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 cementbased 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 wellknown 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.
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Abbreviations
 A _{ i }, A _{0}, A _{1} :

A(0), A(t _{0}), A(t _{1}) of any time function A(t)
 a :

Radius of the injection hole
 b :

Radius of the mix at the front
 c :

Yield stress or cohesion
 f :

Ratio of the viscosity to its initial value
 f, f′:

Local function and derivative defined in Appendix
 F :

Driving relation
 F :

Function
 F ^{(0)}, F ^{(1)} :

Zero order, firstorder approximation of F
 F _{ a }, F _{ b }, F _{ c } :

Local function defined in Appendix
 G :

Driving energy
 H :

Halfthickness of the planar fracture
 L, L _{0} :

Spread of the mix and initial spread
 Li_{2} :

Dilogarithm function
 N :

Number of fractures
 n :

Integer index used in Appendix
 p :

Pressure
 P :

Injection pressure
 Q :

Flow rate
 r :

Radial coordinate
 R :

Reach
 S :

Span
 Sp:

Spence’s function
 T :

Maturity time
 t :

Time
 t _{0} :

Initial time
 t _{G} :

Gel time
 t _{P} :

Pot life
 u :

Dummy variable
 V :

Volume of the mix
 w :

Width
 W :

Work
 x :

Dummy variable in Appendix
 z :

Local variable in Appendix
 α :

Fitting parameter
 β :

Ratio of the pseudocore thickness to the fracture thickness
 η :

Dynamic viscosity
 η _{i} :

Initial dynamic viscosity
 π :

Pi
 ξ :

Ratio of the spread to the span
 ξ _{0} :

Spread ratio at t _{0}
 ξ _{ a } :

Ratio of the injection hole radius to the spread
 ξ _{ r } :

Ratio of the spread to the reach
 ξ _{ r0} :

Ratio of the spread to the reach at t _{0}
 ξ _{ ra } :

Ratio of the radius of the injection hole to the reach
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Appendices
Appendix 1: Time–Spread Closed Form
The integration of the righthand side of Eq. (50) leads to
with,
in which Sp is Spence’s function. Spence’s function is
Spence’s function has different appellations since it has many discoverers or rediscoverers (Zagier 2006). Spence’s function can be calculated using the dilogarithm function that is defined in the interval (−1, 1) as
Their relations are
Appendix 2: Time–Spread FirstOrder Approximation
The firstorder approximations of Eq. (50) in the interval (0, ξ) is
in which
The integration is straightforward and leads to F ^{(1)} = F ^{(0)} + ∆F with
F ^{(0)} is the zeroorder approximation.
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El Tani, M., Stille, H. Grout Spread and Injection Period of Silica Solution and Cement Mix in Rock Fractures. Rock Mech Rock Eng 50, 2365–2380 (2017). https://doi.org/10.1007/s0060301712378
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DOI: https://doi.org/10.1007/s0060301712378