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ECMOR XIV - 14th European Conference on the Mathematics of Oil Recovery
- Conference date: September 8-11, 2014
- Location: Catania, Sicily, Italy
- Published: 08 September 2014
61 - 80 of 136 results
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Geostatistically-consistent History Matching
Authors E.S. Zakirov, I.M. Indrupskiy, O.V. Lubimova and I.M. ShiriaevSummaryHistory matching of 3D model is a necessary step in reservoir simulation practice. This procedure is tedious and time-consuming despite all the contemporary attempts and achievements in integration of multi-scale, heterogeneous and diverse data during 3D geological model creation. Anyway, matching of a 3D simulation model to dynamic data is often conducted without paying adequate attention to peculiarities of underlying 3D geologic model. Often reservoir engineer modifies model parameters without due regard for geological view of the reservoir concerned and underlying geostatistical information. Consequently there is a huge gap between methods and means used for 3D geological model creation and the way it is utilized in 3D reservoir simulation and history matching. On the other hand, several popular math-based approaches to history matching which try to honor geological concepts of a 3D model by adjusting ensembles of model realizations are still impractical for everyday use in terms of robustness and amount of computational work.
The authors plan to present an approach to automatic history matching of 3D simulation model based on adjustment of parameters of underlying geostatistical model. Namely, range, sill and nugget of a variogram, as well as major and minor directions, could serve as control parameters to be updated, meanwhile strictly honoring static input data. Corresponding derivatives were deduced for all popular forms of variogram models. Objective function gradients are calculated with the aid of modern methods of optimal control theory. Inverse problem is treated with highest possible generality, videlicet facial heterogeneity and variogram anisotropy are assumed taking place. Either parameters of log permeability to porosity relation within each facie or well permeability multipliers are also identified through inverse problem solution. Although the algorithm is implemented in in-house SimMatch reservoir simulation and history matching software, a dedicated Petrel plugin has also been developed to assist integration of the approach in practical 3D modeling and reservoir engineering workflow.
The paper is accompanied with synthetic tests confirming robustness of the proposed method as well as full-field example.
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A Derivative Free Optimization Method Adapted to Partially Separable Functions for History Matching Problems
Authors B. Marteau, D. Ding and L. DumasSummaryHistory matching is a challenging optimization problem that involves numerous evaluations of a very expensive objective function through the simulation of complex fluids flows inside an oil reservoir. Typically, the gradient of such a function is not always available. Therefore derivative free optimization methods such as Powell’s NEWUOA are often chosen to try to solve such a problem. One way to reduce the number of evaluations of the objective function is to exploit its specific structure, for example its partial separability. A function F:x->F(x1,…,xp) is said to be partially separable if there exists some subfunctions fi such that F=f1+…+fn and that for all i, fi depends only on pi
In this paper, we study history matching of geostatistics realizations. With an appropriate parameterization by introducing local parameters, one can generally consider the history matching objective function to be partially separable. In fact, if the parameters are localized in space, for example the local gradual deformation parameters or parameters linked to a specific well, we can easily consider that some parameters do not influence certain subfunctions fi which are associated to the wells. We propose in this paper an efficient optimization technique for partially separable functions and show an application of this derivative free optimization method to three cases of oil reservoir. The first one is a simple case constructed to easily identify the partial separability of the objective function while the other two are modified versions of the PUNQ and Brugge cases. Compared to the classic derivative free methods, we were able to significantly reduce the number of objective function evaluations while having a better matching of the production data. Our results confirm the validity of the partial separability assumption on the tested cases and show that our method is well suited to history matching problems.
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On Bias Correction for Parameter Estimation Problems with Applications to Model Updating for Geosteering
Authors Y. Chen, X. Luo and E.H. VefringSummaryIn the geosteering process, a local geomodel is continuously updated around the drill bit to guide the planning of the remaining well trajectory. It was shown that the ensemble-based data assimilation techniques were useful to perform the geomodel updating using directional resistivity data. In ensemble-based methods for parameter estimation, however, it is often assumed that the model error has zero mean and is uncorrelated in time. As a result, the model parameters will be over corrected when the model error has nonzero mean. There exist many studies on modeling and estimation of model error in state estimation problems using the ensemble-base methods. But very few studies have been on parameter estimation problems because it is often difficult to distinguish the effect of model error and the effect of poorly known model parameters.
In this study, we investigate techniques to correct bias from model errors when the ensemble-base methods are used for parameter estimation, i.e. estimation of the geomodel for geosteering. The model parameters to be estimated include the depth of the top reservoir surface and reservoir thickness. We consider model error due to the bias in the resistivity model and due to small scale heterogeneities that are below model resolution. We show that when the model error contributes significantly to the data mismatch, explicit estimation of model errors is necessary to obtain a reasonable estimate of the reservoir boundaries. Typically a state augmentation approach is effective for joint model error and model parameter estimation. When the number of data is much larger than the number of model error parameters, a weighted least square approach is also reliable for model error estimation. The weighted least square approach does not require a prior probabilistic model for the model error parameters, but requires explicit knowledge of the data sensitivity to the model error parameters. The performance of both approaches depends on the selection of the model error parameters.
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Truncation Map Estimation for the Truncated Bigaussian Model Based on Bivariate Unit-lag Probabilities
Authors A. Astrakova and D.S. OliverSummaryThe truncated plurigaussian model is often used to simulate the spatial distribution of random categorical variables such as geological facies.
The problems addressed in this paper are the estimation of parameters of the truncation map for the truncated plurigaussian model, and improvements in the method for conditioning the latent Gaussian random variables to observations of categorical variables.
Unlike standard truncation maps, in this paper a colored Voronoi tessellation with number of nodes, locations of nodes, and category associated with each node all treated as unknowns in the optimization. Parameters were adjusted to match categorical bivariate unit-lag probabilities, which were obtained from a larger pattern joint distribution estimates from the Bayesian maximum-entropy approach conditioned to the unit-lag probabilities. The distribution of categorical variables generated from the estimated truncation map was close to the target unit-lag bivariate probabilities.
The conditioning of the latent Gaussian fields to log-data is also generalized for the case when the truncated bigaussian model is governed by a colored Vorono”i tessellation of the truncation map. Compared to the standard Gibbs sampler, the new approach gives better mixing properties for large amount of closely correlated data observations.
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Fats Model Update (FMU) Implementation for the Norne Field
Authors T. Popa, R.G. Hanea, M.O. Iwajomo and A.W. HeeminkSummaryFast model update (FMU) is a concept that is becoming more recognised and adopted in the oil and gas field. It is an integrated workflow which maintains consistency from structural modelling to flow simulation while history matching on some observed data. In this work, we have developed an assisted history matching workflow that enables the updates of the structural and the geological model parameters using dynamic production, pressure, and seismic data on a real North Sea field, the Norne field.
The assisted history matching framework uses an ensemble of realisations, which is also used to represent the uncertainty, to efficiently and automatically update both the structural parameters and properties in a consistent integrated workflow. The reduced uncertainty after an update can improve our knowledge of the reservoir and aid decision making in IOR strategies.
In this paper, we demonstrated that by using an ensemble iterated smoother as an history matching algorithm and a set of observations on seismic, pressure, and production data on the Norne field, one could consistently update the geological structure and parameters and improve the knowledge of the reservoir properties to achieve better prediction capabilities.
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Ensemble Based Data Assimilation to Reduce Hierarchical Architectural and Structural Uncertainties on Real Geomodels
By A. AbadpourSummary3D distribution of petrophysical parameters (porosity, permeability, relative permeability coefficient, etc) constitutes one of the key reservoir uncertainties. Facies are commonly used as high order controls for petrophysical parameters. While the petrophysical characteristics of each facies can be evaluated at wells, their spatial distribution is very often poorly known, typically derived only from indirect means (such as 3D seismic). This makes the distribution of different facies type in the reservoir model one of the important (if not the main) sources of uncertainty. Of particular importance are the discrete petrophysical properties such as relative permeability tables that are often directly linked to facies.
Facies are not the only categorical parameters in a complex geomodel and normally there is a hierarchy to follow from architectural elements, to facies, rocktypes and finally petrophysical values. Apart from the lowest level of parameters in this hierarchy the rest are discrete and often non-sort-able parameters. As a consequence, adjusting their uncertainty with a methodology like ensemble Kalman filter is not straightforward. Simply assimilating on discrete values there is a high probability of finding non discrete ones which won’t be obvious to associate to the reservoir cells.
Level set methodology proposed in ( Lorentzen et. al. 2011 ) to address this issue and calculate the closest distance to the boundary of facies which has been elaborated by benefiting from variogram information ( Abadpour et al. 2013 ) deployed in this work to address the uncertainty on above mentioned four level of hierarchy (architectural elements, facies, rocktypes and petrophysical parameters in multi realization of geomodels) on top of grid structural uncertainties of a real reservoir model with a synthetic production profile.
After achieving a satisfactory match quality and harmonious final realizations the ensemble of posterior models has been challenged from the geostatistical and geological point of view to show the validity of the process. Furthermore geological and geostatistical biases has been also introduced in the prior models to evaluate the capacity of the filtering technique to reach the truth model via inverse procedure.
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A Reduced Order Surrogate Model for Optimal Reservoir Management
Authors M. Fragoso, B. Horowitz and J.R.P. RodriguesSummaryThis work investigates the application in a Multifidelity Sequential Approximate Optmization (MSAO) framework of a proxy model that uses Trajectory Piecewise Linearization (TPWL) as a numerical complexity reduction technique and Proper Orthogonal Decomposition (POD) as the dimensional reduction technique. TPWL linearizes the reservoir simulation problem around previously converged states stored from a set of training simulations. Therefore, it is a semi-intrusive technique since it only needs simulation state maps and their derivatives to be exported. In this work TPWL stability is assured through the use of Petrov-Galerkin projection on the POD process.
The reservoir simulator used in this work is based on a fully implicit formulation in terms of mass fractions instead of saturations. The mass fraction formulation does not appear to affect the accuracy of TPWL when compared to the literature. Speedups in the order of five hundred and very good accuracy were obtained.
This work tests a new strategy for the number of required initial simulations. Instead of several simulations, it was used only one and considered the possibility of retraining if necessary. The need for retraining, the retraining process itself and how to assimilate new information into the TPWL model are studied in detail.
The problem in study is to determine optimum sequence of well controls to be used during the concession period in order to maximize NPV. The optimization algorithm proposed is based on a trust region framework. It uses TPWL with Kriging correction as a proxy model and decides when it must be retrained. A small number of simulations inside the trust region must be performed in order to build a Kriging correction model. Different schemes to perform these simulations were evaluated in order to increase the efficiency of the method. Different interpolation polynomials to be used in the Kriging correction also considered. The methodology was applied to synthetic reservoir models with good results.
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Reservoir Uncertainty-tolerant, Proactive Control of Intelligent Wells
Authors M. Haghighat Sefat, K.M. Muradov, A.H. Elsheikh and D.R. DaviesSummaryIntelligent wells (I-wells) can provide a layer-by-layer production and injection control. The flow control flexibility relies on the real-time operation of multiple, downhole Interval Control Valves (ICVs) installed across the well completion intervals. Proactive control of I-wells, with its ambition of creating an optimal, operational strategy of ICVs for the full well lifetime, is a computationally demanding optimisation problem. Reservoir uncertainty adds an additional level of complexity related to the reliability of the reservoir model’s prediction. This, along with the large number of control variables involved in the proactive I-well control, make the traditional meta-heuristic optimisation approaches (e.g. Genetic Algorithms (GA)) inefficient.
A stochastic search algorithm based on the Simultaneous Perturbation Stochastic Approximation (SPSA) is employed to solve the proactive, I-well control problem while a utility function approach is used to define the objective function to allow for the uncertainty in the reservoir’s description. The utility function accounts for both the expectation and variance of the Net Present Value (NPV) function while using the same control for the different reservoir model realisations. This approach to identifying a robust control strategy is an improvement compared with the traditional methods that rely on either a single realisation or on the mathematical expectation of the objective function.
The proposed robust optimisation framework is compared with the traditional methods on an uncertain reservoir model. It is found that the optimal control provided by ICVs inherently reduces the variance during the mean optimisation approach. The initial stages of optimisation when the control variables were far from the optimum value did not show conflicting behaviour when attempting to increase the mean and reduce the variance individually. However, conflict was observed between these two objectives during the later stages of optimisation as the optimum value of the augmented objective function is approached. A utility function approach is shown to be an efficient procedure for identifying the control scenario with the maximum expectation of added-value at an adjustable level of risk.
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Simulation Study of Combined Deployment of Smart Completion and Hot Water Injection in a Heavy Oil Reservoir
Authors A.A. Khrulenko and A.B. ZolotukhinSummaryIn this work we investigated possible synergy due to combined deployment and optimization of two IOR techniques (smart completion and hot water injection).
We used a generic simulation model of a heavy oil reservoir. The ensemble of stochastic realizations of reservoir properties was used to represent the geological uncertainties. The model consists of four heterogeneous stacked layers developed in a commingled manner by means of 9-spot pattern.
After several years of cold water injection, smart completion for the injector and hot water injection were applied and optimized both separately and simultaneously.
One random realization was selected from the ensemble. For this realization we optimized:
- case of hot water injection (by means of Pattern Search method);
- case of injector smart completion installation (by means of EnOpt method);
- case of combined deployment of both IOR techniques;
A complex NPV-based criterion was set as objective function for all cases to account for oil production, water injection and expenses associated with water heating.
The simulation results showed:
- Hot water injection increased (as compared with the uncontrolled cold water injection) the NPV criterion by 1.4% and oil recovery by 1.76%;
- Smart injector case yielded the increment of 3.29% (for NPV) and 1.27% (for oil recovery);
- Simultaneous deployment of the smart injector and the hot water injection gave a gain of 8.7% (for NPV) and 4.67% (for oil recovery);
I.e. the results showed a nonlinear increase in both for NPV and oil recovery and proved the synergy due to the combined deployment of two IOR methods.
Then we carried out robust optimization (EnOpt) of the same simulation cases for the ensemble of realizations to estimate the uncertainty impact.
Results of this study also showed the synergy. The uncertainty impact to the synergy is thoroughly discussed in the paper.
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Gas-condensate Field Optimization by Optimal Candidate Selection for Gas Injection
Authors A.I. Ermolaev, A.M. Kuvichko and I.A. TrubachevaSummaryOne of the main technologies for gas-condensate field optimization is a gas injection process. It allows to prevent loss liquid condensate in a reservoir, so the condensate recovery increases.
For effective gas injection it is reasonable to select a group of candidates to be switched from producers to gas injectors. A location of such candidates must help to decrease the size of zones with low field pressure.
The problem is formulated as following. One is to define a subset of wells to be switched into gas injectors. Optimal subset of candidates is such a subset that will optimally minimize the sizes of areas with low field pressure.
Algorithms of discrete programming proposed to solve the formulated problem. These algorithms to select wells close to areas with low pressure, but also to spread the injectors and to select economically reasonable candidates. Different optimization criteria might be used. Found set of producers and injectors becomes an optimal solution of the formulated problem.
Proposed algorithms help to avoid a brute-force or full-search procedures for selecting an optimal candidate subsets. It is reasonable when the number of wells is quite large.
Examples show tests of proposed algorithms. Recovery factors comparison show an efficiency of proposed algorithms.
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Robust Optimisation Using Spectral High Dimensional Model Representation - An Application to CO2 Sequestration Strategy
Authors R. Petvipusit, A.H. Elsheikh, P.R. King and M.J. BluntSummarySuccessful CO2 sequestration relies on operation strategies that maximise performance criteria in the presence of uncertainties. Designing optimal injection strategies under geological uncertainty requires multiple simulation runs at different geological models, rendering it computationally expensive. A surrogate model has been successfully used in several studies to reduce the computational burden by approximating the input-output relationships of the simulator with a limited number of simulation runs. However, building the surrogate is a challenging problem since the cost of building the surrogate increases exponentially with dimension.
In the current work, we propose the use of Adaptive Sparse Grid Interpolation coupled with High Dimensional Model Representation (ASGI-HDMR) to build a surrogate of high-dimensional problems. This surrogate is then used to assist with finding robust CO2 injection strategies. High Dimensional Model Representation (HDMR) is an ANOVA like technique, which is based on the fact that high-order interactions amongst the input variables may not necessarily have an impact on the output variable; the combination of low-order correlations of the input variables can represent the model in high-dimensional problem. Adaptive Sparse Grid Interpolation (ASGI) is a novel surrogate technique that allows automatic refinement in the dimension where added resolution is needed (dimensional adaptivity).
The proposed technique is evaluated on several benchmark functions and on the PUNQ-S3 reservoir model that is based on a real field. For the PUNQ-S3 model, robust CO2 injection strategies were estimated efficiently using the combined ASGI-HDMR technique. Based on our numerical results, ASGI-HDMR is a promising approach since it requires significantly fewer forward runs in building an accurate surrogate model for high-dimensional problems in comparison to ASGI without coupling with HDMR. Hence, the ASGI-HDMR enables efficient construction of the surrogates for high-dimensional problems with several wells and over multiple control periods. The impact of complex and high-dimensional control strategies to the performance criteria is shown in our finding.
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Global Optimization Based on Sparse Grid Surrogate Models for Black-box Expensive Functions
Authors F. Delbos, L. Dumas and E. EchagueSummaryIn the context of oil industry, many problems consist in a minimization process of a computationally expensive function with bound constraints. The values of the objective function are in general the output of a complex simulator for which we don’t have any explicit expression. And the absence of any information on the function gradient narrows the resolution field to algorithms using no first or second order derivatives.
There exists many different approaches in derivative free optimization, among which the most popular are space partitioning methods like DIRECT, direct search methods like Nelder Mead or MADS but also evolutionary algorithms like genetic algorithms, evolution strategies or other similar methods. The main drawback for all these methods is that they can suffer from a poor convergence rate and a high computational cost, especially for high dimensional cases. However, they can succeed in finding a global optimum where all other classical methods fail.
In this work we consider surrogate based optimization which has already been widely used for many years. A surrogate model is a framework used to minimize a function by sequentially building and minimizing a simpler model (surrogate) of the original function. In this work we build a new surrogate model by using the sparse grid interpolation method. Basically, the sparse grid approach is a grid error-controlled hierarchical approximation method which neglects the basis functions with the smallest supports. This approach was introduced in 1963 by Smolyak in order to evaluate integrals in high dimensions. It was then applied for PDE approximations but also for optimization and more recently for sensitivity analysis.
Compared to the first optimization algorithm based on sparse grids proposed by Klimke et al, a local refinement step is constructed here in order to explore the more promising regions. Moreover, no optimization steps are performed over the objective function, which reduces significantly the number of function evaluations employed.
In this talk we present our GOSGrid optimization algorithm based on sparse grids. We also compare it with other derivative free global algorithms. The comparison is based on a parameter estimation problem in reservoir engineering.
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Value of Information in Closed-loop Reservoir Management
Authors E.G.D. Barros, J.D. Jansen and P.M.J. Van den HofSummaryThis paper proposes a new methodology to perform value of information (VOI) analysis within a closed-loop reservoir management (CLRM) framework. The workflow combines tools such as robust optimization and history matching in an environment of uncertainty characterization. The approach is illustrated with two simple examples: an analytical reservoir toy model based on decline curves and a waterflooding problem in a two-dimensional five-spot reservoir. The results are compared with previous work on other measures of information valuation, and we show that our method is a more complete, although also more computationally intensive, approach to VOI analysis in a CLRM framework. We recommend it to be used as the reference for the development of more practical and less computationally demanding tools for VOI assessment in real fields.
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Interface Control Volume Finite Element Method for Modelling Fluid Flow in Heterogeneous Porous Media
Authors A.S. Abushaikha, M.J. Blunt and O.R. GosselinSummaryModelling fluid flow in highly heterogeneous and fractured reservoirs is a challenging task. These reservoirs typically have a complex structure with large and sharp variations in their material properties. Node Control Volume Finite Element (NCVFE) has been used to model those types of reservoirs at the fracture scale for the last decade. However, since the control volumes are constructed around the nodes and the material properties are assigned on elements, there is a loss of accuracy and associated fluid smearing when modelling multi-phase flows. We present a new numerical method to improve the modelling of multi-phase fluid flow in these reservoirs, called Interface Control Volume Finite Element (ICVFE). The method drastically decreases the smearing effects observed with other CVFE methods, such as NCVFE, while being mass conservative and numerically consistent. The pressure is computed at the interfaces of elements, and the control volumes are constructed around them, instead of at the element nodes. This assures that a control volume straddles, at most, two elements, which decreases the fluid smearing between neighbouring elements when large variations in their material properties are present. Lowest order Raviart-Thomas vectorial basis functions are used for the pressure calculation, and Lagrange basis functions are used to compute fluxes. The method is a combination of Mixed Hybrid Finite Element (MHFE) and FE methods. Its accuracy and convergence are tested using three dimensional tetrahedral elements to represent heterogeneous and fractured reservoirs. Our new approach is shown to be more accurate than current methods in the literature.
Significance
- The ICVFE produces less unphysical flows than NCVFE while honouring the material properties of the domain.
- It also models more accurate fluid saturation profiles than NCVFE.
- The ICVFE method defines the primary variables (pressure and saturation) on the interfaces of elements. Therefore, it computes a high resolution of the primary variables over the finite element mesh (the number of interfaces is larger than the number of elements). This down-scaling is attractive and convenient since the truncated numerical errors decrease with the increase of degrees of freedom, and conventionally this is achieved by refining the mesh.
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The Finite-element-centered Finite-volume Discretization Method (FECFVM) for Multiphase Transport in Porous Media with Sharp Material Discontinuities
Authors S. Bazrafkan, S.K. Matthai and J.E. MindelSummarySimulation of coupled pressure-, buoyancy- and capillary driven multiphase fluid flow in geological media is challenging as they are aggregates of many rock types with sharp interfaces and extremely variable flow properties. Geological features have variable orientations, complex intersections, and aspect ratios up to >1000:1. Hence, realistic discretization is only possible with adaptively refined unstructured grids. Finite-element schemes for such meshes are not locally conservative and therefore ill suited for the simulation of transport processes. To overcome this restriction we developed a hybrid Finite-Element Node-Centered Finite Volume scheme (FEFVM). In this method, however, interface finite-volumes include multiple materials and smearing of transport variables occurs.
Here we present a new hybrid discretization technique that permits computation of conservative interelement fluxes. Finite elements are simultaneously used as finite volumes for a piecewise constant discretization of transport variables. This Finite Element-Centered Finite Volume Method (FECFVM) has the advantage that saturation discontinuities at material interfaces are honored without having to add extra degrees of freedom as in the embedded discontinuity method (DFEFVM), our earlier solution resolving the smearing problem. The FECFVM brings additional benefits regarding upwinding. The implementation presented here further allows representation of fractures or sand lenses by lower dimensional elements. This simplifies model construction, meshing, and fracture aperture modeling. It also speeds up computations. gradients allow accurate computation of interface fluxes and capillary effects on global pressure can be considered because it can be discontinuous.
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Compositional Flow Modeling Using Multi-point Flux Mixed Finite Element Method
Authors G. Singh and M.F. WheelerSummaryWe present a general compositional formulation using multi-point flux mixed finite elements (MFMFE) on general hexahedral grids. The mixed finite element framework allows for local mass conservation, accurate flux approximation and a more general treatment of boundary conditions. The multi-point flux inherent in MFMFE scheme allows the usage of a full permeability tensor. The form preserves convergence properties similar to single and two-phase flow formulations presented by (Wheeler and Yotov (2006)). The proposed formulation allows for black oil, single and multi-phase incompressible, slightly compressible and incompressible flow models thereby utilizing the same design for different fluid systems. We also discuss the impact of choice of several scaling factors on convergence rate for a wide array of systems with varying fluid property description. The applications areas of interest include gas flooding, CO2 sequestration, contaminant removal and groundwater remediation.
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Multiphase Flows - Nonlinear Monotone FV Scheme and Dynamic Grids
Authors K.D. Nikitin, K.M. Terekhov and Y.V. VassilevskiSummaryWe consider development of the nonlinear monotone FV scheme and its application to two- and three-phase flow models. The scheme is applicable for full anisotropic discontinuous permeability tensors and arbitrary conformal polyhedral cells.
The nonlinear scheme is compared with conventional linear approaches: two-point and O-scheme multipoint flux approximations. The new nonlinear scheme has a number of important advantages over the traditional linear discretizations.
Compared to the linear TPFA, the nonlinear scheme demonstrates low sensitivity to grid distortions and provides appropriate approximation in case of full anisotropic permeability tensor. For non-orthogonal grids or full anisotropic permeability tensors the conventional linear TPFA 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 TPFA, yet it is rather competitive.
Compared to MPFA, the new scheme provides sparser algebraic systems and thus is less computational expensive. Moreover, it is monotone which means that the discrete solution preserves the non-negativity of the differential solution.
We consider using of the dynamic octree-based grids for better recovery of pressure gradient and saturation fronts propagation.
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Discretization of Flow Diagnostics on Stratigraphic and Unstructured Grids
Authors A.F. Rasmussen and K.A. LieSummaryFlow diagnostics tools yield quantitative information about the flow behaviour of a model, based on controlled numerical flow experiments. We consider a family of flow diagnostic measures that are constructed based on a single pressure solution and can be used to quickly establish flow patterns and well-allocation factors. This offers a means to rank, compare, and validate reservoir models, upscaling procedures, and production scenarios that is significantly less computationally expensive than full-featured multiphase flow simulations.
All flow diagnostic measures considered herein are defined from time-of-flight and tracer partitions. From these basic quantities, one can compute many interesting diagnostics such as: tracer partitions, drainage and swept regions, well-pair connections, well allocation factors, flow-and-storage-capacity (F-Phi) diagrams, sweep efficiency, and Lorenz heterogeneity coefficients. Time-of-flight and tracers are often associated with streamlines, but can equally well be computed from a standard Eulerian discretization. Herein, we discuss two improved discretizations, a multidimensional upwind method and a higher-order discontinuous Galerkin method, that both are applicable to a large family of general, polyhedral grids. We validate the methods, compare their accuracy, and investigates how the improved accuracy impacts flow-diagnostic measures and to what extent this is important for their use in various workflows.
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Spatiotemporal Adaptive Multiscale Multiphysics Simulations of Two-phase Flow
More LessSummaryWe present a spatiotemporal adaptive multiscale algorithm, which is based on the Multiscale Finite Volume method. The algorithm offers a very efficient framework to deal with multiphysics problems and to couple regions with different spatial resolution. We employ the method to simulate two-phase flow through porous media. At the fine scale, we consider a pore-scale description of the flow based on the Volume Of Fluid method. In order to construct a global problem that describes the coarse-scale behavior, the equations are averaged numerically with respect to auxiliary control volumes, and a Darcy-like coarse-scale model is obtained. The space adaptivity is based on the idea that a fine-scale description is only required in the front region, whereas the resolution can be coarsened elsewhere. Temporal adaptivity relies on the fact that the fine-scale and the coarse-scale problems can be solved with different temporal resolution (longer time steps can be used at the coarse scale). By simulating drainage under unstable flow conditions, we show that the method is able to capture the coarse-scale behavior outside the front region and to reproduce complex fluid patterns in the front region.
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The Transmissibility of Partially Connected Cells
Authors M.S. Islam and T. ManzocchiSummaryFlow simulators assume that the transmissibility between two cells is proportional to their connection area. We show that this assumption is incorrect for partially connected cells, and assess the significance of this previously ignored error.
Faulted reservoir flow simulation models built using corner-point geometry contain partially connected cells across faults. Partial connections are an inevitable consequence of miss-alignments of grid-cells resulting from the fault displacement, and it is not possible to eliminate them without compromising either the sedimentary layering or the across-fault juxtaposition geometry. Across-fault cell connections vary in shape from triangular to hexagonal, and have widely-varying fractional connection areas (the area of the connection expressed as a fraction of the area of the cell face).
Using high-resolution flow simulation models of the volume between the centres of partially juxtaposed grid-blocks, we examine systematically the magnitude of the transmissibility error. For two cells, the error is greater when the fractional connection areas are smaller, and the kv:kh and cell length:height ratios are larger. For a realistic cell aspect ratio of 60:1, kv:kh ratio of 0.1, and fractional connection area of 0.2, tortuous flow within the cells results in a transmissibility that is about five times greater than the simulator assumption. The errors decrease when fault rock is present between the cells, and when angular miss-alignments between the cells are larger.
Analysis demonstrates that the transmissibility between partially juxtaposed cells is influenced not only by the geometry and properties of the two cells in question, but also by the surrounding cells, and the error is larger in more heterogeneous sequences. Because of the complexity of the dependencies there is no analytical solution. A wider recognition of the problem, combined with our analysis of its magnitude, may aid a better appreciation of fault-related transmissibility uncertainties.
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