A rapid intelligent multi-fidelity surrogate-assisted multi-objective optimization method for water-flooding reservoir production optimization

Abstract

In recent years, the surrogate-assisted method, which is one of the most promising ways for reducing the computational burden of numerical simulation–based water-flooding reservoir production optimization, has been widely applied by researchers worldwide. Meanwhile, there are different fidelity models that can be applied to extract samples for establishing a surrogate model. However, most of the researches neglected the potential synergies between low-fidelity (LF) and high-fidelity (HF) samples. In this study, therefore, a rapid intelligent multi-fidelity support vector regression (SVR) model–assisted multi-objective production optimization method, namely, MFSVR-MOPO, is proposed to lessen the computational burden of the numerical simulation–based production optimization. The uniqueness of this proposed method is that the LF and HF samples are mapped to a high-dimension space by the kernel function of the SVR, and the relationships between variables and objectives are evaluated by a linear model. Moreover, the gray wolf algorithm was applied to search for the optimal hyperparameters of the SVR model so as to improve the evaluation accuracy. Two frequently used production optimization synthetic reservoirs with different scales were studied to illustrate the effectiveness and accuracy of the MFSVR-MOPO method. The results showed that the MFSVR-MOPO method performed comparably with the HF model–based method in terms of convergence and diversity, but reduced the number of simulation runs on the two reservoirs by approximately 108 times and 50 times, respectively.

Appendix. Procedure of training SVR models

The linear relationship between input (φ(x)) and output (y) in the SVR model is as follows:

$$y=b+{\omega}^T\varphi \left(\mathbf{x}\right)$$

(29)

where ω represents the coefficient vector and b is a constant scalar which could be easily obtained by solving the following cost function J optimization problem:

$$minJ=\frac{1}{2}{\upomega}^T\upomega +\frac{1}{2}c\sum_{k=1}^{N_s}{\left[{y}_k-{\upomega}^T\varphi \left({x}_k\right)-b\right]}^2$$

(30)

In Eq. 30 the training error is defined as the least-square error between predicted and real values. Thus, Eq. 30 is equivalent to:

$$minJ=\frac{1}{2}{\upomega}^T\upomega +\frac{1}{2}c\sum_{k=1}^{N_s}{e_k}^2$$

(31)

Subject to

$${e}_k={y}_k-{w}^T\varphi \left({x}_k\right)-b$$

(32)

The Lagrangian function is described as follows:

$$L\left(w,b,e,\alpha \right)=\frac{1}{2}{w}^Tw+\frac{1}{2}c\sum_{k=1}^{N_s}{e_k}^2-\sum_{k=1}^{N_S}{\alpha}_k\left[{w}^T\varphi \left({x}_k\right)+b+{e}_k-{y}_k\right]$$

(33)

By solving the solution ∇L = 0 of Eqs. 31 and 32, the following linear systems are obtained:

$$\frac{\partial L}{\partial w}=0\to w=\sum_{k=1}^{N_s}{\alpha}_k\varphi \left({x}_k\right)$$

(34)

$$\frac{\partial L}{\partial b}=0\to \sum_{k=1}^{N_s}{\alpha}_k=0$$

(35)

$$\frac{\partial L}{\partial {e}_k}=0\to {\alpha}_k=\gamma {e}_k,k=1,2,\cdots , {N}_s$$

(36)

$${\displaystyle \begin{array}{c}\frac{\partial L}{\partial {\alpha}_k}=0\to {w}^T\varphi \left({x}_k\right)+b+{e}_k-{y}_k=0,k=1,2,\cdots , {N}_s\\ {}\end{array}}$$

(37)

Simultaneously solving Eqs. 34 and 37 yields:

$${y}_k=\sum_{l=1}^{N_s}{\alpha}_l\varphi {\left({x}_l\right)}^T\varphi \left({x}_k\right)+b+{e}_k$$

(38)

Combining Eqs. 36 and 38 yields:

$${y}_k=\sum_{l=1}^{N_s}{\alpha}_l\varphi {\left({x}_l\right)}^T\varphi \left({x}_k\right)+b+\frac{\alpha_k}{\gamma }$$

(39)

We define ${1}_{N_S}={\left[1,1,\cdots , 1\right]}_{N_S}^T$, $Y={\left[{y}_1,{y}_2,\cdots , {y}_{N_S}\right]}^T$, $\alpha ={\left[{\alpha}_1,{\alpha}_2,\cdots , {\alpha}_{N_S}\right]}^T$, and Ω_{K, I} = φ(x_k)^Tφ(x_l).

Equation 39 thus becomes:

$${1}_{N_s}b+\left(\Omega +\frac{1}{\gamma }I\right)\alpha =Y$$

(40)

Combining Eqs. 35 and 40 yields:

$$\left[\begin{array}{c}0{1}_{N_S}^T\\ {}{1}_{N_S}\kern0.5em {\varOmega}_{N_S\times {N}_S}+\frac{1}{\gamma }I\end{array}\right]\left[\begin{array}{c}b\\ {}\alpha \end{array}\right]=\left[\begin{array}{c}0\\ {}Y\end{array}\right]$$

(41)

where

$${\varOmega}_{K,I}=\varphi {\left({x}_k\right)}^T\varphi \left({x}_l\right)$$

(42)

Using the kernel function to replace the inner product of φ gives:

$$\varphi {\left({x}_k\right)}^T\varphi \left({x}_l\right)=K\left(x{}_k,{x}_l\right)$$

(43)

In this study, the RBF kernel was used. The following equation is thus obtained:

$$K\left({\underline{x}}_{\underline{k}},\underline{x}\right)=\mathit{\exp}\left(-{\left\Vert {\underline{x}}_{\underline{k}}-\underline{x}\right\Vert}_2^2/{\sigma}^2\right)$$

(44)

Multiplying Eq. 40 by ${1}_{N_S}^T{\left(\Omega +\frac{1}{\gamma }I\right)}^{-1}$ yields:

$${1}_{N_S}^T{\left(\Omega +\frac{1}{\gamma }I\right)}^{-1}{1}_{N_s}b+{1}_{N_S}^T\alpha ={1}_{N_S}^T{\left(\Omega +\frac{1}{\gamma }I\right)}^{-1}Y$$

(45)

Replacing the relationship ${1}_N^T\alpha =0$, b is solved from Eq. 45 to obtain:

$$b=\frac{1_{N_S}^T{\left(\Omega +\frac{1}{\gamma }I\right)}^{-1}Y}{1_{N_S}^T{\left(\Omega +\frac{1}{\gamma }I\right)}^{-1}{1}_{N_s}}$$

(46)

Considering Eq. 40, α is given as:

$$\alpha ={\left(\Omega +\frac{1}{\gamma }I\right)}^{-1}Y-b{\left(\Omega +\frac{1}{\gamma }I\right)}^{-1}{1}_{N_s}$$

(47)

Combining Eqs. 29 and 34, and then applying Eq. 43, the function y(x) is obtained as:

$$\hat{y}(x)=\sum_{k=1}^{N_s}{\alpha}_kK\left({x}_k,x\right)+b$$

(48)