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Principal component analysis for reservoir uncertainty reduction

Abstract

Reservoir monitoring considering all measurements and simulator outcomes available nowadays can become a complex task. The data integration and mainly the proper use of the big datasets is a challenge, especially in full field studies. This scenario of increasing data availability is an ongoing process due to new measurement technologies, high computational power and the reservoir characterization complexity. We propose to identify reservoir measurements that best represent the overall reservoir behavior using the Principal Component Analysis mathematical procedure. In addition, this procedure allows a reduction of the dataset dimension for a faster and more efficient reservoir analysis. Latin Hypercube sampling is used to sample the reservoir attribute range and the principal component of the measurements are integrated to identify the attribute interval that minimizes the simulation mismatch. The methodology is applied to a reservoir simulation model with 20 uncertainty attributes. Three study tests were performed using different percentiles in the likelihood distribution, which can conservatively or severely reduce the attribute ranges. The method achieved a coverage of approximately 95 % of the problem variability using five out of fifteen original principal components. Reservoir uncertainties were reduced and most of the simulated measurements had a significant history matching improvement.

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Abbreviations

m :

Number of observations (models)

n :

Number of variables

X :

Original data

b :

Principal component

Y :

New data (called Score)

M :

Raw data matrix

M ADJ :

Adjusted matrix

C :

Covariance matrix

N :

New matrix

E :

Matrix containing the eigenvectors

E T :

Transpose of the E matrix containing the eigenvectors

w :

Reservoir uncertainty attributes

K rW :

Water relative permeability

S W :

Water saturation

S Wir :

Irreducible water saturation

S OR :

Residual oil saturation

E xpoW :

Coefficient of Corey’s equation for water relative permeability

k rw*:

Maximum value of the water relative permeability in Corey’s equation

K x :

Effective permeability in X direction

K Z :

Effective permeability in Z direction

Ψ :

Match quality indicator

h :

History data

s :

Simulated data

d :

Number of data points

PCA:

Principal component analysis

PC:

Principal component

KPCA:

Kernel principal component analysis

DCT:

Discrete cosine transform

LSP:

Least square projection

LL:

Lower limit

ProjClus:

Projection by Clustering

SM:

Simulated measurement

SqE:

Square error

SqER:

Square error ratio

UP:

Upper limit

LHC:

Latin hypercube

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Correspondence to André Carlos Bertolini.

Additional information

Technical Editor: Marcelo A. Trindade.

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Bertolini, A.C., Schiozer, D.J. Principal component analysis for reservoir uncertainty reduction. J Braz. Soc. Mech. Sci. Eng. 38, 1345–1355 (2016). https://doi.org/10.1007/s40430-015-0377-6

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Keywords

  • Uncertainty reduction
  • History matching
  • Reservoir simulation
  • Principal components
  • Mismatch