Grout Spread and Injection Period of Silica Solution and Cement Mix in Rock Fractures

Abstract

A systematic presentation of the analytic relations of grout spread to the time period is established. They are divided following the nature of the flow, the property of the mix and the driving process. This includes channel flow between parallel plates and radial flow between parallel discs, nonlinear Newtonian fluids like silica solution, polyurethane and epoxy, and Bingham material like cement-based grout, and three grouting processes at a constant flow rate, constant pressure and constant energy. The analytic relations for the constant energy process are new and complete the relations of the constant flow rate and constant pressure processes. The well-known statement that refusal cannot be obtained during finite time for any injected material at a constant flow rate or constant injection pressure is extended to include the energy process. The term refusal pressure or energy cannot be supported for stop criteria. Stop criteria have to be defined considering confirmed relation of the spread to the time period and of the flow rate to the pressure and spread. It is shown that it is always possible to select a grouting process along which the work will exceed any predefined energy, the consequence of which is that jacking is related to the applied forces and not to the injected energy. Furthermore, a clarification is undertaken concerning the radial flow rate of a Bingham material since there are two different formulations. Their difference is explained and quantified. Finally, it is shown that the applied Lugeon theory is not supported by the analytic relations and needs to be substantially modified.

Appendices

Appendix 1: Time–Spread Closed Form

The integration of the right-hand side of Eq. (50) leads to

$$F = \frac{{\xi_{a} - 2}}{9}(F_{a} + F_{b} ) + \frac{{1 + \xi_{a} }}{3}F_{c}$$

(67)

with,

$$F_{a} = - {\text{Sp}}\left( {\frac{2 + \xi }{{2 - \xi_{a} }}} \right) + \ln \frac{{\left| {2 - \xi_{a} } \right|}}{{\xi_{a} }}\ln (2 + \xi )$$

(68)

$$F_{b} = {\text{Sp}}\left( {\frac{1 - \xi }{{1 + \xi_{a} }}} \right) - \ln \frac{{1 + \xi_{a} }}{{\xi_{a} }}\ln (1 - \xi )$$

(69)

$$F_{c} = \frac{1}{1 - \xi }\ln \frac{{\xi + \xi_{a} }}{{\xi_{a} }} - \frac{1}{{1 + \xi_{a} }}\ln \frac{{\xi + \xi_{a} }}{1 - \xi }$$

(70)

in which Sp is Spence’s function. Spence’s function is

$${\text{Sp}}(z) = - \int\limits_{0}^{z} {\frac{{\ln \left| {1 - x} \right|}}{x}{\text{d}}x}$$

(71)

Spence’s function has different appellations since it has many discoverers or re-discoverers (Zagier 2006). Spence’s function can be calculated using the dilogarithm function that is defined in the interval (−1, 1) as

$${\text{Li}}_{2} (z) = \sum\limits_{n = 1}^{\infty } {\frac{{z^{n} }}{{n^{2} }}}$$

(72)

Their relations are

$${\text{Sp}}(z) = {\text{Li}}_{2} (z)\quad - 1 \le z \le 1$$

(73)

$${\text{Sp}}(z) = \frac{{\pi^{2} }}{3} - \frac{1}{2}\ln^{2} (z) - {\text{Li}}_{2} (1/z)\quad z \ge 1$$

(74)

$${\text{Sp}}(z) = - \frac{{\pi^{2} }}{6} - \frac{1}{2}\ln^{2} \left| z \right| - {\text{Li}}_{2} (1/z)\quad z \le - 1$$

(75)

Appendix 2: Time–Spread First-Order Approximation

The first-order approximations of Eq. (50) in the interval (0, ξ) is

$$F^{(1)} (\xi ) = \int\limits_{0}^{\xi } {(f(\xi ) + (x - \xi )f^{\prime } (\xi ))\frac{x}{{(1 - x)^{2} (2 + x)}}{\text{d}}x}$$

(76)

in which

$$f(x) = \left( {1 + \frac{{\xi_{a} }}{x}} \right)\ln \left( {1 + \frac{x}{{\xi_{a} }}} \right)$$

(77)

The integration is straightforward and leads to F ⁽¹⁾ = F ⁽⁰⁾ + ∆F with

$$F^{(0)} = \frac{1}{3}\left( {1 + \frac{{\xi_{a} }}{\xi }} \right)\ln \left( {1 + \frac{\xi }{{\xi_{a} }}} \right)\left[ {\frac{\xi }{1 - \xi } + \frac{2}{3}\ln \frac{2(1 - \xi )}{2 + \xi }} \right]$$

(78)

$$\Delta F = \frac{1}{\xi }\left( {1 - \frac{{\xi_{a} }}{\xi }\ln \left( {1 + \frac{\xi }{{\xi_{a} }}} \right)} \right)\left[ {\frac{4 + 2\xi }{9}\ln \frac{2 + \xi }{2} + \frac{2\xi - 5}{9}\ln \frac{1}{1 - \xi } + \frac{\xi }{3}} \right]$$

(79)

F ⁽⁰⁾ is the zero-order approximation.