Upscaling relative permeabilities in a structured porous medium

Mature sedimentary basins are characterized by large numbers of abandoned wells. In the Alberta Basin of western Canada, more than 100,000 abandoned wells exist, while about 500,000 may exist in the state of Texas in the United States. Mature sedimentary basins are attractive locations for deep injection of fluids for the purpose of storage and disposal. Injected fluids include hazardous liquid wastes, municipal liquid wastes, and, possibly, carbon dioxide. Abandoned wells represent a critical leakage pathway in these injection operations. While there are many aspects to the problem of potential leakage through abandoned wells, one of the interesting computational aspects is the representation of upscaled constitutive relationships in grid blocks that contain abandoned wells. One of the most important constitutive functions for proper estimation of leakage potential is the relative permeability function. In the simple case of a single borehole in a uniform rock matrix, with both the rock and the borehole material having identical (local) relative permeability functions, the upscaled relative permeability must be changed radically to capture the proper leakage. Standard petroleum reservoir pseudofunctions for relative permeability capture the general features of the upscaled function, but they still produce errors of several hundred percent in the leakage estimation. We present detailed three-dimensional numerical simulations and associated upscaled calculations to demonstrate the proper form for the upscaled relative permeability, and provide a new derivation of pseudofunctions to capture the leakage behavior in upscaled models.