Solution to Boundary Value Problem Relevant in Upscaling

Three boundary value problems for Laplace's equation related to 2D Darcy flow through a gap in barrier are considered. Solutions to two of the problems are obtained, and an exact formula for the effective permeability represented by a gap in an infinitely thin barrier is found. Deriving the effective penneability represented by a gap in a barrier with positive thickness is more difficult, and the problem is here reduced to the task of determining the period matrix of a hyperelliptic Riemann surface of genus three from the knowledge of the periods of a certain analytic differential on the surface.