Sensitivity analysis for the optimization of hydraulic properties using transient pressure and flow data

Analysis of the sensitivity of measurable physical variables (i.e., pressure and flow) to material properties is very useful to reduce uncertainty in estimation of the properties. The subsurface fluid flow through a porous rock is characterized by two hydraulic properties, hydraulic permeability k and specific storage S_s. The pore pressure diffusion is controlled by the hydraulic diffusivity implicating the ratio of the transmitting ability to the storage capacity. Both values of k and S_s can be determined only from a transient pore pressure curve. However, the hydraulic properties (k and S_s) are defined in the relation between pressure (gradient) and flow (rate). The permeability is a proportional parameter between volumetric flux of fluid and pressure gradient. The specific storage denotes the proportionality constant between the increment of fluid content and the pore pressure. Thus, the flow (rate) measurement is also beneficial to obtain the hydraulic properties. This paper presents the sensitivity analysis of pressure and flow data to the properties. One-dimensional pressure diffusion test was conducted on a cylindrical sample of shale subjected to a triaxial confining stress. No flow was allowed at the downstream for the boundary condition. A sudden increase in the upstream pressure followed by a constant pressure leads to the downstream pressure rising transiently until it reached an equilibration to the constant upstream pressure. During the transient period, the upstream fluid flows into the sample because of the time-dependent pressure gradient. Both the downstream pressure and the upstream flow were measured to estimate the hydraulic parameters using curve fittings. The downstream pressure curve fitting yields 4.1×10^-20 m² for the permeability and 1.2×10^-11 Pa^-1 for the specific storage. However, these values for the best fit of the pressure data yield a completely different flow curve from the flow measurements. The permeability and the specific storage obtained from the best fit of the flow curve are 2.6×10^-19 m² and 2.1×10^-10 Pa^-1, respectively. The theoretical pressure curve using these values fits well (not the best) the measured data. Conclusively, the flow data is more sensitive to the hydraulic properties than the pressure data. The flow analysis yields less uncertainty in the optimization of hydraulic properties than the pressure analysis.