ECMOR XI - 11th European Conference on the Mathematics of Oil Recovery

We have suggested running multiple EnKF runs with a smaller ensemble size, and have presented a methodology for determining the optimal ensemble batch size for a synthetic 2D model. The optimal combination of ensemble size and number of EnKF runs is clearly case dependent. However, our results suggest that for a given number of forward model runs (n*m), it will be better to perform several EnKF runs with a smaller ensemble size, than one run with a larger number of ensemble members. Technical contributions: 1) Improvement of the EnKF methodology for characterization of posterior pdf by performing multiple runs with smaller ensemble sizes instead of one large run 2) A methodology for optimal choice of ensemble batch size (n) and number of EnKF runs (m) for a fixed total number of ensemble members (m*n). 3) Developed a methodology for uncertainty estimation of the posterior CDF using EnKF.

The ensemble Kalman filter (EnKF) is a Monte Carlo method for data assimilation and assessment of uncertainties during reservoir characterization and performance forecasting. The method is based on a low-rank approximation to the system covariance matrix calculated from an ensemble which may be orders of magnitude smaller than the number of state variables. In practical applications, the ensemble size has to be kept relatively small. This may lead to poor approximation of the cross-covariance matrix, and sampling errors can result in spurious correlations and incorrect changes in the state variables. Also, since the rank of the covariance matrix cannot be larger than the number of ensemble members, the number of degrees of freedom may be too low when a large number of measurements are assimilated, such as with 4D seismic data. In this work, we have investigated the shortcomings of a straightforward EnKF implementation for small ensemble size, relative to a large number of measurements. This is done by considering a single update of a simple linear model and comparing the EnKF update to the traditional Kalman filter (or Kriging) solution, which in this case is exact. The quality of the EnKF update is assessed by considering the mean and variance of the updated state variable, as well as various error norms and the eigen-spectrum of the covariance matrix. Even for this simple model, spurious long-range correlation, ensemble collapse, etc. are clearly seen as the number of measurements increases for a given ensemble size. For a traditional implementation of EnKF, the ensemble size have to be much larger than the number of measurements to obtain an accurate solution, and the solution gets worse when the measurement uncertainty is reduced.

WAG flooding of an oil reservoir can give rise to large regions of three-phase flow, where the flow parameters, i.e. capillary pressure and relative permeability, are history dependent. This means that three-phase capillary pressure and relative permeability data have to be updated during the flow to account accurately for hysteresis. The idea of this work is to connect a pore-scale model that calculates capillary pressure and relative permeability for given saturations to a three-phase reservoir simulator. This will allow us to calculate the actual saturation paths based on pore-scale physics. The pore-scale model comprises a bundle of cylindrical capillary tubes of different radii and wettability, which are randomly distributed according to the given density functions. Within the bundle the capillary pressure controls the displacement sequence, and for given capillary pressures it is therefore possible to find the corresponding phase saturations in the bundle. However, for using the pore-scale model in the reservoir simulator it is required to obtain capillary pressure and relative permeability from saturation data, rather than the other way around. We hence invert the capillary bundle model iteratively to find the capillary pressures for given saturations. Depending on the required accuracy, these calculations can be time consuming, especially when the behaviour changes between two-phase and three-phase. A capillary bundle is completely accessible, so there will not be any trapped or residual saturations. In principle a more complex network model including residual saturations could be used. Incorporation of the bundle model into the simulator demonstrates the effects of consistent pore-scale based three-phase capillary pressure and relative permeability for different wettability on the continuum, i.e. reservoir scale. This also shows under which conditions pore-scale displacement paths can be reproduced by the macro-scale model.

Recent technological advancements in reservoir management, namely the use of smart-wells, i.e., controlled valves deployed downhole, has made significant impact in the improvement of oil recovery for depleted as well as new oil fields. Inspired by classical feedback control techniques, the oil industry started developing closed-loop optimal reservoir management schemes using optimal control and model updating. Optimization techniques, such as model predictive control (MPC) has been successfully applied in the downstream end of the production line, and in general in the process industry. This is due to the fact that MPC is a model based controller design procedure, which can handle processes with stability issues and time-delays, together with a framework to incorporate constraints into the design. However, in the upstream side of the production, MPC has not gained attention until recently, mainly due to the large-scale nature of the optimization problem. In this paper, we apply real-time optimal control techniques to reservoir management, and in particular to reservoir production. Based on the success of model-based optimization to the process industry, we aim to use MPC schemes to increase the potential for greater oil recovery, and therefore, enhanced reservoir management and profitability. MPC offers a robust control implementation together with constraint handling capabilities. At first, a short survey on MPC is presented and the dynamics of an oil-reservoir is introduced together with the basic equations for flow in porous media. Then, linear MPC is applied and the focus is directed to the generation of low-order reservoir models using subspace identification methods. Lastly, due to the highly nonlinear behavior of the reservoir models, nonlinear MPC (NMPC) schemes is suggested. Comparisons will be provided through a set of realistic simulations using an in-house simulator.

In a reservoir characterization perspective it is important to introduce a consistent parameterization to make an improvement of the reservoir model and provide more reliable predictions of the future production. In this paper we present a methodology for history matching and uncertainty quantification of reservoir simulation models using the Ensemble Kalman filter (EnKF). In the updating sequence, we propose a method for estimating coarse-scale relative permeability curves, based on a Corey function representation. In addition traditionally static (permeability, porosity) and dynamic (pressure, saturation) variables are adjusted. During the assimilation we used the oil production rate, gas-oil ratio and water-cut as history data. The EnKF is applied on a StatoilHydro operated reservoir field in the North Sea, where the relative permeability is identified to be highly uncertain. In this work a Corey function parameterization is used to estimate the relative permeability, and in the history matching process the Corey exponent, the end point saturation and the end point of the relative permeability curve are treated as poorly known parameters. The influence of the relative permeability has been investigated on real field application, and the results of the field study show that a significant improvement in the history match can be achieved by additionally updating the coarse-scale relative permeability properties. The final estimated ensemble shows a reduction in uncertainty for the relative permeability curves, which demonstrate the capability of the EnKF to quantify the uncertainty in the reservoir model. Furthermore, the final estimated ensemble is used to predict the future production performance of the reservoir. Technical contributions: The paper shows that the EnKF is capable to adjust relative permeability curves, based on the information contained in the assimilated production measurements, and it is shown that estimating coarse-scale relative permeability may be crucial to obtain satisfactory history matching results in real field applications.

Reservoir characterization needs the integration of various data, especially dynamic data, which requires history matching the measured production and 4D seismic data. However, reservoir facies and heterogeneities are generated with a geostatistical model, and random realizations cannot generally match dynamic data. To constrain the realizations by using measured dynamic data, it is necessary to parameterize the reservoir model, especially geostatistical realizations, and apply an optimization procedure by minimizing an objective function. However, there are only a few methods available to parameterize geostatistical realizations, and they are not always efficient. In this paper, we propose a local parameterization method which allows to improve history matching for better reservoir characterization. The method of gradual deformation, which allows to change continuously geostatistical realization from one to another, has been increasingly used in history matching. The domain of gradual deformation is generally delimited by gridblocks. In this paper, we present first a technique of spatial combination, which can combine geostatistical realizations on arbitrary domains to form a new one by keeping geostatistical consistency. Then, we present a technique of local parameterization to change continuously geometrical forms or sizes of the domains. This parameterization of domains leads to continuous change of geostatistical realizations with the technique of spatial combination. By analyzing values of objective function on each well or 4D seismic region, we can define local domains to be considered in the optimization. By changing forms and sizes of these domains, the proposed local parameterization method can select automatically best realizations in appropriate domains in order to improve the assistant history matching. Several examples of single or two-phase flows are presented with parameterization for radial or elliptical domains. The efficiencies of the local parameterization approach for history matching are illustrated through these examples.

Capillary imbibition is one of the main recovery mechanisms of naturally fractured reservoirs where fracture fluid imbibes, by capillary forces, in the matrix and the matrix fluid is transferred to the fracture. Simulating counter-current imbibition in dual-medium models is a challenging task. The semi-steady state approach has been used in Warren and Root based transfer functions for the past forty years. However, it eliminates the speed of early time recovery and assigns average property values in matrix and fracture. In this paper, we eliminate the semi-steady state approach in matrix capillary imbibition by making the transfer function depend on time, space and two recovery periods (early and late time). We make it depend on space by dividing the invaded face into two equal sub-faces, each with its own capillary pressure, relative permeability and location. Then, the two contributions are summed up to equal one mass conservation equation for each matrix cell. In early time recovery, the saturation front moves laterally in the matrix, until it reaches the no-flux boundary. The distance of invasion is calculated using an integral of the inverse capillary pressure curve, the saturation values of previous time step, and the distance between the invaded face and the no-flux boundary. Then, new capillary pressure, relative permeability and location values are assigned to each sub-face; where the transfer of fluid is calculated. When the saturation front reaches the no-flux boundary, at start of late time, it moves vertically until the Pc at the no-flux boundary equals the Pc at the invaded face. The capillary pressure and relative permeability of the sub-faces are calculated using integral of the inverse of capillary pressure curve and the saturation value of previous time step. Our approach matched the results of fine-grid single-porosity models under various parameters of capillary pressure, matrix shape and mobility. It also outperformed the results of three transfer functions: Gilman & Kazemi, Quandalle & Sabathier, and the General Transfer Function proposed by Hu & Blunt.

During wellbore drilling operations and successive oil/gas production activities, both the natural stresses and the original thermodynamic equilibrium of virgin rock formations are altered, and, as a result, borehole instability phenomena can arise with time. Instability phenomena can be so severe to determine the wellbore abandonment because of its complete collapse. During the design phase of drilling operations, the adoption of a wellbore instability modeling approach is fundamental to systematically take into consideration time-dependent alterations of the initial natural equilibrium. In this paper the possibility of investigating both the stress-strain and the thermo-dynamic formation behavior through a fully coupled thermo-poro-elasto-plastic approach is discussed. The numerical solution adopted for the model implementation is presented. In the fully coupled approach, porous flow and stress-strain calculations are performed together: the whole system is discretised on one grid domain and solved simultaneously for both the thermodynamic variables (such as: temperature and pressure), and the geomechanical response (such as: displacements). This approach has the advantage of internal consistency and stability; furthermore, it preserves second order convergence of nonlinear iterations. In order to implement the plastic analysis an iteratively coupled approach was adopted in the fully coupled routine. According to the iterative coupling technique, the model basic equations (porous flow and rock deformation) and the plastic behaviour equations are solved separately and sequentially at each non-linear iteration. This is achieved by the solution of two nested Newton-Raphson cycles at each time-step. The iterative coupling approach corresponds to an implicit treatment of the plastic variables, essential to preserve the stability of the elasto-plastic solution. Wellbore stability analyses are also presented to prove the effectiveness of the proposed model to investigate the potential impact of time-dependent phenomena on the well drilling design.

From a real Norwegian North Sea field case we show simulation results where 4D seismic data are incorporated in the history matching process of reservoir simulation models. It has been shown that the Ensemble Kalman Filter technique, a Monte Carlo type Bayesian sequential inversion method, is capable of performing this task. In addition to predicted model states, data uncertainties are provided. In fact, seismic data may not only relate to one (e.g. 3D), but to more instances in time (4D or time-lapse seismic data where their differences are of concern). Furthermore they may be investigated in different domains, depending on their sensitivity to production related changes in the reservoir. In the given case study we observe an emerging fluid contact due to gas injection and corresponding travel-time shifts of seismic events on the real data. We model these effects by generating synthetic seismic sections, but in an environment which allows to incorporate Eclipse simulated variables, namely fluid saturations, densities and pressures. The Compound model builder, an interface shared by geophysical and reservoir engineering data represents our environment for integrated seismic and geologic modeling. The basic reservoir variables to be updated in this study are the static variables porosity and permeability and the dynamic variables pressures and saturations. These variables are optimized by minimizing a joint misfit function consisting of production data, such as oil-production rate, water-cut and gas-oil ratio and seismic data, which is stacked amplitude data. The benefit of including seismic data lies in a better overall reservoir description especially in areas not sampled by observations of production data.

We show how to derive compositional models from balance models including source terms representing mass transfer between phases. Mass transfer rate is taken proportional to the deviation from thermodynamic equilibrium. In the balance models, the mass transfer is very fast and local thermodynamic equilibrium is quickly attained. The derivation is done by means of an asymptotic expansion where the small parameter is the time scale of mass transfer relative to the hydrodynamical time scale. The new theory is illustrated by an example of thermal flow of steam, nitrogen and water in a porous medium, which can be useful in for soil remediation.