Relaxation-induced flow of an Ellis fluid in a smooth fracture
Authors: Ciriello Valentina , Lenci Alessandro , Longo Sandro , Di Federico Vittorio 
Published: March 2021

&lt;p&gt;The process known as hydrofracturing, aimed at improving reservoir productivity, is complex and includes several steps. During the first, high-pressure injection, a network of fractures and cracks is created in the stimulated zone; then proppant is introduced into the network to open the supporting fractures; when the injection stops, the pressure drops and elastic relaxation of the fluid-driven fractures pushes the fracturing fluid back into the injection well. Once recovered, these fluids are typically processed for reuse due to their versatility and economic value; in addition, unrecovered fluid tends to compromise fracture conductivity or migrate into the subsurface environment. Optimizing the recovery rate is critical regardless of the reservoir product (oil, gas, heat). Because the goal is to create fractures that remain open, inevitably some of the fluid is not drained.&lt;/p&gt;&lt;p&gt;The rheology of fracturing fluids is typically described by a non-Newtonian rheology, showing a nonlinear relationship between stress and strain; this allows for flexibility and several design goals to be achieved at the same time.&lt;/p&gt;&lt;p&gt;We adopt a conceptual model to represent the fracture medium, consisting of a single planar fracture with relaxing walls, exerting a force on the fluid proportional to &lt;em&gt;h&lt;sub&gt;&amp;#955;&lt;/sub&gt;&lt;/em&gt;, with &lt;em&gt;h&lt;/em&gt; the time-varying aperture and &lt;em&gt;&amp;#955;&lt;/em&gt; a non-negative exponent; an overload of &lt;em&gt;f&lt;/em&gt;&lt;sub&gt;0&lt;/sub&gt; on the fracture can help slow or accelerate the closure process. The fracture is in a vertical plane perpendicular to a horizontal hole or in a horizontal plane perpendicular to a vertical hole. At time &lt;em&gt;t&lt;/em&gt; = 0, pressure &lt;em&gt;p&lt;/em&gt;&lt;sub&gt;e&lt;/sub&gt; at the outlet begins to act, the elastic response of the wall compresses the fluid and forces a backflow to the outlet as a result of the no-flow boundary condition at &lt;em&gt;x&lt;/em&gt;=&lt;em&gt;L&lt;/em&gt;. Gravity effects are absent in horizontal fractures and negligible with respect to pressure gradients for fractures in any other plane.&amp;#160;&amp;#160;&lt;/p&gt;&lt;p&gt;Fluid rheology is described by the three-parameter Ellis model, which well represents the typical shear-thinning rheology of hydro-fracturing fluids and the Newtonian and power-law coupling behavior at low and high shear rates, respectively.&lt;/p&gt;&lt;p&gt;Under viscous flow and lubrication approximation, the time-varying aperture and discharge rate, the space- and time-varying pressure field, and the time to drain a given fraction of the fracture volume are derived as a function of geometry (length and initial aperture), elastic wall parameters, fluid properties, outlet pressure and overload. The parameters of the problem are combined in a dimensionless number &lt;em&gt;N&lt;/em&gt; that tunes the interplay between Newtonian and power-law rheology. The late-time behavior of the system is practically independent of the rheology, since the Newtonian nature of the fluid prevails at low shear stress. In particular, the aperture and discharge scale are asymptotic with time as &lt;em&gt;t &lt;/em&gt;&amp;#8733; 1/(2+&amp;#955;) and &lt;em&gt;t &lt;/em&gt;&amp;#8733; 1/(3+&amp;#955;) for &lt;em&gt;p&lt;/em&gt;&lt;sub&gt;e&lt;/sub&gt;-&lt;em&gt;f&lt;/em&gt;&lt;sub&gt;0&lt;/sub&gt;=0; otherwise, the aperture tends to a constant, residual value proportional to (&lt;em&gt;p&lt;/em&gt;&lt;sub&gt;e&lt;/sub&gt;-&lt;em&gt;f&lt;/em&gt;&lt;sub&gt;0&lt;/sub&gt;)&lt;sup&gt;&amp;#955;&lt;/sup&gt;. A case study with equally spaced fractures adopting realistic geometric, mechanical and rheological parameters is examined: two fluids normally used in fracking technology exhibit completely different behaviors, with backflow dynamics and drainage times initially not dissimilar, and subsequently varying by orders of magnitude.&lt;/p&gt;