## 2 Problem Statement  
The well placement optimization problem belongs to a typical optimization problem with constraints. Its basic form is shown in Equation (1) where the well position variable \(x\) and the production control variable \(u\) belong to different continuous spaces R. \(d_{min}\) represents the minimum distance between all pairs of wells, including new and old wells. \(d\) represents the required minimum well distance. The subscripts \(l\) and \(u\) represent the lower and upper bounds of the variable. In this study, we base the well position on a grid-based geological model, so the well position still needs to be discretized during processing. This study only considers vertical wells, so the position variables represent locations are expressed through the \((x,y)\) coordinate. The production control variables are composed of production and injection rates in multiple time steps, with each time step being 1 year. The final dimension setting can be represented as Equations (2) and (3).  
\[\widetilde{x},\widetilde{u}=argminf(x,u)\ \ \text{$St$}.\left\{\begin{array}{l l}x\ \in\ R^{n_{x}}\ \text{and}\ \ u\in R^{n_{u}}\\ x_{l}\leq x\leq x_{u}\ \text{and}\ u_{l}\leq u\leq u_{u}\\ d_{min}\geq d\end{array}\right. \tag{1}\]  
\[nx=2\times nw \tag{2}\]  
\[nu=ts\times nw \tag{3}\]  
where \(n_{u}\) refers to the number of wells (including both injection and production wells), and \(t_{s}\) represents the number of time steps. The objective function f is a fitness function used to evaluate the fitness of the current variables, and the final goal is to select the variables with the minimum fitness. For oil reservoir development, we aim to maximize the return, which is the net present value (NPV). Therefore, we adopt -NPV as the fitness function, as shown in Equation (4).  
\[f=-NPV=-\sum_{i=1}^{T}\frac{Q_{0}P_{0}+Q_{w}P_{w}-C_{o}}{\left(1+I\right)^{i}}- C_{f} \tag{4}\]  
where \(Q_{0}\) represents the current oil production, \(Q_{w}\) represents the current water production, \(P_{0}\) is the unit price of oil, \(P_{w}\)  
straints. Constraint 1 states that multiple wells represented by each sample must maintain a certain distance from each other, as shown in Equation (5).  
\[\parallel x_{i}-x_{j}\parallel\geq d_{min} \tag{5}\]  
where \(d_{min}\) represents the minimum allowable distance between the well coordinates indicated by the variables, while \(x_{i}\) and \(x_{j}\) denote the coordinates of any two wells within a sample.  
Equation (6) means the sampled well positions may not necessarily be valid well locations due to the irregular boundary of active grid areas in the geological model and only the well positions within valid regions can yield effective objective function values.  
\[Validity(x)=\begin{cases}0&if\ x\in S\\ 1&if\ x\
otin\ S\end{cases} \tag{6}\]  
where \(S\) represents the effective data sampling space.  
Additionally, because the deployment of well locations in oil and gas reservoir development involves a significant amount of prior knowledge and experience, the well location deployment follows a prior distribution \(p\) as shown in Equation (7). However, since this \(p\) is determined by the actual characteristics of the reservoir distribution, its probability density function cannot be explicitly formulated.  
\[x_{best}\sim p(x) \tag{7}\]  
where \(p\) represents the non-Gaussian prior probability distribution and \(x_{best}\) represents the optimal solution.