The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

History matching is a type of inverse problem in which observed reservoir behavior is used to estimate reservoir model variables that caused the behavior. Obtaining even a single history-matched reservoir model requires a substantial amount of effort, but the past decade has seen remarkable progress in the ability to generate reservoir simulation models that match large amounts of production data. Progress can be partially attributed to an increase in computational power, but the widespread adoption of geostatistics and Monte Carlo methods has also contributed indirectly. In this review paper, we will summarize key developments in history matching and then review many of the accomplishments of the past decade, including developments in reparameterization of the model variables, methods for computation of the sensitivity coefficients, and methods for quantifying uncertainty. An attempt has been made to compare representative procedures and to identify possible limitations of each.

Recent progress on reservoir history matching: a review

Among the current problems that hydrogeologists face, perhaps there is none as challenging as the characterization of fractured rock (Faybishenko and Benson 2000). This paper discusses issues associated with the quantification of flow and transport through fractured rocks on scales not exceeding those typically associated with single- and multi-well pressure (or flow) and tracer tests. As much of the corresponding literature has focused on fractured crystalline rocks and hard sedimentary rocks such as sandstones, limestones (karst is excluded) and chalk, so by default does this paper. Direct quantification of flow and transport in such rocks is commonly done on the basis of fracture geometric data coupled with pressure (or flow) and tracer tests, which therefore form the main focus. Geological, geophysical and geochemical (including isotope) data are critical for the qualitative conceptualization of flow and transport in fractured rocks, and are being gradually incorporated in quantitative flow and transport models, in ways that this paper unfortunately cannot describe but in passing. The hydrogeology of fractured aquifers and other earth science aspects of fractured rock hydrology merit separate treatments. All evidence suggests that rarely can one model flow and transport in a fractured rock consistently by treating it as a uniform or mildly nonuniform isotropic continuum. Instead, one must generally account for the highly erratic heterogeneity, directional dependence, dual or multicomponent nature and multiscale behavior of fractured rocks. One way is to depict the rock as a network of discrete fractures (with permeable or impermeable matrix blocks) and another as a nonuniform (single, dual or multiple) continuum. A third way is to combine these into a hybrid model of a nonuniform continuum containing a relatively small number of discrete dominant features. In either case the description can be deterministic or stochastic. The paper contains a brief assessment of these trends in light of recent experimental and theoretical findings, ending with a short list of prospects and challenges for the future.

Trends, prospects and challenges in quantifying flow and transport through fractured rocks

Summary With the advances in smart well technology, substantially higher oil recovery can be achieved by intelligently managing the operations in a closed-loop optimization framework. The closed-loop optimization consists of two parts: geological model updating and production optimization. Both of these parts require gradient information to minimize or maximize an objective function: squared data mismatch or the net present value (or other quantities depending on financial goals), respectively. Alternatively, an ensemblebased method can acquire the gradient information through the correlations provided by the ensemble. Computation of the optimal controls in this way is nearly independent of the number of control variables, reservoir simulator, and simulation solver. In this paper, we propose an ensemble-based closed-loop optimization method that combines a novel ensemble-based optimization scheme (EnOpt) with the ensemble Kalman filter (EnKF). EnKF has recently been found suitable for sequential data assimilation in large-scale nonlinear dynamics. It adjusts reservoir model variables to honor observations and propagates uncertainty in time. EnOpt optimizes the expectation of the net present value based on the updated reservoir models. The proposed method is fairly robust, completely adjoint-free and can be readily used with any reservoir simulator. The ensemble-based closed-loop optimization method is illustrated with a waterflood example subject to uncertain reservoir description. Results are compared with other possible reservoir operation scenarios, such as wells with no controls, reactive control, and optimization with known geology. The comparison shows that the ensemble-based closed-loop optimization is able to history match the main geological features and increase the net present value to a level comparable with the hypothetical case of optimizing based on known geology.

Efficient Ensemble-Based Closed-Loop Production Optimization

A closed-loop circulating system for drilling wells has control of the flow rates in and out of the wellbore. Kicks and fluid losses are quickly controlled by adjusting the backpressure. Kick tolerance and tripping margins are eliminated by real-time determination of pore and fracture pressure. The system can incorporate a rotating BOP and can be used with underbalanced drilling.

Drilling system and method

A method of simulating subsea mudlift drilling well control operations using a computer system. The method includes simulating a drilling circulation system. The simulation includes simulating drilling the wellbore at a selected rate of penetration, and the simulating drilling a wellbore includes simulating drilling selected earth formations. A kick is simulated at a selected depth in the wellbore, and the kick is simulated as a two-phase mixture including drilling fluid and a formation fluid. Controlling the kick is then simulated, and wellbore parameters are displayed via a graphical user interface connected to the computer system. The simulating drilling the wellbore is then repeated after the kick has been controlled.

Method for simulating subsea mudlift drilling and well control operations

Streamline Assisted Ensemble Kalman Filter for Rapid and Continuous Reservoir Model Updating

The SAGD process has already been implemented for commercial production in Alberta Oil Sands areas in Western Canada since early 2000. SAGD performance is very sensitive to reservoir heterogeneities such as shale barriers, bottom and/or top water zones, and a gas cap. In the SAGD process, low permeability zones such as shale layers may act as a flow barrier depending on their size, vertical and horizontal locations, and continuity throughout the reservoir thus making it very important to understand and characterize the effect of shale layers. In this study, the impact of various sizes and locations of shale barriers have been investigated through two-dimensional hypothetical simulation models. The various simulation models have been designed to investigate the shale size and vertical location in both BIP (shale between the injector and producer) and AP (shale above the producer) cases. Two different types of models were designed to look at the effect of flow path existence between the injector and producer: type-A is designated as having a no flow path directly above the producer and type-B has a flow path directly above the producer. The simulation results show that type-A has a greater impact than type-B on SAGD performance especially for the BIP case. Small shale sizes of 3 and 5 m have a slight impact on performance; however, cases with 10 m shale have a greater impact due to the disruption of gravity drainge to the producer. Type-A BIP may require a longer pre-heating period for successful SAGD operation. Generally, shale barriers of 5 to 25 m are not critical for an AP case regardless of vertical location of shale barriers; however shale barriers greater than 50 m may act as a barrier and reduce the effective pay thickness of the reservoir depending upon the its vertical location.

Shale Barrier Effects on the SAGD Performance

Many authors have used theories of anomalous diffusion in the field of physics to interpret fractally fractured reservoirs. However, their solutions cannot separate the two fractal dimensions df and dw , and their results are not valid in the early time range. In this study, a new general constant-flow-rate solution, which is applicable to whole spatio-temporal ranges, is derived using the Fox H-function. Sensitivity analyses were carried out for various cases. The solution developed here permits the use of early production data. By combining early and late time data, it is possible to determine the two fractal dimensions separately. From the intrinsic property of the Fox H-function, this model makes it possible to capture the history and nonlocality of transport. Therefore, the model provides a general methodology for characterizing the pressure behavior of fractally fractured reservoirs.

Pressure Behavior of Transport in Fractal Porous Media Using a Fractional Calculus Approach

[1] Aperture distributions have been measured using replicas of apertures to investigate the effect of variable aperture on solute transport in five transparent fracture samples. The coefficient of variation (Cv), correlation length (λ), and anisotropy ratio (AR) were used to quantify the aperture distributions. Step-type tracer tests showed that dispersion is significantly influenced by both λ and Cv. Dispersivity (α) increases with λ and Cv. The magnitude of α is proportional to a polynomial function of natural logarithm of Cv2 and is linearly proportional to λ and the total sill measured from the semivariogram. The effect of AR is insignificant to those of the other parameters on α. Despite assumptions of the small perturbation theory developed by Gelhar [1993], the prediction of α from the theory is similar to α measured in this study. If the small perturbation theory is valid and reliable data for the tailing in a breakthrough curve are obtained, the analysis of the curve by the moment method gives a more reliable result than by the conventional fitting of a one-dimensional analytical solution to the curve.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2001WR001246

Jonggeun Choe

Papers

Method for simulating subsea mudlift drilling and well control operations

Streamline Assisted Ensemble Kalman Filter for Rapid and Continuous Reservoir Model Updating

Shale Barrier Effects on the SAGD Performance

Pressure Behavior of Transport in Fractal Porous Media Using a Fractional Calculus Approach

Experimental analysis on the effects of variable apertures on tracer transport