Abstract
This paper presents a non-intrusive subdomain POD-TPWL (SD POD-TPWL) for reservoir history matching through integrating domain decomposition (DD), proper orthogonal decomposition (POD), radial basis function (RBF) interpolation, and the trajectory piecewise linearization (TPWL). It is an efficient approach for model reduction and linearization of general non-linear time-dependent dynamical systems without accessing to the legacy source code. In the subdomain POD-TPWL algorithm, firstly, a sequence of snapshots over the entire computational domain is saved and then partitioned into subdomains. From the local sequence of snapshots over each subdomain, a number of local basis vectors is formed using POD, and then the RBF interpolation is used to estimate the derivative matrices for each subdomain. Finally, those derivative matrices are substituted into a POD-TPWL algorithm to form a reduced-order linear model in each subdomain. This reduced-order linear model makes the implementation of the adjoint easy and results in an efficient adjoint-based parameter estimation procedure. Comparisons with the classic finite-difference-based history matching show that our proposed subdomain POD-TPWL approach is obtaining comparable results. The number of full-order model simulations required is roughly 2–3 times the number of uncertain parameters. Using different background parameter realizations, our approach efficiently generates an ensemble of calibrated models without additional full-order model simulations.
Highlights
History matching is the process of calibrating uncertain reservoir model parameters such as gridblock permeabilities, porosities, faults multipliers and facies distributions, Electronic supplementary material The online version of this article contains supplementary material, which is available to authorized users.Extended author information available on the last page of the article.through minimization of a cost function that quantifies the misfit between simulated and observed data
In order to avoid model intrusion and numerous full-order simulations, we propose to incorporate domain decomposition (DD) and radial basis function (RBF) interpolation into Proper Orthogonal Decomposition (POD)-trajectory piecewise linearization (TPWL)
We propose to use RBF interpolation to obtain the derivative matrices that are required by the POD-TPWL
Summary
If the gradient of the cost function with respect to parameters can be computed using the adjoint of the reservoir model, history matching problems can be efficiently solved using a gradient-based minimization algorithm [11]. POD provides as means to project the high-dimensional states into an optimal lower-dimensional subspace The basis of this subspace is obtained by performing a Singular Value Decomposition (SVD) of a snapshot matrix containing the solution states at selected time steps (snapshots) computed from training simulations. Radial basis function interpolation is used to approximately estimate these derivative matrices These derivative matrices are substituted into POD-TPWL algorithm to form a subdomain reduced-order linear model. Domain decomposition has the abilities to efficiently capture localized physical features [10] and has the potential to improve the derivative estimation by local low-dimensional RBF interpolation which will be described in the subsections
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