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ECMOR XIII - 13th European Conference on the Mathematics of Oil Recovery
- Conference date: 10 Sep 2012 - 13 Sep 2012
- Location: Biarritz, France
- ISBN: 978-90-73834-30-9
- Published: 10 September 2012
1 - 100 of 114 results
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A Mortar Method Based on NURBS for Curve Interfaces
Authors A. Rodriguez, H. Florez and M.F. WheelerThe Mortar Finite Element Method (MFEM) has been demonstrated to be a powerful technique in order to formulate a weak continuity condition at the interface of sub-domains in which different meshes, i.e. non-conforming or hybrid, and / or variational approximations are used. This is particularly suitable when coupling different physics on different domains, such as elasticity and poro-elasticity, for example, in the context of coupled flow and geomechanics. In this area precisely, geometrical aspects play also a role. It is very expensive, from the computational standpoint, having the same mesh for flow and mechanics. Tensor product meshes are usually propagated from the reservoir in a conforming way into its surroundings, which makes non-conforming discretizations a highly attractive option for these cases. In order to tackle these general sub-domains problems, a MFEM scheme on curve interfaces based on Non-Uniform Rational B-Splines (NURBS) curves and surfaces is presented in this paper. The goal is having a more robust geometrical representation for mortar spaces which allows gluing non-conforming interfaces on realistic three-dimensional geometries. The resulting mortar saddle point problem will be decoupled by means of standard Domain Decomposition techniques such as Dirichlet-Neumann and Neumann-Neumann, in order to exploit current parallel machine architectures. Three-dimensional examples ranging from near-wellbore applications to field level subsidence computations show that the proposed scheme can handle problems of practical interest. In order to facilitate the implementation of complex workflows, an advanced Python wrapper interface that allows programming capabilities have been implemented. Extensions to couple elasticity and plasticity, which seems very promising in order to speed up computations involving poroplasticity, will be also discussed.
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Errors in the Upstream Mobility Scheme for Counter-Current Two-Phase Flow With Discontinuous Permeabilities
Authors T.S. Mykkeltvedt, I. Aavatsmark and S. TveitThe upstream mobility scheme (UM) is widely used to solve hyperbolic conservation laws numerically. When applied to a homogeneous porous medium this scheme has been shown convergent. When heterogeneities are introduced through the permeability, the flux function attains a spatial discontinuity. In earlier works UM for some examples of countercurrent flow has been shown to perform badly. We have looked at the performance of UM for the counter-current flow of CO2 and brine in a 1D vertical column. The solutions computed from UM are compared to the physically relevant solution found by the modified Godunov flux approximation. Through four examples we show that UM may not converge to the physically correct solution. The scheme is ill-conditioned since a small perturbation in the permeability may give a large difference in the solution. Without knowledge of the physically correct solution it is impossible to rule out the solution produced by UM. Even if UM performs well in most cases, we emphasize that there exists systems where the scheme approximates a completely different solution than the physically relevant one. Since this scheme is widely used in reservoir simulation it is important to be aware of that the scheme can perform this badly.
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A Rigid Element Method for Building Structural Reservoir Models
Authors G. Laurent, G. Caumon, M. Jessell and J.J. RoyerMost current approaches for building structural reservoir models focus on geometrical aspects and consistency with seismic and well data. Few approaches account for the validity of 3D geological models regarding structural compatibility. It may be done using restoration to check the kinematics or mechanics. This is generally performed a posteriori, which also provides critical insights on the basin/reservoir history, but requires significant modeling efforts. This paper presents an approach introducing a first-order kinematic and mechanical consistency at the early stages of the structural modeling. Because the full deformation path is generally poorly constrained, we suggest using simplified approaches to generate plausible structures and assess first-order deformations, making efficiency and robustness more important than physical accuracy. A mechanical deformable model based on rigid elements linked by a non-linear energy has been adapted to geological problems. The optimal deformation is obtained by minimizing the total energy with appropriate boundary conditions. Last, the displacement field is transferred to the geological objects embedded into the rigid elements. With this approach, 3D structural models can be obtained by successively modeling the tectonic events. The underlying tectonic history of resulting models is explicitly controlled by the interpreter and can be used to study structural uncertainties.
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Predicting Faults from Curvatures of Deformed Geological Layers Viewed as Thin Plates.
By J.J. RoyerContinuous media theory in physics uses the Von Karman's theory to describe the shape, strains and stresses of thin plates, non Euclidian thin shells or surfaces. Given a set of boundary conditions, it relates geometrical shape parameters such as the Gaussian and the mean curvatures, the physical properties of the materials such as the Young's modulus and Poisson's ratio to the bending (or flexural slip) and stretching (or pure shearing) energy terms. Layered geological structures, especially reservoir bearing structures, have typically larger lateral extents compared to their thickness, and can be considered in a first approximation as thin plates regarding their mechanical behavior. Moreover, during sedimentation the top of the sedimentary pile can be generally considered as smooth developable surfaces in the depositional space, which are then deformed during their burial history under tectonic events. This idea is used to suggest a method for identifying the probability of finding sub-seismic faults in thin geological structures or reservoirs. This paper presents theoretical results that relate the curvatures of the top or bottom surfaces of geological structures and reservoirs. Bending and stretching energy terms are used as structural attributes to predict fracturing or the deformation style.
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A Gabriel-Delaunay Triangulation of Complex Fractured Media for Multiphase Flow Simulations
By H. MustaphaFractured reservoirs are complex domains where discrete fractures are constraining boundaries. The discrete fractures are discretized into intersected edges during a grid generation process. Delaunay triangulations are often used to represent complex structures. However, a Delaunay triangulation of a fractured medium generally does not conform to the fracture boundaries. Recovering the fracture elements may violate the Delaunay empty-circle (2D) criterion. Refining the triangulation is not a practical solution in complex fractured media. This paper presents a new approach that combines both Gabriel and Delaunay triangulations. The Gabriel condition of edge-empty-circle is locally employed to quantify the quality of the fracture edges in 2D. The fracture edges violating the Gabriel criterion are released in a first stage. After that, a Delaunay triangulation quality is generated considering the rest of the fracture constraints. The released fracture edges are then approximated by the edges of the Delaunay triangles. The final representation of fractures might be slightly different, but a very accurate solution is always maintained. The method is near optimal and has the capability to generate fine grids and to offer an accurate good-quality grid. Numerical examples are presented to assess the performance and efficiency of the proposed method.
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Numerical Prediction of Relative Permeability in Water-Wet Naturally Fractured Reservoir Rocks
Authors S.K. Matthai, S. Bazrafkan, P. Lang and C. MilliotteThe grid-block scale ensemble relative permeability, kri of fractured porous rock with appreciable matrix permeability is of decisive interest to reservoir simulation and the prediction of production, injector-producer water breakthrough, and ultimate recovery. While the dynamic behaviour of naturally fractured reservoirs (NFR) already provides many clues about (pseudo) kri on the inter-well length scale, such data are difficult to interpret because, in the subsurface, the exact fracture geometry is unknown. Here we present numerical simulation results from discrete fracture and matrix (DFM) unstructured grid hybrid FEM-FVM simulation models, predicting the shape of fracture-matrix kri curves. In contrast to earlier work (Matthai et al. 2007, Nick and Matthai, 2011), we also simulate capillary fracture matrix transfer (CFMT) and without relying the frequently made simplifying assumption that fracture saturation reflects fracture-matrix capillary pressure equilibrium. We also use a novel discretization of saturation which permits jump discontinuities to develop across the fracture-matrix interface. This increased physical realism permits – for the first time - to test the Matthai and Nick (2009) semi-analytical model of the flow rate dependence of relative permeability, ensuing from CFMT. The sensitivity analysis presented here constrains the CMFT-related flow rate dependence of kri and illustrates how it manifests itself in two geometries of layer-restricted well-developed fracture patterns mapped in the field. In a companion paper (Lang et al.), also investigate the dependence of kri on fracture aperture as computed using discrete element analysis for plausible states of in situ stress. Our results indicate that fracture-matrix ensemble relative permeability is matched – for fast flow rates – by the semi-analytic model of Matthai and Nick (2009). For slow rates the strong impact of CFMT leads to significantly different behaviour requiring a more elaborate treatment.
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Flows in Discrete Fracture Networks: from Fine Scale Explicit Simulations to Network Models and Reservoir Simulators
Authors B. Noetinger, M.D. Delorme, A.F. Fourno and N.K. KhvoenkovaModelling flows in fractured reservoirs is becoming essential, due to the increasing number of fractured reservoirs to be exploited. Building fluid flow simulations keeping explicit Discrete Fracture Network (DFN) models that capture well the highly localized nature of flow in fractured reservoirs is a challenging issue. A successful solution will be of considerable help for setting up EOR schemes. A rigorous workflow handling 3D DFN simulations to standard large scale simulations must be set up. We show that it is possible to build an exact approximation scheme using an original Galerkin projection technique and a quasi steady state approximation (simulation time greater that a typical diffusion time over one fracture). At the lowest order, the resulting set of equations to be solved has the structure of a resistor/capacitor network The associated mass and transmissibility matrices can be computed explicitly solving steady state boundary value Laplace equations over each fracture. Considering millions fracture models remains impossible in this context. Using geometrical considerations, we have developed accelerated algorithms allowing to treat such cases in an acceptable time on a standard computer. Validations were done with high resolution reference calculations. The theoretical aspects and validation tests will be adressed during the presentation.
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Single Porosity Model for Fractured Formations
Authors P.YU. Tomin and A.K. Pergamentdeveloped. Analogous to work of G. Dagan & P. Indelman, the energy criterion is used for upscaling of absolute permeability. The fine-scale energy equality to approximated value corresponding to tensor coefficients is required for cells containing fractures. The resulting effective tensor is symmetric and physically consistent since the flux approximation is assured. Two classes of methods are applied to determine the pseudo relative permeability tensor. First one is the stationary capillary equilibrium method which is applicable in capillary trapping zones far from wells. Furthermore, analysis of relations between phase and absolute permeability tensors is carried out using this method. Samples of relative permeability curves are obtained for media with orthotropic and monocline symmetries. The influence of connectivity property on the functions is shown and the saturation dependence of direction of principal axes for phase permeability tensor is investigated. Thereby the misalignment of phase and absolute permeability tensors is shown. The second class is a dynamic pseudo-function approach which uses the multiscale method for water flooding simulation. The method combines the Fedorenko finite superelement method and the Samarskii support operator method and belongs to the high-resolution methods class. The technique developed allows to incorporate fractures of complex geometry, accurately accounts the anisotropy for two-phase flows, and as opposed to dual parameters model doesn’t require the connectivity of fractures system and avoids doubling the number of unknowns. The method is successfully applied for simulation of the China and West Siberia fractured reservoirs.
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Diagnosis and Quantification of Stress-Sensitive Reservoir Behaviour from Pressure and Rate Transient Data
By R.A. ArcherClassical analytical and numerical techniques for simulation of fluid flow in petroleum reservoirs typically assume permeability is independent of pressure. In naturally fractured and low permeability systems the reservoir permeability may depend on the stress state of the reservoir which means the diffusivity equation that governs single phase flow in the reservoir becomes nonlinear. Stress-sensitive behaviour is particularly relevant to the development of tight gas and other unconventional resources. This work develops a set of tools to diagnose and quantify stress sensitivity through analysis of transient pressure or flow rate data. The work builds on analytical solutions for radial flow in a stress-sensitive medium presented by Friedel and Voigt (SPE 122768, 2009), and for the linear flow case presented by Archer (AFMC 17, 2010). The radial flow solution uses the Boltzmann transform whereas the linear flow solution is based on the use of the Cole-Hopf transform. High resolution numerical solutions are also used to complement these analytical solutions. Where appropriate pseudo-pressures are used to take account of the pressure dependence of gas properties on pressure. This paper considers both transient pressure and rate solutions and develops a range of type curve formats to demonstrate how production from stress-sensitive reservoirs differs from conventional reservoirs when plotted in traditional well test format (log-log plot of pressure and pressure derivative), as a p/z plot (for the gas case), as a rate versus cumulative plot, and as “Blasingame” type curves in the including the normalised rate, rate-integral, and rate-integral-derivative formats. This suite of tools can be used in a diagnostic manner to identify whether stress-sensitive behaviour is occurring, to quantify the errors that may be made in permeability estimates if stress-sensitive behaviour is ignored, and to estimate the impact of stress-sensitivity on ultimate recovery from a well.
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A Spectal Approach to Conditional Simulation
Authors I.R. Minniakhmetov and A.H. Pergamentcovariance matrix representing grid point’s correlation. For the large fields the Cholesky factorization can be computationally expensive. In this work we present an alternative approach, based on the usage of spectral representation of a conditional process. It is shown that covariance of two arbitrary spectral components could be factorized into functions of corresponding harmonics. In this case the Cholesky decomposition could be considerably simplified. The advantage of the presented approach is its accuracy and computational simplicity.
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Quantitative Use of Different Seismic Attributes in Reservoir Modeling
Authors T. Feng, J. Skjervheim and G. EvensenAccurate reservoir models are essential for reservoir management. Optimal use of all available data is crucial. Traditionally, reservoir properties have been conditioned to the dynamic production data from the wells. Seismic data, on the other hand, is only used in a qualitative manner. Quantitative use of seismic data is sparse and research based. To use seismic data quantitatively in the reservoir-modeling process, an integrated workflow need to be established such that the forward modeling of the synthetic seismic data and the preferable measured seismic data can be incorporated in the conditioning process. The different modeling regimes, such as reservoir flow simulation, rock physics, and seismic wave propagation, are involved in getting from reservoir flow properties to seismic signals. Hence, different seismic attributes from different levels can be used in the conditioning process. In this work, our focus is to test and demonstrate an integrated workflow for quantitative use of different seismic attributes in history matching. The history matching concept will be formulated in a Bayesian setting through ensemble based algorithms. The uncertainty of model is represented with an ensemble of realizations. A field case study is used to demonstrate the importance of different seismic attributes in the conditioning process.
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Using Two-point Geo-statistics Reservoir Model Parameters Reduction
By J. LeguijtUsing two point geo-statistics reservoir model parameters reduction. An algorithm has been developed to constrain gridded reservoir models that are used with assisted history matching with geo-statistical information and at the same time reduce the number of variables that are needed to describe the model. Gridded models, as used within most reservoir modelling packages, may consist of 10^5 up to 10^6 grid blocks. A covariance matrix which can be used to constrain the model with a variogram (two point statistics) would consist of 10^10 up to 10^12 coefficients and a direct principal component decomposition of is beyond the capability of current computer systems. A common way to reduce the number of variables is using the members of an ensemble of models from a geo-statistical simulation as basis vectors for a subspace. When a history match is obtained with a model that is constrained to this subspace, this model will have a decently looking continuity behaviour. There is however no guarantee that this subspace contains the directions that correspond with the eigenvectors of the covariance matrix with the largest eigenvalues. This can be demonstrated with a simple simulation and is theoretically described by the Wishart distribution. It is possible to construct a set of orthonormal basis vectors that contains the directions that correspond with the eigenvectors of the covariance matrix with the significantly large eigenvalues. The number of basis vectors may still be rather large but it is mainly determined by the size of the model and the range of the variogram. From an eigenvector decomposition of this covariance matrix, a very good approximation can be obtained of the eigenvectors with a significant large eigenvalues. As the small eigenvalues can be neglected, the number of eigenvectors needed to describe the model is approximately 10^2, which results in a significant parameter reduction.
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Numerical Comparison of Ensemble Kalman Filter and Randomized Maximum Likelihood
Authors K. Fossum, T. Mannseth, D. Oliver and H.J. SkaugIn recent years, more traditional history matching methods have been increasingly challenged by sequential data assimilation techniques such as the ensemble Kalman filter (EnKF). There are strong similarities between EnKF and the non-sequential method, randomized maximum likelihood (RML). For a linear forward model the two methods are equal, for a nonlinear forward model there arises some differences (in addition to sequential/batch data assimilation): RML can be iterative, while EnKF is not; RML uses realization-specific gradients/sensitivities to change a model realization while EnKF uses the same covariance for all realizations. We assess the sampling capabilities of RML and EnKF for a weakly nonlinear forward model. Results are compared to a Markov chain Monte Carlo (McMC) method, which samples correctly from the posterior. Our aim is to clarify which of the above mentioned differences between RML and EnKF has the biggest impact on the sampling capabilities. We apply the methods to a two-phase reservoir models small enough to be suitable for McMC. The assessment of RML and EnKF is performed by comparing history matching capabilities, and properties of their posterior distributions to those of the posterior distributions obtained with McMC.
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Smooth Multi-scale Parameterization for Integration of Seismic and Production Data Using Second-generation Wavelets
Authors T. Gentilhomme, T. Mannseth, D. Oliver, G. Caumon and R. MoyenIn this paper, we use the second-generation wavelet transform as multi-scale smooth parameterization technique for history matching of seismic derived models using an ensemble based optimization method (batch-enRML). The construction of the second generation wavelet is presented and their advantages compared to first generation wavelets are discussed. Then, these wavelets are applied to a realistic 3D faulted reservoir model. Their ability to represent correctly this model with a large compression ratio is demonstrated. Finally, using the SGW re-parameterization, we set the basis for a new adaptive multi-scale inversion method, which aims at limiting the increase of the mismatch to seismic data of the seismic-derived realizations by selecting relevant parameters. Efficiency of the method is discussed through a 2D synthetic example.
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Distance Parameterization for Efficient Seismic History Matching with the Ensemble Kalman Filter
Authors O. Leeuwenburgh and R. ArtsThe Ensemble Kalman Filter (EnKF), in combination with travel-time parameterization, provides a robust and flexible method for quantitative multi-model history matching to time-lapse seismic data. A disadvantage of the parameterization in terms of travel-times is that it requires simulation of models beyond the update time. A new distance parameterization is proposed for fronts, or more generally, for isolines of arbitrary seismic attributes, that circumvents the necessity of additional simulation time. An accurate Fast Marching Method for solution of the Eikonal equation in Cartesian grids is used to calculate distances between observed and simulated fronts which are subsequently used as innovations in the EnKF. Experiments are presented that demonstrate the functioning of the method in synthetic 1D and 2D cases that include uncertain model properties, and merging or multiple secondary fronts. Results are compared with those resulting from direct use of saturation data. The proposed algorithm significantly reduces the number of data while still capturing the essential information, it removes the need for seismic inversion when the oil-water front is identified only, and it produces a more favorable distribution of simulated data, leading to improved functioning of the EnKF.
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Preventing Ensemble Collapse and Preserving Geostatistical Variability Across the Ensemble with the Subspace EnKF
More LessOne of the key issues of the EnKF is the well known problem of ensemble collapse, which is particularly evident for small ensembles. This results in an artificial reduction of variability across the ensemble. The second, more important problem is that the EnKF is theoretically appropriate only if all ensemble members belong to the same multi-Gaussian random field (geological/geostatistical model). This is an important problem because for most real fields, we have more than one geological scenario, and ideally, we would like to obtain one or more history-matched models for each geological scenario. In this work, we propose the subspace EnKF to alleviate both problems mentioned above. The basic idea of the subspace EnKF is to constrain the different ensemble members to different subspaces of the same or different random field. This is accomplished by parameterizing the random fields and modifying the EnKF formulation with the gradients of the parameterizations. The subspace EnKF prevents ensemble collapse, providing a better quantification of uncertainty, and more importantly, retains key geological characteristics of the initial ensemble, even when each ensemble member belongs to a different geological model. The approach is demonstrated on a synthetic example with a multi-Gaussian permeability field.
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Multi-objective Scheme of Estimation of Distribution Algorithm for History-Matching
Authors A. Abdollahzadeh, A. Reynolds, M. Christie, D. Corne, G. Williams and B. DaviesHistory matching is one of the key challenges of efficient reservoir management. In history matching, evolutionary algorithms are used to explore the global parameter search space for multiple good fitting models. General critiques of these algorithms include high computational demands, as well as low diversity of multiple models. Estimation of distribution algorithms are a class of evolutionary algorithms in which new candidate solutions are obtained by sampling a probability distribution created from the population. In previous works, we studied estimation of distribution algorithms for history matching and showed that good results can been obtained by using a single misfit function. Multiobjective optimisation algorithms use the concepts of dominance and the Pareto front to find a set of optimal trade-offs between the competing objectives of minimising misfit. In this paper, we apply a multiobjective estimation of distribution algorithm to history matching of firstly a well-known synthetic reservoir simulation model and secondly a real North Sea reservoir. We will show that one can achieve higher solution diversity and in some cases better quality solutions by taking multiple objectives. In addition, multiobjective optimisation algorithms are less sensitive to parameter tuning and provide trade-offs between objectives that give more insights into history matching problem.
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Data Assimilation Using the EnKF for 2-D Markov Chain Models
Authors Y. Zhang, D.S. Oliver, Y. Chen and H.J. SkaugThe ensemble Kalman filter (EnKF) is well-suited to update gaussian variables and can be used for updating continuous nongaussian variables either directly or after transformation. Categorical variables such as facies type are much more difficult for history matching, especially when the variables have complex transitional dependencies. In a previous paper we described a method for updating third order Markov chain models in one dimension using the ENKF, where its efficiency partially depends on the Viterbi algorithm that is not directly applicable in higher dimensions. In this paper, we develop a data assimilation method for updating categorical models using an approximation to the joint probability of facies types (Allard et al 2011) that can be used in a sequential algorithm without iteration. The ensemble of realizations after updating can be used to efficiently approximate the likelihood of the variables, while the categorical model provides an approximation to the transition probabilities. We demonstrate the approach with conditioning two synthetic channel models with two facies types to both linear and nonlinear observations. Our results show the distribution of facies after data assimilation honors data much better than before assimilation, and the transitions among facies are consistent with the prior model.
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An Iterative Version of the Adaptive Gaussian Mixture Filter
Authors A.S. Stordal and R. LorentzenThe adaptive Gaussian mixture filter (AGM) was introduced as a robust filter technique for large scale applications and an alternative to the well known ensemble Kalman filter (EnKF). The bias of AGM is determined by two parameters, one adaptive weight parameter and one predetermined bandwidth parameter which decides the size of the linear update. The bandwidth parameter must often be selected significantly different from zero in order to make large enough linear updates to match the data, at the expense of bias in the estimates. In the iterative AGM we introduce here we take advantage of the fact that the history matching problem is usually estimation of parameters. If the prior distribution of parameters is close to the posterior distribution, it is possible to match the observations with a small bandwidth parameter. Hence the bias of the filter solution is small. In order to obtain this scenario we iteratively run the AGM throughout the data history with a very small bandwidth to create a new prior distribution from the updated samples after each iteration. After a few iterations, nearly all samples from the previous iteration match the data and the above scenario is achieved.
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Ensemble Kalman Filter Data Assimilation to Condition a Real Reservoir Models to Well Test Observation
By A. AbadpourRecently a significant effort has been made to characterize reservoir models benefiting from Ensemble Kalman filter as data assimilation technique. EnKF proved to be a powerful tool to deal with almost any sort of measurement also to be capable of handling different type of uncertainty in the simulation models and and being affordable from the computational point of view. Lately the technique has been deployed to assimilate on pressure transient and production logging data to update permeabilities and estimate layer skin factor. In the present paper EnKF methodology was used to characterize an offshore reservoir model against the well test pressure data as well as the pressure derivative to adjust cell by cell petrophysical properties, and the skin factor in each well perforation. The results showed that using the derivative observations to calibrate the uncertain parameters helps improving the quality of the match not only in the predicted derivative but also in better forecasting the pressure measurements. The importance of assimilation on skin as well as recalculation of well connection factors revealed. Moreover a new distance based localisation scheme based on the well drainage zone has been introduced to help reducing unnecessary changes in the model.
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New Formulation of the Objective Function for Better Incorporation of 4D Seismic Data into Reservoir Models
Authors R. Derfoul, S. Da Veiga, C. Gout and C. Le GuyaderTo build consistent reservoir models, 4D seismic data are an invaluable source of information on fluid displacements and geology over extensive areas of the reservoir. In this paper, we focus on the integration of such data to improve the obtained optimal model in a history matching process. However, this is a challenging task involving a proper definition of the objective function. The objective function computes the discrepancy between observed data and responses computed by the reservoir model. Classical formulations based on the least square mismatch are not adapted to deal with complex, noisy and numerous data such as 4D inverted seismic data. In this paper, we study the integration of seismic data in order to improve the optimal model obtained by the history matching process. The main focus of this paper is to define an experimental methodology to compare and classify seismic matching methods. In particular, we propose an efficient algorithm which focuses on the main trends in a seismic cube. This new algorithm is investigated in the context of seismic data, and its potential is demonstrated on several history matching reservoir examples.
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Time Lapse Inversion Workflow Constrained by Reservoir Grid Parameterization
By P. ThoreTime lapse seismic provides key information for assisted history matching. Qualitatively geo-bodies extracted from 4D data reflects the front for a given flow event (e.g. the water flood due to an injector). But 4D data can also provide quantitative information relative to dynamic parameters. We propose a novel workflow for quantitative use of geophysical data for AHM which consists in three steps: 1. Model-Based-Inversion which keeps the layer parameterization of the reservoir grid. This parameterization introduces high and low frequencies missing in the seismic bandwidth. 2. Pressure and saturation inversion constrained by dynamic information and handling uncertainty on data and model. 3. Direct mapping of seismically derived information into the reservoir grid (without using any time to depth conversion). That solves the main problem inherent with the vertical change of support from the (regular) seismic grid to the (irregular layer-based) reservoir grid. Our paper is illustrated with real data examples.
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Optimal Choice of a Surveillance Operation Using Information Theory
Authors A.C. Reynolds and D.H. LeWe consider the problem of choosing among a suite of potential reservoir surveillance operations. We frame the problem in terms of two questions: (1.) Which surveillance operation is the most useful? (2.) What is the expected value of the reduction in uncertainty in the reservoir variable J (e.g. cumulative oil production) that would be achieved if we were to conduct each surveillance operation to collect and history-match the data obtained? Note that the objective is to answer these questions with an uncertain reservoir description and without any actual measurements. We propose a procedure based on information theory to answer these questions. Question 1 is answered by calculating the mutual information between J and the vector of observed data. Question 2 is answered by estimating the expected value of the standard deviation (or P90-P10) of J in the posterior model from the conditional entropy of J. We apply the proposed method to two simple problems, a nonlinear toy problem and a simple water flooding problem. The results are verified by an exhaustive history matching procedure, which is reasonably rigorous but very computationally demanding. We find that the mutual information approach is a fast and reliable alternative to the history matching approach.
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Application of the Adaptive Gaussian Mixture Filter to History Match a Real Field Case
Authors R. Valestrand, G. Nævdal and A.S. StordalOver the last decade the ensemble Kalman filter (EnKF) has attracted attention as a promising method for solving the reservoir history matching problem: Updating model parameters so that the model output matches the measured production data. The method possesses unique qualities such as; it provides real time update and uncertainty quantification of the estimate, it can estimate any physical property at hand. The method does, however, have its limitations; in particular it is derived based on an assumption of a Gaussianity. A recent method proposed to improve upon the original EnKF method is the Adaptive Gaussian mixture filter (AGM). The AGM loosens up the requirements of a linear and Gaussian model by making smaller linear updates and including importance weights associated with each ensemble member at computational costs as low as EnKF. In this paper we present results where the AGM algorithm is combined with localization. To validate the performance of AGM the result is compared with the EnKF, with and without localization. From the results, we are able to distinguish the performance of the different filters. In particular all the methods provide good history match, but we see that the AGM stands out by better honoring the original geostatistics.
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Neural Networks and their Derivatives for History Matching and other Seismic, Basin and Reservoir Optimization Problems
Authors J. Bruyelle and D.R. GuérillotDescription: In geosciences, complex forward problems met in geophysics, petroleum system analysis and reservoir engineering problems often requires replacing these forward problems by proxies, and these proxies are used for optimizations problems. For instance, History Matching of observed field data requires a so large number of reservoir simulation runs (especially when using geostatistical geological models) that it is often impossible to use the full reservoir simulator. Therefore, several techniques have been proposed to mimic the reservoir simulations using proxies. Due to the use of experimental approach, most of authors propose to use second order polynomials. In this paper we demonstrate that: (1) Neural networks can also be second order polynomials. Therefore, the use of a neural network as a proxy is much more flexible and adaptable to the non linearity of the problem to be solved; (2) First order and second order derivatives of the neural network can be obtained providing gradients and hessian for optimizers. For the first point, a complete description of a neural network equivalent to a second order polynomial will be given. For inverse problems met in seismic inversion, well by well production data, optimal well locations, source rock generation, etc., most of the time, gradient methods are used for finding an optimal solution. The paper will describe how to calculate these gradients from a neural network built as a proxy. When needed, the hessian can also be obtained from the neural network approach. Application: On a real case study, the ability of neural networks to reproduce complex phenomena (water-cuts, production rates. etc.) is showed. Comparisons with second polynomials (and kriging methods) will be done demonstrating the superiority of the neural network approach as soon as non linearity behaviors are present in the responses of the simulator. The gradients and the hessian of the neural network will be compared to those of the real response function. Results and conclusions: (1) Neural Network can replace advantageously polynomial and kriging approaches as proxies for inverse problems and uncertainty analysis, (2) A neural network giving a bilinear polynomial will be explicitly given, (3) Gradients and Hessian of neural network can be calculated and use by optimizers. Keywords: Proxies, History Matching, Gradient Methods, Optimizers, Basin Modelling, Seismic Inversion, Uncertainty Analysis, Hessian
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North Sea Chalk Reservoir – Seismic History Matching Workflow
Authors H. Sudan, E. Tolstukhin and A. JanssenThis presentation outlines an integrated workflow that incorporates 4D seismic data into the North-Sea Chalk Reservoir history matching process. Successful application and associated benefits of the workflow process are also presented. A number of 4D seismic surveys have been acquired over this field between 1989 and 2008 and this data is becoming a quantitative tool for describing the spatial distribution of reservoir properties and compaction. The seismic monitoring data is used to optimize the waterflood by providing water movement insights and subsequently improve infill well placement. Reservoir depletion and water injection in this field lead to rock compaction and fluid substitution. These changes are revealed in space and time through 4D seismic differences. Inconsistencies between predicted 4D differences (calculated from reservoir model output) and actual 4D differences are therefore used to identify reservoir model shortcomings. This process is captured using the following workflow: prepare and upscale a geologic model; simulate fluid flow and associated rock-physics using a reservoir model; generate a synthetic 4D seismic response from fluid and rock-physics forecasts; and update the reservoir model to better match actual production data and 4D seismic observations. The above-mentioned Seismic History Matching (SHM) workflow employs rock-physics modeling to quantitatively constrain the reservoir model and develop a simulated 4D seismic response. Different parameterization techniques and seismic misfit formulations were validated and used to calibrate and update the reservoir model. This workflow updates the parameters in the closed loop system through minimization of a misfit function by using a customized Particle Swarm Optimization Algorithm. In summary, the Seismic History Matching workflow is a multi-disciplinary process that requires strong collaboration between geological, geomechanical, geophysical and reservoir engineering disciplines to optimize reservoir management.
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Efficient Solution of the Optimization Problem in Model-reduced Gradient-based History Matching
Authors S. Szklarz, M. Rojas and M. KaletaAdjusting parameters in reservoir models by minimizing the discrepancy between the model's predictions and actual measurements is a popular approach known as history matching. One of the most effective techniques is gradient-based history matching. For reservoir models, the number of grid blocks and therefore, the size of the problem can become very large. In recent years, model-order reduction techniques aiming to replace large, complex dynamic systems with lower-dimension models have been incorporated into history matching. In both gradient-based history matching and model-reduced gradient-based history matching, first-order optimization methods are used in order to minimize the mismatch between simulated well-production data and observed production. In this work, we investigate the performance of some optimization methods on the minimization problem in model-reduced gradient-based history matching. The methods were tested on the history matching of a small reservoir model with synthetic measurements. Our results show that fast first-order techniques such as the spectral projected gradient method can compete with the popular quasi-Newton BFGS approach.
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Deterministic Linear Bayesian Updating of State and Model Parameters
Authors O. Pajonk, B.V. Rosić and H.G. MatthiesBayesian estimation has become an important topic for inverse problems in the context of hydrocarbon recovery. The conceptual and computational advantages due to direct integration with uncertainty quantification workflows are appealing. Especially, linear Bayesian techniques like the ensemble Kalman filter (EnKF) have been successfully used in numerous cases. However, such techniques have difficulties in some applications which are often caused by sampling errors, a limited ensemble size, or the sometimes large number of required samples. In this work we present and discuss a closely related linear Bayesian technique which is based on orthogonal expansions of the stochastic spectrum of the involved random variables and random fields. Basically being a family of fully deterministic implementations of the well-known projection theorem of Hilbert spaces, the technique is conceptually simple, yet powerful. Since they are fully deterministic, these methods avoid all sampling errors. First combined parameter and state estimation results with a low-dimensional chaotic model are presented, using a specific choice of orthogonal expansion. These are compared to results obtained with EnSRF, since it is a close relative to these spectral estimation methods. Challenges and opportunities for applications to the inverse problem of identification for hydrocarbon reservoirs are discussed.
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A New Global Upscaling Technique for 3D Unstructured Grids
Authors M. Karimi-Fard and L.J. DurlofskyNew procedures for unstructured coarse-model generation are presented and applied. The underlying fine-grid model is considered to be unstructured, and the coarse-model cells are defined as groupings of fine-grid cells. The key flow quantity that must be computed for the coarse model is the upscaled transmissibility for each cell-to-cell connection. We introduce a global upscaling procedure for this computation. The method first requires several (minimum of three) global single-phase flow solutions. Appropriately defined linear combinations of these solutions are used to compute each upscaled transmissibility. This approach circumvents some of the limitations of existing (local and global) upscaling procedures. It also enables transmissibility to be quickly computed for a number of different coarse grids without performing any additional pressure solutions. Results are presented for an idealized two-phase flow problem. The fine grid contains nearly 200,000 cells, and coarse models of varying resolution are considered. Accurate results for total injector-producer flow rate are observed for all grid-resolution levels for the three different well configurations considered. Oil rate as a function of time is shown to improve in accuracy with increasing resolution, and is quite accurate for a model of about 10,000 cells.
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Grid Adaption for Upscaling and Multiscale Method
Authors K.-A. Lie, J.R. Natvig, S. Krogstad, Y. Yang and X.H. WuA Dirichlet-Neumann representation method was recently proposed for upscaling. The method expresses coarse fluxes as linear functions of multiple discrete pressure values along the boundary and at the center of each coarse block. The number of pressure values can be adjusted to improve the accuracy of simulation results, and in particular to resolve important fine-scale details. Improvement over existing approaches is substantial especially for reservoirs that contain high permeability streaks or channels. Multiscale methods obtain fine-scale fluxes or pressures at the cost of solving a coarsened problem, but can also be utilized for flexible upscaling. We compare the DNR and a multiscale mixed finite-element method. Both can be expressed in mixed form, with local stiffness matrices obtained as inner products of basis functions with fine-scale subresolution determined from local flow problems. Piecewise linear Dirichlet boundary conditions are used for DNR and piecewise constant Neumann conditions for MsMFE. Adding discrete pressure points in the DNR method corresponds to subdividing coarse faces and hence increasing the number of basis functions in the MsMFE method. The methods show similar accuracy for 2D Cartesian cases, but the MsMFE method is more straightforward to formulate in 3D and implement for general grids.
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Reduced-order Modeling for Thermal Recovery Processes
Authors M.A.H. Rousset, C.K. Huang, H. Klie and L.J. DurlofskyThermal recovery typically entails higher costs than conventional oil recovery, so the application of computational optimization techniques may be beneficial. Optimization, however, requires many simulations, which incurs substantial computational cost. Here we apply a model-order reduction technique, which aims at large reductions in computational requirements. The technique considered, trajectory piecewise linearization (TPWL), entails the representation of new solutions in terms of linearizations around previously simulated (and saved) training solutions. The linearized representation is projected into a low-dimensional space, with the projection matrix constructed through proper orthogonal decomposition of solution `snapshots' generated in a training step. We consider two idealized problems, specifically primary production of oil driven by downhole heaters, and a simplified model for steam assisted gravity drainage, where water and steam are treated as a single `effective' phase. The strong temperature dependence of oil viscosity is included in both cases. TPWL test-case results for these systems demonstrate that the method can provide accurate predictions relative to full-order reference solutions. The overhead associated with TPWL model construction is equivalent to the computation time for several full-order simulations (the precise overhead depends on the number of training runs). Observed runtime speedups are very substantial -- over two orders of magnitude.
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Enabling Optimal Production Strategies under Uncertainties via Non-Intrusive Model Reduction Methods
Authors H. Klie, H. Chen, Q. Wang and K. WillcoxThe present work proposes an alternative approach to generate nonlinear reduced order models for optimization and control under uncertainty without explicit knowledge of all the equations governing the physics of the simulation. Hence, the proposed method is amenable for legacy simulation codes. In order to cope with the lack of physical information in conjunction with the inherent curse of dimensionality associated with the number of parameter coefficients, control and state variables of the problem, we combine the projection operators obtained from the Proper Orthogonal Decomposition with neural net interpolation. In this way, the proposed Black-Box Stencil Interpolation Method (BSIM) is capable of exploiting both spatial and temporal variable locality. The method can be seen as a competitive but non-intrusive alternative to the Trajectory Piece-Wise Linear method and the Discrete Empirical Interpolation Method (DEIM) both recently proposed in the literature. We illustrate the capabilities of BSIM on a suite of different black-oil and compositional field models subject to multiple well controls under geological uncertainty. We show that the results are comparable in accuracy to DEIM despite the non-intrusive character of BSIM.
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Reservoir Management Using Two-stage Optimization with Streamline Simulation
Authors T. Wen, M.R. Thiele, D. Echeverría Ciaurri, K. Aziz and Y. YeWaterflooding is a common secondary oil recovery process. Performance of waterfloods in mature fields with a significant number of wells can be improved with minimal infrastructure investment by optimizing injection/production rates of individual wells. However, a major bottleneck in the optimization framework is the large number of reservoir flow simulations often required. In this work we propose a new method based on streamline-derived information that significantly reduces these computational costs in addition to making use of the computational efficiency of streamline simulation itself. We seek to maximize the long-term net present value of a waterflood by determining optimal individual well rates, given an expected albeit uncertain oil price and a total fluid injection volume. We approach the optimization problem by decomposing it into two stages which can be implemented in a computationally efficient manner. The two-stage streamline-based optimization approach can be an effective technique when applied to reservoirs with a large number of wells in need of an efficient waterflooding strategy over a 5 to 15 year period.
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Response Surface Approaches for Large Decision Trees: Decision Making Under Uncertainty
By H. GrossTraditionally, the connection between simulation and decision analysis is done by using simulation outputs as inputs to decision algorithms. We propose to use simulation input uncertainties directly in decision algorithms by extending existing probabilistic reservoir simulation tools (experimental design, proxy models), and existing decision analysis tools (decision trees, Pareto fronts). This approach addresses questions on field development options under uncertainty (facility sizing, completion decisions or data collection campaigns). When linking probabilistic simulation with decision analysis, three practical problems arose. First, the number of reservoir uncertainties creates huge decision trees. We solve this problem by creating composite solutions, with some branches evaluated exhaustively, and others evaluated with calibrated response surfaces. Then, assumption of independence between uncertainties, often encountered, was too restrictive for practical uses. We thus specify probabilities on all uncertainty branches. Last, we must handle multiple decision drivers and understand the consequences of decisions on several metrics. We have therefore implemented multi-objective optimization capabilities. The technique developed here extends beyond the capability of existing decision analysis and uncertainty quantification tools. Its practical value is demonstrated on two field problems, and proves useful to identify optimal decision paths.
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A Workflow for Decision Making Under Uncertainty
Authors D. Busby, S. Da Veiga and S. TouzaniWe propose a workflow for decision making under uncertainty aiming at comparing different development plan scenarios under uncertainty. The approach applies to mature fields where the residual uncertainty is estimated using a probabilstic inversion approach. Moreover a robust optimization method is discussed to optimize controllable parameters in the presence of uncertainty. The key elements of this approach are the use of response surface models to reduce the very high number of simulator model evaluations needed. To build efficient and reliable response surfaces for this application we discuss an experimental design method for correlated input variables where the correlation is induced by the probabilistic inversion process. For the problem of optimization under uncertainty an iterative approach is proposed aiming at refining the response surface iteratively such as to reduce effectively approximation errors and converging faster to the true solution. The workflow is illustrated on a realistic test case of a mature field where the approach is used to compare two new development plan scenarios both in terms of expectation and of risk mitigation and to optimize well position parameters in the presence of uncertainty.
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Estimation of Production Rates Using Transient Well Flow Modeling and the Auxiliary Particle Filter
Authors R. Lorentzen, A.S. Stordal, G. Nævdal, H.A. Karlsen and H.J. SkaugImproved recovery of oil from existing petroleum fields is increasingly important. A better representation of production zone information leads to better flowrate control and reservoir management. In order to achieve this, it is possible to utilize the fact that smart wells with multiple zones and laterals are more common, and they may be equipped with permanent instrumentation and control. Today, accurate flowrate measurements or estimates for each zone are lacking, and existing tools are often limited to steady-state models with no uncertainty analysis. Here we combine a transient well flow model and estimation techniques, into a tool for interpretation of wellbore measurements. The estimation technique applied here is the auxiliary sequential importance resampling (ASIR) filter, which has the advantage of being more robust than the traditional particle filter (PF). The ASIR filter is used to tune the output of specific stochastic models of the flowrates. To do this tuning we have chosen a regime type model for the flowrates. More specifically, the model implies that the flowrate process changes structure governed by an underlying Markov jump process. Using this type of models makes us capable of capturing both smooth transitions as well as more abrupt changes of the flowrates.
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Generalized Field Development Optimization: Coupled Well-Placement and Control under Geologic Uncertainty
Authors B. Jafarpour and L. LiWell placement optimization is often formulated as an integer-programming problem and is typically carried out assuming known well control settings. Similarly, finding optimal well controls is usually formulated and solved as a control problem in which the well locations are fixed. Solving each problem independently without accounting for the coupling between them leads to suboptimal solutions. We propose to solve the coupled well placement and control optimization problems for improved production performance. We present two alternative methods: i) sequential solution of the decoupled well placement and control subproblems where each subproblem is resolved after updating the decision variables of the other subporoblem from the previous step; ii) simultaneous solution by concurrently changing well locations and controls during the iterations using a generalized stochastic approximation simultaneous perturbations algorithm. The first approach allows for application of well-established methods in the literature to solve each subproblem individually while the second approach requires development of new methods to solve mix-integer optimization problems. We consider field development optimization under geologic uncertainty and discuss computationally efficient approximate solution techniques for robust optimization under ensemble model representations. Several numerical experiments with the PUNQ and a layer of the SPE10 benchmark models demonstrate the applicability of these methods.
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A Derivative-Free Methodology with Local and Global Search for the Joint Optimization of Well Location and Control
Authors O.J. Isebor, L.J. Durlofsky and D. Echeverría CiaurriIn oil field development, the optimal location for a new well depends on how it is to be operated. Thus, it is generally suboptimal to treat the well location and well control optimization problems sequentially. Rather, they should be considered as a joint problem. In this work, we present noninvasive, derivative-free, easily-parallelizable procedures to solve this joint optimization problem. Specifically, we consider Particle Swarm Optimization (PSO), a heuristic global stochastic search algorithm, Mesh Adaptive Direct Search (MADS), a local search procedure, and a hybrid PSO-MADS technique that combines the advantages of both methods. Nonlinear constraints are handled through use of filter-based treatments that seek to minimize both the objective function and constraint violation. We also introduce a formulation to determine the optimal number of wells, in addition to their locations and controls, by associating a binary variable (drill/do not drill) with each well. Example cases of varying complexity, which include bound constraints, nonlinear constraints, and the determination of the number of wells, are presented. The PSO-MADS hybrid procedure is shown to consistently outperform both standalone PSO and MADS when solving the joint problem. The joint approach is also observed to provide superior performance relative to a sequential procedure.
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Well Placement Optimization under Uncertainty with CMA-ES Using the Neighborhood
Authors Z. Bouzarkouna, D.Y. Ding and A. AugerIn the well placement problem, as well as in other field development optimization problems, geological uncertainty is a key source of risk affecting the viability of field development projects. Well placement problems under geological uncertainty are formulated as optimization problems in which the objective function is evaluated using a reservoir simulator on a number of possible geological realizations. In this paper, we present a new approach to handle geological uncertainty for the well placement problem with a reduced number of reservoir simulations. The proposed approach uses already simulated well configurations in the neighborhood of each well configuration for the objective function evaluation. We use thus only one single reservoir simulation performed on a randomly chosen realization together with the neighborhood to estimate the objective function instead of using multiple simulations on multiple realizations. This approach is combined with the stochastic optimizer CMA-ES. The proposed approach is shown on the benchmark reservoir case PUNQ-S3 to be able to capture the geological uncertainty using a smaller number of reservoir simulations. This approach is compared to the reference approach using all the possible realizations for each well configuration, and shown to be able to reduce significantly the number of reservoir simulations (around 80%).
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Optimization of Well Trajectory under Uncertainty for Proactive Geosteering
Authors Y. Chen, R.J. Lorentzen and E.H. VefringVarious logging-while-drilling (LWD) and seismic-while-drilling (SWD) tools offer opportunities to obtain geological information near the bottom-hole-assembly during the drilling process. These real-time in-situ data provide relatively high-resolution information around and possibly ahead of the drilling path compared to the data from a surface seismic survey. The use of this in-situ data offers substantial potential for improved recovery through continuous optimization of the remaining well path while drilling. We show an automated workflow for proactive geosteering through continuous updating of the estimates of the earth model and robust optimization of the remaining well path under uncertainty. A synthetic example is shown to illustrate the proposed workflow. The estimate of the reservoir surfaces, reservoir thickness, and the depth of the initial oil-water contact and their associated uncertainty are obtained through the ensemble Kalman filter using directional resistivity measurements. A robust optimization is used to compute the well position that minimizes the average cost function evaluated on the ensemble of geological models estimated from the EnKF.
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Adjoint-Based Optimization of a Foam EOR Process
Authors J.F.B.M. Kraaijevanger, M. Namdar Zanganeh, H.W. Buurman, J.D. Jansen and W.R. RossenWe apply adjoint-based optimization to a Surfactant-Alternating-Gas foam process using a linear foam model introducing gradual changes in gas mobility and a nonlinear foam model giving abrupt changes in gas mobility as function of oil and water saturations and surfactant concentration. For the linear foam model, the objective function is a relatively smooth function of the switching time. For the nonlinear foam model, the objective function exhibits many small-scale fluctuations. As a result, a gradient-based optimization routine could have difficulty finding the optimal switching time. For the nonlinear foam model, extremely small time steps were required in the forward integration to converge to an accurate solution to the semi-discrete (discretized in space, continuous in time) problem. The semi-discrete solution still had strong oscillations in gridblock properties associated with the steep front moving through the reservoir. In addition, an extraordinarily tight tolerance was required in the backward integration to obtain accurate adjoints. We believe the small-scale oscillations in the objective function result from the large oscillations in gridblock properties associated with the front moving through the reservoir. Other EOR processes, including surfactant EOR and near-miscible flooding, have similar sharp changes, and may present similar challenges to gradient-based optimization.
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High Order Adjoint Derivatives using ESDIRK Methods for Oil Reservoir Production Optimization
Authors A. Capolei, E.H. Stenby and J.B. JørgensenIn production optimization, computation of the gradients is the computationally expensive step. We improve the computational efficiency of such algorithms by improving the gradient computation using high-order ESDIRK (Explicit Singly Diagonally Implicit Runge-Kutta) temporal integration methods and continuous adjoints . The high order integration scheme allows larger time steps and therefore faster solution times. We compare gradient computation by the continuous adjoint method to the discrete adjoint method and the finite-difference method. The methods are implemented for a two phase flow reservoir simulator. Computational experiments demonstrate that the accuracy of the sensitivities obtained by the adjoint methods are comparable to the accuracy obtained by the finite difference method. The continuous adjoint method is able to use a different time grid than the forward integration. Therefore, it can compute these sensitivities much faster than the discrete adjoint method and the finite-difference method. On the other hand, the discrete adjoint method produces the gradients of the numerical schemes, which is beneficial for the numerical optimization algorithm. Computational experiments show that when the time steps are controlled in a certain range, the continuous adjoint method produces gradients sufficiently accurate for the optimization algorithm and somewhat faster than the discrete adjoint method.
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Simultaneous Optimization of Well Placement and Control Using a Hybrid Global-local Strategy
Authors T. D. Humphries, R.D. Haynes and L.A. JamesOptimal placement and control of wells is essential to ensuring maximal net present value (NPV) or total oil recovery when developing an oil field. The majority of academic literature treats optimal placement and control as two separate problems; however, treating the problems simultaneously may allow us to achieve better results. The objective function (i.e. NPV) in this joint problem tends to vary nonsmoothly as positional parameters are varied, but smoothly in the control parameters. This suggests an approach that utilizes both global and local optimization techniques. In this paper we address the placement and control optimization problem simultaneously with two approaches combining a global search strategy (particle swarm optimization, or PSO), which operates over all variables, along with a local generalized pattern search (GPS) strategy, which operates primarily on the control parameters. The first approach is a hybrid PSO/GPS algorithm which optimizes over all positional and control variables simultaneously, while the second approach decouples the problem into separate placement and control problems, and attempts to solve them sequentially. Simulation experiments show that both approaches tend to outperform PSO in simple problems, while the decoupled approach may be the most suitable for more complicated cases.
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Ensemble Based Multi-Objective Production Optimization of Smart Wells
Authors R.M. Fonseca, O. Leeuwenburgh and J.D. JansenIn a recent study two hierarchical multi-objective methods were suggested to include short-term targets in life-cycle production optimization. However this previous study has two limitations: 1) the adjoint formulation is used to obtain gradient information, requiring simulator source code access and an extensive implementation effort, and 2) one of the two proposed methods relies on the Hessian matrix which is obtained by a computationally expensive method. In order to overcome the first of these limitations, we used ensemble-based optimization (EnOpt). EnOpt does not require source code access and is relatively easy to implement. To address the second limitation, we used the BFGS algorithm to obtain an approximation of the Hessian matrix. We performed experiments in which a water flood was optimized in a geologically realistic multi-layer sector model. The controls were inflow control valve settings at pre-defined time intervals. Undiscounted Net Present Value (NPV) and highly discounted NPV were the long-term and short-term objective functions used. We obtained an increase of approximately 14% in the secondary objective for a decrease of only 0.2-0.5% in the primary objective. The study demonstrates that ensemble-based multi-objective optimization can achieve results of practical value in a computationally efficient manner.
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Mathematical Modeling of Microbial Processes for Oil Recovery
Authors J. Monteagudo and C. HuangMicrobial recovery processes involve the usage of microorganisms, either indigenous or injected into the reservoir, to produce metabolic reactions that trigger a variety of mechanisms conducting to the production of hydrocarbons and/or enhanced oil recovery. In this work we have developed a mathematical model that accounts for several mechanisms involved both in the Microbial Gas Generation (MGG) and Microbial Enhanced Oil Recovery (MEOR) processes. This involves a kinetics model that predicts the cell growth and the metabolite production of gas, bio-surfactants and bio-polymers. Additionally, the model considers the reduction of the residual oil saturation due to the bio-surfactant and the change of water viscosity by the bio-polymer. An adsorption model depicts the retention of solutes in the aqueous phase thus altering the porosity and permeability. The model was implemented in a full-field 3-D compositional and black-oil reservoir simulator. We performed validations against experimental data available in the literature and then used the model to simulate MGG and MEOR processes with synthetic field cases. Sensitivity studies were conducted to assess the influence of the microbial kinetic model parameters in the predictions.
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A 2D Model for the Effect of Gas Diffusion on Mobility of Foam for EOR
Authors L.E. Nonnekes, S.J. Cox and W.R. RossenTransport of gas across liquid films between bubbles is cited as one reason why CO2 foams for enhanced oil recovery (EOR) are usually weaker than N2 foams and why steam foams are weaker than foams of steam mixed with N2. We examine here the effect of inter-bubble gas diffusion on flowing bubbles in a simplified model of a porous medium (a periodically constricted tube in 2D) and in particular its effect on the bubble-size distribution and capillary resistance to flow. Bubbles somewhat smaller than a pore disappear by diffusion as the bubbles move. For bubbles larger than a pore, as expected in EOR, diffusion does not affect bubble size. Instead, diffusion actually increases capillary resistance to flow (i.e. makes foam stronger): lamellae spend more time in positions where lamella curvature resists movement. When fit to pressures and diffusion and convection rates representative of field application of foams, diffusion is not expected to alter the bubble-size distribution in a foam, but instead modestly increases the resistance to flow. The reason for the apparent weakness of CO2 foam therefore evidently lies in factors other than CO2's large diffusion rate through foam.
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Using Dimensionless Numbers to Assess EOR in Heterogeneous Reservoirs
Authors B. Rashid, O. Fagbowore and A.H. MuggeridgeDimensionless numbers such as mobility ratio and the viscous to gravity ratio provide a convenient way of assessing the flow regime and thus ranking performance when designing secondary and tertiary oil recovery processes. Until recently, however, their application has been limited to homogeneous reservoirs due to a) the lack of a robust heterogeneity index and b) the fact that the viscous to gravity ratio depends upon reservoir permeability and thus heterogeneity. In this paper we present 3D phase diagrams showing how recovery and breakthrough time depend upon mobility ratio, viscous-to-gravity ratio and heterogeneity. We review the literature on the application of dimensionless numbers to identify flow regime in oil recovery processes and select a recently developed heterogeneity index based upon vorticity to characterize heterogeneity. The index has been previously verified using heterogeneous reservoir descriptions taken from SPE10 model 2. We use the phase-diagrams to identify dominant flow regimes and provide criteria based on the dimensionless numbers for identifying those flow regimes when assessing alternative EOR processes.
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Molecular Dynamics as a Tool to Deal with Thermogravitation
Authors G. Galliero and F.M. MontelAbstract: A precise description of the initial state of a petroleum reservoir is crucial to optimize its development plan. This relies on an accurate modeling of the spatial distribution of the fluid components within the reservoir which is mainly influenced by gravitational segregation and thermo-diffusion phenomena (thermogravitation). An alternative to the classical thermodynamic modelling to provide further information on thermogravitation without the need of any EoS or any correlation to describe transport properties is to use Non-Equilibrium Molecular Dynamics (NEMD) simulations on systems representing an idealized 1D reservoir fluid column. We will show how such a molecular based approach can shed light on some the underlying physical mechanism (evolution/stability) of the thermo-gravitational process in idealized situations. In particular, it will be shown, on a n-alkane mixture and a acid gas mixture, that the thermodiffusion effect can affects the vertical distribution of the different compounds as much as segregation with the same characteristic time and can even lead to an unstable (i.e. convective) situation in a CO2 rich reservoir.
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Modeling Compositional Compressible Two-phase Flow in Porous Media by the Concept of the Global Pressure
Authors B. Amaziane, M. Jurak and A. Zgaljic-KekoThe modeling of multiphase flow in porous formations is important for both the management of petroleum reservoirs and environmental remediation. More recently, modeling multiphase flow received an increasing attention in connection with the disposal of radioactive waste and sequestration of CO2. In this talk, we will discuss a new formulation for modeling compositional compressible two-phase flow in porous media such as immiscible gas injection in oil reservoirs or gas migration through engineered and geological barriers in a deep repository for radioactive waste . The focus is on the problems arising due to Newton-Raphson's flash calculations and the phase appearance and disappearance . Compositional compressible two-phase flows in porous media are usually modeled by the mass balance law written for each component, Darcy-Muscat's law, and the thermodynamic equilibrium between the phases . The obtained equations represent a set of highly coupled nonlinear partial differential equations. In order to model both saturated and unsaturated zones, one has to change the main unknowns of the system. In the saturated zones, the pressure and the saturation of one of the phases are commonly chosen as the main unknowns, whereas in the unsaturated zones the saturation may be replaced by the mass density of one of the component in its phase. To avoid changing the main unknowns, and to make the system coupling weaker, we derive a new formulation of the compositional compressible liquid and gas flow. The formulation considers gravity, capillary effects and diffusivity of each component. The main feature of this formulation is the introduction of a new variable called the global pressure. The derived system is written in terms of the global pressure and the total gas mass density that partially decouples the equations and is able to model the flows both in the saturated and unsaturated zones with no changes of the primary unknowns. The mathematical structure is well defined: the system consists of two nonlinear degenerate parabolic equations. The derived formulation is fully equivalent to the original equations and is more suitable for mathematical and numerical analysis. The accuracy and effectiveness of the new formulation is demonstrated through numerical results.
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Thermal Adaptive Implicit Method: Temperature Stability Criteria
Authors J. Maes and A. MoncorgéWe present new linear-stability criteria for the Thermal Adaptive Implicit Method (TAIM) for thermal multiphasic compositional displacement. The analysis is applied to the mass and energy equations. Moncorgé and Tchelepi’s work (2009) is based on the assumption of divergence-free total velocity, and accounts for compressibility effects. Our analysis shows that the criteria proposed do not guarantee oscillation-free numerical solutions in case of displacement that involves steep temperature and saturation fronts. We derive new criteria that result from the analysis of a simplified coupled pressure-temperature linearized system, obtained by decoupling from saturations and compositions unknowns. The new criteria explains instabilities that were undetected by the previous analysis. Moreover, we demonstrate through scaling analysis and numerical examples that for most problems of practical interest, a simple temperature stability criterion obtained by assuming incompressible multiphase flow is quite robust. The relationship between the full and simplified stability criteria is analyzed in detail. The methodology is demonstrated using several thermal-compositional examples, including Steam Assisted Gravity Drainage.
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Analytical Front Tracking in Numerical Modelling of Two-phase Flow in Porous Media
Authors I. Panfilova, J. Rihet and M. Panfilovuities in phase saturation are the obligatory attribute of any solution. Any numerical method should contain specific procedures capable to treat the discontinuities. We propose a specific two-scale numerical method which is based on replacing the saturation field by the field of discrete “elementary fronts”, whose movement is calculated on the basis of an algorithm similar to the dynamic invasion percolation. The pressure field is calculated within a macroscopic grid (scale l), while the movement of fronts is calculated inside each macroscopic cell, so that a step of the front motion h may be much lover than l. The equation of saturation transport becomes mono-dimensional within a cell and has the analytical solution. This solution gives the analytical relation for the front velocity. The time step for front motion is selected in such a way that the most rapid front would reach the limit of the corresponding macroscopic cell. Respectively the time step is variable and may be very small. When the elementary front reaches the inlet of the cell, the conditions of its penetration in the neighbouring cells are verified, including the connectivity of the displacing phase, the capillary counter-force, and so on. The connectivity of phase clusters is calculated on the basis of a special iterative algorithm developed by the group. The validity of such a method is proved theoretically: taking into account the very slow variation of the saturation far from the fronts, it is possible to replace the saturation field by a piece-wise constant approximation in the overall domain. Then the problem is reduced to the movement of the discontinuity surface. The advantage of the present method is its absolute physical and numerical stability, so that it can be applied to model the unstable displacement and analyse the fingering process. We illustrate the possibility of the method by simulating several examples of the unstable flow as (i) the gravity driven NAPL penetration in an aquifer (the Reyleigh-Taylor instability) and the (ii) displacement of heavy oil by gas (Saffman-Taylor instability), abd comparing them with experimental results.
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Trust-region Based Nonlinear Solver for Counter-current Two-phase Flow in Heterogeneous Porous Media
Authors X. Wang and H. TchelepiWe describe a new nonlinear solver for immiscible two-phase flow where viscous, buoyancy, and capillary forces are significant. The flux function is a nonlinear function of saturation and typically has inflection points and a unit-flux point. The non-convexity of flux function is a major source of convergence difficulty for nonlinear solvers. We describe a modified Newton solver that employs trust-regions of the flux function to guide Newton iterations and solution updating. The flux function is divided into saturation trust regions. The delineation of these regions is dictated by the inflection and unit-flux points. Newton update is performed such that two successive iterations cannot cross any trust-region boundary. If a crossing is detected, we "chop back" the saturation value at the appropriate trust-region boundary. This development is a significant generalization of the inflection-point approach of Jenny et al. (JCP, 2009) for viscous dominated flows. Mathematically we prove the global convergence of the trust-region based nonlinear solver. Numerically we test it for multiphase flow and transport in large-scale heterogeneous problem. Using our new nonlinear solver, we achieved significant reduction in the total Newton iterations by more than an order of magnitude together with a corresponding reduction in the overall computational cost.
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Bio-reactive Two-phase Transport and Population Dynamics in Underground Storage of Hydrogen: Natural Self-organisation
More LessTwo new research projects on hydrogen underground storage have been submitted in France (ANR - POWELTECH) and Germany (H2STORE), with collaboration of Kazakhstan National University.
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Pore-to-reservoir Modelling of Three-phase Flow Processes in Mixed-wet Carbonate Reservoirs
Authors M.I.J. van Dijke, A. Al-Dhahli and S. GeigerCarbonate reservoirs have structural heterogeneities at all length-scales (triple porosity: pore-vug-fracture) and tend to be mixed- to oil-wet. The interplay of structural and wettability heterogeneities impacts the sweep efficiency and oil recovery. The choice of an enhanced oil recovery process and the prediction of oil recovery require a sound understanding of the fundamental controls on fluid flow in mixed- to oil-wet carbonate rocks, as well as physically robust flow functions, i.e. relative permeability and capillary pressure functions. Obtaining these flow functions is a challenging task, especially when three fluid phases coexist, such as during water-alternating-gas injection (WAG). We have developed a new three-phase flow pore-network model, which comprises a novel thermodynamic criterion for formation and collapse of oil layers that strongly depends on the fluid spreading behaviour and the rock wettability. The criterion affects in particular the oil relative permeability at low oil saturations and the accurate prediction of residual oil saturations. Additionally, multiple displacement chains have been implemented, where injection of one phase at the inlet triggers a chain of interface displacements throughout the network. This allows accurate modelling of the mobilization of the many disconnected phase clusters that arise during higher order WAG cycles. Pore-networks extracted from pore-space reconstruction methods and CT images are used as input for the pore-scale simulations and the model comprises a constrained set of parameters that can be tuned to mimic the wetting state of a given reservoir. Three-phase flow functions generated from networks with carbonate pore geometries and connectivities have been used in a heterogeneous carbonate reservoir model and we demonstrate their impact on the sweep efficiency after gas injection and WAG for a range of realistic wettability scenarios. We also show that the network generated flow functions give distinctly different recovery curves compared to recoveries for traditional three-phase flow relative permeability functions, such as Stone’s.
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Modeling and Simulation of Shale Gas Production in Multi-Staged Hydraulic-Fractured Formations
More LessShale gas production is effectively enhanced by multi-staged hydraulic fracturing from horizontal wells. The characteristics of the generated fracture networks are crucial to estimating shale gas production rate and consequently determine the economics of shale gas projects. The location and geometry of hydraulic fractures are reasonably well known; whereas the secondary fractures, generated during the fracturing process, are numerous and can only be described by a stochastic framework. We thus propose three groups of fractures to be modeled: (1) hydraulic fractures whose location and geometry can be deterministically approximated, (2) smaller induced/natural fracture subset connected between hydraulic fractures, and (3) disconnected small scale (natural or induced) fractures. As the permeability contrast between fractures and micro or nano pores in shale is very large, the gas production rate will be controlled by the diffusion process that feeds gas from shale to fracture networks and by the pressure-drop propagation mechanism in the formation. The transport of gas from micro or nano pores to the fracture network comprises two mechanisms: (1) molecular (or density) diffusion and (2) convective flow due to gas compressibility. We derive a simple numerical solution for the advection/diffusion equation, coupled with statistical distribution of micro and nano pores.
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A Mathematical Model for Interpretation of Brine-Dependent Spontaneous Imbibition Experiments
Authors P.Ø. Andersen and S. EvjeIn this paper we consider a mathematical model that seeks to explain possible mechanisms for brine-dependent oil recovery in chalk. It is well documented through lab experiments that the brine composition has a strong impact on oil recovery. In particular, the role of the divalent ions (Ca2+, Mg2+ and SO24−) present in seawater have been extensively studied. Also the effect of salinity which is mainly controlled by the monovalent ions Na+ and Cl− has been carefully investigated. It has been observed that chemical reactions occur between rock and brine when seawater or seawater-like brines are injected or diffuse into chalk at high temperature. Different chemical mechanisms are involved like ion exchange, adsorption, and precipitation/dissolution of minerals such as calcite, magnesite and anhydrite. Hence, these experiments suggest that for spontaneous imbibition tests the produced oil is a result of an interplay between capillary forces and the imposed water-rock chemistry. We are interested in formulating a theory for this observed behavior based on a proper combination of geochemical and two-phase model components. The mathematical model we present couples geochemical reactive transport with the capillary forces trapping the oil. When a brine different from the formation brine enters pore space the water-rock chemistry induces changes on the rock surface. It is suggested that this leads to correspond- ing changes of the wetting state as represented by relative permeability and capillary pressure curves. Different hypothesis concerning the possible link between geochemical changes of the rock-surface and changes of wetting state are explored. Specifically, we employ the model to dis- cuss some previously published lab experiments where systematic variations in Ca2+ and SO24− in imbibing and initial brine were explored. The model suggests that at 70◦C neither dissolution nor precipitation are the main contributors for wettability alteration. Rather, a conceptual sulfate adsorption mechanism coupled to the surface activity of calcium readily explain how adding more sulfate and calcium to the system would increase oil recovery. Hence, we demonstrate how the model can be used as a tool for systematic investigations aiming at identifying key mechanisms important for mobilization of oil as a function of brine composition.
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Tie-simplex-based Nonlinear Solver for Mass-variables Compositional Formulation
Authors D.V. Voskov and H.A. Tchelepig flux function in parameterized compositional space is developed for general-purpose compositional simulation. This solver takes full advantage of the hyperbolic nature of the transport equations of compositional problem. Since compositional recovery processes evolve along a few ‘key’ tie-simplexes, the flux functions (fractional flow curves) parameterized along these tie-simplexes play a dominant role in the evolution of the solution. For a given nonlinear iteration, the flux functions associated with the parameterized tie-simplex are segmented into trust regions which includes appearance and disappearance of phases, changes in mobility of phases, and the inflection point of flux function. These regions are used to guide the evolution of the composition unknowns on nonlinear iteration since they delineate convex regions of the flux function, where convergence of the Newton iterations is guaranteed. Several challenging compositional problems are used to test the robustness and efficiency of this tie-simplex-based nonlinear solver. The convergence rate of the new nonlinear solver is always better than our standard safeguarded Newton method, which employs heuristics on maximum changes in the variables. We demonstrate that for aggressive time stepping, the new nonlinear solver converges within a fewer number of Newton iterations.
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Compositional Formulation Based on Piece-wise Linear Representation in Tie-simplex Space
Authors R. Zaydullin, D.V. Voskov and H.A. TchelepiCompositional formulations are necessary for numerical simulation of EOR (Enhanced Oil Recovery) processes, such as gas and steam injection. The coupling of the nonlinear conservation laws of multiphase flow and transport with the thermodynamic equilibrium relations poses significant challenges for compositional simulation. We describe a new framework, in which the thermodynamic phase behavior is cast in tie-simplex space as a function of composition, pressure and phase fraction. This parameter space is then used to specify the base nonlinear variables for fully-implicit compositional simulation. The compositional space is discretized using tie-lines. Thus, all the thermodynamic properties become piece-wise linear functions in this space. The numerical implementation employs multilinear interpolation of the phase behavior using adaptively constructed tie-line tables. The computation of the phase behavior in the course of a compositional simulation then becomes an iteration-free procedure and does not require any EoS (flashes or phase-stability tests) computations. The efficiency and accuracy of the method are demonstrated for several multidimensional compositional problems for both miscible and immiscible displacements. For the tested problems, the proposed method reduces the computational cost of the thermodynamic calculations significantly compared with standard EOS-based approaches. Moreover, the method shows better nonlinear convergence behavior for near-miscible gas injection displacements.
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Method of Negative Saturation and Interface Stabilization for Multiphase Compositional Flow in Porous Media
Authors M. Panfilov, M. Ghesmoune and A. AbadpourVarious EOR methods lead to the appearance of various zones with different number of phases and different thermodynamic state. They are separated by specific surfaces called the interfaces of phase transition. Consecutively, the flow equations are also different in various zones and cannot be deduced from each other by continuous degeneration, which imposes serious difficulties in numerical modelling. We suggest a new conceptual mathematical method based on the replacement of real single-phase fluid by an imaginary multiphase muticomponent continuum having fictitious properties. As the result, the fluid over all zones becomes three-phase and can be described by uniform three-phase hydro- and thermodynamic equations, which allows applying the direct numerical simulation. The equivalence principle determines the physical properties of the fictitious multiphase fluid, as well as the structure of the uniform multiphase equations. It also proves that the saturation of each phase may become negative in non-equilibrium zones, which becomes the efficient method of tracking the interface and the number of phases at any point. The method was developed by the authors for two-phase case. In the present paper the new version is developed for three-phase case. Several examples of simulation are presented.
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Vertex-centred Discretization of Multiphase Compositional Darcy Flows on General Meshes
Authors C. Guichard, R. Eymard, R. Herbin, R. Masson and P. SamierThis paper introduces a vertex centred discretization on general 3D meshes of multiphase Darcy flows in heterogeneous anisotropic porous media. The model accounts for the coupling of the mass balance of each component with the pore volume conservation and the thermodynamical equilibrium. The conservative spatial discretization of the Darcy fluxes is based on the Vertex Approximate Gradient scheme (VAG) which is unconditionally coercive for arbitrary meshes and permeability tensors. The stencil of this vertex-centred scheme typically comprises 27 points on topologically Cartesian meshes. On tetrahedral meshes, the number of unknowns is considerably reduced, by typically a factor five, compared with usual cell-centred MultiPoint Fluxes Approximations, which is a key asset for multiphase flow simulations on unstructured meshes. An adaptive choice of the pore volume at the vertices ensures the accuracy of the discretization even for coarse meshes on highly heterogeneous media. This approach can easily be implemented on existing reservoir simulators using a graph of transmissibilities for the computation of the fluxes. The efficiency of our approach is exhibited on several two phase and three phase Darcy flow examples. In particular it includes the nearwell injection of miscible CO2 in a saline aquifer taking into account the precipitation of salt.
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Comparison of a Finite Element Method and a Finite Volume Method for Flow on General Grids in 3D
Authors H. Hægland, I. Aavatsmark, C. Guichard, R. Masson and R. KaufmannWe compare the recently developed Vertex Approximate Gradient (VAG) scheme developed in [R. Eymard et al., ESAIM: Mathematical Modelling and Numerical Analysis, 46(2), 2012] and the multipoint flux approximations (MPFA) O- and L-methods on 3D irregular meshes. It is found that the VAG scheme converges for a wider range of problems than the MPFA methods, however when the MPFA-methods converge, the convergence rate in flux is better than for the VAG method.
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A Monotone Non-linear Finite Volume Method for Advection-diffusion Equations and Multiphase Flows
Authors K. Nikitin and Y. VassilevskiWe present a new nonlinear monotone finite volume method for diffusion and convection-diffusion equations and its application to two-phase black oil models. We consider full anisotropic discontinuous diffusion/permeability tensors and discontinuous velocity fields on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which reduces to the conventional 7-point stencil for cubic meshes and diagonal tensors. The approximation of the advective flux is based on the second-order upwind method with the specially designed minimal nonlinear correction. We show that the quality of the discrete flux in a reservoir simulator has great effect on the front behavior and the water-breakthrough time. We compare the new nonlinear two-point flux discretization with the conventional linear two-point scheme. The new nonlinear scheme has a number of important advantages over the traditional linear discretization. First, it demonstrates low sensitivity to grid distortions. Second, it provides appropriate approximation in the case of full anisotropic permeability tensor. For non-orthogonal grids or full anisotropic permeability tensors the conventional linear scheme provides no approximation, while the nonlinear flux is still first-order accurate. The computational work for the new method is higher than the one for the conventional dicretization, yet it is rather competitive.
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Voronoi Grids Conformal to 3D Structural Features
Authors R. Merland, B. Lévy and G. CaumonWhen simulating flow in a reservoir, errors due to upscaling can have a significant impact on the quality of results. To reduce these errors, the cells of the simulation grid should be as homogeneous as possible, hence conform to horizons and faults. In this paper, we optimize the coordinates of the 3D Voronoi seeds so that cell facets honor the structural features. These features are modeled by piecewise linear complex (PLC). The optimization consists in minimizing a function made of two parts: • A barycentric function, called Centroidal Voronoi Tessellation (CVT) function, which ensures that the cells will be of good quality by maximizing their compactness. • A conformal function, which measures the proportion of cells that is on the "wrong side" of the structural features (if the cell is cut in two by a structural feature, the "good side" contains the Voronoi seed). The novelty in this paper concerns the method of cutting cells by structural features which are locally approximated inside the Voronoi cells. These methods used jointly with an adaptive gradient solver allow dealing with complex 3D geological cases, presented in the paper.
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Adaptive Fully Implicit Multi-scale Meshless Multi-point Flux Method for Fluid Flow in Heterogeneous Porous Media
By A. LukyanovA sequential fully implicit multi-scale meshless multi-point flux method (MS-MMPFA) for nonlinear hyperbolic partial differential equations of fluid flow in heterogeneous porous media is described in this paper. The method extends the recently proposed the meshless multi-point flux approximation (MMPFA) for general fluid flow in porous media [Lukyanov, “Meshless Upscaling Method and its Application to a Fluid Flow in Porous Media”, Proceeding ECMOR XII, 2010] by utilizing advantages of the existing multi-scale finite volume (MSFV) schemes. The MMPFA is based on a gradient approximation commonly used in meshless method and combined with the mixed corrections which ensure linear completeness. In corrected meshless method, the domain boundaries and field variables at the boundaries are approximated with the default accuracy of the method. The MMPFA method was successfully tested for a number of problems where it was clearly shown that the MMPFA gives a good agreement with analytical solutions for a given number of particles. However, the level of detail and range of property variability included in reservoir characterization models leads to a large number of particles to be considered in MMPFA method. In this paper this problem is resolved using a sequential fully implicit MS-MMPFA method. The results are presented, discussed.
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CVD-MPFA Based Multiscale Formulation on Structured and Unstructured Grids
Authors E.T. Parramore, M. G. Edwards, M. Pal and S. LamineSubsurface reservoirs generally have complex geological and geometrical features, such as faults fractures, pinchouts, shales and layers defined on varying length scales. In addition the effect of heterogeneity leads to further multiscale features that cannot be modelled with desired precision on relatively coarse meshes. This has lead to development of multiscale methods over the last decade. This paper focuses on methods for fine scale modelling and presents development of multiscale methods in an unstructured grid framework with particular emphasis on the numerical flux approximation. Families of Darcy-flux approximations have been developed for consistent approximation of the general tensor pressure equation arising from Darcy’s law together with mass conservation. The schemes are control-volume distributed (CVD) with pressure and rock properties sharing the same location in a given control-volume and are comprised of a multipoint flux family formulation (CVD-MPFA). The schemes are used to develop a CVD-MPFA based multiscale formulation applicable to both structured and unstructured grids in two-dimensions. Performance of the Darcy-flux approximations are compared in the multiscale modelling environment on a range of grid types resulting from both structured and unstructured grids. The methods are applied to domains with homogeneous and heterogeneous permeability fields involving a range of test cases. The effects of quadrature range of the schemes is tested. Boundary condition constraints and consequences of basis function formulation, together with implications of scheme and grid type are presented. The development of a CVD-MPFA based multiscale formulation leads to a novel approach for fine scale modelling. The results demonstrate the benefits of the new formulation.
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Multiscale Method for Two and Three-phase Flow Simulation in Subsurface Petroleum Reservoirs
Authors M. Pal, S. Lamine, K.A. Lie and S. KrogstadMultiscale simulation is a new and promising approach that enables simulation of detailed geological model and the retention of level of detail and heterogeneity that would not be possible via conventional upscaling methods. Most multiscale methods are developed from a sequential formulation, in which flow (pressure-flux) and transport (saturation) equations are solved in separate steps. The flow equation is solved using a set of special multiscale basis functions that attempt to incorporate the effects of sub-grid geological heterogeneity into a global flow equation formulated on a coarsened grid. The multiscale basis functions are computed numerically by solving local flow problems, and can be used to construct conservative fluxes on the coarsened as well as the original fine grid. Herein, we consider one particular multiscale method, the multiscale mixed finite-element method, and discuss how it can be extended to account for capillary pressure effects. The method is evaluated for computational efficiency and accuracy on a series of models with a high degree of realism, including spatially dependent relative permeability and capillary effects, gravity, and highly heterogeneous rock properties specified on representative corner-point grids.
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A Framework for Hybrid Simulations of Two-phase Flow in Porous Media
Authors I. Lunati, P. Tomin, A. Ferrari and R. KuenzeIn the last decade multiscale methods have proven efficient in solving large reservoir-scale problems with satisfactory accuracy. Computational efficiency is achieved by splitting the original problem into a set of local problems coupled through a global coarse problem. Although these techniques are usually employed for problems in which the fine-scale processes are described by Darcy’s law, they can also be applied to pore-scale simulations and used as a mathematical framework for hybrid methods that couples a Darcy and pore scales. In this work, we consider a pore-scale description of fine-scale processes. The Navier-Stokes equations are numerically solved in the pore geometry to compute the velocity field and obtain generalized permeabilities. In the case of two-phase flow, the dynamics of the phase interface is described by the volume of fluid method with the continuum surface force model. The MsFV method is employed to construct an algorithm that couples a Darcy macro-scale description with a pore-scale description at the fine scale. The hybrid simulations results presented are in good agreement with the fine-scale reference solutions. As the reconstruction of the fine-scale details can be done adaptively, the presented method offers a flexible framework for hybrid modeling.
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An Unconditionally Stable Splitting Method Using Reordering for Simulating Polymer Injection
Authors H. M. Nilsen, K.A. Lie, A.F. Rasmussen and X. RaynaudWe present an unconditionally stable algorithm for sequential solution of flow and transport that can be used for efficient simulation of polymer injection modeled as a two-phase system with rock compressibility and equal fluid compressibilities. Our formulation gives a set of nonlinear transport equations that can be discretized with standard implicit upwind methods to conserve mass and volume independent of the time step. The resulting nonlinear system of discrete transport equations can, in the absence of gravity and capillary forces, be permute to lower triangular form by using a simple topological sorting method from graph theory. This gives a nonlinear Gauss--Seidel method that computes the solution cell by cell with local iteration control. The single-cell systems can be reduced to a nested set of scalar nonlinear equations that can easily be bracketed and solved with standard gradient or root-bracketing methods. The resulting method gives orders-of-magnitude reduction in runtimes and increases the feasible time-step sizes. Hence, sequential splitting combined with standard upwind discretizations can become a viable alternative to streamline methods for speeding up simulation of advection-dominated systems.
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Algebraic Multiscale Linear Solver for Heterogeneous Elliptic Problems
Authors Y. Wang, H. Hajibeygi and H. TchelepiAn Algebraic Multiscale Solver (AMS) for the pressure system of equations arising from incompressible flow in heterogeneous porous media is developed. The algorithm allows for several independent preconditioning stages to deal with the full spectrum of errors. In addition to the fine-scale system of equations, AMS requires information about the superimposed (dual) coarse grid to construct a wirebasket reordered system. The primal coarse grid is used in the construction of a conservative coarse-scale operator and in the reconstruction of a conservative fine-scale velocity field. The convergence properties of AMS are studied for various combinations including (1) the MultiScale Finite-Element (MSFE) method, (2) the MultiScale Finite-Volume (MSFV) method, (3) Correction Functions (CF), (4) Block Incomplete LU factorization with zero fill-in (BILU), and (5) point-wise Incomplete LU factorization with zero fill-in (ILU). The reduced-problem boundary condition, which is used for localization, is investigated. For a wide range of test cases, the performance of the different preconditioning options is analyzed. It is found that the best overall performance is obtained by combining MSFE and ILU as the global and local preconditioners, respectively. Comparison between AMS and the widely used SAMG solver illustrates that they are comparable, especially for very large heterogeneous problems.
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How Fast Is Your Newton-Like Nonlinear Solver?
Authors R.M. Younis and H.A. TchelepiThis work answers the question for any Newton-like solver that is applied to nonlinear residual systems arising during the course of implicit Reservoir Simulations. We start by developing a mathematical foundation that characterizes the asymptotic convergence rate of infinite dimensional Newton methods applied to continuous form reservoir simulation problems. Using the fact that finite dimensional (discretized) methods are related to their infinite dimensional counterparts through the approximation accuracy of the underlying numerical discretization scheme, we translate the infinite dimensional characterizations to the finite dimensional world. The analysis reveals the asymptotic scaling relations between nonlinear convergence rate and time-step and mesh size. In particular, we show a constant scaling relation for elliptic problems, a set of super-linear relations for hyperbolic situations, and for mixed parabolic problems. Numerical examples are used to illustrate the theoretical results, and we compare the direct convergence results from this work to those obtained using existing convergence monitoring methods. This work should be of interest to any simulation practitioner or developer who previously relied on text-book quadratic local convergence rate characterizations that did not hold in simulation practice and that perhaps are never even observed. The practical applications of this work are in time-step selection for convergence, generalizing single cell safeguarding tactics, and building insight into asymptotic acceleration methods.
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Fast Linear Solver for Pressure Computation in Layered Domains
Authors P. van Slingerland and C. VuikAccurate simulation of fluid pressures in layered reservoirs with strong permeability contrasts is a challenging problem. For this purpose, the Discontinuous Galerkin (DG) method has become increasingly popular. Unfortunately, standard linear solvers are usually too inefficient for the aforementioned application. To increase the efficiency of the Conjugate Gradient (CG) method for linear systems resulting from Symmetric Interior Penalty (discontinuous) Galerkin (SIPG) discretizations, we have cast an existing two-level preconditioner into the deflation framework. The main idea is to use coarse corrections based on the DG solution with polynomial degree p=0. This paper provides a numerical comparison of the performance of both two-level methods in terms of scalability and overall efficiency. Furthermore, it studies the influence of the SIPG penalty parameter, the smoother, damping of the smoother, and the strategy for solving the coarse systems. We have found that the penalty parameter can best be chosen diffusion-dependent. In that case, both two-level methods yield fast and scalable convergence. Whether preconditioning or deflation is to be favored depends on the choice for the smoother and on the damping of the smoother. Altogether, both two-level methods can contribute to faster and more accurate fluid pressure simulations.
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Coupled Geomechanics and Flow in Fractured Porous Media
Authors T. T. Garipov, K.A. Levonyan, M. Karimi-Fard and H.A. TchelepiThe effects of geomechanics on the reservoir response can be important, and this is especially true for naturally fractured formations. Modeling the mechanical deformation of naturally fractured formations poses significant numerical challenges, and accurate coupling between mechanical deformation and flow adds to the challenge. We describe a simulation framework for coupled mechanics and flow based on a Discrete Fracture Model (DFM). An important aspect is that the mechanics and flow problems share the same unstructured DFM grid. The geomechanical model is based on the classical Biot theory. The Barton-Bandis model is used to describe the fracture mechanical response. For the flow problem, we use Darcy’s law and mass conservation for slightly compressible fluids. The fractured formation is discretized using DFM, which leads to complex unstructured grids. Three standard elements (hexahedrons, tetrahedrons and wedges) are used to represent the volumes of the matrix, and the fractures are represented using lower dimensional objects (triangles or quadrangles). The Galerkin finite-element method is used for the mechanics, and a DFM finite-volume method is used the flow equations. Two different coupling strategies are considered: the fully implicit method and the fixed-stress sequential-implicit scheme. Several examples of fractured porous media are used to illustrate our methodology.
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Coupled Flow-deformation Simulations of Realistic Hydraulic Fractured Systems
Authors A.A. Rodriguez, H. Florez and J. MonteagudoAccurate modeling of fractures growth / propagation and their induced perturbation in the stress field suggests the need for coupled flow and fracture mechanics simulations. In order to tackle these challenges, an integrated workflow that considers multiple complex non-planar fractures within a coupled simulation framework will be presented here. A symmetric Galerkin Boundary Element Method (SGBEM) developed by Rungamornrat et al. (SPE 96968), which treats the elasticity problems arising from the presence of a fracture in an unbounded domain, is used to simulate fracture evolution. Fractures generated by the SGBEM are gridded using a triangular mesh and embeded inside a box where boundary conditions for both flow and mechanics are imposed. Using the surface mesh and a triangulation of the box are used as constraints to the volume discretization. In this work we perform calculations of the fracture stress shadow using a FEM approach along the volume tetrahedral grid described above. This is done by spliting the nodes that lie on the fracture an imposing the corresponding displacement boundary conditions in agreement with the results obtained from the SGBEM code. Flow calculations are performed using a control volume finite element approach which allows the incorporation of discrete fractures.
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Stress Dependent Anisotropy of Relative Permeabilities in Naturally Fractured Reservoirs
Authors P. Lang, S. Steinecker, S. Bazr Afkan and S.K. MatthäiRelative permeabilities of fracture networks as used in dual-continua simulations determine predicted producer behavior and ultimately a field’s achievable recovery. We present numerically derived ensemble (upscaled) relative permeability curves as obtained from discrete fracture and matrix (DFM) imbibition simulations. Our flow simulations are based on unstructured finite element grids and fully capable to account for capillary forces which determine the fluid transfer between fractures and adjacent matrix. Joint aperture distributions are obtained for various trends of maximum horizontal stress using finite element analysis assuming a matrix obeying linear-elasticity and accounting for fracture dilation due to normal stress and displacement. Results obtained from two-phase flow simulations show that relative permeability curves for the case of dominant fracture flow and medium to high flow rates cannot be matched by conventional analytic relationships. A strong anisotropy of relative permeability curve is found - not only as a result of fracture set orientation and degree of percolation, but very much due to the stress dependent ratio between matrix and fracture flow. This result reflects the ability of displacing phase to invade small fractures dependent on stress induced opening/closing. Fracture surface area where capillary transfer processes take place hence strongly depends on stress orientation.
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Numerical Convergence Study of Iterative Coupling for Coupled Flow and Geomechanics
Authors M.F. Wheeler, A.M. Mikelić and B.W. WangIn this paper we consider algorithms that will enable scientists and engineers to readily model complex processes in porous media taking into account fluid motion and the accompanying solid deformations. Numerous field applications would benefit from a better understanding and integration of porous flow and solid deformation. Important applications in environmental engineering and petroleum engineering include carbon sequestration, surface subsidence, pore collapse, cavity generation, hydraulic fracturing, thermal fracturing, wellbore collapse, sand production, fault activation, and waste disposal, while similar issues arise in biosciences and chemical sciences as well. Here we consider solving iteratively the coupling of flow and mechanics. We employ mixed finite element method for flow and a continuous Galerkin method for elasticity. For single phase flow, we demonstrate the convergence and convergence rates for two widely used schemes, the undrained split and the fixed stress split. We discuss the extension of the fixed stress iterative coupling scheme to an equation of state compositional flow model coupled with elasticity and a single phase poroelasticity model on general hexahedral grids. Computational results are presented which include parallel simulation of carbon sequestration in saline aquifer, and single phase poroelasticity examples on an unstructured wellbore grid and an unstructured reservoir grid.
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A Numerical Method for Chemical Equilibrium Calculations in Multiphase Systems
Authors A.M.M. Leal, M.J. Blunt and T.C. LaForceWe present a method for calculating chemical equilibria of general multiphase systems. The method is based on a stoichiometric approach, which uses Newton's method to solve a system of mass-balance and mass-action equations. A stabilisation procedure is developed to promote convergence of the calculation when a presupposed phase in the chemical system is absent in the equilibrium state. The formulation of the chemical equilibrium problem is developed by presuming no specific details of the involved phases and species. As a consequence, the method is flexible and general enough so that the calculation can be customised with a combination of thermodynamic models that are appropriate for the problem of interest. Finally, we show the use of the method to solve relevant geochemical equilibrium problems found in modelling of carbon storage in highly saline aquifers.
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Simulation of Near-Well Pressure Build-up in Models of CO2 Injection
Authors G.E. Pickup, M. Jin and E.J. MackayReservoir simulation plays an important role in predicting the outcome of a CO2 storage project, although it is challenging to simulate all the processes that arise. In particular, we need to predict the build-up of pressure in the near well region to be able to estimate the optimum injection rate whilst ensuring that the formation and overlying caprock are not fractured. In this work, we compare simulations of horizontal homogeneous models, with both 1D radial and 2D Cartesian grids, with analytical calculations of pressure build-up. Our results show that several inaccuracies arise when using too coarse a grid, due to the inability to resolve the shock fronts adequately. In a coarse cell, the amount of dissolution is over-estimated and the gas saturation builds up slowly. The presence of a large cell with intermediate gas saturation gives rise to a peak in the pressure build-up curve (due to low mobility). The pressure eventually reduces to the “correct” value when the dry-out region forms. However, if injection ceases before this time, the final pressure will be over-estimated. As the grid size is reduced, these effects become less severe.
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Upscaled Models for CO2 Migration in Geological Formations with Structural Heterogeneity
Authors S.E. Gasda, H.M. Nilsen and H.K. DahleGeological carbon sequestration involves large-scale CO2 migration and immobilization within geometrically heterogeneous storage formations. Recent modeling studies have shown that structural features along the upper boundary of a storage formation can significantly decrease updip CO2 migration speed and increase structural trapping. This impact depends on caprock roughness, which can be present at different spatial scales--from seismic-resolution features such as domes, traps, and spill points to centimeter-scale rugosity observed at outcrops. The ability to resolve all relevant features within large-scale domains is not always practical, and thus upscaled modeling approaches may be required. We propose an alternative modeling approach, the VE model, which is based on the vertical equilibrium assumption. This type of simulator is well suited for modeling CO2 migration in gravity-dominated systems. The Utsira Formation is one such system due to the strong buoyancy effects are observed in the seismic data. We use 4D seismic data and our VE modeling tool to understand the physical parameters that control CO2 migration in the Utsira. Given the uncertainty in some important parameters--CO2 density, porosity, and topography of the top Utsira--we determine the range of uncertainty in CO2 and rock properties that is supported by the data.
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Mixed Multiscale Methods for Compressible Flow
Authors K.-A. Lie, S. Krogstad and B. SkaflestadMultiscale methods are a robust and accurate alternative to traditional upscaling methods. Multiscale methods solve local problems to numerically construct a set of basis functions that later can be used to compute global solutions that describe the flow on both the coarse computational scale and the underlying fine parameter scale. This way, one is able to account for both effective coarse-scale properties and sub-scale variations. The methods are particularly efficient when the flow field must be updated repeatedly. Because temporal changes in the flow equations are moderate compared to the spatial variability, it is seldom necessary to recompute basis functions each time the global flow field is recomputed. Herein, we discuss and compare two ways of extending a multiscale mixed method that was originally developed for incompressible flow to compressible flow. The first approach is based upon a mixed residual formulation with a fine-scale domain-decomposition corrector. The second approach is to associate more than one basis function for each coarse face and coarse cell and use bootstrapping to dynamically build a basis function dictionary that spans the evolving flow patterns. We present and discuss several numerical examples, from simplified 1D cases to 3D cases with realistic reservoir geometries.
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GAMPACK (GPU Accelerated Algebraic Multigrid Package)
Authors K.P. Esler, V. Natoli and A. SamardzicIn reservoir simulation, the elliptic character of the pressure subsystem and the inhomogeneous permeability field result in extremely slow convergence for conventional iterative solvers. Algebraic multigrid (AMG) methods address this challenge by constructing a multilevel hierarchy of matrices that naturally adapts to the permeability channels of the underlying geology. Preconditioning with AMG allows difficult cases with millions of unknowns to be solved in just a few iterations. In just a few years, graphical processing units (GPUs) have progressed from a research curiosity to a productivity workhorse by reducing time-to-solution and overall hardware cost. The highly irregular computation patterns of AMG, however, require new approaches to adapt to the many-core paradigm. The construction of the coarse matrix hierarchy and grid transfer operators poses a particular challenge for GPU acceleration. We show that by carefully selecting algorithms with sufficient fine-grained parallelism, and implementing them with novel approaches, it is possible to substantially accelerate both the setup and solve stages. We present GAMPACK, a library for accelerated AMG, and show that on a single GPU it can typically reduce the total setup and solve time by a factor of over 5, when compared to a widely-used AMG solver running on 8 Xeon cores.
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Multi-core and GPU Parallelization of a General Purpose Reservoir Simulator
Authors Y. Zhou and H. TchelepiWe describe our multi-threading parallelization strategy of a general-purpose reservoir simulator (GPRS) based on a flexible Automatic Differentiation (AD) framework. Parallel Jacobian construction is achieved with a thread-safe extension of our AD library. For linear solution, we use a two-stage CPR (Constrained Pressure Residual) preconditioning strategy, combining the parallel multigrid solver XSAMG and the Block Jacobi technique with Block ILU(0) applied locally. The speedup of the full SPE 10 problem (1.1M cells) is about 5.0X on a dual quad-core Nehalem node. We then discuss the GPU parallelization of Nested Factorization (NF). The Massively Parallel NF (MPNF) algorithm was first introduced by Appleyard et al. (2011), where the 3D structured grid is divided into kernels, and each kernel is assigned a color such that no neighbouring kernels share the same color. Then, parallelism is exploited in the concurrent solution of all kernels with the same color. The most important aspects of our algorithm are: 1) coalesced memory access via special ordering of the matrix elements, and 2) application of the twisted factorization technique that further improves parallelism. With a 512-core Tesla M2090 GPU, the speedup of the full SPE10 problem is about 26X for single precision, and 19X for double precision.
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HPC-Based Optimal Well Placement
Authors A.M. Kuvichko and A.I. ErmolaevThis paper studies the mathematical aspects of well location and related optimization problems. These problems are formulated in terms or integer programming. Optimal solutions found are to formulate initial sets of cases and to improve the efficiency of oil and gas recovery. Described optimization algorithms are presented as high-scalable parallel programs making a vast majority of cases to be considered. Considered integer programming problems are extremely large-scale problems. The matrix structure, the number of feasible solutions, etc. was taken into account. A new fast algorithm for the generalized assignment (transportation) problem has been designed. Programs implementing this algorithm for CPU and GPU were tested and the results presented. A high scalability and good speedup achieved. Performed tests had also shown better timing results comparing to well-known common algorithms. It is reasonable to use the approach studied in the paper to design a set of appropriate initial cases for the small fields or fields with a complex geology. Presented workflow finds optimal well positions for a field or its part. A Brugge field has been taken as a test case. An improvement of production and NPV achieved the comparison between an initial and an optimal case presented.
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A Multilevel Multiscale Finite Volume Method
More LessThe Multiscale Finite Volume (MsFV) method has been developed to efficiently solve reservoir-scale problems while conserving fine-scale details. The method employs two grid levels: a fine grid and a coarse grid. The latter is used to calculate a coarse solution to the original problem, which is interpolated to the fine mesh. The coarse system is constructed from the fine-scale problem using restriction and prolongation operators that are obtained by introducing appropriate localization assumptions. Through a successive reconstruction step, the MsFV method is able to provide an approximate, but fully conservative fine-scale velocity field. For very large problems (e.g. one billion cell model), a two-level algorithm can remain computational expensive. Depending on the upscaling factor, the computational expense comes either from the costs associated with the solution of the coarse problem or from the construction of the local interpolators (basis functions). To ensure numerical efficiency in the former case, the MsFV concept can be reapplied to the coarse problem, leading to a new, coarser level of discretization. One challenge in the use of a multilevel MsFV technique is to find an efficient reconstruction step to obtain a conservative fine-scale velocity field. In this work, we introduce a three-level Multiscale Finite Volume method (MlMsFV) and give a detailed description of the reconstruction step. Complexity analyses of the original MsFV method and the new MlMsFV method are discussed, and their performances in terms of accuracy and efficiency are compared.
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The Gravitational Instability of a Diffusive Boundary Layer; Towards a Theoretical Minimum for Time of Onset of Convecti
More LessIn this paper we extend previous work in on the linearized analysis of gravitational instability of a diffusive boundary layer in a semi-infinite anisotropic homogenous porous medium. We express the time derivative of the square of the standard L^2-norm of a given perturbation as a time dependent quadratic form on an appropriate Hilbert space . Numerical analysis of the spectra of these quadratic forms give rise to results qualitatively similar to previous results in the litterature. We demonstrate that after the time of instability only perturbations having a non-zero projection onto a one-dimensional subspace of are unstable. We also find that the space of neutrally stable perturbations before onset of instability form a large subspace of the space of possible perturbations, where numerical analysis strongly indicate that this subspace is infinite dimensional. Error estimates for a certain part of the numerical analysis are not yet rigorous. In particular, estimating the spectrum of unbounded linear operators using finite matrix approximations still lacks a theoretical basis. However, the largest eigenvalues of larger and larger matrices approximating the operator converge quickly to well defined values, and it is conjectured that the given critical values are the correct ones for the problem at hand.
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Efficiency of Dissolution Trapping in Geological Carbon Storage
Authors M.T. Elenius, J.M. Nordbotten and H. KalischDuring geological storage of carbon dioxide (CO2), several mechanisms contribute to safe storage by immobilizing the CO2 in the injection formation. It has been shown that dissolution into resident brine can be one of the major contributors. The injected supercritical CO2 is buoyant, but dissolved CO2 increases brine density and therefore reduces the tendency for upward CO2 migration. The density increase with dissolved CO2 leads to convective mixing of the brine, thereby enabling more CO2 to dissolve. It is important to quantify the efficiency of CO2 dissolution, and therefore the efficiency of convective mixing. In previous work, we have shown that convective mixing can be considerably enhanced when taking into acccount the interaction between the two-phase region (supercritical CO2 and brine), and the single-phase brine region. Bounds on this impact were obtained for onset times, wavelengths of unstable fingers, and dissolution rates. The maximum increase in the dissolution rate was found to be large, when interaction with the plume was considered. In this paper, we use stability analysis to further study the dissolution in more detail. We make technical contributions to the field of stability analysis and in obtaining more reliable estimates of the efficiency of dissolution trapping.
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Evolution of Seismic Responses due to CO2 Injection in Carbonates Including Chemical Reactions and Rock-Physics Model
Authors L.G. Rodrigues, J.P. Nunes and D.R. GuérillotThe increase in worldwide activities related to CO2 injection in geological formations, for both EOR/CO2 and CCS projects, has pushed oil companies and universities to enhance the modeling of these processes for their better (1) designing and (2) monitoring. The objective of this paper is to describe new improvements for these two aspects through a multi-scale methodology of simulations from laboratory experiments to full-field modeling passing by studies around the wells including new rock-physics model. When injecting CO2 in carbonate rocks, one of the most critical aspects is to understand the complex chemical reactions occurring between the acidic fluid formed by the CO2, the in-situ water (connate and aquifer) and the carbonate matrix depending on its mineralogy. The mathematical formulation of the simultaneous thermodynamic equilibrium and the chemical reactions will be described completely. An original construction of the rock-physic model developed for this multi-phase flow based an effective medium theory will be described. A specific power law equation will be proposed to fit the relation between porosity and permeability obtained in the laboratory for carbonate rocks replacing the classically used Kozeny-Carman equation not valid in our case. To improve the quality of the forecasts at the entire reservoir level, simulations at different scales are performed and used sequentially. Results of sensitivity studies with various rock and mineralogy characteristics showing the impact on (a) the porosity and permeability field on the CO2 segregation, (b) the pH evolution in space and time, (c) the synthetic seismograms, will be described. This paper demonstrates the practicality of the modeling approach and software tools to address the design and monitoring of CO2 injected in a geological formation for CO2/EOR and or CCS processes. In particular, it helps the geophysicists and reservoir engineers based on the geological description of the reservoir to design injection plans for EOR or CCS processes defining plans for well tests and seismic campaigns around the wells and in between them (2D or 3D) whether such changes may be observable as a function of time. Technical contributions: 1. Multi-phase flows on realistic carbonate reservoirs with multi-phase thermodynamic equilibrium and geochemical reaction, 2. New rock-physic model for the evolution of the density and velocities used to construct surface seismic responses, 3. Methodology to improve the quality of the full-field forecast checking the results of the simulations results at the laboratory scale and generating synthetic seismograms to design seismic campaigns for monitoring.
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Classification of Digital Rocks by Machine Learning
Authors J. Ma, Z. Jiang, Q. Tian and G.D. CouplesThe availability of high-resolution 3D digital rocks in ever increasing quantities calls for intelligent Machine Learning (ML) techniques to classify them according to diverse characteristics of their pore structures. If stable classes could be identified, they would aid us to develop better models for rock typing, to gain sounder understanding of the links between the pore structures and the fluid flow behaviours and to develop predictive models of effective flow properties with many potential applications in the petroleum industry and beyond. We reported an approach that the authors developed for classifying digital samples. There, the pore structure is characterised by topological and geometrical attributes obtained from topology-preserved pore networks for each sample. Each attribute is then represented as a 1st-order tensor and normalised so that it is comparable for images sampled at different scales and resolutions. Machine learning techniques are then used to carry out actual classification from a training dataset containing labelled and unlabelled samples. The viability and extendibility of this approach are discussed. We show that this approach can be implemented to classify samples in progressive, recursive and regressive manners, and can be extended to develop correlation between the classes of samples and their fluid flow properties.
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A Flow and Transport Model in Porous Media for Microbial EOR Studies at Core Scale
Authors M. A. Diaz-Viera and J.R. Hernandez-PerezThe oil fields at their initial operation stage produce using basically its natural energy which is known as a primary recovery. As the reservoir loses energy in order to maintain the pressure it requires the injection of gas or water, which is called a secondary recovery. When the secondary recovery process becomes ineffective it is necessary to apply a more sophisticated approach such as steam injection, chemicals, etc. These are known as enhanced oil recovery methods. Some important oil fields in Mexico are entering the third stage. For the optimal design of oil recovery methods it is required to perform a variety of laboratory tests under controlled conditions to model the fundamental recovery mechanisms for a given recovery method in a specific reservoir. However, the laboratory tests commonly have a number of drawbacks, which include among others that they are very sophisticated, time consuming, expensive and always not enough to cover the whole range of field conditions involved. A proper modeling of the laboratory tests would be decisive in the interpretation, analysis and understanding of recovery mechanisms as well as in obtaining the relevant parameters for the subsequent implementation of recovery processes at the well and the reservoir scale. In this work we present a flow and transport model which was implemented using the finite element method to simulate, analyze and interpret MEOR processes at core scale under laboratory conditions. The flow model is biphasic and is based on the oil phase pressure and total velocity formulation given by Chen Z. et al. 2006, in which the capillary pressure, relative permeabilities, the effects of gravity and the dynamic porosity and permeability modification due to the clogging-declogging phenomena (adsorption-desorption of microorganisms) are taken in account. Whereas, the transport model consists of two phases (water-biofilm) and three components (microorganisms, nutrients and bioproducts). The transport model includes physical-chemical-biological phenomena such as advection, diffusion, dispersion, adsorption-desorption, growth and decay of microorganisms. Adsorption of nutrients is implemented through a linear adsorption isotherm. The effects of the bioproducts on the residual oil saturation are also included. From the methodological point of view, each stage of model development (conceptual, mathematical, numerical and computational) is described. Finally, the resulting coupled flow and transport model is numerically validated in a case study of oil displacement by the injection of water follows by the injection of water with microorganisms and nutrients. The oil recovery evaluation considering different scenarios is shown.
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Pore-scale Single and Two-phase Transport in Real Porous Medium
Authors I.I. Bogdanov, J. Kpahou and F. GuertonSince long time it has been recognized that the typical pore size is a fundamental scale in understanding of transport phenomena and determination of global transport properties of porous media. In a similar way like the Navier-Stokes equations may be used at certain limit to derive the Darcy law and define single phase transport properties, the modified Navier-Stokes model might be used to determine medium two-phase flow properties. Instead of using a regularization technique to capture the interface (cf. VOF or level-set functions approach), which may affect the modelling results in a non-trivial way, the diffuse interface method offers a thermodynamic treatment of phase “mixing” zone. As a result, it is a good choice for a numerical technique, handling the morphological changes of the interface which is of great importance for modelling of such a kind. Like zero-order approximation which is at the same time the classical theory assumptions case, the two-phase flow properties (e.g. phase relative permeabilities) are simply two ultimate single phase flow configurations, one per each phase. In both cases only volumes occupied by one fluid are considered so that wetting and capillary properties becomes very important, probably along with the process history as they all are responsible for particular fluid distribution in pore space. Taking advantage of recent advancements in X-ray computed micro-tomography (μCT), the reconstructed real porous medium samples (Bentheimer sandstone) are used for direct numerical simulations (DNS) of single and two-phase transport problems. Main model parameters - capillary, Reynolds, Cahn and Peclet numbers - are defined for each flow case. Emphasis is made on characterization of different steps and features of methodology based on μCT measurements, geometrical reconstruction, grid generation and computational models. The contribution of DNS to understanding of transport phenomena in real media becomes increasingly important factor of porous medium description efforts.
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Multi-scale Simulation of Permeability Fields and History-matching
Authors C. Gardet, M. Le Ravalec and E. GloaguenThe prediction of fluid flows within oil reservoirs or gas storage sites or aquifers requires the characterization of its petro-physical properties, i.e., facies, porosity, permeability, etc. This issue can be addressed through history-matching which calls for the determination of a three-dimensional model representing the studied reservoir. In a nutshell, a model is a grid populated by petrophysical properties. These ones have to be sequentially adjusted until the flow responses simulated for the resulting reservoir model reproduce the available dynamic data: pressures, flow rates, water cuts, 4D-seismic... A difficulty usually disregarded is that these data provide information about petrophysical properties at different scales. Referring to sequential simulation, we propose a method for generating multiscale realizations of both continuous or discrete random fields. These ones are then used to populate reservoir models with the required petrophysical properties. The integration of multiscale simulation within history-matching provides new facilities and makes it possible to incorporate dynamic data at different scales of resolution. When combined with geostatistical parameterization techniques as the gradual deformation method, it gives the essential ability to adjust the reservoir model at various scales. In addition, the overall history-matching process becomes more efficient as targeting the appropriate scale entails an economical parameterization of the model, i.e., the coarser the scale, the smaller the number of unknown parameters. Last, we present a numerical application case to highlight the advantages of the method for conditioning permeability models to dynamic data. For simplicity, we focus on two-scale processes. The coarse scale describes the variations in the mean while the fine scale characterizes local variations around the mean. We investigate the relationships between data resolution and parameterization.
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Upscaling of Vertically Heterogeneous Reservoirs
Authors A. Stovas and Y. RoganovAn accurate description of a reservoir is crucial to the management of production and efficiency of oil recovery. Reservoir modeling is an important step in a reservoir’s future performance, which is in direct proportion to reservoir management, risk analysis and making key economic decisions. Saturation and pressure changes, and porosity and permeability distributions are the most common parameters to estimate in the oil industry. In order to reduce the number of parameters in reservoir description, the different upscaling techniques have been used. At the rock physics level, the Gassmann and Hertz-Mindlin theories are applied in order to incorporate the fluid substitution and pressure changes, respectively. The most popular method at the elastic level is the Backus (1962) averaging. This method is based on the zero frequency limit of seismic wave field in a vertically heterogeneous structure. We extend the Backus averaging for the low-frequency regime by using the Baker-Campbell-Hausdorff series (Serre, 1965). That allows us to compute the frequency dependent effective medium parameters. These parameters can be used in seismic modeling and inversion with band-limited seismic wavelet.
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Homogenization of Relative Permeabilities Curves for Two-phase Flow in Porous Media Using an Optimization Method
Authors F. McKee, C. Preux and C. BerthonGrid coarsening remains essential in practical reservoir studies in order to get acceptable simulation time. This implies being able to upscale two-phase flow in particular the relative permeability. Upscaling can be divided in two stages: homogenization and mesh changing. Optimization gets involved here in the homogenization part. We proceed by identification between fine grid simulation on both representative heterogeneous regions and homogeneous equivalent region. We start with a mesh containing heterogeneous rock type. Each rock type has its own relative permeability curve and these curves are homogenized throughout the mesh with a unique relative permeability curve. In order to do this, the exit oil flow rates from the heterogeneous rock type case are considered as a reference solution. We then simulate the same flow except for the unique effective relative permeability curve. The exit oil flow rates from the two simulations are extracted to build a least squares objective function. The effective relative permeability curve (kr) is the main parameter of the optimization problem : the end points of a Brooks-Corey relative permeabilities model are used to look for a minimum objective function value.
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Mathematical Model of Horizontal Well Acidizing
Authors S.U. Zhuchkov and R.D. KanevskayaHorizontal wells are widely used to increase reservoir development efficiency. The most important factor of successful use of such wells is their capability to preserve reservoir properties in the vicinity of horizontal well. Acid treatment leads to dissolution of rock matrix and rock particles, which can plug flow channels. Therefore it is often used to recover permeability and intensify oil production. Acidizing of horizontal wells requires a special approach. The efficiency of stimulation depends on reagent distribution along the borehole, depth of acid penetration and reaction kinetics. To evaluate these characteristics it is necessary to work with adequate mathematical models giving the possibility to plan the acid treatments. These models should take into account the specific character of fluid flow close to horizontal well, pressure loss along the well, influence of gravity and heterogeneity of reservoir properties. The modeling of rock matrix dissolution should be carried out. Mathematical two-phase multicomponent model describing the flow of acid solution close to horizontal well is presented. The effects related to chemical reactions and fluid flow in well are taken into account. Calculation results for different cases are presented.
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Mixture Models for Sampling Conditional Facies Realizations from Multiple Training Images
Authors B. Jafarpour and M. KhodabakhshiMultiple point statistics (MPS) provides a systematic approach for pattern-based simulation of geologic objects from a conceptual training image (TI). The TI encodes the higher-order spatial statistics of the expected connectivity structures through stationary patterns representing the underlying geologic features. The pattern-imitating nature of MPS simulation implies that the simulated facies inherit the spatial structure of the general features in the TI. This property makes the MPS approach very sensitive to uncertainty in the prior TI. Since TIs are constructed using uncertain data and imperfect assumptions, multiple TIs may be necessary to account for the uncertainty and full range of structural variability in facies descriptions. We present a Bayesian mixture modeling approach for adaptively sampling conditional facies from multiple uncertain TIs using a probability conditioning method (PCM). Using the PCM, we invert the flow data to obtain a facies probability map for drawing conditional facies realizations from each TI. The number of samples drawn from each TI is proportional to the weight assigned to them. The TI weights are assigned based on the predictive performance of its corresponding conditional facies realizations. We demonstrate the suitability of the proposed method using numerical experiments in fluvial formations.
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Optimization of Dynamic 3D Hex-dominant Mesh Adapted for Basins Simulation Using the Smoothing Laplacian 2D
Authors B. Yahiaoui, H. Borouchaki, A. Benali and C. BennisTo improve a dynamic hex-dominant mesh for basins, a particular optimization in shape is proposed in this article. The aim is to return a mesh as regular as possible on $xy$ coordinates and align the $z$ coordinates. This optimization must complete an existing approach generate a hex-dominant mesh to improve these generated elements. To do this optimization for the $xy$ coordinate a Transformation called Smoothing Laplacian is applied. After, an iterative method which transforms some connections between layers in verticals. And it’s possible to conclude that this kind of optimization can be improved to have any shape wanted.
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Quantification of Uncertainty in Reservoir Connectivity for Field Test Evaluation
By H. OkanoProduction forecasts for petroleum reservoirs are essentially uncertain due to the lack of data. The unknown parameters are calibrated so that the simulated profile can match the observed data. A Bayesian framework has been applied to the evaluation of CO2 injection test in a tight oil reservoir. The observed data used for history-matching include the bottom-hole flowing pressure at the injector well and the gas composition at the wellhead of the producer wells. The key is starting with a simple model, because it is much quicker to adjust large-scale heterogeneity in a simple model than in a detailed model. The in-place volumes and connectivity between the wells have been calibrated in the simple models using a stochastic sampling method called the Neighbourhood Approximation algorithm. The aim of our study is to quantify uncertainty of reservoir connectivity. A Bayesian framework along with Markov Chain Monte Carlo and Neighbourhood Approximation in parameter space is used to calculate the posterior probability. We showed the best fit model for the gas breakthrough and the P10-90 envelopes in the forecast of the CO2 mole fraction in the produced gas. Our results contribute to the evaluation of the pilot test for a continuous CO2 injection.
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Development of Iterative Algorithms of Increased Convergence and Accuracy for Multiphase Flow Simulation
Authors D. Yu. Maksimov and M.A. FilatovMost of commercial simulators for multiphase flow in porous media use implicit or adaptively implicit discretization schemes which allow for rather large time steps. It is important that in the iterative process the equations being approximated may change, e.g. in cases of well target change, counterflow. To increase stability of nonlinear iterations convergence, we propose a method relying on the control of residual norm decreasing and taking into account the features of the problem under consideration. Final correction of recurrent solution increment is carried out after direct calculation of the residual with corresponding approximation before and after the point where equations change. Correction of the increment which tries to retain form of equations being approximated (e.g. well mode) makes convergence more stable and additionally allows avoiding convergence to nonphysical solution. In the paper a number of possibilities for safeguard retaining of approximation type from the previous iteration is pointed out for situations (e.g. for low filtration rate) where it has to be changed by algorithm in the strict sense, given constraints being controlled to obtain “physical” solution. Several simulation results are presented, demonstrating the robustness and effectiveness of the proposed method for challenging problems.
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Permeability Change Estimation from Microseismic Event Activity Variations
Authors S.B. Turuntaev, O.Y. Melchaeva, E.V. Zenchenko and E.I. Eremeevaactivated by the pore pressure change. It was found, that the probability distribution of these “potential fractures” can be approximated by a Weibull distribution. It was shown that it is possible to solve the inverse problem of defining local permeability from registered microseismic activity variation in a particular volume of porous medium.
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Modeling the Feasibility of Gas-Water or Gas-Oil Contact Control by Microgravity Monitoring during Enhanced Oil Recovery
By J. MrlinaMonitoring of fluids in reservoirs has become an essential tool for the control of active hydrocarbon fields, including EOR process. Repeated microgravity (time-lapse gravity, 4D gravity) can determine especially gas-water or gas-oil contact displacement in time. The technique can also be used in industrial and construction areas, contrary to 4D seismic and electromagnetics. The efficiency of the technique has been already proven by successful time-lapse gravity surveys, e.g. in Alaska, France, Italy, Oman and Qatar. Based on experience from water penetration to a sandstone formation in Egypt, gravity modelling was performed to simulate gas - water/oil contact movement in reservoirs related to pumping, water-flooding, etc. Various reservoir parameters were changed - depth, thickness, geometry, porosity and density. Gas or steam injecting/pumping was investigated, too. It was found that such processes can be observed by time-lapse microgravity, but the success depends on local geological conditions and reservoir parameters. New complex feasibility parameter cF was developed and established in graphic and tabular forms based on the time- and space-domain 3D and 4D gravity modelling. This parameter should provide fast pre-survey estimation of the effectiveness of microgravity monitoring. This procedure has not been developed before.
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Interactive Sketch-based Estimation of Stimulated Volume in Unconventional Reservoirs Using Microseismic Data
Authors Y. Hajizadeh, R. Amorim, N. Boroumand, E. Vital Brazil, D. Eaton and M. Costa SousaThe development of unconventional reservoirs has received tremendous attention from energy companies in recent years. Due to the low permeability nature of these resources, a hydraulic fracturing is often applied to stimulate the near-well region to enable economic production. The injection pressure, as it propagates, creates fractures that generate microseismic events. The monitoring of such events has become an important tool to better understand hydraulic fracture geometry, to estimate stimulated reservoir volume, to refine fracture treatment, and to optimize long-term field development. In the estimation of Stimulated Reservoir Volume (SRV) from microseismic data, recent literature highlights the importance of using time and uncertainty to achieve a more accurate estimation, as well as the influence of more complex geometries in understanding the microseismic event cloud. However, the current methods do not take any of these factors into consideration. In this work, we propose two different approaches to estimate the SRV that integrate spatial correlation together with time to obtain more accurate volume estimations. The first method is called alpha-shapes which is a generalization of the well-known shrink-wrap algorithm. The second approach is the density-based region reconstruction which considers the density of the microseismic samples in the space to reconstruct the SRV. The density-based approach uses radial basis function with Gaussian kernels to account for uncertainty in microseismic events. In addition to these two methods, we also developed a sketch-based tool to assist the users in filtering microseismic events that are visibly wrong. We molded these two approaches to allow for direct user changes to the final volume through sketch-based tools, and thus giving the expert the ability to guide the SRV estimation and to create "what-if" scenarios for a better understanding of the microseismic data. We also integrated the developed tools in this work with an interactive tabletop multitouch display to create a collaborative work environment for the experts.
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