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Ensemble Smoother with Enhanced Initial Samples for Inverse Modeling of Subsurface Flow Problems

  • Research Article-Petroleum Engineering
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Abstract

Ensemble-based data assimilation methods have been extensively investigated for inverse problems of fluid flow in porous media. However, when the permeability field is characterized by fine-scale gridblocks, the problem can be ill-posed and result in non-unique solutions. To address this issue, the principal component analysis with truncation was presented, but it may lead to biased estimation. In this paper, we propose to keep all eigenfunctions without truncation and add an additional sorting step after principal component analysis: sorting the initial samples according to the dimensional variability and assigning the dimensions with large variances to the leading eigenfunctions. The estimation is expected to be more accurate as the subspace spanned by the ensemble favors the dominant components. The proposed method is tested for multiple synthetic flow and transport cases. The results show that it provides more accurate estimation of the permeability fields and generates better history matching and prediction results for the production data (by 10–15%) than the results from the standard ensemble smoother, with the same computational cost. This sorting approach can be readily extended to the ensemble Kalman filter as well, for inverse modeling and estimating reservoir properties.

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Abbreviations

C :

Covariance

f :

Eigenfunction

h :

Hydraulic head, m

k :

Absolute permeability, mD

krα :

Relative permeability for a phase fluid

N :

Normal distribution

p c :

Capillary pressure, psia

p α :

Pressure of αα phase fluid, psia

q α :

Source/sink term, kg/s

S α :

Saturation of the α phase fluid

S or :

Residual oil saturation

S wc :

Irreducible water saturation

t :

Time, day

u α :

Velocity of α phase fluid, m/s

x :

Location in space, m

Y :

Log-permeability, mD

λ :

Eigenvalue

ξ :

Independent random variable

η :

Correlation length, ft

σ 2 :

Variance

μ α :

Viscosity of α phase fluid, Pa·s

ρ α :

Density of α phase fluid, kg/m3

ϕ :

Porosity

RMSE:

Root mean square error

BHP:

Bottom hole pressure, psia or bar

OPR:

Oil production rate, bbl/day or m3/day

WPR:

Water production rate, bbl/day or m3/day

GOR:

Gas–oil ratio

WCT:

Water cut

FOPT:

Field oil production total, bbl/day or m3/day

FGPT:

Field gas production total, bbl/day or m3/day

FWPT:

Field water production total, bbl/day or m3/day

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  1. College of Petroleum Engineering, China University of Petroleum (Bei**g), Bei**g, 102249, China

    Qinzhuo Liao

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Correspondence to Qinzhuo Liao.

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Liao, Q. Ensemble Smoother with Enhanced Initial Samples for Inverse Modeling of Subsurface Flow Problems. Arab J Sci Eng 48, 9535–9548 (2023). https://doi.org/10.1007/s13369-022-07343-x

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