ABSTRACT
Fractured media are very heterogeneous systems where occur complex physical and chemical processes to model. One of the possible approaches to conceptualize this type of massifs is the Discrete Fracture Network (DFN). Donado et al. (2005) modeled flow and transport in a granitic batholith based on this approach and found good fitting with hydraulic and tracer tests, but the computational cost was excessive due to a gigantic amount of elements to model. We present in this work a methodology based on percolation theory (Berkowitz, 2002) for reducing the number of elements and in consequence, to reduce the bandwidth of the conductance matrix and the execution time of each network.
DFN poses as an excellent representation of all the set of fractures of the media, but not all the fractures of the media are part of the conductive network. Percolation theory is used to identify which nodes or fractures are not conductive, based on the occupation probability or percolation threshold. In a fractured system, connectivity determines the flow pattern in the fractured rock mass. This volume of fluid is driven through connection paths formed by the fractures, when the permeability of the rock is negligible compared to the fractures. In a population of distributed fractures, each of this that has no intersection with any connected fracture does not contribute to generate a flow field. This algorithm also permits us to erase these elements however they are water conducting and hence, refine even more the backbone of the network.
This percolation theory seeks to find a network of conductive fracture smaller than the original, but without departing from the actual behavior of the fluid in a fractured medium and thus improve the calibration of the flow inverse modeling done with TRANSINIV. Donado (2000) used 100 Different generations Fracture Network (DFN) that were optimized in this study using percolation theory. In each of the networks calibrate hydrodynamic parameters as hydraulic conductivity Κ and specific storage coefficient Ss, for each of the five families of fractures (tectonic defined criteria), yielding a total of 10 parameters to estimate, at each generations.
The 100 DFNs used have the following characteristics: (i) they are not a trellis system, this means that a 3D system fractures do not follow a path of any geometric shape known as a cube, diamond, or honeycomb; (ii) from any node can leave many items to other nodes (Fig. 1); (iii) the length of failure of these fracture networks are not standardized and is constant, this means that on average, the minimum length is 23 m and found the maximum length is 955 m; (iv) the fracture networks are contained in a real finite system known limits, with approximate dimensions of a rectangular 710 × 1040 × 576 m. With these features the fractal dimension of this large cluster (which determines the spatial distribution of network connectivity) and the fractal dimension of the backbone (which determines the flow path) are different from those predicted by the classical theory of percolation.
Since the effect of the distribution of fault orientation changes the value of the percolation threshold, but not the universal laws of classical percolation theory, the latter is applicable to such networks. Under these conditions, percolation theory permits to reduce the number of elements (90% in average) that form clusters of the 100 DFNs, preserving the so-called backbone. In this way the calibration runs in these networks changed from several hours to just a second obtaining much better results (Fig. 2).
Green line represents the previous results using all the fractures and purple line shows new results with optimized fractures.
