Homogenization techniques are used to derive a double porosity model for single phase flow in a reservoir with a preferred direction of fracture. The equations in the microscopic model are the usual ones derived from Darcy's law in the fractures and matrix (rock). The permeability coefficients over the matrix domain are scaled, using a parameter
,
based on the
fracture direction in the reservoir. The parameter
represents the size of the parts of the
matrix blocks that are being homogenized and the scaling
preserves the physics of the flow between matrix and fracture
as the blocks shrink. Convergence to the macroscopic model
is shown by extracting the weak limits of the microscopic
model solutions. The limit (macroscopic) model consists of
Darcy flow equations in the matrix blocks and fracture sheet,
with additional terms in the fracture sheet equation.
Together, these terms represent the fluid exchange between the
matrix blocks and the fracture sheet.
