- Concrete leveling
Concrete leveling is a procedure that attempts to correct an uneven concrete surface by altering the foundation that the surface sits upon. It is a cheaper alternative to having replacement concrete poured, and commonly performed at small businesses and private homes. In 1977, the term concrete leveling was coined by Randall Greene in Cleveland, Ohio.  He created the phrase to convey his company's ability to both raise and lower concrete to correct the insufficent grade of a slab. 
Sidewalks, patios and garage floors are most often made of concrete. When the concrete is a few inches thick and poured on insufficient foundation and/or without steel reinforcement it may crack and subside. Sometimes water runoff from rain or flood can wash away dirt upon which the concrete slab is resting, leading to cracking and subsiding. In addition, tree roots and other subsurface obstructions can make concrete slabs rise, so that they no longer align to the adjacent slabs.
Slabjacking can both raise the old cracked slab back to its original position and create a new foundation of cement mortar or sand mix by injecting the mortar under the slab through a hole, under pressure. The viscosity of the mortar will keep it from flowing back through the hole until it sets.
Accounts of raising large concrete slabs through the use of hydraulic pressure go back almost a century. Mudjacking or slabjacking has been in common use for about 50 years. Generally a portable pump is carried to the location of the block to be raised. A hole of up to 3 inches in diameter is drilled into the block. Varying combinations of soil, sand, cement, or other materials, are mixed and then injected under the sunken concrete block, causing it to rise.
Problems associated with slabjacking involve: containment of the mess caused by excess mud or cementeous material in the area to be raised; drilling of large holes that can weaken the block, and allow material to flow too quickly causing cracking of the slabs. HMI is not conducive to filling large void areas.
Modern methods use smaller holes to avoid weakening the concrete slab, or raising the blocks too quickly. A highly dense crushed limestone is sometimes mixed with moderate amounts of cement, and can be pumped slowly and safely through hoses that are connected directly to the pumping truck, with little or no destruction of landscaping or surrounding structures.
This site shows before and after pictures of slabjacking. 
Closed cell polymer foam
Foam leveling utilizes two part closed cell polymer expanding foam injected through a hole less than one inch in diameter, typically 5/8". Although the material injected at a higher psi rate than traditional cementious grouts the pressure is not what causes the lifting. The expansion of the injected material below the slab surface performs the actual lifting action. Material injected below a slab to be lifted will first find weak soils, expanding into them in such a manner as to consolidate and cause sub-soils to become more dense and fill any voids below the slab. One inherent property of expanding foams is that they will follow the path of least resistance, expanding in all directions. Another inherent property includes reaching a hydro-insensitive or hydrophobic state when cured with 100% cure times as little as 30 minutes. Closed cell polymer foams offer benefits which go beyond the goal of leveling hard surfaces. They will not retain moisture, which in northern climates can cause frost heaving. They are not subject to erosion once in place. Their fast cure time allows for immediate use when application is complete. Their light weight, 3 to 8 lbs. per cubic ft. vs. 100 to 120 lbs. per cubic ft. for cementious grout will not cause further settlement. Foams will retain their cured shape and volume indefinitely reducing the possibility of new voids forming below grade to nearly zero unless acted upon by some outside cause. Some closed cell polymer foams have baseline lifting capabilities of 6,000 lbs per sq. ft. and leveling procedures have been performed in which loads as high as 125 tons have been lifted and stabilized in a surface area of less than 900 sq. ft.
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