## Derivation of the double porosity model of single phase flow via homogenization theory.(English)Zbl 0698.76106

Summary: A general form of the double porosity model of single phase flow in a naturally fractured reservoir is derived from homogenization theory. The microscopic model consists of the usual equations describing Darcy flow in a reservoir, except that the porosity and permeability coefficients are highly discontinuous. Over the matrix domain, the coefficients are scaled by a parameter $$\epsilon$$ representing the size of the matrix blocks. This scaling preserves the physics of the flow in the matrix as $$\epsilon$$ tends to zero. An effective macroscopic limit model is obtained that includes the usual Darcy equations in the matrix blocks and a similar equation for the fracture system that contains a term representing a source of fluid from the matrix. The convergence is shown by extracting weak limits in appropriate Hilbert spaces. A dilation operator is utilized to see the otherwise vanishing physics in the matrix blocks as $$\epsilon$$ tends to zero.

### MSC:

 76S05 Flows in porous media; filtration; seepage 35Q99 Partial differential equations of mathematical physics and other areas of application
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