Approximate and precise mathematical equations to determine the grouting maximum pressure and grouting intensity number of cement grout in joint rocks to prevent hydraulic jacking

Cement grout is grouted in rock joints of the foundations of dams to prevent water leakage and strengthen the foundation. One of the techniques used to control the grouting operation is the grouting intensity number (GIN), which is equal to the sum of the grouting pressure and volume and represents the grouting energy. Pressure in the rock joints creates a hydraulic lifting force that should not exceed its allowable limit so as not to lead hydraulic jacking and failure. The maximum pressure and grouting intensity number are determined accordingly. Simplifying the plan of the grouted joint, which is modeled as a thin cylinder with a radius of the grouting extension radius, the present study determines the relationship between hydraulic lifting force with grouting pressure and grouting intensity number. In this regard, the joint aperture coefficient was defined and determined based on the rock permeability (the amount of joint aperture). Then, following the geometry of rock-soil mass at the top of the joint, which is considered as a truncated cone with a β angle, for the first time approximate and exact mathematical equations were obtained to determine the permissible hydraulic lifting force (based on the approximate and exact formulas of truncated cone volume). Besides, its approximate and exact mathematical equations were compared. Then, following the principle that the hydraulic lifting force should not exceed the allowable limit, the maximum pressure and grouting intensity number were determined. Finally, by defining two parameters the normal pressure and the normal spreading length, the maximum normal pressure was set so that the hydraulic lifting force to be within the allowable range and prevent hydraulic jacking and failure