A Gauss-Newton Trust-Region Solver for Large-Scale History-Matching Problems

SPE Journal ◽  
2017 ◽  
Vol 22 (06) ◽  
pp. 1999-2011 ◽  
Author(s):  
Guohua Gao ◽  
Hao Jiang ◽  
Paul van Hagen ◽  
Jeroen C. Vink ◽  
Terence Wells
Keyword(s):  
Reservoir Simulation ◽  
History Matching ◽  
Large Scale ◽  
Computational Cost ◽  
Trust Region ◽  
Uncertain Parameters ◽  
Memory Usage ◽  
New Approach ◽  
Matching Problems ◽  
Cpu Time

Summary Solving the Gauss-Newton trust-region subproblem (TRS) with traditional solvers involves solving a symmetric linear system with dimensions the same as the number of uncertain parameters, and it is extremely computational expensive for history-matching problems with a large number of uncertain parameters. A new trust-region (TR) solver is developed to save both memory usage and computational cost, and its performance is compared with the well-known direct TR solver using factorization and iterative TR solver using the conjugate-gradient approach. With application of the matrix inverse lemma, the original TRS is transformed to a new problem that involves solving a linear system with the number of observed data. For history-matching problems in which the number of uncertain parameters is much larger than the number of observed data, both memory usage and central-processing-unit (CPU) time can be significantly reduced compared with solving the original problem directly. An auto-adaptive power-law transformation technique is developed to transform the original strong nonlinear function to a new function that behaves more like a linear function. Finally, the Newton-Raphson method with some modifications is applied to solve the TRS. The proposed approach is applied to find best-match solutions in Bayesian-style assisted-history-matching (AHM) problems. It is first validated on a set of synthetic test problems with different numbers of uncertain parameters and different numbers of observed data. In terms of efficiency, the new approach is shown to significantly reduce both the computational cost and memory usage compared with the direct TR solver of the GALAHAD optimization library (see http://www.galahad.rl.ac.uk/doc.html). In terms of robustness, the new approach is able to reduce the risk of failure to find the correct solution significantly, compared with the iterative TR solver of the GALAHAD optimization library. Our numerical results indicate that the new solver can solve large-scale TRSs with reasonably small amounts of CPU time (in seconds) and memory (in MB). Compared with the CPU time and memory used for completing one reservoir simulation run for the same problem (in hours and in GB), the cost for finding the best-match parameter values using our new TR solver is negligible. The proposed approach has been implemented in our in-house reservoir simulation and history-matching system, and has been validated on a real-reservoir-simulation model. This illustrates the main result of this paper: the development of a robust Gauss-Newton TR approach, which is applicable for large-scale history-matching problems with negligible extra cost in CPU and memory.

A Gauss–Newton Trust Region Solver for Large Scale History Matching Problems

2017 ◽  
Author(s):  
Guohua Gao ◽  
Hao Jiang ◽  
Paul Van Hagen ◽  
Jeroen C. Vink ◽  
Terence Wells
Keyword(s):  
History Matching ◽  
Large Scale ◽  
Trust Region ◽  
Matching Problems

Reduced-Degrees-of-Freedom Gaussian-Mixture-Model Fitting for Large-Scale History-Matching Problems

SPE Journal ◽  
2019 ◽  
Vol 25 (01) ◽  
pp. 037-055
Author(s):  
Guohua Gao ◽  
Hao Jiang ◽  
Chaohui Chen ◽  
Jeroen C. Vink ◽  
Yaakoub El Khamra ◽  
...  
Keyword(s):  
Gaussian Mixture Model ◽  
Mixture Model ◽  
History Matching ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Computational Cost ◽  
Gaussian Mixture ◽  
New Method ◽  
Matching Problem ◽  
Matching Problems

Summary It has been demonstrated that the Gaussian-mixture-model (GMM) fitting method can construct a GMM that more accurately approximates the posterior probability density function (PDF) by conditioning reservoir models to production data. However, the number of degrees of freedom (DOFs) for all unknown GMM parameters might become huge for large-scale history-matching problems. A new formulation of GMM fitting with a reduced number of DOFs is proposed in this paper to save memory use and reduce computational cost. The performance of the new method is benchmarked against other methods using test problems with different numbers of uncertain parameters. The new method performs more efficiently than the full-rank GMM fitting formulation, reducing the memory use and computational cost by a factor of 5 to 10. Although it is less efficient than the simple GMM approximation dependent on local linearization (L-GMM), it achieves much higher accuracy, reducing the error by a factor of 20 to 600. Finally, the new method together with the parallelized acceptance/rejection (A/R) algorithm is applied to a synthetic history-matching problem for demonstration.

Reduced Degrees of Freedom Gaussian Mixture Model Fitting for Large Scale History Matching Problems

2019 ◽  
Author(s):  
Guohua Gao ◽  
Hao Jiang ◽  
Chaohui Chen ◽  
Jeroen C. Vink ◽  
Yaakoub El Khamra ◽  
...  
Keyword(s):  
Gaussian Mixture Model ◽  
Mixture Model ◽  
History Matching ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Model Fitting ◽  
Gaussian Mixture ◽  
Matching Problems

An Improved Implementation of the LBFGS Algorithm for Automatic History Matching

SPE Journal ◽  
2006 ◽  
Vol 11 (01) ◽  
pp. 5-17 ◽  
Author(s):  
Guohua Gao ◽  
Albert C. Reynolds
Keyword(s):  
Optimization Algorithm ◽  
History Matching ◽  
Large Scale ◽  
Search Algorithm ◽  
Hessian Matrix ◽  
Computational Experiments ◽  
Model Parameters ◽  
Matching Problems ◽  
Sensitivity Coefficients ◽  
Levenberg Marquardt

Summary For large scale history matching problems, where it is not feasible to compute individual sensitivity coefficients, the limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) is an efficient optimization algorithm, (Zhang and Reynolds, 2002; Zhang, 2002). However, computational experiments reveal that application of the original implementation of LBFGS may encounter the following problems:converge to a model which gives an unacceptable match of production data;generate a bad search direction that either leads to false convergence or a restart with the steepest descent direction which radically reduces the convergence rate;exhibit overshooting and undershooting, i.e., converge to a vector of model parameters which contains some abnormally high or low values of model parameters which are physically unreasonable. Overshooting and undershooting can occur even though all history matching problems are formulated in a Bayesian framework with a prior model providing regularization. We show that the rate of convergence and the robustness of the algorithm can be significantly improved by:a more robust line search algorithm motivated by the theoretical result that the Wolfe conditions should be satisfied;an application of a data damping procedure at early iterations orenforcing constraints on the model parameters. Computational experiments also indicate thata simple rescaling of model parameters prior to application of the optimization algorithm can improve the convergence properties of the algorithm although the scaling procedure used can not be theoretically validated. Introduction Minimization of a smooth objective function is customarily done using a gradient based optimization algorithm such as the Gauss- Newton (GN) method or Levenberg-Marquardt (LM) algorithm. The standard implementations of these algorithms (Tan and Kalogerakis, 1991; Wu et al., 1999; Li et al., 2003), however, require the computation of all sensitivity coefficients in order to formulate the Hessian matrix. We are interested in history matching problems where the number of data to be matched ranges from a few hundred to several thousand and the number of reservoir variables or model parameters to be estimated or simulated ranges from a few hundred to a hundred thousand or more. For the larger problems in this range, the computer resources required to compute all sensitivity coefficients would prohibit the use of the standard Gauss- Newton and Levenberg-Marquardt algorithms. Even for the smallest problems in this range, computation of all sensitivity coefficients may not be feasible as the resulting GN and LM algorithms may require the equivalent of several hundred simulation runs. The relative computational efficiency of GN, LM, nonlinear conjugate gradient and quasi-Newton methods have been discussed in some detail by Zhang and Reynolds (2002) and Zhang (2002).

Development of a Two-Dimensional Thermal Model for Li-Ion Battery Pack With Experimental Validation

2019 ◽  
Vol 12 (2) ◽  
Author(s):  
Haoting Wang ◽  
Ning Liu ◽  
Lin Ma
Keyword(s):  
Real Time ◽  
Large Scale ◽  
Experimental Validation ◽  
Computational Cost ◽  
Battery Pack ◽  
Two Dimensional ◽  
Li Ion Battery ◽  
New Approach ◽  
Battery Packs ◽  
Li Ion

Abstract This paper reports the development of a two-dimensional two states (2D2S) model for the analysis of thermal behaviors of Li-ion battery packs and its experimental validation. This development was motivated by the need to fill a niche in our current modeling capabilities: the need to analyze 2D temperature (T) distributions in large-scale battery packs in real time. Past models were predominately developed to either provide detailed T information with high computational cost or provide real-time analysis but only 1D lumped T information. However, the capability to model 2D T field in real time is desirable in many applications ranging from the optimal design of cooling strategies to onboard monitoring and control. Therefore, this work developed a new approach to provide this desired capability. The key innovations in our new approach involved modeling the whole battery pack as a complete thermal-fluid network and at the same time calculating only two states (surface and core T) for each cell. Modeling the whole pack as a complete network captured the interactions between cells and enabled the accurate resolution of the 2D T distribution. Limiting the calculation to only the surface and core T controlled the computational cost at a manageable level and rendered the model suitable for packs at large scale with many cells.

scholarly journals Precomputing Datalog Evaluation Plans in Large-Scale Scenarios

2019 ◽  
Vol 19 (5-6) ◽  
pp. 1073-1089
Author(s):  
ALESSIO FIORENTINO ◽  
NICOLA LEONE ◽  
MARCO MANNA ◽  
SIMONA PERRI ◽  
JESSICA ZANGARI
Keyword(s):  
Web Services ◽  
Semantic Web ◽  
Large Scale ◽  
Semantic Web Services ◽  
Memory Usage ◽  
Memory Consumption ◽  
New Approach ◽  
Large Databases ◽  
Execution Times

AbstractWith the more and more growing demand for semantic Web services over large databases, an efficient evaluation of Datalog queries is arousing a renewed interest among researchers and industry experts. In this scenario, to reduce memory consumption and possibly optimize execution times, the paper proposes novel techniques to determine an optimal indexing schema for the underlying database together with suitable body-orderings for the Datalog rules. The new approach is compared with the standard execution plans implemented in DLV over widely used ontological benchmarks. The results confirm that the memory usage can be significantly reduced without paying any cost in efficiency.

scholarly journals Assessment of multilevel ensemble-based data assimilation for reservoir history matching

2019 ◽  
Vol 24 (1) ◽  
pp. 217-239
Author(s):  
Kristian Fossum ◽  
Trond Mannseth ◽  
Andreas S. Stordal
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
History Matching ◽  
Computational Cost ◽  
Ensemble Size ◽  
Matching Problems ◽  
Reservoir Models ◽  
Reservoir History Matching ◽  
Computational Resources

AbstractMultilevel ensemble-based data assimilation (DA) as an alternative to standard (single-level) ensemble-based DA for reservoir history matching problems is considered. Restricted computational resources currently limit the ensemble size to about 100 for field-scale cases, resulting in large sampling errors if no measures are taken to prevent it. With multilevel methods, the computational resources are spread over models with different accuracy and computational cost, enabling a substantially increased total ensemble size. Hence, reduced numerical accuracy is partially traded for increased statistical accuracy. A novel multilevel DA method, the multilevel hybrid ensemble Kalman filter (MLHEnKF) is proposed. Both the expected and the true efficiency of a previously published multilevel method, the multilevel ensemble Kalman filter (MLEnKF), and the MLHEnKF are assessed for a toy model and two reservoir models. A multilevel sequence of approximations is introduced for all models. This is achieved via spatial grid coarsening and simple upscaling for the reservoir models, and via a designed synthetic sequence for the toy model. For all models, the finest discretization level is assumed to correspond to the exact model. The results obtained show that, despite its good theoretical properties, MLEnKF does not perform well for the reservoir history matching problems considered. We also show that this is probably caused by the assumptions underlying its theoretical properties not being fulfilled for the multilevel reservoir models considered. The performance of MLHEnKF, which is designed to handle restricted computational resources well, is quite good. Furthermore, the toy model is utilized to set up a case where the assumptions underlying the theoretical properties of MLEnKF are fulfilled. On that case, MLEnKF performs very well and clearly better than MLHEnKF.

History Matching Using the Ensemble Kalman Filter on a North Sea Field Case

SPE Journal ◽  
2008 ◽  
Vol 13 (04) ◽  
pp. 382-391 ◽  
Author(s):  
Vibeke Eilwn J. Haugen ◽  
Geir Naevdal ◽  
Lars-Joergen Natvik ◽  
Geir Evensen ◽  
Aina M. Berg ◽  
...  
Keyword(s):  
Simulation Model ◽  
North Sea ◽  
Reservoir Simulation ◽  
Ensemble Kalman Filter ◽  
History Matching ◽  
Large Scale ◽  
Field Model ◽  
Model Systems ◽  
Production Data ◽  
Reservoir Simulation Model

Summary This paper applies the ensemble Kalman filter (EnKF) to history match a North Sea field model. This is, as far as we know, one of the first published studies in which the EnKF is applied in a realistic setting using real production data. The reservoir-simulation model has approximately 45,000 active grid cells, and 5 years of production data are assimilated. The estimated parameters consist of the permeability and porosity fields, and the results are compared with a model previously established using a manual history-matching procedure. It was found that the EnKF estimate improved the match to the production data. This study, therefore, supported previous findings when using synthetic models that the EnKF may provide a useful tool for history matching reservoir parameters such as the permeability and porosity fields. Introduction The EnKF developed by Evensen (1994, 2003, 2007) is a statistical method suitable for data assimilation in large-scale nonlinear models. It is a Monte Carlo method, where model uncertainty is represented by an ensemble of realizations. The prediction of the estimate and uncertainty is performed by ensemble integration using the reservoir-simulation model. The method provides error estimates at any time based on information from the ensemble. When production data are available, a variance-minimizing scheme is used to update the realizations. The EnKF provides a general and model-independent formulation and can be used to improve the estimates of both the parameters and variables in the model. The method has previously been applied in a number of applications [e.g., in dynamical ocean models (Haugen and Evensen 2002), in model systems describing the ocean ecosystems (Natvik and Evensen 2003a, 2003b), and in applications within meteorology (Houtekamer et al. 2005)]. This shows that the EnKF is capable of handling different types of complex- and nonlinear-model systems. The method was first introduced into the petroleum industry in studies related to well-flow modeling (Lorentzen et al. 2001, 2003). Nævdal et al. (2002) used the EnKF in a reservoir application to estimate model permeability focusing on a near-well reservoir model. They showed that there could be a great benefit from using the EnKF to improve the model through parameter estimation, and that this could lead to improved predictions. Nævdal et al. (2005) showed promising results estimating the permeability as a continuous field variable in a 2D field-like example. Gu and Oliver (2005) examined the EnKF for combined parameter and state estimation in a standardized reservoir test case. Gao et al. (2006) compared the EnKF with the randomized-maximum-likelihood method and pointed out several similarities between the methods. Liu and Oliver (2005a, 2005b) examined the EnKF for facies estimation in a reservoir-simulation model. This is a highly nonlinear problem where the probability-density function for the petrophysical properties becomes multimodal, and it is not clear how the EnKF can best handle this. A method was proposed in which the facies distribution for each ensemble member is represented by two normal distributed Gaussian fields using a method called truncated pluri-Gaussian simulation (Lantuéjoul 2002). Wen and Chen (2006) provided another discussion on the EnKF for estimation of the permeability field in a 2D reservoir-simulation model and examined the effect of the ensemble size. Lorentzen et al. (2005) focused on the sensitivity of the results with respect to the choice of initial ensemble using the PUNQ-S3. Skjervheim et al. (2007) used the EnKF to assimilate seismic 4D data. It was shown that the EnKF can handle these large data sets and that a positive impact could be found despite the high noise level in the data. The EnKF has some important advantages when compared to traditional assisted history-matching methods; the result is an ensemble of history-matched models that are all possible model realizations. The data are processed sequentially in time, meaning that new data are easily accounted for when they arrive. The method allows for simultaneous estimation of a huge number of poorly known parameters such as fields of properties defined in each grid cell. By analyzing the EnKF update equations, it is seen that the actual degrees of freedom in the estimation problem are limited equal to the ensemble size. One is still able to update the most important features of large-scale models. A limitation of the EnKF is the fact that its computations are based on first- and second-order moments, and there are problems that are difficult to handle, particularly when the probability distributions are multimodal (e.g., when representing a bimodal channel facies distribution). This paper considers the use of the EnKF for estimating dynamic and static parameters, focusing on permeability and porosity, in a field model of a StatoilHydro-operated field in the North Sea. The largest uncertainty in the model is expected to be related to the permeability values, especially in the upper part of the reservoir where the uncertainty may be as large as 30%.

A New Parameterization Method for Large-Scale Reservoir History

2015 ◽  
Vol 733 ◽  
pp. 156-160
Author(s):  
Xia Yan ◽  
Jun Li ◽  
Hui Zhao
Keyword(s):  
Monte Carlo ◽  
Objective Function ◽  
History Matching ◽  
Large Scale ◽  
Monte Carlo Algorithm ◽  
Reservoir Properties ◽  
Matching Problems ◽  
Parameterization Method ◽  
Gradient Information ◽  
Unconditional Model

A novel and simple parameterization method using an ensemble of unconditional model realizations is applied to decrease the dimension of the misfit objective function in large-scale history matching problems. The major advantage of this parameterization method is that the singular value decomposition (SVD) calculation is completely avoided, which saves time and cost for huge matrix decomposition and the eigenvectors computations in parameterization process. After objective function transforms from a higher dimension to a lower dimension by parameterization, a Monte Carlo approach is introduced to evaluate the gradient information in the lower domain. Unlike the adjoint-gradient algorithms, the gradient in our method is estimated by Monte Carlo stochastic method, which can be easily coupled with different numerical simulator and avoid complicated adjoint code. When the estimated gradient information is obtained, any gradient-based algorithm can be implemented for optimizing the objective function. The Monte Carlo algorithm combined with the parameterization method is applied to Brugge reservoir field. The result shows that our present method gives a good estimation of reservoir properties and decreases the geological uncertainty without SVD but with a lower final objective function value, which provides a more efficient and useful way for history matching in large scale field.

An Improved Adjoint-Sensitivity Computation for Multiphase Flow Using Wavelets

SPE Journal ◽  
2012 ◽  
Vol 17 (02) ◽  
pp. 402-417 ◽  
Author(s):  
A.A.. A. Awotunde ◽  
R.N.. N. Horne
Keyword(s):  
Inverse Problem ◽  
Multiphase Flow ◽  
History Matching ◽  
Large Scale ◽  
Computational Cost ◽  
Sensitivity Matrix ◽  
Adjoint Sensitivity ◽  
Sensitivity Coefficients ◽  
Large Scale Problems ◽  
Gradient Based

Summary In history matching, one of the challenges in the use of gradient-based Newton algorithms (e.g., Gauss-Newton and Leven-berg-Marquardt) in solving the inverse problem is the huge cost associated with the computation of the sensitivity matrix. Although the Newton type of algorithm gives faster convergence than most other gradient-based inverse solution algorithms, its use is limited to small- and medium-scale problems in which the sensitivity coefficients are easily and quickly computed. Modelers often use less-efficient algorithms (e.g., conjugate-gradient and quasi-Newton) to model large-scale problems because these algorithms avoid the direct computation of sensitivity coefficients. To find a direction of descent, such algorithms often use less-precise curvature information that would be contained in the gradient of the objective function. Using a sensitivity matrix gives more-complete information about the curvature of the function; however, this comes with a significant computational cost for large-scale problems. An improved adjoint-sensitivity computation is presented for time-dependent partial-differential equations describing multiphase flow in hydrocarbon reservoirs. The method combines the wavelet parameterization of data space with adjoint-sensitivity formulation to reduce the cost of computing sensitivities. This reduction in cost is achieved by reducing the size of the linear system of equations that are typically solved to obtain the sensitivities. This cost-saving technique makes solving an inverse problem with algorithms (e.g., Levenberg-Marquardt and Gauss-Newton) viable for large multiphase-flow history-matching problems. The effectiveness of this approach is demonstrated for two numerical examples involving multiphase flow in a reservoir with several production and injection wells.

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