# Effect of Threshold Pressure Gradients on Control Areas Determination of Production Well in CBM Reservoirs

Authors: Ruifei Wang; Jiaosheng Yang; Meizhu Wang; Yang Zhao; Weiqing Chen
Journal: Advances in Polymer Technology
 (2019)

---

## Abstract

Understanding the mechanism of water drainage and gas recovery is the burning issue for Coalbed Methane (CBM) reservoir development. In the process of exploitation, threshold pressure gradients (TPG) is an important factor affecting the control areas, which related to the low-permeability and complex water saturation of CBM reservoirs. In this paper, a new flow model of CBM has been established considering the TPG and gas desorption. Then we carried out a series of experiments and fitted out a new relational expression between TPG and permeability and water saturation, which shows that TPG is negatively correlated with permeability and positively correlated with water saturation. After that, we analyzed the influence of TPG and desorption on the control radius and illustrated a case study. The results show that TPG and desorption effect both can slow down the rate of pressure reduction. The case study indicates that the control radius of target well groups ranges from 55 m to 136.7 m The average control radius and gas TPG are 91.3 m 0.0082 MPa/m respectively. Furthermore, we classify the wells into 5 categories, which are mainly distributed in III (80∼100 m). Finally, we suggest using well pattern infilling in region II and III and hydraulic fracturing method for region IV to increase the utilization area and the sustainability for the target area. This study provides a quick and reasonable prediction of control radius in CBM reservoir with different water saturation for further adjustment suggestion and sustainable development.


---

## Body

## 1. Introduction

In recent years, coalbed methane (CBM), as an unconventional and clean natural gas resource, has received lots of attention. The effective exploitation of CBM is of great significance to the full utilization of energy, the improvement of energy structure, and the sustainable development of energy [1, 2]. Meanwhile, the successful CBM recovery can help reduce the risk of the mine explosion and mitigate the greenhouse effect induced by the methane emission, which can achieve the sustainable development of environment and energy [3–9].Among the researches on the CBM development, the coal is usually characterized as a dual-porosity system [10, 11], which refers to a matrix-cleat system. The coal matrix, as the primary porosity system, is comprised of micropores and stores 70–95% of total gas in the adsorbed form [12]. The cleat system, on the other hand, is composed of macropores and usually saturated with water [13, 14]. By pumping out water from cleats, the formation pressure of coalbed will decrease, which results in the desorption of CBM and triggers its flow. With the continuous drainage, the relative permeability of water in coal seam decreases, and the relative permeability of gas increases. The water production of CBM wells decreases gradually, while gas production increases gradually and tends to be stable. Therefore, the main production stage of coalbed methane wells is gas–water two-phase flow, whose duration determines the economic benefits of the whole coalbed methane wells.Up to now, many models have been established to study the CBM development. Some researchers focus on the production characteristics or behavior of CBM reservoirs. Zhao et al. investigated the influence of six factors on gas production quantitatively based on the grey system theory, and then put forward the optimum development strategy for CBM reservoirs in terms of those factors [15]. Lv et al. studied the temporal and spatial production characteristics of CBM wells to determine the dominant factors by using bivariate correlation analysis and gray system theory [16]. Wang et al. set up a numerical model to investigate the combined effect of directional compaction, nondarcy flow and anisotropic swelling on well production rates [17]. Gong et al. proposed a new workflow to perform a 2-D coalbed methane recovery simulation with a discrete fracture model to better simulate the distribution of actual cleats. They then utilized this workflow model to study the enhancement of methane recovery when injecting CO2[18]. Vishal et al. set up a numerical model to investigate the role of sorption time in the production behavior of CBM under carbon dioxide injection [19].At the same time, some scholars pay more attention to the flow mechanism of gas using model establishment [20–23]. Liu et al. proposed a new semi-empirical model for describing the entire methane diffusion process，which is more effective in describing non-linear gas diffusion behavior in the coal matrix than the Fick model for the studied coals [24]. Pillalamarry et al. used a unique model to evaluate the sorption and diffusion properties of methane and study the relationship between the diffusion coefficient and pressure. Sun et al. proposed a semi-analytical model to quantify the effective gas/water phase permeability and analyze the effect of critical desorption pressure, gas desorption capacity, stress dependence, and matrix shrinkage on effective permeability [25].Pore analysis shows that the pore size of the CBM reservoir is mainly nano/micro-scale. The interfacial effects will be obvious at this scale, and a stable adsorption water film will be formed on the solid wall, even the whole pore. The existence of water in the CBM pores will increase the flow resistance and weaken the slippage effect of the gas, which can change the flow characteristics of the gas. At this point, water just like a wall that obstructs the flow of gas. There must be an additional pressure gradient to overcome the resistance of the adsorption layer, as shown in Figure1. We call this pressure gradient as the threshold pressure gradient (TPG) of gas flow [26].Figure 1
Gas transport mechanism in pores of CBM reservoirs containing water.However, the previously modeling efforts did not consider the impact of TPG on the development of CBM. Some scholars have noticed TPG when they research fluid flow in pore systems [27–32]. Miller and Low firstly introduced the conception of TPG when he studied the mechanism of water flow in clay systems [31]. He figured that the threshold gradient needs to be overcome for water flow. Furthermore, some researchers held that the TPG phenomena for gas flow are explicit in ultra-low-permeability reservoirs, especially when water also exists in the flow path. Tian et al. performed the TPG experimental investigations using the air bubble method, and the results showed that the TPG exists at the connate water saturation, and increases exponentially with either an increase in dimensionless water saturation or decrease in permeability [33]. In our previous work, according to the experimental results, it can be concluded that containing water is the prerequisite of the existence of TPG for gas flow in porous media. TPG increases with higher water saturation and lower absolute permeability [32–34].In this paper, the effect of gas TPG and desorption on an effective control radius of the CBM reservoir with different water saturation is studied by experiment and theoretical analysis. Firstly, we established a new flow model of CBM considering the TPG and gas diffusion. Secondly, we carried out a series of experiments to fit out the gas TPG expression. Then we analyzed the influence of TPG and desorption on the control radius. Finally, we have conducted a case study to analyze the utilization control radius of the target area. This study provides a quick and reasonable prediction of control radius in CBM reservoir different water saturation for further adjustment suggestions and sustainable development.


## 2. Study Area and Geological Background

China is not only a country of high coal production and consumption but also a country with abundant CBM resources. The block in this paper is located in certain basin in the southeast of Shanxi Province in China as shown in Figure2 [35]. Since 2011, the block has been explored for CBM development. The common well types in these areas are vertical well.Figure 2
Location of the target block in China and two study areas with corresponding wells distributions.In this block, two important and productive coal seams for CBM recovery are the No. 3 coal seam from Shanxi Formation and No. 15 coal seam from Taiyuan Formation. Some geological details about both coal seams are summarized in Table1. Laboratory tests show that the average permeability of Nos. 3 and 15 coal seams in this block are around 0.06 mD and 0.01 mD respectively, indicating low-permeability traits for both coal seams. The initial water saturation of Nos. 3 and 15 coal seams are about 15.6% and 36.9% separately. According to statistical data from appraisal wells in the target block, the No. 3 coal seam contributes more to the overall CBM development than the No. 15 coal seam. Therefore, in this study, the No. 3 coal seam is selected as the target coal seam for the investigation of control radius in two study areas.Table 1
Some geological information of the Nos. 3 and 15 coal seams.
Erathem
System
Formation
Formation thickness
Coal seam thickness
Paleozoic
Lower permian
Lower shihezi
40–110 m
—
Shanxi
30–70 m
No. 3 : 4 –6.6 m
Upper carboniferous
Taiyuan
80–130 m
No. 15 : 2.2 –10 m
Middle carboniferous
Benxi
0–45 m
—


## 3. Mathematical Model

### 3.1. Governing Equation

To simplify the model, it is assumed that water is standing. The mathematical model describing the whole process of nonDarcy flow for single-phase gas flow considering desorption is presented as follows [26]:Mass conservation equation:(1)
∂
∂
t
ρ
g
ϕ
+
d
i
v
ρ
g
v
→
−
q
d
=
0
,Motion equation:(2)
v
→
=
k
μ
∇
p
−
G
g
,Real gas state equation:(3)
p
V
=
n
Z
R
T
,whereZ is the compressibility factor, which is a function of temperature T and volume V. k is permeability, μ is the viscosity of gas, ρ
g is the density of gas, ϕ is porosity, p is the pressure of the reservoir, G
g is the TPG of gas, and q
d is the adsorption of gas.According to Equations (3), the density of the gas is expressed as follows:(4)
ρ
g
=
p
M
R
T
Z
.Similarly, the gas density under standard condition is(5)
ρ
gsc
=
p
sc
M
R
T
sc
Z
sc
.Substituting Equations (5) into (4), we obtain(6)
ρ
g
=
T
sc
Z
sc
ρ
gsc
p
sc
·
p
T
Z
.Isothermal compressibility of gas is defined as(7)
C
ρ
=
−
d
V
/
V
d
p
=
−
1
V
·
d
V
d
p
=
1
p
−
1
Z
·
∂
Z
∂
p
.Combining with Equation (6) and (7), the time term of Equation (1) can be expressed as(8)
∂
ρ
g
ϕ
∂
t
=
T
sc
Z
sc
ρ
gsc
ϕ
p
sc
T
1
p
−
1
Z
p
·
∂
Z
p
∂
p
·
p
Z
p
∂
p
∂
t
=
T
sc
Z
sc
ρ
gsc
ϕ
μ
p
p
sc
T
C
ρ
·
p
μ
p
Z
p
∂
p
∂
t
.Similarly, according to Equation (2), the second term of Equation (1) in the x direction can be expressed as [26](9)
∂
ρ
g
v
x
∂
x
=
∂
∂
x
T
sc
Z
sc
ρ
gsc
p
sc
·
p
T
Z
·
k
μ
p
∂
p
∂
x
−
G
g
=
T
sc
Z
sc
ρ
gsc
k
p
sc
T
∂
∂
x
p
μ
p
Z
p
·
∂
p
∂
x
−
G
g
·
C
ρ
p
μ
p
Z
p
·
∂
p
∂
x
.Introducing pseudo-pressure function,(10)
m
=
2
∫
0
p
p
μ
p
Z
p
d
p
.Then(11)
d
m
d
p
=
2
·
p
μ
p
Z
p
.For convenient engineering applications,μ
p
Z
p in Equation (11) can be replaced by μ
Z
¯, which is the value of μ
p
Z
p under average pressure and constant temperature. Then, integrating Equation (11), we obtain(12)
m
=
1
μ
Z
¯
·
p
2
.Substituting Equation (11) into Equation (8), we have(13)
∂
ρ
g
ϕ
∂
t
=
1
2
·
T
sc
Z
sc
ρ
gsc
ϕ
μ
p
p
sc
T
C
ρ
·
∂
m
∂
t
.Substituting Equation (11) into Equation (9), we have(14)
∂
ρ
g
v
x
∂
x
=
1
2
·
T
sc
Z
sc
ρ
gsc
k
p
sc
T
∂
2
m
∂
x
2
−
G
g
·
C
ρ
∂
m
∂
x
.Similarly, the following formulas in they and z directions are obtained(15)
∂
ρ
g
v
y
∂
y
=
1
2
·
T
sc
Z
sc
ρ
gsc
k
p
sc
T
∂
2
m
∂
y
2
−
G
g
·
C
ρ
∂
m
∂
y
.
(16)
∂
ρ
g
v
z
∂
z
=
1
2
·
T
sc
Z
sc
ρ
gsc
k
p
sc
T
∂
2
m
∂
z
2
−
G
g
·
C
ρ
∂
m
∂
z
.When the Hamilton operator and Laplace operator are introduced, we have(17)
∂
ρ
g
v
x
∂
x
+
∂
ρ
g
v
y
∂
y
+
∂
ρ
g
v
z
∂
z
−
q
d
=
1
2
·
T
s
c
Z
s
c
ρ
g
s
c
k
p
s
c
T
∇
2
m
−
C
ρ
G
g
∇
m
−
q
d
.According to Equation (13) and (17), Equation (1) can be transformed into(18)
−
∇
2
m
−
C
ρ
G
g
∇
m
+
2
p
sc
T
T
sc
Z
sc
ρ
gsc
k
q
d
=
ϕ
μ
p
C
ρ
k
·
∂
m
∂
t
.The gas pressure derivative is defined as(19)
η
=
k
ϕ
μ
p
C
ρ
.Then(20)
−
∇
2
m
+
C
ρ
G
g
∇
m
+
2
p
sc
T
T
sc
Z
sc
ρ
gsc
k
q
d
=
1
η
∂
m
∂
t
.


### 3.2. Analytical Solution of Nondarcy Radial Flow

Assuming thatG
c
=
C
ρ
G
g, for steady plane radial flow, Equation (20) can be transformed into following ordinary differential equations(21)
d
2
m
d
r
2
+
1
r
·
d
m
d
r
−
G
c
d
m
d
r
−
2
p
sc
T
T
sc
Z
sc
ρ
gsc
k
q
d
=
0
.Equations of constant pressure boundary conditions are given as follows.(22)
r
=
r
w
,
m
=
m
w
,
r
=
r
e
,
m
=
m
e
.Combining with Equation (22), the analytical solution of Equation (21) can be obtained.(23)
m
r
=
−
q
d
∗
ln
G
c
·
r
G
c
2
−
q
d
∗
r
G
c
−
C
1
E
i
G
c
·
r
+
C
2
,where(24)
q
d
∗
=
2
p
sc
T
T
sc
Z
sc
ρ
gsc
k
·
q
d
,
C
1
=
m
e
−
m
w
G
c
2
+
q
d
∗
·
ln
r
e
/
r
w
+
q
d
∗
·
G
c
r
e
−
r
w
G
c
2
·
E
i
G
c
·
r
w
−
E
i
G
c
·
r
e
,
C
2
=
m
w
+
q
d
∗
ln
G
c
·
r
w
G
c
2
+
q
d
∗
r
w
G
c
+
C
1
·
E
i
G
c
·
r
w
.According to Equation (12), m
e
=
1
/
μ
Z
¯
·
p
e
2 and m
w
=
1
/
μ
Z
¯
·
p
w
2, the pressure distribution is shown as Equation (25).(25)
p
r
=
−
q
d
∗
ln
G
c
·
r
G
c
2
−
q
d
∗
r
G
c
−
C
1
E
i
G
c
·
r
+
C
2
·
μ
Z
¯
.According to Equation (25), we can obtain the formation pressure gradient. Comparing with the gas TPG, when d
p
/
d
r
=
G
g, the corresponding radius will be equal to the control radius.


## 3.1. Governing Equation

To simplify the model, it is assumed that water is standing. The mathematical model describing the whole process of nonDarcy flow for single-phase gas flow considering desorption is presented as follows [26]:Mass conservation equation:(1)
∂
∂
t
ρ
g
ϕ
+
d
i
v
ρ
g
v
→
−
q
d
=
0
,Motion equation:(2)
v
→
=
k
μ
∇
p
−
G
g
,Real gas state equation:(3)
p
V
=
n
Z
R
T
,whereZ is the compressibility factor, which is a function of temperature T and volume V. k is permeability, μ is the viscosity of gas, ρ
g is the density of gas, ϕ is porosity, p is the pressure of the reservoir, G
g is the TPG of gas, and q
d is the adsorption of gas.According to Equations (3), the density of the gas is expressed as follows:(4)
ρ
g
=
p
M
R
T
Z
.Similarly, the gas density under standard condition is(5)
ρ
gsc
=
p
sc
M
R
T
sc
Z
sc
.Substituting Equations (5) into (4), we obtain(6)
ρ
g
=
T
sc
Z
sc
ρ
gsc
p
sc
·
p
T
Z
.Isothermal compressibility of gas is defined as(7)
C
ρ
=
−
d
V
/
V
d
p
=
−
1
V
·
d
V
d
p
=
1
p
−
1
Z
·
∂
Z
∂
p
.Combining with Equation (6) and (7), the time term of Equation (1) can be expressed as(8)
∂
ρ
g
ϕ
∂
t
=
T
sc
Z
sc
ρ
gsc
ϕ
p
sc
T
1
p
−
1
Z
p
·
∂
Z
p
∂
p
·
p
Z
p
∂
p
∂
t
=
T
sc
Z
sc
ρ
gsc
ϕ
μ
p
p
sc
T
C
ρ
·
p
μ
p
Z
p
∂
p
∂
t
.Similarly, according to Equation (2), the second term of Equation (1) in the x direction can be expressed as [26](9)
∂
ρ
g
v
x
∂
x
=
∂
∂
x
T
sc
Z
sc
ρ
gsc
p
sc
·
p
T
Z
·
k
μ
p
∂
p
∂
x
−
G
g
=
T
sc
Z
sc
ρ
gsc
k
p
sc
T
∂
∂
x
p
μ
p
Z
p
·
∂
p
∂
x
−
G
g
·
C
ρ
p
μ
p
Z
p
·
∂
p
∂
x
.Introducing pseudo-pressure function,(10)
m
=
2
∫
0
p
p
μ
p
Z
p
d
p
.Then(11)
d
m
d
p
=
2
·
p
μ
p
Z
p
.For convenient engineering applications,μ
p
Z
p in Equation (11) can be replaced by μ
Z
¯, which is the value of μ
p
Z
p under average pressure and constant temperature. Then, integrating Equation (11), we obtain(12)
m
=
1
μ
Z
¯
·
p
2
.Substituting Equation (11) into Equation (8), we have(13)
∂
ρ
g
ϕ
∂
t
=
1
2
·
T
sc
Z
sc
ρ
gsc
ϕ
μ
p
p
sc
T
C
ρ
·
∂
m
∂
t
.Substituting Equation (11) into Equation (9), we have(14)
∂
ρ
g
v
x
∂
x
=
1
2
·
T
sc
Z
sc
ρ
gsc
k
p
sc
T
∂
2
m
∂
x
2
−
G
g
·
C
ρ
∂
m
∂
x
.Similarly, the following formulas in they and z directions are obtained(15)
∂
ρ
g
v
y
∂
y
=
1
2
·
T
sc
Z
sc
ρ
gsc
k
p
sc
T
∂
2
m
∂
y
2
−
G
g
·
C
ρ
∂
m
∂
y
.
(16)
∂
ρ
g
v
z
∂
z
=
1
2
·
T
sc
Z
sc
ρ
gsc
k
p
sc
T
∂
2
m
∂
z
2
−
G
g
·
C
ρ
∂
m
∂
z
.When the Hamilton operator and Laplace operator are introduced, we have(17)
∂
ρ
g
v
x
∂
x
+
∂
ρ
g
v
y
∂
y
+
∂
ρ
g
v
z
∂
z
−
q
d
=
1
2
·
T
s
c
Z
s
c
ρ
g
s
c
k
p
s
c
T
∇
2
m
−
C
ρ
G
g
∇
m
−
q
d
.According to Equation (13) and (17), Equation (1) can be transformed into(18)
−
∇
2
m
−
C
ρ
G
g
∇
m
+
2
p
sc
T
T
sc
Z
sc
ρ
gsc
k
q
d
=
ϕ
μ
p
C
ρ
k
·
∂
m
∂
t
.The gas pressure derivative is defined as(19)
η
=
k
ϕ
μ
p
C
ρ
.Then(20)
−
∇
2
m
+
C
ρ
G
g
∇
m
+
2
p
sc
T
T
sc
Z
sc
ρ
gsc
k
q
d
=
1
η
∂
m
∂
t
.


## 3.2. Analytical Solution of Nondarcy Radial Flow

Assuming thatG
c
=
C
ρ
G
g, for steady plane radial flow, Equation (20) can be transformed into following ordinary differential equations(21)
d
2
m
d
r
2
+
1
r
·
d
m
d
r
−
G
c
d
m
d
r
−
2
p
sc
T
T
sc
Z
sc
ρ
gsc
k
q
d
=
0
.Equations of constant pressure boundary conditions are given as follows.(22)
r
=
r
w
,
m
=
m
w
,
r
=
r
e
,
m
=
m
e
.Combining with Equation (22), the analytical solution of Equation (21) can be obtained.(23)
m
r
=
−
q
d
∗
ln
G
c
·
r
G
c
2
−
q
d
∗
r
G
c
−
C
1
E
i
G
c
·
r
+
C
2
,where(24)
q
d
∗
=
2
p
sc
T
T
sc
Z
sc
ρ
gsc
k
·
q
d
,
C
1
=
m
e
−
m
w
G
c
2
+
q
d
∗
·
ln
r
e
/
r
w
+
q
d
∗
·
G
c
r
e
−
r
w
G
c
2
·
E
i
G
c
·
r
w
−
E
i
G
c
·
r
e
,
C
2
=
m
w
+
q
d
∗
ln
G
c
·
r
w
G
c
2
+
q
d
∗
r
w
G
c
+
C
1
·
E
i
G
c
·
r
w
.According to Equation (12), m
e
=
1
/
μ
Z
¯
·
p
e
2 and m
w
=
1
/
μ
Z
¯
·
p
w
2, the pressure distribution is shown as Equation (25).(25)
p
r
=
−
q
d
∗
ln
G
c
·
r
G
c
2
−
q
d
∗
r
G
c
−
C
1
E
i
G
c
·
r
+
C
2
·
μ
Z
¯
.According to Equation (25), we can obtain the formation pressure gradient. Comparing with the gas TPG, when d
p
/
d
r
=
G
g, the corresponding radius will be equal to the control radius.


## 4. Experimental Study

Due to the existence of TPG, gas–water two-phase flow shows typical low-velocity nonDarcy flow characteristics [21–23]. In this paper, we designed experiments to validate and obtain the relationship between TPG and the permeability, water saturation.


### 4.1. Experimental Method and Procedures

The coal core samples were obtained from the No. 3 coal seam in the study area. The diameter of each core sample is 2.5 cm on average, and the length ranges from 4.5 cm to 6 cm. In this paper, the porosity and the permeability of coal samples were measured using a fully automatic pore-permeability simultaneous measuring instrument. The permeability in this paper is gas log permeability. The average porosity for core samples of No. 3 coal seam is 4.3%. The gas used in this experiment was methane with a purity of more than 99.99%. The water was prepared in the laboratory according to some actual parameters of formation water from No. 3 coal seam, as shown in Table2. After that, we use special core displacement equipment to inject gas into the different permeability cores with different water saturation, and calculate the influence of water cut and permeability on gas injection displacement by recording gas production rate, then determine the TPG of coal-cores with water, the parameters of two groups of experiments are shown in Table 3. Figure 3 shows the schematic of the experimental equipment and core samples.Table 2
Parameters of formation water from the No. 3 coal seam.
Ion concentration, mg/L
Total salinity, mg/L
Water type
Na++K +
Mg2+
Ca2+
Cl−
SO4
2−
HCO3
−
488.16
3.98
4.82
245.59
17.83
884.93
1645.31
NaHCO3Table 3
The permeability and water saturation of the cores for the two groups of experiments.
TPG with permeability (with the same water saturation)
Core number
1
2
3
4
5
6
7
8
9
Permeability (mD)
0.023
0.036
0.043
0.052
0.140
0.154
0.188
0.251
0.326
TPG with water saturation (with the same permeability)
Core number
1
2
3
4
5
6
7
8
Water saturation (%)
15.24
24.56
30.26
35.30
42.10
46.82
52.81
56.18Figure 3
Schematic of TPG experimental equipment and core samples.


### 4.2. Experimental Results

In the experiments, we have obtained the gas flow rate under different permeability and water saturation. Figure4(a) shows the relationship between the gas flow rate and the pressure square difference under different permeability. With the results, we calculated the gas TPGs under different permeability as shown in Figure 4(b) (blue points), and then obtained the fitting curve and the new fitting formula as shown in Equation (26). The results show that TPG is negatively correlated with permeability, and the larger the permeability, the smaller the TPG. In the same way, we obtained the fitting curve as shown in Figure 5, and the new fitting formula of the relationship between gas TPG and water saturation as shown in Equation (27). The results show that TPG is positively correlated with water saturation. It also can be found that the fitting formulas match the data distribution perfectly, and all the correlative factors (R
2) are above 0.9.(26)
G
g
=
c
S
w
d
=
0.0053
S
w
1.2923
.
(27)
G
g
=
a
K
b
=
0.1345
K
−
0.641
.Figure 4
(a) Relationship between gas flow rate and pressure square difference under different permeability. (b) The correlation between permeability and TPG.
(a)
(b)Figure 5
(a) Relationship between gas flow rate and pressure square difference under different water saturation. (b) The correlation between water saturation and TPG.
(a)
(b)According to the analysis and fitting of the test results, the relationship between gas TPG and permeability, water saturation of high-rank coal seams in the target block is shown in Equation (28).(28)
G
g
=
0.6556
·
K
−
1.131
·
S
w
3.27
+
0.2381
.Combining Equation (28) and Equation (25), we can obtain new pressure function, as shown in Equation (29).(29)
p
r
=
−
q
d
∗
ln
C
ρ
0.6556
·
K
−
1.131
·
S
w
3.27
+
0.2381
·
r
C
ρ
0.6556
·
K
−
1.131
·
S
w
3.27
+
0.2381
2
−
q
d
∗
r
C
ρ
0.6556
·
K
−
1.131
·
S
w
3.27
+
0.2381
−
C
1
E
i
C
ρ
0.6556
·
K
−
1.131
·
S
w
3.27
+
0.2381
·
r
+
C
2
·
μ
Z
¯
.


## 4.1. Experimental Method and Procedures

The coal core samples were obtained from the No. 3 coal seam in the study area. The diameter of each core sample is 2.5 cm on average, and the length ranges from 4.5 cm to 6 cm. In this paper, the porosity and the permeability of coal samples were measured using a fully automatic pore-permeability simultaneous measuring instrument. The permeability in this paper is gas log permeability. The average porosity for core samples of No. 3 coal seam is 4.3%. The gas used in this experiment was methane with a purity of more than 99.99%. The water was prepared in the laboratory according to some actual parameters of formation water from No. 3 coal seam, as shown in Table2. After that, we use special core displacement equipment to inject gas into the different permeability cores with different water saturation, and calculate the influence of water cut and permeability on gas injection displacement by recording gas production rate, then determine the TPG of coal-cores with water, the parameters of two groups of experiments are shown in Table 3. Figure 3 shows the schematic of the experimental equipment and core samples.Table 2
Parameters of formation water from the No. 3 coal seam.
Ion concentration, mg/L
Total salinity, mg/L
Water type
Na++K +
Mg2+
Ca2+
Cl−
SO4
2−
HCO3
−
488.16
3.98
4.82
245.59
17.83
884.93
1645.31
NaHCO3Table 3
The permeability and water saturation of the cores for the two groups of experiments.
TPG with permeability (with the same water saturation)
Core number
1
2
3
4
5
6
7
8
9
Permeability (mD)
0.023
0.036
0.043
0.052
0.140
0.154
0.188
0.251
0.326
TPG with water saturation (with the same permeability)
Core number
1
2
3
4
5
6
7
8
Water saturation (%)
15.24
24.56
30.26
35.30
42.10
46.82
52.81
56.18Figure 3
Schematic of TPG experimental equipment and core samples.


## 4.2. Experimental Results

In the experiments, we have obtained the gas flow rate under different permeability and water saturation. Figure4(a) shows the relationship between the gas flow rate and the pressure square difference under different permeability. With the results, we calculated the gas TPGs under different permeability as shown in Figure 4(b) (blue points), and then obtained the fitting curve and the new fitting formula as shown in Equation (26). The results show that TPG is negatively correlated with permeability, and the larger the permeability, the smaller the TPG. In the same way, we obtained the fitting curve as shown in Figure 5, and the new fitting formula of the relationship between gas TPG and water saturation as shown in Equation (27). The results show that TPG is positively correlated with water saturation. It also can be found that the fitting formulas match the data distribution perfectly, and all the correlative factors (R
2) are above 0.9.(26)
G
g
=
c
S
w
d
=
0.0053
S
w
1.2923
.
(27)
G
g
=
a
K
b
=
0.1345
K
−
0.641
.Figure 4
(a) Relationship between gas flow rate and pressure square difference under different permeability. (b) The correlation between permeability and TPG.
(a)
(b)Figure 5
(a) Relationship between gas flow rate and pressure square difference under different water saturation. (b) The correlation between water saturation and TPG.
(a)
(b)According to the analysis and fitting of the test results, the relationship between gas TPG and permeability, water saturation of high-rank coal seams in the target block is shown in Equation (28).(28)
G
g
=
0.6556
·
K
−
1.131
·
S
w
3.27
+
0.2381
.Combining Equation (28) and Equation (25), we can obtain new pressure function, as shown in Equation (29).(29)
p
r
=
−
q
d
∗
ln
C
ρ
0.6556
·
K
−
1.131
·
S
w
3.27
+
0.2381
·
r
C
ρ
0.6556
·
K
−
1.131
·
S
w
3.27
+
0.2381
2
−
q
d
∗
r
C
ρ
0.6556
·
K
−
1.131
·
S
w
3.27
+
0.2381
−
C
1
E
i
C
ρ
0.6556
·
K
−
1.131
·
S
w
3.27
+
0.2381
·
r
+
C
2
·
μ
Z
¯
.


## 5. Results and Discussion

### 5.1. Calculation Parameters

In this paper, we compare the pressure distribution of the target block with the present model under the same reservoir condition. The result shows that the present model has good agreement with the field test data. After that, we analyzed the influence of TPG and desorption on formation pressure and drainage radius, and obtain the effective utilization of reservoirs with different water saturation. The basic parameters used in the calculation process are all from the real values of the target block, which are shown in Table4.Table 4
The basic parameters of the target block.
Parameters
Value (unit)
The formation pressurep
e
10 (MPa)
The production well pressurep
w
3 (MPa)
Isothermal compression coefficientC
ρ
0.01 (MPa−1)
Gas viscosityμ
0.028 (mPa∙s)
Compressibility factorZ
0.89
The radius of pressure reliefr
e
300 (m)
The radius of the wellborer
w
0.1 (m)
Desorption amountsq
d
0.03 (kg/(m3s))


### 5.2. Effect of TPG

Figure6(a) shows the relationship between the formation pressure and the drainage radius under different gas TPG. Figure 6(b) is the relationship between formation pressure gradient and drainage radius. The results show that the control radius is increased with the decrease of TPG, the bigger of the TPG, the smaller the pressure drop rate. When the TPG is 0.005~0.015 MPa/m, the control radius is from 57 m to 138 m, which guides the development and the design of well pattern layout.Figure 6
Effect of TPG on formation pressure and formation pressure gradient. (a) Relationship between formation pressure and drainage radius under different TPG. (b) Relationship between formation pressure gradient and drainage radius under different TPG.
(a)
(b)


### 5.3. Effect of Desorption

In this section, we analyzed the influence of desorption effect and the desorption quantity on the control radius. Figure7(a) is the relationship between the formation pressure and the drainage radius under different desorption quantity. And Figure 7(b) shows the relationship between formation pressure gradient and drainage radius. The result shows that the desorption effect slows down the rate of pressure reduction. In addition, comparing with control radius without desorption effect, the promotion efficiency is 4.3%, 14.3%, and 25.7% when the desorption quantity is 0.01 kg/m3 · s, 0.03 kg/m3 · s and 0.05 kg/m3 · s respectively. It can be concluded that the existence of desorption promotes the sustainable development of CBM.Figure 7
Effect of desorption on formation pressure and formation pressure gradient. (a) Relationship between formation pressure and drainage radius under different desorption quantity. (b) Relationship between formation pressure gradient and drainage radius under different desorption quantity.
(a)
(b)


### 5.4. Control Radius of the Target Well Groups

In order to distinguish the production capacity of gas wells and facilitate the adjustment of production measures in the later period production, wells are classified based on the control radius in the field production process of the target block. The specific classification standard is shown in Table5.Table 5
The classification standard of wells based on control radius.
Control radius (m)
Well classification
<50
V
50~80
IV
80~100
III
100~120
II
>120
IWe calculated the effective control radius of two typical vertical well groups in the target block using the present model, which has 8 wells for each well group. Then classified the wells according to Table5. The results are shown in Table 6. The values of G
g in the table are obtained through core experiments.Table 6
The parameters and the calculation results of two study areas.
Study area
Well number
P
e (MPa)
P
w (MPa)
S
w (%)
TPG (MPa/m)
Control radius
Well classification
Study area 1
1
9.58
5.57
6.4
0.0057
136.7
I
2
9.72
5.37
11.6
0.0087
87.5
III
3
9.84
5.02
15
0.0115
55
IV
4
9.89
5.35
9.5
0.0074
103.6
II
5
9.96
5.85
12.5
0.0094
87.1
III
6
9.91
5.92
11.5
0.0087
93.7
III
7
10
5.06
9.4
0.0073
101.4
II
8
10.01
5.13
11.8
0.0089
80.6
III
Study area 2
1
9.09
4.11
9.1
0.0071
85.7
III
2
9.5
5.24
6.5
0.0058
112.5
II
3
9.22
5.7
11.5
0.0087
91.3
III
4
9.01
4.21
9.1
0.0071
87.2
III
5
9.4
5.44
12.3
0.0093
82.2
III
6
9.04
4.59
11.8
0.0089
79.6
IV
7
9.43
4.93
9.1
0.0071
91.6
III
8
9.44
5.65
12.1
0.0091
85.7
IIIBy analyzing the results in Table6, we can know that the control radius of these two blocks is mainly distributed in III (80~100 m). The average control radius is 91.3 m, the average gas TPG is 0.0082 MPa/m. From Table 6 it can be also proved that the lower the water saturation, the larger the control radius, which will be better development effects. Figure 8 is the histogram of the control radius distribution. From the figure, we can obtain that the effective utilization of study area 1 is better than study area 1.In order to analyze the utilization of the target area more intuitively, we calculated and drew the isogram of the control radius based on the actual well location as shown in Figure9. The results indicate the utilization scope of the target and provide the basis for later adjustment measures. For example, in order to increase the utilization area and sustainable development ability we can use well pattern infilling in region II and III based on engineering practices. For region IV, the control radius can be increased by hydraulic fracturing.Figure 8
The histogram of the control radius distribution.Figure 9
The isogram of the control radius. (a) Study area 1. (b) Study area 2.
(a)
(b)


## 5.1. Calculation Parameters

In this paper, we compare the pressure distribution of the target block with the present model under the same reservoir condition. The result shows that the present model has good agreement with the field test data. After that, we analyzed the influence of TPG and desorption on formation pressure and drainage radius, and obtain the effective utilization of reservoirs with different water saturation. The basic parameters used in the calculation process are all from the real values of the target block, which are shown in Table4.Table 4
The basic parameters of the target block.
Parameters
Value (unit)
The formation pressurep
e
10 (MPa)
The production well pressurep
w
3 (MPa)
Isothermal compression coefficientC
ρ
0.01 (MPa−1)
Gas viscosityμ
0.028 (mPa∙s)
Compressibility factorZ
0.89
The radius of pressure reliefr
e
300 (m)
The radius of the wellborer
w
0.1 (m)
Desorption amountsq
d
0.03 (kg/(m3s))


## 5.2. Effect of TPG

Figure6(a) shows the relationship between the formation pressure and the drainage radius under different gas TPG. Figure 6(b) is the relationship between formation pressure gradient and drainage radius. The results show that the control radius is increased with the decrease of TPG, the bigger of the TPG, the smaller the pressure drop rate. When the TPG is 0.005~0.015 MPa/m, the control radius is from 57 m to 138 m, which guides the development and the design of well pattern layout.Figure 6
Effect of TPG on formation pressure and formation pressure gradient. (a) Relationship between formation pressure and drainage radius under different TPG. (b) Relationship between formation pressure gradient and drainage radius under different TPG.
(a)
(b)


## 5.3. Effect of Desorption

In this section, we analyzed the influence of desorption effect and the desorption quantity on the control radius. Figure7(a) is the relationship between the formation pressure and the drainage radius under different desorption quantity. And Figure 7(b) shows the relationship between formation pressure gradient and drainage radius. The result shows that the desorption effect slows down the rate of pressure reduction. In addition, comparing with control radius without desorption effect, the promotion efficiency is 4.3%, 14.3%, and 25.7% when the desorption quantity is 0.01 kg/m3 · s, 0.03 kg/m3 · s and 0.05 kg/m3 · s respectively. It can be concluded that the existence of desorption promotes the sustainable development of CBM.Figure 7
Effect of desorption on formation pressure and formation pressure gradient. (a) Relationship between formation pressure and drainage radius under different desorption quantity. (b) Relationship between formation pressure gradient and drainage radius under different desorption quantity.
(a)
(b)


## 5.4. Control Radius of the Target Well Groups

In order to distinguish the production capacity of gas wells and facilitate the adjustment of production measures in the later period production, wells are classified based on the control radius in the field production process of the target block. The specific classification standard is shown in Table5.Table 5
The classification standard of wells based on control radius.
Control radius (m)
Well classification
<50
V
50~80
IV
80~100
III
100~120
II
>120
IWe calculated the effective control radius of two typical vertical well groups in the target block using the present model, which has 8 wells for each well group. Then classified the wells according to Table5. The results are shown in Table 6. The values of G
g in the table are obtained through core experiments.Table 6
The parameters and the calculation results of two study areas.
Study area
Well number
P
e (MPa)
P
w (MPa)
S
w (%)
TPG (MPa/m)
Control radius
Well classification
Study area 1
1
9.58
5.57
6.4
0.0057
136.7
I
2
9.72
5.37
11.6
0.0087
87.5
III
3
9.84
5.02
15
0.0115
55
IV
4
9.89
5.35
9.5
0.0074
103.6
II
5
9.96
5.85
12.5
0.0094
87.1
III
6
9.91
5.92
11.5
0.0087
93.7
III
7
10
5.06
9.4
0.0073
101.4
II
8
10.01
5.13
11.8
0.0089
80.6
III
Study area 2
1
9.09
4.11
9.1
0.0071
85.7
III
2
9.5
5.24
6.5
0.0058
112.5
II
3
9.22
5.7
11.5
0.0087
91.3
III
4
9.01
4.21
9.1
0.0071
87.2
III
5
9.4
5.44
12.3
0.0093
82.2
III
6
9.04
4.59
11.8
0.0089
79.6
IV
7
9.43
4.93
9.1
0.0071
91.6
III
8
9.44
5.65
12.1
0.0091
85.7
IIIBy analyzing the results in Table6, we can know that the control radius of these two blocks is mainly distributed in III (80~100 m). The average control radius is 91.3 m, the average gas TPG is 0.0082 MPa/m. From Table 6 it can be also proved that the lower the water saturation, the larger the control radius, which will be better development effects. Figure 8 is the histogram of the control radius distribution. From the figure, we can obtain that the effective utilization of study area 1 is better than study area 1.In order to analyze the utilization of the target area more intuitively, we calculated and drew the isogram of the control radius based on the actual well location as shown in Figure9. The results indicate the utilization scope of the target and provide the basis for later adjustment measures. For example, in order to increase the utilization area and sustainable development ability we can use well pattern infilling in region II and III based on engineering practices. For region IV, the control radius can be increased by hydraulic fracturing.Figure 8
The histogram of the control radius distribution.Figure 9
The isogram of the control radius. (a) Study area 1. (b) Study area 2.
(a)
(b)


## 6. Conclusion

In this paper, a new flow model of CBM with different water saturation has been established considering the TPG and gas desorption. We have obtained the pressure distribution and control radius of CBM reservoirs. After that, we carried out a series of experiments to study the relationship between TPG and permeability and water saturation and fitted out the new relational expression between them. Then, we analyzed the influence of TPG and desorption on the control areas and the sustainable development ability. Finally, we selected 2 well groups from the target block to analyzed the actual utilization situation using the present model.The results show that the present model has good agreement with the field test data. The control radius is increased with the decrease of TPG, the bigger of the TPG, the smaller the pressure drop rate. Besides, the desorption effect slows down the rate of pressure reduction, and the promotion efficiency is 25.7% when the desorption quantity is 0.05 kg/m3·s.Through the case study, we obtain that the control radius of target well groups ranges from 55 m to 136.7 m. The average control radius and gas TPG are 91.3 m 0.0082 MPa/m respectively. In addition, the lower the water saturation is, and the larger the control radius is. After that, we classify the wells into 5 categories according to the control radius, which is mainly distributed in III (80~100 m). Based on engineering practices, we suggest using well pattern infilling to increase the utilization area in region II and III, and utilizing hydraulic fracturing method for region IV.This study provides a quick and reasonable prediction of control radius for the CBM reservoir with different water saturation and gives guidance for further adjustment measures and sustainable development.



---

*Source: 3517642-2019-12-18.xml*