On Modeling Flow in Fractal Media form Fractional Continuum Mechanics and Fractal Geometry

In this work we present a model for radial flow in highly heterogenous porous media. Heterogeneity is modeled by means of fractal geometry, a heterogeneous medium is regarded as fractal if its Hausdorff dimension is non-integral. Our purpose is to present a derivation of the model consistent with continuum mechanics, capable to describe anomalous diffusion as observed in some naturally fractured reservoirs. Consequently, we introduce fractional mass and a generalized Gauss theorem to obtain a continuity equation in fractal media. A generalized Darcy law for flux completes the model. Then we develop the basic equation for Well test analysis as is applied in petroleum engineering. Finally, the equation is solved by Laplace transform and inverted numerically to illustrate anomalous diffusion. In this case by showing that the flow rate from fractal systems is smaller than that from the Euclidean system.