Hydraulic diffusivity estimations for US shale gas reservoirs with Gaussian method: Implications for pore-scale diffusion processes in underground repositories

This paper first presents so-called unified Gaussian solutions for the spatial advance of diffusion transients triggered by a sudden change in pressure, molecular mass concentration and/or temperature. The mathematical description with a Gaussian solution for the pressure transient is similar to that for molecular diffusion and quantifies the diffusion of pressure into the reservoir space due to a change in molecular density initiated at the well intervention point. The resulting pressure gradients due to the pressure transient quantify, via Darcy's Law, the fluid-particle velocity resulting from that gradient everywhere in the reservoir. Also based on the Gaussian pressure transient, a Gaussian decline curve fitting formula is derived, uniquely scaled by the hydraulic diffusivity. The physics-based, Gaussian decline curve equation was utilized to match 30-year production data from 68 counties in four major US shale gas plays to compute their hydraulic diffusivities. The average hydraulic diffusivities of Marcellus, Haynesville-Bossier, Barnett and Utica shale are 7.43 × 10⁻⁹ m² s⁻¹, 7.9 × 10⁻⁹ m² s⁻¹, 12.3 × 10⁻⁹ m² s⁻¹, and 59.0 × 10⁻⁹ m² s⁻¹, respectively. The empirical history-matched estimates of the pressure-gradient-driven diffusion rates in shale are similar or faster than the shale diffusion-rates measured in the laboratory. It can be assumed that the empirical diffusion rate accounts for the integrated effects of Darcy and non-Darcy flow. Computation of the Gaussian Péclet number in gas plays confirms that the advective flux is much faster than the combined Fickean and non-Fickean mass transport rates. The implications for gas recovery from shale formations, and secure disposal of nuclear waste in the subsurface shale repositories (wellbores and cavities) are discussed. In particular, our field estimations being faster than the laboratory diffusion rates calls for caution because mass transport from leaking containers at disposal sites would diffuse several orders of magnitude faster than suggested by the slower laboratory rates.