Abstract
In the development of coalbed methane, the reasonable expansion speed of pressure drop directly affects the effective desorption range and the ability to maintain stable production continuously. Currently, the existing pressure drop propagation models either only adopt the gas pressure propagation formula throughout the gas production stage or neglect the pressure fluctuations in the early stage of gas production, which does not conform to the actual situation. Formulas for calculating the gas saturation at different times and different radii are established. On this basis, a composite model of the combined propagation of water pressure and gas pressure is established. Furthermore, a desorption funnel expansion model that takes into account the dynamic changes of relative permeability and gas saturation over time and space is established, and the pressure drop propagation law applicable to the whole process of gas production is obtained. It is found that during the stage of simultaneous production of gas and water, the damage caused by stress sensitivity to permeability is lower than the positive effect produced by the matrix shrinkage effect. Moreover, the decline amplitude of the bottom hole flowing pressure can be determined according to the highest point of the Rg value curve of the boundary line, so as to maintain a relatively high desorption efficiency. The model in this paper can characterize the impact of the dynamic changes of gas and water on the expansion of the desorption funnel and can evaluate the changes of parameters such as reservoir pressure drop and relative permeability more accurately, laying a foundation for adjusting production measures.
Keyword: Coalbed methane, Composite pressure drop propagation model, Desorption funnel model
Subject terms: Energy science and technology, Engineering
Introduction
The proven reserves of coalbed methane(CBM) in the world account for more than 30% of the total proven natural gas reserves in the world1. CBM plays a pivotal role in the energy field. Coal reservoirs in China are characterized by low pressure, low permeability and low saturation2,3. CBM exists in the coal matrix in an adsorbed form4, and pressure reduction through drainage is required to achieve desorption, diffusion and seepage for production5. In the process of pressure reduction, the reasonable expansion speed of pressure drop directly affects the effective desorption range and the ability to maintain stable production continuously6,7. When the expansion speed of pressure drop is too large, it will cause stress sensitivity of the reservoir, increase the degree of fracture closure, and cause irreversible damage to permeability8, resulting in a significant limitation on the drainage volume and failure to achieve the expected goals; when the expansion speed of pressure drop is relatively small, it will delay the gas production time and increase production costs. Especially in the stage of simultaneous production of gas and water, there are numerous factors affecting the pressure drop propagation. Appropriate evaluation indicators need to be selected and appropriate propagation models need to be established to achieve the expected gas production goals.
Currently, the pressure drop expansion law is mainly obtained by constructing gas–water two-phase seepage equations and using numerical simulation methods to solve the models9. Liu et al.10used the planar radial seepage equation to represent the distribution law of the pressure drop funnel and gave an index representing the shape of the pressure drop funnel by the ratio of the drainage radius to the depth of the pressure drop funnel. Wang et al.11 gave the formula for the single-well drainage radius and characterized the shape of the pressure drop funnel by the ratio of the drainage radius to the longitudinal pressure drop size of the pressure drop funnel. However, the maximum distance of pressure drop expansion and the desorption range were not distinguished12. Liu et al.13 believed that the production pressure difference in the two-phase production stage is equal to the sum of water pressure and gas pressure and conforms to the law of gas planar radial flow, and gave a pressure distribution formula including the radius of the pressure drop funnel, but did not distinguish between the desorption range and the funnel outside the desorption range14. Zhao et al.15 combined the seepage equation with the material balance equation to calculate the propagation law of the pressure drop funnel. Clarkson16, Xu et al.17divided the undersaturated coal reservoir into the drainage area and the desorption area with the critical desorption pressure as the boundary. Sun et al.18 combined the material balance equation with reservoir parameters and production data to solve the variation law of the average reservoir pressure. Liu et al.19 established a pressure distribution model according to the theory of oil and gas seepage and using Darcy’s law, and gave the formula for the drainage radius. Hu et al.20established the gas–water seepage equation according to Darcy’s law and the continuity equation, comprehensively considered the change of permeability, and obtained the reservoir pressure drop model under the condition of dual porosity, then approximated the wellbore flow as a pseudo-steady flow, and simultaneously considered the influence of coalbed permeability changes with matrix shrinkage and effective stress changes during the drainage process21. Yan et al.22 established prediction models for the propagation of different types of pressure drop funnels in undersaturated CBM reservoirs.
In summary, the current pressure drop propagation models can be mainly divided into three categories (Table 1).
Table 1.
Comparison of characteristics of existing methods.
| Number | Method Name | Method Process | Disadvantages |
|---|---|---|---|
| 1 | Pseudo steady state method | Combine the continuity equation with Darcy’s formula, adopt the pseudo-pressure form for the gas phase, replace the pressure at each point in the reservoir with the average pressure, and adopt the pseudo-steady approximation method | The model results are complex. Solved by numerical simulation methods, which feature large computation volume and complicated processes30 |
| 2 | Staged simplification method | Simplify the production stage into the high water content stage and the high gas content stage, and respectively adopt the formula of water pressure propagation(WPP) and the formula of gas pressure propagation(GPP) | The impact of changes in gas content on pressure drop propagation is not considered, which does not conform to the actual process30 |
| 3 | empirical formula method | Combines empirical formulas with the production data of specific research blocks | Difficult to obtain universal conclusions |
In response to the shortcomings of the above methods, this paper abandons the traditional pseudo-pressure and pseudo-steady state, and a desorption funnel model considering the dynamic changes of permeability and gas saturation was established. It first establishes the gas saturation equations at different times and different mining radii, and then establishes a desorption funnel expansion model that considers the dynamic changes of gas-phase permeability, water-phase permeability and water saturation. Furthermore, the pressure drop propagation law applicable to the whole process from the start of gas production to the attenuation of gas production is obtained.
Proposition of the Method
Currently, in the stage of simultaneous production of gas and water, the radius of the desorption funnel still uses the pressure drop funnel formula in the drainage stage instead, and no relationship has been established with gas production parameters. In the early stage of gas production, the desorption funnel and the pressure drop funnel exist simultaneously, and there are multiple flow patterns from the wellbore to the boundary. In the desorption area, when desorption just begins, the amount of gas is relatively small, and the WPP can still be used. However, when the gas content increases, both WPP and GPP will exist simultaneously. Different gas saturation levels in the reservoir result in different laws followed by pressure propagation. Nevertheless, at present, either the GPP is uniformly adopted, or the pressure drop propagation model is established only for the stable stage while ignoring the fluctuating stage, which does not conform to the actual situation. There is currently no research on how to select an appropriate pressure propagation formula according to the dynamic changes of water saturation. Besides, the dynamic changes of gas-phase permeability and water-phase permeability also need to be considered.
In response to this, in the gas–water two-phase flow stage, the entire reservoir is regarded as a composite flow of fluids. The desorption area near the wellbore is a gas–water two-phase flow, and outside the desorption area, part of it is single-phase water flow, and part has not yet flowed. The range from the wellbore to the edge of the desorption area is called the desorption funnel radius, within which there are both WPP and GPP. The range from the wellbore to the boundary of the single-phase water flow is called the radius of the pressure drop funnel.
(1) At the initial stage of gas production, due to the small amount of gas, the entire desorption area conforms to WPP. When the gas production amount is relatively large and the gas saturation in some areas is relatively high, it begins to conform to GPP. Since the WPP speed is inherently higher than the GPP speed, continuing to use the WPP will deviate from the actual situation. Meanwhile, due to the supply and pressurization effect, the average reservoir pressure calculated by the GPP will change from a rapid decline to a gradually stable state, and the gas saturation corresponding to the inflection point moment is the demarcation point between the two pressure propagation methods.
Therefore, within the desorption area, a composite pressure drop propagation model(CPDPM) combining GPP and WPP is established, and outside the desorption area, the single-phase drainage pressure propagation formula is adopted.
(2) The relationship between water saturation and pressure as well as relative permeability is introduced, along with the concept of composite mobility. Furthermore, the influence of the dynamic changes of gas-phase and water-phase contents is taken into account in the calculation formula of the desorption funnel.
Establishment of the CPDPM in the Desorption Area
In the desorption area, the calculation formula for the radius of the desorption funnel is derived from the pressure conductivity coefficient in seepage mechanics, as shown in Eq. (1):
 |
1 |
where, Rde represents the radius of the desorption funnel, 10−3m, t represents the production time, s, k represents the absolute permeability, 10−3μm2, φ represents the porosity, decimal; μ represents the fluid viscosity, mPa·s, Ct represents the comprehensive compression coefficient, MPa−1.
In Eq. (1), the changes in permeability and viscosity are the results of the combined action of the gas and water phases, that is:
 |
2 |
In Eq. (2), each parameter is the average value within the range of the desorption funnel. In the development of CBM, the relationship between permeability and porosity can be established by using the matchstick model, that is:
 |
3 |
Substitute Eq. (2) and (3) into Eq. (1), and we can obtain:
 |
4 |
In Eqs. (1)–(4), kw and kg represent the effective permeabilities of the water phase and the gas phase respectively, 10−3μm2, krw and krg represent the relative permeabilities of the water phase and the gas phase respectively, μw and μg represent the viscosities of the water phase and the gas phase respectively, mPa·s, k0 represents the initial permeability, 10−3μm2, φ0 represents the initial porosity, decimal.
In the desorption area, the reservoir is simultaneously affected by stress sensitivity and the matrix shrinkage effect caused by gas desorption. The latter has a positive and favorable effect on the porosity and permeability conditions. The porosity under the combined action of these two factors is shown in Eq. (5) 23.
 |
5 |
In Eq. (5), Cp represents the cleat compressibility coefficient of coal rock, MPa−1, pe represents the original reservoir pressure, MPa, 
represents the density of coal rock, g/cm3, BL represents the ratio of the density of CBM under standard surface conditions to that of the condensed CBM underground, which is equal to 0.0018016, VL represents the Langmuir volume, cm3/g, pL represents the Langmuir pressure, MPa, pd represents the critical desorption pressure, MPa, p represents the reservoir pressure when the porosity changes to φ, MPa.
Substitute Eq. (5) into Eq. (4) to obtain the formula for the radius of the desorption funnel, as shown in Eq. (6). Since the pressure and porosity in Eq. (5) can represent either the average values or the single-point values at a certain position. And each parameter in Eq. (6) represents the average value within the desorption funnel. In order to distinguish it from the single-point values mentioned below, a bar is added.
 |
6 |
In the past, various average values were given artificially based on production experience, and it was difficult to make rapid responses and adjustments as production changed. According to the variation law of gas saturation in both spatial and temporal dimensions in this paper, the single-point values at different radii are obtained first, and then the corresponding average values are obtained.
(1) Single-point values of relative permeability.
Based on the known relative permeability curve, the corresponding parameters are fitted by the Corey equation to obtain the calculation formula. For the situation where there is no relative permeability curve, the expression of relative permeability applicable to low-permeability CBM reservoirs given by Liu et al.24 can also be adopted.
 |
7 |
 |
8 |
where, Swc represents the irreducible water saturation, decimal, Sg represents the gas saturation, decimal.
In Eqs. (7) and (8), for a specific research block, the irreducible water saturation can be obtained through experiments, and the variable is the gas saturation. The sum of the water saturation and the gas saturation is 1. Therefore, once the water saturation is obtained, the gas saturation will also be obtained.
(2) Single-point values of water saturation.
The water saturation during the stage of simultaneous production of gas and water changes with time and space. The literature25 gives the relationship between the pressure at each point at different radii of the coal seam and the water saturation.
 |
9 |
In Eqs. (9),
 |
10 |
In Eqs. (9) and (10), 
and 
are the water saturations at different radii at time t and t + 1 respectively, decimal. pt and pt+1 are the pressures at different radii at time t and t + 1 respectively, MPa. WGMR is the water–gas ratio. Cg, Cd and Cw are the gas compressibility coefficient, the desorption compressibility coefficient and the water compressibility coefficient respectively, 1/MPa, and the sum of these four parameters is equal to the Ct. Psc represents the pressure under standard conditions, MPa. Zsc and Z represent the gas deviation factors under standard conditions and formation conditions respectively. Tsc and T represent the temperatures under standard conditions and in the formation respectively, K. Vm is the volume constant of the coal seam gas, cm3/g. b is the pressure constant, 1/MPa.
Before solving Eq. (9), it is also necessary to obtain values such as the gas viscosity and the gas deviation factor. These values are calculated by the methods in the literature26.
The solution process of Eq. (9) is as follows: From the moment when desorption and gas production start, the pressure of a certain infinitesimal element close to the wellbore is equal to the critical desorption pressure, that is, the pressure pt of this infinitesimal element at time t is equal to pd. At this time, the water saturation is approximately equal to 1. Taking the critical desorption pressure and the corresponding water saturation as the initial values at time t, the various parameter values at time t + 1 are solved, and then the values at other times are solved in turn. After that, the values of other infinitesimal elements at different radii are solved. The specific steps include:
1) Solve the water saturation 
of a single infinitesimal element at time t + 1.
Obtain the expressions (11) of A/B and B/C according to Eq. (10).
 |
11 |
At time t, since 
is approximately equal to 1, the effective permeability of the gas phase is approximately equal to 0. Therefore, WGMR = + ∞. Then the values of A/B and B/C at time t can be obtained, as shown in Eq. (12):
 |
12 |
At time t + 1, the pressure pt+1can be obtained through the pressure distribution formula. Substitute the values of 
pt+1, 
pt, 
Swt, A/B and B/C into Eq. (9) to obtain the water saturation at time t + 1.
2) Solve the values of WGMR, A/B and B/C
At time t + 1, the pressure 
is already less than 
and 
is no longer equal to 1. So, WGMR is no longer an infinite value. This value needs to be solved at each subsequent moment, and then the corresponding values of A/B and B/C can be obtained.
a. Obtain the pressure pt+2 at time t + 2 through the pressure distribution formula. Then calculate the gas viscosity, density and deviation factor respectively when the pressure is pt+1. The viscosity of water is determined through experiments.
b. Calculate the value of WGMR through Eq. (10). For the situation where there is no relative permeability curve, substitute Eqs. (7) and (8) into Eq. (10) to obtain:
 |
13 |
In Eq. (13), the water saturation at time t + 1 has already been obtained. After calculating the value of WGMR, the values of A/B and B/C can be obtained.
Substitute the values of WGMR, A/B, B/C,
,
and 
into Eq. (9) to calculate the water saturation at time t + 2.Repeat steps 2) and 3) until time t + n, and the corresponding pressure pt+n and water saturation
are obtained. Change the position of the infinitesimal element to obtain the water saturation values of other infinitesimal elements.
By solving the single-point values, use Eq. (14) to obtain the average relative permeability and the average gas viscosity.
 |
14 |
(3) Average reservoir pressure in the desorption zone based on GPP and WPP.
The literature27 presents the expression and assumptions for GPP in the desorption zone.
 |
15 |
where, pd represents the critical desorption pressure, MPa. Referring to Eq. (15), the WPP within the desorption zone is presented as follows:
 |
16 |
When only GPP is adopted, the average reservoir pressure in the desorption zone can be obtained by using Eq. (15) as follows:
 |
17 |
The numerator of Eq. (17) needs to be solved by means of variable transformation.
 |
Therefore, we obtain:
 |
18 |
In Eq. (18), the final integral part is a transcendental equation and cannot be directly integrated. Expand it for calculation. Generally, when three terms are expanded, the accuracy can fully meet the actual requirements.
 |
19 |
Substitute Eq. (19) into Eq. (18) to obtain the sum of the pressures in the desorption zone, and then calculate the average reservoir pressure within the radius range of the desorption funnel.
When only WPP is adopted, the average reservoir pressure in the desorption zone solved by using Eq. (16) can be found in Eq. (17).
 |
20 |
Both Eq. (17) and (20) contain the unknown quantity of the desorption funnel radius, which can be solved by the iterative method.
The production data of a CBM well in a certain block was selected, and the variation process of the average reservoir pressure under the conditions of using the GPP and WPP separately was plotted, as shown in Fig. 1.
Fig. 1.
Average reservoir pressures calculated by three propagation models.
It can be seen from Fig. 1 that although the changing trends of the average reservoir pressures obtained by the two types of pressure propagation are both gradually decreasing, their patterns differ significantly. For GPP, the absolute reduction value of the average reservoir pressure in the early stage of production is relatively small, but the reduction rate is relatively large, and then it maintains a stable state and decreases slowly. Compared with GPP, WPP shows a significant decrease in the initial stage, and then maintains a stable downward trend. This also verifies the conclusion that the propagation speed of water pressure is higher than that of gas pressure.
Analysis of the above results shows that neither of the pressure propagation methods alone can reflect the actual production process. When the reservoir just starts to desorb, a very small amount of desorbed gas flows with water in the form of bubbles, and the reservoir remains under WPP. Hence, in the initial stage, the calculation results of WPP are more accurate, while those of GPP are relatively high. When the amount of desorbed gas increases to a certain level, the casing pressure rises, the bottom hole flowing pressure(BHFP) drops, and the average reservoir pressure continues to decline. Meanwhile, the supply pressure boost effect28 occurs, which further supplements part of the decreased pressure, making the average reservoir pressure show a stable and slow decrease on the whole. This is basically consistent with the calculation results of GPP. It can be seen that in the initial stage, the results of WPP are more accurate. However, when the amount of desorbed gas increases to the point where the supply pressure boost effect occurs, the results of GPP are more in line with the actual situation. At this time, WPP still maintains a constant pressure drop pattern, which is clearly inconsistent with the actual situation. Therefore, this paper will establish a CPDPM that takes into account the combined effects of water pressure and gas pressure.
(5) Composite Pressure Drop Propagation Model.
The CPDPM includes two dimensions, namely the time dimension and the spatial dimension. In terms of the time dimension, it can be divided into three sub-stages. In the first sub-stage, the overall gas content in the reservoir is relatively low, and the pressure drop still conforms to WPP. In the second sub-stage, both WPP and GPP exist. In the third sub-stage, the average gas saturation is relatively high, and the entire reservoir is suitable for GPP.
As for the spatial dimension, it only applies to the second sub-stage. The reservoir is spatially divided into two parts. The part closer to the wellbore has a lower pressure and a relatively high gas saturation, so it is applicable to GPP, while the part far away from the wellbore is suitable for WPP. The boundary line between GPP and WPP moves from the wellbore towards the boundary and changes dynamically. The model is shown in Fig. 2. The assumptions of this model are the same as those of the formulas for gas pressure propagation and water pressure propagation. It should be noted that this model is an innovative model that connects water pressure propagation and gas pressure propagation by adding a dynamic parameter, the demarcation point Rg, to the calculation formulas of the original water pressure propagation model and gas pressure propagation model. It is not an original model.
Fig. 2.
Schematic diagram of the CPDPM.
In Fig. 2, the lower part represents the distribution of the desorption funnel at different times, and the upper part shows the corresponding changes in gas saturation. Sg0 is the critical value of gas saturation in the research block, indicating the minimum gas saturation required for GPP. At the first two moments ① and ②, the gas saturation at all points within the desorption range is lower than Sg0. Therefore, the WPP is adopted for all. At moment ③, the gas saturation curve intersects with Sg0, and the intersection point is Rg1. Between the wellbore and Rg1, the GPP is already applicable. Similarly, at moment ④, the range conforming to the GPP has expanded to Rg2. At this time, within the desorption range from outside Rg2 to Rde4, the WPP is still applicable. Therefore, within this desorption range, the CPDPM is adopted as a whole.
The amount of desorbed gas is related to the law of reservoir pressure reduction. Therefore, by using the GPP to draw a graph showing the variation law of the average reservoir pressure over time and finding the inflection point of pressure change in the graph, the time required to enter the stable stage under the current pressure propagation mode can be determined. Based on this time node and combined with the GPP, the specific critical value of gas saturation can be determined. Taking Fig. 1 as an example, near 400 days, the average reservoir pressure enters a stable state. At this time, the average reservoir pressure is 3.18 MPa, and the water saturation is determined to be 58%, that is, the gas saturation is 42%. This value is taken as the demarcation point between gas pressure and water pressure. When the gas saturation is lower than 42%, WPP is adopted, and when it exceeds this value, GPP is adopted.
The core of the CPDPM is divided into two stages. When the overall gas saturation of the reservoir is lower than Sg0, the reservoir is divided into two regions, and GPP (15) and WPP (16) are used respectively. When the overall gas saturation of the reservoir exceeds Sg0, GPP is adopted entirely. Assuming that the demarcation point at different times is Rg, the average reservoir pressure formula of the CPDPM is:
 |
21 |
In Eq. (21), the solution process of the first part is the same as that of Eq. (17). The difference lies in the upper limit of integration. For 
, we can further obtain:
 |
22 |
 |
23 |
The second part in Eq. (21) is:
 |
24 |
The solution process of the CPDPM in the desorption zone is as follows:
Identify the inflection point of the average reservoir pressure change calculated by GPP. The value calculated by WPP corresponding to this inflection point is the actual average reservoir pressure at this moment, and the gas saturation corresponding to this actual average reservoir pressure is the critical value Sg0.
Given the BHFP and the production duration, assume the value of the desorption funnel Rde, and substitute it into the WPP (16) to obtain the pressure distribution and the average reservoir pressure.
3) Based on the obtained pressure distribution, the gas saturation at each pressure point is calculated. When the difference between the gas saturation at all points within the entire desorption funnel and Sg0 fails to meet the precision requirements, use the WPP to obtain the average relative permeability. Then substitute the obtained results into the desorption funnel radius formula, and compare the calculated value with the assumed value. If the precision requirements are not met, substitute the calculated value into the desorption funnel radius formula and continue the calculation until the precision requirements are satisfied. Subsequently, calculate the pressure drop funnel radius. After that, continue to solve the desorption funnel for the next time period until at a certain time node, the gas saturation at a certain location is greater than or close to Sg0, indicating that part of the reservoir will conform to GPP at this time, and then stop the calculation of WPP.
4) Based on the pressure distribution obtained in step 3), starting from the wellbore where the gas saturation is the highest, use the GPP to calculate the gas saturation. When the gas saturation at the radius r is close to Sg0, this position corresponds to Rg. After that, use the WPP to calculate the pressure and gas saturation at each point within the range greater than Rg until reaching the boundary of the assumed Rde, and then obtain the average relative permeability and average viscosity within the entire desorption funnel. Next, substitute Rg and the assumed Rde into the CPDPM to obtain the average reservoir pressure, and use the desorption funnel radius formula to get the calculated value. Compare the calculated value with the assumed value until the precision requirements are met. Subsequently, calculate the pressure drop funnel radius. Meanwhile, use the obtained single-point gas saturation to calculate the average gas saturation. When the average gas saturation is less than Sg0, that is, only some points exceed Sg0, repeat step 4) to calculate the next desorption funnel and pressure drop funnel radius. Otherwise, when the average gas saturation is greater than Sg0, indicating a relatively high gas saturation, only use the GPP for calculation until reaching the abandonment pressure.
Using the same basic parameters, the calculation results of the CPDPM are shown in Fig. 1 and compared with those of WPP and GPP. It can be seen from Fig. 1 that when the production lasts for more than 400 days, it generally conforms to the WPP. After that, it begins to show a change pattern similar to that of GPP, with the difference lying in the varying degrees of reservoir pressure reduction. Although the starting points of pressure reduction for the CPDPM and WPP are almost the same, their change patterns are completely different. Overall, the CPDPM is equivalent to the superimposed effect of WPP and GPP.
Establishment of the Pressure Drop Funnel Model
After the reservoir starts to desorb, the desorption funnel and the pressure drop funnel expand outward simultaneously. Within the desorption funnel, since the pressure is not completely propagated by WPP, the pressure drop propagation formula outside the desorption funnel needs to be modified to ensure continuous connection with Eq. (16). Therefore, the pressure distribution is improved as follows:
 |
25 |
In Eq. (25), Rdr represents the radius of the pressure drop funnel. Where, this equation takes the position of the desorption funnel radius as the starting point and the radius of the pressure drop funnel as the endpoint. The entire pressure propagation process within the desorption funnel radius is included in the parameter of Rde. It can not only connect well with the pressure propagation within the desorption funnel but also meet the characteristics of pressure propagation outside the desorption funnel. The average reservoir pressure within the range of Rdr can be seen in Eq. (26).
 |
26 |
The first two terms of Eq. (26) are solved by using Eq. (22) and (24) respectively, and the result of the third term is as follows:
 |
27 |
The above equations represent the average reservoir pressure within the entire range of the pressure drop funnel. The desorption funnel radius included therein has been obtained, and then the radius of the pressure drop funnel can be further obtained.
The desorption efficiency 
determines the level of production efficiency, Which is defined as the desorption amount of per ton of coal under unit pressure drop29, and it can be represented as the first derivative of the desorption equation:
 |
28 |
In the factor analysis below, the desorption efficiency, permeability and average reservoir pressure will be combined to comprehensively evaluate the law of pressure drop propagation.
Model verification and factor analysis
(1) Evaluation of Model Accuracy.
Since it is impossible to directly measure the radius of the pressure drop funnel, the formation pressure distribution, the change in permeability and other parameters on site, the results of the model in this paper will be compared with those of the commercial numerical simulation software (CMG) for evaluation. The same basic parameters are input into both models to calculate the radius of the pressure drop funnel and the permeability at each production moment. The basic data comes from an unfractured exploration well in the Shizhuang South Block, as shown in Table 2. In the CMG software, the grid type is set as radial grid, and the boundary distance is 300m. In the near-well area of 0-10m, the number of grids is 20, and the grid size increases geometrically from 0.1m to 1m. In the far-well area of 10-300m, the number of grids is 100, and the grid size increases geometrically from 1 to 10m. Vertically, it is divided into 3 layers: 1 roof layer + 1 coal seam layer + 1 floor layer.
Table 2.
Basic parameters for numerical calculation.
| Original formation pressure | Well radius | Coal seam thickness | Initial permeability | Critical desorption pressure | Formation water viscosity | Coal rock density |
|---|---|---|---|---|---|---|
| 7 MPa | 0.108 m | 7.8 m | 0.5 × 10−3um2 | 4 MPa | 0.98 m·Pas | 1.4 g/cm3 |
| Coal seam cleat compression coefficient | Initial porosity | Desorption Langmuir volume | Desorption Langmuir pressure | Reservoir temperature | Comprehensive compression coefficient | |
| 0.05 MPa−1 | 0.05 | 25.48cm3/g | 1.38 MPa | 28.3℃ | 0.059 MPa−1 |
The expansion law of the desorption funnel is solved respectively by three methods, as shown in Figs. 3 - 5.
Fig. 3.
Desorption funnel expansion process.
Fig. 5.
Desorption funnel radius by CPDPM.
The model in this paper has calculated the changes in reservoir pressure during the process from the start of desorption to 400 days of production. As can be seen from Fig. 3, before 120 days, the pressure drop propagation conformed to WPP. After that, fluctuating points appeared on the pressure drop funnel curve, indicating that it entered the composite propagation stage where both gas pressure and water pressure coexisted, and the Rg continued to advance towards the deeper part of the formation, that is, the range of the region where the gas saturation was higher than Sg0 was constantly expanding. Meanwhile, the degree of fluctuation at the fluctuating points became more intense. This is because, during the advancement of the fluctuating points, the overall pressure of the reservoir gradually decreased, the amount of gas desorption increased, and the total gas content in the entire reservoir became higher, getting closer to GPP. Therefore, compared with formation water, the high compressibility characteristic of gas led to greater pressure fluctuations. The so-called “fluctuating points” in this paper refer to the phenomenon that the desorption funnel curve is not smooth due to the instability of gas saturation before the initial desorption of gas reaches dynamic equilibrium.
The desorption funnel radius at 400 days of production with was calculated through numerical simulation. It was found that except for the absence of the above-mentioned fluctuating points and the desorption funnel radius being slightly lower than that of the model in this paper, the distribution curves of the two were almost the same, which demonstrated the accuracy of the model in calculating the expansion of the desorption funnel.
In Fig. 4, the pressure expansion and gas saturation distribution of GPP and CPDPM from the start of desorption to 600 days of production were calculated. It can be seen that between the wellbore and the pressure fluctuation point, the laws of pressure drop expansion and gas saturation are the same. After the fluctuation point, the calculation results deviate from each other. Especially, the volatility of the gas saturation calculated by the CPDPM is more intense. This is because as the pressure propagates deeper into the formation, the gas saturation gradually decreases and the water saturation increases. It is precisely this transition from a gas-dominated state to a water-dominated state that leads to intensified pressure fluctuations, making the conversion between gas desorption and adsorption more frequent and further resulting in more drastic changes in the gas content, which also conforms to the actual production situation. When GPP is adopted entirely, such fluctuations will be ignored, causing the calculation results to deviate from the actual situation. This is consistent with the explanation of the fluctuation point in Fig. 3.
Fig. 4.
Comparison between GPP and CPDPM.
In Fig. 5, the stage where the CPDPM is in the pressure drop propagation is mainly presented from the perspective of production time. Generally speaking, the durations of both the individual WPP and the combined GPP and WPP are relatively short. However, the desorption funnel radius calculated by the simple GPP has had a large gap from the calculation result of the WPP in the CPDPM right from the start. Furthermore, with the superimposition of the stage of combined gas and water propagation, the difference between the two has been further enlarged.
The verification of the calculation results of the CPDPM for the unfractured reservoir indicates that the model in this paper has good accuracy.
(2) Sensitivity Factor Analysis of the Model
The core of the CPDPM lies in two aspects. On the one hand, it determines the starting point and the ending point of the combined propagation of gas pressure and water pressure. On the other hand, it determines the position of the demarcation line between gas pressure and water pressure.
The starting point and the ending point are determined by the average gas saturation of the reservoir. This is an inevitable process in the development of CBM through pressure reduction and desorption, which belongs to the concept of time and will not have a substantial impact on the final production capacity. However, the position of the demarcation line Rg belongs to the concept of space. Different production conditions will result in different position outcomes, which will further lead to differences in the control ranges of gas pressure and water pressure and will have an impact on the desorption efficiency and production. Therefore, the sensitivity analysis will be carried out around the position of the demarcation line, mainly analyzing the pressure drop propagation process during the stage when WPP and GPP coexist.
① Influence of Different Rg Positions on Permeability Distribution.
Select the parameters of the unfractured exploration well and set the original permeability K0 = 0.32mD. Calculate the actual permeability change values corresponding to different Rg, as shown in Fig. 6.
Fig. 6.
Permeability distribution corresponding to different Rg.
As can be seen from Fig. 6, when it is still in the later stage of the WPP phase, the reservoir permeability near the wellbore has already exceeded the original permeability, indicating that the matrix shrinkage effect caused by gas desorption has surpassed the stress sensitivity damage and has had a positive and beneficial impact on the permeability. When the first fluctuation point appears, the permeability near the wellbore has reached 0.42 mD, but the overall average permeability of the reservoir is 0.28 mD, which is lower than the original permeability, suggesting that WPP is still the main propagation mode at this time. Therefore, the BHFP cannot be rapidly reduced at this time to prevent the formations far away from the wellbore from being damaged by stress sensitivity. After the Rg reaches 100 m, the permeability in 70% of the desorption funnel range has been positively improved. At this time, due to the increase in the overall gas saturation of the reservoir, a certain supply and pressure boost effect has been generated, the expansion speed of the desorption funnel has gradually slowed down, the pressure drop has gradually transitioned to GPP, and the reservoir has begun to enter the constant-pressure steady-state production stage.
② Influence of Different Rg Positions on Average Reservoir Pressure.
Keep the basic parameters unchanged, change the original permeability, and evaluate the influence of the position of the Rg on the average pressure under reservoirs with different permeabilities, as shown in Fig. 7.
Fig. 7.
Effect of Rg position on mean pressure.
As can be seen from Fig. 7, for reservoirs with different permeabilities, as the position of the Rg advances, the average reservoir pressure first decreases at a relatively fast speed and then increases steadily, presenting an overall concave arc shape. This result further corroborates the conclusion about the supply and pressure boost effect. On the other hand, as the permeability changes from high to low, the concave point gradually shifts to the right, indicating that the decreasing rate of the average reservoir pressure also changes from high to low. This is because the flow resistance in high-permeability reservoirs is small, the pressure drop propagation speed is fast, and the pressure reduction efficiency is high, which leads to high desorption efficiency and makes the supply and pressure boost effect appear earlier.
③ Influence of Different Rg Positions on Desorption Efficiency.
Keep the basic parameters unchanged, change the original permeability, and evaluate the influence of the Rg position on the desorption efficiency under reservoirs with different permeabilities, as shown in Fig. 8.
Fig. 8.
Effect of Rg position on desorption efficiency.
It can be found from Fig. 8 that the desorption efficiency has a corresponding relationship with the average reservoir pressure in Fig. 7 and also follows a similar variation pattern, except that the change trend is opposite. The desorption efficiency increases rapidly at first with the advancement of the Rg and begins to decrease after reaching the highest point, which is also due to the supply and pressure boost effect. The lower the permeability is, the farther the Rg where the desorption efficiency reaches the highest point is from the wellbore. Meanwhile, the highest point is also lower than that of high-permeability reservoirs, indicating that low permeability not only leads to a low pressure drop propagation speed but also results in a low overall desorption efficiency of the reservoir.
According to the variation laws of the average reservoir pressure and desorption efficiency, for reservoirs with different permeabilities, different advancing rates of the Rg should be controlled during the composite propagation process. When the desorption efficiency approaches its highest point, the BHFP should be kept stable to further slow down the reduction of the reservoir pressure, delay the forward advancement of the Rg, minimize the decrease of the desorption efficiency as much as possible, and maintain the continuous and stable gas production. For this reason, the relationship between the BHFP and the Rg position is studied.
④ Influence of the BHFP on the Expansion of the Rg.
Keep the basic parameters unchanged and calculate the relationships between different BHFPs and the Rg under three conditions of the original permeability, as shown in Fig. 9.
Fig. 9.
Effect of BHFP on Rg expansion.
As can be seen from Fig. 9, with the decrease of the BHFP, the Rg presents a variation law of increasing slowly at first and then rapidly. This is because the BHFP is relatively high at the beginning, the production pressure difference is small, and the advancement of the Rg is slow. After that, the production pressure difference becomes larger and the advancing speed of the Rg increases.
For reservoirs with relatively low permeability, such as 0.3 mD and 0.5 mD in this case, when the BHFP drops to 2.3 MPa, there will be a certain value of the Rg. That is to say, at this time, the gas saturation of a small amount of reservoirs near the wellbore exceeds the saturation cut-off value Sg0, and the pressure drop enters the stage where gas pressure and water pressure propagate together from the pure WPP stage.
However, for the 0.8 mD reservoir, a significant value of the Rg appears within a very short time after entering the stage of simultaneous production of gas and water. Therefore, for relatively high-permeability reservoirs, the WPP time is relatively short or even negligible, which needs to be noted in practical applications.
It can be seen from model verification and analysis of influencing factors that the desorption funnel calculated by using the CPDPM is obviously more accurate than simply using GPP or WPP. In addition, as the Rg value in the CPDPM advances towards the deep part of the formation, the average reservoir pressure shows a trend of decreasing rapidly at first and then increasing slowly. The change direction of the desorption efficiency is just the opposite of that of the average reservoir pressure. This indicates that when the Rg value advances to a certain extent, it is necessary to slow down the decline range of the BHFP and maintain a relatively high desorption efficiency.
Conclusions
This paper proposes a surrogate model for calculating the radius of coalbed methane desorption funnels, which can characterize the impact of dynamic changes in gas and water on the expansion of the desorption funnel and overcomes the defects of existing methods, including the results of Staged simplification method are inconsistent with reality, the insufficient universality of empirical methods, and the large computational workload and complex processes of numerical simulation methods. The following conclusions are drawn:
(1) A CPDPM with dynamic changes in both time and spatial dimensions during the stage of gas production has been established. This model combines GPP and WPP. According to the gas saturation at different times and different spatial positions, it can automatically select WPP, GPP, or a combination of the two methods. It can cover the entire cycle of the stage of gas production according to changes in gas content and has stronger applicability. The results show that during the gas production, the damage to permeability caused by stress sensitivity is lower than the positive effect generated by the matrix shrinkage effect.
(2) Based on the value of the Rg in the CPDPM, the decline range of the BHFP can be determined, and then a relatively high desorption efficiency can be maintained.
Acknowledgements
This work was supported by the Key R&D Projects in Gansu Province(Grant No. 23YFGM0001), and the Science and Technology Program of Qingyang City, China (Grant No. QY-STK-2023A-012).
Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
Footnotes
Publisher’s note
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Data Availability Statement
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.