bluuebunny/crossref_metadata_2025 huggingface dataset:
Detailed Near-Well Darcy-Forchheimer Flow Modeling and Upscaling on Unstructured 3D Grids
Authors: Karimi-Fard M. , Durlofsky L. J.
Published: February 2009

Computing well performance accurately is very important for reservoir management and optimization. Standard large-scale reservoir simulators, however, are not able to resolve detailed near-well physics such as the interaction between the local pressure gradient, reservoir heterogeneity, and well completion. This is because these effects occur on scales that are very small compared to typical simulation grid blocks. In this work, we describe a modeling procedure for representing the well and near-well region that is able to resolve flow down to the scale of the completion and perforations. Our computations involve single-phase flow simulations on unstructured 3D tetrahedral grids. The Darcy-Forchheimer equation is solved in order to include non-Darcy effects, which can be important in the near-well region where fluid velocities are high. The new modeling capability is applied to quantify the impact of completion type – open hole or perforated cased hole – on well productivity. Next, in order to capture these detailed effects in large-scale simulation models, a new upscaling procedure is developed and applied. The approach can be categorized as a local-global upscaling method which uses global coarse-scale simulations to provide key parameters for the local upscaling computations. An example for a case with significant non-Darcy effects demonstrates the high degree of accuracy achievable using this technique.
A Novel Approach to Integrate Dual Porosity Model and Full Permeability Tensor Representation in Fractures
Authors: Tarahhom Farhad , Sepehrnoori Kamy , Marcondes Francisco
Published: February 2009

Predicting fluid flow behavior in naturally fractured reservoirs is a challenging area in petroleum engineering. Two classes of models used to describe flow and transport phenomena in fracture reservoirs are discrete and continuum (i.e. dual porosity) models. The discrete model is appealing from a modeling point of view, but the huge computational demand and burden in porting the fractures into the computational grid are its shortcomings. On the other hand, the diagonal representation of permeability, which is customarily used in a dual porosity model, is valid only for the cases where fractures are parallel to one of the principal axes. This assumption cannot adequately describe flow characteristics where there is variation in fracture spacing, length, and orientation. To overcome this shortcoming, the principle of the full permeability tensor in the discrete fracture network can be incorporated into the dual porosity model. Hence, the dual porosity model can retain the real fracture system characteristics.

Expelling oil from matrix blocks into fractures based on water imbibition leaves behind significant amounts of oil in the matrix blocks in the form of residual oil. Chemical flooding is an excellent means to expel more oil from the matrix into fractures. A fully implicit parallel, compositional chemical dual porosity model in conjunction with full permeability tensor for the fracture system has been developed. The model is capable of simulating large-scale chemical flooding processes. The matrix blocks are discretized into both rectangular rings and vertical layers to offer a better resolution of transient flow.

The developed model was successfully verified against the UTCHEM simulator. Results show excellent agreements for a variety of flooding processes. Full permeability tensor results were also verified with a discrete fracture simulator.

This study leads to a conclusion that the full permeability tensor representation is essential to accurately simulate fluid flow in heterogeneous and anisotropic fracture systems.
Development of a Three Phase, Fully Implicit, Parallel Chemical Flood Simulator
Authors: Najafabadi N. Fathi , Han C. , Delshad M. , Sepehrnoori K.
Published: February 2009

Field-scale applications of chemical flooding become more attractive with higher oil prices. Several pilot and commercial scale chemical floods are currently in operation or design. Economic feasibility of such projects relies on how cost-effectively the remaining oil volume is recovered. Robust design and optimization are essential for technical success and profitability. The main objective in chemical flooding design is to keep the surfactant in Type III near the optimum salinity. Salinity gradient design is a robust design since it can compensate for heterogeneity and reservoir uncertainties and guarantees the surfactant in Type III for a longer time compared to other designs. A comprehensive surfactant phase behavior model is required to take into account the salinity gradient design with all possible phase transitions.

The development discussed in this paper enables accurate modeling and optimization of chemical flooding designs for realistic field-scale projects where a salinity gradient exists naturally or is imposed by design. The parallel processing capability combined with the fully implicit scheme allows high-resolution full field scale chemical flooding simulations. Comprehensive surfactant phase behavior as a function of water salinity has been implemented into a fully implicit and parallel equation of state (EOS) compositional simulator, GPAS. The fully implicit simulation results are validated against an IMPES chemical flooding simulator.

Accurate simulation of chemical processes requires fine grid resolution which adds a large computational overhead for modeling field-scale projects. Our enhanced simulator will allow for high resolution, field-scale design and optimization of chemical flooding projects. This is the first fully implicit, chemical-EOS compositional simulator with a comprehensive surfactant phase behavior that can take into account the effect of salinity and the resulting two or three phase flow with efficient parallel scalability.
A New Solution Procedure for a Fully Coupled Geomechanics and Compositional Reservoir Simulator
Authors: Pan Feng , Sepehrnoori Kamy , Chin L. Y.
Published: February 2009

Traditional reservoir simulators cannot capture the complicated interactions between fluid production and reservoir rock deformation during hydrocarbon recovery. In particular, in a recovery process when both phase behavior and deformation of reservoir rock play critical roles, a coupled geomechanics and compositional reservoir model can rigorously capture the above physical relations between solid and fluids and thereby perform more precise history matchings and predictions for better well planning and reservoir management decisions.

This paper considers a fully coupled geomechanics and compositional modeling process. A novel solution procedure for the associated fully coupled Jacobian system is presented using the Schur complement technique. The proposed approach has the following features: (1) it has similar convergence properties as the fully implicit method, (2) combined with Krylov subspace iterative solvers, it can deal with symmetric and unsymmetric problems, and (3) it can be easily parallelized to solve large-scale reservoir simulation problems.

In this work, a parallel, fully implicit, equation of state compositional reservoir simulator was fully coupled with a geomechanics FEM module. For each Newton iteration, an original two by two block Jacobian system is converted to a Schur complement system, and then the associated Schur complement system is solved using a BiCGstab(l) iterative method; subsequently all the unknowns for the reservoir simulator and the geomechanics module are obtained at the same time. The above partition solution procedure is equivalent to a block Gaussian elimination; hence its convergence is similar to a fully implicit coupled procedure.

The proposed approach is implemented to solve two- and three-dimensional compositional reservoir simulation case studies considering geomechanics effects. We also compare the results between the fully coupled and the iteratively coupled approaches. It is shown that this approach presents a viable alternative in solution methods for solving general coupled physical problems.
Stochastic History Matching of a Deepwater Turbidite Reservoir
Authors: Alpak Faruk O. , van Kats Florian , Hohl Detlef
Published: February 2009

A novel stochastic framework is described that facilities automatic history matching and uncertainty quantification workflows. The underlying algorithm of the framework combines Design of Experiments (DoE) and Markov Chain Monte Carlo (MCMC) inference techniques. A generic singular-value decomposition technique creates a Response Surface Model (RSM) from a DoE-based selection of simulations. Ensuing RSM serves as a rapid proxy to full-physics reservoir simulation. The data integration and uncertainty quantification framework extends current working practice by sampling Bayes' posterior parameter probability distribution with a rigorous MCMC method using the Metropolis-Hastings algorithm. This improves uncertainty estimates in the history-matching mode, and provides a unified platform for scenario management and analysis in the forecast uncertainty quantification mode.

The stochastic framework is tested for robustness and efficiency on a real field case. A history matching study is carried out for a complex deepwater turbidite reservoir involving multiple geologic realizations. Within the context of the study, the stochastic framework delivered (a) a complete description of the high-impact model parameters dominating the quality of the match, (b) multiple models honoring the historical production data, (c) reduced uncertainty ranges for the history-matching parameters, and (d) a quantitative measure of reliability for different measurement types. The outcome of the multi-realization history matching study reveals a better reservoir connectivity and an increased net-sand volume compared to the maximum a-posteriori (MAP) prediction attained by a commercial assisted history-matching software. A well-by-well analysis of the most probable model reveals that the match is comparable or superior to the ones delivered by the commercial tool. The field test outcome demonstrates the reliability, expediency, rigor, and computational efficiency of our stochastic history-matching and recovery forecasting framework.
Estimation of Depths of Fluid Contacts by History Matching Using Iterative Ensemble Kalman Smoothers
Authors: Wang Yudou , Li Gaoming , Reynolds A. C.
Published: February 2009

With the ensemble Kalman filter (EnKF) or smoother (EnKS), it is easy to adjust a wide variety of model parameters by assimilation of dynamic data. We focus on the case where reallizations and estimates of the depths of the initial fluid contacts as well as gridblock rock property fields are generated by matching production data with the EnKS. The objective is to account for uncertainty in the depths of the contacts and provide improved estimates of the depths by conditioning reservoir models to production data. The efficiency of EnKF and EnKS arises because data are assimilated sequentially in time and so "history matching data" requires only one forward run of the reservoir simulator for each ensemble member. For EnKS and EnKF to yield reliable characterizations of the uncertainty in model parameters and future performance predictions, the updated reservoir simulation variables, e.g., saturations and pressures, must be statistically consistent with the realizations of these variables that would be obtained by rerunning the simulator from time zero using the updated model parameters. This statistical consistency can only be established under assumptions of Gaussianity and linearity that do not normally hold. Here, we explore the use of iterative EnKS methods that are statistically consistent to improve the performance of EnKS.
Use of Reduced-Order Modeling Procedures for Production Optimization
Authors: Cardoso M. A. , Durlofsky L. J.
Published: February 2009

The determination of optimal well settings is very demanding computationally, as the simulation model must be run many times during the course of the optimization. For this reason, reduced-order modeling procedures, which are a family of techniques that enable high efficient simulations, are very useful for optimizing problems. In this paper we describe a recently developed reduced-order modeling technique which has been used in other application areas, we describe a recently developed reduced-order modeling technique which has been used in other application areas, the trajectory piecewise linear (TPWL) procedure, and incorporate it in optimization computations. The TPWL methodology represents solutions encountered during the optimization runs in terms of Taylor series expansions around previously simulated states. This requires a small number of preprocessing simulations using the full (high-fidelity) model, during which pressure and saturation states and Jacobian matrices are saved. These states and matrices are then projected into a low-dimensional space using proper orthogonal decomposition. Simulations in this reduced space can be performed very efficiently; in this work we observe runtime speedups of a factor 450.

We assess the TPWL representation for simulations of waterflood in a heterogeneous 3D model containing over 20,000 grid blocks and six wells. The high degree of accuracy of the TPWL model is first demonstrated for several testing simulations in which producer will settings differ from those used in the preprocessing runs. The TPWL representations are then used in several optimizations involving the determination of optimal bottomhole pressures in four producers at six different times (24 control variables). Both gradient-based and generalized pattern search optimization algorithms are considered. Results for optimized net present value using TPWL are shown to be in consistently close agreement with those computed using high-fidelity simulations. Most significantly, when the optimal well settings obtained using TPWL are applied in high-fidelity models, the resulting net present values are within 0.8% of the values determined using the high-fidelity simulations. Our overall conclusion is that the TPWL representation is very well suited for use in production optimization problems.
A New Method for Thermodynamic Equilibrium Computation of Systems With an Arbitrary Number of Phases
Authors: Fraces Gasmi Cédric , Voskov Denis V. , Tchelepi Hamdi A.
Published: February 2009

Reservoir recovery processes involve complex mass and heat transfer between the injected fluid and the resident rock-fluid system. Thermal-compositional reservoir simulators can be used to plan such displacement processes, in which the phase behavior is computed with an Equation of State (EoS). These thermodynamic-equilibrium computations include phase-stability tests and flash calculations, and can consume a significant fraction of the total simulation time, especially for highly detailed reservoir models and a large number of components. Here, we propose a general Compositional Space Parameterization (CSP) method for complex mixtures, especially those where more than two fluid phases can coexist in parts of the parameter space. For a given pressure (P), temperature (T) and overall composition, a unique tie-simplex (tie-line for two phases, tie-triangle for three phases, etc.) can be defined. For a particular composition at P and T, the tie-simplex provides the necessary phase equilibrium information (i.e., phase state and phase compositions). For compositional flow simulation, a set of tie-simplexes can be calculated in a preprocessing step, or adaptively constructed during the simulation. The tie-simplex representation can be used to replace standard phase-equilibrium calculations completely, or it can be used as an initial guess for standard EoS calculations. Challenging examples with two and three phases are presented to validate this tie-simplex CSP approach. Standard EoS methods, which are widely used in industrial compositional simulators, are compared with CSP-based simulations for problems with large numbers of components and complex two- and three-phase behaviors spanning wide ranges of pressure and temperature. The numerical experiments indicate that our multi-dimensional tie-simplex representation combined with linear pressure and temperature interpolation in tie-simplex space, which is implemented as an adaptive tabulation strategy, leads to highly robust and efficient computations of the phase behavior associated with compositional flow simulation.
Optimization of Residual Gas and Solubility Trapping for CO2 Storage in Saline Aquifers
Authors: Nghiem Long , Yang Chaodong , Shrivastava Vijay , Kohse Bruce , Hassam Mohamed , Chen Dingjun , Card Colin
Published: February 2009

The two most important trapping mechanisms for CO2 storage in saline aquifers, namely residual gas trapping and solubility trapping, are examined. It is shown that a water injector located above the CO2 injector can be used to enhance both residual gas trapping and solubility trapping. Using simulation, the optimal location and operating conditions for the water injector are determined for both a low-permeability aquifer and a high-permeability aquifer. The analysis is performed using the concept of Pareto optimal solutions for multiple-objective optimization. It is shown that for a low-permeability aquifer, the use of a water injector can significantly increase the amount of residual gas trapping, which results in a substantial increase in the total trapped amount. For a high-permeability aquifer, the use of a water injector above the CO2 injector does not increase CO2 trapping.
Stability, Accuracy and Efficiency of Sequential Methods for Coupled Flow and Geomechanics
Authors: Kim J. , Tchelepi H. A. , Juanes R.
Published: February 2009

We perform detailed stability and convergence analyses of iteratively coupled solution methods for coupled fluid flow and reservoir geomechanics. We analyse four different operator-split strategies: two schemes in which the mechanical problem is solved first (drained and undrained splits), and two schemes where the flow problem is solved first (fixed-strain and fixed-stress splits). We use the Van Neumann method to obtain sharp stability criteria. The drained and fixed-strain splits, which are commonly used, are only conditionally stable. Their stability limits are independent of time step size and depend only on the strength of the coupling between flow and mechanics. This implies that problems with strong coupling cannot be solved by the drained or fixed-strain split methods. Moreover, oscillations in the solution by the drained and fixed-strain split methods can occur even when the stability limit is honored. For problems where the deformation may be plastic in nature, the drained and fixed-strain sequential schemes suffer from severe stability problems when the system enters the plastic regime. However, the undrained and fixed-stress splitting methods are unconditionally stable regardless of the coupling strength, and they do not suffer from oscillations. We also show that the drained and fixed-strain split methods with a fixed number of iterations are inconsistent. That is, they converge to the wrong solution as the time step size goes to zero. On the other hand, the fully coupled, undrained, and fixed-stress methods are consistent even in the case of a single iteration per time step. While both the undrained and fixed-stress schemes are stable, the fixed-stress method is more accurate for a given number of iterations than the undrained method. As a result, we strongly recommend the fixed-stress split. These results have immediate and widespread applicability in the design of reservoir flow simulators that account for geomechanics.
Using Model Error With the Ensemble Kalman Filter for More Accurate Model Updates and Predictions
Authors: Aanonsen Sigurd Ivar
Published: February 2009

The EnKF has become increasingly popular for updating and history matching reservoir simulation models. Originally the filter was developed for sequential conditioning of state variables to dynamic data. It was later extended to solve the combined state and parameter estimation problem. Within petroleum, the focus has been mainly on parameter estimation, and although an unknown model error is present in all real reservoir studies, model error is commonly neglected, assuming that all error is due to uncertainty in the estimated parameters. In this paper we investigate the effect of a model error on the EnKF behaviour for a single-phase permeability estimation problem, and whether it is possible to improve the solution and predictions by including an unknown model error in the EnKF procedure. Numerical upscaling errors are chosen for the test case. However, the procedure should be applicable for any type of unknown model errors. It is demonstrated that with model error present, but not included in the analysis, it is not possible to match the data. The permeability estimate becomes unstable, and the predictions are very bad. However, by adding an unknown model error, data can be matched, and the parameter estimates may be significantly improved. Both uncorrelated and correlated errors are tested. Correlated errors are included as additional unknowns in the EnKF state vector. Predictions, and short term predictions in particular, are significantly improved.
Front-Tracking Methods for Use in Streamline Simulation of Compressible Flow
Authors: Nilsen Halvor Møll , Lie Knut-Andreas
Published: February 2009

Streamline methods are gaining popularity in the industry by providing fast desktop simulation of large reservoir models or multiple realizations. Traditionally, streamline simulation has been associated with simplified physics. but recent advances have demonstrated its potential also for compressible three-phase or component flaws. However, streamline simulation is still most efficient for two-phase incompressible flow, for which one can utilize a particularly efficient front-tracking method to solve 1-D transport equations along streamlines that is unconditionally stable and independent of the strongly irregular time-of-flight grid. In a recent paper (Nilson and Lie 2008), we presented, for the first time, front-tracking methods for simulating 1-D compressible two-phase flow. We also developed two methods that were particularly efficient for solving compressible flow in which one phase is incompressible, motivated by the simulation of CO2 injection. Here we apply these methods to streamline simulation of 3-D models, including a real-life model of a North Sea formation, which is under consideration as a potential target for CO2 deposition. Our numerical results demonstrate that streamlines and front tracking together give very efficient simulation of compressible flow. Similar ideas can also be applied for dual-porosity models, but this is not investigated in great detail herein.
A Multiscale Mixed Finite Element Method For Vuggy and Naturally Fractured Reservoirs
Authors: Gulbransen Astrid Fossum , Hauge Vera Louise , Lie Knut-Andreas
Published: February 2009

Vugs, caves, and fracture can significantly alter the effective permeability of carbonate reservoirs and should be accurately accounted for in a geomodel. Accurate modeling of the interaction between free-flow and porous regions is essential for flow simulations and detailed production engineering calculations. However, flow simulation of such reservoirs is very challenging because of the co-existence of porous and free-flow regions on multiple scales that need to be coupled.

Multiscale methods are conceptually well-suited for this type of modeling as they allow varying resolution and provide a systematic procedure for coarsening and refinement. However, to date there are hardly no multiscale methods developed for problems with both free-flow and porous regions. Our work is a first step to make a uniform multiscale framework, where we develop a multiscale mixed finite-element (MsMFE) method for detailed modeling of vuggy and naturally-fractured reservoirs. The MsMFE method uses a standard Darcy model to approximate pressure and fluxes on a coarse grid, but capatures fine-scale effects through basis functions determined from numerical solutions of local Stokes-Brinkman flow problems on the underlying fine-scale geocellular grid. The Stokes-Brinkman equations give a unified approach to simulating free-flow and porous regions using a single system of equations, avoid explicit interface modeling, and reduce to Darcy or Stokes flow by appropriate choices of parameters.

In the paper, the MsMFE solutions are compared with fine-scale Stokes-Brinkman solutions for test cases including both short- and long-range fractures. The results demonstrate how fine-scale flow in fracture networks can be represented within a coarse-scale Darcy flow model by using multiscale elements computed solving the Stokes-Brinkman equations. The results indicate that the MsMFE method is a promising path toward direct simulation of highly detailed geocellular models of vuggy and naturally-fractured reservoirs.
Adjoint Multiscale Mixed Finite Elements
Authors: Krogstad S. , Hauge V. L. , Gulbransen A. F.
Published: February 2009

We develop an adjoint model for a simulator consisting of a multiscale pressure solver and a saturation solver that works on flow adapted grids. The multiscale method solves the pressure on a coarse grid that is close to uniform in index space and incorporates fine-grid effects through numerically computed basis functions. The transport solver works on a coarse grid adapted by a fine-grid velocity field obtained by the multiscale solver. Both the multiscale solver for pressure and the flow-based coarsening approach for transport have earlier shown the ability to produce accurate results for high degree of coarsening. We present results for a complex realistic model to demonstrate that control settings based on optimization of our multiscale flow-based model closely matches or even outperforms those found by using a fine-grid model. For additional speed-up, we develop mappings used for rapid system updates during the time-stepping procedure. As a result, no fine-grid quantities are required during simulations and all fine grid computations (multiscale basis functions, generation of coarse transport grid and coarse mappings) become a preprocessing step. The combined methodology enables optimization of water flooding on a complex model with 45,000 grid-cells in a few minutes.
Interpretation of Partitioning Interwell Tracer Tests Using EnKF with Coarse Scale Constraints
Authors: Devegowda Deepak , Akella Santha , Datta-Gupta Akhil , Efendiev Yalchin
Published: February 2009

The success of enhanced recovery processes depends largely on the accurate characterization of the location and distribution of the target oil. The Partitioning Interwell Tracer Test (PITT) is designed to identify the residual oil saturation during waterflooding. In this test a conservative and a partitioning tracer are injected into the reservoir. The conservative tracer travels with the water phase whereas the partitioning tracer dissolves in both water and oil. The separation between the conservative and the partitioning tracer at the producer can be used to infer the residual oil saturation distribution.

The interpretation of partitioning tracer tests in the presence of mobile oil is considerably more difficult because the separation of the tracer response depends not only on the partitioning coefficients but also on the spatial distribution of the saturation itself. We propose a two-stage Ensemble Kalman Filter (EnKF) that utilizes the tracer data in conjunction with coarse-scale saturation constraints to identify the fine-scale saturation distribution and associated uncertainties. The coarse-scale saturation distribution can be conveniently obtained via an inversion of the waterflood response, for example water-cut data at the producers. The tracer data are first used to condition the ensemble of fine scale model realizations which are further improved via the coarse-scale constraint. The scale-decomposition of the inverse problem significantly improves the performance of the EnKF and prevents filter divergence as more data are assimilated.

We demonstrate the effectiveness of our approach using two and three-dimensional examples. First, we derive the coarse-scale saturation from a streamline-based inversion of the waterflood response. Next, the two-stage EnKF is used to interpret the PITT using the coarse-scale saturation constraint. A performance comparison of our proposed approach with the standard EnKF clearly indicates better characterization of the reservoir heterogeneities and improved estimates of the spatial distribution of bypassed oil.
Partitioners for Parallelizing Reservoir Simulations
Authors: Shuttleworth Robert , Maliassov Serguei , Zhou Hui
Published: February 2009

Computer simulations of oil and gas reservoirs require the construction and solution of large, sparse linear systems with millions or even tens of millions of unknowns at each time step or Newton iteration. For efficiency, these tasks are often parallelized. While the parallel construction of the Jacobian matrix and right hand side vectors mainly depends on the load balance between the processors, the parallel solution of the resulting linear system is sensitive to the way the system is partitioned amongst processors during the solution process. We evaluate several different strategies to parallelize the solution of the linear system on both shared and distributed memory architectures, using a strategy to optimize both the load balance of processors and the convergence of the linear solver. We discuss partitioning strategies based on multi-level graph partitioning using an approach for optimal vertex/edge weighting for reservoir simulation models based on unstructured, 2½D finite volume spatial discretization. We present parallel scalability and timing results obtained from field studies, using ExxonMobil's proprietary reservoir simulator, EMpower.
Multiscale Mimetic Solvers for Efficient Streamline Simulation of Fractured Reservoirs
Authors: Natvig J. , Skaflestad B. , Bratvedt F. , Bratvedt K. , Lie K.-A. , Laptev V. , Khataniar S. K.
Published: February 2009

Advances in reservoir characterization and modeling have given the industry improved ability to build detailed geological models of petroleum reservoirs. These models are characterized by complex shapes and structures with discontinuous material properties that span many orders of magnitude. Models that represent fractures explicitly as volumetric objects pose a particular challenge to standard simulation technology with regard to accuracy and computational efficiency.

We present a new simulation approach based on streamlines in combination with a new multiscale mimetic pressure solver with improved capabilities for complex fractured reservoirs. The multiscale solver approximates the flux as a linear combination of numerically computed basis functions defined over a coarsened simulation grid consisting of collections of cells from the geological model. Here, we use a mimetic multipoint flux approximation to compute the multiscale basis functions. This method has limited sensitivity to grid distortions. The multiscale technology is very robust with respect to fine-scale models containing geological objects such as fractures and fracture corridors. The methodology is very flexible in the choice of the coarse grids introduced to reduce the computational cost of each pressure solve. This can have a large impact on iterative modeling workflows.
Energy Minimizing Bases for Efficient Multiscale Modeling and Linear Solvers in Reservoir Simulation
Authors: Dubois Olivier , Mishev Ilya D. , Zikatanov Ludmil
Published: February 2009

We demonstrate the applicability of energy minimizing bases for two-phase flow simulations, and we highlight several advantageous properties that they possess. We show how they can be implemented to obtain efficient serial (algebraic multigrid) and parallel (Additive Schwarz with coarse space correction) linear solvers for large-scale heterogeneous problems. We also confirm with experiments that using such bases for coarsening a problem produces a numerical solution of quality comparable to that of other available multiscale techniques. Finally, we also indicate how different coarsening levels can be used in different regions, in order to keep a finer resolution near certain areas of interest and significantly coarser resolution elsewhere. A multilevel approach is adopted to efficiently construct the basis functions for the coarser resolution.
Dynamic Data Integration and Quantification of Prediction Uncertainty Using Statistical Moment Equations
Authors: Likanapaisal P. , Li L. , Tchelepi H. A.
Published: February 2009

A probabilistic framework for dynamic data integration (history matching) has become accepted practice. The idea is to build an ensemble of reservoir models, all of which being consistent with the geologic scenario and also honoring all available (static and dynamic) information. In addition to data assimilation, the probabilistic framework provides an assessment of the prediction uncertainty due to incomplete knowledge of the reservoir description. Methods based on Monte Carlo Simulation (MCS) are widely used. This is driven by the generality and simplicity of MCS. As a black-box approach, only pre/post-processing of conventional flow simulations is needed. To achieve reasonable accuracy in estimating the statistical moments of flow performance predictions, however, large numbers of realizations are usually necessary. Here, we use an alternative direct approach for model calibration and uncertainty quantification. Specifically, we describe a Statistical Moment Equations (SME) framework for both the forward and inverse problems. In the SME method, the equations governing the statistical moments of the quantities of interest (e.g., pressure, saturation) are derived and solved directly. Immiscible two-phase flow problems are investigated, where we assume that in addition to statistical information (and a few measurements) about the permeability field, measurements of pressure, saturation, and flow rate are available at a few locations and several times. For the forward problem, the flow (pressure and total-velocity) equations are solved on regular grid, while a streamline-based strategy is used to solve the transport moments. A Kriging-based inversion algorithm, in which the statistical moments of permeability are conditioned directly based on available dynamic data, is used. We analyze the behaviors of the saturation moments and their evolution as they are conditioned on measurements, in both space and time. Moreover, we compare the SME inversion scheme with Kalman filter approaches (an MCS inversion approach) for dynamic data integration, from a mathematical framework perspective.
Comparison of Stochastic Sampling Algorithms for Uncertainty Quantification
Authors: Mohamed L. , Christie M. , Demyanov V.
Published: February 2009

History matching and uncertainty quantification are two important research topics in reservoir simulation currently. In the Bayesian approach, we start with prior information about a reservoir – for example from analogue outcrop data – and update our reservoir models with observations, for example from production data or time lapse seismic. The goal of this activity is often to generate multiple models that match the history and use the models to quantify uncertainties in predictions of reservoir performance. A critical aspect of generating multiple history matched models is the sampling algorithm used to generate the models. Algorithms that have been studied include gradient methods, genetic algorithms, the Ensemble Kalman Filter, and others.

This paper investigates the efficiency of three stochastic sampling algorithms: Hamiltonian Monte Carlo (HMC) algorithm, Particle Swarm Optimization (PSO) algorithm and the Neighborhood Algorithm (NA). HMC is a Markov Chain Monte Carlo (MCMC) technique that uses Hamiltonian dynamics to achieve larger jumps than are possible with other MCMC techniques. PSO is a swarm intelligence algorithm that uses similar dynamics to HMC to guide the search, but incorporates acceleration and damping parameters to provide rapid convergence to possible multiple minima. The Neighbourhood Algorithm is a sampling technique that uses the properties of Voronoi cells in high dimensions to achieve multiple history matched models.

The algorithms are compared by generating multiple history matched reservoir models, and comparing the p10 – p50 – p90 uncertainty bounds produced by each algorithm. We show that all the algorithms are able to find equivalent match qualities for this example, but that some algorithms are able to find good fitting results quickly, whereas others are able to find a more diverse set of models in parameter space. The effects of the different sampling of model parameter space are compared in terms of the p10 - p50 - p90 uncertainty bounds in forecast oil rate.

These results show that algorithms based on Hamiltonian dynamics and swarm intelligence concepts have the potential to be effective tools in uncertainty quantification in the oil industry.
Adaptively-Localized-Continuation-Newton: Reservoir Simulation Nonlinear Solvers That Converge All the Time
Authors: Younis R. M. , Tchelepi H. A. , Aziz K.
Published: February 2009

Growing interest in understanding, predicting, and controlling advanced oil recovery methods emphasizes the importance of numerical methods that exploit the nature of the underlying physics. The Fully Implicit Method offers unconditional stability in the sense of discrete approximations. This stability comes at the expense of transferring the inherent physical stiffness onto the coupled nonlinear residual equations which need to be solved at each time-step. Current reservoir simulators apply safe-guarded variants of Newton's method, and often can neither guarantee convergence, nor provide estimates of the relation between convergence rate and time-step size. In practice, time-step chops become necessary, and are guided heuristically. With growing complexity, such as in thermally reactive compositional models, this can lead to substantial losses in computational effort, and prohibitively small time-steps. We establish an alternate class of nonlinear iteration that both converges, and associates a time-step to each iteration. Moreover, the linear solution process within each iteration is performed locally.

By casting the nonlinear residual for a given time-step as an initial-value-problem, we formulate a solution process that associates a time-step size with each iteration. Subsequently, no iterations are wasted, and a solution is always attainable. Moreover, we show that the rate of progression is as rapid as a standard Newton counterpart whenever it does converge. Finally, by exploiting the local nature of nonlinear waves that is typical to all multi-phase problems, we establish a linear solution process that performs computation only where necessary. That is, given a linear convergence tolerance, we identify the minimal subset of solution components that will change by more than the specified tolerance. Using this a priori criterion, each linear step solves a reduced system of equations. Several challenging examples are presented, and the results demonstrate the robustness of the proposed method as well as its performance.
Geomechanics of Heterogeneous Bitumen Carbonates
Authors: Freeman T. T. , Chalaturnyk R. J. , Bogdanov I. I.
Published: February 2009

Around 26% of Alberta's bitumen resources are held in carbonate formations. Due to the lack of efficient technologies to extract bitumen from carbonates these resources are currently classified as "inaccessible". Among possible technologies, thermal recovery methods will be a strong option for recovery of bitumen from carbonates. Amongst thermal recovery methods, Steam Assisted Gravity Drainage (SAGD) is one of the best methods originally developed and successfully used for oil sands in Alberta. However, because of heterogeneity, lower permeability and deeper position of bitumen carbonates, steam injection method may not be as effective in bitumen carbonate reservoirs as they are in oil sands. Alternate heating methods such as electrical ohmic heating and higher frequency (radio frequency and microwave) electromagnetic methods are strong alternative options.

During a thermal recovery process, stress state within the reservoir is significantly altered due the thermal stresses and steam pressure. The new stress state can impact the multiphase flow of steam and bitumen by enhanced permeability or stimulated fractures. All the previous studies on geomechanics of thermal recovery methods have been focused on SAGD and that only for oil sands and because of fundamental differences between bitumen carbonates and oil sands the results are not applicable for carbonates. Change of permeability in carbonates under elevated stresses would not necessarily be caused by shear failure and it could emerge by connecting the isolated vugs or stimulation of the existing fractures.

A fully coupled electro-thermo-hydro-mechanical (ETHM) model based on multiphysics platform is used to study the simultaneous interaction between non-isothermal multiphase flow in the carbonates and stresses and strains in the reservoir. The advantage of multiphysics modeling by COMSOL Multiphysics is that being a general-purpose simulator it offers a possibility for multi-physical coupling via the wide choice of application modes such as electrical or chemical. Heterogeneity of the carbonate is modeled through a probabilistic distribution of mechanical and hydraulic properties to account for inherent variability in carbonate properties.
A Flow Based Lumping Approach for Compositional Reservoir Simulation
Authors: Rastegar Reza , Jessen Kristian
Published: February 2009

In this paper, we present a new and automatic lumping methodology to improve the accuracy and CPU efficiency of compositional reservoir simulation. Typical equation of state (EOS) based models for prediction of PVT experiments employ a large number of components. Such detailed EOS models are subsequently lumped into a reduced number of pseudocomponents for use in compositional reservoir simulation. Traditional lumping approaches do not include any information regarding the mixture compositions will form during a gas injection process. These mixture compositions can differ significantly from those investigated by standard PVT and swelling test experiments. Here we demonstrate the benefit of including displacement characteristics obtained from semi-analytical displacement calculations in the selection of pseudo-components for compositional reservoir simulation and propose an automated scheme for grouping/lumping.

Components with close proximity in terms of specific velocities and composition space are lumped subject to the observed variation in equilibrium K values as defined by displacement dynamics. Lumping is performed starting from the most detailed fluid description in a sequential manner to gauge the quality of the fluid description as we reduce the number of components in the fluid description. Examples are provided for multicomponent reservoir fluids and CO2 injection where a substantial amount of data is available (PVT, swelling test and slimtube MMP). We demonstrate that the proposed lumping strategy provides for additional control of the predictive quality of the reduced EOS model that may not be possible with currently available lumping approaches.

A majority of oil fields currently under production are candidates for EOR processes such as tertiary gas floods. The work presented in this paper is directly applicable to the study and design of such EOR processes. In addition, this is to our best knowledge the first paper that combines displacement dynamics from the analytical theory of gas injection processes with reservoir fluid characterization towards improved accuracy/efficiency of compositional reservoir simulation.
Efficient General Formulation Approach For Modeling Complex Physics
Authors: Cao H. , Crumpton P. I. , Schrader M. L.
Published: February 2009

This paper describes a general formulation for phase-component partitioning that can accommodate any number of phases and components, any component existing in any phase, and requires no special ordering of phases or components. This type of formulation is desirable for flexibility in reservoir simulation but has not previously been used in commercial simulators due to its complexity and inefficiencies in both memory and speed. Here we describe an efficient natural-variable-based general formulation approach, which handles general partitioning of phase-component, consumes no extra memory, and only has a small amount of CPU overhead. This general formulation approach was developed as part of a next generation reservoir simulation project (DeBaun et al., 2005), for "next generation" is the ability to accommodate fluid models currently used in reservoir simulation as well as models that will be developed in the future. With this general formulation approach, we can model most reservoir physics with a single, unified code base, rather than custom one-off extensions of a more limited one. Currently we have tested and validated our approach on standard black-oil and compositional models, thermal steam model, and 4-phase 4-component Todd-Longstaff model.

We describe our natural variable based nonlinear system which incorporates general phase-component partitioning. We analyze the complexity of our model and explain implementation details required to minimize, and in some cases eliminate, CPU and memory overheads. We conclude with the validation of our approach and performance comparisons and analysis (all against standard implementations) for various types of field case studies.
Phase Equilibrium Computations Are No Longer the Bottleneck in Thermal Compositional EoS Based Simulation
Authors: Iranshahr A. , Voskov D. V. , Tchelepi H. A.
Published: February 2009

Thermodynamic equilibrium computations are usually the most time consuming component in compositional reservoir flow simulation. A Compositional Space Adaptive Tabulation (CSAT) approach has been developed as a preconditioner for Equation of State (EoS) computations in isothermal compositional simulation. The compositional space is decomposed into sub- and super-critical regions. In the sub-critical region, we adaptively parameterize the compositional space using a small number of tie-lines, which are assembled into a table. The critical surface is parameterized and used to identify super-critical compositions. The phase equilibrium information for a composition is interpolated as a function of pressure using the tie-line table. We extend the CSAT approach to thermal problems. Given an overall composition at a fixed temperature, the boundary between sub- and super-critical pressures is represented by the critical tie-line and the corresponding minimal critical pressure. A small set of sub-critical tie-lines is computed and stored for a given temperature. This process is repeated for the pressure and temperature ranges of interest, and a coarse (regular) tie-line table is constructed. Close to the critical boundary, a refined tie-line table is used. A combination of regular and refined interpolation improves the robustness of the tie-line search procedure and the overall efficiency of the computations. Several challenging problems, including an unstructured heterogeneous discrete fracture field model with 26 components, are used to demonstrate the robustness and efficiency of this general tie-line based parameterization method. Our results indicate that CSAT provides very accurate treatment of the near-critical region. Moreover, the computational efficiency of the method is at least an order of magnitude better than that of standard EoS-based reservoir simulation approaches. We also show the efficiency gains relative to standard techniques as a function of the number of gridblocks in the simulation model.