History matching production data and uncertainty assessment with an efficient TSVD parameterization algorithm

Shirangi, Mehrdad G.

doi:10.1016/j.petrol.2013.11.025

Cited by 30 publications

(21 citation statements)

References 16 publications

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“…Here γ > 0 is the adjustable LM damping factor and J ∈ R N d ×N m is the Jacobian matrix which defines an approximate local linear mapping from x to d(x). In [3,43,44,45] J is referred to as the dimensionless sensitivity matrix. For a large number of observations N d and N m , generating the often dense Jacobian and solving (5.2) becomes costly.…”

Section: Levenberg-marquardt Updatementioning

confidence: 99%

“…The work of [3] discussed advantages of using randomized 2-view or 1-view methods for speeding up the LM approach for inverting geothermal reservoir models. The approaches presented in [3] are based on the work of [43,44,45], who used an iterative Lanczos approach to generate a TSVD of J . After forming an approximate TSVD of J , an approximate LM update can be found according to [3,43,44,45]…”

Section: Approximate Truncated Updatesmentioning

confidence: 99%

“…Following [43,44,45], [3] chose to gradually increase p between LM iterations, since using a large p at early iterations can be computationally wasteful and can result in bad convergence characteristics. The study by [3] demonstrated that using randomized low-rank approximations can work well and can be considerably faster than using a standard iterative Lanczos method.…”

Section: Approximate Truncated Updatesmentioning

confidence: 99%

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Pass-Efficient Randomized Algorithms for Low-Rank Matrix Approximation Using Any Number of Views

Bjarkason¹

2019

SIAM J. Sci. Comput.

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This paper describes practical randomized algorithms for low-rank matrix approximation that accommodate any budget for the number of views of the matrix. The presented algorithms, which are aimed at being as pass efficient as needed, expand and improve on popular randomized algorithms targeting efficient low-rank reconstructions. First, a more flexible subspace iteration algorithm is presented that works for any views v ≥ 2, instead of only allowing an even v. Secondly, we propose more general and more accurate singlepass algorithms. In particular, we propose a more accurate memory efficient single-pass method and a more general single-pass algorithm which, unlike previous methods, does not require prior information to assure near peak performance. Thirdly, combining ideas from subspace and single-pass algorithms, we present a more passefficient randomized block Krylov algorithm, which can achieve a desired accuracy using considerably fewer views than that needed by a subspace or previously studied block Krylov methods. However, the proposed accuracy enhanced block Krylov method is restricted to large matrices that are either accessed a few columns or rows at a time. Recommendations are also given on how to apply the subspace and block Krylov algorithms when estimating either the dominant left or right singular subspace of a matrix, or when estimating a normal matrix, such as those appearing in inverse problems. Computational experiments are carried out that demonstrate the applicability and effectiveness of the presented algorithms. Motivation.The primary motivation for this study was to develop algorithms to speed up inverse methods used to estimate parameters in models describing subsurface flow in geothermal reservoirs [36,3]. Inverting models describing complex geophysical processes, such as fluid flow in the subsurface, frequently involves matching a large data set using highly parameterized computational models. Running the model commonly involves solving an expensive and nonlinear forward problem. Despite the possible nonlinearity of the forward problem, the link between the model parameters and simulated observations is often described in terms of a Jacobian matrix J ∈ R N d ×N m , which locally linearizes the relationship between the parameters and observations. The size of J is therefore determined by the (large) parameter and observation spaces. In this case, explicitly forming J is out of the question since at best it involves solving N m direct problems (linearized forward simulations) or N d adjoint problems (linearized backward simulations) [6,23,35,38]. Nevertheless, the information contained in J can be helpful for the purpose of inverting the model using nonlinear inversion methods such as a Gauss-Newton or Levenberg-Marquardt approach, and for quantifying uncertainty.Using adjoint simulation, direct simulation and randomized algorithms, the necessary information can be extracted from J without ever explicitly forming the large matrix J . Bjarkason et al. [3] showed that inversion of a nonlinea...

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Section: Levenberg-marquardt Updatementioning

confidence: 99%

Section: Approximate Truncated Updatesmentioning

confidence: 99%

Section: Approximate Truncated Updatesmentioning

confidence: 99%

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Pass-Efficient Randomized Algorithms for Low-Rank Matrix Approximation Using Any Number of Views

Bjarkason¹

2019

SIAM J. Sci. Comput.

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“…In the context of inverse problem theory (Tarantola, 2005), generating multiple calibrated (history-matched) models correspond to a sampling of the posterior probability density function (pdf). A common approach for solving this problem is the randomized maximum likelihood (RML) (Kitanidis, 1995;Shirangi, 2014). With RML, a sample from the posterior pdf is generated by minimizing an objective function that quantifies the mismatch between observed and predicted data.…”

Section: Clfd Model Calibration For Channelized Modelsmentioning

confidence: 99%

“…The model calibration methods typically involve a challenging minimization problem as this problem is usually ill-posed and the number of unknown model parameters can be very large. The ill-posedness of model calibration can be mitigated by reducing the number of parameters through an appropriate parameterization such as TSVD (Shirangi, 2011(Shirangi, , 2014Shirangi and Emerick, 2016;Bjarkason et al, 2017;Dickstein et al, 2017), ensemble-based methods (Rafiee and Reynolds, 2017;Rafiee, 2017;Rafiee and Reynolds, 2018), and PCA (Vo and Durlofsky, 2016). Durlofsky (2015, 2014) presented a differentiable PCA-based parameterization (O-PCA) that enables application of efficient gradient-based approaches for model calibration of complex channelized systems.…”

Section: Introductionmentioning

confidence: 99%

Closed-loop field development with multipoint geostatistics and statistical performance assessment

Shirangi

2019

Journal of Computational Physics

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Closed-loop field development (CLFD) optimization is a comprehensive framework for optimal development of subsurface resources. CLFD involves three major steps: 1) optimization of full development plan based on current set of models, 2) drilling new wells and collecting new spatial and temporal (production) data, 3) model calibration based on all data. This process is repeated until the optimal number of wells is drilled. This work introduces a new CLFD implementation for complex systems described by multipoint geostatistics (MPS). Model calibration is accomplished in two steps: conditioning to spatial data by a geostatistical simulation method, and conditioning to production data by optimization-based PCA. A statistical procedure (TruMAP) is presented to assess the performance of CLFD. For performance assessment by TruMAP, the methodology is applied to an oil reservoir example for 25 different true-model cases. Application of a single-step of CLFD, improved the true NPV in 64%-80% of cases. The full CLFD procedure (with three steps) improved the true NPV in 96% of cases, with an average improvement of 37%. These results indicate the effectiveness of performing multiple steps of closed-loop optimization. This massive computational experiment involved about 9.5 million reservoir simulation runs that took about 320,000 CPU hours.

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Randomized Truncated SVD Levenberg‐Marquardt Approach to Geothermal Natural State and History Matching

Bjarkason

Maclaren

O’Sullivan

et al. 2018

Water Resources Research

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The Levenberg‐Marquardt (LM) method is commonly used for inverting models used to describe geothermal, groundwater, or oil and gas reservoirs. In previous studies, LM parameter updates have been made tractable for highly parameterized inverse problems with large data sets by applying matrix factorization methods or iterative linear solvers to approximately solve the update equations. Some studies have shown that basing model updates on the truncated singular value decomposition (TSVD) of a dimensionless sensitivity matrix achieved using Lanczos iteration can speed up the inversion of reservoir models. Lanczos iterations only require the sensitivity matrix times a vector and its transpose times a vector, which are found efficiently using adjoint and direct simulations without the expense of forming a large sensitivity matrix. Nevertheless, Lanczos iteration has the drawback of being a serial process, requiring a separate adjoint solve and direct solve every Lanczos iteration. Randomized methods, developed for low‐rank matrix approximation of large matrices, are more efficient alternatives to the standard Lanczos method. Here we develop LM variants which use randomized methods to find a TSVD of a dimensionless sensitivity matrix when updating parameters. The randomized approach offers improved efficiency by enabling simultaneous solution of all adjoint and direct problems for a parameter update.

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History matching production data and uncertainty assessment with an efficient TSVD parameterization algorithm

Cited by 30 publications

References 16 publications

Pass-Efficient Randomized Algorithms for Low-Rank Matrix Approximation Using Any Number of Views

Pass-Efficient Randomized Algorithms for Low-Rank Matrix Approximation Using Any Number of Views

Closed-loop field development with multipoint geostatistics and statistical performance assessment

Randomized Truncated SVD Levenberg‐Marquardt Approach to Geothermal Natural State and History Matching

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