Abstract
We study the dynamics of proppants carried by fluid driven into an evolving penny-shaped fracture. During injection the slurry is modelled using a frictional rheology that takes into account the shear-induced migration and jamming of the proppants. Making pragmatic assumptions of negligible toughness and cross-fracture fluid slip, we find self-similar solutions supporting a range of proppant concentration profiles. In particular, we define an effective viscosity, which equates the fracture evolution of a slurry flow with a given proppant volume fraction, to a Newtonian flow with a particular viscosity. Using this framework, we are able to make predictions about the geometry of the growing fracture and the significance of tip screen-out. We are also able to model the closure of the fracture, after fluid pressure is released, and explore the effect of proppant concentration on the residual geometry of the fracture. The results have important applications to industrial fracking and geological dike formation by hot, intruding magma.
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