Production Performance Analysis and Fracture Volume Parameter Inversion of Deep Coalbed Methane Wells

Abstract

Deep coalbed methane development faces technical challenges, such as high in situ stress and low permeability. The dynamic evolution of fractures after hydraulic fracturing and the flowback mechanism are crucial for optimizing productivity. This paper focuses on the inversion of post-fracturing fracture volume parameters and dynamic analysis of the flowback in deep coalbed methane wells, with 89 vertical wells in the eastern margin of the Ordos Basin as the research objects, conducting systematic studies. Firstly, through the analysis of the double-logarithmic curve of normalized pressure and material balance time, the quantitative inversion of the volume of propped fractures and unpropped secondary fractures was realized. Using Pearson correlation coefficients to screen characteristic parameters, four machine learning models (Ridge Regression, Decision Tree, Random Forest, and AdaBoost) were constructed for fracture volume inversion prediction. The results show that the Random Forest model performed the best, with a test set R² of 0.86 and good generalization performance, so it was selected as the final prediction model. With the help of the SHAP model to analyze the influence of each characteristic parameter, it was found that the total fluid volume into the well, proppant intensity, minimum horizontal in situ stress, and elastic modulus were the main driving factors, all of which had threshold effects and exerted non-linear influences on fracture volume. The interaction of multiple parameters was explored by the Partial Dependence Plot (PDP) method, revealing the synergistic mechanism of geological and engineering parameters. For example, a high elastic modulus can enhance the promoting effect of fluid volume into the well and proppant intensity. There is a critical threshold of 2600 m³ in the interaction between the total fluid volume into the well and the minimum horizontal in situ stress. These findings provide a theoretical basis and technical support for optimizing fracturing operation parameters and efficient development of deep coalbed methane.

1. Introduction

Deep coalbed methane resources are abundant, with a resource area of 3868.67 square kilometers and a resource abundance of 150–250 million cubic meters per square kilometer. The main 8 # and 9 # coal seams have a thickness of 8–14 m and a gas content of 8–27 cubic meters, with a total resource of over 920 billion cubic meters. It is expected that 359.4 billion cubic meters will be newly discovered in the future, accounting for 55% of the total newly discovered onshore reserves. Deep coalbed methane generally refers to coalbed methane resources with a burial depth greater than 1500 m. As a crucial component of unconventional natural gas, it boasts enormous development potential. Compared with shallow coalbed methane, deep coalbed methane reservoirs are characterized by a higher formation pressure, a greater adsorbed gas content, and a more complex stress environment, which pose more significant technical challenges to their development. Hydraulic fracturing is currently a key technical means to enhance the production of deep coalbed methane. Through the propagation and propping of artificial fractures, it can effectively improve the permeability of reservoirs, thereby enhancing the efficiency of gas desorption and seepage. However, the dynamic evolution laws of fractures after fracturing in deep coalbed methane, the propping effect, and the fluid–fracture interaction mechanism during the flowback process remain unclear, requiring in-depth research. In particular, the flowback stage, as a critical link connecting fracturing operations and production, contains key information, such as the conductivity of fracture networks and the distribution state of proppants in its dynamic data. By establishing a quantitative relationship model between flowback parameters and fracture volume, rapid inversion and evaluation of post-fracturing fractures can be realized.

The post-fracturing flowback process serves as a critical step in evaluating fracturing effectiveness and optimizing production systems. Flowback data, including the fluid return volume, gas production rate, and pressure variations, can reflect the closure dynamics of fracture networks, the distribution state of proppants, and the degree of reservoir damage. Recent advancements in flowback data inversion methods, which couple fluid flow equations, fracture mechanical closure models, and proppant transport equations, enable the quantitative analysis of fracture volume parameters. Through the inversion analysis of flowback data, fracture volume parameters can be indirectly obtained, thereby facilitating the optimization of fracturing designs and production strategies. Therefore, research on flowback inversion methods for fracture volume parameters in deep coalbed methane reservoirs holds significant importance for enhancing fracturing efficiency and reducing development costs.

In terms of research on flowback inversion methods, existing technologies can be primarily categorized into three types: first, the dynamic analysis method based on the flowback volume–time curve, which inverts the fracture volume through curve fitting; second, the tracer interpretation method based on the chemical composition of flowback fluid, which calculates the effective fracture volume using tracer breakthrough curves; and third, the multi-parameter inversion method coupled with geomechanics, which integrates multi-source data such as pressure, flow rate, and microseismic signals to establish inversion models. Among these, the dynamic analysis method, characterized by convenient data acquisition and high computational efficiency, possesses significant advantages in field applications and, thus, serves as the core research method in this paper.

In recent years, scholars, both domestically and internationally, have conducted extensive research on the optimization of hydraulic fracturing in deep coalbed methane reservoirs. In terms of fracturing process optimization, Han et al. [1] pointed out that hydraulic fracturing in deep coalbeds must consider the mechanical properties of coal rock and the influence of natural fractures and recommended the use of high-displacement, low-proppant-concentration fracturing techniques to enhance fracture complexity. Through numerical simulations, Zhang et al. [2] employed numerical simulation methodologies integrating fracture propagation modeling and production performance analysis to emphasize the role of key parameters in platform well fracturing and their linkage to objective functions. Qiao et al. [3] investigated fracture propagation mechanisms in deep coalbed methane (CBM) reservoirs, systematically analyzing the influence of the bedding density, stress ratio, rock friction coefficient, and fracturing parameters on vertical fracture development and post-fracturing productivity. Zhang et al. [4] conducted fracture prediction in a deep CBM reservoir in the Ordos Basin based on Gray-Level Run-Length Matrix (GLRLM) texture attributes, providing critical insights for the efficient development and utilization of deep CBM resources. Wang et al. [5] explored macromolecular structures and nanoporosity through a series of laboratory experiments, revealing distinct responses of vitrinite and inertinite to thermal metamorphism and enhancing the understanding of differential nanopore structure evolution during coalification. Song et al. [6] analyzed various field test data from CBM wells and identified distinct geological characteristics at different depths: shallow coal reservoirs exhibit dominant horizontal in situ stress, whereas deep coal reservoirs are characterized by predominant vertical in situ stress. Ezulike et al. [7] proposed the concept of dynamic relative permeability, the findings of which enable the integration of flowback data with established relative permeability curves to characterize the evolution of phase permeability for each fluid phase within fractures during the flowback process. Abbasi et al. [8] proposed a dynamic inversion method combining flowback volume and pressure data, which improved the timeliness of fracture parameter inversion. Flowback analysis is an important means of evaluating fracturing effectiveness, whose core lies in inferring fracture closure through flowback dynamics. Barree et al. [9] established a relationship model between flowback volume and fracture closure, noting that the early flowback rate can reflect fracture conductivity. Abbasi et al. [8] proposed using flowback data to invert the proppant distribution but failed to consider the high stress sensitivity of unconventional reservoir. Xu et al. [10] analyzed the influencing factors of proppant flowback during flowback and put forward methods to optimize the flowback rate. Zhang et al. [11], considering the characteristics of deep coalbed methane, established a flowback model incorporating stress sensitivity effects, which improved inversion accuracy. Despite the certain progress achieved in existing research, in light of the unique characteristics of deep coalbed methane, there remain three aspects of inadequacies: first, some flowback inversion models fail to fully account for the impact of unsupported fractures within reservoir fractures after fracturing; second, due to the complexity and large quantity of on-site construction parameters as well as formation reservoir parameters, the correlation between each factor and post-fracturing fractures is not sufficiently clear.

In summary, against the backdrop and requirements of deep coalbed methane development—such as its significant strategic significance, heavy production increment tasks, rapid development pace, numerous fracturing operations, and preliminary stage of technical exploration—it is urgent to conduct research on post-fracturing evaluation methods for deep coal seams. This research aims to systematically evaluate the implementation effect of the fracturing development stage of deep coalbed methane in a timely manner. By inverting post-fracturing fractures using flowback stage data, the fracture state and reservoir stimulation degree can be determined. Furthermore, data acquisition of the post-fracturing fracture volume can be realized by introducing surrogate models, and machine learning can be applied to analyze the correlation between various characteristic parameters and post-fracturing fractures based on on-site construction and geological parameters. According to the obtained results combined with on-site data, the initial flowback or fracturing strategies can be adjusted promptly, which holds important guiding significance for promoting the efficient development of deep coalbed methane blocks.

2. Production Performance Analysis

2.1. Division of Production and Development Stages

Distinct from medium and shallow coalbed methane, deep coalbed methane is characterized by unique occurrence properties of “high gas content, high saturation, and high free gas proportion”, along with low porosity and low permeability, thus exhibiting distinctive production performance. As of the end of May 2025, the cumulative production of deep coal seams in the Shenfu Block has exceeded 580 million cubic meters, with an average stable gas production of 12,000 cubic meters per day for horizontal wells and 3200 cubic meters per day for vertical/directional wells. According to the normalized production curves of multiple typical vertical wells in Linxing-Shengfu deep coalbed methane at the eastern margin of the Ordos Basin (Figure 1), its production phase can be categorized into four stages: the flowback period, production regulation period, stable production period, and decline period [12].

Stage 1: The Flowback Period. As the initial phase subsequent to hydraulic fracturing operations, the core mission of the flowback period is to expel the fracturing fluid injected during stimulation, thereby creating space for gas desorption and seepage. This period typically lasts 1–3 months, with its specific duration contingent upon the fracturing fluid volume, reservoir permeability, and fracture conductivity. In the early stage, the produced fluids are dominated by fracturing fluid, which may carry minor amounts of coal fines and proppant residues. During this phase, the bottomhole flowing pressure remains at a relatively high level, while gas production remains extremely low as the desorption conditions have not been met. As the fluid is continuously discharged, the pressure within the fracture network is gradually relieved. By the end of this period, the liquid production reaches its peak, after which it gradually decreases due to the intensified tendency of fracture closure and the diminished fluid supply capacity of the reservoir.

Stage 2: The Production Regulation Period. The production regulation period serves as a critical transitional phase from “liquid-dominated production” to “gas-dominated production,” typically lasting 3 to 12 months, marking the entry of deep coalbed methane development into a substantial gas production stage. As a large volume of fracturing fluid is expelled, the bottomhole flowing pressure drops below the critical desorption pressure, enabling free gas in the reservoir to be the first to be produced through the fracture network, with the initial daily production of a single well reaching thousands of cubic meters. The core characteristic of this stage is characterized by “one decrease and one increase”. The liquid production declines continuously, with the coal fines content in the liquid decreasing; in contrast, the gas production rises gradually, accompanied by an increase in methane purity in the gas composition. Meanwhile, the bottomhole flowing pressure exhibits a stepwise decreasing trend, but the rate of decline slows down significantly compared with the flowback period. This phenomenon occurs because gas production gradually replaces liquid discharge as the primary mode of pressure relief.

Stage 3: The Stable Production Period. The stable production period constitutes the core phase for achieving economically viable development of deep coalbed methane, where its duration and production level directly determine the economic efficiency of a single well. During this stage, the produced gas is dominated by adsorbed gas, accounting for over 60% of the total gas production. This is attributed to the continuous decline in bottomhole flowing pressure, which drives the adsorbed gas within the coal matrix to desorb under the action of pressure difference, subsequently diffusing into the fracture network and seeping into the wellbore. A prominent feature of this stage is the relatively stable gas production, with high and steady daily output per well, while the liquid production drops to a low level. Additionally, the liquid properties gradually transition from fracturing flowback fluid to formation water. Despite the stable gas production, the bottomhole flowing pressure continues to decrease slowly. This is due to the continuous depletion of reservoir energy and the gradual closure of fractures under high in situ stress, which leads to a progressive attenuation of fracture conductivity.

Stage 4: The Decline Period. The decline period represents the late stage of deep coalbed methane development, typically lasting 5 to 10 years, and is characterized by a prolonged and gradual decline in gas production with a progressively decelerating decline rate. At this stage, the formation pressure has dropped to a relatively low level, significantly slowing down the desorption rate of adsorbed gas. Meanwhile, the fracture network experiences a substantial reduction in conductivity due to long-term closure and proppant crushing. Liquid production further decreases, even leading to intermittent flow stoppages, while the gas–liquid ratio reaches its peak before slowly declining. The decline rate of gas production gradually decreases from an initial 15–20% per annum to 5–10% per annum. This phenomenon arises because the remaining gas is primarily sourced from low-permeability regions or deep within the coal matrix, where migration paths are elongated and resistance is increased.

In the development of deep coalbed methane, flowback stands as a pivotal technological process, referring to the artificial intervention-driven removal of fluids such as fracturing fluid and formation water from coalbed methane wells. This process aims to reduce wellbore pressure, restore reservoir permeability, and thereby enhance gas desorption efficiency. A scientifically sound flowback regime must balance the fluid discharge rate and reservoir damage, while flowback data analysis serves as the critical link connecting engineering practice and reservoir response. Through data-driven, refined management and control, the efficiency of deep coalbed methane development can be significantly improved, development risks mitigated, and a scientific basis provided for the efficient utilization of complex reservoirs, making it a core technical support for achieving the goal of “fewer wells with higher production.” This paper focuses on data collection and analysis during the flowback stage to invert the post-fracturing fracture volume of vertical wells in deep coalbed methane reservoirs. Research on post-fracturing fracture volume can provide a theoretical foundation for decision making regarding fracturing schemes and optimization of flowback regimes for vertical deep coalbed methane wells. Additionally, it offers a certain reference value for evaluating reservoir stimulation effects in deep coalbed methane reservoirs.

2.2. Analysis of Characteristics in Flowback Stage

In the production process of petroleum engineering, the flowback stage is a critical link subsequent to fracturing operations, characterized by multi-dimensional features. During the initial flowback phase, a large volume of residual fracturing fluid is discharged rapidly, with high flow rates but minimal oil and gas production. The fracturing fluid flowback efficiency is significantly influenced by reservoir permeability, typically below 30% for low-permeability reservoirs and potentially exceeding 50% for high-permeability ones. Entering the main flowback period, fracturing fluid and formation fluids are discharged in a mixed state, with the flow rate gradually declining steadily. A substantial volume of gas begins to be produced, accompanied by significant changes in fluid composition: the water quality transitions from turbid and viscous to stable, while the salinity of the formation water and gas content continue to increase. In the stable flowback period, the produced fluids are dominated by formation water, oil, and gas, reflecting the original productivity of the reservoir.

During the initial flowback stage of coalbed methane (CBM) well production, the aqueous phase is the dominant seepage phase. Therefore, by processing the data from the flowback stage of CBM, reservoir parameters can be preliminarily inverted using aqueous-phase Rate Transient Analysis (RTA). This inversion of CBM reservoir and fracture parameters can guide fracturing operations in similar reservoirs, and the inverted data can also provide data support for the establishment of numerical models and the optimization of stimulation measures. In the process of aqueous-phase flowback, the flow stages of aqueous-phase production can be divided based on the material balance time and the slope of the logarithmic curve of normalized pressure. The calculation methods for normalized pressure and material balance time are as follows:

Material balance time:

$${\mathrm{t}}_{\mathrm{ca}}={\displaystyle \frac{{\displaystyle \sum _{t=1}^n{q}_{w,n}}}{q_{w,n}}}$$

(1)

Normalized pressure:

$$p_n={\displaystyle \frac{p_i-p_{wf}}{q_{w,n}}}$$

(2)

wherein q_w,n is the daily water production (m³/d); p_i is the initial bottomhole flowing pressure at the start of flowback (MPa); and p_wf is the current bottomhole flowing pressure at the flowback moment (MPa).

A characteristic curve is plotted in the log–log coordinate system of normalized pressure and material balance time, as shown in Figure 2. Based on the slope, the fluid flow regime can be relatively easily determined. The seepage of the aqueous phase from the stimulated zone to the artificial fractures is divided into three typical flow regimes: a straight line with a slope of 1/4 represents the fracture-stimulated zone bilinear flow; a straight line with a slope of 1/2 corresponds to the stimulated zone linear flow; and a straight line with a slope of 1 indicates the stimulated zone boundary-dominated flow. The slope characteristics of these typical flow regimes are all manifested in the log–log coordinate system of normalized pressure and material balance time.

3. Dynamic Analysis of Post-Fracturing Flowback in Deep Coalbed Methane Wells

3.1. Establishment of Mathematical Models

Complex fractures serve as pathways for fracturing fluid flowback and coalbed methane production. Therefore, to study the flowback characteristics of coalbed methane, it is first necessary to clarify the distribution of fracturing fluid within complex fractures. As illustrated in Figure 3, after the completion of hydraulic fracturing, the complex fracture structure consists of hydraulically propped main fractures and unsupported secondary fractures. Among these, some secondary fractures are connected to the propped fractures and participate in the flowback of fracturing fluid, while another portion of secondary fractures, due to excessive closure at their locations, lose connectivity with the propped fractures, resulting in the retention of part of the fracturing fluid.

3.2. Model Assumptions

The effective pore volume of fractures involved in flowback mainly includes hydraulically propped fractures and connected, unsupported secondary fractures. Therefore, based on the wellhead flowback data, the hydraulic fractures in the entire well section are regarded as a whole. For the fracture storage stage during flowback, the entire effective fracture system participating in flowback is assumed to be a “reservoir model” [13], with other corresponding assumptions as follows:

(1)

At the moment hydraulic fracturing is completed, the entire fracture system is filled with fracturing fluid;

(2)

The wellbore storage effect is neglected;

(3)

It is assumed that the fluid is slightly compressible, with consideration given to the closure effects of both propped fractures and unsupported fractures.

3.3. Mathematical Model for Predicting Effective Pore Volume of Fractures

According to the material balance equation, the volume of fluid discharged during the initial surface flowback is equal to the volume reduction within the “reservoir,” as shown in Equations (1) and (2).

$$p_{fi}-p_f={\displaystyle \frac{Q_w{B}_w}{(V_{hf}{C}_{t-hf}+V_{sf}{C}_{t-sf})B_{wi}}}$$

(3)

$$C_{t-hf}=C_{hf}+C_w$$

(4)

$$C_{t-sf}=C_{sf}+C_w$$

(5)

wherein P_f is the average pressure in fractures (MPa); Q_w is the cumulative surface fluid production (m3); B_w is the fluid formation volume factor (m³/m³); V_hf is the effective pore volume of propped fractures (m³); C_t−hf is the total compressibility of propped fractures (MPa⁻¹); V_sf is the effective pore volume of unsupported secondary fractures (m³); C_t−sf is the total compressibility of unsupported secondary fractures (MPa⁻¹); B_wi is the initial fluid formation volume factor (m³/m³); C_hf is the compressibility of propped fractures (MPa⁻¹); C_w is the fluid compressibility (MPa⁻¹); and C_sf is the compressibility of unsupported secondary fractures (MPa⁻¹).

Based on the preceding analysis, the log–log curve of normalized pressure (RNP) and material balance time (t_MB) is employed herein to identify the fracture storage stage [14]. The calculation formulas for RNP and t_MB are given in Equations (6) and (7).

$$RN{P}_w={\displaystyle \frac{p_{fi}-p_{wf}}{q_w}}$$

(6)

$$t_{MB}={\displaystyle \frac{Q_w}{q_w}}$$

(7)

wherein w is the aqueous phase; p_fi is the initial average pressure in fractures (MPa); p_wf is the bottomhole flowing pressure (MPa); q_w is the aqueous phase production rate (m³/d); t_MB is the material balance time (d); and Q_w is the cumulative aqueous phase production (m³). Since the actual pressure in fractures is difficult to accurately obtain, it is assumed herein that the average pressure in fractures during the initial flowback stage is approximately equal to the bottomhole flowing pressure, so p_fi is equal to the initial bottomhole flowing pressure p_wfi at the well opening.

When the flowback enters the fracture storage stage, it exhibits the characteristic of pseudo-steady-state flow with a slope of 1 [15]. Based on the assumptions of the closed reservoir model, the liquid production index within the fractures is approximately constant at this point, as shown in Equation (8).

$$p_f-p_{fi}={\displaystyle \frac{q_w}{J_w}}$$

(8)

wherein q_w is the surface fluid production rate (m³/d); J_w is the liquid production index (m³/(MPa·d)).

Adding the left and right sides of Equations (7) and (8), respectively, yields

$$p_{fi}-p_{wf}={\displaystyle \frac{Q_w{B}_w}{(V_{hf}{C}_{t-hf}+V_{sf}{C}_{t-sf})B_{wi}}}+{\displaystyle \frac{q_w}{J_w}}$$

(9)

After rearrangement, the following is obtained:

$${\displaystyle \frac{p_{fi}-p_{wf}}{q_w}}={\displaystyle \frac{B_w}{(V_{hf}{C}_{t-hf}+V_{sf}{C}_{t-sf})B_{wi}}}·{\displaystyle \frac{Q_w}{q_w}}+{\displaystyle \frac{1}{J_w}}$$

(10)

Substituting Equations (8) and (9) into Equation (10) yields

$$RN{P}_w={\displaystyle \frac{Bw}{(V_{hf}{C}_{t-hf}+V_{sf}{C}_{t-sf})B_{wi}}}{t}_{MB}+{\displaystyle \frac{1}{Jw}}$$

(11)

Let ζ be the slope of RNP_w versus t_MB in the Cartesian coordinate system, and b be its intercept on the vertical axis; then, Equation (11) can be rearranged as follows:

$$RN{P}_w=\xi ·t_{MB}+\mathrm{b}$$

(12)

$$\xi ={\displaystyle \frac{B\mathrm{w}}{(V_{hf}{C}_{t-hf}+V_{sf}{C}_{t-sf})B_{wi}}}$$

(13)

$$b={\displaystyle \frac{1}{J_w}}$$

(14)

By collecting field flowback pressure and flow rate data, a series of values for RNP_w and t_MB can be calculated using Equations (8) and (9). The fracture storage stage is identified from the log–log curve of RNP_w versus t_MB, and, subsequently, the slope value ζ of the fracture storage stage is read from its Cartesian coordinate curve. Meanwhile, since the variation in the water volume factor with pressure is minimal, it is assumed herein that B_wi = B_w. Therefore, Equation (14) can be rearranged as follows:

$$V_{hf}{C}_{t-hf}+V_{sf}{C}_{t-sf}={\displaystyle \frac{1}{\xi }}$$

(15)

Thus, the calculation formula for the effective fracture pore volume is derived, i.e., Equation (15), which contains four unknowns: V_hf, C_t−hf, V_sf, and C_t−sf. Furthermore, the water compressibility C_w can be obtained from field test reports. The fracture compressibility can be read from the chart proposed by Aguilera (1999) [16].

MINER refers to the volume fraction of mineral fillings in the fracture pore space. Williams-Kovacs et al. [17] and Fu et al. [18] regarded the filling minerals in the Aguilera model as proppants in fractures, and a similar treatment is adopted in this paper. It is assumed that the sand placement pattern in propped fractures conforms to the particle packing model, with the porosity of propped fractures being approximately 50% and that of unsupported fractures 100%; thus, the corresponding MINER values are set to 50% and 0%, respectively. After determining the MINER values in fractures, the bottomhole flowing pressure corresponding to the fracture storage stage is read from the log–log curve of RNPw and t_MB. Assuming P_f ≈ P_wf [18], the net closure pressure is calculated, and finally, the corresponding fracture compressibility values C_t−hf and C_t−sf are read from the chart. The relevant calculation formulas are as follows:

$${\mathrm{P}}_{\mathrm{n}}{=\mathrm{P}}_{\mathrm{c}}{-\mathrm{P}}_{\mathrm{wf}}$$

(16)

$$MINER=1-{\mathsf{\Phi }}_f$$

(17)

wherein P_n is the net closure pressure in fractures (MPa); P_c is the formation closure stress (MPa); and Φ_f is the fracture porosity. Substituting the fluid and compressibility coefficients into Equations (16) and (17) yields the values of C_t−hf and C_t−sf.

In the initial stage of flowback after well fracturing, the pressure within fractures remains near the closure pressure. At this point, the fracture aperture is relatively large, so the apparent volume of the total proppant injected into the well can be considered as the total volume of propped fractures at this stage. Furthermore, the effective pore volume of propped fractures formed after CBM well fracturing can be estimated, as shown in the following formula:

$${\mathrm{V}}_{\mathrm{hf}}{=\mathrm{V}}_{\mathrm{p}}$$

(18)

wherein Vp is the total volume of proppant injected into the well (apparent volume) (m³). Substituting the values of ξ, V_hf, C_t−hf, and C_t−sf into Equation (18) in sequence allows the calculation of the effective pore volume of unsupported fractures formed after coalbed methane well fracturing, as follows:

$$V_{sf}=({\displaystyle \frac{1}{\xi }}-V_{hf}{C}_{t-hf})/C_{t-sf}$$

(19)

At this point, the model for calculating the effective fracture pore volume has been established, and the calculation workflow is shown in Figure 4.

Based on the flow chart in Figure 4, the effective pore volume of fractures in deep coalbed methane vertical wells is calculated. The ratio of the total effective fracture volume of a single well to the total volume of fracturing fluid injected into the well is the fracturing fluid efficiency [19].

$$\eta ={\displaystyle \frac{V_{\mathrm{hf}}+V_{\mathrm{sf}}}{V_{\mathrm{inj}}}}\times 100\%$$

(20)

wherein is the η-fracturing fluid efficiency (%); Vinj is the total volume of fracturing fluid injected into the well (m3).

3.4. Fracture Complexity Index

The effective fracture complexity is a key parameter for describing the effect of volume fracturing. To characterize the complexity of effective fractures that contribute to fluid production during the flowback process, the fracture complexity index (FCI) proposed by Ghanbari et al. [20] is introduced for quantitative interpretation, which refers to the percentage of the effective secondary fracture volume Vsf in the total effective fracture volume (V_hf + Vsf).

A higher FCI value indicates a more complex network of effective fractures. The FCI can be calculated using Equation (21).

$$FCI={\displaystyle \frac{V_{\mathrm{sf}}}{{\mathrm{V}}_{\mathrm{hf}}+V_{\mathrm{sf}}}}\times 100\%$$

(21)

wherein FCI is the fracture complexity index (%).

4. Fracture Inversion Prediction Model and Analysis of Influencing Factors

4.1. Acquisition and Preprocessing of Datasets

In the field of deep coalbed methane development, fracturing stimulation is a core technical means to enhance single-well productivity, and accurately analyzing the influencing factors of post-fracturing fracture volume is crucial for optimizing fracturing design and improving development efficiency. To systematically study the key controlling factors of post-fracturing fracture volume in deep coalbed methane vertical wells, this research adopted the Pearson correlation coefficient to quantitatively evaluate the impact degree of each parameter on the fracturing stimulation effect and further identify the variables that significantly affect fracture volume.

This study took 89 vertical development wells in a block on the eastern margin of the Ordos Basin as the research objects. The coal seams in this block are characterized by deep burial, low permeability, and complex fracturing processes, making it urgent to clarify the influencing factors of fracturing effectiveness through scientific evaluation. First, as shown in

Table 1. Data table of characteristic parameters.

	Characteristic Parameter	Abbreviation	Unit	Range
Geological parameters	Minimum horizontal in situ stress	Sh min	MPa	23.1~47.7
	Young’s modulus	E	GPa	1~19.9
	Poisson’s ratio	ν		0.24~0.44
Engineering parameters	Actual proppant volume	Vp	m3	100.1~501.4
	Proppant intensity	Pi	m3/m	6.9~40.1
	Pre-pad fluid proportion	PFP	m3/m3	0.02~0.24
	Fracturing displacement	Pr	m3/min	6.7~22
	Total fluid injected into well	Vinj	m3	1022.6~4213.2
Evaluation indicators	Propped fracture volume	Vhf	m3	100.1~501.4
	Unpropped fracture volume	Vsf	m3	79.6~646.1
	Total fracture volume	VF	m3	179.7~1146.3
	Fracture complexity index	FCI	%	0.24~0.72

, the fracturing operation and geological parameters of each well were systematically collected and sorted out, including minimum horizontal in situ stress (Sh min), Young’s modulus (E), Poisson’s ratio, fracturing fluid dosage (Vinj), proppant concentration (PH), operation displacement (Q), proppant volume, fluid volume, etc. Meanwhile, post-fracturing flowback data were recorded in detail, such as key indicators like flowback rate, flowback fluid volume, and flowback time. Based on the above data, advanced numerical simulation and inversion technologies were applied to accurately calculate the post-fracturing fracture volume corresponding to each well.

To provide a scientific basis for subsequent optimization of the coalbed methane fracturing design scheme in this block, improvement of fracturing operation effects, and promotion of the efficient development of deep coalbed methane, after obtaining the basic data, this study used the Pearson correlation coefficient to conduct a correlation analysis between the fracture volume and each characteristic parameter, and preliminarily screened out the parameters that potentially affect the fracture volume.

Furthermore, combined with machine learning methods, a deep learning model was constructed for feature learning of the data. Through model training and optimization, the complex nonlinear relationships between parameters were deeply explored. Finally, the key factors that significantly affect the post-fracturing fracture volume of deep coalbed methane vertical wells were extracted, as shown in Figure 5.

The above Figure 5 shows the correlation matrix between the total post-fracturing fracture volume and characteristic parameters. According to correlation theory, it is generally considered that a correlation coefficient between 0 and 0.3 indicates a weak correlation, between 0.3 and 0.6 indicates a moderate correlation, and between 0.6 and 1 indicates a strong correlation.

Based on the preliminary analysis using the Pearson correlation coefficient, the degree of association between each characteristic parameter and the total fracture volume in the fracturing of vertical wells for deep coalbed methane in the eastern margin of the Ordos Basin was clarified. This study found that the actual fluid volume, actual proppant volume, and proppant intensity showed a strong correlation with the total fracture volume (absolute value of the correlation coefficient > 0.6), indicating that these construction parameters play a dominant role in the scale of fracture propagation; the construction displacement had a moderate correlation with the total fracture volume (0.3 < absolute value of the correlation coefficient ≤ 0.6), and its degree of influence was between weak and strong, while parameters such as Young’s modulus, pre-pad fluid proportion, minimum horizontal in situ stress, and Poisson’s ratio showed a weak correlation (absolute value of the correlation coefficient ≤ 0.3), meaning that their direct impact on the fracture volume was relatively limited.

To further analyze the action mechanisms of different types of parameters, the research team systematically divided the characteristic parameters into two major categories: geological parameters and construction parameters. Among them, geological parameters included indicators reflecting the mechanical properties of the reservoir, such as Young’s modulus of the coal seam, Poisson’s ratio, and minimum horizontal in situ stress. These parameters are determined by stratum sedimentation and tectonic evolution and directly affect the fracture initiation and propagation capabilities. Construction parameters included fracturing engineering parameters such as the actual fluid volume, actual proppant volume, proppant intensity, construction displacement, and pre-pad fluid proportion, and their values can be adjusted through process optimization.

Using the pairplot function of the Seaborn library in Python 3.6, correlation visualization analyses were carried out separately for geological parameters, construction parameters, and the total post-fracturing fracture volume. By drawing a matrix of scatter plots between each pair of parameters, the linear and nonlinear relationships between each parameter were intuitively presented. As shown in Figure 6, within the construction parameter group, the actual fluid volume and the actual proppant volume had a significant synergistic effect, and their combined action had a multiplier effect on the growth of the fracture volume; in the geological parameter group, there was a negative coupling relationship between the minimum horizontal in situ stress and Young’s modulus. In a high-stress environment, the rigidity of the coal seam increases, resulting in restricted fracture propagation. The pairplot graphs of the two types of parameters not only quantify the influence of individual parameters but also reveal the synergistic action mechanisms of parameter combinations, providing multi-dimensional data support for the construction of a prediction model for the fracture volume of deep coalbed methane fracturing.

The characteristic parameters were unevenly distributed, with significant differences in magnitude. Therefore, to eliminate the influence of dimensions, accelerate the convergence speed of the algorithm, and improve the accuracy of the model, the data were normalized according to Formula (22) before model training.

$$x={\displaystyle \frac{x_i-x_{\min }}{x_{\max }-x_{\min }}}$$

(22)

wherein $x$ is the normalized characteristic parameter data; $x_i$ represents the i-th sample data of the characteristic parameter; $x_{\max }$ is the maximum value of the characteristic parameter data; and $x_{\min }$ is the minimum value of the characteristic parameter data.

Considering that the input with multicollinearity among characteristic parameters is likely to cause overfitting in model learning and training, the Pearson correlation coefficient method was used to calculate the pairwise correlation between features. One of the features with strong correlation was removed to ensure the effectiveness of the selected parameters for predicting gas content. The calculation of the correlation coefficient is shown in Formula (23):

$$r=\frac{{\displaystyle \sum _{i=1}^n}(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{{\displaystyle \sum _{i=1}^n}{(x_i-\overline{x})}^2}\sqrt{{\displaystyle \sum _{i=1}^n}{(y_i-\overline{y})}^2}}$$

(23)

wherein $r$ represents the correlation coefficient; ${\mathrm{x}}_{\mathrm{i}}$ and ${\mathrm{y}}_i$ denote the i-th sample data of the characteristic parameter; $\overline{\mathrm{x}}$ and $\overline{\mathrm{y}}$ represent the mean value of the first characteristic parameter; and n is the number of samples.

The correlation coefficient r is defined as the ratio of the covariance of two features to the product of their standard deviations, with its value ranging from −1 to 1. A value of 1 indicates a perfect positive correlation, while −1 indicates a perfect negative correlation, and 0 means no linear correlation. Different geological–engineering parameters have certain influences on explaining fracture volume, and there are also interactions between various characteristic parameters. To more intuitively display the correlation between characteristic parameters and inverted fracture volume, a heatmap of correlation coefficients was plotted, as shown in Figure 5. The analysis results show that the maximum correlation coefficient was 0.99, and the minimum was −0.47, indicating that there was a significant redundancy phenomenon among the selected characteristic parameters. Among them, the actual proppant volume had a strong correlation with construction displacement, proppant intensity, and propped fracture volume. In order to retain more parameters to capture potential relationships, after removing the actual proppant volume, seven remaining geological–engineering parameters were obtained. There was a strong correlation between the four evaluation indicators: propped fracture volume, unpropped fracture volume, total fracture volume, and fracture complexity index. In this paper, the total fracture volume was selected as an example of the prediction target.

In addition to the total post-fracturing fracture volume, the fracture complexity index (FCI) is also an important parameter for evaluating fracturing effectiveness. By quantifying the proportion of fracture volume that actually contributes to fluid production during the flowback process, the FCI breaks through the limitation of traditionally only focusing on the total fracture volume and more accurately reflects the connectivity efficiency between the fracture network and reservoir fluids. To explore the key controlling factors for improving the FCI, this study collected and sorted out fracturing operation parameters and geological parameters of 89 vertical development wells in a block on the eastern margin of the Ordos Basin, including fracturing fluid dosage, proppant concentration, operation displacement, minimum horizontal in situ stress, etc. As shown in Figure 7, the fracture complexity was correlated with pumping rate, actual proppant volume, proppant-carrying ratio, and minimum horizontal in situ stress of the reservoir, among which the pumping rate and actual proppant volume had a relatively high correlation with fracture complexity.

4.2. Dataset Division and Evaluation Indicators

The prediction performance of a machine learning model on unknown data points is primarily influenced by its generalization ability. Therefore, the model training process must include a systematic evaluation of generalization performance. The key step is to reasonably divide the original dataset into a training set and a test set to ensure that the knowledge learned by the model from the training set can be effectively transferred to unseen data. The test set must be strictly kept independent and not involved in any training process to avoid overfitting. The generalization ability of a model is usually quantified by its prediction accuracy on the test set. The dataset was allocated in an 8:2 ratio, where 80% of the data was used to train the model, and the remaining 20% was used to validate the generalization performance of the model. To ensure consistency in the distribution of geological or engineering parameters, the train_test_split () function from sklearn and the model_selection module of Python were used.

To evaluate the performance of the model in predicting fracture volume, it was necessary to assess the model’s predictive capability. In this study, four evaluation metrics were selected: mean squared error (MSE), root-mean-squared error (RMSE), mean absolute error (MAE), and coefficient of determination (R²). These metrics comprehensively evaluated the model’s ability to predict fracture volume. Among them, MSE, RMSE, and MAE were used to examine the model’s prediction accuracy, with smaller values indicating better performance. R² was used to examine the goodness of fit of the model, and a value closer to 1 indicated a better fit. The specific calculation formulas are as follows:

$$\mathrm{MSE}={\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{(y_i-{\widehat{y}}_i)}^2$$

(24)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{(y_i-{\widehat{y}}_i)}^2}$$

(25)

$$\mathrm{MAE}={\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\left|y_i-{\widehat{y}}_i\right|$$

(26)

$${\mathrm{R}}^2=1-{\displaystyle \frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{(y_i-{\widehat{y}}_i)}^2}{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{(y_i-\overline{\mathrm{y}})}^2}}$$

(27)

wherein n represents the number of coalbed methane content samples; ${\widehat{y}}_i$, $\widehat{y}$ both denote the predicted value of the i-th sample.

4.3. Optimized Model

Based on the seven geological–engineering characteristic parameters screened in the previous stage as input variables, regression prediction models were constructed using the Ridge Regression, Decision Tree, Random Forest (RF), and AdaBoost algorithms, respectively, to conduct research on fracture volume inversion prediction and explore the adaptability of different models to the prediction task. Figure 8 shows the performance of different inversion prediction models on the test set. The model effect was measured by the fitting degree between the predicted volume and the actual volume and R² (the coefficient of determination). The closer R² was to 1, the better the model’s fitting degree and prediction ability. As shown in

Table 2. Comparison of effects of different models on training set and test set.

Model	Train MSE	Train RMSE	Train MAE	Train R2	Test MSE	Test RMSE	Test MAE	Test R2
Random Forest	603.60	24.57	17.74	0.97	3193.39	56.51	42.87	0.86
AdaBoost	4283.98	65.45	58.01	0.80	5736.14	75.74	66.88	0.78
Decision Tree	585.06	24.19	14.26	0.97	6613.79	81.33	49.35	0.77
Ridge	9991.27	99.96	81.86	0.53	10,926.06	104.53	83.80	0.55

, the Random Forest model achieved an R² as high as 0.86, with data points closely clustered around the fitting line, demonstrating excellent fitting results. Benefiting from the integration of multiple Decision Trees and the random feature selection mechanism, it effectively reduced the risk of overfitting and enhanced the model’s ability to capture the overall data patterns. It exhibited excellent stability and accuracy in volume prediction, with predicted values for different actual volumes closer to the true values.

Random Forest (RF) is a bagging ensemble learning algorithm. A Decision Tree can be regarded as a collection of if-then rules: each possible path from the root node to the leaf node of the decision tree corresponds to a rule, and the judgment conditions of each rule are determined by the internal nodes on the path. The value of the node determines the output of each rule.

The main idea of RF is to build multiple decision trees in parallel on the basis of bootstrapping and introduce random feature selection in the training of decision trees. When constructing a single decision tree, the RF algorithm randomly splits a subset of the current feature set for each node of the decision tree and then selects the optimal splitting feature from this subset. This process ensures that the diversity of RF decision trees comes not only from random sampling but also from the random division of features. Therefore, by increasing the differences between decision trees, the generalization performance of the final ensemble model can be further improved.

Through a comprehensive comparison, the Random Forest regression model exhibited the highest R² value on the test set, with the optimal data-fitting effect. It more fully explored the mapping relationship between geological–engineering characteristic parameters and volume and demonstrated excellent generalization performance. Therefore, in the volume inversion prediction task of this study, the Random Forest model was selected as the final prediction model, providing more reliable algorithmic support for subsequent volume inversion based on geological–engineering characteristics.

4.4. Interpretability Analysis Based on SHAP (Shapley Additive Explanations)

The SHAP (Shapley Additive Explanations) model is used to interpret the predictions of machine learning models. Based on the Shapley value in game theory, it is employed to determine the contribution of each feature to the overall result of the game. The SHAP explanation value distribution and the ranking of average absolute importance of the fracture inversion model are shown in Figure 9. Each data point in the figure represents a fractured well, and the features are sorted from top to bottom according to their importance. The horizontal axis represents the SHAP value of the model, and the different colors of the points indicate the magnitude of the feature values: the closer to green, the larger the feature value; the closer to dark red, the smaller the feature value. A positive SHAP value indicates that the feature has a positive impact on production, while a negative SHAP value indicates that the feature has a negative impact on the final production. The combined RF-SHAP method was used to analyze the relative importance of each driving factor in the inversion of fracturing fracture volume parameters for deep coalbed methane wells, as well as the direction of action of these driving factors, as shown in Figure 4. The driving factors were in the following order: total fluid volume into the well, proppant intensity, minimum horizontal in situ stress, Young’s modulus, Poisson’s ratio, construction displacement, and pre-pad fluid proportion. The results show that the total fluid volume into the well was the most influential factor on the total fracture volume, while the pre-pad fluid proportion had the smallest impact.

When considering the driving directions, the dominant factors were the total fluid volume injected into the well, proppant intensity, minimum horizontal in situ stress, and elastic modulus. However, these factors did not simply promote or inhibit fracture generation in a single way. Generally speaking, there were varying degrees of autocorrelation between the driving factors of the total fracture volume, which complicates the individual analysis of these factors. The SHAP method was a more effective approach for isolating the influence of specific factors and identifying the trends in total fracture volume changes caused by these factors, as shown in Figure 5. This study focused on four key driving factors that have a significant impact on the interpretation of total fracture volume and conducted a detailed analysis of their influences. Overall, the response of the total fracture volume to these driving factors presented a non-linear pattern. For example, theoretically, during the fracturing process, the injection of fracturing fluid is beneficial for fracture initiation and the extension of the main fracture. However, as can be seen in Figure 10, the influence of the total fluid volume injected into the well on the total fracture volume had a significant threshold effect, with the threshold being approximately 2300 m³. When the total fluid volume injected into the well was greater than 2300 m³, it made a positive contribution to the prediction of the fracture inversion model, indicating that the total fluid volume injected into the well promoted the increase in the total fracture volume. Nevertheless, when the total fluid volume injected into the well was less than 2300 m³, it had a negative impact, thereby inhibiting the total fracture volume. The proppant intensity, minimum horizontal in situ stress, and elastic modulus also exhibited obvious threshold effects.

4.5. Factor Interaction Analysis

To clarify the mechanism by which the interaction of multiple parameters influences the total fracture volume, the Partial Dependence Plot (PDP) method was employed to construct an interaction analysis framework dominated by geological parameters (minimum horizontal in situ stress and elastic modulus) and engineering parameters (total fluid volume injected into the well and proppant intensity) (Figure 11). The PDP can be realized by importing sklearn. inspection. partial_dependence in the Python modules. This framework systematically explores the laws governing the effect of parameter coupling on the total fracture volume. The results show that when the elastic modulus interacted with the total fluid volume injected into the well and proppant intensity, the inhibitory effect on fracture volume weakened and the promoting effect strengthened in the region with larger values within the high elastic modulus interval. This is because rocks with a high elastic modulus exhibit significant brittle characteristics; fluid pressure transmission and proppant embedment are more likely to induce fracturing, and the energy input from engineering parameters and propping constraints can be efficiently converted into a driving force for fracture extension. A critical threshold of approximately 2600 m³ existed in the interaction between the total fluid volume injected into the well and the minimum horizontal in situ stress. When the interaction was below this threshold, the inhibition of fracture volume intensified, and the promoting effect diminished; when above this threshold, the inhibition weakened, and the promoting effect became prominent. Essentially, under low-fluid-volume conditions, the fluid energy is insufficient to overcome the constraint of in situ stress, hindering fracture propagation; a high fluid volume can provide sufficient energy to “offset” the inhibition of in situ stress, driving the continuous growth of the fracture volume.

In addition, when proppant intensity interacted with the minimum horizontal in situ stress and total fluid volume injected into the well, the adaptability of parameter combinations significantly affected the fracture volume: a high proppant intensity requires sufficient fluid volume to break through in situ stress limitations, ensuring that proppants can effectively expand fracture space; a low proppant intensity is relatively less sensitive to fluid volume and in situ stress, reflecting that the function of proppants in “maintaining fracture opening” needs to be dynamically balanced with “energy input (fluid volume) and stress constraint (in situ stress)”. Through the PDP-based analysis of multi-parameter interactions, this study accurately depicted the synergistic mechanism of geological–engineering parameters, providing a theoretical basis for optimizing fracturing operation parameters and facilitating improvements in reservoir stimulation volume and development efficiency.

In summary, the entire contribution of this work can be presented as follows: First, this paper estimated the effective fracture volume based on flowback data by integrating the analysis of the double-logarithmic curve of normalized pressure and material balance time, enabling the quantitative inversion of the volumes of post-fracturing propped fractures and unpropped secondary fractures. It overcame the limitations of traditional methods that rely on long-term production data or high-cost monitoring. Second, following data curation of geological, construction, and production metrics from fracturing wells, feature importance was ranked using Random Forest with the SHAP (Shapley Additive Explanations) model. Third, the multi-parameter interaction analysis based on the Partial Dependence Plot (PDP) was used to clarify the mechanism by which the interaction of multiple parameters influences the total fracture volume and provide a clear direction for the optimization of fracturing parameters. We added this in the revised manuscript.

Here, we can provide two possible suggestions for further research or improvements to the model. First, the source of data can be added by other methods. In the study, we used the flowback method to estimate the fracture parameters. Actually, in the whole-life production of a well, the fracturing data (pumping pressure) and well testing data can also be used to estimate the fracture parameter and supply the training samples. Second, several regression prediction models were constructed using the Ridge Regression, Decision Tree, Random Forest (RF), and AdaBoost algorithms to conduct research on fracture volume inversion prediction and explore the adaptability of different models to the prediction task. Actually, ensemble learning methods can be employed to combine these regression models to improve the model accuracy.

5. Conclusions

(1)

By virtue of flowback volume, pressure, and gas production data, combined with the material balance equation and the double-logarithmic curve method, the effective pore volumes of propped fractures and unpropped secondary fractures can be inversely calculated. This method establishes an effective fracture volume calculation model based on flowback data and integrates the analysis of the double-logarithmic curve of normalized pressure and material balance time, enabling the quantitative inversion of the volumes of post-fracturing propped fractures and unpropped secondary fractures. It provides key parameters for optimizing fracturing design and overcomes the limitations of traditional methods that rely on long-term production data or high-cost monitoring.

(2)

Through Pearson correlation coefficient analysis and comparison of machine learning models, it was found that the total fluid volume injected into the well, sanding intensity, minimum horizontal in situ stress, and elastic modulus are the dominant factors affecting the total fracture volume, with significant non-linear relationships (for example, there was a threshold effect of approximately 2300 m³ for the total fluid volume injected into the well; below this value, fracture propagation was inhibited, and above it, fracture propagation was promoted). Among various models, the Random Forest (RF) model performed the best, with the coefficient of determination (R²) of the test set reaching 0.86. It can effectively capture the complex mapping relationship between geological–engineering parameters and fracture volume. Moreover, through the mechanism of integrating multiple Decision Trees and random feature selection, it significantly reduces the risk of overfitting, and its stability and prediction accuracy are superior to those of the AdaBoost, Decision Tree, and Ridge Regression models.

(3)

The multi-parameter interaction analysis based on the Partial Dependence Plot (PDP) showed that a high elastic modulus can enhance the promoting effect of the injected fluid volume and sanding intensity on fracture propagation because it is more prone to fracture initiation through fluid pressure transmission and proppant embedment. There is a critical threshold of approximately 2600 m³ in the interaction between the total injected fluid volume and minimum horizontal in situ stress. When the fluid volume exceeds this threshold, the fluid energy can offset the constraint of in situ stress, driving the continuous growth of fracture volume. A high sanding intensity needs to be coordinated with sufficient fluid volume to break through the limitation of in situ stress, while a low sanding intensity has low sensitivity to fluid volume and in situ stress, requiring a balance between the “fracture width maintenance” function of proppants and the dynamic relationship between energy and stress. These mechanisms provide a clear direction for the optimization of fracturing parameters.