Introduction

Lost circulation is one of the inevitable problems in drilling engineering, causing economic losses of billions of dollars annually (Sun et al. 2021). In order to address the safe drilling issues in formations with narrow pressure windows or lost circulation formations, numerous experts and scholars have proposed unconventional drilling techniques, among which dual gradient drilling technology is one (Stave et al. 2014; Yang et al. 2022). Currently, offshore dual gradient drilling technology is developing rapidly and has certain commercial application value. Offshore dual gradient drilling technology includes lifting drilling fluid through subsea mud pumps, riserless drilling, and dual-density drilling. Onshore, due to limited underground space and cost constraints, dual gradient drilling technology development is relatively lagging. It typically achieves dual gradient drilling through gas injection. Commonly used onshore dual gradient drilling technologies include parasitic pipe gas injection and concentric pipe gas injection drilling technology (Dou et al. 2013; Gonzalez et al. 2013).

Phillips Petroleum Company conducted a parasitic annulus air-injection drilling test in Gallatin County, Montana (Westermark 1986). By injecting air into the annulus space through a parasitic air-injection pipe and setting the appropriate air/mud ratio and depth of air injection, the equivalent density of bottom-hole pressure could be reduced to 0.719. Guo and Rajtar (1995) proposed a simple method to calculate the ratio of drilling fluid to gas for aerated drilling, which can be used for bottom-hole pressure calculations. Lopes and Bourgoyne (1997) developed a steady-state numerical model to calculate parameters such as gas injection rate, maximum drilling fluid density, and riser diameter for gas injection dual gradient drilling in deep water. Li (2007) systematically described the process flow, equipment, application scope, and development direction of the low-pressure drilling technology with dual-wall drill pipe. Zimuzor et al. (2010) used gas injection drilling with parasitic pipe to solve wellbore leakage and stuck pipe problems in the Piceance Basin. This technology controlled the annulus circulating density by injecting air through a parasitic pipe in annulus, successfully reducing the risk of wellbore leakage in the Piceance Basin. Dou et al. (2013) referenced the Hasan-Kabir model (1986) to analyze the influence of gas injection rate and back pressure at the wellhead on bottom-hole pressure in annulus air-injection drilling with parasitic pipe. It was found that as the gas injection rate increased, the bottom-hole pressure initially decreased rapidly and then slowly increased, indicating the existence of a critical gas injection rate. Ma et al. (2014) simulated the temperature distribution in the riser of gas injection dual gradient drilling. The results showed that an increase in the inlet temperature correspondingly increased the temperature in the annulus. Su et al. (2018) studied the physical process of gas migration in the riser of gas injection dual gradient drilling. The research results showed that a high drilling fluid flow rate could eliminate large bubbles and stabilize wellbore pressure. Wang et al. (2019) designed a new deepwater dual gradient drilling method based on downhole separation. The research results showed that optimizing the wellbore pressure profile could adapt to narrower pressure window and effectively avoid complex downhole accidents caused by improper wellbore pressure.

In summary, the previous research on the construction parameters of annular gas injection dual gradient drilling has mainly focused on engineering applications, wellbore fluid modeling, and drilling parameter design. Parameter design optimization is usually based on empirical formulas. There has been limited research on optimizing construction parameters, and the optimization objectives have been single-focused, considering only wellbore pressure safety without taking cost into account. Therefore, further research is needed to establish a method for optimizing construction parameters for annular gas injection dual gradient drilling that simultaneously considers cost and wellbore safety conditions. This will provide theoretical support for the promotion of this technology to address severe wellbore leakage in drilling operations.

This paper presents an onshore dual gradient drilling using dual-wall drill pipes for annulus gas injection. The process flow is shown in Fig. 1. This technique regulates the bottom-hole pressure through control of drilling fluid displacement, gas injection rate, gas injection depth, and wellhead back pressure. Compared to the aerated mud drilling: (1) It effectively avoids the impact of gas on screw drill tools and MWD; (2) as the gas injection depth is relatively shallower, the gas injection pressure is lower; and (3) gas injection into the casing helps prevent erosion on the mud cake, thereby promoting wellbore stability. In comparison with aerated drilling with a parasitic pipe, it eliminates the issues associated with parasitic pipe blockage, as well as the challenges of repairing and replacing the pipe. GA (Chande and Sinha 2013) is an optimization algorithm based on the principles of biological evolution. Its basic idea is to simulate the processes of natural selection, genetic variation, and mutation in biological evolution to search and optimize the solution space. The advantages of GA lie in their wide applicability, strong parallel processing capabilities, independence from constraints and differentiability, robustness, and global search ability (Ong et al. 2019). GA can be particularly useful when the search space is large and the objective function is not well-behaved or has multiple local optima. In this study, a genetic algorithm is employed, with drilling fluid displacement, gas injection rate, gas injection depth, and wellhead back pressure as decision variables, and the safety of bottom-hole pressure and construction cost as objective functions for multi-objective optimization. It provides an optimal combination of parameters with low cost, laying a foundation for applying this technology to solve drilling problems in severely lost circulation formations.

Fig. 1
figure 1

Schematic diagram of annulus aerated dual gradient drilling with dual-wall drill pipe

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In response to the issues of using empirical formulas to optimize parameters, limited optimization methods, and single optimization objectives in the previous studies on annulus aerated dual gradient drilling, this study has developed a multi-objective optimization method for parameters. The gaps addressed in this study include the following areas:

  1. 1)

    For the first time, a multi-objective optimization method based on genetic algorithm for parameters of annulus aerated dual gradient drilling was established with the objectives of wellbore safety and construction cost.

  2. 2)

    The impact of different selection operators on genetic algorithm was analyzed, and the efficiency of the optimization method could be further enhanced by improving the selection operator.

  3. 3)

    Several algorithms were compared to evaluate the parameter optimization algorithm in terms of effectiveness, efficiency, and accuracy for algorithm improvement.

Multiphase flow model for annulus aerated dual gradient drilling

Assumption

  1. 1)

    Fluid is a continuous infinitesimal material system, and the pressure and flow velocity of the fluid vary continuously.

  2. 2)

    Drilling fluid is an incompressible fluid.

  3. 3)

    The fluid flows axially along the wellbore in one-dimensional direction.

Continuity equation

Take the microelement along the axial direction of the annulus space to form a control volume (Meng et al. 2015). Analyze the components flowing into and out of the control volume per unit time:

(1） Gas phase

$$\frac{{\partial \left( {\rho_{g} v_{g} AE_{g} } \right)}}{\partial z} + \frac{{\partial \left( {\rho_{g} AE_{g} } \right)}}{\partial t} = q_{g}$$
(1)

(2） Liquid phase

$$\frac{{\partial \left( {\rho_{l} v_{l} AE_{l} } \right)}}{\partial z} + \frac{{\partial \left( {\rho_{l} AE_{l} } \right)}}{\partial t} = 0$$
(2)

(3） Cuttings phase

$$\frac{{\partial \left( {\rho_{s} v_{s} AE_{s} } \right)}}{\partial z} + \frac{{\partial \left( {\rho_{s} AE_{s} } \right)}}{\partial t} = 0$$
(3)

where ρg is the density of gas under the condition of wellbore temperature and pressure, kg/m3; vg is the velocity of the aerated gas in the wellbore annulus, m/s; A is the cross-sectional area of wellbore annulus, m2; Eg is the void fraction of wellbore annulus section, dimensionless; qg is the mass rate of aerated gas at the injection point, kg/(s⋅m), and qg = 0 if there is no gas injection point in the control element; ρl is the density of drilling fluid, kg/m3; vl is the velocity of drilling fluid, m/s; El is the liquid fraction of wellbore annulus section, dimensionless; ρs is the density of cuttings, kg/m3; vs is the velocity of cuttings, m/s; and Es is the cuttings fraction of wellbore annulus section, dimensionless.

Momentum equation

The force analysis of the annulus microelement control body is shown in Fig. 2. According to the law of momentum, the momentum equation of gas, liquid, and solid three-phase flow can be written as follows:

$$\begin{gathered} \frac{\partial }{\partial t}\left( {AE_{g} \rho_{g} v_{g} + AE_{l} \rho_{l} v_{l} + AE_{s} \rho_{s} v_{s} } \right) + \frac{\partial }{\partial z}\left( {AE_{g} \rho_{g} v_{g}^{2} + AE_{l} \rho_{l} v_{l}^{2} + AE_{s} \rho_{s} v_{s}^{2} } \right) \hfill \\ \; + Ag\cos \theta \left( {\rho_{g} E_{g} + \rho_{l} E_{l} + \rho_{s} E_{s} } \right)\Sigma + \frac{{\partial \left( {AP} \right)}}{\partial z} + \frac{{\partial \left( {AF_{r} } \right)}}{\partial z} = 0 \hfill \\ \end{gathered}$$
(4)

where g is gravitational acceleration, m/s2; θ is the well deviation angle, ◦; P is annulus pressure, Pa; and Fr is the annulus friction pressure drop, Pa.

Fig. 2
figure 2

Force analysis of the annulus microelement control body

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Energy equation

Compared with non-gas injection drilling, the wellbore heat exchange of gas injection drilling becomes more complex. There is not only heat exchange between the formation and the wellbore fluid, but also heat exchange caused by the gas in the annulus. Therefore, the energy equation for the annulus multiphase can be expressed as follows:

$$\begin{gathered} \frac{\partial }{\partial t}\left( {\rho_{g} E_{g} C_{pg} T_{a} A + \rho_{l} E_{l} C_{pl} T_{a} A} \right) - \frac{\partial }{\partial z}\left( {m_{g} C_{pg} T_{a} + m_{l} C_{pl} T_{a} } \right) \hfill \\ \; = \frac{2}{X^{\prime}}\left( {T_{f} - T_{a} } \right) - \frac{2}{Y^{\prime}}\left( {T_{a} - T_{p} } \right) \hfill \\ \end{gathered}$$
(5)

where Cpg and Cpl are the specific heat of gas and liquid, respectively, J/(kg⋅◦C); Ta, Tf, and Tp are the temperature in annulus, formation, and drill pipe, ◦C; mg and ml are mass flow rate of gas phase and liquid phase, kg/s; X´ is the comprehensive heat transfer factor between formation and annulus fluid, ◦C⋅m⋅s/J; and Y´ is the heat transfer factor between annulus fluid and drill string fluid, ◦C⋅m⋅s/J.

Optimization of parameters design for annulus aerated dual gradient drilling based on genetic algorithm

Formulation of the optimization problem

In the issue of parameters optimization for annulus aerated dual gradient drilling, the most important objective is to ensure the safety of the bottom-hole pressure through parameter combination while minimizing drilling costs as much as possible.

Decision variables

In the multi-objective optimization problem mentioned above, decision variables include drilling fluid displacement, gas injection rate, gas injection depth, and wellhead back pressure.

Objective functions

(1) The first objective function is the bottom-hole safety factor:

$$P_{{{\text{safe}}}} \left( {Q_{l} ,Q_{g} ,H_{g} ,P_{h} } \right) = \frac{{\left| {P\left( {Q_{l} ,Q_{g} ,H_{g} ,P_{h} } \right) - \left( {P_{p} + \frac{{\left( {P_{f} - P_{p} } \right)}}{2}} \right)} \right|}}{{P_{p} + \frac{{\left( {P_{f} - P_{p} } \right)}}{2}}}$$
(6)

where Psafe is the bottom-hole safety factor, dimensionless; Pp is the formation pore pressure, Pa; Ql is the flow rate of drilling fluid, m3/s; Qg is the injection rate of aerated gas under standard conditions, m3/s; Hg is the depth of gas injection point, m; Ph is the wellhead back pressure, Pa; Pf is the formation fracture pressure, Pa; and P is the bottom-hole pressure under specific parameter combination, Pa.

Assuming that P = Pf, the bottom-hole safety factor becomes the maximum value, represented by P*safe:

$$P_{{{\text{safe}}}}^{*} = \frac{{\left| {P_{f} - P_{p} } \right|}}{{P_{f} + P_{p} }}$$
(7)

According to Eqs.(6) and (7), it is obvious that the smaller the Psafe value is, the higher the safety of the well bore would be.

(2) The second objective function is the cost index:

$$\begin{gathered} C_{\cos t} \left( {Q_{l} ,Q_{g} ,H_{g} ,P_{h} } \right) = \left( {\left\lfloor {\frac{{Q_{l} }}{{Q_{l0}^{{}} }}} \right\rfloor + 1} \right) \times C_{l} + \left( {\left\lfloor {\frac{{Q_{g} }}{{Q_{g0}^{{}} }}} \right\rfloor + 1} \right) \times C_{{{\text{air}}}} \hfill \\ + \left( {\left\lfloor {\frac{{H_{g} + H_{g1}^{{}} }}{{H_{g0}^{{}} }}} \right\rfloor + 1} \right) \times C_{{{\text{ddp}}}} + \frac{{P_{h} }}{{P_{{{\text{atm}}}} }} \times C_{h} \hfill \\ \end{gathered}$$
(8)

where Cl is the daily cost of one drilling fluid pump, $/d; Cair is the daily expenses of one air compressor, $/d; Ql0 is the maximum displacement of one drilling fluid pump, m3/s; Qg0 is the maximum displacement of one air compressor, m3/s; Hg0 is the length of one dual-wall drill pipe, m; Hg1 is the estimated drilling footage of the drill trip, m; Cddp is the daily rental cost of one dual-wall drill pipe, $/d; Patm is the standard atmospheric pressure, Pa; and Ch is the daily usage cost of the wellhead back pressure equipment, $/d.

Constraints

There are some constraints that limit the search space in the considered optimization problem.

(1) The allowed ranges of drilling fluid displacement:

$$Q_{l\min } \le Q_{l} \le Q_{l\max }$$
(9)

The minimum drilling fluid displacement is determined by meeting the minimum requirement for borehole cuttings transport, while the maximum drilling fluid displacement is determined by the capacity of the drilling fluid pumps.

(2) The allowed ranges of gas injection rate:

$$Q_{g\min } \le Q_{g} \le Q_{g\max }$$
(10)

The minimum and maximum values of the gas injection velocity are determined by the air compressor.

(3) The limit depths of gas injection point:

$$H_{g\min } \le H_{g} \le H_{g\max }$$
(11)

The minimum depth for gas injection is the length of a dual-wall drill pipe, while the maximum injection depth is determined by the maximum gas injection pressure provided by the air compressor.

(4) Wellhead back pressure:

$$P_{h\min } \le P_{h} \le P_{h\max }$$
(12)

The minimum wellhead back pressure is atmospheric pressure, while the maximum value is determined by the capacity of the throttle valve.

IRSGA for parameters optimization

The traditional methods make it difficult to perform comprehensive constrained optimization of parameters design for annulus aerated dual gradient drilling. In this paper, an improved genetic algorithm is proposed for multi-objective optimization of parameters design for annulus aerated dual gradient drilling.

Initialization of population

Each individual consists of four gene segments, with each gene segment representing a different decision variable. A certain number of individuals are randomly generated as the initial population, with each individual representing a potential solution to the optimization problem. Based on the form of the objective function, the population individuals can be defined as follows:

$$a_{i} = \left[ {Q_{il} ,Q_{ig} ,H_{ig} ,P_{ih} } \right]$$
(13)

The size of population is represented by N; therefore, the first population is A1:

$$A_{1} = \left[ {a_{1} ,a_{2} , \ldots a_{i} \ldots a_{N - 1} ,a_{N} } \right]^{{\text{T}}}$$
(14)

Fitness value

According to Eqs.(6) and (8), the fitness value is expressed as follows:

$${\text{Fitness}}_{i} = \frac{1}{{e^{{P_{{{\text{safe}}}} \left( {Q_{il} ,Q_{ig} ,H_{ig} ,P_{ih} } \right)}} }} \times w_{1} + \frac{1}{{e^{{C_{\cos t} \left( {Q_{il} ,Q_{ig} ,H_{ig} ,P_{ih} } \right)}} }} \times w_{2}$$
(15)

where w1 and w2 are weight coefficient. And w1 + w2 = 1.

The higher the fitness value, the better the individual in the population, and the more likely it is to be selected by the selection operator.

The sum of fitness values for the i individuals in contemporary generation is defined as C_Fitnessi:

$$C\_{\text{Fitness}}_{i} = C\_{\text{Fitness}}_{i - 1} + {\text{Fitness}}_{i}$$
(16)
$$C\_{\text{Fitness}}_{N} = \sum\limits_{1}^{N} {{\text{Fitness}}_{i} }$$
(17)

Selection operator

The selection operator in genetic algorithm selects a portion of individuals from the current population as parents for the next generation according to their fitness evaluation values. Common selection operators include roulette wheel selection, tournament selection, and rank selection (Mao et al. 2020; Pandey 2016). Original roulette wheel selection selects only one parent individual at a time, which may result in better individuals being selected too much and accelerating convergence, or poorer individuals being selected too much and reducing evolutionary efficiency. In this paper, an improved roulette wheel selection operator is proposed, which selects multiple parent individuals at a time for crossover and mutation operations. This approach helps introduce more genetic information and diversity, and enhances the algorithm's exploration capability.

Steps for the improved roulette selection operator are as follows:

① Firstly, let M = C_FitnessN / Nselect, and randomly generate a real number RX(RX ∈ [0, M]);

② Secondly, let j=1(j∈[1, Nselect]), i=1(i∈[1, N]), b0=0 and C_Fitness0=0;

③ Thirdly, define bj = RX + (j-1) × M;

④ Determine whether bj-1 < fitnessi ≤ bj is valid;

⑤ If step ④ is false, let i = i + 1, and turn to step ④;

⑥ If step ④ is true, ai is selected as a parent individual. Then, let i = i + 1, j = j + 1, and jump to step ③. Loop until Nselect parent individuals are selected.

Cross recombination

To ensure the diversity of the population and improve the global search capability, the Nselect parent individuals are subjected to crossover operations:

① Choose two individuals, denoted as a1 and a2, from the Nselect parent individuals. Randomly generate a real number RX(RX ∈ [0, C_FitnessN]). If RX > P_Cross, b1 = a1 and b2 = a2. Turn to step ④.

② If RX ≤ P_Cross, a1 and a2 perform cross recombination. The parent individuals a1 and a2 are scaled by removing the decimal point, resulting in c1 and c2. Binary encoding is then applied to c1 and c2, producing binary gene fragments d1 and d2. A random crossover point within the gene fragments is selected. The gene fragment preceding the crossover point in d1 and the gene fragment following the same crossover point in d2 are copied to create the offspring individual e1;

③ Similarly, the gene fragment following the same crossover point in d1 and the gene fragment preceding the crossover point in d2 are copied to create the offspring individual e2. Decimal encoding is then applied to e1 and e2, resulting in f1 and f2. Finally, f1 and f2 are scaled to obtain the new offspring individuals b1 and b2. As depicted in Fig. 3, two new offspring individuals, b1 and b2, are created, inheriting the genetic traits of their respective parents.

Fig. 3
figure 3

Schematic diagram of cross recombination

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④ Repeat the steps ①–③ until all the Nselect parent individuals are executed.

Where P_Cross is the probability of cross recombination.

Table 1 illustrates an example of the cross recombination operation. The parent individuals a1 and a2 consist of four gene fragments, representing drilling fluid displacement, gas injection rate, gas injection depth, and wellhead back pressure. The specific steps are as follows: (1) Scale the parent individuals a1 and a2 by removing the decimal point, resulting in c1 and c2. (2) Perform binary encoding on the gene fragments of c1 and c2, obtaining d1 and d2. The length of each gene segment is determined by the maximum value of the corresponding parameter during the encoding process. The maximum values of the parameters are presented in Table 3. The binary encoding lengths for each gene segment are 11, 13, 18, and 7, respectively. If the length of the converted binary gene segment is insufficient, it will be padded with zeros. (3) A cross point is randomly selected. In this example, the total length of the parent gene is 49, and the cross point is set at 30. The gene fragments in the parent individuals d1 and d2 are exchanged to generate binary offspring e1 and e2. (4) The binary offspring e1 and e2 are decoded to obtain f1 and f2. Finally, the target offspring b1 and b2 are obtained by scaling f1 and f2. If any of the parameters in the target offspring exceed the extreme value, the extreme value will be used instead.

Table 1 Example of cross recombination
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Mutation

In order to further enhance the exploration ability of solution space, mutation operations are performed on Nselect populations generated through crossover operations to introduce new gene disturbances:

① For each parent individual in the selected population Nselect, generate a random real number RX (RX ∈ [0,1]). If RX > P_Mutate, the parent individual does not undergo mutation. Proceed to step ④;

② If RX ≤ P_Mutate, the parent individual undergoes mutation. After scaling and binary encoding, a random location on the binary gene fragment is selected for mutation. If the selected location is originally 0, it changes to 1; otherwise, it becomes 0;

③The mutated binary offspring is then decoded and scaled to obtain the target offspring individual. If the parameters of the offspring individual exceed the extreme values, the extreme values will be used instead;

④Repeat steps ①–③ until all the Nselect individuals have been processed.

Where P_Mutate is the probability of mutation.

Update of population

Using the elite reservation strategy, the optimal individual of the t-th generation, selected based on the fitness function, is copied Nc times (Nc = (1-Gap) × N). These copies serve as newly generated offspring individuals. The newly generated offspring individuals Nc, along with the offspring individuals Nselect, form the population of the (t + 1)-th generation. Continue iterating until the maximum number of iterations is reached. Then, terminate the iteration process and output the optimal solution.

Where t is the number of the iterations; and Nc is the count of the optimal clones.

Procedure of IRSGA

The procedure of optimizing the parameters for annulus aerated dual gradient drilling using IRSGA is shown in Algorithm 1. The other relevant calculation functions are shown in Algorithms 2 ~ 7.

Algorithm 1
figure a

IRSGA for parameters optimization

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Algorithm 2
figure b

Function initialize population for IRSGA

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Algorithm 3
figure c

Function evaluate fitness for IRSGA

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Algorithm 4
figure d

Function selection for IRSGA

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Algorithm 5
figure e

Function crossover for IRSGA

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Algorithm 6
figure f

Function mutation for IRSGA

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Algorithm 7
figure g

Function update population for IRSGA

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Results and discussion

Case data

The optimization parameters are derived from well FY-X in the Fuyuan block of Tarim Oilfield. The current depth of the well is 7150 m, with a formation pore pressure of 68.3 MPa and a fracture pressure of 71.7 MPa. The drill string consists of a combination of 5-inch dual-wall drill pipe and 3 1/2-inch drill pipe. The annulus channel of the dual-wall drill pipe is utilized as a gas injection pipeline. The density of the drilling fluid is 1200 kg/m3. Please refer to Table 2 for further details on the wellbore structure.

Table 2 Wellbore geometry
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According to the technological requirements of annulus aerated dual gradient drilling, the parameters must adhere to specific constraints, outlined in Table 3:

Table 3 Constraints on parameters
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The involved computing environment is Inter Core TM i7–1165G7 running Windows 10 with 2.80-GHz processor and 16.00-GB RAM. The experiments are developed by C#.

Results of IRSGA optimization

This article conducts multi-objective optimization of the parameters for annulus aerated dual gradient drilling utilizing the IRSGA. The parameters involved in the genetic algorithm are provided in Table 4.

Table 4 Parameters involved in IRSGA
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Figure 4 shows the variation of population fitness and average fitness with the increase in iteration steps in the IRSGA process of the parameters optimization for annulus aerated dual gradient drilling. The larger the fitness value, the better the individual is, representing a safer and lower cost wellbore. The average fitness value represents the degree of evolution of the population. From Fig. 4, it can be observed that starting from the 30th generation, the population tends to converge. Afterward, the minor fluctuations in the average fitness values represent mutations generated by a small number of individuals during the iteration process, which helps enhance exploration of the solution space. Figures 5 and 6 depict the variations in the bottom-hole pressure and cost coefficient under the optimal parameter combinations for each generation during the IRSGA optimization process. It can be seen that the final bottom-hole pressure stabilizes at 70 MPa, and the cost coefficient stabilizes at 2670.99 $/d.

Fig. 4
figure 4

The variation curve of the fitness and average fitness values with increasing iteration

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Fig. 5
figure 5

The variation curve of the bottom-hole pressure with increasing iteration

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Fig. 6
figure 6

The variation curve of the cost index with increasing iteration

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Figure 7 represents the variations of the optimal parameter combinations for each generation during the IRSGA optimization process. It is obvious that before the 20th generation, the optimal parameter combinations keep evolving. However, after that, the algorithm converges, and the optimal parameter combinations stabilize without further changes. Therefore, the final optimized parameters are obtained, which ensure that the bottom-hole pressure in the drilling process remains within the safe pressure window while minimizing the construction cost. Compared with the parameters designed with the empirical gas–liquid ratio method (Liu 2022), the cost coefficient is reduced by 36.46%. The detailed results are shown in Table 5.

Fig. 7
figure 7

The variations of the optimal parameter combinations during the IRSGA process

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Table 5 Comparison of optimization results for IRSGA
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Results of comparison

Comparison of different genetic algorithms

In this study, we conducted a comparative analysis of the parameter optimization results of the IRSGA with ORSGA, RSGA, and TSGA. We evaluated the on-line and off-line performance and analyzed the impact of different selection operators on the optimization performance of genetic algorithms. The relevant parameters are shown in Table 6. Twenty simulations were conducted for each of the four genetic algorithms mentioned above. The optimization results are shown in Table 7.

Table 6 Parameters of the genetic algorithm
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Table 7 Optimization results of the four genetic algorithms
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John Holland (2013) proposed on-line performance and off-line performance to evaluate the performance of genetic algorithms.

(1) The on-line performance Pon-line:

$$P_{{\text{on - line}}} = \frac{1}{T}\sum\limits_{t = 1}^{T} {\left( {\frac{1}{N}\sum\limits_{i = 1}^{N} {f\left( {a_{i} ,t} \right)} } \right)}$$
(18)

The on-line performance represents the changes in the average fitness value of the population, mainly describing the overall performance and evolutionary ability of the population.

(2) The off-line performance Poff-line:

$$P_{{\text{off - line}}} = \frac{1}{T}\sum\limits_{i = 1}^{T} {f^{*} \left( {a_{i} ,t} \right)}$$
(19)

The off-line performance represents the changes in the fitness values of the best individual in the population, mainly describing the individual's evolutionary ability.

$$f^{*} \left( {a_{i} ,t} \right) = \max \left\{ {f\left( {a_{1} ,t} \right),f\left( {a_{2} ,t} \right) \ldots \ldots ,f\left( {a_{N} ,t} \right)} \right\}$$
(20)

where f*(ai,t) is the best individual fitness value in the current population.

In the experiment, Figs. 8 and 9 compared the Pon-line and Poff-line for the four different genetic algorithms. From the graphs, we can analyze the following points: (1) The selection operator has a significant impact on the performance of genetic algorithms. Improving the selection operator can enhance the algorithm's performance. (2) IRSGA demonstrates good performance in both searching for the optimal individual and the overall evolution of the population. It is first quartile, median and third quartile of Pon-line and Poff-line are larger compared to the other algorithms. Additionally, the interquartile range is smaller, indicating a smaller fluctuation range. (3) RSGA and TSGA yield similar results. However, when the population size is larger, RSGA requires more computational resources for sorting operations.

Fig. 8
figure 8

Box plot of different genetic algorithms about on-line performance

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Fig. 9
figure 9

Box plot of different genetic algorithms about off-line performance

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Comparison of different optimization algorithms

A comparative analysis was conducted between the IRSGA and the PSO (Iweh and Akupan 2023) as well as the SAA (Uday Sankar et al. 2023). The comparative experiments covered both small-scale and large-scale instances. The population size was set to 100 for the small-scale instances and 200 for the large-scale instances. Tables 8 and 9 present the optimization results of fitness function values for small-scale and large-scale instances in the IRSGA, PSO algorithm, and SAA. The "Fitness" and "Ave Fitness" represent the final fitness value and the average fitness value for each instance. Table 10 shows the optimization computation time for small-scale and large-scale instances of IRSGA, PSO, and SAA. The astringency of the instance is determined by Eq. (21). Instances that satisfy Eq. (21) are considered globally convergent, denoted as GC = 1. Otherwise, they are considered locally convergent, denoted as LC = 1. The accuracy evaluation of the algorithms is performed based on Eq. (22), and the results are presented in Table 11.

$$\left| {\frac{{{\text{Fitness}}_{ij} - {\text{Max}}\_{\text{Fitness}}}}{{{\text{Max}}\_{\text{Fitness}}}}} \right| \le \varepsilon$$
(21)

where Fitnessij refers to the fitness value for the j-th algorithm in the i-th case. Max_Fitness is the limit of the fitness function value. Max_Fitness = 0.6; ε = 0.01.

$$AR_{j} = \frac{{m - \sum\limits_{{i = 1}}^{m} {LC_{{IJ}} } }}{m} \times 100\%$$
(22)

where ARj is the accuracy rate of the j-th algorithm. The m represents the number of algorithm runs, with m = 20. LCij denotes the local convergence of the optimization algorithm, while GCij represents the global convergence.

Table 8 The optimization results for small-scale instances of IRSGA, PSO, and SAA
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Table 9 The optimization results for large-scale instances of IRSGA, PSO, and SAA
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Table 10 The optimization time of IRSGA, PSO, and SAA
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Table 11 The optimization astringency and accuracy results of IRSGA, PSO, and SAA
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Tables 8 and 9 show that the IRSGA demonstrates good effectiveness and stability in terms of fitness value and average fitness value. Table 10 shows the computation time of each algorithm for both small- and large-scale instances. For small-scale instances, the average computation time for IRSGA optimization is 245.93 s, while for PSO, it is 209.03 s, and for SAA, it is 806.90 s. For large-scale instances, the average computation time for IRSGA optimization is 502.65 s, for PSO, it is 419.22 s, and for SAA, it is 1609.42 s. There is little difference in computational efficiency between IRSGA and PSO, while SAA exhibits the lowest computational efficiency. In terms of accuracy, the IRSGA achieves the highest optimization accuracy. For small-scale instances, the AR of IRSGA is 90%, while for large-scale instances, it increases to 95%. In conclusion, the IRSGA proposed in this paper demonstrates excellent performance in terms of solution effectiveness, computational efficiency, and algorithm accuracy.

On-line and off-line evaluation between IRSGA and PSO

As a result of the low computational efficiency of the SAA, further discussion was abandoned. Instead, we conducted the on-line and off-line performance evaluation of the PSO algorithm and IRSGA. The on-line and off-line performance for the optimization results of both algorithms are shown in Table 12. Furthermore, a comparison of the on-line and off-line performance is presented in Table 13.

Table 12 Optimization results of on-line performance and off-line performance
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Table 13 Comparison results of on-line performance and off-line performance
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In terms of on-line performance, the median and third quartile values of PSO and IRSGA are relatively similar. However, the minimum and first quartile values of PSO are significantly smaller than those of IRSGA, as illustrated in Fig. 10. This indicates that the overall evolutionary performance of the IRSGA is superior to that of PSO. Regarding off-line performance, the maximum and third quartile values of PSO are approximate to those of IRSGA. Surprisingly, even the median value of PSO is slightly larger. However, the minimum and first quartile values of PSO are considerably smaller than those of IRSGA. Consequently, the individual evolution performance of the PSO algorithm is inferior to that of IRSGA. In summary, the IRSGA excels in both individual and overall evolution, demonstrating smaller evolutionary fluctuations and greater stability.

Fig. 10
figure 10

Box plot of PSO and IRSGA about on-line and off-line performance

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Conclusions and future works

In this paper, we proposed a new genetic algorithm-based optimization method called IRSGA for the parameters optimization of annulus aerated dual gradient drilling. We conducted a comprehensive comparative analysis and evaluation on the optimization performance of this algorithm, leading to the following conclusions:

  1. (1)

    The IRSGA offers distinct advantages in maintaining bottom-hole pressure within the safe pressure window and simultaneously reducing drilling costs. Compared to traditional gas–liquid ratio parameter design methods, the implementation of the IRSGA can result in a substantial cost reduction of 36.46%.

  2. (2)

    Improving the selection operator can significantly enhance the performance of genetic algorithms. In comparison with ORSGA, RSGA, and TSGA, the IRSGA proposed in this paper demonstrates better convergence and stability.

  3. (3)

    In terms of optimization stability, efficiency, and computational accuracy, IRSGA performs better than PSO and SAA, especially in large-scale computations, where its computational accuracy exceeds 95%.

  4. (4)

    On the aspect of on-line and off-line performance, the IRSGA performs better than the PSO algorithm.

However, there are still limitations and areas for improvement in this study. Local optima have not been entirely avoided, and there is a need to enhance computational efficiency. Moving forward, there are several directions for future research. Firstly, it would be beneficial to further explore the impact of population size on convergence ability. Additionally, improvements can be made to the crossover and mutation operations in order to enhance the algorithm's global exploration ability. Lastly, it is significative to investigate the influence of population renewal strategies on the overall performance of the algorithm.