Analytical solutions for leakage rates through abandoned wells

Disposal of waste fluids via injection into deep saline aquifers is practiced in a variety of industries. Injection takes place in sedimentary basins that often have a history of oil and gas exploration and production, which means that wells other than those used for waste disposal may exist in the vicinity of the injection site. These existing wells provide possible pathways for leakage of waste fluids toward the shallow subsurface and the land surface. For single-phase flows of liquids with essentially constant properties, the equations governing the system are linear, and solutions may be written using the superposition principle. Because leakage through existing wells produces a time-varying flux rate, the solution of the governing equations involves convolution integrals. Previous solutions have addressed the problem of one injection well, one existing (passive) well, and a simple geometry of two aquifers separated by an aquitard by use of Laplace transforms. Even for this simple case, inversion of the transform is difficult. Solutions involving more than one passive well have not been developed. Nor have solutions been developed for more than two aquifers and one aquitard. Realistic injection cases often involve layered systems with multiple aquifers and aquitards, as well as multiple passive wells, sometimes numbering in the hundreds. Solutions for the general case of multiple aquifers and wells may be developed through introduction of approximations to the well function and appropriate simplification of the convolution integral. Such a solution is computationally simple. Comparison to solutions using the full (Laplace transform) solution indicates that the new solution procedure produces excellent results. Application of the new solution to a case of multiple passive wells shows that the cumulative leakage flux in the passive wells is not a simple sum of the single-well case, owing to leakage-induced drawdown around the passive wells. In addition, application to the case of multiple aquifers and aquitards demonstrates the importance of leakage into intervening aquifers as a mechanism to mitigate leakage into shallow zones, a process referred to as the "elevator model." The new analytical solution provides a tool to analyze practical injection problems and forms a foundation on which more complex solutions, such as those involving injection of a nonaqueous fluid into a deep brine formation, may be based.