Non-intrusive subdomain POD-TPWL for reservoir history matching

Abstract

This paper presents a non-intrusive subdomain POD-TPWL (SD POD-TPWL) algorithm for reservoir data assimilation through integrating domain decomposition (DD), 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 timedependent dynamical systems without intruding the legacy source code. In the subdomain POD-TPWL algorithm, firstly, a sequence of snapshots over the entire computational domain are 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 resulting in an efficient adjoint-based parameter estimation procedure. The performance of the new adjoint-based parameter estimation algorithm has been assessed through several synthetic cases. 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.

Surrogate‐Based Joint Estimation of Subsurface Geological and Relative Permeability Parameters for High‐Dimensional Inverse Problem by Use of Smooth Local Parameterization

“…In terms of representing the nonlinearity, our approach construct local linear model at each out iteration and then update the linear model for each iterations. This alternative inner‐outer loop strategy that constructs local linear surrogates and updates them continuously could progressively capture the nonlinearity and approach to the optimal solution (Altaf et al., 2009; Kaleta et al., 2011; Vermeulen & Heemink, 2006; Xiao et al., 2018). In addition, in the presence of global approximations the order of the polynomial can be quite high, and therefore, the number of coefficients and consequently numerical simulations grows exponentially, while the local representations would only require a limited degree (this is indeed why our proposed method works by only employing a linear local reconstruction).…”

“…In projection-based ROMs, a set of reduced basis is extracted from the simulation snapshots using an unsupervised learning technique, for example, proper orthogonal decomposition (POD) (Altaf et al, 2009;Vermeulen & Heemink, 2006) and the full-order model operator is projected onto the subspace spanned by the reduced basis, which can significantly reduce the degrees of freedom of the system (Kaleta et al, 2011;Xiao et al, 2018). In order to overcome the computational burden associated with the application of Monte Carlo methods to the groundwater flow equation with random hydraulic conductivity, Pasetto et al (2013) present a model-order reduction technique that the high-dimensional model equations are projected onto subspace based on the Galerkin projection.…”

Model‐Reduced Adjoint‐Based Inversion Using Deep‐Learning: Example of Geological Carbon Sequestration Modeling

“…The high dimension of dynamic equations, for example, $${\mathbf{f}}^{n}\hspace*{.5em}\in \hspace*{.5em}{R}^{{N}_{g}}\to {R}^{{N}_{g}}$$, limits the application of the adjoint‐based method to actual models. This limitation has motivated the development of a reduced‐order modeling strategy previously (Heijn & Markovinovic, 2003), where the high‐dimensional model state variables x n and model parameters m are reduced simultaneously (Xiao et al., 2019, 2021b).…”

Deep‐Learning‐Based Adjoint State Method: Methodology and Preliminary Application to Inverse Modeling

“…Thus, many efforts have been done to make the implementation of the adjoint model practically tractable. Among them, surrogate modeling technique is identified to be one of the most effective ways (Altaf et al., 2009; Kaleta et al., 2011; Xiao et al., 2019).…”