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Managing geological uncertainty in expensive reservoir simulation optimization

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Abstract

A method to manage geological uncertainty as part of an expensive simulation-based optimization process is presented. When the number of realizations representing the uncertainty is high, the computational cost to optimize the system can be considerable, and often prohibitively, as each forward evaluation is expensive to evaluate. To overcome this limitation, an iterative procedure is developed that selects a subset of realizations, based on a binary nonlinear optimization subproblem, to match the statistical properties of the target function at known sample points. This results in a reduced-order model that is optimized in place of the full system at a much lower computational cost. The result is validated over the ensemble of all realizations giving rise to one new sample point per iteration. The process repeats until the stipulated stopping conditions are met. Demonstration of the proposed method on a publicly available realistic reservoir model with 50 realizations shows that comparable results to full optimization can be obtained but far more efficiently.

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Abbreviations

AROM:

- adaptive reduced-order modeling

CDF:

- cumulative distribution function

ES:

- evolutionary strategy

FOPT:

- field oil production total

FWPT:

- field water production total

GA:

- genetic algorithm

MAE:

- maximum absolute error

MSE:

- mean squared error

NPV:

- net present value

PSO:

- particle swarm optimization

RO:

- retrospective optimization

RBF:

- radial-basis function

RMSE:

- root mean squared error

SSE:

- sum of squared error

SM3D:

- standard meter cubed per day unit

TS:

- tabu search

B g :

- gas injection cost by unit

B w :

- water injection cost by unit

C g :

- gas production cost by unit

C o :

- oil production cost by unit

C w :

- water production cost by unit

D :

- set of samples with metric values \(\mathbb {R}^{[ S\times (n+m)]}\)

𝜖 m :

- model mismatch error measure

\({\epsilon _{m}^{T}}\) :

- model mismatch error measure threshold

𝜖 s :

- best known solution error measure

\({\epsilon _{s}^{T}}\) :

- best known solution error measure threshold

𝜖 μ :

- error measure of the first moment (mean)

𝜖 σ :

- error measure of the second moment (std-dev)

E :

- function of error measures

f k :

- utility-based objective function

f M :

- reduced-model objective value

f S :

- full-model objective value

F :

- simulation-based NPV function

F b e s t :

- best known solution objective value

i :

- variable count

j :

- realization count

k :

- sample count

K g :

- gas injection quantity

K w :

- water injection quantity

λ :

- confidence factor, ∈ [0 1]

m :

- number of realizations

\(\bar {m}\) :

- number of selected realizations

\(\bar {m}_{i}^{L}\) :

- lower bound on the number of selected realizations

\(\bar {m}_{i}^{U}\) :

- upper bound on the number of selected realizations

M :

- reduced-order simulation-based NPV function

μ k :

- k-th sample mean

\(\bar \mu _{k}\) :

- k-th sample mean estimate

n :

- number of control variables

n p :

- number of producer wells

n q :

- number of injector wells

n v :

- number of partitions

N :

- set of all realizations by index

Ω :

- profit component of NPV function

Ψ :

- cost component of NPV function

P o :

- oil production value by unit

P g :

- gas production value by unit

Q o :

- oil quantity

Q g :

- gas quantity

Q w :

- water quantity

ρ j :

- j th realization

ρ :

- set of discrete uncertainties (realizations)

\(\bar {\boldsymbol {\rho }}\) :

- compact set selected of realizations

r :

- discount rate

R b e s t :

- best known solution realization values, \(\mathbb {R}^{m}\)

R :

- set of sample metric values, \(\mathbb {R}^{[ S\times m ]}\)

σ k :

- k th sample standard deviation

\(\bar \sigma _{k}\) :

- k th sample standard deviation estimate

S :

- number of samples

t :

- incremental simulation time period

T :

- time horizon (years)

U :

- array of selected realizations, \(\mathbb {B}^{m}\)

\(\hat {U}\) :

- solution array of selected realizations, \(\mathbb {B}^{m}\)

x i :

- i th variable

\({x_{i}^{L}}\) :

- i th variable lower bound

\({x_{i}^{U}}\) :

- i th variable upper bound

X :

- set of control variables, \(\mathbb {R}^{n}\)

X b e s t :

- best known solution, \(\mathbb {R}^{n}\)

X :

- set of samples, \(\mathbb {R}^{[ S\times n ]}\)

w k :

- k th sample weight, ∈ [0 1]

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Acknowledgment

Many thanks to William Bailey (Principal Scientist, Schlumberger-Doll Research) for his insights about the Olympus Challenge problem. I am also grateful to the anonymous reviewers for their suggestions on this manuscript.

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  1. Schlumberger-Doll Research, Cambridge, MA, 02139, USA

    Kashif Rashid

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Appendix

Appendix

The continuous time net present value (NPV) objective function used in this study is stated below for completeness:

$$ F(X) = {{\int}_{0}^{t}} e^{-rt} [~{\varOmega}(X,t) - {\varPhi}(X,T) ~] dt $$
(1)

where Ω(X,t) and Φ(X,t) represent the revenue and cost streams as a function of time t respectively, given as:

$$ \begin{array}{@{}rcl@{}} {\varOmega}(X,t) &=& P_{o}Q_{o}(X,t) + P_{g}Q_{g}(X,t) \\ {\varPhi}(X,t) &=& C_{o}Q_{o}(X,t) + C_{g}Q_{g}(X,t) + C_{w}Q_{w}(X,t) {\dots} \\ && + B_{g}K_{g}(X,t) + B_{w}K_{w}(X,t) + D_{t} \end{array} $$

where X is the set of control variables \(\in \mathbb {R}^{n}\), T is the simulation time period, and r is the discount rate. The oil, gas, and water rates are given by Qo,Qg, and Qw. The unit market price for oil and gas is denoted Po and Pg, respectively, while the costs associated with unit production of oil, gas and water are given by Co,Cg, and Cw. Kg and Kw indicate the gas and water injection rates, with unit costs Bg and Bw, respectively. Lastly, a fixed cost per time-step is given by Dt. Note that only those items pertinent to the model are used, as listed in Table ?? and that discrete-time NPV is anticipated in the Olympus Challenge.

Thus, following the protocols of the well control problem stipulated by TNO (all wells come on stream at t = 0), the optimized solution (Table ??) equivalent E(NPV) = $587,875,800 with optimal rig location [524,134.8 618,0130.9] and drilling cost of $464,101,461.

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Rashid, K. Managing geological uncertainty in expensive reservoir simulation optimization. Comput Geosci 24, 2043–2057 (2020). https://doi.org/10.1007/s10596-019-09895-8

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