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Gradient-based Pareto optimal history matching for noisy data of multiple types

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Abstract

The advantages of the simultaneous integration of production and time-lapse seismic data for history matching have been demonstrated in a number of previous studies. Production data provide accurate observations at particular spatial locations (wells), while seismic data enable global, though filtered/noisy, estimates of state variables. In this work, we present an efficient computational tool for bi-objective history matching, in which data misfits for both production and seismic measurements are minimized using an adjoint-gradient approach. This enables us to obtain a set of Pareto optimal solutions defining the optimal trade-off between production and seismic data misfits (which are, to some extent, conflicting). The impact of noise structure and noise level on Pareto optimal solutions is investigated in detail. We discuss the existence of the “best” trade-off solution, or least-conflicting posterior model, which corresponds to the history-matched model that is expected to provide the least-conflicting forecast of future reservoir performance. The overall framework is successfully applied in 2D and 3D compositional simulation problems to provide a single least-conflicting posterior model and, for the 2D case, multiple samples from the posterior distribution using the randomized maximum likelihood method.

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Abbreviations

g :

governing system

x :

state variables

u :

uncertain model parameters

t 0 :

beginning of history matching period

t f :

prediction horizon

\(\mathcal {J}\) :

objective function

G :

model response

d obs :

observed data

\(\mathcal {C}_{D}\) :

measurement error covariance

\(\mathcal {L}\) :

augmented objective function

\(\mathcal {R}\) :

model mismatch and/or regularization term

\(\hat {\mathbf {u}}\) :

optimal solution

\(\mathcal {J}_{\text {prod}}\) :

injection/production rate data mismatch

N obs :

number of observations

\({N}_{\text {well}}^{q}\) :

number of wells providing phase rate data

N p :

number of phases

C j, p :

diagonal elements of covariance matrix

q j, p :

simulated phase injection/production rates

\(\tilde {q}_{j,p}\) :

observed phase injection/production rates

D p :

phase relative density

\(\mathcal {J}_{\text {seism}}\) :

seismic data mismatch

N block :

number of blocks with seismic data

C p :

diagonal elements of covariance matrix

s j, p :

simulated phase saturation

\(\tilde {s}_{j,p}\) :

observed phase saturation

ξ :

PCA variables

Φ:

PCA basis matrix

\(\bar {\mathbf {u}}\) :

prior mean

N r :

number of prior realizations

\(\hat {\boldsymbol {\xi }}\) :

optimal PCA variable solution

\(\alpha _{\mathcal {R}}\) :

model mismatch weight

\(\mathcal {C}_{M}\) :

spatial covariance matrix

u :

random prior realization

ξ :

random realization of PCA variables

d true :

true model response

Σ:

standard deviations of measurement errors

σ i :

component of Σ

γ q :

production data noise parameter

γ s :

seismic data noise parameter

ϕ :

porosity

\({s}_{g}^{\text {true}}\) :

true gas saturation

\({\mathcal {J}}_{i}^{u}\) :

utopian point

α i :

articulation coefficients

\(\mathcal {J}_{\text {mo}}\) :

bi-objective function

\({\mathcal {J}}_{\text {prod}}^{0}\) :

normalization coefficient of production term

\({\mathcal {J}}_{\text {seism}}^{0}\) :

normalization coefficient of seismic term

\({\mathcal {J}}_{\text {prod}}^{*}\) :

upper bound of normalized production term

\({\mathcal {J}}_{\text {seism}}^{*}\) :

upper bound of normalized seismic term

α :

bi-objective articulation coefficient

s :

lower bound of history matching trajectory

S :

upper bound of history matching trajectory

\({\mathcal {J}}_{1}^{\min }\) :

minimum of \(\mathcal {J} _1\)

\({\mathcal {J}}_{2}^{\min }\) :

minimum of \(\mathcal {J} _2\)

𝜖 :

relaxation parameter

\({\mathbf {d}}_{\text {obs}}^{k}\) :

observed data for noise realization k

κ :

estimated curvature of Pareto front

Δx :

x-dimension of grid block

Δy :

y-dimension of grid block

Δz :

z-dimension of grid block

μ :

log-permeability

\({\mathbf {u}}_{\text {true}}^{\textsc {pca}} \) :

projection of the true model into PCA space

u true :

true model

\(\hat {{\Phi }}^{-1}\) :

pseudo-inverse of Φ

\({\mathbf {d}}_{\text {obs}}^{\textsc {pca}}\) :

observed data corresponding to \({\mathbf {u}}_{\text {true}}^{\textsc {pca}}\)

S j :

relative misfit of CO2 rate for well j

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Acknowledgements

We thank Tapan Mukerji (Stanford University) for useful discussions regarding time-lapse seismic data.

Funding

This study received financial support from the industrial affiliates of the Stanford University Reservoir Simulation Research (SUPRI–B) and Smart Fields Consortia.

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Correspondence to Oleg Volkov.

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Volkov, O., Bukshtynov, V., Durlofsky, L.J. et al. Gradient-based Pareto optimal history matching for noisy data of multiple types. Comput Geosci 22, 1465–1485 (2018). https://doi.org/10.1007/s10596-018-9766-0

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