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Automatic history matching considering surrogate-based optimization and Karhunen–Loève expansions

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Abstract

In oil reservoir engineering one stage of great interest is the production history matching process, in which the numerical model is adjusted to reproduce the observed field production. The objective is to find the model parameters which minimize the difference between calculated and observed fluid production rates. However, the consideration of the rock properties along every cell of the numerical model that represents the reservoir as design variables would lead to an unfeasible large number of variables and consequently, a very difficult problem to be solved. To overcome that, we will assume that the permeability field will be stochastic and represented by a spectral decomposition through the Karhunen–Loève expansion (KLE). In this work both linear and nonlinear forms for KLE expansions will be investigated. The mathematical formulation of this problems leads to nonsmooth and high-cost objective functions leading to a difficult problem to be tackled. To overcome that, surrogate models are built and incorporated in a sequential optimization strategy. Results are presented and the potential of the developed tool is discussed.

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Abbreviations

C :

Covariance matrix

d obs :

Observed cumulative oil production

E :

Eigenvector matrix

f error :

Objective function

f:

Calculated cumulative oil production

\( \hat{f}\left( x \right) \) :

Kriging model

F :

High-order space

y :

Permeability realization vector

n :

Time step

n dv :

The total number of design variables

N r :

Number of permeability realizations

N c :

Number of reservoir grid elements

Np :

Number of producers wells

t :

Total time step number

K :

Kernel matrix

W :

Covariance matrix of the transformed realizations

α :

Eigenvector of kernel matrix

Λ :

Eigenvalue diagonal matrix

ξ :

Random numbers vector

λ :

Eigenvalue of covariance matrix

ν :

Eigenvector of covariance matrix

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Acknowledgments

The authors acknowledge the financial support for this research given by ANP (National Agency of Petroleum, Brazil), CNPQ and PETROBRAS.

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Authors and Affiliations

  1. Department of Control and Industrial Processes, Federal Institute of Education, Science and Technology of Pernambuco, Recife, PE, 50740-540, Brazil

    José Dásio de Lira Jr.

  2. Mechanical Engineering Department, Federal University of Pernambuco, Recife, PE, 50670-901, Brazil

    Ramiro B. Willmersdorf

  3. Civil Engineering Department, Federal University of Pernambuco, Recife, PE, 50670-901, Brazil

    Silvana M. B. Afonso & Bernardo Horowitz

Authors
  1. José Dásio de Lira Jr.
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  2. Ramiro B. Willmersdorf
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  3. Silvana M. B. Afonso
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  4. Bernardo Horowitz
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Correspondence to José Dásio de Lira Jr..

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Technical Editor: Celso Kazuyuki Morooka.

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de Lira, J.D., Willmersdorf, R.B., Afonso, S.M.B. et al. Automatic history matching considering surrogate-based optimization and Karhunen–Loève expansions. J Braz. Soc. Mech. Sci. Eng. 36, 919–928 (2014). https://doi.org/10.1007/s40430-014-0128-0

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  • DOI: https://doi.org/10.1007/s40430-014-0128-0

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