Although a simplification of the real world, analytical solutions play an important role in modelling of flow of grouts in jointed rock. Two analytical solutions for 2D radial flow of a Bingham fluid between smooth parallel plates are currently being used, both with simplifying assumptions (El Tani, 2012, Gustafson et al, 2013; Zou et al, 2020).

The main objective of the present study has been to assess which of the two solutions is correct and thereby recommended for use in the grouting community. The solution used in this study assumes a constant plug flow region, similar to 1D flow in channels and pipes, in order to fulfil the requirement of mass balance (Eq. of continuity).

The other available solution (Dai & Bird, 1981; Hässler, 1991, Gustafson et al, 2013; Hoang et al, 2021) implies a radius dependent plug flow region and in order to fulfil mass balance introduces a vertical velocity component (e.g. Hoang et al, 2021).

One cannot conclude from the current work that one solution is right and the other wrong. They are both simplifications and it is interesting to note that for practical purposes, e.g. for grouting simulations, it does not matter which one you use as they both generate similar results. However, it should be stressed that the solution presented in this study has simpler mathematical expressions that makes it easier to use by practitioners involved in grouting.

This study also introduces numerical simulations, as a complement to the analytical solutions, in order to determine the effect of rough fractures, water filled fractures, channel networks and rheological properties of the used grout.

The research presented in this report was mainly carried out at the Department of Sustainable Development, Environmental Sciences and Engineering (SEED), at KTH Royal Institute of Technology. The work was supervised by Prof. Vladimir Cvetkovic, KTH, and Dr. Ulf Håkansson, Skanska Sweden AB. Their support, encouragement, and valuable insights of the work are greatly appreciated.

The input from the project reference group is also gratefully acknowledged. The following have participated:

Thomas Dalmalm and Christian Butron (Swedish Transport Administration), Ann Emmelin och Mikael Creütz (Golder), Magnus Zetterlund (Norconsult), Patrik Vidstrand (SKB), Tommy Ellison (Besab), Johan Wiklund (Incipientus) and Per Tengborg (BeFo). 

Stockholm,  2021

Per Tengborg