Abstract
In the water flooding reservoir management, engineers aim to make an optimal production strategy to maximize both the net present value (NPV) in life-cycle and the short-term NPV. Previous researches of multi-objective optimization of long-term and short-term find that the short-term production optimization decreases the long-term NPV and the gradient calculation using adjoint gradient is hard to be applied. Therefore, we implement a multi-objective optimization method in which the long-term NPV is lumped into the short-term objective function as a constraint in order not to decrease the life-cycle NPV, linear and nonlinear constraints are lumped into the augmented Lagrangian objective function. Also, we propose a new gradient-free algorithm general stochastic approximation algorithm (GSA) to obtain the better approximate gradient. We proved that the research direction of GSA algorithm is always uphill, the simultaneous perturbation stochastic approximation algorithm (SPSA) and the ensemble-based optimization algorithm (EnOpt) are also two special forms of GSA algorithm. Toy problem results show that GSA has a more efficient approximation gradient than SPSA. Optimization results of a 2D reservoir show that the increase of the long-term NPV is about 86.5% and the increase of the short-term NPV is about 18.8%. The field case of Brugge field also shows that the long-term and short-term NPV increase by 64.7 and 48.9%.
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Abbreviations
- \( b \) :
-
The annual discount rate(%)
- \( C_{U} \) :
-
The auto covariance matrix of u
- \( C_{U,L} \) :
-
The covariance matrix between u and L
- \( c_{L} \) :
-
The constraints of the long-term
- \( c_{j} \) :
-
Expression for the jth inequality equation
- \( c_{v} \) :
-
Variable of violating constraints
- \( e_{i} \) :
-
The expression for the ith equality equation
- \( g \) :
-
The true gradient
- \( \hat{g} \) :
-
The approximation gradient
- \( J \) :
-
Object function of production optimization($)
- \( L \) :
-
The augmented Lagrange function
- \( M \) :
-
The introduced optimized matrix
- \( m \) :
-
The vector consisting from the geologic parameters
- \( n_{c} \) :
-
The number of equality constraints
- \( n_{e} \) :
-
The number of inequality constraints
- \( N_{L} \) :
-
The total number of simulation timesteps of long-term
- \( N_{P} \) :
-
The total number of producers
- \( N_{S} \) :
-
The total number of simulation timesteps of short-term
- \( N_{I} \) :
-
The total number of water-injection wells
- \( N_{u} \) :
-
The dimension of control vector
- \( N_{e} \) :
-
Number of perturbation models
- \( q_{o,j}^{n} \) :
-
The average oil-production rate over the nth timestep of the jth producer (STB/d)
- \( q_{w,j}^{n} \) :
-
The water-production rate over the nth timestep of the jth producer (STB/d)
- \( q_{{_{wi,i} }}^{n} \) :
-
The water-injection rate over the nth timestep of the ith water injector (STB/d)
- \( r_{o} \) :
-
Oil price ($/L3)
- \( r_{w} \) :
-
Water production cost ($/L3)
- \( r_{wi} \) :
-
Water injection cost($/L3)
- \( t^{n} \) :
-
Nth step cumulative calculation time
- \( T \) :
-
The total time steps reservoir production control
- \( u \) :
-
Nu-dimensional column vector composed by control variable
- \( u* \) :
-
Optimal control vector of the long-term
- \( u_{k}^{low} \) :
-
Lower bound over the kth control parameters
- \( u_{k}^{up} \) :
-
Upper bound over the kth control parameters
- \( u_{i}^{l} \) :
-
The ith control variables realization of lth step
- \( u_{k}^{l} \) :
-
The control vector in the lth inner loop of the kth outer loop
- \( y \) :
-
The vector consisting from the status parameters
- \( y* \) :
-
Optimal \( y \) after the long-term optimization
- \( \Delta t^{n} \) :
-
Step of nth time step
- \( \Delta \) :
-
Random perturbation vector
- \( \lambda_{e,i} \) :
-
The Lagrange multipliers for the ith equation
- \( \lambda_{c,j} \) :
-
The Lagrange multipliers for the jth equation
- \( \lambda_{l} \) :
-
The step size for search step l
- \( \mu \) :
-
The penalty factor
- \( v_{j} \) :
-
The slack variable which transformed the inequality constraint into an equality constraint
- \( \phi_{j} \) :
-
Equality constraint function
- \( \varphi_{j} \) :
-
Inequality constraint function
- \( \gamma \) :
-
The perturbation step
- \( \tau ,\bar{\gamma } \) :
-
The reduction factor
- \( \alpha_{l} \) :
-
The iteration step size
- \( \alpha_{\eta } ,\beta_{\eta } \) :
-
The intermediate variable
- \( \bar{\eta },\eta^{*} \) :
-
The variable used to detect the violation of the constraints
- \( i \) :
-
Number index
- \( j \) :
-
Number index
- \( k \) :
-
The index of the outer loop iteration
- \( l \) :
-
Iteration step number index
- \( o \) :
-
Oil index
- \( r \) :
-
Rock index
- \( s \) :
-
The gradient index of SPSA method
- \( w \) :
-
Water index
- \( \text{P} \) :
-
Producer index
- \( \text{I} \) :
-
Injector index
- \( k \) :
-
The index of the out loop iteration
- \( l \) :
-
The index of the inner loop iteration
- \( i \) :
-
The realization model
- \( t^{n} \) :
-
Cumulative time up to the nth simulator timestep
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Acknowledgements
Hui Zhao would like to express his gratitude to the Project of Open Fund of State Key Laboratory of Offshore Oil Exploitation (Grant No. CCL2015RCPS0223RNN), the National Natural Science Foundation of China (No. 51674039), the China Important National Science & Technology Specific Projects (Grant No. 2016ZX05014) and the Yangtze Youth Talents found for their generous financial support of the research. Lin Cao also acknowledges the National Natural Science Foundation of China (Grant No. 51604035).
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Zhao, H., Zhang, Y., Cao, L. et al. Constrained short-term and long-term multi-objective production optimization using general stochastic approximation algorithm. Cluster Comput 22 (Suppl 3), 6267–6281 (2019). https://doi.org/10.1007/s10586-018-1965-x
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DOI: https://doi.org/10.1007/s10586-018-1965-x