundamental importance, more so than the mean, for example. Plausibility of the models is best judged by examining individual realizations to see if the spatial variability appears to be reasonable and if the variability of ensemble members appears to be reasonable.
Discussion
This paper illustrates the application of a computer-assisted history-matching technology to a real field with moderate complexity. The goal was to show how to set up a history match quickly to obtain useful results, including an estimate of uncertainty. Neither the inclusion of large numbers of parameters nor the use of multiple iterations was detrimental to the results. In fact, the final realizations appeared to be reasonable models of the reservoir. In cases in which the magnitude of values is somewhat outside the expected range, it was suggested that these would normally be accepted as indicators of the need for additional parameters. Although the number of variables was gradually increased after reviewing results from several data-assimilation runs, access to the advice of a geologist or geophysicist was not available to allow the addition of faults that were not included in the original model.
Clearly, the initial uncertainty in several variables has been underestimated. In particular, no uncertainty in geologic structure was included. To some extent, the authors suggest that the lack of uncertainty in the structure has been compensated by including uncertainty in the depth of the oil/water contact, but this cannot capture the true uncertainty. There was also a strong indication that additional faults should have been added and that the existing faults should have been subdivided into smaller segments with individual multipliers. Finally, uncertainty in skin damage along the length of the producing intervals for the horizontal wells was not considered. The authors of this paper believe that a more general and robust localization method is one of the more pressing needs for practical implementation of the ensemble-based data-assimilation methods by nonexperts.
This article, written by Special Publications Editor Adam Wilson, contains highlights of paper SPE 164902, “History Matching of the Norne Full-Field Model Using an Iterative Ensemble Smoother,” by Yan Chen, SPE, International Research Institute of Stavanger, and Dean S. Oliver, SPE, Uni Center for Integrated Petroleum Research, prepared for the 2013 EAGE Annual Conference and Exhibition/SPE Europec, London, 10–13 June. The paper has been peer reviewed and is scheduled for publication in the SPE Reservoir Evaluation & Engineering journal.