Eduardo Gildin

Local-global multiscale model reduction for flows in high-contrast heterogeneous media

In this paper, we study model reduction for multiscale problems in heterogeneous high-contrast media. Our objective is to combine local model reduction techniques that are based on recently introduced spectral multiscale finite element methods (see [19]) with global model reduction methods such as balanced truncation approaches implemented on a coarse grid. Local multiscale methods considered in this paper use special eigenvalue problems in a local domain to systematically identify important features of the solution. In particular, our local approaches are capable of homogenizing localized features and representing them with one basis function per coarse node that are used in constructing a weight function for the local eigenvalue problem. Global model reduction based on balanced truncation methods is used to identify important global coarse-scale modes. This provides a substantial CPU savings as Lyapunov equations are solved for the coarse system. Typical local multiscale methods are designed to find an approximation of the solution for any given coarse-level inputs. In many practical applications, a goal is to find a reduced basis when the input space belongs to a smaller dimensional subspace of coarse-level inputs. The proposed approaches provide efficient model reduction tools in this direction. Our numerical results show that, only with a careful choice of the number of degrees of freedom for local multiscale spaces and global modes, one can achieve a balanced and optimal result.

Fast Multiscale Reservoir Simulations using POD-DEIM Model Reduction

In this paper, we present a global-local model reduction for fast multiscale reservoir simulations in highly heterogeneous porous media with applications to optimization and history matching. Our proposed approach identifies a low dimensional structure of the solution space. We introduce an auxiliary variable (the velocity field) in our model reduction that allows achieving a high degree of model reduction. The latter is due to the fact that the velocity field is conservative for any low-order reduced model in our framework. Because a typical global model reduction based on POD is a Galerkin finite element method, and thus it can not guarantee local mass conservation. This can be observed in numerical simulations that use finite volume based approaches. Discrete Empirical Interpolation Method (DEIM) is used to approximate the nonlinear functions of fine-grid functions in Newton iterations. This approach allows achieving the computational cost that is independent of the fine grid dimension. POD snapshots are inexpensively computed using local model reduction techniques based on Generalized Multiscale Finite Element Method (GMsFEM) which provides (1) a hierarchical approximation of snapshot vectors (2) adaptive computations by using coarse grids (3) inexpensive global POD operations in a small dimensional spaces on a coarse grid. By balancing the errors of the global and local reduced-order models, our new methodology can provide an error bound in simulations. Our numerical results, utilizing a two-phase immiscible flow, show a substantial speed-up and we compare our results to the standard POD-DEIM in finite volume setup.

Heterogeneous reservoir characterization using efficient parameterization through higher order SVD (HOSVD)

Parameter estimation through reduced-order modeling play a pivotal role in designing real-time optimization schemes for the Oil and Gas upstream sector through the closed-loop reservoir management framework. Reservoir models are in general complex, nonlinear, and large-scale, i.e., large number of states and unknown parameters. Consequently, model reduction techniques are of great interest in reducing the computational burden in reservoir modeling and simulation. Furthermore, de-correlating system parameters in all history matching and reservoir characterization problems is an important task due to its effects on reducing ill-posedness of the system. In this paper, we utilize the higher order singular value decomposition (HOSVD) to reparameterize reservoir characteristics, e.g. permeability, and perform several forward reservoir simulations by the resulted reduced order map as an input. To acquire statistical consistency we repeat all experiments for a set of 1000 samples using both HOSVD and Proper orthogonal decomposition (POD). In addition, we provide RMSE analysis for a better understanding in process of comparing HOSVD and POD. Results show that HOSVD provide a better performance in a RMSE point of view.

Permeability Parametrization Using Higher Order Singular Value Decomposition (HOSVD)

Model reduction is of highly interest in many science and engineering fields where the order of original system is such high that makes it difficult to work with. In fact, model reduction or parametrization defined as reducing the dimensionality of original model to a lower one to make a costly efficient model. In addition, in all history matching problem, in order to reduce the ill-posed ness of the problem, it is necessary to de-correlate the parameters. Proper orthogonal decomposition (POD) as an optimal transformation is widely used in parameterization. To obtain the bases for POD, it is necessary to vectorize the original replicates. Therefore, the higher order statistical information is lost due to slicing the replicates. Another approach that deals with the replicates as they are, is high order singular value decomposition (HOSVD). In the present work permeability maps dimension is reduced using HOSVD image compression method. Unknown permeability maps are also estimated using HOSVD and results of both parts compared to those of SVD.

A New Model Reduction Technique Applied to Reservoir Simulation

One 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.

Complexity Reduction of Multiphase Flows in Heterogeneous Porous Media

In this paper, we apply mode decomposition and interpolatory projection methods to speed up simulations of two-phase flows in heterogeneous porous media. We propose intrusive and nonintrusive model-reduction approaches that enable a significant reduction in the size of the subsurface flow problem while capturing the behavior of the fully resolved solutions. In one approach, we use the dynamic mode decomposition. This approach does not require any modification of the reservoir simulation code but rather postprocesses a set of global snapshots to identify the dynamically relevant structures associated with the flow behavior. In the second approach, we project the governing equations of the velocity and the pressure fields on the subspace spanned by their properorthogonal-decomposition modes. Furthermore, we use the discrete empirical interpolation method to approximate the mobilityrelated term in the global-system assembly and then reduce the online computational cost and make it independent of the fine grid. To show the effectiveness and usefulness of the aforementioned approaches, we consider the SPE-10 benchmark permeability field, and present a numerical example in two-phase flow. One can efficiently use the proposed model-reduction methods in the context of uncertainty quantification and production optimization.

Complexity reduction of multi-phase flows in heterogeneous porous media

In this paper, we apply mode decomposition and interpolatory projection methods to speed up simulations of two-phase flows in highly heterogeneous porous media. We propose intrusive and non-intrusive model reduction approaches that enable a significant reduction in the dimension of the flow problem size while capturing the behavior of the fully-resolved solutions. In one approach, we employ the dynamic mode decomposition (DMD) and the discrete empirical interpolation method (DEIM). This approach does not require any modification of the reservoir simulation code but rather postprocesses a set of global snapshots to identify the dynamically-relevant structures associated with the flow behavior. In a second approach, we project the governing equations of the velocity and the pressure fields on the subspace spanned by their proper orthogonal decomposition (POD) modes. Furthermore, we use DEIM to approximate the mobility related term in the global system assembly and then reduce the online computational cost and make it independent of the fine grid. To show the effectiveness and usefulness of the aforementioned approaches, we consider the SPE 10 benchmark permeability field and present a variety of numerical examples of two-phase flow and transport. The proposed model reduction methods can be efficiently used when performing uncertainty quantification or optimization studies and history matching. Introduction High fidelity reservoir simulation models have been shown to yield better predictions in optimization problems and in planning for new reservoir developments in green fields. In addition, exploring the capabilities of real-time surveillance data to improve the accuracy of such models has led to a new paradigm in reservoir monitoring, namely the closed-loop reservoir management (Gildin and Lopez , 2011; Jansen et al. , 2009). Even in the case of unconventional reservoirs, numerical reservoir simulation has been used with several modifications with the current models, in particular in the nature of the flow. To this end, natural fractures and pore-space networks have been incorporated in the simulation process to account for flow in the fractures and matrix (Moridis et al. , 2010; King , 2010). Despite the great advances in reservoir modeling tools and the advent of high-performance computing (HPC), highfidelity physics-based numerical simulation still remains a challenging step in understanding the physics of the reservoir due to the large scale nature of the discretized of the underlying partial differential equations. Computationally intensive simulations, such as in the case of history matching, optimization and uncertainty quantification, become impractical to be performed in a timely-manner if real-time data needs to be assimilated into the model (Gildin and Lopez , 2011; Jansen et al. , 2009). A variety of complexity reduction techniques have been proposed to ease this problem and reduce the computational cost in the optimization under the uncertainty paradigm (Antoulas , 2005; Heijn et al. , 2004; Gildin , 2010). In general, they can be classified in three broader areas depending if one is dealing with the forward simulations (production optimization) or the inverse problem (parameter estimation) (Oliver et al. , 2008): • Reduction of the cost of forward simulations: surrogate models, reduced-order models, multiscale, upscaling • Reduction of the input space dimension: parameterizations, direct cosine transformation, sparsity-based, polynomial chaos SPE 167

Hybrid Optimization for Closed-Loop Reservoir Management

The closed-loop optimization paradigm of an oil field can increase oil recovery and reduce water production, maximizing economic gains. One way to improve the management of a field involves designing optimal production strategies by means of dynamically adjusting the production flow rates or bottom-hole pressures over the reservoir life-cycle and operation. A major difficulty occurs in optimizing production of all wells, according to constraint along the production of a field. These optimal control strategies are often difficult to be realized in practice due to the large number of control variables to be adjusted during the optimization process, requiring large amount of computational infrastructure in place. These challenges become even more evident with larger number of wells and with complex large-scale reservoirs. For these reasons, this work proposes a new hierarchical hybrid optimization framework employing model order reduction techniques in a closed-loop fashion. This paper proposes the use of proper orthogonal decomposition (POD) with the discrete empirical interpolation method (DEIM), to reduce the computational effort, and to perform local optimization by means of gradient-based approach by using forward and adjoint models followed by aggressive line search process. This approach was applied in the UNISIM-I-D benchmark case, testing the performance of the optimization proposed in a complex reservoir with several producer and injector wells, whose conventional optimization would require a high computational cost. The results showed an improvement in reservoir management by means of additional gains in terms of NPV, and through the proposed robust optimization algorithm, we show advantages of the operation of the wells and in the reduction in the computational efforts necessary to attain optimal solutions. The efficiency of the gradient-based approach coupled with model order reduction can be combined in future entire optimization workflow with global optimum algorithms like Fast Genetic Algorithm.

Dimensionality reduction for production optimization using polynomial approximations

The objective of this paper is to introduce a novel paradigm to reduce the computational effort in waterflooding global optimization problems while realizing smooth well control trajectories amenable for practical deployments in the field. In order to overcome the problems of slow convergence and non-smooth impractical control strategies, often associated with gradient-free optimization (GFO) methods, we introduce a generalized approach which represent the controls by smooth polynomial approximations either by a polynomial function or by a piecewise polynomial interpolation, which we denote as function control method (FCM) and interpolation control method (ICM), respectively. Using these approaches, we aim to optimize the coefficients of the selected functions or the interpolation points in order to represent the well-control trajectories along a time horizon. Our results demonstrate significant computational savings, due to a substantial reduction in the number of control parameters, as we seek the optimal polynomial coefficients or the interpolation points to describe the control trajectories as opposed to directly searching for the optimal control values (bottom hole pressure) at each time interval. We demonstrate the efficiency of the method on two and three-dimensional models, where we found the optimal variables using a parallel dynamic-neighborhood particle swarm optimization (PSO). We compared our FCM-PSO and ICM-PSO to the traditional formulation solved by both gradient-free and gradient-based methods. In all comparisons, both FCM and ICM show very good to superior performances.

Tensor based geology preserving reservoir parameterization with Higher Order Singular Value Decomposition (HOSVD)

Parameter estimation through robust parameterization techniques has been addressed in many works associated with history matching and inverse problems. Reservoir models are in general complex, nonlinear, and large-scale with respect to the large number of states and unknown parameters. Thus, having a practical approach to replace the original set of highly correlated unknown parameters with non-correlated set of lower dimensionality, that captures the most significant features comparing to the original set, is of high importance. Furthermore, de-correlating systems parameters while keeping the geological description intact is critical to control the ill-posedness nature of such problems. We introduce the advantages of a new low dimensional parameterization approach for reservoir characterization applications utilizing multilinear algebra based techniques like higher order singular value decomposition (HOSVD). In tensor based approaches like HOSVD, 2D permeability images are treated as they are, i.e., the data structure is kept as it is, whereas in conventional dimensionality reduction algorithms like SVD data has to be vectorized. Hence, compared to classical methods, higher redundancy reduction with less information loss can be achieved through decreasing present redundancies in all dimensions. In other words, HOSVD approximation results in a better compact data representation with respect to least square sense and geological consistency in comparison with classical algorithms. We examined the performance of the proposed parameterization technique against SVD approach on the SPE10 benchmark reservoir model as well as synthetic channelized permeability maps to demonstrate the capability of the proposed method. Moreover, to acquire statistical consistency, we repeat all experiments for a set of 1000 unknown geological samples and provide comparison using RMSE analysis. Results prove that, for a fixed compression ratio, the performance of the proposed approach outperforms that of conventional methods perceptually and in terms of least square measure. HighlightsIntroduced new permeability parameterization using High Order Singular Value Decomposition (HOSVD).HOSVD yield reduced computational time as compared to classical SVD based on the same compression ratio.The new methodology improves geological description by capturing all important features (spatial) as compared to SVD methods.The HOSVD method is general for any type of reservoir parameter.HOSVD can be applied in the optimization under the uncertainty paradigm.