|Abstract:|| A Discrete Fracture Network model describes a geological reservoir as a system of inter-secting planar polygons representing the network of fractures in the underground. Fracture
intersections are called traces. The quantity of interest is the flow potential, called hydraulic
head, given by the sum of pressure and elevation. The hydraulic head is ruled by Darcy's law in each fracture, with additional matching conditions which ensure hydraulic head continuity and flux balance at fracture intersections. Thanks to these matching conditions, the hydraulic head is continuous across traces but jumps of gradients may occur as a consequence of flux exchange between intersecting fractures. Hence, traces are typically interfaces of discontinuities for the gradient of the solution.
Standard finite element methods or mixed finite elements are widely used for obtaining a numerical solution also in this context, but they require mesh elements to conform with the traces in order to
correctly describe the irregular behavior of the solution. This poses a severe limitation, since realistic fracture networks are typically very intricate, with fractures intersecting each other with arbitrary orientation, position, density and dimension. A conforming meshing process may result infeasible, or might
generate a poor quality mesh, since a coupled meshing process on all the fractures of the system may lead to elongated elements.
In [1, 2, 3] the authors propose a PDE-constrained optimization approach to flow simulations on arbitrary DFNs, in which neither fracture/fracture nor fracture/trace mesh conformity is required. The method is based on the minimization of a quadratic functional constrained by the state equations describing the
ow on the fractures. The approach totally circumvents the problem of mesh generation, without any need of geometrical modification tailored on the DFN (e.g., fracture or trace removal or displacement).
As far as space discretization on the fractures is concerned, several different discretization strategies can be used and mixed in order to improve approximation properties and provide high levels of accuracy, especially at fracture intersections, where a non-smooth behavior of the solution is expected.
In particular, Extended Finite Elements and Virtual Elements have been effectively used when dealing with complex DFN configurations.
By using a gradient based method for the minimization of the functional, the solution of the flow equations on each fracture of the network is carried on independently of the solution on the other fractures. This in turn, together with the meshing process independently performed on each fracture, allows in a natural way for parallel implementation of the overall method, thus providing an efficient handling of problem size and complexity. This is of paramount importance both for addressing computations on huge networks, and for performing massive simulations for uncertainty quantification in stochastically generated networks.