The narrow fracture approximation by channeled flow

The singular problem of non-stationary Darcy flow in a region containing a narrow channel of width O ( ϵ ) and high permeability O ( 1 ϵ ) is shown to be well approximated by a problem with flow concentrated on a weighted Sobolev space over a lower-dimensional interface. The convergence of the solution as ϵ → 0 is proved for both the stationary case and the corresponding initial-boundary-value problem. The structure of the limiting problems is dependent on the rate of taper of the channel at its extremities.