  The quasi-brittle nature of rocks challenges the basic assumptions of linear
hydraulic fracture mechanics (LHFM): linear elastic fracture mechanics and
smooth parallel plates lubrication fluid flow. We relax these hypotheses and
investigate the growth of a plane-strain hydraulic fracture in an impermeable
medium accounting for a rough cohesive zone and a fluid lag. In addition to a
dimensionless toughness and the time-scale of coalescence of the fluid and
fracture fronts as in the LHFM case, the solution now also depends on the
in-situ-to-cohesive stress ratio and the intensity of the flow deviation
induced by aperture roughness. The solution is appropriately described by a
nucleation time-scale, which delineates the fracture growth into a nucleation
phase, an intermediate stage and a late time stage where convergence toward
LHFM predictions finally occurs. A highly non-linear hydro-mechanical coupling
takes place as the fluid front enters the rough cohesive zone which itself
evolves during the nucleation and intermediate stages. This coupling leads to
significant additional viscous flow dissipation. As a result, the fracture
evolution deviates from LHFM solutions with shorter fracture lengths, larger
widths and net pressures. These deviations ultimately decrease at late times as
the lag and cohesive zone fractions both become smaller. The deviations
increase with larger dimensionless toughness and in-situ-to-cohesive stress
ratio, as both further localize viscous dissipation near the fluid front
located in the rough cohesive zone. The convergence toward LHFM can occur at
very late time for realistic values of in-situ-to-cohesive stress ratio
encountered at depth. The impact of a rough cohesive zone appears to be
prominent for laboratory experiments and short in-situ injections in
quasi-brittle rocks with ultimately a larger energy demand compared to LHFM
predictions.
