A Constrained Proper Orthogonal Decomposition Model for Upscaling Permeability

T. A. Onimisi* (Corresponding Author), B. O. Lashore, L. T. Akanji, J. L. M. A. Gomes

*Corresponding author for this work

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Abstract Reservoir modelling and simulation are vital components of modern reservoir management processes. Despite the advances in computing power and the advent of smart technologies, the implementation of model-based operational/control strategies has been limited by the inherent complexity of reservoir models. Thus, reduce order models that not only reduce the cost of the implementation but also provide geological consistent prediction are essential. This paper introduces reduced-order models based on the proper orthogonal decomposition (POD) coupled with linear interpolation for upscaling. First, using POD-based models, low rank approximate (LRA) are obtained by projecting the high dimensional permeability dataset to a low dimensional subspace spanned by its trajectories to decorrelate the dataset. Next, the LRA is integrated into the interpolation algorithm to generate upscaled values. This technique is highly scalable since low-rank approximations are achieved by the variation in the number of modes used for reconstruction. To test the validity and reliability of the model, we show its application to the practical dataset from SPE10 benchmark case2. From statistics of the error analysis, the classical POD algorithm seems to be more preferred for LRA; however, since non-negativity of the permeability data set is a priority, the constrained POD (Non-negative POD) algorithm described in this paper is more appropriate. Finally, we compared the POD-based models to a traditional industry-standard upscaling technique (e.g., arithmetic mean) to highlight our model benefits/performance. Results show that the POD-based models, particularly the Non-negative POD model, produce considerably less error than the arithmetic mean model in the upscaling process. This article is protected by copyright. All rights reserved.
Original languageEnglish
Pages (from-to)899-916
Number of pages18
JournalInternational Journal for Numerical Methods in Fluids
Issue number6
Early online date29 Dec 2022
Publication statusPublished - 1 Jun 2023

Bibliographical note

Temiloluwa A. Onimisi would like to acknowledge Petroleum Development Trust Fund Nigeria, for funding this PhD research project.

Research Funding
Petroleum Development Trust Fund Nigeria

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.


  • Reduce Order Modeling
  • Proper Orthogonal Decomposition
  • Upscaling
  • Interpolation
  • Multifluid Flows


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