  Displacement flows are common in hydraulic fracturing, as fracking fluids of
different composition are injected sequentially in the fracture. The injection
of an immiscible fluid at the center of a liquid-filled fracture results in the
growth of the fracture and the outward displacement of the interface between
the two liquids. We study the dynamics of the fluid-driven fracture which is
controlled by the competition between viscous, elastic, and toughness-related
stresses. We use a model experiment to characterize the dynamics of the
fracture for a range of mechanical properties of the fractured material and
fracturing fluids. We form the liquid-filled pre-fracture in an elastic brittle
matrix of gelatin. The displacing liquid is then injected. We record the radius
and aperture of the fracture, and the position of the interface between the two
liquids. In a typical experiment, the axisymmetric radial viscous flow is
accommodated by the elastic deformation and fracturing of the matrix. We model
the coupling between elastic deformation, viscous dissipation, and fracture
propagation and recover the two fracturing regimes identified for single fluid
injection. For the viscous-dominated and toughness-dominated regimes, we derive
scaling equations that describe the crack growth due to a displacement flow and
show the influence of the pre-existing fracture on the crack dynamics through a
finite initial volume and an average viscosity of the fluid in the fracture.
