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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
1 - 100 of 136 results
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Ordering-based Nonlinear Solver for Fully-implicit Simulation
Authors F.P. Hamon and H.A. TchelepiSummaryThe Fully-Implicit method (FIM) is often the method of choice for the temporal discretization of the partial differential equations governing multiphase flow in porous media, especially when nonlinearity is severe. The FIM offers unconditional stability, but requires the solution of large coupled systems of nonlinear algebraic equations. Newton-based methods – often with damping heuristics - are employed to solve the nonlinear systems. However, Newton-based solvers can suffer from convergence problems, especially for large time steps in the presence of highly nonlinear flow physics. To overcome such convergence problems, the timestep is reduced, and the Newton steps are restarted from the solution of the previous (converged) time step. Recently, potential ordering and the reduced-Newton method were used to solve immiscible three-phase flow in the presence of buoyancy and capillary effects (e.g., Kwok & Tchelepi, JCP, 2007 ). Here, we extend the potential-ordering method to interphase mass transfer. Specifically, we deal with the black-oil model with variable bubble-point pressure. The convergence properties and the computational efficiency of the potential-ordering nonlinear solver are superior to existing damped Newton methods.
The nonlinear iteration process starts with the latest pressure field. Here, we use the Algebraic MultiGrid (AMG) from Fraunhofer (Stueben, International Multigrid Conference, 1983). Based on the latest pressure (potential) distribution, a directed graph is formed, in which nodes represent grid cells (control volumes) and edges represent phase fluxes between cells. As proposed by Natvig & Lie (ECMOR, 2008), and Shahvali & Tchelepi ( SPE RSS, 2013 ), Tarjan’s strongly connected components algorithm is used to order the nonlinear discrete system into a block triangular form. For the transport step, the potential ordering is used to update the saturations/compositions, one cell (control volume) at a time – from highest to lowest phase potential. The algorithm deals effectively with mass transfer between the liquid and gas phases, including phase disappearance (e.g., gas going back in solution) and reappearance (e.g., gas leaving solution), as a function of pressure and composition.
The new nonlinear ordering-based approach was implemented in Stanford’s general-purpose research simulator (AD-GPRS), and was applied to challenging Black-Oil problems using highly heterogeneous permeability fields (e.g., layers of the SPE10 test case). Detailed robustness and performance comparisons of the potential based solver with state-of-the-art nonlinear/linear solvers (e.g. damped Newton with CPR-based AMG) are presented for variable bubble-point black-oil problems using highly detailed 3D heterogeneous models. The results show that for large time steps (e.g., corresponding to fluid throughput – CFL – numbers on the order of hundreds to thousands), our nonlinear ordering-based solver reduces the number of nonlinear iterations significantly, which also leads to gains in the overall computational cost.
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Nonlinear Analysis of Newton-based Solvers for Multiphase Transport in Porous Media with Viscous and Buoyancy Forces
Authors B. Li and H.A. TchelepiSummaryNumerical simulation of multiphase flow in porous geological formations is widely used for subsurface flow management, including oil and gas production. Large-scale simulation is often limited by the computational speed, where nonlinear convergence is one of the main bottlenecks. The nonlinearity of flow properties, the coupling of driving forces for fluid migration, and the heterogeneities of the formation are three main causes for convergence difficulties. Here, we analyze the nonlinearity of two-phase transport in porous media, and we propose an efficient nonlinear solver based on this understanding.
We focus on immiscible, incompressible, two-phase transport in the presence of viscous and buoyancy forces. We investigate the nonlinearity of the discrete transport equation obtained from finite-volume discretization with Single-Point Upstream weighting (SPU), which is the industry standard. In particular, we study the discretized numerical flux expressed as a function of the upstream and downstream saturations of the total velocity. We analyze the locations and complexity of the unit-flux, zero-flux, and inflection lines of the numerical-flux saturation space. The unit- and zero-flux lines, referred to as kinks, correspond to a switch in the flow directions of the different phases, and if SPU is used, then the numerical flux is not differentiable at those points. These kinks and inflection lines are major sources of nonlinear convergence difficulties, especially when their locations in the numerical flux depend on both saturations in the upstream and downstream cells. Our analysis of the discrete (numerical) flux offers a theoretical basis of the convergence challenges associated with multi-cell problems and serves as a foundation for developing efficient nonlinear solvers.
With this understanding, we propose a nonlinear solution scheme that is a significant refinement of the works of Jenny et al. (2009) and Wang and Tchelepi (2013) . We divide the flux function into saturation ‘trust regions’ delineated by the kinks and inflection lines. Determining the boundary of each trust region is straightforward, and it only needs to be computed once in a preprocessing step before performing a simulation. The Newton updates are performed such that two successive iterations do not cross any trust-region boundary. If a crossing is detected, the saturation value is chopped back to the boundary. Our saturation chopping algorithms captures the inflection lines of the numerical flux much more accurately than the treatment of Wang and Tchelepi (2013) . The nonlinear convergence behavior is analyzed using numerical examples, and significant improvements over existing trust-region nonlinear solvers are demonstrated.
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Nearwell Local Space and Time Refinement for Multi-phase Porous Media Flows
Authors W. Kheriji, R. Masson and A. MoncorgéSummaryNearwell regions in reservoir simulations usually require fine space and time scales due to several physical processes such as higher Darcy velocities, the coupling of the stationary well model with the transient reservoir model, high non linearities due to phase appearance (typically gas), complex physics such as formation damage models.
If Local Grid Refinement is commonly used in reservoir simulations in the nearwell regions, current commercial simulators still make use of a single time stepping on the whole reservoir domain. It results that the time step is globally constrained both by the nearwell small refined cells and by the high Darcy velocities and high non linearities in the nearwell region. A Local Time Stepping with a small time step in the nearwell regions and a larger time step in the reservoir region is clearly a promising field of investigation in order to save CPU time.
It is a difficult topic in the context of reservoir simulation due to the implicit time integration, and to the coupling between a mainly elliptic or parabolic unknown, the pressure, and mainly hyperbolic unknowns, the saturations and compositions.
Our proposed approach is based on a Schwarz Domain Decomposition (DDM) Robin-Neumann algorithm using a full overlap at the coarse level to speed up the convergence of the iterative process. The matching conditions at the nearwell reservoir interfaces use optimized Robin conditions for the pressure and Dirichlet conditions for the saturations and compositions. At the well interfaces, a Neumann condition is imposed for the pressure (assuming to fix ideas that the well condition is a fixed pressure) and input Dirichlet conditions are imposed for saturations and compositions. The optimization of the Robin coefficients can be done on a pressure equation only using existing theory for elliptic/parabolic equations while the algorithm is applied on fully implicit discretization of multi-phase Darcy flows.
Numerical experiments on 3D test cases including gas injection and gas-condensate reservoir exhibit the efficiency of the method both in terms of improved accuracy compared with the classical sequential windowing algorithm, and in terms of convergence of the DDM algorithm using Robin coefficients optimized once and for all on the single phase flow equation only.
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An Efficient Solver for Nonlinear Multiphase Flow Based on Adaptive Coupling of Flow and Transport
Authors Z. Li and M. ShahvaliSummaryFully Implicit discretization of flow and transport equations gives rise to a system of coupled nonlinear equations that is typically solved using standard Newton method. For a given timestep size, even if the Newton-based iterative procedure converges, the cost associated with updating all the unknowns simultaneously can be quite expensive. Conventional sequential-implicit strategies can be used to reduce the cost, but they suffer from severe restrictions on the allowable timestep size.
In this paper we formulate, verify and analyze the computational efficiency of a new nonlinear solution strategy. The crux of the proposed algorithm is the use of a hybrid strategy to treat the co-current and counter-current flow regions differently. At each Newton iteration, we first update the pressure variables by solving the Schur complement reduced form of the equations. To avoid costly computation of the reduced Jacobian, we employ phase-based potential ordering, giving rise to a lower triangular Jacobian that can be used to compute the reduced Jacobian efficiently. After updating pressures, we decompose the domain into components in a way that can be sorted and traversed from upstream to downstream. A component in the co-current flow regions is made up of a single cell, whereas a component made up of multiple connected cells indicates counter-current flow in which pressure and saturation are tightly coupled. Marching down from upstream to downstream, for a single-cell component we update saturation nonlinearly by solving the scalar saturation equation(s). For a multi-cell component we discard the pressure of the corresponding cells obtained in the first step, and then perform a simultaneous linear update of the saturation and pressure variables by solving the local linear system. Once all of the components are visited and updated, the iteration is over.
We present a variety of challenging numerical examples in 2-D and 3-D in the presence of strong gravity and heterogeneity. Our results show that as compared with standard Newton method, the proposed hybrid solver has a lower computational cost per iteration without compromising the allowable timestep size. The computational efficiency gain depends on the number and size of the components which vary over the course of iterations. At best, pressure and saturation updates are fully decoupled and saturation variables are updated sequentially one at a time. We present a rigorous complexity analysis of the algorithm for linear solvers with arbitrary order of complexity.
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Realizing the Potential of GPUs for Reservoir Simulation
Authors K. Esler, K. Mukundakrishnan, V. Natoli, J. Shumway, Y. Zhang and J. GilmanSummaryHigher stakes from deep-ocean drilling, increasing complexity from unconventional reservoirs, and an overarching desire for a higher-fidelity subsurface description have led to a demand for reservoir simulators capable of modelling many millions of cells in minutes. Recent advances in heterogeneous computing hardware offer the promise of faster simulation times, particularly through the use of GPUs. Thus far, efforts to take advantage of hardware accelerators have been primarily focused on linear solvers and, in particular, simple preconditioners which often sacrifice rapid convergence for the sake of easy parallelism. This relatively weak convergence, the remaining unaccelerated code paths, and communication bottlenecks have prevented dramatic reductions in run time. A comprehensive approach, however, built from the ground up for accelerators, can deliver on the hardware’s promise to meet industry demand for fast, scalable reservoir simulation.
We present the results of our efforts to fully accelerate reservoir simulations on multiple GPUs in an extended black-oil formulation discretized using a fully-implicit finite volume method. We implement all major computational aspects, including property evaluation, Jacobian construction, and robust solvers/ preconditioners on GPUs. The CPR-AMG preconditioner we employ allows low iteration count and near-optimal order(N) scaling of computational effort with system size. This combination of algorithms and hardware enables the simulation of fine-scale models with many millions of cells in minutes on a single workstation without any upscaling of the original problem. We discuss the algorithms and methods we employ, give performance and accuracy results on a range of benchmark problems and real assets, and discuss the strong and weak scaling behavior of performance with model size and GPU count. This work was supported by the Marathon Oil Corporation.
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Scalable Algebraic Multiscale Linear Solver for Large-scale Reservoir Simulation
Authors A. Manea, J. Sewall and H. TchelepiSummaryThe scalability of the Algebraic Multiscale Solver (AMS) ( Wang et al., 2014 ) for the heterogeneous pressure system that arises from incompressible flow in porous media is analyzed and experimentally demonstrated in parallel computing environments. The major steps of AMS are highly parallel, but the solver overall scalability is strongly tied to the choice of parameters and algorithms involved in each step. These choices additionally impact the convergence properties of the solver. The balance between computational scalability and convergence rate is carefully considered, to ensure high overall performance while maintaining robustness.
The basis-function kernel, which dominates the setup phase, and the local smoother, which dominates the solution phase, are studied in detail. In addition, the balance between convergence rate and scalability as a function of the coarsening ratio, Cr, is investigated.
Based on this analysis, the performance of a scalable AMS implementation is tested using highly heterogeneous problems derived from the SPE10 benchmark ( Christie et al., 2001 ) and geostatistically generated. The problems range in size from a few million to more than a 100 million cells. The solver robustness and scalability is demonstrated on modern multi-core systems and compared with the widely used parallel SAMG solver ( Stüben, 2012 ).
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Meshless Subdomain Deflation Vectors in the Preconditioned Krylov Subspace Iterative Solvers
Authors A. Lukyanov, J. van der Linden, T.B. Jönsthövel and C. VuikSummaryIn reservoir models, the numerical domains are large and as a consequence a robust preconditioned iterative solver applied to the sparse linear system is required. Due to large contrasts either in the permeability field or grid aspect ratio, a large difference in the extreme eigenvalues of the resulting matrix is observable. This leads to slow convergence of iterative methods. A preconditioned Krylov subspace method such as the preconditioned GMRES method can significantly improve the convergence and robustness. Deflation based preconditioners were successfully applied for the problems with discontinuous jumps in coefficients. This paper considers the Deflated Preconditioned GMRES method for solving such systems. The deflation technique proposed in this paper uses a meshless approximation method to construct a priori the deflation space. We justify our approach through numerical experiments on both academic and realistic test problems which show improved convergence rates. For a number of cases, the fundamentals, potential, and parallel computational aspects will be presented.
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Asynchronous Multirate Newton - A Class of Nonlinear Solver that Adaptively Localizes Computation
Authors S. Sheth and R.M. YounisSummaryLocality is inherent to all transient flow and transport phenomena. Moreover, the superposition of the two disparate spatiotemporal scales that underlie flow and transport leads to a problem of dimensionality. Numerous Adaptive discretization methods have been devised to exploit an a priori understanding of locality. While such methods have provided various degrees of success, they are fundamentally restricted by the fidelity of the discretization under aggressive adaptivity. This work seeks a novel class of nonlinear solvers which are proposed to adapt the level of computation to precisely match that of the underlying spatiotemporal change, without affecting the underlying discretization model. We devise a class of nonlinear iteration that on the one hand, converges as rapidly as the best available safeguarding method (e. g. trust-region), while on the other, performing a number of operations that is at most equal to the number of cells that experience changes over an iteration.
We achieve this by developing an Asynchronous Multirate numerical integration of the Newton Flow differential system. At each nonlinear iteration, the domain of interest is partitioned adaptively in two or more levels of disjoint subdomains on the basis of the predicted rate of change of state variables. On one extreme, there are only two levels of partitions; cells that will experience a nonzero change, and cells will not. This two-level solver is equivalent to the adaptively localized solution of the linear Jacobian system. On the other extreme, the domain is decomposed into multiple disjoint subdomains. Under this scheme, the subdomains are solved sequentially using a dynamic partitioning strategy in the order from fastest to slowest. We present detailed computational results focused on general multiphase flow models. The performance improvement directly depends on the extent of locality present in the model. On the two-level end of the spectrum, the convergence rate of the proposed method is unadulterated while the performance is improved by an order of magnitude in computational time.
On the multilevel end of the spectrum, while additional performance gains are obtained for transport components, the convergence rate for pressure requires more costly synchronization strategies. This method looks very promising for the simulation of extremely complex models where well controls change dynamically.
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Construction of Multiscale Preconditioners on Stratigraphic Grids
By O. MøynerSummaryA large number of multiscale methods have been developed based on the same basic concept: Solve localized flow problems to estimate the local effects of fine-scale petrophysical properties. Use the resulting multiscale basis functions to pose a global flow problem a coarser grid. Reconstruct conservative fine-scale approximations from the coarse-scale solution. By extending the basic concept with iteration cycles and additional local stages, one can systematically drive the fine-scale residual towards machine precision. Posed algebraically, this can be seen as a set of restriction operators for computing a reduced global problem and a set of prolongation operators for constructing conservative fine-scale approximations.
Such multiscale finite-volume methods have been extensively developed for Cartesian grids in the literature. The industry, however, uses very complex with unstructured connections and degenerate cell geometries to represent realistic structural frameworks and stratigraphic architectures. A successful multiscale method should therefore be able to handle unstructured polyhedral grids, both on the fine and coarse scale, and be as flexible as possible to enable automatic coarse partitionings that adapt to wells and geological features in a way that ensures optimal accuracy for a chosen level of coarsening.
Herein, we will discuss a compare a set of prolongation operators that can be combined with finite-element or finite-volume restriction operators to form different multiscale finite-volume methods. We consider the MsFV prolongation operator (developed on a dual coarse grid with unitary at coarse block vertices), the more recent MsTPFA operator (developed on primal grid with unitary flux across coarse block faces), as well as a simplified constant prolongation operator. The methods will be compared on a variety of test cases ranging from simple synthetic grids to highly complex, real-world, field models. Our discussion will focus on flexibility wrt (coarse) grids and tendency of creating oscillatory approximations. In addition, we will look at various methods for improving the methods’ convergence properties when used as preconditioners, as well as for generating novel prolongation operators.
This is relevant for oil recovery because:
- Multiscale methods may provide a way to significantly speed up reservoir simulation and make previously intractable problems possible to solve.
- The extension of such methods to industry standard grids used for reservoir modelling enables the evaluation of the methods on real world models
- The construction of basis functions for multiscale methods may have direct connections to the process of upscaling rock derived properties such as transmissibility
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Adaptive Algebraic Multiscale Solver for Compressible Flow in Heterogeneous Porous Media
Authors M. Ţene, H. Hajibeygi, Y. Wang and H.A. TchelepiSummaryAn adaptive Algebraic Multiscale Solver for Compressible (C-AMS) flow in heterogeneous oil reservoirs is developed. Based on the recently developed AMS [ Wang et al., 2014 ] for incompressible linear flows, the C-AMS extends the algebraic formulation of the multiscale methods for compressible (nonlinear) flows. Several types of basis functions (incompressible and compressible with and without accumulation) are considered to construct the prolongation operator. As for the restriction operator, C-AMS allows for both MSFV and MSFE methods. Furthermore, to resolve high-frequency errors, Correction Functions and ILU(0) are considered. The best C-AMS procedure is determined among these various strategies, on the basis of the CPU time for three-dimensional heterogeneous problems. The C-AMS is adaptive in all aspects of prolongation, restriction, and conservative reconstruction operators for time-dependent compressible flow problems. In addition, it is also adaptive in terms of linear-system update. Though the C-AMS is a conservative multiscale solver (i.e., only a few iterations are employed infrequently in order to maintain high-quality results), a benchmark study is performed to investigate its efficiency against an industrial-grade Algebraic Multigrid (AMG) solver, SAMG [ Stuben, 2010 ]. This comparative study illustrates that the C-AMS is quite efficient for compressible flow simulations in large-scale heterogeneous 3D reservoirs.
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Monotone Multiscale Finite Volume Method for Flow in Heterogeneous Porous Media
Authors Y. Wang, H. Hajibeygi and H.A. TchelepiSummaryThe MultiScale Finite-Volume (MSFV) method is known to produce non-monotone solutions. The causes of the non-monotone solutions are identified and connected to the local flux across the boundaries of primal coarse cells induced by the basis functions. We propose a monotone MSFV (m-MSFV) method based on a local stencil-fix that guarantees monotonicity of the coarse-scale operator, and thus the resulting approximate fine-scale solution. Detection of non-physical transmissibility coefficients that lead to non-monotone solutions is achieved using local information only and is performed algebraically. For these ‘critical’ primal coarse-grid interfaces, a monotone local flux approximation, specifically, a Two-Point Flux Approximation (TPFA), is employed. Alternatively, a local linear boundary condition is used for the basis functions to reduce the degree of non-monotonicity. The local nature of the two strategies allows for ensuring monotonicity in local sub-regions, where the non-physical transmissibility occurs. For practical applications, an adaptive approach based on normalized positive off-diagonal coarse-scale transmissibility coefficients is developed. Based on the histogram of these normalized coefficients, one can remove the large peaks by applying the proposed modifications only for a small fraction of the primal coarse grids. Though the m-MSFV approach can guarantee monotonicity of the solutions to any desired level, numerical results illustrate that employing the m-MSFV modifications only for a small fraction of the domain can significantly reduce the non-monotonicity of the conservative MSFV solutions.
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A Constrained Pressure Residual Multiscale (CPR-MS) Compositional Solver
Authors M. Cusini, A.A. Lukyanov, J. Natvig and H. HajibeygiSummaryUnconventional Reservoir simulations involve several challenges not only arising from geological heterogeneities, but also from strong nonlinear physical coupling terms. All exiting upscaling and multiscale methods rely on a classical sequential formulation to treat the coupling between the nonlinear flow-transport equations. Unfortunately, the sequential strategies become severely inefficient when the flow and transport equations are strongly coupled. Examples of these cases include compositional displacements, and processes with strong capillarity effects. To extend the applicability of the multiscale methods for these challenging cases, in this paper, we propose a Constrained Pressure Residual Multiscale (CPR-MS) method. In the CPR-MS method, the CPR strategy is used to formulate the pressure equation, the approximate conservative solution of which is obtained by employing a few iterations of the iterative multiscale procedure. Several local- (ILU(k), BILU(k), etc.) and global-stage (Multiscale Finite Volume, MSFV, and Multiscale Finite Element, MSFE) solvers with different localization conditions (Linear BC, Reduced Problem BC, etc.) are employed in order to find an optimum strategy for the highly nonlinear compositional displacements. Numerical results for a wide range of test cases are presented, discussed and future studies are outlined.
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Improved Estimation of the Stochastic Gradient with Quasi-Monte Carlo Methods
More LessSummaryIn practical reservoir management, although the intent is generally to maximize some key quantity (e.g., net present value or NPV, reserves, cumulative oil production etc.), the operating well parameters (e.g., rate and/or pressure) are seldom, if ever, determined using formal optimization techniques. The usual approach to do so is through a manual process or by simply reacting to key well events (e.g., water breakthrough, or water cut reaching a threshold values). These are either quite time consuming or very likely to provide suboptimal results. Existing optimization tools have not found much use for solving this problem, as they are not efficient enough for applications to large scale simulation models of real fields. Towards this end, both adjoint and ensemble based optimization techniques have recently received significant attention as viable means for practical control optimization of large scale simulation models.
Although adjoints are the most efficient approach for accurate gradient calculation, they are difficult to implement as they require significant changes to the simulator code. The stochastic gradient, although not as efficient, is much easier to implement as it is non-intrusive and treats the simulator as a black box. In this work, we propose the application of quasi-Monte Carlo methods for improving the efficiency and accuracy of calculation of the stochastic gradient compared to current methods. While the existing approaches rely on Monte Carlo sampling which has an error convergence rate proportional to the square root of the number of ensemble members, quasi-Monte Carlo sampling has a better convergence rate proportional directly to the number of ensemble members. In particular, we apply the Sobol sequence for sampling the ensemble members, which demonstrates better convergence compared to other quasi-Monte Carlo sampling techniques. The results are demonstrated on synthetic and real models and also compared to the true gradient obtained using adjoints. In general, more than 30% improvement was obtained in the accuracy of the stochastic gradient calculated with Sobol sampling over standard Monte Carlo sampling, resulting in faster convergence of the gradient based optimization algorithm used in this work (sequential quadratic programming).
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Optimization-based Framework for Geological Scenario Determination Using Parameterized Training Images
Authors M.A.H. Rousset and L.J. DurlofskySummaryIn many reservoir-modeling applications, geological uncertainty is treated by considering multiple realizations generated from a specified geological scenario. In reality, however, the geological scenario itself is uncertain, and the use of qualitative criteria for its specification may lead to inaccuracy in flow predictions. In this work, we introduce a systematic procedure for the determination of the most likely a posteriori geological scenario. As is common in geomodeling applications, the scenario is defined in terms of a training image (TI). We introduce continuous parameterizations for uncertain TI attributes such as channel thickness and orientation, and then determine optimal values for these and other key model quantities. Optimality is defined here in terms of the level of agreement between observed data and flow results for realizations generated from a given geological scenario. The optimum scenario is determined using Particle Swarm Optimization. In the second step of the methodology, a set of specific realizations, which provide closer agreement with observed data, is identified using a rejection-sampling method. The workflow is applied to a synthetic channelized system, and the procedure is repeated using several different ‘true’ reservoir realizations to gauge its performance with data from realizations that are more or less representative of the true scenario. Values for TI parameters found by our procedure are shown to be in reasonable agreement with those of the true scenario in all cases considered. In addition, following the identification of specific realizations using rejection sampling, predicted flow results are shown to be of similar quality to those from the true scenario.
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Geologically Consistent Seismic History Matching Workflow for Ekofisk Chalk Reservoir
Authors E. Tolstukhin, L.Y. Hu and H.H. SudanSummaryReservoir surveillance using 4D seismic has become a valuable resource for managing decisions under uncertainty. This paper highlights an integrated workflow that preserves geological consistency while calibrating a reservoir model using 4D seismic and production data.
We demonstrate a successful application of this integrated approach on the Ekofisk chalk reservoir in the North Sea.
Geological and seismic consistency is preserved by using reservoir model perturbation techniques based on Multi-Point Statistics (MPS) Morphing concept, incorporation of 4D seismic data, rock physics forward modeling, and simulation model update using a computer assisted history matching procedure.
Uncertain geological parameters were updated in a loop using a proxy-based optimization algorithm through minimization of an objective function that contained both production and 4D seismic misfits. The presented approach dynamically coupled all elements of seismic to simulation workflow.
The interpretation of 4D seismic attributes from consequent time-lapse surveys assisted in tracking injection water front advancement in the reservoir. This information was incorporated in the history matching process that resulted in calibrated models with updated fracture network distributions. These multiple calibrated models will provide valuable insights for future well planning in the region and provide options to optimize future well targets under uncertainty.
The integrated workflow provided a quantitative mechanism to improve the predictability of the flow model.
The approach yields improved reservoir management by encouraging multi-disciplinary collaboration between geological, geomechanical, geophysical and reservoir engineering disciplines.
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Toward an Enhanced Bayesian Estimation Framework for Multiphase Flow Soft-sensing
Authors X. Luo, R. Lorentzen, A. Stordal and G. NævdalSummarySmart wells are advanced operation facilities used in modern fields. Typically, a smart well is equipped with downhole sensors that collect and transmit, for instance, pressure and temperature data in order to monitor well and reservoir conditions in the field. For economical reasons, however, the number of downhole sensors is limited. Therefore, they may not be able to provide complete information about the properties of the fluids, e.g., the flow rates, in places other than the locations of the sensors. In order to evaluate fluid properties in the well, one needs to estimate them based on the collected data from the sensors. Such an exercise is often called “soft sensing” or “soft metering” (see, for example, Lorentzen et al., 2010 ).
In this work the authors study the multiphase flow soft-sensing problem based on the framework used in Lorentzen et al. (2013). There are three functional modules in this framework, namely, a transient well flow model that describes the response of certain physical variables in a well, for instance, temperature and pressure, to the flow rates entering and leaving the well zones; a Markov jump process that is designed to capture the potential abrupt changes in the flow rates; and an estimation method that is adopted to estimate the flow rates in the Markov jump process, based on the measurements from downhole sensors.
In Lorentzen et al. (2013), the variances of the flow rates in the Markov jump process are chosen manually. To fill this gap, in the current work two approaches are proposed in order to optimize the variance estimation. Through a numerical example, we show that, when the estimation framework is used in conjunction with these two proposed variance-estimation approaches, it can achieve good performance in terms of matching both the measurements of the physical sensors and the true underlying flow rates.
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Reservoir Characterization in an Underground Gas Storage Field Using Joint Inversion of Flow and Geodetic Data
Authors B. Jha, F. Bottazzi, R. Wojcik, M. Coccia, N. Bechor, D. McLaughlin, T. Herring, B.H. Hager and R. JuanesSummaryCharacterization of reservoir properties like porosity and permeability in reservoir models typically relies on history matching of production data, well pressure data, and possibly other fluid-dynamical data.
Calibrated (history-matched) reservoir models are then used for forecasting production, and designing effective strategies for improved oil and gas recovery. Here, we perform data assimilation of both flow data and deformation data for joint inversion of reservoir properties. Given the coupled nature of the process, joint inversion requires efficient simulation tools of coupled reservoir flow and mechanical deformation. We apply our coupled simulation tool to a real underground gas storage field in Italy. We simulate the initial gas production period, and several decades of seasonal natural gas storage and production. We perform a probabilistic estimation of rock properties by joint inversion of ground deformation data from geodetic measurements and fluid flow data from wells. Using an efficient implementation of the Ensemble Kalman Smoother as the estimator and our coupled multiphase flow and geomechanics simulator as the forward model, we show that incorporating deformation data leads to a significant reduction of uncertainty in the prior distributions of rock properties such as porosity, permeability, and pore compressibility.
Research significance
- We perform joint inversion of flow and surface deformation data for parameter estimation in a real field with complex production-injection history based on the Bayesian inference model and coupled multiphase flow and geomechanics simulation.
- We develop a computationally efficient implementation of the Ensemble Kalman method for uncertainty reduction.
- We quantify the value of information from surface deformation data in uncertainty reduction in prior distributions.
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Beyond the Probability Map - Representation of Posterior Facies Probability
Authors Y. Zhang, D.S. Oliver and Y. ChenSummaryGeologic facies distributions are commonly represented in geomodels by categorical variables that are intrinsically non-Gaussian and thus difficult to calibrate in ensemble Kalman filter-like algorithms.
For certain types of stochastic models such as the truncated plurigaussian, it is possible to directly update model variables in such a way that the resulting realizations appear to be samples from the posterior. For other types of models, this is not possible. One common approach has been to invert flow data using the ensemble Kalman filter (EnKF) to obtain “probability maps” which are then used to condition facies realizations. Data matches obtained in this method are generally poor, however, because the probability map neglects important joint probabilities of model parameters imposed by flow data. In this paper, we propose a data assimilation method with a post-processing step that resembles the post-smoothed maximum-likelihood (ML) reconstruction method described in Nuyts et al. (2005 ). Disregarding the categorical feature of the facies model, reservoir properties are first updated using an EnKF-like assimilation method to honor flow data. In the post-processing step a penalty term forcing model variables to take discrete values is jointly minimized with the distance to the posterior realizations to solve for facies models that match data. The distance to posterior realizations is quantified using the ensemble representation of the posterior covariance, which represents the joint probability of model parameters. The matrix inversion lemma is used in solving the minimization problem to avoid inversion of the covariance.
The ability of the ensemble to accurately represent information in data is demonstrated on two linear examples and a nonlinear reservoir flow example. Comparison is made with approaches that use only the probability map to represent the assimilated data. The results show better data matches obtained with the proposed method and reflect the importance of the information captured by the updated ensemble from the data with respect to the joint probabilities of model variables.
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Integration of Markov Mesh Models and Ensemble Data Assimilation in Reservoirs with Complex Geology
Authors M. Panzeri, E. Della Rossa, L. Dovera, M. Riva and A. GuadagniniSummaryWe present a methodology conducive to updating both facies and petrophysical properties of a reservoir models set characterized by a complex architecture within the context of a history matching procedure based on the Ensemble Kalman Filter (EnKF). Spatial distribution of facies is handled by means of a Markov Mesh (MM) model. The latter is adopted because of its ability to reproduce detailed facies geometries and spatial patterns and can be integrated in a consistent probabilistic Bayesian framework. The MM model is then integrated into a history matching procedure which is based on the EnKF scheme. We test the proposed methodology by means of a synthetic reservoir in the presence of two distinct facies. We analyze the accuracy and computational efficiency of our algorithm with respect to the standard EnKF both in terms of history matching quality and forecast prediction capabilities. The results show that the integration of MM in the data assimilation scheme allows obtaining realistic geological shapes for spatial facies distribution and an improved estimation of petrophysical properties. In addition, the updated ensemble correctly captures the production range in the long term.
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Petro-physical Properties Bias Around Uncertain Facies Boundary - Analysis, Correction by Conditional Property Interface Filler (CoPIF) and Impact on Ensemble-based Assisted History Match
More LessSummaryWith the advent of Ensemble methods for Assisted History Matching of reservoir models, it has become increasingly common to use initial ensembles of reservoir models consisting of a fixed (certain) grid geometry and variable (uncertain) 3D facies distributions and 3D petro-physical property distributions conditional upon facies. In such process, 3D facies realizations are often derived by modifying facies boundary locations around certain known 3D features (often from seismic interpretation).
In the context of a fixed grid, distributions of petro-physical parameters are biased because all the models in the initial ensemble are such that facies boundaries are located at grid boundary locations; in reality, the facies boundary can be located anywhere, and the petro-physical properties of cells containing facies boundaries differ from strictly facies-dependent statistics. The bias results, typically, in excessive variance of petro-physical parameters in initial ensembles in cells which possibly belong to different facies. Such variance bias has detrimental consequences on iterative inversion processes, incrementally introducing a cumulative error on the match term.
A 3D Facies Boundary Filter has been developed to statistically correct this bias over ensembles of initial models. The filter relies on the construction of a refined grid and simple averaging of petro-physical properties and is based on a hypothesis of piecewise linearity of facies boundaries between cell centres. In any ensemble approach, a filtered ensemble would be used to compute unbiased updates and such updates applied upon un-filtered ensembles. The filter helps obtain better matched models (closer to initial ensemble and/or, in synthetic cases, the truth) and ultimately determines better performance forecast. The bias will be illustrated through a simple 1D example, a 3D synthetic dataset and a real 3D field model. The principle of the filter will be detailed. The impact of the filter on the petro-physical statistics will be illustrated as well as its impact within the context of EnKF Assisted History Match.
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Bias Aware Data Assimilation in Reservoir Characterization
Authors M. Glegola, R.G. Hanea and G. KaletaSummaryIn reservoir characterization, modern reservoir modeling and Assisted History Matching aim at delivering integrated models with quantified uncertainty, constrained on all relevant data.
Traditionally, the reservoir model is updated using only the dynamic production data from the wells. Recently, more and more efforts are made to use Geophysical Reservoir Monitoring (GRM) data in history matching, as these types of data can provide valuable information about the reservoir characteristics and geological formations over the whole field.
Time-lapse (4D) gravimetry is a direct measure of a subsurface mass flow and can provide valuable information in this context. It offers an attractive aerial monitoring technique for reservoirs containing fluids with high density contrasts, e.g., gas and water or oil and steam. The method is especially promising for shallow reservoirs as the 4D signal will be stronger for large and shallow reservoirs, compared to smaller and deeper reservoirs.
In reservoir history matching, often an assumption is made that the forward model predictions and the observations are unbiased, i.e., there are no systematic errors. In this study we investigate the added value of gravimetric observations for gas field monitoring and aquifer support estimation, under the assumption that both model and observations are biased.
We perform a numerical study with a realistic 3D gas field model which contains a large and complex aquifer system. The aquifer support along with other reservoir parameters, such as porosities, permeabilities, reservoir top and bottom horizons etc., are jointly estimated using the Ensemble Smoother (ES).
We show that the influence of the observation bias and/or the model bias on assimilation results can be severe and may lead to large errors in the estimations of the states/parameters. By using bias-aware data assimilation methodology, the bias can be estimated separately from the state, and we show that the deteriorating bias influence on the assimilation results to a large extent can be mitigated.
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On an Alternative Implementation of the Iterative Ensemble Smoother and its Application to Reservoir Facies Estimation
Authors X. Luo, Y. Chen, R. Valestrand, A. Stordal, R. Lorentzen and G. NævdalSummaryFor data assimilation problems there are different ways in using available observations. While certain data assimilation algorithms, for instance, the ensemble Kalman filter (EnKF, see, for example, Aanonsen et al., 2009 ) assimilate the observations sequentially in time, other data assimilation algorithms may instead collect the observations at different time instants and assimilate them simultaneously. In general such algorithms can be classified as smoothers. In this aspect, the ensemble smoother (ES, see, for example, Evensen and van Leeuwen, 2000 ) can be considered as an smoother counterpart of the EnKF.
The EnKF has been widely used for reservoir data assimilation problems since its introduction to the community of petroleum engineering ( Nævdal et al., 2002 ). The applications of the ES to reservoir data assimilation problems are also investigated recently. Compared to the EnKF, the ES has certain technical advantages, including, for instance, avoiding the restarts associated with each update step in the EnKF and also having fewer variables to update, which may result in a significant reduction in simulation time, while providing similar assimilation results to those obtained by the EnKF ( Skjervheim and Evensen, 2011 ).
To further improve the performance of the ES, some iterative ensemble smoothers are suggested in the literature, in which the iterations are carried out in the forms of certain iterative optimization algorithms, e. g., the Gaussian-Newton ( Chen and Oliver, 2012 ) or the Levenberg-Marquardt method ( Chen and Oliver, 2013 ; Emerick and Reynolds, 2012 ), or in the context of adaptive Gaussian mixture (AGM, see Stordal and Lorentzen, 2013).
In this contribution we show that the iteration formulae used in Chen and Oliver (2013) ; Emerick and Reynolds (2012) can also be derived from the regularized Levenberg-Marquardt (RLM) algorithm in inverse problems theory ( Engl et al., 2000 ), with certain linearization approximations introduced to the RLM. This does not only lead to an alternative theoretical tool in understanding and analyzing the behaviour of the aforementioned iterative ES, but also provide insights and guidelines for further developments of the iterative ES algorithm. As an example, we show that an alternative implementation of the iterative ES can be derived based on the RLM algorithm. For illustration, we apply this alternative algorithm to a facies estimation problem previously investigated in Lorentzen et al. (2012) , and compare its performance to that of the (approximate) iterative ES used in Chen and Oliver (2013) .
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Complex Geology Estimation Using the Iterative Adaptive Gaussian Mixture (IAGM)
Authors B. Sebacher, A. Stordal and R.G. HaneaSummaryIn the past years the multi-point geostatistical (MPS) simulation geo-models have been used successfully, creating realistic geological instances(facies
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Evaluation of Ordered Sequential Assimilation for Improved EnKF Sampling
Authors K. Fossum and T. MannsethSummaryEnsemble based data assimilation (DA) methods, such as the (sequential) ensemble Kalman filter (EnKF) and the (non-sequential) ensemble smoother (ES), can both be utilized for solving the inverse problem of estimating poorly known parameters from data consisting of noisy observations of some dynamical system. For cases where we have non-linear data, i.e., when there is a non-linear relationship between the parameters and the dynamical model, both DA methods give inexact results. Moreover, several studies have revealed that for non-linear cases the EnKF and ES give different approximation errors.
We recently conducted a thorough investigation of sequential and non-sequential assimilation schemes. The investigation showed that, for a series of weakly non-linear data, sequential assimilation is favorable to non-sequential assimilation. In addition, analytical, and numerical, evidence showed that by ordering data after ascending non-linearity, one reduces the approximation error for the sequential scheme.
Ordering of data will, however, not reduce the approximation error for all cases. It is clear that for a sequence of highly non-linear data the approximate methods, independent of how the data are ordered, will fail. Likewise, if the data has little variation in non-linearity, nothing is gain by ordered sequential assimilation. In this work, we investigate, by simple toy models, for which range of data non-linearity there is a potential advantage of ordered sequential assimilation.
Furthermore, considering a 2D reservoir case, we evaluate the non-linearity for a collection of production data and production strategies. For each numerical setup, we assess the benefit from ordered sequential assimilation of the data, and we compare the results with results obtained by the toy models.
The assimilation schemes are assessed by comparing their history matching capabilities, and by measuring the stochastic distances between their posterior distributions and the posterior distribution obtained by Markov chain Monte Carlo algorithm. Throughout, the non-linearity is evaluated by a stochastic non-linearity measure.
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Percolation Modeling of Relative Permeability Hysteresis Including Surface and Rheological Effects
Authors V. Kadet and A. GalechyanSummaryThe phenomenon of relative permeability hysteresis is observed during the process of developing the oil field by methods where the flow direction changes. In this case the displacement of oil by water changes into the displacement of water by oil and vice versa. This work is devoted to modeling of relative permeability hysteresis for drainage and imbibition based on percolation theory.
The phenomenon of active oil components adsorption on the rockforming minerals is considered as the first mechanism of hysteresis origin. In the process of drainage this causes surface hydrophobization of initially hydrophilic rock which leads to each phase relative permeability change. To describe this phenomenon percolation model for media with microheterogeneous wettability is used. The second mechanism is fluid rheological properties change, caused by the fluids mixing during drainage. It is described by percolation model for fluids with different rheological properties.
Obtained numerical solution is represented as relative permeability curves and is qualitatively confirmed by the experimental data. The behavior of relative permeability hysteresis is analyzed for various differential radius distribution curves, capillary network coordination numbers, saturation models, hydrophobization degree and fluid rheological properties. It allows to establish general tendencies of relative permeability hysteresis behavior. Introduced methodology can be put into practice for relative permeability calculation in any porous media to reduce the time spent. Also this approach can be embedded in hydrodynamic modeling programs to consider the relative permeability hysteresis effect.
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Four-component Gas/Oil Displacement with Constant Pressure Boundaries
Authors H. Nekouie, L.A. James and T.E. JohansenSummaryThis paper presents the analytical solution of four-component gas/oil displacements under constant pressure boundary conditions. All the previous studies in gas/oil displacement problems have been accomplished under the assumption of constant flux boundaries. In practice however, gas flooding projects are often conducted with constant injection pressure and constant producing well pressure. In this work, a novel generation of Buckley-Leverett’s classic fractional flow theory is applied to solve the problem of four-component gas/oil displacements under constant pressure boundaries.
Conservation of mass in a one-dimensional, dispersion-free medium, for a four-component gas/oil displacement system leads to a set of partial differential equations. The solution of the corresponding initial value problem under constant flux boundary conditions consists of rarefaction waves, shock waves and constant states connecting the injection state to the production state. In incompressible systems with constant pressure boundaries, the total volumetric flux is a function of time and hence, the classical Buckley-Leverett theory is not valid. However, the saturation wave structure obtained from the constant flux boundary condition problem can be used in the solution of the associated problem with constant pressure boundaries by determining the flux analytically as a function of time.
The solution for a four-component gas/oil displacement case study is presented. The determination of time dependent volumetric flux from the solution of the constant flux problem is demonstrated. Results are also obtained using a numerical approach and are compared to the analytical results. This indicates that the analytical solution is indistinguishable from the numerical solution as the number of grid blocks in the numerical method approaches infinity. However, a very fine grid is needed for an acceptable solution.
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Modeling of Black Oil PVT-data
Authors K. Potsch, P. Toplack and T. GumpenbergerSummaryReservoir engineers are in need of information about fluid properties of reservoir fluid before calculating reserves or production scenarios. Mature fields may require reviews of older data sets that are sometimes scarce. The so-called PVT properties (black oil or compositional) are generated in either an in-house or an external lab. Prior to their use, these data sets need to be checked for their correctness and consistency. Modelling with correlations for estimating some of the properties or equations of state (EOS) provides only limited insight. First, they are not applicable for each reservoir fluid. Due to the variety of chemical composition every fluid is unique. Secondly, the correlations are purely numerical, lack non-dimensionality and consider physics only to a limited extent. Black oils separate below saturation pressure into a vapour and a liquid phase. The gas phase, consisting predominantly of the lighter compo-nents, increases with decreasing pressure. In other words, the higher the pressure the more gas is in solution. It influences other quantities, like formation volume factor Bo, oil com-pressibility Cpo and oil viscosity µo. This paper analyzes how the components of the gas phase contribute to the PVT-properties mentioned. It is assumed that the light components assume a certain volume in the liquid phase which is dependent on temperature and pressure. Additionally, the shape of the heavier components plays a role. As the light and heavy molecules in the mixture try to assume a minimal volume, the conversion factor from the va-pour to the liquid volume of the light components varies to some degree. The parameters (conversion factors) necessary to model Bo, Cpo and µo are extracted from experimental data. Mathematically, it is a minimization problem where the variables need to be positive. The solution is sought with a simplex algorithm. Once the parameters are determined, an estimate of Bo, Cpo and µo can be calculated, and plausibility and consistency of lab PVT-data can be carried out. This approach provides a valuable tool for the reservoir engineer in assessing the quality of PVT-data.
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Pore-scale Modelling of Shale Gas Permeability Considering Shale Gas Adsorption
Authors X. Zhao, J. Ma and G. CouplesSummaryShale gas permeability needs to be estimated in order to predict the quality of shale gas reservoirs and to develop shale gas production strategies. With advances in high-resolution imaging technology, one can characterise the pore space of a gas shale sample, which typically contains pores ranging from micrometers to nanometers, and to construct a pore-space model to simulate the gas flow numerically and to calculate the permeability. Gas flow has long been known to behave differently in such a confined space, and the smaller the pores the larger discrepancy is generally expected between gas and liquid (e.g. water) permeability. Since shale gas molecules stored mainly in nano-metre pores in kerogens by gas adsorption, adsorbed gas molecules, of half-nanometres in diameter, could reduce the pore size for free gas flow substantially and so alter the gas permeability significantly.
In this work, we extended a model for modelling shale gas flow to account for the gas adsorption effect. We adopted the Langmuir single-layer adsorption model to the multiple layers. We analysed the gas adsorption impact on the permeability on a cylindrical pore analytically, and on a shale sample whose pore space are represented as a node-and-bond pore network, using our network flow model ( Ma et al., 2014 ). The results revealed that the adsorption effect depends strongly on the gas pressure and the radii of pores. Given that low gas pressure increases gas slippage at pore surfaces and decreases the thickness of the adsorption layers then, consequently, enhances the permeability, undesirable operation conditions could lead to an earlier decline of gas production.
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Finite Volume Method for Modelling Gas Flow in Shale
Authors P. Lorinczi, A.D. Burns, D. Lesnic, Q.J. Fisher, A.J. Crook, C. Grattoni and K. RybalcenkoSummaryGas flow in shale is a complex phenomenon and is currently being investigated using a variety of modelling and experimental approaches. A range of flow mechanisms need to be taken into account when describing gas flow in shale including continuum, slip, transitional flow and Knudsen diffusion. A finite volume method (FVM) is presented to mathematically model these flow mechanisms. The approach incorporates the Knudsen number as well as the gas adsorption isotherm, allowing different flow mechanisms to be taken into account as well as methane sorption on organic matter. The approach is applicable to non-linear diffusion problems, in which the permeability and fluid density both depend on the scalar variable, the pressure. The FVM is fully conservative, as it obeys exact conservation laws in a discrete sense integrated over finite volumes. The method is validated first on unsteady-state problems for which analytical or numerical solutions are available. The approach is then applied for solving pressure-pulse decay tests and a comparison with an alternative finite element numerical solution is made. Results for practical laboratory pressure-pulse decay tests of samples with very low permeability are also presented.
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A Mathematical Model for Flow in Multi-stage Hydraulic Fracture Systems Using Fractal Theory
Authors W. Wang, Y. Su and M. ShahvaliSummaryMulti-stage hydraulic fracturing has received considerable attention for production from unconventional resources. One of the key technologies that made development of unconventional shale formations possible is the creation of complex fracture network systems via interaction of hydraulic fractures, natural and induced fractures. Currently, most modeling approaches for multi-stage hydraulically fractured wells are based on diffusivity flow in several distinct scales (matrix/fracture), in which the network of fractures is assumed to be connected and equivalent to a homogeneous medium of Euclidean geometry. In this paper we incorporate a more detailed description of complex fracture networks to improve the pressure transient analysis of hydraulically fractured shale formations. Specifically, we employ a Fractal Diffusivity approach in which characteristics of flow in a dual-continuum porous medium is taken into consideration using fractal theory. In our dual-mechanism Fractal Diffusivity approach, we represent the average porosity and permeability of the fracture network using the fractal porosity-permeability relations. We use a trilinear flow mathematical model to represent the flow in hydraulic fractures, in the formation between the fractures, and in the formation away from the hydraulic fractures. To solve the equations at different regions, we prescribe proper boundary conditions and use Laplace transformation and numerical inversion from Laplace domain to time domain. Using numerical simulation, we validate the new semi-analytical solutions (Fractal Fracture Diffusivity solution) for flow in finite-conductivity multi-staged fractured reservoirs. We perform sensitivity analysis and show that the solution mostly depends on the value of the fractal parameters chosen. Moreover, we generate type curves for well bore pressure and pressure derivatives for multiple transverse fractures for a variety of external boundary conditions and show that the proposed mathematical model is more general than the dual porosity trilinear flow models. We also show applications of the proposed model in flow regime diagnostics.
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A Study of Hydraulic Fracturing Clean-up Efficiency in Unconventional Gas Reservoirs Using Statistical Approaches
Authors H.R. Nasriani, M. Jamiolahmady, E. Alajmi and P. GhahriSummaryHydraulic fracturing is widely used to improve well productivity especially in unconventional reservoirs. This costly operation, however, sometimes underperforms. One of the main reasons for this poor performance is poor clean-up efficiency of injected fracturing fluid (FF).
In this work, a parametric study of FF clean-up efficiency of hydraulic fractured vertical wells was performed with 49152 simulations (in 12 sets) consisting of injection, soaking and production periods.
Due to the large number of required simulations, that were conducted using a commercial reservoir simulator, a developed computer code was used to automatically read input data, run simulations and creates output data. In each set (consisting of 4096 runs), simultaneous impacts of 12 parameters (fracture permeability, matrix permeability and capillary pressure, end points and exponents of Corey gas and FF relative permeability curve in both matrix and fracture)were studied. To sample the variables domain and analyse results, two-level full factorial experimental design and linear surface model describing dependency of gas production loss (GPL), compared to 100% clean-up, to pertinent parameters at three production periods (10, 30 and 365 days) were considered and supported by the tornado charts of fitted equations, frequency of simulations with given GPL and FF saturation maps.
Results indicate that generally parameters controlling FF mobility within fracture had greatest impact on GPL reduction. However in sets with very low matrix permeability especially when applied pressure drop during production is low, the effect of fluid mobility in the matrix on GPL is more pronounced, in other words, it is important how gas and FF flow within matrix rather than how fast fracture is cleaned. In tighter gas formations, generally more GPL and slower clean-up was observed. The effect of matrix capillary pressure on GPL reduction was more pronounced when drawdown was very low and/or soaking time was extended. This observation was more profound in tighter formations, i.e. for these formations, the effect of a change in drawdown and/or soaking time on matrix capillary pressure and GPL was more pronounced.
These findings can be used to make better decisions on the performance and optimised design of hydraulic fracturing, which is a costly but widely used stimulation technique for unconventional low permeability gas reservoirs.
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Ion Exchange and Osmotic Mechanisms of Low Salinity Water Flooding and Modeling of Oil Displacement in Clay Reservoirs
Authors V.V. Kadet and P.S. ChagirovSummaryMost reservoir sands contain clay minerals. It is the well known fact that fresh water injected into the clay reservoirs causes swelling of clays. The swelling clays partly block the capillary openings in the sand and therefore reduce the rate of flow to the well bore. In addition clay minerals are susceptible to destruction of its molecular structure by exposure to waters [1]. Clay particles emerged from swelling process can block capillary openings as well. However, a great number of laboratory tests [ 1 ] showed that enhanced oil recovery can be obtained when performing a low salinity waterflooding (LSW). Despite increasing interest in LSW, none of the proposed mechanisms have so far been accepted as the “true”, none of the mathematical models of LSW have been created.
Mathematical modeling is based on analysis of electrokinetic and physicochemical effects at micro-level.
This process includes description of electro-osmotic flow in a capillary, ion-exchange process in diffusion layer of a capillary and also osmotic swelling of clays. Porous medium is generally modeled by the parallel conducting chains [ 2 ] bound up with interconnecting capillaries so that current could flow into the other chain. The main characteristic of this capillary system is described by the probability density function f(r). After the all micro- processes having been described, we go on with modeling at macro-level by measuring reservoir and two-phase flow characteristics (porosity, permeability, relative permeabilities, capillary pressure curves) depending on clay factor and mineralization of injected water.
Rapoport-Leas model has been chosen to estimate efficiency of oil displacement. This model allows us to take into account capillary pressure taking place during low salinity waterflooding. Salt transport in porous medium is described by convective diffusion equation which includes ion-exchange reaction rate, diffusivity and hydrodynamic dispersion.
The results of the calculations show the growth of oil production rate, water cut decrease and as a consequence an increase in recovery factor when performing LSW. The results fit well with experimental data.
- Tang, G., Morrow, N. R. Influence of brine composition and fines migration on crude oil/brine/rock interactions and oil recovery. Journal of Petroleum Science and Engineering, 1999, 99–111.
- VI. Selyakov and VV Kadet, Percolation Models for Transport in Porous Media With Applications to Reservoir Engineering. Kluwer Academic Publishers. Dordrecht/Boston/London, 1996, 241 p.
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High-order Simulation of Foam Enhanced Oil Recovery
Authors J.M. van der Meer, D.E.A. van Odyck, P. Wirnsberger and J.D. JansenSummaryIf secondary hydrocarbon recovery methods fail because of the occurrence of gravity override or viscous fingering one can turn to an enhanced oil recovery method like the injection of foam. The generation of foam can be described by a set of partial differential equations with strongly nonlinear functions, which impose challenges for the numerical modeling.
To analyze the effect of foam on viscous fingering, we study the dynamics of a simple foam model based on the Buckley-Leverett equation. Whereas the Buckley-Leverett flux is a smooth function of water saturation, the foam will cause a rapid increase of the flux function over a very small saturation scale. Consequently its derivatives can become extremely large and impose a severe constraint on the time step due to the CFL condition.
Until now, the methods applied to foam EOR processes are only first-order accurate and do not incorporate stabilization near the foam front as far as we know. In order to improve the accuracy near the foam front we make use of total variation diminishing schemes that preserve the numerical stability of the solution. Two dimensional simulations, including gravity, will shed light on the conditions under which foam might exhibit viscous fingering behavior.
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Investigation of Saturation Dependency of Oil Relative Permeability during WAG Process through Linear and Non-linear PCA
Authors E. Ranaee, G. Porta, M. Riva and A. GuadagniniSummaryWe characterize three-phase relative permeability data sets available in the literature in terms of basic descriptive statistics, bivariate correlation, as well as linear (PCA), nonlinear (NLPCA) and hierarchical principal component analyses (h-NLPCA). These studies are viewed in the context of the assessment of three-phase oil relative permeabilities for water alternating gas injection (WAG) protocols, where a proper (qualitative and quantitative) analysis of the dependence of observed three-phase oil relative permeability data on fluid saturations is of critical relevance for practical applications. Here, we focus on the characterization of the dependence of three-phase oil relative permeability on an identifiable set of Principal Components. We analyze the relationship between observed core scale three-phase oil relative permeability and input variables which are typically employed in the application of existing effective (pseudo-empirical) models. Input variables include saturations of fluids, saturations ending points, as well as two-phase relative permeabilities obtained from oil-water and oil-gas environments. The use of available prior information about saturation ending points is also discussed in the framework of Constrained Principal Component Analysis (CPCA). Our results show that: (i) the degree of nonlinearity displayed by the relationship between the input variables and three-phase oil relative permeability is in contrast with the fundamental assumptions underlying existing empirical models; (ii) a sigmoid-based empirical model can effectively characterize three-phase oil relative permeability as a function of fluid saturations, saturation ending points and oil relative permeability data collected under two-phase conditions.
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Barodiffusive Extension of Three-phase Flow Model for NegSat Method and Regularization of Three-phase Instability
Authors M. Panfilov, I. Panfilova and A. NaimanovaSummaryTwo fundamental problems of three-phase compositional flow in underground reservoirs have been solved by introducing the similar technique of barodiffusive extension of the classical three-phase model. It consists of introducing of the pseudo barodiffusion terms that are proportional to the weighted sum of the gradients of phase pressures, due to which one can change the direction of the fluxes of individual chemical components.
First of all, this technique enabled us to complete the method of negative saturations for three-phase flow, which was developed to describe the situations when various zones of reservoir contain different number of phases. The method consists of replacing the true fluid by a fictitious three-phase fluid having specific properties that satisfy the equivalence principle. Two fundamental problems, non resolved in preceding publications, concern (a) the replacement of a two-phase fluid by three phases, and (b) the extension to the case when overcritical zones appear. We have shown that the main difficulty in establishing the equivalence between two-phase and three-phase fluids consists of the non-colinear fluxes of chemical components in a two-phase flow. To reach the vectorial equivalence between fluxes, we have introduced the pseudo barodiffusion in the fictitious three-phase fluid. The barodiffusion coefficients and the directions of the fluxes result from the equivalence conditions in a unique way. The same technique provides the solution for the case when the flow contains the zones occupied by overcritical fluid.
In the case of ideal mixing within the phases without capillarity, the flow equations can be converted to the system of conservation laws with respect to the saturations or total concentrations. However the uniform flow equations are non-classical due to the terms of pseudo barodiffusion. The analysis has revealed that the barodiffusion terms ensure the hyperbolic character of the system. Consequently, the well known physical instability that arises in three-phase flow due the loss of hyperbolicity, does not appear in our extended barodiffusive model. Thus, the introduction of the small barodiffusion is the way to suppress the appearance of three-phase instability.
To ensure the numerical stability, we applied the monotone upwind high-order scheme for conservation laws with predictor-corrector. We have calculated several cases of miscible gas injection into the reservoir containing initially oil and water, and proved the good convergence of the result obtained compared to the simulations performed by Eclipse compositional and other techniques.
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Steady State Upscaling of Polymer Flooding
Authors S.T. Hilden, K.A. Lie and X. RaynaudSummaryUpscaling of parameters involved in single and two-phase flow has been researched quite extensively, and several methods for performing upscaling are known and understood. Less work has been done related to upscaling of enhanced oil recovery simulations. This is what we investigate, and in particular, we consider upscaling of parameters related to polymer flooding, which is the process in which large polymer molecules are added to the injected water to enhance its ability to push hydrocarbons through the reservoir. Herein, the polymer flooding process is described as a two-phase, immiscible system that in addition to a Todd-Longstaff mixing model includes permeability reduction, polymer adsorption, and dead pore space.
Effective parameters are computed by running simulations until a steady-state is reached and then performing upscaling based on the fluxes. This method is used by a major oil company as part of an established work flow for single and two-phase upscaling, and it is therefore natural to try to extend the method to polymer flooding. The upscaling is performed on the meter scale, where the steady-state assumption best can be justified. The procedure involves first performing single-phase upscaling of the absolute permeability, then two-phase upscaling of relative permeabilities, and finally, upscaling of the parameters involved in polymer flooding. The new upscaling method is verified against an analytical solution and validated on two synthetic models that include real data.
Results show that the permeability reduction factor, which only depends on polymer concentration in the fine-scale model, will generally also depend on water saturation in the upscaled model. This introduces addition computational costs in the simulation, since the property evaluations now require extensive use of lookup-tables and interpolation. We therefore suggest making simplifications in order to reduce the complexity.
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Image Based Pore-scale Models of Flow through Porous Media - Oil Recovery Applications
Authors I.I. Bogdanov, J. Kpahou and I. BondinoSummaryThe importance of pore-scale flow models for practical applications is widely recognized. Due to recent advances in computed microtomography (μCT) the reconstructed samples are now used for direct numerical simulations (DNS) of the flow. This technique gives a unique opportunity for non-destructive characterization; nevertheless a typical study encounters several challenges. The discussion of the most difficult steps of modeling methodology is our first objective.
The description of dynamic phase distribution and behavior of the fluid interface is a problem of primary importance. A regularization technique may affect the results in non-trivial ways; instead the diffuse-interface method offers a thermodynamic description of phase “mixing” zone and handles the morphological changes of interface and related physical effects. A series of model tests including the juxtaposition to analytical solutions for capillary channel flow, estimation of spurious velocity around a droplet and others, are presented. The quantitative demonstration of the method is our second objective.
Among numerous oil recovery applications one can mention the transport properties determination for different physical environment, the study of fluids entrapment/mobilization, the flow patterns at different capillary numbers and viscosity ratios, the emulsion and foamy oil flow, etc. Here we address the analysis of viscous fingering dynamics (oil-water systems, 2D synthetic medium) and the 3D stationary configurations of single and two-phase flow in real porous samples at different Reynolds, Cahn and capillary numbers. In particular, the computations based on μCT image reconstruction aims at the examination of fluid irreducible saturations. This constitutes our third objective. A discussion on the possibilities and limits of the model in quantitative characterization of porous materials is offered. Contribution of the pore-scale DNS to reservoir characterization becomes an increasingly important factor for numerous practical oil recovery applications.
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The Significance of a Detailed Phase Description in Simulations of Surfactant Flooding
Authors A.M. Al Sofi and A.M. Al KhatibSummaryTwo simulation approaches for modelling surfactant floods exist in the literature. The main difference between the two is the description of the surfactant phase behavior. The first (detailed) approach includes a thorough representation of the surfactant ternary phase behaviour while the second (simplified) approach ignores the formation of a middle phase microemulsion. Several reasons support the use of a simplified two-phase approach including the commercial availability of this option, the ease of incorporating such option in existing waterflood simulators, and the relative ease of generating input data.
Therefore, the objective of this study is to investigate whether the two approaches differ in terms of their predictions. In other words, we ultimately want to know whether a simplified two-phase simulation approach is suitable for the evaluation and design of a given surfactant formulation in any reservoir and/or operational settings or whether we must account for the ternary phase behaviour. For this purpose, we use the University of Texas Chemical Flooding Simulator (UTCHEM) for evaluating both the simplified and detailed modelling options. Simplified models are also built in UTCHEM by diminishing the salinity window. This option was chosen in order to use the same simulator suite for the evaluation of both the detailed and simplified assumptions.
In this work, we first use a detailed surfactant three-phase simulation model that was previously generated in UTCHEM using laboratory data and calibrated against coreflood experiments to generate three simplified surfactant two-phase pseudo models that are equivalent in 1D. Their equivalency in 1D is demonstrated using analysis of variance (ANOVA). We later design two simulation-based experiments to evaluate the suitability of the simplified models for field-scale predictions. Essentially, we divide the problem into two slightly simpler parts. The first experiment looks at the evaluation of a surfactant flood under uncertainty and the second looks at the optimisation of the surfactant injection scheme under a single deterministic realisation. For each of those two simulation-based experiments, we use a 4 × 4 Graeco-Latin square design requiring 16 simulation runs. Beside the surfactant simulation model, three factors are investigated in each of those experiments. For the robust evaluation experiment, the additional factors are permeability, adsorption, and initiation. For the optimisation experiment, the additional factors are surfactant slug size, surfactant concentration, and the injection rate.
ANOVA results of both experiments suggest the surfactant models do not differ significantly. This conclusion is supported by Tukey comparisons and the main effects plots. Therefore, the results suggest that a surfactant two-phase model can reasonably approximate the actual ternary phase behaviour of surfactants. Consequently, such simplified two-phase models can be used to obtain reliable predictions for field scale simulations.
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Recovery of Light Oil by Air Injection
Authors N. Khoshnevis Gargar, A.A. Mailybaev, D. Marchesin and J. BruiningSummaryIn this paper we review the results of analytical, numerical and experimental studies related to air injection into porous medium containing initially light oil, water and gas at medium pressure conditions. The new combustion mechanism is described, where the process of the medium temperature oxidation interacts with the oil vaporization/condensation, resulting in a resonant combustion wave structure. We discuss bifurcations of combustion regimes with a change of reservoir parameters, and analyze the effectiveness of the proposed technique for recovery in light oil reservoirs.
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Analysis of Heat Loss Effects on Thermal Pressure Falloff Tests
Authors A. Jahanbani Ghahfarokhi and J. KleppeSummaryAnalysis of pressure falloff tests gives initial estimates of swept volume, essential for the evaluation of a thermal recovery process. The analysis is based on a two-zone composite reservoir model with highly contrasting fluid mobilities, where the swept zone is assumed to behave as a closed reservoir for a short period exhibiting pseudo steady state behavior.
The upward buckling of the pressure derivative curves at late times in some cases could not be explained using the conventional composite models. This issue and some of the errors associated with the estimation of swept volume may possibly be related to heat loss which could have significant effects on the pressure behavior and dominate the pseudo steady state flow.
A model for the analysis of falloff tests with significant heat loss was suggested by Stanislav et al. (1989) . However, there are limitations in the application of this approach to practical steam falloff tests. Moreover, permeability should be known in advance for further analysis.
In this paper, a modified method of analysis considering heat loss is discussed which makes the flow regime identification easier and removes some of the practical limitations. Results of the analysis show improvement over the estimates obtained by other methods.
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An Exploration of Porosity-permeability Relationships in 3D Fracture Network Using the Lattice-Boltzmann Method
By R.A. ArcherSummaryThis paper explores the relationship of permeability to fracture aperature, density and mineralisation in synthetic dual porosity media. Fluid flow is simulated using the Lattice-Boltzmann method which allows for detailed fracture geometries to considered in 3D. The geometry of the matrix part of the media is built using a Voronoi based approach similar to the work in 2D by Newman and Yin (2013) .
Stress sensitivity can be an important phenomena in dual porosity media. Reductions in reservoir pressure as production proceeds cause decreases in porosity and consequentially decreases in permeability. This effect is explored by considering the impact of fracture compressibility.
While it is not anticipated that full field modelling at this scale will be undertaken in the foreeable future, modelling such as this allows insight to defining appropriate REVs and into upscaling.
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Modelling of Irreversible Deformation near the Tip of a Crack in a Porous Domain Containing Oil and Gas
Authors A. Lukyanov, N. Chugunov, V.M. Sadovskii and O.V. SadovskayaSummaryThermomechanical processes observing in deformable solids under intensive dynamic or quasi-static loadings consist of coupled mechanical, thermal and fracturing stages. The fracturing processes involve formation, motion and interaction of defects in crystals, phase transitions, breaking of bonds between atoms, accumulation of micro-structural damages (pores, cracks), etc. Irreversible deformations, zones of adiabatic shear micro-fractures are caused by these processes. A dynamic fracturing is a complicated multistage process, which includes appearance, evolution and confluence of micro-defects and formation of embryonic micro-cracks, pores that can grow and lead to the breaking-up of bodies with formation of free surfaces. This results in a need to use more advanced mathematical and numerical techniques.
This talk presents modeling of irreversible deformation near the tip of a crack in a porous domain containing oil and gas during the hydraulic fracturing process. The governing equations for a porous domain containing oil and gas are based on constructing mathematical model of thermo-visco-elasto-plastic media with micro-defects (micro-pores) filled with another phase (e.g., oil or/and gas). The micropores can change their size during the process of dynamical irreversible deformation. The existing pores can expand or collapse. The model was created by using the fundamental thermodynamic principles and, therefore, it is a thermodynamically consistent model. All the processes (i.e., irreversible deformation, fracturing, micro-damaging, heat transfer) within a porous domain are strongly coupled. Therefore, explicit normalized-corrected meshless method is used to solve the resulting governing PDEs. The flexibility of the proposed technique allows running efficiently using a great number of micro- and macro-fractures. The results are presented, discussed and future studies are outlined.
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Reexamining CO2 Storage Capacity and Utilization of the Utsira Formation
Authors O.A. Andersen, H.M. Nilsen and K.A. LieSummaryIn this work we provide estimates of CO2 storage capacity of the Utsira Formation using the recently provided datasets from the Norwegian Petroleum Directorate, taking CO2 density variation into account. We also investigate strategies on how to realize as much as possible of this potential in large-scale injection scenarios. We base our study on the assumption that the limiting factor for CO2 storage at Utsira is the efficient use of available trapping mechanisms, with pressure buildup being of secondary concern. We consider the trapping mechanisms considered most important for the medium-to-long term (structural, residual and dissolution trapping), using a combination of modeling tools based on the Matlab Reservoir Simulation Toolbox (MRST).
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Reservoir Modeling through Fast Wavelet-based Stochastic Simulation
Authors H.M. Mustapha, S. Chatterjee and R. DimitrakopoulosSummaryStochastic simulation of complex geology is addressed through discrete wavelet transformation (DWT) that handles multiscale spatial characteristics in datasets and training images. The simulation of the proposed approach is performed on the frequency (wavelet) domain. A multiscale, multipoint simulation algorithm is proposed in this paper in which the scaling image and wavelet images across the scale are simulated jointly. The inverse DWT reconstructs simulated realizations of an attribute of interest in the space domain. The proposed algorithm reduces the computational time required for simulating large domain as compared to spatial domain multipoint simulation algorithm since the simulation is performed in the wavelet domain in which numbers of nodes to be simulated are significantly less as compared to spatial domain nodes. The algorithm is tested with an exhaustive dataset using unconditional simulation in two-dimensional data set. The realisations generated perform well and reproduce the statistics of the training image. The study conducted comparing the spatial domain filtersim multiplepoint simulation algorithm suggests that the proposed multiscale, multipoint algorithm generates equally good realisations at much lower computational cost.
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Shape Factor for Dual-permeability Reservoir Simulation - Effect of Non-uniform Flow in Fracture Network
Authors J. Gong and W.R. RossenSummaryThe flow properties of naturally fractured reservoirs are dominated by flow through the fractures. In a previous study we showed that even a well-connected fracture network behaves as one near the percolation threshold in some cases: i.e., most fractures can be eliminated but still form a percolating sub-network with virtually the same permeability as the original fracture network. In this study, we focus on the influence of eliminating unimportant fractures on the inferred characteristic matrix-block size. We model a two-dimensional fractured reservoir in which the fractures are well-connected. The fractures obey a power-law length distribution, as observed in natural fracture networks. For the aperture distribution, because information from the subsurface is limited, we test a number of cases: narrow and broad log-normal and power-law distributions and one where aperture correlates with fracture length. The matrix blocks in fractured reservoirs are of varying sizes and shapes; we calculate the characteristic matrix-block size from the ratio of matrix-block area to its perimeter. We test different criteria to determine the critical sub-network, such as aperture, flow simulation results, etc. We show how the characteristic matrix-block size increases from the original fracture network to the critical sub-network. An implication of this work is that the matrix-block size, or shape factor, used in dual-porosity or dual-permeability waterflood or EOR simulations should be based not on the entire fracture population but on the sub-network that carries almost of all the flow.
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Estimating Geo-modeling Control-parameters from Historical Data by Means of the EnKF
Authors R. Valestrand and D.R. RenardSummaryEstimating geo-modeling control-parameters from historical data by means of the EnKF.
Over the last decade the ensemble Kalman filter (EnKF) and other offspring ensemble based methods (here also referred to as EnKF), have attracted attention as promising methods for solving the reservoir history matching problem. The EnKF has successfully been used to estimate flow properties, such as permeability and porosity, of each grid cell in a history matching loop. For more complex problems, such as facies estimation problems, there are challenges. The direct use of EnKF to estimate e.g. porosity and permeability of a facies field could lead to a good history match, but, it would ruin the geological realism of facies fields, i.e., the boundaries between facies would be smeared out. Several papers have addressed this problem by estimating facies boundaries instead of, or in addition to, the petrophysical properties, in order to maintain the geological realism. The facies boundaries are typically reparameterized using variables that can be characterized by Gaussian to suit the assumption of EnKF.
In this work we attack the problem from a different angle; we perform ‘‘the Big-Loop’’ update, i.e., the geomodel control-parameters are updated using production data and updated facies models are generated with updated control-parameters, using the same geo-modeling work-flow, so that geological realism is naturally preserved. The implementation is referred to as the Big-Loop as we update the geological models not only reservoir models. To perform the investigation we have coupled a geostatistical simulator, a black oil flow simulator and the EnKF. The Big-Loop is tested on examples with facies models generated using the truncated PluriGaussian method. Based on the results found in this work the following can be stated:
* The update models satisfy geological constraints imposed in the geo-modeling work-flow. The match to historical data is improved by updating both local and global geo-parameters,
* It seems beneficial to include both global and local geo-parameters in the Big-Loop approach compared to only local or only global parameters.
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Multi-objective History Matching of a Deltaic Reservoir with Non-stationary Geostatistical Modelling
Authors M.H. da Silva Caetano Caeiro, V. Demyanov and A. SoaresSummaryHistory matching with a single objective function reflects only some global aggregated reservoir match quality and is not flexible enough to distinguish between local effects and provide a spread of diverse models for the forecast. On the contrary, multi-objective optimization (MOO) can distinguish between the contributions to the goodness to fit from some local parts of the model. History matching with MOO results in more diverse matched solutions and more robust prediction. Use of MOO in history matching also provides extra flexibility in matching local parts of the model especially in non-stationary cases. In this work we show how MOO improves the match quality of a non-stationary geostatistical model of a deltaic reservoir. Ensemble of multiple history match models achieved with MOO feature of better quality local matches. . Complex reservoirs descriptions with non-stationary characteristics are hard to match due to their high intrinsic heterogeneity and uncertainty associated with non-stationary description. Direct sequential simulation with local anisotropy correction (DSS-LA) provides a way to account for large scale trends, which are subject to uncertainty. The implementation of local anisotropy tackles the problem of non-stationary of the geostatistical model by imposing a trend in spatial correlation structure. The anisotropy change follows the trend in spatial correlation range and orientation across the reservoir model. While the uncertainty in the trend and spatial correlation is resolved through history matching using multi-objective optimisation.
The proposed adaptive stochastic sampling framework integrates DSS-LA with the multi-objective history matching. Multiple optima matched models of porosity and permeability were obtained allowing the uncertainty assessment of the anisotropy model parameters.
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Improved Realism of Channel Morphology in Object Modelling with Analogue Data Constraints
Authors A.K. Kuznetsova, J.A. Almeida and P. LegoinhaSummaryPreserving realism of geological structures is a challenge for the reservoir engineer when history matching. Object based models offer the chance of enforcing realism but are hard to constrain to observed data.
In this paper we present a novel approach for generating channels that is easier to match to observed data than the standard object-based approach. Our new technique uses Object Modelling as the first step, mimicking the geometry of fluvial sand channels. Object Modelling enables us to constrain channel geometry with analogue information of channel local orientation and dimension. However, the morphology of the output models is smoothed and the transition channel/no-channel zones are associated with uncertainty as they are not constrained by a variogram or multi-point statistics. The second step, Probability Field Simulation, adds realism in a final output model imposing a variogram to the transition zones. Probability Field Simulation preserves the channel geometry of the object models and shows a significant CPU advantage compared with currently available approaches.
The new method was applied to a synthetic problem in which 3 different channel scenarios were considered: low density, high density, and high density with thin channels. The results show that the proposed method can handle this range of channel densities, and we can therefore assume that the approach could be implemented for different channel density data. A further important finding was that there is no significant difference between the CPU time for each case.
In summary, our new technique is able to constrain object models to a variety of observed data types such as channels size, density and orientation, while showing significant CPU saving and preserving the geological realism of the matched model.
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Mathematical Modelling and Numerical Simulation of the Well-reservoir Coupling Flow
Authors C.R. Maliska, M.P. Tada, A.F.C. Silva and A.B. SopranoSummaryNormally, when reservoirs specialist solve the multiphase flow of oil, gas and water in petroleum reservoirs the existing wells are not treated with the required details. Well trajectories are poorly defined with respect to the reservoir grid and the well models for capturing the large gradients near the well are too simplified, among other simplifications. It is also true that when well specialists solve the flow in the well the reservoir is considered as a constant pressure body. However, the modern technologies for increasing oil production is focused on the design of the well completion, the so-called “smart wells”. These are complex devices installed along the column which are able to control and creating flow patterns from the reservoir to the well, increasing the production. The design of such completions required the knowledge of the flow characteristics near the well, what is only possible by solving the coupled flow between well and reservoir.
This paper presents a mathematical model for the oil, gas and water in the reservoir and in the well, following by the development of a numerical scheme for solving the coupled flow using a finite volume method. Full permeability tensor is considered, taking into account heterogeneities and anisotropies, with compressible fluids and rock. The unknowns are the mass fractions of the components, opposed to the saturation formulation, which has the numerical difficulty in dealing with the disappearance of the gas phase. For the well a drift flux method is employed solving a 1D flow along the horizontal well using a very stable approach for the pressure-velocity coupling in the well. A Newton-like method is used for both, reservoir and well variables, but using a segregated strategy for the well/reservoir coupling. This has proven to be suitable due to the large differences in the spatial and time scales of the problems. Several three-dimensional coupled multiphase flow were solved, including near well analysis with fractures, demonstrating the capabilities of the method.
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Use of Composite Elastic Modulus to Predict Inflow Performance
By T.A. JelmertSummaryThe proposed methodology is valid for any problem described by a diffusivity equation. As an application of the more general technique, we investigate fluid flow in a stress-sensitive reservoir. In case of production, the permeability, viscosity and fluid density may decrease in the near wellbore region. The reservoir thickness may also decrease. The objective of the present study is to quantify such changes and point out the effect on the inflow performance relationship.
The flow equation for stress-sensitive reservoirs may be highly non-linear. A non-linear variable shows up as a quadratic pressure gradient term in the diffusivity equation, Matthews and Russel (1967) . A logarithmic pressure transform may reduce the effect of the quadratic gradient term. The method involves the assumption of a constant elastic modulus. The equivalent assumption is that the variable may be approximated by an exponential function of pressure. Such variations have some experimental support. Many studies investigate the effect of a single pressure dependent variable. Then, a constant permeability modulus or compressibility is assumed. We propose a composite elastic modulus to investigate the simultaneous effect of an arbitrary number of pressure dependent variables in the transport term. The methodology depends on the assumption that every pressure-dependent variable may be approximated by an exponential function.
We investigate steady state flow, then the linearization of the diffusivity equation in terms of the transformed variable is complete, and analytical solutions are readily available. Solutions in terms of pressure are obtained by the inverse transform. Due to the non-linearities, the reservoir behaves different during production and injection. The pressure sensitivity depends on the value of the composite modulus and may be negligible. We provide equations to estimate the error incurred by neglecting the pressure sensitivity. The proposed methodology may be extended to time-dependent problems, but with reduced accuracy. Perturbations techniques are available to improve the accuracy.
Conclusions:
A composite elastic modulus has been proposed. The effect of non-linear terms may be estimated one by one and in combination.
A transformation based on the modulus will linearize the steady state diffusivity equation and the boundary conditions.
The non-linear pressure solutions may be obtained by the inverse transformation.
The degree of non-linearity may be characterized by the numerical value of the composite elastic modulus. Increasing values leads to decreased performance.
By use of L’Hospitals rule, we find that the conventional reservoir solution (without Stress-sensitivity) is included in the generalized equation as limiting behavior.
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Modeling of Multiphase Flow in Elastic Porous Media Based on Thermodynamically Compatible Systems Theory
Authors E.I. Romenskiy, Y.V. Perepechko and G.V. ReshetovaSummaryThe new computational model for multiphase flow in deforming elastic porous media is proposed. The derivation of the model is based on the thermodynamically compatible hyperbolic systems of conservation laws theory. The flow of the mixture of compressible fluids in the elastic medium is supposed to be a continuum, in which the multiphase character of flow is taken into account. This phenomenological approach of continuum mechanics modelling allows us to formulate the system of governing equations in a divergent form, which is advantageous for the mathematical study of the different problems and for the development of advanced numerical methods.
We present a thermodynamically compatible model for the flow of fluids mixture in elastic porous medium. The governing equations comprise balance laws for phase masses, total momentum and total energy supplemented by the equations for relative velocities in divergent form. The high accuracy Runge-Kutta-WENO numerical method for solving equations of the model is presened along the numerical test problem.
The proposed model and developed numerical framework can be used in the wide range of oil recovery problems. Examples are: tracking oil/water interfaces in oil reservoirs, modeling of flows in the well surrounding formation.
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Estimating Bottom Hole Damage Zone Parameters Based on Mathematical Model of Thermo-hydrodynamic Processes
Authors L.A. Gaidukov, D.V. Posvyanskii and R.R. TukhvatullinaSummaryIn order to find the optimal parameters of workovers for recovering well productivity it is important to know the properties of the bottom hole damage zone, such as its radius and permeability. The effect of the damage zone can be described by the skin factor, which can be indirectly estimated by well test analysis. Temperature and pressure are the most frequently observed physical parameters in well testing. During transient tests, both the pressure and temperature are measured at the well bottom hole. Typically, only pressure transient data is considered, and pressure transient data analysis is used to estimate the skin-factor. However, due to well bore effects, it is impossible to identify the structure of the damage zone. The reservoir pressure and temperature both recover after a well shuts. Temperature changes in the reservoir are induced by heat convection, heat conduction and the Joule-Thomson effect. The flow rate of these physical processes depends on the properties of both the reservoir and the damage zone. By considering the differences from the pressure buildup curve, temperature transient data can be used to estimate the radius of the damage zone, which can help give more accurate well test interpretation [1] . The mathematical model for well tests analysis is based on the solution of diffusivity equation for pressure and heat transfer equation for temperature. In [2] Green function technique was used to solve the pressure equation. In this study we present joint solution of the equations for the pressure and temperature dynamics. The calculations take into account the dependence of Joule-Thompson coefficient on reservoir pressure. Calculations were performed for the interpretation of well test data in a gas producing well. The results are qualitatively consistent with field examples.
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Modeling Of Flow in a Fracture Inside Porous Medium
Authors D.YU. Khanukaeva and A. N. FilippovSummaryThe modeling of oil and gas production processes such as hydraulic fracturing or oil displacement requires coupled description of free flow and flow in a porous medium.
In this work we used Stokes’ equation for free viscous flow coupled with Brinkman’s equation for liquid flow in a porous medium. The latter is differential equation of the same order as Stokes’ equation. Both equations together with continuity equation and proper boundary conditions present mathematically correct formulation of the coupled flow boundary value problem. For two-dimensional and axial symmetric geometry we showed analytically that the solution of Brinkman equation totally coincides with Darcy’s velocity in the whole region of a porous medium, except for a thin layer adjacent to impermeable wall. The thickness of the mentioned layer depends on specific permeability of porous material.
Stokes-Brinkman model was used for simulation of flows in finite dead-end channel with porous walls. This model is aimed at the simulation of flow through a hydraulic fracture or flow in a natural crack inside a porous medium. The results were compared with experiments fulfilled at the Institute of Geospheres Dynamics RAS and have demonstrated excellent agreement.
The work is partially supported by RFBR (grant #14-08-00893a) and CRDF-Global (grant #FSAX-14-60158-0).
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A New Model Reduction Technique Applied to Reservoir Simulation
Authors M.G. Ghasemi and E.G. GildinSummaryOne of the challenges in reservoir simulation is the study and analysis of large scale models with complex geology and multiphase fluid for considering real life applications. Even with recent increase in the computation power, the fast and reliable simulation of the fine scale models is still resource-intensive and hardly possible. Particularly, in optimization and field planning, it is necessary to simulate the system for varying input parameters. Here, model order reduction (MOR) can be used to significantly accelerate the repeated simulation. Although theory as well as numerical method for linear systems is quite well-established, for nonlinear systems, e.g. reservoir simulation, it is still a challenging problem.
We apply a recently introduced approach for nonlinear model order reduction to reservoir simulation. In order to overcome the issue of nonlinearity, we introduce the bilinear form of the reservoir model. The bilinear approximation is a simple form of the parent system and it is linear in the input and linear in the state but it not linear in both jointly. This technique is independent of input of the systems, and thus is applicable for wide range of input parameters without any training. Also, the formulation allows certain properties of the original models to be preserved in the reduced order models. The basic tools known from tensor theory are applied to allow for a more efficient computation of the reduced-order model as well as the possibility of constructing two-sided projection methods which are theoretically shown to yield more accurate reduced-order models.
Examples are presented to illustrate this recent approach for the case of two phase flow modeling, and comparisons are made with the case of linearized models and the full nonlinear models. We discuss the model reduction techniques to be applied to the two-phase flow system. We conclude the paper with some remarks and point out two ways to generalize the findings of this paper as a future work.
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Using Percolation Theory to Estimate Recovery from Poorly Connected Sands Using Pressure Depletion
Authors H. Wen, P.R. King, A.H. Muggeridge and E.S. VittoratosSummaryIn conventional waterflooding of low to intermediate net to gross reservoirs there is always some oil unswept even in the sands connected to both injection and production wells. This is oil trapped in “dangling ends”: flow units only poorly connected to the main flow path. In many cases the unswept volumes can be very large, depending on the properties of the reservoir and fluids and the well locations.
In this paper we show how percolation theory can be used to estimate the volumes of oil recovered and those left behind in these dangling ends following a conventional waterflood, without recourse to large scale simulation. Percolation theory is a general mathematical framework for connectivity and has been used previously to investigate the connectivity of flow units. The structure of these connected clusters in terms of backbones and dangling ends has not been previously studied. The results are also used to estimate the recovery of the unswept oil from dangling ends by a waterflood with a voidage replacement ratio <1.
We use a simple model of stochastically-distributed sandbodies to describe the reservoir. Many realizations for a range of net to gross ratio values and sandbody:system sizes were generated. In each realization the clusters connecting the injection and production wells were identified. These spanning clusters were subdivided into backbones and dangling ends. The volume fractions of the backbone and dangling end were then obtained. The statistical average and standard deviation of the volumes association with these clusters were obtained from the ensemble of realisations. These were used to determine the percolation scaling relationships in terms of simple algebraic formulae that cover the whole range of net to gross ratio and system sizes.
Our results show that the fraction of dangling ends can reach 20% of the clusters, and 80% among the spanning clusters, indicating a major proportion of the oil would be unswept by conventional waterflood. Recovery is improved for voidage replacement ratios less than 1 when the dangling ends are drained by oil expansion and solution gas drive.
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The Interplay between Viscous and Gravity Forces in Two-phase Stratified Reservoir Crossflow
More LessSummaryIn stratified reservoir models, crossflow describes how communicating layers of different petrophysical properties allow fluid to pass between them. Viscous crossflow is usually thought to be driven by pressure differentials due to permeability differences between layers, and gravitational crossflow is driven by density differences between phases. In this work, numerical simulations of basic two-dimensional models are presented to demonstrate the effect of gravity-driven crossflow.
Stratified reservoir models are very relevant today, particularly as the importance of offshore carbonate reservoirs grow, with their highly laterally homogenous and vertically cyclical beds with enormous ranges of permeability. Even in reservoirs which are not completely stratified, high-permeability streaks are common, and the crossflow between them and the adjacent low-permeability layers can be vital for maximising the recovery factor.
The results of this investigation serve to demonstrate that gravitational crossflow from high to low permeability layers may not necessarily improve a two-phase displacement sweep, when the low permeability layer is not bound by an impermeable layer beneath. This suggests that, in order to build up physically meaningful analytical models of crossflow during two-phase displacement in stratified reservoirs, a greater understanding of the relative importance of viscous and gravity forces is often required.
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Upgridding of Geological Models Based on Equations of Two-phase Flow
Authors S.P. Rodionov, O.N. Pichugin, L.N. Sokolyuk and Y. V. ShirshovSummaryThis paper presents an upgridding procedure that makes use of an analytical solution for a two-phase flow equation in the fine scale model. Both horizontal and vertical flow in fine scale cells are taken into account using appropriately defined boundary conditions. The method takes into account two-phase flow properties, cell sizes, diagonal tensor of absolute permeability, porosity, initial conditions and boundary conditions. We attempt to mimic the discretisation error using the reservoir heterogeneity concept. We discuss some of the difficulties encountered in applying the published methods to realistic reservoir models. We present several simple numerical cases to prove the advantages of the new method over other upgridding methods.
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Enhanced Decision Making for Chemical EOR Processes under Uncertainty - Applying the LSPC Method
Authors A. Alkhatib and P. KingSummaryThe Least Squares Monte Carlo method is a decision evaluation method that can capture the value of flexibility of a process. This method was shown to provide us with some insight into the effect of uncertainty on decision making and to help us capture the upside potential or mitigate the downside effects for a chemical EOR process. The method is a stochastic approximate dynamic programming approach to decision making. It is based on a forward simulation coupled with a recursive algorithm which produces the near-optimal policy. It relies on Monte Carlo simulation to produce convergent results. This incurs a significant computational requirement when using this method to evaluate decisions for reservoir engineering problems because this requires running many reservoir simulations.
The objective of this study was to enhance the performance of the Least Squares Monte Carlo method by improving the sampling method used to generate the technical uncertainties used in producing the production profiles. The probabilistic collocation method has been proven to be a robust and efficient uncertainty quantification method. It approximates the random input distributions using polynomial chaos expansions and produces a proxy polynomial for the output parameter requiring a limited number of model responses that is conditional on the number of random inputs and the order of the approximation desired. The resulting proxy can then be used to generate the different statistical moments with negligible computational requirement. By using the sampling methods of the probabilistic collocation method to approximate the sampling of the technical uncertainties, it is possible to significantly reduce the computational requirement of running the decision evaluation method. Thus we introduce the least square probabilistic collocation method.
Both methods are then applied to chemical EOR problems using a number of stylized reservoir models. The technical uncertainties considered include the residual oil saturation to chemical flooding, surfactant and polymer adsorption and the viscosity multiplier of the polymer. The economic uncertainties considered were the oil price and the surfactant and polymer price. Both methods were applied using three reservoir case studies: a simple homogeneous model, the PUNQ-S3 model and a modified portion of the SPE10 model. The results show that using the sampling techniques of the probabilistic collocation method produced relatively accurate responses compared with the original method.
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Hidden Information in Ill-posed Inverse Problems
Authors S.S. Kahrobaei, M. Mansoori, G.J.P. Joosten, P.M.J. Van den Hof and J.D. JansenSummaryIt is well known that parameter updating of large-scale numerical reservoir flow models (a.k.a. ‘computer assisted history matching’) is an ill-posed inverse problem. Typically the number of uncertain parameters in a reservoir flow model is very large whereas the available information for estimating these parameters is limited. The classic solution to this problem is to regularize the unknowns, e.g. by penalizing deviations from a prior model. Attempts to estimate all uncertain parameters from production data without regularization typically lead to unrealistically high parameter values and therefore to updated parameter fields that have little or no geological realism. However, it has been suggested that the application of unregularized reservoir parameter estimation may still add value, because it, sometimes, gives an indication of the location of significant missing features in the model. We investigated under which conditions this perceived added value might occur. We conducted several twin experiments and applied unregularized parameter estimation to update uncertain parameters in a simple two-dimensional reservoir model that contained a major deficiency in the form of a missing high or low permeability feature. We found that in case of low-permeability barriers or high-permeability streaks it is indeed sometimes possible to localize the position of the model deficiency. To further analyze this behavior we conducted one-dimensional experiments using a transfer function formalism to characterize the identifiability of the location and magnitude of model deficiencies (flow barriers).
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Efficient Inference of Reservoir Parameter Distribution Utilizing Higher Order SVD Reparameterization
More LessSummaryReservoir parameter inference is a challenging problem to many of the reservoir simulation workflows, especially when it comes to real reservoirs with high degree of complexity and non-linearity, and high dimensionality. In a history matching problem that adapts the reservoir properties grid blocks, the inverse problem leads to an ill-posed and very costly optimization schemes. In this case, it is very important to perform geologically consistent reservoir parameter adjustments as data is being assimilated in the history matching process. Therefore, ways to reduce the number of reservoir parameters need to be sought after.
In this paper, we introduce the advantages of a new parameterization method utilizing higher order singular value decomposition (HOSVD) which is not only computationally more efficient than other known dimensionality reduction methods such as, SVD and DCT, but also provides a consistent model in terms of reservoir geology. HOSVD power is due to its ability to supply a reliable low-dimensional reconstructed model while keeping higher order statistical information and geological characteristics of reservoir model. In HOSVD, we take the snapshots in a 2D or 3D approach, i.e., do not vectorize original replicates, and stack them up into a tensor form, i.e. a multi-way array in multilinear algebra which leads to implementing tensor decomposition. Technically, we performed HOSVD to find the best lower rank approximation of this tensor that is an optimization problem utilizing alternating least square method. This results in a more consistent reduced basis.
We applied this novel parameterization method to the SPE10 benchmark reservoir model to show its promising parameterization performance. We illustrate its advantages by comparing its performance to the regular SVD (PCA) in a history matching framework using EnKF, as well as characterization performance of the ensemble-based history matching approaches along with HOSVD. Overall, HOSVD outperforms SVD in terms of reconstruction and estimation performance.
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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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The Hyperbolic Nature of a System of Equations Describing Three-phase Flows in Wellbores
Authors K. Sinkov, P. Spesivtsev and A.A. OsiptsovSummaryThe problem of the formulation of compressible, isothermal, multi-phase (3-phase) flow in wellbores is considered. One such approach is that provided by the drift-flux model. According to this model, in the three-phase case (typically, oil, water, and gas) the governing system of equations consists of three continuity equations, one for each phase, and a single equation for the conservation of momentum for the mixture. The system is closed by equations-of-state and algebraic relations for determining the individual phase velocities. The detailed characteristic analysis of both two- and three-phase problems is carried out to determine the domains where the system of equations is hyperbolic and, therefore, whether they are suitably well-posed, stable and robust for application in the numerical solution of this type of hyperbolic problem. The transient, steady-state, and non-inertial forms of the momentum conservation equation encountered in the literature are considered. The influence of mass exchange terms, responsible for the solubility of the gas-phase in liquid on the hyperbolicity is also studied. The analysis demonstrated that the system is found to be hyperbolic in the two-phase case and conditionally hyperbolic in the three-phase case, with eigenvalues being functions of the problem variables. It is also shown that the mixture momentum equation can be transformed to the so-called “advection” equation for pressure which possesses a real eigenvalue. The analysis presented suggests recommendations on the domains to which hyperbolicity is valid to a system of equations and on the specification of boundary conditions for the drift-flux model.
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Thermal Simulation of CO2 Storage with Generalized Cubic Equation of State
Authors I. Aavatsmark, B.K. Kometa and S.E. GasdaSummaryRelationships between pressure, temperature and density are generally described by an equation of state. For CO2, the Span-Wagner equation is generally assumed to give the best fit to measured data. The high accuracy of this equation does not come without a cost. Therefore, this equation of state is mostly used as a reference for comparisons with other formulations.
In reservoir simulations, cubic equations of state such as the Peng-Robinson and the Soave-Redlich-Kwong equations are widely used. They are fairly accurate, and computation of the solution is fast. In this paper, a generalized cubic equation of state is introduced. This equation is computationally precisely as efficient as the traditional equations of state. With the generalized equation of state, improved approximations of the density of CO2 in predefined temperature-pressure domains may be obtained. The parameters of the generalized cubic equation of state are determined through comparison with the Span-Wagner equation.
We show applications of the generalized cubic equation of state for different temperature-pressure domains. When compared with the Peng-Robinson equation, the root mean square density deviation is reduced by a factor 2 for domains containing the critical point, and a facto 7 for supercritical domains. Similarly, thermal simulations with the generalized cubic equation of state show large improvements in density and improvements in saturation close to the CO2 front.
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Negative-saturation Approach for Non-isothermal Compositional Three-phase Flow Simulations in Porous Media
Authors H. Salimi and J. BruiningSummaryThis article deals with developing a three-phase solution approach, called the three-phase non-isothermal negative-saturation (NegSat3) solution approach. The NegSat3 solution approach solves efficiently any non-isothermal compositional-flow problem in porous media that involves phase transitions between different phase states when the maximum number of phases is less than four. The advantage of the solution approach is that it circumvents using different equations and primary variables for single-phase, two-phase, and three-phase regions in porous media. Consequently, the NegSat3 solution approach avoids switches and the ensuing-unstable procedure. The NegSat3 solution approach can be implemented efficiently in numerical simulators to deal with modeling issues for thermal recovery processes, CO2-sequestration process, and for multicontact miscible gas injection in oil reservoirs if the number of phases is less than four. We illustrate the NegSat3 solution approach by way of example to steam injection in a 1D heavy-oil reservoir. The solution is compared with a standard numerical solution that is analytically verified by the method of characteristics, and shows excellent agreement. The results show that the oil recovery depends critically on whether the boiling temperature of the volatile oil is around the water boiling temperature, or much below or above it. These boiling-temperature ranges give rise to three different types of wave structures. When the boiling temperature of the volatile oil is near the boiling temperature of water, the striking result is that the speed of the evaporation front is equal or somewhat larger than the speed of the steam-condensation front. Thus, the volatile oil condenses at the location where the steam condenses too, yielding virtually complete oil recovery. Conversely, if the boiling temperature is too high or too low, there is incomplete recovery.
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Modelling In-situ Upgrading of Heavy Oil Using Operator Splitting Methods
Authors J. Maes, A.H. Muggeridge, M.D. Jackson, M. Quintard and A. LapeneSummaryThe In-Situ Upgrading (ISU) of bitumen and oil shale is a very challenging process to model numerically because a large number of components need to be modelled using a system of equations that are both highly non-linear and strongly coupled. Operator splitting methods are one way of potentially improving computational performance. Each numerical operator in a process is modelled separately, allowing the best solution method to be used for the given numerical operator. A significant drawback to the approach is that decoupling the governing equations introduces an additional source of numerical error, known as splitting error. Obviously the best splitting method for modelling a given process is the one that minimises the splitting error whilst improving computational performance over that obtained from using a fully implicit approach.
Although operator splitting has been widely used for the modelling of reactive-transport problems, it has not yet been applied to the modelling of ISU. One reason is that it is not clear which operator splitting technique to use. Numerous such techniques are described in the literature and each leads to a different splitting error. While this error has been extensively analysed for linear operators for a wide range of methods, the results observed cannot be extended to general non-linear systems. It is therefore not clear which of these techniques is most appropriate for the modelling of ISU.
In this paper we investigate the application of various operator splitting techniques to the modelling of the ISU of bitumen and oil shale. The techniques were tested on a simplified model of the physical system in which a solid or heavy liquid component is decomposed by pyrolysis into lighter liquid and gas components. The operator splitting techniques examined include the Sequential Split Operator (SSO), the Strang-Marchuk split operator (SMSO) and the Iterative Split Operator (ISO). They were evaluated on various test cases by considering the evolution of the discretization error as a function of the size of the time-step compared with the results obtained from a fully implicit simulation. We observed that the error was minimum for a splitting scheme where the thermal conduction was performed first, followed by the chemical reaction step and finally the heat and mass convection operator (SSO-CKA). This method was then applied to a more realistic model of the ISU of bitumen with multiple components.
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Vertically Integrated Models with Coupled Thermal Processes
Authors S.E. Gasda, W.G.G. Gray and H.K.D. DahleSummaryCO2 storage in geological formations involves coupled processes that affect the migration and ultimate fate of injected CO2 over multiple length and time scales. For example, coupling of thermal and mechanical process has implications for storage security, including thermally induced fracturing and loss of caprock integrity in the near wellbore environment. This may occur when CO2 is injected at a different temperature than reservoir conditions, e.g. Snøhvit injection, potentially leading to large temperature, density and volume changes within the plume over space and time. In addition, thermally induced density changes also impacts plume buoyancy that may affect large-scale migration patterns in gravity-driven systems such as Utsira storage site. This interaction becomes particularly important at temperatures and pressures near the critical point. Therefore, coupling thermal processes with fluid flow should be considered in order to correctly capture plume migration and trapping within the reservoir.
A practical modeling approach for CO2 storage at the field scale is the vertical-equilibrium (VE) model, which solves partially integrated conservation equations for flow in two lateral dimensions. This class of models is well suited for strongly segregated flows, as can be the case for CO2 injection. In this paper, we extend the classical VE model to non-isothermal systems by vertically integrating the coupled heat transport equations, focusing on the thermal processes that most impact the CO2 plume. The model allows for heat exchange between the CO2 plume and the surrounding environment assuming thermal equilibrium across the plume thickness for relatively thin plumes. We investigate the validity of simplifying assumptions required to reconstruct the fine-scale thermal structure from the coarse-scale model solution. The model concept is verified for relatively simple systems. The results of this work demonstrate the potential for reduced models to advance our understanding of the impact of thermal processes in realistic storage systems.
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Numerical Aspects of Polymer Flood Modeling
Authors T.S. Mykkeltvedt, I. Aavatsmark and K.A. LieSummaryWe discuss the application of modern high-resolution schemes to a hyperbolic system that models polymer flooding. This system consists of a pair of non-strictly hyperbolic conservation laws. In general, high-resolution schemes are often used for model problems where high accuracy is required in the presence of shocks or discontinuities. Polymer flooding is a difficult process to model, especially since the dynamics of the flow lead to concentration fronts that are not self-sharpening. Because the water viscosity is strongly affected by the polymer concentration, it is crucial to capture polymer fronts accurately to resolve the nonlinear displacement mechanism correctly and its efficiency for enhanced recover.
The main objective of this work is to compare different first- and higher-order methods in terms of how the discontinuities are treated. The discussion will focus on the validity, convergence and robustness of the schemes. Especially, different initial conditions and the inclusion of adsorption and permeability reduction can change not only the solution, but also the behavior of the different numerical methods. We show that these effects also can influence the applicability of a solver and we investigate of how suitable different numerical methods are for different polymer flooding situations.
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Non-equilibrium Based Compositional Simulation of Carbonated Water Injection EOR Technique
Authors J. Foroozesh, M. Jamiolahmady, M. Sohrabi and S. IrelandSummaryCarbonated water injection (CWI) is an augmented water injection strategy, which has great potentials for EOR and CO2 storage purposes. When carbonated water, CO2 enriched water, is injected into oil reservoirs, due to higher CO2 solubility in oil compared to that in water, CO2 migrates from carbonated water into oil. This improves oil mobility, due to swelling and viscosity reduction, which consequently increases oil production.
Our core flood experiments show that during CWI, CO2 is transferred and distributed between water and oil gradually and thermodynamic equilibrium is not reached. Available compositional reservoir simulators, which are based on the instantaneous equilibrium assumption, do not capture the actual physics of CWI. In this work, a non-equilibrium based compositional simulator was developed to simulate the performed core flood CWI experiments more realistically. The developed two-phase flow simulator is currently suitable for one dimensional core experiments. It includes a mass transfer term to capture the kinetics of CO2 transfer between phases. Governing equations were derived based on the water, oil and CO2 components balance, and solved using the fully implicit finite difference technique. Black-oil (without mass transfer) and compositional (with mass transfer) modes of the simulator can be used for simulation of conventional water injection (WI) and CWI, respectively. A genetic algorithm based optimization software was also developed that can be linked to the simulator to history match the available production data and obtain the unknown parameters of the model.
The simulator was used to model WI and CWI coreflood experiments conducted on a water-wet sandstone core fully saturated by Decane (with well-defined fluid properties). First, the WI experiment was simulated when an oil-water relative permeability (kro-w) was obtained by history matching of WI production data. The WI test was re-simulated by ECLIPSE100 (E100) commercial simulator using optimized kro-w. E100’s predictions of production data reasonably matched model’s results, which verify its integrity. Next, the obtained kro-w was used for simulation of CWI. Mass transfer coefficient, as the only remaining unknown parameter, was tuned to match the available CWI production data leading to an acceptable match. The simulator shows promising potentials for simulation and better understanding of CWI for practical field applications. Moreover, the structure of this simulator offers a solid foundation for other EOR methods where kinetics of mass transfer is important.
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Numerical Simulation of Mutually Soluble Solvent-aided Spontaneous Imbibition in Fractured Reservoirs
Authors M. Chahardowli, R. Farajzadeh and H. BruiningSummaryMutually-soluble solvents can enhance oil recovery both in completely and partially water wet fractured reservoirs. When a strongly or partially water-wet matrix is surrounded by an immiscible wetting phase in the fracture, spontaneous imbibition is the most important production mechanism. Initially, the solvent moves with the imbibing brine into the core. However, upon contact with oil, diffusion occurs and the solvent is transported in the oleic phase. Through the migration of the mutually soluble component from the aqueous phase into the oleic phase, oil properties and/or rock-fluid interactions are modified. The hypothesis in this work is that a mutually-soluble solvent improves the ultimate recovery and the imbibition rate in partially and completely water-wet cores. The main recovery mechanisms are the wettability change of the partially water-wet cores, oil swelling and oil viscosity reduction in both partially and completely water-wet cores.
This paper considers the numerical modelling of experiments that use Amott imbibition cells to study solvent enhanced spontaneous imbibition in completely water-wet and partially water-wet cores. In the first stage of the experiment, the completely water-wet core was exposed to brine. In a second stage, the core was put in another Amott cell that was filled with a solvent/ brine mixture. The extra recovery by a solvent/brine mixture strongly depends on the residual oil saturation after brine imbibition and it is relatively insensitive to the permeability of the core or the oil viscosity. We implemented the swelling mechanism, the oil viscosity reduction mechanism, and the IFT reduction mechanism in the numerical model. Our studies show that the most important production mechanism in the completely water wet system is the oil swelling and the second most important mechansism is the oil viscosity reduction. The effect of the IFT reduction in the oil production is not significant. The numerical results show an improvement of 10%.
For the partially water-wet samples, we also started with exposing the core to pure brine without solvent. Contrary to the completely water-wet samples, the numerical results show a significant increase in recovery rate when the sample is transferred to another Amott cell where it is exposed to a mixture of solvent and brine. The main modification in simulating the recovery process is that now a wettability change mechanism is taken into account. Our numerical studies show that the contribution of the wettability change is the main contributor to the oil recovery, which is enhanced by 35 %.
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Microbial Enhanced Oil Recovery - A Mathematical Study of the Potential of Spore-forming Bacteria
Authors S.M. Nielsen, I. Nesterov and A.A. ShapiroSummaryMicrobial enhanced oil recovery (MEOR) utilizes microbes for enhancing the recovery by two main mechanisms: 1) reduction of oil-water interfacial tension (IFT) by produced biosurfactant, and 2) selective plugging by microbes and metabolic products. A mathematical model for MEOR has been built. Our model organism is a surfactant-producing spore-forming bacterium that shows a great potential for MEOR. Application of spore-forming bacteria is an advantageous novelty of the present approach. A spore is a dormant, very resistant version of the cell. Sporulation (turning bacteria into spores) occurs when bacteria are exposed to stresses, such as starvation. The remains of a sporulated cell serve as a substrate for the rest bacteria. An inverse process, reactivation, occurs when spores sense favorable conditions.
The model accounts for growth, substrate consumption, surfactant production, attachment/filtering out, sporulation and reactivation. The mathematical setup is a set of 1D transport equations involving reactions and attachment. Characteristic sigmoidal curves are used to describe sporulation and reactivation in response to substrate concentrations. Surfactant decreases IFT, modifying the relative permeabilities and thus decreasing the fraction of water in the flow. Attachment of bacteria reduces the pore space available for flow, i.e. the effective porosity and permeability. Clogging of specific areas may occur.
In order to obtain sufficient local concentrations of surfactant, substantial amounts of substrate should be supplied; however, massive growth of bacteria increases the risk for clogging at the well inlet areas, causing injectivity loss. In such areas starvation may cause sporulation, reducing the risk of clogging. Substrate released during sporulation can be utilized by attached vegetative bacteria and they will continue growing and producing surfactant, which prolongs the effect of the injected substrate. The simulation scenarios show that application of the spore-forming bacteria gives a higher total production of surfactant and the reduced risk of clogging, leading to an increased period of production and a higher oil recovery.
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Unstructured Adaptive Mesh Refinement for Flow in Heterogeneous Porous Media
Authors M. Karimi-Fard and L.J. DurlofskySummaryThe spatial resolution required in a simulation model depends on the type and degree of geological heterogeneity, along with the flow physics and well locations. The use of adaptive mesh refinement (AMR) enables the grid to adapt dynamically during the course of the simulation, which facilitates the efficient use of computational resources. In this work we present a finite-volume-based AMR procedure that is suitable for complex reservoir models such as those characterized by fractures or channels. The approach uses a fully unstructured hierarchical grid to accurately represent complex geological features. An important aspect of the method is the use of flow-based global transmissibility upscaling, which provides accurate cell-to-cell transmissibilities at any level of coarsening. The performance of the method is illustrated for problems involving two-dimensional channelized and fractured systems. High levels of solution accuracy relative to reference fine-scale results are observed for both cases, and the AMR models are shown to consistently use less than 30% of the cells in the underlying fine-grid representation. For fractured systems, the method essentially provides a dual-continuum coarse-scale representation, and can thus be viewed as a dual-continuum-AMR technique.
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Indirect Unstructured Hex-dominant Mesh Generation Using Tetrahedra Recombination
Authors A.B. Botella, B.L. Lévy and G.C. CaumonSummaryCorner-point gridding of reservoir and basin models is widely used but generally yields approximations in the geological interfaces representation in flow simulation. This paper introduces an indirect method to generate a hexdominant mesh conformal to 3D geological surfaces suitable for Finite-Element and Control-Volume Finite-Element simulations. By indirect, we mean that the method first generates an unstructured tetrahedral mesh whose tetrahedra are then merged into primitives (hexahedra, prisms and pyramids). More specifically, we focus on determining the optimal set of primitives that can be recombined from a given tetrahedral mesh. First, we detect in the tetrahedral mesh all the feasible volumetric primitives using a pattern-matching algorithm that we re-visit and extend with configurations that account for degenerated tetrahedra (slivers). Then, we observe that selecting the optimal set of primitives among the feasible ones can be formalized as a maximum weighted independent set problem, known to be N P-Complete. We propose five heuristic optimizations to find a reasonable set of primitives in a practical time. All the tetrahedra of each selected primitive are then merged to build the final unstructured hex-dominant mesh. This method is demonstrated on complex 3D geological models including a discrete fracture network.
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Boundary Aligned Grid Generation and CVD-MPFA - Cell-centred Versus Cell-vertex on Unstructured Grids
Authors S. Manzoor, M.G. Edwards, A. Dogru and T.M. Al-ShaalanSummaryDevelopment of boundary aligned grid generation is presented together with comparative performance of cell-vertex versus cell-centred CVD-MPFA finite-volume formulations using equivalent degrees of freedom. When generating structured or unstructured grids for reservoir simulation, classical key constraints involve boundary aligned grid generation with control-volume boundaries aligned with solid walls and geological features such as layers, shale barriers, fractures, faults, pinchouts and multilateral wells.
The schemes used are control-volume distributed (CVD) with flow variables and rock properties sharing the same control-volume location and are comprised of a multipoint flux family formulation (CVD-MPFA). Consequently a natural choice is for primal grid cells to act as control-volumes, then grid generation can be performed with primal grid cell boundaries being aligned with key interior constraint boundaries. This naturally leads to cell-centred approximation, where flow variables and rock properties are associated with grid cell centres.
The alternative is to employ cell-vertex approximation which uses far fewer approximation points on a given unstructured grid. In this case control-volumes are constructed around primal grid vertices. The grid generation process is less straight forward since control-volumes must be constrained to satisfy interior boundary alignment. A novel grid generation procedure is proposed that automates control-volume boundary alignment and yields a Voronoi mesh. The actual grid is then generated such that dual boundaries are aligned with key internal constraint boundaries and cell-vertex approximation becomes the natural choice. In this case flow variables and rock properties are associated with grid cell vertices and their dual control-volumes.
The relative benefits of both types of approximation is made clear in terms of flow resolution and degrees of freedom required.
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A Volume-conserving Representation of Cell Faces in Corner Point Grids
More LessSummaryCorner point grids are currently the standard representation for reservoir simulation grids. Cell faces in corner point grids are traditionally represented as bilinear surfaces where the edges between the corner points all are straight lines. This representation has the disadvantage that at pillar-gridded faults with varying dip the cell faces on either side will not precisely match, giving overlapping cells or gaps between cells. We propose an alternative representation for the cell faces. The four vertical cell faces are still represented as bilinear surfaces, but instead of having linear edges between the cell corners along the top and bottom faces, we propose a representation where all horizontal intersections through the grid give straight lines between the grid pillars, giving column faces whose shape is independent of the corner point locations of the individual grid cells. This ensures that the grid columns match up and that there are no gaps or overlapping volumes between grid cells. This representation gives a local parameterization for the whole grid column, and the top and bottom grid cell surfaces are modelled as bilinear within this domain. Using this representation we get a parameterization of the grid cell which we use to calculate the cell volume.
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Ensemble Level Upscaling for Compositional Flow Simulation, with Application to Uncertainty Quantification
Authors H. Li and L.J. DurlofskySummaryUncertainty quantification is typically accomplished by simulating multiple geological realizations, which can be very expensive computationally if the flow process is complicated and the models are highly resolved. Upscaling procedures can be applied to reduce computational demands, though it is essential that the resulting coarse-model predictions correspond closely to reference fine-scale solutions. In this work, we develop an ensemble level upscaling (EnLU) procedure for compositional systems, which enables the efficient generation of multiple coarse models for use in uncertainty quantification. This requires us to first develop an accurate global compositional upscaling method for individual realizations. This global upscaling entails transmissibility and relative permeability upscaling, along with the computation of α-factors to capture component fluxes. A procedure for iterating on coarse-scale α-factors is introduced, and this is shown to improve accuracy. In EnLU, global upscaling is applied for only a few selected realizations. For 90% or more of the realizations, upscaled functions are assigned statistically based on quickly-computed flow and permeability attributes. A sequential Gaussian co-simulation procedure is incorporated to provide coarse models that honor the spatial correlation structure of the upscaled properties. The resulting EnLU procedure is applied for multiple realizations of 2D models, for both Gaussian and channelized permeability fields. Results demonstrate that EnLU provides P10, P50 and P90 results for phase and component production rates that are in close agreement with reference fine-scale results. Less accuracy is observed in realization-by-realization comparisons, though the models are still much more accurate than those generated using standard coarsening procedures.
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Analytical Expressions for Upscaled Relative Permeabilities in Three-phase Flow
Authors E. Bianchi Janetti, M. Riva and A. GuadagniniSummaryWe present analytical solutions for the relative permeabilities governing a Darcy scale description of three-phase flow of immiscible fluids. We consider flow taking place within a capillary tube with circular cross-section for two settings corresponding to (a) a water wet and (b) an oil wet configuration. Momentum transfer between the moving phases, which leads to the phenomenon of viscous coupling, is explicitly accounted by imposing continuity of velocity and shear stress at the fluid-fluid interfaces. The macroscopic model describing the system at the Darcy scale includes three-phase effective relative permeabilities, Kij,r, accounting for the flow rate of the ith-phase due to the presence of the jth-phase. These effective coefficients are function of phases saturation, fluids viscosity and wettability of the medium. Our results show that (i) the relative permeability Kii,r of the wetting phase is only a function of its own saturation while Kii,r of the non-wetting phases depend on the saturations of all the fluids; (ii) viscous coupling effects (elucidated by Kij,r with i ≠ j) can be relevant in water wet and oil wet systems.
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An Upscaling Methodology for EOR
Authors A.H. Muggeridge and P. HongtongSummaryThe outcome of immiscible enhanced oil recovery (EOR) processes such as low salinity water injection and polymer flooding can be very sensitive to geological heterogeneity. In some cases the resulting reduction in macroscopic sweep (versus a waterflood) can be more than the improvement in microscopic displacement efficiency. It is therefore important to be able to model the impact of this heterogeneity during simulation studies. This may require an upscaling step if the geological heterogeneity is smaller than the simulation grid block size or simply to compensate for numerical diffusion on the coarse grid.
This paper proposes an upscaling methodology that can be applied to different EOR processes and demonstrates its application to low salinity water injection examples. The methodology involves a hierarchy of upscaling steps. First the absolute permeability is upscaled with the objective of predicting the correct changes in pressure. This may also involve near well bore upscaling. Next pseudo relative permeability curves are generated to capture the shock front behaviour. In this study we compare results obtained from using traditional pore volume weighted pseudos with those obtained using pseudos determined analytically using Buckley-Leverett theory. Finally the simulator models relevant to the EOR process of interest are upscaled. In low salinity waterflooding this is the choice of low and high salinity thresholds.
The upscaled models are in better agreement with the fine grid models in terms of pressure, water saturation production and production of either salinity or polymer than are the outputs of coarse grid models without upscaling. The results using the analytical pseudo relative permeabilities are comparable to those obtained from simulations using the pore volume weighted pseudo in many of the cases tested. These models are obviously less time-consuming and complex to generate as they do not need the engineer to run a full fine grid simulation of the EOR process to calculate the pseudo relative permeabilities.
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A Fast and Accurate Upscaling of Transmissivities for Field Scale Reservoir Simulation
Authors D. Guérillot and J. BruyelleSummaryDescription:
For representing reservoir heterogeneities, most of geological models need several tens of million of cells.
Even though reservoir simulators are quicker and quicker, it is most of the time requested to average the properties representing these heterogeneities on larger cells for simulating the multi-phase flow.
Methods to average reservoir simulation have been studied for several years by many authors. For averaging absolute permeabilities, two main approaches are met: 1) averaging first the permeabilities and using the averaged value when discretizing the equations, leading to average transmissivities, or 2) calculating directly the averaged transmissivities from the high resolution permeabilities.
This paper is in line with the second approach. Compared to previous work, we combined the finite volume principles with algebraic methods providing upper and lower bounds of the upscaled transmissivities considering adjacent volumes on both sides of the boundaries between cells in a manner that the contrast of heterogeneities is better conserved.
Comparisons with current upscaling approaches will confirm that this approach is more accurate than calculating transmissivities from upscaled permeabilities.
The competitive advantages of the algebraic methods are that the upscaling step will be fast and the use of bounds provided by algebraic method allows controlling the quality of the upscaling and gridding.
The main innovation of the presented method is that the calculations of transmissivities are performed on smaller local domain than those proposed in previous work. This approach increases contrast of transmissivity and thus gains in accuracy for field scale flow simulation in heterogeneous media.
Applications:
This approach had been successfully implemented in an industrial E&P software platform and tested on actual fields.
Results and conclusions:
This method calculating upscaled transmissivities is the natural end point of any workflow devoted to the geological modeling to determine the petrophysical parameters of the field scale simulation grid.
Being based on algebraic method, this approach is fast and a control of its quality is possible.
Technical contributions:
- An innovative method calculating on smaller domains the upscaled transmissivities is proposed to increase the accuracy of reservoir simulations.
- This fast approach avoid calculation of local fluid flow simulation everywhere
- This approach allows controlling the quality of upscaling and gridding.
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Reservoir Management Optimization Using Calibrated Transmissibility Upscaling
Authors S. Krogstad, H.M. Nilsen and X. RaynaudSummaryOptimizing the objectives of long-term reservoir management typically requires a high number of forward reservoir simulations. Two important remedies to reduce the runtime (and make the optimization problem manageable) are model reduction/upscaling and efficient computation of gradients. Adjoint methods are generally considered to be the most efficient means for obtaining gradients.
Furthermore, there has been much interest in reduced-order modelling (e.g., based on POD) for reservoir management optimization. Very promising results have been reported for models with somewhat limited complexity. Application to industry-standard cases is inhibited in part by the invasive nature of the approach with respect to simulator code, and the fact that the number of required basis functions is correlated with the degree of nonlinear dynamics.
In this work, we utilize ideas from POD to compute upscaled transmissibilities for a coarse model, rather than using the method directly to build a basis for the fine-scale state space. The proposed coarsening strategy takes as input any number of fine-scale states (pressure fields from e.g., a previous simulation), and produces a coarse-scale model calibrated to the specific flow scenario(s) dictated by the input. We argue that compared to traditional general-purpose upscaling approaches, much more aggressive coarsening can be applied in this type of scenario-specific upscaling. Utilizing a fully-implicit, three-phase, black-oil simulator with adjoint capabilities, we investigate the performance of our methodology by optimizing the net-present-value (NPV) for a real-field model. We consider multiple coarsening strategies and coarsening factors, and conclude that for the model considered, at least two orders of magnitude speed-up can be achieved whilst retaining sufficient accuracy.
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Data-driven Model Inference and its Application to Optimal Control under Reservoir Uncertainty
More LessSummaryMany real-world oil reservoir simulations rely on either legacy code that is difficult to modify or commercial software without available source code. Such black-box simulators solve a set of nonlinear partial differential equations and generally involve the use of many unknown variables, functions or types of discretization methods. For the purpose of carrying out history matching or optimal control in reservoir management studies, black-box simulators are usually employed as function evaluators within the context of derivative-free optimization methods. However, these methods can be computationally unaffordable when the number of parameters or controls becomes high dimensional. The present work describes an efficient optimization methodology for black-box simulations consisting of inferring and calibrating a “twin model” representation of the black-box simulator. The twin model is a non-intrusive model that mirrors the behavior of the black-box simulation using data assimilation techniques. Once the inferred twin model is available, adjoint operators based on gradient-driven techniques can be easily computed to perform efficient optimization. Computational experiments illustrate the significant reduction of simulations to estimate the optimal water injection strategy under various geological uncertainty scenarios when compared with traditional approaches using gradient-free methods.
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Evaluation of Non-intrusive Generalized Polynomial Chaos Expansion in the Context of Reservoir Simulation
Authors O. Pajonk, F.C. Yanou Ngongang and R. Schulze-RiegertSummaryUncertainty quantification workflows which map input uncertainties to output uncertainties are an important part of many reservoir simulation applications. Examples are estimation of prediction uncertainties including history data, reserves estimation, or optimization under uncertainty. Over the last few years, proxy modeling techniques have turned out to be essential for improving the efficiency of the employed methods, with one example being stochastic sampling techniques like Markov Chain Monte Carlo. An established proxy model family is the generalized Polynomial Chaos Expansion (gPCE). Some applications to reservoir simulation exist - also in the context of history matching.
In this work we focus on a non-intrusive, efficient way to construct such gPCE representations for multi-dimensional input uncertainties: sparse-grid spectral projection – also known as Smolyak grid. In standard approaches, which derive their multi-dimensional cubature rules from full tensor-products of one-dimensional rules, the number of required points grows exponentially with the input dimension. This causes a significant computational effort since each point represents a forward simulation. On the other hand, sparse-grid techniques used in this work show polynomial scaling behavior by “thinning out” the full tensor-product. However, in order to avoid possible numerical instabilities of this technique a design of simulation cases needs to be applied. Practical consequences and guidelines for real applications will be described in detail.
The numerical framework for constructing gPCEs is applied to uncertainty quantification workflows in reservoir simulation. We investigate performance criteria for representing key performance indicators using gPCEs, e.g., cumulative properties like oil production total or transient properties such as bottom hole pressure. Practical application designs are derived and described.
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