Hydraulic fracturing is a process aimed at improving the productivity of oil, gas or geothermal reservoirs. During hydrofracturing, backflow follows injection and represents the second phase of the process, when part of the fracturing fluid returns from fractures to well, and from well to surface. A conceptual model is presented to grasp the essential features of the phenomenon, conceiving the draining subsurface domain as a planar and rigid fracture. Backflow against an outlet pressure in the injection well is induced by the relaxation of the fracture wall, exerting a force on the fluid proportional to $h^{\lambda },$ with $h$ the time-variable aperture and $\lambda$ a non-negative exponent; an overload on the fracture may contribute to slowing or accelerating the closure process. The fluid rheology is described by the three-parameter Ellis constitutive equation, well representing the shear-thinning rheology typical of hydrofracturing fluids and coupling Newtonian and power-law behaviour. The interplay between these tendencies is modulated by a dimensionless number $N$ encapsulating most problem parameters; the range of variation of $N$ is discussed and found to vary around unity. The time-variable aperture and discharge rate, the space-time variable pressure field, and the time to drain a specified fraction of the fracture volume are derived as functions of geometry (length and initial aperture), wall elastic parameters, fluid properties, outlet pressure $p_e$ and overload $f_0$. The late-time behaviour of the system is practically independent from rheology as the Newtonian nature of the fluid prevails at low shear stress. In particular, aperture and discharge scale asymptotically with time as $t^{-1/(\lambda +2)}$ and $t^{-1/(\lambda +3)}$ for $p_e-f_0=0$; else, the aperture tends to a constant, residual value proportional to ${(p_e-f_0)}^{\lambda }$. A case study with equally spaced fractures adopting realistic geometric, mechanical and rheological parameters is examined: two fluids normally used in fracking technology show completely different behaviours, with backflow dynamics and drainage times initially not dissimilar, later varying by orders of magnitude.

水力压裂是旨在提高石油，天然气或地热油藏生产率的过程。在加氢压裂过程中，注水后会发生回流，这代表了过程的第二阶段，此时部分压裂液从裂缝返回井眼，再从井眼返回地表。提出了一个概念模型来掌握该现象的基本特征，将排水的地下区域构想为平面的和刚性的裂缝。裂缝壁的松弛引起注入井中逆于出口压力的回流，从而在流体上施加与压力成正比的力。$H^{\lambda }，$ 和 $H$ 随时间变化的光圈和 $\lambda$非负指数；骨折的过载可能会导致减缓或加速闭合过程。流体流变学由三参数Ellis本构方程描述，很好地代表了水力压裂流体典型的剪切稀化流变学，并结合了牛顿和幂律特性。这些趋势之间的相互作用由无量纲数调制$ñ$封装大多数问题参数；变化范围$ñ$进行了讨论，发现在统一性方面有所不同。随时间变化的孔径和排出速率，时空可变压力场以及排空指定体积的裂缝的时间是根据几何形状（长度和初始孔径），壁弹性参数，流体性质，出口确定的函数压力$p_{Ë}$ 和超载 $F_0$。该系统的后期行为实际上与流变学无关，因为流体的牛顿性质在低剪切应力下占主导地位。特别是，光圈和放电随时间渐近缩放${Ť}^{-\mathrm{1个}/（\lambda +\mathrm{2个}）}$ 和 ${Ť}^{-\mathrm{1个}/（\lambda +3）}$ 为了 $p_{Ë}-F_0=0$; 否则，光圈趋向于一个恒定的残差值，该残差值与${（p_{Ë}-F_0）}^{\lambda }$。研究了一个采用现实的几何，机械和流变参数的等间距裂缝案例研究：压裂技术中通常使用的两种流体表现出完全不同的行为，回流动态和排水时间最初没有不同，后来又变化了几个数量级。