A family of methods for DFN flow simulations with non-conforming meshes

A novel approach for simulating subsurface flow in a discrete fracture network (DFN) is presented. The method relies on the reformulation of the problem as a PDE-constrained optimization problem. The minimization of a properly defined functional is adopted to enforce fluid flow continuity and conservation at fracture intersections. The minimization is subject to PDE-constraints, given by the flow equations defined on each fracture. The discretization of the underlying flow equations is performed with finite element based methods.
Since matching conditions are not explicitly imposed, non-conforming meshes at fracture intersections can be used: this crucial point allows for a meshing process which can be independently performed on each fracture, thus being extremely reliable and computationally inexpensive. The figure displays a detail of the computational mesh used for the simulation, highlighting mesh non-conformities at fracture intersections.
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.
References
[1] S. Berrone, S. Pieraccini, S. Scialo' A PDE-constrained optimization formulation for discrete fracture network flows, SIAM J. Sci. Comput., 2013, 35, B487–B510.
[2] S. Berrone, S. Pieraccini, S. Scialo' On simulations of discrete fracture network flows with an optimization-based extended finite element method, SIAM J. Sci. Comput., 2013, 35, A908–A935.
[3] S. Berrone, S. Pieraccini, S. Scialo' An optimization approach for large scale simulations of discrete fracture network flows, J. Comput. Phys., 2014, 256, 838–853.
[4] M. F. Benedetto, S. Berrone, S. Pieraccini, S. Scialo' The Virtual Element Method for Discrete Fracture Network simulations, submitted.