Pressure pulse-decay tests in a dual-continuum medium: Late-time behavior

The pressure pulse-decay test has been one of the most widely used techniques to measure permeability for low-permeability rocks including shale, largely because relatively simple analytical solutions to fluid flow in a rock sample (characterized as a single continuum) exist for analyzing late-time test data. This work develops a new analytical solution for analyzing late-time pulse-decay test data for dual-continuum rock samples, motivated by the possibility that shale matrix (not including fractures) may exhibit dual- or multiple-continuum gas-flow behavior owing to its wide spread pore size distributions and property differences between organic and inorganic components of the shale matrix. This issue has important implications for characterizing and modeling gas flow in shale, because a dual-continuum medium involves more flow parameters and requires more complex modeling approaches than a single-continuum medium. Our new analytical solution shows that the relationship between gas pressure measurements from a pulse-decay test and permeability remains the same for both single- and dual-continuum systems, suggesting that shale-matrix permeability data in the literature, previously obtained from pulse-decay tests, are valid although the corresponding shale samples ought to be treated as dual-continuum instead of a single continuum system. In addition, our analytical solution relates mass transfer (between the two continua) to pressure change during late-time of a test. This relationship, with a clear and intuitive physical interpretation rooted in the mass balance principle, provides a simple and practical analysis technique to identify dual-continuum behavior from pulse-decay test data and to estimate mass transfer coefficient between the two continua if the dual-continuum behavior exists. The usefulness of the analytical solution is demonstrated by satisfactory matches with several pulse-decay test data sets. The data analysis results also indicate that not all shale samples exhibit dual-continuum gas-flow behavior regardless of their make-up and complex pore structures.