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.

本文首先提出了所谓的统一高斯解决方案，用于由压力、分子质量浓度和/或温度的突然变化引发的扩散瞬态的空间推进。压力瞬态的高斯解的数学描述类似于分子扩散的数学描述，并且量化了由于在井干预点开始的分子密度变化而导致的压力扩散到储层空间。由于压力瞬变而产生的压力梯度通过达西定律量化了由储层中各处的该梯度产生的流体-粒子速度。同样基于高斯压力瞬态，推导出高斯下降曲线拟合公式，由水力唯一缩放扩散性。基于物理的高斯递减曲线方程被用来匹配来自美国四个主要页岩气区的 68 个县的 30 年生产数据，以计算它们的水力扩散率。Marcellus、Haynesville-Bossier、Barnett 和 Utica 页岩的平均水力扩散率为 7.43 × 10 ^-9  m ²  s ^-1、7.9 × 10 ^-9  m ²  s ^-1、12.3 × 10 ^-9  m ²  s ^-1和59.0 × 10 ^-9 米² 秒^-1， 分别。页岩中压力梯度驱动的扩散速率的经验历史匹配估计与实验室测量的页岩扩散速率相似或更快。可以假设经验扩散率解释了达西流和非达西流的综合效应。天然气层中高斯 Péclet 数的计算证实平流通量比 Fickean 和非 Fickean 质量传输速率的组合快得多。对页岩地层天然气回收和安全处置的影响讨论了地下页岩储存库（井筒和空腔）中的核废料。特别是，我们的现场估计比实验室扩散速率更快需要谨慎，因为从处置场所泄漏容器的质量传输将比较慢的实验室速率所建议的扩散快几个数量级。