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

Petroleum Reservoir Simulation

Acquisition, optimization and interpretation of X-ray computed tomographic imagery: applications to the geosciences

This paper presents the results of the Tenth SPE Comparative Solution Project on Upscaling. Two problems were chosen. The first problem was a small 2D gas injection problem, chosen so that the fine grid could be computed easily, and both upscaling and pseudoisation methods could be used. The second problem was a waterflood of a large geostatistical model chosen so that it was hard (though not impossible) to compute the true fine grid solution. Nine participants provided results for one or both problems. Introduction The SPE Comparative Solution Projects provide a vehicle for independent comparison of methods and a recognized suite of test datasets for specific problems. The previous nine comparative solution projects – 9 have focussed on black-oil, compositional, dual porosity, thermal or miscible simulations, as well as horizontal wells and gridding techniques. The aim of the tenth comparative solution project was to compare upgridding and upscaling approaches for two problems. Full details of the project, and data files available for downloading can be found on the project web site. The first problem was a simple 2000 cell 2D vertical cross section. The tasks specified were to apply upscaling or pseudoization methods and obtain solutions for a specified coarse grid, and a coarse grid selected by the participant. The second problem was a 3D waterflood of a 1.1 million cell geostatistical model. This model was chosen to be sufficiently detailed that it would be hard, though not impossible, to run the fine grid solution and use classical pseudoisation methods. We will not review the large number of upscaling approaches here. For a detailed description of these methods see any of the reviews of upscaling and pseudoisation techniques, for example – . Description of Problems Model 1 The model is a 2-phase (oil and gas) model that has a simple 2D vertical cross-sectional geometry with no dipping or faults. The dimensions of the model are 762 meters long by 7.62 meters wide by 15.24 meters thick. The fine scale grid is 100 x 1 x 20 with uniform size for each of the grid blocks. The top of the model is at 0.0 metres with initial pressure at this point of 100 psia. Initially the model is fully saturated with oil (no connate water). The permeability distribution is a correlated geostatistically generated field, shown in Fig 1. The fluids are assumed to be incompressible and immiscible. The fine grid relative permeabilities are shown in Fig 2. Capillary pressure was assumed to be negligible in this case. Gas was injected from an injector located at the left of the model and dead oil was produced from a well on the right of the model. Both wells have a well internal diameter of 1.0 ft and are completed vertically throughout the model. The injection rate was set to give a frontal velocity of 0.3 m/d (about 1 foot/day or 6.97 m per day), and the producer is set to produce at a constant bottom pressure limit of 95 psia. The reference depth for the bottom hole pressure is at 0.0 meters (top of the model). The tasks specified were to apply upscaling or pseudoization method in the following scenarios: 1. 2D – 2D uniform 5 x 1 x 5 coarse grid model 2. 2D – 2D nonuniform coarsening. Max 100 cells. Directional pseudo relative permeabilities were allowed if necessary. Model 2 This model has a sufficiently fine grid to make use of any method that relies on having the full fine grid solution almost impossible. The model has a simple geometry, with no top structure or faults. The reason for this choice is to provide maximum flexibility in selection of upscaled grids. SPE 66599 Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques M A Christie, SPE, Heriot-Watt University, and M J Blunt, SPE, Imperial College 2 M A CHRISTIE & M J BLUNT SPE 66599 At the fine geological model scale, the model is described on a regular cartesian grid. The model dimensions are 1200 x 2200 x 170 (ft). The top 70 ft (35 layers) represents the Tarbert formation, and the bottom 100 ft (50 layers) represents Upper Ness. The fine scale cell size is 20 ft x 10 ft x 2 ft. The fine scale model has 60 x 220 x 85 cells (1.122x10 cells). The porosity distribution is shown in Fig 3. The model consists of part of a Brent sequence. The model was originally generated for use in the PUNQ project. The vertical permeability of the model was altered from the original: originally the model had a uniform kv/kh across the whole domain. The model used here has a kv/kh of 0.3 in the channels, and a kv/kh of 10 -3 in the background. The top part of the model is a Tarbert formation, and is a representation of a prograding near shore environment. The lower part (Upper Ness) is fluvial. Participants and Methods Chevron Results were submitted for model 2 using CHEARS, Chevron’s in house reservoir simulator. They used the parallel version and the serial version for the fine grid model, and the serial version for the scaled-up model. Coats Engineering Inc Runs were submitted for both model 1 and model 2. The simulation results were generated using SENSOR. The simulator runs used the conventional 5or 7-point finite difference formulation, zero capillary pressure, and no directional relative permeability. GeoQuest A solution was submitted for model 2 only, with coarse grid runs performed using ECLIPSE 100. The full fine grid model was run using FRONTSIM, a streamline simulator, to check the accuracy of the upscaling. The coarse grid models were constructed using FloGrid, GeoQuest’s gridding and upscaling application. Landmark Landmark submitted entries for both model 1 and model 2 using the VIP simulator. The fine grid for model 2 was run using parallel VIP. Phillips Petroleum Solutions were submitted for both model 1 and model 2. The simulator used was SENSOR. Roxar Entries were submitted for both model 1 and model 2. The simulation results presented were generated using Roxar’s Black Oil, Implicit Simulator, Nextwell. The upscaled grid properties were generated using Roxar’s Geological Modelling software, RMS, in particular the RMSsimgrid option. Streamsim Streamsim submitted an entry for model 2 only. Simulations were run using 3DSL, a streamline based simulator. TotalFinaElf TotalFinaElf submitted a solution for model 2 only. The simulator used for the results presented was ECLIPSE; results were checked using the streamline code 3DSL. University of New South Wales The University of New South Wales submitted results for model 1 only using CMG’s IMEX simulator. Results Model 1 Fine Grid Solution All participants were able to compute the fine grid solution, and the solutions from the different simulators used were very close, as shown in Fig 4. The University of New South Wales fine grid solution departs slightly from the other fine grid solutions; it was not possible to track down the source of this discrepancy in the short time between receiving this solution and the paper submission deadline. Upscaled Solutions Participants were asked to generate solutions on a 5 x 5 grid, and on a grid of their choice with a maximum of 100 cells. The reason for the choice of the 5 x 5 grid was that, with that grid size, the coarse grid boundaries fall on high permeability streaks which is generally a problem for upscaling methods which don’t compute the fine grid solution. The solutions submitted for the 5 x 5 grid used single phase upscaling only (Roxar), or single phase upscaling plus regression based pseudoisation of relative permeabilities (Coats, Phillips, Landmark). The solutions with pseudo relative permeabilities are very close to the fine grid solution, and Roxar’s solution using only single phase upscaling shows a significant discrepancy (Fig 5). A second set of solutions was presented by some participants (shown in Fig 6). Here Roxar used single phase upscaling in conjunction with a streamline approach to generate local grid refinements (with a total of 96 cells) which captured the details of the flow in the early, mid, and late-time regions. Coats showed that good results could also be obtained with homogeneous absolute permeability and no alteration of relative permeability, and Phillips showed that good results could be obtained from a 6 x 2 grid. The University of New South Wales solution was based on a global upscaling and upgridding approach which attempts to minimize the variance of permeability within a cell. Their solution is close to their fine grid solution, although the difference between their fine grid solution and the other fine grid solutions tends to make their method appear to perform less well. Model 2 Fine Grid Solution Five participants provided fine grid results as well as an upscaled solution. Landmark and Chevron ran the full fine grid on a parallel reservoir simulator. GeoQuest and SPE 66599 TENTH SPE COMPARATIVE SOLUTION PROJECT: A COMPARISON OF UPSCALING TECHNIQUES 3 Streamsim provided results using streamline codes (TotalFinaElf also provided streamline results using 3DSL. We have not shown their production curves as they are the same as Streamsim’s). A comparison of the fine grid results is shown in Fig 7, 8, 9, 10. All the figures shows very good agreement between all four fine grid submissions. Although only producer 1 well plots are shown here for reasons of space, plots of the remaining well rates and watercuts show equally high levels of agreement between the four fine grid solutions. The differences that occur likely to be due to either different time steps early on, where the production rate is very sensitive to the transient pressure response, or to different treatment of the injection well, which was at the corner of four cells in the fine model, leading to different injectivity indices. Upscaled Solutions There were two methodologies used to generate the upscaled solutions. Some participants used finer scale information in some way, and then history matched a coarser grid to the finer grid results. Others made no use of fine sc

Tenth SPE Comparative Solution Project: a comparison of upscaling techniques

Annual Energy Outlook

A history-dependent model for saturation functions, combined with a three-dimensional, three-phase, semi-implicit reservoir simulator, has been developed. In water-coning simulations with variable rates, for waterflooding in the presence of free gas saturations, and for gas-cap shrinkage, use of hysteresis in saturation functions shows results significantly different from those obtained by conventional methods. To some extent, the model is based upon remembering the saturation history of the reservoir. In doing this, smooth transitions of both relative permeabilities and capillary pressures from drainage-to-imbibition or imbibition-to-drainage states are allowed. In addition, the effect of trapped gas or oil saturations on relative permeabilities and capillary pressures is accounted for. Tests of the model indicate that simulation with hysteresis is a stable procedure requiring little more computation time and storage than normal simulations. In addition, results of these tests agree qualitatively with experimental and field results.

Reservoir Simulation With History-Dependent Saturation Functions

The ninth SPE Comparative Solution Project presented in the following paper provides a reexamination of black-oil simulation based on a model of moderate size (9,000 cells) and with a high degree of heterogeneity provided by a geostatistically-based permeability field. Nine participants provided data for the comparison which is based on a dipping reservoir with twenty-five somewhat randomly placed producers and a single water injector. Results showed that significant agreement could be achieved for this problem on the basis of total production rates, saturations, and reservoir pressures. On the other hand, rates for some individual wells did show variations of as much as bottomhole pressures. All participants were able to simulate the study in fewer than sixty time steps with an average of 4-5 Newton iterations per step. In addition, the results showed that this moderate-sized problem could be simulated in only a few minutes in a workstation environment for the two plus years of data.

Ninth SPE Comparative Solution Project: A Reexamination of Black-Oil Simulation

The state of the art of modeling fluid flow in shale reservoirs is dominated by dual-porosity models which divide the reservoirs into matrix blocks that significantly contribute to fluid storage and fracture networks which principally control flow capacity. However, recent extensive microscopic studies reveal that there exist massive micro- and nano-pore systems in shale matrices. Because of this, the actual flow mechanisms in shale reservoirs are considerably more complex than can be simulated by the conventional dual-porosity models and Darcy’s law. Therefore, a model capturing multiple pore scales and flow can provide a better understanding of the complex flow mechanisms occurring in these reservoirs. This paper presents a micro-scale multiple-porosity model for fluid flow in shale reservoirs by capturing the dynamics occurring in three porosity systems: inorganic matter, organic matter (mainly kerogen), and natural fractures. Inorganic and organic portions of shale matrix are treated as sub-blocks with different attributes, such as wettability and pore structures. In kerogen, gas desorption and diffusion are the dominant physics. Since the flow regimes are sensitive to pore size, the effects of nano-pores and micro-pores in kerogen are incorporated into the simulator. The multiple-porosity model is built upon a unique tool for simulating general multiple-porosity systems in which several porosity systems may be tied to each other through arbitrary connectivities. This new model allows us to better understand complex flow mechanisms and eventually is extended into the reservoir scale through upscaling techniques. Sensitivity studies on the contributions of the different flow mechanisms and kerogen properties give some insight as to their importance. Results also include a comparison of the conventional dual-porosity treatment and show that significant differences in fluid distributions and dynamics are obtained with the improved multiple-porosity simulation.

Beyond Dual-Porosity Modeling for the Simulation of Complex Flow Mechanisms in Shale Reservoirs

Beyond dual-porosity modeling for the simulation of complex flow mechanisms in shale reservoirs

This paper presents the results of comparisons between both four-component miscible flood simulators and fully compositional reservoir simulation models from seven different participants for a series of three test cases. These cases varied from scenarios dominated by immiscible conditions to scenarios in which minimum miscibliity pressure was maintained or exceeded throughout the simulations. In general, agreement between the models was good. For a test case in which reservoir pressure was maintained above the minimum miscibility pressure, agreement between simulators, with the assumption of complete miXing of solvent and oil, and compositional simulators was excellent based on cumulative oil production as a function of cumulative water injection. For cases in which immiscible conditions dominated, the four-component models tended to be pessimistic compared to fully compositional models because condensible liqUids were not considered to be carried in the gaseous phase in the four-component simulations. Relative permeability treatment, especially near the injection well, tended to the timing of recovery and . injectant br'eakt:hI-murlJ.. The simulation of gas or solvent injection into a volatile oil reservoir can be modeled by approximating the phase behavior with four components oil, water, free/solution gas, and injection gas (solvent)as described by Todd and This process can also be modeled by accurately simulating the phase behavior with n-components whose K-values are complex functions of pressure, temperature. and composition. A precise set of rules of when one may approximate the displacement process with four components and when one must use the fully and i lustrations at end of Paper. compositional formulation is not generally available. There is much discussion in the technical community of exactly this problem. but all too often the decision of which model is used comes from time, money, computer, or data availability or purely subjective reasons. Thus, this comparative solution project has attempted to present an opportunity for the petroleum simulation community to investigate some aspects of this question and at the same time provide an attempt to validate two types of reservoir simulators under certain conditions. As was said in the Fourth SPE Solution Project,2 "good agreement between results from different simulators for the same problem does not insure validity of any of the results, (but) a lack of agreement does give cause for some concern." This paper represents the fifth in a series of comparative solution problems which have been open for participation by oil companies, research institutes, and consult~ts_ The first study was conducted by and consisted of a three-dimensional, two-phase, black-oil simulation. Chappelear and a study of three-phase. single weI radial cross-sectional coning simulations. A compositional, three-phase study of gas cycling in a retrograde gas condensate reservoir comprised the third comparative solution project organized by Kenyon and Behie. 5 The most recent comparative solution problem conducted by Aziz. Ramesh, and was a two-dimensional radial steam injection (thermal) simulation. The object of this paper is to present the simulation problems and selected results as submitted by the participants and to discuss any large differences which exist in the results. Seven participants were involved in this project. An attempt has been made to describe the problems and the input to the simulators in such a fashion that all of the appropriate variables for each participant

John Killough

Papers

Reservoir Simulation With History-Dependent Saturation Functions

Ninth SPE Comparative Solution Project: A Reexamination of Black-Oil Simulation

Beyond Dual-Porosity Modeling for the Simulation of Complex Flow Mechanisms in Shale Reservoirs

Beyond dual-porosity modeling for the simulation of complex flow mechanisms in shale reservoirs

Fifth Comparative Solution Project: Evaluation of Miscible Flood Simulators