Study on the Expansion Law of Pressure Drop Funnel in Unsaturated Low-Permeability Coalbed Methane Wells

Abstract

In China, most medium- and shallow-depth coalbed methane (CBM) reservoirs are in the middle to late stages of development. Exploiting CBM in unsaturated low-permeability reservoirs remains particularly challenging. This study investigates the evolution of reservoir pressure in rock strata during CBM extraction from a low-permeability coal seam in the Ordos Basin. By integrating the seepage equation, material balance equation, and fluid pressure theory, we establish a theoretical and numerical model of reservoir pressure dynamics under varying bottom-hole flowing pressures. The three-dimensional surface of reservoir pressure is characterized by the formation of a stable pressure drop funnel. The results show that gas–liquid flow capacity is significantly constrained in low-permeability reservoirs. A slower drainage control rate facilitates the formation of stable seepage channels and promotes the expansion of the seepage radius. Under ultra-low permeability (0.5 mD) to low permeability (2.5 mD) conditions, controlling the bottom-hole flowing pressure below the average value aids the effective expansion of the pressure drop funnel. Numerical simulations indicate that the seepage and desorption radii expand more effectively under low decline rates in low-permeability zones. Calculations based on production data reveal that, under ultra-low permeability conditions, Well V₁ exhibits a narrower and more elongated pressure drop funnel than Well V₂, which operates in a low permeability zone. Furthermore, well interference has a lesser effect on the expansion of the pressure drop funnel under ultra-low permeability conditions. These differences in the steady-state morphology of the pressure drop funnel ultimately lead to variations in production capacity. These findings provide a theoretical foundation and practical guidance for the rational development of low-permeability CBM reservoirs.

1. Introduction

Coalbed methane (CBM) is an unconventional natural gas resource found in coal seams, with global reserves estimated at approximately 260 trillion cubic meters. In 2022, China’s CBM production reached nearly 10 billion cubic meters, and by 2023, production increased to 11.77 billion cubic meters, marking a year-on-year growth of 20.5%. CBM now accounts for about 5% of China’s domestic natural gas supply, contributing an incremental share of 18%, making it a vital addition to the country’s energy supply [1,2]. Recently, medium and shallow CBM production in China has become a significant portion of total CBM output, being the primary focus of current CBM development [3,4]. Key production bases of CBM in China include the Qinshui Basin and the eastern edge of the Ordos Basin, which contain an estimated resource volume of 3.28 trillion cubic meters and 10.72 trillion cubic meters, respectively. However, China’s shallow and medium CBM production faces challenges such as low single-well constant volume and poor comprehensive development returns. To address these issues and ensure the successful implementation of the deep CBM development strategy in the 14th Five-Year Plan, further research into the fundamental geological theories of CBM is crucial [5]. China’s CBM development is at a pivotal stage, characterized by both a “climbing period” and a “strategic opportunity period”, which calls for technological breakthroughs to transition from medium-shallow to deep layers [6]. There is a pressing need to enhance the development and utilization technologies for medium and shallow CBM to improve extraction efficiency and to lay a solid technical and theoretical foundation for the exploration and development of deep coalbed methane.

CBM mining primarily involves water discharge from the coalbed to reduce the pressure within the coal reservoir. Lowering the reservoir pressure to below the critical desorption pressure of the coal allows adsorbed gas to desorb from the coal’s microporous surface into a free state. This gas then diffuses from the micropores into the fracture system, driven by the concentration gradient. Under the influence of differential pressure, the gas moves through the fractures into the borehole for extraction [7,8]. The process of discharge and extraction in CBM wells is essentially a fluid flow process within the coal reservoir. Based on fluid mechanics principles, such as Darcy’s law, groundwater moves from high-pressure areas to low-pressure areas, forming a pressure drop funnel. In the actual discharge process, various factors, such as ground stress, initiation pressure of fluid flow in pore cracks, permeability, strata anisotropy, well interference, and discharge discontinuities, affect the pressure drop funnel patterns within the coal reservoir. These factors lead to diverse pressure drop funnel shapes and behaviors.

Typically, pressure drop funnels are circular or elliptical in shape, with the wellbore at the center, expanding in all directions [9]. The pressure drop is more significant near the wellbore, making the pressure drop funnel steeper, while further from the wellbore the pressure drop gradually decreases and the funnel becomes flatter. In well group development, when the distance between wells is smaller than the radius of influence of each well, the boundaries of the pressure drop funnels intersect during their expansion. This results in interference between wells, and multiple funnels overlap, creating a more complex pressure distribution. Vertically, the expansion of pressure drop funnels is influenced by the heterogeneity of coal seams and the presence of water-insulating or weakly permeable layers in the top and bottom slabs, which complicate the vertical expansion. Radially, if the permeability of the seams varies significantly, the vertical expansion of pressure drops becomes uneven. For instance, when the top and bottom slabs are relatively watertight, pressure drop is slower near these slabs, while it is faster in the middle of the coal seam [10]. Liu et al. expanded the study of pressure behavior to multi-well conditions, analyzing various boundary topographies and the process of pressure field expansion. They concluded that topographic boundaries have little effect on pressure propagation and distribution in the early stages of mining discharge [11]. Yang et al. developed a multi-well seepage model using simulations, which revealed that reservoir pressure distribution is influenced by the number of drainage wells. Multiple wells increase the inter-well interference effect and expand the pressure drop funnel area [12]. Li et al. emphasized that due to the pressure-sensitive nature of coal reservoirs, a rapid decrease in bottomhole pressure does not effectively propagate the pressure drop. Instead, slow and controlled pressure reduction should be the prerequisite for successful coalbed methane drainage [13].

Coal seams with high permeability allow for better mobility of both groundwater and coalbed methane, resulting in faster and more extensive expansion of pressure drop funnels. In contrast, coal seams with low permeability restrict fluid flow, slowing the expansion of pressure drop funnels and limiting their extent [14]. Gray quantified the change in permeability due to decreased coal reservoir pressure during coalbed methane drainage and extraction, demonstrating that the stress-sensitive effect caused by pressure reduction decreases coal seam permeability. He was the first to quantitatively investigate the role and impact of this stress-sensitive effect on permeability changes in coal reservoirs [15]. Sang used the coal rock matrix compression coefficient, pore compression coefficient, and gas concentration to characterize coal rock morphology and, through the relationship between coal rock morphology and permeability, described the evolving permeability of the coal reservoir during drainage and mining [16]. Wang et al. divided coalbed methane well drainage and extraction into different stages based on actual production scenarios [17]. Provided the reservoir remains undamaged, rapid pressure reduction is employed to increase the pressure difference between the wellbore and the reservoir, facilitating the prompt propagation of the pressure drop. This allows the pressure to quickly fall below the critical desorption pressure, enabling effective pressure reduction and desorption, which enhances the well production capacity.

Ge et al. used software simulations to model reservoir pressure changes, achieving good alignment with actual pressure drops. Their findings revealed that the changes in the pressure drop funnel and wellbore flow pressure are highly consistent [18]. Managing bottomhole flow pressure is critical to the expansion of the pressure drop funnel [19,20]. Peng et al. established a dual-porosity dual-permeability numerical model with varying bottomhole flow pressure decline rates to study the optimization of production through drainage speed control. The research found that controlling different drainage speeds during various development stages can enhance production capacity and protect the reservoir. An excessively high bottomhole flow pressure decline rate does not effectively increase gas production rates; instead, it hinders the expansion of the effective gas-producing area, leading to a shorter stable production period [21].

Controlling the expansion morphology of the pressure drop funnel is of significant engineering importance for the development of gas reservoirs under varying geological conditions, production regimes, and development schemes, as it directly affects reservoir development efficiency and economic benefits [22]. The introduction of artificial fractures greatly enhances the reservoir permeability, with the internal permeability of these fractures being much higher than that of the matrix, particularly in low-permeability reservoirs where this disparity is even more pronounced [23]. This significant permeability contrast causes the expansion morphology of the pressure drop funnel to shift from the traditional circular shape to an elliptical or radial-elliptical shape. However, the permeability enhancement effect of hydraulic fractures is limited beyond the vertical or tip regions of the fractures. Under low-permeability conditions, gas production efficiency remains strongly influenced by seepage capacity. When seepage is significantly restricted, an elliptical pressure drop funnel with a narrow, elongated, and steeply tapered morphology tends to form.

As demonstrated above, the pressure drop funnel in CBM reservoirs is a specific manifestation influenced by the geological characteristics of the reservoir and the development method. Numerous studies have shown that drainage parameters, such as drainage volume, drainage rate, and bottomhole flow pressure, significantly affect the morphology of the pressure drop funnel. Properly adjusted drainage parameters can not only facilitate the stable formation and expansion of the pressure drop funnel but also improve the recovery rate of coalbed methane. However, a comprehensive analysis of the pressure transfer process and the dynamic behavior of coal reservoirs is still lacking. Research on the morphological characteristics of the pressure drop funnel and methods to control its expansion is crucial for the efficient development of coalbed methane.

In this study, the evolution of reservoir pressure in rock strata during coalbed methane (CBM) extraction from low-permeability coal seams in the Ordos Basin is investigated using monitoring technologies and numerical simulation methods. This study relied on laboratory permeability experimental results to determine the stress sensitivity coefficient for low-permeability coal samples, which is then used to derive a permeability correction model through theoretical analysis. By integrating the seepage equation, material balance equation, and fluid mechanics equation, the researchers developed a theoretical and numerical model to simulate the evolution of reservoir pressure during CBM extraction. The findings of this research provided insights into the expansion mechanism of the pressure drop funnel in low-permeability reservoirs. These results contribute to a deeper understanding of the reservoir behavior during CBM extraction and offer valuable theoretical support for the development and management of similar gas reservoirs.

2. Engineering Overview

The Baode coal field is located in Baode County, Shanxi Province (Figure 1). The surface is predominantly covered by loess, with the development of peaks, valleys, and ravines. The terrain is characterized by a high eastern and low western elevation. The structural position is situated in the northern section of the Jinxin Nappe Belt on the eastern margin of the Ordos Basin, the northern part of the Hedong coal field, and the western side of the Lvliang Mountains. The overall structural form is simple, characterized by a westward-dipping monocline with a near north–south strike, and there is no significant development of faults or folds. Baode coal field is mainly dominated by fractures with width less than 5 μm and length less than 1 mm, and the fractures are cross-distributed with matrix pores, which is a typical double pore reservoir with complex pore structure. In Baode north coal field, the main coal seams include 4+5^# and 8+9^# coal seams, as shown in Figure 2, and the overall tectonic elevation range is −1713.06 m~864.10 m. The vertical depth of 4+5^# coal seams ranges from 251.58 m to 2483.55 m, and the vertical depth of 8+9^# coal seams ranges from 297.77 m to 2563.68 m. The thickness of 4+5^# and 8+9^# coal seams are 3.29 m~15.64 m and 3.04 m~18.69 m, respectively.

The studied coal field demonstrates characteristics typical of medium-shallow CBM reservoirs, with limited coal metamorphism, classifying it as medium- to low-rank coal. Medium-shallow CBM reservoirs are generally associated with low gas saturation, and in the northern Baode coal field, the gas saturation is effectively zero. The reservoir porosity ranges from 1.68% to 10.02%, with an average of 3.21%, placing it in the medium-low porosity category. The gas permeability spans from 0.21 mD to 16.18 mD, with an average of 2.68 mD, categorizing it as a low-permeability reservoir.

For the 4+5^# coal seam, porosity ranges from 2.12% to 7.85% (Figure 3), while the 8+9^# coal seam has a porosity range of 1.68% to 10.02%. The permeability of the 4+5^# coal seam varies from 0.210 to 6.028 mD (Figure 4), while the permeability of the 8+9^# coal seam ranges from 0.205 mD to 16.176 mD. The porosity of the 4+5^# coal seam is generally higher than that of the 8+9^# coal seam. For the 4+5^# coal seam, porosity is higher in the northeast and southeast, while for the 8+9^# coal seam, porosity is higher in the northeast, southeast, and southwest. The permeability of the 4+5^# coal seam is also higher compared to the 8+9^# coal seam, with a more uniform distribution. The porosity and permeability in the central nosedive area are more evenly distributed.

The elastic modulus distribution of the reservoir in the Baode north coal field is 2.93~15.26 GPa. The elastic modulus of the 4+5^# coal seam is 3.59~15.26 GPa, with an average value of 8.19 GPa; the elastic modulus of the 8+9^# coal seam is 2.93~14.81 GPa, with an average value of 6.70 GPa. The Poisson’s ratio distribution of the Baode north coal field is 0.25~0.42. The Poisson’s ratio of the 4+5^# coal seam is 0.25~0.42, with an average value of 0.36. The Poisson’s ratio of the 8+9^# coal seam is 0.28~0.42, with an average value of 0.38. The elastic modulus of the 4+5^# and 8+9^# coal seams is higher in the northwest groove area, and the elastic modulus of the 4+5^# coal seam in the western groove and northeast is higher. Similar to the distribution characteristics of elastic modulus, the Poisson’s ratios of 4+5^# and 8+9^# coal seams are lower in the northwest groove area, and the Poisson’s ratio in the western groove and northeast of the 4+5^# coal seam is lower. The vertical reservoir pressure is generally positively correlated with depth, reaching a maximum of 23 MPa in the northwestern region and a minimum of 2.45 MPa in the eastern region, where the depth is shallowest. Gas content distribution ranges from 1.6 m³/t to 12.2 m³/t (Figure 5), with an average gas content of 8.3 m³/t. The gas content of the 4+5^# coal seam ranges from 1.6 m³/t to 11.3 m³/t, with an average of 7.8 m³/t, while the gas content of the 8+9^# coal seam ranges from 2.8 m³/t to 12.2 m³/t, with an average of 8.5 m³/t. Overall, the gas content of the 8+9^# coal seam is higher than that of the 4+5^# coal seam. The distribution of gas content within each seam is positively correlated with the vertical burial depth.

3. Unsaturated Low Permeability Coalbed Methane Reservoir Characteristics

3.1. Pore Volume Pressure Sensitivity Factor

The increase in effective stress during pressure drop leads to compression of the pore structure, and porosity and permeability show non-stationary changes of multiplicative and exponential characteristics, defining the derivative of the pore structure with respect to the change in reservoir pressure as the pressure sensitivity coefficient of the pore volume (C_f). Porosity and permeability can be expressed with the effective stress change relationship equation as:

$$\left\{\begin{array}{l}porosity \phi ={\phi }_i\left[1-C_f(P_i-P)\right]\\ permeability k=k_i^{1-C_f(P_i-P)}\end{array}\right.$$

(1)

where: φ_i, k_i and p_i are, respectively, reservoir initial porosity, permeability, and reservoir pressure.

Cylindrical overburden pore permeability test samples (with a diameter of 2.5 cm and a length of 5.0 cm) were selected from the study coal field. The initial porosity and permeability of the samples were 3.45% and 2.68 mD, respectively. Various levels of confining pressure and pore pressure were applied to these samples. To simulate formation water conditions, deionized water with a viscosity of approximately 1.0 mPa·s was used as the test fluid, and Darcy’s law was applied to measure the permeability. The resulting permeability data and the corresponding applied stress values are presented in

Table 1. Results of indoor testing of overburden permeability.

Lithology	Depth of Burial (m)	Weights (g)	Lengths (cm)	Caliber (cm)	Confinement (MPa)	Pore Pressure (MPa)	Net Surrounding Pressure (MPa)	Permeability (mD)
Coal	1024.09	27.08	4.40	2.50	2.75	0.46	2.40	22.190
					4.85	0.67	4.28	9.133
					7.82	0.91	7.02	3.913
					9.90	1.06	8.94	2.507
					14.62	1.09	13.63	1.695
					19.75	1.10	18.75	1.239
					24.72	1.10	23.72	0.758
					23.92	1.10	22.92	0.752
					19.95	1.10	18.94	1.239
					15.20	1.11	14.19	1.344
					10.27	1.12	9.25	1.621
					8.16	1.12	7.14	1.780
					5.19	1.12	4.17	3.444
					3.18	0.76	2.52	7.786
					2.75	0.46	2.40	22.190
					4.85	0.67	4.28	9.133

.

The experimental results show that the permeability of the coal samples is significantly influenced by effective stress. Under a net confining pressure of 2.40 MPa, changes in the pore structure occur, and the permeability reaches 22.190 mD. The initial reservoir pressure at the sampling location is 6.9 MPa, with a vertical stress of 15.6 MPa. The effective stress, calculated as the vertical stress minus the reservoir pressure, is 8.7 MPa. Consequently, the initial effective stress and permeability of the coal samples are 8.7 MPa and 2.68 mD, respectively. The net confining pressure in the test is analogous to the effective stress and is denoted as P′. The relationship between permeability and effective stress is represented by the fitting Equation (2), as follows:

$$k_{n+1}=k_n^{1+C_f(P_{n+1}^{\prime }-P_n^{\prime })}$$

(2)

With initial conditions of k₀ = 0.268 mD and p₀ = 7.7 MPa, the least-squares fitting method is applied to obtain C_f = −0.09725 MPa⁻¹. The results indicate a negative correlation between permeability and pressure, demonstrating that effective stress has a more significant impact on the pore structure of the reservoir. These findings are used to numerically solve the theoretical model of CBM production in the target coal field.

3.2. Reservoir Adsorption Properties

The primary form of CBM in unsaturated CBM reservoirs is in the adsorption state. Coal’s rich pore structure and large specific surface area allow gas molecules, such as methane, to be adsorbed onto the coal surface. The bonding force between CBM in the adsorption state and coal is strong, and the amount of adsorption is closely related to factors such as the coal’s metamorphic degree, pore structure, temperature, and pressure.

In low permeability unsaturated CBM reservoirs, the migration of gas and water is controlled by permeability, resulting in slower gas movement and larger time constants for desorption and adsorption during production. The permeability of the reference reservoir averages only 2.7 mD, with a large surface area and porosity, which allows for a higher maximum volume of adsorbable gas in the reservoir.

Isothermal adsorption tests were conducted on coal rock samples (a–l) collected from 12 distinct locations within the coal seams of the Baode north coal field. After removing moisture from the coal samples, their methane adsorption capacities were measured at various pressures, with the test maintained at a constant temperature of 40 °C. Methane gas was introduced into the 12 coal samples under conditions of 8 MPa to 10 MPa, and the data on unit volume adsorption capacity versus pressure obtained from the tests were fitted using the Langmuir equation to generate the isothermal adsorption curves, as shown in Figure 6. As shown in Figure 6, we found there are four sets of tests where the Langmuir volume (V_L) is less than 12, four sets where the Langmuir volume (V_L) is greater than 15, and four sets where the Langmuir volume (V_L) falls between 12 and 15.

The Langmuir volume (V_L) for each sample was determined when adsorption reached saturation during the test. This parameter represents the maximum volume of gas that can be adsorbed per unit mass of material. The Plummer pressure (P_L) for each sample was evaluated by reducing the pressure until the adsorption capacity reached half of its maximum value. This parameter indicates the pressure at which the equilibrium adsorption capacity of the gas is half of its maximum adsorption capacity. Both V_L and P_L are essential for constructing isothermal adsorption curves that describe the gas adsorption behavior. The test results show that the V_L and P_L for the air-dried samples ranged from 7.92 m³/t to 19.52 m³/t and from 1.98 MPa to 4.95 MPa, respectively, with average values of 13.55 m³/t and 3.04 MPa.

To further investigate the impact of moisture (M_ad) occupying pore space and minerals (A_ad) with no adsorption capacity in the coal samples, this study included sample selection, pretreatment, component analysis, and data correction to more accurately evaluate the adsorption characteristics of the coal. After considering the proportions of moisture and ash in the coal sample mass, the Langmuir volume and Langmuir pressure on a dry ash-free basis were calculated. The Langmuir volume and Langmuir pressure values for both the air-dried and dry ash-free bases are presented in

Table 2. Indoor isothermal adsorption test.

Test Sample	Test Parameters	Lancefield Volume VL (m3/t)	Langmuir Pressure PL (MPa)
Sample a	Air-dried basis	7.92	2.28
	Dry ash-free basis	11.17	2.28
Sample b	Air-dried basis	10.79	1.98
	Dry ash-free basis	12.82	1.98
Sample c	Air-dried basis	11.83	1.99
	Dry ash-free basis	14.05	1.99
Sample d	Air-dried basis	12.82	4.95
	Dry ash-free basis	18.60	4.95
Sample e	Air-dried basis	18.19	3.08
	Dry ash-free basis	20.76	3.08
Sample f	Air-dried basis	19.52	3.32
	Dry ash-free basis	22.39	3.32
Sample g	Air-dried basis	14.30	3.45
	Dry ash-free basis	17.34	3.45
Sample h	Air-dried basis	8.47	2.05
	Dry ash-free basis	10.29	2.05
Sample i	Air-dried basis	15.74	3.84
	Dry ash-free basis	19.91	3.84
Sample j	Air-dried basis	12.55	3.67
	Dry ash-free basis	16.80	3.67
Sample k	Air-dried basis	16.44	3.18
	Dry ash-free basis	18.52	3.18
Sample l	Air-dried basis	14.08	2.69
	Dry ash-free basis	16.15	2.69

.

The Langmuir Isotherm is a classical model used to describe the adsorption behavior of gases on solid surfaces. It primarily explains the relationship between the amount of gas adsorbed and the gas pressure at adsorption equilibrium on a homogeneous surface. The Langmuir equation represents the isothermal adsorption curve that shows how adsorption capacity changes with pressure at a given temperature, as follows:

$$q={\displaystyle \frac{q_{m}CP}{P_L+P}}$$

(3)

where: q is the adsorption volume, the volume of adsorbed gas per unit mass of coal base, (m³/t); q_m is the maximum adsorption volume, the maximum volume of adsorbed gas per unit mass of coal base, (m³/t); P_L is the Langmuir pressure, the equilibrium reservoir pressure corresponding to the adsorption volume when the adsorption volume is reached q_m/2, (MPa); P is the reservoir pressure at which the desorption–adsorption inter-conversion reaches the steady state, (MPa); and c is the correction factor of the coal base taking into consideration of coal rock compositions.

In the isothermal adsorption tests conducted on an air-dried basis, certain errors are introduced due to the influence of trace volatile matter (V_ad) in the coal samples. The volatile matter content varies among different coal samples. For the 12 samples, the adsorption capacity-pressure experimental data under air-dried conditions, along with the Langmuir volume, Langmuir pressure, and Langmuir equation data, are processed. Using the least squares method for fitting, the coal-specific correction coefficients were derived, as presented in

Table 3. The coal base correction coefficients for the isothermal adsorption curves of 12 samples.

Test Sample	a	b	c	d	e	f	g	h	i	j	k	l
c/%	99.41	99.58	99.56	99.84	99.68	99.63	99.80	99.75	99.71	99.94	99.97	100.05

.

The steady-state wellbore flow pressure distribution was derived from statistical field production data, and the wellbore flow pressure curves for Well No. 9, which is in normal production, were plotted, as shown in Figure 7. The steady-state wellbore flow pressure in the coal field ranges from 0.2 MPa to 10.2 MPa, which is lower than the original reservoir pressure distribution due to the effects of gas production. It is evident that the flow pressure is more significantly affected in the near-wellbore area. The normal production of the No. 9 production wells (Well-A~Well-I) causes a more noticeable regional pressure drop compared to the rest of the area, forming irregular pressure drop funnels of various shapes. By analyzing the bottoming flow pressure curve of Well No. 9, it is found that the curve consists of two phases: dynamic and steady-state. The bottoming flow pressure reaches a steady state approximately 2100 days after production begins. This bottoming flow pressure is a critical internal factor influencing production. As a key internal boundary condition, the dynamic change process is essentially linear, with pressure and time being negatively correlated, as described by the equation y = −0.0018x + 5.1025.

4. Analysis of the Expansion Pattern of Pressure Drop Funnels in Low-Permeability Reservoirs

A theoretical model of reservoir pressure was established based on the seepage equation, continuity equation, and desorption/adsorption equation, and the model was optimized to obtain an accurate description of the reservoir properties by considering the effective stress and matrix shrinkage effects, as follows:

$$\left\{\begin{array}{l}water: P_w={\displaystyle \frac{1}{C_f}}{log}_{ki}\left(\left(k_i^{C_f{P}_i}-k_i^{C_f{P}_{cd}}\right){\displaystyle \frac{ln\left(\frac{r}{r_i}\right)}{ln\left(\frac{r_i}{r_{cd}}\right)}}+k_i^{C_f{P}_i}\right)\\ gas: r{P}_g{k}_i^{P_g}{e}^{\epsilon max\left({\displaystyle \frac{P_L}{P_L+P_g}}\right)}{\displaystyle \frac{\partial P_g}{{\partial }_r}}=P_{cd}{e}^{\epsilon max\left({\displaystyle \frac{P_L}{P_L+P_g}}\right)}{\displaystyle \frac{\left(k_i^{C_f{P}_i}-k_i^{C_f{P}_{cd}}\right)}{C_{f}ln\left(\frac{r_i}{r_{cd}}\right)}}\end{array}\right.$$

(4)

Using MATLAB 2022 software, the theoretical model was solved by inputting the model and its parameters to simulate production conditions. In this model, P_wf is a dynamic variable that changes over time until it reaches a steady state. Three different decline rates for bottomhole flow pressure and five distinct initial permeability conditions were controlled. The production time was determined by dividing the bottomhole flow pressure drop by the decline rate.

The final steady-state bottomhole flow pressure averages around 0.8 MPa, assuming no dynamic changes in bottomhole flow pressure. The numerical solution of the theoretical model was established under steady-state bottomhole flow pressure conditions. However, it is important to note that the permeability of low-permeability reservoirs is typically less than 10 mD, and when there are variations in permeability distribution, the relative percentage increase or decrease compared to the original benchmark becomes significant, which creates challenges for production control.

In the studied coal field, the permeability distribution of the 8+9^# coal seam is relatively high, with a maximum value of up to 12 mD. The dynamic boundary conditions for permeability in the theoretical model are controlled at values of 0.5 mD, 1.0 mD, 2.5 mD, 5.0 mD, and 10 mD for solving the model.

As shown in Figure 7, for the nine reference wells, bottomhole flow pressure discharge control is fitted, the dynamic change process of bottomhole flow pressure is often expressed as the P_wf pressure drop for time t, the initial bottomhole flow pressure is basically equal to the original reservoir pressure P_i, and the steady state bottomhole flow pressure is about 1 MPa on average, that is P_wf = P_i + α_t. The solution process of each model is controlled by three different conditions of the bottomhole flow pressure decline rate, namely α₀ = −0.0018 MPa/d, α₁ = −0.0010 MPa/d, and α₂ = −0.0030 MPa/d. Among these, α₀ = −0.0018 MPa/d is the value closest to the bottomhole flow pressure decline rate in the actual production process, hereinafter referred to as the production decline rate. α₁ = −0.0010 MPa/d and α₂ = −0.0030 MPa/d are referred to as the larger decline rate and smaller decline rate, respectively. The solving results for five groups of different permeability conditions were plotted as pressure drop curves using MATLAB 2022 software, as shown in Figure 8.

The reservoir pressure under different bottom hole flow pressure reduction conditions was solved under the control conditions of initial pressure of 6.2 MPa and critical desorption pressure of 5.0 MPa (

Table 4. Pressure drop funnel theoretical model solution results.

Permeability (mD)	α0 = −0.0018 MPa/d	α1 = −0.0010 MPa/d	α2 = −0.0030 MPa/d
0.5	39	50	14
1.0	50	55	35
2.5	59	58	56
5.0	75	65	87
10.0	91	72	123

).

The solution results of the pressure drop funnel theoretical model show the following trends:

1. When (k = 0.5), the pressure drop curve expands most effectively for the smaller corresponding pressure drop rate. The control points for the desorption radius, corresponding to the critical desorption pressure, are as follows (from left to right): larger rate of drop, production rate of drop, and smaller rate of drop.

2. When (k = 1.0), the pressure drop curves for the production and smaller drop rates are closer, while the curve for the larger drop rate is less extended. The control points for the desorption radius on the curves at the critical desorption pressure are (from left to right): larger drop rate, production drop rate, and smaller drop rate.

3. When (k = 2.5), the permeability is more representative of the average permeability in most of the reference production coal field. At this point, the three pressure drop curves are essentially similar, and the control points for the desorption radius, at the critical desorption pressure, are (from left to right): larger rate of desorption, smaller rate of desorption, and production rate of desorption.

4. When (k = 5.0), the distribution of the pressure drop curve shows a positive correlation with the wellbore flow pressure drop rate. The largest effect is observed in the pressure drop curve for the larger wellbore flow pressure drop rate, where the desorption radius is greater than 300 m. The control points for the desorption radius on the curve, at the critical desorption pressure, are (from left to right): smaller drop rate, production drop rate, and larger drop rate.

5. When (k = 10.0), the differences between the pressure drop curves for different drop rates become more apparent. Both the production drop rate and larger drop rate curves exceed 300 m in desorption radius. The control points for the desorption radius at the critical desorption pressure on the curve are (from left to right): smaller drop rate, production drop rate, and larger drop rate.

5. Numerical Simulation of Pressure Drop Funnel Production in Low-Permeability Reservoirs

The geological parameters and boundary conditions of the numerical model of reservoir production were determined by integrating geological data and engineering understanding. Geological condition parameters include permeability (k), porosity (φ), water saturation (S_w), reservoir pressure (P_i), gas content (C_g), as shown in

Table 5. Range of values for geological condition parameters.

Parameter	Range	Unit
Permeability k	0.21~16.18	mD
Porosity φ	1.68–10.02	%
Water saturation Sw	1	—
Reservoir pressure Pi	2.5~23.1	MPa
Gas content Cg	1.6~12.2	m3/t

.

Based on the boundary conditions derived from the pressure drop funnel theory model (

Table 6. Theoretical model solving fixed boundary conditions.

Parameters	Range of Values	Unit
k	0.21~16.18	mD
φ	1.68~10.02	%
Sw	1	-
Pi	2.5~23.1	MPa
Cg	1.6~12.2	m3/t
VL	7.92~19.52	m3/t
PL	1.98~4.95	MPa
Cf	−0.09725	MPa−1
εmax	0.01	-
c	99.74	%
Pwf	1	MPa

), further numerical simulations were conducted to validate the model. A homogeneous ideal CBM production model was established using the Eclipse 2006 simulation software, with a model area of 600 m × 600 m (Figure 9). To accurately simulate the desorption-seepage process of CBM, the numerical model was defined as a dual-porosity single-permeability grid. This grid type accounts for the different storage and flow characteristics of the coal matrix and fractures in the reservoir. For the numerical production calculations, the decline rates of the bottomhole flow pressure were controlled under varying permeability conditions. The permeability values used for the simulations included 0.5 mD, 1.0 mD, 2.5 mD, 5.0 mD, and 10.0 mD. The corresponding wellbore flow pressure decline rates were set at 0.0018 MPa/d, 0.001 MPa/d, and 0.003 MPa/d to simulate different production scenarios and reservoir conditions.

Based on the boundary conditions such as gas content and initial reservoir pressure, along with the isothermal adsorption curve, the critical desorption pressure in the numerical model is approximately 5.0 MPa. The numerical calculation results of CBM production corresponding to the reservoir pressure less than or equal to 5.0 is the desorption area. The radius of desorption is quantitatively shown as the radius of the circular desorption region in the pressure drop funnel surface diagram under different production conditions. As shown in Figure 10 and Figure 11, the results of the calculations at k = 0.5 mD and k = 2.5 mD are plotted as a 3D surface diagram of the pressure drop funnel, and the points with P = 5.0 in the diagram are connected as circles to quantify the desorption radius.

When k = 2.5 mD, the pressure drop funnel numerical simulation and formation of a three-dimensional stress surface map using MATLAB 2022 software for different P_wf drop rate boundary conditions and the desorption region boundary connection on the stress map to get the calculated value of desorption radius, production drop rate, smaller drop rate, and larger drop rate production control conditions to form the radius of desorption were 40 m, 50 m, and 15 m, respectively.

When k = 2.5 mD, numerical calculations of pressure drop funnels were performed for boundary conditions with different P_wf rate of descent, and based on the stress surface diagrams, the desorption radii were obtained as 60 m, 58 m, and 55 m for the production control conditions of production rate of descent, smaller rate of descent, and larger rate of desorption, respectively.

In addition to permeability values of 0.5 mD and 2.5 mD, numerical gas production calculations were also conducted under three other permeability conditions (1.0 mD, 5.0 mD, and 10.0 mD). The desorption radius values were obtained by analyzing the calculation results, as shown in

Table 7. Comparison of quantitative desorption radius–pressure drop expansion theory solutions.

Permeability (mD)	α0 = −0.0018 MPa/d	α1 = −0.0010 MPa/d	α2 = −0.0030 MPa/d	Theoretical Model Solution (Desorption Radius from Smallest to Largest)
0.5	40	50	15	Larger rate of descent, production rate of descent, smaller rate of descent
1.0	49	54	35	Larger rate of descent, production rate of descent, smaller rate of descent
2.5	60	58	55	Larger rate of descent, smaller rate of descent, production rate of descent
5.0	73	66	85	Smaller rate of descent, production rate of descent, larger rate of descent
10.0	90	71	120	Smaller rate of descent, production rate of descent, larger rate of descent

.

Through controlling five sets of numerical calculations with different permeability values, the numerical results for the range of pressure drop were obtained. By combining the theoretical model of reservoir pressure to solve for the extended range of the pressure drop curve, and statistically analyzing the numerical simulation and the theoretical model solutions for the desorption radius, a comparative analysis of the pressure drop range between the theoretical model and numerical simulation was established. The results from the theoretical model solution are shown in Table 4, with the bottom hole flowing pressure drop rates being α₀ = −0.0018 MPa/d, α₁ = −0.0010 MPa/d, and α₂ = −0.0030 MPa/d, respectively.

From the theoretical reservoir pressure model and numerical calculation results, it can be observed that in low-permeability reservoirs, the decline rate of bottomhole flow pressure (BHP) significantly affects gas production. In reservoirs with permeabilities of 0.5 mD, 1.0 mD, 2.5 mD, 5.0 mD, and 10.0 mD, the boundary conditions of BHP with a wider propagation range of pressure drop curves and a larger desorption radius are α₁, α₁, α₀, α₂, and α₂, respectively, where α₀ < α₁ < α₂. The pressure drop propagation at k = 0.5 mD, k = 1.0 mD, and k = 2.5 mD is slightly more effective with α₁ and α₀ than with α₂. When k = 2.5 mD, the effects of α₁, α₀, and α₂ on pressure drop extension are similar. However, when k = 5.0 mD and k = 10.0 mD, α₂ leads to better production and pressure drop propagation than α₁ and α₀. A comparative analysis of the single well production pressure drop funnel, theoretical model solution results, and numerical simulation results is shown in Figure 12.

The variation curve of the desorption radius with the bottomhole flow pressure drop rate indicates that the theoretical model solution and numerical simulation results are nearly identical, suggesting that the theoretical model of the pressure drop funnel is accurate. The results of both the theoretical model and numerical simulation are presented. The flow of both gas and liquid between wells can be influenced when the bottomhole pressure drop rate is high, under conditions of 0.5 mD permeability. In this case, the calculated desorption radii are only 14 m and 15 m when the steady-state pressure drop funnel is formed. Under the conditions of 0.5 mD and 1.0 mD, the desorption radius with drainage control and smaller deceleration is the optimal value across multiple control boundary conditions, with average desorption radii of approximately 52.5 m and 52 m, respectively. At 2.5 mD, the desorption radius under production deceleration control is slightly better than other deceleration conditions, reaching 59 m and 60 m. At 5.0 mD and 10.0 mD, due to favorable permeability conditions, the pressure drop expansion effect is positively correlated with the bottomhole flow pressure drop rate within a certain range. With a high pressure drop rate, the desorption radius from the theoretical model is 87 m and 123 m, respectively, while the desorption radius from the numerical simulation results is 85 m and 120 m, respectively. Gas flow in low-permeability reservoirs is restricted, and the rate of decline in wellbore flow pressure directly affects the rate of gas flow. An excessively rapid decline in pressure may lead to a quick decrease in reservoir pressure, reducing the effective gas flow.

The results from the theoretical model of a low-permeability reservoir, based on reservoir parameters, and the ideal numerical production model indicate that the effective expansion of the pressure drop funnel is significantly influenced by the rate of decline in the bottomhole flow pressure. The differences observed in gas production between high- and low-producing wells in the reference research coal field can be explained by integrating existing production data. The expansion effect of the pressure drop funnel partially reflects the effectiveness of coalbed methane production. The analysis of the pressure drop expansion and gas production effect in two wells (V₁ and V₂) within the target zone of the study coal field reveals a significant discrepancy in the gas production between these two adjacent wells. The permeability of the main reservoirs in the target area ranges from 0.2 mD to 3.5 mD, with the distribution shown in Figure 13. The V₁ and V₂ wells are 378 m apart; the permeability of the grid containing well V₁ is 0.4 mD, while the grid containing well V₂ has a permeability of 2.6 mD.

The gas content in the target zone is approximately 7.6 m³/t, and the critical desorption pressure is around 4.8 MPa. The bottomhole flow pressure and daily gas production curves for the two wells are shown in Figure 14. The decline rates of the bottomhole flow pressure are approximately 0.0091 MPa/d and 0.0104 MPa/d. The daily gas production can reach up to 268 m³ and 6521 m³, with average values of about 254 m³ and 2749 m³. The initial reservoir pressure is 7.8 MPa, and the critical desorption pressure is 5.0 MPa. The well spacing is 378 m. The theoretical pressure drop funnel models were established for well V₁, with a permeability of 0.4 mD and a bottomhole flow pressure drop rate of 0.0027 MPa/d, and well V₂, with a permeability of 2.6 mD and a bottomhole flow pressure drop rate of 0.0023 MPa/d. The pressure drop curves for each well under the corresponding production conditions were derived, resulting in desorption radii of 19 m and 61 m for wells V₁ and V₂, respectively.

A well-production numerical model was established, and the pressure drop funnel surface diagram was generated. When the bottomhole flow pressure reaches a steady-state condition, the reservoir pressure distribution is as shown in Figure 15. At this point, the desorption radii of wells V₁ and V₂ are approximately 40 m and 180 m, respectively. The analysis of the pressure drop funnel shape and gas production characteristics was based on the theoretical model solution and the numerical calculation results from the coordinated production of the two wells. The discrepancy between the theoretical model results and the collaborative production numerical results is attributed to the overlapping pressure drop areas during production, which create inter-well interference. This interference causes an increase in the desorption radius compared to single-well production. Additionally, the drainage of the non-recoverable inter-well interference area significantly enhances the pressure reduction control effect. However, the gain from inter-well interference varies depending on the type of pressure drop funnel.

For Well V₁, the numerical and theoretical model results indicate desorption radii of 40 m and 19 m, respectively, with a gain of about 2.10 times the original radius during collaborative production. For Well V₂, the results show desorption radii of 180 m and 61 m, respectively, with a gain of approximately 2.95 times the original radius. Gas production under steady-state flow pressure is controlled by managing the drainage rate and production sleeve pressure. Under steady-state conditions, the initial pressure drop funnel is narrower. An excessively rapid decline in flow pressure at the well’s bottom, particularly in low-permeability conditions, can lead to a reduction in pore structure due to effective stress effects, thereby reducing effective permeability. This results in a slower rate of lateral expansion and consequently lower gas production.

Rapid declines in wellbore flow pressure typically cause more significant changes in the reservoir pressure gradient. In low-permeability gas reservoirs, such changes can destabilize gas flow and reduce productivity. In high-permeability reservoirs or at high-pressure points in low- to medium-permeability reservoirs, production ramp-up improvements can be achieved by controlling the rate of decline in bottomhole flow pressure. However, it is crucial to prevent instability in gas flow and structural damage, which can result from a rapid pressure drop rate that may cause premature contact with the production boundary.

The results of the theoretical model solution and the Eclipse numerical simulation can be mutually verified. However, actual reservoirs typically exhibit heterogeneity that differs from the assumptions of this study, such as fractures, faults, and lithological variations. These factors can cause deviations in the morphology and expansion patterns of the pressure drop funnel compared to the theoretical model. Additionally, this method simplifies the boundary and initial conditions, whereas the boundary conditions (e.g., well pattern arrangement or production regime) and initial pressure distribution of actual reservoirs may be more complex. Therefore, in practical applications, it is necessary to refine and calibrate the model by integrating geological characteristics and production data to enhance its predictive accuracy and applicability.

6. Conclusions

(1) The pressure sensitivity coefficient of permeability was obtained by fitting the results from overburden pore permeability tests, and the Langmuir pressure and Langmuir volume specific to the studied coal field were determined through isothermal adsorption experiments. These provide key parameters for reservoir characteristic analysis.

(2) Both the pressure drop funnel expansion theoretical model and the numerical validation model demonstrate that under coalbed methane production conditions with ultra-low permeability (0.5 mD) to low permeability (2.5 mD), the expansion effect of the pressure drop funnel is optimal when the bottomhole flow pressure reaches a steady state at a slower decline rate. A slower pressure decline rate facilitates drainage control, promotes the expansion of the seepage radius, and results in smoother pressure drop curves.

(3) This study on pressure drop funnel expansion is based on a homogeneous assumption. Future research should incorporate heterogeneous characteristics such as fractures, faults, and lithological variations to investigate their impact on pressure drop funnel expansion. Furthermore, the findings on the expansion mechanism of pressure drop funnels in medium-shallow coalbed methane reservoirs should be extended to deep coal-bed methane development, exploring the expansion patterns of pressure drop funnels under high-pressure and high-temperature conditions.