Up-Scaling of Double Porosity Fractured Media Using Continuous-Time Random Walks Methods

We present a new application of continuous time random walks (CTRW) methods to model fluid flows in fractured rocks. The proposed method allows large scale equivalent permeability tensors and matrix/fractures exchange function to be computed from high resolution maps of fractured porous media. Knowing these parameters allows us to carry out large scale simulations of flows governed by the dual porosity equations of Warren and Root. A direct connection between the exchange function and the time correlation function of the presence in the fractures of a particle undergoing a suitable Brownian motion over the whole medium is derived. This connection allows us to develop an efficient numerical method to compute the transient exchange term within the complete range of time scales of interest. It also gives an alternative probabilistic interpretation of the Warren and Root model. For the sake of simplicity, in the present paper, the method will only be developed to Cartesian structured grids, although it can be adapted for unstructured grids highly suited to describing complex fracture networks.