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\begin{document}

\title{Limiting Regimes of a Three-Dimensional Buoyant Hydraulic Fracture
Emerging from a Point Source}



\author[1]{Andreas Möri}%
\author[1]{Brice Lecampion}%
\affil[1]{Swiss Federal Institute of Technology Lausanne}%


\vspace{-1em}



  
  \date{May 17, 2024}


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\selectlanguage{english}
\begin{abstract}
Buoyant hydraulic fractures occur in nature as magmatic dikes and sills.
In industrial applications like well stimulation, the emergence of
buoyant fractures is undesirable and often limited by the injected
volume and/or variation of in-situ stress. This class of tensile
fractures is governed by a buoyancy force resulting from the density
contrast between the surrounding solid and the fracturing fluid. We
focus here on fluid releases from a point source in an impermeable
elastic media with homogeneous rock and fluid properties. The resulting
buoyant force is thus constant. We combine scaling arguments and planar
3D hydraulic fracture growth simulations {[}1{]} to fully understand the
emergence as well as the different propagation regimes of buoyant
fractures. For a continuous release, a family of solutions dependent on
a dimensionless-viscosity Mk^ \selectlanguage{english}exists. In the limit of large toughness
(Mk^[?] \selectlanguage{english}1), we retrieve a finger-like shape {[}2{]}. The stable breadth of
the tail is generally akin to the PKN approximation presented in
{[}2{]}. The limit of a viscosity-dominated buoyant fracture (Mk^[?] \selectlanguage{english}1)
has no stabilized breadth and exhibits a teardrop shape. For the case of
a finite fluid volume release, a dimensionless buoyancy Bk\selectlanguage{ngerman}¯ controls if
a buoyant fracture emerges (Bk¯\selectlanguage{english}[?] 1) or stops and remains at depth
(Bk-\textless{}1). For a finite release, a single large-time solution
corresponding to the solution of {[}2{]} exists. Detailed
characterization of the fracture evolution requires separation between
the cases where the buoyant transition occurs during or after the
release (see attached Fig. 1). For natural configurations, the emerging
buoyant fractures are typically viscosity-dominated, which may explain
the reported discrepancy between field and laboratory measurements of
rock fracture toughness. Representative values of industrial
single-entry hydraulic fracturing treatments lead to buoyant fractures
under homogeneous conditions, which indicate the critical importance of
stress and material heterogeneities in the containment of buoyant
fractures at depth. {[}1{]} Haseeb Zia and Brice Lecampion. Pyfrac: A
planar 3d hydraulic fracture simulator. Computer Physics Communications,
page 107368, 2020. {[}2{]} L.N. Germanovich, D. I. Garagash, Murdoch,
L., and Robinowitz M. Gravity-driven hydraulic fractures. In AGU Fall
meeting, 2014.%
\end{abstract}\selectlanguage{ngerman}%



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