Theoretical Analysis of One-Dimensional Pressure Diffusion from a Constant Upstream Pressure to a Constant Downstream Storage

Abstract

The one-dimensional diffusion equation was solved to understand the pressure and flow behaviors along a cylindrical rock specimen for experimental boundary conditions of constant upstream pressure and constant downstream storage. The solution consists of a time-constant asymptotic part and a transient part that is a negative exponential function of time. This means that the transient flow exponentially decays with time and is eventually followed by a steady-state condition. For a given rock sample, the transient stage is shortest when the downstream storage is minimized. For this boundary condition, a simple equation was derived from the analytic solution to determine the hydraulic permeability from the initial flow rate during the transient stage. The specific storage of a rock sample can be obtained simply from the total flow into the sample during the entire transient stage if there is no downstream storage. In theory, both of these hydraulic properties could be obtained simultaneously from transient-flow stage measurements without a complicated curve fitting or inversion process. Sensitivity analysis showed that the derived permeability is more reliable for lower-permeability rock samples. In conclusion, the constant head method with no downstream storage might be more applicable to extremely low-permeability rocks if the upstream flow rate is measured precisely upstream.