Fast Joint Optimization of Well Placement and Control Strategy Based on Prior Experience and Quasi-Affine Transformation

Abstract

Well placement optimization is one of the most important means to control the decline of oilfields and improve the recovery rate in the development process of deep and heterogeneous reservoirs, such as deep buried carbonate oil reservoirs. However, the mapping relationship from deployed well positions to actual profits is non-linear and multi-modal. At the same time, the injection and production relationship of new wells also affects the contribution of well positions to final profits. Currently, common algorithms include gradient-based and heuristic non-gradient algorithms, which have advantages, but face problems of high computational complexity, slow optimization speed, and difficulty in convergence. We propose an evolutionary algorithm for well placement optimization in carbonate reservoirs. This algorithm improves well placement optimization and computational speed by constraining the sampling process to effective sampling spaces, integrating prior knowledge to enhance sampling efficiency, strengthening local optima exploration, and utilizing parallel computing. Additionally, it refines the optimized variable content based on actual control factors, enhancing the algorithm’s robustness in practical applications. A case study from a carbonate reservoir in northwestern China demonstrated that this algorithm not only improved the performance by 50% compared to the classic DE algorithm but also achieved 15% higher optimization effectiveness than the current state-of-the-art algorithms.

1. Introduction

Carbonate rock formations are the most important type of oil and gas reservoir. It is estimated that approximately 60% of the world’s oil reserves are located in carbonate reservoirs [1]. According to a report by Schlumberger (2019) [2] of conventional oil reserves in the Middle East are located in carbonate reservoirs. In the case of the Tarim oil field’s carbonate rock formations, the recovery rate is less than 20%, and the utilization of the reserves is low, making the deployment of highly efficient new wells of great significance for increasing production. However, well position optimization is a major challenge, as on the one hand, the inhomogeneity of the reservoir will result in serious non-smooth, non-continuous, and non-convex cost functions containing multiple local optimal values [3]. On the other hand, well position optimization problems are often based on reservoir numerical simulation and the scope of the option is determined by the scale and size of the model. To find a more extensive and accurate global optimal solution, large-scale models with small grid sizes are often required, which greatly slows down the convergence speed.

The currently commonly used well position optimization algorithms mainly include two categories: gradient-based algorithms and non-gradient-based algorithms. Gradient-based algorithms include mixed integer programming (MIP) [4], multivariate interpolation algorithm (Pan, n.d.), simultaneous perturbation stochastic approximation [5], and the gradient-based finite difference method Li and Jafarpour (2012), which all rely on gradients to find a better location. However, due to over-reliance on gradients and difficulties in calculating an accurate convergence direction, these methods are easily trapped in local optima. Therefore, due to the nature of the problem [5], gradient-based techniques are not very useful for optimizing well positions.

Compared to gradient-based algorithms, derivative-free methods are less prone to getting trapped in local optima [6]. Many nature-inspired derivative-free methods, such as genetic algorithms [7], particle swarm optimization (PSO) [8], differential evolution (DE) [9], bat algorithm (BA) [10], and ant colony optimization (ACO) [11], among others, have been used for well placement optimization problems. These algorithms do not require gradient information, thus do not rely on derivative computation. Moreover, local search algorithms, such as generalized pattern search (GPS) [12] and Hooke–Jeeves directed search (HJDS) [13], have also been utilized. These derivative-free optimization techniques have shown good performance when derivatives are not available or difficult to obtain for optimization problems. In order to achieve better performance, many researchers have combined the advantages of different algorithms to develop many hybrid strategies [14] for solving well placement optimization problems.

As the size of the reservoir model increases, the sampling interval range of well deployments also becomes larger, and these algorithms all face the same problem of decreasing computational efficiency and convergence speed. Based on this situation, some research has been conducted to integrate human experience into well deployment work. Harb [15] proposed the black hole concept to constrain the location of new wells and maintain a certain distance between new and old wells. Zou [16] proposed using the effective boundary obtained from geological information to constrain the position of horizontal wells and help them quickly reach a more optimal position. Wang [17] used a convolutional network with physical formula constraints and an evolutionary algorithm for well placement optimization. Nakajima [18] Emerick [19] used geological model attribute maps as a reference to optimize the position of horizontal wells. Ding [20] directly used a two-dimensional graph defined by human experience as a mutation standard to guide the mutation process. These methods have all to some extent improved the efficiency of sample variation and allowed the sample to change in the direction of human-perceived optimality in each iteration.

As understanding of development deepens, many researchers have recognized that the maximization of NPV requires the joint optimization of well placement and production, resulting in an increase in the number of variables being optimized. Earlier joint optimization methods, such as Bellout [21], sought to solve the problem of high dimensionality by optimizing well placement and production separately through iterative methods. Swmnani [22] reduced the dimensionality problem by setting the production rate as a fixed value and minimizing the increase in dimensions due to the production rate. Additionally, this method used graph theory to control the sparsity of the well network and ensure sufficient well spacing between new and existing wells. Semnani [23] utilized quantum physics concepts to define samples and achieve dimensionality reduction. Qinyang [24] proposed an algorithm named SDEA that uses the Kriging model as a surrogate and they conducted low-cost population optimization and achieved excellent optimization results.

For deep carbonate reservoirs, not only do they face the difficulties encountered in the well placement optimization process for other reservoirs, but they also have additional specific difficulties of their own. On the one hand, there are many factors that affect the well placement optimization process. Many reservoirs in western China are buried deep (over 6000 m), drilling cycles are long Mazzullo and Harris, and the development process inevitably requires consideration of the production delay caused by drilling. At the same time, for weathering crust reservoirs, the plane has a certain connectivity and there is a phenomenon of a water injection effect [25]. Therefore, the deployment of new wells must coordinate with the type and rate of wells. On the other hand, classical algorithms have a low sampling efficiency and slow convergence speed for heterogeneous reservoirs such as carbonate reservoirs. In carbonate reservoirs, the main reserves are mainly in the form of cave and fracture-vuggy facies [26], and these only accounts for a small part of the reservoir volume, with most still belonging to a matrix with too little contribution to production. Therefore, most areas within the carbonate reservoir belong to non-productive regions and random sampling will lead to low sampling efficiency. Furthermore, the mapping from input to output is completed through a numerical simulator, so the input–output mapping cannot be completed in parallel by matrix. However, the population size is large, and it will greatly prolong the computation time if individuals run one by one.

Moreover, due to the inclusion of well coordinates and injection–production strategies in the sample features, the dimensionality of the sample features unavoidably increases. As the number of wells to be predicted increases, the dimensionality of the features will become even more significant and may result in expensive high-dimensional problems. In such circumstances, enhancing cooperation among samples within the population becomes an important objective since efficient sample collaboration can reduce the population size and the number of iterations, thus enabling faster convergence of the computational model. In their work, Liu [27] proposed a fuzzy adaptive differential evolution approach, which employed fuzzy logic controllers to adapt the mutation and crossover parameters. Brest [28] presented a self-adaptive parameter control scheme that modified the F and Cr values using evolution. Qin [29] introduced a strategy adaption differential evolution algorithm in which parameters were updated by learning from past experiences, and an evolution strategy pool was used to select evolution strategies with updated parameters. Li [30] proposed a DEEP evolutionary algorithm, which utilized an evolution path to enhance the optimization performance of the original DE. It is worth noting that among these DE variants, mutation schemes and parameter control schemes have received more attention from researchers than crossover schemes. The quasi-affine transformation technique proposed by Meng [31] utilizes an affine function to replace the original mutation method, reducing the time complexity of population-based algorithms such as differential evolution (DE) and providing better performance within a limited number of iterations.

Overall, the aforementioned method has three deficits. First, the variables related to the objective function in the well location optimization process of previous studies are not comprehensive enough, resulting in outcomes that are insufficient for identifying feasible optimal locations in the field. Second, fracture-vuggy reservoirs contain a large number of invalid grid areas. Simply improving the uniformity of the sampling distribution from a general perspective leads to low sampling efficiency because most of the population individuals sampled in this way have high fitness values, which are not very useful in the entire optimization process. Finally, the insufficient exploration of local optima may lead to missing the global optimum, necessitating repeated exploration later and wasting convergence time.

This paper proposes a new evolutionary algorithm improvement strategy based on the geological characteristics of carbonate reservoirs. First, it incorporates the injection–production regime of the first three years into the optimization variables to ensure that the obtained optimal well locations are robust in the field. Second, it introduces an a priori graph technique to encourage more sampling well locations in favorable areas based on human experience, thus improving sampling efficiency. It also proposes a relocation technique to ensure that each sample corresponds to a well location in an active grid, ensuring the reference value of each sample calculation and further enhancing the sampling efficiency. Finally, it adopts a quasi-affine transformation technique to strengthen the exploration around local optima, avoiding inefficient repetitions. Additionally, this paper uses parallel optimization methods to improve computer utilization and further accelerate the computation process. The application of this algorithm in actual carbonate reservoirs showed that its computational performance is better than SDEA, which is one of the current state-of-the-art (SOTA) algorithms.

In the following Section 2, this paper introduces the mathematical representation of this task. The Section 3 describes the basic framework of the algorithm, which is the computational process of the differential evolution (DE) algorithm. The Section 4 presents the specific innovative design ideas of the algorithm. The Section 5 describes the computational process of the entire algorithm. The Section 6 uses a specific carbonate reservoir case to illustrate the advantages of the algorithm. The final section is the conclusion, summarizing the contributions and application conditions, and proposing future research plans.

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 $\left(x,y\right)$ 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).

$$\tilde{x},\tilde{u}=argminf\left(x,u\right) St.\left\{\begin{array}{c}{x \in R}^{n_x} and u\in R^{n_u}\\ x_l\le x\le x_u and u_l\le u\le u_u\\ d_{min}\ge d\end{array}\right.$$

(1)

$$nx=2\times nw$$

(2)

$$nu=ts\times nw$$

(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).

$$f=-NPV=-{\sum }_{i=1}^T{\displaystyle \frac{Q_0{P}_0+Q_w{P}_w-C_o}{{\left(1+I\right)}^i}}-C_f$$

(4)

where $Q_0$ represents the current oil production, $Q_w$ represents the current water production, $P_0$ is the unit price of oil, $P_w$ is the unit injection cost, $C_o$ is operating expenses, $C_f$ is fixed capital expenditure, $I$ is the internal rate of return, and $T$ is the number of years since production started. It should be noted that we aim to design new wells at different time steps to reflect the time consumption and order for drilling.

Therefore, we assume that only one well can be completed and put into production each year; for example, the number of new wells deployed in three years is 3. Then, $x=\left(x_1,y_1\right),\left(x_2,y_2\right),\left(x_3,y_3\right)$…, where the subscripts 1, 2, 3 indicate the year. The final optimized well position coordinates $x$ will not only reflect the best well position but also show the best drilling sequence. In addition, in terms of production control, we hope to not only optimize the injection rate but also optimize the well type. Therefore, we set $u_l=-u_u$, which not only allows us to control the injection rate through quantity but also optimize the well type by changing the sign. Other variables that may affect NPV are considered constants.

The above Equations (1)–(4) represent the general mathematical optimization problem. Additionally, for the well placement in carbonate reservoirs, there are two specific constraints. Constraint 1 states that multiple wells represented by each sample must maintain a certain distance from each other, as shown in Equation (5).

$$\parallel x_i-x_j\parallel \ge d_{min}$$

(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.

$$Validity\left(x\right)=\left\{\begin{array}{c}0 if x\in S\\ 1 if x \notin S\end{array}\right.$$

(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.

$$x_{best}\sim p\left(x\right)$$

(7)

where $p$ represents the non-Gaussian prior probability distribution and $x_{best}$ represents the optimal solution.

3. Differential Evolution Algorithm

The differential evolution (DE) algorithm is a crucial component of our methodology. It is a population-based optimization algorithm introduced by Storn and Price in 1997, renowned for its simplicity and effectiveness in solving complex optimization problems. Here, we present a detailed explanation of the canonical DE algorithm to facilitate a comprehensive understanding of its implementation in our work.

3.1. Basic Principles

The DE algorithm operates through a population of candidate solutions, iteratively improving them by employing the following key steps:

3.1.1. Initialization

A population of N candidate solutions (individuals) is randomly generated within the defined bounds of the problem’s search space.

3.1.2. Mutation

For each target vector $\mathrm{x}$ in the population, a mutant vector $\mathrm{v}$ is generated by adding the weighted difference between two randomly selected vectors to a third vector, as shown in Equation (8):

$${\mathrm{v}}_{\mathrm{i}}={\mathrm{x}}_{\mathrm{r}1}+F\cdot \left({\mathrm{x}}_{\mathrm{r}2}-{\mathrm{x}}_{\mathrm{r}3}\right)$$

(8)

where $r_1,r_2,r_3$ are distinct indices randomly chosen from the population, and $F$ is a mutation factor typically within the range [0, 2].

3.1.3. Crossover

The mutant vector undergoes crossover with the target vector to produce a trial vector $\mathrm{u}$, as shown in Equation (9).

$$u_{i,j}=\left\{\begin{array}{c}{v}_{i,j} \mathrm{i}\mathrm{f}\left(r_j\le C_r or r_j=j_{rand}\right)\\ x_{i,j} else\end{array}\right.$$

(9)

where $C_r$ is the crossover probability, $r_j$ is a random number between 0 and 1, and $j_{rand}$ is a randomly chosen index to ensure at least one parameter is inherited from the mutant vector.

3.1.4. Selection

The trial vector is compared to the target vector, and the one with the better objective function value is selected for the next generation, as shown in Equation (10).

$$x_i^{i+1}=\left\{\begin{array}{c}{u}_i\hspace{1em}\mathrm{i}\mathrm{f}\hspace{0.17em}f\left(u_i\right)\le f\left(x_i\right)\\ x_i\hspace{1em}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$$

(10)

3.2. Implementation

In our research, the differential evolution (DE) algorithm serves as the foundational structure for our methodological improvements in optimizing well placement within reservoir development tasks. The canonical DE algorithm, with its systematic steps of initialization, mutation, crossover, and selection, provides a robust framework upon which we have built our enhancements. We have meticulously examined the standard DE parameters N (population size), F (mutation factor), and $C_r$ (crossover probability) to understand their roles and ensure a robust starting point for our modifications.

4. Fast Optimization Method

4.1. Quasi-Affine Transformation

In this paper, we utilized a quasi-affine transformation with its form Equation (11) to replace the mutation process for better cooperation among individuals. Here, $M$ represents the cooperative search matrix, and $M$ is the inverse binary operation of all elements in matrix $M$. The inverse value of non-zero elements is zero, while the inverse value of the zero element is one. $M$ is transformed from a lower triangular matrix with all elements set to one, as for example in Equation (12). The transformation for M is carried out through NumPy to randomly shuffle the elements of each row in a matrix and then randomly shuffle the rows themselves. The symbol ⨂ in Equation (9) denotes component-wise multiplication. $X$ denotes all individuals in the population, where $X_i=\left[x_1,x_2,\dots ,x_D\right]$ represents the feature vector of the ith individual and $X={\left[X_1,X_2,\dots ,X_n\right]}^T$, where n is a number denoting the population size. If the ith individual achieves the best fitness value, it is denoted as $X_{gi}$. Here, $X_g$ represents the global best matrix comprising n $X_{gi}$, where $X_g={\left[X_{gi},X_{gi},\dots ,X_{gi}\right]}^T$. The variable $c$ denotes the coefficient factor or step size of the differential matrix, which is obtained as a result of subtracting matrix $X_{r2}$ from matrix $X_{r1}$.The matrices $X_{r1}$ and $X_{r2}$ are generated by randomly permuting the row vectors of matrix $X$. The primary objective for proposing matrix $B$ is to produce vibrations around the globally best solution identified in each generation. Here, $X$ denotes the current generation, while $B$ denotes the vibration position around the global best.

$$\left\{\begin{array}{c}B=X_g+c\ast \left(X_{r1}-X_{r2}\right)\\ M ☉ X+X ☉ B \end{array}\right.$$

(11)

$$\left[\begin{array}{cccc}1& & & \\ 1& 1& & \\ 1& 1& 1& \\ \dots & & & \\ 1& 1& \dots & 1\end{array}\right] ~ \left[\begin{array}{cccc}1& 1& 1& \\ 1& 1& \dots & 1\\ 1& 1& & \\ 1& & & \\ \dots & 1& 1& 1\end{array}\right]=M$$

(12)

4.2. Well Re-Location

Due to the fact that the sampling for a reservoir geological model is often from a square region, while the boundary shape of the reservoir geological model is irregular, individual sampling may sometimes occur in dead grids or in grids with a lower probability of deployment, especially for models like carbonate rock, where the proportion of grids representing vugs are very small compared with the total number of grids. Thus, most of the individuals sampled in the early stage will be in regions where deployment is not possible. Here, we refer to grids representing karst caves and other flows as “high-active grids”, and grids outside of the matrix or boundaries that do not flow much as “low-active grids”. Note that low-active grids include both “dead grids” and some “active grids” with low permeability. To solve this problem and improve the sampling efficiency, we use the there-location technique, as shown in Equation (13).

$$i_{new},j_{new}=\arg \underset{i,j}{\min }Eu\left(G_{i_{init},j_{init}},G_{i,j}\right)$$

(13)

where $G_{i_{init},j_{init}}$ represents the grid such that the well located at the initial sampling coordinate $G_{i,j}$ represents the grid at the $\left(i,j\right)$ coordinate, and Eu() represents the Euclidean distance between the two coordinates. $i_{new},j_{new}$ represents the final determined new well location. When the well is deployed in the dead grid, this technology automatically recognizes and calculates the distance from nearby active grids to the location, and then selects the nearest location as the there-location. Finally, the sampling point is moved to the nearest active grid, as shown in Figure 1. Here, the green grids represent active grids, the purple grids represent the grid where the well is initially located, and the numbers inside the grids represent the distance between current grids and the grid where the well is located.

4.3. Prior Map

The prior map is calculated by two attribute maps and one well spacing map and is presented in the form of probability after standardization. $A=norm\left(S\ast P\ast W\right)$, where $S$ represents the distribution of the saturation at the moment prior to the deployment of a new well, $P$ represents the log-processed permeability field, and $W$ represents the permitted well spacing map, which calculates the minimum distance from each grid point to the remaining wells as shown in Equation (14). Here, $well1, well2,\mathrm{a}\mathrm{n}\mathrm{d} well3$ are all deployed wells. $W$ represents the 2D map of the reservoir gird model, with the value in each grid equal to the minimum Euclidean distance $Eu\left(\right)$ from the planar map at $\left(i,j\right)$ coordinates to all deployed well locations. The purpose of this design is to obtain the distance between each grid location and the nearest well, thus ensuring a certain distance between the newly deployed well and all existing wells during deployment.

$$W_{i,j}=min\left(Eu\left(W_{i,j},wel{l}_1\right)+Eu\left(W_{i,j},wel{l}_2\right)+\cdots +Eu\left(W_{i,j},wel{l}_3\right)\right)$$

(14)

The prior well spacing diagram is shown in Figure 2, where blue circles indicate the completed well positions and green circles represent random location within the active grids. The double arrow line represents the distance between two locations, and the arrow indicates the distance from the corresponding well indicated by the heat map below. It is assumed that there are already three old wells (the blue cycles), and a plan to deploy a new well. Then, we need to calculate the distance from the remaining old wells to all active grids, as shown in the heatmaps in Figure 2 to be more precise. Figure 2a represents the well location and Figure 2b–d represents the distance from a single deployed well. Figure 2e shows the finished well spacing map W when considering all deployed wells using the min() operation. It is worth noticing that the true reservoir region in fact is continuous, but the calculation of the distance in the continuous space requires calculating an infinite number of points. Therefore, relying on the grid map, we can discretize the position space, thus limiting the number of distance calculations needed.

4.4. Batch Parallel

As the population size and design variable dimensions increase, new requirements for the computational efficiency of the population arise. Parallel technology is proposed to meet this need and it can be divided into two methods, multi-threading and multi-process, as shown in Figure 3. Multi-threading parallel refers to task switching between different cores, reducing input–output latency, with a certain speed-up effect for I/O-intensive tasks, but with no significant increase in speed. Multi-process parallel is a technique that uses multiple independent processes to perform tasks in parallel. It achieves fast task processing by dividing large tasks into smaller tasks and executing these smaller tasks on multiple processes. As a result, we employed a multi-process technique to run our algorithm.

Precisely, the multi-process technique can (1) speed up task processing: multi-process parallel takes full advantage of multi-core processors, greatly improving task processing speed. (2) Improve system resource utilization: multi-process parallel can fully utilize system resources and avoid system efficiency degradation caused by excessive resource utilization by a single process. (3) Enhance program stability: multi-process parallel uses independent processes to execute tasks, so a process failure will not affect other processes, thus improving the stability of the program. The speed of task processing by multi-process parallel can be represented by the Amdahl, as shown in Equation (15), where T(n) represents the time when n processes are used to process a task, T(1) represents the time when a single process is used to process a task, and p represents the proportion of tasks that cannot be processed in parallel.

$$T\left(n\right)=T\left(1\right)/\left(1-p+p/n\right)$$

(15)

This paper proposes a multi-batch parallel technique using the “spawn” method, which is a sort of multi-process technique. This method divides each generation of population individuals into multiple batches, each containing multiple samples. When calculating the fitness value, multiple samples in each batch simultaneously enter the numerical simulator for calculation and they do not interfere with each other. The calculation speed is determined by the number of batches instead of the number of individuals in the population. The reason for using batches instead of the whole population is that the population size is huge, while the number of actual computer CPUs is relatively small. If all individuals within the population are computed at once, it may cause memory stack overflow and error, ending the calculation. The fewer the number of parallel batches, the faster the calculation results of the population, but it is necessary to pay attention to the number of individuals in each batch to ensure it does not exceed the number of CPU cores.

5. Workflow

Since the method presented in this paper is intended for enhancing non-gradient-based algorithms, the calculation process of an improved DE algorithm is demonstrated based on the DE algorithm as the fundamental framework, as shown in Figure 4. Here, the green part represents the original differential evolution algorithm process nodes, and the yellow part represents the newly proposed or optimized calculation process nodes. Specifically, for each generation of the population, the first step is to perform well activity validation. For wells located in low-active grids, we use a re-location technique to move them to high-active grids to increase their usefulness in subsequent processes. After completing the selection of individuals within the population, the entire population is put into the numerical simulator in batches for parallel computation. After the numerical simulator completes the calculation, we can not only obtain the fitness value of individuals in the population, but also obtain the dynamic and static properties of each grid, such as saturation. Based on these properties, the prior map is generated. Now, we use quasi-affine transformation to mutate the individuals in the population and crossover them with the population before mutation according to the probability calculated by the prior map. Then, we verify these crossed individuals are in low-active grids and relocate them again. Next, the crossed population is also processed in batches in the numerical simulator to obtain their fitness values in parallel. Finally, we compare the fitness values of the original population and the crossed population. Based on the greedy principle, we replace the individuals in the original population with those from the crossed population that have a better fitness value and proceed to the next generation.

6. Case Study

In this section, a local region of a weathering crust oil reservoir in the Basin in western China is selected as a research object to demonstrate the superiority of our method.

6.1. Data Preparation

6.1.1. Geological Model Information Display

The oilfield belongs to a typical weathering-shell oilfield, and the geological model is built primarily based on seismic data and various attribute bodies. The development experience shows these attribute body models have a high degree of consistency with development results, ensuring the accuracy of the geological model. The model contains one old well (LG7-2C) and a total of 79,200 grids where the grid dimensions in the X, Y, Z directions are 36, 22, and 100, respectively. The entire model is shown in Figure 5. For the parameters in an objective function, such as prices, they are set to be consistent with the actual field conditions. The detailed prices are shown in

Table 1. The cost of each operation.

Item Name	Price	Unit	Description
oil	60	$/bbl	price of production oil
water	5	$/bbl	cost of disposing of water
water_inj	10	$/bbl	cost of injecting water
drilling	10	M$	cost of drilling well

and these values are derived from actual statistical results of reservoir development in the target block. The data fluctuate yearly, so we have used the 2023 average values as the reference for this optimization.

6.1.2. Algorithm Hyperparameter Setting

To increase the computation speed, we set the population size to 50, meaning that each generation contains 50 individuals. Further, we use 300 as the total number of iterations to ensure that each optimization process has converged to an optimal. Regarding the batch parallel, since each node has 40 cores in the CPU, to reserve a certain computational space, we set the parallel number to 25, meaning each batch contains 25 individuals and the entire population will be divided into 2 batches to complete the task. Additionally, for the original DE algorithm we aimed to compare, we chose the commonly used value of 0.3 for factor c.

6.1.3. Optimization Variable Presentation

We plan to build three new wells in the case where we already have an old well. The location characteristics of the new wells will be represented by x and y coordinates, thus the location coordinates of the three wells can be represented as (x₁, x₂, x₃, y₁, y₂, y₃), totaling six features, which also imply an open sequence of wells. Additionally, considering the actual process of oilfield development, which generally needs a series of procedures such as application, approval, investment, and drilling per well, we set the well open spacing as one year, which is consistent with the time step length. Furthermore, we optimized the well types and rate of production and injection simultaneously. To reduce the number of features, we used positive and negative signs to indicate the well type and specific values to indicate the rate and thus merge them into single features. As a result, the production or injection features for wells at each time step are represented by $q$, where $q\_1$ to $q\_3$ are the production information of the three new wells, $q$ is the production information of the old well, and the superscript indicates the time step. In order to study the long-term benefits, we calculate the NPV of the first three years after the first well opened and take one year as one time step. Hence, the production information can be represented as $q$. Finally, all optimization variables including the placement coordinates of the three new wells, the deployment sequence, the well type and corresponding rate of wells (including the new wells and old wells) are collected. The range of each variable refers to the experience on site and is shown in

Table 2. The range of optimization variables.

Variable	Range	Unit	Description
x1, x2, x3	[1, 36]	grid number	well location x coordinate
y1, y2, y3	[1, 22]	grid number	well location y coordinate
$q$	[−50, 50]	m3/d	injection and production rate

.

6.2. Application of Fast Optimization Approach

6.2.1. Quasi-Affine Transformation

The quasi-affine transformation is utilized with an M-matrix to implement individual mutations within a population. Specifically, when there are 50 individuals with 18 features in the population, the entire M-matrix has a dimension of $R^{n\times d}$, where n is the population size and d is the dimension of individuals. In practical applications, since the number of n is much larger than d, M is constructed by the method of random ordering after concatenating multiple D-dimensional lower triangular matrices and takes the form of Equation (16). Through this approach, the mutated individuals are more likely to explore the area around the global solution, which enhances cooperation among individuals and accelerates the convergence speed.

$$\left[\begin{array}{c}1\\ 1\\ 1\\ \dots \\ 1\\ \dots \\ 1\\ 1\\ 1\\ \dots \\ 1\end{array}\begin{array}{c}\\ 1\\ 1\\ \\ 1\\ \\ \\ 1\\ 1\\ \\ 1\end{array}\begin{array}{c}\\ \\ 1\\ \\ \dots \\ \\ \\ \\ 1\\ \\ \dots \end{array}\begin{array}{c}\\ \\ \\ \\ 1\\ \\ \\ \\ \\ 1\\ 1\end{array}\right]~\left[\begin{array}{c}1\\ 1\\ 1\\ \dots \\ 1\\ \dots \\ 1\\ 1\\ 1\\ \dots \\ 1\end{array}\begin{array}{c}\\ 1\\ 1\\ \\ 1\\ \\ 1\\ \\ 1\\ \\ \end{array}\begin{array}{c}\\ 1\\ \dots \\ \\ \dots \\ \\ \\ \\ 1\\ \\ \end{array}\begin{array}{c}\\ \\ 1\\ \\ 1\\ \\ \\ \\ \\ 1\\ \end{array}\right]=M$$

(16)

6.2.2. Re-Location

Given that we are studying the placement of wells on a plane, we compressed the three-dimensional reservoir model into two dimensions to define the activation region. The compression method involves numerical averaging and threshold truncation of the vertical direction grid. Specifically, for each planar point, the permeability of the vertical grids at its location is averaged, and a fixed value is then used to determine whether it belongs to the activation region. In this paper, we logarithmically averaged the vertical permeability values and used 0.01 as the threshold. Furthermore, for carbonate rock reservoirs, as the flow mainly occurs in the caves and fractures, we designated the caves and fractures as the high-active grids and the matrix as the low-active grids. As seen from the results in Figure 6a, the high-active grid only represents a small portion of the total grid. When the well position is deployed in the low-active region, we employed the automatic re-location technique to re-activate the well, as marked by the red circle in Figure 6b.

6.2.3. Prior Map

We utilize log-permeability fields, well spacing maps, and dynamic saturation fields as the foundational elements for prior image construction. These are then transformed into a probability distribution range through a 0–1 standardization method and serve as the crossover probability. The process is shown in Figure 7.

6.2.4. Batch Parallel

Since the number of individuals in the entire population exceeds the number of computer cores, we divided the population into two batches for calculating the objective function values using the simulator. Once both batches have been fully processed, they are merged back into a complete population. As there is no data interaction between individuals during the objective function calculation within the population, we fully utilize the computer’s CPU cores for computing individual fitness values. Specifically, each individual is computed using a single core. Given that our computer has 48 CPU cores, we can simultaneously calculate the fitness values of 48 samples. To ensure reliability, we used only 30 cores for the calculations, reserving the remaining cores for other tasks to prevent system overload and potential crashes.

6.3. Result Study

After optimization was completed, the optimization curve of the average fitness of the population in 300 iterations was obtained first, and it was compared with the results of the original DE algorithm, as shown in Figure 8. It is evident that our method has a much faster convergence rate compared with the original DE algorithm and produces better variable values that are related to a higher NPV. To be specific, starting from the 30th iteration, the rate of increase in population fitness of the two methods starts to differ until it reaches the maximum gap at the 50th iteration. The gap between the two even reaches 20% of the original DE algorithm result. Our method starts to converge and gradually stabilizes around 250 iterations, while the original DE algorithm begins to converge at a relatively low NPV value from 150 iterations. By 300 iterations, both algorithms have stabilized and our algorithm’s optimal solution has produced a net present value that is approximately 50% higher than the result of the original DE algorithm. Additionally, it can be observed that the SDEA algorithm, due to its use of Latin hypercube sampling, initially has better objective function values than our algorithm. However, in the later stages, since it primarily focuses on exploring the region around optimal samples and relies on surrogate models that cannot accurately simulate the real simulator, its exploration capability diminishes. As a result, the objective function improvement significantly slows down, and the final convergence results are not as good as those of our algorithm.

To further clarify the cause of this difference, we compared the well-deployment maps based on permeability fields. Based on the permeability field, our method for optimal well placement design is illustrated in Figure 9, where the well is placed at the grids with high permeability, indicating its presence in a cave or a fracture. Additionally, the best drilling sequence should prioritize deploying well sites in large, underexploited caves that are far away from old wells in order to maximize the exploration of the remaining oil reserves. As the distant reserves are utilized, new wells can then be deployed to exploit areas with lower permeability or smaller reserves. In contrast, although the well sites are located in high permeability and high saturation regions, well sites designed by the original DE and SDEA algorithm are located at the edge of caves or fissure bodies and this explains why the NPV of both algorithms is lower compared to our algorithm.

Furthermore, we compared the optimization results of well type and injection-production rate and presented them in the form of a heat map, as shown in Figure 10, where positive values represent production wells, negative values represent injection wells, and the final values represent the corresponding rate. The results indicate that our optimization approach adopts a strategy of full production with phase-wise regulation of the production rate to create an asynchronous fluctuation of production rate from these wells, thus activating the flow of residual oil at the corner of the pore. In comparison, the original DE algorithm tends to first replenish energy and then produce at full capacity, which results in a low initial production rate and more dead oil at some pores. Although the strategy proposed by the SDEA algorithm reduces the occurrence of water injection events, the true oil production potential of the reservoir is not fully realized.

Finally, due to the utilization of batch parallel technology in our method, our algorithm effectively saves time. As demonstrated by the time consumption comparison, as shown in Figure 11, our method’s time consumption is slightly higher than the SDEA algorithm but significantly lower than that of the original DE algorithm and is only one twentieth of the original DE algorithm’s calculation time.

6.4. Ablation Test

The most useful of these techniques is the prior map technique, as shown by the yellow curve in Figure 12. This is because the dominance map ensures that the sample is more referenced to artificial experience. As artificial experience has a good big picture view, it can better avoid individuals falling into the trap of local optimal solutions. The second important technique is the quasi-affine transformation, which clearly allows the mean fitness to increase in a fluctuating manner, shown by the purple curve in Figure 12. This is due to good collaboration between individuals, as they identify the local search area based on the information provided by the global optimal solution, and then the individuals divide the work within this local area to explore the local better solutions through the factor c. The M-matrix allows the individuals to fully exchange information with each other about the advantages of the current optimal solution, and thus the overall fitness value of the individuals is improved significantly. The overall fitness of individuals is greatly improved. In addition, this technique is more useful than the dominance map for some reservoirs where the dominance map is not obvious. In contrast, the re-location technique is less important than the first two, but also has a very significant improvement on the algorithm, as seen in the blue curve in Figure 12. This is because the effective sampling area of carbonate reservoirs is relatively small (a small number of active grids) and irregularly distributed, and it is easy for a large number of individuals to fall on a dead grid when uniform selection or mutation is carried out. The there-location technique therefore ensures that these individuals are returned to an active grid so that their fitness value changes from 0 to a valid value. The batch parallelism technique has little effect on the results, as shown in the pink curve in Figure 12. As this technique mainly allows individuals within a population to carry out calculations in parallel, it does not directly affect the selection and mutation of individuals and therefore has little effect on the convergence process.

To better explain the advantages in our method, we also displayed the historical individual points generated by the original DE, as shown in Figure 13 and by algorithms that are deleted versions of our algorithm, as shown in Figure 14. From the distribution of the individual points, it can be clearly seen that (1) the sample points generated by the original DE algorithm present an overall upward trend, and all of the sample points present a uniform distribution within a certain range. (2) The samples in the SDEA algorithm mainly concentrate around the optimal solution. Although it exhibits good exploratory behavior in the initial stages, its exploratory capacity diminishes over time. The rapid convergence of the sample point distribution leads to the potential global optimal solution regions being missed. (3) The quasi-affine transformation method creates individuals that are closer to optimal solutions instead of evenly distributed within the prescribed range due to the fact that the method encourages individuals to explore around the global optimal solution with some probability as they mutate. This allows individuals of each generation to find locally better solutions while taking advantage of the good features of the current optimal solution. At the same time, this M-matrix allows the population to have some probability of interacting with some of the features of the global optimal solution, in different multiple dimensions, allowing individuals to refer to the best direction of convergence when updating. (4) With the addition of the there-location technique, the distribution of points begins to become uneven, and a larger proportion of samples deviate toward the optimal solution at the current iteration, which is due to the there-location technique forcing individuals located in low-active grids to transfer to the nearest high-active grid. Additionally, some individuals seem to form multiple lines. This is due to the limited grid of active regions, where many samples are mapped to the same boundary grid when mapped from non-active regions to active regions, leading to little change in the sample fitness value. (5) With the addition of prior map technology, the points further converge toward the optimal solution direction. The prior map not only encourages the points to be in the high-active region, but also has a certain probability of making the individual mutate to more advantageous caves, which show a higher probability in the prior map.

7. Conclusions

This paper addressed the unique boundary constraints encountered during the optimization process of carbonate reservoirs by employing a series of techniques, including relocation, to enhance the sampling efficiency. Additionally, the prior map method was utilized to integrate field experience into the algorithm, accelerating the convergence speed of the computations. Techniques such as Radon transform and multi-core parallel processing were employed to further improve the computational speed. This method is suitable for optimization tasks with irregular sample effective ranges and substantial prior knowledge, allowing for acceleration and enhancement of the final optimization results based on the evolutionary algorithm process. In the future, machine learning methods can be employed to surrogate the numerical simulation process using samples obtained from extensive sampling, further speeding up the computation process. It is even possible to use proxy models [32] from other reservoirs to enhance the proxy effect in the current reservoir through migration.