New Mechanical Model of Slotting–Directional Hydraulic Fracturing and Experimental Study for Coalbed Methane Development

Abstract

When hydraulic fractures do not expand in the direction required by a project, it is difficult to enhance coal seam permeability effectively. Slotting–directional hydraulic fracturing (SDHF) has been proposed as a possible alternative, but there is not enough theoretical mechanism to guide the construction. Based on preliminary study of the directional mechanism of single slotted hole, we established a new slotting–directional hydraulic fracturing (NSDHF) mechanical model by using the complex function theory of elasticity, and the influence of stress interference between adjacent slotted holes and non-uniform pore water pressure was considered. We carried out true triaxial double slotted holes SDHF experiments and used large-scale nondestructive computer tomography to scan the fractured samples to ensure accurate measurement of directional distance. The measured directional distance was used to verify the NSDHF model; the maximum deviation was 5.1%. Taking the experimental data in this paper as example, the stress interference between adjacent slotted holes decreased the fracture directional distance by 20.3%, and the non-uniform pore water pressure increased the fracture directional distance by 47.6%. NSDHF mechanical model realized the quantitative description of the influence of non-uniform pore water pressure on directional distance. The contribution of non-uniform pore water pressure to directional distance accounted for more than 25% of the total directional distance; the effect of non-uniform pore water pressure on fracture direction distance was almost twice the stress interference between adjacent slotted holes. The verified NSDHF model was used to study further the influence of horizontal stress difference, azimuth of slotted hole, slotting size and fluid injection pressure on directional distance with different slotting spacing. The work discussed in this paper will contribute to promoting and apply SDHF on a large scale in low-permeability coal mines.

Appendix: Total Stresses Around the Slotted Holes

$$\begin{aligned} \omega^{\prime}(\xi ) & = c\left( {1 - \frac{m}{{\xi^{2} }}} \right); \\ \overline{\omega (\xi ) \, } & = c\left( {\frac{{\rho^{2} }}{\xi } + \frac{m\xi }{{\mathop \rho \nolimits^{2} }}} \right); \\ \overline{{\omega^{\prime}(\xi )}} & = c\left( {1 - \frac{{m\xi^{2} }}{{\rho^{4} }}} \right); \\ \varphi_{w} (\xi ) & = - \frac{{p_{w} cm}}{\xi }; \\ \psi_{w} (\xi ) & = - \frac{{p_{w} c}}{\xi } - \frac{{p_{w} cm(1 + m\xi^{2} )}}{{(\xi^{3} - m)}} \\ \varphi_{\sigma 2} (\xi ) & = \frac{{\sigma_{1} c}}{4}\left( {\xi + \frac{{2e^{{2i\alpha_{1} }} - m}}{\xi + d}} \right) + \frac{{\sigma_{3} c}}{4}\left( {\xi + \frac{{2e^{{2i\alpha_{3} }} - m}}{\xi + d}} \right); \\ \psi_{\sigma 2} (\xi ) & = \frac{{\sigma_{1} c}}{2}\left[ { - \left( {\xi e^{{ - 2i\alpha_{1} }} + \frac{{me^{{ - 2i\alpha_{1} }} }}{\xi + d}} \right) - \frac{{1 + m^{2} - 2m\cos 2\alpha_{1} }}{\xi + d} + \frac{{(1 + m^{2} )(\cos 2\alpha_{1} - m)}}{{(\xi + d)^{3} - m(\xi + d)}} + \frac{{(m - 2\cos 2\alpha_{1} )d}}{{2(\xi + d)^{2} }}} \right] \\ & \quad + \frac{{\sigma_{3} c}}{2}\left[ { - \left( {\xi e^{{ - 2i\alpha_{3} }} + \frac{{me^{{ - 2i\alpha_{3} }} }}{\xi + d}} \right) - \frac{{1 + m^{2} - 2m\cos 2\alpha_{3} }}{\xi + d} + \frac{{(1 + m^{2} )(\cos 2\alpha_{3} - m)}}{{(\xi + d)^{3} - m(\xi + d)}} + \frac{{(m - 2\cos 2\alpha_{3} )d}}{{2(\xi + d)^{2} }}} \right] \\ \varphi_{w2}^{2} (\xi ) & = p_{w} c\left[ {\frac{1}{1/\xi + d} - \frac{md}{{(1/\xi + d)^{2} }} + \frac{{m + m^{2} (1/\xi + d)}}{{(1/\xi + d)^{2} - m}} - \frac{{m + m^{2} d^{2} }}{{d^{3} - md}} + \frac{m - 1}{d}} \right] \\ \psi_{w2}^{2} (\xi ) & = p_{w} c\left[ {\begin{array}{*{20}l} {\frac{m}{1/\xi + d} - \frac{m}{d} + \frac{{m + m^{3} }}{{2(\sqrt m + d)^{2} (\sqrt m - \xi )}} - \frac{{(\xi + m\xi^{3} )}}{{\xi^{2} - m}}} \hfill \\ {\quad \left( {\begin{array}{*{20}l} {\frac{1}{{(\xi d + 1)^{2} }} - \frac{2\xi md}{{(1/\xi + d)^{3} }} + \frac{{m + m^{2} (1/\xi + d)}}{{\xi^{2} (m - (1/\xi + d)^{2} )}}} \hfill \\ {\quad + \frac{{(2m + 2m^{2} (1/\xi + d))(1/\xi + d)^{2} }}{{\xi^{2} (m - (1/\xi + d)^{2} )^{2} }}} \hfill \\ \end{array} } \right)} \hfill \\ \end{array} } \right] \\ \varphi_{\sigma 2}^{2} (\xi ) & = \frac{{\sigma_{1} c}}{2}\left[ {\begin{array}{*{20}l} {\frac{{e^{{ - 2i\alpha_{1} }} }}{\xi } + \frac{{1 + m^{2} - 2me^{{2i\alpha_{1} }} + me^{{ - 2i\alpha_{1} }} }}{1/\xi + d} - \frac{{2md - 3e^{{2i\alpha_{1} }} d}}{{2(1/\xi + d)^{2} }} - \frac{{(1 + m^{2} )(e^{{2i\alpha_{1} }} - m)}}{{(1/\xi + d)^{3} - m(1/\xi + d)}}} \hfill \\ {\quad + \frac{{(1 + m^{2} )(e^{{2i\alpha_{1} }} - m)}}{{d^{3} - md}} + \frac{{m - 2e^{{2i\alpha_{1} }} - 2me^{{ - 2i\alpha_{1} }} - 2 - 2m^{2} + 4me^{{2i\alpha_{1} }} }}{2d}} \hfill \\ \end{array} } \right] \\ & \quad + \frac{{\sigma_{3} c}}{2}\left[ {\begin{array}{*{20}l} {\frac{{e^{{ - 2i\alpha_{3} }} }}{\xi } + \frac{{1 + m^{2} - 2me^{{2i\alpha_{3} }} + me^{{ - 2i\alpha_{3} }} }}{1/\xi + d} - \frac{{2md - 3e^{{2i\alpha_{3} }} d}}{{2(1/\xi + d)^{2} }} - \frac{{(1 + m^{2} )(e^{{2i\alpha_{3} }} - m)}}{{(1/\xi + d)^{3} - m(1/\xi + d)}}} \hfill \\ {\quad + \frac{{(1 + m^{2} )(e^{{2i\alpha_{3} }} - m)}}{{d^{3} - md}} + \frac{{m - 2e^{{2i\alpha_{3} }} - 2me^{{ - 2i\alpha_{3} }} - 2 - 2m^{2} + 4me^{{2i\alpha_{3} }} }}{2d}} \hfill \\ \end{array} } \right] \\ \psi_{\sigma 2}^{2} (\xi ) & = \frac{{\sigma_{1} c}}{2}\left[ {\begin{array}{*{20}l} {\frac{1}{2\xi } - \frac{{2e^{{2i\alpha_{1} }} - m}}{2/\xi + 2d} + \frac{{2e^{{2i\alpha_{1} }} - m}}{2d} + \frac{{(1 + m^{2} )}}{4(\sqrt m - \xi )}\left( {1 - \frac{{2e^{{2i\alpha_{1} }} - m}}{{(\sqrt m + d)^{2} }}} \right)} \hfill \\ {\quad - \frac{{\xi + m\xi^{3} }}{{\xi^{2} - m}}\left( {\begin{array}{*{20}l} { - \frac{{e^{{ - 2i\alpha_{1} }} }}{{\xi^{2} }} + \frac{{1 + m^{2} - 2me^{{2i\alpha_{1} }} + me^{{ - 2i\alpha_{1} }} }}{{(\xi d + 1)^{2} }} - \frac{{\xi (2md - 3e^{{2i\alpha_{1} }} d)}}{{(\xi d + 1)^{3} }}} \hfill \\ {\quad + \frac{{(m^{2} + 1)(m - e^{{2i\alpha_{1} }} )(3(d + 1/\xi )^{2} - m)}}{{\xi^{2} (m(d + 1/\xi ) - (d + 1/\xi )^{3} )^{2} }}} \hfill \\ \end{array} } \right)} \hfill \\ \end{array} } \right] \\ & \quad + \frac{{\sigma_{3} c}}{2}\left[ {\begin{array}{*{20}l} {\frac{1}{2\xi } - \frac{{2e^{{2i\alpha_{3} }} - m}}{2/\xi + 2d} + \frac{{2e^{{2i\alpha_{3} }} - m}}{2d} + \frac{{(1 + m^{2} )}}{4(\sqrt m - \xi )}\left( {1 - \frac{{2e^{{2i\alpha_{3} }} - m}}{{(\sqrt m + d)^{2} }}} \right)} \hfill \\ {\quad - \frac{{\xi + m\xi^{3} }}{{\xi^{2} - m}}\left( {\begin{array}{*{20}l} { - \frac{{e^{{ - 2i\alpha_{3} }} }}{{\xi^{2} }} + \frac{{1 + m^{2} - 2me^{{2i\alpha_{3} }} + me^{{ - 2i\alpha_{3} }} }}{{(\xi d + 1)^{2} }} - \frac{{\xi (2md - 3e^{{2i\alpha_{3} }} d)}}{{(\xi d + 1)^{3} }}} \hfill \\ {\quad + \frac{{(m^{2} + 1)(m - e^{{2i\alpha_{3} }} )(3(d + 1/\xi )^{2} - m)}}{{\xi^{2} (m(d + 1/\xi ) - (d + 1/\xi )^{3} )^{2} }}} \hfill \\ \end{array} } \right)} \hfill \\ \end{array} } \right]. \\ \end{aligned}$$