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ECMOR XIV  14th European Conference on the Mathematics of Oil Recovery
 Conference date: September 811, 2014
 Location: Catania, Sicily, Italy
 Published: 08 September 2014
1  20 of 136 results


Orderingbased Nonlinear Solver for Fullyimplicit Simulation
Authors F.P. Hamon and H.A. TchelepiSummaryThe FullyImplicit 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. Newtonbased methods – often with damping heuristics  are employed to solve the nonlinear systems. However, Newtonbased 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 reducedNewton method were used to solve immiscible threephase flow in the presence of buoyancy and capillary effects (e.g., Kwok & Tchelepi, JCP, 2007 ). Here, we extend the potentialordering method to interphase mass transfer. Specifically, we deal with the blackoil model with variable bubblepoint pressure. The convergence properties and the computational efficiency of the potentialordering 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 orderingbased approach was implemented in Stanford’s generalpurpose research simulator (ADGPRS), and was applied to challenging BlackOil 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 stateoftheart nonlinear/linear solvers (e.g. damped Newton with CPRbased AMG) are presented for variable bubblepoint blackoil 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 orderingbased solver reduces the number of nonlinear iterations significantly, which also leads to gains in the overall computational cost.



Nonlinear Analysis of Newtonbased 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. Largescale 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 twophase transport in porous media, and we propose an efficient nonlinear solver based on this understanding.
We focus on immiscible, incompressible, twophase transport in the presence of viscous and buoyancy forces. We investigate the nonlinearity of the discrete transport equation obtained from finitevolume discretization with SinglePoint 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 unitflux, zeroflux, and inflection lines of the numericalflux saturation space. The unit and zeroflux 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 multicell 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 trustregion 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 trustregion nonlinear solvers are demonstrated.



Nearwell Local Space and Time Refinement for Multiphase 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) RobinNeumann 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 multiphase Darcy flows.
Numerical experiments on 3D test cases including gas injection and gascondensate 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.



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 Newtonbased iterative procedure converges, the cost associated with updating all the unknowns simultaneously can be quite expensive. Conventional sequentialimplicit 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 cocurrent and countercurrent 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 phasebased 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 cocurrent flow regions is made up of a single cell, whereas a component made up of multiple connected cells indicates countercurrent flow in which pressure and saturation are tightly coupled. Marching down from upstream to downstream, for a singlecell component we update saturation nonlinearly by solving the scalar saturation equation(s). For a multicell 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 2D and 3D 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.



Realizing the Potential of GPUs for Reservoir Simulation
Authors K. Esler, K. Mukundakrishnan, V. Natoli, J. Shumway, Y. Zhang and J. GilmanSummaryHigher stakes from deepocean drilling, increasing complexity from unconventional reservoirs, and an overarching desire for a higherfidelity 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 blackoil formulation discretized using a fullyimplicit finite volume method. We implement all major computational aspects, including property evaluation, Jacobian construction, and robust solvers/ preconditioners on GPUs. The CPRAMG preconditioner we employ allows low iteration count and nearoptimal order(N) scaling of computational effort with system size. This combination of algorithms and hardware enables the simulation of finescale 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.



Scalable Algebraic Multiscale Linear Solver for Largescale 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 basisfunction 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 multicore systems and compared with the widely used parallel SAMG solver ( Stüben, 2012 ).



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.



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. trustregion), 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 twolevel 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 twolevel 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.



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 finescale petrophysical properties. Use the resulting multiscale basis functions to pose a global flow problem a coarser grid. Reconstruct conservative finescale approximations from the coarsescale solution. By extending the basic concept with iteration cycles and additional local stages, one can systematically drive the finescale 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 finescale approximations.
Such multiscale finitevolume 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 finiteelement or finitevolume restriction operators to form different multiscale finitevolume 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, realworld, 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



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 (CAMS) flow in heterogeneous oil reservoirs is developed. Based on the recently developed AMS [ Wang et al., 2014 ] for incompressible linear flows, the CAMS 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, CAMS allows for both MSFV and MSFE methods. Furthermore, to resolve highfrequency errors, Correction Functions and ILU(0) are considered. The best CAMS procedure is determined among these various strategies, on the basis of the CPU time for threedimensional heterogeneous problems. The CAMS is adaptive in all aspects of prolongation, restriction, and conservative reconstruction operators for timedependent compressible flow problems. In addition, it is also adaptive in terms of linearsystem update. Though the CAMS is a conservative multiscale solver (i.e., only a few iterations are employed infrequently in order to maintain highquality results), a benchmark study is performed to investigate its efficiency against an industrialgrade Algebraic Multigrid (AMG) solver, SAMG [ Stuben, 2010 ]. This comparative study illustrates that the CAMS is quite efficient for compressible flow simulations in largescale heterogeneous 3D reservoirs.



Monotone Multiscale Finite Volume Method for Flow in Heterogeneous Porous Media
Authors Y. Wang, H. Hajibeygi and H.A. TchelepiSummaryThe MultiScale FiniteVolume (MSFV) method is known to produce nonmonotone solutions. The causes of the nonmonotone 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 (mMSFV) method based on a local stencilfix that guarantees monotonicity of the coarsescale operator, and thus the resulting approximate finescale solution. Detection of nonphysical transmissibility coefficients that lead to nonmonotone solutions is achieved using local information only and is performed algebraically. For these ‘critical’ primal coarsegrid interfaces, a monotone local flux approximation, specifically, a TwoPoint Flux Approximation (TPFA), is employed. Alternatively, a local linear boundary condition is used for the basis functions to reduce the degree of nonmonotonicity. The local nature of the two strategies allows for ensuring monotonicity in local subregions, where the nonphysical transmissibility occurs. For practical applications, an adaptive approach based on normalized positive offdiagonal coarsescale 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 mMSFV approach can guarantee monotonicity of the solutions to any desired level, numerical results illustrate that employing the mMSFV modifications only for a small fraction of the domain can significantly reduce the nonmonotonicity of the conservative MSFV solutions.



A Constrained Pressure Residual Multiscale (CPRMS) 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 flowtransport 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 (CPRMS) method. In the CPRMS 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 globalstage (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.



Improved Estimation of the Stochastic Gradient with QuasiMonte 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 nonintrusive and treats the simulator as a black box. In this work, we propose the application of quasiMonte 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, quasiMonte 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 quasiMonte 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).



Optimizationbased Framework for Geological Scenario Determination Using Parameterized Training Images
Authors M.A.H. Rousset and L.J. DurlofskySummaryIn many reservoirmodeling 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 rejectionsampling 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.



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 MultiPoint 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 proxybased 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 timelapse 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 multidisciplinary collaboration between geological, geomechanical, geophysical and reservoir engineering disciplines.



Toward an Enhanced Bayesian Estimation Framework for Multiphase Flow Softsensing
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 softsensing 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 varianceestimation approaches, it can achieve good performance in terms of matching both the measurements of the physical sensors and the true underlying flow rates.



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 fluiddynamical data.
Calibrated (historymatched) 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 productioninjection 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.



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 nonGaussian and thus difficult to calibrate in ensemble Kalman filterlike 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 postprocessing step that resembles the postsmoothed maximumlikelihood (ML) reconstruction method described in Nuyts et al. (2005 ). Disregarding the categorical feature of the facies model, reservoir properties are first updated using an EnKFlike assimilation method to honor flow data. In the postprocessing 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.



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.



Petrophysical Properties Bias Around Uncertain Facies Boundary  Analysis, Correction by Conditional Property Interface Filler (CoPIF) and Impact on Ensemblebased 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 petrophysical 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 petrophysical 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 petrophysical properties of cells containing facies boundaries differ from strictly faciesdependent statistics. The bias results, typically, in excessive variance of petrophysical 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 petrophysical 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 unfiltered 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 petrophysical statistics will be illustrated as well as its impact within the context of EnKF Assisted History Match.
