Establishment and Determination of Matrix Shape Factor for Tectonic Coal: Theoretical, Experimental and Simulation Study

Abstract

Diffusion is an important process in coalbed methane migration. In addition to diffusion coefficient, the size and shape of coal particles also have influences on methane adsorption and desorption. Thus, the matrix shape factor is a key parameter affecting the methane diffusion rate in coal matrix. The matrix shape factor of intact coal has been studied by many scholars, but that of the tectonic coal has not been established. In this paper, the theoretical expression of matrix shape factor of the tectonic coal is proposed for the first time, which is $\alpha_{N} = 480Nb/\left( {(27 - 2N)a^{3} } \right)$. The experimental test of gas emission characteristics of the standard tectonic coal sample is conducted, and the numerical simulation of gas emission is also carried out under the same conditions as the experiment. By matching the numerically calculated gas volumes with the experimental values, the approximate matrix shape factor of the tectonic coal sample can be obtained, and the value is 1.83 × 10⁶ m⁻². The matrix scale and fracture aperture are further calculated, which are 0.518 mm and 3.702 μm, respectively. On this basis, the effects of the matrix scale and fracture aperture on coalbed methane migration are analyzed, and the stress relief measures to enhance the methane recovery of tectonic coal seam are proposed.

Article Highlights

• (1) Theoretical expression of matrix shape factor of tectonic coal is proposed for the first time.

• (2) Values of shape factor, matrix size and fracture aperture are determined.

• (3) Stress relief measures to enhance methane recovery in tectonic coal seam are proposed.

Appendices

Appendix 1: Derivation of Matrix Shape Factors for Intact Coal

For the three-dimensional diffusion of intact coal, the amount of methane diffusing to six faces can be considered consistent, and one-sixth of them is taken as the research object (see Fig. 

Fig. 12

Schematic diagram of methane diffusion in intact coal: a three-dimensional diffusion, b two-dimensional diffusion, c one-dimensional diffusion

12a). The methane concentration at each point on the cross section perpendicular to the x-axis is set to be equal, and it can be expressed as the concentration at the center point of the cross section:

$$ c(x,y,z) = c\left( {x, - \frac{a}{2},\frac{a}{2}} \right). $$

(20)

Suppose that the methane concentration $c$ in the matrix follows a parabolic distribution during the diffusion process, and the following equation can be obtained:

$$ c = b_{1} x^{2} + b_{2} $$

(21)

where $b_{1}$ and $b_{2}$ are undetermined coefficients. Assuming that the gaseous methane concentration at the matrix boundary is equal to the methane concentration in the fractures, then,

$$ b_{2} = c_{{\text{f}}} - \frac{{b_{1} a^{2} }}{4} $$

(22)

The area of the cross section is $4x^{2}$, and the volume of the research domain is $a^{3} /6$. Then, the integral average value of methane concentration in the matrix can be obtained:

$$ \overline{c} = \frac{{\int_{0}^{a/2} {4x^{2} c{\text{d}}x} }}{{a^{3} /6}}{ = }\frac{24}{{a^{3} }}\int\limits_{0}^{a/2} {(b_{1} x^{2} + b_{2} )x^{2} {\text{d}}x} = \frac{{3b_{1} a^{2} }}{20} + b_{2} . $$

(23)

Combining Eqs. (22) and (23), we can obtain

$$ \overline{c} = \frac{{3b_{1} a^{2} }}{20} + c_{{\text{f}}} - \frac{{b_{1} a^{2} }}{4} = c_{{\text{f}}} - \frac{{b_{1} a^{2} }}{10}. $$

(24)

Then,

$$ b_{1} = \frac{{10(c_{{\text{f}}} - \overline{c} )}}{{a^{2} }}. $$

(25)

According to Fick’s first law, the diffusion flux through the contact surface between the matrix and fractures is

$$ J = - D\left( {\frac{\partial c}{{\partial x}}} \right)_{{x = \frac{a}{2}}} = - Dab_{1} = \frac{{10D(\overline{c} - c_{{\text{f}}} )}}{a}. $$

(26)

Therefore, under the condition of three-dimensional diffusion, the diffused gas amount per volume of coal matrix in unit time can be obtained as follows:

$$ Q_{3} = J\frac{S}{V} = \frac{{10D(\overline{c} - c_{{\text{f}}} )}}{a} \cdot \frac{{a^{2} }}{{a^{3} /6}} = \frac{{60D(\overline{c} - c_{{\text{f}}} )}}{{a^{2} }}. $$

(27)

Combined with Eq. (1), the matrix shape factor for the intact coal under three-dimensional diffusion condition is

$$ \alpha_{3} = \frac{60}{{a^{2} }}. $$

(28)

For the two-dimensional diffusion, methane diffusion through four faces is consistent, and one-fourth of the matrix is taken as the research object (see Fig. 12b). Equations (20), (21) and (22) are still valid. The areas of the cross section is $2ax$, and the volume of the research domain is $a^{3} /4$. Then, the integral average value of methane concentration in the matrix can be obtained:

$$ \overline{c} = \frac{{\int_{0}^{a/2} {2axc{\text{d}}x} }}{{a^{3} /4}}{ = }\frac{8}{{a^{2} }}\int_{0}^{a/2} {(b_{1} x^{2} + b_{2} )x{\text{d}}x} = \frac{{b_{1} a^{2} }}{8} + b_{2} . $$

(29)

Combining Eqs. (22) and (29), we can obtain

$$ \overline{c} = \frac{{b_{1} a^{2} }}{8} + c_{f} - \frac{{b_{1} a^{2} }}{4} = c_{f} - \frac{{b_{1} a^{2} }}{8}. $$

(30)

Then,

$$ b_{1} = \frac{{8(c_{{\text{f}}} - \overline{c} )}}{{a^{2} }}. $$

(31)

Similar to Eq. (26), the diffusion flux through the contact surface between the matrix and fractures is

$$ J = - Dab_{1} = \frac{{8D(\overline{c} - c_{{\text{f}}} )}}{a} $$

(32)

Then, the gas amount for two-dimensional diffusion can be obtained:

$$ Q_{2} = J\frac{S}{V} = \frac{{8D(\overline{c} - c_{{\text{f}}} )}}{a} \cdot \frac{{a^{2} }}{{a^{3} /4}} = \frac{{32D(\overline{c} - c_{{\text{f}}} )}}{{a^{2} }}. $$

(33)

Thus, the matrix shape factor for two-dimensional diffusion of intact coal is

$$ \alpha_{2} = \frac{32}{{a^{2} }}. $$

(34)

For the one-dimensional diffusion, methane diffusion through two faces is consistent, and a half of the matrix is taken as the research object (see Fig. 12c). Equations (20), (21) and (22) are still valid. The areas of the cross section is $a^{2}$, and the volume of the research domain is $a^{3} /2$. Then, the integral average value of methane concentration in the matrix can be obtained:

$$ \overline{c} = \frac{{\int_{0}^{a/2} {a^{2} c{\text{d}}x} }}{{a^{3} /2}}{ = }\frac{2}{a}\int\limits_{0}^{a/2} {(b_{1} x^{2} + b_{2} ){\text{d}}x} = \frac{{b_{1} a^{2} }}{12} + b_{2} . $$

(35)

Combining Eqs. (22) and (35), we can obtain

$$ \overline{c} = \frac{{b_{1} a^{2} }}{12} + c_{{\text{f}}} - \frac{{b_{1} a^{2} }}{4} = c_{{\text{f}}} - \frac{{b_{1} a^{2} }}{6}. $$

(36)

Then,

$$ b_{1} = \frac{{6(c_{f} - \overline{c} )}}{{a^{2} }}. $$

(37)

Similar to Eq. (26), the diffusion flux through the contact surface between the matrix and fractures is

$$ J = - Dab_{1} = \frac{{6D(\overline{c} - c_{{\text{f}}} )}}{a}. $$

(38)

Then, the gas amount for one-dimensional diffusion can be obtained:

$$ Q_{1} = J\frac{S}{V} = \frac{{6D(\overline{c} - c_{{\text{f}}} )}}{a} \cdot \frac{{a^{2} }}{{a^{3} /2}} = \frac{{12D(\overline{c} - c_{{\text{f}}} )}}{{a^{2} }}. $$

(39)

Thus, the matrix shape factor for one-dimensional diffusion of intact coal is

$$ \alpha_{1} = \frac{12}{{a^{2} }}. $$

(40)

Appendix 2: Derivation of Matrix Shape Factors for Two- and One-Dimensional Diffusion of Tectonic Coal

For the two-dimensional diffusion, methane diffusion through eight edges is consistent, and one-eighth of the matrix is taken as the research object (see Fig. 2b). Equations (3), (4) and (5) are still valid. The areas of the two cross sections are $2x(a/2 - x)$ and $(2x + a)(a/2 - x)/2$, and the volume of the research domain is $a^{3} /8$. Then, the integral average value of methane concentration in the matrix can be obtained:

$$ \overline{c} = \frac{{\int_{0}^{a/2} {c[2x(a/2 - x) + (2x + a)(a/2 - x)/2]{\text{d}}x} }}{{a^{3} /8}}{ = }\frac{8}{{a^{3} }}\int\limits_{0}^{a/2} {(b_{1} x^{2} + b_{2} )\left( {\frac{{a^{2} }}{4} + ax - 3x^{2} } \right){\text{d}}x} = \frac{{7b_{1} a^{2} }}{120} + b_{2} . $$

(41)

Combining Eqs. (5) and (41), we can obtain

$$ \overline{c} = \frac{{7b_{1} a^{2} }}{120} + c_{{\text{f}}} - \frac{{b_{1} a^{2} }}{4} = c_{{\text{f}}} - \frac{{23b_{1} a^{2} }}{120}. $$

(42)

Then,

$$ b_{1} = \frac{{120(c_{{\text{f}}} - \overline{c} )}}{{23a^{2} }}. $$

(43)

Similar to Eq. (9), the diffusion flux through the contact surface between the matrix and fractures is

$$ J = - D\left( {\frac{\partial c}{{\partial x}}} \right)_{{x = \frac{a}{2}}} = - Dab_{1} = \frac{{120D(\overline{c} - c_{{\text{f}}} )}}{23a}. $$

(44)

Then, the gas amount for two-dimensional diffusion can be obtained:

$$ Q_{2} = J\frac{S}{V} = \frac{{120D(\overline{c} - c_{{\text{f}}} )}}{23a} \cdot \frac{ab}{{a^{3} /8}} = \frac{{960Db(\overline{c} - c_{{\text{f}}} )}}{{23a^{3} }}. $$

(45)

Thus, the matrix shape factor for two-dimensional diffusion of tectonic coal is

$$ \alpha_{2} = \frac{960b}{{23a^{3} }}. $$

(46)

For the one-dimensional diffusion, methane diffusion through four edges is consistent, and one-fourth of the matrix is taken as the research object (see Fig. 2c). Equations (3), (4) and (5) are still valid. The area of each cross section is $a(a/2 - x)$, and the volume of the research domain is $a^{3} /4$. Then, the integral average value of methane concentration in the matrix can be obtained:

$$ \overline{c} = \frac{{\int_{0}^{a/2} {2[a(a/2 - x)c]{\text{d}}x} }}{{a^{3} /4}}{ = }\frac{4}{{a^{3} }}\int\limits_{0}^{a/2} {(b_{1} x^{2} + b_{2} )(a^{2} - 2ax){\text{d}}x} = \frac{{b_{1} a^{2} }}{24} + b_{2} . $$

(47)

Combining Eqs. (5) and (47), we can obtain

$$ \overline{c} = \frac{{b_{1} a^{2} }}{24} + c_{{\text{f}}} - \frac{{b_{1} a^{2} }}{4} = c_{{\text{f}}} - \frac{{5b_{1} a^{2} }}{24}. $$

(48)

Then,

$$ b_{1} = \frac{{24(c_{{\text{f}}} - \overline{c} )}}{{5a^{2} }}. $$

(49)

The diffusion flux through the contact surface between the matrix and fractures is

$$ J = - D\left( {\frac{\partial c}{{\partial x}}} \right)_{{x = \frac{a}{2}}} = - Dab_{1} = \frac{{24D(\overline{c} - c_{{\text{f}}} )}}{5a}. $$

(50)

Then, the gas amount for one-dimensional diffusion can be obtained:

$$ Q_{1} = J\frac{S}{V} = \frac{{24D(\overline{c} - c_{{\text{f}}} )}}{5a} \cdot \frac{ab}{{a^{3} /4}} = \frac{{96Db(\overline{c} - c_{{\text{f}}} )}}{{5a^{3} }}. $$

(51)

Thus, the matrix shape factor for one-dimensional diffusion of tectonic coal is

$$ \alpha_{1} = \frac{96b}{{5a^{3} }} $$

(52)