Solution for a plane strain rough‐walled hydraulic fracture driven by turbulent fluid through impermeable rock

Abstract

Summary The impact of turbulent flow on plane strain fluid‐driven crack propagation is an important but still poorly understood consideration in hydraulic fracture modeling. The changes that hydraulic fracturing has experienced over the past decade, especially in the area of fracturing fluids, have played a major role in the transition of the typical fluid regime from laminar to turbulent flow. Motivated by the increasing preponderance of high‐rate, water‐driven hydraulic fractures with high Reynolds number, we present a semianalytical solution for the propagation of a plane strain hydraulic fracture driven by a turbulent fluid in an impermeable formation. The formulation uses a power law relationship between the Darcy‐Weisbach friction factor and the scale of the fracture roughness, where one specific manifestation of this generalized friction factor is the classical Gauckler‐Manning‐Strickler approximation for turbulent flow in a rough‐walled channel. Conservation of mass, elasticity, and crack propagation are also solved simultaneously. We obtain a semianalytical solution using an orthogonal polynomial series. An approximate closed‐form solution is enabled by a choice of orthogonal polynomials embedding the near‐tip asymptotic behavior and thus giving very rapid convergence; a precise solution is obtained with 2 terms of the series. By comparison with numerical simulations, we show that the transition region between the laminar and turbulent regimes can be relatively small so that full solutions can often be well approximated by either a fully laminar or fully turbulent solution.

Inertial and turbulent flow in hydro‐mechanically coupled KGD‐like fractures

“…However, the results confirm the findings of past studies and find that turbulence is important and will create substantially different solutions as soon as it is invoked. 28,32,33 Inertia remains negligible even in the turbulent case. Care must be taken in considering the results of Poiseuille flow-based models when considering large flow rates, low viscosity fluids or small formations.…”

“…Preliminary investigations into axisymmetric fractures suggest that inertia has a significant role 51,52 and the null result returned here will not be applicable. Further verification of the model could be achieved through comparisons with analytical solutions for laminar and turbulent propagation in fractures 33 . The model is generally not expected to exactly recover the analytical solution due to the different governing equations, including the cohesive law and the turbulent friction factor.…”

“…Turbulence has seen more widespread study than inertia in hydraulic stimulation. Turbulence has been shown to have significant impacts on fracture propagation and geometry under a variety of different conditions 28–34 . Due to the increased hydraulic resistance, turbulence creates shorter and wider fractures.…”

Slickwater hydraulic fracture propagation: near-tip and radial geometry solutions

“…The influence of turbulent flow on hydraulic fracture propagation has been investigated in the context of magmatic dykes (Emerman, Turcotte & Spence 1986; Lister & Kerr 1991), glacier drainage (Tsai & Rice 2010, 2012) as well as industrial hydraulic fracturing (Perkins & Kern 1961). Fracture growth solutions under turbulent conditions have been obtained recently for simple geometries (plane strain, radial, blade-like fractures: see Zia & Lecampion (2017), Zolfaghari, Meyer & Bunger (2017), Zolfaghari & Bunger (2018 a , b ), Zolfaghari, Dontsov & Bunger (2018)). The influence of turbulence was also studied early on with respect to gas-driven fracture propagation (Nilson 1981) where it was noted that the flow will transition from turbulent to laminar condition inside the fracture as one moves closer to the fracture tip.…”

Fully coupled hydraulic fracture simulation using the improved element partition method

“…The HF simulation has many methods, eg, numerical manifold and coupled finite element/finite volume . Recently, Zolfaghari et al developed a semianalytical method to investigate water‐driven hydraulic fractures with high Reynolds number.…”