Analytical solution of grouting fracturing height for the post-grouted bored piles using the Herschel–Bulkley model with time-varying viscosity

Abstract

According to the cylindrical cavity expansion theory, the equation calculating the lateral displacement of soil along the pile shaft is obtained. Combined with the Herschel–Bulkley model with time-varying viscosity and the uniform flow equation, the analytical solution of the grouting fracturing height (GFH) for the post-grouted at the pile tip and considering the time-varying viscosity is derived under certain assumptions. For the problem of post grouting in the layered soil, the iterative calculation method of the GFH is given. The influence of the grouting pressure, pile length, pile radius, thickness of mud skin, rheological index, initial consistency coefficient, time-varying coefficient, and grouting time for grouting on the GFH is analyzed with examples. Finally, the problems of time-varying viscosity and grouting pressure loss are discussed.

Appendix

Combining Eqs. (9) and (12) in the text, separating them, and then integrating along the y direction, we get the following:

$$ \upsilon =-y{\left(\frac{\tau -{\tau}_0}{K(t)}\right)}^{\frac{1}{n}}-B, $$

(25)

where B is the integral constant.

Substituting the boundary condition Eq. (17) in the text into Eq. (25), the following is obtained:

$$ B=-\frac{u+b}{2}{\left(\frac{\tau -{\tau}_0}{K(t)}\right)}^{\frac{1}{n}}. $$

(26)

Substituting Eq. (26) into Eq. (25), we obtain

$$ \upsilon =-y{\left(\frac{\tau -{\tau}_0}{K(t)}\right)}^{\frac{1}{n}}+\frac{u+b}{2}{\left(\frac{\tau -{\tau}_0}{K(t)}\right)}^{\frac{1}{n}}. $$

(27)

Substituting Eq. (10) in the text into Eq. (27), we have

$$ \upsilon =-y{\left(\frac{\tau -{\tau}_0}{C_0{e}^{wt}}\right)}^{\frac{1}{n}}+\frac{u+b}{2}{\left(\frac{\uptau -{\tau}_0}{C_0{e}^{wt}}\right)}^{\frac{1}{n}}. $$

(28)

To facilitate the derivation and calculation of the following formula, Eq. (28) is converted into integral form, and we get

$$ \upsilon ={\int}_y^{\frac{u+b}{2}}{\left(\frac{\tau -{\tau}_0}{C_0{e}^{wt}}\right)}^{\frac{1}{n}}\mathrm{d}y. $$

(29)

Combining Eqs. (15) and (16) in the text, we can obtain

$$ y=\frac{\tau }{\tau_e}\left(u+b\right). $$

(30)

Substituting Eq. (30) into Eq. (29), we transform the integral variable, and we will have

$$ \upsilon =\frac{u+b}{\tau_e}{\int}_{\tau}^{\frac{1}{2}{\tau}_e}{\left(\frac{\tau -{\tau}_0}{C_0{e}^{wt}}\right)}^{\frac{1}{n}}\mathrm{d}\tau . $$

(31)

From Eq. (31), we obtain

$$ \upsilon =\frac{u+b}{\tau_e}{\left(\frac{1}{C_0{e}^{wt}}\right)}^{\frac{1}{n}}\frac{n}{n+1}\left[{\left(\frac{1}{2}{\tau}_e-{\tau}_0\right)}^{\frac{n+1}{n}}-{\left(\tau -{\tau}_0\right)}^{\frac{n+1}{n}}\right]. $$

(32)

Substituting Eqs. (15) and (16) in the text into Eq. (32), we get

$$ \upsilon =\frac{u+b}{\tau_e}{\left(\frac{1}{C_0{e}^{wt}}\right)}^{\frac{1}{n}}\frac{n}{n+1}\left[{\left(\frac{\Delta p\left(u+b\right)}{4h}-{\tau}_0\right)}^{\frac{n+1}{n}}-{\left(\frac{\Delta py}{2h}-{\tau}_0\right)}^{\frac{n+1}{n}}\right]. $$

(33)

The unit time flow q is

$$ q={\int}_{-\frac{u+b}{2}}^{\frac{u+b}{2}}L\bullet \upsilon \mathrm{d}y=2L{\int}_0^{\frac{u+b}{2}}\upsilon \mathrm{d}y, $$

(34)

where

$$ L=2\pi {r}_0, $$

(35)

in which L is the length of the crack.

Substituting Eq. (35) into Eq. (34), we obtain

$$ q=2L{\int}_0^{\frac{u+b}{2}}\upsilon \mathrm{d}y=4\pi {r}_0{\int}_0^{\frac{u+b}{2}}\upsilon \mathrm{d}y. $$

(36)

Substituting Eq. (33) into Eq. (36), we can get Eq. (18) in the text.