Summary The use of ensemble Kalman filter techniques for continuous updating of reservoir model is demonstrated. The ensemble Kalman filter technique is introduced, and thereafter applied to a simplified 2-D field model, which are generated by using a single horizontal layer from a North Sea field model. By assimilating measured production data, the reservoir model is continuously updated. The updated models give improved forecasts and the forecasts improve as more data is included. Both dynamic variables, such as pressure and saturations, and static variables, such as the permeability, are updated in the reservoir model. Introduction In the management of reservoirs, it is important to utilize all available data in order to make accurate forecasts. For short time forecasts, in particular, it is important that the initial values are consistent with recent measurements. The ensemble Kalman filter1 is a Monte Carlo approach, which is promising with respect to achieving this goal through continuous model updating and reservoir monitoring. In this paper, the ensemble Kalman filter is utilized to update both static parameters, such as the permeability, and dynamic variables, such as the pressure and saturation of the reservoir model. The filter computations are based on an ensemble of realizations of the reservoir model, and when new measurements are available, new updates are obtained by combining the model predictions with the new measurements. Statistics about the model uncertainty is built from the ensemble. When new measurements become available, the filter is used to update all the realizations of the reservoir model. This means that an ensemble of updated realizations of the reservoir model is always available. The ensemble Kalman filter has previously been successfully applied for large-scale nonlinear models in oceanography2 and hydrology3. In those applications, only dynamic variables were tuned. Tuning of model parameters and dynamic variables was done simultaneously in a well flow model used for underbalanced drilling4. In two previous papers5,6, the filter has been used to update static parameters in near-well reservoir models, by tuning the permeability field. In this paper, the filter has been further developed to tune the permeability for simplified real field reservoir simulation models. We present results from a synthetic, simplified real field model. The measurements are well bottom-hole pressures, water cuts and gas/oil ratios. A synthetic model gives the possibility of comparing the solution obtained by the filter to the true solution, and the performance of the filter can be evaluated. It is shown how the reservoir model is updated as new measurements becomes available, and that good forecasts are obtained. The convergence of the reservoir properties to the true solution as more measurements becomes available is investigated. Since the members of the ensemble are updated independently of each other, the method is very suitable for parallel processing. It is also conceptually straightforward to extend the methodology to update other reservoir properties than the permeability. Based on the updated ensemble of models, production forecasts and reservoir management studies may be performed on a single "average" model, which is always consistent with the latest measurements. Alternatively, the entire ensemble may be applied to estimate the uncertainties in the forecasts. Updating reservoir models with ensemble Kalman filter The Kalman filter was originally developed to update the states of linear systems to take into account available measurements7. In our case, the system is a reservoir model, using black oil, and three phases (water, oil and gas).For this model, the solution variables of the system are the pressure and the water saturation, in addition to a third solution variable that depends on the oil and gas saturation. If the gas saturation is zero, the third solution variable becomes the solution gas/oil ratio, if the oil saturation is zero it becomes the vapor oil/gas ratio. Otherwise the third solution variable is the gas saturation. The states of this system are the values of the solution variables for each grid block of the simulation model. This model is non-linear.

There is a potential for large improvements in reservoir management by using optimization and model updating techniques in a closed-loop fashion. Here we demonstrate how the combination of the ensemble Kalman filter technique for continuous model updating with an automated adjoint-based water flood optimization algorithm leads to significant improvements in net present value (NPV) of the water flooding process. Using the ensemble Kalman filter, both static parameters (permeabilities) and dynamic variables (pressures and saturations) are updated in the reservoir model as new production measurements become available. Other properties are assumed known in advance. At the start of the production process, in the absence of information on the permeability distribution, an optimal control strategy based on a homogeneous reservoir is used. Subsequently production data are at regular intervals assimilated with the ensemble Kalman filter, resulting in an updated estimate of the reservoir pressures, saturations and permeability field. Based on these updated parameters an optimal water flooding strategy is determined for the remainder of the production process. This process of model updating and optimization is continued over the life of the reservoir. The methodology is applied to two synthetic examples, enabling comparison with traditional production strategies. Significant improvement in NPV, acceleration of oil production, cumulative oil recovery, and reduction of water production were realized. Results were close to those obtained with water flood optimization based on an a-priori known reservoir description. For one example the improvement in cumulative oil recovery is about 44 %, which is quite close to the improvement obtained for an a-priori known reservoir description. Introduction There exists a potential for large improvements in reservoir management by using the measurement and control opportunities nowadays available in the oil industry. One way to fully exploit the control valves is to optimize their settings in order to maximize the net present value (NPV) of the production process. Various studies conducted on numerical reservoir models showed that significant improvements in the production process may be feasible by dynamically controlling the valves. The exact scope varies with geological features, well operating constraints, and well architecture[R1]. Furthermore, the scope depends on whether the objective is to optimize injection and production rates on the short term, or to optimize the production process over the producing life of the reservoir. A literature overview is given in Ref 1. A limitation of these studies is that they are conducted on reservoir models with all properties known a-priori. In reality, the valve settings have to be computed using the available information there is on the reservoir. Before production starts, a reservoir model has to be built on data from seismic, well tests, core samples, etc. Because of geological uncertainties such models are usually only a very crude approximation of reality, and model parameters, such as permeabilities and porosities are only known with a large degree of uncertainty. Therefore the predictive value of such reservoir models is limited and tends to deteriorate over time. To improve its predictive capacity the model may be adapted such that predicted results approach measured production data. In the oil industry this process is general referred to as ‘history matching’. Because production data are used for model updating, the management of a reservoir based on history-matched models could be considered a closed-loop process. Unfortunately, traditional history matching suffers from a number of drawbacks:It is usually only performed on a campaign basis, typically after periods of years.The matching techniques are usually ad-hoc and involve manual adjustment of model parameters, instead of systematic parameter updating.Uncertainties in the state variables, model parameters and measured data are usually not explicitly taken into account.The resulting history-matched models often violate essential geological constraints.

A study of the propagation and interaction of two collinear finite amplitude sound beams, presented in a previous paper [Naze Tjo/tta et al., “Propagation and interaction of two collinear finite amplitude sound beams,” J. Acoust. Soc. Am. 88, 2859–2870 (1990)] is extended to include the effects of focusing. The validity of the parabolic equation when applied to strongly focused sound beams is discussed. Equations and algorithms based on a transformed parabolic equation are presented and used to compute the interaction between two collinear, focused sound beams, and between one plane wave and a focused sound beam. Numerical results are shown, with special emphasis on parametric generation and parametric reception of sound.

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractThe accurate prediction of the downhole pressures and the returning flow rates in low-head drilling (LHD) and underbalanced drilling operations (UBD) is a major concern in the oil industry. The present work shows an original formulation of a dynamic two-phase flow model based on the classic driftflux set of conservation equations. However, differently from the traditional approach, the closure of the system is obtained by using measured data acquired during the execution of the operation. The innovative concept consists of utilizing mechanistic models that are dependent on unknown parameters. These parameters and the model state are calculated and updated on a regular basis to minimize the differences between model predictions and measured data.The potential of this new approach is discussed by presenting some transient examples of application. The current study represents a continuation of previous work on real time data interpretation 1 , and a step in the development of a methodology for the introduction of learning while drilling process in the hydraulic design and follow-up of LHD and UBD operations.

Technological advances have resulted in use of smart wells, which are typically equipped with remotely operated downhole chokes. We present an approach for controlling these chokes so that the water flooding is optimized. The optimization problem is done by maximizing either total cumulative oil production or net present value. The new methodology presented here avoids the limitations related to using optimal control as no adjoint equations are needed and the model equations are treated as a "black box". In the new approach the ensemble Kalman filter is used as an optimization routine, and the methodology is compared to the Partial Enumeration Method. We demonstrate the methodologies on a simple synthetic reservoir with five layers of different permeabilities. The conclusions from this work are that the ensemble Kalman filter approach is working robustly, and the results are in agreement with, or superior to, the results obtained with the Partial Enumeration Method and a reference solution. Introduction Controlling downhole choke settings in smart wells for optimal water flooding represents a great challenge. Traditionally, the solution of this problem has been to apply optimal control. Optimal control falls under the category of gradient-based optimization, and does require the construction and solution of an adjoint set of equations. This approach was pursued in the work by Brouwer and Jansen[1] and Virnovsky.[2] These papers also contain references to other works within this area. A disadvantage with the adjoint approach is that explicit knowledge of the model equations is necessary. In addition, extensive programming is needed to implement the equations. A remedy for the latter drawback was suggested by Sarma et. al.[3] Here an approach was introduced which simplified the calculation of the adjoint equations. However, the approach requires specific forms of the cost function and a fully implicit forward model. We introduce a new approach for solving the optimization problem which is completely independent of the model equations used. That is, the model is treated as a "black box". The approach is not gradient-based, so no implementation of adjoint equations is necessary. Here, the methodology is used to optimize either net present value (NPV), or total cumulative oil production. In addition, for further validation, the methodology is compared to the Partial Enumeration Method (see Wang[4]), which is a discrete non-gradient based method.