Page 231 - AJWEP-22-5

Rheological properties of coal gangue slurry

It is assumed that under injection pressure, the Where:
slurry is injected from the injection hole into the (i) r represents the radius of the slurry spread (m).
fracture and undergoes planar radial flow parallel to the (ii) Δr is the increment of the spread radius per unit
fracture plane. The injection pressure at the injection time.
hole is denoted as P , and the scouring force exerted (iii) Δθ is the angle formed by the increment of the
0
by the water flow on the slurry is denoted as P . 33,34 A spread radius per unit time.
w
schematic diagram of the slurry diffusion in inclined (iv) ρ is the density of the slurry (kg/m ).
3
fractures is shown in Figure 11. An arbitrary differential (v) α is the angle of the crack inclination; higher-order
element is selected from Figure 11 for stress analysis, as infinitesimal terms are neglected.
illustrated in Figure 12. The formula for the power-law The equation can be simplified and organized as
fluid fracture grouting diffusion model is presented in follows:
Equation I: d dP P

dP P w w g sin (II)
PP w r z PP w r r z dz dr
r

dr With respect to z, integration yields:

dP P r d

PP w rz z dP P w z C (III)

w
dr 2 dz dr g sin 1
r r
r

r gsinn rz
2 Where C is a constant determined by boundary
1
r r r conditions, which are then substituted into the

0
2 equation.
(I) du
0 C 0 (IV)
1
dz z0

dP P
w g sin z (V)

dr
Additionally, the flow variation equation for a
Newtonian fluid is generally expressed as:
du n
k (VI)
dz
Substituting Equation VI into Equation V yields:
Figure 11. Schematic diagram illustrating slurry

diffusion in inclined fractures du n dP P
w
k g sin
z (VII)
dz dr
Integration with respect to z yields:
1

n 1 dP P n n1
w
u g sin
z n C (VIII)

2

n 1 k dr
Based on the boundary conditions:
n1
b
Figure 12. Stress analysis of a slurry element within u b 0 C n (IX)
an inclined fracture z 2 2
2
Volume 22 Issue 5 (2025) 225 doi: 10.36922/AJWEP025210162