Abstract
Advanced description of pressure drop funnel is crucial in coalbed methane (CBM) production because of dewatering and depressurization methods. Improving the precision of the pressure drop funnel description facilitates obtaining the actual production status and productivity potential, both pivotal for responsible development plans. The study presents a semianalytical model that integrates pressure profiles and material balance equations, incorporating inner and outer boundary conditions, and dynamic reservoir characteristics. The pressure propagation characteristics in undersaturated coal reservoirs are described during the production life of CBM wells, and the model is validated using two wells with different production characteristics. The results indicate that the effect of water saturation on the expansion of the drainage radius surpasses that of the desorption radius, demonstrating a more precise prediction of the production boundary when dynamic water saturation is considered. Additionally, a rapid drop rate of bottomhole flowing pressure triggers simultaneous propagation of the drainage and desorption radii, resulting in a smaller production boundary and fewer well-controlled resources. Conversely, an appropriate production strategy results in a larger drainage radius and lower boundary pressure before massive gas desorption, thereby facilitating efficient propagation of the pressure drop funnel. Moreover, the pressure drop funnel characterized by the model can compute the dynamic CBM resources and recovery efficiency of a single well, providing a valuable basis for assessing productivity potential. In summary, this model offers a time-saving and practical tool for describing the dynamic pressure drop funnel in various CBM production stages and promoting efficient development for undersaturated CBM reservoirs.
1. Introduction
Currently, CBM has entered the stage of commercial development in China, with abundant CBM resources gradually becoming a significant component of the energy sector.1,2 Unlike conventional oil and gas reservoirs, coal reservoirs exhibit unique adsorption properties.3−5 Typically, drainage depressurization disrupts the original pressure balance in the reservoir, shifting the dynamic equilibrium of CBM adsorption/desorption toward desorption. Consequently, desorbed CH4 migrates along the pressure gradient to regions of lower energy.6,7 CBM production involves a series of complicated processes, including desorption, diffusion, and seepage, with two noteworthy radii: the drainage radius and the desorption radius. These two radii expand outward with drainage and massive CBM desorption, providing a visual representation of pressure propagation, namely pressure drop funnel.8 An advanced description of the pressure drop funnel helps identify CBM production, which is crucial for guiding engineering measures and determining development strategies.
Several methods have been employed to measure and evaluate coal reservoir pressure, typically categorized into three types: field experiments, numerical simulations, and mathematical models. Pressure recovery well testing is a common method in field experiments that can partially alleviate reliance on geological parameters.9,10 Unfortunately, field testing is not feasible for all CBM wells due to cost considerations. Moreover, this method necessitates shutting in the well for pressure measurement, disrupting production continuity and complicating accurate pressure determination, especially in low-permeability reservoirs like CBM reservoirs. Therefore, it is not widely available in CBM fields. In contrast, numerical simulation is increasingly popular for predicting production potential and characterizing pressure propagation.11,12 However, numerical simulation is susceptible to human factors due to the inherent characteristics of historical matching, leading to controversy regarding its accuracy.
Mathematical models, which can quantify pressure propagation and characterize the pressure drop funnel from a three-dimensional perspective, are highly favored by scholars.13−16 Previous models relied on material balance equations (MBE) to estimate average reservoir pressure, with additional factors considered to enhance their accuracy, such as free gas, dynamic porosity, and drainage area.17−20 Despite their simplicity and efficiency, these models only provide a rough estimate of average reservoir pressure rather than accurate pressure drop funnels, limiting their effectiveness as a guide to CBM production. Recent advancements in the field have led to the development of more sophisticated models, such as substituting pressure distribution formulas of gas–water phases into MBE in undersaturated coal reservoirs. This approach allows for a finely defined relationship between pressure, position, and production time. For instance, a model proposed by Li incorporates pressure perturbation area and skin factor to represent the pressure propagation range and reservoir reconstruction effect, respectively.21 This model enables the simultaneous calculation of the dynamic propagation of average reservoir pressure and pressure drop funnel. However, hydrofracturing is prevalent in low-permeability CBM fields in China, because it serves as a crucial stimulation treatment to form narrow fractures along both directions of principal stress to augment original permeability.22 Therefore, the influence of artificial fractures on pressure propagation cannot be disregarded. To account for this influence, conformal transformation was innovatively utilized to derive the pressure state equations of gas/water phases, and a mathematic model was established in combination with MBE, which has been fully applied to the quantitative optimization of drainage rate before CBM desorption.23,24 Subsequently, Sun et al. expanded on this basis, integrating multiple dynamic factors such as porosity, permeability, and water saturation, which play crucial roles in the fluid migration process in coal, effectively improving the accuracy of the model.25−27 The above models lack accurate definition of production boundaries, resulting in the prediction of pressure drop funnels being within a fixed range, usually assumed to be half the well spacing. This work is important and complex, as it is influenced by reservoir physical characteristics, artificial reconstruction effects, production strategies, etc. Unrealistic assumptions frequently lead to distorted pressure drop funnels, which in turn misguide subsequent work, including well pattern design and productivity potential evaluation. Accordingly, in our previous research, a semianalytical mathematical model and related calculation procedures have been proposed, where the production boundary, whether the interwell pressure interference boundary or the damage boundary, can be predicted scientifically.8 In CBM well patterns, interwell pressure interference occurs due to pressure propagation between two or more wells, accelerating depressurization across the formation. For CBM wells with inadequate production strategies or engineering reconstruction, pressure propagation remains limited within the damage boundary. Notably, the model did not comprehensively account for dynamic water saturation, nor did it characterize the pressure drop funnels after reaching the boundary, warranting further in-depth investigation.
In conclusion, current analytical models struggle to accurately describe the characteristics of pressure propagation during CBM development. Their limitations are primarily evident in the following aspects: (1) inaccurate prediction of production boundaries, particularly in understanding the impact of dynamic water saturation on the pressure distribution of gas and water phases; (2) failure to account for various complex factors, such as varying production rates, production strategies, and dynamic geological parameters; (3) imprecise definition of the spreading mechanism of the pressure drop funnel in each production stage. In response, the prediction model for production boundaries is optimized by considering dynamic water saturation, and the production stages of CBM wells can be scientifically categorized based on the relationship between production boundaries and strategies. Subsequently, a mathematical model is established, and a corresponding calculation program is designed, which is implemented for CBM wells with diverse production characteristics. Finally, by integrating the propagation behavior of pressure drop funnels and the production curves of CBM wells, we offer insights into the optimization of production strategies and the evaluation of production potential.
2. Descriptions and Assumptions
2.1. Descriptions of Production Scenarios
The production life of CBM wells in undersaturated coal reservoirs is commonly classified into three stages: single-phase water flow, gas–water two-phase flow, and single-phase gas flow.28 This classification is contingent upon the relationship between inner and outer boundary conditions and the critical desorption pressure (CDP). Typically, the inner boundary condition is determined by the production strategy, often involving the regulation of bottomhole flow pressure (BHFP) or gas/water production. The outer boundary condition defines the pressure propagation range, typically indicating the production boundary. Consequently, the three production stages of CBM wells in undersaturated coal reservoirs can be further subdivided into five scenarios (Figure 1).
-
1.
Scenario a: During the dewatering process, BHFP gradually decreases but remains higher than CDP, signifying the absence of CBM desorption. Meanwhile, the limited expansion of the drainage radius means that pressure propagation has not yet reached the boundary.
-
2.
Scenario b: BHFP remains higher than CDP, while the pressure drop funnel reaches the outer boundary in the drainage area, signifying a decrease in reservoir pressure at the boundary due to depressurization. This scenario occurs when intentionally slowing down the drop rate of BHFP or in coal reservoirs with high permeability.
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3.
Scenario c: In contrast to Scenario b, BHFP rapidly falls below CDP during the outward expansion of the drainage radius, triggering CBM desorption. Simultaneously, both the drainage and desorption radii expand within the coal seam, indicating the coexistence of desorption and drainage areas in this scenario. According to relative permeability theory, water and gas perpetually compete for the seepage channel in fractures and cleat systems, thereby inhibiting each other’s pressure propagation.6,10
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4.
Scenario d: Continuing from scenario B, BHFP continues to drop and gas desorption initiates. Concurrently, the reservoir pressure at the boundary progressively decreases to CDP within the drainage area, while the desorption radius propagates to the outer boundary in the desorption area. This scenario enables the determination of both the desorption radius and the boundary pressure.
-
5.
Scenario e: Both the drainage and desorption radii reach the outer boundary, and the overall coal reservoir is in the desorption area. In this scenario, only the reservoir pressure at the boundary decreases with production, enabling complete CBM desorption and production. This scenario lasts until the end of the production life.
Figure 1.
Propagation scenarios of the pressure drop funnel for CBM wells in undersaturated coal reservoir.
2.2. Assumptions
The dynamic propagation of the pressure drop funnel is characterized based on the following assumptions: (1) The heterogeneity of coal seams and the overflow recharge of aquifers are ignored; (2) CBM and formation water are considered as ideal gas and slightly compressible fluid, and follow the Langmuir equation and Darcy’s law, respectively; (3) A vertical well with infinitely conductive hydraulic fractures is located at the center of the coal reservoir; (4) the outer boundary condition is confined by the interwell pressure interference or damage range, while the inner boundary condition is artificially adjusted by BHFP; (5) Some geological parameters, such as porosity and water saturation, are dynamic with depressurization, and can be defined as follows.
The effects of effective stress and matrix shrinkage provide opposing mechanisms for dynamic porosity. The effective stress effect operates throughout the entire depressurization process, compressing the coal matrix and resulting in porosity damage. Conversely, the matrix shrinkage effect occurs after gas desorption, resulting in porosity recovery. The superposition of these two opposing effects determines the final change in porosity:29
 |
1 |
where the cleat compressibility, sensitive to pressure, is expressed as a linear relationship:30
 |
2 |
In traditional models, the water saturation is presumed to decrease directly from the initial value to the irreducible value under a pressure gradient, a representation clearly inconsistent with empirical observations. Notably, water saturation exhibits specific dynamic mechanisms in different areas: in the drainage area, changes in water saturation are primarily prompted by coal compressibility and water compressibility, which can be disregarded; conversely, spatial occupation resulting from CBM desorption emerges as the dominant factor in the desorption area, significantly impacting dynamic water saturation. Consequently, a mathematical model characterizes the relationship between water saturation and pressure, providing a comprehensive representation of dynamic profiles.31
 |
3 |
 |
4 |
 |
5 |
In the drainage area:
 |
6 |
In the desorption area:
 |
7 |
where the desorption compressibility is expressed as follows:32
 |
8 |
WGMR is water–gas mobility ratio, and can be expressed as
 |
9 |
 |
10 |
3. Mathematical Model Construction
3.1. Pressure Profiles in Different Production Scenarios
Pressure profiles provide a tangible representation of the pressure drop funnel. These profiles can be expressed as a continuous succession of steady-state flow in each area:27
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11 |
 |
12 |
The above equations are suitable for CBM wells without fracturing stimulation. However, for fractured CBM wells, the pressure propagation speed along the fracture direction significantly accelerates, resulting in elliptical propagation characteristics instead of approximately circular ones. To simplify calculations, a conformal transformation is employed:23
 |
13 |
By substituting eq 13 into eqs 11, 12, the pressure profiles are transformed into linear geometry:
 |
14 |
 |
15 |
Considering the subdivided production stages, key parameters, such as the drainage and desorption radii, boundary pressure, CDP, and BHFP, are substituted into eqs 14 and 15 correspondingly. The pressure profiles in various production scenarios are finally described as follows:
Scenario a:
 |
16 |
Scenario b:
 |
17 |
Scenario c:
 |
18 |
Scenario d:
 |
19 |
Scenario e:
 |
20 |
Compared to ξi and ξcd, ξwf is infinitesimal and set to zero. Notably, although the two radii in linear geology are easy to obtain, the calculation results in elliptical geology are more intuitive. Therefore, corresponding fronts of drainage/desorption areas are transformed into elliptical geometry upon completion of the calculations:
 |
21 |
 |
22 |
In conclusion, the pressure profiles have been established to describe the pressure drop funnels in each production scenario.
3.2. Material Balance Equations (MBE) During CBM Development
MBE is guided by the volume conservation principle, and the production mechanisms of the gas phase and water phase are completely different in CBM development. In gas MBE, desorbed gas and free gas dominate due to the infinitesimal proportion of dissolved gas.33 Therefore, gas production is expressed as
 |
23 |
In the absence of external water recharge, water production is a complicated process in undersaturated coal reservoirs because of two primary mechanisms. (1) The decrease in reservoir pressure can lead to pore compression and elastic expansion of water, resulting in a small amount of water production (Wp1). This process continues during depressurization.34 (2) Gas desorbed from the coal matrix gradually occupies the pore space, leading to a decrease in water saturation. This portion of the formation water flows through the cleat systems and ultimately produces from the wellbore (Wp2). Therefore, water production is expressed as
 |
24 |
Upon integrating the pressure profiles in various production scenarios, the gas and water MBE are expressed in differential form.
Scenario a,b:
 |
25 |
Scenario c,d,e:
 |
26 |
 |
27 |
 |
28 |
The above MBE are established under elliptical geology. To combine the pressure profiles and the MBE in the same dimension, the Jacobian matrix is employed and expressed as23
 |
29 |
Substituting eq 29 to eq 28, then we have
 |
30 |
Substituting eq 30 into eqs 25, 26 and integrating, the cumulative gas and water production can be calculated:
Scenario a,b:
 |
31 |
Scenario c,d,e:
 |
32 |
where
 |
33 |
Substituting eqs 1–10, 16–20 into eqs 31–33, the mathematical model is ultimately established. Calculation procedures must be devised to ensure efficient utilization of the model.
3.3. Technical Route and Calculation Procedures
The calculation procedures consist of the following steps (Figure 2):
Figure 2.
Technology roadmap of the mathematic model.
3.3.1. Determining the Calculation Method
The unknowns in the model are desorption radius, drainage radius, and boundary pressure, which determine the pressure profile in each scenario. Due to the difficulty of solving the unknowns analytically, an iterative method is the optimal approach to apply. Cumulative gas and water production can be obtained by substituting known quantities (geological and production parameters) and assumed unknowns into the model. The assumed values are continuously optimized until the error between the calculated results and the actual data is less than 1%. The monotonous increase in cumulative gas/water production indicates the convergence of the method and unique solutions for the unknowns.
3.3.2. Characterizing the Dynamic Behavior of Geological Parameters
According to the MBE, the dynamic behaviors of geological parameters are pivotal in gas/water migration and pressure propagation. The fundamental geological parameters are substituted into eqs 1–10 to depict the dynamic behaviors of porosity and water saturation; these dynamic curves are utilized in subsequent calculations.
3.3.3. Calculating Production Boundary
Production boundaries include both pressure interference boundary and damage boundary, indicating that pressure propagation cannot continue to expand outward after reaching the boundary. It is imperative to calculate the actual boundaries based on real production data and geological parameters rather than making assumptions. The model in scenarios (a, c) is utilized to derive the curves of drainage radius and desorption radius, respectively. The intersection of these curves indicates the location of the production boundary and its corresponding time. For a detailed discussion of the calculation method and principles, readers are referred to our previous research.8
3.3.4. Subdividing Production Scenarios
The determination of production scenarios is contingent on inner and outer boundary conditions. The relationship between BHFP and CDP determines the occurrence of CBM desorption, leading to the division of production scenarios into the single-phase water flow stage and the gas desorption stage. Furthermore, based on the relationship between pressure propagation radii and the production boundary, these two stages can be further subdivided into scenarios (a, b) and scenarios (c, d, e), respectively.
3.3.5. Program Calculation Process
Gas/water migration and pressure propagation mechanisms vary among scenarios, necessitating corresponding operational procedures for the model.
Scenario a and Scenario b: CBM development initiates from scenario a in undersaturated coal reservoirs, transitioning to scenario b as drainage radius reaches the production boundary. Consequently, values for drainage radius and boundary pressure are assumed in scenario a and scenario b, respectively, and pressure profiles in the drainage area are described. Subsequently, cumulative water production in the drainage area is calculated correspondingly, and the accurate drainage radius or boundary pressure is obtained through iterative optimization, ultimately characterizing the pressure drop funnel in the drainage area.
Scenario c and Scenario d: Both the desorption area and the drainage area coexist in these two scenarios, with gas being solely produced from the desorption area while water is produced from both areas. Initially, assuming a desorption radius establishes a pressure profile in the desorption area, and calculates the cumulative gas production. Iterative optimization is employed to determine the accurate desorption radius by matching cumulative gas production, followed by calculating cumulative water production in the desorption area. Subsequently, in scenario c and d, values for the drainage radius and the boundary pressure are assumed respectively, and pressure profiles, along with cumulative water production in the drainage area, are obtained. These assumed values are optimized when the sum of cumulative water production in both desorption and drainage areas matches the actual water production. The pressure drop funnels in these scenarios are finally characterized according to the accurate pressure profiles in these two areas. It is pertinent to highlight that scenarios b and c are incompatible and cannot occur simultaneously during the production life of the same well. Nevertheless, they will ultimately turn to scenario e.
Scenario e: Only the boundary pressure needs to be assumed, enabling the determination of a pressure profile and further calculation of cumulative gas production. The accurate boundary pressure is determined through iterative optimization, ultimately characterizing the pressure drop funnel in this scenario.
The analysis comprehensively delineates the propagation characteristics of the pressure drop funnels throughout the entire production period in detail. The clarity of the calculation procedures is enhanced by the accompanying flowchart (Figure 3).
Figure 3.
Flowchart of the calculation procedures.
4. Results and Discussion
4.1. Well Selection
The anthracite coal reservoir labeled as 3# within the Shanxi Formation of the Shizhuangnan Block has undergone development for several decades, with its geological conditions extensively documented in previous studies (Figures 4 and 5).35−40 Hydraulic fracturing is commonly employed in the block due to its low permeability, resulting in the formation of narrow fractures with a typical half-length of approximately 60 m, based on microseismic detection and fracturing simulation.41,42
Figure 4.
Location and production characteristics of the target wells: (a) location of the target wells; (b) production characteristics of the target wells.
Figure 5.
Stratigraphic column of the study area.
This work focuses on two vertical wells selected as case studies to analyze the propagation of pressure drop funnels (Figure 4). Geological parameters are acquired through experimental testing and refined by numerical simulation, revealing slight discrepancies between these two wells (Table 1). The production characteristics are typical, with average gas production rates of 1096 and 960 m3/day, respectively. Water production is primarily concentrated in the initial stage, with cumulative water production of 850 and 730 m3, respectively. Despite the similarity in average gas production rate and cumulative water production, significant differences exist in the gas production characteristics. The initial daily gas production of well A is relatively low, gradually increasing to 1800 m3/day after 1600 days, which suggests sustainable gas production in the later stage. In contrast, the daily gas production of well B declines after reaching its peak production of 2000 m3/day, currently dropping below 1000 m3/day. The difference in production characteristics is attributed to varying production strategies. BHFP of well A gradually decreases at production commencement and reaches the abandonment pressure after 1000 days, whereas that of well B drops rapidly within the first 100 days. The productivity difference between these two wells signifies varying characteristics in the propagation of pressure drop funnels, which is representative of the majority of CBM wells in the study area.
Table 1. Actual Geological Parameters of the Target Wells.
| parameters | well A | well B | parameters | well A | well B |
|---|---|---|---|---|---|
| VL(m3/t) | 30 | 36 | Pi(MPa) | 3 | 3.3 |
| φi | 1% | 1% | Pcd (MPa) | 2.3 | 2.1 |
| Krg* | 0.5 | 0.7 | PL (MPa) | 1 | 1.5 |
| l | 2 | 2 | Swi | 0.97 | 0.95 |
| m | 2.5 | 3 | Swc | 0.6 | 0.6 |
| Cw(MPa–1) | 0.00045 | 0.00045 | Lf (m) | 60 | 60 |
| εmax | 0.7% | 0.7% | rwf (m) | 0.1 | 0.1 |
| h (m) | 6 | 6 | T (K) | 300 | 300 |
| μw (mPa s) | 0.85 | 0.85 | ρ (t/m3) | 1.4 | 1.4 |
| μg (mPa s) | 0.01134 | 0.01134 |
4.2. Advantage Research for Production Boundary
The accurate calculation of production boundaries serves as a fundamental element for the advanced characterization of pressure drop funnels. The dynamic porosity and water saturation, as stipulated in the assumptions, are taken into account. Substituting parameters such as initial pressure and porosity of the coal reservoir, critical desorption pressure, cleat compressibility, maximum volumetric strain and pressure profiles into eqs 1 and 2 allows for plotting the relationships between porosity and pressure. Similarly, substituting parameters such as relative permeability, desorption compressibility, cleat compressibility and gas compressibility into eqs 3–10 allows for plotting the relationships between water saturation and pressure (Figure 6). The dynamic porosity curves of the two wells exhibit a “rebound type” behavior, ultimately reverting to their original values. For well A, water saturation continuously decreases when the pressure is below CDP, signifying water production with the expansion of the desorption radius, followed by a reduction in water production after the desorption radius reaches the boundary. Well B experiences a lesser reduction in water saturation compared to well A, suggesting that water is predominantly extracted before overall gas desorption within the outer boundary.
Figure 6.
Calculation results of dynamic porosity/water saturation of the target wells.
The investigation focuses on assessing the impact of dynamic water saturation on production boundary prediction, utilizing well A as a case study, and conducting a comparative analysis of the outcomes of two models. The results reveal distinct differences in the effects of dynamic water saturation on the expansion of the drainage and desorption radii (Figure 7). Incorporating dynamic water saturation, as opposed to irreducible water saturation, reduces the volume of CBM remaining in the pores, resulting in an increased desorption radius at the same gas production rate. However, this portion of gas volume is considerably smaller than the total CBM production volume, resulting in a minimal deviation in the expansion of the desorption radius when considering dynamic water saturation rather than the irreducible value. Contrastively, the effect mechanism of water saturation on the expansion of drainage radius varies in different areas. Due to the approximately constant water saturation before gas desorption, the curves in the expansion of the drainage radius are nearly coincidental in the drainage area. Subsequently, the decrease in water saturation caused by gas desorption significantly contributes to water production in the desorption area. Incorporating irreducible water saturation, instead of dynamic water saturation, overestimates the recoverable water in the pores, resulting in a smaller drainage radius and production boundary at the same water production rate. Notably, WP2 constitutes a significant portion of cumulative water production, thus emphasizing the importance of dynamic water saturation in predicting the drainage radius.
Figure 7.
Comparison of production boundary prediction for the well A in linear geometry based on different water saturation.
Consequently, employing dynamic water saturation allows for a more extensive production boundary, enhancing accuracy and credibility.
4.3. Sample Calculation
According to the proposed model, the production boundary is initially predicted, and the pressure profiles can be further described based on the actual production data and geological parameters. Since drainage and desorption radii are calculated in linear geology due to hydraulic fracturing, they should be transformed into elliptical geometry by eqs 21 and 22 for a more intuitive analysis. The calculation results for these two wells show significant differences (Table 2, Figure 8). The major production boundary of well A extended to 108.6 m in elliptical geometry, approximately half of the well spacing (120 m), indicating the formation of pressure interference with the adjacent well; the desorption radius gradually increases and ultimately reaches the production boundary after 800 days. In contrast, the production boundary of well B is smaller, measuring less than 100 m in elliptical geometry, significantly below half of the well spacing, indicative of obstruction of fractures or cleat systems during the production process. This boundary is regarded as the damage boundary. The desorption radius increases at a faster rate, reaching the damage boundary in about 400 days.
Table 2. Calculation Results of Production Boundary in Linear and Elliptical Geometries.
| production boundary in linear geometry (dimensionless) | major production boundary in elliptical geometry (m) | minor production boundary in elliptical geometry (m) | |
|---|---|---|---|
| well A | 1.2 | 108.6 | 90.6 |
| well B | 1.05 | 96.2 | 75.2 |
| optimized well B | 1.31 | 119.3 | 103.1 |
Figure 8.
Extension of drainage radius and desorption radius for the target wells in elliptical geometry: (a) well A; (b) well B.
Dynamic pressure drop funnels are constructed to characterize pressure propagation based on the production boundary (Table 3, Figure 9). In the case of well A, the drainage radius reaches the boundary, and the boundary pressure decreases before BHFP drops below CDP. Then, the desorption radius expands as BHFP drops following gas desorption initiation. The dynamic behaviors of water saturation and pressure propagation result in a gradual occupation of the pores and cleat systems by desorption gas in the desorption area, causing a decrease in water saturation in the coal reservoir and in water production in the well A. The boundary pressure continuously decreases after interwell pressure interference, leading to increased gas production, while daily water production is low or almost nonexistent in the later stage due to a minor change in water saturation.
Table 3. Calculation Results of Pressure Drop Funnel for the Well A.
| time (day) | cumulative water production WP (m3) | cumulative gas production GP (m3) | BHFP Pwf (MPa) | drainage radius in linear geometry ξi (dimensionless) | desorption radius in linear geometry ξi (dimensionless) | reservoir boundary pressure Pe (MPa) |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 3 | 0 | 0 | 3 |
| 10 | 7 | 0 | 2.9 | 0.8 | 0 | 3 |
| 20 | 10 | 0 | 2.8 | 1 | 0 | 3 |
| 30 | 24 | 0 | 2.6 | 1.1 | 0 | 3 |
| 40 | 36 | 0 | 2.5 | 1.2 | 0 | 3 |
| 50 | 69 | 0 | 2.3 | 1.2 | 0 | 2.86 |
| 100 | 190 | 32,932 | 1.4 | 1.2 | 0.25 | 2.4 |
| 200 | 231 | 82,242 | 1 | 1.2 | 0.4 | 2.4 |
| 300 | 321 | 152,830 | 0.9 | 1.2 | 0.66 | 2.3 |
| 400 | 385 | 201,121 | 0.85 | 1.2 | 0.78 | 2.3 |
| 500 | 451 | 254,883 | 0.8 | 1.2 | 0.9 | 2.3 |
| 600 | 557 | 316,023 | 0.8 | 1.2 | 1.05 | 2.3 |
| 800 | 702 | 477,897 | 0.5 | 1.2 | 1.2 | 2.3 |
| 1000 | 744 | 658,115 | 0.2 | 1.2 | 1.2 | 2.12 |
| 1200 | 771 | 873,219 | 0.15 | 1.2 | 1.2 | 1.89 |
| 1400 | 785 | 1,086,484 | 0.15 | 1.2 | 1.2 | 1.7 |
| 1600 | 800 | 1,317,075 | 0.15 | 1.2 | 1.2 | 1.5 |
| 1800 | 813 | 1,645,649 | 0.15 | 1.2 | 1.2 | 1.25 |
| 2000 | 822 | 1,978,565 | 0.15 | 1.2 | 1.2 | 1.05 |
| 2200 | 825 | 2,223,602 | 0.15 | 1.2 | 1.2 | 0.9 |
| 2400 | 825 | 2,464,518 | 0.15 | 1.2 | 1.2 | 0.79 |
Figure 9.
Propagation of pressure drop funnel and water saturation for the well A in linear geometry.
The pressure drop funnel for well B exhibited different characteristics (Figure 10). The rapid drop of BHFP results in gas desorption during the propagation of the drainage radius. Then the boundary depressurization and desorption radius expansion are carried out simultaneously. These phenomena lead to the transition of the production life into scenario e after only 300 days. Additionally, there is minimal water production after the desorption radius reached the boundary due to a slight change in water saturation, and gas production decreases in scenario e due to a smaller damage boundary.
Figure 10.
Propagation of pressure drop funnel and water saturation for the well B in linear geometry.
As shown above, the proposed model can be utilized to calculate the production boundary (pressure interference boundary or damage boundary) in combination with dynamic water saturation and to describe the pressure drop funnel with various production characteristics. Notably, differences in production characteristics correspond to different types of pressure drop funnels, warranting in-depth study of their underlying causes.
4.4. Production Strategy Optimization for CBM Wells
Production strategies, adjusted by humans, are closely related to characteristics of pressure propagation and CBM production. For instance, an inappropriate production strategy leads to rapid degeneration of gas production in well B rather than its sustenance. Therefore, it is crucial to optimize the production strategy and determine the appropriate type of pressure drop funnel.
Early stage implementation of a rapid BHFP drop for well B results in gas desorption near the wellbore, which potentially induce a gas-lock effect and subsequently restrict the expansion of the drainage radius. The production life of well B experienced the scenarios of (a, c, d, and e). In order to achieve broader production boundary expansion and enhance gas production sustainability, the production strategy is optimized by reducing the BHFP drop rate before 300 days (Figure 11a).
Figure 11.
Production strategy optimization for the well B: (a) comparison of actual and optimized BHFP; (b) comparison between actual and optimized gas production.
The numerical simulation software Comet3.0 is employed to conduct a comprehensive analysis of production strategy optimization. In the simulation, history matching with actual BHFP proves valuable for refining geological parameters, and production prediction with optimized BHFP is utilized to validate the reliability of the optimization criteria. As shown in Figure 11b, the optimized daily gas production displays slower growth and a lower peak value than the actual value, yet it exceeds the actual daily gas production after 200 days. Importantly, the period of stable production is prolonged, and decline in daily gas production is less pronounced for the optimized case. As a result, a more extensive production boundary is reached, exceeding 120 m, albeit requiring a longer time of approximately 600 days (Table 2, Figure 12).
Figure 12.
Extension of the drainage radius and desorption radius for the optimized well B: (a) in elliptical geometry; (b) in the top view.
A comparison of the characteristics of the pressure drop funnel and dynamic water saturation between these two cases reveals that artificially adjusting the production strategy alters the production life type (Table 4, Figures 10 and 13). Decreasing the drop rate of BHFP in the early stage promotes pressure propagation and depressurization in the drainage area, facilitating the extensive expansion of the desorption radius. Consequently, the large desorption area provides abundant CBM resources for the CBM wells. Conversely, an inappropriate production strategy constrains the expansion of the production boundary, resulting in fewer CBM resources within the scope of pressure propagation. This is the primary reason for the decline in the actual daily gas production of well B.
Table 4. Calculation Results of Pressure Drop Funnel for the Well B.
| time (day) | BHFP Pwf (MPa) | drainage radius in linear geometry ξi (dimensionless) | desorption radius in linear geometry ξcd (dimensionless) | reservoir boundary pressure Pe (MPa) | ||||
|---|---|---|---|---|---|---|---|---|
| actual | optimized | actual | optimized | actual | optimized | actual | optimized | |
| 0 | 3.30 | 3.30 | 0 | 0 | 0 | 0 | 3.3 | 3.3 |
| 10 | 2.70 | 2.80 | 0.3 | 0.3 | 0 | 0 | 3.3 | 3.3 |
| 20 | 2.35 | 2.60 | 0.7 | 0.6 | 0 | 0 | 3.3 | 3.3 |
| 30 | 2.20 | 2.44 | 0.98 | 0.8 | 0 | 0 | 3.3 | 3.3 |
| 40 | 1.90 | 2.22 | 1 | 1 | 0.05 | 0 | 3.3 | 3.3 |
| 50 | 1.53 | 2.10 | 1.05 | 1.2 | 0.15 | 0 | 3.1 | 3.3 |
| 100 | 0.72 | 1.55 | 1.05 | 1.31 | 0.37 | 0.16 | 2.9 | 3 |
| 200 | 0.62 | 0.91 | 1.05 | 1.31 | 0.75 | 0.46 | 2.6 | 2.2 |
| 300 | 0.54 | 0.66 | 1.05 | 1.31 | 1 | 0.75 | 2.3 | 2.1 |
| 400 | 0.40 | 0.40 | 1.05 | 1.31 | 1.05 | 0.95 | 2 | 2.1 |
| 500 | 0.20 | 0.20 | 1.05 | 1.31 | 1.05 | 1.12 | 1.9 | 2.1 |
| 600 | 0.20 | 0.20 | 1.05 | 1.31 | 1.05 | 1.3 | 1.8 | 2.1 |
| 800 | 0.20 | 0.20 | 1.05 | 1.31 | 1.05 | 1.31 | 1.6 | 1.85 |
| 1000 | 0.20 | 0.20 | 1.05 | 1.31 | 1.05 | 1.31 | 1.4 | 1.65 |
| 1200 | 0.20 | 0.20 | 1.05 | 1.31 | 1.05 | 1.31 | 1.25 | 1.47 |
| 1400 | 0.20 | 0.20 | 1.05 | 1.31 | 1.05 | 1.31 | 1.1 | 1.3 |
| 1600 | 0.20 | 0.20 | 1.05 | 1.31 | 1.05 | 1.31 | 0.95 | 1.16 |
| 1800 | 0.20 | 0.20 | 1.05 | 1.31 | 1.05 | 1.31 | 0.8 | 1.05 |
| 2000 | 0.20 | 0.20 | 1.05 | 1.31 | 1.05 | 1.31 | 0.65 | 0.95 |
| 2200 | 0.20 | 0.20 | 1.05 | 1.31 | 1.05 | 1.31 | 0.55 | 0.85 |
| 2400 | 0.20 | 0.20 | 1.05 | 1.31 | 1.05 | 1.31 | 0.47 | 0.78 |
| 2600 | 0.20 | 0.20 | 1.05 | 1.31 | 1.05 | 1.31 | 0.41 | 0.71 |
Figure 13.
Optimized pressure drop funnel and water saturation for the well B in linear geometry.
In conclusion, adjusting BHFP during the production process to steer the production life toward scenario (a, b, d, and e) rather than scenario (a, c, d, and e) is beneficial for efficient development of CBM (Figures 14 and 15).
Figure 14.
Dynamical characteristics of actual pressure drop funnel for the well B.
Figure 15.
Dynamical characteristics of the optimized pressure drop funnel for the well B.
4.5. Productivity Potential Prediction
In addition to production strategy optimization, dynamic propagation of the pressure drop funnel can predict the productivity potential of CBM wells. This is because the dynamic resources of CBM wells are closely tied to reservoir pressure and the production boundary. Two cases of well B are analyzed, and dynamic CBM resources are compared correspondingly (Table 5, Figure 16). The well-controlled resources increase as the desorption radius extends until it reaches the production boundary, enabling the determination of recovery efficiency and residual CBM resources based on gas production.
Table 5. Calculation Results of CBM Resources for the Well B.
| time (day) | well-controlled resources (m3) | cumulative gas production (m3) | residual resources (m3) | recovery efficiency (dimensionless) | ||||
|---|---|---|---|---|---|---|---|---|
| actual | optimized | actual | optimized | actual | optimized | actual | optimized | |
| 200 | 212,4842 | 1,053,186 | 273,323 | 128,080 | 1,851,519 | 925,106 | 12.86% | 12.16% |
| 400 | 4,013,475 | 3,261,352 | 538,364 | 423,461 | 3,475,111 | 2,837,891 | 13.41% | 12.98% |
| 600 | 4,013,475 | 6,817,223 | 784,110 | 751,436 | 3,229,365 | 6,065,787 | 19.54% | 11.02% |
| 800 | 4,013,475 | 6,817,223 | 993,242 | 1,100,362 | 3,020,233 | 5,716,861 | 24.75% | 16.14% |
| 1000 | 4,013,475 | 6,817,223 | 1,188,000 | 1,44,2836 | 2,825,475 | 5,374,387 | 29.60% | 21.16% |
| 1500 | 4,013,475 | 6,817,223 | 1,673,400 | 2,259,061 | 2,340,075 | 4,558,162 | 41.69% | 33.14% |
| 2000 | 4,013,475 | 6,817,223 | 2,145,610 | 2,939,681 | 1,867,865 | 3,877,542 | 53.46% | 43.12% |
| 2500 | 4,013,475 | 6,817,223 | 2,533,943 | 3,499,521 | 1,479,532 | 3,317,702 | 63.14% | 51.33% |
Figure 16.
Comparison of the actual and optimized productivity potential for well B.
Substantial differences are apparent in the dynamic CBM resources between the two cases. CBM resources and cumulative gas production with the actual production strategy increase more rapidly than in the optimized case during the early stage. However, the well-controlled resources are lower, amounting to even less than 4.1 million m3. The actual recovery efficiency of the well is 63.14%, indicating a likely rapid decline in daily gas production in the near future as it nears the end of its lifespan. Contrastively, CBM resources and cumulative gas production with the optimized production strategy surpass the actual values before 800 days, with optimized well-controlled resources reaching about 7 million m3. Importantly, the optimized residual resources exceed twice the actual value, with a recovery efficiency of approximately 51.33%, suggesting the sustainability of gas production under the optimized conditions.
Although the rapid BHFP drop rate causes peak production in the initial stage, massive gas desorption near the wellbore impedes the extraction of distant formation water, severely inhibiting efficient propagation of the pressure drop funnel. As a result, the coal reservoir suffers irreversible damage and the well-controlled resources become restricted. Reconstruction is a prevalent method employed to resolve issues stemming from inappropriate production strategies. The objective is to penetrate obstructed fractures and cleat systems, thereby extending the production boundary. These studies have emerged as a significant focus in contemporary CBM development.43,44 In summary, this study offers a fresh perspective and an effective method for describing the pressure drop funnel and assessing the productivity potential of CBM wells.
5. Conclusions
-
1.
This contribution presents a semianalytical mathematical model that integrates pressure profiles and MBE to characterize pressure propagation, incorporating factors such as actual production boundaries, production strategies, artificial fractures, and dynamic porosity and water saturation distribution. This model offers a time-saving and practical tool for describing the dynamic pressure drop funnel in various CBM production stages and promoting efficient development of undersaturated CBM reservoirs.
-
2.
The value of water saturation has an important impact on boundary prediction, as it affects the error in the calculation results of drainage radius and desorption radius significantly. Considering irreducible water saturation, the predicted expansion of the desorption radius will be slightly larger, while that of the drainage radius will be significantly smaller. Consequently, the predicted production boundary will be smaller than the actual value. Accounting for dynamic water saturation can rectify this discrepancy, leading to a more accurate prediction of the production boundary.
-
3.
The optimization criterion for production strategy is proposed based on the model. It is recommended to reduce the drop rate of BHFP for a larger drainage radius and lower boundary pressure in the drainage area before the occurrence of massive gas desorption, thereby promoting the expansion of the desorption radius and the formation of interwell pressure interference. Sufficient well-controlled resources will support high and stable production of CBM wells.
Acknowledgments
This research was funded by the National Natural Science Foundation of China (grant numbers 42302201, 42272197) and Fundamental Research Program of Shanxi Province (grant number 202103021223034).
Glossary
Nomenclature
- Bg
gas volume coefficient (dimensionless)
- Bgi
gas volume coefficient at initial pressure (dimensionless)
- Bw
water formation volume coefficient (dimensionless)
- Cf
coal compressibility (MPa–1)
- Cg
gas phase compressibility (MPa–1)
- Cw
water phase compressibility (MPa–1)
- Cd
desorption compressibility (MPa–1)
- GP
cumulative volume of gas production (m3)
- Gd
cumulative volume of desorption gas (m3)
- Gf
volume of free gas under initial conditions (m3)
- Gr
volume of gas remained in the cleat system (m3)
- h
coal seam thickness (m)
gas phase permeability under irreducible water saturation (fraction)
- Lf
half-length of hydraulic fracture (m)
- l
Corey constant for gas phase (dimensionless)
- m
Corey constant for water phase (dimensionless)
- P
reservoir pressure (MPa)
- Pi
initial reservoir pressure (MPa)
- Pn
the formation pressure at step n (MPa)
- Pn+1
the formation pressure at step n + 1 (MPa)
- Pe
reservoir boundary pressure (MPa)
- Pcd
critical desorption pressure (MPa)
- PL
Langmuir pressure (MPa)
- Pg
pressure profile in the desorption area (MPa)
- Pw
pressure profile in the drainage area (MPa)
- Psc
pressure under standard condition (MPa)
- Pwf
bottom hole flowing pressure (BHFP) (MPa)
- Re
distance of reservoir boundary (m)
- rwf
wellbore radius (m)
- ri
drainage radius (m)
- rcd
desorption radius (m)
- r
pressure propagation radius (m)
- Swc
irreducible water saturation of reservoir (fraction)
the water saturation at step n (dimensionless)
the water saturation at step n + 1 (dimensionless)
- T
formation temperature (K)
- Tsc
temperature under standard condition (K)
- μw
water viscosity (mPa s)
- μg
gas viscosity (mPa s)
- VL
Langmuir volume (m3/t)
- WP
cumulative water production (m3)
- Wp1
water production in the drainage area (m3)
- Wp2
water production in the desorption area (m3)
- x, y
Cartesian coordinate (m)
- xi
major semiaxis of the boundary equipotential line of drainage area
- xcd
major semiaxis of the boundary equipotential line of desorption area
- yi
minor semiaxis of the boundary equipotential line of drainage area
- ycd
minor semiaxis of the boundary equipotential line of desorption area
- Z
compressibility factor at reservoir condition (fraction)
- Zsc
compressibility factor at standard condition (fraction)
Glossary
Greek letters
- φi
initial porosity of coal reservoir (dimensionless)
- φ1
the dynamic porosity in the drainage area (dimensionless)
- φ2
the dynamic porosity in the desorption area (dimensionless)
- εmax
maximum volumetric strain (dimensionless)
- ξ, η
elliptical coordinates (dimensionless)
- ξwf
distance of drainage tunnel (dimensionless)
- ξi
drainage radius in linear geometry (dimensionless)
- ξcd
desorption radius in linear geometry (dimensionless)
- ρ
coal density (g/cm3)
The authors declare no competing financial interest.
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