Uncertainty of relative permeability to describe Two-Phase flow in geological disposal system

Relative permeability k_r is a practical tool to describe two-phase flow in the performance assessment of a geological disposal system of radioactive waste. So far to avoid thermal alteration of an engineered barrier system such as bentoniie, the maximum temperature in the conceptual design of a Japanese geological disposal system has been limited to <373 K. However, for a limited time period, even if the temperature exceeds 373 K or the boiling point at the underground level, the robustness of the system is expected to be sufficient. An upward revision of the permissible maximum temperature would reduce the total space of the repository and would result in-more effective use of the space. Therefore, when two-phase flow is also considered, a more reliable estimate of the thermal impact on the repository system is needed. In general, the fluid flow velocities of two phases are described by Darcy's law including the relative permeabilities defined as the functions of liquid-water saturation (or steam saturation), e.g., Corey's equations. However, such saturation (e.g., liquid-water saturation S_w) is not always uniformly distributed in the grid cells of the numerical implementation. In this study, the uncertainty of k_r due to the distribution of S_w was examined by using various kinds of probability density functions (pdf's). The results suggest that the apparent k_r value can be numerically described by the arithmetic mean, the standard deviation, and the skewness of S_w. (In other words, the apparent value of k_r does not depend on the types of pdf's.) Since the value of S_w is in the range of 0 to 1, the standard deviation and the skewness are limited. Therefore, the apparent values of k_r also are in a limited range. Using the Lagrange multiplier method, this study examined the ranges of the k_r value for each arithmetic mean of saturation S_wa. Furthermore, by considering both the frequency distribution and the spatial distribution of saturation, this study quantitatively shows the degree of uncertainty of relative-permeability curves. These curves can explain the scattered data of two-phase-flow experiments.