A Multi-Period Model of Compressor Scheme Optimization for the Shale Gas Gathering and Transportation System

Abstract

In the process of shale gas production, with the change in gas productive parameters, the pressurization demand for the shale gas gathering and transportation system (SGGTS) also changes, which affects the choice of pressurizing location and timing. Our purpose is to effectively respond to the impact of parameter changes during shale gas production and to better select the pressurization schemes. Therefore, we considered the modularization of the compressors and established a mixed-integer nonlinear programming (MINLP) model to minimize the total cost of the SGGTS. Taking an actual shale gas field as an example, by discretizing the time during a given production period to solve the model under multi-period and single-period conditions, the optimal pressurization scheme for the SGGTS in the specified production period is obtained. It indicates that the results obtained under a multi-period condition are more conducive to actual production. Compared with the results obtained under the single-period condition, the cumulative cost obtained in the multi-period condition is reduced by 17.19%. By deploying the MINLP model in the specified production period, the pressurization demand is met in each time period. This greatly improves the utilization rate of modular compressors, reduces the total cost, and improves the economic benefits of the SGGTS.

1. Introduction

1.1. Background

In a contemporary energy landscape, natural gas has emerged as a competitive and predominant energy vector (Zhou et al., 2021) [1]. Yet, as disparities between supply and demand in the natural gas sector continue to escalate, conventional gas resources have consistently fallen short of satisfying prolonged market requirements (Wei et al., 2021) [2]. Shale gas, recognized as a significant unconventional reservoir of natural gas, is progressively gaining heightened attention. Encapsulated within shale formations, this gas can be extracted. Notably, China boasts considerable reserves of this exploitable resource. The formation and accumulation of shale gas have unique characteristics, such as the thick and widely distributed shale source rocks found in basins. Compared with conventional natural gas resources, shale gas resources have the advantages of a longer exploitation lifetime and production period, as well as a higher economic worth.

Shale gas production typically exhibits cyclical and seasonal fluctuations, with initial output and pressure significantly higher than other production stages. This might lead to the need for compressor solutions to adapt to continuously changing gas flow. As the production rate decreases, the compressor may need to adjust its operating mode or be reconfigured to accommodate a lower output, while a higher operational capacity is required during high production. This production characteristic also impacts the efficiency of the compressor operation. Under different production conditions, while a single compressor can meet the production needs, its power or energy consumption might not be in the most optimal state. Therefore, we need different types of compressors and operating parameters to ensure the best efficiency. Additionally, in shale gas production, there may be peak demand periods, requiring extra boosting to meet instantaneous high requirements. Compressor solutions must consider how to fulfill these additional demands during peak periods while avoiding excessive operation.

Hence, within the ambit of shale gas development, there exists a pronounced decline in the production capacity of shale gas, leading to continual alterations in equipment load rates and assorted operational states. Such dynamics critically influence decisions concerning compressor typology, the location of pressurization, and its timing. For the efficacious extraction of shale gas, an optimized operational strategy and a dynamic blueprint for the shale gas gathering and transportation system (SGGTS) are imperative. Modular compressors present a solution adept at countering these evolving productive parameter shifts. Distinct from their conventional counterparts, modular compressors are compact, highly mobile units fabricated as independent modules, allowing swift on-site assembly. As pressurization venues transition, these units can be promptly disassembled and relocated, thereby circumventing compressor resource waste and augmenting utilization efficiency. This approach not only fosters sustained and stable shale gas production but also curtails production expenditures, enhancing the economic viability of shale gas fields. Consequently, delving into optimal pressurization strategies within the SGGTS that employ modular compressors is of paramount significance.

Integrating modular facilities into traditional compressor optimization schemes requires considering the diversity of compressors, management, control, and coordination with existing systems. By knowing the pressure, flow rate, network structure, pipeline length, pipeline diameter, shale gas molar composition, adiabatic compression efficiency, and compressor mechanical efficiency, we can determine the installation location, quantity, and rearrangement of the compressors and obtain the optimal combination of boosting methods, boosting locations, SGGTS compressor operating power, and compressor type. Representing these complex operational dynamics through mathematical models is a long-term and intricate task.

1.2. Related Work

As shale gas gains prominence within the global energy matrix, scholarly interest in its development has surged. International academic circles have embarked on multifaceted investigations into shale gas production nuances, registering noteworthy findings. A salient distinction between the extraction methodologies of shale gas and its conventional counterpart lies in the deployment of horizontal drilling coupled with high-volume hydraulic fracturing (Vengosh et al., 2013) [3]. Advancements in horizontal drilling techniques have catalyzed the economically viable recovery of shale gas. In this realm, Yuan et al. (2013) [4] introduced an innovative predictive technique for ascertaining geostress within shale strata, geared towards bolstering drilling stability during horizontal well construction in shale gas repositories. Similarly, Lin et al. (2021) [5] advanced an optimal schematic for shale gas volumetric fracturing, predicated upon a robust assessment of fracturing outcomes. Real-Miranda et al. (2022) [6] employed stochastic dynamic programming techniques to determine the equivalent insurance premium for extraction personnel during the gas well extraction process. On another front, Cafaro et al. (2018) [7] put forth a continuous-time optimization paradigm tailored for orchestrating multiple re-fracture interventions across a shale gas well’s lifecycle. Complementarily, Drouven et al. (2017) [8] elucidated an optimization matrix, aiming to discern the feasibility of a well undergoing re-fracturing within its operational tenure. Meanwhile, Grossmann et al. (2014) [9] propounded a cutting-edge mixed-integer optimization framework dedicated to the strategic design of shale gas infrastructure, targeting the meticulous calibration of well counts, gas compressor capacities, and judicious planning of freshwater requisites for both drilling and fracturing processes. The aforementioned research primarily focuses on the optimization studies of underground processes and techniques, with relatively less consideration given to surface gathering and transportation systems. To achieve more comprehensive optimization and resource utilization, there is a need to systematically analyze and model the entire shale gas extraction process, including the surface operations.

The trajectory of protracted shale gas development has been an increasingly focal subject in contemporary academic literature. The intricate design and execution of long-term shale gas field development present substantial challenges, attributed to multifaceted developmental operations and an extensive gamut of potential deployment sites. In this context, Tavallali et al. (2013) [10] developed a mixed-integer nonlinear model by integrating subsurface, surface, and oil wells, it provides the optimal number of new producers, their locations, and optimal production plan over a given planning horizon. Cafaro and Grossmann (2014) [11] introduced a mixed-integer non-linear programming (MINLP) framework tailored for the sustainable, long-term strategic planning of the shale gas supply chain. This model adeptly determines optimal parameters, such as well counts per site, natural gas processing plant capacity, compressor potency, and freshwater requisitions for hydraulic fracturing and drilling processes, with an overarching objective to amplify the net present value. Subsequently, Yang et al. (2015) [12] proffered a novel mixed-integer linear programming (MILP) framework devised to fine-tune capital allocation decisions concerning water resources for shale gas extraction, leveraging a discrete-time rendition of a state tasks network. Further enriching the discourse, Drouven and Grossmann (2016) [13] delineated a large-scale, non-convex MINLP framework accentuating quality-centric, enduring shale gas ventures. Adding another dimension, Cafaro et al. (2016) [14] promulgated a discrete-time, multi-temporal MILP model, meticulously outlining the prospects of successive re-fracture interventions throughout a well’s operational lifespan. In a complementary study, Mahmoud et al. (2020) [15] embarked on a holistic assessment of both prolonged and short-term outputs, pivoting on specific shale gas reservoir and completion parameters. Their findings underscored that, while completion parameters influence both temporal output dimensions, it is the reservoir parameters that predominantly steer the long-term production dynamics. Rounding off this cascade of contributions, Peng et al. (2021) [16] proposed an innovative MILP framework for the strategic design and roll-out of enduring shale gas field projects. In tandem, they also presented a solution pool-centric, dual-layer decomposition algorithm to effectively navigate this model. The aforementioned studies focus on the entire process of shale gas extraction and transportation. They utilize various types of mathematical planning methods to address issues related to shale gas development and production, covering areas such as supply chain planning, water resource allocation, and production optimization. These studies have provided essential insights for the long-term development of the shale gas field. However, they have not considered the factor of modular facilities. Modular facilities are gradually playing a pivotal role in modern shale gas development. Through modular facilities, production flexibility can be enhanced, adaptability to production parameter fluctuations can be ensured, and resource waste can be reduced.

Modular facilities, characterized by their technical and economic merits alongside adaptive operability, are increasingly permeating diverse sectors. Delving into this theme, Gao and You (2017) [17] advanced a novel mixed-integer nonlinear fractional programming model to ascertain if modular manufacturing might serve as a pivotal disruptor in designing and operationalizing the shale gas supply chain, with a vision of achieving both economic and ecological sustainability, Similarly, Yang and You [18] conducted a comparative techno-economic and environmental analysis of different shale gas monetization scenarios and showed that modular generation is more advantageous. Tackling capacity planning within a modular infrastructure for shale gas endeavors, Allen et al. (2018) [19] constructed a multi-stage stochastic model that factored in ambiguous production projections. Conversely, the MILP framework conceived by Hong et al. (2020) [20] seamlessly amalgamated decisions spanning shale gas extraction, conveyance, and processing with modular infrastructure deliberations. Their empirical evaluations revealed that the dynamic schema of modular infrastructure adeptly counterbalances productivity oscillations. Furthermore, in 2021, Hong et al. [21] crafted an MILP optimization paradigm, aiming to optimize the spatial arrangement of modular amenities within shale gas domains, thereby enhancing operational efficiency and financial feasibility. Yang and You’s study in 2017 honed in on modular methanol production as a potential strategy for plant repositioning, concluding its superior economic competitiveness vis-à-vis traditional shale gas methodologies. Baldea et al. (2017) [22] proffered a comprehensive synthesis of the various catalysts spurring modular production, subsequently assessing modular production blueprints predicated on a unique metric: the value density of raw material resources coupled with product market dynamics. Pakizer et al. (2020) [23] championed the integration of modular water frameworks with existing centralized infrastructures, thereby sculpting hybrid systems. Lier and Grünewald (2011) [24] undertook a financial juxtaposition of modular and conventional large-scale chemical plants by evaluating their respective net present values. Meanwhile, Palys et al. (2018) [25] introduced modular methodologies into ammonia synthesis, which, as per their findings, engendered diminished supply chain expenditures and augmented utilization of renewable assets relative to continuous operations. Lastly, Tan and Barton (2015) [26] proposed a layered optimization construct tailored to guide optimal temporal allocation and operational resolutions, empowering decision-makers to monetize associated or confined gas through mobile installations. Subsequently, Tan and Barton (2016) [27] further improved the optimization framework to further illustrate the advantages of mobile devices in coping with uncertainty by taking into account the uncertainty of future parameters.

The aforementioned studies have incorporated modular facilities into their respective optimization systems and explored the economic and ecological sustainability of modular facilities in various fields. These studies offer insights into how modular facilities can enhance various industrial processes and supply chains. Traditional fixed facilities are difficult to disassemble once they are installed, which makes it difficult to flexibly adapt to changes in production capacity and to work continuously in high-efficiency work areas (Zhou et al., 2019) [28]. In contrast, modular facilities are better able to adapt to changing production capacities due to their compact structure, ease of disassembly, ease of movement, and low development risk. The academic and engineering communities have recognized the importance of modular infrastructure for shale gas production. However, modular compressors are usually ignored in related research on SGGTS pressurization. The compressor is crucial in the shale gas production process, as it plays an essential role in boosting and transporting the shale gas. Therefore, we establish an MINLP model in this paper, considering the modularity of compressors. and take an actual field as an example to verify the model.

1.3. Contributions

The main contributions of this work include:

• Establishing an MINLP model to solve the pressurization optimization problem in the shale gas field
• Considering the compressor modularization to deal with the influence of the pressurizing location changing with the fluctuation in the productive parameters
• Verifying the model through a practical case

1.4. Paper Organization

The paper proceeds as follows: Section 2 describes the problem; Section 3 introduces the mathematical model; Section 4 introduces the algorithm; Section 5 provides a case study to prove the validity of the model; and Section 6 summarizes the conclusions.

2. Problem Description

The schematic diagram of the optimal pressurization scheme and the rearrangement of modular compressors in the shale gas production process is shown in Figure 1. The rearrangement here is the movement of modular compressors from one location to another over different time periods. In this example, three different shale gas production periods are included. The first period is in the early stage of shale gas production, and there is no need to pressurize the SGGTS. The second period is in the early stage of pressurization, and it is necessary to purchase modular compressors to pressurize some of the platforms. In the third period, the mid-to-late stage, when the pressurizing positions of the SGGTS change, it is not only necessary to purchase more modular compressors but also to rearrange modular compressors. All decisions at each time period are aimed at minimizing the total cost of modular compressors over the entire production period.

The optimization problem of pressurization and rearrangement based on modular compressors is that we have to already know the pressure, flow rate, network structure, pipeline length, internal diameter of the pipeline, molar composition of the shale gas, adiabatic compression efficiency, and compressor mechanical efficiency in order to determine the installation location, quantity, and rearrangement of compressors and then obtain the optimal combined pressurization mode, pressurizing position, compressor operating power, and compressor type of the SGGTS. As shown in Figure 2, there are four types of pressurization modes.

• Platform pressurization is to arrange the compressor at the platform, as shown in Figure 2a.
• Area pressurization is to arrange the compressor at the point of the region to pressurize the low-pressure area, as shown in Figure 2b.
• Station pressurization is to arrange the compressor at the station, as shown in Figure 2c.
• Well pressurization is to arrange the compressor to pressurize the single well or the same half-branch well inside the platform, as shown in Figure 2d.

The objective is to provide the pressurization scheme and modular compressor rearrangement for a shale gas production cycle. To facilitate modeling and improve model-solving efficiency, the shale gas production cycle is typically discretized into a series of short time intervals, and within each interval, the system’s operating and production conditions are considered constant to better understand and manage the system’s operation. The following assumptions are made:

• The shale gas production time can be divided into a series of short, discrete periods (Allen et al., 2018) [19].
• The shale gas production parameters do not change within a unit time period.
• During the specified shale gas production period, the modular compressors will not be damaged or require additional maintenance costs.
• The lead time for purchasing modular compressors is not taken into consideration.

3. Mathematical Formulation

3.1. Objective Function

The objective function is the minimum total compressor cost, as shown in Equation (1). The purchase cost ($F_{cc}$), operation cost ($F_{oc}$), and rearrangement cost ($F_{mc}$) are shown in Equations (2)–(4), respectively.

$$\min F=F_{cc}+F_{oc}+F_{mc}$$

(1)

$$F_{cc}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{y\in Y}{\displaystyle \sum _{i\in I}{b}_{i,y,t}{z}_{i,y,t}{\alpha }_{i,y,t}}}}$$

(2)

$$F_{oc}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in I}{N}_{i,t}{r}_{i,t}}}$$

(3)

$$F_{mc}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{y\in Y}{\displaystyle \sum _{i\in I}{u}_{i,y,t}{\phi }_{i,y,t}}}}$$

(4)

In the formula, $r_{i,t}$ is the operating price, 10⁴ Yuan/kW; $b_{i,y,t}$ is the location variable; $z_{i,y,t}$ is the type y compressor number installed, set; $N_{i,t}$ is the operating power, kW; $u_{i,y,t}$ is the type y compressor number rearranged, set; ${\phi }_{i,y,t}$ is the rearrangement price, 10⁴ Yuan/set; and ${\alpha }_{i,y,t}$ is the compressor price, 10⁴ Yuan/set.

3.2. Constraint

3.2.1. Pipeline Pressure and Flowrate Constraints

The pipeline pressure and flowrate constraints include pressure drop constraints, maximum pressure constraints, minimum pressure constraints, and flow constraints, as shown in Equations (5)–(8), respectively. The pipeline starting pressure ($p_{i|(i,j),t}$) cannot exceed the maximum pressure ($p_{\max }^{}$). The pipeline end pressure ($p_{j|(i,j),t}^{}$) cannot be less than the minimum pressure ($p_{\min }^{}$). The node flow rate ($q_{i|(i,j),t}^{}$, $q_{j|(i,j),t}^{}$) of the pipeline (i, j) is equal to the pipeline flow rate ($Q_{i,j,t}$).

$$Q_{i,j,t}=5033.11d_{i,j}^{8/3}{\left[\frac{{(p_{i|(i,j),t}^{})}^2-{(p_{j|(i,j),t}^{})}^2}{Z\Delta T_{i,j,t}^{}{L}_{i,j}}\right]}^{0.5} \forall (i,j)\in A,i\ne j,\forall t\in T$$

(5)

$$p_{i|(i,j),t}^{}\le p_{\max }^{} \forall (i,j)\in A,i\ne j,\forall t\in T$$

(6)

$$p_{j|(i,j),t}^{}\ge p_{\min }^{} \forall (i,j)\in A,i\ne j,\forall t\in T$$

(7)

$$q_{i|(i,j),t}^{}=Q_{i,j,t}=q_{j|(i,j),t}^{} \forall (i,j)\in A,i\ne j,\forall t\in T$$

(8)

In the formula, $d_{i,j}$ is the pipeline diameter, cm; $Z$ is the compressibility factor; $\Delta$ is the relative density; $T_{i,j,t}^{}$ is the temperature in the pipeline (i, j), K; and $L_{i,j}$ is the pipeline length, km.

3.2.2. Compressor Constraints

The compressor constraints contain suction pressure ($p_{i,t}^{in}$) constraints, discharge pressure ($p_{i,t}^{out}$) constraints, compression ratio (${\epsilon }_{i,t}^{}$) constraints, compressor power ($N_{i,t}$) constraints, and compressor number ($z_{i,y,t}$) constraints, as shown in Equations (9)–(13), respectively.

$$p_{i,t}=p_{i,t}^{in} \forall i\in I,\forall t\in T$$

(9)

$$p_{i,t}^{out}=p_{i|(i,j),t}^{} \forall (i,j)\in A,i\ne j,\forall t\in T$$

(10)

$${\epsilon }_{i,t}^{}=\frac{p_{i,t}^{out}}{p_{i,t}^{in}} \forall i\in I,\forall t\in T$$

(11)

$$N_{i,t}=16.745p_{i,t}^{in}{q}_{i,t}\frac{k}{k-1}({\epsilon }_{i,t}^{\frac{k-1}{k}}-1)\frac{Z_{i,t}^{in}+Z_{i,t}^{out}}{2Z_{i,t}^{in}}\frac{1}{\eta }$$

(12)

$$z_{i,y,t}=\frac{N_{i,t}}{N_y} \forall i\in I,\forall y\in Y,\forall t\in T$$

(13)

In the formula, $p_{i,t}^{}$ is the node i pressure, MPa; $q_{i,t}$ is the node i flow rate, m³/min; $k$ is the gas-specific heat ratio; $\eta$ is the compression efficiency; and $N_y$ is the maximum operating power, kW/set.

3.2.3. Well or Platform Throttling Constraints

The throttling at the well is to control well pressure ($p_{w,t}$), as shown in Equation (14). $p_{w|(w,g),t}$ is the pressure before throttling at the well. Similarly, platform throttling is to control the platform pressure ($p_{g,t}$) to a certain value, as shown in Equation (15). $p_{g|(g,j),t}$ is the pressure before throttling at the platform.

$${\delta }_{w,t}^{}=\frac{p_{w|(w,g),t}^{}}{p_{w,t}} \forall w\in W,\forall (w,g)\in WGA,\forall t\in T$$

(14)

$${\delta }_{g,t}^{}=\frac{p_{g|(g,j),t}^{}}{p_{g,t}} \forall g\in G,\forall (g,j)\in GIA,\forall t\in T$$

(15)

In the formula, ${\delta }_{w,t}^{}$ is the well w throttle ratio and ${\delta }_{g,t}^{}$ is the platform g throttle ratio.

3.2.4. Pressure Balance Constraints

The pressure balance constraints need to be considered at platforms, area points, and stations, as shown in Equations (16)–(18), respectively. $\tau {(g)}^{+}$ is the set of wells at the platform g. The smallest well pressure is denoted by $p_{\tau {(g)}^{+},t}^{\min }$. The pressure balance constraint is used to construct the relationship between the starting pressure ($p_{s|(i,s),t}$) and ending pressure ($p_{n|(i,n),t}$) of the pipeline and the facility point pressure ($p_{s,t}$).

$$p_{\tau {(g)}^{+},t}^{\min }=p_{g,t} \forall g\in G,\forall t\in T$$

(16)

$$p_{n|(i,n),t}=p_{n,t} \forall n\in N,\forall (i,n)\in INA,\forall t\in T$$

(17)

$$p_{s|(i,s),t}=p_{s,t} \forall s\in S,\forall (i,s)\in ISA,\forall t\in T$$

(18)

3.2.5. Well Flow Rate Constraints

Well throttling has some effects on gas productivity. The relationship between flow rate and throttling is shown in Equation (19). The flow rate after well throttling needs to be controlled, as shown in Equation (20), and the maximum flow rate of the nozzle flow $q_{sc,t}^{\max }$ is shown in Equation (21). If wells do not need throttling or pressurizing, the flow rate is the initial flow rate $q_{w,t}^0$, as shown in Equation (22).

$$q_{sc,t}=\frac{0.408p_{1,t}{d}^2}{\sqrt{\Delta T_{1,t}{Z}_{1,t}}}\sqrt{\frac{\varpi }{\varpi -1}\left[{(\frac{p_{2,t}}{p_{1,t}})}^{\frac{2}{\varpi }}-{(\frac{p_{2,t}}{p_{1,t}})}^{\frac{\varpi +1}{\varpi }}\right]}$$

(19)

$$q_{w,t}^1= \left\{\begin{array}{cc}{q}_{sc,t}& q_{sc,t}\le q_{sc,t}^{\max }\\ q_{sc,t}^{\max }& q_{sc,t}>q_{sc,t}^{\max }\end{array}\right. \forall w\in W,\forall t\in T$$

(20)

$$q_{sc,t}^{\max }=\frac{0.408p_{1,t}{d}^2}{\sqrt{\Delta T_{1,t}{Z}_{1,t}}}\sqrt{\frac{\varpi }{\varpi -1}\left[{(\frac{2}{\varpi +1})}^{\frac{2}{\varpi -1}}-{(\frac{2}{\varpi +1})}^{\frac{\varpi +1}{\varpi -1}}\right]}$$

(21)

$$q_{w,t}=\left\{\begin{array}{cc}{q}_{w,t}^0& {\delta }_{w,t}^{}\ge 1\\ q_{w,t}^1& {\delta }_{w,t}^{}<1\end{array}\right. \forall w\in W,\forall t\in T$$

(22)

In the formula, $q_{sc,t}$ is the flow rate through the nozzle in standard condition, m³/d; $p_{1,t}$ is the pressure in front of the nozzle, MPa; $p_{2,t}$ is the pressure behind the nozzle, MPa; $d$ is the nozzle hole diameter, mm; $T_{1,t}$ is the temperature in front of the nozzle, K; $Z_{1,t}$ is the compressibility factor under temperature T₁ and pressure p₁; and $\varpi$ is the gas adiabatic index (take $\varpi =1.3$ for calculation).

Where $q_{sc,t}^{\max }$ is the maximum flow rate through the nozzle, m³/d.

3.2.6. Compressor Rearrangement Constraints

The decision-making difference between time t and time t − 1, where 0 represents no need to rearrange the compressor, 1 represents the need to install the compressor, and −1 represents the need to rearrange the compressor, is shown in Equation (23). The number of identical elements in the set is shown in Equations (24)–(26). The expressions of compressor installation and rearrangement number at time t are shown in Equations (27) and (28), respectively.

$$B_{i,t,y}^{}=b_{i,t,y}^{}-b_{i,t-1,y}^{} \forall i\in I,\forall t\in T,\forall y\in Y$$

(23)

$$PoslNu{m}_t={\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in I}(B_{i,t,y}=1)}} \forall i\in I,\forall t\in T,\forall y\in Y$$

(24)

$$NeglNu{m}_t={\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in I}(B_{i,t,y}=-1)}} \forall i\in I,\forall t\in T,\forall y\in Y$$

(25)

$$Nu{m}_t^{P-N}=PoslNu{m}_t-NeglNu{m}_t$$

(26)

$$o_{i,t}^c=\left\{\begin{array}{cc}Nu{m}_t^{P-N}& Nu{m}_t^{P-N}\ge 1\\ 0& Nu{m}_t^{P-N}<1\end{array}\right. \forall i\in I,\forall t\in T$$

(27)

$$o_{i,t}^u=\left\{\begin{array}{cc}NeglNu{m}_t& Nu{m}_t^{P-N}\ge 1\\ PoslNu{m}_t& Nu{m}_t^{P-N}<1\end{array}\right. \forall i\in I,\forall t\in T$$

(28)

In the formula, $B_{i,t,y}^{}$ is the decision difference between the compressor location at time t and time t − 1; $PoslNu{m}_t$ is the number of 1 defined in the set; $NeglNu{m}_t$ is the number of −1 defined in the set; and $Nu{m}_t^{P-N}$ is the difference value.

4. Solution Approach

The optimization framework for the SGGTS is characterized as an MINLP model, encompassing both 0–1 integer variables and non-linear constraints. The realm of MINLP problem solvers is vast, boasting tools like COUENNE (Belotti, 2009) [29], DICOPT (Duran and Grossmann, 1987) [30], CONOPT (Drud, 1985) [31], and SCIP (Kronqvist et al., 2018) [32], among others. Notably, SCIP stands out as one of the premier high-speed solvers for mixed-integer programs and MINLPs. Furthermore, it serves as an architectural scaffold for constrained integer programs, employing the branch-and-bound methodology. SCIP’s inherent mechanism employs linear programming relaxations paired with cutting planes to delineate a robust dual boundary. Simultaneously, it integrates constraint programming to manage diverse (non-linear) constraints and propagation mechanisms to refine the variable domain. This solver’s design grants comprehensive oversight over the solution trajectory and facilitates access to granular information, delving deep into the solution’s core. Given its capabilities, this research harnesses SCIP to resolve the designated MINLP challenges. A visual representation of the optimization model’s solution process is depicted in Figure 3. First, determine the pressurization location of the SGGTS and establish the optimization model. Then obtain the known data of the SGGTS, including the network structure, number of wells, production time, pipeline parameters, etc. Program the collections, variables, parameters, equations, etc., in GAMS, and finally solve it using SCIP.

5. Case Study

5.1. Basic Parameters

The pipeline network in the shale gas field is a branch system, as shown in Figure 4. The pipeline network includes a total of 51 wells, 13 platforms, 8 area points, and 1 station. The gas collected from the wellheads is gathered at the platforms and then transported to the station through the gathering pipeline. After processing, it is exported. The production parameters in the 4-year production period are selected for research. The specified production period is discretely divided into 48 periods of monthly units, which are named M1, M2, …, and M48. The number and production time of wells at each platform are shown in

Table 1. Production parameters.

Number	Platform	Well Number	Production Time	Number	Platform	Well Number	Production Time
1	H1	8	M1	8	H8	3	M19
2	H2	4	M1	9	H9	6	M19
3	H3	3	M15	10	H11	3	M10
4	H4	3	M16	11	H19	3	M16
5	H5	4	M21	12	H21	3	M1
6	H6	8	M1	13	H13	4	M1
7	H7	1	M24	-	-	-	-

. Pipeline parameters are represented in

Table 2. Pipeline parameters.

Pipeline	Starting	End	Diameter (mm)	Length (km)
1	H1	GS	200	0.2
2	H2	GS	200	1.4
3	H3	GS	200	1.5
4	H4	GS	200	4.4
5	H5	H11	200	2.7
6	H6	H5	200	3.2
7	H7	H6	200	2.8
8	H8	H7	200	2.1
9	H9	H8	200	1.3
10	H11	H3	200	2.6
11	H13	H2	200	6.2
12	H19	GS	200	8.2
13	H21	H19	200	5.5

. The well production data is shown in Figure 5. The station pressure is 6.3 MPa. The gas composition is shown in

Table 3. Shale gas composition.

Composition	CH4	C2H6	C3+	CO2	H2S	He	H2	N2
Mol (%)	99.09	0.44	0.02	0.27	0.00	0.02	0.00	0.16

. The purchase and rearrangement costs of modular compressors of different powers are shown in

Table 4. Compressor price.

Number	Compressor Power	Purchase Price (104 Yuan/Set)	Rearrangement Price (104 Yuan/Set)
1	315 kW	63	5
2	500 kW	96	5
3	710 kW	136	5
4	900 kW	170	5
5	1000 kW	190	5
6	1250 kW	240	5
7	1400 kW	270	5

. The electricity cost is 0.67 Yuan/(kW·h).

5.2. Result Analysis

5.2.1. Iterative Solution

The model is solved under multi-period and single-period conditions, respectively, and the solution parameters under a single-period condition are shown in

Table 5. Iterative of a single-period condition.

Time Period (Month)	CPU Time (s)	Iteration Count	Time Period (Month)	CPU Time (s)	Iteration Count
14	0.20	696	32	0.15	1140
15	0.10	737	33	0.17	1050
16	0.09	900	34	0.60	1164
17	0.09	944	35	0.20	1062
18	0.11	1021	36	0.66	864
19	0.48	819	37	0.53	1179
20	0.38	1091	38	0.75	1239
21	0.44	864	39	0.58	1304
22	0.72	1100	40	0.17	1451
23	0.41	1044	41	0.11	1317
24	0.52	1118	42	0.15	715
25	0.22	1039	43	0.14	1197
26	0.11	1236	44	0.18	1227
27	0.13	1107	45	0.18	1274
28	0.32	1189	46	0.17	1285
29	0.48	763	47	0.11	1147
30	0.14	1496	48	0.16	1051
31	0.18	1076	-	-	-

. Under single-cycle conditions, the system was solved the fastest in the 17th month, taking 0.09 s and 944 iterations to complete, and the slowest in the 18th month, taking 0.75 s and 1239 iterations to complete. Due to the different number of constraints and variables in different cycles, the complexity of problem-solving varies, resulting in different solution times and numbers of iterations.

The iterative solution process of the model under a multi-period condition is shown in Figure 6. The original problem has 11,053 variables, 12,827 constraints, and 32,628 non-zero elements. The total computation time is 1039.63 s, with a total of 4.255 million iterations. It takes 2.1 s to preprocess the original problem, which reduces the original variables to 4872 and constraints to 6945. After preprocessing, the iteration starts, and the whole solution process is carried out under two boundary conditions: the dual bound and the primal bound. Initially, an initial value is found under the dual bound conditions after 2.8 s, and then the initial value gradually decreases. In the 57.5th second, the initial value of the feasible solution is found under the primal bound conditions, and the value gradually increases. As the values under the two bound conditions get closer to each other, the relative error of the two values reaches 9.91% in the 1039.63rd second (the SCIP error is limited to 10%), and the program stops iterating and outputs the result.

5.2.2. Type and Quantity of Modular Compressors

The change in the number of modular compressors over time is shown in Figure 7. Different compressor power levels are represented by different shapes. To ensure stable production in the shale gas field within the specified production period, a total of 13 modular compressors, including 5 power types, are required under a multi-period condition, and a total of 26 modular compressors, including 7 power types, are required under a single-period condition. It is evident that a multi-period condition requires fewer compressors and fewer power types. However, as production progresses, the requirement for compressor power becomes larger and larger, and the number of compressors also increases. Not only are the required power types of modular compressors different in various pressurizing periods, but the number of modular compressors with the same power also changes. This requires managers to reasonably rearrange modular compressors according to the changes in pressurization demand to avoid idle facilities.

5.2.3. Pressurization Scheme

The pressurization scheme of modular compressors for the SGGTS during the specified production period is shown in Figure 8. The colors of different cells represent modular compressors of different powers, and the numbers inside the cells represent the number of modular compressors. This shale gas field enters the pressurization stage at M14. Due to the constant changes in pressurization demand, the purchase and rearrangement of modular compressors are needed continuously. Figure 8a shows the pressurization scheme for each period under the multi-period condition, with a total of 18 rearrangements for the modular compressor during the specified production period. Figure 8b shows the pressurization scheme for each period under the single-period condition, with a total of 33 rearrangements for the modular compressor during the specified production period. It indicates that the results obtained under a multi-period condition are more stable than under a single-period condition, both in terms of the change in the pressurizing position and the rearrangement frequency of the modular compressor.

5.2.4. Operating Power

The operating power of the modular compressor for each pressurizing location over different periods is shown in Figure 9. The shale gas field does not need pressurization in the first 13 months of the specified production period and will need pressurization in different positions for the next 35 months to ensure stable production. It can be easily discovered that the requirement for larger operating power of modular compressors is concentrated in the stations and area points, while the operating power of the modular compressors at the platform is small. Figure 9a shows the variation in the operating power of the modular compressors under the multi-period condition. Compared with the single-period condition shown in Figure 9b, the variation in the operating power of the modular compressor is more stable.

5.2.5. Economic Comparisons

Figure 10 shows the cumulative cost of modular compressors under single-period and multi-period conditions. When entering the early pressurizing period, the cumulative total cost under the single-period condition is relatively lower. That is because the solution under the single-period condition considers the lowest total cost of the month, while the solution under the multi-period condition considers the lowest total cost for the entire period. As production progresses, the economic advantages under the multi-period condition become more and more pronounced. The cumulative cost under single-period and multi-period conditions within four years is 13,920.778 × 10⁴ Yuan and 11,527.532 × 10⁴ Yuan, respectively. The cumulative cost under multi-period conditions is reduced by 17.19%. Therefore, it can be concluded that the solution under multi-period conditions can lead to a more economical modular compressor pressurization scheme, which can help reduce the investment cost of shale gas field production.

6. Conclusions

• Concentrating on the changing characteristics of the pressurization process of shale gas production and considering the compressor modularization, this study established an MINLP model for the pressurization and rearrangement optimization of the SGGTS. The reliability of the model was verified by studying a practical example.
• By deploying the modular compressor, the idleness and waste of the modular compressor are avoided, the utilization rate of the modular compressor is greatly improved, and the production investment cost is also reduced to a certain extent, improving the economic benefits of the shale gas field.
• Under a multi-period condition, it requires fewer compressors and types, and the change in pressurizing position and rearrangement frequency of modular compressors are more stable, as is the change in compressor operating power. In addition, the investment cost of shale gas field production can be reduced. After solving under multi-period conditions, the cumulative cost is reduced by 17.19%.
• Compared with the previous work, our model has made great progress. In future work, we will consider the related uncertainty factors to establish an optimization model for the pressurization and rearrangement of the SGGTS to produce a pressurization scheme with more application values.