Upper- and lower-bound solutions for sectional penetration grouting pressure under time-dependent viscosity of slurry

Under the time-dependent viscosity of slurry, the grouting pressure (p) and the grouting rate (q) are both time-dependent, which makes design and calculation more difficult in grouting project. Firstly, the engineering definitions of grouting pressure (P) and grouting rate (Q) for a sectional grouting under the double-time-variable of p-q are introduced by using the defined integral. Secondly, according to the theory of time-dependent viscosity and the Darcy's law, an analytical model for the penetration grouting under time-dependent viscosity of slurry considering the double-time-variable of p-q is established (including physical equation, geometrical equation and boundary condition), with which the complex process of penetration grouting under time-dependent viscosity of slurry is discovered, and its solutions are discussed systematically. Thirdly, the theoretical solutions for the sectional grouting pressure P are deduced based on its engineering definition and the above analytical model, and with the property of integral inequality and the systematic theoretical derivation, the general limit solutions (including upper- and lower-bound solutions) of P are obtained, and their scientificity and universality are discussed. Finally, considering the actual engineering needs, the special upper- and lower-bound solutions of P for the two modes of spherical and cylindrical diffusions under exponential time-dependent viscosity function are further discussed.