This
paper describes the combination of a Perkins–Kern–Nordgren
(PKN) type model for hydraulic fracture propagation with a method
for calculation of proppant transport and settling in order to simulate
the dynamic growth, stress induced closure, and final geometry of
vertically oriented two-dimensional fractures. A mathematical model
is developed to describe the fracture growth, fluid flow, and proppant
movement along with proppant settling and bank formation. A particle
tracking method which uses the concept of pseudoparticles to represent
the proppant phase is used for the computation of solids distribution
within the fracture and the proppant bank growth. A technique for
periodically combining the elements of the computational grid allows
for reduced simulation time. Using the computational model, contraction
of fracture dimensions after the end of pumping can be simulated in
order to determine the final shape of the propped fracture. A sensitivity
analysis was conducted to study the effects of pumping rate, inlet
proppant concentration, and proppant particle size on the final fracture
condition. To evaluate the efficacy of different treatment designs,
the resulting geometry of the propped fracture dimensions and the
achieved conductivities were compared. Based on the simulation results
obtained, specific recommendations on how to avoid premature tip screen-out
and achieve desired fracture conductivity are presented.
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