^{1}

^{2}

^{1}

^{3}

^{4}

^{1}

^{5}

^{6}

^{3}

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

An accurate evaluation of coal rock fracture conductivity is an important prerequisite for predicting the productivity of CBM wells. Coal rock is soft and fragile, with low elastic modulus and high Poisson ratio. In the process of fracturing flowback, the contact deformation between proppant and fracture wall will affect the fracture conductivity when the proppant is embedded in the coal rock; thus the calculation method of plate fracture conductivity is no longer suitable for the evaluation of coal rock. Based on the contact deformation theory of elastic mechanics, a method for calculating contact deformation of proppant in fracture is proposed. Considering the effect of the deformation and embedded depth of proppant and the tortuosity of pore flow channel between proppant particles on fracture conductivity, a model for calculating fracture conductivity of coal rock fractures under three kinds of proppant arrangement (Model 4-1, Model 3-1, and Model 2-1) is established. Comparison of calculation results of theoretical model and experiments confirmed that the arrangement of proppant in coal rock fracture is closest to Model 3-1, and the influence of mechanical parameters of coal rock and proppant on fracture conductivity is calculated and analyzed by this theoretical model. The study shows that the coal rock fracture conductivity is affected little by Poisson’s ratio of coal rock and proppant, which is greatly influenced by the elastic modulus of them, and the effect of particle size of proppant is especially significant.

Hydraulic fracturing is the primary means of stimulating CBM wells and has been widely used in exploration of coalbed methane [

Researchers have carried out relevant experimental research on fracture conductivity of coal rock, in terms of proppant embedment, crushing and wall compaction, and the damage of fracture fluid residue to fracture conductivity. Gao [

The schematic diagram of proppant 1 and proppant 2 under closure pressure is shown in Figure

Schematic diagram of contact deformation between proppants.

The theoretical relationship between the two proppants is verified by the following relationship [

When

Schematic diagram of proppants embedment.

The value of

Combining (

According to Hooke’s law, when the proppant interacts with the wall of coal rock, it will cause the compacted deformation. It is assumed that the effective thickness of coal seam is

The deformation is caused by the embedment, so the thickness of coal rock without embedding is still initial value. Combining (

Assuming that, under the closure pressure, the number of proppant layers between two walls of coal rock fracture is

That is,

The total number of proppants embedded in the seam is denoted by

A is constant; when

B is constant; when

Combining the definition of width and porosity of fractures [

The effective radius of flow channels and the tortuosity of coal rock fractures can be, respectively, expressed as

The permeability of coal rock with embedment is

In this paper, a model for calculating the fracture conductivity of coal rock under 3 kinds of arrangement modes (Model 4-1, Model 3-1, and Model 2-1) are established. The schematic diagram is shown in Figure

Schematic diagram of 3 kinds of proppant arrangements.

Model 4-1

Model 3-1

Model 2-1

In the different arrangements of proppant, the five, four, and three proppants are, respectively, taken as the study object. The center line constitutes a cube. The upper and lower proppants deform under the closure pressure

Simplified diagram under three arrangement modes.

Model 4-1

Model 3-1

Model 2-1

According to the space geometry theory, the equation for calculating the distance between two deformed proppants can obtained by Figure

As shown in Figure

Schematic diagram of stress unit in the same layer.

The proppant is arranged closely, and the center line forms a square whose diameter is

The external force

By the definition of fracture diversion capacity, the fracture conductivity of coal rock

With (

First, the theoretical model is used to calculate the theoretical value of the fracture conductivity of coal rock under the same conditions as the experimental parameters in the laboratory. The number of proppants layer

^{3};

^{2}.

The proppant in this paper is taken from Daqing oilfield, the density and concentration are, respectively, 1.692 g/cm^{3} and 10 kg/m^{2}, the total length of diversion chamber is 17.7 cm, and its end is semicircle with diameter 3.81 cm. The equivalent particle size of 20 ~ 40 mesh is calculated to be 0.6106 × 10^{-4 }m by using geometric weighted average method, and the corresponding sand layer is calculated as 12 layers when the concentration is 10 kg/m^{2}.

Parameters such as elastic modulus and Poisson’s ratio of coal and proppant are provided by Daqing oilfield, and the experimental parameters are shown in Table

Experimental parameters of coal rock and proppant.

Concentration of proppant (kg·m^{−2}) | Diameter of proppant (m) | Closure pressure (MPa) | Elastic modulus of proppant (GPa) | Poisson’s ratio of proppant | Elastic modulus of coal rock (GPa) | Poisson’s ratio of coal rock | Height of fracture (m) | Length of fracture (m) |
---|---|---|---|---|---|---|---|---|

10 | 6.106×10^{−4} | 10~30 | 20.338 | 0.142 | 1.205 | 0.221 | 0.0381 | 0.127 |

After the test of coal rock fracture conductivity, the coal plate is shown in Figure

Coal plate shape before and after fracture conductivity test.

The average value of fracture conductivity of each pressure test point in the same period is taken as the test result of the fracture conductivity of coal rock; the theoretical results of fracture conductivity of coal rock under different pressures and the comparison curves with experimental results are shown in Table

Comparison of experimental and theoretical results of coal rock fracture conductivity.

10MPa | 15MPa | 20MPa | 25MPa | 30MPa | |
---|---|---|---|---|---|

Model 2-1 | 200.185 | 183.888 | 170.467 | 159.175 | 149.466 |

Model 3-1 | 110.027 | 96.936 | 85.905 | 77.268 | 69.955 |

Model 4-1 | 95.554 | 81.275 | 69.917 | 60.770 | 53.067 |

Experimental data | 117.641 | 101.434 | 90.317 | 79.398 | 73.697 |

Comparison curves of fracture conductivity between theoretical and experimental results

Comparison curves of fracture conductivity between theoretical and experimental results (experimental data of Meng et al. 2014)

The two groups of comparison curves of fracture conductivity between theoretical and experimental results in Figures

Through the previous model verification, the theoretical model of this paper is found to be more consistent with Model 3-1, so the sensitivity analysis of this paper has been based on Model 3-1, and the change of fracture conductivity with the various factors is also based on the Model 3-1.

The theoretical values of fracture conductivity of coal rock are calculated under the condition that Table

Fracture conductivity under different Poisson’s ratio of coal rock.

Poisson’s ratio of coal rock | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 |
---|---|---|---|---|---|

Fracture conductivity of coal rock/(^{2}·cm) | 86.090 | 86.164 | 86.260 | 86.379 | 86.522 |

Relationship curve of fracture conductivity and Poisson’s ratio of coal rock.

It can be seen from the diagram that the influence of coal rock Poisson’s ratio on fracture conductivity is very small. With the increase of coal rock Poisson’s ratio, the fracture conductivity increases slightly, but the increase rate is less than 1%. Therefore, in the field of construction error within the allowable range, the impact is negligible.

In order to study the influence of elastic modulus of coal rock on fracture conductivity, the theoretical values of fracture conductivity of coal rock complex fracturing system under the condition that the other calculation parameters of Table

Fracture conductivity under different elastic modulus of coal rock.

Elastic modulus of coal rock / GPa | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|

Fracture conductivity of coal rock/(^{2}⋅cm) | 95.628 | 102.207 | 106.070 | 108.614 | 110.417 |

The relationship between fracture conductivity and elastic modulus of coal rock is obtained, as shown in Figure

Relationship curve of fracture conductivity and elastic modulus of coal rock.

Similarly, keeping the other calculated parameters in Table

As can be seen from Figure

Relationship curve of fracture conductivity and Poisson’s ratio of proppant.

Relationship curve of fracture conductivity and elastic modulus of coal rock.

The theoretical values are calculated at 20 MPa when the other calculated parameters in Table

It can be seen from Figure

Relationship curve of fracture conductivity and proppant diameter.

The accurate calculation of coal rock conductivity can provide important theoretical basis for CBM wells fracturing scheme design and production prediction after fracturing. Coal rock is soft and brittle with the characteristics of low modulus and high Poisson’s ratio; coal powder is easy to fall off and plug fractures in the process of fracturing. And under the closure pressure, proppant deformation and embedment are prone to occur, which leads to significant changes in fracture width and porosity and changes in fracture conductivity. On the other hand, different arrangements of proppants have a significant effect on the fracture width and porosity under the closure pressure, thus affecting and changing the conductivity of fractures. This paper properly considered the special mechanical characteristics of coal rock and the influence of proppants arrangements and embedments. The paper studied the fracture conductivity of coal rock both from theoretical and experimental aspects. In the past, the research on coal rock fracture conductivity basically used the method of experimental evaluation, Gao [

The authors declare that there are no conflicts of interest regarding the publication of this article.

The research was supported by Natural Science for Youth Foundation of China (no. 51504068), University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province (UNPYSCT-2016124), and Northeast Petroleum University Scientific Research Start Funds Subsidization Project and National Natural Science Foundation of China (no. 51490650).